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+{"_id": "0", "title": "stacks-perfect-lemma-shriek-derived", "text": "Let $\\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas \\ref{stacks-cohomology-lemma-lisse-etale} and \\ref{stacks-cohomology-lemma-lisse-etale-modules}. \\begin{enumerate} \\item The functor $g_! : \\textit{Ab}(\\mathcal{X}_{lisse,\\etale}) \\to \\textit{Ab}(\\mathcal{X}_\\etale)$ has a left derived functor $$ Lg_! : D(\\mathcal{X}_{lisse,\\etale}) \\longrightarrow D(\\mathcal{X}_\\etale) $$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$. \\item The functor $g_! :  \\textit{Mod}(\\mathcal{X}_{lisse,\\etale}, \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\to \\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}})$ has a left derived functor $$ Lg_! : D(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\longrightarrow D(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) $$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$. \\item The functor $g_! : \\textit{Ab}(\\mathcal{X}_{flat,fppf}) \\to \\textit{Ab}(\\mathcal{X}_{fppf})$ has a left derived functor $$ Lg_! : D(\\mathcal{X}_{flat, fppf}) \\longrightarrow D(\\mathcal{X}_{fppf}) $$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$. \\item The functor $g_! : \\textit{Mod}(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\to \\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})$ has a left derived functor $$ Lg_! : D(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}}) \\longrightarrow D(\\mathcal{O}_\\mathcal{X}) $$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$. \\end{enumerate} Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark \\ref{sites-cohomology-remark-when-derived-shriek-equal}."}
+{"_id": "1", "title": "stacks-perfect-lemma-lisse-etale-functorial-derived", "text": "With assumptions and notation as in Cohomology of Stacks, Lemma \\ref{stacks-cohomology-lemma-lisse-etale-functorial}. We have $$ g^{-1} \\circ Rf_* = Rf'_* \\circ (g')^{-1} \\quad\\text{and}\\quad L(g')_! \\circ (f')^{-1} = f^{-1} \\circ Lg_! $$ on unbounded derived categories (both for the case of modules and for the case of abelian sheaves)."}
+{"_id": "2", "title": "stacks-perfect-lemma-higher-shriek-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma \\ref{stacks-cohomology-lemma-lisse-etale}. \\begin{enumerate} \\item Let $\\mathcal{H}$ be a quasi-coherent $\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module  on the lisse-\\'etale site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$ the sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\\mathcal{X}$. \\item Let $\\mathcal{H}$ be a quasi-coherent $\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module  on the flat-fppf site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$ the sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\\mathcal{X}$. \\end{enumerate}"}
+{"_id": "3", "title": "stacks-perfect-lemma-compare-etale-fppf-QCoh", "text": "Let $\\mathcal{X}$ be an algebraic stack. The comparison morphism $\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$ induces a commutative diagram $$ \\xymatrix{ D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] & D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] & D(\\mathcal{O}_\\mathcal{X}) \\\\ D_{\\mathcal{P}_\\mathcal{X}}( \\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) \\ar[r] \\ar[u]^{\\epsilon^*} & D_{\\mathcal{M}_\\mathcal{X}}( \\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) \\ar[r] \\ar[u]^{\\epsilon^*} & D(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) \\ar[u]^{\\epsilon^*} } $$ Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence $$ D_{\\mathcal{M}_\\mathcal{X}}( \\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) / D_{\\mathcal{P}_\\mathcal{X}}( \\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) \\longrightarrow D_\\QCoh(\\mathcal{O}_\\mathcal{X}) $$"}
+{"_id": "4", "title": "stacks-perfect-lemma-derived-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item Let $\\mathcal{F}^\\bullet$ be an object of $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \\ref{stacks-cohomology-lemma-lisse-etale} for the lisse-\\'etale site we have \\begin{enumerate} \\item $g^{-1}\\mathcal{F}^\\bullet$ is in $D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$, \\item $g^{-1}\\mathcal{F}^\\bullet = 0$ if and only if $\\mathcal{F}^\\bullet$ is in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$, \\item $Lg_!\\mathcal{H}^\\bullet$ is in $D_{\\mathcal{M}_\\mathcal{X}}( \\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ for $\\mathcal{H}^\\bullet$ in $D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$, and \\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$ \\xymatrix{ D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} & D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\ar@<1ex>[l]^-{Lg_!} } $$ \\end{enumerate} \\item Let $\\mathcal{F}^\\bullet$ be an object of $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \\ref{stacks-cohomology-lemma-lisse-etale} for the flat-fppf site we have \\begin{enumerate} \\item $g^{-1}\\mathcal{F}^\\bullet$ is in $D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})$, \\item $g^{-1}\\mathcal{F}^\\bullet = 0$ if and only if $\\mathcal{F}^\\bullet$ is in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$, \\item $Lg_!\\mathcal{H}^\\bullet$ is in $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ for $\\mathcal{H}^\\bullet$ in $D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$, and \\item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$ \\xymatrix{ D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} & D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{Lg_!} } $$ \\end{enumerate} \\end{enumerate}"}
+{"_id": "5", "title": "stacks-perfect-lemma-bousfield-colocalization", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$. There exists a canonical distinguished triangle $$ E' \\to E \\to P \\to E'[1] $$ in $D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ such that $P$ is in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$ and $$ \\Hom_{D(\\mathcal{O}_\\mathcal{X})}(E', P') = 0 $$ for all $P'$ in $D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$."}
+{"_id": "6", "title": "stacks-perfect-proposition-derived-direct-image-quasi-coherent", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $Rf_*$ induces a commutative diagram $$ \\xymatrix{ D^{+}_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] \\ar[d]^{Rf_*} & D^{+}_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] \\ar[d]^{Rf_*} & D(\\mathcal{O}_\\mathcal{X}) \\ar[d]^{Rf_*} \\\\ D^{+}_{\\mathcal{P}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] & D^{+}_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] & D(\\mathcal{O}_\\mathcal{Y}) } $$ and hence induces a functor $$ Rf_{\\QCoh, *} : D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\longrightarrow D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{Y}) $$ on quotient categories. Moreover, the functor $R^if_\\QCoh$ of Cohomology of Stacks, Proposition \\ref{stacks-cohomology-proposition-direct-image-quasi-coherent} are equal to $H^i \\circ Rf_{\\QCoh, *}$ with $H^i$ as in (\\ref{equation-Hi-quasi-coherent})."}
+{"_id": "9", "title": "spaces-more-morphisms-theorem-topological-invariance", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is integral, universally injective and surjective. The functor $$ V \\longmapsto V_X = X \\times_Y V $$ defines an equivalence of categories $Y_{spaces, \\etale} \\to X_{spaces, \\etale}$."}
+{"_id": "10", "title": "spaces-more-morphisms-theorem-openness-flatness", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume $f$ is locally of finite presentation and that $\\mathcal{F}$ is an $\\mathcal{O}_X$-module which is locally of finite presentation. Then $$ \\{x \\in |X| : \\mathcal{F}\\text{ is flat over }Y\\text{ at }x\\} $$ is open in $|X|$."}
+{"_id": "11", "title": "spaces-more-morphisms-theorem-criterion-flatness-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $Z$, \\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and \\item $Y$ is locally of finite type over $Z$. \\end{enumerate} Let $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of $x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$, and \\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is flat over $Y$ at $x$. \\end{enumerate} Moreover, the set of points $x$ where (1) and (2) hold is open in $\\text{Supp}(\\mathcal{F})$."}
+{"_id": "12", "title": "spaces-more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X$, $Y$, $Z$ locally Noetherian, and \\item $\\mathcal{F}$ a coherent $\\mathcal{O}_X$-module. \\end{enumerate} Let $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of $x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $Z$ at $x$ and the restriction of $\\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$, and \\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is flat over $Y$ at $x$. \\end{enumerate}"}
+{"_id": "13", "title": "spaces-more-morphisms-theorem-stein-factorization-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\ & Y & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is proper with connected geometric fibres, \\item the morphism $\\pi : Y' \\to Y$ is finite, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$, \\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and \\item $Y'$ is the normalization of $Y$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. \\end{enumerate}"}
+{"_id": "14", "title": "spaces-more-morphisms-theorem-stein-factorization-general", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\ & Y & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is proper with connected geometric fibres, \\item the morphism $\\pi : Y' \\to Y$ is integral, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$, \\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and \\item $Y'$ is the normalization of $Y$ in $X$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-normalization-X-in-Y}). \\end{enumerate}"}
+{"_id": "15", "title": "spaces-more-morphisms-theorem-flatten-module", "text": "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X$ be an algebraic space over $B$. Let $\\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \\subset B$ be a quasi-compact open subspace. Assume \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ is locally of finite presentation over $B$, \\item $\\mathcal{F}$ is a module of finite type, \\item $\\mathcal{F}_U$ is of finite presentation, and \\item $\\mathcal{F}_U$ is flat over $U$. \\end{enumerate} Then there exists a $U$-admissible blowup $B' \\to B$ such that the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an $\\mathcal{O}_{X \\times_B B'}$-module of finite presentation and flat over $B'$."}
+{"_id": "16", "title": "spaces-more-morphisms-theorem-grothendieck-existence", "text": "In Situation \\ref{situation-existence} the functor (\\ref{equation-completion-functor-proper-over-A}) is an equivalence."}
+{"_id": "17", "title": "spaces-more-morphisms-lemma-radicial-implies-universally-injective", "text": "A radicial morphism of algebraic spaces is universally injective."}
+{"_id": "18", "title": "spaces-more-morphisms-lemma-when-universally-injective-radicial", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a universally injective morphism of algebraic spaces over $S$. \\begin{enumerate} \\item If $f$ is decent then $f$ is radicial. \\item If $f$ is quasi-separated then $f$ is radicial. \\item If $f$ is locally separated then $f$ is radicial. \\end{enumerate}"}
+{"_id": "21", "title": "spaces-more-morphisms-lemma-ui-case", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact monomorphism of algebraic spaces such that for every $T \\to Y$ the map $$ \\mathcal{O}_T \\to f_{T,*}\\mathcal{O}_{X \\times_Y T} $$ is injective. Then $f$ is an isomorphism (and hence representable by schemes)."}
+{"_id": "23", "title": "spaces-more-morphisms-lemma-etale-conormal", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion. Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a scheme. Set $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$. Then $$ \\mathcal{C}_{Z/X}|_{Z_U} = \\mathcal{C}_{Z_U/U} $$ canonically and functorially in $U$."}
+{"_id": "24", "title": "spaces-more-morphisms-lemma-conormal-functorial", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of $\\mathcal{O}_Z$-modules $$ f^*\\mathcal{C}_{Z'/X'} \\longrightarrow \\mathcal{C}_{Z/X} $$"}
+{"_id": "25", "title": "spaces-more-morphisms-lemma-conormal-functorial-more", "text": "Let $S$ be a scheme. The conormal sheaf of Definition \\ref{definition-conormal-sheaf}, and its functoriality of Lemma \\ref{lemma-conormal-functorial} satisfy the following properties: \\begin{enumerate} \\item If $Z \\to X$ is an immersion of schemes over $S$, then the conormal sheaf agrees with the one from Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf}. \\item If in Lemma \\ref{lemma-conormal-functorial} all the spaces are schemes, then the map $f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ is the same as the one constructed in Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}. \\item Given a commutative diagram $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} \\ar[d]_{f'} & X' \\ar[d]^{g'} \\\\ Z'' \\ar[r]^{i''} & X'' } $$ then the map $(f' \\circ f)^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z/X}$ is the same as the composition of $f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ with the pullback by $f$ of $(f')^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z'/X'}$ \\end{enumerate}"}
+{"_id": "26", "title": "spaces-more-morphisms-lemma-conormal-functorial-flat", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a fibre product diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. Then the canonical map $f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of Lemma \\ref{lemma-conormal-functorial} is surjective. If $g$ is flat, then it is an isomorphism."}
+{"_id": "27", "title": "spaces-more-morphisms-lemma-transitivity-conormal", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence $$ i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ where the maps come from Lemma \\ref{lemma-conormal-functorial} and $i : Z \\to Y$ is the first morphism."}
+{"_id": "28", "title": "spaces-more-morphisms-lemma-etale-conormal-algebra", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a scheme. Set $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$. Then $$ \\mathcal{C}_{Z/X, *}|_{Z_U} = \\mathcal{C}_{Z_U/U, *} $$ canonically and functorially in $U$."}
+{"_id": "29", "title": "spaces-more-morphisms-lemma-conormal-algebra-functorial", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of graded $\\mathcal{O}_Z$-algebras $$ f^*\\mathcal{C}_{Z'/X', *} \\longrightarrow \\mathcal{C}_{Z/X, *} $$"}
+{"_id": "32", "title": "spaces-more-morphisms-lemma-localize-differentials", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Consider any commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_\\psi & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ where the vertical arrows are \\'etale morphisms of algebraic spaces. Then $$ \\Omega_{X/Y}|_{U_\\etale} = \\Omega_{U/V} $$ In particular, if $U$, $V$ are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes $U \\to V$."}
+{"_id": "33", "title": "spaces-more-morphisms-lemma-module-differentials-quasi-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then $\\Omega_{X/Y}$ is a quasi-coherent $\\mathcal{O}_X$-module."}
+{"_id": "34", "title": "spaces-more-morphisms-lemma-functoriality-differentials", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ be a commutative diagram of algebraic spaces. The map $f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map $f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules $\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}. $$ It is uniquely characterized by the property that $f^*\\text{d}_{X/Y}(t)$ mapsto $\\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\\mathcal{O}_X$."}
+{"_id": "35", "title": "spaces-more-morphisms-lemma-check-functoriality-differentials", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\ Y'' \\ar[r] & Y' \\ar[r] & Y } $$ be a commutative diagram of algebraic spaces over $S$. Then we have $$ c_{f \\circ g} = c_g \\circ g^* c_f $$ as maps $(f \\circ g)^*\\Omega_{X/Y} \\to \\Omega_{X''/Y''}$."}
+{"_id": "36", "title": "spaces-more-morphisms-lemma-triangle-differentials", "text": "Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to B$ be morphisms of algebraic spaces over $S$. Then there is a canonical exact sequence $$ f^*\\Omega_{Y/B} \\to \\Omega_{X/B} \\to \\Omega_{X/Y} \\to 0 $$ where the maps come from applications of Lemma \\ref{lemma-functoriality-differentials}."}
+{"_id": "38", "title": "spaces-more-morphisms-lemma-differentials-relative-immersion", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \\to X$ be an immersion of algebraic spaces over $B$. There is a canonical exact sequence $$ \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0 $$ where the first arrow is induced by $\\text{d}_{X/B}$ and the second arrow comes from Lemma \\ref{lemma-functoriality-differentials}."}
+{"_id": "40", "title": "spaces-more-morphisms-lemma-base-change-differentials", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$. Let $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$. Let $X' = X_{Y'}$ be the base change of $X$. Denote $g' : X' \\to X$ the projection. Then the map $$ (g')^*\\Omega_{X/Y} \\to \\Omega_{X'/Y'} $$ of Lemma \\ref{lemma-functoriality-differentials} is an isomorphism."}
+{"_id": "41", "title": "spaces-more-morphisms-lemma-differential-product", "text": "Let $S$ be a scheme. Let $f : X \\to B$ and $g : Y \\to B$ be morphisms of algebraic spaces over $S$ with the same target. Let $p : X \\times_B Y \\to X$ and $q : X \\times_B Y \\to Y$ be the projection morphisms. The maps from Lemma \\ref{lemma-functoriality-differentials} $$ p^*\\Omega_{X/B} \\oplus q^*\\Omega_{Y/B} \\longrightarrow \\Omega_{X \\times_B Y/B} $$ give an isomorphism."}
+{"_id": "43", "title": "spaces-more-morphisms-lemma-finite-presentation-differentials", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\\Omega_{X/Y}$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "44", "title": "spaces-more-morphisms-lemma-smooth-omega-finite-locally-free", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a smooth morphism of algebraic spaces over $S$. Then the module of differentials $\\Omega_{X/Y}$ is finite locally free."}
+{"_id": "45", "title": "spaces-more-morphisms-lemma-topological-invariance", "text": "With assumption and notation as in Theorem \\ref{theorem-topological-invariance} the equivalence of categories $Y_{spaces, \\etale} \\to X_{spaces, \\etale}$ restricts to equivalences of categories $Y_\\etale \\to X_\\etale$ and $Y_{affine, \\etale} \\to X_{affine, \\etale}$."}
+{"_id": "46", "title": "spaces-more-morphisms-lemma-first-order-thickening-maps", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \\subset X'$ and $Y \\subset Y'$ be thickenings of algebraic spaces over $B$. Let $f : X \\to Y$ be a morphism of algebraic spaces over $B$. Given any map of $\\mathcal{O}_B$-algebras $$ \\alpha : f_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\to \\mathcal{O}_{X'} $$ such that $$ \\xymatrix{ f_{spaces, \\etale}^{-1}\\mathcal{O}_Y \\ar[r]_-{f^\\sharp} \\ar[r] & \\mathcal{O}_X \\\\ f_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\ar[r]^-\\alpha \\ar[u]^{i_Y^\\sharp} & \\mathcal{O}_{X'} \\ar[u]_{i_X^\\sharp} } $$ commutes, there exists a unique morphism of $(f, f')$ of thickenings over $B$ such that $\\alpha = (f')^\\sharp$."}
+{"_id": "48", "title": "spaces-more-morphisms-lemma-first-order-thickening-surjective", "text": "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces over $S$. Let $U$ be an affine object of $X_{spaces, \\etale}$. Then $$ \\Gamma(U, \\mathcal{O}_{X'}) \\to \\Gamma(U, \\mathcal{O}_X) $$ is surjective where we think of $\\mathcal{O}_{X'}$ as a sheaf on $X_{spaces, \\etale}$ via (\\ref{equation-fundamental-equivalence})."}
+{"_id": "49", "title": "spaces-more-morphisms-lemma-thickening-scheme", "text": "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces over $S$. If $X$ is (representable by) a scheme, then so is $X'$."}
+{"_id": "50", "title": "spaces-more-morphisms-lemma-thickening-equivalence", "text": "Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces over $S$. The functor $$ V' \\longmapsto V = X \\times_{X'} V' $$ defines an equivalence of categories $X'_\\etale \\to X_\\etale$."}
+{"_id": "51", "title": "spaces-more-morphisms-lemma-first-order-thickening", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. Consider a short exact sequence $$ 0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0 $$ of sheaves on $X_\\etale$ where $\\mathcal{A}$ is a sheaf of $f^{-1}\\mathcal{O}_B$-algebras, $\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\\mathcal{O}_B$-algebras, and $\\mathcal{I}$ is its kernel. If \\begin{enumerate} \\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and \\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module \\end{enumerate} then there exists a first order thickening $X \\subset X'$ over $B$ and an isomorphism $\\mathcal{O}_{X'} \\to \\mathcal{A}$ of $f^{-1}\\mathcal{O}_B$-algebras compatible with the surjections to $\\mathcal{O}_X$."}
+{"_id": "52", "title": "spaces-more-morphisms-lemma-base-change-thickening", "text": "Let $S$ be a scheme. Let $Y \\subset Y'$ be a thickening of algebraic spaces over $S$. Let $X' \\to Y'$ be a morphism and set $X = Y \\times_{Y'} X'$. Then $(X \\subset X') \\to (Y \\subset Y')$ is a morphism of thickenings. If $Y \\subset Y'$ is a first (resp.\\ finite order) thickening, then $X \\subset X'$ is a first (resp.\\ finite order) thickening."}
+{"_id": "53", "title": "spaces-more-morphisms-lemma-composition-thickening", "text": "Let $S$ be a scheme. If $X \\subset X'$ and $X' \\subset X''$ are thickenings of algebraic spaces over $S$, then so is $X \\subset X''$."}
+{"_id": "54", "title": "spaces-more-morphisms-lemma-descending-property-thickening", "text": "The property of being a thickening is fpqc local. Similarly for first order thickenings."}
+{"_id": "55", "title": "spaces-more-morphisms-lemma-thicken-property-morphisms", "text": "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a morphism of thickenings of algebraic spaces over $S$. Then \\begin{enumerate} \\item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \\item $f$ is quasi-compact if and only if $f'$ quasi-compact, \\item $f$ is universally closed if and only if $f'$ is universally closed, \\item $f$ is integral if and only if $f'$ is integral, \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\item $f$ is universally injective if and only if $f'$ is universally injective, \\item $f$ is universally open if and only if $f'$ is universally open, \\item $f$ is representable if and only if $f'$ is representable, and \\item add more here. \\end{enumerate}"}
+{"_id": "56", "title": "spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian", "text": "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a morphism of thickenings of algebraic spaces over $S$ such that $X = Y \\times_{Y'} X'$. If $X \\subset X'$ is a finite order thickening, then \\begin{enumerate} \\item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \\item $f$ is locally of finite type if and only if $f'$ is locally of finite type, \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\item $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$, \\item $f$ is unramified if and only if $f'$ is unramified, \\item $f$ is proper if and only if $f'$ is proper, \\item $f$ is a finite morphism if and only if $f'$ is an finite morphism, \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\item $f$ is an immersion if and only if $f'$ is an immersion, and \\item add more here. \\end{enumerate}"}
+{"_id": "57", "title": "spaces-more-morphisms-lemma-properties-that-extend-over-thickenings", "text": "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a morphism of thickenings of algebraic spaces over $S$. Assume $f$ and $f'$ are locally of finite type and $X = Y \\times_{Y'} X'$. Then \\begin{enumerate} \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\item $f$ is finite if and only if $f'$ is finite, \\item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$, \\item $f$ is unramified if and only if $f'$ is unramified, \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\item $f$ is an immersion if and only if $f'$ is an immersion, \\item $f$ is proper if and only if $f'$ is proper, and \\item add more here. \\end{enumerate}"}
+{"_id": "58", "title": "spaces-more-morphisms-lemma-picard-group-first-order-thickening", "text": "Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening of algebraic spaces over $S$ with ideal sheaf $\\mathcal{I}$. Then there is a canonical exact sequence $$ \\xymatrix{ 0 \\ar[r] & H^0(X, \\mathcal{I}) \\ar[r] & H^0(X', \\mathcal{O}_{X'}^*) \\ar[r] & H^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\ & H^1(X, \\mathcal{I}) \\ar[r] & \\Pic(X') \\ar[r] & \\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\ & H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots } $$ of abelian groups."}
+{"_id": "59", "title": "spaces-more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "text": "Let $i : Z \\to X$ be an immersion of algebraic spaces. The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$ has the following universal property: Given any commutative diagram $$ \\xymatrix{ Z \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\ X & T' \\ar[l]_b } $$ where $T \\subset T'$ is a first order thickening over $X$, there exists a unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of thickenings over $X$."}
+{"_id": "60", "title": "spaces-more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "text": "Let $i : Z \\to X$ be an immersion of algebraic spaces. Let $Z \\subset Z'$ be the first order infinitesimal neighbourhood of $Z$ in $X$. Then the diagram $$ \\xymatrix{ Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\ Z \\ar[r] & X } $$ induces a map of conormal sheaves $\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by Lemma \\ref{lemma-conormal-functorial}. This map is an isomorphism."}
+{"_id": "61", "title": "spaces-more-morphisms-lemma-formally-etale-is-combination", "text": "Let $S$ be a scheme. Let $a : F \\to G$ be a transformation of functors $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Then $a$ is formally \\'etale if and only if $a$ is both formally smooth and formally unramified."}
+{"_id": "62", "title": "spaces-more-morphisms-lemma-composition-formally-smooth-etale-unramified", "text": "Composition. \\begin{enumerate} \\item A composition of formally smooth transformations of functors is formally smooth. \\item A composition of formally \\'etale transformations of functors is formally \\'etale. \\item A composition of formally unramified transformations of functors is formally unramified. \\end{enumerate}"}
+{"_id": "63", "title": "spaces-more-morphisms-lemma-base-change-formally-smooth-etale-unramified", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : H \\to G$ be transformations of functors. Consider the fibre product diagram $$ \\xymatrix{ H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\ H \\ar[r]^b & G } $$ \\begin{enumerate} \\item If $a$ is formally smooth, then the base change $a'$ is formally smooth. \\item If $a$ is formally \\'etale, then the base change $a'$ is formally \\'etale. \\item If $a$ is formally unramified, then the base change $a'$ is formally unramified. \\end{enumerate}"}
+{"_id": "64", "title": "spaces-more-morphisms-lemma-representable-property-formally-property", "text": "Let $S$ be a scheme. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a representable transformation of functors. \\begin{enumerate} \\item If $a$ is smooth then $a$ is formally smooth. \\item If $a$ is \\'etale, then $a$ is formally \\'etale. \\item If $a$ is unramified, then $a$ is formally unramified. \\end{enumerate}"}
+{"_id": "65", "title": "spaces-more-morphisms-lemma-etale-on-top", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : G \\to H$ be transformations of functors. Assume that $a$ is representable, surjective, and \\'etale. \\begin{enumerate} \\item If $b$ is formally smooth, then $b \\circ a$ is formally smooth. \\item If $b$ is formally \\'etale, then $b \\circ a$ is formally \\'etale. \\item If $b$ is formally unramified, then $b \\circ a$ is formally unramified. \\end{enumerate} Conversely, consider a solid commutative diagram $$ \\xymatrix{ G \\ar[d]_b & T \\ar[d]^i \\ar[l] \\\\ H & T' \\ar[l] \\ar@{-->}[lu] } $$ with $T'$ an affine scheme over $S$ and $i : T \\to T'$ a closed immersion defined by an ideal of square zero. \\begin{enumerate} \\item[(4)] If $b \\circ a$ is formally smooth, then for every $t \\in T$ there exists an \\'etale morphism of affines $U' \\to T'$ and a morphism $U' \\to G$ such that $$ \\xymatrix{ G \\ar[d]_b & T \\ar[l] & T \\times_{T'} U' \\ar[d] \\ar[l]\\\\ H & T' \\ar[l] & U' \\ar[llu] \\ar[l] } $$ commutes and $t$ is in the image of $U' \\to T'$. \\item[(5)] If $b \\circ a$ is formally unramified, then there exists at most one dotted arrow in the diagram above, i.e., $b$ is formally unramified. \\item[(6)] If $b \\circ a$ is formally \\'etale, then there exists exactly one dotted arrow in the diagram above, i.e., $b$ is formally \\'etale. \\end{enumerate}"}
+{"_id": "66", "title": "spaces-more-morphisms-lemma-formally-permanence", "text": "Let $S$ be a scheme. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : G \\to H$ be transformations of functors. Assume $b$ is formally unramified. \\begin{enumerate} \\item If $b \\circ a$ is formally unramified then $a$ is formally unramified. \\item If $b \\circ a$ is formally \\'etale then $a$ is formally \\'etale. \\item If $b \\circ a$ is formally smooth then $a$ is formally smooth. \\end{enumerate}"}
+{"_id": "67", "title": "spaces-more-morphisms-lemma-formally-unramified", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is formally unramified, \\item for every diagram $$ \\xymatrix{ U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \\'etale the morphism of schemes $\\psi$ is formally unramified (as in More on Morphisms, Definition \\ref{more-morphisms-definition-formally-unramified}), and \\item for one such diagram with surjective vertical arrows the morphism $\\psi$ is formally unramified. \\end{enumerate}"}
+{"_id": "68", "title": "spaces-more-morphisms-lemma-formally-unramified-not-affine", "text": "Let $S$ be a scheme. If $f : X \\to Y$ is a formally unramified morphism of algebraic spaces over $S$, then given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of algebraic spaces over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition \\ref{definition-formally-unramified} the condition that $T$ be an affine scheme may be dropped."}
+{"_id": "69", "title": "spaces-more-morphisms-lemma-composition-formally-unramified", "text": "A composition of formally unramified morphisms is formally unramified."}
+{"_id": "70", "title": "spaces-more-morphisms-lemma-base-change-formally-unramified", "text": "A base change of a formally unramified morphism is formally unramified."}
+{"_id": "71", "title": "spaces-more-morphisms-lemma-characterize-formally-unramified", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is formally unramified, and \\item $\\Omega_{X/Y} = 0$. \\end{enumerate}"}
+{"_id": "72", "title": "spaces-more-morphisms-lemma-unramified-formally-unramified", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is unramified, \\item the morphism $f$ is locally of finite type and $\\Omega_{X/Y} = 0$, and \\item the morphism $f$ is locally of finite type and formally unramified. \\end{enumerate}"}
+{"_id": "73", "title": "spaces-more-morphisms-lemma-universally-injective-unramified", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is unramified and a monomorphism, \\item $f$ is unramified and universally injective, \\item $f$ is locally of finite type and a monomorphism, \\item $f$ is universally injective, locally of finite type, and formally unramified. \\end{enumerate} Moreover, in this case $f$ is also representable, separated, and locally quasi-finite."}
+{"_id": "74", "title": "spaces-more-morphisms-lemma-characterize-closed-immersion", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is a closed immersion, \\item $f$ is universally closed, unramified, and a monomorphism, \\item $f$ is universally closed, unramified, and universally injective, \\item $f$ is universally closed, locally of finite type, and a monomorphism, \\item $f$ is universally closed, universally injective, locally of finite type, and formally unramified. \\end{enumerate}"}
+{"_id": "75", "title": "spaces-more-morphisms-lemma-check-universal-first-order-thickening", "text": "Let $S$ be a scheme. Let $h : Z \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset Z'$ be a first order thickening over $X$. The following are equivalent \\begin{enumerate} \\item $Z \\subset Z'$ is a universal first order thickening, \\item for any diagram (\\ref{equation-universal-first-order-thickening}) with $T'$ a scheme a unique dotted arrow exists making the diagram commute, and \\item for any diagram (\\ref{equation-universal-first-order-thickening}) with $T'$ an affine scheme a unique dotted arrow exists making the diagram commute. \\end{enumerate}"}
+{"_id": "76", "title": "spaces-more-morphisms-lemma-universal-thickening-over-formally-etale", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$. If $Z \\subset Z'$ is a universal first order thickening of $Z$ over $Y$ and $Y \\to X$ is formally \\'etale, then $Z \\subset Z'$ is a universal first order thickening of $Z$ over $X$."}
+{"_id": "77", "title": "spaces-more-morphisms-lemma-etale-morphism-of-universal-thickenings", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$. Assume $Z \\to Y$ is \\'etale. \\begin{enumerate} \\item If $Y \\subset Y'$ is a universal first order thickening of $Y$ over $X$, then the unique \\'etale morphism $Z' \\to Y'$ such that $Z = Y \\times_{Y'} Z'$ (see Theorem \\ref{theorem-topological-invariance}) is a universal first order thickening of $Z$ over $X$. \\item If $Z \\to Y$ is surjective and $(Z \\subset Z') \\to (Y \\subset Y')$ is an \\'etale morphism of first order thickenings over $X$ and $Z'$ is a universal first order thickening of $Z$ over $X$, then $Y'$ is a universal first order thickening of $Y$ over $X$. \\end{enumerate}"}
+{"_id": "78", "title": "spaces-more-morphisms-lemma-universal-thickening", "text": "Let $S$ be a scheme. Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces over $S$. There exists a universal first order thickening $Z \\subset Z'$ of $Z$ over $X$."}
+{"_id": "80", "title": "spaces-more-morphisms-lemma-universal-thickening-unramified", "text": "Let $S$ be a scheme. Let $Z \\to X$ be a formally unramified morphism of algebraic spaces over $S$. Then the universal first order thickening $Z'$ is formally unramified over $X$."}
+{"_id": "81", "title": "spaces-more-morphisms-lemma-universal-thickening-functorial", "text": "Let $S$ be a scheme Consider a commutative diagram of algebraic spaces over $S$ $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\ W \\ar[r]^{h'} & Y } $$ with $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal first order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal first order thickening of $W$ over $Y$. There exists a canonical morphism $(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into the following commutative diagram $$ \\xymatrix{ & & & Z' \\ar[ld] \\ar[d]^{f'} \\\\ Z \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\ W \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y } $$ In particular the morphism $(f, f')$ of thickenings induces a morphism of conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$."}
+{"_id": "82", "title": "spaces-more-morphisms-lemma-universal-thickening-fibre-product", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\ W \\ar[r]^{h'} & Y } $$ be a fibre product diagram of algebraic spaces over $S$ with $h'$ formally unramified. Then $h$ is formally unramified and if $W \\subset W'$ is the universal first order thickening of $W$ over $Y$, then $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal first order thickening of $Z$ over $X$. In particular the canonical map $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of Lemma \\ref{lemma-universal-thickening-functorial} is surjective."}
+{"_id": "84", "title": "spaces-more-morphisms-lemma-universal-thickening-localize", "text": "Taking the universal first order thickenings commutes with \\'etale localization. More precisely, let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces over a base scheme $S$. Let $$ \\xymatrix{ V \\ar[d] \\ar[r] & U \\ar[d] \\\\ Z \\ar[r] & X } $$ be a commutative diagram with \\'etale vertical arrows. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $V \\to U$ is formally unramified and the universal first order thickening $V'$ of $V$ over $U$ is \\'etale over $Z'$. In particular, $\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$."}
+{"_id": "85", "title": "spaces-more-morphisms-lemma-differentials-universally-unramified", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces over $B$. Let $Z \\subset Z'$ be the universal first order thickening of $Z$ over $X$ with structure morphism $h' : Z' \\to X$. The canonical map $$ \\text{d}h' : (h')^*\\Omega_{X/B} \\to \\Omega_{Z'/B} $$ induces an isomorphism $h^*\\Omega_{X/B} \\to \\Omega_{Z'/B} \\otimes \\mathcal{O}_Z$."}
+{"_id": "86", "title": "spaces-more-morphisms-lemma-universally-unramified-differentials-sequence", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces over $B$. There is a canonical exact sequence $$ \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0. $$ The first arrow is induced by $\\text{d}_{Z'/B}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$."}
+{"_id": "87", "title": "spaces-more-morphisms-lemma-two-unramified-morphisms", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\ & Y } $$ be a commutative diagram of algebraic spaces over $S$ where $i$ and $j$ are formally unramified. Then there is a canonical exact sequence $$ \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0 $$ where the first arrow comes from Lemma \\ref{lemma-universal-thickening-functorial} and the second from Lemma \\ref{lemma-universally-unramified-differentials-sequence}."}
+{"_id": "89", "title": "spaces-more-morphisms-lemma-formally-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is formally \\'etale, \\item for every diagram $$ \\xymatrix{ U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \\'etale the morphism of schemes $\\psi$ is formally \\'etale (as in More on Morphisms, Definition \\ref{more-morphisms-definition-formally-etale}), and \\item for one such diagram with surjective vertical arrows the morphism $\\psi$ is formally \\'etale. \\end{enumerate}"}
+{"_id": "91", "title": "spaces-more-morphisms-lemma-composition-formally-etale", "text": "A composition of formally \\'etale morphisms is formally \\'etale."}
+{"_id": "92", "title": "spaces-more-morphisms-lemma-base-change-formally-etale", "text": "A base change of a formally \\'etale morphism is formally \\'etale."}
+{"_id": "95", "title": "spaces-more-morphisms-lemma-etale-formally-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is \\'etale, and \\item the morphism $f$ is locally of finite presentation and formally \\'etale. \\end{enumerate}"}
+{"_id": "96", "title": "spaces-more-morphisms-lemma-difference-derivation", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings of algebraic spaces over $B$. Let $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$ be two morphisms of thickenings over $B$. Assume that \\begin{enumerate} \\item $a = b$, and \\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ (Lemma \\ref{lemma-conormal-functorial}) are equal. \\end{enumerate} Then the map $(a')^\\sharp - (b')^\\sharp$ factors as $$ \\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D} a_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'} $$ where $D$ is an $\\mathcal{O}_B$-derivation."}
+{"_id": "97", "title": "spaces-more-morphisms-lemma-action-by-derivations", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(a, a') : (X \\subset X') \\to (Y \\subset Y')$ be a morphism of first order thickenings over $B$. Let $$ \\theta : a^*\\Omega_{Y/B} \\to \\mathcal{C}_{X/X'} $$ be an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs $(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that (1) and (2) of Lemma \\ref{lemma-difference-derivation} hold and the derivation $D$ and $\\theta$ are related by Equation (\\ref{equation-D})."}
+{"_id": "98", "title": "spaces-more-morphisms-lemma-sheaf", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings over $B$. Assume given a morphism $a : X \\to Y$ and a map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of $\\mathcal{O}_X$-modules. For an object $U'$ of $(X')_{spaces, \\etale}$ with $U = X \\times_{X'} U'$ consider morphisms $a' : U' \\to Y'$ such that \\begin{enumerate} \\item $a'$ is a morphism over $B$, \\item $a'|_U = a|_U$, and \\item the induced map $a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$ is the restriction of $A$ to $U$. \\end{enumerate} Then the rule \\begin{equation} \\label{equation-sheaf} U' \\mapsto \\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\} \\end{equation} defines a sheaf of sets on $(X')_{spaces, \\etale}$."}
+{"_id": "99", "title": "spaces-more-morphisms-lemma-action-sheaf", "text": "Same notation and assumptions as in Lemma \\ref{lemma-sheaf}. We identify sheaves on $X$ and $X'$ via (\\ref{equation-equivalence-etale-spaces}). There is an action of the sheaf $$ \\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/B}, \\mathcal{C}_{X/X'}) $$ on the sheaf (\\ref{equation-sheaf}). Moreover, the action is simply transitive for any object $U'$ of $(X')_{spaces, \\etale}$ over which the sheaf (\\ref{equation-sheaf}) has a section."}
+{"_id": "100", "title": "spaces-more-morphisms-lemma-action-by-derivations-etale-localization", "text": "Let $S$ be a scheme. Consider a commutative diagram of first order thickenings $$ \\vcenter{ \\xymatrix{ (T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & & (X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\ (T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & & (X_1 \\subset X_1') } } \\quad \\begin{matrix} \\text{and a commutative} \\\\ \\text{diagram} \\end{matrix} \\quad \\vcenter{ \\xymatrix{ X_2' \\ar[r] \\ar[d] & B_2 \\ar[d] \\\\ X_1' \\ar[r] & B_1 } } $$ of algebraic spaces over $S$ with $X_2 \\to X_1$ and $B_2 \\to B_1$ \\'etale. For any $\\mathcal{O}_{T_1}$-linear map $\\theta_1 : a_1^*\\Omega_{X_1/B_1} \\to \\mathcal{C}_{T_1/T'_1}$ let $\\theta_2$ be the composition $$ \\xymatrix{ a_2^*\\Omega_{X_2/B_2} \\ar@{=}[r] & h^*a_1^*\\Omega_{X_1/B_1} \\ar[r]^-{h^*\\theta_1} & h^*\\mathcal{C}_{T_1/T'_1} \\ar[r] & \\mathcal{C}_{T_2/T'_2} } $$ (equality sign is explained in the proof). Then the diagram $$ \\xymatrix{ T_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\ T_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1 } $$ commutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$ are as in Remark \\ref{remark-action-by-derivations}."}
+{"_id": "101", "title": "spaces-more-morphisms-lemma-deform", "text": "Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a morphism of first order thickenings of algebraic spaces over $S$. Assume that $f$ is flat. Then the following are equivalent \\begin{enumerate} \\item $f'$ is flat and $X = Y \\times_{Y'} X'$, and \\item the canonical map $f^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "103", "title": "spaces-more-morphisms-lemma-deform-property", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (B \\subset B') } $$ of thickenings of algebraic spaces over $S$. Assume $B \\subset B'$ is a finite order thickening, $X'$ flat over $B'$, $X = B \\times_{B'} X'$, and $Y = B \\times_{B'} Y'$. Then \\begin{enumerate} \\item $f$ is representable if and only if $f'$ is representable, \\label{item-representable} \\item $f$ is flat if and only if $f'$ is flat, \\label{item-flat} \\item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \\label{item-isomorphism} \\item $f$ is an open immersion if and only if $f'$ is an open immersion, \\label{item-open-immersion} \\item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \\label{item-quasi-compact} \\item $f$ is universally closed if and only if $f'$ is universally closed, \\label{item-universally-closed} \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\label{item-separated} \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\label{item-monomorphism} \\item $f$ is surjective if and only if $f'$ is surjective, \\label{item-surjective} \\item $f$ is universally injective if and only if $f'$ is universally injective, \\label{item-universally-injective} \\item $f$ is affine if and only if $f'$ is affine, \\label{item-affine} \\item \\label{item-finite-type} $f$ is locally of finite type if and only if $f'$ is locally of finite type, \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\label{item-quasi-finite} \\item \\label{item-finite-presentation} $f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation, \\item \\label{item-relative-dimension-d} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $f$ is universally open if and only if $f'$ is universally open, \\label{item-universally-open} \\item $f$ is syntomic if and only if $f'$ is syntomic, \\label{item-syntomic} \\item $f$ is smooth if and only if $f'$ is smooth, \\label{item-smooth} \\item $f$ is unramified if and only if $f'$ is unramified, \\label{item-unramified} \\item $f$ is \\'etale if and only if $f'$ is \\'etale, \\label{item-etale} \\item $f$ is proper if and only if $f'$ is proper, \\label{item-proper} \\item $f$ is integral if and only if $f'$ is integral, \\label{item-integral} \\item $f$ is finite if and only if $f'$ is finite, \\label{item-finite} \\item \\label{item-finite-locally-free} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \\item add more here. \\end{enumerate}"}
+{"_id": "104", "title": "spaces-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (B \\subset B') } $$ of thickenings of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $Y' \\to B'$ is locally of finite type, \\item $X' \\to B'$ is flat and locally of finite presentation, \\item $f$ is flat, and \\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$. \\end{enumerate} Then $f'$ is flat and for all $y' \\in |Y'|$ in the image of $|f'|$ the morphism $Y' \\to B'$ is flat at $y'$."}
+{"_id": "105", "title": "spaces-more-morphisms-lemma-deform-property-fp-over-ft", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (B \\subset B') } $$ of thickenings of algebraic spaces over $S$. Assume $Y' \\to B'$ locally of finite type, $X' \\to B'$ flat and locally of finite presentation, $X = B \\times_{B'} X'$, and $Y = B \\times_{B'} Y'$. Then \\begin{enumerate} \\item $f$ is representable if and only if $f'$ is representable, \\label{item-representable-fp-over-ft} \\item $f$ is flat if and only if $f'$ is flat, \\label{item-flat-fp-over-ft} \\item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \\label{item-isomorphism-fp-over-ft} \\item $f$ is an open immersion if and only if $f'$ is an open immersion, \\label{item-open-immersion-fp-over-ft} \\item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \\label{item-quasi-compact-fp-over-ft} \\item $f$ is universally closed if and only if $f'$ is universally closed, \\label{item-universally-closed-fp-over-ft} \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\label{item-separated-fp-over-ft} \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\label{item-monomorphism-fp-over-ft} \\item $f$ is surjective if and only if $f'$ is surjective, \\label{item-surjective-fp-over-ft} \\item $f$ is universally injective if and only if $f'$ is universally injective, \\label{item-universally-injective-fp-over-ft} \\item $f$ is affine if and only if $f'$ is affine, \\label{item-affine-fp-over-ft} \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\label{item-quasi-finite-fp-over-ft} \\item \\label{item-relative-dimension-d-fp-over-ft} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $f$ is universally open if and only if $f'$ is universally open, \\label{item-universally-open-fp-over-ft} \\item $f$ is syntomic if and only if $f'$ is syntomic, \\label{item-syntomic-fp-over-ft} \\item $f$ is smooth if and only if $f'$ is smooth, \\label{item-smooth-fp-over-ft} \\item $f$ is unramified if and only if $f'$ is unramified, \\label{item-unramified-fp-over-ft} \\item $f$ is \\'etale if and only if $f'$ is \\'etale, \\label{item-etale-fp-over-ft} \\item $f$ is proper if and only if $f'$ is proper, \\label{item-proper-fp-over-ft} \\item $f$ is finite if and only if $f'$ is finite, \\label{item-finite-fp-over-ft} \\item \\label{item-finite-locally-free-fp-over-ft} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \\item add more here. \\end{enumerate}"}
+{"_id": "106", "title": "spaces-more-morphisms-lemma-composition-formally-smooth", "text": "A composition of formally smooth morphisms is formally smooth."}
+{"_id": "108", "title": "spaces-more-morphisms-lemma-formally-etale-unramified-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. Then $f$ is formally \\'etale if and only if $f$ is formally smooth and formally unramified."}
+{"_id": "109", "title": "spaces-more-morphisms-lemma-helper-formally-smooth", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. If the vertical arrows are \\'etale and $f$ is formally smooth, then $\\psi$ is formally smooth."}
+{"_id": "110", "title": "spaces-more-morphisms-lemma-smooth-formally-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is smooth. \\item The morphism $f$ is locally of finite presentation, and formally smooth. \\end{enumerate}"}
+{"_id": "111", "title": "spaces-more-morphisms-lemma-smooth-strong-lift", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X \\ar[d] & T \\ar[l] \\ar[d] \\\\ Y & T' \\ar[l] } $$ of algebraic spaces over $S$ where $X \\to Y$ is smooth and $T \\to T'$ is a thickening. Then there exists an \\'etale covering $\\{T'_i \\to T'\\}$ such that we can find the dotted arrow in $$ \\xymatrix{ X \\ar[d] & T \\ar[l] \\ar[d] & T \\times_{T'} T'_i \\ar[l] \\ar[d] \\\\ Y & T' \\ar[l] & T'_i \\ar[l] \\ar@{..>}[llu] } $$ making the diagram commute (for all $i$)."}
+{"_id": "112", "title": "spaces-more-morphisms-lemma-formally-smooth-sheaf-differentials", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a formally smooth morphism of algebraic spaces over $S$. Then $\\Omega_{X/Y}$ is locally projective on $X$."}
+{"_id": "113", "title": "spaces-more-morphisms-lemma-h1-is-zero", "text": "Let $T$ be an affine scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_T$-modules on $T_\\etale$. Consider the internal hom sheaf $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$ on $T_\\etale$. If $\\mathcal{F}$ is locally projective, then $H^1(T_\\etale, \\mathcal{H}) = 0$."}
+{"_id": "114", "title": "spaces-more-morphisms-lemma-formally-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is formally smooth, \\item for every diagram $$ \\xymatrix{ U \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \\'etale the morphism of schemes $\\psi$ is formally smooth (as in More on Morphisms, Definition \\ref{more-morphisms-definition-formally-unramified}), and \\item for one such diagram with surjective vertical arrows the morphism $\\psi$ is formally smooth. \\end{enumerate}"}
+{"_id": "119", "title": "spaces-more-morphisms-lemma-lifting-along-artinian-at-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$. Assume that $Y$ is locally Noetherian and $f$ locally of finite type. The following are equivalent: \\begin{enumerate} \\item $f$ is smooth at $x$, \\item for every solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\ Y & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu] } $$ where $B' \\to B$ is a surjection of local rings with $\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the closed point of $\\Spec(B)$ to $x$ there exists a dotted arrow making the diagram commute, and \\item same as in (2) but with $B' \\to B$ ranging over small extensions (see Algebra, Definition \\ref{algebra-definition-small-extension}). \\end{enumerate}"}
+{"_id": "121", "title": "spaces-more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $Z \\subset Y$ be a closed subspace with $n$th infinitesimal neighbourhood $Z_n \\subset Y$. Set $X_n = Z_n \\times_Y X$. \\begin{enumerate} \\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$. \\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$ is \\'etale at every point of $f^{-1}(Z)$. \\end{enumerate}"}
+{"_id": "122", "title": "spaces-more-morphisms-lemma-NL-etale-localization", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ U \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\ X \\ar[r]^f & Y } $$ of algebraic spaces over $S$ with $p$ and $q$ \\'etale. Then there is a canonical identification $\\NL_{X/Y}|_{U_\\etale} = \\NL_{U/V}$ in $D(\\mathcal{O}_U)$."}
+{"_id": "123", "title": "spaces-more-morphisms-lemma-NL-compare-spaces-schemes", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$. Then there is a canonical identification $\\NL_{X/Y} = \\epsilon^*\\NL_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$ where $\\epsilon$ is as in Derived Categories of Spaces, Section \\ref{spaces-perfect-section-derived-quasi-coherent-etale} and $\\NL_{X_0/Y_0}$ is as in More on Morphisms, Definition \\ref{more-morphisms-definition-netherlander}."}
+{"_id": "124", "title": "spaces-more-morphisms-lemma-netherlander-quasi-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The cohomology sheaves of the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$."}
+{"_id": "125", "title": "spaces-more-morphisms-lemma-netherlander-fp", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\\NL_{X/Y}$ is \\'etale locally on $X$ quasi-isomorphic to a complex $$ \\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots $$ of quasi-coherent $\\mathcal{O}_X$-modules with $\\mathcal{F}^0$ of finite presentation and $\\mathcal{F}^{-1}$ of finite type."}
+{"_id": "126", "title": "spaces-more-morphisms-lemma-NL-formally-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is formally smooth, \\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$ is locally projective. \\end{enumerate}"}
+{"_id": "129", "title": "spaces-more-morphisms-lemma-flat-locus-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $g$ is flat, $f$ is locally of finite presentation, and $\\mathcal{F}$ is locally of finite presentation. Then $$ \\{x' \\in |X'| : (g')^*\\mathcal{F}\\text{ is flat over }Y'\\text{ at }x'\\} $$ is the inverse image of the open subset of Theorem \\ref{theorem-openness-flatness} under the continuous map $|g'| : |X'| \\to |X|$."}
+{"_id": "130", "title": "spaces-more-morphisms-lemma-flat-on-fibres-at-point", "text": "In the situation above the following are equivalent \\begin{enumerate} \\item Pick a geometric point $\\overline{x}$ of $X$ lying over $x$. Set $\\overline{y} = f \\circ \\overline{x}$ and $\\overline{z} = g \\circ \\overline{x}$. Then the module $\\mathcal{F}_{\\overline{x}}/ \\mathfrak m_{\\overline{z}}\\mathcal{F}_{\\overline{x}}$ is flat over $\\mathcal{O}_{Y, \\overline{y}}/ \\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}$. \\item Pick a morphism $x : \\Spec(K) \\to X$ in the equivalence class of $x$. Set $z = g \\circ x$, $X_z = \\Spec(K) \\times_{z, Z} X$, $Y_z = \\Spec(K) \\times_{z, Z} Y$, and $\\mathcal{F}_z$ the pullback of $\\mathcal{F}$ to $X_z$. Then $\\mathcal{F}_z$ is flat at $x$ over $Y_z$ (as defined in Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-flat-module}). \\item Pick a commutative diagram $$ \\xymatrix{ & & & U \\ar[llld]_a \\ar[rr] \\ar[dr] & & V \\ar[llld]_>>>>>>>b \\ar[dl] \\\\ X \\ar[rr]_f \\ar[dr]_g & & Y \\ar[dl]^h &  & W \\ar[llld]_c \\\\ & Z } $$ where $U, V, W$ are schemes, and $a, b, c$ are \\'etale, and a point $u \\in U$ mapping to $x$. Let $w \\in W$ be the image of $u$. Let $\\mathcal{F}_w$ be the pullback of $\\mathcal{F}$ to the fibre $U_w$ of $U \\to W$ at $w$. Then $\\mathcal{F}_w$ is flat over $V_w$ at $u$. \\end{enumerate}"}
+{"_id": "132", "title": "spaces-more-morphisms-lemma-base-change-criterion-flatness-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $Z$, \\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, \\item $\\mathcal{F}$ is flat over $Z$, and \\item $Y$ is locally of finite type over $Z$. \\end{enumerate} Then the set $$ A = \\{x \\in |X| : \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}. $$ is open in $|X|$ and its formation commutes with arbitrary base change: If $Z' \\to Z$ is a morphism of algebraic spaces, and $A'$ is the set of points of $X' = X \\times_Z Z'$ where $\\mathcal{F}' = \\mathcal{F} \\times_Z Z'$ is flat over $Y' = Y \\times_Z Z'$, then $A'$ is the inverse image of $A$ under the continuous map $|X'| \\to |X|$."}
+{"_id": "133", "title": "spaces-more-morphisms-lemma-base-change-flatness-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $Z$, \\item $X$ is flat over $Z$, and \\item $Y$ is locally of finite type over $Z$. \\end{enumerate} Then the set $$ \\{x \\in |X| : X\\text{ flat at }x \\text{ over }Y\\}. $$ is open in $|X|$ and its formation commutes with arbitrary base change $Z' \\to Z$."}
+{"_id": "134", "title": "spaces-more-morphisms-lemma-flat-and-free-at-point-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $x \\in |X|$ with image $y \\in |Y|$. If $\\mathcal{F}$ is flat at $x$ over $Y$, then the following are equivalent \\begin{enumerate} \\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a flat $\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module, \\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a free $\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module, \\item $\\mathcal{F}_{\\overline{y}}$ is finite free in an \\'etale neighbourhood of $\\overline{x}$ in $X_{\\overline{y}}$, and \\item $\\mathcal{F}$ is finite free in an \\'etale neighbourhood of $x$ in $X$. \\end{enumerate} Here $\\overline{x}$ is a geometric point of $X$ lying over $x$ and $\\overline{y} = f \\circ \\overline{x}$."}
+{"_id": "135", "title": "spaces-more-morphisms-lemma-finite-free-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite presentation. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module flat over $Y$. Then the set $$ \\{x \\in |X| : \\mathcal{F}\\text{ free in an \\'etale neighbourhood of }x\\} $$ is open in $|X|$ and its formation commutes with arbitrary base change $Y' \\to Y$."}
+{"_id": "137", "title": "spaces-more-morphisms-lemma-flatness-over-Noetherian-ring", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $X$ be an algebraic space locally of finite presentation over $S = \\Spec(A)$. For $n \\geq 1$ set $S_n = \\Spec(A/I^n)$ and $X_n = S_n \\times_S X$. Let $\\mathcal{F}$ be coherent $\\mathcal{O}_X$-module. If for every $n \\geq 1$ the pullback $\\mathcal{F}_n$ of $\\mathcal{F}$ to $X$ is flat over $S_n$, then the (open) locus where $\\mathcal{F}$ is flat over $X$ contains the inverse image of $V(I)$ under $X \\to S$."}
+{"_id": "138", "title": "spaces-more-morphisms-lemma-integral-closure-smooth-pullback", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a smooth morphism of algebraic spaces over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The integral closure of $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$ where $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$."}
+{"_id": "139", "title": "spaces-more-morphisms-lemma-normalization-smooth-localization", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Y_2 \\ar[r] \\ar[d] & Y_1 \\ar[d]^f \\\\ X_2 \\ar[r]^\\varphi & X_1 } $$ be a fibre square of algebraic spaces over $S$. Assume $f$ is quasi-compact and quasi-separated and $\\varphi$ is smooth. Let $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$. Then $X_2' \\cong X_2 \\times_{X_1} X_1'$."}
+{"_id": "140", "title": "spaces-more-morphisms-lemma-CM-local-ring-fibre", "text": "The property of morphisms of germs of schemes \\begin{align*} & \\mathcal{P}((X, x) \\to (S, s)) = \\\\ & \\text{the local ring } \\mathcal{O}_{X_s, x} \\text{ of the fibre is Noetherian and Cohen-Macaulay} \\end{align*} is \\'etale local on the source-and-target (Descent, Definition \\ref{descent-definition-local-source-target-at-point})."}
+{"_id": "141", "title": "spaces-more-morphisms-lemma-CM", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is Cohen-Macaulay, \\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$ where $V$ is a scheme, the fibres of $X_V \\to V$ are Cohen-Macaulay algebraic spaces, and \\item $f$ is flat and for any \\'etale morphism $V \\to Y$ where $V$ is a scheme, the fibres of $X_V \\to V$ are Cohen-Macaulay algebraic spaces. \\end{enumerate} Given $x \\in |X|$ with image $y \\in |Y|$ the following are equivalent \\begin{enumerate} \\item[(a)] $f$ is Cohen-Macaulay at $x$, and \\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$ is flat and $\\mathcal{O}_{X, \\overline{x}}/ \\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Cohen-Macaulay. \\end{enumerate}"}
+{"_id": "142", "title": "spaces-more-morphisms-lemma-composition-CM", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. Assume that the fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian. Let $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$. \\begin{enumerate} \\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$. \\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay. \\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$. \\item If $f \\circ g$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$. \\end{enumerate}"}
+{"_id": "145", "title": "spaces-more-morphisms-lemma-flat-finite-presentation-CM-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $$ W = \\{x \\in |X| : f\\text{ is Cohen-Macaulay at }x\\} $$ Then $W$ is open in $|X|$ and the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : Y' \\to Y$, consider the base change $f' : X' \\to Y'$ of $f$ and the projection $g' : X' \\to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$."}
+{"_id": "146", "title": "spaces-more-morphisms-lemma-lfp-CM-relative-dimension", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is locally of finite presentation and Cohen-Macaulay. Then there exist open and closed subschemes $X_d \\subset X$ such that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to Y$ has relative dimension $d$."}
+{"_id": "147", "title": "spaces-more-morphisms-lemma-gorenstein-local-ring-fibre", "text": "The property of morphisms of germs of schemes \\begin{align*} & \\mathcal{P}((X, x) \\to (S, s)) = \\\\ & \\text{the local ring } \\mathcal{O}_{X_s, x} \\text{ of the fibre is Noetherian and Gorenstein} \\end{align*} is \\'etale local on the source-and-target (Descent, Definition \\ref{descent-definition-local-source-target-at-point})."}
+{"_id": "148", "title": "spaces-more-morphisms-lemma-gorenstein", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is Gorenstein, \\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$ where $V$ is a scheme, the fibres of $X_V \\to V$ are Gorenstein algebraic spaces, and \\item $f$ is flat and for any \\'etale morphism $V \\to Y$ where $V$ is a scheme, the fibres of $X_V \\to V$ are Gorenstein algebraic spaces. \\end{enumerate} Given $x \\in |X|$ with image $y \\in |Y|$ the following are equivalent \\begin{enumerate} \\item[(a)] $f$ is Gorenstein at $x$, and \\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$ is flat and $\\mathcal{O}_{X, \\overline{x}}/ \\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Gorenstein. \\end{enumerate}"}
+{"_id": "149", "title": "spaces-more-morphisms-lemma-composition-gorenstein", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. Assume that the fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian. Let $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$. \\begin{enumerate} \\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$, then $g \\circ f$ is Gorenstein at $x$. \\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein. \\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$, then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$. \\item If $f \\circ g$ is Gorenstein and $f$ is flat, then $f$ is Gorenstein and $g$ is Gorenstein at every point in the image of $f$. \\end{enumerate}"}
+{"_id": "152", "title": "spaces-more-morphisms-lemma-flat-finite-presentation-gorenstein-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $$ W = \\{x \\in |X| : f\\text{ is Gorenstein at }x\\} $$ Then $W$ is open in $|X|$ and the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : Y' \\to Y$, consider the base change $f' : X' \\to Y'$ of $f$ and the projection $g' : X' \\to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$."}
+{"_id": "153", "title": "spaces-more-morphisms-lemma-slice", "text": "Let $S$ be a scheme. Consider a cartesian diagram $$ \\xymatrix{ X \\ar[d] & F \\ar[l]^p \\ar[d] \\\\ Y & \\Spec(k) \\ar[l] } $$ where $X \\to Y$ is a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation, and where $k$ is a field over $S$. Let $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$ and $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence in the local ring $\\mathcal{O}_{F, \\overline{z}}$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism $$ V(f_1, \\ldots, f_r) \\longrightarrow Y $$ is flat and locally of finite presentation."}
+{"_id": "154", "title": "spaces-more-morphisms-lemma-geometrically-reduced-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. The following are equivalent \\begin{enumerate} \\item for some morphism $\\Spec(k) \\to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is geometrically reduced over $k$, \\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is geometrically reduced over $k$, \\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class of $y$ the algebraic space $X_k$ is reduced. \\end{enumerate}"}
+{"_id": "155", "title": "spaces-more-morphisms-lemma-base-change-fibres-geometrically-reduced", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$ be morphisms of algebraic spaces over $S$. Denote $f' : X' \\to Y'$ the base change of $f$ by $g$. Then \\begin{align*} \\{y' \\in |Y'| : \\text{the fibre of }f' : X' \\to Y'\\text{ at }y' \\text{ is geometrically reduced}\\} \\\\ = g^{-1}(\\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y \\text{ is geometrically reduced}\\}). \\end{align*}"}
+{"_id": "156", "title": "spaces-more-morphisms-lemma-geometrically-reduced-constructible", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is quasi-compact and locally of finite presentation. Then the set $$ E = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y \\text{ is geometrically reduced}\\} $$ is \\'etale locally constructible."}
+{"_id": "157", "title": "spaces-more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "text": "Let $X$ be an algebraic space over a discrete valuation ring $R$ whose structure morphism $X \\to \\Spec(R)$ is proper and flat. If the special fibre is reduced, then both $X$ and the generic fibre $X_\\eta$ are reduced."}
+{"_id": "158", "title": "spaces-more-morphisms-lemma-geometrically-reduced-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat, proper, and of finite presentation, then the set $$ E = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y \\text{ is geometrically reduced}\\} $$ is open in $|Y|$."}
+{"_id": "159", "title": "spaces-more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $$ n_{X/Y} : |Y| \\to \\{0, 1, 2, 3, \\ldots, \\infty\\} $$ be the function which associates to $y \\in Y$ the number of connected components of $X_k$ where $\\Spec(k) \\to Y$ is in the equivalence class of $y$ with $k$ algebraically closed. This is well defined and if $g : Y' \\to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \\circ g $$ where $X' \\to Y'$ is the base change of $f$."}
+{"_id": "160", "title": "spaces-more-morphisms-lemma-dimension-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \\in |Y|$. The following quantities are the same \\begin{enumerate} \\item the minimal integer $d$ such that $f$ has relative dimension $\\leq d$ at every $x \\in |X|$ mapping to $y$, \\item the dimension of the algebraic space $X_k = \\Spec(k) \\times_Y X$ for any morphism $\\Spec(k) \\to Y$ in the equivalence class defining $y$. \\end{enumerate}"}
+{"_id": "161", "title": "spaces-more-morphisms-lemma-base-change-dimension-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of algebraic spaces over $S$. Let $$ n_{X/Y} : |Y| \\to \\{0, 1, 2, 3, \\ldots, \\infty\\} $$ be the function which associates to $y \\in |Y|$ the integer discussed in Lemma \\ref{lemma-dimension-fibre}. If $g : Y' \\to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \\circ |g| $$ where $X' \\to Y'$ is the base change of $f$."}
+{"_id": "162", "title": "spaces-more-morphisms-lemma-dimension-fibres-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of finite presentation of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is lower semi-continuous."}
+{"_id": "163", "title": "spaces-more-morphisms-lemma-dimension-fibres-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is upper semi-continuous."}
+{"_id": "164", "title": "spaces-more-morphisms-lemma-dimension-fibres-proper-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat, finitely presented morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is locally constant."}
+{"_id": "165", "title": "spaces-more-morphisms-lemma-construct-glueing", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic spaces over $S$. Let $y \\in |Y|$ be a point which can be represented by a closed immersion $y : \\Spec(k) \\to Y$. Then there exists a factorization $X \\to X' \\to Y$ of $f$ such that \\begin{enumerate} \\item $X' \\to Y$ is integral, \\item $X \\to X'$ is an isomorphism over $X' \\setminus X'_y$, \\item $X'_y$ has a unique point $x'$ with $\\kappa(x') = k$. \\end{enumerate} Moreover, if $f$ is finite and $Y$ is locally Noetherian, then $X' \\to Y$ is finite."}
+{"_id": "167", "title": "spaces-more-morphisms-lemma-etale-splits-off-quasi-finite-part", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. Let $x_1, \\ldots, x_n \\in |X|$ mapping to $y$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is separated, \\item $f$ is quasi-finite at $x_1, \\ldots, x_n$, and \\item $f$ is quasi-compact or $Y$ is decent. \\end{enumerate} Then there exists an \\'etale morphism $(U, u) \\to (Y, y)$ of pointed algebraic spaces and a decomposition $$ U \\times_Y X = W \\amalg V $$ into open and closed subspaces such that the morphism $V \\to U$ is finite, every point of the fibre of $|V| \\to |U|$ over $u$ maps to an $x_i$, and the fibre of $|W| \\to |U|$ over $u$ contains no point mapping to an $x_i$."}
+{"_id": "169", "title": "spaces-more-morphisms-lemma-finite-type-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is of finite type and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\ & Y & } $$ Then there exists an open subspace $U' \\subset Y'$ such that \\begin{enumerate} \\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and \\item $(f')^{-1}(U') \\subset X$ is the set of points at which $f$ is quasi-finite. \\end{enumerate}"}
+{"_id": "170", "title": "spaces-more-morphisms-lemma-quasi-finite-separated-quasi-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\ & Y & } $$ Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral. In particular $f$ is quasi-affine."}
+{"_id": "171", "title": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated and assume that $Y$ is quasi-compact and quasi-separated. Then there exists a factorization $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\ & Y & } $$ where $j$ is a quasi-compact open immersion and $\\pi$ is finite."}
+{"_id": "172", "title": "spaces-more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "text": "With notation and hypotheses as in Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}. Assume moreover that $f$ is locally of finite presentation. Then we can choose the factorization such that $T$ is finite and of finite presentation over $Y$."}
+{"_id": "173", "title": "spaces-more-morphisms-lemma-characterize-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is finite, \\item $f$ is proper and locally quasi-finite, \\item $f$ is proper and $|X_k|$ is a discrete space for every morphism $\\Spec(k) \\to Y$ where $k$ is a field, \\item $f$ is universally closed, separated, locally of finite type and $|X_k|$ is a discrete space for every morphism $\\Spec(k) \\to Y$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "174", "title": "spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. Assume \\begin{enumerate} \\item $f$ is proper, and \\item $f$ is quasi-finite at all $x \\in |X|$ lying over $y$ (Decent Spaces, Lemma \\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}). \\end{enumerate} Then there exists an open neighbourhood $V \\subset Y$ of $y$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite."}
+{"_id": "175", "title": "spaces-more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "text": "\\begin{slogan} Collapsing a fibre of a proper family forces nearby ones to collapse too. \\end{slogan} Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\ & B } $$ be a commutative diagram of morphism of algebraic spaces over $S$. Let $b \\in B$ and let $\\Spec(k) \\to B$ be a morphism in the equivalence class of $b$. Assume \\begin{enumerate} \\item $X \\to B$ is a proper morphism, \\item $Y \\to B$ is separated and locally of finite type, \\item one of the following is true \\begin{enumerate} \\item the image of $|X_k| \\to |Y_k|$ is finite, \\item the image of $|f|^{-1}(\\{b\\})$ in $|Y|$ is finite and $B$ is decent. \\end{enumerate} \\end{enumerate} Then there is an open subspace $B' \\subset B$ containing $b$ such that $X_{B'} \\to Y_{B'}$ factors through a closed subspace $Z \\subset Y_{B'}$ finite over $B'$."}
+{"_id": "176", "title": "spaces-more-morphisms-lemma-stein-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a universally closed and quasi-separated morphism of algebraic spaces over $S$. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\ & Y & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated, and surjective, \\item the morphism $\\pi : Y' \\to Y$ is integral, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$, \\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and \\item $Y'$ is the normalization of $Y$ in $X$ as defined in Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-normalization-X-in-Y}. \\end{enumerate} Formation of the factorization $f = \\pi \\circ f'$ commutes with flat base change."}
+{"_id": "177", "title": "spaces-more-morphisms-lemma-stein-universally-closed-residue-fields", "text": "In Lemma \\ref{lemma-stein-universally-closed} assume in addition that $f$ is locally of finite type and $Y$ affine. Then for $y \\in Y$ the fibre $\\pi^{-1}(\\{y\\}) = \\{y_1, \\ldots, y_n\\}$ is finite and the field extensions $\\kappa(y_i)/\\kappa(y)$ are finite."}
+{"_id": "178", "title": "spaces-more-morphisms-lemma-characterize-geometrically-connected-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$. Then $X_{\\overline{y}}$ is connected, if and only if for every \\'etale neighbourhood $(V, \\overline{v}) \\to (Y, \\overline{y})$ where $V$ is a scheme the base change $X_V \\to V$ has connected fibre $X_v$."}
+{"_id": "180", "title": "spaces-more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is proper, flat, and of finite presentation, then the function $n_{X/Y} : |Y| \\to \\mathbf{Z}$ counting the number of geometric connected components of fibres of $f$ (Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}) is lower semi-continuous."}
+{"_id": "181", "title": "spaces-more-morphisms-lemma-proper-flat-geom-red", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item the geometric fibres of $f$ are reduced. \\end{enumerate} Then the function $n_{X/S} : |Y| \\to \\mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ (Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}) is locally constant."}
+{"_id": "182", "title": "spaces-more-morphisms-lemma-stein-factorization-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. Let $X \\to Y' \\to Y$ be the Stein factorization of $f$ (Theorem \\ref{theorem-stein-factorization-general}). If $f$ is of finite presentation, flat, with geometrically reduced fibres (Definition \\ref{definition-geometrically-reduced-fibre}), then $Y' \\to Y$ is finite \\'etale."}
+{"_id": "183", "title": "spaces-more-morphisms-lemma-split-off-proper-part-henselian", "text": "Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space separated and of finite type over $A$. Set $X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$. Let $Y \\subset X_0$ be an open and closed subspace such that $Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed subspace $W \\subset X$ which is proper over $A$ with $W \\times_{\\Spec(A)} \\Spec(A/I) = Y$."}
+{"_id": "184", "title": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $V \\subset Y$ be an open subspace. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite type and flat over $Y$, \\item $V \\to Y$ is quasi-compact and scheme theoretically dense, \\item $\\mathcal{F}|_{f^{-1}V}$ is of finite presentation. \\end{enumerate} Then $\\mathcal{F}$ is of finite presentation."}
+{"_id": "185", "title": "spaces-more-morphisms-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $V \\subset Y$ be an open subspace. Assume \\begin{enumerate} \\item $f$ is locally of finite type and flat, \\item $V \\to Y$ is quasi-compact and scheme theoretically dense, \\item $f|_{f^{-1}V} : f^{-1}V \\to V$ is locally of finite presentation. \\end{enumerate} Then $f$ is of locally of finite presentation."}
+{"_id": "186", "title": "spaces-more-morphisms-lemma-flat-fp-dimension-over-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite type. Let $V \\subset Y$ be an open subspace such that $|V| \\subset |Y|$ is dense and such that $X_V \\to V$ has relative dimension $\\leq d$. If also either \\begin{enumerate} \\item $f$ is locally of finite presentation, or \\item $V \\to Y$ is quasi-compact, \\end{enumerate} then $f : X \\to Y$ has relative dimension $\\leq d$."}
+{"_id": "187", "title": "spaces-more-morphisms-lemma-proper-flat-finite-over-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and proper. Let $V \\to Y$ be an open subspace with $|V| \\subset |Y|$ dense such that $X_V \\to V$ is finite. If also either $f$ is locally of finite presentation or $V \\to Y$ is quasi-compact, then $f$ is finite."}
+{"_id": "188", "title": "spaces-more-morphisms-lemma-zariski", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $V \\subset Y$ be an open subspace. If \\begin{enumerate} \\item $f$ is separated, locally of finite type, and flat, \\item $f^{-1}(V) \\to V$ is an isomorphism, and \\item $V \\to Y$ is quasi-compact and scheme theoretically dense, \\end{enumerate} then $f$ is an open immersion."}
+{"_id": "189", "title": "spaces-more-morphisms-lemma-push-ideal", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\varphi : W \\to X$ be a quasi-compact separated \\'etale morphism. Let $U \\subset X$ be a quasi-compact open subspace. Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be a finite type quasi-coherent sheaf of ideals such that $V(\\mathcal{I}) \\cap \\varphi^{-1}(U) = \\emptyset$. Then there exists a finite type quasi-coherent sheaf of ideals $\\mathcal{J} \\subset \\mathcal{O}_X$ such that \\begin{enumerate} \\item $V(\\mathcal{J}) \\cap U = \\emptyset$, and \\item $\\varphi^{-1}(\\mathcal{J})\\mathcal{O}_W = \\mathcal{I} \\mathcal{I}'$ for some finite type quasi-coherent ideal $\\mathcal{I}' \\subset \\mathcal{O}_W$. \\end{enumerate}"}
+{"_id": "190", "title": "spaces-more-morphisms-lemma-flat-after-blowing-up", "text": "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X$ be an algebraic space over $B$. Let $U \\subset B$ be a quasi-compact open subspace. Assume \\begin{enumerate} \\item $X \\to B$ is of finite type and quasi-separated, and \\item $X_U \\to U$ is flat and locally of finite presentation. \\end{enumerate} Then there exists a $U$-admissible blowup $B' \\to B$ such that the strict transform of $X$ is flat and of finite presentation over $B'$."}
+{"_id": "192", "title": "spaces-more-morphisms-lemma-zariski-after-blowup", "text": "Let $S$ be a scheme. Let $\\varphi : X \\to B$ be a morphism of algebraic spaces over $S$. Assume $\\varphi$ is of finite type with $B$ quasi-compact and quasi-separated. Let $U \\subset B$ be a quasi-compact open subspace such that $\\varphi^{-1}U \\to U$ is an isomorphism. Then there exists a $U$-admissible blowup $B' \\to B$ such that $U$ is scheme theoretically dense in $B'$ and such that the strict transform $X'$ of $X$ is isomorphic to an open subspace of $B'$."}
+{"_id": "193", "title": "spaces-more-morphisms-lemma-dominate-modification-by-blowup", "text": "Let $S$ be a scheme. Let $\\varphi : X \\to B$ be a proper morphism of algebraic spaces over $S$. Assume $B$ quasi-compact and quasi-separated. Let $U \\subset B$ be a quasi-compact open subspace such that $\\varphi^{-1}U \\to U$ is an isomorphism. Then there exists a $U$-admissible blowup $B' \\to B$ which dominates $X$, i.e., such that there exists a factorization $B' \\to X \\to B$ of the blowup morphism."}
+{"_id": "195", "title": "spaces-more-morphisms-lemma-find-common-blowups", "text": "Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \\to X_1$ and $U \\to X_2$ be open immersions of algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $Y$. Then there exists a commutative diagram $$ \\xymatrix{ X_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\ X_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2 } $$ of algebraic spaces over $Y$ where $X_i' \\to X_i$ is a $U$-admissible blowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite type over $Y$."}
+{"_id": "196", "title": "spaces-more-morphisms-lemma-blowup-to-find-embedding", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $U \\subset X$ be an open subspace. Assume \\begin{enumerate} \\item $U$ is quasi-compact, \\item $Y$ is quasi-compact and quasi-separated, \\item there exists an immersion $U \\to \\mathbf{P}^n_Y$ over $Y$, \\item $f$ is of finite type and separated. \\end{enumerate} Then there exists a commutative diagram $$ \\xymatrix{ & U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\ X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\ & Y & \\mathbf{P}^n_Y \\ar[l] } $$ where the arrows with source $U$ are open immersions, $X' \\to X$ is a $U$-admissible blowup, $X' \\to Z'$ is an open immersion, $Z' \\to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \\to Z$ is a $U$-admissible blowup and $Z \\to \\mathbf{P}^n_Y$ is a closed immersion."}
+{"_id": "197", "title": "spaces-more-morphisms-lemma-chow-noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated, of finite type, and $Y$ Noetherian. Then there exists a dense open subspace $U \\subset X$ and a commutative diagram $$ \\xymatrix{ & U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\ X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\ & Y & \\mathbf{P}^n_Y \\ar[l] } $$ where the arrows with source $U$ are open immersions, $X' \\to X$ is a $U$-admissible blowup, $X' \\to Z'$ is an open immersion, $Z' \\to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \\to Z$ is a $U$-admissible blowup and $Z \\to \\mathbf{P}^n_Y$ is a closed immersion."}
+{"_id": "198", "title": "spaces-more-morphisms-lemma-chow-noetherian-separated", "text": "\\begin{reference} \\cite[IV Theorem 3.1]{Kn} \\end{reference} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated of finite type, and $Y$ separated and Noetherian. Then there exists a commutative diagram $$ \\xymatrix{ X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_Y \\ar[ld] \\\\ & Y } $$ where $X' \\to X$ is a $U$-admissible blowup for some dense open $U \\subset X$ and the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion."}
+{"_id": "201", "title": "spaces-more-morphisms-lemma-inverse-systems-abelian", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. \\begin{enumerate} \\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian. \\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$ can be checked \\'etale locally. \\item For any flat morphism $f : X' \\to X$ of Noetherian algebraic spaces the functor $f^* : \\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(X', f^{-1}\\mathcal{I}\\mathcal{O}_{X'})$ is exact. \\end{enumerate}"}
+{"_id": "203", "title": "spaces-more-morphisms-lemma-exact", "text": "The functor (\\ref{equation-completion-functor}) is exact."}
+{"_id": "204", "title": "spaces-more-morphisms-lemma-completion-internal-hom", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Set $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$. Then $$ \\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) = \\Mor_{\\textit{Coh}(X, \\mathcal{I})} (\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge). $$"}
+{"_id": "205", "title": "spaces-more-morphisms-lemma-fully-faithful", "text": "In Situation \\ref{situation-existence}. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Assume that the intersection of the supports of $\\mathcal{F}$ and $\\mathcal{G}$ is proper over $\\Spec(A)$. Then the map $$ \\Mor_{\\textit{Coh}(\\mathcal{O}_X)}(\\mathcal{F}, \\mathcal{G}) \\longrightarrow \\Mor_{\\textit{Coh}(X, \\mathcal{I})} (\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge) $$ coming from (\\ref{equation-completion-functor}) is a bijection. In particular, (\\ref{equation-completion-functor-proper-over-A}) is fully faithful."}
+{"_id": "206", "title": "spaces-more-morphisms-lemma-existence-easy", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module, $(\\mathcal{F}_n)$ an object of $\\textit{Coh}(X, \\mathcal{I})$, and $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$ a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$. Then there exists a unique (up to unique isomorphism) triple $(\\mathcal{F}, a, \\beta)$ where \\begin{enumerate} \\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module, \\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$, \\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and \\item $\\alpha = a^\\wedge \\circ \\beta$. \\end{enumerate}"}
+{"_id": "207", "title": "spaces-more-morphisms-lemma-existence-tricky", "text": "In Situation \\ref{situation-existence}. Let $\\mathcal{K} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $X_e \\subset X$ be the closed subspace cut out by $\\mathcal{K}^e$. Let $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$. Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})$. Assume \\begin{enumerate} \\item the functor $\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_{X_e}) \\to \\textit{Coh}_{\\text{support proper over } A}(X_e, \\mathcal{I}_e)$ is an equivalence for all $e \\geq 1$, and \\item there exists an object $\\mathcal{H}$ of $\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_X)$ and a map $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$. \\end{enumerate} Then $(\\mathcal{F}_n)$ is in the essential image of (\\ref{equation-completion-functor-proper-over-A})."}
+{"_id": "208", "title": "spaces-more-morphisms-lemma-inverse-systems-push-pull", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable proper morphism of Noetherian algebraic spaces over $S$. Let $\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$ be quasi-coherent sheaves of ideals. Assume $f$ is an isomorphism over $V = Y \\setminus V(\\mathcal{K})$. Set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$. Let $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$, let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let $\\beta : (f^*\\mathcal{G}_n)  \\to \\mathcal{F}^\\wedge$ be an isomorphism in $\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map $$ \\alpha : (\\mathcal{G}_n) \\longrightarrow (f_*\\mathcal{F})^\\wedge $$ in $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$."}
+{"_id": "209", "title": "spaces-more-morphisms-lemma-algebraize-formal-closed-subscheme", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Suppose given a commutative diagram $$ \\xymatrix{ Z_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\ X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots } $$ of algebraic spaces with cartesian squares. Assume that \\begin{enumerate} \\item $Z_1 \\to X_1$ is a closed immersion, and \\item $Z_1 \\to S_1$ is proper. \\end{enumerate} Then there exists a closed immersion of algebraic spaces $Z \\to X$ such that $Z_n = Z \\times_S S_n$ for all $n \\geq 1$. Moreover, $Z$ is proper over $S$."}
+{"_id": "211", "title": "spaces-more-morphisms-lemma-algebraize-morphism", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be algebraic spaces over $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and $Y_n = Y \\times_S S_n$. Suppose given a compatible system of commutative diagrams $$ \\xymatrix{ & & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\ X_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\ & S_n \\ar[rru] } $$ Assume that \\begin{enumerate} \\item $X \\to S$ is proper, and \\item $Y \\to S$ is separated of finite type. \\end{enumerate} Then there exists a unique morphism of algebraic spaces $g : X \\to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$."}
+{"_id": "212", "title": "spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring. Set $S = \\Spec(A)$ and $S_n = \\Spec(A/\\mathfrak m^n)$. Consider a commutative diagram $$ \\xymatrix{ X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] & \\ldots \\\\ S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots } $$ of algebraic spaces with cartesian squares. If $\\dim(X_1) \\leq 1$, then there exists a projective morphism of schemes $X \\to S$ and isomorphisms $X_n \\cong X \\times_S S_n$ compatible with $i_n$."}
+{"_id": "213", "title": "spaces-more-morphisms-lemma-projective-over-complete", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a complete Noetherian local ring. Let $X$ be an algebraic space over $\\Spec(A)$. If $X \\to \\Spec(A)$ is proper and $\\dim(X_\\kappa) \\leq 1$, then $X$ is a scheme projective over $A$."}
+{"_id": "214", "title": "spaces-more-morphisms-lemma-representable-etale-local-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the target. Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Consider commutative diagrams $$ \\xymatrix{ X \\times_Y V \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ where $V$ is a scheme and $V \\to Y$ is \\'etale. The following are equivalent \\begin{enumerate} \\item for any diagram as above the projection $X \\times_Y V \\to V$ has property $\\mathcal{P}$, and \\item for some diagram as above with $V \\to Y$ surjective the projection $X \\times_Y V \\to V$ has property $\\mathcal{P}$. \\end{enumerate} If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\\mathcal{P}$."}
+{"_id": "215", "title": "spaces-more-morphisms-lemma-regular-quasi-regular-immersion", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. We have the following implications: $i$ is Koszul-regular $\\Rightarrow$ $i$ is $H_1$-regular $\\Rightarrow$ $i$ is quasi-regular."}
+{"_id": "218", "title": "spaces-more-morphisms-lemma-quasi-regular-immersion", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied \\begin{enumerate} \\item $i$ is locally of finite presentation, \\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and \\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism. \\end{enumerate}"}
+{"_id": "223", "title": "spaces-more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $m \\in \\mathbf{Z}$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. With notation as explained in Remark \\ref{remark-match-relative-pseudo-coherence} the following are equivalent: \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$, \\item for some commutative diagram as in (1) with $U \\to X$ surjective, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$, \\item for every commutative diagram as in (1) with $U$ and $V$ affine the complex $R\\Gamma(U, E)$ of $\\mathcal{O}_X(U)$-modules is $m$-pseudo-coherent relative to $\\mathcal{O}_Y(V)$. \\end{enumerate}"}
+{"_id": "224", "title": "spaces-more-morphisms-lemma-relative-pseudo-coherent-is-moot", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $E$ in $D_\\QCoh(\\mathcal{O}_X)$. If $f$ is flat and locally of finite presentation, then the following are equivalent \\begin{enumerate} \\item $E$ is pseudo-coherent relative to $Y$, and \\item $E$ is pseudo-coherent on $X$. \\end{enumerate}"}
+{"_id": "227", "title": "spaces-more-morphisms-lemma-pseudo-coherent-finite-presentation", "text": "A pseudo-coherent morphism is locally of finite presentation."}
+{"_id": "234", "title": "spaces-more-morphisms-lemma-perfect-proper-perfect-direct-image", "text": "Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \\to Y$ be a perfect proper morphism of algebraic spaces. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$."}
+{"_id": "237", "title": "spaces-more-morphisms-lemma-flat-base-change-lci", "text": "A flat base change of a local complete intersection morphism is a local complete intersection morphism."}
+{"_id": "238", "title": "spaces-more-morphisms-lemma-composition-lci", "text": "A composition of local complete intersection morphisms is a local complete intersection morphism."}
+{"_id": "239", "title": "spaces-more-morphisms-lemma-flat-lci", "text": "\\begin{slogan} Syntomic equals flat plus lci (for algebraic spaces). \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is flat and a local complete intersection morphism, and \\item $f$ is syntomic. \\end{enumerate}"}
+{"_id": "244", "title": "spaces-more-morphisms-lemma-base-change-lci-fibres", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces over $S$. Assume that both $p$ and $q$ are flat and locally of finite presentation. Then there exists an open subspace $U(f) \\subset X$ such that $|U(f)| \\subset |X|$ is the set of points where $f$ is Koszul. Moreover, for any morphism of algebraic spaces $Z' \\to Z$, if $f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then $U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$."}
+{"_id": "245", "title": "spaces-more-morphisms-lemma-unramified-lci", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free on $X$."}
+{"_id": "246", "title": "spaces-more-morphisms-lemma-transitivity-conormal-lci", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be formally unramified morphisms of algebraic spaces over $S$. Assume that $Z \\to Y$ is a local complete intersection morphism. The exact sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ of Lemma \\ref{lemma-transitivity-conormal} is short exact."}
+{"_id": "247", "title": "spaces-more-morphisms-lemma-where-unramified", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that $p$ is locally of finite type and closed. Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified."}
+{"_id": "248", "title": "spaces-more-morphisms-lemma-where-unramified-universally-injective", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is locally of finite type, \\item $p$ is closed, and \\item $p_2 : X \\times_Y X \\to Z$ is closed. \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified and universally injective."}
+{"_id": "249", "title": "spaces-more-morphisms-lemma-where-closed-immersion", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is locally of finite type, \\item $p$ is universally closed, and \\item $q : Y \\to Z$ is separated. \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a closed immersion."}
+{"_id": "250", "title": "spaces-more-morphisms-lemma-where-flat", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is locally of finite presentation, \\item $p$ is flat, \\item $p$ is closed, and \\item $q$ is locally of finite type. \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is flat."}
+{"_id": "251", "title": "spaces-more-morphisms-lemma-where-surjective-flat", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is locally of finite presentation, \\item $p$ is flat, \\item $p$ is closed, \\item $q$ is locally of finite type, and \\item $q$ is closed. \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is surjective and flat."}
+{"_id": "252", "title": "spaces-more-morphisms-lemma-where-isomorphism", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is locally of finite presentation, \\item $p$ is flat, \\item $p$ is universally closed, \\item $q$ is locally of finite type, \\item $q$ is closed, and \\item $q$ is separated. \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is an isomorphism."}
+{"_id": "253", "title": "spaces-more-morphisms-lemma-where-lci", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces. Assume that \\begin{enumerate} \\item $p$ is flat and locally of finite presentation, \\item $p$ is closed, and \\item $q$ is flat and locally of finite presentation, \\end{enumerate} Then there exists an open subspace $W \\subset Z$ such that a morphism $Z' \\to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a local complete intersection morphism."}
+{"_id": "254", "title": "spaces-more-morphisms-lemma-case-of-tor-independence", "text": "Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces $$ \\xymatrix{ Z' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\ X' \\ar[r] & B' } $$ over $S$. Let $B \\to B'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $B$. Assume $Y' \\to B'$ and $Z' \\to X'$ are flat. Then $X \\times_B Y$ and $Z'$ are Tor independent over $X' \\times_{B'} Y'$."}
+{"_id": "255", "title": "spaces-more-morphisms-lemma-derived-chow", "text": "Let $A$ be a ring. Let $X$ be a separated algebraic space of finite presentation over $A$. Let $x \\in |X|$. Then there exist an $n \\geq 0$, a closed subspace $Z \\subset X \\times_A \\mathbf{P}^n_A$, a point $z \\in |Z|$, an open $V \\subset \\mathbf{P}^n_A$, and an object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that \\begin{enumerate} \\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation, \\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$, set $W = c^{-1}(V)$, \\item the restriction of $b : Z \\to X$ to $W$ is \\'etale, $z \\in W$, and $b(z) = x$, \\item $E|_{X \\times_A V} \\cong (b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$, \\item $E$ is pseudo-coherent and supported on $Z$. \\end{enumerate}"}
+{"_id": "256", "title": "spaces-more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "text": "Let $X/A$, $x \\in |X|$, and $n, Z, z, V, E$ be as in Lemma \\ref{lemma-derived-chow}. For any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have $$ Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(W \\to V)_*K|_W $$ where $p : X \\times_A \\mathbf{P}^n_A \\to X$ and $q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are the projections and where the morphism $W \\to V$ is the finitely presented closed immersion $c|_W : W \\to V$."}
+{"_id": "257", "title": "spaces-more-morphisms-lemma-characterize-pseudo-coherent", "text": "Let $A$ be a ring. Let $X$ be an algebraic space separated and of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$ (Definition \\ref{definition-relative-pseudo-coherence})."}
+{"_id": "258", "title": "spaces-more-morphisms-lemma-characterize-pseudo-coh-improved", "text": "Let $A$ be a ring. Let $X$ be an algebraic space separated and of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If  $R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every perfect  $E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$."}
+{"_id": "259", "title": "spaces-more-morphisms-lemma-affine-locally-rel-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent: \\begin{enumerate} \\item $E$ is $Y$-perfect, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r]_g & V \\ar[d] \\\\ X \\ar[r]^f & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex $E|_U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition \\ref{perfect-definition-relatively-perfect}, \\item for some commutative diagram as in (2) with $U \\to X$ surjective, the complex $E|_U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition \\ref{perfect-definition-relatively-perfect}, \\item for every commutative diagram as in (2) with $U$ and $V$ affine the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_Y(V)$-perfect. \\end{enumerate}"}
+{"_id": "261", "title": "spaces-more-morphisms-lemma-perfect-relatively-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. A perfect object of $D(\\mathcal{O}_X)$ is $Y$-perfect. If $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$ is $Y$-perfect if $K$ is perfect and $M$ is $Y$-perfect."}
+{"_id": "262", "title": "spaces-more-morphisms-lemma-base-change-relatively-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$. Set $X' = Y' \\times_Y X$ and denote $g' : X' \\to X$ the projection. If $K \\in D(\\mathcal{O}_X)$ is $Y$-perfect, then $L(g')^*K$ is $Y'$-perfect."}
+{"_id": "263", "title": "spaces-more-morphisms-lemma-relative-descend-homomorphisms", "text": "In Situation \\ref{situation-relative-descent}. Let $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$. Set $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$ and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map $$ \\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i) \\longrightarrow \\Hom_{D(\\mathcal{O}_X)}(K, L) $$ is an isomorphism if $K_0$ is pseudo-coherent and $L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally) finite tor dimension as an object of $D((X_0 \\to Y_0)^{-1}\\mathcal{O}_{Y_0})$"}
+{"_id": "264", "title": "spaces-more-morphisms-lemma-descend-relatively-perfect", "text": "In Situation \\ref{situation-relative-descent} the category of $Y$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories of $Y_i$-perfect objects of $D(\\mathcal{O}_{X_i})$."}
+{"_id": "265", "title": "spaces-more-morphisms-lemma-derived-pushforward-rel-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat, proper, and of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $Y$-perfect. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change."}
+{"_id": "266", "title": "spaces-more-morphisms-lemma-compute-ext-rel-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $E, K \\in D(\\mathcal{O}_X)$. Assume \\begin{enumerate} \\item $Y$ is quasi-compact and quasi-separated, \\item $f$ is proper, flat, and of finite presentation, \\item $E$ is $Y$-perfect, \\item $K$ is pseudo-coherent. \\end{enumerate} Then there exists a pseudo-coherent $L \\in D(\\mathcal{O}_Y)$ such that $$ Rf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_Y) $$ and the same is true after arbitrary base change: given $$ \\vcenter{ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } } \\quad\\quad \\begin{matrix} \\text{cartesian, then we have } \\\\ Rf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\ = R\\SheafHom(Lg^*L, \\mathcal{O}_{Y'}) \\end{matrix} $$"}
+{"_id": "267", "title": "spaces-more-morphisms-lemma-bounded-on-fibres", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ such that the structure morphism $f : X \\to S$ is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is $S$-perfect, and \\item $E$ is locally bounded below and for every point $s \\in S$ the object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$ is locally bounded below. \\end{enumerate}"}
+{"_id": "268", "title": "spaces-more-morphisms-lemma-characterize-relatively-perfect", "text": "Let $A$ be a ring. Let $X$ be an algebraic space separated, of finite presentation, and flat over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If  $R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in $D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is $\\Spec(A)$-perfect."}
+{"_id": "269", "title": "spaces-more-morphisms-lemma-diagonal-picard-flat-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat, proper morphism of finite presentation of algebraic spaces over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. For a morphism $g : Y' \\to Y$ consider the base change diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ Assume $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is an isomorphism for all $g : Y' \\to Y$. Then there exists an immersion $j : Z \\to Y$ of finite presentation such that a morphism $g : Y' \\to Y$ factors through $Z$ if and only if there exists a finite locally free $\\mathcal{O}_{Y'}$-module $\\mathcal{N}$ with $(f')^*\\mathcal{N} \\cong (g')^*\\mathcal{L}$."}
+{"_id": "270", "title": "spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc", "text": "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of algebraic spaces over $S$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each $Lf_i^*E$ is $m$-pseudo-coherent."}
+{"_id": "271", "title": "spaces-more-morphisms-lemma-tor-amplitude-descends-fppf", "text": "Let $S$ be a scheme. Let $\\{g_i : Y_i \\to Y\\}$ be an fpqc covering of algebraic spaces over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces and set $X_i = Y_i \\times_Y X$ with projections $f_i : X_i \\to Y_i$ and $g'_i : X_i \\to X$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$. Then the following are equivalent \\begin{enumerate} \\item $E$ has tor amplitude in $[a, b]$ as an object of $D(f^{-1}\\mathcal{O}_Y)$, and \\item $L(g'_i)^*E$ has tor amplitude in $[a, b]$ as a object of $D(f_i^{-1}\\mathcal{O}_{Y_i})$ for all $i$. \\end{enumerate} Also true if ``tor amplitude in $[a, b]$'' is replaced by ``locally finite tor dimension''."}
+{"_id": "272", "title": "spaces-more-morphisms-lemma-thickening-pseudo-coherent", "text": "Let $S$ be a scheme. Let $i : X \\to X'$ be a finite order thickening of algebraic spaces. Let $K' \\in D(\\mathcal{O}_{X'})$ be an object such that $K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent."}
+{"_id": "273", "title": "spaces-more-morphisms-lemma-thickening-relatively-perfect", "text": "Let $S$ be a scheme. Consider a cartesian diagram $$ \\xymatrix{ X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\ Y \\ar[r]^j & Y' } $$ of algebraic spaces over $S$. Assume $X' \\to Y'$ is flat and locally of finite presentation and $Y \\to Y'$ is a finite order thickening. Let $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect."}
+{"_id": "274", "title": "spaces-more-morphisms-lemma-henselian-relatively-perfect", "text": "Let $(R, I)$ be a pair consisting of a ring and an ideal $I$ contained in the Jacobson radical. Set $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$. Let $X$ be an algebraic space over $R$ whose structure morphism $f : X \\to S$ is proper, flat, and of finite presentation. Denote $X_0 = S_0 \\times_S X$. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent. If the derived restriction $E_0$ of $E$ to $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect."}
+{"_id": "276", "title": "spaces-more-morphisms-lemma-nodal-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is at-worst-nodal of relative dimension $1$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is at-worst-nodal of relative dimension $1$, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is at-worst-nodal of relative dimension $1$, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is at-worst-nodal of relative dimension $1$, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is at-worst-nodal of relative dimension $1$, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is at-worst-nodal of relative dimension $1$, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is at-worst-nodal of relative dimension $1$, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is at-worst-nodal of relative dimension $1$. \\end{enumerate}"}
+{"_id": "277", "title": "spaces-more-morphisms-lemma-locus-where-nodal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Then there is a maximal open subspace $X' \\subset X$ such that $f|_{X'} : X' \\to Y$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $X'$ commutes with arbitrary base change."}
+{"_id": "315", "title": "algebra-theorem-chevalley", "text": "Suppose that $R \\to S$ is of finite presentation. The image of a constructible subset of $\\Spec(S)$ in $\\Spec(R)$ is constructible."}
+{"_id": "316", "title": "algebra-theorem-nullstellensatz", "text": "Let $k$ be a field. \\begin{enumerate} \\item \\label{item-finite-kappa} For any maximal ideal $\\mathfrak m \\subset k[x_1, \\ldots, x_n]$ the field extension $k \\subset \\kappa(\\mathfrak m)$ is finite. \\item \\label{item-polynomial-ring-Jacobson} Any radical ideal $I \\subset k[x_1, \\ldots, x_n]$ is the intersection of maximal ideals containing it. \\end{enumerate} The same is true in any finite type $k$-algebra."}
+{"_id": "317", "title": "algebra-theorem-uncountable-nullstellensatz", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra generated over $k$ by the elements $\\{x_i\\}_{i \\in I}$. Assume the cardinality of $I$ is smaller than the cardinality of $k$. Then \\begin{enumerate} \\item for all maximal ideals $\\mathfrak m \\subset S$ the field extension $k \\subset \\kappa(\\mathfrak m)$ is algebraic, and \\item $S$ is a Jacobson ring. \\end{enumerate}"}
+{"_id": "318", "title": "algebra-theorem-lazard", "text": "Let $M$ be an $R$-module.  Then $M$ is flat if and only if it is the colimit of a directed system of free finite $R$-modules."}
+{"_id": "319", "title": "algebra-theorem-universally-exact-criteria", "text": "Let $$ 0 \\to M_1 \\xrightarrow{f_1} M_2 \\xrightarrow{f_2} M_3 \\to 0 $$ be an exact sequence of $R$-modules.  The following are equivalent: \\begin{enumerate} \\item The sequence $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ is universally exact. \\item For every finitely presented $R$-module $Q$, the sequence $$ 0 \\to M_1 \\otimes_R Q \\to M_2 \\otimes_R Q \\to M_3 \\otimes_R Q \\to 0 $$ is exact. \\item Given elements $x_i \\in M_1$ $(i = 1, \\ldots, n)$, $y_j \\in M_2$ $(j = 1, \\ldots, m)$, and $a_{ij} \\in R$ $(i = 1, \\ldots, n, j = 1, \\ldots, m)$ such that for all $i$ $$ f_1(x_i) = \\sum\\nolimits_j a_{ij} y_j, $$ there exists $z_j \\in M_1$ $(j =1, \\ldots, m)$ such that for all $i$, $$ x_i = \\sum\\nolimits_j a_{ij} z_j . $$ \\item Given a commutative diagram of $R$-module maps $$ \\xymatrix{ R^n \\ar[r] \\ar[d] &  R^m \\ar[d] \\\\ M_1 \\ar[r]^{f_1}        &  M_2 } $$ where $m$ and $n$ are integers, there exists a map $R^m \\to M_1$ making the top triangle commute. \\item For every finitely presented $R$-module $P$, the $R$-module map $\\Hom_R(P, M_2) \\to \\Hom_R(P, M_3)$ is surjective. \\item The sequence $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ is the colimit of a directed system of split exact sequences of the form $$ 0 \\to M_{1} \\to M_{2, i} \\to M_{3, i} \\to 0 $$ where the $M_{3, i}$ are finitely presented. \\end{enumerate}"}
+{"_id": "320", "title": "algebra-theorem-kaplansky-direct-sum", "text": "Suppose $M$ is a direct sum of countably generated $R$-modules.  If $P$ is a direct summand of $M$, then $P$ is also a direct sum of countably generated $R$-modules."}
+{"_id": "321", "title": "algebra-theorem-projective-direct-sum", "text": "\\begin{slogan} Any projective module is a direct sum of countably generated projective modules. \\end{slogan} If $P$ is a projective $R$-module, then $P$ is a direct sum of countably generated projective $R$-modules."}
+{"_id": "322", "title": "algebra-theorem-projective-free-over-local-ring", "text": "\\begin{slogan} Projective modules over local rings are free. \\end{slogan} If $P$ is a projective module over a local ring $R$, then $P$ is free."}
+{"_id": "323", "title": "algebra-theorem-projectivity-characterization", "text": "Let $M$ be an $R$-module.  Then $M$ is projective if and only it satisfies: \\begin{enumerate} \\item $M$ is flat, \\item $M$ is Mittag-Leffler, \\item $M$ is a direct sum of countably generated $R$-modules. \\end{enumerate}"}
+{"_id": "324", "title": "algebra-theorem-ffdescent-projectivity", "text": "Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module.  If the $S$-module $M \\otimes_R S$ is projective, then $M$ is projective."}
+{"_id": "325", "title": "algebra-theorem-main-theorem", "text": "Let $R$ be a ring. Let $R \\to S$ be a finite type $R$-algebra. Let $S' \\subset S$ be the integral closure of $R$ in $S$. Let $\\mathfrak q \\subset S$ be a prime of $S$. If $R \\to S$ is quasi-finite at $\\mathfrak q$ then there exists a $g \\in S'$, $g \\not \\in \\mathfrak q$ such that $S'_g \\cong S_g$."}
+{"_id": "326", "title": "algebra-theorem-openness-flatness", "text": "Let $R$ be a ring. Let $R \\to S$ be a ring map of finite presentation. Let $M$ be a finitely presented $S$-module. The set $$ \\{ \\mathfrak q \\in \\Spec(S) \\mid M_{\\mathfrak q}\\text{ is flat over }R\\} $$ is open in $\\Spec(S)$."}
+{"_id": "327", "title": "algebra-theorem-cohen-structure-theorem", "text": "Let $(R, \\mathfrak m)$ be a complete local ring. \\begin{enumerate} \\item $R$ has a coefficient ring (see Definition \\ref{definition-coefficient-ring}), \\item if $\\mathfrak m$ is a finitely generated ideal, then $R$ is isomorphic to a quotient $$ \\Lambda[[x_1, \\ldots, x_n]]/I $$ where $\\Lambda$ is either a field or a Cohen ring. \\end{enumerate}"}
+{"_id": "328", "title": "algebra-lemma-snake", "text": "\\begin{reference} \\cite[III, Lemma 3.3]{Cartan-Eilenberg} \\end{reference} Suppose given a commutative diagram $$ \\xymatrix{ & X \\ar[r] \\ar[d]^\\alpha & Y \\ar[r] \\ar[d]^\\beta & Z \\ar[r] \\ar[d]^\\gamma & 0 \\\\ 0 \\ar[r] & U \\ar[r] & V \\ar[r] & W } $$ of abelian groups with exact rows, then there is a canonical exact sequence $$ \\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma) \\to \\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma) $$ Moreover, if $X \\to Y$ is injective, then the first map is injective, and if $V \\to W$ is surjective, then the last map is surjective."}
+{"_id": "329", "title": "algebra-lemma-lift-map", "text": "Let $R$ be a ring. Let $\\alpha : R^{\\oplus n} \\to M$ and $\\beta : N \\to M$ be module maps. If $\\Im(\\alpha) \\subset \\Im(\\beta)$, then there exists an $R$-module map $\\gamma : R^{\\oplus n} \\to N$ such that $\\alpha = \\beta \\circ \\gamma$."}
+{"_id": "330", "title": "algebra-lemma-extension", "text": "Let $R$ be a ring. Let $$ 0 \\to M_1 \\to M_2 \\to M_3 \\to 0 $$ be a short exact sequence of $R$-modules. \\begin{enumerate} \\item If $M_1$ and $M_3$ are finite $R$-modules, then $M_2$ is a finite $R$-module. \\item If $M_1$ and $M_3$ are finitely presented $R$-modules, then $M_2$ is a finitely presented $R$-module. \\item If $M_2$ is a finite $R$-module, then $M_3$ is a finite $R$-module. \\item If $M_2$ is a finitely presented $R$-module and $M_1$ is a finite $R$-module, then $M_3$ is a finitely presented $R$-module. \\item If $M_3$ is a finitely presented $R$-module and $M_2$ is a finite $R$-module, then $M_1$ is a finite $R$-module. \\end{enumerate}"}
+{"_id": "331", "title": "algebra-lemma-trivial-filter-finite-module", "text": "\\begin{slogan} Finite modules have filtrations such that successive quotients are cyclic modules. \\end{slogan} Let $R$ be a ring, and let $M$ be a finite $R$-module. There exists a filtration by $R$-submodules $$ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M $$ such that each quotient $M_i/M_{i-1}$ is isomorphic to $R/I_i$ for some ideal $I_i$ of $R$."}
+{"_id": "332", "title": "algebra-lemma-finite-over-subring", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. If $M$ is finite as an $R$-module, then $M$ is finite as an $S$-module."}
+{"_id": "333", "title": "algebra-lemma-compose-finite-type", "text": "The notions finite type and finite presentation have the following permanence properties. \\begin{enumerate} \\item A composition of ring maps of finite type is of finite type. \\item A composition of ring maps of finite presentation is of finite presentation. \\item Given $R \\to S' \\to S$ with $R \\to S$ of finite type, then $S' \\to S$ is of finite type. \\item Given $R \\to S' \\to S$, with $R \\to S$ of finite presentation, and $R \\to S'$ of finite type, then $S' \\to S$ is of finite presentation. \\end{enumerate}"}
+{"_id": "334", "title": "algebra-lemma-finite-presentation-independent", "text": "Let $R \\to S$ be a ring map of finite presentation. For any surjection $\\alpha : R[x_1, \\ldots, x_n] \\to S$ the kernel of $\\alpha$ is a finitely generated ideal in $R[x_1, \\ldots, x_n]$."}
+{"_id": "335", "title": "algebra-lemma-finitely-presented-over-subring", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume $R \\to S$ is of finite type and $M$ is finitely presented as an $R$-module. Then $M$ is finitely presented as an $S$-module."}
+{"_id": "336", "title": "algebra-lemma-finite-module-over-finite-extension", "text": "Let $R \\to S$ be a finite ring map. Let $M$ be an $S$-module. Then $M$ is finite as an $R$-module if and only if $M$ is finite as an $S$-module."}
+{"_id": "337", "title": "algebra-lemma-finite-transitive", "text": "Suppose that $R \\to S$ and $S \\to T$ are finite ring maps. Then $R \\to T$ is finite."}
+{"_id": "338", "title": "algebra-lemma-finite-finite-type", "text": "Let $\\varphi : R \\to S$ be a ring map. \\begin{enumerate} \\item If $\\varphi$ is finite, then $\\varphi$ is of finite type. \\item If $S$ is of finite presentation as an $R$-module, then $\\varphi$ is of finite presentation. \\end{enumerate}"}
+{"_id": "339", "title": "algebra-lemma-colimit", "text": "Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the preordered set $I$. The colimit of the system $(M_i, \\mu_{ij})$ is the quotient $R$-module $(\\bigoplus_{i\\in I} M_i) /Q$ where $Q$ is the $R$-submodule generated by all elements $$ \\iota_i(x_i) - \\iota_j(\\mu_{ij}(x_i)) $$ where $\\iota_i : M_i \\to \\bigoplus_{i\\in I} M_i$ is the natural inclusion. We denote the colimit $M = \\colim_i M_i$. We denote $\\pi : \\bigoplus_{i\\in I} M_i \\to M$ the projection map and $\\phi_i = \\pi \\circ \\iota_i : M_i \\to M$."}
+{"_id": "340", "title": "algebra-lemma-directed-colimit", "text": "Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the preordered set $I$. Assume that $I$ is directed. The colimit of the system $(M_i, \\mu_{ij})$ is canonically isomorphic to the module $M$ defined as follows: \\begin{enumerate} \\item as a set let $$ M = \\left(\\coprod\\nolimits_{i \\in I} M_i\\right)/\\sim $$ where for $m \\in M_i$ and $m' \\in M_{i'}$ we have $$ m \\sim m' \\Leftrightarrow \\mu_{ij}(m) = \\mu_{i'j}(m')\\text{ for some }j \\geq i, i' $$ \\item as an abelian group for $m \\in M_i$ and $m' \\in M_{i'}$ we define the sum of the classes of $m$ and $m'$ in $M$ to be the class of $\\mu_{ij}(m) + \\mu_{i'j}(m')$ where $j \\in I$ is any index with $i \\leq j$ and $i' \\leq j$, and \\item as an $R$-module define for $m \\in M_i$ and $x \\in R$ the product of $x$ and the class of $m$ in $M$ to be the class of $xm$ in $M$. \\end{enumerate} The canonical maps $\\phi_i : M_i \\to M$ are induced by the canonical maps $M_i \\to \\coprod_{i \\in I} M_i$."}
+{"_id": "341", "title": "algebra-lemma-zero-directed-limit", "text": "Let $(M_i, \\mu_{ij})$ be a directed system. Let $M = \\colim M_i$ with $\\mu_i : M_i \\to M$. Then, $\\mu_i(x_i) = 0$ for $x_i \\in M_i$ if and only if there exists $j \\geq i$ such that $\\mu_{ij}(x_i) = 0$."}
+{"_id": "342", "title": "algebra-lemma-homomorphism-limit", "text": "Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be systems of $R$-modules over the same preordered set. A morphism of systems $\\Phi = (\\phi_i)$ from $(M_i, \\mu_{ij})$ to $(N_i, \\nu_{ij})$ induces a unique homomorphism $$ \\colim \\phi_i : \\colim M_i \\longrightarrow \\colim N_i $$ such that $$ \\xymatrix{ M_i \\ar[r] \\ar[d]_{\\phi_i} & \\colim M_i \\ar[d]^{\\colim \\phi_i} \\\\ N_i \\ar[r] & \\colim N_i } $$ commutes for all $i \\in I$."}
+{"_id": "343", "title": "algebra-lemma-directed-colimit-exact", "text": "\\begin{slogan} Filtered colimits are exact. Directed colimits are exact. \\end{slogan} Let $I$ be a directed set. Let $(L_i, \\lambda_{ij})$, $(M_i, \\mu_{ij})$, and $(N_i, \\nu_{ij})$ be systems of $R$-modules over $I$. Let $\\varphi_i : L_i \\to M_i$ and $\\psi_i : M_i \\to N_i$ be morphisms of systems over $I$. Assume that for all $i \\in I$ the sequence of $R$-modules $$ \\xymatrix{ L_i \\ar[r]^{\\varphi_i} & M_i \\ar[r]^{\\psi_i} & N_i } $$ is a complex with homology $H_i$. Then the $R$-modules $H_i$ form a system over $I$, the sequence of $R$-modules $$ \\xymatrix{ \\colim_i L_i \\ar[r]^\\varphi & \\colim_i M_i \\ar[r]^\\psi & \\colim_i N_i } $$ is a complex as well, and denoting $H$ its homology we have $$ H = \\colim_i H_i. $$"}
+{"_id": "344", "title": "algebra-lemma-almost-directed-colimit-exact", "text": "Let $\\mathcal{I}$ be an index category satisfying the assumptions of Categories, Lemma \\ref{categories-lemma-split-into-directed}. Then taking colimits of diagrams of abelian groups over $\\mathcal{I}$ is exact (i.e., the analogue of Lemma \\ref{lemma-directed-colimit-exact} holds in this situation)."}
+{"_id": "345", "title": "algebra-lemma-localization-zero", "text": "The localization $S^{-1}A$ is the zero ring if and only if $0\\in S$."}
+{"_id": "346", "title": "algebra-lemma-localization-and-modules", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. The category of $S^{-1}R$-modules is equivalent to the category of $R$-modules $N$ with the property that every $s \\in S$ acts as an automorphism on $N$."}
+{"_id": "347", "title": "algebra-lemma-universal-property-localization-module", "text": "Let $R$ be a ring. Let $S \\subset R$ a multiplicative subset. Let $M$, $N$ be $R$-modules. Assume all the elements of $S$ act as automorphisms on $N$. Then the canonical map $$ \\Hom_R(S^{-1}M, N) \\longrightarrow \\Hom_R(M, N) $$ induced by the localization map, is an isomorphism."}
+{"_id": "348", "title": "algebra-lemma-localization-colimit", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. Let $M$ be an $R$-module. Then $$ S^{-1}M = \\colim_{f \\in S} M_f $$ where the preorder on $S$ is given by $f \\geq f' \\Leftrightarrow f = f'f''$ for some $f'' \\in R$ in which case the map $M_{f'} \\to M_f$ is given by $m/(f')^e \\mapsto m(f'')^e/f^e$."}
+{"_id": "349", "title": "algebra-lemma-localize-quotient-modules", "text": "Localization respects quotients, i.e. if $N$ is a submodule of $M$, then $S^{-1}(M/N)\\simeq (S^{-1}M)/(S^{-1}N)$."}
+{"_id": "350", "title": "algebra-lemma-submodule-localization", "text": "Any submodule $N'$ of $S^{-1}M$ is of the form $S^{-1}N$ for some $N\\subset M$. Indeed one can take $N$ to be the inverse image of $N'$ in $M$."}
+{"_id": "352", "title": "algebra-lemma-hom-exact", "text": "Exactness and $\\Hom_R$. Let $R$ be a ring. \\begin{enumerate} \\item Let $M_1 \\to M_2 \\to M_3 \\to 0$ be a complex of $R$-modules. Then $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if $0 \\to \\Hom_R(M_3, N) \\to \\Hom_R(M_2, N) \\to \\Hom_R(M_1, N)$ is exact for all $R$-modules $N$. \\item  Let $0 \\to M_1 \\to M_2 \\to M_3$ be a complex of $R$-modules. Then $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if $0 \\to \\Hom_R(N, M_1) \\to \\Hom_R(N, M_2) \\to \\Hom_R(N, M_3)$ is exact for all $R$-modules $N$. \\end{enumerate}"}
+{"_id": "353", "title": "algebra-lemma-hom-from-finitely-presented", "text": "Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $N$ be an $R$-module. \\begin{enumerate} \\item For $f \\in R$ we have $\\Hom_R(M, N)_f = \\Hom_{R_f}(M_f, N_f) = \\Hom_R(M_f, N_f)$, \\item for a multiplicative subset $S$ of $R$ we have $$ S^{-1}\\Hom_R(M, N) = \\Hom_{S^{-1}R}(S^{-1}M, S^{-1}N) = \\Hom_R(S^{-1}M, S^{-1}N). $$ \\end{enumerate}"}
+{"_id": "354", "title": "algebra-lemma-characterize-finite-module-hom", "text": "Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $N$ is a finite $R$-module, \\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map $\\colim \\Hom_R(N, M_i) \\to \\Hom_R(K, M)$ is injective. \\end{enumerate}"}
+{"_id": "355", "title": "algebra-lemma-module-colimit-fp", "text": "Let $R$ be a ring and let $M$ be an $R$-module. Then $M$ is the colimit of a directed system $(M_i, \\mu_{ij})$ of $R$-modules with all $M_i$ finitely presented $R$-modules."}
+{"_id": "357", "title": "algebra-lemma-tensor-product", "text": "Let $M, N$ be $R$-modules. Then there exists a pair $(T, g)$ where $T$ is an $R$-module, and $g : M \\times N \\to T$ an $R$-bilinear mapping, with the following universal property: For any $R$-module $P$ and any $R$-bilinear mapping $f : M \\times N \\to P$, there exists a unique $R$-linear mapping $\\tilde{f} : T \\to P$ such that $f = \\tilde{f} \\circ g$. In other words, the following diagram commutes: $$ \\xymatrix{ M \\times N \\ar[rr]^f \\ar[dr]_g & & P\\\\ & T \\ar[ur]_{\\tilde f} } $$ Moreover, if $(T, g)$ and $(T', g')$ are two pairs with this property, then there exists a unique isomorphism $j : T \\to T'$ such that $j\\circ g = g'$."}
+{"_id": "358", "title": "algebra-lemma-flip-tensor-product", "text": "Let $M, N, P$ be $R$-modules, then the bilinear maps \\begin{align*} (x, y) & \\mapsto y \\otimes x\\\\ (x + y, z) & \\mapsto x \\otimes z + y \\otimes z\\\\ (r, x) & \\mapsto rx \\end{align*} induce unique isomorphisms \\begin{align*} M \\otimes_R N & \\to N \\otimes_R M, \\\\ (M\\oplus N)\\otimes_R P & \\to (M \\otimes_R P)\\oplus(N \\otimes_R P),  \\\\ R \\otimes_R M & \\to M \\end{align*}"}
+{"_id": "359", "title": "algebra-lemma-multilinear", "text": "Let $M_1, \\ldots, M_r$ be $R$-modules. Then there exists a pair $(T, g)$ consisting of an $R$-module T and an $R$-multilinear mapping $g : M_1\\times \\ldots \\times M_r \\to T$ with the universal property: For any $R$-multilinear mapping $f : M_1\\times \\ldots \\times M_r \\to P$ there exists a unique $R$-module homomorphism $f' : T \\to P$ such that $f'\\circ g = f$. Such a module $T$ is unique up to unique isomorphism. We denote it $M_1\\otimes_R \\ldots \\otimes_R M_r$ and we denote the universal multilinear map $(m_1, \\ldots, m_r) \\mapsto m_1 \\otimes \\ldots \\otimes m_r$."}
+{"_id": "360", "title": "algebra-lemma-transitive", "text": "The homomorphisms $$ (M \\otimes_R N)\\otimes_R P \\to M \\otimes_R N \\otimes_R P \\to M \\otimes_R (N \\otimes_R P) $$ such that $f((x \\otimes y)\\otimes z) = x \\otimes y \\otimes z$ and $g(x \\otimes y \\otimes z) = x \\otimes (y \\otimes z)$, $x\\in M, y\\in N, z\\in P$ are well-defined and are isomorphisms."}
+{"_id": "361", "title": "algebra-lemma-tensor-with-bimodule", "text": "For $A$-module $M$, $B$-module $P$ and $(A, B)$-bimodule $N$, the modules $(M \\otimes_A N)\\otimes_B P$ and $M \\otimes_A(N \\otimes_B P)$ can both be given $(A, B)$-bimodule structure, and moreover $$ (M \\otimes_A N)\\otimes_B P \\cong M \\otimes_A(N \\otimes_B P). $$"}
+{"_id": "362", "title": "algebra-lemma-hom-from-tensor-product", "text": "For any three $R$-modules $M, N, P$, $$ \\Hom_R(M \\otimes_R N, P) \\cong \\Hom_R(M, \\Hom_R(N, P)) $$"}
+{"_id": "363", "title": "algebra-lemma-tensor-products-commute-with-limits", "text": "Let $(M_i, \\mu_{ij})$ be a system over the preordered set $I$. Let $N$ be an $R$-module. Then $$ \\colim (M_i \\otimes N) \\cong (\\colim M_i)\\otimes N. $$ Moreover, the isomorphism is induced by the homomorphisms $\\mu_i \\otimes 1: M_i \\otimes N \\to M \\otimes N$ where $M = \\colim_i M_i$ with natural maps $\\mu_i : M_i \\to M$."}
+{"_id": "364", "title": "algebra-lemma-tensor-product-exact", "text": "Let \\begin{align*} M_1\\xrightarrow{f} M_2\\xrightarrow{g} M_3 \\to 0 \\end{align*} be an exact sequence of $R$-modules and homomorphisms, and let $N$ be any $R$-module. Then the sequence \\begin{equation} \\label{equation-2ndex} M_1\\otimes N\\xrightarrow{f \\otimes 1} M_2\\otimes N \\xrightarrow{g \\otimes 1} M_3\\otimes N \\to 0 \\end{equation} is exact. In other words, the functor $- \\otimes_R N$ is {\\it right exact}, in the sense that tensoring each term in the original right exact sequence preserves the exactness."}
+{"_id": "365", "title": "algebra-lemma-tensor-finiteness", "text": "Let $R$ be a ring. Let $M$ and $N$ be $R$-modules. \\begin{enumerate} \\item If $N$ and $M$ are finite, then so is $M \\otimes_R N$. \\item If $N$ and $M$ are finitely presented, then so is $M \\otimes_R N$. \\end{enumerate}"}
+{"_id": "366", "title": "algebra-lemma-tensor-localization", "text": "Let $M$ be an $R$-module. Then the $S^{-1}R$-modules $S^{-1}M$ and $S^{-1}R \\otimes_R M$ are canonically isomorphic, and the canonical isomorphism $f : S^{-1}R \\otimes_R M \\to S^{-1}M$ is given by $$ f((a/s) \\otimes m) = am/s, \\forall a \\in R, m \\in M, s \\in S $$"}
+{"_id": "367", "title": "algebra-lemma-tensor-product-localization", "text": "Let $M, N$ be $R$-modules, then there is a canonical $S^{-1}R$-module isomorphism $f : S^{-1}M \\otimes_{S^{-1}R}S^{-1}N \\to S^{-1}(M \\otimes_R N)$, given by $$ f((m/s)\\otimes(n/t)) = (m \\otimes n)/st $$"}
+{"_id": "368", "title": "algebra-lemma-free-tensor-algebra", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. If $M$ is a free $R$-module, so is each symmetric and exterior power."}
+{"_id": "369", "title": "algebra-lemma-presentation-sym-exterior", "text": "Let $R$ be a ring. Let $M_2 \\to M_1 \\to M \\to 0$ be an exact sequence of $R$-modules. There are exact sequences $$ M_2 \\otimes_R \\text{Sym}^{n - 1}(M_1) \\to \\text{Sym}^n(M_1) \\to \\text{Sym}^n(M) \\to 0 $$ and similarly $$ M_2 \\otimes_R \\wedge^{n - 1}(M_1) \\to \\wedge^n(M_1) \\to \\wedge^n(M) \\to 0 $$"}
+{"_id": "370", "title": "algebra-lemma-present-sym-wedge", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $x_i$, $i \\in I$ be a given system of generators of $M$ as an $R$-module. Let $n \\geq 2$. There exists a canonical exact sequence $$ \\bigoplus_{1 \\leq j_1 < j_2 \\leq n} \\bigoplus_{i_1, i_2 \\in I} \\text{T}^{n - 2}(M) \\oplus \\bigoplus_{1 \\leq j_1 < j_2 \\leq n} \\bigoplus_{i \\in I} \\text{T}^{n - 2}(M) \\to \\text{T}^n(M) \\to \\wedge^n(M) \\to 0 $$ where the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the first summand maps to \\begin{align*} \\underbrace{ m_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2} }_{\\text{with } x_{i_1} \\text{ and } x_{i_2} \\text{ occupying slots } j_1 \\text{ and } j_2 \\text{ in the tensor}} \\\\ + \\underbrace{ m_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2} }_{\\text{with } x_{i_2} \\text{ and } x_{i_1} \\text{ occupying slots } j_1 \\text{ and } j_2 \\text{ in the tensor}} \\end{align*} and $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the second summand maps to $$ \\underbrace{ m_1 \\otimes \\ldots \\otimes x_i \\otimes \\ldots \\otimes x_i \\otimes \\ldots \\otimes m_{n - 2} }_{\\text{with } x_{i} \\text{ and } x_{i} \\text{ occupying slots } j_1 \\text{ and } j_2 \\text{ in the tensor}} $$ There is also a canonical exact sequence $$ \\bigoplus_{1 \\leq j_1 < j_2 \\leq n} \\bigoplus_{i_1, i_2 \\in I} \\text{T}^{n - 2}(M) \\to \\text{T}^n(M) \\to \\text{Sym}^n(M) \\to 0 $$ where the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ maps to \\begin{align*} \\underbrace{ m_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2} }_{\\text{with } x_{i_1} \\text{ and } x_{i_2} \\text{ occupying slots } j_1 \\text{ and } j_2 \\text{ in the tensor}} \\\\ - \\underbrace{ m_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2} }_{\\text{with } x_{i_2} \\text{ and } x_{i_1} \\text{ occupying slots } j_1 \\text{ and } j_2 \\text{ in the tensor}} \\end{align*}"}
+{"_id": "371", "title": "algebra-lemma-colimit-tensor-algebra", "text": "\\begin{slogan} Taking tensor algebras commutes with filtered colimits. \\end{slogan} Let $R$ be a ring. Let $M_i$ be a directed system of $R$-modules. Then $\\colim_i \\text{T}(M_i) = \\text{T}(\\colim_i M_i)$ and similarly for the symmetric and exterior algebras."}
+{"_id": "373", "title": "algebra-lemma-base-change-finiteness", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Let $R \\to R'$ be a ring map and let $S' = S \\otimes_R R'$ and $M' = M \\otimes_R R'$ be the base changes. \\begin{enumerate} \\item If $M$ is a finite $S$-module, then the base change $M'$ is a finite $S'$-module. \\item If $M$ is an $S$-module finite presentation, then the base change $M'$ is an $S'$-module of finite presentation. \\item If $R \\to S$ is of finite type, then the base change $R' \\to S'$ is of finite type. \\item If $R \\to S$ is of finite presentation, then the base change $R' \\to S'$ is of finite presentation. \\end{enumerate}"}
+{"_id": "374", "title": "algebra-lemma-adjoint-tensor-restrict", "text": "Let $R \\to S$ be a ring map. The functors $\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction) and $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto M \\otimes_R S$ (base change) are adjoint functors. In a formula $$ \\Hom_R(M, N_R) = \\Hom_S(M \\otimes_R S, N) $$"}
+{"_id": "375", "title": "algebra-lemma-adjoint-hom-restrict", "text": "Let $R \\to S$ be a ring map. The functors $\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction) and $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto \\Hom_R(S, M)$ are adjoint functors. In a formula $$ \\Hom_R(N_R, M) = \\Hom_S(N, \\Hom_R(S, M)) $$"}
+{"_id": "376", "title": "algebra-lemma-hom-from-tensor-product-variant", "text": "Let $R \\to S$ be a ring map. Given $S$-modules $M, N$ and an $R$-module $P$ we have $$ \\Hom_R(M \\otimes_S N, P) = \\Hom_S(M, \\Hom_R(N, P)) $$"}
+{"_id": "377", "title": "algebra-lemma-product-ideals-in-prime", "text": "Let $R$ be a ring, $I$ and $J$ two ideals and $\\mathfrak p$ a prime ideal containing the product $IJ$. Then $\\mathfrak{p}$ contains $I$ or $J$."}
+{"_id": "378", "title": "algebra-lemma-silly", "text": "\\begin{slogan} 1. In an affine scheme if a finite number of points are contained in an open subset then they are contained in a smaller principal open subset. 2. Affine opens are cofinal among the neighborhoods of a given finite set of an affine scheme \\end{slogan} Let $R$ be a ring. Let $I_i \\subset R$, $i = 1, \\ldots, r$, and $J \\subset R$ be ideals. Assume \\begin{enumerate} \\item $J \\not\\subset I_i$ for $i = 1, \\ldots, r$, and \\item all but two of $I_i$ are prime ideals. \\end{enumerate} Then there exists an $x \\in J$, $x\\not\\in I_i$ for all $i$."}
+{"_id": "379", "title": "algebra-lemma-silly-silly", "text": "Let $R$ be a ring. Let $x \\in R$, $I \\subset R$ an ideal, and $\\mathfrak p_i$, $i = 1, \\ldots, r$ be prime ideals. Suppose that $x + I \\not \\subset \\mathfrak p_i$ for $i = 1, \\ldots, r$. Then there exists an $y \\in I$ such that $x + y \\not \\in \\mathfrak p_i$ for all $i$."}
+{"_id": "380", "title": "algebra-lemma-chinese-remainder", "text": "Let $R$ be a ring. \\begin{enumerate} \\item If $I_1, \\ldots, I_r$ are ideals such that $I_a + I_b = R$ when $a \\not = b$, then $I_1 \\cap \\ldots \\cap I_r = I_1I_2\\ldots I_r$ and $R/(I_1I_2\\ldots I_r) \\cong R/I_1 \\times \\ldots \\times R/I_r$. \\item If $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ are pairwise distinct maximal ideals then $\\mathfrak m_a + \\mathfrak m_b = R$ for $a \\not = b$ and the above applies. \\end{enumerate}"}
+{"_id": "381", "title": "algebra-lemma-matrix-left-inverse", "text": "Let $R$ be a ring. Let $n \\geq m$. Let $A$ be an $n \\times m$ matrix with coefficients in $R$. Let $J \\subset R$ be the ideal generated by the $m \\times m$ minors of $A$. \\begin{enumerate} \\item For any $f \\in J$ there exists a $m \\times n$ matrix $B$ such that $BA = f 1_{m \\times m}$. \\item If $f \\in R$ and $BA = f 1_{m \\times m}$ for some $m \\times n$ matrix $B$, then $f^m \\in J$. \\end{enumerate}"}
+{"_id": "382", "title": "algebra-lemma-matrix-right-inverse", "text": "Let $R$ be a ring. Let $n \\geq m$. Let $A = (a_{ij})$ be an $n \\times m$ matrix with coefficients in $R$, written in block form as $$ A = \\left( \\begin{matrix} A_1 \\\\ A_2 \\end{matrix} \\right) $$ where $A_1$ has size $m \\times m$. Let $B$ be the adjugate (transpose of cofactor) matrix to $A_1$. Then $$ AB =  \\left( \\begin{matrix} f 1_{m \\times m} \\\\ C \\end{matrix} \\right) $$ where $f = \\det(A_1)$ and $c_{ij}$ is (up to sign) the determinant of the $m \\times m$ minor of $A$ corresponding to the rows $1, \\ldots, \\hat j, \\ldots, m, i$."}
+{"_id": "383", "title": "algebra-lemma-map-cannot-be-injective", "text": "\\begin{slogan} A map of finite free modules cannot be injective if the source has rank bigger than the target. \\end{slogan} Let $R$ be a nonzero ring. Let $n \\geq 1$. Let $M$ be an $R$-module generated by $< n$ elements. Then any $R$-module map $f : R^{\\oplus n} \\to M$ has a nonzero kernel."}
+{"_id": "384", "title": "algebra-lemma-rank", "text": "\\begin{slogan} The rank of a finite free module is well defined. \\end{slogan} Let $R$ be a nonzero ring. Let $n, m \\geq 0$ be integers. If $R^{\\oplus n}$ is isomorphic to $R^{\\oplus m}$ as $R$-modules, then $n = m$."}
+{"_id": "385", "title": "algebra-lemma-charpoly", "text": "Let $R$ be a ring. Let $A = (a_{ij})$ be an $n \\times n$ matrix with coefficients in $R$. Let $P(x) \\in R[x]$ be the characteristic polynomial of $A$ (defined as $\\det(x\\text{id}_{n \\times n} - A)$). Then $P(A) = 0$ in $\\text{Mat}(n \\times n, R)$."}
+{"_id": "386", "title": "algebra-lemma-charpoly-module", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $\\varphi : M \\to M$ be an endomorphism. Then there exists a monic polynomial $P \\in R[T]$ such that $P(\\varphi) = 0$ as an endomorphism of $M$."}
+{"_id": "387", "title": "algebra-lemma-charpoly-module-ideal", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be a finite $R$-module. Let $\\varphi : M \\to M$ be an endomorphism such that $\\varphi(M) \\subset IM$. Then there exists a monic polynomial $P = t^n + a_1 t^{n - 1} + \\ldots + a_n \\in R[T]$ such that $a_j \\in I^j$ and $P(\\varphi) = 0$ as an endomorphism of $M$."}
+{"_id": "388", "title": "algebra-lemma-fun", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $\\varphi : M \\to M$ be a surjective $R$-module map. Then $\\varphi$ is an isomorphism."}
+{"_id": "389", "title": "algebra-lemma-Zariski-topology", "text": "Let $R$ be a ring. \\begin{enumerate} \\item The spectrum of a ring $R$ is empty if and only if $R$ is the zero ring. \\item Every nonzero ring has a maximal ideal. \\item Every nonzero ring has a minimal prime ideal. \\item Given an ideal $I \\subset R$ and a prime ideal $I \\subset \\mathfrak p$ there exists a prime $I \\subset \\mathfrak q \\subset \\mathfrak p$ such that $\\mathfrak q$ is minimal over $I$. \\item If $T \\subset R$, and if $(T)$ is the ideal generated by $T$ in $R$, then $V((T)) = V(T)$. \\item If $I$ is an ideal and $\\sqrt{I}$ is its radical, see basic notion (\\ref{item-radical-ideal}), then $V(I) = V(\\sqrt{I})$. \\item Given an ideal $I$ of $R$ we have $\\sqrt{I} = \\bigcap_{I \\subset \\mathfrak p} \\mathfrak p$. \\item If $I$ is an ideal then $V(I) = \\emptyset$ if and only if $I$ is the unit ideal. \\item If $I$, $J$ are ideals of $R$ then $V(I) \\cup V(J) = V(I \\cap J)$. \\item If $(I_a)_{a\\in A}$ is a set of ideals of $R$ then $\\cap_{a\\in A} V(I_a) = V(\\cup_{a\\in A} I_a)$. \\item If $f \\in R$, then $D(f) \\amalg V(f) = \\Spec(R)$. \\item If $f \\in R$ then $D(f) = \\emptyset$ if and only if $f$ is nilpotent. \\item If $f = u f'$ for some unit $u \\in R$, then $D(f) = D(f')$. \\item If $I \\subset R$ is an ideal, and $\\mathfrak p$ is a prime of $R$ with $\\mathfrak p \\not\\in V(I)$, then there exists an $f \\in R$ such that $\\mathfrak p \\in D(f)$, and $D(f) \\cap V(I) = \\emptyset$. \\item If $f, g \\in R$, then $D(fg) = D(f) \\cap D(g)$. \\item If $f_i \\in R$ for $i \\in I$, then $\\bigcup_{i\\in I} D(f_i)$ is the complement of $V(\\{f_i \\}_{i\\in I})$ in $\\Spec(R)$. \\item If $f \\in R$ and $D(f) = \\Spec(R)$, then $f$ is a unit. \\end{enumerate}"}
+{"_id": "390", "title": "algebra-lemma-spec-functorial", "text": "\\begin{slogan} Functoriality of the spectrum \\end{slogan} Suppose that $\\varphi : R \\to R'$ is a ring homomorphism. The induced map $$ \\Spec(\\varphi) : \\Spec(R') \\longrightarrow \\Spec(R), \\quad \\mathfrak p' \\longmapsto \\varphi^{-1}(\\mathfrak p') $$ is continuous for the Zariski topologies. In fact, for any element $f \\in R$ we have $\\Spec(\\varphi)^{-1}(D(f)) = D(\\varphi(f))$."}
+{"_id": "391", "title": "algebra-lemma-spec-localization", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. The map $R \\to S^{-1}R$ induces via the functoriality of $\\Spec$ a homeomorphism $$ \\Spec(S^{-1}R) \\longrightarrow \\{\\mathfrak p \\in \\Spec(R) \\mid S \\cap \\mathfrak p = \\emptyset \\} $$ where the topology on the right hand side is that induced from the Zariski topology on $\\Spec(R)$. The inverse map is given by $\\mathfrak p \\mapsto S^{-1}\\mathfrak p$."}
+{"_id": "392", "title": "algebra-lemma-standard-open", "text": "Let $R$ be a ring. Let $f \\in R$. The map $R \\to R_f$ induces via the functoriality of $\\Spec$ a homeomorphism $$ \\Spec(R_f) \\longrightarrow D(f) \\subset \\Spec(R). $$ The inverse is given by $\\mathfrak p \\mapsto \\mathfrak p \\cdot R_f$."}
+{"_id": "393", "title": "algebra-lemma-spec-closed", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. The map $R \\to R/I$ induces via the functoriality of $\\Spec$ a homeomorphism $$ \\Spec(R/I) \\longrightarrow V(I) \\subset \\Spec(R). $$ The inverse is given by $\\mathfrak p \\mapsto \\mathfrak p / I$."}
+{"_id": "394", "title": "algebra-lemma-in-image", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak p$ be a prime of $R$. The following are equivalent \\begin{enumerate} \\item $\\mathfrak p$ is in the image of $\\Spec(S) \\to \\Spec(R)$, \\item $S \\otimes_R \\kappa(\\mathfrak p) \\not = 0$, \\item $S_{\\mathfrak p}/\\mathfrak p S_{\\mathfrak p} \\not = 0$, \\item $(S/\\mathfrak pS)_{\\mathfrak p} \\not = 0$, and \\item $\\mathfrak p = \\varphi^{-1}(\\mathfrak pS)$. \\end{enumerate}"}
+{"_id": "395", "title": "algebra-lemma-quasi-compact", "text": "\\begin{slogan} The spectrum of a ring is quasi-compact \\end{slogan} Let $R$ be a ring. The space $\\Spec(R)$ is quasi-compact."}
+{"_id": "396", "title": "algebra-lemma-topology-spec", "text": "Let $R$ be a ring. The topology on $X = \\Spec(R)$ has the following properties: \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ has a basis for the topology consisting of quasi-compact opens, and \\item the intersection of any two quasi-compact opens is quasi-compact. \\end{enumerate}"}
+{"_id": "397", "title": "algebra-lemma-characterize-local-ring", "text": "Let $R$ be a ring. The following are equivalent: \\begin{enumerate} \\item $R$ is a local ring, \\item $\\Spec(R)$ has exactly one closed point, \\item $R$ has a maximal ideal $\\mathfrak m$ and every element of $R \\setminus \\mathfrak m$ is a unit, and \\item $R$ is not the zero ring and for every $x \\in R$ either $x$ or $1 - x$ is invertible or both. \\end{enumerate}"}
+{"_id": "399", "title": "algebra-lemma-contained-in-radical", "text": "Let $R$ be a ring with Jacobson radical $\\text{rad}(R)$. Let $I \\subset R$ be an ideal. The following are equivalent \\begin{enumerate} \\item $I \\subset \\text{rad}(R)$, and \\item every element of $1 + I$ is a unit in $R$. \\end{enumerate} In this case every element of $R$ which maps to a unit of $R/I$ is a unit."}
+{"_id": "401", "title": "algebra-lemma-NAK", "text": "\\begin{reference} \\cite[1.M Lemma (NAK) page 11]{MatCA} \\end{reference} \\begin{history} We quote from \\cite{MatCA}: ``This simple but important lemma is due to T.~Nakayama, G.~Azumaya and W.~Krull. Priority is obscure, and although it is usually called the Lemma of Nakayama, late Prof.~Nakayama did not like the name.'' \\end{history} Let $R$ be a ring with Jacobson radical $\\text{rad}(R)$. Let $M$ be an $R$-module. Let $I \\subset R$ be an ideal. \\begin{enumerate} \\item \\label{item-nakayama} If $IM = M$ and $M$ is finite, then there exists a $f \\in 1 + I$ such that $fM = 0$. \\item If $IM = M$, $M$ is finite, and $I \\subset \\text{rad}(R)$, then $M = 0$. \\item If $N, N' \\subset M$, $M = N + IN'$, and $N'$ is finite, then there exists a $f \\in 1 + I$ such that $fM \\subset N$ and $M_f = N_f$. \\item If $N, N' \\subset M$, $M = N + IN'$, $N'$ is finite, and $I \\subset \\text{rad}(R)$, then $M = N$. \\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is surjective, and $M$ is finite, then there exists a $f \\in 1 + I$ such that $N_f \\to M_f$ is surjective. \\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is surjective, $M$ is finite, and $I \\subset \\text{rad}(R)$, then $N \\to M$ is surjective. \\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$ and $M$ is finite, then there exists an $f \\in 1 + I$ such that $x_1, \\ldots, x_n$ generate $M_f$ over $R_f$. \\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$, $M$ is finite, and $I \\subset \\text{rad}(R)$, then $M$ is generated by $x_1, \\ldots, x_n$. \\item If $IM = M$, $I$ is nilpotent, then $M = 0$. \\item If $N, N' \\subset M$, $M = N + IN'$, and $I$ is nilpotent then $M = N$. \\item If $N \\to M$ is a module map, $I$ is nilpotent, and $N/IN \\to M/IM$ is surjective, then $N \\to M$ is surjective. \\item If $\\{x_\\alpha\\}_{\\alpha \\in A}$ is a set of elements of $M$ which generate $M/IM$ and $I$ is nilpotent, then $M$ is generated by the $x_\\alpha$. \\end{enumerate}"}
+{"_id": "402", "title": "algebra-lemma-when-surjective-local", "text": "Let $A \\to B$ be a local homomorphism of local rings. Assume \\begin{enumerate} \\item $B$ is finite as an $A$-module, \\item $\\mathfrak m_B$ is a finitely generated ideal, \\item $A \\to B$ induces an isomorphism on residue fields, and \\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$ is surjective. \\end{enumerate} Then $A \\to B$ is surjective."}
+{"_id": "403", "title": "algebra-lemma-idempotent-spec", "text": "Let $R$ be a ring. Let $e \\in R$ be an idempotent. In this case $$ \\Spec(R) = D(e) \\amalg D(1-e). $$"}
+{"_id": "404", "title": "algebra-lemma-spec-product", "text": "Let $R_1$ and $R_2$ be rings. Let $R = R_1 \\times R_2$. The maps $R \\to R_1$, $(x, y) \\mapsto x$ and $R \\to R_2$, $(x, y) \\mapsto y$ induce continuous maps $\\Spec(R_1) \\to \\Spec(R)$ and $\\Spec(R_2) \\to \\Spec(R)$. The induced map $$ \\Spec(R_1) \\amalg \\Spec(R_2) \\longrightarrow \\Spec(R) $$ is a homeomorphism. In other words, the spectrum of $R = R_1\\times R_2$ is the disjoint union of the spectrum of $R_1$ and the spectrum of $R_2$."}
+{"_id": "405", "title": "algebra-lemma-disjoint-decomposition", "text": "Let $R$ be a ring. For each $U \\subset \\Spec(R)$ which is open and closed there exists a unique idempotent $e \\in R$ such that $U = D(e)$. This induces a 1-1 correspondence between open and closed subsets $U \\subset \\Spec(R)$ and idempotents $e \\in R$."}
+{"_id": "406", "title": "algebra-lemma-characterize-spec-connected", "text": "Let $R$ be a nonzero ring. Then $\\Spec(R)$ is connected if and only if $R$ has no nontrivial idempotents."}
+{"_id": "407", "title": "algebra-lemma-ideal-is-squared-union-connected", "text": "Let $R$ be a ring. Let $I$ be a finitely generated ideal. Assume that $I = I^2$. Then $V(I)$ is open and closed in $\\Spec(R)$, and $R/I \\cong R_e$ for some idempotent $e \\in R$."}
+{"_id": "409", "title": "algebra-lemma-connected-component", "text": "Let $R$ be a ring. A connected component of $\\Spec(R)$ is of the form $V(I)$, where $I$ is an ideal generated by idempotents such that every idempotent of $R$ either maps to $0$ or $1$ in $R/I$."}
+{"_id": "410", "title": "algebra-lemma-characterize-zero-local", "text": "Let $R$ be a ring. \\begin{enumerate} \\item For an element $x$ of an $R$-module $M$ the following are equivalent \\begin{enumerate} \\item $x = 0$, \\item $x$ maps to zero in $M_\\mathfrak p$ for all $\\mathfrak p \\in \\Spec(R)$, \\item $x$ maps to zero in $M_{\\mathfrak m}$ for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate} In other words, the map $M \\to \\prod_{\\mathfrak m} M_{\\mathfrak m}$ is injective. \\item Given an $R$-module $M$ the following are equivalent \\begin{enumerate} \\item $M$ is zero, \\item $M_{\\mathfrak p}$ is zero for all $\\mathfrak p \\in \\Spec(R)$, \\item $M_{\\mathfrak m}$ is zero for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate} \\item Given a complex $M_1 \\to M_2 \\to M_3$ of $R$-modules the following are equivalent \\begin{enumerate} \\item $M_1 \\to M_2 \\to M_3$ is exact, \\item for every prime $\\mathfrak p$ of $R$ the localization $M_{1, \\mathfrak p} \\to M_{2, \\mathfrak p} \\to M_{3, \\mathfrak p}$ is exact, \\item for every maximal ideal $\\mathfrak m$ of $R$ the localization $M_{1, \\mathfrak m} \\to M_{2, \\mathfrak m} \\to M_{3, \\mathfrak m}$ is exact. \\end{enumerate} \\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent \\begin{enumerate} \\item $f$ is injective, \\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is injective for all primes $\\mathfrak p$ of $R$, \\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is injective for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate} \\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent \\begin{enumerate} \\item $f$ is surjective, \\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is surjective for all primes $\\mathfrak p$ of $R$, \\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is surjective for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate} \\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent \\begin{enumerate} \\item $f$ is bijective, \\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is bijective for all primes $\\mathfrak p$ of $R$, \\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is bijective for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "411", "title": "algebra-lemma-cover", "text": "\\begin{slogan} Zariski-local properties of modules and algebras \\end{slogan} Let $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be an $R$-algebra. Suppose that $f_1, \\ldots, f_n$ is a finite list of elements of $R$ such that $\\bigcup D(f_i) = \\Spec(R)$, in other words $(f_1, \\ldots, f_n) = R$. \\begin{enumerate} \\item If each $M_{f_i} = 0$ then $M = 0$. \\item If each $M_{f_i}$ is a finite $R_{f_i}$-module, then $M$ is a finite $R$-module. \\item If each $M_{f_i}$ is a finitely presented $R_{f_i}$-module, then $M$ is a finitely presented $R$-module. \\item Let $M \\to N$ be a map of $R$-modules. If $M_{f_i} \\to N_{f_i}$ is an isomorphism for each $i$ then $M \\to N$ is an isomorphism. \\item Let $0 \\to M'' \\to M \\to M' \\to 0$ be a complex of $R$-modules. If $0 \\to M''_{f_i} \\to M_{f_i} \\to M'_{f_i} \\to 0$ is exact for each $i$, then $0 \\to M'' \\to M \\to M' \\to 0$ is exact. \\item If each $R_{f_i}$ is Noetherian, then $R$ is Noetherian. \\item If each $S_{f_i}$ is a finite type $R$-algebra, so is $S$. \\item If each $S_{f_i}$ is of finite presentation over $R$, so is $S$. \\end{enumerate}"}
+{"_id": "412", "title": "algebra-lemma-cover-upstairs", "text": "Let $R \\to S$ be a ring map. Suppose that $g_1, \\ldots, g_n$ is a finite list of elements of $S$ such that $\\bigcup D(g_i) = \\Spec(S)$ in other words $(g_1, \\ldots, g_n) = S$. \\begin{enumerate} \\item If each $S_{g_i}$ is of finite type over $R$, then $S$ is of finite type over $R$. \\item If each $S_{g_i}$ is of finite presentation over $R$, then $S$ is of finite presentation over $R$. \\end{enumerate}"}
+{"_id": "413", "title": "algebra-lemma-cover-module", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be elements of $R$ generating the unit ideal. Let $M$ be an $R$-module. The sequence $$ 0 \\to M \\xrightarrow{\\alpha} \\bigoplus\\nolimits_{i = 1}^n M_{f_i} \\xrightarrow{\\beta} \\bigoplus\\nolimits_{i, j = 1}^n M_{f_i f_j} $$ is exact, where $\\alpha(m) = (m/1, \\ldots, m/1)$ and $\\beta(m_1/f_1^{e_1}, \\ldots, m_n/f_n^{e_n}) = (m_i/f_i^{e_i} - m_j/f_j^{e_j})_{(i, j)}$."}
+{"_id": "414", "title": "algebra-lemma-standard-covering", "text": "Let $R$ be a ring, and let $f_1, f_2, \\ldots f_n\\in R$ generate the unit ideal in $R$. Then the following sequence is exact: $$ 0 \\longrightarrow R \\longrightarrow \\bigoplus\\nolimits_i R_{f_i} \\longrightarrow \\bigoplus\\nolimits_{i, j}R_{f_if_j} $$ where the maps $\\alpha : R \\longrightarrow \\bigoplus_i R_{f_i}$ and $\\beta : \\bigoplus_i R_{f_i} \\longrightarrow \\bigoplus_{i, j} R_{f_if_j}$ are defined as $$ \\alpha(x) = \\left(\\frac{x}{1}, \\ldots, \\frac{x}{1}\\right) \\text{ and } \\beta\\left(\\frac{x_1}{f_1^{r_1}}, \\ldots, \\frac{x_n}{f_n^{r_n}}\\right) = \\left(\\frac{x_i}{f_i^{r_i}}-\\frac{x_j}{f_j^{r_j}}~\\text{in}~R_{f_if_j}\\right). $$"}
+{"_id": "415", "title": "algebra-lemma-disjoint-implies-product", "text": "Let $R$ be a ring. If $\\Spec(R) = U \\amalg V$ with both $U$ and $V$ open then $R \\cong R_1 \\times R_2$ with $U \\cong \\Spec(R_1)$ and $V \\cong \\Spec(R_2)$ via the maps in Lemma \\ref{lemma-spec-product}. Moreover, both $R_1$ and $R_2$ are localizations as well as quotients of the ring $R$."}
+{"_id": "416", "title": "algebra-lemma-when-injective-covering", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$. Let $M$ be an $R$-module. Then $M \\to \\bigoplus M_{f_i}$ is injective if and only if $$ M \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} M, \\quad m \\longmapsto (f_1m, \\ldots, f_nm) $$ is injective."}
+{"_id": "417", "title": "algebra-lemma-glue-modules", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$. Suppose we are given the following data: \\begin{enumerate} \\item For each $i$ an $R_{f_i}$-module $M_i$. \\item For each pair $i, j$ an $R_{f_if_j}$-module isomorphism $\\psi_{ij} : (M_i)_{f_j} \\to (M_j)_{f_i}$. \\end{enumerate} which satisfy the ``cocycle condition'' that all the diagrams $$ \\xymatrix{ (M_i)_{f_jf_k} \\ar[rd]_{\\psi_{ij}} \\ar[rr]^{\\psi_{ik}} & & (M_k)_{f_if_j} \\\\ & (M_j)_{f_if_k} \\ar[ru]_{\\psi_{jk}} } $$ commute (for all triples $i, j, k$). Given this data define $$ M = \\Ker\\left( \\bigoplus\\nolimits_{1 \\leq i \\leq n} M_i \\longrightarrow \\bigoplus\\nolimits_{1 \\leq i, j \\leq n} (M_i)_{f_j} \\right) $$ where $(m_1, \\ldots, m_n)$ maps to the element whose $(i, j)$th entry is $m_i/1 - \\psi_{ji}(m_j/1)$. Then the natural map $M \\to M_i$ induces an isorphism $M_{f_i} \\to M_i$. Moreover $\\psi_{ij}(m/1) = m/1$ for all $m \\in M$ (with obvious notation)."}
+{"_id": "418", "title": "algebra-lemma-minimal-prime-reduced-ring", "text": "Let $\\mathfrak p$ be a minimal prime of a ring $R$. Every element of the maximal ideal of $R_{\\mathfrak p}$ is nilpotent. If $R$ is reduced then $R_{\\mathfrak p}$ is a field."}
+{"_id": "419", "title": "algebra-lemma-reduced-ring-sub-product-fields", "text": "Let $R$ be a reduced ring. Then \\begin{enumerate} \\item $R$ is a subring of a product of fields, \\item $R \\to \\prod_{\\mathfrak p\\text{ minimal}} R_{\\mathfrak p}$ is an embedding into a product of fields, \\item $\\bigcup_{\\mathfrak p\\text{ minimal}} \\mathfrak p$ is the set of zerodivisors of $R$. \\end{enumerate}"}
+{"_id": "420", "title": "algebra-lemma-total-ring-fractions", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset consisting of nonzerodivisors. Then $Q(R) \\cong Q(S^{-1}R)$. In particular $Q(R) \\cong Q(Q(R))$."}
+{"_id": "421", "title": "algebra-lemma-total-ring-fractions-no-embedded-points", "text": "Let $R$ be a ring. Assume that $R$ has finitely many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and that $\\mathfrak q_1 \\cup \\ldots \\cup \\mathfrak q_t$ is the set of zerodivisors of $R$. Then the total ring of fractions $Q(R)$ is equal to $R_{\\mathfrak q_1} \\times \\ldots \\times R_{\\mathfrak q_t}$."}
+{"_id": "422", "title": "algebra-lemma-irreducible", "text": "Let $R$ be a ring. \\begin{enumerate} \\item For a prime $\\mathfrak p \\subset R$ the closure of $\\{\\mathfrak p\\}$ in the Zariski topology is $V(\\mathfrak p)$. In a formula $\\overline{\\{\\mathfrak p\\}} = V(\\mathfrak p)$. \\item The irreducible closed subsets of $\\Spec(R)$ are exactly the subsets $V(\\mathfrak p)$, with $\\mathfrak p \\subset R$ a prime. \\item The irreducible components (see Topology, Definition \\ref{topology-definition-irreducible-components}) of $\\Spec(R)$ are  exactly the subsets $V(\\mathfrak p)$, with $\\mathfrak p \\subset R$ a minimal prime. \\end{enumerate}"}
+{"_id": "423", "title": "algebra-lemma-spec-spectral", "text": "The spectrum of a ring is a spectral space, see Topology, Definition \\ref{topology-definition-spectral-space}."}
+{"_id": "424", "title": "algebra-lemma-irreducible-components-containing-x", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. \\begin{enumerate} \\item the set of irreducible closed subsets of $\\Spec(R)$ passing through $\\mathfrak p$ is in one-to-one correspondence with primes $\\mathfrak q \\subset R_{\\mathfrak p}$. \\item The set of irreducible components of $\\Spec(R)$ passing through $\\mathfrak p$ is in one-to-one correspondence with minimal primes $\\mathfrak q \\subset R_{\\mathfrak p}$. \\end{enumerate}"}
+{"_id": "425", "title": "algebra-lemma-standard-open-containing-maximal-point", "text": "Let $R$ be a ring. Let $\\mathfrak p$ be a minimal prime of $R$. Let $W \\subset \\Spec(R)$ be a quasi-compact open not containing the point $\\mathfrak p$. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that $D(f) \\cap W = \\emptyset$."}
+{"_id": "426", "title": "algebra-lemma-ring-with-only-minimal-primes", "text": "Let $R$ be a ring. Let $X = \\Spec(R)$ as a topological space. The following are equivalent \\begin{enumerate} \\item $X$ is profinite, \\item $X$ is Hausdorff, \\item $X$ is totally disconnected. \\item every quasi-compact open of $X$ is closed, \\item there are no nontrivial inclusions between its prime ideals, \\item every prime ideal is a maximal ideal, \\item every prime ideal is minimal, \\item every standard open $D(f) \\subset X$ is closed, and \\item add more here. \\end{enumerate}"}
+{"_id": "427", "title": "algebra-lemma-colon", "text": "Let $R$ be a ring. For a principal ideal $J \\subset R$, and for any ideal $I \\subset J$ we have $I = J (I : J)$."}
+{"_id": "428", "title": "algebra-lemma-simple", "text": "Let $R$ be a ring. Let $S$ be a multiplicative subset of $R$. An ideal $I \\subset R$ which is maximal with respect to the property that $I \\cap S = \\emptyset$ is prime."}
+{"_id": "429", "title": "algebra-lemma-cohen", "text": "Let $R$ be a ring. \\begin{enumerate} \\item An ideal $I \\subset R$ maximal with respect to not being finitely generated is prime. \\item If every prime ideal of $R$ is finitely generated, then every ideal of $R$ is finitely generated\\footnote{Later we will say that $R$ is Noetherian.}. \\end{enumerate}"}
+{"_id": "432", "title": "algebra-lemma-qc-open", "text": "Let $U \\subset \\Spec(R)$ be open. The following are equivalent: \\begin{enumerate} \\item $U$ is retrocompact in $\\Spec(R)$, \\item $U$ is quasi-compact, \\item $U$ is a finite union of standard opens, and \\item there exists a finitely generated ideal $I \\subset R$ such that $X \\setminus V(I) = U$. \\end{enumerate}"}
+{"_id": "433", "title": "algebra-lemma-affine-map-quasi-compact", "text": "Let $\\varphi : R \\to S$ be a ring map. The induced continuous map $f : \\Spec(S) \\to \\Spec(R)$ is quasi-compact. For any constructible set $E \\subset \\Spec(R)$ the inverse image $f^{-1}(E)$ is constructible in $\\Spec(S)$."}
+{"_id": "434", "title": "algebra-lemma-constructible", "text": "Let $R$ be a ring. A subset of $\\Spec(R)$ is constructible if and only if it can be written as a finite union of subsets of the form $D(f) \\cap V(g_1, \\ldots, g_m)$ for $f, g_1, \\ldots, g_m \\in R$."}
+{"_id": "435", "title": "algebra-lemma-constructible-is-image", "text": "Let $R$ be a ring and let $T \\subset \\Spec(R)$ be constructible. Then there exists a ring map $R \\to S$ of finite presentation such that $T$ is the image of $\\Spec(S)$ in $\\Spec(R)$."}
+{"_id": "436", "title": "algebra-lemma-open-fp", "text": "Let $R$ be a ring. Let $f$ be an element of $R$. Let $S = R_f$. Then the image of a constructible subset of $\\Spec(S)$ is constructible in $\\Spec(R)$."}
+{"_id": "437", "title": "algebra-lemma-closed-fp", "text": "Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$. Let $S = R/I$. Then the image of a constructible of $\\Spec(S)$ is constructible in $\\Spec(R)$."}
+{"_id": "438", "title": "algebra-lemma-affineline-open", "text": "Let $R$ be a ring. The map $\\Spec(R[x]) \\to \\Spec(R)$ is open, and the image of any standard open is a quasi-compact open."}
+{"_id": "439", "title": "algebra-lemma-characteristic-polynomial-prime", "text": "Let $R \\to A$ be a ring homomorphism. Assume $A \\cong R^{\\oplus n}$ as an $R$-module. Let $f \\in A$. The multiplication map $m_f: A \\to A$ is $R$-linear and hence has a characteristic polynomial $P(T) = T^n + r_{n-1}T^{n-1} + \\ldots + r_0 \\in R[T]$. For any prime $\\mathfrak{p} \\in \\Spec(R)$, $f$ acts nilpotently on $A \\otimes_R \\kappa(\\mathfrak{p})$ if and only if $\\mathfrak p \\in V(r_0, \\ldots, r_{n-1})$."}
+{"_id": "440", "title": "algebra-lemma-affineline-special", "text": "Let $R$ be a ring. Let $f, g \\in R[x]$ be polynomials. Assume the leading coefficient of $g$ is a unit of $R$. There exists elements $r_i\\in R$, $i = 1\\ldots, n$ such that the image of $D(f) \\cap V(g)$ in $\\Spec(R)$ is $\\bigcup_{i = 1, \\ldots, n} D(r_i)$."}
+{"_id": "441", "title": "algebra-lemma-generic-finite-presentation", "text": "Let $R \\subset S$ be an inclusion of domains. Assume that $R \\to S$ is of finite type. There exists a nonzero $f \\in R$, and a nonzero $g \\in S$ such that $R_f \\to S_{fg}$ is of finite presentation."}
+{"_id": "442", "title": "algebra-lemma-characterize-image-finite-type", "text": "Let $R \\to S$ be a finite type ring map. Denote $X = \\Spec(R)$ and $Y = \\Spec(S)$. Write $f : Y \\to X$ the induced map of spectra. Let $E \\subset Y = \\Spec(S)$ be a constructible set. If a point $\\xi \\in X$ is in $f(E)$, then $\\overline{\\{\\xi\\}} \\cap f(E)$ contains an open dense subset of $\\overline{\\{\\xi\\}}$."}
+{"_id": "443", "title": "algebra-lemma-surjective-spec-radical-ideal", "text": "Let $\\varphi : R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item The map $\\Spec(S) \\to \\Spec(R)$ is surjective. \\item For any ideal $I \\subset R$ the inverse image of $\\sqrt{IS}$ in $R$ is equal to $\\sqrt{I}$. \\item For any radical ideal $I \\subset R$ the inverse image of $IS$ in $R$ is equal to $I$. \\item For every prime $\\mathfrak p$ of $R$ the inverse image of $\\mathfrak p S$ in $R$ is $\\mathfrak p$. \\end{enumerate} In this case the same is true after any base change: Given a ring map $R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ has the equivalent properties (1), (2), (3) as well."}
+{"_id": "444", "title": "algebra-lemma-domain-image-dense-set-points-generic-point", "text": "Let $R$ be a domain. Let $\\varphi : R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item The ring map $R \\to S$ is injective. \\item The image $\\Spec(S) \\to \\Spec(R)$ contains a dense set of points. \\item There exists a prime ideal $\\mathfrak q \\subset S$ whose inverse image in $R$ is $(0)$. \\end{enumerate}"}
+{"_id": "445", "title": "algebra-lemma-injective-minimal-primes-in-image", "text": "Let $R \\subset S$ be an injective ring map. Then $\\Spec(S) \\to \\Spec(R)$ hits all the minimal primes."}
+{"_id": "446", "title": "algebra-lemma-image-dense-generic-points", "text": "Let $R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item The kernel of $R \\to S$ consists of nilpotent elements. \\item The minimal primes of $R$ are in the image of $\\Spec(S) \\to \\Spec(R)$. \\item The image of $\\Spec(S) \\to \\Spec(R)$ is dense in $\\Spec(R)$. \\end{enumerate}"}
+{"_id": "447", "title": "algebra-lemma-minimal-prime-image-minimal-prime", "text": "Let $R \\to S$ be a ring map. If a minimal prime $\\mathfrak p \\subset R$ is in the image of $\\Spec(S) \\to \\Spec(R)$, then it is the image of a minimal prime."}
+{"_id": "448", "title": "algebra-lemma-Noetherian-permanence", "text": "\\begin{slogan} Noetherian property is stable by passage to finite type extension and localization. \\end{slogan} Any finitely generated ring over a Noetherian ring is Noetherian. Any localization of a Noetherian ring is Noetherian."}
+{"_id": "449", "title": "algebra-lemma-Noetherian-power-series", "text": "If $R$ is a Noetherian ring, then so is the formal power series ring $R[[x_1, \\ldots, x_n]]$."}
+{"_id": "450", "title": "algebra-lemma-obvious-Noetherian", "text": "Any finite type algebra over a field is Noetherian. Any finite type algebra over $\\mathbf{Z}$ is Noetherian."}
+{"_id": "451", "title": "algebra-lemma-Noetherian-finite-type-is-finite-presentation", "text": "Let $R$ be a Noetherian ring. \\begin{enumerate} \\item Any finite $R$-module is of finite presentation. \\item Any finite type $R$-algebra is of finite presentation over $R$. \\end{enumerate}"}
+{"_id": "452", "title": "algebra-lemma-Noetherian-topology", "text": "If $R$ is a Noetherian ring then $\\Spec(R)$ is a Noetherian topological space, see Topology, Definition \\ref{topology-definition-noetherian}."}
+{"_id": "453", "title": "algebra-lemma-Noetherian-irreducible-components", "text": "\\begin{slogan} A Noetherian affine scheme has finitely many generic points. \\end{slogan} If $R$ is a Noetherian ring then $\\Spec(R)$ has finitely many irreducible components. In other words $R$ has finitely many minimal primes."}
+{"_id": "454", "title": "algebra-lemma-Noetherian-base-change-finite-type", "text": "Let $R \\to S$ be a ring map. Let $R \\to R'$ be of finite type. If $S$ is Noetherian, then the base change $S' = R' \\otimes_R S$ is Noetherian."}
+{"_id": "455", "title": "algebra-lemma-Noetherian-field-extension", "text": "Let $k$ be a field and let $R$ be a Noetherian $k$-algebra. If $K/k$ is a finitely generated field extension then $K \\otimes_k R$ is Noetherian."}
+{"_id": "456", "title": "algebra-lemma-subring-of-local-ring", "text": "Let $R$ be a ring and $\\mathfrak p \\subset R$ be a prime. There exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that $R_f \\to R_\\mathfrak p$ is injective in each of the following cases \\begin{enumerate} \\item $R$ is a domain, \\item $R$ is Noetherian, or \\item $R$ is reduced and has finitely many minimal primes. \\end{enumerate}"}
+{"_id": "457", "title": "algebra-lemma-surjective-endo-noetherian-ring-is-iso", "text": "Any surjective endomorphism of a Noetherian ring is an isomorphism."}
+{"_id": "458", "title": "algebra-lemma-locally-nilpotent", "text": "Let $R \\to R'$ be a ring map and let $I \\subset R$ be a locally nilpotent ideal. Then $IR'$ is a locally nilpotent ideal of $R'$."}
+{"_id": "459", "title": "algebra-lemma-locally-nilpotent-unit", "text": "Let $R$ be a ring and let $I \\subset R$ be a locally nilpotent ideal. An element $x$ of $R$ is a unit if and only if the image of $x$ in $R/I$ is a unit."}
+{"_id": "460", "title": "algebra-lemma-Noetherian-power", "text": "\\begin{slogan} An ideal in a Noetherian ring is nilpotent if each element of the ideal is nilpotent. \\end{slogan} Let $R$ be a Noetherian ring. Let $I, J$ be ideals of $R$. Suppose $J \\subset \\sqrt{I}$. Then $J^n \\subset I$ for some $n$. In particular, in a Noetherian ring the notions of ``locally nilpotent ideal'' and ``nilpotent ideal'' coincide."}
+{"_id": "461", "title": "algebra-lemma-lift-idempotents", "text": "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. Then $R \\to R/I$ induces a bijection on idempotents."}
+{"_id": "462", "title": "algebra-lemma-lift-idempotents-noncommutative", "text": "Let $A$ be a possibly noncommutative algebra. Let $e \\in A$ be an element such that $x = e^2 - e$ is nilpotent. Then there exists an idempotent of the form $e' = e + x(\\sum a_{i, j}e^ix^j) \\in A$ with $a_{i, j} \\in \\mathbf{Z}$."}
+{"_id": "463", "title": "algebra-lemma-lift-nth-roots", "text": "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. Let $n \\geq 1$ be an integer which is invertible in $R/I$. Then \\begin{enumerate} \\item the $n$th power map $1 + I \\to 1 + I$, $1 + x \\mapsto (1 + x)^n$ is a bijection, \\item a unit of $R$ is a $n$th power if and only if its image in $R/I$ is an $n$th power. \\end{enumerate}"}
+{"_id": "464", "title": "algebra-lemma-invert-closed-quotient", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. Assume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$ is closed. Then $S^{-1}R \\cong R/I$ for some ideal $I \\subset R$."}
+{"_id": "466", "title": "algebra-lemma-field-finite-type-over-domain", "text": "Let $R$ be a ring. Let $K$ be a field. If $R \\subset K$ and $K$ is of finite type over $R$, then there exists an $f \\in R$ such that $R_f$ is a field, and $R_f \\subset K$ is a finite field extension."}
+{"_id": "467", "title": "algebra-lemma-finite-type-field-Jacobson", "text": "Any algebra of finite type over a field is Jacobson."}
+{"_id": "469", "title": "algebra-lemma-jacobson", "text": "A ring $R$ is Jacobson if and only if $\\Spec(R)$ is Jacobson, see Topology, Definition \\ref{topology-definition-space-jacobson}."}
+{"_id": "470", "title": "algebra-lemma-characterize-jacobson", "text": "Let $R$ be a ring. If $R$ is not Jacobson there exist a prime $\\mathfrak p \\subset R$, an element $f \\in R$ such that the following hold \\begin{enumerate} \\item $\\mathfrak p$ is not a maximal ideal, \\item $f \\not \\in \\mathfrak p$, \\item $V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$, and \\item $(R/\\mathfrak p)_f$ is a field. \\end{enumerate} On the other hand, if $R$ is Jacobson, then for any pair $(\\mathfrak p, f)$ such that (1) and (2) hold the set $V(\\mathfrak p) \\cap D(f)$ is infinite."}
+{"_id": "471", "title": "algebra-lemma-pid-jacobson", "text": "The ring $\\mathbf{Z}$ is a Jacobson ring. More generally, let $R$ be a ring such that \\begin{enumerate} \\item $R$ is a domain, \\item $R$ is Noetherian, \\item any nonzero prime ideal is a maximal ideal, and \\item $R$ has infinitely many maximal ideals. \\end{enumerate} Then $R$ is a Jacobson ring."}
+{"_id": "472", "title": "algebra-lemma-finite-residue-extension-closed", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak m \\subset R$ be a maximal ideal. Let $\\mathfrak q \\subset S$ be a prime ideal lying over $\\mathfrak m$ such that $\\kappa(\\mathfrak m) \\subset \\kappa(\\mathfrak q)$ is an algebraic field extension. Then $\\mathfrak q$ is a maximal ideal of $S$."}
+{"_id": "473", "title": "algebra-lemma-dimension", "text": "Suppose that $k$ is a field and suppose that $V$ is a nonzero vector space over $k$. Assume the dimension of $V$ (which is a cardinal number) is smaller than the cardinality of $k$. Then for any linear operator $T : V \\to V$ there exists some monic polynomial $P(t) \\in k[t]$ such that $P(T)$ is not invertible."}
+{"_id": "474", "title": "algebra-lemma-base-change-Jacobson", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. For any field extension $k \\subset K$ whose cardinality is larger than the cardinality of $S$ we have \\begin{enumerate} \\item for every maximal ideal $\\mathfrak m$ of $S_K$ the field $\\kappa(\\mathfrak m)$ is algebraic over $K$, and \\item $S_K$ is a Jacobson ring. \\end{enumerate}"}
+{"_id": "475", "title": "algebra-lemma-Jacobson-invert-element", "text": "Let $R$ be a Jacobson ring. Let $f \\in R$. The ring $R_f$ is Jacobson and maximal ideals of $R_f$ correspond to maximal ideals of $R$ not containing $f$."}
+{"_id": "476", "title": "algebra-lemma-Jacobson-mod-ideal", "text": "Let $R$ be a Jacobson ring. Let $I \\subset R$ be an ideal. The ring $R/I$ is Jacobson and maximal ideals of $R/I$ correspond to maximal ideals of $R$ containing $I$."}
+{"_id": "477", "title": "algebra-lemma-silly-jacobson", "text": "Let $R$ be a Jacobson ring. Let $K$ be a field. Let $R \\subset K$ and $K$ is of finite type over $R$. Then $R$ is a field and $K/R$ is a finite field extension."}
+{"_id": "479", "title": "algebra-lemma-image-finite-type-map-Jacobson-rings", "text": "Let $R \\to S$ be a finite type ring map of Jacobson rings. Denote $X = \\Spec(R)$ and $Y = \\Spec(S)$. Write $f : Y \\to X$ the induced map of spectra. Let $E \\subset Y = \\Spec(S)$ be a constructible set. Denote with a subscript ${}_0$ the set of closed points of a topological space. \\begin{enumerate} \\item We have $f(E)_0 = f(E_0) = X_0 \\cap f(E)$. \\item A point $\\xi \\in X$ is in $f(E)$ if and only if $\\overline{\\{\\xi\\}} \\cap f(E_0)$ is dense in $\\overline{\\{\\xi\\}}$. \\end{enumerate}"}
+{"_id": "481", "title": "algebra-lemma-characterize-integral-element", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $y \\in S$. If there exists a finite $R$-submodule $M$ of $S$ such that $1 \\in M$ and $yM \\subset M$, then $y$ is integral over $R$."}
+{"_id": "482", "title": "algebra-lemma-finite-is-integral", "text": "A finite ring extension is integral."}
+{"_id": "483", "title": "algebra-lemma-characterize-integral", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $s_1, \\ldots, s_n$ be a finite set of elements of $S$. In this case $s_i$ is integral over $R$ for all $i = 1, \\ldots, n$ if and only if there exists an $R$-subalgebra $S' \\subset S$ finite over $R$ containing all of the $s_i$."}
+{"_id": "484", "title": "algebra-lemma-characterize-finite-in-terms-of-integral", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is finite, \\item $R \\to S$ is integral and of finite type, and \\item there exist $x_1, \\ldots, x_n \\in S$ which generate $S$ as an algebra over $R$ such that each $x_i$ is integral over $R$. \\end{enumerate}"}
+{"_id": "485", "title": "algebra-lemma-integral-transitive", "text": "\\begin{slogan} A composition of integral ring maps is integral \\end{slogan} Suppose that $R \\to S$ and $S \\to T$ are integral ring maps. Then $R \\to T$ is integral."}
+{"_id": "486", "title": "algebra-lemma-integral-closure-is-ring", "text": "Let $R \\to S$ be a ring homomorphism. The set $$ S' = \\{s \\in S \\mid s\\text{ is integral over }R\\} $$ is an $R$-subalgebra of $S$."}
+{"_id": "487", "title": "algebra-lemma-finite-product-integral", "text": "Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$. Let $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively. Then an element $s = (s_1, \\ldots, s_n) \\in S$ is integral over $R$ if and only if each $s_i$ is integral over $R_i$."}
+{"_id": "488", "title": "algebra-lemma-finite-product-integral-closure", "text": "Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$. Denote the integral closure of $R_i$ in $S_i$ by $S'_i$. Further let $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively. Then the integral closure of $R$ in $S$ is the product of the $S'_i$. In particular $R \\to S$ is integrally closed if and only if each $R_i \\to S_i$ is integrally closed."}
+{"_id": "489", "title": "algebra-lemma-integral-closure-localize", "text": "Integral closure commutes with localization: If $A \\to B$ is a ring map, and $S \\subset A$ is a multiplicative subset, then the integral closure of $S^{-1}A$ in $S^{-1}B$ is $S^{-1}B'$, where $B' \\subset B$ is the integral closure of $A$ in $B$."}
+{"_id": "490", "title": "algebra-lemma-integral-closure-stalks", "text": "\\begin{slogan} An element of an algebra over a ring is integral over the ring if and only if it is locally integral at every prime ideal of the ring. \\end{slogan} Let $\\varphi : R \\to S$ be a ring map. Let $x \\in S$. The following are equivalent: \\begin{enumerate} \\item $x$ is integral over $R$, and \\item for every prime ideal $\\mathfrak p \\subset R$ the element $x \\in S_{\\mathfrak p}$ is integral over $R_{\\mathfrak p}$. \\end{enumerate}"}
+{"_id": "491", "title": "algebra-lemma-base-change-integral", "text": "\\begin{slogan} Integrality and finiteness are preserved under base change. \\end{slogan} Let $R \\to S$ and $R \\to R'$ be ring maps. Set $S' = R' \\otimes_R S$. \\begin{enumerate} \\item If $R \\to S$ is integral so is $R' \\to S'$. \\item If $R \\to S$ is finite so is $R' \\to S'$. \\end{enumerate}"}
+{"_id": "492", "title": "algebra-lemma-integral-local", "text": "Let $R \\to S$ be a ring map. Let $f_1, \\ldots, f_n \\in R$ generate the unit ideal. \\begin{enumerate} \\item If each $R_{f_i} \\to S_{f_i}$ is integral, so is $R \\to S$. \\item If each $R_{f_i} \\to S_{f_i}$ is finite, so is $R \\to S$. \\end{enumerate}"}
+{"_id": "493", "title": "algebra-lemma-integral-permanence", "text": "Let $A \\to B \\to C$ be ring maps. \\begin{enumerate} \\item If $A \\to C$ is integral so is $B \\to C$. \\item If $A \\to C$ is finite so is $B \\to C$. \\end{enumerate}"}
+{"_id": "494", "title": "algebra-lemma-integral-closure-transitive", "text": "Let $A \\to B \\to C$ be ring maps. Let $B'$ be the integral closure of $A$ in $B$, let $C'$ be the integral closure of $B'$ in $C$. Then $C'$ is the integral closure of $A$ in $C$."}
+{"_id": "495", "title": "algebra-lemma-integral-overring-surjective", "text": "Suppose that $R \\to S$ is an integral ring extension with $R \\subset S$. Then $\\varphi : \\Spec(S) \\to \\Spec(R)$ is surjective."}
+{"_id": "496", "title": "algebra-lemma-integral-under-field", "text": "Let $R$ be a ring. Let $K$ be a field. If $R \\subset K$ and $K$ is integral over $R$, then $R$ is a field and $K$ is an algebraic extension. If $R \\subset K$ and $K$ is finite over $R$, then $R$ is a field and $K$ is a finite algebraic extension."}
+{"_id": "497", "title": "algebra-lemma-integral-over-field", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra over $k$. \\begin{enumerate} \\item If $S$ is a domain and finite dimensional over $k$, then $S$ is a field. \\item If $S$ is integral over $k$ and a domain, then $S$ is a field. \\item If $S$ is integral over $k$ then every prime of $S$ is a maximal ideal (see Lemma \\ref{lemma-ring-with-only-minimal-primes} for more consequences). \\end{enumerate}"}
+{"_id": "498", "title": "algebra-lemma-integral-no-inclusion", "text": "Suppose $R \\to S$ is integral. Let $\\mathfrak q, \\mathfrak q' \\in \\Spec(S)$ be distinct primes having the same image in $\\Spec(R)$. Then neither $\\mathfrak q \\subset \\mathfrak q'$ nor $\\mathfrak q' \\subset \\mathfrak q$."}
+{"_id": "499", "title": "algebra-lemma-finite-finite-fibres", "text": "Suppose $R \\to S$ is finite. Then the fibres of $\\Spec(S) \\to \\Spec(R)$ are finite."}
+{"_id": "500", "title": "algebra-lemma-integral-going-up", "text": "Let $R \\to S$ be a ring map such that $S$ is integral over $R$. Let $\\mathfrak p \\subset \\mathfrak p' \\subset R$ be primes. Let $\\mathfrak q$ be a prime of $S$ mapping to $\\mathfrak p$. Then there exists a prime $\\mathfrak q'$ with $\\mathfrak q \\subset \\mathfrak q'$ mapping to $\\mathfrak p'$."}
+{"_id": "501", "title": "algebra-lemma-finite-finitely-presented-extension", "text": "Let $R \\to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. Then $M$ is finitely presented as an $R$-module if and only if $M$ is finitely presented as an $S$-module."}
+{"_id": "503", "title": "algebra-lemma-integral-closure-in-normal", "text": "Let $R \\to S$ be a ring map. If $S$ is a normal domain, then the integral closure of $R$ in $S$ is a normal domain."}
+{"_id": "504", "title": "algebra-lemma-almost-integral", "text": "Let $R$ be a domain with fraction field $K$. If $u, v \\in K$ are almost integral over $R$, then so are $u + v$ and $uv$. Any element $g \\in K$ which is integral over $R$ is almost integral over $R$. If $R$ is Noetherian then the converse holds as well."}
+{"_id": "505", "title": "algebra-lemma-localize-normal-domain", "text": "Any localization of a normal domain is normal."}
+{"_id": "506", "title": "algebra-lemma-PID-normal", "text": "A principal ideal domain is normal."}
+{"_id": "507", "title": "algebra-lemma-prepare-polynomial-ring-normal", "text": "Let $R$ be a domain with fraction field $K$. Suppose $f = \\sum \\alpha_i x^i$ is an element of $K[x]$. \\begin{enumerate} \\item If $f$ is integral over $R[x]$ then all $\\alpha_i$ are integral over $R$, and \\item If $f$ is almost integral over $R[x]$ then all $\\alpha_i$ are almost integral over $R$. \\end{enumerate}"}
+{"_id": "508", "title": "algebra-lemma-polynomial-domain-normal", "text": "Let $R$ be a normal domain. Then $R[x]$ is a normal domain."}
+{"_id": "509", "title": "algebra-lemma-power-series-over-Noetherian-normal-domain", "text": "Let $R$ be a Noetherian normal domain. Then $R[[x]]$ is a Noetherian normal domain."}
+{"_id": "510", "title": "algebra-lemma-normality-is-local", "text": "Let $R$ be a domain. The following are equivalent: \\begin{enumerate} \\item The domain $R$ is a normal domain, \\item for every prime $\\mathfrak p \\subset R$ the local ring $R_{\\mathfrak p}$ is a normal domain, and \\item for every maximal ideal $\\mathfrak m$ the ring $R_{\\mathfrak m}$ is a normal domain. \\end{enumerate}"}
+{"_id": "511", "title": "algebra-lemma-normal-ring-integrally-closed", "text": "A normal ring is integrally closed in its total ring of fractions."}
+{"_id": "512", "title": "algebra-lemma-localization-normal-ring", "text": "A localization of a normal ring is a normal ring."}
+{"_id": "513", "title": "algebra-lemma-polynomial-ring-normal", "text": "Let $R$ be a normal ring. Then $R[x]$ is a normal ring."}
+{"_id": "514", "title": "algebra-lemma-finite-product-normal", "text": "A finite product of normal rings is normal."}
+{"_id": "515", "title": "algebra-lemma-characterize-reduced-ring-normal", "text": "Let $R$ be a ring. Assume $R$ is reduced and has finitely many minimal primes. Then the following are equivalent: \\begin{enumerate} \\item $R$ is a normal ring, \\item $R$ is integrally closed in its total ring of fractions, and \\item $R$ is a finite product of normal domains. \\end{enumerate}"}
+{"_id": "516", "title": "algebra-lemma-colimit-normal-ring", "text": "Let $(R_i, \\varphi_{ii'})$ be a directed system (Categories, Definition \\ref{definition-directed-system}) of rings. If each $R_i$ is a normal ring so is $R = \\colim_i R_i$."}
+{"_id": "517", "title": "algebra-lemma-characterize-integral-ideal", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. Let $A = \\sum I^nt^n \\subset R[t]$ be the subring of the polynomial ring generated by $R \\oplus It \\subset R[t]$. An element $s \\in S$ is integral over $I$ if and only if the element $st \\in S[t]$ is integral over $A$."}
+{"_id": "518", "title": "algebra-lemma-integral-over-ideal-is-submodule", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. The set of elements of $S$ which are integral over $I$ form a $R$-submodule of $S$. Furthermore, if $s \\in S$ is integral over $R$, and $s'$ is integral over $I$, then $ss'$ is integral over $I$."}
+{"_id": "519", "title": "algebra-lemma-integral-integral-over-ideal", "text": "Suppose $\\varphi : R \\to S$ is integral. Suppose $I \\subset R$ is an ideal. Then every element of $IS$ is integral over $I$."}
+{"_id": "520", "title": "algebra-lemma-polynomials-divide", "text": "Let $K$ be a field. Let $n, m \\in \\mathbf{N}$ and $a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1} \\in K$. If the polynomial $x^n + a_{n - 1}x^{n - 1} + \\ldots + a_0$ divides the polynomial $x^m + b_{m - 1} x^{m - 1} + \\ldots + b_0$ in $K[x]$ then \\begin{enumerate} \\item $a_0, \\ldots, a_{n - 1}$ are integral over any subring $R_0$ of $K$ containing the elements $b_0, \\ldots, b_{m - 1}$, and \\item each $a_i$ lies in $\\sqrt{(b_0, \\ldots, b_{m-1})R}$ for any subring $R \\subset K$ containing the elements $a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1}$. \\end{enumerate}"}
+{"_id": "521", "title": "algebra-lemma-minimal-polynomial-normal-domain", "text": "Let $R \\subset S$ be an inclusion of domains. Assume $R$ is normal. Let $g \\in S$ be integral over $R$. Then the minimal polynomial of $g$ has coefficients in $R$."}
+{"_id": "522", "title": "algebra-lemma-flat-intersect-ideals", "text": "Let $R$ be a ring. Let $I, J \\subset R$ be ideals. Let $M$ be a flat $R$-module. Then $IM \\cap JM = (I \\cap J)M$."}
+{"_id": "523", "title": "algebra-lemma-colimit-flat", "text": "Let $R$ be a ring. Let $\\{M_i, \\varphi_{ii'}\\}$ be a directed system of flat $R$-modules. Then $\\colim_i M_i$ is a flat $R$-module."}
+{"_id": "524", "title": "algebra-lemma-composition-flat", "text": "A composition of (faithfully) flat ring maps is (faithfully) flat. If $R \\to R'$ is (faithfully) flat, and $M'$ is a (faithfully) flat $R'$-module, then $M'$ is a (faithfully) flat $R$-module."}
+{"_id": "525", "title": "algebra-lemma-flat", "text": "Let $M$ be an $R$-module. The following are equivalent: \\begin{enumerate} \\item \\label{item-flat} $M$ is flat over $R$. \\item \\label{item-injective} for every injection of $R$-modules $N \\subset N'$ the map $N \\otimes_R M \\to N'\\otimes_R M$ is injective. \\item \\label{item-f-ideal} for every ideal $I \\subset R$ the map $I \\otimes_R M \\to R \\otimes_R M = M$ is injective. \\item \\label{item-ffg-ideal} for every finitely generated ideal $I \\subset R$ the map $I \\otimes_R M \\to R \\otimes_R M = M$ is injective. \\end{enumerate}"}
+{"_id": "526", "title": "algebra-lemma-colimit-rings-flat", "text": "Let $\\{R_i, \\varphi_{ii'}\\}$ be a system of rings over the directed set $I$. Let $R = \\colim_i R_i$. Let $M$ be an $R$-module such that $M$ is flat as an $R_i$-module for all $i$. Then $M$ is flat as an $R$-module."}
+{"_id": "527", "title": "algebra-lemma-flat-base-change", "text": "Suppose that $M$ is (faithfully) flat over $R$, and that $R \\to R'$ is a ring map. Then $M \\otimes_R R'$ is (faithfully) flat over $R'$."}
+{"_id": "528", "title": "algebra-lemma-flatness-descends", "text": "Let $R \\to R'$ be a faithfully flat ring map. Let $M$ be a module over $R$, and set $M' = R' \\otimes_R M$. Then $M$ is flat over $R$ if and only if $M'$ is flat over $R'$."}
+{"_id": "529", "title": "algebra-lemma-flatness-descends-more-general", "text": "Let $R$ be a ring. Let $S \\to S'$ be a flat map of $R$-algebras. Let $M$ be a module over $S$, and set $M' = S' \\otimes_S M$. \\begin{enumerate} \\item If $M$ is flat over $R$, then $M'$ is flat over $R$. \\item If $S \\to S'$ is faithfully flat, then $M$ is flat over $R$ if and only if $M'$ is flat over $R$. \\end{enumerate}"}
+{"_id": "530", "title": "algebra-lemma-flat-permanence", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. If $M$ is flat as an $R$-module and faithfully flat as an $S$-module, then $R \\to S$ is flat."}
+{"_id": "531", "title": "algebra-lemma-flat-eq", "text": "A module $M$ over $R$ is flat if and only if every relation in $M$ is trivial."}
+{"_id": "532", "title": "algebra-lemma-flat-tor-zero", "text": "Suppose that $R$ is a ring, $0 \\to M'' \\to M' \\to M \\to 0$ a short exact sequence, and $N$ an $R$-module. If $M$ is flat then $N \\otimes_R M'' \\to N \\otimes_R M'$ is injective, i.e., the sequence $$ 0 \\to N \\otimes_R M'' \\to N \\otimes_R M' \\to N \\otimes_R M \\to 0 $$ is a short exact sequence."}
+{"_id": "533", "title": "algebra-lemma-flat-ses", "text": "Suppose that $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of $R$-modules. If $M'$ and $M''$ are flat so is $M$. If $M$ and $M''$ are flat so is $M'$."}
+{"_id": "534", "title": "algebra-lemma-easy-ff", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is faithfully flat, and \\item $M$ is flat and for all $R$-module homomorphisms $\\alpha : N \\to N'$ we have $\\alpha = 0$ if and only if $\\alpha \\otimes \\text{id}_M = 0$. \\end{enumerate}"}
+{"_id": "535", "title": "algebra-lemma-ff", "text": "\\begin{slogan} A flat module is faithfully flat if and only if it has nonzero fibers. \\end{slogan} Let $M$ be a flat $R$-module. The following are equivalent: \\begin{enumerate} \\item $M$ is faithfully flat, \\item for every nonzero $R$-module $N$, then tensor product $M \\otimes_R N$ is nonzero, \\item for all $\\mathfrak p \\in \\Spec(R)$ the tensor product $M \\otimes_R \\kappa(\\mathfrak p)$ is nonzero, and \\item for all maximal ideals $\\mathfrak m$ of $R$ the tensor product $M \\otimes_R \\kappa(\\mathfrak m) = M/{\\mathfrak m}M$ is nonzero. \\end{enumerate}"}
+{"_id": "536", "title": "algebra-lemma-ff-rings", "text": "Let $R \\to S$ be a flat ring map. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is faithfully flat, \\item the induced map on $\\Spec$ is surjective, and \\item any closed point $x \\in \\Spec(R)$ is in the image of the map $\\Spec(S) \\to \\Spec(R)$. \\end{enumerate}"}
+{"_id": "537", "title": "algebra-lemma-local-flat-ff", "text": "A flat local ring homomorphism of local rings is faithfully flat."}
+{"_id": "538", "title": "algebra-lemma-flat-localization", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. \\begin{enumerate} \\item The localization $S^{-1}R$ is a flat $R$-algebra. \\item If $M$ is an $S^{-1}R$-module, then $M$ is a flat $R$-module if and only if $M$ is a flat $S^{-1}R$-module. \\item Suppose $M$ is an $R$-module. Then $M$ is a flat $R$-module if and only if $M_{\\mathfrak p}$ is a flat $R_{\\mathfrak p}$-module for all primes $\\mathfrak p$ of $R$. \\item Suppose $M$ is an $R$-module. Then $M$ is a flat $R$-module if and only if $M_{\\mathfrak m}$ is a flat $R_{\\mathfrak m}$-module for all maximal ideals $\\mathfrak m$ of $R$. \\item Suppose $R \\to A$ is a ring map, $M$ is an $A$-module, and $g_1, \\ldots, g_m \\in A$ are elements generating the unit ideal of $A$. Then $M$ is flat over $R$ if and only if each localization $M_{g_i}$ is flat over $R$. \\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module. Then $M$ is a flat $R$-module if and only if the localization $M_{\\mathfrak q}$ is a flat $R_{\\mathfrak p}$-module (with $\\mathfrak p$ the prime of $R$ lying under $\\mathfrak q$) for all primes $\\mathfrak q$ of $A$. \\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module. Then $M$ is a flat $R$-module if and only if the localization $M_{\\mathfrak m}$ is a flat $R_{\\mathfrak p}$-module (with $\\mathfrak p = R \\cap \\mathfrak m$) for all maximal ideals $\\mathfrak m$ of $A$. \\end{enumerate}"}
+{"_id": "539", "title": "algebra-lemma-flat-going-down", "text": "Let $R \\to S$ be flat. Let $\\mathfrak p \\subset \\mathfrak p'$ be primes of $R$. Let $\\mathfrak q' \\subset S$ be a prime of $S$ mapping to $\\mathfrak p'$. Then there exists a prime $\\mathfrak q \\subset \\mathfrak q'$ mapping to $\\mathfrak p$."}
+{"_id": "540", "title": "algebra-lemma-colimit-faithfully-flat", "text": "Let $R$ be a ring. Let $\\{S_i, \\varphi_{ii'}\\}$ be a directed system of faithfully flat $R$-algebras. Then $S = \\colim_i S_i$ is a faithfully flat $R$-algebra."}
+{"_id": "541", "title": "algebra-lemma-support-zero", "text": "\\begin{slogan} A module over a ring has empty support if and only if it is the trivial module. \\end{slogan} Let $R$ be a ring. Let $M$ be an $R$-module. Then $$ M = (0) \\Leftrightarrow \\text{Supp}(M) = \\emptyset. $$"}
+{"_id": "542", "title": "algebra-lemma-annihilator-flat-base-change", "text": "Let $R \\to S$ be a flat ring map. Let $M$ be an $R$-module and $m \\in M$. Then $\\text{Ann}_R(m) S = \\text{Ann}_S(m \\otimes 1)$. If $M$ is a finite $R$-module, then $\\text{Ann}_R(M) S = \\text{Ann}_S(M \\otimes_R S)$."}
+{"_id": "543", "title": "algebra-lemma-support-closed", "text": "Let $R$ be a ring and let $M$ be an $R$-module. If $M$ is finite, then $\\text{Supp}(M)$ is closed. More precisely, if $I = \\text{Ann}(M)$ is the annihilator of $M$, then $V(I) = \\text{Supp}(M)$."}
+{"_id": "544", "title": "algebra-lemma-support-base-change", "text": "Let $R \\to R'$ be a ring map and let $M$ be a finite $R$-module. Then $\\text{Supp}(M \\otimes_R R')$ is the inverse image of $\\text{Supp}(M)$."}
+{"_id": "545", "title": "algebra-lemma-support-element", "text": "Let $R$ be a ring, let $M$ be an $R$-module, and let $m \\in M$. Then $\\mathfrak p \\in V(\\text{Ann}(m))$ if and only if $m$ does not map to zero in $M_\\mathfrak p$."}
+{"_id": "546", "title": "algebra-lemma-support-finite-presentation-constructible", "text": "Let $R$ be a ring and let $M$ be an $R$-module. If $M$ is a finitely presented $R$-module, then $\\text{Supp}(M)$ is a closed subset of $\\Spec(R)$ whose complement is quasi-compact."}
+{"_id": "547", "title": "algebra-lemma-support-quotient", "text": "Let $R$ be a ring and let $M$ be an $R$-module. \\begin{enumerate} \\item If $M$ is finite then the support of $M/IM$ is $\\text{Supp}(M) \\cap V(I)$. \\item If $N \\subset M$, then $\\text{Supp}(N) \\subset \\text{Supp}(M)$. \\item If $Q$ is a quotient module of $M$ then $\\text{Supp}(Q) \\subset \\text{Supp}(M)$. \\item If $0 \\to N \\to M \\to Q \\to 0$ is a short exact sequence then $\\text{Supp}(M) = \\text{Supp}(Q) \\cup \\text{Supp}(N)$. \\end{enumerate}"}
+{"_id": "548", "title": "algebra-lemma-open-going-down", "text": "Let $R \\to S$ be a ring map. If the induced map $\\varphi : \\Spec(S) \\to \\Spec(R)$ is open, then $R \\to S$ satisfies going down."}
+{"_id": "549", "title": "algebra-lemma-going-up-down-specialization", "text": "Let $R \\to S$ be a ring map. \\begin{enumerate} \\item $R \\to S$ satisfies going down if and only if generalizations lift along the map $\\Spec(S) \\to \\Spec(R)$, see Topology, Definition \\ref{topology-definition-lift-specializations}. \\item $R \\to S$ satisfies going up if and only if specializations lift along the map $\\Spec(S) \\to \\Spec(R)$, see Topology, Definition \\ref{topology-definition-lift-specializations}. \\end{enumerate}"}
+{"_id": "550", "title": "algebra-lemma-going-up-down-composition", "text": "Suppose $R \\to S$ and $S \\to T$ are ring maps satisfying going down. Then so does $R \\to T$. Similarly for going up."}
+{"_id": "551", "title": "algebra-lemma-image-stable-specialization-closed", "text": "Let $R \\to S$ be a ring map. Let $T \\subset \\Spec(R)$ be the image of $\\Spec(S)$. If $T$ is stable under specialization, then $T$ is closed."}
+{"_id": "552", "title": "algebra-lemma-going-up-closed", "text": "Let $R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item Going up holds for $R \\to S$, and \\item the map $\\Spec(S) \\to \\Spec(R)$ is closed. \\end{enumerate}"}
+{"_id": "553", "title": "algebra-lemma-constructible-stable-specialization-closed", "text": "Let $R$ be a ring. Let $E \\subset \\Spec(R)$ be a constructible subset. \\begin{enumerate} \\item If $E$ is stable under specialization, then $E$ is closed. \\item If $E$ is stable under generalization, then $E$ is open. \\end{enumerate}"}
+{"_id": "554", "title": "algebra-lemma-same-image", "text": "Let $k$ be a field, and let $R$, $S$ be $k$-algebras. Let $S' \\subset S$ be a sub $k$-algebra, and let $f \\in S' \\otimes_k R$. In the commutative diagram $$ \\xymatrix{ \\Spec((S \\otimes_k R)_f) \\ar[rd] \\ar[rr] & & \\Spec((S' \\otimes_k R)_f) \\ar[ld] \\\\ & \\Spec(R) & } $$ the images of the diagonal arrows are the same."}
+{"_id": "555", "title": "algebra-lemma-map-into-tensor-algebra-open", "text": "Let $k$ be a field. Let $R$ and $S$ be $k$-algebras. The map $\\Spec(S \\otimes_k R) \\to \\Spec(R)$ is open."}
+{"_id": "556", "title": "algebra-lemma-unique-prime-over-localize-below", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak p \\subset R$ be a prime. Assume that \\begin{enumerate} \\item there exists a unique prime $\\mathfrak q \\subset S$ lying over $\\mathfrak p$, and \\item either \\begin{enumerate} \\item going up holds for $R \\to S$, or \\item going down holds for $R \\to S$ and there is at most one prime of $S$ above every prime of $R$. \\end{enumerate} \\end{enumerate} Then $S_{\\mathfrak p} = S_{\\mathfrak q}$."}
+{"_id": "557", "title": "algebra-lemma-going-down-flat-module", "text": "Let $R \\to S$ be a ring map. Let $N$ be a finite $S$-module flat over $R$. Endow $\\text{Supp}(N) \\subset \\Spec(S)$ with the induced topology. Then generalizations lift along $\\text{Supp}(N) \\to \\Spec(R)$."}
+{"_id": "558", "title": "algebra-lemma-subextensions-are-separable", "text": "Let $k \\subset K$ be a separable field extension. For any subextension $k \\subset K' \\subset K$ the field extension $k \\subset K'$ is separable."}
+{"_id": "559", "title": "algebra-lemma-generating-finitely-generated-separable-field-extensions", "text": "Let $k \\subset K$ be a separably generated, and finitely generated field extension. Set $r = \\text{trdeg}_k(K)$. Then there exist elements $x_1, \\ldots, x_{r + 1}$ of $K$ such that \\begin{enumerate} \\item $x_1, \\ldots, x_r$ is a transcendence basis of $K$ over $k$, \\item $K = k(x_1, \\ldots, x_{r + 1})$, and \\item $x_{r + 1}$ is separable over $k(x_1, \\ldots, x_r)$. \\end{enumerate}"}
+{"_id": "560", "title": "algebra-lemma-make-separably-generated", "text": "Let $k \\subset K$ be a finitely generated field extension. There exists a diagram $$ \\xymatrix{ K \\ar[r] & K' \\\\ k \\ar[u] \\ar[r] & k' \\ar[u] } $$ where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field extensions such that $k' \\subset K'$ is a separably generated field extension."}
+{"_id": "561", "title": "algebra-lemma-subalgebra-separable", "text": "Elementary properties of geometrically reduced algebras. Let $k$ be a field. Let $S$ be a $k$-algebra. \\begin{enumerate} \\item If $S$ is geometrically reduced over $k$ so is every $k$-subalgebra. \\item If all finitely generated $k$-subalgebras of $S$ are geometrically reduced, then $S$ is geometrically reduced. \\item A directed colimit of geometrically reduced $k$-algebras is geometrically reduced. \\item If $S$ is geometrically reduced over $k$, then any localization of $S$ is geometrically reduced over $k$. \\end{enumerate}"}
+{"_id": "562", "title": "algebra-lemma-geometrically-reduced-permanence", "text": "Let $k$ be a field. If $R$ is geometrically reduced over $k$, and $S \\subset R$ is a multiplicative subset, then the localization $S^{-1}R$ is geometrically reduced over $k$. If $R$ is geometrically reduced over $k$, then $R[x]$ is geometrically reduced over $k$."}
+{"_id": "563", "title": "algebra-lemma-limit-argument", "text": "Let $k$ be a field. Let $R$, $S$ be $k$-algebras. \\begin{enumerate} \\item If $R \\otimes_k S$ is nonreduced, then there exist finitely generated subalgebras $R' \\subset R$, $S' \\subset S$ such that $R' \\otimes_k S'$ is not reduced. \\item If $R \\otimes_k S$ contains a nonzero zerodivisor, then there exist finitely generated subalgebras $R' \\subset R$, $S' \\subset S$ such that $R' \\otimes_k S'$ contains a nonzero zerodivisor. \\item If $R \\otimes_k S$ contains a nontrivial idempotent, then there exist finitely generated subalgebras $R' \\subset R$, $S' \\subset S$ such that $R' \\otimes_k S'$ contains a nontrivial idempotent. \\end{enumerate}"}
+{"_id": "564", "title": "algebra-lemma-geometrically-reduced-any-reduced-base-change", "text": "Let $k$ be a field. Let $S$ be a geometrically reduced $k$-algebra. Let $R$ be any reduced $k$-algebra. Then $R \\otimes_k S$ is reduced."}
+{"_id": "565", "title": "algebra-lemma-separable-extension-preserves-reducedness", "text": "Let $k$ be a field. Let $S$ be a reduced $k$-algebra. Let $k \\subset K$ be either a separable field extension, or a separably generated field extension. Then $K \\otimes_k S$ is reduced."}
+{"_id": "566", "title": "algebra-lemma-generic-points-geometrically-reduced", "text": "Let $k$ be a field and let $S$ be a $k$-algebra. Assume that $S$ is reduced and that $S_{\\mathfrak p}$ is geometrically reduced for every minimal prime $\\mathfrak p$ of $S$. Then $S$ is geometrically reduced."}
+{"_id": "567", "title": "algebra-lemma-separable-algebraic-diagonal", "text": "Let $k'/k$ be a separable algebraic extension. Then there exists a multiplicative subset $S \\subset k' \\otimes_k k'$ such that the multiplication map $k' \\otimes_k k' \\to k'$ is identified with $k' \\otimes_k k' \\to S^{-1}(k' \\otimes_k k')$."}
+{"_id": "568", "title": "algebra-lemma-geometrically-reduced-over-separable-algebraic", "text": "Let $k \\subset k'$ be a separable algebraic field extension. Let $A$ be an algebra over $k'$. Then $A$ is geometrically reduced over $k$ if and only if it is geometrically reduced over $k'$."}
+{"_id": "569", "title": "algebra-lemma-characterize-separable-field-extensions", "text": "Let $k$ be a field of characteristic $p > 0$. Let $k \\subset K$ be a field extension. The following are equivalent: \\begin{enumerate} \\item $K$ is separable over $k$, \\item the ring $K \\otimes_k k^{1/p}$ is reduced, and \\item $K$ is geometrically reduced over $k$. \\end{enumerate}"}
+{"_id": "570", "title": "algebra-lemma-separably-generated-separable", "text": "A separably generated field extension is separable."}
+{"_id": "571", "title": "algebra-lemma-geometrically-reduced-finite-purely-inseparable-extension", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. The following are equivalent: \\begin{enumerate} \\item $k' \\otimes_k S$ is reduced for every finite purely inseparable extension $k'$ of $k$, \\item $k^{1/p} \\otimes_k S$ is reduced, \\item $k^{perf} \\otimes_k S$ is reduced, where $k^{perf}$ is the perfect closure of $k$, \\item $\\overline{k} \\otimes_k S$ is reduced, where $\\overline{k}$ is the algebraic closure of $k$, and \\item $S$ is geometrically reduced over $k$. \\end{enumerate}"}
+{"_id": "573", "title": "algebra-lemma-make-separable", "text": "Let $k \\subset K$ be a finitely generated field extension. There exists a diagram $$ \\xymatrix{ K \\ar[r] & K' \\\\ k \\ar[u] \\ar[r] & k' \\ar[u] } $$ where $k \\subset k'$, $K \\subset K'$ are finite purely inseparable field extensions such that $k' \\subset K'$ is a separable field extension. In this situation we can assume that $K' = k'K$ is the compositum, and also that $K' = (k' \\otimes_k K)_{red}$."}
+{"_id": "574", "title": "algebra-lemma-perfection", "text": "\\begin{slogan} Every field has a unique perfect closure. \\end{slogan} For every field $k$ there exists a purely inseparable extension $k \\subset k'$ such that $k'$ is perfect. The field extension $k \\subset k'$ is unique up to unique isomorphism."}
+{"_id": "575", "title": "algebra-lemma-perfect-reduced", "text": "Let $k$ be a perfect field. Any reduced $k$ algebra is geometrically reduced over $k$. Let $R$, $S$ be $k$-algebras. Assume both $R$ and $S$ are reduced. Then the $k$-algebra $R \\otimes_k S$ is reduced."}
+{"_id": "576", "title": "algebra-lemma-surjective-locally-nilpotent-kernel", "text": "Let $\\varphi : R \\to S$ be a surjective map with locally nilpotent kernel. Then $\\varphi$ induces a homeomorphism of spectra and isomorphisms on residue fields. For any ring map $R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ is surjective with locally nilpotent kernel."}
+{"_id": "577", "title": "algebra-lemma-powers-field", "text": "\\begin{reference} \\cite[Lemma 3.1.6]{Alper-adequate} \\end{reference} Let $k \\subset k'$ be a field extension. The following are equivalent \\begin{enumerate} \\item for each $x \\in k'$ there exists an $n > 0$ such that $x^n \\in k$, and \\item $k' = k$ or $k$ and $k'$ have characteristic $p > 0$ and either $k'/k$ is a purely inseparable extension or $k$ and $k'$ are algebraic extensions of $\\mathbf{F}_p$. \\end{enumerate}"}
+{"_id": "578", "title": "algebra-lemma-powers", "text": "Let $\\varphi : R \\to S$ be a ring map. If \\begin{enumerate} \\item for any $x \\in S$ there exists $n > 0$ such that $x^n$ is in the image of $\\varphi$, and \\item $\\Ker(\\varphi)$ is locally nilpotent, \\end{enumerate} then $\\varphi$ induces a homeomorphism on spectra and induces residue field extensions satisfying the equivalent conditions of Lemma \\ref{lemma-powers-field}."}
+{"_id": "579", "title": "algebra-lemma-2-3-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such that $x^2, x^3 \\in \\varphi(R)$, and \\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent, \\end{enumerate} Then $\\varphi$ induces isomorphisms on residue fields and a homeomorphism of spectra. For any ring map $R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b)."}
+{"_id": "580", "title": "algebra-lemma-help-with-powers", "text": "Let $p$ be a prime number. Let $n, m > 0$ be two integers. There exists an integer $a$ such that $(x + y)^{p^a}, p^a(x + y) \\in \\mathbf{Z}[x^{p^n}, p^nx, y^{p^m}, p^my]$."}
+{"_id": "581", "title": "algebra-lemma-p-ring-map-field", "text": "Let $k \\subset k'$ be a field extension. Let $p$ be a prime number. The following are equivalent \\begin{enumerate} \\item $k'$ is generated as a field extension of $k$ by elements $x$ such that there exists an $n > 0$ with $x^{p^n} \\in k$ and $p^nx \\in k$, and \\item $k = k'$ or the characteristic of $k$ and $k'$ is $p$ and $k'/k$ is purely inseparable. \\end{enumerate}"}
+{"_id": "582", "title": "algebra-lemma-p-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $p$ be a prime number. Assume \\begin{enumerate} \\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such that there exists an $n > 0$ with $x^{p^n} \\in \\varphi(R)$ and $p^nx \\in \\varphi(R)$, and \\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent, \\end{enumerate} Then $\\varphi$ induces a homeomorphism of spectra and induces residue field extensions satisfying the equivalent conditions of Lemma \\ref{lemma-p-ring-map-field}. For any ring map $R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b)."}
+{"_id": "583", "title": "algebra-lemma-radicial", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ induces an injective map of spectra, \\item $\\varphi$ induces purely inseparable residue field extensions. \\end{enumerate} Then for any ring map $R \\to R'$ properties (1) and (2) are true for $R' \\to R' \\otimes_R S$."}
+{"_id": "584", "title": "algebra-lemma-radicial-integral", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is integral, \\item $\\varphi$ induces an injective map of spectra, \\item $\\varphi$ induces purely inseparable residue field extensions. \\end{enumerate} Then $\\varphi$ induces a homeomorphism from $\\Spec(S)$ onto a closed subset of $\\Spec(R)$ and for any ring map $R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$."}
+{"_id": "585", "title": "algebra-lemma-radicial-integral-bijective", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is integral, \\item $\\varphi$ induces an bijective map of spectra, \\item $\\varphi$ induces purely inseparable residue field extensions. \\end{enumerate} Then $\\varphi$ induces a homeomorphism on spectra and for any ring map $R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$."}
+{"_id": "586", "title": "algebra-lemma-universally-bijective", "text": "Let $\\varphi : R \\to S$ be a ring map such that \\begin{enumerate} \\item the kernel of $\\varphi$ is locally nilpotent, and \\item $S$ is generated as an $R$-algebra by elements $x$ such that there exist $n > 0$ and a polynomial $P(T) \\in R[T]$ whose image in $S[T]$ is $(T - x)^n$. \\end{enumerate} Then $\\Spec(S) \\to \\Spec(R)$ is a homeomorphism and $R \\to S$ induces purely inseparable extensions of residue fields. Moreover, conditions (1) and (2) remain true on arbitrary base change."}
+{"_id": "587", "title": "algebra-lemma-flat-fibres-irreducible", "text": "Let $R \\to S$ be a ring map. Assume \\begin{enumerate} \\item[(a)] $\\Spec(R)$ is irreducible, \\item[(b)] $R \\to S$ is flat, \\item[(c)] $R \\to S$ is of finite presentation, \\item[(d)] the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ have irreducible spectra for a dense collection of primes $\\mathfrak p$ of $R$. \\end{enumerate} Then $\\Spec(S)$ is irreducible. This is true more generally with (b) $+$ (c) replaced by ``the map $\\Spec(S) \\to \\Spec(R)$ is open''."}
+{"_id": "588", "title": "algebra-lemma-separably-closed-irreducible", "text": "Let $k$ be a separably closed field. Let $R$, $S$ be $k$-algebras. If $R$, $S$ have a unique minimal prime, so does $R \\otimes_k S$."}
+{"_id": "589", "title": "algebra-lemma-geometrically-irreducible", "text": "Let $k$ be a field. Let $R$ be a $k$-algebra. The following are equivalent \\begin{enumerate} \\item for every field extension $k \\subset k'$ the spectrum of $R \\otimes_k k'$ is irreducible, \\item for every finite separable field extension $k \\subset k'$ the spectrum of $R \\otimes_k k'$ is irreducible, \\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible where $\\overline{k}$ is the separable algebraic closure of $k$, and \\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible where $\\overline{k}$ is the algebraic closure of $k$. \\end{enumerate}"}
+{"_id": "590", "title": "algebra-lemma-separably-closed-irreducible-implies-geometric", "text": "Let $k$ be a field. Let $R$ be a $k$-algebra. If $k$ is separably algebraically closed then $R$ is geometrically irreducible over $k$ if and only if the spectrum of $R$ is irreducible."}
+{"_id": "591", "title": "algebra-lemma-subalgebra-geometrically-irreducible", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. \\begin{enumerate} \\item If $S$ is geometrically irreducible over $k$ so is every $k$-subalgebra. \\item If all finitely generated $k$-subalgebras of $S$ are geometrically irreducible, then $S$ is geometrically irreducible. \\item A directed colimit of geometrically irreducible $k$-algebras is geometrically irreducible. \\end{enumerate}"}
+{"_id": "592", "title": "algebra-lemma-geometrically-irreducible-any-base-change", "text": "Let $k$ be a field. Let $S$ be a geometrically irreducible $k$-algebra. Let $R$ be any $k$-algebra. The map $$ \\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R) $$ induces a bijection on irreducible components."}
+{"_id": "593", "title": "algebra-lemma-field-extension-geometrically-irreducible", "text": "Let $K/k$ be a field extension. If $k$ is algebraically closed in $K$, then $K$ is geometrically irreducible over $k$."}
+{"_id": "594", "title": "algebra-lemma-geometrically-irreducible-transitive", "text": "Let $K/k$ be a geometrically irreducible field extension. Let $S$ be a geometrically irreducible $K$-algebra. Then $S$ is geometrically irreducible over $k$."}
+{"_id": "595", "title": "algebra-lemma-geometrically-irreducible-base-change-transcendental", "text": "Let $K/k$ be a field extension. The following are equivalent \\begin{enumerate} \\item $K$ is geometrically irreducible over $k$, and \\item the induced extension $K(t)/k(t)$ of purely transcendental extensions is geometrically irreducible. \\end{enumerate}"}
+{"_id": "596", "title": "algebra-lemma-geometrically-irreducible-add-transcendental", "text": "Let $K/L/M$ be a tower of fields with $L/M$ geometrically irreducible. Let $x \\in K$ be transcendental over $L$. Then $L(x)/M(x)$ is geometrically irreducible."}
+{"_id": "597", "title": "algebra-lemma-geometrically-irreducible-separable-elements", "text": "Let $K/k$ be a field extension. The following are equivalent \\begin{enumerate} \\item $K/k$ is geometrically irreducible, and \\item every element $\\alpha \\in K$ separably algebraic over $k$ is in $k$. \\end{enumerate}"}
+{"_id": "598", "title": "algebra-lemma-make-geometrically-irreducible", "text": "Let $K/k$ be a field extension. Consider the subextension $K/k'/k$ consisting of elements separably algebraic over $k$. Then $K$ is geometrically irreducible over $k'$. If $K/k$ is a finitely generated field extension, then $[k' : k] < \\infty$."}
+{"_id": "599", "title": "algebra-lemma-Galois-orbit", "text": "Let $k \\subset K$ be an extension of fields. Let $k \\subset \\overline{k}$ be a separable algebraic closure. Then $\\text{Gal}(\\overline{k}/k)$ acts transitively on the primes of $\\overline{k} \\otimes_k K$."}
+{"_id": "600", "title": "algebra-lemma-separably-closed-connected", "text": "Let $k$ be a separably algebraically closed field. Let $R$, $S$ be $k$-algebras. If $\\Spec(R)$, and $\\Spec(S)$ are connected, then so is $\\Spec(R \\otimes_k S)$."}
+{"_id": "602", "title": "algebra-lemma-separably-closed-connected-implies-geometric", "text": "Let $k$ be a field. Let $R$ be a $k$-algebra. If $k$ is separably algebraically closed then $R$ is geometrically connected over $k$ if and only if the spectrum of $R$ is connected."}
+{"_id": "603", "title": "algebra-lemma-subalgebra-geometrically-connected", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. \\begin{enumerate} \\item If $S$ is geometrically connected over $k$ so is every $k$-subalgebra. \\item If all finitely generated $k$-subalgebras of $S$ are geometrically connected, then $S$ is geometrically connected. \\item A directed colimit of geometrically connected $k$-algebras is geometrically connected. \\end{enumerate}"}
+{"_id": "605", "title": "algebra-lemma-geometrically-integral", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. In this case $S$ is geometrically integral over $k$ if and only if $S$ is geometrically irreducible as well as geometrically reduced over $k$."}
+{"_id": "607", "title": "algebra-lemma-geometrically-integral-any-integral-base-change", "text": "Let $k$ be a field. Let $S$ be a geometrically integral $k$-algebra. Let $R$ be a $k$-algebra and an integral domain. Then $R \\otimes_k S$ is an integral domain."}
+{"_id": "608", "title": "algebra-lemma-dominate", "text": "Let $K$ be a field. Let $A \\subset K$ be a local subring. Then there exists a valuation ring with fraction field $K$ dominating $A$."}
+{"_id": "609", "title": "algebra-lemma-valuation-ring-x-or-x-inverse", "text": "Let $A$ be a valuation ring with maximal ideal $\\mathfrak m$ and fraction field $K$. Let $x \\in K$. Then either $x \\in A$ or $x^{-1} \\in A$ or both."}
+{"_id": "610", "title": "algebra-lemma-x-or-x-inverse-valuation-ring", "text": "Let $A \\subset K$ be a subring of a field $K$ such that for all $x \\in K$ either $x \\in A$ or $x^{-1} \\in A$ or both. Then $A$ is a valuation ring with fraction field $K$."}
+{"_id": "611", "title": "algebra-lemma-colimit-valuation-rings", "text": "\\begin{slogan} Valuation rings are stable under filtered direct limits \\end{slogan} Let $I$ be a directed set. Let $(A_i, \\varphi_{ij})$ be a system of valuation rings over $I$. Then $A = \\colim A_i$ is a valuation ring."}
+{"_id": "612", "title": "algebra-lemma-valuation-ring-cap-field", "text": "Let $K \\subset L$ be an extension of fields. If $B \\subset L$ is a valuation ring, then $A = K \\cap B$ is a valuation ring."}
+{"_id": "613", "title": "algebra-lemma-valuation-ring-cap-field-finite", "text": "Let $K \\subset L$ be an algebraic extension of fields. If $B \\subset L$ is a valuation ring with fraction field $L$ and not a field, then $A = K \\cap B$ is a valuation ring and not a field."}
+{"_id": "614", "title": "algebra-lemma-make-valuation-rings", "text": "Let $A$ be a valuation ring. For any prime ideal $\\mathfrak p \\subset A$ the quotient $A/\\mathfrak p$ is a valuation ring. The same is true for the localization $A_\\mathfrak p$ and in fact any localization of $A$."}
+{"_id": "615", "title": "algebra-lemma-stack-valuation-rings", "text": "Let $A'$ be a valuation ring with residue field $K$. Let $A$ be a valuation ring with fraction field $K$. Then $C = \\{\\lambda \\in A' \\mid \\lambda \\bmod \\mathfrak m_{A'} \\in A\\}$ is a valuation ring."}
+{"_id": "616", "title": "algebra-lemma-valuation-ring-normal", "text": "Let $A$ be a valuation ring. Then $A$ is a normal domain."}
+{"_id": "617", "title": "algebra-lemma-find-valuation-rings", "text": "Let $A$ be a normal domain with fraction field $K$. \\begin{enumerate} \\item For every $x \\in K$, $x \\not \\in A$ there exists a valuation ring $A \\subset V \\subset K$ with fraction field $K$ such that $x \\not \\in V$. \\item If $A$ is local, we can moreover choose $V$ which dominates $A$. \\end{enumerate} In other words, $A$ is the intersection of all valuation rings in $K$ containing $A$ and if $A$ is local, then $A$ is the intersection of all valuation rings in $K$ dominating $A$."}
+{"_id": "618", "title": "algebra-lemma-valuation-group", "text": "Let $A$ be a valuation ring with field of fractions $K$. Set $\\Gamma = K^*/A^*$ (with group law written additively). For $\\gamma, \\gamma' \\in \\Gamma$ define $\\gamma \\geq \\gamma'$ if and only if $\\gamma - \\gamma'$ is in the image of $A - \\{0\\} \\to \\Gamma$. Then $(\\Gamma, \\geq)$ is a totally ordered abelian group."}
+{"_id": "619", "title": "algebra-lemma-properties-valuation", "text": "Let $A$ be a valuation ring. The valuation $v : A -\\{0\\} \\to \\Gamma_{\\geq 0}$ has the following properties: \\begin{enumerate} \\item $v(a) = 0 \\Leftrightarrow a \\in A^*$, \\item $v(ab) = v(a) + v(b)$, \\item $v(a + b) \\geq \\min(v(a), v(b))$. \\end{enumerate}"}
+{"_id": "620", "title": "algebra-lemma-characterize-valuation-ring", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item $A$ is a valuation ring, \\item $A$ is a local domain and every finitely generated ideal of $A$ is principal. \\end{enumerate}"}
+{"_id": "621", "title": "algebra-lemma-valuation-valuation-ring", "text": "Let $(\\Gamma, \\geq)$ be a totally ordered abelian group. Let $K$ be a field. Let $v : K^* \\to \\Gamma$ be a homomorphism of abelian groups such that $v(a + b) \\geq \\min(v(a), v(b))$ for $a, b \\in K$ with $a, b, a + b$ not zero. Then $$ A = \\{ x \\in K \\mid x = 0 \\text{ or } v(x) \\geq 0 \\} $$ is a valuation ring with value group $\\Im(v) \\subset \\Gamma$, with maximal ideal $$ \\mathfrak m = \\{ x \\in K \\mid x = 0 \\text{ or } v(x) > 0 \\} $$ and with group of units $$ A^* = \\{ x \\in K^* \\mid v(x) = 0 \\}. $$"}
+{"_id": "622", "title": "algebra-lemma-ideals-valuation-ring", "text": "Let $A$ be a valuation ring. Ideals in $A$ correspond $1 - 1$ with ideals of $\\Gamma$. This bijection is inclusion preserving, and maps prime ideals to prime ideals."}
+{"_id": "623", "title": "algebra-lemma-valuation-ring-Noetherian-discrete", "text": "A valuation ring is Noetherian if and only if it is a discrete valuation ring or a field."}
+{"_id": "624", "title": "algebra-lemma-Noetherian-basic", "text": "Let $R$ be a Noetherian ring. Any finite $R$-module is of finite presentation. Any submodule of a finite $R$-module is finite. The ascending chain condition holds for $R$-submodules of a finite $R$-module."}
+{"_id": "625", "title": "algebra-lemma-Artin-Rees", "text": "Suppose that $R$ is Noetherian, $I \\subset R$ an ideal. Let $N \\subset M$ be finite $R$-modules. There exists a constant $c > 0$ such that $I^n M \\cap N  =  I^{n-c}(I^cM \\cap N)$ for all $n \\geq c$."}
+{"_id": "626", "title": "algebra-lemma-map-AR", "text": "Suppose that $0 \\to K \\to M \\xrightarrow{f} N$ is an exact sequence of finitely generated modules over a Noetherian ring $R$. Let $I \\subset R$ be an ideal. Then there exists a $c$ such that $$ f^{-1}(I^nN) = K + I^{n-c}f^{-1}(I^cN) \\quad\\text{and}\\quad f(M) \\cap I^nN \\subset f(I^{n - c}M) $$ for all $n \\geq c$."}
+{"_id": "627", "title": "algebra-lemma-intersect-powers-ideal-module-zero", "text": "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be a proper ideal. Let $M$ be a finite $R$-module. Then $\\bigcap_{n \\geq 0} I^nM = 0$."}
+{"_id": "628", "title": "algebra-lemma-intersection-powers-ideal-module", "text": "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal. Let $M$ be a finite $R$-module. Let $N = \\bigcap_n I^n M$. \\begin{enumerate} \\item For every prime $\\mathfrak p$, $I \\subset \\mathfrak p$ there exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that $N_f = 0$. \\item If $I$ is contained in the Jacobson radical of $R$, then $N = 0$. \\end{enumerate}"}
+{"_id": "629", "title": "algebra-lemma-Artin-Tate", "text": "Let $R$ be a Noetherian ring. Let $S$ be a finitely generated $R$-algebra. If $T \\subset S$ is an $R$-subalgebra such that $S$ is finitely generated as a $T$-module, then $T$ is of finite type over $R$."}
+{"_id": "630", "title": "algebra-lemma-finite-length-finite", "text": "\\begin{slogan} Modules of finite length are finite. \\end{slogan} Let $R$ be a ring. Let $M$ be an $R$-module. If $\\text{length}_R(M) < \\infty$ then $M$ is a finite $R$-module."}
+{"_id": "631", "title": "algebra-lemma-length-additive", "text": "\\begin{slogan} Length is additive in short exact sequences. \\end{slogan} If $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of modules over $R$ then the length of $M$ is the sum of the lengths of $M'$ and $M''$."}
+{"_id": "632", "title": "algebra-lemma-length-infinite", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$. Let $M$ be an $R$-module. \\begin{enumerate} \\item If $M$ is a finite module and $\\mathfrak m^n M \\not = 0$ for all $n\\geq 0$, then $\\text{length}_R(M) = \\infty$. \\item If $M$ has finite length then $\\mathfrak m^nM = 0$ for some $n$. \\end{enumerate}"}
+{"_id": "633", "title": "algebra-lemma-length-independent", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. We always have $\\text{length}_R(M) \\geq \\text{length}_S(M)$. If $R \\to S$ is surjective then equality holds."}
+{"_id": "634", "title": "algebra-lemma-dimension-is-length", "text": "Let $R$ be a ring with maximal ideal $\\mathfrak m$. Suppose that $M$ is an $R$-module with $\\mathfrak m M  =  0$. Then the length of $M$ as an $R$-module agrees with the dimension of $M$ as a $R/\\mathfrak m$ vector space. The length is finite if and only if $M$ is a finite $R$-module."}
+{"_id": "636", "title": "algebra-lemma-length-finite", "text": "Let $R$ be a ring with finitely generated maximal ideal $\\mathfrak m$. (For example $R$ Noetherian.) Suppose that $M$ is a finite $R$-module with $\\mathfrak m^n M  =  0$ for some $n$. Then $\\text{length}_R(M) < \\infty$."}
+{"_id": "637", "title": "algebra-lemma-characterize-length-1", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The following are equivalent: \\begin{enumerate} \\item $M$ is simple, \\item $\\text{length}_R(M) = 1$, and \\item $M \\cong R/\\mathfrak m$ for some maximal ideal $\\mathfrak m \\subset R$. \\end{enumerate}"}
+{"_id": "638", "title": "algebra-lemma-simple-pieces", "text": "Let $R$ be a ring. Let $M$ be a finite length $R$-module. Choose any maximal chain of submodules $$ 0 = M_0 \\subset M_1 \\subset M_2 \\subset \\ldots \\subset M_n = M $$ with $M_i \\not = M_{i-1}$, $i = 1, \\ldots, n$. Then \\begin{enumerate} \\item $n = \\text{length}_R(M)$, \\item each $M_i/M_{i-1}$ is simple, \\item each $M_i/M_{i-1}$ is of the form $R/\\mathfrak m_i$ for some maximal ideal $\\mathfrak m_i$, \\item given a maximal ideal $\\mathfrak m \\subset R$ we have $$ \\# \\{i \\mid \\mathfrak m_i = \\mathfrak m\\} = \\text{length}_{R_{\\mathfrak m}} (M_{\\mathfrak m}). $$ \\end{enumerate}"}
+{"_id": "639", "title": "algebra-lemma-pushdown-module", "text": "Let $A$ be a local ring with maximal ideal $\\mathfrak m$. Let $B$ be a semi-local ring with maximal ideals $\\mathfrak m_i$, $i = 1, \\ldots, n$. Suppose that $A \\to B$ is a homomorphism such that each $\\mathfrak m_i$ lies over $\\mathfrak m$ and such that $$ [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)] < \\infty. $$ Let $M$ be a $B$-module of finite length. Then $$ \\text{length}_A(M) = \\sum\\nolimits_{i = 1, \\ldots, n} [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)] \\text{length}_{B_{\\mathfrak m_i}}(M_{\\mathfrak m_i}), $$ in particular $\\text{length}_A(M) < \\infty$."}
+{"_id": "640", "title": "algebra-lemma-pullback-module", "text": "Let $A \\to B$ be a flat local homomorphism of local rings. Then for any $A$-module $M$ we have $$ \\text{length}_A(M) \\text{length}_B(B/\\mathfrak m_AB) = \\text{length}_B(M \\otimes_A B). $$ In particular, if $\\text{length}_B(B/\\mathfrak m_AB) < \\infty$ then $M$ has finite length if and only if $M \\otimes_A B$ has finite length."}
+{"_id": "641", "title": "algebra-lemma-pullback-transitive", "text": "Let $A \\to B \\to C$ be flat local homomorphisms of local rings. Then $$ \\text{length}_B(B/\\mathfrak m_A B) \\text{length}_C(C/\\mathfrak m_B C) = \\text{length}_C(C/\\mathfrak m_A C) $$"}
+{"_id": "642", "title": "algebra-lemma-finite-dimensional-algebra", "text": "Suppose $R$ is a finite dimensional algebra over a field. Then $R$ is Artinian."}
+{"_id": "643", "title": "algebra-lemma-artinian-finite-nr-max", "text": "If $R$ is Artinian then $R$ has only finitely many maximal ideals."}
+{"_id": "644", "title": "algebra-lemma-artinian-radical-nilpotent", "text": "Let $R$ be Artinian. The Jacobson radical of $R$ is a nilpotent ideal."}
+{"_id": "645", "title": "algebra-lemma-product-local", "text": "Any ring with finitely many maximal ideals and locally nilpotent Jacobson radical is the product of its localizations at its maximal ideals. Also, all primes are maximal."}
+{"_id": "646", "title": "algebra-lemma-artinian-finite-length", "text": "A ring $R$ is Artinian if and only if it has finite length as a module over itself. Any such ring $R$ is both Artinian and Noetherian, any prime ideal of $R$ is a maximal ideal, and $R$ is equal to the (finite) product of its localizations at its maximal ideals."}
+{"_id": "647", "title": "algebra-lemma-composition-essentially-of-finite-type", "text": "The class of ring maps which are essentially of finite type is preserved under composition. Similarly for essentially of finite presentation."}
+{"_id": "648", "title": "algebra-lemma-base-change-essentially-of-finite-type", "text": "The class of ring maps which are essentially of finite type is preserved by base change. Similarly for essentially of finite presentation."}
+{"_id": "649", "title": "algebra-lemma-essentially-of-finite-type-into-artinian-local", "text": "Let $R \\to S$ be a ring map. Assume $S$ is an Artinian local ring with maximal ideal $\\mathfrak m$. Then \\begin{enumerate} \\item $R \\to S$ is finite if and only if $R \\to S/\\mathfrak m$ is finite, \\item $R \\to S$ is of finite type if and only if $R \\to S/\\mathfrak m$ is of finite type. \\item $R \\to S$ is essentially of finite type if and only if the composition $R \\to S/\\mathfrak m$ is essentially of finite type. \\end{enumerate}"}
+{"_id": "650", "title": "algebra-lemma-localization-at-closed-point-special-fibre", "text": "Let $\\varphi : R \\to S$ be essentially of finite type with $R$ and $S$ local (but not necessarily $\\varphi$ local). Then there exists an $n$ and a maximal ideal $\\mathfrak m \\subset R[x_1, \\ldots, x_n]$ lying over $\\mathfrak m_R$ such that $S$ is a localization of a quotient of $R[x_1, \\ldots, x_n]_\\mathfrak m$."}
+{"_id": "651", "title": "algebra-lemma-length-K0", "text": "If $R$ is an Artinian local ring then the length function defines a natural abelian group homomorphism $\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$."}
+{"_id": "652", "title": "algebra-lemma-K0-product", "text": "Let $R = R_1 \\times R_2$. Then $K_0(R) = K_0(R_1) \\times K_0(R_2)$ and $K'_0(R) = K'_0(R_1) \\times K'_0(R_2)$"}
+{"_id": "653", "title": "algebra-lemma-K0prime-Artinian", "text": "Let $R$ be an Artinian local ring. The map $\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$ of Lemma \\ref{lemma-length-K0} is an isomorphism."}
+{"_id": "654", "title": "algebra-lemma-K0-local", "text": "Let $(R, \\mathfrak m)$ be a local ring. Every finite projective $R$-module is finite free. The map $\\text{rank}_R : K_0(R) \\to \\mathbf{Z}$ defined by $[M] \\to \\text{rank}_R(M)$ is well defined and an isomorphism."}
+{"_id": "656", "title": "algebra-lemma-graded-NAK", "text": "Let $S$ be a graded ring. Let $M$ be a graded $S$-module. \\begin{enumerate} \\item If $S_+M = M$ and $M$ is finite, then $M = 0$. \\item If $N, N' \\subset M$ are graded submodules, $M = N + S_+N'$, and $N'$ is finite, then $M = N$. \\item If $N \\to M$ is a map of graded modules, $N/S_+N \\to M/S_+M$ is surjective, and $M$ is finite, then $N \\to M$ is surjective. \\item If $x_1, \\ldots, x_n \\in M$ are homogeneous and generate $M/S_+M$ and $M$ is finite, then $x_1, \\ldots, x_n$ generate $M$. \\end{enumerate}"}
+{"_id": "657", "title": "algebra-lemma-uple-generated-degree-1", "text": "Let $S$ be a graded ring, which is finitely generated over $S_0$. Then for all sufficiently divisible $d$ the algebra $S^{(d)}$ is generated in degree $1$ over $S_0$."}
+{"_id": "659", "title": "algebra-lemma-Z-graded", "text": "Let $S$ be a $\\mathbf{Z}$-graded ring containing a homogeneous invertible element of positive degree. Then the set $G \\subset \\Spec(S)$ of $\\mathbf{Z}$-graded primes of $S$ (with induced topology) maps homeomorphically to $\\Spec(S_0)$."}
+{"_id": "660", "title": "algebra-lemma-topology-proj", "text": "Let $S = \\oplus_{d \\geq 0} S_d$ be a graded ring. \\begin{enumerate} \\item The sets $D_{+}(f)$ are open in $\\text{Proj}(S)$. \\item We have $D_{+}(ff') = D_{+}(f) \\cap D_{+}(f')$. \\item Let $g = g_0 + \\ldots + g_m$ be an element of $S$ with $g_i \\in S_i$. Then $$ D(g) \\cap \\text{Proj}(S) = (D(g_0) \\cap \\text{Proj}(S)) \\cup \\bigcup\\nolimits_{i \\geq 1} D_{+}(g_i). $$ \\item Let $g_0\\in S_0$ be a homogeneous element of degree $0$. Then $$ D(g_0) \\cap \\text{Proj}(S) = \\bigcup\\nolimits_{f \\in S_d, \\ d\\geq 1} D_{+}(g_0 f). $$ \\item The open sets $D_{+}(f)$ form a basis for the topology of $\\text{Proj}(S)$. \\item Let $f \\in S$ be homogeneous of positive degree. The ring $S_f$ has a natural $\\mathbf{Z}$-grading. The ring maps $S \\to S_f \\leftarrow S_{(f)}$ induce homeomorphisms $$ D_{+}(f) \\leftarrow \\{\\mathbf{Z}\\text{-graded primes of }S_f\\} \\to \\Spec(S_{(f)}). $$ \\item There exists an $S$ such that $\\text{Proj}(S)$ is not quasi-compact. \\item The sets $V_{+}(I)$ are closed. \\item Any closed subset $T \\subset \\text{Proj}(S)$ is of the form $V_{+}(I)$ for some homogeneous ideal $I \\subset S$. \\item For any graded ideal $I \\subset S$ we have $V_{+}(I) = \\emptyset$ if and only if $S_{+} \\subset \\sqrt{I}$. \\end{enumerate}"}
+{"_id": "661", "title": "algebra-lemma-proj-prime", "text": "Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\\mathfrak p$ be an element of $\\text{Proj}(S)$. Let $f \\in S$ be a homogeneous element of positive degree such that $f \\not \\in \\mathfrak p$, i.e., $\\mathfrak p \\in D_{+}(f)$. Let $\\mathfrak p' \\subset S_{(f)}$ be the element of $\\Spec(S_{(f)})$ corresponding to $\\mathfrak p$ as in Lemma \\ref{lemma-topology-proj}. Then $S_{(\\mathfrak p)} = (S_{(f)})_{\\mathfrak p'}$ and compatibly $M_{(\\mathfrak p)} = (M_{(f)})_{\\mathfrak p'}$."}
+{"_id": "662", "title": "algebra-lemma-graded-silly", "text": "Suppose $S$ is a graded ring, $\\mathfrak p_i$, $i = 1, \\ldots, r$ homogeneous prime ideals and $I \\subset S_{+}$ a graded ideal. Assume $I \\not\\subset \\mathfrak p_i$ for all $i$. Then there exists a homogeneous element $x\\in I$ of positive degree such that $x\\not\\in \\mathfrak p_i$ for all $i$."}
+{"_id": "663", "title": "algebra-lemma-smear-out", "text": "Let $S$ be a graded ring. Let $\\mathfrak p \\subset S$ be a prime. Let $\\mathfrak q$ be the homogeneous ideal of $S$ generated by the homogeneous elements of $\\mathfrak p$. Then $\\mathfrak q$ is a prime ideal of $S$."}
+{"_id": "664", "title": "algebra-lemma-graded-ring-minimal-prime", "text": "Let $S$ be a graded ring. \\begin{enumerate} \\item Any minimal prime of $S$ is a homogeneous ideal of $S$. \\item Given a homogeneous ideal $I \\subset S$ any minimal prime over $I$ is homogeneous. \\end{enumerate}"}
+{"_id": "665", "title": "algebra-lemma-dehomogenize-finite-type", "text": "Let $R$ be a ring. Let $S$ be a graded $R$-algebra. Let $f \\in S_{+}$ be homogeneous. Assume that $S$ is of finite type over $R$. Then \\begin{enumerate} \\item the ring $S_{(f)}$ is of finite type over $R$, and \\item for any finite graded $S$-module $M$ the module $M_{(f)}$ is a finite $S_{(f)}$-module. \\end{enumerate}"}
+{"_id": "666", "title": "algebra-lemma-homogenize", "text": "Let $R$ be a ring. Let $R'$ be a finite type $R$-algebra, and let $M$ be a finite $R'$-module. There exists a graded $R$-algebra $S$, a graded $S$-module $N$ and an element $f \\in S$ homogeneous of degree $1$ such that \\begin{enumerate} \\item $R' \\cong S_{(f)}$ and $M \\cong N_{(f)}$ (as modules), \\item $S_0 = R$ and $S$ is generated by finitely many elements of degree $1$ over $R$, and \\item $N$ is a finite $S$-module. \\end{enumerate}"}
+{"_id": "667", "title": "algebra-lemma-S-plus-generated", "text": "Let $S$ be a graded ring. A set of homogeneous elements $f_i \\in S_{+}$ generates $S$ as an algebra over $S_0$ if and only if they generate $S_{+}$ as an ideal of $S$."}
+{"_id": "668", "title": "algebra-lemma-graded-Noetherian", "text": "A graded ring $S$ is Noetherian if and only if $S_0$ is Noetherian and $S_{+}$ is finitely generated as an ideal of $S$."}
+{"_id": "669", "title": "algebra-lemma-numerical-polynomial-functorial", "text": "If $A \\to A'$ is a homomorphism of abelian groups and if $f : n \\mapsto f(n) \\in A$ is a numerical polynomial, then so is the composition."}
+{"_id": "670", "title": "algebra-lemma-numerical-polynomial", "text": "Suppose that $f: n \\mapsto f(n) \\in A$ is defined for all $n$ sufficiently large and suppose that $n \\mapsto f(n) - f(n-1)$ is a numerical polynomial. Then $f$ is a numerical polynomial."}
+{"_id": "671", "title": "algebra-lemma-graded-module-fg", "text": "If $M$ is a finitely generated graded $S$-module, and if $S$ is finitely generated over $S_0$, then each $M_n$ is a finite $S_0$-module."}
+{"_id": "672", "title": "algebra-lemma-quotient-smaller-d", "text": "Let $k$ be a field. Suppose that $I \\subset k[X_1, \\ldots, X_d]$ is a nonzero graded ideal. Let $M = k[X_1, \\ldots, X_d]/I$. Then the numerical polynomial $n \\mapsto \\dim_k(M_n)$ (see Example \\ref{example-hilbert-function}) has degree $ < d - 1$ (or is zero if $d = 1$)."}
+{"_id": "673", "title": "algebra-lemma-differ-finite", "text": "Suppose that $M' \\subset M$ are finite $R$-modules with finite length quotient. Then there exists a constants $c_1, c_2$ such that for all $n \\geq c_2$ we have $$ c_1 + \\chi_{I, M'}(n - c_2) \\leq \\chi_{I, M}(n) \\leq c_1 + \\chi_{I, M'}(n) $$"}
+{"_id": "674", "title": "algebra-lemma-hilbert-ses", "text": "Suppose that $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of finite $R$-modules. Then there exists a submodule $N \\subset M'$ with finite colength $l$ and $c \\geq 0$ such that $$ \\chi_{I, M}(n) = \\chi_{I, M''}(n) + \\chi_{I, N}(n - c) + l $$ and $$ \\varphi_{I, M}(n) = \\varphi_{I, M''}(n) + \\varphi_{I, N}(n - c) $$ for all $n \\geq c$."}
+{"_id": "675", "title": "algebra-lemma-hilbert-change-I", "text": "Suppose that $I$, $I'$ are two ideals of definition for the Noetherian local ring $R$. Let $M$ be a finite $R$-module. There exists a constant $a$ such that $\\chi_{I, M}(n) \\leq \\chi_{I', M}(an)$ for $n \\geq 1$."}
+{"_id": "677", "title": "algebra-lemma-differ-finite-chi", "text": "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of definition. Let $M$ be a finite $R$-module which does not have finite length. If $M' \\subset M$ is a submodule with finite colength, then $\\chi_{I, M} - \\chi_{I, M'}$ is a polynomial of degree $<$ degree of either polynomial."}
+{"_id": "678", "title": "algebra-lemma-hilbert-ses-chi", "text": "Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of definition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of finite $R$-modules. Then \\begin{enumerate} \\item if $M'$ does not have finite length, then $\\chi_{I, M} - \\chi_{I, M''} - \\chi_{I, M'}$ is a numerical polynomial of degree $<$ the degree of $\\chi_{I, M'}$, \\item $\\max\\{ \\deg(\\chi_{I, M'}), \\deg(\\chi_{I, M''}) \\} = \\deg(\\chi_{I, M})$, and \\item $\\max\\{d(M'), d(M'')\\} = d(M)$, \\end{enumerate}"}
+{"_id": "679", "title": "algebra-lemma-dimension-height", "text": "The Krull dimension of $R$ is the supremum of the heights of its (maximal) primes."}
+{"_id": "680", "title": "algebra-lemma-Noetherian-dimension-0", "text": "A Noetherian ring of dimension $0$ is Artinian. Conversely, any Artinian ring is Noetherian of dimension zero."}
+{"_id": "681", "title": "algebra-lemma-dimension-0-d-0", "text": "Let $R$ be a Noetherian local ring. Then $\\dim(R) = 0 \\Leftrightarrow d(R) = 0$."}
+{"_id": "682", "title": "algebra-lemma-height-1", "text": "Let $R$ be a local Noetherian ring. The following are equivalent: \\begin{enumerate} \\item \\label{item-dim-1} $\\dim(R) = 1$, \\item \\label{item-d-1} $d(R) = 1$, \\item \\label{item-Vx} there exists an $x \\in \\mathfrak m$, $x$ not nilpotent such that $V(x) = \\{\\mathfrak m\\}$, \\item \\label{item-x} there exists an $x \\in \\mathfrak m$, $x$ not nilpotent such that $\\mathfrak m = \\sqrt{(x)}$, and \\item \\label{item-ideal-1} there exists an ideal of definition generated by $1$ element, and no ideal of definition is generated by $0$ elements. \\end{enumerate}"}
+{"_id": "683", "title": "algebra-lemma-minimal-over-1", "text": "Let $R$ be a Noetherian ring. Let $x \\in R$. \\begin{enumerate} \\item If $\\mathfrak p$ is minimal over $(x)$ then the height of $\\mathfrak p$ is $0$ or $1$. \\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and $\\mathfrak q$ is minimal over $(\\mathfrak p, x)$, then there is no prime strictly between $\\mathfrak p$ and $\\mathfrak q$. \\end{enumerate}"}
+{"_id": "686", "title": "algebra-lemma-elements-generate-ideal-definition", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Suppose $x_1, \\ldots, x_d \\in \\mathfrak m$ generate an ideal of definition and $d = \\dim(R)$. Then $\\dim(R/(x_1, \\ldots, x_i)) = d - i$ for all $i = 1, \\ldots, d$."}
+{"_id": "687", "title": "algebra-lemma-Noetherian-local-domain-dim-2-infinite-opens", "text": "Let $R$ be a Noetherian local domain of dimension $\\geq 2$. A nonempty open subset $U \\subset \\Spec(R)$ is infinite."}
+{"_id": "689", "title": "algebra-lemma-finite-type-algebra-finite-nr-primes", "text": "Let $S$ be a nonzero finite type algebra over a field $k$. Then $\\dim(S) = 0$ if and only if $S$ has finitely many primes."}
+{"_id": "690", "title": "algebra-lemma-noetherian-dim-1-Jacobson", "text": "Noetherian Jacobson rings. \\begin{enumerate} \\item Any Noetherian domain $R$ of dimension $1$ with infinitely many primes is Jacobson. \\item Any Noetherian ring such that every prime $\\mathfrak p$ is either maximal or contained in infinitely many prime ideals is Jacobson. \\end{enumerate}"}
+{"_id": "691", "title": "algebra-lemma-filter-Noetherian-module", "text": "Let $R$ be a Noetherian ring, and let $M$ be a finite $R$-module. There exists a filtration by $R$-submodules $$ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M $$ such that each quotient $M_i/M_{i-1}$ is isomorphic to $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$ of $R$."}
+{"_id": "692", "title": "algebra-lemma-filter-primes-in-support", "text": "Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in Lemma \\ref{lemma-filter-Noetherian-module}. Then $\\text{Supp}(M) = \\bigcup V(\\mathfrak p_i)$ and in particular $\\mathfrak p_i \\in \\text{Supp}(M)$."}
+{"_id": "693", "title": "algebra-lemma-support-point", "text": "Suppose that $R$ is a Noetherian local ring with maximal ideal $\\mathfrak m$. Let $M$ be a nonzero finite $R$-module. Then $\\text{Supp}(M) = \\{ \\mathfrak m\\}$ if and only if $M$ has finite length over $R$."}
+{"_id": "694", "title": "algebra-lemma-Noetherian-power-ideal-kills-module", "text": "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal. Let $M$ be a finite $R$-module. Then $I^nM = 0$ for some $n \\geq 0$ if and only if $\\text{Supp}(M) \\subset V(I)$."}
+{"_id": "695", "title": "algebra-lemma-filter-minimal-primes-in-support", "text": "Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in Lemma \\ref{lemma-filter-Noetherian-module}. The minimal elements of the set $\\{\\mathfrak p_i\\}$ are the minimal elements of $\\text{Supp}(M)$. The number of times a minimal prime $\\mathfrak p$ occurs is $$ \\#\\{i \\mid \\mathfrak p_i = \\mathfrak p\\} = \\text{length}_{R_\\mathfrak p} M_{\\mathfrak p}. $$"}
+{"_id": "696", "title": "algebra-lemma-support-dimension-d", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Then $d(M) = \\dim(\\text{Supp}(M))$."}
+{"_id": "698", "title": "algebra-lemma-ass-support", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Then $\\text{Ass}(M) \\subset \\text{Supp}(M)$."}
+{"_id": "699", "title": "algebra-lemma-ass", "text": "Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of $R$-modules. Then $\\text{Ass}(M') \\subset \\text{Ass}(M)$ and $\\text{Ass}(M) \\subset \\text{Ass}(M') \\cup \\text{Ass}(M'')$. Also $\\text{Ass}(M' \\oplus M'') = \\text{Ass}(M') \\cup \\text{Ass}(M'')$."}
+{"_id": "700", "title": "algebra-lemma-ass-filter", "text": "Let $R$ be a ring, and $M$ an $R$-module. Suppose there exists a filtration by $R$-submodules $$ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M $$ such that each quotient $M_i/M_{i-1}$ is isomorphic to $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$ of $R$. Then $\\text{Ass}(M) \\subset \\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$."}
+{"_id": "701", "title": "algebra-lemma-finite-ass", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Then $\\text{Ass}(M)$ is finite."}
+{"_id": "702", "title": "algebra-lemma-ass-zero", "text": "\\begin{slogan} Over a Noetherian ring each nonzero module has an associated prime. \\end{slogan} Let $R$ be a Noetherian ring. Let $M$ be an $R$-module. Then $$ M = (0) \\Leftrightarrow \\text{Ass}(M) = \\emptyset. $$"}
+{"_id": "703", "title": "algebra-lemma-ass-minimal-prime-support", "text": "Let $R$ be a Noetherian ring. Let $M$ be an $R$-module. Any $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements of $\\text{Supp}(M)$ is an element of $\\text{Ass}(M)$."}
+{"_id": "704", "title": "algebra-lemma-ass-zero-divisors", "text": "Let $R$ be a Noetherian ring. Let $M$ be an $R$-module. The union $\\bigcup_{\\mathfrak q \\in \\text{Ass}(M)} \\mathfrak q$ is the set of elements of $R$ which are zerodivisors on $M$."}
+{"_id": "706", "title": "algebra-lemma-ass-functorial", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Then $\\Spec(\\varphi)(\\text{Ass}_S(M)) \\subset \\text{Ass}_R(M)$."}
+{"_id": "707", "title": "algebra-lemma-ass-functorial-Noetherian", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. If $S$ is Noetherian, then $\\Spec(\\varphi)(\\text{Ass}_S(M)) = \\text{Ass}_R(M)$."}
+{"_id": "708", "title": "algebra-lemma-ass-quotient-ring", "text": "Let $R$ be a ring. Let $I$ be an ideal. Let $M$ be an $R/I$-module. Via the canonical injection $\\Spec(R/I) \\to \\Spec(R)$ we have $\\text{Ass}_{R/I}(M) = \\text{Ass}_R(M)$."}
+{"_id": "709", "title": "algebra-lemma-associated-primes-localize", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\mathfrak p \\subset R$ be a prime. \\begin{enumerate} \\item If $\\mathfrak p \\in \\text{Ass}(M)$ then $\\mathfrak pR_{\\mathfrak p} \\in \\text{Ass}(M_{\\mathfrak p})$. \\item If $\\mathfrak p$ is finitely generated then the converse holds as well. \\end{enumerate}"}
+{"_id": "710", "title": "algebra-lemma-localize-ass", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. Via the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$ we have \\begin{enumerate} \\item $\\text{Ass}_R(S^{-1}M) = \\text{Ass}_{S^{-1}R}(S^{-1}M)$, \\item $\\text{Ass}_R(M) \\cap \\Spec(S^{-1}R) \\subset \\text{Ass}_R(S^{-1}M)$, and \\item if $R$ is Noetherian this inclusion is an equality. \\end{enumerate}"}
+{"_id": "711", "title": "algebra-lemma-localize-ass-nonzero-divisors", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. Assume that every $s \\in S$ is a nonzerodivisor on $M$. Then $$ \\text{Ass}_R(M) = \\text{Ass}_R(S^{-1}M). $$"}
+{"_id": "712", "title": "algebra-lemma-ideal-nonzerodivisor", "text": "Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$. Let $I \\subset \\mathfrak m$ be an ideal. Let $M$ be a finite $R$-module. The following are equivalent: \\begin{enumerate} \\item There exists an $x \\in I$ which is not a zerodivisor on $M$. \\item We have $I \\not \\subset \\mathfrak q$ for all $\\mathfrak q \\in \\text{Ass}(M)$. \\end{enumerate}"}
+{"_id": "713", "title": "algebra-lemma-zero-at-ass-zero", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. If $R$ is Noetherian the map $$ M \\longrightarrow \\prod\\nolimits_{\\mathfrak p \\in \\text{Ass}(M)} M_{\\mathfrak p} $$ is injective."}
+{"_id": "714", "title": "algebra-lemma-symbolic-power-associated", "text": "Let $R$ be a Noetherian ring. Let $\\mathfrak p$ be a prime ideal. Let $n > 0$. Then $\\text{Ass}(R/\\mathfrak p^{(n)}) = \\{\\mathfrak p\\}$."}
+{"_id": "715", "title": "algebra-lemma-symbolic-power-flat-extension", "text": "Let $R \\to S$ be flat ring map. Let $\\mathfrak p \\subset R$ be a prime such that $\\mathfrak q = \\mathfrak p S$ is a prime of $S$. Then $\\mathfrak p^{(n)} S = \\mathfrak q^{(n)}$."}
+{"_id": "716", "title": "algebra-lemma-compare-relative-assassins", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Let $A$, $A'$, $A_{fin}$, $B$, and $B_{fin}$ be the subsets of $\\Spec(S)$ introduced above. \\begin{enumerate} \\item We always have $A = A'$. \\item We always have $A_{fin} \\subset A$, $B_{fin} \\subset B$, $A_{fin} \\subset A'_{fin} \\subset B_{fin}$ and $A \\subset B$. \\item If $S$ is Noetherian, then $A = A_{fin}$ and $B = B_{fin}$. \\item If $N$ is flat over $R$, then $A = A_{fin} = A'_{fin}$ and $B = B_{fin}$. \\item If $R$ is Noetherian and $N$ is flat over $R$, then all of the sets are equal, i.e., $A = A' = A_{fin} = A'_{fin} = B = B_{fin}$. \\end{enumerate}"}
+{"_id": "717", "title": "algebra-lemma-bourbaki", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $R$-module, and let $N$ be an $S$-module. If $N$ is flat as $R$-module, then $$ \\text{Ass}_S(M \\otimes_R N) \\supset \\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)} \\text{Ass}_S(N/\\mathfrak pN) $$ and if $R$ is Noetherian then we have equality."}
+{"_id": "718", "title": "algebra-lemma-post-bourbaki", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is flat as an $R$-module and $R$ is a domain with fraction field $K$. Then $$ \\text{Ass}_S(N) = \\text{Ass}_S(N \\otimes_R K) = \\text{Ass}_{S \\otimes_R K}(N \\otimes_R K) $$ via the canonical inclusion $\\Spec(S \\otimes_R K) \\subset \\Spec(S)$."}
+{"_id": "719", "title": "algebra-lemma-bourbaki-fibres", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $R$-module, and let $N$ be an $S$-module. Assume $N$ is flat as $R$-module. Then $$ \\text{Ass}_S(M \\otimes_R N) \\supset \\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)} \\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p)) $$ where we use Remark \\ref{remark-fundamental-diagram} to think of the spectra of fibre rings as subsets of $\\Spec(S)$. If $R$ is Noetherian then this inclusion is an equality."}
+{"_id": "720", "title": "algebra-lemma-weakly-ass-local", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\mathfrak p$ be a prime of $R$. The following are equivalent: \\begin{enumerate} \\item $\\mathfrak p$ is weakly associated to $M$, \\item $\\mathfrak pR_{\\mathfrak p}$ is weakly associated to $M_{\\mathfrak p}$, and \\item $M_{\\mathfrak p}$ contains an element whose annihilator has radical equal to $\\mathfrak pR_{\\mathfrak p}$. \\end{enumerate}"}
+{"_id": "721", "title": "algebra-lemma-reduced-weakly-ass-minimal", "text": "For a reduced ring the weakly associated primes of the ring are the minimal primes."}
+{"_id": "722", "title": "algebra-lemma-weakly-ass", "text": "Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of $R$-modules. Then $\\text{WeakAss}(M') \\subset \\text{WeakAss}(M)$ and $\\text{WeakAss}(M) \\subset \\text{WeakAss}(M') \\cup \\text{WeakAss}(M'')$."}
+{"_id": "723", "title": "algebra-lemma-weakly-ass-zero", "text": "\\begin{slogan} Every nonzero module has a weakly associated prime. \\end{slogan} Let $R$ be a ring. Let $M$ be an $R$-module. Then $$ M = (0) \\Leftrightarrow \\text{WeakAss}(M) = \\emptyset $$"}
+{"_id": "724", "title": "algebra-lemma-weakly-ass-support", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Then $$ \\text{Ass}(M) \\subset \\text{WeakAss}(M) \\subset \\text{Supp}(M). $$"}
+{"_id": "725", "title": "algebra-lemma-weakly-ass-zero-divisors", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The union $\\bigcup_{\\mathfrak q \\in \\text{WeakAss}(M)} \\mathfrak q$ is the set elements of $R$ which are zerodivisors on $M$."}
+{"_id": "726", "title": "algebra-lemma-weakly-ass-minimal-prime-support", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Any $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements of $\\text{Supp}(M)$ is an element of $\\text{WeakAss}(M)$."}
+{"_id": "727", "title": "algebra-lemma-ass-weakly-ass", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\mathfrak p$ be a prime ideal of $R$ which is finitely generated. Then $$ \\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow \\mathfrak p \\in \\text{WeakAss}(M). $$ In particular, if $R$ is Noetherian, then $\\text{Ass}(M) = \\text{WeakAss}(M)$."}
+{"_id": "728", "title": "algebra-lemma-weakly-ass-reverse-functorial", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Then we have $\\Spec(\\varphi)(\\text{WeakAss}_S(M)) \\supset \\text{WeakAss}_R(M)$."}
+{"_id": "729", "title": "algebra-lemma-weakly-ass-finite-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Denote $f : \\Spec(S) \\to \\Spec(R)$ the associated map on spectra. If $\\varphi$ is a finite ring map, then $$ \\text{WeakAss}_R(M) = f(\\text{WeakAss}_S(M)). $$"}
+{"_id": "731", "title": "algebra-lemma-localize-weakly-ass", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. Via the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$ we have $\\text{WeakAss}_R(S^{-1}M) = \\text{WeakAss}_{S^{-1}R}(S^{-1}M)$ and $$ \\text{WeakAss}(M) \\cap \\Spec(S^{-1}R) = \\text{WeakAss}(S^{-1}M). $$"}
+{"_id": "732", "title": "algebra-lemma-localize-weakly-ass-nonzero-divisors", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. Assume that every $s \\in S$ is a nonzerodivisor on $M$. Then $$ \\text{WeakAss}(M) = \\text{WeakAss}(S^{-1}M). $$"}
+{"_id": "733", "title": "algebra-lemma-zero-at-weakly-ass-zero", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The map $$ M \\longrightarrow \\prod\\nolimits_{\\mathfrak p \\in \\text{WeakAss}(M)} M_{\\mathfrak p} $$ is injective."}
+{"_id": "735", "title": "algebra-lemma-weakly-ass-change-fields", "text": "Let $K/k$ be a field extension. Let $R$ be a $k$-algebra. Let $M$ be an $R$-module. Let $\\mathfrak q \\subset R \\otimes_k K$ be a prime lying over $\\mathfrak p \\subset R$. If $\\mathfrak q$ is weakly associated to $M \\otimes_k K$, then $\\mathfrak p$ is weakly associated to $M$."}
+{"_id": "736", "title": "algebra-lemma-remove-embedded-primes", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Consider the set of $R$-submodules $$ \\{ K \\subset M \\mid \\text{Supp}(K) \\text{ nowhere dense in } \\text{Supp}(M) \\}. $$ This set has a maximal element $K$ and the quotient $M' = M/K$ has the following properties \\begin{enumerate} \\item $\\text{Supp}(M) = \\text{Supp}(M')$, \\item $M'$ has no embedded associated primes, \\item for any $f \\in R$ which is contained in all embedded associated primes of $M$ we have $M_f \\cong M'_f$. \\end{enumerate}"}
+{"_id": "737", "title": "algebra-lemma-remove-embedded-primes-localize", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. For any $f \\in R$ we have $(M')_f = (M_f)'$ where $M \\to M'$ and $M_f \\to (M_f)'$ are the quotients constructed in Lemma \\ref{lemma-remove-embedded-primes}."}
+{"_id": "738", "title": "algebra-lemma-no-embedded-primes-endos", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module without embedded associated primes. Let $I = \\{x \\in R \\mid xM = 0\\}$. Then the ring $R/I$ has no embedded primes."}
+{"_id": "739", "title": "algebra-lemma-permute-xi", "text": "Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module. Let $x_1, \\ldots, x_c$ be an $M$-regular sequence. Then any permutation of the $x_i$ is a regular sequence as well."}
+{"_id": "740", "title": "algebra-lemma-flat-increases-depth", "text": "Let $R, S$ be local rings. Let $R \\to S$ be a flat local ring homomorphism. Let $x_1, \\ldots, x_r$ be a sequence in $R$. Let $M$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $x_1, \\ldots, x_r$ is an $M$-regular sequence in $R$, and \\item the images of $x_1, \\ldots, x_r$ in $S$ form a $M \\otimes_R S$-regular sequence. \\end{enumerate}"}
+{"_id": "741", "title": "algebra-lemma-regular-sequence-in-neighbourhood", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Let $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_r$ be a sequence in $R$ whose image in $R_{\\mathfrak p}$ forms an $M_{\\mathfrak p}$-regular sequence. Then there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ such that the image of $x_1, \\ldots, x_r$ in $R_g$ forms an $M_g$-regular sequence."}
+{"_id": "742", "title": "algebra-lemma-join-regular-sequences", "text": "Let $A$ be a ring. Let $I$ be an ideal generated by a regular sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is a regular sequence in $A$."}
+{"_id": "743", "title": "algebra-lemma-regular-sequence-short-exact-sequence", "text": "Let $R$ be a ring. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence of $R$-modules. Let $f_1, \\ldots, f_r \\in R$. If $f_1, \\ldots, f_r$ is $M_1$-regular and $M_3$-regular, then $f_1, \\ldots, f_r$ is $M_2$-regular."}
+{"_id": "744", "title": "algebra-lemma-regular-sequence-powers", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $f_1, \\ldots, f_r \\in R$ and $e_1, \\ldots, e_r > 0$ integers. Then $f_1, \\ldots, f_r$ is an $M$-regular sequence if and only if $f_1^{e_1}, \\ldots, f_r^{e_r}$ is an $M$-regular sequence."}
+{"_id": "745", "title": "algebra-lemma-regular-sequence-in-polynomial-ring", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ which do not generate the unit ideal. The following are equivalent: \\begin{enumerate} \\item any permutation of $f_1, \\ldots, f_r$ is a regular sequence, \\item any subsequence of $f_1, \\ldots, f_r$ (in the given order) is a regular sequence, and \\item $f_1x_1, \\ldots, f_rx_r$ is a regular sequence in the polynomial ring $R[x_1, \\ldots, x_r]$. \\end{enumerate}"}
+{"_id": "746", "title": "algebra-lemma-regular-quasi-regular", "text": "Let $R$ be a ring. \\begin{enumerate} \\item A regular sequence $f_1, \\ldots, f_c$ of $R$ is a quasi-regular sequence. \\item Suppose that $M$ is an $R$-module and that $f_1, \\ldots, f_c$ is an $M$-regular sequence. Then $f_1, \\ldots, f_c$ is an $M$-quasi-regular sequence. \\end{enumerate}"}
+{"_id": "747", "title": "algebra-lemma-flat-base-change-quasi-regular", "text": "Let $R \\to R'$ be a flat ring map. Let $M$ be an $R$-module. Suppose that $f_1, \\ldots, f_r \\in R$ form an $M$-quasi-regular sequence. Then the images of $f_1, \\ldots, f_r$ in $R'$ form a $M \\otimes_R R'$-quasi-regular sequence."}
+{"_id": "748", "title": "algebra-lemma-quasi-regular-sequence-in-neighbourhood", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Let $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_c$ be a sequence in $R$ whose image in $R_{\\mathfrak p}$ forms an $M_{\\mathfrak p}$-quasi-regular sequence. Then there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ such that the image of $x_1, \\ldots, x_c$ in $R_g$ forms an $M_g$-quasi-regular sequence."}
+{"_id": "749", "title": "algebra-lemma-truncate-quasi-regular", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $f_1, \\ldots, f_c \\in R$ be an $M$-quasi-regular sequence. For any $i$ the sequence $\\overline{f}_{i + 1}, \\ldots, \\overline{f}_c$ of $\\overline{R} = R/(f_1, \\ldots, f_i)$ is an $\\overline{M} = M/(f_1, \\ldots, f_i)M$-quasi-regular sequence."}
+{"_id": "750", "title": "algebra-lemma-quasi-regular-regular", "text": "Let $(R, \\mathfrak m)$ be a local Noetherian ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \\ldots, f_c \\in \\mathfrak m$ be an $M$-quasi-regular sequence. Then $f_1, \\ldots, f_c$ is an $M$-regular sequence."}
+{"_id": "751", "title": "algebra-lemma-quasi-regular-on-quotient", "text": "Let $R$ be a ring. Let $J = (f_1, \\ldots, f_r)$ be an ideal of $R$. Let $M$ be an $R$-module. Set $\\overline{R} = R/\\bigcap_{n \\geq 0} J^n$, $\\overline{M} = M/\\bigcap_{n \\geq 0} J^nM$, and denote $\\overline{f}_i$ the image of $f_i$ in $\\overline{R}$. Then $f_1, \\ldots, f_r$ is $M$-quasi-regular if and only if $\\overline{f}_1, \\ldots, \\overline{f}_r$ is $\\overline{M}$-quasi-regular."}
+{"_id": "752", "title": "algebra-lemma-affine-blowup", "text": "Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$. Let $R' = R[\\frac{I}{a}]$ be the affine blowup algebra. Then \\begin{enumerate} \\item the image of $a$ in $R'$ is a nonzerodivisor, \\item $IR' = aR'$, and \\item $(R')_a = R_a$. \\end{enumerate}"}
+{"_id": "753", "title": "algebra-lemma-blowup-base-change", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal and $a \\in I$. Set $J = IS$ and let $b \\in J$ be the image of $a$. Then $S[\\frac{J}{b}]$ is the quotient of $S \\otimes_R R[\\frac{I}{a}]$ by the ideal of elements annihilated by some power of $b$."}
+{"_id": "754", "title": "algebra-lemma-affine-blowup-quotient-description", "text": "Let $R$ be a ring. Let $I = (a_1, \\ldots, a_n)$ be an ideal of $R$. Let $a = a_1$. Then there is a surjection $$ R[x_2, \\ldots, x_n]/(a x_2 - a_2, \\ldots, a x_n - a_n) \\longrightarrow \\textstyle{R[\\frac{I}{a}]} $$ whose kernel is the $a$-power torsion in the source."}
+{"_id": "755", "title": "algebra-lemma-blowup-in-principal", "text": "Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$. Set $R' = R[\\frac{I}{a}]$. If $f \\in R$ is such that $V(f) = V(I)$, then $f$ maps to a nonzerodivisor in $R'$ and $R'_f = R'_a = R_a$."}
+{"_id": "756", "title": "algebra-lemma-blowup-add-principal", "text": "Let $R$ be a ring, $I \\subset R$ an ideal, $a \\in I$, and $f \\in R$. Set $R' = R[\\frac{I}{a}]$ and $R'' = R[\\frac{fI}{fa}]$. Then there is a surjective $R$-algebra map $R' \\to R''$ whose kernel is the set of $f$-power torsion elements of $R'$."}
+{"_id": "757", "title": "algebra-lemma-blowup-reduced", "text": "\\begin{slogan} Being reduced is invariant under blowup \\end{slogan} If $R$ is reduced then every (affine) blowup algebra of $R$ is reduced."}
+{"_id": "758", "title": "algebra-lemma-blowup-domain", "text": "Let $R$ be a domain, $I \\subset R$ an ideal, and $a \\in I$ a nonzero element. Then the affine blowup algebra $R[\\frac{I}{a}]$ is a domain."}
+{"_id": "760", "title": "algebra-lemma-valuation-ring-colimit-affine-blowups", "text": "Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$. Let $R \\subset A \\subset K$ be a valuation ring which dominates $R$. Then $$ A = \\colim R[\\textstyle{\\frac{I}{a}}] $$ is a directed colimit of affine blowups $R \\to R[\\frac{I}{a}]$ with the following properties \\begin{enumerate} \\item $a \\in I \\subset \\mathfrak m$, \\item $I$ is finitely generated, and \\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$ is not zero. \\end{enumerate}"}
+{"_id": "761", "title": "algebra-lemma-resolution-by-finite-free", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item There exists an exact complex $$ \\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0. $$ with $F_i$ free $R$-modules. \\item If $R$ is Noetherian and $M$ finite over $R$, then we can choose the complex such that $F_i$ is finite free. In other words, we can find an exact complex $$ \\ldots \\to R^{\\oplus n_2} \\to R^{\\oplus n_1} \\to R^{\\oplus n_0} \\to M \\to 0. $$ \\end{enumerate}"}
+{"_id": "762", "title": "algebra-lemma-homotopic-equal-homology", "text": "Any two homotopic maps of complexes induce the same maps on (co)homology groups."}
+{"_id": "763", "title": "algebra-lemma-compare-resolutions", "text": "Let $R$ be a ring. Let $M \\to N$ be a map of $R$-modules. Let $N_\\bullet \\to N$ be an arbitrary resolution. Let $$ \\ldots \\to F_2 \\to F_1 \\to F_0 \\to M $$ be a complex of $R$-modules where each $F_i$ is a free $R$-module. Then \\begin{enumerate} \\item there exists a map of complexes $F_\\bullet \\to N_\\bullet$ such that $$ \\xymatrix{ F_0 \\ar[r] \\ar[d] & M \\ar[d] \\\\ N_0 \\ar[r] & N } $$ is commutative, and \\item any two maps $\\alpha, \\beta : F_\\bullet \\to N_\\bullet$ as in (1) are homotopic. \\end{enumerate}"}
+{"_id": "764", "title": "algebra-lemma-ext-welldefined", "text": "Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules. Suppose that $F_{\\bullet}$ is a free resolution of the module $M_1$, and $G_{\\bullet}$ is a free resolution of the module $M_2$. Let $\\varphi : M_1 \\to M_2$ be a module map. Let $\\alpha : F_{\\bullet} \\to G_{\\bullet}$ be a map of complexes inducing $\\varphi$ on $M_1 = \\Coker(d_{F, 1}) \\to M_2 = \\Coker(d_{G, 1})$, see Lemma \\ref{lemma-compare-resolutions}. Then the induced maps $$ H^i(\\alpha) : H^i(\\Hom_R(F_{\\bullet}, N)) \\longrightarrow H^i(\\Hom_R(G_{\\bullet}, N)) $$ are independent of the choice of $\\alpha$. If $\\varphi$ is an isomorphism, so are all the maps $H^i(\\alpha)$. If $M_1 = M_2$, $F_\\bullet = G_\\bullet$, and $\\varphi$ is the identity, so are all the maps $H_i(\\alpha)$."}
+{"_id": "765", "title": "algebra-lemma-long-exact-seq-ext", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $0 \\to N' \\to N \\to N'' \\to 0$ be a short exact sequence. Then we get a long exact sequence $$ \\begin{matrix} 0 \\to \\Hom_R(M, N') \\to \\Hom_R(M, N) \\to \\Hom_R(M, N'') \\\\ \\phantom{0\\ } \\to \\Ext^1_R(M, N') \\to \\Ext^1_R(M, N) \\to \\Ext^1_R(M, N'') \\to \\ldots \\end{matrix} $$"}
+{"_id": "766", "title": "algebra-lemma-reverse-long-exact-seq-ext", "text": "Let $R$ be a ring. Let $N$ be an $R$-module. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence. Then we get a long exact sequence $$ \\begin{matrix} 0 \\to \\Hom_R(M'', N) \\to \\Hom_R(M, N) \\to \\Hom_R(M', N) \\\\ \\phantom{0\\ } \\to \\Ext^1_R(M'', N) \\to \\Ext^1_R(M, N) \\to \\Ext^1_R(M', N) \\to \\ldots \\end{matrix} $$"}
+{"_id": "767", "title": "algebra-lemma-annihilate-ext", "text": "Let $R$ be a ring. Let $M$, $N$ be $R$-modules. Any $x\\in R$ such that either $xN = 0$, or $xM = 0$ annihilates each of the modules $\\Ext^i_R(M, N)$."}
+{"_id": "768", "title": "algebra-lemma-ext-noetherian", "text": "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules. Then $\\Ext^i_R(M, N)$ is a finite $R$-module for all $i$."}
+{"_id": "769", "title": "algebra-lemma-depth-weak-sequence", "text": "Let $R$ be a ring, $I \\subset R$ an ideal, and $M$ a finite $R$-module. Then $\\text{depth}_I(M)$ is equal to the supremum of the lengths of sequences $f_1, \\ldots, f_r \\in I$ such that $f_i$ is a nonzerodivisor on $M/(f_1, \\ldots, f_{i - 1})M$."}
+{"_id": "770", "title": "algebra-lemma-bound-depth", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Then $\\dim(\\text{Supp}(M)) \\geq \\text{depth}(M)$."}
+{"_id": "771", "title": "algebra-lemma-depth-finite-noetherian", "text": "Let $R$ be a Noetherian ring, $I \\subset R$ an ideal, and $M$ a finite nonzero $R$-module such that $IM \\not = M$. Then $\\text{depth}_I(M) < \\infty$."}
+{"_id": "772", "title": "algebra-lemma-depth-ext", "text": "Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$. Let $M$ be a nonzero finite $R$-module. Then $\\text{depth}(M)$ is equal to the smallest integer $i$ such that $\\Ext^i_R(R/\\mathfrak m, M)$ is nonzero."}
+{"_id": "773", "title": "algebra-lemma-depth-in-ses", "text": "Let $R$ be a local Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$ be a short exact sequence of nonzero finite $R$-modules. \\begin{enumerate} \\item $\\text{depth}(N) \\geq \\min\\{\\text{depth}(N'), \\text{depth}(N'')\\}$ \\item $\\text{depth}(N'') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N') - 1\\}$ \\item $\\text{depth}(N') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N'') + 1\\}$ \\end{enumerate}"}
+{"_id": "774", "title": "algebra-lemma-depth-drops-by-one", "text": "Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module. \\begin{enumerate} \\item If $x \\in \\mathfrak m$ is a nonzerodivisor on $M$, then $\\text{depth}(M/xM) = \\text{depth}(M) - 1$. \\item Any $M$-regular sequence $x_1, \\ldots, x_r$ can be extended to an $M$-regular sequence of length $\\text{depth}(M)$. \\end{enumerate}"}
+{"_id": "775", "title": "algebra-lemma-inherit-minimal-primes", "text": "Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module. Let $x \\in \\mathfrak m$, $\\mathfrak p \\in \\text{Ass}(M)$, and $\\mathfrak q$ minimal over $\\mathfrak p + (x)$. Then $\\mathfrak q \\in \\text{Ass}(M/x^nM)$ for some $n \\geq 1$."}
+{"_id": "776", "title": "algebra-lemma-depth-dim-associated-primes", "text": "Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module. For $\\mathfrak p \\in \\text{Ass}(M)$ we have $\\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$."}
+{"_id": "777", "title": "algebra-lemma-depth-localization", "text": "Let $R$ be a local Noetherian ring and $M$ a finite $R$-module. For a prime ideal $\\mathfrak p \\subset R$ we have $\\text{depth}(M_\\mathfrak p) + \\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$."}
+{"_id": "778", "title": "algebra-lemma-depth-goes-down-finite", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $R \\to S$ be a finite ring map. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ be the maximal ideals of $S$. Let $N$ be a finite $S$-module. Then $$ \\min\\nolimits_{i = 1, \\ldots, n} \\text{depth}(N_{\\mathfrak m_i}) = \\text{depth}_\\mathfrak m(N) $$"}
+{"_id": "779", "title": "algebra-lemma-flat-base-change-ext", "text": "Given a flat ring map $R \\to R'$, an $R$-module $M$, and an $R'$-module $N'$ the natural map $$ \\Ext^i_{R'}(M \\otimes_R R', N') \\to \\text{Ext}^i_R(M, N') $$ is an isomorphism for $i \\geq 0$."}
+{"_id": "780", "title": "algebra-lemma-split-injection-after-completion", "text": "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal contained in the Jacobson radical of $R$. Let $N \\to M$ be a homomorphism of finite $R$-modules. Suppose that there exists arbitrarily large $n$ such that $N/I^nN \\to M/I^nM$ is a split injection. Then $N \\to M$ is a split injection."}
+{"_id": "782", "title": "algebra-lemma-long-exact-sequence-tor", "text": "Let $R$ be a ring and let $M$ be an $R$-module. Suppose that $0 \\to N' \\to N \\to N'' \\to 0$ is a short exact sequence of $R$-modules. There exists a long exact sequence $$ \\text{Tor}_1^R(M, N') \\to \\text{Tor}_1^R(M, N) \\to \\text{Tor}_1^R(M, N'') \\to M \\otimes_R N' \\to M \\otimes_R N \\to M \\otimes_R N'' \\to 0 $$"}
+{"_id": "783", "title": "algebra-lemma-no-spectral-sequence", "text": "Let $(A_{\\bullet, \\bullet}, d, \\delta)$ be a double complex such that \\begin{enumerate} \\item Each row $A_{\\bullet, j}$ is a resolution of $R(A)_j$. \\item Each column $A_{i, \\bullet}$ is a resolution of $U(A)_i$. \\end{enumerate} Then there are canonical isomorphisms $$ H_i(R(A)_\\bullet) \\cong H_i(U(A)_\\bullet). $$ The isomorphisms are functorial with respect to morphisms of double complexes with the properties above."}
+{"_id": "785", "title": "algebra-lemma-tor-noetherian", "text": "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules. Then $\\text{Tor}_p^R(M, N)$ is a finite $R$-module for all $p$."}
+{"_id": "786", "title": "algebra-lemma-characterize-flat", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. The following are equivalent: \\begin{enumerate} \\item The module $M$ is flat over $R$. \\item For all $i > 0$ the functor $\\text{Tor}_i^R(M, -)$ is zero. \\item The functor $\\text{Tor}_1^R(M, -)$ is zero. \\item For all ideals $I \\subset R$ we have $\\text{Tor}_1^R(M, R/I) = 0$. \\item For all finitely generated ideals $I \\subset R$ we have $\\text{Tor}_1^R(M, R/I) = 0$. \\end{enumerate}"}
+{"_id": "787", "title": "algebra-lemma-flat-base-change-tor", "text": "Given a flat ring map $R \\to R'$ and $R$-modules $M$, $N$ the natural $R$-module map $\\text{Tor}_i^R(M, N)\\otimes_R R' \\to \\text{Tor}_i^{R'}(M \\otimes_R R', N \\otimes_R R')$ is an isomorphism for all $i$."}
+{"_id": "788", "title": "algebra-lemma-tor-commutes-filtered-colimits", "text": "Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of $R$-modules. Let $N$ be an $R$-module. Then $\\text{Tor}_n^R(M, N) = \\colim \\text{Tor}_n^R(M_i, N)$ for all $n$."}
+{"_id": "789", "title": "algebra-lemma-characterize-projective", "text": "Let $R$ be a ring. Let $P$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $P$ is projective, \\item $P$ is a direct summand of a free $R$-module, and \\item $\\Ext^1_R(P, M) = 0$ for every $R$-module $M$. \\end{enumerate}"}
+{"_id": "791", "title": "algebra-lemma-direct-sum-projective", "text": "A direct sum of projective modules is projective."}
+{"_id": "792", "title": "algebra-lemma-lift-projective-module", "text": "Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal. Let $\\overline{P}$ be a projective $R/I$-module. Then there exists a projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$."}
+{"_id": "794", "title": "algebra-lemma-lift-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $I$ is nilpotent, \\item $M/IM$ is a projective $R/I$-module, \\item $M$ is a flat $R$-module. \\end{enumerate} Then $M$ is a projective $R$-module."}
+{"_id": "795", "title": "algebra-lemma-finite-projective", "text": "Let $R$ be a ring and let $M$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is finitely presented and $R$-flat, \\item $M$ is finite projective, \\item $M$ is a direct summand of a finite free $R$-module, \\item $M$ is finitely presented and for all $\\mathfrak p \\in \\Spec(R)$ the localization $M_{\\mathfrak p}$ is free, \\item $M$ is finitely presented and for all maximal ideals $\\mathfrak m \\subset R$ the localization $M_{\\mathfrak m}$ is free, \\item $M$ is finite and locally free, \\item $M$ is finite locally free, and \\item $M$ is finite, for every prime $\\mathfrak p$ the module $M_{\\mathfrak p}$ is free, and the function $$ \\rho_M : \\Spec(R) \\to \\mathbf{Z}, \\quad \\mathfrak p \\longmapsto \\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p) $$ is locally constant in the Zariski topology. \\end{enumerate}"}
+{"_id": "796", "title": "algebra-lemma-finite-projective-reduced", "text": "Let $R$ be a reduced ring and let $M$ be an $R$-module. Then the equivalent conditions of Lemma \\ref{lemma-finite-projective} are also equivalent to \\begin{enumerate} \\item[(9)] $M$ is finite and the function $\\rho_M : \\Spec(R) \\to \\mathbf{Z}$, $\\mathfrak p \\mapsto \\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)$ is locally constant in the Zariski topology. \\end{enumerate}"}
+{"_id": "797", "title": "algebra-lemma-finite-flat-local", "text": "(Warning: see Remark \\ref{remark-warning}.) Suppose $R$ is a local ring, and $M$ is a finite flat $R$-module. Then $M$ is finite free."}
+{"_id": "798", "title": "algebra-lemma-finite-projective-descends", "text": "Let $R \\to S$ be a flat local homomorphism of local rings. Let $M$ be a finite $R$-module. Then $M$ is finite projective over $R$ if and only if $M \\otimes_R S$ is finite projective over $S$."}
+{"_id": "799", "title": "algebra-lemma-locally-free-semi-local-free", "text": "Let $R$ be a semi-local ring. Let $M$ be a finite locally free module. If $M$ has constant rank, then $M$ is free. In particular, if $R$ has connected spectrum, then $M$ is free."}
+{"_id": "800", "title": "algebra-lemma-semi-local-module-basis-in-submodule", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and infinite residue field. Let $R \\to S$ be a ring map. Let $M$ be an $S$-module and let $N \\subset M$ be an $R$-submodule. Assume \\begin{enumerate} \\item $S$ is semi-local and $\\mathfrak mS$ is contained in the Jacobson radical of $S$, \\item $M$ is a finite free $S$-module, and \\item $N$ generates $M$ as an $S$-module. \\end{enumerate} Then $N$ contains an $S$-basis of $M$."}
+{"_id": "801", "title": "algebra-lemma-evaluation-map-iso-finite-projective", "text": "Let $R$ be ring. Let $L$, $M$, $N$ be $R$-modules. The canonical map $$ \\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L) $$ is an isomorphism if $M$ is finite projective."}
+{"_id": "802", "title": "algebra-lemma-map-between-finite", "text": "Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules with $N$ a finite $R$-module. Then we have the equality \\begin{align*} U & = \\{\\mathfrak p \\subset R \\mid \\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p} \\text{ is surjective}\\} \\\\ & = \\{\\mathfrak p \\subset R \\mid \\varphi \\otimes \\kappa(\\mathfrak p) : M \\otimes \\kappa(\\mathfrak p) \\to N \\otimes \\kappa(\\mathfrak p) \\text{ is surjective}\\} \\end{align*} and $U$ is an open subset of $\\Spec(R)$. Moreover, for any $f \\in R$ such that $D(f) \\subset U$ the map $M_f \\to N_f$ is surjective."}
+{"_id": "803", "title": "algebra-lemma-map-between-finitely-presented", "text": "Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules with $M$ finite and $N$ finitely presented. Then $$ U = \\{\\mathfrak p \\subset R \\mid \\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p} \\text{ is an isomorphism}\\} $$ is an open subset of $\\Spec(R)$."}
+{"_id": "804", "title": "algebra-lemma-cokernel-flat", "text": "Let $R$ be a ring. Let $\\varphi : P_1 \\to P_2$ be a map of finite projective modules. Then \\begin{enumerate} \\item The set $U$ of primes $\\mathfrak p \\in \\Spec(R)$ such that $\\varphi \\otimes \\kappa(\\mathfrak p)$ is injective is open and for any $f\\in R$ such that $D(f) \\subset U$ we have \\begin{enumerate} \\item $P_{1, f} \\to P_{2, f}$ is injective, and \\item the module $\\Coker(\\varphi)_f$ is finite projective over $R_f$. \\end{enumerate} \\item The set $W$ of primes $\\mathfrak p \\in \\Spec(R)$ such that $\\varphi \\otimes \\kappa(\\mathfrak p)$ is surjective is open and for any $f\\in R$ such that $D(f) \\subset W$ we have \\begin{enumerate} \\item $P_{1, f} \\to P_{2, f}$ is surjective, and \\item the module $\\Ker(\\varphi)_f$ is finite projective over $R_f$. \\end{enumerate} \\item The set $V$ of primes $\\mathfrak p \\in \\Spec(R)$ such that $\\varphi \\otimes \\kappa(\\mathfrak p)$ is an isomorphism is open and for any $f\\in R$ such that $D(f) \\subset V$ the map $\\varphi : P_{1, f} \\to P_{2, f}$ is an isomorphism of modules over $R_f$. \\end{enumerate}"}
+{"_id": "805", "title": "algebra-lemma-flat-factors-free", "text": "Let $M$ be an $R$-module.  The following are equivalent: \\begin{enumerate} \\item $M$ is flat. \\item If $f: R^n \\to M$ is a module map and $x \\in \\Ker(f)$, then there are module maps $h: R^n \\to R^m$ and $g: R^m \\to M$ such that $f = g \\circ h$ and $x \\in \\Ker(h)$. \\item Suppose $f: R^n \\to M$ is a module map, $N \\subset \\Ker(f)$ any submodule, and $h: R^n \\to R^{m}$ a map such that $N \\subset \\Ker(h)$ and $f$ factors through $h$.  Then given any $x \\in \\Ker(f)$ we can find a map $h': R^n \\to R^{m'}$ such that $N + Rx \\subset \\Ker(h')$ and $f$ factors through $h'$. \\item If $f: R^n \\to M$ is a module map and $N \\subset \\Ker(f)$ is a finitely generated submodule, then there are module maps $h: R^n \\to R^m$ and $g: R^m \\to M$ such that $f = g \\circ h$ and $N \\subset \\Ker(h)$. \\end{enumerate}"}
+{"_id": "806", "title": "algebra-lemma-flat-factors-fp", "text": "Let $M$ be an $R$-module.  Then $M$ is flat if and only if the following condition holds: if $P$ is a finitely presented $R$-module and $f: P \\to M$ a module map, then there is a free finite $R$-module $F$ and module maps $h: P \\to F$ and $g: F \\to M$ such that $f = g \\circ h$."}
+{"_id": "807", "title": "algebra-lemma-flat-surjective-hom", "text": "Let $M$ be an $R$-module.  Then $M$ is flat if and only if the following condition holds: for every finitely presented $R$-module $P$, if $N \\to M$ is a surjective $R$-module map, then the induced map $\\Hom_R(P, N) \\to \\Hom_R(P, M)$ is surjective."}
+{"_id": "808", "title": "algebra-lemma-universally-exact-split", "text": "Let $$ 0 \\to M_1 \\to M_2 \\to M_3 \\to 0 $$ be an exact sequence of $R$-modules.  Suppose $M_3$ is of finite presentation. Then $$ 0 \\to M_1 \\to M_2 \\to M_3 \\to 0 $$ is universally exact if and only if it is split."}
+{"_id": "809", "title": "algebra-lemma-flat-universally-injective", "text": "Let $M$ be an $R$-module.  Then $M$ is flat if and only if any exact sequence of $R$-modules $$ 0 \\to M_1 \\to M_2 \\to M \\to 0 $$ is universally exact."}
+{"_id": "810", "title": "algebra-lemma-ui-flat-domain", "text": "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a universally exact sequence of $R$-modules, and suppose $M_2$ is flat. Then $M_1$ and $M_3$ are flat."}
+{"_id": "811", "title": "algebra-lemma-universally-injective-tensor", "text": "Let $R$ be a ring. Let $M \\to M'$ be a universally injective $R$-module map. Then for any $R$-module $N$ the map $M \\otimes_R N \\to M' \\otimes_R N$ is universally injective."}
+{"_id": "812", "title": "algebra-lemma-composition-universally-injective", "text": "Let $R$ be a ring. A composition of universally injective $R$-module maps is universally injective."}
+{"_id": "813", "title": "algebra-lemma-universally-injective-permanence", "text": "Let $R$ be a ring. Let $M \\to M'$ and $M' \\to M''$ be $R$-module maps. If their composition $M \\to M''$ is universally injective, then $M \\to M'$ is universally injective."}
+{"_id": "814", "title": "algebra-lemma-faithfully-flat-universally-injective", "text": "Let $R \\to S$ be a faithfully flat ring map. Then $R \\to S$ is universally injective as a map of $R$-modules. In particular $R \\cap IS = I$ for any ideal $I \\subset R$."}
+{"_id": "815", "title": "algebra-lemma-universally-injective-check-stalks", "text": "Let $R \\to S$ be a ring map. Let $M \\to M'$ be a map of $S$-modules. The following are equivalent \\begin{enumerate} \\item $M \\to M'$ is universally injective as a map of $R$-modules, \\item for each prime $\\mathfrak q$ of $S$ the map $M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective as a map of $R$-modules, \\item for each maximal ideal $\\mathfrak m$ of $S$ the map $M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective as a map of $R$-modules, \\item for each prime $\\mathfrak q$ of $S$ the map $M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective as a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the inverse image of $\\mathfrak q$ in $R$, and \\item for each maximal ideal $\\mathfrak m$ of $S$ the map $M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective as a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the inverse image of $\\mathfrak m$ in $R$. \\end{enumerate}"}
+{"_id": "816", "title": "algebra-lemma-universally-injective-localize", "text": "Let $\\varphi : A \\to B$ be a ring map. Let $S \\subset A$ and $S' \\subset B$ be multiplicative subsets such that $\\varphi(S) \\subset S'$. Let $M \\to M'$ be a map of $B$-modules. \\begin{enumerate} \\item If $M \\to M'$ is universally injective as a map of $A$-modules, then $(S')^{-1}M \\to (S')^{-1}M'$ is universally injective as a map of $A$-modules and as a map of $S^{-1}A$-modules. \\item If $M$ and $M'$ are $(S')^{-1}B$-modules, then $M \\to M'$ is universally injective as a map of $A$-modules if and only if it is universally injective as a map of $S^{-1}A$-modules. \\end{enumerate}"}
+{"_id": "817", "title": "algebra-lemma-check-universally-injective-into-flat", "text": "Let $R$ be a ring and let $M \\to M'$ be a map of $R$-modules. If $M'$ is flat, then $M \\to M'$ is universally injective if and only if $M/IM \\to M'/IM'$ is injective for every finitely generated ideal $I$ of $R$."}
+{"_id": "818", "title": "algebra-lemma-finite-projective-again", "text": "Let $M$ be an $R$-module.  Then $M$ is finite projective if and only if $M$ is finitely presented and flat."}
+{"_id": "819", "title": "algebra-lemma-descend-properties-modules", "text": "Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. Then \\begin{enumerate} \\item if the $S$-module $M \\otimes_R S$ is of finite type, then $M$ is of finite type, \\item if the $S$-module $M \\otimes_R S$ is of finite presentation, then $M$ is of finite presentation, \\item if the $S$-module $M \\otimes_R S$ is flat, then $M$ is flat, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "820", "title": "algebra-lemma-direct-sum-devissage", "text": "Let $M$ be an $R$-module.  If $(M_{\\alpha})_{\\alpha \\in S}$ is a direct sum d\\'evissage of $M$, then $M \\cong \\bigoplus_{\\alpha + 1 \\in S} M_{\\alpha + 1}/M_{\\alpha}$."}
+{"_id": "821", "title": "algebra-lemma-Kaplansky-devissage", "text": "Let $M$ be an $R$-module.  Then $M$ is a direct sum of countably generated $R$-modules if and only if it admits a Kaplansky d\\'evissage."}
+{"_id": "822", "title": "algebra-lemma-projective-free", "text": "Let $R$ be a ring.  Then every projective $R$-module is free if and only if every countably generated projective $R$-module is free."}
+{"_id": "823", "title": "algebra-lemma-freeness-criteria", "text": "Let $M$ be a countably generated $R$-module.  Suppose any direct summand $N$ of $M$ satisfies: any element of $N$ is contained in a free direct summand of $N$.  Then $M$ is free."}
+{"_id": "824", "title": "algebra-lemma-projective-freeness-criteria", "text": "Let $P$ be a projective module over a local ring $R$.  Then any element of $P$ is contained in a free direct summand of $P$."}
+{"_id": "825", "title": "algebra-lemma-ML-limit-nonempty", "text": "Let $(A_i, \\varphi_{ji})$ be a directed inverse system over $I$.  Suppose $I$ is countable.  If $(A_i, \\varphi_{ji})$ is Mittag-Leffler and the $A_i$ are nonempty, then $\\lim A_i$ is nonempty."}
+{"_id": "826", "title": "algebra-lemma-ML-exact-sequence", "text": "Let $$ 0 \\to A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i \\to 0 $$ be an exact sequence of directed inverse systems of abelian groups over $I$. Suppose $I$ is countable.  If $(A_i)$ is Mittag-Leffler, then $$ 0 \\to \\lim A_i \\to \\lim B_i \\to \\lim C_i\\to 0 $$ is exact."}
+{"_id": "827", "title": "algebra-lemma-Mittag-Leffler", "text": "Let $R$ be a ring. Let $0 \\to K_i \\to L_i \\to M_i \\to 0$ be short exact sequences of $R$-modules, $i \\geq 1$ which fit into maps of short exact sequences $$ \\xymatrix{ 0 \\ar[r] & K_i \\ar[r] & L_i \\ar[r] & M_i \\ar[r] & 0 \\\\ 0 \\ar[r] & K_{i + 1} \\ar[r] \\ar[u] & L_{i + 1} \\ar[r] \\ar[u] & M_{i + 1} \\ar[r] \\ar[u] & 0} $$ If for every $i$ there exists a $c = c(i) \\geq i$ such that $\\Im(K_c \\to K_i) = \\Im(K_j \\to K_i)$ for all $j \\geq c$, then the sequence $$ 0 \\to \\lim K_i \\to \\lim L_i \\to \\lim M_i \\to 0 $$ is exact."}
+{"_id": "828", "title": "algebra-lemma-domination-fp", "text": "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules. Then $g$ dominates $f$ if and only if for any finitely presented $R$-module $Q$, we have $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R \\text{id}_Q)$."}
+{"_id": "829", "title": "algebra-lemma-domination-universally-injective", "text": "Let $f : M \\to N$ and $g : M \\to M'$ be maps of $R$-modules. Consider the pushout of $f$ and $g$, $$ \\xymatrix{ M  \\ar[r]_f \\ar[d]_g & N \\ar[d]^{g'} \\\\ M' \\ar[r]^{f'} & N' } $$ Then $g$ dominates $f$ if and only if $f'$ is universally injective."}
+{"_id": "830", "title": "algebra-lemma-domination", "text": "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules. Suppose $\\Coker(f)$ is of finite presentation.  Then $g$ dominates $f$ if and only if $g$ factors through $f$, i.e.\\ there exists a module map $h: N \\to M'$ such that $g = h \\circ f$."}
+{"_id": "831", "title": "algebra-lemma-tensor-ML-modules", "text": "If $R$ is a ring and $M$, $N$ are Mittag-Leffler modules over $R$, then $M \\otimes_R N$ is a Mittag-Leffler module."}
+{"_id": "833", "title": "algebra-lemma-restrict-ML-modules", "text": "Let $R \\to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. If $M$ is a Mittag-Leffler module over $S$ then $M$ is a Mittag-Leffler module over $R$."}
+{"_id": "834", "title": "algebra-lemma-mod-ideal-ML-modules", "text": "Let $R$ be a ring. Let $S = R/I$ for some finitely generated ideal $I$. Let $M$ be an $S$-module. Then $M$ is a Mittag-Leffler module over $R$ if and only if $M$ is a Mittag-Leffler module over $S$."}
+{"_id": "835", "title": "algebra-lemma-kernel-tensored-fp", "text": "Let $M$ be an $R$-module, $P$ a finitely presented $R$-module, and $f: P \\to M$ a map.  Let $Q$ be an $R$-module and suppose $x \\in \\Ker(P \\otimes Q \\to M \\otimes Q)$.  Then there exists a finitely presented $R$-module $P'$ and a map $f': P \\to P'$ such that $f$ factors through $f'$ and $x \\in \\Ker(P \\otimes Q \\to P' \\otimes Q)$."}
+{"_id": "836", "title": "algebra-lemma-minimal-contains", "text": "Let $M$ be a flat Mittag-Leffler module over $R$. Let $F$ be an $R$-module and let $x \\in F \\otimes_R M$. Then there exists a smallest submodule $F' \\subset F$ such that $x \\in F' \\otimes_R M$."}
+{"_id": "837", "title": "algebra-lemma-pure-submodule-ML", "text": "Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a universally exact sequence of $R$-modules.  Then: \\begin{enumerate} \\item If $M_2$ is Mittag-Leffler, then $M_1$ is Mittag-Leffler. \\item If $M_1$ and $M_3$ are Mittag-Leffler, then $M_2$ is Mittag-Leffler. \\end{enumerate}"}
+{"_id": "839", "title": "algebra-lemma-colimit-universally-injective-ML", "text": "If $M = \\colim M_i$ is the colimit of a directed system of Mittag-Leffler $R$-modules $M_i$ with universally injective transition maps, then $M$ is Mittag-Leffler."}
+{"_id": "840", "title": "algebra-lemma-direct-sum-ML", "text": "If $M = \\bigoplus_{i \\in I} M_i$ is a direct sum of $R$-modules, then $M$ is Mittag-Leffler if and only if each $M_i$ is Mittag-Leffler."}
+{"_id": "842", "title": "algebra-lemma-coherent", "text": "Let $R$ be a ring. \\begin{enumerate} \\item A finite submodule of a coherent module is coherent. \\item Let $\\varphi : N \\to M$ be a homomorphism from a finite module to a coherent module. Then $\\Ker(\\varphi)$ is finite. \\item Let $\\varphi : N \\to M$ be a homomorphism of coherent modules. Then $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are coherent modules. \\item Given a short exact sequence of $R$-modules $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ if two out of three are coherent so is the third. \\end{enumerate}"}
+{"_id": "843", "title": "algebra-lemma-coherent-ring", "text": "Let $R$ be a ring. If $R$ is coherent, then a module is coherent if and only if it is finitely presented."}
+{"_id": "844", "title": "algebra-lemma-Noetherian-coherent", "text": "A Noetherian ring is a coherent ring."}
+{"_id": "845", "title": "algebra-lemma-flat-ML-criterion", "text": "Let $M$ be a flat $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is Mittag-Leffler, and \\item if $F$ is a finite free $R$-module and $x \\in F \\otimes_R M$, then there exists a smallest submodule $F'$ of $F$ such that $x \\in F' \\otimes_R M$. \\end{enumerate}"}
+{"_id": "846", "title": "algebra-lemma-product-over-Noetherian-ring", "text": "Let $R$ be a Noetherian ring and $A$ a set. Then $M = R^A$ is a flat and Mittag-Leffler $R$-module."}
+{"_id": "847", "title": "algebra-lemma-power-series-ML", "text": "Let $R$ be a Noetherian ring and $n$ a positive integer.  Then the $R$-module $M = R[[t_1, \\ldots, t_n]]$ is flat and Mittag-Leffler."}
+{"_id": "848", "title": "algebra-lemma-ML-countable-colimit", "text": "Let $M$ be an $R$-module.  Write $M = \\colim_{i \\in I} M_i$ where $(M_i, f_{ij})$ is a directed system of finitely presented $R$-modules.  If $M$ is Mittag-Leffler and countably generated, then there is a directed countable subset $I' \\subset I$ such that $M \\cong \\colim_{i \\in I'} M_i$."}
+{"_id": "849", "title": "algebra-lemma-ML-countable", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Assume $M$ is Mittag-Leffler and countably generated. For any $R$-module map $f : P \\to M$ with $P$ finitely generated there exists an endomorphism $\\alpha : M \\to M$ such that \\begin{enumerate} \\item $\\alpha : M \\to M$ factors through a finitely presented $R$-module, and \\item $\\alpha \\circ f = f$. \\end{enumerate}"}
+{"_id": "850", "title": "algebra-lemma-countgen-projective", "text": "Let $M$ be an $R$-module.  If $M$ is flat, Mittag-Leffler, and countably generated, then $M$ is projective."}
+{"_id": "851", "title": "algebra-lemma-ML-ui-descent", "text": "Let $f: M \\to N$ be universally injective map of $R$-modules.  Suppose $M$ is a direct sum of countably generated $R$-modules, and suppose $N$ is flat and Mittag-Leffler.  Then $M$ is projective."}
+{"_id": "853", "title": "algebra-lemma-ascend-properties-modules", "text": "Let $R \\to S$ be a ring map.  Let $M$ be an $R$-module.  Then: \\begin{enumerate} \\item If $M$ is flat, then the $S$-module $M \\otimes_R S$ is flat. \\item If $M$ is Mittag-Leffler, then the $S$-module $M \\otimes_R S$ is Mittag-Leffler. \\item If $M$ is a direct sum of countably generated $R$-modules, then the $S$-module $M \\otimes_R S$ is a direct sum of countably generated $S$-modules. \\item If $M$ is projective, then the $S$-module $M \\otimes_R S$ is projective. \\end{enumerate}"}
+{"_id": "854", "title": "algebra-lemma-ffdescent-ML", "text": "\\begin{reference} Email from Juan Pablo Acosta Lopez dated 12/20/14. \\end{reference} Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the $S$-module $M \\otimes_R S$ is Mittag-Leffler, then $M$ is Mittag-Leffler."}
+{"_id": "855", "title": "algebra-lemma-ffdescent-countable-projectivity", "text": "Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module.  If the $S$-module $M \\otimes_R S$ is countably generated and projective, then $M$ is countably generated and projective."}
+{"_id": "856", "title": "algebra-lemma-lift-countably-generated-submodule", "text": "Let $R \\to S$ be a ring map, let $M$ be an $R$-module, and let $Q$ be a countably generated $S$-submodule of $M \\otimes_R S$.  Then there exists a countably generated $R$-submodule $P$ of $M$ such that $\\Im(P \\otimes_R S \\to M \\otimes_R S)$ contains $Q$."}
+{"_id": "857", "title": "algebra-lemma-adapted-submodule", "text": "Let $R \\to S$ be a ring map, and let $M$ be an $R$-module.  Suppose $M \\otimes_R S = \\bigoplus_{i \\in I} Q_i$ is a direct sum of countably generated $S$-modules $Q_i$.  If $N$ is a countably generated submodule of $M$, then there is a countably generated submodule $N'$ of $M$ such that $N' \\supset N$ and $\\Im(N' \\otimes_R S \\to M \\otimes_R S) = \\bigoplus_{i \\in I'} Q_i$ for some subset $I' \\subset I$."}
+{"_id": "858", "title": "algebra-lemma-completion-generalities", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\varphi : M \\to N$ be a map of $R$-modules. \\begin{enumerate} \\item If $M/IM \\to N/IN$ is surjective, then $M^\\wedge \\to N^\\wedge$ is surjective. \\item If $M \\to N$ is surjective, then $M^\\wedge \\to N^\\wedge$ is surjective. \\item If $0 \\to K \\to M \\to N \\to 0$ is a short exact sequence of $R$-modules and $N$ is flat, then $0 \\to K^\\wedge \\to M^\\wedge \\to N^\\wedge \\to 0$ is a short exact sequence. \\item The map $M \\otimes_R R^\\wedge \\to M^\\wedge$ is surjective for any finite $R$-module $M$. \\end{enumerate}"}
+{"_id": "859", "title": "algebra-lemma-hathat-finitely-generated", "text": "\\begin{reference} \\cite[Theorem 15]{Matlis}. The slick proof given here is from an email of Bjorn Poonen dated Nov 5, 2016. \\end{reference} Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$. Let $M$ be an $R$-module. Then \\begin{enumerate} \\item the completion $M^\\wedge$ is $I$-adically complete, and \\item $I^nM^\\wedge = \\Ker(M^\\wedge \\to M/I^nM) = (I^nM)^\\wedge$ for all $n \\geq 1$. \\end{enumerate} In particular $R^\\wedge$ is $I$-adically complete, $I^nR^\\wedge = (I^n)^\\wedge$, and $R^\\wedge/I^nR^\\wedge = R/I^n$."}
+{"_id": "860", "title": "algebra-lemma-completion-differ-by-torsion", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $0 \\to M \\to N \\to Q \\to 0$ be an exact sequence of $R$-modules such that $Q$ is annihilated by a power of $I$. Then completion produces an exact sequence $0 \\to M^\\wedge \\to N^\\wedge \\to Q \\to 0$."}
+{"_id": "862", "title": "algebra-lemma-radical-completion", "text": "Let $R$ be a ring, let $I \\subset R$ be an ideal, and let $R^\\wedge = \\lim R/I^n$. \\begin{enumerate} \\item any element of $R^\\wedge$ which maps to a unit of $R/I$ is a unit, \\item any element of $1 + I$ maps to an invertible element of $R^\\wedge$, \\item any element of $1 + IR^\\wedge$ is invertible in $R^\\wedge$, and \\item the ideals $IR^\\wedge$ and $\\Ker(R^\\wedge \\to R/I)$ are contained in the Jacobson radical of $R^\\wedge$. \\end{enumerate}"}
+{"_id": "863", "title": "algebra-lemma-when-surjective-to-completion", "text": "Let $A$ be a ring. Let $I = (f_1, \\ldots, f_r)$ be a finitely generated ideal. If $M \\to \\lim M/f_i^nM$ is surjective for each $i$, then $M \\to \\lim M/I^nM$ is surjective."}
+{"_id": "864", "title": "algebra-lemma-complete-by-sub", "text": "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals. If $M$ is $J$-adically complete and $I$ is finitely generated, then $M$ is $I$-adically complete."}
+{"_id": "865", "title": "algebra-lemma-change-ideal-completion", "text": "Let $R$ be a ring. Let $I$, $J$ be ideals of $R$. Assume there exist integers $c, d > 0$ such that $I^c \\subset J$ and $J^d \\subset I$. Then completion with respect to $I$ agrees with completion with respect to $J$ for any $R$-module. In particular an $R$-module $M$ is $I$-adically complete if and only if it is $J$-adically complete."}
+{"_id": "866", "title": "algebra-lemma-quotient-complete", "text": "Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $I$-adically complete $R$-module, and let $K \\subset M$ be an $R$-submodule. The following are equivalent \\begin{enumerate} \\item $K = \\bigcap (K + I^nM)$ and \\item $M/K$ is $I$-adically complete. \\end{enumerate}"}
+{"_id": "867", "title": "algebra-lemma-when-finite-module-complete-over-complete-ring", "text": "Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $R$-module. If (a) $R$ is $I$-adically complete, (b) $M$ is a finite $R$-module, and (c) $\\bigcap I^nM = (0)$, then $M$ is $I$-adically complete."}
+{"_id": "868", "title": "algebra-lemma-finite-over-complete-ring", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $R$ is $I$-adically complete, \\item $\\bigcap_{n \\geq 1} I^nM = (0)$, and \\item $M/IM$ is a finite $R/I$-module. \\end{enumerate} Then $M$ is a finite $R$-module."}
+{"_id": "869", "title": "algebra-lemma-completion-tensor", "text": "Let $I$ be an ideal of a Noetherian ring $R$. Denote ${}^\\wedge$ completion with respect to $I$. \\begin{enumerate} \\item If $K \\to N$ is an injective map of finite $R$-modules, then the map on completions $K^\\wedge \\to N^\\wedge$ is injective. \\item If $0 \\to K \\to N \\to M \\to 0$ is a short exact sequence of finite $R$-modules, then $0 \\to K^\\wedge \\to N^\\wedge \\to M^\\wedge \\to 0$ is a short exact sequence. \\item If $M$ is a finite $R$-module, then $M^\\wedge = M \\otimes_R R^\\wedge$. \\end{enumerate}"}
+{"_id": "870", "title": "algebra-lemma-completion-flat", "text": "Let $I$ be a ideal of a Noetherian ring $R$. Denote ${}^\\wedge$ completion with respect to $I$. \\begin{enumerate} \\item The ring map $R \\to R^\\wedge$ is flat. \\item The functor $M \\mapsto M^\\wedge$ is exact on the category of finitely generated $R$-modules. \\end{enumerate}"}
+{"_id": "871", "title": "algebra-lemma-completion-faithfully-flat", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset \\mathfrak m$ be an ideal. Denote $R^\\wedge$ the completion of $R$ with respect to $I$. The ring map $R \\to R^\\wedge$ is faithfully flat. In particular the completion with respect to $\\mathfrak m$, namely $\\lim_n R/\\mathfrak m^n$ is faithfully flat."}
+{"_id": "872", "title": "algebra-lemma-completion-complete", "text": "Let $R$ be a Noetherian ring. Let $I$ be an ideal of $R$. Let $M$ be an $R$-module. Then the completion $M^\\wedge$ of $M$ with respect to $I$ is $I$-adically complete, $I^n M^\\wedge = (I^nM)^\\wedge$, and $M^\\wedge/I^nM^\\wedge = M/I^nM$."}
+{"_id": "873", "title": "algebra-lemma-completion-Noetherian", "text": "Let $I$ be an ideal of a ring $R$. Assume \\begin{enumerate} \\item $R/I$ is a Noetherian ring, \\item $I$ is finitely generated. \\end{enumerate} Then the completion $R^\\wedge$ of $R$ with respect to $I$ is a Noetherian ring complete with respect to $IR^\\wedge$."}
+{"_id": "874", "title": "algebra-lemma-completion-Noetherian-Noetherian", "text": "Let $R$ be a Noetherian ring. Let $I$ be an ideal of $R$. The completion $R^\\wedge$ of $R$ with respect to $I$ is Noetherian."}
+{"_id": "875", "title": "algebra-lemma-finite-after-completion", "text": "Let $R \\to S$ be a local homomorphism of local rings $(R, \\mathfrak m)$ and $(S, \\mathfrak n)$. Let $R^\\wedge$, resp.\\ $S^\\wedge$ be the completion of $R$, resp.\\ $S$ with respect to $\\mathfrak m$, resp.\\ $\\mathfrak n$. If $\\mathfrak m$ and $\\mathfrak n$ are finitely generated and $\\dim_{\\kappa(\\mathfrak m)} S/\\mathfrak mS < \\infty$, then \\begin{enumerate} \\item $S^\\wedge$ is equal to the $\\mathfrak m$-adic completion of $S$, and \\item $S^\\wedge$ is a finite $R^\\wedge$-module. \\end{enumerate}"}
+{"_id": "876", "title": "algebra-lemma-completion-finite-extension", "text": "Let $R$ be a Noetherian ring. Let $R \\to S$ be a finite ring map. Let $\\mathfrak p \\subset R$ be a prime and let $\\mathfrak q_1, \\ldots, \\mathfrak q_m$ be the primes of $S$ lying over $\\mathfrak p$ (Lemma \\ref{lemma-finite-finite-fibres}). Then $$ R_\\mathfrak p^\\wedge \\otimes_R S = (S_\\mathfrak p)^\\wedge = S_{\\mathfrak q_1}^\\wedge \\times \\ldots \\times S_{\\mathfrak q_m}^\\wedge $$ where the $(S_\\mathfrak p)^\\wedge$ is the completion with respect to $\\mathfrak p$ and the local rings $R_\\mathfrak p$ and $S_{\\mathfrak q_i}$ are completed with respect to their maximal ideals."}
+{"_id": "877", "title": "algebra-lemma-split-completed-sequence", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $0 \\to K \\to P \\to M \\to 0$ be a short exact sequence of $R$-modules. If $M$ is flat over $R$ and $M/IM$ is a projective $R/I$-module, then the sequence of $I$-adic completions $$ 0 \\to K^\\wedge \\to P^\\wedge \\to M^\\wedge \\to 0 $$ is a split exact sequence."}
+{"_id": "878", "title": "algebra-lemma-complete-modulo-nilpotent", "text": "Let $A$ be a Noetherian ring. Let $I, J \\subset A$ be ideals. If $A$ is $I$-adically complete and $A/I$ is $J$-adically complete, then $A$ is $J$-adically complete."}
+{"_id": "879", "title": "algebra-lemma-limit-complete-pre", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring. Let $(M_n)$ be an inverse system of $A$-modules with $I^n M_n = 0$. Then $M = \\lim M_n$ is $I$-adically complete."}
+{"_id": "880", "title": "algebra-lemma-limit-complete", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring. Let $(M_n)$ be an inverse system of $A$-modules with $M_n = M_{n + 1}/I^nM_{n + 1}$. Then $M/I^nM = M_n$ and $M$ is $I$-adically complete."}
+{"_id": "881", "title": "algebra-lemma-finiteness-graded", "text": "Let $A$ be a Noetherian graded ring. Let $I \\subset A_+$ be a homogeneous ideal. Let $(N_n)$ be an inverse system of finite graded $A$-modules with $N_n = N_{n + 1}/I^n N_{n + 1}$. Then there is a finite graded $A$-module $N$ such that $N_n = N/I^nN$ as graded modules for all $n$."}
+{"_id": "882", "title": "algebra-lemma-daniel-litt", "text": "Let $A$ be a graded ring. Let $I \\subset A_+$ be a homogeneous ideal. Denote $A' = \\lim A/I^n$. Let $(G_n)$ be an inverse system of graded $A$-modules with $G_n$ annihilated by $I^n$. Let $M$ be a graded $A$-module and let $\\varphi_n : M \\to G_n$ be a compatible system of graded $A$-module maps. If the induced map $$ \\varphi : M \\otimes_A A' \\longrightarrow \\lim G_n $$ is an isomorphism, then $M_d \\to \\lim G_{n, d}$ is an isomorphism for all $d \\in \\mathbf{Z}$."}
+{"_id": "883", "title": "algebra-lemma-mod-injective", "text": "Suppose that $R \\to S$ is a local homomorphism of Noetherian local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Let $M$ be a flat $R$-module and $N$ a finite $S$-module. Let $u : N \\to M$ be a map of $R$-modules. If $\\overline{u} : N/\\mathfrak m N \\to M/\\mathfrak m M$ is injective then $u$ is injective. In this case $M/u(N)$ is flat over $R$."}
+{"_id": "884", "title": "algebra-lemma-grothendieck", "text": "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Suppose $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$. Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$."}
+{"_id": "885", "title": "algebra-lemma-grothendieck-regular-sequence", "text": "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Suppose $f_1, \\ldots, f_c$ is a sequence of elements of $S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$ form a regular sequence in $S/{\\mathfrak m}S$. Then $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each of the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$."}
+{"_id": "886", "title": "algebra-lemma-free-fibre-flat-free", "text": "Let $R \\to S$ be a local homomorphism of Noetherian local rings. Let $\\mathfrak m$ be the maximal ideal of $R$. Let $M$ be a finite $S$-module. Suppose that (a) $M/\\mathfrak mM$ is a free $S/\\mathfrak mS$-module, and (b) $M$ is flat over $R$. Then $M$ is free and $S$ is flat over $R$."}
+{"_id": "887", "title": "algebra-lemma-complex-exact-mod", "text": "Let $R \\to S$ be a local homomorphism of local Noetherian rings. Let $\\mathfrak m$ be the maximal ideal of $R$. Let $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$ be a finite complex of finite $S$-modules. Assume that each $F_i$ is $R$-flat, and that the complex $0 \\to F_e/\\mathfrak m F_e \\to F_{e-1}/\\mathfrak m F_{e-1} \\to \\ldots \\to F_0 / \\mathfrak m F_0$ is exact. Then $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$ is exact, and moreover the module $\\Coker(F_1 \\to F_0)$ is $R$-flat."}
+{"_id": "888", "title": "algebra-lemma-prepare-local-criterion-flatness", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $\\kappa = R/\\mathfrak m$. Let $M$ be an $R$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$, then for every finite length $R$-module $N$ we have $\\text{Tor}_1^R(N, M) = 0$."}
+{"_id": "889", "title": "algebra-lemma-local-criterion-flatness", "text": "Let $R \\to S$ be a local homomorphism of local Noetherian rings. Let $\\mathfrak m$ be the maximal ideal of $R$, and let $\\kappa = R/\\mathfrak m$. Let $M$ be a finite $S$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$, then $M$ is flat over $R$."}
+{"_id": "890", "title": "algebra-lemma-what-does-it-mean", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. If $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$ then \\begin{enumerate} \\item $M/I^nM$ is flat over $R/I^n$ for all $n \\geq 1$, and \\item for any module $N$ which is annihilated by $I^m$ for some $m \\geq 0$ we have $\\text{Tor}_1^R(N, M) = 0$. \\end{enumerate} In particular, if $I$ is nilpotent, then $M$ is flat over $R$."}
+{"_id": "891", "title": "algebra-lemma-what-does-it-mean-again", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. \\begin{enumerate} \\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I/I^2 \\to IM/I^2M$ is injective, then $M/I^2M$ is flat over $R/I^2$. \\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I^n/I^{n + 1} \\to I^nM/I^{n + 1}M$ is injective for $n = 1, \\ldots, k$, then $M/I^{k + 1}M$ is flat over $R/I^{k + 1}$. \\end{enumerate}"}
+{"_id": "892", "title": "algebra-lemma-variant-local-criterion-flatness", "text": "Let $R \\to S$ be a local homomorphism of Noetherian local rings. Let $I \\not = R$ be an ideal in $R$. Let $M$ be a finite $S$-module. If $\\text{Tor}_1^R(M, R/I) = 0$ and $M/IM$ is flat over $R/I$, then $M$ is flat over $R$."}
+{"_id": "893", "title": "algebra-lemma-flat-module-powers", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R$ is a Noetherian ring, \\item $S$ is a Noetherian ring, \\item $M$ is a finite $S$-module, and \\item for each $n \\geq 1$ the module $M/I^n M$ is flat over $R/I^n$. \\end{enumerate} Then for every $\\mathfrak q \\in V(IS)$ the localization $M_{\\mathfrak q}$ is flat over $R$. In particular, if $S$ is local and $IS$ is contained in its maximal ideal, then $M$ is flat over $R$."}
+{"_id": "894", "title": "algebra-lemma-surjective-on-tor-one", "text": "Let $R \\to R' \\to R''$ be ring maps. Let $M$ be an $R$-module. Suppose that $M \\otimes_R R'$ is flat over $R'$. Then the natural map $\\text{Tor}_1^R(M, R') \\otimes_{R'} R'' \\to \\text{Tor}_1^R(M, R'')$ is onto."}
+{"_id": "895", "title": "algebra-lemma-surjective-on-tor-one-trivial", "text": "Let $R \\to R'$ be a ring map. Let $I \\subset R$ be an ideal and $I' = IR'$. Let $M$ be an $R$-module and set $M' = M \\otimes_R R'$. The natural map $\\text{Tor}_1^R(R'/I', M) \\to \\text{Tor}_1^{R'}(R'/I', M')$ is surjective."}
+{"_id": "896", "title": "algebra-lemma-another-variant-local-criterion-flatness", "text": "Let $$ \\xymatrix{ S \\ar[r] & S' \\\\ R \\ar[r] \\ar[u] & R' \\ar[u] } $$ be a commutative diagram of local homomorphisms of local Noetherian rings. Let $I \\subset R$ be a proper ideal. Let $M$ be a finite $S$-module. Denote $I' = IR'$ and $M' = M \\otimes_S S'$. Assume that \\begin{enumerate} \\item $S'$ is a localization of the tensor product $S \\otimes_R R'$, \\item $M/IM$ is flat over $R/I$, \\item $\\text{Tor}_1^R(M, R/I) \\to \\text{Tor}_1^{R'}(M', R'/I')$ is zero. \\end{enumerate} Then $M'$ is flat over $R'$."}
+{"_id": "897", "title": "algebra-lemma-criterion-flatness-fibre-Noetherian", "text": "Let $R$, $S$, $S'$ be Noetherian local rings and let $R \\to S \\to S'$ be local ring homomorphisms. Let $\\mathfrak m \\subset R$ be the maximal ideal. Let $M$ be an $S'$-module. Assume \\begin{enumerate} \\item The module $M$ is finite over $S'$. \\item The module $M$ is not zero. \\item The module $M/\\mathfrak m M$ is a flat $S/\\mathfrak m S$-module. \\item The module $M$ is a flat $R$-module. \\end{enumerate} Then $S$ is flat over $R$ and $M$ is a flat $S$-module."}
+{"_id": "898", "title": "algebra-lemma-base-change-flat-up-down", "text": "Let $$ \\xymatrix{ S \\ar[r] & S' \\\\ R \\ar[r] \\ar[u] & R' \\ar[u] } $$ be a commutative diagram of local homomorphisms of local rings. Assume that $S'$ is a localization of the tensor product $S \\otimes_R R'$. Let $M$ be an $S$-module and set $M' = S' \\otimes_S M$. \\begin{enumerate} \\item If $M$ is flat over $R$ then $M'$ is flat over $R'$. \\item If $M'$ is flat over $R'$ and $R \\to R'$ is flat then $M$ is flat over $R$. \\end{enumerate} In particular we have \\begin{enumerate} \\item[(3)] If $S$ is flat over $R$ then $S'$ is flat over $R'$. \\item[(4)] If $R' \\to S'$ and $R \\to R'$ are flat then $S$ is flat over $R$. \\end{enumerate}"}
+{"_id": "899", "title": "algebra-lemma-yet-another-variant-local-criterion-flatness", "text": "Consider a commutative diagram of local rings and local homomorphisms $$ \\xymatrix{ S \\ar[r] & S' \\\\ R \\ar[r] \\ar[u] & R' \\ar[u] } $$ Let $M$ be a finite $S$-module. Assume that \\begin{enumerate} \\item the horizontal arrows are flat ring maps \\item $M$ is flat over $R$, \\item $\\mathfrak m_R R' = \\mathfrak m_{R'}$, \\item $R'$ and $S'$ are Noetherian. \\end{enumerate} Then $M' = M \\otimes_R S'$ is flat over $R'$."}
+{"_id": "900", "title": "algebra-lemma-local-artinian-basis-when-flat", "text": "Let $(R, \\mathfrak m)$ be a local ring with nilpotent maximal ideal $\\mathfrak m$. Let $M$ be a flat $R$-module. If $A$ is a set and $x_\\alpha \\in M$, $\\alpha \\in A$ is a collection of elements of $M$, then the following are equivalent: \\begin{enumerate} \\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis for the vector space $M/\\mathfrak mM$ over $R/\\mathfrak m$, and \\item $\\{x_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M$ over $R$. \\end{enumerate}"}
+{"_id": "901", "title": "algebra-lemma-local-artinian-characterize-flat", "text": "Let $R$ be a local ring with nilpotent maximal ideal. Let $M$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is flat over $R$, \\item $M$ is a free $R$-module, and \\item $M$ is a projective $R$-module. \\end{enumerate}"}
+{"_id": "902", "title": "algebra-lemma-lift-basis", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Let $A$ be a set and let $x_\\alpha \\in M$, $\\alpha \\in A$ be a collection of elements of $M$. Assume \\begin{enumerate} \\item $I$ is nilpotent, \\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M/IM$ over $R/I$, and \\item $\\text{Tor}_1^R(R/I, M) = 0$. \\end{enumerate} Then $M$ is free on $\\{x_\\alpha\\}_{\\alpha \\in A}$ over $R$."}
+{"_id": "903", "title": "algebra-lemma-prepare-lift-flatness", "text": "Let $\\varphi : R \\to R'$ be a ring map. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $M/IM$ is flat over $R/I$, and \\item $R' \\otimes_R M$ is flat over $R'$. \\end{enumerate} Set $I_2 = \\varphi^{-1}(\\varphi(I^2)R')$. Then $M/I_2M$ is flat over $R/I_2$."}
+{"_id": "904", "title": "algebra-lemma-lift-flatness", "text": "Let $\\varphi : R \\to R'$ be a ring map. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $I$ is nilpotent, \\item $R \\to R'$ is injective, \\item $M/IM$ is flat over $R/I$, and \\item $R' \\otimes_R M$ is flat over $R'$. \\end{enumerate} Then $M$ is flat over $R$."}
+{"_id": "906", "title": "algebra-lemma-descent-flatness-injective-map-artinian-rings", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $R$ is Artinian \\item $R \\to S$ is injective, and \\item $M \\otimes_R S$ is a flat $S$-module. \\end{enumerate} Then $M$ is a flat $R$-module."}
+{"_id": "907", "title": "algebra-lemma-criterion-flatness-fibre-nilpotent", "text": "Let $$ \\xymatrix{ S \\ar[rr] & & S' \\\\ & R \\ar[lu] \\ar[ru] } $$ be a commutative diagram in the category of rings. Let $I \\subset R$ be a nilpotent ideal and $M$ an $S'$-module. Assume \\begin{enumerate} \\item The module $M/IM$ is a flat $S/IS$-module. \\item The module $M$ is a flat $R$-module. \\end{enumerate} Then $M$ is a flat $S$-module and $S_{\\mathfrak q}$ is flat over $R$ for every $\\mathfrak q \\subset S$ such that $M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero."}
+{"_id": "908", "title": "algebra-lemma-add-trivial-complex", "text": "Suppose $R$ is a ring. Let $$ \\ldots \\xrightarrow{\\varphi_{i + 1}} R^{n_i} \\xrightarrow{\\varphi_i} R^{n_{i-1}} \\xrightarrow{\\varphi_{i-1}} \\ldots $$ be a complex of finite free $R$-modules. Suppose that for some $i$ some matrix coefficient of the map $\\varphi_i$ is invertible. Then the displayed complex is isomorphic to the direct sum of a complex $$ \\ldots \\to R^{n_{i + 2}} \\xrightarrow{\\varphi_{i + 2}} R^{n_{i + 1}} \\to R^{n_i - 1} \\to R^{n_{i - 1} - 1} \\to R^{n_{i - 2}} \\xrightarrow{\\varphi_{i - 2}} R^{n_{i - 3}} \\to \\ldots $$ and the complex $\\ldots \\to 0 \\to R \\to R \\to 0 \\to \\ldots$ where the map $R \\to R$ is the identity map."}
+{"_id": "909", "title": "algebra-lemma-exact-depth-zero-local", "text": "In Situation \\ref{situation-complex}. Suppose $R$ is a local Noetherian ring with maximal ideal $\\mathfrak m$. Assume $\\mathfrak m \\in \\text{Ass}(R)$, in other words $R$ has depth $0$. Suppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$ is exact at $R^{n_e}, \\ldots, R^{n_1}$. Then the complex is isomorphic to a direct sum of trivial complexes."}
+{"_id": "911", "title": "algebra-lemma-trivial-case-exact", "text": "In Situation \\ref{situation-complex}, suppose the complex is isomorphic to a direct sum of trivial complexes. Then we have \\begin{enumerate} \\item the maps $\\varphi_i$ have rank $r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$, \\item for all $i$, $1 \\leq i \\leq e - 1$ we have $\\text{rank}(\\varphi_{i + 1}) + \\text{rank}(\\varphi_i) = n_i$, \\item each $I(\\varphi_i) = R$. \\end{enumerate}"}
+{"_id": "912", "title": "algebra-lemma-div-x-exact-one-less", "text": "In Situation \\ref{situation-complex}. Suppose $R$ is a local ring with maximal ideal $\\mathfrak m$. Suppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$ is exact at $R^{n_e}, \\ldots, R^{n_1}$. Let $x \\in \\mathfrak m$ be a nonzerodivisor. The complex $0 \\to (R/xR)^{n_e} \\to \\ldots \\to (R/xR)^{n_1}$ is exact at $(R/xR)^{n_e}, \\ldots, (R/xR)^{n_2}$."}
+{"_id": "913", "title": "algebra-lemma-acyclic", "text": "\\begin{reference} \\cite[Lemma 1.8]{Peskine-Szpiro} \\end{reference} Let $R$ be a local Noetherian ring. Let $0 \\to M_e \\to M_{e-1} \\to \\ldots \\to M_0$ be a complex of finite $R$-modules. Assume $\\text{depth}(M_i) \\geq i$. Let $i$ be the largest index such that the complex is not exact at $M_i$. If $i > 0$ then $\\Ker(M_i \\to M_{i-1})/\\Im(M_{i + 1} \\to M_i)$ has depth $\\geq 1$."}
+{"_id": "914", "title": "algebra-lemma-good-element", "text": "Notation and assumptions as above. If $g$ is good with respect to $(M, f_1, \\ldots, f_d)$, then (a) $g$ is a nonzerodivisor on $M$, and (b) $M/gM$ is Cohen-Macaulay with maximal regular sequence $f_1, \\ldots, f_{d - 1}$."}
+{"_id": "915", "title": "algebra-lemma-CM-one-g", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a Cohen-Macaulay module over $R$. Suppose $g \\in \\mathfrak m$ is such that $\\dim(\\text{Supp}(M) \\cap V(g)) = \\dim(\\text{Supp}(M)) - 1$. Then (a) $g$ is a nonzerodivisor on $M$, and (b) $M/gM$ is Cohen-Macaulay of depth one less."}
+{"_id": "917", "title": "algebra-lemma-CM-over-quotient", "text": "Let $R \\to S$ be a surjective homomorphism of Noetherian local rings. Let $N$ be a finite $S$-module. Then $N$ is Cohen-Macaulay as an $S$-module if and only if $N$ is Cohen-Macaulay as an $R$-module."}
+{"_id": "918", "title": "algebra-lemma-CM-ass-minimal-support", "text": "\\begin{reference} \\cite[Chapter 0, Proposition 16.5.4]{EGA} \\end{reference} Let $R$ be a Noetherian local ring. Let $M$ be a finite Cohen-Macaulay $R$-module. If $\\mathfrak p \\in \\text{Ass}(M)$, then $\\dim(R/\\mathfrak p) = \\dim(\\text{Supp}(M))$ and $\\mathfrak p$ is a minimal prime in the support of $M$. In particular, $M$ has no embedded associated primes."}
+{"_id": "919", "title": "algebra-lemma-maximal-chain-maximal-CM", "text": "\\begin{slogan} In a local Cohen-Macaulay ring, any maximal chain of prime ideals has length equal to the dimension. \\end{slogan} Let $R$ be a Noetherian local ring. Assume there exists a Cohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$. Then any maximal chain of ideals $\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$ has length $n = \\dim(R)$."}
+{"_id": "920", "title": "algebra-lemma-dim-formula-maximal-CM", "text": "Suppose $R$ is a Noetherian local ring. Assume there exists a Cohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$. Then for a prime $\\mathfrak p \\subset R$ we have $$ \\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p). $$"}
+{"_id": "921", "title": "algebra-lemma-localize-CM-module", "text": "Suppose $R$ is a Noetherian local ring. Let $M$ be a Cohen-Macaulay module over $R$. For any prime $\\mathfrak p \\subset R$ the module $M_{\\mathfrak p}$ is Cohen-Macaulay over $R_\\mathfrak p$."}
+{"_id": "922", "title": "algebra-lemma-maximal-CM-polynomial-algebra", "text": "Let $R$ be a Noetherian ring. Let $M$ be a Cohen-Macaulay module over $R$. Then $M \\otimes_R R[x_1, \\ldots, x_n]$ is a Cohen-Macaulay module over $R[x_1, \\ldots, x_n]$."}
+{"_id": "923", "title": "algebra-lemma-reformulate-CM", "text": "\\begin{slogan} Regular sequences in Cohen-Macaulay local rings are characterized by cutting out something of the correct dimension. \\end{slogan} Let $R$ be a Noetherian local Cohen-Macaulay ring with maximal ideal $\\mathfrak m $. Let $x_1, \\ldots, x_c \\in \\mathfrak m$ be elements. Then $$ x_1, \\ldots, x_c \\text{ is a regular sequence } \\Leftrightarrow \\dim(R/(x_1, \\ldots, x_c)) = \\dim(R) - c $$ If so $x_1, \\ldots, x_c$ can be extended to a regular sequence of length $\\dim(R)$ and each quotient $R/(x_1, \\ldots, x_i)$ is a Cohen-Macaulay ring of dimension $\\dim(R) - i$."}
+{"_id": "924", "title": "algebra-lemma-maximal-chain-CM", "text": "Let $R$ be Noetherian local. Suppose $R$ is Cohen-Macaulay of dimension $d$. Any maximal chain of ideals $\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$ has length $n = d$."}
+{"_id": "925", "title": "algebra-lemma-CM-dim-formula", "text": "Suppose $R$ is a Noetherian local Cohen-Macaulay ring of dimension $d$. For any prime $\\mathfrak p \\subset R$ we have $$ \\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p). $$"}
+{"_id": "926", "title": "algebra-lemma-localize-CM", "text": "Suppose $R$ is a Cohen-Macaulay local ring. For any prime $\\mathfrak p \\subset R$ the ring $R_{\\mathfrak p}$ is Cohen-Macaulay as well."}
+{"_id": "927", "title": "algebra-lemma-CM-polynomial-algebra", "text": "Suppose $R$ is a Noetherian Cohen-Macaulay ring. Any polynomial algebra over $R$ is Cohen-Macaulay."}
+{"_id": "928", "title": "algebra-lemma-dimension-shift", "text": "Let $R$ be a Noetherian local Cohen-Macaulay ring of dimension $d$. Let $0 \\to K \\to R^{\\oplus n} \\to M \\to 0$ be an exact sequence of $R$-modules. Then either $M = 0$, or $\\text{depth}(K) > \\text{depth}(M)$, or $\\text{depth}(K) = \\text{depth}(M) = d$."}
+{"_id": "929", "title": "algebra-lemma-mcm-resolution", "text": "Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$. Let $M$ be a finite $R$ module of depth $e$. There exists an exact complex $$ 0 \\to K \\to F_{d-e-1} \\to \\ldots \\to F_0 \\to M \\to 0 $$ with each $F_i$ finite free and $K$ maximal Cohen-Macaulay."}
+{"_id": "930", "title": "algebra-lemma-find-sequence-image-regular", "text": "Let $\\varphi : A \\to B$ be a map of local rings. Assume that $B$ is Noetherian and Cohen-Macaulay and that $\\mathfrak m_B = \\sqrt{\\varphi(\\mathfrak m_A) B}$. Then there exists a sequence of elements $f_1, \\ldots, f_{\\dim(B)}$ in $A$ such that $\\varphi(f_1), \\ldots, \\varphi(f_{\\dim(B)})$ is a regular sequence in $B$."}
+{"_id": "931", "title": "algebra-lemma-catenary", "text": "A ring $R$ is catenary if and only if the topological space $\\Spec(R)$ is catenary (see Topology, Definition \\ref{topology-definition-catenary})."}
+{"_id": "932", "title": "algebra-lemma-localization-catenary", "text": "Any localization of a catenary ring is catenary. Any localization of a Noetherian universally catenary ring is universally catenary."}
+{"_id": "933", "title": "algebra-lemma-universally-catenary", "text": "Let $A$ be a Noetherian universally catenary ring. Any $A$-algebra essentially of finite type over $A$ is universally catenary."}
+{"_id": "934", "title": "algebra-lemma-catenary-check-local", "text": "Let $R$ be a ring. The following are equivalent \\begin{enumerate} \\item $R$ is catenary, \\item $R_\\mathfrak p$ is catenary for all prime ideals $\\mathfrak p$, \\item $R_\\mathfrak m$ is catenary for all maximal ideals $\\mathfrak m$. \\end{enumerate} Assume $R$ is Noetherian. The following are equivalent \\begin{enumerate} \\item $R$ is universally catenary, \\item $R_\\mathfrak p$ is universally catenary for all prime ideals $\\mathfrak p$, \\item $R_\\mathfrak m$ is universally catenary for all maximal ideals $\\mathfrak m$. \\end{enumerate}"}
+{"_id": "935", "title": "algebra-lemma-quotient-catenary", "text": "Any quotient of a catenary ring is catenary. Any quotient of a Noetherian universally catenary ring is universally catenary."}
+{"_id": "936", "title": "algebra-lemma-catenary-check-irreducible", "text": "Let $R$ be a Noetherian ring. \\begin{enumerate} \\item $R$ is catenary if and only if $R/\\mathfrak p$ is catenary for every minimal prime $\\mathfrak p$. \\item $R$ is universally catenary if and only if $R/\\mathfrak p$ is universally catenary for every minimal prime $\\mathfrak p$. \\end{enumerate}"}
+{"_id": "937", "title": "algebra-lemma-CM-ring-catenary", "text": "A Noetherian Cohen-Macaulay ring is universally catenary. More generally, if $R$ is a Noetherian ring and $M$ is a Cohen-Macaulay $R$-module with $\\text{Supp}(M) = \\Spec(R)$, then $R$ is universally catenary."}
+{"_id": "939", "title": "algebra-lemma-regular-graded", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $d$. The graded ring $\\bigoplus \\mathfrak m^n / \\mathfrak m^{n + 1}$ is isomorphic to the graded polynomial algebra $\\kappa[X_1, \\ldots, X_d]$."}
+{"_id": "940", "title": "algebra-lemma-regular-domain", "text": "Any regular local ring is a domain."}
+{"_id": "941", "title": "algebra-lemma-regular-ring-CM", "text": "Let $R$ be a regular local ring and let $x_1, \\ldots, x_d$ be a minimal set of generators for the maximal ideal $\\mathfrak m$. Then $x_1, \\ldots, x_d$ is a regular sequence, and each $R/(x_1, \\ldots, x_c)$ is a regular local ring of dimension $d - c$. In particular $R$ is Cohen-Macaulay."}
+{"_id": "942", "title": "algebra-lemma-regular-quotient-regular", "text": "Let $R$ be a regular local ring. Let $I \\subset R$ be an ideal such that $R/I$ is a regular local ring as well. Then there exists a minimal set of generators $x_1, \\ldots, x_d$ for the maximal ideal $\\mathfrak m$ of $R$ such that $I = (x_1, \\ldots, x_c)$ for some $0 \\leq c \\leq d$."}
+{"_id": "943", "title": "algebra-lemma-free-mod-x", "text": "Let $R$ be a Noetherian local ring. Let $x \\in \\mathfrak m$. Let $M$ be a finite $R$-module such that $x$ is a nonzerodivisor on $M$ and $M/xM$ is free over $R/xR$. Then $M$ is free over $R$."}
+{"_id": "944", "title": "algebra-lemma-regular-mcm-free", "text": "Let $R$ be a regular local ring. Any maximal Cohen-Macaulay module over $R$ is free."}
+{"_id": "945", "title": "algebra-lemma-regular-mod-x", "text": "Suppose $R$ is a Noetherian local ring. Let $x \\in \\mathfrak m$ be a nonzerodivisor such that $R/xR$ is a regular local ring. Then $R$ is a regular local ring. More generally, if $x_1, \\ldots, x_r$ is a regular sequence in $R$ such that $R/(x_1, \\ldots, x_r)$ is a regular local ring, then $R$ is a regular local ring."}
+{"_id": "946", "title": "algebra-lemma-colimit-regular", "text": "Let $(R_i, \\varphi_{ii'})$ be a directed system of local rings whose transition maps are local ring maps. If each $R_i$ is a regular local ring and $R = \\colim R_i$ is Noetherian, then $R$ is a regular local ring."}
+{"_id": "947", "title": "algebra-lemma-epimorphism", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is an epimorphism, \\item the two ring maps $S \\to S \\otimes_R S$ are equal, \\item either of the ring maps $S \\to S \\otimes_R S$ is an isomorphism, and \\item the ring map $S \\otimes_R S \\to S$ is an isomorphism. \\end{enumerate}"}
+{"_id": "948", "title": "algebra-lemma-composition-epimorphism", "text": "The composition of two epimorphisms of rings is an epimorphism."}
+{"_id": "949", "title": "algebra-lemma-base-change-epimorphism", "text": "If $R \\to S$ is an epimorphism of rings and $R \\to R'$ is any ring map, then $R' \\to R' \\otimes_R S$ is an epimorphism."}
+{"_id": "950", "title": "algebra-lemma-permanence-epimorphism", "text": "If $A \\to B \\to C$ are ring maps and $A \\to C$ is an epimorphism, so is $B \\to C$."}
+{"_id": "951", "title": "algebra-lemma-epimorphism-local", "text": "Let $R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is an epimorphism, and \\item $R_{\\mathfrak p} \\to S_{\\mathfrak p}$ is an epimorphism for each prime $\\mathfrak p$ of $R$. \\end{enumerate}"}
+{"_id": "952", "title": "algebra-lemma-finite-epimorphism-surjective", "text": "\\begin{slogan} A ring map is surjective if and only if it is a finite epimorphism. \\end{slogan} Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is an epimorphism and finite, and \\item $R \\to S$ is surjective. \\end{enumerate}"}
+{"_id": "953", "title": "algebra-lemma-faithfully-flat-epimorphism", "text": "A faithfully flat epimorphism is an isomorphism."}
+{"_id": "954", "title": "algebra-lemma-epimorphism-over-field", "text": "If $k \\to S$ is an epimorphism and $k$ is a field, then $S = k$ or $S = 0$."}
+{"_id": "956", "title": "algebra-lemma-relations", "text": "Let $R$ be a ring. Let $M$, $N$ be $R$-modules. Let $\\{x_i\\}_{i \\in I}$ be a set of generators of $M$. Let $\\{y_j\\}_{j \\in J}$ be a set of generators of $N$. Let $\\{m_j\\}_{j \\in J}$ be a family of elements of $M$ with $m_j = 0$ for all but finitely many $j$. Then $$ \\sum\\nolimits_{j \\in J} m_j \\otimes y_j = 0 \\text{ in } M \\otimes_R N $$ is equivalent to the following: There exist $a_{i, j} \\in R$ with $a_{i, j} = 0$ for all but finitely many pairs $(i, j)$ such that \\begin{align*} m_j & = \\sum\\nolimits_{i \\in I} a_{i, j} x_i \\quad\\text{for all } j \\in J, \\\\ 0 & = \\sum\\nolimits_{j \\in J} a_{i, j} y_j \\quad\\text{for all } i \\in I. \\end{align*}"}
+{"_id": "957", "title": "algebra-lemma-kernel-difference-projections", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $g \\in S$. The following are equivalent: \\begin{enumerate} \\item $g \\otimes 1 = 1 \\otimes g$ in $S \\otimes_R S$, and \\item there exist $n \\geq 0$ and elements $y_i, z_j \\in S$ and $x_{i, j} \\in R$ for $1 \\leq i, j \\leq n$ such that \\begin{enumerate} \\item $g = \\sum_{i, j \\leq n} x_{i, j} y_i z_j$, \\item for each $j$ we have $\\sum x_{i, j}y_i \\in \\varphi(R)$, and \\item for each $i$ we have $\\sum x_{i, j}z_j \\in \\varphi(R)$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "959", "title": "algebra-lemma-epimorphism-modules", "text": "Let $R \\to S$ be an epimorphism of rings. Let $N_1, N_2$ be $S$-modules. Then $\\Hom_S(N_1, N_2) = \\Hom_R(N_1, N_2)$. In other words, the restriction functor $\\text{Mod}_S \\to \\text{Mod}_R$ is fully faithful."}
+{"_id": "960", "title": "algebra-lemma-pure", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. The following are equivalent: \\begin{enumerate} \\item $I$ is pure, \\item for every ideal $J \\subset R$ we have $J \\cap I = IJ$, \\item for every finitely generated ideal $J \\subset R$ we have $J \\cap I = JI$, \\item for every $x \\in R$ we have $(x) \\cap I = xI$, \\item for every $x \\in I$ we have $x = yx$ for some $y \\in I$, \\item for every $x_1, \\ldots, x_n \\in I$ there exists a $y \\in I$ such that $x_i = yx_i$ for all $i = 1, \\ldots, n$, \\item for every prime $\\mathfrak p$ of $R$ we have $IR_{\\mathfrak p} = 0$ or $IR_{\\mathfrak p} = R_{\\mathfrak p}$, \\item $\\text{Supp}(I) = \\Spec(R) \\setminus V(I)$, \\item $I$ is the kernel of the map $R \\to (1 + I)^{-1}R$, \\item $R/I \\cong S^{-1}R$ as $R$-algebras for some multiplicative subset $S$ of $R$, and \\item $R/I \\cong (1 + I)^{-1}R$ as $R$-algebras. \\end{enumerate}"}
+{"_id": "961", "title": "algebra-lemma-pure-ideal-determined-by-zero-set", "text": "\\begin{slogan} Pure ideals are determined by their vanishing locus. \\end{slogan} Let $R$ be a ring. If $I, J \\subset R$ are pure ideals, then $V(I) = V(J)$ implies $I = J$."}
+{"_id": "962", "title": "algebra-lemma-pure-open-closed-specializations", "text": "Let $R$ be a ring. The rule $I \\mapsto V(I)$ determines a bijection $$ \\{I \\subset R \\text{ pure}\\} \\leftrightarrow \\{Z \\subset \\Spec(R)\\text{ closed and closed under generalizations}\\} $$"}
+{"_id": "963", "title": "algebra-lemma-finitely-generated-pure-ideal", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. The following are equivalent \\begin{enumerate} \\item $I$ is pure and finitely generated, \\item $I$ is generated by an idempotent, \\item $I$ is pure and $V(I)$ is open, and \\item $R/I$ is a projective $R$-module. \\end{enumerate}"}
+{"_id": "964", "title": "algebra-lemma-finite-flat-module-finitely-presented", "text": "Let $R$ be a ring. The following are equivalent: \\begin{enumerate} \\item every $Z \\subset \\Spec(R)$ which is closed and closed under generalizations is also open, and \\item any finite flat $R$-module is finite locally free. \\end{enumerate}"}
+{"_id": "965", "title": "algebra-lemma-Schanuel", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Suppose that $$ 0 \\to K \\xrightarrow{c_1} P_1 \\xrightarrow{p_1} M \\to 0 \\quad\\text{and}\\quad 0 \\to L \\xrightarrow{c_2} P_2 \\xrightarrow{p_2} M \\to 0 $$ are two short exact sequences, with $P_i$ projective. Then $K \\oplus P_2 \\cong L \\oplus P_1$. More precisely, there exist a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & K \\oplus P_2 \\ar[r]_{(c_1, \\text{id})} \\ar[d] & P_1 \\oplus P_2 \\ar[r]_{(p_1, 0)} \\ar[d] & M \\ar[r] \\ar@{=}[d] & 0 \\\\ 0 \\ar[r] & P_1 \\oplus L \\ar[r]^{(\\text{id}, c_2)} & P_1 \\oplus P_2 \\ar[r]^{(0, p_2)} & M \\ar[r] & 0 } $$ whose vertical arrows are isomorphisms."}
+{"_id": "966", "title": "algebra-lemma-independent-resolution", "text": "Let $R$ be a ring. Suppose that $M$ is an $R$-module of projective dimension $d$. Suppose that $F_e \\to F_{e-1} \\to \\ldots \\to F_0 \\to M \\to 0$ is exact with $F_i$ projective and $e \\geq d - 1$. Then the kernel of $F_e \\to F_{e-1}$ is projective (or the kernel of $F_0 \\to M$ is projective in case $e = 0$)."}
+{"_id": "967", "title": "algebra-lemma-what-kind-of-resolutions", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \\geq 0$. The following are equivalent \\begin{enumerate} \\item $M$ has projective dimension $\\leq d$, \\item there exists a resolution $0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$ with $P_i$ projective, \\item for some resolution $\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with $P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$ is projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if $d = 1$, or $M$ is projective if $d = 0$, \\item for any resolution $\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with $P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$ is projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if $d = 1$, or $M$ is projective if $d = 0$. \\end{enumerate}"}
+{"_id": "968", "title": "algebra-lemma-what-kind-of-resolutions-local", "text": "Let $R$ be a local ring. Let $M$ be an $R$-module. Let $d \\geq 0$. The equivalent conditions (1) -- (4) of Lemma \\ref{lemma-what-kind-of-resolutions} are also equivalent to \\begin{enumerate} \\item[(5)] there exists a resolution $0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$ with $P_i$ free. \\end{enumerate}"}
+{"_id": "969", "title": "algebra-lemma-what-kind-of-resolutions-Noetherian", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Let $d \\geq 0$. The equivalent conditions (1) -- (4) of Lemma \\ref{lemma-what-kind-of-resolutions} are also equivalent to \\begin{enumerate} \\item[(6)] there exists a resolution $0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$ with $P_i$ finite projective. \\end{enumerate}"}
+{"_id": "970", "title": "algebra-lemma-what-kind-of-resolutions-Noetherian-local", "text": "Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module. Let $d \\geq 0$. The equivalent conditions (1) -- (4) of Lemma \\ref{lemma-what-kind-of-resolutions}, condition (5) of Lemma \\ref{lemma-what-kind-of-resolutions-local}, and condition (6) of Lemma \\ref{lemma-what-kind-of-resolutions-Noetherian} are also equivalent to \\begin{enumerate} \\item[(7)] there exists a resolution $0 \\to F_d \\to F_{d - 1} \\to \\ldots \\to F_0 \\to M \\to 0$ with $F_i$ finite free. \\end{enumerate}"}
+{"_id": "971", "title": "algebra-lemma-projective-dimension-ext", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $n \\geq 0$. The following are equivalent \\begin{enumerate} \\item $M$ has projective dimension $\\leq n$, \\item $\\Ext^i_R(M, N) = 0$ for all $R$-modules $N$ and all $i \\geq n + 1$, and \\item $\\Ext^{n + 1}_R(M, N) = 0$ for all $R$-modules $N$. \\end{enumerate}"}
+{"_id": "972", "title": "algebra-lemma-exact-sequence-projective-dimension", "text": "Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of $R$-modules. \\begin{enumerate} \\item If $M$ has projective dimension $\\leq n$ and $M''$ has projective dimension $\\leq n + 1$, then $M'$ has projective dimension $\\leq n$. \\item If $M'$ and $M''$ have projective dimension $\\leq n$ then $M$ has projective dimension $\\leq n$. \\item If $M'$ has projective dimension $\\leq n$ and $M$ has projective dimension $\\leq n + 1$ then $M''$ has projective dimension $\\leq n + 1$. \\end{enumerate}"}
+{"_id": "973", "title": "algebra-lemma-colimit-projective-dimension", "text": "Let $R$ be a ring. Suppose we have a module $M = \\bigcup_{e \\in E} M_e$ where the $M_e$ are submodules well-ordered by inclusion. Assume the quotients $M_e/\\bigcup\\nolimits_{e' < e} M_{e'}$ have projective dimension $\\leq n$. Then $M$ has projective dimension $\\leq n$."}
+{"_id": "974", "title": "algebra-lemma-finite-gl-dim", "text": "Let $R$ be a ring. The following are equivalent \\begin{enumerate} \\item $R$ has finite global dimension $\\leq n$, \\item every finite $R$-module has projective dimension $\\leq n$, and \\item every cyclic $R$-module $R/I$ has projective dimension $\\leq n$. \\end{enumerate}"}
+{"_id": "975", "title": "algebra-lemma-localize-finite-gl-dim", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. \\begin{enumerate} \\item If $M$ has projective dimension $\\leq n$, then $S^{-1}M$ has projective dimension $\\leq n$ over $S^{-1}R$. \\item If $R$ has finite global dimension $\\leq n$, then $S^{-1}R$ has finite global dimension $\\leq n$. \\end{enumerate}"}
+{"_id": "976", "title": "algebra-lemma-finite-gl-dim-primes", "text": "Let $R$ be a Noetherian ring. Then $R$ has finite global dimension if and only if there exists an integer $n$ such that for all maximal ideals $\\mathfrak m$ of $R$ the ring $R_{\\mathfrak m}$ has global dimension $\\leq n$."}
+{"_id": "977", "title": "algebra-lemma-length-resolution-residue-field", "text": "Suppose that $R$ is a Noetherian local ring with maximal ideal $\\mathfrak m$ and residue field $\\kappa$. In this case the projective dimension of $\\kappa$ is $\\geq \\dim_\\kappa \\mathfrak m / \\mathfrak m^2$."}
+{"_id": "978", "title": "algebra-lemma-dim-gl-dim", "text": "Let $R$ be a Noetherian local ring. Suppose that the residue field $\\kappa$ has finite projective dimension $n$ over $R$. In this case $\\dim(R) \\geq n$."}
+{"_id": "979", "title": "algebra-lemma-localization-of-regular-local-is-regular", "text": "A Noetherian local ring $R$ is a regular local ring if and only if it has finite global dimension. In this case $R_{\\mathfrak p}$ is a regular local ring for all primes $\\mathfrak p$."}
+{"_id": "980", "title": "algebra-lemma-finite-gl-dim-finite-dim-regular", "text": "Let $R$ be a Noetherian ring. The following are equivalent: \\begin{enumerate} \\item $R$ has finite global dimension $n$, \\item there exists an integer $n$ such that all the localizations $R_{\\mathfrak m}$ at maximal ideals are regular of dimension $\\leq n$ with equality for at least one $\\mathfrak m$, and \\item there exists an integer $n$ such that all the localizations $R_{\\mathfrak p}$ at prime ideals are regular of dimension $\\leq n$ with equality for at least one $\\mathfrak p$. \\end{enumerate}"}
+{"_id": "981", "title": "algebra-lemma-flat-under-regular", "text": "Let $R \\to S$ be a local homomorphism of local Noetherian rings. Assume that $R \\to S$ is flat and that $S$ is regular. Then $R$ is regular."}
+{"_id": "982", "title": "algebra-lemma-dimension-going-up", "text": "Suppose $R \\to S$ is a ring map satisfying either going up, see Definition \\ref{definition-going-up-down}, or going down see Definition \\ref{definition-going-up-down}. Assume in addition that $\\Spec(S) \\to \\Spec(R)$ is surjective. Then $\\dim(R) \\leq \\dim(S)$."}
+{"_id": "983", "title": "algebra-lemma-going-up-maximal-on-top", "text": "Suppose that $R \\to S$ is a ring map with the going up property, see Definition \\ref{definition-going-up-down}. If $\\mathfrak q \\subset S$ is a maximal ideal. Then the inverse image of $\\mathfrak q$ in $R$ is a maximal ideal too."}
+{"_id": "984", "title": "algebra-lemma-integral-dim-up", "text": "Suppose that $R \\to S$ is a ring map such that $S$ is integral over $R$. Then $\\dim (R) \\geq \\dim(S)$, and every closed point of $\\Spec(S)$ maps to a closed point of $\\Spec(R)$."}
+{"_id": "985", "title": "algebra-lemma-integral-sub-dim-equal", "text": "Suppose $R \\subset S$ and $S$ integral over $R$. Then $\\dim(R) = \\dim(S)$."}
+{"_id": "986", "title": "algebra-lemma-dimension-base-fibre-total", "text": "Let $R \\to S$ be a homomorphism of Noetherian rings. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$. Then $$ \\dim(S_{\\mathfrak q}) \\leq \\dim(R_{\\mathfrak p}) + \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}). $$"}
+{"_id": "987", "title": "algebra-lemma-dimension-base-fibre-equals-total", "text": "Let $R \\to S$ be a homomorphism of Noetherian rings. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$. Assume the going down property holds for $R \\to S$ (for example if $R \\to S$ is flat, see Lemma \\ref{lemma-flat-going-down}). Then $$ \\dim(S_{\\mathfrak q}) = \\dim(R_{\\mathfrak p}) + \\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}). $$"}
+{"_id": "988", "title": "algebra-lemma-flat-over-regular-with-regular-fibre", "text": "Let $R \\to S$ be a local homomorphism of local Noetherian rings. Assume \\begin{enumerate} \\item $R$ is regular, \\item $S/\\mathfrak m_RS$ is regular, and \\item $R \\to S$ is flat. \\end{enumerate} Then $S$ is regular."}
+{"_id": "989", "title": "algebra-lemma-finite-flat-over-regular-CM", "text": "Let $R \\to S$ be a local homomorphism of Noetherian local rings. Assume $R$ Cohen-Macaulay. If $S$ is finite flat over $R$, or if $S$ is flat over $R$ and $\\dim(S) \\leq \\dim(R)$, then $S$ is Cohen-Macaulay and $\\dim(R) = \\dim(S)$."}
+{"_id": "990", "title": "algebra-lemma-dimension-formula", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak p$ of $R$. Assume that \\begin{enumerate} \\item $R$ is Noetherian, \\item $R \\to S$ is of finite type, \\item $R$, $S$ are domains, and \\item $R \\subset S$. \\end{enumerate} Then we have $$ \\text{height}(\\mathfrak q) \\leq \\text{height}(\\mathfrak p) + \\text{trdeg}_R(S) - \\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) $$ with equality if $R$ is universally catenary."}
+{"_id": "991", "title": "algebra-lemma-finite-in-codim-1", "text": "Let $A \\to B$ be a ring map. Assume \\begin{enumerate} \\item $A \\subset B$ is an extension of domains, \\item the induced extension of fraction fields is finite, \\item $A$ is Noetherian, and \\item $A \\to B$ is of finite type. \\end{enumerate} Let $\\mathfrak p \\subset A$ be a prime of height $1$. Then there are at most finitely many primes of $B$ lying over $\\mathfrak p$ and they all have height $1$."}
+{"_id": "992", "title": "algebra-lemma-dim-affine-space", "text": "Let $\\mathfrak m$ be a maximal ideal in $k[x_1, \\ldots, x_n]$. The ideal $\\mathfrak m$ is generated by $n$ elements. The dimension of $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is $n$. Hence $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is a regular local ring of dimension $n$."}
+{"_id": "993", "title": "algebra-lemma-dimension-height-polynomial-ring", "text": "Let $k$ be a field. Let $\\mathfrak p \\subset \\mathfrak q \\subset k[x_1, \\ldots, x_n]$ be a pair of primes. Any maximal chain of primes between $\\mathfrak p$ and $\\mathfrak q$ has length $\\text{height}(\\mathfrak q) - \\text{height}(\\mathfrak p)$."}
+{"_id": "994", "title": "algebra-lemma-dimension-spell-it-out", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra which is an integral domain. Then $\\dim(S) = \\dim(S_{\\mathfrak m})$ for any maximal ideal $\\mathfrak m$ of $S$. In words: every maximal chain of primes has length equal to the dimension of $S$."}
+{"_id": "995", "title": "algebra-lemma-dimension-at-a-point-finite-type-over-field", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $X = \\Spec(S)$. Let $\\mathfrak p \\subset S$ be a prime ideal and let $x \\in X$ be the corresponding point. The following numbers are equal \\begin{enumerate} \\item $\\dim_x(X)$, \\item $\\max \\dim(Z)$ where the maximum is over those irreducible components $Z$ of $X$ passing through $x$, and \\item $\\min \\dim(S_{\\mathfrak m})$ where the minimum is over maximal ideals $\\mathfrak m$ with $\\mathfrak p \\subset \\mathfrak m$. \\end{enumerate}"}
+{"_id": "996", "title": "algebra-lemma-dimension-closed-point-finite-type-field", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $X = \\Spec(S)$. Let $\\mathfrak m \\subset S$ be a maximal ideal and let $x \\in X$ be the associated closed point. Then $\\dim_x(X) = \\dim(S_{\\mathfrak m})$."}
+{"_id": "997", "title": "algebra-lemma-disjoint-decomposition-CM-algebra", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra. Assume that $S$ is Cohen-Macaulay. Then $\\Spec(S) = \\coprod T_d$ is a finite disjoint union of open and closed subsets $T_d$ with $T_d$ equidimensional (see Topology, Definition \\ref{topology-definition-equidimensional}) of dimension $d$. Equivalently, $S$ is a product of rings $S_d$, $d = 0, \\ldots, \\dim(S)$ such that every maximal ideal $\\mathfrak m$ of $S_d$ has height $d$."}
+{"_id": "998", "title": "algebra-lemma-helper", "text": "Let $n \\in \\mathbf{N}$. Let $N$ be a finite nonempty set of multi-indices $\\nu = (\\nu_1, \\ldots, \\nu_n)$. Given $e = (e_1, \\ldots, e_n)$ we set $e \\cdot \\nu = \\sum e_i\\nu_i$. Then for $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n$ we have: If $\\nu, \\nu' \\in N$ then $$ (e \\cdot \\nu = e \\cdot \\nu') \\Leftrightarrow (\\nu = \\nu') $$"}
+{"_id": "999", "title": "algebra-lemma-helper-polynomial", "text": "Let $R$ be a ring. Let $g \\in R[x_1, \\ldots, x_n]$ be an element which is nonconstant, i.e., $g \\not \\in R$. For $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n = 1$ the polynomial $$ g(x_1 + x_n^{e_1}, x_2 + x_n^{e_2}, \\ldots, x_{n - 1} + x_n^{e_{n - 1}}, x_n) = ax_n^d + \\text{lower order terms in }x_n $$ where $d > 0$ and $a \\in R$ is one of the nonzero coefficients of $g$."}
+{"_id": "1000", "title": "algebra-lemma-one-relation", "text": "Let $k$ be a field. Let $S = k[x_1, \\ldots, x_n]/I$ for some proper ideal $I$. If $I \\not = 0$, then there exist $y_1, \\ldots, y_{n-1} \\in k[x_1, \\ldots, x_n]$ such that $S$ is finite over $k[y_1, \\ldots, y_{n-1}]$. Moreover we may choose $y_i$ to be in the $\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$ generated by $x_1, \\ldots, x_n$."}
+{"_id": "1001", "title": "algebra-lemma-Noether-normalization", "text": "Let $k$ be a field. Let $S = k[x_1, \\ldots, x_n]/I$ for some ideal $I$. If $I \\neq (1)$, there exist $r\\geq 0$, and $y_1, \\ldots, y_r \\in k[x_1, \\ldots, x_n]$ such that (a) the map $k[y_1, \\ldots, y_r] \\to S$ is injective, and (b) the map $k[y_1, \\ldots, y_r] \\to S$ is finite. In this case the integer $r$ is the dimension of $S$. Moreover we may choose $y_i$ to be in the $\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$ generated by $x_1, \\ldots, x_n$."}
+{"_id": "1002", "title": "algebra-lemma-Noether-normalization-at-point", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra and denote $X = \\Spec(S)$. Let $\\mathfrak q$ be a prime of $S$, and let $x \\in X$ be the corresponding point. There exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $\\dim(S_g) = \\dim_x(X) =: d$ and such that there exists a finite injective map $k[y_1, \\ldots, y_d] \\to S_g$."}
+{"_id": "1003", "title": "algebra-lemma-refined-Noether-normalization", "text": "Let $k$ be a field. Let $\\mathfrak q \\subset k[x_1, \\ldots, x_n]$ be a prime ideal. Set $r = \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$. Then there exists a finite ring map $\\varphi : k[y_1, \\ldots, y_n] \\to k[x_1, \\ldots, x_n]$ such that $\\varphi^{-1}(\\mathfrak q) = (y_{r + 1}, \\ldots, y_n)$."}
+{"_id": "1004", "title": "algebra-lemma-Noether-normalization-over-a-domain", "text": "Let $R \\to S$ be an injective finite type ring map. Assume $R$ is a domain. Then there exists an integer $d$ and a factorization $$ R \\to R[y_1, \\ldots, y_d] \\to S' \\to S $$ by injective maps such that $S'$ is finite over $R[y_1, \\ldots, y_d]$ and such that $S'_f \\cong S_f$ for some nonzero $f \\in R$."}
+{"_id": "1005", "title": "algebra-lemma-dimension-prime-polynomial-ring", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra which is an integral domain. Let $K$ be the field of fractions of $S$. Let $r = \\text{trdeg}(K/k)$ be the transcendence degree of $K$ over $k$. Then $\\dim(S) = r$. Moreover, the local ring of $S$ at every maximal ideal has dimension $r$."}
+{"_id": "1006", "title": "algebra-lemma-tr-deg-specialization", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak q \\subset \\mathfrak q' \\subset S$ be distinct prime ideals. Then $\\text{trdeg}_k\\ \\kappa(\\mathfrak q') < \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$."}
+{"_id": "1007", "title": "algebra-lemma-dimension-at-a-point-finite-type-field", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra. Let $X = \\Spec(S)$. Let $\\mathfrak p \\subset S$ be a prime ideal, and let $x \\in X$ be the corresponding point. Then we have $$ \\dim_x(X) = \\dim(S_{\\mathfrak p}) + \\text{trdeg}_k\\ \\kappa(\\mathfrak p). $$"}
+{"_id": "1008", "title": "algebra-lemma-codimension", "text": "Let $k$ be a field. Let $S' \\to S$ be a surjection of finite type $k$ algebras. Let $\\mathfrak p \\subset S$ be a prime ideal, and let $\\mathfrak p'$ be the corresponding prime ideal of $S'$. Let $X = \\Spec(S)$, resp.\\ $X' = \\Spec(S')$, and let $x \\in X$, resp. $x'\\in X'$ be the point corresponding to $\\mathfrak p$, resp.\\ $\\mathfrak p'$. Then $$ \\dim_{x'} X' - \\dim_x X = \\text{height}(\\mathfrak p') - \\text{height}(\\mathfrak p). $$"}
+{"_id": "1009", "title": "algebra-lemma-dimension-preserved-field-extension", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $k \\subset K$ be a field extension. Then $\\dim(S) = \\dim(K \\otimes_k S)$."}
+{"_id": "1010", "title": "algebra-lemma-dimension-at-a-point-preserved-field-extension", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Set $X = \\Spec(S)$. Let $k \\subset K$ be a field extension. Set $S_K = K \\otimes_k S$, and $X_K = \\Spec(S_K)$. Let $\\mathfrak q \\subset S$ be a prime corresponding to $x \\in X$ and let $\\mathfrak q_K \\subset S_K$ be a prime corresponding to $x_K \\in X_K$ lying over $\\mathfrak q$. Then $\\dim_x X = \\dim_{x_K} X_K$."}
+{"_id": "1011", "title": "algebra-lemma-inequalities-under-field-extension", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $k \\subset K$ be a field extension. Set $S_K = K \\otimes_k S$. Let $\\mathfrak q \\subset S$ be a prime and let $\\mathfrak q_K \\subset S_K$ be a prime lying over $\\mathfrak q$. Then $$ \\dim (S_K \\otimes_S \\kappa(\\mathfrak q))_{\\mathfrak q_K} = \\dim (S_K)_{\\mathfrak q_K} - \\dim S_\\mathfrak q = \\text{trdeg}_k \\kappa(\\mathfrak q) - \\text{trdeg}_K \\kappa(\\mathfrak q_K) $$ Moreover, given $\\mathfrak q$ we can always choose $\\mathfrak q_K$ such that the number above is zero."}
+{"_id": "1013", "title": "algebra-lemma-generic-flatness-Noetherian", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $R$ is a domain, \\item $R \\to S$ is of finite type, and \\item $M$ is a finite type $S$-module. \\end{enumerate} Then there exists a nonzero $f \\in R$ such that $M_f$ is a free $R_f$-module."}
+{"_id": "1014", "title": "algebra-lemma-generic-flatness-finitely-presented", "text": "\\begin{slogan} Generic freeness. \\end{slogan} Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R$ is a domain, \\item $R \\to S$ is of finite presentation, and \\item $M$ is an $S$-module of finite presentation. \\end{enumerate} Then there exists a nonzero $f \\in R$ such that $M_f$ is a free $R_f$-module."}
+{"_id": "1015", "title": "algebra-lemma-generic-flatness", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R$ is a domain, \\item $R \\to S$ is of finite type, and \\item $M$ is a finite type $S$-module. \\end{enumerate} Then there exists a nonzero $f \\in R$ such that \\begin{enumerate} \\item[(a)] $M_f$ and $S_f$ are free as $R_f$-modules, and \\item[(b)] $S_f$ is a finitely presented $R_f$-algebra and $M_f$ is a finitely presented $S_f$-module. \\end{enumerate}"}
+{"_id": "1016", "title": "algebra-lemma-generic-flatness-locus-extension", "text": "Let $R \\to S$ be a ring map. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence of $S$-modules. Then $$ U(R \\to S, M_1) \\cap U(R \\to S, M_3) \\subset U(R \\to S, M_2). $$"}
+{"_id": "1017", "title": "algebra-lemma-generic-flatness-locus-localize", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Let $f \\in R$. Using the identification $\\Spec(R_f) = D(f)$ we have $U(R_f \\to S_f, M_f) = D(f) \\cap U(R \\to S, M)$."}
+{"_id": "1018", "title": "algebra-lemma-generic-flatness-locus-reduce", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Let $U \\subset \\Spec(R)$ be a dense open. Assume there is a covering $U = \\bigcup_{i \\in I} D(f_i)$ of opens such that $U(R_{f_i} \\to S_{f_i}, M_{f_i})$ is dense in $D(f_i)$ for each $i \\in I$. Then $U(R \\to S, M)$ is dense in $\\Spec(R)$."}
+{"_id": "1019", "title": "algebra-lemma-generic-flatness-reduced", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R \\to S$ is of finite type, \\item $M$ is a finite $S$-module, and \\item $R$ is reduced. \\end{enumerate} Then there exists a subset $U \\subset \\Spec(R)$ such that \\begin{enumerate} \\item $U$ is open and dense in $\\Spec(R)$, \\item for every $u \\in U$ there exists an $f \\in R$ such that $u \\in D(f) \\subset U$ and such that we have \\begin{enumerate} \\item $M_f$ and $S_f$ are free over $R_f$, \\item $S_f$ is a finitely presented $R_f$-algebra, and \\item $M_f$ is a finitely presented $S_f$-module. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1020", "title": "algebra-lemma-dominate-by-dimension-1", "text": "Let $R$ be a local Noetherian domain with fraction field $K$. Assume $R$ is not a field. Then there exist $R \\subset R' \\subset K$ with \\begin{enumerate} \\item $R'$ local Noetherian of dimension $1$, \\item $R \\to R'$ a local ring map, i.e., $R'$ dominates $R$, and \\item $R \\to R'$ essentially of finite type. \\end{enumerate}"}
+{"_id": "1021", "title": "algebra-lemma-hart-serre-loc-thm", "text": "\\begin{reference} This is taken from a forthcoming paper by J\\'anos Koll\\'ar entitled ``Variants of normality for Noetherian schemes''. \\end{reference} Let $(R, \\mathfrak m)$ be a local Noetherian ring. Then exactly one of the following holds: \\begin{enumerate} \\item $(R, \\mathfrak m)$ is Artinian, \\item $(R, \\mathfrak m)$ is regular of dimension $1$, \\item $\\text{depth}(R) \\geq 2$, or \\item there exists a finite ring map $R \\to R'$ which is not an isomorphism whose kernel and cokernel are annihilated by a power of $\\mathfrak m$ such that $\\mathfrak m$ is not an associated prime of $R'$. \\end{enumerate}"}
+{"_id": "1022", "title": "algebra-lemma-nonregular-dimension-one", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$. Assume $R$ is Noetherian, has dimension $1$, and that $\\dim(\\mathfrak m/\\mathfrak m^2) > 1$. Then there exists a ring map $R \\to R'$ such that \\begin{enumerate} \\item $R \\to R'$ is finite, \\item $R \\to R'$ is not an isomorphism, \\item the kernel and cokernel of $R \\to R'$ are annihilated by a power of $\\mathfrak m$, and \\item $\\mathfrak m$ is not an associated prime of $R'$. \\end{enumerate}"}
+{"_id": "1023", "title": "algebra-lemma-characterize-dvr", "text": "Let $A$ be a ring. The following are equivalent. \\begin{enumerate} \\item The ring $A$ is a discrete valuation ring. \\item The ring $A$ is a valuation ring and Noetherian but not a field. \\item The ring $A$ is a regular local ring of dimension $1$. \\item The ring $A$ is a Noetherian local domain with maximal ideal $\\mathfrak m$ generated by a single nonzero element. \\item The ring $A$ is a Noetherian local normal domain of dimension $1$. \\end{enumerate} In this case if $\\pi$ is a generator of the maximal ideal of $A$, then every element of $A$ can be uniquely written as $u\\pi^n$, where $u \\in A$ is a unit."}
+{"_id": "1024", "title": "algebra-lemma-finite-length", "text": "Let $R$ be a domain with fraction field $K$. Let $M$ be an $R$-submodule of $K^{\\oplus r}$. Assume $R$ is local Noetherian of dimension $1$. For any nonzero $x \\in R$ we have $\\text{length}_R(R/xR) < \\infty$ and $$ \\text{length}_R(M/xM) \\leq r \\cdot \\text{length}_R(R/xR). $$"}
+{"_id": "1025", "title": "algebra-lemma-finite-extension-residue-fields-dimension-1", "text": "Let $R \\to S$ be a homomorphism of domains inducing an injection of fraction fields $K \\subset L$. If $R$ is Noetherian local of dimension $1$ and $[L : K] < \\infty$ then \\begin{enumerate} \\item each prime ideal $\\mathfrak n_i$ of $S$ lying over the maximal ideal $\\mathfrak m$ of $R$ is maximal, \\item there are finitely many of these, and \\item $[\\kappa(\\mathfrak n_i) : \\kappa(\\mathfrak m)] < \\infty$ for each $i$. \\end{enumerate}"}
+{"_id": "1026", "title": "algebra-lemma-finite-length-global", "text": "Let $R$ be a domain with fraction field $K$. Let $M$ be an $R$-submodule of $K^{\\oplus r}$. Assume $R$ is Noetherian of dimension $1$. For any nonzero $x \\in R$ we have $\\text{length}_R(M/xM) < \\infty$."}
+{"_id": "1027", "title": "algebra-lemma-krull-akizuki", "text": "Let $R$ be a domain with fraction field $K$. Let $K \\subset L$ be a finite extension of fields. Assume $R$ is Noetherian and $\\dim(R) = 1$. In this case any ring $A$ with $R \\subset A \\subset L$ is Noetherian."}
+{"_id": "1028", "title": "algebra-lemma-exists-dvr", "text": "Let $R$ be a Noetherian local domain with fraction field $K$. Assume that $R$ is not a field. Let $K \\subset L$ be a finitely generated field extension. Then there exists discrete valuation ring $A$ with fraction field $L$ which dominates $R$."}
+{"_id": "1029", "title": "algebra-lemma-easy-divisibility", "text": "Let $R$ be a domain. Let $x, y \\in R$. Then $x$, $y$ are associates if and only if $(x) = (y)$."}
+{"_id": "1030", "title": "algebra-lemma-factorization-exists", "text": "Let $R$ be a domain. Consider the following conditions: \\begin{enumerate} \\item The ring $R$ satisfies the ascending chain condition for principal ideals. \\item Every nonzero, nonunit element $a \\in R$ has a factorization $a = b_1 \\ldots b_k$ with each $b_i$ an irreducible element of $R$. \\end{enumerate} Then (1) implies (2)."}
+{"_id": "1031", "title": "algebra-lemma-characterize-UFD", "text": "Let $R$ be a domain. Assume every nonzero, nonunit factors into irreducibles. Then $R$ is a UFD if and only if every irreducible element is prime."}
+{"_id": "1032", "title": "algebra-lemma-characterize-UFD-height-1", "text": "Let $R$ be a Noetherian domain. Then $R$ is a UFD if and only if every height $1$ prime ideal is principal."}
+{"_id": "1033", "title": "algebra-lemma-invert-prime-elements", "text": "\\begin{reference} \\cite[Lemma 2]{Nagata-UFD} \\end{reference} Let $A$ be a domain. Let $S \\subset A$ be a multiplicative subset generated by prime elements. Let $x \\in A$ be irreducible. Then \\begin{enumerate} \\item the image of $x$ in $S^{-1}A$ is irreducible or a unit, and \\item $x$ is prime if and only if the image of $x$ in $S^{-1}A$ is a unit or a prime element in $S^{-1}A$. \\end{enumerate} Moreover, then $A$ is a UFD if and only if every element of $A$ has a factorization into irreducibles and $S^{-1}A$ is a UFD."}
+{"_id": "1034", "title": "algebra-lemma-UFD-ascending-chain-condition-principal-ideals", "text": "A UFD satisfies the ascending chain condition for principal ideals."}
+{"_id": "1035", "title": "algebra-lemma-factoring-in-polynomial", "text": "Let $R$ be a domain. Assume $R$ has the ascending chain condition for principal ideals. Then the same property holds for a polynomial ring over $R$."}
+{"_id": "1036", "title": "algebra-lemma-polynomial-ring-UFD", "text": "A polynomial ring over a UFD is a UFD. In particular, if $k$ is a field, then $k[x_1, \\ldots, x_n]$ is a UFD."}
+{"_id": "1037", "title": "algebra-lemma-UFD-normal", "text": "A unique factorization domain is normal."}
+{"_id": "1038", "title": "algebra-lemma-PID-UFD", "text": "A principal ideal domain is a unique factorization domain."}
+{"_id": "1039", "title": "algebra-lemma-PID-dedekind", "text": "A PID is a Dedekind domain."}
+{"_id": "1040", "title": "algebra-lemma-product-ideals-principal", "text": "\\begin{slogan} A product of ideals is an invertible module iff both factors are. \\end{slogan} Let $A$ be a ring. Let $I$ and $J$ be nonzero ideals of $A$ such that $IJ = (f)$ for some nonzerodivisor $f \\in A$. Then $I$ and $J$ are finitely generated ideals and finitely locally free of rank $1$ as $A$-modules."}
+{"_id": "1041", "title": "algebra-lemma-characterize-Dedekind", "text": "Let $R$ be a ring. The following are equivalent \\begin{enumerate} \\item $R$ is a Dedekind domain, \\item $R$ is a Noetherian domain, and for every maximal ideal $\\mathfrak m$ the local ring $R_{\\mathfrak m}$ is a discrete valuation ring, and \\item $R$ is a Noetherian, normal domain, and $\\dim(R) \\leq 1$. \\end{enumerate}"}
+{"_id": "1042", "title": "algebra-lemma-integral-closure-Dedekind", "text": "Let $A$ be a Noetherian domain of dimension $1$ with fraction field $K$. Let $K \\subset L$ be a finite extension. Let $B$ be the integral closure of $A$ in $L$. Then $B$ is a Dedekind domain and $\\Spec(B) \\to \\Spec(A)$ is surjective, has finite fibres, and induces finite residue field extensions."}
+{"_id": "1043", "title": "algebra-lemma-ord-additive", "text": "Let $R$ be a semi-local Noetherian ring of dimension $1$. If $a, b \\in R$ are nonzerodivisors then $$ \\text{length}_R(R/(ab)) = \\text{length}_R(R/(a)) + \\text{length}_R(R/(b)) $$ and these lengths are finite."}
+{"_id": "1044", "title": "algebra-lemma-compare-lattices", "text": "Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. \\begin{enumerate} \\item If $M$ is a lattice in $V$ and $M \\subset M' \\subset V$ is an $R$-submodule of $V$ containing $M$ then the following are equivalent \\begin{enumerate} \\item $M'$ is a lattice, \\item $\\text{length}_R(M'/M)$ is finite, and \\item $M'$ is finitely generated. \\end{enumerate} \\item If $M$ is a lattice in $V$ and $M' \\subset M$ is an $R$-submodule of $M$ then $M'$ is a lattice if and only if $\\text{length}_R(M/M')$ is finite. \\item If $M$, $M'$ are lattices in $V$, then so are $M \\cap M'$ and $M + M'$. \\item If $M \\subset M' \\subset M'' \\subset V$ are lattices in $V$ then $$ \\text{length}_R(M''/M) = \\text{length}_R(M'/M) + \\text{length}_R(M''/M'). $$ \\item If $M$, $M'$, $N$, $N'$ are lattices in $V$ and $N \\subset M \\cap M'$, $M + M' \\subset N'$, then we have \\begin{eqnarray*} & & \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M')\\\\ & = & \\text{length}_R(M/N) - \\text{length}_R(M'/N) \\\\ & = & \\text{length}_R(M + M' / M') - \\text{length}_R(M + M'/M) \\\\ & = & \\text{length}_R(N' / M') - \\text{length}_R(N'/M) \\end{eqnarray*} \\end{enumerate}"}
+{"_id": "1045", "title": "algebra-lemma-properties-distance-function", "text": "Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. This distance function has the property that $$ d(M, M'') = d(M, M') + d(M', M'') $$ whenever given three lattices $M$, $M'$, $M''$ of $V$. In particular we have $d(M, M') = - d(M', M)$."}
+{"_id": "1046", "title": "algebra-lemma-order-vanishing-determinant", "text": "Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. Let $\\varphi : V \\to V$ be a $K$-linear isomorphism. For any lattice $M \\subset V$ we have $$ d(M, \\varphi(M)) = \\text{ord}_R(\\det(\\varphi)) $$"}
+{"_id": "1047", "title": "algebra-lemma-finite-extension-dim-1", "text": "Let $A \\to B$ be a ring map. Assume \\begin{enumerate} \\item $A$ is a Noetherian local domain of dimension $1$, \\item $A \\subset B$ is a finite extension of domains. \\end{enumerate} Let $L/K$ be the corresponding finite extension of fraction fields. Let $y \\in L^*$ and $x = \\text{Nm}_{L/K}(y)$. In this situation $B$ is semi-local. Let $\\mathfrak m_i$, $i = 1, \\ldots, n$ be the maximal ideals of $B$. Then $$ \\text{ord}_A(x) = \\sum\\nolimits_i [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)] \\text{ord}_{B_{\\mathfrak m_i}}(y) $$ where $\\text{ord}$ is defined as in Definition \\ref{definition-ord}."}
+{"_id": "1048", "title": "algebra-lemma-isolated-point", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra. Let $\\mathfrak q$ be a prime of $S$. The following are equivalent: \\begin{enumerate} \\item $\\mathfrak q$ is an isolated point of $\\Spec(S)$, \\item $S_{\\mathfrak q}$ is finite over $k$, \\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$ such that $D(g) = \\{ \\mathfrak q \\}$, \\item $\\dim_{\\mathfrak q} \\Spec(S) = 0$, \\item $\\mathfrak q$ is a closed point of $\\Spec(S)$ and $\\dim(S_{\\mathfrak q}) = 0$, and \\item the field extension $k \\subset \\kappa(\\mathfrak q)$ is finite and $\\dim(S_{\\mathfrak q}) = 0$. \\end{enumerate} In this case $S = S_{\\mathfrak q} \\times S'$ for some finite type $k$-algebra $S'$. Also, the element $g$ as in (3) has the property $S_{\\mathfrak q} = S_g$."}
+{"_id": "1049", "title": "algebra-lemma-isolated-point-fibre", "text": "\\begin{slogan} Equivalent conditions for isolated points in fibres \\end{slogan} Let $R \\to S$ be a ring map of finite type. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. Let $F = \\Spec(S \\otimes_R \\kappa(\\mathfrak p))$ be the fibre of $\\Spec(S) \\to \\Spec(R)$, see Remark \\ref{remark-fundamental-diagram}. Denote $\\overline{\\mathfrak q} \\in F$ the point corresponding to $\\mathfrak q$. The following are equivalent \\begin{enumerate} \\item $\\overline{\\mathfrak q}$ is an isolated point of $F$, \\item $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is finite over $\\kappa(\\mathfrak p)$, \\item there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that the only prime of $D(g)$ mapping to $\\mathfrak p$ is $\\mathfrak q$, \\item $\\dim_{\\overline{\\mathfrak q}}(F) = 0$, \\item $\\overline{\\mathfrak q}$ is a closed point of $F$ and $\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$, and \\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite and $\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$. \\end{enumerate}"}
+{"_id": "1050", "title": "algebra-lemma-quasi-finite", "text": "Let $R \\to S$ be a finite type ring map. Then $R \\to S$ is quasi-finite if and only if for all primes $\\mathfrak p \\subset R$ the fibre $S \\otimes_R \\kappa(\\mathfrak p)$ is finite over $\\kappa(\\mathfrak p)$."}
+{"_id": "1051", "title": "algebra-lemma-quasi-finite-local", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. Let $f \\in R$, $f \\not \\in \\mathfrak p$ and $g \\in S$, $g \\not \\in \\mathfrak q$. Then $R \\to S$ is quasi-finite at $\\mathfrak q$ if and only if $R_f \\to S_{fg}$ is quasi-finite at $\\mathfrak qS_{fg}$."}
+{"_id": "1052", "title": "algebra-lemma-four-rings", "text": "Let $$ \\xymatrix{ S \\ar[r] & S' & & \\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\ R \\ar[u] \\ar[r] &  R' \\ar[u] & & \\mathfrak p \\ar@{-}[r] \\ar@{-}[u] & \\mathfrak p' \\ar@{-}[u] } $$ be a commutative diagram of rings with primes as indicated. Assume $R \\to S$ of finite type, and $S \\otimes_R R' \\to S'$ surjective. If $R \\to S$ is quasi-finite at $\\mathfrak q$, then $R' \\to S'$ is quasi-finite at $\\mathfrak q'$."}
+{"_id": "1053", "title": "algebra-lemma-quasi-finite-composition", "text": "A composition of quasi-finite ring maps is quasi-finite."}
+{"_id": "1054", "title": "algebra-lemma-quasi-finite-base-change", "text": "Let $R \\to S$ be a ring map of finite type. Let $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$. \\begin{enumerate} \\item The set $\\{\\mathfrak q' \\mid R' \\to S' \\text{ quasi-finite at }\\mathfrak q'\\}$ is the inverse image of the corresponding set of $\\Spec(S)$ under the canonical map $\\Spec(S') \\to \\Spec(S)$. \\item If $\\Spec(R') \\to \\Spec(R)$ is surjective, then $R \\to S$ is quasi-finite if and only if $R' \\to S'$ is quasi-finite. \\item Any base change of a quasi-finite ring map is quasi-finite. \\end{enumerate}"}
+{"_id": "1055", "title": "algebra-lemma-quasi-finite-permanence", "text": "Let $A \\to B$ and $B \\to C$ be ring homomorphisms such that $A \\to C$ is of finite type. Let $\\mathfrak r$ be a prime of $C$ lying over $\\mathfrak q \\subset B$ and $\\mathfrak p \\subset A$. If $A \\to C$ is quasi-finite at $\\mathfrak r$, then $B \\to C$ is quasi-finite at $\\mathfrak r$."}
+{"_id": "1056", "title": "algebra-lemma-generically-finite", "text": "Let $R \\to S$ be a ring map of finite type. Let $\\mathfrak p \\subset R$ be a minimal prime. Assume that there are at most finitely many primes of $S$ lying over $\\mathfrak p$. Then there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ such that the ring map $R_g \\to S_g$ is finite."}
+{"_id": "1057", "title": "algebra-lemma-make-integral-trivial", "text": "Let $\\varphi : R \\to S$ be a ring map. Suppose $t \\in S$ satisfies the relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$. Then $\\varphi(a_n)t$ is integral over $R$."}
+{"_id": "1058", "title": "algebra-lemma-make-integral-trick", "text": "Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be a ring map. Let $t \\in S$. Assume that (a) $t$ is integral over $R[x]$, and (b) there exists a monic $p \\in R[x]$ such that $t \\varphi(p) \\in \\Im(\\varphi)$. Then there exists a $q \\in R[x]$ such that $t - \\varphi(q)$ is integral over $R$."}
+{"_id": "1059", "title": "algebra-lemma-combine-lemmas", "text": "Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be a ring map. Let $t \\in S$. Assume $t$ is integral over $R[x]$. Let $p \\in R[x]$, $p = a_0 + a_1x + \\ldots + a_k x^k$ such that $t \\varphi(p) \\in \\Im(\\varphi)$. Then there exists a $q \\in R[x]$ and $n \\geq 0$ such that $\\varphi(a_k)^n t - \\varphi(q)$ is integral over $R$."}
+{"_id": "1060", "title": "algebra-lemma-leading-coefficient-in-J", "text": "In Situation \\ref{situation-one-transcendental-element}. Suppose $u \\in S$, $a_0, \\ldots, a_k \\in R$, $u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in J$. Then there exists an $m \\geq 0$ such that $u \\varphi(a_k)^m \\in J$."}
+{"_id": "1061", "title": "algebra-lemma-all-coefficients-in-J", "text": "In Situation \\ref{situation-one-transcendental-element}. Suppose $u \\in S$, $a_0, \\ldots, a_k \\in R$, $u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in \\sqrt{J}$. Then $u \\varphi(a_i) \\in \\sqrt{J}$ for all $i$."}
+{"_id": "1062", "title": "algebra-lemma-reduced-strongly-transcendental-minimal-prime", "text": "Suppose $R \\subset S$ is an inclusion of reduced rings and suppose that $x \\in S$ is strongly transcendental over $R$. Let $\\mathfrak q \\subset S$ be a minimal prime and let $\\mathfrak p = R \\cap \\mathfrak q$. Then the image of $x$ in $S/\\mathfrak q$ is strongly transcendental over the subring $R/\\mathfrak p$."}
+{"_id": "1063", "title": "algebra-lemma-domains-transcendental-not-quasi-finite", "text": "Suppose $R\\subset S$ is an inclusion of domains and let $x \\in S$. Assume $x$ is (strongly) transcendental over $R$ and that $S$ is finite over $R[x]$. Then $R \\to S$ is not quasi-finite at any prime of $S$."}
+{"_id": "1064", "title": "algebra-lemma-reduced-strongly-transcendental-not-quasi-finite", "text": "Suppose $R \\subset S$ is an inclusion of reduced rings. Assume $x \\in S$ be strongly transcendental over $R$, and $S$ finite over $R[x]$. Then $R \\to S$ is not quasi-finite at any prime of $S$."}
+{"_id": "1065", "title": "algebra-lemma-quasi-finite-monogenic", "text": "Let $R$ be a ring. Let $S = R[x]/I$. Let $\\mathfrak q \\subset S$ be a prime. Assume $R \\to S$ is quasi-finite at $\\mathfrak q$. Let $S' \\subset S$ be the integral closure of $R$ in $S$. Then there exists an element $g \\in S'$, $g \\not\\in \\mathfrak q$ such that $S'_g \\cong S_g$."}
+{"_id": "1066", "title": "algebra-lemma-quasi-finite-open", "text": "Let $R \\to S$ be a finite type ring map. The set of points $\\mathfrak q$ of $\\Spec(S)$ at which $S/R$ is quasi-finite is open in $\\Spec(S)$."}
+{"_id": "1067", "title": "algebra-lemma-quasi-finite-open-integral-closure", "text": "Let $R \\to S$ be a finite type ring map. Suppose that $S$ is quasi-finite over $R$. Let $S' \\subset S$ be the integral closure of $R$ in $S$. Then \\begin{enumerate} \\item $\\Spec(S) \\to \\Spec(S')$ is a homeomorphism onto an open subset, \\item if $g \\in S'$ and $D(g)$ is contained in the image of the map, then $S'_g \\cong S_g$, and \\item there exists a finite $R$-algebra $S'' \\subset S'$ such that (1) and (2) hold for the ring map $S'' \\to S$. \\end{enumerate}"}
+{"_id": "1068", "title": "algebra-lemma-quasi-finite-extension-dim-1", "text": "Let $A \\subset B$ be an extension of domains. Assume \\begin{enumerate} \\item $A$ is a local Noetherian ring of dimension $1$, \\item $A \\to B$ is of finite type, and \\item the induced extension $L/K$ of fraction fields is finite. \\end{enumerate} Then $B$ is semi-local. Let $x \\in \\mathfrak m_A$, $x \\not = 0$. Let $\\mathfrak m_i$, $i = 1, \\ldots, n$ be the maximal ideals of $B$. Then $$ [L : K]\\text{ord}_A(x) \\geq \\sum\\nolimits_i [\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)] \\text{ord}_{B_{\\mathfrak m_i}}(x) $$ where $\\text{ord}$ is defined as in Definition \\ref{definition-ord}. We have equality if and only if $A \\to B$ is finite."}
+{"_id": "1069", "title": "algebra-lemma-essentially-finite-type-fibre-dim-zero", "text": "Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism of local rings. Assume \\begin{enumerate} \\item $R \\to S$ is essentially of finite type, \\item $\\kappa(\\mathfrak m_R) \\subset \\kappa(\\mathfrak m_S)$ is finite, and \\item $\\dim(S/\\mathfrak m_RS) = 0$. \\end{enumerate} Then $S$ is the localization of a finite $R$-algebra."}
+{"_id": "1070", "title": "algebra-lemma-completion-at-quasi-finite-prime", "text": "Let $R \\to S$ be a ring map, $\\mathfrak q$ a prime of $S$ lying over $\\mathfrak p$ in $R$. If \\begin{enumerate} \\item $R$ is Noetherian, \\item $R \\to S$ is of finite type, and \\item $R \\to S$ is quasi-finite at $\\mathfrak q$, \\end{enumerate} then $R_\\mathfrak p^\\wedge \\otimes_R S = S_\\mathfrak q^\\wedge \\times B$ for some $R_\\mathfrak p^\\wedge$-algebra $B$."}
+{"_id": "1071", "title": "algebra-lemma-quasi-finite-over-polynomial-algebra", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$ be a prime. Suppose that $\\dim_{\\mathfrak q}(S/R) = n$. There exists a $g \\in S$, $g \\not\\in \\mathfrak q$ such that $S_g$ is quasi-finite over a polynomial algebra $R[t_1, \\ldots, t_n]$."}
+{"_id": "1072", "title": "algebra-lemma-refined-quasi-finite-over-polynomial-algebra", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$ of $R$. Assume \\begin{enumerate} \\item $R \\to S$ is of finite type, \\item $\\dim_{\\mathfrak q}(S/R) = n$, and \\item $\\text{trdeg}_{\\kappa(\\mathfrak p)}\\kappa(\\mathfrak q) = r$. \\end{enumerate} Then there exist $f \\in R$, $f \\not \\in \\mathfrak p$, $g \\in S$, $g \\not\\in \\mathfrak q$ and a quasi-finite ring map $$ \\varphi : R_f[x_1, \\ldots, x_n] \\longrightarrow S_g $$ such that $\\varphi^{-1}(\\mathfrak qS_g) = (\\mathfrak p, x_{r + 1}, \\ldots, x_n)R_f[x_{r + 1}, \\ldots, x_n]$"}
+{"_id": "1073", "title": "algebra-lemma-dimension-inequality-quasi-finite", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. If $R \\to S$ is quasi-finite at $\\mathfrak q$, then $\\dim(S_{\\mathfrak q}) \\leq \\dim(R_{\\mathfrak p})$."}
+{"_id": "1074", "title": "algebra-lemma-dimension-quasi-finite-over-polynomial-algebra", "text": "\\begin{slogan} A quasi-finite cover of affine n-space has dimension at most n. \\end{slogan} Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Suppose there is a quasi-finite $k$-algebra map $k[t_1, \\ldots, t_n] \\subset S$. Then $\\dim(S) \\leq n$."}
+{"_id": "1075", "title": "algebra-lemma-dimension-fibres-bounded-open-upstairs", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$ be a prime. Suppose that $\\dim_{\\mathfrak q}(S/R) = n$. There exists an open neighbourhood $V$ of $\\mathfrak q$ in $\\Spec(S)$ such that $\\dim_{\\mathfrak q'}(S/R) \\leq n$ for all $\\mathfrak q' \\in V$."}
+{"_id": "1076", "title": "algebra-lemma-dimension-fibres-bounded-open-upstairs-base-change", "text": "Let $R \\to S$ be a finite type ring map. Let $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$ and denote $f : \\Spec(S') \\to \\Spec(S)$ the associated map on spectra. Let $n \\geq 0$. The inverse image $f^{-1}(\\{\\mathfrak q \\in \\Spec(S) \\mid \\dim_{\\mathfrak q}(S/R) \\leq n\\})$ is equal to $\\{\\mathfrak q' \\in \\Spec(S') \\mid \\dim_{\\mathfrak q'}(S'/R') \\leq n\\}$."}
+{"_id": "1077", "title": "algebra-lemma-dimension-fibres-bounded-quasi-compact-open-upstairs", "text": "Let $R \\to S$ be a ring homomorphism of finite presentation. Let $n \\geq 0$. The set $$ V_n = \\{\\mathfrak q \\in \\Spec(S) \\mid \\dim_{\\mathfrak q}(S/R) \\leq n\\} $$ is a quasi-compact open subset of $\\Spec(S)$."}
+{"_id": "1078", "title": "algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres", "text": "Let $R$ be a valuation ring with residue field $k$ and field of fractions $K$. Let $S$ be a domain containing $R$ such that $S$ is of finite type over $R$. If $S \\otimes_R k$ is not the zero ring then $$ \\dim(S \\otimes_R k) = \\dim(S \\otimes_R K) $$ In fact, $\\Spec(S \\otimes_R k)$ is equidimensional."}
+{"_id": "1079", "title": "algebra-lemma-finite-type-descends", "text": "Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map. Set $S' = R'\\otimes_R S$. Then $R \\to S$ is of finite type if and only if $R' \\to S'$ is of finite type."}
+{"_id": "1080", "title": "algebra-lemma-finite-presentation-descends", "text": "Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map. Set $S' = R'\\otimes_R S$. Then $R \\to S$ is of finite presentation if and only if $R' \\to S'$ is of finite presentation."}
+{"_id": "1081", "title": "algebra-lemma-construct-fp-module", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\subset R$ be a multiplicative subset. Set $R' = S^{-1}(R/I) = S^{-1}R/S^{-1}I$. \\begin{enumerate} \\item For any finite $R'$-module $M'$ there exists a finite $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$. \\item For any finitely presented $R'$-module $M'$ there exists a finitely presented $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$. \\end{enumerate}"}
+{"_id": "1082", "title": "algebra-lemma-construct-fp-module-from-localization", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. Let $M$ be an $R$-module. \\begin{enumerate} \\item If $S^{-1}M$ is a finite $S^{-1}R$-module then there exists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an isomorphism $S^{-1}M' \\to S^{-1}M$. \\item If $S^{-1}M$ is a finitely presented $S^{-1}R$-module then there exists an $R$-module $M'$ of finite presentation and a map $M' \\to M$ which induces an isomorphism $S^{-1}M' \\to S^{-1}M$. \\end{enumerate}"}
+{"_id": "1083", "title": "algebra-lemma-construct-fp-module-from-stalk", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $M$ be an $R$-module. \\begin{enumerate} \\item If $M_{\\mathfrak p}$ is a finite $R_{\\mathfrak p}$-module then there exists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an isomorphism $M'_{\\mathfrak p} \\to M_{\\mathfrak p}$. \\item If $M_{\\mathfrak p}$ is a finitely presented $R_{\\mathfrak p}$-module then there exists an $R$-module $M'$ of finite presentation and a map $M' \\to M$ which induces an isomorphism $M'_{\\mathfrak p} \\to M_{\\mathfrak p}$. \\end{enumerate}"}
+{"_id": "1084", "title": "algebra-lemma-local-isomorphism", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. Assume \\begin{enumerate} \\item $S$ is of finite presentation over $R$, \\item $\\varphi$ induces an isomorphism $R_\\mathfrak p \\cong S_\\mathfrak q$. \\end{enumerate} Then there exist $f \\in R$, $f \\not \\in \\mathfrak p$ and an $R_f$-algebra $C$ such that $S_f \\cong R_f \\times C$ as $R_f$-algebras."}
+{"_id": "1085", "title": "algebra-lemma-isomorphic-local-rings", "text": "Let $R$ be a ring. Let $S$, $S'$ be of finite presentation over $R$. Let $\\mathfrak q \\subset S$ and $\\mathfrak q' \\subset S'$ be primes. If $S_{\\mathfrak q} \\cong S'_{\\mathfrak q'}$ as $R$-algebras, then there exist $g \\in S$, $g \\not \\in \\mathfrak q$ and $g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that $S_g \\cong S'_{g'}$ as $R$-algebras."}
+{"_id": "1086", "title": "algebra-lemma-finite-type-mod-nilpotent", "text": "Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal. Let $S$ be an $R$-algebra such that $R/I \\to S/IS$ is of finite type. Then $R \\to S$ is of finite type."}
+{"_id": "1087", "title": "algebra-lemma-surjective-mod-locally-nilpotent", "text": "Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. Let $S \\to S'$ be an $R$-algebra map such that $S \\to S'/IS'$ is surjective and such that $S'$ is of finite type over $R$. Then $S \\to S'$ is surjective."}
+{"_id": "1088", "title": "algebra-lemma-isomorphism-modulo-ideal", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$ be an $R$-algebra map. Let $IS \\subset \\mathfrak q \\subset S$ be a prime ideal. Assume that \\begin{enumerate} \\item $S \\to S'$ is surjective, \\item $S_\\mathfrak q/IS_\\mathfrak q \\to S'_\\mathfrak q/IS'_\\mathfrak q$ is an isomorphism, \\item $S$ is of finite type over $R$, \\item $S'$ of finite presentation over $R$, and \\item $S'_\\mathfrak q$ is flat over $R$. \\end{enumerate} Then $S_g \\to S'_g$ is an isomorphism for some $g \\in S$, $g \\not \\in \\mathfrak q$."}
+{"_id": "1089", "title": "algebra-lemma-isomorphism-modulo-locally-nilpotent", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$ be an $R$-algebra map. Assume that \\begin{enumerate} \\item $I$ is locally nilpotent, \\item $S/IS \\to S'/IS'$ is an isomorphism, \\item $S$ is of finite type over $R$, \\item $S'$ of finite presentation over $R$, and \\item $S'$ is flat over $R$. \\end{enumerate} Then $S \\to S'$ is an isomorphism."}
+{"_id": "1090", "title": "algebra-lemma-ring-colimit-fp-category", "text": "Let $R \\to A$ be a ring map. Consider the category $\\mathcal{I}$ of all diagrams of $R$-algebra maps $A' \\to A$ with $A'$ finitely presented over $R$. Then $\\mathcal{I}$ is filtered, and the colimit of the $A'$ over $\\mathcal{I}$ is isomorphic to $A$."}
+{"_id": "1091", "title": "algebra-lemma-ring-colimit-fp", "text": "Let $R \\to A$ be a ring map. There exists a directed system $A_\\lambda$ of $R$-algebras of finite presentation such that $A = \\colim_\\lambda A_\\lambda$. If $A$ is of finite type over $R$ we may arrange it so that all the transition maps in the system of $A_\\lambda$ are surjective."}
+{"_id": "1092", "title": "algebra-lemma-characterize-finite-presentation", "text": "Let $\\varphi : R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $\\varphi$ is of finite presentation, \\item for every directed system $A_\\lambda$ of $R$-algebras the map $$ \\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow \\Hom_R(S, \\colim_\\lambda A_\\lambda) $$ is bijective, and \\item for every directed system $A_\\lambda$ of $R$-algebras the map $$ \\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow \\Hom_R(S, \\colim_\\lambda A_\\lambda) $$ is surjective. \\end{enumerate}"}
+{"_id": "1093", "title": "algebra-lemma-when-colimit", "text": "Let $R \\to \\Lambda$ be a ring map. Let $\\mathcal{E}$ be a set of $R$-algebras such that each $A \\in \\mathcal{E}$ is of finite presentation over $R$. Then the following two statements are equivalent \\begin{enumerate} \\item $\\Lambda$ is a filtered colimit of elements of $\\mathcal{E}$, and \\item for any $R$ algebra map $A \\to \\Lambda$ with $A$ of finite presentation over $R$ we can find a factorization $A \\to B \\to \\Lambda$ with $B \\in \\mathcal{E}$. \\end{enumerate}"}
+{"_id": "1094", "title": "algebra-lemma-module-map-property-in-colimit", "text": "Let $A$ be a ring and let $M, N$ be $A$-modules. Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit of $A$-algebras. \\begin{enumerate} \\item If $M$ is a finite $A$-module, and $u, u' : M \\to N$ are $A$-module maps such that $u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ then for some $i$ we have $u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$. \\item If $N$ is a finite $A$-module and $u : M \\to N$ is an $A$-module map such that $u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is surjective, then for some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$ is surjective. \\item If $N$ is a finitely presented $A$-module, and $v : N \\otimes_A R \\to M \\otimes_A R$ is an $R$-module map, then there exists an $i$ and an $R_i$-module map $v_i : N \\otimes_A R_i \\to M \\otimes_A R_i$ such that $v = v_i \\otimes 1$. \\item If $M$ is a finite $A$-module, $N$ is a finitely presented $A$-module, and $u : M \\to N$ is an $A$-module map such that $u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is an isomorphism, then for some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$ is an isomorphism. \\end{enumerate}"}
+{"_id": "1095", "title": "algebra-lemma-colimit-category-fp-modules", "text": "Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit of rings. Then the category of finitely presented $R$-modules is the colimit of the categories of finitely presented $R_\\lambda$-modules. More precisely \\begin{enumerate} \\item Given a finitely presented $R$-module $M$ there exists a $\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-module $M_\\lambda$ such that $M \\cong M_\\lambda \\otimes_{R_\\lambda} R$. \\item Given a $\\lambda \\in \\Lambda$, finitely presented $R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and an $R$-module map $\\varphi : M_\\lambda \\otimes_{R_\\lambda} R \\to N_\\lambda \\otimes_{R_\\lambda} R$, then there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-module map $\\varphi_\\mu : M_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to N_\\lambda \\otimes_{R_\\lambda} R_\\mu$ such that $\\varphi = \\varphi_\\mu \\otimes 1_R$. \\item Given a $\\lambda \\in \\Lambda$, finitely presented $R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and $R$-module maps $\\varphi_\\lambda, \\psi_\\lambda : M_\\lambda \\to N_\\lambda$ such that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then $\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some $\\mu \\geq \\lambda$. \\end{enumerate}"}
+{"_id": "1096", "title": "algebra-lemma-algebra-map-property-in-colimit", "text": "Let $A$ be a ring and let $B, C$ be $A$-algebras. Suppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit of $A$-algebras. \\begin{enumerate} \\item If $B$ is a finite type $A$-algebra, and $u, u' : B \\to C$ are $A$-algebra maps such that $u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ then for some $i$ we have $u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$. \\item If $C$ is a finite type $A$-algebra and $u : B \\to C$ is an $A$-algebra map such that $u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is surjective, then for some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$ is surjective. \\item If $C$ is of finite presentation over $A$ and $v : C \\otimes_A R \\to B \\otimes_A R$ is an $R$-algebra map, then there exists an $i$ and an $R_i$-algebra map $v_i : C \\otimes_A R_i \\to B \\otimes_A R_i$ such that $v = v_i \\otimes 1$. \\item If $B$ is a finite type $A$-algebra, $C$ is a finitely presented $A$-algebra, and $u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is an isomorphism, then for some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$ is an isomorphism. \\end{enumerate}"}
+{"_id": "1097", "title": "algebra-lemma-colimit-category-fp-algebras", "text": "Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit of rings. Then the category of finitely presented $R$-algebras is the colimit of the categories of finitely presented $R_\\lambda$-algebras. More precisely \\begin{enumerate} \\item Given a finitely presented $R$-algebra $A$ there exists a $\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-algebra $A_\\lambda$ such that $A \\cong A_\\lambda \\otimes_{R_\\lambda} R$. \\item Given a $\\lambda \\in \\Lambda$, finitely presented $R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and an $R$-algebra map $\\varphi : A_\\lambda \\otimes_{R_\\lambda} R \\to B_\\lambda \\otimes_{R_\\lambda} R$, then there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-algebra map $\\varphi_\\mu : A_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to B_\\lambda \\otimes_{R_\\lambda} R_\\mu$ such that $\\varphi = \\varphi_\\mu \\otimes 1_R$. \\item Given a $\\lambda \\in \\Lambda$, finitely presented $R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and $R_\\lambda$-algebra maps $\\varphi_\\lambda, \\psi_\\lambda : A_\\lambda \\to B_\\lambda$ such that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then $\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some $\\mu \\geq \\lambda$. \\end{enumerate}"}
+{"_id": "1098", "title": "algebra-lemma-limit-no-condition-local", "text": "Suppose $R \\to S$ is a local homomorphism of local rings. There exists a directed set $(\\Lambda, \\leq)$, and a system of local homomorphisms $R_\\lambda \\to S_\\lambda$ of local rings such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is essentially of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is essentially of finite type over $R_\\lambda$. \\end{enumerate}"}
+{"_id": "1099", "title": "algebra-lemma-limit-essentially-finite-type", "text": "Suppose $R \\to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite type over $R$. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of local homomorphisms $R_\\lambda \\to S_\\lambda$ of local rings such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is essentially of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is essentially of finite type over $R_\\lambda$. \\item For each $\\lambda \\leq \\mu$ the map $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ presents $S_\\mu$ as the localization of a quotient of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$. \\end{enumerate}"}
+{"_id": "1100", "title": "algebra-lemma-limit-essentially-finite-presentation", "text": "Suppose $R \\to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite presentation over $R$. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of local homomorphism $R_\\lambda \\to S_\\lambda$ of local rings such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is essentially of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is essentially of finite type over $R_\\lambda$. \\item For each $\\lambda \\leq \\mu$ the map $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ presents $S_\\mu$ as the localization of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$ at a prime ideal. \\end{enumerate}"}
+{"_id": "1101", "title": "algebra-lemma-limit-module-essentially-finite-presentation", "text": "Suppose $R \\to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite presentation over $R$. Let $M$ be a finitely presented $S$-module. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of local homomorphisms $R_\\lambda \\to S_\\lambda$ of local rings together with $S_\\lambda$-modules $M_\\lambda$, such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. The colimit of the system $M_\\lambda$ is $M$. \\item Each $R_\\lambda$ is essentially of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is essentially of finite type over $R_\\lambda$. \\item Each $M_\\lambda$ is finite over $S_\\lambda$. \\item For each $\\lambda \\leq \\mu$ the map $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ presents $S_\\mu$ as the localization of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$ at a prime ideal. \\item For each $\\lambda \\leq \\mu$ the map $M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$ is an isomorphism. \\end{enumerate}"}
+{"_id": "1102", "title": "algebra-lemma-limit-no-condition", "text": "Suppose $R \\to S$ is a ring map. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of ring maps $R_\\lambda \\to S_\\lambda$ such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is of finite type over $R_\\lambda$. \\end{enumerate}"}
+{"_id": "1103", "title": "algebra-lemma-limit-integral", "text": "Suppose $R \\to S$ is a ring map. Assume that $S$ is integral over $R$. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of ring maps $R_\\lambda \\to S_\\lambda$ such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is of finite over $R_\\lambda$. \\end{enumerate}"}
+{"_id": "1106", "title": "algebra-lemma-limit-module-finite-presentation", "text": "Suppose $R \\to S$ is a ring map. Assume that $S$ is of finite presentation over $R$. Let $M$ be a finitely presented $S$-module. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of ring maps $R_\\lambda \\to S_\\lambda$ together with $S_\\lambda$-modules $M_\\lambda$, such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. The colimit of the system $M_\\lambda$ is $M$. \\item Each $R_\\lambda$ is of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is of finite type over $R_\\lambda$. \\item Each $M_\\lambda$ is finite over $S_\\lambda$. \\item For each $\\lambda \\leq \\mu$ the map $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ is an isomorphism. \\item For each $\\lambda \\leq \\mu$ the map $M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$ is an isomorphism. \\end{enumerate} In particular, for every $\\lambda \\in \\Lambda$ we have $$ M = M_\\lambda \\otimes_{S_\\lambda} S = M_\\lambda \\otimes_{R_\\lambda} R. $$"}
+{"_id": "1107", "title": "algebra-lemma-CM-over-regular-flat", "text": "\\begin{slogan} Miracle flatness \\end{slogan} Let $R \\to S$ be a local homomorphism of Noetherian local rings. Assume \\begin{enumerate} \\item $R$ is regular, \\item $S$ Cohen-Macaulay, \\item $\\dim(S) = \\dim(R) + \\dim(S/\\mathfrak m_R S)$. \\end{enumerate} Then $R \\to S$ is flat."}
+{"_id": "1108", "title": "algebra-lemma-flat-over-regular", "text": "Let $R \\to S$ be a homomorphism of Noetherian local rings. Assume that $R$ is a regular local ring and that a regular system of parameters maps to a regular sequence in $S$. Then $R \\to S$ is flat."}
+{"_id": "1109", "title": "algebra-lemma-colimit-eventually-flat", "text": "Let $R \\to S$, $M$, $\\Lambda$, $R_\\lambda \\to S_\\lambda$, $M_\\lambda$ be as in Lemma \\ref{lemma-limit-module-essentially-finite-presentation}. Assume that $M$ is flat over $R$. Then for some $\\lambda \\in \\Lambda$ the module $M_\\lambda$ is flat over $R_\\lambda$."}
+{"_id": "1110", "title": "algebra-lemma-mod-injective-general", "text": "Suppose that $R \\to S$ is a local homomorphism of local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Let $u : M \\to N$ be a map of $S$-modules. Assume \\begin{enumerate} \\item $S$ is essentially of finite presentation over $R$, \\item $M$, $N$ are finitely presented over $S$, \\item $N$ is flat over $R$, and \\item $\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective. \\end{enumerate} Then $u$ is injective, and $N/u(M)$ is flat over $R$."}
+{"_id": "1111", "title": "algebra-lemma-grothendieck-general", "text": "Suppose that $R \\to S$ is a local ring homomorphism of local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Suppose \\begin{enumerate} \\item $S$ is essentially of finite presentation over $R$, \\item $S$ is flat over $R$, and \\item $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$. \\end{enumerate} Then $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$."}
+{"_id": "1112", "title": "algebra-lemma-grothendieck-regular-sequence-general", "text": "Suppose that $R \\to S$ is a local ring homomorphism of local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Suppose \\begin{enumerate} \\item $R \\to S$ is essentially of finite presentation, \\item $R \\to S$ is flat, and \\item $f_1, \\ldots, f_c$ is a sequence of elements of $S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$ form a regular sequence in $S/{\\mathfrak m}S$. \\end{enumerate} Then $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each of the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$."}
+{"_id": "1114", "title": "algebra-lemma-criterion-flatness-fibre", "text": "Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring homomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$ be the maximal ideal. Assume \\begin{enumerate} \\item The ring maps $R \\to S$ and $R \\to S'$ are essentially of finite presentation. \\item The module $M$ is of finite presentation over $S'$. \\item The module $M$ is not zero. \\item The module $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module. \\item The module $M$ is a flat $R$-module. \\end{enumerate} Then $S$ is flat over $R$ and $M$ is a flat $S$-module."}
+{"_id": "1115", "title": "algebra-lemma-criterion-flatness-fibre-fp-over-ft", "text": "Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring homomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$ be the maximal ideal. Assume \\begin{enumerate} \\item $R \\to S'$ is essentially of finite presentation, \\item $R \\to S$ is essentially of finite type, \\item $M$ is of finite presentation over $S'$, \\item $M$ is not zero, \\item $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module, and \\item $M$ is a flat $R$-module. \\end{enumerate} Then $S$ is essentially of finite presentation and flat over $R$ and $M$ is a flat $S$-module."}
+{"_id": "1116", "title": "algebra-lemma-criterion-flatness-fibre-locally-nilpotent", "text": "Let $$ \\xymatrix{ S \\ar[rr] & & S' \\\\ & R \\ar[lu] \\ar[ru] } $$ be a commutative diagram in the category of rings. Let $I \\subset R$ be a locally nilpotent ideal and $M$ an $S'$-module. Assume \\begin{enumerate} \\item $R \\to S$ is of finite type, \\item $R \\to S'$ is of finite presentation, \\item $M$ is a finitely presented $S'$-module, \\item $M/IM$ is flat as a $S/IS$-module, and \\item $M$ is flat as an $R$-module. \\end{enumerate} Then $M$ is a flat $S$-module and $S_\\mathfrak q$ is flat and essentially of finite presentation over $R$ for every $\\mathfrak q \\subset S$ such that $M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero."}
+{"_id": "1117", "title": "algebra-lemma-CM-dim-finite-type", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $f_1, \\ldots, f_i$ be elements of $S$. Assume that $S$ is Cohen-Macaulay and equidimensional of dimension $d$, and that $\\dim V(f_1, \\ldots, f_i) \\leq d - i$. Then equality holds and $f_1, \\ldots, f_i$ forms a regular sequence in $S_{\\mathfrak q}$ for every prime $\\mathfrak q$ of $V(f_1, \\ldots, f_i)$."}
+{"_id": "1118", "title": "algebra-lemma-open-regular-sequence", "text": "Suppose that $R \\to S$ is a ring map which is finite type, flat. Let $d$ be an integer such that all fibres $S \\otimes_R \\kappa(\\mathfrak p)$ are Cohen-Macaulay and equidimensional of dimension $d$. Let $f_1, \\ldots, f_i$ be elements of $S$. The set $$ \\{ \\mathfrak q \\in V(f_1, \\ldots, f_i) \\mid f_1, \\ldots, f_i \\text{ are a regular sequence in } S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q} \\text{ where }\\mathfrak p = R \\cap \\mathfrak q \\} $$ is open in $V(f_1, \\ldots, f_i)$."}
+{"_id": "1119", "title": "algebra-lemma-exact-on-fibres-open", "text": "Let $R \\to S$ is a ring map. Consider a finite homological complex of finite free $S$-modules: $$ F_{\\bullet} : 0 \\to S^{n_e} \\xrightarrow{\\varphi_e} S^{n_{e-1}} \\xrightarrow{\\varphi_{e-1}} \\ldots \\xrightarrow{\\varphi_{i + 1}} S^{n_i} \\xrightarrow{\\varphi_i} S^{n_{i-1}} \\xrightarrow{\\varphi_{i-1}} \\ldots \\xrightarrow{\\varphi_1} S^{n_0} $$ For every prime $\\mathfrak q$ of $S$ consider the complex $\\overline{F}_{\\bullet, \\mathfrak q} = F_{\\bullet, \\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p$ is inverse image of $\\mathfrak q$ in $R$. Assume there exists an integer $d$ such that $R \\to S$ is finite type, flat with fibres $S \\otimes_R \\kappa(\\mathfrak p)$ Cohen-Macaulay of dimension $d$. The set $$ \\{\\mathfrak q \\in \\Spec(S) \\mid \\overline{F}_{\\bullet, \\mathfrak q}\\text{ is exact}\\} $$ is open in $\\Spec(S)$."}
+{"_id": "1120", "title": "algebra-lemma-where-CM", "text": "Let $S$ be a finite type algebra over a field $k$. Let $\\varphi : k[y_1, \\ldots, y_d] \\to S$ be a quasi-finite ring map. As subsets of $\\Spec(S)$ we have $$ \\{ \\mathfrak q \\mid S_{\\mathfrak q} \\text{ flat over }k[y_1, \\ldots, y_d]\\} = \\{ \\mathfrak q \\mid S_{\\mathfrak q} \\text{ CM and }\\dim_{\\mathfrak q}(S/k) = d\\} $$ For notation see Definition \\ref{definition-relative-dimension}."}
+{"_id": "1121", "title": "algebra-lemma-finite-type-over-field-CM-open", "text": "Let $S$ be a finite type algebra over a field $k$. The set of primes $\\mathfrak q$ such that $S_{\\mathfrak q}$ is Cohen-Macaulay is open in $S$."}
+{"_id": "1122", "title": "algebra-lemma-generic-CM", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra. The set of Cohen-Macaulay primes forms a dense open $U \\subset \\Spec(S)$."}
+{"_id": "1123", "title": "algebra-lemma-dim-not-zero-exists-nonzerodivisor-nonunit", "text": "Let $k$ be a field. Let $S$ be a finite type $k$ algebra. If $\\dim(S) > 0$, then there exists an element $f \\in S$ which is a nonzerodivisor and a nonunit."}
+{"_id": "1124", "title": "algebra-lemma-finite-presentation-flat-CM-locus-open", "text": "Let $R$ be a ring. Let $R \\to S$ be of finite presentation and flat. For any $d \\geq 0$ the set $$ \\left\\{ \\begin{matrix} \\mathfrak q \\in \\Spec(S) \\text{ such that setting }\\mathfrak p = R \\cap \\mathfrak q \\text{ the fibre ring}\\\\ S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q} \\text{ is Cohen-Macaulay} \\text{ and } \\dim_{\\mathfrak q}(S/R) = d \\end{matrix} \\right\\} $$ is open in $\\Spec(S)$."}
+{"_id": "1125", "title": "algebra-lemma-generic-CM-flat-finite-presentation", "text": "Let $R$ be a ring. Let $R \\to S$ be flat of finite presentation. The set of primes $\\mathfrak q$ such that the fibre ring $S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$, with $\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay is open and dense in every fibre of $\\Spec(S) \\to \\Spec(R)$."}
+{"_id": "1126", "title": "algebra-lemma-extend-field-CM-locus", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $k \\subset K$ be a field extension, and set $S_K = K \\otimes_k S$. Let $\\mathfrak q \\subset S$ be a prime of $S$. Let $\\mathfrak q_K \\subset S_K$ be a prime of $S_K$ lying over $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay if and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay."}
+{"_id": "1127", "title": "algebra-lemma-CM-locus-commutes-base-change", "text": "Let $R$ be a ring. Let $R \\to S$ be of finite type. Let $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$. Denote $f : \\Spec(S') \\to \\Spec(S)$ the map associated to the ring map $S \\to S'$. Set $W$ equal to the set of primes $\\mathfrak q$ such that the fibre ring $S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$, $\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay, and let $W'$ denote the analogue for $S'/R'$. Then $W' = f^{-1}(W)$."}
+{"_id": "1128", "title": "algebra-lemma-relative-dimension-CM", "text": "Let $R$ be a ring. Let $R \\to S$ be a ring map which is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write $S = S_0 \\times \\ldots \\times S_n$ as a product of $R$-algebras $S_d$ such that each $S_d$ satisfies (a), (b), (c) and has all fibres equidimensional of dimension $d$."}
+{"_id": "1129", "title": "algebra-lemma-universal-omega", "text": "\\begin{slogan} Maps out of the module of differentials are the same as derivations. \\end{slogan} The module of differentials of $S$ over $R$ has the following universal property. The map $$ \\Hom_S(\\Omega_{S/R}, M) \\longrightarrow \\text{Der}_R(S, M), \\quad \\alpha \\longmapsto \\alpha \\circ \\text{d} $$ is an isomorphism of functors."}
+{"_id": "1130", "title": "algebra-lemma-colimit-differentials", "text": "Let $I$ be a directed set. Let $(R_i \\to S_i, \\varphi_{ii'})$ be a system of ring maps over $I$, see Categories, Section \\ref{categories-section-posets-limits}. Then we have $$ \\Omega_{S/R} = \\colim_i \\Omega_{S_i/R_i}. $$ where $R \\to S = \\colim (R_i \\to S_i)$."}
+{"_id": "1131", "title": "algebra-lemma-trivial-differential-surjective", "text": "Suppose that $R \\to S$ is surjective. Then $\\Omega_{S/R} = 0$."}
+{"_id": "1132", "title": "algebra-lemma-differential-surjective", "text": "In diagram (\\ref{equation-functorial-omega}), suppose that $S \\to S'$ is surjective with kernel $I \\subset S$. Then $\\Omega_{S/R} \\to \\Omega_{S'/R'}$ is surjective with kernel generated as an $S$-module by the elements $\\text{d}a$, where $a \\in S$ is such that $\\varphi(a) \\in \\beta(R')$. (This includes in particular the elements $\\text{d}(i)$, $i \\in I$.)"}
+{"_id": "1133", "title": "algebra-lemma-exact-sequence-differentials", "text": "Let $A \\to B \\to C$ be ring maps. Then there is a canonical exact sequence $$ C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0 $$ of $C$-modules."}
+{"_id": "1134", "title": "algebra-lemma-differentials-localize", "text": "Let $\\varphi : A \\to B$ be a ring map. \\begin{enumerate} \\item If $S \\subset A$ is a multiplicative subset mapping to invertible elements of $B$, then $\\Omega_{B/A} = \\Omega_{B/S^{-1}A}$. \\item If $S \\subset B$ is a multiplicative subset then $S^{-1}\\Omega_{B/A} = \\Omega_{S^{-1}B/A}$. \\end{enumerate}"}
+{"_id": "1135", "title": "algebra-lemma-differential-seq", "text": "In diagram (\\ref{equation-functorial-omega}), suppose that $S \\to S'$ is surjective with kernel $I \\subset S$, and assume that $R' = R$. Then there is a canonical exact sequence of $S'$-modules $$ I/I^2 \\longrightarrow \\Omega_{S/R} \\otimes_S S' \\longrightarrow \\Omega_{S'/R} \\longrightarrow 0 $$ The leftmost map is characterized by the rule that $f \\in I$ maps to $\\text{d}f \\otimes 1$."}
+{"_id": "1136", "title": "algebra-lemma-differential-seq-split", "text": "In diagram (\\ref{equation-functorial-omega}), suppose that $S \\to S'$ is surjective with kernel $I \\subset S$, and assume that $R' = R$. Moreover, assume that there exists an $R$-algebra map $S' \\to S$ which is a right inverse to $S \\to S'$. Then the exact sequence of $S'$-modules of Lemma \\ref{lemma-differential-seq} turns into a short exact sequence $$ 0 \\longrightarrow I/I^2 \\longrightarrow \\Omega_{S/R} \\otimes_S S' \\longrightarrow \\Omega_{S'/R} \\longrightarrow 0 $$ which is even a split short exact sequence."}
+{"_id": "1137", "title": "algebra-lemma-differential-mod-power-ideal", "text": "Let $R \\to S$ be a ring map. Let $I \\subset S$ be an ideal. Let $n \\geq 1$ be an integer. Set $S' = S/I^{n + 1}$. The map $\\Omega_{S/R} \\to \\Omega_{S'/R}$ induces an isomorphism $$ \\Omega_{S/R} \\otimes_S S/I^n \\longrightarrow \\Omega_{S'/R} \\otimes_{S'} S/I^n. $$"}
+{"_id": "1138", "title": "algebra-lemma-differentials-base-change", "text": "Suppose that we have ring maps $R \\to R'$ and $R \\to S$. Set $S' = S \\otimes_R R'$, so that we obtain a diagram (\\ref{equation-functorial-omega}). Then the canonical map defined above induces an isomorphism $\\Omega_{S/R} \\otimes_R R' = \\Omega_{S'/R'}$."}
+{"_id": "1139", "title": "algebra-lemma-differentials-diagonal", "text": "Let $R \\to S$ be a ring map. Let $J = \\Ker(S \\otimes_R S \\to S)$ be the kernel of the multiplication map. There is a canonical isomorphism of $S$-modules $\\Omega_{S/R} \\to J/J^2$, $a \\text{d} b \\mapsto a \\otimes b - ab \\otimes 1$."}
+{"_id": "1140", "title": "algebra-lemma-differentials-polynomial-ring", "text": "If $S = R[x_1, \\ldots, x_n]$, then $\\Omega_{S/R}$ is a finite free $S$-module with basis $\\text{d}x_1, \\ldots, \\text{d}x_n$."}
+{"_id": "1141", "title": "algebra-lemma-differentials-finitely-presented", "text": "Suppose $R \\to S$ is of finite presentation. Then $\\Omega_{S/R}$ is a finitely presented $S$-module."}
+{"_id": "1142", "title": "algebra-lemma-differentials-finitely-generated", "text": "Suppose $R \\to S$ is of finite type. Then $\\Omega_{S/R}$ is finitely generated $S$-module."}
+{"_id": "1143", "title": "algebra-lemma-de-rham-complex", "text": "Let $A \\to B$ be a ring map. Let $\\pi : \\Omega_{B/A} \\to \\Omega$ be a surjective $B$-module map. Denote $\\text{d} : B \\to \\Omega$ the composition of $\\pi$ with the universal derivation $\\text{d}_{B/A} : B \\to \\Omega_{B/A}$. Set $\\Omega^i = \\wedge_B^i(\\Omega)$. Assume that the kernel of $\\pi$ is generated, as a $B$-module, by elements $\\omega \\in \\Omega_{B/A}$ such that $\\text{d}_{B/A}(\\omega) \\in \\Omega_{B/A}^2$ maps to zero in $\\Omega^2$. Then there is a de Rham complex $$ \\Omega^0 \\to \\Omega^1 \\to \\Omega^2 \\to \\ldots $$ whose differential is defined by the rule $$ \\text{d} : \\Omega^p \\to \\Omega^{p + 1},\\quad \\text{d}\\left(f_0\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p\\right) = \\text{d}f_0 \\wedge \\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p $$"}
+{"_id": "1144", "title": "algebra-lemma-composition-differential-operators", "text": "Let $R \\to S$ be a ring map. Let $L, M, N$ be $S$-modules. If $D : L \\to M$ and $D' : M \\to N$ are differential operators of order $k$ and $k'$, then $D' \\circ D$ is a differential operator of order $k + k'$."}
+{"_id": "1145", "title": "algebra-lemma-module-principal-parts", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Let $k \\geq 0$. There exists an $S$-module $P^k_{S/R}(M)$ and a canonical isomorphism $$ \\text{Diff}^k_{S/R}(M, N) = \\Hom_S(P^k_{S/R}(M), N) $$ functorial in the $S$-module $N$."}
+{"_id": "1146", "title": "algebra-lemma-sequence-of-principal-parts", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. There is a canonical short exact sequence $$ 0 \\to \\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M) \\to M \\to 0 $$ functorial in $M$ called the {\\it sequence of principal parts}."}
+{"_id": "1147", "title": "algebra-lemma-differentials-de-rham-complex-order-1", "text": "Let $A \\to B$ be a ring map. The differentials $\\text{d} : \\Omega^i_{B/A}  \\to \\Omega^{i + 1}_{B/A}$ are differential operators of order $1$."}
+{"_id": "1148", "title": "algebra-lemma-check-differential-operators", "text": "Let $A \\to B$ be a ring map. Let $g_i \\in B$, $i \\in I$ be a set of generators for $B$ as an $A$-algebra. Let $M, N$ be $B$-modules. Let $D : M \\to N$ be an $A$-linear map. In order to show that $D$ is a differential operator of order $k$ it suffices to show that $D \\circ g_i - g_i \\circ D$ is a differential operator of order $k - 1$ for $i \\in I$."}
+{"_id": "1149", "title": "algebra-lemma-invert-system-differential-operators", "text": "Let $A \\to B$ be a ring map. Let $M, N$ be $B$-modules. Let $S \\subset B$ be a multiplicative subset. Any differential operator $D : M \\to N$ of order $k$ extends uniquely to a differential operator $E : S^{-1}M \\to S^{-1}N$ of order $k$."}
+{"_id": "1150", "title": "algebra-lemma-base-change-differential-operators", "text": "Let $R \\to A$ and $R \\to B$ be ring maps. Let $M$ and $M'$ be $A$-modules. Let $D : M \\to M'$ be a differential operator of order $k$ with respect to $R \\to A$. Let $N$ be any $B$-module. Then the map $$ D \\otimes \\text{id}_N : M \\otimes_R N \\to M' \\otimes_R N $$ is a differential operator of order $k$ with respect to $B \\to A \\otimes_R B$."}
+{"_id": "1151", "title": "algebra-lemma-NL-homotopy", "text": "Suppose given a diagram (\\ref{equation-functoriality-NL}). Let $\\alpha : P \\to S$ and $\\alpha' : P' \\to S'$ be presentations. \\begin{enumerate} \\item There exists a morphism of presentations from $\\alpha$ to $\\alpha'$. \\item Any two morphisms of presentations induce homotopic morphisms of complexes $\\NL(\\alpha) \\to \\NL(\\alpha')$. \\item The construction is compatible with compositions of morphisms of presentations (see proof for exact statement). \\item If $R \\to R'$ and $S \\to S'$ are isomorphisms, then for any map $\\varphi$ of presentations from $\\alpha$ to $\\alpha'$ the induced map $\\NL(\\alpha) \\to \\NL(\\alpha')$ is a homotopy equivalence and a quasi-isomorphism. \\end{enumerate} In particular, comparing $\\alpha$ to the canonical presentation (\\ref{equation-canonical-presentation}) we conclude there is a quasi-isomorphism $\\NL(\\alpha) \\to \\NL_{S/R}$ well defined up to homotopy and compatible with all functorialities (up to homotopy)."}
+{"_id": "1152", "title": "algebra-lemma-NL-polynomial-algebra", "text": "Let $A \\to B$ be a polynomial algebra. Then $\\NL_{B/A}$ is homotopy equivalent to the chain complex $(0 \\to \\Omega_{B/A})$ with $\\Omega_{B/A}$ in degree $0$."}
+{"_id": "1153", "title": "algebra-lemma-exact-sequence-NL", "text": "Let $A \\to B \\to C$ be ring maps. Choose a presentation $\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation $\\beta : B[y_t, t \\in T] \\to C$ with kernel $J$. Let $\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel $K$. Then we get a canonical commutative diagram $$ \\xymatrix{ 0 \\ar[r] & \\Omega_{A[x_s]/A} \\otimes C \\ar[r] & \\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] & \\Omega_{B[y_t]/B} \\otimes C \\ar[r] & 0 \\\\ & I/I^2 \\otimes C \\ar[r] \\ar[u] & K/K^2 \\ar[r] \\ar[u] & J/J^2 \\ar[r] \\ar[u] & 0 } $$ with exact rows. We get the following exact sequence of homology groups $$ H_1(\\NL_{B/A} \\otimes_B C) \\to H_1(L_{C/A}) \\to H_1(L_{C/B}) \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0 $$ of $C$-modules extending the sequence of Lemma \\ref{lemma-exact-sequence-differentials}. If $\\text{Tor}_1^B(\\Omega_{B/A}, C) = 0$, then $H_1(\\NL_{B/A} \\otimes_B C) = H_1(L_{B/A}) \\otimes_B C$."}
+{"_id": "1154", "title": "algebra-lemma-NL-surjection", "text": "Let $A \\to B$ be a surjective ring map with kernel $I$. Then $\\NL_{B/A}$ is homotopy equivalent to the chain complex $(I/I^2 \\to 0)$ with $I/I^2$ in degree $1$. In particular $H_1(L_{B/A}) = I/I^2$."}
+{"_id": "1155", "title": "algebra-lemma-application-NL", "text": "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so also $B \\to C$ is). Denote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$. Then the sequence $$ I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0 $$ is exact."}
+{"_id": "1156", "title": "algebra-lemma-change-base-NL", "text": "Let $R \\to S$ be a ring map. Let $\\alpha : P \\to S$ be a presentation. Let $R \\to R'$ be a flat ring map. Let $\\alpha' : P \\otimes_R R' \\to S' = S \\otimes_R R'$ be the induced presentation. Then $\\NL(\\alpha) \\otimes_R R' = \\NL(\\alpha) \\otimes_S S' = \\NL(\\alpha')$. In particular, the canonical map $$ \\NL_{S/R} \\otimes_S S' \\longrightarrow \\NL_{S \\otimes_R R'/R'} $$ is a homotopy equivalence if $R \\to R'$ is flat."}
+{"_id": "1157", "title": "algebra-lemma-colimits-NL", "text": "Let $R_i \\to S_i$ be a system of ring maps over the directed set $I$. Set $R = \\colim R_i$ and $S = \\colim S_i$. Then $\\NL_{S/R} = \\colim \\NL_{S_i/R_i}$."}
+{"_id": "1159", "title": "algebra-lemma-NL-localize-bottom", "text": "Let $S \\subset A$ is a multiplicative subset of $A$. Let $S^{-1}A \\to B$ be a ring map. Then $\\NL_{B/A} \\to \\NL_{B/S^{-1}A}$ is a homotopy equivalence."}
+{"_id": "1160", "title": "algebra-lemma-principal-localization-NL", "text": "\\begin{slogan} The formation of the naive cotangent complex commutes with localization at an element. \\end{slogan} Let $A \\to B$ be a ring map. Let $g \\in B$. Suppose $\\alpha : P \\to B$ is a presentation with kernel $I$. Then a presentation of $B_g$ over $A$ is the map $$ \\beta : P[x] \\longrightarrow B_g $$ extending $\\alpha$ and sending $x$ to $1/g$. The kernel $J$ of $\\beta$ is generated by $I$ and the element $f x - 1$ where $f \\in P$ is an element mapped to $g \\in B$ by $\\alpha$. In this situation we have \\begin{enumerate} \\item $J/J^2 = (I/I^2)_g \\oplus B_g (f x - 1)$, \\item $\\Omega_{P[x]/A} \\otimes_{P[x]} B_g = \\Omega_{P/A} \\otimes_P B_g \\oplus B_g \\text{d}x$, \\item $\\NL(\\beta) \\cong \\NL(\\alpha) \\otimes_B B_g \\oplus (B_g \\xrightarrow{g} B_g)$ \\end{enumerate} Hence the canonical map $\\NL_{B/A} \\otimes_B B_g \\to \\NL_{B_g/A}$ is a homotopy equivalence."}
+{"_id": "1161", "title": "algebra-lemma-localize-NL", "text": "Let $A \\to B$ be a ring map. Let $S \\subset B$ be a multiplicative subset. The canonical map $\\NL_{B/A} \\otimes_B S^{-1}B \\to \\NL_{S^{-1}B/A}$ is a quasi-isomorphism."}
+{"_id": "1162", "title": "algebra-lemma-sum-two-terms", "text": "Let $R$ be a ring. Let $A_1 \\to A_0$, and $B_1 \\to B_0$ be two term complexes. Suppose that there exist morphisms of complexes $\\varphi : A_\\bullet \\to B_\\bullet$ and $\\psi : B_\\bullet \\to A_\\bullet$ such that $\\varphi \\circ \\psi$ and $\\psi \\circ \\varphi$ are homotopic to the identity maps. Then $A_1 \\oplus B_0 \\cong B_1 \\oplus A_0$ as $R$-modules."}
+{"_id": "1163", "title": "algebra-lemma-conormal-module", "text": "Let $R \\to S$ be a ring map of finite type. For any presentations $\\alpha : R[x_1, \\ldots, x_n] \\to S$, and $\\beta : R[y_1, \\ldots, y_m] \\to S$ we have $$ I/I^2 \\oplus S^{\\oplus m} \\cong J/J^2 \\oplus S^{\\oplus n} $$ as $S$-modules where $I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$."}
+{"_id": "1164", "title": "algebra-lemma-conormal-module-localize", "text": "Let $R \\to S$ be a ring map of finite type. Let $g \\in S$. For any presentations $\\alpha : R[x_1, \\ldots, x_n] \\to S$, and $\\beta : R[y_1, \\ldots, y_m] \\to S_g$ we have $$ (I/I^2)_g \\oplus S^{\\oplus m}_g \\cong J/J^2 \\oplus S_g^{\\oplus n} $$ as $S_g$-modules where $I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$."}
+{"_id": "1165", "title": "algebra-lemma-localize-lci", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $g \\in S$. \\begin{enumerate} \\item If $S$ is a global complete intersection so is $S_g$. \\item If $S$ is a local complete intersection so is $S_g$. \\end{enumerate}"}
+{"_id": "1166", "title": "algebra-lemma-lci-CM", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. If $S$ is a local complete intersection, then $S$ is a Cohen-Macaulay ring."}
+{"_id": "1167", "title": "algebra-lemma-lci", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak q$ be a prime of $S$. Choose any presentation $S = k[x_1, \\ldots, x_n]/I$. Let $\\mathfrak q'$ be the prime of $k[x_1, \\ldots, x_n]$ corresponding to $\\mathfrak q$. Set $c = \\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q)$, in other words $\\dim_{\\mathfrak q}(S) = n - c$ (see Lemma \\ref{lemma-codimension}). The following are equivalent \\begin{enumerate} \\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $S_g$ is a global complete intersection over $k$. \\item The ideal $I_{\\mathfrak q'} \\subset k[x_1, \\ldots, x_n]_{\\mathfrak q'}$ can be generated by $c$ elements. \\item The conormal module $(I/I^2)_{\\mathfrak q}$ can be generated by $c$ elements over $S_{\\mathfrak q}$. \\item The conormal module $(I/I^2)_{\\mathfrak q}$ is a free $S_{\\mathfrak q}$-module of rank $c$. \\item The ideal $I_{\\mathfrak q'}$ can be generated by a regular sequence in the regular local ring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$. \\end{enumerate} In this case any $c$ elements of $I_{\\mathfrak q'}$ which generate $I_{\\mathfrak q'}/\\mathfrak q'I_{\\mathfrak q'}$ form a regular sequence in the local ring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$."}
+{"_id": "1168", "title": "algebra-lemma-ci-well-defined", "text": "Let $A \\to B \\to C$ be surjective local ring homomorphisms. Assume $A$ and $B$ are regular local rings. The following are equivalent \\begin{enumerate} \\item $\\Ker(A \\to C)$ is generated by a regular sequence, \\item $\\Ker(A \\to C)$ is generated by $\\dim(A) - \\dim(C)$ elements, \\item $\\Ker(B \\to C)$ is generated by a regular sequence, and \\item $\\Ker(B \\to C)$ is generated by $\\dim(B) - \\dim(C)$ elements. \\end{enumerate}"}
+{"_id": "1169", "title": "algebra-lemma-lci-local", "text": "Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type over $k$. The following are equivalent: \\begin{enumerate} \\item $S$ is a complete intersection over $k$, \\item for any surjection $R \\to S$ with $R$ a regular local ring essentially of finite presentation over $k$ the ideal $\\Ker(R \\to S)$ can be generated by a regular sequence, \\item for some surjection $R \\to S$ with $R$ a regular local ring essentially of finite presentation over $k$ the ideal $\\Ker(R \\to S)$ can be generated by $\\dim(R) - \\dim(S)$ elements, \\item there exists a global complete intersection $A$ over $k$ and a prime $\\mathfrak a$ of $A$ such that $S \\cong A_{\\mathfrak a}$, and \\item there exists a local complete intersection $A$ over $k$ and a prime $\\mathfrak a$ of $A$ such that $S \\cong A_{\\mathfrak a}$. \\end{enumerate}"}
+{"_id": "1170", "title": "algebra-lemma-lci-at-prime", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak q$ be a prime of $S$. The following are equivalent: \\begin{enumerate} \\item The local ring $S_{\\mathfrak q}$ is a complete intersection ring (Definition \\ref{definition-lci-local-ring}). \\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $S_g$ is a local complete intersection over $k$. \\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $S_g$ is a global complete intersection over $k$. \\item For any presentation $S = k[x_1, \\ldots, x_n]/I$ with $\\mathfrak q' \\subset k[x_1, \\ldots, x_n]$ corresponding to $\\mathfrak q$ any of the equivalent conditions (1) -- (5) of Lemma \\ref{lemma-lci} hold. \\end{enumerate}"}
+{"_id": "1171", "title": "algebra-lemma-lci-global", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. The following are equivalent: \\begin{enumerate} \\item The ring $S$ is a local complete intersection over $k$. \\item All local rings of $S$ are complete intersection rings over $k$. \\item All localizations of $S$ at maximal ideals are complete intersection rings over $k$. \\end{enumerate}"}
+{"_id": "1172", "title": "algebra-lemma-lci-field-change-local", "text": "Let $k \\subset K$ be a field extension. Let $S$ be a finite type algebra over $k$. Let $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$ and let $\\mathfrak q$ be the corresponding prime of $S$. Then $S_{\\mathfrak q}$ is a complete intersection over $k$ (Definition \\ref{definition-lci-local-ring}) if and only if $(S_K)_{\\mathfrak q_K}$ is a complete intersection over $K$."}
+{"_id": "1173", "title": "algebra-lemma-lci-field-change", "text": "Let $k \\to K$ be a field extension. Let $S$ be a finite type $k$-algebra. Then $S$ is a local complete intersection over $k$ if and only if $S \\otimes_k K$ is a local complete intersection over $K$."}
+{"_id": "1174", "title": "algebra-lemma-lci-permanence-initial", "text": "Let $$ \\xymatrix{ B & S \\ar[l] \\\\ A \\ar[u] & R \\ar[l] \\ar[u] } $$ be a commutative square of local rings. Assume \\begin{enumerate} \\item $R$ and $\\overline{S} = S/\\mathfrak m_R S$ are regular local rings, \\item $A = R/I$ and $B = S/J$ for some ideals $I$, $J$, \\item $J \\subset S$ and $\\overline{J} = J/\\mathfrak m_R \\cap J \\subset \\overline{S}$ are generated by regular sequences, and \\item $A \\to B$ and $R \\to S$ are flat. \\end{enumerate} Then $I$ is generated by a regular sequence."}
+{"_id": "1176", "title": "algebra-lemma-base-change-syntomic", "text": "Any base change of a syntomic map is syntomic."}
+{"_id": "1177", "title": "algebra-lemma-local-syntomic", "text": "Let $R \\to S$ be a ring map. Suppose we have $g_1, \\ldots g_m \\in S$ which generate the unit ideal such that each $R \\to S_{g_i}$ is syntomic. Then $R \\to S$ is syntomic."}
+{"_id": "1178", "title": "algebra-lemma-huber", "text": "Let $S$ be a finitely presented $R$-algebra which has a presentation $S = R[x_1, \\ldots, x_n]/I$ such that $I/I^2$ is free over $S$. Then $S$ has a presentation $S = R[y_1, \\ldots, y_m]/(f_1, \\ldots, f_c)$ such that $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$ is free with basis given by the classes of $f_1, \\ldots, f_c$."}
+{"_id": "1179", "title": "algebra-lemma-adjoin-roots", "text": "Suppose that $A$ is a ring, and $P(x) = x^n + b_1 x^{n-1} + \\ldots + b_n \\in A[x]$ is a monic polynomial over $A$. Then there exists a syntomic, finite locally free, faithfully flat ring extension $A \\subset A'$ such that $P(x) = \\prod_{i = 1, \\ldots, n} (x - \\beta_i)$ for certain $\\beta_i \\in A'$."}
+{"_id": "1180", "title": "algebra-lemma-base-change-relative-global-complete-intersection", "text": "Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection (Definition \\ref{definition-relative-global-complete-intersection}) \\begin{enumerate} \\item For any $R \\to R'$ the base change $R' \\otimes_R S = R'[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative global complete intersection. \\item For any $g \\in S$ which is the image of $h \\in R[x_1, \\ldots, x_n]$ the ring $S_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1)$ is a relative global complete intersection. \\item If $R \\to S$ factors as $R \\to R_f \\to S$ for some $f \\in R$. Then the ring $S = R_f[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative global complete intersection over $R_f$. \\end{enumerate}"}
+{"_id": "1181", "title": "algebra-lemma-localize-relative-complete-intersection", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$. We will find $h \\in R[x_1, \\ldots, x_n]$ which maps to $g \\in S$ such that $$ S_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1) $$ is a relative global complete intersection with a presentation as in Definition \\ref{definition-relative-global-complete-intersection} in each of the following cases: \\begin{enumerate} \\item Let $I \\subset R$ be an ideal. If the fibres of $\\Spec(S/IS) \\to \\Spec(R/I)$ have dimension $n - c$, then we can find $(h, g)$ as above such that $g$ maps to $1 \\in S/IS$. \\item Let $\\mathfrak p \\subset R$ be a prime. If $\\dim(S \\otimes_R \\kappa(\\mathfrak p)) = n - c$, then we can find $(h, g)$ as above such that $g$ maps to a unit of $S \\otimes_R \\kappa(\\mathfrak p)$. \\item Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. If $\\dim_{\\mathfrak q}(S/R) = n - c$, then we can find $(h, g)$ as above such that $g \\not \\in \\mathfrak q$. \\end{enumerate}"}
+{"_id": "1182", "title": "algebra-lemma-relative-global-complete-intersection-Noetherian", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection (Definition \\ref{definition-relative-global-complete-intersection}). There exist a finite type $\\mathbf{Z}$-subalgebra $R_0 \\subset R$ such that $f_i \\in R_0[x_1, \\ldots, x_n]$ and such that $$ S_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) $$ is a relative global complete intersection."}
+{"_id": "1183", "title": "algebra-lemma-relative-global-complete-intersection-conormal", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection (Definition \\ref{definition-relative-global-complete-intersection}). For every prime $\\mathfrak q$ of $S$, let $\\mathfrak q'$ denote the corresponding prime of $R[x_1, \\ldots, x_n]$. Then \\begin{enumerate} \\item $f_1, \\ldots, f_c$ is a regular sequence in the local ring $R[x_1, \\ldots, x_n]_{\\mathfrak q'}$, \\item each of the rings $R[x_1, \\ldots, x_n]_{\\mathfrak q'}/(f_1, \\ldots, f_i)$ is flat over $R$, and \\item the $S$-module $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$ is free with basis given by the elements $f_i \\bmod (f_1, \\ldots, f_c)^2$. \\end{enumerate}"}
+{"_id": "1184", "title": "algebra-lemma-relative-global-complete-intersection", "text": "A relative global complete intersection is syntomic, i.e., flat."}
+{"_id": "1185", "title": "algebra-lemma-syntomic", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$ of $R$. The following are equivalent: \\begin{enumerate} \\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is syntomic. \\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$ such that $S_g$ is a relative global complete intersection over $R$. \\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$, such that $R \\to S_g$ is of finite presentation, the local ring map $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat, and the local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is a complete intersection ring over $\\kappa(\\mathfrak p)$ (see Definition \\ref{definition-lci-local-ring}). \\end{enumerate}"}
+{"_id": "1186", "title": "algebra-lemma-syntomic-presentation-ideal-mod-squares", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/I$ for some finitely generated ideal $I$. If $g \\in S$ is such that $S_g$ is syntomic over $R$, then $(I/I^2)_g$ is a finite projective $S_g$-module."}
+{"_id": "1187", "title": "algebra-lemma-composition-syntomic", "text": "Let $R \\to S$, $S \\to S'$ be ring maps. \\begin{enumerate} \\item If $R \\to S$ and $S \\to S'$ are syntomic, then $R \\to S'$ is syntomic. \\item If $R \\to S$ and $S \\to S'$ are relative global complete intersections, then $R \\to S'$ is a relative global complete intersection. \\end{enumerate}"}
+{"_id": "1188", "title": "algebra-lemma-lift-syntomic", "text": "Let $R$ be a ring and let $I \\subset R$ be an ideal. Let $R/I \\to \\overline{S}$ be a syntomic map. Then there exists elements $\\overline{g}_i \\in \\overline{S}$ which generate the unit ideal of $\\overline{S}$ such that each $\\overline{S}_{g_i} \\cong S_i/IS_i$ for some relative global complete intersection $S_i$ over $R$."}
+{"_id": "1191", "title": "algebra-lemma-base-change-smooth", "text": "\\begin{slogan} Smoothness is preserved under base change \\end{slogan} Let $R \\to S$ be a smooth ring map. Let $R \\to R'$ be any ring map. Then the base change $R' \\to S' = R' \\otimes_R S$ is smooth."}
+{"_id": "1192", "title": "algebra-lemma-smooth-over-field", "text": "Let $k$ be a field. Let $S$ be a smooth $k$-algebra. Then $S$ is a local complete intersection."}
+{"_id": "1193", "title": "algebra-lemma-standard-smooth", "text": "Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) = R[x_1, \\ldots, x_n]/I$ be a standard smooth algebra. Then \\begin{enumerate} \\item the ring map $R \\to S$ is smooth, \\item the $S$-module $\\Omega_{S/R}$ is free on $\\text{d}x_{c + 1}, \\ldots, \\text{d}x_n$, \\item the $S$-module $I/I^2$ is free on the classes of $f_1, \\ldots, f_c$, \\item for any $g \\in S$ the ring map $R \\to S_g$ is standard smooth, \\item for any ring map $R \\to R'$ the base change $R' \\to R'\\otimes_R S$ is standard smooth, \\item if $f \\in R$ maps to an invertible element in $S$, then $R_f \\to S$ is standard smooth, and \\item the ring $S$ is a relative global complete intersection over $R$. \\end{enumerate}"}
+{"_id": "1194", "title": "algebra-lemma-compose-standard-smooth", "text": "A composition of standard smooth ring maps is standard smooth."}
+{"_id": "1195", "title": "algebra-lemma-smooth-syntomic", "text": "Let $R \\to S$ be a smooth ring map. There exists an open covering of $\\Spec(S)$ by standard opens $D(g)$ such that each $S_g$ is standard smooth over $R$. In particular $R \\to S$ is syntomic."}
+{"_id": "1196", "title": "algebra-lemma-smooth-at-point", "text": "Let $R \\to S$ be of finite presentation. Let $\\mathfrak q$ be a prime of $S$. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is smooth at $\\mathfrak q$, \\item $H_1(L_{S/R})_\\mathfrak q = 0$ and $\\Omega_{S/R, \\mathfrak q}$ is a finite free $S_\\mathfrak q$-module, \\item $H_1(L_{S/R})_\\mathfrak q = 0$ and $\\Omega_{S/R, \\mathfrak q}$ is a projective $S_\\mathfrak q$-module, and \\item $H_1(L_{S/R})_\\mathfrak q = 0$ and $\\Omega_{S/R, \\mathfrak q}$ is a flat $S_\\mathfrak q$-module. \\end{enumerate}"}
+{"_id": "1197", "title": "algebra-lemma-locally-smooth", "text": "\\begin{slogan} A ring map is smooth if and only if it is smooth at all primes of the target \\end{slogan} Let $R \\to S$ be a ring map. Then $R \\to S$ is smooth if and only if $R \\to S$ is smooth at every prime $\\mathfrak q$ of $S$."}
+{"_id": "1198", "title": "algebra-lemma-compose-smooth", "text": "A composition of smooth ring maps is smooth."}
+{"_id": "1199", "title": "algebra-lemma-relative-global-complete-intersection-smooth", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection. Let $\\mathfrak q \\subset S$ be a prime. Then $R \\to S$ is smooth at $\\mathfrak q$ if and only if there exists a subset $I \\subset \\{1, \\ldots, n\\}$ of cardinality $c$ such that the polynomial $$ g_I = \\det (\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c, \\ i \\in I}. $$ does not map to an element of $\\mathfrak q$."}
+{"_id": "1200", "title": "algebra-lemma-flat-fibre-smooth", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$ of $R$. Assume \\begin{enumerate} \\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$ such that $R \\to S_g$ is of finite presentation, \\item the local ring homomorphism $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat, \\item the fibre $S \\otimes_R \\kappa(\\mathfrak p)$ is smooth over $\\kappa(\\mathfrak p)$ at the prime corresponding to $\\mathfrak q$. \\end{enumerate} Then $R \\to S$ is smooth at $\\mathfrak q$."}
+{"_id": "1201", "title": "algebra-lemma-flat-base-change-locus-smooth", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $R \\to R'$ be a flat ring map. Denote $S' = R' \\otimes_R S$ the base change. Let $U \\subset \\Spec(S)$ be the set of primes at which $R \\to S$ is smooth. Let $V \\subset \\Spec(S')$ the set of primes at which $R' \\to S'$ is smooth. Then $V$ is the inverse image of $U$ under the map $f : \\Spec(S') \\to \\Spec(S)$."}
+{"_id": "1202", "title": "algebra-lemma-smooth-field-change-local", "text": "Let $k \\subset K$ be a field extension. Let $S$ be a finite type algebra over $k$. Let $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$ and let $\\mathfrak q$ be the corresponding prime of $S$. Then $S$ is smooth over $k$ at $\\mathfrak q$ if and only if $S_K$ is smooth at $\\mathfrak q_K$ over $K$."}
+{"_id": "1203", "title": "algebra-lemma-lift-smooth", "text": "Let $R$ be a ring and let $I \\subset R$ be an ideal. Let $R/I \\to \\overline{S}$ be a smooth ring map. Then there exists elements $\\overline{g}_i \\in \\overline{S}$ which generate the unit ideal of $\\overline{S}$ such that each $\\overline{S}_{g_i} \\cong S_i/IS_i$ for some (standard) smooth ring $S_i$ over $R$."}
+{"_id": "1204", "title": "algebra-lemma-base-change-fs", "text": "Let $R \\to S$ be a formally smooth ring map. Let $R \\to R'$ be any ring map. Then the base change $S' = R' \\otimes_R S$ is formally smooth over $R'$."}
+{"_id": "1205", "title": "algebra-lemma-compose-formally-smooth", "text": "A composition of formally smooth ring maps is formally smooth."}
+{"_id": "1206", "title": "algebra-lemma-polynomial-ring-formally-smooth", "text": "A polynomial ring over $R$ is formally smooth over $R$."}
+{"_id": "1207", "title": "algebra-lemma-characterize-formally-smooth", "text": "Let $R \\to S$ be a ring map. Let $P \\to S$ be a surjective $R$-algebra map from a polynomial ring $P$ onto $S$. Denote $J \\subset P$ the kernel. Then $R \\to S$ is formally smooth if and only if there exists an $R$-algebra map $\\sigma : S \\to P/J^2$ which is a right inverse to the surjection $P/J^2 \\to S$."}
+{"_id": "1208", "title": "algebra-lemma-characterize-formally-smooth-again", "text": "Let $R \\to S$ be a ring map. Let $P \\to S$ be a surjective $R$-algebra map from a polynomial ring $P$ onto $S$. Denote $J \\subset P$ the kernel. Then $R \\to S$ is formally smooth if and only if the sequence $$ 0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to \\Omega_{S/R} \\to 0 $$ of Lemma \\ref{lemma-differential-seq} is a split exact sequence."}
+{"_id": "1209", "title": "algebra-lemma-ses-formally-smooth", "text": "Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is formally smooth. Then the sequence $$ 0 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0 $$ of Lemma \\ref{lemma-exact-sequence-differentials} is a split short exact sequence."}
+{"_id": "1210", "title": "algebra-lemma-differential-seq-formally-smooth", "text": "Let $A \\to B \\to C$ be ring maps with $A \\to C$ formally smooth and $B \\to C$ surjective with kernel $J \\subset B$. Then the exact sequence $$ 0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0 $$ of Lemma \\ref{lemma-differential-seq} is split exact."}
+{"_id": "1211", "title": "algebra-lemma-application-NL-formally-smooth", "text": "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so also $B \\to C$ is) and $A \\to B$ formally smooth. Denote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$. Then the sequence $$ 0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0 $$ of Lemma \\ref{lemma-application-NL} is split exact."}
+{"_id": "1212", "title": "algebra-lemma-lift-formal-smoothness", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. Assume \\begin{enumerate} \\item $I^2 = 0$, \\item $R \\to S$ is flat, and \\item $R/I \\to S/IS$ is formally smooth. \\end{enumerate} Then $R \\to S$ is formally smooth."}
+{"_id": "1213", "title": "algebra-lemma-finite-presentation-fs-Noetherian", "text": "Let $R \\to S$ be a smooth ring map. Then there exists a subring $R_0 \\subset R$ of finite type over $\\mathbf{Z}$ and a smooth ring map $R_0 \\to S_0$ such that $S \\cong R \\otimes_{R_0} S_0$."}
+{"_id": "1215", "title": "algebra-lemma-descent-formally-smooth", "text": "Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map. Set $S' = S \\otimes_R R'$. Then $R \\to S$ is formally smooth if and only if $R' \\to S'$ is formally smooth."}
+{"_id": "1216", "title": "algebra-lemma-smooth-strong-lift", "text": "Let $R \\to S$ be a smooth ring map. Given a commutative solid diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/I \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ where $I \\subset A$ is a locally nilpotent ideal, a dotted arrow exists which makes the diagram commute."}
+{"_id": "1217", "title": "algebra-lemma-triangle-differentials-smooth", "text": "Given ring maps $A \\to B \\to C$ with $B \\to C$ smooth, then the sequence $$ 0 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0 $$ of Lemma \\ref{lemma-exact-sequence-differentials} is exact."}
+{"_id": "1218", "title": "algebra-lemma-differential-seq-smooth", "text": "Let $A \\to B \\to C$ be ring maps with $A \\to C$ smooth and $B \\to C$ surjective with kernel $J \\subset B$. Then the exact sequence $$ 0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0 $$ of Lemma \\ref{lemma-differential-seq} is split exact."}
+{"_id": "1219", "title": "algebra-lemma-application-NL-smooth", "text": "Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so also $B \\to C$ is) and $A \\to B$ smooth. Denote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$. Then the sequence $$ 0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0 $$ of Lemma \\ref{lemma-application-NL} is exact."}
+{"_id": "1221", "title": "algebra-lemma-rank-omega", "text": "Let $k$ be an algebraically closed field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak m \\subset S$ be a maximal ideal. Then $$ \\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m) = \\dim_{\\kappa(\\mathfrak m)} \\mathfrak m/\\mathfrak m^2. $$"}
+{"_id": "1222", "title": "algebra-lemma-characterize-smooth-kbar", "text": "Let $k$ be an algebraically closed field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak m \\subset S$ be a maximal ideal. The following are equivalent: \\begin{enumerate} \\item The ring $S_{\\mathfrak m}$ is a regular local ring. \\item We have $\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m) \\leq \\dim(S_{\\mathfrak m})$. \\item We have $\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m) = \\dim(S_{\\mathfrak m})$. \\item There exists a $g \\in S$, $g \\not \\in \\mathfrak m$ such that $S_g$ is smooth over $k$. In other words $S/k$ is smooth at $\\mathfrak m$. \\end{enumerate}"}
+{"_id": "1223", "title": "algebra-lemma-characterize-smooth-over-field", "text": "Let $k$ be any field. Let $S$ be a finite type $k$-algebra. Let $X = \\Spec(S)$. Let $\\mathfrak q \\subset S$ be a prime corresponding to $x \\in X$. The following are equivalent: \\begin{enumerate} \\item The $k$-algebra $S$ is smooth at $\\mathfrak q$ over $k$. \\item We have $\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q) \\leq \\dim_x X$. \\item We have $\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q) = \\dim_x X$. \\end{enumerate} Moreover, in this case the local ring $S_{\\mathfrak q}$ is regular."}
+{"_id": "1224", "title": "algebra-lemma-computation-differential", "text": "Let $k$ be a field. Let $R$ be a Noetherian local ring containing $k$. Assume that the residue field $\\kappa = R/\\mathfrak m$ is a finitely generated separable extension of $k$. Then the map $$ \\text{d} : \\mathfrak m/\\mathfrak m^2 \\longrightarrow \\Omega_{R/k} \\otimes_R \\kappa(\\mathfrak m) $$ is injective."}
+{"_id": "1225", "title": "algebra-lemma-separable-smooth", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak q \\subset S$ be a prime. Assume $\\kappa(\\mathfrak q)$ is separable over $k$. The following are equivalent: \\begin{enumerate} \\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$. \\item The ring $S_{\\mathfrak q}$ is regular. \\end{enumerate}"}
+{"_id": "1226", "title": "algebra-lemma-characteristic-zero", "text": "Let $R \\to S$ be a $\\mathbf{Q}$-algebra map. Let $f \\in S$ be such that $\\Omega_{S/R} = S \\text{d}f \\oplus C$ for some $S$-submodule $C$. Then \\begin{enumerate} \\item $f$ is not nilpotent, and \\item if $S$ is a Noetherian local ring, then $f$ is a nonzerodivisor in $S$. \\end{enumerate}"}
+{"_id": "1227", "title": "algebra-lemma-characteristic-zero-local-smooth", "text": "Let $k$ be a field of characteristic $0$. Let $S$ be a finite type $k$-algebra. Let $\\mathfrak q \\subset S$ be a prime. The following are equivalent: \\begin{enumerate} \\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$. \\item The $S_{\\mathfrak q}$-module $\\Omega_{S/k, \\mathfrak q}$ is (finite) free. \\item The ring $S_{\\mathfrak q}$ is regular. \\end{enumerate}"}
+{"_id": "1228", "title": "algebra-lemma-smooth-at-generic-point", "text": "Let $R \\to S$ be an injective finite type ring map with $R$ and $S$ domains. Then $R \\to S$ is smooth at $\\mathfrak q = (0)$ if and only if the induced extension $L/K$ of fraction fields is separable."}
+{"_id": "1229", "title": "algebra-lemma-smooth-test-artinian", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime ideal of $S$ lying over $\\mathfrak p \\subset R$. Assume $R$ is Noetherian and $R \\to S$ of finite type. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is smooth at $\\mathfrak q$, \\item for every surjection of local $R$-algebras $(B', \\mathfrak m') \\to (B, \\mathfrak m)$ with $\\Ker(B' \\to B)$ having square zero and every solid commutative diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & B \\\\ R \\ar[r] \\ar[u] & B' \\ar[u] } $$ such that $\\mathfrak q = S \\cap \\mathfrak m$ there exists a dotted arrow making the diagram commute, \\item same as in (2) but with $B' \\to B$ ranging over small extensions, and \\item same as in (2) but with $B' \\to B$ ranging over small extensions such that in addition $S \\to B$ induces an isomorphism $\\kappa(\\mathfrak q) \\cong \\kappa(\\mathfrak m)$. \\end{enumerate}"}
+{"_id": "1230", "title": "algebra-lemma-etale-standard-smooth", "text": "Any \\'etale ring map is standard smooth. More precisely, if $R \\to S$ is \\'etale, then there exists a presentation $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ such that the image of $\\det(\\partial f_j/\\partial x_i)$ is invertible in $S$."}
+{"_id": "1231", "title": "algebra-lemma-etale", "text": "Results on \\'etale ring maps. \\begin{enumerate} \\item The ring map $R \\to R_f$ is \\'etale for any ring $R$ and any $f \\in R$. \\item Compositions of \\'etale ring maps are \\'etale. \\item A base change of an \\'etale ring map is \\'etale. \\item The property of being \\'etale is local: Given a ring map $R \\to S$ and elements $g_1, \\ldots, g_m \\in S$ which generate the unit ideal such that $R \\to S_{g_j}$ is \\'etale for $j = 1, \\ldots, m$ then $R \\to S$ is \\'etale. \\item Given $R \\to S$ of finite presentation, and a flat ring map $R \\to R'$, set $S' = R' \\otimes_R S$. The set of primes where $R' \\to S'$ is \\'etale is the inverse image via $\\Spec(S') \\to \\Spec(S)$ of the set of primes where $R \\to S$ is \\'etale. \\item An \\'etale ring map is syntomic, in particular flat. \\item If $S$ is finite type over a field $k$, then $S$ is \\'etale over $k$ if and only if $\\Omega_{S/k} = 0$. \\item Any \\'etale ring map $R \\to S$ is the base change of an \\'etale ring map $R_0 \\to S_0$ with $R_0$ of finite type over $\\mathbf{Z}$. \\item Let $A = \\colim A_i$ be a filtered colimit of rings. Let $A \\to B$ be an \\'etale ring map. Then there exists an \\'etale ring map $A_i \\to B_i$ for some $i$ such that $B \\cong A \\otimes_{A_i} B_i$. \\item Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$. Let $S^{-1}A \\to B'$ be \\'etale. Then there exists an \\'etale ring map $A \\to B$ such that $B' \\cong S^{-1}B$. \\end{enumerate}"}
+{"_id": "1232", "title": "algebra-lemma-etale-over-field", "text": "Let $k$ be a field. A ring map $k \\to S$ is \\'etale if and only if $S$ is isomorphic as a $k$-algebra to a finite product of finite separable extensions of $k$."}
+{"_id": "1233", "title": "algebra-lemma-etale-at-prime", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$ in $R$. If $S/R$ is \\'etale at $\\mathfrak q$ then \\begin{enumerate} \\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$ is the maximal ideal of the local ring $S_{\\mathfrak q}$, and \\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite separable. \\end{enumerate}"}
+{"_id": "1234", "title": "algebra-lemma-etale-quasi-finite", "text": "An \\'etale ring map is quasi-finite."}
+{"_id": "1235", "title": "algebra-lemma-characterize-etale", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over a prime $\\mathfrak p$ of $R$. If \\begin{enumerate} \\item $R \\to S$ is of finite presentation, \\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat \\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal of the local ring $S_{\\mathfrak q}$, and \\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite separable, \\end{enumerate} then $R \\to S$ is \\'etale at $\\mathfrak q$."}
+{"_id": "1236", "title": "algebra-lemma-map-between-etale", "text": "Let $R \\to S$ and $R \\to S'$ be \\'etale. Then any $R$-algebra map $S' \\to S$ is \\'etale."}
+{"_id": "1237", "title": "algebra-lemma-surjective-flat-finitely-presented", "text": "Let $\\varphi : R \\to S$ be a ring map. If $R \\to S$ is surjective, flat and finitely presented then there exist an idempotent $e \\in R$ such that $S = R_e$."}
+{"_id": "1238", "title": "algebra-lemma-lift-etale", "text": "\\begin{slogan} \\'Etale ring maps lift along surjections of rings \\end{slogan} Let $R$ be a ring and let $I \\subset R$ be an ideal. Let $R/I \\to \\overline{S}$ be an \\'etale ring map. Then there exists an \\'etale ring map $R \\to S$ such that $\\overline{S} \\cong S/IS$ as $R/I$-algebras."}
+{"_id": "1239", "title": "algebra-lemma-lift-etale-infinitesimal", "text": "Consider a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & J \\ar[r] & B' \\ar[r] & B \\ar[r] & 0 \\\\ 0 \\ar[r] & I \\ar[r] \\ar[u] & A' \\ar[r] \\ar[u] & A \\ar[r] \\ar[u] & 0 } $$ with exact rows where $B' \\to B$ and $A' \\to A$ are surjective ring maps whose kernels are ideals of square zero. If $A \\to B$ is \\'etale, and $J = I \\otimes_A B$, then $A' \\to B'$ is \\'etale."}
+{"_id": "1240", "title": "algebra-lemma-factor-mod-lift-etale", "text": "Let $R$ be a ring. Let $f \\in R[x]$ be a monic polynomial. Let $\\mathfrak p$ be a prime of $R$. Let $f \\bmod \\mathfrak p = \\overline{g} \\overline{h}$ be a factorization of the image of $f$ in $\\kappa(\\mathfrak p)[x]$. If $\\gcd(\\overline{g}, \\overline{h}) = 1$, then there exist \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, and \\item a factorization $f = g h$ in $R'[x]$ \\end{enumerate} such that \\begin{enumerate} \\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$, \\item $\\overline{g} = g \\bmod \\mathfrak p'$, $\\overline{h} = h \\bmod \\mathfrak p'$, and \\item the polynomials $g, h$ generate the unit ideal in $R'[x]$. \\end{enumerate}"}
+{"_id": "1241", "title": "algebra-lemma-standard-etale", "text": "Let $R \\to R[x]_g/(f)$ be standard \\'etale. \\begin{enumerate} \\item The ring map $R \\to R[x]_g/(f)$ is \\'etale. \\item For any ring map $R \\to R'$ the base change $R' \\to R'[x]_g/(f)$ of the standard \\'etale ring map $R \\to R[x]_g/(f)$ is standard \\'etale. \\item Any principal localization of $R[x]_g/(f)$ is standard \\'etale over $R$. \\item A composition of standard \\'etale maps is {\\bf not} standard \\'etale in general. \\end{enumerate}"}
+{"_id": "1242", "title": "algebra-lemma-make-etale-map-prescribed-residue-field", "text": "Let $R$ be a ring. Let $\\mathfrak p$ be a prime of $R$. Let $\\kappa(\\mathfrak p) \\subset L$ be a finite separable field extension. There exists an \\'etale ring map $R \\to R'$ together with a prime $\\mathfrak p'$ lying over $\\mathfrak p$ such that the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak p')$ is isomorphic to $\\kappa(\\mathfrak p) \\subset L$."}
+{"_id": "1243", "title": "algebra-lemma-standard-etale-finite-flat-Zariski", "text": "Let $R \\to S$ be a standard \\'etale morphism. There exists a ring map $R \\to S'$ with the following properties \\begin{enumerate} \\item $R \\to S'$ is finite, finitely presented, and flat (in other words $S'$ is finite projective as an $R$-module), \\item $\\Spec(S') \\to \\Spec(R)$ is surjective, \\item for every prime $\\mathfrak q \\subset S$, lying over $\\mathfrak p \\subset R$ and every prime $\\mathfrak q' \\subset S'$ lying over $\\mathfrak p$ there exists a $g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that the ring map $R \\to S'_{g'}$ factors through a map $\\varphi : S \\to S'_{g'}$ with $\\varphi^{-1}(\\mathfrak q'S'_{g'}) = \\mathfrak q$. \\end{enumerate}"}
+{"_id": "1244", "title": "algebra-lemma-etale-finite-flat-zariski", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is \\'etale, and \\item $\\Spec(S) \\to \\Spec(R)$ is surjective. \\end{enumerate} Then there exists a ring map $R \\to S'$ such that \\begin{enumerate} \\item $R \\to S'$ is finite, finitely presented, and flat (in other words it is finite projective as an $R$-module), \\item $\\Spec(S') \\to \\Spec(R)$ is surjective, \\item for every prime $\\mathfrak q' \\subset S'$ there exists a $g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that the ring map $R \\to S'_{g'}$ factors as $R \\to S \\to S'_{g'}$. \\end{enumerate}"}
+{"_id": "1245", "title": "algebra-lemma-produce-finite", "text": "Let $R \\to S' \\to S$ be ring maps. Let $\\mathfrak p \\subset R$ be a prime. Let $g \\in S'$ be an element. Assume \\begin{enumerate} \\item $R \\to S'$ is integral, \\item $R \\to S$ is finite type, \\item $S'_g \\cong S_g$, and \\item $g$ invertible in $S' \\otimes_R \\kappa(\\mathfrak p)$. \\end{enumerate} Then there exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that $R_f \\to S_f$ is finite."}
+{"_id": "1246", "title": "algebra-lemma-etale-makes-quasi-finite-finite-one-prime", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p \\subset R$. Assume $R \\to S$ finite type and quasi-finite at $\\mathfrak q$. Then there exists \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, \\item a product decomposition $$ R' \\otimes_R S = A \\times B $$ \\end{enumerate} with the following properties \\begin{enumerate} \\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$, \\item $R' \\to A$ is finite, \\item $A$ has exactly one prime $\\mathfrak r$ lying over $\\mathfrak p'$, and \\item $\\mathfrak r$ lies over $\\mathfrak q$. \\end{enumerate}"}
+{"_id": "1247", "title": "algebra-lemma-etale-makes-quasi-finite-finite", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak p \\subset R$ be a prime. Assume $R \\to S$ finite type. Then there exists \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, \\item a product decomposition $$ R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B $$ \\end{enumerate} with the following properties \\begin{enumerate} \\item we have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$, \\item each $A_i$ is finite over $R'$, \\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over $\\mathfrak p'$, and \\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$. \\end{enumerate}"}
+{"_id": "1248", "title": "algebra-lemma-etale-makes-quasi-finite-finite-variant", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak p \\subset R$ be a prime. Assume $R \\to S$ finite type. Then there exists \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, \\item a product decomposition $$ R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B $$ \\end{enumerate} with the following properties \\begin{enumerate} \\item each $A_i$ is finite over $R'$, \\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over $\\mathfrak p'$, \\item the finite field extensions $\\kappa(\\mathfrak p') \\subset \\kappa(\\mathfrak r_i)$ are purely inseparable, and \\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$. \\end{enumerate}"}
+{"_id": "1250", "title": "algebra-lemma-trick", "text": "Let $R$ be a ring. Let $f \\in R[x]$ be a monic polynomial. Let $R \\to B$ be a ring map. If $h \\in B[x]/(f)$ is integral over $R$, then the element $f' h$ can be written as $f'h = \\sum_i b_i x^i$ with $b_i \\in B$ integral over $R$."}
+{"_id": "1251", "title": "algebra-lemma-integral-closure-commutes-etale", "text": "Let $R \\to S$ be an \\'etale ring map. Let $R \\to B$ be any ring map. Let $A \\subset B$ be the integral closure of $R$ in $B$. Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in $S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is an isomorphism."}
+{"_id": "1252", "title": "algebra-lemma-integral-closure-commutes-smooth", "text": "Let $R \\to S$ be a smooth ring map. Let $R \\to B$ be any ring map. Let $A \\subset B$ be the integral closure of $R$ in $B$. Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in $S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is an isomorphism."}
+{"_id": "1253", "title": "algebra-lemma-integral-closure-commutes-colim-smooth", "text": "Let $R \\to S$ and $R \\to B$ be ring maps. Let $A \\subset B$ be the integral closure of $R$ in $B$. Let $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in $S \\otimes_R B$. If $S$ is a filtered colimit of smooth $R$-algebras, then the canonical map $S \\otimes_R A \\to A'$ is an isomorphism."}
+{"_id": "1254", "title": "algebra-lemma-characterize-formally-unramified", "text": "Let $R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is formally unramified, \\item the module of differentials $\\Omega_{S/R}$ is zero. \\end{enumerate}"}
+{"_id": "1255", "title": "algebra-lemma-formally-unramified-local", "text": "Let $R \\to S$ be a ring map. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is formally unramified, \\item $R \\to S_{\\mathfrak q}$ is formally unramified for all primes $\\mathfrak q$ of $S$, and \\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is formally unramified for all primes $\\mathfrak q$ of $S$ with $\\mathfrak p = R \\cap \\mathfrak q$. \\end{enumerate}"}
+{"_id": "1256", "title": "algebra-lemma-formally-unramified-localize", "text": "Let $A \\to B$ be a formally unramified ring map. \\begin{enumerate} \\item For $S \\subset A$ a multiplicative subset, $S^{-1}A \\to S^{-1}B$ is formally unramified. \\item For $S \\subset B$ a multiplicative subset, $A \\to S^{-1}B$ is formally unramified. \\end{enumerate}"}
+{"_id": "1258", "title": "algebra-lemma-universal-thickening", "text": "Let $R \\to S$ be a formally unramified ring map. There exists a surjection of $R$-algebras $S' \\to S$ whose kernel is an ideal of square zero with the following universal property: Given any commutative diagram $$ \\xymatrix{ S \\ar[r]_a & A/I \\\\ R \\ar[r]^b \\ar[u] & A \\ar[u] } $$ where $I \\subset A$ is an ideal of square zero, there is a unique $R$-algebra map $a' : S' \\to A$ such that $S' \\to A \\to A/I$ is equal to $S' \\to S \\to A/I$."}
+{"_id": "1259", "title": "algebra-lemma-universal-thickening-quotient", "text": "Let $I \\subset R$ be an ideal of a ring. The universal first order thickening of $R/I$ over $R$ is the surjection $R/I^2 \\to R/I$. The conormal module of $R/I$ over $R$ is $C_{(R/I)/R} = I/I^2$."}
+{"_id": "1260", "title": "algebra-lemma-universal-thickening-localize", "text": "Let $A \\to B$ be a formally unramified ring map. Let $\\varphi : B' \\to B$ be the universal first order thickening of $B$ over $A$. \\begin{enumerate} \\item Let $S \\subset A$ be a multiplicative subset. Then $S^{-1}B' \\to S^{-1}B$ is the universal first order thickening of $S^{-1}B$ over $S^{-1}A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/S^{-1}A}$. \\item Let $S \\subset B$ be a multiplicative subset. Then $S' = \\varphi^{-1}(S)$ is a multiplicative subset in $B'$ and $(S')^{-1}B' \\to S^{-1}B$ is the universal first order thickening of $S^{-1}B$ over $A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/A}$. \\end{enumerate} Note that the lemma makes sense by Lemma \\ref{lemma-formally-unramified-localize}."}
+{"_id": "1261", "title": "algebra-lemma-differentials-universal-thickening", "text": "Let $R \\to A  \\to B$ be ring maps. Assume $A \\to B$ formally unramified. Let $B' \\to B$ be the universal first order thickening of $B$ over $A$. Then $B'$ is formally unramified over $A$, and the canonical map $\\Omega_{A/R} \\otimes_A B \\to \\Omega_{B'/R} \\otimes_{B'} B$ is an isomorphism."}
+{"_id": "1262", "title": "algebra-lemma-formally-etale-etale", "text": "Let $R \\to S$ be a ring map of finite presentation. The following are equivalent: \\begin{enumerate} \\item $R \\to S$ is formally \\'etale, \\item $R \\to S$ is \\'etale. \\end{enumerate}"}
+{"_id": "1263", "title": "algebra-lemma-colimit-formally-etale", "text": "Let $R$ be a ring. Let $I$ be a directed set. Let $(S_i, \\varphi_{ii'})$ be a system of $R$-algebras over $I$. If each $R \\to S_i$ is formally \\'etale, then $S = \\colim_{i \\in I} S_i$ is formally \\'etale over $R$"}
+{"_id": "1264", "title": "algebra-lemma-localization-formally-etale", "text": "Let $R$ be a ring. Let $S \\subset R$ be any multiplicative subset. Then the ring map $R \\to S^{-1}R$ is formally \\'etale."}
+{"_id": "1265", "title": "algebra-lemma-formally-unramified-unramified", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is formally unramified and of finite type, and \\item $R \\to S$ is unramified. \\end{enumerate} Moreover, also the following are equivalent \\begin{enumerate} \\item $R \\to S$ is formally unramified and of finite presentation, and \\item $R \\to S$ is G-unramified. \\end{enumerate}"}
+{"_id": "1266", "title": "algebra-lemma-unramified", "text": "Properties of unramified and G-unramified ring maps. \\begin{enumerate} \\item The base change of an unramified ring map is unramified. The base change of a G-unramified ring map is G-unramified. \\item The composition of unramified ring maps is unramified. The composition of G-unramified ring maps is G-unramified. \\item Any principal localization $R \\to R_f$ is G-unramified and unramified. \\item If $I \\subset R$ is an ideal, then $R \\to R/I$ is unramified. If $I \\subset R$ is a finitely generated ideal, then $R \\to R/I$ is G-unramified. \\item An \\'etale ring map is G-unramified and unramified. \\item If $R \\to S$ is of finite type (resp.\\ finite presentation), $\\mathfrak q \\subset S$ is a prime and $(\\Omega_{S/R})_{\\mathfrak q} = 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$. \\item If $R \\to S$ is of finite type (resp.\\ finite presentation), $\\mathfrak q \\subset S$ is a prime and $\\Omega_{S/R} \\otimes_S \\kappa(\\mathfrak q) = 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$. \\item If $R \\to S$ is of finite type (resp.\\ finite presentation), $\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and $(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)})_{\\mathfrak q} = 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$. \\item If $R \\to S$ is of finite type (resp.\\ presentation), $\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and $(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)}) \\otimes_{S \\otimes_R \\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) = 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$. \\item If $R \\to S$ is a ring map, $g_1, \\ldots, g_m \\in S$ generate the unit ideal and $R \\to S_{g_j}$ is unramified (resp.\\ G-unramified) for $j = 1, \\ldots, m$, then $R \\to S$ is unramified (resp.\\ G-unramified). \\item If $R \\to S$ is a ring map which is unramified (resp.\\ G-unramified) at every prime of $S$, then $R \\to S$ is unramified (resp.\\ G-unramified). \\item If $R \\to S$ is G-unramified, then there exists a finite type $\\mathbf{Z}$-algebra $R_0$ and a G-unramified ring map $R_0 \\to S_0$ and a ring map $R_0 \\to R$ such that $S = R \\otimes_{R_0} S_0$. \\item If $R \\to S$ is unramified, then there exists a finite type $\\mathbf{Z}$-algebra $R_0$ and an unramified ring map $R_0 \\to S_0$ and a ring map $R_0 \\to R$ such that $S$ is a quotient of $R \\otimes_{R_0} S_0$. \\end{enumerate}"}
+{"_id": "1267", "title": "algebra-lemma-diagonal-unramified", "text": "Let $R \\to S$ be a ring map. If $R \\to S$ is unramified, then there exists an idempotent $e \\in S \\otimes_R S$ such that $S \\otimes_R S \\to S$ is isomorphic to $S \\otimes_R S \\to (S \\otimes_R S)_e$."}
+{"_id": "1268", "title": "algebra-lemma-unramified-at-prime", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$ in $R$. If $S/R$ is unramified at $\\mathfrak q$ then \\begin{enumerate} \\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$ is the maximal ideal of the local ring $S_{\\mathfrak q}$, and \\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite separable. \\end{enumerate}"}
+{"_id": "1269", "title": "algebra-lemma-unramified-quasi-finite", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q$ be a prime of $S$. If $R \\to S$ is unramified at $\\mathfrak q$ then $R \\to S$ is quasi-finite at $\\mathfrak q$. In particular, an unramified ring map is quasi-finite."}
+{"_id": "1270", "title": "algebra-lemma-characterize-unramified", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over a prime $\\mathfrak p$ of $R$. If \\begin{enumerate} \\item $R \\to S$ is of finite type, \\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal of the local ring $S_{\\mathfrak q}$, and \\item the field extension $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is finite separable, \\end{enumerate} then $R \\to S$ is unramified at $\\mathfrak q$."}
+{"_id": "1271", "title": "algebra-lemma-etale-flat-unramified-finite-presentation", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is \\'etale, \\item $R \\to S$ is flat and G-unramified, and \\item $R \\to S$ is flat, unramified, and of finite presentation. \\end{enumerate}"}
+{"_id": "1272", "title": "algebra-lemma-characterize-etale-over-polynomial-ring", "text": "Let $k$ be a field. Let $$ \\varphi : k[x_1, \\ldots, x_n] \\to A, \\quad x_i \\longmapsto a_i $$ be a finite type ring map. Then $\\varphi$ is \\'etale if and only if we have the following two conditions: (a) the local rings of $A$ at maximal ideals have dimension $n$, and (b) the elements $\\text{d}(a_1), \\ldots, \\text{d}(a_n)$ generate $\\Omega_{A/k}$ as an $A$-module."}
+{"_id": "1274", "title": "algebra-lemma-etale-makes-unramified-closed", "text": "\\begin{slogan} In an unramified ring map, one can separate the points in a fiber by passing to an \\'etale neighbourhood. \\end{slogan} Let $R \\to S$ be a ring map. Let $\\mathfrak p$ be a prime of $R$. If $R \\to S$ is unramified then there exist \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$. \\item a product decomposition $$ R' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B $$ \\end{enumerate} with the following properties \\begin{enumerate} \\item $R' \\to A_i$ is surjective, \\item $\\mathfrak p'A_i$ is a prime of $A_i$ lying over $\\mathfrak p'$, and \\item there is no prime of $B$ lying over $\\mathfrak p'$. \\end{enumerate}"}
+{"_id": "1275", "title": "algebra-lemma-uniqueness", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $f \\in R[T]$. Let $a, b \\in R$ such that $f(a) = f(b) = 0$, $a = b \\bmod \\mathfrak m$, and $f'(a) \\not \\in \\mathfrak m$. Then $a = b$."}
+{"_id": "1276", "title": "algebra-lemma-characterize-henselian", "text": "\\begin{slogan} Characterizations of henselian local rings \\end{slogan} Let $(R, \\mathfrak m, \\kappa)$ be a local ring. The following are equivalent \\begin{enumerate} \\item $R$ is henselian, \\item for every $f \\in R[T]$ and every root $a_0 \\in \\kappa$ of $\\overline{f}$ such that $\\overline{f'}(a_0) \\not = 0$ there exists an $a \\in R$ such that $f(a) = 0$ and $a_0 = \\overline{a}$, \\item for any monic $f \\in R[T]$ and any factorization $\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$, \\item for any monic $f \\in R[T]$ and any factorization $\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and moreover $\\deg_T(g) = \\deg_T(g_0)$, \\item for any $f \\in R[T]$ and any factorization $\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$, \\item for any $f \\in R[T]$ and any factorization $\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and moreover $\\deg_T(g) = \\deg_T(g_0)$, \\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$ lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$ there exists a section $\\tau : S \\to R$ of $R \\to S$, \\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$ lying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$ there exists a section $\\tau : S \\to R$ of $R \\to S$ with $\\mathfrak q = \\tau^{-1}(\\mathfrak m)$, \\item any finite $R$-algebra is a product of local rings, \\item any finite $R$-algebra is a finite product of local rings, \\item any finite type $R$-algebra $S$ can be written as $A \\times B$ with $R \\to A$ finite and $R \\to B$ not quasi-finite at any prime lying over $\\mathfrak m$, \\item any finite type $R$-algebra $S$ can be written as $A \\times B$ with $R \\to A$ finite such that each irreducible component of $\\Spec(B \\otimes_R \\kappa)$ has dimension $\\geq 1$, and \\item any quasi-finite $R$-algebra $S$ can be written as $S = A \\times B$ with $R \\to A$ finite such that $B \\otimes_R \\kappa = 0$. \\end{enumerate}"}
+{"_id": "1277", "title": "algebra-lemma-finite-over-henselian", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring. \\begin{enumerate} \\item If $R \\subset S$ is a finite ring extension then $S$ is a finite product of henselian local rings. \\item If $R \\subset S$ is a finite local homomorphism of local rings, then $S$ is a henselian local ring. \\item If $R \\to S$ is a finite type ring map, and $\\mathfrak q$ is a prime of $S$ lying over $\\mathfrak m$ at which $R \\to S$ is quasi-finite, then $S_{\\mathfrak q}$ is henselian. \\item If $R \\to S$ is quasi-finite then $S_{\\mathfrak q}$ is henselian for every prime $\\mathfrak q$ lying over $\\mathfrak m$. \\end{enumerate}"}
+{"_id": "1278", "title": "algebra-lemma-mop-up", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring. Any finite type $R$-algebra $S$ can be written as $S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local and finite over $R$ and $R \\to B$ not quasi-finite at any prime of $B$ lying over $\\mathfrak m$."}
+{"_id": "1279", "title": "algebra-lemma-mop-up-strictly-henselian", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring. Any finite type $R$-algebra $S$ can be written as $S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local and finite over $R$ and $\\kappa \\subset \\kappa(\\mathfrak m_{A_i})$ finite purely inseparable and $R \\to B$ not quasi-finite at any prime of $B$ lying over $\\mathfrak m$."}
+{"_id": "1280", "title": "algebra-lemma-henselian-cat-finite-etale", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring. The category of finite \\'etale ring extensions $R \\to S$ is equivalent to the category of finite \\'etale algebras $\\kappa \\to \\overline{S}$ via the functor $S \\mapsto S/\\mathfrak mS$."}
+{"_id": "1282", "title": "algebra-lemma-complete-henselian", "text": "\\begin{slogan} Complete local rings are Henselian by Newton's method \\end{slogan} Let $(R, \\mathfrak m, \\kappa)$ be a complete local ring, see Definition \\ref{definition-complete-local-ring}. Then $R$ is henselian."}
+{"_id": "1283", "title": "algebra-lemma-local-dimension-zero-henselian", "text": "\\begin{slogan} Local rings of dimension zero are henselian. \\end{slogan} Let $(R, \\mathfrak m)$ be a local ring of dimension $0$. Then $R$ is henselian."}
+{"_id": "1284", "title": "algebra-lemma-map-into-henselian", "text": "Let $R \\to S$ be a ring map with $S$ henselian local. Given \\begin{enumerate} \\item an \\'etale ring map $R \\to A$, \\item a prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p = R \\cap \\mathfrak m_S$, \\item a $\\kappa(\\mathfrak p)$-algebra map $\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$, \\end{enumerate} then there exists a unique homomorphism of $R$-algebras $f : A \\to S$ such that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and $f \\bmod \\mathfrak q = \\tau$."}
+{"_id": "1285", "title": "algebra-lemma-strictly-henselian-solutions", "text": "Let $\\varphi : R \\to S$ be a local homomorphism of strictly henselian local rings. Let $P_1, \\ldots, P_n \\in R[x_1, \\ldots, x_n]$ be polynomials such that $R[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n)$ is \\'etale over $R$. Then the map $$ R^n \\longrightarrow S^n, \\quad (h_1, \\ldots, h_n) \\longmapsto (\\varphi(h_1), \\ldots, \\varphi(h_n)) $$ induces a bijection between $$ \\{ (r_1, \\ldots, r_n) \\in R^n \\mid P_i(r_1, \\ldots, r_n) = 0, \\ i = 1, \\ldots, n \\} $$ and $$ \\{ (s_1, \\ldots, s_n) \\in S^n \\mid P'_i(s_1, \\ldots, s_n) = 0, \\ i = 1, \\ldots, n \\} $$ where $P'_i \\in S[x_1, \\ldots, x_n]$ are the images of the $P_i$ under $\\varphi$."}
+{"_id": "1286", "title": "algebra-lemma-split-ML-henselian", "text": "Let $R$ be a henselian local ring. Any countably generated Mittag-Leffler module over $R$ is a direct sum of finitely presented $R$-modules."}
+{"_id": "1287", "title": "algebra-lemma-base-change-colimit-etale", "text": "Let $R \\to A$ and $R \\to R'$ be ring maps. If $A$ is a filtered colimit of \\'etale ring maps, then so is $R' \\to R' \\otimes_R A$."}
+{"_id": "1288", "title": "algebra-lemma-composition-colimit-etale", "text": "Let $A \\to B \\to C$ be ring maps. If $A \\to B$ is a filtered colimit of \\'etale ring maps and $B \\to C$ is a filtered colimit of \\'etale ring maps, then $A \\to C$ is a filtered colimit of \\'etale ring maps."}
+{"_id": "1289", "title": "algebra-lemma-colimit-colimit-etale", "text": "Let $R$ be a ring. Let $A = \\colim A_i$ be a filtered colimit of $R$-algebras such that each $A_i$ is a filtered colimit of \\'etale $R$-algebras. Then $A$ is a filtered colimit of \\'etale $R$-algebras."}
+{"_id": "1290", "title": "algebra-lemma-colimits-of-etale", "text": "Let $R$ be a ring. Let $A \\to B$ be an $R$-algebra homomorphism. If $A$ and $B$ are filtered colimits of \\'etale $R$-algebras, then $B$ is a filtered colimit of \\'etale $A$-algebras."}
+{"_id": "1291", "title": "algebra-lemma-map-into-henselian-colimit", "text": "Let $R \\to S$ be a ring map with $S$ henselian local. Given \\begin{enumerate} \\item an $R$-algebra $A$ which is a filtered colimit of \\'etale $R$-algebras, \\item a prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p = R \\cap \\mathfrak m_S$, \\item a $\\kappa(\\mathfrak p)$-algebra map $\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$, \\end{enumerate} then there exists a unique homomorphism of $R$-algebras $f : A \\to S$ such that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and $f \\bmod \\mathfrak q = \\tau$."}
+{"_id": "1292", "title": "algebra-lemma-uniqueness-henselian", "text": "Let $R$ be a ring. Given a commutative diagram of ring maps $$ \\xymatrix{ S \\ar[r] & K \\\\ R \\ar[u] \\ar[r] & S' \\ar[u] } $$ where $S$, $S'$ are henselian local, $S$, $S'$ are filtered colimits of \\'etale $R$-algebras, $K$ is a field and the arrows $S \\to K$ and  $S' \\to K$ identify $K$ with the residue field of both $S$ and $S'$. Then there exists an unique $R$-algebra isomorphism $S \\to S'$ compatible with the maps to $K$."}
+{"_id": "1293", "title": "algebra-lemma-colimit-henselian", "text": "A filtered colimit of henselian local rings along local homomorphisms is henselian."}
+{"_id": "1294", "title": "algebra-lemma-henselization", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. There exists a local ring map $R \\to R^h$ with the following properties \\begin{enumerate} \\item $R^h$ is henselian, \\item $R^h$ is a filtered colimit of \\'etale $R$-algebras, \\item $\\mathfrak m R^h$ is the maximal ideal of $R^h$, and \\item $\\kappa = R^h/\\mathfrak m R^h$. \\end{enumerate}"}
+{"_id": "1295", "title": "algebra-lemma-strict-henselization", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $\\kappa \\subset \\kappa^{sep}$ be a separable algebraic closure. There exists a commutative diagram $$ \\xymatrix{ \\kappa \\ar[r] & \\kappa \\ar[r] & \\kappa^{sep} \\\\ R \\ar[r] \\ar[u] & R^h \\ar[r] \\ar[u] & R^{sh} \\ar[u] } $$ with the following properties \\begin{enumerate} \\item the map $R^h \\to R^{sh}$ is local \\item $R^{sh}$ is strictly henselian, \\item $R^{sh}$ is a filtered colimit of \\'etale $R$-algebras, \\item $\\mathfrak m R^{sh}$ is the maximal ideal of $R^{sh}$, and \\item $\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$. \\end{enumerate}"}
+{"_id": "1297", "title": "algebra-lemma-henselian-functorial", "text": "Let $R \\to S$ be a local map of local rings. Let $R \\to R^h$ and $S \\to S^h$ be the henselizations. There exists a unique local ring map $R^h \\to S^h$ fitting into the commutative diagram $$ \\xymatrix{ R^h \\ar[r]_f & S^h \\\\ R \\ar[u] \\ar[r] & S \\ar[u] } $$"}
+{"_id": "1298", "title": "algebra-lemma-henselization-different", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Consider the category of pairs $(S, \\mathfrak q)$ where $R \\to S$ is \\'etale and $\\mathfrak q$ is a prime lying over $\\mathfrak p$ such that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$. This category is filtered and $$ (R_{\\mathfrak p})^h = \\colim_{(S, \\mathfrak q)} S = \\colim_{(S, \\mathfrak q)} S_{\\mathfrak q} $$ canonically."}
+{"_id": "1299", "title": "algebra-lemma-henselian-functorial-improve", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. Let $R \\to R^h$ and $S \\to S^h$ be the henselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$. The local ring map $R^h \\to S^h$ of Lemma \\ref{lemma-henselian-functorial} identifies $S^h$ with the henselization of $R^h \\otimes_R S$ at the unique prime lying over $\\mathfrak m^h$ and $\\mathfrak q$."}
+{"_id": "1300", "title": "algebra-lemma-quasi-finite-henselization", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$. Assume $R \\to S$ is quasi-finite at $\\mathfrak q$. The commutative diagram $$ \\xymatrix{ R_{\\mathfrak p}^h \\ar[r] & S_{\\mathfrak q}^h \\\\ R_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u] } $$ of Lemma \\ref{lemma-henselian-functorial} identifies $S_{\\mathfrak q}^h$ with the localization of $R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$ at the prime generated by $\\mathfrak q$."}
+{"_id": "1301", "title": "algebra-lemma-quotient-henselization", "text": "\\begin{slogan} Henselization is compatible with quotients. \\end{slogan} Let $R$ be a local ring with henselization $R^h$. Let $I \\subset \\mathfrak m_R$. Then $R^h/IR^h$ is the henselization of $R/I$."}
+{"_id": "1302", "title": "algebra-lemma-strictly-henselian-functorial-prepare", "text": "Let $\\varphi : R \\to S$ be a local map of local rings. Let $S/\\mathfrak m_S \\subset \\kappa^{sep}$ be a separable algebraic closure. Let $S \\to S^{sh}$ be the strict henselization of $S$ with respect to $S/\\mathfrak m_S \\subset \\kappa^{sep}$. Let $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$ be a prime of $A$ lying over $\\mathfrak m_R$. Given any commutative diagram $$ \\xymatrix{ \\kappa(\\mathfrak q) \\ar[r]_{\\phi} & \\kappa^{sep} \\\\ R/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u] } $$ there exists a unique morphism of rings $f : A \\to S^{sh}$ fitting into the commutative diagram $$ \\xymatrix{ A \\ar[r]_f & S^{sh} \\\\ R \\ar[u] \\ar[r]^{\\varphi} & S \\ar[u] } $$ such that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$ and the induced map $\\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is the given one."}
+{"_id": "1303", "title": "algebra-lemma-strictly-henselian-functorial", "text": "Let $R \\to S$ be a local map of local rings. Choose separable algebraic closures $R/\\mathfrak m_R \\subset \\kappa_1^{sep}$ and $S/\\mathfrak m_S \\subset \\kappa_2^{sep}$. Let $R \\to R^{sh}$ and $S \\to S^{sh}$ be the corresponding strict henselizations. Given any commutative diagram $$ \\xymatrix{ \\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\ R/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u] } $$ There exists a unique local ring map $R^{sh} \\to S^{sh}$ fitting into the commutative diagram $$ \\xymatrix{ R^{sh} \\ar[r]_f & S^{sh} \\\\ R \\ar[u] \\ar[r] & S \\ar[u] } $$ and inducing $\\phi$ on the residue fields of $R^{sh}$ and $S^{sh}$."}
+{"_id": "1304", "title": "algebra-lemma-strict-henselization-different", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $\\kappa(\\mathfrak p) \\subset \\kappa^{sep}$ be a separable algebraic closure. Consider the category of triples $(S, \\mathfrak q, \\phi)$ where $R \\to S$ is \\'etale, $\\mathfrak q$ is a prime lying over $\\mathfrak p$, and $\\phi : \\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is a $\\kappa(\\mathfrak p)$-algebra map. This category is filtered and $$ (R_{\\mathfrak p})^{sh} = \\colim_{(S, \\mathfrak q, \\phi)} S = \\colim_{(S, \\mathfrak q, \\phi)} S_{\\mathfrak q} $$ canonically."}
+{"_id": "1305", "title": "algebra-lemma-strictly-henselian-functorial-improve", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$. Choose separable algebraic closures $\\kappa(\\mathfrak p) \\subset \\kappa_1^{sep}$ and $\\kappa(\\mathfrak q) \\subset \\kappa_2^{sep}$. Let $R^{sh}$ and $S^{sh}$ be the corresponding strict henselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$. Given any commutative diagram $$ \\xymatrix{ \\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\ \\kappa(\\mathfrak p) \\ar[r]^{\\varphi} \\ar[u] & \\kappa(\\mathfrak q) \\ar[u] } $$ The local ring map $R^{sh} \\to S^{sh}$ of Lemma \\ref{lemma-strictly-henselian-functorial} identifies $S^{sh}$ with the strict henselization of $R^{sh} \\otimes_R S$ at a prime lying over $\\mathfrak q$ and the maximal ideal $\\mathfrak m^{sh} \\subset R^{sh}$."}
+{"_id": "1306", "title": "algebra-lemma-quasi-finite-strict-henselization", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$. Let $\\kappa(\\mathfrak q) \\subset \\kappa^{sep}$ be a separable algebraic closure. Assume $R \\to S$ is quasi-finite at $\\mathfrak q$. The commutative diagram $$ \\xymatrix{ R_{\\mathfrak p}^{sh} \\ar[r] & S_{\\mathfrak q}^{sh} \\\\ R_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u] } $$ of Lemma \\ref{lemma-strictly-henselian-functorial} identifies $S_{\\mathfrak q}^{sh}$ with a localization of $R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$."}
+{"_id": "1307", "title": "algebra-lemma-quotient-strict-henselization", "text": "Let $R$ be a local ring with strict henselization $R^{sh}$. Let $I \\subset \\mathfrak m_R$. Then $R^{sh}/IR^{sh}$ is a strict henselization of $R/I$."}
+{"_id": "1308", "title": "algebra-lemma-sh-from-h-map", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$ such that $\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak q)$ is an isomorphism. Choose a separable algebraic closure $\\kappa^{sep}$ of $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$. Then $$ (S_\\mathfrak q)^{sh} = (S_\\mathfrak q)^h \\otimes_{(R_\\mathfrak p)^h} (R_\\mathfrak p)^{sh} $$"}
+{"_id": "1309", "title": "algebra-lemma-criterion-no-embedded-primes", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. The following are equivalent: \\begin{enumerate} \\item $M$ has no embedded associated prime, and \\item $M$ has property $(S_1)$. \\end{enumerate}"}
+{"_id": "1310", "title": "algebra-lemma-criterion-reduced", "text": "\\begin{slogan} Reduced equals R0 plus S1. \\end{slogan} Let $R$ be a Noetherian ring. The following are equivalent: \\begin{enumerate} \\item $R$ is reduced, and \\item $R$ has properties $(R_0)$ and $(S_1)$. \\end{enumerate}"}
+{"_id": "1311", "title": "algebra-lemma-criterion-normal", "text": "\\begin{reference} \\cite[IV, Theorem 5.8.6]{EGA} \\end{reference} \\begin{slogan} Normal equals R1 plus S2. \\end{slogan} Let $R$ be a Noetherian ring. The following are equivalent: \\begin{enumerate} \\item $R$ is a normal ring, and \\item $R$ has properties $(R_1)$ and $(S_2)$. \\end{enumerate}"}
+{"_id": "1312", "title": "algebra-lemma-regular-normal", "text": "A regular ring is normal."}
+{"_id": "1313", "title": "algebra-lemma-normal-domain-intersection-localizations-height-1", "text": "Let $R$ be a Noetherian normal domain with fraction field $K$. Then \\begin{enumerate} \\item for any nonzero $a \\in R$ the quotient $R/aR$ has no embedded primes, and all its associated primes have height $1$ \\item $$ R = \\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1} R_{\\mathfrak p} $$ \\item For any nonzero $x \\in K$ the quotient $R/(R \\cap xR)$ has no embedded primes, and all its associates primes have height $1$. \\end{enumerate}"}
+{"_id": "1314", "title": "algebra-lemma-characterize-separable-algebraic-field-extensions", "text": "Let $k \\subset K$ be a finitely generated field extension. The following are equivalent \\begin{enumerate} \\item $K$ is a finite separable field extension of $k$, \\item $\\Omega_{K/k} = 0$, \\item $K$ is formally unramified over $k$, \\item $K$ is unramified over $k$, \\item $K$ is formally \\'etale over $k$, \\item $K$ is \\'etale over $k$. \\end{enumerate}"}
+{"_id": "1315", "title": "algebra-lemma-derivative-zero-pth-power", "text": "Let $k$ be a perfect field of characteristic $p > 0$. Let $K/k$ be an extension. Let $a \\in K$. Then $\\text{d}a = 0$ in $\\Omega_{K/k}$ if and only if $a$ is a $p$th power."}
+{"_id": "1316", "title": "algebra-lemma-size-extension-pth-roots", "text": "Let $k$ be a field of characteristic $p > 0$. Let $a_1, \\ldots, a_n \\in k$ be elements such that $\\text{d}a_1, \\ldots, \\text{d}a_n$ are linearly independent in $\\Omega_{k/\\mathbf{F}_p}$. Then the field extension $k(a_1^{1/p}, \\ldots, a_n^{1/p})$ has degree $p^n$ over $k$."}
+{"_id": "1317", "title": "algebra-lemma-separable-differentials", "text": "Let $k$ be a field of characteristic $p > 0$. The following are equivalent: \\begin{enumerate} \\item the field extension $K/k$ is separable (see Definition \\ref{definition-separable-field-extension}), and \\item the map $K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$ is injective. \\end{enumerate}"}
+{"_id": "1318", "title": "algebra-lemma-formally-smooth-implies-separable", "text": "Let $k \\subset K$ be an extension of fields. If $K$ is formally smooth over $k$, then $K$ is a separable extension of $k$."}
+{"_id": "1319", "title": "algebra-lemma-characterize-formally-smooth-field-extension", "text": "Let $k \\subset K$ be an extension of fields. Then $K$ is formally smooth over $k$ if and only if $H_1(L_{K/k}) = 0$."}
+{"_id": "1320", "title": "algebra-lemma-formally-smooth-extensions-easy", "text": "Let $k \\subset K$ be an extension of fields. \\begin{enumerate} \\item If $K$ is purely transcendental over $k$, then $K$ is formally smooth over $k$. \\item If $K$ is separable algebraic over $k$, then $K$ is formally smooth over $k$. \\item If $K$ is separable over $k$, then $K$ is formally smooth over $k$. \\end{enumerate}"}
+{"_id": "1321", "title": "algebra-lemma-fields-are-formally-smooth", "text": "\\begin{slogan} Formally smooth equals separable for field extensions. \\end{slogan} Let $k$ be a field. \\begin{enumerate} \\item If the characteristic of $k$ is zero, then any extension field of $k$ is formally smooth over $k$. \\item If the characteristic of $k$ is $p > 0$, then $k \\subset K$ is formally smooth if and only if it is a separable field extension. \\end{enumerate}"}
+{"_id": "1322", "title": "algebra-lemma-localization-smooth-separable", "text": "Let $k \\subset K$ be a finitely generated field extension. Then $K$ is separable over $k$ if and only if $K$ is the localization of a smooth $k$-algebra."}
+{"_id": "1323", "title": "algebra-lemma-colimit-syntomic", "text": "Let $k \\subset K$ be a field extension. Then $K$ is a filtered colimit of global complete intersection algebras over $k$. If $K/k$ is separable, then $K$ is a filtered colimit of smooth algebras over $k$."}
+{"_id": "1324", "title": "algebra-lemma-flat-local-given-residue-field", "text": "Let $(R, \\mathfrak m, k)$ be a local ring. Let $k \\subset K$ be a field extension. There exists a local ring $(R', \\mathfrak m', k')$, a flat local ring map $R \\to R'$ such that $\\mathfrak m' = \\mathfrak mR'$ and such that $k \\subset k'$ is isomorphic to $k \\subset K$."}
+{"_id": "1325", "title": "algebra-lemma-colimit-finite-etale-given-residue-field", "text": "Let $(R, \\mathfrak m, k)$ be a local ring. If $k \\subset K$ is a separable algebraic extension, then there exists a directed set $I$ and a system of finite \\'etale extensions $R \\subset R_i$, $i \\in I$ of local rings such that $R' = \\colim R_i$ has residue field $K$ (as extension of $k$)."}
+{"_id": "1326", "title": "algebra-lemma-finite-free-given-residue-field-extension", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime and let $\\kappa(\\mathfrak p) \\subset L$ be a finite extension of fields. Then there exists a finite free ring map $R \\to S$ such that $\\mathfrak q = \\mathfrak pS$ is prime and $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q)$ is isomorphic to the given extension $\\kappa(\\mathfrak p) \\subset L$."}
+{"_id": "1327", "title": "algebra-lemma-quotient-complete-local", "text": "Let $R$ be a Noetherian complete local ring. Any quotient of $R$ is also a Noetherian complete local ring. Given a finite ring map $R \\to S$, then $S$ is a product of Noetherian complete local rings."}
+{"_id": "1328", "title": "algebra-lemma-complete-local-ring-Noetherian", "text": "Let $(R, \\mathfrak m)$ be a complete local ring. If $\\mathfrak m$ is a finitely generated ideal then $R$ is Noetherian."}
+{"_id": "1329", "title": "algebra-lemma-cohen-rings-exist", "text": "Let $p$ be a prime number. Let $k$ be a field of characteristic $p$. There exists a Cohen ring $\\Lambda$ with $\\Lambda/p\\Lambda \\cong k$."}
+{"_id": "1330", "title": "algebra-lemma-cohen-ring-formally-smooth", "text": "Let $p > 0$ be a prime. Let $\\Lambda$ be a Cohen ring with residue field of characteristic $p$. For every $n \\geq 1$ the ring map $$ \\mathbf{Z}/p^n\\mathbf{Z} \\to \\Lambda/p^n\\Lambda $$ is formally smooth."}
+{"_id": "1331", "title": "algebra-lemma-regular-complete-containing-coefficient-field", "text": "Let $(R, \\mathfrak m)$ be a Noetherian complete local ring. Assume $R$ is regular. \\begin{enumerate} \\item If $R$ contains either $\\mathbf{F}_p$ or $\\mathbf{Q}$, then $R$ is isomorphic to a power series ring over its residue field. \\item If $k$ is a field and $k \\to R$ is a ring map inducing an isomorphism $k \\to R/\\mathfrak m$, then $R$ is isomorphic as a $k$-algebra to a power series ring over $k$. \\end{enumerate}"}
+{"_id": "1332", "title": "algebra-lemma-complete-local-Noetherian-domain-finite-over-regular", "text": "Let $(R, \\mathfrak m)$ be a Noetherian complete local domain. Then there exists a $R_0 \\subset R$ with the following properties \\begin{enumerate} \\item $R_0$ is a regular complete local ring, \\item $R_0 \\subset R$ is finite and induces an isomorphism on residue fields, \\item $R_0$ is either isomorphic to $k[[X_1, \\ldots, X_d]]$ where $k$ is a field or $\\Lambda[[X_1, \\ldots, X_d]]$ where $\\Lambda$ is a Cohen ring. \\end{enumerate}"}
+{"_id": "1333", "title": "algebra-lemma-localize-N", "text": "Let $R$ be a domain. If $R$ is N-1 then so is any localization of $R$. Same for N-2."}
+{"_id": "1334", "title": "algebra-lemma-Japanese-local", "text": "Let $R$ be a domain. Let $f_1, \\ldots, f_n \\in R$ generate the unit ideal. If each domain $R_{f_i}$ is N-1 then so is $R$. Same for N-2."}
+{"_id": "1335", "title": "algebra-lemma-quasi-finite-over-Noetherian-japanese", "text": "Let $R$ be a domain. Let $R \\subset S$ be a quasi-finite extension of domains (for example finite). Assume $R$ is N-2 and Noetherian. Then $S$ is N-2."}
+{"_id": "1336", "title": "algebra-lemma-Laurent-ring-N-1", "text": "Let $R$ be a Noetherian domain. If $R[z, z^{-1}]$ is N-1, then so is $R$."}
+{"_id": "1337", "title": "algebra-lemma-finite-extension-N-2", "text": "Let $R$ be a Noetherian domain, and let $R \\subset S$ be a finite extension of domains. If $S$ is N-1, then so is $R$. If $S$ is N-2, then so is $R$."}
+{"_id": "1338", "title": "algebra-lemma-Noetherian-normal-domain-finite-separable-extension", "text": "Let $R$ be a Noetherian normal domain with fraction field $K$. Let $K \\subset L$ be a finite separable field extension. Then the integral closure of $R$ in $L$ is finite over $R$."}
+{"_id": "1339", "title": "algebra-lemma-Noetherian-normal-domain-insep-extension", "text": "Let $R$ be a Noetherian normal domain with fraction field $K$ of characteristic $p > 0$. Let $a \\in K$ be an element such that there exists a derivation $D : R \\to R$ with $D(a) \\not = 0$. Then the integral closure of $R$ in $L = K[x]/(x^p - a)$ is finite over $R$."}
+{"_id": "1340", "title": "algebra-lemma-domain-char-zero-N-1-2", "text": "A Noetherian domain whose fraction field has characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese)."}
+{"_id": "1341", "title": "algebra-lemma-domain-char-p-N-1-2", "text": "Let $R$ be a Noetherian domain with fraction field $K$ of characteristic $p > 0$. Then $R$ is N-2 if and only if for every finite purely inseparable extension $K \\subset L$ the integral closure of $R$ in $L$ is finite over $R$."}
+{"_id": "1342", "title": "algebra-lemma-polynomial-ring-N-2", "text": "Let $R$ be a Noetherian domain. If $R$ is N-1 then $R[x]$ is N-1. If $R$ is N-2 then $R[x]$ is N-2."}
+{"_id": "1343", "title": "algebra-lemma-openness-normal-locus", "text": "Let $R$ be a Noetherian domain. If there exists an $f \\in R$ such that $R_f$ is normal then $$ U = \\{\\mathfrak p \\in \\Spec(R) \\mid R_{\\mathfrak p} \\text{ is normal}\\} $$ is open in $\\Spec(R)$."}
+{"_id": "1344", "title": "algebra-lemma-characterize-N-1", "text": "Let $R$ be a Noetherian domain. Then $R$ is N-1 if and only if the following two conditions hold \\begin{enumerate} \\item there exists a nonzero $f \\in R$ such that $R_f$ is normal, and \\item for every maximal ideal $\\mathfrak m \\subset R$ the local ring $R_{\\mathfrak m}$ is N-1. \\end{enumerate}"}
+{"_id": "1345", "title": "algebra-lemma-tate-japanese", "text": "\\begin{reference} \\cite[Theorem 23.1.3]{EGA} \\end{reference} Let $R$ be a ring. Let $x \\in R$. Assume \\begin{enumerate} \\item $R$ is a normal Noetherian domain, \\item $R/xR$ is a domain and N-2, \\item $R \\cong \\lim_n R/x^nR$ is complete with respect to $x$. \\end{enumerate} Then $R$ is N-2."}
+{"_id": "1346", "title": "algebra-lemma-power-series-over-N-2", "text": "Let $R$ be a ring. If $R$ is Noetherian, a domain, and N-2, then so is $R[[x]]$."}
+{"_id": "1347", "title": "algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure", "text": "Let $R$ be a Nagata ring. Let $R \\to S$ be essentially of finite type with $S$ reduced. Then the integral closure of $R$ in $S$ is finite over $R$."}
+{"_id": "1348", "title": "algebra-lemma-check-universally-japanese", "text": "Let $R$ be a ring. To check that $R$ is universally Japanese it suffices to show: If $R \\to S$ is of finite type, and $S$ a domain then $S$ is N-1."}
+{"_id": "1349", "title": "algebra-lemma-universally-japanese", "text": "If $R$ is universally Japanese then any algebra essentially of finite type over $R$ is universally Japanese."}
+{"_id": "1350", "title": "algebra-lemma-quasi-finite-over-nagata", "text": "Let $R$ be a Nagata ring. If $R \\to S$ is a quasi-finite ring map (for example finite) then $S$ is a Nagata ring also."}
+{"_id": "1351", "title": "algebra-lemma-nagata-localize", "text": "A localization of a Nagata ring is a Nagata ring."}
+{"_id": "1352", "title": "algebra-lemma-nagata-local", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$ generate the unit ideal. \\begin{enumerate} \\item  If each $R_{f_i}$ is universally Japanese then so is $R$. \\item  If each $R_{f_i}$ is Nagata then so is $R$. \\end{enumerate}"}
+{"_id": "1353", "title": "algebra-lemma-Noetherian-complete-local-Nagata", "text": "A Noetherian complete local ring is a Nagata ring."}
+{"_id": "1354", "title": "algebra-lemma-analytically-unramified-easy", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. \\begin{enumerate} \\item If $R$ is analytically unramified, then $R$ is reduced. \\item If $R$ is analytically unramified, then each minimal prime of $R$ is analytically unramified. \\item If $R$ is reduced with minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and each $\\mathfrak q_i$ is analytically unramified, then $R$ is analytically unramified. \\item If $R$ is analytically unramified, then the integral closure of $R$ in its total ring of fractions $Q(R)$ is finite over $R$. \\item If $R$ is a domain and analytically unramified, then $R$ is N-1. \\end{enumerate}"}
+{"_id": "1355", "title": "algebra-lemma-codimension-1-analytically-unramified", "text": "Let $R$ be a Noetherian local ring. Let $\\mathfrak p \\subset R$ be a prime. Assume \\begin{enumerate} \\item $R_{\\mathfrak p}$ is a discrete valuation ring, and \\item $\\mathfrak p$ is analytically unramified. \\end{enumerate} Then for any associated prime $\\mathfrak q$ of $R^\\wedge/\\mathfrak pR^\\wedge$ the local ring $(R^\\wedge)_{\\mathfrak q}$ is a discrete valuation ring."}
+{"_id": "1356", "title": "algebra-lemma-criterion-analytically-unramified", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local domain. Let $x \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $x \\not = 0$, \\item $R/xR$ has no embedded primes, and \\item for each associated prime $\\mathfrak p \\subset R$ of $R/xR$ we have \\begin{enumerate} \\item the local ring $R_{\\mathfrak p}$ is regular, and \\item $\\mathfrak p$ is analytically unramified. \\end{enumerate} \\end{enumerate} Then $R$ is analytically unramified."}
+{"_id": "1357", "title": "algebra-lemma-local-nagata-domain-analytically-unramified", "text": "Let $(R, \\mathfrak m)$ be a local ring. If $R$ is Noetherian, a domain, and Nagata, then $R$ is analytically unramified."}
+{"_id": "1358", "title": "algebra-lemma-local-nagata-and-analytically-unramified", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. The following are equivalent \\begin{enumerate} \\item $R$ is Nagata, \\item for $R \\to S$ finite with $S$ a domain and $\\mathfrak m' \\subset S$ maximal the local ring $S_{\\mathfrak m'}$ is analytically unramified, \\item for $(R, \\mathfrak m) \\to (S, \\mathfrak m')$ finite local homomorphism with $S$ a domain, then $S$ is analytically unramified. \\end{enumerate}"}
+{"_id": "1359", "title": "algebra-lemma-nagata-pth-roots", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local domain which is Nagata and has fraction field of characteristic $p$. If $a \\in A$ has a $p$th root in $A^\\wedge$, then $a$ has a $p$th root in $A$."}
+{"_id": "1360", "title": "algebra-lemma-apply-grothendieck-module", "text": "\\begin{reference} \\cite[IV, Proposition 6.3.1]{EGA} \\end{reference} We have $$ \\text{depth}(M \\otimes_R N) = \\text{depth}(M) + \\text{depth}(N/\\mathfrak m_RN) $$ where $R \\to S$ is a local homomorphism of local Noetherian rings, $M$ is a finite $R$-module, and $N$ is a finite $S$-module flat over $R$."}
+{"_id": "1361", "title": "algebra-lemma-apply-grothendieck", "text": "Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian local rings. Then $$ \\text{depth}(S) = \\text{depth}(R) + \\text{depth}(S/\\mathfrak m_RS). $$"}
+{"_id": "1362", "title": "algebra-lemma-CM-goes-up", "text": "Let $R \\to S$ be a flat local homomorphism of local Noetherian rings. Then the following are equivalent \\begin{enumerate} \\item $S$ is Cohen-Macaulay, and \\item $R$ and $S/\\mathfrak m_RS$ are Cohen-Macaulay. \\end{enumerate}"}
+{"_id": "1363", "title": "algebra-lemma-Sk-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $S$ is Noetherian, \\item $\\varphi$ is flat, \\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are $(S_k)$, and \\item $R$ has property $(S_k)$. \\end{enumerate} Then $S$ has property $(S_k)$."}
+{"_id": "1364", "title": "algebra-lemma-Rk-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $S$ is Noetherian \\item $\\varphi$ is flat, \\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ have property $(R_k)$, and \\item $R$ has property $(R_k)$. \\end{enumerate} Then $S$ has property $(R_k)$."}
+{"_id": "1365", "title": "algebra-lemma-reduced-goes-up-noetherian", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $S$ is Noetherian \\item $\\varphi$ is flat, \\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are reduced, \\item $R$ is reduced. \\end{enumerate} Then $S$ is reduced."}
+{"_id": "1366", "title": "algebra-lemma-reduced-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is smooth, \\item $R$ is reduced. \\end{enumerate} Then $S$ is reduced."}
+{"_id": "1367", "title": "algebra-lemma-normal-goes-up-noetherian", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $S$ is Noetherian, \\item $\\varphi$ is flat, \\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are normal, and \\item $R$ is normal. \\end{enumerate} Then $S$ is normal."}
+{"_id": "1368", "title": "algebra-lemma-normal-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is smooth, \\item $R$ is normal. \\end{enumerate} Then $S$ is normal."}
+{"_id": "1369", "title": "algebra-lemma-regular-goes-up", "text": "\\begin{slogan} Regularity ascends along smooth maps of rings. \\end{slogan} Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is smooth, \\item $R$ is a regular ring. \\end{enumerate} Then $S$ is regular."}
+{"_id": "1370", "title": "algebra-lemma-descent-Noetherian", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is Noetherian. \\end{enumerate} Then $R$ is Noetherian."}
+{"_id": "1371", "title": "algebra-lemma-descent-reduced", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is reduced. \\end{enumerate} Then $R$ is reduced."}
+{"_id": "1372", "title": "algebra-lemma-descent-normal", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is a normal ring. \\end{enumerate} Then $R$ is a normal ring."}
+{"_id": "1373", "title": "algebra-lemma-descent-regular", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is a regular ring. \\end{enumerate} Then $R$ is a regular ring."}
+{"_id": "1374", "title": "algebra-lemma-descent-Sk", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is Noetherian and has property $(S_k)$. \\end{enumerate} Then $R$ is Noetherian and has property $(S_k)$."}
+{"_id": "1375", "title": "algebra-lemma-descent-Rk", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is faithfully flat, and \\item $S$ is Noetherian and has property $(R_k)$. \\end{enumerate} Then $R$ is Noetherian and has property $(R_k)$."}
+{"_id": "1376", "title": "algebra-lemma-descent-nagata", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R \\to S$ is smooth and surjective on spectra, and \\item $S$ is a Nagata ring. \\end{enumerate} Then $R$ is a Nagata ring."}
+{"_id": "1377", "title": "algebra-lemma-geometrically-normal", "text": "Let $k$ be a field. Let $A$ be a $k$-algebra. The following properties of $A$ are equivalent: \\begin{enumerate} \\item $k' \\otimes_k A$ is a normal ring for every field extension $k'/k$, \\item $k' \\otimes_k A$ is a normal ring for every finitely generated field extension $k'/k$, \\item $k' \\otimes_k A$ is a normal ring for every finite purely inseparable extension $k'/k$, \\item $k^{perf} \\otimes_k A$ is a normal ring. \\end{enumerate} Here normal ring is defined in Definition \\ref{definition-ring-normal}."}
+{"_id": "1378", "title": "algebra-lemma-localization-geometrically-normal-algebra", "text": "\\begin{slogan} Localization preserves geometric normality. \\end{slogan} Let $k$ be a field. A localization of a geometrically normal $k$-algebra is geometrically normal."}
+{"_id": "1379", "title": "algebra-lemma-separable-field-extension-geometrically-normal", "text": "Let $k$ be a field. Let $K/k$ be a separable field extension. Then $K$ is geometrically normal over $k$."}
+{"_id": "1380", "title": "algebra-lemma-geometrically-normal-tensor-normal", "text": "Let $k$ be a field. Let $A, B$ be $k$-algebras. Assume $A$ is geometrically normal over $k$ and $B$ is a normal ring. Then $A \\otimes_k B$ is a normal ring."}
+{"_id": "1381", "title": "algebra-lemma-geometrically-normal-over-separable-algebraic", "text": "Let $k \\subset k'$ be a separable algebraic field extension. Let $A$ be an algebra over $k'$. Then $A$ is geometrically normal over $k$ if and only if it is geometrically normal over $k'$."}
+{"_id": "1382", "title": "algebra-lemma-geometrically-regular", "text": "Let $k$ be a field. Let $A$ be a $k$-algebra. Assume $A$ is Noetherian. The following properties of $A$ are equivalent: \\begin{enumerate} \\item $k' \\otimes_k A$ is regular for every finitely generated field extension $k \\subset k'$, and \\item $k' \\otimes_k A$ is regular for every finite purely inseparable extension $k \\subset k'$. \\end{enumerate} Here regular ring is as in Definition \\ref{definition-regular}."}
+{"_id": "1383", "title": "algebra-lemma-geometrically-regular-descent", "text": "\\begin{slogan} Geometric regularity descends through faithfully flat maps of algebras \\end{slogan} Let $k$ be a field. Let $A \\to B$ be a faithfully flat $k$-algebra map. If $B$ is geometrically regular over $k$, so is $A$."}
+{"_id": "1384", "title": "algebra-lemma-geometrically-regular-goes-up", "text": "Let $k$ be a field. Let $A \\to B$ be a smooth ring map of $k$-algebras. If $A$ is geometrically regular over $k$, then $B$ is geometrically regular over $k$."}
+{"_id": "1385", "title": "algebra-lemma-geometrically-regular-over-subfields", "text": "Let $k$ be a field. Let $A$ be an algebra over $k$. Let $k = \\colim k_i$ be a directed colimit of subfields. If $A$ is geometrically regular over each $k_i$, then $A$ is geometrically regular over $k$."}
+{"_id": "1386", "title": "algebra-lemma-geometrically-regular-over-separable-algebraic", "text": "Let $k \\subset k'$ be a separable algebraic field extension. Let $A$ be an algebra over $k'$. Then $A$ is geometrically regular over $k$ if and only if it is geometrically regular over $k'$."}
+{"_id": "1387", "title": "algebra-lemma-tensor-fields-CM", "text": "Let $k$ be a field and let $k \\subset K$ and $k \\subset L$ be two field extensions such that one of them is a field extension of finite type. Then $K \\otimes_k L$ is a Noetherian Cohen-Macaulay ring."}
+{"_id": "1388", "title": "algebra-lemma-CM-geometrically-CM", "text": "Let $k$ be a field. Let $S$ be a Noetherian $k$-algebra. Let $k \\subset K$ be a finitely generated field extension, and set $S_K = K \\otimes_k S$. Let $\\mathfrak q \\subset S$ be a prime of $S$. Let $\\mathfrak q_K \\subset S_K$ be a prime of $S_K$ lying over $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay if and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay."}
+{"_id": "1389", "title": "algebra-lemma-flat-finite-presentation-limit-flat", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume that \\begin{enumerate} \\item $R \\to S$ is of finite presentation, \\item $M$ is a finitely presented $S$-module, and \\item $M$ is flat over $R$. \\end{enumerate} In this case we have the following: \\begin{enumerate} \\item There exists a finite type $\\mathbf{Z}$-algebra $R_0$ and a finite type ring map $R_0 \\to S_0$ and a finite $S_0$-module $M_0$ such that $M_0$ is flat over $R_0$, together with a ring maps $R_0 \\to R$ and $S_0 \\to S$ and an $S_0$-module map $M_0 \\to M$ such that $S \\cong R \\otimes_{R_0} S_0$ and $M = S \\otimes_{S_0} M_0$. \\item If $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is written as a directed colimit, then there exists a $\\lambda$ and a ring map $R_\\lambda \\to S_\\lambda$ of finite presentation, and an $S_\\lambda$-module $M_\\lambda$ of finite presentation such that $M_\\lambda$ is flat over $R_\\lambda$ and such that $S = R \\otimes_{R_\\lambda} S_\\lambda$ and $M = S \\otimes_{S_{\\lambda}} M_\\lambda$. \\item If $$ (R \\to S, M) = \\colim_{\\lambda \\in \\Lambda} (R_\\lambda \\to S_\\lambda, M_\\lambda) $$ is written as a directed colimit such that \\begin{enumerate} \\item $R_\\mu \\otimes_{R_\\lambda} S_\\lambda \\to S_\\mu$ and $S_\\mu \\otimes_{S_\\lambda} M_\\lambda \\to M_\\mu$ are isomorphisms for $\\mu \\geq \\lambda$, \\item $R_\\lambda \\to S_\\lambda$ is of finite presentation, \\item $M_\\lambda$ is a finitely presented $S_\\lambda$-module, \\end{enumerate} then for all sufficiently large $\\lambda$ the module $M_\\lambda$ is flat over $R_\\lambda$. \\end{enumerate}"}
+{"_id": "1390", "title": "algebra-lemma-descend-faithfully-flat-finite-presentation", "text": "Let $R \\to A \\to B$ be ring maps. Assume $A \\to B$ faithfully flat of finite presentation. Then there exists a commutative diagram $$ \\xymatrix{ R \\ar[r] \\ar@{=}[d] & A_0 \\ar[d] \\ar[r] & B_0 \\ar[d] \\\\ R \\ar[r] & A \\ar[r] & B } $$ with $R \\to A_0$ of finite presentation, $A_0 \\to B_0$ faithfully flat of finite presentation and $B = A \\otimes_{A_0} B_0$."}
+{"_id": "1391", "title": "algebra-lemma-colimit-finite", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is finite, \\item $C_0$ is of finite type over $B_0$. \\end{enumerate} Then there exists an $i \\geq 0$ such that the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is finite."}
+{"_id": "1392", "title": "algebra-lemma-colimit-surjective", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is surjective, \\item $C_0$ is of finite type over $B_0$. \\end{enumerate} Then for some $i \\geq 0$ the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is surjective."}
+{"_id": "1393", "title": "algebra-lemma-colimit-unramified", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is unramified, \\item $C_0$ is of finite type over $B_0$. \\end{enumerate} Then for some $i \\geq 0$ the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is unramified."}
+{"_id": "1394", "title": "algebra-lemma-colimit-isomorphism", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is an isomorphism, \\item $B_0 \\to C_0$ is of finite presentation. \\end{enumerate} Then for some $i \\geq 0$ the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is an isomorphism."}
+{"_id": "1395", "title": "algebra-lemma-colimit-etale", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is \\'etale, \\item $B_0 \\to C_0$ is of finite presentation. \\end{enumerate} Then for some $i \\geq 0$ the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is \\'etale."}
+{"_id": "1396", "title": "algebra-lemma-colimit-smooth", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is smooth, \\item $B_0 \\to C_0$ is of finite presentation. \\end{enumerate} Then for some $i \\geq 0$ the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is smooth."}
+{"_id": "1397", "title": "algebra-lemma-colimit-lci", "text": "Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings. Let $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras. Assume \\begin{enumerate} \\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is syntomic (resp.\\ a relative global complete intersection), \\item $C_0$ is of finite presentation over $B_0$. \\end{enumerate} Then there exists an $i \\geq 0$ such that the map $A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is syntomic (resp.\\ a relative global complete intersection)."}
+{"_id": "1398", "title": "algebra-lemma-fppf-fpqf", "text": "Let $R \\to S$ be a faithfully flat ring map of finite presentation. Then there exists a commutative diagram $$ \\xymatrix{ S \\ar[rr] & & S' \\\\ & R \\ar[lu] \\ar[ru] } $$ where $R \\to S'$ is quasi-finite, faithfully flat and of finite presentation."}
+{"_id": "1399", "title": "algebra-proposition-universal-property-localization", "text": "Let $f : A \\to B$ be a ring map that sends every element in $S$ to a unit of $B$. Then there is a unique homomorphism $g : S^{-1}A \\to B$ such that the following diagram commutes. $$ \\xymatrix{ A \\ar[rr]^{f} \\ar[dr] & & B \\\\ & S^{-1}A \\ar[ur]_g } $$"}
+{"_id": "1400", "title": "algebra-proposition-localize-twice", "text": "Let $\\overline{S}$ be the image of $S$ in $S'^{-1}A$, then $(SS')^{-1}A$ is isomorphic to $\\overline{S}^{-1}(S'^{-1}A)$."}
+{"_id": "1401", "title": "algebra-proposition-localize-twice-module", "text": "View $S'^{-1}M$ as an $A$-module, then $S^{-1}(S'^{-1}M)$ is isomorphic to $(SS')^{-1}M$."}
+{"_id": "1402", "title": "algebra-proposition-localization-exact", "text": "\\begin{slogan} Localization is exact. \\end{slogan} Let $L\\xrightarrow{u} M\\xrightarrow{v} N$ be an exact sequence of $R$-modules. Then $S^{-1}L \\to S^{-1}M \\to S^{-1}N$ is also exact."}
+{"_id": "1403", "title": "algebra-proposition-localize-quotient", "text": "Let $I$ be an ideal of $A$, $S$ a multiplicative set of $A$. Then $S^{-1}I$ is an ideal of $S^{-1}A$ and $\\overline{S}^{-1}(A/I)$ is isomorphic to $S^{-1}A/S^{-1}I$, where $\\overline{S}$ is the image of $S$ in $A/I$."}
+{"_id": "1404", "title": "algebra-proposition-oka", "text": "If $\\mathcal{F}$ is an Oka family of ideals, then any maximal element of the complement of $\\mathcal{F}$ is prime."}
+{"_id": "1405", "title": "algebra-proposition-Jacobson-permanence", "text": "Let $R$ be a Jacobson ring. Let $R \\to S$ be a ring map of finite type. Then \\begin{enumerate} \\item The ring $S$ is Jacobson. \\item The map $\\Spec(S) \\to \\Spec(R)$ transforms closed points to closed points. \\item For $\\mathfrak m' \\subset S$ maximal lying over $\\mathfrak m \\subset R$ the field extension $\\kappa(\\mathfrak m')/\\kappa(\\mathfrak m)$ is finite. \\end{enumerate}"}
+{"_id": "1406", "title": "algebra-proposition-going-down-normal-integral", "text": "Let $R \\subset S$ be an inclusion of domains. Assume $R$ is normal and $S$ integral over $R$. Let $\\mathfrak p \\subset \\mathfrak p' \\subset R$ be primes. Let $\\mathfrak q'$ be a prime of $S$ with $\\mathfrak p' = R \\cap \\mathfrak q'$. Then there exists a prime $\\mathfrak q$ with $\\mathfrak q \\subset \\mathfrak q'$ such that $\\mathfrak p = R \\cap \\mathfrak q$. In other words: the going down property holds for $R \\to S$, see Definition \\ref{definition-going-up-down}."}
+{"_id": "1407", "title": "algebra-proposition-fppf-open", "text": "Let $R \\to S$ be flat and of finite presentation. Then $\\Spec(S) \\to \\Spec(R)$ is open. More generally this holds for any ring map $R \\to S$ of finite presentation which satisfies going down."}
+{"_id": "1408", "title": "algebra-proposition-graded-hilbert-polynomial", "text": "Suppose that $S$ is a Noetherian graded ring and $M$ a finite graded $S$-module. Consider the function $$ \\mathbf{Z} \\longrightarrow K'_0(S_0), \\quad n \\longmapsto [M_n] $$ see Lemma \\ref{lemma-graded-module-fg}. If $S_{+}$ is generated by elements of degree $1$, then this function is a numerical polynomial."}
+{"_id": "1409", "title": "algebra-proposition-hilbert-function-polynomial", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Let $I \\subset R$ be an ideal of definition. The Hilbert function $\\varphi_{I, M}$ and the function $\\chi_{I, M}$ are numerical polynomials."}
+{"_id": "1410", "title": "algebra-proposition-dimension-zero-ring", "text": "Let $R$ be a ring. The following are equivalent: \\begin{enumerate} \\item $R$ is Artinian, \\item $R$ is Noetherian and $\\dim(R) = 0$, \\item $R$ has finite length as a module over itself, \\item $R$ is a finite product of Artinian local rings, \\item $R$ is Noetherian and $\\Spec(R)$ is a finite discrete topological space, \\item $R$ is a finite product of Noetherian local rings of dimension $0$, \\item $R$ is a finite product of Noetherian local rings $R_i$ with $d(R_i) = 0$, \\item $R$ is a finite product of Noetherian local rings $R_i$ whose maximal ideals are nilpotent, \\item $R$ is Noetherian, has finitely many maximal ideals and its Jacobson radical ideal is nilpotent, and \\item $R$ is Noetherian and there are no strict inclusions among its primes. \\end{enumerate}"}
+{"_id": "1411", "title": "algebra-proposition-dimension", "text": "Let $R$ be a local Noetherian ring. Let $d \\geq 0$ be an integer. The following are equivalent: \\begin{enumerate} \\item \\label{item-dim-d} $\\dim(R) = d$, \\item \\label{item-d-d} $d(R) = d$, \\item \\label{item-ideal-d} there exists an ideal of definition generated by $d$ elements, and no ideal of definition is generated by fewer than $d$ elements. \\end{enumerate}"}
+{"_id": "1412", "title": "algebra-proposition-minimal-primes-associated-primes", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. The following sets of primes are the same: \\begin{enumerate} \\item The minimal primes in the support of $M$. \\item The minimal primes in $\\text{Ass}(M)$. \\item For any filtration $0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_{n-1} \\subset M_n = M$ with $M_i/M_{i-1} \\cong R/\\mathfrak p_i$ the minimal primes of the set $\\{\\mathfrak p_i\\}$. \\end{enumerate}"}
+{"_id": "1413", "title": "algebra-proposition-ffdescent-finite-projectivity", "text": "Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module. If the $S$-module $M \\otimes_R S$ is finite projective, then $M$ is finite projective."}
+{"_id": "1414", "title": "algebra-proposition-ML-characterization", "text": "Let $M$ be an $R$-module.  Let $(M_i, f_{ij})$ be a directed system of finitely presented $R$-modules, indexed by $I$, such that $M = \\colim M_i$.  Let $f_i: M_i \\to M$ be the canonical map.  The following are equivalent: \\begin{enumerate} \\item For every finitely presented $R$-module $P$ and module map $f: P \\to M$, there exists a finitely presented $R$-module $Q$ and a module map $g: P \\to Q$ such that $g$ and $f$ dominate each other, i.e., $\\Ker(f \\otimes_R \\text{id}_N) = \\Ker(g \\otimes_R \\text{id}_N)$ for every $R$-module $N$. \\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i \\to M_j$ dominates $f_i: M_i \\to M$. \\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i \\to M_j$ factors through $f_{ik}: M_i \\to M_k$ for all $k \\geq i$. \\item For every $R$-module $N$, the inverse system $(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler. \\item For $N = \\prod_{s \\in I} M_s$, the inverse system $(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler. \\end{enumerate}"}
+{"_id": "1415", "title": "algebra-proposition-fg-tensor", "text": "Let $M$ be an $R$-module.  The following are equivalent: \\begin{enumerate} \\item $M$ is finitely generated. \\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right) \\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is surjective. \\item For every $R$-module $Q$ and every set $A$, the canonical map $M \\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is surjective. \\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to M^{A}$ is surjective. \\end{enumerate}"}
+{"_id": "1416", "title": "algebra-proposition-fp-tensor", "text": "Let $M$ be an $R$-module.  The following are equivalent: \\begin{enumerate} \\item $M$ is finitely presented. \\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right) \\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is bijective. \\item For every $R$-module $Q$ and every set $A$, the canonical map $M \\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is bijective. \\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to M^{A}$ is bijective. \\end{enumerate}"}
+{"_id": "1417", "title": "algebra-proposition-ML-tensor", "text": "Let $M$ be an $R$-module.  The following are equivalent: \\begin{enumerate} \\item $M$ is Mittag-Leffler. \\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the canonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right) \\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is injective. \\end{enumerate}"}
+{"_id": "1418", "title": "algebra-proposition-characterize-coherent", "text": "\\begin{reference} This is \\cite[Theorem 2.1]{Chase}. \\end{reference} Let $R$ be a ring. The following are equivalent \\begin{enumerate} \\item $R$ is coherent, \\item any product of flat $R$-modules is flat, and \\item for every set $A$ the module $R^A$ is flat. \\end{enumerate}"}
+{"_id": "1419", "title": "algebra-proposition-what-exact", "text": "\\begin{reference} \\cite[Corollary 1]{WhatExact} \\end{reference} In Situation \\ref{situation-complex}, suppose $R$ is a local Noetherian ring. The following are equivalent \\begin{enumerate} \\item $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$ is exact at $R^{n_e}, \\ldots, R^{n_1}$, and \\item for all $i$, $1 \\leq i \\leq e$ the following two conditions are satisfied: \\begin{enumerate} \\item $\\text{rank}(\\varphi_i) = r_i$ where $r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$, \\item $I(\\varphi_i) = R$, or $I(\\varphi_i)$ contains a regular sequence of length $i$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1420", "title": "algebra-proposition-CM-module", "text": "Let $R$ be a Noetherian local ring, with maximal ideal $\\mathfrak m$. Let $M$ be a Cohen-Macaulay module over $R$ whose support has dimension $d$. Suppose that $g_1, \\ldots, g_c$ are elements of $\\mathfrak m$ such that $\\dim(\\text{Supp}(M/(g_1, \\ldots, g_c)M)) = d - c$. Then $g_1, \\ldots, g_c$ is an $M$-regular sequence, and can be extended to a maximal $M$-regular sequence."}
+{"_id": "1421", "title": "algebra-proposition-regular-finite-gl-dim", "text": "Let $R$ be a regular local ring of dimension $d$. Every finite $R$-module $M$ of depth $e$ has a finite free resolution $$ 0 \\to F_{d-e} \\to \\ldots \\to F_0 \\to M \\to 0. $$ In particular a regular local ring has global dimension $\\leq d$."}
+{"_id": "1422", "title": "algebra-proposition-finite-gl-dim-regular", "text": "A Noetherian local ring whose residue field has finite projective dimension is a regular local ring. In particular a Noetherian local ring of finite global dimension is a regular local ring."}
+{"_id": "1423", "title": "algebra-proposition-Auslander-Buchsbaum", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module which has finite projective dimension $\\text{pd}_R(M)$. Then we have $$ \\text{depth}(R) = \\text{pd}_R(M) + \\text{depth}(M) $$"}
+{"_id": "1424", "title": "algebra-proposition-finite-gl-dim-polynomial-ring", "text": "A polynomial algebra in $n$ variables over a field is a regular ring. It has global dimension $n$. All localizations at maximal ideals are regular local rings of dimension $n$."}
+{"_id": "1425", "title": "algebra-proposition-characterize-formally-smooth", "text": "Let $R \\to S$ be a ring map. Consider a formally smooth $R$-algebra $P$ and a surjection $P \\to S$ with kernel $J$. The following are equivalent \\begin{enumerate} \\item $S$ is formally smooth over $R$, \\item for some $P \\to S$ as above there exists a section to $P/J^2 \\to S$, \\item for all $P \\to S$ as above there exists a section to $P/J^2 \\to S$, \\item for some $P \\to S$ as above the sequence $0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact, \\item for all $P \\to S$ as above the sequence $0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact, and \\item the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a projective $S$-module placed in degree $0$. \\end{enumerate}"}
+{"_id": "1426", "title": "algebra-proposition-smooth-formally-smooth", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is of finite presentation and formally smooth, \\item $R \\to S$ is smooth. \\end{enumerate}"}
+{"_id": "1427", "title": "algebra-proposition-etale-locally-standard", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime. If $R \\to S$ is \\'etale at $\\mathfrak q$, then there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is standard \\'etale."}
+{"_id": "1428", "title": "algebra-proposition-unramified-locally-standard", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime. If $R \\to S$ is unramified at $\\mathfrak q$, then there exist \\begin{enumerate} \\item a $g \\in S$, $g \\not \\in \\mathfrak q$, \\item a standard \\'etale ring map $R \\to S'$, and \\item a surjective $R$-algebra map $S' \\to S_g$. \\end{enumerate}"}
+{"_id": "1429", "title": "algebra-proposition-characterize-separable-field-extensions", "text": "Let $k \\subset K$ be a field extension. If the characteristic of $k$ is zero then \\begin{enumerate} \\item $K$ is separable over $k$, \\item $K$ is geometrically reduced over $k$, \\item $K$ is formally smooth over $k$, \\item $H_1(L_{K/k}) = 0$, and \\item the map $K \\otimes_k \\Omega_{k/\\mathbf{Z}} \\to \\Omega_{K/\\mathbf{Z}}$ is injective. \\end{enumerate} If the characteristic of $k$ is $p > 0$, then the following are equivalent: \\begin{enumerate} \\item $K$ is separable over $k$, \\item the ring $K \\otimes_k k^{1/p}$ is reduced, \\item $K$ is geometrically reduced over $k$, \\item the map $K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$ is injective, \\item $H_1(L_{K/k}) = 0$, and \\item $K$ is formally smooth over $k$. \\end{enumerate}"}
+{"_id": "1430", "title": "algebra-proposition-nagata-universally-japanese", "text": "Let $R$ be a ring. The following are equivalent: \\begin{enumerate} \\item $R$ is a Nagata ring, \\item any finite type $R$-algebra is Nagata, and \\item $R$ is universally Japanese and Noetherian. \\end{enumerate}"}
+{"_id": "1431", "title": "algebra-proposition-ubiquity-nagata", "text": "The following types of rings are Nagata and in particular universally Japanese: \\begin{enumerate} \\item fields, \\item Noetherian complete local rings, \\item $\\mathbf{Z}$, \\item Dedekind domains with fraction field of characteristic zero, \\item finite type ring extensions of any of the above. \\end{enumerate}"}
+{"_id": "1584", "title": "moduli-curves-theorem-stable-reduction", "text": "\\begin{reference} \\cite[Corollary 2.7]{DM} \\end{reference} Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$ and genus $g \\geq 2$. Then \\begin{enumerate} \\item there exists an extension of discrete valuation rings $R \\subset R'$ inducing a finite separable extension of fraction fields $K'/K$ and a stable family of curves $Y \\to \\Spec(R')$ of genus $g$ with $Y_{K'} \\cong C_{K'}$ over $K'$, and \\item there exists a finite separable extension $L/K$ and a stable family of curves $Y \\to \\Spec(A)$ of genus $g$ where $A \\subset L$ is the integral closure of $R$ in $L$ such that $Y_L \\cong C_L$ over $L$. \\end{enumerate}"}
+{"_id": "1587", "title": "moduli-curves-lemma-polarized-curves-in-polarized", "text": "The morphism $\\textit{PolarizedCurves} \\to \\Polarizedstack$ is an open and closed immersion."}
+{"_id": "1588", "title": "moduli-curves-lemma-polarized-curves-over-curves", "text": "The morphism $\\textit{PolarizedCurves} \\to \\Curvesstack$ is smooth and surjective."}
+{"_id": "1589", "title": "moduli-curves-lemma-etale-locally-scheme", "text": "Let $X \\to S$ be a family of curves. Then there exists an \\'etale covering $\\{S_i \\to S\\}$ such that $X_i = X \\times_S S_i$ is a scheme. We may even assume $X_i$ is H-projective over $S_i$."}
+{"_id": "1590", "title": "moduli-curves-lemma-curves-diagonal-separated-fp", "text": "The diagonal of $\\Curvesstack$ is separated and of finite presentation."}
+{"_id": "1591", "title": "moduli-curves-lemma-curves-qs-lfp", "text": "The morphism $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and locally of finite presentation."}
+{"_id": "1592", "title": "moduli-curves-lemma-DM-curves", "text": "There exist an open substack $\\Curvesstack^{DM} \\subset \\Curvesstack$ with the following properties \\begin{enumerate} \\item $\\Curvesstack^{DM} \\subset \\Curvesstack$ is the maximal open substack which is DM, \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{DM}$, \\item the group algebraic space $\\mathit{Aut}_S(X)$ is unramified over $S$, \\end{enumerate} \\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{DM}$, \\item $\\mathit{Aut}(X)$ is geometrically reduced over $k$ and has dimension $0$, \\item $\\mathit{Aut}(X) \\to \\Spec(k)$ is unramified. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1593", "title": "moduli-curves-lemma-in-DM-locus-vector-fields", "text": "Let $X$ be a proper scheme over a field $k$ of dimension $\\leq 1$. Then properties (3)(a), (b), (c) are also equivalent to $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$."}
+{"_id": "1594", "title": "moduli-curves-lemma-CM-curves", "text": "There exist an open substack $\\Curvesstack^{CM} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{CM}$, \\item the morphism $X \\to S$ is Cohen-Macaulay, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{CM}$, \\item $X$ is Cohen-Macaulay. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1595", "title": "moduli-curves-lemma-CM-1-curves", "text": "There exist an open substack $\\Curvesstack^{CM, 1} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{CM, 1}$, \\item the morphism $X \\to S$ is Cohen-Macaulay and has relative dimension $1$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-relative-dimension}), \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{CM, 1}$, \\item $X$ is Cohen-Macaulay and $X$ is equidimensional of dimension $1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1596", "title": "moduli-curves-lemma-pre-genus-curves", "text": "There exist an open substack $\\Curvesstack^{h0, 1} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{h0, 1}$, \\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after arbitrary base change, and the fibres of $f$ have dimension $1$, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{h0, 1}$, \\item $H^0(X, \\mathcal{O}_X) = k$ and $\\dim(X) = 1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1598", "title": "moduli-curves-lemma-genus", "text": "Let $f : X \\to S$ be a family of curves such that $\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$ for all $s \\in S$, i.e., the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{h0, 1}$ (Lemma \\ref{lemma-pre-genus-curves}). Then \\begin{enumerate} \\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds universally, \\item $R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module, \\item for any morphism $h : S' \\to S$ if $f' : X' \\to S'$ is the base change, then $h^*(R^1f_*\\mathcal{O}_X) = R^1f'_*\\mathcal{O}_{X'}$. \\end{enumerate}"}
+{"_id": "1599", "title": "moduli-curves-lemma-pre-genus-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\Curvesstack^{h0, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack_g $$ where each $\\Curvesstack_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack_g$, \\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after arbitrary base change, the fibres of $f$ have dimension $1$, and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack_g$, \\item $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and the genus of $X$ is $g$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1600", "title": "moduli-curves-lemma-geometrically-reduced-curves", "text": "There exist an open substack $\\Curvesstack^{geomred} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{geomred}$, \\item the fibres of the morphism $X \\to S$ are geometrically reduced (More on Morphisms of Spaces, Definition \\ref{spaces-more-morphisms-definition-geometrically-reduced-fibre}), \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{geomred}$, \\item $X$ is geometrically reduced over $k$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1601", "title": "moduli-curves-lemma-geomred-in-CM", "text": "We have $\\Curvesstack^{geomred} \\subset \\Curvesstack^{CM}$ as open substacks of $\\Curvesstack$."}
+{"_id": "1602", "title": "moduli-curves-lemma-geometrically-reduced-connected-1-curves", "text": "There exist an open substack $\\Curvesstack^{grc, 1} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{grc, 1}$, \\item the geometric fibres of the morphism $X \\to S$ are reduced, connected, and have dimension $1$, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{grc, 1}$, \\item $X$ is geometrically reduced, geometrically connected, and has dimension $1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1603", "title": "moduli-curves-lemma-geomredcon-in-h0-1", "text": "We have $\\Curvesstack^{grc, 1} \\subset \\Curvesstack^{h0, 1}$ as open substacks of $\\Curvesstack$. In particular, given a family of curves $f : X \\to S$ whose geometric fibres are reduced, connected and of dimension $1$, then $R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module whose formation commutes with arbitrary base change."}
+{"_id": "1604", "title": "moduli-curves-lemma-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\Curvesstack^{grc, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack^{grc, 1}_g $$ where each $\\Curvesstack^{grc, 1}_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{grc, 1}_g$, \\item the geometric fibres of the morphism $f : X \\to S$ are reduced, connected, of dimension $1$ and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{grc, 1}_g$, \\item $X$ is geometrically reduced, geometrically connected, has dimension $1$, and has genus $g$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1605", "title": "moduli-curves-lemma-gorenstein-curves", "text": "There exist an open substack $\\Curvesstack^{Gorenstein} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{Gorenstein}$, \\item the morphism $X \\to S$ is Gorenstein, \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{Gorenstein}$, \\item $X$ is Gorenstein. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1606", "title": "moduli-curves-lemma-gorenstein-1-curves", "text": "There exist an open substack $\\Curvesstack^{Gorenstein, 1} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{Gorenstein, 1}$, \\item the morphism $X \\to S$ is Gorenstein and has relative dimension $1$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-relative-dimension}), \\end{enumerate} \\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{Gorenstein, 1}$, \\item $X$ is Gorenstein and $X$ is equidimensional of dimension $1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1607", "title": "moduli-curves-lemma-lci-curves", "text": "There exist an open substack $\\Curvesstack^{lci} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{lci}$, \\item $X \\to S$ is a local complete intersection morphism, and \\item $X \\to S$ is a syntomic morphism. \\end{enumerate} \\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{lci}$, \\item $X$ is a local complete intersection over $k$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1608", "title": "moduli-curves-lemma-isolated-sings-curves", "text": "There exist an open substack $\\Curvesstack^{+} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{+}$, \\item the singular locus of $X \\to S$ endowed with any/some closed subspace structure is finite over $S$. \\end{enumerate} \\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{+}$, \\item $X \\to \\Spec(k)$ is smooth except at finitely many points. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1609", "title": "moduli-curves-lemma-in-smooth-locus", "text": "In the situation above the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through the open where $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is smooth, \\item the deformation category $\\Deformationcategory_X$ is unobstructed. \\end{enumerate}"}
+{"_id": "1610", "title": "moduli-curves-lemma-big-smooth-part-curves", "text": "The open substack $$ \\Curvesstack^{lci+} = \\Curvesstack^{lci} \\cap \\Curvesstack^{+} \\subset \\Curvesstack $$ has the following properties \\begin{enumerate} \\item $\\Curvesstack^{lci+} \\to \\Spec(\\mathbf{Z})$ is smooth, \\item given a family of curves $X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{lci+}$, \\item $X \\to S$ is a local complete intersection morphism and the singular locus of $X \\to S$ endowed with any/some closed subspace structure is finite over $S$, \\end{enumerate} \\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{lci+}$, \\item $X$ is a local complete intersection over $k$ and $X \\to \\Spec(k)$ is smooth except at finitely many points. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1611", "title": "moduli-curves-lemma-smooth-curves", "text": "There exist an open substacks $$ \\Curvesstack^{smooth, 1} \\subset \\Curvesstack^{smooth} \\subset \\Curvesstack $$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$, \\item $f$ is smooth, resp.\\ smooth of relative dimension $1$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$, \\item $X$ is smooth over $k$, resp.\\ $X$ is smooth over $k$ and $X$ is equidimensional of dimension $1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1613", "title": "moduli-curves-lemma-smooth-curves-h0", "text": "There exist an open substack $\\Curvesstack^{smooth, h0} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{smooth}$, \\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after any base change, and $f$ is smooth of relative dimension $1$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{smooth, h0}$, \\item $X$ is smooth, $\\dim(X) = 1$, and $k = H^0(X, \\mathcal{O}_X)$, \\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically connected, \\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically integral, and \\item $X_{\\overline{k}}$ is a smooth curve. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1614", "title": "moduli-curves-lemma-smooth-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\mathcal{M} = \\coprod\\nolimits_{g \\geq 0} \\mathcal{M}_g $$ where each $\\mathcal{M}_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\mathcal{M}_g$, \\item $X \\to S$ is smooth, $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after any base change, and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\mathcal{M}_g$, \\item $X$ is smooth, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and $X$ has genus $g$, \\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically connected, and $X$ has genus $g$, \\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically integral, and $X$ has genus $g$, and \\item $X_{\\overline{k}}$ is a smooth curve of genus $g$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1616", "title": "moduli-curves-lemma-smooth-dense", "text": "The inclusion $$ |\\Curvesstack^{smooth}| \\subset |\\Curvesstack^{lci+}| $$ is that of an open dense subset."}
+{"_id": "1617", "title": "moduli-curves-lemma-nodal-curves", "text": "There exist an open substack $\\Curvesstack^{nodal} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{nodal}$, \\item $f$ is at-worst-nodal of relative dimension $1$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{nodal}$, \\item the singularities of $X$ are at-worst-nodal and $X$ is equidimensional of dimension $1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1618", "title": "moduli-curves-lemma-nodal-curves-smooth", "text": "The morphism $\\Curvesstack^{nodal} \\to \\Spec(\\mathbf{Z})$ is smooth."}
+{"_id": "1619", "title": "moduli-curves-lemma-CM-dualizing", "text": "Let $X \\to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}). Then $\\omega_{X/S}^\\bullet = \\omega_{X/S}[1]$ where $\\omega_{X/S}$ is a pseudo-coherent $\\mathcal{O}_X$-module flat over $S$ whose formation commutes with arbitrary base change."}
+{"_id": "1621", "title": "moduli-curves-lemma-prestable-curves", "text": "There exist an open substack $\\Curvesstack^{prestable} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{prestable}$, \\item $X \\to S$ is a prestable family of curves, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{prestable}$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, and $k = H^0(X, \\mathcal{O}_X)$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1622", "title": "moduli-curves-lemma-prestable-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\Curvesstack^{prestable} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack^{prestable}_g $$ where each $\\Curvesstack^{prestable}_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{prestable}_g$, \\item $X \\to S$ is a prestable family of curves and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{prestable}_g$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and the genus of $X$ is $g$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1624", "title": "moduli-curves-lemma-semistable", "text": "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 1$. Let $s \\in S$ be a point of the base scheme. Let $m \\geq 2$. The following are equivalent \\begin{enumerate} \\item $X_s$ does not have a rational tail (Algebraic Curves, Example \\ref{curves-example-rational-tail}), and \\item $f^*f_*\\omega_{X/S}^{\\otimes m} \\to \\omega_{X/S}^{\\otimes m}$, is surjective over $f^{-1}(U)$ for some $s \\in U \\subset S$ open. \\end{enumerate}"}
+{"_id": "1625", "title": "moduli-curves-lemma-semistable-curves", "text": "There exist an open substack $\\Curvesstack^{semistable} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{semistable}$, \\item $X \\to S$ is a semistable family of curves, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{semistable}$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 1$, and $X$ has no rational tails, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}^{\\otimes m}$ is globally generated for $m \\geq 2$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1628", "title": "moduli-curves-lemma-stable", "text": "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 2$. Let $s \\in S$ be a point of the base scheme. The following are equivalent \\begin{enumerate} \\item $X_s$ does not have a rational tail and does not have a rational bridge (Algebraic Curves, Examples \\ref{curves-example-rational-tail} and \\ref{curves-example-rational-bridge}), and \\item $\\omega_{X/S}$ is ample on $f^{-1}(U)$ for some $s \\in U \\subset S$ open. \\end{enumerate}"}
+{"_id": "1629", "title": "moduli-curves-lemma-stable-curves", "text": "There exist an open substack $\\Curvesstack^{stable} \\subset \\Curvesstack$ such that \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{stable}$, \\item $X \\to S$ is a stable family of curves, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{stable}$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 2$, and $X$ has no rational tails or bridges, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}$ is ample. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1630", "title": "moduli-curves-lemma-stable-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\overline{\\mathcal{M}} = \\coprod\\nolimits_{g \\geq 2} \\overline{\\mathcal{M}}_g $$ where each $\\overline{\\mathcal{M}}_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\overline{\\mathcal{M}}_g$, \\item $X \\to S$ is a stable family of curves and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\overline{\\mathcal{M}}_g$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$ has no rational tails or bridges. \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $\\omega_{X_s}$ is ample. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1631", "title": "moduli-curves-lemma-stable-curves-smooth", "text": "The morphisms $\\overline{\\mathcal{M}} \\to \\Spec(\\mathbf{Z})$ and $\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$ are smooth."}
+{"_id": "1632", "title": "moduli-curves-lemma-stable-curves-deligne-mumford", "text": "The stacks $\\overline{\\mathcal{M}}$ and $\\overline{\\mathcal{M}}_g$ are open substacks of $\\Curvesstack^{DM}$. In particular, $\\overline{\\mathcal{M}}$ and $\\overline{\\mathcal{M}}_g$ are DM (Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-absolute-separated}) as well as Deligne-Mumford stacks (Algebraic Stacks, Definition \\ref{algebraic-definition-deligne-mumford})."}
+{"_id": "1633", "title": "moduli-curves-lemma-smooth-dense-in-stable", "text": "Let $g \\geq 2$. The inclusion $$ |\\mathcal{M}_g| \\subset |\\overline{\\mathcal{M}}_g| $$ is that of an open dense subset."}
+{"_id": "1634", "title": "moduli-curves-lemma-contract", "text": "Let $S$ be a scheme and $s \\in S$ a point. Let $f : X \\to S$ and $g : Y \\to S$ be families of curves. Let $c : X \\to Y$ be a morphism over $S$. If $c_{s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_s}$ and $R^1c_{s, *}\\mathcal{O}_{X_s} = 0$, then after replacing $S$ by an open neighbourhood of $s$ we have $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$ and this remains true after base change by any morphism $S' \\to S$."}
+{"_id": "1635", "title": "moduli-curves-lemma-contract-basic-uniqueness", "text": "Let $S$ be a scheme and $s \\in S$ a point. Let $f : X \\to S$ and $g_i : Y_i \\to S$, $i = 1, 2$ be families of curves. Let $c_i : X \\to Y_i$ be morphisms over $S$. Assume there is an isomorphism $Y_{1, s} \\cong Y_{2, s}$ of fibres compatible with $c_{1, s}$ and $c_{2, s}$. If $c_{1, s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\\mathcal{O}_{X_s} = 0$, then there exist an open neighbourhood $U$ of $s$ and an isomorphism $Y_{1, U} \\cong Y_{2, U}$ of families of curves over $U$ compatible with the given isomorphism of fibres and with $c_1$ and $c_2$."}
+{"_id": "1636", "title": "moduli-curves-lemma-contract-basic", "text": "Let $f : X \\to S$ be a family of curves. Let $s \\in S$ be a point. Let $h_0 : X_s \\to Y_0$ be a morphism to a proper scheme $Y_0$ over $\\kappa(s)$ such that $h_{0, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_0}$ and $R^1h_{0, *}\\mathcal{O}_{X_s} = 0$. Then there exist an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$, a family of curves $Y \\to U$, and a morphism $h : X_U \\to Y$ over $U$ whose fibre in $u$ is isomorphic to $h_0$."}
+{"_id": "1637", "title": "moduli-curves-lemma-contract-prestable-to-stable", "text": "Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 1$. There is a factorization $X \\to Y \\to S$ of $f$ where $g : Y \\to S$ is a stable family of curves and $c : X \\to Y$ has the following properties \\begin{enumerate} \\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$ and this remains true after base change by any morphism $S' \\to S$, and \\item for any $s \\in S$ the morphism $c_s : X_s \\to Y_s$ is the contraction of rational tails and bridges discussed in Algebraic Curves, Section \\ref{curves-section-contracting-to-stable}. \\end{enumerate} Moreover $c : X \\to Y$ is unique up to unique isomorphism."}
+{"_id": "1639", "title": "moduli-curves-lemma-stable-reduction", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $K = H^0(C, \\mathcal{O}_C)$ having genus $g \\geq 2$. The following are equivalent \\begin{enumerate} \\item $C$ has semistable reduction (Semistable Reduction, Definition \\ref{models-definition-semistable}), or \\item there is a stable family of curves over $R$ with generic fibre $C$. \\end{enumerate}"}
+{"_id": "1640", "title": "moduli-curves-lemma-unique-stable-model", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth proper curve over $K$ with $K = H^0(C, \\mathcal{O}_C)$ and genus $g$. If $X$ and $X'$ are models of $C$ (Semistable Reduction, Section \\ref{models-section-models}) and $X$ and $X'$ are stable families of genus $g$ curves over $R$, then there exists an unique isomorphism $X \\to X'$ of models."}
+{"_id": "1641", "title": "moduli-curves-lemma-stable-separated", "text": "Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is separated."}
+{"_id": "1642", "title": "moduli-curves-lemma-stable-quasi-compact", "text": "Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is quasi-compact."}
+{"_id": "1650", "title": "dpa-lemma-silly", "text": "Let $A$ be a ring. Let $I$ be an ideal of $A$. \\begin{enumerate} \\item If $\\gamma$ is a divided power structure\\footnote{Here and in the following, $\\gamma$ stands short for a sequence of maps $\\gamma_1, \\gamma_2, \\gamma_3, \\ldots$ from $I$ to $I$.} on $I$, then $n! \\gamma_n(x) = x^n$ for $n \\geq 1$, $x \\in I$. \\end{enumerate} Assume $A$ is torsion free as a $\\mathbf{Z}$-module. \\begin{enumerate} \\item[(2)] A divided power structure on $I$, if it exists, is unique. \\item[(3)] If $\\gamma_n : I \\to I$ are maps then $$ \\gamma\\text{ is a divided power structure} \\Leftrightarrow n! \\gamma_n(x) = x^n\\ \\forall x \\in I, n \\geq 1. $$ \\item[(4)] The ideal $I$ has a divided power structure if and only if there exists a set of generators $x_i$ of $I$ as an ideal such that for all $n \\geq 1$ we have $x_i^n \\in (n!)I$. \\end{enumerate}"}
+{"_id": "1651", "title": "dpa-lemma-check-on-generators", "text": "Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $\\gamma_n : I \\to I$, $n \\geq 1$ be a sequence of maps. Assume \\begin{enumerate} \\item[(a)] (1), (3), and (4) of Definition \\ref{definition-divided-powers} hold for all $x, y \\in I$, and \\item[(b)] properties (2) and (5) hold for $x$ in some set of generators of $I$ as an ideal. \\end{enumerate} Then $\\gamma$ is a divided power structure on $I$."}
+{"_id": "1652", "title": "dpa-lemma-two-ideals", "text": "Let $A$ be a ring with two ideals $I, J \\subset A$. Let $\\gamma$ be a divided power structure on $I$ and let $\\delta$ be a divided power structure on $J$. Then \\begin{enumerate} \\item $\\gamma$ and $\\delta$ agree on $IJ$, \\item if $\\gamma$ and $\\delta$ agree on $I \\cap J$ then they are the restriction of a unique divided power structure $\\epsilon$ on $I + J$. \\end{enumerate}"}
+{"_id": "1653", "title": "dpa-lemma-nil", "text": "Let $p$ be a prime number. Let $A$ be a ring, let $I \\subset A$ be an ideal, and let $\\gamma$ be a divided power structure on $I$. Assume $p$ is nilpotent in $A/I$. Then $I$ is locally nilpotent if and only if $p$ is nilpotent in $A$."}
+{"_id": "1654", "title": "dpa-lemma-limits", "text": "The category of divided power rings has all limits and they agree with limits in the category of rings."}
+{"_id": "1655", "title": "dpa-lemma-a-version-of-brown", "text": "Let $\\mathcal{C}$ be the category of divided power rings. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item there exists a cardinal $\\kappa$ such that for every $f \\in F(A, I, \\gamma)$ there exists a morphism $(A', I', \\gamma') \\to (A, I, \\gamma)$ of $\\mathcal{C}$ such that $f$ is the image of $f' \\in F(A', I', \\gamma')$ and $|A'| \\leq \\kappa$, and \\item $F$ commutes with limits. \\end{enumerate} Then $F$ is representable, i.e., there exists an object $(B, J, \\delta)$ of $\\mathcal{C}$ such that $$ F(A, I, \\gamma) = \\Hom_\\mathcal{C}((B, J, \\delta), (A, I, \\gamma)) $$ functorially in $(A, I, \\gamma)$."}
+{"_id": "1656", "title": "dpa-lemma-colimits", "text": "The category of divided power rings has all colimits."}
+{"_id": "1657", "title": "dpa-lemma-gamma-extends", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring map. If $\\gamma$ extends to $B$ then it extends uniquely. Assume (at least) one of the following conditions holds \\begin{enumerate} \\item $IB = 0$, \\item $I$ is principal, or \\item $A \\to B$ is flat. \\end{enumerate} Then $\\gamma$ extends to $B$."}
+{"_id": "1658", "title": "dpa-lemma-kernel", "text": "Let $(A, I, \\gamma)$ be a divided power ring. \\begin{enumerate} \\item If $\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a homomorphism of divided power rings, then $\\Ker(\\varphi) \\cap I$ is preserved by $\\gamma_n$ for all $n \\geq 1$. \\item Let $\\mathfrak a \\subset A$ be an ideal and set $I' = I \\cap \\mathfrak a$. The following are equivalent \\begin{enumerate} \\item $I'$ is preserved by $\\gamma_n$ for all $n > 0$, \\item $\\gamma$ extends to $A/\\mathfrak a$, and \\item there exist a set of generators $x_i$ of $I'$ as an ideal such that $\\gamma_n(x_i) \\in I'$ for all $n > 0$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1660", "title": "dpa-lemma-extend-to-completion", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $p$ be a prime. If $p$ is nilpotent in $A/I$, then \\begin{enumerate} \\item the $p$-adic completion $A^\\wedge = \\lim_e A/p^eA$ surjects onto $A/I$, \\item the kernel of this map is the $p$-adic completion $I^\\wedge$ of $I$, and \\item each $\\gamma_n$ is continuous for the $p$-adic topology and extends to $\\gamma_n^\\wedge : I^\\wedge \\to I^\\wedge$ defining a divided power structure on $I^\\wedge$. \\end{enumerate} If moreover $A$ is a $\\mathbf{Z}_{(p)}$-algebra, then \\begin{enumerate} \\item[(4)] for $e$ large enough the ideal $p^eA \\subset I$ is preserved by the divided power structure $\\gamma$ and $$ (A^\\wedge, I^\\wedge, \\gamma^\\wedge) = \\lim_e (A/p^eA, I/p^eA, \\bar\\gamma) $$ in the category of divided power rings. \\end{enumerate}"}
+{"_id": "1661", "title": "dpa-lemma-divided-power-polynomial-algebra", "text": "Let $(A, I, \\gamma)$ be a divided power ring. There exists a unique divided power structure $\\delta$ on $$ J = IA\\langle x_1, \\ldots, x_t \\rangle + A\\langle x_1, \\ldots, x_t \\rangle_{+} $$ such that \\begin{enumerate} \\item $\\delta_n(x_i) = x_i^{[n]}$, and \\item $(A, I, \\gamma) \\to (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$ is a homomorphism of divided power rings. \\end{enumerate} Moreover, $(A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$ has the following universal property: A homomorphism of divided power rings $\\varphi : (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta) \\to (C, K, \\epsilon)$ is the same thing as a homomorphism of divided power rings $A \\to C$ and elements $k_1, \\ldots, k_t \\in K$."}
+{"_id": "1663", "title": "dpa-lemma-dpdga-good", "text": "Let $(A, \\text{d}, \\gamma)$ and $(B, \\text{d}, \\gamma)$ be as in Definition \\ref{definition-divided-powers-dga}. Let $f : A \\to B$ be a map of differential graded algebras compatible with divided power structures. Assume \\begin{enumerate} \\item $H_k(A) = 0$ for $k > 0$, and \\item $f$ is surjective. \\end{enumerate} Then $\\gamma$ induces a divided power structure on the graded $R$-algebra $H(B)$."}
+{"_id": "1664", "title": "dpa-lemma-base-change-div", "text": "Let $(A, \\text{d}, \\gamma)$ be as in Definition \\ref{definition-divided-powers-dga}. Let $R \\to R'$ be a ring map. Then $\\text{d}$ and $\\gamma$ induce similar structures on $A' = A \\otimes_R R'$ such that $(A', \\text{d}, \\gamma)$ is as in Definition \\ref{definition-divided-powers-dga}."}
+{"_id": "1665", "title": "dpa-lemma-extend-differential", "text": "Let $(A, \\text{d}, \\gamma)$ be as in Definition \\ref{definition-divided-powers-dga}. Let $d \\geq 1$ be an integer. Let $A\\langle T \\rangle$ be the graded divided power polynomial algebra on $T$ with $\\deg(T) = d$ constructed in Example \\ref{example-adjoining-odd} or \\ref{example-adjoining-even}. Let $f \\in A_{d - 1}$ be an element with $\\text{d}(f) = 0$. There exists a unique differential $\\text{d}$ on $A\\langle T\\rangle$ such that $\\text{d}(T) = f$ and such that $\\text{d}$ is compatible with the divided power structure on $A\\langle T \\rangle$."}
+{"_id": "1666", "title": "dpa-lemma-tate-resolution", "text": "Let $R \\to S$ be a homomorphism of commutative rings. There exists a factorization $$ R \\to A \\to S $$ with the following properties: \\begin{enumerate} \\item $(A, \\text{d}, \\gamma)$ is as in Definition \\ref{definition-divided-powers-dga}, \\item $A \\to S$ is a quasi-isomorphism (if we endow $S$ with the zero differential), \\item $A_0 = R[x_j: j\\in J] \\to S$ is any surjection of a polynomial ring onto $S$, and \\item $A$ is a graded divided power polynomial algebra over $R$. \\end{enumerate} The last condition means that $A$ is constructed out of $A_0$ by successively adjoining a set of variables $T$ in each degree $> 0$ as in Example \\ref{example-adjoining-odd} or \\ref{example-adjoining-even}. Moreover, if $R$ is Noetherian and $R\\to S$ is of finite type, then $A$ can be taken to have only finitely many generators in each degree."}
+{"_id": "1667", "title": "dpa-lemma-tate-resoluton-pseudo-coherent-ring-map", "text": "Let $R \\to S$ be a pseudo-coherent ring map (More on Algebra, Definition \\ref{more-algebra-definition-pseudo-coherent-perfect}). Then Lemma \\ref{lemma-tate-resolution} holds, with the resolution $A$ of $S$ having finitely many generators in each degree."}
+{"_id": "1668", "title": "dpa-lemma-uniqueness-tate-resolution", "text": "Let $R$ be a commutative ring. Suppose that $(A, \\text{d}, \\gamma)$ and $(B, \\text{d}, \\gamma)$ are as in Definition \\ref{definition-divided-powers-dga}. Let $\\overline{\\varphi} : H_0(A) \\to H_0(B)$ be an $R$-algebra map. Assume \\begin{enumerate} \\item $A$ is a graded divided power polynomial algebra over $R$. \\item $H_k(B) = 0$ for $k > 0$. \\end{enumerate} Then there exists a map $\\varphi : A \\to B$ of differential graded $R$-algebras compatible with divided powers that lifts $\\overline{\\varphi}$."}
+{"_id": "1670", "title": "dpa-lemma-get-derivation", "text": "Let $R$ be a ring. Let $(A, \\text{d}, \\gamma)$ be as in Definition \\ref{definition-divided-powers-dga}. Let $R' \\to R$ be a surjection of rings whose kernel has square zero and is generated by one element $f$. If $A$ is a graded divided power polynomial algebra over $R$ with finitely many variables in each degree, then we obtain a derivation $\\theta : A/IA \\to A/IA$ where $I$ is the annihilator of $f$ in $R$."}
+{"_id": "1671", "title": "dpa-lemma-compute-theta", "text": "Assumption and notation as in Lemma \\ref{lemma-get-derivation}. Suppose $S = H_0(A)$ is isomorphic to $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ for some $n$, $m$, and $f_j \\in R[x_1, \\ldots, x_n]$. Moreover, suppose given a relation $$ \\sum r_j f_j = 0 $$ with $r_j \\in R[x_1, \\ldots, x_n]$. Choose $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ lifting $r_j, f_j$. Write $\\sum r'_j f'_j = gf$ for some $g \\in R/I[x_1, \\ldots, x_n]$. If $H_1(A) = 0$ and all the coefficients of each $r_j$ are in $I$, then there exists an element $\\xi \\in H_2(A/IA)$ such that $\\theta(\\xi) = g$ in $S/IS$."}
+{"_id": "1672", "title": "dpa-lemma-not-finite-pd", "text": "Let $R' \\to R$ be a surjection of Noetherian rings whose kernel has square zero and is generated by one element $f$. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$. Let $\\sum r_j f_j = 0$ be a relation in $R[x_1, \\ldots, x_n]$. Assume that \\begin{enumerate} \\item each $r_j$ has coefficients in the annihilator $I$ of $f$ in $R$, \\item for some lifts $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ we have $\\sum r'_j f'_j = gf$ where $g$ is not nilpotent in $S$. \\end{enumerate} Then $S$ does not have finite tor dimension over $R$ (i.e., $S$ is not a perfect $R$-algebra)."}
+{"_id": "1673", "title": "dpa-lemma-injective", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset J \\subset A$ be proper ideals. If $A/J$ has finite tor dimension over $A/I$, then $I/\\mathfrak m I \\to J/\\mathfrak m J$ is injective."}
+{"_id": "1674", "title": "dpa-lemma-regular-sequence", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset J \\subset A$ be proper ideals. Assume \\begin{enumerate} \\item $A/J$ has finite tor dimension over $A/I$, and \\item $J$ is generated by a regular sequence. \\end{enumerate} Then $I$ is generated by a regular sequence and $J/I$ is generated by a regular sequence."}
+{"_id": "1675", "title": "dpa-lemma-perfect-map-ci", "text": "Let $R \\to S$ be a local ring map of Noetherian local rings. Let $I \\subset R$ and $J \\subset S$ be ideals with $IS \\subset J$. If $R \\to S$ is flat and $S/\\mathfrak m_RS$ is regular, then the following are equivalent \\begin{enumerate} \\item $J$ is generated by a regular sequence and $S/J$ has finite tor dimension as a module over $R/I$, \\item $J$ is generated by a regular sequence and $\\text{Tor}^{R/I}_p(S/J, R/\\mathfrak m_R)$ is nonzero for only finitely many $p$, \\item $I$ is generated by a regular sequence and $J/IS$ is generated by a regular sequence in $S/IS$. \\end{enumerate}"}
+{"_id": "1677", "title": "dpa-lemma-quotient-regular-ring-by-regular-sequence", "text": "Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime. Let $f_1, \\ldots, f_r \\in \\mathfrak p$ be a regular sequence. Then the completion of $$ A = (R/(f_1, \\ldots, f_r))_\\mathfrak p = R_\\mathfrak p/(f_1, \\ldots, f_r)R_\\mathfrak p $$ is a complete intersection in the sense defined above."}
+{"_id": "1678", "title": "dpa-lemma-quotient-regular-ring", "text": "Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime. Let $I \\subset \\mathfrak p$ be an ideal. Set $A = (R/I)_\\mathfrak p = R_\\mathfrak p/I_\\mathfrak p$. The following are equivalent \\begin{enumerate} \\item the completion of $A$ is a complete intersection in the sense above, \\item $I_\\mathfrak p \\subset R_\\mathfrak p$ is generated by a regular sequence, \\item the module $(I/I^2)_\\mathfrak p$ can be generated by $\\dim(R_\\mathfrak p) - \\dim(A)$ elements, \\item add more here. \\end{enumerate}"}
+{"_id": "1679", "title": "dpa-lemma-ci-good", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $\\mathfrak p \\subset A$ be a prime ideal. If $A$ is a complete intersection, then $A_\\mathfrak p$ is a complete intersection too."}
+{"_id": "1681", "title": "dpa-lemma-check-lci-agrees", "text": "Let $S$ be a finite type algebra over a field $k$. \\begin{enumerate} \\item for a prime $\\mathfrak q \\subset S$ the local ring $S_\\mathfrak q$ is a complete intersection in the sense of Algebra, Definition \\ref{algebra-definition-lci-local-ring} if and only if $S_\\mathfrak q$ is a complete intersection in the sense of Definition \\ref{definition-lci}, and \\item $S$ is a local complete intersection in the sense of Algebra, Definition \\ref{algebra-definition-lci-field} if and only if $S$ is a local complete intersection in the sense of Definition \\ref{definition-lci}. \\end{enumerate}"}
+{"_id": "1682", "title": "dpa-lemma-avramov", "text": "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent \\begin{enumerate} \\item $B$ is a complete intersection, \\item $A$ and $B/\\mathfrak m_A B$ are complete intersections. \\end{enumerate}"}
+{"_id": "1683", "title": "dpa-lemma-ci-map-well-defined", "text": "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings. The following are equivalent \\begin{enumerate} \\item for some good factorization $A \\to S \\to B$ the kernel of $S \\to B$ is generated by a regular sequence, and \\item for every good factorization $A \\to S \\to B$ the kernel of $S \\to B$ is generated by a regular sequence. \\end{enumerate}"}
+{"_id": "1684", "title": "dpa-lemma-well-defined-if-you-can-find-good-factorization", "text": "Consider a commutative diagram $$ \\xymatrix{ S \\ar[r] & B \\\\ & A \\ar[lu] \\ar[u] } $$ of Noetherian local rings with $S \\to B$ surjective, $A \\to S$ flat, and $S/\\mathfrak m_A S$ a regular local ring. The following are equivalent \\begin{enumerate} \\item $\\Ker(S \\to B)$ is generated by a regular sequence, and \\item $A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism as defined above. \\end{enumerate}"}
+{"_id": "1685", "title": "dpa-lemma-finite-type-lci-map", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is a local complete intersection in the sense of More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}, \\item for every prime $\\mathfrak q \\subset B$ and with $\\mathfrak p = A \\cap \\mathfrak q$ the ring map $(A_\\mathfrak p)^\\wedge \\to (B_\\mathfrak q)^\\wedge$ is a complete intersection homomorphism in the sense defined above. \\end{enumerate}"}
+{"_id": "1687", "title": "dpa-lemma-local-perfect-diagonal", "text": "Let $A \\to B$ be a local ring homomorphism of Noetherian local rings such that $B$ is flat and essentially of finite type over $A$. If $$ B \\otimes_A B \\longrightarrow B $$ is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \\otimes_A B$, then $B$ is the localization of a smooth $A$-algebra."}
+{"_id": "1688", "title": "dpa-lemma-perfect-diagonal", "text": "Let $A \\to B$ be a flat finite type ring map of Noetherian rings. If $$ B \\otimes_A B \\longrightarrow B $$ is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \\otimes_A B$, then $B$ is a smooth $A$-algebra."}
+{"_id": "1689", "title": "dpa-lemma-free-summand-in-ideal-finite-proj-dim", "text": "\\begin{reference} \\cite{Vasconcelos} \\end{reference} Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of finite projective dimension over $R$. If $F \\subset I/I^2$ is a direct summand isomorphic to $R/I$, then there exists a nonzerodivisor $x \\in I$ such that the image of $x$ in $I/I^2$ generates $F$."}
+{"_id": "1690", "title": "dpa-lemma-vasconcelos", "text": "\\begin{reference} Local version of \\cite[Theorem 1.1]{Vasconcelos} \\end{reference} Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of finite projective dimension over $R$. If $F \\subset I/I^2$ is a direct summand free of rank $r$, then there exists a regular sequence $x_1, \\ldots, x_r \\in I$ such that $x_1 \\bmod I^2, \\ldots, x_r \\bmod I^2$ generate $F$."}
+{"_id": "1691", "title": "dpa-lemma-perfect-NL-lci", "text": "Let $A \\to B$ be a perfect (More on Algebra, Definition \\ref{more-algebra-definition-pseudo-coherent-perfect}) ring homomorphism of Noetherian rings. Then the following are equivalent \\begin{enumerate} \\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$, \\item $\\NL_{B/A}$ is a perfect object of $D(B)$ with tor-amplitude in $[-1, 0]$, and \\item $A \\to B$ is a local complete intersection (More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}). \\end{enumerate}"}
+{"_id": "1692", "title": "dpa-lemma-flat-fp-NL-lci", "text": "Let $A \\to B$ be a flat ring map of finite presentation. Then the following are equivalent \\begin{enumerate} \\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$, \\item $\\NL_{B/A}$ is a perfect object of $D(B)$ with tor-amplitude in $[-1, 0]$, \\item $A \\to B$ is syntomic (Algebra, Definition \\ref{algebra-definition-lci}), and \\item $A \\to B$ is a local complete intersection (More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}). \\end{enumerate}"}
+{"_id": "1693", "title": "dpa-proposition-avramov", "text": "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings. Then the following are equivalent \\begin{enumerate} \\item $B^\\wedge$ is a complete intersection, \\item $A^\\wedge$ and $(B/\\mathfrak m_A B)^\\wedge$ are complete intersections. \\end{enumerate}"}
+{"_id": "1694", "title": "dpa-proposition-avramov-map", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Then the following are equivalent \\begin{enumerate} \\item $B$ is a complete intersection and $\\text{Tor}^A_p(B, A/\\mathfrak m_A)$ is nonzero for only finitely many $p$, \\item $A$ is a complete intersection and $A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism in the sense defined above. \\end{enumerate}"}
+{"_id": "1695", "title": "dpa-proposition-regular-ideal", "text": "\\begin{reference} Variant of \\cite[Corollary 1]{Vasconcelos}. See also \\cite{Iyengar} and \\cite{Ferrand-lci}. \\end{reference} Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal which has finite projective dimension and such that $I/I^2$ is finite locally free over $R/I$. Then $I$ is a regular ideal (More on Algebra, Definition \\ref{more-algebra-definition-regular-ideal})."}
+{"_id": "1706", "title": "moduli-lemma-coherent-diagonal-affine-fp", "text": "The diagonal of $\\Cohstack_{X/B}$ over $B$ is affine and of finite presentation."}
+{"_id": "1707", "title": "moduli-lemma-coherent-qs-lfp", "text": "The morphism $\\Cohstack_{X/B} \\to B$ is quasi-separated and locally of finite presentation."}
+{"_id": "1709", "title": "moduli-lemma-coherent-functorial", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Then $\\pi_*$ induces a morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$."}
+{"_id": "1710", "title": "moduli-lemma-coherent-open", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Then the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of Lemma \\ref{lemma-coherent-functorial} is an open immersion."}
+{"_id": "1712", "title": "moduli-lemma-open-P", "text": "In Situation \\ref{situation-numerical} the stack $\\Cohstack^P_{X/B}$ is algebraic and $$ \\Cohstack^P_{X/B} \\longrightarrow \\Cohstack_{X/B} $$ is a flat closed immersion. If $I$ is finite or $B$ is locally Noetherian, then $\\Cohstack^P_{X/B}$ is an open and closed substack of $\\Cohstack_{X/B}$."}
+{"_id": "1713", "title": "moduli-lemma-finite-list-perfect-objects", "text": "Let $f : X \\to B$ be as in the introduction to this section. Let $E_1, \\ldots, E_r \\in D(\\mathcal{O}_X)$ be perfect. Let $I = \\mathbf{Z}^{\\oplus r}$ and consider the map $$ I \\longrightarrow D(\\mathcal{O}_X),\\quad (n_1, \\ldots, n_r) \\longmapsto E_1^{\\otimes n_1} \\otimes \\ldots \\otimes E_r^{\\otimes n_r} $$ Let $P : I \\to \\mathbf{Z}$ be a map. Then $\\Cohstack^P_{X/B} \\subset \\Cohstack_{X/B}$ as defined in Situation \\ref{situation-numerical} is an open and closed substack."}
+{"_id": "1714", "title": "moduli-lemma-quot-diagonal-closed", "text": "The diagonal of $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is a closed immersion. If $\\mathcal{F}$ is of finite type, then the diagonal is a closed immersion of finite presentation."}
+{"_id": "1715", "title": "moduli-lemma-quot-s-lfp", "text": "The morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is separated. If $\\mathcal{F}$ is of finite presentation, then it is also locally of finite presentation."}
+{"_id": "1716", "title": "moduli-lemma-quot-existence-part", "text": "Assume $X \\to B$ is proper as well as of finite presentation and $\\mathcal{F}$ quasi-coherent of finite type. Then $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-valuative-criterion})."}
+{"_id": "1717", "title": "moduli-lemma-quot-functorial", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\pi_*$ induces a morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$."}
+{"_id": "1718", "title": "moduli-lemma-quot-open", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine open immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then the morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of Lemma \\ref{lemma-quot-functorial} is an open immersion."}
+{"_id": "1719", "title": "moduli-lemma-quot-better-open", "text": "Let $B$ be an algebraic space. Let $j : X \\to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module and set $\\mathcal{F} = j^*\\mathcal{G}$. Then there is an open immersion $$ \\Quotfunctor_{\\mathcal{F}/X/B} \\longrightarrow \\Quotfunctor_{\\mathcal{G}/Y/B} $$ of algebraic spaces over $B$."}
+{"_id": "1720", "title": "moduli-lemma-quot-closed", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then the morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of Lemma \\ref{lemma-quot-functorial} is an isomorphism."}
+{"_id": "1721", "title": "moduli-lemma-quot-quotient", "text": "Let $X \\to B$ be as in the introduction to this section. Let $\\mathcal{F} \\to \\mathcal{G}$ be a surjection of quasi-coherent $\\mathcal{O}_X$-modules. Then there is a canonical closed immersion $\\Quotfunctor_{\\mathcal{G}/X/B} \\to \\Quotfunctor_{\\mathcal{F}/X/B}$."}
+{"_id": "1722", "title": "moduli-lemma-quot-tensor-invertible", "text": "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then tensoring with $\\mathcal{L}$ defines an isomophism $$ \\Quotfunctor_{\\mathcal{F}/X/B} \\longrightarrow \\Quotfunctor_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B} $$ Given a numerical polynomial $P(t)$, then setting $P'(t) = P(t + 1)$ this map induces an isomorphism $\\Quotfunctor^P_{\\mathcal{F}/X/B} \\longrightarrow \\Quotfunctor^{P'}_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}$ of open and closed substacks."}
+{"_id": "1723", "title": "moduli-lemma-quot-power-invertible", "text": "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $$ \\Quotfunctor^{P, \\mathcal{L}}_{\\mathcal{F}/X/B} = \\Quotfunctor^{P', \\mathcal{L}^{\\otimes n}}_{\\mathcal{F}/X/B} $$ where $P'(t) = P(nt)$."}
+{"_id": "1724", "title": "moduli-lemma-quot-Pn", "text": "Let $n \\geq 0$, $r \\geq 1$, $P \\in \\mathbf{Q}[t]$. The algebraic space $$ X = \\Quotfunctor^P_{\\mathcal{O}^{\\oplus r}_{\\mathbf{P}^n_\\mathbf{Z}}/ \\mathbf{P}^n_\\mathbf{Z}/\\mathbf{Z}} $$ parametrizing quotients of $\\mathcal{O}_{\\mathbf{}P^n_\\mathbf{Z}}^{\\oplus r}$ with Hilbert polynomial $P$ is proper over $\\Spec(\\mathbf{Z})$."}
+{"_id": "1725", "title": "moduli-lemma-quot-Pn-over-base", "text": "Let $B$ be an algebraic space. Let $X = B \\times \\mathbf{P}^n_\\mathbf{Z}$. Let $\\mathcal{L}$ be the pullback of $\\mathcal{O}_{\\mathbf{P}^n}(1)$ to $X$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation. The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$ parametrizing quotients of $\\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\\mathcal{L}$ is proper over $B$."}
+{"_id": "1726", "title": "moduli-lemma-quot-proper-over-base", "text": "Let $f : X \\to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$ parametrizing quotients of $\\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\\mathcal{L}$ is proper over $B$."}
+{"_id": "1727", "title": "moduli-lemma-quot-qc-over-base", "text": "Let $f : X \\to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$ parametrizing quotients of $\\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\\mathcal{L}$ is separated of finite presentation over $B$."}
+{"_id": "1728", "title": "moduli-lemma-hilb-diagonal-closed", "text": "The diagonal of $\\Hilbfunctor_{X/B} \\to B$ is a closed immersion of finite presentation."}
+{"_id": "1731", "title": "moduli-lemma-hilb-open", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\\pi$ induces an open immersion $\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$."}
+{"_id": "1733", "title": "moduli-lemma-hilb-proper-over-base", "text": "Let $f : X \\to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\\Hilbfunctor^P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\\mathcal{L}$ is proper over $B$."}
+{"_id": "1734", "title": "moduli-lemma-hilb-qc-over-base", "text": "Let $f : X \\to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module ample on $X/B$, see Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample}. The algebraic space $\\Hilbfunctor^P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\\mathcal{L}$ is separated of finite presentation over $B$."}
+{"_id": "1735", "title": "moduli-lemma-pic-diagonal-affine-fp", "text": "The diagonal of $\\Picardstack_{X/B}$ over $B$ is affine and of finite presentation."}
+{"_id": "1736", "title": "moduli-lemma-pic-qs-lfp", "text": "The morphism $\\Picardstack_{X/B} \\to B$ is quasi-separated and locally of finite presentation."}
+{"_id": "1738", "title": "moduli-lemma-pic-inertia", "text": "Assume $f_{T, *}\\mathcal{O}_{X_T} \\cong \\mathcal{O}_T$ for all schemes $T$ over $B$. Then the inertia stack of $\\Picardstack_{X/B}$ is equal to $\\mathbf{G}_m \\times \\Picardstack_{X/B}$."}
+{"_id": "1739", "title": "moduli-lemma-pic-curves-smooth", "text": "Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to the other assumptions in this section. Then $\\Picardstack_{X/B} \\to B$ is smooth."}
+{"_id": "1740", "title": "moduli-lemma-pic-gerbe-over-pic-functor", "text": "The morphism $\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$ turns the Picard stack into a gerbe over the Picard functor."}
+{"_id": "1741", "title": "moduli-lemma-pic-functor-diagonal-qc-immersion", "text": "The diagonal of $\\Picardfunctor_{X/B}$ over $B$ is a quasi-compact immersion."}
+{"_id": "1745", "title": "moduli-lemma-Mor-diagonal-closed", "text": "The diagonal of $\\mathit{Mor}_B(Y, X) \\to B$ is a closed immersion of finite presentation."}
+{"_id": "1746", "title": "moduli-lemma-Mor-s-lfp", "text": "The morphism $\\mathit{Mor}_B(Y, X) \\to B$ is separated and locally of finite presentation."}
+{"_id": "1747", "title": "moduli-lemma-Isom-in-Mor", "text": "With $B, X, Y$ as in the introduction of this section, in addition assume $X \\to B$ is proper. Then the subfunctor $\\mathit{Isom}_B(Y, X) \\subset \\mathit{Mor}_B(Y, X)$ of isomorphisms is an open subspace."}
+{"_id": "1748", "title": "moduli-lemma-Mor-qc-over-base", "text": "With $B, X, Y$ as in the introduction of this section, let $\\mathcal{L}$ be ample on $X/B$ and let $\\mathcal{N}$ be ample on $Y/B$. See Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample}. Let $P$ be a numerical polynomial. Then $$ \\mathit{Mor}^{P, \\mathcal{M}}_B(Y, X) \\longrightarrow B $$ is separated and of finite presentation where $\\mathcal{M} = \\text{pr}_1^*\\mathcal{N} \\otimes_{\\mathcal{O}_{Y \\times_B X}} \\text{pr}_2^*\\mathcal{L}$."}
+{"_id": "1749", "title": "moduli-lemma-polarized-diagonal-separated-fp", "text": "The diagonal of $\\Polarizedstack$ is separated and of finite presentation."}
+{"_id": "1750", "title": "moduli-lemma-polarized-qs-lfp", "text": "The morphism $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and locally of finite presentation."}
+{"_id": "1751", "title": "moduli-lemma-bounded-polarized", "text": "Let $n \\geq 1$ be an integer and let $P$ be a numerical polynomial. Let $$ T \\subset |\\Polarizedstack| $$ be a subset with the following property: for every $\\xi \\in T$ there exists a field $k$ and an object $(X, \\mathcal{L})$ of $\\Polarizedstack$ over $k$ representing $\\xi$ such that \\begin{enumerate} \\item the Hilbert polynomial of $\\mathcal{L}$ on $X$ is $P$, and \\item there exists a closed immersion $i : X \\to \\mathbf{P}^n_k$ such that $i^*\\mathcal{O}_{\\mathbf{P}^n}(1) \\cong \\mathcal{L}$. \\end{enumerate} Then $T$ is a Noetherian topological space, in particular quasi-compact."}
+{"_id": "1752", "title": "moduli-lemma-complexes-diagonal-affine-fp", "text": "The diagonal of $\\Complexesstack_{X/B}$ over $B$ is affine and of finite presentation."}
+{"_id": "1757", "title": "derived-lemma-composition-zero", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. Then $g \\circ f = 0$, $h \\circ g = 0$ and $f[1] \\circ h = 0$."}
+{"_id": "1758", "title": "derived-lemma-representable-homological", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. For any object $W$ of $\\mathcal{D}$ the functor $\\Hom_\\mathcal{D}(W, -)$ is homological, and the functor $\\Hom_\\mathcal{D}(-, W)$ is cohomological."}
+{"_id": "1759", "title": "derived-lemma-third-isomorphism-triangle", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $$ (a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h') $$ be a morphism of distinguished triangles. If two among $a, b, c$ are isomorphisms so is the third."}
+{"_id": "1760", "title": "derived-lemma-third-map-square-zero", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $$ (0, b, 0), (0, b', 0) : (X, Y, Z, f, g, h) \\to (X, Y, Z, f, g, h) $$ be endomorphisms of a distinguished triangle. Then $bb' = 0$."}
+{"_id": "1762", "title": "derived-lemma-cone-triangle-unique-isomorphism", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$. There exists a distinguished triangle $(X, Y, Z, f, g, h)$ which is unique up to (nonunique) isomorphism of triangles. More precisely, given a second such distinguished triangle $(X, Y, Z', f, g', h')$ there exists an isomorphism $$ (1, 1, c) : (X, Y, Z, f, g, h) \\longrightarrow (X, Y, Z', f, g', h') $$"}
+{"_id": "1763", "title": "derived-lemma-uniqueness-third-arrow", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $$ (a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h') $$ be a morphism of distinguished triangles. If one of the following conditions holds \\begin{enumerate} \\item $\\Hom(Y, X') = 0$, \\item $\\Hom(Z, Y') = 0$, \\item $\\Hom(X, X') = \\Hom(Z, X') = 0$, \\item $\\Hom(Z, X') = \\Hom(Z, Z') = 0$, or \\item $\\Hom(X[1], Z') = \\Hom(Z, X') = 0$ \\end{enumerate} then $b$ is the unique morphism from $Y \\to Y'$ such that $(a, b, c)$ is a morphism of triangles."}
+{"_id": "1764", "title": "derived-lemma-third-object-zero", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $f$ is an isomorphism, \\item $(X, Y, 0, f, 0, 0)$ is a distinguished triangle, and \\item for any distinguished triangle $(X, Y, Z, f, g, h)$ we have $Z = 0$. \\end{enumerate}"}
+{"_id": "1765", "title": "derived-lemma-direct-sum-triangles", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ be triangles. The following are equivalent \\begin{enumerate} \\item $(X \\oplus X', Y \\oplus Y', Z \\oplus Z', f \\oplus f', g \\oplus g', h \\oplus h')$ is a distinguished triangle, \\item both $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ are distinguished triangles. \\end{enumerate}"}
+{"_id": "1766", "title": "derived-lemma-split", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. \\begin{enumerate} \\item If $h = 0$, then there exists a right inverse $s : Z \\to Y$ to $g$. \\item For any right inverse $s : Z \\to Y$ of $g$ the map $f \\oplus s : X \\oplus Z \\to Y$ is an isomorphism. \\item For any objects $X', Z'$ of $\\mathcal{D}$ the triangle $(X', X' \\oplus Z', Z', (1, 0), (0, 1), 0)$ is distinguished. \\end{enumerate}"}
+{"_id": "1767", "title": "derived-lemma-when-split", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $f$ has a kernel, \\item $f$ has a cokernel, \\item $f$ is the isomorphic to a composition $K \\oplus Z \\to Z \\to Z \\oplus Q$ of a projection and coprojection for some objects $K, Z, Q$ of $\\mathcal{D}$. \\end{enumerate}"}
+{"_id": "1769", "title": "derived-lemma-projectors-have-images-triangulated", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. If $\\mathcal{D}$ has countable products, then $\\mathcal{D}$ is Karoubian. If $\\mathcal{D}$ has countable coproducts, then $\\mathcal{D}$ is Karoubian."}
+{"_id": "1770", "title": "derived-lemma-easier-axiom-four", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. In order to prove TR4 it suffices to show that given any pair of composable morphisms $f : X \\to Y$ and $g : Y \\to Z$ there exist \\begin{enumerate} \\item isomorphisms $i : X' \\to X$, $j : Y' \\to Y$ and $k : Z' \\to Z$, and then setting $f' = j^{-1}fi : X' \\to Y'$ and $g' = k^{-1}gj : Y' \\to Z'$ there exist \\item distinguished triangles $(X', Y', Q_1, f', p_1, d_1)$, $(X', Z', Q_2, g' \\circ f', p_2, d_2)$ and $(Y', Z', Q_3, g', p_3, d_3)$, such that the assertion of TR4 holds. \\end{enumerate}"}
+{"_id": "1771", "title": "derived-lemma-triangulated-subcategory", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Assume that $\\mathcal{D}'$ is an additive full subcategory of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item there exists a set of triangles $\\mathcal{T}'$ such that $(\\mathcal{D}', \\mathcal{T}')$ is a pre-triangulated subcategory of $\\mathcal{D}$, \\item $\\mathcal{D}'$ is preserved under $[1], [-1]$ and given any morphism $f : X \\to Y$ in $\\mathcal{D}'$ there exists a distinguished triangle $(X, Y, Z, f, g, h)$ in $\\mathcal{D}$ such that $Z$ is isomorphic to an object of $\\mathcal{D}'$. \\end{enumerate} In this case $\\mathcal{T}'$ as in (1) is the set of distinguished triangles $(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ such that $X, Y, Z \\in \\Ob(\\mathcal{D}')$. Finally, if $\\mathcal{D}$ is a triangulated category, then (1) and (2) are also equivalent to \\begin{enumerate} \\item[(3)] $\\mathcal{D}'$ is a triangulated subcategory. \\end{enumerate}"}
+{"_id": "1773", "title": "derived-lemma-exact-equivalence", "text": "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a fully faithful exact functor of pre-triangulated categories. Then a triangle $(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ is distinguished if and only if $(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished in $\\mathcal{D}'$."}
+{"_id": "1774", "title": "derived-lemma-composition-exact", "text": "Let $\\mathcal{D}, \\mathcal{D}', \\mathcal{D}''$ be pre-triangulated categories. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ and $F' : \\mathcal{D}' \\to \\mathcal{D}''$ be exact functors. Then $F' \\circ F$ is an exact functor."}
+{"_id": "1775", "title": "derived-lemma-exact-compose-homological-functor", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $\\mathcal{A}$ be an abelian category. Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor. \\begin{enumerate} \\item Let $\\mathcal{D}'$ be a pre-triangulated category. Let $F : \\mathcal{D}' \\to \\mathcal{D}$ be an exact functor. Then the composition $H \\circ F$ is a homological functor as well. \\item Let $\\mathcal{A}'$ be an abelian category. Let $G : \\mathcal{A} \\to \\mathcal{A}'$ be an exact functor. Then $G \\circ H$ is a homological functor as well. \\end{enumerate}"}
+{"_id": "1776", "title": "derived-lemma-exact-compose-delta-functor", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A}$ be an abelian category. Let $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor. \\begin{enumerate} \\item Let $\\mathcal{D}'$ be a triangulated category. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor. Then the composition $F \\circ G$ is a $\\delta$-functor as well. \\item Let $\\mathcal{A}'$ be an abelian category. Let $H : \\mathcal{A}' \\to \\mathcal{A}$ be an exact functor. Then $G \\circ H$ is a $\\delta$-functor as well. \\end{enumerate}"}
+{"_id": "1777", "title": "derived-lemma-compose-delta-functor-homological", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor. Let $H : \\mathcal{D} \\to \\mathcal{B}$ be a homological functor. Assume that $H^{-1}(G(A)) = 0$ for all $A$ in $\\mathcal{A}$. Then the collection $$ \\{H^n \\circ G, H^n(\\delta_{A \\to B \\to C})\\}_{n \\geq 0} $$ is a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see Homology, Definition \\ref{homology-definition-cohomological-delta-functor}."}
+{"_id": "1778", "title": "derived-lemma-localization-conditions", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a set of morphisms of $\\mathcal{D}$ and assume that axioms MS1, MS5, MS6 hold (see Categories, Definition \\ref{categories-definition-multiplicative-system} and Definition \\ref{definition-localization}). Then MS2 holds."}
+{"_id": "1779", "title": "derived-lemma-triangle-functor-localize", "text": "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of pre-triangulated categories.  Let $$ S = \\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid F(f)\\text{ is an isomorphism}\\} $$ Then $S$ is a saturated (see Categories, Definition \\ref{categories-definition-saturated-multiplicative-system}) multiplicative system compatible with the triangulated structure on $\\mathcal{D}$."}
+{"_id": "1780", "title": "derived-lemma-homological-functor-localize", "text": "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor between a pre-triangulated category and an abelian category. Let $$ S = \\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid H^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\} $$ Then $S$ is a saturated (see Categories, Definition \\ref{categories-definition-saturated-multiplicative-system}) multiplicative system compatible with the triangulated structure on $\\mathcal{D}$."}
+{"_id": "1781", "title": "derived-lemma-universal-property-localization", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated category. Let $Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ be the localization functor, see Proposition \\ref{proposition-construct-localization}. \\begin{enumerate} \\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into an abelian category $\\mathcal{A}$ such that $H(s)$ is an isomorphism for all $s \\in S$, then the unique factorization $H' : S^{-1}\\mathcal{D} \\to \\mathcal{A}$ such that $H = H' \\circ Q$ (see Categories, Lemma \\ref{categories-lemma-properties-left-localization}) is a homological functor too. \\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into a pre-triangulated category $\\mathcal{D}'$ such that $F(s)$ is an isomorphism for all $s \\in S$, then the unique factorization $F' : S^{-1}\\mathcal{D} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$ (see Categories, Lemma \\ref{categories-lemma-properties-left-localization}) is an exact functor too. \\end{enumerate}"}
+{"_id": "1782", "title": "derived-lemma-kernel-localization", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated structure. Let $Z$ be an object of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $Q(Z) = 0$ in $S^{-1}\\mathcal{D}$, \\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that $0 : Z \\to Z'$ is an element of $S$, \\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that $0 : Z' \\to Z$ is an element of $S$, and \\item there exists an object $Z'$ and a distinguished triangle $(X, Y, Z \\oplus Z', f, g, h)$ such that $f \\in S$. \\end{enumerate} If $S$ is saturated, then these are also equivalent to \\begin{enumerate} \\item[(5)] the morphism $0 \\to Z$ is an element of $S$, \\item[(6)] the morphism $Z \\to 0$ is an element of $S$, \\item[(7)] there exists a distinguished triangle $(X, Y, Z, f, g, h)$ such that $f \\in S$. \\end{enumerate}"}
+{"_id": "1783", "title": "derived-lemma-limit-triangles", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $S$ be a saturated multiplicative system in $\\mathcal{D}$ that is compatible with the triangulated structure. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle in $\\mathcal{D}$. Consider the category of morphisms of triangles $$ \\mathcal{I} = \\{(s, s', s'') : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h') \\mid s, s', s'' \\in S\\} $$ Then $\\mathcal{I}$ is a filtered category and the functors $\\mathcal{I} \\to X/S$, $\\mathcal{I} \\to Y/S$, and $\\mathcal{I} \\to Z/S$ are cofinal."}
+{"_id": "1784", "title": "derived-lemma-triangle-functor-kernel", "text": "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $\\mathcal{D}''$ be the full subcategory of $\\mathcal{D}$ with objects $$ \\Ob(\\mathcal{D}'') = \\{X \\in \\Ob(\\mathcal{D}) \\mid F(X) = 0\\} $$ Then $\\mathcal{D}''$ is a strictly full saturated pre-triangulated subcategory of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then $\\mathcal{D}''$ is a triangulated subcategory."}
+{"_id": "1785", "title": "derived-lemma-homological-functor-kernel", "text": "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of a pre-triangulated category into an abelian category. Let $\\mathcal{D}'$ be the full subcategory of $\\mathcal{D}$ with objects $$ \\Ob(\\mathcal{D}') = \\{X \\in \\Ob(\\mathcal{D}) \\mid H(X[n]) = 0\\text{ for all }n \\in \\mathbf{Z}\\} $$ Then $\\mathcal{D}'$ is a strictly full saturated pre-triangulated subcategory of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then $\\mathcal{D}'$ is a triangulated subcategory."}
+{"_id": "1786", "title": "derived-lemma-homological-functor-bounded", "text": "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of a pre-triangulated category into an abelian category. Let $\\mathcal{D}_H^{+}, \\mathcal{D}_H^{-}, \\mathcal{D}_H^b$ be the full subcategory of $\\mathcal{D}$ with objects $$ \\begin{matrix} \\Ob(\\mathcal{D}_H^{+}) = \\{X \\in \\Ob(\\mathcal{D}) \\mid H(X[n]) = 0\\text{ for all }n \\ll 0\\} \\\\ \\Ob(\\mathcal{D}_H^{-}) = \\{X \\in \\Ob(\\mathcal{D}) \\mid H(X[n]) = 0\\text{ for all }n \\gg 0\\} \\\\ \\Ob(\\mathcal{D}_H^b) = \\{X \\in \\Ob(\\mathcal{D}) \\mid H(X[n]) = 0\\text{ for all }|n| \\gg 0\\} \\end{matrix} $$ Each of these is a strictly full saturated pre-triangulated subcategory of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then each is a triangulated subcategory."}
+{"_id": "1787", "title": "derived-lemma-construct-multiplicative-system", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated subcategory. Set \\begin{equation} \\label{equation-multiplicative-system} S = \\left\\{ \\begin{matrix} f \\in \\text{Arrows}(\\mathcal{D}) \\text{ such that there exists a distinguished triangle }\\\\ (X, Y, Z, f, g, h) \\text{ of }\\mathcal{D}\\text{ with } Z\\text{ isomorphic to an object of }\\mathcal{D}' \\end{matrix} \\right\\} \\end{equation} Then $S$ is a multiplicative system compatible with the triangulated structure on $\\mathcal{D}$. In this situation the following are equivalent \\begin{enumerate} \\item $S$ is a saturated multiplicative system, \\item $\\mathcal{D}'$ is a saturated triangulated subcategory. \\end{enumerate}"}
+{"_id": "1788", "title": "derived-lemma-universal-property-quotient", "text": "\\begin{slogan} The universal property of the Verdier quotient. \\end{slogan} Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$ be a full triangulated subcategory of $\\mathcal{D}$. Let $Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$ be the quotient functor. \\begin{enumerate} \\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into an abelian category $\\mathcal{A}$ such that $\\mathcal{B} \\subset \\Ker(H)$ then there exists a unique factorization $H' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{A}$ such that $H = H' \\circ Q$ and $H'$ is a homological functor too. \\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into a pre-triangulated category $\\mathcal{D}'$ such that $\\mathcal{B} \\subset \\Ker(F)$ then there exists a unique factorization $F' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$ and $F'$ is an exact functor too. \\end{enumerate}"}
+{"_id": "1789", "title": "derived-lemma-kernel-quotient", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$ be a full triangulated subcategory. The kernel of the quotient functor $Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$ is the strictly full subcategory of $\\mathcal{D}$ whose objects are $$ \\Ob(\\Ker(Q)) = \\left\\{ \\begin{matrix} Z \\in \\Ob(\\mathcal{D}) \\text{ such that there exists a }Z' \\in \\Ob(\\mathcal{D}) \\\\ \\text{ such that }Z \\oplus Z'\\text{ is isomorphic to an object of }\\mathcal{B} \\end{matrix} \\right\\} $$ In other words it is the smallest strictly full saturated triangulated subcategory of $\\mathcal{D}$ containing $\\mathcal{B}$."}
+{"_id": "1790", "title": "derived-lemma-operations", "text": "Let $\\mathcal{D}$ be a triangulated category. The operations described above have the following properties \\begin{enumerate} \\item $S(\\mathcal{B}(S))$ is the ``saturation'' of $S$, i.e., it is the smallest saturated multiplicative system in $\\mathcal{D}$ containing $S$, and \\item $\\mathcal{B}(S(\\mathcal{B}))$ is the ``saturation'' of $\\mathcal{B}$, i.e., it is the smallest strictly full saturated triangulated subcategory of $\\mathcal{D}$ containing $\\mathcal{B}$. \\end{enumerate} In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in $\\mathcal{D}$ compatible with the triangulated structure on $\\mathcal{D}$ and the (partially ordered) set of strictly full saturated triangulated subcategories of $\\mathcal{D}$."}
+{"_id": "1791", "title": "derived-lemma-acyclic-general", "text": "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor from a triangulated category $\\mathcal{D}$ to an abelian category $\\mathcal{A}$, see Definition \\ref{definition-homological}. The subcategory $\\Ker(H)$ of $\\mathcal{D}$ is a strictly full saturated triangulated subcategory of $\\mathcal{D}$ whose corresponding saturated multiplicative system (see Lemma \\ref{lemma-operations}) is the set $$ S = \\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid H^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\}. $$ The functor $H$ factors through the quotient functor $Q : \\mathcal{D} \\to \\mathcal{D}/\\Ker(H)$."}
+{"_id": "1792", "title": "derived-lemma-adjoint-is-exact", "text": "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor between triangulated categories. If $F$ admits a right adjoint $G: \\mathcal{D'} \\to \\mathcal{D}$, then $G$ is also an exact functor."}
+{"_id": "1793", "title": "derived-lemma-fully-faithful-adjoint-kernel-zero", "text": "Let $\\mathcal{D}$, $\\mathcal{D}'$ be triangulated categories. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ and $G : \\mathcal{D}' \\to \\mathcal{D}$ be functors. Assume that \\begin{enumerate} \\item $F$ and $G$ are exact functors, \\item $F$ is fully faithful, \\item $G$ is a right adjoint to $F$, and \\item the kernel of $G$ is zero. \\end{enumerate} Then $F$ is an equivalence of categories."}
+{"_id": "1794", "title": "derived-lemma-functorial-cone", "text": "Suppose that $$ \\xymatrix{ K_1^\\bullet \\ar[r]_{f_1} \\ar[d]_a & L_1^\\bullet \\ar[d]^b \\\\ K_2^\\bullet \\ar[r]^{f_2} & L_2^\\bullet } $$ is a diagram of morphisms of complexes which is commutative up to homotopy. Then there exists a morphism $c : C(f_1)^\\bullet \\to C(f_2)^\\bullet$ which gives rise to a morphism of triangles $(a, b, c) : (K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1) \\to (K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2)$ of $K(\\mathcal{A})$."}
+{"_id": "1795", "title": "derived-lemma-map-from-cone", "text": "Suppose that $f: K^\\bullet \\to L^\\bullet$ and $g : L^\\bullet \\to M^\\bullet$ are morphisms of complexes such that $g \\circ f$ is homotopic to zero. Then \\begin{enumerate} \\item $g$ factors through a morphism $C(f)^\\bullet \\to M^\\bullet$, and \\item $f$ factors through a morphism $K^\\bullet \\to C(g)^\\bullet[-1]$. \\end{enumerate}"}
+{"_id": "1796", "title": "derived-lemma-make-commute-map", "text": "Let $\\mathcal{A}$ be an additive category. Let $$ \\xymatrix{ A^\\bullet \\ar[r]_f \\ar[d]_a & B^\\bullet \\ar[d]^b \\\\ C^\\bullet \\ar[r]^g & D^\\bullet } $$ be a diagram of morphisms of complexes commuting up to homotopy. If $f$ is a termwise split injection, then $b$ is homotopic to a morphism which makes the diagram commute. If $g$ is a split surjection, then $a$ is homotopic to a morphism which makes the diagram commute."}
+{"_id": "1797", "title": "derived-lemma-make-injective", "text": "Let $\\mathcal{A}$ be an additive category. Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of complexes of $\\mathcal{A}$. There exists a factorization $$ \\xymatrix{ K^\\bullet \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha & \\tilde L^\\bullet \\ar[r]^\\pi & L^\\bullet } $$ such that \\begin{enumerate} \\item $\\tilde \\alpha$ is a termwise split injection (see Definition \\ref{definition-termwise-split-map}), \\item there is a map of complexes $s : L^\\bullet \\to \\tilde L^\\bullet$ such that $\\pi \\circ s = \\text{id}_{L^\\bullet}$ and such that $s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L^\\bullet}$. \\end{enumerate} Moreover, if both $K^\\bullet$ and $L^\\bullet$ are in $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$, then so is $\\tilde L^\\bullet$."}
+{"_id": "1798", "title": "derived-lemma-make-surjective", "text": "Let $\\mathcal{A}$ be an additive category. Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of complexes of $\\mathcal{A}$. There exists a factorization $$ \\xymatrix{ K^\\bullet \\ar[r]^i \\ar@/_1pc/[rr]_\\alpha & \\tilde K^\\bullet \\ar[r]^{\\tilde \\alpha} & L^\\bullet } $$ such that \\begin{enumerate} \\item $\\tilde \\alpha$ is a termwise split surjection (see Definition \\ref{definition-termwise-split-map}), \\item there is a map of complexes $s : \\tilde K^\\bullet \\to K^\\bullet$ such that $s \\circ i = \\text{id}_{K^\\bullet}$ and such that $i \\circ s$ is homotopic to $\\text{id}_{\\tilde K^\\bullet}$. \\end{enumerate} Moreover, if both $K^\\bullet$ and $L^\\bullet$ are in $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$, then so is $\\tilde K^\\bullet$."}
+{"_id": "1800", "title": "derived-lemma-nilpotent", "text": "Let $\\mathcal{A}$ be an additive category. Let $0 \\to A_i^\\bullet \\to B_i^\\bullet \\to C_i^\\bullet \\to 0$, $i = 1, 2, 3$ be termwise split exact sequences of complexes. Let $b : B_1^\\bullet \\to B_2^\\bullet$ and $b' : B_2^\\bullet \\to B_3^\\bullet$ be morphisms of complexes such that $$ \\vcenter{ \\xymatrix{ A_1^\\bullet \\ar[d]_0 \\ar[r] & B_1^\\bullet \\ar[r] \\ar[d]_b & C_1^\\bullet \\ar[d]_0 \\\\ A_2^\\bullet \\ar[r] & B_2^\\bullet \\ar[r] & C_2^\\bullet } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ A_2^\\bullet \\ar[d]^0 \\ar[r] & B_2^\\bullet \\ar[r] \\ar[d]^{b'} & C_2^\\bullet \\ar[d]^0 \\\\ A_3^\\bullet \\ar[r] & B_3^\\bullet \\ar[r] & C_3^\\bullet } } $$ commute in $K(\\mathcal{A})$. Then $b' \\circ b = 0$ in $K(\\mathcal{A})$."}
+{"_id": "1801", "title": "derived-lemma-third-isomorphism", "text": "Let $\\mathcal{A}$ be an additive category. Let $f_1 : K_1^\\bullet \\to L_1^\\bullet$ and $f_2 : K_2^\\bullet \\to L_2^\\bullet$ be morphisms of complexes. Let $$ (a, b, c) : (K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1) \\longrightarrow (K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2) $$ be any morphism of triangles of $K(\\mathcal{A})$. If $a$ and $b$ are homotopy equivalences then so is $c$."}
+{"_id": "1802", "title": "derived-lemma-the-same-up-to-isomorphisms", "text": "Let $\\mathcal{A}$ be an additive category. \\begin{enumerate} \\item Given a termwise split sequence of complexes $(\\alpha : A^\\bullet \\to B^\\bullet, \\beta : B^\\bullet \\to C^\\bullet, s^n, \\pi^n)$ there exists a homotopy equivalence $C(\\alpha)^\\bullet \\to C^\\bullet$ such that the diagram $$ \\xymatrix{ A^\\bullet \\ar[r] \\ar[d] & B^\\bullet \\ar[d] \\ar[r] & C(\\alpha)^\\bullet \\ar[r]_{-p} \\ar[d] & A^\\bullet[1] \\ar[d] \\\\ A^\\bullet \\ar[r] & B^\\bullet \\ar[r] & C^\\bullet \\ar[r]^\\delta & A^\\bullet[1] } $$ defines an isomorphism of triangles in $K(\\mathcal{A})$. \\item Given a morphism of complexes $f : K^\\bullet \\to L^\\bullet$ there exists an isomorphism of triangles $$ \\xymatrix{ K^\\bullet \\ar[r] \\ar[d] & \\tilde L^\\bullet \\ar[d] \\ar[r] & M^\\bullet \\ar[r]_{\\delta} \\ar[d] & K^\\bullet[1] \\ar[d] \\\\ K^\\bullet \\ar[r] & L^\\bullet \\ar[r] & C(f)^\\bullet \\ar[r]^{-p} & K^\\bullet[1] } $$ where the upper triangle is the triangle associated to a termwise split exact sequence $K^\\bullet \\to \\tilde L^\\bullet \\to M^\\bullet$. \\end{enumerate}"}
+{"_id": "1803", "title": "derived-lemma-sequence-maps-split", "text": "Let $\\mathcal{A}$ be an additive category. Let $A_1^\\bullet \\to A_2^\\bullet \\to \\ldots \\to A_n^\\bullet$ be a sequence of composable morphisms of complexes. There exists a commutative diagram $$ \\xymatrix{ A_1^\\bullet \\ar[r] & A_2^\\bullet \\ar[r] & \\ldots \\ar[r] & A_n^\\bullet \\\\ B_1^\\bullet \\ar[r] \\ar[u] & B_2^\\bullet \\ar[r] \\ar[u] & \\ldots \\ar[r] & B_n^\\bullet \\ar[u] } $$ such that each morphism $B_i^\\bullet \\to B_{i + 1}^\\bullet$ is a split injection and each $B_i^\\bullet \\to A_i^\\bullet$ is a homotopy equivalence. Moreover, if all $A_i^\\bullet$ are in $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$, then so are the $B_i^\\bullet$."}
+{"_id": "1804", "title": "derived-lemma-rotate-triangle", "text": "Let $\\mathcal{A}$ be an additive category. Let $(\\alpha : A^\\bullet \\to B^\\bullet, \\beta : B^\\bullet \\to C^\\bullet, s^n, \\pi^n)$ be a termwise split sequence of complexes. Let $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$ be the associated triangle. Then the triangle $(C^\\bullet[-1], A^\\bullet, B^\\bullet, \\delta[-1], \\alpha, \\beta)$ is isomorphic to the triangle $(C^\\bullet[-1], A^\\bullet, C(\\delta[-1])^\\bullet, \\delta[-1], i, p)$."}
+{"_id": "1805", "title": "derived-lemma-rotate-cone", "text": "Let $\\mathcal{A}$ be an additive category. Let $f : K^\\bullet \\to L^\\bullet$ be a morphism of complexes. The triangle $(L^\\bullet, C(f)^\\bullet, K^\\bullet[1], i, p, f[1])$ is the triangle associated to the termwise split sequence $$ 0 \\to L^\\bullet \\to C(f)^\\bullet \\to K^\\bullet[1] \\to 0 $$ coming from the definition of the cone of $f$."}
+{"_id": "1806", "title": "derived-lemma-two-split-injections", "text": "Let $\\mathcal{A}$ be an additive category. Suppose that $\\alpha : A^\\bullet \\to B^\\bullet$ and $\\beta : B^\\bullet \\to C^\\bullet$ are split injections of complexes. Then there exist distinguished triangles $(A^\\bullet, B^\\bullet, Q_1^\\bullet, \\alpha, p_1, d_1)$, $(A^\\bullet, C^\\bullet, Q_2^\\bullet, \\beta \\circ \\alpha, p_2, d_2)$ and $(B^\\bullet, C^\\bullet, Q_3^\\bullet, \\beta, p_3, d_3)$ for which TR4 holds."}
+{"_id": "1807", "title": "derived-lemma-bounded-triangulated-subcategories", "text": "Let $\\mathcal{A}$ be an additive category. The categories $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$ are full triangulated subcategories of $K(\\mathcal{A})$."}
+{"_id": "1809", "title": "derived-lemma-improve-distinguished-triangle-homotopy", "text": "Let $\\mathcal{A}$ be an additive category. Let $(A^\\bullet, B^\\bullet, C^\\bullet, a, b, c)$ be a distinguished triangle in $K(\\mathcal{A})$. Then there exists an isomorphic distinguished triangle $(A^\\bullet, (B')^\\bullet, C^\\bullet, a', b', c)$ such that $0 \\to A^n \\to (B')^n \\to C^n \\to 0$ is a split short exact sequence for all $n$."}
+{"_id": "1810", "title": "derived-lemma-cohomology-homological", "text": "Let $\\mathcal{A}$ be an abelian category. The functor $$ H^0 : K(\\mathcal{A}) \\longrightarrow \\mathcal{A} $$ is homological."}
+{"_id": "1811", "title": "derived-lemma-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. The full subcategory $\\text{Ac}(\\mathcal{A})$ of $K(\\mathcal{A})$ consisting of acyclic complexes is a strictly full saturated triangulated subcategory of $K(\\mathcal{A})$. The corresponding saturated multiplicative system (see Lemma \\ref{lemma-operations}) of $K(\\mathcal{A})$ is the set $\\text{Qis}(\\mathcal{A})$ of quasi-isomorphisms. In particular, the kernel of the localization functor $Q : K(\\mathcal{A}) \\to \\text{Qis}(\\mathcal{A})^{-1}K(\\mathcal{A})$ is $\\text{Ac}(\\mathcal{A})$ and the functor $H^0$ factors through $Q$."}
+{"_id": "1812", "title": "derived-lemma-complex-cohomology-bounded", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex. \\begin{enumerate} \\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$, then there exists a quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $L^\\bullet$ bounded below. \\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$, then there exists a quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with $M^\\bullet$ bounded above. \\item If $H^n(K^\\bullet) = 0$ for all $|n| \\gg 0$, then there exists a commutative diagram of morphisms of complexes $$ \\xymatrix{ K^\\bullet \\ar[r] & L^\\bullet \\\\ M^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u] } $$ where all the arrows are quasi-isomorphisms, $L^\\bullet$ bounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded complex. \\end{enumerate}"}
+{"_id": "1813", "title": "derived-lemma-bounded-derived", "text": "Let $\\mathcal{A}$ be an abelian category. The subcategories $\\text{Ac}^{+}(\\mathcal{A})$, $\\text{Ac}^{-}(\\mathcal{A})$, resp.\\ $\\text{Ac}^b(\\mathcal{A})$ are strictly full saturated triangulated subcategories of $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, resp.\\ $K^b(\\mathcal{A})$. The corresponding saturated multiplicative systems (see Lemma \\ref{lemma-operations}) are the sets $\\text{Qis}^{+}(\\mathcal{A})$, $\\text{Qis}^{-}(\\mathcal{A})$, resp.\\ $\\text{Qis}^b(\\mathcal{A})$. \\begin{enumerate} \\item The kernel of the functor $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$ is $\\text{Ac}^{+}(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{+}(\\mathcal{A})/\\text{Ac}^{+}(\\mathcal{A}) = \\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{A}) $$ \\item The kernel of the functor $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$ is $\\text{Ac}^{-}(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{-}(\\mathcal{A})/\\text{Ac}^{-}(\\mathcal{A}) = \\text{Qis}^{-}(\\mathcal{A})^{-1}K^{-}(\\mathcal{A}) \\longrightarrow D^{-}(\\mathcal{A}) $$ \\item The kernel of the functor $K^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$ is $\\text{Ac}^b(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^b(\\mathcal{A})/\\text{Ac}^b(\\mathcal{A}) = \\text{Qis}^b(\\mathcal{A})^{-1}K^b(\\mathcal{A}) \\longrightarrow D^b(\\mathcal{A}) $$ \\end{enumerate}"}
+{"_id": "1814", "title": "derived-lemma-derived-canonical-delta-functor", "text": "Let $\\mathcal{A}$ be an abelian category. The functor $\\text{Comp}(\\mathcal{A}) \\to D(\\mathcal{A})$ defined has the natural structure of a $\\delta$-functor, with $$ \\delta_{A^\\bullet \\to B^\\bullet \\to C^\\bullet} = - p \\circ q^{-1} $$ with $p$ and $q$ as explained above. The same construction turns the functors $\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$, $\\text{Comp}^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$, and $\\text{Comp}^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$ into $\\delta$-functors."}
+{"_id": "1815", "title": "derived-lemma-derived-compare-triangles-ses", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ 0 \\ar[r] & A^\\bullet \\ar[r] \\ar[d] & B^\\bullet \\ar[r] \\ar[d] & C^\\bullet \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & D^\\bullet \\ar[r] & E^\\bullet \\ar[r] & F^\\bullet \\ar[r] & 0 } $$ be a commutative diagram of morphisms of complexes such that the rows are short exact sequences of complexes, and the vertical arrows are quasi-isomorphisms. The $\\delta$-functor of Lemma \\ref{lemma-derived-canonical-delta-functor} above maps the short exact sequences $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ and $0 \\to D^\\bullet \\to E^\\bullet \\to F^\\bullet \\to 0$ to isomorphic distinguished triangles."}
+{"_id": "1817", "title": "derived-lemma-trick-vanishing-composition", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ K_0^\\bullet \\to K_1^\\bullet \\to \\ldots \\to K_n^\\bullet $$ be maps of complexes such that \\begin{enumerate} \\item $H^i(K_0^\\bullet) = 0$ for $i > 0$, \\item $H^{-j}(K_j^\\bullet) \\to H^{-j}(K_{j + 1}^\\bullet)$ is zero. \\end{enumerate} Then the composition $K_0^\\bullet \\to K_n^\\bullet$ factors through $\\tau_{\\leq -n}K_n^\\bullet \\to K_n^\\bullet$ in $D(\\mathcal{A})$. Dually, given maps of complexes $$ K_n^\\bullet \\to K_{n - 1}^\\bullet \\to \\ldots \\to K_0^\\bullet $$ such that \\begin{enumerate} \\item $H^i(K_0^\\bullet) = 0$ for $i < 0$, \\item $H^j(K_{j + 1}^\\bullet) \\to H^j(K_j^\\bullet)$ is zero, \\end{enumerate} then the composition $K_n^\\bullet \\to K_0^\\bullet$ factors through $K_n^\\bullet \\to \\tau_{\\geq n}K_n^\\bullet$ in $D(\\mathcal{A})$."}
+{"_id": "1818", "title": "derived-lemma-filtered-cohomology-homological", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item The functor $K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\text{Gr}(\\mathcal{A})$, $K^\\bullet \\longmapsto H^0(\\text{gr}(K^\\bullet))$ is homological. \\item The functor $K(\\text{Fil}^f(\\mathcal{A})) \\rightarrow \\mathcal{A}$, $K^\\bullet \\longmapsto H^0(\\text{gr}^p(K^\\bullet))$ is homological. \\item The functor $K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\mathcal{A}$, $K^\\bullet \\longmapsto H^0((\\text{forget }F)K^\\bullet)$ is homological. \\end{enumerate}"}
+{"_id": "1819", "title": "derived-lemma-filtered-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. The full subcategory $\\text{FAc}(\\mathcal{A})$ of $K(\\text{Fil}^f(\\mathcal{A}))$ consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of $K(\\text{Fil}^f(\\mathcal{A}))$. The corresponding saturated multiplicative system (see Lemma \\ref{lemma-operations}) of $K(\\text{Fil}^f(\\mathcal{A}))$ is the set $\\text{FQis}(\\mathcal{A})$ of filtered quasi-isomorphisms. In particular, the kernel of the localization functor $$ Q : K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\text{FQis}(\\mathcal{A})^{-1}K(\\text{Fil}^f(\\mathcal{A})) $$ is $\\text{FAc}(\\mathcal{A})$ and the functor $H^0 \\circ \\text{gr}$ factors through $Q$."}
+{"_id": "1821", "title": "derived-lemma-filtered-complex-cohomology-bounded", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$. \\begin{enumerate} \\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n < a$, then there exists a filtered quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $L^n = 0$ for all $n < a$. \\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n > b$, then there exists a filtered quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with $M^n = 0$ for all $n > b$. \\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $|n| \\gg 0$, then there exists a commutative diagram of morphisms of complexes $$ \\xymatrix{ K^\\bullet \\ar[r] & L^\\bullet \\\\ M^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u] } $$ where all the arrows are filtered quasi-isomorphisms, $L^\\bullet$ bounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded complex. \\end{enumerate}"}
+{"_id": "1823", "title": "derived-lemma-derived-functor", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $f : X \\to Y$ be a morphism of $\\mathcal{D}$. \\begin{enumerate} \\item If $RF$ is defined at $X$ and $Y$ then there exists a unique morphism $RF(f) : RF(X) \\to RF(Y)$ between the values such that for any commutative diagram $$ \\xymatrix{ X \\ar[d]_f \\ar[r]_s & X' \\ar[d]^{f'} \\\\ Y \\ar[r]^{s'} & Y' } $$ with $s, s' \\in S$ the diagram $$ \\xymatrix{ F(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & RF(X) \\ar[d] \\\\ F(Y) \\ar[r] & F(Y') \\ar[r] & RF(Y) } $$ commutes. \\item If $LF$ is defined at $X$ and $Y$ then there exists a unique morphism $LF(f) : LF(X) \\to LF(Y)$ between the values such that for any commutative diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_s & X \\ar[d]^f \\\\ Y' \\ar[r]^{s'} & Y } $$ with $s, s'$ in $S$ the diagram $$ \\xymatrix{ LF(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & F(X) \\ar[d] \\\\ LF(Y) \\ar[r] & F(Y') \\ar[r] & F(Y) } $$ commutes. \\end{enumerate}"}
+{"_id": "1824", "title": "derived-lemma-derived-inverts", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $s : X \\to Y$ be an element of $S$. \\begin{enumerate} \\item $RF$ is defined at $X$ if and only if it is defined at $Y$. In this case the map $RF(s) : RF(X) \\to RF(Y)$ between values is an isomorphism. \\item $LF$ is defined at $X$ if and only if it is defined at $Y$. In this case the map $LF(s) : LF(X) \\to LF(Y)$ between values is an isomorphism. \\end{enumerate}"}
+{"_id": "1825", "title": "derived-lemma-derived-shift", "text": "\\begin{slogan} Derived functors are compatible with shifts \\end{slogan} Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$. \\begin{enumerate} \\item $RF$ is defined at $X$ if and only if it is defined at $X[n]$. In this case there is a canonical isomorphism $RF(X)[n]= RF(X[n])$ between values. \\item $LF$ is defined at $X$ if and only if it is defined at $X[n]$. In this case there is a canonical isomorphism $LF(X)[n] \\to LF(X[n])$ between values. \\end{enumerate}"}
+{"_id": "1826", "title": "derived-lemma-2-out-of-3-defined", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$. If $RF$ is defined at two out of three of $X, Y, Z$, then it is defined at the third. Moreover, in this case $$ (RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h)) $$ is a distinguished triangle in $\\mathcal{D}'$. Similarly for $LF$."}
+{"_id": "1827", "title": "derived-lemma-direct-sum-defined", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $X, Y$ be objects of $\\mathcal{D}$. \\begin{enumerate} \\item If $RF$ is defined at $X$ and $Y$, then $RF$ is defined at $X \\oplus Y$. \\item If $\\mathcal{D}'$ is Karoubian and $RF$ is defined at $X \\oplus Y$, then $RF$ is defined at both $X$ and $Y$. \\end{enumerate} In either case we have $RF(X \\oplus Y) = RF(X) \\oplus RF(Y)$. Similarly for $LF$."}
+{"_id": "1828", "title": "derived-lemma-computes-shift", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$. \\begin{enumerate} \\item $X$ computes $RF$ if and only if $X[n]$ computes $RF$. \\item $X$ computes $LF$ if and only if $X[n]$ computes $LF$. \\end{enumerate}"}
+{"_id": "1829", "title": "derived-lemma-2-out-of-3-computes", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$. If $X, Y$ compute $RF$ then so does $Z$. Similar for $LF$."}
+{"_id": "1831", "title": "derived-lemma-existence-computes", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. \\begin{enumerate} \\item If for every object $X \\in \\Ob(\\mathcal{D})$ there exists an arrow $s : X \\to X'$ in $S$ such that $X'$ computes $RF$, then $RF$ is everywhere defined. \\item If for every object $X \\in \\Ob(\\mathcal{D})$ there exists an arrow $s : X' \\to X$ in $S$ such that $X'$ computes $LF$, then $LF$ is everywhere defined. \\end{enumerate}"}
+{"_id": "1832", "title": "derived-lemma-find-existence-computes", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. If there exists a subset $\\mathcal{I} \\subset \\Ob(\\mathcal{D})$ such that \\begin{enumerate} \\item for all $X \\in \\Ob(\\mathcal{D})$ there exists $s : X \\to X'$ in $S$ with $X' \\in \\mathcal{I}$, and \\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{I}$ the map $F(s) : F(X) \\to F(X')$ is an isomorphism, \\end{enumerate} then $RF$ is everywhere defined and every $X \\in \\mathcal{I}$ computes $RF$. Dually, if there exists a subset $\\mathcal{P} \\subset \\Ob(\\mathcal{D})$ such that \\begin{enumerate} \\item for all $X \\in \\Ob(\\mathcal{D})$ there exists $s : X' \\to X$ in $S$ with $X' \\in \\mathcal{P}$, and \\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{P}$ the map $F(s) : F(X) \\to F(X')$ is an isomorphism, \\end{enumerate} then $LF$ is everywhere defined and every $X \\in \\mathcal{P}$ computes $LF$."}
+{"_id": "1833", "title": "derived-lemma-compose-derived-functors-general", "text": "Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be triangulated categories. Let $S$, resp.\\ $S'$ be a saturated multiplicative system in $\\mathcal{A}$, resp.\\ $\\mathcal{B}$ compatible with the triangulated structure. Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$ be exact functors. Denote $F' : \\mathcal{A} \\to (S')^{-1}\\mathcal{B}$ the composition of $F$ with the localization functor. \\begin{enumerate} \\item If $RF'$, $RG$, $R(G \\circ F)$ are everywhere defined, then there is a canonical transformation of functors $t : R(G \\circ F) \\longrightarrow RG \\circ RF'$. \\item If $LF'$, $LG$, $L(G \\circ F)$ are everywhere defined, then there is a canonical transformation of functors $t : LG \\circ LF' \\to L(G \\circ F)$. \\end{enumerate}"}
+{"_id": "1835", "title": "derived-lemma-subcategory-left-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$ such that every object of $\\mathcal{A}$ is a quotient of an element of $\\mathcal{P}$. Let $a \\in \\mathbf{Z}$. \\begin{enumerate} \\item Given $K^\\bullet$ with $K^n = 0$ for $n > a$ there exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$ with $P^n \\in \\mathcal{P}$ and $P^n \\to K^n$ surjective for all $n$ and $P^n = 0$ for $n > a$. \\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$ for $n > a$ there exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$ with $P^n \\in \\mathcal{P}$ for all $n$ and $P^n = 0$ for $n > a$. \\end{enumerate}"}
+{"_id": "1836", "title": "derived-lemma-subcategory-right-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$ such that every object of $\\mathcal{A}$ is a subobject of an element of $\\mathcal{I}$. Let $a \\in \\mathbf{Z}$. \\begin{enumerate} \\item Given $K^\\bullet$ with $K^n = 0$ for $n < a$ there exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$ with $K^n \\to I^n$ injective and $I^n \\in \\mathcal{I}$ for all $n$ and $I^n = 0$ for $n < a$, \\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$ for $n < a$ there exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$ with $I^n \\in \\mathcal{I}$ and $I^n = 0$ for $n < a$. \\end{enumerate}"}
+{"_id": "1837", "title": "derived-lemma-subcategory-right-acyclics", "text": "In Situation \\ref{situation-classical}. Let $\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset with the following properties: \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a subobject of an element of $\\mathcal{I}$, \\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of $\\mathcal{A}$ with $P, Q \\in \\mathcal{I}$, then $R \\in \\mathcal{I}$, and $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact. \\end{enumerate} Then every object of $\\mathcal{I}$ is acyclic for $RF$."}
+{"_id": "1838", "title": "derived-lemma-subcategory-left-acyclics", "text": "In Situation \\ref{situation-classical}. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset with the following properties: \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a quotient of an element of $\\mathcal{P}$, \\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of $\\mathcal{A}$ with $Q, R \\in \\mathcal{P}$, then $P \\in \\mathcal{P}$, and $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact. \\end{enumerate} Then every object of $\\mathcal{P}$ is acyclic for $LF$."}
+{"_id": "1839", "title": "derived-lemma-negative-vanishing", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories. Let $K^\\bullet$ be a complex of $\\mathcal{A}$ and $a \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $H^i(K^\\bullet) = 0$ for all $i < a$ and $RF$ is defined at $K^\\bullet$, then $H^i(RF(K^\\bullet)) = 0$ for all $i < a$. \\item If $RF$ is defined at $K^\\bullet$ and $\\tau_{\\leq a}K^\\bullet$, then $H^i(RF(\\tau_{\\leq a}K^\\bullet)) = H^i(RF(K^\\bullet))$ for all $i \\leq a$. \\end{enumerate}"}
+{"_id": "1840", "title": "derived-lemma-left-exact-higher-derived", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. \\begin{enumerate} \\item We have $R^iF = 0$ for $i < 0$, \\item $R^0F$ is left exact, \\item the map $F \\to R^0F$ is an isomorphism if and only if $F$ is left exact. \\end{enumerate}"}
+{"_id": "1841", "title": "derived-lemma-F-acyclic", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. Let $A$ be an object of $\\mathcal{A}$. \\begin{enumerate} \\item $A$ is right acyclic for $F$ if and only if $F(A) \\to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$, \\item if $F$ is left exact, then $A$ is right acyclic for $F$ if and only if $R^iF(A) = 0$ for all $i > 0$. \\end{enumerate}"}
+{"_id": "1843", "title": "derived-lemma-right-derived-delta-functor", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. \\begin{enumerate} \\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical structure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see Homology, Definition \\ref{homology-definition-cohomological-delta-functor}. \\item If every object of $\\mathcal{A}$ is a subobject of a right acyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a universal $\\delta$-functor, see Homology, Definition \\ref{homology-definition-universal-delta-functor}. \\end{enumerate}"}
+{"_id": "1844", "title": "derived-lemma-leray-acyclicity", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. Let $A^\\bullet$ be a bounded below complex of $F$-acyclic objects. The canonical map $$ F(A^\\bullet) \\longrightarrow RF(A^\\bullet) $$ is an isomorphism in $D^{+}(\\mathcal{B})$, i.e., $A^\\bullet$ computes $RF$."}
+{"_id": "1845", "title": "derived-lemma-right-derived-exact-functor", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor of abelian categories. Then \\begin{enumerate} \\item every object of $\\mathcal{A}$ is right acyclic for $F$, \\item $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined, \\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ is everywhere defined, \\item every complex computes $RF$, in other words, the canonical map $F(K^\\bullet) \\to RF(K^\\bullet)$ is an isomorphism for all complexes, and \\item $R^iF = 0$ for $i \\not = 0$. \\end{enumerate}"}
+{"_id": "1846", "title": "derived-lemma-cohomology-in-serre-subcategory", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$ be a weak Serre subcategory. The category $D_\\mathcal{B}(\\mathcal{A})$ is a strictly full saturated triangulated subcategory of $D(\\mathcal{A})$. Similarly for the bounded versions."}
+{"_id": "1847", "title": "derived-lemma-derived-of-quotient", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory. Then $D(\\mathcal{A}) \\to D(\\mathcal{A}/\\mathcal{B})$ is essentially surjective."}
+{"_id": "1848", "title": "derived-lemma-quotient-by-serre-easy", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory. Suppose that the functor $v : \\mathcal{A} \\to \\mathcal{A}/\\mathcal{B}$ has a left adjoint $u : \\mathcal{A}/\\mathcal{B} \\to \\mathcal{A}$ such that $vu \\cong \\text{id}$. Then $$ D(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A}) = D(\\mathcal{A}/\\mathcal{B}) $$ and similarly for the bounded versions."}
+{"_id": "1849", "title": "derived-lemma-fully-faithful-embedding", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory. Assume that for every surjection $X \\to Y$ with $X \\in \\Ob(\\mathcal{A})$ and $Y \\in \\Ob(\\mathcal{B})$ there exists $X' \\subset X$, $X' \\in \\Ob(\\mathcal{B})$ which surjects onto $Y$. Then the functor $D^-(\\mathcal{B}) \\to D^-_\\mathcal{B}(\\mathcal{A})$ of (\\ref{equation-compare}) is an equivalence."}
+{"_id": "1850", "title": "derived-lemma-cohomology-bounded-below", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex of $\\mathcal{A}$. \\begin{enumerate} \\item If $K^\\bullet$ has an injective resolution then $H^n(K^\\bullet) = 0$ for $n \\ll 0$. \\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then there exists a quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $L^\\bullet$ bounded below. \\end{enumerate}"}
+{"_id": "1851", "title": "derived-lemma-injective-resolutions-exist", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough injectives. \\begin{enumerate} \\item Any object of $\\mathcal{A}$ has an injective resolution. \\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then $K^\\bullet$ has an injective resolution. \\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then there exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$ with $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is injective. \\end{enumerate}"}
+{"_id": "1852", "title": "derived-lemma-acyclic-is-zero", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be an acyclic complex. Let $I^\\bullet$ be bounded below and consisting of injective objects. Any morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero."}
+{"_id": "1853", "title": "derived-lemma-morphisms-lift", "text": "Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \\xymatrix{ K^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^\\beta \\\\ I^\\bullet } $$ where $I^\\bullet$ is bounded below and consists of injective objects, and $\\alpha$ is a quasi-isomorphism. \\begin{enumerate} \\item There exists a map of complexes $\\beta$ making the diagram commute up to homotopy. \\item If $\\alpha$ is injective in every degree then we can find a $\\beta$ which makes the diagram commute. \\end{enumerate}"}
+{"_id": "1854", "title": "derived-lemma-morphisms-equal-up-to-homotopy", "text": "Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \\xymatrix{ K^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^{\\beta_i} \\\\ I^\\bullet } $$ where $I^\\bullet$ is bounded below and consists of injective objects, and $\\alpha$ is a quasi-isomorphism. Any two morphisms $\\beta_1, \\beta_2$ making the diagram commute up to homotopy are homotopic."}
+{"_id": "1855", "title": "derived-lemma-morphisms-into-injective-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I^\\bullet$ be bounded below complex consisting of injective objects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then $$ \\Mor_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) = \\Mor_{D(\\mathcal{A})}(L^\\bullet, I^\\bullet). $$"}
+{"_id": "1856", "title": "derived-lemma-injective-resolution-ses", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough injectives. For any short exact sequence $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ of $\\text{Comp}^{+}(\\mathcal{A})$ there exists a commutative diagram in $\\text{Comp}^{+}(\\mathcal{A})$ $$ \\xymatrix{ 0 \\ar[r] & A^\\bullet \\ar[r] \\ar[d] & B^\\bullet \\ar[r] \\ar[d] & C^\\bullet \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & I_1^\\bullet \\ar[r] & I_2^\\bullet \\ar[r] & I_3^\\bullet \\ar[r] & 0 } $$ where the vertical arrows are injective resolutions and the rows are short exact sequences of complexes. In fact, given any injective resolution $A^\\bullet \\to I^\\bullet$ we may assume $I_1^\\bullet = I^\\bullet$."}
+{"_id": "1858", "title": "derived-lemma-projective-resolutions-exist", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough projectives. \\begin{enumerate} \\item Any object of $\\mathcal{A}$ has a projective resolution. \\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$ then $K^\\bullet$ has a projective resolution. \\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n > a$, then there exists a projective resolution $\\alpha : P^\\bullet \\to K^\\bullet$ with $P^n = 0$ for $n > a$ such that each $\\alpha^n : P^n \\to K^n$ is surjective. \\end{enumerate}"}
+{"_id": "1859", "title": "derived-lemma-projective-into-acyclic-is-zero", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be an acyclic complex. Let $P^\\bullet$ be bounded above and consisting of projective objects. Any morphism $P^\\bullet \\to K^\\bullet$ is homotopic to zero."}
+{"_id": "1860", "title": "derived-lemma-morphisms-lift-projective", "text": "Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \\xymatrix{ K^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\ P^\\bullet \\ar[u] \\ar@{-->}[ru]_\\beta } $$ where $P^\\bullet$ is bounded above and consists of projective objects, and $\\alpha$ is a quasi-isomorphism. \\begin{enumerate} \\item There exists a map of complexes $\\beta$ making the diagram commute up to homotopy. \\item If $\\alpha$ is surjective in every degree then we can find a $\\beta$ which makes the diagram commute. \\end{enumerate}"}
+{"_id": "1862", "title": "derived-lemma-morphisms-from-projective-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $P^\\bullet$ be bounded above complex consisting of projective objects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then $$ \\Mor_{K(\\mathcal{A})}(P^\\bullet, L^\\bullet) = \\Mor_{D(\\mathcal{A})}(P^\\bullet, L^\\bullet). $$"}
+{"_id": "1864", "title": "derived-lemma-precise-vanishing", "text": "Let $\\mathcal{A}$ be an abelian category. Let $P^\\bullet$, $K^\\bullet$ be complexes. Let $n \\in \\mathbf{Z}$. Assume that \\begin{enumerate} \\item $P^\\bullet$ is a bounded complex consisting of projective objects, \\item $P^i = 0$ for $i < n$, and \\item $H^i(K^\\bullet) = 0$ for $i \\geq n$. \\end{enumerate} Then $\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet) = \\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet) = 0$."}
+{"_id": "1865", "title": "derived-lemma-lift-map", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\beta : P^\\bullet \\to L^\\bullet$ and $\\alpha : E^\\bullet \\to L^\\bullet$ be maps of complexes. Let $n \\in \\mathbf{Z}$. Assume \\begin{enumerate} \\item $P^\\bullet$ is a bounded complex of projectives and $P^i = 0$ for $i < n$, \\item $H^i(\\alpha)$ is an isomorphism for $i > n$ and surjective for $i = n$. \\end{enumerate} Then there exists a map of complexes $\\gamma : P^\\bullet \\to E^\\bullet$ such that $\\alpha \\circ \\gamma$ and $\\beta$ are homotopic."}
+{"_id": "1866", "title": "derived-lemma-injective-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I \\in \\Ob(\\mathcal{A})$ be an injective object. Let $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$. \\begin{enumerate} \\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$ for any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$ into any triangulated category $\\mathcal{D}$. \\item $I$ is right acyclic for any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$. \\end{enumerate}"}
+{"_id": "1867", "title": "derived-lemma-enough-injectives-right-derived", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. \\begin{enumerate} \\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$ into a triangulated category $\\mathcal{D}$ the right derived functor $$ RF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D} $$ is everywhere defined. \\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an abelian category $\\mathcal{B}$ the right derived functor $$ RF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B}) $$ is everywhere defined. \\end{enumerate}"}
+{"_id": "1868", "title": "derived-lemma-right-derived-properties", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor. \\begin{enumerate} \\item The functor $RF$ is an exact functor $D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$. \\item The functor $RF$ induces an exact functor $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$. \\item The functor $RF$ induces a $\\delta$-functor $\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$. \\item The functor $RF$ induces a $\\delta$-functor $\\mathcal{A} \\to D^{+}(\\mathcal{B})$. \\end{enumerate}"}
+{"_id": "1869", "title": "derived-lemma-higher-derived-functors", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor. \\begin{enumerate} \\item For any short exact sequence $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ of complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there is an associated long exact sequence $$ \\ldots \\to H^i(RF(A^\\bullet)) \\to H^i(RF(B^\\bullet)) \\to H^i(RF(C^\\bullet)) \\to H^{i + 1}(RF(A^\\bullet)) \\to \\ldots $$ \\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$ are zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$. \\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective. \\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see Homology, Definition \\ref{homology-definition-universal-delta-functor}) from $\\mathcal{A}$ to $\\mathcal{B}$. \\end{enumerate}"}
+{"_id": "1870", "title": "derived-lemma-cartan-eilenberg", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $K^\\bullet$ be a bounded below complex. There exists a Cartan-Eilenberg resolution of $K^\\bullet$."}
+{"_id": "1871", "title": "derived-lemma-two-ss-complex-functor", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of abelian categories. Let $K^\\bullet$ be a bounded below complex of $\\mathcal{A}$. Let $I^{\\bullet, \\bullet}$ be a Cartan-Eilenberg resolution for $K^\\bullet$. The spectral sequences $({}'E_r, {}'d_r)_{r \\geq 0}$ and $({}''E_r, {}''d_r)_{r \\geq 0}$ associated to the double complex $F(I^{\\bullet, \\bullet})$ satisfy the relations $$ {}'E_1^{p, q} = R^qF(K^p) \\quad \\text{and} \\quad {}''E_2^{p, q} = R^pF(H^q(K^\\bullet)) $$ Moreover, these spectral sequences are bounded, converge to $H^*(RF(K^\\bullet))$, and the associated induced filtrations on $H^n(RF(K^\\bullet))$ are finite."}
+{"_id": "1872", "title": "derived-lemma-compose-derived-functors", "text": "Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$ be left exact functors. Assume $\\mathcal{A}$, $\\mathcal{B}$ have enough injectives. The following are equivalent \\begin{enumerate} \\item $F(I)$ is right acyclic for $G$ for each injective object $I$ of $\\mathcal{A}$, and \\item the canonical map $$ t : R(G \\circ F) \\longrightarrow RG \\circ RF. $$ is isomorphism of functors from $D^{+}(\\mathcal{A})$ to $D^{+}(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "1873", "title": "derived-lemma-grothendieck-spectral-sequence", "text": "With assumptions as in Lemma \\ref{lemma-compose-derived-functors} and assuming the equivalent conditions (1) and (2) hold. Let $X$ be an object of $D^{+}(\\mathcal{A})$. There exists a spectral sequence $(E_r, d_r)_{r \\geq 0}$ consisting of bigraded objects $E_r$ of $\\mathcal{C}$ with $d_r$ of bidegree $(r, - r + 1)$ and with $$ E_2^{p, q} = R^pG(H^q(RF(X))) $$ Moreover, this spectral sequence is bounded, converges to $H^*(R(G \\circ F)(X))$, and induces a finite filtration on each $H^n(R(G \\circ F)(X))$."}
+{"_id": "1874", "title": "derived-lemma-resolution-functor", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Given a resolution functor $(j, i)$ there is a unique way to turn $j$ into a functor and $i$ into a $2$-isomorphism producing a $2$-commutative diagram $$ \\xymatrix{ K^{+}(\\mathcal{A}) \\ar[rd] \\ar[rr]_j & & K^{+}(\\mathcal{I}) \\ar[ld] \\\\ & D^{+}(\\mathcal{A}) } $$ where $\\mathcal{I}$ is the full additive subcategory of $\\mathcal{A}$ consisting of injective objects."}
+{"_id": "1875", "title": "derived-lemma-into-derived-category", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough injectives. Then a resolution functor $j$ exists and is unique up to unique isomorphism of functors."}
+{"_id": "1877", "title": "derived-lemma-resolution-functor-quasi-inverse", "text": "Let $\\mathcal{A}$ be an abelian category which has enough injectives. Let $j$ be a resolution functor. Write $Q : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$ for the natural functor. Then $j = j' \\circ Q$ for a unique functor $j' : D^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$ which is quasi-inverse to the canonical functor $K^{+}(\\mathcal{I}) \\to D^{+}(\\mathcal{A})$."}
+{"_id": "1878", "title": "derived-lemma-functorial-injective-resolutions", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has functorial injective embeddings, see Homology, Definition \\ref{homology-definition-functorial-injective-embedding}. \\begin{enumerate} \\item There exists a functor $inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$ such that $s \\circ inj = \\text{id}$. \\item For any functor $inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$ such that $s \\circ inj = \\text{id}$ we obtain a resolution functor, see Definition \\ref{definition-localization-functor}. \\end{enumerate}"}
+{"_id": "1879", "title": "derived-lemma-right-derived-functor", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor into an abelian category. Let $(i, j)$ be a resolution functor, see Definition \\ref{definition-localization-functor}. The right derived functor $RF$ of $F$ fits into the following $2$-commutative diagram $$ \\xymatrix{ D^{+}(\\mathcal{A}) \\ar[rd]_{RF} \\ar[rr]^{j'} & & K^{+}(\\mathcal{I}) \\ar[ld]^F \\\\ & D^{+}(\\mathcal{B}) } $$ where $j'$ is the functor from Lemma \\ref{lemma-resolution-functor-quasi-inverse}."}
+{"_id": "1880", "title": "derived-lemma-filtered-injective", "text": "Let $\\mathcal{A}$ be an abelian category. An object $I$ of $\\text{Fil}^f(\\mathcal{A})$ is filtered injective if and only if there exist $a \\leq b$, injective objects $I_n$, $a \\leq n \\leq b$ of $\\mathcal{A}$ and an isomorphism $I \\cong \\bigoplus_{a \\leq n \\leq b} I_n$ such that $F^pI = \\bigoplus_{n \\geq p} I_n$."}
+{"_id": "1881", "title": "derived-lemma-split-strict-monomorphism", "text": "Let $\\mathcal{A}$ be an abelian category. Any strict monomorphism $u : I \\to A$ of $\\text{Fil}^f(\\mathcal{A})$ where $I$ is a filtered injective object is a split injection."}
+{"_id": "1882", "title": "derived-lemma-injective-property-filtered-injective", "text": "Let $\\mathcal{A}$ be an abelian category. Let $u : A \\to B$ be a strict monomorphism of $\\text{Fil}^f(\\mathcal{A})$ and $f : A \\to I$ a morphism from $A$ into a filtered injective object in $\\text{Fil}^f(\\mathcal{A})$. Then there exists a morphism $g : B \\to I$ such that $f = g \\circ u$."}
+{"_id": "1883", "title": "derived-lemma-strict-monomorphism-into-filtered-injective", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists a strict monomorphism $A \\to I$ where $I$ is a filtered injective object."}
+{"_id": "1884", "title": "derived-lemma-filtered-injective-right-resolution-single-object", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists a filtered quasi-isomorphism $A[0] \\to I^\\bullet$ where $I^\\bullet$ is a complex of filtered injective objects with $I^n = 0$ for $n < 0$."}
+{"_id": "1885", "title": "derived-lemma-filtered-injective-right-resolution-map", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $f : A \\to B$ be a morphism of $\\text{Fil}^f(\\mathcal{A})$. Given filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and $B[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then there exists a commutative diagram $$ \\xymatrix{ A[0] \\ar[r] \\ar[d] & B[0] \\ar[d] \\\\ I^\\bullet \\ar[r] & J^\\bullet } $$"}
+{"_id": "1886", "title": "derived-lemma-filtered-injective-right-resolution-ses", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence in $\\text{Fil}^f(\\mathcal{A})$. Given filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and $C[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then there exists a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & A[0] \\ar[r] \\ar[d] & B[0] \\ar[r] \\ar[d] & C[0] \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & I^\\bullet \\ar[r] & M^\\bullet \\ar[r] & J^\\bullet \\ar[r] & 0 } $$ where the lower row is a termwise split sequence of complexes."}
+{"_id": "1887", "title": "derived-lemma-right-resolution-by-filtered-injectives", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. For every $K^\\bullet \\in K^{+}(\\text{Fil}^f(\\mathcal{A}))$ there exists a filtered quasi-isomorphism $K^\\bullet \\to I^\\bullet$ with $I^\\bullet$ bounded below, each $I^n$ a filtered injective object, and each $K^n \\to I^n$ a strict monomorphism."}
+{"_id": "1888", "title": "derived-lemma-filtered-acyclic-is-zero", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet, I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$. Assume $K^\\bullet$ is filtered acyclic and $I^\\bullet$ bounded below and consisting of filtered injective objects. Any morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero: $\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet) = 0$."}
+{"_id": "1889", "title": "derived-lemma-morphisms-into-filtered-injective-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$ be a bounded below complex consisting of filtered injective objects. \\begin{enumerate} \\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ in $K(\\text{Fil}^f(\\mathcal{A}))$ be a filtered quasi-isomorphism. Then the map $$ \\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet) \\to \\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet) $$ is bijective. \\item Let $L^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$. Then $$ \\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet) = \\Hom_{DF(\\mathcal{A})}(L^\\bullet, I^\\bullet). $$ \\end{enumerate}"}
+{"_id": "1891", "title": "derived-lemma-ss-filtered-derived", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let $T : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor. Assume $\\mathcal{A}$ has enough injectives. Let $(K^\\bullet, F)$ be an object of $\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$. There exists a spectral sequence $(E_r, d_r)_{r\\geq 0}$ consisting of bigraded objects $E_r$ of $\\mathcal{B}$ and $d_r$ of bidegree $(r, - r + 1)$ and with $$ E_1^{p, q} = R^{p + q}T(\\text{gr}^p(K^\\bullet)) $$ Moreover, this spectral sequence is bounded, converges to $R^*T(K^\\bullet)$, and induces a finite filtration on each $R^nT(K^\\bullet)$. The construction of this spectral sequence is functorial in the object $K^\\bullet$ of $\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$ and the terms $(E_r, d_r)$ for $r \\geq 1$ do not depend on any choices."}
+{"_id": "1892", "title": "derived-lemma-compute-ext-resolutions", "text": "Let $\\mathcal{A}$ be an abelian category. Let $X^\\bullet, Y^\\bullet \\in \\Ob(K(\\mathcal{A}))$. \\begin{enumerate} \\item Let $Y^\\bullet \\to I^\\bullet$ be an injective resolution (Definition \\ref{definition-injective-resolution}). Then $$ \\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) = \\Hom_{K(\\mathcal{A})}(X^\\bullet, I^\\bullet[i]). $$ \\item Let $P^\\bullet \\to X^\\bullet$ be a projective resolution (Definition \\ref{definition-projective-resolution}). Then $$ \\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) = \\Hom_{K(\\mathcal{A})}(P^\\bullet[-i], Y^\\bullet). $$ \\end{enumerate}"}
+{"_id": "1893", "title": "derived-lemma-negative-exts", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item Let $X$, $Y$ be objects of $D(\\mathcal{A})$. Given $a, b \\in \\mathbf{Z}$ such that $H^i(X) = 0$ for $i > a$ and $H^j(Y) = 0$ for $j < b$, we have $\\Ext^n_\\mathcal{A}(X, Y) = 0$ for $n < b - a$ and $$ \\Ext^{b - a}_\\mathcal{A}(X, Y) = \\Hom_\\mathcal{A}(H^a(X), H^b(Y)) $$ \\item Let $A, B \\in \\Ob(\\mathcal{A})$. For $i < 0$ we have $\\Ext^i_\\mathcal{A}(B, A) = 0$. We have $\\Ext^0_\\mathcal{A}(B, A) = \\Hom_\\mathcal{A}(B, A)$. \\end{enumerate}"}
+{"_id": "1894", "title": "derived-lemma-yoneda-extension", "text": "Let $\\mathcal{A}$ be an abelian category with objects $A$, $B$. Any element in $\\Ext^i_\\mathcal{A}(B, A)$ is $\\delta(E)$ for some degree $i$ Yoneda extension of $B$ by $A$. Given two Yoneda extensions $E$, $E'$ of the same degree then $E$ is equivalent to $E'$ if and only if $\\delta(E) = \\delta(E')$."}
+{"_id": "1895", "title": "derived-lemma-ext-1", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$, $B$ be objects of $\\mathcal{A}$. Then $\\Ext^1_\\mathcal{A}(B, A)$ is the group $\\Ext_\\mathcal{A}(B, A)$ constructed in Homology, Definition \\ref{homology-definition-ext-group}."}
+{"_id": "1896", "title": "derived-lemma-higher-ext-zero", "text": "Let $\\mathcal{A}$ be an abelian category and let $p \\geq 0$. If $\\Ext^p_\\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\\mathcal{A}$, then $\\Ext^i_\\mathcal{A}(B, A) = 0$ for $i \\geq p$ and any pair of objects $A$, $B$ of $\\mathcal{A}$."}
+{"_id": "1897", "title": "derived-lemma-ext-2-zero", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\Ext^2_\\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\\mathcal{A}$. Then any object $K$ of $D^b(\\mathcal{A})$ is isomorphic to the direct sum of its cohomologies: $K \\cong \\bigoplus H^i(K)[-i]$."}
+{"_id": "1898", "title": "derived-lemma-K-bounded-derived", "text": "Let $\\mathcal{A}$ be an abelian category. Then there is a canonical identification $K_0(D^b(\\mathcal{A})) = K_0(\\mathcal{A})$ of zeroth $K$-groups."}
+{"_id": "1899", "title": "derived-lemma-map-K", "text": "Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated categories. Then $F$ induces a group homomorphism $K_0(\\mathcal{D}) \\to K_0(\\mathcal{D}')$."}
+{"_id": "1900", "title": "derived-lemma-homological-map-K", "text": "Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor from a triangulated category to an abelian category. Assume that for any $X$ in $\\mathcal{D}$ only a finite number of the objects $H(X[i])$ are nonzero in $\\mathcal{A}$. Then $H$ induces a group homomorphism $K_0(\\mathcal{D}) \\to K_0(\\mathcal{A})$ sending $[X]$ to $\\sum (-1)^i[H(X[i])]$."}
+{"_id": "1901", "title": "derived-lemma-DBA-map-K", "text": "Let $\\mathcal{B}$ be a weak Serre subcategory of the abelian category $\\mathcal{A}$. Then there are canonical maps $$ K_0(\\mathcal{B}) \\longrightarrow K_0(D^b_\\mathcal{B}(\\mathcal{A})) \\longrightarrow K_0(\\mathcal{B}) $$ whose composition is zero. The second arrow sends the class $[X]$ of the object $X$ to the element $\\sum (-1)^i[H^i(X)]$ of $K_0(\\mathcal{B})$."}
+{"_id": "1902", "title": "derived-lemma-bilinear-map-K", "text": "Let $\\mathcal{D}$, $\\mathcal{D}'$, $\\mathcal{D}''$ be triangulated categories. Let $$ \\otimes : \\mathcal{D} \\times \\mathcal{D}' \\longrightarrow \\mathcal{D}'' $$ be a functor such that for fixed $X$ in $\\mathcal{D}$ the functor $X \\otimes - : \\mathcal{D}' \\to \\mathcal{D}''$ is an exact functor and for fixed $X'$ in $\\mathcal{D}'$ the functor $- \\otimes X' : \\mathcal{D} \\to \\mathcal{D}''$ is an exact functor. Then $\\otimes$ induces a bilinear map $K_0(\\mathcal{D}) \\times K_0(\\mathcal{D}') \\to K_0(\\mathcal{D}'')$ which sends $([X], [X'])$ to $[X \\otimes X']$."}
+{"_id": "1903", "title": "derived-lemma-special-direct-system", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset. Assume $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and every object of $\\mathcal{A}$ is a quotient of an element of $\\mathcal{P}$. Let $K^\\bullet$ be a complex. There exists a commutative diagram $$ \\xymatrix{ P_1^\\bullet \\ar[d] \\ar[r] & P_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\ \\tau_{\\leq 1}K^\\bullet \\ar[r] & \\tau_{\\leq 2}K^\\bullet \\ar[r] & \\ldots } $$ in the category of complexes such that \\begin{enumerate} \\item the vertical arrows are quasi-isomorphisms and termwise surjective, \\item $P_n^\\bullet$ is a bounded above complex with terms in $\\mathcal{P}$, \\item the arrows $P_n^\\bullet \\to P_{n + 1}^\\bullet$ are termwise split injections and each cokernel $P^i_{n + 1}/P^i_n$ is an element of $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "1904", "title": "derived-lemma-special-inverse-system", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset. Assume $\\mathcal{I}$ contains $0$, is closed under (finite) products, and every object of $\\mathcal{A}$ is a subobject of an element of $\\mathcal{I}$. Let $K^\\bullet$ be a complex. There exists a commutative diagram $$ \\xymatrix{ \\ldots \\ar[r] & \\tau_{\\geq -2}K^\\bullet \\ar[r] \\ar[d] & \\tau_{\\geq -1}K^\\bullet \\ar[d] \\\\ \\ldots \\ar[r] & I_2^\\bullet \\ar[r] & I_1^\\bullet } $$ in the category of complexes such that \\begin{enumerate} \\item the vertical arrows are quasi-isomorphisms and termwise injective, \\item $I_n^\\bullet$ is a bounded below complex with terms in $\\mathcal{I}$, \\item the arrows $I_{n + 1}^\\bullet \\to I_n^\\bullet$ are termwise split surjections and $\\Ker(I^i_{n + 1} \\to I^i_n)$ is an element of $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "1905", "title": "derived-lemma-pre-derived-adjoint-functors-general", "text": "In the situation above assume $F$ is right adjoint to $G$. Let $K \\in \\Ob(\\mathcal{D})$ and $M \\in \\Ob(\\mathcal{D}')$. If $RF$ is defined at $K$ and $LG$ is defined at $M$, then there is a canonical isomorphism $$ \\Hom_{(S')^{-1}\\mathcal{D}'}(M, RF(K)) = \\Hom_{S^{-1}\\mathcal{D}}(LG(M), K) $$ This isomorphism is functorial in both variables on the triangulated subcategories of $S^{-1}\\mathcal{D}$ and $(S')^{-1}\\mathcal{D}'$ where $RF$ and $LG$ are defined."}
+{"_id": "1906", "title": "derived-lemma-pre-derived-adjoint-functors", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$ be functors of abelian categories such that $F$ is a right adjoint to $G$. Let $K^\\bullet$ be a complex of $\\mathcal{A}$ and let $M^\\bullet$ be a complex of $\\mathcal{B}$. If $RF$ is defined at $K^\\bullet$ and $LG$ is defined at $M^\\bullet$, then there is a canonical isomorphism $$ \\Hom_{D(\\mathcal{B})}(M^\\bullet, RF(K^\\bullet)) = \\Hom_{D(\\mathcal{A})}(LG(M^\\bullet), K^\\bullet) $$ This isomorphism is functorial in both variables on the triangulated subcategories of $D(\\mathcal{A})$ and $D(\\mathcal{B})$ where $RF$ and $LG$ are defined."}
+{"_id": "1907", "title": "derived-lemma-derived-adjoint-functors", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$ be functors of abelian categories such that $F$ is a right adjoint to $G$. If the derived functors $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ and $LG : D(\\mathcal{B}) \\to D(\\mathcal{A})$ exist, then $RF$ is a right adjoint to $LG$."}
+{"_id": "1908", "title": "derived-lemma-K-injective", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I^\\bullet$ be a complex. The following are equivalent \\begin{enumerate} \\item $I^\\bullet$ is K-injective, \\item for every quasi-isomorphism $M^\\bullet \\to N^\\bullet$ the map $$ \\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet) \\to \\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) $$ is bijective, and \\item for every complex $N^\\bullet$ the map $$ \\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet) \\to \\Hom_{D(\\mathcal{A})}(N^\\bullet, I^\\bullet) $$ is an isomorphism. \\end{enumerate}"}
+{"_id": "1909", "title": "derived-lemma-triangle-K-injective", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, L, M, f, g, h)$ be a distinguished triangle of $K(\\mathcal{A})$. If two out of $K$, $L$, $M$ are K-injective complexes, then the third is too."}
+{"_id": "1910", "title": "derived-lemma-bounded-below-injectives-K-injective", "text": "Let $\\mathcal{A}$ be an abelian category. A bounded below complex of injectives is K-injective."}
+{"_id": "1911", "title": "derived-lemma-product-K-injective", "text": "Let $\\mathcal{A}$ be an abelian category. Let $T$ be a set and for each $t \\in T$ let $I_t^\\bullet$ be a K-injective complex. If $I^n = \\prod_t I_t^n$ exists for all $n$, then $I^\\bullet$ is a K-injective complex. Moreover, $I^\\bullet$ represents the product of the objects $I_t^\\bullet$ in $D(\\mathcal{A})$."}
+{"_id": "1912", "title": "derived-lemma-K-injective-defined", "text": "Let $\\mathcal{A}$ be an abelian category. Let $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ be an exact functor of triangulated categories. Then $RF$ is defined at every complex in $K(\\mathcal{A})$ which is quasi-isomorphic to a K-injective complex. In fact, every K-injective complex computes $RF$."}
+{"_id": "1915", "title": "derived-lemma-adjoint-preserve-K-injectives", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume \\begin{enumerate} \\item $u$ is right adjoint to $v$, and \\item $v$ is exact. \\end{enumerate} Then $u$ transforms K-injective complexes into K-injective complexes."}
+{"_id": "1916", "title": "derived-lemma-replace-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $d : \\Ob(\\mathcal{A}) \\to \\{0, 1, 2, \\ldots, \\infty\\}$ be a function. Assume that \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a subobject of an object $A$ with $d(A) = 0$, \\item $d(A \\oplus B) \\leq \\max \\{d(A), d(B)\\}$ for $A, B \\in \\mathcal{A}$, and \\item if $0 \\to A \\to B \\to C \\to 0$ is short exact, then $d(C) \\leq \\max\\{d(A) - 1, d(B)\\}$. \\end{enumerate} Let $K^\\bullet$ be a complex such that $n + d(K^n)$ tends to $-\\infty$ as $n \\to -\\infty$. Then there exists a quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $d(L^n) = 0$ for all $n \\in \\mathbf{Z}$."}
+{"_id": "1917", "title": "derived-lemma-unbounded-right-derived", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of abelian categories. Assume \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a subobject of an object which is right acyclic for $F$, \\item there exists an integer $n \\geq 0$ such that $R^nF = 0$, \\end{enumerate} Then \\begin{enumerate} \\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists, \\item any complex consisting of right acyclic objects for $F$ computes $RF$, \\item any complex is the source of a quasi-isomorphism into a complex consisting of right acyclic objects for $F$, \\item for $E \\in D(\\mathcal{A})$ \\begin{enumerate} \\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism for $i \\leq a$, \\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - n + 1}E))$ is an isomorphism for $i \\geq b$, \\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some $-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$ for $i \\not \\in [a, b + n - 1]$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1920", "title": "derived-lemma-direct-sums", "text": "Let $\\mathcal{A}$ be an abelian category. If $\\mathcal{A}$ has exact countable direct sums, then $D(\\mathcal{A})$ has countable direct sums. In fact given a collection of complexes $K_i^\\bullet$ indexed by a countable index set $I$ the termwise direct sum $\\bigoplus K_i^\\bullet$ is the direct sum of $K_i^\\bullet$ in $D(\\mathcal{A})$."}
+{"_id": "1921", "title": "derived-lemma-compute-colimit", "text": "Let $\\mathcal{A}$ be an abelian category. Assume colimits over $\\mathbf{N}$ exist and are exact. Then countable direct sums exists and are exact. Moreover, if $(A_n, f_n)$ is a system over $\\mathbf{N}$, then there is a short exact sequence $$ 0 \\to \\bigoplus A_n \\to \\bigoplus A_n \\to \\colim A_n \\to 0 $$ where the first map in degree $n$ is given by $1 - f_n$."}
+{"_id": "1922", "title": "derived-lemma-colim-hocolim", "text": "Let $\\mathcal{A}$ be an abelian category. Let $L_n^\\bullet$ be a system of complexes of $\\mathcal{A}$. Assume colimits over $\\mathbf{N}$ exist and are exact in $\\mathcal{A}$. Then the termwise colimit $L^\\bullet = \\colim L_n^\\bullet$ is a homotopy colimit of the system in $D(\\mathcal{A})$."}
+{"_id": "1923", "title": "derived-lemma-cohomology-of-hocolim", "text": "Let $\\mathcal{D}$ be a triangulated category having countable direct sums. Let $\\mathcal{A}$ be an abelian category with exact colimits over $\\mathbf{N}$. Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor commuting with countable direct sums. Then $H(\\text{hocolim} K_n) = \\colim H(K_n)$ for any system of objects of $\\mathcal{D}$."}
+{"_id": "1924", "title": "derived-lemma-commutes-with-countable-sums", "text": "Let $\\mathcal{D}$ be a triangulated category with countable direct sums. Let $K \\in \\mathcal{D}$ be an object such that for every countable set of objects $E_n \\in \\mathcal{D}$ the canonical map $$ \\bigoplus \\Hom_\\mathcal{D}(K, E_n) \\longrightarrow \\Hom_\\mathcal{D}(K, \\bigoplus E_n) $$ is a bijection. Then, given any system $L_n$ of $\\mathcal{D}$ over $\\mathbf{N}$ whose derived colimit $L = \\text{hocolim} L_n$ exists we have that $$ \\colim \\Hom_\\mathcal{D}(K, L_n) \\longrightarrow \\Hom_\\mathcal{D}(K, L) $$ is a bijection."}
+{"_id": "1925", "title": "derived-lemma-products", "text": "Let $\\mathcal{A}$ be an abelian category with exact countable products. Then \\begin{enumerate} \\item $D(\\mathcal{A})$ has countable products, \\item countable products $\\prod K_i$ in $D(\\mathcal{A})$ are obtained by taking termwise products of any complexes representing the $K_i$, and \\item $H^p(\\prod K_i) = \\prod H^p(K_i)$. \\end{enumerate}"}
+{"_id": "1926", "title": "derived-lemma-inverse-limit-bounded-below", "text": "Let $\\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $(K_n)$ be an inverse system of $D^+(\\mathcal{A})$. Then $R\\lim K_n$ exists."}
+{"_id": "1927", "title": "derived-lemma-difficulty-K-injectives", "text": "Let $\\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $K^\\bullet$ be a complex. Let $I_n^\\bullet$ be the inverse system of bounded below complexes of injectives produced by Lemma \\ref{lemma-special-inverse-system}. Then $I^\\bullet = \\lim I_n^\\bullet$ exists, is K-injective, and the following are equivalent \\begin{enumerate} \\item the map $K^\\bullet \\to I^\\bullet$ is a quasi-isomorphism, \\item the canonical map $K^\\bullet \\to R\\lim \\tau_{\\geq -n}K^\\bullet$ is an isomorphism in $D(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "1929", "title": "derived-lemma-associativity-star", "text": "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$ we have $(\\mathcal{A} \\star \\mathcal{B}) \\star \\mathcal{C} = \\mathcal{A} \\star (\\mathcal{B} \\star \\mathcal{C})$."}
+{"_id": "1930", "title": "derived-lemma-smd-star", "text": "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories $\\mathcal{A}$, $\\mathcal{B}$ we have $smd(\\mathcal{A}) \\star smd(\\mathcal{B}) \\subset smd(\\mathcal{A} \\star \\mathcal{B})$ and $smd(smd(\\mathcal{A}) \\star smd(\\mathcal{B})) = smd(\\mathcal{A} \\star \\mathcal{B})$."}
+{"_id": "1931", "title": "derived-lemma-add-star", "text": "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories $\\mathcal{A}$, $\\mathcal{B}$ the full subcategories $add(\\mathcal{A}) \\star add(\\mathcal{B})$ and $smd(add(\\mathcal{A}))$ are closed under direct sums."}
+{"_id": "1932", "title": "derived-lemma-cone-n", "text": "Let $\\mathcal{T}$ be a triangulated category. Given a full subcategory $\\mathcal{A}$ for $n \\geq 1$ the subcategory $$ \\mathcal{C}_n = smd(add(\\mathcal{A})^{\\star n}) = smd(add(\\mathcal{A}) \\star \\ldots \\star add(\\mathcal{A})) $$ defined above is a strictly full subcategory of $\\mathcal{T}$ closed under direct sums and direct summands and $\\mathcal{C}_{n + m} = smd(\\mathcal{C}_n \\star \\mathcal{C}_m)$ for all $n, m \\geq 1$."}
+{"_id": "1933", "title": "derived-lemma-in-cone-n", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D} = D(\\mathcal{A})$. Let $\\mathcal{E} \\subset \\Ob(\\mathcal{A})$ be a subset which we view as a subset of $\\Ob(\\mathcal{D})$ also. Let $K$ be an object of $\\mathcal{D}$. \\begin{enumerate} \\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$ and $H^i(K) \\in \\mathcal{E}$ if $i \\in [a, b]$. Then $K$ is in $smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$. \\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$ and $H^i(K) \\in smd(add(\\mathcal{E}))$ if $i \\in [a, b]$. Then $K$ is in $smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$. \\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$ with $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in \\mathcal{E}$ for $i \\in [a, b]$. Then $K$ is in $smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$. \\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$ with $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in smd(add(\\mathcal{E}))$ for $i \\in [a, b]$. Then $K$ is in $smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$. \\end{enumerate}"}
+{"_id": "1934", "title": "derived-lemma-forward-cone-n", "text": "Let $\\mathcal{T}$ be a triangulated category. Let $H : \\mathcal{T} \\to \\mathcal{A}$ be a homological functor to an abelian category $\\mathcal{A}$. Let $a \\leq b$ and $\\mathcal{E} \\subset \\Ob(\\mathcal{T})$ be a subset such that $H^i(E) = 0$ for $E \\in \\mathcal{E}$ and $i \\not \\in [a, b]$. Then for $X \\in smd(add(\\mathcal{E}[-m, m])^{\\star n})$ we have $H^i(X) = 0$ for $i \\not \\in [-m + na, m + nb]$."}
+{"_id": "1935", "title": "derived-lemma-generated-by-E-explicit", "text": "Let $\\mathcal{T}$ be a triangulated category. Let $E$ be an object of $\\mathcal{T}$. For $n \\geq 1$ we have $$ \\langle E \\rangle_n = smd(\\langle E \\rangle_1 \\star \\ldots \\star \\langle E \\rangle_1) = smd({\\langle E \\rangle_1}^{\\star n}) = \\bigcup\\nolimits_{m \\geq 1} smd(add(E[-m, m])^{\\star n}) $$ For $n, n' \\geq 1$ we have $\\langle E \\rangle_{n + n'} = smd(\\langle E \\rangle_n \\star \\langle E \\rangle_{n'})$."}
+{"_id": "1936", "title": "derived-lemma-find-smallest-containing-E", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\\mathcal{D}$. The subcategory $$ \\langle E \\rangle = \\bigcup\\nolimits_n \\langle E \\rangle_n = \\bigcup\\nolimits_{n, m \\geq 1} smd(add(E[-m, m])^{\\star n}) $$ is a strictly full, saturated, triangulated subcategory of $\\mathcal{D}$ and it is the smallest such subcategory of $\\mathcal{D}$ containing the object $E$."}
+{"_id": "1937", "title": "derived-lemma-right-orthogonal", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $E, K$ be objects of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $\\Hom(E, K[i]) = 0$ for all $i \\in \\mathbf{Z}$, \\item $\\Hom(E', K) = 0$ for all $E' \\in \\langle E \\rangle$. \\end{enumerate}"}
+{"_id": "1939", "title": "derived-lemma-classical-generator-strong-generator", "text": "Let $\\mathcal{D}$ be a triangulated category which has a strong generator. Let $E$ be an object of $\\mathcal{D}$. If $E$ is a classical generator of $\\mathcal{D}$, then $E$ is a strong generator."}
+{"_id": "1940", "title": "derived-lemma-compact-objects-subcategory", "text": "Let $\\mathcal{D}$ be a (pre-)triangulated category with direct sums. Then the compact objects of $\\mathcal{D}$ form the objects of a Karoubian, saturated, strictly full, (pre-)triangulated subcategory $\\mathcal{D}_c$ of $\\mathcal{D}$."}
+{"_id": "1941", "title": "derived-lemma-write-as-colimit", "text": "Let $\\mathcal{D}$ be a triangulated category with direct sums. Let $E_i$, $i \\in I$ be a family of compact objects of $\\mathcal{D}$ such that $\\bigoplus E_i$ generates $\\mathcal{D}$. Then every object $X$ of $\\mathcal{D}$ can be written as $$ X = \\text{hocolim} X_n $$ where $X_1$ is a direct sum of shifts of the $E_i$ and each transition morphism fits into a distinguished triangle $Y_n \\to X_n \\to X_{n + 1} \\to Y_n[1]$ where $Y_n$ is a direct sum of shifts of the $E_i$."}
+{"_id": "1942", "title": "derived-lemma-factor-through", "text": "With assumptions and notation as in Lemma \\ref{lemma-write-as-colimit}. If $C$ is a compact object and $C \\to X_n$ is a morphism, then there is a factorization $C \\to E \\to X_n$ where $E$ is an object of $\\langle E_{i_1} \\oplus \\ldots \\oplus E_{i_t} \\rangle$ for some $i_1, \\ldots, i_t \\in I$."}
+{"_id": "1943", "title": "derived-lemma-brown", "text": "\\begin{reference} \\cite[Theorem 3.1]{Neeman-Grothendieck}. \\end{reference} Let $\\mathcal{D}$ be a triangulated category with direct sums which is compactly generated. Let $H : \\mathcal{D} \\to \\textit{Ab}$ be a contravariant cohomological functor which transforms direct sums into products. Then $H$ is representable."}
+{"_id": "1944", "title": "derived-lemma-pre-prepare-adjoint", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A} \\subset \\mathcal{D}$ be a full subcategory invariant under all shifts. Consider a distinguished triangle $$ X \\to Y \\to Z \\to X[1] $$ of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $Z$ is in $\\mathcal{A}^\\perp$, and \\item $\\Hom(A, X) = \\Hom(A, Y)$ for all $A \\in \\Ob(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "1945", "title": "derived-lemma-orthogonal-triangulated", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A} \\subset \\mathcal{D}$ be a full subcategory invariant under all shifts. Then both the right orthogonal $\\mathcal{A}^\\perp$ and the left orthogonal ${}^\\perp\\mathcal{A}$ of $\\mathcal{A}$ are strictly full, saturated\\footnote{Definition \\ref{definition-saturated}.}, triangulated subcagories of $\\mathcal{D}$."}
+{"_id": "1946", "title": "derived-lemma-prepare-adjoint", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A}$ be a full triangulated subcategory of $\\mathcal{D}$. For an object $X$ of $\\mathcal{D}$ consider the property $P(X)$: there exists a distinguished triangle $A \\to X \\to B \\to A[1]$ in $\\mathcal{D}$ with $A$ in $\\mathcal{A}$ and $B$ in $\\mathcal{A}^\\perp$. \\begin{enumerate} \\item If $X_1 \\to X_2 \\to X_3 \\to X_1[1]$ is a distinguished triangle and $P$ holds for two out of three, then it holds for the third. \\item If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \\oplus X_2$. \\end{enumerate}"}
+{"_id": "1947", "title": "derived-lemma-right-adjoint", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A} \\subset \\mathcal{D}$ be a full triangulated subcategory. The following are equivalent \\begin{enumerate} \\item the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$ has a right adjoint, and \\item for every $X$ in $\\mathcal{D}$ there exists a distinguished triangle $$ A \\to X \\to B \\to X'[1] $$ in $\\mathcal{D}$ with $A \\in \\Ob(\\mathcal{A})$ and $B \\in \\Ob(\\mathcal{A}^\\perp)$. \\end{enumerate} If this holds, then $\\mathcal{A}$ is saturated (Definition \\ref{definition-saturated}) and if $\\mathcal{A}$ is strictly full in $\\mathcal{D}$, then $\\mathcal{A} = {}^\\perp(\\mathcal{A}^\\perp)$."}
+{"_id": "1948", "title": "derived-lemma-left-adjoint", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A} \\subset \\mathcal{D}$ be a full triangulated subcategory. The following are equivalent \\begin{enumerate} \\item the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$ has a left adjoint, and \\item for every $X$ in $\\mathcal{D}$ there exists a distinguished triangle $$ B \\to X \\to A \\to K[1] $$ in $\\mathcal{D}$ with $A \\in \\Ob(\\mathcal{A})$ and $B \\in \\Ob({}^\\perp\\mathcal{A})$. \\end{enumerate} If this holds, then $\\mathcal{A}$ is saturated (Definition \\ref{definition-saturated}) and if $\\mathcal{A}$ is strictly full in $\\mathcal{D}$, then $\\mathcal{A} = ({}^\\perp\\mathcal{A})^\\perp$."}
+{"_id": "1949", "title": "derived-lemma-postnikov-system-small-cases", "text": "Let $\\mathcal{D}$ be a triangulated category. Consider Postnikov systems for complexes of length $n$. \\begin{enumerate} \\item For $n = 0$ Postnikov systems always exist and any morphism (\\ref{equation-map-complexes}) of complexes extends to a unique morphism of Postnikov systems. \\item For $n = 1$ Postnikov systems always exist and any morphism (\\ref{equation-map-complexes}) of complexes extends to a (nonunique) morphism of Postnikov systems. \\item For $n = 2$ Postnikov systems always exist but morphisms (\\ref{equation-map-complexes}) of complexes in general do not extend to morphisms of Postnikov systems. \\item For $n > 2$ Postnikov systems do not always exist. \\end{enumerate}"}
+{"_id": "1950", "title": "derived-lemma-maps-postnikov-systems-vanishing", "text": "Let $\\mathcal{D}$ be a triangulated category. Given a map (\\ref{equation-map-complexes}) consider the condition \\begin{equation} \\label{equation-P} \\Hom(X_i[i - j - 1], X'_j) = 0 \\text{ for }i > j + 1 \\end{equation} Then \\begin{enumerate} \\item If we have a Postnikov system for $X'_n \\to X'_{n - 1} \\to \\ldots \\to X'_0$ then property (\\ref{equation-P}) implies that $$ \\Hom(X_i[i - j - 1], Y'_j) = 0 \\text{ for }i > j + 1 $$ \\item If we are given Postnikov systems for both complexes and we have (\\ref{equation-P}), then the map extends to a (nonunique) map of Postnikov systems. \\end{enumerate}"}
+{"_id": "1952", "title": "derived-lemma-existence-postnikov-system", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $X_n \\to X_{n - 1} \\to \\ldots \\to X_0$ be a complex in $\\mathcal{D}$. If $$ \\Hom(X_i[i - j - 2], X_j) = 0 \\text{ for }i > j + 2 $$ then there exists a Postnikov system. If we have $$ \\Hom(X_i[i - j - 1], X_j) = 0 \\text{ for }i > j + 1 $$ then any two Postnikov systems are isomorphic."}
+{"_id": "1953", "title": "derived-lemma-essentially-constant", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(A_i)$ be an inverse system in $\\mathcal{D}$. Then $(A_i)$ is essentially constant (see Categories, Definition \\ref{categories-definition-essentially-constant-diagram}) if and only if there exists an $i$ and for all $j \\geq i$ a direct sum decomposition $A_j = A \\oplus Z_j$ such that (a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum decompositions and identity on $A$, (b) for all $j \\geq i$ there exists some $j' \\geq j$ such that $Z_{j'} \\to Z_j$ is zero."}
+{"_id": "1954", "title": "derived-lemma-essentially-constant-2-out-of-3", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $$ A_n \\to B_n \\to C_n \\to A_n[1] $$ be an inverse system of distinguished triangles in $\\mathcal{D}$. If $(A_n)$ and $(C_n)$ are essentially constant, then $(B_n)$ is essentially constant and their values fit into a distinguished triangle $A \\to B \\to C \\to A[1]$ such that for some $n \\geq 1$ there is a map $$ \\xymatrix{ A_n \\ar[d] \\ar[r] & B_n \\ar[d] \\ar[r] & C_n \\ar[d] \\ar[r] & A_n[1] \\ar[d] \\\\ A \\ar[r] & B \\ar[r] & C \\ar[r] & A[1] } $$ of distinguished triangles which induces an isomorphism $\\lim_{n' \\geq n} A_{n'} \\to A$ and similarly for $B$ and $C$."}
+{"_id": "1955", "title": "derived-lemma-essentially-constant-cohomology", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A_n$ be an inverse system of objects of $D(\\mathcal{A})$. Assume \\begin{enumerate} \\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$ for $i \\not \\in [a, b]$, and \\item the inverse systems $H^i(A_n)$ of $\\mathcal{A}$ are essentially constant for all $i \\in \\mathbf{Z}$. \\end{enumerate} Then $A_n$ is an essentially constant system of $D(\\mathcal{A})$ whose value $A$ satisfies that $H^i(A)$ is the value of the constant system $H^i(A_n)$ for each $i \\in \\mathbf{Z}$."}
+{"_id": "1956", "title": "derived-lemma-pro-isomorphism", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $$ A_n \\to B_n \\to C_n \\to A_n[1] $$ be an inverse system of distinguished triangles. If the system $C_n$ is pro-zero (essentially constant with value $0$), then the maps $A_n \\to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$ and the pro-object $(B_n)$."}
+{"_id": "1957", "title": "derived-lemma-pro-isomorphism-bis", "text": "Let $\\mathcal{A}$ be an abelian category. $$ A_n \\to B_n $$ be an inverse system of maps of $D(\\mathcal{A})$. Assume \\begin{enumerate} \\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$ and $H^i(B_n) = 0$ for $i \\not \\in [a, b]$, and \\item the inverse system of maps $H^i(A_n) \\to H^i(B_n)$ of $\\mathcal{A}$ define an isomorphism of pro-objects of $\\mathcal{A}$ for all $i \\in \\mathbf{Z}$. \\end{enumerate} Then the maps $A_n \\to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$ and the pro-object $(B_n)$."}
+{"_id": "1958", "title": "derived-proposition-9", "text": "Let $\\mathcal{D}$ be a triangulated category. Any commutative diagram $$ \\xymatrix{ X \\ar[r] \\ar[d] & Y \\ar[d] \\\\ X' \\ar[r] & Y' } $$ can be extended to a diagram $$ \\xymatrix{ X \\ar[r] \\ar[d] & Y \\ar[r] \\ar[d] & Z \\ar[r] \\ar[d] & X[1] \\ar[d] \\\\ X' \\ar[r] \\ar[d] & Y' \\ar[r] \\ar[d] & Z' \\ar[r] \\ar[d] & X'[1] \\ar[d] \\\\ X'' \\ar[r] \\ar[d] & Y'' \\ar[r] \\ar[d] & Z'' \\ar[r] \\ar[d] & X''[1] \\ar[d] \\\\ X[1] \\ar[r] & Y[1] \\ar[r] & Z[1] \\ar[r] & X[2] } $$ where all the squares are commutative, except for the lower right square which is anticommutative. Moreover, each of the rows and columns are distinguished triangles. Finally, the morphisms on the bottom row (resp.\\ right column) are obtained from the morphisms of the top row (resp.\\ left column) by applying $[1]$."}
+{"_id": "1959", "title": "derived-proposition-construct-localization", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated structure. Then there exists a unique structure of a pre-triangulated category on $S^{-1}\\mathcal{D}$ such that the localization functor $Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ is exact. Moreover, if $\\mathcal{D}$ is a triangulated category, so is $S^{-1}\\mathcal{D}$."}
+{"_id": "1960", "title": "derived-proposition-homotopy-category-triangulated", "text": "Let $\\mathcal{A}$ be an additive category. The category $K(\\mathcal{A})$ of complexes up to homotopy with its natural translation functors and distinguished triangles as defined above is a triangulated category."}
+{"_id": "1961", "title": "derived-proposition-derived-functor", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. \\begin{enumerate} \\item The full subcategory $\\mathcal{E}$ of $\\mathcal{D}$ consisting of objects at which $RF$ is defined is a strictly full triangulated subcategory of $\\mathcal{D}$. \\item We obtain an exact functor $RF : \\mathcal{E} \\longrightarrow \\mathcal{D}'$ of triangulated categories. \\item Elements of $S$ with either source or target in $\\mathcal{E}$ are morphisms of $\\mathcal{E}$. \\item The functor $S_\\mathcal{E}^{-1}\\mathcal{E} \\to S^{-1}\\mathcal{D}$ is a fully faithful exact functor of triangulated categories. \\item Any element of $S_\\mathcal{E} = \\text{Arrows}(\\mathcal{E}) \\cap S$ is mapped to an isomorphism by $RF$. \\item We obtain an exact functor $$ RF : S_\\mathcal{E}^{-1}\\mathcal{E} \\longrightarrow \\mathcal{D}'. $$ \\item If $\\mathcal{D}'$ is Karoubian, then $\\mathcal{E}$ is a saturated triangulated subcategory of $\\mathcal{D}$. \\end{enumerate} A similar result holds for $LF$."}
+{"_id": "1962", "title": "derived-proposition-enough-acyclics", "text": "\\begin{slogan} A functor on an Abelian categories is extended to the (bounded below or above) derived category by resolving with a complex that is acyclic for that functor. \\end{slogan} Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of abelian categories. \\begin{enumerate} \\item If every object of $\\mathcal{A}$ injects into an object acyclic for $RF$, then $RF$ is defined on all of $K^{+}(\\mathcal{A})$ and we obtain an exact functor $$ RF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B}) $$ see (\\ref{equation-everywhere}). Moreover, any bounded below complex $A^\\bullet$ whose terms are acyclic for $RF$ computes $RF$. \\item If every object of $\\mathcal{A}$ is quotient of an object acyclic for $LF$, then $LF$ is defined on all of $K^{-}(\\mathcal{A})$ and we obtain an exact functor $$ LF : D^{-}(\\mathcal{A}) \\longrightarrow D^{-}(\\mathcal{B}) $$ see (\\ref{equation-everywhere}). Moreover, any bounded above complex $A^\\bullet$ whose terms are acyclic for $LF$ computes $LF$. \\end{enumerate}"}
+{"_id": "1964", "title": "derived-proposition-left-derived-exists", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor of abelian categories. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset. Assume \\begin{enumerate} \\item $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and every object of $\\mathcal{A}$ is a quotient of an element of $\\mathcal{P}$, \\item for any bounded above acyclic complex $P^\\bullet$ of $\\mathcal{A}$ with $P^n \\in \\mathcal{P}$ for all $n$ the complex $F(P^\\bullet)$ is exact, \\item $\\mathcal{A}$ and $\\mathcal{B}$ have colimits of systems over $\\mathbf{N}$, \\item colimits over $\\mathbf{N}$ are exact in both $\\mathcal{A}$ and $\\mathcal{B}$, and \\item $F$ commutes with colimits over $\\mathbf{N}$. \\end{enumerate} Then $LF$ is defined on all of $D(\\mathcal{A})$."}
+{"_id": "1965", "title": "derived-proposition-generator-versus-classical-generator", "text": "Let $\\mathcal{D}$ be a triangulated category with direct sums. Let $E$ be a compact object of $\\mathcal{D}$. The following are equivalent \\begin{enumerate} \\item $E$ is a classical generator for $\\mathcal{D}_c$ and $\\mathcal{D}$ is compactly generated, and \\item $E$ is a generator for $\\mathcal{D}$. \\end{enumerate}"}
+{"_id": "1966", "title": "derived-proposition-brown", "text": "\\begin{reference} \\cite[Theorem 4.1]{Neeman-Grothendieck}. \\end{reference} Let $\\mathcal{D}$ be a triangulated category with direct sums which is compactly generated. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated categories which transforms direct sums into direct sums. Then $F$ has an exact right adjoint."}
+{"_id": "2031", "title": "cohomology-theorem-proper-base-change", "text": "\\begin{reference} \\cite[Expose V bis, 4.1.1]{SGA4} \\end{reference} Consider a cartesian square of topological spaces $$ \\xymatrix{ X' = Y' \\times_Y X \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ Assume that $f$ is proper and separated. Let $E$ be an object of $D^+(X)$. Then the base change map $$ g^{-1}Rf_*E \\longrightarrow Rf'_*(g')^{-1}E $$ of Lemma \\ref{lemma-base-change-map-flat-case} is an isomorphism in $D^+(Y')$."}
+{"_id": "2032", "title": "cohomology-theorem-glueing-bbd-general", "text": "\\begin{reference} Special case of \\cite[Theorem 3.2.4]{BBD} without boundedness assumption. \\end{reference} In Situation \\ref{situation-locally-given} assume \\begin{enumerate} \\item $X = \\bigcup_{U \\in \\mathcal{B}} U$, \\item for $U, V \\in \\mathcal{B}$ we have $U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$, \\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$ for $i < 0$. \\end{enumerate} Then there exists an object $K$ of $D(\\mathcal{O}_X)$ and isomorphisms $\\rho_U : K|_U \\to K_U$ in $D(\\mathcal{O}_U)$ for $U \\in \\mathcal{B}$ such that $\\rho^U_V \\circ \\rho_U|_V = \\rho_V$ for all $V \\subset U$ with $U, V \\in \\mathcal{B}$. The pair $(K, \\rho_U)$ is unique up to unique isomorphism."}
+{"_id": "2033", "title": "cohomology-lemma-trivial-torsor", "text": "Let $X$ be a topological space. Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if $\\mathcal{F}(X) \\not = \\emptyset$."}
+{"_id": "2034", "title": "cohomology-lemma-torsors-h1", "text": "Let $X$ be a topological space. Let $\\mathcal{H}$ be an abelian sheaf on $X$. There is a canonical bijection between the set of isomorphism classes of $\\mathcal{H}$-torsors and $H^1(X, \\mathcal{H})$."}
+{"_id": "2035", "title": "cohomology-lemma-h1-extensions", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. There is a canonical bijection $$ \\Ext^1_{\\textit{Mod}(\\mathcal{O}_X)}(\\mathcal{O}_X, \\mathcal{F}) \\longrightarrow H^1(X, \\mathcal{F}) $$ which associates to the extension $$ 0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O}_X \\to 0 $$ the image of $1 \\in \\Gamma(X, \\mathcal{O}_X)$ in $H^1(X, \\mathcal{F})$."}
+{"_id": "2036", "title": "cohomology-lemma-h1-invertible", "text": "Let $(X, \\mathcal{O}_X)$ be a locally ringed space. There is a canonical isomorphism $$ H^1(X, \\mathcal{O}_X^*) = \\Pic(X). $$ of abelian groups."}
+{"_id": "2037", "title": "cohomology-lemma-cohomology-of-open", "text": "Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace. \\begin{enumerate} \\item If $\\mathcal{I}$ is an injective $\\mathcal{O}_X$-module then $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module. \\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have $H^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}|_U)$. \\end{enumerate}"}
+{"_id": "2038", "title": "cohomology-lemma-kill-cohomology-class-on-covering", "text": "Let $X$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $U \\subset X$ be an open subspace. Let $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$. Then there exists an open covering $U = \\bigcup_{i\\in I} U_i$ such that $\\xi|_{U_i} = 0$ for all $i \\in I$."}
+{"_id": "2039", "title": "cohomology-lemma-describe-higher-direct-images", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a $\\mathcal{O}_X$-module. The sheaves $R^if_*\\mathcal{F}$ are the sheaves associated to the presheaves $$ V \\longmapsto H^i(f^{-1}(V), \\mathcal{F}) $$ with restriction mappings as in Equation (\\ref{equation-restriction-mapping}). There is a similar statement for $R^if_*$ applied to a bounded below complex $\\mathcal{F}^\\bullet$."}
+{"_id": "2040", "title": "cohomology-lemma-localize-higher-direct-images", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $V \\subset Y$ be an open subspace. Denote $g : f^{-1}(V) \\to V$ the restriction of $f$. Then we have $$ R^pg_*(\\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\\mathcal{F})|_V $$ There is a similar statement for the derived image $Rf_*\\mathcal{F}^\\bullet$ where $\\mathcal{F}^\\bullet$ is a bounded below complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2041", "title": "cohomology-lemma-injective-restriction-surjective", "text": "\\begin{slogan} Local sections in injective sheaves can be extended globally. \\end{slogan} Let $X$ be a ringed space. Let $U' \\subset U \\subset X$ be open subspaces. For any injective $\\mathcal{O}_X$-module $\\mathcal{I}$ the restriction mapping $\\mathcal{I}(U) \\to \\mathcal{I}(U')$ is surjective."}
+{"_id": "2042", "title": "cohomology-lemma-mayer-vietoris", "text": "Let $X$ be a ringed space. Suppose that $X = U \\cup V$ is a union of two open subsets. For every $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists a long exact cohomology sequence $$ 0 \\to H^0(X, \\mathcal{F}) \\to H^0(U, \\mathcal{F}) \\oplus H^0(V, \\mathcal{F}) \\to H^0(U \\cap V, \\mathcal{F}) \\to H^1(X, \\mathcal{F}) \\to \\ldots $$ This long exact sequence is functorial in $\\mathcal{F}$."}
+{"_id": "2043", "title": "cohomology-lemma-relative-mayer-vietoris", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Suppose that $X = U \\cup V$ is a union of two open subsets. Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and $c = f|_{U \\cap V} : U \\cap V \\to Y$. For every $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists a long exact sequence $$ 0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to c_*(\\mathcal{F}|_{U \\cap V}) \\to R^1f_*\\mathcal{F} \\to \\ldots $$ This long exact sequence is functorial in $\\mathcal{F}$."}
+{"_id": "2044", "title": "cohomology-lemma-cech-h0", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is an abelian sheaf and \\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ the natural map $$ \\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F}) $$ is bijective. \\end{enumerate}"}
+{"_id": "2045", "title": "cohomology-lemma-cech-trivial", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If $U_i = U$ for some $i \\in I$, then the extended {\\v C}ech complex $$ \\mathcal{F}(U) \\to \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ obtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by the canonical map of $\\mathcal{F}(U)$ into $\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$ is homotopy equivalent to $0$."}
+{"_id": "2046", "title": "cohomology-lemma-cech-exact-presheaves", "text": "The functor given by Equation (\\ref{equation-cech-functor}) is an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor})."}
+{"_id": "2047", "title": "cohomology-lemma-cech-cohomology-delta-functor-presheaves", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. The functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$ form a $\\delta$-functor from the abelian category of presheaves of $\\mathcal{O}_X$-modules to the category of $\\mathcal{O}_X(U)$-modules (see Homology, Definition \\ref{homology-definition-cohomological-delta-functor})."}
+{"_id": "2048", "title": "cohomology-lemma-cech-map-into", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. Denote $j_{i_0\\ldots i_p} : U_{i_0 \\ldots i_p} \\to X$ the open immersion. Consider the chain complex $K(\\mathcal{U})_\\bullet$ of presheaves of $\\mathcal{O}_X$-modules $$ \\ldots \\to \\bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\\mathcal{O}_{U_{i_0i_1i_2}} \\to \\bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\\mathcal{O}_{U_{i_0i_1}} \\to \\bigoplus_{i_0} (j_{i_0})_{p!}\\mathcal{O}_{U_{i_0}} \\to 0 \\to \\ldots $$ where the last nonzero term is placed in degree $0$ and where the map $$ (j_{i_0\\ldots i_{p + 1}})_{p!}\\mathcal{O}_{U_{i_0\\ldots i_{p + 1}}} \\longrightarrow (j_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}})_{p!} \\mathcal{O}_{U_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}}} $$ is given by $(-1)^j$ times the canonical map. Then there is an isomorphism $$ \\Hom_{\\mathcal{O}_X}(K(\\mathcal{U})_\\bullet, \\mathcal{F}) = \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ functorial in $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}_X))$."}
+{"_id": "2049", "title": "cohomology-lemma-homology-complex", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. Let $\\mathcal{O}_\\mathcal{U} \\subset \\mathcal{O}_X$ be the image presheaf of the map $\\bigoplus j_{p!}\\mathcal{O}_{U_i} \\to \\mathcal{O}_X$. The chain complex $K(\\mathcal{U})_\\bullet$ of presheaves of Lemma \\ref{lemma-cech-map-into} above has homology presheaves $$ H_i(K(\\mathcal{U})_\\bullet) = \\left\\{ \\begin{matrix} 0 & \\text{if} & i \\not = 0 \\\\ \\mathcal{O}_\\mathcal{U} & \\text{if} & i = 0 \\end{matrix} \\right. $$"}
+{"_id": "2050", "title": "cohomology-lemma-cech-cohomology-derived-presheaves", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering of $U \\subset X$. The {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$ are canonically isomorphic as a $\\delta$-functor to the right derived functors of the functor $$ \\check{H}^0(\\mathcal{U}, -) : \\textit{PMod}(\\mathcal{O}_X) \\longrightarrow \\text{Mod}_{\\mathcal{O}_X(U)}. $$ Moreover, there is a functorial quasi-isomorphism $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\longrightarrow R\\check{H}^0(\\mathcal{U}, \\mathcal{F}) $$ where the right hand side indicates the right derived functor $$ R\\check{H}^0(\\mathcal{U}, -) : D^{+}(\\textit{PMod}(\\mathcal{O}_X)) \\longrightarrow D^{+}(\\mathcal{O}_X(U)) $$ of the left exact functor $\\check{H}^0(\\mathcal{U}, -)$."}
+{"_id": "2051", "title": "cohomology-lemma-injective-trivial-cech", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module. Then $$ \\check{H}^p(\\mathcal{U}, \\mathcal{I}) = \\left\\{ \\begin{matrix} \\mathcal{I}(U) & \\text{if} & p = 0 \\\\ 0 & \\text{if} & p > 0 \\end{matrix} \\right. $$"}
+{"_id": "2052", "title": "cohomology-lemma-cech-cohomology", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. There is a transformation $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -) \\longrightarrow R\\Gamma(U, -) $$ of functors $\\textit{Mod}(\\mathcal{O}_X) \\to D^{+}(\\mathcal{O}_X(U))$. In particular this provides canonical maps $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for $\\mathcal{F}$ ranging over $\\textit{Mod}(\\mathcal{O}_X)$."}
+{"_id": "2053", "title": "cohomology-lemma-cech-h1", "text": "Let $X$ be a topological space. Let $\\mathcal{H}$ be an abelian sheaf on $X$. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering. The map $$ \\check{H}^1(\\mathcal{U}, \\mathcal{H}) \\longrightarrow H^1(X, \\mathcal{H}) $$ is injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{H})$ via the bijection of Lemma \\ref{lemma-torsors-h1} with the set of isomorphism classes of $\\mathcal{H}$-torsors which restrict to trivial torsors over each $U_i$."}
+{"_id": "2054", "title": "cohomology-lemma-include", "text": "Let $X$ be a ringed space. Consider the functor $i : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)$. It is a left exact functor with right derived functors given by $$ R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) : U \\longmapsto H^p(U, \\mathcal{F}) $$ see discussion in Section \\ref{section-locality}."}
+{"_id": "2055", "title": "cohomology-lemma-cech-spectral-sequence", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F})) $$ converging to $H^{p + q}(U, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$."}
+{"_id": "2056", "title": "cohomology-lemma-cech-spectral-sequence-application", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be a covering. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume that $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}) = 0$ for all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$. Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$ as $\\mathcal{O}_X(U)$-modules."}
+{"_id": "2057", "title": "cohomology-lemma-ses-cech-h1", "text": "Let $X$ be a ringed space. Let $$ 0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0 $$ be a short exact sequence of $\\mathcal{O}_X$-modules. Let $U \\subset X$ be an open subset. If there exists a cofinal system of open coverings $\\mathcal{U}$ of $U$ such that $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$, then the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is surjective."}
+{"_id": "2058", "title": "cohomology-lemma-cech-vanish", "text": "\\begin{slogan} If higher {\\v C}ech cohomology of an abelian sheaf vanishes for all open covers, then higher cohomology vanishes. \\end{slogan} Let $X$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module such that $$ \\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0 $$ for all $p > 0$ and any open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ of an open of $X$. Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any open $U \\subset X$."}
+{"_id": "2059", "title": "cohomology-lemma-cech-vanish-basis", "text": "(Variant of Lemma \\ref{lemma-cech-vanish}.) Let $X$ be a ringed space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume there exists a set of open coverings $\\text{Cov}$ with the following properties: \\begin{enumerate} \\item For every $\\mathcal{U} \\in \\text{Cov}$ with $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ we have $U, U_i \\in \\mathcal{B}$ and every $U_{i_0 \\ldots i_p} \\in \\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ the open coverings of $U$ occurring in $\\text{Cov}$ is a cofinal system of open coverings of $U$. \\item For every $\\mathcal{U} \\in \\text{Cov}$ we have $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$. \\end{enumerate} Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$."}
+{"_id": "2060", "title": "cohomology-lemma-pushforward-injective", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item $\\check{H}^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$ for all $p > 0$ and any open covering $\\mathcal{V} : V = \\bigcup_{j \\in J} V_j$ of $Y$. \\item $H^p(V, f_*\\mathcal{I}) = 0$ for all $p > 0$ and every open $V \\subset Y$. \\end{enumerate} In other words, $f_*\\mathcal{I}$ is right acyclic for $\\Gamma(V, -)$ (see Derived Categories, Definition \\ref{derived-definition-derived-functor}) for any $V \\subset Y$ open."}
+{"_id": "2061", "title": "cohomology-lemma-pushforward-injective-flat", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Assume $f$ is flat. Then $f_*\\mathcal{I}$ is an injective $\\mathcal{O}_Y$-module for any injective $\\mathcal{O}_X$-module $\\mathcal{I}$."}
+{"_id": "2062", "title": "cohomology-lemma-cohomology-products", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $I$ be a set. For $i \\in I$ let  $\\mathcal{F}_i$ be an $\\mathcal{O}_X$-module. Let $U \\subset X$ be open. The canonical map $$ H^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i) \\longrightarrow \\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i) $$ is an isomorphism for $p = 0$ and injective for $p = 1$."}
+{"_id": "2063", "title": "cohomology-lemma-injective-flasque", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Then any injective $\\mathcal{O}_X$-module is flasque."}
+{"_id": "2064", "title": "cohomology-lemma-flasque-acyclic", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any flasque $\\mathcal{O}_X$-module is acyclic for $R\\Gamma(X, -)$ as well as $R\\Gamma(U, -)$ for any open $U$ of $X$."}
+{"_id": "2065", "title": "cohomology-lemma-flasque-acyclic-cech", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $\\mathcal{U} : U = \\bigcup U_i$ be an open covering. If $\\mathcal{F}$ is flasque, then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for $p > 0$."}
+{"_id": "2066", "title": "cohomology-lemma-flasque-acyclic-pushforward", "text": "Let $(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is flasque, then $R^pf_*\\mathcal{F} = 0$ for $p > 0$."}
+{"_id": "2067", "title": "cohomology-lemma-vanishing-ravi", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Assume the restriction mappings $\\mathcal{F}(U) \\to \\mathcal{F}(U')$ are surjective for $U'$ an arbitrary union of opens of the form $U_{i_0 \\ldots i_p}$. Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F})$ vanishes for $p > 0$."}
+{"_id": "2068", "title": "cohomology-lemma-before-Leray", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. There is a commutative diagram $$ \\xymatrix{ D^{+}(X) \\ar[rr]_-{R\\Gamma(X, -)} \\ar[d]_{Rf_*} & & D^{+}(\\mathcal{O}_X(X)) \\ar[d]^{\\text{restriction}} \\\\ D^{+}(Y) \\ar[rr]^-{R\\Gamma(Y, -)} & & D^{+}(\\mathcal{O}_Y(Y)) } $$ More generally for any $V \\subset Y$ open and $U = f^{-1}(V)$ there is a commutative diagram $$ \\xymatrix{ D^{+}(X) \\ar[rr]_-{R\\Gamma(U, -)} \\ar[d]_{Rf_*} & & D^{+}(\\mathcal{O}_X(U)) \\ar[d]^{\\text{restriction}} \\\\ D^{+}(Y) \\ar[rr]^-{R\\Gamma(V, -)} & & D^{+}(\\mathcal{O}_Y(V)) } $$ See also Remark \\ref{remark-elucidate-lemma} for more explanation."}
+{"_id": "2069", "title": "cohomology-lemma-modules-abelian", "text": "Let $X$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. \\begin{enumerate} \\item The cohomology groups $H^i(U, \\mathcal{F})$ for $U \\subset X$ open of $\\mathcal{F}$ computed as an $\\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \\item Let $f : X \\to Y$ be a morphism of ringed spaces. The higher direct images $R^if_*\\mathcal{F}$ of $\\mathcal{F}$ computed as an $\\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \\end{enumerate} There are similar statements in the case of bounded below complexes of $\\mathcal{O}_X$-modules."}
+{"_id": "2070", "title": "cohomology-lemma-Leray", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_X$-modules. There is a spectral sequence $$ E_2^{p, q} = H^p(Y, R^qf_*(\\mathcal{F}^\\bullet)) $$ converging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$."}
+{"_id": "2071", "title": "cohomology-lemma-apply-Leray", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then $H^p(X, \\mathcal{F}) = H^p(Y, f_*\\mathcal{F})$ for all $p$. \\item If $H^p(Y, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^q(X, \\mathcal{F}) = H^0(Y, R^qf_*\\mathcal{F})$ for all $q$. \\end{enumerate}"}
+{"_id": "2072", "title": "cohomology-lemma-higher-direct-images-compose", "text": "\\begin{slogan} The total derived functor of a composition is the composition of the total derived functors. \\end{slogan} Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. In this case $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors from $D^{+}(X) \\to D^{+}(Z)$."}
+{"_id": "2073", "title": "cohomology-lemma-relative-Leray", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. There is a spectral sequence with $$ E_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F}) $$ converging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$. This spectral sequence is functorial in $\\mathcal{F}$, and there is a version for bounded below complexes of $\\mathcal{O}_X$-modules."}
+{"_id": "2074", "title": "cohomology-lemma-functoriality", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{G}^\\bullet$, resp.\\ $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_Y$-modules, resp.\\ $\\mathcal{O}_X$-modules. Let $\\varphi : \\mathcal{G}^\\bullet \\to f_*\\mathcal{F}^\\bullet$ be a morphism of complexes. There is a canonical morphism $$ \\mathcal{G}^\\bullet \\longrightarrow Rf_*(\\mathcal{F}^\\bullet) $$ in $D^{+}(Y)$. Moreover this construction is functorial in the triple $(\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet, \\varphi)$."}
+{"_id": "2075", "title": "cohomology-lemma-functoriality-cech", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\varphi : f^*\\mathcal{G} \\to \\mathcal{F}$ be an $f$-map from an $\\mathcal{O}_Y$-module $\\mathcal{G}$ to an $\\mathcal{O}_X$-module $\\mathcal{F}$. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and $\\mathcal{V} : Y = \\bigcup_{j \\in J} V_j$ be open coverings. Assume that $\\mathcal{U}$ is a refinement of $f^{-1}\\mathcal{V} : X = \\bigcup_{j \\in J} f^{-1}(V_j)$. In this case there exists a commutative diagram $$ \\xymatrix{ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] & R\\Gamma(X, \\mathcal{F}) \\\\ \\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}) \\ar[r] \\ar[u]^\\gamma & R\\Gamma(Y, \\mathcal{G}) \\ar[u] } $$ in $D^{+}(\\mathcal{O}_X(X))$ with horizontal arrows given by Lemma \\ref{lemma-cech-cohomology} and right vertical arrow by (\\ref{equation-functorial-derived}). In particular we get commutative diagrams of cohomology groups $$ \\xymatrix{ \\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\ar[r] & H^p(X, \\mathcal{F}) \\\\ \\check{H}^p(\\mathcal{V}, \\mathcal{G}) \\ar[r] \\ar[u]^\\gamma & H^p(Y, \\mathcal{G}) \\ar[u] } $$ where the right vertical arrow is (\\ref{equation-functorial})"}
+{"_id": "2076", "title": "cohomology-lemma-cech-always", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf. Then the map $\\check{H}^1(X, \\mathcal{F}) \\to H^1(X, \\mathcal{F})$ defined in (\\ref{equation-cech-to-cohomology}) is an isomorphism."}
+{"_id": "2077", "title": "cohomology-lemma-cech-Hausdorff-quasi-compact", "text": "Let $X$ be a Hausdorff and quasi-compact topological space. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Then the map $\\check{H}^n(X, \\mathcal{F}) \\to H^n(X, \\mathcal{F})$ defined in (\\ref{equation-cech-to-cohomology}) is an isomorphism for all $n$."}
+{"_id": "2078", "title": "cohomology-lemma-cohomology-of-closed", "text": "\\begin{reference} \\cite[Expose V bis, 4.1.3]{SGA4} \\end{reference} Let $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. For every abelian sheaf $\\mathcal{F}$ on $X$ the canonical map $$ \\colim H^p(U, \\mathcal{F}) \\longrightarrow H^p(Z, \\mathcal{F}|_Z) $$ where the colimit is over open neighbourhoods $U$ of $Z$ in $X$ is an isomorphism."}
+{"_id": "2079", "title": "cohomology-lemma-base-change-map-flat-case", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a commutative diagram of ringed spaces. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_X$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map $$ g^*Rf_*\\mathcal{F}^\\bullet \\longrightarrow R(f')_*(g')^*\\mathcal{F}^\\bullet $$ in $D^{+}(S')$."}
+{"_id": "2080", "title": "cohomology-lemma-proper-base-change", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $y \\in Y$. Assume that \\begin{enumerate} \\item $f$ is closed, \\item $f$ is separated, and \\item $f^{-1}(y)$ is quasi-compact. \\end{enumerate} Then for $E$ in $D^+(\\mathcal{O}_X)$ we have $(Rf_*E)_y = R\\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$ in $D^+(\\mathcal{O}_{Y, y})$."}
+{"_id": "2081", "title": "cohomology-lemma-proper-base-change-sheaves-of-sets", "text": "Consider a cartesian square of topological spaces $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ Assume that $f$ is proper and separated. Then $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$ for any sheaf of sets $\\mathcal{F}$ on $X$."}
+{"_id": "2082", "title": "cohomology-lemma-quasi-separated-cohomology-colimit", "text": "Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties: \\begin{enumerate} \\item there exists a basis of quasi-compact open subsets, and \\item the intersection of any two quasi-compact opens is quasi-compact. \\end{enumerate} Then for any directed system $(\\mathcal{F}_i, \\varphi_{ii'})$ of sheaves of $\\mathcal{O}_X$-modules and for any quasi-compact open $U \\subset X$ the canonical map $$ \\colim_i H^q(U, \\mathcal{F}_i) \\longrightarrow H^q(U, \\colim_i \\mathcal{F}_i) $$ is an isomorphism for every $q \\geq 0$."}
+{"_id": "2083", "title": "cohomology-lemma-colimit", "text": "In the situation discussed above. Let $i \\in \\Ob(\\mathcal{I})$ and let $U_i \\subset X_i$ be quasi-compact open. Then $$ \\colim_{a : j \\to i} H^p(f_a^{-1}(U_i), \\mathcal{F}_j) = H^p(p_i^{-1}(U_i), \\mathcal{F}) $$ for all $p \\geq 0$. In particular we have $H^p(X, \\mathcal{F}) = \\colim H^p(X_i, \\mathcal{F}_i)$."}
+{"_id": "2084", "title": "cohomology-lemma-cohomology-and-closed-immersions", "text": "Let $i : Z \\to X$ be a closed immersion of topological spaces. For any abelian sheaf $\\mathcal{F}$ on $Z$ we have $H^p(Z, \\mathcal{F}) = H^p(X, i_*\\mathcal{F})$."}
+{"_id": "2085", "title": "cohomology-lemma-irreducible-constant-cohomology-zero", "text": "Let $X$ be an irreducible topological space. Then $H^p(X, \\underline{A}) = 0$ for all $p > 0$ and any abelian group $A$."}
+{"_id": "2086", "title": "cohomology-lemma-subsheaf-of-constant-sheaf", "text": "\\begin{reference} \\cite[Page 168]{Tohoku}. \\end{reference} Let $X$ be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let $\\mathcal{F} \\subset \\underline{\\mathbf{Z}}$ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration $$ (0) = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset \\mathcal{F}_n = \\mathcal{F} $$ by abelian subsheaves such that for each $0 < i \\leq n$ there exists a short exact sequence $$ 0 \\to j'_!\\underline{\\mathbf{Z}}_V \\to j_!\\underline{\\mathbf{Z}}_U \\to \\mathcal{F}_i/\\mathcal{F}_{i - 1} \\to 0 $$ with $j : U \\to X$ and $j' : V \\to X$ the inclusion of quasi-compact opens into $X$."}
+{"_id": "2090", "title": "cohomology-lemma-cohomology-of-neighbourhoods-of-closed", "text": "Let $X$ be a spectral space. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Let $E \\subset X$ be a quasi-compact subset. Let $W \\subset X$ be the set of points of $X$ which specialize to a point of $E$. \\begin{enumerate} \\item $H^p(W, \\mathcal{F}|_W) = \\colim H^p(U, \\mathcal{F})$ where the colimit is over quasi-compact open neighbourhoods of $E$, \\item $H^p(W \\setminus E, \\mathcal{F}|_{W \\setminus E}) = \\colim H^p(U \\setminus E, \\mathcal{F}|_{U \\setminus E})$ if $E$ is a constructible subset. \\end{enumerate}"}
+{"_id": "2092", "title": "cohomology-lemma-vanishing-for-profinite", "text": "Let $X$ be a profinite topological space. Then $H^q(X, \\mathcal{F}) = 0$ for all $q > 0$ and all abelian sheaves $\\mathcal{F}$."}
+{"_id": "2093", "title": "cohomology-lemma-ordered-alternating", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c$ is a morphism of complexes. In fact it induces an isomorphism $$ c : \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ of complexes."}
+{"_id": "2094", "title": "cohomology-lemma-project-to-ordered", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $\\pi : \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is a morphism of complexes. It induces an isomorphism $$ \\pi : \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ of complexes which is a left inverse to the morphism $c$."}
+{"_id": "2095", "title": "cohomology-lemma-alternating-usual", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c \\circ \\pi$ is homotopic to the identity on $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$. In particular the inclusion map $\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is a homotopy equivalence."}
+{"_id": "2096", "title": "cohomology-lemma-alternating-cech-trivial", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If $U_i = U$ for some $i \\in I$, then the extended alternating {\\v C}ech complex $$ \\mathcal{F}(U) \\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ obtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by the canonical map of $\\mathcal{F}(U)$ into $\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$ is homotopy equivalent to $0$. Similarly, for any total ordering on $I$ the extended ordered {\\v C}ech complex $$ \\mathcal{F}(U) \\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ is homotopy equivalent to $0$."}
+{"_id": "2097", "title": "cohomology-lemma-covering-resolution", "text": "Let $X$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering of $X$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Denote $\\mathcal{F}_{i_0 \\ldots i_p}$ the restriction of $\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$. There exists a complex ${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$ of $\\mathcal{O}_X$-modules with $$ {\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) = \\prod\\nolimits_{i_0 \\ldots i_p} (j_{i_0 \\ldots i_p})_* \\mathcal{F}_{i_0 \\ldots i_p} $$ and differential $d : {\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) \\to {\\mathfrak C}^{p + 1}(\\mathcal{U}, \\mathcal{F})$ as in Equation (\\ref{equation-d-cech}). Moreover, there exists a canonical map $$ \\mathcal{F} \\to {\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ which is a quasi-isomorphism, i.e., ${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is a resolution of $\\mathcal{F}$."}
+{"_id": "2098", "title": "cohomology-lemma-cech-complex-complex", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering. For a bounded below complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical map $$ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)) \\longrightarrow R\\Gamma(X, \\mathcal{F}^\\bullet) $$ functorial in $\\mathcal{F}^\\bullet$ and compatible with (\\ref{equation-global-sections-to-cech}) and (\\ref{equation-transformation}). There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_2^{p, q} = H^p(\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}^\\bullet))) $$ converging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$."}
+{"_id": "2099", "title": "cohomology-lemma-cech-complex-complex-computes", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_X$-modules. If $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}^q) = 0$ for all $i > 0$ and all $p, i_0, \\ldots, i_p, q$, then the map $ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)) \\to R\\Gamma(X, \\mathcal{F}^\\bullet) $ of Lemma \\ref{lemma-cech-complex-complex} is an isomorphism."}
+{"_id": "2100", "title": "cohomology-lemma-compute-sign-cup-product-boundaries", "text": "In the situation above, assume {\\v C}ech cohomology agrees with cohomology for the sheaves $\\mathcal{F}_i^p$ and $\\mathcal{G}_j^q$. Let $a_3 \\in H^n(X, \\mathcal{F}_3^\\bullet)$ and $b_1 \\in H^m(X, \\mathcal{G}_1^\\bullet)$. Then we have $$ \\gamma_1( \\partial a_3 \\cup b_1) = (-1)^{n + 1} \\gamma_3( a_3 \\cup \\partial b_1) $$ in $H^{n + m}(X, \\mathcal{H}^\\bullet)$ where $\\partial$ indicates the boundary map on cohomology associated to the short exact sequences of complexes above."}
+{"_id": "2101", "title": "cohomology-lemma-boundary-derivation-over-cup-product", "text": "Let $X$ be a topological space. Let $\\mathcal{O}' \\to \\mathcal{O}$ be a surjection of sheaves of rings whose kernel $\\mathcal{I} \\subset \\mathcal{O}'$ has square zero. Then $M = H^1(X, \\mathcal{I})$ is a $R = H^0(X, \\mathcal{O})$-module and the boundary map $\\partial : R \\to M$ associated to the short exact sequence $$ 0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0 $$ is a derivation (Algebra, Definition \\ref{algebra-definition-derivation})."}
+{"_id": "2102", "title": "cohomology-lemma-derived-tor-exact", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. The functors $$ K(\\textit{Mod}(\\mathcal{O}_X)) \\longrightarrow K(\\textit{Mod}(\\mathcal{O}_X)), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet) $$ and $$ K(\\textit{Mod}(\\mathcal{O}_X)) \\longrightarrow K(\\textit{Mod}(\\mathcal{O}_X)), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet) $$ are exact functors of triangulated categories."}
+{"_id": "2103", "title": "cohomology-lemma-K-flat-quasi-isomorphism", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}^\\bullet$ be a K-flat complex. Then the functor $$ K(\\textit{Mod}(\\mathcal{O}_X)) \\longrightarrow K(\\textit{Mod}(\\mathcal{O}_X)), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms."}
+{"_id": "2104", "title": "cohomology-lemma-check-K-flat-stalks", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. Then $\\mathcal{K}^\\bullet$ is K-flat if and only if for all $x \\in X$ the complex $\\mathcal{K}_x^\\bullet$ of $\\mathcal{O}_{X, x}$-modules is K-flat (More on Algebra, Definition \\ref{more-algebra-definition-K-flat})."}
+{"_id": "2105", "title": "cohomology-lemma-tensor-product-K-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. If $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes of $\\mathcal{O}_X$-modules, then $\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)$ is a K-flat complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2106", "title": "cohomology-lemma-K-flat-two-out-of-three", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$ be a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}_X))$. If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "2108", "title": "cohomology-lemma-pullback-K-flat", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback of a K-flat complex of $\\mathcal{O}_Y$-modules is a K-flat complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2109", "title": "cohomology-lemma-bounded-flat-K-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. A bounded above complex of flat $\\mathcal{O}_X$-modules is K-flat."}
+{"_id": "2110", "title": "cohomology-lemma-colimit-K-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$ be a system of K-flat complexes. Then $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat."}
+{"_id": "2111", "title": "cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}_X$-modules there exists a commutative diagram of complexes of $\\mathcal{O}_X$-modules $$ \\xymatrix{ \\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] & \\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\ \\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] & \\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots } $$ with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms are direct sums of $\\mathcal{O}_X$-modules of the form $j_{U!}\\mathcal{O}_U$, and (3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are termwise split injections whose cokernels are direct sums of $\\mathcal{O}_X$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map $\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism."}
+{"_id": "2112", "title": "cohomology-lemma-K-flat-resolution", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For any complex $\\mathcal{G}^\\bullet$ there exists a $K$-flat complex $\\mathcal{K}^\\bullet$ whose terms are flat $\\mathcal{O}_X$-modules and a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$ which is termwise surjective."}
+{"_id": "2113", "title": "cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a quasi-isomorphism of K-flat complexes of $\\mathcal{O}_X$-modules. For every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules the induced map $$ \\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) : \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{P}^\\bullet) \\longrightarrow \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{Q}^\\bullet) $$ is a quasi-isomorphism."}
+{"_id": "2114", "title": "cohomology-lemma-flat-tor-zero", "text": "\\begin{slogan} Tor measures the deviation of flatness. \\end{slogan} Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module, and \\item $\\text{Tor}_1^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) = 0$ for every $\\mathcal{O}_X$-module $\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "2115", "title": "cohomology-lemma-factor-through-K-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes of $\\mathcal{O}_X$-modules. If $\\mathcal{K}^\\bullet$ is K-flat, then there exist a complex $\\mathcal{N}^\\bullet$ and maps of complexes $b : \\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet$ and $c : \\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ such that \\begin{enumerate} \\item $\\mathcal{N}^\\bullet$ is K-flat, \\item $c$ is a quasi-isomorphism, \\item $a$ is homotopic to $c \\circ b$. \\end{enumerate} If the terms of $\\mathcal{K}^\\bullet$ are flat, then we may choose $\\mathcal{N}^\\bullet$, $b$, and $c$ such that the same is true for $\\mathcal{N}^\\bullet$."}
+{"_id": "2116", "title": "cohomology-lemma-derived-base-change", "text": "The construction above is independent of choices and defines an exact functor of triangulated categories $Lf^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$."}
+{"_id": "2117", "title": "cohomology-lemma-derived-pullback-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. Then $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors $D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$."}
+{"_id": "2118", "title": "cohomology-lemma-pullback-tensor-product", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism $$ Lf^*( \\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y}^{\\mathbf{L}} \\mathcal{G}^\\bullet ) = Lf^*\\mathcal{F}^\\bullet  \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} Lf^*\\mathcal{G}^\\bullet  $$ for $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(X))$."}
+{"_id": "2119", "title": "cohomology-lemma-variant-derived-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism $$ \\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} Lf^*\\mathcal{G}^\\bullet = \\mathcal{F}^\\bullet  \\otimes_{f^{-1}\\mathcal{O}_Y}^{\\mathbf{L}} f^{-1}\\mathcal{G}^\\bullet  $$ for $\\mathcal{F}^\\bullet$ in $D(X)$ and $\\mathcal{G}^\\bullet$ in $D(Y)$."}
+{"_id": "2120", "title": "cohomology-lemma-tensor-pull-compatibility", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$ be complexes of $\\mathcal{O}_Y$-modules. The diagram $$ \\xymatrix{ Lf^*(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} \\mathcal{M}^\\bullet) \\ar[r] \\ar[d] & Lf^*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{M}^\\bullet) \\ar[d] \\\\ Lf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{M}^\\bullet \\ar[d] & f^*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y} \\mathcal{M}^\\bullet) \\ar[d] \\\\ f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^*\\mathcal{M}^\\bullet \\ar[r] & \\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{M}^\\bullet) } $$ commutes."}
+{"_id": "2121", "title": "cohomology-lemma-adjoint", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma \\ref{lemma-derived-base-change} are adjoint: $$ \\Hom_{D(X)}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet) = \\Hom_{D(Y)}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet) $$ bifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(X))$ and $\\mathcal{G}^\\bullet \\in \\Ob(D(Y))$."}
+{"_id": "2122", "title": "cohomology-lemma-derived-pushforward-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. Then $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_Z)$."}
+{"_id": "2123", "title": "cohomology-lemma-adjoints-push-pull-compatibility", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. The diagram $$ \\xymatrix{ Lf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] & f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\ Lf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] & \\mathcal{K}^\\bullet } $$ coming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes, and adjunction $Lf^* \\circ Rf_* \\to \\text{id}$ commutes in $D(\\mathcal{O}_X)$."}
+{"_id": "2124", "title": "cohomology-lemma-spectral-sequence-filtered-object", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a filtered complex of $\\mathcal{O}_X$-modules. There exists a canonical spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded $\\Gamma(X, \\mathcal{O}_X)$-modules with $d_r$ of bidegree $(r, -r + 1)$ and $$ E_1^{p, q} = H^{p + q}(X, \\text{gr}^p\\mathcal{F}^\\bullet) $$ If for every $n$ we have $$ H^n(X, F^p\\mathcal{F}^\\bullet) = 0\\text{ for }p \\gg 0 \\quad\\text{and}\\quad H^n(X, F^p\\mathcal{F}^\\bullet) = H^n(X, \\mathcal{F}^\\bullet)\\text{ for }p \\ll 0 $$ then the spectral sequence is bounded and converges to $H^*(X, \\mathcal{F}^\\bullet)$."}
+{"_id": "2125", "title": "cohomology-lemma-relative-spectral-sequence-filtered-object", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}^\\bullet$ be a filtered complex of $\\mathcal{O}_X$-modules. There exists a canonical spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded $\\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and $$ E_1^{p, q} = R^{p + q}f_*\\text{gr}^p\\mathcal{F}^\\bullet $$ If for every $n$ we have $$ R^nf_*F^p\\mathcal{F}^\\bullet = 0 \\text{ for }p \\gg 0 \\quad\\text{and}\\quad R^nf_*F^p\\mathcal{F}^\\bullet = R^nf_*\\mathcal{F}^\\bullet \\text{ for }p \\ll 0 $$ then the spectral sequence is bounded and converges to $Rf_*\\mathcal{F}^\\bullet$."}
+{"_id": "2126", "title": "cohomology-lemma-godement-resolution", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution $$ 0 \\to \\mathcal{F} \\to f_*f^*\\mathcal{F} \\to f_*f^*f_*f^*\\mathcal{F} \\to f_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots $$ functorial in $\\mathcal{F}$ such that each term $f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque $\\mathcal{O}_X$-module and such that for all $x \\in X$ the map $$ \\mathcal{F}_x[0] \\to \\Big( (f_*f^*\\mathcal{F})_x \\to (f_*f^*f_*f^*\\mathcal{F})_x \\to (f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots \\Big) $$ is a homotopy equivalence in the category of complexes of $\\mathcal{O}_{X, x}$-modules."}
+{"_id": "2127", "title": "cohomology-lemma-godement-resolution-bounded-below", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_X$-modules. There exists a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ where $\\mathcal{F}^\\bullet$ be a bounded below complex of flasque $\\mathcal{O}_X$-modules and for all $x \\in X$ the map $\\mathcal{F}^\\bullet_x \\to \\mathcal{G}^\\bullet_x$ is a homotopy equivalence in the category of complexes of $\\mathcal{O}_{X, x}$-modules."}
+{"_id": "2128", "title": "cohomology-lemma-second-cup-equals-first", "text": "This construction gives the cup product."}
+{"_id": "2129", "title": "cohomology-lemma-cup-compatible-with-naive", "text": "In the situation above the following diagram commutes $$ \\xymatrix{ f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} f_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d] & Rf_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\ \\text{Tot}( f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_Y} f_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} & Rf_*(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{M}^\\bullet) \\ar[d] \\\\ f_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet) \\ar[r] & Rf_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet) } $$"}
+{"_id": "2130", "title": "cohomology-lemma-diagrams-commute", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$ be bounded below complexes of $\\mathcal{O}_X$-modules. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering Then $$ \\xymatrix{ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet)) \\otimes_A^\\mathbf{L} \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet)) \\ar[d] \\ar[r] & R\\Gamma(X, \\mathcal{K}^\\bullet) \\otimes_A^\\mathbf{L} R\\Gamma(X, \\mathcal{M}^\\bullet) \\ar[d]^\\mu \\\\ \\text{Tot}( \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet)) \\otimes_A \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet))) \\ar[d]^{(\\ref{equation-needs-signs})} & R\\Gamma(X, \\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{M}^\\bullet) \\ar[d] \\\\ \\text{Tot}( \\check{\\mathcal{C}}^\\bullet({\\mathcal U}, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet) )) \\ar[r] & R\\Gamma(X, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)) } $$ where the horizontal arrows are the ones in Lemma \\ref{lemma-cech-complex-complex} commutes in $D(A)$."}
+{"_id": "2131", "title": "cohomology-lemma-cup-product-associative", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The relative cup product of Remark \\ref{remark-cup-product} is associative in the sense that the diagram $$ \\xymatrix{ Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[r] \\ar[d] & Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[d] \\\\ Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[r] & Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L}  L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) } $$ is commutative in $D(\\mathcal{O}_Y)$ for all $K, L, M$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2132", "title": "cohomology-lemma-cup-product-commutative", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The relative cup product of Remark \\ref{remark-cup-product} is commutative in the sense that the diagram $$ \\xymatrix{ Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi & Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\ Rf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*K \\ar[r] & Rf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K) } $$ is commutative in $D(\\mathcal{O}_Y)$ for all $K, L$ in $D(\\mathcal{O}_X)$. Here $\\psi$ is the commutativity constraint on the derived category (Lemma \\ref{lemma-symmetric-monoidal-derived})."}
+{"_id": "2134", "title": "cohomology-lemma-restrict-K-injective-to-open", "text": "Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace. The restriction of a K-injective complex of $\\mathcal{O}_X$-modules to $U$ is a K-injective complex of $\\mathcal{O}_U$-modules."}
+{"_id": "2135", "title": "cohomology-lemma-unbounded-cohomology-of-open", "text": "Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace. For $K$ in $D(\\mathcal{O}_X)$ we have $H^p(U, K) = H^p(U, K|_U)$."}
+{"_id": "2136", "title": "cohomology-lemma-sheafification-cohomology", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$. The sheafification of $$ U \\mapsto H^q(U, K) = H^q(U, K|_U) $$ is the $q$th cohomology sheaf $H^q(K)$ of $K$."}
+{"_id": "2137", "title": "cohomology-lemma-restrict-direct-image-open", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Given an open subspace $V \\subset Y$, set $U = f^{-1}(V)$ and denote $g : U \\to V$ the induced morphism. Then $(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2138", "title": "cohomology-lemma-Leray-unbounded", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Then $R\\Gamma(Y, -) \\circ Rf_* = R\\Gamma(X, -)$ as functors $D(\\mathcal{O}_X) \\to D(\\Gamma(Y, \\mathcal{O}_Y))$. More generally for $V \\subset Y$ open and $U = f^{-1}(V)$ we have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$."}
+{"_id": "2139", "title": "cohomology-lemma-unbounded-describe-higher-direct-images", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $K$ be in $D(\\mathcal{O}_X)$. Then $H^i(Rf_*K)$ is the sheaf associated to the presheaf $$ V \\mapsto H^i(f^{-1}(V), K) = H^i(V, Rf_*K) $$"}
+{"_id": "2140", "title": "cohomology-lemma-modules-abelian-unbounded", "text": "Let $X$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$ and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$. \\begin{enumerate} \\item For any open $U \\subset X$ there is a canonical map $R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$ which is an isomorphism in $D(\\textit{Ab})$. \\item Let $f : X \\to Y$ be a morphism of ringed spaces. There is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which is an isomorphism in $D(\\underline{\\mathbf{Z}}_Y)$. \\end{enumerate}"}
+{"_id": "2141", "title": "cohomology-lemma-adjoint-lower-shriek-restrict", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $U \\subset X$ be an open subset. Denote $j : (U, \\mathcal{O}_U) \\to (X, \\mathcal{O}_X)$ the corresponding open immersion. The restriction functor $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_U)$ is a right adjoint to extension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O}_X)$."}
+{"_id": "2142", "title": "cohomology-lemma-K-injective-flat", "text": "Let $f : X \\to Y$ be a flat morphism of ringed spaces. If $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}_X$-modules, then $f_*\\mathcal{I}^\\bullet$ is K-injective as a complex of $\\mathcal{O}_Y$-modules."}
+{"_id": "2143", "title": "cohomology-lemma-exact-sequence-lower-shriek", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be the union of two open subspaces. For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ j_{U \\cap V!}E|_{U \\cap V} \\to j_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to  j_{U \\cap V!}E|_{U \\cap V}[1] $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2144", "title": "cohomology-lemma-exact-sequence-j-star", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be the union of two open subspaces. For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ E \\to  Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to Rj_{U \\cap V, *}E|_{U \\cap V} \\to E[1] $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2145", "title": "cohomology-lemma-mayer-vietoris-hom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be the union of two open subspaces of $X$. For objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a Mayer-Vietoris sequence $$ \\xymatrix{ & \\ldots \\ar[r] & \\Ext^{-1}(E_{U \\cap V}, F_{U \\cap V}) \\ar[lld] \\\\ \\Hom(E, F) \\ar[r] & \\Hom(E_U, F_U) \\oplus \\Hom(E_V, F_V) \\ar[r] & \\Hom(E_{U \\cap V}, F_{U \\cap V}) } $$ where the subscripts denote restrictions to the relevant opens and the $\\Hom$'s and $\\Ext$'s are taken in the relevant derived categories."}
+{"_id": "2146", "title": "cohomology-lemma-unbounded-mayer-vietoris", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Suppose that $X = U \\cup V$ is a union of two open subsets. For an object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ R\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to R\\Gamma(U \\cap V, E) \\to R\\Gamma(X, E)[1] $$ and in particular a long exact cohomology sequence $$ \\ldots \\to H^n(X, E) \\to H^n(U, E) \\oplus H^0(V, E) \\to H^n(U \\cap V, E) \\to H^{n + 1}(X, E) \\to \\ldots $$ The construction of the distinguished triangle and the long exact sequence is functorial in $E$."}
+{"_id": "2147", "title": "cohomology-lemma-unbounded-relative-mayer-vietoris", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Suppose that $X = U \\cup V$ is a union of two open subsets. Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and $c = f|_{U \\cap V} : U \\cap V \\to Y$. For every object $E$ of $D(\\mathcal{O}_X)$ there exists a distinguished triangle $$ Rf_*E \\to Ra_*(E|_U) \\oplus Rb_*(E|_V) \\to Rc_*(E|_{U \\cap V}) \\to Rf_*E[1] $$ This triangle is functorial in $E$."}
+{"_id": "2148", "title": "cohomology-lemma-pushforward-restriction", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an open subspace. Let $T \\subset X$ be a closed subset contained in $U$. \\begin{enumerate} \\item If $E$ is an object of $D(\\mathcal{O}_X)$ whose cohomology sheaves are supported on $T$, then $E \\to Rj_*(E|_U)$ is an isomorphism. \\item If $F$ is an object of $D(\\mathcal{O}_U)$ whose cohomology sheaves are supported on $T$, then $j_!F \\to Rj_*F$ is an isomorphism. \\end{enumerate}"}
+{"_id": "2150", "title": "cohomology-lemma-cohomology-with-support-sheaf-on-support", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. \\begin{enumerate} \\item $R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X|_Z)$ is right adjoint to $i_* : D(\\mathcal{O}_X|_Z) \\to D(\\mathcal{O}_X)$. \\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $R\\mathcal{H}_Z(i_*K) = K$. \\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_X|_Z$-modules on $Z$. Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$. \\end{enumerate}"}
+{"_id": "2151", "title": "cohomology-lemma-complexes-with-support-on-closed", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. \\begin{enumerate} \\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $i_*K$ in $D_Z(\\mathcal{O}_X)$. \\item The functor $i_* : D(\\mathcal{O}_X|_Z) \\to D_Z(\\mathcal{O}_X)$ is an equivalence with quasi-inverse $i^{-1}|_{D_Z(\\mathcal{O}_X)} = R\\mathcal{H}_Z|_{D_Z(\\mathcal{O}_X)}$. \\item The functor $i_* \\circ R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D_Z(\\mathcal{O}_X)$ is right adjoint to the inclusion functor $D_Z(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "2152", "title": "cohomology-lemma-sections-with-support-K-injective", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. If $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}_X$-modules, then $\\mathcal{H}_Z(\\mathcal{I}^\\bullet)$ is K-injective complex of $\\mathcal{O}_X|_Z$-modules."}
+{"_id": "2153", "title": "cohomology-lemma-local-to-global-sections-with-support", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. Then $R\\Gamma(Z, - ) \\circ R\\mathcal{H}_Z = R\\Gamma_Z(X, - )$ as functors $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X(X))$."}
+{"_id": "2154", "title": "cohomology-lemma-triangle-sections-with-support", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. Let $U = X \\setminus Z$. There is a distinguished triangle $$ R\\Gamma_Z(X, K) \\to R\\Gamma(X, K) \\to R\\Gamma(U, K) \\to R\\Gamma_Z(X, K)[1] $$ in $D(\\mathcal{O}_X(X))$ functorial for $K$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2155", "title": "cohomology-lemma-triangle-sections-with-support-sheaves", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. Denote $j : U = X \\setminus Z \\to X$ the inclusion of the complement. There is a distinguished triangle $$ i_*R\\mathcal{H}_Z(K) \\to K \\to Rj_*(K|_U) \\to i_*R\\mathcal{H}_Z(K)[1] $$ in $D(\\mathcal{O}_X)$ functorial for $K$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2156", "title": "cohomology-lemma-sections-support-in-closed-disjoint-open", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$ be a closed subset. Let $j : U \\to X$ be the inclusion of an open subset with $U \\cap Z = \\emptyset$. Then $R\\mathcal{H}_Z(Rj_*K) = 0$ for all $K$ in $D(\\mathcal{O}_U)$."}
+{"_id": "2157", "title": "cohomology-lemma-sections-support-abelian-unbounded", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$ be a closed subset. Let $K$ be an object of $D(\\mathcal{O}_X)$ and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$. \\begin{enumerate} \\item There is a canonical map $R\\Gamma_Z(X, K) \\to R\\Gamma_Z(X, K_{ab})$ which is an isomorphism in $D(\\textit{Ab})$. \\item There is a canonical map $R\\mathcal{H}_Z(K) \\to R\\mathcal{H}_Z(K_{ab})$ which is an isomorphism in $D(\\underline{\\mathbf{Z}}_Z)$. \\end{enumerate}"}
+{"_id": "2158", "title": "cohomology-lemma-support-cup-product", "text": "With notation as in Remark \\ref{remark-support-cup-product} the diagram $$ \\xymatrix{ H^i(X, K) \\times H^j_Z(X, M) \\ar[r] \\ar[d] & H^{i + j}_Z(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\ H^i(X, K) \\times H^j(X, M) \\ar[r] & H^{i + j}(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) } $$ commutes where the top horizontal arrow is the cup product of Remark \\ref{remark-support-cup-product-global}."}
+{"_id": "2159", "title": "cohomology-lemma-support-functorial", "text": "With notation and assumptions as in Remark \\ref{remark-support-functorial} the diagram $$ \\xymatrix{ H^p_Z(X, K) \\ar[r] \\ar[d] & H^p_{Z'}(X, Lf^*K) \\ar[d] \\\\ H^p(X, K) \\ar[r] & H^p(X', Lf^*K) } $$ commutes. Here the top horizontal arrow comes from the identifications $H^p_Z(X, K) = H^p(Z, R\\mathcal{H}_Z(K))$ and $H^p_{Z'}(X', Lf^*K) = H^p(Z', R\\mathcal{H}_{Z'}(K'))$, the pullback map $H^p(Z, R\\mathcal{H}_Z(K)) \\to H^p(Z', L(f|_{Z'})^*R\\mathcal{H}_Z(K))$, and the map constructed in Remark \\ref{remark-support-functorial}."}
+{"_id": "2160", "title": "cohomology-lemma-RGamma-commutes-with-Rlim", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For $U \\subset X$ open the functor $R\\Gamma(U, -)$ commutes with $R\\lim$. Moreover, there are short exact sequences $$ 0 \\to R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to \\lim H^m(U, K_n) \\to 0 $$ for any inverse system $(K_n)$ in $D(\\mathcal{O}_X)$ and any $m \\in \\mathbf{Z}$."}
+{"_id": "2161", "title": "cohomology-lemma-Rf-commutes-with-Rlim", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Then $Rf_*$ commutes with $R\\lim$, i.e., $Rf_*$ commutes with derived limits."}
+{"_id": "2162", "title": "cohomology-lemma-inverse-limit-is-derived-limit", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n)$ be an inverse system of $\\mathcal{O}_X$-modules. Let $\\mathcal{B}$ be a set of opens of $X$. Assume \\begin{enumerate} \\item every open of $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$, \\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$ for $U \\in \\mathcal{B}$. \\end{enumerate} Then $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have $H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$."}
+{"_id": "2163", "title": "cohomology-lemma-cohomology-derived-limit-injective", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system in $D(\\mathcal{O}_X)$. Let $x \\in X$ and $m \\in \\mathbf{Z}$. Assume there exist an integer $n(x)$ and a fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$ such that for $U \\in \\mathfrak{U}_x$ \\begin{enumerate} \\item $R^1\\lim H^{m - 1}(U, K_n) = 0$, and \\item $H^m(U, K_n) \\to H^m(U, K_{n(x)})$ is injective for $n \\geq n(x)$. \\end{enumerate} Then the map on stalks $H^m(R\\lim K_n)_x \\to H^m(K_{n(x)})_x$ is injective."}
+{"_id": "2164", "title": "cohomology-lemma-is-limit-per-point", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$. Assume that for every $x \\in X$ there exist a function $p(x, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and a fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$ such that $$ H^p(U, H^{m - p}(E)) = 0 \\text{ for } U \\in \\mathfrak{U}_x \\text{ and } p > p(x, m) $$ Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O}_X)$."}
+{"_id": "2165", "title": "cohomology-lemma-is-limit-spaltenstein", "text": "\\begin{reference} \\cite[Proposition 3.13]{Spaltenstein} \\end{reference} Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$. Assume that for every $x \\in X$ there exist an integer $d_x \\geq 0$ and a fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$ such that $$ H^p(U, H^q(E)) = 0 \\text{ for } U \\in \\mathfrak{U}_x,\\ p > d_x, \\text{ and }q < 0 $$ Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O}_X)$."}
+{"_id": "2166", "title": "cohomology-lemma-is-limit", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$. Assume there exist a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$ and a set $\\mathcal{B}$ of opens of $X$ such that \\begin{enumerate} \\item every open in $X$ has a covering whose members are elements of $\\mathcal{B}$, and \\item $H^p(U, H^{m - p}(E)) = 0$ for $p > p(m)$ and $U \\in \\mathcal{B}$. \\end{enumerate} Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O}_X)$."}
+{"_id": "2167", "title": "cohomology-lemma-is-limit-dimension", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$. Assume there exist an integer $d \\geq 0$ and a basis $\\mathcal{B}$ for the topology of $X$ such that $$ H^p(U, H^q(E)) = 0 \\text{ for } U \\in \\mathcal{B},\\ p > d, \\text{ and }q < 0 $$ Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O}_X)$."}
+{"_id": "2168", "title": "cohomology-lemma-cohomology-over-U-trivial", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{B}$ be a set of opens of $X$. Assume \\begin{enumerate} \\item every open of $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and $U \\in \\mathcal{B}$. \\end{enumerate} Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$ and $U \\in \\mathcal{B}$."}
+{"_id": "2169", "title": "cohomology-lemma-derived-limit-suitable-system", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system of objects of $D(\\mathcal{O}_X)$. Let $\\mathcal{B}$ be a set of opens of $X$. Assume \\begin{enumerate} \\item every open of $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have \\begin{enumerate} \\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$, \\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$. \\end{enumerate} \\end{enumerate} Then $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$."}
+{"_id": "2170", "title": "cohomology-lemma-K-injective", "text": "In the situation described above. Denote $\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf. Let $\\mathcal{B}$ be a set of open subsets of $X$. Let $d \\in \\mathbf{N}$. Assume \\begin{enumerate} \\item every open in $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$ for $p > d$ and $q < 0$\\footnote{It suffices if $\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for $p > p(m)$, see Lemma \\ref{lemma-is-limit}.}. \\end{enumerate} Then (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism."}
+{"_id": "2172", "title": "cohomology-lemma-alternating-cech-complex-complex", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite open covering. For a complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical map $$ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet)) \\longrightarrow R\\Gamma(X, \\mathcal{F}^\\bullet) $$ functorial in $\\mathcal{F}^\\bullet$ and compatible with (\\ref{equation-global-sections-to-alternating-cech})."}
+{"_id": "2173", "title": "cohomology-lemma-alternating-cech-complex-complex-ss", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite open covering. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. Let $\\mathcal{B}$ be a set of open subsets of $X$. Assume \\begin{enumerate} \\item every open in $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item we have $U_{i_0\\ldots i_p} \\in \\mathcal{B}$ for all $i_0, \\ldots, i_p \\in I$, \\item for every $U \\in \\mathcal{B}$ and $p > 0$ we have \\begin{enumerate} \\item $H^p(U, \\mathcal{F}^q) = 0$, \\item $H^p(U, \\Coker(\\mathcal{F}^{q - 1} \\to \\mathcal{F}^q)) = 0$, and \\item $H^p(U, H^q(\\mathcal{F})) = 0$. \\end{enumerate} \\end{enumerate} Then the map $$ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet)) \\longrightarrow R\\Gamma(X, \\mathcal{F}^\\bullet) $$ of Lemma \\ref{lemma-alternating-cech-complex-complex} is an isomorphism in $D(\\textit{Ab})$."}
+{"_id": "2174", "title": "cohomology-lemma-compose", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}_X$-modules there is an isomorphism $$ \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)) = \\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}_X$-modules functorial in $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$."}
+{"_id": "2175", "title": "cohomology-lemma-composition", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical morphism $$ \\text{Tot}\\left( \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\otimes_{\\mathcal{O}_X} \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet) \\right) \\longrightarrow \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}_X$-modules."}
+{"_id": "2176", "title": "cohomology-lemma-diagonal-better", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical morphism $$ \\text{Tot}\\left( \\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet) \\right) \\longrightarrow \\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)) $$ of complexes of $\\mathcal{O}_X$-modules functorial in all three complexes."}
+{"_id": "2177", "title": "cohomology-lemma-diagonal", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical morphism $$ \\mathcal{K}^\\bullet \\longrightarrow \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)) $$ of complexes of $\\mathcal{O}_X$-modules functorial in both complexes."}
+{"_id": "2178", "title": "cohomology-lemma-evaluate-and-more", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}_X$-modules there is a canonical morphism $$ \\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet) \\longrightarrow \\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}_X$-modules functorial in all three complexes."}
+{"_id": "2179", "title": "cohomology-lemma-RHom-into-K-injective", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}_X$-modules. Let $\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. Then $$ H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) = \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U) $$ for all $U \\subset X$ open."}
+{"_id": "2181", "title": "cohomology-lemma-RHom-from-K-flat-into-K-injective", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}_X$-modules. Let $\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}_X$-modules. Then $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$ is a K-injective complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2183", "title": "cohomology-lemma-internal-hom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of $D(\\mathcal{O}_X)$. With the construction as described above there is a canonical isomorphism $$ R\\SheafHom(K, R\\SheafHom(L, M)) = R\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, M) $$ in $D(\\mathcal{O}_X)$ functorial in $K, L, M$ which recovers (\\ref{equation-internal-hom}) by taking $H^0(X, -)$."}
+{"_id": "2184", "title": "cohomology-lemma-restriction-RHom-to-U", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\\mathcal{O}_X)$. The construction of $R\\SheafHom(K, L)$ commutes with restrictions to opens, i.e., for every open $U$ we have $R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$."}
+{"_id": "2185", "title": "cohomology-lemma-RHom-triangulated", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The bifunctor $R\\SheafHom(- , -)$ transforms distinguished triangles into distinguished triangles in both variables."}
+{"_id": "2186", "title": "cohomology-lemma-internal-hom-composition", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in $D(\\mathcal{O}_X)$ there is a canonical morphism $$ R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(K, M) $$ in $D(\\mathcal{O}_X)$ functorial in $K, L, M$."}
+{"_id": "2187", "title": "cohomology-lemma-internal-hom-diagonal-better", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in $D(\\mathcal{O}_X)$ there is a canonical morphism $$ K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L) \\longrightarrow R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) $$ in $D(\\mathcal{O}_X)$ functorial in $K, L, M$."}
+{"_id": "2189", "title": "cohomology-lemma-dual", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L$ be an object of $D(\\mathcal{O}_X)$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O}_X)$. For $M$ in $D(\\mathcal{O}_X)$ there is a canonical map \\begin{equation} \\label{equation-eval} M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee \\longrightarrow R\\SheafHom(L, M) \\end{equation} which induces a canonical map $$ H^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee) \\longrightarrow \\Hom_{D(\\mathcal{O}_X)}(L, M) $$ functorial in $M$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2190", "title": "cohomology-lemma-internal-hom-evaluate", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of $D(\\mathcal{O}_X)$. There is a canonical morphism $$ R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\longrightarrow R\\SheafHom(R\\SheafHom(K, L), M) $$ in $D(\\mathcal{O}_X)$ functorial in $K, L, M$."}
+{"_id": "2191", "title": "cohomology-lemma-glue", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be the union of two open subspaces of $X$. Suppose given \\begin{enumerate} \\item an object $A$ of $D(\\mathcal{O}_U)$, \\item an object $B$ of $D(\\mathcal{O}_V)$, and \\item an isomorphism $c : A|_{U \\cap V} \\to B|_{U \\cap V}$. \\end{enumerate} Then there exists an object $F$ of $D(\\mathcal{O}_X)$ and isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such that $c = g|_{U \\cap V} \\circ f^{-1}|_{U \\cap V}$. Moreover, given \\begin{enumerate} \\item an object $E$ of $D(\\mathcal{O}_X)$, \\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$, \\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$,  \\end{enumerate} such that $$ a|_{U \\cap V}  = b|_{U \\cap V} \\circ c. $$ Then there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$ whose restriction to $U$ is $a \\circ f$ and whose restriction to $V$ is $b \\circ g$."}
+{"_id": "2192", "title": "cohomology-lemma-vanishing-and-glueing", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{B}$ be a basis for the topology on $Y$. \\begin{enumerate} \\item Assume $K$ is in $D(\\mathcal{O}_X)$ such that for $V \\in \\mathcal{B}$ we have $H^i(f^{-1}(V), K) = 0$ for $i < 0$. Then $Rf_*K$ has vanishing cohomology sheaves in negative degrees, $H^i(f^{-1}(V), K) = 0$ for $i < 0$ for all opens $V \\subset Y$, and the rule $V \\mapsto H^0(f^{-1}V, K)$ is a sheaf on $Y$. \\item Assume $K, L$ are in $D(\\mathcal{O}_X)$ such that for $V \\in \\mathcal{B}$ we have $\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$. Then $\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$ for all opens $V \\subset Y$ and the rule $V \\mapsto \\Hom(K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$. \\end{enumerate}"}
+{"_id": "2193", "title": "cohomology-lemma-uniqueness", "text": "In Situation \\ref{situation-locally-given} assume \\begin{enumerate} \\item $X = \\bigcup_{U \\in \\mathcal{B}} U$ and for $U, V \\in \\mathcal{B}$ we have $U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$, \\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$ for $i < 0$. \\end{enumerate} If a solution $(K, \\rho_U)$ exists, then it is unique up to unique isomorphism and moreover $\\Ext^i(K, K) = 0$ for $i < 0$."}
+{"_id": "2194", "title": "cohomology-lemma-solution-in-finite-case", "text": "In Situation \\ref{situation-locally-given} assume \\begin{enumerate} \\item $X = U_1 \\cup \\ldots \\cup U_n$ with $U_i \\in \\mathcal{B}$, \\item for $U, V \\in \\mathcal{B}$ we have $U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$, \\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$ for $i < 0$. \\end{enumerate} Then a solution exists and is unique up to unique isomorphism."}
+{"_id": "2195", "title": "cohomology-lemma-glueing-increasing-union", "text": "Let $X$ be a ringed space. Let $E$ be a well ordered set and let $$ X = \\bigcup\\nolimits_{\\alpha \\in E} W_\\alpha $$ be an open covering with $W_\\alpha \\subset W_{\\alpha + 1}$ and $W_\\alpha = \\bigcup_{\\beta < \\alpha} W_\\beta$ if $\\alpha$ is not a successor. Let $K_\\alpha$ be an object of $D(\\mathcal{O}_{W_\\alpha})$ with $\\Ext^i(K_\\alpha, K_\\alpha) = 0$ for $i < 0$. Assume given isomorphisms $\\rho_\\beta^\\alpha :  K_\\alpha|_{W_\\beta} \\to K_\\beta$ in $D(\\mathcal{O}_{W_\\beta})$ for all $\\beta < \\alpha$ with $\\rho_\\gamma^\\alpha = \\rho_\\gamma^\\beta \\circ \\rho^\\alpha_\\beta|_{W_\\gamma}$ for $\\gamma < \\beta < \\alpha$. Then there exists an object $K$ in $D(\\mathcal{O}_X)$ and isomorphisms $K|_{W_\\alpha} \\to K_\\alpha$ for $\\alpha \\in E$ compatible with the isomorphisms $\\rho_\\beta^\\alpha$."}
+{"_id": "2196", "title": "cohomology-lemma-cone", "text": "The cone on a morphism of strictly perfect complexes is strictly perfect."}
+{"_id": "2197", "title": "cohomology-lemma-tensor", "text": "The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect."}
+{"_id": "2198", "title": "cohomology-lemma-strictly-perfect-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. If $\\mathcal{F}^\\bullet$ is a strictly perfect complex of $\\mathcal{O}_Y$-modules, then $f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2199", "title": "cohomology-lemma-local-lift-map", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given a solid diagram of $\\mathcal{O}_X$-modules $$ \\xymatrix{ \\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\ & \\mathcal{G} \\ar[u]_p } $$ with $\\mathcal{E}$ a direct summand of a finite free $\\mathcal{O}_X$-module and $p$ surjective, then a dotted arrow making the diagram commute exists locally on $X$."}
+{"_id": "2200", "title": "cohomology-lemma-local-homotopy", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ be a morphism of complexes of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$ acyclic. Then $\\alpha$ is locally on $X$ homotopic to zero. \\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ be a morphism of complexes of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$ for $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$. Then $\\alpha$ is locally on $X$ homotopic to zero. \\end{enumerate}"}
+{"_id": "2201", "title": "cohomology-lemma-lift-through-quasi-isomorphism", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given a solid diagram of complexes of $\\mathcal{O}_X$-modules $$ \\xymatrix{ \\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\ & \\mathcal{G}^\\bullet \\ar[u]_f } $$ with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for $j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then a dotted arrow making the diagram commute up to homotopy exists locally on $X$."}
+{"_id": "2202", "title": "cohomology-lemma-local-actual", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect. \\begin{enumerate} \\item For any element $\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ there exists an open covering $X = \\bigcup U_i$ such that $\\alpha|_{U_i}$ is given by a morphism of complexes $\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$. \\item Given a morphism of complexes $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ whose image in the group $\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is zero, there exists an open covering $X = \\bigcup U_i$ such that $\\alpha|_{U_i}$ is homotopic to zero. \\end{enumerate}"}
+{"_id": "2203", "title": "cohomology-lemma-Rhom-strictly-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect. Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is represented by the complex $\\mathcal{H}^\\bullet$ with terms $$ \\mathcal{H}^n = \\bigoplus\\nolimits_{n = p + q} \\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p) $$ and differential as described in Section \\ref{section-internal-hom}."}
+{"_id": "2204", "title": "cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}_X$-modules with \\begin{enumerate} \\item $\\mathcal{F}^n = 0$ for $n \\ll 0$, \\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and \\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite free $\\mathcal{O}_X$-module. \\end{enumerate} Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is represented by the complex $\\mathcal{H}^\\bullet$ with terms $$ \\mathcal{H}^n = \\bigoplus\\nolimits_{n = p + q} \\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p) $$ and differential as described in Section \\ref{section-internal-hom}."}
+{"_id": "2205", "title": "cohomology-lemma-pseudo-coherent-independent-representative", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. \\begin{enumerate} \\item If there exists an open covering $X = \\bigcup U_i$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$, and maps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in $D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$ and $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent. \\item If $E$ is $m$-pseudo-coherent, then any complex representing $E$ is $m$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "2206", "title": "cohomology-lemma-pseudo-coherent-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent."}
+{"_id": "2207", "title": "cohomology-lemma-cone-pseudo-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space and $m \\in \\mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O}_X)$. \\begin{enumerate} \\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent then $M$ is $m$-pseudo-coherent. \\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent. \\item If $L$ is $(m + 1)$-pseudo-coherent and $M$ is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "2208", "title": "cohomology-lemma-tensor-pseudo-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\\mathcal{O}_X)$. \\begin{enumerate} \\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is $t$-pseudo-coherent with $t = \\max(m + a, n + b)$. \\item If $K$ and $L$ are pseudo-coherent, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is pseudo-coherent. \\end{enumerate}"}
+{"_id": "2209", "title": "cohomology-lemma-summands-pseudo-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$. If $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) in $D(\\mathcal{O}_X)$ so are $K$ and $L$."}
+{"_id": "2210", "title": "cohomology-lemma-complex-pseudo-coherent-modules", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$. Let $\\mathcal{F}^\\bullet$ be a (locally) bounded above complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is $(m - i)$-pseudo-coherent for all $i$. Then $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent."}
+{"_id": "2211", "title": "cohomology-lemma-cohomology-pseudo-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$. If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then $E$ is $m$-pseudo-coherent."}
+{"_id": "2212", "title": "cohomology-lemma-finite-cohomology", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}_X$-module. \\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented $\\mathcal{O}_X$-module. \\end{enumerate}"}
+{"_id": "2214", "title": "cohomology-lemma-last-one-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$ be a bounded above complex of flat $\\mathcal{O}_X$-modules with tor-amplitude in $[a, b]$. Then $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat $\\mathcal{O}_X$-module."}
+{"_id": "2215", "title": "cohomology-lemma-tor-amplitude", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent \\begin{enumerate} \\item $E$ has tor-amplitude in $[a, b]$. \\item $E$ is represented by a complex $\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}_X$-modules with $\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "2216", "title": "cohomology-lemma-tor-amplitude-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$."}
+{"_id": "2217", "title": "cohomology-lemma-tor-amplitude-stalk", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent \\begin{enumerate} \\item $E$ has tor-amplitude in $[a, b]$. \\item for every $x \\in X$ the object $E_x$ of $D(\\mathcal{O}_{X, x})$ has tor-amplitude in $[a, b]$. \\end{enumerate}"}
+{"_id": "2218", "title": "cohomology-lemma-cone-tor-amplitude", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$. \\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$. \\item If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$, then $K$ has tor-amplitude in $[a + 1, b + 1]$. \\end{enumerate}"}
+{"_id": "2219", "title": "cohomology-lemma-tensor-tor-amplitude", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of $D(\\mathcal{O}_X)$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ has tor amplitude in $[a + c, b + d]$."}
+{"_id": "2220", "title": "cohomology-lemma-summands-tor-amplitude", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $a, b \\in \\mathbf{Z}$. For $K$, $L$ objects of $D(\\mathcal{O}_X)$ if $K \\oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$."}
+{"_id": "2221", "title": "cohomology-lemma-perfect-independent-representative", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. \\begin{enumerate} \\item If there exists an open covering $X = \\bigcup U_i$ and strictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$ such that $\\mathcal{E}_i^\\bullet$ represents $E|_{U_i}$ in $D(\\mathcal{O}_{U_i})$, then $E$ is perfect. \\item If $E$ is perfect, then any complex representing $E$ is perfect. \\end{enumerate}"}
+{"_id": "2222", "title": "cohomology-lemma-perfect-on-locally-ringed", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. Assume that all stalks $\\mathcal{O}_{X, x}$ are local rings. Then the following are equivalent \\begin{enumerate} \\item $E$ is perfect, \\item there exists an open covering $X = \\bigcup U_i$ such that $E|_{U_i}$ can be represented by a finite complex of finite locally free $\\mathcal{O}_{U_i}$-modules, and \\item there exists an open covering $X = \\bigcup U_i$ such that $E|_{U_i}$ can be represented by a finite complex of finite free $\\mathcal{O}_{U_i}$-modules. \\end{enumerate}"}
+{"_id": "2223", "title": "cohomology-lemma-perfect-precise", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect."}
+{"_id": "2224", "title": "cohomology-lemma-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is perfect, and \\item $E$ is pseudo-coherent and locally has finite tor dimension. \\end{enumerate}"}
+{"_id": "2225", "title": "cohomology-lemma-perfect-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$. If $E$ is perfect in $D(\\mathcal{O}_Y)$, then $Lf^*E$ is perfect in $D(\\mathcal{O}_X)$."}
+{"_id": "2226", "title": "cohomology-lemma-two-out-of-three-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O}_X)$. If two out of three of $K, L, M$ are perfect then the third is also perfect."}
+{"_id": "2228", "title": "cohomology-lemma-summands-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. If $K \\oplus L$ is a perfect object of $D(\\mathcal{O}_X)$, then so are $K$ and $L$."}
+{"_id": "2229", "title": "cohomology-lemma-pushforward-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an open subspace. Let $E$ be a perfect object of $D(\\mathcal{O}_U)$ whose cohomology sheaves are supported on a closed subset $T \\subset U$ with $j(T)$ closed in $X$. Then $Rj_*E$ is a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "2230", "title": "cohomology-lemma-symmetric-monoidal-cat-complexes", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The category of complexes of $\\mathcal{O}_X$-modules with tensor product defined by $\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet = \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$ is a symmetric monoidal category (for sign rules, see More on Algebra, Section \\ref{more-algebra-section-sign-rules})."}
+{"_id": "2232", "title": "cohomology-lemma-internal-hom-evaluate-isom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M \\in D(\\mathcal{O}_X)$. If $K$ is perfect, then the map $$ R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\longrightarrow R\\SheafHom(R\\SheafHom(K, L), M) $$ of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism."}
+{"_id": "2233", "title": "cohomology-lemma-dual-perfect-complex", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then $K^\\vee = R\\SheafHom(K, \\mathcal{O}_X)$ is a perfect object too and $(K^\\vee)^\\vee \\cong K$. There are functorial isomorphisms $$ M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee = R\\SheafHom(K, M) $$ and $$ H^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee) = \\Hom_{D(\\mathcal{O}_X)}(K, M) $$ for $M$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2234", "title": "cohomology-lemma-symmetric-monoidal-derived", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The derived category $D(\\mathcal{O}_X)$ is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section \\ref{more-algebra-section-sign-rules})."}
+{"_id": "2235", "title": "cohomology-lemma-left-dual-derived", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object of $D(\\mathcal{O}_X)$. If $M$ has a left dual in the monoidal category $D(\\mathcal{O}_X)$ (Categories, Definition \\ref{categories-definition-dual}) then $M$ is perfect and the left dual is as constructed in Example \\ref{example-dual-derived}."}
+{"_id": "2236", "title": "cohomology-lemma-colim-and-lim-of-duals", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O}_X)$. Let $K = \\text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \\ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(\\mathcal{O}_X)$ we have $$ R\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K_n^\\vee $$ where $(K_n^\\vee)$ is the inverse system of dual perfect complexes."}
+{"_id": "2237", "title": "cohomology-lemma-ext-composition-is-cup", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ and $E$ be objects of $D(\\mathcal{O}_X)$ with $E$ perfect. The diagram $$ \\xymatrix{ H^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee) \\times H^0(X, E) \\ar[r] \\ar[d] & H^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) \\ar[d] \\\\ \\Hom_X(E, K) \\times H^0(X, E) \\ar[r] & H^0(X, K) } $$ commutes where the top horizontal arrow is the cup product, the right vertical arrow uses $\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$ (Example \\ref{example-dual-derived}), the left vertical arrow uses Lemma \\ref{lemma-dual-perfect-complex}, and the bottom horizontal arrow is the obvious one."}
+{"_id": "2238", "title": "cohomology-lemma-category-summands-finite-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $R = \\Gamma(X, \\mathcal{O}_X)$. The category of $\\mathcal{O}_X$-modules which are summands of finite free $\\mathcal{O}_X$-modules is equivalent to the category of finite projective $R$-modules."}
+{"_id": "2239", "title": "cohomology-lemma-invertible-derived", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $M$ is invertible in $D(\\mathcal{O}_X)$, see Categories, Definition \\ref{categories-definition-invertible}, and \\item there is a locally finite direct product decomposition $$ \\mathcal{O}_X = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n $$ and for each $n$ there is an invertible $\\mathcal{O}_n$-module $\\mathcal{H}^n$ (Modules, Definition \\ref{modules-definition-invertible}) and $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O}_X)$. \\end{enumerate} If (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O}_X)$. If $\\mathcal{O}_{X, x}$ is a local ring for all $x \\in X$ these condition are also equivalent to \\begin{enumerate} \\item[(3)] there exists an open covering $X = \\bigcup U_i$ and for each $i$ an integer $n_i$ such that $M|_{U_i}$ is represented by an invertible $\\mathcal{O}_{U_i}$-module placed in degree $n_i$. \\end{enumerate}"}
+{"_id": "2240", "title": "cohomology-lemma-when-jshriek-compact", "text": "Let $X$ be a ringed space. Let $j : U \\to X$ be the inclusion of an open. The $\\mathcal{O}_X$-module $j_!\\mathcal{O}_U$ is a compact object of $D(\\mathcal{O}_X)$ if there exists an integer $d$ such that \\begin{enumerate} \\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and \\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$ commute with direct sums. \\end{enumerate}"}
+{"_id": "2242", "title": "cohomology-lemma-injective-tensor-finite-locally-free", "text": "Let $X$ be a ringed space. Let $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module. Let $\\mathcal{E}$ be an $\\mathcal{O}_X$-module. Assume $\\mathcal{E}$ is finite locally free on $X$, see Modules, Definition \\ref{modules-definition-locally-free}. Then $\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I}$ is an injective $\\mathcal{O}_X$-module."}
+{"_id": "2243", "title": "cohomology-lemma-projection-formula", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $\\mathcal{E}$ be an $\\mathcal{O}_Y$-module. Assume $\\mathcal{E}$ is finite locally free on $Y$, see Modules, Definition \\ref{modules-definition-locally-free}. Then there exist isomorphisms $$ \\mathcal{E} \\otimes_{\\mathcal{O}_Y} R^qf_*\\mathcal{F} \\longrightarrow R^qf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) $$ for all $q \\geq 0$. In fact there exists an isomorphism $$ \\mathcal{E} \\otimes_{\\mathcal{O}_Y} Rf_*\\mathcal{F} \\longrightarrow Rf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) $$ in $D^{+}(Y)$ functorial in $\\mathcal{F}$."}
+{"_id": "2244", "title": "cohomology-lemma-projection-formula-perfect", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $E \\in D(\\mathcal{O}_X)$ and $K \\in D(\\mathcal{O}_Y)$. If $K$ is perfect, then $$ Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K = Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K) $$ in $D(\\mathcal{O}_Y)$."}
+{"_id": "2245", "title": "cohomology-lemma-projection-formula-closed-immersion", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces such that $f$ is a homeomorphism onto a closed subset. Then (\\ref{equation-projection-formula-map}) is an isomorphism always."}
+{"_id": "2246", "title": "cohomology-proposition-vanishing-Noetherian", "text": "\\begin{reference} \\cite[Theorem 3.6.5]{Tohoku}. \\end{reference} Let $X$ be a Noetherian topological space. If $\\dim(X) \\leq d$, then $H^p(X, \\mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\\mathcal{F}$ on $X$."}
+{"_id": "2247", "title": "cohomology-proposition-cohomological-dimension-spectral", "text": "\\begin{reference} Part (1) is the main theorem of \\cite{Scheiderer}. \\end{reference} Let $X$ be a spectral space of Krull dimension $d$. Let $\\mathcal{F}$ be an abelian sheaf on $X$. \\begin{enumerate} \\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$, \\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective for every quasi-compact open $U \\subset X$, \\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ and any constructible closed subset $Z \\subset X$. \\end{enumerate}"}
+{"_id": "2286", "title": "stacks-introduction-lemma-key-fact", "text": "The functor $\\Sch^{opp} \\to \\textit{Sets}$, $T \\mapsto \\{(a, a', \\alpha)\\text{ as above}\\}$ is representable by a scheme $S \\times_{\\mathcal{M}_{1, 1}} S'$."}
+{"_id": "2287", "title": "stacks-introduction-lemma-Weierstrass-smooth-cover", "text": "The morphism $W \\xrightarrow{(E_W, f_W, 0_W)} \\mathcal{M}_{1, 1}$ is smooth and surjective."}
+{"_id": "2292", "title": "restricted-theorem-dilatations-general", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. Let $U \\subset X$ be the open subspace with $|U| = |X| \\setminus T$. The completion functor (\\ref{equation-completion-functor}) $$ \\left\\{ \\begin{matrix} \\text{morphisms of algebraic spaces}\\\\ f : X' \\to X\\text{ which are locally}\\\\ \\text{of finite type and such that}\\\\ f^{-1}U \\to U\\text{ is an isomorphism} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{morphisms }g : W \\to X_{/T}\\\\ \\text{of formal algebraic spaces}\\\\ \\text{with }W\\text{ locally Noetherian}\\\\ \\text{and }g\\text{ rig-\\'etale} \\end{matrix} \\right\\} $$ sending $f : X' \\to X$ to $f_{/T} : X'_{/T'} \\to X_{/T}$ is an equivalence."}
+{"_id": "2293", "title": "restricted-theorem-dilatations", "text": "\\begin{reference} \\cite[Theorem 3.2]{ArtinII} \\end{reference} Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. Let $\\mathfrak X = X_{/T}$ be the formal completion of $X$ along $T$. Let $$ \\mathfrak f : \\mathfrak X' \\to \\mathfrak X $$ be a formal modification (Definition \\ref{definition-formal-modification}). Then there exists a unique proper morphism $f : X' \\to X$ which is an isomorphism over the complement of $T$ in $X$ whose completion $f_{/T}$ recovers $\\mathfrak f$."}
+{"_id": "2294", "title": "restricted-lemma-topologically-finite-type", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. The functor $$ \\mathcal{C} \\longrightarrow \\mathcal{C}',\\quad (B_n) \\longmapsto B = \\lim B_n $$ is a quasi-inverse to (\\ref{equation-from-complete-to-systems}). The completions $A[x_1, \\ldots, x_r]^\\wedge$ are in $\\mathcal{C}'$ and any object of $\\mathcal{C}'$ is of the form $$ B = A[x_1, \\ldots, x_r]^\\wedge / J $$ for some ideal $J \\subset A[x_1, \\ldots, x_r]^\\wedge$."}
+{"_id": "2295", "title": "restricted-lemma-topologically-finite-type-Noetherian", "text": "\\begin{reference} \\cite[Proposition 7.5.5]{EGA1} \\end{reference} Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Then \\begin{enumerate} \\item every object of the category $\\mathcal{C}'$ (\\ref{equation-C-prime}) is Noetherian, \\item if $B \\in \\Ob(\\mathcal{C}')$ and $J \\subset B$ is an ideal, then $B/J$ is an object of $\\mathcal{C}'$, \\item for a finite type $A$-algebra $C$ the $I$-adic completion $C^\\wedge$ is in $\\mathcal{C}'$, \\item in particular the completion $A[x_1, \\ldots, x_r]^\\wedge$ is in $\\mathcal{C}'$. \\end{enumerate}"}
+{"_id": "2296", "title": "restricted-lemma-NL-up-to-homotopy", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal. Let $B$ be an object of (\\ref{equation-C-prime}). The naive cotangent complex $\\NL_{B/A}^\\wedge$ is well defined in $K(B)$."}
+{"_id": "2297", "title": "restricted-lemma-NL-is-completion", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal. Let $A \\to B$ be a finite type ring map. Choose a presentation $\\alpha : A[x_1, \\ldots, x_n] \\to B$. Then $\\NL_{B^\\wedge/A}^\\wedge = \\lim \\NL(\\alpha) \\otimes_B B^\\wedge$ as complexes and $\\NL_{B^\\wedge/A}^\\wedge = \\NL_{B/A} \\otimes_B^\\mathbf{L} B^\\wedge$ in $D(B^\\wedge)$."}
+{"_id": "2298", "title": "restricted-lemma-NL-is-limit", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal. Let $B$ be an object of (\\ref{equation-C-prime}). Then \\begin{enumerate} \\item the pro-objects $\\{\\NL_{B/A}^\\wedge \\otimes_B B/I^nB\\}$ and $\\{\\NL_{B_n/A_n}\\}$ of $D(B)$ are strictly isomorphic (see proof for elucidation), \\item $\\NL_{B/A}^\\wedge = R\\lim \\NL_{B_n/A_n}$ in $D(B)$. \\end{enumerate} Here $B_n$ and $A_n$ are as in Section \\ref{section-two-categories}."}
+{"_id": "2299", "title": "restricted-lemma-NL-base-change", "text": "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. Then there is a canonical map $$ \\NL_{B_1/A_1} \\otimes_{B_2} B_1 \\to \\NL_{B_2/A_2} $$ which induces and isomorphism on $H^0$ and a surjection on $H^{-1}$."}
+{"_id": "2300", "title": "restricted-lemma-exact-sequence-NL", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be a ideal. Let $B \\to C$ be morphism of (\\ref{equation-C-prime}). Then there is an exact sequence $$ \\xymatrix{ C \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] & H^0(\\NL_{C/A}^\\wedge) \\ar[r] & H^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\ H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] & H^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] & H^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu] } $$ See proof for elucidation."}
+{"_id": "2301", "title": "restricted-lemma-transitive-lci-at-end", "text": "With assumptions as in Lemma \\ref{lemma-exact-sequence-NL} assume that $B/I^nB \\to C/I^nC$ is a local complete intersection homomorphism for all $n$. Then $H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge)$ is injective."}
+{"_id": "2302", "title": "restricted-lemma-equivalent-with-artin-smooth", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). Write $B = A[x_1, \\ldots, x_r]^\\wedge/J$ (Lemma \\ref{lemma-topologically-finite-type-Noetherian}) and let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$ be its naive cotangent complex (\\ref{equation-NL}). The following are equivalent \\begin{enumerate} \\item $B$ is rig-smooth over $(A, I)$, \\item the object $\\NL_{B/A}^\\wedge$ of $D(B)$ satisfies the equivalent conditions (1) -- (4) of More on Algebra, Lemma \\ref{more-algebra-lemma-ext-1-annihilated} with respect to the ideal $IB$, \\item there exists a $c \\geq 0$ such that for all $a \\in I^c$ there is a map $h : \\bigoplus B\\text{d}x_i \\to J/J^2$ such that $a : J/J^2 \\to J/J^2$ is equal to $h \\circ \\text{d}$, \\item there exist $b_1, \\ldots, b_s \\in B$ such that $V(b_1, \\ldots, b_s) \\subset V(IB)$ and such that for every $l = 1, \\ldots, s$ there exist $m \\geq 0$, $f_1, \\ldots, f_m \\in J$, and subset $T \\subset \\{1, \\ldots, n\\}$ with $|T| = m$ such that \\begin{enumerate} \\item $\\det_{i \\in T, j \\leq m}(\\partial f_j/ \\partial x_i)$ divides $b_l$ in $B$, and \\item $b_l J \\subset (f_1, \\ldots, f_m) + J^2$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2303", "title": "restricted-lemma-rig-smooth", "text": "Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be a finite type $A$-algebra. \\begin{enumerate} \\item If $\\Spec(B) \\to \\Spec(A)$ is smooth over $\\Spec(A) \\setminus V(I)$, then $B^\\wedge$ is rig-smooth over $(A, I)$. \\item If $B^\\wedge$ is rig-smooth over $(A, I)$, then there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is smooth over $\\Spec(A) \\setminus V(I)$. \\end{enumerate}"}
+{"_id": "2304", "title": "restricted-lemma-zero-ext-1-after-modding-out", "text": "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. Let $f_1 \\in B_1$ with image $f_2 \\in B_2$. If $\\Ext^1_{B_1}(\\NL_{B_1/A_1}^\\wedge, N_1)$ is annihilated by $f_1$ for every $B_1$-module $N_1$, then $\\Ext^1_{B_2}(\\NL_{B_2/A_2}^\\wedge, N_2)$ is annihilated by $f_2$ for every $B_2$-module $N_2$."}
+{"_id": "2305", "title": "restricted-lemma-base-change-rig-smooth-homomorphism", "text": "Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$ be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$ as in Remark \\ref{remark-base-change}. If $B_1$ is rig-smooth over $(A_1, I_1)$, then $B_2$ is rig-smooth over $(A_2, I_2)$."}
+{"_id": "2306", "title": "restricted-lemma-get-morphism-general-better", "text": "Assume given the following data \\begin{enumerate} \\item an integer $c \\geq 0$, \\item an ideal $I$ of a Noetherian ring $A$, \\item $B$ in (\\ref{equation-C-prime}) for $(A, I)$ such that $I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ for any $B$-module $N$, \\item a Noetherian $I$-adically complete $A$-algebra $C$; denote $d = d(\\text{Gr}_I(C))$ and $q_0 = q(\\text{Gr}_I(C))$ the integers found in Local Cohomology, Section \\ref{local-cohomology-section-uniform}, \\item an integer $n \\geq \\max(q_0 + (d + 1)c, 2(d + 1)c + 1)$, and \\item an $A$-algebra homomorphism $\\psi_n : B \\to C/I^nC$. \\end{enumerate} Then there exists a map $\\varphi : B \\to C$ of $A$-algebras such that $\\psi_n \\bmod I^{n - (d + 1)c} = \\varphi \\bmod I^{n - (d + 1)c}$."}
+{"_id": "2307", "title": "restricted-lemma-get-morphism-nonzerodivisor", "text": "Let $I = (a)$ be a principal ideal of a Noetherian ring $A$. Let $B$ be an object of (\\ref{equation-C-prime}). Assume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ is annihilated by $a^c$ for all $B$-modules $N$. Let $C$ be an $I$-adically complete $A$-algebra such that $a$ is a nonzerodivisor on $C$. Let $n > 2c$. For any $A$-algebra map $\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra map $\\varphi : B \\to C$ such that $\\psi_n \\bmod a^{n - c}C = \\varphi \\bmod a^{n - c}C$."}
+{"_id": "2308", "title": "restricted-lemma-get-morphism-principal", "text": "Let $I = (a)$ be a principal ideal of a Noetherian ring $A$. Let $B$ be an object of (\\ref{equation-C-prime}). Assume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ is annihilated by $a^c$ for all $B$-modules $N$. Let $C$ be an $I$-adically complete $A$-algebra. Assume given an integer $d \\geq 0$ such that $C[a^\\infty] \\cap a^dC = 0$. Let $n > \\max(2c, c + d)$. For any $A$-algebra map $\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra map $\\varphi : B \\to C$ such that $\\psi_n \\bmod a^{n - c} = \\varphi \\bmod a^{n - c}$."}
+{"_id": "2309", "title": "restricted-lemma-close-enough", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $r \\geq 0$ and write $P = A[x_1, \\ldots, x_r]$ the $I$-adic completion. Consider a resolution $$ P^{\\oplus t} \\xrightarrow{K} P^{\\oplus m} \\xrightarrow{g_1, \\ldots, g_m} P \\to B \\to 0 $$ of a quotient of $P$. Assume $B$ is rig-smooth over $(A, I)$. Then there exists an integer $n$ such that for any complex $$ P^{\\oplus t} \\xrightarrow{K'} P^{\\oplus m} \\xrightarrow{g'_1, \\ldots, g'_m} P $$ with $g_i - g'_i \\in I^nP$ and $K - K' \\in I^n\\text{Mat}(m \\times t, P)$ there exists an isomorphism $B \\to B'$ of $A$-algebras where $B' = P/(g'_1, \\ldots, g'_m)$."}
+{"_id": "2310", "title": "restricted-lemma-algebraize-easy", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $C^h$ be the henselization of a finite type $A$-algebra $C$ with respect to the ideal $IC$. Let $J \\subset C^h$ be an ideal. Then there exists a finite type $A$-algebra $B$ such that $B^\\wedge \\cong (C^h/J)^\\wedge$."}
+{"_id": "2312", "title": "restricted-lemma-presentation-rig-smooth", "text": "Let $A$ be a ring. Let $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$ and set $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$. Assume $m \\leq n$ and set $g = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$. Then \\begin{enumerate} \\item $g$ annihilates $\\Ext^1_B(\\NL_{B/A}, N)$ for every $B$-module $N$, \\item if $n = m$, then multiplication by $g$ on $\\NL_{B/A}$ is $0$ in $D(B)$. \\end{enumerate}"}
+{"_id": "2313", "title": "restricted-lemma-approximate-presentation-rig-smooth", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object of (\\ref{equation-C-prime}). Let $B = A[x_1, \\ldots, x_r]^\\wedge/J$ be a presentation. Assume there exists an element $b \\in B$, $0 \\leq m \\leq r$, and $f_1, \\ldots, f_m \\in J$ such that \\begin{enumerate} \\item $V(b) \\subset V(IB)$ in $\\Spec(B)$, \\item the image of $\\Delta = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$ in $B$ divides $b$, and \\item $b J \\subset (f_1, \\ldots, f_m) + J^2$. \\end{enumerate} Then there exists a finite type $A$-algebra $C$ and an $A$-algebra isomorphism $B \\cong C^\\wedge$."}
+{"_id": "2314", "title": "restricted-lemma-equivalent-with-artin", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). Write $B = A[x_1, \\ldots, x_r]^\\wedge/J$ (Lemma \\ref{lemma-topologically-finite-type-Noetherian}) and let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$ be its naive cotangent complex (\\ref{equation-NL}). The following are equivalent \\begin{enumerate} \\item $B$ is rig-\\'etale over $(A, I)$, \\item \\label{item-zero-on-NL} there exists a $c \\geq 0$ such that for all $a \\in I^c$ multiplication by $a$ on $\\NL_{B/A}^\\wedge$ is zero in $D(B)$, \\item \\label{item-zero-on-cohomology-NL} there exits a $c \\geq 0$ such that $H^i(\\NL_{B/A}^\\wedge)$, $i = -1, 0$ is annihilated by $I^c$, \\item \\label{item-zero-on-cohomology-NL-truncations} there exists a $c \\geq 0$ such that $H^i(\\NL_{B_n/A_n})$, $i = -1, 0$ is annihilated by $I^c$ for all $n \\geq 1$ where $A_n = A/I^n$ and $B_n = B/I^nB$, \\item \\label{item-condition-artin-pre-pre} for every $a \\in I$ there exists a $c \\geq 0$ such that \\begin{enumerate} \\item $a^c$ annihilates $H^0(\\NL_{B/A}^\\wedge)$, and \\item there exist $f_1, \\ldots, f_r \\in J$ such that $a^c J \\subset (f_1, \\ldots, f_r) + J^2$. \\end{enumerate} \\item \\label{item-condition-artin-pre} for every $a \\in I$ there exist $f_1, \\ldots, f_r \\in J$ and $c \\geq 0$ such that \\begin{enumerate} \\item $\\det_{1 \\leq i, j \\leq r}(\\partial f_j/\\partial x_i)$ divides $a^c$ in $B$, and \\item $a^c J \\subset (f_1, \\ldots, f_r) + J^2$. \\end{enumerate} \\item \\label{item-condition-artin} choosing generaters $f_1, \\ldots, f_t$ for $J$ we have \\begin{enumerate} \\item the Jacobian ideal of $B$ over $A$, namely the ideal in $B$ generated by the $r \\times r$ minors of the matrx $(\\partial f_j/\\partial x_i)_{1 \\leq i \\leq r, 1 \\leq j \\leq t}$, contains the ideal $I^cB$ for some $c$, and \\item the Cramer ideal of $B$ over $A$, namely the ideal in $B$ generated by the image in $B$ of the $r$th Fitting ideal of $J$ as an $A[x_1, \\ldots, x_r]^\\wedge$-module, contains $I^cB$ for some $c$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2315", "title": "restricted-lemma-rig-etale-rig-smooth", "text": "Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). If $B$ is rig-\\'etale over $(A, I)$, then $B$ is rig-smooth over $(A, I)$."}
+{"_id": "2316", "title": "restricted-lemma-rig-etale", "text": "Let $A$ be a Noetherian ring and let $I$ be an ideal. Let $B$ be a finite type $A$-algebra. \\begin{enumerate} \\item If $\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$, then $B^\\wedge$ satisfies the equivalent conditions of Lemma \\ref{lemma-equivalent-with-artin}. \\item If $B^\\wedge$ satisfies the equivalent conditions of Lemma \\ref{lemma-equivalent-with-artin}, then there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is \\'etale over $\\Spec(A) \\setminus V(I)$. \\end{enumerate}"}
+{"_id": "2317", "title": "restricted-lemma-zero-after-modding-out", "text": "Let $(A_1, I_1) \\to (A_2, I_2)$ be as in Remark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian. Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$. If multiplication by $f_1 \\in B_1$ on $\\NL^\\wedge_{B_1/A_1}$ is zero in $D(B_1)$, then multiplication by the image $f_2 \\in B_2$ on $\\NL^\\wedge_{B_2/A_2}$ is zero in $D(B_2)$."}
+{"_id": "2318", "title": "restricted-lemma-base-change-rig-etale-homomorphism", "text": "Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$ be an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$. Let $B_2$ be the base change of $B_1$ as in Remark \\ref{remark-base-change}. If $B_1$ is rig-\\'etale over $(A_1, I_1)$, then $B_2$ is rig-\\'etale over $(A_2, I_2)$."}
+{"_id": "2319", "title": "restricted-lemma-fully-faithful-etale-over-complement", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $B$ be a finite type $A$-algebra such that $\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$. Let $C$ be a Noetherian $A$-algebra. Then any $A$-algebra map $B^\\wedge \\to C^\\wedge$ of $I$-adic completions comes from a unique $A$-algebra map $$ B \\longrightarrow C^h $$ where $C^h$ is the henselization of the pair $(C, IC)$ as in More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}. Moreover, any $A$-algebra homomorphism $B \\to C^h$ factors through some \\'etale $C$-algebra $C'$ such that $C/IC \\to C'/IC'$ is an isomorphism."}
+{"_id": "2320", "title": "restricted-lemma-lift-approximation", "text": "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal. Let $J \\subset A$ be a nilpotent ideal. Consider a commutative diagram $$ \\xymatrix{ C \\ar[r] & C_0 \\ar@{=}[r] & C/JC \\\\ & B_0 \\ar[u] \\\\ A \\ar[r] \\ar[uu] & A_0 \\ar[u] \\ar@{=}[r] & A/J } $$ whose vertical arrows are of finite type such that \\begin{enumerate} \\item $\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$, \\item $\\Spec(B_0) \\to \\Spec(A_0)$ is \\'etale over $\\Spec(A_0) \\setminus V(IA_0)$, and \\item $B_0 \\to C_0$ is \\'etale and induces an isomorphism $B_0/IB_0 = C_0/IC_0$. \\end{enumerate} Then we can fill in the diagram above to a commutative diagram $$ \\xymatrix{ C \\ar[r] & C/JC \\\\ B \\ar[u] \\ar[r] & B_0 \\ar[u] \\\\ A \\ar[r] \\ar[u] & A/J \\ar[u] } $$ with $A \\to B$ of finite type, $B/JB = B_0$, $B \\to C$ \\'etale, and $\\Spec(B) \\to \\Spec(A)$ \\'etale over $\\Spec(A) \\setminus V(I)$."}
+{"_id": "2321", "title": "restricted-lemma-approximate-principal", "text": "\\begin{reference} The rig-\\'etale case of \\cite[III Theorem 7]{Elkik} \\end{reference} Let $A$ be a Noetherian ring and $I = (a)$ a principal ideal. Let $B$ be an object of (\\ref{equation-C-prime}) which is rig-\\'etale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \\cong C^\\wedge$."}
+{"_id": "2322", "title": "restricted-lemma-approximate", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}) which is rig-\\'etale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \\cong C^\\wedge$."}
+{"_id": "2323", "title": "restricted-lemma-approximate-by-etale-over-complement", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $B$ be an $I$-adically complete $A$-algebra with $A/I \\to B/IB$ of finite type. The equivalent conditions of Lemma \\ref{lemma-equivalent-with-artin} are also equivalent to \\begin{enumerate} \\item[(8)] \\label{item-algebraize} there exists a finite type $A$-algebra $C$ such that $\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$ and such that $B \\cong C^\\wedge$. \\end{enumerate}"}
+{"_id": "2324", "title": "restricted-lemma-finite-type", "text": "Let $A$ and $B$ be adic topological rings which have a finitely generated ideal of definition. Let $\\varphi : A \\to B$ be a continuous ring homomorphism. The following are equivalent: \\begin{enumerate} \\item $\\varphi$ is adic and $B$ is topologically of finite type over $A$, \\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$, \\item there exists an ideal of definition $I \\subset A$ such that the topology on $B$ is the $I$-adic topology and there exist an ideal of definition $I' \\subset A$ such that $A/I' \\to B/I'B$ is of finite type, \\item for all ideals of definition $I \\subset A$ the topology on $B$ is the $I$-adic topology and $A/I \\to B/IB$ is of finite type, \\item there exists an ideal of definition $I \\subset A$ such that the topology on $B$ is the $I$-adic topology and $B$ is in the category (\\ref{equation-C-prime}), \\item for all ideals of definition $I \\subset A$ the topology on $B$ is the $I$-adic topology and $B$ is in the category (\\ref{equation-C-prime}), \\item $B$ as a topological $A$-algebra is the quotient of $A\\{x_1, \\ldots, x_r\\}$ by a closed ideal, \\item $B$ as a topological $A$-algebra is the quotient of $A[x_1, \\ldots, x_r]^\\wedge$ by a closed ideal where $A[x_1, \\ldots, x_r]^\\wedge$ is the completion of $A[x_1, \\ldots, x_r]$ with respect to some ideal of definition of $A$, and \\item add more here. \\end{enumerate} Moreover, these equivalent conditions define a local property of morphisms of $\\text{WAdm}^{adic*}$ as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}."}
+{"_id": "2326", "title": "restricted-lemma-composition-finite-type", "text": "Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in Lemma \\ref{lemma-finite-type}. Then $P$ is stable under composition as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-composition-variant-adic-star}."}
+{"_id": "2327", "title": "restricted-lemma-finite-type-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally adic* formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to an arrow of $\\textit{WAdm}^{adic*}$ which is adic and topologically of finite type, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to an arrow of $\\textit{WAdm}^{adic*}$ which is adic and topologically of finite type, \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to an arrow of $\\textit{WAdm}^{adic*}$ which is adic and topologically of finite type, and \\item $f$ is locally of finite type. \\end{enumerate}"}
+{"_id": "2328", "title": "restricted-lemma-finite-type-red", "text": "For an arrow $\\varphi : A \\to B$ in $\\text{WAdm}^{count}$ consider the property $P(\\varphi)=$``the induced ring homomorphism $A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type'' where $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ are the ideals of topologically nilpotent elements. Then $P$ is a local property as defined in Formal Spaces, Situation \\ref{formal-spaces-situation-local-property}."}
+{"_id": "2330", "title": "restricted-lemma-composition-finite-type-red", "text": "Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in Lemma \\ref{lemma-finite-type-red}. Then $P$ is stable under composition (Formal Spaces, Situation \\ref{formal-spaces-situation-composition-local-property})."}
+{"_id": "2331", "title": "restricted-lemma-finite-type-finite-type-red", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{count}$. If $\\varphi$ is taut and topologically of finite type, then $\\varphi$ satisfies the condition defined in Lemma \\ref{lemma-finite-type-red}."}
+{"_id": "2332", "title": "restricted-lemma-Noetherian-finite-type-red", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$ satisfying the condition defined in Lemma \\ref{lemma-finite-type-red}. Then $A \\to B$ is topologically of finite type."}
+{"_id": "2334", "title": "restricted-lemma-finite-type-red-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to an arrow of $\\textit{WAdm}^{count}$ satisfying the property defined in Lemma \\ref{lemma-finite-type-red}, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to an arrow of $\\textit{WAdm}^{count}$ satisfying the property defined in Lemma \\ref{lemma-finite-type-red}, \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to an arrow of $\\textit{WAdm}^{count}$ satisfying the property defined in Lemma \\ref{lemma-finite-type-red}, and \\item the morphism $f_{red} : X_{red} \\to Y_{red}$ is locally of finite type. \\end{enumerate}"}
+{"_id": "2335", "title": "restricted-lemma-flat-axioms", "text": "The property $P(\\varphi)=$``$\\varphi$ is flat'' on arrows of $\\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}."}
+{"_id": "2336", "title": "restricted-lemma-base-change-flat-continuous", "text": "Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$ defined in Lemma \\ref{lemma-flat-axioms}. Denote $Q$ the property defined in Lemma \\ref{lemma-finite-type-red} viewed as a property of arrows of $\\textit{WAdm}^{Noeth}$. Denote $R$ the property defined in Lemma \\ref{lemma-finite-type} viewed as a property of arrows of $\\textit{WAdm}^{Noeth}$. Then \\begin{enumerate} \\item $P$ is stable under base change by $Q$ (Formal Spaces, Remark \\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}), and \\item $P + R$ is stable under base change (Formal Spaces, Remark \\ref{formal-spaces-remark-base-change-variant-Noetherian}). \\end{enumerate}"}
+{"_id": "2337", "title": "restricted-lemma-composition-flat-continuous", "text": "Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$ defined in Lemma \\ref{lemma-flat-axioms}. Then $P$ is stable under composition (Formal Spaces, Remark \\ref{formal-spaces-remark-composition-variant-Noetherian})."}
+{"_id": "2338", "title": "restricted-lemma-flat-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is flat, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a flat map in $\\textit{WAdm}^{Noeth}$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a flat map in $\\textit{WAdm}^{Noeth}$, and \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a flat map in $\\textit{WAdm}^{Noeth}$. \\end{enumerate}"}
+{"_id": "2341", "title": "restricted-lemma-representable-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is representable by algebraic spaces and flat in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}, then $f$ is flat in the sense of Definition \\ref{definition-flat}."}
+{"_id": "2342", "title": "restricted-lemma-rig-point", "text": "Let $A$ be a Noetherian adic topological ring. Let $\\mathfrak q \\subset A$ be a prime ideal. The following are equivalent \\begin{enumerate} \\item for some ideal of definition $I \\subset A$ we have $I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal with respect to this property, \\item for some ideal of definition $I \\subset A$ the prime $\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$, \\item for any ideal of definition $I \\subset A$ we have $I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal with respect to this property, \\item for any ideal of definition $I \\subset A$ the prime $\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$, \\item $\\dim(A/\\mathfrak q) = 1$ and for some ideal of definition $I \\subset A$ we have $I \\not \\subset \\mathfrak q$, \\item $\\dim(A/\\mathfrak q) = 1$ and for any ideal of definition $I \\subset A$ we have $I \\not \\subset \\mathfrak q$, \\item $\\dim(A/\\mathfrak q) = 1$ and the induced topology on $A/\\mathfrak q$ is nontrivial, \\item $A/\\mathfrak q$ is a $1$-dimensional Noetherian complete local domain whose maximal ideal is the radical of the image of any ideal of definition of $A$, and \\item add more here. \\end{enumerate}"}
+{"_id": "2343", "title": "restricted-lemma-rig-closed-point-relative-residue-field", "text": "Let $\\varphi : A \\to B$ in $\\textit{WAdm}^{Noeth}$. Denote $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ the ideals of topologically nilpotent elements. Assume $A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type. Let $\\mathfrak q \\subset B$ be rig-closed. The residue field $\\kappa$ of the local ring $B/\\mathfrak q$ is a finite type $A/\\mathfrak a$-algebra."}
+{"_id": "2344", "title": "restricted-lemma-rig-closed-point-relative", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\\mathfrak q \\subset B$ be rig-closed. Let $\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$. Let $\\mathfrak a \\subset A$ be the ideal of topologically nilpotent elements. The following are equivalent \\begin{enumerate} \\item the residue field $\\kappa$ of $B/\\mathfrak q$ is finite over $A/\\mathfrak a$, \\item $\\mathfrak p \\subset A$ is rig-closed, \\item $A/\\mathfrak p \\subset B/\\mathfrak q$ is a finite extension of rings. \\end{enumerate}"}
+{"_id": "2345", "title": "restricted-lemma-rig-closed-jacobson", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\\mathfrak q \\subset B$ be rig-closed. If $A/I$ is Jacobson for some ideal of definition $I \\subset A$, then $\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$ is rig-closed."}
+{"_id": "2347", "title": "restricted-lemma-rig-closed-point-in-localization", "text": "Let $A$ be an adic Noetherian topological ring. Let $\\mathfrak p \\subset A$ be a prime ideal. Let $f \\in A$ be an element mapping to a unit in $A/\\mathfrak p$. Then $$ \\mathfrak p A_{\\{f\\}} = \\mathfrak p(A_f)^\\wedge = \\mathfrak p \\otimes_A (A_f)^\\wedge = (\\mathfrak p_f)^\\wedge $$ is a prime ideal with quotient $$ A/\\mathfrak p = (A/\\mathfrak p) \\otimes_A (A_f)^\\wedge = (A_f)^\\wedge / \\mathfrak p (A_f)^\\wedge = A_{\\{f\\}}/\\mathfrak p A_{\\{f\\}} $$"}
+{"_id": "2348", "title": "restricted-lemma-rig-closed-point-after-localization", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\\mathfrak q \\subset B$ be rig-closed. There exists an $f \\in A$ which maps to a unit in $B/\\mathfrak q$ such that we obtain a diagram $$ \\vcenter{ \\xymatrix{ B \\ar[r] & B_{\\{f\\}} \\\\ A \\ar[r] \\ar[u]_\\varphi & A_{\\{f\\}} \\ar[u]_{\\varphi_{\\{f\\}}} } } \\quad\\text{with primes}\\quad \\vcenter{ \\xymatrix{ \\mathfrak q \\ar@{-}[r] \\ar@{-}[d] & \\mathfrak q' \\ar@{-}[d] \\ar@{=}[r] & \\mathfrak q B_{\\{f\\}} \\\\ \\mathfrak p \\ar@{-}[r] & \\mathfrak p' } } $$ such that $\\mathfrak p'$ is rig-closed, i.e., the map $A_{\\{f\\}} \\to B_{\\{f\\}}$ and the prime ideals $\\mathfrak q'$ and $\\mathfrak p'$ satisfy the equivalent conditions of Lemma \\ref{lemma-rig-closed-point-relative}."}
+{"_id": "2349", "title": "restricted-lemma-rig-closed-point-variables", "text": "Let $A$ be a Noetherian adic topological ring. Denote $A\\{x_1, \\ldots, x_n\\}$ the restricted power series over $A$. Let $\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ be a prime ideal. Set $\\mathfrak q' = A[x_1, \\ldots, x_n] \\cap \\mathfrak q$ and $\\mathfrak p = A \\cap \\mathfrak q$. If $\\mathfrak q$ and $\\mathfrak p$ are rig-closed, then the map $$ A[x_1, \\ldots, x_n]_{\\mathfrak q'} \\to A\\{x_1, \\ldots, x_n\\}_\\mathfrak q $$ defines an isomorphism on completions with respect to their maximal ideals."}
+{"_id": "2350", "title": "restricted-lemma-rig-closed-point-etale", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$. Assume $\\varphi$ is adic, topologically of finite type, flat, and $A/I \\to B/IB$ is \\'etale for some (resp.\\ any) ideal of definition $I \\subset A$. Let $\\mathfrak q \\subset B$ be rig-closed such that $\\mathfrak p = A \\cap \\mathfrak q$ is rig-closed as well. Then $\\mathfrak p B_\\mathfrak q = \\mathfrak q B_\\mathfrak q$."}
+{"_id": "2351", "title": "restricted-lemma-fibre-regular", "text": "Let $A$ be an adic Noetherian topological ring. Let $\\mathfrak p \\subset A$ be a rig-closed prime. For any $n \\geq 1$ the ring map $$ A/\\mathfrak p \\longrightarrow A\\{x_1, \\ldots, x_n\\} \\otimes_A A/\\mathfrak p = A/\\mathfrak p\\{x_1, \\ldots, x_n\\} $$ is regular. In particular, the algebra $A\\{x_1, \\ldots, x_n\\} \\otimes_A \\kappa(\\mathfrak p)$ is geometrically regular over $\\kappa(\\mathfrak p)$."}
+{"_id": "2352", "title": "restricted-lemma-naively-rig-flat-continuous", "text": "Let $\\varphi : A \\to B$ be a morphism in $\\textit{WAdm}^{adic*}$ (Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}). Assume $\\varphi$ is adic. The following are equivalent: \\begin{enumerate} \\item $B_f$ is flat over $A$ for all topologically nilpotent $f \\in A$, \\item $B_g$ is flat over $A$ for all topologically nilpotent $g \\in B$, \\item $B_\\mathfrak q$ is flat over $A$ for all primes $\\mathfrak q \\subset B$ which do not contain an ideal of definition, \\item $B_\\mathfrak q$ is flat over $A$ for every rig-closed prime $\\mathfrak q \\subset B$, and \\item add more here. \\end{enumerate}"}
+{"_id": "2354", "title": "restricted-lemma-rig-flat-base-change", "text": "Let $\\varphi : A \\to B$ and $A \\to C$ be arrows of $\\textit{WAdm}^{Noeth}$. Assume $\\varphi$ is rig-flat and $A \\to C$ adic and topologically of finite type. Then $C \\to B \\widehat{\\otimes}_A C$ is rig-flat."}
+{"_id": "2355", "title": "restricted-lemma-rig-flat-local-etale", "text": "Consider a commutative diagram $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'} } $$ in $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically of finite type. Assume $A \\to A'$ and $B \\to B'$ are flat. Let $I \\subset A$ be an ideal of definition. If $\\varphi$ is rig-flat and $A/I \\to A'/IA'$ is \\'etale, then $\\varphi'$ is rig-flat."}
+{"_id": "2356", "title": "restricted-lemma-rig-flat-local-down", "text": "Consider a commutative diagram $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'} } $$ in $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically of finite type. Assume $A \\to A'$ flat and $B \\to B'$ faithfully flat. If $\\varphi'$ is rig-flat, then $\\varphi$ is rig-flat."}
+{"_id": "2357", "title": "restricted-lemma-rig-flat-axioms", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-flat'' on arrows of $\\textit{WAdm}^{adic*}$ is a local property as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}."}
+{"_id": "2358", "title": "restricted-lemma-composition-rig-flat-continuous", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-flat'' on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-composition-variant-Noetherian}."}
+{"_id": "2362", "title": "restricted-lemma-rig-smooth-continuous", "text": "Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$ (Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}). The following are equivalent: \\begin{enumerate} \\item[(a)] $A \\to B$ satisfies the equivalent conditions of Lemma \\ref{lemma-finite-type} and there exists an ideal of definition $I \\subset B$ such that $B$ is rig-smooth over $(A, I)$, and \\item[(b)] $A \\to B$ satisfies the equivalent conditions of Lemma \\ref{lemma-finite-type} and for all ideals of definition $I \\subset A$ the algebra $B$ is rig-smooth over $(A, I)$. \\end{enumerate}"}
+{"_id": "2363", "title": "restricted-lemma-rig-smooth-axioms", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-smooth'' on arrows of $\\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}."}
+{"_id": "2364", "title": "restricted-lemma-base-change-rig-smooth-continuous", "text": "Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-smooth'' and $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}."}
+{"_id": "2365", "title": "restricted-lemma-composition-rig-smooth-continuous", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-smooth'' on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-composition-variant-Noetherian}."}
+{"_id": "2366", "title": "restricted-lemma-rig-smooth-rig-flat", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$. If $\\varphi$ is rig-smooth, then $\\varphi$ is rig-flat, and for any presentation $B = A\\{x_1, \\ldots, x_n\\}/J$ and prime $J \\subset \\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ not containing an ideal of definition the ideal $J_\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}_\\mathfrak q$ is generated by a regular sequence."}
+{"_id": "2367", "title": "restricted-lemma-exact-sequence-NL-rig-smooth", "text": "Let $A \\to B \\to C$ be arrows in $\\textit{WAdm}^{Noeth}$ which are adic and topologically of finite type. If $B \\to C$ is rig-smooth, then the kernel of the map $$ H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge) $$ (see Lemma \\ref{lemma-exact-sequence-NL}) is annihilated by an ideal of definition."}
+{"_id": "2368", "title": "restricted-lemma-rig-smooth-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is rig-smooth, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-smooth map in $\\textit{WAdm}^{Noeth}$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a rig-smooth map in $\\textit{WAdm}^{Noeth}$, and \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a rig-smooth map in $\\textit{WAdm}^{Noeth}$. \\end{enumerate}"}
+{"_id": "2369", "title": "restricted-lemma-base-change-rig-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-smooth and $g$ is adic, then the base change $X \\times_Y Z \\to Z$ is rig-smooth."}
+{"_id": "2372", "title": "restricted-lemma-rig-etale-continuous", "text": "Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$ (Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}). The following are equivalent: \\begin{enumerate} \\item[(a)] $A \\to B$ satisfies the equivalent conditions of Lemma \\ref{lemma-finite-type} and there exists an ideal of definition $I \\subset B$ such that $B$ is rig-\\'etale over $(A, I)$, and \\item[(b)] $A \\to B$ satisfies the equivalent conditions of Lemma \\ref{lemma-finite-type} and for all ideals of definition $I \\subset A$ the algebra $B$ is rig-\\'etale over $(A, I)$. \\end{enumerate}"}
+{"_id": "2373", "title": "restricted-lemma-rig-etale-axioms", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' on arrows of $\\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}."}
+{"_id": "2374", "title": "restricted-lemma-base-change-rig-etale-continuous", "text": "Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' and $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}."}
+{"_id": "2375", "title": "restricted-lemma-composition-rig-etale-continuous", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' on arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-composition-variant-Noetherian}."}
+{"_id": "2376", "title": "restricted-lemma-permanence-rig-etale-continuous", "text": "The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' on arrows of $\\textit{WAdm}^{Noeth}$ has the cancellation property as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-permanence-variant-Noetherian}."}
+{"_id": "2377", "title": "restricted-lemma-rig-etale-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is rig-\\'etale, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$, and \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$. \\end{enumerate}"}
+{"_id": "2379", "title": "restricted-lemma-rig-etale-rig-smooth-morphism", "text": "A rig-\\'etale morphism of locally Noetherian formal algebraic spaces is rig-smooth."}
+{"_id": "2380", "title": "restricted-lemma-base-change-rig-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-\\'etale and $g$ is adic, then the base change $X \\times_Y Z \\to Z$ is rig-\\'etale."}
+{"_id": "2382", "title": "restricted-lemma-rig-etale-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be a morphism of locally Noetherian formal algebraic spaces over $S$. If $g \\circ f$ and $g$ are rig-\\'etale, then so is $f$."}
+{"_id": "2383", "title": "restricted-lemma-rig-etale-alternative-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $g \\circ f$ is rig-\\'etale and $g$ is an adic monomorphism, then $f$ is rig-\\'etale."}
+{"_id": "2384", "title": "restricted-lemma-closed-immersion-rig-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \\begin{enumerate} \\item $f$ is rig-smooth, \\item $f$ is rig-\\'etale, \\item for every affine formal algebraic space $V$ and every morphism $V \\to Y$ which is representable by algebraic spaces and \\'etale the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism $B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $I(J/J^2) = 0$ for some ideal of definition $I$ of $B$. \\end{enumerate}"}
+{"_id": "2385", "title": "restricted-lemma-composition-rig-surjective", "text": "\\begin{slogan} Rig-surjectivity of locally finite type morphisms is preserved under composition \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally Noetherian and $f$ and $g$ locally of finite type. Then if $f$ and $g$ are rig-surjective, so is $g \\circ f$."}
+{"_id": "2386", "title": "restricted-lemma-base-change-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms of formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally Noetherian and $f$ and $g$ locally of finite type. If $f$ is rig-surjective, then the base change $Z \\times_Y X \\to Z$ is too."}
+{"_id": "2387", "title": "restricted-lemma-rig-surjective-alternative-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms locally of finite type of locally Noetherian formal algebraic spaces over $S$. If $g \\circ f$ is rig-surjective and $g$ is a monomorphism, then $f$ is rig-surjective."}
+{"_id": "2388", "title": "restricted-lemma-permanence-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ locally Noetherian and $f$ and $g$ locally of finite type. If $g \\circ f : X \\to Z$ is rig-surjective, so is $g : Y \\to Z$."}
+{"_id": "2389", "title": "restricted-lemma-etale-covering-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces which is representable by algebraic spaces, \\'etale, and surjective. Then $f$ is rig-surjective."}
+{"_id": "2390", "title": "restricted-lemma-upshot", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let $\\{g_i : Y_i \\to Y\\}$ be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and \\'etale such that $\\coprod g_i$ is surjective. Then $f$ is rig-surjective if and only if each $f_i : X \\times_Y Y_i \\to Y_i$ is rig-surjective."}
+{"_id": "2391", "title": "restricted-lemma-faithfully-flat-rig-surjective", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $B$ be an $I$-adically complete $A$-algebra. If $A/I^n \\to B/I^nB$ is of finite type and flat for all $n$ and faithfully flat for $n = 1$, then $\\text{Spf}(B) \\to \\text{Spf}(A)$ is rig-surjective."}
+{"_id": "2393", "title": "restricted-lemma-monomorphism-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume $X$ and $Y$ are locally Noetherian, $f$ locally of finite type, and $f$ a monomorphism. Then $f$ is rig surjective if and only if every adic morphism $\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete valuation ring factors through $X$."}
+{"_id": "2394", "title": "restricted-lemma-closed-immersion-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \\begin{enumerate} \\item $f$ is rig-surjective, and \\item for every affine formal algebraic space $V$ and every morphism $V \\to Y$ which is representable by algebraic spaces and \\'etale the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism $B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $IJ^n = 0$ for some ideal of definition $I$ of $B$ and some $n \\geq 1$. \\end{enumerate}"}
+{"_id": "2395", "title": "restricted-lemma-closed-immersion-rig-smooth-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a closed immersion. The following are equivalent \\begin{enumerate} \\item $f$ is rig-smooth and rig-surjective, \\item $f$ is rig-\\'etale and rig-surjective, and \\item for every affine formal algebraic space $V$ and every morphism $V \\to Y$ which is representable by algebraic spaces and \\'etale the morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism $B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following property: $IJ = 0$ for some ideal of definition $I$ of $B$. \\end{enumerate}"}
+{"_id": "2396", "title": "restricted-lemma-rig-etale-descent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. Assume \\begin{enumerate} \\item $g$ is locally of finite type, \\item $f$ is rig-smooth (resp.\\ rig-\\'etale) and rig-surjective, \\item $g \\circ f$ is rig-smooth (resp.\\ rig-\\'etale) \\end{enumerate} then $g$ is rig-smooth (resp.\\ rig-\\'etale)."}
+{"_id": "2397", "title": "restricted-lemma-flat-locus", "text": "Let $X$ be a locally Noetherian formal algebraic space over a complete discrete valuation ring $A$. Then there exists a closed immersion $X' \\to X$ of formal algebraic spaces such that $X'$ is flat over $A$ and such that any morphism $Y \\to X$ of locally Noetherian formal algebraic spaces with $Y$ flat over $A$ factors through $X'$."}
+{"_id": "2398", "title": "restricted-lemma-flat-and-diagonal-rig-surjective", "text": "Let $X$ be a locally Noetherian formal algebraic space which is locally of finite type over a complete discrete valuation ring $A$. Let $X' \\subset X$ be as in Lemma \\ref{lemma-flat-locus}. If $X \\to X \\times_{\\text{Spf}(A)} X$ is rig-\\'etale and rig-surjective, then $X' = \\text{Spf}(A)$ or $X' = \\emptyset$."}
+{"_id": "2399", "title": "restricted-lemma-rig-monomorphism-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume \\begin{enumerate} \\item $X$ and $Y$ are locally Noetherian, \\item $f$ locally of finite type, \\item $\\Delta_f : X \\to X \\times_Y X$ is rig-\\'etale and rig-surjective. \\end{enumerate} Then $f$ is rig surjective if and only if every adic morphism $\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete valuation ring lifts to a morphism $\\text{Spf}(R) \\to X$."}
+{"_id": "2400", "title": "restricted-lemma-map-completions-finite-type", "text": "In the situation above. If $f$ is locally of finite type, then $f_{/T}$ is locally of finite type."}
+{"_id": "2401", "title": "restricted-lemma-map-completions-etale", "text": "In the situation above. If $f$ is \\'etale, then $f_{/T}$ is \\'etale."}
+{"_id": "2402", "title": "restricted-lemma-closed-immersion-gives-closed-immersion", "text": "In the situation above. If $f$ is a closed immersion, then $f_{/T}$ is a closed immersion."}
+{"_id": "2403", "title": "restricted-lemma-proper-gives-proper", "text": "In the situation above. If $f$ is proper, then $f_{/T}$ is proper."}
+{"_id": "2404", "title": "restricted-lemma-quasi-compact-gives-quasi-compact", "text": "In the situation above. If $f$ is quasi-compact, then $f_{/T}$ is quasi-compact."}
+{"_id": "2405", "title": "restricted-lemma-quasi-separated-gives-quasi-separated", "text": "In the situation above. If $f$ is (quasi-)separated, then $f_{/T}$ is too."}
+{"_id": "2406", "title": "restricted-lemma-smooth-gives-rig-smooth", "text": "In the situation above. If $X$ is locally Noetherian, $f$ is locally of finite type, and $U' \\to U$ is smooth, then $f_{/T}$ is rig-smooth."}
+{"_id": "2407", "title": "restricted-lemma-etale-gives-rig-etale", "text": "In the situation above. If $X$ is locally Noetherian, $f$ is locally of finite type, and $U' \\to U$ is \\'etale, then $f_{/T}$ is rig-\\'etale."}
+{"_id": "2408", "title": "restricted-lemma-completion-proper-surjective-rig-surjective", "text": "In the situation above. If $X$ is locally Noetherian, $f$ is proper, and $U' \\to U$ is surjective, then $f_{/T}$ is rig-surjective."}
+{"_id": "2409", "title": "restricted-lemma-separated-mono-open-diagonal-rig-surjective", "text": "In the situation above. If $X$ is locally Noetherian, $f$ is separated and locally of finite type, and $U' \\to U$ is a monomorphism, then $\\Delta_{f_{/T}}$ is rig-surjective."}
+{"_id": "2410", "title": "restricted-lemma-modification-gives-formal-modification", "text": "Let $S$, $f : X' \\to X$, $T \\subset |X|$, $U \\subset X$, $T' \\subset |X'|$, and $U' \\subset X'$ be as in Section \\ref{section-completion-functor}. If $X$ is locally Noetherian, $f$ is proper, and $U' \\to U$ is an isomorphism, then $f_{/T} : X'_{/T'} \\to X_{/T}$ is a formal modification."}
+{"_id": "2411", "title": "restricted-lemma-base-change-formal-modification", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$ which is a formal modification. Then for any adic morphism $Y' \\to Y$ of locally Noetherian formal algebraic spaces, the base change $f' : X \\times_Y Y' \\to Y'$ is a formal modification."}
+{"_id": "2412", "title": "restricted-lemma-algebraize-rig-etale-affine", "text": "Let $T \\subset X$ be a closed subset of a Noetherian affine scheme $X$. Let $W$ be a Noetherian affine formal algebraic space. Let $g : W \\to X_{/T}$ be a rig-\\'etale morphism. Then there exists an affine scheme $X'$ and a finite type morphism $f : X' \\to X$ \\'etale over $X \\setminus T$ such that there is an isomorphism $X'_{/f^{-1}T} \\cong W$ compatible with $f_{/T}$ and $g$. Moreover, if $W \\to X_{/T}$ is \\'etale, then $X' \\to X$ is \\'etale."}
+{"_id": "2413", "title": "restricted-lemma-algebraize-morphism-rig-etale", "text": "Assume we have \\begin{enumerate} \\item Noetherian affine schemes $X$, $X'$, and $Y$, \\item a closed subset $T \\subset |X|$, \\item a morphism $f : X' \\to X$ locally of finite type and \\'etale over $X \\setminus T$, \\item a morphism $h : Y \\to X$, \\item a morphism $\\alpha : Y_{/T} \\to X'_{/T}$ over $X_{/T}$ (see proof for notation). \\end{enumerate} Then there exists an \\'etale morphism $b : Y' \\to Y$ of affine schemes which induces an isomorphism $b_{/T} : Y'_{/T} \\to Y_{/T}$ and a morphism $a : Y' \\to X'$ over $X$ such that $\\alpha = a_{/T} \\circ b_{/T}^{-1}$."}
+{"_id": "2414", "title": "restricted-lemma-factor", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of algebraic spaces. Let $T \\subset |X|$ be closed. Assume that \\begin{enumerate} \\item $X$ is locally Noetherian, \\item $g$ is a monomorphism and locally of finite type, \\item $f|_{X \\setminus T} : X \\setminus T \\to Y$ factors through $g$, and \\item $f_{/T} : X_{/T} \\to Y$ factors through $g$, \\end{enumerate} then $f$ factors through $g$."}
+{"_id": "2415", "title": "restricted-lemma-faithful-general", "text": "Let $S$ be a scheme. Let $X$, $W$ be algebraic spaces over $S$ with $X$ locally Noetherian. Let $T \\subset |X|$ be a closed subset. Let $a, b : X \\to W$ be morphisms of algebraic spaces over $S$ such that $a|_{X \\setminus T} = b|_{X \\setminus T}$ and such that $a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to W$. Then $a = b$."}
+{"_id": "2416", "title": "restricted-lemma-faithful", "text": "Let $S$ be a scheme. Let $X$, $Y$ be locally Noetherian algebraic spaces over $S$. Let $T \\subset |X|$ and $T' \\subset |Y|$ be closed subsets. Let $a, b : X \\to Y$ be morphisms of algebraic spaces over $S$ such that $a|_{X \\setminus T} = b|_{X \\setminus T}$, such that $|a|(T) \\subset T'$ and $|b|(T) \\subset T'$, and such that $a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to Y_{/T'}$. Then $a = b$."}
+{"_id": "2417", "title": "restricted-lemma-equivalence-relation", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. Let $s, t : R \\to U$ be two morphisms of algebraic spaces over $X$. Assume \\begin{enumerate} \\item $R$, $U$ are locally of finite type over $X$, \\item the base change of $s$ and $t$ to $X \\setminus T$ is an \\'etale equivalence relation, and \\item the formal completion $(t_{/T}, s_{/T}) : R_{/T} \\to U_{/T} \\times_{X_{/T}} U_{/T}$ is an equivalence relation too (see proof for notation). \\end{enumerate} Then $(t, s) : R \\to U \\times_X U$ is an \\'etale equivalence relation."}
+{"_id": "2418", "title": "restricted-lemma-smash-away-from-T", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$ and let $T \\subset |X|$ be a closed subset. Let $f : X' \\to X$ be a morphism of algebraic spaces which is locally of finite type and \\'etale outside of $T$. There exists a factorization $$ X' \\longrightarrow X'' \\longrightarrow X $$ of $f$ with the following properties: $X'' \\to X$ is locally of finite type, $X'' \\to X$ is an isomorphism over $X \\setminus T$, and $X'_{/T} \\to X''_{/T}$ is an isomorphism (see proof for notation)."}
+{"_id": "2419", "title": "restricted-lemma-functoriality-completion-functor", "text": "In the situation above, let $X_1 \\to X$ be a morphism of algebraic spaces with $X_1$ locally Noetherian. Denote $T_1 \\subset |X_1|$ the inverse image of $T$ and $U_1 \\subset X_1$ the inverse image of $U$. We denote \\begin{enumerate} \\item $\\mathcal{C}_{X, T}$ the category whose objects are morphisms of algebraic spaces $f : X' \\to X$ which are locally of finite type and such that $U' = f^{-1}U \\to U$ is an isomorphism, \\item $\\mathcal{C}_{X_1, T_1}$ the category whose objects are morphisms of algebraic spaces $f_1 : X_1' \\to X_1$ which are locally of finite type and such that $f_1^{-1}U_1 \\to U_1$ is an isomorphism, \\item $\\mathcal{C}_{X_{/T}}$ the category whose objects are morphisms $g : W \\to X_{/T}$ of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-\\'etale, \\item $\\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are morphisms $g_1 : W_1 \\to X_{1, /T_1}$ of formal algebraic spaces with $W_1$ locally Noetherian and $g_1$ rig-\\'etale. \\end{enumerate} Then the diagram $$ \\xymatrix{ \\mathcal{C}_{X, T} \\ar[d] \\ar[r] & \\mathcal{C}_{X_{/T}} \\ar[d] \\\\ \\mathcal{C}_{X_1, T_1} \\ar[r] & \\mathcal{C}_{X_{1, /T_1}} } $$ is commutative where the horizonal arrows are given by (\\ref{equation-completion-functor}) and the vertical arrows by base change along $X_1 \\to X$ and along $X_{1, /T_1} \\to X_{/T}$."}
+{"_id": "2420", "title": "restricted-lemma-completion-functor-fully-faithful", "text": "In the situation above. Let $f : X' \\to X$ be a morphism of algebraic spaces which is locally of finite type and an isomorphism over $U$. Let $g : Y \\to X$ be a morphism with $Y$ locally Noetherian. Then completion defines a bijection $$ \\Mor_X(Y, X') \\longrightarrow \\Mor_{X_{/T}}(Y_{/T}, X'_{/T}) $$ In particular, the functor (\\ref{equation-completion-functor}) is fully faithful."}
+{"_id": "2421", "title": "restricted-lemma-dilatations-affine", "text": "In the situation above. Assume $X$ is affine. Then the functor (\\ref{equation-completion-functor}) is an equivalence."}
+{"_id": "2422", "title": "restricted-lemma-output-quasi-compact", "text": "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general} let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$. Then $f$ is quasi-compact if and only if $g$ is quasi-compact."}
+{"_id": "2423", "title": "restricted-lemma-output-quasi-separated", "text": "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general} let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$. Then $f$ is quasi-separated if and only if $g$ is so."}
+{"_id": "2424", "title": "restricted-lemma-output-separated", "text": "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general} let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$. Then $f$ is separated $\\Leftrightarrow$ $g$ is separated and $\\Delta_g : W \\to W \\times_{X_{/T}} W$ is rig-surjective."}
+{"_id": "2425", "title": "restricted-lemma-output-proper", "text": "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general} let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$. Then $f$ is proper if and only if $g$ is a formal modification (Definition \\ref{definition-formal-modification})."}
+{"_id": "2426", "title": "restricted-lemma-output-etale", "text": "With assumptions and notation as in Theorem \\ref{theorem-dilatations-general} let $f : X' \\to X$ correspond to $g : W \\to X_{/T}$. Then $f$ is \\'etale if and only if $g$ is \\'etale."}
+{"_id": "2428", "title": "restricted-lemma-Noetherian-local-ring", "text": "Let $A \\to B$ be a ring homomorphism of Noetherian rings inducing an isomorphism on $I$-adic completions for some ideal $I \\subset A$ (for example if $B$ is the $I$-adic completion of $A$). Then base change defines an equivalence of categories between the category (\\ref{equation-modification}) for $(A, I)$ with the category (\\ref{equation-modification}) for $(B, IB)$."}
+{"_id": "2430", "title": "restricted-lemma-equivalence-to-completion", "text": "Let $A \\to B$ be a local map of local Noetherian rings such that \\begin{enumerate} \\item $A \\to B$ is flat, \\item $\\mathfrak m_B = \\mathfrak m_A B$, and \\item $\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$ \\end{enumerate} Then the base change functor from the category (\\ref{equation-modification}) for $(A, \\mathfrak m_A)$ to the category (\\ref{equation-modification}) for $(B, \\mathfrak m_B)$ is an equivalence."}
+{"_id": "2432", "title": "restricted-proposition-approximate", "text": "Let $I$ be an ideal of a Noetherian G-ring $A$. Let $B$ be an object of (\\ref{equation-C-prime}). If $B$ is rig-smooth over $(A, I)$, then there exists a finite type $A$-algebra $C$ and an isomorphism $B \\cong C^\\wedge$ of $A$-algebras."}
+{"_id": "2433", "title": "restricted-proposition-glue-modification", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \\subset |X|$ be a closed subset with complementary open subspace $U \\subset X$. Let $f : X' \\to X$ be a proper morphism of algebraic spaces such that $f^{-1}(U) \\to U$ is an isomorphism. For any algebraic space $W$ over $S$ the map $$ \\Mor_S(X, W) \\longrightarrow \\Mor_S(X', W) \\times_{\\Mor_S(X'_{/T}, W)} \\Mor_S(X_{/T}, W) $$ is bijective."}
+{"_id": "2457", "title": "more-groupoids-lemma-sheaf-differentials", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The sheaf of differentials of $R$ seen as a scheme over $U$ via $t$ is a quotient of the pullback via $t$ of the conormal sheaf of the immersion $e : U \\to R$. In a formula: there is a canonical surjection $t^*\\mathcal{C}_{U/R} \\to \\Omega_{R/U}$. If $s$ is flat, then this map is an isomorphism."}
+{"_id": "2458", "title": "more-groupoids-lemma-first-order-structure-c", "text": "The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition $$ I/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2 $$ where the second arrow comes from the equality $J = (I \\otimes B + B \\otimes I)C$. The map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$."}
+{"_id": "2459", "title": "more-groupoids-lemma-local-source", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. Let $g : U' \\to U$ be a morphism of schemes. Denote $h$ the composition $$ \\xymatrix{ h : U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} & R \\ar[r]_s & U. } $$ Let $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms of schemes. Assume \\begin{enumerate} \\item $\\mathcal{R} \\Rightarrow \\mathcal{Q}$, \\item $\\mathcal{Q}$ is preserved under base change and composition, \\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a largest open $W(\\mathcal{P}, f) \\subset X$ such that $f|_{W(\\mathcal{P}, f)}$ has $\\mathcal{P}$, and \\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$, and any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have $Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where $f' : X_{Y'} \\to Y'$ is the base change of $f$. \\end{enumerate} If $s, t$ have $\\mathcal{R}$ and $g$ has $\\mathcal{Q}$, then there exists an open subscheme $W \\subset U'$ such that $W \\times_{g, U, t} R = W(\\mathcal{P}, h)$."}
+{"_id": "2460", "title": "more-groupoids-lemma-property-invariant", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$\\footnote{The fact that $fpqc$ is missing is not a typo.}. Let $\\mathcal{P}$ be a property of morphisms of schemes which is $\\tau$-local on the target (Descent, Definition \\ref{descent-definition-property-morphisms-local}). Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the $\\tau$-topology. Let $W \\subset U$ be the maximal open subscheme such that $s|_{s^{-1}(W)} : s^{-1}(W) \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant, see Groupoids, Definition \\ref{groupoids-definition-invariant-open}."}
+{"_id": "2462", "title": "more-groupoids-lemma-two-fibres", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $r, r' \\in R$ with $t(r) = t(r')$ in $U$. Set $u = s(r)$, $u' = s(r')$. Denote $F_u = s^{-1}(u)$ and $F_{u'} = s^{-1}(u')$ the scheme theoretic fibres. \\begin{enumerate} \\item There exists a common field extension $\\kappa(u) \\subset k$, $\\kappa(u') \\subset k$ and an isomorphism $(F_u)_k \\cong (F_{u'})_k$. \\item We may choose the isomorphism of (1) such that a point lying over $r$ maps to a point lying over $r'$. \\item If the morphisms $s$, $t$ are flat then the morphisms of germs $s : (R, r) \\to (U, u)$ and $s : (R, r') \\to (U, u')$ are flat locally on the base isomorphic. \\item If the morphisms $s$, $t$ are \\'etale (resp.\\ smooth, syntomic, or flat and locally of finite presentation) then the morphisms of germs $s : (R, r) \\to (U, u)$ and $s : (R, r') \\to (U, u')$ are locally on the base isomorphic in the \\'etale (resp.\\ smooth, syntomic, or fppf) topology. \\end{enumerate}"}
+{"_id": "2463", "title": "more-groupoids-lemma-make-CM", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Assume $s$ and $t$ are flat and locally of finite presentation. Then there exists an open $U' \\subset U$ such that \\begin{enumerate} \\item $t^{-1}(U') \\subset R$ is the largest open subscheme of $R$ on which the morphism $s$ is Cohen-Macaulay, \\item $s^{-1}(U') \\subset R$ is the largest open subscheme of $R$ on which the morphism $t$ is Cohen-Macaulay, \\item the morphism $t|_{s^{-1}(U')} : s^{-1}(U') \\to U$ is surjective, \\item the morphism $s|_{t^{-1}(U')} : t^{-1}(U') \\to U$ is surjective, and \\item the restriction $R' = s^{-1}(U') \\cap t^{-1}(U')$ of $R$ to $U'$ defines a groupoid $(U', R', s', t', c')$ which has the property that the morphisms $s'$ and $t'$ are Cohen-Macaulay and locally of finite presentation. \\end{enumerate}"}
+{"_id": "2464", "title": "more-groupoids-lemma-restrict-preserves-type", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. \\begin{enumerate} \\item If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type. \\item If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation. \\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat. \\item Add more here. \\end{enumerate}"}
+{"_id": "2465", "title": "more-groupoids-lemma-restrict-property", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$, and let $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$. If $\\mathcal{P}$ is a property of morphisms of schemes such that \\begin{enumerate} \\item $h$ has property $\\mathcal{P}$, and \\item $\\mathcal{P}$ is preserved under base change, \\end{enumerate} then $s', t'$ have property $\\mathcal{P}$."}
+{"_id": "2466", "title": "more-groupoids-lemma-double-restrict", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes. Set $g'' = g \\circ g'$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. Let $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$, let $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and let $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$. The following diagram is commutative $$ \\xymatrix{ U'' \\times_{g', U', t} R' \\ar[d]^{h'} & (U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R) \\ar[l] \\ar[r] \\ar[d] & U'' \\times_{g'', U, t} R \\ar[d]_{h''} \\\\ U' & U' \\times_{g, U, t} R \\ar[l]_{\\text{pr}_0} \\ar[r]^h & U } $$ with both squares cartesian where the left upper horizontal arrow is given by the rule $$ \\begin{matrix} (U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R) & \\longrightarrow & U'' \\times_{g', U', t} R' \\\\ ((u', r_0), (u'', r_1)) & \\longmapsto & (u'', (c(r_1, i(r_0)), (g'(u''), u'))) \\end{matrix} $$ with notation as explained in the proof."}
+{"_id": "2467", "title": "more-groupoids-lemma-double-restrict-property", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes. Set $g'' = g \\circ g'$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. Let $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$, let $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and let $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$. Let $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic, \\linebreak[0] fppf, \\linebreak[0] fpqc\\}$. Let $\\mathcal{P}$ be a property of morphisms of schemes which is preserved under base change, and which is local on the target for the $\\tau$-topology. If \\begin{enumerate} \\item $h(U' \\times_U R)$ is open in $U$, \\item $\\{h : U' \\times_U R \\to h(U' \\times_U R)\\}$ is a $\\tau$-covering, \\item $h'$ has property $\\mathcal{P}$, \\end{enumerate} then $h''$ has property $\\mathcal{P}$. Conversely, if \\begin{enumerate} \\item[(a)] $\\{t : R \\to U\\}$ is a $\\tau$-covering, \\item[(d)] $h''$ has property $\\mathcal{P}$, \\end{enumerate} then $h'$ has property $\\mathcal{P}$."}
+{"_id": "2470", "title": "more-groupoids-lemma-groupoid-on-field-homogeneous", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. For any points $r, r' \\in R$ there exists a field extension $k \\subset k'$ and points $r_1, r_2 \\in R \\times_{s, \\Spec(k)} \\Spec(k')$ and a diagram $$ \\xymatrix{ R & R \\times_{s, \\Spec(k)} \\Spec(k') \\ar[l]_-{\\text{pr}_0} \\ar[r]^\\varphi & R \\times_{s, \\Spec(k)} \\Spec(k') \\ar[r]^-{\\text{pr}_0} & R } $$ such that $\\varphi$ is an isomorphism of schemes over $\\Spec(k')$, we have $\\varphi(r_1) = r_2$, $\\text{pr}_0(r_1) = r$, and $\\text{pr}_0(r_2) = r'$."}
+{"_id": "2471", "title": "more-groupoids-lemma-restrict-groupoid-on-field", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. Let $k \\subset k'$ be a field extension, $U' = \\Spec(k')$ and let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram $$ \\xymatrix{ R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] & R \\times_{s, U} U' \\ar[r] \\ar[d] & U' \\ar[d] \\\\ U' \\times_{U, t} R \\ar[d] \\ar[r] & R \\ar[r]^s \\ar[d]_t & U \\\\ U' \\ar[r] & U } $$ all the morphisms are surjective, flat, and universally open. The dotted arrow $R' \\to R$ is in addition affine."}
+{"_id": "2472", "title": "more-groupoids-lemma-groupoid-on-field-explain-points", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. For any point $r \\in R$ there exist \\begin{enumerate} \\item a field extension $k \\subset k'$ with $k'$ algebraically closed, \\item a point $r' \\in R'$ where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $\\Spec(k') \\to \\Spec(k)$ \\end{enumerate} such that \\begin{enumerate} \\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and \\item the maps $s', t' : R' \\to \\Spec(k')$ induce isomorphisms $k' \\to \\kappa(r')$. \\end{enumerate}"}
+{"_id": "2473", "title": "more-groupoids-lemma-groupoid-on-field-move-point", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. If $r \\in R$ is a point such that $s, t$ induce isomorphisms $k \\to \\kappa(r)$, then the map $$ R \\longrightarrow R, \\quad x \\longmapsto c(r, x) $$ (see proof for precise notation) is an automorphism $R \\to R$ which maps $e$ to $r$."}
+{"_id": "2475", "title": "more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. By abuse of notation denote $e \\in R$ the image of the identity morphism $e : U \\to R$. Then \\begin{enumerate} \\item every local ring $\\mathcal{O}_{R, r}$ of $R$ has a unique minimal prime ideal, \\item there is exactly one irreducible component $Z$ of $R$ passing through $e$, and \\item $Z$ is geometrically irreducible over $k$ via either $s$ or $t$. \\end{enumerate}"}
+{"_id": "2476", "title": "more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. Assume $s, t$ are locally of finite type. Then \\begin{enumerate} \\item $R$ is equidimensional, \\item $\\dim(R) = \\dim_r(R)$ for all $r \\in R$, \\item for any $r \\in R$ we have $\\text{trdeg}_{s(k)}(\\kappa(r)) = \\text{trdeg}_{t(k)}(\\kappa(r))$, and \\item for any closed point $r \\in R$ we have $\\dim(R) = \\dim(\\mathcal{O}_{R, r})$. \\end{enumerate}"}
+{"_id": "2477", "title": "more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(k)$ with $k$ a field. Assume $s, t$ are locally of finite type. Then $\\dim(R) = \\dim(G)$ where $G$ is the stabilizer group scheme of $R$."}
+{"_id": "2480", "title": "more-groupoids-lemma-open-image-is-closed", "text": "Notation and assumptions as in Situation \\ref{situation-morphism-groupoids-on-field}. If $a(R_1)$ is open in $R_2$, then $a(R_1)$ is closed in $R_2$."}
+{"_id": "2481", "title": "more-groupoids-lemma-map-groupoids-on-field-image", "text": "Notation and assumptions as in Situation \\ref{situation-morphism-groupoids-on-field}. Let $Z \\subset R_2$ be the reduced closed subscheme (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $a : R_1 \\to R_2$. Then $c_2(Z \\times_{s_2, U, t_2} Z) \\subset Z$ set theoretically."}
+{"_id": "2482", "title": "more-groupoids-lemma-map-groupoids-on-perfect-field-image", "text": "Notation and assumptions as in Situation \\ref{situation-morphism-groupoids-on-field}. Assume that $k$ is perfect. Let $Z \\subset R_2$ be the reduced closed subscheme (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $a : R_1 \\to R_2$. Then $$ (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) $$ is a groupoid scheme over $S$."}
+{"_id": "2483", "title": "more-groupoids-lemma-locally-closed-image-is-closed", "text": "Notation and assumptions as in Situation \\ref{situation-morphism-groupoids-on-field}. If the image $a(R_1)$ is a locally closed subset of $R_2$ then it is a closed subset."}
+{"_id": "2484", "title": "more-groupoids-lemma-quasi-compact-map-groupoids-on-field-image", "text": "Notation and assumptions as in Situation \\ref{situation-morphism-groupoids-on-field}. Assume that $a : R_1 \\to R_2$ is a quasi-compact morphism. Let $Z \\subset R_2$ be the scheme theoretic image (see Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}) of $a : R_1 \\to R_2$. Then $$ (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) $$ is a groupoid scheme over $S$."}
+{"_id": "2485", "title": "more-groupoids-lemma-groupoid-on-field-image", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field. Let $Z \\subset U \\times_S U$ be the reduced closed subscheme (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $j = (t, s) : R \\to U \\times_S U$. Then $\\text{pr}_{02}(Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z) \\subset Z$ set theoretically."}
+{"_id": "2486", "title": "more-groupoids-lemma-groupoid-on-perfect-field-image", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a perfect field. Let $Z \\subset U \\times_S U$ be the reduced closed subscheme (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}) whose underlying topological space is the closure of the image of $j = (t, s) : R \\to U \\times_S U$. Then $$ (U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z, \\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z}) $$ is a groupoid scheme over $S$."}
+{"_id": "2488", "title": "more-groupoids-lemma-slice", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \\to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \\in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \\ref{morphisms-definition-finite-type-point}. With notation as in Situation \\ref{situation-slice}, set $$ d_1 = \\dim(G_u), \\quad d_2 = \\dim_{e(u)}(F_u). $$ If $d_2 > d_1$, then there exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that (with notation as in Situation \\ref{situation-slice}) \\begin{enumerate} \\item $g$ is an immersion \\item $u \\in U'$, \\item $g$ is locally of finite presentation, \\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$ is Cohen-Macaulay at $(u, e(u))$, and \\item we have $\\dim_{e'(u)}(F'_u) = d_2 - 1$. \\end{enumerate}"}
+{"_id": "2489", "title": "more-groupoids-lemma-max-slice", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \\to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \\in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \\ref{morphisms-definition-finite-type-point}. With notation as in Situation \\ref{situation-slice} there exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that \\begin{enumerate} \\item $g$ is an immersion, \\item $u \\in U'$, \\item $g$ is locally of finite presentation, \\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$ is Cohen-Macaulay and locally of finite presentation, \\item the morphisms $s', t' : R' \\to U'$ are Cohen-Macaulay and locally of finite presentation, and \\item $\\dim_{e(u)}(F'_u) = \\dim(G'_u)$. \\end{enumerate}"}
+{"_id": "2490", "title": "more-groupoids-lemma-max-slice-quasi-finite", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \\to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \\in U$ be a finite type point of the scheme $U$, see Morphisms, Definition \\ref{morphisms-definition-finite-type-point}. Assume that $G \\to U$ is locally quasi-finite. With notation as in Situation \\ref{situation-slice} there exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that \\begin{enumerate} \\item $g$ is an immersion, \\item $u \\in U'$, \\item $g$ is locally of finite presentation, \\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$ is flat, locally of finite presentation, and locally quasi-finite, and \\item the morphisms $s', t' : R' \\to U'$ are flat, locally of finite presentation, and locally quasi-finite. \\end{enumerate}"}
+{"_id": "2491", "title": "more-groupoids-lemma-quasi-finite-over-base", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$. Assume that \\begin{enumerate} \\item $U \\to S$ is locally of finite type, \\item $U \\to S$ is quasi-finite at $u$, \\item $U \\to S$ is separated, \\item $R \\to S$ is separated, \\item $s$, $t$ are flat and locally of finite presentation, and \\item $s^{-1}(\\{u\\})$ is finite. \\end{enumerate} Then there exists an \\'etale neighbourhood $(S', p') \\to (S, p)$ with $\\kappa(p) = \\kappa(p')$ and a base change diagram $$ \\xymatrix{ R' \\amalg W' \\ar@{=}[r] & S' \\times_S R \\ar[r] \\ar@<2ex>[d]^{s'} \\ar@<-2ex>[d]_{t'} & R \\ar@<1ex>[d]^s \\ar@<-1ex>[d]_t \\\\ U' \\amalg W \\ar@{=}[r] & S' \\times_S U \\ar[r] \\ar[d] & U \\ar[d] \\\\  & S' \\ar[r] & S } $$ where the equal signs are decompositions into open and closed subschemes such that \\begin{enumerate} \\item[(a)] there exists a point $u'$ of $U'$ mapping to $u$ in $U$, \\item[(b)] the fibre $(U')_{p'}$ equals $t'\\big((s')^{-1}(\\{u'\\})\\big)$ set theoretically, \\item[(c)] the fibre $(R')_{p'}$ equals $(s')^{-1}\\big((U')_{p'}\\big)$ set theoretically, \\item[(d)] the schemes $U'$ and $R'$ are finite over $S'$, \\item[(e)] we have $s'(R') \\subset U'$ and $t'(R') \\subset U'$, \\item[(f)] we have $c'(R' \\times_{s', U', t'} R') \\subset R'$ where $c'$ is the base change of $c$, and \\item[(g)] the morphisms $s', t', c'$ determine a groupoid structure by taking the system $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \\times_{s', U', t'} R'})$. \\end{enumerate}"}
+{"_id": "2492", "title": "more-groupoids-lemma-quasi-finite-over-base-j-proper", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$. Assume assumptions (1) -- (6) of Lemma \\ref{lemma-quasi-finite-over-base} hold as well as \\begin{enumerate} \\item[(7)] $j : R \\to U \\times_S U$ is universally closed\\footnote{In view of the other conditions this is equivalent to requiring $j$ to be proper.}. \\end{enumerate} Then we can choose $(S', p') \\to (S, p)$ and decompositions $S' \\times_S U = U' \\amalg W$ and $S' \\times_S R = R' \\amalg W'$ and $u' \\in U'$ such that (a) -- (g) of Lemma \\ref{lemma-quasi-finite-over-base} hold as well as \\begin{enumerate} \\item[(h)] $R'$ is the restriction of $S' \\times_S R$ to $U'$. \\end{enumerate}"}
+{"_id": "2493", "title": "more-groupoids-lemma-finite-stratify", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists a sequence of $R$-invariant closed subschemes $$ U = Z_0 \\supset Z_1 \\supset Z_2 \\supset \\ldots $$ such that $\\bigcap Z_r = \\emptyset$ and such that $s^{-1}(Z_{r - 1}) \\setminus s^{-1}(Z_r) \\to Z_{r - 1} \\setminus Z_r$ is finite locally free of rank $r$."}
+{"_id": "2494", "title": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists an open subscheme $W \\subset U$ and a closed subscheme $W' \\subset W$ such that \\begin{enumerate} \\item $W$ and $W'$ are $R$-invariant, \\item $U = t(s^{-1}(\\overline{W}))$ set theoretically, \\item $W$ is a thickening of $W'$, and \\item the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$ are finite locally free. \\end{enumerate}"}
+{"_id": "2495", "title": "more-groupoids-lemma-finite-flat-over-almost-dense-subscheme-addendum", "text": "In Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme} assume in addition that $s$ and $t$ are of finite presentation. Then \\begin{enumerate} \\item the morphism $W' \\to W$ is of finite presentation, and \\item if $u \\in U$ is a point whose $R$-orbit consists of generic points of irreducible components of $U$, then $u \\in W$. \\end{enumerate}"}
+{"_id": "2496", "title": "more-groupoids-lemma-invariant-affine-open-around-generic-point", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite and of finite presentation and $U$ quasi-separated. Let $u_1, \\ldots, u_m \\in U$ be points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \\subset V \\subset U$ such that \\begin{enumerate} \\item $u_1, \\ldots, u_m \\in V'$, \\item $V$ is open in $U$, \\item $V'$ and $V$ are affine, \\item $V' \\subset V$ is a thickening of finite presentation, \\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free. \\end{enumerate}"}
+{"_id": "2498", "title": "more-groupoids-lemma-find-affine-integral", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$ with $s, t$ integral. Let $g : U' \\to U$ be an integral morphism such that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. If $u' \\in U'$ is contained in an $R'$-invariant affine open, then the image $u \\in U$ is contained in an $R$-invariant affine open of $U$."}
+{"_id": "2499", "title": "more-groupoids-lemma-find-almost-invariant-function", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of finite presentation. Let $u_1, \\ldots, u_m \\in U$ be points whose $R$-orbits consist of generic points of irreducible components of $U$. Let $j : U \\to \\Spec(A)$ be an immersion. Let $I \\subset A$ be an ideal such that $j(U) \\cap V(I) = \\emptyset$ and $V(I) \\cup j(U)$ is closed in $\\Spec(A)$. Then there exists an $h \\in I$ such that $j^{-1}D(h)$ is an $R$-invariant affine open subscheme of $U$ containing $u_1, \\ldots, u_m$."}
+{"_id": "2500", "title": "more-groupoids-lemma-no-specializations-map-to-same-point", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite, and $u, u' \\in R$ are distinct points in the same orbit, then $u'$ is not a specialization of $u$."}
+{"_id": "2501", "title": "more-groupoids-lemma-get-affine", "text": "Let $j : V \\to \\Spec(A)$ be a quasi-compact immersion of schemes. Let $f \\in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \\cap V(f)$ is closed. Then $V$ is affine."}
+{"_id": "2502", "title": "more-groupoids-lemma-find-affine-codimension-1", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \\in U$. Assume \\begin{enumerate} \\item $s, t$ are finite morphisms, \\item $U$ is separated and locally Noetherian, \\item $\\dim(\\mathcal{O}_{U, u'}) \\leq 1$ for every point $u'$ in the orbit of $u$. \\end{enumerate} Then $u$ is contained in an $R$-invariant affine open of $U$."}
+{"_id": "2503", "title": "more-groupoids-lemma-sits-in-functions", "text": "Let $X$ be an ind-quasi-affine scheme. Let $E \\subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \\Gamma(E, \\mathcal{O}_X|_E)$ and $Y = \\Spec(A)$. Then the canonical morphism $$ j : (E, \\mathcal{O}_X|_E) \\longrightarrow (Y, \\mathcal{O}_Y) $$ of Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} determines an isomorphism $(E, \\mathcal{O}_X|_E) \\to (E', \\mathcal{O}_Y|_{E'})$ where $E' \\subset Y$ is an intersection of quasi-compact opens. If $W \\subset E$ is open in $X$, then $j(W)$ is open in $Y$."}
+{"_id": "2504", "title": "more-groupoids-lemma-affine-base-change", "text": "Suppose given a cartesian diagram $$ \\xymatrix{ X \\ar[d]_f \\ar[r] & \\Spec(B) \\ar[d] \\\\ Y \\ar[r] & \\Spec(A) } $$ of schemes. Let $E \\subset Y$ be an intersection of a nonempty family of quasi-compact opens of $Y$. Then $$ \\Gamma(f^{-1}(E), \\mathcal{O}_X|_{f^{-1}(E)}) = \\Gamma(E, \\mathcal{O}_Y|_E) \\otimes_A B $$ provided $Y$ is quasi-separated and $A \\to B$ is flat."}
+{"_id": "2505", "title": "more-groupoids-lemma-ind-quasi-affine", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering. Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$, see Descent, Definition \\ref{descent-definition-descent-datum-for-family-of-morphisms}.  If each morphism $V_i \\to X_i$ is ind-quasi-affine, then the descent datum is effective."}
+{"_id": "2509", "title": "examples-lemma-lim-not-quasi-compact", "text": "There exists an inverse system of quasi-compact topological spaces over $\\mathbf{N}$ whose limit is not quasi-compact."}
+{"_id": "2510", "title": "examples-lemma-noncomplete-completion", "text": "There exists a local ring $R$ and a maximal ideal $\\mathfrak m$ such that the completion $R^\\wedge$ of $R$ with respect to $\\mathfrak m$ has the following properties \\begin{enumerate} \\item $R^\\wedge$ is local, but its maximal ideal is not equal to $\\mathfrak m R^\\wedge$, \\item $R^\\wedge$ is not a complete local ring, and \\item $R^\\wedge$ is not $\\mathfrak m$-adically complete as an $R$-module. \\end{enumerate}"}
+{"_id": "2511", "title": "examples-lemma-noncomplete-quotient", "text": "There exists a ring $R$ complete with respect to a principal ideal $I$ and a principal ideal $J$ such that $R/J$ is not $I$-adically complete."}
+{"_id": "2512", "title": "examples-lemma-completion-not-exact", "text": "\\begin{slogan} Completion is neither left nor right exact in general. \\end{slogan} Completion is not an exact functor in general; it is not even right exact in general. This holds even when $I$ is finitely generated on the category of finitely presented modules."}
+{"_id": "2513", "title": "examples-lemma-complete-modules-not-abelian", "text": "Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal. The category of $I$-adically complete $R$-modules has kernels and cokernels but is not abelian in general."}
+{"_id": "2514", "title": "examples-lemma-derived-complete-modules", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal. The category $\\mathcal{C}$ of derived complete modules is abelian and the inclusion functor $F : \\mathcal{C} \\to \\text{Mod}_A$ is exact and commutes with arbitrary limits. If $I$ is finitely generated, then $\\mathcal{C}$ has arbitrary direct sums and colimits, but $F$ does not commute with these in general. Finally, filtered colimits are not exact in $\\mathcal{C}$ in general, hence $\\mathcal{C}$ is not a Grothendieck abelian category."}
+{"_id": "2515", "title": "examples-lemma-countable-fg-tensor", "text": "Let $R$ be a ring. Let $M$ be an $R$-module which is countable. Then $M$ is a finite $R$-module if and only if $M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is surjective."}
+{"_id": "2516", "title": "examples-lemma-countable-fp-tensor", "text": "Let $R$ be a countable ring. Let $M$ be a countable $R$-module. Then $M$ is finitely presented if and only if the canonical map $M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is an isomorphism."}
+{"_id": "2518", "title": "examples-lemma-completion-polynomial-ring-not-flat", "text": "There exists a ring such that the completion $R[[x]]$ of $R[x]$ at $(x)$ is not flat over $R$ and a fortiori not flat over $R[x]$."}
+{"_id": "2521", "title": "examples-lemma-nonflat-completion-localization", "text": "There exists a ring $A$ complete with respect to a principal ideal $I$ and an element $f \\in A$ such that the $I$-adic completion $A_f^\\wedge$ of $A_f$ is not flat over $A$."}
+{"_id": "2522", "title": "examples-lemma-quasi-coherent-not-abelian", "text": "The category of quasi-coherent\\footnote{With quasi-coherent modules as defined above. Due to how things are setup in the Stacks project, this is really the correct definition; as seen above our definition agrees with what one would naively have defined to be quasi-coherent modules on $\\text{Spf}(A)$, namely complete $A$-modules.} modules on a formal algebraic space $X$ is not abelian in general, even if $X$ is a Noetherian affine formal algebraic space."}
+{"_id": "2523", "title": "examples-lemma-strange-regular-sequence", "text": "There exists a local ring $R$ and a regular sequence $x, y, z$ (in the maximal ideal) such that there exists a nonzero element $\\delta \\in R/zR$ with $x\\delta = y\\delta = 0$."}
+{"_id": "2525", "title": "examples-lemma-nonreduced-recompletion", "text": "There exists a local Noetherian $2$-dimensional domain $(B, \\mathfrak m)$ complete with respect to a principal ideal $I = (b)$ and an element $f \\in \\mathfrak m$, $f \\not \\in I$ such that the $I$-adic completion $C = (B_f)^\\wedge$ of the principal localization $B_f$ is nonreduced and even such that $C_b = C[1/b] = (B_f)^\\wedge[1/b]$ is nonreduced."}
+{"_id": "2527", "title": "examples-lemma-quasi-affine-normalization-not-quasi-affine", "text": "Let $k$ be a field. There exists a variety $X$ whose normalization is quasi-affine but which is itself not quasi-affine."}
+{"_id": "2528", "title": "examples-lemma-complement-of-affine-does-not-contain-qc-dense-open", "text": "Nonexistence quasi-compact opens of affines: \\begin{enumerate} \\item There exist an affine scheme $S$ and affine open $U \\subset S$ such that there is no quasi-compact open $V \\subset S$ with $U \\cap V = \\emptyset$ and $U \\cup V$ dense in $S$. \\item There exists an affine scheme $S$ and a closed point $s \\in S$ such that $S \\setminus \\{s\\}$ does not contain a quasi-compact dense open. \\end{enumerate}"}
+{"_id": "2529", "title": "examples-lemma-no-dense-separated-quasi-compact-open-in-qcqs", "text": "There exists a quasi-compact and quasi-separated scheme $X$ which does not contain a separated quasi-compact dense open."}
+{"_id": "2530", "title": "examples-lemma-nonexistence-qc-dense-open-subscheme", "text": "There exists a quasi-compact and quasi-separated algebraic space which does not contain a quasi-compact dense open subscheme."}
+{"_id": "2531", "title": "examples-lemma-cannot-embed-into-affine", "text": "There exists a finite type morphism of algebraic spaces $Y \\to X$ with $Y$ affine and $X$ quasi-separated, such that there does not exist an immersion $Y \\to \\mathbf{A}^n_X$ over $X$."}
+{"_id": "2532", "title": "examples-lemma-pushforward-quasi-coherent", "text": "Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent} is sharp in the sense that one can neither drop the assumption of quasi-compactness nor the assumption of quasi-separatedness."}
+{"_id": "2533", "title": "examples-lemma-locally-principal-not-invertible", "text": "There exists a domain $A$ and a nonzero ideal $I \\subset A$ such that $I_\\mathfrak q \\subset A_\\mathfrak q$ is a principal ideal for all primes $\\mathfrak q \\subset A$ but $I$ is not an invertible $A$-module."}
+{"_id": "2534", "title": "examples-lemma-finite-flat-non-projective", "text": "Strange flat modules. \\begin{enumerate} \\item There exists a ring $R$ and a finite flat $R$-module $M$ which is not projective. \\item There exists a closed immersion which is flat but not open. \\end{enumerate}"}
+{"_id": "2536", "title": "examples-lemma-ideal-projective-not-locally-free", "text": "There exists a ring $R$ and an ideal $I$ such that $I$ is projective as an $R$-module but not locally free as an $R$-module."}
+{"_id": "2537", "title": "examples-lemma-chow-group-product", "text": "Let $K$ be a field. Let $C_i$, $i = 1, \\ldots, n$ be smooth, projective, geometrically irreducible curves over $K$. Let $P_i \\in C_i(K)$ be a rational point and let $Q_i \\in C_i$ be a point such that $[\\kappa(Q_i) : K] = 2$. Then $[P_1 \\times \\ldots \\times P_n]$ is nonzero in $\\CH_0(U_1 \\times_K \\ldots \\times_K U_n)$ where $U_i = C_i \\setminus \\{Q_i\\}$."}
+{"_id": "2538", "title": "examples-lemma-projective-not-locally-free", "text": "There exists a countable ring $R$ and a projective module $M$ which is a direct sum of countably many locally free rank $1$ modules such that $M$ is not locally free."}
+{"_id": "2539", "title": "examples-lemma-zero-dimensional-flat-ideal", "text": "\\begin{slogan} Zero dimensional ring with flat ideal. \\end{slogan} There exists a local ring $R$ with a unique prime ideal and a nonzero ideal $I \\subset R$ which is a flat $R$-module"}
+{"_id": "2540", "title": "examples-lemma-epi-not-surjective", "text": "There exists an epimorphism of local rings of dimension $0$ which is not a surjection."}
+{"_id": "2541", "title": "examples-lemma-example-raynaud-gruson", "text": "There exists a local ring $A$, a finite type ring map $A \\to B$ and a prime $\\mathfrak q$ lying over $\\mathfrak m_A$ such that $B_{\\mathfrak q}$ is flat over $A$, and for any element $g \\in B$, $g \\not \\in \\mathfrak q$ the ring $B_g$ is neither finitely presented over $A$ nor flat over $A$."}
+{"_id": "2542", "title": "examples-lemma-finite-type-flat-not-finitely-presented", "text": "There exist examples of \\begin{enumerate} \\item a flat finite type ring map with geometrically irreducible complete intersection fibre rings which is not of finite presentation, \\item a flat finite type ring map with geometrically connected, geometrically reduced, dimension 1, complete intersection fibre rings which is not of finite presentation, \\item a proper flat morphism of schemes $X \\to S$ each of whose fibres is isomorphic to either $\\mathbf{P}^1_s$ or to the vanishing locus of $X_1X_2$ in $\\mathbf{P}^2_s$ which is not of finite presentation, and \\item a proper flat morphism of schemes $X \\to S$ each of whose fibres is isomorphic to either $\\mathbf{P}^1_s$ or $\\mathbf{P}^2_s$ which is not of finite presentation. \\end{enumerate}"}
+{"_id": "2543", "title": "examples-lemma-topology-finite-type", "text": "There exists a local homomorphism $A \\to B$ of local domains which is essentially of finite type and such that $A/\\mathfrak m_A \\to B/\\mathfrak m_B$ is finite such that for every prime $\\mathfrak q \\not = \\mathfrak m_B$ of $B$ the ring map $A \\to B/\\mathfrak q$ is not the localization of a quasi-finite ring map."}
+{"_id": "2544", "title": "examples-lemma-pure-not-universally-pure", "text": "There exists a morphism of affine schemes of finite presentation $X \\to S$ and an $\\mathcal{O}_X$-module $\\mathcal{F}$ of finite presentation such that $\\mathcal{F}$ is pure relative to $S$, but not universally pure relative to $S$."}
+{"_id": "2545", "title": "examples-lemma-formally-smooth-nonflat", "text": "There exists a formally smooth ring map which is not flat."}
+{"_id": "2546", "title": "examples-lemma-formally-etale-not-presented", "text": "There exist formally \\'etale nonflat ring maps."}
+{"_id": "2547", "title": "examples-lemma-formally-etale-nontrivial-cotangent-complex", "text": "There exists a formally \\'etale surjective ring map $A \\to B$ with $L_{B/A}$ not equal to zero."}
+{"_id": "2549", "title": "examples-lemma-completion-etale", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring of prime characteristic $p > 0$ such that $[\\kappa : \\kappa^p] < \\infty$. Then the  canonical map  $A \\to A^\\wedge$ to the completion of $A$ is flat and formally unramified. However, if $A$ is regular but not excellent, then this map is not formally \\'etale."}
+{"_id": "2551", "title": "examples-lemma-not-generated-idempotents", "text": "There exists an affine scheme $X = \\Spec(A)$ and a closed subscheme $T \\subset X$ such that $T$ is Zariski locally on $X$ cut out by ideals generated by idempotents, but $T$ is not cut out by an ideal generated by idempotents."}
+{"_id": "2552", "title": "examples-lemma-not-ind-etale", "text": "There is a ring map $A \\to B$ which identifies local rings but which is not ind-\\'etale. A fortiori it is not ind-Zariski."}
+{"_id": "2553", "title": "examples-lemma-nonflasque", "text": "There exists an affine scheme $X = \\Spec(A)$ and an injective $A$-module $J$ such that $\\widetilde{J}$ is not a flasque sheaf on $X$. Even the restriction $\\Gamma(X, \\widetilde{J}) \\to \\Gamma(U, \\widetilde{J})$ with $U$ a standard open need not be surjective."}
+{"_id": "2554", "title": "examples-lemma-nonvanishing", "text": "There exists an affine scheme $X = \\Spec(A)$ whose underlying topological space is Noetherian and an injective $A$-module $I$ such that $\\widetilde{I}$ has nonvanishing $H^1$ on some quasi-compact open $U$ of $X$."}
+{"_id": "2555", "title": "examples-lemma-non-separated-group-scheme", "text": "There exists a flat group scheme of finite type over the affine line which is not separated."}
+{"_id": "2556", "title": "examples-lemma-non-quasi-separated-group-scheme", "text": "There exists a flat group scheme of finite type over the infinite dimensional affine space which is not quasi-separated."}
+{"_id": "2557", "title": "examples-lemma-non-flat-group-scheme", "text": "There exists a group scheme $G$ over a base $S$ whose identity component is flat over $S$ but which is not flat over $S$."}
+{"_id": "2558", "title": "examples-lemma-non-separated-group-space", "text": "There exists a group algebraic space of finite type over a field which is not separated (and not even quasi-separated or locally separated)."}
+{"_id": "2559", "title": "examples-lemma-specializations-fibre-etale", "text": "There exists an \\'etale morphism of algebraic spaces $f : X \\to Y$ and a nontrivial specialization of points $x \\leadsto x'$ in $|X|$ with $f(x) = f(x')$ in $|Y|$."}
+{"_id": "2560", "title": "examples-lemma-torsors-principal-spaces-not-equal", "text": "Let $S$ be a scheme. Let $G$ be a group scheme over $S$. The stack $G\\textit{-Principal}$ classifying principal homogeneous $G$-spaces (see Examples of Stacks, Subsection \\ref{examples-stacks-subsection-principal-homogeneous-spaces}) and the stack $G\\textit{-Torsors}$ classifying fppf $G$-torsors (see Examples of Stacks, Subsection \\ref{examples-stacks-subsection-fppf-torsors}) are not equivalent in general."}
+{"_id": "2561", "title": "examples-lemma-BG-algebraic", "text": "Let $k$ be a field. Let $G$ be an affine group scheme over $k$. If the stack $[\\Spec(k)/G]$ has a smooth covering by a scheme, then $G$ is of finite type over $k$."}
+{"_id": "2562", "title": "examples-lemma-limit-preserving-on-objects-not-limit-preserving", "text": "Let $S$ be a nonempty scheme. There exists a stack in groupoids $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ such that $p$ is limit preserving on objects, but $\\mathcal{X}$ is not limit preserving."}
+{"_id": "2563", "title": "examples-lemma-not-algebraic", "text": "There exists a functor $F : \\Sch^{opp} \\to \\textit{Sets}$ which satisfies the sheaf condition for the fpqc topology, has representable diagonal $\\Delta : F \\to F \\times F$, and such that there exists a surjective, flat, universally open, quasi-compact morphism $U \\to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space."}
+{"_id": "2564", "title": "examples-lemma-sheaf-zero-on-low-dimension", "text": "There exists a sheaf of abelian groups $G$ on $\\Sch_\\etale$ with the following properties \\begin{enumerate} \\item $G(X) = 0$ whenever $\\dim(X) < n$, \\item $G(X)$ is not zero if $\\dim(X) \\geq n$, and \\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$. \\end{enumerate}"}
+{"_id": "2565", "title": "examples-lemma-weird-sheaf", "text": "There exists a sheaf of abelian groups $G$ on $\\Sch_\\etale$ with the following properties \\begin{enumerate} \\item $G(\\Spec(k)) = 0$ whenever $k$ is a field, \\item $G$ is limit preserving, \\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$, and \\item $G$ is not zero. \\end{enumerate}"}
+{"_id": "2566", "title": "examples-lemma-lisse-etale-not-functorial", "text": "The lisse-\\'etale site is not functorial, even for morphisms of schemes."}
+{"_id": "2567", "title": "examples-lemma-not-a-morphism-of-sites-noetherian-to-all", "text": "With $S = \\Spec(\\mathbf{F}_p)$ the inclusion functor $(\\textit{Noetherian}/S)_{fppf} \\to (\\Sch/S)_{fppf}$ does not define a morphism of sites."}
+{"_id": "2568", "title": "examples-lemma-is-limit", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{X})$ whose cohomology sheaves are locally quasi-coherent (Sheaves on Stacks, Definition \\ref{stacks-sheaves-definition-locally-quasi-coherent}) and satisfy the flat base change property (Cohomology of Stacks, Definition \\ref{stacks-cohomology-definition-flat-base-change}). Then there exists a distinguished triangle $$ K \\to \\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to \\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to K[1] $$ in $D(\\mathcal{O}_\\mathcal{X})$. In other words, $K$ is the derived limit of its canonical truncations."}
+{"_id": "2570", "title": "examples-lemma-push-not-OK", "text": "A quasi-compact and quasi-separated morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks need not induce a functor $Rf_* : D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$."}
+{"_id": "2571", "title": "examples-lemma-big-abelian-category", "text": "There exists a ``big'' abelian category $\\mathcal{A}$ whose $\\Ext$-groups are proper classes."}
+{"_id": "2572", "title": "examples-lemma-example-schematically-dense-missing-weakly-associated-point", "text": "There exists a reduced scheme $X$ and a schematically dense open $U \\subset X$ such that some weakly associated point $x \\in X$ is not in $U$."}
+{"_id": "2574", "title": "examples-lemma-non-descending-property-projective", "text": "The properties \\begin{enumerate} \\item[] $\\mathcal{P}(f) =$``$f$ is projective'', and \\item[] $\\mathcal{P}(f) =$``$f$ is quasi-projective'' \\end{enumerate} are not Zariski local on the base. A fortiori, they are not fpqc local on the base."}
+{"_id": "2576", "title": "examples-lemma-family-of-curves-not-scheme", "text": "There exists a field $k$ and a family of curves $X \\to \\mathbf{A}^1_k$ such that $X$ is not a scheme."}
+{"_id": "2577", "title": "examples-lemma-no-derived-base-change", "text": "Let $R \\to R'$ and $R \\to A$ be ring maps. In general there does not exist a functor $T : D(A) \\to D(A \\otimes_R R')$ of triangulated categories such that an $A$-module $M$ gives an object $T(M)$ of $D(A \\otimes_R R')$ which maps to $M \\otimes_R^\\mathbf{L} R'$ under the map $D(A \\otimes_R R') \\to D(R')$."}
+{"_id": "2578", "title": "examples-lemma-no-good-representatif-compact-object", "text": "There exists a differential graded algebra $(A, \\text{d})$ and a compact object $E$ of $D(A, \\text{d})$ such that $E$ cannot be represented by a finite and graded projective differential graded $A$-module."}
+{"_id": "2580", "title": "examples-lemma-torsors-over-two-dimensional-regular", "text": "Let $W$ be a two dimensional regular integral Noetherian scheme with function field $K$. Let $G \\to W$ be an abelian scheme. Then the map $H^1_{fppf}(W, G) \\to H^1_{fppf}(\\Spec(K), G)$ is injective."}
+{"_id": "2581", "title": "examples-lemma-torsors-over-field-torsion", "text": "Let $G$ be a smooth commutative group algebraic space over a field $K$. Then $H^1_{fppf}(\\Spec(K), G)$ is torsion."}
+{"_id": "2583", "title": "examples-lemma-non-formal-effectiveness", "text": "Let $k$ be an algebraically closed field which is not the closure of a finite field. Let $A$ be an abelian variety over $k$. Let $\\mathcal{X} = [\\Spec(k)/A]$. There exists an inverse system of $k$-algebras $R_n$ with surjective transition maps whose kernels are locally nilpotent and a system $(\\xi_n)$ of $\\mathcal{X}$ lying over the system $(\\Spec(R_n))$ such that this system is not effective in the sense of Artin's Axioms, Remark \\ref{artin-remark-strong-effectiveness}."}
+{"_id": "2585", "title": "examples-lemma-affine-not-mcquillan", "text": "There exists an affine formal algebraic space which is not McQuillan."}
+{"_id": "2586", "title": "examples-lemma-affine-formal-functions-do-not-separate-points", "text": "There exists an affine formal algebraic space $X$ whose regular functions do not separate points, in the following sense: If we write $X = \\colim X_\\lambda$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-affine-formal-algebraic-space} then $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ is a field, but $X_{red}$ has infinitely many points."}
+{"_id": "2587", "title": "examples-lemma-representable-morphism-affine-formal-not-mcquillan-top", "text": "There exists a representable morphism $f : X \\to Y$ of affine formal algebraic spaces with $Y$ McQuillan, but $X$ not McQuillan."}
+{"_id": "2588", "title": "examples-lemma-weird-flat-map", "text": "There exists a commutative ring $A$ and a flat $A$-algebra $B$ which cannot be written as a filtered colimit of finitely presented flat $A$-algebras. In fact, we may either choose $A$ to be a finite type $\\mathbf{F}_p$-algebra or a $1$-dimensional Noetherian local ring with residue field of characteristic $0$."}
+{"_id": "2589", "title": "examples-lemma-colimit-topology", "text": "There exists a system $G_1 \\to G_2 \\to G_3 \\to \\ldots$ of (abelian) topological groups such that $\\colim G_n$ taken in the category of topological spaces is different from $\\colim G_n$ taken in the category of topological groups."}
+{"_id": "2590", "title": "examples-lemma-universally-submersive-not-V", "text": "There exists a morphism $X \\to Y$ of affine schemes which is universally submersive such that $\\{X \\to Y\\}$ is not a V covering."}
+{"_id": "2591", "title": "examples-lemma-non-fpqc-descent", "text": "There exists a ring $A$ and an infinite family of flat ring maps $\\{A \\to A_i\\}_{i \\in I}$ such that for every $A$-module $M$  $$ M = \\text{Equalizer}\\left( \\xymatrix{ \\prod\\nolimits_{i \\in I} M \\otimes_A A_i \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod\\nolimits_{i, j \\in I} M \\otimes_A A_i \\otimes_A A_j } \\right) $$ but there is no finite subfamily where the same thing is true."}
+{"_id": "2593", "title": "examples-proposition-nonalghomstack", "text": "The stack $\\mathcal{X} = \\underline{\\Mor}_S(X, [S/A])$ is not algebraic."}
+{"_id": "2594", "title": "examples-proposition-localization-and-serre-quotients", "text": "Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$. Let $\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its Serre subcategory of modules for which any element is annihilated by some element of $S$. Then there is a canonical equivalence $\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{S^{-1}A}$."}
+{"_id": "2595", "title": "examples-proposition-quotient-by-torsion-modules", "text": "Let $A$ be a ring. Let $Q(A)$ denote its total quotient ring (as in Algebra, Example \\ref{algebra-example-localize-at-prime}). Let $\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its Serre subcategory of torsion modules. Let $\\text{Mod}_{Q(A)}$ denote the category of $Q(A)$-modules. Then there is a canonical equivalence $\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{Q(A)}$."}
+{"_id": "2601", "title": "bootstrap-theorem-bootstrap", "text": "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item the presheaf $F$ is a sheaf, \\item the diagonal morphism $F  \\to F \\times F$ is representable by algebraic spaces, and \\item there exists an algebraic space $X$ and a map $X \\to F$ which is surjective, and \\'etale. \\end{enumerate} Then $F$ is an algebraic space."}
+{"_id": "2602", "title": "bootstrap-theorem-final-bootstrap", "text": "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Any one of the following conditions implies that $F$ is an algebraic space: \\begin{enumerate} \\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \\to U \\times_S U$ is an equivalence relation, \\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \\to U \\times_S U$ is an equivalence relation, \\item $F$ is a sheaf and there exists an algebraic space $U$ and a morphism $U \\to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation, \\item $F$ is a sheaf and there exists a scheme $U$ and a morphism $U \\to F$ which is representable (by algebraic spaces or schemes), surjective, flat and locally of finite presentation, \\item $F$ is a sheaf, $\\Delta_F$ is representable by algebraic spaces, and there exists an algebraic space $U$ and a morphism $U \\to F$ which is surjective, flat, and locally of finite presentation, or \\item $F$ is a sheaf, $\\Delta_F$ is representable, and there exists a scheme $U$ and a morphism $U \\to F$ which is surjective, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "2604", "title": "bootstrap-lemma-base-change-transformation", "text": "\\begin{slogan} A base change of a representable by algebraic spaces morphism of presheaves is representable by algebraic spaces. \\end{slogan} Let $S$ be a scheme. Let $$ \\xymatrix{ G' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\ G' \\ar[r] & G } $$ be a fibre square of presheaves on $(\\Sch/S)_{fppf}$. If $a$ is representable by algebraic spaces so is $a'$."}
+{"_id": "2605", "title": "bootstrap-lemma-representable-by-spaces-transformation-to-sheaf", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be representable by algebraic spaces. If $G$ is a sheaf, then so is $F$."}
+{"_id": "2607", "title": "bootstrap-lemma-representable-by-spaces-over-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be representable by algebraic spaces. If $G$ is an algebraic space, then so is $F$."}
+{"_id": "2608", "title": "bootstrap-lemma-representable-by-spaces", "text": "Let $S$ be a scheme. Let $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$. Suppose $a : F \\to G$ is representable by algebraic spaces. If $X$ is an algebraic space over $S$, and $X \\to G$ is a map of presheaves then $X \\times_G F$ is an algebraic space."}
+{"_id": "2609", "title": "bootstrap-lemma-composition-transformation", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ F \\ar[r]^a & G \\ar[r]^b & H } $$ be maps of presheaves on $(\\Sch/S)_{fppf}$. If $a$ and $b$ are representable by algebraic spaces, so is $b \\circ a$."}
+{"_id": "2610", "title": "bootstrap-lemma-product-transformations", "text": "Let $S$ be a scheme. Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable by algebraic spaces. Then $$ a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2 $$ is a representable by algebraic spaces."}
+{"_id": "2611", "title": "bootstrap-lemma-representable-by-spaces-permanence", "text": "Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be transformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Assume \\begin{enumerate} \\item $\\Delta : G \\to G \\times_H G$ is representable by algebraic spaces, and \\item $b \\circ a : F \\to H$ is representable by algebraic spaces. \\end{enumerate} Then $a$ is representable by algebraic spaces."}
+{"_id": "2612", "title": "bootstrap-lemma-glueing-sheaves", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $F$ is a sheaf for the Zariski topology on $(\\Sch/S)_{fppf}$, \\item there exists an index set $I$ and subfunctors $F_i \\subset F$ such that \\begin{enumerate} \\item each $F_i$ is an fppf sheaf, \\item each $F_i \\to F$ is representable by algebraic spaces, \\item $\\coprod F_i \\to F$ becomes surjective after fppf sheafification. \\end{enumerate} \\end{enumerate} Then $F$ is an fppf sheaf."}
+{"_id": "2613", "title": "bootstrap-lemma-base-change-transformation-property", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-property-transformation}. Let $$ \\xymatrix{ G' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\ G' \\ar[r] & G } $$ be a fibre square of presheaves on $(\\Sch/S)_{fppf}$. If $a$ is representable by algebraic spaces and has $\\mathcal{P}$ so does $a'$."}
+{"_id": "2614", "title": "bootstrap-lemma-composition-transformation-property", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-property-transformation}, and assume $\\mathcal{P}$ is stable under composition. Let $$ \\xymatrix{ F \\ar[r]^a & G \\ar[r]^b & H } $$ be maps of presheaves on $(\\Sch/S)_{fppf}$. If $a$, $b$ are representable by algebraic spaces and has $\\mathcal{P}$ so does $b \\circ a$."}
+{"_id": "2615", "title": "bootstrap-lemma-product-transformations-property", "text": "Let $S$ be a scheme. Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable by algebraic spaces. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-property-transformation} which is stable under composition. If $a_1$ and $a_2$ have property $\\mathcal{P}$ so does $a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$."}
+{"_id": "2616", "title": "bootstrap-lemma-transformations-property-implication", "text": "Let $S$ be a scheme. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a transformation of functors representable by algebraic spaces. Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in Definition \\ref{definition-property-transformation}. Suppose that for any morphism $f : X \\to Y$ of algebraic spaces over $S$ we have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$. If $a$ has property $\\mathcal{P}$, then $a$ has property $\\mathcal{P}'$."}
+{"_id": "2617", "title": "bootstrap-lemma-surjective-flat-locally-finite-presentation", "text": "Let $S$ be a scheme. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves. Let $a : F \\to G$ be representable by algebraic spaces, flat, locally of finite presentation, and surjective. Then $a : F \\to G$ is surjective as a map of sheaves."}
+{"_id": "2618", "title": "bootstrap-lemma-representable-diagonal", "text": "\\begin{slogan} The diagonal of a presheaf is representable by algebraic spaces if and only if every map from a scheme to the presheaf is representable by algebraic spaces. \\end{slogan} Let $S$ be a scheme. If $F$ is a presheaf on $(\\Sch/S)_{fppf}$. The following are equivalent: \\begin{enumerate} \\item $\\Delta_F : F \\to F \\times F$ is representable by algebraic spaces, \\item for every scheme $T$ any map $T \\to F$ is representable by algebraic spaces, and \\item for every algebraic space $X$ any map $X \\to F$ is representable by algebraic spaces. \\end{enumerate}"}
+{"_id": "2619", "title": "bootstrap-lemma-after-fppf-sep-lqf", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ E \\ar[r]_a \\ar[d]_f & F \\ar[d]^g \\\\ H \\ar[r]^b & G } $$ be a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$, so $E = H \\times_G F$. If \\begin{enumerate} \\item $g$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation, and \\item $a$ is representable by algebraic spaces, separated, and locally quasi-finite \\end{enumerate} then $b$ is representable (by schemes) as well as separated and locally quasi-finite."}
+{"_id": "2620", "title": "bootstrap-lemma-bootstrap-diagonal", "text": "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item the presheaf $F$ is a sheaf, \\item there exists an algebraic space $X$ and a map $X \\to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation. \\end{enumerate} Then $\\Delta_F$ is representable (by schemes)."}
+{"_id": "2622", "title": "bootstrap-lemma-better-finding-opens", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism. Assume \\begin{enumerate} \\item the composition $$ \\xymatrix{ U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h & R \\ar[r]_s & U } $$ has an open image $W \\subset U$, and \\item the resulting map $h : U' \\times_{g, U, t} R \\to W$ defines a surjection of sheaves in the fppf topology. \\end{enumerate} Let $R' = R|_{U'}$ be the restriction of $R$ to $U$. Then the map of quotient sheaves $$ U'/R' \\to U/R $$ in the fppf topology is representable, and is an open immersion."}
+{"_id": "2623", "title": "bootstrap-lemma-slice-equivalence-relation", "text": "Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be an equivalence relation on schemes over $S$. Assume $s, t : R \\to U$ are flat and locally of finite presentation. Then there exists an equivalence relation $j' : R' \\to U'\\times_S U'$ on schemes over $S$, and an isomorphism $$ U'/R' \\longrightarrow U/R $$ induced by a morphism $U' \\to U$ which maps $R'$ into $R$ such that $s', t' : R \\to U$ are flat, locally of finite presentation and locally quasi-finite."}
+{"_id": "2624", "title": "bootstrap-lemma-divide-subgroupoid", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $P \\to R$ be monomorphism of schemes. Assume that \\begin{enumerate} \\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t}P})$ is a groupoid scheme, \\item $s|_P, t|_P : P \\to U$ are finite locally free, \\item $j|_P : P \\to U \\times_S U$ is a monomorphism. \\item $U$ is affine, and \\item $j : R \\to U \\times_S U$ is separated and locally quasi-finite, \\end{enumerate} Then $U/P$ is representable by an affine scheme $\\overline{U}$, the quotient morphism $U \\to \\overline{U}$ is finite locally free, and $P = U \\times_{\\overline{U}} U$. Moreover, $R$ is the restriction of a groupoid scheme $(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$ on $\\overline{U}$ via the quotient morphism $U \\to \\overline{U}$."}
+{"_id": "2625", "title": "bootstrap-lemma-locally-algebraic-space", "text": "\\begin{slogan} The definition of an algebraic space is fppf local. \\end{slogan} Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$. Assume that \\begin{enumerate} \\item $F$ is a sheaf, \\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and \\item $\\coprod_{i \\in I} F_i$ is an algebraic space (see Spaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}). \\end{enumerate} Then $F$ is an algebraic space."}
+{"_id": "2626", "title": "bootstrap-lemma-locally-algebraic-space-finite-type", "text": "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$. Assume that \\begin{enumerate} \\item $F$ is a sheaf, \\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and \\item the morphisms $F_i \\to S_i$ are of finite type. \\end{enumerate} Then $F$ is an algebraic space."}
+{"_id": "2627", "title": "bootstrap-lemma-descend-algebraic-space", "text": "\\begin{slogan} Fppf descent data for algebraic spaces are effective. \\end{slogan} Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic spaces over $S$. \\begin{enumerate} \\item If $I$ is countable\\footnote{The restriction on countablility can be ignored by those who do not care about set theoretical issues. We can allow larger index sets here if we can bound the size of the algebraic spaces which we are descending. See for example Lemma \\ref{lemma-locally-algebraic-space-finite-type}.}, then any descent datum for algebraic spaces relative to $\\{X_i \\to X\\}$ is effective. \\item Any descent datum $(Y_i, \\varphi_{ij})$ relative to $\\{X_i \\to X\\}_{i \\in I}$ (Descent on Spaces, Definition \\ref{spaces-descent-definition-descent-datum-for-family-of-morphisms}) with $Y_i \\to X_i$ of finite type is effective. \\end{enumerate}"}
+{"_id": "2628", "title": "bootstrap-lemma-representable-by-spaces-cover", "text": "Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be transformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Assume \\begin{enumerate} \\item $F, G, H$ are sheaves, \\item $a : F \\to G$ is representable by algebraic spaces, flat, locally of finite presentation, and surjective, and \\item $b \\circ a : F \\to H$ is representable by algebraic spaces. \\end{enumerate} Then $b$ is representable by algebraic spaces."}
+{"_id": "2629", "title": "bootstrap-lemma-quotient-stack-isom", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in Groupoids in Spaces, Definition \\ref{spaces-groupoids-definition-quotient-stack} (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$ is an algebraic space."}
+{"_id": "2630", "title": "bootstrap-lemma-covering-quotient", "text": "Let $S$ be a scheme. Consider an algebraic space $F$ of the form $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \\to U \\times_S U$ is an equivalence relation. Then $U \\to F$ is surjective, flat, and locally of finite presentation."}
+{"_id": "2631", "title": "bootstrap-lemma-quotient-free-action", "text": "Let $S$ be a scheme. Let $X \\to B$ be a morphism of algebraic spaces over $S$. Let $G$ be a group algebraic space over $B$ and let $a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$. If \\begin{enumerate} \\item $a$ is a free action, and \\item $G \\to B$ is flat and locally of finite presentation, \\end{enumerate} then $X/G$ (see Groupoids in Spaces, Definition \\ref{spaces-groupoids-definition-quotient-sheaf}) is an algebraic space and $X \\to X/G$ is surjective, flat, and locally of finite presentation."}
+{"_id": "2632", "title": "bootstrap-lemma-descent-torsor", "text": "Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$. Let $G$ be a group algebraic space over $S$, and denote $G_i = G_{S_i}$ the base changes. Suppose given \\begin{enumerate} \\item for each $i \\in I$ an fppf $G_i$-torsor $X_i$ over $S_i$, and \\item for each $i, j \\in I$ a $G_{S_i \\times_S S_j}$-equivariant isomorphism $\\varphi_{ij} : X_i \\times_S S_j \\to S_i \\times_S X_j$ satisfying the cocycle condition over every $S_i \\times_S S_j \\times_S S_j$. \\end{enumerate} Then there exists an fppf $G$-torsor $X$ over $S$ whose base change to $S_i$ is isomorphic to $X_i$ such that we recover the descent datum $\\varphi_{ij}$."}
+{"_id": "2633", "title": "bootstrap-lemma-spaces-etale", "text": "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object of $\\Sch_\\alpha$. Let $$ F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets} $$ be a presheaf with the following properties: \\begin{enumerate} \\item $F$ is a sheaf for the \\'etale topology, \\item the diagonal $\\Delta : F \\to F \\times F$ is representable, and \\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$ and $U \\to F$ which is surjective and \\'etale. \\end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}."}
+{"_id": "2634", "title": "bootstrap-lemma-spaces-etale-locally-representable", "text": "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object of $\\Sch_\\alpha$. Let $$ F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets} $$ be a presheaf with the following properties: \\begin{enumerate} \\item $F$ is a sheaf for the \\'etale topology, \\item there exists an algebraic space $U$ over $S$ and a map $U \\to F$ which is representable by algebraic spaces, surjective, and \\'etale. \\end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}."}
+{"_id": "2635", "title": "bootstrap-lemma-spaces-etale-smooth-cover", "text": "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object of $\\Sch_\\alpha$.  $$ F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets} $$ be a presheaf with the following properties: \\begin{enumerate} \\item $F$ is a sheaf for the \\'etale topology, \\item the diagonal $\\Delta : F \\to F \\times F$ is representable by algebraic spaces, and \\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$ and $U \\to F$ which is surjective and smooth. \\end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}."}
+{"_id": "2639", "title": "spaces-perfect-theorem-approximation", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then approximation by perfect complexes holds on $X$."}
+{"_id": "2640", "title": "spaces-perfect-theorem-bondal-van-den-Bergh", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The category $D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ such that for  $E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent \\begin{enumerate} \\item $E = 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "2641", "title": "spaces-perfect-theorem-DQCoh-is-Ddga", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Then there exist a differential graded algebra $(E, \\text{d})$ with only a finite number of nonzero cohomology groups $H^i(E)$ such that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent to $D(E, \\text{d})$."}
+{"_id": "2642", "title": "spaces-perfect-lemma-restrict-direct-image-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Given an \\'etale morphism $V \\to Y$, set $U = V \\times_Y X$ and denote $g : U \\to V$ the projection morphism. Then $(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2643", "title": "spaces-perfect-lemma-epsilon-flat", "text": "The morphism $\\epsilon$ of (\\ref{equation-epsilon}) is a flat morphism of ringed sites. In particular the functor $\\epsilon^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_\\etale)$ is exact. Moreover, if $\\epsilon^*\\mathcal{F} = 0$, then $\\mathcal{F} = 0$."}
+{"_id": "2644", "title": "spaces-perfect-lemma-derived-quasi-coherent-small-etale-site", "text": "Let $X$ be a scheme. The functor $\\epsilon^* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_\\etale)$ defined above is an equivalence."}
+{"_id": "2645", "title": "spaces-perfect-lemma-check-quasi-coherence-on-covering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is in $D_\\QCoh(\\mathcal{O}_X)$, \\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is an affine scheme $\\varphi^*E$ is an object of $D_\\QCoh(\\mathcal{O}_U)$, \\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme $\\varphi^*E$ is an object of $D_\\QCoh(\\mathcal{O}_U)$, \\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme such that $\\varphi^*E$ is an object of $D_\\QCoh(\\mathcal{O}_U)$, and \\item there exists a surjective \\'etale morphism of algebraic spaces $f : Y \\to X$ such that $Lf^*E$ is an object of $D_\\QCoh(\\mathcal{O}_Y)$. \\end{enumerate}"}
+{"_id": "2646", "title": "spaces-perfect-lemma-quasi-coherence-direct-sums", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $D_\\QCoh(\\mathcal{O}_X)$ has direct sums."}
+{"_id": "2647", "title": "spaces-perfect-lemma-Rlim-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $(K_n)$ be an inverse system of $D_\\QCoh(\\mathcal{O}_X)$ with derived limit $K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$ is surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$. Then \\begin{enumerate} \\item $H^q(K) = \\lim H^q(K_n)$, \\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and \\item for every affine open $U \\subset X$ we have $H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$. \\end{enumerate}"}
+{"_id": "2648", "title": "spaces-perfect-lemma-quasi-coherence-pullback", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. The functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "2649", "title": "spaces-perfect-lemma-quasi-coherence-tensor-product", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$ the derived tensor product $K \\otimes^\\mathbf{L} L$ is in $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "2650", "title": "spaces-perfect-lemma-nice-K-injective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Then the canonical map $E \\to R\\lim \\tau_{\\geq -n}E$ is an isomorphism\\footnote{In particular, $E$ has a K-injective representative as in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-K-injective}.}."}
+{"_id": "2651", "title": "spaces-perfect-lemma-application-nice-K-injective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$ be a functor and $N \\geq 0$ an integer. Assume that \\begin{enumerate} \\item $F$ is left exact, \\item $F$ commutes with countable direct products, \\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$ quasi-coherent. \\end{enumerate} Then for $E \\in D_\\QCoh(\\mathcal{O}_X)$ \\begin{enumerate} \\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism for $i \\leq a$, \\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism for $i \\geq b$, \\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some $-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$ for $i \\not \\in [a, b + N - 1]$. \\end{enumerate}"}
+{"_id": "2652", "title": "spaces-perfect-lemma-quasi-coherence-direct-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$. \\item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$ such that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$ with $H^m(E) = 0$ for $m > 0$ we have $H^m(Rf_*E) = 0$ for $m \\geq N$. \\item In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$ such that for every morphism of algebraic spaces $Y' \\to Y$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \\to Y'$ is the base change of $f$. \\end{enumerate}"}
+{"_id": "2653", "title": "spaces-perfect-lemma-quasi-coherence-pushforward-direct-sums", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. Then $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ commutes with direct sums."}
+{"_id": "2654", "title": "spaces-perfect-lemma-affine-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. Then $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ reflects isomorphisms."}
+{"_id": "2655", "title": "spaces-perfect-lemma-affine-morphism-pull-push", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. For $E$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have $Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} f_*\\mathcal{O}_X$."}
+{"_id": "2656", "title": "spaces-perfect-lemma-closed-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T \\subset |X|$ be a closed subset. The following are equivalent \\begin{enumerate} \\item the morphism $Z \\to Y$ is proper if $Z$ is the reduced induced algebraic space structure on $T$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-reduced-induced-space}), \\item for some closed subspace $Z \\subset X$ with $|Z| = T$ the morphism $Z \\to Y$ is proper, and \\item for any closed subspace $Z \\subset X$ with $|Z| = T$ the morphism $Z \\to Y$ is proper. \\end{enumerate}"}
+{"_id": "2657", "title": "spaces-perfect-lemma-closed-closed-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T' \\subset T \\subset |X|$ be closed subsets. If $T$ is proper over $Y$, then the same is true for $T'$."}
+{"_id": "2658", "title": "spaces-perfect-lemma-base-change-closed-proper-over-base", "text": "Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces over $S$ $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ with $f$ locally of finite type. If $T$ is a closed subset of $|X|$ proper over $Y$, then $|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$."}
+{"_id": "2659", "title": "spaces-perfect-lemma-functoriality-closed-proper-over-base", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces which are locally of finite type over $B$. \\begin{enumerate} \\item If $Y$ is separated over $B$ and $T \\subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$. \\item If $f$ is universally closed and $T \\subset |X|$ is a closed subset proper over $B$, then $|f|(T)$ is a closed subset of $Y$ proper over $B$. \\item If $f$ is proper and $T \\subset |Y|$ is a closed subset proper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$ proper over $B$. \\end{enumerate}"}
+{"_id": "2660", "title": "spaces-perfect-lemma-union-closed-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T_i \\subset |X|$, $i = 1, \\ldots, n$ be closed subsets. If $T_i$, $i = 1, \\ldots, n$ are proper over $Y$, then the same is true for $T_1 \\cup \\ldots \\cup T_n$."}
+{"_id": "2661", "title": "spaces-perfect-lemma-module-support-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item the support of $\\mathcal{F}$ is proper over $Y$, \\item the scheme theoretic support of $\\mathcal{F}$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-scheme-theoretic-support}) is proper over $Y$, and \\item there exists a closed subspace $Z \\subset X$ and a finite type, quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ such that (a) $Z \\to Y$ is proper, and (b) $(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$. \\end{enumerate}"}
+{"_id": "2664", "title": "spaces-perfect-lemma-support-proper-over-base-pushforward", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module with support proper over $Y$. Then $R^pf_*\\mathcal{F}$ is a coherent $\\mathcal{O}_Y$-module for all $p \\geq 0$."}
+{"_id": "2665", "title": "spaces-perfect-lemma-direct-image-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $Y$ for all $i$. Then $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$."}
+{"_id": "2666", "title": "spaces-perfect-lemma-direct-image-coherent-bdd-below", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian. Let $E$ be an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$."}
+{"_id": "2667", "title": "spaces-perfect-lemma-coherent-internal-hom", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then $R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "2668", "title": "spaces-perfect-lemma-ext-finite", "text": "Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$. For $L$ in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules $\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite."}
+{"_id": "2669", "title": "spaces-perfect-lemma-make-more-elementary-distinguished-squares", "text": "Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be an elementary distinguished square of algebraic spaces over $S$. \\begin{enumerate} \\item If $V' \\subset V$ and $U \\subset U' \\subset W$ are open subspaces and $W' = U' \\cup f(V')$ then $(U' \\subset W', f|_{V'} : V' \\to W')$ is an elementary distinguished square. \\item If $p : W' \\to W$ is a morphism of algebraic spaces, then $(p^{-1}(U) \\subset W', V \\times_W W' \\to W')$ is an elementary distinguished square. \\item If $S' \\to S$ is a morphism of schemes, then $(S' \\times_S U \\subset S' \\times_S W, S' \\times_S V \\to S' \\times_S W)$ is an elementary distinguished square. \\end{enumerate}"}
+{"_id": "2670", "title": "spaces-perfect-lemma-induction-principle", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P$ be a property of the quasi-compact and quasi-separated objects of $X_{spaces, \\etale}$. Assume that \\begin{enumerate} \\item $P$ holds for every affine object of $X_{spaces, \\etale}$, \\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$ such that \\begin{enumerate} \\item $W$ is a quasi-compact and quasi-separated object of $X_{spaces, \\etale}$, \\item $U$ is quasi-compact, \\item $V$ is affine, and \\item $P$ holds for $U$, $V$, and $U \\times_W V$, \\end{enumerate} then $P$ holds for $W$. \\end{enumerate} Then $P$ holds for every quasi-compact and quasi-separated object of $X_{spaces, \\etale}$ and in particular for $X$."}
+{"_id": "2671", "title": "spaces-perfect-lemma-induction-principle-separated", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$. Let $P$ be a property of the elements of $\\mathcal{B}$. Assume that \\begin{enumerate} \\item every $W \\in \\mathcal{B}$ is quasi-compact and quasi-separated, \\item if $W \\in \\mathcal{B}$ and $U \\subset W$ is quasi-compact open, then $U \\in \\mathcal{B}$, \\item if $V \\in \\Ob(X_{spaces, \\etale})$ is affine, then (a) $V \\in \\mathcal{B}$ and (b) $P$ holds for $V$, \\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$ such that \\begin{enumerate} \\item $W \\in \\mathcal{B}$, \\item $U$ is quasi-compact, \\item $V$ is affine, and \\item $P$ holds for $U$, $V$, and $U \\times_W V$, \\end{enumerate} then $P$ holds for $W$. \\end{enumerate} Then $P$ holds for every $W \\in \\mathcal{B}$."}
+{"_id": "2672", "title": "spaces-perfect-lemma-induction-principle-enlarge", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \\subset X$ be a quasi-compact open subspace. Let $P$ be a property of quasi-compact open subspaces of $X$. Assume that \\begin{enumerate} \\item $P$ holds for $W$, and \\item for every elementary distinguished square $(W_1 \\subset W_2, f : V \\to W_2)$ where  such that \\begin{enumerate} \\item $W_1$, $W_2$ are quasi-compact open subspaces of $X$, \\item $W \\subset W_1$, \\item $V$ is affine, and \\item $P$ holds for $W_1$, \\end{enumerate} then $P$ holds for $W_2$. \\end{enumerate} Then $P$ holds for $X$."}
+{"_id": "2673", "title": "spaces-perfect-lemma-exact-sequence-lower-shriek", "text": "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. \\begin{enumerate} \\item For a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have a short exact sequence $$ 0 \\to j_{U \\times_X V!}\\mathcal{F}|_{U \\times_X V} \\to j_{U!}\\mathcal{F}|_U \\oplus j_{V!}\\mathcal{F}|_V \\to \\mathcal{F} \\to 0 $$ \\item For an object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ j_{U \\times_X V!}E|_{U \\times_X V} \\to j_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to  j_{U \\times_X V!}E|_{U \\times_X V}[1] $$ in $D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "2674", "title": "spaces-perfect-lemma-exact-sequence-j-star", "text": "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. \\begin{enumerate} \\item For every sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have a short exact sequence $$ 0 \\to \\mathcal{F} \\to j_{U, *}\\mathcal{F}|_U \\oplus j_{V, *}\\mathcal{F}|_V \\to j_{U \\times_X V, *}\\mathcal{F}|_{U \\times_X V} \\to 0 $$ \\item For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ E \\to  Rj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to Rj_{U \\times_X V, *}E|_{U \\times_X V} \\to E[1] $$ in $D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "2675", "title": "spaces-perfect-lemma-unbounded-relative-mayer-vietoris", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square. Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and $c = f|_{U \\times_X V} : U \\times_X V \\to Y$ the restrictions. For every object $E$ of $D(\\mathcal{O}_X)$ there exists a distinguished triangle $$ Rf_*E \\to Ra_*(E|_U) \\oplus Rb_*(E|_V) \\to Rc_*(E|_{U \\times_X V}) \\to Rf_*E[1] $$ in $D(\\mathcal{O}_Y)$. This triangle is functorial in $E$."}
+{"_id": "2676", "title": "spaces-perfect-lemma-mayer-vietoris-hom", "text": "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. For objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a Mayer-Vietoris sequence $$ \\xymatrix{ & \\ldots \\ar[r] & \\Ext^{-1}(E_{U \\times_X V}, F_{U \\times_X V}) \\ar[lld] \\\\ \\Hom(E, F) \\ar[r] & \\Hom(E_U, F_U) \\oplus \\Hom(E_V, F_V) \\ar[r] & \\Hom(E_{U \\times_X V}, F_{U \\times_X V}) } $$ where the subscripts denote restrictions to the relevant opens and the $\\Hom$'s are taken in the relevant derived categories."}
+{"_id": "2677", "title": "spaces-perfect-lemma-unbounded-mayer-vietoris", "text": "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. For an object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished triangle $$ R\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to R\\Gamma(U \\times_X V, E) \\to R\\Gamma(X, E)[1] $$ and in particular a long exact cohomology sequence $$ \\ldots \\to H^n(X, E) \\to H^n(U, E) \\oplus H^n(V, E) \\to H^n(U \\times_X V, E) \\to H^{n + 1}(X, E) \\to \\ldots $$ The construction of the distinguished triangle and the long exact sequence is functorial in $E$."}
+{"_id": "2678", "title": "spaces-perfect-lemma-restrict-lower-shriek", "text": "Let $S$ be a scheme. Let $j : U \\to X$ be a \\'etale morphism of algebraic spaces over $S$. Given an \\'etale morphism $V \\to Y$, set $W = V \\times_X U$ and denote $j_W : W \\to V$ the projection morphism. Then $(j_!E)|_V = j_{W!}(E|_W)$ for $E$ in $D(\\mathcal{O}_U)$."}
+{"_id": "2679", "title": "spaces-perfect-lemma-pushforward-with-support-in-open", "text": "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. Set $T = |X| \\setminus |U|$. \\begin{enumerate} \\item If $E$ is an object of $D(\\mathcal{O}_X)$ supported on $T$, then (a) $E \\to Rj_*(E|_V)$ and (b) $j_!(E|_V) \\to E$ are isomorphisms. \\item If $F$ is an object of $D(\\mathcal{O}_V)$ supported on $j^{-1}T$, then (a) $F \\to (j_!F)|_V$, (b) $(Rj_*F)|_V \\to F$, and (c) $j_!F \\to Rj_*F$ are isomorphisms. \\end{enumerate}"}
+{"_id": "2680", "title": "spaces-perfect-lemma-glue", "text": "Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. Suppose given \\begin{enumerate} \\item an object $A$ of $D(\\mathcal{O}_U)$, \\item an object $B$ of $D(\\mathcal{O}_V)$, and \\item an isomorphism $c : A|_{U \\times_X V} \\to B|_{U \\times_X V}$. \\end{enumerate} Then there exists an object $F$ of $D(\\mathcal{O}_X)$ and isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such that $c = g|_{U \\times_X V} \\circ f^{-1}|_{U \\times_X V}$. Moreover, given \\begin{enumerate} \\item an object $E$ of $D(\\mathcal{O}_X)$, \\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$, \\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$, \\end{enumerate} such that $$ a|_{U \\times_X V}  = b|_{U \\times_X V} \\circ c. $$ Then there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$ whose restriction to $U$ is $a \\circ f$ and whose restriction to $V$ is $b \\circ g$."}
+{"_id": "2681", "title": "spaces-perfect-lemma-affine-pushforward", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. Then $f_*$ defines a derived functor $f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$. This functor has the property that $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ commutes."}
+{"_id": "2682", "title": "spaces-perfect-lemma-flat-pushforward-coherator", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact, quasi-separated, and flat. Then, denoting $$ \\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y)) $$ the right derived functor of $f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ we have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$."}
+{"_id": "2683", "title": "spaces-perfect-lemma-affine-coherator", "text": "Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$. Set $A = \\Gamma(X, \\mathcal{O}_X)$. Then \\begin{enumerate} \\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$ is the functor which sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module associated to the $A$-module $\\Gamma(X, \\mathcal{F})$, \\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$ is the functor which sends $E$ to the complex of quasi-coherent $\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$, \\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor $RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}). \\end{enumerate}"}
+{"_id": "2684", "title": "spaces-perfect-lemma-argument-proves", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Suppose that for every \\'etale morphism $j : V \\to W$ with $W \\subset X$ quasi-compact open and $V$ affine the right derived functor $$ \\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_W)) $$ of the left exact functor $j_* : \\QCoh(\\mathcal{O}_V) \\to \\QCoh(\\mathcal{O}_W)$ fits into a commutative diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_V)) \\ar[d]_\\Phi \\ar[r]_{i_V} & D_\\QCoh(\\mathcal{O}_V) \\ar[d]^{Rj_*} \\\\ D(\\QCoh(\\mathcal{O}_W)) \\ar[r]^{i_W} & D_\\QCoh(\\mathcal{O}_W) } $$ Then the functor (\\ref{equation-compare}) $$ D(\\QCoh(\\mathcal{O}_X)) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$."}
+{"_id": "2686", "title": "spaces-perfect-lemma-affine-injective-colimit-direct-sum-pushforwards-artin", "text": "Let $S$ be a Noetherian affine scheme. Every injective object of $\\QCoh(\\mathcal{O}_S)$ is a filtered colimit $\\colim_i \\mathcal{F}_i$ of quasi-coherent sheaves of the form $$ \\mathcal{F}_i = (Z_i \\to S)_*\\mathcal{G}_i $$ where $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$ is a coherent module on $Z_i$."}
+{"_id": "2687", "title": "spaces-perfect-lemma-injective-colimit-direct-sum-pushforwards-artin", "text": "Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space over $S$. Every injective object of $\\QCoh(\\mathcal{O}_X)$ is a direct summand of a filtered colimit $\\colim_i \\mathcal{F}_i$ of quasi-coherent sheaves of the form $$ \\mathcal{F}_i = (Z_i \\to X)_*\\mathcal{G}_i $$ where $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$ is a coherent module on $Z_i$."}
+{"_id": "2689", "title": "spaces-perfect-lemma-injective-pushforward", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If $\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$, then \\begin{enumerate} \\item $H^p(U, \\mathcal{J}|_U) = 0$ for $p > 0$ and for every quasi-compact and quasi-separated algebraic space $U$ \\'etale over $X$, \\item for any morphism $f : X \\to Y$ of algebraic spaces over $S$ with $Y$ quasi-separated we have $R^pf_*\\mathcal{J} = 0$ for $p > 0$. \\end{enumerate}"}
+{"_id": "2690", "title": "spaces-perfect-lemma-Noetherian-pushforward", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of Noetherian algebraic spaces over $S$. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$ such that the diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ commutes."}
+{"_id": "2691", "title": "spaces-perfect-lemma-descend-finite-type", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is of finite type as an $\\mathcal{O}_X$-module, and \\item $\\epsilon^*\\mathcal{F}$ is of finite type as an $\\mathcal{O}_\\etale$-module on the small \\'etale site of $X$. \\end{enumerate} Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2692", "title": "spaces-perfect-lemma-descend-pseudo-coherent", "text": "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent, and \\item $\\epsilon^*E$ is $m$-pseudo-coherent on the small \\'etale site of $X$. \\end{enumerate} Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2693", "title": "spaces-perfect-lemma-descend-tor-amplitude", "text": "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item $E$ has tor amplitude in $[a, b]$ if and only if $\\epsilon^*E$ has tor amplitude in $[a, b]$. \\item $E$ has finite tor dimension if and only if $\\epsilon^*E$ has finite tor dimension. \\end{enumerate} Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2694", "title": "spaces-perfect-lemma-tor-dimension-rel", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $E$ be an object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item $E$ as an object of $D(f^{-1}\\mathcal{O}_Y)$ has tor amplitude in $[a, b]$ if and only if $\\epsilon^*E$ has tor amplitude in $[a, b]$ as an object of $D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$. \\item $E$ locally has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_Y)$ if and only if $\\epsilon^*E$ locally has finite tor dimension as an object of $D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$. \\end{enumerate} Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2695", "title": "spaces-perfect-lemma-descend-perfect", "text": "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\\mathcal{O}_X)$ if and only if $\\epsilon^*E$ is a perfect object of $D(\\mathcal{O}_\\etale)$. Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2696", "title": "spaces-perfect-lemma-pseudo-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $E$ is an $m$-pseudo-coherent object of $D(\\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$. If $E$ is pseudo-coherent, then $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "2697", "title": "spaces-perfect-lemma-identify-pseudo-coherent-noetherian", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. For $m \\in \\mathbf{Z}$ the following are equivalent \\begin{enumerate} \\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and \\item $E$ is $m$-pseudo-coherent. \\end{enumerate} In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "2698", "title": "spaces-perfect-lemma-tor-qc-qs", "text": "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$. The following are equivalent \\begin{enumerate} \\item $E$ has tor amplitude in $[a, b]$, and \\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$ we have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$ for $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "2699", "title": "spaces-perfect-lemma-descend-RHom", "text": "Let $X$ be a scheme. Let $E, F$ be objects of $D(\\mathcal{O}_X)$. Assume either \\begin{enumerate} \\item $E$ is pseudo-coherent and $F$ lies in $D^+(\\mathcal{O}_X)$, or \\item $E$ is perfect and $F$ arbitrary, \\end{enumerate} then there is a canonical isomorphism $$ \\epsilon^*R\\SheafHom(E, F) \\longrightarrow R\\SheafHom(\\epsilon^*E, \\epsilon^*F) $$ Here $\\epsilon$ is as in (\\ref{equation-epsilon})."}
+{"_id": "2700", "title": "spaces-perfect-lemma-quasi-coherence-internal-hom", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $L, K$ be objects of $D(\\mathcal{O}_X)$. If either \\begin{enumerate} \\item $L$ in $D^+_\\QCoh(\\mathcal{O}_X)$ and $K$ is pseudo-coherent, \\item $L$ in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ is perfect, \\end{enumerate} then $R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "2701", "title": "spaces-perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$. The map $$ K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L) \\longrightarrow R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) $$ of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \\begin{enumerate} \\item $M$ perfect, or \\item $K$ is perfect, or \\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "2702", "title": "spaces-perfect-lemma-pushforward-perfect", "text": "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary distinguished square of algebraic space over $S$. Let $E$ be a perfect object of $D(\\mathcal{O}_V)$ supported on $j^{-1}(T)$ where $T = |X| \\setminus |U|$. Then $Rj_*E$ is a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "2703", "title": "spaces-perfect-lemma-open", "text": "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. Let $T$ be a closed subset of $|X| \\setminus |U|$ and let $(T, E, m)$ be a triple as in Definition \\ref{definition-approximation-holds}. If \\begin{enumerate} \\item approximation holds for $(j^{-1}T, E|_V, m)$, and \\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$, \\end{enumerate} then approximation holds for $(T, E, m)$."}
+{"_id": "2704", "title": "spaces-perfect-lemma-approximation-affine", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is representable by an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds} such that there exists an integer $r \\geq 0$ with \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent, \\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$, \\item $X \\setminus T$ is the union of $r$ affine opens. \\end{enumerate} In particular, approximation by perfect complexes holds for affine schemes."}
+{"_id": "2705", "title": "spaces-perfect-lemma-induction-step", "text": "Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $U$ quasi-compact, $V$ affine, and $U \\times_X V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation by perfect complexes holds on $X$."}
+{"_id": "2706", "title": "spaces-perfect-lemma-lift-map-from-perfect-complex-with-support", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W \\subset X$ be a quasi-compact open. Let $T \\subset |X|$ be a closed subset such that $X \\setminus T \\to X$ is a quasi-compact morphism. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $\\alpha : P \\to E|_W$ be a map where $P$ is a perfect object of $D(\\mathcal{O}_W)$ supported on $T \\cap W$. Then there exists a map $\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$ supported on $T$ such that $P$ is a direct summand of $R|_W$ in $D(\\mathcal{O}_W)$ compatible $\\alpha$ and $\\beta|_W$."}
+{"_id": "2707", "title": "spaces-perfect-lemma-direct-summand-of-a-restriction", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $W$ be a quasi-compact open subspace of $X$. Let $P$ be a perfect object of $D(\\mathcal{O}_W)$. Then $P$ is a direct summand of the restriction of a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "2708", "title": "spaces-perfect-lemma-generator-with-support", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$ is quasi-compact. With notation as above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a single perfect object."}
+{"_id": "2709", "title": "spaces-perfect-lemma-compact-is-perfect-with-support", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$ is quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact if and only if it is perfect as an object of $D(\\mathcal{O}_X)$."}
+{"_id": "2711", "title": "spaces-perfect-lemma-tensor-with-QCoh-complex", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K^\\bullet$ be a complex of $\\mathcal{O}_X$-modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$ be the endomorphism differential graded algebra. Then the functor $$ - \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X) $$ of Differential Graded Algebra, Lemma \\ref{dga-lemma-tensor-with-complex-derived} has image contained in $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "2712", "title": "spaces-perfect-lemma-ext-from-perfect-into-bounded-QCoh", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item there exist integers $a \\leq b$ such that $\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$ for $L \\in D_\\QCoh(\\mathcal{O}_X)$ with $H^i(L) = 0$ for $i \\in [a, b]$, and \\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$ is zero for all but finitely many $n$. \\end{enumerate}"}
+{"_id": "2713", "title": "spaces-perfect-lemma-pseudo-coherent-hocolim", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K \\in D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $K$ is pseudo-coherent, and \\item $K = \\text{hocolim} K_n$ where $K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$. \\end{enumerate}"}
+{"_id": "2715", "title": "spaces-perfect-lemma-better-coherator", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$ has a right adjoint."}
+{"_id": "2716", "title": "spaces-perfect-lemma-pushforward-better-coherator", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. If the right adjoints $DQ_X$ and $DQ_Y$ of the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then $$ Rf_* \\circ DQ_X = DQ_Y \\circ Rf_* $$"}
+{"_id": "2717", "title": "spaces-perfect-lemma-boundedness-better-coherator", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_X$ of Lemma \\ref{lemma-better-coherator} has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\\mathcal{O}_X)$ with $H^i(U, K) = 0$ for $U$ affine \\'etale over $X$ and $i \\not \\in [a, b]$, then the cohomology sheaves $H^i(DQ_X(K))$ are zero for $i \\not \\in [a, b + N]$."}
+{"_id": "2718", "title": "spaces-perfect-lemma-cohomology-base-change", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have $$ Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K) $$"}
+{"_id": "2719", "title": "spaces-perfect-lemma-tor-independent", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. The following are equivalent \\begin{enumerate} \\item $X$ and $Y$ are Tor independent over $B$, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & W \\ar[d] & V \\ar[d] \\ar[l] \\\\ X \\ar[r] & B & Y \\ar[l] } $$ with \\'etale vertical arrows $U$ and $V$ are Tor independent over $W$, \\item for some commutative diagram as in (2) with (a) $W \\to B$ \\'etale surjective, (b) $U \\to X \\times_B W$ \\'etale surjective, (c) $V \\to Y \\times_B W$ \\'etale surjective, the spaces $U$ and $V$ are Tor independent over $W$, and \\item for some commutative diagram as in (3) with $U$, $V$, $W$ schemes, the schemes $U$ and $V$ are Tor independent over $W$ in the sense of Derived Categories of Schemes, Definition \\ref{perfect-definition-tor-independent}. \\end{enumerate}"}
+{"_id": "2720", "title": "spaces-perfect-lemma-compare-base-change", "text": "Let $S$ be a scheme. Let $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Consider the base change diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ If $X$ and $Y'$ are Tor independent over $Y$, then for all $E \\in D_\\QCoh(\\mathcal{O}_X)$ we have $Rf'_*L(g')^*E = Lg^*Rf_*E$."}
+{"_id": "2721", "title": "spaces-perfect-lemma-affine-morphism-and-hom-out-of-perfect", "text": "Let $g : S' \\to S$ be a morphism of affine schemes. Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ of quasi-compact and quasi-separated algebraic spaces. Assume $g$ and $f$ Tor independent. Write $S = \\Spec(R)$ and $S' = \\Spec(R')$. For $M, K \\in D(\\mathcal{O}_X)$ the canonical map $$ R\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R' \\longrightarrow R\\Hom_{X'}(L(g')^*M, L(g')^*K) $$ in $D(R')$ is an isomorphism in the following two cases \\begin{enumerate} \\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or \\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent, $K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$. \\end{enumerate}"}
+{"_id": "2722", "title": "spaces-perfect-lemma-tor-independence-and-tor-amplitude", "text": "Let $S$ be a scheme. Consider a cartesian square of algebraic spaces $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ over $S$. Assume $g$ and $f$ Tor independent. \\begin{enumerate} \\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_Y$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{Y'}$-modules. \\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat over $Y$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "2723", "title": "spaces-perfect-lemma-single-complex-base-change-condition", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. The following are equivalent \\begin{enumerate} \\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc}) is an isomorphism, \\item for any commutative diagram $$ \\xymatrix{ & U \\ar[d] \\ar[rd]^a \\\\ V' \\ar[r] \\ar[rd]^c & V \\ar[rd]^b & X \\ar[d]^f \\\\ & Y' \\ar[r]^g & Y } $$ with $a, b, c$ \\'etale, $U, V, V'$ schemes, and with $U' = V' \\times_V U$ the equivalent conditions of Derived Categories of Schemes, Lemma \\ref{lemma-single-complex-base-change-condition} hold for $(U \\to X)^*K$ and $(U' \\to X')^*K'$, and \\item there is some diagram as in (2) with $U' \\to X'$ surjective. \\end{enumerate}"}
+{"_id": "2724", "title": "spaces-perfect-lemma-single-complex-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If \\begin{enumerate} \\item the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold, and \\item $f$ is quasi-compact and quasi-separated, \\end{enumerate} then the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$ is an isomorphism."}
+{"_id": "2725", "title": "spaces-perfect-lemma-single-complex-base-change-condition-inherited", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of algebraic spaces over $S$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold, then \\begin{enumerate} \\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$, \\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and \\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$ pseudo-coherent the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$. \\end{enumerate}"}
+{"_id": "2726", "title": "spaces-perfect-lemma-base-change-tensor", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$ be a bounded above complex of quasi-coherent $\\mathcal{O}_X$-modules flat over $Y$. Then formation of $$ Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet) $$ commutes with arbitrary base change (see proof for precise statement)."}
+{"_id": "2727", "title": "spaces-perfect-lemma-base-change-RHom", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules. If \\begin{enumerate} \\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above, and $\\mathcal{G}^n$ is flat over $Y$, or \\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded, and $\\mathcal{G}^n$ is flat over $Y$, \\end{enumerate} then formation of $$ Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet) $$ commutes with arbitrary base change (see proof for precise statement)."}
+{"_id": "2728", "title": "spaces-perfect-lemma-perfect-direct-image", "text": "Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ such that \\begin{enumerate} \\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item the support of $H^i(E)$ is proper over $Y$ for all $i$, \\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_Y)$. \\end{enumerate} Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$."}
+{"_id": "2729", "title": "spaces-perfect-lemma-tensor-perfect", "text": "Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \\to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent $\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$ is a perfect object of $D(\\mathcal{O}_B)$."}
+{"_id": "2730", "title": "spaces-perfect-lemma-ext-perfect", "text": "Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \\to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent $\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then $K = Rf_*R\\SheafHom(E, \\mathcal{G})$ is a perfect object of $D(\\mathcal{O}_B)$."}
+{"_id": "2731", "title": "spaces-perfect-lemma-compute-tensor-perfect", "text": "Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}. Then there are functorial isomorphisms $$ H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}) \\longrightarrow H^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} (\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})) $$ for $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof)."}
+{"_id": "2732", "title": "spaces-perfect-lemma-compute-ext-perfect", "text": "Assumption and notation as in Lemma \\ref{lemma-ext-perfect}. Then there are functorial isomorphisms $$ H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof)."}
+{"_id": "2733", "title": "spaces-perfect-lemma-compute-ext", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$, $E \\in D(\\mathcal{O}_X)$, and $\\mathcal{F}^\\bullet$ a complex of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $B$ is Noetherian, \\item $f$ is locally of finite type and quasi-separated, \\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item $\\mathcal{G}^\\bullet$ is a bounded complex of coherent $\\mathcal{O}_X$-module flat over $B$ with support proper over $B$. \\end{enumerate} Then the following two statements are true \\begin{enumerate} \\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$ of $D(\\mathcal{O}_B)$ and functorial maps $$ \\alpha^i_\\mathcal{F} : \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) \\longrightarrow H^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof) such that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$, and \\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_B)$ and functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_B}(L, \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps. \\end{enumerate}"}
+{"_id": "2734", "title": "spaces-perfect-lemma-descend-homomorphisms", "text": "In Situation \\ref{situation-descent}. Let $E_0$ and $K_0$ be objects of $D(\\mathcal{O}_{X_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \\geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map $$ \\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(E_i, K_i) \\longrightarrow \\Hom_{D(\\mathcal{O}_X)}(E, K) $$ is an isomorphism if either \\begin{enumerate} \\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$, or \\item $E_0$ is pseudo-coherent and $K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "2735", "title": "spaces-perfect-lemma-perfect-on-limit", "text": "In Situation \\ref{situation-descent} the category of perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories of perfect objects of $D(\\mathcal{O}_{X_i})$."}
+{"_id": "2736", "title": "spaces-perfect-lemma-base-change-tensor-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented $\\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet) $$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change."}
+{"_id": "2737", "title": "spaces-perfect-lemma-base-change-tensor-pseudo-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object. Let $\\mathcal{G}^\\bullet$ be a bounded above complex of finitely presented $\\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet) $$ is a pseudo-coherent object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change."}
+{"_id": "2738", "title": "spaces-perfect-lemma-flat-proper-perfect-direct-image-general", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of finite presentation of algebraic spaces over $S$. \\begin{enumerate} \\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change. \\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change. \\end{enumerate}"}
+{"_id": "2739", "title": "spaces-perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat proper morphism of finite presentation of algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change."}
+{"_id": "2740", "title": "spaces-perfect-lemma-pullback-and-limits", "text": "Let $R$ be a ring. Let $X$ be an algebraic space and let $f : X \\to \\Spec(R)$ be proper, flat, and of finite presentation. Let $(M_n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map $$ \\mathcal{O}_X \\otimes_R (\\lim M_n) \\longrightarrow \\lim \\mathcal{O}_X \\otimes_R M_n $$ induces an isomorphism from the source to $DQ_X$ applied to the target."}
+{"_id": "2741", "title": "spaces-perfect-lemma-perfect-enough", "text": "Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is quasi-compact and quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$. If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$, then $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$."}
+{"_id": "2742", "title": "spaces-perfect-lemma-base-change-RHom-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented $\\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then $$ K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet) $$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change."}
+{"_id": "2743", "title": "spaces-perfect-lemma-jump-loci", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent (for example perfect). For any $i \\in \\mathbf{Z}$ consider the function $$ \\beta_i : |X| \\longrightarrow \\{0, 1, 2, \\ldots\\} $$ defined above. Then we have \\begin{enumerate} \\item formation of $\\beta_i$ commutes with arbitrary base change, \\item the functions $\\beta_i$ are upper semi-continuous, and \\item the level sets of $\\beta_i$ are \\'etale locally constructible. \\end{enumerate}"}
+{"_id": "2744", "title": "spaces-perfect-lemma-jump-loci-geometric", "text": "Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \\to Y$ is flat, proper, and of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $Y$. For fixed $i \\in \\mathbf{Z}$ consider the function $$ \\beta_i : |Y| \\to \\{0, 1, 2, \\ldots\\},\\quad y \\longmapsto \\dim_{\\kappa(y)} H^i(X_y, \\mathcal{F}_y) $$ Then we have \\begin{enumerate} \\item formation of $\\beta_i$ commutes with arbitrary base change, \\item the functions $\\beta_i$ are upper semi-continuous, and \\item the level sets of $\\beta_i$ are locally constructible in $Y$. \\end{enumerate}"}
+{"_id": "2746", "title": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Given $i, r \\in \\mathbf{Z}$, there exists an open subspace $U \\subset X$ characterized by the following \\begin{enumerate} \\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free $\\mathcal{O}_U$-module of rank $r$, \\item a morphism $f : Y \\to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$. \\end{enumerate}"}
+{"_id": "2747", "title": "spaces-perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is proper, flat, and of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $Y$. Fix $i, r \\in \\mathbf{Z}$. Then there exists an open subspace $V \\subset Y$ with the following property: A morphism $T \\to Y$ factors through $V$ if and only if $Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$."}
+{"_id": "2748", "title": "spaces-perfect-lemma-locally-closed-where-H0-locally-free", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\in D(\\mathcal{O}_X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$. Let $r \\geq 0$. Then there exists a locally closed subspace $j : Z \\to X$ characterized by the following \\begin{enumerate} \\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and \\item a morphism $f : Y \\to X$ factors through $Z$ if and only if for all morphisms $g : Y' \\to Y$ the $\\mathcal{O}_{Y'}$-module $H^a(L(f \\circ g)^*E)$ is locally free of rank $r$. \\end{enumerate} Moreover, $j : Z \\to X$ is of finite presentation and we have \\begin{enumerate} \\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then $H^a(Lf^*E) = g^*H^a(Lj^*E)$, \\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in |X|$, then $j$ is a closed immersion and given $f : Y \\to X$ the following are equivalent \\begin{enumerate} \\item $f : Y \\to X$ factors through $Z$, \\item $H^0(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$, \\end{enumerate} and if $r = 1$ these are also equivalent to \\begin{enumerate} \\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2749", "title": "spaces-perfect-lemma-proper-flat-h0", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item for a morphism $\\Spec(k) \\to Y$ where $k$ is a field, we have $k = H^0(X_k, \\mathcal{O}_{X_k})$. \\end{enumerate} Then we have \\begin{enumerate} \\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after any base change, \\item[(b)] \\'etale locally on $Y$ we have $$ Rf_*\\mathcal{O}_X = \\mathcal{O}_Y \\oplus P $$ in $D(\\mathcal{O}_Y)$ where $P$ is perfect of tor amplitude in $[1, \\infty)$. \\end{enumerate}"}
+{"_id": "2750", "title": "spaces-perfect-lemma-proper-flat-geom-red-connected", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item the geometric fibres of $f$ are reduced and connected. \\end{enumerate} Then $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and this holds after any base change."}
+{"_id": "2751", "title": "spaces-perfect-lemma-countable-cohomology", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$ such that the cohomology sheaves $H^i(K)$ have countable sets of sections over affine schemes \\'etale over $X$. Then for any quasi-compact and quasi-separated \\'etale morphism $U \\to X$ and any perfect object $E$ in $D(\\mathcal{O}_X)$ the sets $$ H^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U) $$ are countable."}
+{"_id": "2752", "title": "spaces-perfect-lemma-countable", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume the sets of sections of $\\mathcal{O}_X$ over affines \\'etale over $X$ are countable. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of $D(\\mathcal{O}_X)$, and \\item the cohomology sheaves $H^i(K)$ have countable sets of sections over affines \\'etale over $X$. \\end{enumerate}"}
+{"_id": "2753", "title": "spaces-perfect-lemma-computing-sections-as-colim", "text": "Let $A$ be a ring. Let $f : U \\to X$ be a flat morphism of algebraic spaces of finite presentation over $A$. Then \\begin{enumerate} \\item there exists an inverse system of perfect objects $L_n$ of $D(\\mathcal{O}_X)$ such that $$ R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K) $$ in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and \\item there exists a system of perfect objects $E_n$ of $D(\\mathcal{O}_X)$ such that $$ R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K) $$ in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "2754", "title": "spaces-perfect-lemma-bounded-truncation", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$. Let $a \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$. \\end{enumerate}"}
+{"_id": "2755", "title": "spaces-perfect-lemma-bounded-below-truncation", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$. Let $a \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$. \\end{enumerate}"}
+{"_id": "2756", "title": "spaces-perfect-proposition-quasi-compact-affine-diagonal", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$ with affine diagonal. Then the functor (\\ref{equation-compare}) $$ D(\\QCoh(\\mathcal{O}_X)) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$."}
+{"_id": "2758", "title": "spaces-perfect-proposition-compact-is-perfect", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. An object of $D_\\QCoh(\\mathcal{O}_X)$ is compact if and only if it is perfect."}
+{"_id": "2759", "title": "spaces-perfect-proposition-detecting-bounded-above", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates $D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$, \\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$, \\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$, \\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$, \\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$, \\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^-(\\mathbf{Z})$, \\item for every perfect object $P$ of $D(\\mathcal{O}_X)$ \\begin{enumerate} \\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$, \\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^-(\\mathbf{Z})$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2760", "title": "spaces-perfect-proposition-detecting-bounded-below", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates $D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$, \\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$, \\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$, \\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$, \\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$, \\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^+(\\mathbf{Z})$, \\item for every perfect object $P$ of $D(\\mathcal{O}_X)$ \\begin{enumerate} \\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$, \\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^+(\\mathbf{Z})$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2780", "title": "dualizing-theorem-local-duality", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $\\omega_A^\\bullet$ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $Z = V(\\mathfrak m) \\subset \\Spec(A)$. Denote ${}^\\wedge$ derived completion with respect to $\\mathfrak m$. Then $$ R\\Hom_A(K, \\omega_A^\\bullet)^\\wedge \\cong R\\Hom_A(R\\Gamma_Z(K), E[0]) $$ for $K$ in $D(A)$."}
+{"_id": "2781", "title": "dualizing-lemma-essential", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item If $A \\subset B$ and $B \\subset C$ are essential extensions, then $A \\subset C$ is an essential extension. \\item If $A \\subset B$ is an essential extension and $C \\subset B$ is a subobject, then $A \\cap C \\subset C$ is an essential extension. \\item If $A \\to B$ and $B \\to C$ are essential surjections, then $A \\to C$ is an essential surjection. \\item Given an essential surjection $f : A \\to B$ and a surjection $A \\to C$ with kernel $K$, the morphism $C \\to B/f(K)$ is an essential surjection. \\end{enumerate}"}
+{"_id": "2782", "title": "dualizing-lemma-union-essential-extensions", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $E = \\colim E_i$ be a filtered colimit of $R$-modules. Suppose given a compatible system of essential injections $M \\to E_i$ of $R$-modules. Then $M \\to E$ is an essential injection."}
+{"_id": "2783", "title": "dualizing-lemma-essential-extension", "text": "Let $R$ be a ring. Let $M \\subset N$ be $R$-modules. The following are equivalent \\begin{enumerate} \\item $M \\subset N$ is an essential extension, \\item for all $x \\in N$ nonzero there exists an $f \\in R$ such that $fx \\in M$ and $fx \\not = 0$. \\end{enumerate}"}
+{"_id": "2785", "title": "dualizing-lemma-injective-flat", "text": "Let $R \\to S$ be a flat ring map. If $E$ is an injective $S$-module, then $E$ is injective as an $R$-module."}
+{"_id": "2786", "title": "dualizing-lemma-injective-epimorphism", "text": "Let $R \\to S$ be an epimorphism of rings. Let $E$ be an $S$-module. If $E$ is injective as an $R$-module, then $E$ is an injective $S$-module."}
+{"_id": "2787", "title": "dualizing-lemma-hom-injective", "text": "Let $R \\to S$ be a ring map. If $E$ is an injective $R$-module, then $\\Hom_R(S, E)$ is an injective $S$-module."}
+{"_id": "2788", "title": "dualizing-lemma-essential-extensions-in-injective", "text": "Let $R$ be a ring. Let $I$ be an injective $R$-module. Let $E \\subset I$ be a submodule. The following are equivalent \\begin{enumerate} \\item $E$ is injective, and \\item for all $E \\subset E' \\subset I$ with $E \\subset E'$ essential we have $E = E'$. \\end{enumerate} In particular, an $R$-module is injective if and only if every essential extension is trivial."}
+{"_id": "2789", "title": "dualizing-lemma-sum-injective-modules", "text": "Let $R$ be a Noetherian ring. A direct sum of injective modules is injective."}
+{"_id": "2790", "title": "dualizing-lemma-localization-injective-modules", "text": "Let $R$ be a Noetherian ring. Let $S \\subset R$ be a multiplicative subset. If $E$ is an injective $R$-module, then $S^{-1}E$ is an injective $S^{-1}R$-module."}
+{"_id": "2791", "title": "dualizing-lemma-injective-module-divide", "text": "Let $R$ be a Noetherian ring. Let $I$ be an injective $R$-module. \\begin{enumerate} \\item Let $f \\in R$. Then $E = \\bigcup I[f^n] = I[f^\\infty]$ is an injective submodule of $I$. \\item Let $J \\subset R$ be an ideal. Then the $J$-power torsion submodule $I[J^\\infty]$ is an injective submodule of $I$. \\end{enumerate}"}
+{"_id": "2792", "title": "dualizing-lemma-injective-dimension-over-polynomial-ring", "text": "Let $A$ be a Noetherian ring. Let $E$ be an injective $A$-module. Then $E \\otimes_A A[x]$ has injective-amplitude $[0, 1]$ as an object of $D(A[x])$. In particular, $E \\otimes_A A[x]$ has finite injective dimension as an $A[x]$-module."}
+{"_id": "2793", "title": "dualizing-lemma-projective-cover-unique", "text": "Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover of $M$ exists, then it is unique up to isomorphism."}
+{"_id": "2796", "title": "dualizing-lemma-injective-hull-unique", "text": "Let $R$ be a ring. Let $M$, $N$ be $R$-modules and let $M \\to E$ and $N \\to E'$ be injective hulls. Then \\begin{enumerate} \\item for any $R$-module map $\\varphi : M \\to N$ there exists an $R$-module map $\\psi : E \\to E'$ such that $$ \\xymatrix{ M \\ar[r] \\ar[d]_\\varphi & E \\ar[d]^\\psi \\\\ N \\ar[r] & E' } $$ commutes, \\item if $\\varphi$ is injective, then $\\psi$ is injective, \\item if $\\varphi$ is an essential injection, then $\\psi$ is an isomorphism, \\item if $\\varphi$ is an isomorphism, then $\\psi$ is an isomorphism, \\item if $M \\to I$ is an embedding of $M$ into an injective $R$-module, then there is an isomorphism $I \\cong E \\oplus I'$ compatible with the embeddings of $M$, \\end{enumerate} In particular, the injective hull $E$ of $M$ is unique up to isomorphism."}
+{"_id": "2797", "title": "dualizing-lemma-indecomposable-injective", "text": "Let $R$ be a ring. Let $E$ be an indecomposable injective $R$-module. Then \\begin{enumerate} \\item $E$ is the injective hull of any nonzero submodule of $E$, \\item the intersection of any two nonzero submodules of $E$ is nonzero, \\item $\\text{End}_R(E, E)$ is a noncommutative local ring with maximal ideal those $\\varphi : E \\to E$ whose kernel is nonzero, and \\item the set of zerodivisors on $E$ is a prime ideal $\\mathfrak p$ of $R$ and $E$ is an injective $R_\\mathfrak p$-module. \\end{enumerate}"}
+{"_id": "2798", "title": "dualizing-lemma-injective-hull-indecomposable", "text": "Let $\\mathfrak p \\subset R$ be a prime of a ring $R$. Let $E$ be the injective hull of $R/\\mathfrak p$. Then \\begin{enumerate} \\item $E$ is indecomposable, \\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$, \\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$ over the ring $R_\\mathfrak p$. \\end{enumerate}"}
+{"_id": "2799", "title": "dualizing-lemma-indecomposable-injective-noetherian", "text": "Let $R$ be a Noetherian ring. Let $E$ be an indecomposable injective $R$-module. Then there exists a prime ideal $\\mathfrak p$ of $R$ such that $E$ is the injective hull of $\\kappa(\\mathfrak p)$."}
+{"_id": "2800", "title": "dualizing-lemma-finite", "text": "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\\kappa$. For every finite $R$-module $M$ we have $$ \\text{length}_R(M) = \\text{length}_R(\\Hom_R(M, E)) $$ In particular, the injective hull $E$ of $\\kappa$ is a finite $R$-module."}
+{"_id": "2801", "title": "dualizing-lemma-evaluate", "text": "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\\kappa$. For any finite $R$-module $M$ the evaluation map $$ M \\longrightarrow \\Hom_R(\\Hom_R(M, E), E) $$ is an isomorphism. In particular $R = \\Hom_R(E, E)$."}
+{"_id": "2802", "title": "dualizing-lemma-duality", "text": "Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring. Let $E$ be an injective hull of $\\kappa$. The functor $D(-) = \\Hom_R(-, E)$ induces an exact anti-equivalence $\\text{Mod}^{fg}_R \\to \\text{Mod}^{fg}_R$ and $D \\circ D \\cong \\text{id}$."}
+{"_id": "2803", "title": "dualizing-lemma-duality-torsion-cotorsion", "text": "Assumptions and notation as in Lemma \\ref{lemma-duality}. Let $I \\subset R$ be an ideal and $M$ a finite $R$-module. Then $$ D(M[I]) = D(M)/ID(M) \\quad\\text{and}\\quad D(M/IM) = D(M)[I] $$"}
+{"_id": "2804", "title": "dualizing-lemma-quotient", "text": "Let $R \\to S$ be a surjective map of local rings with kernel $I$. Let $E$ be the injective hull of the residue field of $R$ over $R$. Then $E[I]$ is the injective hull of the residue field of $S$ over $S$."}
+{"_id": "2805", "title": "dualizing-lemma-torsion-submodule-sum-injective-hulls", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $E$ be the injective hull of $\\kappa$. Let $M$ be a $\\mathfrak m$-power torsion $R$-module with $n = \\dim_\\kappa(M[\\mathfrak m]) < \\infty$. Then $M$ is isomorphic to a submodule of $E^{\\oplus n}$."}
+{"_id": "2806", "title": "dualizing-lemma-union-artinian", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\\kappa$ over $R$. Let $E_n$ be an injective hull of $\\kappa$ over $R/\\mathfrak m^n$. Then $E = \\bigcup E_n$ and $E_n = E[\\mathfrak m^n]$."}
+{"_id": "2807", "title": "dualizing-lemma-compare", "text": "Let $R \\to S$ be a flat local homomorphism of local Noetherian rings such that $R/\\mathfrak m_R \\cong S/\\mathfrak m_R S$. Then the injective hull of the residue field of $R$ is the injective hull of the residue field of $S$."}
+{"_id": "2808", "title": "dualizing-lemma-endos", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\\kappa$ over $R$. Then $\\Hom_R(E, E)$ is canonically isomorphic to the completion of $R$."}
+{"_id": "2809", "title": "dualizing-lemma-injective-hull-has-dcc", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\\kappa$ over $R$. Then $E$ satisfies the descending chain condition."}
+{"_id": "2810", "title": "dualizing-lemma-describe-categories", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\\kappa$. \\begin{enumerate} \\item For an $R$-module $M$ the following are equivalent: \\begin{enumerate} \\item $M$ satisfies the ascending chain condition, \\item $M$ is a finite $R$-module, and \\item there exist $n, m$ and an exact sequence $R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$. \\end{enumerate} \\item For an $R$-module $M$ the following are equivalent: \\begin{enumerate} \\item $M$ satisfies the descending chain condition, \\item $M$ is $\\mathfrak m$-power torsion and $\\dim_\\kappa(M[\\mathfrak m]) < \\infty$, and \\item there exist $n, m$ and an exact sequence $0 \\to M \\to E^{\\oplus n} \\to E^{\\oplus m}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "2811", "title": "dualizing-lemma-adjoint", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. The functor $R\\Gamma_I$ is right adjoint to the functor $D(I^\\infty\\text{-torsion}) \\to D(A)$."}
+{"_id": "2812", "title": "dualizing-lemma-local-cohomology-ext", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. For any object $K$ of $D(A)$ we have $$ R\\Gamma_I(K) = \\text{hocolim}\\ R\\Hom_A(A/I^n, K) $$ in $D(A)$ and $$ R^q\\Gamma_I(K) = \\colim_n \\Ext_A^q(A/I^n, K) $$ as modules for all $q \\in \\mathbf{Z}$."}
+{"_id": "2813", "title": "dualizing-lemma-bad-local-cohomology-vanishes", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $K^\\bullet$ be a complex of $A$-modules such that $f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some $f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then $R\\Gamma_I(K^\\bullet) = 0$."}
+{"_id": "2814", "title": "dualizing-lemma-not-equal", "text": "Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $M$ and $N$ be $I$-power torsion modules. \\begin{enumerate} \\item $\\Hom_{D(A)}(M, N) = \\Hom_{D({I^\\infty\\text{-torsion}})}(M, N)$, \\item $\\Ext^1_{D(A)}(M, N) = \\Ext^1_{D({I^\\infty\\text{-torsion}})}(M, N)$, \\item $\\Ext^2_{D({I^\\infty\\text{-torsion}})}(M, N) \\to \\Ext^2_{D(A)}(M, N)$ is not surjective in general, \\item (\\ref{equation-compare-torsion}) is not an equivalence in general. \\end{enumerate}"}
+{"_id": "2815", "title": "dualizing-lemma-local-cohomology-adjoint", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. There exists a right adjoint $R\\Gamma_Z$ (\\ref{equation-local-cohomology}) to the inclusion functor $D_{I^\\infty\\text{-torsion}}(A) \\to D(A)$. In fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have $$ R\\Gamma_Z(K) = (A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to \\ldots \\to A_{f_1\\ldots f_r}) \\otimes_A^\\mathbf{L} K $$ functorially in $K \\in D(A)$."}
+{"_id": "2816", "title": "dualizing-lemma-local-cohomology-and-restriction", "text": "Let $A \\to B$ be a ring homomorphism and let $I \\subset A$ be a finitely generated ideal. Set $J = IB$. Set $Z = V(I)$ and $Y = V(J)$. Then $$ R\\Gamma_Z(M_A) = R\\Gamma_Y(M)_A $$ functorially in $M \\in D(B)$. Here $(-)_A$ denotes the restriction functors $D(B) \\to D(A)$ and $D_{J^\\infty\\text{-torsion}}(B) \\to D_{I^\\infty\\text{-torsion}}(A)$."}
+{"_id": "2817", "title": "dualizing-lemma-torsion-change-rings", "text": "Let $A \\to B$ be a ring homomorphism and let $I \\subset A$ be a finitely generated ideal. Set $J = IB$. Let $Z = V(I)$ and $Y = V(J)$. Then $$ R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} B = R\\Gamma_Y(K \\otimes_A^\\mathbf{L} B) $$ functorially in $K \\in D(A)$."}
+{"_id": "2818", "title": "dualizing-lemma-local-cohomology-vanishes", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $K^\\bullet$ be a complex of $A$-modules such that $f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some $f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then $R\\Gamma_Z(K^\\bullet) = 0$."}
+{"_id": "2819", "title": "dualizing-lemma-torsion-tensor-product", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. For $K, L \\in D(A)$ we have $$ R\\Gamma_Z(K \\otimes_A^\\mathbf{L} L) = K \\otimes_A^\\mathbf{L} R\\Gamma_Z(L) = R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} L = R\\Gamma_Z(K) \\otimes_A^\\mathbf{L} R\\Gamma_Z(L) $$ If $K$ or $L$ is in $D_{I^\\infty\\text{-torsion}}(A)$ then so is $K \\otimes_A^\\mathbf{L} L$."}
+{"_id": "2820", "title": "dualizing-lemma-local-cohomology-ss", "text": "Let $A$ be a ring and let $I, J \\subset A$ be finitely generated ideals. Set $Z = V(I)$ and $Y = V(J)$. Then $Z \\cap Y = V(I + J)$ and $R\\Gamma_Y \\circ R\\Gamma_Z = R\\Gamma_{Y \\cap Z}$ as functors $D(A) \\to D_{(I + J)^\\infty\\text{-torsion}}(A)$. For $K \\in D^+(A)$ there is a spectral sequence $$ E_2^{p, q} = H^p_Y(H^q_Z(K)) \\Rightarrow H^{p + q}_{Y \\cap Z}(K) $$ as in Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}."}
+{"_id": "2821", "title": "dualizing-lemma-torsion-flat-change-rings", "text": "Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a finitely generated ideal such that $A/I = B/IB$. Then base change and restriction induce quasi-inverse equivalences $D_{I^\\infty\\text{-torsion}}(A) = D_{(IB)^\\infty\\text{-torsion}}(B)$."}
+{"_id": "2823", "title": "dualizing-lemma-local-cohomology-noetherian", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. \\begin{enumerate} \\item the adjunction $R\\Gamma_I(K) \\to K$ is an isomorphism for $K \\in D_{I^\\infty\\text{-torsion}}(A)$, \\item the functor (\\ref{equation-compare-torsion}) $D(I^\\infty\\text{-torsion}) \\to D_{I^\\infty\\text{-torsion}}(A)$ is an equivalence, \\item the transformation of functors (\\ref{equation-compare-torsion-functors}) is an isomorphism, in other words $R\\Gamma_I(K) = R\\Gamma_Z(K)$ for $K \\in D(A)$. \\end{enumerate}"}
+{"_id": "2824", "title": "dualizing-lemma-compute-local-cohomology-noetherian", "text": "Let $A$ be a Noetherian ring and let $I = (f_1, \\ldots, f_r)$ be an ideal of $A$. Set $Z = V(I) \\subset \\Spec(A)$. There are canonical isomorphisms $$ R\\Gamma_I(A) \\to (A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to \\ldots \\to A_{f_1\\ldots f_r}) \\to R\\Gamma_Z(A) $$ in $D(A)$. If $M$ is an $A$-module, then we have similarly $$ R\\Gamma_I(M) \\cong (M \\to \\prod\\nolimits_{i_0} M_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \\to \\ldots \\to M_{f_1\\ldots f_r}) \\cong R\\Gamma_Z(M) $$ in $D(A)$."}
+{"_id": "2825", "title": "dualizing-lemma-local-cohomology-change-rings", "text": "If $A \\to B$ is a homomorphism of Noetherian rings and $I \\subset A$ is an ideal, then in $D(B)$ we have $$ R\\Gamma_I(A) \\otimes_A^\\mathbf{L} B = R\\Gamma_Z(A) \\otimes_A^\\mathbf{L} B = R\\Gamma_Y(B) = R\\Gamma_{IB}(B) $$ where $Y = V(IB) \\subset \\Spec(B)$."}
+{"_id": "2826", "title": "dualizing-lemma-depth", "text": "Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and let $M$ be a finite $A$-module such that $IM \\not = M$. Then the following integers are equal: \\begin{enumerate} \\item $\\text{depth}_I(M)$, \\item the smallest integer $i$ such that $\\Ext_A^i(A/I, M)$ is nonzero, and \\item the smallest integer $i$ such that $H^i_I(M)$ is nonzero. \\end{enumerate} Moreover, we have $\\Ext^i_A(N, M) = 0$ for $i < \\text{depth}_I(M)$ for any finite $A$-module $N$ annihilated by a power of $I$."}
+{"_id": "2827", "title": "dualizing-lemma-depth-in-ses", "text": "Let $A$ be a Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$ be a short exact sequence of finite $A$-modules. Let $I \\subset A$ be an ideal. \\begin{enumerate} \\item $\\text{depth}_I(N) \\geq \\min\\{\\text{depth}_I(N'), \\text{depth}_I(N'')\\}$ \\item $\\text{depth}_I(N'') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N') - 1\\}$ \\item $\\text{depth}_I(N') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N'') + 1\\}$ \\end{enumerate}"}
+{"_id": "2828", "title": "dualizing-lemma-depth-drops-by-one", "text": "Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and let $M$ a finite $A$-module with $IM \\not = M$. \\begin{enumerate} \\item If $x \\in I$ is a nonzerodivisor on $M$, then $\\text{depth}_I(M/xM) = \\text{depth}_I(M) - 1$. \\item Any $M$-regular sequence $x_1, \\ldots, x_r$ in $I$ can be extended to an $M$-regular sequence in $I$ of length $\\text{depth}_I(M)$. \\end{enumerate}"}
+{"_id": "2829", "title": "dualizing-lemma-depth-CM", "text": "Let $R$ be a Noetherian local ring. If $M$ is a finite Cohen-Macaulay $R$-module and $I \\subset R$ a nontrivial ideal. Then $$ \\text{depth}_I(M) = \\dim(\\text{Supp}(M)) - \\dim(\\text{Supp}(M/IM)). $$"}
+{"_id": "2830", "title": "dualizing-lemma-depth-flat-CM", "text": "Let $R \\to S$ be a flat local ring homomorphism of Noetherian local rings. Denote $\\mathfrak m \\subset R$ the maximal ideal. Let $I \\subset S$ be an ideal. If $S/\\mathfrak mS$ is Cohen-Macaulay, then $$ \\text{depth}_I(S) \\geq \\dim(S/\\mathfrak mS) - \\dim(S/\\mathfrak mS + I) $$"}
+{"_id": "2831", "title": "dualizing-lemma-divide-by-torsion", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $M$ be an $A$-module. Let $Z = V(I)$. Then $H^0_I(M) = H^0_Z(M)$. Let $N$ be the common value and set $M' = M/N$. Then \\begin{enumerate} \\item $H^0_I(M') = 0$ and $H^p_I(M) = H^p_I(M')$ and $H^p_I(N) = 0$ for all $p > 0$, \\item $H^0_Z(M') = 0$ and $H^p_Z(M) = H^p_Z(M')$ and $H^p_Z(N) = 0$ for all $p > 0$. \\end{enumerate}"}
+{"_id": "2832", "title": "dualizing-lemma-complete-and-local", "text": "\\begin{slogan} Results of this nature are sometimes referred to as Greenlees-May duality. \\end{slogan} Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $R\\Gamma_Z$ be as in Lemma \\ref{lemma-local-cohomology-adjoint}. Let ${\\ }^\\wedge$ denote derived completion as in More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion}. For an object $K$ in $D(A)$ we have $$ R\\Gamma_Z(K^\\wedge) = R\\Gamma_Z(K) \\quad\\text{and}\\quad (R\\Gamma_Z(K))^\\wedge = K^\\wedge $$ in $D(A)$."}
+{"_id": "2833", "title": "dualizing-lemma-compare-RHom", "text": "With notation as in Lemma \\ref{lemma-complete-and-local}. For objects $K, L$ in $D(A)$ there is a canonical isomorphism $$ R\\Hom_A(K^\\wedge, L^\\wedge) \\longrightarrow R\\Hom_A(R\\Gamma_Z(K), R\\Gamma_Z(L)) $$ in $D(A)$."}
+{"_id": "2834", "title": "dualizing-lemma-completion-local", "text": "Let $I$ and $J$ be ideals in a Noetherian ring $A$. Let $M$ be a finite $A$-module. Set $Z =V(J)$. Consider the derived $I$-adic completion $R\\Gamma_Z(M)^\\wedge$ of local cohomology. Then \\begin{enumerate} \\item we have $R\\Gamma_Z(M)^\\wedge = R\\lim R\\Gamma_Z(M/I^nM)$, and \\item there are short exact sequences $$ 0 \\to R^1\\lim H^{i - 1}_Z(M/I^nM) \\to H^i(R\\Gamma_Z(M)^\\wedge) \\to \\lim H^i_Z(M/I^nM) \\to 0 $$ \\end{enumerate} In particular $R\\Gamma_Z(M)^\\wedge$ has vanishing cohomology in negative degrees."}
+{"_id": "2835", "title": "dualizing-lemma-completion-local-H0", "text": "With notation and hypotheses as in Lemma \\ref{lemma-completion-local} assume $A$ is $I$-adically complete. Then $$ H^0(R\\Gamma_Z(M)^\\wedge) = \\colim H^0_{V(J')}(M) $$ where the filtered colimit is over $J' \\subset J$ such that $V(J') \\cap V(I) = V(J) \\cap V(I)$."}
+{"_id": "2836", "title": "dualizing-lemma-right-adjoint", "text": "Let $A \\to B$ be a ring homomorphism. The functor $R\\Hom(B, -)$ constructed above is right adjoint to the restriction functor $D(B) \\to D(A)$."}
+{"_id": "2837", "title": "dualizing-lemma-composition-right-adjoints", "text": "Let $A \\to B \\to C$ be ring maps. Then $R\\Hom(C, -) \\circ R\\Hom(B, -) : D(A) \\to D(C)$ is the functor $R\\Hom(C, -) : D(A) \\to D(C)$."}
+{"_id": "2838", "title": "dualizing-lemma-RHom-ext", "text": "Let $\\varphi : A \\to B$ be a ring homomorphism. For $K$ in $D(A)$ we have $$ \\varphi_*R\\Hom(B, K) = R\\Hom_A(B, K) $$ where $\\varphi_* : D(B) \\to D(A)$ is restriction. In particular $R^q\\Hom(B, K) = \\Ext_A^q(B, K)$."}
+{"_id": "2839", "title": "dualizing-lemma-exact-support-coherent", "text": "With notation as above, assume $A \\to B$ is a finite ring map of Noetherian rings. Then $R\\Hom(B, -)$ maps $D^+_{\\textit{Coh}}(A)$ into $D^+_{\\textit{Coh}}(B)$."}
+{"_id": "2840", "title": "dualizing-lemma-RHom-dga", "text": "In Situation \\ref{situation-resolution} the functor $R\\Hom(A, -)$ is equal to the composition of $R\\Hom(E, -) : D(R) \\to D(E, \\text{d})$ and the equivalence $- \\otimes^\\mathbf{L}_E A : D(E, \\text{d}) \\to D(A)$."}
+{"_id": "2841", "title": "dualizing-lemma-RHom-is-tensor", "text": "In Situation \\ref{situation-resolution} assume that \\begin{enumerate} \\item $E$ viewed as an object of $D(R)$ is compact, and \\item $N = \\Hom^\\bullet_R(E^\\bullet, R)$ computes $R\\Hom(E, R)$. \\end{enumerate} Then $R\\Hom(E, -) : D(R) \\to D(E)$ is isomorphic to $K \\mapsto K \\otimes_R^\\mathbf{L} N$."}
+{"_id": "2842", "title": "dualizing-lemma-RHom-is-tensor-special", "text": "In Situation \\ref{situation-resolution} assume $A$ is a perfect $R$-module. Then $$ R\\Hom(A, -) : D(R) \\to D(A) $$ is given by $K \\mapsto K \\otimes_R^\\mathbf{L} M$ where $M = R\\Hom(A, R) \\in D(A)$."}
+{"_id": "2843", "title": "dualizing-lemma-compute-for-effective-Cartier-algebraic", "text": "Let $R \\to A$ be a surjective ring map whose kernel $I$ is an invertible $R$-module. The functor $R\\Hom(A, -) : D(R) \\to D(A)$ is isomorphic to $K \\mapsto K \\otimes_R^\\mathbf{L} N[-1]$ where $N$ is inverse of the invertible $A$-module $I \\otimes_R A$."}
+{"_id": "2844", "title": "dualizing-lemma-check-base-change-is-iso", "text": "In the situation above, the map (\\ref{equation-base-change}) is an isomorphism if and only if the map $$ R\\Hom_R(A, K) \\otimes_R^\\mathbf{L} R' \\longrightarrow R\\Hom_R(A, K \\otimes_R^\\mathbf{L} R') $$ of More on Algebra, Lemma \\ref{more-algebra-lemma-internal-hom-diagonal-better} is an isomorphism."}
+{"_id": "2845", "title": "dualizing-lemma-flat-bc-surjection", "text": "Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$. Assume \\begin{enumerate} \\item $A$ is pseudo-coherent as an $R$-module, \\item $R'$ has finite tor dimension as an $R$-module (for example $R \\to R'$ is flat), \\item $A$ and $R'$ are tor independent over $R$. \\end{enumerate} Then (\\ref{equation-base-change}) is an isomorphism for $K \\in D^+(R)$."}
+{"_id": "2846", "title": "dualizing-lemma-bc-surjection", "text": "Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$. Assume \\begin{enumerate} \\item $A$ is perfect as an $R$-module, \\item $A$ and $R'$ are tor independent over $R$. \\end{enumerate} Then (\\ref{equation-base-change}) is an isomorphism for all $K \\in D(R)$."}
+{"_id": "2847", "title": "dualizing-lemma-finite-ext-into-bounded-injective", "text": "Let $A$ be a Noetherian ring. Let $K, L \\in D_{\\textit{Coh}}(A)$ and assume $L$ has finite injective dimension. Then $R\\Hom_A(K, L)$ is in $D_{\\textit{Coh}}(A)$."}
+{"_id": "2848", "title": "dualizing-lemma-dualizing", "text": "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing complex, then the functor $$ D : K \\longmapsto R\\Hom_A(K, \\omega_A^\\bullet) $$ is an anti-equivalence $D_{\\textit{Coh}}(A) \\to D_{\\textit{Coh}}(A)$ which exchanges $D^+_{\\textit{Coh}}(A)$ and $D^-_{\\textit{Coh}}(A)$ and induces an anti-equivalence $D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$. Moreover $D \\circ D$ is isomorphic to the identity functor."}
+{"_id": "2849", "title": "dualizing-lemma-equivalence-comes-from-invertible", "text": "Let $A$ be a Noetherian ring. Let $F : D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$ be an $A$-linear equivalence of categories. Then $F(A)$ is an invertible object of $D(A)$."}
+{"_id": "2850", "title": "dualizing-lemma-dualizing-unique", "text": "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ and $(\\omega'_A)^\\bullet$ are dualizing complexes, then $(\\omega'_A)^\\bullet$ is quasi-isomorphic to $\\omega_A^\\bullet \\otimes_A^\\mathbf{L} L$ for some invertible object $L$ of $D(A)$."}
+{"_id": "2851", "title": "dualizing-lemma-dualizing-localize", "text": "Let $A$ be a Noetherian ring. Let $B = S^{-1}A$ be a localization. If $\\omega_A^\\bullet$ is a dualizing complex, then $\\omega_A^\\bullet \\otimes_A B$ is a dualizing complex for $B$."}
+{"_id": "2852", "title": "dualizing-lemma-dualizing-glue", "text": "Let $A$ be a Noetherian ring. Let $f_1, \\ldots, f_n \\in A$ generate the unit ideal. If $\\omega_A^\\bullet$ is a complex of $A$-modules such that $(\\omega_A^\\bullet)_{f_i}$ is a dualizing complex for $A_{f_i}$ for all $i$, then $\\omega_A^\\bullet$ is a dualizing complex for $A$."}
+{"_id": "2853", "title": "dualizing-lemma-dualizing-finite", "text": "Let $A \\to B$ be a finite ring map of Noetherian rings. Let $\\omega_A^\\bullet$ be a dualizing complex. Then $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$."}
+{"_id": "2854", "title": "dualizing-lemma-dualizing-quotient", "text": "Let $A \\to B$ be a surjective homomorphism of Noetherian rings. Let $\\omega_A^\\bullet$ be a dualizing complex. Then $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$."}
+{"_id": "2855", "title": "dualizing-lemma-dualizing-polynomial-ring", "text": "Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing complex, then $\\omega_A^\\bullet \\otimes_A A[x]$ is a dualizing complex for $A[x]$."}
+{"_id": "2856", "title": "dualizing-lemma-find-function", "text": "Let $A$ be a Noetherian ring. Let $\\omega_A^\\bullet$ be a dualizing complex. Let $\\mathfrak m \\subset A$ be a maximal ideal and set $\\kappa = A/\\mathfrak m$. Then $R\\Hom_A(\\kappa, \\omega_A^\\bullet) \\cong \\kappa[n]$ for some $n \\in \\mathbf{Z}$."}
+{"_id": "2857", "title": "dualizing-lemma-normalized-finite", "text": "Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak m', \\kappa')$ be a finite local map of Noetherian local rings. Let $\\omega_A^\\bullet$ be a normalized dualizing complex. Then $\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ is a normalized dualizing complex for $B$."}
+{"_id": "2858", "title": "dualizing-lemma-normalized-quotient", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $A \\to B$ be surjective. Then $\\omega_B^\\bullet = R\\Hom_A(B, \\omega_A^\\bullet)$ is a normalized dualizing complex for $B$."}
+{"_id": "2859", "title": "dualizing-lemma-equivalence-finite-length", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $F$ be an $A$-linear self-equivalence of the category of finite length $A$-modules. Then $F$ is isomorphic to the identity functor."}
+{"_id": "2860", "title": "dualizing-lemma-dualizing-finite-length", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $E$ be an injective hull of $\\kappa$. Then there exists a functorial isomorphism $$ R\\Hom_A(N, \\omega_A^\\bullet) = \\Hom_A(N, E)[0] $$ for $N$ running through the finite length $A$-modules."}
+{"_id": "2861", "title": "dualizing-lemma-sitting-in-degrees", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite $A$-module and let $d = \\dim(\\text{Supp}(M))$. Then \\begin{enumerate} \\item if $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero, then $i \\in \\{-d, \\ldots, 0\\}$, \\item the dimension of the support of $\\Ext^i_A(M, \\omega_A^\\bullet)$ is at most $-i$, \\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that $\\Ext^{-\\delta}_A(M, \\omega_A^\\bullet) \\not = 0$. \\end{enumerate}"}
+{"_id": "2862", "title": "dualizing-lemma-local-CM", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite $A$-module. The following are equivalent \\begin{enumerate} \\item $M$ is Cohen-Macaulay, \\item $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero for a single $i$, \\item $\\Ext^{-i}_A(M, \\omega_A^\\bullet)$ is zero for $i \\not = \\dim(\\text{Supp}(M))$. \\end{enumerate} Denote $CM_d$ the category of finite Cohen-Macaulay $A$-modules of depth $d$. Then $M \\mapsto \\Ext^{-d}_A(M, \\omega_A^\\bullet)$ defines an anti-auto-equivalence of $CM_d$."}
+{"_id": "2863", "title": "dualizing-lemma-dualizing-artinian", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. If $\\dim(A) = 0$, then $\\omega_A^\\bullet \\cong E[0]$ where $E$ is an injective hull of the residue field."}
+{"_id": "2864", "title": "dualizing-lemma-divide-by-finite-length-ideal", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex. Let $I \\subset \\mathfrak m$ be an ideal of finite length. Set $B = A/I$. Then there is a distinguished triangle $$ \\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\Hom_A(I, E)[0] \\to \\omega_B^\\bullet[1] $$ in $D(A)$ where $E$ is an injective hull of $\\kappa$ and $\\omega_B^\\bullet$ is a normalized dualizing complex for $B$."}
+{"_id": "2865", "title": "dualizing-lemma-divide-by-nonzerodivisor", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $f \\in \\mathfrak m$ be a nonzerodivisor. Set $B = A/(f)$. Then there is a distinguished triangle $$ \\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\omega_A^\\bullet \\to \\omega_B^\\bullet[1] $$ in $D(A)$ where $\\omega_B^\\bullet$ is a normalized dualizing complex for $B$."}
+{"_id": "2866", "title": "dualizing-lemma-nonvanishing-generically-local", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $\\mathfrak p$ be a minimal prime of $A$ with $\\dim(A/\\mathfrak p) = e$. Then $H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero if and only if $i = -e$."}
+{"_id": "2867", "title": "dualizing-lemma-nonvanishing-generically", "text": "Let $A$ be a Noetherian ring. Let $\\mathfrak p$ be a minimal prime of $A$. Then $H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero for exactly one $i$."}
+{"_id": "2868", "title": "dualizing-lemma-quotient-function", "text": "Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing complex. Let $A \\to B$ be a surjective ring map and let $\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ be the dualizing complex for $B$ of Lemma \\ref{lemma-dualizing-quotient}. Then we have $$ \\delta_{\\omega_B^\\bullet} = \\delta_{\\omega_A^\\bullet}|_{\\Spec(B)} $$"}
+{"_id": "2869", "title": "dualizing-lemma-dimension-function", "text": "Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing complex. The function $\\delta = \\delta_{\\omega_A^\\bullet}$ defined above is a dimension function (Topology, Definition \\ref{topology-definition-dimension-function})."}
+{"_id": "2870", "title": "dualizing-lemma-universally-catenary", "text": "Let $A$ be a Noetherian ring which has a dualizing complex. Then $A$ is universally catenary of finite dimension."}
+{"_id": "2871", "title": "dualizing-lemma-depth-dualizing-module", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $d = \\dim(A)$ and $\\omega_A = H^{-d}(\\omega_A^\\bullet)$. Then \\begin{enumerate} \\item the support of $\\omega_A$ is the union of the irreducible components of $\\Spec(A)$ of dimension $d$, \\item $\\omega_A$ satisfies $(S_2)$, see Algebra, Definition \\ref{algebra-definition-conditions}. \\end{enumerate}"}
+{"_id": "2872", "title": "dualizing-lemma-local-cohomology-of-dualizing", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $\\omega_A^\\bullet$ be a normalized dualizing complex. Let $Z = V(\\mathfrak m) \\subset \\Spec(A)$. Then $E = R^0\\Gamma_Z(\\omega_A^\\bullet)$ is an injective hull of $\\kappa$ and $R\\Gamma_Z(\\omega_A^\\bullet) = E[0]$."}
+{"_id": "2873", "title": "dualizing-lemma-special-case-local-duality", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $\\omega_A^\\bullet$ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $K \\in D_{\\textit{Coh}}(A)$. Then $$ \\Ext^{-i}_A(K, \\omega_A^\\bullet)^\\wedge = \\Hom_A(H^i_{\\mathfrak m}(K), E) $$ where ${}^\\wedge$ denotes $\\mathfrak m$-adic completion."}
+{"_id": "2874", "title": "dualizing-lemma-depth-in-terms-dualizing-complex", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Then $\\text{depth}(A)$ is equal to the smallest integer $\\delta \\geq 0$ such that $H^{-\\delta}(\\omega_A^\\bullet) \\not = 0$."}
+{"_id": "2875", "title": "dualizing-lemma-apply-CM", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$ and dualizing module $\\omega_A = H^{-\\dim(A)}(\\omega_A^\\bullet)$. The following are equivalent \\begin{enumerate} \\item $A$ is Cohen-Macaulay, \\item $\\omega_A^\\bullet$ is concentrated in a single degree, and \\item $\\omega_A^\\bullet = \\omega_A[\\dim(A)]$. \\end{enumerate} In this case $\\omega_A$ is a maximal Cohen-Macaulay module."}
+{"_id": "2876", "title": "dualizing-lemma-has-dualizing-module-CM", "text": "Let $A$ be a Noetherian ring. If there exists a finite $A$-module $\\omega_A$ such that $\\omega_A[0]$ is a dualizing complex, then $A$ is Cohen-Macaulay."}
+{"_id": "2877", "title": "dualizing-lemma-CM-open", "text": "Let $A$ be a Noetherian ring with dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite $A$-module. Then $$ U = \\{\\mathfrak p \\in \\Spec(A) \\mid M_\\mathfrak p\\text{ is Cohen-Macaulay}\\} $$ is an open subset of $\\Spec(A)$ whose intersection with $\\text{Supp}(M)$ is dense."}
+{"_id": "2879", "title": "dualizing-lemma-gorenstein-CM", "text": "A Gorenstein ring is Cohen-Macaulay."}
+{"_id": "2880", "title": "dualizing-lemma-regular-gorenstein", "text": "A regular local ring is Gorenstein. A regular ring is Gorenstein."}
+{"_id": "2881", "title": "dualizing-lemma-gorenstein", "text": "Let $A$ be a Noetherian ring. \\begin{enumerate} \\item If $A$ has a dualizing complex $\\omega_A^\\bullet$, then \\begin{enumerate} \\item $A$ is Gorenstein $\\Leftrightarrow$ $\\omega_A^\\bullet$ is an invertible object of $D(A)$, \\item $A_\\mathfrak p$ is Gorenstein $\\Leftrightarrow$ $(\\omega_A^\\bullet)_\\mathfrak p$ is an invertible object of $D(A_\\mathfrak p)$, \\item $\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Gorenstein}\\}$ is an open subset. \\end{enumerate} \\item If $A$ is Gorenstein, then $A$ has a dualizing complex if and only if $A[0]$ is a dualizing complex. \\end{enumerate}"}
+{"_id": "2882", "title": "dualizing-lemma-gorenstein-ext", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Then $A$ is Gorenstein if and only if $\\Ext^i_A(\\kappa, A)$ is zero for $i \\gg 0$."}
+{"_id": "2883", "title": "dualizing-lemma-gorenstein-divide-by-nonzerodivisor", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$ be a nonzerodivisor. Set $B = A/(f)$. Then $A$ is Gorenstein if and only if $B$ is Gorenstein."}
+{"_id": "2884", "title": "dualizing-lemma-gorenstein-lci", "text": "If $A \\to B$ is a local complete intersection homomorphism of rings and $A$ is a Noetherian Gorenstein ring, then $B$ is a Gorenstein ring."}
+{"_id": "2885", "title": "dualizing-lemma-flat-under-gorenstein", "text": "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings. The following are equivalent \\begin{enumerate} \\item $B$ is Gorenstein, and \\item $A$ and $B/\\mathfrak m_A B$ are Gorenstein. \\end{enumerate}"}
+{"_id": "2886", "title": "dualizing-lemma-tor-injective-hull", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local Gorenstein ring of dimension $d$. Let $E$ be the injective hull of $\\kappa$. Then $\\text{Tor}_i^A(E, \\kappa)$ is zero for $i \\not = d$ and $\\text{Tor}_d^A(E, \\kappa) = \\kappa$."}
+{"_id": "2887", "title": "dualizing-lemma-flat-unramified", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Let $\\omega_A^\\bullet$ be a normalized dualizing complex. If $A \\to B$ is flat and $\\mathfrak m_A B = \\mathfrak m_B$, then $\\omega_A^\\bullet \\otimes_A B$ is a normalized dualizing complex for $B$."}
+{"_id": "2888", "title": "dualizing-lemma-flat-iso-mod-I", "text": "Let $A \\to B$ be a flat map of Noetherian rings. Let $I \\subset A$ be an ideal such that $A/I = B/IB$ and such that $IB$ is contained in the Jacobson radical of $B$. Let $\\omega_A^\\bullet$ be a dualizing complex. Then $\\omega_A^\\bullet \\otimes_A B$ is a dualizing complex for $B$."}
+{"_id": "2889", "title": "dualizing-lemma-completion-henselization-dualizing", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $\\omega_A^\\bullet$ be a dualizing complex. \\begin{enumerate} \\item $\\omega_A^\\bullet \\otimes_A A^h$ is a dualizing complex on the henselization $(A^h, I^h)$ of the pair $(A, I)$, \\item $\\omega_A^\\bullet \\otimes_A A^\\wedge$ is a dualizing complex on the $I$-adic completion $A^\\wedge$, and \\item if $A$ is local, then $\\omega_A^\\bullet \\otimes_A A^h$, resp.\\ $\\omega_A^\\bullet \\otimes_A A^{sh}$ is a dualzing complex on the henselization, resp.\\ strict henselization of $A$. \\end{enumerate}"}
+{"_id": "2890", "title": "dualizing-lemma-ubiquity-dualizing", "text": "The following types of rings have a dualizing complex: \\begin{enumerate} \\item fields, \\item Noetherian complete local rings, \\item $\\mathbf{Z}$, \\item Dedekind domains, \\item any ring which is obtained from one of the rings above by taking an algebra essentially of finite type, or by taking an ideal-adic completion, or by taking a henselization,  or by taking a strict henselization. \\end{enumerate}"}
+{"_id": "2891", "title": "dualizing-lemma-formal-fibres-gorenstein", "text": "Properties (A), (B), (C), (D), and (E) of More on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres} hold for $P(k \\to R) =$``$R$ is a Gorenstein ring''."}
+{"_id": "2892", "title": "dualizing-lemma-dualizing-gorenstein-formal-fibres", "text": "Let $A$ be a Noetherian local ring. If $A$ has a dualizing complex, then the formal fibres of $A$ are Gorenstein."}
+{"_id": "2895", "title": "dualizing-lemma-shriek-boundedness", "text": "Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings. \\begin{enumerate} \\item $\\varphi^!$ maps $D^+(R)$ into $D^+(A)$ and $D^+_{\\textit{Coh}}(R)$ into $D^+_{\\textit{Coh}}(A)$. \\item if $\\varphi$ is perfect, then $\\varphi^!$ maps $D^-(R)$ into $D^-(A)$, $D^-_{\\textit{Coh}}(R)$ into $D^-_{\\textit{Coh}}(A)$, and $D^b_{\\textit{Coh}}(R)$ into $D^b_{\\textit{Coh}}(A)$. \\end{enumerate}"}
+{"_id": "2896", "title": "dualizing-lemma-shriek-dualizing-algebraic", "text": "Let $\\varphi$ be a finite type homomorphism of Noetherian rings. If $\\omega_R^\\bullet$ is a dualizing complex for $R$, then $\\varphi^!(\\omega_R^\\bullet)$ is a dualizing complex for $A$."}
+{"_id": "2897", "title": "dualizing-lemma-flat-bc", "text": "Let $R \\to R'$ be a flat homomorphism of Noetherian rings. Let $\\varphi : R \\to A$ be a finite type ring map. Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$. Then we have a functorial maps $$ \\varphi^!(K) \\otimes_A^\\mathbf{L} A' \\longrightarrow (\\varphi')^!(K \\otimes_R^\\mathbf{L} R') $$ for $K$ in $D(R)$ which are isomorphisms for $K \\in D^+(R)$."}
+{"_id": "2898", "title": "dualizing-lemma-bc", "text": "Let $R \\to R'$ be a homomorphism of Noetherian rings. Let $\\varphi : R \\to A$ be a perfect ring map (More on Algebra, Definition \\ref{more-algebra-definition-pseudo-coherent-perfect}) such that $R'$ and $A$ are tor independent over $R$. Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$. Then we have a functorial isomorphism $$ \\varphi^!(K) \\otimes_A^\\mathbf{L} A' = (\\varphi')^!(K \\otimes_R^\\mathbf{L} R') $$ for $K$ in $D(R)$."}
+{"_id": "2899", "title": "dualizing-lemma-bc-flat", "text": "Let $R \\to R'$ be a homomorphism of Noetherian rings. Let $\\varphi : R \\to A$ be flat of finite type. Let $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$. Then we have a functorial isomorphism $$ \\varphi^!(K) \\otimes_A^\\mathbf{L} A' = (\\varphi')^!(K \\otimes_R^\\mathbf{L} R') $$ for $K$ in $D(R)$."}
+{"_id": "2900", "title": "dualizing-lemma-composition-shriek-algebraic", "text": "Let $A \\xrightarrow{a} B \\xrightarrow{b} C$ be finite type homomorphisms of Noetherian rings. Then there is a transformation of functors $b^! \\circ a^! \\to (b \\circ a)^!$ which is an isomorphism on $D^+(A)$."}
+{"_id": "2901", "title": "dualizing-lemma-upper-shriek-finite", "text": "Let $\\varphi : R \\to A$ be a finite map of Noetherian rings. Then $\\varphi^!$ is isomorphic to the functor $R\\Hom(A, -) : D(R) \\to D(A)$ from Section \\ref{section-trivial}."}
+{"_id": "2902", "title": "dualizing-lemma-upper-shriek-localize", "text": "Let $R$ be a Noetherian ring and let $f \\in R$. If $\\varphi$ denotes the map $R \\to R_f$, then $\\varphi^!$ is isomorphic to $- \\otimes_R^\\mathbf{L} R_f$. More generally, if $\\varphi : R \\to R'$ is a map such that $\\Spec(R') \\to \\Spec(R)$ is an open immersion, then $\\varphi^!$ is isomorphic to $- \\otimes_R^\\mathbf{L} R'$."}
+{"_id": "2903", "title": "dualizing-lemma-upper-shriek-is-tensor-functor", "text": "Let $\\varphi : R \\to A$ be a perfect homomorphism of Noetherian rings (for example $\\varphi$ is flat of finite type). Then $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\varphi^!(R)$ for $K \\in D(R)$."}
+{"_id": "2904", "title": "dualizing-lemma-relative-dualizing-if-have-omega", "text": "Let $\\varphi : A \\to B$ be a finite type homomorphism of Noetherian rings. Let $\\omega_A^\\bullet$ be a dualizing complex for $A$. Set $\\omega_B^\\bullet = \\varphi^!(\\omega_A^\\bullet)$. Denote $D_A(K) = R\\Hom_A(K, \\omega_A^\\bullet)$ for $K \\in D_{\\textit{Coh}}(A)$ and $D_B(L) = R\\Hom_B(L, \\omega_B^\\bullet)$ for $L \\in D_{\\textit{Coh}}(B)$. Then there is a functorial isomorphism $$ \\varphi^!(K) = D_B(D_A(K) \\otimes_A^\\mathbf{L} B) $$ for $K \\in D_{\\textit{Coh}}(A)$."}
+{"_id": "2905", "title": "dualizing-lemma-base-change-relative-algebraic", "text": "Let $R \\to R'$ be a homomorphism of Noetherian rings. Let $R \\to A$ be of finite type. Set $A' = A \\otimes_R R'$. If \\begin{enumerate} \\item $R \\to R'$ is flat, or \\item $R \\to A$ is flat, or \\item $R \\to A$ is perfect and $R'$ and $A$ are tor independent over $R$, \\end{enumerate} then there is an isomorphism $\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} A' \\to \\omega^\\bullet_{A'/R'}$ in $D(A')$."}
+{"_id": "2906", "title": "dualizing-lemma-relative-dualizing-algebraic", "text": "Let $\\varphi : R \\to A$ be a flat finite type map of Noetherian rings. Then \\begin{enumerate} \\item $\\omega_{A/R}^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$ and $R$-perfect (More on Algebra, Definition \\ref{more-algebra-definition-relatively-perfect}), \\item $A \\to R\\Hom_A(\\omega_{A/R}^\\bullet, \\omega_{A/R}^\\bullet)$ is an isomorphism, and \\item for every map $R \\to k$ to a field the base change $\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} (A \\otimes_R k)$ is a dualizing complex for $A \\otimes_R k$. \\end{enumerate}"}
+{"_id": "2907", "title": "dualizing-lemma-base-change-dualizing-over-field", "text": "Let $K/k$ be an extension of fields. Let $A$ be a finite type $k$-algebra. Let $A_K = A \\otimes_k K$. If $\\omega_A^\\bullet$ is a dualizing complex for $A$, then $\\omega_A^\\bullet \\otimes_A A_K$ is a dualizing complex for $A_K$."}
+{"_id": "2908", "title": "dualizing-lemma-lci-shriek", "text": "Let $\\varphi : R \\to A$ be a local complete intersection homomorphism of Noetherian rings. Then $\\omega_{A/R}^\\bullet$ is an invertible object of $D(A)$ and $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\omega_{A/R}^\\bullet$ for all $K \\in D(R)$."}
+{"_id": "2909", "title": "dualizing-lemma-gorenstein-shriek", "text": "Let $\\varphi : R \\to A$ be a flat finite type homomorphism of Noetherian rings. The following are equivalent \\begin{enumerate} \\item the fibres $A \\otimes_R \\kappa(\\mathfrak p)$ are Gorenstein for all primes $\\mathfrak p \\subset R$, and \\item $\\omega_{A/R}^\\bullet$ is an invertible object of $D(A)$, see More on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}. \\end{enumerate}"}
+{"_id": "2910", "title": "dualizing-lemma-shriek-normalized", "text": "Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings. Assume $R$ local and let $\\mathfrak m \\subset A$ be a maximal ideal lying over the maximal ideal of $R$. If $\\omega_R^\\bullet$ is a normalized dualizing complex for $R$, then $\\varphi^!(\\omega_R^\\bullet)_\\mathfrak m$ is a normalized dualizing complex for $A_\\mathfrak m$."}
+{"_id": "2911", "title": "dualizing-lemma-relative-dualizing-trivial-vanishing", "text": "Let $R \\to A$ be a finite type homomorphism of Noetherian rings. Let $\\mathfrak q \\subset A$ be a prime ideal lying over $\\mathfrak p \\subset R$. Then $$ H^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0 \\Rightarrow - d \\leq i $$ where $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$ over $\\mathfrak p$ at the point $\\mathfrak q$."}
+{"_id": "2912", "title": "dualizing-lemma-relative-dualizing-flat-vanishing", "text": "Let $R \\to A$ be a flat finite type homomorphism of Noetherian rings. Let $\\mathfrak q \\subset A$ be a prime ideal lying over $\\mathfrak p \\subset R$. Then $$ H^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0 \\Rightarrow - d \\leq i \\leq 0 $$ where $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$ over $\\mathfrak p$ at the point $\\mathfrak q$. If all fibres of $\\Spec(A) \\to \\Spec(R)$ have dimension $\\leq d$, then $\\omega_{A/R}^\\bullet$ has tor amplitude in $[-d, 0]$ as a complex of $R$-modules."}
+{"_id": "2913", "title": "dualizing-lemma-relative-dualizing-CM-vanishing", "text": "Let $R \\to A$ be a finite type homomorphism of Noetherian rings. Let $\\mathfrak p \\subset R$ be a prime ideal. Assume \\begin{enumerate} \\item $R_\\mathfrak p$ is Cohen-Macaulay, and \\item for any minimal prime $\\mathfrak q \\subset A$ we have $\\text{trdeg}_{\\kappa(R \\cap \\mathfrak q)} \\kappa(\\mathfrak q) \\leq r$. \\end{enumerate} Then $$ H^i(\\omega_{A/R}^\\bullet)_\\mathfrak p \\not = 0 \\Rightarrow - r \\leq i $$ and $H^{-r}(\\omega_{A/R}^\\bullet)_\\mathfrak p$ is $(S_2)$ as an $A_\\mathfrak p$-module."}
+{"_id": "2914", "title": "dualizing-lemma-descent", "text": "Let $A \\to B$ be a faithfully flat map of Noetherian rings. If $K \\in D(A)$ and $K \\otimes_A^\\mathbf{L} B$ is a dualizing complex for $B$, then $K$ is a dualizing complex for $A$."}
+{"_id": "2915", "title": "dualizing-lemma-descent-ascent", "text": "Let $\\varphi : A \\to B$ be a homomorphism of Noetherian rings. Assume \\begin{enumerate} \\item $A \\to B$ is syntomic and induces a surjective map on spectra, or \\item $A \\to B$ is a faithfully flat local complete intersection, or \\item $A \\to B$ is faithfully flat of finite type with Gorenstein fibres. \\end{enumerate} Then $K \\in D(A)$ is a dualizing complex for $A$ if and only if $K \\otimes_A^\\mathbf{L} B$ is a dualizing complex for $B$."}
+{"_id": "2916", "title": "dualizing-lemma-injective-hull-goes-up", "text": "Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak n, l)$ be a flat local homorphism of Noetherian rings such that $\\mathfrak n = \\mathfrak m B$. If $E$ is the injective hull of $\\kappa$, then $E \\otimes_A B$ is the injective hull of $l$."}
+{"_id": "2917", "title": "dualizing-lemma-injective-goes-up", "text": "Let $\\varphi : A \\to B$ be a flat homorphism of Noetherian rings such that for all primes $\\mathfrak q \\subset B$ we have $\\mathfrak p B_\\mathfrak q = \\mathfrak qB_\\mathfrak q$ where $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$, for example if $\\varphi$ is \\'etale. If $I$ is an injective $A$-module, then $I \\otimes_A B$ is an injective $B$-module."}
+{"_id": "2918", "title": "dualizing-lemma-uniqueness-relative-dualizing", "text": "Let $R \\to A$ be a flat ring map of finite presentation. Any two relative dualizing complexes for $R \\to A$ are isomorphic."}
+{"_id": "2919", "title": "dualizing-lemma-relative-dualizing-noetherian", "text": "Let $\\varphi : R \\to A$ be a flat finite type ring map of Noetherian rings. Then the relative dualizing complex $\\omega_{A/R}^\\bullet = \\varphi^!(R)$ of Section \\ref{section-relative-dualizing-complexes-Noetherian} is a relative dualizing complex in the sense of Definition \\ref{definition-relative-dualizing-complex}."}
+{"_id": "2920", "title": "dualizing-lemma-base-change-relative-dualizing", "text": "Let $R \\to A$ be a flat ring map of finite presentation. Then \\begin{enumerate} \\item there exists a relative dualizing complex $K$ in $D(A)$, and \\item for any ring map $R \\to R'$ setting $A' = A \\otimes_R R'$ and $K' = K \\otimes_A^\\mathbf{L} A'$, then $K'$ is a relative dualizing complex for $R' \\to A'$. \\end{enumerate} Moreover, if $$ \\xi : A \\longrightarrow K \\otimes_A^\\mathbf{L} (A \\otimes_R A) $$ is a generator for the cyclic module $\\Hom_{D(A \\otimes_R A)}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$ then in (2) the derived base change of $\\xi$ by $A \\otimes_R A \\to A' \\otimes_{R'} A'$ is a generator for the cyclic module $\\Hom_{D(A' \\otimes_{R'} A')}(A', K' \\otimes_{A'}^\\mathbf{L} (A' \\otimes_{R'} A'))$"}
+{"_id": "2921", "title": "dualizing-lemma-relative-dualizing-RHom", "text": "Let $R \\to A$ be a flat ring map of finite presentation. Let $K$ be a relative dualizing complex. Then $A \\to R\\Hom_A(K, K)$ is an isomorphism."}
+{"_id": "2922", "title": "dualizing-lemma-relative-dualizing-composition", "text": "Let $R \\to A \\to B$ be a ring maps which are flat and of finite presentation. Let $K_{A/R}$ and $K_{B/A}$ be relative dualizing complexes for $R \\to A$ and $A \\to B$. Then $K = K_{A/R} \\otimes_A^\\mathbf{L} K_{B/A}$ is a relative dualizing complex for $R \\to B$."}
+{"_id": "2923", "title": "dualizing-proposition-structure-injectives-noetherian", "text": "Let $R$ be a Noetherian ring. Every injective module is a direct sum of indecomposable injective modules. Every indecomposable injective module is the injective hull of the residue field at a prime."}
+{"_id": "2924", "title": "dualizing-proposition-matlis", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring. Let $E$ be an injective hull of $\\kappa$ over $R$. The functor $D(-) = \\Hom_R(-, E)$ induces an anti-equivalence $$ \\left\\{ \\begin{matrix} R\\text{-modules with the} \\\\ \\text{descending chain condition} \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} R\\text{-modules with the} \\\\ \\text{ascending chain condition} \\end{matrix} \\right\\} $$ and we have $D \\circ D = \\text{id}$ on either side of the equivalence."}
+{"_id": "2925", "title": "dualizing-proposition-torsion-complete", "text": "\\begin{reference} This is a special case of \\cite[Theorem 1.1]{Porta-Liran-Yekutieli}. \\end{reference} Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. The functors $R\\Gamma_Z$ and ${\\ }^\\wedge$ define quasi-inverse equivalences of categories $$ D_{I^\\infty\\text{-torsion}}(A) \\leftrightarrow D_{comp}(A, I) $$"}
+{"_id": "2926", "title": "dualizing-proposition-dualizing-essentially-finite-type", "text": "Let $A$ be a Noetherian ring which has a dualizing complex. Then any $A$-algebra essentially of finite type over $A$ has a dualizing complex."}
+{"_id": "2938", "title": "properties-lemma-locally-constructible", "text": "Let $X$ be a scheme. A subset $E$ of $X$ is locally constructible in $X$ if and only if $E \\cap U$ is constructible in $U$ for every affine open $U$ of $X$."}
+{"_id": "2939", "title": "properties-lemma-generic-point-in-constructible", "text": "Let $X$ be a scheme and let $E \\subset X$ be a locally constructible subset. Let $\\xi \\in X$ be a generic point of an irreducible component of $X$. \\begin{enumerate} \\item If $\\xi \\in E$, then an open neighbourhood of $\\xi$ is contained in $E$. \\item If $\\xi \\not \\in E$, then an open neighbourhood of $\\xi$ is disjoint from $E$. \\end{enumerate}"}
+{"_id": "2940", "title": "properties-lemma-quasi-separated-quasi-compact-open-retrocompact", "text": "Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$."}
+{"_id": "2941", "title": "properties-lemma-quasi-compact-quasi-separated-spectral", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space."}
+{"_id": "2942", "title": "properties-lemma-constructible-quasi-compact-quasi-separated", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible."}
+{"_id": "2943", "title": "properties-lemma-retrocompact", "text": "Let $X$ be a scheme. A subset $E$ of $X$ is retrocompact in $X$ if and only if $E \\cap U$ is quasi-compact for every affine open $U$ of $X$."}
+{"_id": "2944", "title": "properties-lemma-stratification-locally-finite-constructible", "text": "A partition $X = \\coprod_{i \\in I} X_i$ of a scheme $X$ with retrocompact parts is locally finite if and only if the parts are locally constructible."}
+{"_id": "2945", "title": "properties-lemma-characterize-reduced", "text": "Let $X$ be a scheme. The following are equivalent. \\begin{enumerate} \\item The scheme $X$ is reduced, see Schemes, Definition \\ref{schemes-definition-reduced}. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\Gamma(U_i, \\mathcal{O}_X)$ is reduced. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is reduced. \\item For every open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is reduced. \\end{enumerate}"}
+{"_id": "2946", "title": "properties-lemma-characterize-irreducible", "text": "Let $X$ be a scheme. The following are equivalent. \\begin{enumerate} \\item The scheme $X$ is irreducible. \\item There exists an affine open covering $X = \\bigcup_{i \\in I} U_i$ such that $I$ is not empty, $U_i$ is irreducible for all $i \\in I$, and $U_i \\cap U_j \\not = \\emptyset$ for all $i, j \\in I$. \\item The scheme $X$ is nonempty and every nonempty affine open $U \\subset X$ is irreducible. \\end{enumerate}"}
+{"_id": "2947", "title": "properties-lemma-characterize-integral", "text": "A scheme $X$ is integral if and only if it is reduced and irreducible."}
+{"_id": "2948", "title": "properties-lemma-locally-P", "text": "Let $X$ be a scheme. Let $P$ be a local property of rings. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is locally $P$. \\item For every affine open $U \\subset X$ the property $P(\\mathcal{O}_X(U))$ holds. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ satisfies $P$. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is locally $P$. \\end{enumerate} Moreover, if $X$ is locally $P$ then every open subscheme is locally $P$."}
+{"_id": "2950", "title": "properties-lemma-properties-local", "text": "The following properties of a ring $R$ are local. \\begin{enumerate} \\item (Cohen-Macaulay.) The ring $R$ is Noetherian and CM, see Algebra, Definition \\ref{algebra-definition-ring-CM}. \\item (Regular.) The ring $R$ is Noetherian and regular, see Algebra, Definition \\ref{algebra-definition-regular}. \\item (Absolutely Noetherian.) The ring $R$ is of finite type over $Z$. \\item Add more here as needed.\\footnote{But we only list those properties here which we have not already dealt with separately somewhere else.} \\end{enumerate}"}
+{"_id": "2951", "title": "properties-lemma-locally-Noetherian", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is locally Noetherian. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is Noetherian. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Noetherian. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is locally Noetherian. \\end{enumerate} Moreover, if $X$ is locally Noetherian then every open subscheme is locally Noetherian."}
+{"_id": "2952", "title": "properties-lemma-immersion-into-noetherian", "text": "Any immersion $Z \\to X$ with $X$ locally Noetherian is quasi-compact."}
+{"_id": "2953", "title": "properties-lemma-locally-Noetherian-quasi-separated", "text": "A locally Noetherian scheme is quasi-separated."}
+{"_id": "2954", "title": "properties-lemma-Noetherian-topology", "text": "A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition \\ref{topology-definition-noetherian}."}
+{"_id": "2955", "title": "properties-lemma-locally-closed-in-Noetherian", "text": "Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian."}
+{"_id": "2956", "title": "properties-lemma-Noetherian-irreducible-components", "text": "A Noetherian scheme has a finite number of irreducible components."}
+{"_id": "2957", "title": "properties-lemma-morphism-Noetherian-schemes-quasi-compact", "text": "Any morphism of schemes $f : X \\to Y$ with $X$ Noetherian is quasi-compact."}
+{"_id": "2958", "title": "properties-lemma-locally-Noetherian-closed-point", "text": "Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point."}
+{"_id": "2959", "title": "properties-lemma-locally-Noetherian-specialization-dvr", "text": "Let $X$ be a locally Noetherian scheme. Let $x' \\leadsto x$ be a specialization of points of $X$. Then \\begin{enumerate} \\item there exists a discrete valuation ring $R$ and a morphism $f : \\Spec(R) \\to X$ such that the generic point $\\eta$ of $\\Spec(R)$ maps to $x'$ and the special point maps to $x$, and \\item given a finitely generated field extension $\\kappa(x') \\subset K$ we may arrange it so that the extension $\\kappa(x') \\subset \\kappa(\\eta)$ induced by $f$ is isomorphic to the given one. \\end{enumerate}"}
+{"_id": "2960", "title": "properties-lemma-thin-infinite-sequence", "text": "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite subset. Then there exists an infinite subset $T' \\subset T$ such that there are no nontrivial specializations among the points $T'$."}
+{"_id": "2961", "title": "properties-lemma-maximal-points", "text": "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be a subset. Let $T_0 \\subset T$ be the set of $t \\in T$ such that there is no nontrivial specialization $t' \\leadsto t$ with $t' \\in T'$. Then (a) there are no specializations among the points of $T_0$, (b) every point of $T$ is a specialization of a point of $T_0$, and (c) the closures of $T$ and $T_0$ are the same."}
+{"_id": "2962", "title": "properties-lemma-countable-dense-subset", "text": "Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite dense subset. Then there exist a countable subset $E \\subset T$ which is dense in $S$."}
+{"_id": "2963", "title": "properties-lemma-affine-jacobson", "text": "An affine scheme $\\Spec(R)$ is Jacobson if and only if the ring $R$ is Jacobson."}
+{"_id": "2964", "title": "properties-lemma-locally-jacobson", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is Jacobson. \\item The scheme $X$ is ``locally Jacobson'' in the sense of Definition \\ref{definition-locally-P}. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is Jacobson. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Jacobson. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is Jacobson. \\end{enumerate} Moreover, if $X$ is Jacobson then every open subscheme is Jacobson."}
+{"_id": "2965", "title": "properties-lemma-complement-closed-point-Jacobson", "text": "Examples of Noetherian Jacobson schemes. \\begin{enumerate} \\item If $(R, \\mathfrak m)$ is a Noetherian local ring, then the punctured spectrum $\\Spec(R) \\setminus \\{\\mathfrak m\\}$ is a Jacobson scheme. \\item If $R$ is a Noetherian ring with Jacobson radical $\\text{rad}(R)$ then $\\Spec(R) \\setminus V(\\text{rad}(R))$ is a Jacobson scheme. \\item If $(R, I)$ is a Zariski pair (More on Algebra, Definition \\ref{more-algebra-definition-zariski-pair}) with $R$ Noetherian, then $\\Spec(R) \\setminus V(I)$ is a Jacobson scheme. \\end{enumerate}"}
+{"_id": "2966", "title": "properties-lemma-locally-normal", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is normal. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is normal. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is normal. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is normal. \\end{enumerate} Moreover, if $X$ is normal then every open subscheme is normal."}
+{"_id": "2967", "title": "properties-lemma-normal-reduced", "text": "A normal scheme is reduced."}
+{"_id": "2969", "title": "properties-lemma-normal-locally-finite-nr-irreducibles", "text": "Let $X$ be a scheme such that any quasi-compact open has a finite number of irreducible components. The following are equivalent: \\begin{enumerate} \\item $X$ is normal, and \\item $X$ is a disjoint union of normal integral schemes. \\end{enumerate}"}
+{"_id": "2970", "title": "properties-lemma-normal-Noetherian", "text": "Let $X$ be a Noetherian scheme. The following are equivalent: \\begin{enumerate} \\item $X$ is normal, and \\item $X$ is a finite disjoint union of normal integral schemes. \\end{enumerate}"}
+{"_id": "2971", "title": "properties-lemma-normal-locally-Noetherian", "text": "Let $X$ be a locally Noetherian scheme. The following are equivalent: \\begin{enumerate} \\item $X$ is normal, and \\item $X$ is a disjoint union of integral normal schemes. \\end{enumerate}"}
+{"_id": "2972", "title": "properties-lemma-normal-integral-sections", "text": "\\begin{slogan} The ring of functions on a normal scheme is normal. \\end{slogan} Let $X$ be an integral normal scheme. Then $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain."}
+{"_id": "2973", "title": "properties-lemma-characterize-Cohen-Macaulay", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item $X$ is Cohen-Macaulay, \\item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and \\item $X$ is locally Noetherian and for any closed point $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is Cohen-Macaulay. \\end{enumerate}"}
+{"_id": "2975", "title": "properties-lemma-characterize-regular", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item $X$ is regular, \\item $X$ is locally Noetherian and all of its local rings are regular, and \\item $X$ is locally Noetherian and for any closed point $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is regular. \\end{enumerate}"}
+{"_id": "2977", "title": "properties-lemma-regular-normal", "text": "A regular scheme is normal."}
+{"_id": "2978", "title": "properties-lemma-dimension", "text": "Let $X$ be a scheme. The following are equal \\begin{enumerate} \\item The dimension of $X$. \\item The supremum of the dimensions of the local rings of $X$. \\item The supremum of $\\dim_x(X)$ for $x \\in X$. \\end{enumerate}"}
+{"_id": "2979", "title": "properties-lemma-codimension-local-ring", "text": "Let $X$ be a scheme. Let $Y \\subset X$ be an irreducible closed subset. Let $\\xi \\in Y$ be the generic point. Then $$ \\text{codim}(Y, X) = \\dim(\\mathcal{O}_{X, \\xi}) $$ where the codimension is as defined in Topology, Definition \\ref{topology-definition-codimension}."}
+{"_id": "2980", "title": "properties-lemma-generic-point", "text": "Let $X$ be a scheme. Let $x \\in X$. Then $x$ is a generic point of an irreducible component of $X$ if and only if $\\dim(\\mathcal{O}_{X, x}) = 0$."}
+{"_id": "2981", "title": "properties-lemma-locally-Noetherian-dimension-0", "text": "A locally Noetherian scheme of dimension $0$ is a disjoint union of spectra of Artinian local rings."}
+{"_id": "2983", "title": "properties-lemma-catenary-local", "text": "Let $S$ be a scheme. The following are equivalent \\begin{enumerate} \\item $S$ is catenary, \\item there exists an open covering of $S$ all of whose members are catenary schemes, \\item for every affine open $\\Spec(R) = U \\subset S$ the ring $R$ is catenary, and \\item there exists an affine open covering $S = \\bigcup U_i$ such that each $U_i$ is the spectrum of a catenary ring. \\end{enumerate} Moreover, in this case any locally closed subscheme of $S$ is catenary as well."}
+{"_id": "2987", "title": "properties-lemma-scheme-CM-iff-all-Sk", "text": "Let $X$ be a locally Noetherian scheme. Then $X$ is Cohen-Macaulay if and only if $X$ has $(S_k)$ for all $k \\geq 0$."}
+{"_id": "2988", "title": "properties-lemma-criterion-reduced", "text": "Let $X$ be a locally Noetherian scheme. Then $X$ is reduced if and only if $X$ has properties $(S_1)$ and $(R_0)$."}
+{"_id": "2989", "title": "properties-lemma-criterion-normal", "text": "Let $X$ be a locally Noetherian scheme. Then $X$ is normal if and only if $X$ has properties $(S_2)$ and $(R_1)$."}
+{"_id": "2990", "title": "properties-lemma-normal-dimension-1-regular", "text": "Let $X$ be a locally Noetherian scheme which is normal and has dimension $\\leq 1$. Then $X$ is regular."}
+{"_id": "2991", "title": "properties-lemma-normal-dimension-2-Cohen-Macaulay", "text": "Let $X$ be a locally Noetherian scheme which is normal and has dimension $\\leq 2$. Then $X$ is Cohen-Macaulay."}
+{"_id": "2992", "title": "properties-lemma-nagata-locally-Noetherian", "text": "A Nagata scheme is locally Noetherian."}
+{"_id": "2993", "title": "properties-lemma-locally-Japanese", "text": "Let $X$ be an integral scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is Japanese. \\item For every affine open $U \\subset X$ the domain $\\mathcal{O}_X(U)$ is Japanese. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Japanese. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is Japanese. \\end{enumerate} Moreover, if $X$ is Japanese then every open subscheme is Japanese."}
+{"_id": "2995", "title": "properties-lemma-locally-nagata", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is Nagata. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is Nagata. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Nagata. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is Nagata. \\end{enumerate} Moreover, if $X$ is Nagata then every open subscheme is Nagata."}
+{"_id": "2998", "title": "properties-lemma-normal-geometrically-unibranch", "text": "A normal scheme is geometrically unibranch."}
+{"_id": "3000", "title": "properties-lemma-number-of-branches-irreducible-components", "text": "Let $X$ be a scheme and $x \\in X$. Let $X_i$, $i \\in I$ be the irreducible components of $X$ passing through $x$. Then the number of (geometric) branches of $X$ at $x$ is the sum over $i \\in I$ of the number of (geometric) branches of $X_i$ at $x$."}
+{"_id": "3001", "title": "properties-lemma-number-of-branches-1", "text": "Let $X$ be a scheme. Let $x \\in X$. \\begin{enumerate} \\item The number of branches of $X$ at $x$ is $1$ if and only if $X$ is unibranch at $x$. \\item The number of geometric branches of $X$ at $x$ is $1$ if and only if $X$ is geometrically unibranch at $x$. \\end{enumerate}"}
+{"_id": "3002", "title": "properties-lemma-finite-type-module", "text": "Let $X = \\Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\\mathcal{O}_X$-modules $\\widetilde M$ is a finite type $\\mathcal{O}_X$-module if and only if $M$ is a finite $R$-module."}
+{"_id": "3003", "title": "properties-lemma-finite-presentation-module", "text": "Let $X = \\Spec(R)$ be an affine scheme. The quasi-coherent sheaf of $\\mathcal{O}_X$-modules $\\widetilde M$ is an $\\mathcal{O}_X$-module of finite presentation if and only if $M$ is an $R$-module of finite presentation."}
+{"_id": "3004", "title": "properties-lemma-invert-f-sections", "text": "\\begin{slogan} Sections of quasi-coherent sheaves have only meromorphic singularities at infinity. \\end{slogan} Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. Denote $X_f \\subset X$ the open where $f$ is invertible, see Schemes, Lemma \\ref{schemes-lemma-f-open}. If $X$ is quasi-compact and quasi-separated, the canonical map $$ \\Gamma(X, \\mathcal{O}_X)_f \\longrightarrow \\Gamma(X_f, \\mathcal{O}_X) $$ is an isomorphism. Moreover, if $\\mathcal{F}$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules the map $$ \\Gamma(X, \\mathcal{F})_f \\longrightarrow \\Gamma(X_f, \\mathcal{F}) $$ is an isomorphism."}
+{"_id": "3005", "title": "properties-lemma-invert-s-sections", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $X$ is quasi-compact, then (\\ref{equation-module-invert-s}) is injective, and \\item if $X$ is quasi-compact and quasi-separated, then (\\ref{equation-module-invert-s}) is an isomorphism. \\end{enumerate} In particular, the canonical map $$ \\Gamma_*(X, \\mathcal{L})_{(s)} \\longrightarrow \\Gamma(X_s, \\mathcal{O}_X),\\quad a/s^n \\longmapsto a \\otimes s^{-n} $$ is an isomorphism if $X$ is quasi-compact and quasi-separated."}
+{"_id": "3007", "title": "properties-lemma-quasi-coherent-quasi-affine", "text": "Let $A$ be a ring and let $U \\subset \\Spec(A)$ be a quasi-compact open subscheme. For $\\mathcal{F}$ quasi-coherent on $U$ the canonical map $$ \\widetilde{H^0(U, \\mathcal{F})}|_U \\to \\mathcal{F} $$ is an isomorphism."}
+{"_id": "3008", "title": "properties-lemma-invert-f-affine", "text": "Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. Assume $X$ is quasi-compact and quasi-separated and assume that $X_f$ is affine. Then the canonical morphism $$ j : X \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X)) $$ from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} induces an isomorphism of $X_f = j^{-1}(D(f))$ onto the standard affine open $D(f) \\subset \\Spec(\\Gamma(X, \\mathcal{O}_X))$."}
+{"_id": "3009", "title": "properties-lemma-quasi-affine", "text": "Let $X$ be a scheme. Then $X$ is quasi-affine if and only if the canonical morphism $$ X \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X)) $$ from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} is a quasi-compact open immersion."}
+{"_id": "3010", "title": "properties-lemma-cartesian-diagram-quasi-affine", "text": "Let $U \\to V$ be an open immersion of quasi-affine schemes. Then $$ \\xymatrix{ U \\ar[d] \\ar[rr]_-j & & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\ U \\ar[r] & V \\ar[r]^-{j'} & \\Spec(\\Gamma(V, \\mathcal{O}_V)) } $$ is cartesian."}
+{"_id": "3011", "title": "properties-lemma-quasi-affine-presentation", "text": "Let $X$ be a quasi-affine scheme. There exists an integer $n \\geq 0$, an affine scheme $T$, and a morphism $T \\to X$ such that for every morphism $X' \\to X$ with $X'$ affine the fibre product $X' \\times_X T$ is isomorphic to $\\mathbf{A}^n_{X'}$ over $X'$."}
+{"_id": "3012", "title": "properties-lemma-flat-module", "text": "\\begin{slogan} Flatness is the same for modules and sheaves. \\end{slogan} Let $X = \\Spec(R)$ be an affine scheme. Let $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module."}
+{"_id": "3013", "title": "properties-lemma-locally-free-module", "text": "Let $X = \\Spec(R)$ be an affine scheme. Let $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\\mathcal{F}$ is a (finite) locally free $\\mathcal{O}_X$-module of if and only if $M$ is a (finite) locally free $R$-module."}
+{"_id": "3014", "title": "properties-lemma-finite-locally-free", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module of finite presentation, \\item $\\mathcal{F}$ is $\\mathcal{O}_X$-module of finite presentation and for all $x \\in X$ the stalk $\\mathcal{F}_x$ is a free $\\mathcal{O}_{X, x}$-module, \\item $\\mathcal{F}$ is a locally free, finite type $\\mathcal{O}_X$-module, \\item $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module, and \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, for every $x \\in X$ the stalk $\\mathcal{F}_x$ is a free $\\mathcal{O}_{X, x}$-module, and the function $$ \\rho_\\mathcal{F} : X \\to \\mathbf{Z}, \\quad x \\longmapsto \\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ is locally constant in the Zariski topology on $X$. \\end{enumerate}"}
+{"_id": "3016", "title": "properties-lemma-locally-projective", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is locally projective, and \\item there exists an affine open covering $X = \\bigcup U_i$ such that the $\\mathcal{O}_X(U_i)$-module $\\mathcal{F}(U_i)$ is projective for every $i$. \\end{enumerate} In particular, if $X = \\Spec(A)$ and $\\mathcal{F} = \\widetilde{M}$ then $\\mathcal{F}$ is locally projective if and only if $M$ is a projective $A$-module."}
+{"_id": "3017", "title": "properties-lemma-locally-projective-pullback", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. If $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$ is locally projective on $X$."}
+{"_id": "3018", "title": "properties-lemma-extend-trivial", "text": "Let $j : U \\to X$ be a quasi-compact open immersion of schemes. \\begin{enumerate} \\item Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$. \\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\\mathcal{H}$ of $\\mathcal{F}$ such that $\\mathcal{H}|_U = \\mathcal{G}$ as subsheaves of $\\mathcal{F}|_U$. \\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism of $\\mathcal{O}_U$-modules. There exists a quasi-coherent sheaf $\\mathcal{H}$ of $\\mathcal{O}_X$-modules and a map $\\psi : \\mathcal{H} \\to \\mathcal{F}$ such that $\\mathcal{H}|_U = \\mathcal{G}$ and that $\\psi|_U = \\varphi$. \\end{enumerate}"}
+{"_id": "3019", "title": "properties-lemma-extend", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent $\\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$ which is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$."}
+{"_id": "3020", "title": "properties-lemma-quasi-coherent-colimit-finite-type", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of $\\mathcal{O}_X$-modules is the directed colimit of its quasi-coherent $\\mathcal{O}_X$-submodules which are of finite type."}
+{"_id": "3021", "title": "properties-lemma-extend-finite-presentation", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be a quasi-compact open. Let $\\mathcal{G}$ be an $\\mathcal{O}_U$-module which is of finite presentation. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism of $\\mathcal{O}_U$-modules. Then there exists an $\\mathcal{O}_X$-module $\\mathcal{G}'$ of finite presentation, and a morphism of $\\mathcal{O}_X$-modules $\\varphi' : \\mathcal{G}' \\to \\mathcal{F}$ such that $\\mathcal{G}'|_U = \\mathcal{G}$ and such that $\\varphi'|_U = \\varphi$."}
+{"_id": "3022", "title": "properties-lemma-lift-finite-presentation", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open. Let $\\mathcal{G}$ be an $\\mathcal{O}_U$-module. \\begin{enumerate} \\item If $\\mathcal{G}$ is quasi-coherent and of finite type, then there exists a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{G}'$ of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$. \\item If $\\mathcal{G}$ is of finite presentation, then there exists an $\\mathcal{O}_X$-module $\\mathcal{G}'$ of finite presentation such that $\\mathcal{G}'|_U = \\mathcal{G}$. \\end{enumerate}"}
+{"_id": "3023", "title": "properties-lemma-directed-colimit-diagram-finite-presentation", "text": "\\begin{slogan} Quasi-coherent modules on quasi-compact and quasi-separated schemes are filtered colimits of finitely presented modules. \\end{slogan} Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. There exist \\begin{enumerate} \\item a filtered index category $\\mathcal{I}$ (see Categories, Definition \\ref{categories-definition-directed}), \\item a diagram $\\mathcal{I} \\to \\textit{Mod}(\\mathcal{O}_X)$ (see Categories, Section \\ref{categories-section-limits}), $i \\mapsto \\mathcal{F}_i$, \\item morphisms of $\\mathcal{O}_X$-modules $\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$ \\end{enumerate} such that each $\\mathcal{F}_i$ is of finite presentation and such that the morphisms $\\varphi_i$ induce an isomorphism $$ \\colim_i \\mathcal{F}_i = \\mathcal{F}. $$"}
+{"_id": "3024", "title": "properties-lemma-directed-colimit-finite-presentation", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. There exist \\begin{enumerate} \\item a directed set $I$ (see Categories, Definition \\ref{categories-definition-directed-set}), \\item a system $(\\mathcal{F}_i, \\varphi_{ii'})$ over $I$ in $\\textit{Mod}(\\mathcal{O}_X)$ (see Categories, Definition \\ref{categories-definition-system-over-poset}) \\item morphisms of $\\mathcal{O}_X$-modules $\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$ \\end{enumerate} such that each $\\mathcal{F}_i$ is of finite presentation and such that the morphisms $\\varphi_i$ induce an isomorphism $$ \\colim_i \\mathcal{F}_i = \\mathcal{F}. $$"}
+{"_id": "3025", "title": "properties-lemma-finite-directed-colimit-surjective-maps", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Then we can write $\\mathcal{F} = \\colim \\mathcal{F}_i$ with $\\mathcal{F}_i$ of finite presentation and all transition maps $\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective."}
+{"_id": "3027", "title": "properties-lemma-algebra-directed-colimit-finite-presentation", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra. There exist \\begin{enumerate} \\item a directed set $I$ (see Categories, Definition \\ref{categories-definition-directed-set}), \\item a system $(\\mathcal{A}_i, \\varphi_{ii'})$ over $I$ in the category of $\\mathcal{O}_X$-algebras, \\item morphisms of $\\mathcal{O}_X$-algebras $\\varphi_i : \\mathcal{A}_i \\to \\mathcal{A}$ \\end{enumerate} such that each $\\mathcal{A}_i$ is a quasi-coherent $\\mathcal{O}_X$-algebra of finite presentation and such that the morphisms $\\varphi_i$ induce an isomorphism $$ \\colim_i \\mathcal{A}_i = \\mathcal{A}. $$"}
+{"_id": "3028", "title": "properties-lemma-algebra-directed-colimit-finite-type", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra. Then $\\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\\mathcal{O}_X$-subalgebras."}
+{"_id": "3029", "title": "properties-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{A}$ be a finite quasi-coherent $\\mathcal{O}_X$-algebra. Then $\\mathcal{A} = \\colim \\mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\\mathcal{O}_X$-algebras such that all transition maps $\\mathcal{A}_{i'} \\to \\mathcal{A}_i$ are surjective."}
+{"_id": "3030", "title": "properties-lemma-integral-algebra-directed-colimit-finite", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{A}$ be an integral quasi-coherent $\\mathcal{O}_X$-algebra. Then \\begin{enumerate} \\item $\\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\\mathcal{O}_X$-subalgebras, and \\item $\\mathcal{A}$ is a direct colimit of finite and finitely presented quasi-coherent $\\mathcal{O}_X$-algebras. \\end{enumerate}"}
+{"_id": "3031", "title": "properties-lemma-set-of-iso-classes", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be a cardinal. There exists a set $T$ and a family $(\\mathcal{F}_t)_{t \\in T}$ of $\\kappa$-generated $\\mathcal{O}_X$-modules such that every $\\kappa$-generated $\\mathcal{O}_X$-module is isomorphic to one of the $\\mathcal{F}_t$."}
+{"_id": "3032", "title": "properties-lemma-colimit-kappa", "text": "Let $X$ be a scheme. There exists a cardinal $\\kappa$ such that every quasi-coherent module $\\mathcal{F}$ is the directed colimit of its quasi-coherent $\\kappa$-generated quasi-coherent subsheaves."}
+{"_id": "3033", "title": "properties-lemma-quasi-coherent-finite-type-ideals", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be an open subscheme. The following are equivalent: \\begin{enumerate} \\item $U$ is retrocompact in $X$, \\item $U$ is quasi-compact, \\item $U$ is a finite union of affine opens, and \\item there exists a finite type quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ such that $X \\setminus U = V(\\mathcal{I})$ (set theoretically). \\end{enumerate}"}
+{"_id": "3034", "title": "properties-lemma-sections-annihilated-by-ideal", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$ which associates to every open $U \\subset X$ $$ \\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}s = 0\\} $$ Assume $\\mathcal{I}$ is of finite type. Then \\begin{enumerate} \\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules, \\item on any affine open $U \\subset X$ we have $\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and \\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$. \\end{enumerate}"}
+{"_id": "3036", "title": "properties-lemma-sections-supported-on-closed-subset", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$ which associates to every open $U \\subset X$ $$ \\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\text{the support of }s\\text{ is contained in }Z \\cap U\\} $$ If $X \\setminus Z$ is a retrocompact open of $X$, then \\begin{enumerate} \\item for an affine open $U \\subset X$ there exist a finitely generated ideal $I \\subset \\mathcal{O}_X(U)$ such that $Z \\cap U = V(I)$, \\item for $U$ and $I$ as in (1) we have $\\mathcal{F}'(U) = \\{x \\in \\mathcal{F}(U) \\mid I^nx = 0 \\text{ for some } n\\}$, \\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "3037", "title": "properties-lemma-push-sections-supported-on-closed-subset", "text": "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $Z \\subset Y$ be a closed subset such that $Y \\setminus Z$ is retrocompact in $Y$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the subsheaf of sections supported in $f^{-1}Z$. Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf of sections supported in $Z$."}
+{"_id": "3038", "title": "properties-lemma-sections-over-quasi-compact-open-in-affine", "text": "Let $A$ be a ring. Let $I \\subset A$ be a finitely generated ideal. Let $M$ be an $A$-module. Then there is a canonical map $$ \\colim_n \\Hom_A(I^n, M) \\longrightarrow \\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}). $$ This map is always injective. If for all $x \\in M$ we have $Ix = 0 \\Rightarrow x = 0$ then this map is an isomorphism. In general, set $M_n = \\{x \\in M \\mid I^nx = 0\\}$, then there is an isomorphism $$ \\colim_n \\Hom_A(I^n, M/M_n) \\longrightarrow \\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}). $$"}
+{"_id": "3039", "title": "properties-lemma-sections-over-quasi-compact-open", "text": "Let $X$ be a quasi-compact scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $Z \\subset X$ be the closed subscheme defined by $\\mathcal{I}$ and set $U = X \\setminus Z$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The canonical map $$ \\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n, \\mathcal{F}) \\longrightarrow \\Gamma(U, \\mathcal{F}) $$ is injective. Assume further that $X$ is quasi-separated. Let $\\mathcal{F}_n \\subset \\mathcal{F}$ be subsheaf of sections annihilated by $\\mathcal{I}^n$. The canonical map $$ \\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n, \\mathcal{F}/\\mathcal{F}_n) \\longrightarrow \\Gamma(U, \\mathcal{F}) $$ is an isomorphism."}
+{"_id": "3040", "title": "properties-lemma-ample-power-ample", "text": "\\begin{reference} \\cite[II Proposition 4.5.6(i)]{EGA} \\end{reference} Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $n \\geq 1$. Then $\\mathcal{L}$ is ample if and only if $\\mathcal{L}^{\\otimes n}$ is ample."}
+{"_id": "3041", "title": "properties-lemma-ample-on-closed", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. For any closed subscheme $Z \\subset X$ the restriction of $\\mathcal{L}$ to $Z$ is ample."}
+{"_id": "3042", "title": "properties-lemma-affine-cap-s-open", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. For any affine $U \\subset X$ the intersection $U \\cap X_s$ is affine."}
+{"_id": "3043", "title": "properties-lemma-ample-tensor-globally-generated", "text": "\\begin{reference} \\cite[II Proposition 4.5.6(ii)]{EGA} \\end{reference} Let $X$ be a scheme. Let $\\mathcal{L}$ and $\\mathcal{M}$ be invertible $\\mathcal{O}_X$-modules. If \\begin{enumerate} \\item $\\mathcal{L}$ is ample, and \\item the open sets $X_t$ where $t \\in \\Gamma(X, \\mathcal{M}^{\\otimes m})$ for $m > 0$ cover $X$, \\end{enumerate} then $\\mathcal{L} \\otimes \\mathcal{M}$ is ample."}
+{"_id": "3044", "title": "properties-lemma-affine-s-opens", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume the open sets $X_s$, where $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ and $n \\geq 1$, form a basis for the topology on $X$. Then among those opens, the open sets $X_s$ which are affine form a basis for the topology on $X$."}
+{"_id": "3045", "title": "properties-lemma-affine-s-opens-cover-quasi-separated", "text": "Let $X$ be a scheme and $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume for every point $x$ of $X$ there exists $n \\geq 1$ and $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that $x \\in X_s$ and $X_s$ is affine. Then $X$ is separated."}
+{"_id": "3046", "title": "properties-lemma-ample-separated", "text": "Let $X$ be a scheme. If there exists an ample invertible sheaf on $X$ then $X$ is separated."}
+{"_id": "3047", "title": "properties-lemma-map-into-proj", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Set $S = \\Gamma_*(X, \\mathcal{L})$ as a graded ring. If every point of $X$ is contained in one of the open subschemes $X_s$, for some $s \\in S_{+}$ homogeneous, then there is a canonical morphism of schemes $$ f : X \\longrightarrow Y = \\text{Proj}(S), $$ to the homogeneous spectrum of $S$ (see Constructions, Section \\ref{constructions-section-proj}). This morphism has the following properties \\begin{enumerate} \\item $f^{-1}(D_{+}(s)) = X_s$ for any $s \\in S_{+}$ homogeneous, \\item there are $\\mathcal{O}_X$-module maps $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$ compatible with multiplication maps, see Constructions, Equation (\\ref{constructions-equation-multiply}), \\item the composition $S_n \\to \\Gamma(Y, \\mathcal{O}_Y(n)) \\to \\Gamma(X, \\mathcal{L}^{\\otimes n})$ is the identity map, and \\item for every $x \\in X$ there is an integer $d \\geq 1$ and an open neighbourhood $U \\subset X$ of $x$ such that $f^*\\mathcal{O}_Y(dn)|_U \\to \\mathcal{L}^{\\otimes dn}|_U$ is an isomorphism for all $n \\in \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "3048", "title": "properties-lemma-map-into-proj-quasi-compact", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Set $S = \\Gamma_*(X, \\mathcal{L})$. Assume (a) every point of $X$ is contained in one of the open subschemes $X_s$, for some $s \\in S_{+}$ homogeneous, and (b) $X$ is quasi-compact. Then the canonical morphism of schemes $f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj} above is quasi-compact with dense image."}
+{"_id": "3049", "title": "properties-lemma-ample-immersion-into-proj", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Set $S = \\Gamma_*(X, \\mathcal{L})$. Assume $\\mathcal{L}$ is ample. Then the canonical morphism of schemes $f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj} is an open immersion with dense image."}
+{"_id": "3050", "title": "properties-lemma-open-in-proj-ample", "text": "Let $X$ be a scheme. Let $S$ be a graded ring. Assume $X$ is quasi-compact, and assume there exists an open immersion $$ j : X \\longrightarrow Y = \\text{Proj}(S). $$ Then $j^*\\mathcal{O}_Y(d)$ is an invertible ample sheaf for some $d > 0$."}
+{"_id": "3051", "title": "properties-lemma-ample-on-locally-closed", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $i : X' \\to X$ be a morphism of schemes. Assume at least one of the following conditions holds \\begin{enumerate} \\item $i$ is a quasi-compact immersion, \\item $X'$ is quasi-compact and $i$ is an immersion, \\item $i$ is quasi-compact and induces a homeomorphism between $X'$ and $i(X')$, \\item $X'$ is quasi-compact and $i$ induces a homeomorphism between $X'$ and $i(X')$. \\end{enumerate} Then $i^*\\mathcal{L}$ is ample on $X'$."}
+{"_id": "3052", "title": "properties-lemma-ample-on-product", "text": "Let $S$ be a quasi-separated scheme. Let $X$, $Y$ be schemes over $S$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module and let $\\mathcal{N}$ be an ample invertible $\\mathcal{O}_Y$-module. Then $\\mathcal{M} = \\text{pr}_1^*\\mathcal{L} \\otimes_{\\mathcal{O}_{X \\times_S Y}} \\text{pr}_2^*\\mathcal{N}$ is an ample invertible sheaf on $X \\times_S Y$."}
+{"_id": "3053", "title": "properties-lemma-quasi-affine-O-ample", "text": "Let $X$ be a scheme. Then $X$ is quasi-affine if and only if $\\mathcal{O}_X$ is ample."}
+{"_id": "3054", "title": "properties-lemma-quasi-affine-locally-closed", "text": "Let $X$ be a quasi-affine scheme. For any quasi-compact immersion $i : X' \\to X$ the scheme $X'$ is quasi-affine."}
+{"_id": "3055", "title": "properties-lemma-characterize-affine", "text": "Let $X$ be a scheme. Suppose that there exist finitely many elements $f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$ such that \\begin{enumerate} \\item each $X_{f_i}$ is an affine open of $X$, and \\item the ideal generated by $f_1, \\ldots, f_n$ in $\\Gamma(X, \\mathcal{O}_X)$ is equal to the unit ideal. \\end{enumerate} Then $X$ is affine."}
+{"_id": "3056", "title": "properties-lemma-ample-gcd-is-one", "text": "In Situation \\ref{situation-ample}. The canonical morphism $f : X \\to Y$ maps $X$ into the open subscheme $W = W_1 \\subset Y$ where $\\mathcal{O}_Y(1)$ is invertible and where all multiplication maps $\\mathcal{O}_Y(n) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m) \\to \\mathcal{O}_Y(n + m)$ are isomorphisms (see Constructions, Lemma \\ref{constructions-lemma-where-invertible}). Moreover, the maps $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$ are all isomorphisms."}
+{"_id": "3057", "title": "properties-lemma-ample-quasi-coherent", "text": "In Situation \\ref{situation-ample}. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Set $M = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F})$ as a graded $S$-module. There are isomorphisms $$ f^*\\widetilde{M} \\longrightarrow \\mathcal{F} $$ functorial in $\\mathcal{F}$ such that $M_0 \\to \\Gamma(\\text{Proj}(S), \\widetilde{M}) \\to \\Gamma(X, \\mathcal{F})$ is the identity map."}
+{"_id": "3058", "title": "properties-lemma-proj-quasi-coherent", "text": "Let $S$ be a graded ring such that $X = \\text{Proj}(S)$ is quasi-compact. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Set $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as a graded $S$-module, see Constructions, Section \\ref{constructions-section-invertible-on-proj}. The map $$ \\widetilde{M} \\longrightarrow \\mathcal{F} $$ of Constructions, Lemma \\ref{constructions-lemma-comparison-proj-quasi-coherent} is an isomorphism. If $X$ is covered by standard opens $D_+(f)$ where $f$ has degree $1$, then the induced maps $M_n \\to \\Gamma(X, \\mathcal{F}(n))$ are the identity maps."}
+{"_id": "3059", "title": "properties-lemma-maximal-points-affine", "text": "Let $X$ be a quasi-separated scheme. Let $Z_1, \\ldots, Z_n$ be pairwise distinct irreducible components of $X$, see Topology, Section \\ref{topology-section-irreducible-components}. Let $\\eta_i \\in Z_i$ be their generic points, see Schemes, Lemma \\ref{schemes-lemma-scheme-sober}. There exist affine open neighbourhoods $\\eta_i \\in U_i$ such that $U_i \\cap U_j = \\emptyset$ for all $i \\not = j$. In particular, $U = U_1 \\cup \\ldots \\cup U_n$ is an affine open containing all of the points $\\eta_1, \\ldots, \\eta_n$."}
+{"_id": "3060", "title": "properties-lemma-quasi-compact-dense-open-separated", "text": "Let $X$ be a quasi-compact scheme. There exists a dense open $V \\subset X$ which is separated."}
+{"_id": "3061", "title": "properties-lemma-point-and-maximal-points-affine", "text": "Let $X$ be a quasi-separated scheme. Let $Z_1, \\ldots, Z_n$ be pairwise distinct irreducible components of $X$. Let $\\eta_i \\in Z_i$ be their generic points. Let $x \\in X$ be arbitrary. There exists an affine open $U \\subset X$ containing $x$ and all the $\\eta_i$."}
+{"_id": "3062", "title": "properties-lemma-ample-finite-set-in-affine", "text": "Let $X$ be a scheme. Assume either \\begin{enumerate} \\item The scheme $X$ is quasi-affine. \\item The scheme $X$ is isomorphic to a locally closed subscheme of an affine scheme. \\item There exists an ample invertible sheaf on $X$. \\item The scheme $X$ is isomorphic to a locally closed subscheme of $\\text{Proj}(S)$ for some graded ring $S$. \\end{enumerate} Then for any finite subset $E \\subset X$ there exists an affine open $U \\subset X$ with $E \\subset U$."}
+{"_id": "3063", "title": "properties-lemma-ample-finite-set-in-principal-affine", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$. Let $$ E \\subset W \\subset X $$ with $E$ finite and $W$ open in $X$. Then there exists an $n > 0$ and a section $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that $X_s$ is affine and $E \\subset X_s \\subset W$."}
+{"_id": "3065", "title": "properties-lemma-ring-affine-open-injective-local-ring", "text": "Let $X$ be a scheme and $x \\in X$ a point. There exists an affine open neighbourhood $U \\subset X$ of $x$ such that the canonical map $\\mathcal{O}_X(U) \\to \\mathcal{O}_{X, x}$ is injective in each of the following cases: \\begin{enumerate} \\item $X$ is integral, \\item $X$ is locally Noetherian, \\item $X$ is reduced and has a finite number of irreducible components. \\end{enumerate}"}
+{"_id": "3066", "title": "properties-proposition-coherator", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$ has enough injectives and all limits. \\item The inclusion functor $\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$ has a right adjoint\\footnote{This functor is sometimes called the {\\it coherator}.} $$ Q : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X) $$ such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "3067", "title": "properties-proposition-characterize-ample", "text": "Let $X$ be a quasi-compact scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Set $S = \\Gamma_*(X, \\mathcal{L})$. The following are equivalent: \\begin{enumerate} \\item \\label{item-ample} $\\mathcal{L}$ is ample, \\item \\label{item-immersion} the open sets $X_s$, with $s \\in S_{+}$ homogeneous, cover $X$ and the associated morphism $X \\to \\text{Proj}(S)$ is an open immersion, \\item \\label{item-s-basis} the open sets $X_s$, with $s \\in S_{+}$ homogeneous, form a basis for the topology of $X$, \\item \\label{item-s-affine-basis} the open sets $X_s$, with $s \\in S_{+}$ homogeneous, which are affine form a basis for the topology of $X$, \\item \\label{item-qc-gg} for every quasi-coherent sheaf $\\mathcal{F}$ on $X$ the sum of the images of the canonical maps $$ \\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) \\otimes_{\\mathbf{Z}} \\mathcal{L}^{\\otimes -n} \\longrightarrow \\mathcal{F} $$ with $n \\geq 1$ equals $\\mathcal{F}$, \\item \\label{item-qc-i-gg} same property as (\\ref{item-qc-gg}) with $\\mathcal{F}$ ranging over all quasi-coherent sheaves of ideals, \\item \\label{item-c-gg} $X$ is quasi-separated and for every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$ there exists an integer $n_0$ such that $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$ is globally generated for all $n \\geq n_0$, \\item \\label{item-c-q} $X$ is quasi-separated and for every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$ there exist integers $n > 0$, $k \\geq 0$ such that $\\mathcal{F}$ is a quotient of a direct sum of $k$ copies of $\\mathcal{L}^{\\otimes - n}$, and \\item \\label{item-c-i-q} same as in (\\ref{item-c-q}) with $\\mathcal{F}$ ranging over all sheaves of ideals of finite type on $X$. \\end{enumerate}"}
+{"_id": "3093", "title": "criteria-theorem-bootstrap", "text": "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If \\begin{enumerate} \\item $\\mathcal{X}$ is representable by an algebraic space, and \\item $F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation, \\end{enumerate} then $\\mathcal{Y}$ is an algebraic stack."}
+{"_id": "3094", "title": "criteria-theorem-flat-groupoid-gives-algebraic-stack", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the quotient stack $[U/R]$ is an algebraic stack over $S$."}
+{"_id": "3095", "title": "criteria-lemma-etale-permanence", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ are representable by algebraic spaces and \\'etale so is $\\mathcal{X} \\to \\mathcal{Y}$."}
+{"_id": "3097", "title": "criteria-lemma-flat-finite-presentation-surjective-diagonal", "text": "Let $S$ be a scheme. Let $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If \\begin{enumerate} \\item $\\mathcal{U}$ is representable by an algebraic space, and \\item $u$ is representable by algebraic spaces, surjective, flat and locally of finite presentation, \\end{enumerate} then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ representable by algebraic spaces."}
+{"_id": "3098", "title": "criteria-lemma-second-diagonal", "text": "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent \\begin{enumerate} \\item $\\Delta_\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$ is representable by algebraic spaces, \\item for every $1$-morphism $\\mathcal{V} \\to \\mathcal{X} \\times \\mathcal{X}$ with $\\mathcal{V}$ representable (by a scheme) the fibre product $\\mathcal{Y} = \\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}} \\mathcal{V}$ has diagonal representable by algebraic spaces. \\end{enumerate}"}
+{"_id": "3099", "title": "criteria-lemma-base-change-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving on objects, then so is the base change $p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$ of $p$ by $q$."}
+{"_id": "3100", "title": "criteria-lemma-composition-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ and $q$ are limit preserving on objects, then so is the composition $q \\circ p$."}
+{"_id": "3101", "title": "criteria-lemma-representable-by-spaces-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \\begin{enumerate} \\item $p$ is limit preserving on objects, and \\item $p$ is locally of finite presentation (see Algebraic Stacks, Definition \\ref{algebraic-definition-relative-representable-property}). \\end{enumerate}"}
+{"_id": "3102", "title": "criteria-lemma-open-immersion-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $p$ is representable by algebraic spaces and an open immersion. Then $p$ is limit preserving on objects."}
+{"_id": "3103", "title": "criteria-lemma-check-representable-limit-preserving", "text": "Let $S$ be a scheme. Let $\\kappa = \\text{size}(T)$ for some $T \\in \\Ob((\\Sch/S)_{fppf})$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ such that \\begin{enumerate} \\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, \\item for an affine scheme $V$ locally of finite presentation over $S$ and $y \\in \\Ob(\\mathcal{Y}_V)$ the fibre product $(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ is representable by an algebraic space of size $\\leq \\kappa$\\footnote{The condition on size can be dropped by those ignoring set theoretic issues.}, \\item $\\mathcal{X}$ and $\\mathcal{Y}$ are stacks for the Zariski topology. \\end{enumerate} Then $f$ is representable by algebraic spaces."}
+{"_id": "3104", "title": "criteria-lemma-check-property-limit-preserving", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition \\ref{algebraic-definition-relative-representable-property}. If \\begin{enumerate} \\item $f$ is representable by algebraic spaces, \\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, \\item for an affine scheme $V$ locally of finite presentation over $S$ and $y \\in \\mathcal{Y}_V$ the resulting morphism of algebraic spaces $f_y : F_y \\to V$, see Algebraic Stacks, Equation (\\ref{algebraic-equation-representable-by-algebraic-spaces}), has property $\\mathcal{P}$. \\end{enumerate} Then $f$ has property $\\mathcal{P}$."}
+{"_id": "3105", "title": "criteria-lemma-base-change-formally-smooth", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p : \\mathcal{X} \\to \\mathcal{Y}$ is formally smooth on objects, then so is the base change $p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$ of $p$ by $q$."}
+{"_id": "3106", "title": "criteria-lemma-composition-formally-smooth", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ and $q$ are formally smooth on objects, then so is the composition $q \\circ p$."}
+{"_id": "3107", "title": "criteria-lemma-representable-by-spaces-formally-smooth", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \\begin{enumerate} \\item $p$ is formally smooth on objects, and \\item $p$ is formally smooth (see Algebraic Stacks, Definition \\ref{algebraic-definition-relative-representable-property}). \\end{enumerate}"}
+{"_id": "3108", "title": "criteria-lemma-base-change-surjective", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p : \\mathcal{X} \\to \\mathcal{Y}$ is surjective on objects, then so is the base change $p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$ of $p$ by $q$."}
+{"_id": "3109", "title": "criteria-lemma-composition-surjective", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ and $q$ are surjective on objects, then so is the composition $q \\circ p$."}
+{"_id": "3110", "title": "criteria-lemma-representable-by-spaces-surjective", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \\begin{enumerate} \\item $p$ is surjective on objects, and \\item $p$ is surjective (see Algebraic Stacks, Definition \\ref{algebraic-definition-relative-representable-property}). \\end{enumerate}"}
+{"_id": "3113", "title": "criteria-lemma-diagonals-and-algebraic-morphisms", "text": "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If $F$ is algebraic and $\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$ is representable by algebraic spaces, then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces."}
+{"_id": "3114", "title": "criteria-lemma-surjection-space-of-sections", "text": "Let $Z \\to U$ be a finite morphism of schemes. Let $W$ be an algebraic space and let $W \\to Z$ be a surjective \\'etale morphism. Then there exists a surjective \\'etale morphism $U' \\to U$ and a section $$ \\sigma : Z_{U'} \\to W_{U'} $$ of the morphism $W_{U'} \\to Z_{U'}$."}
+{"_id": "3115", "title": "criteria-lemma-space-of-sections", "text": "Let $Z \\to U$ be a finite locally free morphism of schemes. Let $W$ be an algebraic space and let $W \\to Z$ be an \\'etale morphism. Then the functor $$ F : (\\Sch/U)_{fppf}^{opp} \\longrightarrow \\textit{Sets}, $$ defined by the rule $$ U' \\longmapsto F(U') = \\{\\sigma : Z_{U'} \\to W_{U'}\\text{ section of }W_{U'} \\to Z_{U'}\\} $$ is an algebraic space and the morphism $F \\to U$ is \\'etale."}
+{"_id": "3116", "title": "criteria-lemma-hom-functor-sheaf", "text": "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of algebraic spaces over $S$. Then \\begin{enumerate} \\item $\\mathit{Mor}_B(Z, X)$ is a sheaf on $(\\Sch/S)_{fppf}$. \\item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Mor}_B(Z, X)) = \\{(a, b)\\text{ as in }(\\ref{equation-hom})\\} $$ \\end{enumerate}"}
+{"_id": "3117", "title": "criteria-lemma-base-change-hom-functor", "text": "Let $S$ be a scheme. Let $Z \\to B$, $X \\to B$, and $B' \\to B$ be morphisms of algebraic spaces over $S$. Set $Z' = B' \\times_B Z$ and $X' = B' \\times_B X$. Then $$ \\mathit{Mor}_{B'}(Z', X') = B' \\times_B \\mathit{Mor}_B(Z, X) $$ in $\\Sh((\\Sch/S)_{fppf})$."}
+{"_id": "3118", "title": "criteria-lemma-etale-covering-hom-functor", "text": "Let $S$ be a scheme. Let $Z \\to B$ and $X' \\to X \\to B$ be morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X' \\to X$ is \\'etale, and \\item $Z \\to B$ is finite locally free. \\end{enumerate} Then $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is representable by algebraic spaces and \\'etale. If $X' \\to X$ is also surjective, then $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is surjective."}
+{"_id": "3119", "title": "criteria-lemma-restriction-of-scalars-sheaf", "text": "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of algebraic spaces over $S$. Then \\begin{enumerate} \\item $\\text{Res}_{Z/B}(X)$ is a sheaf on $(\\Sch/S)_{fppf}$. \\item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\text{Res}_{Z/B}(X)) = \\{(a, b)\\text{ as in }(\\ref{equation-pairs})\\} $$ \\end{enumerate}"}
+{"_id": "3122", "title": "criteria-lemma-fibre-diagram", "text": "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of algebraic spaces over $S$. The following diagram $$ \\xymatrix{ \\mathit{Mor}_B(Z, X) \\ar[r] & \\mathit{Mor}_B(Z, Z) \\\\ \\text{Res}_{Z/B}(X) \\ar[r] \\ar[u] & B \\ar[u]_{\\text{id}_Z} } $$ is a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$."}
+{"_id": "3123", "title": "criteria-lemma-map-hilbert", "text": "Consider a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[r]_G \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\ \\mathcal{Y}' \\ar[r]^H & \\mathcal{Y} } $$ of stacks in groupoids over $(\\Sch/S)_{fppf}$ with a given $2$-isomorphism $\\gamma : H \\circ F' \\to F \\circ G$. In this situation we obtain a canonical $1$-morphism $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$. This morphism is compatible with the forgetful $1$-morphisms of Examples of Stacks, Equation (\\ref{examples-stacks-equation-diagram-hilbert-d-stack})."}
+{"_id": "3124", "title": "criteria-lemma-cartesian-map-hilbert", "text": "In the situation of Lemma \\ref{lemma-map-hilbert} assume that the given square is $2$-cartesian. Then the diagram $$ \\xymatrix{ \\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\ar[r] \\ar[d] & \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\ar[d] \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ is $2$-cartesian."}
+{"_id": "3125", "title": "criteria-lemma-etale-covering-hilbert", "text": "In the situation of Lemma \\ref{lemma-map-hilbert} assume \\begin{enumerate} \\item $\\mathcal{Y}' = \\mathcal{Y}$ and $H = \\text{id}_\\mathcal{Y}$, \\item $G$ is representable by algebraic spaces and \\'etale. \\end{enumerate} Then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by algebraic spaces and \\'etale. If $G$ is also surjective, then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective."}
+{"_id": "3127", "title": "criteria-lemma-relative-hilbert", "text": "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume that $\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$ is representable by algebraic spaces. Then $$ \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\longrightarrow \\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y} $$ see Examples of Stacks, Equation (\\ref{examples-stacks-equation-diagram-hilbert-d-stack}) is representable by algebraic spaces."}
+{"_id": "3128", "title": "criteria-lemma-representable-on-top", "text": "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{X}' \\to \\mathcal{X}$ be $1$-morphisms of stacks in groupoids over $(\\Sch/S)_{fppf}$. If $G$ is representable by algebraic spaces, then the $1$-morphism $$ \\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\longrightarrow \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) $$ is representable by algebraic spaces."}
+{"_id": "3129", "title": "criteria-lemma-limit-preserving", "text": "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces and locally of finite presentation. Then $$ p : \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y} $$ is limit preserving on objects."}
+{"_id": "3130", "title": "criteria-lemma-represent-FAd", "text": "The functor in groupoids $FA_d$ defined in (\\ref{equation-FAd}) is isomorphic (!) to the functor in groupoids which associates to a scheme $T$ the category with \\begin{enumerate} \\item set of objects is $X(T)$, \\item set of morphisms is $G(T) \\times X(T)$, \\item $s : G(T) \\times X(T) \\to X(T)$ is the projection map, \\item $t : G(T) \\times X(T) \\to X(T)$ is $a(T)$, and \\item composition $G(T) \\times X(T) \\times_{s, X(T), t} G(T) \\times X(T) \\to G(T) \\times X(T)$ is given by $((g, m), (g', m')) \\mapsto (gg', m')$. \\end{enumerate}"}
+{"_id": "3131", "title": "criteria-lemma-hilbert-stack-of-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $\\mathcal{H}_d(X)$ is an algebraic stack."}
+{"_id": "3132", "title": "criteria-lemma-hilbert-stack-relative-space", "text": "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$ such that \\begin{enumerate} \\item $\\mathcal{X}$ is representable by an algebraic space, and \\item $F$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. \\end{enumerate} Then $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is an algebraic stack."}
+{"_id": "3133", "title": "criteria-lemma-lci-locus-stack-in-groupoids", "text": "Let $S$ be a scheme. Fix a $1$-morphism $F : \\mathcal{X} \\longrightarrow \\mathcal{Y}$ of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$ is a stack in groupoids and the inclusion functor $$ \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\longrightarrow \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) $$ is representable and an open immersion."}
+{"_id": "3134", "title": "criteria-lemma-lci-unobstructed", "text": "Let $U \\subset U'$ be a first order thickening of affine schemes. Let $X'$ be an algebraic space flat over $U'$. Set $X = U \\times_{U'} X'$. Let $Z \\to U$ be finite locally free of degree $d$. Finally, let $f : Z \\to X$ be unramified and a local complete intersection morphism. Then there exists a commutative diagram $$ \\xymatrix{ (Z \\subset Z') \\ar[rd] \\ar[rr]_{(f, f')} & & (X \\subset X') \\ar[ld] \\\\ & (U \\subset U') } $$ of algebraic spaces over $U'$ such that $Z' \\to U'$ is finite locally free of degree $d$ and $Z = U \\times_{U'} Z'$."}
+{"_id": "3135", "title": "criteria-lemma-lci-formally-smooth", "text": "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, and locally of finite presentation. Then $$ p : \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y} $$ is formally smooth on objects."}
+{"_id": "3136", "title": "criteria-lemma-lci-surjective", "text": "Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic spaces, flat, surjective, and locally of finite presentation. Then $$ \\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\longrightarrow \\mathcal{Y} $$ is surjective on objects."}
+{"_id": "3137", "title": "criteria-lemma-flat-quotient-flat-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the morphism $\\mathcal{S}_U \\to [U/R]$ is flat, locally of finite presentation, and surjective."}
+{"_id": "3138", "title": "criteria-lemma-quotient-algebraic", "text": "Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The quotient stack $[U/R]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $g : U' \\to U$ such that \\begin{enumerate} \\item the composition $U' \\times_{g, U, t} R \\to R \\xrightarrow{s} U$ is a surjection of sheaves, and \\item the morphisms $s', t' : R' \\to U'$ are flat and locally of finite presentation where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $g$. \\end{enumerate}"}
+{"_id": "3139", "title": "criteria-lemma-group-quotient-algebraic", "text": "Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$. The quotient stack $[X/G]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $\\varphi : X' \\to X$ such that \\begin{enumerate} \\item $G \\times_B X' \\to X$, $(g, x') \\mapsto a(g, \\varphi(x'))$ is a surjection of sheaves, and \\item the two projections $X'' \\to X'$ of the algebraic space $X''$ given by the rule $$ T \\longmapsto \\{(x'_1, g, x'_2) \\in (X' \\times_B G \\times_B X')(T) \\mid \\varphi(x'_1) = a(g, \\varphi(x'_2))\\} $$ are flat and locally of finite presentation. \\end{enumerate}"}
+{"_id": "3140", "title": "criteria-lemma-BG-algebraic", "text": "\\begin{slogan} Gerbes are algebraic if and only if the associated groups are flat and locally of finite presentation \\end{slogan} Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. Then the quotient stack $[B/G]$ is an algebraic stack if and only if $G$ is flat and locally of finite presentation over $B$."}
+{"_id": "3141", "title": "criteria-lemma-stacks-etale", "text": "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see Sheaves on Stacks, Section \\ref{stacks-sheaves-section-sheaves} and Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object of $\\Sch_\\alpha$. Let $$ p : \\mathcal{X} \\to \\Sch_\\alpha/S $$ be a category fibred in groupoids with the following properties: \\begin{enumerate} \\item $\\mathcal{X}$ is a stack in groupoids over $(\\Sch/S)_\\etale$, \\item the diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces\\footnote{Here we can either mean sheaves in the \\'etale topology whose diagonal is representable and which have an \\'etale surjective covering by a scheme or algebraic spaces as defined in Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}. Namely, by Bootstrap, Lemma \\ref{bootstrap-lemma-spaces-etale} there is no difference.}, and \\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$ and a $1$-morphism $(\\Sch/U)_\\etale \\to \\mathcal{X}$ which is surjective and smooth. \\end{enumerate} Then $\\mathcal{X}$ is an algebraic stack in the sense of Algebraic Stacks, Definition \\ref{algebraic-definition-algebraic-stack}."}
+{"_id": "3142", "title": "criteria-proposition-hom-functor-algebraic-space", "text": "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of algebraic spaces over $S$. If $Z \\to B$ is finite locally free then $\\mathit{Mor}_B(Z, X)$ is an algebraic space."}
+{"_id": "3143", "title": "criteria-proposition-restriction-of-scalars-algebraic-space", "text": "Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of algebraic spaces over $S$. If $Z \\to B$ is finite locally free then $\\text{Res}_{Z/B}(X)$ is an algebraic space."}
+{"_id": "3144", "title": "criteria-proposition-finite-hilbert-point", "text": "The stack $\\mathcal{H}_d$ is equivalent to the quotient stack $[X/G]$ described above. In particular $\\mathcal{H}_d$ is an algebraic stack."}
+{"_id": "3146", "title": "quot-theorem-coherent-algebraic", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation, separated, and flat\\footnote{This assumption is not necessary. See Section \\ref{section-not-flat}.}. Then $\\Cohstack_{X/B}$ is an algebraic stack over $S$."}
+{"_id": "3147", "title": "quot-theorem-coherent-algebraic-general", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\\Cohstack_{X/B}$ is an algebraic stack over $S$."}
+{"_id": "3151", "title": "quot-lemma-hom-sheaf", "text": "In Situation \\ref{situation-hom} the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$  satisfies the sheaf property for the fpqc topology."}
+{"_id": "3152", "title": "quot-lemma-extend-hom-to-spaces", "text": "In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$. We have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})) = \\{(h, u) \\mid h : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\} $$ where $\\mathcal{F}_T, \\mathcal{G}_T$ denote the pullbacks of $\\mathcal{F}$ and $\\mathcal{G}$ to the algebraic space $X \\times_{B, h} T$."}
+{"_id": "3153", "title": "quot-lemma-hom-sheaf-in-X", "text": "In Situation \\ref{situation-hom} let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf covering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$ be an fppf covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$ the pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. Similarly define $\\mathcal{G}_i$ and $\\mathcal{G}_{ijk}$. For every scheme $T$ over $B$ the diagram $$ \\xymatrix{ \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T) \\ar[r] & \\prod\\nolimits_i \\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G}_i)(T) \\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*} & \\prod\\nolimits_{i, j, k} \\mathit{Hom}(\\mathcal{F}_{ijk}, \\mathcal{G}_{ijk})(T) } $$ presents the first arrow as the equalizer of the other two."}
+{"_id": "3156", "title": "quot-lemma-cohomology-perfect-complex", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $K$ be a pseudo-coherent object of $D(\\mathcal{O}_B)$. \\begin{enumerate} \\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaf $H^{-1}(Lg^*K)$ is zero, then the functor $$ (\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad (g : T \\to B) \\longmapsto H^0(T, H^0(Lg^*K)) $$ is an algebraic space affine and of finite presentation over $B$. \\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaves $H^i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect, $K$ locally has tor amplitude in $[0, b]$, and the functor $$ (\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad (g : T \\to B) \\longmapsto H^0(T, Lg^*K) $$ is an algebraic space affine and of finite presentation over $B$. \\end{enumerate}"}
+{"_id": "3157", "title": "quot-lemma-noetherian-hom", "text": "In Situation \\ref{situation-hom} assume that \\begin{enumerate} \\item $B$ is a Noetherian algebraic space, \\item $f$ is locally of finite type and quasi-separated, \\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and \\item $\\mathcal{G}$ is a finite type $\\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \\end{enumerate} Then the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is an algebraic space affine and of finite presentation over $B$."}
+{"_id": "3161", "title": "quot-lemma-coherent-diagonal", "text": "In Situation \\ref{situation-coherent}. Denote $\\mathcal{X} = \\Cohstack_{X/B}$. Then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces."}
+{"_id": "3162", "title": "quot-lemma-coherent-stack", "text": "In Situation \\ref{situation-coherent} the functor $p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$ is a stack in groupoids."}
+{"_id": "3163", "title": "quot-lemma-coherent-limits", "text": "In Situation \\ref{situation-coherent} assume that $B \\to S$ is locally of finite presentation. Then $p : \\Cohstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})."}
+{"_id": "3164", "title": "quot-lemma-coherent-RS-star", "text": "In Situation \\ref{situation-coherent}. Let $$ \\xymatrix{ Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\ Y \\ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \\Cohstack_{X/B, Y'} \\longrightarrow \\Cohstack_{X/B, Y} \\times_{\\Cohstack_{X/B, Z}} \\Cohstack_{X/B, Z'} $$ is an equivalence."}
+{"_id": "3165", "title": "quot-lemma-coherent-over-first-order-thickening", "text": "Let $$ \\xymatrix{ X \\ar[d] \\ar[r]_i & X' \\ar[d] \\\\ T \\ar[r] & T' } $$ be a cartesian square of algebraic spaces where $T \\to T'$ is a first order thickening. Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module flat over $T'$. Set $\\mathcal{F} = i^*\\mathcal{F}'$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{X'}$-module of finite presentation, \\item $\\mathcal{F}'$ is an $\\mathcal{O}_{X'}$-module of finite presentation, \\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite presentation, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, \\end{enumerate}"}
+{"_id": "3166", "title": "quot-lemma-coherent-tangent-space", "text": "In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian scheme and $B \\to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\\Spec(k), g_0, \\mathcal{G}_0)$ be an object of $\\mathcal{X} = \\Cohstack_{X/B}$ over $k$. Then the spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$ (Artin's Axioms, Section \\ref{artin-section-tangent-spaces}) are finite dimensional."}
+{"_id": "3167", "title": "quot-lemma-coherent-existence", "text": "In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian scheme and that $f : X \\to B$ is separated. Let $\\mathcal{X} = \\Cohstack_{X/B}$. Then the functor Artin's Axioms, Equation (\\ref{artin-equation-approximation}) is an equivalence."}
+{"_id": "3168", "title": "quot-lemma-coherent-defo-thy", "text": "In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian scheme, $S = B$, and $f : X \\to B$ is flat. Let $\\mathcal{X} = \\Cohstack_{X/B}$. Then we have openness of versality for $\\mathcal{X}$ (see Artin's Axioms, Definition \\ref{artin-definition-openness-versality})."}
+{"_id": "3169", "title": "quot-lemma-q-sheaf", "text": "In Situation \\ref{situation-q}. The functors $\\text{Q}_{\\mathcal{F}/X/B}$ and $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$ satisfy the sheaf property for the fpqc topology."}
+{"_id": "3170", "title": "quot-lemma-extend-q-to-spaces", "text": "In Situation \\ref{situation-q}. Let $T$ be an algebraic space over $S$. We have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\text{Q}_{\\mathcal{F}/X/B}) = \\left\\{ \\begin{matrix} (h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where } h : T \\to B \\text{ and}\\\\ \\mathcal{Q}\\text{ is quasi-coherent and flat over }T \\end{matrix} \\right\\} $$ where $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$ to the algebraic space $X \\times_{B, h} T$. Similarly, we have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\text{Q}^{fp}_{\\mathcal{F}/X/B}) = \\left\\{ \\begin{matrix} (h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where } h : T \\to B \\text{ and}\\\\ \\mathcal{Q}\\text{ is of finite presentation and flat over }T \\end{matrix} \\right\\} $$"}
+{"_id": "3172", "title": "quot-lemma-q-limit-preserving", "text": "In Situation \\ref{situation-q} assume also that (a) $f$ is quasi-compact and quasi-separated and (b) $\\mathcal{F}$ is of finite presentation. Then the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$ is limit preserving in the following sense: If $T = \\lim T_i$ is a directed limit of affine schemes over $B$, then $\\text{Q}^{fp}_{\\mathcal{F}/X/B}(T) = \\colim \\text{Q}^{fp}_{\\mathcal{F}/X/B}(T_i)$."}
+{"_id": "3173", "title": "quot-lemma-q-RS-star", "text": "In Situation \\ref{situation-q}. Let $$ \\xymatrix{ Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\ Y \\ar[r] & Y' } $$ be a pushout in the category of schemes over $B$ where $Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the natural map $$ Q_{\\mathcal{F}/X/B}(Y') \\longrightarrow Q_{\\mathcal{F}/X/B}(Y) \\times_{Q_{\\mathcal{F}/X/B}(Z)} Q_{\\mathcal{F}/X/B}(Z') $$ is bijective. If $X \\to B$ is locally of finite presentation, then the same thing is true for $Q^{fp}_{\\mathcal{F}/X/B}$."}
+{"_id": "3174", "title": "quot-lemma-quot-sheaf", "text": "In Situation \\ref{situation-quot}. The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$ satisfies the sheaf property for the fpqc topology."}
+{"_id": "3175", "title": "quot-lemma-extend-quot-to-spaces", "text": "In Situation \\ref{situation-quot}. Let $T$ be an algebraic space over $S$. We have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\Quotfunctor_{\\mathcal{F}/X/B}) = \\left\\{ \\begin{matrix} (h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where } h : T \\to B \\text{ and}\\\\ \\mathcal{Q}\\text{ is of finite presentation and}\\\\ \\text{flat over }T\\text{ with support proper over }T \\end{matrix} \\right\\} $$ where $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$ to the algebraic space $X \\times_{B, h} T$."}
+{"_id": "3176", "title": "quot-lemma-hilb-is-quot", "text": "In Situation \\ref{situation-hilb} we have $\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$."}
+{"_id": "3177", "title": "quot-lemma-extend-hilb-to-spaces", "text": "In Situation \\ref{situation-hilb}. Let $T$ be an algebraic space over $S$. We have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\Hilbfunctor_{X/B}) = \\left\\{ \\begin{matrix} (h, Z)\\text{ where }h : T \\to B,\\ Z \\subset X_T \\\\ \\text{finite presentation, flat, proper over }T \\end{matrix} \\right\\} $$ where $X_T = X \\times_{B, h} T$."}
+{"_id": "3178", "title": "quot-lemma-picard-stack-open-in-coh", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$ which is flat, of finite presentation, and proper. The natural map $$ \\Picardstack_{X/B} \\longrightarrow \\Cohstack_{X/B} $$ is representable by open immersions."}
+{"_id": "3179", "title": "quot-lemma-pic-over-pic", "text": "In Situation \\ref{situation-pic} the functor $\\Picardfunctor_{X/B}$ is the sheafification of the functor $T \\mapsto \\Ob(\\Picardstack_{X/B, T})/\\cong$."}
+{"_id": "3180", "title": "quot-lemma-flat-geometrically-connected-fibres", "text": "In Situation \\ref{situation-pic}. If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$, then $$ 0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T) $$ is an exact sequence for all $T$."}
+{"_id": "3184", "title": "quot-lemma-flat-geometrically-connected-fibres-with-section-functor-stack", "text": "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section. If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$, then $\\Picardstack_{X/B, \\sigma} \\to (\\Sch/S)_{fppf}$ is fibred in setoids with set of isomorphism classes over $T$ given by $$ \\coprod\\nolimits_{h : T \\to B} \\Ker(\\sigma_T^* : \\Pic(X \\times_{B, h} T) \\to \\Pic(T)) $$"}
+{"_id": "3185", "title": "quot-lemma-diagonal-pic", "text": "With assumptions and notation as in Proposition \\ref{proposition-pic-functor}. Then the diagonal $\\Picardfunctor_{X/B} \\to \\Picardfunctor_{X/B} \\times_B \\Picardfunctor_{X/B}$ is representable by immersions. In other words, $\\Picardfunctor_{X/B} \\to B$ is locally separated."}
+{"_id": "3186", "title": "quot-lemma-Mor-into-Hilb", "text": "Let $S$ be a scheme. Consider morphisms of algebraic spaces $Z \\to B$ and $X \\to B$ over $S$. If $X \\to B$ is separated and $Z \\to B$ is of finite presentation, flat, and proper, then there is a natural injective transformation of functors $$ \\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B} $$ which maps a morphism $f : Z_T \\to X_T$ to its graph."}
+{"_id": "3187", "title": "quot-lemma-Mor-into-Hilb-open", "text": "Assumption and notation as in Lemma \\ref{lemma-Mor-into-Hilb}. The transformation $\\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B}$ is representable by open immersions."}
+{"_id": "3188", "title": "quot-lemma-spaces-fibred-in-groupoids", "text": "The category $\\Spacesstack'_{ft}$ is fibred in groupoids over $\\Sch_{fppf}$. The same is true for $\\Spacesstack'_{fp, flat, proper}$."}
+{"_id": "3189", "title": "quot-lemma-spaces-diagonal", "text": "The diagonal $$ \\Delta : \\Spacesstack'_{fp, flat, proper} \\longrightarrow \\Spacesstack'_{fp, flat, proper} \\times \\Spacesstack'_{fp, flat, proper} $$ is representable by algebraic spaces."}
+{"_id": "3190", "title": "quot-lemma-spaces-stack", "text": "The category $\\Spacesstack'_{ft}$ is a stack in groupoids over $\\Sch_{fppf}$. The same is true for $\\Spacesstack'_{fp, flat, proper}$."}
+{"_id": "3191", "title": "quot-lemma-extend-spaces-to-spaces", "text": "Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections \\ref{algebraic-section-split}, \\ref{algebraic-section-representable-by-algebraic-spaces}, and \\ref{algebraic-section-stacks-spaces}). We have an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{morphisms of algebraic spaces }\\\\ X \\to T\\text{ of finite type} \\end{matrix} \\right\\} \\longrightarrow \\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Spacesstack'_{ft}) $$ and an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{morphisms of algebraic spaces }X \\to T\\\\ \\text{of finite presentation, flat, and proper} \\end{matrix} \\right\\} \\longrightarrow \\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Spacesstack'_{fp, flat, proper}) $$"}
+{"_id": "3192", "title": "quot-lemma-spaces-limits", "text": "The stack $p'_{fp, flat, proper} : \\Spacesstack'_{fp, flat, proper} \\to \\Sch_{fppf}$ is limit preserving (Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})."}
+{"_id": "3193", "title": "quot-lemma-spaces-RS-star", "text": "Let $$ \\xymatrix{ T \\ar[r] \\ar[d] & T' \\ar[d] \\\\ S \\ar[r] & S' } $$ be a pushout in the category of schemes where $T \\to T'$ is a thickening and $T \\to S$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \\begin{matrix} \\Spacesstack'_{fp, flat, proper, S'} \\\\ \\downarrow \\\\ \\Spacesstack'_{fp, flat, proper, S} \\times_{\\Spacesstack'_{fp, flat, proper, T}} \\Spacesstack'_{fp, flat, proper, T'} \\end{matrix} $$ is an equivalence."}
+{"_id": "3194", "title": "quot-lemma-spaces-tangent-space", "text": "Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of $\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$ over $\\Spec(k)$. \\begin{enumerate} \\item If $k$ is of finite type over $\\mathbf{Z}$, then the vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$ (see Artin's Axioms, Section \\ref{artin-section-tangent-spaces}) are finite dimensional, and \\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$ (see Artin's Axioms, Section \\ref{artin-section-inf}) are finite dimensional. \\end{enumerate}"}
+{"_id": "3195", "title": "quot-lemma-spaces-defo-thy", "text": "The stack in groupoids $\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$ satisfies openness of versality over $\\Spec(\\mathbf{Z})$. Similarly, after base change (Remark \\ref{remark-spaces-base-change}) openness of versality holds over any Noetherian base scheme $S$."}
+{"_id": "3196", "title": "quot-lemma-polarized-fibred-in-groupoids", "text": "The category $\\Polarizedstack$ is fibred in groupoids over $\\Spacesstack'_{fp, flat, proper}$. The category $\\Polarizedstack$ is fibred in groupoids over $\\Sch_{fppf}$."}
+{"_id": "3197", "title": "quot-lemma-polarized-stack", "text": "The category $\\Polarizedstack$ is a stack in groupoids over $\\Spacesstack'_{fp, flat, proper}$ (endowed with the inherited topology, see Stacks, Definition \\ref{stacks-definition-topology-inherited}). The category $\\Polarizedstack$ is a stack in groupoids over $\\Sch_{fppf}$."}
+{"_id": "3198", "title": "quot-lemma-extend-polarized-to-spaces", "text": "Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$ denote the corresponding algebraic stack (Algebraic Stacks, Sections \\ref{algebraic-section-split}, \\ref{algebraic-section-representable-by-algebraic-spaces}, and \\ref{algebraic-section-stacks-spaces}). We have an equivalence of categories $$ \\left\\{ \\begin{matrix} (X \\to T, \\mathcal{L})\\text{ where }X \\to T\\text{ is a morphism}\\\\ \\text{of algebraic spaces, is proper, flat, and of}\\\\ \\text{finite presentation and }\\mathcal{L}\\text{ ample on }X/T \\end{matrix} \\right\\} \\longrightarrow \\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Polarizedstack) $$"}
+{"_id": "3199", "title": "quot-lemma-polarized-to-spaces-algebraic", "text": "The functor (\\ref{equation-over-proper-spaces}) defines a $1$-morphism $$ \\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper} $$ of stacks in groupoids over $\\Sch_{fppf}$ which is algebraic in the sense of Criteria for Representability, Definition \\ref{criteria-definition-algebraic}."}
+{"_id": "3200", "title": "quot-lemma-polarized-diagonal", "text": "The diagonal $$ \\Delta : \\Polarizedstack \\longrightarrow \\Polarizedstack \\times \\Polarizedstack $$ is representable by algebraic spaces."}
+{"_id": "3201", "title": "quot-lemma-polarized-limits", "text": "The stack in groupoids $\\Polarizedstack$ is limit preserving (Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})."}
+{"_id": "3202", "title": "quot-lemma-polarized-RS-star", "text": "In Situation \\ref{situation-coherent}. Let $$ \\xymatrix{ T \\ar[r] \\ar[d] & T' \\ar[d] \\\\ S \\ar[r] & S' } $$ be a pushout in the category of schemes where $T \\to T'$ is a thickening and $T \\to S$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \\Polarizedstack_{S'} \\longrightarrow \\Polarizedstack_S \\times_{\\Polarizedstack_T} \\Polarizedstack_{T'} $$ is an equivalence."}
+{"_id": "3204", "title": "quot-lemma-polarized-strong-effectiveness", "text": "\\begin{slogan} Grothendieck's algebraization theorem continues to hold in the non-Noetherian setting if one assumes flatness and finite presentation. \\end{slogan} Let $(R_n)$ be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set $R = \\lim R_n$. Set $S_n = \\Spec(R_n)$ and $S = \\Spec(R)$. Consider a commutative diagram $$ \\xymatrix{ X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] & \\ldots \\\\ S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots } $$ of schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$ where each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and $\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$ is an isomorphism. If \\begin{enumerate} \\item $X_n \\to S_n$ is proper, flat, of finite presentation, and \\item $\\mathcal{L}_1$ is ample on $X_1$ \\end{enumerate} then there exists a morphism of schemes $X \\to S$ proper, flat, and of finite presentation and an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and isomorphisms $X_n \\cong X \\times_S S_n$ and $\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with the morphisms $i_n$ and $\\varphi_n$."}
+{"_id": "3205", "title": "quot-lemma-polarized-existence", "text": "Consider the stack $\\Polarizedstack$ over the base scheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective."}
+{"_id": "3206", "title": "quot-lemma-polarized-defo-thy", "text": "The stack in groupoids $\\Polarizedstack$ satisfies openness of versality over $\\Spec(\\mathbf{Z})$. Similarly, after base change (Remark \\ref{remark-polarized-base-change}) openness of versality holds over any Noetherian base scheme $S$."}
+{"_id": "3209", "title": "quot-lemma-curves-diagonal", "text": "The diagonal $$ \\Delta : \\Curvesstack \\longrightarrow \\Curvesstack \\times \\Curvesstack $$ is representable by algebraic spaces."}
+{"_id": "3210", "title": "quot-lemma-curves-limits", "text": "The stack $\\Curvesstack \\to \\Sch_{fppf}$ is limit preserving (Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})."}
+{"_id": "3211", "title": "quot-lemma-curves-RS-star", "text": "Let $$ \\xymatrix{ T \\ar[r] \\ar[d] & T' \\ar[d] \\\\ S \\ar[r] & S' } $$ be a pushout in the category of schemes where $T \\to T'$ is a thickening and $T \\to S$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \\Curvesstack_{S'} \\longrightarrow \\Curvesstack_S \\times_{\\Curvesstack_T} \\Curvesstack_{T'} $$ is an equivalence."}
+{"_id": "3213", "title": "quot-lemma-curves-existence", "text": "Consider the stack $\\Curvesstack$ over the base scheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective."}
+{"_id": "3214", "title": "quot-lemma-curves-defo-thy", "text": "The stack in groupoids $\\mathcal{X} = \\Curvesstack$ satisfies openness of versality over $\\Spec(\\mathbf{Z})$. Similarly, after base change (Remark \\ref{remark-curves-base-change}) openness of versality holds over any Noetherian base scheme $S$."}
+{"_id": "3215", "title": "quot-lemma-curves-open-and-closed-in-spaces", "text": "The $1$-morphism (\\ref{equation-curves-over-proper-spaces}) $$ \\Curvesstack \\longrightarrow \\Spacesstack'_{fp, flat, proper} $$ is representable by open and closed immersions."}
+{"_id": "3216", "title": "quot-lemma-complexes-open-neg-exts-vanishing", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. Let $K, E \\in D(\\mathcal{O}_X)$. Assume $K$ is pseudo-coherent and $E$ is $Y$-perfect (More on Morphisms of Spaces, Definition \\ref{spaces-more-morphisms-definition-relatively-perfect}). For a field $k$ and a morphism $y : \\Spec(k) \\to Y$ denote $K_y$, $E_y$ the pullback to the fibre $X_y$. \\begin{enumerate} \\item There is an open $W \\subset Y$ characterized by the property $$ y \\in |W| \\Leftrightarrow \\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y) = 0 \\text{ for }i < 0. $$ \\item For any morphism $V \\to Y$ factoring through $W$ we have $$ \\Ext^i_{\\mathcal{O}_{X_V}}(K_V, E_V) = 0 \\quad\\text{for}\\quad i < 0 $$ where $X_V$ is the base change of $X$ and $K_V$ and $E_V$ are the derived pullbacks of $K$ and $E$ to $X_V$. \\item The functor $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(K_V, E_V)$ is a sheaf on $(\\textit{Spaces}/W)_{fppf}$ representable by an algebraic space affine and of finite presentation over $W$. \\end{enumerate}"}
+{"_id": "3217", "title": "quot-lemma-complexes", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. Let $E \\in D(\\mathcal{O}_X)$. Assume \\begin{enumerate} \\item $E$ is $S$-perfect (More on Morphisms of Spaces, Definition \\ref{spaces-more-morphisms-definition-relatively-perfect}), and \\item for every point $s \\in S$ we have $$ \\Ext^i_{\\mathcal{O}_{X_s}}(E_s, E_s) = 0 \\quad\\text{for}\\quad i < 0 $$ where $E_s$ is the pullback to the fibre $X_s$. \\end{enumerate} Then \\begin{enumerate} \\item[(a)] (1) and (2) are preserved by arbitrary base change $V \\to Y$, \\item[(b)] $\\Ext^i_{\\mathcal{O}_{X_V}}(E_V, E_V) = 0$ for $i < 0$ and all $V$ over $Y$, \\item[(c)] $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(E_V, E_V)$ is representable by an algebraic space affine and of finite presentation over $Y$. \\end{enumerate} Here $X_V$ is the base change of $X$ and $E_V$ is the derived pullback of $E$ to $X_V$."}
+{"_id": "3219", "title": "quot-lemma-complexes-diagonal", "text": "In Situation \\ref{situation-complexes}. Denote $\\mathcal{X} = \\Complexesstack_{X/B}$. Then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces."}
+{"_id": "3220", "title": "quot-lemma-complexes-stack", "text": "In Situation \\ref{situation-complexes} the functor $p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$ is a stack in groupoids."}
+{"_id": "3221", "title": "quot-lemma-complexes-limits", "text": "In Situation \\ref{situation-complexes} assume that $B \\to S$ is locally of finite presentation. Then $p : \\Complexesstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving (Artin's Axioms, Definition \\ref{artin-definition-limit-preserving})."}
+{"_id": "3222", "title": "quot-lemma-complexes-RS-star", "text": "In Situation \\ref{situation-complexes}. Let $$ \\xymatrix{ Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\ Y \\ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $Z \\to Z'$ is a finite order thickening and $Z \\to Y$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Then the functor on fibre categories $$ \\Complexesstack_{X/B, Y'} \\longrightarrow \\Complexesstack_{X/B, Y} \\times_{\\Complexesstack_{X/B, Z}} \\Complexesstack_{X/B, Z'} $$ is an equivalence."}
+{"_id": "3223", "title": "quot-lemma-complexes-tangent-space", "text": "In Situation \\ref{situation-complexes} assume that $S$ is a locally Noetherian scheme and $B \\to S$ is locally of finite presentation. Let $k$ be a finite type field over $S$ and let $x_0 = (\\Spec(k), g_0, E_0)$ be an object of $\\mathcal{X} = \\Complexesstack_{X/B}$ over $k$. Then the spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$ (Artin's Axioms, Section \\ref{artin-section-tangent-spaces}) are finite dimensional."}
+{"_id": "3224", "title": "quot-lemma-complexes-strong-effectiveness", "text": "In Situation \\ref{situation-complexes} assume $B = S$ is locally Noetherian. Then strong formal effectiveness in the sense of Artin's Axioms, Remark \\ref{artin-remark-strong-effectiveness} holds for $p : \\Complexesstack_{X/S} \\to (\\Sch/S)_{fppf}$."}
+{"_id": "3225", "title": "quot-proposition-hom", "text": "In Situation \\ref{situation-hom} assume that \\begin{enumerate} \\item $f$ is of finite presentation, and \\item $\\mathcal{G}$ is a finitely presented $\\mathcal{O}_X$-module, flat over $B$, with support proper over $B$. \\end{enumerate} Then the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is an algebraic space affine over $B$. If $\\mathcal{F}$ is of finite presentation, then $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is of finite presentation over $B$."}
+{"_id": "3226", "title": "quot-proposition-isom", "text": "In Situation \\ref{situation-hom} assume that \\begin{enumerate} \\item $f$ is of finite presentation, and \\item $\\mathcal{F}$ and $\\mathcal{G}$ are finitely presented $\\mathcal{O}_X$-modules, flat over $B$, with support proper over $B$. \\end{enumerate} Then the functor $\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$ is an algebraic space affine of finite presentation over $B$."}
+{"_id": "3227", "title": "quot-proposition-quot", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. If $f$ is of finite presentation and separated, then $\\Quotfunctor_{\\mathcal{F}/X/B}$ is an algebraic space. If $\\mathcal{F}$ is of finite presentation, then $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is locally of finite presentation."}
+{"_id": "3228", "title": "quot-proposition-hilb", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and separated, then $\\Hilbfunctor_{X/B}$ is an algebraic space locally of finite presentation over $B$."}
+{"_id": "3229", "title": "quot-proposition-pic", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. If $f$ is flat, of finite presentation, and proper, then $\\Picardstack_{X/B}$ is an algebraic stack."}
+{"_id": "3230", "title": "quot-proposition-pic-functor", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $f$ is flat, of finite presentation, and proper, and \\item $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all schemes $T$ over $B$. \\end{enumerate} Then $\\Picardfunctor_{X/B}$ is an algebraic space."}
+{"_id": "3231", "title": "quot-proposition-Mor", "text": "Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of algebraic spaces over $S$. Assume $X \\to B$ is of finite presentation and separated and $Z \\to B$ is of finite presentation, flat, and proper. Then $\\mathit{Mor}_B(Z, X)$ is an algebraic space locally of finite presentation over $B$."}
+{"_id": "3246", "title": "spaces-more-cohomology-theorem-proper-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$. Then the base change map $$ g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F} $$ is an isomorphism."}
+{"_id": "3247", "title": "spaces-more-cohomology-lemma-compare-cohomology-other-topologies", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{\\etale, fppf, ph\\}$ (add more here). The inclusion functor $$ (\\Sch/S)_\\tau \\longrightarrow (\\textit{Spaces}/S)_\\tau $$ is a special cocontinuous functor (Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}) and hence identifies topoi."}
+{"_id": "3248", "title": "spaces-more-cohomology-lemma-surjective-proper", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then $\\mathcal{F} \\to f_*f^{-1}\\mathcal{F}$ is injective with image the equalizer of the two maps $f_*f^{-1}\\mathcal{F} \\to g_*g^{-1}\\mathcal{F}$ where $g$ is the structure morphism $g : Y \\times_X Y \\to X$."}
+{"_id": "3249", "title": "spaces-more-cohomology-lemma-h0-proper-over-henselian-pair", "text": "Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$ such that the structure morphism $f : X \\to \\Spec(A)$ is proper. Let $i : X_0 \\to X$ be the inclusion of $X \\times_{\\Spec(A)} \\Spec(A/I)$. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i^{-1}\\mathcal{F})$."}
+{"_id": "3251", "title": "spaces-more-cohomology-lemma-proper-base-change-f-star", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \\times_Y X$ with projections $f' : X' \\to Y'$ and $g' : X' \\to X$. Let $\\mathcal{F}$ be any sheaf on $X_\\etale$. Then $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$."}
+{"_id": "3253", "title": "spaces-more-cohomology-lemma-proper-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves. Then the base change map $g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$ is an isomorphism."}
+{"_id": "3256", "title": "spaces-more-cohomology-lemma-describe-pullback", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then $\\pi_X^{-1}\\mathcal{F}$ is given by the rule $$ (\\pi_X^{-1}\\mathcal{F})(Y) = \\Gamma(Y_\\etale, f_{small}^{-1}\\mathcal{F}) $$ for $f : Y \\to X$ in $(\\textit{Spaces}/X)_\\etale$. Moreover, $\\pi_Y^{-1}\\mathcal{F}$ satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings."}
+{"_id": "3257", "title": "spaces-more-cohomology-lemma-compare-injectives", "text": "Let $S$ be a scheme. Let $Y \\to X$ be a morphism of $(\\textit{Spaces}/S)_\\etale$. \\begin{enumerate} \\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(Y_\\etale)$, \\item $\\mathcal{I}|_{X_\\etale}$ is injective in $\\textit{Ab}(X_\\etale)$, \\end{enumerate} \\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}(Y_\\etale)$, \\item $\\mathcal{I}^\\bullet|_{X_\\etale}$ is a K-injective complex in $\\textit{Ab}(X_\\etale)$, \\end{enumerate} \\end{enumerate} The corresponding statements for modules do not hold."}
+{"_id": "3258", "title": "spaces-more-cohomology-lemma-compare-higher-direct-image", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have $ (Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale}) $ in $D(X_\\etale)$. \\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have $ (Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale}) $ in $D(\\textit{Mod}(X_\\etale, \\mathcal{O}_X))$. \\end{enumerate} More generally, let $g : X' \\to X$ be an object of $(\\textit{Spaces}/X)_\\etale$. Consider the fibre product $$ \\xymatrix{ Y' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\ X' \\ar[r]^g & X } $$ Then \\begin{enumerate} \\item[(3)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have $i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(X'_\\etale)$. \\item[(4)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have $i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$. \\item[(5)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\\textit{Spaces}/X')_\\etale)$. \\item[(6)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$. \\end{enumerate}"}
+{"_id": "3260", "title": "spaces-more-cohomology-lemma-cohomological-descent-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \\in D(X_\\etale)$ the map $$ K \\longrightarrow R\\pi_{X, *}\\pi_X^{-1}K $$ is an isomorphism where $\\pi_X : \\Sh((\\textit{Spaces}/X)_\\etale) \\to \\Sh(X_\\etale)$ is as above."}
+{"_id": "3261", "title": "spaces-more-cohomology-lemma-compare-higher-direct-image-proper", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$. Then we have \\begin{enumerate} \\item $\\pi_X^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_Y^{-1}$ as functors $\\Sh(Y_\\etale) \\to \\Sh((\\textit{Spaces}/X)_\\etale)$, \\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$ for $K$ in $D^+(Y_\\etale)$ whose cohomology sheaves are torsion, and \\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$ for all $K$ in $D(Y_\\etale)$ if $f$ is finite. \\end{enumerate}"}
+{"_id": "3262", "title": "spaces-more-cohomology-lemma-comparison-fppf-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. \\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\end{enumerate}"}
+{"_id": "3263", "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \\in D^+(X_\\etale)$ the maps $$ \\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K \\quad\\text{and}\\quad K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ are isomorphisms with $a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above."}
+{"_id": "3265", "title": "spaces-more-cohomology-lemma-push-pull-fppf-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi $$ \\xymatrix{ \\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & & \\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & & \\Sh((\\textit{Spaces}/Y)_\\etale) } $$ and $$ \\xymatrix{ \\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & & \\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{a_Y} \\\\ \\Sh(X_\\etale) \\ar[rr]^{f_{small}} & & \\Sh(Y_\\etale) } $$ with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$."}
+{"_id": "3268", "title": "spaces-more-cohomology-lemma-descent-sheaf-fppf-etale", "text": "In Lemma \\ref{lemma-push-pull-fppf-etale} assume $f$ is flat, locally of finite presentation, and surjective. Then the functor $$ \\Sh(Y_\\etale) \\longrightarrow \\left\\{ (\\mathcal{G}, \\mathcal{H}, \\alpha) \\middle| \\begin{matrix} \\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\ \\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H} \\text{ an isomorphism} \\end{matrix} \\right\\} $$ sending $\\mathcal{F}$ to $(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence."}
+{"_id": "3269", "title": "spaces-more-cohomology-lemma-review-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item The rule $$ \\mathcal{F}^a : (\\textit{Spaces}/X)_\\etale \\longrightarrow \\textit{Ab},\\quad (f : Y \\to X) \\longmapsto \\Gamma(Y, f^*\\mathcal{F}) $$ satisfies the sheaf condition for fpqc and a fortiori fppf and \\'etale coverings, \\item $\\mathcal{F}^a = \\pi_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_\\etale$, \\item $\\mathcal{F}^a = a_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_{fppf}$, \\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence between quasi-coherent $\\mathcal{O}_X$-modules and quasi-coherent modules on $((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$, \\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence between quasi-coherent $\\mathcal{O}_X$-modules and quasi-coherent modules on $((\\textit{Spaces}/X)_{fppf}, \\mathcal{O})$, \\item we have $\\epsilon_{X, *}a_X^*\\mathcal{F} = \\pi_X^*\\mathcal{F}$ and $a_{X, *}a_X^*\\mathcal{F} = \\mathcal{F}$, \\item we have $R^i\\epsilon_{X, *}(a_X^*\\mathcal{F}) = 0$ and $R^ia_{X, *}(a_X^*\\mathcal{F}) = 0$ for $i > 0$. \\end{enumerate}"}
+{"_id": "3270", "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module the maps $$ \\pi_X^*\\mathcal{F} \\longrightarrow R\\epsilon_{X, *}(a_X^*\\mathcal{F}) \\quad\\text{and}\\quad \\mathcal{F} \\longrightarrow Ra_{X, *}(a_X^*\\mathcal{F}) $$ are isomorphisms."}
+{"_id": "3271", "title": "spaces-more-cohomology-lemma-vanishing-adequate", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules. Set $$ \\mathcal{H}_\\etale = \\Ker(\\pi_X^*\\mathcal{F}_2 \\to \\pi_X^*\\mathcal{F}_3)/ \\Im(\\pi_X^*\\mathcal{F}_1 \\to \\pi_X^*\\mathcal{F}_2) $$ on $(\\textit{Spaces}/X)_\\etale$ and set $$ \\mathcal{H}_{fppf} = \\Ker(a_X^*\\mathcal{F}_2 \\to a_X^*\\mathcal{F}_3)/ \\Im(a_X^*\\mathcal{F}_1 \\to a_X^*\\mathcal{F}_2) $$ on $(\\textit{Spaces}/X)_{fppf}$. Then $\\mathcal{H}_\\etale = \\epsilon_{X, *}\\mathcal{H}_{fppf}$ and $$ H^p_\\etale(U, \\mathcal{H}_\\etale) = H^p_{fppf}(U, \\mathcal{H}_{fppf}) = 0 $$ for $p > 0$ and any affine object $U$ of $(\\textit{Spaces}/X)_\\etale$."}
+{"_id": "3273", "title": "spaces-more-cohomology-lemma-comparison-ph-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$. \\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ torsion we have $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\end{enumerate}"}
+{"_id": "3274", "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-ph", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves the maps $$ \\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K \\quad\\text{and}\\quad K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ are isomorphisms with $a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$ as above."}
+{"_id": "3276", "title": "spaces-more-cohomology-lemma-push-pull-ph-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi $$ \\xymatrix{ \\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & & \\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & & \\Sh((\\textit{Spaces}/Y)_\\etale) } $$ and $$ \\xymatrix{ \\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & & \\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{a_Y} \\\\ \\Sh(X_\\etale) \\ar[rr]^{f_{small}} & & \\Sh(Y_\\etale) } $$ with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$."}
+{"_id": "3277", "title": "spaces-more-cohomology-lemma-proper-push-pull-ph-etale", "text": "In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have \\begin{enumerate} \\item $a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$, and \\item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves. \\end{enumerate}"}
+{"_id": "3278", "title": "coherent-theorem-formal-functions", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \\geq 0$. The system of maps $$ H^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F}) \\longrightarrow H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ define an isomorphism of limits $$ H^p(X, \\mathcal{F})^\\wedge \\longrightarrow \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ where the left hand side is the completion of the $A$-module $H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see Algebra, Section \\ref{algebra-section-completion}. Moreover, this is in fact a homeomorphism for the limit topologies."}
+{"_id": "3279", "title": "coherent-theorem-grothendieck-existence", "text": "\\begin{reference} \\cite[III Theorem 5.1.5]{EGA} \\end{reference} Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $X$ be a separated, finite type scheme over $A$. Then the functor (\\ref{equation-completion-functor-proper-over-A}) $$ \\textit{Coh}_{\\text{support proper over }A}(\\mathcal{O}_X) \\longrightarrow \\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I}) $$ is an equivalence."}
+{"_id": "3280", "title": "coherent-theorem-algebraization", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Set $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Consider a commutative diagram $$ \\xymatrix{ X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] & \\ldots \\\\ S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots } $$ of schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$ where each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and $\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$ is an isomorphism. If \\begin{enumerate} \\item $X_1 \\to S_1$ is proper, and \\item $\\mathcal{L}_1$ is ample on $X_1$ \\end{enumerate} then there exists a proper morphism of schemes $X \\to S$ and an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and isomorphisms $X_n \\cong X \\times_S S_n$ and $\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with the morphisms $i_n$ and $\\varphi_n$."}
+{"_id": "3281", "title": "coherent-lemma-cech-cohomology-quasi-coherent-trivial", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{U} : U = \\bigcup_{i = 1}^n D(f_i)$ be a standard open covering of an affine open of $X$. Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$."}
+{"_id": "3282", "title": "coherent-lemma-quasi-coherent-affine-cohomology-zero", "text": "\\begin{slogan} Serre vanishing: Higher cohomology vanishes on affine schemes for quasi-coherent modules. \\end{slogan} Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. For any affine open $U \\subset X$ we have $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$."}
+{"_id": "3283", "title": "coherent-lemma-relative-affine-vanishing", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $f$ is affine then $R^if_*\\mathcal{F} = 0$ for all $i > 0$."}
+{"_id": "3284", "title": "coherent-lemma-relative-affine-cohomology", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $H^i(X, \\mathcal{F}) = H^i(S, f_*\\mathcal{F})$ for all $i \\geq 0$."}
+{"_id": "3285", "title": "coherent-lemma-affine-diagonal", "text": "Let $X$ be a scheme. The following are equivalent \\begin{enumerate} \\item $X$ has affine diagonal $\\Delta : X \\to X \\times X$, \\item for $U, V \\subset X$ affine open, the intersection $U \\cap V$ is affine, and \\item there exists an open covering $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ such that $U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all $i_0, \\ldots, i_p \\in I$. \\end{enumerate} In particular this holds if $X$ is separated."}
+{"_id": "3286", "title": "coherent-lemma-cech-cohomology-quasi-coherent", "text": "Let $X$ be a scheme. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering such that $U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all $i_0, \\ldots, i_p \\in I$. In this case for any quasi-coherent sheaf $\\mathcal{F}$ we have $$ \\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(X, \\mathcal{F}) $$ as $\\Gamma(X, \\mathcal{O}_X)$-modules for all $p$."}
+{"_id": "3287", "title": "coherent-lemma-quasi-compact-h1-zero-covering", "text": "\\begin{reference} \\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1 (d') and IV (1.7.17)]{EGA} \\end{reference} \\begin{slogan} Serre's criterion for affineness. \\end{slogan} Let $X$ be a scheme. Assume that \\begin{enumerate} \\item $X$ is quasi-compact, \\item for every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ we have $H^1(X, \\mathcal{I}) = 0$. \\end{enumerate} Then $X$ is affine."}
+{"_id": "3288", "title": "coherent-lemma-quasi-separated-h1-zero-covering", "text": "\\begin{reference} \\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1]{EGA} \\end{reference} \\begin{slogan} Serre's criterion for affineness. \\end{slogan} Let $X$ be a scheme. Assume that \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ is quasi-separated, and \\item $H^1(X, \\mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals $\\mathcal{I}$ of finite type. \\end{enumerate} Then $X$ is affine."}
+{"_id": "3289", "title": "coherent-lemma-quasi-compact-h1-zero-invertible", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume that \\begin{enumerate} \\item $X$ is quasi-compact, \\item for every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ there exists an $n \\geq 1$ such that $H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$. \\end{enumerate} Then $\\mathcal{L}$ is ample."}
+{"_id": "3290", "title": "coherent-lemma-criterion-affine-morphism", "text": "Let $f : X \\to Y$ be a quasi-compact morphism with $X$ and $Y$ quasi-separated. If $R^1f_*\\mathcal{I} = 0$ for every quasi-coherent sheaf of ideals $\\mathcal{I}$ on $X$, then $f$ is affine."}
+{"_id": "3291", "title": "coherent-lemma-induction-principle", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that \\begin{enumerate} \\item $P$ holds for every affine open of $X$, \\item if $U$ is quasi-compact open, $V$ affine open, $P$ holds for $U$, $V$, and $U \\cap V$, then $P$ holds for $U \\cup V$. \\end{enumerate} Then $P$ holds for every quasi-compact open of $X$ and in particular for $X$."}
+{"_id": "3292", "title": "coherent-lemma-vanishing-nr-affines", "text": "\\begin{slogan} For schemes with affine diagonal, the cohomology of quasi-coherent modules vanishes in degrees bigger than the number of affine opens needed in a covering. \\end{slogan} Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Let $t = t(X)$ be the minimal number of affine opens needed to cover $X$. Then $H^n(X, \\mathcal{F}) = 0$ for all $n \\geq t$ and all quasi-coherent sheaves $\\mathcal{F}$."}
+{"_id": "3293", "title": "coherent-lemma-affine-diagonal-universal-delta-functor", "text": "Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Then \\begin{enumerate} \\item given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists an embedding $\\mathcal{F} \\to \\mathcal{F}'$ of quasi-coherent $\\mathcal{O}_X$-modules such that $H^p(X, \\mathcal{F}') = 0$ for all $p \\geq 1$, and \\item $\\{H^n(X, -)\\}_{n \\geq 0}$ is a universal $\\delta$-functor from $\\QCoh(\\mathcal{O}_X)$ to $\\textit{Ab}$. \\end{enumerate}"}
+{"_id": "3294", "title": "coherent-lemma-vanishing-nr-affines-quasi-separated", "text": "Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \\cup \\ldots \\cup U_t$ be an affine open covering. Set $$ d = \\max\\nolimits_{I \\subset \\{1, \\ldots, t\\}} \\left(|I| + t(\\bigcap\\nolimits_{i \\in I} U_i)\\right) $$ where $t(U)$ is the minimal number of affines needed to cover the scheme $U$. Then $H^n(X, \\mathcal{F}) = 0$ for all $n \\geq d$ and all quasi-coherent sheaves $\\mathcal{F}$."}
+{"_id": "3295", "title": "coherent-lemma-quasi-coherence-higher-direct-images", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. \\begin{enumerate} \\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the higher direct images $R^pf_*\\mathcal{F}$ are quasi-coherent on $S$. \\item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$ such that $R^pf_*\\mathcal{F} = 0$ for all $p \\geq n$ and any quasi-coherent sheaf $\\mathcal{F}$ on $X$. \\item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$ such that for every morphism of schemes $S' \\to S$ we have $R^p(f')_*\\mathcal{F}' = 0$ for $p \\geq n$ and any quasi-coherent sheaf $\\mathcal{F}'$ on $X'$. Here $f' : X' = S' \\times_S X \\to S'$ is the base change of $f$. \\end{enumerate}"}
+{"_id": "3296", "title": "coherent-lemma-quasi-coherence-higher-direct-images-application", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. Assume $S$ is affine. For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $$ H^q(X, \\mathcal{F}) = H^0(S, R^qf_*\\mathcal{F}) $$ for all $q \\in \\mathbf{Z}$."}
+{"_id": "3297", "title": "coherent-lemma-affine-base-change", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ is affine. In this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is a quasi-coherent sheaf, and for every base change diagram (\\ref{equation-base-change-diagram}) we have $$ g^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}. $$"}
+{"_id": "3298", "title": "coherent-lemma-flat-base-change-cohomology", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module with pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \\geq 0$ \\begin{enumerate} \\item the base change map of Cohomology, Lemma \\ref{cohomology-lemma-base-change-map-flat-case} is an isomorphism $$ g^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}', $$ \\item if $S = \\Spec(A)$ and $S' = \\Spec(B)$, then $H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$. \\end{enumerate}"}
+{"_id": "3299", "title": "coherent-lemma-finite-locally-free-base-change-cohomology", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ Y \\ar[d]_{g} \\ar[r]_h & X \\ar[d]^f \\\\ \\Spec(B) \\ar[r] & \\Spec(A) } $$ Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module with pullback $\\mathcal{G} = h^*\\mathcal{F}$. If $B$ is a finite locally free $A$-module, then $H^i(X, \\mathcal{F}) \\otimes_A B = H^i(Y, \\mathcal{G})$."}
+{"_id": "3300", "title": "coherent-lemma-colimit-cohomology", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F} = \\colim \\mathcal{F}_i$ be a filtered colimit of quasi-coherent sheaves on $X$. Then for any $p \\geq 0$ we have $$ R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i. $$"}
+{"_id": "3301", "title": "coherent-lemma-separated-case-relative-cech", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal (e.g., if $X$ and $S$ are separated). In this case we can compute $Rf_*\\mathcal{F}$ as follows: \\begin{enumerate} \\item Choose a finite affine open covering $\\mathcal{U} : X = \\bigcup_{i = 1, \\ldots, n} U_i$. \\item For $i_0, \\ldots, i_p \\in \\{1, \\ldots, n\\}$ denote $f_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to S$ the restriction of $f$ to the intersection $U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$. \\item Set $\\mathcal{F}_{i_0 \\ldots i_p}$ equal to the restriction of $\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$. \\item Set $$ \\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F}) = \\bigoplus\\nolimits_{i_0 \\ldots i_p} f_{i_0 \\ldots i_p *} \\mathcal{F}_{i_0 \\ldots i_p} $$ and define differentials $d : \\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F}) \\to \\check{\\mathcal{C}}^{p + 1}(\\mathcal{U}, f, \\mathcal{F})$ as in Cohomology, Equation (\\ref{cohomology-equation-d-cech}). \\end{enumerate} Then the complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})$ is a complex of quasi-coherent sheaves on $S$ which comes equipped with an isomorphism $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F}) \\longrightarrow Rf_*\\mathcal{F} $$ in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent sheaf $\\mathcal{F}$."}
+{"_id": "3302", "title": "coherent-lemma-base-change-complex", "text": "With notation as in diagram (\\ref{equation-base-change-diagram}). Assume $f : X \\to S$ and $\\mathcal{F}$ satisfy the hypotheses of Lemma \\ref{lemma-separated-case-relative-cech}. Choose a finite affine open covering $\\mathcal{U} : X = \\bigcup U_i$ of $X$. There is a canonical isomorphism $$ g^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F}) \\longrightarrow Rf'_*\\mathcal{F}' $$ in $D^{+}(S')$. Moreover, if $S' \\to S$ is affine, then in fact $$ g^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F}) = \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}') $$ with $\\mathcal{U}' : X' = \\bigcup U_i'$ where $U_i' = (g')^{-1}(U_i) = U_{i, S'}$ is also affine."}
+{"_id": "3303", "title": "coherent-lemma-hypercoverings", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume that $f$ is quasi-compact and quasi-separated and that $S$ is quasi-compact and separated. There exists a bounded below complex $\\mathcal{K}^\\bullet$ of quasi-coherent $\\mathcal{O}_S$-modules with the following property: For every morphism $g : S' \\to S$ the complex $g^*\\mathcal{K}^\\bullet$ is a representative for $Rf'_*\\mathcal{F}'$ with notation as in diagram (\\ref{equation-base-change-diagram})."}
+{"_id": "3304", "title": "coherent-lemma-cohomology-projective-space-over-ring", "text": "\\begin{reference} \\cite[III Proposition 2.1.12]{EGA} \\end{reference} Let $R$ be a ring. Let $n \\geq 0$ be an integer. We have $$ H^q(\\mathbf{P}^n, \\mathcal{O}_{\\mathbf{P}^n_R}(d)) = \\left\\{ \\begin{matrix} (R[T_0, \\ldots, T_n])_d & \\text{if} & q = 0 \\\\ 0 & \\text{if} & q \\not = 0, n \\\\ \\left(\\frac{1}{T_0 \\ldots T_n} R[\\frac{1}{T_0}, \\ldots, \\frac{1}{T_n}]\\right)_d & \\text{if} & q = n \\end{matrix} \\right. $$ as $R$-modules."}
+{"_id": "3305", "title": "coherent-lemma-identify-functorially", "text": "The identifications of Equation (\\ref{equation-identify}) are compatible with base change w.r.t.\\ ring maps $R \\to R'$. Moreover, for any $f \\in R[T_0, \\ldots, T_n]$ homogeneous of degree $m$ the map multiplication by $f$ $$ \\mathcal{O}_{\\mathbf{P}^n_R}(d) \\longrightarrow \\mathcal{O}_{\\mathbf{P}^n_R}(d + m) $$ induces the map on the cohomology group via the identifications of Equation (\\ref{equation-identify}) which is multiplication by $f$ for $H^0$ and the contragredient of multiplication by $f$ $$ (R[T_0, \\ldots, T_n])_{-n - 1 - (d + m)} \\longrightarrow (R[T_0, \\ldots, T_n])_{-n - 1 - d} $$ on $H^n$."}
+{"_id": "3306", "title": "coherent-lemma-cohomology-projective-space-over-base", "text": "Let $S$ be a scheme. Let $n \\geq 0$ be an integer. Consider the structure morphism $$ f : \\mathbf{P}^n_S \\longrightarrow S. $$ We have $$ R^qf_*(\\mathcal{O}_{\\mathbf{P}^n_S}(d)) = \\left\\{ \\begin{matrix} (\\mathcal{O}_S[T_0, \\ldots, T_n])_d & \\text{if} & q = 0 \\\\ 0 & \\text{if} & q \\not = 0, n \\\\ \\SheafHom_{\\mathcal{O}_S}( (\\mathcal{O}_S[T_0, \\ldots, T_n])_{- n - 1 - d}, \\mathcal{O}_S) & \\text{if} & q = n \\end{matrix} \\right. $$"}
+{"_id": "3307", "title": "coherent-lemma-cohomology-projective-bundle", "text": "Let $S$ be a scheme. Let $n \\geq 1$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_S$-module of constant rank $n + 1$. Consider the structure morphism $$ \\pi : \\mathbf{P}(\\mathcal{E}) \\longrightarrow S. $$ We have $$ R^q\\pi_*(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)) = \\left\\{ \\begin{matrix} \\text{Sym}^d(\\mathcal{E}) & \\text{if} & q = 0 \\\\ 0 & \\text{if} & q \\not = 0, n \\\\ \\SheafHom_{\\mathcal{O}_S}( \\text{Sym}^{- n - 1 - d}(\\mathcal{E}) \\otimes_{\\mathcal{O}_S} \\wedge^{n + 1}\\mathcal{E}, \\mathcal{O}_S) & \\text{if} & q = n \\end{matrix} \\right. $$ These identifications are compatible with base change and isomorphism between locally free sheaves."}
+{"_id": "3308", "title": "coherent-lemma-coherent-Noetherian", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is coherent, \\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module, \\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module, \\item for any affine open $\\Spec(A) = U \\subset X$ we have $\\mathcal{F}|_U = \\widetilde M$ with $M$ a finite $A$-module, and \\item there exists an affine open covering $X = \\bigcup U_i$, $U_i = \\Spec(A_i)$ such that each $\\mathcal{F}|_{U_i} = \\widetilde M_i$ with $M_i$ a finite $A_i$-module. \\end{enumerate} In particular $\\mathcal{O}_X$ is coherent, any invertible $\\mathcal{O}_X$-module is coherent, and more generally any finite locally free $\\mathcal{O}_X$-module is coherent."}
+{"_id": "3309", "title": "coherent-lemma-coherent-abelian-Noetherian", "text": "Let $X$ be a locally Noetherian scheme. The category of coherent $\\mathcal{O}_X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\\mathcal{O}_X$-modules are coherent. Any extension of coherent sheaves is coherent."}
+{"_id": "3310", "title": "coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Any quasi-coherent submodule of $\\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\\mathcal{F}$ is coherent."}
+{"_id": "3311", "title": "coherent-lemma-tensor-hom-coherent", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. The $\\mathcal{O}_X$-modules $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$ and $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are coherent."}
+{"_id": "3312", "title": "coherent-lemma-local-isomorphism", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism of $\\mathcal{O}_X$-modules. Let $x \\in X$. \\begin{enumerate} \\item If $\\mathcal{F}_x = 0$ then there exists an open neighbourhood $U \\subset X$ of $x$ such that $\\mathcal{F}|_U = 0$. \\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is injective, then there exists an open neighbourhood $U \\subset X$ of $x$ such that $\\varphi|_U$ is injective. \\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is surjective, then there exists an open neighbourhood $U \\subset X$ of $x$ such that $\\varphi|_U$ is surjective. \\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is bijective, then there exists an open neighbourhood $U \\subset X$ of $x$ such that $\\varphi|_U$ is an isomorphism. \\end{enumerate}"}
+{"_id": "3313", "title": "coherent-lemma-map-stalks-local-map", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Let $x \\in X$. Suppose $\\psi : \\mathcal{G}_x \\to \\mathcal{F}_x$ is a map of $\\mathcal{O}_{X, x}$-modules. Then there exists an open neighbourhood $U \\subset X$ of $x$ and a map $\\varphi : \\mathcal{G}|_U \\to \\mathcal{F}|_U$ such that $\\varphi_x = \\psi$."}
+{"_id": "3314", "title": "coherent-lemma-coherent-support-closed", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $\\text{Supp}(\\mathcal{F})$ is closed, and $\\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\\mathcal{F}$, see Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}."}
+{"_id": "3315", "title": "coherent-lemma-i-star-equivalence", "text": "Let $i : Z \\to X$ be a closed immersion of locally Noetherian schemes. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$ and the category of coherent $\\mathcal{O}_Z$-modules."}
+{"_id": "3316", "title": "coherent-lemma-finite-pushforward-coherent", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and $f_*\\mathcal{F}$ is coherent if $\\mathcal{F}$ is coherent."}
+{"_id": "3317", "title": "coherent-lemma-coherent-support-dimension-0", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$. Then $\\mathcal{F}$ is generated by global sections and $H^i(X, \\mathcal{F}) = 0$ for $i > 0$."}
+{"_id": "3318", "title": "coherent-lemma-pushforward-coherent-on-open", "text": "Let $X$ be a scheme. Let $j : U \\to X$ be the inclusion of an open. Let $T \\subset X$ be a closed subset contained in $U$. If $\\mathcal{F}$ is a coherent $\\mathcal{O}_U$-module with $\\text{Supp}(\\mathcal{F}) \\subset T$, then $j_*\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module."}
+{"_id": "3319", "title": "coherent-lemma-acc-coherent", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The ascending chain condition holds for quasi-coherent submodules of $\\mathcal{F}$. In other words, given any sequence $$ \\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F} $$ of quasi-coherent submodules, then $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$."}
+{"_id": "3320", "title": "coherent-lemma-power-ideal-kills-sheaf", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals corresponding to a closed subscheme $Z \\subset X$. Then there is some $n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$ if and only if $\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically)."}
+{"_id": "3321", "title": "coherent-lemma-Artin-Rees", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \\geq 0$ such that for all $n \\geq c$ we have $$ \\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G}) = \\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G} $$"}
+{"_id": "3322", "title": "coherent-lemma-homs-over-open", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-module. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Denote $Z \\subset X$ the corresponding closed subscheme and set $U = X \\setminus Z$. There is a canonical isomorphism $$ \\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F}) \\longrightarrow \\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U). $$ In particular we have an isomorphism $$ \\colim_n \\Hom_{\\mathcal{O}_X}( \\mathcal{I}^n, \\mathcal{F}) \\longrightarrow \\Gamma(U, \\mathcal{F}). $$"}
+{"_id": "3323", "title": "coherent-lemma-extend-coherent", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Let $U \\subset X$ be open and let $\\varphi : \\mathcal{F}|_U \\to \\mathcal{G}|_U$ be an $\\mathcal{O}_U$-module map. Then there exists a coherent submodule $\\mathcal{F}' \\subset \\mathcal{F}$ agreeing with $\\mathcal{F}$ over $U$ such that $\\varphi$ extends to $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}$."}
+{"_id": "3324", "title": "coherent-lemma-hom-into-depth", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules and $x \\in X$. \\begin{enumerate} \\item If $\\mathcal{G}_x$ has depth $\\geq 1$, then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$ has depth $\\geq 1$. \\item If $\\mathcal{G}_x$ has depth $\\geq 2$, then $\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$ has depth $\\geq 2$. \\end{enumerate}"}
+{"_id": "3327", "title": "coherent-lemma-prepare-filter-support", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Suppose that $\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed. Then there exists a short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0 $$ with $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and $\\text{Supp}(\\mathcal{G}) \\subset Z$."}
+{"_id": "3328", "title": "coherent-lemma-prepare-filter-irreducible", "text": "Let $X$ be a Noetherian scheme. Let $i : Z \\to X$ be an integral closed subscheme. Let $\\xi \\in Z$ be the generic point. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\\mathcal{F}_\\xi$ is annihilated by $\\mathfrak m_\\xi$. Then there exists an integer $r \\geq 0$ and a sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ and an injective map of coherent sheaves $$ i_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F} $$ which is an isomorphism in a neighbourhood of $\\xi$."}
+{"_id": "3329", "title": "coherent-lemma-coherent-filter", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration $$ 0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset \\mathcal{F}_m = \\mathcal{F} $$ by coherent subsheaves such that for each $j = 1, \\ldots, m$ there exists an integral closed subscheme $Z_j \\subset X$ and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$ such that $$ \\mathcal{F}_j/\\mathcal{F}_{j - 1} \\cong (Z_j \\to X)_* \\mathcal{I}_j $$"}
+{"_id": "3330", "title": "coherent-lemma-property-initial", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \\begin{enumerate} \\item For any short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0 $$ if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$ then so does $\\mathcal{F}$. \\item For every integral closed subscheme $Z \\subset X$ and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$ for $i_*\\mathcal{I}$. \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "3331", "title": "coherent-lemma-property-irreducible", "text": "Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible closed subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ with support contained in $Z_0$ such that \\begin{enumerate} \\item For any short exact sequence of coherent sheaves if two out of three of them have property $\\mathcal{P}$ then so does the third. \\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$, $Z \\not = Z_0$ and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$. \\item There exists some coherent sheaf $\\mathcal{G}$ on $X$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z_0$, \\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, \\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and \\item property $\\mathcal{P}$ holds for $\\mathcal{G}$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf $\\mathcal{F}$ on $X$ whose support is contained in $Z_0$."}
+{"_id": "3332", "title": "coherent-lemma-property", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that \\begin{enumerate} \\item For any short exact sequence of coherent sheaves if two out of three of them have property $\\mathcal{P}$ then so does the third. \\item For every integral closed subscheme $Z \\subset X$ with generic point $\\xi$ there exists some coherent sheaf $\\mathcal{G}$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z$, \\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, \\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and \\item property $\\mathcal{P}$ holds for $\\mathcal{G}$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "3333", "title": "coherent-lemma-property-irreducible-higher-rank-cohomological", "text": "Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible closed subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that \\begin{enumerate} \\item For any short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0 $$ if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$ then so does $\\mathcal{F}$. \\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for some $r \\geq 1$, then it holds for $\\mathcal{F}$. \\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$, $Z \\not = Z_0$ and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$. \\item There exists some coherent sheaf $\\mathcal{G}$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z_0$, \\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and \\item for every quasi-coherent sheaf of ideals $\\mathcal{J} \\subset \\mathcal{O}_X$ such that $\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent subsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with $\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that $\\mathcal{P}$ holds for $\\mathcal{G}'$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf $\\mathcal{F}$ on $X$ whose support is contained in $Z_0$."}
+{"_id": "3334", "title": "coherent-lemma-property-higher-rank-cohomological", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that \\begin{enumerate} \\item For any short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0 $$ if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$ then so does $\\mathcal{F}$. \\item If $\\mathcal{P}$ holds  for $\\mathcal{F}^{\\oplus r}$ for some $r \\geq 1$, then it holds for $\\mathcal{F}$. \\item For every integral closed subscheme $Z \\subset X$ with generic point $\\xi$ there exists some coherent sheaf $\\mathcal{G}$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z$, \\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and \\item for every quasi-coherent sheaf of ideals $\\mathcal{J} \\subset \\mathcal{O}_X$ such that $\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent subsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with $\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that $\\mathcal{P}$ holds for $\\mathcal{G}'$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "3335", "title": "coherent-lemma-finite-morphism-Noetherian", "text": "Let $f : Y \\to X$ be a morphism of schemes. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $Z \\subset X$ be an integral closed subscheme with generic point $\\xi$. Then there exists a coherent sheaf $\\mathcal{F}$ on $Y$ such that the support of $f_*\\mathcal{F}$ is equal to $Z$ and $(f_*\\mathcal{F})_\\xi$ is annihilated by $\\mathfrak m_\\xi$."}
+{"_id": "3336", "title": "coherent-lemma-affine-morphism-projection-ideal", "text": "Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If the morphism $f$ is affine then $\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$."}
+{"_id": "3337", "title": "coherent-lemma-image-affine-finite-morphism-affine-Noetherian", "text": "Let $f : Y \\to X$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ finite, \\item $f$ surjective, \\item $Y$ affine, and \\item $X$ Noetherian. \\end{enumerate} Then $X$ is affine."}
+{"_id": "3338", "title": "coherent-lemma-coherent-projective", "text": "Let $R$ be a Noetherian ring. Let $n \\geq 0$ be an integer. For every coherent sheaf $\\mathcal{F}$ on $\\mathbf{P}^n_R$ we have the following: \\begin{enumerate} \\item There exists an $r \\geq 0$ and $d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection $$ \\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_{\\mathbf{P}^n_R}(d_j) \\longrightarrow \\mathcal{F}. $$ \\item We have $H^i(\\mathbf{P}^n_R, \\mathcal{F}) = 0$ unless $0 \\leq i \\leq n$. \\item For any $i$ the cohomology group $H^i(\\mathbf{P}^n_R, \\mathcal{F})$ is a finite $R$-module. \\item If $i > 0$, then $H^i(\\mathbf{P}^n_R, \\mathcal{F}(d)) = 0$ for all $d$ large enough. \\item For any $k \\in \\mathbf{Z}$ the graded $R[T_0, \\ldots, T_n]$-module $$ \\bigoplus\\nolimits_{d \\geq k} H^0(\\mathbf{P}^n_R, \\mathcal{F}(d)) $$ is a finite $R[T_0, \\ldots, T_n]$-module. \\end{enumerate}"}
+{"_id": "3339", "title": "coherent-lemma-coherent-on-proj", "text": "Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Set $X = \\text{Proj}(A)$. Then $X$ is a Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item There exists an $r \\geq 0$ and $d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection $$ \\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j) \\longrightarrow \\mathcal{F}. $$ \\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite $A_0$-module. \\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough. \\item For any $k \\in \\mathbf{Z}$ the graded $A$-module $$ \\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d)) $$ is a finite $A$-module. \\end{enumerate}"}
+{"_id": "3340", "title": "coherent-lemma-recover-tail-graded-module", "text": "Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Let $M$ be a finite graded $A$-module. Set $X = \\text{Proj}(A)$ and let $\\widetilde{M}$ be the quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$. The maps $$ M_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n)) $$ from Constructions, Lemma \\ref{constructions-lemma-apply-modules} are isomorphisms for all sufficiently large $n$."}
+{"_id": "3341", "title": "coherent-lemma-coherent-on-proj-general", "text": "Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. Then $X$ is a Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item There exists an $r \\geq 0$ and $d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection $$ \\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j) \\longrightarrow \\mathcal{F}. $$ \\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite $A_0$-module. \\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough. \\item For any $k \\in \\mathbf{Z}$ the graded $A$-module $$ \\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d)) $$ is a finite $A$-module. \\end{enumerate}"}
+{"_id": "3342", "title": "coherent-lemma-recover-tail-graded-module-general", "text": "Let $A$ be a Noetherian graded ring and let $d$ be the lcm of generators of $A$ over $A_0$. Let $M$ be a finite graded $A$-module. Set $X = \\text{Proj}(A)$ and let $\\widetilde{M}$ be the quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$. Let $k \\in \\mathbf{Z}$. \\begin{enumerate} \\item $N' = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M(n)})$ is a finite $A$-module, \\item $N = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M}(n))$ is a finite $A$-module, \\item there is a canonical map $N \\to N'$, \\item if $k$ is small enough there is a canonical map $M \\to N'$, \\item the map $M_n \\to N'_n$ is an isomorphism for $n \\gg 0$, \\item $N_n \\to N'_n$ is an isomorphism for $d | n$. \\end{enumerate}"}
+{"_id": "3343", "title": "coherent-lemma-coherent-proper-ample", "text": "Let $R$ be a Noetherian ring. Let $X \\to \\Spec(R)$ be a proper morphism. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item The graded ring $A = \\bigoplus_{d \\geq 0} H^0(X, \\mathcal{L}^{\\otimes d})$ is a finitely generated $R$-algebra. \\item There exists an $r \\geq 0$ and $d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection $$ \\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{L}^{\\otimes d_j} \\longrightarrow \\mathcal{F}. $$ \\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite $R$-module. \\item If $p > 0$, then $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$ for all $d$ large enough. \\item For any $k \\in \\mathbf{Z}$ the graded $A$-module $$ \\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) $$ is a finite $A$-module. \\end{enumerate}"}
+{"_id": "3344", "title": "coherent-lemma-kill-by-twisting", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume that \\begin{enumerate} \\item $S$ is Noetherian, \\item $f$ is proper, \\item $\\mathcal{F}$ is coherent, and \\item $\\mathcal{L}$ is relatively ample on $X/S$. \\end{enumerate} Then there exists an $n_0$ such that for all $n \\geq n_0$ we have $$ R^pf_*\\left(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}\\right) = 0 $$ for all $p > 0$."}
+{"_id": "3345", "title": "coherent-lemma-locally-projective-pushforward", "text": "Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a locally projective morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module for all $i \\geq 0$."}
+{"_id": "3346", "title": "coherent-lemma-vanshing-gives-ample", "text": "\\begin{reference} \\cite[III Proposition 2.6.1]{EGA} \\end{reference} Let $R$ be a Noetherian ring. Let $f : X \\to \\Spec(R)$ be a proper morphism. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X$ (this is equivalent to many other things, see Properties, Proposition \\ref{properties-proposition-characterize-ample} and Morphisms, Lemma \\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}), \\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists an $n_0 \\geq 0$ such that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$ and $p > 0$, and \\item for every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$, there exists an $n \\geq 1$ such that $H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n}) = 0$. \\end{enumerate}"}
+{"_id": "3347", "title": "coherent-lemma-surjective-finite-morphism-ample", "text": "Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of schemes proper over $R$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective. Then $\\mathcal{L}$ is ample if and only if $f^*\\mathcal{L}$ is ample."}
+{"_id": "3348", "title": "coherent-lemma-invert-s-cohomology", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $X$ is quasi-compact and quasi-separated, the canonical map $$ H^p_*(X, \\mathcal{L}, \\mathcal{F})_{(s)} \\longrightarrow H^p(X_s, \\mathcal{F}) $$ which maps $\\xi/s^n$ to $s^{-n}\\xi$ is an isomorphism."}
+{"_id": "3350", "title": "coherent-lemma-ample-on-reduction", "text": "Let $i : Z \\to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\\mathcal{L}$ is ample on $Z$, if and only if $\\mathcal{L}$ is ample on $X$."}
+{"_id": "3352", "title": "coherent-lemma-affine-in-presence-ample", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $n_0$ be an integer. If $H^p(X, \\mathcal{L}^{\\otimes -n}) = 0$ for $n \\geq n_0$ and $p > 0$, then $X$ is affine."}
+{"_id": "3353", "title": "coherent-lemma-affine-if-quasi-affine", "text": "Let $X$ be a quasi-affine scheme. If $H^p(X, \\mathcal{O}_X) = 0$ for $p > 0$, then $X$ is affine."}
+{"_id": "3354", "title": "coherent-lemma-chow-Noetherian", "text": "\\begin{reference} \\cite[II Theorem 5.6.1(a)]{EGA} \\end{reference} Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a separated morphism of finite type. Then there exists an $n \\geq 0$ and a diagram $$ \\xymatrix{ X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\ & S & } $$ where $X' \\to \\mathbf{P}^n_S$ is an immersion, and $\\pi : X' \\to X$ is proper and surjective. Moreover, we may arrange it such that there exists a dense open subscheme $U \\subset X$ such that $\\pi^{-1}(U) \\to U$ is an isomorphism."}
+{"_id": "3355", "title": "coherent-lemma-proper-over-affine-cohomology-finite", "text": "Let $S = \\Spec(A)$ with $A$ a Noetherian ring. Let $f : X \\to S$ be a proper morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $H^i(X, \\mathcal{F})$ is finite $A$-module for all $i \\geq 0$."}
+{"_id": "3356", "title": "coherent-lemma-graded-finiteness", "text": "Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \\to \\Spec(A)$ be a proper morphism. Set $\\mathcal{B} = f^*\\widetilde B$. Let $\\mathcal{F}$ be a quasi-coherent graded $\\mathcal{B}$-module of finite type. \\begin{enumerate} \\item For every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$ is a finite $B$-module. \\item If $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module, then there exists an integer $d_0$ such that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes d}) = 0$ for all $p > 0$ and $d \\geq d_0$. \\end{enumerate}"}
+{"_id": "3357", "title": "coherent-lemma-cohomology-powers-ideal-times-F", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Set $B = \\bigoplus_{n \\geq 0} I^n$. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then for every $p \\geq 0$ the graded $B$-module $\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is a finite $B$-module."}
+{"_id": "3359", "title": "coherent-lemma-cohomology-powers-ideal-application", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then for every $p \\geq 0$ there exists an integer $c \\geq 0$ such that \\begin{enumerate} \\item the multiplication map $I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$ is surjective for all $n \\geq c$, \\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$ is contained in the submodule $I^{m - e} H^p(X, I^n\\mathcal{F})$ where $e = \\max(0, c - n)$ for $n + m \\geq c$, $n, m \\geq 0$, \\item we have $$ \\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})) = \\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F})) $$ for $n \\geq c$, \\item there are maps $I^nH^p(X, \\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F})$ for $n \\geq c$ such that the compositions $$ H^p(X, I^n\\mathcal{F}) \\to  I^{n - c}H^p(X, \\mathcal{F}) \\to H^p(X, I^{n - 2c}\\mathcal{F}) $$ and $$ I^nH^p(X, \\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F}) \\to I^{n - 2c}H^p(X, \\mathcal{F}) $$ for $n \\geq 2c$ are the canonical ones, and \\item the inverse systems $(H^p(X, I^n\\mathcal{F}))$ and $(I^nH^p(X, \\mathcal{F}))$ are pro-isomorphic. \\end{enumerate}"}
+{"_id": "3360", "title": "coherent-lemma-ML-cohomology-powers-ideal", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \\geq 0$. There exists a $c \\geq 0$ such that \\begin{enumerate} \\item for all $n \\geq c$ we have $$ \\Ker(H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) \\subset I^{n - c}H^p(X, \\mathcal{F}). $$ \\item the inverse system $$ \\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}} $$ satisfies the Mittag-Leffler condition (see Homology, Definition \\ref{homology-definition-Mittag-Leffler}), and \\item we have $$ \\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) = \\Im(H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) $$ for all $k \\geq n + c$. \\end{enumerate}"}
+{"_id": "3361", "title": "coherent-lemma-spell-out-theorem-formal-functions", "text": "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then $$ H^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ for all $p \\geq 0$."}
+{"_id": "3362", "title": "coherent-lemma-formal-functions-stalk", "text": "Given a morphism of schemes $f : X \\to Y$ and a quasi-coherent sheaf $\\mathcal{F}$ on $X$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ proper, and \\item $\\mathcal{F}$ coherent. \\end{enumerate} Let $y \\in Y$ be a point. Consider the infinitesimal neighbourhoods $$ \\xymatrix{ X_n = \\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\times_Y X \\ar[r]_-{i_n} \\ar[d]_{f_n} & X \\ar[d]^f \\\\ \\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\ar[r]^-{c_n} & Y } $$ of the fibre $X_1 = X_y$ and set $\\mathcal{F}_n = i_n^*\\mathcal{F}$. Then we have $$ \\left(R^pf_*\\mathcal{F}\\right)_y^\\wedge \\cong \\lim_n H^p(X_n, \\mathcal{F}_n) $$ as $\\mathcal{O}_{Y, y}^\\wedge$-modules."}
+{"_id": "3363", "title": "coherent-lemma-higher-direct-images-zero-finite-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $y \\in Y$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ is proper, and \\item $f^{-1}(\\{y\\})$ is finite. \\end{enumerate} Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have $(R^pf_*\\mathcal{F})_y = 0$ for all $p > 0$."}
+{"_id": "3364", "title": "coherent-lemma-higher-direct-images-zero-above-dimension-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $y \\in Y$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ is proper, and \\item $\\dim(X_y) = d$. \\end{enumerate} Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have $(R^pf_*\\mathcal{F})_y = 0$ for all $p > d$."}
+{"_id": "3365", "title": "coherent-lemma-characterize-finite", "text": "(For a more general version see More on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}.) Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is finite, and \\item $f$ is proper with finite fibres. \\end{enumerate}"}
+{"_id": "3366", "title": "coherent-lemma-proper-finite-fibre-finite-in-neighbourhood", "text": "\\begin{slogan} A proper morphism is finite in a neighbourhood of a finite fiber. \\end{slogan} (For a more general version see More on Morphisms, Lemma \\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}.) Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume \\begin{enumerate} \\item $S$ is locally Noetherian, \\item $f$ is proper, and \\item $f^{-1}(\\{s\\})$ is a finite set. \\end{enumerate} Then there exists an open neighbourhood $V \\subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite."}
+{"_id": "3367", "title": "coherent-lemma-ample-on-fibre", "text": "Let $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample on $X_y$. Then there exists a $d_0$ such that for all $d \\geq d_0$ we have $$ R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0 \\text{ for }p > 0 $$ and the map $$ f_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y \\longrightarrow H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d}) $$ is surjective."}
+{"_id": "3368", "title": "coherent-lemma-ample-in-neighbourhood", "text": "(For a more general version see More on Morphisms, Lemma \\ref{more-morphisms-lemma-ample-in-neighbourhood}.) Let $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \\subset Y$ of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$."}
+{"_id": "3369", "title": "coherent-lemma-perfect-direct-image", "text": "Let $A$ be a Noetherian ring and set $S = \\Spec(A)$. Let $f : X \\to S$ be a proper morphism of schemes. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module flat over $S$. Then \\begin{enumerate} \\item $R\\Gamma(X, \\mathcal{F})$ is a perfect object of $D(A)$, and \\item for any ring map $A \\to A'$ the base change map $$ R\\Gamma(X, \\mathcal{F}) \\otimes_A^{\\mathbf{L}} A' \\longrightarrow R\\Gamma(X_{A'}, \\mathcal{F}_{A'}) $$ is an isomorphism. \\end{enumerate}"}
+{"_id": "3370", "title": "coherent-lemma-inverse-systems-affine", "text": "If $X = \\Spec(A)$ is the spectrum of a Noetherian ring and $\\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal $I \\subset A$, then $\\textit{Coh}(X, \\mathcal{I})$ is equivalent to the category of finite $A^\\wedge$-modules where $A^\\wedge$ is the completion of $A$ with respect to $I$."}
+{"_id": "3371", "title": "coherent-lemma-inverse-systems-abelian", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. \\begin{enumerate} \\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian. \\item For $U \\subset X$ open the restriction functor $\\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(U, \\mathcal{I}|_U)$ is exact. \\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$ may be checked by restricting to the members of an open covering of $X$. \\end{enumerate}"}
+{"_id": "3372", "title": "coherent-lemma-inverse-systems-surjective", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. A map $(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ is surjective in $\\textit{Coh}(X, \\mathcal{I})$ if and only if $\\mathcal{F}_1 \\to \\mathcal{G}_1$ is surjective."}
+{"_id": "3373", "title": "coherent-lemma-exact", "text": "The functor (\\ref{equation-completion-functor}) is exact."}
+{"_id": "3374", "title": "coherent-lemma-completion-internal-hom", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Set $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$. Then $$ \\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) = \\Mor_{\\textit{Coh}(X, \\mathcal{I})} (\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge). $$"}
+{"_id": "3375", "title": "coherent-lemma-existence-easy", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module. Let $(\\mathcal{F}_n)$ an object of $\\textit{Coh}(X, \\mathcal{I})$. \\begin{enumerate} \\item If $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$ is a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\\mathcal{F}, a, \\beta)$ where \\begin{enumerate} \\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module, \\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$, \\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and \\item $\\alpha = a^\\wedge \\circ \\beta$. \\end{enumerate} \\item If $\\alpha : \\mathcal{G}^\\wedge \\to (\\mathcal{F}_n)$ is a map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\\mathcal{F}, a, \\beta)$ where \\begin{enumerate} \\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module, \\item $a : \\mathcal{G} \\to \\mathcal{F}$ is an $\\mathcal{O}_X$-module map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$, \\item $\\beta : \\mathcal{F}^\\wedge \\to (\\mathcal{F}_n)$ is an isomorphism, and \\item $\\alpha = \\beta \\circ a^\\wedge$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3377", "title": "coherent-lemma-finite-over-rees-algebra", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $(\\mathcal{F}_n)$ is an object of $\\textit{Coh}(X, \\mathcal{I})$ then $\\bigoplus \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)$ is a finite type, graded, quasi-coherent $\\bigoplus \\mathcal{I}^n/\\mathcal{I}^{n + 1}$-module."}
+{"_id": "3378", "title": "coherent-lemma-inverse-systems-pullback", "text": "Let $f : X \\to Y$ be a morphism of Noetherian schemes. Let $\\mathcal{J} \\subset \\mathcal{O}_Y$ be a quasi-coherent sheaf of ideals and set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$. Then there is a right exact functor $$ f^* : \\textit{Coh}(Y, \\mathcal{J}) \\longrightarrow \\textit{Coh}(X, \\mathcal{I}) $$ which sends $(\\mathcal{G}_n)$ to $(f^*\\mathcal{G}_n)$. If $f$ is flat, then $f^*$ is an exact functor."}
+{"_id": "3379", "title": "coherent-lemma-inverse-systems-pullback-equivalence", "text": "Let $f : X' \\to X$ be a morphism of Noetherian schemes. Let $Z \\subset X$ be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic inverse image. Let $\\mathcal{I} \\subset \\mathcal{O}_X$, $\\mathcal{I}' \\subset \\mathcal{O}_{X'}$ be the corresponding quasi-coherent sheaves of ideals. If $f$ is flat and the induced morphism $Z' \\to Z$ is an isomorphism, then the pullback functor $f^* : \\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(X', \\mathcal{I}')$ (Lemma \\ref{lemma-inverse-systems-pullback}) is an equivalence."}
+{"_id": "3380", "title": "coherent-lemma-inverse-systems-ideals-equivalence", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of ideals. If $V(\\mathcal{I}) = V(\\mathcal{J})$ is the same closed subset of $X$, then $\\textit{Coh}(X, \\mathcal{I})$ and $\\textit{Coh}(X, \\mathcal{J})$ are equivalent."}
+{"_id": "3381", "title": "coherent-lemma-fully-faithful", "text": "Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \\to \\Spec(A)$ be a proper morphism. Let $\\mathcal{I} = I\\mathcal{O}_X$. Then the functor (\\ref{equation-completion-functor}) is fully faithful."}
+{"_id": "3382", "title": "coherent-lemma-vanishing-projective", "text": "Let $A$ be Noetherian ring and $I \\subset A$ and ideal. Let $f : X \\to \\Spec(A)$ be a proper morphism and let $\\mathcal{L}$ be an $f$-ample invertible sheaf. Let $\\mathcal{I} = I\\mathcal{O}_X$. Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(X, \\mathcal{I})$. Then there exists an integer $d_0$ such that $$ H^1(X, \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n) \\otimes \\mathcal{L}^{\\otimes d} ) = 0 $$ for all $n \\geq 0$ and all $d \\geq d_0$."}
+{"_id": "3383", "title": "coherent-lemma-existence-projective", "text": "Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \\to \\Spec(A)$ be a projective morphism. Let $\\mathcal{I} = I\\mathcal{O}_X$. Then the functor (\\ref{equation-completion-functor}) is an equivalence."}
+{"_id": "3384", "title": "coherent-lemma-existence-tricky", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $X_e \\subset X$ be the closed subscheme cut out by $\\mathcal{K}^e$. Let $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$. Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(X, \\mathcal{I})$. Assume \\begin{enumerate} \\item the functor $\\textit{Coh}(\\mathcal{O}_{X_e}) \\to \\textit{Coh}(X_e, \\mathcal{I}_e)$ is an equivalence for all $e \\geq 1$, and \\item there exists a coherent sheaf $\\mathcal{H}$ on $X$ and a map $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$. \\end{enumerate} Then $(\\mathcal{F}_n)$ is in the essential image of (\\ref{equation-completion-functor})."}
+{"_id": "3385", "title": "coherent-lemma-inverse-systems-push-pull", "text": "Let $Y$ be a Noetherian scheme. Let $\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$ be quasi-coherent sheaves of ideals. Let $f : X \\to Y$ be a proper morphism which is an isomorphism over $V = Y \\setminus V(\\mathcal{K})$. Set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$. Let $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$, let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let $\\beta : (f^*\\mathcal{G}_n)  \\to \\mathcal{F}^\\wedge$ be an isomorphism in $\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map $$ \\alpha : (\\mathcal{G}_n) \\longrightarrow (f_*\\mathcal{F})^\\wedge $$ in $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$."}
+{"_id": "3386", "title": "coherent-lemma-closed-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $Z \\subset X$ be a closed subset. The following are equivalent \\begin{enumerate} \\item the morphism $Z \\to S$ is proper if $Z$ is endowed with the reduced induced closed subscheme structure (Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}), \\item for some closed subscheme structure on $Z$ the morphism $Z \\to S$ is proper, \\item for any closed subscheme structure on $Z$ the morphism $Z \\to S$ is proper. \\end{enumerate}"}
+{"_id": "3387", "title": "coherent-lemma-closed-closed-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $Y \\subset Z \\subset X$ be closed subsets. If $Z$ is proper over $S$, then the same is true for $Y$."}
+{"_id": "3388", "title": "coherent-lemma-base-change-closed-proper-over-base", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ with $f$ locally of finite type. If $Z$ is a closed subset of $X$ proper over $S$, then $(g')^{-1}(Z)$ is a closed subset of $X'$ proper over $S'$."}
+{"_id": "3389", "title": "coherent-lemma-functoriality-closed-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes which are locally of finite type over $S$. \\begin{enumerate} \\item If $Y$ is separated over $S$ and $Z \\subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$. \\item If $f$ is universally closed and $Z \\subset X$ is a closed subset proper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$. \\item If $f$ is proper and $Z \\subset Y$ is a closed subset proper over $S$, then $f^{-1}(Z)$ is a closed subset of $X$ proper over $S$. \\end{enumerate}"}
+{"_id": "3390", "title": "coherent-lemma-union-closed-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $Z_i \\subset X$, $i = 1, \\ldots, n$ be closed subsets. If $Z_i$, $i = 1, \\ldots, n$ are proper over $S$, then the same is true for $Z_1 \\cup \\ldots \\cup Z_n$."}
+{"_id": "3391", "title": "coherent-lemma-module-support-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item the support of $\\mathcal{F}$ is proper over $S$, \\item the scheme theoretic support of $\\mathcal{F}$ (Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}) is proper over $S$, and \\item there exists a closed subscheme $Z \\subset X$ and a finite type, quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ such that (a) $Z \\to S$ is proper, and (b) $(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$. \\end{enumerate}"}
+{"_id": "3394", "title": "coherent-lemma-support-proper-over-base-pushforward", "text": "Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module with support proper over $S$. Then $R^pf_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module for all $p \\geq 0$."}
+{"_id": "3395", "title": "coherent-lemma-systems-with-proper-support", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a finite type morphism. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. The following are Serre subcategories of $\\textit{Coh}(X, \\mathcal{I})$ \\begin{enumerate} \\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$ consisting of those objects $(\\mathcal{F}_n)$ such that the support of $\\mathcal{F}_1$ is proper over $S$, \\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$ consisting of those objects $(\\mathcal{F}_n)$ such that there exists a closed subscheme $Z \\subset X$ proper over $S$ with $\\mathcal{I}_Z \\mathcal{F}_n = 0$ for all $n \\geq 1$. \\end{enumerate}"}
+{"_id": "3396", "title": "coherent-lemma-algebraize-formal-closed-subscheme", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a separated morphism of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Suppose given a commutative diagram $$ \\xymatrix{ Z_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\ X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots } $$ of schemes with cartesian squares. Assume that \\begin{enumerate} \\item $Z_1 \\to X_1$ is a closed immersion, and \\item $Z_1 \\to S_1$ is proper. \\end{enumerate} Then there exists a closed immersion of schemes $Z \\to X$ such that $Z_n = Z \\times_S S_n$. Moreover, $Z$ is proper over $S$."}
+{"_id": "3397", "title": "coherent-lemma-algebraize-formal-scheme-finite-over-proper", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a separated morphism of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Suppose given a commutative diagram $$ \\xymatrix{ Y_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\ X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots } $$ of schemes with cartesian squares. Assume that \\begin{enumerate} \\item $Y_n \\to X_n$ is a finite morphism, and \\item $Y_1 \\to S_1$ is proper. \\end{enumerate} Then there exists a finite morphism of schemes $Y \\to X$ such that $Y_n = Y \\times_S S_n$. Moreover, $Y$ is proper over $S$."}
+{"_id": "3398", "title": "coherent-lemma-algebraize-morphism", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be schemes over $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and $Y_n = Y \\times_S S_n$. Suppose given a compatible system of commutative diagrams $$ \\xymatrix{ & & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\ X_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\ & S_n \\ar[rru] } $$ Assume that \\begin{enumerate} \\item $X \\to S$ is proper, and \\item $Y \\to S$ is separated of finite type. \\end{enumerate} Then there exists a unique morphism of schemes $g : X \\to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$."}
+{"_id": "3399", "title": "coherent-proposition-coherent-modules-on-proj", "text": "Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Set $X = \\text{Proj}(A)$. The functor $M \\mapsto \\widetilde M$ induces an equivalence $$ \\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, torsion} \\longrightarrow \\textit{Coh}(\\mathcal{O}_X) $$ whose quasi-inverse is given by $\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$."}
+{"_id": "3400", "title": "coherent-proposition-coherent-modules-on-proj-general", "text": "Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. The functor $M \\mapsto \\widetilde M$ induces an equivalence $$ \\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, irrelevant} \\longrightarrow \\textit{Coh}(\\mathcal{O}_X) $$ whose quasi-inverse is given by $\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$."}
+{"_id": "3401", "title": "coherent-proposition-proper-pushforward-coherent", "text": "\\begin{reference} \\cite[III Theorem 3.2.1]{EGA} \\end{reference} Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a proper morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module for all $i \\geq 0$."}
+{"_id": "3402", "title": "coherent-proposition-existence-proper", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $f : X \\to \\Spec(A)$ be a proper morphism of schemes. Set $\\mathcal{I} = I\\mathcal{O}_X$. Then the functor (\\ref{equation-completion-functor}) is an equivalence."}
+{"_id": "3410", "title": "formal-defos-theorem-miniversal-object-existence", "text": "Let $\\mathcal{F}$ be a predeformation category. Consider the following conditions \\begin{enumerate} \\item $\\mathcal{F}$ has a minimal versal formal object satisfying (\\ref{equation-bijective}), \\item $\\mathcal{F}$ has a minimal versal formal object satisfying (\\ref{equation-bijective-orbits}), \\item the following conditions hold: \\begin{enumerate} \\item $\\mathcal{F}$ satisfies (S1). \\item $\\mathcal{F}$ satisfies (S2). \\item $\\dim_k T\\mathcal{F}$ is finite. \\end{enumerate} \\end{enumerate} We always have $$ (1) \\Rightarrow (3) \\Rightarrow (2). $$ If $k' \\subset k$ is separable, then all three are equivalent."}
+{"_id": "3411", "title": "formal-defos-theorem-Schlessinger-prorepresentability", "text": "Let $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor. Then $F$ is prorepresentable if and only if (a) $F$ is a deformation functor, (b) $\\dim_k TF$ is finite, and (c) $\\gamma : \\text{Der}_\\Lambda(k, k) \\to TF$ is injective."}
+{"_id": "3412", "title": "formal-defos-theorem-presentation-deformation-groupoid", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Then $\\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$ if and only if the following conditions hold: \\begin{enumerate} \\item $\\mathcal{F}$ is a deformation category. \\item $\\dim_k T\\mathcal{F}$ is finite. \\item $\\dim_k \\text{Inf}(\\mathcal{F})$ is finite. \\end{enumerate}"}
+{"_id": "3414", "title": "formal-defos-lemma-factor-small-extension", "text": "Let $f: B \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$. Then $f$ can be factored as a composition of small extensions."}
+{"_id": "3415", "title": "formal-defos-lemma-length", "text": "Let $A$ be a local $\\Lambda$-algebra with residue field $k$. Let $M$ be an $A$-module. Then $[k : k'] \\text{length}_A(M) = \\text{length}_\\Lambda(M)$. In the classical case we have $\\text{length}_A(M) = \\text{length}_\\Lambda(M)$."}
+{"_id": "3416", "title": "formal-defos-lemma-surjective", "text": "Let $A \\to B$ be a ring map in $\\mathcal{C}_\\Lambda$. The following are equivalent \\begin{enumerate} \\item $f$ is surjective, \\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$ is surjective, and \\item $\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2) \\to \\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$ is surjective. \\end{enumerate}"}
+{"_id": "3417", "title": "formal-defos-lemma-fiber-product-CLambda", "text": "Let $f_1 : A_1 \\to A$ and $f_2 : A_2 \\to A$ be ring maps in $\\mathcal{C}_\\Lambda$. Then: \\begin{enumerate} \\item If $f_1$ or $f_2$ is surjective, then $A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$. \\item If $f_2$ is a small extension, then so is $A_1 \\times_A A_2 \\to A_1$. \\item If the field extension $k' \\subset k$ is separable, then $A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$. \\end{enumerate}"}
+{"_id": "3418", "title": "formal-defos-lemma-essential-surjection-mod-squares", "text": "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$. The following are equivalent \\begin{enumerate} \\item $f$ is an essential surjection, \\item the map $B/\\mathfrak m_B^2 \\to A/\\mathfrak m_A^2$ is an essential surjection, and \\item the map $B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is an essential surjection. \\end{enumerate}"}
+{"_id": "3419", "title": "formal-defos-lemma-H1-separable-case", "text": "There is a canonical map $$ \\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow H_1(L_{k/\\Lambda}). $$ If $k' \\subset k$ is separable (for example if the characteristic of $k$ is zero), then this map induces an isomorphism $\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k = H_1(L_{k/\\Lambda})$. If $k = k'$ (for example in the classical case), then $\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 = H_1(L_{k/\\Lambda})$. The composition $$ \\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow H_1(L_{k/\\Lambda}) \\longrightarrow \\mathfrak m_A/\\mathfrak m_A^2 $$ comes from the canonical map $\\mathfrak m_\\Lambda \\to \\mathfrak m_A$."}
+{"_id": "3420", "title": "formal-defos-lemma-essential-surjection", "text": "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$. Notation as in (\\ref{equation-sequence-functorial}). \\begin{enumerate} \\item The equivalent conditions of Lemma \\ref{lemma-essential-surjection-mod-squares} characterizing when $f$ is surjective are also equivalent to \\begin{enumerate} \\item $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is surjective, and \\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to \\Omega_{A/\\Lambda} \\otimes_A k$ is surjective. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $f$ is an essential surjection, \\item the map $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is an isomorphism, and \\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to \\Omega_{A/\\Lambda} \\otimes_A k$ is an isomorphism. \\end{enumerate} \\item If $k/k'$ is separable, then $f$ is an essential surjection if and only if the map $\\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to \\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is an isomorphism. \\item If $f$ is a small extension, then $f$ is not essential if and only if $f$ has a section $s: A \\to B$ in $\\mathcal{C}_\\Lambda$ with $f \\circ s = \\text{id}_A$. \\end{enumerate}"}
+{"_id": "3421", "title": "formal-defos-lemma-surjective-cotangent-space", "text": "Let $f: R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$. The following are equivalent \\begin{enumerate} \\item $f$ is surjective, \\item the map $\\mathfrak m_R/\\mathfrak m_R^2 \\to \\mathfrak m_S/\\mathfrak m_S^2$ is surjective, and \\item the map $\\mathfrak m_R/(\\mathfrak m_\\Lambda R + \\mathfrak m_R^2) \\to \\mathfrak m_S/(\\mathfrak m_\\Lambda S + \\mathfrak m_S^2)$ is surjective. \\end{enumerate}"}
+{"_id": "3422", "title": "formal-defos-lemma-CLambdahat-pushouts", "text": "The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits pushouts."}
+{"_id": "3423", "title": "formal-defos-lemma-CLambdahat-coproducts", "text": "The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits coproducts of pairs of objects."}
+{"_id": "3424", "title": "formal-defos-lemma-derivations-finite", "text": "Let $S$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$. Then $\\dim_k \\text{Der}_\\Lambda(S, k) < \\infty$."}
+{"_id": "3425", "title": "formal-defos-lemma-derivations-surjective", "text": "Let $f : R \\to S$ be a morphism of $\\widehat{\\mathcal{C}}_\\Lambda$. If $\\text{Der}_\\Lambda(S, k) \\to \\text{Der}_\\Lambda(R, k)$ is injective, then $f$ is surjective."}
+{"_id": "3426", "title": "formal-defos-lemma-m-adic-topology", "text": "Let $R$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$. Let $(J_n)$ be a decreasing sequence of ideals such that $\\mathfrak m_R^n \\subset J_n$. Set $J = \\bigcap J_n$. Then the sequence $(J_n/J)$ defines the $\\mathfrak m_{R/J}$-adic topology on $R/J$."}
+{"_id": "3427", "title": "formal-defos-lemma-limit-artinian", "text": "Let $\\ldots \\to A_3 \\to A_2 \\to A_1$ be a sequence of surjective ring maps in $\\mathcal{C}_\\Lambda$. If $\\dim_k (\\mathfrak m_{A_n}/\\mathfrak m_{A_n}^2)$ is bounded, then $S = \\lim A_n$ is an object in $\\widehat{\\mathcal{C}}_\\Lambda$ and the ideals $I_n = \\Ker(S \\to A_n)$ define the $\\mathfrak m_S$-adic topology on $S$."}
+{"_id": "3428", "title": "formal-defos-lemma-power-series", "text": "Let $R', R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. Suppose that $R = R' \\oplus I$ for some ideal $I$ of $R$. Let $x_1, \\ldots, x_r \\in I$ map to a basis of $I/\\mathfrak m_R I$. Set $S = R'[[X_1, \\ldots, X_r]]$ and consider the $R'$-algebra map $S \\to R$ mapping $X_i$ to $x_i$. Assume that for every $n \\gg 0$ the map $S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$ has a left inverse in $\\mathcal{C}_\\Lambda$. Then $S \\to R$ is an isomorphism."}
+{"_id": "3429", "title": "formal-defos-lemma-completion-cofibred", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids. Then $\\widehat{p} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{C}}_\\Lambda$ is a category cofibered in groupoids."}
+{"_id": "3430", "title": "formal-defos-lemma-formal-objects-different-filtration", "text": "In the situation above, $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$ is equivalent to the category $\\widehat{\\mathcal{F}}(R)$."}
+{"_id": "3431", "title": "formal-defos-lemma-smoothness-small-extensions", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$.  Then $\\varphi$ is smooth if the condition in Definition \\ref{definition-smooth-morphism} is assumed to hold only for small extensions $B \\to A$."}
+{"_id": "3432", "title": "formal-defos-lemma-smooth-morphism-power-series", "text": "Let $R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$. Then the induced morphism $\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$ is smooth if and only if $S$ is a power series ring over $R$."}
+{"_id": "3433", "title": "formal-defos-lemma-smooth-properties", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{G} \\to \\mathcal{H}$ be morphisms of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$. \\begin{enumerate} \\item If $\\varphi$ and $\\psi$ are smooth, then $\\psi \\circ \\varphi$ is smooth. \\item If $\\varphi$ is essentially surjective and $\\psi \\circ \\varphi$ is smooth, then $\\psi$ is smooth. \\item If $\\mathcal{G}' \\to \\mathcal{G}$ is a morphism of categories cofibered in groupoids and $\\varphi$ is smooth, then $\\mathcal{F} \\times_\\mathcal{G} \\mathcal{G}' \\to \\mathcal{G}'$ is smooth. \\end{enumerate}"}
+{"_id": "3434", "title": "formal-defos-lemma-smooth-morphism-essentially-surjective", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a smooth morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$.  Assume $\\varphi : \\mathcal{F}(k) \\to \\mathcal{G}(k)$ is essentially surjective. Then $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\widehat{\\varphi} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{G}}$ are essentially surjective."}
+{"_id": "3435", "title": "formal-defos-lemma-versal-object-quasi-initial", "text": "Let $\\mathcal{F}$ be a predeformation category. Let $\\xi$ be a versal formal object of $\\mathcal{F}$. For any formal object $\\eta$ of $\\widehat{\\mathcal{F}}$, there exists a morphism $\\xi \\to \\eta$."}
+{"_id": "3436", "title": "formal-defos-lemma-smooth", "text": "Let $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are equivalent \\begin{enumerate} \\item $\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is smooth, \\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology, \\item $\\Lambda \\to R$ is flat and $R \\otimes_\\Lambda k'$ is geometrically regular over $k'$, and \\item $\\Lambda \\to R$ is flat and $k' \\to R \\otimes_\\Lambda k'$ is formally smooth in the $\\mathfrak m_R$-adic topology. \\end{enumerate} In the classical case, these are also equivalent to \\begin{enumerate} \\item[(5)] $R$ is isomorphic to $\\Lambda[[x_1, \\ldots, x_n]]$ for some $n$. \\end{enumerate}"}
+{"_id": "3437", "title": "formal-defos-lemma-smooth-power-series-classical", "text": "Let $\\mathcal{F}$ be a predeformation category. Let $\\xi$ be a versal formal object of $\\mathcal{F}$ lying over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is unobstructed, and \\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology. \\end{enumerate} In the classical case these are also equivalent to \\begin{enumerate} \\item[(3)] $R \\cong \\Lambda[[x_1, \\ldots, x_n]]$ for some $n$.\\ \\end{enumerate}"}
+{"_id": "3438", "title": "formal-defos-lemma-exists-smooth", "text": "There exists an $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$ such that the equivalent conditions of Lemma \\ref{lemma-smooth} hold and moreover $H_1(L_{k/\\Lambda}) = \\mathfrak m_R/\\mathfrak m_R^2$ and $\\Omega_{R/\\Lambda} \\otimes_R k = \\Omega_{k/\\Lambda}$."}
+{"_id": "3440", "title": "formal-defos-lemma-S2-extensions", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$. If $\\mathcal{F}$ satisfies (S2), then the condition of (S2) also holds when $k[\\epsilon]$ is replaced by $k[V]$ for any finite dimensional $k$-vector space $V$."}
+{"_id": "3441", "title": "formal-defos-lemma-S1-S2-associated-functor", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. \\begin{enumerate} \\item If $\\mathcal{F}$ satisfies (S1), then so does $\\overline{\\mathcal{F}}$. \\item If $\\mathcal{F}$ satisfies (S2), then so does $\\overline{\\mathcal{F}}$ provided at least one of the following conditions is satisfied \\begin{enumerate} \\item $\\mathcal{F}$ is a predeformation category, \\item the category $\\mathcal{F}(k)$ is a set or a setoid, or \\item for any morphism $x_\\epsilon \\to x_0$ of $\\mathcal{F}$ lying over $k[\\epsilon] \\to k$ the pushforward map $\\text{Aut}_{k[\\epsilon]}(x_\\epsilon) \\to \\text{Aut}_k(x_0)$ is surjective. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3442", "title": "formal-defos-lemma-S1-S2-localize", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Let $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ constructed in Remark \\ref{remark-localize-cofibered-groupoid}. \\begin{enumerate} \\item If $\\mathcal{F}$ satisfies (S1), then so does $\\mathcal{F}_{x_0}$. \\item If $\\mathcal{F}$ satisfies (S2), then so does $\\mathcal{F}_{x_0}$. \\end{enumerate}"}
+{"_id": "3443", "title": "formal-defos-lemma-lifting-section", "text": "Let $p: \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids. Consider a diagram of $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{ y \\ar[r] \\ar[d]_a & x_\\epsilon \\ar[d]_e \\\\ x \\ar[r]^d        & x_0 } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{ A \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\ A \\ar[r] & k. } } $$ in $\\mathcal{C}_\\Lambda$. Assume $\\mathcal{F}$ satisfies (S2). Then there exists a morphism $s : x \\to y$ with $a \\circ s = \\text{id}_x$ if and only if there exists a morphism $s_\\epsilon : x \\to x_\\epsilon$ with $e \\circ s_\\epsilon = d$."}
+{"_id": "3444", "title": "formal-defos-lemma-lifting-along-small-extension", "text": "Consider a commutative diagram in a predeformation category $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{ y \\ar[r] \\ar[d] & x_2 \\ar[d]^{a_2} \\\\ x_1 \\ar[r]^{a_1}        & x } } \\quad\\text{lying over} \\vcenter{ \\xymatrix{ A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d]^{f_2} \\\\ A_1 \\ar[r]^{f_1} & A } } $$ in $\\mathcal{C}_\\Lambda$ where $f_2 : A_2 \\to A$ is a small extension. Assume there is a map $h : A_1 \\to A_2$ such that $f_2 = f_1 \\circ h$. Let $I = \\Ker(f_2)$. Consider the ring map $$ g : A_1 \\times_A A_2 \\longrightarrow k[I] = k \\oplus I, \\quad (u, v) \\longmapsto \\overline{u} \\oplus (v - h(u)) $$ Choose a pushforward $y \\to g_*y$. Assume $\\mathcal{F}$ satisfies (S2). If there exists a morphism $x_1 \\to g_*y$, then there exists a morphism $b: x_1 \\to x_2$ such that $a_1 =  a_2 \\circ b$."}
+{"_id": "3445", "title": "formal-defos-lemma-linear-functor", "text": "Let $L: \\text{Mod}^{fg}_R \\to \\textit{Sets}$, resp.\\ $L: \\text{Mod}_R \\to \\textit{Sets}$ be a functor.  Suppose $L(0)$ is a one element set and $L$ preserves finite products.  Then there exists a unique $R$-linear functor $\\widetilde{L} : \\text{Mod}^{fg}_R \\to \\text{Mod}_R$, resp.\\  $\\widetilde{L} : \\text{Mod}^{fg}_R \\to \\text{Mod}_R$, such that $$ \\vcenter{ \\xymatrix{ & \\text{Mod}_R \\ar[dr]^{\\text{forget}} &   \\\\ \\text{Mod}^{fg}_R  \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} &  & \\textit{Sets} } } \\quad\\text{resp.}\\quad \\vcenter{ \\xymatrix{ & \\text{Mod}_R \\ar[dr]^{\\text{forget}} &   \\\\ \\text{Mod}_R  \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} &  & \\textit{Sets} } } $$ commutes."}
+{"_id": "3446", "title": "formal-defos-lemma-morphism-linear-functors", "text": "Let $L_1, L_2: \\text{Mod}^{fg}_R \\to \\textit{Sets}$ be functors that take $0$ to a one element set and preserve finite products. Let $t : L_1 \\to L_2$ be a morphism of functors. Then $t$ induces a morphism $\\widetilde{t} : \\widetilde{L}_1 \\to \\widetilde{L}_2$ between the functors guaranteed by Lemma \\ref{lemma-linear-functor}, which is given simply by $\\widetilde{t}_M = t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$ for each $M \\in \\Ob(\\text{Mod}^{fg}_R)$. In other words, $t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$ is a map of $R$-modules."}
+{"_id": "3447", "title": "formal-defos-lemma-linear-functor-over-field", "text": "Let $K$ be a field. Let $L: \\text{Mod}^{fg}_K \\to \\text{Mod}_K$ be a $K$-linear functor.  Then $L$ is isomorphic to the functor $L(K) \\otimes_K - : \\text{Mod}^{fg}_K \\to \\text{Mod}_K$."}
+{"_id": "3448", "title": "formal-defos-lemma-preserves-products", "text": "Let $R$ be an $S$-algebra. Then the functor $\\text{Mod}_R \\to S\\text{-Alg}/R$ described above preserves finite products."}
+{"_id": "3449", "title": "formal-defos-lemma-tangent-space-functor", "text": "Let $R$ be an $S$-algebra, and let $\\mathcal{C}$ be a strictly full subcategory of $S\\text{-Alg}/R$ containing $R[M]$ for all $M \\in \\Ob(\\text{Mod}^{fg}_R)$. Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor. Suppose that $F(R)$ is a one element set and that for any $M, N \\in \\Ob(\\text{Mod}^{fg}_R)$, the induced map $$ F(R[M] \\times_R R[N]) \\to F(R[M]) \\times F(R[N]) $$ is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M \\in \\Ob(\\text{Mod}^{fg}_R)$."}
+{"_id": "3450", "title": "formal-defos-lemma-morphism-tangent-spaces", "text": "Let $F, G: \\mathcal{C} \\to \\textit{Sets}$ be functors satisfying the hypotheses of Lemma \\ref{lemma-tangent-space-functor}. Let $t : F \\to G$ be a morphism of functors. For any $M \\in \\Ob(\\text{Mod}^{fg}_R)$, the map $t_{R[M]}: F(R[M]) \\to G(R[M])$ is a map of $R$-modules, where $F(R[M])$ and $G(R[M])$ are given the $R$-module structure from Lemma \\ref{lemma-tangent-space-functor}. In particular, $t_{R[\\epsilon]} : TF \\to TG$ is a map of $R$-modules."}
+{"_id": "3451", "title": "formal-defos-lemma-tangent-space-tensor", "text": "Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor satisfying the hypotheses of Lemma \\ref{lemma-tangent-space-functor}. Assume $R = K$ is a field. Then $F(K[V]) \\cong TF \\otimes_K V$ for any finite dimensional $K$-vector space $V$."}
+{"_id": "3452", "title": "formal-defos-lemma-tangent-space-vector-space", "text": "Let $\\mathcal{F}$ be a predeformation category such that $\\overline{\\mathcal{F}}$ satisfies (S2)\\footnote{For example if $\\mathcal{F}$ satisfies (S2), see Lemma \\ref{lemma-S1-S2-associated-functor}.}. Then $T \\mathcal{F}$ has a natural $k$-vector space structure. For any finite dimensional vector space $V$ we have $\\overline{\\mathcal{F}}(k[V]) = T\\mathcal{F} \\otimes_k V$ functorially in $V$."}
+{"_id": "3453", "title": "formal-defos-lemma-k-linear-differential", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of predeformation categories. Assume $\\overline{\\mathcal{F}}$ and $\\overline{\\mathcal{G}}$ both satisfy (S2). Then $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ is $k$-linear."}
+{"_id": "3454", "title": "formal-defos-lemma-action-linear", "text": "Let $\\mathcal{F}$ be a predeformation category over $\\mathcal{C}_\\Lambda$. If $\\overline{\\mathcal{F}}$ has (S2) then the maps $\\gamma_V$ are $k$-linear and we have $a_V(D, x) = x + \\gamma_V(D)$."}
+{"_id": "3455", "title": "formal-defos-lemma-versal-object-S1", "text": "Let $\\mathcal{F}$ be a predeformation category. Assume $\\mathcal{F}$ has a versal formal object. Then $\\mathcal{F}$ satisfies (S1)."}
+{"_id": "3456", "title": "formal-defos-lemma-versal-criterion", "text": "Let $\\mathcal{F}$ be a predeformation category satisfying (S1) and (S2). Let $\\xi$ be a formal object of $\\mathcal{F}$ corresponding to $\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$, see Remark \\ref{remark-formal-objects-yoneda}. Then $\\xi$ is versal if and only if the following two conditions hold: \\begin{enumerate} \\item the map $d\\underline{\\xi} : T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to T\\mathcal{F}$ on tangent spaces is surjective, and \\item given a diagram in $\\widehat{\\mathcal{F}}$ $$ \\vcenter{ \\xymatrix{             &  y \\ar[d] \\\\ \\xi \\ar[r]  &  x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{          &   B  \\ar[d]^{f} \\\\ R \\ar[r] &   A } } $$ in $\\widehat{\\mathcal{C}}_\\Lambda$ with $B \\to A$ a small extension of Artinian rings, then there exists a ring map $R \\to B$ such that $$ \\xymatrix{          &   B  \\ar[d]^{f} \\\\ R \\ar[ur] \\ar[r] &   A } $$ commutes. \\end{enumerate}"}
+{"_id": "3457", "title": "formal-defos-lemma-largest-closed-where-lift", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ which has (S1). Let $B \\to A$ be a surjection in $\\mathcal{C}_\\Lambda$ with kernel $I$ annihilated by $\\mathfrak m_B$. Let $x \\in \\mathcal{F}(A)$. The set of ideals $$ \\mathcal{J} = \\{ J \\subset I \\mid \\text{there exists an }y \\to x\\text{ lying over }B/J \\to A\\} $$ has a smallest element."}
+{"_id": "3458", "title": "formal-defos-lemma-versal-object-existence", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$. Assume the following conditions hold: \\begin{enumerate} \\item $\\mathcal{F}$ is a predeformation category. \\item $\\mathcal{F}$ satisfies (S1). \\item $\\mathcal{F}$ satisfies (S2). \\item $\\dim_k T\\mathcal{F}$ is finite. \\end{enumerate} Then $\\mathcal{F}$ has a versal formal object."}
+{"_id": "3459", "title": "formal-defos-lemma-smallest-where-descends", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ which has (S1). \\begin{enumerate} \\item For $y \\to x$ in $\\mathcal{F}$ a minimal object in $\\mathcal{S}_y$ maps to a minimal object of $\\mathcal{S}_x$. \\item For $y \\to x$ in $\\mathcal{F}$ lying over a surjection $f : B \\to A$ in $\\mathcal{C}_\\Lambda$ every minimal object of $\\mathcal{S}_x$ is the image of a minimal object of $\\mathcal{S}_y$. \\end{enumerate}"}
+{"_id": "3460", "title": "formal-defos-lemma-smallest-where-descends-versal", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object of $\\mathcal{F}$ lying over $R$. There exists a morphism $\\xi' \\to \\xi$ lying over $R' \\subset R$ with the following minimality properties \\begin{enumerate} \\item for every $f : R \\to A$ with $A \\in \\Ob(\\mathcal{C}_\\Lambda)$ the pushforwards $$ \\vcenter{ \\xymatrix{ \\xi' \\ar[d] \\ar[r] & x' \\ar[d] \\\\ \\xi \\ar[r] & x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{ R' \\ar[d] \\ar[r] & f(R') \\ar[d] \\\\ R \\ar[r] & A } } $$ produce a minimal object $x' \\to x$ of $\\mathcal{S}_x$, and \\item for any morphism of formal objects $\\xi'' \\to \\xi'$ the corresponding morphism $R'' \\to R'$ is surjective. \\end{enumerate}"}
+{"_id": "3461", "title": "formal-defos-lemma-descends-versal", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object of $\\mathcal{F}$ lying over $R$. Let $\\xi' \\to \\xi$ be a morphism of formal objects lying over $R' \\subset R$ as constructed in Lemma \\ref{lemma-smallest-where-descends-versal}. Then $$ R \\cong R'[[x_1, \\ldots, x_r]] $$ is a power series ring over $R'$. Moreover, $\\xi'$ is a versal formal object too."}
+{"_id": "3462", "title": "formal-defos-lemma-minimal-versal", "text": "Let $\\mathcal{F}$ be a predeformation category which has a versal formal object. Then \\begin{enumerate} \\item $\\mathcal{F}$ has a minimal versal formal object, \\item minimal versal objects are unique up to isomorphism, and \\item any versal object is the pushforward of a minimal versal object along a power series ring extension. \\end{enumerate}"}
+{"_id": "3464", "title": "formal-defos-lemma-miniversal-object-existence-1", "text": "Let $\\mathcal{F}$ be a predeformation category. Let $\\xi$ be a versal formal object of $\\mathcal{F}$ such that (\\ref{equation-bijective}) holds. Then \\begin{enumerate} \\item $\\mathcal{F}$ satisfies (S1). \\item $\\mathcal{F}$ satisfies (S2). \\item $\\dim_k T\\mathcal{F}$ is finite. \\end{enumerate}"}
+{"_id": "3465", "title": "formal-defos-lemma-construct-bijective-orbits", "text": "Let $\\mathcal{F}$ be a predeformation category satisfying (S2) which has a versal formal object. Then its minimal versal formal object satisfies (\\ref{equation-bijective-orbits})."}
+{"_id": "3466", "title": "formal-defos-lemma-RS-fiber-square", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Given a commutative diagram in $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{ y \\ar[r] \\ar[d] & x_2 \\ar[d]   \\\\ x_1 \\ar[r]      & x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{ A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\ A_1 \\ar[r]      & A. } } $$ with $A_2 \\to A$ surjective, then it is a fiber square."}
+{"_id": "3467", "title": "formal-defos-lemma-RS-small-extension", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$. Then $\\mathcal{F}$ satisfies (RS) if the condition in Definition \\ref{definition-RS} is assumed to hold only when $A_2 \\to A$ is a small extension."}
+{"_id": "3468", "title": "formal-defos-lemma-RS-2-categorical", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ satisfies (RS), \\item the functor $\\mathcal{F}(A_1 \\times_A A_2) \\to \\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$ see (\\ref{equation-compare}) is an equivalence of categories whenever $A_2 \\to A$ is surjective, and \\item same as in (2) whenever $A_2 \\to A$ is a small extension. \\end{enumerate}"}
+{"_id": "3469", "title": "formal-defos-lemma-RS-implies-S1-S2", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. The condition (RS) for $\\mathcal{F}$ implies both (S1) and (S2) for $\\mathcal{F}$."}
+{"_id": "3470", "title": "formal-defos-lemma-RS-associated-functor", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). The following conditions are equivalent: \\begin{enumerate} \\item $\\overline{\\mathcal{F}}$ satisfies (RS). \\item Let $f_1: A_1 \\to A$ and $f_2: A_2 \\to A$ be ring maps in $\\mathcal{C}_\\Lambda$ with $f_2$ surjective. The induced map of sets of isomorphism classes $$ \\overline{\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)} \\to \\overline{\\mathcal{F}}(A_1) \\times_{\\overline{\\mathcal{F}}(A)} \\overline{\\mathcal{F}}(A_2) $$ is injective. \\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a surjective ring map $A' \\to A$, the map $\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective. \\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a small extension $A' \\to A$, the map $\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective. \\end{enumerate}"}
+{"_id": "3471", "title": "formal-defos-lemma-localize-RS", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Let $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over $\\mathcal{C}_\\Lambda$ constructed in Remark \\ref{remark-localize-cofibered-groupoid}. If $\\mathcal{F}$ satisfies (RS), then so does $\\mathcal{F}_{x_0}$. In particular, $\\mathcal{F}_{x_0}$ is a deformation category."}
+{"_id": "3472", "title": "formal-defos-lemma-RS-fiber-product-morphisms", "text": "Let $$ \\xymatrix{ \\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\ar[r] \\ar[d] & \\mathcal{G} \\ar[d]^g \\\\ \\mathcal{H} \\ar[r]^f & \\mathcal{F} } $$ be $2$-fibre product of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$. If $\\mathcal{F}, \\mathcal{G}, \\mathcal{H}$ all satisfy (RS), then $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$ satisfies (RS)."}
+{"_id": "3473", "title": "formal-defos-lemma-free-transitive-action", "text": "Let $\\mathcal{F}$ be a deformation category. Let $A' \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$ whose kernel $I$ is annihilated by $\\mathfrak m_{A'}$. Let $x \\in \\Ob(\\mathcal{F}(A))$. If $\\text{Lift}(x, A')$ is nonempty, then there is a free and transitive action of $T\\mathcal{F} \\otimes_k I$ on $\\text{Lift}(x, A')$."}
+{"_id": "3474", "title": "formal-defos-lemma-minimal-smooth-morphism-functors", "text": "Let $F, G: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be deformation functors. Let $\\varphi : F \\to G$ be a smooth morphism which induces an isomorphism $d\\varphi : TF \\to TG$ of tangent spaces. Then $\\varphi$ is an isomorphism."}
+{"_id": "3475", "title": "formal-defos-lemma-Aut-functor-RS", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x \\in \\Ob(\\mathcal{F}(A))$. Then $\\mathit{Aut}(x): \\mathcal{C}_A \\to \\textit{Sets}$ satisfies (RS)."}
+{"_id": "3476", "title": "formal-defos-lemma-Aut-functor-tangent-space", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x \\in \\Ob(\\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to $\\mathcal{F}(k)$. \\begin{enumerate} \\item $T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ has a natural $k$-vector space structure such that addition agrees with composition in $T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$. In particular, composition in $T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ is commutative. \\item There is a canonical isomorphism $T_{\\text{id}_{x_0}} \\mathit{Aut}(x) \\to T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$ of $k$-vector spaces. \\end{enumerate}"}
+{"_id": "3478", "title": "formal-defos-lemma-k-linear-infaut", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories cofibred in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\varphi$ induces a $k$-linear map $\\text{Inf}_{x_0}(\\mathcal{F}) \\to \\text{Inf}_{\\varphi(x_0)}(\\mathcal{G})$."}
+{"_id": "3479", "title": "formal-defos-lemma-lifted-automorphisms-torsor", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a morphism lying over a surjective ring map $A' \\to A$ with kernel $I$ annihilated by $\\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to $\\mathcal{F}(k)$. Then $\\text{Inf}(x'/x)$ has a free and transitive action by $T_{\\text{id}_{x_0}} \\mathit{Aut}(x') \\otimes_k I = \\text{Inf}_{x_0}(\\mathcal{F}) \\otimes_k I$."}
+{"_id": "3480", "title": "formal-defos-lemma-infaut-trivial", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a morphism in $\\mathcal{F}$ lying over a surjective ring map. Let $x_0$ be a pushforward of $x$ to $\\mathcal{F}(k)$. If $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$ then $\\text{Inf}(x'/x) = 0$."}
+{"_id": "3481", "title": "formal-defos-lemma-infdef-trivial", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$ if and only if the natural morphism $\\mathcal{F}_{x_0} \\to \\overline{\\mathcal{F}_{x_0}}$ of categories cofibered in groupoids is an equivalence."}
+{"_id": "3482", "title": "formal-defos-lemma-deformation-categories-fiber-product-morphisms", "text": "Let $f : \\mathcal{H} \\to \\mathcal{F}$ and $g : \\mathcal{G} \\to \\mathcal{F}$ be $1$-morphisms of deformation categories. Then \\begin{enumerate} \\item $\\mathcal{W} = \\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$ is a deformation category, and \\item we have a $6$-term exact sequence of vector spaces $$ 0 \\to \\text{Inf}(\\mathcal{W}) \\to \\text{Inf}(\\mathcal{H}) \\oplus \\text{Inf}(\\mathcal{G}) \\to \\text{Inf}(\\mathcal{F}) \\to T\\mathcal{W} \\to T\\mathcal{H} \\oplus T\\mathcal{G} \\to T\\mathcal{F} $$ \\end{enumerate}"}
+{"_id": "3483", "title": "formal-defos-lemma-map-fibre-products-smooth", "text": "Let $\\mathcal{H}_1 \\to \\mathcal{G}$, $\\mathcal{H}_2 \\to \\mathcal{G}$, and $\\mathcal{G} \\to \\mathcal{F}$ be maps of categories cofibred in groupoids over $\\mathcal{C}_\\Lambda$. Assume \\begin{enumerate} \\item $\\mathcal{F}$ and $\\mathcal{G}$ are deformation categories, \\item $T\\mathcal{G} \\to T\\mathcal{F}$ is injective, and \\item $\\text{Inf}(\\mathcal{G}) \\to \\text{Inf}(\\mathcal{F})$ is surjective. \\end{enumerate} Then $\\mathcal{H}_1 \\times_\\mathcal{G} \\mathcal{H}_2 \\to \\mathcal{H}_1 \\times_\\mathcal{F} \\mathcal{H}_2$ is smooth."}
+{"_id": "3484", "title": "formal-defos-lemma-easy-check-smooth", "text": "Let $f : \\mathcal{F} \\to \\mathcal{G}$ be a map of deformation categories. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$ with image $y_0 \\in \\Ob(\\mathcal{G}(k))$. If \\begin{enumerate} \\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and \\item for every small extension $A' \\to A$ in $\\mathcal{C}_\\Lambda$ and $x \\in \\mathcal{F}(A)$ with image $y \\in \\mathcal{G}(A)$ if there is a lift of $y$ to $A'$, then there is a lift of $x$ to $A'$, \\end{enumerate} then $\\mathcal{F} \\to \\mathcal{G}$ is smooth (and vice versa)."}
+{"_id": "3486", "title": "formal-defos-lemma-groupoid-in-functors-prorep-equivalences", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. \\begin{enumerate} \\item $(U, R, s, t, c)$ is prorepresentable if and only if its completion is representable as a groupoid in functors on $\\widehat{\\mathcal{C}}_\\Lambda$. \\item $(U, R, s, t, c)$ is prorepresentable if and only if $U$ and $R$ are prorepresentable. \\end{enumerate}"}
+{"_id": "3487", "title": "formal-defos-lemma-smooth-quotient-morphism", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. The following are equivalent: \\begin{enumerate} \\item The groupoid in functors $(U, R, s, t, c)$ is smooth. \\item The morphism $s : R \\to U$ is smooth. \\item The morphism $t : R \\to U$ is smooth. \\item The quotient morphism $U \\to [U/R]$ is smooth. \\end{enumerate}"}
+{"_id": "3488", "title": "formal-defos-lemma-smooth-RS-groupoid-in-functors-quotient", "text": "Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$. Assume $U$ and $R$ satisfy (RS). Then $[U/R]$ satisfies (RS)."}
+{"_id": "3489", "title": "formal-defos-lemma-deformation-groupoid-quotient", "text": "Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$. Assume $U$ and $R$ are deformation functors. Then: \\begin{enumerate} \\item The quotient $[U/R]$ is a deformation category. \\item The tangent space of $[U/R]$ is $$ T[U/R] = \\Coker(ds-dt: TR \\to TU). $$ \\item The space of infinitesimal automorphisms of $[U/R]$ is $$ \\text{Inf}([U/R]) = \\Ker(ds \\oplus dt : TR \\to TU \\oplus TU). $$ \\end{enumerate}"}
+{"_id": "3490", "title": "formal-defos-lemma-presentation-construction", "text": "Let $\\mathcal{F}$ be category cofibered in groupoids over a category $\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}$. Define $R, s, t, c$ as follows: \\begin{enumerate} \\item $R : \\mathcal{C} \\to \\textit{Sets}$ is the functor $U \\times_{f, \\mathcal{F}, f} U$. \\item $t, s : R \\to U$ are the first and second projections, respectively. \\item $c : R \\times_{s, U, t} R \\to R$ is the morphism given by projection onto the first and last factors of $U \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$ under the canonical isomorphism $R \\times_{s, U, t} R \\to U \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$. \\end{enumerate} Then $(U, R, s, t, c)$ is a groupoid in functors on $\\mathcal{C}$."}
+{"_id": "3491", "title": "formal-defos-lemma-presentation-morphism", "text": "Let $\\mathcal{F}$ be category cofibered in groupoids over a category $\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on $\\mathcal{C}$ constructed from $f : U \\to \\mathcal{F}$ in Lemma \\ref{lemma-presentation-construction}. Then there is a natural morphism $[f] : [U/R] \\to \\mathcal{F}$ such that: \\begin{enumerate} \\item $[f]: [U/R] \\to \\mathcal{F}$ is fully faithful. \\item $[f]: [U/R] \\to \\mathcal{F}$ is an equivalence if and only if $f : U \\to \\mathcal{F}$ is essentially surjective. \\end{enumerate}"}
+{"_id": "3492", "title": "formal-defos-lemma-smooth-groupoid-in-functors-construction", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor. Let $f : U \\to \\mathcal{F}$ be a smooth morphism of categories cofibered in groupoids. Then: \\begin{enumerate} \\item If $(U, R, s, t, c)$ is the groupoid in functors on $\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in Lemma \\ref{lemma-presentation-construction}, then $(U, R, s, t, c)$ is smooth. \\item If $f : U(k) \\to \\mathcal{F}(k)$ is essentially surjective, then the morphism $[f] : [U/R] \\to \\mathcal{F}$ of Lemma \\ref{lemma-presentation-morphism} is an equivalence. \\end{enumerate}"}
+{"_id": "3493", "title": "formal-defos-lemma-deformation-functor-diagonal", "text": "Let $\\mathcal{F}$ be a deformation category. Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a deformation functor. Let $f: U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids. Then $U \\times_{f, \\mathcal{F}, f} U$ is a deformation functor with tangent space fitting into an exact sequence of $k$-vector spaces $$ 0 \\to \\text{Inf}(\\mathcal{F}) \\to T(U \\times_{f, \\mathcal{F}, f} U) \\to TU \\oplus TU $$"}
+{"_id": "3494", "title": "formal-defos-lemma-prorepresentable-groupoid-in-functors-construction", "text": "Let $\\mathcal{F}$ be a deformation category. Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a prorepresentable functor. Let $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids. Let $(U, R, s, t, c)$ be the groupoid in functors on $\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in Lemma \\ref{lemma-presentation-construction}. If $\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$, then $(U, R, s, t, c)$ is prorepresentable."}
+{"_id": "3495", "title": "formal-defos-lemma-characterize-minimal-groupoid-in-functors", "text": "Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$. \\begin{enumerate} \\item $(U, R, s, t, c)$ is normalized if and only if the morphism $U \\to [U/R]$ induces an isomorphism on tangent spaces, and \\item $(U, R, s, t, c)$ is minimal if and only if the kernel of $TU \\to T[U/R]$ is contained in the image of $\\text{Der}_\\Lambda(k, k) \\to TU$. \\end{enumerate}"}
+{"_id": "3496", "title": "formal-defos-lemma-surjective-morphism-prorepresentable-functor", "text": "Let $U: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a prorepresentable functor. Let $\\varphi : U \\to U$ be a morphism such that $d\\varphi : TU \\to TU$ is an isomorphism.  Then $\\varphi$ is an isomorphism."}
+{"_id": "3497", "title": "formal-defos-lemma-minimal-prorepresentable-groupoid-autoequivalence", "text": "Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$. If $\\varphi : [U/R] \\to [U/R]$ is an equivalence of categories cofibered in groupoids, then $\\varphi$ is an isomorphism."}
+{"_id": "3498", "title": "formal-defos-lemma-minimal-prorepresentable-groupoid-equivalence", "text": "Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth prorepresentable groupoids in functors on $\\mathcal{C}_\\Lambda$. If $\\varphi : [U/R] \\to [U'/R']$ is an equivalence of categories cofibered in groupoids, then $\\varphi$ is an isomorphism."}
+{"_id": "3499", "title": "formal-defos-lemma-minimal-groupoid-in-functors-construction", "text": "Let $\\mathcal{F}$ be a deformation category such that $\\dim_k T\\mathcal{F} <\\infty$ and $\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$. Then there exists a minimal versal formal object $\\xi$ of $\\mathcal{F}$. Say $\\xi$ lies over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. Let $U = \\underline{R}|_{\\mathcal{C}_\\Lambda}$. Let $f = \\underline{\\xi} : U \\to \\mathcal{F}$ be the associated morphism. Let $(U, R, s, t, c)$ be the groupoid in functors on $\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in Lemma \\ref{lemma-presentation-construction}. Then $(U, R, s, t, c)$ is a minimal smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$ and there is an equivalence $[U/R] \\to \\mathcal{F}$."}
+{"_id": "3500", "title": "formal-defos-lemma-minimal-presentations-equivalent", "text": "Let $\\mathcal{F}$ be category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Assume there exist presentations of $\\mathcal{F}$ by minimal smooth prorepresentable groupoids in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$. Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic."}
+{"_id": "3502", "title": "formal-defos-lemma-composition-homotopic", "text": "In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$ are formally homotopic and $g : S \\to S'$ is a morphism, then $g \\circ f_1$ and $g \\circ f_2$ are formally homotopic."}
+{"_id": "3503", "title": "formal-defos-lemma-versal-unique-up-to-homotopy", "text": "Let $\\mathcal{F}$ be a deformation category over $\\mathcal{C}_\\Lambda$ with $\\dim_k T\\mathcal{F} < \\infty$ and $\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$. Let $\\xi$ be a versal formal object lying over $R$. Let $\\eta$ be a formal object lying over $S$. Then any two maps $$ f, g : R \\to S $$ such that $f_*\\xi \\cong \\eta \\cong g_*\\xi$ are formally homotopic."}
+{"_id": "3504", "title": "formal-defos-lemma-homotopic-minimal-prime", "text": "In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$ are formally homotopic and $\\mathfrak p \\subset R$ is a minimal prime ideal, then $f_1(\\mathfrak p)S = f_2(\\mathfrak p)S$ as ideals."}
+{"_id": "3505", "title": "formal-defos-lemma-elementary-properties-change-of-field", "text": "With notation and assumptions as in Situation \\ref{situation-change-of-fields}. \\begin{enumerate} \\item We have $\\overline{\\mathcal{F}_{l/k}} = (\\overline{\\mathcal{F}})_{l/k}$. \\item If $\\mathcal{F}$ is a predeformation category, then $\\mathcal{F}_{l/k}$ is a predeformation category. \\item If $\\mathcal{F}$ satisfies (S1), then $\\mathcal{F}_{l/k}$ satisfies (S1). \\item If $\\mathcal{F}$ satisfies (S2), then $\\mathcal{F}_{l/k}$ satisfies (S2). \\item If $\\mathcal{F}$ satisfies (RS), then $\\mathcal{F}_{l/k}$ satisfies (RS). \\end{enumerate}"}
+{"_id": "3506", "title": "formal-defos-lemma-tangent-space-change-of-field", "text": "With notation and assumptions as in Situation \\ref{situation-change-of-fields}. Assume $\\mathcal{F}$ is a predeformation category and $\\overline{\\mathcal{F}}$ satisfies (S2). Then there is a canonical $l$-vector space isomorphism $$ T\\mathcal{F} \\otimes_k l \\longrightarrow T\\mathcal{F}_{l/k} $$ of tangent spaces."}
+{"_id": "3507", "title": "formal-defos-lemma-inf-aut-change-of-field", "text": "With notation and assumptions as in Situation \\ref{situation-change-of-fields}. Assume $\\mathcal{F}$ is a deformation category. Then there is a canonical $l$-vector space isomorphism $$ \\text{Inf}(\\mathcal{F}) \\otimes_k l \\longrightarrow \\text{Inf}(\\mathcal{F}_{l/k}) $$ of infinitesimal automorphism spaces."}
+{"_id": "3508", "title": "formal-defos-lemma-change-of-fields-smooth", "text": "With notation and assumptions as in Situation \\ref{situation-change-of-fields}. If $\\mathcal{F} \\to \\mathcal{G}$ is a smooth morphism of categories cofibred in groupoids over $\\mathcal{C}_{\\Lambda, k}$, then $\\mathcal{F}_{l/k} \\to \\mathcal{G}_{l/k}$ is a smooth morphism of categories cofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$."}
+{"_id": "3509", "title": "formal-defos-lemma-change-of-field-versal-ring", "text": "With notation and assumptions as in Situation \\ref{situation-change-of-fields}. Let $\\xi$ be a versal formal object for $\\mathcal{F}$ lying over $R \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, k})$. Then there exist \\begin{enumerate} \\item an $S \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, l})$ and a local $\\Lambda$-algebra homomorphism $R \\to S$ which is formally smooth in the $\\mathfrak m_S$-adic topology and induces the given field extension $l/k$ on residue fieds, and \\item a versal formal object of $\\mathcal{F}_{l/k}$ lying over $S$. \\end{enumerate}"}
+{"_id": "3586", "title": "adequate-lemma-adequate-finite-presentation", "text": "Let $A$ be a ring. Let $F$ be an adequate functor on $\\textit{Alg}_A$. If $B = \\colim B_i$ is a filtered colimit of $A$-algebras, then $F(B) = \\colim F(B_i)$."}
+{"_id": "3587", "title": "adequate-lemma-adequate-flat", "text": "Let $A$ be a ring. Let $F$ be an adequate functor on $\\textit{Alg}_A$. If $B \\to B'$ is flat, then $F(B) \\otimes_B B' \\to F(B')$ is an isomorphism."}
+{"_id": "3588", "title": "adequate-lemma-adequate-surjection-from-linear", "text": "Let $A$ be a ring. Let $F$ be an adequate functor on $\\textit{Alg}_A$. Then there exists a surjection $L \\to F$ with $L$ a direct sum of linearly adequate functors."}
+{"_id": "3589", "title": "adequate-lemma-flat-functor-split", "text": "Let $A$ be a ring. Let $F$ be a module-valued functor on $\\textit{Alg}_A$. Assume that for $B \\to B'$ flat the map $F(B) \\otimes_B B' \\to F(B')$ is an isomorphism. Let $B$ be a graded $A$-algebra. Then \\begin{enumerate} \\item $F(B) = \\bigoplus_{k \\in \\mathbf{Z}} F(B)^{(k)}$, and \\item the map $B \\to B_0 \\to B$ induces map $F(B) \\to F(B)$ whose image is contained in $F(B)^{(0)}$. \\end{enumerate}"}
+{"_id": "3590", "title": "adequate-lemma-lift-map", "text": "Let $A$ be a ring. Given a solid diagram $$ \\xymatrix{ 0 \\ar[r] & L \\ar[d]_\\varphi \\ar[r] & \\underline{A^{\\oplus n}} \\ar[r] \\ar@{..>}[ld] & \\underline{A^{\\oplus m}} \\\\ & \\underline{M} } $$ of module-valued functors on $\\textit{Alg}_A$ with exact row there exists a dotted arrow making the diagram commute."}
+{"_id": "3591", "title": "adequate-lemma-cokernel-into-module", "text": "Let $A$ be a ring. Let $\\varphi : F \\to \\underline{M}$ be a map of module-valued functors on $\\textit{Alg}_A$ with $F$ adequate. Then $\\Coker(\\varphi)$ is adequate."}
+{"_id": "3592", "title": "adequate-lemma-cokernel-adequate", "text": "\\begin{slogan} The cokernel of a map of adequate functors on the category of algebras over a ring is adequate. \\end{slogan} Let $A$ be a ring. Let $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$. Then $\\Coker(\\varphi)$ is adequate."}
+{"_id": "3593", "title": "adequate-lemma-kernel-adequate", "text": "Let $A$ be a ring. Let $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$. Then $\\Ker(\\varphi)$ is adequate."}
+{"_id": "3594", "title": "adequate-lemma-colimit-adequate", "text": "Let $A$ be a ring. An arbitrary direct sum of adequate functors on $\\textit{Alg}_A$ is adequate. A colimit of adequate functors is adequate."}
+{"_id": "3595", "title": "adequate-lemma-flat-linear-functor", "text": "Let $A$ be a ring. Let $F, G$ be module-valued functors on $\\textit{Alg}_A$. Let $\\varphi : F \\to G$ be a transformation of functors. Assume \\begin{enumerate} \\item $\\varphi$ is additive, \\item for every $A$-algebra $B$ and $\\xi \\in F(B)$ and unit $u \\in B^*$ we have $\\varphi(u\\xi) = u\\varphi(\\xi)$ in $G(B)$, and \\item for any flat ring map $B \\to B'$ we have $G(B) \\otimes_B B' = G(B')$. \\end{enumerate} Then $\\varphi$ is a morphism of module-valued functors."}
+{"_id": "3596", "title": "adequate-lemma-extension-adequate-key", "text": "Let $A$ be a ring. Let $0 \\to \\underline{M} \\to G \\to L \\to 0$ be a short exact sequence of module-valued functors on $\\textit{Alg}_A$ with $L$ linearly adequate. Then $G$ is adequate."}
+{"_id": "3597", "title": "adequate-lemma-extension-adequate", "text": "Let $A$ be a ring. Let $0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of module-valued functors on $\\textit{Alg}_A$. If $F$ and $H$ are adequate, so is $G$."}
+{"_id": "3598", "title": "adequate-lemma-base-change-adequate", "text": "Let $A \\to A'$ be a ring map. If $F$ is an adequate functor on $\\textit{Alg}_A$, then its restriction $F'$ to $\\textit{Alg}_{A'}$ is adequate too."}
+{"_id": "3599", "title": "adequate-lemma-pushforward-adequate", "text": "Let $A \\to A'$ be a ring map. If $F'$ is an adequate functor on $\\textit{Alg}_{A'}$, then the module-valued functor $F : B \\mapsto F'(A' \\otimes_A B)$ on $\\textit{Alg}_A$ is adequate too."}
+{"_id": "3600", "title": "adequate-lemma-adequate-product", "text": "Let $A = A_1 \\times \\ldots \\times A_n$ be a product of rings. An adequate functor over $A$ is the same thing as a sequence $F_1, \\ldots, F_n$ of adequate functors $F_i$ over $A_i$."}
+{"_id": "3601", "title": "adequate-lemma-adequate-descent", "text": "Let $A \\to A'$ be a ring map and let $F$ be a module-valued functor on $\\textit{Alg}_A$ such that \\begin{enumerate} \\item the restriction $F'$ of $F$ to the category of $A'$-algebras is adequate, and \\item for any $A$-algebra $B$ the sequence $$ 0 \\to F(B) \\to F(B \\otimes_A A') \\to F(B \\otimes_A A' \\otimes_A A') $$ is exact. \\end{enumerate} Then $F$ is adequate."}
+{"_id": "3602", "title": "adequate-lemma-adjoint", "text": "Let $A$ be a ring. For every module-valued functor $F$ on $\\textit{Alg}_A$ there exists a morphism $Q(F) \\to F$ of module-valued functors on $\\textit{Alg}_A$ such that (1) $Q(F)$ is adequate and (2) for every adequate functor $G$ the map $\\Hom(G, Q(F)) \\to \\Hom(G, F)$ is a bijection."}
+{"_id": "3603", "title": "adequate-lemma-enough-injectives", "text": "Let $A$ be a ring. Denote $\\mathcal{P}$ the category of module-valued functors on $\\textit{Alg}_A$ and $\\mathcal{A}$ the category of adequate functors on $\\textit{Alg}_A$. Denote $i : \\mathcal{A} \\to \\mathcal{P}$ the inclusion functor. Denote $Q : \\mathcal{P} \\to \\mathcal{A}$ the construction of Lemma \\ref{lemma-adjoint}. Then \\begin{enumerate} \\item $i$ is fully faithful, exact, and its image is a weak Serre subcategory, \\item $\\mathcal{P}$ has enough injectives, \\item the functor $Q$ is a right adjoint to $i$ hence left exact, \\item $Q$ transforms injectives into injectives, \\item $\\mathcal{A}$ has enough injectives. \\end{enumerate}"}
+{"_id": "3604", "title": "adequate-lemma-tangent-functor", "text": "Let $A$ be a ring. Let $F$ be a module valued functor. For every $B \\in \\Ob(\\textit{Alg}_A)$ and $B$-module $N$ there is a canonical decomposition $$ F(B[N]) = F(B) \\oplus TF(B, N) $$ characterized by the following properties \\begin{enumerate} \\item $TF(B, N) = \\Ker(F(B[N]) \\to F(B))$, \\item there is a $B$-module structure $TF(B, N)$ compatible with $B[N]$-module structure on $F(B[N])$, \\item $TF$ is a functor from the category of pairs $(B, N)$, \\item \\label{item-mult-map} there are canonical maps $N \\otimes_B F(B) \\to TF(B, N)$ inducing a transformation between functors defined on the category of pairs $(B, N)$, \\item $TF(B, 0) = 0$ and the map $TF(B, N) \\to TF(B, N')$ is zero when $N \\to N'$ is the zero map. \\end{enumerate}"}
+{"_id": "3605", "title": "adequate-lemma-tangent-injective", "text": "Let $A$ be a ring. Let $I$ be an injective object of the category of module-valued functors. Then for any $B \\in \\Ob(\\textit{Alg}_A)$ and short exact sequence $0 \\to N_1 \\to N \\to N_2 \\to 0$ of $B$-modules the sequence $$ TI(B, N_1) \\to TI(B, N) \\to TI(B, N_2) \\to 0 $$ is exact."}
+{"_id": "3606", "title": "adequate-lemma-exactness-implies", "text": "Let $A$ be a ring. Let $F$ be a module-valued functor such that for any $B \\in \\Ob(\\textit{Alg}_A)$ the functor $TF(B, -)$ on $B$-modules transforms a short exact sequence of $B$-modules into a right exact sequence. Then \\begin{enumerate} \\item $TF(B, N_1 \\oplus N_2) = TF(B, N_1) \\oplus TF(B, N_2)$, \\item there is a second functorial $B$-module structure on $TF(B, N)$ defined by setting $x \\cdot b = TF(B, b\\cdot 1_N)(x)$ for $x \\in TF(B, N)$ and $b \\in B$, \\item \\label{item-mult-map-linear} the canonical map $N \\otimes_B F(B) \\to TF(B, N)$ of Lemma \\ref{lemma-tangent-functor} is $B$-linear also with respect to the second $B$-module structure, \\item \\label{item-tangent-right-exact} given a finitely presented $B$-module $N$ there is a canonical isomorphism $TF(B, B) \\otimes_B N \\to TF(B, N)$ where the tensor product uses the second $B$-module structure on $TF(B, B)$. \\end{enumerate}"}
+{"_id": "3607", "title": "adequate-lemma-exactness-permanence", "text": "Let $A$ be a ring. For $F$ a module-valued functor on $\\textit{Alg}_A$ say $(*)$ holds if for all $B \\in \\Ob(\\textit{Alg}_A)$ the functor $TF(B, -)$ on $B$-modules transforms a short exact sequence of $B$-modules into a right exact sequence. Let $0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of module-valued functors on $\\textit{Alg}_A$. \\begin{enumerate} \\item If $(*)$ holds for $F, G$ then $(*)$ holds for $H$. \\item If $(*)$ holds for $F, H$ then $(*)$ holds for $G$. \\item If $H' \\to H$ is morphism of module-valued functors on $\\textit{Alg}_A$ and $(*)$ holds for $F$, $G$, $H$, and $H'$, then $(*)$ holds for $G \\times_H H'$. \\end{enumerate}"}
+{"_id": "3608", "title": "adequate-lemma-ext-group-zero-key", "text": "Let $A$ be a ring. Let $M$, $P$ be $A$-modules with $P$ of finite presentation. Then $\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M}) = 0$ for $i > 0$ where $\\mathcal{P}$ is the category of module-valued functors on $\\textit{Alg}_A$."}
+{"_id": "3609", "title": "adequate-lemma-ext-group-zero", "text": "Let $A$ be a ring. Let $M$ be an $A$-module. Let $L$ be a linearly adequate functor on $\\textit{Alg}_A$. Then $\\Ext^i_\\mathcal{P}(L, \\underline{M}) = 0$ for $i > 0$ where $\\mathcal{P}$ is the category of module-valued functors on $\\textit{Alg}_A$."}
+{"_id": "3610", "title": "adequate-lemma-RQ-zero", "text": "With notation as in Lemma \\ref{lemma-enough-injectives} we have $R^pQ(F) = 0$ for all $p > 0$ and any adequate functor $F$."}
+{"_id": "3611", "title": "adequate-lemma-adequate-local", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$. For any affine scheme $\\Spec(A)$ over $S$ the functor $F_{\\mathcal{F}, A}$ is adequate."}
+{"_id": "3612", "title": "adequate-lemma-adequate-affine", "text": "Let $S = \\Spec(A)$ be an affine scheme. The category of adequate $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$ is equivalent to the category of adequate module-valued functors on $\\textit{Alg}_A$."}
+{"_id": "3613", "title": "adequate-lemma-pullback-adequate", "text": "Let $f : T \\to S$ be a morphism of schemes. The pullback $f^*\\mathcal{F}$ of an adequate $\\mathcal{O}$-module $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is an adequate $\\mathcal{O}$-module on $(\\Sch/T)_\\tau$."}
+{"_id": "3614", "title": "adequate-lemma-adequate-characterize", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is adequate, \\item there exists an affine open covering $S = \\bigcup S_i$ and maps of quasi-coherent $\\mathcal{O}_{S_i}$-modules $\\mathcal{G}_i \\to \\mathcal{H}_i$ such that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the kernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$ \\item there exists a $\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ and maps of $\\mathcal{O}_{S_i}$-quasi-coherent modules $\\mathcal{G}_i \\to \\mathcal{H}_i$ such that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the kernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$, \\item there exists a $\\tau$-covering $\\{f_i : S_i \\to S\\}_{i \\in I}$ such that each $f_i^*\\mathcal{F}$ is adequate, \\item for any affine scheme $U$ over $S$ the restriction $\\mathcal{F}|_{(\\Sch/U)_\\tau}$ is the kernel of a map $\\mathcal{G}^a \\to \\mathcal{H}^a$ of quasi-coherent $\\mathcal{O}_U$-modules. \\end{enumerate}"}
+{"_id": "3615", "title": "adequate-lemma-adequate-fpqc", "text": "Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$. For any surjective flat morphism $\\Spec(B) \\to \\Spec(A)$ of affines over $S$ the extended {\\v C}ech complex $$ 0 \\to \\mathcal{F}(\\Spec(A)) \\to \\mathcal{F}(\\Spec(B)) \\to \\mathcal{F}(\\Spec(B \\otimes_A B)) \\to \\ldots $$ is exact. In particular $\\mathcal{F}$ satisfies the sheaf condition for fpqc coverings, and is a sheaf of $\\mathcal{O}$-modules on $(\\Sch/S)_{fppf}$."}
+{"_id": "3616", "title": "adequate-lemma-same-cohomology-adequate", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$. \\begin{enumerate} \\item The restriction $\\mathcal{F}|_{S_{Zar}}$ is a quasi-coherent $\\mathcal{O}_S$-module on the scheme $S$. \\item The restriction $\\mathcal{F}|_{S_\\etale}$ is the quasi-coherent module associated to $\\mathcal{F}|_{S_{Zar}}$. \\item For any affine scheme $U$ over $S$ we have $H^q(U, \\mathcal{F}) = 0$ for all $q > 0$. \\item There is a canonical isomorphism $$ H^q(S, \\mathcal{F}|_{S_{Zar}}) = H^q((\\Sch/S)_\\tau, \\mathcal{F}). $$ \\end{enumerate}"}
+{"_id": "3617", "title": "adequate-lemma-sheafification-adequate", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$. If for every affine scheme $\\Spec(A)$ over $S$ the functor $F_{\\mathcal{F}, A}$ is adequate, then the sheafification of $\\mathcal{F}$ is an adequate $\\mathcal{O}$-module."}
+{"_id": "3618", "title": "adequate-lemma-abelian-adequate", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item The category $\\textit{Adeq}(\\mathcal{O})$ is abelian. \\item The functor $\\textit{Adeq}(\\mathcal{O}) \\to \\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$ is exact. \\item If $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ is a short exact sequence of $\\mathcal{O}$-modules and $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are adequate, then $\\mathcal{F}_2$ is adequate. \\item The category $\\textit{Adeq}(\\mathcal{O})$ has colimits and $\\textit{Adeq}(\\mathcal{O}) \\to \\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$ commutes with them. \\end{enumerate}"}
+{"_id": "3619", "title": "adequate-lemma-direct-image-adequate", "text": "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes. For any adequate $\\mathcal{O}_T$-module on $(\\Sch/T)_\\tau$ the pushforward $f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$ are adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$."}
+{"_id": "3620", "title": "adequate-lemma-parasitic-adequate", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$. The following are equivalent: \\begin{enumerate} \\item $v\\mathcal{F} = 0$, \\item $\\mathcal{F}$ is parasitic, \\item $\\mathcal{F}$ is parasitic for the $\\tau$-topology, \\item $\\mathcal{F}(U) = 0$ for all $U \\subset S$ open, and \\item there exists an affine open covering $S = \\bigcup U_i$ such that $\\mathcal{F}(U_i) = 0$ for all $i$. \\end{enumerate}"}
+{"_id": "3625", "title": "adequate-lemma-right-adjoint-adequate", "text": "Let $U = \\Spec(A)$ be an affine scheme. The inclusion functor $$ \\textit{Adeq}(\\mathcal{O}) \\to \\textit{Mod}((\\Sch/U)_\\tau, \\mathcal{O}) $$ has a right adjoint $A$\\footnote{This is the ``adequator''.}. Moreover, the adjunction mapping $A(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism for every adequate module $\\mathcal{F}$."}
+{"_id": "3626", "title": "adequate-lemma-RA-zero", "text": "Let $U = \\Spec(A)$ be an affine scheme. For any object $\\mathcal{F}$ of $\\textit{Adeq}(\\mathcal{O})$ we have $R^pA(\\mathcal{F}) = 0$ for all $p > 0$ where $A$ is as in Lemma \\ref{lemma-right-adjoint-adequate}."}
+{"_id": "3627", "title": "adequate-lemma-bounded-below", "text": "If $U = \\Spec(A)$ is an affine scheme, then the bounded below version \\begin{equation} \\label{equation-compare-bounded-adequate} D^+(\\textit{Adeq}(\\mathcal{O})) \\longrightarrow D^+_{\\textit{Adeq}}(\\mathcal{O}) \\end{equation} of the functor above is an equivalence."}
+{"_id": "3628", "title": "adequate-lemma-ext-adequate", "text": "Let $U = \\Spec(A)$ be an affine scheme. Let $\\mathcal{F}$ and $\\mathcal{G}$ be adequate $\\mathcal{O}$-modules. For any $i \\geq 0$ the natural map $$ \\Ext^i_{\\textit{Adeq}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G}) \\longrightarrow \\Ext^i_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G}) $$ is an isomorphism."}
+{"_id": "3629", "title": "adequate-lemma-pure-projective", "text": "Let $A$ be a ring. \\begin{enumerate} \\item A module is pure projective if and only if it is a direct summand of a direct sum of finitely presented $A$-modules. \\item For any module $M$ there exists a universally exact sequence $0 \\to N \\to P \\to M \\to 0$ with $P$ pure projective. \\end{enumerate}"}
+{"_id": "3630", "title": "adequate-lemma-pure-injective", "text": "Let $A$ be a ring. For any $A$-module $M$ set $M^\\vee = \\Hom_\\mathbf{Z}(M, \\mathbf{Q}/\\mathbf{Z})$. \\begin{enumerate} \\item For any $A$-module $M$ the $A$-module $M^\\vee$ is pure injective. \\item An $A$-module $I$ is pure injective if and only if the map $I \\to (I^\\vee)^\\vee$ splits. \\item For any module $M$ there exists a universally exact sequence $0 \\to M \\to I \\to N \\to 0$ with $I$ pure injective. \\end{enumerate}"}
+{"_id": "3631", "title": "adequate-lemma-split-universally-exact-sequence", "text": "Let $A$ be a ring. \\begin{enumerate} \\item Let $L \\to M \\to N$ be a universally exact sequence of $A$-modules. Let $K = \\Im(M \\to N)$. Then $K \\to N$ is universally injective. \\item Any universally exact complex can be split into universally exact short exact sequences. \\end{enumerate}"}
+{"_id": "3632", "title": "adequate-lemma-pure-projective-resolutions", "text": "Let $A$ be a ring. \\begin{enumerate} \\item Any $A$-module has a pure projective resolution. \\end{enumerate} Let $M \\to N$ be a map of $A$-modules. Let $P_\\bullet \\to M$ be a pure projective resolution and let $N_\\bullet \\to N$ be a universally exact resolution. \\begin{enumerate} \\item[(2)] There exists a map of complexes $P_\\bullet \\to N_\\bullet$ inducing the given map $$ M = \\Coker(P_1 \\to P_0) \\to \\Coker(N_1 \\to N_0) = N $$ \\item[(3)] two maps $\\alpha, \\beta : P_\\bullet \\to N_\\bullet$ inducing the same map $M \\to N$ are homotopic. \\end{enumerate}"}
+{"_id": "3634", "title": "adequate-lemma-facts-pext", "text": "Let $A$ be a ring. \\begin{enumerate} \\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $N$ is pure injective, \\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $M$ is pure projective, in particular if $M$ is an $A$-module of finite presentation, \\item $\\text{Pext}^i_A(M, N)$ is also the $i$th cohomology module of the complex $\\Hom_A(P_\\bullet, N)$ where $P_\\bullet$ is a pure projective resolution of $M$. \\end{enumerate}"}
+{"_id": "3635", "title": "adequate-lemma-pure-injective-injective-adequate", "text": "Let $A$ be a ring. Let $\\mathcal{A}$ be the category of adequate functors on $\\textit{Alg}_A$. The injective objects of $\\mathcal{A}$ are exactly the functors $\\underline{I}$ where $I$ is a pure injective $A$-module."}
+{"_id": "3650", "title": "spaces-topologies-lemma-zariski", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is a Zariski covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering and for each $i$ we have a Zariski covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a Zariski covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a Zariski covering. \\end{enumerate}"}
+{"_id": "3651", "title": "spaces-topologies-lemma-zariski-etale", "text": "Any Zariski covering is an \\'etale covering."}
+{"_id": "3652", "title": "spaces-topologies-lemma-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is a \\'etale covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering and for each $i$ we have a \\'etale covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a \\'etale covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a \\'etale covering. \\end{enumerate}"}
+{"_id": "3653", "title": "spaces-topologies-lemma-etale-dominates-smooth", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\{X_i \\to X\\}_{i \\in I}$ be a smooth covering of $X$. Then there exists an \\'etale covering $\\{U_j \\to X\\}_{j \\in J}$ of $X$ which refines $\\{X_i \\to X\\}_{i \\in I}$."}
+{"_id": "3654", "title": "spaces-topologies-lemma-put-in-T-etale", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of $(\\textit{Spaces}/S)_\\etale$. The inclusion functor $Y_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$ is cocontinuous and induces a morphism of topoi $$ i_f : \\Sh(Y_\\etale) \\longrightarrow \\Sh((\\textit{Spaces}/X)_\\etale) $$ For a sheaf $\\mathcal{G}$ on $(\\textit{Spaces}/X)_\\etale$ we have the formula $(i_f^{-1}\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers."}
+{"_id": "3655", "title": "spaces-topologies-lemma-at-the-bottom-etale", "text": "Let $S$ be a scheme. Let $X$ be an object of $(\\textit{Spaces}/S)_\\etale$. The inclusion functor $X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$ satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \\pi_X : (\\textit{Spaces}/X)_\\etale \\longrightarrow X_{spaces, \\etale} $$ and a morphism of topoi $$ i_X : \\Sh(X_\\etale) \\longrightarrow \\Sh((\\textit{Spaces}/X)_\\etale) $$ such that $\\pi_X \\circ i_X = \\text{id}$. Moreover, $i_X = i_{\\text{id}_X}$ with $i_{\\text{id}_X}$ as in Lemma \\ref{lemma-put-in-T-etale}. In particular the functor $i_X^{-1} = \\pi_{X, *}$ is described by the rule $i_X^{-1}(\\mathcal{G})(U/X) = \\mathcal{G}(U/X)$."}
+{"_id": "3656", "title": "spaces-topologies-lemma-morphism-big-etale", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in $(\\textit{Spaces}/S)_\\etale$. The functor $$ u : (\\textit{Spaces}/Y)_\\etale \\longrightarrow (\\textit{Spaces}/X)_\\etale, \\quad V/Y \\longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\textit{Spaces}/X)_\\etale \\longrightarrow (\\textit{Spaces}/Y)_\\etale, \\quad (U \\to X) \\longmapsto (U \\times_X Y \\to Y). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\textit{Spaces}/Y)_\\etale) \\longrightarrow \\Sh((\\textit{Spaces}/X)_\\etale) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$. We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "3657", "title": "spaces-topologies-lemma-morphism-big-small-etale", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in $(\\textit{Spaces}/S)_\\etale$. \\begin{enumerate} \\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in Lemma \\ref{lemma-put-in-T-etale} and $i_T$ as in Lemma \\ref{lemma-at-the-bottom-etale}. \\item The functor $X_{spaces, \\etale} \\to T_{spaces, \\etale}$, $(U \\to X) \\mapsto (U \\times_X Y \\to Y)$ is continuous and induces a morphism of sites $$ f_{spaces, \\etale} : Y_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale} $$ The corresponding morphism of small \\'etale topoi is denoted $$ f_{small} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale) $$ We have $f_{small, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$. \\item We have a commutative diagram of morphisms of sites $$ \\xymatrix{ Y_{spaces, \\etale} \\ar[d]_{f_{spaces, \\etale}} & (\\textit{Spaces}/Y)_\\etale \\ar[d]^{f_{big}} \\ar[l]^-{\\pi_Y}\\\\ X_{spaces, \\etale} & (\\textit{Spaces}/X)_\\etale \\ar[l]_-{\\pi_X} } $$ so that $f_{small} \\circ \\pi_Y = \\pi_X \\circ f_{big}$ as morphisms of topoi. \\item We have $f_{small} = \\pi_X \\circ f_{big} \\circ i_Y = \\pi_X \\circ i_f$. \\end{enumerate}"}
+{"_id": "3660", "title": "spaces-topologies-lemma-zariski-etale-smooth", "text": "Any \\'etale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering."}
+{"_id": "3661", "title": "spaces-topologies-lemma-smooth", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is a smooth covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a smooth covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a smooth covering. \\end{enumerate}"}
+{"_id": "3662", "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic", "text": "Any smooth covering is a syntomic covering, and a fortiori, any \\'etale or Zariski covering is a syntomic covering."}
+{"_id": "3663", "title": "spaces-topologies-lemma-syntomic", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is a syntomic covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a syntomic covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a syntomic covering. \\end{enumerate}"}
+{"_id": "3664", "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf", "text": "Any syntomic covering is an fppf covering, and a fortiori, any smooth, \\'etale, or Zariski covering is an fppf covering."}
+{"_id": "3665", "title": "spaces-topologies-lemma-fppf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is an fppf covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fppf covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fppf covering. \\end{enumerate}"}
+{"_id": "3666", "title": "spaces-topologies-lemma-refine-fppf-schemes", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an fppf covering of $X$. Then there exists a refinement $\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an fppf covering such that each $T_i$ is a scheme."}
+{"_id": "3667", "title": "spaces-topologies-lemma-fppf-covering-surjective", "text": "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic spaces over $S$. Then the map of sheaves $$ \\coprod X_i \\longrightarrow X $$ is surjective."}
+{"_id": "3668", "title": "spaces-topologies-lemma-morphism-big-fppf", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. The functor $$ u : (\\textit{Spaces}/Y)_{fppf} \\longrightarrow (\\textit{Spaces}/X)_{fppf}, \\quad V/Y \\longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\textit{Spaces}/X)_{fppf} \\longrightarrow (\\textit{Spaces}/Y)_{fppf}, \\quad (U \\to Y) \\longmapsto (U \\times_X Y \\to Y). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\textit{Spaces}/Y)_{fppf}) \\longrightarrow \\Sh((\\textit{Spaces}/X)_{fppf}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$. We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "3670", "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph", "text": "Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, \\'etale or Zariski covering is a ph covering."}
+{"_id": "3671", "title": "spaces-topologies-lemma-surjective-proper-ph", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism of algebraic spaces over $S$. Then $\\{Y \\to X\\}$ is a ph covering."}
+{"_id": "3673", "title": "spaces-topologies-lemma-characterize-sheaf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a presheaf on $(\\textit{Spaces}/X)_{ph}$. Then $\\mathcal{F}$ is a sheaf if and only if \\begin{enumerate} \\item $\\mathcal{F}$ satisfies the sheaf condition for \\'etale coverings, and \\item if $f : V \\to U$ is a proper surjective morphism of $(\\textit{Spaces}/X)_{ph}$, then $\\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$. \\end{enumerate}"}
+{"_id": "3674", "title": "spaces-topologies-lemma-morphism-big-ph", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. The functor $$ u : (\\textit{Spaces}/Y)_{ph} \\longrightarrow (\\textit{Spaces}/X)_{ph}, \\quad V/Y \\longmapsto V/X $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\textit{Spaces}/X)_{ph} \\longrightarrow (\\textit{Spaces}/Y)_{ph}, \\quad (U \\to Y) \\longmapsto (U \\times_X Y \\to Y). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\textit{Spaces}/Y)_{ph}) \\longrightarrow \\Sh((\\textit{Spaces}/X)_{ph}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$. We have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "3676", "title": "spaces-topologies-lemma-cech-enough", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $P$ be a property of objects in $(\\textit{Spaces}/X)_{fppf}$ such that whenever $\\{U_i \\to U\\}$ is a covering in $(\\textit{Spaces}/X)_{fppf}$, then $$ P(U_{i_0} \\times_U \\ldots \\times_U U_{i_p}) \\text{ for all } p \\geq 0,\\ i_0, \\ldots, i_p \\in I \\Rightarrow P(U) $$ If $P(U)$ for all $U$ affine and flat, locally of finite presentation over $X$, then $P(X)$."}
+{"_id": "3678", "title": "spaces-topologies-lemma-fpqc", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is an fpqc covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering and for each $i$ we have an fpqc covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fpqc covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fpqc covering. \\end{enumerate}"}
+{"_id": "3679", "title": "spaces-topologies-lemma-recognize-fpqc-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a family of morphisms of algebraic spaces with target $X$. Let $U \\to X$ be a surjective \\'etale morphism from a scheme towards $X$. Then $\\{f_i : X_i \\to X\\}_{i \\in I}$ is an fpqc covering of $X$ if and only if $\\{U \\times_X X_i \\to U\\}_{i \\in I}$ is an fpqc covering of $U$."}
+{"_id": "3680", "title": "spaces-topologies-lemma-refine-fpqc-schemes", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an fpqc covering of $X$. Then there exists a refinement $\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an fpqc covering such that each $T_i$ is a scheme."}
+{"_id": "3697", "title": "proetale-lemma-spectral-split", "text": "Let $X$ be a spectral space. Let $X_0 \\subset X$ be the set of closed points. The following are equivalent \\begin{enumerate} \\item Every open covering of $X$ can be refined by a finite disjoint union decomposition $X = \\coprod U_i$ with $U_i$ open and closed in $X$. \\item The composition $X_0 \\to X \\to \\pi_0(X)$ is bijective. \\end{enumerate} Moreover, if $X_0$ is closed in $X$ and every point of $X$ specializes to a unique point of $X_0$, then these conditions are satisfied."}
+{"_id": "3698", "title": "proetale-lemma-closed-subspace-w-local", "text": "Let $X$ be a w-local spectral space. If $Y \\subset X$ is closed, then $Y$ is w-local."}
+{"_id": "3699", "title": "proetale-lemma-silly", "text": "Let $X$ be a spectral space. Let $$ \\xymatrix{ Y \\ar[r] \\ar[d] & T \\ar[d] \\\\ X \\ar[r] & \\pi_0(X) } $$ be a cartesian diagram in the category of topological spaces with $T$ profinite. Then $Y$ is spectral and $T = \\pi_0(Y)$. If moreover $X$ is w-local, then $Y$ is w-local, $Y \\to X$ is w-local, and the set of closed points of $Y$ is the inverse image of the set of closed points of $X$."}
+{"_id": "3700", "title": "proetale-lemma-base-change-local-isomorphism", "text": "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$ be the base change of $B$. \\begin{enumerate} \\item If $A \\to B$ is a local isomorphism, then $A' \\to B'$ is a local isomorphism. \\item If $A \\to B$ identifies local rings, then $A' \\to B'$ identifies local rings. \\end{enumerate}"}
+{"_id": "3701", "title": "proetale-lemma-composition-local-isomorphism", "text": "Let $A \\to B$ and $B \\to C$ be ring maps. \\begin{enumerate} \\item If $A \\to B$ and $B \\to C$ are local isomorphisms, then $A \\to C$ is a local isomorphism. \\item If $A \\to B$ and $B \\to C$ identify local rings, then $A \\to C$ identifies local rings. \\end{enumerate}"}
+{"_id": "3702", "title": "proetale-lemma-local-isomorphism-permanence", "text": "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism. \\begin{enumerate} \\item If $A \\to B$ and $A \\to C$ are local isomorphisms, then $B \\to C$ is a local isomorphism. \\item If $A \\to B$ and $A \\to C$ identify local rings, then $B \\to C$ identifies local rings. \\end{enumerate}"}
+{"_id": "3703", "title": "proetale-lemma-local-isomorphism-implies", "text": "Let $A \\to B$ be a local isomorphism. Then \\begin{enumerate} \\item $A \\to B$ is \\'etale, \\item $A \\to B$ identifies local rings, \\item $A \\to B$ is quasi-finite. \\end{enumerate}"}
+{"_id": "3704", "title": "proetale-lemma-structure-local-isomorphism", "text": "Let $A \\to B$ be a local isomorphism. Then there exist $n \\geq 0$, $g_1, \\ldots, g_n \\in B$, $f_1, \\ldots, f_n \\in A$ such that $(g_1, \\ldots, g_n) = B$ and $A_{f_i} \\cong B_{g_i}$."}
+{"_id": "3705", "title": "proetale-lemma-fully-faithful-spaces-over-X", "text": "Let $p : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$ and $q : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be morphisms of locally ringed spaces. If $\\mathcal{O}_Y = p^{-1}\\mathcal{O}_X$, then $$ \\Mor_{\\text{LRS}/(X, \\mathcal{O}_X)}((Z, \\mathcal{O}_Z), (Y, \\mathcal{O}_Y)) \\longrightarrow \\Mor_{\\textit{Top}/X}(Z, Y),\\quad (f, f^\\sharp) \\longmapsto f $$ is bijective. Here $\\text{LRS}/(X, \\mathcal{O}_X)$ is the category of locally ringed spaces over $X$ and $\\textit{Top}/X$ is the category of topological spaces over $X$."}
+{"_id": "3706", "title": "proetale-lemma-local-isomorphism-fully-faithful", "text": "Let $A$ be a ring. Set $X = \\Spec(A)$. The functor $$ B \\longmapsto \\Spec(B) $$ from the category of $A$-algebras $B$ such that $A \\to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful."}
+{"_id": "3707", "title": "proetale-lemma-base-change-ind-zariski", "text": "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$ be the base change of $B$. If $A \\to B$ is ind-Zariski, then $A' \\to B'$ is ind-Zariski."}
+{"_id": "3708", "title": "proetale-lemma-composition-ind-zariski", "text": "Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$ are ind-Zariski, then $A \\to C$ is ind-Zariski."}
+{"_id": "3709", "title": "proetale-lemma-ind-zariski-permanence", "text": "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism. If $A \\to B$ and $A \\to C$ are ind-Zariski, then $B \\to C$ is ind-Zariski."}
+{"_id": "3710", "title": "proetale-lemma-ind-ind-zariski", "text": "A filtered colimit of ind-Zariski $A$-algebras is ind-Zariski over $A$."}
+{"_id": "3711", "title": "proetale-lemma-ind-zariski-implies", "text": "Let $A \\to B$ be ind-Zariski. Then $A \\to B$ identifies local rings,"}
+{"_id": "3712", "title": "proetale-lemma-localization", "text": "Let $A$ be a ring. Set $X = \\Spec(A)$. Let $Z \\subset X$ be a locally closed subscheme which is of the form $D(f) \\cap V(I)$ for some $f \\in A$ and ideal $I \\subset A$. Then \\begin{enumerate} \\item there exists a multiplicative subset $S \\subset A$ such that $\\Spec(S^{-1}A)$ maps by a homeomorphism to the set of points of $X$ specializing to $Z$, \\item the $A$-algebra $A_Z^\\sim = S^{-1}A$ depends only on the underlying locally closed subset $Z \\subset X$, \\item $Z$ is a closed subscheme of $\\Spec(A_Z^\\sim)$, \\end{enumerate} If $A \\to A'$ is a ring map and $Z' \\subset X' = \\Spec(A')$ is a locally closed subscheme of the same form which maps into $Z$, then there is a unique $A$-algebra map $A_Z^\\sim \\to (A')_{Z'}^\\sim$."}
+{"_id": "3713", "title": "proetale-lemma-refine", "text": "Let $X = \\Spec(A)$ as above. Given any finite stratification $X = \\coprod T_i$ by constructible subsets, there exists a finite subset $E \\subset A$ such that the stratification (\\ref{equation-stratify}) refines $X = \\coprod T_i$."}
+{"_id": "3714", "title": "proetale-lemma-make-w-local", "text": "Let $X = \\Spec(A)$ be an affine scheme. With $A \\to A_w$, $X_w = \\Spec(A_w)$, and $Z \\subset X_w$ as above. \\begin{enumerate} \\item $A \\to A_w$ is ind-Zariski and faithfully flat, \\item $X_w \\to X$ induces a bijection $Z \\to X$, \\item $Z$ is the set of closed points of $X_w$, \\item $Z$ is a reduced scheme, and \\item every point of $X_w$ specializes to a unique point of $Z$. \\end{enumerate} In particular, $X_w$ is w-local (Definition \\ref{definition-w-local})."}
+{"_id": "3715", "title": "proetale-lemma-universal", "text": "Let $A$ be a ring. Let $A \\to A_w$ be the ring map constructed in Lemma \\ref{lemma-make-w-local}. For any ring map $A \\to B$ such that $\\Spec(B)$ is w-local, there is a unique factorization $A \\to A_w \\to B$ such that $\\Spec(B) \\to \\Spec(A_w)$ is w-local."}
+{"_id": "3716", "title": "proetale-lemma-profinite-goes-up", "text": "Let $A$ be a ring such that $\\Spec(A)$ is profinite. Let $A \\to B$ be a ring map. Then $\\Spec(B)$ is profinite in each of the following cases: \\begin{enumerate} \\item if $\\mathfrak q,\\mathfrak q' \\subset B$ lie over the same prime of $A$, then neither $\\mathfrak q \\subset \\mathfrak q'$, nor $\\mathfrak q' \\subset \\mathfrak q$, \\item $A \\to B$ induces algebraic extensions of residue fields, \\item $A \\to B$ is a local isomorphism, \\item $A \\to B$ identifies local rings, \\item $A \\to B$ is weakly \\'etale, \\item $A \\to B$ is quasi-finite, \\item $A \\to B$ is unramified, \\item $A \\to B$ is \\'etale, \\item $B$ is a filtered colimit of $A$-algebras as in (1) -- (8), \\item etc. \\end{enumerate}"}
+{"_id": "3717", "title": "proetale-lemma-localize-along-closed-profinite", "text": "Let $A$ be a ring. Let $V(I) \\subset \\Spec(A)$ be a closed subset which is a profinite topological space. Then there exists an ind-Zariski ring map $A \\to B$ such that $\\Spec(B)$ is w-local, the set of closed points is $V(IB)$, and $A/I \\cong B/IB$."}
+{"_id": "3718", "title": "proetale-lemma-w-local-algebraic-residue-field-extensions", "text": "Let $A$ be a ring such that $X = \\Spec(A)$ is w-local. Let $I \\subset A$ be the radical ideal cutting out the set $X_0$ of closed points in $X$. Let $A \\to B$ be a ring map inducing algebraic extensions on residue fields at primes. Then \\begin{enumerate} \\item every point of $Z = V(IB)$ is a closed point of $\\Spec(B)$, \\item there exists an ind-Zariski ring map $B \\to C$ such that \\begin{enumerate} \\item $B/IB \\to C/IC$ is an isomorphism, \\item the space $Y = \\Spec(C)$ is w-local, \\item the induced map $p : Y \\to X$ is w-local, and \\item $p^{-1}(X_0)$ is the set of closed points of $Y$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3719", "title": "proetale-lemma-construct", "text": "Let $A$ be a ring. Let $X = \\Spec(A)$. Let $T \\subset \\pi_0(X)$ be a closed subset. There exists a surjective ind-Zariski ring map $A \\to B$ such that $\\Spec(B) \\to \\Spec(A)$ induces a homeomorphism of $\\Spec(B)$ with the inverse image of $T$ in $X$."}
+{"_id": "3720", "title": "proetale-lemma-construct-profinite", "text": "Let $A$ be a ring and let $X = \\Spec(A)$. Let $T$ be a profinite space and let $T \\to \\pi_0(X)$ be a continuous map. There exists an ind-Zariski ring map $A \\to B$ such that with $Y = \\Spec(B)$ the diagram $$ \\xymatrix{ Y \\ar[r] \\ar[d] & \\pi_0(Y) \\ar[d] \\\\ X \\ar[r] & \\pi_0(X) } $$ is cartesian in the category of topological spaces and such that $\\pi_0(Y) = T$ as spaces over $\\pi_0(X)$."}
+{"_id": "3721", "title": "proetale-lemma-w-local-morphism-equal-points-stalks-is-iso", "text": "Let $A \\to B$ be ring map such that \\begin{enumerate} \\item $A \\to B$ identifies local rings, \\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local, \\item $\\Spec(B) \\to \\Spec(A)$ is w-local, and \\item $\\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$ is bijective. \\end{enumerate} Then $A \\to B$ is an isomorphism"}
+{"_id": "3722", "title": "proetale-lemma-w-local-morphism-equal-stalks-is-ind-zariski", "text": "Let $A \\to B$ be ring map such that \\begin{enumerate} \\item $A \\to B$ identifies local rings, \\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local, and \\item $\\Spec(B) \\to \\Spec(A)$ is w-local. \\end{enumerate} Then $A \\to B$ is ind-Zariski."}
+{"_id": "3723", "title": "proetale-lemma-w-local-extremally-disconnected", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item every faithfully flat ring map $A \\to B$ identifying local rings has a section, \\item every faithfully flat ind-Zariski ring map $A \\to B$ has a section, and \\item $A$ satisfies \\begin{enumerate} \\item $\\Spec(A)$ is w-local, and \\item $\\pi_0(\\Spec(A))$ is extremally disconnected. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3724", "title": "proetale-lemma-find-Zariski-w-contractible", "text": "Let $A$ be a ring. There exists a faithfully flat, ind-Zariski ring map $A \\to B$ such that $B$ satisfies the equivalent conditions of Lemma \\ref{lemma-w-local-extremally-disconnected}."}
+{"_id": "3725", "title": "proetale-lemma-base-change-ind-etale", "text": "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$ be the base change of $B$. If $A \\to B$ is ind-\\'etale, then $A' \\to B'$ is ind-\\'etale."}
+{"_id": "3726", "title": "proetale-lemma-composition-ind-etale", "text": "Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$ are ind-\\'etale, then $A \\to C$ is ind-\\'etale."}
+{"_id": "3727", "title": "proetale-lemma-ind-ind-etale", "text": "A filtered colimit of ind-\\'etale $A$-algebras is ind-\\'etale over $A$."}
+{"_id": "3728", "title": "proetale-lemma-ind-etale-permanence", "text": "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of ind-\\'etale $A$-algebras. Then $C$ is an ind-\\'etale $B$-algebra."}
+{"_id": "3730", "title": "proetale-lemma-lift-ind-etale", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal. The base change functor $$ \\text{ind-\\'etale }A\\text{-algebras} \\longrightarrow \\text{ind-\\'etale }A/I\\text{-algebras},\\quad C \\longmapsto C/IC $$ has a fully faithful right adjoint $v$. In particular, given an ind-\\'etale $A/I$-algebra $\\overline{C}$ there exists an ind-\\'etale $A$-algebra $C = v(\\overline{C})$ such that $\\overline{C} = C/IC$."}
+{"_id": "3731", "title": "proetale-lemma-first-construction", "text": "Given a ring $A$ there exists a faithfully flat ind-\\'etale $A$-algebra $C$ such that every faithfully flat \\'etale ring map $C \\to B$ has a section."}
+{"_id": "3732", "title": "proetale-lemma-have-sections-quotient", "text": "Let $A$ be a ring such that every faithfully flat \\'etale ring map $A \\to B$ has a section. Then the same is true for every quotient ring $A/I$."}
+{"_id": "3733", "title": "proetale-lemma-have-sections-strictly-henselian", "text": "Let $A$ be a ring such that every faithfully flat \\'etale ring map $A \\to B$ has a section. Then every local ring of $A$ at a maximal ideal is strictly henselian."}
+{"_id": "3734", "title": "proetale-lemma-have-sections-localize", "text": "Let $A$ be a ring such that every faithfully flat \\'etale ring map $A \\to B$ has a section. Let $Z \\subset \\Spec(A)$ be a closed subscheme of the form $D(f) \\cap V(I)$ and let $A \\to A_Z^\\sim$ be as constructed in Lemma \\ref{lemma-localization}. Then every faithfully flat \\'etale ring map $A_Z^\\sim \\to C$ has a section."}
+{"_id": "3735", "title": "proetale-lemma-get-w-local-algebraic-residue-field-extensions", "text": "Let $A \\to B$ be a ring map inducing algebraic extensions on residue fields. There exists a commutative diagram $$ \\xymatrix{ B \\ar[r] & D \\\\ A \\ar[r] \\ar[u] & C \\ar[u] } $$ with the following properties: \\begin{enumerate} \\item $A \\to C$ is faithfully flat and ind-\\'etale, \\item $B \\to D$ is faithfully flat and ind-\\'etale, \\item $\\Spec(C)$ is w-local, \\item $\\Spec(D)$ is w-local, \\item $\\Spec(D) \\to \\Spec(C)$ is w-local, \\item the set of closed points of $\\Spec(D)$ is the inverse image of the set of closed points of $\\Spec(C)$, \\item the set of closed points of $\\Spec(C)$ surjects onto $\\Spec(A)$, \\item the set of closed points of $\\Spec(D)$ surjects onto $\\Spec(B)$, \\item for $\\mathfrak m \\subset C$ maximal the local ring $C_\\mathfrak m$ is strictly henselian. \\end{enumerate}"}
+{"_id": "3736", "title": "proetale-lemma-pro-V-V", "text": "Let $Y$ be an affine scheme. Let $X = \\lim X_i$ be a directed limit of affine schemes over $Y$. The following are equivalent \\begin{enumerate} \\item $\\{X \\to Y\\}$ is a standard V covering (Topologies, Definition \\ref{topologies-definition-standard-V-covering}), and \\item $\\{X_i \\to Y\\}$ is a standard V covering for all $i$. \\end{enumerate}"}
+{"_id": "3737", "title": "proetale-lemma-pro-h-V", "text": "Let $X \\to Y$ be a morphism of affine schemes. The following are equivalent \\begin{enumerate} \\item $\\{X \\to Y\\}$ is a standard V covering (Topologies, Definition \\ref{topologies-definition-standard-V-covering}), \\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$ such that $\\{X_i \\to Y\\}$ is a ph covering for each $i$, and \\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$ such that $\\{X_i \\to Y\\}$ is an h covering for each $i$. \\end{enumerate}"}
+{"_id": "3738", "title": "proetale-lemma-h-limit-preserving", "text": "Let $S$ be a scheme. Let $F$ be a contravariant functor defined on the category of all schemes over $S$. If \\begin{enumerate} \\item $F$ satisfies the sheaf property for the h topology, and \\item $F$ is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving}), \\end{enumerate} then $F$ satisfies the sheaf property for the V topology."}
+{"_id": "3739", "title": "proetale-lemma-w-local-strictly-henselian-extremally-disconnected", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item $A$ is w-contractible, \\item every faithfully flat, ind-\\'etale ring map $A \\to B$ has a section, and \\item $A$ satisfies \\begin{enumerate} \\item $\\Spec(A)$ is w-local, \\item $\\pi_0(\\Spec(A))$ is extremally disconnected, and \\item for every maximal ideal $\\mathfrak m \\subset A$ the local ring $A_\\mathfrak m$ is strictly henselian. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3740", "title": "proetale-lemma-finite-finitely-presented-over-extremally-disconnected", "text": "Let $A \\to B$ be a quasi-finite and finitely presented ring map. If the residue fields of $A$ are separably algebraically closed and $\\Spec(A)$ is extremally disconnected, then $\\Spec(B)$ is extremally disconnected."}
+{"_id": "3741", "title": "proetale-lemma-finite-finitely-presented-over-w-contractible", "text": "Let $A \\to B$ be a finite and finitely presented ring map. If $A$ is w-contractible, so is $B$."}
+{"_id": "3742", "title": "proetale-lemma-localization-w-contractible", "text": "Let $A$ be a ring. Let $Z \\subset \\Spec(A)$ be a closed subset of the form $Z = V(f_1, \\ldots, f_r)$. Set $B = A_Z^\\sim$, see Lemma \\ref{lemma-localization}. If $A$ is w-contractible, so is $B$."}
+{"_id": "3743", "title": "proetale-lemma-recognize-proetale-covering", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \\begin{enumerate} \\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is a pro-\\'etale covering, \\item each $f_i$ is weakly \\'etale and $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering, \\item each $f_i$ is weakly \\'etale and for every affine open $U \\subset T$ there exist quasi-compact opens $U_i \\subset T_i$ which are almost all empty, such that $U = \\bigcup f_i(U_i)$, \\item each $f_i$ is weakly \\'etale and there exists an affine open covering $T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$ there exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$ and quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that $U_\\alpha = \\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$. \\end{enumerate} If $T$ is quasi-separated, these are also equivalent to \\begin{enumerate} \\item[(5)] each $f_i$ is weakly \\'etale, and for every $t \\in T$ there exist $i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$ such that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a (not necessarily open) neighbourhood of $t$ in $T$. \\end{enumerate}"}
+{"_id": "3744", "title": "proetale-lemma-etale-proetale", "text": "Any \\'etale covering and any Zariski covering is a pro-\\'etale covering."}
+{"_id": "3745", "title": "proetale-lemma-proetale", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a pro-\\'etale covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering and for each $i$ we have a pro-\\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a pro-\\'etale covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a pro-\\'etale covering. \\end{enumerate}"}
+{"_id": "3746", "title": "proetale-lemma-proetale-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a pro-\\'etale covering of $T$. Then there exists a pro-\\'etale covering $\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "3748", "title": "proetale-lemma-fibre-products-proetale", "text": "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site containing $S$. Let $\\Sch$ be the category of all schemes. \\begin{enumerate} \\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$, $S_\\proetale$, and $(\\textit{Aff}/S)_\\proetale$ have fibre products agreeing with fibre products in $\\Sch$. \\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$, $S_\\proetale$ have equalizers agreeing with equalizers in $\\Sch$. \\item The categories $(\\Sch/S)_\\proetale$, and $S_\\proetale$ both have a final object, namely $S/S$. \\item The category $\\Sch_\\proetale$ has a final object agreeing with the final object of $\\Sch$, namely $\\Spec(\\mathbf{Z})$. \\end{enumerate}"}
+{"_id": "3749", "title": "proetale-lemma-affine-big-site-proetale", "text": "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site containing $S$. The functor $(\\textit{Aff}/S)_\\proetale \\to (\\Sch/S)_\\proetale$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_\\proetale)$ to $\\Sh((\\Sch/S)_\\proetale)$."}
+{"_id": "3750", "title": "proetale-lemma-put-in-T", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$. The functor $T_\\proetale \\to (\\Sch/S)_\\proetale$ is cocontinuous and induces a morphism of topoi $$ i_f : \\Sh(T_\\proetale) \\longrightarrow \\Sh((\\Sch/S)_\\proetale) $$ For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\proetale$ we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers."}
+{"_id": "3751", "title": "proetale-lemma-at-the-bottom", "text": "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site containing $S$. The inclusion functor $S_\\proetale \\to (\\Sch/S)_\\proetale$ satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale $$ and a morphism of topoi $$ i_S : \\Sh(S_\\proetale) \\longrightarrow \\Sh((\\Sch/S)_\\proetale) $$ such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$ with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule $i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$."}
+{"_id": "3752", "title": "proetale-lemma-morphism-big", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$. The functor $$ u : (\\Sch/T)_\\proetale \\longrightarrow (\\Sch/S)_\\proetale, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_\\proetale \\longrightarrow (\\Sch/T)_\\proetale, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_\\proetale) \\longrightarrow \\Sh((\\Sch/S)_\\proetale) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "3753", "title": "proetale-lemma-morphism-big-small", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\proetale$. \\begin{enumerate} \\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in Lemma \\ref{lemma-put-in-T} and $i_T$ as in Lemma \\ref{lemma-at-the-bottom}. \\item The functor $S_\\proetale \\to T_\\proetale$, $(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces a morphism of topoi $$ f_{small} : \\Sh(T_\\proetale) \\longrightarrow \\Sh(S_\\proetale). $$ We have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. \\item We have a commutative diagram of morphisms of sites $$ \\xymatrix{ T_\\proetale \\ar[d]_{f_{small}} & (\\Sch/T)_\\proetale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\ S_\\proetale & (\\Sch/S)_\\proetale \\ar[l]_{\\pi_S} } $$ so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi. \\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$. \\end{enumerate}"}
+{"_id": "3755", "title": "proetale-lemma-morphism-big-small-cartesian-diagram", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Consider a cartesian diagram $$ \\xymatrix{ T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ in $\\Sch_\\proetale$. Then $i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$ and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$."}
+{"_id": "3757", "title": "proetale-lemma-alternative", "text": "Let $S$ be a scheme. Let $S_{affine, \\proetale}$ denote the full subcategory of $S_\\proetale$ consisting of affine objects. A covering of $S_{affine, \\proetale}$ will be a standard pro-\\'etale covering, see Definition \\ref{definition-standard-proetale}. Then restriction $$ \\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, \\etale}} $$ defines an equivalence of topoi $\\Sh(S_\\proetale) \\cong \\Sh(S_{affine, \\proetale})$."}
+{"_id": "3758", "title": "proetale-lemma-affine-alternative", "text": "Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory of $S_\\proetale$ consisting of affine objects $U$ such that $\\mathcal{O}(S) \\to \\mathcal{O}(U)$ is ind-\\'etale. A covering of $S_{app}$ will be a standard pro-\\'etale covering, see Definition \\ref{definition-standard-proetale}. Then restriction $$ \\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{app}} $$ defines an equivalence of topoi $\\Sh(S_\\proetale) \\cong \\Sh(S_{app})$."}
+{"_id": "3759", "title": "proetale-lemma-proetale-subcanonical", "text": "Let $S$ be a scheme. The topology on each of the pro-\\'etale sites $\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and $(\\textit{Aff}/S)_\\proetale$ is subcanonical."}
+{"_id": "3760", "title": "proetale-lemma-w-contractible-proetale-cover", "text": "Let $T = \\Spec(A)$ be an affine scheme. The following are equivalent \\begin{enumerate} \\item $A$ is w-contractible, and \\item every pro-\\'etale covering of $T$ can be refined by a Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$. \\end{enumerate}"}
+{"_id": "3761", "title": "proetale-lemma-w-contractible-is-weakly-contractible", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in Definition \\ref{definition-big-proetale-site}. Let $T = \\Spec(A)$ be an affine object of $\\Sch_\\proetale$. The following are equivalent \\begin{enumerate} \\item $A$ is w-contractible, \\item $T$ is a weakly contractible (Sites, Definition \\ref{sites-definition-w-contractible}) object of $\\Sch_\\proetale$, and \\item every pro-\\'etale covering of $T$ can be refined by a Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$. \\end{enumerate}"}
+{"_id": "3762", "title": "proetale-lemma-get-many-weakly-contractible", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in Definition \\ref{definition-big-proetale-site}. For every object $T$ of $\\Sch_\\proetale$ there exists a covering $\\{T_i \\to T\\}$ in $\\Sch_\\proetale$ with each $T_i$ affine and the spectrum of a w-contractible ring. In particular, $T_i$ is weakly contractible in $\\Sch_\\proetale$."}
+{"_id": "3763", "title": "proetale-lemma-proetale-enough-w-contractible", "text": "Let $S$ be a scheme. The pro-\\'etale sites $S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and $(\\textit{Aff}/S)_\\proetale$ and if $S$ is affine $S_{app}$ have enough (affine) quasi-compact, weakly contractible objects, see Sites, Definition \\ref{sites-definition-w-contractible}."}
+{"_id": "3764", "title": "proetale-lemma-weakly-contractible-cover", "text": "Let $S$ be a scheme. The pro-\\'etale sites $\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$ have the following property: for any object $U$ there exists a covering $\\{V \\to U\\}$ with $V$ a weakly contractible object. If $U$ is quasi-compact, then we may choose $V$ affine and weakly contractible."}
+{"_id": "3765", "title": "proetale-lemma-w-contractible-hypercovering", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item For every object $U$ of $X_\\proetale$ there exists a hypercovering $K$ of $U$ in $X_\\proetale$ such that each term $K_n$ consists of a single weakly contractible object of $X_\\proetale$ covering $U$. \\item For every quasi-compact and quasi-separated object $U$ of $X_\\proetale$ there exists a hypercovering $K$ of $U$ in $X_\\proetale$ such that each term $K_n$ consists of a single affine and weakly contractible object of $X_\\proetale$ covering $U$. \\end{enumerate}"}
+{"_id": "3766", "title": "proetale-lemma-compute-cohomology", "text": "Let $X$ be a scheme. Let $E \\in D^+(X_\\proetale)$ be represented by a bounded below complex $\\mathcal{E}^\\bullet$ of abelian sheaves. Let $K$ be a hypercovering of $U \\in \\Ob(X_\\proetale)$ with $K_n = \\{U_n \\to U\\}$ where $U_n$ is a weakly contractible object of $X_\\proetale$. Then $$ R\\Gamma(U, E) = \\text{Tot}(s(\\mathcal{E}^\\bullet(K))) $$ in $D(\\textit{Ab})$."}
+{"_id": "3769", "title": "proetale-lemma-presheaf-value-weakly-contractible", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a presheaf of sets on $X_\\proetale$ which sends finite disjoint unions to products. Then $\\mathcal{F}^\\#(W) = \\mathcal{F}(W)$ if $W$ is an affine weakly contractible object of $X_\\proetale$."}
+{"_id": "3770", "title": "proetale-lemma-small-pullback-weakly-contractible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a sheaf of sets on $X_\\proetale$. If $W$ is an affine weakly contractible object of $X_\\proetale$, then $$ f_{small}^{-1}\\mathcal{F}(W) = \\colim_{W \\to V} \\mathcal{F}(V) $$ where the colimit is over morphisms $W \\to V$ over $Y$ with $V \\in Y_\\proetale$."}
+{"_id": "3772", "title": "proetale-lemma-compare-injectives", "text": "Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\proetale$. \\begin{enumerate} \\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\proetale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\proetale)$, \\item $\\mathcal{I}|_{S_\\proetale}$ is injective in $\\textit{Ab}(S_\\proetale)$, \\end{enumerate} \\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}((\\Sch/S)_\\proetale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}(T_\\proetale)$, \\item $\\mathcal{I}^\\bullet|_{S_\\proetale}$ is a K-injective complex in $\\textit{Ab}(S_\\proetale)$, \\end{enumerate} \\end{enumerate}"}
+{"_id": "3775", "title": "proetale-lemma-cohomological-descent-proetale", "text": "Let $S$ be a scheme. For $K \\in D(S_\\proetale)$ the map $$ K \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K $$ is an isomorphism."}
+{"_id": "3780", "title": "proetale-lemma-limit-pullback", "text": "Let $X$ be a scheme. Let $Y = \\lim Y_i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_\\proetale$ with affine transition morphisms. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\epsilon^{-1}\\mathcal{F}(Y) = \\colim \\mathcal{F}(Y_i)$."}
+{"_id": "3781", "title": "proetale-lemma-fully-faithful", "text": "Let $X$ be a scheme. For every sheaf $\\mathcal{F}$ on $X_\\etale$ the adjunction map $\\mathcal{F} \\to \\epsilon_*\\epsilon^{-1}\\mathcal{F}$ is an isomorphism."}
+{"_id": "3782", "title": "proetale-lemma-affine-vanishing", "text": "Let $X$ be an affine scheme. For injective abelian sheaf $\\mathcal{I}$ on $X_\\etale$ we have $H^p(X_\\proetale, \\epsilon^{-1}\\mathcal{I}) = 0$ for $p > 0$."}
+{"_id": "3783", "title": "proetale-lemma-relative-comparison", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $R\\epsilon_*(\\epsilon^{-1}\\mathcal{F}) = \\mathcal{F}$. \\item For $K \\in D^+(X_\\etale)$ the map $K \\to R\\epsilon_*\\epsilon^{-1}K$ is an isomorphism. \\end{enumerate}"}
+{"_id": "3785", "title": "proetale-lemma-compare-cohomology-nonabelian", "text": "Let $X$ be a scheme. Let $\\mathcal{G}$ be a sheaf of (possibly noncommutative) groups on $X_\\etale$. We have $$ H^1(X_\\etale, \\mathcal{G}) = H^1(X_\\proetale, \\epsilon^{-1}\\mathcal{G}) $$ where $H^1$ is defined as the set of isomorphism classes of torsors (see Cohomology on Sites, Section \\ref{sites-cohomology-section-h1-torsors})."}
+{"_id": "3786", "title": "proetale-lemma-compare-derived", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a ring. \\begin{enumerate} \\item The essential image of the fully faithful functor $\\epsilon^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to \\textit{Mod}(X_\\proetale, \\Lambda)$ is a weak Serre subcategory $\\mathcal{C}$. \\item The functor $\\epsilon^{-1}$ defines an equivalence of categories of $D^+(X_\\etale, \\Lambda)$ with $D^+_\\mathcal{C}(X_\\proetale, \\Lambda)$ with question inverse given by $R\\epsilon_*$. \\end{enumerate}"}
+{"_id": "3787", "title": "proetale-lemma-compare-locally-constant", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a ring. The functor $\\epsilon^{-1}$ defines an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{locally constant sheaves}\\\\ \\text{of }\\Lambda\\text{-modules on }X_\\etale\\\\ \\text{of finite presentation} \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{locally constant sheaves}\\\\ \\text{of }\\Lambda\\text{-modules on }X_\\proetale\\\\ \\text{of finite presentation} \\end{matrix} \\right\\} $$"}
+{"_id": "3790", "title": "proetale-lemma-naive-completion", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. The left adjoint to the inclusion functor $D_{comp}(\\mathcal{C}, \\Lambda) \\to D(\\mathcal{C}, \\Lambda)$ of Algebraic and Formal Geometry, Proposition \\ref{algebraization-proposition-derived-completion} sends $K$ to $$ K^\\wedge = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}) $$ In particular, $K$ is derived complete if and only if $K = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})$."}
+{"_id": "3791", "title": "proetale-lemma-pushforward-Noetherian-case", "text": "Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal. Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. Then \\begin{enumerate} \\item $Rf_*$ sends $D_{comp}(\\mathcal{D}, \\Lambda)$ into $D_{comp}(\\mathcal{C}, \\Lambda)$, \\item the map $Rf_* : D_{comp}(\\mathcal{D}, \\Lambda) \\to D_{comp}(\\mathcal{C}, \\Lambda)$ has a left adjoint $Lf_{comp}^* : D_{comp}(\\mathcal{C}, \\Lambda) \\to D_{comp}(\\mathcal{D}, \\Lambda)$ which is $Lf^*$ followed by derived completion, \\item $Rf_*$ commutes with derived completion, \\item for $K$ in $D_{comp}(\\mathcal{D}, \\Lambda)$ we have $Rf_*K = R\\lim Rf_*(K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$. \\item for $M$ in $D_{comp}(\\mathcal{C}, \\Lambda)$ we have $Lf^*_{comp}M = R\\lim Lf^*(M \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$. \\end{enumerate}"}
+{"_id": "3793", "title": "proetale-lemma-morphism-comparison", "text": "Let $f : X \\to Y$ be a morphism of schemes. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. Then we have $f_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} = \\epsilon^{-1}f_{\\etale, *}\\mathcal{F}$. \\item Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then we have $Rf_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} = \\epsilon^{-1}Rf_{\\etale, *}\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "3794", "title": "proetale-lemma-finite", "text": "Let $f : Z \\to X$ be a finite morphism of schemes which is locally of finite presentation. Then $f_{\\proetale, *} : \\textit{Ab}(Z_\\proetale) \\to \\textit{Ab}(X_\\proetale)$ is exact."}
+{"_id": "3795", "title": "proetale-lemma-closed-immersion-affines", "text": "Let $i : Z \\to X$ be a closed immersion morphism of affine schemes. Denote $X_{app}$ and $Z_{app}$ the sites introduced in Lemma \\ref{lemma-affine-alternative}. The base change functor $$ u : X_{app} \\to Z_{app},\\quad U \\longmapsto u(U) = U \\times_X Z $$ is continuous and has a fully faithful left adjoint $v$. For $V$ in $Z_{app}$ the morphism $V \\to v(V)$ is a closed immersion identifying $V$ with $u(v(V)) = v(V) \\times_X Z$ and every point of $v(V)$ specializes to a point of $V$. The functor $v$ is cocontinuous and sends coverings to coverings."}
+{"_id": "3796", "title": "proetale-lemma-closed-immersion-affines-apply", "text": "Let $Z \\to X$ be a closed immersion morphism of affine schemes. The corresponding morphism of topoi $i = i_\\proetale$ is equal to the morphism of topoi associated to the fully faithful cocontinuous functor $v : Z_{app} \\to X_{app}$ of Lemma \\ref{lemma-closed-immersion-affines}. It follows that \\begin{enumerate} \\item $i^{-1}\\mathcal{F}$ is the sheaf associated to the presheaf $V \\mapsto \\mathcal{F}(v(V))$, \\item for a weakly contractible object $V$ of $Z_{app}$ we have $i^{-1}\\mathcal{F}(V) = \\mathcal{F}(v(V))$, \\item $i^{-1} : \\Sh(X_\\proetale) \\to \\Sh(Z_\\proetale)$ has a left adjoint $i^{Sh}_!$, \\item $i^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(Z_\\proetale)$ has a left adjoint $i_!$, \\item $\\text{id} \\to i^{-1}i^{Sh}_!$, $\\text{id} \\to i^{-1}i_!$, and $i^{-1}i_* \\to \\text{id}$ are isomorphisms, and \\item $i_*$, $i^{Sh}_!$ and $i_!$ are fully faithful. \\end{enumerate}"}
+{"_id": "3797", "title": "proetale-lemma-closed-immersion", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Then \\begin{enumerate} \\item $i_\\proetale^{-1}$ commutes with limits, \\item $i_{\\proetale, *}$ is fully faithful, and \\item $i_\\proetale^{-1}i_{\\proetale, *} \\cong \\text{id}_{\\Sh(Z_\\proetale)}$. \\end{enumerate}"}
+{"_id": "3798", "title": "proetale-lemma-thickening", "text": "Let $i : Z \\to X$ be an integral universally injective and surjective morphism of schemes. Then $i_{\\proetale, *}$ and $i_\\proetale^{-1}$ are quasi-inverse equivalences of categories of pro-\\'etale topoi."}
+{"_id": "3799", "title": "proetale-lemma-compute-i-star", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $U \\to X$ be an object of $X_\\proetale$ such that \\begin{enumerate} \\item $U$ is affine and weakly contractible, and \\item every point of $U$ specializes to a point of $U \\times_X Z$. \\end{enumerate} Then $i_\\proetale^{-1}\\mathcal{F}(U \\times_X Z) = \\mathcal{F}(U)$ for all abelian sheaves on $X_\\proetale$."}
+{"_id": "3801", "title": "proetale-lemma-jshriek-comparison", "text": "Let $j : U \\to X$ be an \\'etale morphism of schemes. Let $\\mathcal{G}$ be an abelian sheaf on $U_\\etale$. Then $\\epsilon^{-1} j_!\\mathcal{G} = j_!\\epsilon^{-1}\\mathcal{G}$ as sheaves on $X_\\proetale$."}
+{"_id": "3802", "title": "proetale-lemma-jshriek-zero", "text": "Let $j : U \\to X$ be a weakly \\'etale morphism of schemes. Let $i : Z \\to X$ be a closed immersion such that $U \\times_X Z = \\emptyset$. Let $V \\to X$ be an affine object of $X_\\proetale$ such that every point of $V$ specializes to a point of $V_Z = Z \\times_X V$. Then $j_!\\mathcal{F}(V) = 0$ for all abelian sheaves on $U_\\proetale$."}
+{"_id": "3804", "title": "proetale-lemma-ses-associated-to-open", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let $U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$ the inclusion morphisms. Assume that $j$ is a quasi-compact morphism. For every abelian sheaf on $X_\\proetale$ there is a canonical short exact sequence $$ 0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0 $$ on $X_\\proetale$ where all the functors are for the pro-\\'etale topology."}
+{"_id": "3806", "title": "proetale-lemma-compare-constructible", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. The functor $\\epsilon^{-1}$ defines an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{constructible sheaves of}\\\\ \\Lambda\\text{-modules on }X_\\etale\\\\ \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{constructible sheaves of}\\\\ \\Lambda\\text{-modules on }X_\\proetale\\\\ \\end{matrix} \\right\\} $$ between constructible sheaves of $\\Lambda$-modules on $X_\\etale$ and constructible sheaves of $\\Lambda$-modules on $X_\\proetale$."}
+{"_id": "3807", "title": "proetale-lemma-constructible-serre", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\\Lambda$-modules on $X_\\proetale$ is a weak Serre subcategory of $\\textit{Mod}(X_\\proetale, \\Lambda)$."}
+{"_id": "3808", "title": "proetale-lemma-compare-constructible-derived", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. Let $D_c(X_\\etale, \\Lambda)$, resp.\\ $D_c(X_\\proetale, \\Lambda)$ be the full subcategory of $D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$ consisting of those complexes whose cohomology sheaves are constructible sheaves of $\\Lambda$-modules. Then $$ \\epsilon^{-1} : D_c^+(X_\\etale, \\Lambda) \\longrightarrow D_c^+(X_\\proetale, \\Lambda) $$ is an equivalence of categories."}
+{"_id": "3809", "title": "proetale-lemma-tensor-c", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. Let $K, L \\in D_c^-(X_\\proetale, \\Lambda)$. Then $K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\proetale, \\Lambda)$."}
+{"_id": "3810", "title": "proetale-lemma-compare-truncations-constructible", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. Let $K$ be an object of $D(X_\\proetale, \\Lambda)$. Set $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$. If $K_1$ is in $D^-_c(X_\\proetale, \\Lambda/I)$, then $K_n$ is in $D^-_c(X_\\proetale, \\Lambda/I^n)$ for all $n$."}
+{"_id": "3811", "title": "proetale-lemma-Noetherian-constructible", "text": "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a constructible $\\Lambda$-sheaf on $X_\\proetale$. Then there exists a finite partition $X = \\coprod X_i$ by locally closed subschemes such that the restriction $\\mathcal{F}|_{X_i}$ is lisse."}
+{"_id": "3812", "title": "proetale-lemma-weird", "text": "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian ring and $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\proetale$ such that \\begin{enumerate} \\item $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$, \\item $\\mathcal{F}/I^n\\mathcal{F}$ is a constant sheaf of $\\Lambda/I^n$-modules, \\item $\\mathcal{F}/I\\mathcal{F}$ is of finite type. \\end{enumerate} Then $\\mathcal{F} \\cong \\underline{M}^\\wedge$ where $M$ is a finite $\\Lambda^\\wedge$-module."}
+{"_id": "3813", "title": "proetale-lemma-connected-lisse", "text": "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. If $\\mathcal{F}$ is a lisse constructible $\\Lambda$-sheaf on $X_\\proetale$, then $\\mathcal{F}$ is adic lisse."}
+{"_id": "3815", "title": "proetale-lemma-derived-complete-zero", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a ring and let $I \\subset \\Lambda$ be a finitely generated ideal. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\proetale$. If $\\mathcal{F}$ is derived complete and $\\mathcal{F}/I\\mathcal{F} = 0$, then $\\mathcal{F} = 0$."}
+{"_id": "3816", "title": "proetale-lemma-derived-complete-limit", "text": "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a derived complete sheaf of $\\Lambda$-modules on $X_\\proetale$ with $\\mathcal{F}/I\\mathcal{F}$ a locally constant sheaf of $\\Lambda/I$-modules of finite type. Then there exists an integer $t$ and a surjective map $$ (\\underline{\\Lambda}^\\wedge)^{\\oplus t} \\to \\mathcal{F} $$"}
+{"_id": "3817", "title": "proetale-lemma-describe-constructible-complexes", "text": "In the situation above suppose $K$ is in $D_{cons}(X, \\Lambda)$ and $X$ is quasi-compact. Set $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$. There exist $a, b$ such that \\begin{enumerate} \\item $K = R\\lim K_n$ and $H^i(K) = 0$ for $i \\not \\in [a, b]$, \\item each $K_n$ has tor amplitude in $[a, b]$, \\item each $K_n$ has constructible cohomology sheaves, \\item each $K_n = \\epsilon^{-1}L_n$ for some $L_n \\in D_{ctf}(X_\\etale, \\Lambda/I^n)$ (\\'Etale Cohomology, Definition \\ref{etale-cohomology-definition-ctf}). \\end{enumerate}"}
+{"_id": "3819", "title": "proetale-lemma-weakly-contractible-locally-constant-ML", "text": "Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of $D_{cons}(X, \\Lambda)$ such that $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ is isomorphic in $D(X_\\proetale, \\Lambda/I^n)$ to a complex of constant sheaves of $\\Lambda/I^n$-modules. Then $$ H^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^n) $$ has the Mittag-Leffler condition."}
+{"_id": "3820", "title": "proetale-lemma-connected-adic-lisse", "text": "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. If $K$ is in $D_{cons}(X, \\Lambda)$ such that $K \\otimes_\\Lambda \\underline{\\Lambda/I}$ has locally constant cohomology sheaves, then $K$ is adic lisse (Definition \\ref{definition-adic-constructible})."}
+{"_id": "3821", "title": "proetale-lemma-proetale-induced", "text": "Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in Definition \\ref{definition-big-proetale-site}. Let $T \\in \\Ob(\\Sch_\\proetale)$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary pro-\\'etale covering of $T$. There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_\\proetale$ which refines $\\{T_i \\to T\\}_{i \\in I}$."}
+{"_id": "3823", "title": "proetale-lemma-change-alpha", "text": "Suppose given big sites $\\Sch_\\proetale$ and $\\Sch'_\\proetale$ as in Definition \\ref{definition-big-proetale-site}. Assume that $\\Sch_\\proetale$ is contained in $\\Sch'_\\proetale$. The inclusion functor $\\Sch_\\proetale \\to \\Sch'_\\proetale$ satisfies the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}. There are morphisms of topoi \\begin{eqnarray*} g : \\Sh(\\Sch_\\proetale) & \\longrightarrow & \\Sh(\\Sch'_\\proetale) \\\\ f : \\Sh(\\Sch'_\\proetale) & \\longrightarrow & \\Sh(\\Sch_\\proetale) \\end{eqnarray*} such that $f \\circ g \\cong \\text{id}$. For any object $S$ of $\\Sch_\\proetale$ the inclusion functor $(\\Sch/S)_\\proetale \\to (\\Sch'/S)_\\proetale$ satisfies the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site} also. Hence similarly we obtain morphisms \\begin{eqnarray*} g : \\Sh((\\Sch/S)_\\proetale) & \\longrightarrow & \\Sh((\\Sch'/S)_\\proetale) \\\\ f : \\Sh((\\Sch'/S)_\\proetale) & \\longrightarrow & \\Sh((\\Sch/S)_\\proetale) \\end{eqnarray*} with $f \\circ g \\cong \\text{id}$."}
+{"_id": "3825", "title": "proetale-proposition-maps-wich-identify-local-rings", "text": "Let $A \\to B$ be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map $B \\to B'$ such that $A \\to B'$ is ind-Zariski."}
+{"_id": "3826", "title": "proetale-proposition-weakly-etale", "text": "Let $A \\to B$ be a weakly \\'etale ring map. Then there exists a faithfully flat, ind-\\'etale ring map $B \\to B'$ such that $A \\to B'$ is ind-\\'etale."}
+{"_id": "3827", "title": "proetale-proposition-find-w-contractible", "text": "For every ring $A$ there exists a faithfully flat, ind-\\'etale ring map $A \\to D$ such that $D$ is w-contractible."}
+{"_id": "3828", "title": "proetale-proposition-enough-weakly-contractibles", "text": "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has enough weakly contractible objects. Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal. \\begin{enumerate} \\item The category of derived complete sheaves $\\Lambda$-modules is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{C}, \\Lambda)$. \\item A sheaf $\\mathcal{F}$ of $\\Lambda$-modules satisfies $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ if and only if $\\mathcal{F}$ is derived complete and $\\bigcap I^n\\mathcal{F} = 0$. \\item The sheaf $\\underline{\\Lambda}^\\wedge$ is derived complete. \\item If $\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$ is an inverse system of derived complete sheaves of $\\Lambda$-modules, then $\\lim \\mathcal{F}_n$ is derived complete. \\item An object $K \\in D(\\mathcal{C}, \\Lambda)$ is derived complete if and only if each cohomology sheaf $H^p(K)$ is derived complete. \\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded above if and only if $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ is bounded above. \\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded if $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "3852", "title": "formal-spaces-lemma-fully-faithful", "text": "Choose a category of schemes $\\Sch_\\alpha$ as in Sets, Lemma \\ref{sets-lemma-construct-category}. Given a formal scheme $\\mathfrak X$ let $$ h_\\mathfrak X : (\\Sch_\\alpha)^{opp} \\longrightarrow \\textit{Sets},\\quad h_\\mathfrak X(S) = \\Mor_{\\textit{Formal Schemes}}(S, \\mathfrak X) $$ be its functor of points. Then we have $$ \\Mor_{\\textit{Formal Schemes}}(\\mathfrak X, \\mathfrak Y) = \\Mor_{\\textit{PSh}(\\Sch_\\alpha)}(h_\\mathfrak X, h_\\mathfrak Y) $$ provided the size of $\\mathfrak X$ is not too large."}
+{"_id": "3853", "title": "formal-spaces-lemma-formal-scheme-sheaf-fppf", "text": "\\begin{slogan} Formal schemes are fpqc sheaves \\end{slogan} Let $\\mathfrak X$ be a formal scheme. The functor of points $h_\\mathfrak X$ (see Lemma \\ref{lemma-fully-faithful}) satisfies the sheaf condition for fpqc coverings."}
+{"_id": "3854", "title": "formal-spaces-lemma-closed", "text": "Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module and let $M_\\lambda$, $\\lambda \\in \\Lambda$ be a fundamental system of open submodules. Let $N \\subset M$ be a submodule. The closure of $N$ is $\\bigcap_{\\lambda \\in \\Lambda} (N + M_\\lambda)$."}
+{"_id": "3855", "title": "formal-spaces-lemma-closure", "text": "Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \\subset M$ be a submodule. Then \\begin{enumerate} \\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge$ is exact, and \\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$. \\end{enumerate}"}
+{"_id": "3856", "title": "formal-spaces-lemma-quotient-by-closed", "text": "Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized $R$-module. Let $N \\subset M$ be a closed submodule. If $M$ has a countable fundamental system of neighbourhoods of $0$, then $M/N$ is complete and the map $M \\to M/N$ is open."}
+{"_id": "3857", "title": "formal-spaces-lemma-ses", "text": "\\begin{reference} \\cite[Theorem 8.1]{Ma} \\end{reference} Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \\subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then \\begin{enumerate} \\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge \\to 0$ is exact, \\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$, \\item $M^\\wedge \\to (M/N)^\\wedge$ is open. \\end{enumerate}"}
+{"_id": "3860", "title": "formal-spaces-lemma-topologically-nilpotent", "text": "Let $B$ be a linearly topologized ring. The set of topologically nilpotent elements of $B$ is a closed, radical ideal of $B$. Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings. \\begin{enumerate} \\item If $f \\in A$ is topologically nilpotent, then $\\varphi(f)$ is topologically nilpotent. \\item If $I \\subset A$ consists of topologically nilpotent elements, then the closure of $\\varphi(I)B$ consists of topologically nilpotent elements. \\end{enumerate}"}
+{"_id": "3862", "title": "formal-spaces-lemma-closure-image-ideal", "text": "Let $A \\to B$ be a continuous map of linearly topologized rings. Let $I \\subset A$ be an ideal. The closure of $IB$ is the kernel of $B \\to B \\widehat{\\otimes}_A A/I$."}
+{"_id": "3863", "title": "formal-spaces-lemma-dense-image-surjective", "text": "Let $\\varphi : A \\to B$ be a continuous homomorphism of linearly topologized rings. If \\begin{enumerate} \\item $\\varphi$ is taut, \\item $\\varphi$ has dense image, \\item $A$ is complete, \\item $B$ is separated, and \\item $A$ has a countable fundamental system of neighbourhoods of $0$. \\end{enumerate} Then $\\varphi$ is surjective and open, $B$ is complete, and $B = A/K$ for some closed ideal $K \\subset A$."}
+{"_id": "3864", "title": "formal-spaces-lemma-taut-is-adic", "text": "Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings. Let $I \\subset A$ be an ideal. Assume \\begin{enumerate} \\item $I$ is finitely generated, \\item $A$ has the $I$-adic topology, \\item $B$ is complete, and \\item $\\varphi$ is taut. \\end{enumerate} Then the topology on $B$ is the $I$-adic topology."}
+{"_id": "3865", "title": "formal-spaces-lemma-completed-tensor-product", "text": "Let $B \\to A$ and $B \\to C$ be continuous homomorphisms of linearly topologized rings. \\begin{enumerate} \\item If $A$ and $C$ are weakly pre-admissible, then $A \\widehat{\\otimes}_B C$ is weakly admissible. \\item If $A$ and $C$ are pre-admissible, then $A \\widehat{\\otimes}_B C$ is admissible. \\item If $A$ and $C$ have a countable fundamental system of open ideals, then $A \\widehat{\\otimes}_B C$ has a countable fundamental system of open ideals. \\item If $A$ and $C$ are pre-adic and have finitely generated ideals of definition, then $A \\widehat{\\otimes}_B C$ is adic and has a finitely generated ideal of definition. \\item If $A$ and $C$ are pre-adic Noetherian rings and $B/\\mathfrak b \\to A/\\mathfrak a$ is of finite type where $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ are the ideals of topologically nilpotent elements, then $A \\widehat{\\otimes}_B C$ is adic Noetherian. \\end{enumerate}"}
+{"_id": "3866", "title": "formal-spaces-lemma-diagonal-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. If $X$ is an affine formal algebraic space over $S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$ is representable and a closed immersion."}
+{"_id": "3867", "title": "formal-spaces-lemma-covering-by-thickenings", "text": "Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$ be as in Definition \\ref{definition-affine-formal-algebraic-space}. Then $X_\\lambda \\to X$ is representable and a thickening."}
+{"_id": "3868", "title": "formal-spaces-lemma-factor-through-thickening", "text": "Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$ be as in Definition \\ref{definition-affine-formal-algebraic-space}. If $Y$ is a quasi-compact algebraic space over $S$, then any morphism $Y \\to X$ factors through an $X_\\lambda$."}
+{"_id": "3870", "title": "formal-spaces-lemma-mcquillan-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be an fppf sheaf on $(\\Sch/S)_{fppf}$ which satisfies the set theoretic condition of Remark \\ref{remark-set-theoretic}. The following are equivalent: \\begin{enumerate} \\item there exists a weakly admissible topological ring $A$ over $S$ (see Remark \\ref{remark-mcquillan}) such that $X = \\colim_{I \\subset A\\text{ weak ideal of definition}} \\Spec(A/I)$, \\item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$ such that for a closed immersion $T \\to X$ with $T$ an affine scheme the composition $T \\to \\Spec(A)$ is a closed immersion, \\item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$ such that for a closed immersion $T \\to X$ with $T$ a scheme the composition $T \\to \\Spec(A)$ is a closed immersion, \\item $X$ is an affine formal algebraic space and for some choice of $X = \\colim X_\\lambda$ as in Definition \\ref{definition-affine-formal-algebraic-space} the projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda}) \\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective, \\item $X$ is an affine formal algebraic space and for any choice of $X = \\colim X_\\lambda$ as in Definition \\ref{definition-affine-formal-algebraic-space} the projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda}) \\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective. \\end{enumerate} Moreover, the weakly admissible topological ring is $A = \\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ endowed with its limit topology and the weak ideals of definition classify exactly the morphisms $T \\to X$ which are representable and thickenings."}
+{"_id": "3871", "title": "formal-spaces-lemma-morphism-between-formal-spectra", "text": "Let $S$ be a scheme. Let $A$, $B$ be weakly admissible topological rings over $S$. Any morphism $f : \\text{Spf}(B) \\to \\text{Spf}(A)$ of affine formal algebraic spaces over $S$ is equal to $\\text{Spf}(f^\\sharp)$ for a unique continuous $S$-algebra map $f^\\sharp : A \\to B$."}
+{"_id": "3872", "title": "formal-spaces-lemma-presentation-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a map of presheaves on $(\\Sch/S)_{fppf}$. If $X$ is an affine formal algebraic space and $f$ is representable by algebraic spaces and locally quasi-finite, then $f$ is representable (by schemes)."}
+{"_id": "3873", "title": "formal-spaces-lemma-countable-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item there exists a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$ of thickenings of affine schemes over $S$ such that $X = \\colim X_n$, \\item there exists a choice $X = \\colim X_\\lambda$ as in Definition \\ref{definition-affine-formal-algebraic-space} such that $\\Lambda$ is countable. \\end{enumerate}"}
+{"_id": "3874", "title": "formal-spaces-lemma-implications-between-types", "text": "Let $X$ be an affine formal algebraic space over a scheme $S$. \\begin{enumerate} \\item If $X$ is Noetherian, then $X$ is adic*. \\item If $X$ is adic*, then $X$ is adic. \\item If $X$ is adic, then $X$ is countably indexed. \\item If $X$ is countably indexed, then $X$ is McQuillan. \\end{enumerate}"}
+{"_id": "3875", "title": "formal-spaces-lemma-countably-indexed", "text": "Let $S$ be a scheme. Let $X$ be a presheaf on $(\\Sch/S)_{fppf}$. The following are equivalent \\begin{enumerate} \\item $X$ is a countably indexed affine formal algebraic space, \\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a countable fundamental system of neighbourhoods of $0$, \\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a fundamental system $A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$ of weak ideals of definition, \\item $X = \\text{Spf}(A)$ where $A$ is a complete topological $S$-algebra with a fundamental system of open neighbourhoods of $0$ given by a countable sequence $A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$ of ideals such that $I_n/I_{n + 1}$ is locally nilpotent, and \\item $X = \\text{Spf}(A)$ where $A = \\lim B/J_n$ with the limit topology where $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$ is a sequence of ideals in an $S$-algebra $B$ with $J_n/J_{n + 1}$ locally nilpotent. \\end{enumerate}"}
+{"_id": "3876", "title": "formal-spaces-lemma-characterize-noetherian-affine", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent \\begin{enumerate} \\item $X$ is Noetherian, \\item $X$ is adic* and for some choice of $X = \\colim X_\\lambda$ as in Definition \\ref{definition-affine-formal-algebraic-space} the schemes $X_\\lambda$ are Noetherian, \\item $X$ is adic* and for any closed immersion $T \\to X$ with $T$ a scheme, $T$ is Noetherian. \\end{enumerate}"}
+{"_id": "3877", "title": "formal-spaces-lemma-diagonal-formal-algebraic-space", "text": "Let $S$ be a scheme. If $X$ is a formal algebraic space over $S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$ is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated."}
+{"_id": "3878", "title": "formal-spaces-lemma-space-to-formal-space", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism from an algebraic space over $S$ to a formal algebraic space over $S$. Then $f$ is representable by algebraic spaces."}
+{"_id": "3879", "title": "formal-spaces-lemma-reduction-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. There exists a reduced algebraic space $X_{red}$ and a representable morphism $X_{red} \\to X$ which is a thickening. A morphism $U \\to X$ with $U$ a reduced algebraic space factors uniquely through $X_{red}$."}
+{"_id": "3880", "title": "formal-spaces-lemma-reduction-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces and smooth (for example \\'etale). Then $X_{red} = X \\times_Y Y_{red}$."}
+{"_id": "3881", "title": "formal-spaces-lemma-reduction-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is surjective in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation} if and only if $f_{red} : X_{red} \\to Y_{red}$ is a surjective morphism of algebraic spaces."}
+{"_id": "3882", "title": "formal-spaces-lemma-colimit-is-formal", "text": "Let $S$ be a scheme. Suppose given a directed set $\\Lambda$ and a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$ where each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening. Then $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$ is a formal algebraic space over $S$."}
+{"_id": "3883", "title": "formal-spaces-lemma-completion-affine-is-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be an affine scheme over $S$. Let $T \\subset |X|$ be a closed subset. Then the functor $$ (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad U \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\} $$ is a McQuillan affine formal algebraic space."}
+{"_id": "3884", "title": "formal-spaces-lemma-completion-is-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. Then the functor $$ (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad U \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\} $$ is a formal algebraic space."}
+{"_id": "3885", "title": "formal-spaces-lemma-map-completions-representable", "text": "Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$. Let $T \\subset |X|$ be a closed subset and let $T' = |f|^{-1}(T) \\subset |X'|$. Then $$ \\xymatrix{ X'_{/T'} \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\ X_{/T} \\ar[r] & X } $$ is a cartesian diagram of sheaves. In particular, the morphism $X'_{/T'} \\to X_{/T}$ is representable by algebraic spaces."}
+{"_id": "3886", "title": "formal-spaces-lemma-reduction-completion", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. The reduction $(X_{/T})_{red}$ of the completion $X_{/T}$ of $X$ along $T$ is the reduced induced closed subspace $Z$ of $X$ corresponding to $T$."}
+{"_id": "3887", "title": "formal-spaces-lemma-affine-formal-completion-types", "text": "Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$. Let $T \\subset X$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$. \\begin{enumerate} \\item If $X \\setminus T$ is quasi-compact, i.e., $T$ is constructible, then $X_{/T}$ is adic*. \\item If $T = V(I)$ for some finitely generated ideal $I \\subset A$, then $X_{/T} = \\text{Spf}(A^\\wedge)$ where $A^\\wedge$ is the $I$-adic completion of $A$. \\item If $X$ is Noetherian, then $X_{/T}$ is Noetherian. \\end{enumerate}"}
+{"_id": "3889", "title": "formal-spaces-lemma-etale-covering-by-formal-algebraic-spaces", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of maps of sheaves on $(\\Sch/S)_{fppf}$. Assume (a) $X_i$ is a formal algebraic space over $S$, (b) $X_i \\to X$ is representable by algebraic spaces and \\'etale, and (c) $\\coprod X_i \\to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$."}
+{"_id": "3890", "title": "formal-spaces-lemma-fibre-products-general", "text": "Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$ and let $Z$ be a sheaf whose diagonal is representable by algebraic spaces. Let $X \\to Z$ and $Y \\to Z$ be maps of sheaves. Then $X \\times_Z Y$ is a formal algebraic space."}
+{"_id": "3891", "title": "formal-spaces-lemma-fibre-products", "text": "Let $S$ be a scheme. The category of formal algebraic spaces over $S$ has fibre products."}
+{"_id": "3893", "title": "formal-spaces-lemma-diagonal-morphism-formal-algebraic-spaces", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The diagonal morphism $\\Delta : X \\to X \\times_Y X$ is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated."}
+{"_id": "3894", "title": "formal-spaces-lemma-characterize-quasi-separated", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item the reduction of $X$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-separated algebraic space, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \\times_X V$ is quasi-compact, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine the fibre product $U \\times_X V$ is quasi-compact. \\end{enumerate}"}
+{"_id": "3895", "title": "formal-spaces-lemma-characterize-separated", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item the reduction of $X$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a separated algebraic space, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine the fibre product $U \\times_X V$ is affine and $$ \\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V) \\longrightarrow \\mathcal{O}(U \\times_X V) $$ is surjective. \\end{enumerate}"}
+{"_id": "3896", "title": "formal-spaces-lemma-fibre-product-affines-over-separated", "text": "Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms of formal algebraic spaces over $S$. Assume $Z$ separated. \\begin{enumerate} \\item If $X$ and $Y$ are affine formal algebraic spaces, then so is $X \\times_Z Y$. \\item If $X$ and $Y$ are McQuillan affine formal algebraic spaces, then so is $X \\times_Z Y$. \\item If $X$, $Y$, and $Z$ are McQuillan affine formal algebraic spaces corresponding to the weakly admissible topological $S$-algebras $A$, $B$, and $C$, then $X \\times_Z Y$ corresponds to $A \\widehat{\\otimes}_C B$. \\end{enumerate}"}
+{"_id": "3897", "title": "formal-spaces-lemma-separated-from-separated", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $U \\to X$ be a morphism where $U$ is a separated algebraic space over $S$. Then $U \\to X$ is separated."}
+{"_id": "3898", "title": "formal-spaces-lemma-characterize-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item the reduction of $X$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact algebraic space, \\item we can find $\\{X_i \\to X\\}_{i \\in I}$ as in Definition \\ref{definition-formal-algebraic-space} with $I$ finite, \\item there exists a morphism $Y \\to X$ representable by algebraic spaces which is \\'etale and surjective and where $Y$ is an affine formal algebraic space. \\end{enumerate}"}
+{"_id": "3899", "title": "formal-spaces-lemma-characterize-quasi-compact-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item the induced map $f_{red} : X_{red} \\to Y_{red}$ between reductions (Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact morphism of algebraic spaces, \\item for every quasi-compact scheme $T$ and morphism $T \\to Y$ the fibre product $X \\times_Y T$ is a quasi-compact formal algebraic space, \\item for every affine scheme $T$ and morphism $T \\to Y$ the fibre product $X \\times_Y T$ is a quasi-compact formal algebraic space, and \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X \\times_Y Y_j$ is a quasi-compact formal algebraic space. \\end{enumerate}"}
+{"_id": "3900", "title": "formal-spaces-lemma-quasi-compact-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is quasi-compact in the sense of Definition \\ref{definition-quasi-compact-morphism} if and only if $f$ is quasi-compact in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}."}
+{"_id": "3901", "title": "formal-spaces-lemma-structure-quasi-compact-quasi-separated", "text": "\\begin{reference} \\cite[Proposition 3.32]{Yasuda} \\end{reference} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated formal algebraic space over $S$. Then $X = \\colim X_\\lambda$ for a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$ over a directed set $\\Lambda$ where each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening."}
+{"_id": "3902", "title": "formal-spaces-lemma-characterize-affine", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X$ is an affine formal algebraic space if and only if its reduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}) is affine."}
+{"_id": "3905", "title": "formal-spaces-lemma-permanence-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is representable by algebraic spaces, then $f : X \\to Y$ is representable by algebraic spaces."}
+{"_id": "3906", "title": "formal-spaces-lemma-representable-by-algebraic-spaces-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item the morphism $f$ is representable by algebraic spaces, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, $U \\to X$ is surjective \\'etale, and $U \\to V$ is representable by algebraic spaces, \\item for any commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces and the vertical arrows are representable by algebraic spaces, the morphism $U \\to V$ is representable by algebraic spaces, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that $X_{ji} \\to Y_j$ is representable by algebraic spaces for each $j$ and $i$, \\item there exist a covering $\\{X_i \\to X\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces, such that $X_i \\to Y_i$ is representable by algebraic spaces, and \\item add more here. \\end{enumerate}"}
+{"_id": "3907", "title": "formal-spaces-lemma-algebraic-space-over-affine-formal", "text": "Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space over $S$. Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space."}
+{"_id": "3909", "title": "formal-spaces-lemma-affine-representable-by-algebraic-spaces", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then $f$ is representable (by schemes) and affine."}
+{"_id": "3910", "title": "formal-spaces-lemma-property-goes-up-affine-morphism", "text": "Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space. Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is representable and affine. Then \\begin{enumerate} \\item $X$ is an affine formal algebraic space. \\item if $Y$ is countably indexed, then $X$ is countably indexed. \\item if $Y$ is adic*, then $X$ is adic*, \\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \\end{enumerate}"}
+{"_id": "3911", "title": "formal-spaces-lemma-property-goes-up-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then \\begin{enumerate} \\item if $Y$ is countably indexed, then $X$ is countably indexed. \\item if $Y$ is adic*, then $X$ is adic*, \\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \\end{enumerate}"}
+{"_id": "3912", "title": "formal-spaces-lemma-representable-affine", "text": "Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of weakly admissible topological rings over $S$. The following are equivalent \\begin{enumerate} \\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$ is representable by algebraic spaces, \\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$ is representable (by schemes), \\item $\\varphi$ is taut, see Definition \\ref{definition-taut}. \\end{enumerate}"}
+{"_id": "3913", "title": "formal-spaces-lemma-etale", "text": "Let $S$ be a scheme. Let $Y$ be a McQuillan affine formal algebraic space over $S$, i.e., $Y = \\text{Spf}(B)$ for some weakly admissible topological $S$-algebra $B$. Then there is an equivalence of categories between \\begin{enumerate} \\item the category of morphisms $f : X \\to Y$ of affine formal algebraic spaces which are representable by algebraic spaces and \\'etale, and \\item the category of topological $B$-algebras of the form $A^\\wedge$ where $A$ is an \\'etale $B$-algebra and $A^\\wedge = \\lim A/JA$ with $J \\subset B$ running over the weak ideals of definition of $B$. \\end{enumerate} The equivalence is given by sending $A^\\wedge$ to  $X = \\text{Spf}(A^\\wedge)$. In particular, any $X$ as in (1) is McQuillan."}
+{"_id": "3914", "title": "formal-spaces-lemma-etale-surjective", "text": "With notation and assumptions as in Lemma \\ref{lemma-etale} let $f : X \\to Y$ correspond to $B \\to A^\\wedge$. The following are equivalent \\begin{enumerate} \\item $f : X \\to Y$ is surjective, \\item $B \\to A$ is faithfully flat, \\item for every weak ideal of definition $J \\subset B$ the ring map $B/J \\to A/JA$ is faithfully flat, and \\item for some weak ideal of definition $J \\subset B$ the ring map $B/J \\to A/JA$ is faithfully flat. \\end{enumerate}"}
+{"_id": "3915", "title": "formal-spaces-lemma-iff-countably-indexed", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then $X$ is countably indexed if and only if $Y$ is countably indexed."}
+{"_id": "3916", "title": "formal-spaces-lemma-iff-adic-star", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then $X$ is adic* if and only if $Y$ is adic*."}
+{"_id": "3917", "title": "formal-spaces-lemma-iff-noetherian", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and (locally) of finite type. Then $X$ is Noetherian if and only if $Y$ is Noetherian."}
+{"_id": "3918", "title": "formal-spaces-lemma-type-local", "text": "Let $S$ be a scheme. Let $P \\in \\{countably\\ indexed, adic*, Noetherian\\}$. Let $X$ be a formal algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item if $Y$ is an affine formal algebraic space and $f : Y \\to X$ is representable by algebraic spaces and \\'etale, then $Y$ has property $P$, \\item for some $\\{X_i \\to X\\}_{i \\in I}$ as in Definition \\ref{definition-formal-algebraic-space} each $X_i$ has property $P$. \\end{enumerate}"}
+{"_id": "3919", "title": "formal-spaces-lemma-formal-completion-types", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. Let $X_{/T}$ be the formal completion of $X$ along $T$. \\begin{enumerate} \\item If $X \\setminus T \\to X$ is quasi-compact, then $X_{/T}$ is locally adic*. \\item If $X$ is locally Noetherian, then $X_{/T}$ is locally Noetherian. \\end{enumerate}"}
+{"_id": "3920", "title": "formal-spaces-lemma-types-fibre-products", "text": "Let $S$ be a scheme. Let $X \\to Y$ and $Z \\to Y$ be morphisms of formal algebraic space over $S$. Then \\begin{enumerate} \\item If $X$ and $Z$ are locally countably indexed, then $X \\times_Y Z$ is locally countably indexed. \\item If $X$ and $Z$ are locally adic*, then $X \\times_Y Z$ is locally adic*. \\item If $X$ and $Z$ are locally Noetherian and $X_{red} \\to Y_{red}$ is locally of finite type, then $X \\times_Y Z$ is locally Noetherian. \\end{enumerate}"}
+{"_id": "3921", "title": "formal-spaces-lemma-structure-locally-noetherian", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian formal algebraic space over $S$. Then $X = \\colim X_n$ for a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$ of finite order thickenings of locally Noetherian algebraic spaces over $S$ where $X_1 = X_{red}$ and $X_n$ is the $n$th infinitesimal neighbourhood of $X_1$ in $X_m$ for all $m \\geq n$."}
+{"_id": "3922", "title": "formal-spaces-lemma-completion-in-sub", "text": "Let $A \\in \\Ob(\\textit{WAdm})$. Let $A \\to A'$ be a ring map (no topology). Let $(A')^\\wedge = \\lim_{I \\subset A\\text{ w.i.d}} A'/IA'$ be the object of $\\textit{WAdm}$ constructed in Example \\ref{example-representable-morphism-from-completion}. \\begin{enumerate} \\item If $A$ is in $\\textit{WAdm}^{count}$, so is $(A')^\\wedge$. \\item If $A$ is in $\\textit{WAdm}^{adic*}$, so is $(A')^\\wedge$. \\item If $A$ is in $\\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then $(A')^\\wedge$ is in $\\textit{WAdm}^{Noeth}$. \\end{enumerate}"}
+{"_id": "3923", "title": "formal-spaces-lemma-property-defines-property-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. Let $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$. The following are equivalent \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a morphism of $\\textit{WAdm}^{count}$ with property $P$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a morphism of $\\textit{WAdm}^{count}$ with property $P$, and \\item there exist a covering $\\{X_i \\to X\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a morphism of $\\textit{WAdm}^{count}$ with property $P$. \\end{enumerate}"}
+{"_id": "3924", "title": "formal-spaces-lemma-base-change-property-morphisms", "text": "Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$ which is stable under base change. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} then so does $\\text{pr}_2 : X \\times_Y Z \\to Z$."}
+{"_id": "3925", "title": "formal-spaces-lemma-composition-property-morphisms", "text": "Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$ which is stable under composition. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ and $g$ satisfies the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} then so does $g \\circ f : X \\to Z$."}
+{"_id": "3927", "title": "formal-spaces-lemma-representable-property-rings", "text": "Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$. The following are equivalent \\begin{enumerate} \\item[(a)] $B \\to A$ is taut (Definition \\ref{definition-taut}), \\item[(b)] for $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$ a fundamental system of weak ideals of definitions there exist a commutative diagram $$ \\xymatrix{ A \\ar[r] & \\ldots \\ar[r] & A_3 \\ar[r] & A_2 \\ar[r] & A_1 \\\\ B \\ar[r] \\ar[u] & \\ldots \\ar[r] & B/J_3 \\ar[r] \\ar[u] & B/J_2 \\ar[r] \\ar[u] & B/J_1 \\ar[u] } $$ such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and $A = \\lim A_n$ as topological ring. \\end{enumerate} Moreover, these equivalent conditions define a local property, i.e., they satisfy axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3})."}
+{"_id": "3928", "title": "formal-spaces-lemma-representable-local-property", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a taut map $B \\to A$ of $\\textit{WAdm}^{count}$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a taut ring map in $\\textit{WAdm}^{count}$, \\item there exist a covering $\\{X_i \\to X\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a taut ring map in $\\textit{WAdm}^{count}$, and \\item $f$ is representable by algebraic spaces. \\end{enumerate}"}
+{"_id": "3929", "title": "formal-spaces-lemma-adic-homomorphism", "text": "Let $A$ and $B$ be pre-adic topological rings. Let $\\varphi : A \\to B$ be a continuous ring homomorphism. \\begin{enumerate} \\item If $\\varphi$ is adic, then $\\varphi$ is taut. \\item If $B$ is complete, $A$ has a finitely generated ideal of definition, and $\\varphi$ is taut, then $\\varphi$ is adic. \\end{enumerate} In particular the conditions ``$\\varphi$ is adic'' and ``$\\varphi$ is taut'' are equivalent on the category $\\textit{WAdm}^{adic*}$."}
+{"_id": "3931", "title": "formal-spaces-lemma-composition-finite-type", "text": "The composition of finite type morphisms is of finite type. The same holds for locally of finite type."}
+{"_id": "3932", "title": "formal-spaces-lemma-base-change-finite-type", "text": "A base change of a finite type morphism is finite type. The same holds for locally of finite type."}
+{"_id": "3934", "title": "formal-spaces-lemma-finite-type-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item the morphism $f$ is locally of finite type, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces and \\'etale, $U \\to X$ is surjective, and $U \\to V$ is locally of finite type, \\item for any commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are formal algebraic spaces and vertical arrows representable by algebraic spaces and \\'etale, the morphism $U \\to V$ is locally of finite type, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that $X_{ji} \\to Y_j$ is locally of finite type for each $j$ and $i$, \\item there exist a covering $\\{X_i \\to X\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, such that $X_i \\to Y_i$ is locally of finite type, and \\item add more here. \\end{enumerate}"}
+{"_id": "3935", "title": "formal-spaces-lemma-locally-finite-type-locally-noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. If $Y$ is locally Noetherian and $f$ locally of finite type, then $X$ is locally Noetherian."}
+{"_id": "3936", "title": "formal-spaces-lemma-fibre-product-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms of formal algebraic spaces over $S$. If $Z$ is locally Noetherian and $f$ locally of finite type, then $Z \\times_Y X$ is locally Noetherian."}
+{"_id": "3937", "title": "formal-spaces-lemma-composition-surjective", "text": "The composition of two surjective morphisms is a surjective morphism."}
+{"_id": "3938", "title": "formal-spaces-lemma-base-change-surjective", "text": "A base change of a surjective morphism is a surjective morphism."}
+{"_id": "3939", "title": "formal-spaces-lemma-characterize-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is surjective, \\item for every scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a surjective morphism of formal algebraic spaces, \\item for every affine scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a surjective morphism of formal algebraic spaces, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X \\times_Y Y_j \\to Y_j$ is a surjective morphism of formal algebraic spaces, \\item there exists a surjective morphism $Z \\to Y$ of formal algebraic spaces such that $X \\times_Y Z \\to Z$ is surjective, and \\item add more here. \\end{enumerate}"}
+{"_id": "3940", "title": "formal-spaces-lemma-composition-monomorphism", "text": "The composition of two monomorphisms is a monomorphism."}
+{"_id": "3941", "title": "formal-spaces-lemma-base-change-monomorphism", "text": "A base change of a monomorphism is a monomorphism."}
+{"_id": "3942", "title": "formal-spaces-lemma-characterize-monomorphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is a monomorphism, \\item for every scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a monomorphism of formal algebraic spaces, \\item for every affine scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a monomorphism of formal algebraic spaces, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X \\times_Y Y_j \\to Y_j$ is a monomorphism of formal algebraic spaces, and \\item there exists a family of morphisms $\\{Y_j \\to Y\\}$ such that $\\coprod Y_j \\to Y$ is a surjection of sheaves on $(\\Sch/S)_{fppf}$ such that each $X \\times_Y Y_j \\to Y_j$ is a monomorphism for all $j$, \\item there exists a morphism $Z \\to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \\times_Y Z \\to X$ is a monomorphism, and \\item add more here. \\end{enumerate}"}
+{"_id": "3943", "title": "formal-spaces-lemma-composition-closed-immersion", "text": "The composition of two closed immersions is a closed immersion."}
+{"_id": "3944", "title": "formal-spaces-lemma-base-change-closed-immersion", "text": "A base change of a closed immersion is a closed immersion."}
+{"_id": "3945", "title": "formal-spaces-lemma-characterize-closed-immersion", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is a closed immersion, \\item for every scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a closed immersion, \\item for every affine scheme $T$ and morphism $T \\to Y$ the projection $X \\times_Y T \\to T$ is a closed immersion, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X \\times_Y Y_j \\to Y_j$ is a closed immersion, and \\item there exists a morphism $Z \\to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \\times_Y Z \\to X$ is a closed immersion, and \\item add more here. \\end{enumerate}"}
+{"_id": "3946", "title": "formal-spaces-lemma-closed-immersion-into-McQuillan", "text": "Let $S$ be a scheme. Let $X$ be a McQuillan affine formal algebraic space over $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic spaces over $S$. Then $Y$ is a McQuillan affine formal algebraic space and $f$ corresponds to a continuous homomorphism $A \\to B$ of weakly admissible topological $S$-algebras which is taut, has closed kernel, and has dense image."}
+{"_id": "3947", "title": "formal-spaces-lemma-monomorphism-iso-over-red", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces. Assume \\begin{enumerate} \\item $f$ is representable by algebraic spaces, \\item $f$ is a monomorphism, \\item the inclusion $Y_{red} \\to Y$ factors through $f$, and \\item $f$ is locally of finite type or $Y$ is locally Noetherian. \\end{enumerate} Then $f$ is a closed immersion."}
+{"_id": "3948", "title": "formal-spaces-lemma-topologically-finite-type-finite-type", "text": "Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of weakly admissible topological rings over $S$. The following are equivalent \\begin{enumerate} \\item $\\text{Spf}(\\varphi) : Y = \\text{Spf}(B) \\to \\text{Spf}(A) = X$ is of finite type, \\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$. \\end{enumerate}"}
+{"_id": "3949", "title": "formal-spaces-lemma-category-affine-over", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Assume $X$ is McQuillan and let $A$ be the weakly admissible topological ring associated to $X$. Then there is an anti-equivalence of categories between \\begin{enumerate} \\item the category $\\mathcal{C}$ introduced above, and \\item the category of maps $Y \\to X$ of finite type of affine formal algebraic spaces. \\end{enumerate}"}
+{"_id": "3950", "title": "formal-spaces-lemma-closed-immersion-into-countably-indexed", "text": "Let $S$ be a scheme. Let $X$ be a countably indexed affine formal algebraic space over $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic spaces over $S$. Then $Y$ is a countably indexed affine formal algebraic space and $f$ corresponds to $A \\to A/K$ where $A$ is an object of $\\textit{WAdm}^{count}$ (Section \\ref{section-morphisms-rings}) and $K \\subset A$ is a closed ideal."}
+{"_id": "3951", "title": "formal-spaces-lemma-quotient-restricted-power-series", "text": "Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$, see Section \\ref{section-morphisms-rings}. The following are equivalent \\begin{enumerate} \\item[(a)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for every weak ideal of definition $J \\subset B$ where $I \\subset A$ is the closure of $JA$, \\item[(b)] $B \\to A$ is taut and $B/J_\\lambda \\to A/I_\\lambda$ is of finite type for a cofinal system $(J_\\lambda)$ of weak ideals of definition of $B$ where $I_\\lambda \\subset A$ is the closure of $J_\\lambda A$, \\item[(c)] $B \\to A$ is taut and $A$ is topologically of finite type over $B$, \\item[(d)] $A$ is isomorphic as a topological $B$-algebra to a quotient of $B\\{x_1, \\ldots, x_n\\}$ by a closed ideal. \\end{enumerate} Moreover, these equivalent conditions define a local property, i.e., they satisfy Axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3})."}
+{"_id": "3952", "title": "formal-spaces-lemma-quotient-restricted-power-series-admissible", "text": "In Lemma \\ref{lemma-quotient-restricted-power-series} if $B$ is admissible (for example adic), then the equivalent conditions (a) -- (d) are also equivalent to \\begin{enumerate} \\item[(e)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for some ideal of definition $J \\subset B$ where $I \\subset A$ is the closure of $JA$. \\end{enumerate}"}
+{"_id": "3953", "title": "formal-spaces-lemma-representable-affine-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of affine formal algebraic spaces. Assume $Y$ countably indexed. The following are equivalent \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is of finite type, \\item $f$ corresponds to a morphism $B \\to A$ of $\\textit{WAdm}^{count}$ (Section \\ref{section-morphisms-rings}) satisfying the equivalent conditions of Lemma \\ref{lemma-quotient-restricted-power-series}. \\end{enumerate}"}
+{"_id": "3954", "title": "formal-spaces-lemma-finite-type-local-property", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a morphism of $\\textit{WAdm}^{count}$ which is taut and topologically of finite type, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a morphism of $\\textit{WAdm}^{count}$ which is taut and topologically of finite type, \\item there exist a covering $\\{X_i \\to X\\}$ as in Definition \\ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a morphism of $\\textit{WAdm}^{count}$ which is, taut and topologically of finite type, and \\item $f$ is locally of finite type. \\end{enumerate}"}
+{"_id": "3955", "title": "formal-spaces-lemma-base-change-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated-morphism} are stable under base change."}
+{"_id": "3956", "title": "formal-spaces-lemma-fibre-product-after-map", "text": "Let $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$ be morphisms of formal algebraic spaces over $S$. Consider the induced morphism $i : X \\times_Z Y \\to X \\times_T Y$. Then \\begin{enumerate} \\item $i$ is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism, \\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and \\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact. \\end{enumerate}"}
+{"_id": "3958", "title": "formal-spaces-lemma-separated-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. Let $\\mathcal{P}$ be any of the separation axioms of Definition \\ref{definition-separated-morphism}. The following are equivalent \\begin{enumerate} \\item $f$ is $\\mathcal{P}$, \\item for every scheme $Z$ and morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the formal algebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see Definition \\ref{definition-separated}), \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that the base change $Y_j \\times_Y X \\to Y_j$ has $\\mathcal{P}$ for all $j$. \\end{enumerate}"}
+{"_id": "3959", "title": "formal-spaces-lemma-proper-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is proper, \\item for every scheme $Z$ and morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is proper, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is proper, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the formal algebraic space $Z \\times_Y X$ is an algebraic space proper over $Z$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Definition \\ref{definition-formal-algebraic-space} such that the base change $Y_j \\times_Y X \\to Y_j$ is proper for all $j$. \\end{enumerate}"}
+{"_id": "3960", "title": "formal-spaces-lemma-base-change-proper", "text": "Proper morphisms of formal algebraic spaces are preserved by base change."}
+{"_id": "3961", "title": "formal-spaces-lemma-sheaf-fpqc", "text": "\\begin{slogan} Formal algebraic spaces are fpqc sheaves \\end{slogan} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X$ satisfies the sheaf property for the fpqc topology."}
+{"_id": "3962", "title": "formal-spaces-lemma-map-into-affine", "text": "Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$-algebra. Let $X$ be an affine scheme over $S$. Then the natural map $$ \\Mor_S(\\Spec(A), X) \\longrightarrow \\Mor_S(\\text{Spf}(A), X) $$ is bijective."}
+{"_id": "3963", "title": "formal-spaces-lemma-map-into-scheme", "text": "Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$-algebra such that $A/I$ is a local ring for some weak ideal of definition $I \\subset A$. Let $X$ be a scheme over $S$. Then the natural map $$ \\Mor_S(\\Spec(A), X) \\longrightarrow \\Mor_S(\\text{Spf}(A), X) $$ is bijective."}
+{"_id": "3964", "title": "formal-spaces-lemma-map-into-algebraic-space", "text": "Let $S$ be a scheme. Let $R$ be a complete local Noetherian $S$-algebra. Let $X$ be an algebraic space over $S$. Then the natural map $$ \\Mor_S(\\Spec(R), X) \\longrightarrow \\Mor_S(\\text{Spf}(R), X) $$ is bijective."}
+{"_id": "3965", "title": "formal-spaces-lemma-adic-into-completion", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset such that $X \\setminus T \\to X$ is quasi-compact. Let $R$ be a complete local Noetherian $S$-algebra. Then an adic morphism $p : \\text{Spf}(R) \\to X_{/T}$ corresponds to a unique morphism $g : \\Spec(R) \\to X$ such that $g^{-1}(T) = \\{\\mathfrak m_R\\}$."}
+{"_id": "3967", "title": "formal-spaces-lemma-etale-morphism-topoi", "text": "Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. Assume $f$ is representable by algebraic spaces and \\'etale. In this case there is a cocontinuous functor $j : X_\\etale \\to Y_\\etale$. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}. Moreover, $j$ is continuous as well, hence Sites, Lemma \\ref{sites-lemma-when-shriek} applies."}
+{"_id": "3968", "title": "formal-spaces-lemma-affine-identify-affine-etale", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Then $X_{affine, \\etale}$ is equivalent to the category whose objects are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that \\begin{enumerate} \\item $U$ is an affine formal algebraic space, \\item $\\varphi$ is representable by algebraic spaces and \\'etale. \\end{enumerate}"}
+{"_id": "3969", "title": "formal-spaces-lemma-affine-etale-mcquillan", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Assume $X$ is McQuillan, i.e., equal to $\\text{Spf}(A)$ for some weakly admissible topological $S$-algebra $A$. Then $(X_{affine, \\etale})^{opp}$ is equivalent to the category whose \\begin{enumerate} \\item objects are $A$-algebras of the form $B^\\wedge = \\lim B/JB$ where $A \\to B$ is an \\'etale ring map and $J$ runs over the weak ideals of definition of $A$, and \\item morphisms are continuous $A$-algebra homomorphisms. \\end{enumerate}"}
+{"_id": "3970", "title": "formal-spaces-lemma-identify-spaces-etale", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{spaces, \\etale}$ is equivalent to the category whose objects are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that $\\varphi$ is representable by algebraic spaces and \\'etale."}
+{"_id": "3971", "title": "formal-spaces-lemma-identify-affine-etale", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{affine, \\etale}$ is equivalent to the category whose objects are morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that \\begin{enumerate} \\item $U$ is an affine formal algebraic space, \\item $\\varphi$ is representable by algebraic spaces and \\'etale. \\end{enumerate}"}
+{"_id": "4021", "title": "pione-theorem-fundamental-group", "text": "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point of $X$. \\begin{enumerate} \\item The fibre functor $F_{\\overline{x}}$ defines an equivalence of categories $$ \\textit{F\\'Et}_X \\longrightarrow \\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets} $$ \\item Given a second geometric point $\\overline{x}'$ of $X$ there exists an isomorphism $t : F_{\\overline{x}} \\to F_{\\overline{x}'}$. This gives an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(X, \\overline{x}')$ compatible with the equivalences in (1). This isomorphism is independent of $t$ up to inner conjugation. \\item Given a morphism $f : X \\to Y$ of connected schemes denote $\\overline{y} = f \\circ \\overline{x}$. There is a canonical continuous homomorphism $$ f_* : \\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y}) $$ such that the diagram $$ \\xymatrix{ \\textit{F\\'Et}_Y \\ar[r]_{\\text{base change}} \\ar[d]_{F_{\\overline{y}}} & \\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\ \\textit{Finite-}\\pi_1(Y, \\overline{y})\\textit{-Sets} \\ar[r]^{f_*} & \\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets} } $$ is commutative. \\end{enumerate}"}
+{"_id": "4024", "title": "pione-lemma-sheaves-point", "text": "Let $K$ be a field. Let $K^{sep}$ be a separable closure of $K$. Consider the profinite group $G = \\text{Gal}(K^{sep}/K)$. The functor $$ \\begin{matrix} \\text{schemes \\'etale over }K & \\longrightarrow & G\\textit{-Sets} \\\\ X/K & \\longmapsto & \\Mor_{\\Spec(K)}(\\Spec(K^{sep}), X) \\end{matrix} $$ is an equivalence of categories."}
+{"_id": "4025", "title": "pione-lemma-aut-inverse-limit", "text": "Let $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor. The map (\\ref{equation-embedding-product}) identifies $\\text{Aut}(F)$ with a closed subgroup of $\\prod_{X \\in \\Ob(\\mathcal{C})} \\text{Aut}(F(X))$. In particular, if $F(X)$ is finite for all $X$, then $\\text{Aut}(F)$ is a profinite group."}
+{"_id": "4026", "title": "pione-lemma-single-out-profinite", "text": "Let $G$ be a topological group. The automorphism group of the functor (\\ref{equation-forgetful}) endowed with its profinite topology from Lemma \\ref{lemma-aut-inverse-limit} is the profinite completion of $G$."}
+{"_id": "4027", "title": "pione-lemma-second-fundamental-functor", "text": "Let $G$ be a topological group. Let $F : \\textit{Finite-}G\\textit{-Sets} \\to \\textit{Sets}$ be an exact functor with $F(X)$ finite for all $X$. Then $F$ is isomorphic to the functor (\\ref{equation-forgetful})."}
+{"_id": "4028", "title": "pione-lemma-epi-mono", "text": "Let $(\\mathcal{C}, F)$ be a Galois category. Let $X \\to Y \\in \\text{Arrows}(\\mathcal{C})$. Then \\begin{enumerate} \\item $F$ is faithful, \\item $X \\to Y$ is a monomorphism $\\Leftrightarrow F(X) \\to F(Y)$ is injective, \\item $X \\to Y$ is an epimorphism $\\Leftrightarrow F(X) \\to F(Y)$ is surjective, \\item an object $A$ of $\\mathcal{C}$ is initial if and only if $F(A) = \\emptyset$, \\item an object $Z$ of $\\mathcal{C}$ is final if and only if $F(Z)$ is a singleton, \\item if $X$ and $Y$ are connected, then $X \\to Y$ is an epimorphism, \\item \\label{item-one-element} if $X$ is connected and $a, b : X \\to Y$ are two morphisms then $a = b$ as soon as $F(a)$ and $F(b)$ agree on one element of $F(X)$, \\item if $X = \\coprod_{i = 1, \\ldots, n} X_i$ and $Y = \\coprod_{j = 1, \\ldots, m} Y_j$ where $X_i$, $Y_j$ are connected, then there is map $\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}$ such that $X \\to Y$ comes from a collection of morphisms $X_i \\to Y_{\\alpha(i)}$. \\end{enumerate}"}
+{"_id": "4029", "title": "pione-lemma-galois", "text": "Let $(\\mathcal{C}, F)$ be a Galois category. For any connected object $X$ of $\\mathcal{C}$ there exists a Galois object $Y$ and a morphism $Y \\to X$."}
+{"_id": "4030", "title": "pione-lemma-tame", "text": "\\begin{reference} Compare with \\cite[Definition 7.2.4]{BS}. \\end{reference} Let $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$ be as in Example \\ref{example-from-C-F-to-G-sets}. For any connected $X$ in $\\mathcal{C}$ the action of $G$ on $F(X)$ is transitive."}
+{"_id": "4031", "title": "pione-lemma-functoriality-galois", "text": "Let $(\\mathcal{C}, F)$ and $(\\mathcal{C}', F')$ be Galois categories. Let $H : \\mathcal{C} \\to \\mathcal{C}'$ be an exact functor. There exists an isomorphism $t : F' \\circ H \\to F$. The choice of $t$ determines a continuous homomorphism $h : G' = \\text{Aut}(F') \\to \\text{Aut}(F) = G$ and a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{C} \\ar[r]_H \\ar[d] & \\mathcal{C}' \\ar[d] \\\\ \\textit{Finite-}G\\textit{-Sets} \\ar[r]^h & \\textit{Finite-}G'\\textit{-Sets} } $$ The map $h$ is independent of $t$ up to an inner automorphism of $G$. Conversely, given a continuous homomorphism $h : G' \\to G$ there is an exact functor $H : \\mathcal{C} \\to \\mathcal{C}'$ and an isomorphism $t$ recovering $h$ as above."}
+{"_id": "4032", "title": "pione-lemma-functoriality-galois-surjective", "text": "In diagram (\\ref{equation-translation}) the following are equivalent \\begin{enumerate} \\item $h : G' \\to G$ is surjective, \\item $H : \\mathcal{C} \\to \\mathcal{C}'$ is fully faithful, \\item if $X \\in \\Ob(\\mathcal{C})$ is connected, then $H(X)$ is connected, \\item if $X \\in \\Ob(\\mathcal{C})$ is connected and there is a morphism $*' \\to H(X)$ in $\\mathcal{C}'$, then there is a morphism $* \\to X$, and \\item for any object $X$ of $\\mathcal{C}$ the map $\\Mor_\\mathcal{C}(*, X) \\to \\Mor_{\\mathcal{C}'}(*', H(X))$ is bijective. \\end{enumerate} Here $*$ and $*'$ are final objects of $\\mathcal{C}$ and $\\mathcal{C}'$."}
+{"_id": "4033", "title": "pione-lemma-composition-trivial", "text": "In diagram (\\ref{equation-translation}) the following are equivalent \\begin{enumerate} \\item $h \\circ h'$ is trivial, and \\item the image of $H' \\circ H$ consists of objects isomorphic to finite coproducts of final objects. \\end{enumerate}"}
+{"_id": "4034", "title": "pione-lemma-functoriality-galois-ses", "text": "In diagram (\\ref{equation-translation}) the following are equivalent \\begin{enumerate} \\item the sequence $G'' \\xrightarrow{h'} G' \\xrightarrow{h} G \\to 1$ is exact in the following sense: $h$ is surjective, $h \\circ h'$ is trivial, and $\\Ker(h)$ is the smallest closed normal subgroup containing $\\Im(h')$, \\item $H$ is fully faithful and an object $X'$ of $\\mathcal{C}'$ is in the essential image of $H$ if and only if $H'(X')$ is isomorphic to a finite coproduct of final objects, and \\item $H$ is fully faithful, $H \\circ H'$ sends every object to a finite coproduct of final objects, and for an object $X'$ of $\\mathcal{C}'$ such that $H'(X')$ is a finite coproduct of final objects there exists an object $X$ of $\\mathcal{C}$ and an epimorphism $H(X) \\to X'$. \\end{enumerate}"}
+{"_id": "4035", "title": "pione-lemma-functoriality-galois-injective", "text": "In diagram (\\ref{equation-translation}) the following are equivalent \\begin{enumerate} \\item $h'$ is injective, and \\item for every connected object $X''$ of $\\mathcal{C}''$ there exists an object $X'$ of $\\mathcal{C}'$ and a diagram $$ X'' \\leftarrow Y'' \\rightarrow H(X') $$ in $\\mathcal{C}''$ where $Y'' \\to X''$ is an epimorphism and $Y'' \\to H(X')$ is a monomorphism. \\end{enumerate}"}
+{"_id": "4036", "title": "pione-lemma-functoriality-galois-normal", "text": "In diagram (\\ref{equation-translation}) the following are equivalent \\begin{enumerate} \\item the image of $h'$ is normal, and \\item for every connected object $X'$ of $\\mathcal{C}'$ such that there is a morphism from the final object of $\\mathcal{C}''$ to $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct of final objects. \\end{enumerate}"}
+{"_id": "4037", "title": "pione-lemma-finite-etale-covers-limits-colimits", "text": "Let $X$ be a scheme. The category $\\textit{F\\'Et}_X$ has finite limits and finite colimits and for any morphism $X' \\to X$ the base change functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X'}$ is exact."}
+{"_id": "4038", "title": "pione-lemma-internal-hom-finite-etale", "text": "Let $X$ be a scheme. Given $U, V$ finite \\'etale over $X$ there exists a scheme $W$ finite \\'etale over $X$ such that $$ \\Mor_X(X, W) = \\Mor_X(U, V) $$ and such that the same remains true after any base change."}
+{"_id": "4039", "title": "pione-lemma-finite-etale-connected-galois-category", "text": "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point. The functor $$ F_{\\overline{x}} : \\textit{F\\'Et}_X \\longrightarrow \\textit{Sets},\\quad Y \\longmapsto |Y_{\\overline{x}}| $$ defines a Galois category (Definition \\ref{definition-galois-category})."}
+{"_id": "4040", "title": "pione-lemma-fundamental-group-Galois-group", "text": "Let $K$ be a field and set $X = \\Spec(K)$. Let $\\overline{K}$ be an algebraic closure and denote $\\overline{x} : \\Spec(\\overline{K}) \\to X$ the corresponding geometric point. Let $K^{sep} \\subset \\overline{K}$ be the separable algebraic closure. \\begin{enumerate} \\item The functor of Lemma \\ref{lemma-sheaves-point} induces an equivalence $$ \\textit{F\\'Et}_X \\longrightarrow \\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets}. $$ compatible with $F_{\\overline{x}}$ and the functor $\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets} \\to \\textit{Sets}$. \\item This induces a canonical isomorphism $$ \\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X, \\overline{x}) $$ of profinite topological groups. \\end{enumerate}"}
+{"_id": "4041", "title": "pione-lemma-what-equivalence-gives", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes such that the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$ is an equivalence of categories. In this case \\begin{enumerate} \\item $f$ induces a homeomorphism $\\pi_0(X) \\to \\pi_0(Y)$, \\item if $X$ or equivalently $Y$ is connected, then $\\pi_1(X, \\overline{x}) = \\pi_1(Y, \\overline{y})$. \\end{enumerate}"}
+{"_id": "4042", "title": "pione-lemma-gabber", "text": "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. The functor $$ \\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_Z,\\quad U \\longmapsto U \\times_X Z $$ is an equivalence of categories."}
+{"_id": "4043", "title": "pione-lemma-thickening", "text": "Let $X \\subset X'$ be a thickening of schemes. The functor $$ \\textit{F\\'Et}_{X'} \\longrightarrow \\textit{F\\'Et}_X,\\quad U' \\longmapsto U' \\times_{X'} X $$ is an equivalence of categories."}
+{"_id": "4044", "title": "pione-lemma-finite-etale-on-proper-over-henselian", "text": "Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Then the functor $$ \\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_0},\\quad U \\longmapsto U_0 = U \\times_X X_0 $$ is an equivalence of categories."}
+{"_id": "4045", "title": "pione-lemma-finite-etale-invariant-over-proper", "text": "Let $k \\subset k'$ be an extension of algebraically closed fields. Let $X$ be a proper scheme over $k$. Then the functor $$ U \\longmapsto U_{k'} $$ is an equivalence of categories between schemes finite \\'etale over $X$ and schemes finite \\'etale over $X_{k'}$."}
+{"_id": "4046", "title": "pione-lemma-dense-faithful", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$ then the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$ is faithful."}
+{"_id": "4047", "title": "pione-lemma-same-etale-extensions", "text": "Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$ and let $U = X \\setminus \\{\\mathfrak m\\}$. If the punctured spectrum of the strict henselization of $A$ is connected, then $$ \\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U,\\quad Y \\longmapsto Y \\times_X U $$ is a fully faithful functor."}
+{"_id": "4048", "title": "pione-lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian", "text": "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume \\begin{enumerate} \\item the underlying topological space of $X$ is Noetherian, and \\item for every $x \\in X \\setminus U$ the punctured spectrum of the strict henselization of $\\mathcal{O}_{X, x}$ is connected. \\end{enumerate} Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful."}
+{"_id": "4049", "title": "pione-lemma-retrocompact-dense-open-connected-at-infinity-closed", "text": "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume \\begin{enumerate} \\item $U \\to X$ is quasi-compact, \\item every point of $X \\setminus U$ is closed, and \\item for every $x \\in X \\setminus U$ the punctured spectrum of the strict henselization of $\\mathcal{O}_{X, x}$ is connected. \\end{enumerate} Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful."}
+{"_id": "4050", "title": "pione-lemma-quasi-compact-dense-open-connected-at-infinity", "text": "Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume \\begin{enumerate} \\item every quasi-compact open of $X$ has finitely many irreducible components, \\item for every $x \\in X \\setminus U$ the punctured spectrum of the strict henselization of $\\mathcal{O}_{X, x}$ is connected. \\end{enumerate} Then $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful."}
+{"_id": "4051", "title": "pione-lemma-local-exact-sequence", "text": "Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$. Let $U^{sh}$ be the punctured spectrum of the strict henselization $A^{sh}$ of $A$. Assume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence $$ \\pi_1(U^{sh}, \\overline{u}) \\to \\pi_1(U, \\overline{u}) \\to \\pi_1(X, \\overline{u}) \\to 1 $$ is exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1)."}
+{"_id": "4052", "title": "pione-lemma-irreducible-geometrically-unibranch", "text": "Let $X$ be an irreducible, geometrically unibranch scheme. For any nonempty open $U \\subset X$ the canonical map $$ \\pi_1(U, \\overline{u}) \\longrightarrow \\pi_1(X, \\overline{u}) $$ is surjective. The map (\\ref{equation-inclusion-generic-point}) $\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$ is surjective as well."}
+{"_id": "4054", "title": "pione-lemma-unramified-in-L", "text": "In the situation above the following are equivalent \\begin{enumerate} \\item $X$ is unramified in $L$, \\item $Y \\to X$ is \\'etale, and \\item $Y \\to X$ is finite \\'etale. \\end{enumerate}"}
+{"_id": "4055", "title": "pione-lemma-finite-etale-covering-normal-unramified", "text": "Let $X$ be a normal integral scheme with function field $K$. Let $Y \\to X$ be a finite \\'etale morphism. If $Y$ is connected, then $Y$ is an integral normal scheme and $Y$ is the normalization of $X$ in the function field of $Y$."}
+{"_id": "4056", "title": "pione-lemma-local-exact-sequence-normal", "text": "Let $(A, \\mathfrak m)$ be a normal local ring. Set $X = \\Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$. Let $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$. Then the sequence $$ \\pi_1(\\Spec(K^{sh})) \\to \\pi_1(\\Spec(K)) \\to \\pi_1(X) \\to 1 $$ is exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1)."}
+{"_id": "4057", "title": "pione-lemma-get-algebraic-closure", "text": "Let $A$ be a normal domain whose fraction field $K$ is separably algebraically closed. Let $\\mathfrak p \\subset A$ be a nonzero prime ideal. Then the residue field $\\kappa(\\mathfrak p)$ is algebraically closed."}
+{"_id": "4058", "title": "pione-lemma-normal-local-domain-separablly-closed-fraction-field", "text": "A normal local ring with separably closed fraction field is strictly henselian."}
+{"_id": "4059", "title": "pione-lemma-inertia-base-change", "text": "Let $G$ be a finite group acting on a ring $R$. Let $R^G \\to A$ be a ring map. Let $\\mathfrak q' \\subset A \\otimes_{R^G} R$ be a prime lying over the prime $\\mathfrak q \\subset R$. Then $$ I_\\mathfrak q = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\} $$ is equal to $$ I_{\\mathfrak q'} = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q') = \\mathfrak q'\\text{ and } \\sigma \\bmod \\mathfrak q' = \\text{id}_{\\kappa(\\mathfrak q')}\\} $$"}
+{"_id": "4060", "title": "pione-lemma-inertia-invariants-etale", "text": "Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$ be a prime. Set $$ I = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q \\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_\\mathfrak q\\} $$ Then $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$."}
+{"_id": "4061", "title": "pione-lemma-inertial-invariants-unramified", "text": "Let $A$ be a normal domain with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \\text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak q \\subset B$. Set $$ I = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q \\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\} $$ Then $(B^I)_{B^I \\cap \\mathfrak q}$ is a filtered colimit of \\'etale $A$-algebras."}
+{"_id": "4062", "title": "pione-lemma-identify-inertia", "text": "In the situation described above, via the isomorphism $\\pi_1(U) = \\text{Gal}(K^{sep}/K)$ the diagram (\\ref{equation-inertia-diagram-pione}) translates into the diagram $$ \\xymatrix{ I \\ar[r] \\ar[rd]_1 & D \\ar[d] \\ar[r] & \\text{Gal}(K^{sep}/K) \\ar[d] \\\\ & \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\ar[r] & \\text{Gal}(M/K) } $$ where $K^{sep}/M/K$ is the maximal subextension unramified with respect to $A$. Moreover, the vertical arrows are surjective, the kernel of the left vertical arrow is $I$ and the kernel of the right vertical arrow is the smallest closed normal subgroup of $\\text{Gal}(K^{sep}/K)$ containing $I$."}
+{"_id": "4065", "title": "pione-lemma-structure-decomposition", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m$ be a maximal ideal of $B$. Let $G = \\text{Gal}(L/K)$, $D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$, and $I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m = \\text{id}_{\\kappa(\\mathfrak m)}\\}$. The decomposition group $D$ fits into a canonical exact sequence $$ 1 \\to I \\to D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A) \\to 1 $$ The inertia group $I$ fits into a canonical exact sequence $$ 1 \\to P \\to I \\to I_t \\to 1 $$ such that \\begin{enumerate} \\item $P$ is a normal subgroup of $D$, \\item $P$ is a pro-$p$-group if the characteristic of $\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$ is zero, \\item there is a multiplicatively directed $S \\subset \\mathbf{N}$ such that $\\kappa(\\mathfrak m)$ contains a primitive $n$th root of unity for each $n \\in S$ (elements of $S$ are prime to $p$), \\item there exists a canonical surjective map $$ \\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m)) $$ whose kernel is $P$, which satisfies $\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$ for $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism $I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$. \\end{enumerate}"}
+{"_id": "4067", "title": "pione-lemma-limit", "text": "Let $I$ be a directed set. Let $X_i$ be an inverse system of quasi-compact and quasi-separated schemes over $I$ with affine transition morphisms. Let $X = \\lim X_i$ as in Limits, Section \\ref{limits-section-limits}. Then there is an equivalence of categories $$ \\colim \\textit{F\\'Et}_{X_i} = \\textit{F\\'Et}_X $$ If $X_i$ is connected for all sufficiently large $i$ and $\\overline{x}$ is a geometric point of $X$, then $$ \\pi_1(X, \\overline{x}) = \\lim \\pi_1(X_i, \\overline{x}) $$"}
+{"_id": "4068", "title": "pione-lemma-perfection", "text": "Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme over $k$. Then $X_{k^{perf}}$ is connected and $\\pi_1(X_{k^{perf}}) \\to \\pi_1(X)$ is an isomorphism."}
+{"_id": "4070", "title": "pione-lemma-stein-factorization-etale", "text": "\\begin{reference} \\cite[Expose X, Proposition 1.2, p. 262]{SGA1}. \\end{reference} Let $f : X \\to S$ be a proper morphism of schemes. Let $X \\to S' \\to S$ be the Stein factorization of $f$, see More on Morphisms, Theorem \\ref{more-morphisms-theorem-stein-factorization-general}. If $f$ is of finite presentation, flat, with geometrically reduced fibres, then $S' \\to S$ is finite \\'etale."}
+{"_id": "4071", "title": "pione-lemma-specialization-map-base-change", "text": "Consider a commutative diagram $$ \\xymatrix{ Y \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\ T \\ar[r] & S } $$ of schemes where $f$ and $g$ are proper with geometrically connected fibres. Let $t' \\leadsto t$ be a specialization of points in $T$ and consider a specialization map $sp : \\pi_1(Y_{\\overline{t}'}) \\to \\pi_1(Y_{\\overline{t}})$ as above. Then there is a commutative diagram $$ \\xymatrix{ \\pi_1(Y_{\\overline{t}'}) \\ar[r]_{sp} \\ar[d] & \\pi_1(Y_{\\overline{t}}) \\ar[d] \\\\ \\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & \\pi_1(X_{\\overline{s}}) } $$ of specialization maps where $\\overline{s}$ and $\\overline{s}'$ are the images of $\\overline{t}$ and $\\overline{t}'$."}
+{"_id": "4074", "title": "pione-lemma-specialization-map-discrete-valuation-ring", "text": "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres. Let $s' \\leadsto s$ be a specialization of points of $S$ and let $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$ be a specialization map. If $S$ is Noetherian, then there exists a strictly henselian discrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_R$ defined above."}
+{"_id": "4075", "title": "pione-lemma-restriction-fully-faithful", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Assume the completion functor $$ \\textit{Coh}(\\mathcal{O}_X) \\longrightarrow  \\textit{Coh}(X, \\mathcal{I}),\\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is fully faithful."}
+{"_id": "4076", "title": "pione-lemma-restriction-equivalence", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Assume the completion functor $$ \\textit{Coh}(\\mathcal{O}_X) \\longrightarrow  \\textit{Coh}(X, \\mathcal{I}),\\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is an equivalence."}
+{"_id": "4077", "title": "pione-lemma-restriction-fully-faithful-general", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(X, \\mathcal{I}), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ defines is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$ is fully faithful."}
+{"_id": "4078", "title": "pione-lemma-restriction-equivalence-general", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(X, \\mathcal{I}), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor $$ \\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y $$ is an equivalence."}
+{"_id": "4079", "title": "pione-lemma-restriction-faithful", "text": "Let $X$ be a scheme and let $Y \\subset X$ be a closed subscheme. If every connected component of $X$ meets $Y$, then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is faithful."}
+{"_id": "4080", "title": "pione-lemma-restriction-fully-faithful-special", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme. Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Assume one of the following holds \\begin{enumerate} \\item $X$ is quasi-affine and $\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$ is an isomorphism, or \\item $X$ has an ample invertible module $\\mathcal{L}$ and $\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to \\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$ is an isomorphism for all $m \\gg 0$, or \\item for every finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ the map $\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$ is an isomorphism. \\end{enumerate} Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is fully faithful."}
+{"_id": "4084", "title": "pione-lemma-fill-in-missing", "text": "In Situation \\ref{situation-local-lefschetz}. Let $V \\to U$ be a finite morphism.  Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$, let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of $U$ and $V$ to $X'$. If $Y' \\to X'$ is a finite morphism such that $V' = Y' \\times_{X'} U'$, then there exists a finite morphism $Y \\to X$ such that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$."}
+{"_id": "4085", "title": "pione-lemma-fully-faithful-henselian-completion", "text": "In Situation \\ref{situation-local-lefschetz} assume $A$ is henselian or more generally that $(A, (f))$ is a henselian pair. Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$, let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $U'_0$ be the base changes of $U$ and $U_0$ to $X'$. If $\\textit{F\\'Et}_{U'} \\to \\textit{F\\'Et}_{U'_0}$ is fully faithful, then $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$ is fully faithful."}
+{"_id": "4086", "title": "pione-lemma-fully-faithful-simple", "text": "In Situation \\ref{situation-local-lefschetz}. Assume \\begin{enumerate} \\item[(a)] $A$ has a dualizing complex, \\item[(b)] the pair $(A, (f))$ is henselian, \\item[(c)] one of the following is true \\begin{enumerate} \\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 3$, or \\item[(ii)] for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ we have $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$. \\end{enumerate} \\end{enumerate} Then the restriction functor $\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$ is fully faithful."}
+{"_id": "4087", "title": "pione-lemma-fully-faithful-minimal", "text": "\\begin{reference} \\cite[Corollary 1.11]{Bhatt-local} \\end{reference} In Situation \\ref{situation-local-lefschetz}. Assume \\begin{enumerate} \\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$ are annihilated by a power of $f$, and \\item $A$ is henselian or more generally $(A, (f))$ is a henselian pair. \\end{enumerate} Then the restriction functor $\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$ is fully faithful."}
+{"_id": "4089", "title": "pione-lemma-sections-over-punctured-spec", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$ and let $U = X \\setminus \\{\\mathfrak m\\}$. Let $\\pi : Y \\to X$ be a finite morphism such that $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for all closed points $y \\in Y$. Then $Y$ is the spectrum of $B = \\mathcal{O}_Y(\\pi^{-1}(U))$."}
+{"_id": "4090", "title": "pione-lemma-reformulate-purity", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$ and let $U = X \\setminus \\{\\mathfrak m\\}$. Let $V$ be finite \\'etale over $U$. Assume $A$ has depth $\\geq 2$. The following are equivalent \\begin{enumerate} \\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale, \\item $B = \\Gamma(V, \\mathcal{O}_V)$ is finite \\'etale over $A$. \\end{enumerate}"}
+{"_id": "4091", "title": "pione-lemma-reformulate-purity-normal", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$ and let $U = X \\setminus \\{\\mathfrak m\\}$. Assume $A$ is normal of dimension $\\geq 2$. The functor $$ \\textit{F\\'Et}_U \\longrightarrow \\left\\{ \\begin{matrix} \\text{finite normal }A\\text{-algebras }B\\text{ such} \\\\ \\text{that }\\Spec(B) \\to X\\text{ is \\'etale over }U \\end{matrix} \\right\\}, \\quad V \\longmapsto \\Gamma(V, \\mathcal{O}_V) $$ is an equivalence. Moreover, $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale if and only if $B = \\Gamma(V, \\mathcal{O}_V)$ is finite \\'etale over $A$."}
+{"_id": "4092", "title": "pione-lemma-purity-and-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$ and let $U = X \\setminus \\{\\mathfrak m\\}$. Let $V$ be finite \\'etale over $U$. Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$, let $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of $U$ and $V$ to $X'$. The following are equivalent \\begin{enumerate} \\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale, and \\item $V' = Y' \\times_{X'} U'$ for some $Y' \\to X'$ finite \\'etale. \\end{enumerate}"}
+{"_id": "4094", "title": "pione-lemma-lift-purity-general", "text": "In Situation \\ref{situation-local-lefschetz}. Let $V$ be finite \\'etale over $U$. Assume \\begin{enumerate} \\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$ are annihilated by a power of $f$, \\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$ for some $Y_0 \\to X_0$ finite \\'etale. \\end{enumerate} Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale."}
+{"_id": "4095", "title": "pione-lemma-lift-purity", "text": "In Situation \\ref{situation-local-lefschetz}. Let $V$ be finite \\'etale over $U$. Assume \\begin{enumerate} \\item $A$ has depth $\\geq 3$, \\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$ for some $Y_0 \\to X_0$ finite \\'etale. \\end{enumerate} Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale."}
+{"_id": "4096", "title": "pione-lemma-find-point-codim-1", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring with $\\dim(A) \\geq 1$. Let $f \\in \\mathfrak m$. Then there exist a $\\mathfrak p \\in V(f)$ with $\\dim(A_\\mathfrak p) = 1$."}
+{"_id": "4097", "title": "pione-lemma-ramification-quasi-finite-flat", "text": "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Let $x \\in X$. Assume \\begin{enumerate} \\item $f$ is flat, \\item $f$ is quasi-finite at $x$, \\item $x$ is not a generic point of an irreducible component of $X$, \\item for specializations $x' \\leadsto x$ with $\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$. \\end{enumerate} Then $f$ is \\'etale at $x$."}
+{"_id": "4098", "title": "pione-lemma-local-purity", "text": "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $d \\geq 2$. Set $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$. Then \\begin{enumerate} \\item the functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ is essentially surjective, i.e., purity holds for $A$, \\item any finite $A \\to B$ with $B$ normal which induces a finite \\'etale morphism on punctured spectra is \\'etale. \\end{enumerate}"}
+{"_id": "4099", "title": "pione-lemma-purity", "text": "\\begin{reference} \\cite{Nagata-Purity} and \\cite[Exp. X, Thm. 3.1]{SGA1} \\end{reference} \\begin{history} This result was first stated and proved by Zariski in geometric form in \\cite{Zariski-Purity}. The generalization to nonperfect ground fields by Nagata was published as the next article in the same volume of the Proceedings of the National Academy of Sciences of the United States of America in \\cite{Nagata-Remarks-Purity}. In the following year Nagata proved the result for Noetherian local rings in \\cite{Nagata-Purity}. His proof uses a result of Chow which is a Bertini theorem for complete local domains, see \\cite{Chow-Bertini}; the history of Bertini's theorems is discussed in Kleiman's historical article \\cite{Kleiman-Bertini}. A few years later a completely different proof was found by Auslander, see \\cite{Auslander-Purity}. \\end{history} Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Let $x \\in X$ and set $y = f(x)$. Assume \\begin{enumerate} \\item $\\mathcal{O}_{X, x}$ is normal, \\item $\\mathcal{O}_{Y, y}$ is regular, \\item $f$ is quasi-finite at $x$, \\item $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) \\geq 1$ \\item for specializations $x' \\leadsto x$ with $\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$. \\end{enumerate} Then $f$ is \\'etale at $x$."}
+{"_id": "4100", "title": "pione-lemma-extend-S2", "text": "Let $j : U \\to X$ be an open immersion of locally Noetherian schemes such that $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$ for $x \\not \\in U$. Let $\\pi : V \\to U$ be finite \\'etale. Then \\begin{enumerate} \\item $\\mathcal{B} = j_*\\pi_*\\mathcal{O}_V$ is a reflexive coherent $\\mathcal{O}_X$-algebra, set $Y = \\underline{\\Spec}_X(\\mathcal{B})$, \\item $Y \\to X$ is the unique finite morphism such that $V = Y \\times_X U$ and $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for $y \\in Y \\setminus V$, \\item $Y \\to X$ is \\'etale at $y$ if and only if $Y \\to X$ is flat at $y$, and \\item $Y \\to X$ is \\'etale if and only if $\\mathcal{B}$ is finite locally free as an $\\mathcal{O}_X$-module. \\end{enumerate} Moreover, (a) the construction of $\\mathcal{B}$ and $Y \\to X$ commutes with base change by flat morphisms $X' \\to X$ of locally Noetherian schemes, and (b) if $V' \\to U'$ is a finite \\'etale morphism with $U \\subset U' \\subset X$ open which restricts to $V \\to U$ over $U$, then there is a unique isomorphism $Y' \\times_X U' = V'$ over $U'$."}
+{"_id": "4101", "title": "pione-lemma-extend-pure", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes such that purity holds for $\\mathcal{O}_{X, x}$ for all $x \\not \\in U$. Then $$ \\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U $$ is essentially surjective."}
+{"_id": "4105", "title": "pione-lemma-essentially-surjective-general-better", "text": "In Situation \\ref{situation-local-lefschetz} assume \\begin{enumerate} \\item $A$ has a dualizing complex and is $f$-adically complete, \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\end{enumerate} Then the restriction functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0} $$ is an equivalence."}
+{"_id": "4106", "title": "pione-lemma-essentially-surjective-general", "text": "In Situation \\ref{situation-local-lefschetz} assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite $A$-modules. \\end{enumerate} Then the restriction functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0} $$ is an equivalence."}
+{"_id": "4107", "title": "pione-lemma-equivalence-better", "text": "In Situation \\ref{situation-local-lefschetz} assume \\begin{enumerate} \\item $A$ has a dualizing complex and is $f$-adically complete, \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\item for every maximal ideal $\\mathfrak p \\subset A_f$ purity holds for $(A_f)_\\mathfrak p$. \\end{enumerate} Then the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$ is essentially surjective."}
+{"_id": "4108", "title": "pione-lemma-equivalence", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite $A$-modules, \\item for every maximal ideal $\\mathfrak p \\subset A_f$ purity holds for $(A_f)_\\mathfrak p$. \\end{enumerate} Then the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$ is essentially surjective."}
+{"_id": "4109", "title": "pione-lemma-purity-inherited-by-hypersurface-better", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex and is $f$-adically complete, \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\item for every maximal ideal $\\mathfrak p \\subset A_f$ purity holds for $(A_f)_\\mathfrak p$, and \\item purity holds for $A$. \\end{enumerate} Then purity holds for $A/fA$."}
+{"_id": "4110", "title": "pione-lemma-purity-inherited-by-hypersurface", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite $A$-modules, \\item for every maximal ideal $\\mathfrak p \\subset A_f$ purity holds for $(A_f)_\\mathfrak p$, \\item purity holds for $A$. \\end{enumerate} Then purity holds for $A/fA$."}
+{"_id": "4111", "title": "pione-lemma-fully-faithful-power-series-over-depth2", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$. Let $B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$. Set $Y = \\Spec(B)$ and $Y_0 = V(x_1, \\ldots, x_d)$. For any open subscheme $V \\subset Y$ with $V_0 = V \\cap Y_0$ equal to $Y_0 \\setminus \\{\\mathfrak m_B\\}$ the restriction functor $$ \\textit{F\\'Et}_V \\longrightarrow \\textit{F\\'Et}_{V_0} $$ is fully faithful."}
+{"_id": "4112", "title": "pione-lemma-purity-power-series-over-depth2", "text": "\\begin{slogan} Ramanujam-Samuel for finite \\'etale covers \\end{slogan} Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$. Let $B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$. For any open $V \\subset Y = \\Spec(B)$ which contains \\begin{enumerate} \\item any prime $\\mathfrak q \\subset B$ such that $\\mathfrak q \\cap A \\not = \\mathfrak m$, \\item the prime $\\mathfrak m B$ \\end{enumerate} the functor $ \\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_V $ is an equivalence. In particular purity holds for $B$."}
+{"_id": "4113", "title": "pione-lemma-purity-smooth-over-depth2", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$ be an open subscheme. Assume \\begin{enumerate} \\item $f$ is smooth, \\item $S$ is Noetherian, \\item for $s \\in S$ with $\\text{depth}(\\mathcal{O}_{S, s}) \\leq 1$ we have $X_s = U_s$, \\item $U_s \\subset X_s$ is dense for all $s \\in S$. \\end{enumerate} Then $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ is an equivalence."}
+{"_id": "4114", "title": "pione-lemma-characterize-rational-singularity", "text": "Let $A$ be a Noetherian normal local domain of dimension $2$. Assume $A$ is Nagata, has a dualizing module $\\omega_A$, and has a resolution of singularities $f : X \\to \\Spec(A)$. Let $\\omega_X$ be as in Resolution of Surfaces, Remark \\ref{resolve-remark-dualizing-setup}. If $\\omega_X \\cong \\mathcal{O}_X(E)$ for some effective Cartier divisor $E \\subset X$ supported on the exceptional fibre, then $A$ defines a rational singularity. If $f$ is a minimal resolution, then $E = 0$."}
+{"_id": "4115", "title": "pione-lemma-key-purity-ramification", "text": "Let $f : X \\to \\Spec(A)$ be a finite type morphism. Let $x \\in X$ be a point. Assume \\begin{enumerate} \\item $A$ is an excellent regular local ring, \\item $\\mathcal{O}_{X, x}$ is normal of dimension $2$, \\item $f$ is \\'etale outside of $\\overline{\\{x\\}}$. \\end{enumerate} Then $f$ is \\'etale at $x$."}
+{"_id": "4116", "title": "pione-lemma-purity-ramification", "text": "\\begin{reference} This result for complex spaces can be found on page 170 of \\cite{Fischer}. In general this is \\cite[Theorem 2.4]{Zong} attributed to Gabber. \\end{reference} Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Let $x \\in X$ and set $y = f(x)$. Assume \\begin{enumerate} \\item $\\mathcal{O}_{X, x}$ is normal of dimension $\\geq 1$, \\item $\\mathcal{O}_{Y, y}$ is regular, \\item $f$ is locally of finite type, and \\item for specializations $x' \\leadsto x$ with $\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is \\'etale at $x'$. \\end{enumerate} Then $f$ is \\'etale at $x$."}
+{"_id": "4117", "title": "pione-lemma-structure-cohomology", "text": "Let $(A, \\mathfrak m)$ be a regular local ring which contains a field. Let $f : V \\to \\Spec(A)$ be \\'etale and quasi-compact. Assume that $\\mathfrak m \\not \\in f(V)$ and assume that $g : V \\to \\Spec(A) \\setminus \\{\\mathfrak m\\}$ is affine. Then $H^i(V, \\mathcal{O}_V)$, $i > 0$ is isomorphic to a direct sum of copies of the injective hull of the residue field of $A$."}
+{"_id": "4118", "title": "pione-lemma-conclude", "text": "In the situation of Lemma \\ref{lemma-structure-cohomology} assume that $H^i(V, \\mathcal{O}_V) = 0$ for $i \\geq \\dim(A) - 1$. Then $V$ is affine."}
+{"_id": "4119", "title": "pione-lemma-specialization-map-surjective", "text": "Let $f : X \\to S$ be a flat proper morphism with geometrically connected fibres. Let $s' \\leadsto s$ be a specialization. If $X_s$ is geometrically reduced, then the specialization map $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$ is surjective."}
+{"_id": "4120", "title": "pione-lemma-pullback-tame-codim1", "text": "Let $X' \\to X$ be a morphism of locally Noetherian schemes. Let $U \\subset X$ be a dense open. Assume \\begin{enumerate} \\item $U' = f^{-1}(U)$ is dense open in $X'$, \\item for every prime divisor $Z \\subset X$ with $Z \\cap U = \\emptyset$ the local ring $\\mathcal{O}_{X, \\xi}$ of $X$ at the generic point $\\xi$ of $Z$ is a discrete valuation ring, \\item for every prime divisor $Z' \\subset X'$ with $Z' \\cap U' = \\emptyset$ the local ring $\\mathcal{O}_{X', \\xi'}$ of $X'$ at the generic point $\\xi'$ of $Z'$ is a discrete valuation ring, \\item if $\\xi' \\in X'$ is as in (3), then $\\xi = f(\\xi')$ is as in (2). \\end{enumerate} Then if $f : Y \\to U$ is finite \\'etale and $Y$ is unramified, resp.\\ tamely ramified over $X$ in codimension $1$, then $Y' = Y \\times_X X' \\to U'$ is finite \\'etale and $Y'$ is unramified, resp.\\ tamely ramified over $X'$ in codimension $1$."}
+{"_id": "4121", "title": "pione-lemma-purity-one-divisor", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an effective Cartier divisor such that $D$ is a regular scheme. Let $Y \\to X \\setminus D$ be a finite \\'etale morphism. If $Y$ is unramified over $X$ in codimension $1$, then there exists a finite \\'etale morphism $Y' \\to X$ whose restriction to $X \\setminus D$ is $Y$."}
+{"_id": "4122", "title": "pione-lemma-abhyankar-one-divisor", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an effective Cartier divisor such that $D$ is a regular scheme. Let $Y \\to X \\setminus D$ be a finite \\'etale morphism. If $Y$ is tamely ramified over $X$ in codimension $1$, then \\'etale locally on $X$ the morphism $Y \\to X$ is as given as a finite disjoint union of standard tamely ramified morphisms as described in Example \\ref{example-tamely-ramified}."}
+{"_id": "4123", "title": "pione-lemma-extend-tame-covering-normal", "text": "In the situation of Lemma \\ref{lemma-abhyankar-one-divisor} the normalization of $X$ in $Y$ is a finite locally free morphism $\\pi : Y' \\to X$ such that \\begin{enumerate} \\item the restriction of $Y'$ to $X \\setminus D$ is isomorphic to $Y$, \\item $D' = \\pi^{-1}(D)_{red}$ is an effective Cartier divisor on $Y'$, and \\item $D'$ is a regular scheme. \\end{enumerate} Moreover, \\'etale locally on $X$ the morphism $Y' \\to X$ is a finite disjoint union of morphisms $$ \\Spec(A[x]/(x^e - f)) \\to \\Spec(A) $$ where $A$ is a Noetherian ring, $f \\in A$ is a nonzerodivisor with $A/fA$ regular, and $e \\geq 1$ is invertible in $A$."}
+{"_id": "4124", "title": "pione-lemma-tame-covering-split", "text": "In the situation of Lemma \\ref{lemma-abhyankar-one-divisor} let $Y' \\to X$ be as in Lemma \\ref{lemma-extend-tame-covering-normal}. Let $R$ be a discrete valuation ring with fraction field $K$. Let $$ t : \\Spec(R) \\to X $$ be a morphism such that the scheme theoretic inverse image $t^{-1}D$ is the reduced closed point of $\\Spec(R)$. \\begin{enumerate} \\item If $t|_{\\Spec(K)}$ lifts to a point of $Y$, then we get a lift $t' : \\Spec(R) \\to Y'$ such that $Y' \\to X$ is \\'etale along $t'(\\Spec(R))$. \\item If $\\Spec(K) \\times_X Y$ is isomorphic to a disjoint union of copies of $\\Spec(K)$, then $Y' \\to X$ is finite \\'etale over an open neighbourhood of $t(\\Spec(R))$. \\end{enumerate}"}
+{"_id": "4125", "title": "pione-lemma-extend-covering", "text": "Let $S$ be an integral normal Noetherian scheme with generic point $\\eta$. Let $f : X \\to S$ be a smooth morphism with geometrically connected fibres. Let $\\sigma : S \\to X$ be a section of $f$. Let $Z \\to X_\\eta$ be a finite \\'etale Galois cover (Section \\ref{section-finite-etale-under-galois}) with group $G$ of order invertible on $S$ such that $Z$ has a $\\kappa(\\eta)$-rational point mapping to $\\sigma(\\eta)$. Then there exists a finite \\'etale Galois cover $Y \\to X$ with group $G$ whose restriction to $X_\\eta$ is $Z$."}
+{"_id": "4127", "title": "pione-lemma-lower-invariant", "text": "In the situation above, if $NF(C, \\varphi, \\tau) > 0$, then there exist an \\'etale $k$-algebra map $\\varphi'$ and a surjective $k$-algebra map $\\tau'$ fitting into the commutative diagram $$ \\xymatrix{ & B \\\\ C \\ar[r] & C/\\varphi'(I)C \\ar[u]_{\\tau'} \\\\ k[x_1, \\ldots, x_n] \\ar[u]^{\\varphi'} \\ar[r] & A \\ar[u] \\ar@/_3em/[uu]_\\pi } $$ with $NF(C, \\varphi', \\tau') < NF(C, \\varphi, \\tau)$."}
+{"_id": "4129", "title": "pione-lemma-dominate-affine-space", "text": "Let $k$ be a field of characteristic $p > 0$. Let $Z \\subset \\mathbf{A}^n_k$ be a closed subscheme. Let $Y \\to Z$ be finite \\'etale. There exists a finite \\'etale morphism $f : U \\to \\mathbf{A}^n_k$ such that there is an open and closed immersion $Y \\to f^{-1}(Z)$ over $Z$."}
+{"_id": "4130", "title": "pione-proposition-galois", "text": "\\begin{reference} This is a weak version of \\cite[Expos\\'e V]{SGA1}. The proof is borrowed from \\cite[Theorem 7.2.5]{BS}. \\end{reference} Let $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$ be as in Example \\ref{example-from-C-F-to-G-sets}. The functor $F : \\mathcal{C} \\to \\textit{Finite-}G\\textit{-Sets}$ (\\ref{equation-remember}) an equivalence."}
+{"_id": "4131", "title": "pione-proposition-universal-homeomorphism", "text": "Let $f : X \\to Y$ be a universal homeomorphism of schemes. Then $$ \\textit{F\\'Et}_Y \\longrightarrow \\textit{F\\'Et}_X,\\quad V \\longmapsto V \\times_Y X $$ is an equivalence. Thus if $X$ and $Y$ are connected, then $f$ induces an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})$ of fundamental groups."}
+{"_id": "4132", "title": "pione-proposition-normal", "text": "Let $X$ be a normal integral scheme with function field $K$. Then the canonical map (\\ref{equation-inclusion-generic-point}) $$ \\text{Gal}(K^{sep}/K) = \\pi_1(\\eta, \\overline{\\eta}) \\longrightarrow \\pi_1(X, \\overline{\\eta}) $$ is identified with the quotient map $\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K)$ where $M \\subset K^{sep}$ is the union of the finite subextensions $L$ such that $X$ is unramified in $L$."}
+{"_id": "4133", "title": "pione-proposition-first-homotopy-sequence", "text": "Let $f : X \\to S$ be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume $S$ is connected and let $\\overline{s}$ be a geometric point of $S$. Then there is an exact sequence $$ \\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S) \\to 1 $$ of fundamental groups."}
+{"_id": "4136", "title": "pione-proposition-lefschetz-fully-faithful", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Assume that for all $x \\in X \\setminus Y$ we have $$ \\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 1 $$ Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is fully faithful. In fact, for any open subscheme $V \\subset X$ containing $Y$ the restriction functor $\\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$ is fully faithful."}
+{"_id": "4137", "title": "pione-proposition-lefschetz-equivalence-general", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Let $\\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \\in X \\setminus Y$ we have $$ \\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2 $$ Then the restriction functor $$ \\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y $$ is an equivalence."}
+{"_id": "4139", "title": "pione-proposition-specialization-map-isomorphism", "text": "Let $f : X \\to S$ be a smooth proper morphism with geometrically connected fibres. Let $s' \\leadsto s$ be a specialization. If the characteristic to $\\kappa(s)$ is zero, then the specialization map $$ sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}}) $$ is an isomorphism."}
+{"_id": "4148", "title": "stacks-cohomology-lemma-flat-pullback-quasi-coherent", "text": "If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a flat morphism of algebraic stacks then $f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$ is an exact functor."}
+{"_id": "4149", "title": "stacks-cohomology-lemma-general-pushforward", "text": "Let $\\mathcal{M}$ be a rule which associates to every algebraic stack $\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ such that \\begin{enumerate} \\item $\\mathcal{M}_\\mathcal{X}$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ (see Homology, Definition \\ref{homology-definition-serre-subcategory}) for all algebraic stacks $\\mathcal{X}$, \\item for a smooth morphism of algebraic stacks $f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps $\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$, \\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ is in $\\mathcal{M}_\\mathcal{X}$ if and only if $f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and \\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic stacks such that $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$ into $\\mathcal{M}_\\mathcal{X}$. \\end{enumerate} Then for any quasi-compact and quasi-separated morphism  $f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$ into $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in \\'etale topology.)"}
+{"_id": "4151", "title": "stacks-cohomology-lemma-check-lqc-on-etale-covering", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$ is locally quasi-coherent, then so is $\\mathcal{F}$."}
+{"_id": "4152", "title": "stacks-cohomology-lemma-pushforward-locally-quasi-coherent", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let  $\\mathcal{F}$ be a locally quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$. Then $R^if_*\\mathcal{F}$ (computed in the \\'etale topology) is locally quasi-coherent on $\\mathcal{Y}_\\etale$."}
+{"_id": "4153", "title": "stacks-cohomology-lemma-check-lqc-on-flat-covering", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$ is locally quasi-coherent, then so is $\\mathcal{F}$."}
+{"_id": "4154", "title": "stacks-cohomology-lemma-check-flat-comparison-on-etale-covering", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$. \\begin{enumerate} \\item If $\\mathcal{F}$ has the flat base change property then for any morphism $g : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks, the pullback $g^*\\mathcal{F}$ does too. \\item The full subcategory of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ consisting of modules with the flat base change property is a weak Serre subcategory. \\item  Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of smooth morphisms of algebraic stacks such that $|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each $f_i^*\\mathcal{F}$ has the flat base change property then so does $\\mathcal{F}$. \\item The category of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_\\etale$ with the flat base change property has colimits and they agree with colimits in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "4155", "title": "stacks-cohomology-lemma-flat-comparison", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let  $\\mathcal{F}$ be an object of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ which is locally quasi-coherent and has the flat base change property. Then each $R^if_*\\mathcal{F}$ (computed in the \\'etale topology) has the flat base change property."}
+{"_id": "4156", "title": "stacks-cohomology-lemma-check-lqc-fbc-on-covering", "text": "Let $\\mathcal{X}$ be an algebraic stack. With $\\mathcal{M}_\\mathcal{X}$ the category of locally quasi-coherent modules with the flat base change property. \\begin{enumerate} \\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$, then $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$. \\item  Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$, then $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$. \\end{enumerate}"}
+{"_id": "4157", "title": "stacks-cohomology-lemma-parasitic", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. \\begin{enumerate} \\item If $\\mathcal{F}$ is parasitic and $g : \\mathcal{Y} \\to \\mathcal{X}$ is a flat morphism of algebraic stacks, then $g^*\\mathcal{F}$ is parasitic. \\item For $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$ we have \\begin{enumerate} \\item the $\\tau$ sheafification of a parasitic presheaf of modules is parasitic, and \\item the full subcategory of $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$ consisting of parasitic modules is a Serre subcategory. \\end{enumerate} \\item Suppose $\\mathcal{F}$ is a sheaf for the \\'etale topology. Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of smooth morphisms of algebraic stacks such that $|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each $f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$. \\item Suppose $\\mathcal{F}$ is a sheaf for the fppf topology. Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks such that $|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each $f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "4158", "title": "stacks-cohomology-lemma-pushforward-parasitic", "text": "Let $\\tau \\in \\{\\etale, fppf\\}$. Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$ be a parasitic object of $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. \\begin{enumerate} \\item $H^i_\\tau(\\mathcal{X}, \\mathcal{F}) = 0$ for all $i$. \\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Then $R^if_*\\mathcal{F}$ (computed in $\\tau$-topology) is a parasitic object of $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$. \\end{enumerate}"}
+{"_id": "4159", "title": "stacks-cohomology-lemma-exact-sequence-quasi-coherent-parasitic-cohomology", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}^\\bullet$ be an exact complex in $\\QCoh(\\mathcal{O}_\\mathcal{X})$. Then the cohomology sheaves of $\\mathcal{F}^\\bullet$ in either the \\'etale or the fppf topology are parasitic $\\mathcal{O}_\\mathcal{X}$-modules."}
+{"_id": "4160", "title": "stacks-cohomology-lemma-adjoint", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{M}_\\mathcal{X}$ be the category of locally quasi-coherent modules with the flat base change property, see Proposition \\ref{proposition-lcq-flat-base-change}. The inclusion functor $i : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}$ has a right adjoint $$ Q : \\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X}) $$ such that $Q \\circ i$ is the identity functor."}
+{"_id": "4161", "title": "stacks-cohomology-lemma-adjoint-kernel-parasitic", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $Q : \\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$ be the functor constructed in Lemma \\ref{lemma-adjoint}. \\begin{enumerate} \\item The kernel of $Q$ is exactly the collection of parasitic objects of $\\mathcal{M}_\\mathcal{X}$. \\item For any object $\\mathcal{F}$ of $\\mathcal{M}_\\mathcal{X}$ both the kernel and the cokernel of the adjunction map $Q(\\mathcal{F}) \\to \\mathcal{F}$ are parasitic. \\item The functor $Q$ is exact. \\end{enumerate}"}
+{"_id": "4164", "title": "stacks-cohomology-lemma-lisse-etale", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item The inclusion functor $\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{X}_\\etale$ is fully faithful, continuous and cocontinuous. It follows that \\begin{enumerate} \\item there is a morphism of topoi $$ g : \\Sh(\\mathcal{X}_{lisse,\\etale}) \\longrightarrow \\Sh(\\mathcal{X}_\\etale) $$ with $g^{-1}$ given by restriction, \\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets, \\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and $\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms, \\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves, \\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and \\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$. \\end{enumerate} \\item The inclusion functor $\\mathcal{X}_{flat,fppf} \\to \\mathcal{X}_{fppf}$ is fully faithful, continuous and cocontinuous. It follows that \\begin{enumerate} \\item there is a morphism of topoi $$ g : \\Sh(\\mathcal{X}_{flat,fppf}) \\longrightarrow \\Sh(\\mathcal{X}_{fppf}) $$ with $g^{-1}$ given by restriction, \\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets, \\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and $\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms, \\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves, \\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and \\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4165", "title": "stacks-cohomology-lemma-lisse-etale-modules", "text": "Let $\\mathcal{X}$ be an algebraic stack. Notation as in Lemma \\ref{lemma-lisse-etale}. \\begin{enumerate} \\item There exists a functor $$ g_! : \\textit{Mod}(\\mathcal{X}_{lisse,\\etale}, \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\longrightarrow \\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}}) $$ which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \\text{id}$. \\item There exists a functor $$ g_! : \\textit{Mod}(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\longrightarrow \\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}}) $$ which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \\text{id}$. \\end{enumerate}"}
+{"_id": "4166", "title": "stacks-cohomology-lemma-lisse-etale-structure-sheaf", "text": "Let $\\mathcal{X}$ be an algebraic stack. Notation as in Lemmas \\ref{lemma-lisse-etale} and \\ref{lemma-lisse-etale-modules}. \\begin{enumerate} \\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}} = \\mathcal{O}_\\mathcal{X}$. \\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{flat, fppf}} = \\mathcal{O}_\\mathcal{X}$. \\end{enumerate}"}
+{"_id": "4167", "title": "stacks-cohomology-lemma-parasitic-in-terms-flat-fppf", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module with the flat base change property on $\\mathcal{X}_\\etale$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is parasitic, and \\item $g^*\\mathcal{F} = 0$ where $g : \\Sh(\\mathcal{X}_{lisse,\\etale}) \\to \\Sh(\\mathcal{X}_\\etale)$ is as in Lemma \\ref{lemma-lisse-etale}. \\end{enumerate} \\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_{fppf}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is parasitic, and \\item $g^*\\mathcal{F} = 0$ where $g :  \\Sh(\\mathcal{X}_{flat,fppf}) \\to \\Sh(\\mathcal{X}_{fppf})$ is as in Lemma \\ref{lemma-lisse-etale}. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4168", "title": "stacks-cohomology-lemma-lisse-etale-functorial", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous functor $\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{Y}_{lisse,\\etale}$ which gives a morphism of ringed topoi fitting into the following commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{X}_{lisse,\\etale}) \\ar[r]_{g'} \\ar[d]_{f'} & \\Sh(\\mathcal{X}_\\etale) \\ar[d]^f \\\\ \\Sh(\\mathcal{Y}_{lisse,\\etale}) \\ar[r]^g & \\Sh(\\mathcal{Y}_\\etale) } $$ We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$. \\item If $f$ is flat, then $f$ restricts to a continuous and cocontinuous functor $\\mathcal{X}_{flat,fppf} \\to \\mathcal{Y}_{flat,fppf}$ which gives a morphism of ringed topoi fitting into the following commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{X}_{flat,fppf}) \\ar[r]_{g'} \\ar[d]_{f'} & \\Sh(\\mathcal{X}_{fppf}) \\ar[d]^f \\\\ \\Sh(\\mathcal{Y}_{flat,fppf}) \\ar[r]^g & \\Sh(\\mathcal{Y}_{fppf}) } $$ We have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$. \\end{enumerate}"}
+{"_id": "4169", "title": "stacks-cohomology-lemma-check-qc-on-etale-covering", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$ is quasi-coherent, then so is $\\mathcal{F}$. \\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules on $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$ is quasi-coherent, then so is $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "4170", "title": "stacks-cohomology-lemma-shriek-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack. Notation as in Lemma \\ref{lemma-lisse-etale}. \\begin{enumerate} \\item Let $\\mathcal{H}$ be a quasi-coherent $\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module  on the lisse-\\'etale site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a quasi-coherent module on $\\mathcal{X}$. \\item Let $\\mathcal{H}$ be a quasi-coherent $\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module  on the flat-fppf site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a quasi-coherent module on $\\mathcal{X}$. \\end{enumerate}"}
+{"_id": "4171", "title": "stacks-cohomology-lemma-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{M}_\\mathcal{X}$ be the category of locally quasi-coherent $\\mathcal{O}_\\mathcal{X}$-modules with the flat base change property. \\begin{enumerate} \\item With $g$ as in Lemma \\ref{lemma-lisse-etale} for the lisse-\\'etale site we have \\begin{enumerate} \\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$ \\xymatrix{ \\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} & \\QCoh(\\mathcal{X}_{lisse,\\etale}, \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\ar@<1ex>[l]^-{g_!} } $$ \\item if $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$ then $g^{-1}\\mathcal{F}$ is in $\\QCoh(\\mathcal{X}_{lisse,\\etale}, \\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$ and \\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in Lemma \\ref{lemma-adjoint}. \\end{enumerate} \\item With $g$ as in Lemma \\ref{lemma-lisse-etale} for the flat-fppf site we have \\begin{enumerate} \\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors $$ \\xymatrix{ \\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} & \\QCoh(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{g_!} } $$ \\item if $\\mathcal{F}$ is in $\\mathcal{M}_\\mathcal{X}$ then $g^{-1}\\mathcal{F}$ is in $\\QCoh(\\mathcal{X}_{flat,fppf}, \\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$ and \\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in Lemma \\ref{lemma-adjoint}. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4173", "title": "stacks-cohomology-proposition-lcq-flat-base-change", "text": "Summary of results on locally quasi-coherent modules having the flat base change property. \\begin{enumerate} \\item Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{F}$ is an object of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ which is locally quasi-coherent and has the flat base change property, then $\\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is an object of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. \\item The category of modules which are locally quasi-coherent and have the flat base change property is a weak Serre subcategory $\\mathcal{M}_\\mathcal{X}$ of both $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ and $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$. \\item Pullback $f^*$ along any morphism of algebraic stacks $f : \\mathcal{X} \\to \\mathcal{Y}$ induces a functor $f^* : \\mathcal{M}_\\mathcal{Y} \\to \\mathcal{M}_\\mathcal{X}$. \\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a quasi-compact and quasi-separated morphism of algebraic stacks and $\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{X}$, then \\begin{enumerate} \\item the derived direct image $Rf_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$ can be computed in either the \\'etale or the fppf topology with the same result, and \\item each $R^if_*\\mathcal{F}$ is an object of $\\mathcal{M}_\\mathcal{Y}$. \\end{enumerate} \\item The category $\\mathcal{M}_\\mathcal{X}$ has colimits and they agree with colimits in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ as well as in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "4174", "title": "stacks-cohomology-proposition-direct-image-quasi-coherent", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$ has a right adjoint $$ f_{\\QCoh, *} : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\longrightarrow \\QCoh(\\mathcal{O}_\\mathcal{Y}) $$ which can be defined as the composition $$ \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X} \\xrightarrow{f_*} \\mathcal{M}_\\mathcal{Y} \\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y}) $$ where the functors $f_*$ and $Q$ are as in Proposition \\ref{proposition-lcq-flat-base-change} and Lemma \\ref{lemma-adjoint}. Moreover, if we define $R^if_{\\QCoh, *}$ as the composition $$ \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X} \\xrightarrow{R^if_*} \\mathcal{M}_\\mathcal{Y} \\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y}) $$ then the sequence of functors $\\{R^if_{\\QCoh, *}\\}_{i \\geq 0}$ forms a cohomological $\\delta$-functor."}
+{"_id": "4181", "title": "sites-cohomology-lemma-trivial-torsor", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\\mathcal{C}$. A $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if $\\Gamma(\\mathcal{C}, \\mathcal{F}) \\not = \\emptyset$."}
+{"_id": "4182", "title": "sites-cohomology-lemma-torsors-h1", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{H}$ be an abelian sheaf on $\\mathcal{C}$. There is a canonical bijection between the set of isomorphism classes of $\\mathcal{H}$-torsors and $H^1(\\mathcal{C}, \\mathcal{H})$."}
+{"_id": "4183", "title": "sites-cohomology-lemma-h1-extensions", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$. There is a canonical bijection $$ \\Ext^1_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{O}, \\mathcal{F}) \\longrightarrow H^1(\\mathcal{C}, \\mathcal{F}) $$ which associates to the extension $$ 0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O} \\to 0 $$ the image of $1 \\in \\Gamma(\\mathcal{C}, \\mathcal{O})$ in $H^1(\\mathcal{C}, \\mathcal{F})$."}
+{"_id": "4185", "title": "sites-cohomology-lemma-h1-invertible", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site. There is a canonical isomorphism $$ H^1(\\mathcal{C}, \\mathcal{O}^*) = \\Pic(\\mathcal{O}). $$ of abelian groups."}
+{"_id": "4186", "title": "sites-cohomology-lemma-cohomology-of-open", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. \\begin{enumerate} \\item If $\\mathcal{I}$ is an injective $\\mathcal{O}$-module then $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module. \\item For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ we have $H^p(U, \\mathcal{F}) = H^p(\\mathcal{C}/U, \\mathcal{F}|_U)$. \\end{enumerate}"}
+{"_id": "4187", "title": "sites-cohomology-lemma-cohomology-bigger-site", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume $u$ satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the associated morphism of topoi. For any abelian sheaf $\\mathcal{F}$ on $\\mathcal{D}$ we have isomorphisms $$ R\\Gamma(\\mathcal{C}, g^{-1}\\mathcal{F}) = R\\Gamma(\\mathcal{D}, \\mathcal{F}), $$ in particular $H^p(\\mathcal{C}, g^{-1}\\mathcal{F}) = H^p(\\mathcal{D}, \\mathcal{F})$ and for any $U \\in \\Ob(\\mathcal{C})$ we have isomorphisms $$ R\\Gamma(U, g^{-1}\\mathcal{F}) = R\\Gamma(u(U), \\mathcal{F}), $$ in particular $H^p(U, g^{-1}\\mathcal{F}) = H^p(u(U), \\mathcal{F})$. All of these isomorphisms are functorial in $\\mathcal{F}$."}
+{"_id": "4188", "title": "sites-cohomology-lemma-kill-cohomology-class-on-covering", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Let $U$ be an object of $\\mathcal{C}$. Let $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$. Then there exists a covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ such that $\\xi|_{U_i} = 0$ for all $i \\in I$."}
+{"_id": "4189", "title": "sites-cohomology-lemma-higher-direct-images", "text": "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. For any $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_\\mathcal{C}))$ the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf $$ V \\longmapsto H^i(u(V), \\mathcal{F}) $$"}
+{"_id": "4190", "title": "sites-cohomology-lemma-cech-h0", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{C}$ and \\item for every covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of the site $\\mathcal{C}$ the natural map $$ \\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F}) $$ (see Sites, Section \\ref{sites-section-sheafification}) is bijective. \\end{enumerate}"}
+{"_id": "4191", "title": "sites-cohomology-lemma-cech-exact-presheaves", "text": "The functor given by Equation (\\ref{equation-cech-functor}) is an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor})."}
+{"_id": "4192", "title": "sites-cohomology-lemma-cech-cohomology-delta-functor-presheaves", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. The functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$ form a $\\delta$-functor from the abelian category $\\textit{PAb}(\\mathcal{C})$ to the category of $\\mathbf{Z}$-modules (see Homology, Definition \\ref{homology-definition-cohomological-delta-functor})."}
+{"_id": "4193", "title": "sites-cohomology-lemma-cech-map-into", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. Consider the chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ of abelian presheaves $$ \\ldots \\to \\bigoplus_{i_0i_1i_2} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}} \\to \\bigoplus_{i_0i_1} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\to \\bigoplus_{i_0} \\mathbf{Z}_{U_{i_0}} \\to 0 \\to \\ldots $$ where the last nonzero term is placed in degree $0$ and where the map $$ \\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\times_u U_{i_{p + 1}}} \\longrightarrow \\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\widehat{U_{i_j}} \\ldots \\times_U U_{i_{p + 1}}} $$ is given by $(-1)^j$ times the canonical map. Then there is an isomorphism $$ \\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{F}) = \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ functorial in $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$."}
+{"_id": "4194", "title": "sites-cohomology-lemma-homology-complex", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. The chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ of presheaves of Lemma \\ref{lemma-cech-map-into} above is exact in positive degrees, i.e., the homology presheaves $H_i(\\mathbf{Z}_{\\mathcal{U}, \\bullet})$ are zero for $i > 0$."}
+{"_id": "4195", "title": "sites-cohomology-lemma-complex-tensored-still-exact", "text": "\\begin{slogan} The integral presheaf {\\v C}ech complex is a flat resolution of the constant presheaf of integers. \\end{slogan} Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$. The chain complex $$ \\mathbf{Z}_{\\mathcal{U}, \\bullet} \\otimes_{p, \\mathbf{Z}} \\mathcal{O} $$ is exact in positive degrees. Here $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ is the chain complex of Lemma \\ref{lemma-cech-map-into}, and the tensor product is over the constant presheaf of rings with value $\\mathbf{Z}$."}
+{"_id": "4196", "title": "sites-cohomology-lemma-cech-cohomology-derived-presheaves", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. The {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$ are canonically isomorphic as a $\\delta$-functor to the right derived functors of the functor $$ \\check{H}^0(\\mathcal{U}, -) : \\textit{PAb}(\\mathcal{C}) \\longrightarrow \\textit{Ab}. $$ Moreover, there is a functorial quasi-isomorphism $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\longrightarrow R\\check{H}^0(\\mathcal{U}, \\mathcal{F}) $$ where the right hand side indicates the derived functor $$ R\\check{H}^0(\\mathcal{U}, -) : D^{+}(\\textit{PAb}(\\mathcal{C})) \\longrightarrow D^{+}(\\mathbf{Z}) $$ of the left exact functor $\\check{H}^0(\\mathcal{U}, -)$."}
+{"_id": "4197", "title": "sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf", "text": "Let $\\mathcal{C}$ be a site. An injective abelian sheaf is also injective as an object in the category $\\textit{PAb}(\\mathcal{C})$."}
+{"_id": "4198", "title": "sites-cohomology-lemma-injective-trivial-cech", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. Let $\\mathcal{I}$ be an injective abelian sheaf, i.e., an injective object of $\\textit{Ab}(\\mathcal{C})$. Then $$ \\check{H}^p(\\mathcal{U}, \\mathcal{I}) = \\left\\{ \\begin{matrix} \\mathcal{I}(U) & \\text{if} & p = 0 \\\\ 0 & \\text{if} & p > 0 \\end{matrix} \\right. $$"}
+{"_id": "4200", "title": "sites-cohomology-lemma-cech-h1", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be an abelian sheaf on $\\mathcal{C}$. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. The map $$ \\check{H}^1(\\mathcal{U}, \\mathcal{G}) \\longrightarrow H^1(U, \\mathcal{G}) $$ is injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{G})$ via the bijection of Lemma \\ref{lemma-torsors-h1} with the set of isomorphism classes of $\\mathcal{G}|_U$-torsors which restrict to trivial torsors over each $U_i$."}
+{"_id": "4201", "title": "sites-cohomology-lemma-include", "text": "Let $\\mathcal{C}$ be a site. Consider the functor $i : \\textit{Ab}(\\mathcal{C}) \\to \\textit{PAb}(\\mathcal{C})$. It is a left exact functor with right derived functors given by $$ R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) : U \\longmapsto H^p(U, \\mathcal{F}) $$ see discussion in Section \\ref{section-locality}."}
+{"_id": "4202", "title": "sites-cohomology-lemma-cech-spectral-sequence", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. For any abelian sheaf $\\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F})) $$ converging to $H^{p + q}(U, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$."}
+{"_id": "4203", "title": "sites-cohomology-lemma-cech-spectral-sequence-application", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering. Let $\\mathcal{F} \\in \\Ob(\\textit{Ab}(\\mathcal{C}))$. Assume that $H^i(U_{i_0} \\times_U \\ldots \\times_U U_{i_p}, \\mathcal{F}) = 0$ for all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$. Then $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$."}
+{"_id": "4204", "title": "sites-cohomology-lemma-ses-cech-h1", "text": "Let $\\mathcal{C}$ be a site. Let $$ 0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0 $$ be a short exact sequence of abelian sheaves on $\\mathcal{C}$. Let $U$ be an object of $\\mathcal{C}$. If there exists a cofinal system of coverings $\\mathcal{U}$ of $U$ such that $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$, then the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is surjective."}
+{"_id": "4205", "title": "sites-cohomology-lemma-cech-vanish-collection", "text": "(Variant of Cohomology, Lemma \\ref{cohomology-lemma-cech-vanish}.) Let $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set of coverings of $\\mathcal{C}$ (see Sites, Definition \\ref{sites-definition-site}). Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and $\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$ be subsets. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$. Assume that \\begin{enumerate} \\item For every $\\mathcal{U} \\in \\text{Cov}$, $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ we have $U, U_i \\in \\mathcal{B}$ and every $U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ the coverings of $U$ occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$. \\item For every $\\mathcal{U} \\in \\text{Cov}$ we have $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$. \\end{enumerate} Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$."}
+{"_id": "4206", "title": "sites-cohomology-lemma-existence", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a gerbe over a site whose automorphism sheaves are abelian. Let $\\mathcal{G}$ be the sheaf of abelian groups constructed in Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}. Let $U$ be an object of $\\mathcal{C}$ such that \\begin{enumerate} \\item there exists a cofinal system of coverings $\\{U_i \\to U\\}$ of $U$ in $\\mathcal{C}$ such that $H^1(U_i, \\mathcal{G}) = 0$ and $H^1(U_i \\times_U U_j, \\mathcal{G}) = 0$ for all $i, j$, and \\item $H^2(U, \\mathcal{G}) = 0$. \\end{enumerate} Then there exists an object of $\\mathcal{S}$ lying over $U$."}
+{"_id": "4207", "title": "sites-cohomology-lemma-injective-module-injective-presheaf", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. An injective sheaf of modules is also injective as an object in the category $\\textit{PMod}(\\mathcal{O})$."}
+{"_id": "4208", "title": "sites-cohomology-lemma-include-modules", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Consider the functor $i : \\textit{Mod}(\\mathcal{C}) \\to \\textit{PMod}(\\mathcal{C})$. It is a left exact functor with right derived functors given by $$ R^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) : U \\longmapsto H^p(U, \\mathcal{F}) $$ see discussion in Section \\ref{section-locality}."}
+{"_id": "4209", "title": "sites-cohomology-lemma-injective-module-trivial-cech", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. Let $\\mathcal{I}$ be an injective $\\mathcal{O}$-module, i.e., an injective object of $\\textit{Mod}(\\mathcal{O})$. Then $$ \\check{H}^p(\\mathcal{U}, \\mathcal{I}) = \\left\\{ \\begin{matrix} \\mathcal{I}(U) & \\text{if} & p = 0 \\\\ 0 & \\text{if} & p > 0 \\end{matrix} \\right. $$"}
+{"_id": "4210", "title": "sites-cohomology-lemma-cohomology-modules-abelian-agree", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module, and denote $\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then we have $$ H^i(\\mathcal{C}, \\mathcal{F}_{ab}) = H^i(\\mathcal{C}, \\mathcal{F}) $$ and for any object $U$ of $\\mathcal{C}$ we also have $$ H^i(U, \\mathcal{F}_{ab}) = H^i(U, \\mathcal{F}). $$ Here the left hand side is cohomology computed in $\\textit{Ab}(\\mathcal{C})$ and the right hand side is cohomology computed in $\\textit{Mod}(\\mathcal{O})$."}
+{"_id": "4211", "title": "sites-cohomology-lemma-cohomology-products", "text": "Let $\\mathcal{C}$ be a site. Let $I$ be a set. For $i \\in I$ let  $\\mathcal{F}_i$ be an abelian sheaf on $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$. The canonical map $$ H^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i) \\longrightarrow \\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i) $$ is an isomorphism for $p = 0$ and injective for $p = 1$."}
+{"_id": "4212", "title": "sites-cohomology-lemma-restriction-along-monomorphism-surjective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a : U' \\to U$ be a monomorphism in $\\mathcal{C}$. Then for any injective $\\mathcal{O}$-module $\\mathcal{I}$ the restriction mapping $\\mathcal{I}(U) \\to \\mathcal{I}(U')$ is surjective."}
+{"_id": "4214", "title": "sites-cohomology-lemma-cech-to-cohomology-sheaf-sets", "text": "Let $\\mathcal{C}$ be a site. Let $K' \\to K$ be a map of presheaves of sets on $\\mathcal{C}$ whose sheafification is surjective. Set $K'_p = K' \\times_K \\ldots \\times_K K'$ ($p + 1$-factors). For every abelian sheaf $\\mathcal{F}$ there is a spectral sequence with $E_1^{p, q} = H^q(K'_p, \\mathcal{F})$ converging to $H^{p + q}(K, \\mathcal{F})$."}
+{"_id": "4215", "title": "sites-cohomology-lemma-cohomology-on-sheaf-sets", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\\mathcal{C}$. Consider the morphism of topoi $j : \\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})$, see Sites, Lemma \\ref{sites-lemma-localize-topos-site}. Then $j^{-1}$ preserves injectives and $H^p(K, \\mathcal{F}) = H^p(\\mathcal{C}/K, j^{-1}\\mathcal{F})$ for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "4216", "title": "sites-cohomology-lemma-characterize-limp", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be an abelian sheaf. If \\begin{enumerate} \\item $H^p(U, \\mathcal{F}) = 0$ for $p > 0$ and $U \\in \\Ob(\\mathcal{C})$, and \\item for every surjection $K' \\to K$ of sheaves of sets the extended {\\v C}ech complex $$ 0 \\to H^0(K, \\mathcal{F}) \\to H^0(K', \\mathcal{F}) \\to H^0(K' \\times_K K', \\mathcal{F}) \\to \\ldots $$ is exact, \\end{enumerate} then $\\mathcal{F}$ is totally acyclic (and the converse holds too)."}
+{"_id": "4217", "title": "sites-cohomology-lemma-direct-image-injective-sheaf", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Then for any injective object $\\mathcal{I}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ the pushforward $f_*\\mathcal{I}$ is totally acyclic."}
+{"_id": "4218", "title": "sites-cohomology-lemma-pushforward-injective-flat", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. If $f$ is flat, then $f_*\\mathcal{I}$ is an injective $\\mathcal{O}_\\mathcal{D}$-module for any injective $\\mathcal{O}_\\mathcal{C}$-module $\\mathcal{I}$."}
+{"_id": "4219", "title": "sites-cohomology-lemma-limp-acyclic", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors: \\begin{enumerate} \\item the functor $H^0(U, -)$ for any object $U$ of $\\mathcal{C}$, \\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(K)$ for any presheaf of sets $K$, \\item the functor $\\Gamma(\\mathcal{C}, -)$ of global sections, \\item the functor $f_*$ for any morphism $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ of ringed topoi. \\end{enumerate}"}
+{"_id": "4220", "title": "sites-cohomology-lemma-Leray", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_\\mathcal{C}$-modules. There is a spectral sequence $$ E_2^{p, q} = H^p(\\mathcal{D}, R^qf_*(\\mathcal{F}^\\bullet)) $$ converging to $H^{p + q}(\\mathcal{C}, \\mathcal{F}^\\bullet)$."}
+{"_id": "4221", "title": "sites-cohomology-lemma-apply-Leray", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module. \\begin{enumerate} \\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then $H^p(\\mathcal{C}, \\mathcal{F}) = H^p(\\mathcal{D}, f_*\\mathcal{F})$ for all $p$. \\item If $H^p(\\mathcal{D}, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^q(\\mathcal{C}, \\mathcal{F}) = H^0(\\mathcal{D}, R^qf_*\\mathcal{F})$ for all $q$. \\end{enumerate}"}
+{"_id": "4222", "title": "sites-cohomology-lemma-relative-Leray", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ and $g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to (\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$ be morphisms of ringed topoi. Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module. There is a spectral sequence with $$ E_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F}) $$ converging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$. This spectral sequence is functorial in $\\mathcal{F}$, and there is a version for bounded below complexes of $\\mathcal{O}_\\mathcal{C}$-modules."}
+{"_id": "4223", "title": "sites-cohomology-lemma-base-change-map-flat-case", "text": "Let $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{g'} \\ar[d]_{f'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^g & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ be a commutative diagram of ringed topoi. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_\\mathcal{C}$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map $$ g^*Rf_*\\mathcal{F}^\\bullet \\longrightarrow R(f')_*(g')^*\\mathcal{F}^\\bullet $$ in $D^{+}(\\mathcal{O}_{\\mathcal{D}'})$."}
+{"_id": "4224", "title": "sites-cohomology-lemma-colim-works-over-collection", "text": "Let $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set of coverings of $\\mathcal{C}$ (see Sites, Definition \\ref{sites-definition-site}). Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and $\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$ be subsets. Assume that \\begin{enumerate} \\item For every $\\mathcal{U} \\in \\text{Cov}$ we have $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite, $U, U_i \\in \\mathcal{B}$ and every $U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ the coverings of $U$ occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$. \\end{enumerate} Then the map $$ \\colim_i H^p(U, \\mathcal{F}_i) \\longrightarrow H^p(U, \\colim_i \\mathcal{F}_i) $$ is an isomorphism for every $p \\geq 0$, every $U \\in \\mathcal{B}$, and every filtered diagram $\\mathcal{I} \\to \\textit{Ab}(\\mathcal{C})$."}
+{"_id": "4225", "title": "sites-cohomology-lemma-colim-sites-injective", "text": "Let $\\mathcal{I}$ be a cofiltered index category and let $(\\mathcal{C}_i, f_a)$ be an inverse system of sites over $\\mathcal{I}$ as in Sites, Situation \\ref{sites-situation-inverse-limit-sites}. Set $\\mathcal{C} = \\colim \\mathcal{C}_i$ as in Sites, Lemmas \\ref{sites-lemma-colimit-sites} and \\ref{sites-lemma-compute-pullback-to-limit}. Moreover, assume given \\begin{enumerate} \\item an abelian sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all $i \\in \\Ob(\\mathcal{I})$, \\item for $a : j \\to i$ a map $\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$ of abelian sheaves on $\\mathcal{C}_j$ \\end{enumerate} such that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$ whenever $c = a \\circ b$. Then there exists a map of systems $(\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$ such that $\\mathcal{F}_i \\to \\mathcal{G}_i$ is injective and $\\mathcal{G}_i$ is an injective abelian sheaf."}
+{"_id": "4226", "title": "sites-cohomology-lemma-colimit", "text": "In the situation of Lemma \\ref{lemma-colim-sites-injective} set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Let $i \\in \\Ob(\\mathcal{I})$, $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then $$ \\colim_{a : j \\to i} H^p(u_a(X_i), \\mathcal{F}_j) = H^p(u_i(X_i), \\mathcal{F}) $$ for all $p \\geq 0$."}
+{"_id": "4227", "title": "sites-cohomology-lemma-derived-tor-exact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}$-modules. The functors $$ K(\\textit{Mod}(\\mathcal{O})) \\longrightarrow K(\\textit{Mod}(\\mathcal{O})), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{G}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet) $$ and $$ K(\\textit{Mod}(\\mathcal{O})) \\longrightarrow K(\\textit{Mod}(\\mathcal{O})), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet) $$ are exact functors of triangulated categories."}
+{"_id": "4228", "title": "sites-cohomology-lemma-K-flat-quasi-isomorphism", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}^\\bullet$ be a K-flat complex. Then the functor $$ K(\\textit{Mod}(\\mathcal{O})) \\longrightarrow K(\\textit{Mod}(\\mathcal{O})), \\quad \\mathcal{F}^\\bullet \\longmapsto \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms."}
+{"_id": "4229", "title": "sites-cohomology-lemma-restriction-K-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. If $\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}$-modules, then $\\mathcal{K}^\\bullet|_U$ is a K-flat complex of $\\mathcal{O}_U$-modules."}
+{"_id": "4230", "title": "sites-cohomology-lemma-tensor-product-K-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes of $\\mathcal{O}$-modules, then $\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)$ is a K-flat complex of $\\mathcal{O}$-modules."}
+{"_id": "4231", "title": "sites-cohomology-lemma-K-flat-two-out-of-three", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$ be a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}))$. If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "4232", "title": "sites-cohomology-lemma-K-flat-two-out-of-three-ses", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes such that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}$-modules. If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "4233", "title": "sites-cohomology-lemma-bounded-flat-K-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A bounded above complex of flat $\\mathcal{O}$-modules is K-flat."}
+{"_id": "4234", "title": "sites-cohomology-lemma-colimit-K-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$ be a system of K-flat complexes. Then $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat."}
+{"_id": "4235", "title": "sites-cohomology-lemma-resolution-by-direct-sums-extensions-by-zero", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. For any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}$-modules there exists a commutative diagram of complexes of $\\mathcal{O}$-modules $$ \\xymatrix{ \\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] & \\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\ \\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] & \\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots } $$ with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms are direct sums of $\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$, and (3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are termwise split injections whose cokernels are direct sums of $\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map $\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism."}
+{"_id": "4236", "title": "sites-cohomology-lemma-K-flat-resolution", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. For any complex $\\mathcal{G}^\\bullet$ there exists a $K$-flat complex $\\mathcal{K}^\\bullet$ whose terms are flat $\\mathcal{O}$-modules and a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$ which is termwise surjective."}
+{"_id": "4237", "title": "sites-cohomology-lemma-derived-tor-quasi-isomorphism-other-side", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a quasi-isomorphism of K-flat complexes of $\\mathcal{O}$-modules. For every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}$-modules the induced map $$ \\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) : \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{P}^\\bullet) \\longrightarrow \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{Q}^\\bullet) $$ is a quasi-isomorphism."}
+{"_id": "4238", "title": "sites-cohomology-lemma-flat-tor-zero", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, and \\item $\\text{Tor}_1^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) = 0$ for every $\\mathcal{O}$-module $\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "4239", "title": "sites-cohomology-lemma-K-flat-flat-acyclic", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}^\\bullet$ be a K-flat, acyclic complex with flat terms. Then $\\mathcal{F} = \\Ker(\\mathcal{K}^n \\to \\mathcal{K}^{n + 1})$ is a flat $\\mathcal{O}$-module."}
+{"_id": "4241", "title": "sites-cohomology-lemma-pullback-K-flat", "text": "Let $f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ be a morphism of ringed topoi. Let $\\mathcal{K}^\\bullet$ be a K-flat complex of $\\mathcal{O}$-modules whose terms are flat $\\mathcal{O}$-modules. Then $f^*\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}'$-modules whose terms are flat $\\mathcal{O}'$-modules."}
+{"_id": "4242", "title": "sites-cohomology-lemma-derived-base-change", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. There exists an exact functor $$ Lf^* : D(\\mathcal{O}') \\longrightarrow D(\\mathcal{O}) $$ of triangulated categories so that $Lf^*\\mathcal{K}^\\bullet = f^*\\mathcal{K}^\\bullet$ for any K-flat complex $\\mathcal{K}^\\bullet$ with flat terms and in particular for any bounded above complex of flat $\\mathcal{O}'$-modules."}
+{"_id": "4243", "title": "sites-cohomology-lemma-derived-pullback-composition", "text": "Consider morphisms of ringed topoi $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ and $g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to (\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$. Then $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors $D(\\mathcal{O}_\\mathcal{E}) \\to D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "4244", "title": "sites-cohomology-lemma-pullback-tensor-product", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism $$ Lf^*( \\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'}^{\\mathbf{L}} \\mathcal{G}^\\bullet ) = Lf^*\\mathcal{F}^\\bullet  \\otimes_{\\mathcal{O}}^{\\mathbf{L}} Lf^*\\mathcal{G}^\\bullet  $$ for $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$."}
+{"_id": "4245", "title": "sites-cohomology-lemma-variant-derived-pullback", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism $$ \\mathcal{F}^\\bullet \\otimes_\\mathcal{O}^{\\mathbf{L}} Lf^*\\mathcal{G}^\\bullet = \\mathcal{F}^\\bullet  \\otimes_{f^{-1}\\mathcal{O}_Y}^{\\mathbf{L}} f^{-1}\\mathcal{G}^\\bullet  $$ for $\\mathcal{F}^\\bullet$ in $D(\\mathcal{O})$ and $\\mathcal{G}^\\bullet$ in $D(\\mathcal{O}')$."}
+{"_id": "4246", "title": "sites-cohomology-lemma-check-K-flat-stalks", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}$-modules. \\begin{enumerate} \\item If $\\mathcal{K}^\\bullet$ is K-flat, then for every point $p$ of the site $\\mathcal{C}$ the complex of $\\mathcal{O}_p$-modules $\\mathcal{K}_p^\\bullet$ is K-flat in the sense of More on Algebra, Definition \\ref{more-algebra-definition-K-flat} \\item If $\\mathcal{C}$ has enough points, then the converse is true. \\end{enumerate}"}
+{"_id": "4247", "title": "sites-cohomology-lemma-pullback-K-flat-points", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. If $\\mathcal{C}$ has enough points, then the pullback of a K-flat complex of $\\mathcal{O}'$-modules is a K-flat complex of $\\mathcal{O}$-modules."}
+{"_id": "4248", "title": "sites-cohomology-lemma-tensor-pull-compatibility", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$ be complexes of $\\mathcal{O}_\\mathcal{D}$-modules. The diagram $$ \\xymatrix{ Lf^*(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} \\mathcal{M}^\\bullet) \\ar[r] \\ar[d] & Lf^*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{M}^\\bullet) \\ar[d] \\\\ Lf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} Lf^*\\mathcal{M}^\\bullet \\ar[d] & f^*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{M}^\\bullet) \\ar[d] \\\\ f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} f^*\\mathcal{M}^\\bullet \\ar[r] & \\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}} f^*\\mathcal{M}^\\bullet) } $$ commutes."}
+{"_id": "4249", "title": "sites-cohomology-lemma-adjoint", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of ringed topoi. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma \\ref{lemma-derived-base-change} are adjoint: $$ \\Hom_{D(\\mathcal{O})}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet) = \\Hom_{D(\\mathcal{O}')}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet) $$ bifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(\\mathcal{O}))$ and $\\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$."}
+{"_id": "4250", "title": "sites-cohomology-lemma-derived-pushforward-composition", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ and $g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to (\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$ be morphisms of ringed topoi. Then $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_\\mathcal{E})$."}
+{"_id": "4251", "title": "sites-cohomology-lemma-adjoints-push-pull-compatibility", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}_\\mathcal{C}$-modules. The diagram $$ \\xymatrix{ Lf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] & f^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\ Lf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] & \\mathcal{K}^\\bullet } $$ coming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes, and adjunction $Lf^* \\circ Rf_* \\to \\text{id}$ commutes in $D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "4252", "title": "sites-cohomology-lemma-torsion", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\subset \\textit{Ab}(\\mathcal{C})$ denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor $D(\\mathcal{A}) \\to D_\\mathcal{A}(\\mathcal{C})$ is an equivalence."}
+{"_id": "4253", "title": "sites-cohomology-lemma-restrict-K-injective-to-open", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. The restriction of a K-injective complex of $\\mathcal{O}$-modules to $\\mathcal{C}/U$ is a K-injective complex of $\\mathcal{O}_U$-modules."}
+{"_id": "4254", "title": "sites-cohomology-lemma-unbounded-cohomology-of-open", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. For $K$ in $D(\\mathcal{O})$ we have $H^p(U, K) = H^p(\\mathcal{C}/U, K|_{\\mathcal{C}/U})$."}
+{"_id": "4255", "title": "sites-cohomology-lemma-sheafification-cohomology", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\\mathcal{O})$. The sheafification of $$ U \\mapsto H^q(U, K) = H^q(\\mathcal{C}/U, K|_{\\mathcal{C}/U}) $$ is the $q$th cohomology sheaf $H^q(K)$ of $K$."}
+{"_id": "4256", "title": "sites-cohomology-lemma-restrict-direct-image-open", "text": "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Given $V \\in \\mathcal{D}$, set $U = u(V)$ and denote $g : (\\mathcal{C}/U, \\mathcal{O}_U) \\to (\\mathcal{D}/V, \\mathcal{O}_V)$ the induced morphism of ringed sites (Modules on Sites, Lemma \\ref{sites-modules-lemma-localize-morphism-ringed-sites}). Then $(Rf_*E)|_{\\mathcal{D}/V} = Rg_*(E|_{\\mathcal{C}/U})$ for $E$ in $D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "4257", "title": "sites-cohomology-lemma-Leray-unbounded", "text": "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Then $R\\Gamma(\\mathcal{D}, -) \\circ Rf_* = R\\Gamma(\\mathcal{C}, -)$ as functors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\Gamma(\\mathcal{O}_\\mathcal{D}))$. More generally, for $V \\in \\mathcal{D}$ with $U = u(V)$ we have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$."}
+{"_id": "4258", "title": "sites-cohomology-lemma-unbounded-describe-higher-direct-images", "text": "Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $K$ be in $D(\\mathcal{O}_\\mathcal{C})$. Then $H^i(Rf_*K)$ is the sheaf associated to the presheaf $$ V \\mapsto H^i(u(V), K) = H^i(V, Rf_*K) $$"}
+{"_id": "4259", "title": "sites-cohomology-lemma-modules-abelian-unbounded", "text": "Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site. Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$ and denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_\\mathcal{C})$. \\begin{enumerate} \\item There is a canonical map $R\\Gamma(\\mathcal{C}, K) \\to R\\Gamma(\\mathcal{C}, K_{ab})$ which is an isomorphism in $D(\\textit{Ab})$. \\item For any $U \\in \\mathcal{C}$ there is a canonical map $R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$ which is an isomorphism in $D(\\textit{Ab})$. \\item Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites. There is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which is an isomorphism in $D(\\underline{\\mathbf{Z}}_\\mathcal{D})$. \\end{enumerate}"}
+{"_id": "4260", "title": "sites-cohomology-lemma-adjoint-lower-shriek-restrict", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. Denote $j : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ the corresponding localization morphism. The restriction functor $D(\\mathcal{O}) \\to D(\\mathcal{O}_U)$ is a right adjoint to extension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O})$."}
+{"_id": "4261", "title": "sites-cohomology-lemma-K-injective-flat", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a flat morphism of ringed topoi. If $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}_\\mathcal{C}$-modules, then $f_*\\mathcal{I}^\\bullet$ is K-injective as a complex of $\\mathcal{O}_\\mathcal{D}$-modules."}
+{"_id": "4262", "title": "sites-cohomology-lemma-hom-K-injective", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map of sheaves of rings. If $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}$-modules, then $\\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{I}^\\bullet)$ is a K-injective complex of $\\mathcal{O}'$-modules."}
+{"_id": "4263", "title": "sites-cohomology-lemma-localize-cartesian-square", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ X' \\ar[d] \\ar[r] & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ be a cartesian diagram of $\\mathcal{C}$. Then we have $j_{Y'/Y}^{-1} \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^{-1}$ as functors $D(\\mathcal{C}/X) \\to D(\\mathcal{C}/Y')$."}
+{"_id": "4264", "title": "sites-cohomology-lemma-localize-cartesian-square-modules", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $$ \\xymatrix{ X' \\ar[d] \\ar[r] & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ be a cartesian diagram of $\\mathcal{C}$. Then we have $j_{Y'/Y}^* \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^*$ as functors $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_{Y'})$."}
+{"_id": "4265", "title": "sites-cohomology-lemma-derived-limit-is-ok", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be an object of $D(\\mathcal{C} \\times \\mathbf{N})$. Set $K_n = i_n^{-1}K$ as above. Then $$ R\\lim K \\cong R\\lim K_n $$ in $D(\\mathcal{C})$."}
+{"_id": "4266", "title": "sites-cohomology-lemma-RGamma-commutes-with-Rlim", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The functors $R\\Gamma(\\mathcal{C}, -)$ and $R\\Gamma(U, -)$ for $U \\in \\Ob(\\mathcal{C})$ commute with $R\\lim$. Moreover, there are short exact sequences $$ 0 \\to R^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to \\lim H^m(U, K_n) \\to 0 $$ for any inverse system $(K_n)$ in $D(\\mathcal{O})$ and $m \\in \\mathbf{Z}$. Similar for $H^m(\\mathcal{C}, R\\lim K_n)$."}
+{"_id": "4267", "title": "sites-cohomology-lemma-Rf-commutes-with-Rlim", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. Then $Rf_*$ commutes with $R\\lim$, i.e., $Rf_*$ commutes with derived limits."}
+{"_id": "4268", "title": "sites-cohomology-lemma-inverse-limit-is-derived-limit", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{F}_n)$ be an inverse system of $\\mathcal{O}$-modules. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$, \\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$ for $U \\in \\mathcal{B}$. \\end{enumerate} Then $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have $H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$."}
+{"_id": "4269", "title": "sites-cohomology-lemma-cohomology-derived-limit-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system in $D(\\mathcal{O})$. Let $V \\in \\Ob(\\mathcal{C})$ and $m \\in \\mathbf{Z}$. Assume there exist an integer $n(V)$ and a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that for $\\{V_i \\to V\\} \\in \\text{Cov}_V$ \\begin{enumerate} \\item $R^1\\lim H^{m - 1}(V_i, K_n) = 0$, and \\item $H^m(V_i, K_n) \\to H^m(V_i, K_{n(V)})$ is injective for $n \\geq n(V)$. \\end{enumerate} Then the map on sections $H^m(R\\lim K_n)(V) \\to H^m(K_{n(V)})(V)$ is injective."}
+{"_id": "4270", "title": "sites-cohomology-lemma-is-limit-per-object", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, and \\item for every $V \\in \\mathcal{B}$ there exist a function $p(V, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that $$ H^p(V_i, H^{m - p}(E)) = 0 $$ for all $\\{V_i \\to V\\} \\in \\text{Cov}_V$ and all integers $p, m$ satisfying $p > p(V, m)$. \\end{enumerate} Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O})$."}
+{"_id": "4271", "title": "sites-cohomology-lemma-is-limit-spaltenstein", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, and \\item for every $V \\in \\mathcal{B}$ there exist an integer $d_V \\geq 0$ and a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that $$ H^p(V_i, H^q(E)) = 0 \\text{ for } \\{V_i \\to V\\} \\in \\text{Cov}_V,\\ p > d_V, \\text{ and }q < 0 $$ \\end{enumerate} Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O})$."}
+{"_id": "4272", "title": "sites-cohomology-lemma-is-limit", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$. Assume there exists a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$ and a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \\in \\mathcal{B}$. \\end{enumerate} Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O})$."}
+{"_id": "4273", "title": "sites-cohomology-lemma-is-limit-dimension", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$. Assume there exists an integer $d \\geq 0$ and a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(V, H^q(E)) = 0$ for $p > d$, $q < 0$, and $V \\in \\mathcal{B}$. \\end{enumerate} Then the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O})$."}
+{"_id": "4274", "title": "sites-cohomology-lemma-cohomology-over-U-trivial", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\\mathcal{O})$. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and $U \\in \\mathcal{B}$. \\end{enumerate} Then $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$ and $U \\in \\mathcal{B}$."}
+{"_id": "4275", "title": "sites-cohomology-lemma-derived-limit-suitable-system", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system of objects of $D(\\mathcal{O})$. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have \\begin{enumerate} \\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$, \\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$. \\end{enumerate} \\end{enumerate} Then $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$."}
+{"_id": "4276", "title": "sites-cohomology-lemma-K-injective", "text": "In the situation described above. Denote $\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Let $d \\in \\mathbf{N}$. Assume \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$ for $p > d$ and $q < 0$\\footnote{It suffices if $\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for $p > p(m)$, see Lemma \\ref{lemma-is-limit}.}. \\end{enumerate} Then (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism."}
+{"_id": "4278", "title": "sites-cohomology-lemma-olsson-laszlo", "text": "\\begin{reference} This is \\cite[Proposition 2.1.4]{six-I} with slightly changed hypotheses; it is the analogue of \\cite[Proposition 3.13]{Spaltenstein} for sites. \\end{reference} In Situation \\ref{situation-olsson-laszlo} for any $E \\in D_\\mathcal{A}(\\mathcal{O})$ the canonical map $E \\to R\\lim \\tau_{\\geq -n} E$ is an isomorphism in $D(\\mathcal{O})$."}
+{"_id": "4279", "title": "sites-cohomology-lemma-olsson-laszlo-modified", "text": "In Situation \\ref{situation-olsson-laszlo} let $(K_n)$ be an inverse system in $D_\\mathcal{A}^+(\\mathcal{O})$. Assume that for every $j$ the inverse system $(H^j(K_n))$ in $\\mathcal{A}$ is eventually constant with value $\\mathcal{H}^j$. Then $H^j(R\\lim K_n) = \\mathcal{H}^j$ for all $j$."}
+{"_id": "4280", "title": "sites-cohomology-lemma-olsson-laszlo-map-version-one", "text": "\\begin{reference} This is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed hypotheses. \\end{reference} Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$ be weak Serre subcategories. Assume there is an integer $N$ such that \\begin{enumerate} \\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}, \\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}, \\item $R^pf_*\\mathcal{F} \\in \\Ob(\\mathcal{A}')$ for $p \\geq 0$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$, \\item $R^pf_*\\mathcal{F} = 0$ for $p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$, \\end{enumerate} Then for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ we have \\begin{enumerate} \\item[(a)] $Rf_*K$ is in $D_{\\mathcal{A}'}(\\mathcal{O}')$, \\item[(b)] the map $H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism for $j \\geq N - n$. \\end{enumerate}"}
+{"_id": "4281", "title": "sites-cohomology-lemma-olsson-laszlo-map-version-two", "text": "\\begin{reference} This is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed hypotheses. \\end{reference} Let $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$ be a morphism of ringed sites. assume moreover there is an integer $N$ such that \\begin{enumerate} \\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}, \\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$ and $\\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo-prime}, \\item $R^pf_*\\mathcal{F} = 0$ for $p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$, \\end{enumerate} Then for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ the map $H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism for $j \\geq N - n$."}
+{"_id": "4282", "title": "sites-cohomology-lemma-c-square", "text": "In the situation above, choose a K-injective complex $\\mathcal{I}^\\bullet$ of $\\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes $$ \\mathcal{I}^\\bullet(X) \\xrightarrow{\\alpha} \\mathcal{I}^\\bullet(Z) \\oplus \\mathcal{I}^\\bullet(Y) \\xrightarrow{\\beta} \\mathcal{I}^\\bullet(E) $$ with $\\beta \\circ \\alpha = 0$. Thus a canonical map $$ c^K_{X, Z, Y, E} : \\mathcal{I}^\\bullet(X) \\longrightarrow C(\\beta)^\\bullet[-1] $$ This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle $$ R\\Gamma(X, K) \\to R\\Gamma(Z, K) \\oplus R\\Gamma(Y, K) \\to R\\Gamma(E, K) \\to R\\Gamma(X, K)[1] $$ All of these constructions are functorial in $K$."}
+{"_id": "4283", "title": "sites-cohomology-lemma-two-out-of-three-blow-up-square", "text": "In the situation above, let $K_1 \\to K_2 \\to K_3 \\to K_1[1]$ be a distinguished triangle in $D(\\mathcal{O})$. If $c^{K_i}_{X, Z, Y, E}$ is a quasi-isomorphism for two $i$ out of $\\{1, 2, 3\\}$, then it is a quasi-isomorphism for the third $i$."}
+{"_id": "4284", "title": "sites-cohomology-lemma-square-triangle", "text": "In the situation above assume \\begin{enumerate} \\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and \\item $h_E^\\# \\to h_Y^\\#$ is injective. \\end{enumerate} Then the construction of Lemma \\ref{lemma-c-square} produces a distinguished triangle $$ R\\Gamma(X, K) \\to R\\Gamma(Z, K) \\oplus R\\Gamma(Y, K) \\to R\\Gamma(E, K) \\to R\\Gamma(X, K)[1] $$ functorial for $K$ in $D(\\mathcal{C})$."}
+{"_id": "4286", "title": "sites-cohomology-lemma-downstairs", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{D})$ consisting of objects $K$ such that $$ K \\longrightarrow Rf_*Lf^*K $$ is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\\mathcal{O}_\\mathcal{D})$ and the functor $Lf^* : D' \\to D(\\mathcal{O}_\\mathcal{C})$ is fully faithful."}
+{"_id": "4287", "title": "sites-cohomology-lemma-upstairs", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{C})$ consisting of objects $K$ such that $$ Lf^*Rf_*K \\longrightarrow K $$ is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\\mathcal{O}_\\mathcal{C})$ and the functor $Rf_* : D' \\to D(\\mathcal{O}_\\mathcal{D})$ is fully faithful."}
+{"_id": "4288", "title": "sites-cohomology-lemma-bounded-in-image-upstairs", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$. Assume \\begin{enumerate} \\item $f$ is flat, \\item $K$ is bounded below, \\item $f^*Rf_*H^q(K) \\to H^q(K)$ is an isomorphism. \\end{enumerate} Then $f^*Rf_*K \\to K$ is an isomorphism."}
+{"_id": "4289", "title": "sites-cohomology-lemma-bounded-in-image-downstairs", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{D})$. Assume \\begin{enumerate} \\item $f$ is flat, \\item $K$ is bounded below, \\item $H^q(K) \\to Rf_*f^*H^q(K)$ is an isomorphism. \\end{enumerate} Then $K \\to Rf_*f^*K$ is an isomorphism."}
+{"_id": "4290", "title": "sites-cohomology-lemma-equivalence-bounded", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$ be weak Serre subcategories. Assume \\begin{enumerate} \\item $f$ is flat, \\item $f^*$ induces an equivalence of categories $\\mathcal{A}' \\to \\mathcal{A}$, \\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$. \\end{enumerate} Then $f^* : D_{\\mathcal{A}'}^+(\\mathcal{O}') \\to D_\\mathcal{A}^+(\\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\\mathcal{A}^+(\\mathcal{O}) \\to D_{\\mathcal{A}'}^+(\\mathcal{O}')$."}
+{"_id": "4291", "title": "sites-cohomology-lemma-equivalence-unbounded-one", "text": "\\begin{reference} This is analogous to \\cite[Theorem 2.2.3]{six-I}. \\end{reference} Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$ be weak Serre subcategories. Assume \\begin{enumerate} \\item $f$ is flat, \\item $f^*$ induces an equivalence of categories $\\mathcal{A}' \\to \\mathcal{A}$, \\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$, \\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}, \\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}. \\end{enumerate} Then $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$."}
+{"_id": "4292", "title": "sites-cohomology-lemma-equivalence-unbounded-two", "text": "\\begin{reference} This is analogous to \\cite[Theorem 2.2.3]{six-I}. \\end{reference} Let $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$ be a morphism of ringed sites. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ and $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$ be weak Serre subcategories. Assume \\begin{enumerate} \\item $f$ is flat, \\item $f^*$ induces an equivalence of categories $\\mathcal{A}' \\to \\mathcal{A}$, \\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism for $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$, \\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo}, \\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$ and $\\mathcal{A}$ satisfy the assumption of Situation \\ref{situation-olsson-laszlo-prime}. \\end{enumerate} Then $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$ is an equivalence of categories with quasi-inverse given by $Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$."}
+{"_id": "4293", "title": "sites-cohomology-lemma-compare-topologies-derived-adequate-modules", "text": "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to (\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Let $\\mathcal{A} \\subset \\textit{PMod}(\\mathcal{O})$ be a full subcategory. Assume \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a sheaf for the $\\tau$-topology, \\item $\\mathcal{A}$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_\\tau)$, \\item every object of $\\mathcal{C}$ has a $\\tau'$-covering whose members are elements of $\\mathcal{B}$, and \\item for every $U \\in \\mathcal{B}$ we have $H^p_\\tau(U, \\mathcal{F}) = 0$, $p > 0$ for all $\\mathcal{F} \\in \\mathcal{A}$. \\end{enumerate} Then $\\mathcal{A}$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_{\\tau'})$ and there is an equivalence of triangulated categories $D_\\mathcal{A}(\\mathcal{O}_\\tau) = D_\\mathcal{A}(\\mathcal{O}_{\\tau'})$ given by $\\epsilon^*$ and $R\\epsilon_*$."}
+{"_id": "4294", "title": "sites-cohomology-lemma-descent-squares", "text": "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to (\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above. Let $A$ be a set and for $\\alpha \\in A$ let $$ \\xymatrix{ E_\\alpha \\ar[d] \\ar[r] & Y_\\alpha \\ar[d] \\\\ Z_\\alpha \\ar[r] & X_\\alpha } $$ be a commutative diagram in the category $\\mathcal{C}$. Assume that \\begin{enumerate} \\item a $\\tau'$-sheaf $\\mathcal{F}'$ is a $\\tau$-sheaf if $\\mathcal{F}'(X_\\alpha) = \\mathcal{F}'(Z_\\alpha) \\times_{\\mathcal{F}'(E_\\alpha)}  \\mathcal{F}'(Y_\\alpha)$ for all $\\alpha$, \\item for $K'$ in $D(\\mathcal{O}_{\\tau'})$ in the essential image of $R\\epsilon_*$ the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$ of Lemma \\ref{lemma-c-square} are isomorphisms for all $\\alpha$. \\end{enumerate} Then $K' \\in D^+(\\mathcal{O}_{\\tau'})$ is in the essential image of $R\\epsilon_*$ if and only if the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$ are isomorphisms for all $\\alpha$."}
+{"_id": "4295", "title": "sites-cohomology-lemma-descent-squares-helper", "text": "With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to (\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above. Let $$ \\xymatrix{ E \\ar[d] \\ar[r] & Y \\ar[d] \\\\ Z \\ar[r] & X } $$ be a commutative diagram in the category $\\mathcal{C}$ such that \\begin{enumerate} \\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and \\item $h_E^\\# \\to h_Y^\\#$ is injective \\end{enumerate} where ${}^\\#$ denotes $\\tau$-sheafification. Then for $K' \\in D(\\mathcal{O}_{\\tau'})$ in the essential image of $R\\epsilon_*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma \\ref{lemma-c-square} (using the $\\tau'$-topology) is an isomorphism."}
+{"_id": "4296", "title": "sites-cohomology-lemma-A", "text": "In Situation \\ref{situation-compare} for $X$ in $\\mathcal{C}$ denote $\\mathcal{A}_X$ the objects of $\\textit{Ab}(\\mathcal{C}_\\tau/X)$ of the form $\\epsilon_X^{-1}\\mathcal{F}'$ with $\\mathcal{F}'$ in $\\mathcal{A}'_X$. Then \\begin{enumerate} \\item for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C}_\\tau/X)$ we have $\\mathcal{F} \\in \\mathcal{A}_X \\Leftrightarrow \\epsilon_{X, *}\\mathcal{F} \\in \\mathcal{A}'_X$, and \\item $f_\\tau^{-1}$ sends $\\mathcal{A}_Y$ into $\\mathcal{A}_X$ for any morphism $f : X \\to Y$ of $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "4297", "title": "sites-cohomology-lemma-V-implies-C-general", "text": "In Situation \\ref{situation-compare} assume $(V_n)$ holds. For $f : X \\to Y$ in $\\mathcal{P}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$ we have $R^if_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} = \\epsilon_{Y, *}R^if_{\\tau, *}\\mathcal{F}$ for $i \\leq n$."}
+{"_id": "4298", "title": "sites-cohomology-lemma-V-implies-cohomology-general", "text": "In Situation \\ref{situation-compare} if $(V_n)$ holds, then for $X$ in $\\mathcal{C}$ and $L \\in D(\\mathcal{C}_{\\tau'}/X)$ with $H^i(L) = 0$ for $i < 0$ and $H^i(L)$ in $\\mathcal{A}'_X$ for $0 \\leq i \\leq n$ we have $H^n_{\\tau'}(X, L) = H^n_\\tau(X, \\epsilon_X^{-1}L)$."}
+{"_id": "4299", "title": "sites-cohomology-lemma-V-implies-cohomology-extra-general", "text": "In Situation \\ref{situation-compare} if $(V_n)$ holds, then for $X$ in $\\mathcal{C}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$ the map $H^{n + 1}_{\\tau'}(X, \\epsilon_{X, *}\\mathcal{F}) \\to H^{n + 1}_\\tau(X, \\mathcal{F})$ is injective with image those classes which become trivial on a $\\tau'$-covering of $X$."}
+{"_id": "4300", "title": "sites-cohomology-lemma-make-class-zero-general", "text": "In Situation \\ref{situation-compare} let $f : X \\to Y$ be in $\\mathcal{P}$ such that $\\{X \\to Y\\}$ is a $\\tau$-covering. Let $\\mathcal{F}'$ be in $\\mathcal{A}'_Y$. If $n \\geq 0$ and $$ \\theta \\in \\text{Equalizer}\\left( \\xymatrix{ H^{n + 1}_{\\tau'}(X, \\mathcal{F}') \\ar@<1ex>[r] \\ar@<-1ex>[r] & H^{n + 1}_{\\tau'}(X \\times_Y X, \\mathcal{F}') } \\right) $$ then there exists a $\\tau'$-covering $\\{Y_i \\to Y\\}$ such that $\\theta$ restricts to zero in $H^{n + 1}_{\\tau'}(Y_i \\times_Y X, \\mathcal{F}')$."}
+{"_id": "4301", "title": "sites-cohomology-lemma-induction-step-V-C-general", "text": "In Situation \\ref{situation-compare} we have $(V_n) \\Rightarrow (V_{n + 1})$."}
+{"_id": "4302", "title": "sites-cohomology-lemma-V-C-all-n-general", "text": "In Situation \\ref{situation-compare} we have that $(V_n)$ is true for all $n$. Moreover: \\begin{enumerate} \\item For $X$ in $\\mathcal{C}$ and $K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ the map $K' \\to R\\epsilon_{X, *}(\\epsilon_X^{-1}K')$ is an isomorphism. \\item For $f : X \\to Y$ in $\\mathcal{P}$ and $K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ we have $Rf_{\\tau', *}K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/Y)$ and $\\epsilon_Y^{-1}(Rf_{\\tau', *}K') = Rf_{\\tau, *}(\\epsilon_X^{-1}K')$. \\end{enumerate}"}
+{"_id": "4305", "title": "sites-cohomology-lemma-LC-basic", "text": "The category $\\textit{LC}$ has fibre products and a final object and hence has arbitrary finite limits. Given morphisms $X \\to Z$ and $Y \\to Z$ in $\\textit{LC}$ with $X$ and $Y$ quasi-compact, then $X \\times_Z Y$ is quasi-compact."}
+{"_id": "4306", "title": "sites-cohomology-lemma-qc", "text": "Let $X$ be a Hausdorff and locally quasi-compact space, in other words, an object of $\\textit{LC}$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism in $\\textit{LC}$ then $\\{X' \\to X\\}$ is a qc covering. \\item If $\\{f_i : X_i \\to X\\}_{i\\in I}$ is a qc covering and for each $i$ we have a qc covering $\\{g_{ij} : X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a qc covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a qc covering and $X' \\to X$ is a morphism of $\\textit{LC}$ then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a qc covering. \\end{enumerate}"}
+{"_id": "4307", "title": "sites-cohomology-lemma-proper-surjective-is-qc-covering", "text": "Let $f : X \\to Y$ be a morphism of $\\textit{LC}$. If $f$ is proper and surjective, then $\\{f : X \\to Y\\}$ is a qc covering."}
+{"_id": "4308", "title": "sites-cohomology-lemma-describe-pullback-pi", "text": "Let $X$ be an object of $\\textit{LC}_{qc}$. Let $\\mathcal{F}$ be a sheaf on $X$. The rule $$ \\textit{LC}_{qc}/X \\longrightarrow \\textit{Sets},\\quad (f : Y \\to X) \\longmapsto \\Gamma(Y, f^{-1}\\mathcal{F}) $$ is a sheaf and a fortiori also a sheaf on $\\textit{LC}_{Zar}/X$. This sheaf is equal to $\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{Zar}/X$ and $\\epsilon_X^{-1}\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{qc}/X$."}
+{"_id": "4309", "title": "sites-cohomology-lemma-collect-true-things-Zar", "text": "Let $X$ be an object of $\\textit{LC}_{Zar}$. Then \\begin{enumerate} \\item for $\\mathcal{F} \\in \\textit{Ab}(X)$ we have $H^n_{Zar}(X, \\pi_X^{-1}\\mathcal{F}) = H^n(X, \\mathcal{F})$, \\item $\\pi_{X, *} : \\textit{Ab}(\\textit{LC}_{Zar}/X) \\to \\textit{Ab}(X)$ is exact, \\item the unit $\\text{id} \\to \\pi_{X, *} \\circ \\pi_X^{-1}$ of the adjunction is an isomorphism, and \\item for $K \\in D(X)$ the canonical map $K \\to R\\pi_{X, *} \\pi_X^{-1}K$ is an isomorphism. \\end{enumerate} Let $f : X \\to Y$ be a morphism of $\\textit{LC}_{Zar}$. Then \\begin{enumerate} \\item[(5)] there is a commutative diagram $$ \\xymatrix{ \\Sh(\\textit{LC}_{Zar}/X) \\ar[r]_{f_{Zar}} \\ar[d]_{\\pi_X} & \\Sh(\\textit{LC}_{Zar}/Y) \\ar[d]^{\\pi_Y} \\\\ \\Sh(X_{Zar}) \\ar[r]^f & \\Sh(Y_{Zar}) } $$ of topoi, \\item[(6)] for $L \\in D^+(Y)$ we have $H^n_{Zar}(X, \\pi_Y^{-1}L) = H^n(X, f^{-1}L)$, \\item[(7)] if $f$ is proper, then we have \\begin{enumerate} \\item $\\pi_Y^{-1} \\circ f_* = f_{Zar, *} \\circ \\pi_X^{-1}$ as functors $\\Sh(X) \\to \\Sh(\\textit{LC}_{Zar}/Y)$, \\item $\\pi_Y^{-1} \\circ Rf_* = Rf_{Zar, *} \\circ \\pi_X^{-1}$ as functors $D^+(X) \\to D^+(\\textit{LC}_{Zar}/Y)$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4310", "title": "sites-cohomology-lemma-push-pull-LC", "text": "Let $f : X \\to Y$ be a morphism of $\\textit{LC}_{qc}$. Then there are commutative diagrams of topoi $$ \\vcenter{ \\xymatrix{ \\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{\\epsilon_X} & \\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh(\\textit{LC}_{Zar}/X) \\ar[r]^{f_{Zar}} & \\Sh(\\textit{LC}_{Zar}/Y) } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ \\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{a_X} & \\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{a_Y} \\\\ \\Sh(X) \\ar[r]^f & \\Sh(Y) } } $$ with $a_X = \\pi_X \\circ \\epsilon_X$, $a_Y = \\pi_X \\circ \\epsilon_X$. If $f$ is proper, then $a_Y^{-1} \\circ f_* = f_{qc, *} \\circ a_X^{-1}$."}
+{"_id": "4311", "title": "sites-cohomology-lemma-compare-qc-zar", "text": "Consider the comparison morphism $\\epsilon : \\textit{LC}_{qc} \\to \\textit{LC}_{Zar}$. Let $\\mathcal{P}$ denote the class of proper maps of topological spaces. For $X$ in $\\textit{LC}_{Zar}$ denote $\\mathcal{A}'_X \\subset \\textit{Ab}(\\textit{LC}_{Zar}/X)$ the full subcategory consisting of sheaves of the form $\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X)$. Then (\\ref{item-base-change-P}), (\\ref{item-restriction-A}), (\\ref{item-A-sheaf}), (\\ref{item-A-and-P}), and (\\ref{item-refine-tau-by-P}) of Situation \\ref{situation-compare} hold."}
+{"_id": "4312", "title": "sites-cohomology-lemma-V-C-all-n", "text": "With notation as above. \\begin{enumerate} \\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and an abelian sheaf $\\mathcal{F}$ on $X$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$ and abelian sheaf $\\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_*\\mathcal{F}) = R^if_{qc, *}(a_X^{-1}\\mathcal{F})$ for all $i$. \\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map $\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$ and $K$ in $D^+(X)$ we have $a_Y^{-1}(Rf_*K) = Rf_{qc, *}(a_X^{-1}K)$. \\end{enumerate}"}
+{"_id": "4313", "title": "sites-cohomology-lemma-cohomological-descent-LC", "text": "Let $X$ be an object of $\\textit{LC}_{qc}$. For $K \\in D^+(X)$ the map $$ K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above."}
+{"_id": "4316", "title": "sites-cohomology-lemma-cup-product-associative", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. The relative cup product of Remark \\ref{remark-cup-product} is associative in the sense that the diagram $$ \\xymatrix{ Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*M \\ar[r] \\ar[d] & Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*M \\ar[d] \\\\ Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} M) \\ar[r] & Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L}  L \\otimes_\\mathcal{O}^\\mathbf{L} M) } $$ is commutative in $D(\\mathcal{O}')$ for all $K, L, M$ in $D(\\mathcal{O})$."}
+{"_id": "4317", "title": "sites-cohomology-lemma-cup-product-commutative", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. The relative cup product of Remark \\ref{remark-cup-product} is commutative in the sense that the diagram $$ \\xymatrix{ Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi & Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\ Rf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*K \\ar[r] & Rf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} K) } $$ is commutative in $D(\\mathcal{O}')$ for all $K, L$ in $D(\\mathcal{O})$. Here $\\psi$ is the commutativity constraint on the derived category (Lemma \\ref{lemma-symmetric-monoidal-derived})."}
+{"_id": "4319", "title": "sites-cohomology-lemma-compose", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules there is an isomorphism $$ \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)) = \\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}$-modules functorial in $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$."}
+{"_id": "4320", "title": "sites-cohomology-lemma-composition", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules there is a canonical morphism $$ \\text{Tot}\\left( \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\otimes_\\mathcal{O} \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet) \\right) \\longrightarrow \\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}$-modules."}
+{"_id": "4321", "title": "sites-cohomology-lemma-diagonal-better", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules there is a canonical morphism $$ \\text{Tot}\\left( \\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet) \\right) \\longrightarrow \\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)) $$ of complexes of $\\mathcal{O}$-modules functorial in all three complexes."}
+{"_id": "4322", "title": "sites-cohomology-lemma-diagonal", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules there is a canonical morphism $$ \\mathcal{K}^\\bullet \\longrightarrow \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)) $$ of complexes of $\\mathcal{O}$-modules functorial in both complexes."}
+{"_id": "4323", "title": "sites-cohomology-lemma-evaluate-and-more", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules there is a canonical morphism $$ \\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet) \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet) \\longrightarrow \\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet) $$ of complexes of $\\mathcal{O}$-modules functorial in all three complexes."}
+{"_id": "4324", "title": "sites-cohomology-lemma-RHom-into-K-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}$-modules. Let $\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}$-modules. Then $$ H^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) = \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U) $$ for all $U \\in \\Ob(\\mathcal{C})$. Similarly, $H^0(\\Gamma(\\mathcal{C}, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) = \\Hom_{D(\\mathcal{O}_U)}(L, M)$."}
+{"_id": "4326", "title": "sites-cohomology-lemma-RHom-from-K-flat-into-K-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I}^\\bullet$ be a K-injective complex of $\\mathcal{O}$-modules. Let $\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}$-modules. Then $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$ is a K-injective complex of $\\mathcal{O}$-modules."}
+{"_id": "4328", "title": "sites-cohomology-lemma-internal-hom", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\\mathcal{O})$. With the construction as described above there is a canonical isomorphism $$ R\\SheafHom(K, R\\SheafHom(L, M)) = R\\SheafHom(K \\otimes_\\mathcal{O}^\\mathbf{L} L, M) $$ in $D(\\mathcal{O})$ functorial in $K, L, M$ which recovers (\\ref{equation-internal-hom}) on taking $H^0(\\mathcal{C}, -)$."}
+{"_id": "4329", "title": "sites-cohomology-lemma-restriction-RHom-to-U", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\\mathcal{O})$. The construction of $R\\SheafHom(K, L)$ commutes with restrictions, i.e., for every object $U$ of $\\mathcal{C}$ we have $R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$."}
+{"_id": "4330", "title": "sites-cohomology-lemma-RHom-triangulated", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The bifunctor $R\\SheafHom(- , -)$ transforms distinguished triangles into distinguished triangles in both variables."}
+{"_id": "4331", "title": "sites-cohomology-lemma-internal-hom-evaluate", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\\mathcal{O})$. There is a canonical morphism $$ R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} K \\longrightarrow R\\SheafHom(R\\SheafHom(K, L), M) $$ in $D(\\mathcal{O})$ functorial in $K, L, M$."}
+{"_id": "4332", "title": "sites-cohomology-lemma-internal-hom-composition", "text": "\\begin{slogan} Composition on RSheafHom. \\end{slogan} Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\\mathcal{O})$ there is a canonical morphism $$ R\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(K, M) $$ in $D(\\mathcal{O})$."}
+{"_id": "4333", "title": "sites-cohomology-lemma-internal-hom-diagonal-better", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\\mathcal{O})$ there is a canonical morphism $$ K \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(M, L) \\longrightarrow R\\SheafHom(M, K \\otimes_\\mathcal{O}^\\mathbf{L} L) $$ in $D(\\mathcal{O})$ functorial in $K, L, M$."}
+{"_id": "4335", "title": "sites-cohomology-lemma-dual", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $L$ be an object of $D(\\mathcal{O})$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O})$. For $M$ in $D(\\mathcal{O})$ there is a canonical map \\begin{equation} \\label{equation-eval} M \\otimes^\\mathbf{L}_\\mathcal{O} L^\\vee \\longrightarrow R\\SheafHom(L, M) \\end{equation} which induces a canonical map $$ H^0(\\mathcal{C}, M \\otimes_\\mathcal{O}^\\mathbf{L} L^\\vee) \\longrightarrow \\Hom_{D(\\mathcal{O})}(L, M) $$ functorial in $M$ in $D(\\mathcal{O})$."}
+{"_id": "4336", "title": "sites-cohomology-lemma-pullback-injective-pre-limp", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings and let $\\mathcal{I}$ be an injective $\\mathcal{O}_\\mathcal{D}$-module. Then $H^p(U, g^{-1}\\mathcal{I}) = 0$ for all $p > 0$ and $U \\in \\Ob(\\mathcal{C})$."}
+{"_id": "4337", "title": "sites-cohomology-lemma-existence-derived-lower-shriek", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings and set $\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$. The functor $g_! : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{D})$ (see Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}) has a left derived functor $$ Lg_! : D(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D(\\mathcal{O}_\\mathcal{D}) $$ which is left adjoint to $g^*$. Moreover, for $U \\in \\Ob(\\mathcal{C})$ we have $$ Lg_!(j_{U!}\\mathcal{O}_U) = g_!j_{U!}\\mathcal{O}_U = j_{u(U)!} \\mathcal{O}_{u(U)}. $$ where $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the localization morphism $j_U : \\mathcal{C}/U \\to \\mathcal{C}$ and $j_{u(U)} : \\mathcal{D}/u(U) \\to \\mathcal{D}$."}
+{"_id": "4338", "title": "sites-cohomology-lemma-pullback-injective-limp", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings and let $\\mathcal{I}$ be an injective $\\mathcal{O}_\\mathcal{D}$-module. If $g_!^{Sh} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ commutes with fibre products\\footnote{Holds if $\\mathcal{C}$ has finite connected limits and $u$ commutes with them, see Sites, Lemma \\ref{sites-lemma-preserve-equalizers}.}, then $g^{-1}\\mathcal{I}$ is totally acyclic."}
+{"_id": "4339", "title": "sites-cohomology-lemma-pullback-same-cohomology", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the corresponding morphism of topoi. Let $U \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item For $M$ in $D(\\mathcal{D})$ we have $R\\Gamma(U, g^{-1}M) = R\\Gamma(u(U), M)$. \\item If $\\mathcal{O}_\\mathcal{D}$ is a sheaf of rings and $\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$, then for $M$ in $D(\\mathcal{O}_\\mathcal{D})$ we have $R\\Gamma(U, g^*M) = R\\Gamma(u(U), M)$. \\end{enumerate}"}
+{"_id": "4342", "title": "sites-cohomology-lemma-fibred-category-with-object", "text": "Assumptions and notation as in Situation \\ref{situation-fibred-category}. For $U \\in \\Ob(\\mathcal{C})$ consider the induced morphism of topoi $$ \\pi_U : \\Sh(\\mathcal{C}/U) \\longrightarrow \\Sh(\\mathcal{D}/p(U)) $$ Then there exists a morphism of topoi $$ \\sigma : \\Sh(\\mathcal{D}/p(U)) \\to \\Sh(\\mathcal{C}/U) $$ such that $\\pi_U \\circ \\sigma = \\text{id}$ and $\\sigma^{-1} = \\pi_{U, *}$."}
+{"_id": "4343", "title": "sites-cohomology-lemma-morphism-fibred-categories-with-object", "text": "Assumptions and notation as in Situation \\ref{situation-morphism-fibred-categories}. For $U' \\in \\Ob(\\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and consider the induced morphisms of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C}'/U'), \\mathcal{O}_{U'}) \\ar[rd]_{\\pi'_{U'}} \\ar[rr]_{g'} & & (\\Sh(\\mathcal{C}), \\mathcal{O}_U) \\ar[ld]^{\\pi_U} \\\\ & (\\Sh(\\mathcal{D}/V), \\mathcal{O}_V) } $$ Then there exists a morphism of topoi $$ \\sigma' : \\Sh(\\mathcal{D}/V) \\to \\Sh(\\mathcal{C}'/U'), $$ such that setting $\\sigma = g' \\circ \\sigma'$ we have $\\pi'_{U'} \\circ \\sigma' = \\text{id}$, $\\pi_U \\circ \\sigma = \\text{id}$, $(\\sigma')^{-1} = \\pi'_{U', *}$, and $\\sigma^{-1} = \\pi_{U, *}$."}
+{"_id": "4344", "title": "sites-cohomology-lemma-properties-lower-shriek-fibred-category", "text": "Assumption and notation as in Situation \\ref{situation-morphism-fibred-categories}. \\begin{enumerate} \\item There are left adjoints $g_! : \\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ and $g_!^{\\textit{Ab}} : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$ to $g^* = g^{-1}$ on modules and on abelian sheaves. \\item The diagram $$ \\xymatrix{ \\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{g_!} & \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\ \\textit{Ab}(\\mathcal{C}') \\ar[r]^{g_!^{\\textit{Ab}}} & \\textit{Ab}(\\mathcal{C}) } $$ commutes. \\item There are left adjoints $Lg_! : D(\\mathcal{O}_{\\mathcal{C}'}) \\to D(\\mathcal{O}_\\mathcal{C})$ and $Lg_!^{\\textit{Ab}} : D(\\mathcal{C}') \\to D(\\mathcal{C})$ to $g^* = g^{-1}$ on derived categories of modules and abelian sheaves. \\item The diagram $$ \\xymatrix{ D(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{Lg_!} & D(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\ D(\\mathcal{C}') \\ar[r]^{Lg_!^{\\textit{Ab}}} & D(\\mathcal{C}) } $$ commutes. \\end{enumerate}"}
+{"_id": "4345", "title": "sites-cohomology-lemma-compute-pi-shriek", "text": "Assumptions and notation as in Situation \\ref{situation-fibred-category}. For $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$ the sheaf $\\pi_!\\mathcal{F}$ is the sheaf associated to the presheaf $$ V \\longmapsto \\colim_{\\mathcal{C}_V^{opp}} \\mathcal{F}|_{\\mathcal{C}_V} $$ with restriction maps as indicated in the proof."}
+{"_id": "4346", "title": "sites-cohomology-lemma-initial-final", "text": "Notation and assumptions as in Example \\ref{example-category-to-point}. If $\\mathcal{C}$ has either an initial or a final object, then $L\\pi_! \\circ \\pi^{-1} = \\text{id}$ on $D(\\textit{Ab})$, resp.\\ $D(B)$."}
+{"_id": "4347", "title": "sites-cohomology-lemma-change-of-rings", "text": "Notation and assumptions as in Example \\ref{example-category-to-point}. Let $B \\to B'$ be a ring map. Consider the commutative diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C}), \\underline{B}) \\ar[d]_\\pi & (\\Sh(\\mathcal{C}), \\underline{B'}) \\ar[d]^{\\pi'} \\ar[l]^h \\\\ (*, B) & (*, B') \\ar[l]_f } $$ Then $L\\pi_! \\circ Lh^* = Lf^* \\circ L\\pi'_!$."}
+{"_id": "4348", "title": "sites-cohomology-lemma-compute-by-cosimplicial-resolution", "text": "Notation and assumptions as in Example \\ref{example-category-to-point}. Let $U_\\bullet$ be a cosimplicial object in $\\mathcal{C}$ such that for every $U \\in \\Ob(\\mathcal{C})$ the simplicial set $\\Mor_\\mathcal{C}(U_\\bullet, U)$ is homotopy equivalent to the constant simplicial set on a singleton. Then $$ L\\pi_!(\\mathcal{F}) = \\mathcal{F}(U_\\bullet) $$ in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$."}
+{"_id": "4349", "title": "sites-cohomology-lemma-get-it-now", "text": "Notation and assumptions as in Example \\ref{example-morphism-categories}. If there exists a cosimplicial object $U'_\\bullet$ of $\\mathcal{C}'$ such that Lemma \\ref{lemma-compute-by-cosimplicial-resolution} applies to both $U'_\\bullet$ in $\\mathcal{C}'$ and $u(U'_\\bullet)$ in $\\mathcal{C}$, then we have $L\\pi'_! \\circ g^{-1} = L\\pi_!$ as functors $D(\\mathcal{C}) \\to D(\\textit{Ab})$, resp.\\ $D(\\mathcal{C}, \\underline{B}) \\to D(B)$."}
+{"_id": "4350", "title": "sites-cohomology-lemma-product-categories", "text": "Let $\\mathcal{C}_i$, $i = 1, 2$ be categories. Let $u_i : \\mathcal{C}_1 \\times \\mathcal{C}_2 \\to \\mathcal{C}_i$ be the projection functors. Let $B$ be a ring. Let $g_i : (\\Sh(\\mathcal{C}_1 \\times \\mathcal{C}_2), \\underline{B}) \\to (\\Sh(\\mathcal{C}_i), \\underline{B})$ be the corresponding morphisms of ringed topoi, see Example \\ref{example-morphism-categories}. For $K_i \\in D(\\mathcal{C}_i, B)$ we have $$ L(\\pi_1 \\times \\pi_2)_!( g_1^{-1}K_1 \\otimes_{\\underline{B}}^\\mathbf{L} g_2^{-1}K_2) = L\\pi_{1, !}(K_1) \\otimes_B^\\mathbf{L} L\\pi_{2, !}(K_2) $$ in $D(B)$ with obvious notation."}
+{"_id": "4351", "title": "sites-cohomology-lemma-eilenberg-zilber", "text": "Notation and assumptions as in Example \\ref{example-category-to-point}. If there exists a cosimplicial object $U_\\bullet$ of $\\mathcal{C}$ such that Lemma \\ref{lemma-compute-by-cosimplicial-resolution} applies, then $$ L\\pi_!(K_1 \\otimes^\\mathbf{L}_{\\underline{B}} K_2) = L\\pi_!(K_1) \\otimes^\\mathbf{L}_B L\\pi_!(K_2) $$ for all $K_i \\in D(\\underline{B})$."}
+{"_id": "4352", "title": "sites-cohomology-lemma-O-homology-qis", "text": "Let $\\mathcal{C}$ be a category (endowed with chaotic topology). Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map of sheaves of rings on $\\mathcal{C}$. Assume \\begin{enumerate} \\item there exists a cosimplicial object $U_\\bullet$ in $\\mathcal{C}$ as in Lemma \\ref{lemma-compute-by-cosimplicial-resolution}, and \\item $L\\pi_!\\mathcal{O} \\to L\\pi_!\\mathcal{O}'$ is an isomorphism. \\end{enumerate} For $K$ in $D(\\mathcal{O})$ we have $$ L\\pi_!(K) = L\\pi_!(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}') $$ in $D(\\textit{Ab})$."}
+{"_id": "4353", "title": "sites-cohomology-lemma-compute-left-derived-pi-shriek-pre", "text": "Assumptions and notation as in Situation \\ref{situation-fibred-category}. For $\\mathcal{F}$ in $\\textit{PAb}(\\mathcal{C})$ and $n \\geq 0$ consider the abelian sheaf $L_n(\\mathcal{F})$ on $\\mathcal{D}$ which is the sheaf associated to the presheaf $$ V \\longmapsto H_n(\\mathcal{C}_V, \\mathcal{F}|_{\\mathcal{C}_V}) $$ with restriction maps as indicated in the proof. Then $L_n(\\mathcal{F}) = L_n(\\mathcal{F}^\\#)$."}
+{"_id": "4354", "title": "sites-cohomology-lemma-compute-left-derived-pi-shriek", "text": "Assumptions and notation as in Situation \\ref{situation-fibred-category}. For $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$ and $n \\geq 0$ the sheaf $L_n\\pi_!(\\mathcal{F})$ is equal to the sheaf $L_n(\\mathcal{F})$ constructed in Lemma \\ref{lemma-compute-left-derived-pi-shriek-pre}."}
+{"_id": "4355", "title": "sites-cohomology-lemma-compute-left-derived-g-shriek", "text": "Assumptions and notation as in Situation \\ref{situation-morphism-fibred-categories}. For an abelian sheaf $\\mathcal{F}'$ on $\\mathcal{C}'$ the sheaf $L_ng_!(\\mathcal{F}')$ is the sheaf associated to the presheaf $$ U \\longmapsto H_n(\\mathcal{I}_U, \\mathcal{F}'_U) $$ For notation and restriction maps see proof."}
+{"_id": "4356", "title": "sites-cohomology-lemma-base-change-by-qis", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet \\to \\mathcal{B}_\\bullet$ be a homomorphism of simplicial sheaves of rings on $\\mathcal{C}$. If $L\\pi_!\\mathcal{A}_\\bullet \\to L\\pi_!\\mathcal{B}_\\bullet$ is an isomorphism in $D(\\mathcal{C})$, then we have $$ L\\pi_!(K) = L\\pi_!(K \\otimes^\\mathbf{L}_{\\mathcal{A}_\\bullet} \\mathcal{B}_\\bullet) $$ for all $K$ in $D(\\mathcal{A}_\\bullet)$."}
+{"_id": "4357", "title": "sites-cohomology-lemma-cone", "text": "The cone on a morphism of strictly perfect complexes is strictly perfect."}
+{"_id": "4358", "title": "sites-cohomology-lemma-tensor", "text": "The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect."}
+{"_id": "4359", "title": "sites-cohomology-lemma-strictly-perfect-pullback", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. If $\\mathcal{F}^\\bullet$ is a strictly perfect complex of $\\mathcal{O}_\\mathcal{D}$-modules, then $f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of $\\mathcal{O}_\\mathcal{C}$-modules."}
+{"_id": "4360", "title": "sites-cohomology-lemma-local-lift-map", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. Given a solid diagram of $\\mathcal{O}_U$-modules $$ \\xymatrix{ \\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\ & \\mathcal{G} \\ar[u]_p } $$ with $\\mathcal{E}$ a direct summand of a finite free $\\mathcal{O}_U$-module and $p$ surjective, then there exists a covering $\\{U_i \\to U\\}$ such that a dotted arrow making the diagram commute exists over each $U_i$."}
+{"_id": "4361", "title": "sites-cohomology-lemma-local-homotopy", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. \\begin{enumerate} \\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ be a morphism of complexes of $\\mathcal{O}_U$-modules with $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$ acyclic. Then there exists a covering $\\{U_i \\to U\\}$ such that each $\\alpha|_{U_i}$ is homotopic to zero. \\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ be a morphism of complexes of $\\mathcal{O}_U$-modules with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$ for $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$. Then there exists a covering $\\{U_i \\to U\\}$ such that each $\\alpha|_{U_i}$ is homotopic to zero. \\end{enumerate}"}
+{"_id": "4362", "title": "sites-cohomology-lemma-lift-through-quasi-isomorphism", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. Given a solid diagram of complexes of $\\mathcal{O}_U$-modules $$ \\xymatrix{ \\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\ & \\mathcal{G}^\\bullet \\ar[u]_f } $$ with $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for $j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then there exists a covering $\\{U_i \\to U\\}$ and for each $i$ a dotted arrow over $U_i$ making the diagram commute up to homotopy."}
+{"_id": "4363", "title": "sites-cohomology-lemma-local-actual", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}_U$-modules with $\\mathcal{E}^\\bullet$ strictly perfect. \\begin{enumerate} \\item For any element $\\alpha \\in \\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ there exists a covering $\\{U_i \\to U\\}$ such that $\\alpha|_{U_i}$ is given by a morphism of complexes $\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$. \\item Given a morphism of complexes $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ whose image in the group $\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is zero, there exists a covering $\\{U_i \\to U\\}$ such that $\\alpha|_{U_i}$ is homotopic to zero. \\end{enumerate}"}
+{"_id": "4364", "title": "sites-cohomology-lemma-Rhom-strictly-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}$-modules with $\\mathcal{E}^\\bullet$ strictly perfect. Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is represented by the complex $\\mathcal{H}^\\bullet$ with terms $$ \\mathcal{H}^n = \\bigoplus\\nolimits_{n = p + q} \\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p) $$ and differential as described in Section \\ref{section-internal-hom}."}
+{"_id": "4365", "title": "sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes of $\\mathcal{O}$-modules with \\begin{enumerate} \\item $\\mathcal{F}^n = 0$ for $n \\ll 0$, \\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and \\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite free $\\mathcal{O}$-module. \\end{enumerate} Then the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ is represented by the complex $\\mathcal{H}^\\bullet$ with terms $$ \\mathcal{H}^n = \\bigoplus\\nolimits_{n = p + q} \\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p) $$ and differential as described in Section \\ref{section-internal-hom}."}
+{"_id": "4366", "title": "sites-cohomology-lemma-pseudo-coherent-independent-representative", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. \\begin{enumerate} \\item If $\\mathcal{C}$ has a final object $X$ and if there exist a covering $\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$ of $\\mathcal{O}_{U_i}$-modules, and maps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in $D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$ and $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent. \\item If $E$ is $m$-pseudo-coherent, then any complex of $\\mathcal{O}$-modules representing $E$ is $m$-pseudo-coherent. \\item If for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $E|_{U_i}$ is $m$-pseudo-coherent, then $E$ is $m$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "4367", "title": "sites-cohomology-lemma-pseudo-coherent-pullback", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{C})$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent."}
+{"_id": "4368", "title": "sites-cohomology-lemma-cone-pseudo-coherent", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site and $m \\in \\mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O})$. \\begin{enumerate} \\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent then $M$ is $m$-pseudo-coherent. \\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent. \\item If $L$ is $(m + 1)$-pseudo-coherent and $M$ is $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "4369", "title": "sites-cohomology-lemma-tensor-pseudo-coherent", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\\mathcal{O})$. \\begin{enumerate} \\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is $t$-pseudo-coherent with $t = \\max(m + a, n + b)$. \\item If $K$ and $L$ are pseudo-coherent, then $K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is pseudo-coherent. \\end{enumerate}"}
+{"_id": "4370", "title": "sites-cohomology-lemma-summands-pseudo-coherent", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $m \\in \\mathbf{Z}$. If $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) in $D(\\mathcal{O})$ so are $K$ and $L$."}
+{"_id": "4371", "title": "sites-cohomology-lemma-finite-cohomology", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\\mathcal{O})$. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}$-module. \\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$ for $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented $\\mathcal{O}$-module. \\end{enumerate}"}
+{"_id": "4372", "title": "sites-cohomology-lemma-last-one-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$ be a bounded above complex of flat $\\mathcal{O}$-modules with tor-amplitude in $[a, b]$. Then $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat $\\mathcal{O}$-module."}
+{"_id": "4373", "title": "sites-cohomology-lemma-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent \\begin{enumerate} \\item $E$ has tor-amplitude in $[a, b]$. \\item $E$ is represented by a complex $\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with $\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "4374", "title": "sites-cohomology-lemma-bounded-below-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. Let $a \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $E$ has tor-amplitude in $[a, \\infty]$. \\item $E$ can be represented by a K-flat complex $\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with $\\mathcal{E}^i = 0$ for $i \\not \\in [a, \\infty]$. \\end{enumerate} Moreover, we can choose $\\mathcal{E}^\\bullet$ such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms."}
+{"_id": "4375", "title": "sites-cohomology-lemma-tor-amplitude-pullback", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$."}
+{"_id": "4376", "title": "sites-cohomology-lemma-cone-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$. \\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$. \\item If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$, then $K$ has tor-amplitude in $[a + 1, b + 1]$. \\end{enumerate}"}
+{"_id": "4377", "title": "sites-cohomology-lemma-tensor-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \\otimes_\\mathcal{O}^\\mathbf{L} L$ has tor amplitude in $[a + c, b + d]$."}
+{"_id": "4378", "title": "sites-cohomology-lemma-summands-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a, b \\in \\mathbf{Z}$. For $K$, $L$ objects of $D(\\mathcal{O})$ if $K \\oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$."}
+{"_id": "4379", "title": "sites-cohomology-lemma-bounded", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals. Let $K$ be an object of $D(\\mathcal{O})$. \\begin{enumerate} \\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$ is bounded above, then $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$ is uniformly bounded above for all $n$. \\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$ as an object of $D(\\mathcal{O}/\\mathcal{I})$ has tor amplitude in $[a, b]$, then $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$ as an object of $D(\\mathcal{O}/\\mathcal{I}^n)$ has tor amplitude in $[a, b]$ for all $n$. \\end{enumerate}"}
+{"_id": "4380", "title": "sites-cohomology-lemma-tor-amplitude-stalk", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $E$ has tor amplitude in $[a, b]$, then for every point $p$ of the site $\\mathcal{C}$ the object $E_p$ of $D(\\mathcal{O}_p)$ has tor amplitude in $[a, b]$. \\item If $\\mathcal{C}$ has enough points, then the converse is true. \\end{enumerate}"}
+{"_id": "4381", "title": "sites-cohomology-lemma-perfect-independent-representative", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. \\begin{enumerate} \\item If $\\mathcal{C}$ has a final object $X$ and there exist a covering $\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$ of $\\mathcal{O}_{U_i}$-modules, and isomorphisms  $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in $D(\\mathcal{O}_{U_i})$, then $E$ is perfect. \\item If $E$ is perfect, then any complex representing $E$ is perfect. \\end{enumerate}"}
+{"_id": "4382", "title": "sites-cohomology-lemma-perfect-precise", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. Let $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect."}
+{"_id": "4383", "title": "sites-cohomology-lemma-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. The following are equivalent \\begin{enumerate} \\item $E$ is perfect, and \\item $E$ is pseudo-coherent and locally has finite tor dimension. \\end{enumerate}"}
+{"_id": "4384", "title": "sites-cohomology-lemma-perfect-pullback", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to (\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$. If $E$ is perfect in $D(\\mathcal{O}_\\mathcal{D})$, then $Lf^*E$ is perfect in $D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "4385", "title": "sites-cohomology-lemma-two-out-of-three-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O})$. If two out of three of $K, L, M$ are perfect then the third is also perfect."}
+{"_id": "4388", "title": "sites-cohomology-lemma-symmetric-monoidal-cat-complexes", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. The category of complexes of $\\mathcal{O}$-modules with tensor product defined by $\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet = \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet)$ is a symmetric monoidal category."}
+{"_id": "4390", "title": "sites-cohomology-lemma-dual-perfect-complex", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be a perfect object of $D(\\mathcal{O})$. Then $K^\\vee = R\\SheafHom(K, \\mathcal{O})$ is a perfect object too and $(K^\\vee)^\\vee \\cong K$. There are functorial isomorphisms $$ M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee = R\\SheafHom_\\mathcal{O}(K, M) $$ and $$ H^0(\\mathcal{C}, M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee) = \\Hom_{D(\\mathcal{O})}(K, M) $$ for $M$ in $D(\\mathcal{O})$."}
+{"_id": "4391", "title": "sites-cohomology-lemma-symmetric-monoidal-derived", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The derived category $D(\\mathcal{O})$ is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section \\ref{more-algebra-section-sign-rules})."}
+{"_id": "4392", "title": "sites-cohomology-lemma-left-dual-derived", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\\mathcal{O})$. If $M$ has a left dual in the monoidal category $D(\\mathcal{O})$ (Categories, Definition \\ref{categories-definition-dual}) then $M$ is perfect and the left dual is as constructed in Example \\ref{example-dual-derived}."}
+{"_id": "4393", "title": "sites-cohomology-lemma-colim-and-lim-of-duals", "text": "\\begin{slogan} Trivial duality for systems of perfect objects. \\end{slogan} Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O})$. Let $K = \\text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \\ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(\\mathcal{O})$ we have $$ R\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_\\mathcal{O} K_n^\\vee $$ where $(K_n^\\vee)$ is the inverse system of dual perfect complexes."}
+{"_id": "4394", "title": "sites-cohomology-lemma-category-summands-finite-free", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. Set $R = \\Gamma(\\mathcal{C}, \\mathcal{O})$. The category of $\\mathcal{O}$-modules which are summands of finite free $\\mathcal{O}$-modules is equivalent to the category of finite projective $R$-modules."}
+{"_id": "4396", "title": "sites-cohomology-lemma-projection-formula", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $E \\in D(\\mathcal{O}_\\mathcal{C})$ and $K \\in D(\\mathcal{O}_\\mathcal{D})$. If $K$ is perfect, then $$ Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{D}} K = Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K) $$ in $D(\\mathcal{O}_\\mathcal{D})$."}
+{"_id": "4397", "title": "sites-cohomology-lemma-w-contractible", "text": "Let $\\mathcal{C}$ be a site. Let $U$ be a weakly contractible object of $\\mathcal{C}$. Then \\begin{enumerate} \\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(U)$ is an exact functor $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$, \\item $H^p(U, \\mathcal{F}) = 0$ for every abelian sheaf $\\mathcal{F}$ and all $p \\geq 1$, and \\item for any sheaf of groups $\\mathcal{G}$ any $\\mathcal{G}$-torsor has a section over $U$. \\end{enumerate}"}
+{"_id": "4398", "title": "sites-cohomology-lemma-compact-in-terms-of-generators", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $S \\subset \\Ob(\\mathcal{A})$ be a set of objects such that \\begin{enumerate} \\item any object of $\\mathcal{A}$ is a quotient of a direct sum of elements of $S$, and \\item for any $E \\in S$ the functor $\\Hom_\\mathcal{A}(E, -)$ commutes with direct sums. \\end{enumerate} Then every compact object of $D(\\mathcal{A})$ is a direct summand in $D(\\mathcal{A})$ of a finite complex of finite direct sums of elements of $S$."}
+{"_id": "4400", "title": "sites-cohomology-lemma-when-jshriek-lower-compact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. Assume the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$ commute with direct sums. Then $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a compact object of $D^+(\\mathcal{O})$ in the following sense: if $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is bounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) = \\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$."}
+{"_id": "4401", "title": "sites-cohomology-lemma-when-jshriek-lower-compact-worked-out", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site with set of coverings $\\text{Cov}_\\mathcal{C}$. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and $\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$ be subsets. Assume that \\begin{enumerate} \\item For every $\\mathcal{U} \\in \\text{Cov}$ we have $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite, $U, U_i \\in \\mathcal{B}$ and every $U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ the coverings of $U$ occurring in $\\text{Cov}$ is a cofinal system of coverings of $U$. \\end{enumerate} Then for $U \\in \\mathcal{B}$ the object $j_{U!}\\mathcal{O}_U$ is a compact object of $D^+(\\mathcal{O})$ in the following sense: if $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is bounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) = \\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$."}
+{"_id": "4402", "title": "sites-cohomology-lemma-when-jshriek-compact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. The $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a compact object of $D(\\mathcal{O})$ if there exists an integer $d$ such that \\begin{enumerate} \\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and \\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$ commute with direct sums. \\end{enumerate}"}
+{"_id": "4403", "title": "sites-cohomology-lemma-quasi-compact-weakly-contractible-compact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$ which is quasi-compact and weakly contractible. Then $j_!\\mathcal{O}_U$ is a compact object of $D(\\mathcal{O})$."}
+{"_id": "4405", "title": "sites-cohomology-lemma-locally-constant", "text": "Let $\\mathcal{C}$ be a site with final object $X$. Let $\\Lambda$ be a Noetherian ring. Let $K \\in D^b(\\mathcal{C}, \\Lambda)$ with $H^i(K)$ locally constant sheaves of $\\Lambda$-modules of finite type. Then there exists a covering $\\{U_i \\to X\\}$ such that each $K|_{U_i}$ is represented by a complex of locally constant sheaves of $\\Lambda$-modules of finite type."}
+{"_id": "4406", "title": "sites-cohomology-lemma-map-out-of-locally-constant", "text": "Let $\\mathcal{C}$ be a site with final object $X$. Let $\\Lambda$ be a ring. Let \\begin{enumerate} \\item $K$ a perfect object of $D(\\Lambda)$, \\item a finite complex $K^\\bullet$ of finite projective $\\Lambda$-modules representing $K$, \\item $\\mathcal{L}^\\bullet$ a complex of sheaves of $\\Lambda$-modules, and \\item $\\varphi : \\underline{K} \\to \\mathcal{L}^\\bullet$ a map in $D(\\mathcal{C}, \\Lambda)$. \\end{enumerate} Then there exists a covering $\\{U_i \\to X\\}$ and maps of complexes $\\alpha_i : \\underline{K}^\\bullet|_{U_i} \\to \\mathcal{L}^\\bullet|_{U_i}$ representing $\\varphi|_{U_i}$."}
+{"_id": "4407", "title": "sites-cohomology-lemma-locally-constant-map", "text": "Let $\\mathcal{C}$ be a site with final object $X$. Let $\\Lambda$ be a ring. Let $K, L$ be objects of $D(\\Lambda)$ with $K$ perfect. Let $\\varphi : \\underline{K} \\to \\underline{L}$ be map in $D(\\mathcal{C}, \\Lambda)$. There exists a covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is equal to $\\underline{\\alpha_i}$ for some map $\\alpha_i : K \\to L$ in $D(\\Lambda)$."}
+{"_id": "4408", "title": "sites-cohomology-lemma-locally-constant-tensor-product", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring. Let $K, L \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of $K$ and $L$ are locally constant sheaves of $\\Lambda$-modules of finite type, then the cohomology sheaves of $K \\otimes_\\Lambda^\\mathbf{L} L$ are locally constant sheaves of $\\Lambda$-modules of finite type."}
+{"_id": "4409", "title": "sites-cohomology-lemma-locally-constant-bounded", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal. Let $K \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally constant sheaves of $\\Lambda/I$-modules of finite type, then the cohomology sheaves of $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ are locally constant sheaves of $\\Lambda/I^n$-modules of finite type for all $n \\geq 1$."}
+{"_id": "4410", "title": "sites-cohomology-proposition-enough-weakly-contractibles", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that every $U \\in \\mathcal{B}$ is weakly contractible and every object of $\\mathcal{C}$ has a covering by elements of $\\mathcal{B}$. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Then \\begin{enumerate} \\item A complex $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$ of $\\mathcal{O}$-modules is exact, if and only if $\\mathcal{F}_1(U) \\to \\mathcal{F}_2(U) \\to \\mathcal{F}_3(U)$ is exact for all $U \\in \\mathcal{B}$. \\item Every object $K$ of $D(\\mathcal{O})$ is a derived limit of its canonical truncations: $K = R\\lim \\tau_{\\geq -n} K$. \\item Given an inverse system $\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$ with surjective transition maps, the projection $\\lim \\mathcal{F}_n \\to \\mathcal{F}_1$ is surjective. \\item Products are exact on $\\textit{Mod}(\\mathcal{O})$. \\item Products on $D(\\mathcal{O})$ can be computed by taking products of any representative complexes. \\item If $(\\mathcal{F}_n)$ is an inverse system of $\\mathcal{O}$-modules, then $R^p\\lim \\mathcal{F}_n = 0$ for all $p > 1$ and $$ R^1\\lim \\mathcal{F}_n  = \\Coker(\\prod \\mathcal{F}_n \\to \\prod \\mathcal{F}_n) $$ where the map is $(x_n) \\mapsto (x_n - f(x_{n + 1}))$. \\item If $(K_n)$ is an inverse system of objects of $D(\\mathcal{O})$, then there are short exact sequences $$ 0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(R\\lim K_n) \\to \\lim H^p(K_n) \\to 0 $$ \\end{enumerate}"}
+{"_id": "4441", "title": "fields-theorem-existence-algebraic-closure", "text": "Every field has an algebraic closure."}
+{"_id": "4442", "title": "fields-theorem-galois-theory", "text": "Let $L/K$ be a finite Galois extension with Galois group $G$. Then we have $K = L^G$ and the map $$ \\{\\text{subgroups of }G\\} \\longrightarrow \\{\\text{subextensions }K \\subset M \\subset L\\},\\quad H \\longmapsto L^H $$ is a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$. The normal subgroups $H$ of $G$ correspond exactly to those subextensions $M$ with $M/K$ Galois."}
+{"_id": "4443", "title": "fields-theorem-inifinite-galois-theory", "text": "Let $L/K$ be a Galois extension. Let $G = \\text{Gal}(L/K)$ be the Galois group viewed as a profinite topological group (Lemma \\ref{lemma-galois-profinite}). Then we have $K = L^G$ and the map $$ \\{\\text{closed subgroups of }G\\} \\longrightarrow \\{\\text{subextensions }K \\subset M \\subset L\\},\\quad H \\longmapsto L^H $$ is a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$. The finite subextensions $M$ correspond exactly to the open subgroups $H \\subset G$. The normal closed subgroups $H$ of $G$ correspond exactly to subextensions $M$ Galois over $K$."}
+{"_id": "4444", "title": "fields-lemma-vector-space-is-free", "text": "If $k$ is a field, then every $k$-module is free."}
+{"_id": "4446", "title": "fields-lemma-field-maps-injective", "text": "If $F$ is a field and $R$ is a nonzero ring, then any ring homomorphism $\\varphi : F \\to R$ is injective."}
+{"_id": "4447", "title": "fields-lemma-field-extension-generated-by-one-element", "text": "If a field extension $F/k$ is generated by one element, then it is $k$-isomorphic either to the rational function field $k(t)/k$ or to one of the extensions $k[t]/(P)$ for $P \\in k[t]$ irreducible."}
+{"_id": "4448", "title": "fields-lemma-finite-goes-up", "text": "Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is finite over $F$, then $K$ is finite over $E$."}
+{"_id": "4449", "title": "fields-lemma-finite-finitely-generated", "text": "A finite extension of fields is a finitely generated field extension. The converse is not true."}
+{"_id": "4450", "title": "fields-lemma-multiplicativity-degrees", "text": "Suppose given a tower of fields $F/E/k$. Then $$ [F:k] = [F:E][E:k] $$"}
+{"_id": "4451", "title": "fields-lemma-algebraic-goes-up", "text": "Let $K/E/F$ be a tower of field extensions. \\begin{enumerate} \\item If $\\alpha \\in K$ is algebraic over $F$, then $\\alpha$ is algebraic over $E$. \\item if $K$ is algebraic over $F$, then $K$ is algebraic over $E$. \\end{enumerate}"}
+{"_id": "4452", "title": "fields-lemma-finite-is-algebraic", "text": "A finite extension is algebraic. In fact, an extension $E/k$ is algebraic if and only if every subextension $k(\\alpha)/k$ generated by some $\\alpha \\in E$ is finite."}
+{"_id": "4453", "title": "fields-lemma-algebraic-finitely-generated", "text": "\\begin{slogan} A finitely generated algebraic extension is finite. \\end{slogan} Let $k$ be a field, and let $\\alpha_1, \\alpha_2, \\ldots, \\alpha_n$ be elements of some extension field such that each $\\alpha_i$ is algebraic over $k$. Then the extension $k(\\alpha_1, \\ldots, \\alpha_n)/k$ is finite. That is, a finitely generated algebraic extension is finite."}
+{"_id": "4455", "title": "fields-lemma-algebraic-permanence", "text": "Let $E/k$ and $F/E$ be algebraic extensions of fields. Then $F/k$ is an algebraic extension of fields."}
+{"_id": "4456", "title": "fields-lemma-size-algebraic-extension", "text": "Let $E/F$ be an algebraic extension of fields. Then the cardinality $|E|$ of $E$ is at most $\\max(\\aleph_0, |F|)$."}
+{"_id": "4457", "title": "fields-lemma-subalgebra-algebraic-extension-field", "text": "Let $E/F$ be a finite or more generally an algebraic extension of fields. Any subring $F \\subset R \\subset E$ is a field."}
+{"_id": "4458", "title": "fields-lemma-algebraic-extension-self-map", "text": "Let $E/F$ an algebraic extension of fields. Any $F$-algebra map $f : E \\to E$ is an automorphism."}
+{"_id": "4459", "title": "fields-lemma-degree-minimal-polynomial", "text": "The degree of the minimal polynomial is $[k(\\alpha) : k]$."}
+{"_id": "4460", "title": "fields-lemma-algebraically-closed", "text": "Let $F$ be a field. The following are equivalent \\begin{enumerate} \\item $F$ is algebraically closed, \\item every irreducible polynomial over $F$ is linear, \\item every nonconstant polynomial over $F$ has a root, \\item every nonconstant polynomial over $F$ is a product of linear factors. \\end{enumerate}"}
+{"_id": "4461", "title": "fields-lemma-map-into-algebraic-closure", "text": "Let $F$ be a field. Let $\\overline{F}$ be an algebraic closure of $F$. Let $M/F$ be an algebraic extension. Then there is a morphism of $F$-extensions $M \\to \\overline{F}$."}
+{"_id": "4463", "title": "fields-lemma-relatively-prime-polynomials", "text": "Two polynomials in $k[x]$ are relatively prime precisely when they have no common roots in an algebraic closure $\\overline{k}$ of $k$."}
+{"_id": "4464", "title": "fields-lemma-irreducible-polynomials", "text": "Let $F$ be a field. Let $P \\in F[x]$ be an irreducible polynomial over $F$. Let $P' = \\text{d}P/\\text{d}x$ be the derivative of $P$ with respect to $x$. Then one of the following two cases happens \\begin{enumerate} \\item $P$ and $P'$ are relatively prime, or \\item $P'$ is the zero polynomial. \\end{enumerate} The second case can only happen if $F$ has characteristic $p > 0$. In this case $P(x) = Q(x^q)$ where $q = p^f$ is a power of $p$ and $Q \\in F[x]$ is an irreducible polynomial such that $Q$ and $Q'$ are relatively prime."}
+{"_id": "4465", "title": "fields-lemma-separable-goes-up", "text": "Let $K/E/F$ be a tower of algebraic field extensions. \\begin{enumerate} \\item If $\\alpha \\in K$ is separable over $F$, then $\\alpha$ is separable over $E$. \\item if $K$ is separable over $F$, then $K$ is separable over $E$. \\end{enumerate}"}
+{"_id": "4466", "title": "fields-lemma-recognize-separable", "text": "Let $F$ be a field. An irreducible polynomial $P$ over $F$ is separable if and only if $P$ has pairwise distinct roots in an algebraic closure of $F$."}
+{"_id": "4467", "title": "fields-lemma-nr-roots-unchanged", "text": "Let $F$ be a field and let $\\overline{F}$ be an algebraic closure of $F$. Let $p > 0$ be the characteristic of $F$. Let $P$ be a polynomial over $F$. Then the set of roots of $P$ and $P(x^p)$ in $\\overline{F}$ have the same cardinality (not counting multiplicity)."}
+{"_id": "4468", "title": "fields-lemma-count-embeddings", "text": "In Situation \\ref{situation-finitely-generated} the correspondence $$ \\Mor_F(K, \\overline{F}) \\longrightarrow \\{(\\beta_1, \\ldots, \\beta_n)\\text{ as below}\\}, \\quad \\varphi \\longmapsto (\\varphi(\\alpha_1), \\ldots, \\varphi(\\alpha_n)) $$ is a bijection. Here the right hand side is the set of $n$-tuples $(\\beta_1, \\ldots, \\beta_n)$ of elements of $\\overline{F}$ such that $\\beta_i$ is a root of $P_i^\\varphi$."}
+{"_id": "4469", "title": "fields-lemma-count-embeddings-explicitly", "text": "In Situation \\ref{situation-finitely-generated} we have $|\\Mor_F(K, \\overline{F})| = \\prod_{i = 1}^n \\deg_s(P_i)$."}
+{"_id": "4470", "title": "fields-lemma-separably-generated-separable", "text": "Assumptions and notation as in Situation \\ref{situation-finitely-generated}. If each $P_i$ is separable, i.e., each $\\alpha_i$ is separable over $K_{i - 1}$, then $$ |\\Mor_F(K, \\overline{F})| = [K : F] $$ and the field extension $K/F$ is separable. If one of the $\\alpha_i$ is not separable over $K_{i - 1}$, then $|\\Mor_F(K, \\overline{F})| < [K : F]$."}
+{"_id": "4471", "title": "fields-lemma-separable-equality", "text": "Let $K/F$ be a finite extension of fields. Let $\\overline{F}$ be an algebraic closure of $F$. Then we have $$ |\\Mor_F(K, \\overline{F})| \\leq [K : F] $$ with equality if and only if $K$ is separable over $F$."}
+{"_id": "4472", "title": "fields-lemma-separable-permanence", "text": "Let $E/k$ and $F/E$ be separable algebraic extensions of fields. Then $F/k$ is a separable extension of fields."}
+{"_id": "4473", "title": "fields-lemma-separable-elements", "text": "Let $E/k$ be a field extension. Then the elements of $E$ separable over $k$ form a subextension of $E/k$."}
+{"_id": "4474", "title": "fields-lemma-independence-characters", "text": "Let $L$ be a field. Let $G$ be a monoid, for example a group. Let $\\chi_1, \\ldots, \\chi_n : G \\to L$ be pairwise distinct homomorphisms of monoids where $L$ is regarded as a monoid by multiplication. Then $\\chi_1, \\ldots, \\chi_n$ are $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$ not all zero, then $\\sum \\lambda_i\\chi_i(g) \\not = 0$ for some $g \\in G$."}
+{"_id": "4475", "title": "fields-lemma-sums-of-powers", "text": "Let $L$ be a field. Let $n \\geq 1$ and $\\alpha_1, \\ldots, \\alpha_n \\in L$ pairwise distinct elements of $L$. Then there exists an $e \\geq 0$ such that $\\sum_{i = 1, \\ldots, n} \\alpha_i^e \\not = 0$."}
+{"_id": "4476", "title": "fields-lemma-independence-embeddings", "text": "Let $K/F$ and $L/F$ be field extensions. Let $\\sigma_1, \\ldots, \\sigma_n : K \\to L$ be pairwise distinct morphisms of $F$-extensions. Then $\\sigma_1, \\ldots, \\sigma_n$ are $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$ not all zero, then $\\sum \\lambda_i\\sigma_i(\\alpha) \\not = 0$ for some $\\alpha \\in K$."}
+{"_id": "4477", "title": "fields-lemma-finite-separable-tensor-alg-closed", "text": "Let $K/F$ and $L/F$ be field extensions with $K/F$ finite separable and $L$ algebraically closed. Then the map $$ K \\otimes_F L \\longrightarrow \\prod\\nolimits_{\\sigma \\in \\Hom_F(K, L)} L,\\quad \\alpha \\otimes \\beta \\mapsto (\\sigma(\\alpha)\\beta)_\\sigma $$ is an isomorphism of $L$-algebras."}
+{"_id": "4478", "title": "fields-lemma-take-pth-root", "text": "Let $p$ be a prime number. Let $F$ be a field of characteristic $p$. Let $t \\in F$ be an element which does not have a $p$th root in $F$. Then the polynomial $x^p - t$ is irreducible over $F$."}
+{"_id": "4479", "title": "fields-lemma-purely-inseparable-permanence", "text": "Let $E/k$ and $F/E$ be purely inseparable extensions of fields. Then $F/k$ is a purely inseparable extension of fields."}
+{"_id": "4480", "title": "fields-lemma-purely-inseparable-elements", "text": "Let $E/k$ be a field extension. Then the elements of $E$ purely-inseparable over $k$ form a subextension of $E/k$."}
+{"_id": "4481", "title": "fields-lemma-finite-purely-inseparable", "text": "Let $E/F$ be a finite purely inseparable field extension of characteristic $p > 0$. Then there exists a sequence of elements $\\alpha_1, \\ldots, \\alpha_n \\in E$ such that we obtain a tower of fields $$ E = F(\\alpha_1, \\ldots, \\alpha_n) \\supset F(\\alpha_1, \\ldots, \\alpha_{n - 1}) \\supset \\ldots \\supset F(\\alpha_1) \\supset F $$ such that each intermediate extension is of degree $p$ and comes from adjoining a $p$th root. Namely, $\\alpha_i^p \\in F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ is an element which does not have a $p$th root in $F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ for $i = 1, \\ldots, n$."}
+{"_id": "4482", "title": "fields-lemma-separable-first", "text": "\\begin{slogan} Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one. \\end{slogan} Let $E/F$ be an algebraic field extension. There exists a unique subextension $E/E_{sep}/F$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is purely inseparable."}
+{"_id": "4483", "title": "fields-lemma-separable-degree", "text": "Let $K/F$ be a finite extension. Let $\\overline{F}$ be an algebraic closure of $F$. Then $[K : F]_s = |\\Mor_F(K, \\overline{F})|$."}
+{"_id": "4484", "title": "fields-lemma-multiplicativity-all-degrees", "text": "Suppose given a tower of algebraic field extensions $K/E/F$. Then $$ [K : F]_s = [K : E]_s [E : F]_s \\quad\\text{and}\\quad [K : F]_i = [K : E]_i [E : F]_i $$"}
+{"_id": "4485", "title": "fields-lemma-normal-goes-up", "text": "Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is normal over $F$, then $K$ is normal over $E$."}
+{"_id": "4486", "title": "fields-lemma-intersect-normal", "text": "Let $F$ be a field. Let $M/F$ be an algebraic extension. Let $F \\subset E_i \\subset M$, $i \\in I$ be subextensions with $E_i/F$ normal. Then $\\bigcap E_i$ is normal over $F$."}
+{"_id": "4487", "title": "fields-lemma-separable-first-normal", "text": "Let $E/F$ be a normal algebraic field extension. Then the subextension $E/E_{sep}/F$ of Lemma \\ref{lemma-separable-first} is normal."}
+{"_id": "4488", "title": "fields-lemma-characterize-normal", "text": "Let $E/F$ be an algebraic extension of fields. Let $\\overline{F}$ be an algebraic closure of $F$. The following are equivalent \\begin{enumerate} \\item $E$ is normal over $F$, and \\item for every pair $\\sigma, \\sigma' \\in \\Mor_F(E, \\overline{F})$ we have $\\sigma(E) = \\sigma'(E)$. \\end{enumerate}"}
+{"_id": "4489", "title": "fields-lemma-normally-generated", "text": "Let $E/F$ be an algebraic extension of fields. If $E$ is generated by $\\alpha_i \\in E$, $i \\in I$ over $F$ and if for each $i$ the minimal polynomial of $\\alpha_i$ over $F$ splits completely in $E$, then $E/F$ is normal."}
+{"_id": "4490", "title": "fields-lemma-lift-maps", "text": "Let $L/M/K$ be a tower of algebraic extensions. \\begin{enumerate} \\item If $M/K$ is normal, then any automorphism $\\tau$ of $L/K$ induces an automorphism $\\tau|_M : M \\to M$. \\item If $L/K$ is normal, then any $K$-algebra map $\\sigma : M \\to L$ extends to an automorphism of $L$. \\end{enumerate}"}
+{"_id": "4491", "title": "fields-lemma-normal-and-automorphisms", "text": "Let $E/F$ be a finite extension. We have $$ |\\text{Aut}(E/F)| \\leq [E : F]_s $$ with equality if and only if $E$ is normal over $F$."}
+{"_id": "4492", "title": "fields-lemma-normal-embeddings-differ-by-aut", "text": "Let $L/K$ be an algebraic normal extension of fields. Let $E/K$ be an extension of fields. Then either there is no $K$-embedding from $L$ to $E$ or there is one $\\tau : L \\to E$ and every other one is of the form $\\tau \\circ \\sigma$ where $\\sigma \\in \\text{Aut}(L/K)$."}
+{"_id": "4493", "title": "fields-lemma-splitting-field", "text": "Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial. There exists a smallest field extension $E/F$ such that $P$ splits completely over $E$. Moreover, the field extension $E/F$ is normal and unique up to (nonunique) isomorphism."}
+{"_id": "4494", "title": "fields-lemma-normal-closure", "text": "\\begin{slogan} Existence of normal closure of finite extensions of fields. \\end{slogan} Let $E/F$ be a finite extension of fields. There exists a unique smallest finite extension $K/E$ such that $K$ is normal over $F$."}
+{"_id": "4495", "title": "fields-lemma-normal-closure-inside-normal", "text": "Let $L/K$ be an algebraic normal extension. \\begin{enumerate} \\item If $L/M/K$ is a subextension with $M/K$ finite, then there exists a tower $L/M'/M/K$ with $M'/K$ finite and normal. \\item If $L/M'/M/K$ is a tower with $M/K$ normal and $M'/M$ finite, then there exists a tower $L/M''/M'/M/K$ with $M''/M$ finite and $M''/K$ normal. \\end{enumerate}"}
+{"_id": "4496", "title": "fields-lemma-normal-closure-tensor-product", "text": "Let $L/K$ be a finite extension. Let $M/L$ be the normal closure of $L$ over $K$. Then there is a surjective map $$ L \\otimes_K L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M $$ of $K$-algebras where the number of tensors can be taken $[L : K]_s \\leq [L : K]$."}
+{"_id": "4497", "title": "fields-lemma-cyclic", "text": "Let $A$ be an abelian group of exponent dividing $n$ such that $\\{x \\in A \\mid dx = 0\\}$ has cardinality at most $d$ for all $d | n$. Then $A$ is cyclic of order dividing $n$."}
+{"_id": "4498", "title": "fields-lemma-primitive-element", "text": "Let $E/F$ be a finite extension of fields. The following are equivalent \\begin{enumerate} \\item there exists a primitive element for $E$ over $F$, and \\item there are finitely many subextensions $E/K/F$. \\end{enumerate} Moreover, (1) and (2) hold if $E/F$ is separable."}
+{"_id": "4499", "title": "fields-lemma-characteristic-vs-minimal-polynomial", "text": "Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let $P$ be the minimal polynomial of $\\alpha$ over $K$. Then the characteristic polynomial of the $K$-linear map $\\alpha : L \\to L$ is equal to $P^e$ with $e \\deg(P) = [L : K]$."}
+{"_id": "4500", "title": "fields-lemma-trace-and-norm-from-minimal-polynomial", "text": "Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let $P = x^d + a_1 x^{d - 1} + \\ldots + a_d$ be the minimal polynomial of $\\alpha$ over $K$. Then $$ \\text{Norm}_{L/K}(\\alpha) = (-1)^{[L : K]} a_d^e \\quad\\text{and}\\quad \\text{Trace}_{L/K}(\\alpha) = - e a_1 $$ where $e d = [L : K]$."}
+{"_id": "4501", "title": "fields-lemma-trace-and-norm-linear", "text": "Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional vector space over $L$. Let $\\varphi : V \\to V$ be an $L$-linear map. Then $$ \\text{Trace}_K(\\varphi : V \\to V) = \\text{Trace}_{L/K}(\\text{Trace}_L(\\varphi : V \\to V)) $$ and $$ \\det\\nolimits_K(\\varphi : V \\to V) = \\text{Norm}_{L/K}(\\det\\nolimits_L(\\varphi : V \\to V)) $$"}
+{"_id": "4502", "title": "fields-lemma-trace-and-norm-tower", "text": "Let $M/L/K$ be a tower of finite extensions of fields. Then $$ \\text{Trace}_{M/K} = \\text{Trace}_{L/K} \\circ \\text{Trace}_{M/L} \\quad\\text{and}\\quad \\text{Norm}_{M/K} = \\text{Norm}_{L/K} \\circ \\text{Norm}_{M/L} $$"}
+{"_id": "4503", "title": "fields-lemma-separable-trace-pairing", "text": "Let $L/K$ be a finite extension of fields. The following are equivalent: \\begin{enumerate} \\item $L/K$ is separable, \\item $\\text{Trace}_{L/K}$ is not identically zero, and \\item the trace pairing $Q_{L/K}$ is nondegenerate. \\end{enumerate}"}
+{"_id": "4504", "title": "fields-lemma-finite-Galois", "text": "Let $E/F$ be a finite extension of fields. Then $E$ is Galois over $F$ if and only if $|\\text{Aut}(E/F)| = [E : F]$."}
+{"_id": "4505", "title": "fields-lemma-galois-goes-up", "text": "Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is Galois over $F$, then $K$ is Galois over $E$."}
+{"_id": "4506", "title": "fields-lemma-normal-closure-galois", "text": "Let $L/K$ be a finite separable extension of fields. Let $M$ be the normal closure of $L$ over $K$ (Definition \\ref{definition-normal-closure}). Then $M/K$ is Galois."}
+{"_id": "4507", "title": "fields-lemma-galois-over-fixed-field", "text": "Let $K$ be a field. Let $G$ be a finite group acting faithfully on $K$. Then the extension $K/K^G$ is Galois, we have $[K : K^G] = |G|$, and the Galois group of the extension is $G$."}
+{"_id": "4508", "title": "fields-lemma-ses-galois", "text": "Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are finite Galois. Then we obtain a short exact sequence $$ 1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1 $$ of finite groups."}
+{"_id": "4509", "title": "fields-lemma-galois-profinite", "text": "Let $E/F$ be a Galois extension. Endow $\\text{Gal}(E/F)$ with the coarsest topology such that $$ \\text{Gal}(E/F) \\times E \\longrightarrow E $$ is continuous when $E$ is given the discrete topology. Then \\begin{enumerate} \\item for any topological space $X$ and map $X \\to \\text{Aut}(E/F)$ such that the action $X \\times E \\to E$ is continuous the induced map $X \\to \\text{Gal}(E/F)$ is continuous, \\item this topology turns $\\text{Gal}(E/F)$ into a profinite topological group. \\end{enumerate}"}
+{"_id": "4510", "title": "fields-lemma-galois-infinite", "text": "Let $L/M/K$ be a tower of fields. Assume both $L/K$ and $M/K$ are Galois. Then there is a canonical surjective continuous homomorphism $c : \\text{Gal}(L/K) \\to \\text{Gal}(M/K)$."}
+{"_id": "4511", "title": "fields-lemma-infinite-galois-limit", "text": "Let $L/K$ be a Galois extension with Galois group $G$. Let $\\Lambda$ be the set of finite Galois subextensions, i.e., $\\lambda \\in \\Lambda$ corresponds to $L/L_\\lambda/K$ with $L_\\lambda/K$ finite Galois with Galois group $G_\\lambda$. Define a partial ordering on $\\Lambda$ by the rule $\\lambda \\geq \\lambda'$ if and only if $L_\\lambda \\supset L_{\\lambda'}$. Then \\begin{enumerate} \\item $\\Lambda$ is a directed partially ordered set, \\item $L_\\lambda$ is a system of $K$-extensions over $\\Lambda$ and $L = \\colim L_\\lambda$, \\item $G_\\lambda$ is an inverse system of finite groups over $\\Lambda$, the transition maps are surjective, and $$ G = \\lim_{\\lambda \\in \\Lambda} G_\\lambda $$ as a profinite group, and \\item each of the projections $G \\to G_\\lambda$ is continuous and surjective. \\end{enumerate}"}
+{"_id": "4512", "title": "fields-lemma-ses-infinite-galois", "text": "Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are Galois. Then we obtain a short exact sequence $$ 1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1 $$ of profinite topological groups."}
+{"_id": "4513", "title": "fields-lemma-C-algebraically-closed", "text": "The field $\\mathbf{C}$ is algebraically closed."}
+{"_id": "4514", "title": "fields-lemma-Kummer", "text": "Let $K \\subset L$ be a Galois extension of fields whose Galois group is $\\mathbf{Z}/n\\mathbf{Z}$. Assume moreover that the characteristic of $K$ is prime to $n$ and that $K$ contains a primitive $n$th root of $1$. Then $L = K[z]$ with $z^n \\in K$."}
+{"_id": "4515", "title": "fields-lemma-adjoint-pth-root-unity", "text": "Let $K$ be a field with algebraic closure $\\overline{K}$. Let $p$ be a prime different from the characteristic of $K$. Let $\\zeta \\in \\overline{K}$ be a primitive $p$th root of $1$. Then $K(\\zeta)/K$ is a Galois extension of degree dividing $p - 1$."}
+{"_id": "4516", "title": "fields-lemma-subfields-kummer", "text": "Let $K$ be a field. Let $L/K$ be a finite extension of degree $e$ which is generated by an element $\\alpha$ with $a = \\alpha^e \\in K$. Then any sub extension $L/L'/K$ is generated by $\\alpha^d$ for some $d | e$."}
+{"_id": "4517", "title": "fields-lemma-Artin-Schreier", "text": "Let $K \\subset L$ be a Galois extension of fields of characteristic $p > 0$ with Galois group $\\mathbf{Z}/p\\mathbf{Z}$. Then $L = K[z]$ with $z^p - z \\in K$."}
+{"_id": "4518", "title": "fields-lemma-transcendence-degree", "text": "Let $E/F$ be a field extension. A transcendence basis of $E$ over $F$ exists. Any two transcendence bases have the same cardinality."}
+{"_id": "4519", "title": "fields-lemma-transcendence-degree-tower", "text": "Let $k \\subset K \\subset L$ be field extensions. Then $$ \\text{trdeg}_k(L) = \\text{trdeg}_K(L) + \\text{trdeg}_k(K). $$"}
+{"_id": "4520", "title": "fields-lemma-purely-transcendental-degree", "text": "Let $k'/k$ be a finite extension of fields. Let $k'(x_1, \\ldots, x_r)/k(x_1, \\ldots, x_r)$ be the induced extension of purely transcendental extensions. Then $[k'(x_1, \\ldots, x_r) : k(x_1, \\ldots, x_r)] = [k' : k] < \\infty$."}
+{"_id": "4521", "title": "fields-lemma-algebraic-closure-in-finitely-generated", "text": "Let $k \\subset K$ be a finitely generated field extension. The algebraic closure of $k$ in $K$ is finite over $k$."}
+{"_id": "4522", "title": "fields-lemma-normal-case", "text": "Let $E/F$ be a normal algebraic field extension. There exist subextensions $E / E_{sep} /F$ and $E / E_{insep} / F$ such that \\begin{enumerate} \\item $F \\subset E_{sep}$ is Galois and $E_{sep} \\subset E$ is purely inseparable, \\item $F \\subset E_{insep}$ is purely inseparable and $E_{insep} \\subset E$ is Galois, \\item $E = E_{sep} \\otimes_F E_{insep}$. \\end{enumerate}"}
+{"_id": "4523", "title": "fields-lemma-pth-root", "text": "Let $K$ be a field of characteristic $p > 0$. Let $K \\subset L$ be a separable algebraic extension. Let $\\alpha \\in L$. \\begin{enumerate} \\item If the coefficients of the minimal polynomial of $\\alpha$ over $K$ are $p$th powers in $K$ then $\\alpha$ is a $p$th power in $L$. \\item More generally, if $P \\in K[T]$ is a polynomial such that (a) $\\alpha$ is a root of $P$, (b) $P$ has pairwise distinct roots in an algebraic closure, and (c) all coefficients of $P$ are $p$th powers, then $\\alpha$ is a $p$th power in $L$. \\end{enumerate}"}
+{"_id": "4556", "title": "spaces-limits-lemma-characterize-relative-limit-preserving", "text": "Let $S$ be a scheme. Let $a : F \\to G$ be a transformation of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. The following are equivalent \\begin{enumerate} \\item $a : F \\to G$ is limit preserving, and \\item for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$ the diagram of sets $$ \\xymatrix{ \\colim_i F(T_i) \\ar[r] \\ar[d]_a & F(T) \\ar[d]^a \\\\ \\colim_i G(T_i) \\ar[r] & G(T) } $$ is a fibre product diagram. \\end{enumerate}"}
+{"_id": "4557", "title": "spaces-limits-lemma-composition-locally-finite-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : G \\to H$ be transformations of functors. If $a$ and $b$ are limit preserving, then $$ b \\circ a : F \\longrightarrow H $$ is limit preserving."}
+{"_id": "4558", "title": "spaces-limits-lemma-locally-finite-presentation-permanence", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : G \\to H$ be transformations of functors. If $b \\circ a$ and $b$ are limit preserving, then $a$ is limit preserving."}
+{"_id": "4559", "title": "spaces-limits-lemma-base-change-locally-finite-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : H \\to G$ be transformations of functors. Consider the fibre product diagram $$ \\xymatrix{ H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\ H \\ar[r]^b & G } $$ If $a$ is limit preserving, then the base change $a'$ is limit preserving."}
+{"_id": "4560", "title": "spaces-limits-lemma-fibre-product-locally-finite-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $E, F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : H \\to G$, and $c : G \\to E$ be transformations of functors. If $c$, $c \\circ a$, and $c \\circ b$ are limit preserving, then $F \\times_G H \\to E$ is too."}
+{"_id": "4561", "title": "spaces-limits-lemma-sheafify-finite-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. If $F$ is limit preserving then its sheafification $F^\\#$ is limit preserving."}
+{"_id": "4562", "title": "spaces-limits-lemma-sheaf-finite-presentation", "text": "Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item $F$ is a sheaf, and \\item there exists an fppf covering $\\{U_j \\to S\\}_{j \\in J}$ such that $F|_{(\\Sch/U_j)_{fppf}}$ is limit preserving. \\end{enumerate} Then $F$ is limit preserving."}
+{"_id": "4564", "title": "spaces-limits-lemma-surjection-is-enough", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \\lim_{i \\in I} T_i$ of affine schemes over $S$ the map $$ \\colim X(T_i) \\longrightarrow X(T) \\times_{Y(T)} \\colim Y(T_i) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \\ref{proposition-characterize-locally-finite-presentation} part (2) it suffices to check surjectivity in the criterion of Lemma \\ref{lemma-characterize-relative-limit-preserving}."}
+{"_id": "4565", "title": "spaces-limits-lemma-directed-inverse-system-has-limit", "text": "Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ in the category of algebraic spaces over $S$. If the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine, then the limit $X = \\lim_i X_i$ (as an fppf sheaf) is an algebraic space. Moreover, \\begin{enumerate} \\item each of the morphisms $f_i : X \\to X_i$ is affine, \\item for any $i \\in I$ and any morphism of algebraic spaces $T \\to X_i$ we have $$ X \\times_{X_i} T = \\lim_{i' \\geq i} X_{i'} \\times_{X_i} T. $$ as algebraic spaces over $S$. \\end{enumerate}"}
+{"_id": "4566", "title": "spaces-limits-lemma-space-over-limit", "text": "Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$ with affine transition maps. Let $X = \\lim_i X_i$. Let $0 \\in I$. Suppose that $T \\to X_0$ is a morphism of algebraic spaces. Then $$ T \\times_{X_0} X = \\lim_{i \\geq 0} T \\times_{X_0} X_i $$ as algebraic spaces over $S$."}
+{"_id": "4567", "title": "spaces-limits-lemma-directed-inverse-system-closed-immersions", "text": "Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i}) \\to (Y_i, g_{i'i})$ be a morphism of inverse systems over $I$ of algebraic spaces over $S$. Assume \\begin{enumerate} \\item the morphisms $f_{i'i} : X_{i'} \\to X_i$ are affine, \\item the morphisms $g_{i'i} : Y_{i'} \\to Y_i$ are affine, \\item the morphisms $X_i \\to Y_i$ are closed immersions. \\end{enumerate} Then $\\lim X_i \\to \\lim Y_i$ is a closed immersion."}
+{"_id": "4568", "title": "spaces-limits-lemma-directed-inverse-system-reduced", "text": "Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse systems over $I$ of algebraic spaces over $S$. If $X_i$ is reduced for all $i$, then $X$ is reduced."}
+{"_id": "4569", "title": "spaces-limits-lemma-better-characterize-relative-limit-preserving", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$. The equivalent conditions (1) and (2) of Proposition \\ref{proposition-characterize-locally-finite-presentation} are also equivalent to \\begin{enumerate} \\item[(3)] for every directed limit $T = \\lim T_i$ of quasi-compact and quasi-separated algebraic spaces $T_i$ over $S$ with affine transition morphisms the diagram of sets $$ \\xymatrix{ \\colim_i \\Mor(T_i, X) \\ar[r] \\ar[d] & \\Mor(T, X) \\ar[d] \\\\ \\colim_i \\Mor(T_i, Y) \\ar[r] & \\Mor(T, Y) } $$ is a fibre product diagram. \\end{enumerate}"}
+{"_id": "4570", "title": "spaces-limits-lemma-inverse-limit-sets", "text": "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is decent (for example quasi-separated or locally separated) then $|X| = \\lim_i |X_i|$ as sets."}
+{"_id": "4571", "title": "spaces-limits-lemma-topology-limit", "text": "With same notation and assumptions as in Lemma \\ref{lemma-inverse-limit-sets} we have $|X| = \\lim_i |X_i|$ as topological spaces."}
+{"_id": "4572", "title": "spaces-limits-lemma-limit-nonempty", "text": "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$ is quasi-compact and nonempty, then $|X|$ is nonempty."}
+{"_id": "4573", "title": "spaces-limits-lemma-inverse-limit-irreducibles", "text": "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). Let $x \\in |X|$ with images $x_i \\in |X_i|$. If each $X_i$ is decent, then $\\overline{\\{x\\}} = \\lim_i \\overline{\\{x_i\\}}$ as sets and as algebraic spaces if endowed with reduced induced scheme structure."}
+{"_id": "4575", "title": "spaces-limits-lemma-descend-opens", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. For any quasi-compact open subspace $U \\subset X$ there exists an $i$ and a quasi-compact open $U_i \\subset X_i$ whose inverse image in $X$ is $U$."}
+{"_id": "4576", "title": "spaces-limits-lemma-descend-equality", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. Let $f_0 : Y_0 \\to Z_0$ be a morphism of algebraic spaces over $X_0$. Assume (a) $Y_0 \\to X_0$ and $Z_0 \\to X_0$ are representable, (b) $Y_0$, $Z_0$ quasi-compact and quasi-separated, (c) $f_0$ locally of finite presentation, and (d) $Y_0 \\times_{X_0} X \\to Z_0 \\times_{X_0} X$ an isomorphism. Then there exists an $i \\geq 0$ such that $Y_0 \\times_{X_0} X_i \\to Z_0 \\times_{X_0} X_i$ is an isomorphism."}
+{"_id": "4577", "title": "spaces-limits-lemma-descend-separated", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. If $X$ is separated, then $X_i$ is separated for some $i \\in I$."}
+{"_id": "4578", "title": "spaces-limits-lemma-limit-is-affine", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. If $X$ is affine, then there exists an $i$ such that $X_i$ is affine."}
+{"_id": "4579", "title": "spaces-limits-lemma-limit-is-scheme", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. If $X$ is a scheme, then there exists an $i$ such that $X_i$ is a scheme."}
+{"_id": "4580", "title": "spaces-limits-lemma-finite-type-eventually-closed", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $B$ with affine transition morphisms. Let $Y \\to X$ be a morphism of algebraic spaces over $B$. \\begin{enumerate} \\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and $Y \\to B$ locally of finite type, then $Y \\to X_i$ is a closed immersion for $i$ large enough. \\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to B$ locally of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an immersion for $i$ large enough. \\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact, $X_i \\to B$ locally of finite type, the transition morphisms $X_{i'} \\to X_i$ are closed immersions, and $Y \\to B$ is locally of finite presentation, then $Y \\to X_i$ is an isomorphism for $i$ large enough. \\item If $Y \\to X$ is a monomorphism, $X_i$ quasi-separated, $Y \\to B$ locally of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is a monomorphism for $i$ large enough. \\end{enumerate}"}
+{"_id": "4581", "title": "spaces-limits-lemma-eventually-separated", "text": "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \\begin{enumerate} \\item $Y$ is quasi-separated, \\item $X_i$ is quasi-compact and quasi-separated, \\item the morphism $X \\to Y$ is separated. \\end{enumerate} Then $X_i \\to Y$ is separated for all $i$ large enough."}
+{"_id": "4584", "title": "spaces-limits-lemma-eventually-closed-immersion", "text": "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \\begin{enumerate} \\item $Y$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions, \\item $X_i \\to Y$ locally of finite type \\item $X \\to Y$ is a closed immersion. \\end{enumerate} Then $X_i \\to Y$ is a closed immersion for $i$ large enough."}
+{"_id": "4585", "title": "spaces-limits-lemma-descend-etale", "text": "With notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is \\'etale, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is \\'etale for some $i \\geq 0$."}
+{"_id": "4586", "title": "spaces-limits-lemma-descend-smooth", "text": "With notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is smooth, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is smooth for some $i \\geq 0$."}
+{"_id": "4587", "title": "spaces-limits-lemma-descend-surjective", "text": "With notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is surjective, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is surjective for some $i \\geq 0$."}
+{"_id": "4588", "title": "spaces-limits-lemma-descend-universally-injective", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is universally injective, \\item $f_0$ is locally of finite type, \\end{enumerate} then $f_i$ is universally injective for some $i \\geq 0$."}
+{"_id": "4590", "title": "spaces-limits-lemma-descend-finite", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is finite, \\item $f_0$ is locally of finite type, \\end{enumerate} then $f_i$ is finite for some $i \\geq 0$."}
+{"_id": "4591", "title": "spaces-limits-lemma-descend-closed-immersion", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a closed immersion, \\item $f_0$ is locally of finite type, \\end{enumerate} then $f_i$ is a closed immersion for some $i \\geq 0$."}
+{"_id": "4592", "title": "spaces-limits-lemma-descend-separated-morphism", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If $f$ is separated, then $f_i$ is separated for some $i \\geq 0$."}
+{"_id": "4593", "title": "spaces-limits-lemma-descend-isomorphism", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a isomorphism, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is a isomorphism for some $i \\geq 0$."}
+{"_id": "4595", "title": "spaces-limits-lemma-descend-flat", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module and denote $\\mathcal{F}_i$ the pullback to $X_i$ and $\\mathcal{F}$ the pullback to $X$. If \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $Y$, \\item $\\mathcal{F}_0$ is of finite presentation, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $\\mathcal{F}_i$ is flat over $Y_i$ for some $i \\geq 0$. In particular, if $f_0$ is locally of finite presentation and $f$ is flat, then $f_i$ is flat for some $i \\geq 0$."}
+{"_id": "4596", "title": "spaces-limits-lemma-eventually-proper", "text": "Assumptions and notation as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is proper, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i$ such that $f_i$ is proper."}
+{"_id": "4597", "title": "spaces-limits-lemma-eventually-relative-dimension", "text": "Assumptions and notation as in Situation \\ref{situation-descent-property}. Let $d \\geq 0$. If \\begin{enumerate} \\item $f$ has relative dimension $\\leq d$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-relative-dimension}), and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i$ such that $f_i$ has relative dimension $\\leq d$."}
+{"_id": "4598", "title": "spaces-limits-lemma-descend-finite-presentation", "text": "Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces over $S$. Assume \\begin{enumerate} \\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine, \\item the spaces $X_i$ are quasi-compact and quasi-separated. \\end{enumerate} Let $X = \\lim_i X_i$. Then the category of algebraic spaces of finite presentation over $X$ is the colimit over $I$ of the categories of algebraic spaces of finite presentation over $X_i$."}
+{"_id": "4599", "title": "spaces-limits-lemma-descend-modules-finite-presentation", "text": "With notation and assumptions as in Lemma \\ref{lemma-descend-finite-presentation}. The category of $\\mathcal{O}_X$-modules of finite presentation is the colimit over $I$ of the categories $\\mathcal{O}_{X_i}$-modules of finite presentation."}
+{"_id": "4600", "title": "spaces-limits-lemma-descend-invertible-modules", "text": "With notation and assumptions as in Lemma \\ref{lemma-descend-finite-presentation}. Then any invertible $\\mathcal{O}_X$-module is the pullback of an invertible $\\mathcal{O}_{X_i}$-module for some $i$."}
+{"_id": "4601", "title": "spaces-limits-lemma-colimit-finitely-presented", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Every quasi-coherent $\\mathcal{O}_X$-module is a filtered colimit of finitely presented $\\mathcal{O}_X$-modules."}
+{"_id": "4602", "title": "spaces-limits-lemma-directed-colimit-finite-type", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules."}
+{"_id": "4603", "title": "spaces-limits-lemma-finite-directed-colimit-surjective-maps", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Then we can write $\\mathcal{F} = \\lim \\mathcal{F}_i$ where each $\\mathcal{F}_i$ is an $\\mathcal{O}_X$-module of finite presentation and all transition maps $\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective."}
+{"_id": "4604", "title": "spaces-limits-lemma-algebra-directed-colimit-finite-presentation", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra. Then $\\mathcal{A}$ is a directed colimit of finitely presented quasi-coherent $\\mathcal{O}_X$-algebras."}
+{"_id": "4605", "title": "spaces-limits-lemma-algebra-directed-colimit-finite-type", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra. Then $\\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\\mathcal{O}_X$-subalgebras."}
+{"_id": "4606", "title": "spaces-limits-lemma-finite-algebra-directed-colimit-finite-finitely-presented", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be a finite quasi-coherent $\\mathcal{O}_X$-algebra. Then $\\mathcal{A} = \\colim \\mathcal{A}_i$ is a directed colimit of finite and finitely presented quasi-coherent $\\mathcal{O}_X$-algebras with surjective transition maps."}
+{"_id": "4607", "title": "spaces-limits-lemma-integral-algebra-directed-colimit-finite", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be an integral quasi-coherent $\\mathcal{O}_X$-algebra. Then \\begin{enumerate} \\item $\\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\\mathcal{O}_X$-subalgebras, and \\item $\\mathcal{A}$ is a directed colimit of finite and finitely presented $\\mathcal{O}_X$-algebras. \\end{enumerate}"}
+{"_id": "4608", "title": "spaces-limits-lemma-extend", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \\subset X$ be a quasi-compact open. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent $\\mathcal{O}_U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$ which is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$."}
+{"_id": "4609", "title": "spaces-limits-lemma-relative-approximation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated, and \\item $Y$ is quasi-separated. \\end{enumerate} Then $X = \\lim X_i$ is a limit of a directed inverse system of algebraic spaces $X_i$ of finite presentation over $Y$ with affine transition morphisms over $Y$."}
+{"_id": "4610", "title": "spaces-limits-lemma-affine-morphism-is-limit", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. If $Y$ quasi-compact and quasi-separated, then $X$ is a directed limit $X = \\lim X_i$ with each $X_i$ affine and of finite presentation over $Y$."}
+{"_id": "4611", "title": "spaces-limits-lemma-integral-limit-finite-and-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$ where $X_i$ are finite and of finite presentation over $Y$."}
+{"_id": "4612", "title": "spaces-limits-lemma-finite-in-finite-and-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$ where the transition maps are closed immersions and the objects $X_i$ are finite and of finite presentation over $Y$."}
+{"_id": "4613", "title": "spaces-limits-lemma-closed-is-limit-closed-and-finite-presentation", "text": "\\begin{slogan} Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a closed immersion of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$ where the transition maps are closed immersions and the morphisms $X_i \\to Y$ are closed immersions of finite presentation."}
+{"_id": "4614", "title": "spaces-limits-lemma-quasi-affine-closed-in-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is locally of finite type and quasi-affine, and \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a morphism of finite presentation $f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$."}
+{"_id": "4615", "title": "spaces-limits-lemma-finite-type-closed-in-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume: \\begin{enumerate} \\item $f$ is of locally of finite type. \\item $X$ is quasi-compact and quasi-separated, and \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a morphism of finite presentation $f' : X' \\to Y$ and a closed immersion $X \\to X'$ of algebraic spaces over $Y$."}
+{"_id": "4616", "title": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then $X = \\lim X_i$ is a directed limit of algebraic spaces $X_i$ proper and of finite presentation over $Y$ and with transition morphisms and morphisms $X \\to X_i$ closed immersions."}
+{"_id": "4617", "title": "spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "text": "Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $\\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \\to Y_i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_i \\to X_{i'}$ and $Y_i \\to Y_{i'}$ are affine, such that $f_i$ is proper and of finite presentation, such that $Y_i$ is of finite presentation over $\\mathbf{Z}$, and such that $(X \\to Y) = \\lim (X_i \\to Y_i)$."}
+{"_id": "4618", "title": "spaces-limits-lemma-eventually-proper-support", "text": "Assumptions and notation as in Situation \\ref{situation-descent-property}. Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module. Denote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of $\\mathcal{F}_0$ to $X$ and $X_i$. Assume \\begin{enumerate} \\item $f_0$ is locally of finite type, \\item $\\mathcal{F}_0$ is of finite type, \\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$. \\end{enumerate} Then the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$ for some $i$."}
+{"_id": "4619", "title": "spaces-limits-lemma-embedding-into-affine-over-ls-qs", "text": "Let $S$ be a scheme. Let $f : U \\to X$ be a morphism of algebraic spaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of finite type, and $X$ quasi-separated and locally separated. Then there exists an immersion $U \\to \\mathbf{A}^n_X$ over $X$."}
+{"_id": "4620", "title": "spaces-limits-lemma-embedding-into-affine-over-qs", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation. Then there exists a dense open $V \\subset Y$ and an immersion $V \\to \\mathbf{A}^n_X$."}
+{"_id": "4621", "title": "spaces-limits-lemma-quasi-coherent-finite-type-ideals", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U \\subset X$ be an open subspace. The following are equivalent: \\begin{enumerate} \\item $U \\to X$ is quasi-compact, \\item $U$ is quasi-compact, and \\item there exists a finite type quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ such that $|X| \\setminus |U| = |V(\\mathcal{I})|$. \\end{enumerate}"}
+{"_id": "4622", "title": "spaces-limits-lemma-sections-annihilated-by-ideal", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$ which associates to every object $U$ of $X_\\etale$ the module $$ \\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}s = 0\\} $$ Assume $\\mathcal{I}$ is of finite type. Then \\begin{enumerate} \\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules, \\item for affine $U$ in $X_\\etale$ we have $\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and \\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$. \\end{enumerate}"}
+{"_id": "4624", "title": "spaces-limits-lemma-sections-supported-on-closed-subset", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset and let $U \\subset X$ be the open subspace such that $T \\amalg |U| = |X|$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Consider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$ which associates to every object $\\varphi : W \\to X$ of $X_\\etale$ the module $$ \\mathcal{F}'(W) = \\{s \\in \\mathcal{F}(W) \\mid \\text{the support of }s\\text{ is contained in }|\\varphi|^{-1}(T)\\} $$ If $U \\to X$ is quasi-compact, then \\begin{enumerate} \\item for $W$ affine there exist a finitely generated ideal $I \\subset \\mathcal{O}_X(W)$ such that $|\\varphi|^{-1}(T) = V(I)$, \\item for $W$ and $I$ as in (1) we have $\\mathcal{F}'(W) = \\{x \\in \\mathcal{F}(W) \\mid I^nx = 0 \\text{ for some } n\\}$, \\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "4626", "title": "spaces-limits-lemma-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $Y$ is affine."}
+{"_id": "4627", "title": "spaces-limits-lemma-reduction-scheme", "text": "\\begin{reference} \\cite[3.1.12]{CLO} \\end{reference} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X_{red}$ is a scheme, then $X$ is a scheme."}
+{"_id": "4628", "title": "spaces-limits-lemma-integral-universally-bijective-scheme", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is integral and induces a bijection $|X| \\to |Y|$. Then $X$ is a scheme if and only if $Y$ is a scheme."}
+{"_id": "4629", "title": "spaces-limits-lemma-check-closed-infinitesimally", "text": "Let $S$ be a scheme. Let $f : X \\to B$ and $B' \\to B$ be morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $B' \\to B$ is a closed immersion, \\item $|B'| \\to |B|$ is bijective, \\item $X \\times_B B' \\to B'$ is a closed immersion, and \\item $X \\to B$ is of finite type or $B' \\to B$ is of finite presentation. \\end{enumerate} Then $f : X \\to B$ is a closed immersion."}
+{"_id": "4632", "title": "spaces-limits-lemma-enough-local", "text": "Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism from an algebraic space to a scheme $S$. If for every $x \\in |X|$ with image $s = f(x) \\in S$ the algebraic space $X \\times_S \\Spec(\\mathcal{O}_{S,s})$ is a scheme, then $X$ is a scheme."}
+{"_id": "4634", "title": "spaces-limits-lemma-relative-glueing", "text": "Let $S = U \\cup W$ be an open covering of a scheme. Then the functor $$ FP_S \\longrightarrow FP_U \\times_{FP_{U \\cap W}} FP_W $$ given by base change is an equivalence where $FP_T$ is the category of algebraic spaces of finite presentation over the scheme $T$."}
+{"_id": "4635", "title": "spaces-limits-lemma-glueing-near-closed-point", "text": "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that $U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With $V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is an equivalence of categories $$ FP_S \\longrightarrow FP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})} $$ where $FP_T$ is the category of algebraic spaces of finite presentation over $T$."}
+{"_id": "4637", "title": "spaces-limits-lemma-glueing-near-multiple-closed-points", "text": "Let $S$ be a scheme. Let $s_1, \\ldots, s_n \\in S$ be pairwise distinct closed points such that $U = S \\setminus \\{s_1, \\ldots, s_n\\} \\to S$ is quasi-compact. With $S_i = \\Spec(\\mathcal{O}_{S, s_i})$ and $U_i = S_i \\setminus \\{s_i\\}$ there is an equivalence of categories $$ FP_S \\longrightarrow FP_U \\times_{(FP_{U_1} \\times \\ldots \\times FP_{U_n})} (FP_{S_1} \\times \\ldots \\times FP_{S_n}) $$ where $FP_T$ is the category of algebraic spaces of finite presentation over $T$."}
+{"_id": "4638", "title": "spaces-limits-lemma-excision-modifications", "text": "Let $S$ be a scheme. Consider a separated \\'etale morphism $f : V \\to W$ of algebraic spaces over $S$. Assume there exists a closed subspace $T \\subset W$ such that $f^{-1}T \\to T$ is an isomorphism. Then, with $W^0 = W \\setminus T$ and $V^0 = f^{-1}W^0$ the base change functor $$ \\left\\{ \\begin{matrix} g : X \\to W\\text{ morphism of algebraic spaces} \\\\ g^{-1}(W^0) \\to W^0\\text{ is an isomorphism} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} h : Y \\to V\\text{ morphism of algebraic spaces} \\\\ h^{-1}(V^0) \\to V^0\\text{ is an isomorphism} \\end{matrix} \\right\\} $$ is an equivalence of categories."}
+{"_id": "4639", "title": "spaces-limits-lemma-excision-modifications-properties", "text": "Notation and assumptions as in Lemma \\ref{lemma-excision-modifications}. Let $g : X \\to W$ correspond to $h : Y \\to V$ via the equivalence. Then $g$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if $h$ is so."}
+{"_id": "4641", "title": "spaces-limits-lemma-separate", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of algebraic spaces over $S$. Let $z \\in |Z|$ and let $T \\subset |X \\times_Y Z|$ be a closed subset with $z \\not \\in \\Im(T \\to |Z|)$. If $f$ is quasi-compact, then there exists an \\'etale neighbourhood $(V, v) \\to (Z, z)$, a commutative diagram $$ \\xymatrix{ V \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\ Z \\ar[r]^g & Y, } $$ and a closed subset $T' \\subset |X \\times_Y Z'|$ such that \\begin{enumerate} \\item the morphism $b : Z' \\to Y$ is locally of finite presentation, \\item with $z' = a(v)$ we have $z' \\not \\in \\Im(T' \\to |Z'|)$, and \\item the inverse image of $T$ in $|X \\times_Y V|$ maps into $T'$ via $|X \\times_Y V| \\to |X \\times_Y Z'|$. \\end{enumerate} Moreover, we may assume $V$ and $Z'$ are affine schemes and if $Z$ is a scheme we may assume $V$ is an affine open neighbourhood of $z$."}
+{"_id": "4642", "title": "spaces-limits-lemma-test-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item for every morphism $Z \\to Y$ which is locally of finite presentation the map $|X \\times_Y Z| \\to |Z|$ is closed, and \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $|\\mathbf{A}^n \\times (X \\times_Y V)| \\to |\\mathbf{A}^n \\times V|$ is closed for all $n \\geq 0$. \\end{enumerate}"}
+{"_id": "4645", "title": "spaces-limits-lemma-refined-valuative-criterion-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |U| \\to |X|$ is dense in $|X|$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper."}
+{"_id": "4646", "title": "spaces-limits-lemma-refined-valuative-criterion-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |U| \\to |X|$ is dense in $|X|$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists at most one dotted arrow making the diagram commute, then $f$ is separated."}
+{"_id": "4648", "title": "spaces-limits-lemma-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $X \\to B$ be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an $i \\in I$ and a diagram \\begin{equation} \\label{equation-good-diagram} \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ B \\ar[r] & B_i } } \\end{equation} such that $W \\to B_i$ is of finite type and such that the induced morphism $X \\to B \\times_{B_i} W$ is a closed immersion."}
+{"_id": "4649", "title": "spaces-limits-lemma-limit-from-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $X \\to B$ be a quasi-separated and finite type morphism of algebraic spaces. Given $i \\in I$ and a diagram $$ \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ B \\ar[r] & B_i } } $$ as in (\\ref{equation-good-diagram}) for $i' \\geq i$ let $X_{i'}$ be the scheme theoretic image of $X \\to B_{i'} \\times_{B_i} W$. Then $X = \\lim_{i' \\geq i} X_{i'}$."}
+{"_id": "4650", "title": "spaces-limits-lemma-morphism-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $f : X \\to Y$ be a morphism of algebraic spaces quasi-separated and of finite type over $B$. Let $$ \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ B \\ar[r] & B_{i_1} } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ Y \\ar[r] \\ar[d] & V \\ar[d] \\\\ B \\ar[r] & B_{i_2} } } $$ be diagrams as in (\\ref{equation-good-diagram}). Let $X = \\lim_{i \\geq i_1} X_i$ and $Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding limit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}. Then there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism $$ (f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0} $$ of inverse systems over $(B_i)_{i \\geq i_0}$ such that such that $f = \\lim_{i \\geq i_0} f_i$. If $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$ is a second morphism of inverse systems over $(B_i)_{i \\geq i_0}$ such that such that $f = \\lim_{i \\geq i_0} g_i$ then $f_i = g_i$ for all $i \\gg i_0$."}
+{"_id": "4651", "title": "spaces-limits-lemma-morphism-good-diagram-flat", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is flat and of finite presentation, then there exists an $i_3 > i_0$ such that for $i \\geq i_3$ we have $f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and $X = Y \\times_{Y_{i_3}} X_{i_3}$."}
+{"_id": "4652", "title": "spaces-limits-lemma-morphism-good-diagram-smooth", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is smooth, then there exists an $i_3 > i_0$ such that for $i \\geq i_3$ we have $f_i$ is smooth."}
+{"_id": "4653", "title": "spaces-limits-lemma-morphism-good-diagram-proper", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have $f_i$ is proper."}
+{"_id": "4654", "title": "spaces-limits-lemma-good-diagram-fibre-product", "text": "In Situation \\ref{situation-limit-noetherian} suppose that we have a cartesian diagram $$ \\xymatrix{ X^1 \\ar[r]_p \\ar[d]_q & X^3 \\ar[d]^a \\\\ X^2 \\ar[r]^b & X^4 } $$ of algebraic spaces quasi-separated and of finite type over $B$. For each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram $$ \\xymatrix{ X^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\ B \\ar[r] & B_{i_j} } $$ as in (\\ref{equation-good-diagram}). Let $X^j = \\lim_{i \\geq i_j} X^j_i$ be the corresponding limit descriptions as in Lemma \\ref{lemma-morphism-good-diagram}. Let $(a_i)_{i \\geq i_5}$, $(b_i)_{i \\geq i_6}$, $(p_i)_{i \\geq i_7}$, and $(q_i)_{i \\geq i_8}$ be the corresponding morphisms of inverse systems contructed in Lemma \\ref{lemma-morphism-good-diagram}. Then there exists an $i_9 \\geq \\max(i_5, i_6, i_7, i_8)$ such that for $i \\geq i_9$ we have $a_i \\circ p_i = b_i \\circ q_i$ and such that $$ (q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i $$ is a closed immersion. If $a$ and $b$ are flat and of finite presentation, then there exists an $i_{10} \\geq \\max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \\geq i_{10}$ the last displayed morphism is an isomorphism."}
+{"_id": "4655", "title": "spaces-limits-proposition-characterize-locally-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-locally-finite-presentation}. \\item The morphism $f : X \\to Y$ is limit preserving as a transformation of functors, see Definition \\ref{definition-locally-finite-presentation}. \\end{enumerate}"}
+{"_id": "4656", "title": "spaces-limits-proposition-approximate", "text": "\\begin{reference} Our proof follows closely the proof given in \\cite[Theorem 1.2.2]{CLO}. \\end{reference} Let $X$ be a quasi-compact and quasi-separated algebraic space over $\\Spec(\\mathbf{Z})$. There exist a directed set $I$ and an inverse system of algebraic spaces $(X_i, f_{ii'})$ over $I$ such that \\begin{enumerate} \\item the transition morphisms $f_{ii'}$ are affine \\item each $X_i$ is quasi-separated and of finite type over $\\mathbf{Z}$, and \\item $X = \\lim X_i$. \\end{enumerate}"}
+{"_id": "4657", "title": "spaces-limits-proposition-separated-closed-in-finite-presentation", "text": "Let $S$ be a scheme. $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is of finite type and separated, and \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$."}
+{"_id": "4658", "title": "spaces-limits-proposition-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is surjective and integral, and assume that $X$ is affine. Then $Y$ is affine."}
+{"_id": "4659", "title": "spaces-limits-proposition-there-is-a-scheme-finite-over", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. \\begin{enumerate} \\item There exists a surjective finite morphism $Y \\to X$ of finite presentation where $Y$ is a scheme, \\item given a surjective \\'etale morphism $U \\to X$ we may choose $Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood $V \\subset Y$ such that $V \\to X$ factors through $U$. \\end{enumerate}"}
+{"_id": "4666", "title": "stacks-geometry-lemma-deformation-category", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ is a deformation category and $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$ are finite dimensional $k$-vector spaces."}
+{"_id": "4667", "title": "stacks-geometry-lemma-versal-ring", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then a versal ring to $\\mathcal{X}$ at $x_0$ exists. Given a pair $A$, $A'$ of these, then $A \\cong A'[[t_1, \\ldots, t_r]]$ or $A' \\cong A[[t_1, \\ldots, t_r]]$ as $S$-algebras for some $r$."}
+{"_id": "4668", "title": "stacks-geometry-lemma-versal-ring-field-extension", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $l/k$ be a finite extension of fields and denote $x_{l, 0} : \\Spec(l) \\to \\mathcal{X}$ the induced morphism. Given a versal ring $A$ to $\\mathcal{X}$ at $x_0$ there exists a versal ring $A'$ to $\\mathcal{X}$ at $x_{l, 0}$ such that there is a $S$-algebra map $A \\to A'$ which induces the given field extension $l/k$ and is formally smooth in the $\\mathfrak m_{A'}$-adic topology."}
+{"_id": "4669", "title": "stacks-geometry-lemma-compare-versal-ring-completion", "text": "In Situation \\ref{situation-versal} let $x : U \\to \\mathcal{X}$ be a morphism where $U$ is a scheme locally of finite type over $S$. Let $u_0 \\in U$ be a finite type point. Set $k = \\kappa(u_0)$ and denote $x_0 : \\Spec(k) \\to \\mathcal{X}$ the induced map. The following are equivalent \\begin{enumerate} \\item $x$ is versal at $u_0$ (Artin's Axioms, Definition \\ref{artin-definition-versal}), \\item $\\hat x : \\mathcal{F}_{U, k, u_0} \\to \\mathcal{F}_{\\mathcal{X}, k, x_0}$ is smooth, \\item the formal object associated to $x|_{\\Spec(\\mathcal{O}_{U, u_0}^\\wedge)}$ is versal, and \\item there is an open neighbourhood $U' \\subset U$ of $x$ such that $x|_{U'} : U' \\to \\mathcal{X}$ is smooth. \\end{enumerate} Moreover, in this case the completion $\\mathcal{O}_{U, u_0}^\\wedge$ is a versal ring to $\\mathcal{X}$ at $x_0$."}
+{"_id": "4670", "title": "stacks-geometry-lemma-characterize-smoothness", "text": "In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$. Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. The following are equivalent \\begin{enumerate} \\item $x_0$ is in the smooth locus of $\\mathcal{X} \\to S$ (Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-where-smooth}), \\item $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the $\\mathfrak m_A$-adic topology, and \\item $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ is unobstructed. \\end{enumerate}"}
+{"_id": "4671", "title": "stacks-geometry-lemma-Artin-approximation-by-smooth-morphism", "text": "In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$. Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. If $\\mathcal{O}_{S, s}$ is a G-ring, then we may find a smooth morphism $U \\to \\mathcal{X}$ whose source is a scheme and a point $u_0 \\in U$ with residue field $k$, such that \\begin{enumerate} \\item $\\Spec(k) \\to U \\to \\mathcal{X}$ coincides with the given morphism $x_0$, \\item there is an isomorphism $\\mathcal{O}_{U, u_0}^\\wedge \\cong A$. \\end{enumerate}"}
+{"_id": "4672", "title": "stacks-geometry-lemma-versal-ring-flat", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. Then the morphism $\\Spec(A) \\to \\mathcal{X}$ of Remark \\ref{remark-upgrade} is flat."}
+{"_id": "4673", "title": "stacks-geometry-lemma-map-of-components", "text": "Let $f : U \\to \\mathcal{X}$ be a smooth morphism from a scheme to a locally Noetherian algebraic stack. The closure of the image of any irreducible component of $|U|$ is an irreducible component of $|\\mathcal{X}|$. If $U \\to \\mathcal{X}$ is surjective, then all irreducible components of $|\\mathcal{X}|$ are obtained in this way."}
+{"_id": "4674", "title": "stacks-geometry-lemma-multiplicities", "text": "Let $U \\to X$ be a smooth morphism of locally Noetherian schemes. Let $T'$ is an irreducible component of $U$. Let $T$ be the irreducible component of $X$ obtained as the closure of the image of $T'$. Then $m_{T', U} = m_{T, X}$."}
+{"_id": "4676", "title": "stacks-geometry-lemma-branches", "text": "In the situation of Definition \\ref{definition-formal-branches} there is a canonical surjection from the set of formal branches of $\\mathcal{X}$ through $x_0$ to the set of irreducible components of $|\\mathcal{X}|$ containing $x_0$ in $|\\mathcal{X}|$."}
+{"_id": "4678", "title": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms", "text": "If $f: U \\to X$ is a smooth morphism of locally Noetherian algebraic spaces, and if $u \\in |U|$ with image $x \\in |X|$, then $$ \\dim_u (U) = \\dim_x(X) + \\dim_{u} (U_x) $$ where $\\dim_u (U_x)$ is defined via Definition \\ref{definition-relative-dimension}."}
+{"_id": "4680", "title": "stacks-geometry-lemma-base-change-invariance-of-relative-dimension", "text": "Suppose given a Cartesian square of morphisms of locally Noetherian stacks $$ \\xymatrix{ \\mathcal{T}' \\ar[d]\\ar[r] & \\mathcal{T} \\ar[d] \\\\ \\mathcal{X}' \\ar[r] & \\mathcal{X} } $$ in which the vertical morphisms are locally of finite type. If $t' \\in |\\mathcal{T}'|$, with images $t$, $x'$, and $x$ in $|\\mathcal{T}|$, $|\\mathcal{X}'|$, and $|\\mathcal{X}|$ respectively, then $\\dim_{t'}(\\mathcal{T}'_{x'}) = \\dim_{t}(\\mathcal{T}_x).$"}
+{"_id": "4682", "title": "stacks-geometry-lemma-relative-dimension-is-semi-continuous", "text": "Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type morphism of algebraic stacks. \\begin{enumerate} \\item The function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is upper semi-continuous on $|\\mathcal{T}|$. \\item If $f$ is smooth, then the function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is locally constant on $|\\mathcal{T}|$. \\end{enumerate}"}
+{"_id": "4683", "title": "stacks-geometry-lemma-dimension-achieved-by-finite-type-point", "text": "If $X$ is a finite dimensional scheme, then there exists a closed (and hence finite type) point $x \\in X$ such that $\\dim_x X = \\dim X$."}
+{"_id": "4684", "title": "stacks-geometry-lemma-constancy-of-dimension", "text": "If $X$ is an irreducible, Jacobson, catenary, and locally Noetherian scheme of finite dimension, then $\\dim U = \\dim X$ for every non-empty open subset $U$ of $X$. Equivalently, $\\dim_x X$ is a constant function on $X$."}
+{"_id": "4686", "title": "stacks-geometry-lemma-irreducible-implies-equidimensional", "text": "If $\\mathcal{X}$ is a Jacobson, pseudo-catenary, and locally Noetherian  algebraic stack for which $|\\mathcal{X}|$ is irreducible, then $\\dim_x(\\mathcal{X})$ is a constant function on $|\\mathcal{X}|$."}
+{"_id": "4687", "title": "stacks-geometry-lemma-closed-immersions", "text": "If $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ is a closed immersion of locally Noetherian schemes, and if $z \\in |\\mathcal{Z}|$ has image $x \\in |\\mathcal{X}|$, then $\\dim_z (\\mathcal{Z}) \\leq \\dim_x(\\mathcal{X})$."}
+{"_id": "4692", "title": "stacks-geometry-lemma-dims-of-images-two", "text": "Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks which is quasi-DM, whose source is irreducible and whose target is quasi-separated, and let $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ denote the scheme-theoretic image of $\\mathcal{T}$. Then $\\dim \\mathcal{Z} \\leq \\dim \\mathcal{T}$, and furthermore, exactly one of the following two conditions holds: \\begin{enumerate} \\item for every finite type point $t \\in |T|,$ we have $\\dim_t(\\mathcal{T}_{f(t)}) > 0,$ in which case $\\dim \\mathcal{Z} < \\dim \\mathcal{T}$; or \\item   $\\mathcal{T}$ and $\\mathcal{Z}$ are of the same dimension. \\end{enumerate}"}
+{"_id": "4694", "title": "stacks-geometry-lemma-dimension-local-ring", "text": "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let $x \\in |\\mathcal{X}|$ be a finite type point  Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-finite-type-point}). Let $d \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item there exists a scheme $U$, a smooth morphism $U \\to \\mathcal{X}$, and a finite type point $u \\in U$ mapping to $x$ such that $2\\dim(\\mathcal{O}_{U, \\overline{u}}) - \\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$, and \\item for any scheme $U$, a smooth morphism $U \\to \\mathcal{X}$, and finite type point $u \\in U$ mapping to $x$ we have $2\\dim(\\mathcal{O}_{U, \\overline{u}}) - \\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$. \\end{enumerate} Here $R = U \\times_\\mathcal{X} U$ with projections $s, t : R \\to U$ and diagonal $e : U \\to R$ and $R_u$ is the fibre of $s : R \\to U$ over $u$."}
+{"_id": "4712", "title": "spaces-morphisms-lemma-properties-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism. Then \\begin{enumerate} \\item $\\Delta_{X/Y}$ is representable, \\item $\\Delta_{X/Y}$ is locally of finite type, \\item $\\Delta_{X/Y}$ is a monomorphism, \\item $\\Delta_{X/Y}$ is separated, and \\item $\\Delta_{X/Y}$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "4714", "title": "spaces-morphisms-lemma-base-change-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated} are stable under base change."}
+{"_id": "4715", "title": "spaces-morphisms-lemma-fibre-product-after-map", "text": "\\begin{slogan} The top arrow of a ``magic diagram'' of algebraic spaces has nice immersion-like properties, and under separatedness hypotheses these get stronger. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$ be morphisms of algebraic spaces over $S$. Consider the induced morphism $i : X \\times_Z Y \\to X \\times_T Y$. Then \\begin{enumerate} \\item $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \\item if $Z \\to T$ is locally separated, then $i$ is an immersion, \\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and \\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact. \\end{enumerate}"}
+{"_id": "4716", "title": "spaces-morphisms-lemma-semi-diagonal", "text": "\\begin{slogan} Properties of the graph of a morphism of algebraic spaces as a consequence of separation properties of the target. \\end{slogan} Let $S$ be a scheme. Let $T$ be an algebraic space over $S$. Let $g : X \\to Y$ be a morphism of algebraic spaces over $T$. Consider the graph $i : X \\to X \\times_T Y$ of $g$. Then \\begin{enumerate} \\item $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \\item if $Y \\to T$ is locally separated, then $i$ is an immersion, \\item if $Y \\to T$ is separated, then $i$ is a closed immersion, and \\item if $Y \\to T$ is quasi-separated, then $i$ is quasi-compact. \\end{enumerate}"}
+{"_id": "4718", "title": "spaces-morphisms-lemma-composition-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated} are stable under composition of morphisms."}
+{"_id": "4719", "title": "spaces-morphisms-lemma-separated-over-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item If $Y$ is separated and $f$ is separated, then $X$ is separated. \\item If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated. \\item If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated. \\item If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$. \\item If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$. \\item If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$. \\end{enumerate}"}
+{"_id": "4720", "title": "spaces-morphisms-lemma-compose-after-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. \\begin{enumerate} \\item If $g \\circ f$ is separated then so is $f$. \\item If $g \\circ f$ is locally separated then so is $f$. \\item If $g \\circ f$ is quasi-separated then so is $f$. \\end{enumerate}"}
+{"_id": "4721", "title": "spaces-morphisms-lemma-separated-implies-morphism-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X$ is separated then $X$ is separated over $S$. \\item If $X$ is locally separated then $X$ is locally separated over $S$. \\item If $X$ is quasi-separated then $X$ is quasi-separated over $S$. \\end{enumerate} Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item[(4)] If $X$ is separated over $S$ then $f$ is separated. \\item[(5)] If $X$ is locally separated over $S$ then $f$ is locally separated. \\item[(6)] If $X$ is quasi-separated over $S$ then $f$ is quasi-separated. \\end{enumerate}"}
+{"_id": "4722", "title": "spaces-morphisms-lemma-separated-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{P}$ be any of the separation axioms of Definition \\ref{definition-separated}. The following are equivalent \\begin{enumerate} \\item $f$ is $\\mathcal{P}$, \\item for every scheme $Z$ and morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the algebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see Properties of Spaces, Definition \\ref{spaces-properties-definition-separated}), \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that the base change $V \\times_Y X \\to V$ has $\\mathcal{P}$, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "4724", "title": "spaces-morphisms-lemma-surjective-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is surjective (in the sense of Section \\ref{section-representable}) if and only if $|f| : |X| \\to |Y|$ is surjective."}
+{"_id": "4725", "title": "spaces-morphisms-lemma-surjective-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is surjective, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is surjective, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is surjective, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a surjective morphism, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is surjective, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are surjective \\'etale such that the top horizontal arrow is surjective, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is surjective. \\end{enumerate}"}
+{"_id": "4726", "title": "spaces-morphisms-lemma-composition-surjective", "text": "The composition of surjective morphisms is surjective."}
+{"_id": "4727", "title": "spaces-morphisms-lemma-base-change-surjective", "text": "The base change of a surjective morphism is surjective."}
+{"_id": "4728", "title": "spaces-morphisms-lemma-characterize-representable-universally-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally open (in the sense of Section \\ref{section-representable}), and \\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of topological spaces $|Z \\times_Y X| \\to |Z|$ is open. \\end{enumerate}"}
+{"_id": "4729", "title": "spaces-morphisms-lemma-base-change-universally-open", "text": "The base change of a universally open morphism of algebraic spaces by any morphism of algebraic spaces is universally open."}
+{"_id": "4730", "title": "spaces-morphisms-lemma-composition-open", "text": "The composition of a pair of (universally) open morphisms of algebraic spaces is (universally) open."}
+{"_id": "4731", "title": "spaces-morphisms-lemma-universally-open-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally open, \\item for every scheme $Z$ and every morphism $Z \\to Y$ the projection $|Z \\times_Y X| \\to |Z|$ is open, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the projection $|Z \\times_Y X| \\to |Z|$ is open, and \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a universally open morphism of algebraic spaces, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally open. \\end{enumerate}"}
+{"_id": "4732", "title": "spaces-morphisms-lemma-space-over-field-universally-open", "text": "Let $S$ be a scheme. Let $p : X \\to \\Spec(k)$ be a morphism of algebraic spaces over $S$ where $k$ is a field. Then $p : X \\to \\Spec(k)$ is universally open."}
+{"_id": "4733", "title": "spaces-morphisms-lemma-characterize-representable-universally-submersive", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally submersive (in the sense of Section \\ref{section-representable}), and \\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of topological spaces $|Z \\times_Y X| \\to |Z|$ is submersive. \\end{enumerate}"}
+{"_id": "4734", "title": "spaces-morphisms-lemma-base-change-universally-submersive", "text": "The base change of a universally submersive morphism of algebraic spaces by any morphism of algebraic spaces is universally submersive."}
+{"_id": "4735", "title": "spaces-morphisms-lemma-composition-universally-submersive", "text": "The composition of a pair of (universally) submersive morphisms of algebraic spaces is (universally) submersive."}
+{"_id": "4737", "title": "spaces-morphisms-lemma-quasi-compact-is-quasi-compact", "text": "Let $S$ be a scheme. If $f : X \\to Y$ is a quasi-compact morphism of algebraic spaces over $S$, then the underlying map $|f| : |X| \\to |Y|$ of topological space is quasi-compact."}
+{"_id": "4738", "title": "spaces-morphisms-lemma-base-change-quasi-compact", "text": "The base change of a quasi-compact morphism of algebraic spaces by any morphism of algebraic spaces is quasi-compact."}
+{"_id": "4739", "title": "spaces-morphisms-lemma-composition-quasi-compact", "text": "The composition of a pair of quasi-compact morphisms of algebraic spaces is quasi-compact."}
+{"_id": "4740", "title": "spaces-morphisms-lemma-surjection-from-quasi-compact", "text": "\\begin{slogan} The image of a quasi-compact algebraic space under a surjective morphism is quasi-compact. \\end{slogan} Let $S$ be a scheme. \\begin{enumerate} \\item If $X \\to Y$ is a surjective morphism of algebraic spaces over $S$, and $X$ is quasi-compact then $Y$ is quasi-compact. \\item If $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & Z } $$ is a commutative diagram of morphisms of algebraic spaces over $S$ and $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact. \\end{enumerate}"}
+{"_id": "4741", "title": "spaces-morphisms-lemma-descent-quasi-compact", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $g : Y' \\to Y$ be a universally open and surjective morphism of algebraic spaces such that the base change $f' : X' \\to Y'$ is quasi-compact. Then $f$ is quasi-compact."}
+{"_id": "4742", "title": "spaces-morphisms-lemma-quasi-compact-local", "text": "\\begin{slogan} Quasi-compact morphisms of algebraic spaces are preserved under pullback and local on the target. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is quasi-compact, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism of algebraic spaces $Z \\times_Y X \\to Z$ is quasi-compact, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a quasi-compact morphism of algebraic spaces, and \\item there exists a surjective \\'etale morphism $Y' \\to Y$ of algebraic spaces such that $Y' \\times_Y X \\to Y'$ is a quasi-compact morphism of algebraic spaces, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-compact. \\end{enumerate}"}
+{"_id": "4743", "title": "spaces-morphisms-lemma-quasi-compact-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. If $g \\circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact."}
+{"_id": "4744", "title": "spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence", "text": "Let $f : X \\to Y$ be a morphism of algebraic spaces over a scheme $S$. \\begin{enumerate} \\item If $X$ is quasi-compact and $Y$ is quasi-separated, then $f$ is quasi-compact. \\item If $X$ is quasi-compact and quasi-separated and $Y$ is quasi-separated, then $f$ is quasi-compact and quasi-separated. \\item A fibre product of quasi-compact and quasi-separated algebraic spaces is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "4745", "title": "spaces-morphisms-lemma-characterize-representable-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed (in the sense of Section \\ref{section-representable}), and \\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of topological spaces $|Z \\times_Y X| \\to |Z|$ is closed. \\end{enumerate}"}
+{"_id": "4746", "title": "spaces-morphisms-lemma-base-change-universally-closed", "text": "The base change of a universally closed morphism of algebraic spaces by any morphism of algebraic spaces is universally closed."}
+{"_id": "4747", "title": "spaces-morphisms-lemma-composition-universally-closed", "text": "The composition of a pair of (universally) closed morphisms of algebraic spaces is (universally) closed."}
+{"_id": "4748", "title": "spaces-morphisms-lemma-universally-closed-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item for every scheme $Z$ and every morphism $Z \\to Y$ the projection $|Z \\times_Y X| \\to |Z|$ is closed, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the projection $|Z \\times_Y X| \\to |Z|$ is closed, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a universally closed morphism of algebraic spaces, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally closed. \\end{enumerate}"}
+{"_id": "4749", "title": "spaces-morphisms-lemma-universally-closed-quasi-compact", "text": "Let $S$ be a scheme. A universally closed morphism of algebraic spaces over $S$ is quasi-compact."}
+{"_id": "4750", "title": "spaces-morphisms-lemma-image-universally-closed-separated", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \\to Y$ be a surjective universally closed morphism of algebraic spaces over $B$. \\begin{enumerate} \\item If $X$ is quasi-separated, then $Y$ is quasi-separated. \\item If $X$ is separated, then $Y$ is separated. \\item If $X$ is quasi-separated over $B$, then $Y$ is quasi-separated over $B$. \\item If $X$ is separated over $B$, then $Y$ is separated over $B$. \\end{enumerate}"}
+{"_id": "4751", "title": "spaces-morphisms-lemma-monomorphism", "text": "Let $S$ be a scheme. Let $j : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $j$ is a monomorphism (as in Definition \\ref{definition-monomorphism}), \\item $j$ is a monomorphism in the category of algebraic spaces over $S$, and \\item the diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is an isomorphism. \\end{enumerate}"}
+{"_id": "4752", "title": "spaces-morphisms-lemma-monomorphism-separated", "text": "A monomorphism of algebraic spaces is separated."}
+{"_id": "4753", "title": "spaces-morphisms-lemma-composition-monomorphism", "text": "A composition of monomorphisms is a monomorphism."}
+{"_id": "4754", "title": "spaces-morphisms-lemma-base-change-monomorphism", "text": "The base change of a monomorphism is a monomorphism."}
+{"_id": "4756", "title": "spaces-morphisms-lemma-immersions-monomorphisms", "text": "An immersion of algebraic spaces is a monomorphism. In particular, any immersion is separated."}
+{"_id": "4757", "title": "spaces-morphisms-lemma-monomorphism-toward-field", "text": "Let $S$ be a scheme. Let $k$ be a field and let $Z \\to \\Spec(k)$ be a monomorphism of algebraic spaces over $S$. Then either $Z = \\emptyset$ or $Z = \\Spec(k)$."}
+{"_id": "4758", "title": "spaces-morphisms-lemma-monomorphism-injective-points", "text": "Let $S$ be a scheme. If $X \\to Y$ is a monomorphism of algebraic spaces over $S$, then $|X| \\to |Y|$ is injective."}
+{"_id": "4759", "title": "spaces-morphisms-lemma-compute-pushforward", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $U \\to X$ be a surjective \\'etale morphism from a scheme to $X$. Set $R = U \\times_X U$ and denote $t, s : R \\to U$ the projection morphisms as usual. Denote $a : U \\to Y$ and $b : R \\to Y$ the induced morphisms. For any object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_X)$ there exists an exact sequence $$ 0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\to b_*(\\mathcal{F}|_R) $$ where the second arrow is the difference $t^* - s^*$."}
+{"_id": "4760", "title": "spaces-morphisms-lemma-pushforward", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is quasi-compact and quasi-separated, then $f_*$ transforms quasi-coherent $\\mathcal{O}_X$-modules into quasi-coherent $\\mathcal{O}_Y$-modules."}
+{"_id": "4761", "title": "spaces-morphisms-lemma-closed-immersion-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion), \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion), \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion), \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion), and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion). \\end{enumerate}"}
+{"_id": "4763", "title": "spaces-morphisms-lemma-immersion-when-closed", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Then $|i| : |Z| \\to |X|$ is a homeomorphism onto a locally closed subset, and $i$ is a closed immersion if and only if the image $|i|(|Z|) \\subset |X|$ is a closed subset."}
+{"_id": "4764", "title": "spaces-morphisms-lemma-factor-the-other-way", "text": "Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces over $S$. Assume $Z \\to X$ is quasi-compact. There exists a factorization $Z \\to \\overline{Z} \\to X$ where $Z \\to \\overline{Z}$ is an open immersion and $\\overline{Z} \\to X$ is a closed immersion."}
+{"_id": "4765", "title": "spaces-morphisms-lemma-closed-immersion-ideals", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For every closed immersion $i : Z \\to X$ the sheaf $i_*\\mathcal{O}_Z$ is a quasi-coherent $\\mathcal{O}_X$-module, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective and its kernel is a quasi-coherent sheaf of ideals. The rule $Z \\mapsto \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$ defines an inclusion reversing bijection $$ \\begin{matrix} \\text{closed subspaces}\\\\ Z \\subset X \\end{matrix} \\longrightarrow \\begin{matrix} \\text{quasi-coherent sheaves}\\\\ \\text{of ideals }\\mathcal{I} \\subset \\mathcal{O}_X \\end{matrix} $$ Moreover, given a closed subscheme $Z$ corresponding to the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ a morphism of algebraic spaces $h : Y \\to X$ factors through $Z$ if and only if the map $h^*\\mathcal{I} \\to h^*\\mathcal{O}_X = \\mathcal{O}_Y$ is zero."}
+{"_id": "4766", "title": "spaces-morphisms-lemma-closed-immersion-quasi-compact", "text": "A closed immersion of algebraic spaces is quasi-compact."}
+{"_id": "4767", "title": "spaces-morphisms-lemma-closed-immersion-separated", "text": "A closed immersion of algebraic spaces is separated."}
+{"_id": "4768", "title": "spaces-morphisms-lemma-closed-immersion-push-pull", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. \\begin{enumerate} \\item The functor $$ i_{small, *} : \\Sh(Z_\\etale) \\longrightarrow \\Sh(X_\\etale) $$ is fully faithful and its essential image is those sheaves of sets $\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is isomorphic to $*$, and \\item the functor $$ i_{small, *} : \\textit{Ab}(Z_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale) $$ is fully faithful and its essential image is those abelian sheaves on $X_\\etale$ whose support is contained in $|Z|$. \\end{enumerate} In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$."}
+{"_id": "4769", "title": "spaces-morphisms-lemma-stalk-push-closed", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. Let $\\overline{z}$ be a geometric point of $Z$ with image $\\overline{x}$ in $X$. Then $(i_{small, *}\\mathcal{F})_{\\overline{z}} = \\mathcal{F}_{\\overline{x}}$ for any sheaf $\\mathcal{F}$ on $Z_\\etale$."}
+{"_id": "4770", "title": "spaces-morphisms-lemma-closed-immersion-rings", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$. Let $\\mathcal{B}$ be a sheaf of rings on $Z_\\etale$. Let $\\varphi : \\mathcal{A} \\to i_{small, *}\\mathcal{B}$ be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi $$ f : (\\Sh(Z_\\etale), \\mathcal{B}) \\longrightarrow (\\Sh(X_\\etale), \\mathcal{A}). $$ For a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a sheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map $$ \\mathcal{F} \\otimes_\\mathcal{A} f_*\\mathcal{G} \\longrightarrow f_*(f^*\\mathcal{F} \\otimes_\\mathcal{B} \\mathcal{G}). $$ is an isomorphism."}
+{"_id": "4771", "title": "spaces-morphisms-lemma-i-star-equivalence", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. \\begin{enumerate} \\item For any $\\mathcal{O}_X$-module $\\mathcal{F}$ the adjunction map $\\mathcal{F} \\to i_*i^*\\mathcal{F}$ induces an isomorphism $\\mathcal{F}/\\mathcal{I}\\mathcal{F} \\cong i_*i^*\\mathcal{F}$. \\item The functor $i^*$ is a left inverse to $i_*$, i.e., for any $\\mathcal{O}_Z$-module $\\mathcal{G}$ the adjunction map $i^*i_*\\mathcal{G} \\to \\mathcal{G}$ is an isomorphism. \\item The functor $$ i_* : \\QCoh(\\mathcal{O}_Z) \\longrightarrow \\QCoh(\\mathcal{O}_X) $$ is exact, fully faithful, with essential image those quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$ such that $\\mathcal{I}\\mathcal{F} = 0$. \\end{enumerate}"}
+{"_id": "4772", "title": "spaces-morphisms-lemma-largest-quasi-coherent-subsheaf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}$ be a $\\mathcal{O}_X$-submodule. There exists a unique quasi-coherent $\\mathcal{O}_X$-submodule $\\mathcal{G}' \\subset \\mathcal{G}$ with the following property: For every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{H}$ the map $$ \\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}') \\longrightarrow \\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}) $$ is bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent $\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$."}
+{"_id": "4775", "title": "spaces-morphisms-lemma-scheme-theoretic-union", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Y, Z \\subset X$ be closed subspaces. Let $Y \\cup Z$ be the scheme theoretic union of $Y$ and $Z$. Let $Y \\cap Z$ be the scheme theoretic intersection of $Y$ and $Z$. Then $Y \\to Y \\cup Z$ and $Z \\to Y \\cup Z$ are closed immersions, there is a short exact sequence $$ 0 \\to \\mathcal{O}_{Y \\cup Z} \\to \\mathcal{O}_Y \\times \\mathcal{O}_Z \\to \\mathcal{O}_{Y \\cap Z} \\to 0 $$ of $\\mathcal{O}_Z$-modules, and the diagram $$ \\xymatrix{ Y \\cap Z \\ar[r] \\ar[d] & Y \\ar[d] \\\\ Z \\ar[r] & Y \\cup Z } $$ is cocartesian in the category of algebraic spaces over $S$, i.e., $Y \\cup Z = Y \\amalg_{Y \\cap Z} Z$."}
+{"_id": "4776", "title": "spaces-morphisms-lemma-support-covering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism. Then $$ \\text{Supp}(\\varphi^*\\mathcal{F}) = |\\varphi|^{-1}(\\text{Supp}(\\mathcal{F})) $$ where the left hand side is the support of $\\varphi^*\\mathcal{F}$ as a quasi-coherent module on the scheme $U$."}
+{"_id": "4777", "title": "spaces-morphisms-lemma-support-finite-type", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item The support of $\\mathcal{F}$ is closed. \\item For a geometric point $\\overline{x}$ lying over $x \\in |X|$ we have $$ x \\in \\text{Supp}(\\mathcal{F}) \\Leftrightarrow \\mathcal{F}_{\\overline{x}} \\not = 0 \\Leftrightarrow \\mathcal{F}_{\\overline{x}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}} \\kappa(\\overline{x}) \\not = 0. $$ \\item For any morphism of algebraic spaces $f : Y \\to X$ the pullback $f^*\\mathcal{F}$ is of finite type as well and we have $\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$. \\end{enumerate}"}
+{"_id": "4778", "title": "spaces-morphisms-lemma-scheme-theoretic-support", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. There exists a smallest closed subspace $i : Z \\to X$ such that there exists a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ with $i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover: \\begin{enumerate} \\item If $U$ is a scheme and $\\varphi : U \\to X$ is an \\'etale morphism then $Z \\times_X U$ is the scheme theoretic support of $\\varphi^*\\mathcal{F}$. \\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique isomorphism. \\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type. \\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $|Z|$. \\end{enumerate}"}
+{"_id": "4779", "title": "spaces-morphisms-lemma-scheme-theoretic-image", "text": "\\begin{slogan} The scheme-theoretic image of a morphism of algebraic spaces exists. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. There exists a closed subspace $Z \\subset Y$ such that $f$ factors through $Z$ and such that for any other closed subspace $Z' \\subset Y$ such that $f$ factors through $Z'$ we have $Z \\subset Z'$."}
+{"_id": "4780", "title": "spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $Z \\subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then \\begin{enumerate} \\item the sheaf of ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$ is quasi-coherent, \\item the scheme theoretic image $Z$ is the closed subspace corresponding to $\\mathcal{I}$, \\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of $X \\times_Y V \\to V$ is equal to $Z \\times_Y V$, and \\item the image $|f|(|X|) \\subset |Z|$ is a dense subset of $|Z|$. \\end{enumerate}"}
+{"_id": "4781", "title": "spaces-morphisms-lemma-scheme-theoretic-image-reduced", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then \\begin{enumerate} \\item the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\\overline{|f|(|X|)}$, and \\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of $X \\times_Y V \\to V$ is equal to $Z \\times_Y V$. \\end{enumerate}"}
+{"_id": "4782", "title": "spaces-morphisms-lemma-reach-points-scheme-theoretic-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$. Let $Z$ be the scheme theoretic image of $f$. Let $z \\in |Z|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\ \\Spec(A) \\ar[r] & Z \\ar[r] & Y } $$ such that the closed point of $\\Spec(A)$ maps to $z$."}
+{"_id": "4783", "title": "spaces-morphisms-lemma-factor-factor", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\ X_2 \\ar[r]^{f_2} & Y_2 } $$ be a commutative diagram of algebraic spaces over $S$. Let $Z_i \\subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a commutative diagram $$ \\xymatrix{ X_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\ X_2 \\ar[r] & Z_2 \\ar[r] & Y_2 } $$"}
+{"_id": "4785", "title": "spaces-morphisms-lemma-scheme-theoretically-dense-representable", "text": "Let $S$ be a scheme. Let $W \\subset S$ be a scheme theoretically dense open subscheme (Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}). Let $f : X \\to S$ be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Then $f^{-1}(W)$ is scheme theoretically dense in $X$."}
+{"_id": "4786", "title": "spaces-morphisms-lemma-scheme-theoretically-dense", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \\subset X$ be an open subspace. The following are equivalent \\begin{enumerate} \\item for every \\'etale morphism $\\varphi : V \\to X$ (of algebraic spaces) the scheme theoretic closure of $\\varphi^{-1}(U)$ in $V$ is equal to $V$, \\item there exists a scheme $V$ and a surjective \\'etale morphism $\\varphi : V \\to X$ such that the scheme theoretic closure of $\\varphi^{-1}(U)$ in $V$ is equal to $V$, \\end{enumerate}"}
+{"_id": "4788", "title": "spaces-morphisms-lemma-characterize-scheme-theoretically-dense", "text": "Let $S$ be a scheme. Let $j : U \\to X$ be an open immersion of algebraic spaces over $S$. Then $U$ is scheme theoretically dense in $X$ if and only if $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective."}
+{"_id": "4790", "title": "spaces-morphisms-lemma-quasi-compact-immersion", "text": "Let $S$ be a scheme. Let $h : Z \\to X$ be an immersion of algebraic spaces over $S$. Assume either $Z \\to X$ is quasi-compact or $Z$ is reduced. Let $\\overline{Z} \\subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \\to \\overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subspace of $\\overline{Z}$. Moreover, $Z$ is topologically dense in $\\overline{Z}$."}
+{"_id": "4791", "title": "spaces-morphisms-lemma-equality-of-morphisms", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f, g : X \\to Y$ be morphisms of algebraic spaces over $B$. Let $U \\subset X$ be an open subspace such that $f|_U = g|_U$. If the scheme theoretic closure of $U$ in $X$ is $X$ and $Y \\to B$ is separated, then $f = g$."}
+{"_id": "4793", "title": "spaces-morphisms-lemma-universally-injective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item the map $X(K) \\to Y(K)$ is injective for every field $K$ over $S$ \\item for every morphism $Y' \\to Y$ of algebraic spaces over $S$ the induced map $|Y' \\times_Y X| \\to |Y'|$ is injective, and \\item the diagonal morphism $X \\to X \\times_Y X$ is surjective. \\end{enumerate}"}
+{"_id": "4794", "title": "spaces-morphisms-lemma-base-change-universally-injective", "text": "The base change of a universally injective morphism is universally injective."}
+{"_id": "4795", "title": "spaces-morphisms-lemma-universally-injective-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is universally injective, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is universally injective, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is universally injective, \\item there exists a scheme $Z$ and a surjective morphism $Z \\to Y$ such that $Z \\times_Y X \\to Z$ is universally injective, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally injective. \\end{enumerate}"}
+{"_id": "4796", "title": "spaces-morphisms-lemma-composition-universally-injective", "text": "A composition of universally injective morphisms is universally injective."}
+{"_id": "4798", "title": "spaces-morphisms-lemma-affine-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is representable and affine, \\item $f$ is affine, \\item for every affine scheme $V$ and \\'etale morphism $V \\to Y$ the scheme $X \\times_Y V$ is affine, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is affine, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is affine. \\end{enumerate}"}
+{"_id": "4799", "title": "spaces-morphisms-lemma-composition-affine", "text": "The composition of affine morphisms is affine."}
+{"_id": "4800", "title": "spaces-morphisms-lemma-base-change-affine", "text": "The base change of an affine morphism is affine."}
+{"_id": "4801", "title": "spaces-morphisms-lemma-closed-immersion-affine", "text": "A closed immersion is affine."}
+{"_id": "4802", "title": "spaces-morphisms-lemma-affine-equivalence-algebras", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There is an anti-equivalence of categories $$ \\begin{matrix} \\text{algebraic spaces} \\\\ \\text{affine over }X \\end{matrix} \\longleftrightarrow \\begin{matrix} \\text{quasi-coherent sheaves} \\\\ \\text{of }\\mathcal{O}_X\\text{-algebras} \\end{matrix} $$ which associates to $f : Y \\to X$ the sheaf $f_*\\mathcal{O}_Y$. Moreover, this equivalence is compatible with arbitrary base change."}
+{"_id": "4803", "title": "spaces-morphisms-lemma-affine-equivalence-modules", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be an affine morphism of algebraic spaces over $S$. Let $\\mathcal{A} = f_*\\mathcal{O}_Y$. The functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{category of quasi-coherent}\\\\ \\mathcal{O}_Y\\text{-modules} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{category of quasi-coherent}\\\\ \\mathcal{A}\\text{-modules} \\end{matrix} \\right\\} $$ Moreover, an $\\mathcal{A}$-module is quasi-coherent as an $\\mathcal{O}_X$-module if and only if it is quasi-coherent as an $\\mathcal{A}$-module."}
+{"_id": "4804", "title": "spaces-morphisms-lemma-affine-permanence", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Suppose $g : X \\to Y$ is a morphism of algebraic spaces over $B$. \\begin{enumerate} \\item If $X$ is affine over $B$ and $\\Delta : Y \\to Y \\times_B Y$ is affine, then $g$ is affine. \\item If $X$ is affine over $B$ and $Y$ is separated over $B$, then $g$ is affine. \\item A morphism from an affine scheme to an algebraic space with affine diagonal is affine. \\item A morphism from an affine scheme to a separated algebraic space is affine. \\end{enumerate}"}
+{"_id": "4805", "title": "spaces-morphisms-lemma-Artinian-affine", "text": "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $A$ be an Artinian ring. Any morphism $\\Spec(A) \\to X$ is affine."}
+{"_id": "4807", "title": "spaces-morphisms-lemma-quasi-affine-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is representable and quasi-affine, \\item $f$ is quasi-affine, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is quasi-affine, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-affine. \\end{enumerate}"}
+{"_id": "4808", "title": "spaces-morphisms-lemma-composition-quasi-affine", "text": "The composition of quasi-affine morphisms is quasi-affine."}
+{"_id": "4809", "title": "spaces-morphisms-lemma-base-change-quasi-affine", "text": "The base change of a quasi-affine morphism is quasi-affine."}
+{"_id": "4810", "title": "spaces-morphisms-lemma-characterize-quasi-affine", "text": "Let $S$ be a scheme. A quasi-compact and quasi-separated morphism of algebraic spaces $f : Y \\to X$ is quasi-affine if and only if the canonical factorization $Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ (Remark \\ref{remark-factorization-quasi-compact-quasi-separated}) is an open immersion."}
+{"_id": "4811", "title": "spaces-morphisms-lemma-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Consider commutative diagrams $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are \\'etale. The following are equivalent \\begin{enumerate} \\item for any diagram as above the morphism $h$ has property $\\mathcal{P}$, and \\item for some diagram as above with $a : U \\to X$ surjective the morphism $h$ has property $\\mathcal{P}$. \\end{enumerate} If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for representable morphisms $f$ this is also equivalent to $f$ having property $\\mathcal{P}$ in the sense of Section \\ref{section-representable}."}
+{"_id": "4812", "title": "spaces-morphisms-lemma-local-source-target-at-point", "text": "Let $\\mathcal{Q}$ be a property of morphisms of germs which is \\'etale local on the source-and-target. Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$ be a point of $X$. Consider the diagrams $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } \\quad\\quad \\xymatrix{ u \\ar[d] \\ar[r] & v \\ar[d] \\\\ x \\ar[r] & y } $$ where $U$ and $V$ are schemes, $a, b$ are \\'etale, and $u, v, x, y$ are points of the corresponding spaces. The following are equivalent \\begin{enumerate} \\item for any diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$, and \\item for some diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$. \\end{enumerate} If $X$ and $Y$ are representable, then this is also equivalent to $\\mathcal{Q}((X, x) \\to (Y, y))$."}
+{"_id": "4813", "title": "spaces-morphisms-lemma-local-source-target-global-implies-local", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Consider the property $\\mathcal{Q}$ of morphisms of germs associated to $\\mathcal{P}$ in Descent, Lemma \\ref{descent-lemma-local-source-target-global-implies-local}. Then \\begin{enumerate} \\item $\\mathcal{Q}$ is \\'etale local on the source-and-target. \\item given a morphism of algebraic spaces $f : X \\to Y$ and $x \\in |X|$ the following are equivalent \\begin{enumerate} \\item $f$ has $\\mathcal{Q}$ at $x$, and \\item there is an open neighbourhood $X' \\subset X$ of $x$ such that $X' \\to Y$ has $\\mathcal{P}$. \\end{enumerate} \\item given a morphism of algebraic spaces $f : X \\to Y$ the following are equivalent: \\begin{enumerate} \\item $f$ has $\\mathcal{P}$, \\item for every $x \\in |X|$ the morphism $f$ has $\\mathcal{Q}$ at $x$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4814", "title": "spaces-morphisms-lemma-composition-finite-type", "text": "The composition of finite type morphisms is of finite type. The same holds for locally of finite type."}
+{"_id": "4815", "title": "spaces-morphisms-lemma-base-change-finite-type", "text": "A base change of a finite type morphism is finite type. The same holds for locally of finite type."}
+{"_id": "4816", "title": "spaces-morphisms-lemma-finite-type-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is locally of finite type, \\item for every $x \\in |X|$ the morphism $f$ is of finite type at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite type, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite type, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is locally of finite type, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is locally of finite type, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is locally of finite type, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, $U \\to X$ is surjective, and the top horizontal arrow is locally of finite type, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is locally of finite type. \\end{enumerate}"}
+{"_id": "4817", "title": "spaces-morphisms-lemma-locally-finite-type-locally-noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and $Y$ is locally Noetherian, then $X$ is locally Noetherian."}
+{"_id": "4818", "title": "spaces-morphisms-lemma-permanence-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is locally of finite type, then $f : X \\to Y$ is locally of finite type."}
+{"_id": "4819", "title": "spaces-morphisms-lemma-immersion-locally-finite-type", "text": "An immersion is locally of finite type."}
+{"_id": "4820", "title": "spaces-morphisms-lemma-locally-finite-type-surjective-geometric-points", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. The following are equivalent: \\begin{enumerate} \\item $f$ is surjective, and \\item for every algebraically closed field $k$ over $S$ the induced map $X(k) \\to Y(k)$ is surjective. \\end{enumerate}"}
+{"_id": "4821", "title": "spaces-morphisms-lemma-large-enough", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. \\begin{enumerate} \\item As $k$ ranges over all algebraically closed fields over $S$ the collection of geometric points $\\overline{y} \\in Y(k)$ cover all of $|Y|$. \\item As $k$ ranges over all algebraically closed fields over $S$ with $|k| \\geq \\lambda(Y)$ and $|k| > \\lambda(X)$ the geometric points $\\overline{y} \\in Y(k)$ cover all of $|Y|$. \\item For any geometric point $\\overline{s} : \\Spec(k) \\to S$ where $k$ has cardinality $> \\lambda(X)$ the map $$ X(k) \\longrightarrow |X_s| $$ is surjective. \\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$. For any geometric point $\\overline{s} : \\Spec(k) \\to S$ where $k$ has cardinality $> \\lambda(X)$ the map $$ X(k) \\longrightarrow |X| \\times_{|Y|} Y(k) $$ is surjective. \\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item the map $X \\to Y$ is surjective, \\item for all algebraically closed fields $k$ over $S$ with $|k| > \\lambda(X)$, and $|k| \\geq \\lambda(Y)$ the map $X(k) \\to Y(k)$ is surjective. \\end{enumerate} \\end{enumerate}"}
+{"_id": "4822", "title": "spaces-morphisms-lemma-point-finite-type", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The following are equivalent: \\begin{enumerate} \\item There exists a morphism $\\Spec(k) \\to X$ which is locally of finite type and represents $x$. \\item There exists a scheme $U$, a closed point $u \\in U$, and an \\'etale morphism $\\varphi : U \\to X$ such that $\\varphi(u) = x$. \\end{enumerate}"}
+{"_id": "4823", "title": "spaces-morphisms-lemma-identify-finite-type-points", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have $$ X_{\\text{ft-pts}} = \\bigcup\\nolimits_{\\varphi : U \\to X\\text{ \\'etale }} |\\varphi|(U_0) $$ where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes \\'etale over $X$ or over all affine schemes \\'etale over $X$."}
+{"_id": "4824", "title": "spaces-morphisms-lemma-finite-type-points-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then $f(X_{\\text{ft-pts}}) \\subset Y_{\\text{ft-pts}}$."}
+{"_id": "4825", "title": "spaces-morphisms-lemma-finite-type-points-surjective-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and surjective, then $f(X_{\\text{ft-pts}}) = Y_{\\text{ft-pts}}$."}
+{"_id": "4826", "title": "spaces-morphisms-lemma-enough-finite-type-points", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For any locally closed subset $T \\subset |X|$ we have $$ T \\not = \\emptyset \\Rightarrow T \\cap X_{\\text{ft-pts}} \\not = \\emptyset. $$ In particular, for any closed subset $T \\subset |X|$ we see that $T \\cap X_{\\text{ft-pts}}$ is dense in $T$."}
+{"_id": "4828", "title": "spaces-morphisms-lemma-finite-type-nagata", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $Y$ is Nagata and $f$ locally of finite type then $X$ is Nagata."}
+{"_id": "4830", "title": "spaces-morphisms-lemma-base-change-quasi-finite-locus", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$ be morphisms of algebraic spaces over $S$. Denote $f' : X' \\to Y'$ the base change of $f$ by $g$. Denote $g' : X' \\to X$ the projection. Assume $f$ is locally of finite type. Let $W \\subset |X|$, resp.\\ $W' \\subset |X'|$ be the set of points where $f$, resp.\\ $f'$ is quasi-finite. \\begin{enumerate} \\item $W \\subset |X|$ and $W' \\subset |X'|$ are open, \\item $W' = (g')^{-1}(W)$, i.e., formation of the locus where $f$ is quasi-finite commutes with base change, \\item the base change of a locally quasi-finite morphism is locally quasi-finite, and \\item the base change of a quasi-finite morphism is quasi-finite. \\end{enumerate}"}
+{"_id": "4831", "title": "spaces-morphisms-lemma-composition-quasi-finite", "text": "The composition of quasi-finite morphisms is quasi-finite. The same holds for locally quasi-finite."}
+{"_id": "4832", "title": "spaces-morphisms-lemma-base-change-quasi-finite", "text": "A base change of a quasi-finite morphism is quasi-finite. The same holds for locally quasi-finite."}
+{"_id": "4833", "title": "spaces-morphisms-lemma-locally-quasi-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces. Assume $f$ is locally of finite type. The following are equivalent \\begin{enumerate} \\item $f$ is locally quasi-finite, \\item for every morphism $\\Spec(k) \\to Y$ where $k$ is a field the space $|X_k|$ is discrete. Here $X_k = \\Spec(k) \\times_Y X$. \\end{enumerate}"}
+{"_id": "4834", "title": "spaces-morphisms-lemma-quasi-finite-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is locally quasi-finite, \\item for every $x \\in |X|$ the morphism $f$ is quasi-finite at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally quasi-finite, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally quasi-finite, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is locally quasi-finite, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is locally quasi-finite, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is locally quasi-finite, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is locally quasi-finite, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "4835", "title": "spaces-morphisms-lemma-immersion-quasi-finite", "text": "An immersion is locally quasi-finite."}
+{"_id": "4836", "title": "spaces-morphisms-lemma-permanence-quasi-finite", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$. If $X \\to Z$ is locally quasi-finite, then $X \\to Y$ is locally quasi-finite."}
+{"_id": "4837", "title": "spaces-morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \\in |Y|$. There are at most finitely many points of $|X|$ lying over $y$ at which $f$ is quasi-finite."}
+{"_id": "4838", "title": "spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite."}
+{"_id": "4839", "title": "spaces-morphisms-lemma-composition-finite-presentation", "text": "The composition of morphisms of finite presentation is of finite presentation. The same holds for locally of finite presentation."}
+{"_id": "4840", "title": "spaces-morphisms-lemma-base-change-finite-presentation", "text": "A base change of a morphism of finite presentation is of finite presentation The same holds for locally of finite presentation."}
+{"_id": "4841", "title": "spaces-morphisms-lemma-finite-presentation-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item for every $x \\in |X|$ the morphism $f$ is of finite presentation at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite presentation, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite presentation, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is locally of finite presentation, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is locally of finite presentation, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is locally of finite presentation, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is locally of finite presentation, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is locally of finite presentation. \\end{enumerate}"}
+{"_id": "4842", "title": "spaces-morphisms-lemma-finite-presentation-finite-type", "text": "A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type."}
+{"_id": "4843", "title": "spaces-morphisms-lemma-finite-presentation-noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and $Y$ is Noetherian, then $X$ is Noetherian."}
+{"_id": "4844", "title": "spaces-morphisms-lemma-noetherian-finite-type-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item If $Y$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation. \\item If $Y$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation. \\end{enumerate}"}
+{"_id": "4845", "title": "spaces-morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "text": "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $Y$, then $X$ is quasi-compact and quasi-separated."}
+{"_id": "4846", "title": "spaces-morphisms-lemma-finite-presentation-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$. If $X$ is locally of finite presentation over $Z$, and $Y$ is locally of finite type over $Z$, then $f$ is locally of finite presentation."}
+{"_id": "4847", "title": "spaces-morphisms-lemma-diagonal-morphism-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ with diagonal $\\Delta : X \\to X \\times_Y X$. If $f$ is locally of finite type then $\\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\\Delta$ is of finite presentation."}
+{"_id": "4848", "title": "spaces-morphisms-lemma-open-immersion-locally-finite-presentation", "text": "An open immersion of algebraic spaces is locally of finite presentation."}
+{"_id": "4849", "title": "spaces-morphisms-lemma-closed-immersion-finite-presentation", "text": "A closed immersion $i : Z \\to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$ is of finite type (as an $\\mathcal{O}_X$-module)."}
+{"_id": "4852", "title": "spaces-morphisms-lemma-composition-flat", "text": "The composition of flat morphisms is flat."}
+{"_id": "4853", "title": "spaces-morphisms-lemma-base-change-flat", "text": "The base change of a flat morphism is flat."}
+{"_id": "4854", "title": "spaces-morphisms-lemma-flat-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is flat, \\item for every $x \\in |X|$ the morphism $f$ is flat at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is flat, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is flat, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is flat, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is flat, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is flat, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is flat, and \\item there exists a Zariski coverings $Y = \\bigcup Y_i$ and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is flat. \\end{enumerate}"}
+{"_id": "4855", "title": "spaces-morphisms-lemma-fppf-open", "text": "A flat morphism locally of finite presentation is universally open."}
+{"_id": "4857", "title": "spaces-morphisms-lemma-flat-at-point-etale-local-rings", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\overline{x}$ be a geometric point of $X$ lying over the point $x \\in |X|$. Let $\\overline{y} = f \\circ \\overline{x}$. The following are equivalent \\begin{enumerate} \\item $f$ is flat at $x$, and \\item the map on \\'etale local rings $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$ is flat. \\end{enumerate}"}
+{"_id": "4858", "title": "spaces-morphisms-lemma-flat-morphism-sites", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if the morphism of sites $ (f_{small}, f^\\sharp) : (X_\\etale, \\mathcal{O}_X) \\to (Y_\\etale, \\mathcal{O}_Y) $ associated to $f$ is flat."}
+{"_id": "4859", "title": "spaces-morphisms-lemma-flat-pullback-support", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module with scheme theoretic support $Z \\subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\\mathcal{F}$."}
+{"_id": "4860", "title": "spaces-morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$. Let $V \\to Y$ be a quasi-compact open immersion. If $V$ is scheme theoretically dense in $Y$, then $f^{-1}V$ is scheme theoretically dense in $X$."}
+{"_id": "4861", "title": "spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$. Let $g : V \\to Y$ be a quasi-compact morphism of algebraic spaces. Let $Z \\subset Y$ be the scheme theoretic image of $g$ and let $Z' \\subset X$ be the scheme theoretic image of the base change $V \\times_Y X \\to X$. Then $Z' = f^{-1}Z$."}
+{"_id": "4862", "title": "spaces-morphisms-lemma-flat-at-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item for some commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ where $U$ and $V$ are schemes, $a, b$ are \\'etale, and $u \\in U$ mapping to $x$ the module $a^*\\mathcal{F}$ is flat at $u$ over $V$, \\item the stalk $\\mathcal{F}_{\\overline{x}}$ is flat over the \\'etale local ring $\\mathcal{O}_{Y, \\overline{y}}$ where $\\overline{x}$ is any geometric point lying over $x$ and $\\overline{y} = f \\circ \\overline{x}$. \\end{enumerate}"}
+{"_id": "4863", "title": "spaces-morphisms-lemma-base-change-module-flat", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $x' \\in |X'|$ with image $x \\in |X|$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$ and denote $\\mathcal{F}' = (g')^*\\mathcal{F}$. \\begin{enumerate} \\item If $\\mathcal{F}$ is flat at $x$ over $Y$ then $\\mathcal{F}'$ is flat at $x'$ over $Y'$. \\item If $g$ is flat at $f'(x')$ and $\\mathcal{F}'$ is flat at $x'$ over $Y'$, then $\\mathcal{F}$ is flat at $x$ over $Y$. \\end{enumerate} In particular, if $\\mathcal{F}$ is flat over $Y$, then $\\mathcal{F}'$ is flat over $Y'$."}
+{"_id": "4864", "title": "spaces-morphisms-lemma-composition-module-flat", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in |X|$ with image $y \\in |Y|$. \\begin{enumerate} \\item If $\\mathcal{F}$ is flat at $x$ over $Y$ and $Y$ is flat at $y$ over $Z$, then $\\mathcal{F}$ is flat at $x$ over $Z$. \\item Let $x : \\Spec(K) \\to X$ be a representative of $x$. If \\begin{enumerate} \\item $\\mathcal{F}$ is flat at $x$ over $Y$, \\item $x^*\\mathcal{F} \\not = 0$, and \\item $\\mathcal{F}$ is flat at $x$ over $Z$, \\end{enumerate} then $Y$ is flat at $y$ over $Z$. \\item Let $\\overline{x}$ be a geometric point of $X$ lying over $x$ with image $\\overline{y}$ in $Y$. If $\\mathcal{F}_{\\overline{x}}$ is a faithfully flat $\\mathcal{O}_{Y, \\overline{y}}$-module and $\\mathcal{F}$ is flat at $x$ over $Z$, then $Y$ is flat at $y$ over $Z$. \\end{enumerate}"}
+{"_id": "4865", "title": "spaces-morphisms-lemma-flat-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \\in |X|$ with image $y \\in |Y|$. If $f$ is flat at $x$, then $$ \\mathcal{G}\\text{ flat over }Z\\text{ at }y \\Leftrightarrow f^*\\mathcal{G}\\text{ flat over }Z\\text{ at }x. $$ In particular: If $f$ is surjective and flat, then $\\mathcal{G}$ is flat over $Z$, if and only if $f^*\\mathcal{G}$ is flat over $Z$."}
+{"_id": "4867", "title": "spaces-morphisms-lemma-compare-tr-deg", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $x \\in |X|$ and let $y \\in |Y|$, $z \\in |Z|$ be the images. Assume $X \\to Y$ is locally quasi-finite and $Y \\to Z$ locally of finite type. Then the transcendence degree of $x/z$ is equal to the transcendence degree of $y/z$."}
+{"_id": "4868", "title": "spaces-morphisms-lemma-jacobson-finite-type-points", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, $Y$ is Jacobson (Properties of Spaces, Remark \\ref{spaces-properties-remark-list-properties-local-etale-topology}), and $x \\in |X|$ is a finite type point of $X$, then the transcendence degree of $x/f(x)$ is $0$."}
+{"_id": "4870", "title": "spaces-morphisms-lemma-dimension-fibre-at-a-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$. Assume $f$ is locally of finite type. Then we have $$ \\begin{matrix} \\text{relative dimension of }f\\text{ at }x \\\\ = \\\\ \\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\ + \\\\ \\text{transcendence degree of }x/f(x) \\end{matrix} $$ where the notation is as in Definition \\ref{definition-dimension-fibre}."}
+{"_id": "4871", "title": "spaces-morphisms-lemma-dimension-fibre-at-a-point-additive", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $x \\in |X|$ and set $y = f(x)$. Assume $f$ and $g$ locally of finite type. Then \\begin{enumerate} \\item $$ \\begin{matrix} \\text{relative dimension of }g \\circ f\\text{ at }x \\\\ \\leq \\\\ \\text{relative dimension of }f\\text{ at }x \\\\ + \\\\ \\text{relative dimension of }g\\text{ at }y \\end{matrix} $$ \\item equality holds in (1) if for some morphism $\\Spec(k) \\to Z$ from the spectrum of a field in the class of $g(f(x)) = g(y)$ the morphism $X_k \\to Y_k$ is flat at $x$, for example if $f$ is flat at $x$, \\item $$ \\begin{matrix} \\text{transcendence degree of }x/g(f(x)) \\\\ = \\\\ \\text{transcendence degree of }x/f(x) \\\\ + \\\\ \\text{transcendence degree of }f(x)/g(f(x)) \\end{matrix} $$ \\end{enumerate}"}
+{"_id": "4872", "title": "spaces-morphisms-lemma-dimension-fibre-after-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a fibre product diagram of algebraic spaces over $S$. Let $x' \\in |X'|$. Set $x = g'(x')$. Assume $f$ locally of finite type. Then \\begin{enumerate} \\item $$ \\begin{matrix} \\text{relative dimension of }f\\text{ at }x \\\\ = \\\\ \\text{relative dimension of }f'\\text{ at }x' \\end{matrix} $$ \\item we have $$ \\begin{matrix} \\text{dimension of local ring of the fibre of }f'\\text{ at }x' \\\\ - \\\\ \\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\ = \\\\ \\text{transcendence degree of }x/f(x) \\\\ - \\\\ \\text{transcendence degree of }x'/f'(x') \\end{matrix} $$ and the common value is $\\geq 0$, \\item given $x$ and $y' \\in |Y'|$ mapping to the same $y \\in |Y|$ there exists a choice of $x'$ such that the integer in (2) is $0$. \\end{enumerate}"}
+{"_id": "4873", "title": "spaces-morphisms-lemma-openness-bounded-dimension-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $n \\geq 0$. Assume $f$ is locally of finite type. The set $$ W_n = \\{x \\in |X| \\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\} $$ is open in $|X|$."}
+{"_id": "4875", "title": "spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. Then $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$ at each $x \\in |X|$."}
+{"_id": "4876", "title": "spaces-morphisms-lemma-locally-finite-type-quasi-finite-part", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. Then there exists a canonical open subspace $X' \\subset X$ such that $f|_{X'} : X' \\to Y$ is locally quasi-finite, and such that the relative dimension of $f$ at any $x \\in |X|$, $x \\not \\in |X'|$ is $\\geq 1$. Formation of $X'$ commutes with arbitrary base change."}
+{"_id": "4877", "title": "spaces-morphisms-lemma-quasi-finite-at-point", "text": "Let $S$ be a scheme. Consider a cartesian diagram $$ \\xymatrix{ X \\ar[d] & F \\ar[l]^p \\ar[d] \\\\ Y & \\Spec(k) \\ar[l] } $$ where $X \\to Y$ is a morphism of algebraic spaces over $S$ which is locally of finite type and where $k$ is a field over $S$. Let $z \\in |F|$ be such that $\\dim_z(F) = 0$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism $$ X \\longrightarrow Y $$ is locally quasi-finite."}
+{"_id": "4878", "title": "spaces-morphisms-lemma-dimension-formula-general", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian and $f$ locally of finite type. Let $x \\in |X|$ with image $y \\in |Y|$. Then we have \\begin{align*} & \\text{the dimension of the local ring of }X\\text{ at }x \\leq \\\\ & \\text{the dimension of the local ring of }Y\\text{ at }y + E - \\\\ & \\text{ the transcendence degree of }x/y \\end{align*} Here $E$ is the maximum of the transcendence degrees of $\\xi/f(\\xi)$ where $\\xi \\in |X|$ runs over the points specializing to $x$ at which the local ring of $X$ has dimension $0$."}
+{"_id": "4879", "title": "spaces-morphisms-lemma-alteration-dimension-general", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian and $f$ is locally of finite type. Then $$ \\dim(X) \\leq \\dim(Y) + E $$ where $E$ is the supremum of the transcendence degrees of $\\xi/f(\\xi)$ where $\\xi$ runs through the points at which the local ring of $X$ has dimension $0$."}
+{"_id": "4882", "title": "spaces-morphisms-lemma-syntomic-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is syntomic, \\item for every $x \\in |X|$ the morphism $f$ is syntomic at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is syntomic, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is syntomic, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a syntomic morphism, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is syntomic, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is syntomic, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is syntomic, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is syntomic. \\end{enumerate}"}
+{"_id": "4883", "title": "spaces-morphisms-lemma-syntomic-locally-finite-presentation", "text": "A syntomic morphism is locally of finite presentation."}
+{"_id": "4884", "title": "spaces-morphisms-lemma-syntomic-flat", "text": "A syntomic morphism is flat."}
+{"_id": "4886", "title": "spaces-morphisms-lemma-composition-smooth", "text": "The composition of smooth morphisms is smooth."}
+{"_id": "4887", "title": "spaces-morphisms-lemma-base-change-smooth", "text": "The base change of a smooth morphism is smooth."}
+{"_id": "4888", "title": "spaces-morphisms-lemma-smooth-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is smooth, \\item for every $x \\in |X|$ the morphism $f$ is smooth at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is smooth, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is smooth, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is a smooth morphism, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is smooth, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is smooth, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is smooth, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is smooth. \\end{enumerate}"}
+{"_id": "4889", "title": "spaces-morphisms-lemma-smooth-locally-finite-presentation", "text": "A smooth morphism of algebraic spaces is locally of finite presentation."}
+{"_id": "4890", "title": "spaces-morphisms-lemma-smooth-locally-finite-type", "text": "A smooth morphism of algebraic spaces is locally of finite type."}
+{"_id": "4891", "title": "spaces-morphisms-lemma-smooth-flat", "text": "A smooth morphism of algebraic spaces is flat."}
+{"_id": "4892", "title": "spaces-morphisms-lemma-smooth-syntomic", "text": "A smooth morphism of algebraic spaces is syntomic."}
+{"_id": "4893", "title": "spaces-morphisms-lemma-where-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. There is a maximal open subspace $U \\subset X$ such that $f|_U : U \\to Y$ is smooth. Moreover, formation of this open commutes with base change by \\begin{enumerate} \\item morphisms which are flat and locally of finite presentation, \\item flat morphisms provided $f$ is locally of finite presentation. \\end{enumerate}"}
+{"_id": "4894", "title": "spaces-morphisms-lemma-smoothness-dimension-spaces", "text": "Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme $S$, and let $f : X \\to Y$ be a smooth morphism. For every point $x \\in |X|$ with image $y \\in |Y|$, $$ \\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y) $$ where $\\dim_x(X_y)$ is the relative dimension of $f$ at $x$ as in Definition \\ref{definition-dimension-fibre}."}
+{"_id": "4896", "title": "spaces-morphisms-lemma-composition-unramified", "text": "The composition of unramified morphisms is unramified."}
+{"_id": "4897", "title": "spaces-morphisms-lemma-base-change-unramified", "text": "The base change of an unramified morphism is unramified."}
+{"_id": "4898", "title": "spaces-morphisms-lemma-unramified-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is unramified, \\item for every $x \\in |X|$ the morphism $f$ is unramified at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is unramified, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is unramified, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is an unramified morphism, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is unramified, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is unramified, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is surjective such that the top horizontal arrow is unramified, and \\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$, and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is unramified. \\end{enumerate}"}
+{"_id": "4899", "title": "spaces-morphisms-lemma-unramified-locally-finite-type", "text": "An unramified morphism of algebraic spaces is locally of finite type."}
+{"_id": "4900", "title": "spaces-morphisms-lemma-unramified-quasi-finite", "text": "If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$. In particular, an unramified morphism is locally quasi-finite."}
+{"_id": "4901", "title": "spaces-morphisms-lemma-immersion-unramified", "text": "An immersion of algebraic spaces is unramified."}
+{"_id": "4902", "title": "spaces-morphisms-lemma-diagonal-unramified-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item If $f$ is unramified, then the diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is an open immersion. \\item If $f$ is locally of finite type and $\\Delta_{X/Y}$ is an open immersion, then $f$ is unramified. \\end{enumerate}"}
+{"_id": "4903", "title": "spaces-morphisms-lemma-where-unramified", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\ & Z } $$ of algebraic spaces over $S$. Assume that $X \\to Z$ is locally of finite type. Then there exists an open subspace $U(f) \\subset X$ such that $|U(f)| \\subset |X|$ is the set of points where $f$ is unramified. Moreover, for any morphism of algebraic spaces $Z' \\to Z$, if $f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then $U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$."}
+{"_id": "4904", "title": "spaces-morphisms-lemma-permanence-unramified", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$. If $X \\to Z$ is unramified, then $X \\to Y$ is unramified."}
+{"_id": "4905", "title": "spaces-morphisms-lemma-etale-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is \\'etale, \\item for every $x \\in |X|$ the morphism $f$ is \\'etale at $x$, \\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is \\'etale, \\item for every affine scheme $Z$ and any morphism $Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is \\'etale, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is an \\'etale morphism, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$ is \\'etale, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are \\'etale the top horizontal arrow is \\'etale, \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ surjective such that the top horizontal arrow is \\'etale, and \\item there exist Zariski coverings $Y = \\bigcup Y_i$ and $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism $X_{ij} \\to Y_i$ is \\'etale. \\end{enumerate}"}
+{"_id": "4906", "title": "spaces-morphisms-lemma-composition-etale", "text": "The composition of two \\'etale morphisms of algebraic spaces is \\'etale."}
+{"_id": "4907", "title": "spaces-morphisms-lemma-base-change-etale", "text": "The base change of an \\'etale morphism of algebraic spaces by any morphism of algebraic spaces is \\'etale."}
+{"_id": "4908", "title": "spaces-morphisms-lemma-etale-locally-quasi-finite", "text": "An \\'etale morphism of algebraic spaces is locally quasi-finite."}
+{"_id": "4909", "title": "spaces-morphisms-lemma-etale-smooth", "text": "An \\'etale morphism of algebraic spaces is smooth."}
+{"_id": "4910", "title": "spaces-morphisms-lemma-etale-flat", "text": "An \\'etale morphism of algebraic spaces is flat."}
+{"_id": "4911", "title": "spaces-morphisms-lemma-etale-locally-finite-presentation", "text": "\\begin{slogan} \\'Etale implies locally of finite presentation. \\end{slogan} An \\'etale morphism of algebraic spaces is locally of finite presentation."}
+{"_id": "4912", "title": "spaces-morphisms-lemma-etale-locally-finite-type", "text": "An \\'etale morphism of algebraic spaces is locally of finite type."}
+{"_id": "4913", "title": "spaces-morphisms-lemma-etale-unramified", "text": "An \\'etale morphism of algebraic spaces is unramified."}
+{"_id": "4914", "title": "spaces-morphisms-lemma-etale-permanence", "text": "Let $S$ be a scheme. Let $X, Y$ be algebraic spaces \\'etale over an algebraic space $Z$. Any morphism $X \\to Y$ over $Z$ is \\'etale."}
+{"_id": "4915", "title": "spaces-morphisms-lemma-unramified-flat-lfp-etale", "text": "A locally finitely presented, flat, unramified morphism of algebraic spaces is \\'etale."}
+{"_id": "4916", "title": "spaces-morphisms-lemma-proper-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is proper, \\item for every scheme $Z$ and every morphism $Z \\to Y$ the projection $Z \\times_Y X \\to Z$ is proper, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the projection $Z \\times_Y X \\to Z$ is proper, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is proper, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is proper. \\end{enumerate}"}
+{"_id": "4917", "title": "spaces-morphisms-lemma-base-change-proper", "text": "A base change of a proper morphism is proper."}
+{"_id": "4918", "title": "spaces-morphisms-lemma-composition-proper", "text": "A composition of proper morphisms is proper."}
+{"_id": "4919", "title": "spaces-morphisms-lemma-closed-immersion-proper", "text": "A closed immersion of algebraic spaces is a proper morphism of algebraic spaces."}
+{"_id": "4920", "title": "spaces-morphisms-lemma-universally-closed-permanence", "text": "Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces $$ \\xymatrix{ X \\ar[rr] \\ar[rd] & & Y \\ar[ld] \\\\ & B & } $$ over $S$. \\begin{enumerate} \\item If $X \\to B$ is universally closed and $Y \\to B$ is separated, then the morphism $X \\to Y$ is universally closed. In particular, the image of $|X|$ in $|Y|$ is closed. \\item If $X \\to B$ is proper and $Y \\to B$ is separated, then the morphism $X \\to Y$ is proper. \\end{enumerate}"}
+{"_id": "4921", "title": "spaces-morphisms-lemma-image-proper-is-proper", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces over $B$. If $X$ is universally closed over $B$ and $f$ is surjective then $Y$ is universally closed over $B$. In particular, if also $Y$ is separated and of finite type over $B$, then $Y$ is proper over $B$."}
+{"_id": "4922", "title": "spaces-morphisms-lemma-scheme-theoretic-image-is-proper", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\ & B } $$ be a commutative diagram of morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X \\to B$ is a proper morphism, \\item $Y \\to B$ is separated and locally of finite type, \\end{enumerate} Then the scheme theoretic image $Z \\subset Y$ of $h$ is proper over $B$ and $X \\to Z$ is surjective."}
+{"_id": "4923", "title": "spaces-morphisms-lemma-separated-diagonal-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is separated, \\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is universally closed, and \\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is proper. \\end{enumerate}"}
+{"_id": "4926", "title": "spaces-morphisms-lemma-push-down-solution", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of algebraic spaces over $S$. Suppose given a diagram $$ \\xymatrix{ \\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\ \\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ as in Definition \\ref{definition-valuative-criterion} with $K \\subset K'$ arbitrary. Then the dotted arrow exists making the diagram commute."}
+{"_id": "4927", "title": "spaces-morphisms-lemma-usual-enough", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ satisfies the existence part of the valuative criterion as in Definition \\ref{definition-valuative-criterion}, \\item given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition \\ref{schemes-definition-valuative-criterion}. \\end{enumerate}"}
+{"_id": "4928", "title": "spaces-morphisms-lemma-base-change-valuative-criteria", "text": "The base change of a morphism of algebraic spaces which satisfies the existence part of (resp.\\ uniqueness part of) the valuative criterion by any morphism of algebraic spaces satisfies the existence part of (resp.\\ uniqueness part of) the valuative criterion."}
+{"_id": "4929", "title": "spaces-morphisms-lemma-composition-valuative-criteria", "text": "The composition of two morphisms of algebraic spaces which satisfy the (existence part of, resp.\\ uniqueness part of) the valuative criterion satisfies the (existence part of, resp.\\ uniqueness part of) the valuative criterion."}
+{"_id": "4930", "title": "spaces-morphisms-lemma-quasi-compact-existence-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is quasi-compact, and \\item $f$ satisfies the existence part of the valuative criterion. \\end{enumerate} Then $f$ is universally closed."}
+{"_id": "4931", "title": "spaces-morphisms-lemma-characterize-universally-closed-quasi-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item If $f$ is quasi-separated and universally closed, then $f$ satisfies the existence part of the valuative criterion. \\item If $f$ is quasi-compact and quasi-separated, then $f$ is universally closed if and only if the existence part of the valuative criterion holds. \\end{enumerate}"}
+{"_id": "4932", "title": "spaces-morphisms-lemma-characterize-universally-closed-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and separated. Then the following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item the existence part of the valuative criterion holds as in Definition \\ref{definition-valuative-criterion}, and \\item given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition \\ref{schemes-definition-valuative-criterion}. \\end{enumerate}"}
+{"_id": "4933", "title": "spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$. Let $\\Spec(A) \\to Y$ be a morphism where $A$ is a valuation ring. If the closed point of $\\Spec(A)$ maps to a point of $|Y|$ in the image of $|X| \\to |Y|$, then there exists a commutative diagram $$ \\xymatrix{ \\Spec(A') \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] & Y } $$ where $A \\to A'$ is an extension of valuation rings (More on Algebra, Definition \\ref{more-algebra-definition-extension-valuation-rings})."}
+{"_id": "4934", "title": "spaces-morphisms-lemma-refined-valuative-criterion-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of algebraic spaces over $S$. If \\begin{enumerate} \\item $f$ and $h$ are quasi-compact, \\item $|h|(|U|)$ is dense in $|X|$, and \\end{enumerate} given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r] & X \\ar[d] \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y } $$ where $A$ is a valuation ring with field of fractions $K$ \\begin{enumerate} \\item[(3)] there exists at most one dotted arrow making the diagram commute, and \\item[(4)] there exists an extension $K \\subset K'$ of fields, a valuation ring $A' \\subset K'$ dominating $A$ and a morphism $\\Spec(A') \\to X$ such that the following diagram commutes $$ \\xymatrix{ \\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & U \\ar[r] & X \\ar[d] \\\\ \\Spec(A') \\ar[r] \\ar[rrru] & \\Spec(A) \\ar[rr] & & Y } $$ \\end{enumerate} then $f$ is universally closed. If moreover \\begin{enumerate} \\item[(5)] $f$ is quasi-separated \\end{enumerate} then $f$ is separated and universally closed."}
+{"_id": "4935", "title": "spaces-morphisms-lemma-separated-implies-valuative", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion."}
+{"_id": "4936", "title": "spaces-morphisms-lemma-valuative-criterion-separatedness", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item the morphism $f$ is quasi-separated, and \\item the morphism $f$ satisfies the uniqueness part of the valuative criterion. \\end{enumerate} Then $f$ is separated."}
+{"_id": "4937", "title": "spaces-morphisms-lemma-characterize-separated-and-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and quasi-separated. Then the following are equivalent \\begin{enumerate} \\item $f$ is separated and universally closed, \\item the valuative criterion holds as in Definition \\ref{definition-valuative-criterion}, \\item given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition \\ref{schemes-definition-valuative-criterion}. \\end{enumerate}"}
+{"_id": "4938", "title": "spaces-morphisms-lemma-characterize-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \\begin{enumerate} \\item $f$ is proper, \\item the valuative criterion holds as in Definition \\ref{definition-valuative-criterion}, \\item given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition \\ref{schemes-definition-valuative-criterion}. \\end{enumerate}"}
+{"_id": "4940", "title": "spaces-morphisms-lemma-integral-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is representable and integral (resp.\\ finite), \\item $f$ is integral (resp.\\ finite), \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is integral (resp. finite), and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is integral (resp.\\ finite). \\end{enumerate}"}
+{"_id": "4941", "title": "spaces-morphisms-lemma-composition-integral", "text": "The composition of integral (resp.\\ finite) morphisms is integral (resp.\\ finite)."}
+{"_id": "4942", "title": "spaces-morphisms-lemma-base-change-integral", "text": "The base change of an integral (resp.\\ finite) morphism is integral (resp.\\ finite)."}
+{"_id": "4943", "title": "spaces-morphisms-lemma-finite-integral", "text": "A finite morphism of algebraic spaces is integral. An integral morphism of algebraic spaces which is locally of finite type is finite."}
+{"_id": "4944", "title": "spaces-morphisms-lemma-integral-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is integral, and \\item $f$ is affine and universally closed. \\end{enumerate}"}
+{"_id": "4945", "title": "spaces-morphisms-lemma-finite-quasi-finite", "text": "A finite morphism of algebraic spaces is quasi-finite."}
+{"_id": "4946", "title": "spaces-morphisms-lemma-finite-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is finite, and \\item $f$ is affine and proper. \\end{enumerate}"}
+{"_id": "4947", "title": "spaces-morphisms-lemma-closed-immersion-finite", "text": "A closed immersion is finite (and a fortiori integral)."}
+{"_id": "4948", "title": "spaces-morphisms-lemma-finite-union-finite", "text": "Let $S$ be a scheme. Let $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of algebraic spaces over $S$. Then $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too."}
+{"_id": "4949", "title": "spaces-morphisms-lemma-finite-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. \\begin{enumerate} \\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite. \\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral. \\end{enumerate}"}
+{"_id": "4951", "title": "spaces-morphisms-lemma-finite-locally-free-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is representable and finite locally free, \\item $f$ is finite locally free, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that $V \\times_Y X \\to V$ is finite locally free, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each morphism $f^{-1}(Y_i) \\to Y_i$ is finite locally free. \\end{enumerate}"}
+{"_id": "4952", "title": "spaces-morphisms-lemma-composition-finite-locally-free", "text": "The composition of finite locally free morphisms is finite locally free."}
+{"_id": "4953", "title": "spaces-morphisms-lemma-base-change-finite-locally-free", "text": "The base change of a finite locally free morphism is finite locally free."}
+{"_id": "4954", "title": "spaces-morphisms-lemma-finite-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is finite locally free, \\item $f$ is finite, flat, and locally of finite presentation. \\end{enumerate} If $Y$ is locally Noetherian these are also equivalent to \\begin{enumerate} \\item[(3)] $f$ is finite and flat. \\end{enumerate}"}
+{"_id": "4957", "title": "spaces-morphisms-lemma-integral-closure", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. There exists a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras $\\mathcal{A}' \\subset \\mathcal{A}$ such that for any affine object $U$ of $X_\\etale$ the ring $\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is the integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$."}
+{"_id": "4958", "title": "spaces-morphisms-lemma-properties-normalization", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $Y \\to X' \\to X$ be the normalization of $X$ in $Y$. \\begin{enumerate} \\item If $W \\to X$ is an \\'etale morphism of algebraic spaces over $S$, then $W \\times_X X'$ is the normalization of $W$ in $W \\times_X Y$. \\item If $Y$ and $X$ are representable, then $Y'$ is representable and is canonically isomorphic to the normalization of the scheme $X$ in the scheme $Y$ as constructed in Morphisms, Section \\ref{morphisms-section-normalization}. \\end{enumerate}"}
+{"_id": "4959", "title": "spaces-morphisms-lemma-characterize-normalization", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. The factorization $f = \\nu \\circ f'$, where $\\nu : X' \\to X$ is the normalization of $X$ in $Y$ is characterized by the following two properties: \\begin{enumerate} \\item the morphism $\\nu$ is integral, and \\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$ integral, there exists a commutative diagram $$ \\xymatrix{ Y \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\ X' \\ar[ru]^h \\ar[r]^\\nu & X } $$ for a unique morphism $h : X' \\to Z$. \\end{enumerate} Moreover, in (2) the morphism $h : X' \\to Z$ is the normalization of $Z$ in $Y$."}
+{"_id": "4960", "title": "spaces-morphisms-lemma-normalization-in-reduced", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $X' \\to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$."}
+{"_id": "4961", "title": "spaces-morphisms-lemma-normalization-generic", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \\to X$ be the normalization of $X$ in $Y$. If $x' \\in |X'|$ is a point of codimension $0$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension-local-ring}), then $x'$ is the image of some $y \\in |Y|$ of codimension $0$."}
+{"_id": "4962", "title": "spaces-morphisms-lemma-normalization-in-disjoint-union", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Suppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two algebraic spaces. Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$. Then $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$."}
+{"_id": "4963", "title": "spaces-morphisms-lemma-normalization-in-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact, quasi-separated and universally closed morphisms of algebraic spaces over $S$. Then $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_Y$. In other words, the normalization of $Y$ in $X$ is equal to the factorization $$ X \\longrightarrow \\underline{\\Spec}_Y(f_*\\mathcal{O}_X) \\longrightarrow Y $$ of Remark \\ref{remark-factorization-quasi-compact-quasi-separated}."}
+{"_id": "4964", "title": "spaces-morphisms-lemma-normalization-in-integral", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be an integral morphism of algebraic spaces over $S$. Then the integral closure of $X$ in $Y$ is equal to $Y$."}
+{"_id": "4966", "title": "spaces-morphisms-lemma-prepare-normalization", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item there is a surjective \\'etale morphism $U \\to X$ where $U$ is as scheme such that every quasi-compact open of $U$ has finitely many irreducible components, \\item for every scheme $U$ and every \\'etale morphism $U \\to X$ every quasi-compact open of $U$ has finitely many irreducible components, and \\item for every quasi-compact algebraic space $Y$ \\'etale over $X$ the space $|Y|$ has finitely many irreducible components. \\end{enumerate} If $X$ is representable this means that every quasi-compact open of $X$ has finitely many irreducible components."}
+{"_id": "4967", "title": "spaces-morphisms-lemma-normalization", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-prepare-normalization}. Then there exists an integral morphism of algebraic spaces $$ X^\\nu \\longrightarrow X $$ such that for every scheme $U$ and \\'etale morphism $U \\to X$ the fibre product $X^\\nu \\times_X U$ is the normalization of $U$."}
+{"_id": "4969", "title": "spaces-morphisms-lemma-normalization-normal", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-prepare-normalization}. \\begin{enumerate} \\item The normalization $X^\\nu$ is normal. \\item The morphism $\\nu : X^\\nu \\to X$ is integral and surjective. \\item The map $|\\nu| : |X^\\nu| \\to |X|$ induces a bijection between the sets of points of codimension $0$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension-local-ring}). \\item Let $Z \\to X$ be a morphism. Assume $Z$ is a normal algebraic space and that for $z \\in |Z|$ we have: $z$ has codimension $0$ in $Z \\Rightarrow f(z)$ has codimension $0$ in $X$. Then there exists a unique factorization $Z \\to X^\\nu \\to X$. \\end{enumerate}"}
+{"_id": "4970", "title": "spaces-morphisms-lemma-nagata-normalization", "text": "Let $S$ be a scheme. Let $X$ be a Nagata algebraic space over $S$. The normalization $\\nu : X^\\nu \\to X$ is a finite morphism."}
+{"_id": "4971", "title": "spaces-morphisms-lemma-neighbourhood-scheme", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ V' \\ar[r] \\ar[rd] & T' \\times_T X \\ar[r] \\ar[d] & X \\ar[d] \\\\ & T' \\ar[r] & T } $$ of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $T' \\to T$ is an \\'etale morphism of affine schemes, \\item $X \\to T$ is a separated, locally quasi-finite morphism, \\item $V'$ is an open subspace of $T' \\times_T X$, and \\item $V' \\to T'$ is quasi-affine. \\end{enumerate} In this situation the image $U$ of $V'$ in $X$ is a quasi-compact open subspace of $X$ which is representable."}
+{"_id": "4972", "title": "spaces-morphisms-lemma-locally-quasi-finite-separated-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally quasi-finite and separated, then $f$ is representable."}
+{"_id": "4973", "title": "spaces-morphisms-lemma-etale-universally-injective-open", "text": "\\begin{slogan} Universally injective \\'etale maps are open immersions. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale and universally injective morphism of algebraic spaces over $S$. Then $f$ is an open immersion."}
+{"_id": "4974", "title": "spaces-morphisms-lemma-finite-type-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is representable, of finite type, and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\ & Y & } $$ Then there exists an open subspace $U' \\subset Y'$ such that \\begin{enumerate} \\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and \\item $(f')^{-1}(U') \\subset X$ is the set of points at which $f$ is quasi-finite. \\end{enumerate}"}
+{"_id": "4975", "title": "spaces-morphisms-lemma-quasi-finite-separated-quasi-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-finite and separated. Let $Y'$ be the normalization of $Y$ in $X$. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\ & Y & } $$ Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral. In particular $f$ is quasi-affine."}
+{"_id": "4977", "title": "spaces-morphisms-lemma-base-change-universal-homeomorphism", "text": "The base change of a universal homeomorphism of algebraic spaces by any morphism of algebraic spaces is a universal homeomorphism."}
+{"_id": "4978", "title": "spaces-morphisms-lemma-composition-universal-homeomorphism", "text": "The composition of a pair of universal homeomorphisms of algebraic spaces is a universal homeomorphism."}
+{"_id": "4979", "title": "spaces-morphisms-lemma-reduction-universal-homeomorphism", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The canonical closed immersion $X_{red} \\to X$ (see Properties of Spaces, Definition \\ref{spaces-properties-definition-reduced-induced-space}) is a universal homeomorphism."}
+{"_id": "4980", "title": "spaces-morphisms-lemma-integral-universally-injective-push-pull", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a universally injective, integral morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The functor $$ f_{small, *} : \\Sh(Y_\\etale) \\longrightarrow \\Sh(X_\\etale) $$ is fully faithful and its essential image is those sheaves of sets $\\mathcal{F}$ on $X_\\etale$ whose restriction to $|X| \\setminus f(|Y|)$ is isomorphic to $*$, and \\item the functor $$ f_{small, *} : \\textit{Ab}(Y_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale) $$ is fully faithful and its essential image is those abelian sheaves on $Y_\\etale$ whose support is contained in $f(|Y|)$. \\end{enumerate} In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$."}
+{"_id": "4981", "title": "spaces-morphisms-proposition-generic-flatness-reduced", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $Y$ is reduced, \\item $f$ is of finite type, and \\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module. \\end{enumerate} Then there exists an open dense subspace $W \\subset Y$ such that the base change $X_W \\to W$ of $f$ is flat, locally of finite presentation, and quasi-compact and such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of finite presentation over $\\mathcal{O}_{X_W}$."}
+{"_id": "4983", "title": "spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme", "text": "Let $S$ be a scheme. Let $f : X \\to T$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $T$ is representable, \\item $f$ is locally quasi-finite, and \\item $f$ is separated. \\end{enumerate} Then $X$ is representable."}
+{"_id": "5036", "title": "weil-lemma-composition-correspondences", "text": "We have the following for correspondences: \\begin{enumerate} \\item composition of correspondences is $\\mathbf{Q}$-bilinear and associative, \\item there is a canonical isomorphism $$ \\CH_{-r}(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(X, \\Spec(k)) $$ such that pullback by correspondences corresponds to composition, \\item there is a canonical isomorphism $$ \\CH^r(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(\\Spec(k), X) $$ such that pushforward by correspondences corresponds to composition, \\item composition of correspondences is compatible with pushforward and pullback of cycles. \\end{enumerate}"}
+{"_id": "5037", "title": "weil-lemma-category-correspondences", "text": "Smooth projective schemes over $k$ with correspondences and composition of correspondences as defined above form a graded category over $\\mathbf{Q}$ (Differential Graded Algebra, Definition \\ref{dga-definition-graded-category})."}
+{"_id": "5038", "title": "weil-lemma-contravariant-functor", "text": "There is a contravariant functor from the category of smooth projective schemes over $k$ to the category of correspondences which is the identity on objects and sends $f : Y \\to X$ to the element $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$."}
+{"_id": "5039", "title": "weil-lemma-functor-and-cycles", "text": "Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$. Let $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ be as in Example \\ref{example-graph-correspondence}. Then \\begin{enumerate} \\item pushforward of cycles by the correspondence $[\\Gamma_f]$ agrees with the gysin map $f^! : \\CH^*(Y) \\to \\CH^*(X)$, \\item pullback of cycles by the correspondence $[\\Gamma_f]$ agrees with the pushforward map $f_* : \\CH_*(Y) \\to \\CH_*(X)$, \\item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$, then \\begin{enumerate} \\item pushforward of cycles by the correspondence $[\\Gamma_f^t]$ of Remark \\ref{remark-transpose} corresponds to pushforward of cycles by $f$, and \\item pullback of cycles by the correspondence $[\\Gamma_f^t]$ of Remark \\ref{remark-transpose} corresponds to the gysin map $f^!$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "5040", "title": "weil-lemma-tensor-product", "text": "The tensor product of correspondences defined above turns the category of correspondences into a symmetric monoidal category with unit $\\Spec(k)$."}
+{"_id": "5041", "title": "weil-lemma-prep-dual", "text": "Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$. Assume $X$ and $Y$ equidimensional of dimensions $d$ and $e$. Denote $a = [\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ and $a^t = [\\Gamma_f^t] \\in \\text{Corr}^{d - e}(Y, X)$. Set $\\eta_X = [\\Gamma_{X \\to X \\times X}] \\in \\text{Corr}^0(X \\times X, X)$, $\\eta_Y = [\\Gamma_{Y \\to Y \\times Y}] \\in \\text{Corr}^0(Y \\times Y, Y)$, $[X] \\in \\text{Corr}^{-d}(X, \\Spec(k))$, and $[Y] \\in \\text{Corr}^{-e}(Y, \\Spec(k))$. The diagram $$ \\xymatrix{ X \\otimes Y \\ar[r]_{a \\otimes \\text{id}} \\ar[d]_{\\text{id} \\otimes a^t} & Y \\otimes Y \\ar[r]_{\\eta_Y} & Y \\ar[d]^{[Y]} \\\\ X \\otimes X \\ar[r]^{\\eta_X} & X \\ar[r]^{[X]} & \\Spec(k) } $$ is commutative in the category of correspondences."}
+{"_id": "5045", "title": "weil-lemma-inverse-h2", "text": "With notation as in Example \\ref{example-decompose-P1} \\begin{enumerate} \\item the motive $(X, c_0, 0)$ is isomorphic to the motive $\\mathbf{1} = (\\Spec(k), 1, 0)$. \\item the motive $(X, c_2, 0)$ is isomorphic to the motive $\\mathbf{1}(-1) = (\\Spec(k), 1, -1)$. \\end{enumerate}"}
+{"_id": "5046", "title": "weil-lemma-additive", "text": "The category $M_k$ is additive."}
+{"_id": "5048", "title": "weil-lemma-characterize-motives", "text": "Let $X$, $c_2$ be as in Example \\ref{example-decompose-P1}. Let $\\mathcal{C}$ be a $\\mathbf{Q}$-linear Karoubian symmetric monoidal category. Any $\\mathbf{Q}$-linear functor $$ F : \\left\\{ \\begin{matrix} \\text{smooth projective schemes over }k\\\\ \\text{morphisms are correspondences of degree }0 \\end{matrix} \\right\\} \\longrightarrow \\mathcal{C} $$ of symmetric monoidal categories such that the image of $F(c_2)$ on $F(X)$ is an invertible object, factors uniquely through a functor $F : M_k \\to \\mathcal{C}$ of symmetric monoidal categories."}
+{"_id": "5049", "title": "weil-lemma-dual", "text": "Let $X$ be a smooth projective scheme over $k$ which is equidimensional of dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_k$."}
+{"_id": "5050", "title": "weil-lemma-dual-general", "text": "Every object of $M_k$ has a left dual."}
+{"_id": "5051", "title": "weil-lemma-chow-groups-representable", "text": "Let $k$ be a base field. The functor $\\CH^i(-)$ on the category of motives $M_k$ is representable by $\\mathbf{1}(-i)$, i.e., we have $$ \\CH^i(M) = \\Hom_{M_k}(\\mathbf{1}(-i), M) $$ functorially in $M$ in $M_k$."}
+{"_id": "5052", "title": "weil-lemma-manin", "text": "Let $k$ be a base field. Let $c : M \\to N$ be a morphism of motives. If for every smooth projective scheme $X$ over $k$ the map $c \\otimes 1 : M \\otimes h(X) \\to N \\otimes h(X)$ induces an isomorphism on Chow groups, then $c$ is an isomorphism."}
+{"_id": "5053", "title": "weil-lemma-projective-space-bundle-formula", "text": "In the situation above, the map $$ \\sum\\nolimits_{i = 0, \\ldots, r - 1} c_i : \\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} h(X)(i) \\longrightarrow h(P) $$ is an isomorphism in the category of motives."}
+{"_id": "5054", "title": "weil-lemma-diagonal-projective-bundle", "text": "Let $p : P \\to X$ be as in Lemma \\ref{lemma-projective-space-bundle-formula}. The class $[\\Delta_P]$ of the diagonal of $P$ in $\\CH^*(P \\times P)$ can be written as $$ [\\Delta_P] = \\left(\\sum\\nolimits_{i = 0, \\ldots, r - 1} {r - 1 \\choose i} c_{r - 1 - i}(\\text{pr}_1^*\\mathcal{S}^\\vee) \\cap c_1(\\text{pr}_2^*\\mathcal{O}_P(1))^i\\right) \\cap (p \\times p)^*[\\Delta_X] $$ where $\\mathcal{S}$ is the kernel of the canonical surjection $p^*\\mathcal{E} \\to \\mathcal{O}_P(1)$."}
+{"_id": "5055", "title": "weil-lemma-pushforward-classical", "text": "Assume given (D1) and (D3) satisfying (A). For $f : X \\to Y$ a morphism of smooth projective varieties we have $f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism of smooth projective varieties, then $g_* \\circ f_* = (g \\circ f)_*$."}
+{"_id": "5056", "title": "weil-lemma-degrees-cycles-classical", "text": "Let $H^*$ be a classical Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety of dimension $d$. The diagram $$ \\xymatrix{ \\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] & H^{2d}(X) \\ar[d]^{\\int_X} \\\\ \\CH_0(X) \\ar[r]^\\deg & F } $$ commutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section \\ref{chow-section-degree-zero-cycles}."}
+{"_id": "5057", "title": "weil-lemma-trace-product-classical", "text": "Let $H^*$ be a classical Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory-classical}). Let $X$ and $Y$ be smooth projective varieties. Then $\\int_{X \\times Y} = \\int_X \\otimes \\int_Y$."}
+{"_id": "5058", "title": "weil-lemma-pr2star-classical", "text": "Let $H^*$ be a classical Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory-classical}). Let $X$ and $Y$ be smooth projective varieties. Then $\\text{pr}_{2, *} : H^*(X \\times Y) \\to H^*(Y)$ sends $a \\otimes b$ to $(\\int_X a) b$."}
+{"_id": "5059", "title": "weil-lemma-class-diagonal-classical", "text": "Let $H^*$ be a classical Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety of dimension $d$. Choose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of $H^i(X)$ over $F$. Using K\\\"unneth write $$ \\gamma([\\Delta]) = \\sum\\nolimits_{i = 0, \\ldots, 2d} \\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j} \\quad\\text{in}\\quad \\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X) $$ with $e'_{2d - i, j} \\in H^{2d - i}(X)$. Then $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$."}
+{"_id": "5060", "title": "weil-lemma-square-diagonal-classical", "text": "Let $H^*$ be a classical Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory-classical}). Let $X$ be a smooth projective variety. We have $$ \\sum\\nolimits_{i = 0, \\ldots, 2\\dim(X)} (-1)^i\\dim_F H^i(X) = \\deg([\\Delta] \\cdot [\\Delta]) = \\deg(c_d(\\mathcal{T}_X) \\cap [X]) $$"}
+{"_id": "5061", "title": "weil-lemma-from-functor-to-weil-classical", "text": "Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Consider a $\\mathbf{Q}$-linear functor $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ of symmetric monoidal categories such that $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$. Then we obtain data (D1), (D2), (D3) satisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d)."}
+{"_id": "5062", "title": "weil-lemma-from-weil-to-functor-classical", "text": "Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. Let $H^*$ be a classical Weil cohomology theory. Then we can construct a $\\mathbf{Q}$-linear functor $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ of symmetric monoidal categories such that $H^*(X) = G(h(X))$."}
+{"_id": "5063", "title": "weil-lemma-generated-by-separable", "text": "Let $k$ be a field. Let $X$ be a smooth projective scheme over $k$. Then $\\CH_0(X)$ is generated by classes of closed points whose residue fields are separable over $k$."}
+{"_id": "5064", "title": "weil-lemma-chow-limit", "text": "Let $K/k$ be an algebraic field extension. Let $X$ be a finite type scheme over $k$. Then $\\CH_i(X_K) = \\colim \\CH_i(X_{k'})$ where the colimit is over the subextensions $K/k'/k$ with $k'/k$ finite."}
+{"_id": "5065", "title": "weil-lemma-divide-difference-points", "text": "Let $k$ be a field. Let $X$ be a geometrically irreducible smooth projective scheme over $k$. Let $x, x' \\in X$ be $k$-rational points. Let $n$ be an integer invertible in $k$. Then there exists a finite separable extension $k'/k$ such that the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in $\\CH_0(X_{k'})$."}
+{"_id": "5066", "title": "weil-lemma-kernel-to-closure", "text": "Let $K/k$ be an algebraic extension of fields. Let $X$ be a finite type scheme over $k$. The kernel of the map $\\CH_i(X) \\to \\CH_i(X_K)$ constructed in Lemma \\ref{lemma-chow-limit} is torsion."}
+{"_id": "5068", "title": "weil-lemma-pushforward", "text": "Assume given (D0), (D1), and (D3) satisfying (A). For $f : X \\to Y$ a morphism of nonempty equidimensional smooth projective schemes over $k$ we have $f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism with $Z$ nonempty smooth projective and equidimensional, then $g_* \\circ f_* = (g \\circ f)_*$."}
+{"_id": "5069", "title": "weil-lemma-pr2star", "text": "Assume given (D0), (D1), and (D3) satisfying (A) and (B). Let $X$ and $Y$ be nonempty smooth projective schemes over $k$ equidimensional of dimensions $d$ and $e$. Then $\\text{pr}_{2, *} : H^*(X \\times Y)(d + e) \\to H^*(Y)(e)$ sends $a \\otimes b$ to $(\\int_X a) b$."}
+{"_id": "5070", "title": "weil-lemma-base", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then $H^i(\\Spec(k)) = 0$ for $i \\not = 0$ and there is a unique $F$-algebra isomorphism $F = H^0(\\Spec(k))$. We have $\\gamma([\\Spec(k)]) = 1$ and $\\int_{\\Spec(k)} 1 = 1$."}
+{"_id": "5071", "title": "weil-lemma-unit", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme over $k$. If $X = \\emptyset$, then $H^*(X) = 0$. If $X$ is nonempty, then $\\gamma([X]) = 1$ and $1 \\not = 0$ in $H^0(X)$."}
+{"_id": "5072", "title": "weil-lemma-push-unit", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $i : X \\to Y$ be a closed immersion of nonempty smooth projective equidimensional schemes over $k$. Then $\\gamma([X]) = i_*1$ in $H^{2c}(Y)(c)$ where $c = \\dim(Y) - \\dim(X)$."}
+{"_id": "5073", "title": "weil-lemma-class-diagonal", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme over $k$ equidimensional of dimension $d$. Choose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of $H^i(X)$ over $F$. Using K\\\"unneth write $$ \\gamma([\\Delta]) = \\sum\\nolimits_i \\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j} \\quad\\text{in}\\quad \\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X)(d) $$ with $e'_{2d - i, j} \\in H^{2d - i}(X)(d)$. Then $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$."}
+{"_id": "5074", "title": "weil-lemma-cohomology-P1", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then $H^*(\\mathbf{P}^1_k)$ is $1$-dimensional in dimensions $0$ and $2$ and zero in other degrees."}
+{"_id": "5075", "title": "weil-lemma-weil-additive", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If $X$ and $Y$ are smooth projective schemes over $k$, then $H^*(X \\amalg Y) \\to H^*(X) \\times H^*(Y)$, $a \\mapsto (i^*a, j^*a)$ is an isomorphism where $i$, $j$ are the coprojections."}
+{"_id": "5076", "title": "weil-lemma-from-functor-to-weil", "text": "Let $k$ be a field. Let $F$ be a field of characteristic $0$. Assume given a $\\mathbf{Q}$-linear functor $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ of symmetric monoidal categories such that $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$. Then we obtain data (D0), (D1), (D2), and (D3) satisfying all of (A), (B), and (C) above."}
+{"_id": "5077", "title": "weil-lemma-from-weil-to-functor", "text": "Let $k$ be a field. Let $F$ be a field of characteristic $0$. Given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C) we can construct a $\\mathbf{Q}$-linear functor $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ of symmetric monoidal categories such that $H^*(X) = G(h(X))$."}
+{"_id": "5078", "title": "weil-lemma-trace-disjoint-union", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X, Y$ be nonempty smooth projective schemes both equidimensional of dimension $d$ over $k$. Then $\\int_{X \\amalg Y} = \\int_X + \\int_Y$."}
+{"_id": "5079", "title": "weil-lemma-dim-0", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme of dimension zero over $k$. Then \\begin{enumerate} \\item $H^i(X) = 0$ for $i \\not = 0$, \\item $H^0(X)$ is a finite separable algebra over $F$, \\item $\\dim_F H^0(X) = \\deg(X \\to \\Spec(F))$, \\item $\\int_X : H^0(X) \\to F$ is the trace map, \\item $\\gamma([X]) = 1$, and \\item $\\int_X \\gamma([X]) = \\deg(X \\to \\Spec(k))$. \\end{enumerate}"}
+{"_id": "5080", "title": "weil-lemma-degrees-cycles", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme equidimensional of dimension $d$ over $k$. The diagram $$ \\xymatrix{ \\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] & H^{2d}(X)(d) \\ar[d]^{\\int_X} \\\\ \\CH_0(X) \\ar[r]^\\deg & F } $$ commutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section \\ref{chow-section-degree-zero-cycles}."}
+{"_id": "5082", "title": "weil-lemma-algebra-relations", "text": "Let $F$ be a field of characteristic $0$. Let $F'$ and $F_i$, $i = 1, \\ldots, r$ be finite separable $F$-algebras. Let $A$ be a finite $F$-algebra. Let $\\sigma, \\sigma' : A \\to F'$ and $\\sigma_i : A \\to F_i$ be $F$-algebra maps. Assume $\\sigma$ and $\\sigma'$ surjective. If there is a relation $$ \\text{Tr}_{F'/F} \\circ \\sigma - \\text{Tr}_{F'/F} \\circ \\sigma' = n(\\sum m_i \\text{Tr}_{F_i/F} \\circ \\sigma_i) $$ where $n > 1$ and $m_i$ are integers, then $\\sigma = \\sigma'$."}
+{"_id": "5083", "title": "weil-lemma-relations-classes-points", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k'/k$ be a finite separable extension. Let $X$ be a smooth projective scheme over $k'$. Let $x, x' \\in X$ be $k'$-rational points. If $\\gamma(x) \\not = \\gamma(x')$, then $[x] - [x']$ is not divisible by any integer $n > 1$ in $\\CH_0(X)$."}
+{"_id": "5084", "title": "weil-lemma-classes-points", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k'/k$ be a finite separable extension. Let $X$ be a geometrically irreducible smooth projective scheme over $k'$ of dimension $d$. Then $\\gamma : \\CH_0(X) \\to H^{2d}(X)(d)$ factors through $\\deg : \\CH_0(X) \\to \\mathbf{Z}$."}
+{"_id": "5085", "title": "weil-lemma-injective", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $f : X \\to Y$ be a dominant morphism of irreducible smooth projective schemes over $k$. Then $H^*(Y) \\to H^*(X)$ is injective."}
+{"_id": "5086", "title": "weil-lemma-otimes", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $k''/k'/k$ be finite separable algebras and let $X$ be a smooth projective scheme over $k'$. Then $$ H^*(X) \\otimes_{H^0(\\Spec(k'))} H^0(\\Spec(k'')) = H^*(X \\times_{\\Spec(k')} \\Spec(k'')) $$"}
+{"_id": "5087", "title": "weil-lemma-H-0-separable", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a smooth projective scheme over $k$. Set $k' = \\Gamma(X, \\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item there exist finitely many closed points $x_1, \\ldots, x_r \\in X$ whose residue fields are separable over $k$ such that $H^0(X) \\to H^0(x_1) \\oplus \\ldots \\oplus H^0(x_r)$ is injective, \\item the map $H^0(\\Spec(k')) \\to H^0(X)$ is an isomorphism. \\end{enumerate} If $X$ is equidimensional of dimension $d$, these are also equivalent to \\begin{enumerate} \\item[(3)] the classes of closed points generate $H^{2d}(X)(d)$ as a module over $H^0(X)$. \\end{enumerate} If this is true, then $H^0(X)$ is a finite separable algebra over $F$."}
+{"_id": "5089", "title": "weil-lemma-splitting-principle", "text": "In the situation above. Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}_i$ be a finite collection of locally free $\\mathcal{O}_X$-modules of rank $r_i$. There exists a morphism $p : P \\to X$ in $\\mathcal{C}$ such that \\begin{enumerate} \\item $p^* : A(X) \\to A(P)$ is injective, \\item each $p^*\\mathcal{E}_i$ has a filtration whose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$ are invertible $\\mathcal{O}_P$-modules. \\end{enumerate}"}
+{"_id": "5090", "title": "weil-lemma-chern-classes-E-tensor-L", "text": "Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $$ c^A_i({\\mathcal E} \\otimes {\\mathcal L}) = \\sum\\nolimits_{j = 0}^i \\binom{r - i + j}{j} c^A_{i - j}({\\mathcal E}) \\cup c^A_1({\\mathcal L})^j $$"}
+{"_id": "5091", "title": "weil-lemma-adams-and-chern", "text": "In the situation above let $X \\in \\Ob(\\mathcal{C})$. If $\\psi^2$ is as in Chow Homology, Lemma \\ref{chow-lemma-second-adams-operator} and $c^A$ and $ch^A$ are as in Propositions \\ref{proposition-chern-class} and \\ref{proposition-chern-character} then we have $c^A_i(\\psi^2(\\alpha)) = 2^i c^A_i(\\alpha)$ and $ch^A_i(\\psi^2(\\alpha)) = 2^i ch^A_i(\\alpha)$ for all $\\alpha \\in K_0(\\textit{Vect}(X))$."}
+{"_id": "5092", "title": "weil-lemma-lambda-operations", "text": "Let $X$ be a scheme. There are maps $$ \\lambda^r : K_0(\\textit{Vect}(X)) \\longrightarrow K_0(\\textit{Vect}(X)) $$ which sends $[\\mathcal{E}]$ to $[\\wedge^r(\\mathcal{E})]$ when $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module and which are compatible with pullbacks."}
+{"_id": "5093", "title": "weil-lemma-chern-classes", "text": "Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective $X$ over $k$ a ring homomorphism $$ ch^H : K_0(\\textit{Vec}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} H^{2i}(X)(i) $$ compatible with pullbacks such that $ch^H(\\mathcal{L}) = \\exp(c_1^H(\\mathcal{L}))$ for any invertible $\\mathcal{O}_X$-module $\\mathcal{L}$."}
+{"_id": "5094", "title": "weil-lemma-cycle-classes", "text": "Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective $X$ over $k$ a graded ring homomorphism $$ \\gamma : \\CH^*(X) \\longrightarrow \\bigoplus\\nolimits_{i \\geq 0} H^{2i}(X)(i) $$ compatible with pullbacks such that $ch^H(\\alpha) = \\gamma(ch(\\alpha))$ for $\\alpha$ in $K_0(\\textit{Vect}(X))$."}
+{"_id": "5095", "title": "weil-lemma-divide-pullback-good-blowing-up", "text": "Let $b : X' \\to X$ be the blowing up of a smooth projective scheme over $k$ in a smooth closed subscheme $Z \\subset X$. Picture $$ \\xymatrix{ E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\ Z \\ar[r]^i & X } $$ Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$. Assume every irreducible component of $Z$ has codimension $r$ in $X$. Then there exists a cycle $\\theta \\in \\CH^{r - 1}(X')$ such that $b^![Z] = [E] \\cdot \\theta$ in $\\CH^r(X')$ and $\\pi_*j^!(\\theta) = [Z]$ in $\\CH^r(Z)$."}
+{"_id": "5096", "title": "weil-lemma-A5-A6-imply", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A4) and (A7). Let $X$ be a smooth projective scheme over $k$. Let $Z \\subset X$ be a smooth closed subscheme such that every irreducible component of $Z$ has codimension $r$ in $X$. Assume the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the restriction of an element of $K_0(X)$. If $a \\in H^*(X)$ and $a|_Z = 0$ in $H^*(Z)$, then $\\gamma([Z]) \\cup a = 0$."}
+{"_id": "5097", "title": "weil-lemma-poincare-duality", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (A) of Section \\ref{section-axioms} holds with $\\int_X = \\lambda$ as in axiom (A6)."}
+{"_id": "5098", "title": "weil-lemma-trace-product", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (B) of Section \\ref{section-axioms} holds."}
+{"_id": "5099", "title": "weil-lemma-trace-base", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (C)(d) of Section \\ref{section-axioms} holds."}
+{"_id": "5100", "title": "weil-lemma-ok-for-projective-bundle", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Let $p : P \\to X$ be as in axiom (A3) with $X$ nonempty equidimensional. Then $\\gamma$ commutes with pushforward along $p$."}
+{"_id": "5101", "title": "weil-lemma-integrate-1", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If $k'/k$ is a Galois extension, then we have $\\int_{\\Spec(k')} 1 = [k' : k]$."}
+{"_id": "5102", "title": "weil-lemma-enough", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that $\\gamma$ commutes with pushforward it suffices to show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$."}
+{"_id": "5103", "title": "weil-lemma-grassmanian", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Given integers $0 < l < n$ and a nonempty equidimensional smooth projective scheme $X$ over $k$ consider the projection morphism $p : X \\times \\mathbf{G}(l, n) \\to X$. Then $\\gamma$ commutes with pushforward along $p$."}
+{"_id": "5104", "title": "weil-lemma-enough-better", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that $\\gamma$ commutes with pushforward it suffices to show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed immersion of nonempty smooth projective equidimensional schemes over $k$ such that the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the pullback of a class in $K_0(X)$."}
+{"_id": "5105", "title": "weil-lemma-injective-H0", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If $k''/k'/k$ are finite separable field extensions, then $H^0(\\Spec(k')) \\to H^0(\\Spec(k''))$ is injective."}
+{"_id": "5106", "title": "weil-lemma-pushforward-blowup", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Let $b : X' \\to X$ be a blowing up of a smooth projective scheme $X$ over $k$ which is nonempty equidimensional of dimension $d$ in a nonwhere dense smooth center $Z$. Then $b_*(1) = 1$."}
+{"_id": "5107", "title": "weil-lemma-done", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then the cycle class map $\\gamma$ commutes with pushforward."}
+{"_id": "5108", "title": "weil-lemma-check-over-extension", "text": "Let $k'/k$ be an extension of fields. Let $F'/F$ be an extension of fields of characteristic $0$. Assume given \\begin{enumerate} \\item data (D0), (D1), (D2') for $k$ and $F$ denoted $F(1), H^*, c_1^H$, \\item data (D0), (D1), (D2') for $k'$ and $F'$ denoted $F'(1), (H')^*, c_1^{H'}$, and \\item an isomorphism $F(1) \\otimes_F F' \\to F'(1)$, functorial isomorphisms $H^*(X) \\otimes_F F' \\to (H')^*(X_{k'})$ on the category of smooth projective schemes $X$ over $k$ such that the diagrams $$ \\xymatrix{ \\Pic(X) \\ar[r]_{c_1^H} \\ar[d] & H^2(X)(1) \\ar[d] \\\\ \\Pic(X_{k'}) \\ar[r]^{c_1^{H'}} & (H')^2(X_{k'})(1) } $$ commute. \\end{enumerate} In this case, if $F'(1), (H')^*, c_1^{H'}$ satisfy axioms (A1) -- (A9), then the same is true for $F(1), H^*, c_1^H$."}
+{"_id": "5111", "title": "weil-proposition-chern-class", "text": "In the situation above there is a unique rule which assigns to every $X \\in \\Ob(\\mathcal{C})$ a ``total Chern class'' $$ c^A : K_0(\\textit{Vect}(X)) \\longrightarrow  \\prod\\nolimits_{i \\geq 0} A^i(X) $$ with the following properties \\begin{enumerate} \\item For $X \\in \\Ob(\\mathcal{C})$ we have $c^A(\\alpha + \\beta) = c^A(\\alpha) c^A(\\beta)$ and $c^A(0) = 1$. \\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then $f^* \\circ c^A =  c^A \\circ f^*$. \\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$ we have $c^A([\\mathcal{L}]) = 1 + c_1^A(\\mathcal{L})$. \\end{enumerate}"}
+{"_id": "5112", "title": "weil-proposition-chern-character", "text": "In the situation above assume $A(X)$ is a $\\mathbf{Q}$-algebra for all $X \\in \\Ob(\\mathcal{C})$. Then there is a unique rule which assigns to every $X \\in \\Ob(\\mathcal{C})$ a ``chern character'' $$ ch^A : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(X) $$ with the following properties \\begin{enumerate} \\item $ch^A$ is a ring map for all $X \\in \\Ob(\\mathcal{C})$. \\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then $f^* \\circ ch^A =  ch^A \\circ f^*$. \\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$ we have $ch^A([\\mathcal{L}]) = \\exp(c_1^A(\\mathcal{L}))$. \\end{enumerate}"}
+{"_id": "5113", "title": "weil-proposition-get-weil", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then we have a Weil cohomology theory."}
+{"_id": "5123", "title": "morphisms-theorem-chevalley", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every locally constructible subset is locally constructible."}
+{"_id": "5124", "title": "morphisms-theorem-main-theorem", "text": "Let $f : Y \\to X$ be an affine morphism of schemes. Assume $f$ is of finite type. Let $X'$ be the normalization of $X$ in $Y$. Picture: $$ \\xymatrix{ Y \\ar[rd]_f \\ar[rr]_{f'} & & X' \\ar[ld]^\\nu \\\\ & X & } $$ Then there exists an open subscheme $U' \\subset X'$ such that \\begin{enumerate} \\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and \\item $(f')^{-1}(U') \\subset Y$ is the set of points at which $f$ is quasi-finite. \\end{enumerate}"}
+{"_id": "5125", "title": "morphisms-lemma-closed-immersion", "text": "Let $i : Z \\to X$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $i$ is a closed immersion. \\item For every affine open $\\Spec(R) = U \\subset X$, there exists an ideal $I \\subset R$ such that $i^{-1}(U) = \\Spec(R/I)$ as schemes over $U = \\Spec(R)$. \\item There exists an affine open covering $X = \\bigcup_{j \\in J} U_j$, $U_j = \\Spec(R_j)$ and for every $j \\in J$ there exists an ideal $I_j \\subset R_j$ such that $i^{-1}(U_j) = \\Spec(R_j/I_j)$ as schemes over $U_j = \\Spec(R_j)$. \\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$ and $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective. \\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective, and the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a quasi-coherent sheaf of ideals. \\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective, and the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a sheaf of ideals which is locally generated by sections. \\end{enumerate}"}
+{"_id": "5126", "title": "morphisms-lemma-closed-immersion-ideals", "text": "Let $X$ be a scheme. Let $i : Z \\to X$ and $i' : Z' \\to X$ be closed immersions and consider the ideal sheaves $\\mathcal{I} = \\Ker(i^\\sharp)$ and $\\mathcal{I}' = \\Ker((i')^\\sharp)$ of $\\mathcal{O}_X$. \\begin{enumerate} \\item The morphism $i : Z \\to X$ factors as $Z \\to Z' \\to X$ for some $a : Z \\to Z'$ if and only if $\\mathcal{I}' \\subset \\mathcal{I}$. If this happens, then $a$ is a closed immersion. \\item We have $Z \\cong Z'$ over $X$ if and only if $\\mathcal{I} = \\mathcal{I}'$. \\end{enumerate}"}
+{"_id": "5127", "title": "morphisms-lemma-closed-immersion-bijection-ideals", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{I}$ is locally generated by sections as a sheaf of $\\mathcal{O}_X$-modules, \\item $\\mathcal{I}$ is quasi-coherent as a sheaf of $\\mathcal{O}_X$-modules, and \\item there exists a closed immersion $i : Z \\to X$ of schemes whose corresponding sheaf of ideals $\\Ker(i^\\sharp)$ is equal to $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "5128", "title": "morphisms-lemma-base-change-closed-immersion", "text": "The base change of a closed immersion is a closed immersion."}
+{"_id": "5130", "title": "morphisms-lemma-closed-immersion-quasi-compact", "text": "A closed immersion is quasi-compact."}
+{"_id": "5131", "title": "morphisms-lemma-closed-immersion-separated", "text": "A closed immersion is separated."}
+{"_id": "5132", "title": "morphisms-lemma-immersion-permanence", "text": "Let $Z \\to Y \\to X$ be morphisms of schemes. \\begin{enumerate} \\item If $Z \\to X$ is an immersion, then $Z \\to Y$ is an immersion. \\item If $Z \\to X$ is a quasi-compact immersion and $Y \\to X$ is quasi-separated, then $Z \\to Y$ is a quasi-compact immersion. \\item If $Z \\to X$ is a closed immersion and $Y \\to X$ is separated, then $Z \\to Y$ is a closed immersion. \\end{enumerate}"}
+{"_id": "5133", "title": "morphisms-lemma-factor-quasi-compact-immersion", "text": "Let $h : Z \\to X$ be an immersion. If $h$ is quasi-compact, then we can factor $h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an open immersion and $i : \\overline{Z} \\to X$ a closed immersion."}
+{"_id": "5134", "title": "morphisms-lemma-factor-reduced-immersion", "text": "Let $h : Z \\to X$ be an immersion. If $Z$ is reduced, then we can factor $h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an open immersion and $i : \\overline{Z} \\to X$ a closed immersion."}
+{"_id": "5135", "title": "morphisms-lemma-check-immersion", "text": "Let $f : Y \\to X$ be a morphism of schemes. If for all $y \\in Y$ there is an open subscheme $f(y) \\in U \\subset X$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is an immersion, then $f$ is an immersion."}
+{"_id": "5136", "title": "morphisms-lemma-i-star-equivalence", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $$ i_* : \\QCoh(\\mathcal{O}_Z) \\longrightarrow \\QCoh(\\mathcal{O}_X) $$ is exact, fully faithful, with essential image those quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$."}
+{"_id": "5137", "title": "morphisms-lemma-largest-quasi-coherent-subsheaf", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}$ be a $\\mathcal{O}_X$-submodule. There exists a unique quasi-coherent $\\mathcal{O}_X$-submodule $\\mathcal{G}' \\subset \\mathcal{G}$ with the following property: For every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{H}$ the map $$ \\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}') \\longrightarrow \\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}) $$ is bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent $\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$."}
+{"_id": "5139", "title": "morphisms-lemma-scheme-theoretic-intersection", "text": "Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes. Let $Z \\cap Y$ be the scheme theoretic intersection of $Z$ and $Y$. Then $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions and $$ \\xymatrix{ Z \\cap Y \\ar[r] \\ar[d] & Z \\ar[d] \\\\ Y \\ar[r] & X } $$ is a cartesian diagram of schemes, i.e., $Z \\cap Y = Z \\times_X Y$."}
+{"_id": "5140", "title": "morphisms-lemma-scheme-theoretic-union", "text": "Let $S$ be a scheme. Let $X, Y \\subset S$ be closed subschemes. Let $X \\cup Y$ be the scheme theoretic union of $X$ and $Y$. Let $X \\cap Y$ be the scheme theoretic intersection of $X$ and $Y$. Then $X \\to X \\cup Y$ and $Y \\to X \\cup Y$ are closed immersions, there is a short exact sequence $$ 0 \\to \\mathcal{O}_{X \\cup Y} \\to \\mathcal{O}_X \\times \\mathcal{O}_Y \\to \\mathcal{O}_{X \\cap Y} \\to 0 $$ of $\\mathcal{O}_S$-modules, and the diagram $$ \\xymatrix{ X \\cap Y \\ar[r] \\ar[d] & X \\ar[d] \\\\ Y \\ar[r] & X \\cup Y } $$ is cocartesian in the category of schemes, i.e., $X \\cup Y = X \\amalg_{X \\cap Y} Y$."}
+{"_id": "5141", "title": "morphisms-lemma-support-affine-open", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\\Spec(A) = U \\subset X$ be an affine open, and set $M = \\Gamma(U, \\mathcal{F})$. Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime. The following are equivalent \\begin{enumerate} \\item $\\mathfrak p$ is in the support of $M$, and \\item $x$ is in the support of $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "5142", "title": "morphisms-lemma-support-closed-specialization", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. The support of $\\mathcal{F}$ is closed under specialization."}
+{"_id": "5143", "title": "morphisms-lemma-support-finite-type", "text": "Let $\\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then \\begin{enumerate} \\item The support of $\\mathcal{F}$ is closed. \\item For $x \\in X$ we have $$ x \\in \\text{Supp}(\\mathcal{F}) \\Leftrightarrow \\mathcal{F}_x \\not = 0 \\Leftrightarrow \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) \\not = 0. $$ \\item For any morphism of schemes $f : Y \\to X$ the pullback $f^*\\mathcal{F}$ is of finite type as well and we have $\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$. \\end{enumerate}"}
+{"_id": "5144", "title": "morphisms-lemma-scheme-theoretic-support", "text": "Let $\\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. There exists a smallest closed subscheme $i : Z \\to X$ such that there exists a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ with $i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover: \\begin{enumerate} \\item If $\\Spec(A) \\subset X$ is any affine open, and $\\mathcal{F}|_{\\Spec(A)} = \\widetilde{M}$ then $Z \\cap \\Spec(A) = \\Spec(A/I)$ where $I = \\text{Ann}_A(M)$. \\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique isomorphism. \\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type. \\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $Z$. \\end{enumerate}"}
+{"_id": "5145", "title": "morphisms-lemma-scheme-theoretic-image", "text": "Let $f : X \\to Y$ be a morphism of schemes. There exists a closed subscheme $Z \\subset Y$ such that $f$ factors through $Z$ and such that for any other closed subscheme $Z' \\subset Y$ such that $f$ factors through $Z'$ we have $Z \\subset Z'$."}
+{"_id": "5146", "title": "morphisms-lemma-quasi-compact-scheme-theoretic-image", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then \\begin{enumerate} \\item the sheaf of ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$ is quasi-coherent, \\item the scheme theoretic image $Z$ is the closed subscheme determined by $\\mathcal{I}$, \\item for any open $U \\subset Y$ the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is equal to $Z \\cap U$, and \\item the image $f(X) \\subset Z$ is a dense subset of $Z$, in other words the morphism $X \\to Z$ is dominant (see Definition \\ref{definition-dominant}). \\end{enumerate}"}
+{"_id": "5147", "title": "morphisms-lemma-reach-points-scheme-theoretic-image", "text": "Let $f : X \\to Y$ be a quasi-compact morphism. Let $Z$ be the scheme theoretic image of $f$. Let $z \\in Z$\\footnote{By Lemma \\ref{lemma-quasi-compact-scheme-theoretic-image} set-theoretically $Z$ agrees with the closure of $f(X)$ in $Y$.}. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\ \\Spec(A) \\ar[r] & Z \\ar[r] & Y } $$ such that the closed point of $\\Spec(A)$ maps to $z$. In particular any point of $Z$ is the specialization of a point of $f(X)$."}
+{"_id": "5148", "title": "morphisms-lemma-factor-factor", "text": "Let $$ \\xymatrix{ X_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\ X_2 \\ar[r]^{f_2} & Y_2 } $$ be a commutative diagram of schemes. Let $Z_i \\subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a commutative diagram $$ \\xymatrix{ X_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\ X_2 \\ar[r] & Z_2 \\ar[r] & Y_2 } $$"}
+{"_id": "5149", "title": "morphisms-lemma-scheme-theoretic-image-reduced", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $X$ is reduced, then the scheme theoretic image of $f$ is the reduced induced scheme structure on $\\overline{f(X)}$."}
+{"_id": "5152", "title": "morphisms-lemma-characterize-scheme-theoretically-dense", "text": "Let $j : U \\to X$ be an open immersion of schemes. Then $U$ is scheme theoretically dense in $X$ if and only if $\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective."}
+{"_id": "5153", "title": "morphisms-lemma-intersection-scheme-theoretically-dense", "text": "Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense open subschemes of $X$, then so is $U \\cap V$."}
+{"_id": "5154", "title": "morphisms-lemma-quasi-compact-immersion", "text": "Let $h : Z \\to X$ be an immersion. Assume either $h$ is quasi-compact or $Z$ is reduced. Let $\\overline{Z} \\subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \\to \\overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subscheme of $\\overline{Z}$. Moreover, $Z$ is topologically dense in $\\overline{Z}$."}
+{"_id": "5155", "title": "morphisms-lemma-reduced-scheme-theoretically-dense", "text": "Let $X$ be a reduced scheme and let $U \\subset X$ be an open subscheme. Then the following are equivalent \\begin{enumerate} \\item $U$ is topologically dense in $X$, \\item the scheme theoretic closure of $U$ in $X$ is $X$, and \\item $U$ is scheme theoretically dense in $X$. \\end{enumerate}"}
+{"_id": "5156", "title": "morphisms-lemma-reduced-subscheme-closure", "text": "Let $X$ be a scheme and let $U \\subset X$ be a reduced open subscheme. Then the following are equivalent \\begin{enumerate} \\item the scheme theoretic closure of $U$ in $X$ is $X$, and \\item $U$ is scheme theoretically dense in $X$. \\end{enumerate} If this holds then $X$ is a reduced scheme."}
+{"_id": "5157", "title": "morphisms-lemma-equality-of-morphisms", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \\to Y$ be morphisms of schemes over $S$. Let $U \\subset X$ be an open subscheme such that $f|_U = g|_U$. If the scheme theoretic closure of $U$ in $X$ is $X$ and $Y \\to S$ is separated, then $f = g$."}
+{"_id": "5158", "title": "morphisms-lemma-generic-points-in-image-dominant", "text": "Let $f : X \\to S$ be a morphism of schemes. If every generic point of every irreducible component of $S$ is in the image of $f$, then $f$ is dominant."}
+{"_id": "5159", "title": "morphisms-lemma-quasi-compact-dominant", "text": "\\begin{slogan} Morphisms whose image contains the generic points are dominant \\end{slogan} Let $f : X \\to S$ be a quasi-compact morphism of schemes. Then $f$ is dominant (if and) only if for every irreducible component $Z \\subset S$ the generic point of $Z$ is in the image of $f$."}
+{"_id": "5160", "title": "morphisms-lemma-quasi-compact-generic-point-not-in-image", "text": "Let $f : X \\to S$ be a quasi-compact morphism of schemes. Let $\\eta \\in S$ be a generic point of an irreducible component of $S$. If $\\eta \\not \\in f(X)$ then there exists an open neighbourhood $V \\subset S$ of $\\eta$ such that $f^{-1}(V) = \\emptyset$."}
+{"_id": "5161", "title": "morphisms-lemma-dominant-finite-number-irreducible-components", "text": "Let $f : X \\to S$ be a morphism of schemes. Suppose that $X$ has finitely many irreducible components. Then $f$ is dominant (if and) only if for every irreducible component $Z \\subset S$ the generic point of $Z$ is in the image of $f$. If so, then $S$ has finitely many irreducible components as well."}
+{"_id": "5162", "title": "morphisms-lemma-dominant-between-integral", "text": "Let $f : X \\to Y$ be a morphism of integral schemes. The following are equivalent \\begin{enumerate} \\item $f$ is dominant, \\item $f$ maps the generic point of $X$ to the generic point of $Y$, \\item for some nonempty affine opens $U \\subset X$ and $V \\subset Y$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is injective, \\item for all nonempty affine opens $U \\subset X$ and $V \\subset Y$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is injective, \\item for some $x \\in X$ with image $y = f(x) \\in Y$ the local ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective, and \\item for all $x \\in X$ with image $y = f(x) \\in Y$ the local ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective. \\end{enumerate}"}
+{"_id": "5163", "title": "morphisms-lemma-composition-surjective", "text": "The composition of surjective morphisms is surjective."}
+{"_id": "5164", "title": "morphisms-lemma-when-point-maps-to-pair", "text": "Let $X$ and $Y$ be schemes over a base scheme $S$. Given points $x \\in X$ and $y \\in Y$, there is a point of $X \\times_S Y$ mapping to $x$ and $y$ under the projections if and only if $x$ and $y$ lie above the same point of $S$."}
+{"_id": "5165", "title": "morphisms-lemma-base-change-surjective", "text": "The base change of a surjective morphism is surjective."}
+{"_id": "5167", "title": "morphisms-lemma-universally-injective", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item For every field $K$ the induced map $\\Mor(\\Spec(K), X) \\to \\Mor(\\Spec(K), S)$ is injective. \\item The morphism $f$ is universally injective. \\item The morphism $f$ is radicial. \\item The diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$ is surjective. \\end{enumerate}"}
+{"_id": "5168", "title": "morphisms-lemma-universally-injective-separated", "text": "A universally injective morphism is separated."}
+{"_id": "5169", "title": "morphisms-lemma-base-change-universally-injective", "text": "A base change of a universally injective morphism is universally injective."}
+{"_id": "5170", "title": "morphisms-lemma-composition-universally-injective", "text": "A composition of radicial morphisms is radicial, and so the same holds for the equivalent condition of being universally injective."}
+{"_id": "5172", "title": "morphisms-lemma-characterize-affine", "text": "\\begin{reference} \\cite[II, Corollary 1.3.2]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is affine. \\item There exists an affine open covering $S = \\bigcup W_j$ such that each $f^{-1}(W_j)$ is affine. \\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras $\\mathcal{A}$ and an isomorphism $X \\cong \\underline{\\Spec}_S(\\mathcal{A})$ of schemes over $S$. See Constructions, Section \\ref{constructions-section-spec} for notation. \\end{enumerate} Moreover, in this case $X = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$."}
+{"_id": "5173", "title": "morphisms-lemma-affine-equivalence-algebras", "text": "Let $S$ be a scheme. There is an anti-equivalence of categories $$ \\begin{matrix} \\text{Schemes affine} \\\\ \\text{over }S \\end{matrix} \\longleftrightarrow \\begin{matrix} \\text{quasi-coherent sheaves} \\\\ \\text{of }\\mathcal{O}_S\\text{-algebras} \\end{matrix} $$ which associates to $f : X \\to S$ the sheaf $f_*\\mathcal{O}_X$. Moreover, this equivalence is compatible with arbitrary base change."}
+{"_id": "5174", "title": "morphisms-lemma-affine-equivalence-modules", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{A} = f_*\\mathcal{O}_X$. The functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{category of quasi-coherent}\\\\ \\mathcal{O}_X\\text{-modules} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{category of quasi-coherent}\\\\ \\mathcal{A}\\text{-modules} \\end{matrix} \\right\\} $$ Moreover, an $\\mathcal{A}$-module is quasi-coherent as an $\\mathcal{O}_S$-module if and only if it is quasi-coherent as an $\\mathcal{A}$-module."}
+{"_id": "5175", "title": "morphisms-lemma-composition-affine", "text": "The composition of affine morphisms is affine."}
+{"_id": "5176", "title": "morphisms-lemma-base-change-affine", "text": "The base change of an affine morphism is affine."}
+{"_id": "5177", "title": "morphisms-lemma-closed-immersion-affine", "text": "A closed immersion is affine."}
+{"_id": "5178", "title": "morphisms-lemma-affine-s-open", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. The inclusion morphism $j : X_s \\to X$ is affine."}
+{"_id": "5179", "title": "morphisms-lemma-affine-permanence", "text": "Suppose $g : X \\to Y$ is a morphism of schemes over $S$. \\begin{enumerate} \\item If $X$ is affine over $S$ and $\\Delta : Y \\to Y \\times_S Y$ is affine, then $g$ is affine. \\item If $X$ is affine over $S$ and $Y$ is separated over $S$, then $g$ is affine. \\item A morphism from an affine scheme to a scheme with affine diagonal is affine. \\item A morphism from an affine scheme to a separated scheme is affine. \\end{enumerate}"}
+{"_id": "5180", "title": "morphisms-lemma-morphism-affines-affine", "text": "A morphism between affine schemes is affine."}
+{"_id": "5181", "title": "morphisms-lemma-Artinian-affine", "text": "Let $S$ be a scheme. Let $A$ be an Artinian ring. Any morphism $\\Spec(A) \\to S$ is affine."}
+{"_id": "5182", "title": "morphisms-lemma-get-affine", "text": "Let $j : Y \\to X$ be an immersion of schemes. Assume there exists an open $U \\subset X$ with complement $Z = X \\setminus U$ such that \\begin{enumerate} \\item $U \\to X$ is affine, \\item $j^{-1}(U) \\to U$ is affine, and \\item $j(Y) \\cap Z$ is closed. \\end{enumerate} Then $j$ is affine. In particular, if $X$ is affine, so is $Y$."}
+{"_id": "5184", "title": "morphisms-lemma-quasi-affine-separated", "text": "A quasi-affine morphism is separated and quasi-compact."}
+{"_id": "5185", "title": "morphisms-lemma-characterize-quasi-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is quasi-affine. \\item There exists an affine open covering $S = \\bigcup W_j$ such that each $f^{-1}(W_j)$ is quasi-affine. \\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras $\\mathcal{A}$ and a quasi-compact open immersion $$ \\xymatrix{ X \\ar[rr] \\ar[rd] & & \\underline{\\Spec}_S(\\mathcal{A}) \\ar[dl] \\\\ & S & } $$ over $S$. \\item Same as in (3) but with $\\mathcal{A} = f_*\\mathcal{O}_X$ and the horizontal arrow the canonical morphism of Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}. \\end{enumerate}"}
+{"_id": "5186", "title": "morphisms-lemma-composition-quasi-affine", "text": "The composition of quasi-affine morphisms is quasi-affine."}
+{"_id": "5187", "title": "morphisms-lemma-base-change-quasi-affine", "text": "The base change of a quasi-affine morphism is quasi-affine."}
+{"_id": "5188", "title": "morphisms-lemma-quasi-compact-immersion-quasi-affine", "text": "A quasi-compact immersion is quasi-affine."}
+{"_id": "5190", "title": "morphisms-lemma-quasi-affine-permanence", "text": "Suppose $g : X \\to Y$ is a morphism of schemes over $S$. If $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$, then $g$ is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine."}
+{"_id": "5191", "title": "morphisms-lemma-locally-P", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $P$ be a property of ring maps. Let $U$ be an affine open of $X$, and $V$ an affine open of $S$ such that $f(U) \\subset V$. If $f$ is locally of type $P$ and $P$ is local, then $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$ holds."}
+{"_id": "5192", "title": "morphisms-lemma-locally-P-characterize", "text": "Let $P$ be a local property of ring maps. Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is locally of type $P$. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ we have $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is locally of type $P$. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that $P(\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i))$ holds, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is locally of type $P$ then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is locally of type $P$."}
+{"_id": "5193", "title": "morphisms-lemma-composition-type-P", "text": "Let $P$ be a property of ring maps. Assume $P$ is local and stable under composition. The composition of morphisms locally of type $P$ is locally of type $P$."}
+{"_id": "5194", "title": "morphisms-lemma-base-change-type-P", "text": "Let $P$ be a property of ring maps. Assume $P$ is local and stable under base change. The base change of a morphism locally of type $P$ is locally of type $P$."}
+{"_id": "5195", "title": "morphisms-lemma-properties-local", "text": "The following properties of a ring map $R \\to A$ are local. \\begin{enumerate} \\item (Isomorphism on local rings.) For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$ the ring map $R \\to A$ induces an isomorphism $R_{\\mathfrak p} \\to A_{\\mathfrak q}$. \\item (Open immersion.) For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$, $\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$ induces an isomorphism $R_f \\to A_f$. \\item (Reduced fibres.) For every prime $\\mathfrak p$ of $R$ the fibre ring $A \\otimes_R \\kappa(\\mathfrak p)$ is reduced. \\item (Fibres of dimension at most $n$.) For every prime $\\mathfrak p$ of $R$ the fibre ring $A \\otimes_R \\kappa(\\mathfrak p)$ has Krull dimension at most $n$. \\item (Locally Noetherian on the target.) The ring map $R \\to A$ has the property that $A$ is Noetherian. \\item Add more here as needed\\footnote{But only those properties that are not already dealt with separately elsewhere.}. \\end{enumerate}"}
+{"_id": "5196", "title": "morphisms-lemma-properties-base-change", "text": "The following properties of ring maps are stable under base change. \\begin{enumerate} \\item (Isomorphism on local rings.) For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$ the ring map $R \\to A$ induces an isomorphism $R_{\\mathfrak p} \\to A_{\\mathfrak q}$. \\item (Open immersion.) For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$, $\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$ induces an isomorphism $R_f \\to A_f$. \\item Add more here as needed\\footnote{But only those properties that are not already dealt with separately elsewhere.}. \\end{enumerate}"}
+{"_id": "5197", "title": "morphisms-lemma-properties-composition", "text": "The following properties of ring maps are stable under composition. \\begin{enumerate} \\item (Isomorphism on local rings.) For every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$ the ring map $R \\to A$ induces an isomorphism $R_{\\mathfrak p} \\to A_{\\mathfrak q}$. \\item (Open immersion.) For every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$, $\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$ induces an isomorphism $R_f \\to A_f$. \\item (Locally Noetherian on the target.) The ring map $R \\to A$ has the property that $A$ is Noetherian. \\item Add more here as needed\\footnote{But only those properties that are not already dealt with separately elsewhere.}. \\end{enumerate}"}
+{"_id": "5198", "title": "morphisms-lemma-locally-finite-type-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is locally of finite type. \\item For all affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite type. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is locally of finite type. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is of finite type, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is locally of finite type then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is locally of finite type."}
+{"_id": "5199", "title": "morphisms-lemma-composition-finite-type", "text": "The composition of two morphisms which are locally of finite type is locally of finite type. The same is true for morphisms of finite type."}
+{"_id": "5200", "title": "morphisms-lemma-base-change-finite-type", "text": "The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type."}
+{"_id": "5201", "title": "morphisms-lemma-immersion-locally-finite-type", "text": "A closed immersion is of finite type. An immersion is locally of finite type."}
+{"_id": "5202", "title": "morphisms-lemma-finite-type-noetherian", "text": "Let $f : X \\to S$ be a morphism. If $S$ is (locally) Noetherian and $f$ (locally) of finite type then $X$ is (locally) Noetherian."}
+{"_id": "5203", "title": "morphisms-lemma-finite-type-Noetherian-quasi-separated", "text": "Let $f : X \\to S$ be locally of finite type with $S$ locally Noetherian. Then $f$ is quasi-separated."}
+{"_id": "5204", "title": "morphisms-lemma-permanence-finite-type", "text": "Let $X \\to Y$ be a morphism of schemes over a base scheme $S$. If $X$ is locally of finite type over $S$, then $X \\to Y$ is locally of finite type."}
+{"_id": "5205", "title": "morphisms-lemma-point-finite-type", "text": "Let $S$ be a scheme. Let $k$ be a field. Let $f : \\Spec(k) \\to S$ be a morphism. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is of finite type. \\item The morphism $f$ is locally of finite type. \\item There exists an affine open $U = \\Spec(R)$ of $S$ such that $f$ corresponds to a finite ring map $R \\to k$. \\item There exists an affine open $U = \\Spec(R)$ of $S$ such that the image of $f$ consists of a closed point $u$ in $U$ and the field extension $\\kappa(u) \\subset k$ is finite. \\end{enumerate}"}
+{"_id": "5206", "title": "morphisms-lemma-artinian-finite-type", "text": "Let $S$ be a scheme. Let $A$ be an Artinian local ring with residue field $\\kappa$. Let $f : \\Spec(A) \\to S$ be a morphism of schemes. Then $f$ is of finite type if and only if the composition $\\Spec(\\kappa) \\to \\Spec(A) \\to S$ is of finite type."}
+{"_id": "5207", "title": "morphisms-lemma-identify-finite-type-points", "text": "Let $S$ be a scheme. We have $$ S_{\\text{ft-pts}} = \\bigcup\\nolimits_{U \\subset S\\text{ open}} U_0 $$ where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all opens or over all affine opens of $S$."}
+{"_id": "5208", "title": "morphisms-lemma-finite-type-points-morphism", "text": "Let $f : T \\to S$ be a morphism of schemes. If $f$ is locally of finite type, then $f(T_{\\text{ft-pts}}) \\subset S_{\\text{ft-pts}}$."}
+{"_id": "5209", "title": "morphisms-lemma-finite-type-points-surjective-morphism", "text": "Let $f : T \\to S$ be a morphism of schemes. If $f$ is locally of finite type and surjective, then $f(T_{\\text{ft-pts}}) = S_{\\text{ft-pts}}$."}
+{"_id": "5210", "title": "morphisms-lemma-enough-finite-type-points", "text": "Let $S$ be a scheme. For any locally closed subset $T \\subset S$ we have $$ T \\not = \\emptyset \\Rightarrow T \\cap S_{\\text{ft-pts}} \\not = \\emptyset. $$ In particular, for any closed subset $T \\subset S$ we see that $T \\cap S_{\\text{ft-pts}}$ is dense in $T$."}
+{"_id": "5211", "title": "morphisms-lemma-jacobson-finite-type-points", "text": "Let $S$ be a scheme. The following are equivalent: \\begin{enumerate} \\item the scheme $S$ is Jacobson, \\item $S_{\\text{ft-pts}}$ is the set of closed points of $S$, \\item for all $T \\to S$ locally of finite type closed points map to closed points, and \\item for all $T \\to S$ locally of finite type closed points $t \\in T$ map to closed points $s \\in S$ with $\\kappa(s) \\subset \\kappa(t)$ finite. \\end{enumerate}"}
+{"_id": "5212", "title": "morphisms-lemma-Jacobson-universally-Jacobson", "text": "Let $S$ be a Jacobson scheme. Any scheme locally of finite type over $S$ is Jacobson."}
+{"_id": "5213", "title": "morphisms-lemma-ubiquity-Jacobson-schemes", "text": "The following types of schemes are Jacobson. \\begin{enumerate} \\item Any scheme locally of finite type over a field. \\item Any scheme locally of finite type over $\\mathbf{Z}$. \\item Any scheme locally of finite type over a $1$-dimensional Noetherian domain with infinitely many primes. \\item A scheme of the form $\\Spec(R) \\setminus \\{\\mathfrak m\\}$ where $(R, \\mathfrak m)$ is a Noetherian local ring. Also any scheme locally of finite type over it. \\end{enumerate}"}
+{"_id": "5214", "title": "morphisms-lemma-universally-catenary-local", "text": "Let $S$ be a locally Noetherian scheme. The following are equivalent \\begin{enumerate} \\item $S$ is universally catenary, \\item there exists an open covering of $S$ all of whose members are universally catenary schemes, \\item for every affine open $\\Spec(R) = U \\subset S$ the ring $R$ is universally catenary, and \\item there exists an affine open covering $S = \\bigcup U_i$ such that each $U_i$ is the spectrum of a universally catenary ring. \\end{enumerate} Moreover, in this case any scheme locally of finite type over $S$ is universally catenary as well."}
+{"_id": "5217", "title": "morphisms-lemma-ubiquity-uc", "text": "The following types of schemes are universally catenary. \\begin{enumerate} \\item Any scheme locally of finite type over a field. \\item Any scheme locally of finite type over a Cohen-Macaulay scheme. \\item Any scheme locally of finite type over $\\mathbf{Z}$. \\item Any scheme locally of finite type over a $1$-dimensional Noetherian domain. \\item And so on. \\end{enumerate}"}
+{"_id": "5218", "title": "morphisms-lemma-finite-type-nagata", "text": "Let $f : X \\to S$ be a morphism. If $S$ is Nagata and $f$ locally of finite type then $X$ is Nagata. If $S$ is universally Japanese and $f$ locally of finite type then $X$ is universally Japanese."}
+{"_id": "5219", "title": "morphisms-lemma-ubiquity-nagata", "text": "The following types of schemes are Nagata. \\begin{enumerate} \\item Any scheme locally of finite type over a field. \\item Any scheme locally of finite type over a Noetherian complete local ring. \\item Any scheme locally of finite type over $\\mathbf{Z}$. \\item Any scheme locally of finite type over a Dedekind ring of characteristic zero. \\item And so on. \\end{enumerate}"}
+{"_id": "5220", "title": "morphisms-lemma-J", "text": "Let $X$ be a locally Noetherian scheme. The following are equivalent \\begin{enumerate} \\item $X$ is J-2, \\item there exists an open covering of $X$ all of whose members are J-2 schemes, \\item for every affine open $\\Spec(R) = U \\subset X$ the ring $R$ is J-2, and \\item there exists an affine open covering $S = \\bigcup U_i$ such that each $\\mathcal{O}(U_i)$ is J-2 for all $i$. \\end{enumerate} Moreover, in this case any scheme locally of finite type over $X$ is J-2 as well."}
+{"_id": "5222", "title": "morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Set $s = f(x)$. If $\\kappa(x)/\\kappa(s)$ is an algebraic field extension, then \\begin{enumerate} \\item $x$ is a closed point of its fibre, and \\item if in addition $s$ is a closed point of $S$, then $x$ is a closed point of $X$. \\end{enumerate}"}
+{"_id": "5223", "title": "morphisms-lemma-closed-point-fibre-locally-finite-type", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Set $s = f(x)$. Assume $f$ is locally of finite type. Then $x$ is a closed point of its fibre if and only if $\\kappa(s) \\subset \\kappa(x)$ is a finite field extension."}
+{"_id": "5224", "title": "morphisms-lemma-base-change-closed-point-fibre-locally-finite-type", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $g : S' \\to S$ be any morphism. Denote $f' : X' \\to S'$ the base change. If $x' \\in X'$ maps to a point $x \\in X$ which is closed in $X_{f(x)}$ then $x'$ is closed in $X'_{f'(x')}$."}
+{"_id": "5225", "title": "morphisms-lemma-residue-field-quasi-finite", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Set $s = f(x)$. If $f$ is quasi-finite at $x$, then the residue field extension $\\kappa(s) \\subset \\kappa(x)$ is finite."}
+{"_id": "5226", "title": "morphisms-lemma-quasi-finite-at-point-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Set $s = f(x)$. Let $X_s$ be the fibre of $f$ at $s$. Assume $f$ is locally of finite type. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is quasi-finite at $x$. \\item The point $x$ is isolated in $X_s$. \\item The point $x$ is closed in $X_s$ and there is no point $x' \\in X_s$, $x' \\not = x$ which specializes to $x$. \\item For any pair of affine opens $\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ and $x \\in U$ corresponding to $\\mathfrak q \\subset A$ the ring map $R \\to A$ is quasi-finite at $\\mathfrak q$. \\end{enumerate}"}
+{"_id": "5227", "title": "morphisms-lemma-finite-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, and \\item $f^{-1}(\\{s\\})$ is a finite set. \\end{enumerate} Then $X_s$ is a finite discrete topological space, and $f$ is quasi-finite at each point of $X$ lying over $s$."}
+{"_id": "5228", "title": "morphisms-lemma-locally-quasi-finite-fibres", "text": "\\begin{slogan} Finite type morphisms with discrete fibers are quasi-finite. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Then the following are equivalent \\begin{enumerate} \\item $f$ is locally quasi-finite, \\item for every $s \\in S$ the fibre $X_s$ is a discrete topological space, and \\item for every morphism $\\Spec(k) \\to S$ where $k$ is a field the base change $X_k$ has an underlying discrete topological space. \\end{enumerate}"}
+{"_id": "5229", "title": "morphisms-lemma-quasi-finite-locally-quasi-compact", "text": "Let $f : X \\to S$ be a morphism of schemes. Then $f$ is quasi-finite if and only if $f$ is locally quasi-finite and quasi-compact."}
+{"_id": "5230", "title": "morphisms-lemma-quasi-finite", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is quasi-finite, and \\item $f$ is locally of finite type, quasi-compact, and has finite fibres. \\end{enumerate}"}
+{"_id": "5231", "title": "morphisms-lemma-locally-quasi-finite-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is locally quasi-finite. \\item For every pair of affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is quasi-finite. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is locally quasi-finite. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is quasi-finite, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is locally quasi-finite then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is locally quasi-finite."}
+{"_id": "5232", "title": "morphisms-lemma-composition-quasi-finite", "text": "The composition of two morphisms which are locally quasi-finite is locally quasi-finite. The same is true for quasi-finite morphisms."}
+{"_id": "5233", "title": "morphisms-lemma-base-change-quasi-finite", "text": "\\begin{slogan} (Locally) quasi-finite morphisms are stable under base change. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Let $g : S' \\to S$ be a morphism of schemes. Denote $f' : X' \\to S'$ the base change of $f$ by $g$ and denote $g' : X' \\to X$ the projection. Assume $X$ is locally of finite type over $S$. \\begin{enumerate} \\item Let $U \\subset X$ (resp.\\ $U' \\subset X'$) be the set of points where $f$ (resp.\\ $f'$) is quasi-finite. Then $U' = U \\times_S S' = (g')^{-1}(U)$. \\item The base change of a locally quasi-finite morphism is locally quasi-finite. \\item The base change of a quasi-finite morphism is quasi-finite. \\end{enumerate}"}
+{"_id": "5234", "title": "morphisms-lemma-quasi-finite-at-a-finite-number-of-points", "text": "Let $f : X \\to S$ be a morphism of schemes of finite type. Let $s \\in S$. There are at most finitely many points of $X$ lying over $s$ at which $f$ is quasi-finite."}
+{"_id": "5235", "title": "morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite."}
+{"_id": "5236", "title": "morphisms-lemma-immersion-locally-quasi-finite", "text": "Any immersion is locally quasi-finite."}
+{"_id": "5237", "title": "morphisms-lemma-permanence-quasi-finite", "text": "Let $X \\to Y$ be a morphism of schemes over a base scheme $S$. Let $x \\in X$. If $X \\to S$ is quasi-finite at $x$, then $X \\to Y$ is quasi-finite at $x$. If $X$ is locally quasi-finite over $S$, then $X \\to Y$ is locally quasi-finite."}
+{"_id": "5238", "title": "morphisms-lemma-locally-finite-presentation-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is locally of finite presentation. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is locally of finite presentation. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is of finite presentation, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is locally of finite presentation then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is locally of finite presentation."}
+{"_id": "5239", "title": "morphisms-lemma-composition-finite-presentation", "text": "The composition of two morphisms which are locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation."}
+{"_id": "5240", "title": "morphisms-lemma-base-change-finite-presentation", "text": "The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation."}
+{"_id": "5241", "title": "morphisms-lemma-open-immersion-locally-finite-presentation", "text": "Any open immersion is locally of finite presentation."}
+{"_id": "5242", "title": "morphisms-lemma-quasi-compact-open-immersion-finite-presentation", "text": "Any open immersion is of finite presentation if and only if it is quasi-compact."}
+{"_id": "5243", "title": "morphisms-lemma-closed-immersion-finite-presentation", "text": "\\begin{slogan} Closed immersions of finite presentation correspond to quasi-coherent sheaves of ideals of finite type. \\end{slogan} A closed immersion $i : Z \\to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$ is of finite type (as an $\\mathcal{O}_X$-module)."}
+{"_id": "5244", "title": "morphisms-lemma-finite-presentation-finite-type", "text": "A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type."}
+{"_id": "5245", "title": "morphisms-lemma-noetherian-finite-type-finite-presentation", "text": "\\begin{slogan} Over a locally Noetherian base, finite type is finite presentation. \\end{slogan} Let $f : X \\to S$ be a morphism. \\begin{enumerate} \\item If $S$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation. \\item If $S$ is locally Noetherian and $f$ of finite type then $f$ is of finite presentation. \\end{enumerate}"}
+{"_id": "5246", "title": "morphisms-lemma-finite-presentation-quasi-compact-quasi-separated", "text": "Let $S$ be a scheme which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $S$, then $X$ is quasi-compact and quasi-separated."}
+{"_id": "5247", "title": "morphisms-lemma-finite-presentation-permanence", "text": "Let $f : X \\to Y$ be a morphism of schemes over $S$. \\begin{enumerate} \\item If $X$ is locally of finite presentation over $S$ and $Y$ is locally of finite type over $S$, then $f$ is locally of finite presentation. \\item If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated and locally of finite type over $S$, then $f$ is of finite presentation. \\end{enumerate}"}
+{"_id": "5248", "title": "morphisms-lemma-diagonal-morphism-finite-type", "text": "Let $f : X \\to Y$ be a morphism of schemes with diagonal $\\Delta : X \\to X \\times_Y X$. If $f$ is locally of finite type then $\\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\\Delta$ is of finite presentation."}
+{"_id": "5249", "title": "morphisms-lemma-inverse-image-constructible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $E \\subset Y$ be a subset. If $E$ is (locally) constructible in $Y$, then $f^{-1}(E)$ is (locally) constructible in $X$."}
+{"_id": "5250", "title": "morphisms-lemma-chevalley", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is quasi-compact and locally of finite presentation, and \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then the image of every constructible subset of $X$ is constructible in $Y$."}
+{"_id": "5251", "title": "morphisms-lemma-constructible-containing-open", "text": "Let $X$ be a scheme. Let $x \\in X$. Let $E \\subset X$ be a locally constructible subset. If $\\{x' \\mid x' \\leadsto x\\} \\subset E$, then $E$ contains an open neighbourhood of $x$."}
+{"_id": "5252", "title": "morphisms-lemma-locally-finite-presentation-universally-open", "text": "Let $f : X \\to S$ be a morphism. \\begin{enumerate} \\item If $f$ is locally of finite presentation and generalizations lift along $f$, then $f$ is open. \\item If $f$ is locally of finite presentation and generalizations lift along every base change of $f$, then $f$ is universally open. \\end{enumerate}"}
+{"_id": "5253", "title": "morphisms-lemma-composition-open", "text": "A composition of (universally) open morphisms is (universally) open."}
+{"_id": "5254", "title": "morphisms-lemma-scheme-over-field-universally-open", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. The structure morphism $X \\to \\Spec(k)$ is universally open."}
+{"_id": "5255", "title": "morphisms-lemma-open-generizing", "text": "\\begin{reference} Follows from the implication (a) $\\Rightarrow$ (b) in \\cite[IV, Corollary 1.10.4]{EGA} \\end{reference} Let $\\varphi : X \\to Y$ be a morphism of schemes. If $\\varphi$ is open, then $\\varphi$ is generizing (i.e., generalizations lift along $\\varphi$). If $\\varphi$ is universally open, then $\\varphi$ is universally generizing."}
+{"_id": "5257", "title": "morphisms-lemma-base-change-universally-submersive", "text": "The base change of a universally submersive morphism of schemes by any morphism of schemes is universally submersive."}
+{"_id": "5258", "title": "morphisms-lemma-composition-universally-submersive", "text": "The composition of a pair of (universally) submersive morphisms of schemes is (universally) submersive."}
+{"_id": "5259", "title": "morphisms-lemma-flat-module-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. The following are equivalent \\begin{enumerate} \\item The sheaf $\\mathcal{F}$ is flat over $S$. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the $\\mathcal{O}_S(V)$-module $\\mathcal{F}(U)$ is flat. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the modules $\\mathcal{F}|_{U_i}$ is flat over $V_j$, for all $j\\in J, i\\in I_j$. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that $\\mathcal{F}(U_i)$ is a flat $\\mathcal{O}_S(V_j)$-module, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $\\mathcal{F}$ is flat over $S$ then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $\\mathcal{F}|_U$ is flat over $V$."}
+{"_id": "5260", "title": "morphisms-lemma-flat-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is flat. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is flat. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is flat. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is flat, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is flat then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is flat."}
+{"_id": "5261", "title": "morphisms-lemma-pushforward-flat-affine", "text": "Let $f : X \\to Y$ be an affine morphism of schemes over a base scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is flat over $S$ if and only if $f_*\\mathcal{F}$ is flat over $S$."}
+{"_id": "5262", "title": "morphisms-lemma-composition-module-flat", "text": "Let $X \\to Y \\to Z$ be morphisms of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $y$ in $Y$. If $\\mathcal{F}$ is flat over $Y$ at $x$, and $Y$ is flat over $Z$ at $y$, then $\\mathcal{F}$ is flat over $Z$ at $x$."}
+{"_id": "5263", "title": "morphisms-lemma-composition-flat", "text": "The composition of flat morphisms is flat."}
+{"_id": "5264", "title": "morphisms-lemma-base-change-module-flat", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. Let $g : S' \\to S$ be a morphism of schemes. Denote $g' : X' = X_{S'} \\to X$ the projection. Let $x' \\in X'$ be a point with image $x = g'(x') \\in X$. If $\\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$. In particular, if $\\mathcal{F}$ is flat over $S$, then $(g')^*\\mathcal{F}$ is flat over $S'$."}
+{"_id": "5265", "title": "morphisms-lemma-base-change-flat", "text": "The base change of a flat morphism is flat."}
+{"_id": "5266", "title": "morphisms-lemma-generalizations-lift-flat", "text": "Let $f : X \\to S$ be a flat morphism of schemes. Then generalizations lift along $f$, see Topology, Definition \\ref{topology-definition-lift-specializations}."}
+{"_id": "5267", "title": "morphisms-lemma-fppf-open", "text": "A flat morphism locally of finite presentation is universally open."}
+{"_id": "5268", "title": "morphisms-lemma-pf-flat-module-open", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ locally finite presentation, $\\mathcal{F}$ of finite type, $X = \\text{Supp}(\\mathcal{F})$, and $\\mathcal{F}$ flat over $Y$. Then $f$ is universally open."}
+{"_id": "5269", "title": "morphisms-lemma-fpqc-quotient-topology", "text": "\\begin{reference} \\cite[Expose VIII, Corollaire 4.3]{SGA1} and \\cite[IV, Corollaire 2.3.12]{EGA} \\end{reference} Let $f : X \\to Y$ be a quasi-compact, surjective, flat morphism. A subset $T \\subset Y$ is open (resp.\\ closed) if and only $f^{-1}(T)$ is open (resp.\\ closed). In other words, $f$ is a submersive morphism."}
+{"_id": "5270", "title": "morphisms-lemma-flat-permanence", "text": "Let $h : X \\to Y$ be a morphism of schemes over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \\in X$ with $y = h(x) \\in Y$. If $h$ is flat at $x$, then $$ \\mathcal{G}\\text{ flat over }S\\text{ at }y \\Leftrightarrow h^*\\mathcal{G}\\text{ flat over }S\\text{ at }x. $$ In particular: If $h$ is surjective and flat, then $\\mathcal{G}$ is flat over $S$, if and only if $h^*\\mathcal{G}$ is flat over $S$. If $h$ is surjective and flat, and $X$ is flat over $S$, then $Y$ is flat over $S$."}
+{"_id": "5271", "title": "morphisms-lemma-flat-pullback-support", "text": "Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module with scheme theoretic support $Z \\subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\\mathcal{F}$."}
+{"_id": "5272", "title": "morphisms-lemma-flat-morphism-scheme-theoretically-dense-open", "text": "Let $f : X \\to Y$ be a flat morphism of schemes. Let $V \\subset Y$ be a retrocompact open which is scheme theoretically dense. Then $f^{-1}V$ is scheme theoretically dense in $X$."}
+{"_id": "5273", "title": "morphisms-lemma-flat-base-change-scheme-theoretic-image", "text": "\\begin{slogan} Taking scheme theoretic images commutes with flat base change in the quasi-compact case \\end{slogan} Let $f : X \\to Y$ be a flat morphism of schemes. Let $g : V \\to Y$ be a quasi-compact morphism of schemes. Let $Z \\subset Y$ be the scheme theoretic image of $g$ and let $Z' \\subset X$ be the scheme theoretic image of the base change $V \\times_Y X \\to X$. Then $Z' = f^{-1}Z$."}
+{"_id": "5274", "title": "morphisms-lemma-characterize-flat-closed-immersions", "text": "Let $X$ be a scheme. The rule which associates to a closed subscheme of $X$ its underlying closed subset defines a bijection $$ \\left\\{ \\begin{matrix} \\text{closed subschemes }Z \\subset X \\\\ \\text{such that }Z \\to X\\text{ is flat} \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{closed subsets }Z \\subset X \\\\ \\text{closed under generalizations} \\end{matrix} \\right\\} $$ If $Z \\subset X$ is such a closed subscheme, every morphism of schemes $g : Y \\to X$ with $g(Y) \\subset Z$ set theoretically factors (scheme theoretically) through $Z$."}
+{"_id": "5275", "title": "morphisms-lemma-flat-closed-immersions-finite-presentation", "text": "A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme."}
+{"_id": "5277", "title": "morphisms-lemma-dimension-fibre-at-a-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ and set $s = f(x)$. Assume $f$ is locally of finite type. Then $$ \\dim_x(X_s) = \\dim(\\mathcal{O}_{X_s, x}) + \\text{trdeg}_{\\kappa(s)}(\\kappa(x)). $$"}
+{"_id": "5278", "title": "morphisms-lemma-dimension-fibre-at-a-point-additive", "text": "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes. Let $x \\in X$ and set $y = f(x)$, $s = g(y)$. Assume $f$ and $g$ locally of finite type. Then $$ \\dim_x(X_s) \\leq \\dim_x(X_y) + \\dim_y(Y_s). $$ Moreover, equality holds if $\\mathcal{O}_{X_s, x}$ is flat over $\\mathcal{O}_{Y_s, y}$, which holds for example if $\\mathcal{O}_{X, x}$ is flat over $\\mathcal{O}_{Y, y}$."}
+{"_id": "5279", "title": "morphisms-lemma-dimension-fibre-after-base-change", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a fibre product diagram of schemes. Assume $f$ locally of finite type. Suppose that $x' \\in X'$, $x = g'(x')$, $s' = f'(x')$ and $s = g(s') = f(x)$. Then \\begin{enumerate} \\item $\\dim_x(X_s) = \\dim_{x'}(X'_{s'})$, \\item if $F$ is the fibre of the morphism $X'_{s'} \\to X_s$ over $x$, then $$ \\dim(\\mathcal{O}_{F, x'}) = \\dim(\\mathcal{O}_{X'_{s'}, x'}) - \\dim(\\mathcal{O}_{X_s, x}) = \\text{trdeg}_{\\kappa(s)}(\\kappa(x)) - \\text{trdeg}_{\\kappa(s')}(\\kappa(x')) $$ In particular $\\dim(\\mathcal{O}_{X'_{s'}, x'}) \\geq \\dim(\\mathcal{O}_{X_s, x})$ and $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) \\geq \\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$. \\item given $s', s, x$ there exists a choice of $x'$ such that $\\dim(\\mathcal{O}_{X'_{s'}, x'}) = \\dim(\\mathcal{O}_{X_s, x})$ and $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$. \\end{enumerate}"}
+{"_id": "5280", "title": "morphisms-lemma-openness-bounded-dimension-fibres", "text": "\\begin{reference} \\cite[IV Theorem 13.1.3]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Let $n \\geq 0$. Assume $f$ is locally of finite type. The set $$ U_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\} $$ is open in $X$."}
+{"_id": "5281", "title": "morphisms-lemma-morphism-finite-type-bounded-dimension", "text": "Let $f : X \\to Y$ be a morphism of finite type with $Y$ quasi-compact. Then the dimension of the fibres of $f$ is bounded."}
+{"_id": "5282", "title": "morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $n \\geq 0$. Assume $f$ is locally of finite presentation. The open $$ U_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\} $$ of Lemma \\ref{lemma-openness-bounded-dimension-fibres} is retrocompact in $X$. (See Topology, Definition \\ref{topology-definition-quasi-compact}.)"}
+{"_id": "5283", "title": "morphisms-lemma-dimension-fibre-specialization", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\leadsto x'$ be a nontrivial specialization of points in $X$ lying over the same point $s \\in S$. Assume $f$ is locally of finite type. Then \\begin{enumerate} \\item $\\dim_x(X_s) \\leq \\dim_{x'}(X_s)$, \\item $\\dim(\\mathcal{O}_{X_s, x}) < \\dim(\\mathcal{O}_{X_s, x'})$, and \\item $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) > \\text{trdeg}_{\\kappa(s)}(\\kappa(x'))$. \\end{enumerate}"}
+{"_id": "5284", "title": "morphisms-lemma-base-change-relative-dimension-d", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. If $f$ has relative dimension $d$, then so does any base change of $f$. Same for relative dimension $\\leq d$."}
+{"_id": "5285", "title": "morphisms-lemma-composition-relative-dimension-d", "text": "Let $f : X \\to Y$, $g : Y \\to Z$ be locally of finite type. If $f$ has relative dimension $\\leq d$ and $g$ has relative dimension $\\leq e$ then $g \\circ f$ has relative dimension $\\leq d + e$. If \\begin{enumerate} \\item $f$ has relative dimension $d$, \\item $g$ has relative dimension $e$, and \\item $f$ is flat, \\end{enumerate} then $g \\circ f$ has relative dimension $d + e$."}
+{"_id": "5286", "title": "morphisms-lemma-flat-finite-presentation-CM-fibres-relative-dimension", "text": "\\begin{slogan} Cohen-Macaulay morphisms decompose into clopens of pure relative dimension \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume that \\begin{enumerate} \\item $f$ is flat, \\item $f$ is locally of finite presentation, and \\item for all $s \\in S$ the fibre $X_s$ is Cohen-Macaulay (Properties, Definition \\ref{properties-definition-Cohen-Macaulay}) \\end{enumerate} Then there exist open and closed subschemes $X_d \\subset X$ such that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to S$ has relative dimension $d$."}
+{"_id": "5287", "title": "morphisms-lemma-locally-quasi-finite-rel-dimension-0", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Let $x \\in X$ with $s = f(x)$. Then $f$ is quasi-finite at $x$ if and only if $\\dim_x(X_s) = 0$. In particular, $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$."}
+{"_id": "5288", "title": "morphisms-lemma-rel-dimension-dimension", "text": "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes which is flat, locally of finite type and of relative dimension $d$. For every point $x$ in $X$ with image $y$ in $Y$ we have $\\dim_x(X) = \\dim_y(Y) + d$."}
+{"_id": "5289", "title": "morphisms-lemma-syntomic-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is syntomic. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is syntomic. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is syntomic. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is syntomic, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is syntomic then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is syntomic."}
+{"_id": "5290", "title": "morphisms-lemma-composition-syntomic", "text": "The composition of two morphisms which are syntomic is syntomic."}
+{"_id": "5291", "title": "morphisms-lemma-base-change-syntomic", "text": "The base change of a morphism which is syntomic is syntomic."}
+{"_id": "5292", "title": "morphisms-lemma-open-immersion-syntomic", "text": "Any open immersion is syntomic."}
+{"_id": "5293", "title": "morphisms-lemma-syntomic-locally-finite-presentation", "text": "A syntomic morphism is locally of finite presentation."}
+{"_id": "5294", "title": "morphisms-lemma-syntomic-flat", "text": "A syntomic morphism is flat."}
+{"_id": "5295", "title": "morphisms-lemma-syntomic-open", "text": "A syntomic morphism is universally open."}
+{"_id": "5296", "title": "morphisms-lemma-local-complete-intersection", "text": "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. The following are equivalent: \\begin{enumerate} \\item $X$ is a local complete intersection over $k$, \\item for every $x \\in X$ there exists an affine open $U = \\Spec(R) \\subset X$ neighbourhood of $x$ such that $R \\cong k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a global complete intersection over $k$, and \\item for every $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is a complete intersection over $k$. \\end{enumerate}"}
+{"_id": "5297", "title": "morphisms-lemma-syntomic-locally-standard-syntomic", "text": "Let $f : X  \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s = f(x)$. Let $V \\subset S$ be an affine open neighbourhood of $s$. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is syntomic at $x$. \\item There exist an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ such that $f|_U : U \\to V$ is standard syntomic. \\item The morphism $f$ is of finite presentation at $x$, the local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and $\\mathcal{O}_{X, x}/\\mathfrak m_s \\mathcal{O}_{X, x}$ is a complete intersection over $\\kappa(s)$ (see Algebra, Definition \\ref{algebra-definition-lci-local-ring}). \\end{enumerate}"}
+{"_id": "5298", "title": "morphisms-lemma-syntomic-flat-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and all fibres $X_s$ are local complete intersections, then $f$ is syntomic."}
+{"_id": "5299", "title": "morphisms-lemma-set-points-where-fibres-lci", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the set $$ T = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x} \\text{ is a complete intersection over }\\kappa(f(x))\\} $$ commutes with arbitrary base change: For any morphism $g : S' \\to S$, consider the base change $f' : X' \\to S'$ of $f$ and the projection $g' : X' \\to X$. Then the corresponding set $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$. In particular, if $f$ is assumed flat, and locally of finite presentation then the same holds for the open set of points where $f$ is syntomic."}
+{"_id": "5300", "title": "morphisms-lemma-standard-syntomic-relative-dimension", "text": "Let $R$ be a ring. Let $R \\to A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection. Set $S = \\Spec(R)$ and $X = \\Spec(A)$. Consider the morphism $f : X \\to S$ associated to the ring map $R \\to A$. The function $x \\mapsto \\dim_x(X_{f(x)})$ is constant with value $n - c$."}
+{"_id": "5301", "title": "morphisms-lemma-syntomic-relative-dimension", "text": "Let $f : X \\to S$ be a syntomic morphism. The function $x \\mapsto \\dim_x(X_{f(x)})$ is locally constant on $X$."}
+{"_id": "5302", "title": "morphisms-lemma-syntomic-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective and syntomic, \\item $p$ is syntomic, and \\item $q$ is locally of finite presentation\\footnote{In fact, if $f$ is surjective, flat, and of finite presentation and $p$ is syntomic, then both $q$ and $f$ are syntomic, see Descent, Lemma \\ref{descent-lemma-syntomic-permanence}.}. \\end{enumerate} Then $q$ is syntomic."}
+{"_id": "5303", "title": "morphisms-lemma-affine-conormal", "text": "Let $i : Z \\to X$ be an immersion. The conormal sheaf of $i$ has the following properties: \\begin{enumerate} \\item Let $U \\subset X$ be any open subscheme such that $i$ factors as $Z \\xrightarrow{i'} U \\to X$ where $i'$ is a closed immersion. Let $\\mathcal{I} = \\Ker((i')^\\sharp) \\subset \\mathcal{O}_U$. Then $$ \\mathcal{C}_{Z/X} = (i')^*\\mathcal{I}\\quad\\text{and}\\quad i'_*\\mathcal{C}_{Z/X} = \\mathcal{I}/\\mathcal{I}^2 $$ \\item For any affine open $\\Spec(R) = U \\subset X$ such that $Z \\cap U = \\Spec(R/I)$ there is a canonical isomorphism $\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X}) = I/I^2$. \\end{enumerate}"}
+{"_id": "5304", "title": "morphisms-lemma-conormal-functorial", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of $\\mathcal{O}_Z$-modules $$ f^*\\mathcal{C}_{Z'/X'} \\longrightarrow \\mathcal{C}_{Z/X} $$ characterized by the following property: For every pair of affine opens $(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with $f(U) \\subset U'$ such that $Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$ the induced map $$ \\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X'}) = I'/I'^2 \\longrightarrow I/I^2 = \\Gamma(Z \\cap U, \\mathcal{C}_{Z/X}) $$ is the one induced by the ring map $f^\\sharp : R' \\to R$ which has the property $f^\\sharp(I') \\subset I$."}
+{"_id": "5305", "title": "morphisms-lemma-conormal-functorial-flat", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a fibre product diagram in the category of schemes with $i$, $i'$ immersions. Then the canonical map $f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of Lemma \\ref{lemma-conormal-functorial} is surjective. If $g$ is flat, then it is an isomorphism."}
+{"_id": "5306", "title": "morphisms-lemma-transitivity-conormal", "text": "Let $Z \\to Y \\to X$ be immersions of schemes. Then there is a canonical exact sequence $$ i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ where the maps come from Lemma \\ref{lemma-conormal-functorial} and $i : Z \\to Y$ is the first morphism."}
+{"_id": "5307", "title": "morphisms-lemma-universal-derivation-universal", "text": "Let $f : X \\to S$ be a morphism of schemes. The map $$ \\Hom_{\\mathcal{O}_X}(\\Omega_{X/S}, \\mathcal{F}) \\longrightarrow \\text{Der}_S(\\mathcal{O}_X, \\mathcal{F}),\\quad \\alpha \\longmapsto \\alpha \\circ \\text{d}_{X/S} $$ is an isomorphism of functors $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Sets}$."}
+{"_id": "5308", "title": "morphisms-lemma-differentials-restrict-open", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$, $V \\subset S$ be open subschemes such that $f(U) \\subset V$. Then there is a unique isomorphism $\\Omega_{X/S}|_U = \\Omega_{U/V}$ of $\\mathcal{O}_U$-modules such that $\\text{d}_{X/S}|_U = \\text{d}_{U/V}$."}
+{"_id": "5309", "title": "morphisms-lemma-affine-case-derivation", "text": "Let $R \\to A$ be a ring map. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules on $X = \\Spec(A)$. Set $S = \\Spec(R)$. The rule which associates to an $S$-derivation on $\\mathcal{F}$ its action on global sections defines a bijection between the set of $S$-derivations of $\\mathcal{F}$ and the set of $R$-derivations on $M = \\Gamma(X, \\mathcal{F})$."}
+{"_id": "5310", "title": "morphisms-lemma-differentials-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. For any pair of affine opens $\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ there is a unique isomorphism $$ \\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}. $$ compatible with $\\text{d}_{X/S}$ and $\\text{d} : A \\to \\Omega_{A/R}$."}
+{"_id": "5311", "title": "morphisms-lemma-differentials-diagonal", "text": "Let $f : X \\to S$ be a morphism of schemes. There is a canonical isomorphism between $\\Omega_{X/S}$ and the conormal sheaf of the diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$."}
+{"_id": "5312", "title": "morphisms-lemma-functoriality-differentials", "text": "Let $$ \\xymatrix{ X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\ S' \\ar[r] & S } $$ be a commutative diagram of schemes. The canonical map $\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map $f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a $S$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules $\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'}. $$ It is uniquely characterized by the property that $f^*\\text{d}_{X/S}(h)$ maps to $\\text{d}_{X'/S'}(f^* h)$ for any local section $h$ of $\\mathcal{O}_X$."}
+{"_id": "5313", "title": "morphisms-lemma-triangle-differentials", "text": "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes. Then there is a canonical exact sequence $$ f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0 $$ where the maps come from applications of Lemma \\ref{lemma-functoriality-differentials}."}
+{"_id": "5314", "title": "morphisms-lemma-base-change-differentials", "text": "Let $X \\to S$ be a morphism of schemes. Let $g : S' \\to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \\to X$ the projection. Then the map $$ (g')^*\\Omega_{X/S} \\to \\Omega_{X'/S'} $$ of Lemma \\ref{lemma-functoriality-differentials} is an isomorphism."}
+{"_id": "5315", "title": "morphisms-lemma-differential-product", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. Let $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ be the projection morphisms. The maps from Lemma \\ref{lemma-functoriality-differentials} $$ p^*\\Omega_{X/S} \\oplus q^*\\Omega_{Y/S} \\longrightarrow \\Omega_{X \\times_S Y/S} $$ give an isomorphism."}
+{"_id": "5316", "title": "morphisms-lemma-finite-type-differentials", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is locally of finite type, then $\\Omega_{X/S}$ is a finite type $\\mathcal{O}_X$-module."}
+{"_id": "5317", "title": "morphisms-lemma-finite-presentation-differentials", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\\Omega_{X/S}$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "5318", "title": "morphisms-lemma-immersion-differentials", "text": "If $X \\to S$ is an immersion, or more generally a monomorphism, then $\\Omega_{X/S}$ is zero."}
+{"_id": "5319", "title": "morphisms-lemma-differentials-relative-immersion", "text": "Let $i : Z \\to X$ be an immersion of schemes over $S$. There is a canonical exact sequence $$ \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0 $$ where the first arrow is induced by $\\text{d}_{X/S}$ and the second arrow comes from Lemma \\ref{lemma-functoriality-differentials}."}
+{"_id": "5321", "title": "morphisms-lemma-two-immersions", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are immersions. Then there is a canonical exact sequence $$ \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0 $$ where the first arrow comes from Lemma \\ref{lemma-conormal-functorial} and the second from Lemma \\ref{lemma-differentials-relative-immersion}."}
+{"_id": "5322", "title": "morphisms-lemma-affine-case-differential-operators", "text": "Let $R \\to A$ be a ring map. Denote $f : X \\to S$ the corresponding morphism of affine schemes. Let $\\mathcal{F}$ and $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is quasi-coherent then the map $$ \\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G}) \\to \\text{Diff}^k_{A/R}(\\Gamma(X, \\mathcal{F}), \\Gamma(X, \\mathcal{G})) $$ sending a differential operator to its action on global sections is bijective."}
+{"_id": "5323", "title": "morphisms-lemma-base-change-differential-operators", "text": "Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes. Let $\\mathcal{F}$ and $\\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules. Let $D : \\mathcal{F} \\to \\mathcal{F}'$ be a differential operator of order $k$ on $X/S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Then there is a unique differential operator $$ D' : \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}} \\text{pr}_2^*\\mathcal{G} \\longrightarrow \\text{pr}_1^*\\mathcal{F}' \\otimes_{\\mathcal{O}_{X \\times_S Y}} \\text{pr}_2^*\\mathcal{G} $$ of order $k$ on $X \\times_S Y / Y$ such that $ D'(s \\otimes t) = D(s) \\otimes t $ for local sections $s$ of $\\mathcal{F}$ and $t$ of $\\mathcal{G}$."}
+{"_id": "5324", "title": "morphisms-lemma-smooth-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is smooth. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is smooth. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is smooth. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is smooth, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is smooth then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is smooth."}
+{"_id": "5325", "title": "morphisms-lemma-smooth-flat-smooth-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and all fibres $X_s$ are smooth, then $f$ is smooth."}
+{"_id": "5326", "title": "morphisms-lemma-composition-smooth", "text": "The composition of two morphisms which are smooth is smooth."}
+{"_id": "5327", "title": "morphisms-lemma-base-change-smooth", "text": "The base change of a morphism which is smooth is smooth."}
+{"_id": "5328", "title": "morphisms-lemma-open-immersion-smooth", "text": "Any open immersion is smooth."}
+{"_id": "5329", "title": "morphisms-lemma-smooth-syntomic", "text": "A smooth morphism is syntomic."}
+{"_id": "5330", "title": "morphisms-lemma-smooth-locally-finite-presentation", "text": "A smooth morphism is locally of finite presentation."}
+{"_id": "5331", "title": "morphisms-lemma-smooth-flat", "text": "A smooth morphism is flat."}
+{"_id": "5332", "title": "morphisms-lemma-smooth-open", "text": "A smooth morphism is universally open."}
+{"_id": "5333", "title": "morphisms-lemma-smooth-locally-standard-smooth", "text": "\\begin{slogan} Smooth morphisms are locally standard smooth. \\end{slogan} Let $f : X  \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Let $V \\subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is smooth at $x$. \\item There exist an affine open $U \\subset X$, with $x \\in U$ and $f(U) \\subset V$ such that the induced morphism $f|_U : U \\to V$ is standard smooth. \\end{enumerate}"}
+{"_id": "5334", "title": "morphisms-lemma-smooth-omega-finite-locally-free", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is smooth. Then the module of differentials $\\Omega_{X/S}$ of $X$ over $S$ is finite locally free and $$ \\text{rank}_x(\\Omega_{X/S}) = \\dim_x(X_{f(x)}) $$ for every $x \\in X$."}
+{"_id": "5335", "title": "morphisms-lemma-smooth-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is smooth at $x$. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and $X_s \\to \\Spec(\\kappa(s))$ is smooth at $x$. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$ can be generated by at most $\\dim_x(X_{f(x)})$ elements. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and the $\\kappa(x)$-vector space $$ \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ can be generated by at most $\\dim_x(X_{f(x)})$ elements. \\item There exist affine opens $U \\subset X$, and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the induced morphism $f|_U : U \\to V$ is standard smooth. \\item There exist affine opens $\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding to $\\mathfrak q \\subset A$, and $f(U) \\subset V$ such that there exists a presentation $$ A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) $$ with $$ g = \\det \\left( \\begin{matrix} \\partial f_1/\\partial x_1 & \\partial f_2/\\partial x_1 & \\ldots & \\partial f_c/\\partial x_1 \\\\ \\partial f_1/\\partial x_2 & \\partial f_2/\\partial x_2 & \\ldots & \\partial f_c/\\partial x_2 \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ \\partial f_1/\\partial x_c & \\partial f_2/\\partial x_c & \\ldots & \\partial f_c/\\partial x_c \\end{matrix} \\right) $$ mapping to an element of $A$ not in $\\mathfrak q$. \\end{enumerate}"}
+{"_id": "5336", "title": "morphisms-lemma-set-points-where-fibres-smooth", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$ be the open subscheme of points where $f$, resp.\\ $f'$ is smooth. Then $W' = (g')^{-1}(W)$ if \\begin{enumerate} \\item $f$ is flat and locally of finite presentation, or \\item $f$ is locally of finite presentation and $g$ is flat. \\end{enumerate}"}
+{"_id": "5337", "title": "morphisms-lemma-triangle-differentials-smooth", "text": "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes. Assume $f$ is smooth. Then $$ 0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0 $$ (see Lemma \\ref{lemma-triangle-differentials}) is short exact."}
+{"_id": "5340", "title": "morphisms-lemma-smooth-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective, and smooth, \\item $p$ is smooth, and \\item $q$ is locally of finite presentation\\footnote{In fact this is implied by (1) and (2), see Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}. Moreover, it suffices to assume $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma \\ref{descent-lemma-smooth-permanence}.}. \\end{enumerate} Then $q$ is smooth."}
+{"_id": "5341", "title": "morphisms-lemma-section-smooth-morphism", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\sigma : S \\to X$ be a section of $f$. Let $s \\in S$ be a point such that $f$ is smooth at $x = \\sigma(s)$. Then there exist affine open neighbourhoods $\\Spec(A) = U \\subset S$ of $s$ and $\\Spec(B) = V \\subset X$ of $x$ such that \\begin{enumerate} \\item $f(V) \\subset U$ and $\\sigma(U) \\subset V$, \\item with $I = \\Ker(\\sigma^\\# : B \\to A)$ the module $I/I^2$ is a free $A$-module, and \\item $B^\\wedge \\cong A[[x_1, \\ldots, x_d]]$ as $A$-algebras where $B^\\wedge$ denotes the completion of $B$ with respect to $I$. \\end{enumerate}"}
+{"_id": "5342", "title": "morphisms-lemma-smoothness-dimension", "text": "Let $f : X \\to Y$ be a smooth morphism of locally Noetherian schemes. For every point $x$ in $X$ with image $y$ in $Y$, $$ \\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y), $$ where $X_y$ denotes the fiber over $y$."}
+{"_id": "5343", "title": "morphisms-lemma-unramified-omega-zero", "text": "Let $f : X \\to S$ be a morphism of schemes. Then \\begin{enumerate} \\item $f$ is unramified if and only if $f$ is locally of finite type and $\\Omega_{X/S} = 0$, and \\item $f$ is G-unramified if and only if $f$ is locally of finite presentation and $\\Omega_{X/S} = 0$. \\end{enumerate}"}
+{"_id": "5344", "title": "morphisms-lemma-unramified-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is unramified (resp.\\ G-unramified). \\item For every affine open $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is unramified (resp.\\ G-unramified). \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is unramified (resp.\\ G-unramified). \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is unramified (resp.\\ G-unramified), for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is unramified (resp.\\ G-unramified) then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is unramified (resp.\\ G-unramified)."}
+{"_id": "5345", "title": "morphisms-lemma-composition-unramified", "text": "The composition of two morphisms which are unramified is unramified. The same holds for G-unramified morphisms."}
+{"_id": "5346", "title": "morphisms-lemma-base-change-unramified", "text": "The base change of a morphism which is unramified is unramified. The same holds for G-unramified morphisms."}
+{"_id": "5348", "title": "morphisms-lemma-open-immersion-unramified", "text": "Any open immersion is G-unramified."}
+{"_id": "5349", "title": "morphisms-lemma-closed-immersion-unramified", "text": "A closed immersion $i : Z \\to X$ is unramified. It is G-unramified if and only if the associated quasi-coherent sheaf of ideals $\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$ is of finite type (as an $\\mathcal{O}_X$-module)."}
+{"_id": "5350", "title": "morphisms-lemma-unramified-locally-finite-type", "text": "An unramified morphism is locally of finite type. A G-unramified morphism is locally of finite presentation."}
+{"_id": "5351", "title": "morphisms-lemma-unramified-quasi-finite", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$. In particular, an unramified morphism is locally quasi-finite."}
+{"_id": "5352", "title": "morphisms-lemma-unramified-over-field", "text": "Fibres of unramified morphisms. \\begin{enumerate} \\item Let $X$ be a scheme over a field $k$. The structure morphism $X \\to \\Spec(k)$ is unramified if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$. \\item If $f : X \\to S$ is an unramified morphism then for every $s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\\kappa(s)$. \\end{enumerate}"}
+{"_id": "5353", "title": "morphisms-lemma-unramified-etale-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item If $f$ is unramified then for any $x \\in X$ the field extension $\\kappa(f(x)) \\subset \\kappa(x)$ is finite separable. \\item If $f$ is locally of finite type, and for every $s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\\kappa(s)$ then $f$ is unramified. \\item If $f$ is locally of finite presentation, and for every $s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\\kappa(s)$ then $f$ is G-unramified. \\end{enumerate}"}
+{"_id": "5354", "title": "morphisms-lemma-diagonal-unramified-morphism", "text": "Let $f : X \\to S$ be a morphism. \\begin{enumerate} \\item If $f$ is unramified, then the diagonal morphism $\\Delta : X \\to X \\times_S X$ is an open immersion. \\item If $f$ is locally of finite type and $\\Delta$ is an open immersion, then $f$ is unramified. \\item If $f$ is locally of finite presentation and $\\Delta$ is an open immersion, then $f$ is G-unramified. \\end{enumerate}"}
+{"_id": "5355", "title": "morphisms-lemma-unramified-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Set $s = f(x)$. Assume $f$ is locally of finite type (resp.\\ locally of finite presentation). The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is unramified (resp.\\ G-unramified) at $x$. \\item The fibre $X_s$ is unramified over $\\kappa(s)$ at $x$. \\item The $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$ is zero. \\item The $\\mathcal{O}_{X_s, x}$-module $\\Omega_{X_s/s, x}$ is zero. \\item The $\\kappa(x)$-vector space $$ \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ is zero. \\item We have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$ and the field extension $\\kappa(s) \\subset \\kappa(x)$ is finite separable. \\end{enumerate}"}
+{"_id": "5356", "title": "morphisms-lemma-set-points-where-fibres-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ locally of finite type. Formation of the open set \\begin{align*} T & = \\{x \\in X \\mid X_{f(x)}\\text{ is unramified over }\\kappa(f(x))\\text{ at }x\\} \\\\ & = \\{x \\in X \\mid X\\text{ is unramified over }S\\text{ at }x\\} \\end{align*} commutes with arbitrary base change: For any morphism $g : S' \\to S$, consider the base change $f' : X' \\to S'$ of $f$ and the projection $g' : X' \\to X$. Then the corresponding set $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$. If $f$ is assumed locally of finite presentation then the same holds for the open set of points where $f$ is G-unramified."}
+{"_id": "5357", "title": "morphisms-lemma-unramified-permanence", "text": "Let $f : X \\to Y$ be a morphism of schemes over $S$. \\begin{enumerate} \\item If $X$ is unramified over $S$, then $f$ is unramified. \\item If $X$ is G-unramified over $S$ and $Y$ of finite type over $S$, then $f$ is G-unramified. \\end{enumerate}"}
+{"_id": "5358", "title": "morphisms-lemma-value-at-one-point", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \\to Y$ be morphisms over $S$. Let $x \\in X$. Assume that \\begin{enumerate} \\item the structure morphism $Y \\to S$ is unramified, \\item $f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and \\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(y) \\to \\kappa(x)$ are equal. \\end{enumerate} Then there exists an open neighbourhood of $x$ in $X$ on which $f$ and $g$ are equal."}
+{"_id": "5359", "title": "morphisms-lemma-etale-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is \\'etale. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is \\'etale. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is \\'etale. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is \\'etale, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is \\'etale then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is \\'etale."}
+{"_id": "5360", "title": "morphisms-lemma-composition-etale", "text": "The composition of two morphisms which are \\'etale is \\'etale."}
+{"_id": "5361", "title": "morphisms-lemma-base-change-etale", "text": "The base change of a morphism which is \\'etale is \\'etale."}
+{"_id": "5362", "title": "morphisms-lemma-etale-smooth-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Then $f$ is \\'etale at $x$ if and only if $f$ is smooth and unramified at $x$."}
+{"_id": "5363", "title": "morphisms-lemma-etale-locally-quasi-finite", "text": "An \\'etale morphism is locally quasi-finite."}
+{"_id": "5364", "title": "morphisms-lemma-etale-over-field", "text": "\\begin{slogan} Description of the \\'etale schemes over fields and fibres of \\'etale morphisms. \\end{slogan} Fibres of \\'etale morphisms. \\begin{enumerate} \\item Let $X$ be a scheme over a field $k$. The structure morphism $X \\to \\Spec(k)$ is \\'etale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$. \\item If $f : X \\to S$ is an \\'etale morphism, then for every $s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\\kappa(s)$. \\end{enumerate}"}
+{"_id": "5365", "title": "morphisms-lemma-etale-flat-etale-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and for every $s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\\kappa(s)$, then $f$ is \\'etale."}
+{"_id": "5366", "title": "morphisms-lemma-open-immersion-etale", "text": "Any open immersion is \\'etale."}
+{"_id": "5367", "title": "morphisms-lemma-etale-syntomic", "text": "An \\'etale morphism is syntomic."}
+{"_id": "5368", "title": "morphisms-lemma-etale-locally-finite-presentation", "text": "An \\'etale morphism is locally of finite presentation."}
+{"_id": "5369", "title": "morphisms-lemma-etale-flat", "text": "An \\'etale morphism is flat."}
+{"_id": "5370", "title": "morphisms-lemma-etale-open", "text": "An \\'etale morphism is open."}
+{"_id": "5371", "title": "morphisms-lemma-etale-locally-standard-etale", "text": "Let $f : X  \\to S$ be a morphism of schemes. Let $x \\in X$ be a point. Let $V \\subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is \\'etale at $x$. \\item There exist an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ such that the induced morphism $f|_U : U \\to V$ is standard \\'etale (see Definition \\ref{definition-etale}). \\end{enumerate}"}
+{"_id": "5372", "title": "morphisms-lemma-etale-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is \\'etale at $x$. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and $X_s \\to \\Spec(\\kappa(s))$ is \\'etale at $x$. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and $X_s \\to \\Spec(\\kappa(s))$ is unramified at $x$. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$ is zero. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat and the $\\kappa(x)$-vector space $$ \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ is zero. \\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$ is flat, we have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$ and the field extension $\\kappa(s) \\subset \\kappa(x)$ is finite separable. \\item There exist affine opens $U \\subset X$, and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the induced morphism $f|_U : U \\to V$ is standard smooth of relative dimension $0$. \\item There exist affine opens $\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding to $\\mathfrak q \\subset A$, and $f(U) \\subset V$ such that there exists a presentation $$ A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n) $$ with $$ g = \\det \\left( \\begin{matrix} \\partial f_1/\\partial x_1 & \\partial f_2/\\partial x_1 & \\ldots & \\partial f_n/\\partial x_1 \\\\ \\partial f_1/\\partial x_2 & \\partial f_2/\\partial x_2 & \\ldots & \\partial f_n/\\partial x_2 \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ \\partial f_1/\\partial x_n & \\partial f_2/\\partial x_n & \\ldots & \\partial f_n/\\partial x_n \\end{matrix} \\right) $$ mapping to an element of $A$ not in $\\mathfrak q$. \\item There exist affine opens $U \\subset X$, and $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the induced morphism $f|_U : U \\to V$ is standard \\'etale. \\item There exist affine opens $\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding to $\\mathfrak q \\subset A$, and $f(U) \\subset V$ such that there exists a presentation $$ A = R[x]_Q/(P) = R[x, 1/Q]/(P) $$ with $P, Q \\in R[x]$, $P$ monic and $P' = \\text{d}P/\\text{d}x$ mapping to an element of $A$ not in $\\mathfrak q$. \\end{enumerate}"}
+{"_id": "5373", "title": "morphisms-lemma-flat-unramified-etale", "text": "A morphism is \\'etale at a point if and only if it is flat and G-unramified at that point. A morphism is \\'etale if and only if it is flat and G-unramified."}
+{"_id": "5374", "title": "morphisms-lemma-set-points-where-fibres-etale", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$ be the open subscheme of points where $f$, resp.\\ $f'$ is \\'etale. Then $W' = (g')^{-1}(W)$ if \\begin{enumerate} \\item $f$ is flat and locally of finite presentation, or \\item $f$ is locally of finite presentation and $g$ is flat. \\end{enumerate}"}
+{"_id": "5375", "title": "morphisms-lemma-etale-permanence", "text": "\\begin{slogan} Cancellation law for \\'etale morphisms \\end{slogan} Let $f : X \\to Y$ be a morphism of schemes over $S$. If $X$ and $Y$ are \\'etale over $S$, then $f$ is \\'etale."}
+{"_id": "5376", "title": "morphisms-lemma-etale-permanence-two", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective, and \\'etale, \\item $p$ is \\'etale, and \\item $q$ is locally of finite presentation\\footnote{In fact this is implied by (1) and (2), see Descent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma \\ref{descent-lemma-smooth-permanence}.}. \\end{enumerate} Then $q$ is \\'etale."}
+{"_id": "5377", "title": "morphisms-lemma-smooth-etale-over-affine-space", "text": "\\begin{slogan} Smooth schemes are \\'etale-locally like affine spaces. \\end{slogan} Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$. Let $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$. If $\\varphi$ is smooth at $x$, then there exists an integer $d \\geq 0$ and an affine open $U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$ such that there exists a commutative diagram $$ \\xymatrix{ X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi & \\mathbf{A}^d_V \\ar[ld] \\\\ Y & V \\ar[l] } $$ where $\\pi$ is \\'etale."}
+{"_id": "5378", "title": "morphisms-lemma-ample-power-ample", "text": "Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $n \\geq 1$. Then $\\mathcal{L}$ is $f$-ample if and only if $\\mathcal{L}^{\\otimes n}$ is $f$-ample."}
+{"_id": "5379", "title": "morphisms-lemma-relatively-ample-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. If there exists an $f$-ample invertible sheaf, then $f$ is separated."}
+{"_id": "5380", "title": "morphisms-lemma-characterize-relatively-ample", "text": "\\begin{reference} \\cite[II, Proposition 4.6.3]{EGA} \\end{reference} Let $f : X \\to S$ be a quasi-compact morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. The following are equivalent: \\begin{enumerate} \\item The invertible sheaf $\\mathcal{L}$ is $f$-ample. \\item There exists an open covering $S = \\bigcup V_i$ such that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample relative to $f^{-1}(V_i) \\to V_i$. \\item There exists an affine open covering $S = \\bigcup V_i$ such that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample. \\item There exists a quasi-coherent graded $\\mathcal{O}_S$-algebra $\\mathcal{A}$ and a map of graded $\\mathcal{O}_X$-algebras $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$ such that $U(\\psi) = X$ and $$ r_{\\mathcal{L}, \\psi} : X \\longrightarrow \\underline{\\text{Proj}}_S(\\mathcal{A}) $$ is an open immersion (see Constructions, Lemma \\ref{constructions-lemma-invertible-map-into-relative-proj} for notation). \\item The morphism $f$ is quasi-separated and part (4) above holds with $\\mathcal{A} = f_*(\\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d})$ and $\\psi$ the adjunction mapping. \\item Same as (4) but just requiring $r_{\\mathcal{L}, \\psi}$ to be an immersion. \\end{enumerate}"}
+{"_id": "5381", "title": "morphisms-lemma-ample-over-affine", "text": "\\begin{reference} \\cite[II Corollary 4.6.6]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $S$ affine. Then $\\mathcal{L}$ is $f$-relatively ample if and only if $\\mathcal{L}$ is ample on $X$."}
+{"_id": "5382", "title": "morphisms-lemma-quasi-affine-O-ample", "text": "\\begin{reference} \\cite[II Proposition 5.1.6]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Then $f$ is quasi-affine if and only if $\\mathcal{O}_X$ is $f$-relatively ample."}
+{"_id": "5383", "title": "morphisms-lemma-pullback-ample-tensor-relatively-ample", "text": "Let $f : X \\to Y$ be a morphism of schemes, $\\mathcal{M}$ an invertible $\\mathcal{O}_Y$-module, and $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $\\mathcal{L}$ is $f$-ample and $\\mathcal{M}$ is ample, then $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$ is ample for $a \\gg 0$. \\item If $\\mathcal{M}$ is ample and $f$ quasi-affine, then $f^*\\mathcal{M}$ is ample. \\end{enumerate}"}
+{"_id": "5384", "title": "morphisms-lemma-ample-composition", "text": "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes. Let $\\mathcal{M}$ be an invertible $\\mathcal{O}_Y$-module. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $S$ is quasi-compact, $\\mathcal{M}$ is $g$-ample, and $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$ is $g \\circ f$-ample for $a \\gg 0$."}
+{"_id": "5385", "title": "morphisms-lemma-ample-base-change", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $S' \\to S$ be a morphism of schemes. Let $f' : X' \\to S'$ be the base change of $f$ and denote $\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$. If $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample."}
+{"_id": "5386", "title": "morphisms-lemma-ample-permanence", "text": "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $\\mathcal{L}$ is $g \\circ f$-ample and $f$ is quasi-compact\\footnote{This follows if $g$ is quasi-separated by Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.} then $\\mathcal{L}$ is $f$-ample."}
+{"_id": "5387", "title": "morphisms-lemma-ample-very-ample", "text": "\\begin{reference} \\cite[II, Proposition 4.6.2]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $f$ is quasi-compact and $\\mathcal{L}$ is a relatively very ample invertible sheaf, then $\\mathcal{L}$ is a relatively ample invertible sheaf."}
+{"_id": "5388", "title": "morphisms-lemma-relatively-very-ample-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. If $\\mathcal{L}$ is relatively very ample on $X/S$ then $f$ is separated."}
+{"_id": "5389", "title": "morphisms-lemma-relatively-very-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is quasi-compact. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is relatively very ample on $X/S$, \\item there exists an open covering $S = \\bigcup V_j$ such that $\\mathcal{L}|_{f^{-1}(V_j)}$ is relatively very ample on $f^{-1}(V_j)/V_j$ for all $j$, \\item there exists a quasi-coherent sheaf of graded $\\mathcal{O}_S$-algebras $\\mathcal{A}$ generated in degree $1$ over $\\mathcal{O}_S$ and a map of graded $\\mathcal{O}_X$-algebras $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$ such that $f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective and the associated morphism $r_{\\mathcal{L}, \\psi} : X \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ is an immersion, and \\item $f$ is quasi-separated, the canonical map $\\psi : f^*f_*\\mathcal{L} \\to \\mathcal{L}$ is surjective, and the associated map $r_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(f_*\\mathcal{L})$ is an immersion. \\end{enumerate}"}
+{"_id": "5390", "title": "morphisms-lemma-very-ample-base-change", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $S' \\to S$ be a morphism of schemes. Let $f' : X' \\to S'$ be the base change of $f$ and denote $\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$. If $\\mathcal{L}$ is $f$-very ample, then $\\mathcal{L}'$ is $f'$-very ample."}
+{"_id": "5391", "title": "morphisms-lemma-very-ample-finite-type-over-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume that \\begin{enumerate} \\item the invertible sheaf $\\mathcal{L}$ is very ample on $X/S$, \\item the morphism $X \\to S$ is of finite type, and \\item $S$ is affine. \\end{enumerate} Then there exists an $n \\geq 0$ and an immersion $i : X \\to \\mathbf{P}^n_S$ over $S$ such that $\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$."}
+{"_id": "5392", "title": "morphisms-lemma-quasi-affine-finite-type-over-S", "text": "Let $\\pi : X \\to S$ be a morphism of schemes. Assume that $X$ is quasi-affine and that $\\pi$ is locally of finite type. Then there exist $n \\geq 0$ and an immersion $i : X \\to \\mathbf{A}^n_S$ over $S$."}
+{"_id": "5393", "title": "morphisms-lemma-quasi-projective-finite-type-over-S", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume that \\begin{enumerate} \\item the invertible sheaf $\\mathcal{L}$ is ample on $X$, and \\item the morphism $X \\to S$ is locally of finite type. \\end{enumerate} Then there exists a $d_0 \\geq 1$ such that for every $d \\geq d_0$ there exists an $n \\geq 0$ and an immersion $i : X \\to \\mathbf{P}^n_S$ over $S$ such that $\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$."}
+{"_id": "5394", "title": "morphisms-lemma-finite-type-over-affine-ample-very-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $S$ affine and $f$ of finite type. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X$, \\item $\\mathcal{L}$ is $f$-ample, \\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$, \\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$, \\item for some $d \\geq 1$ there exist $n \\geq 1$ and an immersion $i : X \\to \\mathbf{P}^n_S$ such that $\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$, and \\item for all $d \\gg 1$ there exist $n \\geq 1$ and an immersion $i : X \\to \\mathbf{P}^n_S$ such that $\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$. \\end{enumerate}"}
+{"_id": "5395", "title": "morphisms-lemma-finite-type-ample-very-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $S$ quasi-compact and $f$ of finite type. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is $f$-ample, \\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$, \\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$. \\end{enumerate}"}
+{"_id": "5396", "title": "morphisms-lemma-characterize-very-ample-on-finite-type", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{L}$ is $f$-relatively very ample, and \\item there exist an open covering $S = \\bigcup V_j$, for each $j$ an integer $n_j$, and immersions $$ i_j : X_j = f^{-1}(V_j) = V_j \\times_S X \\longrightarrow \\mathbf{P}^{n_j}_{V_j} $$ over $V_j$ such that $\\mathcal{L}|_{X_j} \\cong i_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$. \\end{enumerate}"}
+{"_id": "5397", "title": "morphisms-lemma-characterize-ample-on-finite-type", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{L}$ is $f$-relatively ample, and \\item there exist an open covering $S = \\bigcup V_j$, for each $j$ an integers $d_j \\geq 1$, $n_j \\geq 0$, and immersions $$ i_j : X_j = f^{-1}(V_j) = V_j \\times_S X \\longrightarrow \\mathbf{P}^{n_j}_{V_j} $$ over $V_j$ such that $\\mathcal{L}^{\\otimes d_j}|_{X_j} \\cong i_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$. \\end{enumerate}"}
+{"_id": "5398", "title": "morphisms-lemma-invertible-add-enough-ample-very-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{N}$, $\\mathcal{L}$ be invertible $\\mathcal{O}_X$-modules. Assume $S$ is quasi-compact, $f$ is of finite type, and $\\mathcal{L}$ is $f$-ample. Then $\\mathcal{N} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$."}
+{"_id": "5399", "title": "morphisms-lemma-base-change-quasi-projective", "text": "A base change of a quasi-projective morphism is quasi-projective."}
+{"_id": "5400", "title": "morphisms-lemma-composition-quasi-projective", "text": "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes. If $S$ is quasi-compact and $f$ and $g$ are quasi-projective, then $g \\circ f$ is quasi-projective."}
+{"_id": "5401", "title": "morphisms-lemma-quasi-projective-properties", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is quasi-projective, or H-quasi-projective or locally quasi-projective, then $f$ is separated of finite type."}
+{"_id": "5402", "title": "morphisms-lemma-H-quasi-projective-quasi-projective", "text": "A H-quasi-projective morphism is quasi-projective."}
+{"_id": "5403", "title": "morphisms-lemma-characterize-locally-quasi-projective", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is locally quasi-projective. \\item There exists an open covering $S = \\bigcup V_j$ such that each $f^{-1}(V_j) \\to V_j$ is H-quasi-projective. \\end{enumerate}"}
+{"_id": "5404", "title": "morphisms-lemma-quasi-affine-finite-type-quasi-projective", "text": "\\begin{reference} \\cite[II, Proposition 5.3.4 (i)]{EGA} \\end{reference} A quasi-affine morphism of finite type is quasi-projective."}
+{"_id": "5406", "title": "morphisms-lemma-universally-closed-local-on-the-base", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is universally closed. \\item There exists an open covering $S = \\bigcup V_j$ such that $f^{-1}(V_j) \\to V_j$ is universally closed for all indices $j$. \\end{enumerate}"}
+{"_id": "5407", "title": "morphisms-lemma-proper-local-on-the-base", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is proper. \\item There exists an open covering $S = \\bigcup V_j$ such that $f^{-1}(V_j) \\to V_j$ is proper for all indices $j$. \\end{enumerate}"}
+{"_id": "5408", "title": "morphisms-lemma-composition-proper", "text": "The composition of proper morphisms is proper. The same is true for universally closed morphisms."}
+{"_id": "5409", "title": "morphisms-lemma-base-change-proper", "text": "The base change of a proper morphism is proper. The same is true for universally closed morphisms."}
+{"_id": "5410", "title": "morphisms-lemma-closed-immersion-proper", "text": "A closed immersion is proper, hence a fortiori universally closed."}
+{"_id": "5411", "title": "morphisms-lemma-image-proper-scheme-closed", "text": "Suppose given a commutative diagram of schemes $$ \\xymatrix{ X \\ar[rr] \\ar[rd] & & Y \\ar[ld] \\\\ & S & } $$ with $Y$ separated over $S$. \\begin{enumerate} \\item If $X \\to S$ is universally closed, then the morphism $X \\to Y$ is universally closed. \\item If $X$ is proper over $S$, then the morphism $X \\to Y$ is proper. \\end{enumerate} In particular, in both cases the image of $X$ in $Y$ is closed."}
+{"_id": "5412", "title": "morphisms-lemma-universally-closed-quasi-compact", "text": "\\begin{reference} Due to Bjorn Poonen. \\end{reference} A universally closed morphism of schemes is quasi-compact."}
+{"_id": "5413", "title": "morphisms-lemma-image-proper-is-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. If $X$ is universally closed over $S$ and $f$ is surjective then $Y$ is universally closed over $S$. In particular, if also $Y$ is separated and locally of finite type over $S$, then $Y$ is proper over $S$."}
+{"_id": "5414", "title": "morphisms-lemma-scheme-theoretic-image-is-proper", "text": "Suppose given a commutative diagram of schemes $$ \\xymatrix{ X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\ & S } $$ Assume \\begin{enumerate} \\item $X \\to S$ is a universally closed (for example proper) morphism, and \\item $Y \\to S$ is separated and locally of finite type. \\end{enumerate} Then the scheme theoretic image $Z \\subset Y$ of $h$ is proper over $S$ and $X \\to Z$ is surjective."}
+{"_id": "5415", "title": "morphisms-lemma-image-universally-closed-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a surjective universally closed morphism of schemes over $S$. \\begin{enumerate} \\item If $X$ is quasi-separated, then $Y$ is quasi-separated. \\item If $X$ is separated, then $Y$ is separated. \\item If $X$ is quasi-separated over $S$, then $Y$ is quasi-separated over $S$. \\item If $X$ is separated over $S$, then $Y$ is separated over $S$. \\end{enumerate}"}
+{"_id": "5416", "title": "morphisms-lemma-characterize-proper", "text": "\\begin{reference} \\cite[II Theorem 7.3.8]{EGA} \\end{reference} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \\begin{enumerate} \\item $f$ is proper, \\item $f$ satisfies the valuative criterion (Schemes, Definition \\ref{schemes-definition-valuative-criterion}), \\item given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute. \\end{enumerate}"}
+{"_id": "5417", "title": "morphisms-lemma-refined-valuative-criterion-universally-closed", "text": "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes. Assume that $f$ and $h$ are quasi-compact and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is universally closed. If moreover $f$ is quasi-separated, then $f$ is separated."}
+{"_id": "5418", "title": "morphisms-lemma-morphism-defined-local-ring", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $s \\in S$ and $x \\in X$, $y \\in Y$ points over $s$. \\begin{enumerate} \\item Let $f, g : X \\to Y$ be morphisms over $S$ such that $f(x) = g(x) = y$ and $f^\\sharp_x = g^\\sharp_x : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$. Then there is an open neighbourhood $U \\subset X$ with $f|_U = g|_U$ in the following cases \\begin{enumerate} \\item $Y$ is locally of finite type over $S$, \\item $X$ is integral, \\item $X$ is locally Noetherian, or \\item $X$ is reduced with finitely many irreducible components. \\end{enumerate} \\item Let $\\varphi : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ be a local $\\mathcal{O}_{S, s}$-algebra map. Then there exists an open neighbourhood $U \\subset X$ of $x$ and a morphism $f : U \\to Y$ mapping $x$ to $y$ with $f^\\sharp_x = \\varphi$ in the following cases \\begin{enumerate} \\item $Y$ is locally of finite presentation over $S$, \\item $Y$ is locally of finite type and $X$ is integral, \\item $Y$ is locally of finite type and $X$ is locally Noetherian, or \\item $Y$ is locally of finite type and $X$ is reduced with finitely many irreducible components. \\end{enumerate} \\end{enumerate}"}
+{"_id": "5419", "title": "morphisms-lemma-extend-across", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $x \\in X$. Let $U \\subset X$ be an open and let $f : U \\to Y$ be a morphism over $S$. Assume \\begin{enumerate} \\item $x$ is in the closure of $U$, \\item $X$ is reduced with finitely many irreducible components or $X$ is Noetherian, \\item $\\mathcal{O}_{X, x}$ is a valuation ring, \\item $Y \\to S$ is proper \\end{enumerate} Then there exists an open $U \\subset U' \\subset X$ containing $x$ and an $S$-morphism $f' : U' \\to Y$ extending $f$."}
+{"_id": "5420", "title": "morphisms-lemma-H-projective", "text": "An H-projective morphism is H-quasi-projective. An H-projective morphism is projective."}
+{"_id": "5421", "title": "morphisms-lemma-characterize-locally-projective", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is locally projective. \\item There exists an open covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is H-projective. \\end{enumerate}"}
+{"_id": "5422", "title": "morphisms-lemma-locally-projective-proper", "text": "A locally projective morphism is proper."}
+{"_id": "5423", "title": "morphisms-lemma-proper-ample-locally-projective", "text": "Let $f : X \\to S$ be a proper morphism of schemes. If there exists an $f$-ample invertible sheaf on $X$, then $f$ is locally projective."}
+{"_id": "5424", "title": "morphisms-lemma-H-projective-composition", "text": "A composition of H-projective morphisms is H-projective."}
+{"_id": "5425", "title": "morphisms-lemma-H-projective-base-change", "text": "A base change of a H-projective morphism is H-projective."}
+{"_id": "5426", "title": "morphisms-lemma-base-change-projective", "text": "A base change of a (locally) projective morphism is (locally) projective."}
+{"_id": "5427", "title": "morphisms-lemma-projective-quasi-projective", "text": "A projective morphism is quasi-projective."}
+{"_id": "5428", "title": "morphisms-lemma-H-quasi-projective-open-H-projective", "text": "Let $f : X \\to S$ be a H-quasi-projective morphism. Then $f$ factors as $X \\to X' \\to S$ where $X \\to X'$ is an open immersion and $X' \\to S$ is H-projective."}
+{"_id": "5429", "title": "morphisms-lemma-quasi-projective-open-projective", "text": "Let $f : X \\to S$ be a quasi-projective morphism with $S$ quasi-compact and quasi-separated. Then $f$ factors as $X \\to X' \\to S$ where $X \\to X'$ is an open immersion and $X' \\to S$ is projective."}
+{"_id": "5430", "title": "morphisms-lemma-projective-is-quasi-projective-proper", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a morphism of schemes. Then \\begin{enumerate} \\item $f$ is projective if and only if $f$ is quasi-projective and proper, and \\item $f$ is H-projective if and only if $f$ is H-quasi-projective and proper. \\end{enumerate}"}
+{"_id": "5431", "title": "morphisms-lemma-composition-projective", "text": "Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes. If $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective, then $g \\circ f$ is projective."}
+{"_id": "5432", "title": "morphisms-lemma-projective-permanence", "text": "Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes. If $g \\circ f$ is projective and $g$ is separated, then $f$ is projective."}
+{"_id": "5433", "title": "morphisms-lemma-projective-over-quasi-projective-is-H-projective", "text": "Let $S$ be a scheme which admits an ample invertible sheaf. Then \\begin{enumerate} \\item any projective morphism $X \\to S$ is H-projective, and \\item any quasi-projective morphism $X \\to S$ is H-quasi-projective. \\end{enumerate}"}
+{"_id": "5434", "title": "morphisms-lemma-proper-ample-is-proj", "text": "Let $f : X \\to S$ be a universally closed morphism. Let $\\mathcal{L}$ be an $f$-ample invertible $\\mathcal{O}_X$-module. Then the canonical morphism $$ r : X \\longrightarrow \\underline{\\text{Proj}}_S \\left( \\bigoplus\\nolimits_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d} \\right) $$ of Lemma \\ref{lemma-characterize-relatively-ample} is an isomorphism."}
+{"_id": "5435", "title": "morphisms-lemma-proper-ample-delete-affine", "text": "Let $f : X \\to S$ be a universally closed morphism. Let $\\mathcal{L}$ be an $f$-ample invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Then $X_s \\to S$ is an affine morphism."}
+{"_id": "5437", "title": "morphisms-lemma-finite-local", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is finite. \\item There exists an affine open covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i)$ is affine and $\\mathcal{O}_S(U_i) \\to \\mathcal{O}_X(f^{-1}(U_i))$ is finite. \\item There exists an open covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is finite. \\end{enumerate} Moreover, if $f$ is finite then for every open subscheme $U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is finite."}
+{"_id": "5438", "title": "morphisms-lemma-finite-integral", "text": "A finite morphism is integral. An integral morphism which is locally of finite type is finite."}
+{"_id": "5439", "title": "morphisms-lemma-composition-finite", "text": "A composition of finite morphisms is finite. Same is true for integral morphisms."}
+{"_id": "5440", "title": "morphisms-lemma-base-change-finite", "text": "A base change of a finite morphism is finite. Same is true for integral morphisms."}
+{"_id": "5441", "title": "morphisms-lemma-integral-universally-closed", "text": "\\begin{slogan} integral $=$ affine $+$ universally closed \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is integral, and \\item $f$ is affine and universally closed. \\end{enumerate}"}
+{"_id": "5442", "title": "morphisms-lemma-integral-fibres", "text": "Let $f : X \\to S$ be an integral morphism. Then every point of $X$ is closed in its fibre."}
+{"_id": "5443", "title": "morphisms-lemma-integral-dimension", "text": "Let $f : X \\to Y$ be an integral morphism. Then $\\dim(X) \\leq \\dim(Y)$. If $f$ is surjective then $\\dim(X) = \\dim(Y)$."}
+{"_id": "5444", "title": "morphisms-lemma-finite-quasi-finite", "text": "A finite morphism is quasi-finite."}
+{"_id": "5445", "title": "morphisms-lemma-finite-proper", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is finite, and \\item $f$ is affine and proper. \\end{enumerate}"}
+{"_id": "5446", "title": "morphisms-lemma-closed-immersion-finite", "text": "A closed immersion is finite (and a fortiori integral)."}
+{"_id": "5447", "title": "morphisms-lemma-finite-union-finite", "text": "Let $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of schemes. Then $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too."}
+{"_id": "5448", "title": "morphisms-lemma-finite-permanence", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms. \\begin{enumerate} \\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite. \\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral. \\end{enumerate}"}
+{"_id": "5449", "title": "morphisms-lemma-finite-monomorphism-closed", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is finite and a monomorphism, then $f$ is a closed immersion."}
+{"_id": "5450", "title": "morphisms-lemma-finite-projective", "text": "A finite morphism is projective."}
+{"_id": "5451", "title": "morphisms-lemma-base-change-universal-homeomorphism", "text": "The base change of a universal homeomorphism of schemes by any morphism of schemes is a universal homeomorphism."}
+{"_id": "5452", "title": "morphisms-lemma-composition-universal-homeomorphism", "text": "The composition of a pair of universal homeomorphisms of schemes is a universal homeomorphism."}
+{"_id": "5453", "title": "morphisms-lemma-homeomorphism-affine", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is a homeomorphism onto a closed subset of $Y$ then $f$ is affine."}
+{"_id": "5454", "title": "morphisms-lemma-universal-homeomorphism", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is a universal homeomorphism, and \\item $f$ is integral, universally injective and surjective. \\end{enumerate}"}
+{"_id": "5455", "title": "morphisms-lemma-reduction-universal-homeomorphism", "text": "Let $X$ be a scheme. The canonical closed immersion $X_{red} \\to X$ (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}) is a universal homeomorphism."}
+{"_id": "5456", "title": "morphisms-lemma-check-closed-infinitesimally", "text": "Let $f : X \\to S$ and $S' \\to S$ be morphisms of schemes. Assume \\begin{enumerate} \\item $S' \\to S$ is a closed immersion, \\item $S' \\to S$ is bijective on points, \\item $X \\times_S S' \\to S'$ is a closed immersion, and \\item $X \\to S$ is of finite type or $S' \\to S$ is of finite presentation. \\end{enumerate} Then $f : X \\to S$ is a closed immersion."}
+{"_id": "5457", "title": "morphisms-lemma-subalgebra-inherits", "text": "Let $A \\to B$ be a ring map such that the induced morphism of schemes $f : \\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, resp.\\ a universal homeomorphism inducing isomorphisms on residue fields, resp.\\ universally closed, resp.\\ universally closed and universally injective. Then for any $A$-subalgebra $B' \\subset B$ the same thing is true for $f' : \\Spec(B') \\to \\Spec(A)$."}
+{"_id": "5458", "title": "morphisms-lemma-colimit-inherits", "text": "Let $A$ be a ring. Let $B = \\colim B_\\lambda$ be a filtered colimit of $A$-algebras. If each $f_\\lambda : \\Spec(B_\\lambda) \\to \\Spec(A)$ is a universal homeomorphism, resp.\\ a universal homeomorphism inducing isomorphisms on residue fields, resp.\\ universally closed, resp.\\ universally closed and universally injective, then the same thing is true for $f : \\Spec(B) \\to \\Spec(A)$."}
+{"_id": "5459", "title": "morphisms-lemma-special-elements-and-localization", "text": "Let $A \\subset B$ be a ring extension. Let $S \\subset A$ be a multiplicative subset. Let $n \\geq 1$ and $b_i \\in B$ for $1 \\leq i \\leq n$. If the set $$ \\{x \\in S^{-1}B \\mid x \\not \\in S^{-1}A\\text{ and } b_i x^i \\in S^{-1}A\\text{ for }i = 1, \\ldots, n\\} $$ is nonempty, then so is $$ \\{x \\in B \\mid x \\not \\in A\\text{ and } b_i x^i \\in A\\text{ for }i = 1, \\ldots, n\\} $$"}
+{"_id": "5460", "title": "morphisms-lemma-nth-and-nplusone-implies-square-and-cube", "text": "Let $A \\subset B$ be a ring extension. If there exists $b \\in B$, $b \\not \\in A$ and an integer $n \\geq 2$ with $b^n \\in A$ and $b^{n + 1} \\in A$, then there exists a $b' \\in B$, $b' \\not \\in A$ with $(b')^2 \\in A$ and $(b')^3 \\in A$."}
+{"_id": "5461", "title": "morphisms-lemma-square-and-cube", "text": "Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism inducing isomorphisms on residue fields. If $A \\not = B$, then there exists a $b \\in B$, $b \\not \\in A$ with $b^2 \\in A$ and $b^3 \\in A$."}
+{"_id": "5462", "title": "morphisms-lemma-pth-power-and-multiple", "text": "Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism. If $A \\not = B$, then either there exists a $b \\in B$, $b \\not \\in A$ with $b^2 \\in A$ and $b^3 \\in A$ or there exists a prime number $p$ and a $b \\in B$, $b \\not \\in A$ with $pb \\in A$ and $b^p \\in A$."}
+{"_id": "5464", "title": "morphisms-lemma-make-universal-homeo", "text": "Let $A$ be a ring. Let $x, y \\in A$. \\begin{enumerate} \\item If $x^3 = y^2$ in $A$, then $A \\to B = A[t]/(t^2 - x, t^3 - y)$ induces bijections on residue fields and a universal homeomorphism on spectra. \\item If there is a prime number $p$ such that $p^px = y^p$ in $A$, then $A \\to B = A[t]/(t^p - x, pt - y)$ induces a universal homeomorphism on spectra. \\end{enumerate}"}
+{"_id": "5465", "title": "morphisms-lemma-universal-homeo-limit", "text": "Let $A \\to B$ be a ring map. \\begin{enumerate} \\item If $A \\to B$ induces a universal homeomorphism on spectra, then $B = \\colim B_i$ is a filtered colimit of finitely presented $A$-algebras $B_i$ such that $A \\to B_i$ induces a universal homeomorphism on spectra. \\item If $A \\to B$ induces isomorphisms on residue fields and a universal homeomorphism on spectra, then $B = \\colim B_i$ is a filtered colimit of finitely presented $A$-algebras $B_i$ such that $A \\to B_i$ induces isomorphisms on residue fields and a universal homeomorphism on spectra. \\end{enumerate}"}
+{"_id": "5466", "title": "morphisms-lemma-seminormal-local-property", "text": "Being seminormal or being absolutely weakly normal is a local property of rings, see Properties, Definition \\ref{properties-definition-property-local}."}
+{"_id": "5467", "title": "morphisms-lemma-locally-seminormal", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is seminormal. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is seminormal. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is seminormal. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is seminormal. \\end{enumerate} Moreover, if $X$ is seminormal then every open subscheme is seminormal. The same statements are true with ``seminormal'' replaced by ``absolutely weakly normal''."}
+{"_id": "5468", "title": "morphisms-lemma-seminormal-reduced", "text": "A seminormal scheme or ring is reduced. A fortiori the same is true for absolutely weakly normal schemes or rings."}
+{"_id": "5469", "title": "morphisms-lemma-seminormalization-ring", "text": "Let $A$ be a ring. \\begin{enumerate} \\item The category of ring maps $A \\to B$ inducing a universal homeomorphism on spectra has a final object $A \\to A^{awn}$. \\item Given $A \\to B$ in the category of (1) the resulting map $B \\to A^{awn}$ is an isomorphism if and only if $B$ is absolutely weakly normal. \\item The category of ring maps $A \\to B$ inducing isomorphisms on residue fields and a universal homeomorphism on spectra has a final object $A \\to A^{sn}$. \\item Given $A \\to B$ in the category of (3) the resulting map $B \\to A^{sn}$ is an isomorphism if and only if $B$ is seminormal. \\end{enumerate} For any ring map $\\varphi : A \\to A'$ there are unique maps $\\varphi^{awn} : A^{awn} \\to (A')^{awn}$ and $\\varphi^{sn} : A^{sn} \\to (A')^{sn}$ compatible with $\\varphi$."}
+{"_id": "5470", "title": "morphisms-lemma-seminormalization", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The category of universal homeomorphisms $Y \\to X$ has an initial object $X^{awn} \\to X$. \\item Given $Y \\to X$ in the category of (1) the resulting morphism $X^{awn} \\to Y$ is an isomorphism if and only if $Y$ is absolutely weakly normal. \\item The category of universal homeomorphisms $Y \\to X$ which induce ismomorphisms on residue fields has an initial object $X^{sn} \\to X$. \\item Given $Y \\to X$ in the category of (3) the resulting morphism $X^{sn} \\to Y$ is an isomorphism if and only if $Y$ is seminormal. \\end{enumerate} For any morphism $h : X' \\to X$ of schemes there are unique morphisms $h^{awn} : (X')^{awn} \\to X^{awn}$ and $h^{sn} : (X')^{sn} \\to X^{sn}$ compatible with $h$."}
+{"_id": "5471", "title": "morphisms-lemma-finite-flat", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is finite locally free, \\item $f$ is finite, flat, and locally of finite presentation. \\end{enumerate} If $S$ is locally Noetherian these are also equivalent to \\begin{enumerate} \\item[(3)] $f$ is finite and flat. \\end{enumerate}"}
+{"_id": "5472", "title": "morphisms-lemma-composition-finite-locally-free", "text": "A composition of finite locally free morphisms is finite locally free."}
+{"_id": "5473", "title": "morphisms-lemma-base-change-finite-locally-free", "text": "A base change of a finite locally free morphism is finite locally free."}
+{"_id": "5474", "title": "morphisms-lemma-finite-locally-free", "text": "Let $f : X \\to S$ be a finite locally free morphism of schemes. There exists a disjoint union decomposition $S = \\coprod_{d \\geq 0} S_d$ by open and closed subschemes such that setting $X_d = f^{-1}(S_d)$ the restrictions $f|_{X_d}$ are finite locally free morphisms $X_d \\to S_d$ of degree $d$."}
+{"_id": "5475", "title": "morphisms-lemma-massage-finite", "text": "Let $f : Y \\to X$ be a finite morphism with $X$ affine. There exists a diagram $$ \\xymatrix{ Z' \\ar[rd] & Y' \\ar[l]^i \\ar[d] \\ar[r] & Y \\ar[d] \\\\  & X' \\ar[r] & X } $$ where \\begin{enumerate} \\item $Y' \\to Y$ and $X' \\to X$ are surjective finite locally free, \\item $Y' = X' \\times_X Y$, \\item $i : Y' \\to Z'$ is a closed immersion, \\item $Z' \\to X'$ is finite locally free, and \\item $Z' = \\bigcup_{j = 1, \\ldots, m} Z'_j$ is a (set theoretic) finite union of closed subschemes, each of which maps isomorphically to $X'$. \\end{enumerate}"}
+{"_id": "5476", "title": "morphisms-lemma-image-nowhere-dense-finite", "text": "Let $f : Y \\to X$ be a finite morphism of schemes. Let $T \\subset Y$ be a closed nowhere dense subset of $Y$. Then $f(T) \\subset X$ is a closed nowhere dense subset of $X$."}
+{"_id": "5477", "title": "morphisms-lemma-rational-map-finite-presentation", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $X$ has finitely many irreducible components with generic points $x_1, \\ldots, x_n$. Let $s_i \\in S$ be the image of $x_i$. Consider the map $$ \\left\\{ \\begin{matrix} S\\text{-rational maps} \\\\ \\text{from }X\\text{ to }Y \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} (y_1, \\varphi_1, \\ldots, y_n, \\varphi_n)\\text{ where } y_i \\in Y\\text{ lies over }s_i\\text{ and}\\\\ \\varphi_i : \\mathcal{O}_{Y, y_i} \\to \\mathcal{O}_{X, x_i} \\text{ is a local }\\mathcal{O}_{S, s_i}\\text{-algebra map} \\end{matrix} \\right\\} $$ which sends $f : U \\to Y$ to the $2n$-tuple with $y_i = f(x_i)$ and $\\varphi_i = f^\\sharp_{x_i}$. Then \\begin{enumerate} \\item If $Y \\to S$ is locally of finite type, then the map is injective. \\item If $Y \\to S$ is locally of finite presentation, then the map is bijective. \\item If $Y \\to S$ is locally of finite type and $X$ reduced, then the map is bijective. \\end{enumerate}"}
+{"_id": "5478", "title": "morphisms-lemma-integral-scheme-rational-functions", "text": "Let $X$ be a scheme with finitely many irreducible components $X_1, \\ldots, X_n$. If $\\eta_i \\in X_i$ is the generic point, then $$ R(X) = \\mathcal{O}_{X, \\eta_1} \\times \\ldots \\times \\mathcal{O}_{X, \\eta_n} $$ If $X$ is reduced this is equal to $\\prod \\kappa(\\eta_i)$. If $X$ is integral then $R(X) = \\mathcal{O}_{X, \\eta} = \\kappa(\\eta)$ is a field."}
+{"_id": "5479", "title": "morphisms-lemma-distinct-local-rings", "text": "Let $X$ be an integral separated scheme. Let $Z_1$, $Z_2$ be distinct irreducible closed subsets of $X$. Let $\\eta_i$ be the generic point of $Z_i$. If $Z_1 \\not\\subset Z_2$, then $\\mathcal{O}_{X, \\eta_1} \\not \\subset \\mathcal{O}_{X, \\eta_2}$ as subrings of $R(X)$. In particular, if $Z_1 = \\{x\\}$ consists of one closed point $x$, there exists a function regular in a neighborhood of $x$ which is not in $\\mathcal{O}_{X, \\eta_{2}}$."}
+{"_id": "5480", "title": "morphisms-lemma-rational-map-from-reduced-to-separated", "text": "Let $X$ and $Y$ be schemes. Assume $X$ reduced and $Y$ separated. Let $\\varphi$ be a rational map from $X$ to $Y$ with domain of definition $U \\subset X$. Then there exists a unique morphism $f : U \\to Y$ representing $\\varphi$. If $X$ and $Y$ are schemes over a separated scheme $S$ and if $\\varphi$ is an $S$-rational map, then $f$ is a morphism over $S$."}
+{"_id": "5481", "title": "morphisms-lemma-birational-integral", "text": "Let $X$ and $Y$ be irreducible schemes. \\begin{enumerate} \\item The schemes $X$ and $Y$ are birational if and only if they have isomorphic nonempty opens. \\item Assume $X$ and $Y$ are schemes over a base scheme $S$. Then $X$ and $Y$ are $S$-birational if and only if there are nonempty opens $U \\subset X$ and $V \\subset Y$ which are $S$-isomorphic. \\end{enumerate}"}
+{"_id": "5483", "title": "morphisms-lemma-birational-generic-fibres", "text": "Let $f : X \\to Y$ be a birational morphism of schemes having finitely many irreducible components. If $y \\in Y$ is the generic point of an irreducible component, then the base change $X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$ is an isomorphism."}
+{"_id": "5484", "title": "morphisms-lemma-birational-birational", "text": "Let $f : X \\to Y$ be a birational morphism of schemes having finitely many irreducible components over a base scheme $S$. Assume one of the following conditions is satisfied \\begin{enumerate} \\item $f$ is locally of finite type and $Y$ reduced, \\item $f$ is locally of finite presentation. \\end{enumerate} Then there exist dense opens $U \\subset X$ and $V \\subset Y$ such that $f(U) \\subset V$ and $f|_U : U \\to V$ is an isomorphism. In particular if $X$ and $Y$ are irreducible, then $X$ and $Y$ are $S$-birational."}
+{"_id": "5485", "title": "morphisms-lemma-criterion-birational-finite-presentation", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be irreducible schemes locally of finite presentation over $S$. Let $x \\in X$ and $y \\in Y$ be the generic points. The following are equivalent \\begin{enumerate} \\item $X$ and $Y$ are $S$-birational, \\item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and \\item $x$ and $y$ map to the same point $s$ of $S$ and $\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are isomorphic as $\\mathcal{O}_{S, s}$-algebras. \\end{enumerate}"}
+{"_id": "5486", "title": "morphisms-lemma-common-open", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be integral schemes locally of finite type over $S$. Let $x \\in X$ and $y \\in Y$ be the generic points. The following are equivalent \\begin{enumerate} \\item $X$ and $Y$ are $S$-birational, \\item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and \\item $x$ and $y$ map to the same point $s \\in S$ and $\\kappa(x) \\cong \\kappa(y)$ as $\\kappa(s)$-extensions. \\end{enumerate}"}
+{"_id": "5487", "title": "morphisms-lemma-generically-finite", "text": "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type. Let $\\eta \\in Y$ be a generic point of an irreducible component of $Y$. The following are equivalent: \\begin{enumerate} \\item the set $f^{-1}(\\{\\eta\\})$ is finite, \\item there exist affine opens $U_i \\subset X$, $i = 1, \\ldots, n$ and $V \\subset Y$ with $f(U_i) \\subset V$, $\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset \\bigcup U_i$ such that each $f|_{U_i} : U_i \\to V$ is finite. \\end{enumerate} If $f$ is quasi-separated, then these are also equivalent to \\begin{enumerate} \\item[(3)] there exist affine opens $V \\subset Y$, and $U \\subset X$ with $f(U) \\subset V$, $\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset U$ such that $f|_U : U \\to V$ is finite. \\end{enumerate} If $f$ is quasi-compact and quasi-separated, then these are also equivalent to \\begin{enumerate} \\item[(4)] there exists an affine open $V \\subset Y$, $\\eta \\in V$ such that $f^{-1}(V) \\to V$ is finite. \\end{enumerate}"}
+{"_id": "5488", "title": "morphisms-lemma-quasi-finiteness-over-generic-point", "text": "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type. Let $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible components of $X$, resp.\\ $Y$. Let $\\eta \\in Y^0$. The following are equivalent \\begin{enumerate} \\item $f^{-1}(\\{\\eta\\}) \\subset X^0$, \\item $f$ is quasi-finite at all points lying over $\\eta$, \\item $f$ is quasi-finite at all $\\xi \\in X^0$ lying over $\\eta$. \\end{enumerate}"}
+{"_id": "5489", "title": "morphisms-lemma-finite-over-dense-open", "text": "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type. Let $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible components of $X$, resp.\\ $Y$. Assume \\begin{enumerate} \\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$, \\item either $f$ is quasi-compact or $f$ is separated. \\end{enumerate} Then there exists a dense open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is finite."}
+{"_id": "5490", "title": "morphisms-lemma-birational-isomorphism-over-dense-open", "text": "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a birational morphism between schemes which have finitely many irreducible components. Assume \\begin{enumerate} \\item either $f$ is quasi-compact or $f$ is separated, and \\item either $f$ is locally of finite type and $Y$ is reduced or $f$ is locally of finite presentation. \\end{enumerate} Then there exists a dense open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is an isomorphism."}
+{"_id": "5491", "title": "morphisms-lemma-finite-degree", "text": "Let $X$, $Y$ be integral schemes. Let $f : X \\to Y$ be locally of finite type. Assume $f$ is dominant. The following are equivalent: \\begin{enumerate} \\item the extension $R(Y) \\subset R(X)$ has transcendence degree $0$, \\item the extension $R(Y) \\subset R(X)$ is finite, \\item there exist nonempty affine opens $U \\subset X$ and $V \\subset Y$ such that $f(U) \\subset V$ and $f|_U : U \\to V$ is finite, and \\item the generic point of $X$ is the only point of $X$ mapping to the generic point of $Y$. \\end{enumerate} If $f$ is separated or if $f$ is quasi-compact, then these are also equivalent to \\begin{enumerate} \\item[(5)] there exists a nonempty affine open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is finite. \\end{enumerate}"}
+{"_id": "5492", "title": "morphisms-lemma-degree-composition", "text": "Let $X$, $Y$, $Z$ be integral schemes. Let $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally of finite type. Assume that $[R(X) : R(Y)] < \\infty$ and $[R(Y) : R(Z)] < \\infty$. Then $$ \\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z). $$"}
+{"_id": "5493", "title": "morphisms-lemma-dimension-formula", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$, and set $s = f(x)$. Assume \\begin{enumerate} \\item $S$ is locally Noetherian, \\item $f$ is locally of finite type, \\item $X$ and $S$ integral, and \\item $f$ dominant. \\end{enumerate} We have \\begin{equation} \\label{equation-dimension-formula} \\dim(\\mathcal{O}_{X, x}) \\leq \\dim(\\mathcal{O}_{S, s}) + \\text{trdeg}_{R(S)}R(X) - \\text{trdeg}_{\\kappa(s)} \\kappa(x). \\end{equation} Moreover, equality holds if $S$ is universally catenary."}
+{"_id": "5494", "title": "morphisms-lemma-dimension-formula-general", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$, and set $s = f(x)$. Assume $S$ is locally Noetherian and $f$ is locally of finite type, We have \\begin{equation} \\label{equation-dimension-formula-general} \\dim(\\mathcal{O}_{X, x}) \\leq \\dim(\\mathcal{O}_{S, s}) + E - \\text{trdeg}_{\\kappa(s)} \\kappa(x). \\end{equation} where $E$ is the maximum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$ where $\\xi$ runs over the generic points of irreducible components of $X$ containing $x$."}
+{"_id": "5495", "title": "morphisms-lemma-dimension-function-propagates", "text": "Let $S$ be a locally Noetherian and universally catenary scheme. Let $\\delta : S \\to \\mathbf{Z}$ be a dimension function. Let $f : X \\to S$ be a morphism of schemes. Assume $f$ locally of finite type. Then the map \\begin{align*} \\delta = \\delta_{X/S} : X & \\longrightarrow \\mathbf{Z} \\\\ x & \\longmapsto \\delta(f(x)) + \\text{trdeg}_{\\kappa(f(x))} \\kappa(x) \\end{align*} is a dimension function on $X$."}
+{"_id": "5496", "title": "morphisms-lemma-alteration-dimension", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that \\begin{enumerate} \\item $Y$ is locally Noetherian, \\item $X$ and $Y$ are integral schemes, \\item $f$ is dominant, and \\item $f$ is locally of finite type. \\end{enumerate} Then we have $$ \\dim(X) \\leq \\dim(Y) + \\text{trdeg}_{R(Y)} R(X). $$ If $f$ is closed\\footnote{For example if $f$ is proper, see Definition \\ref{definition-proper}.} then equality holds."}
+{"_id": "5498", "title": "morphisms-lemma-integral-closure", "text": "Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The subsheaf $\\mathcal{A}' \\subset \\mathcal{A}$ defined by the rule $$ U \\longmapsto \\{f \\in \\mathcal{A}(U) \\mid f_x \\in \\mathcal{A}_x \\text{ integral over } \\mathcal{O}_{X, x} \\text{ for all }x \\in U\\} $$ is a quasi-coherent $\\mathcal{O}_X$-algebra, the stalk $\\mathcal{A}'_x$ is the integral closure of $\\mathcal{O}_{X, x}$ in $\\mathcal{A}_x$, and for any affine open $U \\subset X$ the ring $\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is the integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$."}
+{"_id": "5499", "title": "morphisms-lemma-characterize-normalization", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. The factorization $f = \\nu \\circ f'$, where $\\nu : X' \\to X$ is the normalization of $X$ in $Y$ is characterized by the following two properties: \\begin{enumerate} \\item the morphism $\\nu$ is integral, and \\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$ integral, there exists a commutative diagram $$ \\xymatrix{ Y \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\ X' \\ar[ru]^h \\ar[r]^\\nu & X } $$ for some unique morphism $h : X' \\to Z$. \\end{enumerate} Moreover, the morphism $f' : Y \\to X'$ is dominant and in (2) the morphism $h : X' \\to Z$ is the normalization of $Z$ in $Y$."}
+{"_id": "5500", "title": "morphisms-lemma-functoriality-normalization", "text": "Let $$ \\xymatrix{ Y_2 \\ar[d]_{f_2} \\ar[r] & Y_1 \\ar[d]^{f_1} \\\\ X_2 \\ar[r] & X_1 } $$ be a commutative diagram of morphisms of schemes. Assume $f_1$, $f_2$ quasi-compact and quasi-separated. Let $f_i = \\nu_i \\circ f_i'$, $i = 1, 2$ be the canonical factorizations, where $\\nu_i : X_i' \\to X_i$ is the normalization of $X_i$ in $Y_i$. Then there exists a unique arrow $X'_2 \\to X'_1$ fitting into a commutative diagram $$ \\xymatrix{ Y_2 \\ar[d]_{f_2'} \\ar[r] & Y_1 \\ar[d]^{f_1'} \\\\ X_2' \\ar[d]_{\\nu_2} \\ar[r] & X_1' \\ar[d]^{\\nu_1} \\\\ X_2 \\ar[r] & X_1 } $$"}
+{"_id": "5501", "title": "morphisms-lemma-normalization-localization", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $U \\subset X$ be an open subscheme and set $V = f^{-1}(U)$. Then the normalization of $U$ in $V$ is the inverse image of $U$ in the normalization of $X$ in $Y$."}
+{"_id": "5502", "title": "morphisms-lemma-normalization-is-normalization", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X'$ be the normalization of $X$ in $Y$. Then the normalization of $X'$ in $Y$ is $X'$."}
+{"_id": "5503", "title": "morphisms-lemma-normalization-in-reduced", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \\to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$."}
+{"_id": "5504", "title": "morphisms-lemma-normalization-generic", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \\to X$ be the normalization of $X$ in $Y$. Every generic point of an irreducible component of $X'$ is the image of a generic point of an irreducible component of $Y$."}
+{"_id": "5505", "title": "morphisms-lemma-normalization-in-disjoint-union", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Suppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two schemes. Write $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$. Then $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$."}
+{"_id": "5506", "title": "morphisms-lemma-normalization-in-universally-closed", "text": "Let $f : X \\to S$ be a quasi-compact, quasi-separated and universally closed morphisms of schemes. Then $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_S$. In other words, the normalization of $S$ in $X$ is equal to the factorization $$ X \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X) \\longrightarrow S $$ of Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}."}
+{"_id": "5507", "title": "morphisms-lemma-normalization-in-integral", "text": "Let $f : Y \\to X$ be an integral morphism. Then the normalization of $X$ in $Y$ is equal to $Y$."}
+{"_id": "5508", "title": "morphisms-lemma-normal-normalization", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X'$ be the normalization of $X$ in $Y$. Assume \\begin{enumerate} \\item $Y$ is a normal scheme, \\item quasi-compact opens of $Y$ have finitely many irreducible components. \\end{enumerate} Then $X'$ is a disjoint union of integral normal schemes. Moreover, the morphism $Y \\to X'$ is dominant and induces a bijection of irreducible components."}
+{"_id": "5509", "title": "morphisms-lemma-nagata-normalization-finite-general", "text": "Let $f : X \\to S$ be a morphism. Assume that \\begin{enumerate} \\item $S$ is a Nagata scheme, \\item $f$ is quasi-compact and quasi-separated, \\item quasi-compact opens of $X$ have finitely many irreducible components, \\item if $x \\in X$ is a generic point of an irreducible component, then the field extension $\\kappa(f(x)) \\subset \\kappa(x)$ is finitely generated, and \\item $X$ is reduced. \\end{enumerate} Then the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite."}
+{"_id": "5510", "title": "morphisms-lemma-nagata-normalization-finite", "text": "Let $f : X \\to S$ be a morphism. Assume that \\begin{enumerate} \\item $S$ is a Nagata scheme, \\item $f$ is of finite type, \\item $X$ is reduced. \\end{enumerate} Then the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite."}
+{"_id": "5511", "title": "morphisms-lemma-relative-normalization-normal-codim-1", "text": "Let $f : Y \\to X$ be a finite type morphism of schemes with $Y$ reduced and $X$ Nagata. Let $X'$ be the normalization of $X$ in $Y$. Let $x' \\in X'$ be a point such that \\begin{enumerate} \\item $\\dim(\\mathcal{O}_{X', x'}) = 1$, and \\item the fibre of $Y \\to X'$ over $x'$ is empty. \\end{enumerate} Then $\\mathcal{O}_{X', x'}$ is a discrete valuation ring."}
+{"_id": "5512", "title": "morphisms-lemma-normalization-reduced", "text": "Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. The normalization morphism $\\nu$ factors through the reduction $X_{red}$ and $X^\\nu \\to X_{red}$ is the normalization of $X_{red}$."}
+{"_id": "5513", "title": "morphisms-lemma-description-normalization", "text": "Let $X$ be a reduced scheme such that every quasi-compact open has finitely many irreducible components. Let $\\Spec(A) = U \\subset X$ be an affine open. Then \\begin{enumerate} \\item $A$ has finitely many minimal primes $\\mathfrak q_1, \\ldots, \\mathfrak q_t$, \\item the total ring of fractions $Q(A)$ of $A$ is $Q(A/\\mathfrak q_1) \\times \\ldots \\times Q(A/\\mathfrak q_t)$, \\item the integral closure $A'$ of $A$ in $Q(A)$ is the product of the integral closures of the domains $A/\\mathfrak q_i$ in the fields $Q(A/\\mathfrak q_i)$, and \\item $\\nu^{-1}(U)$ is identified with the spectrum of $A'$ where $\\nu : X^\\nu \\to X$ is the normalization morphism. \\end{enumerate}"}
+{"_id": "5514", "title": "morphisms-lemma-stalk-normalization", "text": "Let $X$ be a scheme such that every quasi-compact open has a finite number of irreducible components. Let $\\nu : X^\\nu \\to X$ be the normalization of $X$. Let $x \\in X$. Then the following are canonically isomorphic as $\\mathcal{O}_{X, x}$-algebras \\begin{enumerate} \\item the stalk $(\\nu_*\\mathcal{O}_{X^\\nu})_x$, \\item the integral closure of $\\mathcal{O}_{X, x}$ in the total ring of fractions of $(\\mathcal{O}_{X, x})_{red}$, \\item the integral closure of $\\mathcal{O}_{X, x}$ in the product of the residue fields of the minimal primes of $\\mathcal{O}_{X, x}$ (and there are finitely many of these). \\end{enumerate}"}
+{"_id": "5515", "title": "morphisms-lemma-normalization-normal", "text": "Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. \\begin{enumerate} \\item The normalization $X^\\nu$ is a disjoint union of integral normal schemes. \\item The morphism $\\nu : X^\\nu \\to X$ is integral, surjective, and induces a bijection on irreducible components. \\item For any integral morphism $\\alpha : X' \\to X$ such that for $U \\subset X$ quasi-compact open the inverse image $\\alpha^{-1}(U)$ has finitely many irreducible components and $\\alpha|_{\\alpha^{-1}(U)} : \\alpha^{-1}(U) \\to U$ is birational\\footnote{This awkward formulation is necessary as we've only defined what it means for a morphism to be birational if the source and target have finitely many irreducible components. It suffices if $X'_{red} \\to X_{red}$ satisfies the condition.} there exists a factorization $X^\\nu \\to X' \\to X$ and $X^\\nu \\to X'$ is the normalization of $X'$. \\item For any morphism $Z \\to X$ with $Z$ a normal scheme such that each irreducible component of $Z$ dominates an irreducible component of $X$ there exists a unique factorization $Z \\to X^\\nu \\to X$. \\end{enumerate}"}
+{"_id": "5516", "title": "morphisms-lemma-normalization-in-terms-of-components", "text": "Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $Z_i \\subset X$, $i \\in I$ be the irreducible components of $X$ endowed with the reduced induced structure. Let $Z_i^\\nu \\to Z_i$ be the normalization. Then $\\coprod_{i \\in I} Z_i^\\nu \\to X$ is the normalization of $X$."}
+{"_id": "5517", "title": "morphisms-lemma-normalization-birational", "text": "Let $X$ be a reduced scheme with finitely many irreducible components. Then the normalization morphism $X^\\nu \\to X$ is birational."}
+{"_id": "5518", "title": "morphisms-lemma-finite-birational-over-normal", "text": "A finite (or even integral) birational morphism $f : X \\to Y$ of integral schemes with $Y$ normal is an isomorphism."}
+{"_id": "5519", "title": "morphisms-lemma-Japanese-normalization", "text": "Let $X$ be an integral, Japanese scheme. The normalization $\\nu : X^\\nu \\to X$ is a finite morphism."}
+{"_id": "5520", "title": "morphisms-lemma-nagata-normalization", "text": "Let $X$ be a Nagata scheme. The normalization $\\nu : X^\\nu \\to X$ is a finite morphism."}
+{"_id": "5521", "title": "morphisms-lemma-quasi-finite-points-open", "text": "\\begin{slogan} The locally quasi-finite locus of a morphism is open \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. The set of points of $X$ where $f$ is quasi-finite is an open $U \\subset X$. The induced morphism $U \\to S$ is locally quasi-finite."}
+{"_id": "5522", "title": "morphisms-lemma-quasi-finite-affine", "text": "Let $f : Y \\to X$ be a morphism of schemes. Assume \\begin{enumerate} \\item $X$ and $Y$ are affine, and \\item $f$ is quasi-finite. \\end{enumerate} Then there exists a diagram $$ \\xymatrix{ Y \\ar[rd]_f \\ar[rr]_j & & Z \\ar[ld]^\\pi \\\\ & X & } $$ with $Z$ affine, $\\pi$ finite and $j$ an open immersion."}
+{"_id": "5524", "title": "morphisms-lemma-characterize-universally-bounded", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n \\geq 0$. The following are equivalent: \\begin{enumerate} \\item the integer $n$ bounds the degrees of the fibres of $f$, and \\item for every morphism $\\Spec(k) \\to Y$, where $k$ is a field, the fibre product $X_k = \\Spec(k) \\times_Y X$ is finite over $k$ of degree $\\leq n$. \\end{enumerate} In this case the fibres of $f$ are universally bounded and the schemes $X_k$ have at most $n$ points. More precisely, if $X_k = \\{x_1, \\ldots, x_t\\}$, then we have $$ n \\geq \\sum\\nolimits_{i = 1, \\ldots, t} [\\kappa(x_i) : k] $$"}
+{"_id": "5525", "title": "morphisms-lemma-finite-locally-free-universally-bounded", "text": "If $f$ is a finite locally free morphism of degree $d$, then $d$ bounds the degree of the fibres of $f$."}
+{"_id": "5526", "title": "morphisms-lemma-composition-universally-bounded", "text": "A composition of morphisms with universally bounded fibres is a morphism with universally bounded fibres. More precisely, assume that $n$ bounds the degrees of the fibres of $f : X \\to Y$ and $m$ bounds the degrees of $g : Y \\to Z$. Then $nm$ bounds the degrees of the fibres of $g \\circ f : X \\to Z$."}
+{"_id": "5527", "title": "morphisms-lemma-base-change-universally-bounded", "text": "A base change of a morphism with universally bounded fibres is a morphism with universally bounded fibres. More precisely, if $n$ bounds the degrees of the fibres of $f : X \\to Y$ and $Y' \\to Y$ is any morphism, then the degrees of the fibres of the base change $f' : Y' \\times_Y X \\to Y'$ is also bounded by $n$."}
+{"_id": "5528", "title": "morphisms-lemma-descent-universally-bounded", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $Y' \\to Y$ be a morphism of schemes, and let $f' : X' = X_{Y'} \\to Y'$ be the base change of $f$. If $Y' \\to Y$ is surjective and $f'$ has universally bounded fibres, then $f$ has universally bounded fibres. More precisely, if $n$ bounds the degree of the fibres of $f'$, then also $n$ bounds the degrees of the fibres of $f$."}
+{"_id": "5529", "title": "morphisms-lemma-immersion-universally-bounded", "text": "An immersion has universally bounded fibres."}
+{"_id": "5530", "title": "morphisms-lemma-etale-universally-bounded", "text": "Let $f : X \\to Y$ be an \\'etale morphism of schemes. Let $n \\geq 0$. The following are equivalent \\begin{enumerate} \\item the integer $n$ bounds the degrees of the fibres, \\item for every field $k$ and morphism $\\Spec(k) \\to Y$ the base change $X_k = \\Spec(k) \\times_Y X$ has at most $n$ points, and \\item for every $y \\in Y$ and every separable algebraic closure $\\kappa(y) \\subset \\kappa(y)^{sep}$ the scheme $X_{\\kappa(y)^{sep}}$ has at most $n$ points. \\end{enumerate}"}
+{"_id": "5531", "title": "morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that \\begin{enumerate} \\item $f$ is locally quasi-finite, and \\item $X$ is quasi-compact. \\end{enumerate} Then $f$ has universally bounded fibres."}
+{"_id": "5532", "title": "morphisms-lemma-universally-bounded-permanence", "text": "Consider a commutative diagram of morphisms of schemes $$ \\xymatrix{ X \\ar[rd]_g \\ar[rr]_f & & Y \\ar[ld]^h \\\\ & Z & } $$ If $g$ has universally bounded fibres, and $f$ is surjective and flat, then also $h$ has universally bounded fibres. More precisely, if $n$ bounds the degree of the fibres of $g$, then also $n$ bounds the degree of the fibres of $h$."}
+{"_id": "5533", "title": "morphisms-proposition-generic-flatness", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $S$ is integral, \\item $f$ is of finite type, and \\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module. \\end{enumerate} Then there exists an open dense subscheme $U \\subset S$ such that $X_U \\to U$ is flat and of finite presentation and such that $\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation over $\\mathcal{O}_{X_U}$."}
+{"_id": "5534", "title": "morphisms-proposition-generic-flatness-reduced", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $S$ is reduced, \\item $f$ is of finite type, and \\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module. \\end{enumerate} Then there exists an open dense subscheme $U \\subset S$ such that $X_U \\to U$ is flat and of finite presentation and such that $\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation over $\\mathcal{O}_{X_U}$."}
+{"_id": "5535", "title": "morphisms-proposition-universal-homeomorphism-equal-residue-fields", "text": "Let $A \\subset B$ be a ring extension. The following are equivalent \\begin{enumerate} \\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism inducing isomorphisms on residue fields, and \\item every finite subset $E \\subset B$ is contained in an extension $$ A[b_1, \\ldots, b_n] \\subset B $$ such that $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$ for $i = 1, \\ldots, n$. \\end{enumerate}"}
+{"_id": "5536", "title": "morphisms-proposition-universal-homeomorphism", "text": "Let $A \\subset B$ be a ring extension. The following are equivalent \\begin{enumerate} \\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and \\item every finite subset $E \\subset B$ is contained in an extension $$ A[b_1, \\ldots, b_n] \\subset B $$ such that for $i = 1, \\ldots, n$ we have \\begin{enumerate} \\item $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$, or \\item there exists a prime number $p$ with $pb_i, b_i^p \\in A[b_1, \\ldots, b_{i - 1}]$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "5605", "title": "smoothing-theorem-popescu", "text": "Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps."}
+{"_id": "5606", "title": "smoothing-theorem-approximation-property", "text": "Let $R$ be a Noetherian local ring. Let $f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$. Suppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution in $R^\\wedge$. If $R$ is a henselian G-ring, then for every integer $N$ there exists a solution $(b_1, \\ldots, b_n) \\in R^n$ in $R$ such that $a_i - b_i \\in \\mathfrak m^NR^\\wedge$."}
+{"_id": "5607", "title": "smoothing-theorem-approximation-property-variant", "text": "Let $R$ be a Noetherian local ring. Let $f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$. Suppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution. If $R$ is a G-ring, then for every integer $N$ there exist \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a maximal ideal $\\mathfrak m' \\subset R'$ lying over $\\mathfrak m$ \\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$ \\end{enumerate} such that $\\kappa(\\mathfrak m) = \\kappa(\\mathfrak m')$ and $a_i - b_i \\in (\\mathfrak m')^NR^\\wedge$."}
+{"_id": "5608", "title": "smoothing-lemma-find-strictly-standard", "text": "Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$. Let $\\mathfrak q \\subset A$ be a prime ideal. Assume $R \\to A$ is smooth at $\\mathfrak q$. Then there exists an $a \\in A$, $a \\not \\in \\mathfrak q$, an integer $c$, $0 \\leq c \\leq \\min(n, m)$, subsets $U \\subset \\{1, \\ldots, n\\}$, $V \\subset \\{1, \\ldots, m\\}$ of cardinality $c$ such that $$ a = a' \\det(\\partial f_j/\\partial x_i)_{j \\in V, i \\in U} $$ for some $a' \\in A$ and $$ a f_\\ell \\in (f_j, j \\in V) + (f_1, \\ldots, f_m)^2 $$ for all $\\ell \\in \\{1, \\ldots, m\\}$."}
+{"_id": "5610", "title": "smoothing-lemma-elkik", "text": "Let $R \\to A$ be a ring map of finite presentation. The singular ideal $H_{A/R}$ is the radical of the ideal generated by strictly standard elements in $A$ over $R$ and also the radical of the ideal generated by elementary standard elements in $A$ over $R$."}
+{"_id": "5611", "title": "smoothing-lemma-strictly-standard-base-change", "text": "Let $R \\to A$ be a ring map of finite presentation. Let $R \\to R'$ be a ring map. If $a \\in A$ is elementary, resp.\\ strictly standard in $A$ over $R$, then $a \\otimes 1$ is elementary, resp.\\ strictly standard in $A \\otimes_R R'$ over $R'$."}
+{"_id": "5612", "title": "smoothing-lemma-final-solve", "text": "Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation over $R$. Assume that $H_{A/R} \\Lambda = \\Lambda$. Then there exists a factorization $A \\to B \\to \\Lambda$ with $B$ smooth over $R$."}
+{"_id": "5613", "title": "smoothing-lemma-improve-presentation", "text": "Let $R$ be a ring and let $A$ be a finitely presented $R$-algebra. There exists finite type $R$-algebra map $A \\to C$ which has a retraction with the following two properties \\begin{enumerate} \\item for each $a \\in A$ such that $R \\to A_a$ is a local complete intersection (More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}) the ring $C_a$ is smooth over $A_a$ and has a presentation $C_a = R[y_1, \\ldots, y_m]/J$ such that $J/J^2$ is free over $C_a$, and \\item for each $a \\in A$ such that $A_a$ is smooth over $R$ the module $\\Omega_{C_a/R}$ is free over $C_a$. \\end{enumerate}"}
+{"_id": "5614", "title": "smoothing-lemma-syntomic-complete-intersection", "text": "Let $R \\to A$ be a syntomic ring map. Then there exists a smooth $R$-algebra map $A \\to C$ with a retraction such that $C$ is a global relative complete intersection over $R$, i.e., $$ C \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) $$ flat over $R$ and all fibres of dimension $n - c$."}
+{"_id": "5615", "title": "smoothing-lemma-smooth-standard-smooth", "text": "Let $R \\to A$ be a smooth ring map. Then there exists a smooth $R$-algebra map $A \\to B$ with a retraction such that $B$ is standard smooth over $R$, i.e., $$ B \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) $$ and $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$ is invertible in $B$."}
+{"_id": "5616", "title": "smoothing-lemma-colimit-standard-smooth", "text": "Let $R \\to \\Lambda$ be a ring map. If $\\Lambda$ is a filtered colimit of smooth $R$-algebras, then $\\Lambda$ is a filtered colimit of standard smooth $R$-algebras."}
+{"_id": "5617", "title": "smoothing-lemma-standard-smooth-include-generators", "text": "Let $R \\to A$ be a standard smooth ring map. Let $E \\subset A$ be a finite subset of order $|E| = n$. Then there exists a presentation $A = R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$ with $c \\geq n$, with $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$ invertible in $A$, and such that $E$ is the set of congruence classes of $x_1, \\ldots, x_n$."}
+{"_id": "5618", "title": "smoothing-lemma-compare-standard", "text": "Let $R \\to A$ be a ring map of finite presentation. Let $a \\in A$. Consider the following conditions on $a$: \\begin{enumerate} \\item $A_a$ is smooth over $R$, \\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is stably free, \\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is free, \\item $A_a$ is standard smooth over $R$, \\item $a$ is strictly standard in $A$ over $R$, \\item $a$ is elementary standard in $A$ over $R$. \\end{enumerate} Then we have \\begin{enumerate} \\item[(a)] (4) $\\Rightarrow$ (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1), \\item[(b)] (6) $\\Rightarrow$ (5), \\item[(c)] (6) $\\Rightarrow$ (4), \\item[(d)] (5) $\\Rightarrow$ (2), \\item[(e)] (2) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are strictly standard in $A$ over $R$, \\item[(f)] (4) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are elementary standard in $A$ over $R$. \\end{enumerate}"}
+{"_id": "5619", "title": "smoothing-lemma-neron-functorial", "text": "In Situation \\ref{situation-neron} N\\'eron's blowup is functorial in the following sense \\begin{enumerate} \\item if $a \\in A$, $a \\not \\in \\mathfrak p$, then N\\'eron's blowup of $A_a$ is $A'_a$, and \\item if $B \\to A$ is a surjection of flat finite type $R$-algebras with kernel $I$, then $A'$ is the quotient of $B'/IB'$ by its $\\pi$-power torsion. \\end{enumerate}"}
+{"_id": "5620", "title": "smoothing-lemma-neron-blowup-smooth", "text": "In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth at $\\mathfrak p$ and that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$ is a separable field extension. Then $R \\to A'$ is smooth at $\\mathfrak p'$ and there is a short exact sequence $$ 0 \\to \\Omega_{A/R} \\otimes_A A'_{\\mathfrak p'} \\to \\Omega_{A'/R, \\mathfrak p'} \\to (A'/\\pi A')_{\\mathfrak p'}^{\\oplus c} \\to 0 $$ where $c = \\dim((A/\\pi A)_\\mathfrak p)$."}
+{"_id": "5621", "title": "smoothing-lemma-neron-when-smooth", "text": "In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth at $\\mathfrak q$ and that we have a surjection of $R$-algebras $B \\to A$ with kernel $I$. Assume $R \\to B$ smooth at $\\mathfrak p_B = (B \\to A)^{-1}\\mathfrak p$. If the cokernel of $$ I/I^2 \\otimes_A \\Lambda \\to \\Omega_{B/R} \\otimes_B \\Lambda $$ is a free $\\Lambda$-module, then $R \\to A$ is smooth at $\\mathfrak p$."}
+{"_id": "5622", "title": "smoothing-lemma-neron-desingularization", "text": "In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth at $\\mathfrak q$ and that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$ is a separable extension of fields. Then after a finite number of affine N\\'eron blowups the algebra $A$ becomes smooth over $R$ at $\\mathfrak p$."}
+{"_id": "5624", "title": "smoothing-lemma-lift-once", "text": "Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal. Assume that \\begin{enumerate} \\item $I^2 = 0$, and \\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras. \\end{enumerate} Let $\\varphi : A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite presentation over $R$. Then there exists a factorization $$ A \\to B/J \\to \\Lambda $$ where $B$ is a smooth $R$-algebra and $J \\subset IB$ is a finitely generated ideal."}
+{"_id": "5625", "title": "smoothing-lemma-lift-twice", "text": "Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal. Assume that \\begin{enumerate} \\item $I^2 = 0$, \\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and \\item $R \\to \\Lambda$ is flat. \\end{enumerate} Let $\\varphi : B \\to \\Lambda$ be an $R$-algebra map with $B$ smooth over $R$. Let $J \\subset IB$ be a finitely generated ideal such that $\\varphi(J) = 0$. Then there exists $R$-algebra maps $$ B \\xrightarrow{\\alpha} B' \\xrightarrow{\\beta} \\Lambda $$ such that $B'$ is smooth over $R$, such that $\\alpha(J) = 0$ and such that $\\beta \\circ \\alpha = \\varphi \\bmod I\\Lambda$."}
+{"_id": "5626", "title": "smoothing-lemma-lifting", "text": "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Suppose we have $R$-algebra maps $R/\\pi^2R \\to \\bar C \\to \\Lambda/\\pi^2\\Lambda$ with $\\bar C$ of finite presentation. Then there exists an $R$-algebra homomorphism $D \\to \\Lambda$ and a commutative diagram $$ \\xymatrix{ R/\\pi^2R \\ar[r] \\ar[d] & \\bar C \\ar[r] \\ar[d] & \\Lambda/\\pi^2\\Lambda \\ar[d] \\\\ R/\\pi R \\ar[r] & D/\\pi D \\ar[r] & \\Lambda/\\pi \\Lambda } $$ with the following properties \\begin{enumerate} \\item[(a)] $D$ is of finite presentation, \\item[(b)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with $\\pi \\not \\in \\mathfrak q$, \\item[(c)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with $\\pi \\in \\mathfrak q$ lying over a prime of $\\bar C$ where $R/\\pi^2 R \\to \\bar C$ is smooth, and \\item[(d)] $\\bar C/\\pi \\bar C \\to D/\\pi D$ is smooth at any prime lying over a prime of $\\bar C$ where $R/\\pi^2R \\to \\bar C$ is smooth. \\end{enumerate}"}
+{"_id": "5627", "title": "smoothing-lemma-desingularize", "text": "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and assume that $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Let $A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite presentation. Assume \\begin{enumerate} \\item the image of $\\pi$ is strictly standard in $A$ over $R$, and \\item there exists a section $\\rho : A/\\pi^4 A \\to R/\\pi^4 R$ which is compatible with the map to $\\Lambda/\\pi^4 \\Lambda$. \\end{enumerate} Then we can find $R$-algebra maps $A \\to B \\to \\Lambda$ with $B$ of finite presentation such that $\\mathfrak a B \\subset H_{B/R}$ where $\\mathfrak a = \\text{Ann}_R(\\text{Ann}_R(\\pi^2)/\\text{Ann}_R(\\pi))$."}
+{"_id": "5628", "title": "smoothing-lemma-desingularize-strictly-standard", "text": "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Let $A \\to \\Lambda$ and $D \\to \\Lambda$ be $R$-algebra maps with $A$ and $D$ of finite presentation. Assume \\begin{enumerate} \\item $\\pi$ is strictly standard in $A$ over $R$, and \\item there exists an $R$-algebra map $A/\\pi^4 A \\to D/\\pi^4 D$ compatible with the maps to $\\Lambda/\\pi^4 \\Lambda$. \\end{enumerate} Then we can find an $R$-algebra map $B \\to \\Lambda$ with $B$ of finite presentation and $R$-algebra maps $A \\to B$ and $D \\to B$ compatible with the maps to $\\Lambda$ such that $H_{D/R}B \\subset H_{B/D}$ and $H_{D/R}B \\subset H_{B/R}$."}
+{"_id": "5629", "title": "smoothing-lemma-desingularize-lifting-apply", "text": "Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Let $A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite presentation and assume $\\pi$ is strictly standard in $A$ over $R$. Let $$ A/\\pi^8A \\to \\bar C \\to \\Lambda/\\pi^8\\Lambda $$ be a factorization with $\\bar C$ of finite presentation. Then we can find a factorization $A \\to B \\to \\Lambda$ with $B$ of finite presentation such that $R_\\pi \\to B_\\pi$ is smooth and such that $$ H_{\\bar C/(R/\\pi^8 R)} \\cdot \\Lambda/\\pi^8\\Lambda \\subset \\sqrt{H_{B/R} \\Lambda} \\bmod \\pi^8\\Lambda. $$"}
+{"_id": "5630", "title": "smoothing-lemma-product", "text": "Let $R_i \\to \\Lambda_i$, $i = 1, 2$ be as in Situation \\ref{situation-global}. If PT holds for $R_i \\to \\Lambda_i$, $i = 1, 2$, then PT holds for $R_1 \\times R_2 \\to \\Lambda_1 \\times \\Lambda_2$."}
+{"_id": "5631", "title": "smoothing-lemma-delocalize-base", "text": "Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation over $R$. Let $S \\subset R$ be a multiplicative set. Let $S^{-1}A \\to B' \\to S^{-1}\\Lambda$ be a factorization with $B'$ smooth over $S^{-1}R$. Then we can find a factorization $A \\to B \\to \\Lambda$ such that some $s \\in S$ maps to an elementary standard element (Definition \\ref{definition-strictly-standard}) in $B$ over $R$."}
+{"_id": "5632", "title": "smoothing-lemma-reduce-to-field", "text": "\\begin{slogan} Proving Popescu approximation reduces to algebras over a field \\end{slogan} If for every Situation \\ref{situation-global} where $R$ is a field PT holds, then PT holds in general."}
+{"_id": "5633", "title": "smoothing-lemma-lift-solution", "text": "Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in Situation \\ref{situation-local}. Let $r \\geq 1$ and $\\pi_1, \\ldots, \\pi_r \\in R$ map to elements of $\\mathfrak q$. Assume \\begin{enumerate} \\item for $i = 1, \\ldots, r$ we have $$ \\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i) = \\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i^2) $$ and $$ \\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i) = \\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i^2) $$ \\item for $i = 1, \\ldots, r$ the element $\\pi_i$ maps to a strictly standard element in $A$ over $R$. \\end{enumerate} Then, if $$ R/(\\pi_1^8, \\ldots, \\pi_r^8)R \\to A/(\\pi_1^8, \\ldots, \\pi_r^8)A \\to \\Lambda/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda \\supset \\mathfrak q/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda $$ can be resolved, so can $R \\to A \\to \\Lambda \\supset \\mathfrak q$."}
+{"_id": "5634", "title": "smoothing-lemma-delocalize-weak", "text": "\\begin{reference} \\cite[Lemma 12.2]{swan} or \\cite[Lemma 2]{popescu-GND} \\end{reference} Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in Situation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$. Assume that $\\mathfrak q$ is minimal over $\\mathfrak h_A$ and that $R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q \\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved. Then there exists a factorization $A \\to C \\to \\Lambda$ with $C$ of finite presentation such that $H_{C/R} \\Lambda \\not \\subset \\mathfrak q$."}
+{"_id": "5635", "title": "smoothing-lemma-delocalize-height-zero", "text": "Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in Situation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$. Assume \\begin{enumerate} \\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$, \\item $R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q \\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved, and \\item $\\dim(\\Lambda_\\mathfrak q) = 0$. \\end{enumerate} Then $R \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved."}
+{"_id": "5636", "title": "smoothing-lemma-ogoma", "text": "Let $A$ be a Noetherian ring and let $M$ be a finite $A$-module. Let $S \\subset A$ be a multiplicative set. If $\\pi \\in A$ and $\\Ker(\\pi : S^{-1}M \\to S^{-1}M) = \\Ker(\\pi^2 : S^{-1}M \\to S^{-1}M)$ then there exists an $s \\in S$ such that for any $n > 0$ we have $\\Ker(s^n\\pi : M \\to M) = \\Ker((s^n\\pi)^2 : M \\to M)$."}
+{"_id": "5637", "title": "smoothing-lemma-find-sequence", "text": "Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal. Let $I \\subset \\mathfrak q$ be a prime. Let $n, e$ be positive integers Assume that $\\mathfrak q^n\\Lambda_\\mathfrak q \\subset I\\Lambda_\\mathfrak q$ and that $\\Lambda_\\mathfrak q$ is a regular local ring of dimension $d$. Then there exists an $n > 0$ and $\\pi_1, \\ldots, \\pi_d \\in \\Lambda$ such that \\begin{enumerate} \\item $(\\pi_1, \\ldots, \\pi_d)\\Lambda_\\mathfrak q = \\mathfrak q\\Lambda_\\mathfrak q$, \\item $\\pi_1^n, \\ldots, \\pi_d^n \\in I$, and \\item for $i = 1, \\ldots, d$ we have $$ \\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i) = \\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i^2). $$ \\end{enumerate}"}
+{"_id": "5638", "title": "smoothing-lemma-resolve-special", "text": "Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in Situation \\ref{situation-local} where \\begin{enumerate} \\item $k$ is a field, \\item $\\Lambda$ is Noetherian, \\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$, \\item $\\Lambda_\\mathfrak q$ is a regular local ring, and \\item the field extension $k \\subset \\kappa(\\mathfrak q)$ is separable. \\end{enumerate} Then $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved."}
+{"_id": "5639", "title": "smoothing-lemma-helper", "text": "Let $k$ be a field of characteristic $p > 0$. Let $(\\Lambda, \\mathfrak m, K)$ be an Artinian local $k$-algebra. Assume that $\\dim H_1(L_{K/k}) < \\infty$. Then $\\Lambda$ is a filtered colimit of Artinian local $k$-algebras $A$ with each map $A \\to \\Lambda$ flat, with $\\mathfrak m_A \\Lambda = \\mathfrak m$, and with $A$ essentially of finite type over $k$."}
+{"_id": "5640", "title": "smoothing-lemma-solution-modulo", "text": "Let $k$ be a field of characteristic $p > 0$. Let $\\Lambda$ be a Noetherian geometrically regular $k$-algebra. Let $\\mathfrak q \\subset \\Lambda$ be a prime ideal. Let $n \\geq 1$ be an integer and let $E \\subset \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$ be a finite subset. Then we can find $m \\geq 0$ and $\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$ with the following properties \\begin{enumerate} \\item setting $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$ we have $\\mathfrak q\\Lambda_\\mathfrak q = \\mathfrak p \\Lambda_\\mathfrak q$ and $k[y_1, \\ldots, y_m]_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is flat, \\item there is a factorization by homomorphisms of local Artinian rings $$ k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p \\to D \\to \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q $$ where the first arrow is essentially smooth and the second is flat, \\item $E$ is contained in $D$ modulo $\\mathfrak q^n\\Lambda_\\mathfrak q$. \\end{enumerate}"}
+{"_id": "5641", "title": "smoothing-lemma-enlarge-solution-modulo", "text": "Let $\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$, $n$, $\\mathfrak q$, $\\mathfrak p$ and $$ k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to D \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q $$ be as in Lemma \\ref{lemma-solution-modulo}. Then for any $\\lambda \\in \\Lambda \\setminus \\mathfrak q$ there exists an integer $q > 0$ and a factorization $$ k[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to D \\to D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q $$ such that $D \\to D'$ is an essentially smooth map of local Artinian rings, the last arrow is flat, and $\\lambda^q$ is in $D'$."}
+{"_id": "5642", "title": "smoothing-lemma-resolve-general", "text": "Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in Situation \\ref{situation-local} where \\begin{enumerate} \\item $k$ is a field of characteristic $p > 0$, \\item $\\Lambda$ is Noetherian and geometrically regular over $k$, \\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$. \\end{enumerate} Then $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved."}
+{"_id": "5643", "title": "smoothing-lemma-approximation-property-variant", "text": "Let $R$ be a Noetherian ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$. Suppose that $(a_1, \\ldots, a_n) \\in ((R_\\mathfrak p)^\\wedge)^n$ is a solution. If $R_\\mathfrak p$ is a G-ring, then for every integer $N$ there exist \\begin{enumerate} \\item an \\'etale ring map $R \\to R'$, \\item a prime ideal $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$ \\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$ \\end{enumerate} such that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and $a_i - b_i \\in (\\mathfrak p')^N(R'_{\\mathfrak p'})^\\wedge$."}
+{"_id": "5644", "title": "smoothing-lemma-henselian-pair", "text": "\\begin{slogan} Approximation for henselian pairs. \\end{slogan} Let $(A, I)$ be a henselian pair with $A$ Noetherian. Let $A^\\wedge$ be the $I$-adic completion of $A$. Assume at least one of the following conditions holds \\begin{enumerate} \\item $A \\to A^\\wedge$ is a regular ring map, \\item $A$ is a Noetherian G-ring, or \\item $(A, I)$ is the henselization (More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}) of a pair $(B, J)$ where $B$ is a Noetherian G-ring. \\end{enumerate} Given $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$ and $\\hat{a}_1, \\ldots, \\hat{a}_n \\in A^\\wedge$ such that $f_j(\\hat{a}_1, \\ldots, \\hat{a}_n) = 0$ for $j = 1, \\ldots, m$, for every $N \\geq 1$ there exist $a_1, \\ldots, a_n \\in A$ such that $\\hat{a}_i - a_i \\in I^N$ and such that $f_j(a_1, \\ldots, a_n) = 0$ for $j = 1, \\ldots, m$."}
+{"_id": "5645", "title": "smoothing-proposition-lift-smooth", "text": "\\begin{slogan} Smooth and syntomic algebras lift along surjections \\end{slogan} Let $R \\to R_0$ be a surjective ring map with kernel $I$. \\begin{enumerate} \\item If $R_0 \\to A_0$ is a syntomic ring map, then there exists a syntomic ring map $R \\to A$ such that $A/IA \\cong A_0$. \\item If $R_0 \\to A_0$ is a smooth ring map, then there exists a smooth ring map $R \\to A$ such that $A/IA \\cong A_0$. \\end{enumerate}"}
+{"_id": "5646", "title": "smoothing-proposition-lift", "text": "\\begin{slogan} Ind-smoothness of an algebra is stable under infinitesimal deformations \\end{slogan} Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal. Assume that \\begin{enumerate} \\item $I$ is nilpotent, \\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and \\item $R \\to \\Lambda$ is flat. \\end{enumerate} Then $\\Lambda$ is a filtered colimit of smooth $R$-algebras."}
+{"_id": "5649", "title": "chow-lemma-additivity-periodic-length", "text": "Let $R$ be a ring. Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \\to (M_1, N_1, \\varphi_1, \\psi_1) \\to (M_2, N_2, \\varphi_2, \\psi_2) \\to (M_3, N_3, \\varphi_3, \\psi_3) \\to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \\varphi_2, \\psi_2) = e_R(M_1, N_1, \\varphi_1, \\psi_1) + e_R(M_3, N_3, \\varphi_3, \\psi_3). $$"}
+{"_id": "5650", "title": "chow-lemma-finite-periodic-length", "text": "Let $R$ be a ring. If $(M, N, \\varphi, \\psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length, then $e_R(M, N, \\varphi, \\psi) = \\text{length}_R(M) - \\text{length}_R(N)$. In particular, if $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length, then $e_R(M, \\varphi, \\psi) = 0$."}
+{"_id": "5651", "title": "chow-lemma-compare-periodic-lengths", "text": "Let $R$ be a ring. Let $f : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$ be a map of $(2, 1)$-periodic complexes whose cohomology modules have finite length. If $\\Ker(f)$ and $\\Coker(f)$ have finite length, then $e_R(M, \\varphi, \\psi) = e_R(M', \\varphi', \\psi')$."}
+{"_id": "5652", "title": "chow-lemma-length-multiplication", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Let $x \\in R$. Assume that \\begin{enumerate} \\item $\\dim(\\text{Supp}(M)) \\leq 1$, and \\item $\\dim(\\text{Supp}(M/xM)) \\leq 0$. \\end{enumerate} Write $\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$. Then $$ e_R(M, 0, x) = \\sum\\nolimits_{i = 1, \\ldots, t} \\text{ord}_{R/\\mathfrak q_i}(x) \\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}). $$"}
+{"_id": "5653", "title": "chow-lemma-additivity-divisors-restricted", "text": "Let $R$ be a Noetherian local ring. Let $x \\in R$. If $M$ is a finite Cohen-Macaulay module over $R$ with $\\dim(\\text{Supp}(M)) = 1$ and $\\dim(\\text{Supp}(M/xM)) = 0$, then $$ \\text{length}_R(M/xM) = \\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i)) \\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}). $$ where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ are the minimal primes of the support of $M$. If $I \\subset R$ is an ideal such that $x$ is a nonzerodivisor on $R/I$ and $\\dim(R/I) = 1$, then $$ \\text{length}_R(R/(x, I)) = \\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i)) \\text{length}_{R_{\\mathfrak q_i}}((R/I)_{\\mathfrak q_i}) $$ where $\\mathfrak q_1, \\ldots, \\mathfrak q_n$ are the minimal primes over $I$."}
+{"_id": "5654", "title": "chow-lemma-powers-period-length-zero", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\varphi : M \\to M$ be an endomorphism and $n > 0$ such that $\\varphi^n = 0$ and such that $\\Ker(\\varphi)/\\Im(\\varphi^{n - 1})$ has finite length as an $R$-module. Then $$ e_R(M, \\varphi^i, \\varphi^{n - i}) = 0 $$ for $i = 0, \\ldots, n$."}
+{"_id": "5655", "title": "chow-lemma-multiply-period-length", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$ with $M$ finite and with cohomology groups of finite length over $R$. Let $x \\in R$ be such that $\\dim(\\text{Supp}(M/xM)) \\leq 0$. Then $$ e_R(M, x\\varphi, \\psi) = e_R(M, \\varphi, \\psi) - e_R(\\Im(\\varphi), 0, x) $$ and $$ e_R(M, \\varphi, x\\psi) = e_R(M, \\varphi, \\psi) + e_R(\\Im(\\psi), 0, x) $$"}
+{"_id": "5656", "title": "chow-lemma-glue-at-max", "text": "Let $A$ be a Noetherian ring. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ be pairwise distinct maximal ideals of $A$. For $i = 1, \\ldots, r$ let $\\varphi_i : A_{\\mathfrak m_i} \\to B_i$ be a ring map whose kernel and cokernel are annihilated by a power of $\\mathfrak m_i$. Then there exists a ring map $\\varphi : A \\to B$ such that \\begin{enumerate} \\item the localization of $\\varphi$ at $\\mathfrak m_i$ is isomorphic to $\\varphi_i$, and \\item $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are annihilated by a power of $\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_r$. \\end{enumerate} Moreover, if each $\\varphi_i$ is finite, injective, or surjective then so is $\\varphi$."}
+{"_id": "5657", "title": "chow-lemma-Noetherian-domain-dim-1-two-elements", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $a, b \\in R$ be nonzerodivisors. There exists a finite ring extension $R \\subset R'$ with $R'/R$ annihilated by a power of $\\mathfrak m$ and nonzerodivisors $t, a', b' \\in R'$ such that $a = ta'$ and $b = tb'$ and $R' = a'R' + b'R'$."}
+{"_id": "5658", "title": "chow-lemma-not-infinitely-divisible", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $a, b \\in R$ be nonzerodivisors with $a \\in \\mathfrak m$. There exists an integer $n = n(R, a, b)$ such that for a finite ring extension $R \\subset R'$ if $b = a^m c$ for some $c \\in R'$, then $m \\leq n$."}
+{"_id": "5659", "title": "chow-lemma-prepare-tame-symbol", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $r \\geq 2$ and let $a_1, \\ldots, a_r \\in A$ be nonzerodivisors not all units. Then there exist \\begin{enumerate} \\item a finite ring extension $A \\subset B$ with $B/A$ annihilated by a power of $\\mathfrak m$, \\item for each of maximal ideal $\\mathfrak m_j \\subset B$ a nonzerodivisor $\\pi_j \\in B_j = B_{\\mathfrak m_j}$, and \\item factorizations $a_i = u_{i, j} \\pi_j^{e_{i, j}}$ in $B_j$ with $u_{i, j} \\in B_j$ units and $e_{i, j} \\geq 0$. \\end{enumerate}"}
+{"_id": "5662", "title": "chow-lemma-norm-down-tame-symbol", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $A \\subset B$ be a finite ring extension with $B/A$ annihilated by a power of $\\mathfrak m$ and $\\mathfrak m$ not an associated prime of $B$. For $a, b \\in A$ nonzerodivisors we have $$ \\partial_A(a, b) = \\prod \\text{Norm}_{\\kappa(\\mathfrak m_j)/\\kappa(\\mathfrak m)}(\\partial_{B_j}(a, b)) $$ where the product is over the maximal ideals $\\mathfrak m_j$ of $B$ and $B_j = B_{\\mathfrak m_j}$."}
+{"_id": "5663", "title": "chow-lemma-tame-symbol-formally-smooth", "text": "Let $(A, \\mathfrak m, \\kappa) \\to (A', \\mathfrak m', \\kappa')$ be a local homomorphism of Noetherian local rings of dimension $1$. If $A \\to A'$ is flat, $\\mathfrak m' = \\mathfrak m A'$, and $\\kappa'/\\kappa$ is separable, then for $a_1, a_2 \\in A$ nonzerodivisors the tame symbol $\\partial_A(a_1, a_2)$ maps to $\\partial_{A'}(a_1, a_2)$."}
+{"_id": "5664", "title": "chow-lemma-perpare-key", "text": "Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring. Let $t \\in \\mathfrak m$ be a nonzerodivisor. Say $V(t) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_r\\}$. Let $A_{\\mathfrak q_i} \\subset B_i$ be a finite ring extension with $B_i/A_{\\mathfrak q_i}$ annihilated by a power of $t$. Then there exists a finite extension $A \\subset B$ of local rings identifying residue fields with $B_i \\cong B_{\\mathfrak q_i}$ and $B/A$ annihilated by a power of $t$."}
+{"_id": "5665", "title": "chow-lemma-key-nonzerodivisors", "text": "Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring. Let $a, b \\in A$ be nonzerodivisors. Then we have $$ \\sum \\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(a, b)) = 0 $$ where the sum is over the height $1$ primes $\\mathfrak q$ of $A$."}
+{"_id": "5666", "title": "chow-lemma-milnor-gersten-low-degree", "text": "\\begin{reference} When $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}. \\end{reference} Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$. Let $f, g \\in K^*$. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height $1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an element of $A^*_{\\mathfrak q}$. Then we have $$ \\sum\\nolimits_{i = 1, \\ldots, t} \\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(f, g)) = 0 $$ We can also write this as $$ \\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1} \\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(f, g)) = 0 $$ since at any height $1$ prime $\\mathfrak q$ of $A$ where $f, g \\in A^*_{\\mathfrak q}$ we have $\\partial_{A_{\\mathfrak q}}(f, g) = 1$."}
+{"_id": "5668", "title": "chow-lemma-multiplicity-finite", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme. \\begin{enumerate} \\item Let $Z' \\subset Z$ be an irreducible component and let $\\xi \\in Z'$ be its generic point. Then $$ \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi} < \\infty $$ \\item If $\\dim_\\delta(Z) \\leq k$ and $\\xi \\in Z$ with $\\delta(\\xi) = k$, then $\\xi$ is a generic point of an irreducible component of $Z$. \\end{enumerate}"}
+{"_id": "5669", "title": "chow-lemma-length-finite", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item The collection of irreducible components of the support of $\\mathcal{F}$ is locally finite. \\item Let $Z' \\subset \\text{Supp}(\\mathcal{F})$ be an irreducible component and let $\\xi \\in Z'$ be its generic point. Then $$ \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi < \\infty $$ \\item If $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$ and $\\xi \\in Z$ with $\\delta(\\xi) = k$, then $\\xi$ is a generic point of an irreducible component of $\\text{Supp}(\\mathcal{F})$. \\end{enumerate}"}
+{"_id": "5670", "title": "chow-lemma-cycle-closed-coherent", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme. If $\\dim_\\delta(Z) \\leq k$, then $[Z]_k = [{\\mathcal O}_Z]_k$."}
+{"_id": "5671", "title": "chow-lemma-additivity-sheaf-cycle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$ be a short exact sequence of coherent sheaves on $X$. Assume that the $\\delta$-dimension of the supports of $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ is $\\leq k$. Then $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$."}
+{"_id": "5672", "title": "chow-lemma-equal-dimension", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$. Then either $f(X)$ is contained in a proper closed subscheme of $Y$, or $f$ is dominant and the extension of function fields $R(Y) \\subset R(X)$ is finite."}
+{"_id": "5673", "title": "chow-lemma-quasi-compact-locally-finite", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $f$ is quasi-compact, and $\\{Z_i\\}_{i \\in I}$ is a locally finite collection of closed subsets of $X$. Then $\\{\\overline{f(Z_i)}\\}_{i \\in I}$ is a locally finite collection of closed subsets of $Y$."}
+{"_id": "5674", "title": "chow-lemma-compose-pushforward", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$, and $Z$ be locally of finite type over $S$. Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$."}
+{"_id": "5675", "title": "chow-lemma-exact-sequence-closed", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$ be closed subschemes such that $X = X_1 \\cup X_2$ set theoretically. For every $k \\in \\mathbf{Z}$ the sequence of abelian groups $$ \\xymatrix{ Z_k(X_1 \\cap X_2) \\ar[r] & Z_k(X_1) \\oplus Z_k(X_2) \\ar[r] & Z_k(X) \\ar[r] & 0 } $$ is exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$."}
+{"_id": "5676", "title": "chow-lemma-cycle-push-sheaf", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a proper morphism of schemes which are locally of finite type over $S$. \\begin{enumerate} \\item Let $Z \\subset X$ be a closed subscheme with $\\dim_\\delta(Z) \\leq k$. Then $$ f_*[Z]_k = [f_*{\\mathcal O}_Z]_k. $$ \\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then $$ f_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k. $$ \\end{enumerate} Note that the statement makes sense since $f_*\\mathcal{F}$ and $f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by Cohomology of Schemes, Proposition \\ref{coherent-proposition-proper-pushforward-coherent}."}
+{"_id": "5677", "title": "chow-lemma-flat-inverse-image-dimension", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. For any closed subset $Z \\subset Y$ we have $$ \\dim_\\delta(f^{-1}(Z)) = \\dim_\\delta(Z) + r. $$ provided $f^{-1}(Z)$ is nonempty. If $Z$ is irreducible and $Z' \\subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$."}
+{"_id": "5678", "title": "chow-lemma-inverse-image-locally-finite", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $\\{Z_i\\}_{i \\in I}$ is a locally finite collection of closed subsets of $Y$. Then $\\{f^{-1}(Z_i)\\}_{i \\in I}$ is a locally finite collection of closed subsets of $X$."}
+{"_id": "5679", "title": "chow-lemma-exact-sequence-open", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $U \\subset X$ be an open subscheme, and denote $i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$. For every $k \\in \\mathbf{Z}$ the sequence $$ \\xymatrix{ Z_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0 } $$ is an exact complex of abelian groups."}
+{"_id": "5680", "title": "chow-lemma-compose-flat-pullback", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X, Y, Z$ be locally of finite type over $S$. Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions $r$ and $s$. Then $g \\circ f$ is flat of relative dimension $r + s$ and $$ f^* \\circ g^* = (g \\circ f)^* $$ as maps $Z_k(Z) \\to Z_{k + r + s}(X)$."}
+{"_id": "5681", "title": "chow-lemma-pullback-coherent", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X, Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. \\begin{enumerate} \\item Let $Z \\subset Y$ be a closed subscheme with $\\dim_\\delta(Z) \\leq k$. Then we have $\\dim_\\delta(f^{-1}(Z)) \\leq k + r$ and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$. \\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$ and $$ f^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r} $$ in $Z_{k + r}(X)$. \\end{enumerate}"}
+{"_id": "5682", "title": "chow-lemma-flat-pullback-proper-pushforward", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a fibre product diagram of schemes locally of finite type over $S$. Assume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$. Then also $f'$ is proper and $g'$ is flat of relative dimension $r$. For any $k$-cycle $\\alpha$ on $X$ we have $$ g^*f_*\\alpha = f'_*(g')^*\\alpha $$ in $Z_{k + r}(Y')$."}
+{"_id": "5683", "title": "chow-lemma-finite-flat", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a finite locally free morphism of degree $d$ (see Morphisms, Definition \\ref{morphisms-definition-finite-locally-free}). Then $f$ is both proper and flat of relative dimension $0$, and $$ f_*f^*\\alpha = d\\alpha $$ for every $\\alpha \\in Z_k(Y)$."}
+{"_id": "5684", "title": "chow-lemma-divisor-delta-dimension", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral. \\begin{enumerate} \\item If $Z \\subset X$ is an integral closed subscheme, then the following are equivalent: \\begin{enumerate} \\item $Z$ is a prime divisor, \\item $Z$ has codimension $1$ in $X$, and \\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$. \\end{enumerate} \\item If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$. \\end{enumerate}"}
+{"_id": "5685", "title": "chow-lemma-finite-in-codimension-one", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\xi \\in Y$ be a point. Assume that \\begin{enumerate} \\item $X$, $Y$ are integral, \\item $Y$ is locally Noetherian \\item $f$ is proper, dominant and $R(Y) \\subset R(X)$ is finite, and \\item $\\dim(\\mathcal{O}_{Y, \\xi}) = 1$. \\end{enumerate} Then there exists an open neighbourhood $V \\subset Y$ of $\\xi$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite."}
+{"_id": "5686", "title": "chow-lemma-flat-pullback-principal-divisor", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \\dim_\\delta(Y)$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $g \\in R(Y)^*$. Then $$ f^*(\\text{div}_Y(g)) = \\text{div}_X(g) $$ in $Z_{n + r - 1}(X)$."}
+{"_id": "5687", "title": "chow-lemma-proper-pushforward-alteration", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$. Let $p : X \\to Y$ be a dominant proper morphism. Let $f \\in R(X)^*$. Set $$ g = \\text{Nm}_{R(X)/R(Y)}(f). $$ Then we have $p_*\\text{div}(f) = \\text{div}(g)$."}
+{"_id": "5688", "title": "chow-lemma-rational-function", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $f \\in R(X)^*$. Let $U \\subset X$ be a nonempty open such that $f$ corresponds to a section $f \\in \\Gamma(U, \\mathcal{O}_X^*)$. Let $Y \\subset X \\times_S \\mathbf{P}^1_S$ be the closure of the graph of $f : U \\to \\mathbf{P}^1_S$. Then \\begin{enumerate} \\item the projection morphism $p : Y \\to X$ is proper, \\item $p|_{p^{-1}(U)} : p^{-1}(U) \\to U$ is an isomorphism, \\item the pullbacks $Y_0 = q^{-1}D_0$ and $Y_\\infty = q^{-1}D_\\infty$ via the morphism $q : Y \\to \\mathbf{P}^1_S$ are defined (Divisors, Definition \\ref{divisors-definition-pullback-effective-Cartier-divisor}), \\item we have $$ \\text{div}_Y(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1} $$ \\item we have $$ \\text{div}_X(f) = p_*\\text{div}_Y(f) $$ \\item if we view $Y_0$ and $Y_\\infty$ as closed subschemes of $X$ via the morphism $p$ then we have $$ \\text{div}_X(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1} $$ \\end{enumerate}"}
+{"_id": "5689", "title": "chow-lemma-curve-principal-divisor", "text": "Let $K$ be any field. Let $X$ be a $1$-dimensional integral scheme endowed with a proper morphism $c : X \\to \\Spec(K)$. Let $f \\in K(X)^*$ be an invertible rational function. Then $$ \\sum\\nolimits_{x \\in X \\text{ closed}} [\\kappa(x) : K] \\text{ord}_{\\mathcal{O}_{X, x}}(f) = 0 $$ where $\\text{ord}$ is as in Algebra, Definition \\ref{algebra-definition-ord}. In other words, $c_*\\text{div}(f) = 0$."}
+{"_id": "5690", "title": "chow-lemma-restrict-to-open", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $U \\subset X$ be an open subscheme, and denote $i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$. Let $k \\in \\mathbf{Z}$. Suppose $\\alpha, \\beta \\in Z_k(X)$. If $\\alpha|_U \\sim_{rat} \\beta|_U$ then there exist a cycle $\\gamma \\in Z_k(Y)$ such that $$ \\alpha \\sim_{rat} \\beta + i_*\\gamma. $$ In other words, the sequence $$ \\xymatrix{ \\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0 } $$ is an exact complex of abelian groups."}
+{"_id": "5691", "title": "chow-lemma-exact-sequence-closed-chow", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$ be closed subschemes such that $X = X_1 \\cup X_2$ set theoretically. For every $k \\in \\mathbf{Z}$ the sequence of abelian groups $$ \\xymatrix{ \\CH_k(X_1 \\cap X_2) \\ar[r] & \\CH_k(X_1) \\oplus \\CH_k(X_2) \\ar[r] & \\CH_k(X) \\ar[r] & 0 } $$ is exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$."}
+{"_id": "5692", "title": "chow-lemma-prepare-flat-pullback-rational-equivalence", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Assume $Y$ integral with $\\dim_\\delta(Y) = k$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Then for $g \\in R(Y)^*$ we have $$ f^*\\text{div}_Y(g) = \\sum n_j i_{j, *}\\text{div}_{X_j}(g \\circ f|_{X_j}) $$ as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible components $X_j$ of $X$ and $n_j$ is the multiplicity of $X_j$ in $X$."}
+{"_id": "5693", "title": "chow-lemma-flat-pullback-rational-equivalence", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles on $Y$. Then $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$."}
+{"_id": "5694", "title": "chow-lemma-proper-pushforward-rational-equivalence", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $p : X \\to Y$ be a proper morphism. Suppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent. Then $p_*\\alpha$ is rationally equivalent to $p_*\\beta$."}
+{"_id": "5695", "title": "chow-lemma-rational-equivalence-family", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $W \\subset X \\times_S \\mathbf{P}^1_S$ be an integral closed subscheme of $\\delta$-dimension $k + 1$. Assume $W \\not = W_0$, and $W \\not = W_\\infty$. Then \\begin{enumerate} \\item $W_0$, $W_\\infty$ are effective Cartier divisors of $W$, \\item $W_0$, $W_\\infty$ can be viewed as closed subschemes of $X$ and $$ [W_0]_k \\sim_{rat} [W_\\infty]_k, $$ \\item for any locally finite family of integral closed subschemes $W_i \\subset X \\times_S \\mathbf{P}^1_S$ of $\\delta$-dimension $k + 1$ with $W_i \\not = (W_i)_0$ and $W_i \\not = (W_i)_\\infty$ we have $\\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k) \\sim_{rat} 0$ on $X$, and \\item for any $\\alpha \\in Z_k(X)$ with $\\alpha \\sim_{rat} 0$ there exists a locally finite family of integral closed subschemes $W_i \\subset X \\times_S \\mathbf{P}^1_S$ as above such that $\\alpha = \\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k)$. \\end{enumerate}"}
+{"_id": "5696", "title": "chow-lemma-closed-subscheme-cross-p1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z$ be a closed subscheme of $X \\times \\mathbf{P}^1$. Assume \\begin{enumerate} \\item $\\dim_\\delta(Z) \\leq k + 1$, \\item $\\dim_\\delta(Z_0) \\leq k$, $\\dim_\\delta(Z_\\infty) \\leq k$, and \\item for any embedded point $\\xi$ (Divisors, Definition \\ref{divisors-definition-embedded}) of $Z$ either $\\xi \\not \\in Z_0 \\cup Z_\\infty$ or $\\delta(\\xi) < k$. \\end{enumerate} Then $[Z_0]_k \\sim_{rat} [Z_\\infty]_k$ as $k$-cycles on $X$."}
+{"_id": "5699", "title": "chow-lemma-cycles-k-group", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The maps $$ Z_k(X) \\longrightarrow K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)), \\quad \\sum n_Z[Z] \\mapsto \\left[\\bigoplus\\nolimits_{n_Z > 0} \\mathcal{O}_Z^{\\oplus n_Z}\\right] - \\left[\\bigoplus\\nolimits_{n_Z < 0} \\mathcal{O}_Z^{\\oplus -n_Z}\\right] $$ and $$ K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)) \\longrightarrow Z_k(X),\\quad \\mathcal{F} \\longmapsto [\\mathcal{F}]_k $$ are mutually inverse isomorphisms."}
+{"_id": "5700", "title": "chow-lemma-finite-cycles-k-group", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. Then $\\pi_* : \\textit{Coh}(X) \\to \\textit{Coh}(Y)$ is an exact functor which sends $\\textit{Coh}_{\\leq k}(X)$ into $\\textit{Coh}_{\\leq k}(Y)$ and induces homomorphisms on $K_0$ of these categories and their quotients. The maps of Lemma \\ref{lemma-cycles-k-group} fit into a commutative diagram $$ \\xymatrix{ Z_k(X) \\ar[d]^{\\pi_*} \\ar[r] & K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)) \\ar[d]^{\\pi_*} \\ar[r] & Z_k(X) \\ar[d]^{\\pi_*} \\\\ Z_k(Y) \\ar[r] & K_0(\\textit{Coh}_{\\leq k}(Y)/\\textit{Coh}_{\\leq k - 1}(Y)) \\ar[r] & Z_k(Y) } $$"}
+{"_id": "5701", "title": "chow-lemma-from-chow-to-K", "text": "Let $X$ be a scheme locally of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. There is a canonical map $$ \\CH_k(X) \\longrightarrow K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)) $$ induced by the map $Z_k(X) \\to K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$ from Lemma \\ref{lemma-cycles-k-group}."}
+{"_id": "5702", "title": "chow-lemma-K-coherent-supported-on-closed", "text": "Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed subscheme. Denote $\\textit{Coh}_Z(X) \\subset \\textit{Coh}(X)$ the Serre subcategory of coherent $\\mathcal{O}_X$-modules whose set theoretic support is contained in $Z$. Then the exact inclusion functor $\\textit{Coh}(Z) \\to \\textit{Coh}_Z(X)$ induces an isomorphism $$ K'_0(Z) = K_0(\\textit{Coh}(Z)) \\longrightarrow K_0(\\textit{Coh}_Z(X)) $$"}
+{"_id": "5703", "title": "chow-lemma-compute-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section. Then $$ \\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1} $$ in $Z_{n - 1}(X)$ and $$ c_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1} $$ in $\\CH_{n - 1}(X)$."}
+{"_id": "5705", "title": "chow-lemma-c1-cap-additive", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$. Then $$ c_1(\\mathcal{L}) \\cap \\alpha  + c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha $$ in $\\CH_k(X)$ for every $\\alpha \\in Z_{k + 1}(X)$. Moreover, $c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$."}
+{"_id": "5706", "title": "chow-lemma-prepare-geometric-cap", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $Y$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module. Let $s \\in \\Gamma(Y, \\mathcal{L})$. Assume \\begin{enumerate} \\item $\\dim_\\delta(Y) \\leq k + 1$, \\item $\\dim_\\delta(Z(s)) \\leq k$, and \\item for every generic point $\\xi$ of an irreducible component of $Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$ induces an injection $\\mathcal{O}_{Y, \\xi} \\to \\mathcal{L}_\\xi$. \\end{enumerate} Write $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the irreducible components of $Y$ of $\\delta$-dimension $k + 1$. Set $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then \\begin{equation} \\label{equation-equal-as-cycles} [Z(s)]_k =  \\sum n_i[Z(s_i)]_k \\end{equation} as $k$-cycles on $Y$."}
+{"_id": "5707", "title": "chow-lemma-geometric-cap", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Y \\subset X$ be a closed subscheme. Let $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$. Assume \\begin{enumerate} \\item $\\dim_\\delta(Y) \\leq k + 1$, \\item $\\dim_\\delta(Z(s)) \\leq k$, and \\item for every generic point $\\xi$ of an irreducible component of $Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$ induces an injection $\\mathcal{O}_{Y, \\xi} \\to (\\mathcal{L}|_Y)_\\xi$\\footnote{For example, this holds if $s$ is a regular section of $\\mathcal{L}|_Y$.}. \\end{enumerate} Then $$ c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k $$ in $\\CH_k(X)$."}
+{"_id": "5708", "title": "chow-lemma-prepare-flat-pullback-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \\dim_\\delta(Y)$. Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$. Then we have $$ f^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i) $$ in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_i \\subset X$ of $\\delta$-dimension $n + r$, the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and $n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$."}
+{"_id": "5709", "title": "chow-lemma-flat-pullback-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Let $\\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha $$ in $\\CH_{k + r - 1}(X)$."}
+{"_id": "5710", "title": "chow-lemma-equal-c1-as-cycles", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a proper morphism. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Let $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$. Assume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$. Then $$ f_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) = [R(X) : R(Y)]\\text{div}_\\mathcal{L}(s). $$ as cycles on $Y$. In particular $$ f_*(c_1(f^*\\mathcal{L}) \\cap [X]) = c_1(\\mathcal{L}) \\cap [Y]. $$"}
+{"_id": "5711", "title": "chow-lemma-pushforward-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \\to Y$ be a proper morphism. Let $\\alpha \\in Z_{k + 1}(X)$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha $$ in $\\CH_k(Y)$."}
+{"_id": "5712", "title": "chow-lemma-key-formula", "text": "In the situation above the cycle $$ \\sum (Z_i \\to X)_*\\left( \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right) $$ is equal to the cycle $$ \\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i)) $$"}
+{"_id": "5713", "title": "chow-lemma-commutativity-on-integral", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\\mathcal{L}$ and a nonzero meromorphic section $t$ of $\\mathcal{N}$. Set $\\alpha = \\text{div}_\\mathcal{L}(s)$ and $\\beta = \\text{div}_\\mathcal{N}(t)$. Then $$ c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{L}) \\cap \\beta $$ in $\\CH_{n - 2}(X)$."}
+{"_id": "5714", "title": "chow-lemma-factors", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be invertible on $X$. The operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$ factors through rational equivalence to give an operation $$ c_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X) $$"}
+{"_id": "5715", "title": "chow-lemma-cap-commutative", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$. For any $\\alpha \\in \\CH_{k + 2}(X)$ we have $$ c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha $$ as elements of $\\CH_k(X)$."}
+{"_id": "5716", "title": "chow-lemma-support-cap-effective-Cartier", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$ in $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$."}
+{"_id": "5717", "title": "chow-lemma-easy-gysin", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. \\begin{enumerate} \\item Let $Z \\subset X$ be a closed subscheme such that $\\dim_\\delta(Z) \\leq k + 1$ and such that $D \\cap Z$ is an effective Cartier divisor on $Z$. Then $i^*[Z]_{k + 1} = [D \\cap Z]_k$. \\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$ and $s : \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ is injective. Then $$ i^*[\\mathcal{F}]_{k + 1} = [i^*\\mathcal{F}]_k $$ in $\\CH_k(D)$. \\end{enumerate}"}
+{"_id": "5718", "title": "chow-lemma-closed-in-X-gysin", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a proper morphism of schemes locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Form the diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ as in Remark \\ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have $i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$ (this makes sense as $f_*$ is defined on the level of cycles)."}
+{"_id": "5719", "title": "chow-lemma-gysin-flat-pullback", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a flat morphism of relative dimension $r$ of schemes locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Form the diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ as in Remark \\ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\\alpha$ on $X$ we have $(i')^*f^*\\alpha = g^*i^*\\alpha'$ in $\\CH_{k + r}(D)$ (this makes sense as $f^*$ is defined on the level of cycles)."}
+{"_id": "5720", "title": "chow-lemma-gysin-factors-general", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \\dim_\\delta(X)$. Let $i : D \\to X$ be an effective Cartier divisor. Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module and let $t$ be a nonzero meromorphic section of $\\mathcal{N}$. Then $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}) \\cap [D]_{n - 1}$ in $\\CH_{n - 2}(D)$."}
+{"_id": "5721", "title": "chow-lemma-gysin-factors", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$."}
+{"_id": "5722", "title": "chow-lemma-gysin-back", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Then $i^*i_* : \\CH_k(D) \\to \\CH_{k - 1}(D)$ sends $\\alpha$ to $c_1(\\mathcal{L}|_D) \\cap \\alpha$."}
+{"_id": "5723", "title": "chow-lemma-gysin-commutes-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module. Then $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$ in $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(X)$."}
+{"_id": "5724", "title": "chow-lemma-gysin-commutes-gysin", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ and $(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in Definition \\ref{definition-gysin-homomorphism}. Then the diagram $$ \\xymatrix{ \\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d]^{j^*} \\\\ \\CH_{k - 1}(D') \\ar[r]^{(j')^*} & \\CH_{k - 2}(D \\cap D') } $$ commutes where each of the maps is a gysin map."}
+{"_id": "5725", "title": "chow-lemma-relative-effective-cartier", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \\to Y$ be a flat morphism of relative dimension $r$. Let $i : D \\to X$ be a relative effective Cartier divisor (Divisors, Definition \\ref{divisors-definition-relative-effective-Cartier-divisor}). Let $\\mathcal{L} = \\mathcal{O}_X(D)$. For any $\\alpha \\in \\CH_{k + 1}(Y)$ we have $$ i^*p^*\\alpha = (p|_D)^*\\alpha $$ in $\\CH_{k + r}(D)$ and $$ c_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha) $$ in $\\CH_{k + r}(X)$."}
+{"_id": "5726", "title": "chow-lemma-pullback-affine-fibres-surjective", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Assume that for every $y \\in Y$, there exists an open neighbourhood $U \\subset Y$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is identified with the morphism $U \\times \\mathbf{A}^r \\to U$. Then $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all $k \\in \\mathbf{Z}$."}
+{"_id": "5727", "title": "chow-lemma-linebundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $$ p : L = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L})) \\longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$."}
+{"_id": "5728", "title": "chow-lemma-linebundle-formulae", "text": "In the situation of Lemma \\ref{lemma-linebundle} denote $o : X \\to L$ the zero section (see proof of the lemma). Then we have \\begin{enumerate} \\item $o(X)$ is the zero scheme of a regular global section of $p^*\\mathcal{L}^{\\otimes -1}$, \\item $o_* : \\CH_k(X) \\to \\CH_k(L)$ as $o$ is a closed immersion, \\item $o^* : \\CH_{k + 1}(L) \\to \\CH_k(X)$ as $o(X)$ is an effective Cartier divisor, \\item $o^* p^* : \\CH_k(X) \\to \\CH_k(X)$ is the identity map, \\item $o_*\\alpha = - p^*(c_1(\\mathcal{L}) \\cap \\alpha)$ for any $\\alpha \\in \\CH_k(X)$, and \\item $o^* o_* : \\CH_k(X) \\to \\CH_{k - 1}(X)$ is equal to the map $\\alpha \\mapsto - c_1(\\mathcal{L}) \\cap \\alpha$. \\end{enumerate}"}
+{"_id": "5729", "title": "chow-lemma-decompose-section", "text": "Let $Y$ be a scheme. Let $\\mathcal{L}_i$, $i = 1, 2$ be invertible $\\mathcal{O}_Y$-modules. Let $s$ be a global section of $\\mathcal{L}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{L}_2$. Denote $i : D \\to X$ the zero scheme of $s$. Then there exists a commutative diagram $$ \\xymatrix{ D_1 \\ar[r]_{i_1} \\ar[d]_{p_1} & L \\ar[d]^p & D_2 \\ar[l]^{i_2} \\ar[d]^{p_2} \\\\ D \\ar[r]^i & Y & D \\ar[l]_i } $$ and sections $s_i$ of $p^*\\mathcal{L}_i$ such that the following hold: \\begin{enumerate} \\item $p^*s = s_1 \\otimes s_2$, \\item $p$ is of finite type and flat of relative dimension $1$, \\item $D_i$ is the zero scheme of $s_i$, \\item $D_i \\cong \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}_{1 - i}^{\\otimes -1})|_D))$ over $D$ for $i = 1, 2$, \\item $p^{-1}D = D_1 \\cup D_2$ (scheme theoretic union), \\item $D_1 \\cap D_2$ (scheme theoretic intersection) maps isomorphically to $D$, and \\item $D_1 \\cap D_2 \\to D_i$ is the zero section of the line bundle $D_i \\to D$ for $i = 1, 2$. \\end{enumerate} Moreover, the formation of this diagram and the sections $s_i$ commutes with arbitrary base change."}
+{"_id": "5731", "title": "chow-lemma-normal-cone-effective-Cartier", "text": "In Situation \\ref{situation-setup} let $X$ be a scheme locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. There exists a commutative diagram $$ \\xymatrix{ D' \\ar[r]_{i'} \\ar[d]_p & X' \\ar[d]^g \\\\ D \\ar[r]^i & X } $$ such that \\begin{enumerate} \\item $p$ and $g$ are of finite type and flat of relative dimension $1$, \\item $p^* : \\CH_k(D) \\to \\CH_{k + 1}(D')$ is injective for all $k$, \\item $D' \\subset X'$ is the zero scheme of a global section $s' \\in \\Gamma(X', \\mathcal{O}_{X'})$, \\item $p^*i^* = (i')^*g^*$ as maps $\\CH_k(X) \\to \\CH_k(D')$. \\end{enumerate} Moreover, these properties remain true after arbitrary base change by morphisms $Y \\to X$ which are locally of finite type."}
+{"_id": "5732", "title": "chow-lemma-flat-pullback-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$ between schemes locally of finite type over $S$. Then the rule that to $Y' \\to Y$ assigns $(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$ is a bivariant class of degree $-r$."}
+{"_id": "5733", "title": "chow-lemma-gysin-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. Then the rule that to $f : X' \\to X$ assigns $(i')^* : \\CH_k(X') \\to \\CH_{k - 1}(D')$ where $D' = D \\times_X X'$ is a bivariant class of degree $1$."}
+{"_id": "5734", "title": "chow-lemma-push-proper-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes locally of finite type over $S$. Let $c \\in A^p(X \\to Z)$ and assume $f$ is proper. Then the rule that to $Z' \\to Z$ assigns $\\alpha \\longmapsto f'_*(c \\cap \\alpha)$ is a bivariant class denoted $f_* \\circ c \\in A^p(Y \\to Z)$."}
+{"_id": "5735", "title": "chow-lemma-cap-c1-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then the rule that to $f : X' \\to X$ assigns $c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$ is a bivariant class of degree $1$."}
+{"_id": "5736", "title": "chow-lemma-c1-center", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item $c_1(\\mathcal{L}) \\in A^1(X)$ is in the center of $A^*(X)$ and \\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$, then $c \\circ c_1(\\mathcal{L}) = c_1(f^*\\mathcal{L}) \\circ c$. \\end{enumerate}"}
+{"_id": "5737", "title": "chow-lemma-vanish-above-dimension", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a finite type scheme over $S$ which has an ample invertible sheaf. Assume $d = \\dim(X) < \\infty$ (here we really mean dimension and not $\\delta$-dimension). Then for any invertible sheaves $\\mathcal{L}_1, \\ldots, \\mathcal{L}_{d + 1}$ on $X$ we have $c_1(\\mathcal{L}_1) \\circ \\ldots \\circ c_1(\\mathcal{L}_{d + 1}) = 0$ in $A^{d + 1}(X)$."}
+{"_id": "5740", "title": "chow-lemma-bivariant-zero", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Let $c \\in A^p(X \\to Y)$. For $Y'' \\to Y' \\to Y$ set $X'' = Y'' \\times_Y X$ and $X' = Y' \\times_Y X$. The following are equivalent \\begin{enumerate} \\item $c$ is zero, \\item $c \\cap [Y'] = 0$ in $\\CH_*(X')$ for every integral scheme $Y'$ locally of finite type over $Y$, and \\item for every integral scheme $Y'$ locally of finite type over $Y$, there exists a proper birational morphism $Y'' \\to Y'$ such that $c \\cap [Y''] = 0$ in $\\CH_*(X'')$. \\end{enumerate}"}
+{"_id": "5741", "title": "chow-lemma-disjoint-decomposition-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Assume we have disjoint union decompositions $X = \\coprod_{i \\in I} X_i$ and $Y = \\coprod_{j \\in J} Y_j$ by open and closed subschemes and a map $a : I \\to J$ of sets such that $f(X_i) \\subset Y_{a(i)}$. Then $$ A^p(X \\to Y) = \\prod\\nolimits_{i \\in I} A^p(X_i \\to Y_{a(i)}) $$"}
+{"_id": "5742", "title": "chow-lemma-cap-projective-bundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle associated to $\\mathcal{E}$. For any $\\alpha \\in \\CH_k(X)$ the element $$ \\pi_*\\left( c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha \\right) \\in \\CH_{k + r - 1 - s}(X) $$ is $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$."}
+{"_id": "5743", "title": "chow-lemma-chow-ring-projective-bundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle associated to $\\mathcal{E}$. The map $$ \\bigoplus\\nolimits_{i = 0}^{r - 1} \\CH_{k + i}(X) \\longrightarrow \\CH_{k + r - 1}(P), $$ $$ (\\alpha_0, \\ldots, \\alpha_{r-1}) \\longmapsto \\pi^*\\alpha_0 + c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1 + \\ldots + c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1} $$ is an isomorphism."}
+{"_id": "5744", "title": "chow-lemma-vectorbundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $$ p : E = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E})) \\longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$."}
+{"_id": "5746", "title": "chow-lemma-determine-intersections", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle associated to $\\mathcal{E}$. For $\\alpha \\in Z_k(X)$ the elements $c_j(\\mathcal{E}) \\cap \\alpha$ are the unique elements $\\alpha_j$ of $\\CH_{k - j}(X)$ such that $\\alpha_0 = \\alpha$ and $$ \\sum\\nolimits_{i = 0}^r (-1)^i c_1(\\mathcal{O}_P(1))^i \\cap \\pi^*(\\alpha_{r - i}) = 0 $$ holds in the Chow group of $P$."}
+{"_id": "5747", "title": "chow-lemma-cap-chern-class-factors-rational-equivalence", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. If $\\alpha \\sim_{rat} \\beta$ are rationally equivalent $k$-cycles on $X$ then $c_j(\\mathcal{E}) \\cap \\alpha = c_j(\\mathcal{E}) \\cap \\beta$ in $\\CH_{k - j}(X)$."}
+{"_id": "5748", "title": "chow-lemma-pushforward-cap-cj", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $p : X \\to Y$ be a proper morphism. Let $\\alpha$ be a $k$-cycle on $X$. Let $\\mathcal{E}$ be a finite locally free sheaf on $Y$. Then $$ p_*(c_j(p^*\\mathcal{E}) \\cap \\alpha) = c_j(\\mathcal{E}) \\cap p_*\\alpha $$"}
+{"_id": "5749", "title": "chow-lemma-flat-pullback-cap-cj", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_j(\\mathcal{E}) \\cap \\alpha) = c_j(f^*\\mathcal{E}) \\cap f^*\\alpha $$"}
+{"_id": "5750", "title": "chow-lemma-cap-chern-class-commutes-with-gysin", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Then $c_j(\\mathcal{E}|_D) \\cap i^*\\alpha = i^*(c_j(\\mathcal{E}) \\cap \\alpha)$ for all $\\alpha \\in \\CH_k(X)$."}
+{"_id": "5751", "title": "chow-lemma-cap-cp-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. Let $0 \\leq p \\leq r$. Then the rule that to $f : X' \\to X$ assigns $c_p(f^*\\mathcal{E}) \\cap - : \\CH_k(X') \\to \\CH_{k - p}(X')$ is a bivariant class of degree $p$."}
+{"_id": "5752", "title": "chow-lemma-cap-commutative-chern", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. Then \\begin{enumerate} \\item $c_j(\\mathcal{E}) \\in A^j(X)$ is in the center of $A^*(X)$ and \\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$, then $c \\circ c_j(\\mathcal{E}) = c_j(f^*\\mathcal{E}) \\circ c$. \\end{enumerate} In particular, if $\\mathcal{F}$ is a second locally free $\\mathcal{O}_X$-module on $X$ of rank $s$, then $$ c_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha = c_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha $$ as elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$."}
+{"_id": "5753", "title": "chow-lemma-chern-classes-E-tensor-L", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Then we have \\begin{equation} \\label{equation-twist} c_i({\\mathcal E} \\otimes {\\mathcal L}) = \\sum\\nolimits_{j = 0}^i \\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j \\end{equation} in $A^*(X)$."}
+{"_id": "5754", "title": "chow-lemma-get-rid-of-trivial-subbundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0 $$ Then we have $$ c_r(\\mathcal{E}) = 0, \\quad c_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1 $$ in $A^*(X)$."}
+{"_id": "5755", "title": "chow-lemma-additivity-invertible-subsheaf", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0 $$ where $\\mathcal{L}$ is an invertible sheaf. Then $$ c(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F}) $$ in $A^*(X)$."}
+{"_id": "5756", "title": "chow-lemma-additivity-chern-classes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Suppose that ${\\mathcal E}$ sits in an exact sequence $$ 0 \\to {\\mathcal E}_1 \\to {\\mathcal E} \\to {\\mathcal E}_2 \\to 0 $$ of finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$. The total Chern classes satisfy $$ c({\\mathcal E}) = c({\\mathcal E}_1) c({\\mathcal E}_2) $$ in $A^*(X)$."}
+{"_id": "5757", "title": "chow-lemma-chern-filter-by-linebundles", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible $\\mathcal{O}_X$-modules on $X$. Let $\\mathcal{E}$ be a locally free rank $\\mathcal{O}_X$-module endowed with a filtration $$ 0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2 \\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E} $$ such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$. Set $c_1({\\mathcal L}_i) = x_i$. Then $$ c(\\mathcal{E}) = \\prod\\nolimits_{i = 1}^r (1 + x_i) $$ in $A^*(X)$."}
+{"_id": "5760", "title": "chow-lemma-degrees-and-numerical-intersections", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \\subset X$ be a closed subscheme of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. Then $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) = \\deg( c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z]_d) $$ where the left hand side is defined in Varieties, Definition \\ref{varieties-definition-intersection-number}. In particular, $$ \\deg_\\mathcal{L}(Z) = \\deg(c_1(\\mathcal{L})^d \\cap [Z]_d) $$ if $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module."}
+{"_id": "5761", "title": "chow-lemma-locally-equidimensional", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Write $\\delta = \\delta_{X/S}$ as in Section \\ref{section-setup}. The following are equivalent \\begin{enumerate} \\item There exists a decomposition $X = \\coprod_{n \\in \\mathbf{Z}} X_n$ into open and closed subschemes such that $\\delta(\\xi) = n$ whenever $\\xi \\in X_n$ is a generic point of an irreducible component of $X_n$. \\item For all $x \\in X$ there exists an open neighbourhood $U \\subset X$ of $x$ and an integer $n$ such that $\\delta(\\xi) = n$ whenever $\\xi \\in U$ is a generic point of an irreducible component of $U$. \\item For all $x \\in X$ there exists an integer $n_x$ such that $\\delta(\\xi) = n_x$ for any generic point $\\xi$ of an irreducible component of $X$ containing $x$. \\end{enumerate} The conditions are satisfied if $X$ is either normal or Cohen-Macaulay\\footnote{In fact, it suffices if $X$ is $(S_2)$. Compare with Local Cohomology, Lemma \\ref{local-cohomology-lemma-catenary-S2-equidimensional}.}."}
+{"_id": "5762", "title": "chow-lemma-splitting-principle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}_i$ be a finite collection of locally free $\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective flat morphism $\\pi : P \\to X$ of relative dimension $d$ such that \\begin{enumerate} \\item for any morphism $f : Y \\to X$ the map $\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and \\item each $\\pi^*\\mathcal{E}_i$ has a filtration whose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$ are invertible ${\\mathcal O}_P$-modules. \\end{enumerate} Moreover, when (1) holds the restriction map $A^*(X) \\to A^*(P)$ (Remark \\ref{remark-pullback-cohomology}) is injective."}
+{"_id": "5763", "title": "chow-lemma-chern-classes-dual", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module with dual $\\mathcal{E}^\\vee$. Then $$ c_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E}) $$ in $A^i(X)$."}
+{"_id": "5764", "title": "chow-lemma-chern-classes-tensor-product", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free $\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have $$ c_1(\\mathcal{E} \\otimes \\mathcal{F}) = r c_1(\\mathcal{F}) + s c_1(\\mathcal{E}) $$ $$ c_2(\\mathcal{E} \\otimes \\mathcal{F}) = r c_2(\\mathcal{F}) + s c_2(\\mathcal{E}) + {r \\choose 2} c_1(\\mathcal{F})^2 + (rs - 1) c_1(\\mathcal{F})c_1(\\mathcal{E}) + {s \\choose 2} c_1(\\mathcal{E})^2 $$ and so on in $A^*(X)$."}
+{"_id": "5765", "title": "chow-lemma-top-chern-class", "text": "In the situation described just above assume $\\dim_\\delta(X') = n$, that $f^*\\mathcal{E}$ has constant rank $r$, that $\\dim_\\delta(Z(s)) \\leq n - r$, and that for every generic point $\\xi \\in Z(s)$ with $\\delta(\\xi) = n - r$ the ideal of $Z(s)$ in $\\mathcal{O}_{X', \\xi}$ is generated by a regular sequence of length $r$. Then $$ c_r(\\mathcal{E}) \\cap [X']_n = [Z(s)]_{n - r} $$ in $\\CH_*(X')$."}
+{"_id": "5766", "title": "chow-lemma-easy-virtual-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $$ 0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0 $$ be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules. Consider the closed embedding $$ i : N' = \\underline{\\text{Spec}}_X(\\text{Sym}((\\mathcal{N}')^\\vee)) \\longrightarrow N = \\underline{\\text{Spec}}_X(\\text{Sym}(\\mathcal{N}^\\vee)) $$ For $\\alpha \\in \\CH_k(X)$ we have $$ i_*(p')^*\\alpha = p^*(c_{top}(\\mathcal{E}) \\cap \\alpha) $$ where $p' : N' \\to X$ and $p : N \\to X$ are the structure morphisms."}
+{"_id": "5767", "title": "chow-lemma-chern-character-additive", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $ 0 \\to \\mathcal{E}_1 \\to \\mathcal{E} \\to \\mathcal{E}_2 \\to 0 $ be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules. Then we have the equality $$ ch(\\mathcal{E}) = ch(\\mathcal{E}_1) + ch(\\mathcal{E}_2) $$ in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have $P_p(\\mathcal{E}) = P_p(\\mathcal{E}_1) + P_p(\\mathcal{E}_2)$ in $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}."}
+{"_id": "5768", "title": "chow-lemma-chern-character-multiplicative", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}_1$ and $\\mathcal{E}_2$ be finite locally free $\\mathcal{O}_X$-modules. Then we have the equality $$ ch(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) = ch(\\mathcal{E}_1) ch(\\mathcal{E}_2) $$ in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have $$ P_p(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) = \\sum\\nolimits_{p_1 + p_2 = p} {p \\choose p_1} P_{p_1}(\\mathcal{E}_1) P_{p_2}(\\mathcal{E}_2) $$ in $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}."}
+{"_id": "5771", "title": "chow-lemma-commutative-chern-perfect", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$ be perfect. If the Chern classes of $E$ are defined then \\begin{enumerate} \\item $c_p(E)$ is in the center of the algebra $A^*(X)$ and \\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$, then $c \\circ c_j(E) = c_j(Lf^*E) \\circ c$. \\end{enumerate}"}
+{"_id": "5772", "title": "chow-lemma-additivity-on-perfect", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $$ E_1 \\to E_2 \\to E_3 \\to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$. If $E_1 \\to E_2$ can be represented be a map of bounded complexes of finite locally free $\\mathcal{O}_X$-modules, then we have $c(E_2) = c(E_1) c(E_3)$, $ch(E_2) = ch(E_1) + ch(E_3)$, and $P_p(E_2) = P_p(E_1) + P_p(E_3)$."}
+{"_id": "5774", "title": "chow-lemma-chern-class-perfect-tensor-invertible", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E$ be a perfect object of $D(\\mathcal{O}_X)$ whose Chern classes are defined. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $$ c_i(E \\otimes \\mathcal{L}) = \\sum\\nolimits_{j = 0}^i \\binom{r - i + j}{j} c_{i - j}(E) c_1(\\mathcal{L})^j $$ provided $E$ has constant rank $r \\in \\mathbf{Z}$."}
+{"_id": "5775", "title": "chow-lemma-chern-classes-perfect-tensor-product", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E$ and $F$ be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes are defined. Then we have $$ c_1(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = r(E) c_1(\\mathcal{F}) + r(F) c_1(\\mathcal{E}) $$ and for $c_2(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F)$ we have the expression $$ r(E) c_2(F) + r(F) c_2(E) + {r(E) \\choose 2} c_1(F)^2 + (r(E)r(F) - 1) c_1(F)c_1(E) + {r(F) \\choose 2} c_1(E)^2 $$ and so on for higher Chern classes in $A^*(X)$. Similarly, we have $ch(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = ch(E) ch(F)$ in $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have $$ P_p(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = \\sum\\nolimits_{p_1 + p_2 = p} {p \\choose p_1} P_{p_1}(E) P_{p_2}(F) $$ in $A^p(X)$."}
+{"_id": "5776", "title": "chow-lemma-silly", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let $E_2 \\in D(\\mathcal{O}_{X_2})$ be a perfect object. Assume \\begin{enumerate} \\item Chern classes of $E_2$ are defined, \\item the restriction $E_2|_{X_1 \\cap X_2}$ is zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1 \\cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$. \\end{enumerate} Then there is a canonical bivariant class $$ P'_p(E_2),\\text{ resp. }c'_p(E_2) \\in A^p(X_2 \\to X) $$ characterized by the property $$ P'_p(E_2) \\cap i_{2, *} \\alpha_2 = P_p(E_2) \\cap \\alpha_2 \\quad\\text{and}\\quad P'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0, $$ respectively $$ c'_p(E_2) \\cap i_{2, *} \\alpha_2 = c_p(E_2) \\cap \\alpha_2 \\quad\\text{and}\\quad c'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0 $$ for $\\alpha_i \\in \\CH_k(X_i)$ and similarly after any base change $X' \\to X$ locally of finite type."}
+{"_id": "5777", "title": "chow-lemma-silly-independent", "text": "In Lemma \\ref{lemma-silly} the bivariant class $P'_p(E_2)$, resp.\\ $c'_p(E_2)$ in $A^p(X_2 \\to X)$ does not depend on the choice of $X_1$."}
+{"_id": "5778", "title": "chow-lemma-silly-silly", "text": "In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a perfect $E \\in D(\\mathcal{O}_X)$ such that $E|_{X_1}$ is zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. If Chern classes of $E$ are defined, then $i_{2, *} \\circ P'_p(E_2) = P_p(E)$, resp.\\ $i_{2, *} \\circ c'_p(E_2) = c_p(E)$ (with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})."}
+{"_id": "5779", "title": "chow-lemma-silly-shrink", "text": "In Lemma \\ref{lemma-silly} suppose we have closed subschemes $X'_2 \\subset X_2$ and $X_1 \\subset X'_1 \\subset X$ such that $X = X'_1 \\cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \\cap X_2}$ is zero, resp.\\ isomorphic to a finite locally free module of rank $< p$ placed in degree $0$. Then we have $(X'_2 \\to X_2)_* \\circ P'_p(E_2|_{X'_2}) = P'_p(E_2)$, resp.\\  $(X'_2 \\to X_2)_* \\circ c'_p(E_2|_{X'_2}) = c_p(E_2)$ (with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})."}
+{"_id": "5780", "title": "chow-lemma-silly-commutes", "text": "In Lemma \\ref{lemma-silly} let $f : Y \\to X$ be locally of finite type and say $c \\in A^*(Y \\to X)$. Then $$ c \\circ P'_p(E_2) = P'_p(Lf_2^*E_2) \\circ c \\quad\\text{resp.}\\quad c \\circ c'_p(E_2) = c'_p(Lf_2^*E_2) \\circ c $$ in $A^*(Y_2 \\to Y)$ where $f_2 : Y_2 \\to X_2$ is the base change of $f$."}
+{"_id": "5781", "title": "chow-lemma-silly-compose", "text": "In Lemma \\ref{lemma-silly} assume $E_2|_{X_1 \\cap X_2}$ is zero. Then \\begin{align*} P'_1(E_2) & = c'_1(E_2), \\\\ P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\\\ P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\\\ P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), \\end{align*} and so on with multiplication as in Remark \\ref{remark-ring-loc-classes}."}
+{"_id": "5782", "title": "chow-lemma-silly-sum-c", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let $E, F \\in D(\\mathcal{O}_X)$ be perfect objects. Assume \\begin{enumerate} \\item Chern classes of $E$ and $F$ are defined, \\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$ are isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-modules of rank $< p$ and $< q$ sitting in cohomological degree $0$. \\end{enumerate} With notation as in Remark \\ref{remark-ring-loc-classes} set $$ c^{(p)}(E) = 1 + c_1(E) + \\ldots + c_{p - 1}(E) + c'_p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \\ldots \\in A^{(p)}(X_2 \\to X) $$ with $c'_p(E|_{X_2})$ as in Lemma \\ref{lemma-silly}. Similarly for $c^{(q)}(F)$ and $c^{(p + q)}(E \\oplus F)$. Then $c^{(p + q)}(E \\oplus F) = c^{(p)}(E)c^{(q)}(F)$ in $A^{(p + q)}(X_2 \\to X)$."}
+{"_id": "5783", "title": "chow-lemma-silly-sum-P", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let $E, F \\in D(\\mathcal{O}_{X_2})$ be perfect objects. Assume \\begin{enumerate} \\item Chern classes of $E$ and $F$ are defined, \\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$ are zero, \\end{enumerate} Denote $P'_p(E), P'_p(F), P'_p(E \\oplus F) \\in A^p(X_2 \\to X)$ for $p \\geq 0$ the classes constructed in Lemma \\ref{lemma-silly}. Then $P'_p(E \\oplus F) = P'_p(E) + P'_p(F)$."}
+{"_id": "5785", "title": "chow-lemma-silly-tensor-product", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $$ X = X_1 \\cup X_2 = X'_1 \\cup X'_2 $$ be two ways of writing $X$ as a set theoretic union of closed subschemes. Let $E$, $E'$ be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes are defined. Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero\\footnote{Presumably there is a variant of this lemma where we only assume these restrictions are isomorphic to a finite locally free modules of rank $< p$ and $< p'$.} for $i = 1, 2$. Denote \\begin{enumerate} \\item $r = P'_0(E) \\in A^0(X_2 \\to X)$ and $r' = P'_0(E') \\in A^0(X'_2 \\to X)$, \\item $\\gamma_p = c'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and $\\gamma'_p = c'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$, \\item $\\chi_p = P'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and $\\chi'_p = P'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$ \\end{enumerate} the classes constructed in Lemma \\ref{lemma-silly}. Then we have $$ c'_1((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) = r \\gamma'_1 + r' \\gamma_1 $$ in $A^1(X_2 \\cap X'_2 \\to X)$ and $$ c'_2((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) = r \\gamma'_2 + r' \\gamma_2 + {r \\choose 2} (\\gamma'_1)^2 + (rr' - 1) \\gamma'_1\\gamma_1 + {r' \\choose 2} \\gamma_1^2 $$ in $A^2(X_2 \\cap X'_2 \\to X)$ and so on for higher Chern classes. Similarly, we have $$ P'_p((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) = \\sum\\nolimits_{p_1 + p_2 = p} {p \\choose p_1} \\chi_{p_1} \\chi'_{p_2} $$ in $A^p(X_2 \\cap X'_2 \\to X)$."}
+{"_id": "5786", "title": "chow-lemma-gysin-at-infty", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $b : W \\to \\mathbf{P}^1_X$ be a proper morphism of schemes which is an isomorphism over $\\mathbf{A}^1_X$. Denote $i_\\infty : W_\\infty \\to W$ the inverse image of the divisor $D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$. Then there is a canonical bivariant class $$ C \\in A^0(W_\\infty \\to X) $$ with the property that $i_{\\infty, *}(C \\cap \\alpha) = i_{0, *}\\alpha$ for $\\alpha \\in \\CH_k(X)$ and similarly after any base change by $X' \\to X$ locally of finite type."}
+{"_id": "5787", "title": "chow-lemma-gysin-at-infty-independent", "text": "In Lemma \\ref{lemma-gysin-at-infty} let $g : W' \\to W$ be a proper morphism which is an isomorphism over $\\mathbf{A}^1_X$. Let $C' \\in A^0(W'_\\infty \\to X)$ and $C \\in A^0(W_\\infty \\to X)$ be the classes constructed in Lemma \\ref{lemma-gysin-at-infty}. Then $g_{\\infty, *} \\circ C' = C$ in $A^0(W_\\infty \\to X)$."}
+{"_id": "5788", "title": "chow-lemma-homomorphism-pre", "text": "In Lemma \\ref{lemma-gysin-at-infty} we have $C \\circ (W_\\infty \\to X)_* \\circ i_\\infty^* = i_\\infty^*$."}
+{"_id": "5789", "title": "chow-lemma-gysin-at-infty-commutes", "text": "In Lemma \\ref{lemma-gysin-at-infty} let $f : Y \\to X$ be a morphism locally of finite type and $c \\in A^*(Y \\to X)$. Then $C \\circ c = c \\circ C$ in $A^*(W_\\infty \\times_X Y)$."}
+{"_id": "5790", "title": "chow-lemma-localized-chern-pre", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme. Let $$ b : W \\longrightarrow \\mathbf{P}^1_X $$ be a proper morphism of schemes. Let $Q \\in D(\\mathcal{O}_W)$ be a perfect object. Denote $W_\\infty \\subset W$ the inverse image of the divisor $D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$. We assume \\begin{enumerate} \\item[(A0)] Chern classes of $Q$ are defined (Section \\ref{section-pre-derived}), \\item[(A1)] $b$ is an isomorphism over $\\mathbf{A}^1_X$, \\item[(A2)] there exists a closed subscheme $T \\subset W_\\infty$ containing all points of $W_\\infty$ lying over $X \\setminus Z$ such that $Q|_T$ is zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_T$-module of rank $< p$ sitting in cohomological degree $0$. \\end{enumerate} Then there exists a canonical bivariant class $$ P'_p(Q),\\text{ resp. }c'_p(Q) \\in A^p(Z \\to X) $$ with $(Z \\to X)_* \\circ P'_p(Q) = P_p(Q|_{X \\times \\{0\\}})$, resp.\\ $(Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{X \\times \\{0\\}})$."}
+{"_id": "5791", "title": "chow-lemma-localized-chern-pre-independent", "text": "In Lemma \\ref{lemma-localized-chern-pre} the bivariant class $P'_p(Q)$, resp.\\ $c'_p(Q)$ is independent of the choice of the closed subscheme $T$. Moreover, given a proper morphism $g : W' \\to W$ which is an isomorphism over $\\mathbf{A}^1_X$, then setting $Q' = g^*Q$ we have $P'_p(Q) = P'_p(Q')$, resp.\\ $c'_p(Q) = c'_p(Q')$."}
+{"_id": "5792", "title": "chow-lemma-homomorphism", "text": "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic to a finite locally free $\\mathcal{O}_T$-module of rank $< p$. Denote $C \\in A^0(W_\\infty \\to X)$ the class of Lemma \\ref{lemma-gysin-at-infty}. Then $$ C \\circ c_p(Q|_{X \\times \\{0\\}}) = C \\circ (Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{W_\\infty}) \\circ C $$"}
+{"_id": "5793", "title": "chow-lemma-homomorphism-commute", "text": "In Lemma \\ref{lemma-localized-chern-pre} let $Y \\to X$ be a morphism locally of finite type and let $c \\in A^*(Y \\to X)$ be a bivariant class. Then $$ P'_p(Q) \\circ c = c \\circ P'_p(Q) \\quad\\text{resp.}\\quad c'_p(Q) \\circ c = c \\circ c'_p(Q) $$ in $A^*(Y \\times_X Z \\to X)$."}
+{"_id": "5794", "title": "chow-lemma-localized-chern-pre-compose", "text": "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. In $A^*(Z \\to X)$ we have \\begin{align*} P'_1(Q) & = c'_1(Q), \\\\ P'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), \\\\ P'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), \\\\ P'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) + 4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q), \\end{align*} and so on with multiplication as in Remark \\ref{remark-ring-loc-classes}."}
+{"_id": "5795", "title": "chow-lemma-localized-chern-pre-sum-c", "text": "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic to a finite locally free $\\mathcal{O}_T$-module of rank $< p$. Assume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$ whose Chern classes are defined with $Q'|_T$ isomorphic to a finite locally free $\\mathcal{O}_T$-module of rank $< p'$ placed in cohomological degree $0$. With notation as in Remark \\ref{remark-ring-loc-classes} set $$ c^{(p)}(Q) = 1 + c_1(Q|_{X \\times \\{0\\}}) + \\ldots + c_{p - 1}(Q|_{X \\times \\{0\\}}) + c'_{p}(Q) + c'_{p + 1}(Q) + \\ldots $$ in $A^{(p)}(Z \\to X)$ with $c'_i(Q)$ for $i \\geq p$ as in Lemma \\ref{lemma-localized-chern-pre}. Similarly for $c^{(p')}(Q')$ and $c^{(p + p')}(Q \\oplus Q')$. Then $c^{(p + p')}(Q \\oplus Q') = c^{(p)}(Q)c^{(p')}(Q')$ in $A^{(p + p')}(Z \\to X)$."}
+{"_id": "5796", "title": "chow-lemma-localized-chern-pre-sum-P", "text": "In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. Assume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$ whose Chern classes are defined such that the restriction $Q'|_T$ is zero. In this case the classes $P'_p(Q), P'_p(Q'), P'_p(Q \\oplus Q') \\in A^p(Z \\to X)$ constructed in Lemma \\ref{lemma-localized-chern-pre} satisfy $P'_p(Q \\oplus Q') = P'_p(Q) + P'_p(Q')$."}
+{"_id": "5798", "title": "chow-lemma-base-change-loc-chern", "text": "In the situation above let $f : X' \\to X$ be a morphism of schemes which is locally of finite type. Denote $E' = Lf^*E$ and $Z' = f^{-1}(Z)$. Then the bivariant class $$ P_p(Z' \\to X', E') \\in A^p(Z' \\to X'), \\quad\\text{resp.}\\quad c_p(Z' \\to X', E') \\in A^p(Z' \\to X') $$ constructed above using $X', Z', E'$ is the restriction (Remark \\ref{remark-restriction-bivariant}) of the bivariant class $P_p(Z \\to X, E) \\in A^p(Z \\to X)$, resp.\\ $c_p(Z \\to X, E) \\in A^p(Z \\to X)$."}
+{"_id": "5799", "title": "chow-lemma-loc-chern-after-pushforward", "text": "In the situation above we have $$ P_p(Z \\to X, E) \\cap i_*\\alpha = P_p(E|_Z) \\cap \\alpha, \\quad\\text{resp.}\\quad c_p(Z \\to X, E) \\cap i_*\\alpha = c_p(E|_Z) \\cap \\alpha $$ in $\\CH_*(Z)$ for any $\\alpha \\in \\CH_*(Z)$."}
+{"_id": "5800", "title": "chow-lemma-loc-chern-disjoint", "text": "In the situation of Definition \\ref{definition-localized-chern} if $\\alpha \\in \\CH_k(X)$ has support disjoint from $Z$, then $P_p(Z \\to X, E) \\cap \\alpha = 0$, resp.\\ $c_p(Z \\to X, E) \\cap \\alpha = 0$."}
+{"_id": "5801", "title": "chow-lemma-loc-chern-shrink-Z", "text": "In the situation of Definition \\ref{definition-localized-chern} assume $Z \\subset Z' \\subset X$ where $Z'$ is a closed subscheme of $X$. Then $P_p(Z' \\to X, E) = (Z \\to Z')_* \\circ P_p(Z \\to X, E)$, resp.\\ $c_p(Z' \\to X, E) = (Z \\to Z')_* \\circ c_p(Z \\to X, E)$ (with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant})."}
+{"_id": "5802", "title": "chow-lemma-loc-chern-agree", "text": "In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a perfect $E \\in D(\\mathcal{O}_X)$ whose Chern classes are defined and whose restriction to $X_1$ is zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. Then the class $P'_p(E_2)$, resp.\\ $c'_p(E_2)$ of Lemma \\ref{lemma-silly} agrees with $P_p(X_2 \\to X, E)$, resp.\\ $c_p(X_2 \\to X, E)$ of Definition \\ref{definition-localized-chern}."}
+{"_id": "5803", "title": "chow-lemma-homomorphism-final", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $b : W \\longrightarrow \\mathbf{P}^1_X$ be a proper morphism of schemes. Let $n \\geq 1$. For $i = 1, \\ldots, n$ let $Z_i \\subset X$ be a closed subscheme, let $Q_i \\in D(\\mathcal{O}_W)$, be a perfect object, let $p_i \\geq 0$ be an integer, and let $T_i \\subset W_\\infty$, $i = 1, \\ldots, n$ be closed. Denote $W_i = b^{-1}(\\mathbf{P}^1_{Z_i})$. Assume \\begin{enumerate} \\item for $i = 1, \\ldots, n$ the assumption of Lemma \\ref{lemma-localized-chern-pre} hold for $b, Z_i, Q_i, T_i, p_i$, \\item $Q_i|_{W \\setminus W_i}$ is zero, resp.\\ isomorphic to a finite locally free module of rank $< p_i$ placed in cohomological degree $0$. \\end{enumerate} Then $P'_{p_n}(Q_n) \\circ \\ldots \\circ P'_{p_1}(Q_1)$ is equal to $$ (W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to Z_n \\cap \\ldots \\cap Z_1)_* \\circ P'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ P'_{p_1}(Q_1|_{W_{1, \\infty}}) \\circ C $$ in $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$, resp.\\ $c'_{p_n}(Q_n) \\circ \\ldots \\circ c'_{p_1}(Q_1)$ is equal to $$ (W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to Z_n \\cap \\ldots \\cap Z_1)_* \\circ c'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ c'_{p_1}(Q_1|_{W_{1, \\infty}}) \\circ C $$ in $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$."}
+{"_id": "5804", "title": "chow-lemma-independent-loc-chern-bQ", "text": "Assume $(S, \\delta), X, Z, b : W \\to \\mathbf{P}^1_X, Q, T, p$ satisfy all the assumptions of Lemma \\ref{lemma-localized-chern-pre}. Finally, let $F \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes are defined such that \\begin{enumerate} \\item the restriction of $Q$ to $b^{-1}(\\mathbf{A}^1_X)$ is isomorphic to the pullback of $F$, and \\item $F|_{X \\setminus Z}$ is zero, resp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$. \\end{enumerate} Then the class $P'_p(Q)$, resp.\\ $c'_p(Q)$ in $A^p(Z \\to X)$ constructed in Lemma \\ref{lemma-localized-chern-pre} is equal to $P_p(Z \\to X, F)$, resp.\\ $c_p(Z \\to X, F)$."}
+{"_id": "5805", "title": "chow-lemma-loc-chern-character", "text": "In the situation of Definition \\ref{definition-localized-chern} assume $E|_{X \\setminus Z}$ is zero. Then \\begin{align*} P_1(Z \\to X, E) & = c_1(Z \\to X, E), \\\\ P_2(Z \\to X, E) & = c_1(Z \\to X, E)^2 - 2c_2(Z \\to X, E), \\\\ P_3(Z \\to X, E) & = c_1(Z \\to X, E)^3 - 3c_1(Z \\to X, E)c_2(Z \\to X, E) + 3c_3(Z \\to X, E), \\end{align*} and so on where the products are taken in the algebra $A^{(1)}(Z \\to X)$ of Remark \\ref{remark-ring-loc-classes}."}
+{"_id": "5807", "title": "chow-lemma-additivity-loc-chern-c", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\to X$ be a closed immersion. Let $$ E_1 \\to E_2 \\to E_3 \\to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$. Assume \\begin{enumerate} \\item $E_3 \\to E_1[1]$ can be represented be a map of bounded complexes of finite locally free $\\mathcal{O}_X$-modules, and \\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$ are isomorphic to finite locally free $\\mathcal{O}_{X \\setminus Z}$-modules of rank $< p_1$ and $< p_3$ placed in degree $0$. \\end{enumerate} With notation as in Remark \\ref{remark-loc-chern-classes} we have $$ c^{(p_1 + p_3)}(Z \\to X, E_2) = c^{(p_1)}(Z \\to X, E_1)c^{(p_3)}(Z \\to X, E_3) $$ in $A^{(p_1 + p_3)}(Z \\to X)$."}
+{"_id": "5808", "title": "chow-lemma-additivity-loc-chern-P", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\to X$ be a closed immersion. Let $$ E_1 \\to E_2 \\to E_3 \\to E_1[1] $$ be a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$. Assume \\begin{enumerate} \\item $E_3 \\to E_1[1]$ can be represented be a map of bounded complexes of finite locally free $\\mathcal{O}_X$-modules, and \\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$ are zero. \\end{enumerate} Then we have $$ P_p(Z \\to X, E_2) = P_p(Z \\to X, E_1) + P_p(Z \\to X, E_3) $$ for all $p \\in \\mathbf{Z}$ and consequently $ch(Z \\to X, E_2) = ch(Z \\to X, E_1) + ch(Z \\to X, E_3)$."}
+{"_id": "5809", "title": "chow-lemma-loc-chern-tensor-product", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $Z_i \\subset X$, $i = 1, 2$ be closed subschemes. Let $F_i$, $i = 1, 2$ be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes are defined. Assume that $F_i|_{X \\setminus Z_i}$ is zero\\footnote{Presumably there is a variant of this lemma where we only assume $F_i|_{X \\setminus Z_i}$ is isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z_i}$-module of rank $< p_i$.} for $i = 1, 2$. Denote $r_i = P_0(Z_i \\to X, F_i) \\in A^0(Z_i \\to X)$. Then we have $$ c_1(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) = r_1 c_1(Z_2 \\to X, F_2) + r_2 c_1(Z_1 \\to X, F_1) $$ in $A^1(Z_1 \\cap Z_2 \\to X)$ and \\begin{align*} c_2(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) & = r_1 c_2(Z_2 \\to X, F_2) + r_2 c_2(Z_1 \\to X, F_1) + \\\\ & {r_1 \\choose 2} c_1(Z_2 \\to X, F_2)^2 + \\\\ & (r_1r_2 - 1) c_1(Z_2 \\to X, F_2)c_1(Z_1 \\to X, F_1) + \\\\ & {r_2 \\choose 2} c_1(Z_1 \\to X, F_1)^2 \\end{align*} in $A^2(Z_1 \\cap Z_2 \\to X)$ and so on for higher Chern classes. Similarly, we have $$ ch(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) = ch(Z_1 \\to X, F_1) ch(Z_2 \\to X, F_2) $$ in $A^*(Z_1 \\cap Z_2 \\to X) \\otimes \\mathbf{Q}$. More precisely, we have $$ P_p(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) = \\sum\\nolimits_{p_1 + p_2 = p} {p \\choose p_1} P_{p_1}(Z_1 \\to X, F_1) P_{p_2}(Z_2 \\to X, F_2) $$ in $A^p(Z_1 \\cap Z_2 \\to X)$."}
+{"_id": "5810", "title": "chow-lemma-pullback-virtual-normal-sheaf", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $$ \\xymatrix{ Z' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\ Z \\ar[r] & X } $$ be a cartesian diagram of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. If $\\mathcal{N}$ is a virtual normal sheaf for $Z$ in $X$, then $\\mathcal{N}' = g^*\\mathcal{N}$ is a virtual normal sheaf for $Z'$ in $X'$."}
+{"_id": "5811", "title": "chow-lemma-construction-gysin", "text": "The construction above defines a bivariant class\\footnote{The notation $A^*(Z \\to X)^\\wedge$ is discussed in Remark \\ref{remark-completion-bivariant}. If $X$ is quasi-compact, then $A^*(Z \\to X)^\\wedge = A^*(Z \\to X)$.} $$ c(Z \\to X, \\mathcal{N}) \\in A^*(Z \\to X)^\\wedge $$ and moreover the construction is compatible with base change as in Lemma \\ref{lemma-pullback-virtual-normal-sheaf}. If $\\mathcal{N}$ has constant rank $r$, then $c(Z \\to X, \\mathcal{N}) \\in A^r(Z \\to X)$."}
+{"_id": "5812", "title": "chow-lemma-gysin-decompose", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Suppose that we have a short exact sequence $0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0$ of finite locally free $\\mathcal{O}_Z$-modules such that the given surjection $\\sigma : \\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ factors through a map $\\sigma' : (\\mathcal{N}')^\\vee \\to \\mathcal{C}_{Z/X}$. Then $$ c(Z \\to X, \\mathcal{N}) = c_{top}(\\mathcal{E}) \\circ c(Z \\to X, \\mathcal{N}') $$ as bivariant classes."}
+{"_id": "5813", "title": "chow-lemma-gysin-excess", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider a cartesian diagram $$ \\xymatrix{ Z' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\ Z \\ar[r] & X } $$ of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. Let $\\mathcal{N}$, resp.\\ $\\mathcal{N}'$ be a virtual normal sheaf for $Z \\subset X$, resp.\\ $Z' \\to X'$. Assume given a short exact sequence $0 \\to \\mathcal{N}' \\to g^*\\mathcal{N} \\to \\mathcal{E} \\to 0$ of finite locally free modules on $Z'$ such that the diagram $$ \\xymatrix{ g^*\\mathcal{N}^\\vee \\ar[r] \\ar[d] & (\\mathcal{N}')^\\vee \\ar[d] \\\\ g^*\\mathcal{C}_{Z/X} \\ar[r] & \\mathcal{C}_{Z'/X'} } $$ commutes. Then we have $$ res(c(Z \\to X, \\mathcal{N})) = c_{top}(\\mathcal{E}) \\circ c(Z' \\to X', \\mathcal{N}') $$ in $A^*(Z' \\to X')^\\wedge$."}
+{"_id": "5814", "title": "chow-lemma-gysin-fundamental", "text": "In the situation described just above assume $\\dim_\\delta(Y) = n$ and that $\\mathcal{C}_{Y \\times_X Z/Z}$ has constant rank $r$. Then $$ c(Z \\to X, \\mathcal{N}) \\cap [Y]_n = c_{top}(\\mathcal{E}) \\cap [Z \\times_X Y]_{n - r} $$ in $\\CH_*(Z \\times_X Y)$."}
+{"_id": "5815", "title": "chow-lemma-gysin-easy", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Let $Y \\to X$ be a morphism which is locally of finite type. Given integers $r$, $n$ assume \\begin{enumerate} \\item $\\mathcal{N}$ is locally free of rank $r$, \\item every irreducible component of $Y$ has $\\delta$-dimension $n$, \\item $\\dim_\\delta(Z \\times_X Y) \\leq n - r$, and \\item for $\\xi \\in Z \\times_X Y$ with $\\delta(\\xi) = n - r$ the local ring $\\mathcal{O}_{Y, \\xi}$ is Cohen-Macaulay. \\end{enumerate} Then $c(Z \\to X, \\mathcal{N}) \\cap [Y]_n = [Z \\times_X Y]_{n - r}$ in $\\CH_{n - r}(Z \\times_X Y)$."}
+{"_id": "5816", "title": "chow-lemma-gysin-agrees", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. The gysin homomorphism $i^*$ viewed as an element of $A^1(D \\to X)$ (see Lemma \\ref{lemma-gysin-bivariant}) is the same as the bivariant class $c(D \\to X, \\mathcal{N}) \\in A^1(D \\to X)$ constructed using $\\mathcal{N} = i^*\\mathcal{L}$ viewed as a virtual normal sheaf for $D$ in $X$."}
+{"_id": "5817", "title": "chow-lemma-gysin-commutes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme with virtual normal sheaf $\\mathcal{N}$. Let $Y \\to X$ be locally of finite type and $c \\in A^*(Y \\to X)$. Then $c$ and $c(Z \\to X, \\mathcal{N})$ commute (Remark \\ref{remark-bivariant-commute})."}
+{"_id": "5818", "title": "chow-lemma-relation-normal-cones", "text": "With notation as above we have $$ o^*[C_ZX]_n = [C_Z Y]_{n - 1} $$ in $\\CH_{n - 1}(Y \\times_{o, C_Y X} C_ZX)$."}
+{"_id": "5819", "title": "chow-lemma-gysin-composition", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $Z \\subset Y \\subset X$ be closed subschemes of a scheme locally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal sheaf for $Z \\subset X$. Let $\\mathcal{N}'$ be a virtual normal sheaf for $Z \\subset Y$. Let $\\mathcal{N}''$ be a virtual normal sheaf for $Y \\subset X$. Assume there is a commutative diagram $$ \\xymatrix{ (\\mathcal{N}'')^\\vee|_Z \\ar[r] \\ar[d] & \\mathcal{N}^\\vee \\ar[r] \\ar[d] & (\\mathcal{N}')^\\vee \\ar[d] \\\\ \\mathcal{C}_{Y/X}|_Z \\ar[r] & \\mathcal{C}_{Z/X} \\ar[r] & \\mathcal{C}_{Z/Y} } $$ where the sequence at the bottom is from More on Morphisms, Lemma \\ref{more-morphisms-lemma-transitivity-conormal} and the top sequence is a short exact sequence. Then $$ c(Z \\to X, \\mathcal{N}) = c(Z \\to Y, \\mathcal{N}') \\circ c(Y \\to X, \\mathcal{N}'') $$ in $A^*(Z \\to X)^\\wedge$."}
+{"_id": "5820", "title": "chow-lemma-compute-koszul", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. Then $$ \\prod\\nolimits_{n = 0, \\ldots, r} c(\\wedge^n \\mathcal{E})^{(-1)^n} = 1 - (r - 1)! c_r(\\mathcal{E}) + \\ldots $$"}
+{"_id": "5822", "title": "chow-lemma-agreement-with-loc-chern", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $i : Z \\to X$ be a regular closed immersion of codimension $r$ between schemes locally of finite type over $S$. Let $\\mathcal{N} = \\mathcal{C}_{Z/X}^\\vee$ be the normal sheaf. If $X$ is quasi-compact and has the resolution property, then $c_t(Z \\to X, i_*\\mathcal{O}_Z) = 0$ for $t = 1, \\ldots, r - 1$ and $$ c_r(Z \\to X, i_*\\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \\to X, \\mathcal{N}) \\quad\\text{in}\\quad A^r(Z \\to X) $$ where $c_t(Z \\to X, i_*\\mathcal{O}_Z)$ is the localized Chern class of Definition \\ref{definition-localized-chern}."}
+{"_id": "5823", "title": "chow-lemma-actual-computation", "text": "In the situation of Lemma \\ref{lemma-agreement-with-loc-chern} say $\\dim_\\delta(X) = n$. Then we have \\begin{enumerate} \\item $c_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for $t = 1, \\ldots, r - 1$, \\item $c_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = (-1)^{r - 1}(r - 1)![Z]_{n - r}$, \\item $ch_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for $t = 0, \\ldots, r - 1$, and \\item $ch_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = [Z]_{n - r}$. \\end{enumerate}"}
+{"_id": "5824", "title": "chow-lemma-second-adams-operator", "text": "Let $X$ be a scheme. There is a ring map $$ \\psi^2 : K_0(\\textit{Vect}(X)) \\longrightarrow K_0(\\textit{Vect}(X)) $$ which sends $[\\mathcal{L}]$ to $[\\mathcal{L}^{\\otimes 2}]$ when $\\mathcal{L}$ is invertible and is compatible with pullbacks."}
+{"_id": "5825", "title": "chow-lemma-minus-adams-operator", "text": "Let $X$ be a scheme. There is a ring map $\\psi^{-1} : K_0(\\textit{Vect}(X)) \\to K_0(\\textit{Vect}(X))$ which sends $[\\mathcal{E}]$ to $[\\mathcal{E}^\\vee]$ when $\\mathcal{E}$ is finite locally free and is compatible with pullbacks."}
+{"_id": "5827", "title": "chow-lemma-perf-Z-regular", "text": "Let $X$ be a Noetherian regular scheme of finite dimension. Let $Z \\subset X$ be a closed subschemes. The maps constructed in Remarks \\ref{remark-perf-Z-cohomology-K} and \\ref{remark-perf-Z-regular} are mutually inverse and we get $K'_0(Z) = K_0(D_{Z, perf}(\\mathcal{O}_X))$."}
+{"_id": "5828", "title": "chow-lemma-K-tensor-Q", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a quasi-compact regular scheme of finite type over $S$ with affine diagonal and $\\delta_{X/S} : X \\to \\mathbf{Z}$ bounded. Then the composition $$ K_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q} \\longrightarrow A^*(X) \\otimes \\mathbf{Q} \\longrightarrow \\CH_*(X) \\otimes \\mathbf{Q} $$ of the map $ch$ from Remark \\ref{remark-chern-character-K} and the map $c \\mapsto c \\cap [X]$ is an isomorphism."}
+{"_id": "5829", "title": "chow-lemma-composition-regular-immersion", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $i : X \\to Y$ and $j : Y \\to Z$ be regular immersions of schemes locally of finite type over $S$. Then $j \\circ i$ is a regular immersion and $(j \\circ i)^! = i^! \\circ j^!$."}
+{"_id": "5832", "title": "chow-lemma-lci-gysin-flat", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. If the gysin map exists for $f$ and $f$ is flat, then $f^!$ is equal to the bivariant class of Lemma \\ref{lemma-flat-pullback-bivariant}."}
+{"_id": "5833", "title": "chow-lemma-lci-gysin-composition", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ and $g : Y \\to Z$ be local complete intersection morphisms of schemes locally of finite type over $S$. Assume the gysin map exists for $g \\circ f$ and $g$. Then the gysin map exists for $f$ and $(g \\circ f)^! = f^! \\circ g^!$."}
+{"_id": "5834", "title": "chow-lemma-lci-gysin-commutes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[d] \\ar[r] & X' \\ar[d] \\ar[r] & X \\ar[d]^f \\\\ Y'' \\ar[r] & Y' \\ar[r] & Y } $$ of schemes locally of finite type over $S$ with both square cartesian. Assume $f : X \\to Y$ is a local complete intersection morphism such that the gysin map exists for $f$. Let $c \\in A^*(Y'' \\to Y')$. Denote $res(f^!) \\in A^*(X' \\to Y')$ the restriction of $f^!$ to $Y'$ (Remark \\ref{remark-restriction-bivariant}). Then $c$ and $res(f^!)$ commute (Remark \\ref{remark-bivariant-commute})."}
+{"_id": "5835", "title": "chow-lemma-lci-gysin-easy", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider a cartesian diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r] & X \\ar[d]^f \\\\ Y' \\ar[r] & Y } $$ of schemes locally of finite type over $S$. Assume \\begin{enumerate} \\item $f$ is a local complete intersection morphism and the gysin map exists for $f$, \\item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of Lemma \\ref{lemma-locally-equidimensional}, \\item for $x' \\in X'$ with images $x$, $y'$, and $y$ in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$ where $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and \\item for every generic point $\\xi \\in X'$ the local ring $\\mathcal{O}_{Y', f'(\\xi)}$ is Cohen-Macaulay. \\end{enumerate} Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in Remark \\ref{remark-fundamental-class}."}
+{"_id": "5836", "title": "chow-lemma-compare-gysin-base-change", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider a cartesian square $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes locally of finite type over $S$. Assume \\begin{enumerate} \\item both $f$ and $f'$ are local complete intersection morphisms, and \\item the gysin map exists for $f$ \\end{enumerate} Then $\\mathcal{C} = \\Ker(H^{-1}((g')^*\\NL_{X/Y}) \\to H^{-1}(\\NL_{X'/Y'}))$ is a finite locally free $\\mathcal{O}_{X'}$-module, the gysin map exists for $f'$, and we have $$ res(f^!) = c_{top}(\\mathcal{C}^\\vee) \\circ (f')^! $$ in $A^*(X' \\to Y')$."}
+{"_id": "5837", "title": "chow-lemma-blow-up-formula", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $i : Z \\to X$ be a regular closed immersion of schemes locally of finite type over $S$. Let $b : X' \\to X$ be the blowing up with center $Z$. Picture $$ \\xymatrix{ E \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\ Z \\ar[r]^i & X } $$ Assume that the gysin map exists for $b$. Then we have $$ res(b^!) = c_{top}(\\mathcal{F}^\\vee) \\circ \\pi^* $$ in $A^*(E \\to Z)$ where $\\mathcal{F}$ is the kernel of the canonical map $\\pi^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{E/X'}$."}
+{"_id": "5840", "title": "chow-lemma-diagonal-identity", "text": "In the situation above we have $\\Delta^! \\circ \\text{pr}_i^! = 1$ in $A^0(X)$."}
+{"_id": "5842", "title": "chow-lemma-chow-cohomology-towards-point", "text": "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Then we have a canonical identification $$ A^p(X \\to \\Spec(k)) = \\CH_{-p}(X) $$ for all $p \\in \\mathbf{Z}$."}
+{"_id": "5843", "title": "chow-lemma-chow-cohomology-towards-point-commutes", "text": "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $c \\in A^p(X \\to \\Spec(k))$. Let $Y \\to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \\in A^q(Y \\to Z)$. Then $c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_k Y \\to X \\times_k Z)$."}
+{"_id": "5844", "title": "chow-lemma-exterior-product-associative", "text": "Exterior product is associative. More precisely, let $k$ be a field, let $X, Y, Z$ be schemes locally of finite type over $k$, let $\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$. Then $(\\alpha \\times \\beta) \\times \\gamma = \\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_k Y \\times_k Z)$."}
+{"_id": "5845", "title": "chow-lemma-associative", "text": "The product defined above is associative. More precisely, let $k$ be a field, let $X$ be smooth over $k$, let $Y, Z, W$ be schemes locally of finite type over $X$, let $\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$. Then $(\\alpha \\cdot \\beta) \\cdot \\gamma = \\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$."}
+{"_id": "5846", "title": "chow-lemma-identify-chow-for-smooth", "text": "Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional of dimension $d$. The map $$ A^p(X) \\longrightarrow \\CH_{d - p}(X),\\quad c \\longmapsto c \\cap [X]_d $$ is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above."}
+{"_id": "5847", "title": "chow-lemma-lci-gysin-product", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes smooth over $k$. Then the gysin map is defined for $f$ and $f^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta$."}
+{"_id": "5848", "title": "chow-lemma-projection-formula", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a proper morphism of schemes smooth over $k$. Then the gysin map is defined for $f$ and $f_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta$."}
+{"_id": "5849", "title": "chow-lemma-intersect-properly", "text": "Let $k$ be a field. Let $X$ be an integral scheme smooth over $k$. Let $Y, Z \\subset X$ be integral closed subschemes. Set $d = \\dim(Y) + \\dim(Z) - \\dim(X)$. Assume \\begin{enumerate} \\item $\\dim(Y \\cap Z) \\leq d$, and \\item $\\mathcal{O}_{Y, \\xi}$ and $\\mathcal{O}_{Z, \\xi}$ are Cohen-Macaulay for every $\\xi \\in Y \\cap Z$ with $\\delta(\\xi) = d$. \\end{enumerate} Then $[Y] \\cdot [Z] = [Y \\cap Z]_d$ in $\\CH_d(X)$."}
+{"_id": "5850", "title": "chow-lemma-intersect-regularly-embedded", "text": "Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \\to X$ be a regular closed immersion. Let $\\alpha \\in \\CH_*(X)$. If $Y$ is equidimensional of dimension $e$, then $\\alpha \\cdot [Y]_e = i_*(i^!(\\alpha))$ in $\\CH_*(X)$."}
+{"_id": "5852", "title": "chow-lemma-exterior-product-well-defined-dim-1", "text": "The map $\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m - 1}(X \\times_S Y)$ is well defined."}
+{"_id": "5853", "title": "chow-lemma-chow-cohomology-towards-base-dim-1", "text": "Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Then we have a canonical identification $$ A^p(X \\to S) = \\CH_{1 - p}(X) $$ for all $p \\in \\mathbf{Z}$."}
+{"_id": "5854", "title": "chow-lemma-chow-cohomology-towards-base-dim-1-commutes", "text": "Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $c \\in A^p(X \\to S)$. Let $Y \\to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \\in A^q(Y \\to Z)$. Then $c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_S Y \\to X \\times_S Z)$."}
+{"_id": "5855", "title": "chow-lemma-exterior-product-associative-dim-1", "text": "Exterior product is associative. More precisely, let $(S, \\delta)$ be as above, let $X, Y, Z$ be schemes locally of finite type over $S$, let $\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$. Then $(\\alpha \\times \\beta) \\times \\gamma = \\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_S Y \\times_S Z)$."}
+{"_id": "5858", "title": "chow-lemma-dimension-base-change", "text": "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally of finite type. Denote $X' \\to S'$ the base change by $S' \\to S$. If $X$ is integral with $\\dim_\\delta(X) = k$, then every irreducible component $Z'$ of $X'$ has $\\dim_{\\delta'}(Z') = k + c$,"}
+{"_id": "5859", "title": "chow-lemma-pullback-coherent-base-change", "text": "In Situation \\ref{situation-setup-base-change} let $X \\to S$ locally of finite type and let $X' \\to S$ be the base change by $S' \\to S$. \\begin{enumerate} \\item Let $Z \\subset X$ be a closed subscheme with $\\dim_\\delta(Z) \\leq k$ and base change $Z' \\subset X'$. Then we have $\\dim_{\\delta'}(Z')) \\leq k + c$ and $[Z']_{k + c} = g^*[Z]_k$ in $Z_{k + c}(X')$. \\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ with $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$ and base change $\\mathcal{F}'$ on $X'$. Then we have $\\dim_\\delta(\\text{Supp}(\\mathcal{F}')) \\leq k + c$ and $g^*[\\mathcal{F}]_k = [\\mathcal{F}']_{k + c}$ in $Z_{k + c}(X')$. \\end{enumerate}"}
+{"_id": "5860", "title": "chow-lemma-pullback-base-change", "text": "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally of finite type and let $X' \\to S'$ be the base change by $S' \\to S$. The map $g^* : Z_k(X) \\to Z_{k + c}(X')$ above factors through rational equivalence to give a map $$ g^* : \\CH_k(X) \\longrightarrow \\CH_{k + c}(X') $$ of chow groups."}
+{"_id": "5861", "title": "chow-lemma-pullback-base-change-pullback", "text": "In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally of finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$. Assume $f : Y \\to X$ is flat of relative dimension $r$. Then $f' : Y' \\to X'$ is flat of relative dimension $r$ and the diagram $$ \\xymatrix{ \\CH_{k + r}(Y) \\ar[r]_{g^*} & \\CH_{k + c + r}(Y') \\\\ \\CH_k(X) \\ar[r]^{g^*} \\ar[u]^{(f')^*} & \\CH_{k + c}(X') \\ar[u]_{f^*} } $$ of chow groups commutes."}
+{"_id": "5862", "title": "chow-lemma-pullback-base-change-pushforward", "text": "In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally of finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$. Assume $f : Y \\to X$ is proper. Then $f' : Y' \\to X'$ is proper and the diagram $$ \\xymatrix{ \\CH_k(Y) \\ar[r]_{g^*} \\ar[d]_{f_*} & \\CH_{k + c}(Y') \\ar[d]^{f'_*} \\\\ \\CH_k(X) \\ar[r]^{g^*} & \\CH_{k + c}(X') } $$ of chow groups commutes."}
+{"_id": "5863", "title": "chow-lemma-pullback-base-change-c1", "text": "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally of finite type and let $X' \\to S'$ be the base change by $S' \\to S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module with base change $\\mathcal{L}'$ on $X'$. Then the diagram $$ \\xymatrix{ \\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_1(\\mathcal{L}) \\cap -} & \\CH_{k + c}(X') \\ar[d]^{c_1(\\mathcal{L}') \\cap -} \\\\ \\CH_{k - 1}(X) \\ar[r]^{g^*} & \\CH_{k + c - 1}(X') } $$ of chow groups commutes."}
+{"_id": "5864", "title": "chow-lemma-pullback-base-change-chern-classes", "text": "In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally of finite type and let $X' \\to S'$ be the base change by $S' \\to S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of rank $r$ with base change $\\mathcal{E}'$ on $X'$. Then the diagram $$ \\xymatrix{ \\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_i(\\mathcal{E}) \\cap -} & \\CH_{k + c}(X') \\ar[d]^{c_i(\\mathcal{E}') \\cap -} \\\\ \\CH_{k - i}(X) \\ar[r]^{g^*} & \\CH_{k + c - i}(X') } $$ of chow groups commutes for all $i$."}
+{"_id": "5865", "title": "chow-lemma-compose-base-change", "text": "Let $(S, \\delta)$, $(S', \\delta')$, $(S'', \\delta'')$ be as in Situation \\ref{situation-setup}. Let $g : S' \\to S$ and $g' : S'' \\to S'$ be flat morphisms of schemes and let $c, c' \\in \\mathbf{Z}$ be integers such that $S, \\delta, S', \\delta', g, c$ and $S', \\delta', S'', g', c'$ are as in Situation \\ref{situation-setup-base-change}. Let $X \\to S$ be locally of finite type and denote $X' \\to S'$ and $X'' \\to S''$ the base changes by $S' \\to S$ and $S'' \\to S$. Then $S, \\delta, S'', \\delta'', g \\circ g', c + c'$ is as in Situation \\ref{situation-setup-base-change} and the maps $g^* : \\CH_k(X) \\to \\CH_{k + c}(X')$ and $(g')^* : \\CH_{k + c}(X') \\to \\CH_{k + c + c'}(X'')$ of Lemma \\ref{lemma-pullback-base-change} compose to give the map $(g \\circ g')^* : \\CH_k(X) \\to \\CH_{k + c + c'}(X'')$ of Lemma \\ref{lemma-pullback-base-change}."}
+{"_id": "5866", "title": "chow-lemma-chow-limit", "text": "In Situation \\ref{situation-setup-base-change} assume $c = 0$ and assume that $S' = \\lim_{i \\in I} S_i$ is a filtered limit of schemes $S_i$ affine over $S$ such that \\begin{enumerate} \\item with $\\delta_i$ equal to $S_i \\to S \\xrightarrow{\\delta} \\mathbf{Z}$ the pair $(S_i, \\delta_i)$ is as in Situation \\ref{situation-setup}, \\item $S_i, \\delta_i, S, \\delta, S \\to S_i, c = 0$ is as in Situation \\ref{situation-setup-base-change}, \\item $S_i, \\delta_i, S_{i'}, \\delta_{i'}, S_i \\to S_{i'}, c = 0$  for $i \\geq i'$ is as in Situation \\ref{situation-setup-base-change}. \\end{enumerate} Then for a quasi-compact scheme $X$ of finite type over $S$ with base change $X'$ and $X_i$ by $S' \\to S$ and $S_i \\to S$ we have $\\CH_k(X') = \\colim \\CH_k(X_i)$."}
+{"_id": "5867", "title": "chow-lemma-dimension-at-most-one", "text": "With notations as above we have $\\dim_\\kappa(\\det_\\kappa(M)) \\leq 1$."}
+{"_id": "5868", "title": "chow-lemma-compare-det", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $\\kappa$. Let $M$ be a finite length $R$-module which is annihilated by $\\mathfrak m$. Let $l = \\dim_\\kappa(M)$. Then the map $$ \\det\\nolimits_\\kappa(M) \\longrightarrow \\wedge^l_\\kappa(M), \\quad [e_1, \\ldots, e_l] \\longmapsto e_1 \\wedge \\ldots \\wedge e_l $$ is an isomorphism."}
+{"_id": "5869", "title": "chow-lemma-determinant-dimension-one", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $\\kappa$. Let $M$ be a finite length $R$-module. The determinant $\\det_\\kappa(M)$ defined above is a $\\kappa$-vector space of dimension $1$. It is generated by the symbol $[f_1, \\ldots, f_l]$ for any admissible sequence such that $\\langle f_1, \\ldots f_l \\rangle = M$."}
+{"_id": "5870", "title": "chow-lemma-det-exact-sequences", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. For every short exact sequence $$ 0 \\to K \\to L \\to M \\to 0 $$ of finite length $R$-modules there exists a canonical isomorphism $$ \\gamma_{K \\to L \\to M} : \\det\\nolimits_\\kappa(K) \\otimes_\\kappa \\det\\nolimits_\\kappa(M) \\longrightarrow \\det\\nolimits_\\kappa(L) $$ defined by the rule on nonzero symbols $$ [e_1, \\ldots, e_k] \\otimes [\\overline{f}_1, \\ldots, \\overline{f}_m] \\longrightarrow [e_1, \\ldots, e_k, f_1, \\ldots, f_m] $$ with the following properties: \\begin{enumerate} \\item For every isomorphism of short exact sequences, i.e., for every commutative diagram $$ \\xymatrix{ 0 \\ar[r] & K \\ar[r] \\ar[d]^u & L \\ar[r] \\ar[d]^v & M \\ar[r] \\ar[d]^w & 0 \\\\ 0 \\ar[r] & K' \\ar[r] & L' \\ar[r] & M' \\ar[r] & 0 } $$ with short exact rows and isomorphisms $u, v, w$ we have $$ \\gamma_{K' \\to L' \\to M'} \\circ (\\det\\nolimits_\\kappa(u) \\otimes \\det\\nolimits_\\kappa(w)) = \\det\\nolimits_\\kappa(v) \\circ \\gamma_{K \\to L \\to M}, $$ \\item for every commutative square of finite length $R$-modules with exact rows and columns $$ \\xymatrix{ & 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] & \\\\ 0 \\ar[r] & A \\ar[r] \\ar[d] & B \\ar[r] \\ar[d] & C \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & D \\ar[r] \\ar[d] & E \\ar[r] \\ar[d] & F \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & G \\ar[r] \\ar[d] & H \\ar[r] \\ar[d] & I \\ar[r] \\ar[d] & 0 \\\\ & 0  & 0  & 0  & } $$ the following diagram is commutative $$ \\xymatrix{ \\det\\nolimits_\\kappa(A) \\otimes \\det\\nolimits_\\kappa(C) \\otimes \\det\\nolimits_\\kappa(G) \\otimes \\det\\nolimits_\\kappa(I) \\ar[dd]_{\\epsilon} \\ar[rrr]_-{\\gamma_{A \\to B \\to C} \\otimes \\gamma_{G \\to H \\to I}} & & & \\det\\nolimits_\\kappa(B) \\otimes \\det\\nolimits_\\kappa(H) \\ar[d]^{\\gamma_{B \\to E \\to H}} \\\\ & & & \\det\\nolimits_\\kappa(E) \\\\ \\det\\nolimits_\\kappa(A) \\otimes \\det\\nolimits_\\kappa(G) \\otimes \\det\\nolimits_\\kappa(C) \\otimes \\det\\nolimits_\\kappa(I) \\ar[rrr]^-{\\gamma_{A \\to D \\to G} \\otimes \\gamma_{C \\to F \\to I}} & & & \\det\\nolimits_\\kappa(D) \\otimes \\det\\nolimits_\\kappa(F) \\ar[u]_{\\gamma_{D \\to E \\to F}} } $$ where $\\epsilon$ is the switch of the factors in the tensor product times $(-1)^{cg}$ with $c = \\text{length}_R(C)$ and $g = \\text{length}_R(G)$, and \\item the map $\\gamma_{K \\to L \\to M}$ agrees with the usual isomorphism if $0 \\to K \\to L \\to M \\to 0$ is actually a short exact sequence of $\\kappa$-vector spaces. \\end{enumerate}"}
+{"_id": "5871", "title": "chow-lemma-uniqueness-det", "text": "Let $(R, \\mathfrak m, \\kappa)$ be any local ring. The functor $$ \\det\\nolimits_\\kappa : \\left\\{ \\begin{matrix} \\text{finite length }R\\text{-modules} \\\\ \\text{with isomorphisms} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} 1\\text{-dimensional }\\kappa\\text{-vector spaces} \\\\ \\text{with isomorphisms} \\end{matrix} \\right\\} $$ endowed with the maps $\\gamma_{K \\to L \\to M}$ is characterized by the following properties \\begin{enumerate} \\item its restriction to the subcategory of modules annihilated by $\\mathfrak m$ is isomorphic to the usual determinant functor (see Lemma \\ref{lemma-compare-det}), and \\item (1), (2) and (3) of Lemma \\ref{lemma-det-exact-sequences} hold. \\end{enumerate}"}
+{"_id": "5872", "title": "chow-lemma-determinant-quotient-ring", "text": "Let $(R', \\mathfrak m') \\to (R, \\mathfrak m)$ be a local ring homomorphism which induces an isomorphism on residue fields $\\kappa$. Then for every finite length $R$-module the restriction $M_{R'}$ is a finite length $R'$-module and there is a canonical isomorphism $$ \\det\\nolimits_{R, \\kappa}(M) \\longrightarrow \\det\\nolimits_{R', \\kappa}(M_{R'}) $$ This isomorphism is functorial in $M$ and compatible with the isomorphisms $\\gamma_{K \\to L \\to M}$ of Lemma \\ref{lemma-det-exact-sequences} defined for $\\det_{R, \\kappa}$ and $\\det_{R', \\kappa}$."}
+{"_id": "5873", "title": "chow-lemma-times-u-determinant", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $u \\in R^*$ be a unit. Let $M$ be a module of finite length over $R$. Denote $u_M : M \\to M$ the map multiplication by $u$. Then $$ \\det\\nolimits_\\kappa(u_M) : \\det\\nolimits_\\kappa(M) \\longrightarrow \\det\\nolimits_\\kappa(M) $$ is multiplication by $\\overline{u}^l$ where $l = \\text{length}_R(M)$ and $\\overline{u} \\in \\kappa^*$ is the image of $u$."}
+{"_id": "5874", "title": "chow-lemma-periodic-determinant-shift", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is exact. Then $$ \\det\\nolimits_\\kappa(M, \\varphi, \\psi) \\det\\nolimits_\\kappa(M, \\psi, \\varphi) = 1. $$"}
+{"_id": "5875", "title": "chow-lemma-periodic-determinant-sign", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $(M, \\varphi, \\varphi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \\varphi, \\varphi)$ is exact. Then $\\text{length}_R(M) = 2 \\text{length}_R(\\Im(\\varphi))$ and $$ \\det\\nolimits_\\kappa(M, \\varphi, \\varphi) = (-1)^{\\text{length}_R(\\Im(\\varphi))} = (-1)^{\\frac{1}{2}\\text{length}_R(M)} $$"}
+{"_id": "5876", "title": "chow-lemma-periodic-determinant-easy-case", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $M$ be a finite length $R$-module. \\begin{enumerate} \\item if $\\varphi : M \\to M$ is an isomorphism then $\\det_\\kappa(M, \\varphi, 0) = \\det_\\kappa(\\varphi)$. \\item if $\\psi : M \\to M$ is an isomorphism then $\\det_\\kappa(M, 0, \\psi) = \\det_\\kappa(\\psi)^{-1}$. \\end{enumerate}"}
+{"_id": "5877", "title": "chow-lemma-periodic-determinant", "text": "Let $R$ be a local ring with residue field $\\kappa$. Suppose that we have a short exact sequence of $(2, 1)$-periodic complexes $$ 0 \\to (M_1, \\varphi_1, \\psi_1) \\to (M_2, \\varphi_2, \\psi_2) \\to (M_3, \\varphi_3, \\psi_3) \\to 0 $$ with all $M_i$ of finite length, and each $(M_1, \\varphi_1, \\psi_1)$ exact. Then $$ \\det\\nolimits_\\kappa(M_2, \\varphi_2, \\psi_2) = \\det\\nolimits_\\kappa(M_1, \\varphi_1, \\psi_1) \\det\\nolimits_\\kappa(M_3, \\varphi_3, \\psi_3). $$ in $\\kappa^*$."}
+{"_id": "5878", "title": "chow-lemma-multiplicativity-determinant", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $M$ be a finite length $R$-module. Let $\\alpha, \\beta, \\gamma$ be endomorphisms of $M$. Assume that \\begin{enumerate} \\item $I_\\alpha = K_{\\beta\\gamma}$, and similarly for any permutation of $\\alpha, \\beta, \\gamma$, \\item $K_\\alpha = I_{\\beta\\gamma}$, and similarly for any permutation of $\\alpha, \\beta, \\gamma$. \\end{enumerate} Then \\begin{enumerate} \\item The triple $(M, \\alpha, \\beta\\gamma)$ is an exact $(2, 1)$-periodic complex. \\item The triple $(I_\\gamma, \\alpha, \\beta)$ is an exact $(2, 1)$-periodic complex. \\item The triple $(M/K_\\beta, \\alpha, \\gamma)$ is an exact $(2, 1)$-periodic complex. \\item We have $$ \\det\\nolimits_\\kappa(M, \\alpha, \\beta\\gamma) = \\det\\nolimits_\\kappa(I_\\gamma, \\alpha, \\beta) \\det\\nolimits_\\kappa(M/K_\\beta, \\alpha, \\gamma). $$ \\end{enumerate}"}
+{"_id": "5879", "title": "chow-lemma-tricky", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $\\alpha : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$ be a morphism of $(2, 1)$-periodic complexes over $R$. Assume \\begin{enumerate} \\item $M$, $M'$ have finite length, \\item $(M, \\varphi, \\psi)$, $(M', \\varphi', \\psi')$ are exact, \\item the maps $\\varphi$, $\\psi$ induce the zero map on $K = \\Ker(\\alpha)$, and \\item the maps $\\varphi$, $\\psi$ induce the zero map on $Q = \\Coker(\\alpha)$. \\end{enumerate} Denote $N = \\alpha(M) \\subset M'$. We obtain two short exact sequences of $(2, 1)$-periodic complexes $$ \\begin{matrix} 0 \\to (N, \\varphi', \\psi') \\to (M', \\varphi', \\psi') \\to (Q, 0, 0) \\to 0 \\\\ 0 \\to (K, 0, 0) \\to (M, \\varphi, \\psi) \\to (N, \\varphi', \\psi') \\to 0 \\end{matrix} $$ which induce two isomorphisms $\\alpha_i : Q \\to K$, $i = 0, 1$. Then $$ \\det\\nolimits_\\kappa(M, \\varphi, \\psi) = \\det\\nolimits_\\kappa(\\alpha_0^{-1} \\circ \\alpha_1) \\det\\nolimits_\\kappa(M', \\varphi', \\psi') $$ In particular, if $\\alpha_0 = \\alpha_1$, then $\\det\\nolimits_\\kappa(M, \\varphi, \\psi) = \\det\\nolimits_\\kappa(M', \\varphi', \\psi')$."}
+{"_id": "5880", "title": "chow-lemma-pre-symbol", "text": "Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Assume $\\varphi, \\psi : M \\to M$ are two injective $A$-module maps, and assume $\\varphi(\\psi(M)) = \\psi(\\varphi(M))$, for example if $\\varphi$ and $\\psi$ commute. Then $\\text{length}_R(M/\\varphi\\psi M) < \\infty$ and $(M/\\varphi\\psi M, \\varphi, \\psi)$ is an exact $(2, 1)$-periodic complex."}
+{"_id": "5882", "title": "chow-lemma-multiplicativity-symbol", "text": "Let $A$ be a Noetherian local ring. Let $a, b, c \\in A$. Let $M$ be a finite $A$-module with $\\dim(\\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$. Then $$ d_M(a, bc) = d_M(a, b) d_M(a, c) $$ and $d_M(a, b)d_M(b, a) = 1$."}
+{"_id": "5883", "title": "chow-lemma-symbol-when-equal", "text": "Let $A$ be a Noetherian local ring and $M$ a finite $A$-module of dimension $1$. Let $a \\in A$ be a nonzerodivisor on $M$. Then $d_M(a, a) = (-1)^{\\text{length}_A(M/aM)}$."}
+{"_id": "5884", "title": "chow-lemma-symbol-when-one-is-a-unit", "text": "Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Let $b \\in A$ be a nonzerodivisor on $M$, and let $u \\in A^*$. Then $$ d_M(u, b) = u^{\\text{length}_A(M/bM)} \\bmod \\mathfrak m_A. $$ In particular, if $M = A$, then $d_A(u, b) = u^{\\text{ord}_A(b)} \\bmod \\mathfrak m_A$."}
+{"_id": "5885", "title": "chow-lemma-symbol-short-exact-sequence", "text": "Let $A$ be a Noetherian local ring. Let $a, b \\in A$. Let $$ 0 \\to M \\to M' \\to M'' \\to 0 $$ be a short exact sequence of $A$-modules of dimension $1$ such that $a, b$ are nonzerodivisors on all three $A$-modules. Then $$ d_{M'}(a, b) = d_M(a, b) d_{M''}(a, b) $$ in $\\kappa^*$."}
+{"_id": "5886", "title": "chow-lemma-symbol-compare-modules", "text": "Let $A$ be a Noetherian local ring. Let $\\alpha : M \\to M'$ be a homomorphism of finite $A$-modules of dimension $1$. Let $a, b \\in A$. Assume \\begin{enumerate} \\item $a$, $b$ are nonzerodivisors on both $M$ and $M'$, and \\item $\\dim(\\Ker(\\alpha)), \\dim(\\Coker(\\alpha)) \\leq 0$. \\end{enumerate} Then $d_M(a, b) = d_{M'}(a, b)$."}
+{"_id": "5887", "title": "chow-lemma-compute-symbol-M", "text": "Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module with $\\dim(\\text{Supp}(M)) = 1$. Let $a, b \\in A$ nonzerodivisors on $M$. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal primes in the support of $M$. Then $$ d_M(a, b) = \\prod\\nolimits_{i = 1, \\ldots, t} d_{A/\\mathfrak q_i}(a, b)^{ \\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})} $$ as elements of $\\kappa^*$."}
+{"_id": "5889", "title": "chow-lemma-symbol-is-steinberg-prepare", "text": "Let $A$ be a Noetherian local ring. Let $a, b \\in A$. Let $M$ be a finite $A$-module of dimension $1$ on which each of $a$, $b$, $b - a$ are nonzerodivisors. Then $$ d_M(a, b - a)d_M(b, b) = d_M(b, b - a)d_M(a, b) $$ in $\\kappa^*$."}
+{"_id": "5891", "title": "chow-lemma-key-lemma", "text": "Let $R$ be a Noetherian local ring. Let $\\mathfrak q \\subset R$ be a prime with $\\dim(R/\\mathfrak q) = 1$. Let $\\varphi : M \\to N$ be a homomorphism of finite $R$-modules. Assume there exist $x_1, \\ldots, x_l \\in M$ and $y_1, \\ldots, y_l \\in M$ with the following properties \\begin{enumerate} \\item $M = \\langle x_1, \\ldots, x_l\\rangle$, \\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle \\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$, \\item $N = \\langle y_1, \\ldots, y_l\\rangle$, and \\item $\\langle y_1, \\ldots, y_i\\rangle / \\langle y_1, \\ldots, y_{i - 1}\\rangle \\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$. \\end{enumerate} Then $\\varphi$ is injective if and only if $\\varphi_{\\mathfrak q}$ is an isomorphism, and in this case we have $$ \\text{length}_R(\\Coker(\\varphi)) = \\text{ord}_{R/\\mathfrak q}(f) $$ where $f \\in \\kappa(\\mathfrak q)$ is the element such that $$ [\\varphi(x_1), \\ldots, \\varphi(x_l)] = f [y_1, \\ldots, y_l] $$ in $\\det_{\\kappa(\\mathfrak q)}(N_{\\mathfrak q})$."}
+{"_id": "5892", "title": "chow-lemma-good-sequence-exists", "text": "Let $R$ be a local Noetherian ring. Let $\\mathfrak q \\subset R$ be a prime ideal. Let $M$ be a finite $R$-module such that $\\mathfrak q$ is one of the minimal primes of the support of $M$. Then there exist $x_1, \\ldots, x_l \\in M$ such that \\begin{enumerate} \\item the support of $M / \\langle x_1, \\ldots, x_l\\rangle$ does not contain $\\mathfrak q$, and \\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle \\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$. \\end{enumerate} Moreover, in this case $l = \\text{length}_{R_\\mathfrak q}(M_\\mathfrak q)$."}
+{"_id": "5895", "title": "chow-lemma-maps-between-coherent-sheaves", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $$ \\xymatrix{ \\ldots \\ar[r] & \\mathcal{F} \\ar[r]^\\varphi & \\mathcal{F} \\ar[r]^\\psi & \\mathcal{F} \\ar[r]^\\varphi & \\mathcal{F} \\ar[r] & \\ldots } $$ be a complex as in Homology, Equation (\\ref{homology-equation-cyclic-complex}). Assume that \\begin{enumerate} \\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$. \\item $\\dim_\\delta(\\text{Supp}(H^i(\\mathcal{F}, \\varphi, \\psi))) \\leq k$ for $i = 0, 1$. \\end{enumerate} Then we have $$ [H^0(\\mathcal{F}, \\varphi, \\psi)]_k \\sim_{rat} [H^1(\\mathcal{F}, \\varphi, \\psi)]_k $$ as $k$-cycles on $X$."}
+{"_id": "5896", "title": "chow-lemma-cycles-rational-equivalence-K-group", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The map $$ \\CH_k(X) \\longrightarrow K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)) $$ from Lemma \\ref{lemma-from-chow-to-K} induces a bijection from $\\CH_k(X)$ onto the image $B_k(X)$ of the map $$ K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)) \\longrightarrow K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)). $$"}
+{"_id": "5897", "title": "chow-lemma-no-embedded-points-modules", "text": "Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $a, b \\in A$. Assume \\begin{enumerate} \\item $\\dim(A) = 1$, \\item both $a$ and $b$ are nonzerodivisors in $A$, \\item $A$ has no embedded primes, \\item $M$ has no embedded associated primes, \\item $\\text{Supp}(M) = \\Spec(A)$. \\end{enumerate} Let $I = \\{x \\in A \\mid x(a/b) \\in A\\}$. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal primes of $A$. Then $(a/b)IM \\subset M$ and $$ \\text{length}_A(M/(a/b)IM) - \\text{length}_A(M/IM) = \\sum\\nolimits_i \\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i}) \\text{ord}_{A/\\mathfrak q_i}(a/b) $$"}
+{"_id": "5898", "title": "chow-lemma-no-embedded-points", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$ be a meromorphic section of $\\mathcal{L}$. Assume \\begin{enumerate} \\item $\\dim_\\delta(X) \\leq k + 1$, \\item $X$ has no embedded points, \\item $\\mathcal{F}$ has no embedded associated points, \\item the support of $\\mathcal{F}$ is $X$, and \\item the section $s$ is regular meromorphic. \\end{enumerate} In this situation let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of denominators of $s$, see Divisors, Definition \\ref{divisors-definition-regular-meromorphic-ideal-denominators}. Then we have the following: \\begin{enumerate} \\item there are short exact sequences $$ \\begin{matrix} 0 & \\to & \\mathcal{I}\\mathcal{F} & \\xrightarrow{1} & \\mathcal{F} & \\to & \\mathcal{Q}_1 & \\to & 0 \\\\ 0 & \\to & \\mathcal{I}\\mathcal{F} & \\xrightarrow{s} & \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L} & \\to & \\mathcal{Q}_2 & \\to & 0 \\end{matrix} $$ \\item the coherent sheaves $\\mathcal{Q}_1$, $\\mathcal{Q}_2$ are supported in $\\delta$-dimension $\\leq k$, \\item the section $s$ restricts to a regular meromorphic section $s_i$ on every irreducible component $X_i$ of $X$ of $\\delta$-dimension $k + 1$, and \\item writing $[\\mathcal{F}]_{k + 1} = \\sum m_i[X_i]$ we have $$ [\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k = \\sum m_i(X_i \\to X)_*\\text{div}_{\\mathcal{L}|_{X_i}}(s_i) $$ in $Z_k(X)$, in particular $$ [\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k = c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1} $$ in $\\CH_k(X)$. \\end{enumerate}"}
+{"_id": "5900", "title": "chow-proposition-K-tensor-Q", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Assume given a closed immersion $X \\to Y$ of schemes locally of finite type over $S$ with $Y$ regular, quasi-compact, affine diagonal, and $\\delta_{Y/S} : Y \\to \\mathbf{Z}$ bounded. Then the composition $$ K'_0(X) \\to K_0(D_{X, perf}(\\mathcal{O}_Y)) \\to A^*(X \\to Y) \\to \\CH_*(X) $$ of the map $\\mathcal{F} \\mapsto \\mathcal{F}[0]$ from Remark \\ref{remark-perf-Z-regular}, the map $ch(X \\to Y, -)$ from Remark \\ref{remark-localized-chern-character-K}, and the map $c \\mapsto c \\cap [Y]$ induces an isomorphism $$ K'_0(X) \\otimes \\mathbf{Q} \\longrightarrow \\CH_*(X) \\otimes \\mathbf{Q} $$ which depends on the choice of $Y$. Moreover, the canonical map $$ \\CH_k(X) \\otimes \\mathbf{Q} \\longrightarrow \\text{gr}_k K'_0(X) \\otimes \\mathbf{Q} $$ (see above) is an isomorphism of $\\mathbf{Q}$-vector spaces for all $k \\in \\mathbf{Z}$."}
+{"_id": "5901", "title": "chow-proposition-compute-bivariant", "text": "\\begin{reference} \\cite[Proposition 17.4.2]{F} \\end{reference} Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes locally of finite type over $S$. If $g$ is smooth of relative dimension $d$, then $A^p(X \\to Y) = A^{p - d}(X \\to Z)$."}
+{"_id": "5902", "title": "chow-proposition-length-determinant-periodic-complex", "text": "Let $R$ be a local Noetherian ring with residue field $\\kappa$. Suppose that $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex over $R$. Assume \\begin{enumerate} \\item $M$ is a finite $R$-module, \\item the cohomology modules of $(M, \\varphi, \\psi)$ are of finite length, and \\item $\\dim(\\text{Supp}(M)) = 1$. \\end{enumerate} Let $\\mathfrak q_i$, $i = 1, \\ldots, t$ be the minimal primes of the support of $M$. Then we have\\footnote{ Obviously we could get rid of the minus sign by redefining $\\det_\\kappa(M, \\varphi, \\psi)$ as the inverse of its current value, see Definition \\ref{definition-periodic-determinant}.} $$ - e_R(M, \\varphi, \\psi) = \\sum\\nolimits_{i = 1, \\ldots, t} \\text{ord}_{R/\\mathfrak q_i}\\left( \\det\\nolimits_{\\kappa(\\mathfrak q_i)} (M_{\\mathfrak q_i}, \\varphi_{\\mathfrak q_i}, \\psi_{\\mathfrak q_i}) \\right) $$"}
+{"_id": "5968", "title": "flat-theorem-finite-type-flat", "text": "\\begin{slogan} The flat locus is open (non-Noetherian version). \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X \\to S$ is locally of finite presentation, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and \\item the set of weakly associated points of $S$ is locally finite in $S$. \\end{enumerate} Then $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$ is open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module of finite presentation and flat over $S$."}
+{"_id": "5969", "title": "flat-theorem-flattening-map", "text": "In Situation \\ref{situation-iso} assume \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, flat over $S$, and pure relative to $S$, and \\item $u$ is surjective. \\end{enumerate} Then $F_{iso}$ is representable by a closed immersion $Z \\to S$. Moreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is of finite presentation."}
+{"_id": "5970", "title": "flat-theorem-flattening-local", "text": "In Situation \\ref{situation-flat-at-point} assume $A$ is henselian, $B$ is essentially of finite type over $A$, and $M$ is a finite $B$-module. Then there exists an ideal $I \\subset A$ such that $A/I$ corepresents the functor $F_{lf}$ on the category $\\mathcal{C}$. In other words given a local homomorphism of local rings $\\varphi : A \\to A'$ with $B' = B \\otimes_A A'$ and $M' = M \\otimes_A A'$ the following are equivalent: \\begin{enumerate} \\item $\\forall \\mathfrak q \\in V(\\mathfrak m_{A'}B' + \\mathfrak m_B B') \\subset \\Spec(B') : M'_{\\mathfrak q}\\text{ is flat over }A'$, and \\item $\\varphi(I) = 0$. \\end{enumerate} If $B$ is essentially of finite presentation over $A$ and $M$ of finite presentation over $B$, then $I$ is a finitely generated ideal."}
+{"_id": "5971", "title": "flat-theorem-check-flatness-at-associated-points", "text": "Let $f : X \\to S$ be locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $x \\in X$ with image $s \\in S$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is flat at $x$ over $S$, and \\item for every $x' \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ which specializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $S$. \\end{enumerate}"}
+{"_id": "5972", "title": "flat-theorem-flat-dimension-n-representable", "text": "In Situation \\ref{situation-flat-dimension-n}. Assume moreover that $f$ is of finite presentation, that $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and that $\\mathcal{F}$ is pure relative to $S$. Then $F_n$ is representable by a monomorphism $Z_n \\to S$ of finite presentation."}
+{"_id": "5975", "title": "flat-theorem-flatten-module", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $\\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \\subset S$ be a quasi-compact open. Assume \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ is locally of finite presentation over $S$, \\item $\\mathcal{F}$ is a module of finite type, \\item $\\mathcal{F}_U$ is of finite presentation, and \\item $\\mathcal{F}_U$ is flat over $U$. \\end{enumerate} Then there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an $\\mathcal{O}_{X \\times_S S'}$-module of finite presentation and flat over $S'$."}
+{"_id": "5976", "title": "flat-theorem-nagata", "text": "\\begin{reference} See \\cite{Lutkebohmert}, \\cite{Conrad-Nagata}, \\cite{Nagata-1}, \\cite{Nagata-2}, \\cite{Nagata-3}, and \\cite{Nagata-4} \\end{reference} Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \\to S$ be a separated, finite type morphism. Then $X$ has a compactification over $S$."}
+{"_id": "5978", "title": "flat-lemma-lift-etale", "text": "Let $i : Z \\to X$ be a closed immersion of affine schemes. Let $Z' \\to Z$ be an \\'etale morphism with $Z'$ affine. Then there exists an \\'etale morphism $X' \\to X$ with $X'$ affine such that $Z' \\cong Z \\times_X X'$ as schemes over $Z$."}
+{"_id": "5979", "title": "flat-lemma-etale-at-point", "text": "Let $$ \\xymatrix{ X \\ar[d] & X' \\ar[l] \\ar[d] \\\\ S & S' \\ar[l] } $$ be a commutative diagram of schemes with $X' \\to X$ and $S' \\to S$ \\'etale. Let $s' \\in S'$ be a point. Then $$ X' \\times_{S'} \\Spec(\\mathcal{O}_{S', s'}) \\longrightarrow X \\times_S \\Spec(\\mathcal{O}_{S', s'}) $$ is \\'etale."}
+{"_id": "5980", "title": "flat-lemma-etale-flat-up-down", "text": "Let $X \\to T \\to S$ be morphisms of schemes with $T \\to S$ \\'etale. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ be a point. Then $$ \\mathcal{F}\\text{ flat over }S\\text{ at }x \\Leftrightarrow \\mathcal{F}\\text{ flat over }T\\text{ at }x $$ In particular $\\mathcal{F}$ is flat over $S$ if and only if $\\mathcal{F}$ is flat over $T$."}
+{"_id": "5981", "title": "flat-lemma-etale-flat-up-down-local-ring", "text": "Let $T \\to S$ be an \\'etale morphism. Let $t \\in T$ with image $s \\in S$. Let $M$ be a $\\mathcal{O}_{T, t}$-module. Then $$ M\\text{ flat over }\\mathcal{O}_{S, s} \\Leftrightarrow M\\text{ flat over }\\mathcal{O}_{T, t}. $$"}
+{"_id": "5982", "title": "flat-lemma-flat-up-down-henselization", "text": "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$ (resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict henselization), see Algebra, Definition \\ref{algebra-definition-henselization}. Let $M^{sh}$ be a $\\mathcal{O}_{S, s}^{sh}$-module. The following are equivalent \\begin{enumerate} \\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}$, \\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^h$, and \\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^{sh}$. \\end{enumerate} If $M^{sh} = M^h \\otimes_{\\mathcal{O}_{S, s}^h} \\mathcal{O}_{S, s}^{sh}$ this is also equivalent to \\begin{enumerate} \\item[(4)] $M^h$ is flat over $\\mathcal{O}_{S, s}$, and \\item[(5)] $M^h$ is flat over $\\mathcal{O}_{S, s}^h$. \\end{enumerate} If $M^h = M \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$ this is also equivalent to \\begin{enumerate} \\item[(6)] $M$ is flat over $\\mathcal{O}_{S, s}$. \\end{enumerate}"}
+{"_id": "5983", "title": "flat-lemma-tor-amplitude-up-down-henselization", "text": "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$ (resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict henselization), see Algebra, Definition \\ref{algebra-definition-henselization}. Let $M^{sh}$ be an object of $D(\\mathcal{O}_{S, s}^{sh})$. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$, \\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$, and \\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^{sh}$. \\end{enumerate} If $M^{sh} = M^h \\otimes_{\\mathcal{O}_{S, s}^h}^\\mathbf{L} \\mathcal{O}_{S, s}^{sh}$ for $M^h \\in D(\\mathcal{O}_{S, s}^h)$ this is also equivalent to \\begin{enumerate} \\item[(4)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$, and \\item[(5)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$. \\end{enumerate} If $M^h = M \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L} \\mathcal{O}_{S, s}^h$ for $M \\in D(\\mathcal{O}_{S, s})$ this is also equivalent to \\begin{enumerate} \\item[(6)] $M$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$. \\end{enumerate}"}
+{"_id": "5984", "title": "flat-lemma-finite-flat-weak-assassin-up-down", "text": "Let $g : T \\to S$ be a finite flat morphism of schemes. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module. Let $t \\in T$ be a point with image $s \\in S$. Then $$ t \\in \\text{WeakAss}(g^*\\mathcal{G}) \\Leftrightarrow s \\in \\text{WeakAss}(\\mathcal{G}) $$"}
+{"_id": "5985", "title": "flat-lemma-etale-weak-assassin-up-down", "text": "Let $h : U \\to S$ be an \\'etale morphism of schemes. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module. Let $u \\in U$ be a point with image $s \\in S$. Then $$ u \\in \\text{WeakAss}(h^*\\mathcal{G}) \\Leftrightarrow s \\in \\text{WeakAss}(\\mathcal{G}) $$"}
+{"_id": "5986", "title": "flat-lemma-weakly-associated-henselization", "text": "Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$ (resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict henselization), see Algebra, Definition \\ref{algebra-definition-henselization}. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module. The following are equivalent \\begin{enumerate} \\item $s$ is a weakly associated point of $\\mathcal{F}$, \\item $\\mathfrak m_s$ is a weakly associated prime of $\\mathcal{F}_s$, \\item $\\mathfrak m_s^h$ is a weakly associated prime of $\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$, and \\item $\\mathfrak m_s^{sh}$ is a weakly associated prime of $\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^{sh}$. \\end{enumerate}"}
+{"_id": "5987", "title": "flat-lemma-sheaf-lives-on-subscheme", "text": "Let $f : X \\to S$ be a finite type morphism of affine schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $s = f(x)$ in $S$. Set $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$. Then there exist a closed immersion $i : Z \\to X$ of finite presentation, and a quasi-coherent finite type $\\mathcal{O}_Z$-module $\\mathcal{G}$ such that $i_*\\mathcal{G} = \\mathcal{F}$ and $Z_s = \\text{Supp}(\\mathcal{F}_s)$."}
+{"_id": "5988", "title": "flat-lemma-elementary-devissage", "text": "Let $f : X \\to S$ be morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $s = f(x)$ in $S$. Set $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$ and $n = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$. Then we can construct \\begin{enumerate} \\item elementary \\'etale neighbourhoods $g : (X', x') \\to (X, x)$, $e : (S', s') \\to (S, s)$, \\item a commutative diagram $$ \\xymatrix{ X \\ar[dd]_f & X' \\ar[dd] \\ar[l]^g & Z' \\ar[l]^i \\ar[d]^\\pi \\\\ & & Y' \\ar[d]^h \\\\ S & S' \\ar[l]_e & S' \\ar@{=}[l] } $$ \\item a point $z' \\in Z'$ with $i(z') = x'$, $y' = \\pi(z')$, $h(y') = s'$, \\item a finite type quasi-coherent $\\mathcal{O}_{Z'}$-module $\\mathcal{G}$, \\end{enumerate} such that the following properties hold \\begin{enumerate} \\item $X'$, $Z'$, $Y'$, $S'$ are affine schemes, \\item $i$ is a closed immersion of finite presentation, \\item $i_*(\\mathcal{G}) \\cong g^*\\mathcal{F}$, \\item $\\pi$ is finite and $\\pi^{-1}(\\{y'\\}) = \\{z'\\}$, \\item the extension $\\kappa(s') \\subset \\kappa(y')$ is purely transcendental, \\item $h$ is smooth of relative dimension $n$ with geometrically integral fibres. \\end{enumerate}"}
+{"_id": "5989", "title": "flat-lemma-devissage-finite-presentation", "text": "Assumptions and notation as in Lemma \\ref{lemma-elementary-devissage}. If $f$ is locally of finite presentation then $\\pi$ is of finite presentation. In this case the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation in a neighbourhood of $x$, \\item $\\mathcal{G}$ is an $\\mathcal{O}_{Z'}$-module of finite presentation in a neighbourhood of $z'$, and \\item $\\pi_*\\mathcal{G}$ is an $\\mathcal{O}_{Y'}$-module of finite presentation in a neighbourhood of $y'$. \\end{enumerate} Still assuming $f$ locally of finite presentation the following are equivalent to each other \\begin{enumerate} \\item[(a)] $\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite presentation, \\item[(b)] $\\mathcal{G}_{z'}$ is an $\\mathcal{O}_{Z', z'}$-module of finite presentation, and \\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is an $\\mathcal{O}_{Y', y'}$-module of finite presentation. \\end{enumerate}"}
+{"_id": "5990", "title": "flat-lemma-devissage-flat", "text": "Assumptions and notation as in Lemma \\ref{lemma-elementary-devissage}. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$, \\item $\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and \\item $\\pi_*\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$. \\end{enumerate} The following are equivalent also \\begin{enumerate} \\item[(a)] $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$, \\item[(b)] $\\mathcal{G}_{z'}$ is flat over $\\mathcal{O}_{S', s'}$, and \\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is flat over $\\mathcal{O}_{S', s'}$. \\end{enumerate}"}
+{"_id": "5991", "title": "flat-lemma-elementary-devissage-variant", "text": "Let $f : X \\to S$ be morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $s = f(x)$ in $S$. Then there exists a commutative diagram of pointed schemes $$ \\xymatrix{ (X, x) \\ar[d]_f & (X', x') \\ar[l]^g \\ar[d] \\\\ (S, s) & (S', s') \\ar[l] \\\\ } $$ such that $(S', s') \\to (S, s)$ and $(X', x') \\to (X, x)$ are elementary \\'etale neighbourhoods, and such that $g^*\\mathcal{F}/X'/S'$ has a one step d\\'evissage at $x'$."}
+{"_id": "5992", "title": "flat-lemma-base-change-one-step", "text": "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in Definition \\ref{definition-one-step-devissage}. Let $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$. Let $(S', s') \\to (S, s)$ be any morphism of pointed schemes. Given this data let $X', Z', Y', i', \\pi'$ be the base changes of $X, Z, Y, i, \\pi$ via $S' \\to S$. Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$ and let $\\mathcal{G}'$ be the pullback of $\\mathcal{G}$ to $Z'$. If $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}')$ is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$."}
+{"_id": "5993", "title": "flat-lemma-base-change-one-step-at-x", "text": "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in Definition \\ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes which induces an isomorphism $\\kappa(s) = \\kappa(s')$. Let $(Z', Y', i', \\pi', \\mathcal{G}')$ be as constructed in Lemma \\ref{lemma-base-change-one-step} and let $x' \\in X'$ (resp.\\ $z' \\in Z'$, $y' \\in Y'$) be the unique point mapping to both $x \\in X$ (resp.\\ $z \\in Z$, $y \\in Y$) and $s' \\in S'$. If $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}', z', y')$ is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$."}
+{"_id": "5994", "title": "flat-lemma-shrink", "text": "With assumption and notation as in Definition \\ref{definition-shrink} we have: \\begin{enumerate} \\item \\label{item-shrink-base} If $S' \\subset S$ is a standard open neighbourhood of $s$, then setting $X' = X_{S'}$, $Z' = Z_{S'}$ and $Y' = Y_{S'}$ we obtain a standard shrinking. \\item \\label{item-shrink-on-Y} Let $W \\subset Y$ be a standard open neighbourhood of $y$. Then there exists a standard shrinking with $Y' = W \\times_S S'$. \\item \\label{item-shrink-on-X} Let $U \\subset X$ be an open neighbourhood of $x$. Then there exists a standard shrinking with $X' \\subset U$. \\end{enumerate}"}
+{"_id": "5995", "title": "flat-lemma-elementary-etale-neighbourhood", "text": "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in Definition \\ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$. Let $$ \\xymatrix{ (Y, y) \\ar[d] & (Y', y') \\ar[l] \\ar[d] \\\\ (S, s) & (S', s') \\ar[l] } $$ be a commutative diagram of pointed schemes such that the horizontal arrows are elementary \\'etale neighbourhoods. Then there exists a commutative diagram $$ \\xymatrix{ & & (X'', x'') \\ar[lld] \\ar[d] & (Z'', z'') \\ar[l] \\ar[lld] \\ar[d] \\\\ (X, x) \\ar[d] & (Z, z) \\ar[l] \\ar[d] & (S'', s'') \\ar[lld] & (Y'', y'') \\ar[lld] \\ar[l] \\\\ (S, s) & (Y, y) \\ar[l] } $$ of pointed schemes with the following properties: \\begin{enumerate} \\item $(S'', s'') \\to (S', s')$ is an elementary \\'etale neighbourhood and the morphism $S'' \\to S$ is the composition $S'' \\to S' \\to S$, \\item $Y''$ is an open subscheme of $Y' \\times_{S'} S''$, \\item $Z'' = Z \\times_Y Y''$, \\item $(X'', x'') \\to (X, x)$ is an elementary \\'etale neighbourhood, and \\item $(Z'', Y'', i'', \\pi'', \\mathcal{G}'', z'', y'')$ is a one step d\\'evissage at $x''$ of the sheaf $\\mathcal{F}''$. \\end{enumerate} Here $\\mathcal{F}''$ (resp.\\ $\\mathcal{G}''$) is the pullback of $\\mathcal{F}$ (resp.\\ $\\mathcal{G}$) via the morphism $X'' \\to X$ (resp.\\ $Z'' \\to Z$) and $i'' : Z'' \\to X''$ and $\\pi'' : Z'' \\to Y''$ are as in the diagram."}
+{"_id": "5996", "title": "flat-lemma-existence-alpha", "text": "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in Definition \\ref{definition-one-step-devissage}. Let $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$. Let $\\xi \\in Y_s$ be the (unique) generic point. Then there exists an integer $r > 0$ and an $\\mathcal{O}_Y$-module map $$ \\alpha : \\mathcal{O}_Y^{\\oplus r} \\longrightarrow \\pi_*\\mathcal{G} $$ such that $$ \\alpha : \\kappa(\\xi)^{\\oplus r} \\longrightarrow (\\pi_*\\mathcal{G})_\\xi \\otimes_{\\mathcal{O}_{Y, \\xi}} \\kappa(\\xi) $$ is an isomorphism. Moreover, in this case we have $$ \\dim(\\text{Supp}(\\Coker(\\alpha)_s)) < \\dim(\\text{Supp}(\\mathcal{F}_s)). $$"}
+{"_id": "5997", "title": "flat-lemma-base-change-complete", "text": "Let $S$, $X$, $\\mathcal{F}$, $s$ be as in Definition \\ref{definition-complete-devissage}. Let $(S', s') \\to (S, s)$ be any morphism of pointed schemes. Let $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$ be a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$. Given this data let $X', Z'_k, Y'_k, i'_k, \\pi'_k$ be the base changes of $X, Z_k, Y_k, i_k, \\pi_k$ via $S' \\to S$. Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$ and let $\\mathcal{G}'_k$ be the pullback of $\\mathcal{G}_k$ to $Z'_k$. Let $\\alpha'_k$ be the pullback of $\\alpha_k$ to $Y'_k$. If $S'$ is affine, then $(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$ is a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$."}
+{"_id": "5998", "title": "flat-lemma-base-change-complete-at-x", "text": "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in Definition \\ref{definition-complete-devissage-at-x}. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes which induces an isomorphism $\\kappa(s) = \\kappa(s')$. Let $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$ be a complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$. Let $(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$ be as constructed in Lemma \\ref{lemma-base-change-complete} and let $x' \\in X'$ (resp.\\ $z'_k \\in Z'$, $y'_k \\in Y'$) be the unique point mapping to both $x \\in X$ (resp.\\ $z_k \\in Z_k$, $y_k \\in Y_k$) and $s' \\in S'$. If $S'$ is affine, then $(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k, z'_k, y'_k)_{k = 1, \\ldots, n}$ is a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$."}
+{"_id": "5999", "title": "flat-lemma-shrink-complete", "text": "With assumption and notation as in Definition \\ref{definition-shrink-complete} we have: \\begin{enumerate} \\item \\label{item-shrink-base-complete} If $S' \\subset S$ is a standard open neighbourhood of $s$, then setting $X' = X_{S'}$, $Z'_k = Z_{S'}$ and $Y'_k = Y_{S'}$ we obtain a standard shrinking. \\item \\label{item-shrink-on-Y-complete} Let $W \\subset Y_n$ be a standard open neighbourhood of $y$. Then there exists a standard shrinking with $Y'_n = W \\times_S S'$. \\item \\label{item-shrink-on-X-complete} Let $U \\subset X$ be an open neighbourhood of $x$. Then there exists a standard shrinking with $X' \\subset U$. \\end{enumerate}"}
+{"_id": "6000", "title": "flat-lemma-existence-complete", "text": "Let $X \\to S$ be a finite type morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$ be a point. There exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and \\'etale morphisms $h_i : Y_i \\to X_{S'}$, $i = 1, \\ldots, n$ such that for each $i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/Y_i/S'$ over $s'$, where $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $Y_i$ and such that $X_s = (X_{S'})_{s'} \\subset \\bigcup h_i(Y_i)$."}
+{"_id": "6001", "title": "flat-lemma-existence-algebra", "text": "Let $R \\to S$ be a finite type ring map. Let $M$ be a finite $S$-module. Let $\\mathfrak q$ be a prime ideal of $S$. There exists an elementary \\'etale localization $R' \\to S', \\mathfrak q', \\mathfrak p'$ of the ring map $R \\to S$ at $\\mathfrak q$ such that there exists a complete d\\'evissage of $(M \\otimes_S S')/S'/R'$ at $\\mathfrak q'$."}
+{"_id": "6002", "title": "flat-lemma-homothety-spectrum", "text": "Let $R \\to S$ be a ring map. Let $N$ be a $S$-module. Assume \\begin{enumerate} \\item $R$ is a local ring with maximal ideal $\\mathfrak m$, \\item $\\overline{S} = S/\\mathfrak m S$ is Noetherian, and \\item $\\overline{N} = N/\\mathfrak m_R N$ is a finite $\\overline{S}$-module. \\end{enumerate} Let $\\Sigma \\subset S$ be the multiplicative subset of elements which are not a zerodivisor on $\\overline{N}$. Then $\\Sigma^{-1}S$ is a semi-local ring whose spectrum consists of primes $\\mathfrak q \\subset S$ contained in an element of $\\text{Ass}_S(\\overline{N})$. Moreover, any maximal ideal of $\\Sigma^{-1}S$ corresponds to an associated prime of $\\overline{N}$ over $\\overline{S}$."}
+{"_id": "6003", "title": "flat-lemma-homothety-universally-injective", "text": "Assumption and notation as in Lemma \\ref{lemma-homothety-spectrum}. Assume moreover that \\begin{enumerate} \\item $S$ is local and $R \\to S$ is a local homomorphism, \\item $S$ is essentially of finite presentation over $R$, \\item $N$ is finitely presented over $S$, and \\item $N$ is flat over $R$. \\end{enumerate} Then each $s \\in \\Sigma$ defines a universally injective $R$-module map $s : N \\to N$, and the map $N \\to \\Sigma^{-1}N$ is $R$-universally injective."}
+{"_id": "6004", "title": "flat-lemma-base-change-universally-flat-local", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Let $S \\to S'$ be a ring map. Assume \\begin{enumerate} \\item $R \\to S$ is a local homomorphism of local rings \\item $S$ is essentially of finite presentation over $R$, \\item $N$ is of finite presentation over $S$, \\item $N$ is flat over $R$, \\item $S \\to S'$ is flat, and \\item the image of $\\Spec(S') \\to \\Spec(S)$ contains all primes $\\mathfrak q$ of $S$ lying over $\\mathfrak m_R$ such that $\\mathfrak q$ is an associated prime of $N/\\mathfrak m_R N$. \\end{enumerate} Then $N \\to N \\otimes_S S'$ is $R$-universally injective."}
+{"_id": "6005", "title": "flat-lemma-base-change-universally-flat", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Let $S \\to S'$ be a ring map. Assume \\begin{enumerate} \\item $R \\to S$ is of finite presentation and $N$ is of finite presentation over $S$, \\item $N$ is flat over $R$, \\item $S \\to S'$ is flat, and \\item the image of $\\Spec(S') \\to \\Spec(S)$ contains all primes $\\mathfrak q$ such that $\\mathfrak q$ is an associated prime of $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p$ is the inverse image of $\\mathfrak q$ in $R$. \\end{enumerate} Then $N \\to N \\otimes_S S'$ is $R$-universally injective."}
+{"_id": "6006", "title": "flat-lemma-universally-injective-local", "text": "Let $(R, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be an $R$-module map. If $M$ is a projective $R$-module, $N$ is a flat $R$-module, and $\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective then $u$ is universally injective."}
+{"_id": "6007", "title": "flat-lemma-invert-universally-injective", "text": "Assumption and notation as in Lemma \\ref{lemma-homothety-spectrum}. Assume moreover that $N$ is projective as an $R$-module. Then each $s \\in \\Sigma$ defines a universally injective $R$-module map $s : N \\to N$, and the map $N \\to \\Sigma^{-1}N$ is $R$-universally injective."}
+{"_id": "6008", "title": "flat-lemma-completed-direct-sum-ML", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian and complete with respect to $I$. The completion $(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$ is flat and Mittag-Leffler."}
+{"_id": "6009", "title": "flat-lemma-lift-ML", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $R$ is Noetherian and $I$-adically complete, \\item $M$ is flat over $R$, and \\item $M/IM$ is a projective $R/I$-module. \\end{enumerate} Then the $I$-adic completion $M^\\wedge$ is a flat Mittag-Leffler $R$-module."}
+{"_id": "6010", "title": "flat-lemma-universally-injective-to-completion", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $R \\to S$ be a ring map, and $N$ an $S$-module. Assume \\begin{enumerate} \\item $R$ is a Noetherian ring, \\item $S$ is a Noetherian ring, \\item $N$ is a finite $S$-module, and \\item for any finite $R$-module $Q$, any $\\mathfrak q \\in \\text{Ass}_S(Q \\otimes_R N)$ satisfies $IS + \\mathfrak q \\not = S$. \\end{enumerate} Then the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$ is universally injective as a map of $R$-modules."}
+{"_id": "6011", "title": "flat-lemma-universally-injective-to-completion-flat", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $R \\to S$ be a ring map, and $N$ an $S$-module. Assume \\begin{enumerate} \\item $R$ is a Noetherian ring, \\item $S$ is a Noetherian ring, \\item $N$ is a finite $S$-module, \\item $N$ is flat over $R$, and \\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$ we have $IS + \\mathfrak q \\not = S$. \\end{enumerate} Then the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$ is universally injective as a map of $R$-modules."}
+{"_id": "6012", "title": "flat-lemma-flat-pure-over-complete-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $R \\to S$ be a ring map, and $N$ an $S$-module. Assume \\begin{enumerate} \\item $R$ is Noetherian and $I$-adically complete, \\item $R \\to S$ is of finite type, \\item $N$ is a finite $S$-module, \\item $N$ is flat over $R$, \\item $N/IN$ is projective as a $R/I$-module, and \\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$ we have $IS + \\mathfrak q \\not = S$. \\end{enumerate} Then $N$ is projective as an $R$-module."}
+{"_id": "6013", "title": "flat-lemma-fibres-irreducible-flat-projective", "text": "Let $R$ be a ring. Let $R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $R \\to S$ is of finite type and flat, and \\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is geometrically integral over $\\kappa(\\mathfrak p)$. \\end{enumerate} Then $S$ is projective as an $R$-module."}
+{"_id": "6014", "title": "flat-lemma-fibres-irreducible-flat-projective-nonnoetherian", "text": "Let $R$ be a ring. Let $R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $R \\to S$ is of finite presentation and flat, and \\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is geometrically integral over $\\kappa(\\mathfrak p)$. \\end{enumerate} Then $S$ is projective as an $R$-module."}
+{"_id": "6015", "title": "flat-lemma-induction-step", "text": "Let $(R, \\mathfrak m)$ be a local ring. Let $R \\to S$ be a finitely presented flat ring map with geometrically integral fibres. Write $\\mathfrak p = \\mathfrak mS$. Let $\\mathfrak q \\subset S$ be a prime ideal lying over $\\mathfrak m$. Let $N$ be a finite $S$-module. There exist $r \\geq 0$ and an $S$-module map $$ \\alpha : S^{\\oplus r} \\longrightarrow N $$ such that $\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$ is an isomorphism. For any such $\\alpha$ the following are equivalent: \\begin{enumerate} \\item $N_{\\mathfrak q}$ is $R$-flat, \\item $\\alpha$ is $R$-universally injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, \\item $\\alpha$ is injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, \\item $\\alpha_{\\mathfrak p}$ is an isomorphism and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and \\item $\\alpha_{\\mathfrak q}$ is injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat. \\end{enumerate}"}
+{"_id": "6016", "title": "flat-lemma-complete-devissage-flat-finite-type-module", "text": "Let $(R, \\mathfrak m)$ be a local ring. Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a finite $S$-module. Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak m$. Assume that $N_{\\mathfrak q}$ is flat over $R$, and assume there exists a complete d\\'evissage of $N/S/R$ at $\\mathfrak q$. Then $N$ is a finitely presented $S$-module, free as an $R$-module, and there exists an isomorphism $$ N \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n} $$ as $R$-modules where each $B_i$ is a smooth $R$-algebra with geometrically irreducible fibres."}
+{"_id": "6017", "title": "flat-lemma-open-in-fibre-where-flat", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $s \\in S$. Then the set $$ \\{x \\in X_s \\mid \\mathcal{F} \\text{ flat over }S\\text{ at }x\\} $$ is open in the fibre $X_s$."}
+{"_id": "6018", "title": "flat-lemma-finite-type-flat-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in X$ with image $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat at $x$ over $S$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme $$ V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'}) $$ which contains the unique point of $X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\\mathcal{F}$ to $V$ is flat over $\\mathcal{O}_{S', s'}$."}
+{"_id": "6019", "title": "flat-lemma-finite-type-flat-along-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme $$ V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'}) $$ which contains the fibre $X_s = X \\times_S s'$ such that the pullback of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module of finite presentation and flat over $\\mathcal{O}_{S', s'}$."}
+{"_id": "6022", "title": "flat-lemma-finite-type-flat-at-point-local", "text": "Let $f : X \\to S$ be a morphism which is locally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. If $x \\in X$ and $\\mathcal{F}$ is flat at $x$ over $S$, then $\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite presentation."}
+{"_id": "6024", "title": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open", "text": "\\begin{slogan} $S$-flat and finite type extensions of finitely presented modules on a (good) open are also $X$-finitely presented. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset S$ be open. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite type and flat over $S$, \\item $U \\subset S$ is retrocompact and scheme theoretically dense, \\item $\\mathcal{F}|_{f^{-1}U}$ is of finite presentation. \\end{enumerate} Then $\\mathcal{F}$ is of finite presentation."}
+{"_id": "6025", "title": "flat-lemma-flat-finite-type-finitely-presented-over-dense-open-X", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset S$ be open. Assume \\begin{enumerate} \\item $f$ is locally of finite type and flat, \\item $U \\subset S$ is retrocompact and scheme theoretically dense, \\item $f|_{f^{-1}U} : f^{-1}U \\to U$ is locally of finite presentation. \\end{enumerate} Then $f$ is of locally of finite presentation."}
+{"_id": "6026", "title": "flat-lemma-flat-finite-presentation-dimension-over-dense-open", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite type. Let $U \\subset S$ be a dense open such that $X_U \\to U$ has relative dimension $\\leq e$, see Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}. If also either \\begin{enumerate} \\item $f$ is locally of finite presentation, or \\item $U \\subset S$ is retrocompact, \\end{enumerate} then $f$ has relative dimension $\\leq e$."}
+{"_id": "6027", "title": "flat-lemma-proper-flat-finite-over-dense-open", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and proper. Let $U \\subset S$ be a dense open such that $X_U \\to U$ is finite. If also either $f$ is locally of finite presentation or $U \\subset S$ is retrocompact, then $f$ is finite."}
+{"_id": "6028", "title": "flat-lemma-zariski", "text": "Let $f : X \\to S$ be a morphism of schemes and $U \\subset S$ an open. If \\begin{enumerate} \\item $f$ is separated, locally of finite type, and flat, \\item $f^{-1}(U) \\to U$ is an isomorphism, and \\item $U \\subset S$ is retrocompact and scheme theoretically dense, \\end{enumerate} then $f$ is an open immersion."}
+{"_id": "6029", "title": "flat-lemma-induction-step-fp", "text": "Let $R$ be a ring. Let $R \\to S$ be a finitely presented flat ring map with geometrically integral fibres. Let $\\mathfrak q \\subset S$ be a prime ideal lying over the prime $\\mathfrak r \\subset R$. Set $\\mathfrak p = \\mathfrak r S$. Let $N$ be a finitely presented $S$-module. There exists $r \\geq 0$ and an $S$-module map $$ \\alpha : S^{\\oplus r} \\longrightarrow N $$ such that $\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$ is an isomorphism. For any such $\\alpha$ the following are equivalent: \\begin{enumerate} \\item $N_{\\mathfrak q}$ is $R$-flat, \\item there exists an $f \\in R$, $f \\not \\in \\mathfrak r$ such that $\\alpha_f : S_f^{\\oplus r} \\to N_f$ is $R_f$-universally injective and a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $\\Coker(\\alpha)_g$ is $R$-flat, \\item $\\alpha_{\\mathfrak r}$ is $R_{\\mathfrak r}$-universally injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat \\item $\\alpha_{\\mathfrak r}$ is injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, \\item $\\alpha_{\\mathfrak p}$ is an isomorphism and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and \\item $\\alpha_{\\mathfrak q}$ is injective and $\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat. \\end{enumerate}"}
+{"_id": "6030", "title": "flat-lemma-complete-devissage-flat-finitely-presented-module", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module flat over $R$. Let $\\mathfrak r \\subset R$ be a prime ideal. Assume there exists a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak r$ such that $$ N_f \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n} $$ as $R$-modules where each $B_i$ is a smooth $R_f$-algebra with geometrically irreducible fibres. Moreover, $N_f$ is projective as an $R_f$-module."}
+{"_id": "6031", "title": "flat-lemma-finite-presentation-flat-along-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, and \\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of schemes $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ S & S' \\ar[l] } $$ such that $g$ is \\'etale, $X_s \\subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module."}
+{"_id": "6034", "title": "flat-lemma-finite-type-flat-at-point-free", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in X$ with image $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat at $x$ over $S$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of pointed schemes $$ \\xymatrix{ (X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\ (S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l] } $$ such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ is \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is affine of finite presentation over $\\mathcal{O}_{S', s'}$, the sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{S', s'}$-module."}
+{"_id": "6035", "title": "flat-lemma-finite-type-flat-at-point-free-variant", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in X$ with image $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat at $x$ over $S$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of pointed schemes $$ \\xymatrix{ (X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\ (S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l] } $$ such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ is \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is affine, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{S', s'}$-module."}
+{"_id": "6036", "title": "flat-lemma-finite-type-flat-along-fibre-free", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of schemes $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ S & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l] } $$ such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ is \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine of finite presentation over $\\mathcal{O}_{S', s'}$, the sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{S', s'}$-module."}
+{"_id": "6037", "title": "flat-lemma-finite-type-flat-along-fibre-free-variant", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of schemes $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ S & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l] } $$ such that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ is \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{S', s'}$-module."}
+{"_id": "6038", "title": "flat-lemma-weak-bourbaki-pre-pre", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module. Let $\\mathfrak q \\subset S$ be a prime ideal lying over $\\mathfrak p \\subset R$. Set $\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$, $\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and $\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$. Then we can find a $g \\in S$ with $g \\not \\in \\mathfrak q$ such that $\\overline{g} \\in \\mathfrak r$ for all $\\mathfrak r \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$ such that $\\mathfrak r \\not \\subset \\overline{\\mathfrak q}$."}
+{"_id": "6039", "title": "flat-lemma-weak-bourbaki-pre", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a finitely presented $S$-module which is flat as an $R$-module. Let $M$ be an $R$-module. Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$. Then $$ \\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N) \\Leftrightarrow \\Big( \\mathfrak p \\in \\text{WeakAss}_R(M) \\text{ and } \\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N}) \\Big) $$ Here $\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$, $\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and $\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$."}
+{"_id": "6040", "title": "flat-lemma-bourbaki-finite-type-general-base-at-point", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be locally of finite type. Let $x \\in X$ with image $s \\in S$. Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$. If $\\mathcal{F}$ is flat at $x$ over $S$, then $$ x \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) \\Leftrightarrow s \\in \\text{WeakAss}_S(\\mathcal{G}) \\text{ and } x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s). $$"}
+{"_id": "6042", "title": "flat-lemma-bourbaki-finite-type-general-base", "text": "Let $f : X \\to S$ be a morphism which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$ which is flat over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have $$ \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) = \\bigcup\\nolimits_{s \\in \\text{WeakAss}_S(\\mathcal{G})} \\text{Ass}_{X_s}(\\mathcal{F}_s) $$"}
+{"_id": "6043", "title": "flat-lemma-finite-type-flat-algebra", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $M$ be a finite $S$-module. Assume $\\text{WeakAss}_S(S)$ is finite. Then $$ U = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\} $$ is open in $\\Spec(S)$ and for every $g \\in S$ such that $D(g) \\subset U$ the localization $M_g$ is a finitely presented $S_g$-module flat over $R$."}
+{"_id": "6044", "title": "flat-lemma-finite-type-flat-X", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Assume the set of weakly associated points of $S$ is locally finite in $S$. Then the set of points $x \\in X$ where $f$ is flat is an open subscheme $U \\subset X$ and $U \\to S$ is flat and locally of finite presentation."}
+{"_id": "6045", "title": "flat-lemma-finite-type-flat-over-integral", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type and flat. If $S$ is integral, then $f$ is locally of finite presentation."}
+{"_id": "6046", "title": "flat-lemma-explain-why-pure", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$. Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Assume \\begin{enumerate} \\item $N$ is projective as an $R$-module, and \\item $S/\\mathfrak mS$ is Noetherian and $N/\\mathfrak mN$ is a finite $S/\\mathfrak mS$-module. \\end{enumerate} Then for any prime $\\mathfrak q \\subset S$ which is an associated prime of $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$ we have $\\mathfrak q + \\mathfrak m S \\not = S$."}
+{"_id": "6048", "title": "flat-lemma-explain-why-pure-direct-sum-finite-modules", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$. Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is isomorphic as an $R$-module to a direct sum of finite $R$-modules. Then for any $R$-module $M$ and for any prime $\\mathfrak q \\subset S$ which is an associated prime of $N \\otimes_R M$ we have $\\mathfrak q + \\mathfrak m S \\not = S$."}
+{"_id": "6050", "title": "flat-lemma-impure-finite-presentation", "text": "In Situation \\ref{situation-pre-pure}. If there exists an impurity of $\\mathcal{F}$ above $s$, then there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ such that $g$ is locally of finite presentation and $t$ a closed point of the fibre of $g$ above $s$."}
+{"_id": "6051", "title": "flat-lemma-impure-limit", "text": "In Situation \\ref{situation-pre-pure}. Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$ above $s$. Assume $T = \\lim_{i \\in I} T_i$ is a directed limit of affine schemes over $S$. Then for some $i$ the triple $(T_i \\to S, t'_i \\leadsto t_i, \\xi_i)$ is an impurity of $\\mathcal{F}$ above $s$."}
+{"_id": "6052", "title": "flat-lemma-quasi-finite-impurity-elementary", "text": "In Situation \\ref{situation-pre-pure}. If there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ with $g$ quasi-finite at $t$, then there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ such that $(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood."}
+{"_id": "6053", "title": "flat-lemma-Noetherian-impurity-quasi-finite", "text": "In Situation \\ref{situation-pre-pure}. Assume that $S$ is locally Noetherian. If there exists an impurity of $\\mathcal{F}$ above $s$, then there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$."}
+{"_id": "6054", "title": "flat-lemma-impurity-on-henselization", "text": "In Situation \\ref{situation-pre-pure}. If there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$ of $\\mathcal{F}$ above $s$ then there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ where $(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood."}
+{"_id": "6056", "title": "flat-lemma-base-change-universally", "text": "Let $f : X \\to S$ be a morphism of schemes which is of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is universally pure along $X_s$, and \\item for every morphism of pointed schemes $(S', s') \\to (S, s)$ the pullback $\\mathcal{F}_{S'}$ is pure along $X_{s'}$. \\end{enumerate} In particular, $\\mathcal{F}$ is universally pure relative to $S$ if and only if every base change $\\mathcal{F}_{S'}$ of $\\mathcal{F}$ is pure relative to $S'$."}
+{"_id": "6057", "title": "flat-lemma-quasi-finite-base-change", "text": "Let $f : X \\to S$ be a morphism of schemes which is of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes. If $S' \\to S$ is quasi-finite at $s'$ and $\\mathcal{F}$ is pure along $X_s$, then $\\mathcal{F}_{S'}$ is pure along $X_{s'}$."}
+{"_id": "6059", "title": "flat-lemma-flat-descend-pure", "text": "Let $f : X \\to S$ be a morphism of schemes which is of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes. Assume $S' \\to S$ is flat at $s'$. \\begin{enumerate} \\item If $\\mathcal{F}_{S'}$ is pure along $X_{s'}$, then $\\mathcal{F}$ is pure along $X_s$. \\item If $\\mathcal{F}_{S'}$ is universally pure along $X_{s'}$, then $\\mathcal{F}$ is universally pure along $X_s$. \\end{enumerate}"}
+{"_id": "6060", "title": "flat-lemma-supported-on-closed", "text": "Let $i : Z \\to X$ be a closed immersion of schemes of finite type over a scheme $S$. Let $s \\in S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent sheaf on $Z$. Then $\\mathcal{F}$ is (universally) pure along $Z_s$ if and only if $i_*\\mathcal{F}$ is (universally) pure along $X_s$."}
+{"_id": "6061", "title": "flat-lemma-proper-pure", "text": "Let $f : X \\to S$ be a morphism of schemes which is of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If the support of $\\mathcal{F}$ is proper over $S$, then $\\mathcal{F}$ is universally pure relative to $S$. \\item If $f$ is proper, then $\\mathcal{F}$ is universally pure relative to $S$. \\item If $f$ is proper, then $X$ is universally pure relative to $S$. \\end{enumerate}"}
+{"_id": "6064", "title": "flat-lemma-affine-locally-projective-pure", "text": "Let $f : X \\to S$ be a finite type, affine morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module such that $f_*\\mathcal{F}$ is locally projective on $S$, see Properties, Definition \\ref{properties-definition-locally-projective}. Then $\\mathcal{F}$ is universally pure over $S$."}
+{"_id": "6065", "title": "flat-lemma-associated-point-specializes", "text": "Let $f : X \\to S$ be a morphism of schemes of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $s \\in S$. Assume that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$. Let $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$ with $f(x') = s'$ such that $s' \\leadsto s$ is a specialization in $S$. If $x'$ specializes to a point of $X_s$, then $x' \\leadsto x$ with $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$."}
+{"_id": "6066", "title": "flat-lemma-criterion", "text": "Let $f : X \\to S$ be a morphism of schemes of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $s \\in S$. Let $(S', s') \\to (S, s)$ be an elementary \\'etale neighbourhood and let $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ S & S' \\ar[l] } $$ be a commutative diagram of morphisms of schemes. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$, \\item $X' \\to S'$ is of finite type, \\item $g^*\\mathcal{F}$ is pure along $X'_{s'}$, \\item $g : X' \\to X$ is \\'etale, and \\item $g(X')$ contains $\\text{Ass}_{X_s}(\\mathcal{F}_s)$. \\end{enumerate} In this situation $\\mathcal{F}$ is pure along $X_s$ if and only if the image of $X' \\to X \\times_S S'$ contains the points of $\\text{Ass}_{X \\times_S S'/S'}(\\mathcal{F} \\times_S S')$ lying over points in $S'$ which specialize to $s'$."}
+{"_id": "6067", "title": "flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$. Assume \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is of finite type, \\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$, and \\item $\\mathcal{F}$ is pure along $X_s$. \\end{enumerate} Then $\\mathcal{F}$ is universally pure along $X_s$."}
+{"_id": "6070", "title": "flat-lemma-flat-finite-presentation-purity-open", "text": "Let $f : X \\to S$ be a morphism of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation flat over $S$. Then the set $$ U = \\{s \\in S \\mid \\mathcal{F}\\text{ is pure along }X_s\\} $$ is open in $S$."}
+{"_id": "6071", "title": "flat-lemma-injectivity-map-source-flat-pure", "text": "Let $f : X \\to S$ be a morphism of finite type. Let $\\mathcal{F}$ be a quasi-coherent sheaf of finite type on $X$. Assume $S$ is local with closed point $s$. Assume $\\mathcal{F}$ is pure along $X_s$ and that $\\mathcal{F}$ is flat over $S$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of quasi-coherent $\\mathcal{O}_X$-modules. Then the following are equivalent \\begin{enumerate} \\item the map on stalks $\\varphi_x$ is injective for all $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$, and \\item $\\varphi$ is injective. \\end{enumerate}"}
+{"_id": "6072", "title": "flat-lemma-flat-finite-presentation-affine-neighbourhood-projective", "text": "Let $f : X \\to S$ be a morphism which is locally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is of finite presentation. Let $x \\in X$ with $s = f(x) \\in S$. If $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and an affine open $U' \\subset X \\times_S S'$ which contains $x' = (x, s')$ such that $\\Gamma(U', \\mathcal{F}|_{U'})$ is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module."}
+{"_id": "6073", "title": "flat-lemma-flat-finite-type-affine-neighbourhood-projective", "text": "Let $f : X \\to S$ be a morphism which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is of finite type. Let $x \\in X$ with $s = f(x) \\in S$. If $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and an affine open $U' \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ which contains $x' = (x, s')$ such that $\\Gamma(U', \\mathcal{F}|_{U'})$ is a free $\\mathcal{O}_{S', s'}$-module."}
+{"_id": "6074", "title": "flat-lemma-flat-finite-type-local-colimit-free", "text": "Let $A \\to B$ be a local ring map of local rings which is essentially of finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module. If $A$ is henselian, then $N$ is a filtered colimit $$ N = \\colim_i F_i $$ of free $A$-modules $F_i$ such that all transition maps $u_i : F_i \\to F_{i'}$ of the system induce injective maps $\\overline{u}_i : F_i/\\mathfrak m_AF_i \\to F_{i'}/\\mathfrak m_AF_{i'}$. Also, $N$ is a Mittag-Leffler $A$-module."}
+{"_id": "6075", "title": "flat-lemma-flat-finite-type-local-valuation-ring-has-content", "text": "Let $A \\to B$ be a local ring map of local rings which is essentially of finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module. If $A$ is a valuation ring, then any element of $N$ has a content ideal $I \\subset A$ (More on Algebra, Definition \\ref{more-algebra-definition-content-ideal})."}
+{"_id": "6076", "title": "flat-lemma-iso-sheaf", "text": "In Situation \\ref{situation-iso}. \\begin{enumerate} \\item Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology. \\item If $f$ is quasi-compact and $\\mathcal{G}$ is of finite type, then $F_{surj}$ is limit preserving. \\item If $f$ is quasi-compact and $\\mathcal{F}$ of finite type, then $F_{zero}$ is limit preserving. \\item If $f$ is quasi-compact, $\\mathcal{F}$ is of finite type, and $\\mathcal{G}$ is of finite presentation, then $F_{iso}$ is limit preserving. \\end{enumerate}"}
+{"_id": "6077", "title": "flat-lemma-flat-at-point", "text": "In Situation \\ref{situation-flat-at-point}. \\begin{enumerate} \\item If $A' \\to A''$ is a flat morphism in $\\mathcal{C}$ then $F_{lf}(A') = F_{lf}(A'')$. \\item If $A \\to B$ is essentially of finite presentation and $M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit preserving: If $\\{A_i\\}_{i \\in I}$ is a directed system of objects of $\\mathcal{C}$, then $F_{lf}(\\colim_i A_i) = \\colim_i F_{lf}(A_i)$. \\end{enumerate}"}
+{"_id": "6078", "title": "flat-lemma-flat-at-point-finite", "text": "In Situation \\ref{situation-flat-at-point}. Let $B \\to C$ is a local map of local $A$-algebras and $N$ a $C$-module. Denote $F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated to the pair $(C, N)$. If $M \\cong N$ as $B$-modules and $B \\to C$ is finite, then $F_{lf} = F'_{lf}$."}
+{"_id": "6079", "title": "flat-lemma-flat-at-point-go-up", "text": "In Situation \\ref{situation-flat-at-point} suppose that $B \\to C$ is a flat local homomorphism of local rings. Set $N = M \\otimes_B C$. Denote $F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated to the pair $(C, N)$. Then $F_{lf} = F'_{lf}$."}
+{"_id": "6080", "title": "flat-lemma-free-at-generic-points", "text": "In Situation \\ref{situation-free-at-generic-points}. \\begin{enumerate} \\item The functor $H_p$ satisfies the sheaf property for the fpqc topology. \\item If $\\mathcal{F}$ is of finite presentation, then functor $H_p$ is limit preserving. \\end{enumerate}"}
+{"_id": "6081", "title": "flat-lemma-pre-flat-dimension-n", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $n \\geq 0$. The following are equivalent \\begin{enumerate} \\item for $s \\in S$ the closed subset $Z \\subset X_s$ of points where $\\mathcal{F}$ is not flat over $S$ (see Lemma \\ref{lemma-open-in-fibre-where-flat}) satisfies $\\dim(Z) < n$, and \\item for $x \\in X$ such that $\\mathcal{F}$ is not flat at $x$ over $S$ we have $\\text{trdeg}_{\\kappa(f(x))}(\\kappa(x)) < n$. \\end{enumerate} If this is true, then it remains true after any base change."}
+{"_id": "6082", "title": "flat-lemma-flat-dimension-n", "text": "In Situation \\ref{situation-flat-dimension-n}. \\begin{enumerate} \\item The functor $F_n$ satisfies the sheaf property for the fpqc topology. \\item If $f$ is quasi-compact and locally of finite presentation and $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is limit preserving. \\end{enumerate}"}
+{"_id": "6084", "title": "flat-lemma-generic-flatness-stratification", "text": "Let $f : X \\to S$ be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation. Then there exists a $t \\geq 0$ and closed subschemes $$ S \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset $$ such that $S_i \\to S$ is defined by a finite type ideal sheaf, $S_0 \\subset S$ is a thickening, and $\\mathcal{F}$ pulled back to $X \\times_S (S_i \\setminus S_{i + 1})$ is flat over $S_i \\setminus S_{i + 1}$."}
+{"_id": "6086", "title": "flat-lemma-universally-separating", "text": "Let $S$ be a scheme. Let $g : X' \\to X$ be a flat morphism of schemes over $S$ with $X$ locally of finite type over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module which is flat over $S$. If $\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$ then the canonical map $$ \\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F} $$ is injective, and remains injective after any base change."}
+{"_id": "6087", "title": "flat-lemma-flattening-module-map", "text": "Let $A$ be a ring. Let $u : M \\to N$ be a surjective map of $A$-modules. If $M$ is projective as an $A$-module, then there exists an ideal $I \\subset A$ such that for any ring map $\\varphi : A \\to B$ the following are equivalent \\begin{enumerate} \\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is an isomorphism, and \\item $\\varphi(I) = 0$. \\end{enumerate}"}
+{"_id": "6088", "title": "flat-lemma-Weil-restriction-closed-subschemes", "text": "Let $f:X\\to S$ be a morphism of schemes which is of finite presentation, flat, and pure. Let $Y$ be a closed subscheme of $X$. Let $F=f_*Y$ be the Weil restriction functor of $Y$ along $f$, defined by $$ F : (\\Sch/S)^{opp} \\to \\textit{Sets}, \\quad T \\mapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if} & Y_T\\to X_T \\text{ is an isomorphism, }\\\\ \\emptyset & \\text{else.} & \\end{matrix} \\right. $$ Then $F$ is representable by a closed immersion $Z\\to S$. Moreover $Z\\to S$ is of finite presentation if $Y\\to S$ is."}
+{"_id": "6089", "title": "flat-lemma-freebie", "text": "Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $E \\subset X_s$ be a subset. There exists a closed subscheme $Z \\subset S$ with the following property: for any morphism of pointed schemes $(T, t) \\to (S, s)$ the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}_T$ is flat over $T$ at all points of the fibre $X_t$ which map to a point of $E \\subset X_s$, and \\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$. \\end{enumerate} Moreover, if $X \\to S$ is locally of finite presentation, $\\mathcal{F}$ is of finite presentation, and $E \\subset X_s$ is closed and quasi-compact, then $Z \\to S$ is of finite presentation."}
+{"_id": "6091", "title": "flat-lemma-reduce-finite-type-injective-into-flat-mod-m", "text": "Let $A_0$ be a local ring. If the lemma holds for every Situation \\ref{situation-mod-injective} with $A = A_0$, with $B$ a localization of a polynomial algebra over $A$, and $N$ of finite presentation over $B$, then the lemma holds for every Situation \\ref{situation-mod-injective} with $A = A_0$."}
+{"_id": "6092", "title": "flat-lemma-henselian-finite-type-injective-into-flat-mod-m", "text": "If in Situation \\ref{situation-mod-injective} the ring $A$ is henselian then the lemma holds."}
+{"_id": "6093", "title": "flat-lemma-upstairs-finite-type-injective-into-flat-mod-m", "text": "Let $A \\to B$ be a local ring homomorphism of local rings which is essentially of finite type. Let $u : N \\to M$ be a $B$-module map. If $N$ is a finite $B$-module, $M$ is flat over $A$, and $\\overline{u} : N/\\mathfrak m_A N \\to M/\\mathfrak m_A M$ is injective, then $u$ is $A$-universally injective, $N$ is of finite presentation over $B$, and $N$ is flat over $A$."}
+{"_id": "6095", "title": "flat-lemma-properties-pure-spreadout", "text": "In (\\ref{equation-star}) if there exists a pure spreadout, then \\begin{enumerate} \\item elements of $N$ have content ideals in $A$, and \\item if $u : N \\to M$ is a morphism to a flat $A$-module $M$ such that $N/\\mathfrak m N \\to M/\\mathfrak m M$ is injective for all maximal ideals $\\mathfrak m$ of $A$, then $u$ is $A$-universally injective. \\end{enumerate}"}
+{"_id": "6096", "title": "flat-lemma-find-pure-spreadout", "text": "In (\\ref{equation-star}) for every $\\mathfrak p \\in \\Spec(A)$ there is a finitely generated ideal $I \\subset \\mathfrak pA_\\mathfrak p$ such that over $A_\\mathfrak p/I$ we have a pure spreadout."}
+{"_id": "6097", "title": "flat-lemma-universally-injective-if-flat", "text": "In (\\ref{equation-star}) assume $N$ is $A$-flat, $M$ is a flat $A$-module, and $u : N \\to M$ is an $A$-module map such that $u \\otimes \\text{id}_{\\kappa(\\mathfrak p)}$ is injective for all $\\mathfrak p \\in \\Spec(A)$. Then $u$ is $A$-universally injective."}
+{"_id": "6098", "title": "flat-lemma-big-intersection-is-zero", "text": "Let $A$ be a local domain which is not a field. Let $S$ be a set of finitely generated ideals of $A$. Assume that $S$ is closed under products and such that $\\bigcup_{I \\in S} V(I)$ is the complement of the generic point of $\\Spec(A)$. Then $\\bigcap_{I \\in S} I = (0)$."}
+{"_id": "6099", "title": "flat-lemma-closed-points-complement", "text": "Let $A$ be a local ring. Let $I, J \\subset A$ be ideals. If $J$ is finitely generated and $I \\subset J^n$ for all $n \\geq 1$, then $V(I)$ contains the closed points of $\\Spec(A) \\setminus V(J)$."}
+{"_id": "6100", "title": "flat-lemma-make-smaller-flatness-ideal", "text": "Let $A$ be a local ring. Let $I \\subset A$ be an ideal. Let $U \\subset \\Spec(A)$ be quasi-compact open. Let $M$ be an $A$-module. Assume that \\begin{enumerate} \\item $M/IM$ is flat over $A/I$, \\item $M$ is flat over $U$, \\end{enumerate} Then $M/I_2M$ is flat over $A/I_2$ where $I_2 = \\Ker(I \\to \\Gamma(U, I/I^2))$."}
+{"_id": "6102", "title": "flat-lemma-free-at-generic-points-representable", "text": "In Situation \\ref{situation-free-at-generic-points}. For each $p \\geq 0$ the functor $H_p$ (\\ref{equation-free-at-generic-points}) is representable by a locally closed immersion $S_p \\to S$. If $\\mathcal{F}$ is of finite presentation, then $S_p \\to S$ is of finite presentation."}
+{"_id": "6103", "title": "flat-lemma-localize-flat-dimension-n", "text": "In Situation \\ref{situation-flat-dimension-n}. Let $h : X' \\to X$ be an \\'etale morphism. Set $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$. Let $F_n'$ be (\\ref{equation-flat-dimension-n}) associated to $(f' : X' \\to S, \\mathcal{F}')$. Then $F_n$ is a subfunctor of $F_n'$ and if $h(X') \\supset \\text{Ass}_{X/S}(\\mathcal{F})$, then $F_n = F'_n$."}
+{"_id": "6104", "title": "flat-lemma-compare-H-F", "text": "Assume that $X \\to S$ is a smooth morphism of affine schemes with geometrically irreducible fibres of dimension $d$ and that $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite presentation. Then $F_d = \\coprod_{p = 0, \\ldots, c} H_p$ for some $c \\geq 0$ with $F_d$ as in (\\ref{equation-flat-dimension-n}) and $H_p$ as in (\\ref{equation-free-at-generic-points})."}
+{"_id": "6105", "title": "flat-lemma-flat-dimension-n-representable", "text": "In Situation \\ref{situation-flat-dimension-n}. Let $s \\in S$ let $d \\geq 0$. Assume \\begin{enumerate} \\item there exists a complete d\\'evissage of $\\mathcal{F}/X/S$ over some point $s \\in S$, \\item $X$ is of finite presentation over $S$, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and \\item $\\mathcal{F}$ is flat in dimensions $\\geq d + 1$ over $S$. \\end{enumerate} Then after possibly replacing $S$ by an open neighbourhood of $s$ the functor $F_d$ (\\ref{equation-flat-dimension-n}) is representable by a monomorphism $Z_d \\to S$ of finite presentation."}
+{"_id": "6106", "title": "flat-lemma-when-universal-flattening", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $f$ is of finite presentation, $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is pure relative to $S$, then there exists a universal flattening $S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is a monomorphism of finite presentation. \\item If $f$ is of finite presentation and $X$ is pure relative to $S$, then there exists a universal flattening $S' \\to S$ of $X$. Moreover $S' \\to S$ is a monomorphism of finite presentation. \\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, then there exists a universal flattening $S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is a monomorphism of finite presentation. \\item If $f$ is proper and of finite presentation then there exists a universal flattening $S' \\to S$ of $X$. \\end{enumerate}"}
+{"_id": "6107", "title": "flat-lemma-compute-what-it-should-be", "text": "In Situation \\ref{situation-existence} consider $$ K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) = DQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n) $$ Then $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\\geq 0$."}
+{"_id": "6108", "title": "flat-lemma-compute-against-perfect", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. For any perfect object $E$ of $D(\\mathcal{O}_X)$ we have \\begin{enumerate} \\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n$ in $D(A_n)$, \\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$ in $D(A_n)$. \\end{enumerate} In both statements $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$."}
+{"_id": "6109", "title": "flat-lemma-relative-pseudo-coherence", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Then $K$ is pseudo-coherent relative to $A$."}
+{"_id": "6110", "title": "flat-lemma-compute-over-affine", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. For any quasi-compact open $U \\subset X$ we have $$ R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n = R\\Gamma(U_n, \\mathcal{F}_n) $$ in $D(A_n)$ where $U_n = U \\cap X_n$."}
+{"_id": "6111", "title": "flat-lemma-finitely-presented", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Denote $X_0 \\subset X$ the closed subset consisting of points lying over the closed subset $\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$. There exists an open $W \\subset X$ containing $X_0$ such that \\begin{enumerate} \\item $H^i(K)|_W$ is zero unless $i = 0$, \\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and \\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$. \\end{enumerate}"}
+{"_id": "6112", "title": "flat-lemma-proper-support", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$ be as in Lemma \\ref{lemma-finitely-presented}. Set $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$, the support of $\\mathcal{F}$ is proper over $A$."}
+{"_id": "6113", "title": "flat-lemma-monomorphism-isomorphism", "text": "Let $A = \\lim A_n$ be a limit of a system of rings whose transition maps are surjective and with locally nilpotent kernels. Let $S = \\Spec(A)$. Let $T \\to S$ be a monomorphism which is locally of finite type. If $\\Spec(A_n) \\to S$ factors through $T$ for all $n$, then $T = S$."}
+{"_id": "6114", "title": "flat-lemma-compute-what-it-should-be-derived", "text": "In Situation \\ref{situation-existence-derived} consider $$ K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) = DQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n) $$ Then $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$."}
+{"_id": "6115", "title": "flat-lemma-compute-against-perfect-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect object $E$ of $D(\\mathcal{O}_X)$ the cohomology $$ M = R\\Gamma(X, K \\otimes^\\mathbf{L} E) $$ is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism $$ R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n $$ in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$."}
+{"_id": "6116", "title": "flat-lemma-relative-pseudo-coherence-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$ is pseudo-coherent on $X$."}
+{"_id": "6117", "title": "flat-lemma-compute-over-affine-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any quasi-compact open $U \\subset X$ we have $$ R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n = R\\Gamma(U_n, K_n) $$ in $D(A_n)$ where $U_n = U \\cap X_n$."}
+{"_id": "6118", "title": "flat-lemma-helper-blowup-affine-space", "text": "Let $R$ be a ring and let $f \\in R$. Let $r\\geq 0$ be an integer. Let $R \\to S$ be a ring map and let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $R \\to S$ is of finite presentation and flat, \\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is geometrically integral over $R$, \\item $M$ is a finite $S$-module, \\item $M_f$ is a finitely presented $S_f$-module, \\item for all $\\mathfrak p \\in R$, $f \\not \\in \\mathfrak p$ with $\\mathfrak q = \\mathfrak pS$ the module $M_{\\mathfrak q}$ is free of rank $r$ over $S_\\mathfrak q$. \\end{enumerate} Then there exists a finitely generated ideal $I \\subset R$ with $V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$ the quotient $$ M' = (M \\otimes_R R')/a\\text{-power torsion} $$ over $S' = S \\otimes_R R'$ satisfies the following: for every prime $\\mathfrak p' \\subset R'$ there exists a $g \\in S'$, $g \\not \\in \\mathfrak p'S'$ such that $M'_g$ is a free $S'_g$-module of rank $r$."}
+{"_id": "6119", "title": "flat-lemma-flatten-module-pre", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \\subset S$ be a quasi-compact open. Assume \\begin{enumerate} \\item $X \\to S$ is affine, of finite presentation, flat, geometrically integral fibres, \\item $\\mathcal{F}$ is a module of finite type, \\item $\\mathcal{F}_U$ is of finite presentation, \\item $\\mathcal{F}$ is flat over $S$ at all generic points of fibres lying over points of $U$. \\end{enumerate} Then there exists a $U$-admissible blowup $S' \\to S$ and an open subscheme $V \\subset X_{S'}$ such that (a) the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ restricts to a finitely locally free $\\mathcal{O}_V$-module and (b) $V \\to S'$ is surjective."}
+{"_id": "6120", "title": "flat-lemma-trick-fitting-ideal", "text": "Let $A \\to C$ be a finite locally free ring map of rank $d$. Let $h \\in C$ be an element such that $C_h$ is \\'etale over $A$. Let $J \\subset C$ be an ideal. Set $I = \\text{Fit}_0(C/J)$ where we think of $C/J$ as a finite $A$-module. Then $IC_h = JJ'$ for some ideal $J' \\subset C_h$. If $J$ is finitely generated so are $I$ and $J'$."}
+{"_id": "6121", "title": "flat-lemma-push-ideal", "text": "Let $A \\to B$ be an \\'etale ring map. Let $a \\in A$ be a nonzerodivisor. Let $J \\subset B$ be a finite type ideal with $V(J) \\subset V(aB)$. For every $\\mathfrak q \\subset B$ there exists a finite type ideal $I \\subset A$ with $V(I) \\subset V(a)$ and $g \\in B$, $g \\not \\in \\mathfrak q$ such that $IB_g = JJ'$ for some finite type ideal $J' \\subset B_g$."}
+{"_id": "6122", "title": "flat-lemma-flatten-module-etale-localize", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \\subset S$ be a quasi-compact open. Assume there exist finitely many commutative diagrams $$ \\xymatrix{ & X_i \\ar[r]_{j_i} \\ar[d] & X \\ar[d] \\\\ S_i^* \\ar[r] & S_i \\ar[r]^{e_i} & S } $$ where \\begin{enumerate} \\item $e_i : S_i \\to S$ are quasi-compact \\'etale morphisms and $S = \\bigcup e_i(S_i)$, \\item $j_i : X_i \\to X$ are \\'etale morphisms and $X = \\bigcup j_i(X_i)$, \\item $S^*_i \\to S_i$ is an $e_i^{-1}(U)$-admissible blowup such that the strict transform $\\mathcal{F}_i^*$ of $j_i^*\\mathcal{F}$ is flat over $S^*_i$. \\end{enumerate} Then there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform of $\\mathcal{F}$ is flat over $S'$."}
+{"_id": "6123", "title": "flat-lemma-flat-after-blowing-up", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \\subset S$ be a quasi-compact open. Assume \\begin{enumerate} \\item $X \\to S$ is of finite type and quasi-separated, and \\item $X_U \\to U$ is flat and locally of finite presentation. \\end{enumerate} Then there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform of $X$ is flat and of finite presentation over $S'$."}
+{"_id": "6124", "title": "flat-lemma-finite-after-blowing-up", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \\subset S$ be a quasi-compact open. Assume \\begin{enumerate} \\item $X \\to S$ is proper, and \\item $X_U \\to U$ is finite locally free. \\end{enumerate} Then there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform of $X$ is finite locally free over $S'$."}
+{"_id": "6125", "title": "flat-lemma-zariski-after-blowup", "text": "Let $\\varphi : X \\to S$ be a separated morphism of finite type with $S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a quasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform $X'$ of $X$ is isomorphic to an open subscheme of $S'$."}
+{"_id": "6126", "title": "flat-lemma-dominate-modification-by-blowup", "text": "Let $\\varphi : X \\to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a quasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \\to S$ which dominates $X$, i.e., such that there exists a factorization $S' \\to X \\to S$ of the blowup morphism."}
+{"_id": "6128", "title": "flat-lemma-compactifications-cofiltered", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a compactifyable scheme over $S$. \\begin{enumerate} \\item[(a)] The category of compactifications of $X$ over $S$ is cofiltered. \\item[(b)] The full subcategory consisting of compactifications $j : X \\to \\overline{X}$ such that $j(X)$ is dense and scheme theoretically dense in $\\overline{X}$ is initial (Categories, Definition \\ref{categories-definition-initial}). \\item[(c)] If $f : \\overline{X}' \\to \\overline{X}$ is a morphism of compactifications of $X$ such that $j'(X)$ is dense in $\\overline{X}'$, then $f^{-1}(j(X)) = j'(X)$. \\end{enumerate}"}
+{"_id": "6129", "title": "flat-lemma-compactifyable", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$ with $Y$ separated and of finite type over $S$ and $X$ compactifyable over $S$. Then $X$ has a compactification over $Y$."}
+{"_id": "6130", "title": "flat-lemma-right-multiplicative-system", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. The collection of morphisms $(u, \\overline{u}) : (X', \\overline{X}') \\to (X, \\overline{X})$ such that $u$ is an isomorphism forms a right multiplicative system (Categories, Definition \\ref{categories-definition-multiplicative-system}) of arrows in the category of compactifications."}
+{"_id": "6132", "title": "flat-lemma-check-separated", "text": "Let $X \\to S$ be a morphism of schemes. If $X = U \\cup V$ is an open cover such that $U \\to S$ and $V \\to S$ are separated and $U \\cap V \\to U \\times_S V$ is closed, then $X \\to S$ is separated."}
+{"_id": "6133", "title": "flat-lemma-separate-disjoint-locally-closed-by-blowing-up", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open. \\begin{enumerate} \\item If $Z_1, Z_2 \\subset X$ are closed subschemes of finite presentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then there exists a $U$-admissible blowing up $X' \\to X$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. \\item If $T_1, T_2 \\subset U$ are disjoint constructible closed subsets, then there is a $U$-admissible blowing up $X' \\to X$ such that the closures of $T_1$ and $T_2$ are disjoint. \\end{enumerate}"}
+{"_id": "6134", "title": "flat-lemma-blowup-iso-along", "text": "Let $f : X \\to Y$ be a proper morphism of quasi-compact and quasi-separated schemes. Let $V \\subset Y$ be a quasi-compact open and $U = f^{-1}(V)$. Let $T \\subset V$ be a closed subset such that $f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$ in $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that the strict transform $f' : X' \\to Y'$ of $f$ is an isomorphism over an open neighbourhood of the closure of $T$ in $Y'$."}
+{"_id": "6135", "title": "flat-lemma-find-common-blowups", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $U \\to X_1$ and $U \\to X_2$ be open immersions of schemes over $S$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $S$. Then there exists a commutative diagram $$ \\xymatrix{ X_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\ X_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2 } $$ of schemes over $S$ where $X_i' \\to X_i$ is a $U$-admissible blowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite type over $S$."}
+{"_id": "6136", "title": "flat-lemma-replaced-by-strict-transform", "text": "Let $X \\to S$ and $Y \\to S$ be morphisms of schemes. Let $U \\subset X$ be an open subscheme. Let $V \\to X \\times_S Y$ be a quasi-compact morphism whose composition with the first projection maps into $U$. Let $Z \\subset X \\times_S Y$ be the scheme theoretic image of $V \\to X \\times_S Y$. Let $X' \\to X$ be a $U$-admissible blowup. Then the scheme theoretic image of $V \\to X' \\times_S Y$ is the strict transform of $Z$ with respect to the blowing up."}
+{"_id": "6137", "title": "flat-lemma-compactification-dominates", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $U$ be a scheme of finite type and separated over $S$. Let $V \\subset U$ be a quasi-compact open. If $V$ has a compactification $V \\subset Y$ over $S$, then there exists a $V$-admissible blowing up $Y' \\to Y$ and an open $V \\subset V' \\subset Y'$ such that $V \\to U$ extends to a proper morphism $V' \\to U$."}
+{"_id": "6138", "title": "flat-lemma-two-compactifications", "text": "Let $S$ be a Noetherian scheme. Let $U$ be a scheme of finite type and separated over $S$. Let $U = U_1 \\cup U_2$ be opens such that $U_1$ and $U_2$ have compactifications over $S$ and such that $U_1 \\cap U_2$ is dense in $U$. Then $U$ has a compactification over $S$."}
+{"_id": "6139", "title": "flat-lemma-equivalence-h-v-locally-finite-presentation", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes with fixed target with $f_i$ locally of finite presentation for all $i$. The following are equivalent \\begin{enumerate} \\item $\\{X_i \\to X\\}$ is a ph covering, and \\item $\\{X_i \\to X\\}$ is a V covering. \\end{enumerate}"}
+{"_id": "6141", "title": "flat-lemma-approximate-h-cover", "text": "Let $X$ be an affine scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an h covering. Then there exists a surjective proper morphism $$ Y \\longrightarrow X $$ of finite presentation (!) and a finite affine open covering $Y = \\bigcup_{j = 1, \\ldots, m} Y_j$ such that $\\{Y_j \\to X\\}_{j = 1, \\ldots, m}$ refines $\\{X_i \\to X\\}_{i \\in I}$."}
+{"_id": "6142", "title": "flat-lemma-zariski-h", "text": "An fppf covering is a h covering. Hence syntomic, smooth, \\'etale, and Zariski coverings are h coverings as well."}
+{"_id": "6143", "title": "flat-lemma-surjective-proper-finite-presentation-h", "text": "Let $f : Y \\to X$ be a surjective proper morphism of schemes which is of finite presentation. Then $\\{Y \\to X\\}$ is an h covering."}
+{"_id": "6145", "title": "flat-lemma-h", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is an h covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering and for each $i$ we have an h covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an h covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an h covering. \\end{enumerate}"}
+{"_id": "6146", "title": "flat-lemma-h-induced", "text": "Let $\\Sch_h$ be a big h site as in Definition \\ref{definition-big-h-site}. Let $T \\in \\Ob(\\Sch_h)$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary h covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_h$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard h covering, then it is tautologically equivalent to a covering of $\\Sch_h$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\\Sch_h$. \\end{enumerate}"}
+{"_id": "6150", "title": "flat-lemma-characterize-sheaf-h", "text": "Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_h$. Then $\\mathcal{F}$ is a sheaf if and only if \\begin{enumerate} \\item $\\mathcal{F}$ satisfies the sheaf condition for Zariski coverings, and \\item if $f : V \\to U$ is proper, surjective, and of finite presentation, then $\\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$. \\end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \\begin{enumerate} \\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine. \\end{enumerate}"}
+{"_id": "6151", "title": "flat-lemma-morphism-big-h", "text": "Let $\\Sch_h$ be a big h site. Let $f : T \\to S$ be a morphism in $\\Sch_h$. The functor $$ u : (\\Sch/T)_h \\longrightarrow (\\Sch/S)_h, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_h \\longrightarrow (\\Sch/T)_h, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_h) \\longrightarrow \\Sh((\\Sch/S)_h) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "6153", "title": "flat-lemma-limit-h-topology", "text": "Let $T$ be an affine scheme which is written as a limit $T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes. \\begin{enumerate} \\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard h covering of $T$, see Definition \\ref{definition-standard-h}. Then there exists an index $i$ and a standard h covering $\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$ whose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$ is isomorphic to $\\mathcal{V}$. \\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard h coverings of $T_i$. If $f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is a morphism of coverings of $T$, then there exists an index $i' \\geq i$ and a morphism $f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to T_{i'} \\times_{T_i} \\mathcal{V}'_i$ whose base change to $T$ is $f$. \\item If $f, g : \\mathcal{V} \\to \\mathcal{V}'_i$ are morphisms of standard h coverings of $T_i$ whose base changes $f_T, g_T$ to $T$ are equal then there exists an index $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$. \\end{enumerate} In other words, the category of standard h coverings of $T$ is the colimit over $I$ of the categories of standard h coverings of $T_i$."}
+{"_id": "6154", "title": "flat-lemma-extend-sheaf-h", "text": "Let $S$ be a scheme contained in a big site $\\Sch_h$. Let $F : (\\Sch/S)_h^{opp} \\to \\textit{Sets}$ be an h sheaf satisfying property (b) of Topologies, Lemma \\ref{topologies-lemma-extend} with $\\mathcal{C} = (\\Sch/S)_h$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all h coverings and is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving})."}
+{"_id": "6157", "title": "flat-lemma-base-change-almost-blow-up", "text": "Consider an almost blow up square (\\ref{equation-almost-blow-up-square}). Let $Y \\to X$ be any morphism. Then the base change $$ \\xymatrix{ Y \\times_X E \\ar[d] \\ar[r] & Y \\times_X X' \\ar[d] \\\\ Y \\times_X Z \\ar[r] & Y } $$ is an almost blow up square too."}
+{"_id": "6158", "title": "flat-lemma-shrink-almost-blow-up", "text": "Consider an almost blow up square (\\ref{equation-almost-blow-up-square}). Let $W \\to X'$ be a closed immersion of finite presentation. The following are equivalent \\begin{enumerate} \\item $X' \\setminus E$ is scheme theoretically contained in $W$, \\item the blowup $X''$ of $X$ in $Z$ is scheme theoretically contained in $W$, \\item the diagram $$ \\xymatrix{ E \\cap W \\ar[d] \\ar[r] & W \\ar[d] \\\\ Z \\ar[r] & X } $$ is an almost blow up square. Here $E \\cap W$ is the scheme theoretic intersection. \\end{enumerate}"}
+{"_id": "6159", "title": "flat-lemma-blow-up-limit-almost-blow-up", "text": "Consider an almost blow up square (\\ref{equation-almost-blow-up-square}) with $X$ quasi-compact and quasi-separated. Then the blowup $X''$ of $X$ in $Z$ can be written as $$ X'' = \\lim X'_i $$ where the limit is over the directed system of closed subschemes $X'_i \\subset X'$ of finite presentation satisfying the equivalent conditions of Lemma \\ref{lemma-shrink-almost-blow-up}."}
+{"_id": "6160", "title": "flat-lemma-almost-blow-up-square", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \\subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Then there exists an almost blow up square as in (\\ref{equation-almost-blow-up-square})."}
+{"_id": "6161", "title": "flat-lemma-almost-blow-up-unique", "text": "Let $X$ be a quasi-compact and quasi-separated scheme and let $Z \\subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Suppose given almost blow up squares (\\ref{equation-almost-blow-up-square}) $$ \\xymatrix{ E_k \\ar[r] \\ar[d] & X_k' \\ar[d] \\\\ Z \\ar[r] & X } $$ for $k = 1, 2$, then there exists an almost blow up square $$ \\xymatrix{ E \\ar[r] \\ar[d] & X' \\ar[d] \\\\ Z \\ar[r] & X } $$ and closed immersions $i_k : X' \\to X'_k$ over $X$ with $E = i_k^{-1}(E_k)$."}
+{"_id": "6162", "title": "flat-lemma-flat-after-almost-blowing-up", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme of finite presentation over $Y$. Let $V \\subset Y$ be a quasi-compact open such that $X_V \\to V$ is flat. Then there exist a commutative diagram $$ \\xymatrix{ E \\ar[ddd] \\ar[rd] & & & D \\ar[lll] \\ar[ddd] \\ar[ld] \\\\ & Y' \\ar[d] & X' \\ar[l] \\ar[d] \\\\ & Y & X \\ar[l] \\\\ Z \\ar[ru] & & & T \\ar[lll] \\ar[lu] } $$ whose right and left hand squares are almost blow up squares, whose lower and top squares are cartesian, such that $Z \\cap V = \\emptyset$, and such that $X' \\to Y'$ is flat (and of finite presentation)."}
+{"_id": "6163", "title": "flat-lemma-blow-up-square-h", "text": "Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$ constructed in Definition \\ref{definition-big-small-h}. Then for any almost blow up square (\\ref{equation-almost-blow-up-square}) in the category $(\\Sch/S)_h$ the diagram $$ \\xymatrix{ \\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\ \\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l] } $$ is cartesian in the category of sets."}
+{"_id": "6164", "title": "flat-lemma-thickening-h", "text": "Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$ constructed in Definition \\ref{definition-big-small-h}. Let $X \\to X'$ be a morphism of $(\\Sch/S)_h$ which is a thickening and of finite presentation. Then $\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective."}
+{"_id": "6165", "title": "flat-lemma-refine-check-h", "text": "Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$ constructed in Definition \\ref{definition-big-small-h}. Then $\\mathcal{F}$ is a sheaf if and only if the following conditions are satisfied \\begin{enumerate} \\item $\\mathcal{F}$ is a sheaf for the Zariski topology, \\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $\\mathcal{F}(Y)$ is the equalizer of the two maps $\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$, \\item $\\mathcal{F}$ turns an almost blow up square as in Example \\ref{example-one-generator} in the category $(\\Sch/S)_h$ into a cartesian diagram of sets, and \\item $\\mathcal{F}$ turns an almost blow up square as in Example \\ref{example-two-generators} in the category $(\\Sch/S)_h$ into a cartesian diagram of sets. \\end{enumerate}"}
+{"_id": "6166", "title": "flat-lemma-refine-check-h-stack", "text": "Let $p : \\mathcal{S} \\to (\\Sch/S)_h$ be a category fibred in groupoids. Then $\\mathcal{S}$ is a stack in groupoids if and only if the following conditions are satisfied \\begin{enumerate} \\item $\\mathcal{S}$ is a stack in groupoids for the Zariski topology, \\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $$ \\mathcal{S}_Y \\longrightarrow \\mathcal{S}_X \\times_{\\mathcal{S}_{X \\times_Y X}} \\mathcal{S}_X $$ is an equivalence of categories, \\item for an almost blow up square as in Example \\ref{example-one-generator} or \\ref{example-two-generators} in the category $(\\Sch/S)_h$ the functor $$ \\mathcal{S}_X \\longrightarrow \\mathcal{S}_Z \\times_{\\mathcal{S}_E} \\mathcal{S}_{X'} $$ is an equivalence of categories. \\end{enumerate}"}
+{"_id": "6167", "title": "flat-lemma-funny-blow-up", "text": "Let $Z, X, X', E$ be an almost blow up square as in Example \\ref{example-two-generators}. Then $H^p(X', \\mathcal{O}_{X'}) = 0$ for $p > 0$ and $\\Gamma(X, \\mathcal{O}_X) \\to \\Gamma(X', \\mathcal{O}_{X'})$ is a surjective map of rings whose kernel is an ideal of square zero."}
+{"_id": "6168", "title": "flat-lemma-h-sheaf-colim-F", "text": "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$. Let $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}. There is a unique sheaf $\\mathcal{F}$ on $(\\Sch/S)_h$ such that $$ \\mathcal{F}(X) = \\colim_F \\Gamma(X, \\mathcal{O}_X) $$ for any quasi-compact and quasi-separated object $X$ of $(\\Sch/S)_h$."}
+{"_id": "6170", "title": "flat-lemma-weak-normalization-ph-sheaf", "text": "Let $(\\Sch/S)_{ph}$ be a site as in Topologies, Definition \\ref{topologies-definition-big-small-ph}. The rule $$ X \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}}) $$ is a sheaf on $(\\Sch/S)_{ph}$."}
+{"_id": "6171", "title": "flat-lemma-weak-normalization-h-sheaf", "text": "Let $S$ be a scheme. Choose a site $(\\Sch/S)_h$ as in Definition \\ref{definition-big-small-h}. The rule $$ X \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}}) $$ is the sheafification of the ``structure sheaf'' $\\mathcal{O}$ on $(\\Sch/S)_h$. Similarly for the ph topology."}
+{"_id": "6172", "title": "flat-lemma-perfect-weankly-normal", "text": "Let $p$ be a prime number. An $\\mathbf{F}_p$-algebra $A$ is absolutely weakly normal if and only if it is perfect."}
+{"_id": "6173", "title": "flat-lemma-char-p", "text": "Let $p$ be a prime number. \\begin{enumerate} \\item If $A$ is an $\\mathbf{F}_p$-algebra, then $\\colim_F A = A^{awn}$. \\item If $S$ is a scheme over $\\mathbf{F}_p$, then the h sheafification of $\\mathcal{O}$ sends a quasi-compact and quasi-separated $X$ to $\\colim_F \\Gamma(X, \\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6174", "title": "flat-lemma-colim-F-Vect", "text": "Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\\mathbf{F}_p$. The category $\\colim_F \\textit{Vect}(S)$ is equivalent to the category of finite locally free modules over the sheaf of rings $\\colim_F \\mathcal{O}_S$ on $S$."}
+{"_id": "6175", "title": "flat-lemma-vector-bundle-I", "text": "Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example \\ref{example-one-generator}. Then the functor $$ \\colim_F \\textit{Vect}(X) \\longrightarrow \\colim_F \\textit{Vect}(Z) \\times_{\\colim_F \\textit{Vect}(E)} \\colim_F \\textit{Vect}(X') $$ is an equivalence."}
+{"_id": "6176", "title": "flat-lemma-vector-bundle-II", "text": "Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example \\ref{example-two-generators}. Then the functor $$ G : \\colim_F \\textit{Vect}(X) \\longrightarrow \\colim_F \\textit{Vect}(Z) \\times_{\\colim_F \\textit{Vect}(E)} \\colim_F \\textit{Vect}(X') $$ is an equivalence."}
+{"_id": "6177", "title": "flat-lemma-trivial-fibres-dvr", "text": "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres where $S$ is the spectrum of a discrete valuation ring. Denote $\\eta \\in S$ the generic point and denote $X_n \\subset X$ the closed subscheme cutout by the $n$th power of a uniformizer on $S$. Then there exists an integer $n$ such that the following is true: any finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ such that $\\mathcal{E}|_{X_\\eta}$ and $\\mathcal{E}|_{X_n}$ are free, is free."}
+{"_id": "6178", "title": "flat-lemma-trivial-over-dvrs", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $f$ is flat and proper and $\\mathcal{O}_S = f_*\\mathcal{O}_X$, \\item $S$ is a normal Noetherian scheme, \\item the pullback of $\\mathcal{E}$ to $X \\times_S \\Spec(\\mathcal{O}_{S, s})$ is free for every codimension $1$ point $s \\in S$. \\end{enumerate} Then $\\mathcal{E}$ is isomorphic to the pullback of a finite locally free $\\mathcal{O}_S$-module."}
+{"_id": "6179", "title": "flat-lemma-fitting-ideals-complex", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent. For every $p, k \\in \\mathbf{Z}$ there is an finite type quasi-coherent sheaf of ideals $\\text{Fit}_{p, k}(E) \\subset \\mathcal{O}_X$ with the following property: for $U \\subset X$ open such that $E|_U$ is isomorphic to $$ \\ldots \\to \\mathcal{O}_U^{\\oplus n_{b - 2}} \\xrightarrow{d_{b - 2}} \\mathcal{O}_U^{\\oplus n_{b - 1}} \\xrightarrow{d_{b - 1}} \\mathcal{O}_U^{\\oplus n_b} \\to 0 \\to \\ldots $$ the restriction $\\text{Fit}_{p, k}(E)|_U$ is generated by the minors of the matrix of $d_p$ of size $$ - k + n_{p + 1} - n_{p + 2} + \\ldots + (-1)^{b - p + 1} n_b $$ Convention: the ideal generated by $r \\times r$-minors is $\\mathcal{O}_U$ if $r \\leq 0$ and the ideal generated by $r \\times r$-minors where $r > \\min(n_p, n_{p + 1})$ is zero."}
+{"_id": "6180", "title": "flat-lemma-blowup-complex", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Let $U \\subset X$ be a scheme theoretically dense open subscheme such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$ for all $i \\in \\mathbf{Z}$. Then there exists a $U$-admissible blowup $b : X' \\to X$ such that $H^i(Lb^*E)$ is a perfect $\\mathcal{O}_{X'}$-module of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$."}
+{"_id": "6181", "title": "flat-lemma-blowup-complex-integral", "text": "Let $X$ be an integral scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then there exists a nonempty open $U \\subset X$ such that $H^i(E|_U)$ is finite locally free of constant rank $r_i$ for all $i \\in \\mathbf{Z}$ and there exists a $U$-admissible blowup $b : X' \\to X$ such that $H^i(Lb^*E)$ is a perfect $\\mathcal{O}_{X'}$-module of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$."}
+{"_id": "6182", "title": "flat-lemma-blowup-pd1-derived", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$. For any blowup $b : X' \\to X$ we have $Lb^*\\mathcal{F} = b^*\\mathcal{F}$ and $b^*\\mathcal{F}$ is a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$."}
+{"_id": "6186", "title": "flat-lemma-eta-stalks", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$. \\begin{enumerate} \\item For $x \\in X$ choose a generator $f \\in \\mathcal{I}_x$. Then the stalk of $\\eta_\\mathcal{I}\\mathcal{F}^\\bullet$ is canonically isomorphic to $\\eta_f\\mathcal{F}^\\bullet_x$. \\item If the $\\mathcal{F}^i$ are quasi-coherent $\\mathcal{O}_X$-modules, then so are the $(\\eta_\\mathcal{I}\\mathcal{F})^i$ and in this case if $U = \\Spec(A) \\subset X$ is affine open and $D \\cap U = V(f)$, then $\\eta_f(\\mathcal{F}^\\bullet(U))$ is canonically isomorphic to $(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)(U)$. \\end{enumerate}"}
+{"_id": "6187", "title": "flat-lemma-eta-first-property", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$. There is a canonical isomorphism $$ \\mathcal{I}^{\\otimes i} \\otimes_{\\mathcal{O}_X} \\left( H^i(\\mathcal{F}^\\bullet)/H^i(\\mathcal{F}^\\bullet)[\\mathcal{I}] \\right) \\longrightarrow H^i(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet) $$ of cohomology sheaves."}
+{"_id": "6188", "title": "flat-lemma-eta-qis", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be a map of complexes of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ and $\\mathcal{G}^i$ are $\\mathcal{I}$-torsion free for all $i$. Then the induced map $\\eta_\\mathcal{I}\\mathcal{F}^\\bullet \\to \\eta_\\mathcal{I}\\mathcal{G}^\\bullet$ is a quasi-isomorphism too."}
+{"_id": "6189", "title": "flat-lemma-eta-tensor-invertible", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $\\eta_\\mathcal{I}(\\mathcal{F}^\\bullet \\otimes \\mathcal{L}) = (\\eta_\\mathcal{I}\\mathcal{F}^\\bullet) \\otimes \\mathcal{L}$."}
+{"_id": "6190", "title": "flat-lemma-complex-and-divisor-blowup", "text": "In Situation \\ref{situation-complex-and-divisor} let $b : X' \\to X$ be the blowing up of the product of the ideals $\\mathcal{J}_i$ from Remark \\ref{remark-complex-and-divisor-ideal}. Denote $D' = b^{-1}D$ with ideal sheaf $\\mathcal{I}' \\subset \\mathcal{O}_{X'}$. Then $$ \\mathcal{Q}^\\bullet = \\eta_{\\mathcal{I}'}b^*\\mathcal{E}^\\bullet $$ is a bounded complex of finite locally free $\\mathcal{O}_{X'}$-modules."}
+{"_id": "6191", "title": "flat-lemma-complex-and-divisor-eta-pull", "text": "In Situation \\ref{situation-complex-and-divisor} let $f : Y \\to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor with ideal sheaf $\\mathcal{J}$. Assume $\\mathcal{J}_i$ as in Remark \\ref{remark-complex-and-divisor-ideal} is invertible for all $i$. Then $f^*(\\eta_\\mathcal{I}\\mathcal{E}^\\bullet) = \\eta_\\mathcal{J}(f^*\\mathcal{E}^\\bullet)$."}
+{"_id": "6192", "title": "flat-lemma-complex-and-divisor-blowup-base-change", "text": "In Situation \\ref{situation-complex-and-divisor} let $f : Y \\to X$ be a morphism of schemes such that the inverse image $f^{-1}D$ is an effective Cartier divisor. Let $X' \\to X$ and $\\mathcal{Q}^\\bullet$, resp.\\ $Y' \\to Y$ and $\\mathcal{Q}_{Y'}^\\bullet$, be as constructed in Lemma \\ref{lemma-complex-and-divisor-blowup} for $D \\subset X$ and $\\mathcal{E}^\\bullet$, resp.\\ $f^{-1}D \\subset Y$ and $f^*\\mathcal{E}^\\bullet$. Then $Y'$ is the strict transform of $Y$ with respect to $X' \\to X$ and $\\mathcal{Q}_{Y'}^\\bullet = (Y' \\to X')^*\\mathcal{Q}^\\bullet$."}
+{"_id": "6193", "title": "flat-lemma-complex-and-divisor-blowup-good", "text": "In Situation \\ref{situation-complex-and-divisor} let $U \\subset X$ be the maximal open subscheme over which the cohomology sheaves of $\\mathcal{E}^\\bullet$ are locally free. Then the blowing up $b : X' \\to X$ of Lemma \\ref{lemma-complex-and-divisor-blowup} is an isomorphism over $U$."}
+{"_id": "6194", "title": "flat-lemma-complex-and-divisor-blowup-T", "text": "In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$, $D' \\subset X'$, and $\\mathcal{Q}^\\bullet$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup}. Let $U \\subset X$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-good}. Then there exists a closed immersion $T \\to D'$ of finite presentation with $D' \\cap b^{-1}(U) \\subset T$ scheme theoretically such that $\\mathcal{Q}^\\bullet|_T$ has finite locally free cohomology sheaves."}
+{"_id": "6195", "title": "flat-lemma-complex-and-divisor-blowup-T-ranks", "text": "In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$, $D' \\subset X'$, and $\\mathcal{Q}^\\bullet$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup}. Given integers $\\rho_i \\geq 0$ almost all zero, let $U' \\subset X$ be the maximal open subscheme where $H^i(\\mathcal{E}^\\bullet)$ is finite locally free of rank $\\rho_i$ for all $i$. Let $T \\subset D'$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-T}. Then there exists an open and closed subscheme $T' \\subset T$ containing $D' \\cap b^{-1}(U')$ scheme theoretically such that $\\mathcal{Q}^\\bullet|_{T'}$ has finite locally free cohomology sheaves $H^i(\\mathcal{Q}^\\bullet|_{T'})$ of rank $\\rho_i$."}
+{"_id": "6196", "title": "flat-lemma-complex-and-divisor-derived", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $U \\subset X$ be the maximal open over which the cohomology sheaves $H^i(E)$ are locally free. There exists a proper morphism $b : X' \\longrightarrow X$ and an object $Q \\in D(\\mathcal{O}_{X'})$ with the following properties \\begin{enumerate} \\item $D' = b^{-1}D$ is an effective Cartier divisor, \\item $Q = L\\eta_{\\mathcal{I}'}Lb^*E$ where $\\mathcal{I}'$ is the ideal sheaf of $D'$, \\item $Q$ is a perfect object of $D(\\mathcal{O}_{X'})$, \\item there exists a closed immersion $T \\to D'$ of finite presentation with $D' \\cap b^{-1}(U) \\subset T$ scheme theoretically such that $Q|_T$ has finite locally free cohomology sheaves, \\item for any open subscheme $V \\subset X$ such that $E|_V$ can be represented by a bounded complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_V$-modules, the base changes of $X' \\to X$, $Q$, $D'$, and $T$ to $V$ are given by the constructions of Lemmas \\ref{lemma-complex-and-divisor-blowup} and \\ref{lemma-complex-and-divisor-blowup-T}. \\end{enumerate}"}
+{"_id": "6197", "title": "flat-lemma-graph-construction", "text": "The construction above has the following properties: \\begin{enumerate} \\item $b$ is an isomorphism over $\\mathbf{P}^1_U \\cup \\mathbf{A}^1_X$, \\item the restriction of $\\mathcal{Q}^\\bullet$ to $\\mathbf{A}^1_X$ is equal to the pullback of $\\mathcal{E}^\\bullet$, \\item there exists a closed immersion $T \\to W_\\infty$ of finite presentation such that $\\infty(U) \\subset T$ scheme theoretically and such that $\\mathcal{Q}^\\bullet|_T$ has finite locally free cohomology sheaves. \\end{enumerate}"}
+{"_id": "6198", "title": "flat-proposition-existence-complete-at-x", "text": "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \\in X$ be a point with image $s \\in S$. There exists a commutative diagram $$ \\xymatrix{ (X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\ (S, s) & (S', s') \\ar[l] } $$ of pointed schemes such that the horizontal arrows are elementary \\'etale neighbourhoods and such that $g^*\\mathcal{F}/X'/S'$ has a complete d\\'evissage at $x$."}
+{"_id": "6199", "title": "flat-proposition-finite-type-flat-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in X$ with image $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat at $x$ over $S$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme $$ V \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'}) $$ which contains the unique point of $X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$ such that the pullback of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module of finite presentation and flat over $\\mathcal{O}_{S', s'}$."}
+{"_id": "6200", "title": "flat-proposition-finite-presentation-flat-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in X$ with image $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, and \\item $\\mathcal{F}$ is flat at $x$ over $S$. \\end{enumerate} Then there exists a commutative diagram of pointed schemes $$ \\xymatrix{ (X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\ (S, s) & (S', s') \\ar[l] } $$ whose horizontal arrows are elementary \\'etale neighbourhoods such that $X'$, $S'$ are affine and such that $\\Gamma(X', g^*\\mathcal{F})$ is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module."}
+{"_id": "6201", "title": "flat-proposition-flat-finite-type-finite-presentation-domain", "text": "Let $R$ be a domain. Let $R \\to S$ be a ring map of finite type. Let $M$ be a finite $S$-module. \\begin{enumerate} \\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra. \\item If $M$ is flat as an $R$-module, then $M$ is finitely presented as an $S$-module. \\end{enumerate}"}
+{"_id": "6202", "title": "flat-proposition-finite-presentation-flat-pure-is-projective", "text": "Let $f : X \\to S$ be an affine, finitely presented morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation, flat over $S$. Then the following are equivalent \\begin{enumerate} \\item $f_*\\mathcal{F}$ is locally projective on $S$, and \\item $\\mathcal{F}$ is pure relative to $S$. \\end{enumerate} In particular, given a ring map $A \\to B$ of finite presentation and a finitely presented $B$-module $N$ flat over $A$ we have: $N$ is projective as an $A$-module if and only if $\\widetilde{N}$ on $\\Spec(B)$ is pure relative to $\\Spec(A)$."}
+{"_id": "6204", "title": "flat-proposition-check-h", "text": "Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$ constructed in Definition \\ref{definition-big-small-h}. Then $\\mathcal{F}$ is a sheaf if and only if the following conditions are satisfied \\begin{enumerate} \\item $\\mathcal{F}$ is a sheaf for the Zariski topology, \\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then $\\mathcal{F}(Y)$ is the equalizer of the two maps $\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$, \\item given an almost blow up square (\\ref{equation-almost-blow-up-square}) with $X$ affine in the category $(\\Sch/S)_h$ the diagram $$ \\xymatrix{ \\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\ \\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l] } $$ is cartesian in the category of sets. \\end{enumerate}"}
+{"_id": "6205", "title": "flat-proposition-h-descent-vector-bundles-p", "text": "Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. Then the category fibred in groupoids $$ p : \\mathcal{S} \\longrightarrow (\\Sch/S)_h $$ whose fibre category over $U$ is the category of finite locally free $\\colim_F \\mathcal{O}_U$-modules over $U$ is a stack in groupoids. Moreover, if $U$ is quasi-compact and quasi-separated, then $\\mathcal{S}_U$ is $\\colim_F \\textit{Vect}(U)$."}
+{"_id": "6238", "title": "curves-theorem-curves-rational-maps", "text": "Let $k$ be a field. The following categories are canonically equivalent \\begin{enumerate} \\item The category of finitely generated field extensions $K/k$ of transcendence degree $1$. \\item The category of curves and dominant rational maps. \\item The category of normal projective curves and nonconstant morphisms. \\item The category of nonsingular projective curves and nonconstant morphisms. \\item The category of regular projective curves and nonconstant morphisms. \\item The category of normal proper curves and nonconstant morphisms. \\end{enumerate}"}
+{"_id": "6239", "title": "curves-lemma-extend-over-dvr", "text": "Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety. Let $U \\subset X$ be a nonempty open and let $f : U \\to Y$ be a morphism. If $x \\in X$ is a closed point such that $\\mathcal{O}_{X, x}$ is a discrete valuation ring, then there exists an open $U \\subset U' \\subset X$ containing $x$ and a morphism of varieties $f' : U' \\to Y$ extending $f$."}
+{"_id": "6240", "title": "curves-lemma-extend-over-normal-curve", "text": "Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety. The set of rational maps from $X$ to $Y$ is the same as the set of morphisms $X \\to Y$."}
+{"_id": "6241", "title": "curves-lemma-flat", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism of curves over $k$. If $Y$ is normal, then $f$ is flat."}
+{"_id": "6242", "title": "curves-lemma-finite", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes over $k$. Assume \\begin{enumerate} \\item $Y$ is separated over $k$, \\item $X$ is proper of dimension $\\leq 1$ over $k$, \\item $f(Z)$ has at least two points for every irreducible component $Z \\subset X$ of dimension $1$. \\end{enumerate} Then $f$ is finite."}
+{"_id": "6245", "title": "curves-lemma-smooth-models", "text": "Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. For a field extension $K/k$ denote $Y_K$ a nonsingular projective model of $(X_K)_{red}$. \\begin{enumerate} \\item If $X$ is proper, then $Y_K$ is the normalization of $X_K$. \\item There exists $K/k$ finite purely inseparable such that $Y_K$ is smooth. \\item Whenever $Y_K$ is smooth\\footnote{Or even geometrically reduced.} we have $H^0(Y_K, \\mathcal{O}_{Y_K}) = K$. \\item Given a commutative diagram $$ \\xymatrix{ \\Omega & K' \\ar[l] \\\\ K \\ar[u] & k \\ar[l] \\ar[u] } $$ of fields such that $Y_K$ and $Y_{K'}$ are smooth, then $Y_\\Omega = (Y_K)_\\Omega = (Y_{K'})_\\Omega$. \\end{enumerate}"}
+{"_id": "6246", "title": "curves-lemma-linear-series", "text": "Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ on $X$. Then there exists a morphism $$ \\varphi : X \\longrightarrow \\mathbf{P}^r_k = \\text{Proj}(k[T_0, \\ldots, T_r]) $$ of varieties over $k$ and a map $\\alpha : \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_k}(1) \\to \\mathcal{L}$ such that $\\varphi^*T_0, \\ldots, \\varphi^*T_r$ are sent to a basis of $V$ by $\\alpha$."}
+{"_id": "6247", "title": "curves-lemma-linear-series-trivial-existence", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. If $X$ has a $\\mathfrak g^r_d$, then $X$ has a $\\mathfrak g^s_d$ for all $0 \\leq s \\leq r$."}
+{"_id": "6248", "title": "curves-lemma-g1d", "text": "Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$. Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^1_d$ on $X$. Then the morphism $\\varphi : X \\to \\mathbf{P}^1_k$ of Lemma \\ref{lemma-linear-series} either \\begin{enumerate} \\item is nonconstant and has degree $\\leq d$, or \\item factors through a closed point of $\\mathbf{P}^1_k$ and in this case $H^0(X, \\mathcal{O}_X) \\not = k$. \\end{enumerate}"}
+{"_id": "6249", "title": "curves-lemma-grd-inequalities", "text": "Let $k$ be a field. Let $X$ be a proper curve over $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $\\mathfrak g^r_d$, then $r \\leq d$. If equality holds, then $H^1(X, \\mathcal{O}_X) = 0$, i.e., the genus of $X$ (Definition \\ref{definition-genus}) is $0$."}
+{"_id": "6250", "title": "curves-lemma-duality-dim-1", "text": "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$. There exists a dualizing complex $\\omega_X^\\bullet$ with the following properties \\begin{enumerate} \\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i = -1, 0$, \\item $\\omega_X = H^{-1}(\\omega_X^\\bullet)$ is a coherent Cohen-Macaulay module whose support is the irreducible components of dimension $1$, \\item for $x \\in X$ closed, the module $H^0(\\omega_{X, x}^\\bullet)$ is nonzero if and only if either \\begin{enumerate} \\item $\\dim(\\mathcal{O}_{X, x}) = 0$ or \\item $\\dim(\\mathcal{O}_{X, x}) = 1$ and $\\mathcal{O}_{X, x}$ is not Cohen-Macaulay, \\end{enumerate} \\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial isomorphisms\\footnote{This property characterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$ up to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.} $$ \\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k) $$ compatible with shifts and distinguished triangles, \\item there are functorial isomorphisms $\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^1(X, \\mathcal{F}), k)$ for $\\mathcal{F}$ quasi-coherent on $X$, \\item if $X \\to \\Spec(k)$ is smooth of relative dimension $1$, then $\\omega_X \\cong \\Omega_{X/k}$. \\end{enumerate}"}
+{"_id": "6251", "title": "curves-lemma-duality-dim-1-CM", "text": "Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $1$. The module $\\omega_X$ of Lemma \\ref{lemma-duality-dim-1} has the following properties \\begin{enumerate} \\item $\\omega_X$ is a dualizing module on $X$ (Duality for Schemes, Section \\ref{duality-section-dualizing-module}), \\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$, \\item there are functorial isomorphisms $\\Ext^i_X(K, \\omega_X[1]) = \\Hom_k(H^{-i}(X, K), k)$ compatible with shifts for $K \\in D_\\QCoh(X)$, \\item there are functorial isomorphisms $\\Ext^{1 + i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{-i}(X, \\mathcal{F}), k)$ for $\\mathcal{F}$ quasi-coherent on $X$. \\end{enumerate}"}
+{"_id": "6252", "title": "curves-lemma-sanity-check-duality", "text": "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$. Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}. \\begin{enumerate} \\item If $X \\to \\Spec(k)$ factors as $X \\to \\Spec(k') \\to \\Spec(k)$ for some field $k'$, then $\\omega_X^\\bullet$ and $\\omega_X$ satisfy properties (4), (5), (6) with $k$ replaced with $k'$. \\item If $K/k$ is a field extension, then the pullback of $\\omega_X^\\bullet$ and $\\omega_X$ to the base change $X_K$ are as in  Lemma \\ref{lemma-duality-dim-1} for the morphism $X_K \\to \\Spec(K)$. \\end{enumerate}"}
+{"_id": "6253", "title": "curves-lemma-closed-immersion-dim-1-CM", "text": "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$. Let $i : Y \\to X$ be a closed immersion. Let $\\omega_X^\\bullet$, $\\omega_X$, $\\omega_Y^\\bullet$, $\\omega_Y$ be as in Lemma \\ref{lemma-duality-dim-1}. Then \\begin{enumerate} \\item $\\omega_Y^\\bullet = R\\SheafHom(\\mathcal{O}_Y, \\omega_X^\\bullet)$, \\item $\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$ and $i_*\\omega_Y = \\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Y, \\omega_X)$. \\end{enumerate}"}
+{"_id": "6254", "title": "curves-lemma-closed-subscheme-reduced-gorenstein", "text": "Let $X$ be a proper scheme over a field $k$ which is Gorenstein, reduced, and equidimensional of dimension $1$. Let $i : Y \\to X$ be a reduced closed subscheme equidimensional of dimension $1$. Let $j : Z \\to X$ be the scheme theoretic closure of $X \\setminus Y$. Then \\begin{enumerate} \\item $Y$ and $Z$ are Cohen-Macaulay, \\item if $\\mathcal{I} \\subset \\mathcal{O}_X$, resp.\\ $\\mathcal{J} \\subset \\mathcal{O}_X$ is the ideal sheaf of $Y$, resp.\\ $Z$ in $X$, then $$ \\mathcal{I} = i_*\\mathcal{I}' \\quad\\text{and}\\quad \\mathcal{J} = j_*\\mathcal{J}' $$ where $\\mathcal{I}' \\subset \\mathcal{O}_Z$, resp.\\ $\\mathcal{J}' \\subset \\mathcal{O}_Y$ is the ideal sheaf of $Y \\cap Z$ in $Z$, resp.\\ $Y$, \\item $\\omega_Y = \\mathcal{J}'(i^*\\omega_X)$ and $i_*(\\omega_Y) = \\mathcal{J}\\omega_X$, \\item $\\omega_Z = \\mathcal{I}'(i^*\\omega_X)$ and $i_*(\\omega_Z) = \\mathcal{I}\\omega_X$, \\item we have the following short exact sequences \\begin{align*} 0 \\to \\omega_X \\to i_*i^*\\omega_X \\oplus j_*j^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\ 0 \\to i_*\\omega_Y \\to \\omega_X \\to j_*j^*\\omega_X \\to 0 \\\\ 0 \\to j_*\\omega_Z \\to \\omega_X \\to i_*i^*\\omega_X \\to 0 \\\\ 0 \\to i_*\\omega_Y \\oplus j_*\\omega_Z \\to \\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\ 0 \\to \\omega_Y \\to i^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\ 0 \\to \\omega_Z \\to j^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\end{align*} \\end{enumerate} Here $\\omega_X$, $\\omega_Y$, $\\omega_Z$ are as in Lemma \\ref{lemma-duality-dim-1}."}
+{"_id": "6255", "title": "curves-lemma-euler", "text": "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$. With $\\omega_X^\\bullet$ and $\\omega_X$ as in Lemma \\ref{lemma-duality-dim-1} we have $$ \\chi(X, \\mathcal{O}_X) = \\chi(X, \\omega_X^\\bullet) $$ If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then $$ \\chi(X, \\mathcal{O}_X) = - \\chi(X, \\omega_X) $$"}
+{"_id": "6256", "title": "curves-lemma-rr", "text": "Let $X$ be a proper scheme over a field $k$ which is Gorenstein and equidimensional of dimension $1$. Let $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}. Then \\begin{enumerate} \\item $\\omega_X$ is an invertible $\\mathcal{O}_X$-module, \\item $\\deg(\\omega_X) = -2\\chi(X, \\mathcal{O}_X)$, \\item for a locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ of constant rank we have $$ \\chi(X, \\mathcal{E}) = \\deg(\\mathcal{E}) - \\textstyle{\\frac{1}{2}} \\text{rank}(\\mathcal{E}) \\deg(\\omega_X) $$ and $\\dim_k(H^i(X, \\mathcal{E})) = \\dim_k(H^{1 - i}(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\omega_X))$ for all $i \\in \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "6257", "title": "curves-lemma-automatic", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Then $X$ is connected, Cohen-Macaulay, and equidimensional of dimension $1$."}
+{"_id": "6258", "title": "curves-lemma-vanishing", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence $$ \\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0 $$ of coherent $\\mathcal{O}_X$-modules with $H^1(X, \\mathcal{Q}) = 0$ (for example if $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$), then either $H^1(X, \\mathcal{F}) = 0$ or $\\mathcal{F} = \\omega_X \\oplus \\mathcal{Q}$."}
+{"_id": "6259", "title": "curves-lemma-vanishing-twist", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve}. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module which is globally generated and not isomorphic to $\\mathcal{O}_X$. Then $H^1(X, \\omega_X \\otimes \\mathcal{L}) = 0$."}
+{"_id": "6260", "title": "curves-lemma-globally-generated", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence $$ \\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0 $$ of coherent $\\mathcal{O}_X$-modules with $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$ and $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$ and such that there is no nonzero submodule $\\mathcal{Q}' \\subset \\mathcal{F}$ such that $\\mathcal{Q}' \\to \\mathcal{Q}$ is injective. Then the submodule of $\\mathcal{F}$ generated by global sections surjects onto $\\mathcal{Q}$."}
+{"_id": "6262", "title": "curves-lemma-tensor-omega-with-globally-generated-invertible", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve}. Let $\\mathcal{L}$ be a very ample invertible $\\mathcal{O}_X$-module with $\\deg(\\mathcal{L}) \\geq 2$. Then $\\omega_X \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ is globally generated."}
+{"_id": "6263", "title": "curves-lemma-criterion-very-ample", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $\\mathcal{L}$ has a regular global section, \\item $H^1(X, \\mathcal{L}) = 0$, and \\item $\\mathcal{L}$ is ample. \\end{enumerate} Then $\\mathcal{L}^{\\otimes 6}$ is very ample on $X$ over $k$."}
+{"_id": "6265", "title": "curves-lemma-genus-base-change", "text": "Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Then $X_{k'}$ is a proper scheme over $k'$ having dimension $1$ and $H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$. Moreover the genus of $X_{k'}$ is equal to the genus of $X$."}
+{"_id": "6266", "title": "curves-lemma-genus-gorenstein", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. If $X$ is Gorenstein, then $$ \\deg(\\omega_X) = 2g - 2 $$ where $g$ is the genus of $X$ and $\\omega_X$ is as in Lemma \\ref{lemma-duality-dim-1}."}
+{"_id": "6267", "title": "curves-lemma-genus-smooth", "text": "Let $X$ be a smooth proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$. Then $$ \\dim_k H^0(X, \\Omega_{X/k}) = g \\quad\\text{and}\\quad \\deg(\\Omega_{X/k}) = 2g - 2 $$ where $g$ is the genus of $X$."}
+{"_id": "6270", "title": "curves-lemma-genus-plane-curve", "text": "Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve} and let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$. Then $H^0(Z, \\mathcal{O}_Z) = k$ and the genus of $Z$ is $(d - 1)(d - 2)/2$ where $d = \\deg(F)$."}
+{"_id": "6271", "title": "curves-lemma-smooth-plane-curve-point-over-separable", "text": "Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve} and let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$. If $Z \\to \\Spec(k)$ is smooth in at least one point and $k$ is infinite, then there exists a closed point $z \\in Z$ contained in the smooth locus such that $\\kappa(z)/k$ is finite separable of degree at most $d$."}
+{"_id": "6272", "title": "curves-lemma-genus-zero-pic", "text": "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ has genus $0$, then every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ of degree $0$ is trivial."}
+{"_id": "6273", "title": "curves-lemma-genus-zero-positive-degree", "text": "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$. Assume $X$ has genus $0$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module of degree $d > 0$. Then we have \\begin{enumerate} \\item $\\dim_k H^0(X, \\mathcal{L}) = d + 1$ and $\\dim_k H^1(X, \\mathcal{L}) = 0$, \\item $\\mathcal{L}$ is very ample and defines a closed immersion into $\\mathbf{P}^d_k$. \\end{enumerate}"}
+{"_id": "6274", "title": "curves-lemma-genus-zero", "text": "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ is Gorenstein and has genus $0$, then $X$ is isomorphic to a plane curve of degree $2$."}
+{"_id": "6276", "title": "curves-lemma-generically-etale", "text": "\\begin{slogan} A morphism of smooth curves is separable iff it is etale almost everywhere \\end{slogan} Let $k$ be a field. Let $f : X \\to Y$ be a morphism of smooth curves over $k$. The following are equivalent \\begin{enumerate} \\item $\\text{d}f : f^*\\Omega_{Y/k} \\to \\Omega_{X/k}$ is nonzero, \\item $\\Omega_{X/Y}$ is supported on a proper closed subset of $X$, \\item there exists a nonempty open $U \\subset X$ such that $f|_U : U \\to Y$ is unramified, \\item there exists a nonempty open $U \\subset X$ such that $f|_U : U \\to Y$ is \\'etale, \\item the extension $k(Y) \\subset k(X)$ of function fields is finite separable. \\end{enumerate}"}
+{"_id": "6277", "title": "curves-lemma-rh", "text": "Let $f : X \\to Y$ be a morphism of smooth proper curves over a field $k$ which satisfies the equivalent conditions of Lemma \\ref{lemma-generically-etale}. If $k = H^0(Y, \\mathcal{O}_Y) = H^0(X, \\mathcal{O}_X)$ and $X$ and $Y$ have genus $g_X$ and $g_Y$, then $$ 2g_X - 2 = (2g_Y - 2) \\deg(f) + \\deg(R) $$ where $R \\subset X$ is the effective Cartier divisor cut out by the different of $f$."}
+{"_id": "6278", "title": "curves-lemma-uniformizer-works", "text": "Let $X \\to \\Spec(k)$ be smooth of relative dimension $1$ at a closed point $x \\in X$. If $\\kappa(x)$ is separable over $k$, then for any uniformizer $s$ in the discrete valuation ring $\\mathcal{O}_{X, x}$ the element $\\text{d}s$ freely generates $\\Omega_{X/k, x}$ over $\\mathcal{O}_{X, x}$."}
+{"_id": "6281", "title": "curves-lemma-purely-inseparable", "text": "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If the extension $k(Y) \\subset k(X)$ of function fields is purely inseparable, then there exists a factorization $$ X = X_0 \\to X_1 \\to \\ldots \\to X_n = Y $$ such that each $X_i$ is a proper nonsingular curve and $X_i \\to X_{i + 1}$ is a degree $p$ morphism with $k(X_{i + 1}) \\subset k(X_i)$ inseparable."}
+{"_id": "6282", "title": "curves-lemma-inseparable-deg-p-smooth", "text": "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \\subset k(X)$ is inseparable of degree $p$, then there is a unique isomorphism $Y = X^{(p)}$ such that $f$ is $F_{X/k}$."}
+{"_id": "6283", "title": "curves-lemma-purely-inseparable-smooth", "text": "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \\subset k(X)$ is purely inseparable, then there is a unique $n \\geq 0$ and a unique isomorphism $Y = X^{(p^n)}$ such that $f$ is the $n$-fold relative Frobenius of $X/k$."}
+{"_id": "6285", "title": "curves-lemma-inseparable-linear-system", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a smooth proper curve over $k$. Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ with $r \\geq 1$. Then one of the following two is true \\begin{enumerate} \\item there exists a $\\mathfrak g^1_d$ whose corresponding morphism $X \\to \\mathbf{P}^1_k$ (Lemma \\ref{lemma-linear-series}) is generically \\'etale (i.e., is as in Lemma \\ref{lemma-generically-etale}), or \\item there exists a $\\mathfrak g^r_{d'}$ on $X^{(p)}$ where $d' \\leq d/p$. \\end{enumerate}"}
+{"_id": "6289", "title": "curves-lemma-no-in-between-over-k", "text": "Let $k$ be an algebraically closed field. Let $k \\subset A$ be a ring extension such that $A$ has exactly two $k$-sub algebras, then either $A = k \\times k$ or $A = k[\\epsilon]$."}
+{"_id": "6290", "title": "curves-lemma-factor-almost-isomorphism", "text": "Let $k$ be an algebraically closed field. Let $f : X' \\to X$ be a finite morphism algebraic $k$-schemes such that $\\mathcal{O}_X \\subset f_*\\mathcal{O}_{X'}$ and such that $f$ is an isomorphism away from a finite set of points. Then there is a factorization $$ X' = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X $$ such that each $X_i \\to X_{i - 1}$ is either the glueing of two points or the squishing of a tangent vector (see Examples \\ref{example-glue-points} and \\ref{example-squish-tangent-vector})."}
+{"_id": "6291", "title": "curves-lemma-glue-points", "text": "Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the glueing of two points $a, b$ as in Example \\ref{example-glue-points}, then there is an exact sequence $$ k^* \\to \\Pic(X) \\to \\Pic(X') \\to 0 $$ The first map is zero if $a$ and $b$ are on different connected components of $X'$ and injective if $X'$ is proper and $a$ and $b$ are on the same connected component of $X'$."}
+{"_id": "6292", "title": "curves-lemma-squish-tangent-vector", "text": "Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the squishing of a tangent vector $\\vartheta$ as in Example \\ref{example-squish-tangent-vector}, then there is an exact sequence $$ (k, +) \\to \\Pic(X) \\to \\Pic(X') \\to 0 $$ and the first map is injective if $X'$ is proper and reduced."}
+{"_id": "6293", "title": "curves-lemma-multicross-algebra", "text": "Let $k$ be a separably closed field. Let $A$ be a $1$-dimensional reduced Nagata local $k$-algebra with residue field $k$. Then $$ \\delta\\text{-invariant }A \\geq \\text{number of branches of }A - 1 $$ If equality holds, then $A^\\wedge$ is as in (\\ref{equation-multicross})."}
+{"_id": "6294", "title": "curves-lemma-multicross", "text": "Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \\in X$. The following are equivalent \\begin{enumerate} \\item $x$ defines a multicross singularity, \\item the $\\delta$-invariant of $X$ at $x$ is the number of branches of $X$ at $x$ minus $1$, \\item there is a sequence of morphisms $U_n \\to U_{n - 1} \\to \\ldots \\to U_0 = U \\subset X$ where $U$ is an open neighbourhood of $x$, where $U_n$ is nonsingular, and where each $U_i \\to U_{i - 1}$ is the glueing of two points as in Example \\ref{example-glue-points}. \\end{enumerate}"}
+{"_id": "6296", "title": "curves-lemma-torsion-picard-smooth-projective", "text": "Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. \\begin{enumerate} \\item If $n \\geq 1$ is invertible in $k$, then $\\Pic(X)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$. \\item If the characteristic of $k$ is $p > 0$, then there exists an integer $0 \\leq f \\leq g$ such that $\\Pic(X)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$ for all $m \\geq 1$. \\end{enumerate}"}
+{"_id": "6297", "title": "curves-lemma-torsion-picard-becomes-visible", "text": "Let $k$ be a field. Let $n$ be prime to the characteristic of $k$. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \\mathcal{O}_X) = k$ and of genus $g$. \\begin{enumerate} \\item If $g = 1$ then there exists a finite separable extension $k'/k$ such that $X_{k'}$ has a $k'$-rational point and $\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2}$. \\item If $g \\geq 2$ then there exists a finite separable extension $k'/k$ with $[k' : k] \\leq (2g - 2)(n^{2g})!$ such that $X_{k'}$ has a $k'$-rational point and $\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$. \\end{enumerate}"}
+{"_id": "6298", "title": "curves-lemma-bound-geometric-genus", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Then $$ g_{geom}(X/k) = \\sum\\nolimits_{C \\subset X} g_{geom}(C/k) $$ where the sum is over irreducible components $C \\subset X$ of dimension $1$."}
+{"_id": "6300", "title": "curves-lemma-genus-goes-down", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $f : Y \\to X$ be a finite morphism such that there exists a dense open $U \\subset X$ over which $f$ is a closed immersion. Then $$ \\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y) $$"}
+{"_id": "6301", "title": "curves-lemma-genus-normalization", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. If $X' \\to X$ is a birational proper morphism, then $$ \\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(X', \\mathcal{O}_{X'}) $$ If $X$ is reduced, $H^0(X, \\mathcal{O}_X) \\to H^0(X', \\mathcal{O}_{X'})$ is surjective, and equality holds, then $X' = X$."}
+{"_id": "6302", "title": "curves-lemma-bound-geometric-genus-curve", "text": "Let $k$ be a field. Let $C$ be a proper curve over $k$. Set $\\kappa = H^0(C, \\mathcal{O}_C)$. Then $$ [\\kappa : k]_s \\dim_\\kappa H^1(C, \\mathcal{O}_C) \\geq g_{geom}(C/k) $$"}
+{"_id": "6304", "title": "curves-lemma-reduced-quotient-regular-ring-dim-2", "text": "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$. Let $I \\subset \\mathfrak m$ be an ideal. \\begin{enumerate} \\item If $A/I$ is reduced, then $I = (0)$, $I = \\mathfrak m$, or $I = (f)$ for some nonzero $f \\in \\mathfrak m$. \\item If $A/I$ has depth $1$, then $I = (f)$ for some nonzero $f \\in \\mathfrak m$.  \\end{enumerate}"}
+{"_id": "6305", "title": "curves-lemma-nodal-algebraic", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local $k$-algebra. The following are equivalent \\begin{enumerate} \\item $\\kappa/k$ is separable, $A$ is reduced, $\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate $q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$ which maps to zero in $\\mathfrak m^2/\\mathfrak m^3$, \\item $\\kappa/k$ is separable, $\\text{depth}(A) = 1$, $\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate $q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$ which maps to zero in $\\mathfrak m^2/\\mathfrak m^3$, \\item $\\kappa/k$ is separable, $A^\\wedge \\cong \\kappa[[x, y]]/(ax^2 + bxy + cy^2)$ as a $k$-algebra where $ax^2 + bxy + cy^2$ is a nondegenerate quadratic form over $\\kappa$. \\end{enumerate}"}
+{"_id": "6306", "title": "curves-lemma-2-branches-delta-1", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a Nagata local $k$-algebra. The following are equivalent \\begin{enumerate} \\item $k \\to A$ is as in Lemma \\ref{lemma-nodal-algebraic}, \\item $\\kappa/k$ is separable, $A$ is reduced of dimension $1$, the $\\delta$-invariant of $A$ is $1$, and $A$ has $2$ geometric branches. \\end{enumerate} If this holds, then the integral closure $A'$ of $A$ in its total ring of fractions has either $1$ or $2$ maximal ideals $\\mathfrak m'$ and the extensions $\\kappa(\\mathfrak m')/k$ are separable."}
+{"_id": "6307", "title": "curves-lemma-fitting-ideal-well-defined", "text": "Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let $I = (f_1, \\ldots, f_m) \\subset A$ be an ideal. For any $r \\geq 0$ the ideal in $A/I$ generated by the $r \\times r$-minors of the matrix $(\\partial f_j/\\partial x_i)$ is independent of the choice of the generators of $I$ or the regular system of parameters $x_1, \\ldots, x_n$ of $A$."}
+{"_id": "6308", "title": "curves-lemma-fitting-ideal", "text": "Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let $I = (f_1, \\ldots, f_m) \\subset \\mathfrak m_A$ be an ideal. The following are equivalent \\begin{enumerate} \\item $k \\to A/I$ is as in Lemma \\ref{lemma-nodal-algebraic}, \\item $A/I$ is reduced and the $(n - 1) \\times (n - 1)$ minors of the matrix $(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$, \\item $\\text{depth}(A/I) = 1$ and the $(n - 1) \\times (n - 1)$ minors of the matrix $(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$. \\end{enumerate}"}
+{"_id": "6309", "title": "curves-lemma-nodal", "text": "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \\in X$ be a closed point. The following are equivalent \\begin{enumerate} \\item $x$ is a node, \\item $k \\to \\mathcal{O}_{X, x}$ is as in Lemma \\ref{lemma-nodal-algebraic}, \\item any $\\overline{x} \\in X_{\\overline{k}}$ mapping to $x$ defines a nodal singularity, \\item $\\kappa(x)/k$ is separable, $\\mathcal{O}_{X, x}$ is reduced, and the first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$ in $\\mathcal{O}_{X, x}$, \\item $\\kappa(x)/k$ is separable, $\\text{depth}(\\mathcal{O}_{X, x}) = 1$, and the first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$ in $\\mathcal{O}_{X, x}$, \\item $\\kappa(x)/k$ is separable and $\\mathcal{O}_{X, x}$ is reduced, has $\\delta$-invariant $1$, and has $2$ geometric branches. \\end{enumerate}"}
+{"_id": "6310", "title": "curves-lemma-split-node", "text": "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \\in X$ be a closed point. The following are equivalent \\begin{enumerate} \\item $x$ is a split node, \\item $x$ is a node and there are exactly two points $x_1, x_2$ of the normalization $X^\\nu$ lying over $x$ with $k = \\kappa(x_1) = \\kappa(x_2)$, \\item $\\mathcal{O}_{X, x}^\\wedge \\cong k[[x, y]]/(xy)$ as a $k$-algebra, and \\item add more here. \\end{enumerate}"}
+{"_id": "6311", "title": "curves-lemma-node-field-extension", "text": "Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic $k$-scheme of dimension $1$. Let $y \\in X_K$ be a point with image $x \\in X$. The following are equivalent \\begin{enumerate} \\item $x$ is a closed point of $X$ and a node, and \\item $y$ is a closed point of $Y$ and a node. \\end{enumerate}"}
+{"_id": "6312", "title": "curves-lemma-node-etale-local", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $1$. Let $Y \\to X$ be an \\'etale morphism. Let $y \\in Y$ be a point with image $x \\in X$. The following are equivalent \\begin{enumerate} \\item $x$ is a closed point of $X$ and a node, and \\item $y$ is a closed point of $Y$ and a node. \\end{enumerate}"}
+{"_id": "6313", "title": "curves-lemma-node-over-separable-extension", "text": "Let $k'/k$ be a finite separable field extension. Let $X$ be a locally algebraic $k'$-scheme of dimension $1$. Let $x \\in X$ be a closed point. The following are equivalent \\begin{enumerate} \\item $x$ is a node, and \\item $x$ is a node when $X$ viewed as a locally algebraic $k$-scheme. \\end{enumerate}"}
+{"_id": "6314", "title": "curves-lemma-nodal-lci", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$. The following are equivalent \\begin{enumerate} \\item the singularities of $X$ are at-worst-nodal, and \\item $X$ is a local complete intersection over $k$ and the closed subscheme $Z \\subset X$ cut out by the first fitting ideal of $\\Omega_{X/k}$ is unramified over $k$. \\end{enumerate}"}
+{"_id": "6315", "title": "curves-lemma-facts-about-nodal-curves", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$ whose singularities are at-worst-nodal. Then $X$ is Gorenstein and geometrically reduced."}
+{"_id": "6316", "title": "curves-lemma-closed-subscheme-nodal-curve", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$ whose singularities are at-worst-nodal. If $Y \\subset X$ is a reduced closed subscheme equidimensional of dimension $1$, then \\begin{enumerate} \\item the singularities of $Y$ are at-worst-nodal, and \\item if $Z \\subset X$ is the scheme theoretic closure of $X \\setminus Y$, then \\begin{enumerate} \\item the scheme theoretic intersection $Y \\cap Z$ is the disjoint union of spectra of finite separable extensions of $k$, \\item each point of $Y \\cap Z$ is a node of $X$, and \\item $Y \\to \\Spec(k)$ is smooth at every point of $Y \\cap Z$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "6317", "title": "curves-lemma-nodal-family", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is flat, locally of finite presentation, every nonempty fibre $X_s$ is equidimensional of dimension $1$, and $X_s$ has at-worst-nodal singularities, and \\item $f$ is syntomic of relative dimension $1$ and the closed subscheme $\\text{Sing}(f) \\subset X$ defined by the first Fitting ideal of $\\Omega_{X/S}$ is unramified over $S$. \\end{enumerate}"}
+{"_id": "6318", "title": "curves-lemma-smooth-relative-dimension-1", "text": "A smooth morphism of relative dimension $1$ is at-worst-nodal of relative dimension $1$."}
+{"_id": "6319", "title": "curves-lemma-base-change-nodal-family", "text": "Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$. Then the same is true for any base change of $f$."}
+{"_id": "6320", "title": "curves-lemma-locus-where-nodal", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Then there is a maximal open subscheme $U \\subset X$ such that $f|_U : U \\to S$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $U$ commutes with arbitrary base change."}
+{"_id": "6321", "title": "curves-lemma-nodal-family-precompose-etale", "text": "Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$. If $Y \\to X$ is an \\'etale morphism, then the composition $g : Y \\to S$ is at-worst-nodal of relative dimension $1$."}
+{"_id": "6322", "title": "curves-lemma-nodal-family-postcompose-etale", "text": "Let $S' \\to S$ be an \\'etale morphism of schemes. Let $f : X \\to S'$ be at-worst-nodal of relative dimension $1$. Then the composition $g : X \\to S$ is at-worst-nodal of relative dimension $1$."}
+{"_id": "6323", "title": "curves-lemma-nodal-family-etale-local-source", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to X\\}$ be an \\'etale covering. The following are equivalent \\begin{enumerate} \\item $f$ is at-worst-nodal of relative dimension $1$, \\item each $U_i \\to S$ is at-worst-nodal of relative dimension $1$. \\end{enumerate} In other words, being at-worst-nodal of relative dimension $1$ is \\'etale local on the source."}
+{"_id": "6325", "title": "curves-lemma-descend-nodal-family", "text": "Let $S = \\lim S_i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_i : X_i \\to Y_i$ be the base change of $f_0$ to $S_i$ and let $f : X \\to Y$ be the base change of $f_0$ to $S$. If \\begin{enumerate} \\item $f$ is at-worst-nodal of relative dimension $1$, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is at-worst-nodal of relative dimension $1$."}
+{"_id": "6326", "title": "curves-lemma-formal-local-structure-nodal-family", "text": "Let $f : T \\to S$ be a morphism of schemes. Let $t \\in T$ with image $s \\in S$. Assume \\begin{enumerate} \\item $f$ is flat at $t$, \\item $\\mathcal{O}_{S, s}$ is Noetherian, \\item $f$ is locally of finite type, \\item $t$ is a split node of the fibre $T_s$. \\end{enumerate} Then there exists an $h \\in \\mathfrak m_s^\\wedge$ and an isomorphism $$ \\mathcal{O}_{T, t}^\\wedge \\cong \\mathcal{O}_{S, s}^\\wedge[[x, y]]/(xy - h) $$ of $\\mathcal{O}_{S, s}^\\wedge$-algebras."}
+{"_id": "6327", "title": "curves-lemma-etale-local-structure-nodal-family", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is at-worst-nodal of relative dimension $1$. Let $x \\in X$ be a point which is a singular point of the fibre $X_s$. Then there exists a commutative diagram of schemes $$ \\xymatrix{ X \\ar[d] & U \\ar[rr] \\ar[l] \\ar[rd] & & W \\ar[r] \\ar[ld] & \\Spec(\\mathbf{Z}[u, v, a]/(uv - a)) \\ar[d] \\\\ S & & V \\ar[ll] \\ar[rr] & & \\Spec(\\mathbf{Z}[a]) } $$ with $X \\leftarrow U$, $S \\leftarrow V$, and $U \\to W$ \\'etale morphisms, and with the right hand square cartesian, such that there exists a point $u \\in U$ mapping to $x$ in $X$."}
+{"_id": "6328", "title": "curves-lemma-h1-nonzero-degree-leq-2g-2", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. If $H^1(X, \\mathcal{I}\\mathcal{L})$ is nonzero, then $$ \\deg(\\mathcal{L}) \\leq 2g - 2 + \\deg(Z) $$ with strict inequality unless $\\mathcal{I}\\mathcal{L} \\cong \\omega_X$."}
+{"_id": "6329", "title": "curves-lemma-degree-more-than-2g-2", "text": "\\begin{reference} \\cite[Lemma 2]{Jongmin} \\end{reference} In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. If $\\deg(\\mathcal{L}) > 2g - 2 + \\deg(Z)$, then $H^1(X, \\mathcal{I}\\mathcal{L}) = 0$ and one of the following possibilities occurs \\begin{enumerate} \\item $H^0(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, or \\item $g = 0$ and $\\deg(\\mathcal{L}) = \\deg(Z) - 1$. \\end{enumerate} In case (2) if $Z = \\emptyset$, then $X \\cong \\mathbf{P}^1_k$ and $\\mathcal{L}$ corresponds to $\\mathcal{O}_{\\mathbf{P}^1}(-1)$."}
+{"_id": "6330", "title": "curves-lemma-degree-more-than-2g", "text": "\\begin{reference} \\cite[Lemma 3]{Jongmin} \\end{reference} In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $\\deg(\\mathcal{L}) \\geq 2g$, then $\\mathcal{L}$ is globally generated."}
+{"_id": "6332", "title": "curves-lemma-vanishing-on-gorenstein", "text": "\\begin{reference} Weak version of \\cite[Lemma 4]{Jongmin} \\end{reference} Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and of dimension $1$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. If $H^1(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, then there exists a reduced connected closed subscheme $Y \\subset X$ of dimension $1$ such that $$ \\deg(\\mathcal{L}|_Y) \\leq -2\\chi(Y, \\mathcal{O}_Y) + \\deg(Z \\cap Y) $$ where $Z \\cap Y$ is the scheme theoretic intersection."}
+{"_id": "6333", "title": "curves-lemma-global-generation", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and of dimension $1$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume that for every reduced connected closed subscheme $Y \\subset X$ of dimension $1$ we have $$ \\deg(\\mathcal{L}|_Y) \\geq 2\\dim_k H^1(Y, \\mathcal{O}_Y) $$ Then $\\mathcal{L}$ is globally generated."}
+{"_id": "6334", "title": "curves-lemma-rational-tail-negative", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \\subset X$ be a rational tail (Example \\ref{example-rational-tail}). Then $\\deg(\\omega_X|_C) < 0$."}
+{"_id": "6335", "title": "curves-lemma-rational-tail-field-extension", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \\subset X$ be a rational tail (Example \\ref{example-rational-tail}). For any field extension $K/k$ the base change $C_K \\subset X_K$ is a finite disjoint union of rational tails."}
+{"_id": "6336", "title": "curves-lemma-no-rational-tail", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. If $X$ does not have a rational tail (Example \\ref{example-rational-tail}), then for every reduced connected closed subscheme $Y \\subset X$, $Y \\not = X$ of dimension $1$ we have $\\deg(\\omega_X|_Y) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y)$."}
+{"_id": "6337", "title": "curves-lemma-no-rational-tail-semiample-genus-geq-2", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Assume $X$ does not have a rational tail (Example \\ref{example-rational-tail}). If \\begin{enumerate} \\item the genus of $X$ is $0$, then $X$ is isomorphic to an irreducible plane conic and $\\omega_X^{\\otimes -1}$ is very ample, \\item the genus of $X$ is $1$, then $\\omega_X \\cong \\mathcal{O}_X$, \\item the genus of $X$ is $\\geq 2$, then $\\omega_X^{\\otimes m}$ is globally generated for $m \\geq 2$. \\end{enumerate}"}
+{"_id": "6338", "title": "curves-lemma-contracting-rational-tails", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Consider a sequence $$ X = X_0 \\to X_1 \\to \\ldots \\to X_n = X' $$ of contractions of rational tails (Example \\ref{example-rational-tail}) until none are left. Then \\begin{enumerate} \\item if the genus of $X$ is $0$, then $X'$ is an irreducible plane conic, \\item if the genus of $X$ is $1$, then $\\omega_{X'} \\cong \\mathcal{O}_X$, \\item if the genus of $X$ is $> 1$, then $\\omega_{X'}^{\\otimes m}$ is globally generated for $m \\geq 2$. \\end{enumerate} If the genus of $X$ is $\\geq 1$, then the morphism $X \\to X'$ is independent of choices and formation of this morphism commutes with base field extensions."}
+{"_id": "6339", "title": "curves-lemma-rational-bridge-zero", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \\subset X$ be a rational bridge (Example \\ref{example-rational-bridge}). Then $\\deg(\\omega_X|_C) = 0$."}
+{"_id": "6340", "title": "curves-lemma-rational-bridge-field-extension", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \\subset X$ be a rational bridge (Example \\ref{example-rational-bridge}). For any field extension $K/k$ the base change $C_K \\subset X_K$ is a finite disjoint union of rational bridges."}
+{"_id": "6341", "title": "curves-lemma-rational-bridge-canonical", "text": "Let $c : X \\to Y$ be the contraction of a rational bridge (Example \\ref{example-rational-bridge}). Then $c^*\\omega_Y \\cong \\omega_X$."}
+{"_id": "6342", "title": "curves-lemma-no-rational-bridge-ample-genus-geq-2", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Assume \\begin{enumerate} \\item the singularities of $X$ are at-worst-nodal, \\item $X$ does not have a rational tail (Example \\ref{example-rational-tail}), \\item $X$ does not have a rational bridge (Example \\ref{example-rational-bridge}), \\item the genus $g$ of $X$ is $\\geq 2$. \\end{enumerate} Then $\\omega_X$ is ample."}
+{"_id": "6343", "title": "curves-lemma-contracting-rational-bridges", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$. Assume the singularities of $X$ are at-worst-nodal and that $X$ has no rational tails. Consider a sequence $$ X = X_0 \\to X_1 \\to \\ldots \\to X_n = X' $$ of contractions of rational bridges (Example \\ref{example-rational-bridge}) until none are left. Then $\\omega_{X'}$ ample. The morphism $X \\to X'$ is independent of choices and formation of this morphism commutes with base field extensions."}
+{"_id": "6344", "title": "curves-lemma-contract-gorenstein-canonical", "text": "Let $k$ be a field. Let $c : X \\to Y$ be a morphism of proper schemes over $k$ Assume \\begin{enumerate} \\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$, \\item $X$ and $Y$ are reduced, Gorenstein, and have dimension $1$, \\item $\\exists\\ m \\in \\mathbf{Z}$ with $H^1(X, \\omega_X^{\\otimes m}) = 0$ and $\\omega_X^{\\otimes m}$ generated by global sections. \\end{enumerate} Then $c^*\\omega_Y \\cong \\omega_X$."}
+{"_id": "6345", "title": "curves-lemma-characterize-contraction-to-stable", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$. Assume the singularities of $X$ are at-worst-nodal. There is a unique morphism (up to unique isomorphism) $$ c : X \\longrightarrow Y $$ of schemes over $k$ having the following properties: \\begin{enumerate} \\item $Y$ is proper over $k$, $\\dim(Y) = 1$, the singularities of $Y$ are at-worst-nodal, \\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$, and \\item $\\omega_Y$ is ample on $Y$. \\end{enumerate}"}
+{"_id": "6346", "title": "curves-lemma-tricanonical", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$. Assume the singularities of $X$ are at-worst-nodal and $\\omega_X$ is ample. Then $\\omega_X^{\\otimes 3}$ is very ample and $H^1(X, \\omega_X^{\\otimes 3}) = 0$."}
+{"_id": "6347", "title": "curves-lemma-smooth-vector-fields", "text": "Let $k$ be an algebraically closed field. Let $X$ be a smooth, proper, connected curve over $k$. Let $g$ be the genus of $X$. \\begin{enumerate} \\item If $g \\geq 2$, then $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$ is zero, \\item if $g = 1$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$ is nonzero, then $D$ does not fix any closed point of $X$, and \\item if $g = 0$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$ is nonzero, then $D$ fixes at most $2$ closed points of $X$. \\end{enumerate}"}
+{"_id": "6348", "title": "curves-lemma-nodal-vector-fields", "text": "Let $k$ be an algebraically closed field. Let $X$ be an at-worst-nodal, proper, connected $1$-dimensional scheme over $k$. Let $\\nu : X^\\nu \\to X$ be the normalization. Let $S \\subset X^\\nu$ be the set of points where $\\nu$ is not an isomorphism. Then $$ \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = \\{D' \\in \\text{Der}_k(\\mathcal{O}_{X^\\nu}, \\mathcal{O}_{X^\\nu}) \\mid D' \\text{ fixes every }x^\\nu \\in S\\} $$"}
+{"_id": "6349", "title": "curves-lemma-stable-vector-fields", "text": "Let $k$ be an algebraically closed field. Let $X$ be an at-worst-nodal, proper, connected $1$-dimensional scheme over $k$. Assume the genus of $X$ is at least $2$ and that $X$ has no rational tails or bridges. Then $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$."}
+{"_id": "6350", "title": "curves-proposition-projective-line", "text": "Let $k$ be a field. Let $X$ be a proper curve over $k$. The following are equivalent \\begin{enumerate} \\item $X \\cong \\mathbf{P}^1_k$, \\item $X$ is smooth and geometrically irreducible over $k$, $X$ has genus $0$, and $X$ has an invertible module of odd degree, \\item $X$ is geometrically integral over $k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree, \\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree, \\item $X$ is geometrically integral over $k$, $X$ has genus $0$, and $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$, \\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$, and $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$, \\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$, \\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$ has closed points $x_1, \\ldots, x_n$ such that $\\mathcal{O}_{X, x_i}$ is normal and $\\gcd([\\kappa(x_i) : k]) = 1$, and \\item add more here. \\end{enumerate}"}
+{"_id": "6351", "title": "curves-proposition-unwind-morphism-smooth", "text": "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a nonconstant morphism of proper smooth curves over $k$. Then we can factor $f$ as $$ X \\longrightarrow X^{(p^n)} \\longrightarrow Y $$ where $X^{(p^n)} \\to Y$ is a nonconstant morphism of proper smooth curves inducing a separable field extension $k(X^{(p^n)})/k(Y)$, we have $$ X^{(p^n)} = X \\times_{\\Spec(k), F_{\\Spec(k)}^n} \\Spec(k), $$ and $X \\to X^{(p^n)}$ is the $n$-fold relative frobenius of $X$."}
+{"_id": "6352", "title": "curves-proposition-torsion-picard-reduced-proper", "text": "Let $k$ be an algebraically closed field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and has dimension $1$. Let $g$ be the genus of $X$ and let $g_{geom}$ be the sum of the geometric genera of the irreducible components of $X$. For any prime $\\ell$ different from the characteristic of $k$ we have $$ \\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell] \\leq g + g_{geom} $$ and equality holds if and only if all the singularities of $X$ are multicross."}
+{"_id": "6366", "title": "etale-cohomology-theorem-sheafification", "text": "Let $\\mathcal{C}$ be a site and $\\mathcal{F}$ a presheaf on $\\mathcal{C}$. \\begin{enumerate} \\item The rule $$ U \\mapsto \\mathcal{F}^+(U) := \\colim_{\\mathcal{U} \\text{ covering of }U} \\check H^0(\\mathcal{U}, \\mathcal{F}) $$ is a presheaf. And the colimit is a directed one. \\item There is a canonical map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$. \\item If $\\mathcal{F}$ is a separated presheaf then $\\mathcal{F}^+$ is a sheaf and the map in (2) is injective. \\item $\\mathcal{F}^+$ is a separated presheaf. \\item $\\mathcal{F}^\\# = (\\mathcal{F}^+)^+$ is a sheaf, and the canonical map induces a functorial isomorphism $$ \\Hom_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G}) = \\Hom_{\\Sh(\\mathcal{C})}(\\mathcal{F}^\\#, \\mathcal{G}) $$ for any $\\mathcal{G} \\in \\Sh(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "6368", "title": "etale-cohomology-theorem-descent-quasi-coherent", "text": "If $\\mathcal{V} = \\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering, then all descent data for quasi-coherent sheaves with respect to $\\mathcal{V}$ are effective."}
+{"_id": "6370", "title": "etale-cohomology-theorem-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\mathcal{C}$ be a site. Assume that \\begin{enumerate} \\item the underlying category $\\mathcal{C}$ is a full subcategory of $\\Sch/S$, \\item any Zariski covering of $T \\in \\Ob(\\mathcal{C})$ can be refined by a covering of $\\mathcal{C}$, \\item $S/S$ is an object of $\\mathcal{C}$, \\item every covering of $\\mathcal{C}$ is an fpqc covering of schemes. \\end{enumerate} Then the presheaf $\\mathcal{O}$ is a sheaf on $\\mathcal{C}$ and any quasi-coherent $\\mathcal{O}$-module on $(\\mathcal{C}, \\mathcal{O})$ is of the form $\\mathcal{F}^a$ for some quasi-coherent sheaf $\\mathcal{F}$ on $S$."}
+{"_id": "6372", "title": "etale-cohomology-theorem-cech-ss", "text": "Let $\\mathcal{C}$ be a site. For any covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U \\in \\Ob(\\mathcal{C})$ and any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$ there is a spectral sequence $$ E_2^{p, q} = \\check H^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F})) \\Rightarrow H^{p+q}(U, \\mathcal{F}), $$ where $\\underline{H}^q(\\mathcal{F})$ is the abelian presheaf $V \\mapsto H^q(V, \\mathcal{F})$."}
+{"_id": "6373", "title": "etale-cohomology-theorem-zariski-fpqc-quasi-coherent", "text": "Let $S$ be a scheme and $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_S$-module. Let $\\mathcal{C}$ be either $(\\Sch/S)_\\tau$ for $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ or $S_\\etale$. Then $$ H^p(S, \\mathcal{F}) = H^p_\\tau(S, \\mathcal{F}^a) $$ for all $p \\geq 0$ where \\begin{enumerate} \\item the left hand side indicates the usual cohomology of the sheaf $\\mathcal{F}$ on the underlying topological space of the scheme $S$, and \\item the right hand side indicates cohomology of the abelian sheaf $\\mathcal{F}^a$ (see Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}) on the site $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "6374", "title": "etale-cohomology-theorem-picard-group", "text": "For any scheme $X$ we have canonical identifications \\begin{align*} H_{fppf}^1(X, \\mathbf{G}_m) & = H^1_{syntomic}(X, \\mathbf{G}_m) \\\\ & = H^1_{smooth}(X, \\mathbf{G}_m) \\\\ & = H_\\etale^1(X, \\mathbf{G}_m) \\\\ & = H^1_{Zar}(X, \\mathbf{G}_m) \\\\ & = \\Pic(X) \\\\ & = H^1(X, \\mathcal{O}_X^*) \\end{align*}"}
+{"_id": "6376", "title": "etale-cohomology-theorem-exactness-stalks", "text": "Let $S$ be a scheme. A map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective (resp.\\ surjective) if and only if the map on stalks $a_{\\overline{s}} : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{G}_{\\overline{s}}$ is injective (resp.\\ surjective) for all geometric points of $S$. A sequence of abelian sheaves on $S_\\etale$ is exact if and only if it is exact on all stalks at geometric points of $S$."}
+{"_id": "6379", "title": "etale-cohomology-theorem-henselian", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. The following are equivalent: \\begin{enumerate} \\item $R$ is henselian, \\item for any $f\\in R[T]$ and any factorization $\\bar f = g_0 h_0$ in $\\kappa[T]$ with $\\gcd(g_0, h_0)=1$, there exists a factorization $f = gh$ in $R[T]$ with $\\bar g = g_0$ and $\\bar h = h_0$, \\item any finite $R$-algebra $S$ is isomorphic to a finite product of local rings finite over $R$, \\item any finite type $R$-algebra $A$ is isomorphic to a product $A \\cong A' \\times C$ where $A' \\cong A_1 \\times \\ldots \\times A_r$ is a product of finite local $R$-algebras and all the irreducible components of $C \\otimes_R \\kappa$ have dimension at least 1, \\item if $A$ is an \\'etale $R$-algebra and $\\mathfrak n$ is a maximal ideal of $A$ lying over $\\mathfrak m$ such that $\\kappa \\cong A/\\mathfrak n$, then there exists an isomorphism $\\varphi : A \\cong R \\times A'$ such that $\\varphi(\\mathfrak n) = \\mathfrak m \\times A' \\subset R \\times A'$. \\end{enumerate}"}
+{"_id": "6380", "title": "etale-cohomology-theorem-henselization", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring and $\\kappa\\subset\\kappa^{sep}$ a separable algebraic closure. There exist canonical flat local ring maps $R \\to R^h \\to R^{sh}$ where \\begin{enumerate} \\item $R^h$, $R^{sh}$ are filtered colimits of \\'etale $R$-algebras, \\item $R^h$ is henselian, $R^{sh}$ is strictly henselian, \\item $\\mathfrak m R^h$ (resp.\\ $\\mathfrak m R^{sh}$) is the maximal ideal of $R^h$ (resp.\\ $R^{sh}$), and \\item $\\kappa = R^h/\\mathfrak m R^h$, and $\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$ as extensions of $\\kappa$. \\end{enumerate}"}
+{"_id": "6381", "title": "etale-cohomology-theorem-fully-faithful", "text": "Let $X$, $Y$ be schemes. Let $$ (g, g^\\#) : (\\Sh(X_\\etale), \\mathcal{O}_X) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{O}_Y) $$ be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes $f : X \\to Y$ such that $(g, g^\\#)$ is isomorphic to $(f_{small}, f_{small}^\\sharp)$. In other words, the construction $$ \\Sch \\longrightarrow \\textit{Locally ringed topoi}, \\quad X \\longrightarrow (X_\\etale, \\mathcal{O}_X) $$ is fully faithful (morphisms up to $2$-isomorphisms on the right hand side)."}
+{"_id": "6382", "title": "etale-cohomology-theorem-etale-topological", "text": "Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S' \\to S$ be a universal homeomorphism. Denote $X'$ (resp.\\ $Y'$) the base change to $S'$. If $X$ is \\'etale over $S$, then the map $$ \\Mor_S(Y, X) \\longrightarrow \\Mor_{S'}(Y', X') $$ is bijective."}
+{"_id": "6383", "title": "etale-cohomology-theorem-topological-invariance", "text": "\\begin{reference} \\cite[IV Theorem 18.1.2]{EGA} \\end{reference} Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is integral, universally injective and surjective (i.e., $f$ is a universal homeomorphism, see Morphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}). The functor $$ V \\longmapsto V_X = X \\times_Y V $$ defines an equivalence of categories $$ \\{ \\text{schemes }V\\text{ \\'etale over }Y \\} \\leftrightarrow \\{ \\text{schemes }U\\text{ \\'etale over }X \\} $$"}
+{"_id": "6384", "title": "etale-cohomology-theorem-colimit", "text": "Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : X_{i'} \\to X_i$. We assume that $X_i$ is quasi-compact and quasi-separated for all $i \\in I$. Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves on $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then $$ \\colim_{i\\in I} H_\\etale^p(X_i, \\mathcal{F}_i) = H_\\etale^p(X, \\mathcal{F}). $$ for all $p \\geq 0$."}
+{"_id": "6385", "title": "etale-cohomology-theorem-higher-direct-images", "text": "Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism of schemes, $\\mathcal{F}$ an abelian sheaf on $X_\\etale$, and $\\overline{s}$ a geometric point of $S$ lying over $s \\in S$. Then $$ \\left(R^nf_* \\mathcal{F}\\right)_{\\overline{s}} = H_\\etale^n( X \\times_S \\Spec(\\mathcal{O}_{S, s}^{sh}), p^{-1}\\mathcal{F}) $$ where $p : X \\times_S \\Spec(\\mathcal{O}_{S, s}^{sh}) \\to X$ is the projection."}
+{"_id": "6386", "title": "etale-cohomology-theorem-equivalence-sheaves-point", "text": "Let $S = \\Spec(K)$ with $K$ a field. Let $\\overline{s}$ be a geometric point of $S$. Let $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group. Taking stalks induces an equivalence of categories $$ \\Sh(S_\\etale) \\longrightarrow G\\textit{-Sets}, \\quad \\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}. $$"}
+{"_id": "6387", "title": "etale-cohomology-theorem-central-simple-algebra", "text": "Let $K$ be a field. For a unital, associative (not necessarily commutative) $K$-algebra $A$ the following are equivalent \\begin{enumerate} \\item $A$ is finite central simple $K$-algebra, \\item $A$ is a finite dimensional $K$-vector space, $K$ is the center of $A$, and $A$ has no nontrivial two-sided ideal, \\item there exists $d \\geq 1$ such that $A \\otimes_K \\bar K \\cong \\text{Mat}(d \\times d, \\bar K)$, \\item there exists $d \\geq 1$ such that $A \\otimes_K K^{sep} \\cong \\text{Mat}(d \\times d, K^{sep})$, \\item there exist $d \\geq 1$ and a finite Galois extension $K \\subset K'$ such that $A \\otimes_{K'} K' \\cong \\text{Mat}(d \\times d, K')$, \\item there exist $n \\geq 1$ and a finite central skew field $D$ over $K$ such that $A \\cong \\text{Mat}(n \\times n, D)$. \\end{enumerate} The integer $d$ is called the {\\it degree} of $A$."}
+{"_id": "6388", "title": "etale-cohomology-theorem-brauer-delta", "text": "Let $K$ be a field with separable algebraic closure $K^{sep}$. The map $\\delta : \\text{Br}(K) \\to H^2(\\text{Gal}(K^{sep}/K), (K^{sep})^*)$ defined above is a group isomorphism."}
+{"_id": "6389", "title": "etale-cohomology-theorem-C1-brauer-group-zero", "text": "Let $K$ be a $C_1$ field. Then $\\text{Br}(K) = 0$."}
+{"_id": "6390", "title": "etale-cohomology-theorem-tsen", "text": "The function field of a variety of dimension $r$ over an algebraically closed field $k$ is $C_r$."}
+{"_id": "6392", "title": "etale-cohomology-theorem-vanishing-cohomology-Gm-curve", "text": "Let $X$ be a smooth curve over an algebraically closed field. Then $$ H_\\etale^q(X, \\mathbf{G}_m) = 0 \\ \\ \\text{ for all } q \\geq 2. $$"}
+{"_id": "6393", "title": "etale-cohomology-theorem-gabber", "text": "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. For any torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(Z, \\mathcal{F}|_Z)$."}
+{"_id": "6394", "title": "etale-cohomology-theorem-vanishing-affine-curves", "text": "If $k$ is an algebraically closed field, $X$ is a separated, finite type scheme of dimension $\\leq 1$ over $k$, and $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then \\begin{enumerate} \\item $H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 2$, \\item $H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $X$ is affine, \\item $H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $p = \\text{char}(k) > 0$ and $\\mathcal{F}$ is $p$-power torsion, \\item $H^q_\\etale(X, \\mathcal{F})$ is finite if $\\mathcal{F}$ is constructible and torsion prime to $\\text{char}(k)$, \\item $H^q_\\etale(X, \\mathcal{F})$ is finite if $X$ is proper and $\\mathcal{F}$ constructible, \\item $H^q_\\etale(X, \\mathcal{F}) \\to H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k \\subset k'$ of algebraically closed fields if $\\mathcal{F}$ is torsion prime to $\\text{char}(k)$, \\item $H^q_\\etale(X, \\mathcal{F}) \\to H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism for any extension $k \\subset k'$ of algebraically closed fields if $X$ is proper, \\item $H^2_\\etale(X, \\mathcal{F}) \\to H^2_\\etale(U, \\mathcal{F})$ is surjective for all $U \\subset X$ open. \\end{enumerate}"}
+{"_id": "6395", "title": "etale-cohomology-theorem-vanishing-curves", "text": "Let $X$ be a finite type, dimension $1$ scheme over an algebraically closed field $k$. Let $\\mathcal{F}$ be a torsion sheaf on $X_\\etale$. Then $$ H_\\etale^q(X, \\mathcal{F}) = 0, \\quad \\forall q \\geq 3. $$ If $X$ affine then also $H_\\etale^2(X, \\mathcal{F}) = 0$."}
+{"_id": "6396", "title": "etale-cohomology-theorem-smooth-base-change", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ where $f$ is smooth and $g$ quasi-compact and quasi-separated. Then $$ f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F} $$ for any $q$ and any abelian sheaf $\\mathcal{F}$ on $T_\\etale$ all of whose stalks at geometric points are torsion of orders invertible on $S$."}
+{"_id": "6397", "title": "etale-cohomology-theorem-proper-base-change", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be a morphism of schemes. Set $X' = Y' \\times_Y X$ and consider the cartesian diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$. Then the base change map $$ g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F} $$ is an isomorphism."}
+{"_id": "6400", "title": "etale-cohomology-lemma-fpqc-sheaves", "text": "Let $\\mathcal{F}$ be a presheaf on $\\Sch/S$. Then $\\mathcal{F}$ satisfies the sheaf property for the fpqc topology if and only if \\begin{enumerate} \\item $\\mathcal{F}$ satisfies the sheaf property with respect to the Zariski topology, and \\item for every faithfully flat morphism $\\Spec(B) \\to \\Spec(A)$ of affine schemes over $S$, the sheaf axiom holds for the covering $\\{\\Spec(B) \\to \\Spec(A)\\}$. Namely, this means that $$ \\xymatrix{ \\mathcal{F}(\\Spec(A)) \\ar[r] & \\mathcal{F}(\\Spec(B)) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\mathcal{F}(\\Spec(B \\otimes_A B)) } $$ is an equalizer diagram. \\end{enumerate}"}
+{"_id": "6401", "title": "etale-cohomology-lemma-representable-sheaf-fpqc", "text": "Any representable presheaf on $\\Sch/S$ satisfies the sheaf condition for the fpqc topology."}
+{"_id": "6403", "title": "etale-cohomology-lemma-descent-modules", "text": "If $A \\to B$ is faithfully flat and $M$ is an $A$-module, then the complex $(B/A)_\\bullet \\otimes_A M$ is exact in positive degrees, and $H^0((B/A)_\\bullet \\otimes_A M) = M$."}
+{"_id": "6408", "title": "etale-cohomology-lemma-hom-injective", "text": "Let $\\mathcal{C}$ be a category. If $\\mathcal{I}$ is an injective object of $\\textit{PAb}(\\mathcal{C})$ and $\\mathcal{U}$ is a family of morphisms with fixed target in $\\mathcal{C}$, then $\\check H^p(\\mathcal{U}, \\mathcal{I}) = 0$ for all $p > 0$."}
+{"_id": "6409", "title": "etale-cohomology-lemma-forget-injectives", "text": "The forgetful functor $\\textit{Ab}(\\mathcal{C})\\to \\textit{PAb}(\\mathcal{C})$ transforms injectives into injectives."}
+{"_id": "6411", "title": "etale-cohomology-lemma-compare-cohomology-big-small", "text": "Let $\\tau \\in \\{\\etale, Zariski\\}$. If $\\mathcal{F}$ is an abelian sheaf defined on $(\\Sch/S)_\\tau$, then the cohomology groups of $\\mathcal{F}$ over $S$ agree with the cohomology groups of $\\mathcal{F}|_{S_\\tau}$ over $S$."}
+{"_id": "6413", "title": "etale-cohomology-lemma-alternative", "text": "Let $S$ be a scheme. Let $S_{affine, \\etale}$ denote the full subcategory of $S_\\etale$ whose objects are those $U/S \\in \\Ob(S_\\etale)$ with $U$ affine. A covering of $S_{affine, \\etale}$ will be a standard \\'etale covering, see Topologies, Definition \\ref{topologies-definition-standard-etale}. Then restriction $$ \\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, \\etale}} $$ defines an equivalence of topoi $\\Sh(S_\\etale) \\cong \\Sh(S_{affine, \\etale})$."}
+{"_id": "6416", "title": "etale-cohomology-lemma-cech-complex", "text": "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$. Let $S$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $(\\Sch/S)_\\tau$, or on $S_\\tau$ in case $\\tau = \\etale$, and let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a standard $\\tau$-covering of this site. Let $V = \\coprod_{i \\in I} U_i$. Then \\begin{enumerate} \\item $V$ is an affine scheme, \\item $\\mathcal{V} = \\{V \\to U\\}$ is an fpqc covering and also a $\\tau$-covering unless $\\tau = Zariski$, \\item the {\\v C}ech complexes $\\check{\\mathcal{C}}^\\bullet (\\mathcal{U}, \\mathcal{F})$ and $\\check{\\mathcal{C}}^\\bullet (\\mathcal{V}, \\mathcal{F})$ agree. \\end{enumerate}"}
+{"_id": "6417", "title": "etale-cohomology-lemma-locality-cohomology", "text": "Let $\\mathcal{C}$ be a site, $\\mathcal{F}$ an abelian sheaf on $\\mathcal{C}$, $U$ an object of $\\mathcal{C}$, $p > 0$ an integer and $\\xi \\in H^p(U, \\mathcal{F})$. Then there exists a covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U$ in $\\mathcal{C}$ such that $\\xi |_{U_i} = 0$ for all $i \\in I$."}
+{"_id": "6419", "title": "etale-cohomology-lemma-kummer-sequence", "text": "If $n\\in \\mathcal{O}_S^*$ then $$ 0 \\to \\mu_{n, S} \\to \\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n} \\mathbf{G}_{m, S} \\to 0 $$ is a short exact sequence of sheaves on both the small and big \\'etale site of $S$."}
+{"_id": "6420", "title": "etale-cohomology-lemma-kummer-sequence-syntomic", "text": "For any $n \\in \\mathbf{N}$ the sequence $$ 0 \\to \\mu_{n, S} \\to \\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n} \\mathbf{G}_{m, S} \\to 0 $$ is a short exact sequence of sheaves on the site $(\\Sch/S)_{fppf}$ and $(\\Sch/S)_{syntomic}$."}
+{"_id": "6422", "title": "etale-cohomology-lemma-cofinal-etale", "text": "Let $S$ be a scheme, and let $\\overline{s}$ be a geometric point of $S$. The category of \\'etale neighborhoods is cofiltered. More precisely: \\begin{enumerate} \\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of $\\overline{s}$ in $S$. Then there exists a third \\'etale neighborhood $(U, \\overline{u})$ and morphisms $(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$. \\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two morphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an \\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism $h : (U'', \\overline{u}'') \\to (U, \\overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \\circ h = h_2 \\circ h$. \\end{enumerate}"}
+{"_id": "6423", "title": "etale-cohomology-lemma-geometric-lift-to-cover", "text": "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$. Let $(U, \\overline{u})$ be an \\'etale neighborhood of $\\overline{s}$. Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U \\}_{i\\in I}$ be an \\'etale covering. Then there exist $i \\in I$ and $\\overline{u}_i : \\overline{s} \\to U_i$ such that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$ is a morphism of \\'etale neighborhoods."}
+{"_id": "6424", "title": "etale-cohomology-lemma-stalk-gives-point", "text": "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$. Consider the functor \\begin{align*} u : S_\\etale & \\longrightarrow \\textit{Sets}, \\\\ U & \\longmapsto |U_{\\overline{s}}| = \\{\\overline{u} \\text{ such that }(U, \\overline{u}) \\text{ is an \\'etale neighbourhood of }\\overline{s}\\}. \\end{align*} Here $|U_{\\overline{s}}|$ denotes the underlying set of the geometric fibre. Then $u$ defines a point $p$ of the site $S_\\etale$ (Sites, Definition \\ref{sites-definition-point}) and its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$ (Sites, Equation \\ref{sites-equation-stalk}) is the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$ defined above."}
+{"_id": "6425", "title": "etale-cohomology-lemma-stalk-exact", "text": "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$. \\begin{enumerate} \\item The stalk functor $\\textit{PAb}(S_\\etale) \\to \\textit{Ab}$, $\\mathcal{F}  \\mapsto  \\mathcal{F}_{\\overline{s}}$ is exact. \\item We have $(\\mathcal{F}^\\#)_{\\overline{s}} = \\mathcal{F}_{\\overline{s}}$ for any presheaf of sets $\\mathcal{F}$ on $S_\\etale$. \\item The functor $\\textit{Ab}(S_\\etale) \\to \\textit{Ab}$, $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$ is exact. \\item Similarly the functors $\\textit{PSh}(S_\\etale) \\to \\textit{Sets}$ and $\\Sh(S_\\etale) \\to \\textit{Sets}$ given by the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ are exact (see Categories, Definition \\ref{categories-definition-exact}) and commute with arbitrary colimits. \\end{enumerate}"}
+{"_id": "6426", "title": "etale-cohomology-lemma-points-small-etale-site", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item Let $p$ be a point of the small \\'etale site $S_\\etale$ of $S$ given by a functor $u : S_\\etale \\to \\textit{Sets}$. Then there exists a geometric point $\\overline{s}$ of $S$ such that $p$ is isomorphic to the point of $S_\\etale$ associated to $\\overline{s}$ in Lemma \\ref{lemma-stalk-gives-point}. \\item Let $p : \\Sh(pt) \\to \\Sh(S_\\etale)$ be a point of the small \\'etale topos of $S$. Then $p$ comes from a geometric point of $S$, i.e., the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is isomorphic to a stalk functor as defined in Definition \\ref{definition-stalk}. \\end{enumerate}"}
+{"_id": "6427", "title": "etale-cohomology-lemma-points-fppf", "text": "Let $S$ be a scheme. All of the following sites have enough points $S_{Zar}$, $S_\\etale$, $(\\Sch/S)_{Zar}$, $(\\textit{Aff}/S)_{Zar}$, $(\\Sch/S)_\\etale$, $(\\textit{Aff}/S)_\\etale$, $(\\Sch/S)_{smooth}$, $(\\textit{Aff}/S)_{smooth}$, $(\\Sch/S)_{syntomic}$, $(\\textit{Aff}/S)_{syntomic}$, $(\\Sch/S)_{fppf}$, and $(\\textit{Aff}/S)_{fppf}$."}
+{"_id": "6428", "title": "etale-cohomology-lemma-support-subsheaf-final", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a subsheaf of the final object of the \\'etale topos of $S$ (see Sites, Example \\ref{sites-example-singleton-sheaf}). Then there exists a unique open $W \\subset S$ such that $\\mathcal{F} = h_W$."}
+{"_id": "6429", "title": "etale-cohomology-lemma-zero-over-image", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$. Let $\\sigma \\in \\mathcal{F}(U)$ be a local section. There exists an open subset $W \\subset U$ such that \\begin{enumerate} \\item $W \\subset U$ is the largest Zariski open subset of $U$ such that $\\sigma|_W = 0$, \\item for every $\\varphi : V \\to U$ in $S_\\etale$ we have $$ \\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W, $$ \\item for every geometric point $\\overline{u}$ of $U$ we have $$ (U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{s}} \\Leftrightarrow \\overline{u} \\in W $$ where $\\overline{s} = (U \\to S) \\circ \\overline{u}$. \\end{enumerate}"}
+{"_id": "6431", "title": "etale-cohomology-lemma-support-sheaf-rings-closed", "text": "The support of a sheaf of rings on $S_\\etale$ is closed."}
+{"_id": "6432", "title": "etale-cohomology-lemma-finite-over-henselian", "text": "If $R$ is henselian and $A$ is a finite $R$-algebra, then $A$ is a finite product of henselian local rings."}
+{"_id": "6433", "title": "etale-cohomology-lemma-describe-etale-local-ring", "text": "\\begin{slogan} The stalk of the structure sheaf of a scheme in the etale topology is the strict henselization. \\end{slogan} Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$ lying over $s \\in S$. Let $\\kappa = \\kappa(s)$ and let $\\kappa \\subset \\kappa^{sep} \\subset \\kappa(\\overline{s})$ denote the separable algebraic closure of $\\kappa$ in $\\kappa(\\overline{s})$. Then there is a canonical identification $$ (\\mathcal{O}_{S, s})^{sh} \\cong \\mathcal{O}_{S, \\overline{s}} $$ where the left hand side is the strict henselization of the local ring $\\mathcal{O}_{S, s}$ as described in Theorem \\ref{theorem-henselization} and right hand side is the stalk of the structure sheaf $\\mathcal{O}_S$ on $S_\\etale$ at the geometric point $\\overline{s}$."}
+{"_id": "6435", "title": "etale-cohomology-lemma-etale-site-locally-ringed", "text": "Let $S$ be a scheme. The small \\'etale site $S_\\etale$ endowed with its structure sheaf $\\mathcal{O}_S$ is a locally ringed site, see Modules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}."}
+{"_id": "6436", "title": "etale-cohomology-lemma-stalk-pullback", "text": "Let $f : X \\to Y$ be a morphism of schemes. \\begin{enumerate} \\item The functor $f^{-1} : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$ is exact. \\item The functor $f^{-1} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition \\ref{categories-definition-exact}. \\item Let $\\overline{x} \\to X$ be a geometric point. Let $\\mathcal{G}$ be a sheaf on $Y_\\etale$. Then there is a canonical identification $$ (f^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}. $$ where $\\overline{y} = f \\circ \\overline{x}$. \\item For any $V \\to Y$ \\'etale we have $f^{-1}h_V = h_{X \\times_Y V}$. \\end{enumerate}"}
+{"_id": "6437", "title": "etale-cohomology-lemma-where-sections-are-equal", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a sheaf of sets on $S_\\etale$. Let $s, t \\in \\mathcal{F}(S)$. Then there exists an open $W \\subset S$ characterized by the following property: A morphism $f : T \\to S$ factors through $W$ if and only if $s|_T = t|_T$ (restriction is pullback by $f_{small}$)."}
+{"_id": "6438", "title": "etale-cohomology-lemma-describe-pullback", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\etale\\}$. Consider the morphism $$ \\pi_S : (\\Sch/S)_\\tau \\longrightarrow S_\\tau $$ of Topologies, Lemma \\ref{topologies-lemma-at-the-bottom} or \\ref{topologies-lemma-at-the-bottom-etale}. Let $\\mathcal{F}$ be a sheaf on $S_\\tau$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule $$ (\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\tau, f_{small}^{-1}\\mathcal{F}) $$ where $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings."}
+{"_id": "6439", "title": "etale-cohomology-lemma-sections-upstairs", "text": "Let $S$ be a scheme. Let $f : T \\to S$ be a morphism such that \\begin{enumerate} \\item $f$ is flat and quasi-compact, and \\item the geometric fibres of $f$ are connected. \\end{enumerate} Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(T, f^{-1}_{small}\\mathcal{F})$."}
+{"_id": "6441", "title": "etale-cohomology-lemma-sections-base-field-extension", "text": "Let $k \\subset K$ be an extension of fields with $k$ separably algebraically closed. Let $S$ be a scheme over $k$. Denote $p : S_K = S \\times_{\\Spec(k)} \\Spec(K) \\to S$ the projection. Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(S_K, p^{-1}_{small}\\mathcal{F})$."}
+{"_id": "6442", "title": "etale-cohomology-lemma-morphism-locally-ringed", "text": "Let $f : X \\to Y$ be a morphism of schemes. The morphism of ringed sites $(f_{small}, f_{small}^\\sharp)$ associated to $f$ is a morphism of locally ringed sites, see Modules on Sites, Definition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}."}
+{"_id": "6443", "title": "etale-cohomology-lemma-2-morphism", "text": "Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a morphism of schemes. Let $t$ be a $2$-morphism from $(f_{small}, f_{small}^\\sharp)$ to itself, see Modules on Sites, Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}. Then $t = \\text{id}$."}
+{"_id": "6444", "title": "etale-cohomology-lemma-faithful", "text": "Let $X$, $Y$ be schemes. Any two morphisms $a, b : X \\to Y$ of schemes for which there exists a $2$-isomorphism $(a_{small}, a_{small}^\\sharp) \\cong (b_{small}, b_{small}^\\sharp)$ in the $2$-category of ringed topoi are equal."}
+{"_id": "6445", "title": "etale-cohomology-lemma-morphism-ringed-etale-topoi-affines", "text": "Let $X$, $Y$ be affine schemes. Let $$ (g, g^\\#) : (\\Sh(X_\\etale), \\mathcal{O}_X) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{O}_Y) $$ be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes $f : X \\to Y$ such that $(g, g^\\#)$ is $2$-isomorphic to $(f_{small}, f_{small}^\\sharp)$, see Modules on Sites, Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}."}
+{"_id": "6446", "title": "etale-cohomology-lemma-property-A-implies", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume (A). \\begin{enumerate} \\item $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ reflects injections and surjections, \\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$ is surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is faithful. \\end{enumerate}"}
+{"_id": "6447", "title": "etale-cohomology-lemma-locally-quasi-finite-A", "text": "Let $f : X \\to Y$ be a separated locally quasi-finite morphism of schemes. Then property (A) above holds."}
+{"_id": "6448", "title": "etale-cohomology-lemma-integral-A", "text": "Let $f : X \\to Y$ be an integral morphism of schemes. Then property (A) holds."}
+{"_id": "6449", "title": "etale-cohomology-lemma-when-push-pull-surjective", "text": "Let $f : X \\to Y$ be a morphism of schemes. Denote $f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ the associated morphism of small \\'etale topoi. Assume at least one of the following \\begin{enumerate} \\item $f$ is integral, or \\item $f$ is separated and locally quasi-finite. \\end{enumerate} Then the functor $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ has the following properties \\begin{enumerate} \\item the map $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$ is always surjective, \\item $f_{small, *}$ is faithful, and \\item $f_{small, *}$ reflects injections and surjections. \\end{enumerate}"}
+{"_id": "6450", "title": "etale-cohomology-lemma-property-B-implies", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume (B) holds. Then the functor $f_{small, *} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ transforms surjections into surjections."}
+{"_id": "6451", "title": "etale-cohomology-lemma-simplify-B", "text": "Let $f : X \\to Y$ be a morphism of schemes. Suppose \\begin{enumerate} \\item $V \\to Y$ is an \\'etale morphism of schemes, \\item $\\{U_i \\to X \\times_Y V\\}$ is an \\'etale covering, and \\item $v \\in V$ is a point. \\end{enumerate} Assume that for any such data there exists an \\'etale neighbourhood $(V', v') \\to (V, v)$, a disjoint union decomposition $X \\times_Y V' = \\coprod W'_i$, and morphisms $W'_i \\to U_i$ over $X \\times_Y V$. Then property (B) holds."}
+{"_id": "6452", "title": "etale-cohomology-lemma-finite-B", "text": "Let $f : X \\to Y$ be a finite morphism of schemes. Then property (B) holds."}
+{"_id": "6453", "title": "etale-cohomology-lemma-integral-B", "text": "Let $f : X \\to Y$ be an integral morphism of schemes. Then property (B) holds."}
+{"_id": "6454", "title": "etale-cohomology-lemma-what-integral", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is integral (for example finite). Then \\begin{enumerate} \\item $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves), \\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$ is surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is faithful and reflects injections and surjections, and \\item $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is exact. \\end{enumerate}"}
+{"_id": "6455", "title": "etale-cohomology-lemma-property-C-implies", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume (C) holds. Then the functor $f_{small, *} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ reflects injections and surjections."}
+{"_id": "6456", "title": "etale-cohomology-lemma-property-C-closed-implies", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that for any $V \\to Y$ \\'etale we have that \\begin{enumerate} \\item $X \\times_Y V \\to V$ has property (C), and \\item $X \\times_Y V \\to V$ is closed. \\end{enumerate} Then the functor $Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$ is almost cocontinuous, see Sites, Definition \\ref{sites-definition-almost-cocontinuous}."}
+{"_id": "6457", "title": "etale-cohomology-lemma-integral-homeo-onto-image-C", "text": "Let $f : X \\to Y$ be an integral morphism of schemes which defines a homeomorphism of $X$ with a closed subset of $Y$. Then property (C) holds."}
+{"_id": "6459", "title": "etale-cohomology-lemma-closed-immersion-almost-full", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $U, U'$ be schemes \\'etale over $X$. Let $h : U_Z \\to U'_Z$ be a morphism over $Z$. Then there exists a diagram $$ \\xymatrix{ U & W \\ar[l]_a \\ar[r]^b & U' } $$ such that $a_Z : W_Z \\to U_Z$ is an isomorphism and $h = b_Z \\circ (a_Z)^{-1}$."}
+{"_id": "6460", "title": "etale-cohomology-lemma-closed-immersion-almost-essentially-surjective", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $V \\to Z$ be an \\'etale morphism of schemes. There exist \\'etale morphisms $U_i \\to X$ and morphisms $U_{i, Z} \\to V$ such that $\\{U_{i, Z} \\to V\\}$ is a Zariski covering of $V$."}
+{"_id": "6461", "title": "etale-cohomology-lemma-stalk-pushforward-closed-immersion", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $\\mathcal{G}$ be a sheaf of sets on $Z_\\etale$. Let $\\overline{x}$ be a geometric point of $X$. Then $$ (i_{small, *}\\mathcal{G})_{\\overline{x}} = \\left\\{ \\begin{matrix} * & \\text{if} & \\overline{x} \\not \\in Z \\\\ \\mathcal{G}_{\\overline{x}} & \\text{if} & \\overline{x} \\in Z \\end{matrix} \\right. $$ where $*$ denotes a singleton set."}
+{"_id": "6462", "title": "etale-cohomology-lemma-monomorphism-big-push-pull", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $f : X \\to Y$ be a monomorphism of schemes. Then the canonical map $f_{big}^{-1}f_{big, *}\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism for any sheaf $\\mathcal{F}$ on $(\\Sch/X)_\\tau$."}
+{"_id": "6463", "title": "etale-cohomology-lemma-closed-immersion-cover-from-below", "text": "Let $f : X \\to Y$ be a closed immersion of schemes. Let $U \\to X$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) morphism. Then there exist syntomic (resp.\\ smooth, resp.\\ \\'etale) morphisms $V_i \\to Y$ and morphisms $V_i \\times_Y X \\to U$ such that $\\{V_i \\times_Y X \\to U\\}$ is a Zariski covering of $U$."}
+{"_id": "6464", "title": "etale-cohomology-lemma-prepare-closed-immersion-almost-cocontinuous", "text": "Let $f : X \\to Y$ be a closed immersion of schemes. Let $\\{U_i \\to X\\}$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering. There exists a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering $\\{V_j \\to Y\\}$ such that for each $j$, either $V_j \\times_Y X = \\emptyset$, or the morphism $V_j \\times_Y X \\to X$ factors through $U_i$ for some $i$."}
+{"_id": "6465", "title": "etale-cohomology-lemma-closed-immersion-almost-cocontinuous", "text": "Let $f : X \\to Y$ be a closed immersion of schemes. Let $\\tau \\in \\{syntomic, smooth, \\etale\\}$. The functor $V \\mapsto X \\times_Y V$ defines an almost cocontinuous functor (see Sites, Definition \\ref{sites-definition-almost-cocontinuous}) $(\\Sch/Y)_\\tau \\to (\\Sch/X)_\\tau$ between big $\\tau$ sites."}
+{"_id": "6466", "title": "etale-cohomology-lemma-closed-immersion-pushforward-exact", "text": "Let $f : X \\to Y$ be a closed immersion of schemes. Let $\\tau \\in \\{syntomic, smooth, \\etale\\}$. \\begin{enumerate} \\item The pushforward $f_{big, *} : \\Sh((\\Sch/X)_\\tau) \\to \\Sh((\\Sch/Y)_\\tau)$ commutes with coequalizers and pushouts. \\item The pushforward $f_{big, *} : \\textit{Ab}((\\Sch/X)_\\tau) \\to \\textit{Ab}((\\Sch/Y)_\\tau)$ is exact. \\end{enumerate}"}
+{"_id": "6468", "title": "etale-cohomology-lemma-compare-structure-sheaves", "text": "Let $X$ be a scheme. Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $\\mathcal{C}_1 \\subset \\mathcal{C}_2 \\subset (\\Sch/X)_\\tau$ be full subcategories with the following properties: \\begin{enumerate} \\item For an object $U/X$ of $\\mathcal{C}_t$, \\begin{enumerate} \\item if $\\{U_i \\to U\\}$ is a covering of $(\\Sch/X)_\\tau$, then $U_i/X$ is an object of $\\mathcal{C}_t$, \\item $U \\times \\mathbf{A}^1/X$ is an object of $\\mathcal{C}_t$. \\end{enumerate} \\item $X/X$ is an object of $\\mathcal{C}_t$. \\end{enumerate} We endow $\\mathcal{C}_t$ with the structure of a site whose coverings are exactly those coverings $\\{U_i \\to U\\}$ of $(\\Sch/X)_\\tau$ with $U \\in \\Ob(\\mathcal{C}_t)$. Then \\begin{enumerate} \\item[(a)] The functor $\\mathcal{C}_1 \\to \\mathcal{C}_2$ is fully faithful, continuous, and cocontinuous. \\end{enumerate} Denote $g : \\Sh(\\mathcal{C}_1) \\to \\Sh(\\mathcal{C}_2)$ the corresponding morphism of topoi. Denote $\\mathcal{O}_t$ the restriction of $\\mathcal{O}$ to $\\mathcal{C}_t$. Denote $g_!$ the functor of Modules on Sites, Definition \\ref{sites-modules-definition-g-shriek}. \\begin{enumerate} \\item[(b)] The canonical map $g_!\\mathcal{O}_1 \\to \\mathcal{O}_2$ is an isomorphism. \\end{enumerate}"}
+{"_id": "6469", "title": "etale-cohomology-lemma-mayer-vietoris", "text": "Let $X$ be a scheme. Suppose that $X = U \\cup V$ is a union of two opens. For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$ there exists a long exact cohomology sequence $$ \\begin{matrix} 0 \\to H^0_\\etale(X, \\mathcal{F}) \\to H^0_\\etale(U, \\mathcal{F}) \\oplus H^0_\\etale(V, \\mathcal{F}) \\to H^0_\\etale(U \\cap V, \\mathcal{F}) \\phantom{\\to \\ldots} \\\\ \\phantom{0} \\to H^1_\\etale(X, \\mathcal{F}) \\to H^1_\\etale(U, \\mathcal{F}) \\oplus H^1_\\etale(V, \\mathcal{F}) \\to H^1_\\etale(U \\cap V, \\mathcal{F}) \\to \\ldots \\end{matrix} $$ This long exact sequence is functorial in $\\mathcal{F}$."}
+{"_id": "6470", "title": "etale-cohomology-lemma-relative-mayer-vietoris", "text": "Let $f : X \\to Y$ be a morphism of schemes. Suppose that $X = U \\cup V$ is a union of two open subschemes. Denote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and $c = f|_{U \\cap V} : U \\cap V \\to Y$. For every abelian sheaf $\\mathcal{F}$ on $X_\\etale$ there exists a long exact sequence $$ 0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to c_*(\\mathcal{F}|_{U \\cap V}) \\to R^1f_*\\mathcal{F} \\to \\ldots $$ on $Y_\\etale$. This long exact sequence is functorial in $\\mathcal{F}$."}
+{"_id": "6471", "title": "etale-cohomology-lemma-colimit-affine-sites", "text": "Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse system of schemes over $I$ with affine transition morphisms. Let $X = \\lim_{i \\in I} X_i$. With notation as in Lemma \\ref{lemma-alternative} we have $$ X_{affine, \\etale} = \\colim (X_i)_{affine, \\etale} $$ as sites in the sense of Sites, Lemma \\ref{sites-lemma-colimit-sites}."}
+{"_id": "6472", "title": "etale-cohomology-lemma-colimit", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$ be a directed set. Let $(\\mathcal{F}_i, \\varphi_{ij})$ be a system of abelian sheaves on $X_\\etale$ over $I$. Then $$ \\colim_{i\\in I} H_\\etale^p(X, \\mathcal{F}_i) = H_\\etale^p(X, \\colim_{i\\in I} \\mathcal{F}_i). $$"}
+{"_id": "6473", "title": "etale-cohomology-lemma-directed-colimit-cohomology", "text": "Let $A$ be a ring, $(I, \\leq)$ a directed set and $(B_i, \\varphi_{ij})$ a system of $A$-algebras. Set $B = \\colim_{i\\in I} B_i$. Let $X \\to \\Spec(A)$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F}$ an abelian sheaf on $X_\\etale$. Denote $Y_i = X \\times_{\\Spec(A)} \\Spec(B_i)$, $Y = X \\times_{\\Spec(A)} \\Spec(B)$, $\\mathcal{G}_i = (Y_i \\to X)^{-1}\\mathcal{F}$ and $\\mathcal{G} = (Y \\to X)^{-1}\\mathcal{F}$. Then $$ H_\\etale^p(Y, \\mathcal{G}) = \\colim_{i\\in I} H_\\etale^p (Y_i, \\mathcal{G}_i). $$"}
+{"_id": "6474", "title": "etale-cohomology-lemma-higher-direct-images", "text": "Let $f: X\\to Y$ be a morphism of schemes and $\\mathcal{F}\\in \\textit{Ab}(X_\\etale)$. Then $R^pf_*\\mathcal{F}$ is the sheaf associated to the presheaf $$ (V \\to Y) \\longmapsto H_\\etale^p(X \\times_Y V, \\mathcal{F}|_{X \\times_Y V}). $$"}
+{"_id": "6475", "title": "etale-cohomology-lemma-relative-colimit", "text": "Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes over $S$ with affine transition morphisms $f_{i'i} : X_{i'} \\to X_i$. We assume the structure morphisms $g_i : X_i \\to S$ and $g : X \\to S$ are quasi-compact and quasi-separated. Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves on $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then $$ \\colim_{i\\in I} R^p g_{i, *} \\mathcal{F}_i = R^p g_* \\mathcal{F} $$ for all $p \\geq 0$."}
+{"_id": "6476", "title": "etale-cohomology-lemma-relative-colimit-general", "text": "Let $I$ be a directed set. Let $g_i : X_i \\to S_i$ be an inverse system of morphisms of schemes over $I$. Assume $g_i$ is quasi-compact and quasi-separated and for $i' \\geq i$ the transition morphisms $f_{i'i} : X_{i'} \\to X_i$ and $h_{i'i} : S_{i'} \\to S_i$ are affine. Let $g : X \\to S$ be the limit of the morphisms $g_i$, see Limits, Section \\ref{limits-section-limits}. Denote $f_i : X \\to X_i$ and $h_i : S \\to S_i$ the projections. Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves on $(X_i, f_{i'i})$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then $$ R^p g_* \\mathcal{F} = \\colim_{i \\in I} h_i^{-1}R^p g_{i, *} \\mathcal{F}_i $$ for all $p \\geq 0$."}
+{"_id": "6477", "title": "etale-cohomology-lemma-linus-hamann", "text": "Let $X = \\lim_{i \\in I} X_i$ be a directed limit of schemes with affine transition morphisms $f_{i'i}$ and projection morphisms $f_i : X \\to X_i$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then \\begin{enumerate} \\item there are canonical maps $\\varphi_{i'i} : f_{i'i}^{-1}f_{i, *}\\mathcal{F} \\to f_{i', *}\\mathcal{F}$ such that $(f_{i, *}\\mathcal{F}, \\varphi_{i'i})$ is a system of sheaves on $(X_i, f_{i'i})$ as in Definition \\ref{definition-inverse-system-sheaves}, and \\item $\\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "6479", "title": "etale-cohomology-lemma-prepare-leray", "text": "Let $f: X \\to Y$ be a morphism and $\\mathcal{I}$ an injective object of $\\textit{Ab}(X_\\etale)$. Let $V \\in \\Ob(Y_\\etale)$. Then \\begin{enumerate} \\item for any covering $\\mathcal{V} = \\{V_j\\to V\\}_{j \\in J}$ we have $\\check H^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$ for all $p > 0$, \\item $f_*\\mathcal{I}$ is acyclic for the functor $\\Gamma(V, -)$, and \\item if $g : Y \\to Z$, then $f_*\\mathcal{I}$ is acyclic for $g_*$. \\end{enumerate}"}
+{"_id": "6480", "title": "etale-cohomology-lemma-vanishing-etale-cohomology-strictly-henselian", "text": "Let $R$ be a strictly henselian local ring. Set $S = \\Spec(R)$ and let $\\overline{s}$ be its closed point. Then the global sections functor $\\Gamma(S, -) : \\textit{Ab}(S_\\etale) \\to \\textit{Ab}$ is exact. In fact we have $\\Gamma(S, \\mathcal{F}) = \\mathcal{F}_{\\overline{s}}$ for any sheaf of sets $\\mathcal{F}$. In particular $$ \\forall p\\geq 1, \\quad H_\\etale^p(S, \\mathcal{F})=0 $$ for all $\\mathcal{F}\\in \\textit{Ab}(S_\\etale)$."}
+{"_id": "6481", "title": "etale-cohomology-lemma-finite-pushforward-commutes-with-base-change", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes with $f$ a finite morphism. For any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have $f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$."}
+{"_id": "6482", "title": "etale-cohomology-lemma-integral-pushforward-commutes-with-base-change", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes with $f$ an integral morphism. For any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have $f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$."}
+{"_id": "6483", "title": "etale-cohomology-lemma-cohomological-descent-finite", "text": "Let $f : X \\to Y$ be a surjective finite morphism of schemes. Set $f_n : X_n \\to Y$ equal to the $(n + 1)$-fold fibre product of $X$ over $Y$. For $\\mathcal{F} \\in \\textit{Ab}(Y_\\etale)$ set $\\mathcal{F}_n = f_{n, *}f_n^{-1}\\mathcal{F}$. There is an exact sequence $$ 0 \\to \\mathcal{F} \\to \\mathcal{F}_0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\ldots $$ on $X_\\etale$. Moreover, there is a spectral sequence $$ E_1^{p, q} = H^q_\\etale(X_p, f_p^{-1}\\mathcal{F}) $$ converging to $H^{p + q}(Y_\\etale, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$."}
+{"_id": "6484", "title": "etale-cohomology-lemma-global-sections-point", "text": "Assumptions and notations as in Theorem \\ref{theorem-equivalence-sheaves-point}. There is a functorial bijection $$ \\Gamma(S, \\mathcal{F}) = (\\mathcal{F}_{\\overline{s}})^G $$"}
+{"_id": "6486", "title": "etale-cohomology-lemma-ext-modules-hom", "text": "Let $G$ be a topological group. Let $R$ be a ring. Let $M$, $N$ be $R\\text{-}G$-modules. If $M$ is finite projective as an $R$-module, then $\\text{Ext}^i(M, N) = H^i(G, M^\\vee \\otimes_R N)$ (for notation see proof)."}
+{"_id": "6487", "title": "etale-cohomology-lemma-finite-dim-group-cohomology", "text": "Let $G$ be a topological group. Let $k$ be a field. Let $V$ be a $k\\text{-}G$-module. If $G$ is topologically finitely generated and $\\dim_k(V) < \\infty$, then $\\dim_k H^1(G, V) < \\infty$."}
+{"_id": "6488", "title": "etale-cohomology-lemma-profinite-group-cohomology-torsion", "text": "Let $G$ be a profinite topological group. Then \\begin{enumerate} \\item $H^i(G, M)$ is torsion for $i > 0$ and any $G$-module $M$, and \\item $H^i(G, M) = 0$ if $M$ is a $\\mathbf{Q}$-vector space. \\end{enumerate}"}
+{"_id": "6489", "title": "etale-cohomology-lemma-equivalence-abelian-sheaves-point", "text": "Let $S = \\Spec(K)$ with $K$ a field. Let $\\overline{s}$ be a geometric point of $S$. Let $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group. The stalk functor induces an equivalence of categories $$ \\textit{Ab}(S_\\etale) \\longrightarrow \\text{Mod}_G, \\quad \\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}. $$"}
+{"_id": "6490", "title": "etale-cohomology-lemma-compare-cohomology-point", "text": "Notation and assumptions as in Lemma \\ref{lemma-equivalence-abelian-sheaves-point}. Let $\\mathcal{F}$ be an abelian sheaf on $\\Spec(K)_\\etale$ which corresponds to the $G$-module $M$. Then \\begin{enumerate} \\item in $D(\\textit{Ab})$ we have a canonical isomorphism $R\\Gamma(S, \\mathcal{F}) = R\\Gamma_G(M)$, \\item $H_\\etale^0(S, \\mathcal{F}) = M^G$, and \\item $H_\\etale^q(S, \\mathcal{F}) = H^q(G, M)$. \\end{enumerate}"}
+{"_id": "6494", "title": "etale-cohomology-lemma-end-unique-up-to-invertible", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ and $\\mathcal{G}$ be finite locally free sheaves of $\\mathcal{O}_S$-modules of positive rank. If there exists an isomorphism $\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{F}) \\cong \\SheafHom_{\\mathcal{O}_S}(\\mathcal{G}, \\mathcal{G})$ of $\\mathcal{O}_S$-algebras, then there exists an invertible sheaf $\\mathcal{L}$ on $S$ such that $\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{L} \\cong \\mathcal{G}$ and such that this isomorphism induces the given isomorphism of endomorphism algebras."}
+{"_id": "6496", "title": "etale-cohomology-lemma-vanishing-affine-char-p-p", "text": "Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$. \\begin{enumerate} \\item If $S$ is affine, then $H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for all $q \\geq 2$. \\item If $S$ is a quasi-compact and quasi-separated scheme of dimension $d$, then $H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for all $q \\geq 2 + d$. \\end{enumerate}"}
+{"_id": "6497", "title": "etale-cohomology-lemma-F-1", "text": "Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional $k$-vector space. Let $F : V \\to V$ be a frobenius linear map, i.e., an additive map such that $F(\\lambda v) = \\lambda^p F(v)$ for all $\\lambda \\in k$ and $v \\in V$. Then $F - 1 : V \\to V$ is surjective with kernel a finite dimensional $\\mathbf{F}_p$-vector space of dimension $\\leq \\dim_k(V)$."}
+{"_id": "6498", "title": "etale-cohomology-lemma-top-cohomology-coherent", "text": "Let $X$ be a separated scheme of finite type over a field $k$. Let $\\mathcal{F}$ be a coherent sheaf of $\\mathcal{O}_X$-modules. Then $\\dim_k H^d(X, \\mathcal{F}) < \\infty$ where $d = \\dim(X)$."}
+{"_id": "6499", "title": "etale-cohomology-lemma-vanishing-variety-char-p-p", "text": "Let $X$ be separated of finite type over an algebraically closed field $k$ of characteristic $p > 0$. Then $H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for $q \\geq dim(X) + 1$."}
+{"_id": "6500", "title": "etale-cohomology-lemma-finiteness-proper-variety-char-p-p", "text": "Let $X$ be a proper scheme over an algebraically closed field $k$ of characteristic $p > 0$. Then \\begin{enumerate} \\item $H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}})$ is a finite $\\mathbf{Z}/p\\mathbf{Z}$-module for all $q$, and \\item $H^q_\\etale(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) \\to H^q_\\etale(X_{k'}, \\underline{\\mathbf{Z}/p\\mathbf{Z}}))$ is an isomorphism if $k \\subset k'$ is an extension of algebraically closed fields. \\end{enumerate}"}
+{"_id": "6501", "title": "etale-cohomology-lemma-pullback-locally-constant", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{G}$ is a locally constant sheaf of sets, abelian groups, or $\\Lambda$-modules on $Y_\\etale$, the same is true for $f^{-1}\\mathcal{G}$ on $X_\\etale$."}
+{"_id": "6502", "title": "etale-cohomology-lemma-pushforward-locally-constant", "text": "Let $f : X \\to Y$ be a finite \\'etale morphism of schemes. If $\\mathcal{F}$ is a (finite) locally constant sheaf of sets, (finite) locally constant sheaf of abelian groups, or (finite type) locally constant sheaf of $\\Lambda$-modules on $X_\\etale$, the same is true for $f_*\\mathcal{F}$ on $Y_\\etale$."}
+{"_id": "6503", "title": "etale-cohomology-lemma-characterize-finite-locally-constant", "text": "Let $X$ be a scheme and $\\mathcal{F}$ a sheaf of sets on $X_\\etale$. Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is finite locally constant, and \\item $\\mathcal{F} = h_U$ for some finite \\'etale morphism $U \\to X$. \\end{enumerate}"}
+{"_id": "6504", "title": "etale-cohomology-lemma-morphism-locally-constant", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of sets on $X_\\etale$. If $\\mathcal{F}$ is finite locally constant, there exists an \\'etale covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant sheaves associated to a map of sets. \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of abelian groups on $X_\\etale$. If $\\mathcal{F}$ is finite locally constant, there exists an \\'etale covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant abelian sheaves associated to a map of abelian groups. \\item Let $\\Lambda$ be a ring. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of $\\Lambda$-modules on $X_\\etale$. If $\\mathcal{F}$ is of finite type, then there exists an \\'etale covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant sheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules. \\end{enumerate}"}
+{"_id": "6505", "title": "etale-cohomology-lemma-kernel-finite-locally-constant", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside $\\Sh(X_\\etale)$. \\item The category of finite locally constant abelian sheaves is a weak Serre subcategory of $\\textit{Ab}(X_\\etale)$. \\item Let $\\Lambda$ be a Noetherian ring. The category of finite type, locally constant sheaves of $\\Lambda$-modules on $X_\\etale$ is a weak Serre subcategory of $\\textit{Mod}(X_\\etale, \\Lambda)$. \\end{enumerate}"}
+{"_id": "6506", "title": "etale-cohomology-lemma-tensor-product-locally-constant", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\\Lambda$-modules on $X_\\etale$ is a locally constant sheaf of $\\Lambda$-modules."}
+{"_id": "6507", "title": "etale-cohomology-lemma-connected-locally-constant", "text": "Let $X$ be a connected scheme. Let $\\Lambda$ be a ring and let $\\mathcal{F}$ be a locally constant sheaf of $\\Lambda$-modules. Then there exists a $\\Lambda$-module $M$ and an \\'etale covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i} \\cong \\underline{M}|_{U_i}$."}
+{"_id": "6508", "title": "etale-cohomology-lemma-locally-constant-on-connected", "text": "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point of $X$. \\begin{enumerate} \\item There is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{finite locally constant}\\\\ \\text{sheaves of sets on }X_\\etale \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{finite }\\pi_1(X, \\overline{x})\\text{-sets} \\end{matrix} \\right\\} $$ \\item There is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{finite locally constant}\\\\ \\text{sheaves of abelian groups on }X_\\etale \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{finite }\\pi_1(X, \\overline{x})\\text{-modules} \\end{matrix} \\right\\} $$ \\item Let $\\Lambda$ be a finite ring. There is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{finite type, locally constant}\\\\ \\text{sheaves of }\\Lambda\\text{-modules on }X_\\etale \\end{matrix} \\right\\} \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{finite }\\pi_1(X, \\overline{x})\\text{-modules endowed}\\\\ \\text{with commuting }\\Lambda\\text{-module structure} \\end{matrix} \\right\\} $$ \\end{enumerate}"}
+{"_id": "6509", "title": "etale-cohomology-lemma-pullback-filtered", "text": "Let $S$ be a connected scheme. Let $\\ell$ be a prime number. Let $\\mathcal{F}$ be a finite type, locally constant sheaf of $\\mathbf{F}_\\ell$-vector spaces on $S_\\etale$. Then there exists a finite \\'etale morphism $f : T \\to S$ of degree prime to $\\ell$ such that $f^{-1}\\mathcal{F}$ has a finite filtration whose successive quotients are $\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_T$."}
+{"_id": "6510", "title": "etale-cohomology-lemma-nonvanishing-inherited", "text": "Let $\\ell$ be a prime number and $n$ an integer $> 0$. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed system of $S$-schemes each $X_i \\to S$ being finite \\'etale of constant degree relatively prime to $\\ell$. The following are equivalent: \\begin{enumerate} \\item there exists an $\\ell$-power torsion sheaf $\\mathcal{G}$ on $S$ such that $H_\\etale^n(S, \\mathcal{G}) \\neq 0$ and \\item there exists an $\\ell$-power torsion sheaf $\\mathcal{F}$ on $X$ such that $H_\\etale^n(X, \\mathcal{F}) \\neq 0$. \\end{enumerate} In fact, given $\\mathcal{G}$ we can take $\\mathcal{F} = g^{-1}\\mathcal{F}$ and given $\\mathcal{F}$ we can take $\\mathcal{G} = g_*\\mathcal{F}$."}
+{"_id": "6511", "title": "etale-cohomology-lemma-reduce-to-l-group", "text": "Let $\\ell$ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \\subset G$ be a maximal pro-$\\ell$ subgroup with $L/K$ being the corresponding field extension. Then $H^n_\\etale(\\Spec(K), \\mathcal{F}) = 0$ for all $\\ell$-power torsion $\\mathcal{F}$ if and only if $H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$."}
+{"_id": "6512", "title": "etale-cohomology-lemma-reduce-to-l-group-higher", "text": "Let $\\ell$ be a prime number and $n$ an integer $> 0$. Let $K$ be a field with $G = Gal(K^{sep}/K)$ and let $H \\subset G$ be a maximal pro-$\\ell$ subgroup  with $L/K$ being the corresponding field extension. Then $H^q_\\etale(\\Spec(K),\\mathcal{F}) = 0$ for $q \\geq n$ and all $\\ell$-torsion sheaves $\\mathcal{F}$ if  and only if $H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$."}
+{"_id": "6513", "title": "etale-cohomology-lemma-algebraically-closed-find-solutions", "text": "Let $k$ be an algebraically closed field. Let $f_1, \\ldots, f_s \\in k[T_1, \\ldots, T_n]$ be homogeneous polynomials of degree $d_1, \\ldots, d_s$ with $d_i > 0$. If $s < n$, then $f_1 = \\ldots = f_s = 0$ have a common nontrivial solution."}
+{"_id": "6514", "title": "etale-cohomology-lemma-curve-brauer-zero", "text": "Let $C$ be a curve over an algebraically closed field $k$. Then the Brauer group of the function field of $C$ is zero: $\\text{Br}(k(C)) = 0$."}
+{"_id": "6515", "title": "etale-cohomology-lemma-cohomology-Gm-function-field-curve", "text": "Let $k$ be an algebraically closed field and $k \\subset K$ a field extension of transcendence degree 1. Then for all $q \\geq 1$, $H_\\etale^q(\\Spec(K), \\mathbf{G}_m) = 0$."}
+{"_id": "6519", "title": "etale-cohomology-lemma-cohomology-smooth-projective-curve", "text": "Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$. Then there are canonical identifications $$ H_\\etale^q(X, \\mu_n) = \\left\\{ \\begin{matrix} \\mu_n(k) & \\text{ if }q = 0, \\\\ \\Pic^0(X)[n] & \\text{ if }q = 1, \\\\ \\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\ 0 & \\text{ if }q \\geq 3. \\end{matrix} \\right. $$ Since $\\mu_n \\cong \\underline{\\mathbf{Z}/n\\mathbf{Z}}$, this gives (noncanonical) identifications $$ H_\\etale^q(X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\cong \\left\\{ \\begin{matrix} \\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 0, \\\\ (\\mathbf{Z}/n\\mathbf{Z})^{2g} & \\text{ if }q = 1, \\\\ \\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\ 0 & \\text{ if }q \\geq 3. \\end{matrix} \\right. $$"}
+{"_id": "6521", "title": "etale-cohomology-lemma-vanishing-cohomology-mu-smooth-curve", "text": "Let $X$ be an affine smooth curve over an algebraically closed field $k$ and $n\\in k^*$. Then \\begin{enumerate} \\item $H_\\etale^0(X, \\mu_n) = \\mu_n(k)$; \\item $H_\\etale^1(X, \\mu_n) \\cong \\left(\\mathbf{Z}/n\\mathbf{Z}\\right)^{2g+r-1}$, where $r$ is the number of points in $\\bar X - X$ for some smooth projective compactification $\\bar X$ of $X$, and \\item for all $q\\geq 2$, $H_\\etale^q(X, \\mu_n) = 0$. \\end{enumerate}"}
+{"_id": "6522", "title": "etale-cohomology-lemma-jshriek-open", "text": "Let $j : U \\to X$ be an open immersion of schemes. For any abelian sheaf $\\mathcal{F}$ on $U_\\etale$, the adjunction mappings $j^{-1}j_*\\mathcal{F} \\to \\mathcal{F}$ and $\\mathcal{F} \\to j^{-1}j_!\\mathcal{F}$ are isomorphisms. In fact, $j_!\\mathcal{F}$ is the unique abelian sheaf on $X_\\etale$ whose restriction to $U$ is $\\mathcal{F}$ and whose stalks at geometric points of $X \\setminus U$ are zero."}
+{"_id": "6523", "title": "etale-cohomology-lemma-shriek-base-change", "text": "Let $f: Y \\to X$ be a morphism of schemes. Let $j: V \\to X$ be an \\'etale morphism. Consider the fibre product $$ \\xymatrix{ V' = Y \\times_X V \\ar[d]_{f'} \\ar[r]_-{j'} & Y \\ar[d]^f \\\\ V \\ar[r]^j & X } $$ Then we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on sheaves of modules."}
+{"_id": "6524", "title": "etale-cohomology-lemma-shriek-into-star-separated-etale", "text": "Let $j : U \\to X$ be separated and \\'etale. Then there is a functorial injective map $j_!\\mathcal{F} \\to j_*\\mathcal{F}$ on abelian sheaves and sheaves of $\\Lambda$-modules."}
+{"_id": "6525", "title": "etale-cohomology-lemma-shriek-equals-star-finite-etale", "text": "Let $j : U \\to X$ be finite and \\'etale. Then the map $j_! \\to j_*$ of Lemma \\ref{lemma-shriek-into-star-separated-etale} is an isomorphism on abelian sheaves and sheaves of $\\Lambda$-modules."}
+{"_id": "6526", "title": "etale-cohomology-lemma-ses-associated-to-open", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let $U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$ the inclusion morphisms. For every abelian sheaf $\\mathcal{F}$ on $X_\\etale$ there is a canonical short exact sequence $$ 0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0 $$ on $X_\\etale$."}
+{"_id": "6527", "title": "etale-cohomology-lemma-constructible-quasi-compact-quasi-separated", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is constructible, \\item there exists an open covering $X = \\bigcup U_i$ such that $\\mathcal{F}|_{U_i}$ is constructible, and \\item there exists a partition $X = \\bigcup X_i$ by constructible locally closed subschemes such that $\\mathcal{F}|_{X_i}$ is finite locally constant. \\end{enumerate} A similar statement holds for abelian sheaves and sheaves of $\\Lambda$-modules if $\\Lambda$ is Noetherian."}
+{"_id": "6529", "title": "etale-cohomology-lemma-constructible-local", "text": "Let $X$ be a scheme. Checking constructibility of a sheaf of sets, abelian groups, $\\Lambda$-modules (with $\\Lambda$ Noetherian) can be done Zariski locally on $X$."}
+{"_id": "6530", "title": "etale-cohomology-lemma-pullback-constructible", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\\Lambda$-modules (with $\\Lambda$ Noetherian) on $Y_\\etale$, the same is true for $f^{-1}\\mathcal{F}$ on $X_\\etale$."}
+{"_id": "6531", "title": "etale-cohomology-lemma-constructible-abelian", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The category of constructible sheaves of sets is closed under finite limits and colimits inside $\\Sh(X_\\etale)$. \\item The category of constructible abelian sheaves is a weak Serre subcategory of $\\textit{Ab}(X_\\etale)$. \\item Let $\\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\\Lambda$-modules on $X_\\etale$ is a weak Serre subcategory of $\\textit{Mod}(X_\\etale, \\Lambda)$. \\end{enumerate}"}
+{"_id": "6532", "title": "etale-cohomology-lemma-tensor-product-constructible", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. The tensor product of two constructible sheaves of $\\Lambda$-modules on $X_\\etale$ is a constructible sheaf of $\\Lambda$-modules."}
+{"_id": "6533", "title": "etale-cohomology-lemma-support-constructible", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. \\begin{enumerate} \\item Let $\\mathcal{F} \\to \\mathcal{G}$ be a map of constructible sheaves of sets on $X_\\etale$. Then the set of points $x \\in X$ where $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$ is surjective, resp.\\ injective, resp.\\ is isomorphic to a given map of sets, is constructible in $X$. \\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$. The support of $\\mathcal{F}$ is constructible. \\item Let $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$. The support of $\\mathcal{F}$ is constructible. \\end{enumerate}"}
+{"_id": "6534", "title": "etale-cohomology-lemma-colimit-constructible", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$ be a filtered colimit of sheaves of sets, abelian sheaves, or sheaves of modules. \\begin{enumerate} \\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of sets, then the ind-object $\\mathcal{F}_i$ is essentially constant with value $\\mathcal{F}$. \\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of abelian groups, then the ind-object $\\mathcal{F}_i$ is essentially constant with value $\\mathcal{F}$. \\item Let $\\Lambda$ be a Noetherian ring. If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of $\\Lambda$-modules, then the ind-object $\\mathcal{F}_i$ is essentially constant with value $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "6535", "title": "etale-cohomology-lemma-etale-stratified-finite", "text": "Let $U \\to X$ be an \\'etale morphism of quasi-compact and quasi-separated schemes (for example an \\'etale morphism of Noetherian schemes). Then there exists a partition $X = \\coprod_i X_i$ by constructible locally closed subschemes such that $X_i \\times_X U \\to X_i$ is finite \\'etale for all $i$."}
+{"_id": "6536", "title": "etale-cohomology-lemma-generically-finite", "text": "Let $f: X \\to Y$ be a morphism of schemes which is quasi-compact, quasi-separated, and locally of finite type. If $\\eta$ is a generic point of an irreducible component of $Y$ such that $f^{-1}(\\eta)$ is finite, then there exists an open $V \\subset Y$ containing $\\eta$ such that $f^{-1}(V) \\to V$ is finite."}
+{"_id": "6537", "title": "etale-cohomology-lemma-decompose-quasi-finite-morphism", "text": "Let $f : Y \\to X$ be a quasi-finite and finitely presented morphism of affine schemes. \\begin{enumerate} \\item There exists a surjective morphism of affine schemes $X' \\to X$ and a closed subscheme $Z' \\subset Y' = X' \\times_X Y$ such that \\begin{enumerate} \\item $Z' \\subset Y'$ is a thickening, and \\item $Z' \\to X'$ is a finite \\'etale morphism. \\end{enumerate} \\item There exists a finite partition $X = \\coprod X_i$ by locally closed, constructible, affine strata, and surjective finite locally free morphisms $X'_i \\to X_i$ such that the reduction of $Y'_i = X'_i \\times_X Y \\to X'_i$ is isomorphic to $\\coprod_{j = 1}^{n_i} (X'_i)_{red} \\to (X'_i)_{red}$ for some $n_i$. \\end{enumerate}"}
+{"_id": "6538", "title": "etale-cohomology-lemma-jshriek-constructible", "text": "Let $j : U \\to X$ be an \\'etale morphism of quasi-compact and quasi-separated schemes. \\begin{enumerate} \\item The sheaf $h_U$ is a constructible sheaf of sets. \\item The sheaf $j_!\\underline{M}$ is a constructible abelian sheaf for a finite abelian group $M$. \\item If $\\Lambda$ is a Noetherian ring and $M$ is a finite $\\Lambda$-module, then $j_!\\underline{M}$ is a constructible sheaf of $\\Lambda$-modules on $X_\\etale$. \\end{enumerate}"}
+{"_id": "6539", "title": "etale-cohomology-lemma-torsion-colimit-constructible", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. Then $\\mathcal{F}$ is a filtered colimit of constructible sheaves of sets. \\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$. Then $\\mathcal{F}$ is a filtered colimit of constructible abelian sheaves. \\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf of $\\Lambda$-modules on $X_\\etale$. Then $\\mathcal{F}$ is a filtered colimit of constructible sheaves of $\\Lambda$-modules. \\end{enumerate}"}
+{"_id": "6540", "title": "etale-cohomology-lemma-check-constructible", "text": "Let $f : X \\to Y$ be a surjective morphism of quasi-compact and quasi-separated schemes. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$. Then $\\mathcal{F}$ is constructible if and only if $f^{-1}\\mathcal{F}$ is constructible. \\item Let $\\mathcal{F}$ be an abelian sheaf on $Y_\\etale$. Then $\\mathcal{F}$ is constructible if and only if $f^{-1}\\mathcal{F}$ is constructible. \\item Let $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be sheaf of $\\Lambda$-modules on $Y_\\etale$. Then $\\mathcal{F}$ is constructible if and only if $f^{-1}\\mathcal{F}$ is constructible. \\end{enumerate}"}
+{"_id": "6541", "title": "etale-cohomology-lemma-pushforward-constructible", "text": "Let $f : X \\to Y$ be a finite \\'etale morphism of schemes. Let $\\Lambda$ be a Noetherian ring. If $\\mathcal{F}$ is a constructible sheaf of sets, constructible sheaf of abelian groups, or constructible sheaf of $\\Lambda$-modules on $X_\\etale$, the same is true for $f_*\\mathcal{F}$ on $Y_\\etale$."}
+{"_id": "6542", "title": "etale-cohomology-lemma-category-constructible-sets", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible sheaves of sets is the full subcategory of $\\Sh(X_\\etale)$ consisting of sheaves $\\mathcal{F}$ which are coequalizers $$ \\xymatrix{ \\mathcal{F}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\mathcal{F}_0 \\ar[r] & \\mathcal{F}} $$ such that $\\mathcal{F}_i$, $i = 0, 1$ is a finite coproduct of sheaves of the form $h_U$ with $U$ a quasi-compact and quasi-separated object of $X_\\etale$."}
+{"_id": "6543", "title": "etale-cohomology-lemma-category-constructible-modules", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\\Lambda$-modules is exactly the category of modules of the form $$ \\Coker\\left( \\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\underline{\\Lambda}_{V_j} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\Lambda}_{U_i} \\right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\\etale$. In fact, we can even assume $U_i$ and $V_j$ affine."}
+{"_id": "6544", "title": "etale-cohomology-lemma-category-constructible-abelian", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The category of constructible abelian sheaves is exactly the category of abelian sheaves of the form $$ \\Coker\\left( \\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\underline{\\mathbf{Z}/m_j\\mathbf{Z}}_{V_j} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\mathbf{Z}/n_i\\mathbf{Z}}_{U_i} \\right) $$ with $V_j$ and $U_i$ quasi-compact and quasi-separated objects of $X_\\etale$ and $m_j$, $n_i$ positive integers. In fact, we can even assume $U_i$ and $V_j$ affine."}
+{"_id": "6545", "title": "etale-cohomology-lemma-constructible-is-compact", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of sets, abelian groups, or $\\Lambda$-modules on $X_\\etale$. Let $\\mathcal{G} = \\colim \\mathcal{G}_i$ be a filtered colimit of sheaves of sets, abelian groups, or $\\Lambda$-modules. Then $$ \\Mor(\\mathcal{F}, \\mathcal{G}) = \\colim \\Mor(\\mathcal{F}, \\mathcal{G}_i) $$ in the category of sheaves of sets, abelian groups, or $\\Lambda$-modules on $X_\\etale$."}
+{"_id": "6546", "title": "etale-cohomology-lemma-finite-pushforward-constructible", "text": "Let $f : X \\to Y$ be a finite and finitely presented morphism of schemes. Let $\\Lambda$ be a Noetherian ring. If $\\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\\Lambda$-modules on $X_\\etale$, then $f_*\\mathcal{F}$ is too."}
+{"_id": "6547", "title": "etale-cohomology-lemma-category-is-colimit", "text": "Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes with affine transition morphisms. We assume that $X_i$ is quasi-compact and quasi-separated for all $i \\in I$. \\begin{enumerate} \\item The category of constructible sheaves of sets on $X_\\etale$ is the colimit of the categories of constructible sheaves of sets on $(X_i)_\\etale$. \\item The category of constructible abelian sheaves on $X_\\etale$ is the colimit of the categories of constructible abelian sheaves on $(X_i)_\\etale$. \\item Let $\\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\\Lambda$-modules on $X_\\etale$ is the colimit of the categories of constructible sheaves of $\\Lambda$-modules on $(X_i)_\\etale$. \\end{enumerate}"}
+{"_id": "6548", "title": "etale-cohomology-lemma-irreducible-subsheaf-constant-zero", "text": "Let $X$ be an irreducible scheme with generic point $\\eta$. \\begin{enumerate} \\item Let $S' \\subset S$ be an inclusion of sets. If we have $\\underline{S'} \\subset \\mathcal{G} \\subset \\underline{S}$ in $\\Sh(X_\\etale)$ and $S' = \\mathcal{G}_{\\overline{\\eta}}$, then $\\mathcal{G} = \\underline{S'}$. \\item Let $A' \\subset A$ be an inclusion of abelian groups. If we have $\\underline{A'} \\subset \\mathcal{G} \\subset \\underline{A}$ in $\\textit{Ab}(X_\\etale)$ and $A' = \\mathcal{G}_{\\overline{\\eta}}$, then $\\mathcal{G} = \\underline{A'}$. \\item Let $M' \\subset M$ be an inclusion of modules over a ring $\\Lambda$. If we have $\\underline{M'} \\subset \\mathcal{G} \\subset \\underline{M}$ in $\\textit{Mod}(X_\\etale, \\underline{\\Lambda})$ and $M' = \\mathcal{G}_{\\overline{\\eta}}$, then $\\mathcal{G} = \\underline{M'}$. \\end{enumerate}"}
+{"_id": "6549", "title": "etale-cohomology-lemma-push-constant-sheaf-from-open", "text": "Let $X$ be an integral normal scheme with function field $K$. Let $E$ be a set. \\begin{enumerate} \\item Let $g : \\Spec(K) \\to X$ be the inclusion of the generic point. Then $g_*\\underline{E} = \\underline{E}$. \\item Let $j : U \\to X$ be the inclusion of a nonempty open. Then $j_*\\underline{E} = \\underline{E}$. \\end{enumerate}"}
+{"_id": "6551", "title": "etale-cohomology-lemma-constructible-over-noetherian-noetherian", "text": "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring. Consider inclusions $$ \\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\mathcal{F}_3 \\subset \\ldots \\subset \\mathcal{F} $$ in the category of sheaves of sets, abelian groups, or $\\Lambda$-modules. If $\\mathcal{F}$ is constructible, then for some $n$ we have $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\mathcal{F}_{n + 2} = \\ldots$."}
+{"_id": "6552", "title": "etale-cohomology-lemma-constructible-maps-into-constant", "text": "Let $X$ be a Noetherian scheme. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$. There exist an injective map of sheaves $$ \\mathcal{F} \\longrightarrow \\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i} $$ where $f_i : Y_i \\to X$ is a finite morphism and $E_i$ is a finite set. \\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$. There exist an injective map of abelian sheaves $$ \\mathcal{F} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i} $$ where $f_i : Y_i \\to X$ is a finite morphism and $M_i$ is a finite abelian group. \\item Let $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$. There exist an injective map of sheaves of modules $$ \\mathcal{F} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i} $$ where $f_i : Y_i \\to X$ is a finite morphism and $M_i$ is a finite $\\Lambda$-module. \\end{enumerate} Moreover, we may assume each $Y_i$ is irreducible, reduced, maps onto an irreducible and reduced closed subscheme $Z_i \\subset X$ such that $Y_i \\to Z_i$ is finite \\'etale over a nonempty open of $Z_i$."}
+{"_id": "6553", "title": "etale-cohomology-lemma-constructible-maps-into-constant-general", "text": "\\begin{reference} \\cite[Exposee IX, Proposition 2.14]{SGA4} \\end{reference} Let $X$ be a quasi-compact and quasi-separated scheme. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$. There exist an injective map of sheaves $$ \\mathcal{F} \\longrightarrow \\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i} $$ where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and $E_i$ is a finite set. \\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$. There exist an injective map of abelian sheaves $$ \\mathcal{F} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i} $$ where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and $M_i$ is a finite abelian group. \\item Let $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$. There exist an injective map of sheaves of modules $$ \\mathcal{F} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i} $$ where $f_i : Y_i \\to X$ is a finite and finitely presented morphism and $M_i$ is a finite $\\Lambda$-module. \\end{enumerate}"}
+{"_id": "6554", "title": "etale-cohomology-lemma-support-in-subset", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be a subset closed under specialization. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$ whose support is contained in $E$. Then $\\mathcal{F} = \\colim \\mathcal{F}_i$ is a filtered colimit of constructible abelian sheaves $\\mathcal{F}_i$ such that for each $i$ the support of $\\mathcal{F}_i$ is contained in a closed subset contained in $E$. \\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf of $\\Lambda$-modules on $X_\\etale$ whose support is contained in $E$. Then $\\mathcal{F} = \\colim \\mathcal{F}_i$ is a filtered colimit of constructible sheaves of $\\Lambda$-modules $\\mathcal{F}_i$ such that for each $i$ the support of $\\mathcal{F}_i$ is contained in a closed subset contained in $E$. \\end{enumerate}"}
+{"_id": "6557", "title": "etale-cohomology-lemma-one-constructible", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a Noetherian ring. Let $K \\in D(X_\\etale, \\Lambda)$ and $b \\in \\mathbf{Z}$ such that $H^b(K)$ is constructible. Then there exist a sheaf $\\mathcal{F}$ which is a finite direct sum of $j_{U!}\\underline{\\Lambda}$ with $U \\in \\Ob(X_\\etale)$ affine and a map $\\mathcal{F}[-b] \\to K$ in $D(X_\\etale, \\Lambda)$ inducing a surjection $\\mathcal{F} \\to H^b(K)$."}
+{"_id": "6558", "title": "etale-cohomology-lemma-bounded-above-c", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a Noetherian ring. Let $K \\in D^-(X_\\etale, \\Lambda)$. Then the following are equivalent \\begin{enumerate} \\item $K$ is in $D_c(X_\\etale, \\Lambda)$, \\item $K$ can be represented by a bounded above complex whose terms are finite direct sums of $j_{U!}\\underline{\\Lambda}$ with $U \\in \\Ob(X_\\etale)$ affine, \\item $K$ can be represented by a bounded above complex of flat constructible sheaves of $\\Lambda$-modules. \\end{enumerate}"}
+{"_id": "6559", "title": "etale-cohomology-lemma-tensor-c", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. Let $K, L \\in D_c^-(X_\\etale, \\Lambda)$. Then $K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\etale, \\Lambda)$."}
+{"_id": "6560", "title": "etale-cohomology-lemma-when-ctf", "text": "Let $\\Lambda$ be a Noetherian ring. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \\in D(X_\\etale, \\Lambda)$. The following are equivalent \\begin{enumerate} \\item $K \\in D_{ctf}(X_\\etale, \\Lambda)$, and \\item $K$ can be represented by a finite complex of constructible flat sheaves of $\\Lambda$-modules. \\end{enumerate} In fact, if $K$ has tor amplitude in $[a, b]$ then we can represent $K$ by a complex $\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b$ with $\\mathcal{F}^p$ a constructible flat sheaf of $\\Lambda$-modules."}
+{"_id": "6563", "title": "etale-cohomology-lemma-connected-ctf-locally-constant", "text": "Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring. Let $K \\in D_{ctf}(X_\\etale, \\Lambda)$ have locally constant cohomology sheaves. Then there exists a finite complex of finite projective $\\Lambda$-modules $M^\\bullet$ and an \\'etale covering $\\{U_i \\to X\\}$ such that $K|_{U_i} \\cong \\underline{M^\\bullet}|_{U_i}$ in $D(U_{i, \\etale}, \\Lambda)$."}
+{"_id": "6564", "title": "etale-cohomology-lemma-torsion-cohomology", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. \\begin{enumerate} \\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then $H^n_\\etale(X, \\mathcal{F})$ is a torsion abelian group for all $n$. \\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then $H^n_\\etale(X, K)$ is a torsion abelian group for all $n$. \\end{enumerate}"}
+{"_id": "6565", "title": "etale-cohomology-lemma-torsion-direct-image", "text": "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. \\begin{enumerate} \\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then $R^nf_*\\mathcal{F}$ is a torsion abelian sheaf on $Y_\\etale$ for all $n$. \\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then $Rf_*K$ is an object of $D^+(Y_\\etale)$ whose cohomology sheaves are torsion abelian sheaves. \\end{enumerate}"}
+{"_id": "6566", "title": "etale-cohomology-lemma-sections-with-support-acyclic", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$. Then $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf on $Z_\\etale$."}
+{"_id": "6568", "title": "etale-cohomology-lemma-cohomology-with-support-triangle", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $j : U \\to X$ be the inclusion of the complement of $Z$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. There is a distinguished triangle $$ i_*R\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to Rj_*(\\mathcal{F}|_U) \\to i_*R\\mathcal{H}_Z(\\mathcal{F})[1] $$ in $D(X_\\etale)$. This produces an exact sequence $$ 0 \\to i_*\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to j_*(\\mathcal{F}|_U) \\to i_*\\mathcal{H}^1_Z(\\mathcal{F}) \\to 0 $$ and isomorphisms $R^pj_*(\\mathcal{F}|_U) \\cong i_*\\mathcal{H}^{p + 1}_Z(\\mathcal{F})$ for $p \\geq 1$."}
+{"_id": "6570", "title": "etale-cohomology-lemma-cohomology-with-support-quasi-coherent", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module and denote $\\mathcal{F}^a$ the associated quasi-coherent sheaf on the small \\'etale site of $X$ (Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}). Then \\begin{enumerate} \\item $H^q_Z(X, \\mathcal{F})$ agrees with $H^q_Z(X_\\etale, \\mathcal{F}^a)$, \\item if the complement of $Z$ is retrocompact in $X$, then $i_*\\mathcal{H}^q_Z(\\mathcal{F}^a)$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules equal to $(i_*\\mathcal{H}^q_Z(\\mathcal{F}))^a$. \\end{enumerate}"}
+{"_id": "6571", "title": "etale-cohomology-lemma-local-rings-strictly-henselian", "text": "Let $S$ be a scheme all of whose local rings are strictly henselian. Then for any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have $H^i(S_\\etale, \\mathcal{F}) = H^i(S_{Zar}, \\mathcal{F})$."}
+{"_id": "6573", "title": "etale-cohomology-lemma-normal-scheme-with-alg-closed-function-field", "text": "Let $X$ be an integral normal scheme with separably closed function field. \\begin{enumerate} \\item A separated \\'etale morphism $U \\to X$ is a disjoint union of open immersions. \\item All local rings of $X$ are strictly henselian. \\end{enumerate}"}
+{"_id": "6574", "title": "etale-cohomology-lemma-Rf-star-zero-normal-with-alg-closed-function-field", "text": "Let $f : X \\to Y$ be a morphism of schemes where $X$ is an integral normal scheme with separably closed function field. Then $R^qf_*\\underline{M} = 0$ for $q > 0$ and any abelian group $M$."}
+{"_id": "6575", "title": "etale-cohomology-lemma-closed-of-affine-normal-scheme-with-alg-closed-function-field", "text": "Let $X$ be an affine integral normal scheme with separably closed function field. Let $Z \\subset X$ be a closed subscheme. Let $V \\to Z$ be an \\'etale morphism with $V$ affine. Then $V$ is a finite disjoint union of open subschemes of $Z$. If $V \\to Z$ is surjective and finite \\'etale, then $V \\to Z$ has a section."}
+{"_id": "6576", "title": "etale-cohomology-lemma-gabber-for-h1-absolutely-algebraically-closed", "text": "Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \\subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^1_\\etale(Z, \\underline{M}) = 0$."}
+{"_id": "6577", "title": "etale-cohomology-lemma-gabber-for-absolutely-algebraically-closed", "text": "Let $X$ be a normal integral affine scheme with separably closed function field. Let $Z \\subset X$ be a closed subscheme. For any finite abelian group $M$ we have $H^q_\\etale(Z, \\underline{M}) = 0$ for $q \\geq 1$."}
+{"_id": "6578", "title": "etale-cohomology-lemma-integral-cover-trivial-cohomology", "text": "Let $X$ be an affine scheme. \\begin{enumerate} \\item There exists an integral surjective morphism $X' \\to X$ such that for every closed subscheme $Z' \\subset X'$, every finite abelian group $M$, and every $q \\geq 1$ we have $H^q_\\etale(Z', \\underline{M}) = 0$. \\item For any closed subscheme $Z \\subset X$, finite abelian group $M$, $q \\geq 1$, and $\\xi \\in H^q_\\etale(Z, \\underline{M})$ there exists a finite surjective morphism $X' \\to X$ of finite presentation such that $\\xi$ pulls back to zero in $H^q_\\etale(X' \\times_X Z, \\underline{M})$. \\end{enumerate}"}
+{"_id": "6579", "title": "etale-cohomology-lemma-efface-cohomology-on-closed-by-finite-cover", "text": "Let $X$ be an affine scheme. Let $\\mathcal{F}$ be a torsion abelian sheaf on  $X_\\etale$. Let $Z \\subset X$ be a closed subscheme. Let $\\xi \\in H^q_\\etale(Z, \\mathcal{F}|_Z)$ for some $q > 0$. Then there exists an injective map $\\mathcal{F} \\to \\mathcal{F}'$ of torsion abelian sheaves on $X_\\etale$ such that the image of $\\xi$ in $H^q_\\etale(Z, \\mathcal{F}'|_Z)$ is zero."}
+{"_id": "6580", "title": "etale-cohomology-lemma-gabber-h0", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \\to X$ be a closed immersion. Assume that \\begin{enumerate} \\item for any sheaf $\\mathcal{F}$ on $X_{Zar}$ the map $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(Z, i^{-1}\\mathcal{F})$ is bijective, and \\item for any finite morphism $X' \\to X$ assumption (1) holds for $Z \\times_X X' \\to X'$. \\end{enumerate} Then for any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i^{-1}_{small}\\mathcal{F})$."}
+{"_id": "6581", "title": "etale-cohomology-lemma-connected-topological", "text": "Let $Z \\subset X$ be a closed subset of a topological space $X$. Assume \\begin{enumerate} \\item $X$ is a spectral space (Topology, Definition \\ref{topology-definition-spectral-space}), and \\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$ is connected (in particular nonempty). \\end{enumerate} If $Z = Z_1 \\amalg Z_2$ with $Z_i$ closed in $Z$, then there exists a decomposition $X = X_1 \\amalg X_2$ with $X_i$ closed in $X$ and $Z_i = Z \\cap X_i$."}
+{"_id": "6582", "title": "etale-cohomology-lemma-h0-topological", "text": "Let $Z \\subset X$ be a closed subset of a topological space $X$. Assume \\begin{enumerate} \\item $X$ is a spectral space (Topology, Definition \\ref{topology-definition-spectral-space}), and \\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$ is connected (in particular nonempty). \\end{enumerate} Then for any sheaf $\\mathcal{F}$ on $X$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$."}
+{"_id": "6583", "title": "etale-cohomology-lemma-h0-henselian-pair", "text": "Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$."}
+{"_id": "6584", "title": "etale-cohomology-lemma-vanishing-restriction-injective", "text": "Let $X$ be a scheme with affine diagonal which can be covered by $n + 1$ affine opens. Let $Z \\subset X$ be a closed subscheme. Let $\\mathcal{A}$ be a torsion sheaf of rings on $X_\\etale$ and let $\\mathcal{I}$ be an injective sheaf of $\\mathcal{A}$-modules on $X_\\etale$. Then $H^q_\\etale(Z, \\mathcal{I}|_Z) = 0$ for $q > n$."}
+{"_id": "6585", "title": "etale-cohomology-lemma-constant-smooth-statements", "text": "In Situation \\ref{situation-what-to-prove} assume $X$ is smooth and $\\mathcal{F} = \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$ for some prime number $\\ell$. Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$."}
+{"_id": "6586", "title": "etale-cohomology-lemma-ses-statements", "text": "Let $k$ be an algebraically closed field. Let $X$ be a separated finite type scheme over $k$ of dimension $\\leq 1$. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0$ be a short exact sequence of torsion abelian sheaves on $X$. If statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}_1$ and $\\mathcal{F}_2$, then they hold for $\\mathcal{F}$."}
+{"_id": "6587", "title": "etale-cohomology-lemma-finite-pushforward-statements", "text": "Let $k$ be an algebraically closed field. Let $f : X \\to Y$ be a finite morphism of separated finite type schemes over $k$ of dimension $\\leq 1$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$. If statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$, then they hold for $f_*\\mathcal{F}$."}
+{"_id": "6588", "title": "etale-cohomology-lemma-even-easier", "text": "In Situation \\ref{situation-what-to-prove} assume $X$ is smooth. Let $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number. Let $\\mathcal{F} = j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$. Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$."}
+{"_id": "6589", "title": "etale-cohomology-lemma-somewhat-easier", "text": "In Situation \\ref{situation-what-to-prove} assume $X$ reduced. Let $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number and $\\mathcal{F} = j_! \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$. Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$."}
+{"_id": "6590", "title": "etale-cohomology-lemma-vanishing-easier", "text": "In Situation \\ref{situation-what-to-prove} assume $X$ reduced. Let $j : U \\to X$ an open immersion with $U$ connected. Let $\\ell$ be a prime number. Let $\\mathcal{G}$ a finite locally constant sheaf of $\\mathbf{F}_\\ell$-vector spaces on $U$. Let $\\mathcal{F} = j_!\\mathcal{G}$. Then statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$."}
+{"_id": "6591", "title": "etale-cohomology-lemma-base-change-dim-1-separably-closed", "text": "Let $k \\subset k'$ be an extension of separably closed fields. Let $X$ be a proper scheme over $k$ of dimension $\\leq 1$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^q_\\etale(X, \\mathcal{F}) \\to H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \\geq 0$."}
+{"_id": "6592", "title": "etale-cohomology-lemma-proper-over-henselian-and-h1", "text": "Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Let $M$ be a finite abelian group. Then $H^1_\\etale(X, \\underline{M}) = H^1_\\etale(X_0, \\underline{M})$."}
+{"_id": "6593", "title": "etale-cohomology-lemma-efface-cohomology-on-fibre-by-finite-cover", "text": "Let $A$ be a henselian local ring. Let $X = \\mathbf{P}^1_A$. Let $X_0 \\subset X$ be the closed fibre. Let $\\ell$ be a prime number. Let $\\mathcal{I}$ be an injective sheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$. Then $H^q_\\etale(X_0, \\mathcal{I}|_{X_0}) = 0$ for $q > 0$."}
+{"_id": "6596", "title": "etale-cohomology-lemma-base-change-Rf-star-colim", "text": "Let $I$ be a directed set. Consider an inverse system of cartesian diagrams of schemes $$ \\xymatrix{ X_i \\ar[d]_{f_i} & Y_i \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\ S_i & T_i \\ar[l]_{g_i} } $$ with affine transition morphisms and with $g_i$ quasi-compact and quasi-separated. Set $X = \\lim X_i$, $S = \\lim S_i$, $T = \\lim T_i$ and $Y = \\lim Y_i$ to obtain the cartesian diagram $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ Let $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves on $(T_i)$ as in Definition \\ref{definition-inverse-system-sheaves}. Set $\\mathcal{F} = \\colim p_i^{-1}\\mathcal{F}_i$ on $T$ where $p_i : T \\to T_i$ is the projection. Then we have the following \\begin{enumerate} \\item If $f_i^{-1}g_{i, *}\\mathcal{F}_i = h_{i, *}e_i^{-1}\\mathcal{F}_i$ for all $i$, then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$. \\item If $\\mathcal{F}_i$ is an abelian sheaf for all $i$ and $f_i^{-1}R^qg_{i, *}\\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$ for all $i$, then $f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "6599", "title": "etale-cohomology-lemma-base-change-f-star-general", "text": "Consider the cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ Assume that $f$ is flat and every object $U$ of $X_\\etale$ has a covering $\\{U_i \\to U\\}$ such that $U_i \\to S$ factors as $U_i \\to V_i \\to S$ with $V_i \\to S$ \\'etale and $U_i \\to V_i$ quasi-compact with geometrically connected fibres. Then for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$."}
+{"_id": "6600", "title": "etale-cohomology-lemma-fppf-reduced-fibres-base-change-f-star", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ where $f$ is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$ for any sheaf $\\mathcal{F}$ on $T_\\etale$."}
+{"_id": "6605", "title": "etale-cohomology-lemma-base-change-q-injective", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Then for every commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\ S & S' \\ar[l] & T \\ar[l]_g } $$ with $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\\mathcal{F}$ on $T_\\etale$ annihilated by $n$ \\begin{enumerate} \\item the base change map $(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}$ is injective, \\item if $\\mathcal{F} \\subset \\mathcal{G}$ where $\\mathcal{G}$ on $T_\\etale$ is annihilated by $n$, then $$ \\Coker\\left( (f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F} \\right) \\subset \\Coker\\left( (f')^{-1}R^qg_*\\mathcal{G}\\to R^qh_*e^{-1}\\mathcal{G} \\right) $$ \\item if in (2) the sheaf $\\mathcal{G}$ is an injective sheaf of $\\mathbf{Z}/n\\mathbf{Z}$-modules, then $$ \\Coker\\left((f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F} \\right) \\subset R^qh_*e^{-1}\\mathcal{G} $$ \\end{enumerate}"}
+{"_id": "6606", "title": "etale-cohomology-lemma-base-change-q-integral-top", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider commutative diagrams $$ \\vcenter{ \\xymatrix{ X \\ar[d]_f & X' \\ar[d]_{f'} \\ar[l] & Y \\ar[l]^h \\ar[d]^e & Y' \\ar[l]^{\\pi'} \\ar[d]^{e'} \\\\ S & S' \\ar[l] & T \\ar[l]_g & T' \\ar[l]_\\pi } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ X' \\ar[d]_{f'} & & Y' \\ar[ll]^{h' = h \\circ \\pi'} \\ar[d]^{e'} \\\\ S' & & T' \\ar[ll]_{g' = g \\circ \\pi} } } $$ where all squares are cartesian, $g$ quasi-compact and quasi-separated, and $\\pi$ is integral surjective. Let $\\mathcal{F}$ be an abelian sheaf on $T_\\etale$ annihilated by $n$ and set $\\mathcal{F}' = \\pi^{-1}\\mathcal{F}$. If the base change map $$ (f')^{-1}R^qg'_*\\mathcal{F}' \\longrightarrow R^qh'_*(e')^{-1}\\mathcal{F}' $$ is an isomorphism, then the base change map $(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$ is an isomorphism."}
+{"_id": "6607", "title": "etale-cohomology-lemma-base-change-q-integral-bottom", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider commutative diagrams $$ \\vcenter{ \\xymatrix{ X \\ar[d]_f & X' \\ar[d]_{f'} \\ar[l] & X'' \\ar[l]^{\\pi'} \\ar[d]_{f''} & Y \\ar[l]^{h'} \\ar[d]^e \\\\ S & S' \\ar[l] & S'' \\ar[l]_\\pi & T \\ar[l]_{g'} } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ X' \\ar[d]_{f'} & & Y \\ar[ll]^{h = h' \\circ \\pi'} \\ar[d]^e \\\\ S' & & T \\ar[ll]_{g = g' \\circ \\pi} } } $$ where all squares are cartesian, $g'$ quasi-compact and quasi-separated, and $\\pi$ is integral. Let $\\mathcal{F}$ be an abelian sheaf on $T_\\etale$ annihilated by $n$. If the base change map $$ (f')^{-1}R^qg_*\\mathcal{F} \\longrightarrow R^qh_*e^{-1}\\mathcal{F} $$ is an isomorphism, then the base change map $(f'')^{-1}R^qg'_*\\mathcal{F} \\to R^qh'_*e^{-1}\\mathcal{F}$ is an isomorphism."}
+{"_id": "6608", "title": "etale-cohomology-lemma-formal-argument", "text": "Let $T$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property for quasi-compact and quasi-separated schemes over $T$. Assume \\begin{enumerate} \\item If $T'' \\to T'$ is a thickening of quasi-compact and quasi-separated schemes over $T$, then $P(T'')$ if and only if $P(T')$. \\item If $T' = \\lim T_i$ is a limit of an inverse system of quasi-compact and quasi-separated schemes over $T$ with affine transition morphisms and $P(T_i)$ holds for all $i$, then $P(T')$ holds. \\item If $Z \\subset T'$ is a closed subscheme with quasi-compact complement $V \\subset T'$ and $P(T')$ holds, then either $P(V)$ or $P(Z)$ holds. \\end{enumerate} Then $P(T)$ implies $P(\\Spec(K))$ for some morphism $\\Spec(K) \\to T$ where $K$ is a field."}
+{"_id": "6609", "title": "etale-cohomology-lemma-base-change-does-not-hold-pre", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume for some $q \\geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X' \\ar[d]_{f'} \\ar[l] & Y \\ar[l]^h \\ar[d]^e \\\\ S & S' \\ar[l] & \\Spec(K) \\ar[l]_g } $$ where $X' = X \\times_S S'$, $Y = X' \\times_{S'} \\Spec(K)$, $K$ is a field, and $\\mathcal{F}$ is an abelian sheaf on $\\Spec(K)$ annihilated by $n$ such that $(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$ is not an isomorphism."}
+{"_id": "6610", "title": "etale-cohomology-lemma-base-change-does-not-hold", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume for some $q \\geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\ S & S' \\ar[l] & \\Spec(K) \\ar[l] } $$ with both squares cartesian, where \\begin{enumerate} \\item $S'$ is affine, integral, and normal with algebraically closed function field, \\item $K$ is algebraically closed and $\\Spec(K) \\to S'$ is dominant (in other words $K$ is an extension of the function field of $S'$) \\end{enumerate} and there exists an integer $d | n$ such that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero."}
+{"_id": "6611", "title": "etale-cohomology-lemma-smooth-base-change-fields", "text": "Let $K/k$ be an extension of fields. Let $X$ be a smooth affine curve over $k$ with a rational point $x \\in X(k)$. Let $\\mathcal{F}$ be an abelian sheaf on $\\Spec(K)$ annihilated by an integer $n$ invertible in $k$. Let $q > 0$ and $$ \\xi \\in H^q(X_K, (X_K \\to \\Spec(K))^{-1}\\mathcal{F}) $$ There exist \\begin{enumerate} \\item finite extensions $K'/K$ and $k'/k$ with $k' \\subset K'$, \\item a finite \\'etale Galois cover $Z \\to X_{k'}$ with group $G$ \\end{enumerate} such that the order of $G$ divides a power of $n$, such that $Z \\to X_{k'}$ is split over $x_{k'}$, and such that $\\xi$ dies in $H^q(Z_{K'}, (Z_{K'} \\to \\Spec(K))^{-1}\\mathcal{F})$."}
+{"_id": "6612", "title": "etale-cohomology-lemma-smooth-base-change-general", "text": "Let $S$ be a scheme. Let $S' = \\lim S_i$ be a directed inverse limit of schemes $S_i$ smooth over $S$ with affine transition morphisms. Let $f : X \\to S$ be quas-compact and quasi-separated and form the fibre square $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ Then $$ g^{-1}Rf_*E = R(f')_*(g')^{-1}E $$ for any $E \\in D^+(X_\\etale)$ whose cohomology sheaves $H^q(E)$ have stalks which are torsion of orders invertible on $S$."}
+{"_id": "6613", "title": "etale-cohomology-lemma-base-change-field-extension", "text": "Let $L/K$ be an extension of fields. Let $g : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes over $K$. Denote $g_L : T_L \\to S_L$ the base change of $g$ to $\\Spec(L)$. Let $E \\in D^+(T_\\etale)$ have cohomology sheaves whose stalks are torsion of orders invertible in $K$. Let $E_L$ be the pullback of $E$ to $(T_L)_\\etale$. Then $Rg_{L, *}E_L$ is the pullback of $Rg_*E$ to $S_L$."}
+{"_id": "6614", "title": "etale-cohomology-lemma-smooth-base-change-separably-closed", "text": "Let $K/k$ be an extension of separably closed fields. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. Let $E \\in D^+(X_\\etale)$ have cohomology sheaves whose stalks are torsion of orders invertible in $k$. Then \\begin{enumerate} \\item the maps $H^q_\\etale(X, E) \\to H^q_\\etale(X_K, E|_{X_K})$ are isomorphisms, and \\item $E \\to R(X_K \\to X)_*E|_{X_K}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "6615", "title": "etale-cohomology-lemma-base-change-does-not-hold-post", "text": "With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds} assume $n$ is invertible on $S$ and that for some $q \\geq 1$ we have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not. Then there exist a commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\ S & S' \\ar[l] & \\Spec(K) \\ar[l] } $$ with both squares cartesian, where $S'$ is affine, integral, and normal with algebraically closed function field $K$ and there exists an integer $d | n$ such that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero."}
+{"_id": "6616", "title": "etale-cohomology-lemma-zariski-h0-proper-over-henselian-pair", "text": "Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism of schemes. Let $Z = X \\times_{\\Spec(A)} \\Spec(A/I)$. For any sheaf $\\mathcal{F}$ on the topological space associated to $X$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$."}
+{"_id": "6617", "title": "etale-cohomology-lemma-h0-proper-over-henselian-pair", "text": "Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism of schemes. Let $i : Z \\to X$ be the closed immersion of $X \\times_{\\Spec(A)} \\Spec(A/I)$ into $X$. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i_{small}^{-1}\\mathcal{F})$."}
+{"_id": "6618", "title": "etale-cohomology-lemma-h0-proper-over-henselian-local", "text": "Let $A$ be a henselian local ring. Let $f : X \\to \\Spec(A)$ be a proper morphism of schemes. Let $X_0 \\subset X$ be the fibre of $f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$."}
+{"_id": "6619", "title": "etale-cohomology-lemma-proper-pushforward-stalk", "text": "Let $f : X \\to S$ be a proper morphism of schemes. Let $\\overline{s} \\to S$ be a geometric point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ the canonical map $$ (f_*\\mathcal{F})_{\\overline{s}} \\longrightarrow \\Gamma(X_{\\overline{s}}, \\mathcal{F}_{\\overline{s}}) $$ is bijective."}
+{"_id": "6620", "title": "etale-cohomology-lemma-proper-base-change-f-star", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be a morphism of schemes. Set $X' = Y' \\times_Y X$ with projections $f' : X' \\to Y'$ and $g' : X' \\to X$. Let $\\mathcal{F}$ be any sheaf on $X_\\etale$. Then $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$."}
+{"_id": "6621", "title": "etale-cohomology-lemma-proper-base-change-in-terms-of-injectives", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. The following are equivalent \\begin{enumerate} \\item cohomology commutes with base change for $f$ (see above), \\item for every prime number $\\ell$ and every injective sheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules $\\mathcal{I}$ on $X_\\etale$ and every diagram (\\ref{equation-base-change-diagram}) where $X' = Y' \\times_Y X$ the sheaves $R^qf'_*(g')^{-1}\\mathcal{I}$ are zero for $q > 0$. \\end{enumerate}"}
+{"_id": "6622", "title": "etale-cohomology-lemma-sandwich", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume \\begin{enumerate} \\item cohomology commutes with base change for $f$, \\item cohomology commutes with base change for $g \\circ f$, and \\item $f$ is surjective. \\end{enumerate} Then cohomology commutes with base change for $g$."}
+{"_id": "6623", "title": "etale-cohomology-lemma-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume \\begin{enumerate} \\item cohomology commutes with base change for $f$, and \\item cohomology commutes with base change for $g$. \\end{enumerate} Then cohomology commutes with base change for $g \\circ f$."}
+{"_id": "6624", "title": "etale-cohomology-lemma-finite", "text": "\\begin{slogan} Proper base change for \\'etale cohomology holds for finite morphisms. \\end{slogan} Let $f : X \\to Y$ be a finite morphism of schemes. Then cohomology commutes with base change for $f$."}
+{"_id": "6625", "title": "etale-cohomology-lemma-reduce-to-P1", "text": "To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it holds for the morphism $\\mathbf{P}^1_S \\to S$ for every scheme $S$."}
+{"_id": "6626", "title": "etale-cohomology-lemma-proper-base-change", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be a morphism of schemes. Set $X' = Y' \\times_Y X$ and denote $f' : X' \\to Y'$ and $g' : X' \\to X$ the projections. Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves. Then the base change map (\\ref{equation-base-change}) $g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$ is an isomorphism."}
+{"_id": "6627", "title": "etale-cohomology-lemma-proper-base-change-stalk", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $\\overline{y} \\to Y$ be a geometric point. \\begin{enumerate} \\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $(R^nf_*\\mathcal{F})_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$. \\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have $(R^nf_*E)_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, E_{\\overline{y}})$. \\end{enumerate}"}
+{"_id": "6629", "title": "etale-cohomology-lemma-cohomological-dimension-proper", "text": "Let $f : X \\to Y$ be a proper morphism of schemes all of whose fibres have dimension $\\leq n$. Then for any abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$ we have $R^qf_*\\mathcal{F} = 0$ for $q > 2n$."}
+{"_id": "6630", "title": "etale-cohomology-lemma-proper-base-change-mod-n", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be a morphism of schemes. Set $X' = Y' \\times_Y X$ and denote $f' : X' \\to Y'$ and $g' : X' \\to X$ the projections. Let $n \\geq 1$ be an integer. Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$. Then the base change map (\\ref{equation-base-change}) $g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$ is an isomorphism."}
+{"_id": "6631", "title": "etale-cohomology-lemma-pull-out-constant", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \\in D^+(X_\\etale)$ and $K \\in D^+(\\mathbf{Z})$. Then $$ R\\Gamma(X, E \\otimes_\\mathbf{Z}^\\mathbf{L} \\underline{K}) = R\\Gamma(X, E) \\otimes_\\mathbf{Z}^\\mathbf{L} K $$"}
+{"_id": "6632", "title": "etale-cohomology-lemma-projection-formula-proper", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves. Let $K \\in D^+(Y_\\etale)$. Then $$ Rf_*E \\otimes_\\mathbf{Z}^\\mathbf{L} K = Rf_*(E \\otimes_\\mathbf{Z}^\\mathbf{L} f^{-1}K) $$ in $D^+(Y_\\etale)$."}
+{"_id": "6633", "title": "etale-cohomology-lemma-cd-limit", "text": "Let $X = \\lim X_i$ be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $\\text{cd}(X) \\leq \\max \\text{cd}(X_i)$."}
+{"_id": "6634", "title": "etale-cohomology-lemma-cd-curve-over-field", "text": "Let $K$ be a field. Let $X$ be a $1$-dimensional affine scheme of finite type over $K$. Then $\\text{cd}(X) \\leq 1 + \\text{cd}(K)$."}
+{"_id": "6635", "title": "etale-cohomology-lemma-cd-field-extension", "text": "Let $L/K$ be a field extension. Then we have $\\text{cd}(L) \\leq \\text{cd}(K) + \\text{trdeg}_K(L)$."}
+{"_id": "6636", "title": "etale-cohomology-lemma-strictly-henselian", "text": "Let $K$ be a field. Let $X$ be a scheme of finite type over $K$. Let $x \\in X$. Set $a = \\text{trdeg}_K(\\kappa(x))$ and $d = \\dim_x(X)$. Then there is a map $$ K(t_1, \\ldots, t_a)^{sep} \\longrightarrow \\mathcal{O}_{X, x}^{sh} $$ such that \\begin{enumerate} \\item the residue field of $\\mathcal{O}_{X, x}^{sh}$ is a purely inseparable extension of $K(t_1, \\ldots, t_a)^{sep}$, \\item $\\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite type $K(t_1, \\ldots, t_a)^{sep}$-algebras of dimension $\\leq d - a$. \\end{enumerate}"}
+{"_id": "6637", "title": "etale-cohomology-lemma-interlude-II", "text": "Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Let $E_a \\subset X$ be the set of points $x \\in X$ with $\\text{trdeg}_K(\\kappa(x)) \\leq a$. Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$ whose support is contained in $E_a$. Then $H^b_\\etale(X, \\mathcal{F}) = 0$ for $b > a + \\text{cd}(K)$."}
+{"_id": "6638", "title": "etale-cohomology-lemma-interlude-I", "text": "Let $f : X \\to Y$ be an affine morphism of schemes of finite type over a field $K$. Let $E_a(X)$ be the set of points $x \\in X$ with $\\text{trdeg}_K(\\kappa(x)) \\leq a$. Let $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$ whose support is contained in $E_a$. Then $R^qf_*\\mathcal{F}$ has support contained in $E_{a - q}(Y)$."}
+{"_id": "6639", "title": "etale-cohomology-lemma-finite-cd", "text": "Let $K$ be a field. \\begin{enumerate} \\item If $f : X \\to Y$ is a morphism of finite type schemes over $K$, then $\\text{cd}(f) < \\infty$. \\item If $\\text{cd}(K) < \\infty$, then $\\text{cd}(X) < \\infty$ for any finite type scheme $X$ over $K$. \\end{enumerate}"}
+{"_id": "6640", "title": "etale-cohomology-lemma-finite-cd-mod-n-direct-sums", "text": "Cohomology and direct sums. Let $n \\geq 1$ be an integer. \\begin{enumerate} \\item Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes with $\\text{cd}(f) < \\infty$. Then the functor $$ Rf_* : D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z}) \\longrightarrow D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z}) $$ commutes with direct sums. \\item Let $X$ be a quasi-compact and quasi-separated scheme with $\\text{cd}(X) < \\infty$. Then the functor $$ R\\Gamma(X, -) : D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z}) \\longrightarrow D(\\mathbf{Z}/n\\mathbf{Z}) $$ commutes with direct sums. \\end{enumerate}"}
+{"_id": "6641", "title": "etale-cohomology-lemma-proper-mod-n-direct-sums", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $n \\geq 1$ be an integer. Then the functor $$ Rf_* : D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z}) \\longrightarrow D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z}) $$ commutes with direct sums."}
+{"_id": "6642", "title": "etale-cohomology-lemma-pull-out-constant-mod-n", "text": "Let $X$ be a quasi-compact and quasi-separated scheme such that $\\text{cd}(X) < \\infty$. Let $n \\geq 1$ be an integer. Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and $K \\in D(\\mathbf{Z}/n\\mathbf{Z})$. Then $$ R\\Gamma(X, E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} \\underline{K}) = R\\Gamma(X, E) \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} K $$"}
+{"_id": "6643", "title": "etale-cohomology-lemma-projection-formula-proper-mod-n", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $n \\geq 1$ be an integer. Let $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and $K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$. Then $$ Rf_*E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} K = Rf_*(E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} f^{-1}K) $$ in $D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$."}
+{"_id": "6644", "title": "etale-cohomology-lemma-kunneth-one-proper", "text": "Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$. Let $Y$ be a quasi-compact and quasi-separated scheme over $k$. \\begin{enumerate} \\item If $E \\in D^+(X_\\etale)$ has torsion cohomology sheaves and $K \\in D^+(Y_\\etale)$, then $$ R\\Gamma(X \\times_{\\Spec(k)} Y, \\text{pr}_1^{-1}E \\otimes_\\mathbf{Z}^\\mathbf{L} \\text{pr}_2^{-1}K ) = R\\Gamma(X, E) \\otimes_\\mathbf{Z}^\\mathbf{L} R\\Gamma(Y, K) $$ \\item If $n \\geq 1$ is an integer, $Y$ is of finite type over $k$, $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and $K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, then $$ R\\Gamma(X \\times_{\\Spec(k)} Y, \\text{pr}_1^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} \\text{pr}_2^{-1}K ) = R\\Gamma(X, E) \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} R\\Gamma(Y, K) $$ \\end{enumerate}"}
+{"_id": "6645", "title": "etale-cohomology-lemma-supported-in-closed-points", "text": "Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$ whose support is contained in the set of closed points of $X$. Then $H^q(X, \\mathcal{F}) = 0$ for $q > 0$ and $\\mathcal{F}$ is globally generated."}
+{"_id": "6646", "title": "etale-cohomology-lemma-vanishing-closed-points", "text": "Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $Q \\in D(X_\\etale)$. Assume that $Q_{\\overline{x}}$ is nonzero only if $x$ is a closed point of $X$. Then $$ Q = 0 \\Leftrightarrow H^i(X, Q) = 0 \\text{ for all }i $$"}
+{"_id": "6647", "title": "etale-cohomology-lemma-kunneth-localize-on-X", "text": "Let $K$ be a field. Let $j : U \\to X$ be an open immersion of schemes of finite type over $K$. Let $Y$ be a scheme of finite type over $K$. Consider the diagram $$ \\xymatrix{ Y \\times_{\\Spec(K)} X \\ar[d]_q  & Y \\times_{\\Spec(K)} U \\ar[l]^h \\ar[d]^p \\\\ X & U \\ar[l]_j } $$ Then the base change map $q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$ is an isomorphism for $\\mathcal{F}$ an abelian sheaf on $U_\\etale$ whose stalks are torsion of orders invertible in $K$."}
+{"_id": "6648", "title": "etale-cohomology-lemma-punctual-base-change", "text": "Let $K$ be a field. For any commutative diagram $$ \\xymatrix{ X \\ar[d] & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\ \\Spec(K) & S' \\ar[l] & T \\ar[l]_g } $$ of schemes over $K$ with $X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\\mathcal{F}$ on $T_\\etale$ whose stalks are torsion of orders invertible in $K$ the base change map $$ (f')^{-1}Rg_*\\mathcal{F} \\longrightarrow Rh_*e^{-1}\\mathcal{F} $$ is an isomorphism."}
+{"_id": "6649", "title": "etale-cohomology-lemma-punctual-base-change-upgrade", "text": "Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$. Consider a commutative diagram $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\ \\Spec(K) & S' \\ar[l] & T \\ar[l]_g } $$ of schemes with $X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and $g$ quasi-compact and quasi-separated. The canonical map $$ p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F \\longrightarrow Rh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F) $$ is an isomorphism if $E$ in $D^+(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ has tor amplitude in $[a, \\infty]$ for some $a \\in \\mathbf{Z}$ and $F$ in $D^+(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$."}
+{"_id": "6650", "title": "etale-cohomology-lemma-punctual-base-change-upgrade-unbounded", "text": "Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$. Consider a commutative diagram $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\ \\Spec(K) & S' \\ar[l] & T \\ar[l]_g } $$ of schemes of finite type over $K$ with $X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$. The canonical map $$ p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F \\longrightarrow Rh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F) $$ is an isomorphism for $E$ in $D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and $F$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$."}
+{"_id": "6653", "title": "etale-cohomology-lemma-compare-injectives", "text": "Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\etale$. \\begin{enumerate} \\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\etale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\etale)$, \\item $\\mathcal{I}|_{S_\\etale}$ is injective in $\\textit{Ab}(S_\\etale)$, \\end{enumerate} \\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}((\\Sch/S)_\\etale)$, then \\begin{enumerate} \\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}(T_\\etale)$, \\item $\\mathcal{I}^\\bullet|_{S_\\etale}$ is a K-injective complex in $\\textit{Ab}(S_\\etale)$, \\end{enumerate} \\end{enumerate} The corresponding statements for modules do not hold."}
+{"_id": "6654", "title": "etale-cohomology-lemma-compare-higher-direct-image", "text": "Let $f : T \\to S$ be a morphism of schemes. \\begin{enumerate} \\item For $K$ in $D((\\Sch/T)_\\etale)$ we have $ (Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale}) $ in $D(S_\\etale)$. \\item For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have $ (Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale}) $ in $D(\\textit{Mod}(S_\\etale, \\mathcal{O}_S))$. \\end{enumerate} More generally, let $g : S' \\to S$ be an object of $(\\Sch/S)_\\etale$. Consider the fibre product $$ \\xymatrix{ T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ Then \\begin{enumerate} \\item[(3)] For $K$ in $D((\\Sch/T)_\\etale)$ we have $i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(S'_\\etale)$. \\item[(4)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have $i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$. \\item[(5)] For $K$ in $D((\\Sch/T)_\\etale)$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\\Sch/S')_\\etale)$. \\item[(6)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$. \\end{enumerate}"}
+{"_id": "6656", "title": "etale-cohomology-lemma-cohomological-descent-etale", "text": "Let $S$ be a scheme. For $K \\in D(S_\\etale)$ the map $$ K \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K $$ is an isomorphism."}
+{"_id": "6657", "title": "etale-cohomology-lemma-compare-higher-direct-image-proper", "text": "Let $f : T \\to S$ be a proper morphism of schemes. Then we have \\begin{enumerate} \\item $\\pi_S^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_T^{-1}$ as functors $\\Sh(T_\\etale) \\to \\Sh((\\Sch/S)_\\etale)$, \\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$ for $K$ in $D^+(T_\\etale)$ whose cohomology sheaves are torsion, \\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$ for $K$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and \\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$ for all $K$ in $D(T_\\etale)$ if $f$ is finite. \\end{enumerate}"}
+{"_id": "6658", "title": "etale-cohomology-lemma-describe-pullback-pi-fppf", "text": "With notation as above. Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule $$ (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad (f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$. In fact this sheaf is equal to $\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and $\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{fppf}$."}
+{"_id": "6659", "title": "etale-cohomology-lemma-push-pull-fppf-etale", "text": "With notation as above. Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_{fppf}$. Then there are commutative diagrams of topoi $$ \\xymatrix{ \\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & & \\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & & \\Sh((\\Sch/Y)_\\etale) } $$ and $$ \\xymatrix{ \\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & & \\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{a_Y} \\\\ \\Sh(X_\\etale) \\ar[rr]^{f_{small}} & & \\Sh(Y_\\etale) } $$ with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$."}
+{"_id": "6660", "title": "etale-cohomology-lemma-proper-push-pull-fppf-etale", "text": "In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have $a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$."}
+{"_id": "6661", "title": "etale-cohomology-lemma-descent-sheaf-fppf-etale", "text": "In Lemma \\ref{lemma-push-pull-fppf-etale} assume $f$ is flat, locally of finite presentation, and surjective. Then the functor $$ \\Sh(Y_\\etale) \\longrightarrow \\left\\{ (\\mathcal{G}, \\mathcal{H}, \\alpha) \\middle| \\begin{matrix} \\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\ \\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H} \\text{ an isomorphism} \\end{matrix} \\right\\} $$ sending $\\mathcal{F}$ to $(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence."}
+{"_id": "6662", "title": "etale-cohomology-lemma-compare-fppf-etale", "text": "Consider the comparison morphism $\\epsilon : (\\Sch/S)_{fppf} \\to (\\Sch/S)_\\etale$. Let $\\mathcal{P}$ denote the class of finite morphisms of schemes. For $X$ in $(\\Sch/S)_\\etale$ denote $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$ the full subcategory consisting of sheaves of the form $\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$. Then Cohomology on Sites, Properties (\\ref{sites-cohomology-item-base-change-P}), (\\ref{sites-cohomology-item-restriction-A}), (\\ref{sites-cohomology-item-A-sheaf}), (\\ref{sites-cohomology-item-A-and-P}), and (\\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \\ref{sites-cohomology-situation-compare} hold."}
+{"_id": "6663", "title": "etale-cohomology-lemma-V-C-all-n-etale-fppf", "text": "With notation as above. \\begin{enumerate} \\item For $X \\in \\Ob((\\Sch/S)_{fppf})$ and an abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\item For a finite morphism $f : X \\to Y$ in $(\\Sch/S)_{fppf}$ and abelian sheaf $\\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\\mathcal{F}) = R^if_{big, fppf, *}(a_X^{-1}\\mathcal{F})$ for all $i$. \\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ the map $\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \\item For a finite morphism $f : X \\to Y$ of schemes and $K$ in $D^+(X_\\etale)$ we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$. \\item For a proper morphism $f : X \\to Y$ of schemes and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$. \\end{enumerate}"}
+{"_id": "6664", "title": "etale-cohomology-lemma-cohomological-descent-etale-fppf", "text": "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ the map $$ K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above."}
+{"_id": "6666", "title": "etale-cohomology-lemma-review-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module on $S_\\etale$. \\begin{enumerate} \\item The rule $$ \\mathcal{F}^a : (\\Sch/S)_\\etale \\longrightarrow \\textit{Ab},\\quad (f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{F}) $$ satisfies the sheaf condition for fppf and a fortiori \\'etale coverings, \\item $\\mathcal{F}^a = \\pi_S^*\\mathcal{F}$ on $(\\Sch/S)_\\etale$, \\item $\\mathcal{F}^a = a_S^*\\mathcal{F}$ on $(\\Sch/S)_{fppf}$, \\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence between quasi-coherent $\\mathcal{O}_S$-modules and quasi-coherent modules on $((\\Sch/S)_\\etale, \\mathcal{O})$, \\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence between quasi-coherent $\\mathcal{O}_S$-modules and quasi-coherent modules on $((\\Sch/S)_{fppf}, \\mathcal{O})$, \\item we have $\\epsilon_{S, *}a_S^*\\mathcal{F} = \\pi_S^*\\mathcal{F}$ and $a_{S, *}a_S^*\\mathcal{F} = \\mathcal{F}$, \\item we have $R^i\\epsilon_{S, *}(a_S^*\\mathcal{F}) = 0$ and $R^ia_{S, *}(a_S^*\\mathcal{F}) = 0$ for $i > 0$. \\end{enumerate}"}
+{"_id": "6668", "title": "etale-cohomology-lemma-describe-pullback-pi-ph", "text": "With notation as above. Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule $$ (\\Sch/S)_{ph} \\longrightarrow \\textit{Sets},\\quad (f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$. In fact this sheaf is equal to $\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and $\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{ph}$."}
+{"_id": "6669", "title": "etale-cohomology-lemma-push-pull-ph-etale", "text": "With notation as above. Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_{ph}$. Then there are commutative diagrams of topoi $$ \\xymatrix{ \\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & & \\Sh((\\Sch/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & & \\Sh((\\Sch/Y)_\\etale) } $$ and $$ \\xymatrix{ \\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & & \\Sh((\\Sch/Y)_{ph}) \\ar[d]^{a_Y} \\\\ \\Sh(X_\\etale) \\ar[rr]^{f_{small}} & & \\Sh(Y_\\etale) } $$ with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$."}
+{"_id": "6671", "title": "etale-cohomology-lemma-compare-ph-etale", "text": "Consider the comparison morphism $\\epsilon : (\\Sch/S)_{ph} \\to (\\Sch/S)_\\etale$. Let $\\mathcal{P}$ denote the class of proper morphisms of schemes. For $X$ in $(\\Sch/S)_\\etale$ denote $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$ the full subcategory consisting of sheaves of the form $\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$ Then Cohomology on Sites, Properties (\\ref{sites-cohomology-item-base-change-P}), (\\ref{sites-cohomology-item-restriction-A}), (\\ref{sites-cohomology-item-A-sheaf}), (\\ref{sites-cohomology-item-A-and-P}), and (\\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \\ref{sites-cohomology-situation-compare} hold."}
+{"_id": "6672", "title": "etale-cohomology-lemma-V-C-all-n-etale-ph", "text": "With notation as above. \\begin{enumerate} \\item For $X \\in \\Ob((\\Sch/S)_{ph})$ and an abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_{ph}$ and abelian torsion sheaf $\\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\\mathcal{F}) = R^if_{big, ph, *}(a_X^{-1}\\mathcal{F})$ for all $i$. \\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves the map $\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \\item For a proper morphism $f : X \\to Y$ of schemes and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$. \\end{enumerate}"}
+{"_id": "6673", "title": "etale-cohomology-lemma-cohomological-descent-etale-ph", "text": "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves the map $$ K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$ as above."}
+{"_id": "6675", "title": "etale-cohomology-lemma-describe-pullback-pi-h", "text": "With notation as above. Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule $$ (\\Sch/S)_h \\longrightarrow \\textit{Sets},\\quad (f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F}) $$ is a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$. In fact this sheaf is equal to $\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and $\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_h$."}
+{"_id": "6676", "title": "etale-cohomology-lemma-push-pull-h-etale", "text": "With notation as above. Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_h$. Then there are commutative diagrams of topoi $$ \\xymatrix{ \\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{\\epsilon_X} & & \\Sh((\\Sch/Y)_h) \\ar[d]^{\\epsilon_Y} \\\\ \\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & & \\Sh((\\Sch/Y)_\\etale) } $$ and $$ \\xymatrix{ \\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{a_X} & & \\Sh((\\Sch/Y)_h) \\ar[d]^{a_Y} \\\\ \\Sh(X_\\etale) \\ar[rr]^{f_{small}} & & \\Sh(Y_\\etale) } $$ with $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$."}
+{"_id": "6678", "title": "etale-cohomology-lemma-compare-h-etale", "text": "Consider the comparison morphism $\\epsilon : (\\Sch/S)_h \\to (\\Sch/S)_\\etale$. Let $\\mathcal{P}$ denote the class of proper morphisms. For $X$ in $(\\Sch/S)_\\etale$ denote $\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$ the full subcategory consisting of sheaves of the form $\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$ Then Cohomology on Sites, Properties (\\ref{sites-cohomology-item-base-change-P}), (\\ref{sites-cohomology-item-restriction-A}), (\\ref{sites-cohomology-item-A-sheaf}), (\\ref{sites-cohomology-item-A-and-P}), and (\\ref{sites-cohomology-item-refine-tau-by-P}) of Cohomology on Sites, Situation \\ref{sites-cohomology-situation-compare} hold."}
+{"_id": "6679", "title": "etale-cohomology-lemma-V-C-all-n-etale-h", "text": "With notation as above. \\begin{enumerate} \\item For $X \\in \\Ob((\\Sch/S)_{h})$ and an abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$ and $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$. \\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_h$ and abelian torsion sheaf $\\mathcal{F}$ on $X$ we have $a_Y^{-1}(R^if_{small, *}\\mathcal{F}) = R^if_{big, h, *}(a_X^{-1}\\mathcal{F})$ for all $i$. \\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves the map $\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism. \\item For a proper morphism $f : X \\to Y$ of schemes and $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_X^{-1}K)$. \\end{enumerate}"}
+{"_id": "6680", "title": "etale-cohomology-lemma-cohomological-descent-etale-h", "text": "Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves the map $$ K \\longrightarrow Ra_{X, *}a_X^{-1}K $$ is an isomorphism with $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$ as above."}
+{"_id": "6682", "title": "etale-cohomology-lemma-glue-etale-sheaf-section", "text": "Let $f : X \\to Y$ be a morphism of schemes which has a section. Then the functor $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ sending $\\mathcal{G}$ in $\\Sh(Y_\\etale)$ to the canonical descent datum is an equivalence of categories."}
+{"_id": "6683", "title": "etale-cohomology-lemma-glue-etale-sheaf-integral-surjective", "text": "Let $f : X \\to Y$ be a surjective integral morphism of schemes. The functor $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "6684", "title": "etale-cohomology-lemma-glue-etale-sheaf-proper-surjective", "text": "Let $f : X \\to Y$ be a surjective proper morphism of schemes. The functor $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "6685", "title": "etale-cohomology-lemma-glue-etale-sheaf-check-after-base-change", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\to Y$ be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors $$ \\Sh(Z_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Z \\to Z\\} $$ and $$ \\Sh((Z \\times_Y Z)_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt } \\{X \\times_Y (Z \\times_Y Z) \\to Z \\times_Y Z\\} $$ are equivalences of categories, then $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence."}
+{"_id": "6686", "title": "etale-cohomology-lemma-glue-etale-sheaf-fppf-cover", "text": "Let $f : X \\to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "6688", "title": "etale-cohomology-lemma-glue-etale-sheaf-modification", "text": "Let $f : X' \\to X$ be a proper morphism of schemes. Let $i : Z \\to X$ be a closed immersion. Set $E = Z \\times_X X'$. Picture $$ \\xymatrix{ E \\ar[d]_g \\ar[r]_j & X' \\ar[d]^f \\\\ Z \\ar[r]^i & X } $$ If $f$ is an isomorphism over $X \\setminus Z$, then the functor $$ \\Sh(X_\\etale) \\longrightarrow \\Sh(X'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale) $$ is an equivalence of categories."}
+{"_id": "6690", "title": "etale-cohomology-lemma-blow-up-square-cohomological-descent", "text": "Let $X$ be a scheme and let $Z \\subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square $$ \\xymatrix{ E \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\ Z \\ar[r]^i & X } $$ For $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have a distinguished triangle $$ K \\to Ri_*(K|_Z) \\oplus Rb_*(K|_{X'}) \\to Rc_*(K|_E) \\to K[1] $$ in $D(X_\\etale)$ where $c = i \\circ \\pi = b \\circ j$."}
+{"_id": "6693", "title": "etale-cohomology-lemma-blow-up-square-h", "text": "With notation as above, if $K$ is in the essential image of $R\\epsilon_*$, then the maps $c^K_{X, Z, X', E}$ of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square} are quasi-isomorphisms."}
+{"_id": "6694", "title": "etale-cohomology-lemma-refine-check-h", "text": "Let $K$ be an object of $D^+((\\Sch/S)_{fppf})$. Then $K$ is in the essential image of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$ if and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism for every almost blow up square as in More on Flatness, Examples \\ref{flat-example-one-generator} and \\ref{flat-example-two-generators}."}
+{"_id": "6695", "title": "etale-cohomology-lemma-h-sheaf-colim-F", "text": "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$. Consider the sheaf $\\mathcal{O}^{perf} = \\colim_F \\mathcal{O}$ on $(\\Sch/S)_{fppf}$. Then $\\mathcal{O}^{perf}$ is in the essential image of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$."}
+{"_id": "6696", "title": "etale-cohomology-proposition-quasi-coherent-sheaf-fpqc", "text": "For any quasi-coherent sheaf $\\mathcal{F}$ on $S$ the presheaf $$ \\begin{matrix} \\mathcal{F}^a : & \\Sch/S & \\to & \\textit{Ab}\\\\ & (f: T \\to S) & \\mapsto & \\Gamma(T, f^*\\mathcal{F}) \\end{matrix} $$ is an $\\mathcal{O}$-module which satisfies the sheaf condition for the fpqc topology."}
+{"_id": "6697", "title": "etale-cohomology-proposition-etale-morphisms", "text": "Facts on \\'etale morphisms. \\begin{enumerate} \\item Let $k$ be a field. A morphism of schemes $U \\to \\Spec(k)$ is \\'etale if and only if $U \\cong \\coprod_{i \\in I} \\Spec(k_i)$ such that for each $i \\in I$ the ring $k_i$ is a field which is a finite separable extension of $k$. \\item Let $\\varphi : U \\to S$ be a morphism of schemes. The following conditions are equivalent: \\begin{enumerate} \\item $\\varphi$ is \\'etale, \\item $\\varphi$ is locally finitely presented, flat, and all its fibres are \\'etale, \\item $\\varphi$ is flat, unramified and locally of finite presentation. \\end{enumerate} \\item A ring map $A \\to B$ is \\'etale if and only if $B \\cong A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ such that $\\Delta = \\det \\left( \\frac{\\partial f_i}{\\partial x_j} \\right)$ is invertible in $B$. \\item The base change of an \\'etale morphism is \\'etale. \\item Compositions of \\'etale morphisms are \\'etale. \\item Fibre products and products of \\'etale morphisms are \\'etale. \\item An \\'etale morphism has relative dimension 0. \\item Let $Y \\to X$ be an \\'etale morphism. If $X$ is reduced (respectively regular) then so is $Y$. \\item \\'Etale morphisms are open. \\item If $X \\to S$ and $Y \\to S$ are \\'etale, then any $S$-morphism $X \\to Y$ is also \\'etale. \\end{enumerate}"}
+{"_id": "6699", "title": "etale-cohomology-proposition-topological-invariance", "text": "Let $X_0 \\to X$ be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then \\begin{enumerate} \\item the \\'etale sites $X_\\etale$ and $(X_0)_\\etale$ are isomorphic, \\item the \\'etale topoi $\\Sh(X_\\etale)$ and $\\Sh((X_0)_\\etale)$ are equivalent, and \\item $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(X_0, \\mathcal{F}|_{X_0})$ for all $q$ and for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$. \\end{enumerate}"}
+{"_id": "6700", "title": "etale-cohomology-proposition-closed-immersion-pushforward", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. \\begin{enumerate} \\item The functor $$ i_{small, *} : \\Sh(Z_\\etale) \\longrightarrow \\Sh(X_\\etale) $$ is fully faithful and its essential image is those sheaves of sets $\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is isomorphic to $*$, and \\item the functor $$ i_{small, *} : \\textit{Ab}(Z_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale) $$ is fully faithful and its essential image is those abelian sheaves on $X_\\etale$ whose support is contained in $Z$. \\end{enumerate} In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$."}
+{"_id": "6701", "title": "etale-cohomology-proposition-integral-universally-injective-pushforward", "text": "Let $f : X \\to Y$ be a morphism of schemes which is integral and universally injective. \\begin{enumerate} \\item The functor $$ f_{small, *} : \\Sh(X_\\etale) \\longrightarrow \\Sh(Y_\\etale) $$ is fully faithful and its essential image is those sheaves of sets $\\mathcal{F}$ on $Y_\\etale$ whose restriction to $Y \\setminus f(X)$ is isomorphic to $*$, and \\item the functor $$ f_{small, *} : \\textit{Ab}(X_\\etale) \\longrightarrow \\textit{Ab}(Y_\\etale) $$ is fully faithful and its essential image is those abelian sheaves on $Y_\\etale$ whose support is contained in $f(X)$. \\end{enumerate} In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$."}
+{"_id": "6702", "title": "etale-cohomology-proposition-leray", "text": "Let $f: X \\to Y$ be a morphism of schemes and $\\mathcal{F}$ an \\'etale sheaf on $X$. Then there is a spectral sequence $$ E_2^{p, q} = H_\\etale^p(Y, R^qf_*\\mathcal{F}) \\Rightarrow H_\\etale^{p+q}(X, \\mathcal{F}). $$"}
+{"_id": "6703", "title": "etale-cohomology-proposition-finite-higher-direct-image-zero", "text": "Let $f : X \\to Y$ be a finite morphism of schemes. \\begin{enumerate} \\item For any geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have $$ (f_*\\mathcal{F})_{\\overline{y}} = \\prod\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}. $$ for $\\mathcal{F}$ in $\\Sh(X_\\etale)$ and $$ (f_*\\mathcal{F})_{\\overline{y}} = \\bigoplus\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}. $$ for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$. \\item For any $q \\geq 1$ we have $R^q f_*\\mathcal{F} = 0$. \\end{enumerate}"}
+{"_id": "6704", "title": "etale-cohomology-proposition-serre-galois", "text": "\\begin{reference} \\cite[Chapter II, Section 3, Proposition 5]{SerreGaloisCohomology} \\end{reference} Let $K$ be a field with separable algebraic closure $K^{sep}$. Assume that for any finite extension $K'$ of $K$ we have $\\text{Br}(K') = 0$. Then \\begin{enumerate} \\item $H^q(\\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \\geq 1$, and \\item $H^q(\\text{Gal}(K^{sep}/K), M) = 0$ for any torsion $\\text{Gal}(K^{sep}/K)$-module $M$ and any $q \\geq 2$, \\end{enumerate}"}
+{"_id": "6705", "title": "etale-cohomology-proposition-describe-jshriek", "text": "Let $j : U \\to X$ be an \\'etale morphism of schemes. Let $\\mathcal{F}$ in $\\textit{Ab}(U_\\etale)$. If $\\overline{x} : \\Spec(k) \\to X$ is a geometric point of $X$, then  $$ (j_!\\mathcal{F})_{\\overline{x}} = \\bigoplus\\nolimits_{\\overline{u} : \\Spec(k) \\to U,\\ j(\\overline{u}) = \\overline{x}} \\mathcal{F}_{\\bar{u}}. $$ In particular, $j_!$ is an exact functor."}
+{"_id": "6706", "title": "etale-cohomology-proposition-constructible-over-noetherian", "text": "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring. \\begin{enumerate} \\item Any sub or quotient sheaf of a constructible sheaf of sets is constructible. \\item The category of constructible abelian sheaves on $X_\\etale$ is a (strong) Serre subcategory of $\\textit{Ab}(X_\\etale)$. In particular, every sub and quotient sheaf of a constructible abelian sheaf on $X_\\etale$ is constructible. \\item The category of constructible sheaves of $\\Lambda$-modules on $X_\\etale$ is a (strong) Serre subcategory of $\\textit{Mod}(X_\\etale, \\Lambda)$. In particular, every submodule and quotient module of a constructible sheaf of $\\Lambda$-modules on $X_\\etale$ is constructible. \\end{enumerate}"}
+{"_id": "6707", "title": "etale-cohomology-proposition-cd-affine", "text": "Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Then we have $\\text{cd}(X) \\leq \\dim(X) + \\text{cd}(K)$."}
+{"_id": "6800", "title": "equiv-theorem-fully-faithful", "text": "\\begin{reference} \\cite[Theorem 2.2]{Orlov-K3}; this is shown in \\cite{Noah} without the assumption that $X$ be projective \\end{reference} Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$ with $X$ projective over $k$. Any $k$-linear fully faithful exact  functor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ is a Fourier-Mukai functor for some kernel in $D_{perf}(\\mathcal{O}_{X \\times Y})$."}
+{"_id": "6802", "title": "equiv-lemma-Serre-functor-exists", "text": "Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear triangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$ for all $X, Y \\in \\Ob(\\mathcal{T})$. The following are equivalent \\begin{enumerate} \\item there exists a $k$-linear equivalence $S : \\mathcal{T} \\to \\mathcal{T}$ and $k$-linear isomorphisms $c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$ functorial in $X, Y \\in \\Ob(\\mathcal{T})$, \\item for every $X \\in \\Ob(\\mathcal{T})$ the functor $Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$ is representable and the functor $Y \\mapsto \\Hom_\\mathcal{T}(Y, X)^\\vee$ is corepresentable. \\end{enumerate}"}
+{"_id": "6805", "title": "equiv-lemma-perfect-for-R", "text": "With $k$, $n$, and $R$ as above, for an object $K$ of $D(R)$ the following are equivalent \\begin{enumerate} \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(K) < \\infty$, and \\item $K$ is a compact object. \\end{enumerate}"}
+{"_id": "6806", "title": "equiv-lemma-coherent-on-projective-space", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_k})$. The following are equivalent \\begin{enumerate} \\item $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_{\\mathbf{P}^n_k})$, \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(\\mathbf{P}^n_k, E \\otimes^\\mathbf{L} K) < \\infty$ for each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_k})$, \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_{\\mathbf{P}^n_k}(E, K) < \\infty$ for each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_k})$, \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(\\mathbf{P}^n_k, K \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_k}(d)) < \\infty$ for $d = 0, 1, \\ldots, n$. \\end{enumerate}"}
+{"_id": "6807", "title": "equiv-lemma-finiteness", "text": "Let $X$ be a scheme proper over a field $k$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and let $E$ in $D(\\mathcal{O}_X)$ be perfect. Then $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$."}
+{"_id": "6808", "title": "equiv-lemma-characterize-dbcoh-projective", "text": "\\begin{reference} \\cite[Lemma 7.46]{Rouquier-dimensions} and implicit in \\cite[Theorem A.1]{BvdB} \\end{reference} Let $X$ be a projective scheme over a field $k$. Let $K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$. The following are equivalent \\begin{enumerate} \\item $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, and \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$ for all perfect $E$ in $D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6809", "title": "equiv-lemma-maps-from-compact-filtered", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated subcategory. Let $X \\in \\Ob(\\mathcal{D})$. The category of arrows $E \\to X$ with $E \\in \\Ob(\\mathcal{D}')$ is filtered."}
+{"_id": "6810", "title": "equiv-lemma-van-den-bergh", "text": "\\begin{reference} \\cite[Lemma 2.14]{CKN} \\end{reference} Let $k$ be a field. Let $\\mathcal{D}$ be a $k$-linear triangulated category which has direct sums and is compactly generated. Denote $\\mathcal{D}_c$ the full subcategory of compact objects. Let $H : \\mathcal{D}_c^{opp} \\to \\text{Vect}_k$ be a $k$-linear cohomological functor such that $\\dim_k H(X) < \\infty$ for all $X \\in \\Ob(\\mathcal{D}_c)$. Then $H$ is isomorphic to the functor $X \\mapsto \\Hom(X, Y)$ for some $Y \\in \\Ob(\\mathcal{D})$."}
+{"_id": "6812", "title": "equiv-lemma-characterize-dbcoh-proper-regular", "text": "Let $X$ be a proper scheme over a field $k$ which is regular. Let $K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$. The following are equivalent \\begin{enumerate} \\item $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D_{perf}(\\mathcal{O}_X)$, and \\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$ for all perfect $E$ in $D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6813", "title": "equiv-lemma-bondal-van-den-bergh", "text": "Let $X$ be a proper scheme over a field $k$ which is regular. \\begin{enumerate} \\item Let $F : D_{perf}(\\mathcal{O}_X)^{opp} \\to \\text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k F(E[n]) < \\infty $$ for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \\mapsto \\Hom_X(E, K)$ for some $K \\in D_{perf}(\\mathcal{O}_X)$. \\item Let $G : D_{perf}(\\mathcal{O}_X) \\to \\text{Vect}_k$ be a $k$-linear homological functor such that $$ \\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k G(E[n]) < \\infty $$ for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $G$ is isomorphic to a functor of the form $E \\mapsto \\Hom_X(K, E)$ for some $K \\in D_{perf}(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6814", "title": "equiv-lemma-always-right-adjoints", "text": "Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. If $X$ is regular, then $k$-linear any exact functor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ has an exact right adjoint and an exact left adjoint."}
+{"_id": "6815", "title": "equiv-lemma-fourier-Mukai-QCoh", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$. The corresponding Fourier-Mukai functor $\\Phi_K$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$ if $K$ is in $D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$ and $X \\to S$ is quasi-compact and quasi-separated."}
+{"_id": "6816", "title": "equiv-lemma-compose-fourier-mukai", "text": "Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume $X \\to S$, $Y \\to S$, and $Z \\to S$ are quasi-compact and quasi-separated. Let $K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$. Let $K' \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S Z})$. Consider the Fourier-Mukai functors $\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ and $\\Phi_{K'} : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Z)$. If $X$ and $Z$ are tor independent over $S$ and $Y \\to S$ is flat, then $$ \\Phi_{K'} \\circ \\Phi_K = \\Phi_{K''} : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(\\mathcal{O}_Z) $$ where $$ K'' = R\\text{pr}_{13, *}( L\\text{pr}_{12}^*K \\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S Z}}^\\mathbf{L} L\\text{pr}_{23}^*K') $$ in $D_\\QCoh(\\mathcal{O}_{X \\times_S Z})$."}
+{"_id": "6817", "title": "equiv-lemma-fourier-mukai", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$. The corresponding Fourier-Mukai functor $\\Phi_K$ sends $D_{perf}(\\mathcal{O}_X)$ into $D_{perf}(\\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \\begin{enumerate} \\item $S$ is Noetherian, $X \\to S$ and $Y \\to S$ are of finite type, $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\\text{pr}_2^{-1}\\mathcal{O}_Y)$, \\item $X \\to S$ is of finite presentation and $K$ can be represented by a bounded complex $\\mathcal{K}^\\bullet$ of finitely presented $\\mathcal{O}_{X \\times_S Y}$-modules, flat over $Y$, with support proper over $Y$, \\item $X \\to S$ is a proper flat morphism of finite presentation and $K$ is perfect, \\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is perfect \\item $X \\to S$ is a proper flat morphism of finite presentation and $K$ is $Y$-perfect, \\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is $Y$-perfect. \\end{enumerate}"}
+{"_id": "6818", "title": "equiv-lemma-fourier-mukai-Coh", "text": "Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type over $S$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$. The corresponding Fourier-Mukai functor $\\Phi_K$ sends $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \\begin{enumerate} \\item the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\\text{pr}_1^{-1}\\mathcal{O}_X)$, \\item $K$ can be represented by a bounded complex $\\mathcal{K}^\\bullet$ of coherent $\\mathcal{O}_{X \\times_S Y}$-modules, flat over $X$, with support proper over $Y$, \\item the support of $H^i(K)$ is proper over $Y$ for all $i$ and $X$ is a regular scheme, \\item $K$ is perfect, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $Y \\to S$ is flat. \\end{enumerate} Furthermore in each case the support condition is automatic if $X \\to S$ is proper."}
+{"_id": "6819", "title": "equiv-lemma-fourier-mukai-right-adjoint", "text": "\\begin{reference} Compare with discussion in \\cite{Rizzardo}. \\end{reference} Let $X \\to S$ and $Y \\to S$ be morphisms of quasi-compact and quasi-separated schemes. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ be a Fourier-Mukai functor with pseudo-coherent kernel $K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$. Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to  D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$ be the right adjoint to $R\\text{pr}_{2, *}$, see Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}. Denote $$ K' = (Y \\times_S X \\to X \\times_S Y)^* R\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}(K, a(\\mathcal{O}_Y)) \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S X}) $$ and denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical map $$ \\Hom_X(M, \\Phi'(N)) \\longrightarrow \\Hom_Y(\\Phi(M), N) $$ functorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in $D_\\QCoh(\\mathcal{O}_Y)$ which is an isomorphism if \\begin{enumerate} \\item $N$ is perfect, or \\item $K$ is perfect and $X \\to S$ is proper flat and of finite presentation. \\end{enumerate}"}
+{"_id": "6820", "title": "equiv-lemma-fourier-mukai-left-adjoint", "text": "\\begin{reference} Compare with discussion in \\cite{Rizzardo}. \\end{reference} Let $S$ be a Noetherian scheme. Let $Y \\to S$ be a flat proper Gorenstein morphism and let $X \\to S$ be a finite type morphism. Denote $\\omega^\\bullet_{Y/S}$ the relative dualizing complex of $Y$ over $S$. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ be a Fourier-Mukai functor with perfect kernel $K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$. Denote $$ K' = (Y \\times_S X \\to X \\times_S Y)^*(K^\\vee \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} L\\text{pr}_2^*\\omega^\\bullet_{Y/S}) \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S X}) $$ and denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical isomorphism $$ \\Hom_Y(N, \\Phi(M)) \\longrightarrow \\Hom_X(\\Phi'(N), M) $$ functorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in $D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "6821", "title": "equiv-lemma-fourier-mukai-flat-proper-over-noetherian", "text": "Let $S$ be a Noetherian scheme. \\begin{enumerate} \\item For $X$, $Y$ proper and flat over $S$ and $K$ in $D_{perf}(\\mathcal{O}_{X \\times_S Y})$ we obtain a Fourier-Mukai functor $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$. \\item For $X$, $Y$, $Z$ proper and flat over $S$, $K \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$, $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S Z})$ the composition $\\Phi_{K'} \\circ \\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Z)$ is equal to $\\Phi_{K''}$ with $K'' \\in D_{perf}(\\mathcal{O}_{X \\times_S Z})$ computed as in Lemma \\ref{lemma-compose-fourier-mukai}, \\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $X \\to S$ is Gorenstein, then $\\Phi_{K'} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a right adjoint to $\\Phi_K$ where $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$ is the pullback of $L\\text{pr}_1^*\\omega_{X/S}^\\bullet \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by $Y \\times_S X \\to X \\times_S Y$. \\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $Y \\to S$ is Gorenstein, then $\\Phi_{K''} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a left adjoint to $\\Phi_K$ where $K'' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$ is the pullback of $L\\text{pr}_2^*\\omega_{Y/S}^\\bullet \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by $Y \\times_S X \\to X \\times_S Y$. \\end{enumerate}"}
+{"_id": "6822", "title": "equiv-lemma-on-product", "text": "Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$ having the resolution property. For any coherent $\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$ there exist a surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$ where $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module and $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module."}
+{"_id": "6824", "title": "equiv-lemma-diagonal-resolution", "text": "Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme over $R$ which has the resolution property. Set $\\mathcal{O}_\\Delta = \\Delta_*(\\mathcal{O}_X)$ where $\\Delta : X \\to X \\times_R X$ is the diagonal of $X/k$. There exists a resolution $$ \\ldots \\to \\mathcal{E}_2 \\boxtimes \\mathcal{G}_2 \\to \\mathcal{E}_1 \\boxtimes \\mathcal{G}_1 \\to \\mathcal{E}_0 \\boxtimes \\mathcal{G}_0 \\to \\mathcal{O}_\\Delta \\to 0 $$ where each $\\mathcal{E}_i$ and $\\mathcal{G}_i$ is a finite locally free $\\mathcal{O}_X$-module."}
+{"_id": "6825", "title": "equiv-lemma-Ext-0-regular", "text": "Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$. Then \\begin{enumerate} \\item for $\\mathcal{F}$, $\\mathcal{G}$ coherent $\\mathcal{O}_X$-modules we have $\\Ext^n_X(\\mathcal{F}, \\mathcal{G}) = 0$ for $n > d$, and \\item for $K, L \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$ if $H^i(K) = 0$ for $i < a + d$ and $H^i(L) = 0$ for $i \\geq a$ then $\\Hom_X(K, L) = 0$. \\end{enumerate}"}
+{"_id": "6826", "title": "equiv-lemma-split-complex-regular", "text": "Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$. If $H^i(K) = 0$ for $a < i < a + d$, then $K = \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + d}K$."}
+{"_id": "6827", "title": "equiv-lemma-diagonal-trick", "text": "Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$. There exist finite locally free $\\mathcal{O}_X$-modules $\\mathcal{E}$ and $\\mathcal{G}$ such that $$ \\mathcal{O}_\\Delta \\in \\langle \\mathcal{E} \\boxtimes \\mathcal{G} \\rangle $$ in $D(\\mathcal{O}_{X \\times X})$ where the notation is as in Derived Categories, Section \\ref{derived-section-generators}."}
+{"_id": "6828", "title": "equiv-lemma-smooth-proper-strong-generator", "text": "Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$. Then $D_{perf}(\\mathcal{O}_X)$ has a strong generator."}
+{"_id": "6829", "title": "equiv-lemma-diagonal-trick-proper", "text": "Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$. There exists integers $m, n \\geq 1$ and a finite locally free $\\mathcal{O}_X$-module $\\mathcal{G}$ such that every coherent $\\mathcal{O}_X$-module is contained in $smd(add(\\mathcal{G}[-m, m])^{\\star n})$ with notation as in Derived Categories, Section \\ref{derived-section-operate-on-full}."}
+{"_id": "6832", "title": "equiv-lemma-functor-quasi-coherent-from-affine", "text": "Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ affine. There is an equivalence of categories between \\begin{enumerate} \\item the category of $R$-linear functors $F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ which are right exact and commute with arbitrary direct sums, and \\item the category $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ \\end{enumerate} given by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh})."}
+{"_id": "6833", "title": "equiv-lemma-functor-quasi-coherent-from-affine-compose", "text": "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine} let $F$ correspond to $\\mathcal{K}$ in $\\QCoh(\\mathcal{O}_{X \\times_R Y})$. We have \\begin{enumerate} \\item If $f : X' \\to X$ is an affine morphism, then $F \\circ f_*$ corresponds to $(f \\times \\text{id}_Y)^*\\mathcal{K}$. \\item If $g : Y' \\to Y$ is a quasi-compact and quasi-separated flat morphism, then $g^* \\circ F$ corresponds to $(\\text{id}_X \\times g)^*\\mathcal{K}$. \\item If $j : V \\to Y$ is an open immersion, then $j^* \\circ F$ corresponds to $\\mathcal{K}|_{X \\times_R V}$. \\end{enumerate}"}
+{"_id": "6834", "title": "equiv-lemma-coh-noetherian-from-affine-flat", "text": "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine} if $F$ is an exact functor, then the corresponding object $\\mathcal{K}$ of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ is flat over $X$."}
+{"_id": "6835", "title": "equiv-lemma-functor-quasi-coherent-from-affine-diagonal", "text": "Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact and that the diagonal morphism of $X$ is affine. There is an equivalence of categories between \\begin{enumerate} \\item the category of $R$-linear exact functors $F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ which commute with arbitrary direct sums, and \\item the full subcategory of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ consisting of $\\mathcal{K}$ such that \\begin{enumerate} \\item $\\mathcal{K}$ is flat over $X$, \\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have $R^q\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}) = 0$ for $q > 0$. \\end{enumerate} \\end{enumerate} given by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh})."}
+{"_id": "6836", "title": "equiv-lemma-persistence-exactness", "text": "Let $R$, $X$, $Y$, and $\\mathcal{K}$ be as in Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal} part (2). Then for any scheme $T$ over $R$ we have $$ R^q\\text{pr}_{13, *}(\\text{pr}_{12}^*\\mathcal{F} \\otimes_{\\mathcal{O}_{T \\times_R X \\times_R Y}} \\text{pr}_{23}^*\\mathcal{K}) = 0 $$ for $\\mathcal{F}$ quasi-coherent on $T \\times_R X$ and $q > 0$."}
+{"_id": "6837", "title": "equiv-lemma-functor-quasi-coherent-from-separated", "text": "In Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal} let $F$ and $\\mathcal{K}$ correspond. If $X$ is separated and flat over $R$, then there is a surjection $\\mathcal{O}_X \\boxtimes F(\\mathcal{O}_X) \\to \\mathcal{K}$."}
+{"_id": "6838", "title": "equiv-lemma-functor-coherent", "text": "Let $X$ and $Y$ be Noetherian schemes. Let $F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$ be a functor. Then $F$ extends uniquely to a functor $\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ which commutes with filtered colimits. If $F$ is additive, then its extension commutes with arbitrary direct sums. If $F$ is exact, left exact, or right exact, so is its extension."}
+{"_id": "6839", "title": "equiv-lemma-characterize-finite", "text": "Let $f : V \\to X$ be a quasi-finite separated morphism of Noetherian schemes. If there exists a coherent $\\mathcal{O}_V$-module $\\mathcal{K}$ whose support is $V$ such that $f_*\\mathcal{K}$ is coherent and $R^qf_*\\mathcal{K} = 0$, then $f$ is finite."}
+{"_id": "6840", "title": "equiv-lemma-functor-coherent-over-field", "text": "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. There is an equivalence of categories between \\begin{enumerate} \\item the category of $k$-linear exact functors $F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$, and \\item the category of coherent $\\mathcal{O}_{X \\times Y}$-modules $\\mathcal{K}$ which are flat over $X$ and have support finite over $Y$ \\end{enumerate} given by sending $\\mathcal{K}$ to the restriction of the functor (\\ref{equation-FM-QCoh}) to $\\textit{Coh}(\\mathcal{O}_X)$."}
+{"_id": "6841", "title": "equiv-lemma-pushforward-invertible-pre", "text": "Let $f : X \\to Y$ be a finite type separated morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent module on $X$ with support finite over $Y$ and with $\\mathcal{L} = f_*\\mathcal{F}$ an invertible $\\mathcal{O}_X$-module. Then there exists a section $s : Y \\to X$ such that $\\mathcal{F} \\cong s_*\\mathcal{L}$."}
+{"_id": "6844", "title": "equiv-lemma-sibling-fully-faithful", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a triangulated category. Let $F, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D}$ be exact functors of triangulated categories. Assume \\begin{enumerate} \\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic where $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and \\item for all $X, Y \\in \\Ob(\\mathcal{A})$ we have $\\Ext^q_\\mathcal{D}(F(X), F(Y)) = 0$ for $q < 0$ (for example if $F$ is fully faithful). \\end{enumerate} Then $F$ and $F'$ are siblings."}
+{"_id": "6845", "title": "equiv-lemma-sibling-faithful", "text": "Let $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}. Then \\begin{enumerate} \\item if $F$ is essentially surjective, then $F'$ is essentially surjective, \\item if $F$ is fully faithful, then $F'$ is fully faithful. \\end{enumerate}"}
+{"_id": "6846", "title": "equiv-lemma-get-fully-faithful", "text": "\\begin{reference} Variant of \\cite[Lemma 2.15]{Orlov-K3} \\end{reference} Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated categories. Let $S \\subset \\Ob(\\mathcal{D})$ be a set of objects. Assume \\begin{enumerate} \\item $F$ has both right and left adjoints, \\item for $K \\in \\mathcal{D}$ if $\\Hom(E, K[i]) = 0$ for all $E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$, \\item for $K \\in \\mathcal{D}$ if $\\Hom(K, E[i]) = 0$ for all $E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$, \\item the map $\\Hom(E, E'[i]) \\to \\Hom(F(E), F(E')[i])$ induced by $F$ is bijective for all $E, E' \\in S$ and $i \\in \\mathbf{Z}$. \\end{enumerate} Then $F$ is fully faithful."}
+{"_id": "6847", "title": "equiv-lemma-duality-at-point", "text": "Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \\in X$ be a closed point. For a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ supported at $x$ choose a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$ supported at $x$ such that $\\mathcal{F}_x$ and $\\mathcal{F}'_x$ are Matlis dual. Then there is an isomorphism $$ \\Hom_X(\\mathcal{F}, M) = H^0(X, M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}'[-d_x]) $$ where $d_x = \\dim(\\mathcal{O}_{X, x})$ functorial in $M$ in $D_{perf}(\\mathcal{O}_X)$."}
+{"_id": "6848", "title": "equiv-lemma-orthogonal-point-sheaf", "text": "Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \\in X$ be a closed point and denote $\\mathcal{O}_x$ the skyscraper sheaf at $x$ with value $\\kappa(x)$. Let $K$ in $D_{perf}(\\mathcal{O}_X)$. \\begin{enumerate} \\item If $\\Ext^i_X(\\mathcal{O}_x, K) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^{i - d_x}(K)|_U = 0$ where $d_x = \\dim(\\mathcal{O}_{X, x})$. \\item If $\\Hom_X(\\mathcal{O}_x, K[i]) = 0$ for all $i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \\item If $\\Ext^i_X(K, \\mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K^\\vee)|_U = 0$. \\item If $\\Hom_X(K, \\mathcal{O}_x[i]) = 0$ for all $i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K)|_U = 0$. \\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$ for $i \\in \\mathbf{Z}$ then $K$ is zero in an open neighbourhood of $x$. \\end{enumerate}"}
+{"_id": "6849", "title": "equiv-lemma-get-fully-faithful-geometric", "text": "Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. Assume $X$ is regular. Then a $k$-linear exact functor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ is fully faithful if and only if for any closed points $x, x' \\in X$ the maps $$ F : \\Ext^i_X(\\mathcal{O}_x, \\mathcal{O}_{x'}) \\longrightarrow \\Ext^i_Y(F(\\mathcal{O}_x), F(\\mathcal{O}_{x'})) $$ are isomorphisms for all $i \\in \\mathbf{Z}$. Here $\\mathcal{O}_x$ is the skyscraper sheaf at $x$ with value $\\kappa(x)$."}
+{"_id": "6850", "title": "equiv-lemma-noah-pre", "text": "\\begin{reference} Email from Noah Olander of Jun 9, 2020 \\end{reference} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$ there is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$. Then $F$ is fully faithful."}
+{"_id": "6851", "title": "equiv-lemma-exact-functor-preserving-Coh", "text": "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ be a $k$-linear exact functor sending $\\textit{Coh}(\\mathcal{O}_X) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $\\textit{Coh}(\\mathcal{O}_Y) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$. Then there exists a Fourier-Mukai functor $F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ whose kernel is a coherent $\\mathcal{O}_{X \\times Y}$-module $\\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$."}
+{"_id": "6852", "title": "equiv-lemma-preserves-Coh", "text": "Let $k$ be a field. Let $X$ be a separated scheme of finite type over $k$ which is regular. Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$ there is an isomorphism of $k$-vector spaces $$ \\Hom_X(\\mathcal{F}, M) = \\Hom_X(\\mathcal{F}, F(M)) $$ functorial in $M$ in $D_{perf}(\\mathcal{O}_X)$. Then there exists an automorphism $f : X \\to X$ over $k$ which induces the identity on the underlying topological space\\footnote{This often forces $f$ to be the identity, see Lemma \\ref{lemma-automorphism}.} and an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ such that $F$ and $F'(M) = f^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}$ are siblings."}
+{"_id": "6853", "title": "equiv-lemma-automorphism", "text": "Let $X$ be a reduced scheme of finite type over a field $k$. Let $f : X \\to X$ be an automorphism over $k$ which induces the identity map on the underlying topological space of $X$. Then \\begin{enumerate} \\item $f^*\\mathcal{F} \\cong \\mathcal{F}$ for every coherent $\\mathcal{O}_X$-module, and \\item if $\\dim(Z) > 0$ for every irreducible component $Z \\subset X$, then $f$ is the identity. \\end{enumerate}"}
+{"_id": "6854", "title": "equiv-lemma-noah", "text": "\\begin{reference} Email from Noah Olander of Jun 8, 2020 \\end{reference} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$ there is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$. Then there exists an automorphism $f : X \\to X$ over $k$ which induces the identity on the underlying topological space\\footnote{This often forces $f$ to be the identity, see Lemma \\ref{lemma-automorphism}.} and an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ such that $F$ and $F'(M) = f^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}$ are siblings."}
+{"_id": "6855", "title": "equiv-lemma-two-functors", "text": "Let $k$ be a field. Let $X$, $Y$ be smooth proper schemes over $k$. Let $F, G : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ be $k$-linear exact functors such that \\begin{enumerate} \\item $F(\\mathcal{F}) \\cong G(\\mathcal{F})$ for any coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$, \\item $F$ is fully faithful, and \\item $G$ is a Fourier-Mukai functor whose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$. \\end{enumerate} Then there exists a Fourier-Mukai functor $F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$ such that $F$ and $F'$ are siblings."}
+{"_id": "6856", "title": "equiv-lemma-fully-faithful", "text": "Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. Given a $k$-linear, exact, fully faithful functor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ there exists a Fourier-Mukai functor $F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$ which is a sibling to $F$."}
+{"_id": "6857", "title": "equiv-lemma-uniqueness", "text": "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $K \\in D_{perf}(\\mathcal{O}_{X \\times X})$. If the Fourier-Mukai functor $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$ is isomorphic to the identity functor, then $K \\cong \\Delta_*\\mathcal{O}_X$ in $_{perf}(\\mathcal{O}_{X \\times X})$."}
+{"_id": "6858", "title": "equiv-lemma-base-change-is-functor", "text": "Let $S' \\to S$ be a morphism of schemes. The rule which sends \\begin{enumerate} \\item a smooth proper scheme $X$ over $S$ to $X' =  S' \\times_S X$, and \\item the isomorphism class of an object $K$ of $D_{perf}(\\mathcal{O}_{X \\times_S Y})$ to the isomorphism class of $L(X' \\times_{S'} Y' \\to X \\times_S Y)^*K$ in $D_{perf}(\\mathcal{O}_{X' \\times_{S'} Y'})$ \\end{enumerate} is a functor from the category defined for $S$ to the category defined for $S'$."}
+{"_id": "6860", "title": "equiv-lemma-base-change-rek", "text": "With notation as in Definition \\ref{definition-relative-equivalence-kernel} let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Let $S_1 \\to S$ be a morphism of schemes. Let $X_1 = S_1 \\times_S X$ and $Y_1 = S_1 \\times_S Y$. Then the pullback $K_1 = L(X_1 \\times_{S_1} Y_1 \\to X \\times_S Y)^*K$ is the Fourier-Mukai kernel of a relative equivalence from $X_1$ to $Y_1$ over $S_1$."}
+{"_id": "6861", "title": "equiv-lemma-descend-rek", "text": "Let $S = \\lim_{i \\in I} S_i$ be a limit of a directed system of schemes with affine transition morphisms $g_{i'i} : S_{i'} \\to S_i$. We assume that $S_i$ is quasi-compact and quasi-separated for all $i \\in I$. Let $0 \\in I$. Let $X_0 \\to S_0$ and $Y_0 \\to S_0$ be smooth proper morphisms. We set $X_i = S_i \\times_{S_0} X_0$ for $i \\geq 0$ and $X = S \\times_{S_0} X_0$ and similarly for $Y_0$. If $K$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$ then for some $i \\geq 0$ there exists a Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$."}
+{"_id": "6862", "title": "equiv-lemma-deform-koszul", "text": "Let $(R, \\mathfrak m, \\kappa) \\to (A, \\mathfrak n, \\lambda)$ be a flat local ring homorphism of local rings which is essentially of finite presentation. Let $\\overline{f}_1, \\ldots, \\overline{f}_r \\in \\mathfrak n/\\mathfrak m A \\subset A/\\mathfrak m A$ be a regular sequence. Let $K \\in D(A)$. Assume \\begin{enumerate} \\item $K$ is perfect, \\item $K \\otimes_A^\\mathbf{L} A/\\mathfrak m A$ is isomorphic in $D(A/\\mathfrak m A)$ to the Koszul complex on $\\overline{f}_1, \\ldots, \\overline{f}_r$. \\end{enumerate} Then $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence $f_1, \\ldots, f_r \\in A$ lifting the given elements $\\overline{f}_1, \\ldots, \\overline{f}_r$. Moreover, $A/(f_1, \\ldots, f_r)$ is flat over $R$."}
+{"_id": "6863", "title": "equiv-lemma-limit-arguments", "text": "Let $R \\to S$ be a finite type flat ring map of Noetherian rings. Let $\\mathfrak q \\subset S$ be a prime ideal lying over $\\mathfrak p \\subset R$. Let $K \\in D(S)$ be perfect. Let $f_1, \\ldots, f_r \\in \\mathfrak q S_\\mathfrak q$ be a regular sequence such that $S_\\mathfrak q/(f_1, \\ldots, f_r)$ is flat over $R$ and such that $K \\otimes_S^\\mathbf{L} S_\\mathfrak q$ is isomorphic to the Koszul complex on $f_1, \\ldots, f_r$. Then there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that \\begin{enumerate} \\item $f_1, \\ldots, f_r$ are the images of $f'_1, \\ldots, f'_r \\in S_g$, \\item $f'_1, \\ldots, f'_r$ form a regular sequence in $S_g$, \\item $S_g/(f'_1, \\ldots, f'_r)$ is flat over $R$, \\item $K \\otimes_S^\\mathbf{L} S_g$ is isomorphic to the Koszul complex on $f_1, \\ldots, f_r$. \\end{enumerate}"}
+{"_id": "6864", "title": "equiv-lemma-isomorphism-in-neighbourhood", "text": "Let $S$ be a Noetherian scheme. Let $s \\in S$. Let $p : X \\to Y$ be a morphism of schemes over $S$. Assume \\begin{enumerate} \\item $Y \\to S$ and $X \\to S$ proper, \\item $X$ is flat over $S$, \\item $X_s \\to Y_s$ an isomorphism. \\end{enumerate} Then there exists an open neighbourhood $U \\subset S$ of $s$ such that the base change $X_U \\to Y_U$ is an isomorphism."}
+{"_id": "6865", "title": "equiv-lemma-no-deformations", "text": "Let $k$ be a field. Let $S$ be a finite type scheme over $k$ with $k$-rational point $s$. Let $Y \\to S$ be a smooth proper morphism. Let $X = Y_s \\times S \\to S$ be the constant family with fibre $Y_s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Assume the restriction $$ L(Y_s \\times_S Y_s \\to X \\times_S Y)^*K \\cong  \\Delta_{Y_s/k, *} \\mathcal{O}_{Y_s} $$ in $D(\\mathcal{O}_{Y_s \\times Y_s})$. Then there is an open neighbourhood $s \\in U \\subset S$ such that $Y|_U$ is isomorphic to $Y_s \\times U$ over $U$."}
+{"_id": "6866", "title": "equiv-lemma-no-deformations-better", "text": "Let $k$ be an algebraically closed field. Let $X$ be a smooth proper scheme over $k$. Let $f : Y \\to S$ be a smooth proper morphism with $S$ of finite type over $k$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X \\times S$ to $Y$ over $S$. Then $S$ can be covered by open subschemes $U$ such that there is a $U$-isomorphism $f^{-1}(U) \\cong Y_0 \\times U$ for some $Y_0$ proper and smooth over $k$."}
+{"_id": "6867", "title": "equiv-lemma-countable-finite-type", "text": "Let $R$ be a countable Noetherian ring. Then the category of schemes of finite type over $R$ is countable."}
+{"_id": "6868", "title": "equiv-lemma-countable-abelian", "text": "Let $\\mathcal{A}$ be a countable abelian category. Then $D^b(\\mathcal{A})$ is countable."}
+{"_id": "6869", "title": "equiv-lemma-countable-perfect", "text": "Let $X$ be a scheme of finite type over a countable Noetherian ring. Then the categories $D_{perf}(\\mathcal{O}_X)$ and $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ are countable."}
+{"_id": "6870", "title": "equiv-lemma-countable-isos", "text": "Let $K$ be an algebraically closed field. Let $S$ be a finite type scheme over $K$. Let $X \\to S$ and $Y \\to S$ be finite type morphisms. There exists a countable set $I$ and for $i \\in I$ a pair $(S_i \\to S, h_i)$ with the following properties \\begin{enumerate} \\item $S_i \\to S$ is a morphism of finite type, set $X_i = X \\times_S S_i$ and $Y_i = Y \\times_S S_i$, \\item $h_i : X_i \\to Y_i$ is an isomorphism over $S_i$, and \\item for any closed point $s \\in S(K)$ if $X_s \\cong Y_s$ over $K = \\kappa(s)$ then $s$ is in the image of $S_i \\to S$ for some $i$. \\end{enumerate}"}
+{"_id": "6871", "title": "equiv-lemma-countable-equivs", "text": "Let $K$ be an algebraically closed field. There exists a countable set $I$ and for $i \\in I$ a pair $(S_i/K, X_i \\to S_i, Y_i \\to S_i, M_i)$ with the following properties \\begin{enumerate} \\item $S_i$ is a scheme of finite type over $K$, \\item $X_i \\to S_i$ and $Y_i \\to S_i$ are proper smooth morphisms of schemes, \\item $M_i \\in D_{perf}(\\mathcal{O}_{X_i \\times_{S_i} Y_i})$ is the Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$, and \\item for any smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ there exists an $i \\in I$ and a $s \\in S_i(K)$ such that $X \\cong (X_i)_s$ and $Y \\cong (Y_i)_s$. \\end{enumerate}"}
+{"_id": "6872", "title": "equiv-proposition-siblings-isomorphic", "text": "\\begin{reference} \\cite[Proposition 2.16]{Orlov-K3} \\end{reference} Let $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}. Assume that $F$ is fully faithful and that $\\mathcal{A}$ has enough negative objects (see above). Then $F$ and $F'$ are isomorphic functors."}
+{"_id": "6873", "title": "equiv-proposition-equivalence", "text": "Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. If $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ is a $k$-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor $F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$ which is an equivalence and a sibling of $F$."}
+{"_id": "6882", "title": "stacks-more-morphisms-theorem-chow-finite-type", "text": "\\begin{reference} This is a result due to Ofer Gabber, see \\cite[Theorem 1.1]{olsson_proper} \\end{reference} Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. Assume \\begin{enumerate} \\item $Y$ is quasi-compact and quasi-separated, \\item $f$ is separated of finite type. \\end{enumerate} Then there exists a commutative diagram $$ \\xymatrix{ \\mathcal{X} \\ar[rd] & X \\ar[l] \\ar[d] \\ar[r] & \\overline{X} \\ar[ld] \\\\ & Y } $$ where $X \\to \\mathcal{X}$ is proper surjective, $X \\to \\overline{X}$ is an open immersion, and $\\overline{X} \\to Y$ is proper morphism of algebraic spaces."}
+{"_id": "6883", "title": "stacks-more-morphisms-theorem-keel-mori", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite. Then there exists a uniform categorical moduli space $$ f : \\mathcal{X} \\longrightarrow M $$ and $f$ is separated, quasi-compact, and a universal homeomorphism."}
+{"_id": "6884", "title": "stacks-more-morphisms-lemma-thickening", "text": "Let $i : \\mathcal{X} \\to \\mathcal{X}'$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $i$ is a thickening of algebraic stacks (abuse of language as above), and \\item $i$ is representable by algebraic spaces and is a thickening in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}. \\end{enumerate} In this case $i$ is a closed immersion and a universal homeomorphism."}
+{"_id": "6885", "title": "stacks-more-morphisms-lemma-base-change-thickening", "text": "Let $\\mathcal{Y} \\subset \\mathcal{Y}'$ be a thickening of algebraic stacks. Let $\\mathcal{X}' \\to \\mathcal{Y}'$ be a morphism of algebraic stacks and set $\\mathcal{X} = \\mathcal{Y} \\times_{\\mathcal{Y}'} \\mathcal{X}'$. Then $(\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$ is a morphism of thickenings. If $\\mathcal{Y} \\subset \\mathcal{Y}'$ is a first order thickening, then $\\mathcal{X} \\subset \\mathcal{X}'$ is a first order thickening."}
+{"_id": "6886", "title": "stacks-more-morphisms-lemma-composition-thickening", "text": "If $\\mathcal{X} \\subset \\mathcal{X}'$ and $\\mathcal{X}' \\subset \\mathcal{X}''$ are thickenings of algebraic stacks, then so is $\\mathcal{X} \\subset \\mathcal{X}''$."}
+{"_id": "6887", "title": "stacks-more-morphisms-lemma-reduced-diagonal", "text": "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Then $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$ is a thickening and the canonical diagram $$ \\xymatrix{ \\mathcal{X} \\ar[r]_-\\Delta \\ar[d] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\ \\mathcal{X}' \\ar[r]^-{\\Delta'} & \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}' } $$ is cartesian."}
+{"_id": "6888", "title": "stacks-more-morphisms-lemma-thickening-diagonals", "text": "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Let $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ and $\\Delta' : \\mathcal{X}' \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$ be the corresponding diagonal morphisms. Then each property from the following list is satisfied by $\\Delta$ if and only if it is satisfied by $\\Delta'$: (a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact, (e) universally closed, (f) integral, (g) quasi-separated, (h) separated, (i) universally injective, (j) universally open, (k) locally quasi-finite, (l) finite, (m) unramified, (n) monomorphism, (o) immersion, (p) closed immersion, and (q) proper."}
+{"_id": "6889", "title": "stacks-more-morphisms-lemma-thickening-properties", "text": "\\begin{reference} \\cite[Theorem 2.2.5]{Conrad-moduli} \\end{reference} Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks. Then \\begin{enumerate} \\item $\\mathcal{X}$ is an algebraic space if and only if $\\mathcal{X}'$ is an algebraic space, \\item $\\mathcal{X}$ is a scheme if and only if $\\mathcal{X}'$ is a scheme, \\item $\\mathcal{X}$ is DM if and only if $\\mathcal{X}'$ is DM, \\item $\\mathcal{X}$ is quasi-DM if and only if $\\mathcal{X}'$ is quasi-DM, \\item $\\mathcal{X}$ is separated if and only if $\\mathcal{X}'$ is separated, \\item $\\mathcal{X}$ is quasi-separated if and only if $\\mathcal{X}'$ is quasi-separated, and \\item add more here. \\end{enumerate}"}
+{"_id": "6890", "title": "stacks-more-morphisms-lemma-thicken-property-morphisms", "text": "Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Then \\begin{enumerate} \\item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \\item $f$ is quasi-compact if and only if $f'$ quasi-compact, \\item $f$ is universally closed if and only if $f'$ is universally closed, \\item $f$ is integral if and only if $f'$ is integral, \\item $f$ is universally injective if and only if $f'$ is universally injective, \\item $f$ is universally open if and only if $f'$ is universally open, \\item $f$ is quasi-DM if and only if $f'$ is quasi-DM, \\item $f$ is DM if and only if $f'$ is DM, \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\item $f$ is representable if and only if $f'$ is representable, \\item $f$ is representable by algebraic spaces if and only if $f'$ is representable by algebraic spaces, \\item add more here. \\end{enumerate}"}
+{"_id": "6891", "title": "stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & & (\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\ & (\\mathcal{B} \\subset \\mathcal{B}') } $$ of thickenings of algebraic stacks. Assume \\begin{enumerate} \\item $\\mathcal{Y}' \\to \\mathcal{B}'$ is locally of finite type, \\item $\\mathcal{X}' \\to \\mathcal{B}'$ is flat and locally of finite presentation, \\item $f$ is flat, and \\item $\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$ and $\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$. \\end{enumerate} Then $f'$ is flat and for all $y' \\in |\\mathcal{Y}'|$ in the image of $|f'|$ the morphism $\\mathcal{Y}' \\to \\mathcal{B}'$ is flat at $y'$."}
+{"_id": "6892", "title": "stacks-more-morphisms-lemma-deform-property-fp-over-ft", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & & (\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\ & (\\mathcal{B} \\subset \\mathcal{B}') } $$ of thickenings of algebraic stacks. Assume $\\mathcal{Y}' \\to \\mathcal{B}'$ locally of finite type, $\\mathcal{X}' \\to \\mathcal{B}'$ flat and locally of finite presentation, $\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$, and $\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$. Then \\begin{enumerate} \\item $f$ is flat if and only if $f'$ is flat, \\label{item-flat-fp-over-ft} \\item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \\label{item-isomorphism-fp-over-ft} \\item $f$ is an open immersion if and only if $f'$ is an open immersion, \\label{item-open-immersion-fp-over-ft} \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\label{item-monomorphism-fp-over-ft} \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\label{item-quasi-finite-fp-over-ft} \\item $f$ is syntomic if and only if $f'$ is syntomic, \\label{item-syntomic-fp-over-ft} \\item $f$ is smooth if and only if $f'$ is smooth, \\label{item-smooth-fp-over-ft} \\item $f$ is unramified if and only if $f'$ is unramified, \\label{item-unramified-fp-over-ft} \\item $f$ is \\'etale if and only if $f'$ is \\'etale, \\label{item-etale-fp-over-ft} \\item $f$ is finite if and only if $f'$ is finite, and \\label{item-finite-fp-over-ft} \\item add more here. \\end{enumerate}"}
+{"_id": "6893", "title": "stacks-more-morphisms-lemma-morphisms-lifts-etale", "text": "For any morphism (\\ref{equation-morphism}) the map $f' : V' \\to U'$ is \\'etale."}
+{"_id": "6894", "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-fibred", "text": "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed in Remark \\ref{remark-gerbe-of-lifts} is fibred in groupoids."}
+{"_id": "6895", "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-stack", "text": "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed in Remark \\ref{remark-gerbe-of-lifts} is a stack in groupoids."}
+{"_id": "6896", "title": "stacks-more-morphisms-lemma-etale-local-lifts", "text": "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks. Let $W$ be an algebraic space and let $W \\to \\mathcal{X}$ be a smooth morphism. There exists an \\'etale covering $\\{W_i \\to W\\}_{i \\in I}$ and for each $i$ a cartesian diagram $$ \\xymatrix{ W_i \\ar[r] \\ar[d] & W_i' \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{X}' } $$ with $W_i' \\to \\mathcal{X}'$ smooth."}
+{"_id": "6897", "title": "stacks-more-morphisms-lemma-etale-local-lifts-isomorphic", "text": "Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks. Consider a commutative diagram $$ \\xymatrix{ W'' \\ar[d]_{x''} & W \\ar[l] \\ar[r] \\ar[d]_x & W' \\ar[d]^{x'} \\\\ \\mathcal{X}' & \\mathcal{X} \\ar[l] \\ar[r] & \\mathcal{X}' } $$ with cartesian squares where $W', W, W''$ are algebraic spaces and the vertical arrows are smooth. Then there exist \\begin{enumerate} \\item an \\'etale covering $\\{f'_k : W'_k \\to W'\\}_{k \\in K}$, \\item \\'etale morphisms $f''_k : W'_k \\to W''$, and \\item $2$-morphisms $\\gamma_k : x'' \\circ f''_k \\to x' \\circ f'_k$ \\end{enumerate} such that (a) $(f'_k)^{-1}(W) = (f''_k)^{-1}(W)$, (b) $f'_k|_{(f'_k)^{-1}(W)} = f''_k|_{(f''_k)^{-1}(W)}$, and (c) pulling back $\\gamma_k$ to the closed subscheme of (a) agrees with the $2$-morphism given by the commutativity of the initial diagram over $W$."}
+{"_id": "6898", "title": "stacks-more-morphisms-lemma-gerbe-of-lifts", "text": "The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed in Remark \\ref{remark-gerbe-of-lifts} is a gerbe."}
+{"_id": "6899", "title": "stacks-more-morphisms-lemma-gerbe-of-lifts-first-order", "text": "In Remark \\ref{remark-gerbe-of-lifts} assume $\\mathcal{X} \\subset \\mathcal{X}'$ is a first order thickening. Then \\begin{enumerate} \\item the automorphism sheaves of objects of the gerbe $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed in Remark \\ref{remark-gerbe-of-lifts} are abelian, and \\item the sheaf of groups $\\mathcal{G}$ constructed in Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts} is a quasi-coherent $\\mathcal{O}_W$-module. \\end{enumerate}"}
+{"_id": "6900", "title": "stacks-more-morphisms-lemma-inf-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$. Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally of finite presentation. Let $A \\to B$ be a flat $S$-algebra homomorphism. Let $x$ be an object of $\\mathcal{X}$ over $A$ and set $y = x|_B$. Then $\\text{Inf}_x(M) \\otimes_A B = \\text{Inf}_y(M \\otimes_A B)$."}
+{"_id": "6901", "title": "stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts", "text": "Let $\\mathcal{X}$ be an algebraic stack over a base scheme $S$. Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally of finite presentation. Let $(A' \\to A, x)$ be a deformation situation. Then the functor $$ F : B' \\longmapsto \\{\\text{lifts of }x|_{B' \\otimes_{A'} A}\\text{ to } B'\\}/\\text{isomorphisms} $$ is a sheaf on the site $(\\textit{Aff}/\\Spec(A'))_{fppf}$ of Topologies, Definition \\ref{topologies-definition-big-small-fppf}."}
+{"_id": "6902", "title": "stacks-more-morphisms-lemma-T-quasi-coherent", "text": "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose structure morphism $\\mathcal{X} \\to S$ is locally of finite presentation. Let $A \\to B$ be a flat $S$-algebra homomorphism. Let $x$ be an object of $\\mathcal{X}$ over $A$. Then $T_x(M) \\otimes_A B = T_y(M \\otimes_A B)$."}
+{"_id": "6903", "title": "stacks-more-morphisms-lemma-local-lift-enough", "text": "Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose structure morphism $\\mathcal{X} \\to S$ is locally of finite presentation. Let $(A' \\to A, x)$ be a deformation situation. If there exists a faithfully flat finitely presented $A'$-algebra $B'$ and an object $y'$ of $\\mathcal{X}$ over $B'$ lifting $x|_{B' \\otimes_{A'} A}$, then there exists an object $x'$ over $A'$ lifting $x$."}
+{"_id": "6904", "title": "stacks-more-morphisms-lemma-reformulate-formal-smoothness", "text": "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is formally smooth (Definition \\ref{definition-formally-smooth}) if and only if for every diagram (\\ref{equation-diagram}) and $\\gamma$ the category of dotted arrows is nonempty."}
+{"_id": "6905", "title": "stacks-more-morphisms-lemma-lift-to-smooth", "text": "Let $T \\to T'$ be a first order thickening of affine schemes. Let $\\mathcal{X}'$ be an algebraic stack over $T'$ whose structure morphism $\\mathcal{X}' \\to T'$ is smooth. Let $x : T \\to \\mathcal{X}'$ be a morphism over $T'$. Then there exists a morphsm $x' : T' \\to \\mathcal{X}'$ over $T'$ with $x'|_T = x$."}
+{"_id": "6906", "title": "stacks-more-morphisms-lemma-smooth-formally-smooth", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is smooth. \\item The morphism $f$ is locally of finite presentation and formally smooth. \\end{enumerate}"}
+{"_id": "6907", "title": "stacks-more-morphisms-lemma-flatten-stack", "text": "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. Let $V \\subset Y$ be an open subspace. Assume \\begin{enumerate} \\item $Y$ is quasi-compact and quasi-separated, \\item $f$ is of finite type and quasi-separated, \\item $V$ is quasi-compact, and \\item $\\mathcal{X}_V$ is flat and locally of finite presentation over $V$. \\end{enumerate} Then there exists a $V$-admissible blowup $Y' \\to Y$ and a closed substack $\\mathcal{X}' \\subset \\mathcal{X}_{Y'}$ with $\\mathcal{X}'_V = \\mathcal{X}_V$ such that $\\mathcal{X}' \\to Y'$ is flat and of finite presentation."}
+{"_id": "6908", "title": "stacks-more-morphisms-lemma-finite-cover-factor", "text": "Let $Y$ be a quasi-compact and quasi-separated algebraic space. Let $V \\subset Y$ be a quasi-compact open. Let $f : \\mathcal{X} \\to V$ be surjective, flat, and locally of finite presentation. Then there exists a finite surjective morphism $g : Y' \\to Y$ such that $V' = g^{-1}(V) \\to Y$ factors Zariski locally through $f$."}
+{"_id": "6911", "title": "stacks-more-morphisms-lemma-refined-valuative-criterion-proper", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$ be morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$. If given any $2$-commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[rr]^-y & & \\mathcal{Y} } $$ where $A$ is a discrete valuation ring with field of fractions $K$ and $\\gamma : y \\circ j \\to f \\circ h \\circ u$ there exist an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$ dominating $A$ such that the category of dotted arrows for the induced diagram $$ \\xymatrix{ \\Spec(K') \\ar[r]_-{x'} \\ar[d]_{j'} & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A') \\ar[r]^-{y'} \\ar@{..>}[ru] & \\mathcal{Y} } $$ with induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$ is nonempty (Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-fill-in-diagram}), then $f$ is proper."}
+{"_id": "6912", "title": "stacks-more-morphisms-lemma-refined-valuative-criterion-separated", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$ be morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$. If given any $2$-commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[rr]^-y \\ar@{..>}[rru] & & \\mathcal{Y} } $$ where $A$ is a discrete valuation ring with field of fractions $K$ and $\\gamma : y \\circ j \\to f \\circ h \\circ u$, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then $f$ is separated."}
+{"_id": "6913", "title": "stacks-more-morphisms-lemma-quotient-compare", "text": "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with $s, t : R \\to U$ flat and locally of finite presentation. Consider the algebraic stack $\\mathcal{X} = [U/R]$. Given an algebraic space $Y$ there is a $1$-to-$1$ correspondence between morphisms $f : \\mathcal{X} \\to Y$ and $R$-invariant morphisms $\\phi : U \\to Y$."}
+{"_id": "6914", "title": "stacks-more-morphisms-lemma-categorical-quotient-compare", "text": "With assumption and notation as in Lemma \\ref{lemma-quotient-compare}. Then $f$ is a (uniform) categorical moduli space if and only if $\\phi$ is a (uniform) categorical quotient. Similarly for moduli spaces in a full subcategory."}
+{"_id": "6915", "title": "stacks-more-morphisms-lemma-check-uniform-categorical-quotient-on-affines", "text": "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. If for every affine scheme $Y'$ and flat morphism $Y' \\to Y$ the base change $f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space, then $f$ is a uniform categorical moduli space."}
+{"_id": "6916", "title": "stacks-more-morphisms-lemma-well-nigh-affine", "text": "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is well-nigh affine, and \\item there exists a groupoid scheme $(U, R, s, t, c)$ with $U$ and $R$ affine and $s, t : R \\to U$ finite locally free such that $\\mathcal{X} = [U/R]$. \\end{enumerate} If true then $\\mathcal{X}$ is quasi-compact, quasi-DM, and separated."}
+{"_id": "6917", "title": "stacks-more-morphisms-lemma-affine-over-well-nigh-affine", "text": "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. \\begin{enumerate} \\item If $\\mathcal{X}$ is an algebraic space, then it is affine. \\item If $\\mathcal{X}' \\to \\mathcal{X}$ is an affine morphism of algebraic stacks, then $\\mathcal{X}'$ is well-nigh affine. \\end{enumerate}"}
+{"_id": "6918", "title": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space", "text": "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. There exists a uniform categorical moduli space $$ f : \\mathcal{X} \\longrightarrow M $$ in the category of affine schemes. Moreover $f$ is separated, quasi-compact, and a universal homeomorphism."}
+{"_id": "6919", "title": "stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale", "text": "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks. Assume $\\mathcal{X}'$ and $\\mathcal{X}$ are well-nigh affine, $h$ is \\'etale, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \\ref{stacks-morphisms-remark-identify-automorphism-groups}). Then there exists a cartesian diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\ M' \\ar[r] & M } $$ where $M' \\to M$ is \\'etale and the vertical arrows are the moduli spaces constructed in Lemma \\ref{lemma-well-nigh-affine-moduli-space}."}
+{"_id": "6920", "title": "stacks-more-morphisms-lemma-moduli-space-finite-affine", "text": "Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. The morphism $$ f : \\mathcal{X} \\longrightarrow M $$ of Lemma \\ref{lemma-well-nigh-affine-moduli-space} is a uniform categorical moduli space."}
+{"_id": "6921", "title": "stacks-more-morphisms-lemma-etale-separated-over-well-nigh-affine", "text": "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks. Assume $\\mathcal{X}$ is well-nigh affine, $h$ is \\'etale, $h$ is separated, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \\ref{stacks-morphisms-remark-identify-automorphism-groups}). Then there exists a cartesian diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\ M' \\ar[r] & M } $$ where $M' \\to M$ is a separated \\'etale morphism of schemes and $\\mathcal{X} \\to M$ is the moduli space constructed in Lemma \\ref{lemma-well-nigh-affine-moduli-space}."}
+{"_id": "6922", "title": "stacks-more-morphisms-lemma-etale-local-finite-inertia", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite. Then there exist a set $I$ and for $i \\in I$ a morphism of algebraic stacks $$ g_i : \\mathcal{X}_i \\longrightarrow \\mathcal{X} $$ with the following properties \\begin{enumerate} \\item $|\\mathcal{X}| = \\bigcup |g_i|(|\\mathcal{X}_i|)$, \\item $\\mathcal{X}_i$ is well-nigh affine, \\item $\\mathcal{I}_{\\mathcal{X}_i} \\to \\mathcal{X}_i \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$ is an isomorphism, and \\item $g_i : \\mathcal{X}_i \\to \\mathcal{X}$ is representable by algebraic spaces, separated, and \\'etale, \\end{enumerate}"}
+{"_id": "6923", "title": "stacks-more-morphisms-lemma-etale-separated-over-keel-mori", "text": "Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks. Assume \\begin{enumerate} \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite, \\item $h$ is \\'etale, separated, and induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \\ref{stacks-morphisms-remark-identify-automorphism-groups}). \\end{enumerate} Then there exists a cartesian diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\ M' \\ar[r] & M } $$ where $M' \\to M$ is a separated \\'etale morphism of algebraic spaces and the vertical arrows are the moduli spaces constructed in Theorem \\ref{theorem-keel-mori}."}
+{"_id": "6924", "title": "stacks-more-morphisms-lemma-keel-mori-finite-type", "text": "Let $p : \\mathcal{X} \\to Y$ be a morphism of an algebraic stack to an algebraic space. Assume \\begin{enumerate} \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite, \\item $Y$ is locally Noetherian, and \\item $p$ is locally of finite type. \\end{enumerate} Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in Theorem \\ref{theorem-keel-mori}. Then $M \\to Y$ is locally of finite type."}
+{"_id": "6925", "title": "stacks-more-morphisms-lemma-keel-mori-diagonal", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite. Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in Theorem \\ref{theorem-keel-mori}. \\begin{enumerate} \\item If $\\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated. \\item If $\\mathcal{X}$ is separated, then $M$ is separated. \\item Add more here, for example relative versions of the above. \\end{enumerate}"}
+{"_id": "6934", "title": "perfect-theorem-approximation", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Then approximation by perfect complexes holds on $X$."}
+{"_id": "6935", "title": "perfect-theorem-bondal-van-den-Bergh", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The category $D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ such that for  $E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent \\begin{enumerate} \\item $E = 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "6936", "title": "perfect-theorem-DQCoh-is-Ddga", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Then there exist a differential graded algebra $(E, \\text{d})$ with only a finite number of nonzero cohomology groups $H^i(E)$ such that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent to $D(E, \\text{d})$."}
+{"_id": "6937", "title": "perfect-lemma-quasi-coherence-direct-sums", "text": "Let $X$ be a scheme. Then $D_\\QCoh(\\mathcal{O}_X)$ has direct sums."}
+{"_id": "6938", "title": "perfect-lemma-Rlim-quasi-coherent", "text": "Let $X$ be a scheme. Let $(K_n)$ be an inverse system of $D_\\QCoh(\\mathcal{O}_X)$ with derived limit $K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$ is surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$. Then \\begin{enumerate} \\item $H^q(K) = \\lim H^q(K_n)$, \\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and \\item for every affine open $U \\subset X$ we have $H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$. \\end{enumerate}"}
+{"_id": "6939", "title": "perfect-lemma-nice-K-injective", "text": "Let $X$ be a scheme. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Then the canonical map $E \\to R\\lim \\tau_{\\geq -n}E$ is an isomorphism\\footnote{In particular, $E$ has a K-injective representative as in Cohomology, Lemma \\ref{cohomology-lemma-K-injective}.}."}
+{"_id": "6940", "title": "perfect-lemma-application-nice-K-injective", "text": "Let $X$ be a scheme. Let $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$ be an additive functor and $N \\geq 0$ an integer. Assume that \\begin{enumerate} \\item $F$ commutes with countable direct products, \\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$ quasi-coherent. \\end{enumerate} Then for $E \\in D_\\QCoh(\\mathcal{O}_X)$ \\begin{enumerate} \\item $H^i(RF(\\tau_{\\leq a}E)) \\to H^i(RF(E))$ is an isomorphism for $i \\leq a$, \\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism for $i \\geq b$, \\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some $-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$ for $i \\not \\in [a, b + N - 1]$. \\end{enumerate}"}
+{"_id": "6941", "title": "perfect-lemma-affine-compare-bounded", "text": "Let $X = \\Spec(A)$ be an affine scheme. All the functors in the diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[rr]_{(\\ref{equation-compare})} & & D_\\QCoh(\\mathcal{O}_X) \\ar[ld]^{R\\Gamma(X, -)} \\\\ & D(A) \\ar[lu]^{\\widetilde{\\ \\ }} } $$ are equivalences of triangulated categories. Moreover, for $E$ in $D_\\QCoh(\\mathcal{O}_X)$ we have $H^0(X, E) = H^0(X, H^0(E))$."}
+{"_id": "6942", "title": "perfect-lemma-affine-K-flat", "text": "Let $X = \\Spec(A)$ be an affine scheme. If $K^\\bullet$ is a K-flat complex of $A$-modules, then $\\widetilde{K^\\bullet}$ is a K-flat complex of $\\mathcal{O}_X$-modules."}
+{"_id": "6943", "title": "perfect-lemma-quasi-coherence-pushforward", "text": "If $f : X \\to Y$ is a morphism of affine schemes given by the ring map $A \\to B$, then the diagram $$ \\xymatrix{ D(B) \\ar[d] \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(A) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ commutes."}
+{"_id": "6944", "title": "perfect-lemma-quasi-coherence-pullback", "text": "Let $f : Y \\to X$ be a morphism of schemes. \\begin{enumerate} \\item The functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$. \\item If $X$ and $Y$ are affine and $f$ is given by the ring map $A \\to B$, then the diagram $$ \\xymatrix{ D(B) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) \\\\ D(A) \\ar[r] \\ar[u]^{- \\otimes_A^\\mathbf{L} B} & D_\\QCoh(\\mathcal{O}_X) \\ar[u]_{Lf^*} } $$ commutes. \\end{enumerate}"}
+{"_id": "6945", "title": "perfect-lemma-quasi-coherence-tensor-product", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item For objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$ the derived tensor product $K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L$ is in $D_\\QCoh(\\mathcal{O}_X)$. \\item If $X = \\Spec(A)$ is affine then $$ \\widetilde{M^\\bullet} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\widetilde{K^\\bullet} = \\widetilde{M^\\bullet \\otimes_A^\\mathbf{L} K^\\bullet} $$ for any pair of complexes of $A$-modules $K^\\bullet$, $M^\\bullet$. \\end{enumerate}"}
+{"_id": "6946", "title": "perfect-lemma-quasi-coherence-direct-image", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. \\begin{enumerate} \\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_S)$. \\item If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$ such that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$ with $H^m(E) = 0$ for $m > 0$ we have $H^m(Rf_*E) = 0$ for $m \\geq N$. \\item In fact, if $S$ is quasi-compact we can find $N = N(X, S, f)$ such that for every morphism of schemes $S' \\to S$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \\to S'$ is the base change of $f$. \\end{enumerate}"}
+{"_id": "6947", "title": "perfect-lemma-acyclicity-lemma", "text": "Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism of schemes. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules each of which is right acyclic for $f_*$. Then $f_*\\mathcal{F}^\\bullet$ represents $Rf_*\\mathcal{F}^\\bullet$ in $D(\\mathcal{O}_S)$."}
+{"_id": "6948", "title": "perfect-lemma-acyclicity-lemma-global", "text": "Let $X$ be a quasi-separated and quasi-compact scheme. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules each of which is right acyclic for $\\Gamma(X, -)$. Then $\\Gamma(X, \\mathcal{F}^\\bullet)$ represents $R\\Gamma(X, \\mathcal{F}^\\bullet)$ in $D(\\Gamma(X, \\mathcal{O}_X)$."}
+{"_id": "6949", "title": "perfect-lemma-spectral-sequence", "text": "Let $X$ be a quasi-separated and quasi-compact scheme. For any object $K$ of $D_\\QCoh(\\mathcal{O}_X)$ the spectral sequence $$ E_2^{i, j} = H^i(X, H^j(K)) \\Rightarrow H^{i + j}(X, K) $$ of Cohomology, Example \\ref{cohomology-example-spectral-sequence} is bounded and converges."}
+{"_id": "6950", "title": "perfect-lemma-quasi-coherence-pushforward-direct-sums", "text": "Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism of schemes. Then $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$ commutes with direct sums."}
+{"_id": "6951", "title": "perfect-lemma-pushforward-affine-morphism", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then $f_*\\mathcal{F}^\\bullet = Rf_*\\mathcal{F}^\\bullet$."}
+{"_id": "6952", "title": "perfect-lemma-affine-morphism", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Then $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$ reflects isomorphisms."}
+{"_id": "6953", "title": "perfect-lemma-affine-morphism-pull-push", "text": "Let $f : X \\to S$ be an affine morphism of schemes. For $E$ in $D_\\QCoh(\\mathcal{O}_S)$ we have $Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_S} f_*\\mathcal{O}_X$."}
+{"_id": "6954", "title": "perfect-lemma-affine-morphism-equivalence", "text": "Let $f : X \\to Y$ be an affine morphism of schemes. Then $f_*$ induces an equivalence $$ \\Phi : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(f_*\\mathcal{O}_X) $$ whose composition with $D_\\QCoh(f_*\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "6957", "title": "perfect-lemma-extended-alternating-zero", "text": "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and $\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations} the complex (\\ref{equation-extended-alternating}) restricts to an acyclic complex over $X \\setminus Z$."}
+{"_id": "6958", "title": "perfect-lemma-extended-alternating-represented", "text": "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and $\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}. If $\\mathcal{F}$ is quasi-coherent, then the complex (\\ref{equation-extended-alternating}) represents $i_* R\\mathcal{H}_Z(\\mathcal{F})$ in $D_Z(\\mathcal{O}_X)$."}
+{"_id": "6959", "title": "perfect-lemma-supported-trivial-vanishing", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset which can locally be cut out by at most $c$ elements of the structure sheaf. Then $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i > c$ and any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$."}
+{"_id": "6960", "title": "perfect-lemma-supported-vanishing", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset which can locally be cut out by a Koszul regular sequence having $c$ elements. Then $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i \\not = c$ for every flat, quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$."}
+{"_id": "6961", "title": "perfect-lemma-supported-map-determinant", "text": "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and $\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}. Let $a_{ji} \\in \\Gamma(X, \\mathcal{O}_X)$ for $1 \\leq i, j \\leq c$ and set $g_j = \\sum_{i = 1, \\ldots, c} a_{ji}f_i$. Assume $g_1, \\ldots, g_c$ scheme theoretically cut out $Z$. If $\\mathcal{F}$ is quasi-coherent, then $$ c_{f_1, \\ldots, f_c} = \\det(a_{ji}) c_{g_1, \\ldots, g_c} $$ where $c_{f_1, \\ldots, f_c}$ and $c_{g_1, \\ldots, g_c}$ are as in Remark \\ref{remark-supported-map-c-equations}."}
+{"_id": "6963", "title": "perfect-lemma-affine-pushforward", "text": "Let $f : X \\to Y$ be an affine morphism of schemes. Then $f_*$ defines a derived functor $f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$. This functor has the property that $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ commutes."}
+{"_id": "6964", "title": "perfect-lemma-flat-pushforward-coherator", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is quasi-compact, quasi-separated, and flat. Then, denoting $$ \\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y)) $$ the right derived functor of $f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ we have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$."}
+{"_id": "6965", "title": "perfect-lemma-affine-coherator", "text": "Let $X = \\Spec(A)$ be an affine scheme. Then \\begin{enumerate} \\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$ is the functor which sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module associated to the $A$-module $\\Gamma(X, \\mathcal{F})$, \\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$ is the functor which sends $E$ to the complex of quasi-coherent $\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$, \\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor $RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}). \\end{enumerate}"}
+{"_id": "6966", "title": "perfect-lemma-argument-proves", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Suppose that for every affine open $U \\subset X$ the right derived functor $$ \\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_X)) $$ of the left exact functor $j_* : \\QCoh(\\mathcal{O}_U) \\to \\QCoh(\\mathcal{O}_X)$ fits into a commutative diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_U)) \\ar[d]_\\Phi \\ar[r]_{i_U} & D_\\QCoh(\\mathcal{O}_U) \\ar[d]^{Rj_*} \\\\ D(\\QCoh(\\mathcal{O}_X)) \\ar[r]^{i_X} & D_\\QCoh(\\mathcal{O}_X) } $$ Then the functor (\\ref{equation-compare}) $$ D(\\QCoh(\\mathcal{O}_X)) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$."}
+{"_id": "6968", "title": "perfect-lemma-injective-quasi-coherent-sheaf-Noetherian", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{J}$ be an injective object of $\\QCoh(\\mathcal{O}_X)$. Then $\\mathcal{J}$ is a flasque sheaf of $\\mathcal{O}_X$-modules."}
+{"_id": "6969", "title": "perfect-lemma-Noetherian-pushforward", "text": "Let $f : X \\to Y$ be a morphism of Noetherian schemes. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$ such that the diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ commutes."}
+{"_id": "6970", "title": "perfect-lemma-alternating-cech-complex", "text": "In Situation \\ref{situation-complex}. Let $M$ be an $A$-module and denote $\\mathcal{F}$ the associated $\\mathcal{O}_X$-module. Then there is a canonical isomorphism of complexes $$ \\colim_e \\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M) \\longrightarrow \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ functorial in $M$."}
+{"_id": "6971", "title": "perfect-lemma-alternating-cech-complex-complex", "text": "In Situation \\ref{situation-complex}. Let $M^\\bullet$ be a complex of $A$-modules and denote $\\mathcal{F}^\\bullet$ the associated complex of $\\mathcal{O}_X$-modules. Then there is a canonical isomorphism of complexes $$ \\colim_e \\text{Tot}(\\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet)) \\longrightarrow \\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)) $$ functorial in $M^\\bullet$."}
+{"_id": "6972", "title": "perfect-lemma-alternating-cech-complex-complex-computes-cohomology", "text": "In Situation \\ref{situation-complex}. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then there is a canonical isomorphism $$ \\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)) \\longrightarrow R\\Gamma(U, \\mathcal{F}^\\bullet) $$ in $D(A)$ functorial in $\\mathcal{F}^\\bullet$."}
+{"_id": "6973", "title": "perfect-lemma-represent-cohomology-class-on-closed", "text": "In Situation \\ref{situation-complex}. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Assume that $H^i(E)|_U = 0$ for $i = - r + 1, \\ldots, 0$. Then given $s \\in H^0(X, E)$ there exists an $e \\geq 0$ and a morphism $K_e \\to E$ such that $s$ is in the image of $H^0(X, K_e) \\to H^0(X, E)$."}
+{"_id": "6974", "title": "perfect-lemma-pseudo-coherent", "text": "Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent object of $D(\\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$ and $H^m(E)$ is a quotient of a quasi-coherent $\\mathcal{O}_X$-module. If $E$ is pseudo-coherent, then $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "6975", "title": "perfect-lemma-pseudo-coherent-affine", "text": "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\\mathcal{O}_X)$. Then $E$ is an $m$-pseudo-coherent (resp.\\ pseudo-coherent) as an object of $D(\\mathcal{O}_X)$ if and only if $M^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) as a complex of $A$-modules."}
+{"_id": "6976", "title": "perfect-lemma-identify-pseudo-coherent-noetherian", "text": "Let $X$ be a Noetherian scheme. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. For $m \\in \\mathbf{Z}$ the following are equivalent \\begin{enumerate} \\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and \\item $E$ is $m$-pseudo-coherent. \\end{enumerate} In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "6977", "title": "perfect-lemma-tor-dimension-affine", "text": "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item $E$ has tor amplitude in $[a, b]$ if and only if $M^\\bullet$ has tor amplitude in $[a, b]$. \\item $E$ has finite tor dimension if and only if $M^\\bullet$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "6978", "title": "perfect-lemma-tor-dimension-rel-affine", "text": "Let $f : X \\to S$ be a morphism of affine schemes corresponding to the ring map $R \\to A$. Let $M^\\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item $E$ as an object of $D(f^{-1}\\mathcal{O}_S)$ has tor amplitude in $[a, b]$ if and only if $M^\\bullet$ has tor amplitude in $[a, b]$ as an object of $D(R)$. \\item $E$ locally has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$ if and only if $M^\\bullet$ has finite tor dimension as an object of $D(R)$. \\end{enumerate}"}
+{"_id": "6979", "title": "perfect-lemma-tor-qc-qs", "text": "Let $X$ be a quasi-separated scheme. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$. The following are equivalent \\begin{enumerate} \\item $E$ has tor amplitude in $[a, b]$, and \\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$ we have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$ for $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "6980", "title": "perfect-lemma-perfect-affine", "text": "Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\\mathcal{O}_X)$ if and only if $M^\\bullet$ is perfect as an object of $D(A)$."}
+{"_id": "6981", "title": "perfect-lemma-quasi-coherence-internal-hom", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item If $L$ is in $D^+_\\QCoh(\\mathcal{O}_X)$ and $K$ in $D(\\mathcal{O}_X)$ is pseudo-coherent, then $R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$ and locally bounded below. \\item If $L$ is in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ in $D(\\mathcal{O}_X)$ is perfect, then $R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$. \\item If $X = \\Spec(A)$ is affine and $K, L \\in D(A)$ then $$ R\\SheafHom(\\widetilde{K}, \\widetilde{L}) = \\widetilde{R\\Hom_A(K, L)} $$ in the following two cases \\begin{enumerate} \\item $K$ is pseudo-coherent and $L$ is bounded below, \\item $K$ is perfect and $L$ arbitrary. \\end{enumerate} \\item If $X = \\Spec(A)$ and $K, L$ are in $D(A)$, then the $n$th cohomology sheaf of $R\\SheafHom(\\widetilde{K}, \\widetilde{L})$ is the sheaf associated to the presheaf $$ X \\supset D(f) \\longmapsto \\Ext^n_{A_f}(K \\otimes_A A_f, L \\otimes_A A_f) $$ for $f \\in A$. \\end{enumerate}"}
+{"_id": "6982", "title": "perfect-lemma-internal-hom-evaluate-tensor-isomorphism", "text": "Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$. The map $$ K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L) \\longrightarrow R\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) $$ of Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \\begin{enumerate} \\item $M$ perfect, or \\item $K$ is perfect, or \\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "6983", "title": "perfect-lemma-coh-to-qcoh", "text": "Let $X$ be a Noetherian scheme. Then the functor $$ D^-(\\textit{Coh}(\\mathcal{O}_X)) \\longrightarrow D^-_{\\textit{Coh}(\\mathcal{O}_X)}(\\QCoh(\\mathcal{O}_X)) $$ is an equivalence."}
+{"_id": "6984", "title": "perfect-lemma-direct-image-coherent", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_S)$."}
+{"_id": "6985", "title": "perfect-lemma-direct-image-coherent-bdd-below", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_S)$."}
+{"_id": "6986", "title": "perfect-lemma-coherent-internal-hom", "text": "Let $X$ be a locally Noetherian scheme. If $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then $R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "6987", "title": "perfect-lemma-perfect-on-noetherian", "text": "Let $X$ be a Noetherian scheme. Let $E$ in $D(\\mathcal{O}_X)$ be perfect. Then \\begin{enumerate} \\item $E$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item if $L$ is in $D_{\\textit{Coh}}(\\mathcal{O}_X)$ then $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and $R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in $D_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item if $L$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ then $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and $R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item if $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ then $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and $R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item if $L$ is in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ then $E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and $R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6988", "title": "perfect-lemma-ext-finite", "text": "Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$. For $L$ in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules $\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite."}
+{"_id": "6989", "title": "perfect-lemma-perfect-on-regular", "text": "Let $X$ be a Noetherian regular scheme of finite dimension. Then every object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ is perfect and conversely every perfect object of $D(\\mathcal{O}_X)$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "6990", "title": "perfect-lemma-tor-amplitude-descends", "text": "Let $f : X \\to Y$ be a surjective flat morphism of schemes (or more generally locally ringed spaces). Let $E \\in D(\\mathcal{O}_Y)$. Let $a, b \\in \\mathbf{Z}$. Then $E$ has tor-amplitude in $[a, b]$ if and only if $Lf^*E$ has tor-amplitude in $[a, b]$."}
+{"_id": "6991", "title": "perfect-lemma-pseudo-coherent-descends-fpqc", "text": "Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each $Lf_i^*E$ is $m$-pseudo-coherent."}
+{"_id": "6994", "title": "perfect-lemma-closed-push-pseudo-coherent", "text": "Let $i : Z \\to X$ be a morphism of ringed spaces such that $i$ is a closed immersion of underlying topological spaces and such that $i_*\\mathcal{O}_Z$ is pseudo-coherent as an $\\mathcal{O}_X$-module. Let $E \\in D(\\mathcal{O}_Z)$. Then $E$ is $m$-pseudo-coherent if and only if $Ri_*E$ is $m$-pseudo-coherent."}
+{"_id": "6996", "title": "perfect-lemma-lift-quasi-coherent", "text": "Let $X$ be a scheme and let $j : U \\to X$ be a quasi-compact open immersion. The functors $$ D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_U) \\quad\\text{and}\\quad D^+_\\QCoh(\\mathcal{O}_X) \\to D^+_\\QCoh(\\mathcal{O}_U) $$ are essentially surjective. If $X$ is quasi-compact, then the functors $$ D^-_\\QCoh(\\mathcal{O}_X) \\to D^-_\\QCoh(\\mathcal{O}_U) \\quad\\text{and}\\quad D^b_\\QCoh(\\mathcal{O}_X) \\to D^b_\\QCoh(\\mathcal{O}_U) $$ are essentially surjective."}
+{"_id": "6997", "title": "perfect-lemma-lift-coherent", "text": "Let $X$ be a Noetherian scheme and let $j : U \\to X$ be an open immersion. The functor $D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ is essentially surjective."}
+{"_id": "6998", "title": "perfect-lemma-lift-pseudo-coherent", "text": "Let $X$ be an affine scheme and let $U \\subset X$ be a quasi-compact open subscheme. For any pseudo-coherent object $E$ of $D(\\mathcal{O}_U)$ there exists a bounded above complex of finite free $\\mathcal{O}_X$-modules  whose restriction to $U$ is isomorphic to $E$."}
+{"_id": "6999", "title": "perfect-lemma-vanishing-ext", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \\in D^b_\\QCoh(\\mathcal{O}_X)$. There exists an integer $n_0 > 0$ such that $\\Ext^n_{D(\\mathcal{O}_X)}(\\mathcal{E}, E) = 0$ for every finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ and every $n \\geq n_0$."}
+{"_id": "7000", "title": "perfect-lemma-lift-perfect-complex-plus-locally-free", "text": "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open. For every perfect object $E$ of $D(\\mathcal{O}_U)$ there exists an integer $r$ and a finite locally free sheaf $\\mathcal{F}$ on $U$ such that $\\mathcal{F}[-r] \\oplus E$ is the restriction of a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "7001", "title": "perfect-lemma-lift-map", "text": "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open. Let $E, E'$ be objects of $D_\\QCoh(\\mathcal{O}_X)$ with $E$ perfect. For every map $\\alpha : E|_U \\to E'|_U$ there exist maps $$ E \\xleftarrow{\\beta} E_1 \\xrightarrow{\\gamma} E' $$ of perfect complexes on $X$ such that $\\beta : E_1 \\to E$ restricts to an isomorphism on $U$ and such that $\\alpha = \\gamma|_U \\circ \\beta|_U^{-1}$. Moreover we can assume $E_1 = E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} I$ for some perfect complex $I$ on $X$."}
+{"_id": "7002", "title": "perfect-lemma-lift-perfect-complex-plus-shift", "text": "Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\\mathcal{O}_U)$ the object $F \\oplus F[1]$ is the restriction of a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "7003", "title": "perfect-lemma-perfect-into-support-on-T", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. For any morphism $\\alpha : E \\to E'$ in $D_\\QCoh(\\mathcal{O}_X)$ such that \\begin{enumerate} \\item $E$ is perfect, and \\item $E'$ is supported on $T = V(f)$ \\end{enumerate} there exists an $n \\geq 0$ such that $f^n \\alpha  = 0$."}
+{"_id": "7004", "title": "perfect-lemma-lift-perfect-complex-plus-shift-support", "text": "Let $X$ be an affine scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. Let $U \\subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\\mathcal{O}_U)$ supported on $T \\cap U$ the object $F \\oplus F[1]$ is the restriction of a perfect object $E$ of $D(\\mathcal{O}_X)$ supported in $T$."}
+{"_id": "7005", "title": "perfect-lemma-lift-map-from-perfect-complex-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open. Let $T \\subset X$ be a closed subset with $X \\setminus T$ retro-compact in $X$. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $\\alpha : P \\to E|_U$ be a map where $P$ is a perfect object of $D(\\mathcal{O}_U)$ supported on $T \\cap U$. Then there exists a map $\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$ supported on $T$ such that $P$ is a direct summand of $R|_U$ in $D(\\mathcal{O}_U)$ compatible $\\alpha$ and $\\beta|_U$."}
+{"_id": "7006", "title": "perfect-lemma-open", "text": "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. Let $(T, E, m)$ be a triple as in Definition \\ref{definition-approximation-holds}. If \\begin{enumerate} \\item $T \\subset U$, \\item approximation holds for $(T, E|_U, m)$, and \\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$, \\end{enumerate} then approximation holds for $(T, E, m)$."}
+{"_id": "7007", "title": "perfect-lemma-approximation-affine", "text": "Let $X$ be an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds} such that there exists an integer $r \\geq 0$ with \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent, \\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$, \\item $X \\setminus T$ is the union of $r$ affine opens. \\end{enumerate} In particular, approximation by perfect complexes holds for affine schemes."}
+{"_id": "7008", "title": "perfect-lemma-induction-step", "text": "Let $X$ be a scheme. Let $X = U \\cup V$ be an open covering with $U$ quasi-compact, $V$ affine, and $U \\cap V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation holds on $X$."}
+{"_id": "7009", "title": "perfect-lemma-direct-summand-of-a-restriction", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U$ be a quasi-compact open subscheme. Let $P$ be a perfect object of $D(\\mathcal{O}_U)$. Then $P$ is a direct summand of the restriction of a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "7010", "title": "perfect-lemma-orthogonal-koszul-complex", "text": "\\begin{reference} \\cite[Proposition 6.1]{Bokstedt-Neeman} \\end{reference} In Situation \\ref{situation-complex} denote $j : U \\to X$ the open immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$ corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$. For $E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent \\begin{enumerate} \\item $E = Rj_*(E|_U)$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(K[n], E) = 0$ for all $n \\in \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "7011", "title": "perfect-lemma-generator-with-support", "text": "\\begin{reference} \\cite[Theorem 6.8]{Rouquier-dimensions} \\end{reference} Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. With notation as above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a single perfect object."}
+{"_id": "7012", "title": "perfect-lemma-nonzero-some-cohomology", "text": "Let $X$ be a scheme and $\\mathcal{L}$ an ample invertible $\\mathcal{O}_X$-module. If $K$ is a nonzero object of $D_\\QCoh(\\mathcal{O}_X)$, then for some $n \\geq 0$ and $p \\in \\mathbf{Z}$ the cohomology group $H^p(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}^{\\otimes n})$ is nonzero."}
+{"_id": "7013", "title": "perfect-lemma-construct-the-next-one", "text": "Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. For every $a \\in \\mathbf{Z}$ there exists an exact complex $$ 0 \\to \\mathcal{O}_X(a) \\to \\ldots \\to \\mathcal{O}_X(a + i)^{\\oplus {n + 1 \\choose i}} \\to \\ldots \\to \\mathcal{O}_X(a + n + 1) \\to 0 $$ of vector bundles on $X$."}
+{"_id": "7014", "title": "perfect-lemma-generator-P1", "text": "Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. Then $$ E = \\mathcal{O}_X \\oplus \\mathcal{O}_X(-1) \\oplus \\ldots \\oplus \\mathcal{O}_X(-n) $$ is a generator (Derived Categories, Definition \\ref{derived-definition-generators}) of $D_\\QCoh(X)$."}
+{"_id": "7015", "title": "perfect-lemma-compact-is-perfect-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact if and only if it is perfect as an object of $D(\\mathcal{O}_X)$."}
+{"_id": "7016", "title": "perfect-lemma-map-from-pseudo-coherent-to-complex-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $U = X \\setminus T$ is quasi-compact. Let $\\alpha : P \\to E$ be a morphism of $D_\\QCoh(\\mathcal{O}_X)$ with either \\begin{enumerate} \\item $P$ is perfect and $E$ supported on $T$, or \\item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below. \\end{enumerate} Then there exists a perfect complex of $\\mathcal{O}_X$-modules $I$ and a map $I \\to \\mathcal{O}_X[0]$ such that $I \\otimes^\\mathbf{L} P \\to E$ is zero and such that $I|_U \\to \\mathcal{O}_U[0]$ is an isomorphism."}
+{"_id": "7017", "title": "perfect-lemma-tensor-with-QCoh-complex", "text": "Let $X$ be a scheme. Let $K^\\bullet$ be a complex of $\\mathcal{O}_X$-modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$ be the endomorphism differential graded algebra. Then the functor $$ - \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X) $$ of Differential Graded Algebra, Lemma \\ref{dga-lemma-tensor-with-complex-derived} has image contained in $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "7018", "title": "perfect-lemma-ext-from-perfect-into-bounded-QCoh", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item there exist integers $a \\leq b$ such that $\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$ for $L \\in D_\\QCoh(\\mathcal{O}_X)$ with $H^i(L) = 0$ for $i \\in [a, b]$, and \\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$ is zero for all but finitely many $n$. \\end{enumerate}"}
+{"_id": "7019", "title": "perfect-lemma-pseudo-coherent-hocolim", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \\in D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $K$ is pseudo-coherent, and \\item $K = \\text{hocolim} K_n$ where $K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$. \\end{enumerate}"}
+{"_id": "7021", "title": "perfect-lemma-Pn-module-category", "text": "\\begin{reference} \\cite{Beilinson} \\end{reference} Let $A$ be a ring. Let $X = \\mathbf{P}^n_A = \\text{Proj}(S)$ where $S = A[X_0, \\ldots, X_n]$. With $P$ as in (\\ref{equation-generator-Pn}) and $R$ as in (\\ref{equation-algebra-for-Pn}) the functor $$ - \\otimes_R^\\mathbf{L} P : D(R) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an $A$-linear equivalence of triangulated categories sending $R$ to $P$."}
+{"_id": "7022", "title": "perfect-lemma-better-coherator", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$ has a right adjoint $DQ_X$."}
+{"_id": "7023", "title": "perfect-lemma-pushforward-better-coherator", "text": "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. If the right adjoints $DQ_X$ and $DQ_Y$ of the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then $$ Rf_* \\circ DQ_X = DQ_Y \\circ Rf_* $$"}
+{"_id": "7024", "title": "perfect-lemma-boundedness-better-coherator", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The functor $DQ_X$ of Lemma \\ref{lemma-better-coherator} has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\\mathcal{O}_X)$ with $H^i(U, K) = 0$ for $U$ affine open in $X$ and $i \\not \\in [a, b]$, then the cohomology sheaves $H^i(DQ_X(K))$ are zero for $i \\not \\in [a, b + N]$."}
+{"_id": "7025", "title": "perfect-lemma-cohomology-base-change", "text": "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ in $D_\\QCoh(\\mathcal{O}_Y)$ the map $$ Rf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K \\longrightarrow Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K) $$ defined in Cohomology, Equation (\\ref{cohomology-equation-projection-formula-map}) is an isomorphism."}
+{"_id": "7026", "title": "perfect-lemma-tor-independent", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes. The following are equivalent \\begin{enumerate} \\item $X$ and $Y$ are tor independent over $S$, and \\item for every affine opens $U \\subset X$, $V \\subset Y$, $W \\subset S$ with $f(U) \\subset W$ and $g(V) \\subset W$ the rings $\\mathcal{O}_X(U)$ and $\\mathcal{O}_Y(V)$ are tor independent over $\\mathcal{O}_S(W)$. \\item there exists an affine open overing $S = \\bigcup W_i$ and for each $i$ affine open coverings $f^{-1}(W_i) = \\bigcup U_{ij}$ and $g^{-1}(W_i) = \\bigcup V_{ik}$ such that the rings $\\mathcal{O}_X(U_{ij})$ and $\\mathcal{O}_Y(V_{ik})$ are tor independent over $\\mathcal{O}_S(W_i)$ for all $i, j, k$. \\end{enumerate}"}
+{"_id": "7027", "title": "perfect-lemma-flat-base-change-tor-independent", "text": "Let $X \\to S$ and $Y \\to S$ be morphisms of schemes. Let $S' \\to S$ be a morphism of schemes and denote $X' = X \\times_S S'$ and $Y' = Y \\times_S S'$. If $X$ and $Y$ are tor independent over $S$ and $S' \\to S$ is flat, then $X'$ and $Y'$ are tor independent over $S'$."}
+{"_id": "7028", "title": "perfect-lemma-compare-base-change", "text": "Let $g : S' \\to S$ be a morphism of schemes. Let $f : X \\to S$ be quasi-compact and quasi-separated. Consider the base change diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ If $X$ and $S'$ are Tor independent over $S$, then for all $E \\in D_\\QCoh(\\mathcal{O}_X)$ we have $Rf'_*L(g')^*E = Lg^*Rf_*E$."}
+{"_id": "7029", "title": "perfect-lemma-affine-morphism-and-hom-out-of-perfect", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ of quasi-compact and quasi-separated schemes. Assume $g$ and $f$ Tor independent and $S = \\Spec(R)$, $S' = \\Spec(R')$ affine. For $M, K \\in D(\\mathcal{O}_X)$ the canonical map $$ R\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R' \\longrightarrow R\\Hom_{X'}(L(g')^*M, L(g')^*K) $$ in $D(R')$ is an isomorphism in the following two cases \\begin{enumerate} \\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or \\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent, $K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$. \\end{enumerate}"}
+{"_id": "7030", "title": "perfect-lemma-tor-independence-and-tor-amplitude", "text": "Consider a cartesian square of schemes $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ Assume $g$ and $f$ Tor independent. \\begin{enumerate} \\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_S$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{S'}$-modules. \\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat over $S$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "7031", "title": "perfect-lemma-compare-base-change-closed-immersion", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ Z' \\ar[r]_{i'} \\ar[d]_g & X' \\ar[d]^f \\\\ Z \\ar[r]^i & X } $$ where $i$ is a closed immersion. If $Z$ and $X'$ are tor independent over $X$, then $Ri'_* \\circ Lg^* = Lf^* \\circ Ri_*$ as functors $D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_{X'})$."}
+{"_id": "7032", "title": "perfect-lemma-kunneth", "text": "In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor-independent over $S$, then (\\ref{equation-kunneth}) is an isomorphism for $K \\in D_\\QCoh(\\mathcal{O}_X)$ and $M \\in D_\\QCoh(\\mathcal{O}_Y)$. If in addition $S = \\Spec(A)$ is affine, then the map (\\ref{equation-kunneth-global}) is an isomorphism."}
+{"_id": "7033", "title": "perfect-lemma-cohomology-de-rham-base-change", "text": "Let $a : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F}^\\bullet$ be a locally bounded complex of $a^{-1}\\mathcal{O}_S$-modules. Assume for all $n \\in \\mathbf{Z}$ the sheaf $\\mathcal{F}^n$ is a flat $a^{-1}\\mathcal{O}_S$-module and $\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module compatible with the given $a^{-1}\\mathcal{O}_S$-module structure (but the differentials in the complex $\\mathcal{F}^\\bullet$ need not be $\\mathcal{O}_X$-linear). Then the following hold \\begin{enumerate} \\item $Ra_*\\mathcal{F}^\\bullet$ is locally bounded, \\item $Ra_*\\mathcal{F}^\\bullet$ is in $D_\\QCoh(\\mathcal{O}_S)$, \\item $Ra_*\\mathcal{F}^\\bullet$ locally has finite tor dimension, \\item $\\mathcal{G} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet = Ra_*(a^{-1}\\mathcal{G} \\otimes_{a^{-1}\\mathcal{O}_S} \\mathcal{F}^\\bullet)$ for $\\mathcal{G} \\in \\QCoh(\\mathcal{O}_S)$, and \\item $K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet = Ra_*(a^{-1}K \\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}^\\bullet)$ for $K \\in D_\\QCoh(\\mathcal{O}_S)$. \\end{enumerate}"}
+{"_id": "7034", "title": "perfect-lemma-K-flat", "text": "Let $f : X \\to Y$ be a morphism of schemes with $Y = \\Spec(A)$ affine. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite affine open covering such that all the finite intersections $U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$ are affine. Let $\\mathcal{F}^\\bullet$ be a bounded complex of $f^{-1}\\mathcal{O}_Y$-modules. Assume for all $n \\in \\mathbf{Z}$ the sheaf $\\mathcal{F}^n$ is a flat $f^{-1}\\mathcal{O}_Y$-module and $\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module compatible with the given $p^{-1}\\mathcal{O}_Y$-module structure (but the differentials in the complex $\\mathcal{F}^\\bullet$ need not be $\\mathcal{O}_X$-linear). Then the complex $\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))$ is K-flat as a complex of $A$-modules."}
+{"_id": "7035", "title": "perfect-lemma-kunneth-single-sheaf", "text": "In the situation above the map (\\ref{equation-kunneth-single-sheaves}) is an isomorphism if $S$ is affine, $\\mathcal{F}$ and $\\mathcal{G}$ are $S$-flat and quasi-coherent and $X$ and $Y$ are quasi-compact with affine diagonal."}
+{"_id": "7036", "title": "perfect-lemma-kunneth-special", "text": "In the situation above the cup product (\\ref{equation-de-rham-kunneth}) is an isomorphism in $D(A)$ if the following assumptions hold \\begin{enumerate} \\item $S = \\Spec(A)$ is affine, \\item $X$ and $Y$ are quasi-compact with affine diagonal, \\item $\\mathcal{F}^\\bullet$ is bounded, \\item $\\mathcal{G}^\\bullet$ is bounded below, \\item $\\mathcal{F}^n$ is $S$-flat, and \\item $\\mathcal{G}^m$ is $S$-flat. \\end{enumerate}"}
+{"_id": "7038", "title": "perfect-lemma-single-complex-base-change-condition", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. The following are equivalent \\begin{enumerate} \\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc}) is an isomorphism, \\item for $U \\subset X$, $V' \\subset S'$ affine open both mapping into the affine open $V \\subset S$ with $U' = V' \\times_V U$ the composition $$ R\\Gamma(U, K) \\otimes_{\\mathcal{O}_S(U)}^\\mathbf{L} \\mathcal{O}_{S'}(V') \\to R\\Gamma(U, K) \\otimes_{\\mathcal{O}_X(U)}^\\mathbf{L} \\mathcal{O}_{X'}(U') \\to R\\Gamma(U', K') $$ is an isomorphism in $D(\\mathcal{O}_{S'}(V'))$, and \\item there is a set $I$ of quadruples $U_i, V_i', V_i, U_i'$, $i \\in I$ as in (2) with $X' = \\bigcup U'_i$. \\end{enumerate}"}
+{"_id": "7039", "title": "perfect-lemma-single-complex-base-change", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If \\begin{enumerate} \\item the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold, and \\item $f$ is quasi-compact and quasi-separated, \\end{enumerate} then the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$ is an isomorphism."}
+{"_id": "7040", "title": "perfect-lemma-single-complex-base-change-condition-inherited", "text": "Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold, then \\begin{enumerate} \\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$, \\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and \\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$ pseudo-coherent the equivalent conditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$. \\end{enumerate}"}
+{"_id": "7041", "title": "perfect-lemma-base-change-tensor", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$ be a bounded above complex of quasi-coherent $\\mathcal{O}_X$-modules flat over $S$. Then formation of $$ Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet) $$ commutes with arbitrary base change (see proof for precise statement)."}
+{"_id": "7042", "title": "perfect-lemma-base-change-RHom", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$ be a complex of quasi-coherent $\\mathcal{O}_X$-modules. If \\begin{enumerate} \\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above, and $\\mathcal{G}^n$ is flat over $S$, or \\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded, and $\\mathcal{G}^n$ is flat over $S$, \\end{enumerate} then formation of $$ Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet) $$ commutes with arbitrary base change (see proof for precise statement)."}
+{"_id": "7043", "title": "perfect-lemma-perfect-direct-image", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ such that \\begin{enumerate} \\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item the support of $H^i(E)$ is proper over $S$ for all $i$, and \\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$. \\end{enumerate} Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$."}
+{"_id": "7044", "title": "perfect-lemma-tensor-perfect", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent $\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)$ is a perfect object of $D(\\mathcal{O}_S)$."}
+{"_id": "7045", "title": "perfect-lemma-ext-perfect", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Let $\\mathcal{G}^\\bullet$ be a bounded complex of coherent $\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)$ is a perfect object of $D(\\mathcal{O}_S)$."}
+{"_id": "7046", "title": "perfect-lemma-flat-proper-perfect-direct-image", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a flat proper morphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$."}
+{"_id": "7047", "title": "perfect-lemma-compute-tensor-perfect", "text": "Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}. Then there are functorial isomorphisms $$ H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}) \\longrightarrow H^i(X, E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} (\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})) $$ for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof)."}
+{"_id": "7048", "title": "perfect-lemma-compute-ext-perfect", "text": "Assumptions and notation as in Lemma \\ref{lemma-ext-perfect}. Then there are functorial isomorphisms $$ H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof)."}
+{"_id": "7049", "title": "perfect-lemma-compute-ext", "text": "Let $f : X \\to S$ be a morphism of schemes, $E \\in D(\\mathcal{O}_X)$ and $\\mathcal{G}^\\bullet$ a complex of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $S$ is Noetherian, \\item $f$ is locally of finite type, \\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, \\item $\\mathcal{G}^\\bullet$ is a bounded complex of coherent $\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. \\end{enumerate} Then the following two statements are true \\begin{enumerate} \\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$ of $D(\\mathcal{O}_S)$ and functorial maps $$ \\alpha^i_\\mathcal{F} : \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) \\longrightarrow H^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof) such that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$ \\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$ and functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(E, \\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}) $$ for $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps. \\end{enumerate}"}
+{"_id": "7050", "title": "perfect-lemma-descend-homomorphisms", "text": "In Situation \\ref{situation-descent}. Let $E_0$ and $K_0$ be objects of $D(\\mathcal{O}_{S_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \\geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map $$ \\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{S_i})}(E_i, K_i) \\longrightarrow \\Hom_{D(\\mathcal{O}_S)}(E, K) $$ is an isomorphism if either \\begin{enumerate} \\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$, or \\item $E_0$ is pseudo-coherent and $K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "7051", "title": "perfect-lemma-descend-perfect", "text": "In Situation \\ref{situation-descent} the category of perfect objects of $D(\\mathcal{O}_S)$ is the colimit of the categories of perfect objects of $D(\\mathcal{O}_{S_i})$."}
+{"_id": "7052", "title": "perfect-lemma-base-change-tensor-perfect", "text": "Let $f : X \\to S$ be a morphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented $\\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet) $$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change."}
+{"_id": "7053", "title": "perfect-lemma-base-change-tensor-pseudo-coherent", "text": "Let $f : X \\to S$ be a morphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object. Let $\\mathcal{G}^\\bullet$ be a bounded above complex of finitely presented $\\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet) $$ is a pseudo-coherent object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change."}
+{"_id": "7054", "title": "perfect-lemma-flat-proper-perfect-direct-image-general", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a proper morphism of finite presentation. \\begin{enumerate} \\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change. \\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change. \\end{enumerate}"}
+{"_id": "7055", "title": "perfect-lemma-flat-proper-pseudo-coherent-direct-image-general", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a flat proper morphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change."}
+{"_id": "7057", "title": "perfect-lemma-base-change-RHom-perfect", "text": "Let $f : X \\to S$ be a morphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented $\\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet) $$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change."}
+{"_id": "7058", "title": "perfect-lemma-jump-loci", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent (for example perfect). For any $i \\in \\mathbf{Z}$ consider the function $$ \\beta_i : X \\longrightarrow \\{0, 1, 2, \\ldots\\},\\quad x \\longmapsto \\dim_{\\kappa(x)} H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x)) $$ Then we have \\begin{enumerate} \\item formation of $\\beta_i$ commutes with arbitrary base change, \\item the functions $\\beta_i$ are upper semi-continuous, and \\item the level sets of $\\beta_i$ are locally constructible in $X$. \\end{enumerate}"}
+{"_id": "7059", "title": "perfect-lemma-chi-locally-constant", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect. The function $$ \\chi_E : X \\longrightarrow \\mathbf{Z},\\quad x \\longmapsto \\sum (-1)^i \\dim_{\\kappa(x)} H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x)) $$ is locally constant on $X$."}
+{"_id": "7060", "title": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Given $i, r \\in \\mathbf{Z}$, there exists an open subscheme $U \\subset X$ characterized by the following \\begin{enumerate} \\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free $\\mathcal{O}_U$-module of rank $r$, \\item a morphism $f : Y \\to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$. \\end{enumerate}"}
+{"_id": "7061", "title": "perfect-lemma-locally-closed-where-H0-locally-free", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$. Let $r \\geq 0$. Then there exists a locally closed subscheme $j : Z \\to X$ characterized by the following \\begin{enumerate} \\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and \\item a morphism $f : Y \\to X$ factors through $Z$ if and only if for all morphisms $g : Y' \\to Y$ the $\\mathcal{O}_{Y'}$-module $H^a(L(f \\circ g)^*E)$ is locally free of rank $r$. \\end{enumerate} Moreover, $j : Z \\to X$ is of finite presentation and we have \\begin{enumerate} \\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then $H^a(Lf^*E) = g^*H^a(Lj^*E)$, \\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in X$, then $j$ is a closed immersion and given $f : Y \\to X$ the following are equivalent \\begin{enumerate} \\item $f : Y \\to X$ factors through $Z$, \\item $H^0(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$, \\end{enumerate} and if $r = 1$ these are also equivalent to \\begin{enumerate} \\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective. \\end{enumerate} \\end{enumerate}"}
+{"_id": "7062", "title": "perfect-lemma-jump-loci-geometric", "text": "Let $f : X \\to S$ be a flat, proper morphism of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$. For fixed $i \\in \\mathbf{Z}$ consider the function $$ \\beta_i : S \\to \\{0, 1, 2, \\ldots\\},\\quad s \\longmapsto \\dim_{\\kappa(s)} H^i(X_s, \\mathcal{F}_s) $$ Then we have \\begin{enumerate} \\item formation of $\\beta_i$ commutes with arbitrary base change, \\item the functions $\\beta_i$ are upper semi-continuous, and \\item the level sets of $\\beta_i$ are locally constructible in $S$. \\end{enumerate}"}
+{"_id": "7063", "title": "perfect-lemma-chi-locally-constant-geometric", "text": "Let $f : X \\to S$ be a flat, proper morphism of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$. The function $$ s \\longmapsto \\chi(X_s, \\mathcal{F}_s) $$ is locally constant on $S$. Formation of this function commutes with base change."}
+{"_id": "7064", "title": "perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric", "text": "Let $f : X \\to S$ be a flat, proper morphism of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$. Fix $i, r \\in \\mathbf{Z}$. Then there exists an open subscheme $U \\subset S$ with the following property: A morphism $T \\to S$ factors through $U$ if and only if $Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$."}
+{"_id": "7065", "title": "perfect-lemma-vanishing-implies-locally-free", "text": "Let $f : X \\to S$ be a morphism of finite presentation. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $S$ with support proper over $S$. If $R^if_*\\mathcal{F} = 0$ for $i > 0$, then $f_*\\mathcal{F}$ is locally free and its formation commutes with arbitrary base change (see proof for explanation)."}
+{"_id": "7066", "title": "perfect-lemma-proper-flat-h0", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item for all $s \\in S$ we have $\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$. \\end{enumerate} Then we have \\begin{enumerate} \\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after any base change, \\item[(b)] locally on $S$ we have $$ Rf_*\\mathcal{O}_X = \\mathcal{O}_S \\oplus P $$ in $D(\\mathcal{O}_S)$ where $P$ is perfect of tor amplitude in $[1, \\infty)$. \\end{enumerate}"}
+{"_id": "7067", "title": "perfect-lemma-proper-flat-geom-red-connected", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item the geometric fibres of $f$ are reduced and connected. \\end{enumerate} Then $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after any base change."}
+{"_id": "7068", "title": "perfect-lemma-proper-idempotent-on-fibre", "text": "Let $f : X \\to S$ be a proper morphism of schemes. Let $s \\in S$ and let $e \\in H^0(X_s, \\mathcal{O}_{X_s})$ be an idempotent. Then $e$ is in the image of the map $(f_*\\mathcal{O}_X)_s \\to H^0(X_s, \\mathcal{O}_{X_s})$."}
+{"_id": "7069", "title": "perfect-lemma-proper-flat-geom-red", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item the fibre $X_s$ is geometrically reduced. \\end{enumerate} Then, after replacing $S$ by an open neighbourhood of $s$, there exists a direct sum decomposition $Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$ in $D(\\mathcal{O}_S)$ where $f_*\\mathcal{O}_X$ is a finite \\'etale $\\mathcal{O}_S$-algebra and $P$ is a perfect of tor amplitude in $[1, \\infty)$."}
+{"_id": "7070", "title": "perfect-lemma-cohomology-over-coherent-ring", "text": "Let $R$ be a coherent ring. Let $X$ be a scheme of finite presentation over $R$. Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation, flat over $R$, with support proper over $R$. Then $H^i(X, \\mathcal{G})$ is a coherent $R$-module."}
+{"_id": "7071", "title": "perfect-lemma-countable-cohomology", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$ such that the cohomology sheaves $H^i(K)$ have countable sets of sections over affine opens. Then for any quasi-compact open $U \\subset X$ and any perfect object $E$ in $D(\\mathcal{O}_X)$ the sets $$ H^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U) $$ are countable."}
+{"_id": "7072", "title": "perfect-lemma-countable", "text": "Let $X$ be a quasi-compact and quasi-separated scheme such that the sets of sections of $\\mathcal{O}_X$ over affine opens are countable. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of $D(\\mathcal{O}_X)$, and \\item the cohomology sheaves $H^i(K)$ have countable sets of sections over affine opens. \\end{enumerate}"}
+{"_id": "7073", "title": "perfect-lemma-computing-sections-as-colim", "text": "Let $A$ be a ring. Let $X$ be a scheme of finite presentation over $A$. Let $f : U \\to X$ be a flat morphism of finite presentation. Then \\begin{enumerate} \\item there exists an inverse system of perfect objects $L_n$ of $D(\\mathcal{O}_X)$ such that $$ R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K) $$ in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and \\item there exists a system of perfect objects $E_n$ of $D(\\mathcal{O}_X)$ such that $$ R\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K) $$ in $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "7074", "title": "perfect-lemma-pseudo-coherent-over-algebra", "text": "Let $A$ be a ring. Let $R$ be a (possibly noncommutative) $A$-algebra which is finite free as an $A$-module. Then any object $M$ of $D(R)$ which is pseudo-coherent in $D(A)$ can be represented by a bounded above complex of finite free (right) $R$-modules."}
+{"_id": "7075", "title": "perfect-lemma-pseudo-coherent-on-projective-space", "text": "Let $A$ be a ring. Let $n \\geq 0$. Let $K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_A})$. The following are equivalent \\begin{enumerate} \\item $K$ is pseudo-coherent, \\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent object of $D(A)$ for each pseudo-coherent object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_A})$, \\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_A})$, \\item $R\\Hom_{\\mathbf{P}^n_A}(E, K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_A})$, \\item $R\\Gamma(\\mathbf{P}^n_A, K \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_A}(d))$ is pseudo-coherent object of $D(A)$ for $d = 0, 1, \\ldots, n$. \\end{enumerate}"}
+{"_id": "7076", "title": "perfect-lemma-perfect-enough", "text": "Let $A$ be a ring. Let $X$ be a scheme over $A$ which is quasi-compact and quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$. If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$, then $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$."}
+{"_id": "7077", "title": "perfect-lemma-affine-locally-rel-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is $S$-perfect, \\item for any affine open $U \\subset X$ mapping into an affine open $V \\subset S$ the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_S(V)$-perfect. \\item there exists an affine open covering $S = \\bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$ such that the complex $R\\Gamma(U_{ij}, E)$ is $\\mathcal{O}_S(V_i)$-perfect. \\end{enumerate}"}
+{"_id": "7079", "title": "perfect-lemma-perfect-relatively-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. A perfect object of $D(\\mathcal{O}_X)$ is $S$-perfect. If $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$ is $S$-perfect if $K$ is perfect and $M$ is $S$-perfect."}
+{"_id": "7080", "title": "perfect-lemma-base-change-relatively-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $g : S' \\to S$ be a morphism of schemes. Set $X' = S' \\times_S X$ and denote $g' : X' \\to X$ the projection. If $K \\in D(\\mathcal{O}_X)$ is $S$-perfect, then $L(g')^*K$ is $S'$-perfect."}
+{"_id": "7081", "title": "perfect-lemma-relative-descend-homomorphisms", "text": "In Situation \\ref{situation-relative-descent}. Let $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$. Set $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$ and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map $$ \\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i) \\longrightarrow \\Hom_{D(\\mathcal{O}_X)}(K, L) $$ is an isomorphism if $K_0$ is pseudo-coherent and $L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally) finite tor dimension as an object of $D((X_0 \\to S_0)^{-1}\\mathcal{O}_{S_0})$"}
+{"_id": "7082", "title": "perfect-lemma-descend-relatively-perfect", "text": "In Situation \\ref{situation-relative-descent} the category of $S$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories of $S_i$-perfect objects of $D(\\mathcal{O}_{X_i})$."}
+{"_id": "7083", "title": "perfect-lemma-derived-pushforward-rel-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat, proper, and of finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $S$-perfect. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change."}
+{"_id": "7084", "title": "perfect-lemma-compute-ext-rel-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $E, K \\in D(\\mathcal{O}_X)$. Assume \\begin{enumerate} \\item $S$ is quasi-compact and quasi-separated, \\item $f$ is proper, flat, and of finite presentation, \\item $E$ is $S$-perfect, \\item $K$ is pseudo-coherent. \\end{enumerate} Then there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$ such that $$ Rf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_S) $$ and the same is true after arbitrary base change: given $$ \\vcenter{ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } } \\quad\\quad \\begin{matrix} \\text{cartesian, then we have } \\\\ Rf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\ = R\\SheafHom(Lg^*L, \\mathcal{O}_{S'}) \\end{matrix} $$"}
+{"_id": "7085", "title": "perfect-lemma-bounded-on-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E$ is $S$-perfect, and \\item $E$ is locally bounded below and for every point $s \\in S$ the object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$ is locally bounded below. \\end{enumerate}"}
+{"_id": "7086", "title": "perfect-lemma-resolution-property-ample", "text": "Let $X$ be a scheme. If $X$ has an ample invertible $\\mathcal{O}_X$-module, then $X$ has the resolution property."}
+{"_id": "7087", "title": "perfect-lemma-resolution-property-ample-relative", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $Y$ is quasi-compact and quasi-separated and has the resolution property, \\item there exists an $f$-ample invertible module on $X$. \\end{enumerate} Then $X$ has the resolution property."}
+{"_id": "7088", "title": "perfect-lemma-resolution-property-goes-up-affine", "text": "Let $f : X \\to Y$ be an affine morphism of schemes with $Y$ quasi-compact and quasi-separated. If $Y$ has the resolution property, so does $X$."}
+{"_id": "7089", "title": "perfect-lemma-resolution-property-finite-number", "text": "Let $X$ be a scheme. Suppose given \\begin{enumerate} \\item a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_m$ \\item finite type quasi-coherent ideals $\\mathcal{I}_j$ with $V(\\mathcal{I}_j) = X \\setminus U_j$ \\end{enumerate} Then $X$ has the resolution property if and only if $\\mathcal{I}_j$ is the quotient of a finite locally free $\\mathcal{O}_X$-module for $j = 1, \\ldots, m$."}
+{"_id": "7090", "title": "perfect-lemma-regular-resolution-property", "text": "Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has the resolution property."}
+{"_id": "7091", "title": "perfect-lemma-resolution-property-descends", "text": "Let $X = \\lim X_i$ be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $X$ has the resolution property if and only if $X_i$ has the resolution properties for some $i$."}
+{"_id": "7092", "title": "perfect-lemma-resolution-property-affine-diagonal", "text": "\\begin{reference} Special case of \\cite[Proposition 1.3]{totaro_resolution}. \\end{reference} Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then $X$ has affine diagonal."}
+{"_id": "7093", "title": "perfect-lemma-construct-strictly-perfect", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of quasi-coherent $\\mathcal{O}_X$-modules representing a perfect object of $D(\\mathcal{O}_X)$. Then there exists a bounded complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules and a quasi-isomorphism $\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$."}
+{"_id": "7094", "title": "perfect-lemma-resolution-property-perfect-complex", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of $D(\\mathcal{O}_X)$ can be represented by a bounded complex of finite locally free $\\mathcal{O}_X$-modules."}
+{"_id": "7095", "title": "perfect-lemma-resolution-property-map-perfect-complex", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$ be finite complexes of finite locally free $\\mathcal{O}_X$-modules. Then any $\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$ can be represented by a diagram $$ \\mathcal{E}^\\bullet \\leftarrow \\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet $$ where $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free $\\mathcal{O}_X$-modules and where $\\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$ is a quasi-isomorphism."}
+{"_id": "7096", "title": "perfect-lemma-resolution-property-homotopy-map-perfect-complex", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$ be finite complexes of finite locally free $\\mathcal{O}_X$-modules. Let $\\alpha^\\bullet, \\beta^\\bullet :\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$ be two maps of complexes defining the same map in $D(\\mathcal{O}_X)$. Then there exists a quasi-isomorphism $\\gamma^\\bullet : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$ where $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free $\\mathcal{O}_X$-modules such that $\\alpha^\\bullet \\circ \\gamma^\\bullet$ and $\\beta^\\bullet \\circ \\gamma^\\bullet$ are homotopic maps of complexes."}
+{"_id": "7097", "title": "perfect-lemma-Noetherian-Kprime", "text": "Let $X$ be a Noetherian scheme. Then $$ K_0(\\textit{Coh}(\\mathcal{O}_X)) = K_0(D^b(\\textit{Coh}(\\mathcal{O}_X)) = K_0(D^b_{\\textit{Coh}}(\\mathcal{O}_X)) $$"}
+{"_id": "7099", "title": "perfect-lemma-Kprime-K", "text": "Let $X$ be a Noetherian regular scheme of finite dimension. Then the map $K_0(X) \\to K'_0(X)$ is an isomorphism."}
+{"_id": "7100", "title": "perfect-lemma-K-is-old-K", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then the map $K_0(\\textit{Vect}(X)) \\to K_0(X)$ is an isomorphism."}
+{"_id": "7101", "title": "perfect-lemma-projection-formula", "text": "Let $f : X \\to Y$ be a proper morphism of locally Noetherian schemes. Then we have $f_*(\\alpha \\cdot f^*\\beta) = f_*\\alpha \\cdot \\beta$ for $\\alpha \\in K'_0(X)$ and $\\beta \\in K_0(Y)$."}
+{"_id": "7102", "title": "perfect-lemma-determinant-two-term-complexes", "text": "Let $X$ be a scheme. There is a functor $$ \\det : \\left\\{ \\begin{matrix} \\text{category of perfect complexes} \\\\ \\text{with tor amplitude in }[-1, 0] \\\\ \\text{morphisms are isomorphisms} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{category of invertible modules} \\\\ \\text{morphisms are isomorphisms} \\end{matrix} \\right\\} $$ In addition, given a rank $0$ perfect object $L$ of $D(\\mathcal{O}_X)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\\delta(L) \\in \\Gamma(X, \\det(L))$ such that for any isomorphism $a : L \\to K$ in $D(\\mathcal{O}_X)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$. Moreover, the construction is affine locally given by the construction of More on Algebra, Section \\ref{more-algebra-section-determinants-complexes}."}
+{"_id": "7103", "title": "perfect-lemma-orthogonal-koszul-first-variant", "text": "In Situation \\ref{situation-complex} denote $j : U \\to X$ the open immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$ corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$. Consider the following conditions \\begin{enumerate} \\item The canonical map $\\tau_{\\geq a}E \\to \\tau_{\\geq a} Rj_*(E|_U)$ is an isomorphism. \\item We have $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\geq a$. \\end{enumerate} Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a + 1$."}
+{"_id": "7104", "title": "perfect-lemma-orthogonal-koszul-second-variant", "text": "In Situation \\ref{situation-complex} denote $j : U \\to X$ the open immersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$ corresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$. Consider the following conditions \\begin{enumerate} \\item The canonical map $\\tau_{\\leq a}E \\to \\tau_{\\leq a} Rj_*(E|_U)$ is an isomorphism, and \\item $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\leq a$. \\end{enumerate} Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a - 1$."}
+{"_id": "7105", "title": "perfect-lemma-bounded-truncation", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.  Let $a \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$. \\end{enumerate}"}
+{"_id": "7106", "title": "perfect-lemma-bounded-below-truncation", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$. Let $a \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and \\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$. \\end{enumerate}"}
+{"_id": "7107", "title": "perfect-proposition-quasi-compact-affine-diagonal", "text": "Let $X$ be a quasi-compact scheme with affine diagonal. Then the functor (\\ref{equation-compare}) $$ D(\\QCoh(\\mathcal{O}_X)) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$."}
+{"_id": "7108", "title": "perfect-proposition-Noetherian", "text": "Let $X$ be a Noetherian scheme. Then the functor (\\ref{equation-compare}) $$ D(\\QCoh(\\mathcal{O}_X)) \\longrightarrow D_\\QCoh(\\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$."}
+{"_id": "7109", "title": "perfect-proposition-represent-cohomology-class-on-open", "text": "In Situation \\ref{situation-complex}. For every object $E$ of $D_\\QCoh(\\mathcal{O}_X)$ the map (\\ref{equation-comparison}) is an isomorphism."}
+{"_id": "7110", "title": "perfect-proposition-DCoh", "text": "Let $X$ be a Noetherian scheme. Then the functors $$ D^-(\\textit{Coh}(\\mathcal{O}_X)) \\longrightarrow D^-_{\\textit{Coh}}(\\mathcal{O}_X) \\quad\\text{and}\\quad D^b(\\textit{Coh}(\\mathcal{O}_X)) \\longrightarrow D^b_{\\textit{Coh}}(\\mathcal{O}_X) $$ are equivalences."}
+{"_id": "7111", "title": "perfect-proposition-compact-is-perfect", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. An object of $D_\\QCoh(\\mathcal{O}_X)$ is compact if and only if it is perfect."}
+{"_id": "7113", "title": "perfect-proposition-detecting-bounded-above", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates  $D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$, \\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$, \\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$, \\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$, \\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$, \\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^-(\\mathbf{Z})$, \\item for every perfect object $P$ of $D(\\mathcal{O}_X)$ \\begin{enumerate} \\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$, \\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^-(\\mathbf{Z})$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "7114", "title": "perfect-proposition-detecting-bounded-below", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates $D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$, \\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$, \\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$, \\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$, \\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$, \\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^+(\\mathbf{Z})$, \\item for every perfect object $P$ of $D(\\mathcal{O}_X)$ \\begin{enumerate} \\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$, \\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$ is in $D^+(\\mathbf{Z})$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "7144", "title": "spaces-flat-theorem-finite-type-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X \\to Y$ is locally of finite presentation, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and \\item the set of weakly associated points of $Y$ is locally finite in $Y$. \\end{enumerate} Then $U = \\{x \\in |X| : \\mathcal{F}\\text{ flat at }x\\text{ over }Y\\}$ is open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module of finite presentation and flat over $Y$."}
+{"_id": "7145", "title": "spaces-flat-theorem-check-flatness-at-associated-points", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $x \\in |X|$ with image $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Consider the conditions \\begin{enumerate} \\item $\\mathcal{F}$ is flat at $x$ over $Y$, and \\item for every $x' \\in F \\cap \\text{Ass}_{X/Y}(\\mathcal{F})$ which specializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $Y$. \\end{enumerate} Then we always have (2) $\\Rightarrow$ (1). If $X$ and $Y$ are decent, then (1) $\\Rightarrow$ (2)."}
+{"_id": "7147", "title": "spaces-flat-theorem-flat-dimension-n-representable", "text": "In Situation \\ref{situation-flat-dimension-n}. Assume moreover that $f$ is of finite presentation, that $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and that $\\mathcal{F}$ is pure relative to $Y$. Then $F_n$ is an algebraic space and $F_n \\to Y$ is a monomorphism of finite presentation."}
+{"_id": "7148", "title": "spaces-flat-theorem-existence", "text": "In Situation \\ref{situation-existence} there exists a finitely presented $\\mathcal{O}_X$-module $\\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\\varphi_n$."}
+{"_id": "7149", "title": "spaces-flat-theorem-existence-derived", "text": "In Situation \\ref{situation-existence-derived} there exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$ such that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\\varphi_n$."}
+{"_id": "7150", "title": "spaces-flat-lemma-impure-limit", "text": "In Situation \\ref{situation-pre-pure}. Let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$ above $y$. Assume $T = \\lim_{i \\in I} T_i$ is a directed limit of affine schemes over $Y$. Then for some $i$ the triple $(T_i \\to Y, t'_i \\leadsto t_i, \\xi_i)$ is an impurity of $\\mathcal{F}$ above $y$."}
+{"_id": "7151", "title": "spaces-flat-lemma-flat-ascent-impurity", "text": "In Situation \\ref{situation-pre-pure}. Let $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed algebraic spaces over $S$. Assume $Y_1 \\to Y$ is flat at $y_1$. If $(T \\to Y, t' \\leadsto t, \\xi)$ is an impurity of $\\mathcal{F}$ above $y$, then there exists an impurity $(T_1 \\to Y_1, t_1' \\leadsto t_1, \\xi_1)$ of the pullback $\\mathcal{F}_1$ of $\\mathcal{F}$ to $X_1 = Y_1 \\times_Y X$ over $y_1$ such that $T_1$ is \\'etale over $Y_1 \\times_Y T$."}
+{"_id": "7152", "title": "spaces-flat-lemma-pure-along-X-y", "text": "In Situation \\ref{situation-pre-pure}. Let $\\overline{y}$ be a geometric point lying over $y$. Let $\\mathcal{O} = \\mathcal{O}_{Y, \\overline{y}}$ be the \\'etale local ring of $Y$ at $\\overline{y}$. Denote $Y^{sh} = \\Spec(\\mathcal{O})$, $X^{sh} = X \\times_Y Y^{sh}$, and $\\mathcal{F}^{sh}$ the pullback of $\\mathcal{F}$ to $X^{sh}$. The following are equivalent \\begin{enumerate} \\item there exists an impurity $(Y^{sh} \\to Y, y' \\leadsto \\overline{y}, \\xi)$ of $\\mathcal{F}$ above $y$, \\item every point of $\\text{Ass}_{X^{sh}/Y^{sh}}(\\mathcal{F}^{sh})$ specializes to a point of the closed fibre $X_{\\overline{y}}$, \\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $y$ such that $(T, t) \\to (Y, y)$ is an \\'etale neighbourhood, and \\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $y$ such that $T \\to Y$ is quasi-finite at $t$. \\end{enumerate}"}
+{"_id": "7153", "title": "spaces-flat-lemma-base-change-universally", "text": "In Situation \\ref{situation-pre-pure}. \\begin{enumerate} \\item $\\mathcal{F}$ is universally pure above $y$, and \\item for every morphism $(Y', y') \\to (Y, y)$ of pointed algebraic spaces the pullback $\\mathcal{F}_{Y'}$ is pure above $y'$. \\end{enumerate} In particular, $\\mathcal{F}$ is universally pure relative to $Y$ if and only if every base change $\\mathcal{F}_{Y'}$ of $\\mathcal{F}$ is pure relative to $Y'$."}
+{"_id": "7154", "title": "spaces-flat-lemma-quasi-finite-base-change", "text": "In Situation \\ref{situation-pre-pure}. Let $(Y', y') \\to (Y, y)$ be a morphism of pointed algebraic spaces. If $Y' \\to Y$ is quasi-finite at $y'$ and $\\mathcal{F}$ is pure above $y$, then $\\mathcal{F}_{Y'}$ is pure above $y'$."}
+{"_id": "7155", "title": "spaces-flat-lemma-flat-descend-pure", "text": "In Situation \\ref{situation-pre-pure}. Let $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed algebraic spaces. Assume $Y_1 \\to Y$ is flat at $y_1$. \\begin{enumerate} \\item If $\\mathcal{F}_{Y_1}$ is pure above $y_1$, then $\\mathcal{F}$ is pure above $y$. \\item If $\\mathcal{F}_{Y_1}$ is universally pure above $y_1$, then $\\mathcal{F}$ is universally pure above $y$. \\end{enumerate}"}
+{"_id": "7156", "title": "spaces-flat-lemma-supported-on-closed", "text": "In Situation \\ref{situation-pre-pure}. Let $i : Z \\to X$ be a closed immersion and assume that $\\mathcal{F} = i_*\\mathcal{G}$ for some finite type, quasi-coherent sheaf $\\mathcal{G}$ on $Z$. Then $\\mathcal{G}$ is (universally) pure above $y$ if and only if $\\mathcal{F}$ is (universally) pure above $y$."}
+{"_id": "7157", "title": "spaces-flat-lemma-proper-pure", "text": "In Situation \\ref{situation-pre-pure}. \\begin{enumerate} \\item If the support of $\\mathcal{F}$ is proper over $Y$, then $\\mathcal{F}$ is universally pure relative to $Y$. \\item If $f$ is proper, then $\\mathcal{F}$ is universally pure relative to $Y$. \\item If $f$ is proper, then $X$ is universally pure relative to $Y$. \\end{enumerate}"}
+{"_id": "7158", "title": "spaces-flat-lemma-existence-complete", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a finite type morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in |Y|$ be a point. There exists an \\'etale morphism $(Y', y') \\to (Y, y)$ with $Y'$ an affine scheme and \\'etale morphisms $h_i : W_i \\to X_{Y'}$, $i = 1, \\ldots, n$ such that for each $i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/W_i/Y'$ over $y'$, where $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $W_i$ and such that $|(X_{Y'})_{y'}| \\subset \\bigcup h_i(W_i)$."}
+{"_id": "7160", "title": "spaces-flat-lemma-bourbaki-finite-type-general-base-at-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $x \\in |X|$ with image $y \\in |Y|$. Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$. If $\\mathcal{F}$ is flat at $x$ over $Y$, then $$ x \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) \\Leftrightarrow y \\in \\text{WeakAss}_Y(\\mathcal{G}) \\text{ and } x \\in \\text{Ass}_{X/Y}(\\mathcal{F}). $$"}
+{"_id": "7161", "title": "spaces-flat-lemma-bourbaki-finite-type-general-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$ which is flat over $Y$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then we have $$ \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) = \\text{Ass}_{X/Y}(\\mathcal{F}) \\cap |f|^{-1}(\\text{WeakAss}_Y(\\mathcal{G})) $$"}
+{"_id": "7162", "title": "spaces-flat-lemma-finite-type-flat-along-fibre-free-variant", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$. \\end{enumerate} Then there exists an \\'etale morphism $(Y', y') \\to (Y, y)$ where $Y'$ is a scheme and a commutative diagram of algebraic spaces $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ Y & \\Spec(\\mathcal{O}_{Y', y'}) \\ar[l] } $$ such that $X' \\to X \\times_Y \\Spec(\\mathcal{O}_{Y', y'})$ is \\'etale, $|X'_{y'}| \\to F$ is surjective, $X'$ is affine, and $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{Y', y'}$-module."}
+{"_id": "7164", "title": "spaces-flat-lemma-finite-presentation-flat-along-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, and \\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$. \\end{enumerate} Then there exists a commutative diagram of algebraic spaces $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ Y & Y' \\ar[l]_h } $$ such that $h$ and $g$ are \\'etale, there is a point $y' \\in |Y'|$ mapping to $y$, we have $F \\subset g(|X'|)$, the algebraic spaces $X'$, $Y'$ are affine, and $\\Gamma(X', g^*\\mathcal{F})$ is a projective $\\Gamma(Y', \\mathcal{O}_{Y'})$-module."}
+{"_id": "7165", "title": "spaces-flat-lemma-associated-point-specializes", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space locally of finite type over $S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$ such that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$. Let $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$. If the closure of $\\{x'\\}$ in $|X|$ meets $|X_s|$, then the closure meets $\\text{Ass}_{X/S}(\\mathcal{F}) \\cap |X_s|$."}
+{"_id": "7166", "title": "spaces-flat-lemma-contains-relative-ass-after-base-change", "text": "Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be a morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$, then for any morphism $Z \\to Y$ we have $\\text{Ass}_{X_Z/Z}(\\mathcal{F}_Z) \\subset g_Z(|X'_Z|)$."}
+{"_id": "7167", "title": "spaces-flat-lemma-pure-on-top", "text": "Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be an \\'etale morphism of algebraic spaces over $Y$. Assume the structure morphisms $X' \\to Y$ and $X \\to Y$ are decent and of finite type. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. \\begin{enumerate} \\item If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$ and $g^*\\mathcal{F}$ is (universally) pure above $y$, then $\\mathcal{F}$ is (universally) pure above $y$. \\item If $\\mathcal{F}$ is pure above $y$, $g(|X'|)$ contains $F$, and $Y$ is affine local with closed point $y$, then $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$. \\item If $\\mathcal{F}$ is pure above $y$, $\\mathcal{F}$ is flat at all points of $F$, $g(|X'|)$ contains $\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap F$, and $Y$ is affine local with closed point $y$, then $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$. \\item Add more here. \\end{enumerate}"}
+{"_id": "7168", "title": "spaces-flat-lemma-finite-type-flat-pure-along-fibre-is-universal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in |Y|$. Assume \\begin{enumerate} \\item $f$ is decent and of finite type, \\item $\\mathcal{F}$ is of finite type, \\item $\\mathcal{F}$ is flat over $Y$ at all points lying over $y$, and \\item $\\mathcal{F}$ is pure above $y$. \\end{enumerate} Then $\\mathcal{F}$ is universally pure above $y$."}
+{"_id": "7169", "title": "spaces-flat-lemma-finite-type-flat-pure-is-universal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a decent, finite type morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Assume $\\mathcal{F}$ is flat over $Y$. In this case $\\mathcal{F}$ is pure relative to $Y$ if and only if $\\mathcal{F}$ is universally pure relative to $Y$."}
+{"_id": "7170", "title": "spaces-flat-lemma-universally-separating", "text": "Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be a flat morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module which is flat over $Y$. If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$ then the canonical map $$ \\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F} $$ is injective, and remains injective after any base change."}
+{"_id": "7171", "title": "spaces-flat-lemma-iso-sheaf", "text": "In Situation \\ref{situation-iso}. Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology."}
+{"_id": "7172", "title": "spaces-flat-lemma-iso-go-up", "text": "In Situation \\ref{situation-iso} let $X' \\to X$ be a flat morphism of algebraic spaces. Denote $u' : \\mathcal{F}' \\to \\mathcal{G}'$ the pullback of $u$ to $X'$. Denote $F'_{iso}$, $F'_{inj}$, $F'_{surj}$, $F'_{zero}$ the functors on $\\Sch/B$ associated to $u'$. \\begin{enumerate} \\item If $\\mathcal{G}$ is of finite type and the image of $|X'| \\to |X|$ contains the support of $\\mathcal{G}$, then $F_{surj} = F'_{surj}$ and $F_{zero} = F'_{zero}$. \\item If $\\mathcal{F}$ is of finite type and the image of $|X'| \\to |X|$ contains the support of $\\mathcal{F}$, then $F_{inj} = F'_{inj}$ and $F_{zero} = F'_{zero}$. \\item If $\\mathcal{F}$ and $\\mathcal{G}$ are of finite type and the image of $|X'| \\to |X|$ contains the supports of $\\mathcal{F}$ and $\\mathcal{G}$, then $F_{iso} = F'_{iso}$. \\end{enumerate}"}
+{"_id": "7173", "title": "spaces-flat-lemma-iso-limits", "text": "In Situation \\ref{situation-iso}. \\begin{enumerate} \\item If $\\mathcal{G}$ is of finite type and the scheme theoretic support of $\\mathcal{G}$ is quasi-compact over $B$, then $F_{surj}$ is limit preserving. \\item If $\\mathcal{F}$ of finite type and the scheme theoretic support of $\\mathcal{F}$ is quasi-compact over $B$, then $F_{zero}$ is limit preserving. \\item If $\\mathcal{F}$ is of finite type, $\\mathcal{G}$ is of finite presentation, and the scheme theoretic supports of $\\mathcal{F}$ and $\\mathcal{G}$ are quasi-compact over $B$, then $F_{iso}$ is limit preserving. \\end{enumerate}"}
+{"_id": "7174", "title": "spaces-flat-lemma-relate-zero-iso", "text": "In Situation \\ref{situation-iso} suppose given an exact sequence $$ \\mathcal{F} \\xrightarrow{u} \\mathcal{G} \\xrightarrow{v} \\mathcal{H} \\to 0 $$ Then we have $F_{v, iso} = F_{u, zero}$ with obvious notation."}
+{"_id": "7175", "title": "spaces-flat-lemma-relate-zero-affine", "text": "In Situation \\ref{situation-iso} suppose given an affine morphism $i : Z \\to X$ and a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{H}$ such that $\\mathcal{G} = i_*\\mathcal{H}$. Let $v : i^*\\mathcal{F} \\to \\mathcal{H}$ be the map adjoint to $u$. Then \\begin{enumerate} \\item $F_{v, zero} = F_{u, zero}$, and \\item if $i$ is a closed immersion, then $F_{v, surj} = F_{u, surj}$. \\end{enumerate}"}
+{"_id": "7178", "title": "spaces-flat-lemma-F-zero-somewhat-closed", "text": "In Situation \\ref{situation-somewhat-closed}. Let $T \\to S$ be a quasi-compact morphism of schemes such that the base change $u_T$ is zero. Then exists a closed subscheme $Z \\subset S$ such that (a) $T \\to S$ factors through $Z$ and (b) the base change $u_Z$ is zero. If $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module and the scheme theoretic support of $\\mathcal{F}$ is quasi-compact, then we can take $Z \\to S$ of finite presentation."}
+{"_id": "7179", "title": "spaces-flat-lemma-F-zero-module-map", "text": "Let $A$ be a ring. Let $u : M \\to N$ be a map of $A$-modules. If $N$ is projective as an $A$-module, then there exists an ideal $I \\subset A$ such that for any ring map $\\varphi : A \\to B$ the following are equivalent \\begin{enumerate} \\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is zero, and \\item $\\varphi(I) = 0$. \\end{enumerate}"}
+{"_id": "7181", "title": "spaces-flat-lemma-F-zero-closed-pure", "text": "In Situation \\ref{situation-iso}. Assume \\begin{enumerate} \\item $f$ is of finite presentation, and \\item $\\mathcal{G}$ is of finite presentation, flat over $B$, and pure relative to $B$. \\end{enumerate} Then $F_{zero}$ is an algebraic space and $F_{zero} \\to B$ is a closed immersion. If $\\mathcal{F}$ is of finite type, then $F_{zero} \\to B$ is of finite presentation."}
+{"_id": "7182", "title": "spaces-flat-lemma-F-zero-closed-proper", "text": "In Situation \\ref{situation-iso}. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation flat over $B$, \\item the support of $\\mathcal{G}$ is proper over $B$. \\end{enumerate} Then the functor $F_{zero}$ is an algebraic space and $F_{zero} \\to B$ is a closed immersion. If $\\mathcal{F}$ is of finite type, then $F_{zero} \\to B$ is of finite presentation."}
+{"_id": "7184", "title": "spaces-flat-lemma-F-surj-open", "text": "In Situation \\ref{situation-iso}. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{G}$ is of finite type, \\item the support of $\\mathcal{G}$ is proper over $B$. \\end{enumerate} Then $F_{surj}$ is an algebraic space and $F_{surj} \\to B$ is an open immersion."}
+{"_id": "7186", "title": "spaces-flat-lemma-pre-flat-dimension-n", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $n \\geq 0$. The following are equivalent \\begin{enumerate} \\item for some commutative diagram $$ \\xymatrix{ U \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with surjective, \\'etale vertical arrows where $U$ and $V$ are schemes, the sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$ in dimensions $\\geq n$ (More on Flatness, Definition \\ref{flat-definition-flat-dimension-n}), \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with \\'etale vertical arrows where $U$ and $V$ are schemes, the sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$ in dimensions $\\geq n$, and \\item for $x \\in |X|$ such that $\\mathcal{F}$ is not flat at $x$ over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dimension-fibre}). \\end{enumerate} If this is true, then it remains true after any base change $Y' \\to Y$."}
+{"_id": "7187", "title": "spaces-flat-lemma-flat-dimension-n", "text": "In Situation \\ref{situation-flat-dimension-n}. \\begin{enumerate} \\item The functor $F_n$ satisfies the sheaf property for the fpqc topology. \\item If $f$ is quasi-compact and locally of finite presentation and $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is limit preserving. \\end{enumerate}"}
+{"_id": "7188", "title": "spaces-flat-lemma-localize-flat-dimension-n", "text": "In Situation \\ref{situation-flat-dimension-n}. Let $h : X' \\to X$ be an \\'etale morphism. Set $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$. Let $F_n'$ be (\\ref{equation-flat-dimension-n}) associated to $(f' : X' \\to Y, \\mathcal{F}')$. Then $F_n$ is a subfunctor of $F_n'$ and if $h(X') \\supset \\text{Ass}_{X/Y}(\\mathcal{F})$, then $F_n = F'_n$."}
+{"_id": "7189", "title": "spaces-flat-lemma-when-universal-flattening", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $f$ is of finite presentation, $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is pure relative to $Y$, then there exists a universal flattening $Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is a monomorphism of finite presentation. \\item If $f$ is of finite presentation and $X$ is pure relative to $Y$, then there exists a universal flattening $Y' \\to Y$ of $X$. Moreover $Y' \\to Y$ is a monomorphism of finite presentation. \\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, then there exists a universal flattening $Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is a monomorphism of finite presentation. \\item If $f$ is proper and of finite presentation then there exists a universal flattening $Y' \\to Y$ of $X$. \\end{enumerate}"}
+{"_id": "7190", "title": "spaces-flat-lemma-compute-what-it-should-be", "text": "In Situation \\ref{situation-existence} consider $$ K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) = DQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n) $$ Then $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\\geq 0$."}
+{"_id": "7191", "title": "spaces-flat-lemma-compute-against-perfect", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. For any perfect object $E$ of $D(\\mathcal{O}_X)$ we have \\begin{enumerate} \\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n$ in $D(A_n)$, \\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$ in $D(A_n)$. \\end{enumerate} In both statements $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$."}
+{"_id": "7192", "title": "spaces-flat-lemma-relative-pseudo-coherence", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Then $K$ is pseudo-coherent relative to $A$."}
+{"_id": "7193", "title": "spaces-flat-lemma-compute-over-affine", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. For any \\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have $$ R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n = R\\Gamma(U_n, \\mathcal{F}_n) $$ in $D(A_n)$ where $U_n = U \\times_X X_n$."}
+{"_id": "7194", "title": "spaces-flat-lemma-finitely-presented", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Denote $X_0 \\subset |X|$ the closed subset consisting of points lying over the closed subset $\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$. There exists an open subspace $W \\subset X$ containing $X_0$ such that \\begin{enumerate} \\item $H^i(K)|_W$ is zero unless $i = 0$, \\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and \\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$. \\end{enumerate}"}
+{"_id": "7195", "title": "spaces-flat-lemma-proper-support", "text": "In Situation \\ref{situation-existence} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$ be as in Lemma \\ref{lemma-finitely-presented}. Set $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$, the support of $\\mathcal{F}$ is proper over $A$."}
+{"_id": "7196", "title": "spaces-flat-lemma-compute-what-it-should-be-derived", "text": "In Situation \\ref{situation-existence-derived} consider $$ K = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) = DQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n) $$ Then $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$."}
+{"_id": "7197", "title": "spaces-flat-lemma-compute-against-perfect-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect object $E$ of $D(\\mathcal{O}_X)$ the cohomology $$ M = R\\Gamma(X, K \\otimes^\\mathbf{L} E) $$ is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism $$ R\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n $$ in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$."}
+{"_id": "7198", "title": "spaces-flat-lemma-relative-pseudo-coherence-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$ is pseudo-coherent on $X$."}
+{"_id": "7199", "title": "spaces-flat-lemma-compute-over-affine-derived", "text": "In Situation \\ref{situation-existence-derived} let $K$ be as in Lemma \\ref{lemma-compute-what-it-should-be-derived}. For any \\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have $$ R\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n = R\\Gamma(U_n, K_n) $$ in $D(A_n)$ where $U_n = U \\times_X X_n$."}
+{"_id": "7200", "title": "spaces-flat-proposition-finite-presentation-flat-at-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in |X|$ with image $y \\in |Y|$. Assume that \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, and \\item $\\mathcal{F}$ is flat at $x$ over $Y$. \\end{enumerate} Then there exists a commutative diagram of pointed schemes $$ \\xymatrix{ (X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\ (Y, y) & (Y', y') \\ar[l] } $$ whose horizontal arrows are \\'etale such that $X'$, $Y'$ are affine and such that $\\Gamma(X', g^*\\mathcal{F})$ is a projective $\\Gamma(Y', \\mathcal{O}_{Y'})$-module."}
+{"_id": "7207", "title": "spaces-chow-lemma-length", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are equivalent \\begin{enumerate} \\item $\\text{length}_{\\mathcal{O}_{X, \\overline{x}}} \\mathcal{F}_{\\overline{x}} = d$ \\item for some \\'etale morphism $U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$ we have $\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$ \\item for any \\'etale morphism $U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$ we have $\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$ \\end{enumerate}"}
+{"_id": "7208", "title": "spaces-chow-lemma-length-closed-immersion", "text": "Let $S$ be a scheme. Let $i : Y \\to X$ be a closed immersion of algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Let $y \\in |Y|$ with image $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{G}$ has length $d$ at $y$, and \\item $i_*\\mathcal{G}$ has length $d$ at $x$. \\end{enumerate}"}
+{"_id": "7209", "title": "spaces-chow-lemma-length-finite", "text": "Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item for some \\'etale morphism $U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$ we have $u$ is a generic point of an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$, \\item for any \\'etale morphism $U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$ we have $u$ is a generic point of an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$, \\item the length of $\\mathcal{F}$ at $x$ is finite and nonzero. \\end{enumerate} If $X$ is decent (equivalently quasi-separated) then these are also equivalent to \\begin{enumerate} \\item[(4)] $x$ is a generic point of an irreducible component of $\\text{Supp}(\\mathcal{F})$. \\end{enumerate}"}
+{"_id": "7210", "title": "spaces-chow-lemma-point-of-max-dimension", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $T \\subset |X|$ be a closed subset and $t \\in T$. If $\\dim_\\delta(T) \\leq k$ and $\\delta(t) = k$, then $t$ is a generic point of an irreducible component of $T$."}
+{"_id": "7211", "title": "spaces-chow-lemma-reformulate-coeff-coherent", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module with $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Let $Z$ be an integral closed subspace of $X$ with $\\dim_\\delta(Z) = k$. Let $\\xi \\in |Z|$ be the generic point. Then the coefficient of $Z$ in $[\\mathcal{F}]_k$ is the length of $\\mathcal{F}$ at $\\xi$."}
+{"_id": "7212", "title": "spaces-chow-lemma-cycle-closed-coherent", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $Y \\subset X$ be a closed subspace. If $\\dim_\\delta(Y) \\leq k$, then $[Y]_k = [i_*\\mathcal{O}_Y]_k$ where $i : Y \\to X$ is the inclusion morphism."}
+{"_id": "7214", "title": "spaces-chow-lemma-proper-image", "text": "In Situation \\ref{situation-setup} let $X,Y/B$ be good and let $f : X \\to Y$ be a morphism over $B$. If $Z \\subset X$ is an integral closed subspace, then there exists a unique integral closed subspace $Z' \\subset Y$ such that there is a commutative diagram $$ \\xymatrix{ Z \\ar[r] \\ar[d] & X \\ar[d]^f \\\\ Z' \\ar[r] & Y } $$ with $Z \\to Z'$ dominant. If $f$ is proper, then $Z \\to Z'$ is proper and surjective."}
+{"_id": "7215", "title": "spaces-chow-lemma-equal-dimension", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good and let $f : X \\to Y$ be a morphism over $B$. Assume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$. Then either $f$ factors through a proper closed subspace of $Y$, or $f$ is dominant and the extension of function fields $R(X) / R(Y)$ is finite."}
+{"_id": "7216", "title": "spaces-chow-lemma-quasi-compact-locally-finite", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a morphism over $B$. Assume $f$ is quasi-compact, and $\\{T_i\\}_{i \\in I}$ is a locally finite collection of closed subsets of $|X|$. Then $\\{\\overline{|f|(T_i)}\\}_{i \\in I}$ is a locally finite collection of closed subsets of $|Y|$."}
+{"_id": "7217", "title": "spaces-chow-lemma-compose-pushforward", "text": "In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good. Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms over $B$. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$."}
+{"_id": "7218", "title": "spaces-chow-lemma-cycle-push-sheaf", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a proper morphism of good algebraic spaces over $B$. \\begin{enumerate} \\item Let $Z \\subset X$ be a closed subspace with $\\dim_\\delta(Z) \\leq k$. Then $$ f_*[Z]_k = [f_*{\\mathcal O}_Z]_k. $$ \\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then $$ f_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k. $$ \\end{enumerate} Note that the statement makes sense since $f_*\\mathcal{F}$ and $f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-proper-pushforward-coherent}."}
+{"_id": "7219", "title": "spaces-chow-lemma-flat-inverse-image-dimension", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a morphism over $B$. Assume $f$ is flat of relative dimension $r$. For any closed subset $T \\subset |Y|$ we have $$ \\dim_\\delta(|f|^{-1}(T)) = \\dim_\\delta(T) + r. $$ provided $|f|^{-1}(T)$ is nonempty. If $Z \\subset Y$ is an integral closed subscheme and $Z' \\subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$."}
+{"_id": "7220", "title": "spaces-chow-lemma-inverse-image-locally-finite", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a morphism over $B$. Assume $\\{T_i\\}_{i \\in I}$ is a locally finite collection of closed subsets of $|Y|$. Then $\\{|f|^{-1}(T_i)\\}_{i \\in I}$ is a locally finite collection of closed subsets of $X$."}
+{"_id": "7221", "title": "spaces-chow-lemma-exact-sequence-open", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $U \\subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$ and denote $i : Y \\to X$ the inclusion morphism. For every $k \\in \\mathbf{Z}$ the sequence $$ \\xymatrix{ Z_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0 } $$ is an exact complex of abelian groups."}
+{"_id": "7222", "title": "spaces-chow-lemma-etale-pullback", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale morphism of good algebraic spaces over $B$. If $Z \\subset Y$ is an integral closed subspace, then $f^*[Z] = \\sum [Z']$ where the sum is over the irreducible components (Remark \\ref{remark-irreducible-component}) of $f^{-1}(Z)$."}
+{"_id": "7223", "title": "spaces-chow-lemma-compose-flat-pullback", "text": "In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good. Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions $r$ and $s$ over $B$. Then $g \\circ f$ is flat of relative dimension $r + s$ and $$ f^* \\circ g^* = (g \\circ f)^* $$ as maps $Z_k(Z) \\to Z_{k + r + s}(X)$."}
+{"_id": "7224", "title": "spaces-chow-lemma-pullback-coherent", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. \\begin{enumerate} \\item Let $Z \\subset Y$ be a closed subspace with $\\dim_\\delta(Z) \\leq k$. Then we have $\\dim_\\delta(f^{-1}(Z)) \\leq k + r$ and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$. \\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$ and $$ f^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r} $$ in $Z_{k + r}(X)$. \\end{enumerate}"}
+{"_id": "7225", "title": "spaces-chow-lemma-flat-pullback-proper-pushforward", "text": "In Situation \\ref{situation-setup} let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a fibre product diagram of good algebraic spaces over $B$. Assume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$. Then also $f'$ is proper and $g'$ is flat of relative dimension $r$. For any $k$-cycle $\\alpha$ on $X$ we have $$ g^*f_*\\alpha = f'_*(g')^*\\alpha $$ in $Z_{k + r}(Y')$."}
+{"_id": "7227", "title": "spaces-chow-lemma-divisor-delta-dimension", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral. \\begin{enumerate} \\item If $Z \\subset X$ is an integral closed subspace, then the following are equivalent: \\begin{enumerate} \\item $Z$ is a prime divisor, \\item $|Z|$ has codimension $1$ in $|X|$, and \\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$. \\end{enumerate} \\item If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$. \\end{enumerate}"}
+{"_id": "7228", "title": "spaces-chow-lemma-etale-pullback-principal-divisor", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale morphism of good algebraic spaces over $B$. Assume $Y$ is integral. Let $g \\in R(Y)^*$. As cycles on $X$ we have $$ f^*(\\text{div}_Y(g)) = \\sum\\nolimits_{X'} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'}) $$ where the sum is over the irreducible components of $X$ (Remark \\ref{remark-irreducible-component})."}
+{"_id": "7229", "title": "spaces-chow-lemma-proper-pushforward-alteration", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Assume $X$, $Y$ are integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$. Let $p : X \\to Y$ be a dominant proper morphism. Let $f \\in R(X)^*$. Set $$ g = \\text{Nm}_{R(X)/R(Y)}(f). $$ Then we have $p_*\\text{div}(f) = \\text{div}(g)$."}
+{"_id": "7230", "title": "spaces-chow-lemma-restrict-to-open", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $U \\subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$ and denote $i : Y \\to X$ the inclusion morphism. Let $k \\in \\mathbf{Z}$. Suppose $\\alpha, \\beta \\in Z_k(X)$. If $\\alpha|_U \\sim_{rat} \\beta|_U$ then there exist a cycle $\\gamma \\in Z_k(Y)$ such that $$ \\alpha \\sim_{rat} \\beta + i_*\\gamma. $$ In other words, the sequence $$ \\xymatrix{ \\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0 } $$ is an exact complex of abelian groups."}
+{"_id": "7231", "title": "spaces-chow-lemma-prepare-flat-pullback-rational-equivalence", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Assume $Y$ integral with $\\dim_\\delta(Y) = k$. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Then for $g \\in R(Y)^*$ we have $$ f^*\\text{div}_Y(g) = \\sum m_{X', X} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'}) $$ as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible components $X'$ of $X$ and $m_{X', X}$ is the multiplicity of $X'$ in $X$."}
+{"_id": "7232", "title": "spaces-chow-lemma-flat-pullback-rational-equivalence", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles on $Y$. Then $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$."}
+{"_id": "7233", "title": "spaces-chow-lemma-proper-pushforward-rational-equivalence", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $p : X \\to Y$ be a proper morphism. Suppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent. Then $p_*\\alpha$ is rationally equivalent to $p_*\\beta$."}
+{"_id": "7234", "title": "spaces-chow-lemma-compute-c1", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section. Then $$ \\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1} $$ in $Z_{n - 1}(X)$ and $$ c_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1} $$ in $\\CH_{n - 1}(X)$."}
+{"_id": "7235", "title": "spaces-chow-lemma-Gm-torsor", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The morphism $$ q : T = \\underline{\\Spec}\\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n}\\right) \\longrightarrow X $$ has the following properties: \\begin{enumerate} \\item $q$ is surjective, smooth, affine, of relative dimension $1$, \\item there is an isomorphism $\\alpha : q^*\\mathcal{L} \\cong \\mathcal{O}_T$, \\item formation of $(q : T \\to X, \\alpha)$ commutes with base change, \\item $q^* : Z_k(X) \\to Z_{k + 1}(T)$ is injective, \\item if $Z \\subset X$ is an integral closed subspace, then $q^{-1}(Z) \\subset T$ is an integral closed subspace, \\item if $Z \\subset X$ is a closed subspace of $X$ of $\\delta$-dimension $\\leq k$, then $q^{-1}(Z)$ is a closed subspace of $T$ of $\\delta$-dimension $\\leq k + 1$ and $q^*[Z]_k = [q^{-1}(Z)]_{k + 1}$, \\item if $\\xi' \\in |T|$ is the generic point of the fibre of $|T| \\to |X|$ over $\\xi$, then the ring map $\\mathcal{O}_{X, \\xi}^h \\to \\mathcal{O}_{T, \\xi'}^h$ is flat, we have $\\mathfrak m_{\\xi'}^h = \\mathfrak m_\\xi^h \\mathcal{O}_{T, \\xi'}^h$, and the residue field extension is purely transcendental of transcendence degree $1$, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "7237", "title": "spaces-chow-lemma-c1-cap-additive", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$. Then $$ c_1(\\mathcal{L}) \\cap \\alpha  + c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha $$ in $\\CH_k(X)$ for every $\\alpha \\in Z_{k - 1}(X)$. Moreover, $c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$."}
+{"_id": "7238", "title": "spaces-chow-lemma-prepare-geometric-cap", "text": "In Situation \\ref{situation-setup} let $Y/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module. Let $s \\in \\Gamma(Y, \\mathcal{L})$ be a regular section and assume $\\dim_\\delta(Y) \\leq k + 1$. Write $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the irreducible components of $Y$ of $\\delta$-dimension $k + 1$. Set $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then \\begin{equation} \\label{equation-equal-as-cycles} [Z(s)]_k =  \\sum n_i[Z(s_i)]_k \\end{equation} as $k$-cycles on $Y$."}
+{"_id": "7239", "title": "spaces-chow-lemma-geometric-cap", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Y \\subset X$ be a closed subscheme with $\\dim_\\delta(Y) \\leq k + 1$ and let $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$ be a regular section. Then $$ c_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k $$ in $\\CH_k(X)$."}
+{"_id": "7240", "title": "spaces-chow-lemma-prepare-flat-pullback-cap-c1", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \\dim_\\delta(Y)$. Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$. Then we have $$ f^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i) $$ in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_i \\subset X$ of $\\delta$-dimension $n + r$, the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and $n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$."}
+{"_id": "7241", "title": "spaces-chow-lemma-flat-pullback-cap-c1", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Let $\\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha $$ in $\\CH_{k + r - 1}(X)$."}
+{"_id": "7242", "title": "spaces-chow-lemma-equal-c1-as-cycles", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a proper morphism. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Assume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$. Let $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$. Then $$ f_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) = [R(X) : R(Y)]\\text{div}_\\mathcal{L}(s). $$ as cycles on $Y$. In particular $$ f_*(c_1(f^*\\mathcal{L}) \\cap [X]) = c_1(\\mathcal{L}) \\cap f_*[Y]. $$"}
+{"_id": "7243", "title": "spaces-chow-lemma-pushforward-cap-c1", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $p : X \\to Y$ be a proper morphism. Let $\\alpha \\in Z_{k + 1}(X)$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ p_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha $$ in $\\CH_k(Y)$."}
+{"_id": "7244", "title": "spaces-chow-lemma-key-formula", "text": "In the situation above the cycle $$ \\sum (Z_i \\to X)_*\\left( \\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right) $$ is equal to the cycle $$ \\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i)) $$"}
+{"_id": "7245", "title": "spaces-chow-lemma-commutativity-on-integral", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\\mathcal{L}$ and a nonzero meromorphic section $t$ of $\\mathcal{N}$. Set $\\alpha = \\text{div}_\\mathcal{L}(s)$ and $\\beta = \\text{div}_\\mathcal{N}(t)$. Then $$ c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{L}) \\cap \\beta $$ in $\\CH_{n - 2}(X)$."}
+{"_id": "7246", "title": "spaces-chow-lemma-factors", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be invertible on $X$. The operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$ factors through rational equivalence to give an operation $$ c_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X) $$"}
+{"_id": "7247", "title": "spaces-chow-lemma-cap-commutative", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$. For any $\\alpha \\in \\CH_{k + 2}(X)$ we have $$ c_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha = c_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha $$ as elements of $\\CH_k(X)$."}
+{"_id": "7248", "title": "spaces-chow-lemma-support-cap-effective-Cartier", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$ in $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$."}
+{"_id": "7249", "title": "spaces-chow-lemma-closed-in-X-gysin", "text": "In Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a proper morphism of good algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Form the diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ as in Remark \\ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have $i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$ (this makes sense as $f_*$ is defined on the level of cycles)."}
+{"_id": "7250", "title": "spaces-chow-lemma-gysin-flat-pullback", "text": "In Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a flat morphism of relative dimension $r$ of good algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Form the diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ as in Remark \\ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\\alpha$ on $X$ we have $(i')^*f^*\\alpha = g^*i^*\\alpha'$ in $\\CH_{k + r}(D)$ (this makes sense as $f^*$ is defined on the level of cycles)."}
+{"_id": "7251", "title": "spaces-chow-lemma-easy-gysin", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. Let $Z \\subset X$ be a closed subscheme such that $\\dim_\\delta(Z) \\leq k + 1$ and such that $D \\cap Z$ is an effective Cartier divisor on $Z$. Then $i^*([Z]_{k + 1}) = [D \\cap Z]_k$."}
+{"_id": "7252", "title": "spaces-chow-lemma-gysin-factors-general", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ integral and $n = \\dim_\\delta(X)$. Let $i : D \\to X$ be an effective Cartier divisor. Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module and let $t$ be a nonzero meromorphic section of $\\mathcal{N}$. Then $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}) \\cap [D]_{n - 1}$ in $\\CH_{n - 2}(D)$."}
+{"_id": "7253", "title": "spaces-chow-lemma-gysin-factors", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in Definition \\ref{definition-gysin-homomorphism}. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$."}
+{"_id": "7254", "title": "spaces-chow-lemma-gysin-commutes-cap-c1", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. Let $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module. Then $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$ in $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(Z)$."}
+{"_id": "7255", "title": "spaces-chow-lemma-gysin-commutes-gysin", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s, i : D \\to X)$ and $(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in Definition \\ref{definition-gysin-homomorphism}. Then the diagram $$ \\xymatrix{ \\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d] \\\\ \\CH_{k - 1}(D') \\ar[r] & \\CH_{k - 2}(D \\cap D') } $$ commutes where each of the maps is a gysin map."}
+{"_id": "7256", "title": "spaces-chow-lemma-relative-effective-cartier", "text": "In Situation \\ref{situation-setup}. Let $X, Y/B$ be good. Let $p : X \\to Y$ be a flat morphism of relative dimension $r$. Let $i : D \\to X$ be a relative effective Cartier divisor (Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relative-effective-Cartier-divisor}). Let $\\mathcal{L} = \\mathcal{O}_X(D)$. For any $\\alpha \\in \\CH_{k + 1}(Y)$ we have $$ i^*p^*\\alpha = (p|_D)^*\\alpha $$ in $\\CH_{k + r}(D)$ and $$ c_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha) $$ in $\\CH_{k + r}(X)$."}
+{"_id": "7257", "title": "spaces-chow-lemma-pullback-affine-fibres-surjective", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a quasi-compact flat morphism over $B$ of relative dimension $r$. Assume that for every $y \\in Y$ we have $X_y \\cong \\mathbf{A}^r_{\\kappa(y)}$. Then $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all $k \\in \\mathbf{Z}$."}
+{"_id": "7258", "title": "spaces-chow-lemma-linebundle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $$ p : L = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L})) \\longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$."}
+{"_id": "7259", "title": "spaces-chow-lemma-cap-c1-bivariant", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then the rule that to $f : X' \\to X$ assigns $c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$ is a bivariant class of degree $1$."}
+{"_id": "7260", "title": "spaces-chow-lemma-flat-pullback-bivariant", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of good algebraic spaces over $B$ which is flat of relative dimension $r$. Then the rule that to $Y' \\to Y$ assigns $(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$ is a bivariant class of degree $-r$."}
+{"_id": "7262", "title": "spaces-chow-lemma-push-proper-bivariant", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of good algebraic spaces over $B$. Let $c \\in A^p(X \\to Z)$ and assume $f$ is proper. Then the rule that to $X' \\to X$ assigns $\\alpha \\longmapsto f_*(c \\cap \\alpha)$ is a bivariant class of degree $p$."}
+{"_id": "7263", "title": "spaces-chow-lemma-c1-center", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $c_1(\\mathcal{L}) \\in A^1(X)$ commutes with every element $c \\in A^p(X)$."}
+{"_id": "7264", "title": "spaces-chow-lemma-bivariant-zero", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $c \\in A^p(X)$. Then $c$ is zero if and only if $c \\cap [Y] = 0$ in $\\CH_*(Y)$ for every integral algebraic space $Y$ locally of finite type over $X$."}
+{"_id": "7265", "title": "spaces-chow-lemma-cap-projective-bundle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle associated to $\\mathcal{E}$. For any $\\alpha \\in \\CH_k(X)$ the element $$ \\pi_*\\left( c_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha \\right) \\in \\CH_{k + r - 1 - s}(X) $$ is $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$."}
+{"_id": "7266", "title": "spaces-chow-lemma-chow-ring-projective-bundle", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle associated to $\\mathcal{E}$. The map $$ \\bigoplus\\nolimits_{i = 0}^{r - 1} \\CH_{k + i}(X) \\longrightarrow \\CH_{k + r - 1}(P), $$ $$ (\\alpha_0, \\ldots, \\alpha_{r-1}) \\longmapsto \\pi^*\\alpha_0 + c_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1 + \\ldots + c_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1} $$ is an isomorphism."}
+{"_id": "7268", "title": "spaces-chow-lemma-segre-classes", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space bundle associated to $\\mathcal{E}$. For every morphism $X' \\to X$ of good algebraic spaces over $B$ there are unique maps $$ c_i(\\mathcal{E}) \\cap - : \\CH_k(X') \\longrightarrow \\CH_{k - i}(X'),\\quad i = 0, \\ldots, r $$ such that for $\\alpha \\in \\CH_k(X')$ we have $c_0(\\mathcal{E}) \\cap \\alpha = \\alpha$ and $$ \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap (\\pi')^*\\left(c_{r - i}(\\mathcal{E}) \\cap \\alpha\\right) = 0 $$ where $\\pi' : P' \\to X'$ is the base change of $\\pi$. Moreover, these maps define a bivariant class $c_i(\\mathcal{E})$ of degree $i$ on $X$."}
+{"_id": "7271", "title": "spaces-chow-lemma-chern-classes-E-tensor-L", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Then we have \\begin{equation} \\label{equation-twist} c_i({\\mathcal E} \\otimes {\\mathcal L}) = \\sum\\nolimits_{j = 0}^i \\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j \\end{equation} in $A^*(X)$."}
+{"_id": "7272", "title": "spaces-chow-lemma-get-rid-of-trivial-subbundle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0 $$ Then we have $$ c_r(\\mathcal{E}) = 0, \\quad c_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1 $$ in $A^*(X)$."}
+{"_id": "7273", "title": "spaces-chow-lemma-additivity-invertible-subsheaf", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0 $$ where $\\mathcal{L}$ is an invertible sheaf. Then $$ c(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F}) $$ in $A^*(X)$."}
+{"_id": "7275", "title": "spaces-chow-lemma-chern-filter-by-linebundles", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible $\\mathcal{O}_X$-modules. Let $\\mathcal{E}$ be a locally free rank $\\mathcal{O}_X$-module endowed with a filtration $$ 0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2 \\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E} $$ such that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$. Set $c_1({\\mathcal L}_i) = x_i$. Then $$ c(\\mathcal{E}) = \\prod\\nolimits_{i = 1}^r (1 + x_i) $$ in $A^*(X)$."}
+{"_id": "7276", "title": "spaces-chow-lemma-splitting-principle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}_i$ be a finite collection of locally free $\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective flat morphism $\\pi : P \\to X$ of relative dimension $d$ such that \\begin{enumerate} \\item for any morphism $f : Y \\to X$ of good algebraic spaces over $B$ the map $\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and \\item each $\\pi^*\\mathcal{E}_i$ has a filtration whose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$ are invertible ${\\mathcal O}_P$-modules. \\end{enumerate}"}
+{"_id": "7277", "title": "spaces-chow-lemma-chern-classes-dual", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module with dual $\\mathcal{E}^\\vee$. Then $$ c_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E}) $$ in $A^i(X)$."}
+{"_id": "7307", "title": "sdga-theorem-qis-into-dg-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. For every differential graded $\\mathcal{A}$-module $\\mathcal{M}$ there exists a quasi-isomorphism $\\mathcal{M} \\to \\mathcal{I}$ where $\\mathcal{I}$ is a graded injective and K-injective differential graded $\\mathcal{A}$-module. Moreover, the construction is functorial in $\\mathcal{M}$."}
+{"_id": "7308", "title": "sdga-lemma-gm-abelian", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra. The category $\\text{Mod}_\\mathcal{A}$ is an abelian category with the following properties \\begin{enumerate} \\item $\\text{Mod}_\\mathcal{A}$ has arbitrary direct sums, \\item $\\text{Mod}_\\mathcal{A}$ has arbitrary colimits, \\item filtered colimit in $\\text{Mod}_\\mathcal{A}$ are exact, \\item $\\text{Mod}_\\mathcal{A}$ has arbitrary products, \\item $\\text{Mod}_\\mathcal{A}$ has arbitrary limits. \\end{enumerate} The functor $$ \\text{Mod}_\\mathcal{A} \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad \\mathcal{M} \\longmapsto \\mathcal{M}^n $$ sending a graded $\\mathcal{A}$-module to its $n$th term commutes with all limits and colimits."}
+{"_id": "7309", "title": "sdga-lemma-tensor-hom-adjunction-gr", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ and $\\mathcal{B}$ be a sheaves of graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right graded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$ be a right graded $\\mathcal{B}$-module. With conventions as above we have $$ \\Hom_{\\text{Mod}_\\mathcal{B}^{gr}}( \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) = \\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}( \\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L})) $$ and $$ \\SheafHom_\\mathcal{B}^{gr}( \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) = \\SheafHom_\\mathcal{A}^{gr}( \\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L})) $$ functorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$."}
+{"_id": "7310", "title": "sdga-lemma-adjunction-push-pull-gr", "text": "In the situation above we have $$ \\Hom_{\\text{Mod}_\\mathcal{B}^{gr}}( \\mathcal{N}, f_*\\mathcal{M}) = \\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}( f^*\\mathcal{N}, \\mathcal{M}) $$"}
+{"_id": "7311", "title": "sdga-lemma-extension-by-zero-graded", "text": "In the situation above we have $$ \\Hom_{\\text{Mod}_\\mathcal{A}^{gr}}( j_!\\mathcal{M}, \\mathcal{N}) = \\Hom_{\\text{Mod}_{\\mathcal{A}_U}^{gr}}( \\mathcal{M}, j^*\\mathcal{N}) $$"}
+{"_id": "7312", "title": "sdga-lemma-tensor-with-extension-by-zero", "text": "In the situation above, let $\\mathcal{M}$ be a right graded $\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left graded $\\mathcal{A}$-module. Then $$ j_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} = j_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U) $$ as graded $\\mathcal{O}$-modules functorially in $\\mathcal{M}$ and $\\mathcal{N}$."}
+{"_id": "7313", "title": "sdga-lemma-gm-grothendieck-abelian", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra. The category $\\text{Mod}_\\mathcal{A}$ is a Grothendieck abelian category."}
+{"_id": "7314", "title": "sdga-lemma-dgm-abelian", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra. The category $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ is an abelian category with the following properties \\begin{enumerate} \\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary direct sums, \\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary colimits, \\item filtered colimit in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ are exact, \\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary products, \\item $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ has arbitrary limits. \\end{enumerate} The forgetful functor $$ \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\longrightarrow \\text{Mod}_\\mathcal{A} $$ sending a differential graded $\\mathcal{A}$-module to its underlying graded module commutes with all limits and colimits."}
+{"_id": "7315", "title": "sdga-lemma-what-makes-a-bimodule-dg", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ and $\\mathcal{B}$ be a sheaves of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{N}$ be a right differential graded $\\mathcal{B}$-module. There is a $1$-to-$1$ correspondence between $(\\mathcal{A}, \\mathcal{B})$-bimodule structures on $\\mathcal{N}$ compatible with the given differential graded $\\mathcal{B}$-module structure and homomorphisms $$ \\mathcal{A} \\longrightarrow \\SheafHom^{dg}_\\mathcal{B}(\\mathcal{N}, \\mathcal{N}) $$ of differential graded $\\mathcal{O}$-algebras."}
+{"_id": "7316", "title": "sdga-lemma-tensor-hom-adjunction-dg", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ and $\\mathcal{B}$ be a sheaves of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right differential graded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$ be a right differential graded $\\mathcal{B}$-module. With conventions as above we have $$ \\Hom_{\\text{Mod}_{(\\mathcal{B}, \\text{d})}^{dg}}( \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) = \\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}( \\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L})) $$ and $$ \\SheafHom_\\mathcal{B}^{dg}( \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) = \\SheafHom_\\mathcal{A}^{dg}( \\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L})) $$ functorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$."}
+{"_id": "7317", "title": "sdga-lemma-adjunction-push-pull-dg", "text": "In the situation above we have $$ \\Hom_{\\text{Mod}_{(\\mathcal{B}, \\text{d})}^{dg}}( \\mathcal{N}, f_*\\mathcal{M}) = \\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}( f^*\\mathcal{N}, \\mathcal{M}) $$"}
+{"_id": "7319", "title": "sdga-lemma-tensor-with-extension-by-zero-dg", "text": "In the situation above, let $\\mathcal{M}$ be a right differential graded $\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left differential graded $\\mathcal{A}$-module. Then $$ j_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} = j_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U) $$ as complexes of $\\mathcal{O}$-modules functorially in $\\mathcal{M}$ and $\\mathcal{N}$."}
+{"_id": "7320", "title": "sdga-lemma-homotopy-direct-sums", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ has direct sums and products."}
+{"_id": "7321", "title": "sdga-lemma-axioms-AB", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The differential graded category $\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}$ satisfies axioms (A) and (B) of Differential Graded Algebra, Section \\ref{dga-section-review}."}
+{"_id": "7322", "title": "sdga-lemma-axiom-C", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The differential graded category $\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}$ satisfies axiom (C) formulated in Differential Graded Algebra, Situation \\ref{dga-situation-ABC}."}
+{"_id": "7324", "title": "sdga-lemma-supply-good", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $U \\in \\Ob(\\mathcal{C})$. Then $j_!\\mathcal{A}_U$ is a good differential graded $\\mathcal{A}$-module."}
+{"_id": "7325", "title": "sdga-lemma-good-admissible-ses", "text": "et $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $0 \\to \\mathcal{P} \\to \\mathcal{P}' \\to \\mathcal{P}'' \\to 0$ be an admissible short exact sequence of differential graded $\\mathcal{A}$-modules. If two-out-of-three of these modules are good, so is the third."}
+{"_id": "7326", "title": "sdga-lemma-good-direct-sum", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. An arbitrary direct sum of good differential graded $\\mathcal{A}$-modules is good. A filtered colimit of good differential graded $\\mathcal{A}$-modules is good."}
+{"_id": "7328", "title": "sdga-lemma-free-graded-module-good", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a differential graded $\\mathcal{A}$-algebra. Let $\\mathcal{S}$ be a sheaf of graded sets on $\\mathcal{C}$. Then the free graded module $\\mathcal{A}[\\mathcal{S}]$ on $\\mathcal{S}$ endowed with differential as in Remark \\ref{remark-sheaf-graded-sets} is a good differential graded $\\mathcal{A}$-module."}
+{"_id": "7329", "title": "sdga-lemma-resolve", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module. There exists a homomorphism $\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules with the following properties \\begin{enumerate} \\item $\\mathcal{P} \\to \\mathcal{M}$ is a quasi-isomorphism, and \\item $\\mathcal{P}$ is good. \\end{enumerate}"}
+{"_id": "7330", "title": "sdga-lemma-acyclic-good", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{P}$ be a good acyclic right differential graded $\\mathcal{A}$-module. \\begin{enumerate} \\item for any differential graded left $\\mathcal{A}$-module $\\mathcal{N}$ the tensor product $\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$ is acyclic, \\item for any morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ of ringed topoi and any differential graded $\\mathcal{O}'$-algebra $\\mathcal{A}'$ and any map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$ of differential graded $f^{-1}\\mathcal{O}$-algebras the pullback $f^*\\mathcal{P}$ is acyclic and good. \\end{enumerate}"}
+{"_id": "7331", "title": "sdga-lemma-dg-hull", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The forgetful functor $F : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\text{Mod}_\\mathcal{A}$ has a left adjoint $G : \\text{Mod}_\\mathcal{A} \\to \\text{Mod}_{(\\mathcal{A}, \\text{d})}$."}
+{"_id": "7332", "title": "sdga-lemma-dg-hull-acyclic", "text": "The functors $F, G$ of Lemma \\ref{lemma-dg-hull} have the following properties. Given a graded $\\mathcal{A}$-module $\\mathcal{N}$ we have \\begin{enumerate} \\item the counit $\\mathcal{N} \\to F(G(\\mathcal{N}))$ is injective, \\item the map $\\overline{\\text{d}} : \\mathcal{N} \\to \\Coker(\\mathcal{N} \\to F(G(\\mathcal{N})))[1]$ is an isomorphism, and \\item $G(\\mathcal{N})$ is an acyclic differential graded $\\mathcal{A}$-module. \\end{enumerate}"}
+{"_id": "7333", "title": "sdga-lemma-characterize-injectives", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of graded algebras on $(\\mathcal{C}, \\mathcal{O})$. There exists a set $T$ and for each $t \\in T$ an injective map $\\mathcal{N}_t \\to \\mathcal{N}'_t$ of graded $\\mathcal{A}$-modules such that an object $\\mathcal{I}$ of $\\text{Mod}_\\mathcal{A}$ is injective if and only if for every solid diagram $$ \\xymatrix{ \\mathcal{N}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\ \\mathcal{N}'_t \\ar@{..>}[ru] } $$ a dotted arrow exists in $\\text{Mod}_\\mathcal{A}$ making the diagram commute."}
+{"_id": "7334", "title": "sdga-lemma-product-graded-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for each $t \\in T$ let $\\mathcal{I}_t$ be a graded injective diffential graded $\\mathcal{A}$-module. Then $\\prod \\mathcal{I}_t$ is a graded injective differential graded $\\mathcal{A}$-module."}
+{"_id": "7335", "title": "sdga-lemma-characterize-graded-injectives-in-dg", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. There exists a set $T$ and for each $t \\in T$ an injective map $\\mathcal{M}_t \\to \\mathcal{M}'_t$ of acyclic differential graded $\\mathcal{A}$-modules such that for an object $\\mathcal{I}$ of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ the following are equivalent \\begin{enumerate} \\item $\\mathcal{I}$ is graded injective, and \\item for every solid diagram $$ \\xymatrix{ \\mathcal{M}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\ \\mathcal{M}'_t \\ar@{..>}[ru] } $$ a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ making the diagram commute. \\end{enumerate}"}
+{"_id": "7336", "title": "sdga-lemma-small-acyclics", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. There exists a set $S$ and for each $s$ an acyclic differential graded $\\mathcal{A}$-module $\\mathcal{M}_s$ such that for every nonzero acyclic differential graded $\\mathcal{A}$-module $\\mathcal{M}$ there is an $s \\in S$ and an injective map $\\mathcal{M}_s \\to \\mathcal{M}$ in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$."}
+{"_id": "7337", "title": "sdga-lemma-product-K-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for each $t \\in T$ let $\\mathcal{I}_t$ be a K-injective diffential graded $\\mathcal{A}$-module. Then $\\prod \\mathcal{I}_t$ is a K-injective differential graded $\\mathcal{A}$-module."}
+{"_id": "7338", "title": "sdga-lemma-first-property-dg-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a K-injective and graded injective object of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$. For every solid diagram in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ $$ \\xymatrix{ \\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\ \\mathcal{M}' \\ar@{..>}[ru] } $$ where $b$ is injective and $\\mathcal{M}$ is acyclic a dotted arrow exists making the diagram commute."}
+{"_id": "7339", "title": "sdga-lemma-second-property-dg-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a K-injective and graded injective object of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$. For every solid diagram in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ $$ \\xymatrix{ \\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\ \\mathcal{M}' \\ar@{..>}[ru] } $$ where $b$ is a quasi-isomorphism a dotted arrow exists making the diagram commute up to homotopy."}
+{"_id": "7340", "title": "sdga-lemma-better-set-of-monos", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. There exists a set $R$ and for each $r \\in R$ an injective map $\\mathcal{M}_r \\to \\mathcal{M}'_r$ of acyclic differential graded $\\mathcal{A}$-modules such that for an object $\\mathcal{I}$ of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ the following are equivalent \\begin{enumerate} \\item $\\mathcal{I}$ is K-injective and graded injective, and \\item for every solid diagram $$ \\xymatrix{ \\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{I} \\\\ \\mathcal{M}'_r \\ar@{..>}[ru] } $$ a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ making the diagram commute. \\end{enumerate}"}
+{"_id": "7341", "title": "sdga-lemma-functor-set-of-monos", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $R$ be a set and for each $r \\in R$ let an injective map $\\mathcal{M}_r \\to \\mathcal{M}'_r$ of acyclic differential graded $\\mathcal{A}$-modules be given. There exists a functor $M : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\text{Mod}_{(\\mathcal{A}, \\text{d})}$ and a natural transformation $j : \\text{id} \\to M$ such that \\begin{enumerate} \\item $j_\\mathcal{M} : \\mathcal{M} \\to M(\\mathcal{M})$ is injective and a quasi-isomorphism, \\item for every solid diagram $$ \\xymatrix{ \\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{M} \\ar[d]^{j_\\mathcal{M}} \\\\ \\mathcal{M}'_r \\ar@{..>}[r] & M(\\mathcal{M}) } $$ a dotted arrow exists in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ making the diagram commute. \\end{enumerate}"}
+{"_id": "7342", "title": "sdga-lemma-cohomology-homological", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The functor  $H^0 : \\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to \\textit{Mod}(\\mathcal{O})$ of Section \\ref{section-modules} factors through a functor $$ H^0 : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to \\textit{Mod}(\\mathcal{O}) $$ which is homological in the sense of Derived Categories, Definition \\ref{derived-definition-homological}."}
+{"_id": "7343", "title": "sdga-lemma-acyclics", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The full subcategory $\\text{Ac}$ of the homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\\text{Mod}_{(A, \\text{d})})$."}
+{"_id": "7344", "title": "sdga-lemma-qis", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Consider the subclass $\\text{Qis} \\subset \\text{Arrows}(K(\\text{Mod}_{(A, \\text{d})}))$ consisting of quasi-isomorphisms. This is a saturated multiplicative system compatible with the triangulated structure on $K(\\text{Mod}_{(A, \\text{d})})$."}
+{"_id": "7347", "title": "sdga-lemma-hom-derived", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ and $\\mathcal{N}$ be differential graded $\\mathcal{A}$-modules. Let $\\mathcal{N} \\to \\mathcal{I}$ be a quasi-isomorphism with $\\mathcal{I}$ a graded injective and K-injective differential graded $\\mathcal{A}$-module. Then $$ \\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{N}) = \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I}) $$"}
+{"_id": "7348", "title": "sdga-lemma-derived-products", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Then \\begin{enumerate} \\item $D(\\mathcal{A}, \\text{d})$ has both direct sums and products, \\item direct sums are obtained by taking direct sums of differential graded $\\mathcal{A}$-modules, \\item products are obtained by taking products of K-injective differential graded modules. \\end{enumerate}"}
+{"_id": "7349", "title": "sdga-lemma-derived-canonical-delta-functor", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The localization functor $\\text{Mod}_{(\\mathcal{A}, \\text{d})} \\to D(\\mathcal{A}, \\text{d})$ has the natural structure of a $\\delta$-functor, with $$ \\delta_{\\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{M}} = - p \\circ q^{-1} $$ with $p$ and $q$ as explained above."}
+{"_id": "7351", "title": "sdga-lemma-derived-tensor-product", "text": "In the situation above, the functor (\\ref{equation-pullback}) composed with the localization functor $K(\\text{Mod}_{(\\mathcal{A}', \\text{d})}) \\to D(\\mathcal{A}', \\text{d})$ has a left derived extension $D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ whose value on a good right differential graded $\\mathcal{B}$-module $\\mathcal{P}$ is $f^*\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$."}
+{"_id": "7352", "title": "sdga-lemma-compose-pullback-tensor", "text": "In Lemma \\ref{lemma-derived-tensor-product} the functor $D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ is equal to $\\mathcal{M} \\mapsto Lf^*\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$."}
+{"_id": "7353", "title": "sdga-lemma-compose-pullback", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and $(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$ be morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and $\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra, $\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let $\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and $\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$ be a homomorphism of differential graded $\\mathcal{O}'$-algebras and $\\mathcal{O}''$-algebras. Then we have $L(g \\circ f)^* = Lf^* \\circ Lg^* : D(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$."}
+{"_id": "7354", "title": "sdga-lemma-tensor-symmetry", "text": "In the situation above, if $\\mathcal{N} \\to \\mathcal{N}'$ is an isomorphism on cohomology sheaves, then $t$ is an isomorphism of functors $(- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}) \\to (- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}')$."}
+{"_id": "7355", "title": "sdga-lemma-good-on-other-side", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras. Let $\\mathcal{N}$ be a  differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. If $\\mathcal{N}$ is good as a left differential graded $\\mathcal{A}$-module, then we have $\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} = \\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$ for all differential graded $\\mathcal{A}$-modules $\\mathcal{M}$."}
+{"_id": "7356", "title": "sdga-lemma-compose-tensor", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$, $\\mathcal{A}'$, $\\mathcal{A}''$ be differential graded $\\mathcal{O}$-algebras. Let $\\mathcal{N}$ and $\\mathcal{N}'$ be a differential graded $(\\mathcal{A}, \\mathcal{A}')$-bimodule and $(\\mathcal{A}', \\mathcal{A}'')$-bimodule. Assume that the canonical map $$ \\mathcal{N} \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}' \\longrightarrow \\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}' $$ in $D(\\mathcal{A}'', \\text{d})$ is a quasi-isomorphism. Then we have $$ (\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}) \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}' = \\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} (\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}') $$ as functors $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$."}
+{"_id": "7357", "title": "sdga-lemma-right-derived", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Then any exact functor $$ T : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\longrightarrow \\mathcal{D} $$ of triangulated categories has a right derived extension $RT : D(\\mathcal{A}, \\text{d}) \\to \\mathcal{D}$ whose value on a graded injective and K-injective differential graded $\\mathcal{A}$-module $\\mathcal{I}$ is $T(\\mathcal{I})$."}
+{"_id": "7358", "title": "sdga-lemma-derived-adjoint-tensor-hom", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras. Let $\\mathcal{N}$ be a  differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Then $$ R\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) : D(\\mathcal{B}, \\text{d}) \\longrightarrow D(\\mathcal{A}, \\text{d}) $$ is right adjoint to $$ - \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} : D(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}) $$"}
+{"_id": "7359", "title": "sdga-lemma-derived-adjoint-push-pull", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let $\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras. Then $$ Rf_* :  D(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}) $$ is right adjoint to $$ Lf^* : D(\\mathcal{B}, \\text{d}) \\longrightarrow D(\\mathcal{A}, \\text{d}) $$"}
+{"_id": "7361", "title": "sdga-lemma-compose-pushforward", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and $(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$ be morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and $\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra, $\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let $\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and $\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$ be a homomorphism of differential graded $\\mathcal{O}'$-algebras and $\\mathcal{O}''$-algebras. Then we have $R(g \\circ f)_* = Rg_* \\circ Rf_* : D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$."}
+{"_id": "7363", "title": "sdga-lemma-pushforward-agrees", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let $\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras. The diagram $$ \\xymatrix{ D(\\mathcal{A}, \\text{d}) \\ar[d]_{Rf_*} \\ar[rr]_{forget} & & D(\\mathcal{O}_\\mathcal{C}) \\ar[d]^{Rf_*} \\\\ D(\\mathcal{B}, \\text{d}) \\ar[rr]^{forget} & & D(\\mathcal{O}_\\mathcal{D}) } $$ commutes."}
+{"_id": "7366", "title": "sdga-lemma-special-good", "text": "In the situation above the differential graded $\\mathcal{O}$-algebra $$ \\mathcal{A} = \\colim \\mathcal{A}_i $$ has the following property: for any morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ of ringed topoi, the pullback $f^*\\mathcal{A}$ is flat as a graded $\\mathcal{O}'$-module and is K-flat as a complex of $\\mathcal{O}'$-modules."}
+{"_id": "7388", "title": "stacks-morphisms-theorem-quasi-DM", "text": "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is quasi-DM, and \\item there exists a scheme $W$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $W \\to \\mathcal{X}$. \\end{enumerate}"}
+{"_id": "7389", "title": "stacks-morphisms-theorem-DM", "text": "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is DM, \\item $\\mathcal{X}$ is Deligne-Mumford, and \\item there exists a scheme $W$ and a surjective \\'etale morphism $W \\to \\mathcal{X}$. \\end{enumerate}"}
+{"_id": "7390", "title": "stacks-morphisms-lemma-isom-locally-finite-type", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\\mathcal{X}$ over $T$. Then the morphism $\\mathit{Isom}_\\mathcal{X}(x, y) \\to T$ is locally of finite type."}
+{"_id": "7391", "title": "stacks-morphisms-lemma-isom-pseudo-torsor-aut", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\\mathcal{X}$ over $T$. Then \\begin{enumerate} \\item $\\mathit{Isom}_\\mathcal{X}(y, y)$ is a group algebraic space over $T$, and \\item $\\mathit{Isom}_\\mathcal{X}(x, y)$ is a pseudo torsor for $\\mathit{Isom}_\\mathcal{X}(y, y)$ over $T$. \\end{enumerate}"}
+{"_id": "7392", "title": "stacks-morphisms-lemma-properties-diagonal", "text": "\\begin{slogan} Diagonals of morphisms of algebraic stacks are representable by algebraic spaces and locally of finite type. \\end{slogan} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Then \\begin{enumerate} \\item $\\Delta_f$ is representable by algebraic spaces, and \\item $\\Delta_f$ is locally of finite type. \\end{enumerate}"}
+{"_id": "7393", "title": "stacks-morphisms-lemma-properties-diagonal-representable", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then \\begin{enumerate} \\item $\\Delta_f$ is representable (by schemes), \\item $\\Delta_f$ is locally of finite type, \\item $\\Delta_f$ is a monomorphism, \\item $\\Delta_f$ is separated, and \\item $\\Delta_f$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "7394", "title": "stacks-morphisms-lemma-representable-separated-diagonal-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \\begin{enumerate} \\item $f$ is separated, \\item $\\Delta_f$ is a closed immersion, \\item $\\Delta_f$ is proper, or \\item $\\Delta_f$ is universally closed. \\end{enumerate}"}
+{"_id": "7395", "title": "stacks-morphisms-lemma-representable-quasi-separated-diagonal-quasi-compact", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \\begin{enumerate} \\item $f$ is quasi-separated, \\item $\\Delta_f$ is quasi-compact, or \\item $\\Delta_f$ is of finite type. \\end{enumerate}"}
+{"_id": "7396", "title": "stacks-morphisms-lemma-representable-locally-separated-diagonal-immersion", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent \\begin{enumerate} \\item $f$ is locally separated, and \\item $\\Delta_f$ is an immersion. \\end{enumerate}"}
+{"_id": "7397", "title": "stacks-morphisms-lemma-trivial-implications", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item If $f$ is separated, then $f$ is quasi-separated. \\item If $f$ is DM, then $f$ is quasi-DM. \\item If $f$ is representable by algebraic spaces, then $f$ is DM. \\end{enumerate}"}
+{"_id": "7398", "title": "stacks-morphisms-lemma-base-change-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated} are stable under base change."}
+{"_id": "7399", "title": "stacks-morphisms-lemma-check-separated-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ has one of the separation properties of Definition \\ref{definition-separated} then so does $f$."}
+{"_id": "7401", "title": "stacks-morphisms-lemma-fibre-product-after-map", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ and $\\mathcal{Z} \\to \\mathcal{T}$ be morphisms of algebraic stacks. Consider the induced morphism $i : \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$. Then \\begin{enumerate} \\item $i$ is representable by algebraic spaces and locally of finite type, \\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is quasi-separated, then $i$ is quasi-separated, \\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is separated, then $i$ is separated, \\item if $\\mathcal{Z} \\to \\mathcal{T}$ is DM, then $i$ is unramified, \\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite, \\item if $\\mathcal{Z} \\to \\mathcal{T}$ is separated, then $i$ is proper, and \\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "7402", "title": "stacks-morphisms-lemma-semi-diagonal", "text": "Let $\\mathcal{T}$ be an algebraic stack. Let $g : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks over $\\mathcal{T}$. Consider the graph $i : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$ of $g$. Then \\begin{enumerate} \\item $i$ is representable by algebraic spaces and locally of finite type, \\item if $\\mathcal{Y} \\to \\mathcal{T}$ is DM, then $i$ is unramified, \\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite, \\item if $\\mathcal{Y} \\to \\mathcal{T}$ is separated, then $i$ is proper, and \\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "7404", "title": "stacks-morphisms-lemma-composition-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated} are stable under composition of morphisms."}
+{"_id": "7405", "title": "stacks-morphisms-lemma-separated-over-separated", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$. \\begin{enumerate} \\item If $\\mathcal{Y}$ is DM over $S$ and $f$ is DM, then $\\mathcal{X}$ is DM over $S$. \\item If $\\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM, then $\\mathcal{X}$ is quasi-DM over $S$. \\item If $\\mathcal{Y}$ is separated over $S$ and $f$ is separated, then $\\mathcal{X}$ is separated over $S$. \\item If $\\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated, then $\\mathcal{X}$ is quasi-separated over $S$. \\item If $\\mathcal{Y}$ is DM and $f$ is DM, then $\\mathcal{X}$ is DM. \\item If $\\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM, then $\\mathcal{X}$ is quasi-DM. \\item If $\\mathcal{Y}$ is separated and $f$ is separated, then $\\mathcal{X}$ is separated. \\item If $\\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated, then $\\mathcal{X}$ is quasi-separated. \\end{enumerate}"}
+{"_id": "7406", "title": "stacks-morphisms-lemma-compose-after-separated", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. \\begin{enumerate} \\item If $g \\circ f$ is DM then so is $f$. \\item If $g \\circ f$ is quasi-DM then so is $f$. \\item If $g \\circ f$ is separated and $\\Delta_g$ is separated, then $f$ is separated. \\item If $g \\circ f$ is quasi-separated and $\\Delta_g$ is quasi-separated, then $f$ is quasi-separated. \\end{enumerate}"}
+{"_id": "7407", "title": "stacks-morphisms-lemma-separated-implies-morphism-separated", "text": "Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$. \\begin{enumerate} \\item $\\mathcal{X}$ is DM $\\Leftrightarrow$ $\\mathcal{X}$ is DM over $S$. \\item $\\mathcal{X}$ is quasi-DM $\\Leftrightarrow$ $\\mathcal{X}$ is quasi-DM over $S$. \\item If $\\mathcal{X}$ is separated, then $\\mathcal{X}$ is separated over $S$. \\item If $\\mathcal{X}$ is quasi-separated, then $\\mathcal{X}$ is quasi-separated over $S$. \\end{enumerate} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$. \\begin{enumerate} \\item[(5)] If $\\mathcal{X}$ is DM over $S$, then $f$ is DM. \\item[(6)] If $\\mathcal{X}$ is quasi-DM over $S$, then $f$ is quasi-DM. \\item[(7)] If $\\mathcal{X}$ is separated over $S$ and $\\Delta_{\\mathcal{Y}/S}$ is separated, then $f$ is separated. \\item[(8)] If $\\mathcal{X}$ is quasi-separated over $S$ and $\\Delta_{\\mathcal{Y}/S}$ is quasi-separated, then $f$ is quasi-separated. \\end{enumerate}"}
+{"_id": "7408", "title": "stacks-morphisms-lemma-properties-covering-imply-diagonal", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $W$ be an algebraic space, and let $f : W \\to \\mathcal{X}$ be a surjective, flat, locally finitely presented morphism. \\begin{enumerate} \\item If $f$ is unramified (i.e., \\'etale, i.e., $\\mathcal{X}$ is Deligne-Mumford), then $\\mathcal{X}$ is DM. \\item If $f$ is locally quasi-finite, then $\\mathcal{X}$ is quasi-DM. \\end{enumerate}"}
+{"_id": "7409", "title": "stacks-morphisms-lemma-monomorphism-separated", "text": "A monomorphism of algebraic stacks is separated and DM. The same is true for immersions of algebraic stacks."}
+{"_id": "7410", "title": "stacks-morphisms-lemma-separation-properties-residual-gerbe", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. Assume the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists. If $\\mathcal{X}$ is DM, resp.\\ quasi-DM, resp.\\ separated, resp.\\ quasi-separated, then so is $\\mathcal{Z}_x$."}
+{"_id": "7411", "title": "stacks-morphisms-lemma-inertia", "text": "Let $\\mathcal{X}$ be an algebraic stack. Then the inertia stack $\\mathcal{I}_\\mathcal{X}$ is an algebraic stack as well. The morphism $$ \\mathcal{I}_\\mathcal{X} \\longrightarrow \\mathcal{X} $$ is representable by algebraic spaces and locally of finite type. More generally, let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Then the relative inertia $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is an algebraic stack and the morphism $$ \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\longrightarrow \\mathcal{X} $$ is representable by algebraic spaces and locally of finite type."}
+{"_id": "7412", "title": "stacks-morphisms-lemma-isom-pseudo-torsor-aut-over-space", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space and let $x_i : Z \\to \\mathcal{X}$, $i = 1, 2$ be morphisms. Then \\begin{enumerate} \\item $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$ is a group algebraic space over $Z$, \\item there is an exact sequence of groups $$ 0 \\to \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2) \\to \\mathit{Isom}_\\mathcal{X}(x_2, x_2) \\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_2, f \\circ x_2) $$ \\item there is a map of algebraic spaces $ \\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2) $ such that for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$ we obtain a cartesian diagram $$ \\xymatrix{ \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) \\ar[d] \\ar[r] & Z \\ar[d]^\\alpha \\\\ \\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\ar[r] & \\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2) } $$ \\item for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$ the algebraic space $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$ is a pseudo torsor for $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$ over $Z$. \\end{enumerate}"}
+{"_id": "7413", "title": "stacks-morphisms-lemma-cartesian-square-inertia", "text": "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ and $f : \\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks. Set $\\mathcal{X}' = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Y}'$. Then both squares in the diagram $$ \\xymatrix{ \\mathcal{I}_{\\mathcal{X}'/\\mathcal{Y}'} \\ar[r] \\ar[d]_{ \\text{Categories, Equation}\\ (\\ref{categories-equation-functorial}) } & \\mathcal{X}' \\ar[r]_{\\pi'} \\ar[d] & \\mathcal{Y}' \\ar[d]^f \\\\ \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[r] & \\mathcal{X} \\ar[r]^\\pi & \\mathcal{Y} } $$ are fibre product squares."}
+{"_id": "7414", "title": "stacks-morphisms-lemma-monomorphism-cartesian-square-inertia", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a monomorphism of algebraic stacks. Then the diagram $$ \\xymatrix{ \\mathcal{I}_\\mathcal{X} \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\mathcal{I}_\\mathcal{Y} \\ar[r] & \\mathcal{Y} } $$ is a fibre product square."}
+{"_id": "7415", "title": "stacks-morphisms-lemma-presentation-inertia", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $[U/R] \\to \\mathcal{X}$ be a presentation. Let $G/U$ be the stabilizer group algebraic space associated to the groupoid $(U, R, s, t, c)$. Then $$ \\xymatrix{ G \\ar[d] \\ar[r] & U \\ar[d] \\\\ \\mathcal{I}_\\mathcal{X} \\ar[r] & \\mathcal{X} } $$ is a fibre product diagram."}
+{"_id": "7416", "title": "stacks-morphisms-lemma-diagonal-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is separated, \\item $\\Delta_{f, 1} = \\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is separated, and \\item $\\Delta_{f, 2} = e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is a closed immersion. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is quasi-separated, \\item $\\Delta_{f, 1} = \\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-separated, and \\item $\\Delta_{f, 2} = e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is a quasi-compact. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is locally separated, \\item $\\Delta_{f, 1} = \\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is locally separated, and \\item $\\Delta_{f, 2} = e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is an immersion. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is unramified, \\item $f$ is DM. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is locally quasi-finite, \\item $f$ is quasi-DM. \\end{enumerate} \\end{enumerate}"}
+{"_id": "7417", "title": "stacks-morphisms-lemma-second-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \\begin{enumerate} \\item the morphism $f$ is representable by algebraic spaces, \\item the second diagonal of $f$ is an isomorphism, \\item the group stack $ \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is trivial over $\\mathcal X$, and \\item for a scheme $T$ and a morphism $x : T \\to \\mathcal{X}$ the kernel of $\\mathit{Isom}_\\mathcal{X}(x, x) \\to \\mathit{Isom}_\\mathcal{Y}(f(x), f(x))$ is trivial. \\end{enumerate}"}
+{"_id": "7418", "title": "stacks-morphisms-lemma-hierarchy", "text": "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is \\begin{enumerate} \\item a monomorphism if and only if $\\Delta_{f, 1}$ is an isomorphism, and \\item representable by algebraic spaces if and only if $\\Delta_{f, 1}$ is a monomorphism. \\end{enumerate} Moreover, the second diagonal $\\Delta_{f, 2}$ is always a monomorphism."}
+{"_id": "7419", "title": "stacks-morphisms-lemma-first-diagonal-separated-second-diagonal-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Then \\begin{enumerate} \\item $\\Delta_{f, 1}$ separated $\\Leftrightarrow$ $\\Delta_{f, 2}$ closed immersion $\\Leftrightarrow$ $\\Delta_{f, 2}$ proper $\\Leftrightarrow$ $\\Delta_{f, 2}$ universally closed, \\item $\\Delta_{f, 1}$ quasi-separated $\\Leftrightarrow$ $\\Delta_{f, 2}$ finite type $\\Leftrightarrow$ $\\Delta_{f, 2}$ quasi-compact, and \\item $\\Delta_{f, 1}$ locally separated $\\Leftrightarrow$ $\\Delta_{f, 2}$ immersion. \\end{enumerate}"}
+{"_id": "7421", "title": "stacks-morphisms-lemma-separated-implies-isom", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a separated (resp.\\ quasi-separated, resp.\\ quasi-DM, resp.\\ DM) morphism of algebraic stacks. Then \\begin{enumerate} \\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms $x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism $$ T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2 $$ is proper (resp.\\ quasi-compact and quasi-separated, resp.\\ locally quasi-finite, resp.\\ unramified), \\item given an algebraic space $T$ and morphisms $x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism $$ \\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow \\mathit{Isom}_\\mathcal{Y}(y_1, y_2) $$ is proper (resp.\\ quasi-compact and quasi-separated, resp.\\ locally quasi-finite, resp.\\ unramified). \\end{enumerate}"}
+{"_id": "7423", "title": "stacks-morphisms-lemma-base-change-quasi-compact", "text": "The base change of a quasi-compact morphism of algebraic stacks by any morphism of algebraic stacks is quasi-compact."}
+{"_id": "7424", "title": "stacks-morphisms-lemma-composition-quasi-compact", "text": "The composition of a pair of quasi-compact morphisms of algebraic stacks is quasi-compact."}
+{"_id": "7425", "title": "stacks-morphisms-lemma-closed-immersion-quasi-compact", "text": "A closed immersion of algebraic stacks is quasi-compact."}
+{"_id": "7426", "title": "stacks-morphisms-lemma-surjection-from-quasi-compact", "text": "Let $$ \\xymatrix{ \\mathcal{X} \\ar[rr]_f \\ar[rd]_p & & \\mathcal{Y} \\ar[dl]^q \\\\ & \\mathcal{Z} } $$ be a $2$-commutative diagram of morphisms of algebraic stacks. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact."}
+{"_id": "7427", "title": "stacks-morphisms-lemma-quasi-compact-permanence", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $g \\circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact."}
+{"_id": "7428", "title": "stacks-morphisms-lemma-quasi-compact-quasi-separated-permanence", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item If $\\mathcal{X}$ is quasi-compact and $\\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact. \\item If $\\mathcal{X}$ is quasi-compact and quasi-separated and $\\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact and quasi-separated. \\item A fibre product of quasi-compact and quasi-separated algebraic stacks is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "7429", "title": "stacks-morphisms-lemma-reach-points-scheme-theoretic-image", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Let $y \\in |\\mathcal{Y}|$ be a point in the closure of the image of $|f|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\Spec(A) \\ar[r] & \\mathcal{Y} } $$ such that the closed point of $\\Spec(A)$ maps to $y$."}
+{"_id": "7430", "title": "stacks-morphisms-lemma-check-quasi-compact-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is quasi-compact, then $f$ is quasi-compact."}
+{"_id": "7431", "title": "stacks-morphisms-lemma-locally-closed-in-noetherian", "text": "Let $j : \\mathcal{X} \\to \\mathcal{Y}$ be an immersion of algebraic stacks. \\begin{enumerate} \\item If $\\mathcal{Y}$ is locally Noetherian, then $\\mathcal{X}$ is locally Noetherian and $j$ is quasi-compact. \\item If $\\mathcal{Y}$ is Noetherian, then $\\mathcal{X}$ is Noetherian. \\end{enumerate}"}
+{"_id": "7432", "title": "stacks-morphisms-lemma-Noetherian-topology", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item If $\\mathcal{X}$ is locally Noetherian then $|\\mathcal{X}|$ is a locally Noetherian topological space. \\item If $\\mathcal{X}$ is quasi-compact and locally Noetherian, then $|\\mathcal{X}|$ is a Noetherian topological space. \\end{enumerate}"}
+{"_id": "7433", "title": "stacks-morphisms-lemma-base-change-affine", "text": "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be an affine morphism of algebraic stacks. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$ is an affine morphism of algebraic stacks."}
+{"_id": "7435", "title": "stacks-morphisms-lemma-base-change-integral", "text": "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be an integral (or finite) morphism of algebraic stacks. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$ is an integral (or finite) morphism of algebraic stacks."}
+{"_id": "7437", "title": "stacks-morphisms-lemma-characterize-representable-universally-open", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is universally open (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the morphism of topological spaces $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is open. \\end{enumerate}"}
+{"_id": "7438", "title": "stacks-morphisms-lemma-base-change-universally-open", "text": "The base change of a universally open morphism of algebraic stacks by any morphism of algebraic stacks is universally open."}
+{"_id": "7439", "title": "stacks-morphisms-lemma-composition-universally-open", "text": "The composition of a pair of (universally) open morphisms of algebraic stacks is (universally) open."}
+{"_id": "7440", "title": "stacks-morphisms-lemma-characterize-representable-universally-submersive", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is universally submersive (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the morphism of topological spaces $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is submersive. \\end{enumerate}"}
+{"_id": "7441", "title": "stacks-morphisms-lemma-base-change-universally-submersive", "text": "The base change of a universally submersive morphism of algebraic stacks by any morphism of algebraic stacks is universally submersive."}
+{"_id": "7442", "title": "stacks-morphisms-lemma-composition-universally-submersive", "text": "The composition of a pair of (universally) submersive morphisms of algebraic stacks is (universally) submersive."}
+{"_id": "7443", "title": "stacks-morphisms-lemma-characterize-representable-universally-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the morphism of topological spaces $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is closed. \\end{enumerate}"}
+{"_id": "7444", "title": "stacks-morphisms-lemma-base-change-universally-closed", "text": "The base change of a universally closed morphism of algebraic stacks by any morphism of algebraic stacks is universally closed."}
+{"_id": "7445", "title": "stacks-morphisms-lemma-composition-universally-closed", "text": "The composition of a pair of (universally) closed morphisms of algebraic stacks is (universally) closed."}
+{"_id": "7446", "title": "stacks-morphisms-lemma-universally-closed-local", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item for every scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$ the projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$ is closed, \\item for every affine scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$ the projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$ is closed, and \\item there exists an algebraic space $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$ such that $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$ is a universally closed morphism of algebraic stacks. \\end{enumerate}"}
+{"_id": "7447", "title": "stacks-morphisms-lemma-characterize-representable-universally-injective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is universally injective (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is injective. \\end{enumerate}"}
+{"_id": "7448", "title": "stacks-morphisms-lemma-base-change-universally-injective", "text": "The base change of a universally injective morphism of algebraic stacks by any morphism of algebraic stacks is universally injective."}
+{"_id": "7449", "title": "stacks-morphisms-lemma-composition-universally-injective", "text": "The composition of a pair of universally injective morphisms of algebraic stacks is universally injective."}
+{"_id": "7450", "title": "stacks-morphisms-lemma-universally-injective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is universally injective, \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is surjective, and \\item for an algebraically closed field, for $x_1, x_2 : \\Spec(k) \\to \\mathcal{X}$, and for a $2$-arrow $\\beta : f \\circ x_1 \\to f \\circ x_2$ there is a $2$-arrow $\\alpha : x_1 \\to x_2$ with $\\beta = \\text{id}_f \\star \\alpha$. \\end{enumerate}"}
+{"_id": "7451", "title": "stacks-morphisms-lemma-universally-injective-point", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a universally injective morphism of algebraic stacks. Let $y : \\Spec(k) \\to \\mathcal{Y}$ be a morphism where $k$ is an algebraically closed field. If $y$ is in the image of $|\\mathcal{X}| \\to |\\mathcal{Y}|$, then there is a morphism $x : \\Spec(k) \\to \\mathcal{X}$ with $y = f \\circ x$."}
+{"_id": "7452", "title": "stacks-morphisms-lemma-universally-injective-local", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \\begin{enumerate} \\item $f$ is universally injective, \\item for every affine scheme $Z$ and any morphism $Z \\to \\mathcal{Y}$ the morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$ is universally injective, and \\item add more here. \\end{enumerate}"}
+{"_id": "7453", "title": "stacks-morphisms-lemma-check-universally-injective-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is universally injective, then $f$ is universally injective."}
+{"_id": "7454", "title": "stacks-morphisms-lemma-characterize-representable-universal-homeomorphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is a universal homeomorphism (Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map of topological spaces $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is a homeomorphism. \\end{enumerate}"}
+{"_id": "7455", "title": "stacks-morphisms-lemma-base-change-universal-homeomorphism", "text": "The base change of a universal homeomorphism of algebraic stacks by any morphism of algebraic stacks is a universal homeomorphism."}
+{"_id": "7456", "title": "stacks-morphisms-lemma-composition-universal-homeomorphism", "text": "The composition of a pair of universal homeomorphisms of algebraic stacks is a universal homeomorphism."}
+{"_id": "7457", "title": "stacks-morphisms-lemma-check-universal-homeomorphism-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is a universal homeomorphism, then $f$ is a universal homeomorphism."}
+{"_id": "7458", "title": "stacks-morphisms-lemma-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Consider commutative diagrams $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ \\mathcal{X} \\ar[r]^f & \\mathcal{Y} } $$ where $U$ and $V$ are algebraic spaces and the vertical arrows are smooth. The following are equivalent \\begin{enumerate} \\item for any diagram as above such that in addition $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is smooth the morphism $h$ has property $\\mathcal{P}$, and \\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective the morphism $h$ has property $\\mathcal{P}$. \\end{enumerate} If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces, then this is also equivalent to $f$ (as a morphism of algebraic spaces) having property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for morphisms $f$ which are representable by algebraic spaces this is also equivalent to $f$ having property $\\mathcal{P}$ in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}."}
+{"_id": "7459", "title": "stacks-morphisms-lemma-composition-finite-type", "text": "The composition of finite type morphisms is of finite type. The same holds for locally of finite type."}
+{"_id": "7460", "title": "stacks-morphisms-lemma-base-change-finite-type", "text": "A base change of a finite type morphism is finite type. The same holds for locally of finite type."}
+{"_id": "7461", "title": "stacks-morphisms-lemma-immersion-locally-finite-type", "text": "An immersion is locally of finite type."}
+{"_id": "7462", "title": "stacks-morphisms-lemma-locally-finite-type-locally-noetherian", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type and $\\mathcal{Y}$ is locally Noetherian, then $\\mathcal{X}$ is locally Noetherian."}
+{"_id": "7463", "title": "stacks-morphisms-lemma-check-finite-type-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite presentation where $W$ is an algebraic space. If the base change $W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is locally of finite type, then $f$ is locally of finite type."}
+{"_id": "7464", "title": "stacks-morphisms-lemma-check-finite-type-precompose", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. Assume $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite type and that $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then $\\mathcal{Y} \\to \\mathcal{Z}$ is locally of finite type."}
+{"_id": "7465", "title": "stacks-morphisms-lemma-finite-type-permanence", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ is locally of finite type, then $f : \\mathcal{X} \\to \\mathcal{Y}$ is locally of finite type."}
+{"_id": "7466", "title": "stacks-morphisms-lemma-point-finite-type", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. The following are equivalent: \\begin{enumerate} \\item There exists a morphism $\\Spec(k) \\to \\mathcal{X}$ which is locally of finite type and represents $x$. \\item There exists a scheme $U$, a closed point $u \\in U$, and a smooth morphism $\\varphi : U \\to \\mathcal{X}$ such that $\\varphi(u) = x$. \\end{enumerate}"}
+{"_id": "7467", "title": "stacks-morphisms-lemma-identify-finite-type-points", "text": "Let $\\mathcal{X}$ be an algebraic stack. We have $$ \\mathcal{X}_{\\text{ft-pts}} = \\bigcup\\nolimits_{\\varphi : U \\to \\mathcal{X}\\text{ smooth}} |\\varphi|(U_0) $$ where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes smooth over $\\mathcal{X}$ or over all affine schemes smooth over $\\mathcal{X}$."}
+{"_id": "7468", "title": "stacks-morphisms-lemma-finite-type-points-morphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type, then $f(\\mathcal{X}_{\\text{ft-pts}}) \\subset \\mathcal{Y}_{\\text{ft-pts}}$."}
+{"_id": "7470", "title": "stacks-morphisms-lemma-enough-finite-type-points", "text": "Let $\\mathcal{X}$ be an algebraic stack. For any locally closed subset $T \\subset |\\mathcal{X}|$ we have $$ T \\not = \\emptyset \\Rightarrow T \\cap \\mathcal{X}_{\\text{ft-pts}} \\not = \\emptyset. $$ In particular, for any closed subset $T \\subset |\\mathcal{X}|$ we see that $T \\cap \\mathcal{X}_{\\text{ft-pts}}$ is dense in $T$."}
+{"_id": "7471", "title": "stacks-morphisms-lemma-point-finite-type-monomorphism", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. The following are equivalent: \\begin{enumerate} \\item $x$ is a finite type point, \\item there exists an algebraic stack $\\mathcal{Z}$ whose underlying topological space $|\\mathcal{Z}|$ is a singleton, and a morphism $f : \\mathcal{Z} \\to \\mathcal{X}$ which is locally of finite type such that $\\{x\\} = |f|(|\\mathcal{Z}|)$, and \\item the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists and the inclusion morphism $\\mathcal{Z}_x \\to \\mathcal{X}$ is locally of finite type. \\end{enumerate}"}
+{"_id": "7472", "title": "stacks-morphisms-lemma-automorphism-group-scheme", "text": "In the situation above $G_x$ is a scheme if one of the following holds \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is quasi-separated \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is locally separated, \\item $\\mathcal{X}$ is quasi-DM, \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-separated, \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally separated, or \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "7473", "title": "stacks-morphisms-lemma-property-automorphism-groups", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point. Let $P$ be a property of algebraic spaces over fields which is invariant under ground field extensions; for example $P(X/k) = X \\to \\Spec(k)\\text{ is finite}$. The following are equivalent \\begin{enumerate} \\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the class of $x$ the automorphism group algebraic space $G_x/k$ has $P$, and \\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the class of $x$ the automorphism group algebraic space $G_x/k$ has $P$. \\end{enumerate}"}
+{"_id": "7474", "title": "stacks-morphisms-lemma-iso-automorphism-groups", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$ be a point. The following are equivalent \\begin{enumerate} \\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the class of $x$ setting $y = f \\circ x$ the map $G_x \\to G_y$ of automorphism group algebraic spaces is an isomorphism, and \\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the class of $x$ setting $y = f \\circ x$ the map $G_x \\to G_y$ of automorphism group algebraic spaces is an isomorphism. \\end{enumerate}"}
+{"_id": "7475", "title": "stacks-morphisms-lemma-properties-diagonal-from-presentation", "text": "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that $s, t : R \\to U$ are flat and locally of finite presentation. Consider the algebraic stack $\\mathcal{X} = [U/R]$ (see above). \\begin{enumerate} \\item If $R \\to U \\times U$ is separated, then $\\Delta_\\mathcal{X}$ is separated. \\item If $U$, $R$ are separated, then $\\Delta_\\mathcal{X}$ is separated. \\item If $R \\to U \\times U$ is locally quasi-finite, then $\\mathcal{X}$ is quasi-DM. \\item If $s, t : R \\to U$ are locally quasi-finite, then $\\mathcal{X}$ is quasi-DM. \\item If $R \\to U \\times U$ is proper, then $\\mathcal{X}$ is separated. \\item If $s, t : R \\to U$ are proper and $U$ is separated, then $\\mathcal{X}$ is separated. \\item Add more here. \\end{enumerate}"}
+{"_id": "7476", "title": "stacks-morphisms-lemma-points-presentation", "text": "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that $s, t : R \\to U$ are flat and locally of finite presentation. Consider the algebraic stack $\\mathcal{X} = [U/R]$ (see above). Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation and $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation."}
+{"_id": "7477", "title": "stacks-morphisms-lemma-slice", "text": "Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \\xymatrix{ U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\ \\mathcal{X} & \\Spec(k) \\ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$ is flat and locally of finite presentation. Let $f_1, \\ldots, f_r \\in \\Gamma(U, \\mathcal{O}_U)$ and $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence in the local ring $\\mathcal{O}_{F, \\overline{z}}$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ V(f_1, \\ldots, f_r) \\longrightarrow \\mathcal{X} $$ is flat and locally of finite presentation."}
+{"_id": "7478", "title": "stacks-morphisms-lemma-quasi-finite-at-point", "text": "Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \\xymatrix{ U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\ \\mathcal{X} & \\Spec(k) \\ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$ is locally of finite type. Let $z \\in |F|$ be such that $\\dim_z(F) = 0$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ U \\longrightarrow \\mathcal{X} $$ is locally quasi-finite."}
+{"_id": "7479", "title": "stacks-morphisms-lemma-DM-residual-gerbe", "text": "Let $\\mathcal{Z}$ be a DM, locally Noetherian, reduced algebraic stack with $|\\mathcal{Z}|$ a singleton. Then there exists a field $k$ and a surjective \\'etale morphism $\\Spec(k) \\to \\mathcal{Z}$."}
+{"_id": "7480", "title": "stacks-morphisms-lemma-etale-at-point", "text": "Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram $$ \\xymatrix{ U \\ar[d] & F \\ar[l]^p \\ar[d] \\\\ \\mathcal{X} & \\Spec(k) \\ar[l] } $$ where $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$ is flat and locally of finite presentation. Let $z \\in |F|$ be such that $F \\to \\Spec(k)$ is unramified at $z$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism $$ U \\longrightarrow \\mathcal{X} $$ is \\'etale."}
+{"_id": "7481", "title": "stacks-morphisms-lemma-DM", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic stacks. Then \\begin{enumerate} \\item For every DM algebraic stack $\\mathcal{Z}$ and morphism $\\mathcal{Z} \\to \\mathcal{Y}$ there exists a scheme and a surjective \\'etale morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$. \\item For every algebraic space $Z$ and morphism $Z \\to \\mathcal{Y}$ there exists a scheme and a surjective \\'etale morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} Z$. \\end{enumerate}"}
+{"_id": "7482", "title": "stacks-morphisms-lemma-open-DM-locus", "text": "Let $\\mathcal{X}$ be an algebraic stack. There exist open substacks $$ \\mathcal{X}'' \\subset \\mathcal{X}' \\subset \\mathcal{X} $$ such that $\\mathcal{X}''$ is DM, $\\mathcal{X}'$ is quasi-DM, and such that these are the largest open substacks with these properties."}
+{"_id": "7483", "title": "stacks-morphisms-lemma-points-DM-locus", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ correspond to $x : \\Spec(k) \\to \\mathcal{X}$. Let $G_x/k$ be the automorphism group algebraic space of $x$. Then \\begin{enumerate} \\item  $x$ is in the DM locus of $\\mathcal{X}$ if and only if $G_x \\to \\Spec(k)$ is unramified, and \\item $x$ is in the quasi-DM locus of $\\mathcal{X}$ if and only if $G_x \\to \\Spec(k)$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "7485", "title": "stacks-morphisms-lemma-base-change-locally-quasi-finite", "text": "A base change of a locally quasi-finite morphism is locally quasi-finite."}
+{"_id": "7486", "title": "stacks-morphisms-lemma-locally-quasi-finite-over-field", "text": "Let $\\mathcal{X} \\to \\Spec(k)$ be a locally quasi-finite morphism where $\\mathcal{X}$ is an algebraic stack and $k$ is a field. Let $f : V \\to \\mathcal{X}$ be a locally quasi-finite morphism where $V$ is a scheme. Then $V \\to \\Spec(k)$ is locally quasi-finite."}
+{"_id": "7487", "title": "stacks-morphisms-lemma-composition-locally-quasi-finite", "text": "A composition of a locally quasi-finite morphisms is locally quasi-finite."}
+{"_id": "7488", "title": "stacks-morphisms-lemma-characterize-quasi-DM", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is quasi-DM, \\item for any morphism $V \\to \\mathcal{Y}$ with $V$ an algebraic space there exists a surjective, flat, locally finitely presented, locally quasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where $U$ is an algebraic space, and \\item there exist algebraic spaces $U$, $V$ and a morphism $V \\to \\mathcal{Y}$ which is surjective, flat, and locally of finite presentation, and a morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite. \\end{enumerate}"}
+{"_id": "7490", "title": "stacks-morphisms-lemma-locally-quasi-finite-permanence", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. Assume that $\\mathcal{X} \\to \\mathcal{Z}$ is locally quasi-finite and $\\mathcal{Y} \\to \\mathcal{Z}$ is quasi-DM. Then $\\mathcal{X} \\to \\mathcal{Y}$ is locally quasi-finite."}
+{"_id": "7494", "title": "stacks-morphisms-lemma-composition-flat", "text": "The composition of flat morphisms is flat."}
+{"_id": "7495", "title": "stacks-morphisms-lemma-base-change-flat", "text": "A base change of a flat morphism is flat."}
+{"_id": "7496", "title": "stacks-morphisms-lemma-descent-flat", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective flat morphism of algebraic stacks. If the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is flat, then $f$ is flat."}
+{"_id": "7497", "title": "stacks-morphisms-lemma-flat-permanence", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is flat and $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and flat, then $\\mathcal{Y} \\to \\mathcal{Z}$ is flat."}
+{"_id": "7498", "title": "stacks-morphisms-lemma-lift-valuation-ring-through-flat-morphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a flat morphism of algebraic stacks. Let $\\Spec(A) \\to \\mathcal{Y}$ be a morphism where $A$ is a valuation ring. If the closed point of $\\Spec(A)$ maps to a point of $|\\mathcal{Y}|$ in the image of $|\\mathcal{X|} \\to |\\mathcal{Y}|$, then there exists a commutative diagram $$ \\xymatrix{ \\Spec(A') \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\Spec(A) \\ar[r] & \\mathcal{Y} } $$ where $A \\to A'$ is an extension of valuation rings (More on Algebra, Definition \\ref{more-algebra-definition-extension-valuation-rings})."}
+{"_id": "7499", "title": "stacks-morphisms-lemma-flat-at-point", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$. Consider commutative diagrams $$ \\vcenter{ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ \\mathcal{X} \\ar[r]^f & \\mathcal{Y} } } \\quad\\text{with points} \\vcenter{ \\xymatrix{ u \\in |U| \\ar[d] \\\\ x \\in |\\mathcal{X}| } } $$ where $U$ and $V$ are algebraic spaces, $b$ is flat, and $(a, h) : U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is flat. The following are equivalent \\begin{enumerate} \\item $h$ is flat at $u$ for one diagram as above, \\item $h$ is flat at $u$ for every diagram as above. \\end{enumerate}"}
+{"_id": "7500", "title": "stacks-morphisms-lemma-composition-finite-presentation", "text": "The composition of finitely presented morphisms is of finite presentation. The same holds for morphisms which are locally of finite presentation."}
+{"_id": "7501", "title": "stacks-morphisms-lemma-base-change-finite-presentation", "text": "A base change of a finitely presented morphism is of finite presentation. The same holds for morphisms which are locally of finite presentation."}
+{"_id": "7503", "title": "stacks-morphisms-lemma-noetherian-finite-type-finite-presentation", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item If $\\mathcal{Y}$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation. \\item If $\\mathcal{Y}$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation. \\end{enumerate}"}
+{"_id": "7504", "title": "stacks-morphisms-lemma-finite-presentation-permanence", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks If $g \\circ f$ is locally of finite presentation and $g$ is locally of finite type, then $f$ is locally of finite presentation."}
+{"_id": "7505", "title": "stacks-morphisms-lemma-diagonal-morphism-finite-type", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks with diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$. If $f$ is locally of finite type then $\\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\\Delta$ is of finite presentation."}
+{"_id": "7507", "title": "stacks-morphisms-lemma-check-property-after-fppf-base-change", "text": "Let $P$ be a property of morphisms of algebraic spaces which is fppf local on the target and preserved by arbitrary base change. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Set $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then $$ (f\\text{ has }P) \\Leftrightarrow (\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P). $$ For the meaning of this statement see Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}."}
+{"_id": "7508", "title": "stacks-morphisms-lemma-descent-property", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and fppf local on the target. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$."}
+{"_id": "7509", "title": "stacks-morphisms-lemma-descent-finite-presentation", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is locally of finite presentation, then $f$ is locally of finite presentation."}
+{"_id": "7510", "title": "stacks-morphisms-lemma-flat-finite-presentation-permanence", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite presentation and $\\mathcal{X} \\to \\mathcal{Y}$ is surjective, flat, and locally of finite presentation, then $\\mathcal{Y} \\to \\mathcal{Z}$ is locally of finite presentation."}
+{"_id": "7511", "title": "stacks-morphisms-lemma-surjective-flat-locally-finite-presentation", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then for every scheme $U$ and object $y$ of $\\mathcal{Y}$ over $U$ there exists an fppf covering $\\{U_i \\to U\\}$ and objects $x_i$ of $\\mathcal{X}$ over $U_i$ such that $f(x_i) \\cong y|_{U_i}$ in $\\mathcal{Y}_{U_i}$."}
+{"_id": "7512", "title": "stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation", "text": "Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$, $j \\in J$ be a family of morphisms of algebraic stacks which are each flat and locally of finite presentation and which are jointly surjective, i.e., $|\\mathcal{X}| = \\bigcup |f_j|(|\\mathcal{X}_j|)$. Then for every scheme $U$ and object $x$ of $\\mathcal{X}$ over $U$ there exists an fppf covering $\\{U_i \\to U\\}_{i \\in I}$, a map $a : I \\to J$, and objects $x_i$ of $\\mathcal{X}_{a(i)}$ over $U_i$ such that $f_{a(i)}(x_i) \\cong y|_{U_i}$ in $\\mathcal{X}_{U_i}$."}
+{"_id": "7513", "title": "stacks-morphisms-lemma-fppf-open", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be flat and locally of finite presentation. Then $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is open."}
+{"_id": "7514", "title": "stacks-morphisms-lemma-descent-quasi-compact", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is quasi-compact, then $f$ is quasi-compact."}
+{"_id": "7515", "title": "stacks-morphisms-lemma-check-separated-on-ui-cover", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be composable morphisms of algebraic stacks with composition $h = g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$. If $f$ is surjective, flat, locally of finite presentation, and universally injective and if $h$ is separated, then $g$ is separated."}
+{"_id": "7516", "title": "stacks-morphisms-lemma-gerbe-over-iso-classes", "text": "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe, then the sheafification of the presheaf $$ (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}, \\quad U \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong $$ is an algebraic space and $\\mathcal{X}$ is a gerbe over it."}
+{"_id": "7517", "title": "stacks-morphisms-lemma-base-change-gerbe", "text": "Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ be a fibre product of algebraic stacks. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$."}
+{"_id": "7519", "title": "stacks-morphisms-lemma-gerbe-descent", "text": "Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ be a fibre product of algebraic stacks. If $\\mathcal{Y}' \\to \\mathcal{Y}$ is surjective, flat, and locally of finite presentation and $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$, then $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$."}
+{"_id": "7521", "title": "stacks-morphisms-lemma-local-structure-gerbe", "text": "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and \\item there exists an algebraic space $U$, a group algebraic space $G$ flat and locally of finite presentation over $U$, and a surjective, flat, and locally finitely presented morphism $U \\to \\mathcal{Y}$ such that $\\mathcal{X} \\times_\\mathcal{Y} U \\cong [U/G]$ over $U$. \\end{enumerate}"}
+{"_id": "7522", "title": "stacks-morphisms-lemma-gerbe-fppf", "text": "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective, flat, and locally of finite presentation."}
+{"_id": "7523", "title": "stacks-morphisms-lemma-gerbe-isom-fppf", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which makes $\\mathcal{X}$ a gerbe over $\\mathcal{Y}$. Then \\begin{enumerate} \\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$ is flat and locally of finite presentation, \\item $\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is surjective, flat, and locally of finite presentation, \\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms $x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism $$ T_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow T_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2 $$ is surjective, flat, and locally of finite presentation, \\item given an algebraic space $T$ and morphisms $x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism $$ \\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow \\mathit{Isom}_\\mathcal{Y}(y_1, y_2) $$ is surjective, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7525", "title": "stacks-morphisms-lemma-gerbe-bijection-points", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ then $f$ is a universal homeomorphism."}
+{"_id": "7526", "title": "stacks-morphisms-lemma-gerbe-diagonal-quasi-compact", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks such that $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$. If $\\Delta_\\mathcal{X}$ is quasi-compact, so is $\\Delta_\\mathcal{Y}$."}
+{"_id": "7527", "title": "stacks-morphisms-lemma-gerbe-residual-gerbe-exists", "text": "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe then for every $x \\in |\\mathcal{X}|$ the residual gerbe of $\\mathcal{X}$ at $x$ exists."}
+{"_id": "7528", "title": "stacks-morphisms-lemma-every-point-in-a-stratum", "text": "Let $\\mathcal{X}$ be an algebraic stack such that $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact. Then there exists a well-ordered index set $I$ and for every $i \\in I$ a reduced locally closed substack $\\mathcal{U}_i \\subset \\mathcal{X}$ such that \\begin{enumerate} \\item each $\\mathcal{U}_i$ is a gerbe, \\item we have $|\\mathcal{X}| = \\bigcup_{i \\in I} |\\mathcal{U}_i|$, \\item $T_i = |\\mathcal{X}| \\setminus \\bigcup_{i' < i} |\\mathcal{U}_{i'}|$ is closed in $|\\mathcal{X}|$ for all $i \\in I$, and \\item $|\\mathcal{U}_i|$ is open in $T_i$. \\end{enumerate} We can moreover arrange it so that either (a) $|\\mathcal{U}_i| \\subset T_i$ is dense, or (b) $\\mathcal{U}_i$ is quasi-compact. In case (a), if we choose $\\mathcal{U}_i$ as large as possible (see proof for details), then the stratification is canonical."}
+{"_id": "7529", "title": "stacks-morphisms-lemma-spectral-qc-diagonal-qc", "text": "Let $\\mathcal{X}$ be a quasi-compact algebraic stack whose diagonal $\\Delta$ is quasi-compact. Then $|\\mathcal{X}|$ is a spectral topological space."}
+{"_id": "7532", "title": "stacks-morphisms-lemma-every-point-residual-gerbe", "text": "Let $\\mathcal{X}$ be an algebraic stack such that $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact. Then the residual gerbe of $\\mathcal{X}$ at $x$ exists for every $x \\in |\\mathcal{X}|$."}
+{"_id": "7533", "title": "stacks-morphisms-lemma-every-point-residual-gerbe-quasi-DM", "text": "Let $\\mathcal{X}$ be a quasi-DM algebraic stack. Then the residual gerbe of $\\mathcal{X}$ at $x$ exists for every $x \\in |\\mathcal{X}|$."}
+{"_id": "7534", "title": "stacks-morphisms-lemma-quotient-etale", "text": "Let $Y$ be an algebraic space. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $Y$. Assume $U \\to Y$ is flat and locally of finite presentation and $R \\to U \\times_Y U$ an open immersion. Then $X = [U/R] = U/R$ is an algebraic space and $X \\to Y$ is \\'etale."}
+{"_id": "7535", "title": "stacks-morphisms-lemma-quasi-splitting-etale", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $P \\subset R$ be an open subspace such that $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a groupoid in algebraic spaces over $S$. Then $$ [U/P] \\longrightarrow [U/R] $$ is a morphism of algebraic stacks which is representable by algebraic spaces, surjective, and \\'etale."}
+{"_id": "7536", "title": "stacks-morphisms-lemma-etale-local-quasi-DM", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is quasi-DM with separated diagonal (equivalently $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and separated). Let $x \\in |\\mathcal{X}|$. Then there exists a morphism of algebraic stacks $$ \\mathcal{U} \\longrightarrow \\mathcal{X} $$ with the following properties \\begin{enumerate} \\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$, \\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and \\'etale, \\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid scheme with $U$, $R$ affine, and $s, t$ finite, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7537", "title": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is quasi-DM with separated diagonal (equivalently $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and separated). Let $x \\in |\\mathcal{X}|$. Assume the automorphism group of $\\mathcal{X}$ at $x$ is finite (Remark \\ref{remark-property-automorphism-groups}). Then there exists a morphism of algebraic stacks $$ g : \\mathcal{U} \\longrightarrow \\mathcal{X} $$ with the following properties \\begin{enumerate} \\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and $g$ induces an isomorphism between automorphism groups at $u$ and $x$ (Remark \\ref{remark-identify-automorphism-groups}), \\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and \\'etale, \\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid scheme with $U$, $R$ affine, and $s, t$ finite, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7539", "title": "stacks-morphisms-lemma-composition-smooth", "text": "The composition of smooth morphisms is smooth."}
+{"_id": "7540", "title": "stacks-morphisms-lemma-base-change-smooth", "text": "A base change of a smooth morphism is smooth."}
+{"_id": "7541", "title": "stacks-morphisms-lemma-descent-smooth", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is smooth, then $f$ is smooth."}
+{"_id": "7542", "title": "stacks-morphisms-lemma-smooth-locally-finite-presentation", "text": "A smooth morphism of algebraic stacks is locally of finite presentation."}
+{"_id": "7543", "title": "stacks-morphisms-lemma-where-smooth", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. There is a maximal open substack $\\mathcal{U} \\subset \\mathcal{X}$ such that $f|_\\mathcal{U} : \\mathcal{U} \\to \\mathcal{Y}$ is smooth. Moreover, formation of this open commutes with \\begin{enumerate} \\item precomposing by smooth morphisms, \\item base change by morphisms which are flat and locally of finite presentation, \\item base change by flat morphisms provided $f$ is locally of finite presentation. \\end{enumerate}"}
+{"_id": "7544", "title": "stacks-morphisms-lemma-smooth-quotient-stack", "text": "Let $X \\to Y$ be a smooth morphism of algebraic spaces. Let $G$ be a group algebraic space over $Y$ which is flat and locally of finite presentation over $Y$. Let $G$ act on $X$ over $Y$. Then the quotient stack $[X/G]$ is smooth over $Y$."}
+{"_id": "7545", "title": "stacks-morphisms-lemma-gerbe-smooth", "text": "Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective and smooth."}
+{"_id": "7546", "title": "stacks-morphisms-lemma-etale-smooth-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is \\'etale-smooth local on the source-and-target. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic stacks. Consider commutative diagrams $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ \\mathcal{X} \\ar[r]^f & \\mathcal{Y} } $$ where $U$ and $V$ are algebraic spaces, $V \\to \\mathcal{Y}$ is smooth, and $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is \\'etale. The following are equivalent \\begin{enumerate} \\item for any diagram as above the morphism $h$ has property $\\mathcal{P}$, and \\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective the morphism $h$ has property $\\mathcal{P}$. \\end{enumerate} If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces, then this is also equivalent to $f$ (as a morphism of algebraic spaces) having property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for morphisms $f$ which are representable by algebraic spaces this is also equivalent to $f$ having property $\\mathcal{P}$ in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}."}
+{"_id": "7548", "title": "stacks-morphisms-lemma-base-change-etale", "text": "A base change of an \\'etale morphism is \\'etale."}
+{"_id": "7551", "title": "stacks-morphisms-lemma-etale-permanence", "text": "Let $\\mathcal{X}, \\mathcal{Y}$ be algebraic stacks \\'etale over an algebraic stack $\\mathcal{Z}$. Any morphism $\\mathcal{X} \\to \\mathcal{Y}$ over $\\mathcal{Z}$ is \\'etale."}
+{"_id": "7558", "title": "stacks-morphisms-lemma-characterize-unramified", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is unramified, and \\item $f$ is locally of finite type and its diagonal is \\'etale. \\end{enumerate}"}
+{"_id": "7563", "title": "stacks-morphisms-lemma-universally-closed-permanence", "text": "Consider a commutative diagram $$ \\xymatrix{ \\mathcal{X} \\ar[rr] \\ar[rd] & & \\mathcal{Y} \\ar[ld] \\\\ & \\mathcal{Z} & } $$ of algebraic stacks. \\begin{enumerate} \\item If $\\mathcal{X} \\to \\mathcal{Z}$ is universally closed and $\\mathcal{Y} \\to \\mathcal{Z}$ is separated, then the morphism $\\mathcal{X} \\to \\mathcal{Y}$ is universally closed. In particular, the image of $|\\mathcal{X}|$ in $|\\mathcal{Y}|$ is closed. \\item If $\\mathcal{X} \\to \\mathcal{Z}$ is proper and $\\mathcal{Y} \\to \\mathcal{Z}$ is separated, then the morphism $\\mathcal{X} \\to \\mathcal{Y}$ is proper. \\end{enumerate}"}
+{"_id": "7564", "title": "stacks-morphisms-lemma-image-proper-is-proper", "text": "Let $\\mathcal{Z}$ be an algebraic stack. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks over $\\mathcal{Z}$. If $\\mathcal{X}$ is universally closed over $\\mathcal{Z}$ and $f$ is surjective then $\\mathcal{Y}$ is universally closed over $\\mathcal{Z}$. In particular, if also $\\mathcal{Y}$ is separated and of finite type over $\\mathcal{Z}$, then $\\mathcal{Y}$ is proper over $\\mathcal{Z}$."}
+{"_id": "7565", "title": "stacks-morphisms-lemma-cover-upstairs", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $g : \\mathcal{W} \\to \\mathcal{X}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then the scheme theoretic image of $f$ exists if and only if the scheme theoretic image of $f \\circ g$ exists and if so then these scheme theoretic images are the same."}
+{"_id": "7566", "title": "stacks-morphisms-lemma-scheme-theoretic-image-existence", "text": "Let $f : \\mathcal{Y} \\to \\mathcal{X}$ be a morphism of algebraic stacks. Then the scheme theoretic image of $f$ exists."}
+{"_id": "7567", "title": "stacks-morphisms-lemma-factor-factor", "text": "Let $$ \\xymatrix{ \\mathcal{X}_1 \\ar[d] \\ar[r]_{f_1} & \\mathcal{Y}_1 \\ar[d] \\\\ \\mathcal{X}_2 \\ar[r]^{f_2} & \\mathcal{Y}_2 } $$ be a commutative diagram of algebraic stacks. Let $\\mathcal{Z}_i \\subset \\mathcal{Y}_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $\\mathcal{Y}_1 \\to \\mathcal{Y}_2$ induces a morphism $\\mathcal{Z}_1 \\to \\mathcal{Z}_2$ and a commutative diagram $$ \\xymatrix{ \\mathcal{X}_1 \\ar[r] \\ar[d] & \\mathcal{Z}_1 \\ar[d] \\ar[r] & \\mathcal{Y}_1 \\ar[d] \\\\ \\mathcal{X}_2 \\ar[r] & \\mathcal{Z}_2 \\ar[r] & \\mathcal{Y}_2 } $$"}
+{"_id": "7568", "title": "stacks-morphisms-lemma-existence-plus-flat-base-change", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Then formation of the scheme theoretic image commutes with flat base change."}
+{"_id": "7569", "title": "stacks-morphisms-lemma-topology-scheme-theoretic-image", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Let $\\mathcal{Z} \\subset \\mathcal{Y}$ be the scheme theoretic image of $f$. Then $|\\mathcal{Z}|$ is the closure of the image of $|f|$."}
+{"_id": "7571", "title": "stacks-morphisms-lemma-cat-dotted-arrows", "text": "In the situation of Definition \\ref{definition-fill-in-diagram} the category of dotted arrows is a groupoid. If $\\Delta_f$ is separated, then it is a setoid."}
+{"_id": "7572", "title": "stacks-morphisms-lemma-cat-dotted-arrows-independent", "text": "In Definition \\ref{definition-fill-in-diagram} assume $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper (for example if $\\mathcal{Y}$ is separated or if $\\mathcal{Y}$ is separated over an algebraic space). Then the category of dotted arrows is independent (up to noncanonical equivalence) of the choice of $\\gamma$ and the existence of a dotted arrow (for some and hence equivalently all $\\gamma$) is equivalent to the existence of a diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^-y \\ar[ru]_a & \\mathcal{Y} } $$ with $2$-commutative triangles (without checking the $2$-morphisms compose correctly)."}
+{"_id": "7573", "title": "stacks-morphisms-lemma-cat-dotted-arrows-base-change", "text": "Assume given a $2$-commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-{x'} \\ar[d]_j & \\mathcal{X}' \\ar[d]^p \\ar[r]_q & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^-{y'} & \\mathcal{Y}' \\ar[r]^g & \\mathcal{Y} } $$ with the right square $2$-cartesian. Choose a $2$-arrow $\\gamma' : y' \\circ j \\to p \\circ x'$. Set $x = q \\circ x'$, $y = g \\circ y'$ and let $\\gamma : y \\circ j \\to f \\circ x$ be the composition of $\\gamma'$ with the $2$-arrow implicit in the $2$-commutativity of the right square. Then the category of dotted arrows for the left square and $\\gamma'$ is equivalent to the category of dotted arrows for the outer rectangle and $\\gamma$."}
+{"_id": "7574", "title": "stacks-morphisms-lemma-cat-dotted-arrows-composition", "text": "Assume given a $2$-commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-x \\ar[dd]_j & \\mathcal{X} \\ar[d]^f \\\\ & \\mathcal{Y} \\ar[d]^g \\\\ \\Spec(A) \\ar[r]^-z & \\mathcal{Z} } $$ Choose a $2$-arrow $\\gamma : z \\circ j \\to g \\circ f \\circ x$. Let $\\mathcal{C}$ be the category of dotted arrows for the outer rectangle and $\\gamma$. Let $\\mathcal{C}'$ be the category of dotted arrows for the square $$ \\xymatrix{ \\Spec(K) \\ar[r]_-{f \\circ x} \\ar[d]_j & \\mathcal{Y} \\ar[d]^g \\\\ \\Spec(A) \\ar[r]^-z & \\mathcal{Z} } $$ and $\\gamma$. There is a canonical functor $\\mathcal{C} \\to \\mathcal{C}'$ which turns $\\mathcal{C}$ into a category fibred in groupoids over $\\mathcal{C}'$ and whose fibre categories are categories of dotted arrows for certain squares of the form $$ \\xymatrix{ \\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^-y & \\mathcal{Y} } $$ and some choice of $y \\circ j \\to f \\circ x$."}
+{"_id": "7577", "title": "stacks-morphisms-lemma-uniqueness-representable", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent \\begin{enumerate} \\item $f$ satisfies the uniqueness part of the valuative criterion, \\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$ the morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies the uniqueness part of the valuative criterion as a morphism of algebraic spaces. \\end{enumerate}"}
+{"_id": "7578", "title": "stacks-morphisms-lemma-base-change-existence", "text": "The base change of a morphism of algebraic stacks which satisfies the existence part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the existence part of the valuative criterion."}
+{"_id": "7579", "title": "stacks-morphisms-lemma-composition-existence", "text": "The composition of morphisms of algebraic stacks which satisfy the existence part of the valuative criterion is another morphism of algebraic stacks which satisfies the existence part of the valuative criterion."}
+{"_id": "7580", "title": "stacks-morphisms-lemma-existence-representable", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent \\begin{enumerate} \\item $f$ satisfies the existence part of the valuative criterion, \\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$ the morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies the existence part of the valuative criterion as a morphism of algebraic spaces. \\end{enumerate}"}
+{"_id": "7581", "title": "stacks-morphisms-lemma-closed-immersion-valuative-criteria", "text": "A closed immersion of algebraic stacks satisfies both the existence and uniqueness part of the valuative criterion."}
+{"_id": "7582", "title": "stacks-morphisms-lemma-setoids-and-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $\\Delta_f$ is quasi-separated and if for every diagram (\\ref{equation-diagram}) and choice of $\\gamma$ as in Definition \\ref{definition-fill-in-diagram} the category of dotted arrows is a setoid, then $\\Delta_f$ is separated."}
+{"_id": "7583", "title": "stacks-morphisms-lemma-helper-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[rr]_-x \\ar[d]_j & & \\mathcal{X} \\ar[d]^{\\Delta_f} \\\\ \\Spec(A) \\ar[rr]^{(a_1, a_2, \\varphi)} \\ar@{..>}[rru] & & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} } $$ where $A$ is a valuation ring with field of fractions $K$. Let $\\gamma : (a_1, a_2, \\varphi) \\circ j \\longrightarrow \\Delta_f \\circ x$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. Then \\begin{enumerate} \\item Writing $\\gamma = (\\alpha_1, \\alpha_2)$ with $\\alpha_i : a_i \\circ j \\to x$ we obtain two dotted arrows $(a_1, \\alpha_1, \\text{id})$ and $(a_2, \\alpha_2, \\varphi)$ in the diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^-{f \\circ a_1} \\ar@{..>}[ru] & \\mathcal{Y} } $$ \\item The category of dotted arrows for the original diagram and $\\gamma$ is a setoid whose set of isomorphism classes of objects equal to the set of morphisms $(a_1, \\alpha_1, \\text{id}) \\to (a_2, \\alpha_2, \\varphi)$ in the category of dotted arrows. \\end{enumerate}"}
+{"_id": "7584", "title": "stacks-morphisms-lemma-uniqueness-and-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is quasi-separated. If $f$ satisfies the uniqueness part of the valuative criterion, then $f$ is separated."}
+{"_id": "7585", "title": "stacks-morphisms-lemma-converse-uniqueness-and-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion."}
+{"_id": "7586", "title": "stacks-morphisms-lemma-quasi-compact-existence-universally-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume \\begin{enumerate} \\item $f$ is quasi-compact, and \\item $f$ satisfies the existence part of the valuative criterion. \\end{enumerate} Then $f$ is universally closed."}
+{"_id": "7587", "title": "stacks-morphisms-lemma-converse-existence-universally-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume \\begin{enumerate} \\item $f$ is quasi-separated, and \\item $f$ is universally closed. \\end{enumerate} Then $f$ satisfies the existence part of the valuative criterion."}
+{"_id": "7588", "title": "stacks-morphisms-lemma-criterion-proper", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent \\begin{enumerate} \\item $f$ is proper, and \\item $f$ satisfies both the uniqueness and existence parts of the valuative criterion. \\end{enumerate}"}
+{"_id": "7591", "title": "stacks-morphisms-lemma-lci-permanence", "text": "Let $$ \\xymatrix{ \\mathcal{X} \\ar[rr]_f \\ar[rd] & & \\mathcal{Y} \\ar[ld] \\\\ & \\mathcal{Z} } $$ be a commutative diagram of morphisms of algebraic stacks. Assume $\\mathcal{Y} \\to \\mathcal{Z}$ is smooth and $\\mathcal{X} \\to \\mathcal{Z}$ is a local complete intersection morphism. Then $f : \\mathcal{X} \\to \\mathcal{Y}$ is a local complete intersection morphism."}
+{"_id": "7592", "title": "stacks-morphisms-lemma-stabilizer-preserving", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism, then $f$ is representable by algebraic spaces."}
+{"_id": "7593", "title": "stacks-morphisms-lemma-aut-iso-unramified", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be an unramified morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism, and \\item $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \\ref{remark-identify-automorphism-groups}) for all $x \\in |\\mathcal{X}|$. \\end{enumerate}"}
+{"_id": "7594", "title": "stacks-morphisms-lemma-stabilizer-preserving-unramified", "text": "\\begin{reference} \\cite[Proposition 3.5]{rydh_quotients} and \\cite[Proposition 2.5]{alper_quotient} \\end{reference} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume \\begin{enumerate} \\item $f$ is representable by algebraic spaces and unramified, and \\item $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper. \\end{enumerate} Then the set of $x \\in |\\mathcal{X}|$ such that $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \\ref{remark-identify-automorphism-groups}) is open. Letting $\\mathcal{U} \\subset \\mathcal{X}$ be the corresponding open substack, the morphism $\\mathcal{I}_\\mathcal{U} \\to \\mathcal{U} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism."}
+{"_id": "7595", "title": "stacks-morphisms-lemma-base-change-stabilizer-preserving", "text": "Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ be a cartesian diagram of algebraic stacks. \\begin{enumerate} \\item Let $x' \\in |\\mathcal{X}'|$ with image $x \\in |\\mathcal{X}|$. If $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$ (Remark \\ref{remark-identify-automorphism-groups}), then $f'$ induces an isomorphism between automorphism groups at $x'$ and $f(x')$. \\item If $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism, then $\\mathcal{I}_{\\mathcal{X}'} \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{I}_{\\mathcal{Y}'}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "7596", "title": "stacks-morphisms-lemma-stabilizer-preserving-points-cartesian", "text": "Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r]^g & \\mathcal{Y} } $$ be a cartesian diagram of algebraic stacks. If $f$ induces an isomorphism between automorphism groups at points (Remark \\ref{remark-identify-automorphism-groups}), then $$ \\Mor(\\Spec(k), \\mathcal{X}') \\longrightarrow \\Mor(\\Spec(k), \\mathcal{Y}') \\times \\Mor(\\Spec(k), \\mathcal{X}) $$ is injective on isomorphism classes for any field $k$."}
+{"_id": "7597", "title": "stacks-morphisms-lemma-etale-iso", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is separated, \\'etale, $f$ induces an isomorphism between automorphism groups at points (Remark \\ref{remark-identify-automorphism-groups}) and for every algebraically closed field $k$ the functor $$ f : \\Mor(\\Spec(k), \\mathcal{X}) \\longrightarrow \\Mor(\\Spec(k), \\mathcal{Y}) $$ is an equivalence. Then $f$ is an isomorphism."}
+{"_id": "7598", "title": "stacks-morphisms-proposition-when-gerbe", "text": "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a gerbe, and \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is flat and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7599", "title": "stacks-morphisms-proposition-when-gerbe-over", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and \\item $f : \\mathcal{X} \\to \\mathcal{Y}$ and $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ are surjective, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7600", "title": "stacks-morphisms-proposition-open-stratum", "text": "Let $\\mathcal{X}$ be a reduced algebraic stack such that $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact. Then there exists a dense open substack $\\mathcal{U} \\subset \\mathcal{X}$ which is a gerbe."}
+{"_id": "7643", "title": "schemes-lemma-isomorphism-locally-ringed", "text": "\\begin{slogan} An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces. \\end{slogan} Let $X$, $Y$ be locally ringed spaces. If $f : X \\to Y$ is an isomorphism of ringed spaces, then $f$ is an isomorphism of locally ringed spaces."}
+{"_id": "7644", "title": "schemes-lemma-open-immersion", "text": "Let $f : X \\to Y$ be an open immersion of locally ringed spaces. Let $j : V = f(X) \\to Y$ be the open subspace of $Y$ associated to the image of $f$. There is a unique isomorphism $f' : X \\cong V$ of locally ringed spaces such that $f = j \\circ f'$."}
+{"_id": "7645", "title": "schemes-lemma-restrict-map-to-opens", "text": "Let $f : X \\to Y$ be a morphism of locally ringed spaces. Let $U \\subset X$, and $V \\subset Y$ be open subsets. Suppose that $f(U) \\subset V$. There exists a unique morphism of locally ringed spaces $f|_U : U \\to V$ such that the following diagram is a commutative square of locally ringed spaces $$ \\xymatrix{ U \\ar[d]_{f|_U} \\ar[r] & X \\ar[d]^f \\\\ V \\ar[r] & Y } $$"}
+{"_id": "7646", "title": "schemes-lemma-closed-local-target", "text": "Let $f : Z \\to X$ be a morphism of locally ringed spaces. In order for $f$ to be a closed immersion it suffices that there exists an open covering $X = \\bigcup U_i$ such that each $f : f^{-1}U_i \\to U_i$ is a closed immersion."}
+{"_id": "7647", "title": "schemes-lemma-closed-immersion", "text": "Let $f : X \\to Y$ be a closed immersion of locally ringed spaces. Let $\\mathcal{I}$ be the kernel of the map $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$. Let $i : Z \\to Y$ be the closed subspace of $Y$ associated to $\\mathcal{I}$. There is a unique isomorphism $f' : X \\cong Z$ of locally ringed spaces such that $f = i \\circ f'$."}
+{"_id": "7648", "title": "schemes-lemma-characterize-closed-subspace", "text": "Let $X$, $Y$ be locally ringed spaces. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals locally generated by sections. Let $i : Z \\to X$ be the associated closed subspace. A morphism $f : Y \\to X$ factors through $Z$ if and only if the map $f^*\\mathcal{I} \\to f^*\\mathcal{O}_X = \\mathcal{O}_Y$ is zero. If this is the case the morphism $g : Y \\to Z$ such that $f = i \\circ g$ is unique."}
+{"_id": "7649", "title": "schemes-lemma-restrict-map-to-closed", "text": "Let $f : X \\to Y$ be a morphism of locally ringed spaces. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of ideals which is locally generated by sections. Let $i : Z \\to Y$ be the closed subspace associated to the sheaf of ideals $\\mathcal{I}$. Let $\\mathcal{J}$ be the image of the map $f^*\\mathcal{I} \\to f^*\\mathcal{O}_Y = \\mathcal{O}_X$. Then this ideal is locally generated by sections. Moreover, let $i' : Z' \\to X$ be the associated closed subspace of $X$. There exists a unique morphism of locally ringed spaces $f' : Z' \\to Z$ such that the following diagram is a commutative square of locally ringed spaces $$ \\xymatrix{ Z' \\ar[d]_{f'} \\ar[r]_{i'} & X \\ar[d]^f \\\\ Z \\ar[r]^{i} & Y } $$ Moreover, this diagram is a fibre square in the category of locally ringed spaces."}
+{"_id": "7650", "title": "schemes-lemma-standard-open", "text": "Let $R$ be a ring. Let $f \\in R$. \\begin{enumerate} \\item If $g\\in R$ and $D(g) \\subset D(f)$, then \\begin{enumerate} \\item $f$ is invertible in $R_g$, \\item $g^e = af$ for some $e \\geq 1$ and $a \\in R$, \\item there is a canonical ring map $R_f \\to R_g$, and \\item there is a canonical $R_f$-module map $M_f \\to M_g$ for any $R$-module $M$. \\end{enumerate} \\item Any open covering of $D(f)$ can be refined to a finite open covering of the form $D(f) = \\bigcup_{i = 1}^n D(g_i)$. \\item If $g_1, \\ldots, g_n \\in R$, then $D(f) \\subset \\bigcup D(g_i)$ if and only if $g_1, \\ldots, g_n$ generate the unit ideal in $R_f$. \\end{enumerate}"}
+{"_id": "7651", "title": "schemes-lemma-spec-sheaves", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\widetilde M$ be the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules associated to $M$. \\begin{enumerate} \\item We have $\\Gamma(\\Spec(R), \\mathcal{O}_{\\Spec(R)}) = R$. \\item We have $\\Gamma(\\Spec(R), \\widetilde M) = M$ as an $R$-module. \\item For every $f \\in R$ we have $\\Gamma(D(f), \\mathcal{O}_{\\Spec(R)}) = R_f$. \\item For every $f\\in R$ we have $\\Gamma(D(f), \\widetilde M) = M_f$ as an $R_f$-module. \\item Whenever $D(g) \\subset D(f)$ the restriction mappings on $\\mathcal{O}_{\\Spec(R)}$ and $\\widetilde M$ are the maps $R_f \\to R_g$ and $M_f \\to M_g$ from Lemma \\ref{lemma-standard-open}. \\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$ be the corresponding point. We have $\\mathcal{O}_{\\Spec(R), x} = R_{\\mathfrak p}$. \\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$ be the corresponding point. We have $\\widetilde M_x = M_{\\mathfrak p}$ as an $R_{\\mathfrak p}$-module. \\end{enumerate} Moreover, all these identifications are functorial in the $R$ module $M$. In particular, the functor $M \\mapsto \\widetilde M$ is an exact functor from the category of $R$-modules to the category of $\\mathcal{O}_{\\Spec(R)}$-modules."}
+{"_id": "7652", "title": "schemes-lemma-morphism-into-affine-where-point-goes", "text": "Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. Let $f \\in \\Mor(X, Y)$ be a morphism of locally ringed spaces. Given a point $x \\in X$ consider the ring maps $$ \\Gamma(Y, \\mathcal{O}_Y) \\xrightarrow{f^\\sharp} \\Gamma(X, \\mathcal{O}_X) \\to \\mathcal{O}_{X, x} $$ Let $\\mathfrak p \\subset \\Gamma(Y, \\mathcal{O}_Y)$ denote the inverse image of $\\mathfrak m_x$. Let $y \\in Y$ be the corresponding point. Then $f(x) = y$."}
+{"_id": "7653", "title": "schemes-lemma-f-open", "text": "Let $X$ be a locally ringed space. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. The set $$ D(f) = \\{x \\in X \\mid \\text{image }f \\not\\in \\mathfrak m_x\\} $$ is open. Moreover $f|_{D(f)}$ has an inverse."}
+{"_id": "7654", "title": "schemes-lemma-f-open-affine", "text": "In Lemma \\ref{lemma-f-open} above, if $X$ is an affine scheme, then the open $D(f)$ agrees with the standard open $D(f)$ defined previously (in Algebra, Definition \\ref{algebra-definition-spectrum-ring})."}
+{"_id": "7655", "title": "schemes-lemma-morphism-into-affine", "text": "\\begin{reference} A reference for this fact is \\cite[II, Err 1, Prop. 1.8.1]{EGA} where it is attributed to J. Tate. \\end{reference} Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. The map $$ \\Mor(X, Y) \\longrightarrow \\Hom(\\Gamma(Y, \\mathcal{O}_Y), \\Gamma(X, \\mathcal{O}_X)) $$ which maps $f$ to $f^\\sharp$ (on global sections) is bijective."}
+{"_id": "7656", "title": "schemes-lemma-category-affine-schemes", "text": "The category of affine schemes is equivalent to the opposite of the category of rings. The equivalence is given by the functor that associates to an affine scheme the global sections of its structure sheaf."}
+{"_id": "7657", "title": "schemes-lemma-standard-open-affine", "text": "Let $Y$ be an affine scheme. Let $f \\in \\Gamma(Y, \\mathcal{O}_Y)$. The open subspace $D(f)$ is an affine scheme."}
+{"_id": "7658", "title": "schemes-lemma-fibre-product-affine-schemes", "text": "The category of affine schemes has finite products, and fibre products. In other words, it has finite limits. Moreover, the products and fibre products in the category of affine schemes are the same as in the category of locally ringed spaces. In a formula, we have (in the category of locally ringed spaces) $$ \\Spec(R) \\times \\Spec(S) = \\Spec(R \\otimes_{\\mathbf{Z}} S) $$ and given ring maps $R \\to A$, $R \\to B$ we have $$ \\Spec(A) \\times_{\\Spec(R)} \\Spec(B) = \\Spec(A \\otimes_R B). $$"}
+{"_id": "7659", "title": "schemes-lemma-disjoint-union-affines", "text": "Let $X$ be a locally ringed space. Assume $X = U \\amalg V$ with $U$ and $V$ open and such that $U$, $V$ are affine schemes. Then $X$ is an affine scheme."}
+{"_id": "7660", "title": "schemes-lemma-compare-constructions", "text": "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. Let $M$ be an $R$-module. There exists a canonical isomorphism between the sheaf $\\widetilde M$ associated to the $R$-module $M$ (Definition \\ref{definition-structure-sheaf}) and the sheaf $\\mathcal{F}_M$ associated to the $R$-module $M$ (Modules, Definition \\ref{modules-definition-sheaf-associated}). This isomorphism is functorial in $M$. In particular, the sheaves $\\widetilde M$ are quasi-coherent. Moreover, they are characterized by the following mapping property $$ \\Hom_{\\mathcal{O}_X}(\\widetilde M, \\mathcal{F}) = \\Hom_R(M, \\Gamma(X, \\mathcal{F})) $$ for any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$. Here a map $\\alpha : \\widetilde M \\to \\mathcal{F}$ corresponds to its effect on global sections."}
+{"_id": "7662", "title": "schemes-lemma-widetilde-pullback", "text": "Let $(X, \\mathcal{O}_X) = (\\Spec(S), \\mathcal{O}_{\\Spec(S)})$, $(Y, \\mathcal{O}_Y) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be affine schemes. Let $\\psi : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of affine schemes, corresponding to the ring map $\\psi^\\sharp : R \\to S$ (see Lemma \\ref{lemma-category-affine-schemes}). \\begin{enumerate} \\item We have $\\psi^* \\widetilde M = \\widetilde{S \\otimes_R M}$ functorially in the $R$-module $M$. \\item We have $\\psi_* \\widetilde N = \\widetilde{N_R}$ functorially in the $S$-module $N$. \\end{enumerate}"}
+{"_id": "7663", "title": "schemes-lemma-quasi-coherent-affine", "text": "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is isomorphic to the sheaf associated to the $R$-module $\\Gamma(X, \\mathcal{F})$."}
+{"_id": "7664", "title": "schemes-lemma-equivalence-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. The functors $M \\mapsto \\widetilde M$ and $\\mathcal{F} \\mapsto \\Gamma(X, \\mathcal{F})$ define quasi-inverse equivalences of categories $$ \\xymatrix{ \\QCoh(\\mathcal{O}_X) \\ar@<1ex>[r] & \\text{Mod-}R \\ar@<1ex>[l] } $$ between the category of quasi-coherent $\\mathcal{O}_X$-modules and the category of $R$-modules."}
+{"_id": "7667", "title": "schemes-lemma-extension-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. Suppose that $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0 $$ is a short exact sequence of sheaves of $\\mathcal{O}_X$-modules. If two out of three are quasi-coherent then so is the third."}
+{"_id": "7668", "title": "schemes-lemma-closed-immersion-affine-case", "text": "\\begin{slogan} For affine schemes, closed immersions correspond to ideals. \\end{slogan} Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. Let $i : Z \\to X$ be any closed immersion of locally ringed spaces. Then there exists a unique ideal $I \\subset R$ such that the morphism $i : Z \\to X$ can be identified with the closed immersion $\\Spec(R/I) \\to \\Spec(R)$ constructed in Example \\ref{example-closed-immersion-affines} above."}
+{"_id": "7669", "title": "schemes-lemma-open-subspace-scheme", "text": "Let $X$ be a scheme. Let $j : U \\to X$ be an open immersion of locally ringed spaces. Then $U$ is a scheme. In particular, any open subspace of $X$ is a scheme."}
+{"_id": "7670", "title": "schemes-lemma-closed-subspace-scheme", "text": "Let $X$ be a scheme. Let $i : Z \\to X$ be a closed immersion of locally ringed spaces. \\begin{enumerate} \\item The locally ringed space $Z$ is a scheme, \\item the kernel $\\mathcal{I}$ of the map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is a quasi-coherent sheaf of ideals, \\item for any affine open $U = \\Spec(R)$ of $X$ the morphism $i^{-1}(U) \\to U$ can be identified with $\\Spec(R/I) \\to \\Spec(R)$ for some ideal $I \\subset R$, and \\item we have $\\mathcal{I}|_U = \\widetilde I$. \\end{enumerate} In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $X$ is a scheme."}
+{"_id": "7671", "title": "schemes-lemma-immersion-when-closed", "text": "Let $f : Y \\to X$ be an immersion of schemes. Then $f$ is a closed immersion if and only if $f(Y) \\subset X$ is a closed subset."}
+{"_id": "7672", "title": "schemes-lemma-scheme-sober", "text": "Let $X$ be a scheme. Any irreducible closed subset of $X$ has a unique generic point. In other words, $X$ is a sober topological space, see Topology, Definition \\ref{topology-definition-generic-point}."}
+{"_id": "7673", "title": "schemes-lemma-basis-affine-opens", "text": "Let $X$ be a scheme. The collection of affine opens of $X$ forms a basis for the topology on $X$."}
+{"_id": "7675", "title": "schemes-lemma-standard-open-two-affines", "text": "Let $X$ be a scheme. Let $U, V$ be affine opens of $X$, and let $x \\in U \\cap V$. There exists an affine open neighbourhood $W$ of $x$ such that $W$ is a standard open of both $U$ and $V$."}
+{"_id": "7676", "title": "schemes-lemma-good-subcover", "text": "Let $X$ be a scheme. Let $X = \\bigcup_i U_i$ be an affine open covering. Let $V \\subset X$ be an affine open. There exists a standard open covering $V = \\bigcup_{j = 1, \\ldots, m} V_j$ (see Definition \\ref{definition-standard-covering}) such that each $V_j$ is a standard open in one of the $U_i$."}
+{"_id": "7677", "title": "schemes-lemma-sheaf-on-affines", "text": "Let $X$ be a scheme. Let $\\mathcal{B}$ be the set of affine opens of $X$. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$, see Sheaves, Definition \\ref{sheaves-definition-presheaf-basis}. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is the restriction of a sheaf on $X$ to $\\mathcal{B}$, \\item $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$, and \\item $\\mathcal{F}(\\emptyset)$ is a singleton and whenever $U = V \\cup W$ with $U, V, W \\in \\mathcal{B}$ and $V, W \\subset U$ standard open (Algebra, Definition \\ref{algebra-definition-Zariski-topology}) the map $$ \\mathcal{F}(U) \\longrightarrow \\mathcal{F}(V) \\times \\mathcal{F}(W) $$ is injective with image the set of pairs $(s, t)$ such that $s|_{V \\cap W} = t|_{V \\cap W}$. \\end{enumerate}"}
+{"_id": "7678", "title": "schemes-lemma-scheme-finite-discrete-affine", "text": "Let $X$ be a scheme whose underlying topological space is a finite discrete set. Then $X$ is affine."}
+{"_id": "7679", "title": "schemes-lemma-reduced", "text": "A scheme $X$ is reduced if and only if $\\mathcal{O}_X(U)$ is a reduced ring for all $U \\subset X$ open."}
+{"_id": "7680", "title": "schemes-lemma-affine-reduced", "text": "An affine scheme $\\Spec(R)$ is reduced if and only if $R$ is reduced."}
+{"_id": "7681", "title": "schemes-lemma-reduced-closed-subscheme", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset. There exists a unique closed subscheme $Z \\subset X$ with the following properties: (a) the underlying topological space of $Z$ is equal to $T$, and (b) $Z$ is reduced."}
+{"_id": "7682", "title": "schemes-lemma-map-into-reduction", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Let $Y$ be a reduced scheme. A morphism $f : Y \\to X$ factors through $Z$ if and only if $f(Y) \\subset Z$ (set theoretically). In particular, any morphism $Y \\to X$ factors as $Y \\to X_{red} \\to X$."}
+{"_id": "7683", "title": "schemes-lemma-morphism-from-spec-local-ring", "text": "Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\\Spec(R) \\to X$ and pairs $(x, \\varphi)$ consisting of a point $x \\in X$ and a local homomorphism of local rings $\\varphi : \\mathcal{O}_{X, x} \\to R$."}
+{"_id": "7684", "title": "schemes-lemma-specialize-points", "text": "Let $X$ be a scheme. Let $x, x' \\in X$ be points of $X$. Then $x' \\in X$ is a generalization of $x$ if and only if $x'$ is in the image of the canonical morphism $\\Spec(\\mathcal{O}_{X, x}) \\to X$."}
+{"_id": "7685", "title": "schemes-lemma-characterize-points", "text": "Let $X$ be a scheme. Points of $X$ correspond bijectively to equivalence classes of morphisms from spectra of fields into $X$. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element $\\Spec(\\kappa(x)) \\to X$."}
+{"_id": "7686", "title": "schemes-lemma-glue", "text": "\\begin{slogan} If you have two locally ringed spaces, and a subspace of the first one is isomorphic to a subspace of the other, then you can glue them together into one big locally ringed space. \\end{slogan} Given any glueing data of locally ringed spaces there exists a locally ringed space $X$ and open subspaces $U_i \\subset X$ together with isomorphisms $\\varphi_i : X_i \\to U_i$ of locally ringed spaces such that \\begin{enumerate} \\item $\\varphi_i(U_{ij}) = U_i \\cap U_j$, and \\item $\\varphi_{ij} = \\varphi_j^{-1}|_{U_i \\cap U_j} \\circ \\varphi_i|_{U_{ij}}$. \\end{enumerate} The locally ringed space $X$ is characterized by the following mapping properties: Given a locally ringed space $Y$ we have \\begin{eqnarray*} \\Mor(X, Y) & = & \\{ (f_i)_{i\\in I} \\mid f_i : X_i \\to Y, \\ f_j \\circ \\varphi_{ij} = f_i|_{U_{ij}}\\} \\\\ f & \\mapsto & (f|_{U_i} \\circ \\varphi_i)_{i \\in I} \\\\ \\Mor(Y, X) & = & \\left\\{ \\begin{matrix} \\text{open covering }Y = \\bigcup\\nolimits_{i \\in I} V_i\\text{ and } (g_i : V_i \\to X_i)_{i \\in I} \\text{ such that}\\\\ g_i^{-1}(U_{ij}) = V_i \\cap V_j \\text{ and } g_j|_{V_i \\cap V_j} = \\varphi_{ij} \\circ g_i|_{V_i \\cap V_j} \\end{matrix} \\right\\} \\\\ g & \\mapsto & V_i = g^{-1}(U_i), \\ g_i = \\varphi_i^{-1} \\circ g|_{V_i} \\end{eqnarray*}"}
+{"_id": "7687", "title": "schemes-lemma-glue-schemes", "text": "\\begin{slogan} Schemes can be glued to give new schemes. \\end{slogan} In Lemma \\ref{lemma-glue} above, assume that all $X_i$ are schemes. Then the resulting locally ringed space $X$ is a scheme."}
+{"_id": "7688", "title": "schemes-lemma-glue-functors", "text": "Let $F$ be a contravariant functor on the category of schemes with values in the category of sets. Suppose that \\begin{enumerate} \\item $F$ satisfies the sheaf property for the Zariski topology, \\item there exists a set $I$ and a collection of subfunctors $F_i \\subset F$ such that \\begin{enumerate} \\item each $F_i$ is representable, \\item each $F_i \\subset F$ is representable by open immersions, and \\item the collection $(F_i)_{i \\in I}$ covers $F$. \\end{enumerate} \\end{enumerate} Then $F$ is representable."}
+{"_id": "7690", "title": "schemes-lemma-fibre-product-affines", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. If $X, Y, S$ are all affine then $X \\times_S Y$ is affine."}
+{"_id": "7691", "title": "schemes-lemma-open-fibre-product", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. Let $X \\times_S Y$, $p$, $q$ be the fibre product. Suppose that $U \\subset S$, $V \\subset X$, $W \\subset Y$ are open subschemes such that $f(V) \\subset U$ and $g(W) \\subset U$. Then the canonical morphism $V \\times_U W \\to X \\times_S Y$ is an open immersion which identifies $V \\times_U W$ with $p^{-1}(V) \\cap q^{-1}(W)$."}
+{"_id": "7692", "title": "schemes-lemma-affine-covering-fibre-product", "text": "\\begin{slogan} Bare-hands construction of fiber products: an affine open cover of a fiber product of schemes can be assembled from compatible affine open covers of the pieces. \\end{slogan} Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. Let $S = \\bigcup U_i$ be any affine open covering of $S$. For each $i \\in I$, let $f^{-1}(U_i) = \\bigcup_{j \\in J_i} V_j$ be an affine open covering of $f^{-1}(U_i)$ and let $g^{-1}(U_i) = \\bigcup_{k \\in K_i} W_k$ be an affine open covering of $g^{-1}(U_i)$. Then $$ X \\times_S Y = \\bigcup\\nolimits_{i \\in I} \\bigcup\\nolimits_{j \\in J_i, \\ k \\in K_i} V_j \\times_{U_i} W_k $$ is an affine open covering of $X \\times_S Y$."}
+{"_id": "7693", "title": "schemes-lemma-points-fibre-product", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. Points $z$ of $X \\times_S Y$ are in bijective correspondence to quadruples $$ (x, y, s, \\mathfrak p) $$ where $x \\in X$, $y \\in Y$, $s \\in S$ are points with $f(x) = s$, $g(y) = s$ and $\\mathfrak p$ is a prime ideal of the ring $\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y)$. The residue field of $z$ corresponds to the residue field of the prime $\\mathfrak p$."}
+{"_id": "7694", "title": "schemes-lemma-fibre-product-immersion", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same target. \\begin{enumerate} \\item If $f : X \\to S$ is a closed immersion, then $X \\times_S Y \\to Y$ is a closed immersion. Moreover, if $X \\to S$ corresponds to the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_S$, then $X \\times_S Y \\to Y$ corresponds to the sheaf of ideals $\\Im(g^*\\mathcal{I} \\to \\mathcal{O}_Y)$. \\item If $f : X \\to S$ is an open immersion, then $X \\times_S Y \\to Y$ is an open immersion. \\item If $f : X \\to S$ is an immersion, then $X \\times_S Y \\to Y$ is an immersion. \\end{enumerate}"}
+{"_id": "7695", "title": "schemes-lemma-base-change-immersion", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an immersion (resp.\\ closed immersion, resp. open immersion) of schemes over $S$. Then any base change of $f$ is an immersion (resp.\\ closed immersion, resp. open immersion)."}
+{"_id": "7696", "title": "schemes-lemma-fibre-topological", "text": "Let $f : X \\to S$ be a morphism of schemes. Consider the diagrams $$ \\xymatrix{ X_s \\ar[r] \\ar[d] & X \\ar[d] & \\Spec(\\mathcal{O}_{S, s}) \\times_S X \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(\\kappa(s)) \\ar[r] & S & \\Spec(\\mathcal{O}_{S, s}) \\ar[r] & S } $$ In both cases the top horizontal arrow is a homeomorphism onto its image."}
+{"_id": "7697", "title": "schemes-lemma-quasi-compact-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f : X \\to S$ is quasi-compact, \\item the inverse image of every affine open is quasi-compact, and \\item there exists some affine open covering $S = \\bigcup_{i \\in I} U_i$ such that $f^{-1}(U_i)$ is quasi-compact for all $i$. \\end{enumerate}"}
+{"_id": "7698", "title": "schemes-lemma-quasi-compact-preserved-base-change", "text": "Being quasi-compact is a property of morphisms of schemes over a base which is preserved under arbitrary base change."}
+{"_id": "7699", "title": "schemes-lemma-composition-quasi-compact", "text": "The composition of quasi-compact morphisms is quasi-compact."}
+{"_id": "7700", "title": "schemes-lemma-closed-immersion-quasi-compact", "text": "A closed immersion is quasi-compact."}
+{"_id": "7701", "title": "schemes-lemma-image-quasi-compact-closed", "text": "Let $f : X \\to S$ be a quasi-compact morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f(X) \\subset S$ is closed, and \\item $f(X) \\subset S$ is stable under specialization. \\end{enumerate}"}
+{"_id": "7702", "title": "schemes-lemma-quasi-compact-closed", "text": "Let $f : X \\to S$ be a quasi-compact morphism of schemes. Then $f$ is closed if and only if specializations lift along $f$, see Topology, Definition \\ref{topology-definition-lift-specializations}."}
+{"_id": "7703", "title": "schemes-lemma-specializations-lift", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item If $f$ is universally closed then specializations lift along any base change of $f$, see Topology, Definition \\ref{topology-definition-lift-specializations}. \\item If $f$ is quasi-compact and specializations lift along any base change of $f$, then $f$ is universally closed. \\end{enumerate}"}
+{"_id": "7704", "title": "schemes-lemma-points-specialize", "text": "Let $S$ be a scheme. Let $s' \\leadsto s$ be a specialization of points of $S$. Then \\begin{enumerate} \\item there exists a valuation ring $A$ and a morphism $f : \\Spec(A) \\to S$ such that the generic point $\\eta$ of $\\Spec(A)$ maps to $s'$ and the special point maps to $s$, and \\item given a field extension $\\kappa(s') \\subset K$ we may arrange it so that the extension $\\kappa(s') \\subset \\kappa(\\eta)$ induced by $f$ is isomorphic to the given extension. \\end{enumerate}"}
+{"_id": "7705", "title": "schemes-lemma-lift-specializations-valuative", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item Specializations lift along any base change of $f$ \\item The morphism $f$ satisfies the existence part of the valuative criterion. \\end{enumerate}"}
+{"_id": "7706", "title": "schemes-lemma-diagonal-affines-closed", "text": "The diagonal morphism of a morphism between affines is closed."}
+{"_id": "7707", "title": "schemes-lemma-diagonal-immersion", "text": "\\begin{slogan} The diagonal morphism for relative schemes is an immersion. \\end{slogan} Let $X$ be a scheme over $S$. The diagonal morphism $\\Delta_{X/S}$ is an immersion."}
+{"_id": "7708", "title": "schemes-lemma-where-are-they-equal", "text": "Let $X$, $Y$ be schemes over $S$. Let $a, b : X \\to Y$ be morphisms of schemes over $S$. There exists a largest locally closed subscheme $Z \\subset X$ such that $a|_Z = b|_Z$. In fact $Z$ is the equalizer of $(a, b)$. Moreover, if $Y$ is separated over $S$, then $Z$ is a closed subscheme."}
+{"_id": "7709", "title": "schemes-lemma-characterize-quasi-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is quasi-separated. \\item For every pair of affine opens $U, V \\subset X$ which map into a common affine open of $S$ the intersection $U \\cap V$ is a finite union of affine opens of $X$. \\item There exists an affine open covering $S = \\bigcup_{i \\in I} U_i$ and for each $i$ an affine open covering $f^{-1}U_i = \\bigcup_{j \\in I_i} V_j$ such that for each $i$ and each pair $j, j' \\in I_i$ the intersection $V_j \\cap V_{j'}$ is a finite union of affine opens of $X$. \\end{enumerate}"}
+{"_id": "7710", "title": "schemes-lemma-characterize-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item If $f$ is separated then for every pair of affine opens $(U, V)$ of $X$ which map into a common affine open of $S$ we have \\begin{enumerate} \\item the intersection $U \\cap V$ is affine. \\item the ring map $\\mathcal{O}_X(U) \\otimes_{\\mathbf{Z}} \\mathcal{O}_X(V) \\to \\mathcal{O}_X(U \\cap V)$ is surjective. \\end{enumerate} \\item If any pair of points $x_1, x_2 \\in X$ lying over a common point $s \\in S$ are contained in affine opens $x_1 \\in U$, $x_2 \\in V$ which map into a common affine open of $S$ such that (a), (b) hold, then $f$ is separated. \\end{enumerate}"}
+{"_id": "7711", "title": "schemes-lemma-fibre-product-after-map", "text": "Let $f : X \\to T$ and $g : Y \\to T$ be morphisms of schemes with the same target. Let $h : T \\to S$ be a morphism of schemes. Then the induced morphism $i : X \\times_T Y \\to X \\times_S Y$ is an immersion. If $T \\to S$ is separated, then $i$ is a closed immersion. If $T \\to S$ is quasi-separated, then $i$ is a quasi-compact morphism."}
+{"_id": "7712", "title": "schemes-lemma-semi-diagonal", "text": "Let $g : X \\to Y$ be a morphism of schemes over $S$. The morphism $i : X \\to X \\times_S Y$ is an immersion. If $Y$ is separated over $S$ it is a closed immersion. If $Y$ is quasi-separated over $S$ it is quasi-compact."}
+{"_id": "7714", "title": "schemes-lemma-separated-permanence", "text": "Permanence properties. \\begin{enumerate} \\item A composition of separated morphisms is separated. \\item A composition of quasi-separated morphisms is quasi-separated. \\item The base change of a separated morphism is separated. \\item The base change of a quasi-separated morphism is quasi-separated. \\item A (fibre) product of separated morphisms is separated. \\item A (fibre) product of quasi-separated morphisms is quasi-separated. \\end{enumerate}"}
+{"_id": "7715", "title": "schemes-lemma-compose-after-separated", "text": "\\begin{slogan} Separated and quasi-separated morphisms satisfy cancellation. \\end{slogan} Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. If $g \\circ f$ is separated then so is $f$. If $g \\circ f$ is quasi-separated then so is $f$."}
+{"_id": "7716", "title": "schemes-lemma-quasi-compact-permanence", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. If $g \\circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact."}
+{"_id": "7717", "title": "schemes-lemma-affine-separated", "text": "An affine scheme is separated. A morphism from an affine scheme to another scheme is separated."}
+{"_id": "7719", "title": "schemes-lemma-separated-implies-valuative", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion."}
+{"_id": "7720", "title": "schemes-lemma-valuative-criterion-separatedness", "text": "\\begin{reference} \\cite[II Proposition 7.2.3]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism. Assume \\begin{enumerate} \\item the morphism $f$ is quasi-separated, and \\item the morphism $f$ satisfies the uniqueness part of the valuative criterion. \\end{enumerate} Then $f$ is separated."}
+{"_id": "7721", "title": "schemes-lemma-monomorphism", "text": "\\begin{slogan} A scheme morphism is a monomorphism iff its diagonal is an isomorphism. \\end{slogan} Let $j : X \\to Y$ be a morphism of schemes. Then $j$ is a monomorphism if and only if the diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is an isomorphism."}
+{"_id": "7722", "title": "schemes-lemma-monomorphism-separated", "text": "A monomorphism of schemes is separated."}
+{"_id": "7723", "title": "schemes-lemma-composition-monomorphism", "text": "A composition of monomorphisms is a monomorphism."}
+{"_id": "7724", "title": "schemes-lemma-base-change-monomorphism", "text": "The base change of a monomorphism is a monomorphism."}
+{"_id": "7726", "title": "schemes-lemma-injective-points-surjective-stalks", "text": "Let $j : X \\to Y$ be a morphism of schemes. If \\begin{enumerate} \\item $j$ is injective on points, and \\item for any $x \\in X$ the ring map $j^\\sharp_x : \\mathcal{O}_{Y, j(x)} \\to \\mathcal{O}_{X, x}$ is surjective, \\end{enumerate} then $j$ is a monomorphism."}
+{"_id": "7727", "title": "schemes-lemma-immersions-monomorphisms", "text": "An immersion of schemes is a monomorphism. In particular, any immersion is separated."}
+{"_id": "7728", "title": "schemes-lemma-subscheme-of-separated-scheme", "text": "Let $f : X \\to S$ be a separated morphism. Any locally closed subscheme $Z \\subset X$ is separated over $S$."}
+{"_id": "7729", "title": "schemes-lemma-mono-towards-spec-field", "text": "Let $k_1, \\ldots, k_n$ be fields. For any monomorphism of schemes $X \\to \\Spec(k_1 \\times \\ldots \\times k_n)$ there exists a subset $I \\subset \\{1, \\ldots, n\\}$ such that $X \\cong \\Spec(\\prod_{i \\in I} k_i)$ as schemes over $\\Spec(k_1 \\times \\ldots \\times k_n)$. More generally, if $X = \\coprod_{i \\in I} \\Spec(k_i)$ is a disjoint union of spectra of fields and $Y \\to X$ is a monomorphism, then there exists a subset $J \\subset I$ such that $Y = \\coprod_{i \\in J} \\Spec(k_i)$."}
+{"_id": "7730", "title": "schemes-lemma-push-forward-quasi-coherent", "text": "Let $f : X \\to S$ be a morphism of schemes. If $f$ is quasi-compact and quasi-separated then $f_*$ transforms quasi-coherent $\\mathcal{O}_X$-modules into quasi-coherent $\\mathcal{O}_S$-modules."}
+{"_id": "7731", "title": "schemes-lemma-characterize-closed-immersions", "text": "Let $f : X \\to Y$ be a morphism of schemes. Suppose that \\begin{enumerate} \\item $f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and \\item $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective. \\end{enumerate} Then $f$ is a closed immersion of schemes."}
+{"_id": "7732", "title": "schemes-lemma-composition-immersion", "text": "A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion."}
+{"_id": "7733", "title": "schemes-proposition-characterize-universally-closed", "text": "Let $f$ be a quasi-compact morphism of schemes. Then $f$ is universally closed if and only if $f$ satisfies the existence part of the valuative criterion."}
+{"_id": "7764", "title": "injectives-theorem-baer-grothendieck", "text": "Let $\\kappa$ be the cardinality of the set of ideals in $R$, and let $\\alpha$ be an ordinal whose cofinality is greater than $\\kappa$. Then $\\mathbf{M}_\\alpha(N)$ is an injective $R$-module, and $N \\to \\mathbf{M}_\\alpha(N)$ is a functorial injective embedding."}
+{"_id": "7765", "title": "injectives-theorem-sheaves-injectives", "text": "The category of sheaves of abelian groups on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition \\ref{homology-definition-functorial-injective-embedding}."}
+{"_id": "7766", "title": "injectives-theorem-sheaves-modules-injectives", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. The category of sheaves of $\\mathcal{O}$-modules on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition \\ref{homology-definition-functorial-injective-embedding}."}
+{"_id": "7767", "title": "injectives-theorem-injective-embedding-grothendieck", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Then $\\mathcal{A}$ has functorial injective embeddings."}
+{"_id": "7768", "title": "injectives-theorem-K-injective-embedding-grothendieck", "text": "\\begin{slogan} Existence of K-injective complexes for Grothendieck abelian categories. \\end{slogan} Let $\\mathcal{A}$ be a Grothendieck abelian category. For every complex $M^\\bullet$ there exists a quasi-isomorphism $M^\\bullet \\to I^\\bullet$ such that $M^n \\to I^n$ is injective and $I^n$ is an injective object of $\\mathcal{A}$ for all $n$ and $I^\\bullet$ is a K-injective complex. Moreover, the construction is functorial in $M^\\bullet$."}
+{"_id": "7769", "title": "injectives-theorem-gabriel-popescu", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Then there exists a (noncommutative) ring $R$ and functors $G : \\mathcal{A} \\to \\text{Mod}_R$ and $F : \\text{Mod}_R \\to \\mathcal{A}$ such that \\begin{enumerate} \\item $F$ is the left adjoint to $G$, \\item $G$ is fully faithful, and \\item $F$ is exact. \\end{enumerate} Moreover, the functors are the ones constructed above."}
+{"_id": "7770", "title": "injectives-lemma-out-of-finite", "text": "Suppose that, in (\\ref{equation-compare}), $\\mathcal{C}$ is the category of sets and $A$ is a {\\it finite set}, then the map is a bijection."}
+{"_id": "7771", "title": "injectives-lemma-criterion-baer", "text": "\\begin{reference} \\cite[Theorem 1]{Baer} \\end{reference} Let $R$ be a ring. An $R$-module $Q$ is injective if and only if in every commutative diagram $$ \\xymatrix{ \\mathfrak{a} \\ar[d] \\ar[r] &  Q \\\\ R \\ar@{-->}[ru] } $$ for $\\mathfrak{a} \\subset R$ an ideal, the dotted arrow exists."}
+{"_id": "7772", "title": "injectives-lemma-construction", "text": "Let $R$ be a ring. \\begin{enumerate} \\item The construction $M \\mapsto (M \\to \\mathbf{M}(M))$ is functorial in $M$. \\item The map $M \\to \\mathbf{M}(M)$ is injective. \\item For any ideal $\\mathfrak{a}$ and any $R$-module map $\\varphi : \\mathfrak a \\to M$ there is an $R$-module map $\\varphi' : R \\to \\mathbf{M}(M)$ such that $$ \\xymatrix{ \\mathfrak{a} \\ar[d] \\ar[r]_\\varphi &  M \\ar[d] \\\\ R \\ar[r]^{\\varphi'} & \\mathbf{M}(M) } $$ commutes. \\end{enumerate}"}
+{"_id": "7774", "title": "injectives-lemma-abelian-sheaves-space", "text": "Let $X$ be a topological space. The category of abelian sheaves on $X$ has enough injectives. In fact it has functorial injective embeddings."}
+{"_id": "7775", "title": "injectives-lemma-sheaves-modules-space", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space, see Sheaves, Section \\ref{sheaves-section-ringed-spaces}. The category of sheaves of $\\mathcal{O}_X$-modules on $X$ has enough injectives. In fact it has functorial injective embeddings."}
+{"_id": "7776", "title": "injectives-lemma-map-into-next-one", "text": "With notation as above. Suppose that $\\mathcal{G}_1 \\to \\mathcal{G}_2$ is an injective map of abelian sheaves on $\\mathcal{C}$. Let $\\alpha$ be an ordinal and let $\\mathcal{G}_1 \\to J_\\alpha(\\mathcal{F})$ be a morphism of sheaves. There exists a morphism $\\mathcal{G}_2 \\to J_{\\alpha + 1}(\\mathcal{F})$ such that the following diagram commutes $$ \\xymatrix{ \\mathcal{G}_1 \\ar[d] \\ar[r] & \\mathcal{G}_2 \\ar[d] \\\\ J_{\\alpha}(\\mathcal{F}) \\ar[r] & J_{\\alpha + 1}(\\mathcal{F}) } $$"}
+{"_id": "7777", "title": "injectives-lemma-map-into-smaller", "text": "Suppose that $\\mathcal{G}_i$, $i\\in I$ is set of abelian sheaves on $\\mathcal{C}$. There exists an ordinal $\\beta$ such that for any sheaf $\\mathcal{F}$, any $i\\in I$, and any map $\\varphi : \\mathcal{G}_i \\to J_\\beta(\\mathcal{F})$ there exists an $\\alpha < \\beta$ such that $ \\varphi $ factors through $J_\\alpha(\\mathcal{F})$."}
+{"_id": "7778", "title": "injectives-lemma-characterize-injectives", "text": "Suppose $\\mathcal{J}$ is a sheaf of abelian groups with the following property: For all $X\\in \\Ob(\\mathcal{C})$, for any abelian subsheaf $\\mathcal{S} \\subset \\mathbf{Z}_X^\\#$ and any morphism $\\varphi : \\mathcal{S} \\to \\mathcal{J}$, there exists a morphism $\\mathbf{Z}_X^\\# \\to \\mathcal{J}$ extending $\\varphi$. Then $\\mathcal{J}$ is an injective sheaf of abelian groups."}
+{"_id": "7779", "title": "injectives-lemma-vee-exact-sheaves", "text": "The functor $\\mathcal{F} \\mapsto \\mathcal{F}^\\vee$ is exact."}
+{"_id": "7780", "title": "injectives-lemma-ev-injective-sheaves", "text": "For any $\\mathcal{O}$-module $\\mathcal{F}$ the evaluation map $ev : \\mathcal{F} \\to (\\mathcal{F}^\\vee)^\\vee$ is injective."}
+{"_id": "7781", "title": "injectives-lemma-JM-injective-sheaves", "text": "Let $\\mathcal{O}$ be a sheaf of rings. For every $\\mathcal{O}$-module $\\mathcal{F}$ the $\\mathcal{O}$-module $J(\\mathcal{F})$ is injective."}
+{"_id": "7782", "title": "injectives-lemma-site-abelian-category", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\text{Cov} = \\{\\{f : V \\to U\\} \\mid f\\text{ is surjective}\\}. $$ Then $(\\mathcal{A}, \\text{Cov})$ is a site, see Sites, Definition \\ref{sites-definition-site}."}
+{"_id": "7783", "title": "injectives-lemma-embedding", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C} = (\\mathcal{A}, \\text{Cov})$ be the site defined in Lemma \\ref{lemma-site-abelian-category}. Then $X \\mapsto h_X$ defines a fully faithful, exact functor $$ \\mathcal{A} \\longrightarrow \\textit{Ab}(\\mathcal{C}). $$ Moreover, the site $\\mathcal{C}$ has enough points."}
+{"_id": "7784", "title": "injectives-lemma-set-of-subobjects", "text": "Let $\\mathcal{A}$ be an abelian category with a generator $U$ and $X$ and object of $\\mathcal{A}$. If $\\kappa$ is the cardinality of $\\Mor(U, X)$ then \\begin{enumerate} \\item There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of $X$ indexed by a cardinal bigger than $\\kappa$. \\item If $\\alpha$ is an ordinal of cofinality $> \\kappa$ then any increasing (or decreasing) sequence of subobjects of $X$ indexed by $\\alpha$ is eventually constant. \\item The cardinality of the set of subobjects of $X$ is $\\leq 2^\\kappa$. \\end{enumerate}"}
+{"_id": "7785", "title": "injectives-lemma-size-goes-down", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. If $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of $\\mathcal{A}$, then $|M'|, |M''| \\leq |M|$."}
+{"_id": "7786", "title": "injectives-lemma-set-iso-classes-bounded-size", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$. \\begin{enumerate} \\item If $|M| \\leq \\kappa$, then $M$ is the quotient of a direct sum of at most $\\kappa$ copies of $U$. \\item For every cardinal $\\kappa$ there exists a set of isomorphism classes of objects $M$ with $|M| \\leq \\kappa$. \\end{enumerate}"}
+{"_id": "7787", "title": "injectives-lemma-characterize-injective", "text": "\\begin{slogan} To check that an object is injective, one only needs to check that lifting holds for subobjects of a generator. \\end{slogan} Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$. An object $I$ of $\\mathcal{A}$ is injective if and only if in every commutative diagram $$ \\xymatrix{ M \\ar[d] \\ar[r] &  I \\\\ U \\ar@{-->}[ru] } $$ for $M \\subset U$ a subobject, the dotted arrow exists."}
+{"_id": "7788", "title": "injectives-lemma-surjection-bounded-size", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$. Let $c$ be the function on cardinals defined by $c(\\kappa) = |\\bigoplus_{\\alpha \\in \\kappa} U|$. If $\\pi : M \\to N$ is a surjection then there exists a subobject $M' \\subset M$ which surjects onto $N$ with $|N'| \\leq c(|N|)$."}
+{"_id": "7789", "title": "injectives-lemma-acyclic-quotient-complexes-bounded-size", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. There exists a cardinal $\\kappa$ such that given any acyclic complex $M^\\bullet$ we have \\begin{enumerate} \\item if $M^\\bullet$ is nonzero, there is a nonzero subcomplex $N^\\bullet$ which is bounded above, acyclic, and $|N^n| \\leq \\kappa$, \\item there exists a surjection of complexes $$ \\bigoplus\\nolimits_{i \\in I} M_i^\\bullet \\longrightarrow M^\\bullet $$ where $M_i^\\bullet$ is bounded above, acyclic, and $|M_i^n| \\leq \\kappa$. \\end{enumerate}"}
+{"_id": "7790", "title": "injectives-lemma-characterize-K-injective", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $\\kappa$ be a cardinal as in Lemma \\ref{lemma-acyclic-quotient-complexes-bounded-size}. Suppose that $I^\\bullet$ is a complex such that \\begin{enumerate} \\item each $I^j$ is injective, and \\item for every bounded above acyclic complex $M^\\bullet$ such that $|M^n| \\leq \\kappa$ we have $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$. \\end{enumerate} Then $I^\\bullet$ is an $K$-injective complex."}
+{"_id": "7791", "title": "injectives-lemma-functorial-homotopies", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $(K_i^\\bullet)_{i \\in I}$ be a set of acyclic complexes. There exists a functor $M^\\bullet \\mapsto \\mathbf{M}^\\bullet(M^\\bullet)$ and a natural transformation $j_{M^\\bullet} : M^\\bullet \\to \\mathbf{M}^\\bullet(M^\\bullet)$ such \\begin{enumerate} \\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and \\item for every $i \\in I$ and $w : K_i^\\bullet \\to M^\\bullet$ the morphism $j_{M^\\bullet} \\circ w$ is homotopic to zero. \\end{enumerate}"}
+{"_id": "7792", "title": "injectives-lemma-functorial-injective", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. There exists a functor $M^\\bullet \\mapsto \\mathbf{N}^\\bullet(M^\\bullet)$ and a natural transformation $j_{M^\\bullet} : M^\\bullet \\to \\mathbf{N}^\\bullet(M^\\bullet)$ such \\begin{enumerate} \\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and \\item for every $n \\in \\mathbf{Z}$ the map $M^n \\to \\mathbf{N}^n(M^\\bullet)$ factors through a subobject $I^n \\subset \\mathbf{N}^n(M^\\bullet)$ where $I^n$ is an injective object of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "7793", "title": "injectives-lemma-grothendieck-brown", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $F : \\mathcal{A}^{opp} \\to \\textit{Sets}$ be a functor. Then $F$ is representable if and only if $F$ commutes with colimits, i.e., $$ F(\\colim_i N_i) = \\lim F(N_i) $$ for any diagram $\\mathcal{I} \\to \\mathcal{A}$, $i \\in \\mathcal{I}$."}
+{"_id": "7794", "title": "injectives-lemma-grothendieck-products", "text": "A Grothendieck abelian category has Ab3*."}
+{"_id": "7795", "title": "injectives-lemma-derived-products", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Then \\begin{enumerate} \\item $D(\\mathcal{A})$ has both direct sums and products, \\item direct sums are obtained by taking termwise direct sums of any complexes, \\item products are obtained by taking termwise products of K-injective complexes. \\end{enumerate}"}
+{"_id": "7796", "title": "injectives-lemma-RF-commutes-with-Rlim", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of abelian categories. Assume \\begin{enumerate} \\item $\\mathcal{A}$ is a Grothendieck abelian category, \\item $\\mathcal{B}$ has exact countable products, and \\item $F$ commutes with countable products. \\end{enumerate} Then $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ commutes with derived limits."}
+{"_id": "7797", "title": "injectives-lemma-K-injective-embedding-filtration", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $K^\\bullet$ be a filtered complex of $\\mathcal{A}$, see Homology, Definition \\ref{homology-definition-filtered-complex}. Then there exists a morphism $j : K^\\bullet \\to J^\\bullet$ of filtered complexes of $\\mathcal{A}$ such that \\begin{enumerate} \\item $J^n$, $F^pJ^n$, $J^n/F^pJ^n$ and $F^pJ^n/F^{p'}J^n$ are injective objects of $\\mathcal{A}$, \\item $J^\\bullet$, $F^pJ^\\bullet$, $J^\\bullet/F^pJ^\\bullet$, and $F^pJ^\\bullet/F^{p'}J^\\bullet$ are K-injective complexes, \\item $j$ induces quasi-isomorphisms $K^\\bullet \\to J^\\bullet$, $F^pK^\\bullet \\to F^pJ^\\bullet$, $K^\\bullet/F^pK^\\bullet \\to J^\\bullet/F^pJ^\\bullet$, and $F^pK^\\bullet/F^{p'}K^\\bullet \\to F^pJ^\\bullet/F^{p'}J^\\bullet$. \\end{enumerate}"}
+{"_id": "7798", "title": "injectives-lemma-represent-by-filtered-complex", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Suppose given an object $E \\in D(\\mathcal{A})$ and an inverse system $\\{E^i\\}_{i \\in \\mathbf{Z}}$ of objects of $D(\\mathcal{A})$ over $\\mathbf{Z}$ together with a compatible system of maps $E^i \\to E$. Picture: $$ \\ldots \\to E^{i + 1} \\to E^i \\to E^{i - 1} \\to \\ldots \\to E $$ Then there exists a filtered complex $K^\\bullet$ of $\\mathcal{A}$ (Homology, Definition \\ref{homology-definition-filtered-complex}) such that $K^\\bullet$ represents $E$ and $F^iK^\\bullet$ represents $E^i$ compatibly with the given maps."}
+{"_id": "7800", "title": "injectives-lemma-gabriel-popescu-left-adjoint", "text": "The functor $G$ above has a left adjoint $F : \\text{Mod}_R \\to \\mathcal{A}$."}
+{"_id": "7801", "title": "injectives-lemma-F-G-monos", "text": "Let $f : M \\to G(A)$ be an injective map in $\\text{Mod}_R$. Then the adjoint map $f' : F(M) \\to A$ is injective too."}
+{"_id": "7802", "title": "injectives-lemma-gabriel-popescu", "text": "\\begin{reference} \\cite[Corollary 4.1]{serpe} \\end{reference} Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $R$, $F$, $G$ be as in the Gabriel-Popescu theorem (Theorem \\ref{theorem-gabriel-popescu}). Then we obtain derived functors $$ RG : D(\\mathcal{A}) \\to D(\\text{Mod}_R) \\quad\\text{and}\\quad F : D(\\text{Mod}_R) \\to D(\\mathcal{A}) $$ such that $F$ is left adjoint to $RG$, $RG$ is fully faithful, and $F \\circ RG = \\text{id}$."}
+{"_id": "7803", "title": "injectives-lemma-brown", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $H : D(\\mathcal{A}) \\to \\textit{Ab}$ be a contravariant cohomological functor which transforms direct sums into products. Then $H$ is representable."}
+{"_id": "7804", "title": "injectives-proposition-modules-are-small", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\kappa$ the cardinality of the set of submodules of $M$. If $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$, then $M$ is $\\alpha$-small with respect to injections."}
+{"_id": "7805", "title": "injectives-proposition-presheaves-injectives", "text": "For abelian presheaves on a category there is a functorial injective embedding."}
+{"_id": "7806", "title": "injectives-proposition-presheaves-modules", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$. The category $\\textit{PMod}(\\mathcal{O})$ of presheaves of $\\mathcal{O}$-modules has functorial injective embeddings."}
+{"_id": "7807", "title": "injectives-proposition-objects-are-small", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\\mathcal{A}$. Let $\\kappa = |M|$. If $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$, then $M$ is $\\alpha$-small with respect to injections."}
+{"_id": "7808", "title": "injectives-proposition-brown", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $\\mathcal{D}$ be a triangulated category. Let $F : D(\\mathcal{A}) \\to \\mathcal{D}$ be an exact functor of triangulated categories which transforms direct sums into direct sums. Then $F$ has an exact right adjoint."}
+{"_id": "7822", "title": "brauer-theorem-wedderburn", "text": "\\begin{slogan} Simple finite algebras over a field are matrix algebras over a skew field. \\end{slogan} Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field."}
+{"_id": "7823", "title": "brauer-theorem-skolem-noether", "text": "Let $A$ be a finite central simple $k$-algebra. Let $B$ be a simple $k$-algebra. Let $f, g : B \\to A$ be two $k$-algebra homomorphisms. Then there exists an invertible element $x \\in A$ such that $f(b) = xg(b)x^{-1}$ for all $b \\in B$."}
+{"_id": "7824", "title": "brauer-theorem-centralizer", "text": "Let $A$ be a finite central simple algebra over $k$, and let $B$ be a simple subalgebra of $A$. Then \\begin{enumerate} \\item the centralizer $C$ of $B$ in $A$ is simple, \\item $[A : k] = [B : k][C : k]$, and \\item the centralizer of $C$ in $A$ is $B$. \\end{enumerate}"}
+{"_id": "7825", "title": "brauer-theorem-splitting", "text": "Let $A$ be a finite central simple $k$-algebra. Let $k \\subset k'$ be a finite field extension. The following are equivalent \\begin{enumerate} \\item $k'$ splits $A$, and \\item there exists a finite central simple algebra $B$ similar to $A$ such that $k' \\subset B$ and $[B : k] = [k' : k]^2$. \\end{enumerate}"}
+{"_id": "7826", "title": "brauer-lemma-rieffel", "text": "Let $A$ be a possibly noncommutative ring with $1$ which contains no nontrivial two-sided ideal. Let $M$ be a nonzero right ideal in $A$, and view $M$ as a right $A$-module. Then $A$ coincides with the bicommutant of $M$."}
+{"_id": "7827", "title": "brauer-lemma-simple-module", "text": "Let $A$ be a $k$-algebra. If $A$ is finite, then \\begin{enumerate} \\item $A$ has a simple module, \\item any nonzero module contains a simple submodule, \\item a simple module over $A$ has finite dimension over $k$, and \\item if $M$ is a simple $A$-module, then $\\text{End}_A(M)$ is a skew field. \\end{enumerate}"}
+{"_id": "7828", "title": "brauer-lemma-centralizer", "text": "Let $A$, $A'$ be $k$-algebras. Let $B \\subset A$, $B' \\subset A'$ be subalgebras with centralizers $C$, $C'$. Then the centralizer of $B \\otimes_k B'$ in $A \\otimes_k A'$ is $C \\otimes_k C'$."}
+{"_id": "7829", "title": "brauer-lemma-center-csa", "text": "Let $A$ be a finite simple $k$-algebra. Then the center $k'$ of $A$ is a finite field extension of $k$."}
+{"_id": "7830", "title": "brauer-lemma-generate-two-sided-sub", "text": "Let $V$ be a $k$ vector space. Let $K$ be a central $k$-algebra which is a skew field. Let $W \\subset V \\otimes_k K$ be a two-sided $K$-sub vector space. Then $W$ is generated as a left $K$-vector space by $W \\cap (V \\otimes 1)$."}
+{"_id": "7831", "title": "brauer-lemma-generate-two-sided-ideal", "text": "Let $A$ be a $k$-algebra. Let $K$ be a central $k$-algebra which is a skew field. Then any two-sided ideal $I \\subset A \\otimes_k K$ is of the form $J \\otimes_k K$ for some two-sided ideal $J \\subset A$. In particular, if $A$ is simple, then so is $A \\otimes_k K$."}
+{"_id": "7832", "title": "brauer-lemma-matrix-algebras", "text": "Let $R$ be a possibly noncommutative ring. Let $n \\geq 1$ be an integer. Let $R_n = \\text{Mat}(n \\times n, R)$. \\begin{enumerate} \\item The functors $M \\mapsto M^{\\oplus n}$ and $N \\mapsto Ne_{11}$ define quasi-inverse equivalences of categories $\\text{Mod}_R \\leftrightarrow \\text{Mod}_{R_n}$. \\item A two-sided ideal of $R_n$ is of the form $IR_n$ for some two-sided ideal $I$ of $R$. \\item The center of $R_n$ is equal to the center of $R$. \\end{enumerate}"}
+{"_id": "7833", "title": "brauer-lemma-simple-module-unique", "text": "Let $A$ be a finite simple $k$-algebra. \\begin{enumerate} \\item There exists exactly one simple $A$-module $M$ up to isomorphism. \\item Any finite $A$-module is a direct sum of copies of a simple module. \\item Two finite $A$-modules are isomorphic if and only if they have the same dimension over $k$. \\item If $A = \\text{Mat}(n \\times n, K)$ with $K$ a finite skew field extension of $k$, then $M = K^{\\oplus n}$ is a simple $A$-module and $\\text{End}_A(M) = K^{op}$. \\item If $M$ is a simple $A$-module, then $L = \\text{End}_A(M)$ is a skew field finite over $k$ acting on the left on $M$, we have $A = \\text{End}_L(M)$, and the centers of $A$ and $L$ agree. Also $[A : k] [L : k] = \\dim_k(M)^2$. \\item For a finite $A$-module $N$ the algebra $B = \\text{End}_A(N)$ is a matrix algebra over the skew field $L$ of (5). Moreover $\\text{End}_B(N) = A$. \\end{enumerate}"}
+{"_id": "7834", "title": "brauer-lemma-tensor-simple", "text": "Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central over $k$. Then $A \\otimes_k A'$ is simple."}
+{"_id": "7835", "title": "brauer-lemma-tensor-central-simple", "text": "The tensor product of finite central simple algebras over $k$ is finite, central, and simple."}
+{"_id": "7837", "title": "brauer-lemma-inverse", "text": "Let $A$ be a finite central simple algebra over $k$. Then $A \\otimes_k A^{op} \\cong \\text{Mat}(n \\times n, k)$ where $n = [A : k]$."}
+{"_id": "7838", "title": "brauer-lemma-similar", "text": "Similarity. \\begin{enumerate} \\item Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over $k$. \\item Every similarity class contains a unique (up to isomorphism) finite central skew field extension of $k$. \\item If $A = \\text{Mat}(n \\times n, K)$ and $B = \\text{Mat}(m \\times m, K')$ for some finite central skew fields $K$, $K'$ over $k$ then $A$ and $B$ are similar if and only if $K \\cong K'$ as $k$-algebras. \\end{enumerate}"}
+{"_id": "7839", "title": "brauer-lemma-brauer-algebraically-closed", "text": "The Brauer group of an algebraically closed field is zero."}
+{"_id": "7842", "title": "brauer-lemma-when-tensor-is-equal", "text": "Let $A$ be a finite central simple algebra over $k$, and let $B$ be a simple subalgebra of $A$. If $B$ is a central $k$-algebra, then $A = B \\otimes_k C$ where $C$ is the (central simple) centralizer of $B$ in $A$."}
+{"_id": "7843", "title": "brauer-lemma-self-centralizing-subfield", "text": "Let $A$ be a finite central simple algebra over $k$. If $K \\subset A$ is a subfield, then the following are equivalent \\begin{enumerate} \\item $[A : k] = [K : k]^2$, \\item $K$ is its own centralizer, and \\item $K$ is a maximal commutative subring. \\end{enumerate}"}
+{"_id": "7844", "title": "brauer-lemma-maximal-subfield", "text": "\\begin{slogan} The dimension of a finite central skew field is the square of the dimension of any maximal subfield. \\end{slogan} Let $A$ be a finite central skew field over $k$. Then every maximal subfield $K \\subset A$ satisfies $[A : k] = [K : k]^2$."}
+{"_id": "7845", "title": "brauer-lemma-maximal-subfield-splits", "text": "A maximal subfield of a finite central skew field $K$ over $k$ is a splitting field for $K$."}
+{"_id": "7847", "title": "brauer-lemma-finite-central-simple-algebra", "text": "Let $k$ be a field. For a $k$-algebra $A$ the following are equivalent \\begin{enumerate} \\item $A$ is finite central simple $k$-algebra, \\item $A$ is a finite dimensional $k$-vector space, $k$ is the center of $A$, and $A$ has no nontrivial two-sided ideal, \\item there exists $d \\geq 1$ such that $A \\otimes_k \\bar k \\cong \\text{Mat}(d \\times d, \\bar k)$, \\item there exists $d \\geq 1$ such that $A \\otimes_k k^{sep} \\cong \\text{Mat}(d \\times d, k^{sep})$, \\item there exist $d \\geq 1$ and a finite Galois extension $k \\subset k'$ such that $A \\otimes_k k' \\cong \\text{Mat}(d \\times d, k')$, \\item there exist $n \\geq 1$ and a finite central skew field $K$ over $k$ such that $A \\cong \\text{Mat}(n \\times n, K)$. \\end{enumerate} The integer $d$ is called the {\\it degree} of $A$."}
+{"_id": "7848", "title": "brauer-proposition-separable-splitting-field", "text": "Consider a finite central skew field $K$ over $k$. There exists a maximal subfield $k \\subset k' \\subset K$ which is separable over $k$. In particular, every Brauer class has a finite separable spitting field."}
+{"_id": "7856", "title": "divisors-lemma-associated-affine-open", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\\Spec(A) = U \\subset X$ be an affine open, and set $M = \\Gamma(U, \\mathcal{F})$. Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime. \\begin{enumerate} \\item If $\\mathfrak p$ is associated to $M$, then $x$ is associated to $\\mathcal{F}$. \\item If $\\mathfrak p$ is finitely generated, then the converse holds as well. \\end{enumerate} In particular, if $X$ is locally Noetherian, then the equivalence $$ \\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow x \\in \\text{Ass}(\\mathcal{F}) $$ holds for all pairs $(\\mathfrak p, x)$ as above."}
+{"_id": "7857", "title": "divisors-lemma-ass-support", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\text{Ass}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$."}
+{"_id": "7858", "title": "divisors-lemma-ses-ass", "text": "Let $X$ be a scheme. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\\text{Ass}(\\mathcal{F}_2) \\subset \\text{Ass}(\\mathcal{F}_1) \\cup \\text{Ass}(\\mathcal{F}_3)$ and $\\text{Ass}(\\mathcal{F}_1) \\subset \\text{Ass}(\\mathcal{F}_2)$."}
+{"_id": "7859", "title": "divisors-lemma-finite-ass", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $\\text{Ass}(\\mathcal{F}) \\cap U$ is finite for every quasi-compact open $U \\subset X$."}
+{"_id": "7860", "title": "divisors-lemma-ass-zero", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $$ \\mathcal{F} = 0 \\Leftrightarrow \\text{Ass}(\\mathcal{F}) = \\emptyset. $$"}
+{"_id": "7861", "title": "divisors-lemma-restriction-injective-open-contains-ass", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $U \\subset X$ is open and $\\text{Ass}(\\mathcal{F}) \\subset U$, then $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective."}
+{"_id": "7862", "title": "divisors-lemma-minimal-support-in-ass", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in \\text{Supp}(\\mathcal{F})$ be a point in the support of $\\mathcal{F}$ which is not a specialization of another point of $\\text{Supp}(\\mathcal{F})$. Then $x \\in \\text{Ass}(\\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is an associated point of $X$."}
+{"_id": "7863", "title": "divisors-lemma-check-injective-on-ass", "text": "Let $X$ be a locally Noetherian scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of quasi-coherent $\\mathcal{O}_X$-modules. Assume that for every $x \\in X$ at least one of the following happens \\begin{enumerate} \\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is injective, or \\item $x \\not \\in \\text{Ass}(\\mathcal{F})$. \\end{enumerate} Then $\\varphi$ is injective."}
+{"_id": "7864", "title": "divisors-lemma-check-isomorphism-via-depth-and-ass", "text": "Let $X$ be a locally Noetherian scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of quasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is coherent and that for every $x \\in X$ one of the following happens \\begin{enumerate} \\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is an isomorphism, or \\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$ and $x \\not \\in \\text{Ass}(\\mathcal{G})$. \\end{enumerate} Then $\\varphi$ is an isomorphism."}
+{"_id": "7865", "title": "divisors-lemma-bourbaki", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have $$ \\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) \\supset \\bigcup\\nolimits_{s \\in \\text{Ass}_S(\\mathcal{G})} \\text{Ass}_{X_s}(\\mathcal{F}_s) $$ and equality holds if $S$ is locally Noetherian (for the notation $\\mathcal{F}_s$ see above)."}
+{"_id": "7867", "title": "divisors-lemma-S1-no-embedded", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then the following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ has no embedded associated points, and \\item $\\mathcal{F}$ has property $(S_1)$. \\end{enumerate}"}
+{"_id": "7868", "title": "divisors-lemma-noetherian-dim-1-CM-no-embedded-points", "text": "Let $X$ be a locally Noetherian scheme of dimension $\\leq 1$. The following are equivalent \\begin{enumerate} \\item $X$ is Cohen-Macaulay, and \\item $X$ has no embedded points. \\end{enumerate}"}
+{"_id": "7870", "title": "divisors-lemma-remove-embedded-points", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent sheaf on $X$. The set of coherent subsheaves $$ \\{ \\mathcal{K} \\subset \\mathcal{F} \\mid \\text{Supp}(\\mathcal{K})\\text{ is nowhere dense in }\\text{Supp}(\\mathcal{F}) \\} $$ has a maximal element $\\mathcal{K}$. Setting $\\mathcal{F}' = \\mathcal{F}/\\mathcal{K}$ we have the following \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{F}') = \\text{Supp}(\\mathcal{F})$, \\item $\\mathcal{F}'$ has no embedded associated points, and \\item there exists a dense open $U \\subset X$ such that $U \\cap \\text{Supp}(\\mathcal{F})$ is dense in $\\text{Supp}(\\mathcal{F})$ and $\\mathcal{F}'|_U \\cong \\mathcal{F}|_U$. \\end{enumerate}"}
+{"_id": "7871", "title": "divisors-lemma-no-embedded-points-endos", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module without embedded associated points. Set $$ \\mathcal{I} = \\Ker(\\mathcal{O}_X \\longrightarrow \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{F})). $$ This is a coherent sheaf of ideals which defines a closed subscheme $Z \\subset X$ without embedded points. Moreover there exists a coherent sheaf $\\mathcal{G}$ on $Z$ such that (a) $\\mathcal{F} = (Z \\to X)_*\\mathcal{G}$, (b) $\\mathcal{G}$ has no associated embedded points, and (c) $\\text{Supp}(\\mathcal{G}) = Z$ (as sets)."}
+{"_id": "7872", "title": "divisors-lemma-weakly-associated-affine-open", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\\Spec(A) = U \\subset X$ be an affine open, and set $M = \\Gamma(U, \\mathcal{F})$. Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime. The following are equivalent \\begin{enumerate} \\item $\\mathfrak p$ is weakly associated to $M$, and \\item $x$ is weakly associated to $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "7873", "title": "divisors-lemma-weakly-ass-support", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $$ \\text{Ass}(\\mathcal{F}) \\subset \\text{WeakAss}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F}). $$"}
+{"_id": "7874", "title": "divisors-lemma-ses-weakly-ass", "text": "Let $X$ be a scheme. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\\text{WeakAss}(\\mathcal{F}_2) \\subset \\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$ and $\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$."}
+{"_id": "7875", "title": "divisors-lemma-weakly-ass-zero", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $$ \\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset $$"}
+{"_id": "7876", "title": "divisors-lemma-restriction-injective-open-contains-weakly-ass", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\text{WeakAss}(\\mathcal{F}) \\subset U \\subset X$ is open, then $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective."}
+{"_id": "7878", "title": "divisors-lemma-ass-weakly-ass", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathfrak m_x$ is a finitely generated ideal of $\\mathcal{O}_{X, x}$, then $$ x \\in \\text{Ass}(\\mathcal{F}) \\Leftrightarrow x \\in \\text{WeakAss}(\\mathcal{F}). $$ In particular, if $X$ is locally Noetherian, then $\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$."}
+{"_id": "7879", "title": "divisors-lemma-weakass-pushforward", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$ be a point which is not in the image of $f$. Then $s$ is not weakly associated to $f_*\\mathcal{F}$."}
+{"_id": "7881", "title": "divisors-lemma-depth-2-hartog", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $j : U \\to X$ be an open subscheme such that for $x \\in X \\setminus U$ we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. Then $$ \\mathcal{F} \\longrightarrow j_*(\\mathcal{F}|_U) $$ is an isomorphism and consequently $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is an isomorphism too."}
+{"_id": "7882", "title": "divisors-lemma-weakass-reduced", "text": "Let $X$ be a reduced scheme. Then the weakly associated points of $X$ are exactly the generic points of the irreducible components of $X$."}
+{"_id": "7883", "title": "divisors-lemma-weakly-ass-reverse-functorial", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then we have $$ \\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F})) $$"}
+{"_id": "7884", "title": "divisors-lemma-ass-functorial-equal", "text": "Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $X$ is locally Noetherian, then we have $$ f(\\text{Ass}_X(\\mathcal{F})) = \\text{Ass}_S(f_*\\mathcal{F}) = \\text{WeakAss}_S(f_*\\mathcal{F}) = f(\\text{WeakAss}_X(\\mathcal{F})) $$"}
+{"_id": "7885", "title": "divisors-lemma-weakly-associated-finite", "text": "Let $f : X \\to S$ be a finite morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$."}
+{"_id": "7886", "title": "divisors-lemma-weakly-ass-pullback", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module. Let $x \\in X$ with $s = f(x)$. If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and $s \\in \\text{WeakAss}_S(\\mathcal{G})$, then $x \\in \\text{WeakAss}(f^*\\mathcal{G})$."}
+{"_id": "7887", "title": "divisors-lemma-weakly-ass-change-fields", "text": "Let $K/k$ be a field extension. Let $X$ be a scheme over $k$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly associated point of the pullback $\\mathcal{F}_K$, then $x$ is a weakly associated point of $\\mathcal{F}$."}
+{"_id": "7888", "title": "divisors-lemma-depth-pushforward", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $s \\in S$. \\begin{enumerate} \\item If $s \\not \\in f(X)$, then $s$ is not weakly associated to $f_*\\mathcal{F}$. \\item If $s \\not \\in f(X)$ and $\\mathcal{O}_{S, s}$ is Noetherian, then $s$ is not associated to $f_*\\mathcal{F}$. \\item If $s \\not \\in f(X)$, $(f_*\\mathcal{F})_s$ is a finite $\\mathcal{O}_{S, s}$-module, and $\\mathcal{O}_{S, s}$ is Noetherian, then $\\text{depth}((f_*\\mathcal{F})_s) \\geq 2$. \\item If $\\mathcal{F}$ is flat over $S$ and $a \\in \\mathfrak m_s$ is a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$. \\item If $\\mathcal{F}$ is flat over $S$ and $a, b \\in \\mathfrak m_s$ is a regular sequence, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$ and $b$ is a nonzerodivisor on $(f_*\\mathcal{F})_s/a(f_*\\mathcal{F})_s$. \\item If $\\mathcal{F}$ is flat over $S$ and $(f_*\\mathcal{F})_s$ is a finite $\\mathcal{O}_{S, s}$-module, then $\\text{depth}((f_*\\mathcal{F})_s) \\geq \\min(2, \\text{depth}(\\mathcal{O}_{S, s}))$. \\end{enumerate}"}
+{"_id": "7890", "title": "divisors-lemma-base-change-relative-assassin", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $g : S' \\to S$ be a morphism of schemes. Consider the base change diagram $$ \\xymatrix{ X' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\ S' \\ar[r]^g & S } $$ and set $\\mathcal{F}' = (g')^*\\mathcal{F}$. Let $x' \\in X'$ be a point with images $x \\in X$, $s' \\in S'$ and $s \\in S$. Assume $f$ locally of finite type. Then $x' \\in \\text{Ass}_{X'/S'}(\\mathcal{F}')$ if and only if $x \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and $x'$ corresponds to a generic point of an irreducible component of $\\Spec(\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x))$."}
+{"_id": "7892", "title": "divisors-lemma-relative-weak-assassin-finite", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $i : Z \\to X$ be a finite morphism. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module. Then $\\text{WeakAss}_{X/S}(i_*\\mathcal{F}) = i(\\text{WeakAss}_{Z/S}(\\mathcal{F}))$."}
+{"_id": "7893", "title": "divisors-lemma-base-change-fitting-ideal", "text": "Let $f : T \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module. Then $f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_T = \\text{Fit}_i(f^*\\mathcal{F})$."}
+{"_id": "7894", "title": "divisors-lemma-fitting-ideal-of-finitely-presented", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_S$-module. Then $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type."}
+{"_id": "7895", "title": "divisors-lemma-on-subscheme-cut-out-by-Fit-0", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module. Let $Z_0 \\subset S$ be the closed subscheme cut out by $\\text{Fit}_0(\\mathcal{F})$. Let $Z \\subset S$ be the scheme theoretic support of $\\mathcal{F}$. Then \\begin{enumerate} \\item $Z \\subset Z_0 \\subset S$ as closed subschemes, \\item $Z = Z_0 = \\text{Supp}(\\mathcal{F})$ as closed subsets, \\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module $\\mathcal{G}_0$ with $$ (Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}. $$ \\end{enumerate}"}
+{"_id": "7896", "title": "divisors-lemma-fitting-ideal-generate-locally", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module. Let $s \\in S$. Then $\\mathcal{F}$ can be generated by $r$ elements in a neighbourhood of $s$ if and only if $\\text{Fit}_r(\\mathcal{F})_s = \\mathcal{O}_{S, s}$."}
+{"_id": "7897", "title": "divisors-lemma-fitting-ideal-finite-locally-free", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module. Let $r \\geq 0$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is finite locally free of rank $r$ \\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and $\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_S$, and \\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and $\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_S$ for $k \\geq r$. \\end{enumerate}"}
+{"_id": "7898", "title": "divisors-lemma-locally-free-rank-r-pullback", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module. The closed subschemes $$ S = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots $$ defined by the Fitting ideals of $\\mathcal{F}$ have the following properties \\begin{enumerate} \\item The intersection $\\bigcap Z_r$ is empty. \\item The functor $(\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule $$ T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by } \\leq r\\text{ sections} \\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ is representable by the open subscheme $S \\setminus Z_r$. \\item The functor $F_r : (\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule $$ T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ is representable by the locally closed subscheme $Z_{r - 1} \\setminus Z_r$ of $S$. \\end{enumerate} If $\\mathcal{F}$ is of finite presentation, then $Z_r \\to S$, $S \\setminus Z_r \\to S$, and $Z_{r - 1} \\setminus Z_r \\to S$ are of finite presentation."}
+{"_id": "7899", "title": "divisors-lemma-finite-presentation-module", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_S$-module of finite presentation. Let $S = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$ be as in Lemma \\ref{lemma-locally-free-rank-r-pullback}. Set $S_r = Z_{r - 1} \\setminus Z_r$. Then $S' = \\coprod_{r \\geq 0} S_r$ represents the functor $$ F_{flat} : \\Sch/S \\longrightarrow \\textit{Sets},\\quad\\quad T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ Moreover, $\\mathcal{F}|_{S_r}$ is locally free of rank $r$ and the morphisms $S_r \\to S$ and $S' \\to S$ are of finite presentation."}
+{"_id": "7900", "title": "divisors-lemma-base-change-and-fitting-ideal-omega", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $X = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$ be the closed subschemes defined by the fitting ideals of $\\Omega_{X/S}$. Then the formation of $Z_i$ commutes with arbitrary base change."}
+{"_id": "7902", "title": "divisors-lemma-d-fitting-ideal-omega-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $d \\geq 0$ be an integer. Assume \\begin{enumerate} \\item $f$ is flat, \\item $f$ is locally of finite presentation, and \\item every nonempty fibre of $f$ is equidimensional of dimension $d$. \\end{enumerate} Let $Z \\subset X$ be the closed subscheme cut out by the $d$th fitting ideal of $\\Omega_{X/S}$. Then $Z$ is exactly the set of points where $f$ is not smooth."}
+{"_id": "7903", "title": "divisors-lemma-torsion-sections", "text": "Let $X$ be an integral scheme with generic point $\\eta$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be nonempty open and $s \\in \\mathcal{F}(U)$. The following are equivalent \\begin{enumerate} \\item for some $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion, \\item for all $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion, \\item the image of $s$ in $\\mathcal{F}_\\eta$ is zero, \\item the image of $s$ in $j_*\\mathcal{F}_\\eta$ is zero, where $j : \\eta \\to X$ is the inclusion morphism. \\end{enumerate}"}
+{"_id": "7904", "title": "divisors-lemma-check-torsion-on-affines", "text": "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is torsion free, \\item for $U \\subset X$ affine open $\\mathcal{F}(U)$ is a torsion free $\\mathcal{O}(U)$-module. \\end{enumerate}"}
+{"_id": "7908", "title": "divisors-lemma-flat-over-integral-integral-fibre", "text": "Let $f : X \\to Y$ be a flat morphism of schemes. If $Y$ is integral and the generic fibre of $f$ is integral, then $X$ is integral."}
+{"_id": "7911", "title": "divisors-lemma-torsion-free-finite-noetherian-domain", "text": "Let $X$ be a locally Noetherian integral scheme with generic point $\\eta$. Let $\\mathcal{F}$ be a nonzero coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is torsion free, \\item $\\eta$ is the only associated prime of $\\mathcal{F}$, \\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$ has property $(S_1)$, and \\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$ has no embedded associated prime. \\end{enumerate}"}
+{"_id": "7912", "title": "divisors-lemma-torsion-free-over-regular-dim-1", "text": "Let $X$ be an integral regular scheme of dimension $\\leq 1$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is torsion free, \\item $\\mathcal{F}$ is finite locally free. \\end{enumerate}"}
+{"_id": "7913", "title": "divisors-lemma-hom-into-torsion-free", "text": "Let $X$ be an integral scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_X$-modules. If $\\mathcal{G}$ is torsion free and $\\mathcal{F}$ is of finite presentation, then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is torsion free."}
+{"_id": "7915", "title": "divisors-lemma-check-reflexive-on-affines", "text": "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is reflexive, \\item for $U \\subset X$ affine open $\\mathcal{F}(U)$ is a reflexive $\\mathcal{O}(U)$-module. \\end{enumerate}"}
+{"_id": "7919", "title": "divisors-lemma-sequence-reflexive", "text": "Let $X$ be an integral locally Noetherian scheme. Let $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}''$ an exact sequence of coherent $\\mathcal{O}_X$-modules. If $\\mathcal{F}'$ is reflexive and $\\mathcal{F}''$ is torsion free, then $\\mathcal{F}$ is reflexive."}
+{"_id": "7920", "title": "divisors-lemma-dual-reflexive", "text": "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. If $\\mathcal{G}$ is reflexive, then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is reflexive."}
+{"_id": "7922", "title": "divisors-lemma-reflexive-S2", "text": "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent reflexive $\\mathcal{O}_X$-module. Let $x \\in X$. \\begin{enumerate} \\item If $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, then $\\text{depth}(\\mathcal{F}_x) \\geq 2$. \\item If $X$ is $(S_2)$, then $\\mathcal{F}$ is $(S_2)$. \\end{enumerate}"}
+{"_id": "7923", "title": "divisors-lemma-reflexive-S2-extend", "text": "Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$ be an open subscheme with complement $Z$. Assume $\\mathcal{O}_{X, z}$ has depth $\\geq 2$ for all $z \\in Z$. Then $j^*$ and $j_*$ define an equivalence of categories between the category of coherent reflexive $\\mathcal{O}_X$-modules and the category of coherent reflexive $\\mathcal{O}_U$-modules."}
+{"_id": "7924", "title": "divisors-lemma-reflexive-over-normal", "text": "Let $X$ be an integral locally Noetherian normal scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is reflexive, \\item $\\mathcal{F}$ is torsion free and has property $(S_2)$, and \\item there exists an open subscheme $j : U \\to X$ such that \\begin{enumerate} \\item every irreducible component of $X \\setminus U$ has codimension $\\geq 2$ in $X$, \\item $j^*\\mathcal{F}$ is finite locally free, and \\item $\\mathcal{F} = j_*j^*\\mathcal{F}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "7925", "title": "divisors-lemma-describe-reflexive-hull", "text": "Let $X$ be an integral locally Noetherian normal scheme with generic point $\\eta$. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Let $T : \\mathcal{G}_\\eta \\to \\mathcal{F}_\\eta$ be a linear map. Then $T$ extends to a map $\\mathcal{G} \\to \\mathcal{F}^{**}$ of $\\mathcal{O}_X$-modules if and only if \\begin{itemize} \\item[(*)] for every $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 1$ we have $$ T\\left(\\Im(\\mathcal{G}_x \\to \\mathcal{G}_\\eta)\\right) \\subset \\Im(\\mathcal{F}_x \\to \\mathcal{F}_\\eta). $$ \\end{itemize}"}
+{"_id": "7926", "title": "divisors-lemma-reflexive-over-regular-dim-2", "text": "Let $X$ be a regular scheme of dimension $\\leq 2$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is reflexive, \\item $\\mathcal{F}$ is finite locally free. \\end{enumerate}"}
+{"_id": "7927", "title": "divisors-lemma-characterize-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $D \\subset S$ be a closed subscheme. The following are equivalent: \\begin{enumerate} \\item The subscheme $D$ is an effective Cartier divisor on $S$. \\item For every $x \\in D$ there exists an affine open neighbourhood $\\Spec(A) = U \\subset S$ of $x$ such that $U \\cap D = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor. \\end{enumerate}"}
+{"_id": "7928", "title": "divisors-lemma-complement-locally-principal-closed-subscheme", "text": "Let $S$ be a scheme. Let $Z \\subset S$ be a locally principal closed subscheme. Let $U = S \\setminus Z$. Then $U \\to S$ is an affine morphism."}
+{"_id": "7929", "title": "divisors-lemma-complement-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor. Let $U = S \\setminus D$. Then $U \\to S$ is an affine morphism and $U$ is scheme theoretically dense in $S$."}
+{"_id": "7930", "title": "divisors-lemma-effective-Cartier-makes-dimension-drop", "text": "Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor. Let $s \\in D$. If $\\dim_s(S) < \\infty$, then $\\dim_s(D) < \\dim_s(S)$."}
+{"_id": "7931", "title": "divisors-lemma-sum-effective-Cartier-divisors", "text": "The sum of two effective Cartier divisors is an effective Cartier divisor."}
+{"_id": "7932", "title": "divisors-lemma-difference-effective-Cartier-divisors", "text": "Let $X$ be a scheme. Let $D, D'$ be two effective Cartier divisors on $X$. If $D \\subset D'$ (as closed subschemes of $X$), then there exists an effective Cartier divisor $D''$ such that $D' = D + D''$."}
+{"_id": "7933", "title": "divisors-lemma-sum-closed-subschemes-effective-Cartier", "text": "Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$ with ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$ defines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$."}
+{"_id": "7934", "title": "divisors-lemma-sum-effective-Cartier-divisors-union", "text": "Let $X$ be a scheme. Let $D, D' \\subset X$ be effective Cartier divisors such that the scheme theoretic intersection $D \\cap D'$ is an effective Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic union of $D$ and $D'$."}
+{"_id": "7935", "title": "divisors-lemma-pullback-locally-principal", "text": "Let $f : S' \\to S$ be a morphism of schemes. Let $Z \\subset S$ be a locally principal closed subscheme. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subscheme of $S'$."}
+{"_id": "7936", "title": "divisors-lemma-pullback-effective-Cartier-defined", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $D \\subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases: \\begin{enumerate} \\item $f(x) \\not \\in D$ for any weakly associated point $x$ of $X$, \\item $X$, $Y$ integral and $f$ dominant, \\item $X$ reduced and $f(\\xi) \\not \\in D$ for any generic point $\\xi$ of any irreducible component of $X$, \\item $X$ is locally Noetherian and $f(x) \\not \\in D$ for any associated point $x$ of $X$, \\item $X$ is locally Noetherian, has no embedded points, and $f(\\xi) \\not \\in D$ for any generic point $\\xi$ of an irreducible component of $X$, \\item $f$ is flat, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "7937", "title": "divisors-lemma-pullback-effective-Cartier-divisors-additive", "text": "Let $f : S' \\to S$ be a morphism of schemes. Let $D_1$, $D_2$ be effective Cartier divisors on $S$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$."}
+{"_id": "7938", "title": "divisors-lemma-conormal-effective-Cartier-divisor", "text": "Let $S$ be a scheme and let $D \\subset S$ be an effective Cartier divisor. Then the conormal sheaf is $\\mathcal{C}_{D/S} = \\mathcal{I}_D|D = \\mathcal{O}_S(-D)|_D$ and the normal sheaf is $\\mathcal{N}_{D/S} = \\mathcal{O}_S(D)|_D$."}
+{"_id": "7939", "title": "divisors-lemma-ses-add-divisor", "text": "Let $X$ be a scheme. Let $D, C \\subset X$ be effective Cartier divisors with $C \\subset D$ and let $D' = D + C$. Then there is a short exact sequence $$ 0 \\to \\mathcal{O}_X(-D)|_C \\to \\mathcal{O}_{D'} \\to \\mathcal{O}_D \\to 0 $$ of $\\mathcal{O}_X$-modules."}
+{"_id": "7940", "title": "divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "text": "Let $S$ be a scheme. Let $D_1$, $D_2$ be effective Cartier divisors on $S$. Let $D = D_1 + D_2$. Then there is a unique isomorphism $$ \\mathcal{O}_S(D_1) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(D_2) \\longrightarrow \\mathcal{O}_S(D) $$ which maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$."}
+{"_id": "7941", "title": "divisors-lemma-pullback-effective-Cartier-divisors", "text": "Let $f : S' \\to S$ be a morphism of schemes. Let $D$ be a effective Cartier divisors on $S$. If the pullback of $D$ is defined then $f^*\\mathcal{O}_S(D) = \\mathcal{O}_{S'}(f^*D)$ and the canonical section $1_D$ pulls back to the canonical section $1_{f^*D}$."}
+{"_id": "7942", "title": "divisors-lemma-regular-section-structure-sheaf", "text": "Let $X$ be a locally ringed space. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. The following are equivalent: \\begin{enumerate} \\item $f$ is a regular section, and \\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, x}$ is a nonzerodivisor. \\end{enumerate} If $X$ is a scheme these are also equivalent to \\begin{enumerate} \\item[(3)] for any affine open $\\Spec(A) = U \\subset X$ the image $f \\in A$ is a nonzerodivisor, \\item[(4)] there exists an affine open covering $X = \\bigcup \\Spec(A_i)$ such that the image of $f$ in $A_i$ is a nonzerodivisor for all $i$. \\end{enumerate}"}
+{"_id": "7943", "title": "divisors-lemma-zero-scheme", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf. Let $s \\in \\Gamma(X, \\mathcal{L})$. \\begin{enumerate} \\item Consider closed immersions $i : Z \\to X$ such that $i^*s \\in \\Gamma(Z, i^*\\mathcal{L})$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set. \\item For any morphism of schemes $f : Y \\to X$ we have $f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if $f$ factors through $Z(s)$. \\item The zero scheme $Z(s)$ is a locally principal closed subscheme. \\item The zero scheme $Z(s)$ is an effective Cartier divisor if and only if $s$ is a regular section of $\\mathcal{L}$. \\end{enumerate}"}
+{"_id": "7944", "title": "divisors-lemma-characterize-OD", "text": "\\begin{slogan} Effective Cartier divisors on a scheme are the same as invertible sheaves with fixed regular global section. \\end{slogan} Let $X$ be a scheme. \\begin{enumerate} \\item If $D \\subset X$ is an effective Cartier divisor, then the canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular. \\item Conversely, if $s$ is a regular section of the invertible sheaf $\\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \\subset X$ and a unique isomorphism $\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$. \\end{enumerate} The constructions $D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$ give mutually inverse maps $$ \\left\\{ \\begin{matrix} \\text{effective Cartier divisors on }X \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{isomorphism classes of pairs }(\\mathcal{L}, s)\\\\ \\text{consisting of an invertible } \\mathcal{O}_X\\text{-module}\\\\ \\mathcal{L}\\text{ and a regular global section }s \\end{matrix} \\right\\} $$"}
+{"_id": "7946", "title": "divisors-lemma-effective-Cartier-in-points", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed subscheme corresponding to the quasi-coherent ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. \\begin{enumerate} \\item If for every $x \\in D$ the ideal $\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$ can be generated by one element, then $D$ is locally principal. \\item If for every $x \\in D$ the ideal $\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$ can be generated by a single nonzerodivisor, then $D$ is an effective Cartier divisor. \\end{enumerate}"}
+{"_id": "7947", "title": "divisors-lemma-effective-Cartier-codimension-1", "text": "Let $X$ be a locally Noetherian scheme. \\begin{enumerate} \\item Let $D \\subset X$ be a locally principal closed subscheme. Let $\\xi \\in D$ be a generic point of an irreducible component of $D$. Then $\\dim(\\mathcal{O}_{X, \\xi}) \\leq 1$. \\item Let $D \\subset X$ be an effective Cartier divisor. Let $\\xi \\in D$ be a generic point of an irreducible component of $D$. Then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$. \\end{enumerate}"}
+{"_id": "7948", "title": "divisors-lemma-integral-effective-Cartier-divisor-dvr", "text": "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an integral closed subscheme which is also an effective Cartier divisor. Then the local ring of $X$ at the generic point of $D$ is a discrete valuation ring."}
+{"_id": "7949", "title": "divisors-lemma-effective-Cartier-divisor-Sk", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$."}
+{"_id": "7950", "title": "divisors-lemma-normal-effective-Cartier-divisor-S1", "text": "Let $X$ be a locally Noetherian normal scheme. Let $D \\subset X$ be an effective Cartier divisor. Then $D$ is $(S_1)$."}
+{"_id": "7951", "title": "divisors-lemma-weil-divisor-is-cartier-UFD", "text": "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be a integral closed subscheme. Assume that \\begin{enumerate} \\item $D$ has codimension $1$ in $X$, and \\item $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in D$. \\end{enumerate} Then $D$ is an effective Cartier divisor."}
+{"_id": "7952", "title": "divisors-lemma-codim-1-part", "text": "Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme. Assume there exist integral effective Cartier divisors $D_i \\subset X$ and a closed subset $Z' \\subset X$ of codimension $\\geq 2$ such that $Z \\subset Z' \\cup \\bigcup D_i$ set-theoretically. Then there exists an effective Cartier divisor of the form $$ D = \\sum a_i D_i \\subset Z $$ such that $D \\to Z$ is an isomorphism away from codimension $2$ in $X$. The existence of the $D_i$ is guaranteed if $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in Z$ or if $X$ is regular."}
+{"_id": "7953", "title": "divisors-lemma-codimension-1-is-effective-Cartier", "text": "Let $Z \\subset X$ be a closed subscheme of a Noetherian scheme. Assume \\begin{enumerate} \\item $Z$ has no embedded points, \\item every irreducible component of $Z$ has codimension $1$ in $X$, \\item every local ring $\\mathcal{O}_{X, x}$, $x \\in Z$ is a UFD or $X$ is regular. \\end{enumerate} Then $Z$ is an effective Cartier divisor."}
+{"_id": "7955", "title": "divisors-lemma-effective-Cartier-divisor-is-a-sum", "text": "Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an effective Cartier divisor. Assume that there exist integral effective Cartier divisors $D_i \\subset X$ such that $D \\subset \\bigcup D_i$ set theoretically. Then $D = \\sum a_i D_i$ for some $a_i \\geq 0$. The existence of the $D_i$ is guaranteed if $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in D$ or if $X$ is regular."}
+{"_id": "7956", "title": "divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier", "text": "\\begin{slogan} On a projective scheme, every line bundle has a regular meromorphic section. \\end{slogan} Let $X$ be a Noetherian scheme which has an ample invertible sheaf. Then every invertible $\\mathcal{O}_X$-module is isomorphic to $$ \\mathcal{O}_X(D - D') = \\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1} $$ for some effective Cartier divisors $D, D'$ in $X$. Moreover, given a finite subset $E \\subset X$ we may choose $D, D'$ such that $E \\cap D = \\emptyset$ and $E \\cap D' = \\emptyset$. If $X$ is quasi-affine, then we may choose $D' = \\emptyset$."}
+{"_id": "7957", "title": "divisors-lemma-wedge-product-ses", "text": "Let $X$ be an integral regular scheme of dimension $2$. Let $i : D \\to X$ be the immersion of an effective Cartier divisor. Let $\\mathcal{F} \\to \\mathcal{F}' \\to i_*\\mathcal{G} \\to 0$ be an exact sequence of coherent $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $\\mathcal{F}, \\mathcal{F}'$ are locally free of rank $r$ on a nonempty open of $X$, \\item $D$ is an integral scheme, \\item $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_D$-module of rank $s$. \\end{enumerate} Then $\\mathcal{L} = (\\wedge^r\\mathcal{F})^{**}$ and $\\mathcal{L}' = (\\wedge^r \\mathcal{F}')^{**}$ are invertible $\\mathcal{O}_X$-modules and $\\mathcal{L}' \\cong \\mathcal{L}(k D)$ for some $k \\in \\{0, \\ldots, \\min(s, r)\\}$."}
+{"_id": "7958", "title": "divisors-lemma-affine-punctured-spec", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The punctured spectrum $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ of $A$ is affine if and only if $\\dim(A) \\leq 1$."}
+{"_id": "7959", "title": "divisors-lemma-complement-affine-open-immersion", "text": "\\begin{reference} \\cite[EGA IV, Corollaire 21.12.7]{EGA4} \\end{reference} Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open subscheme such that the inclusion morphism $U \\to X$ is affine. For every generic point $\\xi$ of an irreducible component of $X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$ has dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure of $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$."}
+{"_id": "7960", "title": "divisors-lemma-complement-affine-open", "text": "Let $X$ be a separated locally Noetherian scheme. Let $U \\subset X$ be an affine open. For every generic point $\\xi$ of an irreducible component of $X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$ has dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure of $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$."}
+{"_id": "7961", "title": "divisors-lemma-complement-open-affine-effective-cartier-divisor", "text": "Let $X$ be a Noetherian separated scheme. Let $U \\subset X$ be a dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$, then there exists an effective Cartier divisor $D \\subset X$ with $U = X \\setminus D$."}
+{"_id": "7962", "title": "divisors-lemma-complement-open-affine-effective-cartier-divisor-bis", "text": "Let $X$ be a Noetherian scheme with affine diagonal. Let $U \\subset X$ be a dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$, then there exists an effective Cartier divisor $D \\subset X$ with $U = X \\setminus D$."}
+{"_id": "7963", "title": "divisors-lemma-finite-trivialize-invertible-upstairs", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $y \\in Y$. There exists an open neighbourhood $V \\subset Y$ of $y$ such that $\\mathcal{L}|_{\\pi^{-1}(V)}$ is trivial."}
+{"_id": "7964", "title": "divisors-lemma-norm-invertible", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes. If there exists a norm of degree $d$ for $\\pi$, then there exists a homomorphism of abelian groups $$ \\text{Norm}_\\pi : \\Pic(X) \\to \\Pic(Y) $$ such that $\\text{Norm}_\\pi(\\pi^*\\mathcal{N}) \\cong \\mathcal{N}^{\\otimes d}$ for all invertible $\\mathcal{O}_Y$-modules $\\mathcal{N}$."}
+{"_id": "7965", "title": "divisors-lemma-norm-map-invertible", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume there exists a norm of degree $d$ for $\\pi$. For any $\\mathcal{O}_X$-linear map $\\varphi : \\mathcal{L} \\to \\mathcal{L}'$ of invertible $\\mathcal{O}_X$-modules there is an $\\mathcal{O}_Y$-linear map $$ \\text{Norm}_\\pi(\\varphi) : \\text{Norm}_\\pi(\\mathcal{L}) \\longrightarrow \\text{Norm}_\\pi(\\mathcal{L}') $$ with $\\text{Norm}_\\pi(\\mathcal{L})$, $\\text{Norm}_\\pi(\\mathcal{L}')$ as in Lemma \\ref{lemma-norm-invertible}. Moreover, for $y \\in Y$ the following are equivalent \\begin{enumerate} \\item $\\varphi$ is zero at a point of $x \\in X$ with $\\pi(x) = y$, and \\item $\\text{Norm}_\\pi(\\varphi)$ is zero at $y$. \\end{enumerate}"}
+{"_id": "7966", "title": "divisors-lemma-norm-ample", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ has an ample invertible sheaf and there exists a norm of degree $d$ for $\\pi$. Then $Y$ has an ample invertible sheaf."}
+{"_id": "7967", "title": "divisors-lemma-norm-quasi-affine", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine and there exists a norm of degree $d$ for $\\pi$. Then $Y$ is quasi-affine."}
+{"_id": "7968", "title": "divisors-lemma-finite-locally-free-has-norm", "text": "Let $\\pi : X \\to Y$ be a finite locally free morphism of degree $d \\geq 1$. Then there exists a canonical norm of degree $d$ whose formation commutes with arbitrary base change."}
+{"_id": "7969", "title": "divisors-lemma-norm-in-normal-case", "text": "Let $\\pi : X \\to Y$ be a finite surjective morphism with $X$ and $Y$ integral and $Y$ normal. Then there exists a norm of degree $[R(X) : R(Y)]$ for $\\pi$."}
+{"_id": "7970", "title": "divisors-lemma-Frobenius-gives-norm-for-reduction", "text": "Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that $p\\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm of degree $p^e$ for $X_{red} \\to X$ where $X_{red}$ is the reduction of $X$."}
+{"_id": "7971", "title": "divisors-lemma-push-down-quasi-affine", "text": "Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes. Assume that $X$ is quasi-affine. If either \\begin{enumerate} \\item $\\pi$ is finite locally free, or \\item $Y$ is an integral normal scheme \\end{enumerate} then $Y$ is quasi-affine."}
+{"_id": "7972", "title": "divisors-lemma-relative-Cartier", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $D \\subset X$ be a closed subscheme. Assume \\begin{enumerate} \\item $D$ is an effective Cartier divisor, and \\item $D \\to S$ is a flat morphism. \\end{enumerate} Then for every morphism of schemes $g : S' \\to S$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = S' \\times_S X$ where $g' : X' \\to X$ is the projection."}
+{"_id": "7973", "title": "divisors-lemma-sum-relative-effective-Cartier-divisor", "text": "Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$ are relative effective Cartier divisor on $X/S$ then so is $D_1 + D_2$ (Definition \\ref{definition-sum-effective-Cartier-divisors})."}
+{"_id": "7974", "title": "divisors-lemma-difference-relative-effective-Cartier-divisor", "text": "Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$ are relative effective Cartier divisor on $X/S$ and $D_1 \\subset D_2$ as closed subschemes, then the effective Cartier divisor $D$ such that $D_2 = D_1 + D$ (Lemma \\ref{lemma-difference-effective-Cartier-divisors}) is a relative effective Cartier divisor on $X/S$."}
+{"_id": "7975", "title": "divisors-lemma-flat-at-x", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $D \\subset X$ be a relative effective Cartier divisor on $X/S$. If $x \\in D$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$."}
+{"_id": "7978", "title": "divisors-lemma-fibre-Cartier", "text": "Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite presentation. Let $Z \\subset X$ be a closed subscheme. Let $x \\in Z$ with image $s \\in S$. \\begin{enumerate} \\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$, then there exists an open $U \\subset X$ and a relative effective Cartier divisor $D \\subset U$ such that $Z \\cap U \\subset D$ and $Z_s \\cap U = D_s$. \\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$, the morphism $Z \\to X$ is of finite presentation, and $Z \\to S$ is flat at $x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$. \\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$ and $Z$ is a locally principal closed subscheme of $X$ in a neighbourhood of $x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$. \\end{enumerate} In particular, if $Z \\to S$ is locally of finite presentation and flat and all fibres $Z_s \\subset X_s$ are effective Cartier divisors, then $Z$ is a relative effective Cartier divisor. Similarly, if $Z$ is a locally principal closed subscheme of $X$ such that all fibres $Z_s \\subset X_s$ are effective Cartier divisors, then $Z$ is a relative effective Cartier divisor."}
+{"_id": "7979", "title": "divisors-lemma-affine-conormal-sheaf", "text": "Let $i : Z \\to X$ be an immersion. The conormal algebra of $i$ has the following properties: \\begin{enumerate} \\item Let $U \\subset X$ be any open such that $i(Z)$ is a closed subset of $U$. Let $\\mathcal{I} \\subset \\mathcal{O}_U$ be the sheaf of ideals corresponding to the closed subscheme $i(Z) \\subset U$. Then $$ \\mathcal{C}_{Z/X, *} = i^*\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right) = i^{-1}\\left( \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1} \\right) $$ \\item For any affine open $\\Spec(R) = U \\subset X$ such that $Z \\cap U = \\Spec(R/I)$ there is a canonical isomorphism $\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *}) = \\bigoplus_{n \\geq 0} I^n/I^{n + 1}$. \\end{enumerate}"}
+{"_id": "7980", "title": "divisors-lemma-conormal-algebra-functorial", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of graded $\\mathcal{O}_Z$-algebras $$ f^*\\mathcal{C}_{Z'/X', *} \\longrightarrow \\mathcal{C}_{Z/X, *} $$ characterized by the following property: For every pair of affine opens $(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with $f(U) \\subset U'$ such that $Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$ the induced map $$ \\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X', *}) = \\bigoplus\\nolimits (I')^n/(I')^{n + 1} \\longrightarrow \\bigoplus\\nolimits_{n \\geq 0} I^n/I^{n + 1} = \\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *}) $$ is the one induced by the ring map $f^\\sharp : R' \\to R$ which has the property $f^\\sharp(I') \\subset I$."}
+{"_id": "7981", "title": "divisors-lemma-conormal-algebra-functorial-flat", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\ Z' \\ar[r]^{i'} & X' } $$ be a fibre product diagram in the category of schemes with $i$, $i'$ immersions. Then the canonical map $f^*\\mathcal{C}_{Z'/X', *} \\to \\mathcal{C}_{Z/X, *}$ of Lemma \\ref{lemma-conormal-algebra-functorial} is surjective. If $g$ is flat, then it is an isomorphism."}
+{"_id": "7982", "title": "divisors-lemma-types-regular-sequences-implications", "text": "Let $X$ be a ringed space. Let $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$. We have the following implications $f_1, \\ldots, f_r$ is a regular sequence $\\Rightarrow$ $f_1, \\ldots, f_r$ is a Koszul-regular sequence $\\Rightarrow$ $f_1, \\ldots, f_r$ is an $H_1$-regular sequence $\\Rightarrow$ $f_1, \\ldots, f_r$ is a quasi-regular sequence."}
+{"_id": "7983", "title": "divisors-lemma-regular-quasi-regular-scheme", "text": "Let $X$ be a ringed space. Let $\\mathcal{J}$ be a sheaf of ideals. We have the following implications: $\\mathcal{J}$ is regular $\\Rightarrow$ $\\mathcal{J}$ is Koszul-regular $\\Rightarrow$ $\\mathcal{J}$ is $H_1$-regular $\\Rightarrow$ $\\mathcal{J}$ is quasi-regular."}
+{"_id": "7984", "title": "divisors-lemma-quasi-regular-ideal", "text": "Let $X$ be a locally ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a sheaf of ideals. Then $\\mathcal{J}$ is quasi-regular if and only if the following conditions are satisfied: \\begin{enumerate} \\item $\\mathcal{J}$ is an $\\mathcal{O}_X$-module of finite type, \\item $\\mathcal{J}/\\mathcal{J}^2$ is a finite locally free $\\mathcal{O}_X/\\mathcal{J}$-module, and \\item the canonical maps $$ \\text{Sym}^n_{\\mathcal{O}_X/\\mathcal{J}}(\\mathcal{J}/\\mathcal{J}^2) \\longrightarrow \\mathcal{J}^n/\\mathcal{J}^{n + 1} $$ are isomorphisms for all $n \\geq 0$. \\end{enumerate}"}
+{"_id": "7985", "title": "divisors-lemma-generate-regular-ideal", "text": "Let $(X, \\mathcal{O}_X)$ be a locally ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a sheaf of ideals. Let $x \\in X$ and $f_1, \\ldots, f_r \\in \\mathcal{J}_x$ whose images give a basis for the $\\kappa(x)$-vector space $\\mathcal{J}_x/\\mathfrak m_x\\mathcal{J}_x$. \\begin{enumerate} \\item If $\\mathcal{J}$ is quasi-regular, then there exists an open neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ form a quasi-regular sequence generating $\\mathcal{J}|_U$. \\item If $\\mathcal{J}$ is $H_1$-regular, then there exists an open neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ form an $H_1$-regular sequence generating $\\mathcal{J}|_U$. \\item If $\\mathcal{J}$ is Koszul-regular, then there exists an open neighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ form an Koszul-regular sequence generating $\\mathcal{J}|_U$. \\end{enumerate}"}
+{"_id": "7986", "title": "divisors-lemma-regular-ideal-sheaf-quasi-coherent", "text": "Any regular, Koszul-regular, $H_1$-regular, or quasi-regular sheaf of ideals on a scheme is a finite type quasi-coherent sheaf of ideals."}
+{"_id": "7987", "title": "divisors-lemma-regular-ideal-sheaf-scheme", "text": "Let $X$ be a scheme. Let $\\mathcal{J}$ be a sheaf of ideals. Then $\\mathcal{J}$ is regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) if and only if for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an affine open neighbourhood $x \\in U \\subset X$, $U = \\Spec(A)$ such that $\\mathcal{J}|_U = \\widetilde{I}$ and such that $I$ is generated by a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \\ldots, f_r \\in A$."}
+{"_id": "7988", "title": "divisors-lemma-Noetherian-scheme-regular-ideal", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of $\\mathcal{O}_X/\\mathcal{J}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{J}_x$ is generated by a regular sequence in $\\mathcal{O}_{X, x}$, \\item $\\mathcal{J}_x$ is generated by a Koszul-regular sequence in $\\mathcal{O}_{X, x}$, \\item $\\mathcal{J}_x$ is generated by an $H_1$-regular sequence in $\\mathcal{O}_{X, x}$, \\item $\\mathcal{J}_x$ is generated by a quasi-regular sequence in $\\mathcal{O}_{X, x}$, \\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a regular sequence in $A$, and \\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a Koszul-regular sequence in $A$, and \\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by an $H_1$-regular sequence in $A$, and \\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such that $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a quasi-regular sequence in $A$, \\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$ is regular, and \\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$ is Koszul-regular, and \\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$ is $H_1$-regular, and \\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$ is quasi-regular. \\end{enumerate} In particular, on a locally Noetherian scheme the notions of regular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree."}
+{"_id": "7989", "title": "divisors-lemma-regular-quasi-regular-immersion", "text": "Let $i : Z \\to X$ be an immersion of schemes. We have the following implications: $i$ is regular $\\Rightarrow$ $i$ is Koszul-regular $\\Rightarrow$ $i$ is $H_1$-regular $\\Rightarrow$ $i$ is quasi-regular."}
+{"_id": "7990", "title": "divisors-lemma-regular-immersion-noetherian", "text": "Let $i : Z \\to X$ be an immersion of schemes. Assume $X$ is locally Noetherian. Then $i$ is regular $\\Leftrightarrow$ $i$ is Koszul-regular $\\Leftrightarrow$ $i$ is $H_1$-regular $\\Leftrightarrow$ $i$ is quasi-regular."}
+{"_id": "7991", "title": "divisors-lemma-flat-base-change-regular-immersion", "text": "Let $i : Z \\to X$ be a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) immersion. Let $X' \\to X$ be a flat morphism. Then the base change $i' : Z \\times_X X' \\to X'$ is a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) immersion."}
+{"_id": "7992", "title": "divisors-lemma-quasi-regular-immersion", "text": "Let $i : Z \\to X$ be an immersion of schemes. Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied \\begin{enumerate} \\item $i$ is locally of finite presentation, \\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and \\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism. \\end{enumerate}"}
+{"_id": "7993", "title": "divisors-lemma-transitivity-conormal-quasi-regular", "text": "Let $Z \\to Y \\to X$ be immersions of schemes. Assume that $Z \\to Y$ is $H_1$-regular. Then the canonical sequence of Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal} $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ is exact and locally split."}
+{"_id": "7994", "title": "divisors-lemma-composition-regular-immersion", "text": "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. \\begin{enumerate} \\item If $i$ and $j$ are regular immersions, so is $j \\circ i$. \\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$. \\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$. \\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \\circ i$ is a quasi-regular immersion. \\end{enumerate}"}
+{"_id": "7995", "title": "divisors-lemma-permanence-regular-immersion", "text": "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. Assume that the sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ of Morphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal} is exact and locally split. \\begin{enumerate} \\item If $j \\circ i$ is a quasi-regular immersion, so is $i$. \\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$. \\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$. \\end{enumerate}"}
+{"_id": "7996", "title": "divisors-lemma-extra-permanence-regular-immersion-noetherian", "text": "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. Pick $z \\in Z$ and denote $y \\in Y$, $x \\in X$ the corresponding points. Assume $X$ is locally Noetherian. The following are equivalent \\begin{enumerate} \\item $i$ is a regular immersion in a neighbourhood of $z$ and $j$ is a regular immersion in a neighbourhood of $y$, \\item $i$ and $j \\circ i$ are regular immersions in a neighbourhood of $z$, \\item $j \\circ i$ is a regular immersion in a neighbourhood of $z$ and the conormal sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ is split exact in a neighbourhood of $z$. \\end{enumerate}"}
+{"_id": "7998", "title": "divisors-lemma-immersion-regular-regular-immersion", "text": "Let $i : Z \\to X$ be an immersion. If $Z$ and $X$ are regular schemes, then $i$ is a regular immersion."}
+{"_id": "7999", "title": "divisors-lemma-relative-regular-immersion", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $i : Z \\subset X$ be an immersion. Assume \\begin{enumerate} \\item $i$ is an $H_1$-regular (resp.\\ quasi-regular) immersion, and \\item $Z \\to S$ is a flat morphism. \\end{enumerate} Then for every morphism of schemes $g : S' \\to S$ the base change $Z' = S' \\times_S Z \\to X' = S' \\times_S X$ is an $H_1$-regular (resp.\\ quasi-regular) immersion."}
+{"_id": "8000", "title": "divisors-lemma-quasi-regular-immersion-flat-at-x", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $Z \\to X$ be a relative quasi-regular immersion. If $x \\in Z$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$."}
+{"_id": "8001", "title": "divisors-lemma-relative-regular-immersion-flat-in-neighbourhood", "text": "Let $X \\to S$ be a morphism of schemes. Let $Z \\to X$ be an immersion. Assume \\begin{enumerate} \\item $X \\to S$ is flat and locally of finite presentation, \\item $Z \\to X$ is a relative quasi-regular immersion. \\end{enumerate} Then $Z \\to X$ is a regular immersion and the same remains true after any base change."}
+{"_id": "8003", "title": "divisors-lemma-fibre-quasi-regular", "text": "Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite presentation. Let $T \\subset X$ be a closed subscheme. Let $x \\in T$ with image $s \\in S$. \\begin{enumerate} \\item If $T_s \\subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, then there exists an open $U \\subset X$ and a relative quasi-regular immersion $Z \\subset U$ such that $Z_s = T_s \\cap U_s$ and $T \\cap U \\subset Z$. \\item If $T_s \\subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, the morphism $T \\to X$ is of finite presentation, and $T \\to S$ is flat at $x$, then we can choose $U$ and $Z$ as in (1) such that $T \\cap U = Z$. \\item If $T_s \\subset X_s$ is a quasi-regular immersion in a neighbourhood of $x$, and $T$ is cut out by $c$ equations in a neighbourhood of $x$, where $c = \\dim_x(X_s) - \\dim_x(T_s)$, then we can choose $U$ and $Z$ as in (1) such that $T \\cap U = Z$. \\end{enumerate} In each case $Z \\to U$ is a regular immersion by Lemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood}. In particular, if $T \\to S$ is locally of finite presentation and flat and all fibres $T_s \\subset X_s$ are quasi-regular immersions, then $T \\to X$ is a relative quasi-regular immersion."}
+{"_id": "8005", "title": "divisors-lemma-lift-regular-immersion-to-smooth", "text": "Let $$ \\xymatrix{ Y \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $X \\to S$ smooth, and $i$, $j$ immersions. If $j$ is a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) immersion, then so is $i$."}
+{"_id": "8006", "title": "divisors-lemma-immersion-lci-into-smooth-regular-immersion", "text": "Let $$ \\xymatrix{ Y \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $Y \\to S$ is syntomic, $X \\to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion."}
+{"_id": "8007", "title": "divisors-lemma-immersion-smooth-into-smooth-regular-immersion", "text": "Let $$ \\xymatrix{ Y \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $Y \\to S$ is smooth, $X \\to S$ smooth, and $i$ an immersion. Then $i$ is a regular immersion."}
+{"_id": "8008", "title": "divisors-lemma-push-regular-immersion-thru-smooth", "text": "Let $$ \\xymatrix{ Y \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $X \\to S$ smooth, and $i$, $j$ immersions. If $i$ is a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular) immersion, then so is $j$."}
+{"_id": "8009", "title": "divisors-lemma-pullback-meromorphic-sections-defined", "text": "Let $f : X \\to Y$ be a morphism of schemes. In each of the following cases pullbacks of meromorphic functions are defined. \\begin{enumerate} \\item every weakly associated point of $X$ maps to a generic point of an irreducible component of $Y$, \\item $X$, $Y$ are integral and $f$ is dominant, \\item $X$ is integral and the generic point of $X$ maps to a generic point of an irreducible component of $Y$, \\item $X$ is reduced and every generic point of every irreducible component of $X$ maps to the generic point of an irreducible component of $Y$, \\item $X$ is locally Noetherian, and any associated point of $X$ maps to a generic point of an irreducible component of $Y$, \\item $X$ is locally Noetherian, has no embedded points and any generic point of an irreducible component of $X$ maps to the generic point of an irreducible component of $Y$, and \\item $f$ is flat. \\end{enumerate}"}
+{"_id": "8010", "title": "divisors-lemma-meromorphic-weakass-finite", "text": "Let $X$ be a scheme such that \\begin{enumerate} \\item[(a)] every weakly associated point of $X$ is a generic point of an irreducible component of $X$, and \\item[(b)] any quasi-compact open has a finite number of irreducible components. \\end{enumerate} Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \\mathcal{K}_X = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} $$ where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map of Schemes, Section \\ref{schemes-section-points}. Moreover \\begin{enumerate} \\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras, \\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf $$ \\mathcal{K}_X(\\mathcal{F}) = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta $$ of meromorphic sections of $\\mathcal{F}$ is quasi-coherent, \\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \\in X$, \\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$ for any $x \\in X$, \\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$ for any affine open $U \\subset X$, \\item the ring of rational functions of $X$ (Morphisms, Definition \\ref{morphisms-definition-rational-function}) is the ring of meromorphic functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$. \\end{enumerate}"}
+{"_id": "8011", "title": "divisors-lemma-meromorphic-sections-pullback", "text": "Let $f : X \\to Y$ be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition \\ref{definition-pullback-meromorphic-sections}). \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules. There is a canonical pullback map $f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to \\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$ for meromorphic sections of $\\mathcal{F}$. \\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. A regular meromorphic section $s$ of $\\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$. \\end{enumerate}"}
+{"_id": "8012", "title": "divisors-lemma-regular-meromorphic-ideal-denominators", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\\mathcal{L}$. Let us denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the sheaf of ideals defined by the rule $$ \\mathcal{I}(V) = \\{f \\in \\mathcal{O}_X(V) \\mid fs \\in \\mathcal{L}(V)\\}. $$ The formula makes sense since $\\mathcal{L}(V) \\subset \\mathcal{K}_X(\\mathcal{L})(V)$. Then $\\mathcal{I}$ is a quasi-coherent sheaf of ideals and we have injective maps $$ 1 : \\mathcal{I} \\longrightarrow \\mathcal{O}_X, \\quad s : \\mathcal{I} \\longrightarrow \\mathcal{L} $$ whose cokernels are supported on closed nowhere dense subsets of $X$."}
+{"_id": "8013", "title": "divisors-lemma-meromorphic-section-restricts-to-zero", "text": "Let $X$ be a quasi-compact scheme. Let $h \\in \\Gamma(X, \\mathcal{O}_X)$ and $f \\in \\Gamma(X, \\mathcal{K}_X)$ such that $f$ restricts to zero on $X_h$. Then $h^n f = 0$ for some $n \\gg 0$."}
+{"_id": "8015", "title": "divisors-lemma-quasi-coherent-K", "text": "Let $X$ be a locally Noetherian scheme having no embedded points. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \\mathcal{K}_X = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} $$ where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map of Schemes, Section \\ref{schemes-section-points}. Moreover \\begin{enumerate} \\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras, \\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf $$ \\mathcal{K}_X(\\mathcal{F}) = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta $$ of meromorphic sections of $\\mathcal{F}$ is quasi-coherent, and \\item the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$. \\end{enumerate}"}
+{"_id": "8018", "title": "divisors-lemma-reduced-finite-irreducible", "text": "Let $X$ be a reduced scheme such that any quasi-compact open has a finite number of irreducible components. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have $$ \\mathcal{K}_X = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta) = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta) $$ where $j_\\eta : \\Spec(\\kappa(\\eta)) \\to X$ is the canonical map of Schemes, Section \\ref{schemes-section-points}. Moreover \\begin{enumerate} \\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras, \\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf $$ \\mathcal{K}_X(\\mathcal{F}) = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta $$ of meromorphic sections of $\\mathcal{F}$ is quasi-coherent, \\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \\in X$, \\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$ for any $x \\in X$, \\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$ for any affine open $U \\subset X$, \\item the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$. \\end{enumerate}"}
+{"_id": "8020", "title": "divisors-lemma-meromorphic-functions-integral-scheme", "text": "Let $X$ be an integral scheme with generic point $\\eta$. We have \\begin{enumerate} \\item the sheaf of meromorphic functions is isomorphic to the constant sheaf with value the function field (see Morphisms, Definition \\ref{morphisms-definition-function-field}) of $X$. \\item for any quasi-coherent sheaf $\\mathcal{F}$ on $X$ the sheaf $\\mathcal{K}_X(\\mathcal{F})$ is isomorphic to the constant sheaf with value $\\mathcal{F}_\\eta$. \\end{enumerate}"}
+{"_id": "8022", "title": "divisors-lemma-components-locally-finite", "text": "Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed subscheme. The collection of irreducible components of $Z$ is locally finite in $X$."}
+{"_id": "8023", "title": "divisors-lemma-divisor-locally-finite", "text": "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$. Then the collections $$ \\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi \\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\} $$ and $$ \\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\} $$ are locally finite in $X$."}
+{"_id": "8024", "title": "divisors-lemma-div-additive", "text": "Let $X$ be a locally Noetherian integral scheme. Let $f, g \\in R(X)^*$. Then $$ \\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g) $$ as Weil divisors on $X$."}
+{"_id": "8025", "title": "divisors-lemma-divisor-meromorphic-locally-finite", "text": "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets $$ \\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi \\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\} $$ and $$ \\{Z \\subset X \\mid Z \\text{ is a prime divisor and } \\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\} $$ are locally finite in $X$."}
+{"_id": "8026", "title": "divisors-lemma-divisor-meromorphic-well-defined", "text": "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero meromorphic sections of $\\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have $$ \\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z] = \\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z] + \\text{div}(f) $$ as Weil divisors."}
+{"_id": "8027", "title": "divisors-lemma-c1-additive", "text": "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible $\\mathcal{O}_X$-modules. Let $s$, resp.\\ $t$ be a nonzero meromorphic section of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\\mathcal{L} \\otimes \\mathcal{N}$, and $$ \\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st) = \\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t) $$ in $\\text{Div}(X)$. In particular, the Weil divisor class of $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum of the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$."}
+{"_id": "8028", "title": "divisors-lemma-normal-c1-injective", "text": "Let $X$ be a locally Noetherian integral scheme. If $X$ is normal, then the map (\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$ is injective."}
+{"_id": "8029", "title": "divisors-lemma-local-rings-UFD-c1-bijective", "text": "Let $X$ be a locally Noetherian integral scheme. Consider the map (\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$. The following are equivalent \\begin{enumerate} \\item the local rings of $X$ are UFDs, and \\item $X$ is normal and $\\Pic(X) \\to \\text{Cl}(X)$ is surjective. \\end{enumerate} In this case $\\Pic(X) \\to \\text{Cl}(X)$ is an isomorphism."}
+{"_id": "8030", "title": "divisors-lemma-in-image-pullback", "text": "Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume that \\begin{enumerate} \\item $X$ is locally Noetherian, \\item $Y$ is locally Noetherian, integral, and normal, \\item $\\varphi$ is flat with integral (hence nonempty) fibres, \\item $\\varphi$ is either quasi-compact or locally of finite type, \\item $\\mathcal{L}$ is trivial when restricted to the generic fibre of $\\varphi$. \\end{enumerate} Then $\\mathcal{L} \\cong \\varphi^*\\mathcal{N}$ for some invertible $\\mathcal{O}_Y$-module $\\mathcal{N}$."}
+{"_id": "8031", "title": "divisors-lemma-closure-effective-cartier-divisor", "text": "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open and let $D \\subset U$ be an effective Cartier divisor. If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$, then there exists an effective Cartier divisor $D' \\subset X$ with $D = U \\cap D'$."}
+{"_id": "8032", "title": "divisors-lemma-extend-invertible-module", "text": "Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module. If $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$, then there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}'$ with $\\mathcal{L} \\cong \\mathcal{L}'|_U$."}
+{"_id": "8033", "title": "divisors-lemma-open-subscheme-UFD", "text": "Let $R$ be a UFD. The Picard groups of the following are trivial. \\begin{enumerate} \\item $\\Spec(R)$ and any open subscheme of it. \\item $\\mathbf{A}^n_R = \\Spec(R[x_1, \\ldots, x_n])$ and any open subscheme of it. \\end{enumerate} In particular, the Picard group of any open subscheme of affine $n$-space $\\mathbf{A}^n_k$ over a field $k$ is trivial."}
+{"_id": "8034", "title": "divisors-lemma-Pic-projective-space-UFD", "text": "Let $R$ be a UFD. The Picard group of $\\mathbf{P}^n_R$ is $\\mathbf{Z}$. More precisely, there is an isomorphism $$ \\mathbf{Z} \\longrightarrow \\Pic(\\mathbf{P}^n_R),\\quad m \\longmapsto \\mathcal{O}_{\\mathbf{P}^n_R}(m) $$ In particular, the Picard group of $\\mathbf{P}^n_k$ of projective space over a field $k$ is $\\mathbf{Z}$."}
+{"_id": "8037", "title": "divisors-lemma-structure-sheaf-Xs", "text": "Let $X$ be an integral locally Noetherian normal scheme. Let $\\mathcal{F}$ be a rank 1 coherent reflexive $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{F})$. Let $$ U = \\{x \\in X \\mid s : \\mathcal{O}_{X, x} \\to \\mathcal{F}_x \\text{ is an isomorphism}\\} $$ Then $j : U \\to X$ is an open subscheme of $X$ and $$ j_*\\mathcal{O}_U = \\colim (\\mathcal{O}_X \\xrightarrow{s} \\mathcal{F} \\xrightarrow{s} \\mathcal{F}^{[2]} \\xrightarrow{s} \\mathcal{F}^{[3]} \\xrightarrow{s} \\ldots) $$ where $\\mathcal{F}^{[1]} = \\mathcal{F}$ and inductively $\\mathcal{F}^{[n + 1]} = (\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{F}^{[n]})^{**}$."}
+{"_id": "8038", "title": "divisors-lemma-Xs-codim-complement", "text": "Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}. If $s$ is nonzero, then every irreducible component of $X \\setminus U$ has codimension $1$ in $X$."}
+{"_id": "8040", "title": "divisors-lemma-relative-proj-quasi-compact", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If one of the following holds \\begin{enumerate} \\item $\\mathcal{A}$ is of finite type as a sheaf of $\\mathcal{A}_0$-algebras, \\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an $\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type $\\mathcal{A}_0$-module, \\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule $\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that $\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$, \\end{enumerate} then $p$ is quasi-compact."}
+{"_id": "8041", "title": "divisors-lemma-relative-proj-finite-type", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is of finite type as a sheaf of $\\mathcal{O}_S$-algebras, then $p$ is of finite type and $\\mathcal{O}_X(d)$ is a finite type $\\mathcal{O}_X$-module."}
+{"_id": "8042", "title": "divisors-lemma-relative-proj-universally-closed", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If $\\mathcal{O}_S \\to \\mathcal{A}_0$ is an integral algebra map\\footnote{In other words, the integral closure of $\\mathcal{O}_S$ in $\\mathcal{A}_0$, see Morphisms, Definition \\ref{morphisms-definition-integral-closure}, equals $\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra, then $p$ is universally closed."}
+{"_id": "8043", "title": "divisors-lemma-relative-proj-proper", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. The following conditions are equivalent \\begin{enumerate} \\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra, \\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module and $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_S$-algebra \\end{enumerate} If these conditions hold, then $p$ is locally projective and in particular proper."}
+{"_id": "8044", "title": "divisors-lemma-relative-proj-projective", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$ and $\\mathcal{A}_1$ is a finite type $\\mathcal{O}_S$-module, then $p$ is projective."}
+{"_id": "8045", "title": "divisors-lemma-relative-proj-flat", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If $\\mathcal{A}_d$ is a flat $\\mathcal{O}_S$-module for $d \\gg 0$, then $p$ is flat and $\\mathcal{O}_X(d)$ is flat over $S$."}
+{"_id": "8046", "title": "divisors-lemma-relative-proj-finite-presentation", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. If $\\mathcal{A}$ is a finitely presented $\\mathcal{O}_S$-algebra, then $p$ is of finite presentation and $\\mathcal{O}_X(d)$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "8047", "title": "divisors-lemma-closed-subscheme-proj", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme. Denote $\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map $$ \\mathcal{A} \\longrightarrow \\bigoplus\\nolimits_{d \\geq 0} p_*\\left((i_*\\mathcal{O}_Z)(d)\\right) $$ If $p$ is quasi-compact, then there is an isomorphism $Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{I})$."}
+{"_id": "8049", "title": "divisors-lemma-closed-subscheme-proj-finite", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme. If $p$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_S$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ such that $Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$."}
+{"_id": "8050", "title": "divisors-lemma-closed-subscheme-proj-finite-type", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme. Let $U \\subset X$ be an open. Assume that \\begin{enumerate} \\item $p$ is quasi-compact, \\item $i$ of finite presentation, \\item $U \\cap p(i(Z)) = \\emptyset$, \\item $U$ is quasi-compact, \\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_S$-module for all $n$. \\end{enumerate} Then there exists a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_S$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a) $Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$ and (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$."}
+{"_id": "8052", "title": "divisors-lemma-blowing-up-affine", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine open subscheme of $X$ and let $I \\subset A$ be the ideal corresponding to $\\mathcal{I}|_U$. If $b : X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$, then there is a canonical isomorphism $$ b^{-1}(U) = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d) $$ of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by spectra of the affine blowup algebras $A[\\frac{I}{a}]$."}
+{"_id": "8053", "title": "divisors-lemma-flat-base-change-blowing-up", "text": "\\begin{slogan} Blowing up commutes with flat base change. \\end{slogan} Let $X_1 \\to X_2$ be a flat morphism of schemes. Let $Z_2 \\subset X_2$ be a closed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian diagram $$ \\xymatrix{ X_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\ X_1 \\ar[r] & X_2 } $$ of schemes."}
+{"_id": "8054", "title": "divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. The blowing up $b : X' \\to X$ of $Z$ in $X$ has the following properties: \\begin{enumerate} \\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$ is an isomorphism, \\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$, \\item there is a canonical isomorphism $\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$ \\end{enumerate}"}
+{"_id": "8055", "title": "divisors-lemma-universal-property-blowing-up", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Let $\\mathcal{C}$ be the full subcategory of $(\\Sch/X)$ consisting of $Y \\to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$ is a final object of $\\mathcal{C}$."}
+{"_id": "8056", "title": "divisors-lemma-characterize-affine-blowup", "text": "Let $b : X' \\to X$ be the blowing up of the scheme $X$ along a closed subscheme $Z$. Let $U = \\Spec(A)$ be an affine open of $X$ and let $I \\subset A$ be the ideal corresponding to $Z \\cap U$. Let $a \\in I$ and let $x' \\in X'$ be a point mapping to a point of $U$. Then $x'$ is a point of the affine open $U' = \\Spec(A[\\frac{I}{a}])$ if and only if the image of $a$ in $\\mathcal{O}_{X', x'}$ cuts out the exceptional divisor."}
+{"_id": "8057", "title": "divisors-lemma-blow-up-effective-Cartier-divisor", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$."}
+{"_id": "8058", "title": "divisors-lemma-blow-up-reduced-scheme", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\\mathcal{I}$ is reduced."}
+{"_id": "8059", "title": "divisors-lemma-blow-up-integral-scheme", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a nonzero quasi-coherent sheaf of ideals. If $X$ is integral, then the blowup $X'$ of $X$ in $\\mathcal{I}$ is integral."}
+{"_id": "8060", "title": "divisors-lemma-blow-up-and-irreducible-components", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Let $b : X' \\to X$ be the blowing up of $X$ along $Z$. Then $b$ induces an bijective map from the set of generic points of irreducible components of $X'$ to the set of generic points of irreducible components of $X$ which are not in $Z$."}
+{"_id": "8061", "title": "divisors-lemma-blow-up-pullback-effective-Cartier", "text": "Let $X$ be a scheme. Let $b : X' \\to X$ be a blowup of $X$ in a closed subscheme. The pullback $b^{-1}D$ is defined for all effective Cartier divisors $D \\subset X$ and pullbacks of meromorphic functions are defined for $b$ (Definitions \\ref{definition-pullback-effective-Cartier-divisor} and \\ref{definition-pullback-meromorphic-sections})."}
+{"_id": "8062", "title": "divisors-lemma-blowing-up-two-ideals", "text": "Let $X$ be a scheme. Let $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $b : X' \\to X$ be the blowing up of $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the blowing up of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$ is canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$."}
+{"_id": "8063", "title": "divisors-lemma-blowing-up-projective", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$ in the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then \\begin{enumerate} \\item $b : X' \\to X$ is a projective morphism, and \\item $\\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf. \\end{enumerate}"}
+{"_id": "8064", "title": "divisors-lemma-composition-finite-type-blowups", "text": "\\begin{slogan} Composition of blowing ups is a blowing up \\end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \\subset X$ be a closed subscheme of finite presentation. Let $b : X' \\to X$ be the blowing up with center $Z$. Let $Z' \\subset X'$ be a closed subscheme of finite presentation. Let $X'' \\to X'$ be the blowing up with center $Z'$. There exists a closed subscheme $Y \\subset X$ of finite presentation, such that \\begin{enumerate} \\item $Y = Z \\cup b(Z')$ set theoretically, and \\item the composition $X'' \\to X$ is isomorphic to the blowing up of $X$ in $Y$. \\end{enumerate}"}
+{"_id": "8065", "title": "divisors-lemma-strict-transform", "text": "In the situation of Definition \\ref{definition-strict-transform}. \\begin{enumerate} \\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subscheme $f^{-1}Z$ of $X$. \\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the strict transform $\\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \\to X \\times_S S'$ of the strict transform of $\\mathcal{F}$ relative to the blowing up $X' \\to X$. \\end{enumerate}"}
+{"_id": "8066", "title": "divisors-lemma-strict-transform-flat", "text": "In the situation of Definition \\ref{definition-strict-transform}. \\begin{enumerate} \\item If $X$ is flat over $S$ at all points lying over $Z$, then the strict transform of $X$ is equal to the base change $X \\times_S S'$. \\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathcal{F}$ is flat over $S$ at all points lying over $Z$, then the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the pullback $\\text{pr}_X^*\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "8067", "title": "divisors-lemma-strict-transform-affine", "text": "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme. Let $b : S' \\to S$ be the blowing up of $Z$ in $S$. Let $g : X \\to Y$ be an affine morphism of schemes over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \\times_S S' \\to Y \\times_S S'$ be the base change of $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$ relative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform of $g_*\\mathcal{F}$."}
+{"_id": "8068", "title": "divisors-lemma-strict-transform-different-centers", "text": "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme. Let $D \\subset S$ be an effective Cartier divisor. Let $Z' \\subset S$ be the closed subscheme cut out by the product of the ideal sheaves of $Z$ and $D$. Let $S' \\to S$ be the blowup of $S$ in $Z$. \\begin{enumerate} \\item The blowup of $S$ in $Z'$ is isomorphic to $S' \\to S$. \\item Let $f : X \\to S$ be a morphism of schemes and let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathcal{F}$ has no nonzero local sections supported in $f^{-1}D$, then the strict transform of $\\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\\mathcal{F}$ relative to the blowing up of $S$ in $Z'$. \\end{enumerate}"}
+{"_id": "8069", "title": "divisors-lemma-strict-transform-composition-blowups", "text": "Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme. Let $b : S' \\to S$ be the blowing up with center $Z$. Let $Z' \\subset S'$ be a closed subscheme. Let $S'' \\to S'$ be the blowing up with center $Z'$. Let $Y \\subset S$ be a closed subscheme such that $Y = Z \\cup b(Z')$ set theoretically and the composition $S'' \\to S$ is isomorphic to the blowing up of $S$ in $Y$. In this situation, given any scheme $X$ over $S$ and $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have \\begin{enumerate} \\item the strict transform of $\\mathcal{F}$ with respect to the blowing up of $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \\to S'$ in $Z'$ of the strict transform of $\\mathcal{F}$ with respect to the blowup $S' \\to S$ of $S$ in $Z$, and \\item the strict transform of $X$ with respect to the blowing up of $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \\to S'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $S' \\to S$ of $S$ in $Z$. \\end{enumerate}"}
+{"_id": "8070", "title": "divisors-lemma-strict-transform-universally-injective", "text": "In the situation of Definition \\ref{definition-strict-transform}. Suppose that $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0 $$ is an exact sequence of quasi-coherent sheaves on $X$ which remains exact after any base change $T \\to S$. Then the strict transforms of $\\mathcal{F}_i'$ relative to any blowup $S' \\to S$ form a short exact sequence $0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too."}
+{"_id": "8071", "title": "divisors-lemma-composition-admissible-blowups", "text": "\\begin{slogan} Admissible blowups are stable under composition. \\end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open subscheme. Let $b : X' \\to X$ be a $U$-admissible blowup. Let $X'' \\to X'$ be a $U$-admissible blowup. Then the composition $X'' \\to X$ is a $U$-admissible blowup."}
+{"_id": "8072", "title": "divisors-lemma-extend-admissible-blowups", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V \\subset X$ be quasi-compact open subschemes. Let $b : V' \\to V$ be a $U \\cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \\to X$ whose restriction to $V$ is $V'$."}
+{"_id": "8073", "title": "divisors-lemma-dominate-admissible-blowups", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$ be a quasi-compact open subscheme. Let $b_i : X_i \\to X$, $i = 1, \\ldots, n$ be $U$-admissible blowups. There exists a $U$-admissible blowup $b : X' \\to X$ such that (a) $b$ factors as $X' \\to X_i \\to X$ for $i = 1, \\ldots, n$ and (b) each of the morphisms $X' \\to X_i$ is a $U$-admissible blowup."}
+{"_id": "8074", "title": "divisors-lemma-separate-disjoint-opens-by-blowing-up", "text": "\\begin{slogan} Separate irreducible components by blowing up. \\end{slogan} Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V$ be quasi-compact disjoint open subschemes of $X$. Then there exist a $U \\cup V$-admissible blowup $b : X' \\to X$ such that $X'$ is a disjoint union of open subschemes $X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and $b^{-1}(V) \\subset X'_2$."}
+{"_id": "8075", "title": "divisors-lemma-blowing-up-denominators", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\\mathcal{L}$. Let $U \\subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\\mathcal{L}$ over $U$. The blowup $b : X' \\to X$ in the ideal of denominators of $s$ is $U$-admissible. There exists an effective Cartier divisor $D \\subset X'$ and an isomorphism $$ b^*\\mathcal{L} = \\mathcal{O}_{X'}(D - E), $$ where $E \\subset X'$ is the exceptional divisor such that the meromorphic section $b^*s$ corresponds, via the isomorphism, to the meromorphic section $1_D \\otimes (1_E)^{-1}$."}
+{"_id": "8076", "title": "divisors-lemma-strict-transform-blowup-fitting-ideal", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed subscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}. Let $S' \\to S$ be the blowup of $S$ in $Z_k$ and let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$. Then $\\mathcal{F}'$ can locally be generated by $\\leq k$ sections."}
+{"_id": "8077", "title": "divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed subscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}. Assume that $\\mathcal{F}$ is locally free of rank $k$ on $S \\setminus Z_k$. Let $S' \\to S$ be the blowup of $S$ in $Z_k$ and let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$. Then $\\mathcal{F}'$ is locally free of rank $k$."}
+{"_id": "8078", "title": "divisors-lemma-blowup-fitting-ideal", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $U \\subset X$ be a scheme theoretically dense open such that $\\mathcal{F}|_U$ is finite locally free of constant rank $r$. Then \\begin{enumerate} \\item the blowup $b : X' \\to X$ of $X$ in the $r$th Fitting ideal of $\\mathcal{F}$ is $U$-admissible, \\item the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ with respect to $b$ is locally free of rank $r$, \\item the kernel $\\mathcal{K}$ of the surjection $b^*\\mathcal{F} \\to \\mathcal{F}'$ is finitely presented and $\\mathcal{K}|_U = 0$, \\item $b^*\\mathcal{F}$ and $\\mathcal{K}$ are perfect $\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$. \\end{enumerate}"}
+{"_id": "8079", "title": "divisors-lemma-filter-after-modification", "text": "Let $X$ be an integral scheme. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. There exists a modification $f : X' \\to X$ such that $f^*\\mathcal{E}$ has a filtration whose successive quotients are invertible $\\mathcal{O}_{X'}$-modules."}
+{"_id": "8080", "title": "divisors-lemma-extend-rational-map-after-modification", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Assume $X$ is Noetherian and $Y$ is proper over $S$. Given an $S$-rational map $f : U \\to Y$ from $X$ to $Y$ there exists a morphism $p : X' \\to X$ and an $S$-morphism $f' : X' \\to Y$ such that \\begin{enumerate} \\item $p$ is proper and $p^{-1}(U) \\to U$ is an isomorphism, \\item $f'|_{p^{-1}(U)}$ is equal to $f \\circ p|_{p^{-1}(U)}$. \\end{enumerate}"}
+{"_id": "8081", "title": "divisors-proposition-push-down-ample", "text": "Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes. Assume that $X$ has an ample invertible $\\mathcal{O}_X$-module. If \\begin{enumerate} \\item $\\pi$ is finite locally free, or \\item $Y$ is an integral normal scheme, or \\item $Y$ is Noetherian, $p\\mathcal{O}_Y = 0$, and $X = Y_{red}$, \\end{enumerate} then $Y$ has an ample invertible $\\mathcal{O}_Y$-module."}
+{"_id": "8124", "title": "spaces-theorem-presentation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \\to U \\times_S U$ be an \\'etale equivalence relation on $U$ over $S$. Then the quotient $U/R$ is an algebraic space, and $U \\to U/R$ is \\'etale and surjective, in other words $(U, R, U \\to U/R)$ is a presentation of $U/R$."}
+{"_id": "8125", "title": "spaces-lemma-morphism-schemes-gives-representable-transformation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$ and let $X$, $Y$ be objects of $(\\Sch/S)_{fppf}$. Let $f : X \\to Y$ be a morphism of schemes. Then $$ h_f : h_X \\longrightarrow h_Y $$ is a representable transformation of functors."}
+{"_id": "8126", "title": "spaces-lemma-composition-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$, $b : G \\to H$ be representable transformations of functors. Then $$ b \\circ a : F \\longrightarrow H $$ is a representable transformation of functors."}
+{"_id": "8127", "title": "spaces-lemma-base-change-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a representable transformation of functors. Let $b : H \\to G$ be any transformation of functors. Consider the fibre product diagram $$ \\xymatrix{ H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\ H \\ar[r]^b & G } $$ Then the base change $a'$ is a representable transformation of functors."}
+{"_id": "8128", "title": "spaces-lemma-product-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable transformations of functors. Then $$ a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2 $$ is a representable transformation of functors."}
+{"_id": "8129", "title": "spaces-lemma-representable-transformation-to-sheaf", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a representable transformation of functors. If $G$ is a sheaf, then so is $F$."}
+{"_id": "8130", "title": "spaces-lemma-representable-transformation-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a representable transformation of functors. Then $\\Delta_{F/G} : F \\to F \\times_G F$ is representable."}
+{"_id": "8131", "title": "spaces-lemma-morphism-schemes-gives-representable-transformation-property", "text": "Let $S$, $X$, $Y$ be objects of $\\Sch_{fppf}$. Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{P}$ be as in Definition \\ref{definition-relative-representable-property}. Then $h_X \\longrightarrow h_Y$ has property $\\mathcal{P}$ if and only if $f$ has property $\\mathcal{P}$."}
+{"_id": "8132", "title": "spaces-lemma-composition-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property} which is stable under composition. Let $a : F \\to G$, $b : G \\to H$ be representable transformations of functors. If $a$ and $b$ have property $\\mathcal{P}$ so does $b \\circ a : F \\longrightarrow H$."}
+{"_id": "8133", "title": "spaces-lemma-base-change-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property}. Let $a : F \\to G$ be a representable transformations of functors. Let $b : H \\to G$ be any transformation of functors. Consider the fibre product diagram $$ \\xymatrix{ H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\ H \\ar[r]^b & G } $$ If $a$ has property $\\mathcal{P}$ then also the base change $a'$ has property $\\mathcal{P}$."}
+{"_id": "8134", "title": "spaces-lemma-descent-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property}. Let $a : F \\to G$ be a representable transformations of functors. Let $b : H \\to G$ be any transformation of functors. Consider the fibre product diagram $$ \\xymatrix{ H \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\ H \\ar[r]^b & G } $$ Assume that $b$ induces a surjective map of fppf sheaves $H^\\# \\to G^\\#$. In this case, if $a'$ has property $\\mathcal{P}$, then also $a$ has property $\\mathcal{P}$."}
+{"_id": "8135", "title": "spaces-lemma-product-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$. Let $a_i : F_i \\to G_i$, $i = 1, 2$ be representable transformations of functors. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property} which is stable under composition. If $a_1$ and $a_2$ have property $\\mathcal{P}$ so does $a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$."}
+{"_id": "8136", "title": "spaces-lemma-representable-transformations-property-implication", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be a representable transformation of functors. Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in Definition \\ref{definition-relative-representable-property}. Suppose that for any morphism of schemes $f : X \\to Y$ we have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$. If $a$ has property $\\mathcal{P}$ then $a$ has property $\\mathcal{P}'$."}
+{"_id": "8137", "title": "spaces-lemma-surjective-flat-locally-finite-presentation", "text": "Let $S$ be a scheme. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves. Let $a : F \\to G$ be representable, flat, locally of finite presentation, and surjective. Then $a : F \\to G$ is surjective as a map of sheaves."}
+{"_id": "8138", "title": "spaces-lemma-representable-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$. The following are equivalent: \\begin{enumerate} \\item the diagonal $F \\to F \\times F$ is representable, \\item for $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $a \\in F(U)$ the map $a : h_U \\to F$ is representable, \\item for every pair $U, V \\in \\Ob((\\Sch/S)_{fppf})$ and any $a \\in F(U)$, $b \\in F(V)$ the fibre product $h_U \\times_{a, F, b} h_V$ is representable. \\end{enumerate}"}
+{"_id": "8139", "title": "spaces-lemma-transformation-diagonal-properties", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property}. If for every $U, V \\in \\Ob((\\Sch/S)_{fppf})$ and $a \\in F(U)$, $b \\in F(V)$ we have \\begin{enumerate} \\item $h_U \\times_{a, F, b} h_V$ is representable, say by the scheme $W$, and \\item the morphism $W \\to U \\times_S V$ corresponding to $h_U \\times_{a, F, b} h_V \\to h_U \\times h_V$ has property $\\mathcal{P}$, \\end{enumerate} then $\\Delta : F \\to F \\times F$ is representable and has property $\\mathcal{P}$."}
+{"_id": "8140", "title": "spaces-lemma-scheme-is-space", "text": "A scheme is an algebraic space. More precisely, given a scheme $T \\in \\Ob((\\Sch/S)_{fppf})$ the representable functor $h_T$ is an algebraic space."}
+{"_id": "8142", "title": "spaces-lemma-fibre-product-spaces-over-sheaf-with-representable-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $H$ be a sheaf on $(\\Sch/S)_{fppf}$ whose diagonal is representable. Let $F, G$ be algebraic spaces over $S$. Let $F \\to H$, $G \\to H$ be maps of sheaves. Then $F \\times_H G$ is an algebraic space."}
+{"_id": "8143", "title": "spaces-lemma-fibre-product-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F \\to H$, $G \\to H$ be morphisms of algebraic spaces over $S$. Then $F \\times_H G$ is an algebraic space, and is a fibre product in the category of algebraic spaces over $S$."}
+{"_id": "8144", "title": "spaces-lemma-coproduct-sheaves-open-and-closed", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ and $G$ be sheaves on $(\\Sch/S)_{fppf}^{opp}$ and denote $F \\amalg G$ the coproduct in the category of sheaves. The map $F \\to F \\amalg G$ is representable by open and closed immersions."}
+{"_id": "8145", "title": "spaces-lemma-representable-sheaf-coproduct-sheaves", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $U \\in \\Ob((\\Sch/S)_{fppf})$. Given a set $I$ and sheaves $F_i$ on $\\Ob((\\Sch/S)_{fppf})$, if $U \\cong \\coprod_{i\\in I} F_i$ as sheaves, then each $F_i$ is representable by an open and closed subscheme $U_i$ and $U \\cong \\coprod U_i$ as schemes."}
+{"_id": "8147", "title": "spaces-lemma-coproduct-algebraic-spaces", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Suppose given a set $I$ and algebraic spaces $F_i$, $i \\in I$. Then $F = \\coprod_{i \\in I} F_i$ is an algebraic space provided $I$, and the $F_i$ are not too ``large'': for example if we can choose surjective \\'etale morphisms $U_i \\to F_i$ such that $\\coprod_{i \\in I} U_i$ is isomorphic to an object of $(\\Sch/S)_{fppf}$, then $F$ is an algebraic space."}
+{"_id": "8148", "title": "spaces-lemma-glueing-algebraic-spaces", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $F$ is a sheaf, \\item there exists an index set $I$ and subfunctors $F_i \\subset F$ such that \\begin{enumerate} \\item each $F_i$ is an algebraic space, \\item each $F_i \\to F$ is representable, \\item each $F_i \\to F$ is an open immersion (see Definition \\ref{definition-relative-representable-property}), \\item the map $\\coprod F_i \\to F$ is surjective as a map of sheaves, and \\item $\\coprod F_i$ is an algebraic space (set theoretic condition, see Lemma \\ref{lemma-coproduct-algebraic-spaces}). \\end{enumerate} \\end{enumerate} Then $F$ is an algebraic space."}
+{"_id": "8149", "title": "spaces-lemma-space-presentation", "text": "Let $F$ be an algebraic space over $S$. Let $f : U \\to F$ be a surjective \\'etale morphism from a scheme to $F$. Set $R = U \\times_F U$. Then \\begin{enumerate} \\item $j : R \\to U \\times_S U$ defines an equivalence relation on $U$ over $S$ (see Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}). \\item the morphisms $s, t : R \\to U$ are \\'etale, and \\item the diagram $$ \\xymatrix{ R \\ar@<1ex>[r] \\ar@<-1ex>[r] & U \\ar[r] & F } $$ is a coequalizer diagram in $\\Sh((\\Sch/S)_{fppf})$. \\end{enumerate}"}
+{"_id": "8150", "title": "spaces-lemma-pullback-etale-equivalence-relation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \\to U \\times_S U$ be an \\'etale equivalence relation on $U$ over $S$. Let $U' \\to U$ be an \\'etale morphism. Let $R'$ be the restriction of $R$ to $U'$, see Groupoids, Definition \\ref{groupoids-definition-restrict-relation}. Then $j' : R' \\to U' \\times_S U'$ is an \\'etale equivalence relation also."}
+{"_id": "8151", "title": "spaces-lemma-finding-opens", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \\to U \\times_S U$ be a pre-relation. Let $g : U' \\to U$ be a morphism. Assume \\begin{enumerate} \\item $j$ is an equivalence relation, \\item $s, t : R \\to U$ are surjective, flat and locally of finite presentation, \\item $g$ is flat and locally of finite presentation. \\end{enumerate} Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. Then $U'/R' \\to U/R$ is representable, and is an open immersion."}
+{"_id": "8152", "title": "spaces-lemma-when-it-works-it-works", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \\to U \\times_S U$ be an \\'etale equivalence relation on $U$ over $S$. If the quotient $U/R$ is an algebraic space, then $U \\to U/R$ is \\'etale and surjective. Hence $(U, R, U \\to U/R)$ is a presentation of the algebraic space $U/R$."}
+{"_id": "8153", "title": "spaces-lemma-presentation-quasi-compact", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \\to U \\times_S U$ be an \\'etale equivalence relation on $U$ over $S$. Assume that $U$ is affine. Then the quotient $F = U/R$ is an algebraic space, and $U \\to F$ is \\'etale and surjective."}
+{"_id": "8154", "title": "spaces-lemma-etale-locally-representable-gives-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be a sheaf on $(\\Sch/S)_{fppf}$ such that there exists $U \\in \\Ob((\\Sch/S)_{fppf})$ and a map $U \\to F$ which is representable, surjective, and \\'etale. Then $F$ is an algebraic space."}
+{"_id": "8155", "title": "spaces-lemma-etale-locally-representable-by-space-gives-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $G$ be an algebraic space over $S$, let $F$ be a sheaf on $(\\Sch/S)_{fppf}$, and let $G \\to F$ be a representable transformation of functors which is surjective and \\'etale. Then $F$ is an algebraic space."}
+{"_id": "8156", "title": "spaces-lemma-representable-over-space", "text": "\\begin{slogan} A functor that admits a representable morphism to an algebraic space is an algebraic space. \\end{slogan} Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $G \\to F$ be a representable transformation of functors. Then $G$ is an algebraic space."}
+{"_id": "8157", "title": "spaces-lemma-representable-morphisms-spaces-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$, $G$ be algebraic spaces over $S$. Let $G \\to F$ be a representable morphism. Let $U \\in \\Ob((\\Sch/S)_{fppf})$, and $q : U \\to F$ surjective and \\'etale. Set $V = G \\times_F U$. Finally, let $\\mathcal{P}$ be a property of morphisms of schemes as in Definition \\ref{definition-relative-representable-property}. Then $G \\to F$ has property $\\mathcal{P}$ if and only if $V \\to U$ has property $\\mathcal{P}$."}
+{"_id": "8158", "title": "spaces-lemma-morphism-sheaves-with-P-effective-descent-etale", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $G \\to F$ be a transformation of presheaves on $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property of morphisms of schemes. Assume \\begin{enumerate} \\item $\\mathcal{P}$ is preserved under any base change, fppf local on the base, and morphisms of type $\\mathcal{P}$ satisfy descent for fppf coverings, see Descent, Definition \\ref{descent-definition-descending-types-morphisms}, \\item $G$ is a sheaf, \\item $F$ is an algebraic space, \\item there exists a $U \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective \\'etale morphism $U \\to F$ such that $V = G \\times_F U$ is representable, and \\item $V \\to U$ has $\\mathcal{P}$. \\end{enumerate} Then $G$ is an algebraic space, $G \\to F$ is representable and has property $\\mathcal{P}$."}
+{"_id": "8159", "title": "spaces-lemma-lift-morphism-presentations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Let $a : F \\to G$ be a morphism. Given any $V \\in \\Ob((\\Sch/S)_{fppf})$ and a surjective \\'etale morphism $q : V \\to G$ there exists a $U \\in \\Ob((\\Sch/S)_{fppf})$ and a commutative diagram $$ \\xymatrix{ U \\ar[d]_p \\ar[r]_\\alpha & V \\ar[d]^q \\\\ F \\ar[r]^a & G } $$ with $p$ surjective and \\'etale."}
+{"_id": "8160", "title": "spaces-lemma-composition-immersions", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. A composition of (closed, resp.\\ open) immersions of algebraic spaces over $S$ is a (closed, resp.\\ open) immersion of algebraic spaces over $S$."}
+{"_id": "8161", "title": "spaces-lemma-base-change-immersions", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. A base change of a (closed, resp.\\ open) immersion of algebraic spaces over $S$ is a (closed, resp.\\ open) immersion of algebraic spaces over $S$."}
+{"_id": "8162", "title": "spaces-lemma-sub-subspaces", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. Let $F_1$, $F_2$ be locally closed subspaces of $F$. If $F_1 \\subset F_2$ as subfunctors of $F$, then $F_1$ is a locally closed subspace of $F_2$. Similarly for closed and open subspaces."}
+{"_id": "8163", "title": "spaces-lemma-properties-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\\Delta : F \\to F \\times F$ be the diagonal morphism. Then \\begin{enumerate} \\item $\\Delta$ is locally of finite type, \\item $\\Delta$ is a monomorphism, \\item $\\Delta$ is separated, and \\item $\\Delta$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "8164", "title": "spaces-lemma-quotient", "text": "Let $U \\to S$ be a morphism of $\\Sch_{fppf}$. Let $G$ be an abstract group. Let $G \\to \\text{Aut}_S(U)$ be a group homomorphism. Assume \\begin{itemize} \\item[(*)] if $u \\in U$ is a point, and $g(u) = u$ for some non-identity element $g \\in G$, then $g$ induces a nontrivial automorphism of $\\kappa(u)$. \\end{itemize} Then $$ j : R = \\coprod\\nolimits_{g \\in G} U \\longrightarrow U \\times_S U, \\quad (g, x) \\longmapsto (g(x), x) $$ is an \\'etale equivalence relation and hence $$ F = U/R $$ is an algebraic space by Theorem \\ref{theorem-presentation}."}
+{"_id": "8165", "title": "spaces-lemma-quotient-finite-separated", "text": "Notation and assumptions as in Lemma \\ref{lemma-quotient}. Assume $G$ is finite. Then \\begin{enumerate} \\item if $U \\to S$ is quasi-separated, then $U/G$ is quasi-separated over $S$, and \\item if $U \\to S$ is separated, then $U/G$ is separated over $S$. \\end{enumerate}"}
+{"_id": "8167", "title": "spaces-lemma-change-big-site", "text": "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$. Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$, see Topologies, Section \\ref{topologies-section-change-alpha}. Let $S$ be an object of $\\Sch_{fppf}$. Let \\begin{align*} g : \\Sh((\\Sch/S)_{fppf}) \\longrightarrow \\Sh((\\Sch'/S)_{fppf}), \\\\ f : \\Sh((\\Sch'/S)_{fppf}) \\longrightarrow \\Sh((\\Sch/S)_{fppf}) \\end{align*} be the morphisms of topoi of Topologies, Lemma \\ref{topologies-lemma-change-alpha}. Let $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$. Then \\begin{enumerate} \\item if $F$ is representable by a scheme $X \\in \\Ob((\\Sch/S)_{fppf})$ over $S$, then $f^{-1}F$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$, and \\item if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic space over $S$ also. \\end{enumerate}"}
+{"_id": "8168", "title": "spaces-lemma-fully-faithful", "text": "Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$. Let $S$ be an object of $\\Sch_{fppf}$. Denote $\\textit{Spaces}/S$ the category of algebraic spaces over $S$ defined using $\\Sch_{fppf}$. Similarly, denote $\\textit{Spaces}'/S$ the category of algebraic spaces over $S$ defined using $\\Sch'_{fppf}$. The construction of Lemma \\ref{lemma-change-big-site} defines a fully faithful functor $$ \\textit{Spaces}/S \\longrightarrow \\textit{Spaces}'/S $$ whose essential image consists of those $X' \\in \\Ob(\\textit{Spaces}'/S)$ such that there exist $U, R \\in \\Ob((\\Sch/S)_{fppf})$\\footnote{Requiring the existence of $R$ is necessary because of our choice of the function $Bound$ in Sets, Equation (\\ref{sets-equation-bound}). The size of the fibre product $U \\times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We can illustrate this by setting $S = \\Spec(A)$, $U = \\Spec(A[x_i, i \\in I])$ and $R = \\coprod_{(\\lambda_i) \\in A^I} \\Spec(A[x_i, y_i]/(x_i - \\lambda_i y_i))$. In this case the size of $R$ grows like $\\kappa^\\kappa$ where $\\kappa$ is the size of $U$.} and morphisms $$ U \\longrightarrow X' \\quad\\text{and}\\quad R \\longrightarrow U \\times_{X'} U $$ in $\\Sh((\\Sch'/S)_{fppf})$ which are surjective as maps of sheaves (for example if the displayed morphisms are surjective and \\'etale)."}
+{"_id": "8169", "title": "spaces-lemma-change-base-scheme", "text": "Suppose given a big site $\\Sch_{fppf}$. Let $g : S \\to S'$ be morphism of $\\Sch_{fppf}$. Let $j : (\\Sch/S)_{fppf} \\to (\\Sch/S')_{fppf}$ be the corresponding localization functor. Let $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$. Then \\begin{enumerate} \\item for a scheme $T'$ over $S'$ we have $j_!F(T'/S') = \\coprod\\nolimits_{\\varphi : T' \\to S} F(T' \\xrightarrow{\\varphi} S),$ \\item if $F$ is representable by a scheme $X \\in \\Ob((\\Sch/S)_{fppf})$, then $j_!F$ is representable by $j(X)$ which is $X$ viewed as a scheme over $S'$, and \\item if $F$ is an algebraic space over $S$, then $j_!F$ is an algebraic space over $S'$, and if $F = U/R$ is a presentation, then $j_!F = j(U)/j(R)$ is a presentation. \\end{enumerate} Let $F'$ be a sheaf of sets on $(\\Sch/S')_{fppf}$. Then \\begin{enumerate} \\item[(4)] for a scheme $T$ over $S$ we have $j^{-1}F'(T/S) = F'(T/S')$, \\item[(5)] if $F'$ is representable by a scheme $X' \\in \\Ob((\\Sch/S')_{fppf})$, then $j^{-1}F'$ is representable, namely by $X'_S = S \\times_{S'} X'$, and \\item[(6)] if $F'$ is an algebraic space, then $j^{-1}F'$ is an algebraic space, and if $F' = U'/R'$ is a presentation, then $j^{-1}F' = U'_S/R'_S$ is a presentation. \\end{enumerate}"}
+{"_id": "8170", "title": "spaces-lemma-category-of-spaces-over-smaller-base-scheme", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. The construction above give an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{category of algebraic}\\\\ \\text{spaces over }S \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{category of pairs }(F', F' \\to S)\\text{ consisting}\\\\ \\text{of an algebraic space }F'\\text{ over }S'\\text{ and a}\\\\ \\text{morphism }F' \\to S\\text{ of algebraic spaces over }S' \\end{matrix} \\right\\} $$"}
+{"_id": "8171", "title": "spaces-lemma-rephrase", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. Let $F'$ be a sheaf on $(\\Sch/S')_{fppf}$. The following are equivalent: \\begin{enumerate} \\item The restriction $F'|_{(\\Sch/S)_{fppf}}$ is an algebraic space over $S$, and \\item the sheaf $h_S \\times F'$ is an algebraic space over $S'$. \\end{enumerate}"}
+{"_id": "8188", "title": "topology-theorem-tychonov", "text": "A product of quasi-compact spaces is quasi-compact."}
+{"_id": "8189", "title": "topology-theorem-characterize-proper", "text": "\\begin{reference} In \\cite[I, p. 75, Theorem 1]{Bourbaki} you can find: (2) $\\Leftrightarrow$ (4). In \\cite[I, p. 77, Proposition 6]{Bourbaki} you can find: (2) $\\Rightarrow$ (1). \\end{reference} Let $f: X\\to Y$ be a continuous map between topological spaces. The following conditions are equivalent: \\begin{enumerate} \\item The map $f$ is quasi-proper and closed. \\item The map $f$ is proper. \\item The map $f$ is universally closed. \\item The map $f$ is closed and $f^{-1}(y)$ is quasi-compact for any $y\\in Y$. \\end{enumerate}"}
+{"_id": "8190", "title": "topology-lemma-Hausdorff", "text": "Let $X$ be a topological space. The following are equivalent: \\begin{enumerate} \\item $X$ is Hausdorff, \\item the diagonal $\\Delta(X) \\subset X \\times X$ is closed. \\end{enumerate}"}
+{"_id": "8191", "title": "topology-lemma-graph-closed", "text": "\\begin{slogan} Graphs of maps to Hausdorff spaces are closed. \\end{slogan} Let $f : X \\to Y$ be a continuous map of topological spaces. If $Y$ is Hausdorff, then the graph of $f$ is closed in $X \\times Y$."}
+{"_id": "8193", "title": "topology-lemma-fibre-product-closed", "text": "Let $X \\to Z$ and $Y \\to Z$ be continuous maps of topological spaces. If $Z$ is Hausdorff, then  $X \\times_Z Y$ is closed in $X \\times Y$."}
+{"_id": "8194", "title": "topology-lemma-separated", "text": "Let $f : X \\to Y$ be continuous map of topological spaces. The following are equivalent: \\begin{enumerate} \\item $f$ is separated, \\item $\\Delta(X) \\subset X \\times_Y X$ is a closed subset, \\item given distinct points $x, x' \\in X$ mapping to the same point of $Y$, there exist disjoint open neighbourhoods of $x$ and $x'$. \\end{enumerate}"}
+{"_id": "8196", "title": "topology-lemma-base-change-separated", "text": "Let $f : X \\to Y$ and $Z \\to Y$ be continuous maps of topological spaces. If $f$ is separated, then $f' : Z \\times_Y X \\to Z$ is separated."}
+{"_id": "8197", "title": "topology-lemma-make-base", "text": "Let $X$ be a set and let $\\mathcal{B}$ be a collection of subsets. Assume that $X = \\bigcup_{B \\in \\mathcal{B}} B$ and that given $x \\in B_1 \\cap B_2$ with $B_1, B_2 \\in \\mathcal{B}$ there is a $B_3 \\in \\mathcal{B}$ with $x \\in B_3 \\subset B_1 \\cap B_2$. Then there is a unique topology on $X$ such that $\\mathcal{B}$ is a basis for this topology."}
+{"_id": "8198", "title": "topology-lemma-refine-covering-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{U} : U = \\bigcup_i U_i$ be an open covering of $U \\subset X$. There exists an open covering $U = \\bigcup V_j$ which is a refinement of $\\mathcal{U}$ such that each $V_j$ is an element of the basis $\\mathcal{B}$."}
+{"_id": "8199", "title": "topology-lemma-subbase", "text": "Let $X$ be a set. Given any collection $\\mathcal{B}$ of subsets of $X$ there is a unique topology on $X$ such that $\\mathcal{B}$ is a subbase for this topology."}
+{"_id": "8200", "title": "topology-lemma-create-map-from-subcollection", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a collection of opens of $X$. Assume $X = \\bigcup_{U \\in \\mathcal{B}} U$ and for $U, V \\in \\mathcal{B}$ we have $U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$. Then there is a continuous map $f : X \\to Y$ of topological spaces such that \\begin{enumerate} \\item for $U \\in \\mathcal{B}$ the image $f(U)$ is open, \\item for $U \\in \\mathcal{B}$ we have $f^{-1}(f(U)) = U$, and \\item the opens $f(U)$, $U \\in \\mathcal{B}$ form a basis for the topology on $Y$. \\end{enumerate}"}
+{"_id": "8201", "title": "topology-lemma-induced", "text": "Let $X$ be a topological space. Let $Y$ be a set and let $f : Y \\to X$ be an injective map of sets. The induced topology on $Y$ is the topology characterized by each of the following statements: \\begin{enumerate} \\item it is the weakest topology on $Y$ such that $f$ is continuous, \\item the open subsets of $Y$ are $f^{-1}(U)$ for $U \\subset X$ open, \\item the closed subsets of $Y$ are the sets $f^{-1}(Z)$ for $Z \\subset X$ closed. \\end{enumerate}"}
+{"_id": "8202", "title": "topology-lemma-quotient", "text": "Let $X$ be a topological space. Let $Y$ be a set and let $f : X \\to Y$ be a surjective map of sets. The quotient topology on $Y$ is the topology characterized by each of the following statements: \\begin{enumerate} \\item it is the strongest topology on $Y$ such that $f$ is continuous, \\item a subset $V$ of $Y$ is open if and only if $f^{-1}(V)$ is open, \\item a subset $Z$ of $Y$ is closed if and only if $f^{-1}(Z)$ is closed. \\end{enumerate}"}
+{"_id": "8203", "title": "topology-lemma-open-morphism-quotient-topology", "text": "Let $f : X \\to Y$ be surjective, open, continuous map of topological spaces. Let $T \\subset Y$ be a subset. Then \\begin{enumerate} \\item $f^{-1}(\\overline{T}) = \\overline{f^{-1}(T)}$, \\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed, \\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and \\item $T \\subset Y$ is locally closed if and only if $f^{-1}(T)$ is locally closed. \\end{enumerate} In particular we see that $f$ is submersive."}
+{"_id": "8204", "title": "topology-lemma-closed-morphism-quotient-topology", "text": "Let $f : X \\to Y$ be surjective, closed, continuous map of topological spaces. Let $T \\subset Y$ be a subset. Then \\begin{enumerate} \\item $\\overline{T} = f(\\overline{f^{-1}(T)})$, \\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed, \\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and \\item $T \\subset Y$ is locally closed if and only if $f^{-1}(T)$ is locally closed. \\end{enumerate} In particular we see that $f$ is submersive."}
+{"_id": "8205", "title": "topology-lemma-image-connected-space", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. If $E \\subset X$ is a connected subset, then $f(E) \\subset Y$ is connected as well."}
+{"_id": "8206", "title": "topology-lemma-connected-components", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item If $T \\subset X$ is connected, then so is its closure. \\item Any connected component of $X$ is closed (but not necessarily open). \\item Every connected subset of $X$ is contained in a unique connected component of $X$. \\item Every point of $X$ is contained in a unique connected component, in other words, $X$ is the union of its connected components. \\end{enumerate}"}
+{"_id": "8207", "title": "topology-lemma-connected-fibres-quotient-topology-connected-components", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that \\begin{enumerate} \\item all fibres of $f$ are connected, and \\item a set $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed. \\end{enumerate} Then $f$ induces a bijection between the sets of connected components of $X$ and $Y$."}
+{"_id": "8208", "title": "topology-lemma-connected-fibres-connected-components", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that (a) $f$ is open, (b) all fibres of $f$ are connected. Then $f$ induces a bijection between the sets of connected components of $X$ and $Y$."}
+{"_id": "8209", "title": "topology-lemma-finite-fibre-connected-components", "text": "Let $f : X \\to Y$ be a continuous map of nonempty topological spaces. Assume that (a) $Y$ is connected, (b) $f$ is open and closed, and (c) there is a point $y\\in Y$ such that the fiber $f^{-1}(y)$ is a finite set. Then $X$ has at most $|f^{-1}(y)|$ connected components. Hence any connected  component $T$ of $X$ is open and closed, and $f(T)$ is a nonempty open and  closed subset of $Y$, which is therefore equal to $Y$."}
+{"_id": "8210", "title": "topology-lemma-space-connected-components", "text": "Let $X$ be a topological space. Let $\\pi_0(X)$ be the set of connected components of $X$. Let $X \\to \\pi_0(X)$ be the map which sends $x \\in X$ to the connected component of $X$ passing through $x$. Endow $\\pi_0(X)$ with the quotient topology. Then $\\pi_0(X)$ is a totally disconnected space and any continuous map $X \\to Y$ from $X$ to a totally disconnected space $Y$ factors through $\\pi_0(X)$."}
+{"_id": "8211", "title": "topology-lemma-locally-connected", "text": "Let $X$ be a topological space. If $X$ is locally connected, then \\begin{enumerate} \\item any open subset of $X$ is locally connected, and \\item the connected components of $X$ are open. \\end{enumerate} So also the connected components of open subsets of $X$ are open. In particular, every point has a fundamental system of open connected neighbourhoods."}
+{"_id": "8212", "title": "topology-lemma-image-irreducible-space", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. If $E \\subset X$ is an irreducible subset, then $f(E) \\subset Y$ is irreducible as well."}
+{"_id": "8213", "title": "topology-lemma-irreducible", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item If $T \\subset X$ is irreducible so is its closure in $X$. \\item Any irreducible component of $X$ is closed. \\item Any irreducible subset of $X$ is contained in an irreducible component of $X$. \\item Every point of $X$ is contained in some irreducible component of $X$, in other words, $X$ is the union of its irreducible components. \\end{enumerate}"}
+{"_id": "8214", "title": "topology-lemma-pick-irreducible-components", "text": "Let $X$ be a topological space and suppose $X = \\bigcup_{i = 1, \\ldots, n} X_i$ where each $X_i$ is an irreducible closed subset of $X$ and no $X_i$ is contained in the union of the other members.  Then each $X_i$ is an irreducible component of $X$ and each irreducible component of $X$ is one of the $X_i$."}
+{"_id": "8215", "title": "topology-lemma-sober-subspace", "text": "Let $X$ be a topological space and let $Y\\subset X$. \\begin{enumerate} \\item If $X$ is Kolmogorov then so is $Y$. \\item Suppose $Y$ is locally closed in $X$. If $X$ is quasi-sober then so is $Y$. \\item Suppose $Y$ is locally closed in $X$. If $X$ is sober then so is $Y$. \\end{enumerate}"}
+{"_id": "8216", "title": "topology-lemma-sober-local", "text": "Let $X$ be a topological space and let $(X_i)_{i\\in I}$ be a covering of $X$. \\begin{enumerate} \\item Suppose $X_i$ is locally closed in $X$ for every $i\\in I$. Then, $X$ is Kolmogorov if and only if $X_i$ is Kolmogorov for every $i\\in I$. \\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is quasi-sober if and only if $X_i$ is quasi-sober for every $i\\in I$. \\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is sober if and  only if $X_i$ is sober for every $i\\in I$. \\end{enumerate}"}
+{"_id": "8217", "title": "topology-lemma-irreducible-on-top", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that (a) $Y$ is irreducible, (b) $f$ is open, and (c) there exists a dense collection of points $y \\in Y$ such that $f^{-1}(y)$ is irreducible. Then $X$ is irreducible."}
+{"_id": "8218", "title": "topology-lemma-irreducible-fibres-irreducible-components", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that (a) $f$ is open, and (b) for every $y \\in Y$ the fibre $f^{-1}(y)$ is irreducible. Then $f$ induces a bijection between irreducible components."}
+{"_id": "8219", "title": "topology-lemma-make-sober", "text": "Let $X$ be a topological space. There is a canonical continuous map $$ c : X \\longrightarrow X' $$ from $X$ to a sober topological space $X'$ which is universal among continuous maps from $X$ to sober topological spaces. Moreover, the assignment $U' \\mapsto c^{-1}(U')$ is a bijection between opens of $X'$ and $X$ which commutes with finite intersections and arbitrary unions. The image $c(X)$ is a Kolmogorov topological space and the map $c : X \\to c(X)$ is universal for maps of $X$ into Kolmogorov spaces."}
+{"_id": "8220", "title": "topology-lemma-Noetherian", "text": "Let $X$ be a Noetherian topological space. \\begin{enumerate} \\item Any subset of $X$ with the induced topology is Noetherian. \\item The space $X$ has finitely many irreducible components. \\item Each irreducible component of $X$ contains a nonempty open of $X$. \\end{enumerate}"}
+{"_id": "8221", "title": "topology-lemma-image-Noetherian", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item If $X$ is Noetherian, then $f(X)$ is Noetherian. \\item If $X$ is locally Noetherian and $f$ open, then $f(X)$ is locally Noetherian. \\end{enumerate}"}
+{"_id": "8222", "title": "topology-lemma-finite-union-Noetherian", "text": "Let $X$ be a topological space. Let $X_i \\subset X$, $i = 1, \\ldots, n$ be a finite collection of subsets. If each $X_i$ is Noetherian (with the induced topology), then $\\bigcup_{i = 1, \\ldots, n}  X_i$ is Noetherian (with the induced topology)."}
+{"_id": "8223", "title": "topology-lemma-locally-Noetherian-locally-connected", "text": "Let $X$ be a locally Noetherian topological space. Then $X$ is locally connected."}
+{"_id": "8224", "title": "topology-lemma-dimension-supremum-local-dimensions", "text": "Let $X$ be a topological space. Then $\\dim(X) = \\sup \\dim_x(X)$ where the supremum runs over the points $x$ of $X$."}
+{"_id": "8225", "title": "topology-lemma-codimension-at-generic-point", "text": "Let $X$ be a topological space. Let $Y \\subset X$ be an irreducible closed subset. Let $U \\subset X$ be an open subset such that $Y \\cap U$ is nonempty. Then $$ \\text{codim}(Y, X) = \\text{codim}(Y \\cap U, U) $$"}
+{"_id": "8226", "title": "topology-lemma-catenary", "text": "Let $X$ be a topological space. The following are equivalent: \\begin{enumerate} \\item $X$ is catenary, \\item $X$ has an open covering by catenary spaces. \\end{enumerate} Moreover, in this case any locally closed subspace of $X$ is catenary."}
+{"_id": "8227", "title": "topology-lemma-catenary-in-codimension", "text": "Let $X$ be a topological space. The following are equivalent: \\begin{enumerate} \\item $X$ is catenary, and \\item for every pair of irreducible closed subsets $Y \\subset Y'$ we have $\\text{codim}(Y, Y') < \\infty$ and for every triple $Y \\subset Y' \\subset Y''$ of irreducible closed subsets we have $$ \\text{codim}(Y, Y'') = \\text{codim}(Y, Y') + \\text{codim}(Y', Y''). $$ \\end{enumerate}"}
+{"_id": "8228", "title": "topology-lemma-composition-quasi-compact", "text": "A composition of quasi-compact maps is quasi-compact."}
+{"_id": "8229", "title": "topology-lemma-closed-in-quasi-compact", "text": "A closed subset of a quasi-compact topological space is quasi-compact."}
+{"_id": "8230", "title": "topology-lemma-quasi-compact-in-Hausdorff", "text": "Let $X$ be a Hausdorff topological space. \\begin{enumerate} \\item If $E \\subset X$ is quasi-compact, then it is closed. \\item If $E_1, E_2 \\subset X$ are disjoint quasi-compact subsets then there exists opens $E_i \\subset U_i$ with $U_1 \\cap U_2 = \\emptyset$. \\end{enumerate}"}
+{"_id": "8231", "title": "topology-lemma-closed-in-compact", "text": "Let $X$ be a quasi-compact Hausdorff space. Let $E \\subset X$. The following are equivalent: (a) $E$ is closed in $X$, (b) $E$ is quasi-compact."}
+{"_id": "8232", "title": "topology-lemma-intersection-closed-in-quasi-compact", "text": "Let $X$ be a quasi-compact topological space. If $\\{Z_\\alpha\\}_{\\alpha \\in A}$ is a collection of closed subsets such that the intersection of each finite subcollection is nonempty, then $\\bigcap_{\\alpha \\in A} Z_\\alpha$ is nonempty."}
+{"_id": "8233", "title": "topology-lemma-image-quasi-compact", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item If $X$ is quasi-compact, then $f(X)$ is quasi-compact. \\item If $f$ is quasi-compact, then $f(X)$ is retrocompact. \\end{enumerate}"}
+{"_id": "8234", "title": "topology-lemma-quasi-compact-closed-point", "text": "Let $X$ be a topological space. Assume that \\begin{enumerate} \\item $X$ is nonempty, \\item $X$ is quasi-compact, and \\item $X$ is Kolmogorov. \\end{enumerate} Then $X$ has a closed point."}
+{"_id": "8235", "title": "topology-lemma-closed-points-quasi-compact", "text": "Let $X$ be a quasi-compact Kolmogorov space. Then the set $X_0$ of closed points of $X$ is quasi-compact."}
+{"_id": "8236", "title": "topology-lemma-connected-component-intersection", "text": "Let $X$ be a topological space. Assume \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ has a basis for the topology consisting of quasi-compact opens, and \\item the intersection of two quasi-compact opens is quasi-compact. \\end{enumerate} For any $x \\in X$ the connected component of $X$ containing $x$ is the intersection of all open and closed subsets of $X$ containing $x$."}
+{"_id": "8237", "title": "topology-lemma-connected-component-intersection-compact-Hausdorff", "text": "Let $X$ be a topological space. Assume $X$ is quasi-compact and Hausdorff. For any $x \\in X$ the connected component of $X$ containing $x$ is the intersection of all open and closed subsets of $X$ containing $x$."}
+{"_id": "8238", "title": "topology-lemma-closed-union-connected-components", "text": "Let $X$ be a topological space. Assume \\begin{enumerate} \\item $X$ is quasi-compact, \\item $X$ has a basis for the topology consisting of quasi-compact opens, and \\item the intersection of two quasi-compact opens is quasi-compact. \\end{enumerate} For a subset $T \\subset X$ the following are equivalent: \\begin{enumerate} \\item[(a)] $T$ is an intersection of open and closed subsets of $X$, and \\item[(b)] $T$ is closed in $X$ and is a union of connected components of $X$. \\end{enumerate}"}
+{"_id": "8239", "title": "topology-lemma-Noetherian-quasi-compact", "text": "Let $X$ be a Noetherian topological space. \\begin{enumerate} \\item The space $X$ is quasi-compact. \\item Any subset of $X$ is retrocompact. \\end{enumerate}"}
+{"_id": "8240", "title": "topology-lemma-quasi-compact-locally-Noetherian-Noetherian", "text": "A quasi-compact locally Noetherian space is Noetherian."}
+{"_id": "8241", "title": "topology-lemma-subbase-theorem", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a subbase for $X$. If every covering of $X$ by elements of $\\mathcal{B}$ has a finite refinement, then $X$ is quasi-compact."}
+{"_id": "8242", "title": "topology-lemma-locally-quasi-compact-Hausdorff", "text": "A Hausdorff space is locally quasi-compact if and only if every point has a quasi-compact neighbourhood."}
+{"_id": "8244", "title": "topology-lemma-relatively-compact-refinement", "text": "Let $X$ be a Hausdorff and quasi-compact space. Let $X = \\bigcup_{i \\in I} U_i$ be an open covering. Then there exists an open covering $X = \\bigcup_{i \\in I} V_i$ such that $\\overline{V_i} \\subset U_i$ for all $i$."}
+{"_id": "8245", "title": "topology-lemma-refine-covering", "text": "Let $X$ be a Hausdorff and quasi-compact space. Let $X = \\bigcup_{i \\in I} U_i$ be an open covering. Suppose given an integer $p \\geq 0$ and for every $(p + 1)$-tuple $i_0, \\ldots, i_p$ of $I$ an open covering $U_{i_0} \\cap \\ldots \\cap U_{i_p} = \\bigcup W_{i_0 \\ldots i_p, k}$. Then there exists an open covering $X = \\bigcup_{j \\in J} V_j$ and a map $\\alpha : J \\to I$ such that $\\overline{V_j} \\subset U_{\\alpha(j)}$ and such that each $V_{j_0} \\cap \\ldots \\cap V_{j_p}$ is contained in $W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$ for some $k$."}
+{"_id": "8247", "title": "topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset", "text": "Let $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. Suppose given an integer $p \\geq 0$, a set $I$, for every $i \\in I$ an open $U_i \\subset X$, and for every $(p + 1)$-tuple $i_0, \\ldots, i_p$ of $I$ an open $W_{i_0 \\ldots i_p} \\subset U_{i_0} \\cap \\ldots \\cap U_{i_p}$ such that \\begin{enumerate} \\item $Z \\subset \\bigcup U_i$, and \\item for every $i_0, \\ldots, i_p$ we have $W_{i_0 \\ldots i_p} \\cap Z = U_{i_0} \\cap \\ldots \\cap U_{i_p} \\cap Z$. \\end{enumerate} Then there exist opens $V_i$ of $X$ such that \\begin{enumerate} \\item $Z \\subset \\bigcup V_i$, \\item $V_i \\subset U_i$ for all $i$, \\item $\\overline{V_i} \\cap Z \\subset U_i$ for all $i$, and \\item $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$ for all $(p + 1)$-tuples $i_0, \\ldots, i_p$. \\end{enumerate}"}
+{"_id": "8248", "title": "topology-lemma-limits", "text": "The category of topological spaces has limits and the forgetful functor to sets commutes with them."}
+{"_id": "8249", "title": "topology-lemma-describe-limits", "text": "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram of topological spaces over $\\mathcal{I}$. Let $X = \\lim X_i$ be the limit with projection maps $f_i : X \\to X_i$. \\begin{enumerate} \\item Any open of $X$ is of the form $\\bigcup_{j \\in J} f_j^{-1}(U_j)$ for some subset $J \\subset I$ and opens $U_j \\subset X_j$. \\item Any quasi-compact open of $X$ is of the form $f_i^{-1}(U_i)$ for some $i$ and some $U_i \\subset X_i$ open. \\end{enumerate}"}
+{"_id": "8250", "title": "topology-lemma-characterize-limit", "text": "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram of topological spaces over $\\mathcal{I}$. Let $X$ be a topological space such that \\begin{enumerate} \\item $X = \\lim X_i$ as a set (denote $f_i$ the projection maps), \\item the sets $f_i^{-1}(U_i)$ for $i \\in \\Ob(\\mathcal{I})$ and $U_i \\subset X_i$ open form a basis for the topology of $X$. \\end{enumerate} Then $X$ is the limit of the $X_i$ as a topological space."}
+{"_id": "8251", "title": "topology-lemma-inverse-limit-quasi-compact", "text": "Let $\\mathcal{I}$ be a category and let $i \\mapsto X_i$ be a diagram over $\\mathcal{I}$ in the category of topological spaces. If each $X_i$ is quasi-compact and Hausdorff, then $\\lim X_i$ is quasi-compact."}
+{"_id": "8252", "title": "topology-lemma-nonempty-limit", "text": "Let $\\mathcal{I}$ be a cofiltered category and let $i \\mapsto X_i$ be a diagram over $\\mathcal{I}$ in the category of topological spaces. If each $X_i$ is quasi-compact, Hausdorff, and nonempty, then $\\lim X_i$ is nonempty."}
+{"_id": "8253", "title": "topology-lemma-constructible", "text": "The collection of constructible sets is closed under finite intersections, finite unions and complements."}
+{"_id": "8254", "title": "topology-lemma-inverse-images-constructibles", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. If the inverse image of every retrocompact open subset of $Y$ is retrocompact in $X$, then inverse images of constructible sets are constructible."}
+{"_id": "8255", "title": "topology-lemma-open-immersion-constructible-inverse-image", "text": "Let $U \\subset X$ be open. For a constructible set $E \\subset X$ the intersection $E \\cap U$ is constructible in $U$."}
+{"_id": "8256", "title": "topology-lemma-quasi-compact-open-immersion-constructible-image", "text": "Let $U \\subset X$ be a retrocompact open. Let $E \\subset U$. If $E$ is constructible in $U$, then $E$ is constructible in $X$."}
+{"_id": "8257", "title": "topology-lemma-collate-constructible", "text": "Let $X$ be a topological space. Let $E \\subset X$ be a subset. Let $X = V_1 \\cup \\ldots \\cup V_m$ be a finite covering by retrocompact opens. Then $E$ is constructible in $X$ if and only if $E \\cap V_j$ is constructible in $V_j$ for each $j = 1, \\ldots, m$."}
+{"_id": "8258", "title": "topology-lemma-intersect-constructible-with-closed", "text": "Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset such that $X \\setminus Z$ is quasi-compact. Then for a constructible set $E \\subset X$ the intersection $E \\cap Z$ is constructible in $Z$."}
+{"_id": "8259", "title": "topology-lemma-intersect-constructible-with-retrocompact", "text": "Let $X$ be a topological space. Let $T \\subset X$ be a subset. Suppose \\begin{enumerate} \\item $T$ is retrocompact in $X$, \\item quasi-compact opens form a basis for the topology on $X$. \\end{enumerate} Then for a constructible set $E \\subset X$ the intersection $E \\cap T$ is constructible in $T$."}
+{"_id": "8260", "title": "topology-lemma-closed-constructible-image", "text": "Let $Z \\subset X$ be a closed subset whose complement is retrocompact open. Let $E \\subset Z$. If $E$ is constructible in $Z$, then $E$ is constructible in $X$."}
+{"_id": "8261", "title": "topology-lemma-constructible-is-retrocompact", "text": "Let $X$ be a topological space. Every constructible subset of $X$ is retrocompact."}
+{"_id": "8262", "title": "topology-lemma-intersect-constructible-with-constructible", "text": "Let $X$ be a topological space. Assume $X$ has a basis consisting of quasi-compact opens. For $E, E'$ constructible in $X$, the intersection $E \\cap E'$ is constructible in $E$."}
+{"_id": "8263", "title": "topology-lemma-constructible-in-constructible", "text": "Let $X$ be a topological space. Assume $X$ has a basis consisting of quasi-compact opens. Let $E$ be constructible in $X$ and $F \\subset E$ constructible in $E$. Then $F$ is constructible in $X$."}
+{"_id": "8264", "title": "topology-lemma-locally-closed-constructible-image", "text": "Let $X$ be a quasi-compact topological space having a basis consisting of quasi-compact opens such that the intersection of any two quasi-compact opens is quasi-compact. Let $T \\subset X$ be a locally closed subset such that $T$ is quasi-compact and $T^c$ is retrocompact in $X$. Then $T$ is constructible in $X$."}
+{"_id": "8265", "title": "topology-lemma-collate-constructible-from-constructible", "text": "Let $X$ be a topological space which has a basis for the topology consisting of quasi-compact opens. Let $E \\subset X$ be a subset. Let $X = E_1 \\cup \\ldots \\cup E_m$ be a finite covering by constructible subsets. Then $E$ is constructible in $X$ if and only if $E \\cap E_j$ is constructible in $E_j$ for each $j = 1, \\ldots, m$."}
+{"_id": "8266", "title": "topology-lemma-generic-point-in-constructible", "text": "Let $X$ be a topological space. Suppose that $Z \\subset X$ is irreducible. Let $E \\subset X$ be a finite union of locally closed subsets (e.g.\\ $E$ is constructible). The following are equivalent \\begin{enumerate} \\item The intersection $E \\cap Z$ contains an open dense subset of $Z$. \\item The intersection $E \\cap Z$ is dense in $Z$. \\end{enumerate} If $Z$ has a generic point $\\xi$, then this is also equivalent to \\begin{enumerate} \\item[(3)] We have $\\xi \\in E$. \\end{enumerate}"}
+{"_id": "8267", "title": "topology-lemma-constructible-Noetherian-space", "text": "Let $X$ be a Noetherian topological space. The constructible sets in $X$ are precisely the finite unions of locally closed subsets of $X$."}
+{"_id": "8268", "title": "topology-lemma-constructible-map-Noetherian", "text": "Let $f : X \\to Y$ be a continuous map of Noetherian topological spaces. If $E \\subset Y$ is constructible in $Y$, then $f^{-1}(E)$ is constructible in $X$."}
+{"_id": "8269", "title": "topology-lemma-characterize-constructible-Noetherian", "text": "Let $X$ be a Noetherian topological space. Let $E \\subset X$ be a subset. The following are equivalent: \\begin{enumerate} \\item $E$ is constructible in $X$, and \\item for every irreducible closed $Z \\subset X$ the intersection $E \\cap Z$ either contains a nonempty open of $Z$ or is not dense in $Z$. \\end{enumerate}"}
+{"_id": "8270", "title": "topology-lemma-constructible-neighbourhood-Noetherian", "text": "Let $X$ be a Noetherian topological space. Let $x \\in X$. Let $E \\subset X$ be constructible in $X$. The following are equivalent: \\begin{enumerate} \\item $E$ is a neighbourhood of $x$, and \\item for every irreducible closed subset $Y$ of $X$ which contains $x$ the intersection $E \\cap Y$ is dense in $Y$. \\end{enumerate}"}
+{"_id": "8271", "title": "topology-lemma-characterize-open-Noetherian", "text": "Let $X$ be a Noetherian topological space. Let $E \\subset X$ be a subset. The following are equivalent: \\begin{enumerate} \\item $E$ is open in $X$, and \\item for every irreducible closed subset $Y$ of $X$ the intersection $E \\cap Y$ is either empty or contains a nonempty open of $Y$. \\end{enumerate}"}
+{"_id": "8272", "title": "topology-lemma-tube", "text": "Let $X$ and $Y$ be topological spaces. Let $A \\subset X$ and $B \\subset Y$ be quasi-compact subsets. Let $A \\times B \\subset W \\subset X \\times Y$ with $W$ open in $X \\times Y$. Then there exists opens $A \\subset U \\subset X$ and $B \\subset V \\subset Y$ such that $U \\times V \\subset W$."}
+{"_id": "8273", "title": "topology-lemma-characterize-quasi-compact", "text": "\\begin{reference} Combination of \\cite[I, p. 75, Lemme 1]{Bourbaki} and \\cite[I, p. 76, Corrolaire 1]{Bourbaki}. \\end{reference} A topological space $X$ is quasi-compact if and only if the projection map $Z \\times X \\to Z$ is closed for any topological space $Z$."}
+{"_id": "8274", "title": "topology-lemma-closed-map", "text": "\\begin{slogan} A map from a compact space to a Hausdorff space is a proper. \\end{slogan} Let $f : X \\to Y$ be a continuous map of topological spaces. If $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is proper."}
+{"_id": "8275", "title": "topology-lemma-bijective-map", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. If $f$ is bijective, $X$ is quasi-compact, and $Y$ is Hausdorff, then $f$ is a homeomorphism."}
+{"_id": "8276", "title": "topology-lemma-jacobson-check-irreducible-closed", "text": "Let $X$ be a topological space. Let $X_0$ be the set of closed points of $X$. Suppose that for every point $x\\in X$ the intersection $X_0 \\cap \\overline{\\{x\\}}$ is dense in $\\overline{\\{x\\}}$. Then $X$ is Jacobson."}
+{"_id": "8277", "title": "topology-lemma-non-jacobson-Noetherian-characterize", "text": "Let $X$ be a Kolmogorov topological space with a basis of quasi-compact open sets. If $X$ is not Jacobson, then there exists a non-closed point $x \\in X$ such that $\\{x\\}$ is locally closed."}
+{"_id": "8278", "title": "topology-lemma-jacobson-local", "text": "Let $X$ be a topological space. Let $X = \\bigcup U_i$ be an open covering. Then $X$ is Jacobson if and only if each $U_i$ is Jacobson. Moreover, in this case $X_0 = \\bigcup U_{i, 0}$."}
+{"_id": "8279", "title": "topology-lemma-jacobson-inherited", "text": "Let $X$ be Jacobson. The following types of subsets $T \\subset X$ are Jacobson: \\begin{enumerate} \\item Open subspaces. \\item Closed subspaces. \\item Locally closed subspaces. \\item Unions of locally closed subspaces. \\item Constructible sets. \\item Any subset $T \\subset X$ which locally on $X$ is a union of locally closed subsets. \\end{enumerate} In each of these cases closed points of $T$ are closed in $X$."}
+{"_id": "8280", "title": "topology-lemma-finite-jacobson", "text": "A finite Jacobson space is discrete."}
+{"_id": "8281", "title": "topology-lemma-jacobson-equivalent-locally-closed", "text": "\\begin{slogan} For Jacobson spaces, closed points see everything about the topology. \\end{slogan} Suppose $X$ is a Jacobson topological space. Let $X_0$ be the set of closed points of $X$. There is a bijective, inclusion preserving correspondence $$ \\{\\text{finite unions loc.\\ closed subsets of } X\\} \\leftrightarrow \\{\\text{finite unions loc.\\ closed subsets of } X_0\\} $$ given by $E \\mapsto E \\cap X_0$. This correspondence preserves the subsets of locally closed, of open and of closed subsets."}
+{"_id": "8282", "title": "topology-lemma-jacobson-equivalent-constructible", "text": "Suppose $X$ is a Jacobson topological space. Let $X_0$ be the set of closed points of $X$. There is a bijective, inclusion preserving correspondence $$ \\{\\text{constructible subsets of } X\\} \\leftrightarrow \\{\\text{constructible subsets of } X_0\\} $$ given by $E \\mapsto E \\cap X_0$. This correspondence preserves the subset of retrocompact open subsets, as well as complements of these."}
+{"_id": "8283", "title": "topology-lemma-open-closed-specialization", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item Any closed subset of $X$ is stable under specialization. \\item Any open subset of $X$ is stable under generalization. \\item A subset $T \\subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization. \\end{enumerate}"}
+{"_id": "8284", "title": "topology-lemma-stable-specialization", "text": "Let $T \\subset X$ be a subset of a topological space $X$. The following are equivalent \\begin{enumerate} \\item $T$ is stable under specialization, and \\item $T$ is a (directed) union of closed subsets of $X$. \\end{enumerate}"}
+{"_id": "8285", "title": "topology-lemma-lift-specialization-composition", "text": "Suppose $f : X \\to Y$ and $g : Y \\to Z$ are continuous maps of topological spaces. If specializations lift along both $f$ and $g$ then specializations lift along $g \\circ f$. Similarly for ``generalizations lift along''."}
+{"_id": "8286", "title": "topology-lemma-lift-specializations-images", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item If specializations lift along $f$, and if $T \\subset X$ is stable under specialization, then $f(T) \\subset Y$ is stable under specialization. \\item If generalizations lift along $f$, and if $T \\subset X$ is stable under generalization, then $f(T) \\subset Y$ is stable under generalization. \\end{enumerate}"}
+{"_id": "8287", "title": "topology-lemma-closed-open-map-specialization", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item If $f$ is closed then specializations lift along $f$. \\item If $f$ is open, $X$ is a Noetherian topological space, each irreducible closed subset of $X$ has a generic point, and $Y$ is Kolmogorov then generalizations lift along $f$. \\end{enumerate}"}
+{"_id": "8288", "title": "topology-lemma-quotient-kolmogorov", "text": "Suppose that $s, t : R \\to U$ and $\\pi : U \\to X$ are continuous maps of topological spaces such that \\begin{enumerate} \\item $\\pi$ is open, \\item $U$ is sober, \\item $s, t$ have finite fibres, \\item generalizations lift along $s, t$, \\item $(t, s)(R) \\subset U \\times U$ is an equivalence relation on $U$ and $X$ is the quotient of $U$ by this equivalence relation (as a set). \\end{enumerate} Then $X$ is Kolmogorov."}
+{"_id": "8289", "title": "topology-lemma-dimension-specializations-lift", "text": "Let $f : X \\to Y$ be a morphism of topological spaces. Suppose that $Y$ is a sober topological space, and $f$ is surjective. If either specializations or generalizations lift along $f$, then $\\dim(X) \\geq \\dim(Y)$."}
+{"_id": "8290", "title": "topology-lemma-characterize-closed-Noetherian", "text": "Let $X$ be a Noetherian sober topological space. Let $E \\subset X$ be a subset of $X$. \\begin{enumerate} \\item If $E$ is constructible and stable under specialization, then $E$ is closed. \\item If $E$ is constructible and stable under generalization, then $E$ is open. \\end{enumerate}"}
+{"_id": "8291", "title": "topology-lemma-dimension-function-catenary", "text": "Let $X$ be a topological space. If $X$ is sober and has a dimension function, then $X$ is catenary. Moreover, for any $x \\leadsto y$ we have $$ \\delta(x) - \\delta(y) = \\text{codim}\\left(\\overline{\\{y\\}}, \\ \\overline{\\{x\\}}\\right). $$"}
+{"_id": "8292", "title": "topology-lemma-dimension-function-unique", "text": "Let $X$ be a topological space. Let $\\delta$, $\\delta'$ be two dimension functions on $X$. If $X$ is locally Noetherian and sober then $\\delta - \\delta'$ is locally constant on $X$."}
+{"_id": "8293", "title": "topology-lemma-locally-dimension-function", "text": "Let $X$ be locally Noetherian, sober and catenary. Then any point has an open neighbourhood $U \\subset X$ which has a dimension function."}
+{"_id": "8294", "title": "topology-lemma-nowhere-dense", "text": "Let $X$ be a topological space. The union of a finite number of nowhere dense sets is a nowhere dense set."}
+{"_id": "8295", "title": "topology-lemma-image-nowhere-dense-open", "text": "Let $X$ be a topological space. Let $U \\subset X$ be an open. Let $T \\subset U$ be a subset. If $T$ is nowhere dense in $U$, then $T$ is nowhere dense in $X$."}
+{"_id": "8296", "title": "topology-lemma-nowhere-dense-local", "text": "Let $X$ be a topological space. Let $X = \\bigcup U_i$ be an open covering. Let $T \\subset X$ be a subset. If $T \\cap U_i$ is nowhere dense in $U_i$ for all $i$, then $T$ is nowhere dense in $X$."}
+{"_id": "8297", "title": "topology-lemma-closed-image-nowhere-dense", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $T \\subset X$ be a subset. If $f$ is a homeomorphism of $X$ onto a closed subset of $Y$ and $T$ is nowhere dense in $X$, then also $f(T)$ is nowhere dense in $Y$."}
+{"_id": "8298", "title": "topology-lemma-open-inverse-image-closed-nowhere-dense", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $T \\subset Y$ be a subset. If $f$ is open and $T$ is a closed nowhere dense subset of $Y$, then also $f^{-1}(T)$ is a closed nowhere dense subset of $X$. If $f$ is surjective and open, then $T$ is closed nowhere dense if and only if $f^{-1}(T)$ is closed nowhere dense."}
+{"_id": "8299", "title": "topology-lemma-profinite", "text": "Let $X$ be a topological space. The following are equivalent \\begin{enumerate} \\item $X$ is a profinite space, and \\item $X$ is Hausdorff, quasi-compact, and totally disconnected. \\end{enumerate} If this is true, then $X$ is a cofiltered limit of finite discrete spaces."}
+{"_id": "8300", "title": "topology-lemma-directed-inverse-limit-profinite", "text": "A limit of profinite spaces is profinite."}
+{"_id": "8301", "title": "topology-lemma-profinite-refine-open-covering", "text": "Let $X$ be a profinite space. Every open covering of $X$ has a refinement by a finite covering $X = \\coprod U_i$ with $U_i$ open and closed."}
+{"_id": "8302", "title": "topology-lemma-pi0-profinite", "text": "Let $X$ be a topological space. If $X$ is quasi-compact and every connected component of $X$ is the intersection of the open and closed subsets containing it, then $\\pi_0(X)$ is a profinite space."}
+{"_id": "8303", "title": "topology-lemma-constructible-hausdorff-quasi-compact", "text": "Let $X$ be a spectral space. The constructible topology is Hausdorff, totally disconnected, and quasi-compact."}
+{"_id": "8304", "title": "topology-lemma-fibres-spectral-map-quasi-compact", "text": "Let $f : X \\to Y$ be a spectral map of spectral spaces. Then \\begin{enumerate} \\item $f$ is continuous in the constructible topology, \\item the fibres of $f$ are quasi-compact, and \\item the image is closed in the constructible topology. \\end{enumerate}"}
+{"_id": "8306", "title": "topology-lemma-spectral-sub", "text": "Let $X$ be a spectral space. Let $E \\subset X$ be closed in the constructible topology (for example constructible or closed). Then $E$ with the induced topology is a spectral space."}
+{"_id": "8307", "title": "topology-lemma-constructible-stable-specialization-closed", "text": "Let $X$ be a spectral space. Let $E \\subset X$ be a subset closed in the constructible topology (for example constructible). \\begin{enumerate} \\item If $x \\in \\overline{E}$, then $x$ is the specialization of a point of $E$. \\item If $E$ is stable under specialization, then $E$ is closed. \\item If $E' \\subset X$ is open in the constructible topology (for example constructible) and stable under generalization, then $E'$ is open. \\end{enumerate}"}
+{"_id": "8309", "title": "topology-lemma-characterize-profinite-spectral", "text": "Let $X$ be a spectral space. The following are equivalent: \\begin{enumerate} \\item $X$ is profinite, \\item $X$ is Hausdorff, \\item $X$ is totally disconnected, \\item every quasi-compact open is closed, \\item there are no nontrivial specializations between points, \\item every point of $X$ is closed, \\item every point of $X$ is the generic point of an irreducible component of $X$, \\item the constructible topology equals the given topology on $X$, and \\item add more here. \\end{enumerate}"}
+{"_id": "8310", "title": "topology-lemma-spectral-pi0", "text": "If $X$ is a spectral space, then $\\pi_0(X)$ is a profinite space."}
+{"_id": "8311", "title": "topology-lemma-product-spectral-spaces", "text": "The product of two spectral spaces is spectral."}
+{"_id": "8312", "title": "topology-lemma-spectral-bijective", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. If \\begin{enumerate} \\item $X$ and $Y$ are spectral, \\item $f$ is spectral and bijective, and \\item generalizations (resp.\\ specializations) lift along $f$. \\end{enumerate} Then $f$ is a homeomorphism."}
+{"_id": "8313", "title": "topology-lemma-directed-inverse-limit-finite-sober-spectral-spaces", "text": "The inverse limit of a directed inverse system of finite sober topological spaces is a spectral topological space."}
+{"_id": "8314", "title": "topology-lemma-spectral-closed-in-product-two-point-space", "text": "Let $W$ be the topological space with two points, one closed, the other not. A topological space is spectral if and only if it is homeomorphic to a subspace of a product of copies of $W$ which is closed in the constructible topology."}
+{"_id": "8315", "title": "topology-lemma-spectral-inverse-limit-finite-sober-spaces", "text": "A topological space is spectral if and only if it is a directed inverse limit of finite sober topological spaces."}
+{"_id": "8316", "title": "topology-lemma-Noetherian-goes-to-spectral", "text": "Let $X$ be a topological space and let $c : X \\to X'$ be the universal map from $X$ to a sober topological space, see Lemma \\ref{lemma-make-sober}. \\begin{enumerate} \\item If $X$ is quasi-compact, so is $X'$. \\item If $X$ is quasi-compact, has a basis of quasi-compact opens, and the intersection of two quasi-compact opens is quasi-compact, then  $X'$ is spectral. \\item If $X$ is Noetherian, then $X'$ is a Noetherian spectral space. \\end{enumerate}"}
+{"_id": "8317", "title": "topology-lemma-inverse-limit-spectral-spaces-quasi-compact", "text": "Let $\\mathcal{I}$ be a category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. \\begin{enumerate} \\item Given subsets $Z_i \\subset X_i$ closed in the constructible topology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$, then $\\lim Z_i$ is quasi-compact. \\item The space $X = \\lim X_i$ is quasi-compact. \\end{enumerate}"}
+{"_id": "8318", "title": "topology-lemma-inverse-limit-spectral-spaces-nonempty", "text": "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. \\begin{enumerate} \\item Given nonempty subsets $Z_i \\subset X_i$ closed in the constructible topology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$, then $\\lim Z_i$ is nonempty. \\item If each $X_i$ is nonempty, then $X = \\lim X_i$ is nonempty. \\end{enumerate}"}
+{"_id": "8319", "title": "topology-lemma-inverse-limit-spectral-spaces-equal", "text": "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$ with projections $p_i : X \\to X_i$. Let $i \\in \\Ob(\\mathcal{I})$ and let $E, F \\subset X_i$ be subsets with $E$ closed in the constructible topology and $F$ open in the constructible topology. Then $p_i^{-1}(E) \\subset p_i^{-1}(F)$ if and only if there is a morphism $a : j \\to i$ in $\\mathcal{I}$ such that $f_a^{-1}(E) \\subset f_a^{-1}(F)$."}
+{"_id": "8320", "title": "topology-lemma-inverse-limit-spectral-spaces-constructible", "text": "Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$ with projections $p_i : X \\to X_i$. Let $E \\subset X$ be a constructible subset. Then there exists an $i \\in \\Ob(\\mathcal{I})$ and a constructible subset $E_i \\subset X_i$ such that $p_i^{-1}(E_i) = E$. If $E$ is open, resp.\\ closed, we may choose $E_i$ open, resp.\\ closed."}
+{"_id": "8321", "title": "topology-lemma-directed-inverse-limit-spectral-spaces", "text": "Let $\\mathcal{I}$ be a cofiltered index category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. Then the inverse limit $X = \\lim X_i$ is a spectral topological space and the projection maps $p_i : X \\to X_i$ are spectral."}
+{"_id": "8322", "title": "topology-lemma-descend-opens", "text": "Let $\\mathcal{I}$ be a cofiltered index category. Let $i \\mapsto X_i$ be a diagram of spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map $f_a : X_j \\to X_i$ is spectral. Set $X = \\lim X_i$ and denote $p_i : X \\to X_i$ the projection. \\begin{enumerate} \\item Given any quasi-compact open $U \\subset X$ there exists an $i \\in \\Ob(\\mathcal{I})$ and a quasi-compact open $U_i \\subset X_i$ such that $p_i^{-1}(U_i) = U$. \\item Given $U_i \\subset X_i$ and $U_j \\subset X_j$ quasi-compact opens such that $p_i^{-1}(U_i) \\subset p_j^{-1}(U_j)$ there exist $k \\in \\Ob(\\mathcal{I})$ and morphisms $a : k \\to i$ and $b : k \\to j$ such that $f_a^{-1}(U_i) \\subset f_b^{-1}(U_j)$. \\item If $U_i, U_{1, i}, \\ldots, U_{n, i} \\subset X_i$ are quasi-compact opens and $p_i^{-1}(U_i) = p_i^{-1}(U_{1, i}) \\cup \\ldots \\cup p_i^{-1}(U_{n, i})$ then $f_a^{-1}(U_i) = f_a^{-1}(U_{1, i}) \\cup \\ldots \\cup f_a^{-1}(U_{n, i})$ for some morphism $a : j \\to i$ in $\\mathcal{I}$. \\item Same statement as in (3) but for intersections. \\end{enumerate}"}
+{"_id": "8323", "title": "topology-lemma-make-spectral-space", "text": "Let $W$ be a subset of a spectral space $X$. The following are equivalent: \\begin{enumerate} \\item $W$ is an intersection of constructible sets and closed under generalizations, \\item $W$ is quasi-compact and closed under generalizations, \\item there exists a quasi-compact subset $E \\subset X$ such that $W$ is the set of points specializing to $E$, \\item $W$ is an intersection of quasi-compact open subsets, \\item \\label{item-intersection-quasi-compact-open} there exists a nonempty set $I$ and quasi-compact opens $U_i \\subset X$, $i \\in I$ such that $W = \\bigcap U_i$ and for all $i, j \\in I$ there exists a $k \\in I$ with $U_k \\subset U_i \\cap U_j$. \\end{enumerate} In this case we have (a) $W$ is a spectral space, (b) $W = \\lim U_i$ as topological spaces, and (c) for any open $U$ containing $W$ there exists an $i$ with $U_i \\subset U$."}
+{"_id": "8324", "title": "topology-lemma-make-spectral-space-minus", "text": "Let $X$ be a spectral space. Let $E \\subset X$ be a constructible subset. Let $W \\subset X$ be the set of points of $X$ which specialize to a point of $E$. Then $W \\setminus E$ is a spectral space. If $W = \\bigcap U_i$ with $U_i$ as in Lemma \\ref{lemma-make-spectral-space} (\\ref{item-intersection-quasi-compact-open}) then $W \\setminus E = \\lim (U_i \\setminus E)$."}
+{"_id": "8325", "title": "topology-lemma-dense-image", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that $f(X)$ is dense in $Y$ and that $Y$ is Hausdorff. Then the cardinality of $Y$ is at most the cardinality of $P(P(X))$ where $P$ is the power set operation."}
+{"_id": "8327", "title": "topology-lemma-image-open-technical", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Assume $f$ is surjective and $f(E) \\not = Y$ for all proper closed subsets $E \\subset X$. Then for $U \\subset X$ open the subset $f(U)$ is contained in the closure of $Y \\setminus f(X \\setminus U)$."}
+{"_id": "8328", "title": "topology-lemma-intersection-empty", "text": "Let $X$ be an extremally disconnected space. If $U, V \\subset X$ are disjoint open subsets, then $\\overline{U}$ and $\\overline{V}$ are disjoint too."}
+{"_id": "8329", "title": "topology-lemma-isomorphism", "text": "Let $f : X \\to Y$ be a continuous map of Hausdorff quasi-compact topological spaces. If $Y$ is extremally disconnected, $f$ is surjective, and $f(Z) \\not = Y$ for every proper closed subset $Z$ of $X$, then $f$ is a homeomorphism."}
+{"_id": "8330", "title": "topology-lemma-find-compact-subset", "text": "Let $f : X \\to Y$ be a continuous surjective map of Hausdorff quasi-compact topological spaces. There exists a quasi-compact subset $E \\subset X$ such that $f(E) = Y$ but $f(E') \\not = Y$ for all proper closed subsets $E' \\subset E$."}
+{"_id": "8331", "title": "topology-lemma-rainwater", "text": "Let $f : X \\to X$ be a surjective continuous selfmap of a Hausdorff topological space. If $f$ is not $\\text{id}_X$, then there exists a proper closed subset $E \\subset X$ such that $X = E \\cup f(E)$."}
+{"_id": "8332", "title": "topology-lemma-existence-projective-cover", "text": "\\begin{slogan} Every quasi-compact Hausdorff space has a canonical extremally disconnected cover \\end{slogan} Let $X$ be a quasi-compact Hausdorff space. There exists a continuous surjection $X' \\to X$ with $X'$ quasi-compact, Hausdorff, and extremally disconnected. If we require that every proper closed subset of $X'$ does not map onto $X$, then $X'$ is unique up to isomorphism."}
+{"_id": "8333", "title": "topology-lemma-topology-quasi-separated-scheme", "text": "Let $X$ be a topological space which \\begin{enumerate} \\item has a basis of the topology consisting of quasi-compact opens, and \\item has the property that the intersection of any two quasi-compact opens is quasi-compact. \\end{enumerate} Then \\begin{enumerate} \\item $X$ is locally quasi-compact, \\item a quasi-compact open $U \\subset X$ is retrocompact, \\item any quasi-compact open $U \\subset X$ has a cofinal system of open coverings $\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with $J$ finite and all $U_j$ and $U_j \\cap U_{j'}$ quasi-compact, \\item add more here. \\end{enumerate}"}
+{"_id": "8334", "title": "topology-lemma-partition-refined-by-stratification", "text": "Let $X$ be a topological space. Let $X = \\coprod X_i$ be a finite partition of $X$. Then there exists a finite stratification of $X$ refining it."}
+{"_id": "8335", "title": "topology-lemma-constructible-partition-refined-by-stratification", "text": "Let $X$ be a topological space. Suppose $X = T_1 \\cup \\ldots \\cup T_n$ is written as a union of constructible subsets. There exists a finite stratification $X = \\coprod X_i$ with each $X_i$ constructible such that each $T_k$ is a union of strata."}
+{"_id": "8337", "title": "topology-lemma-colimits", "text": "The category of topological spaces has colimits and the forgetful functor to sets commutes with them."}
+{"_id": "8338", "title": "topology-lemma-topological-group-limits", "text": "The category of topological groups has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of groups."}
+{"_id": "8339", "title": "topology-lemma-profinite-group", "text": "Let $G$ be a topological group. The following are equivalent \\begin{enumerate} \\item $G$ as a topological space is profinite, \\item $G$ is a limit of a diagram of finite discrete topological groups, \\item $G$ is a cofiltered limit of finite discrete topological groups. \\end{enumerate}"}
+{"_id": "8340", "title": "topology-lemma-topological-group-colimits", "text": "The category of topological groups has colimits and colimits commute with the forgetful functor to the category of groups."}
+{"_id": "8345", "title": "topology-proposition-projective-in-category-hausdorff-qc", "text": "Let $X$ be a Hausdorff, quasi-compact topological space. The following are equivalent \\begin{enumerate} \\item $X$ is extremally disconnected, \\item for any surjective continuous map $f : Y \\to X$ with $Y$ Hausdorff quasi-compact there exists a continuous section, and \\item for any solid commutative diagram $$ \\xymatrix{ & Y \\ar[d] \\\\ X \\ar@{..>}[ru] \\ar[r] & Z } $$ of continuous maps of quasi-compact Hausdorff spaces with $Y \\to Z$ surjective, there is a dotted arrow in the category of topological spaces making the diagram commute. \\end{enumerate}"}
+{"_id": "8387", "title": "hypercovering-theorem-cohomology-hypercoverings", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $i \\geq 0$. The functors \\begin{eqnarray*} \\textit{Ab}(\\mathcal{C}) & \\longrightarrow & \\textit{Ab} \\\\ \\mathcal{F} & \\longmapsto & H^i(X, \\mathcal{F}) \\\\ \\mathcal{F} & \\longmapsto & \\check{H}^i_{\\text{HC}}(X, \\mathcal{F}) \\end{eqnarray*} are canonically isomorphic."}
+{"_id": "8388", "title": "hypercovering-lemma-coprod-prod-SR", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item the category $\\text{SR}(\\mathcal{C})$ has coproducts and $F$ commutes with them, \\item the functor $F : \\text{SR}(\\mathcal{C}) \\to \\textit{PSh}(\\mathcal{C})$ commutes with limits, \\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C})$ has fibre products, \\item if $\\mathcal{C}$ has products of pairs, then $\\text{SR}(\\mathcal{C})$ has products of pairs, \\item if $\\mathcal{C}$ has equalizers, so does $\\text{SR}(\\mathcal{C})$, and \\item if $\\mathcal{C}$ has a final object, so does $\\text{SR}(\\mathcal{C})$. \\end{enumerate} Let $X \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item the category $\\text{SR}(\\mathcal{C}, X)$ has coproducts and $F$ commutes with them, \\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C}, X)$ has finite limits and $F : \\text{SR}(\\mathcal{C}, X) \\to \\textit{PSh}(\\mathcal{C})/h_X$ commutes with them. \\end{enumerate}"}
+{"_id": "8389", "title": "hypercovering-lemma-covering-permanence", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item A composition of coverings in $\\text{SR}(\\mathcal{C})$ is a covering. \\item If $K \\to L$ is a covering in $\\text{SR}(\\mathcal{C})$ and $L' \\to L$ is a morphism, then $L' \\times_L K$ exists and $L' \\times_L K \\to L'$ is a covering. \\item If $\\mathcal{C}$ has products of pairs, and $A \\to B$ and $K \\to L$ are coverings in $\\text{SR}(\\mathcal{C})$, then $A \\times K \\to B \\times L$ is a covering. \\end{enumerate} Let $X \\in \\Ob(\\mathcal{C})$. Then (1) and (2) holds for $\\text{SR}(\\mathcal{C}, X)$ and (3) holds if $\\mathcal{C}$ has fibre products."}
+{"_id": "8390", "title": "hypercovering-lemma-hypercoverings-set", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$. The collection of all hypercoverings of $X$ forms a set."}
+{"_id": "8391", "title": "hypercovering-lemma-hypercovering-F", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Consider the simplicial object $F(K)$ of $\\textit{PSh}(\\mathcal{C})$, endowed with its augmentation to the constant simplicial presheaf $h_X$. \\begin{enumerate} \\item The morphism of presheaves $F(K)_0 \\to h_X$ becomes a surjection after sheafification. \\item The morphism $$ (d^1_0, d^1_1) : F(K)_1 \\longrightarrow F(K)_0 \\times_{h_X} F(K)_0 $$ becomes a surjection after sheafification. \\item For every $n \\geq 1$ the morphism $$ F(K)_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n F(K))_{n + 1} $$ turns into a surjection after sheafification. \\end{enumerate}"}
+{"_id": "8392", "title": "hypercovering-lemma-compare-cosk0", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} \\to \\mathcal{G}$ be a morphism of presheaves of sets. Denote $K$ the simplicial object of $\\textit{PSh}(\\mathcal{C})$ whose $n$th term is the $(n + 1)$st fibre product of $\\mathcal{F}$ over $\\mathcal{G}$, see Simplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}. Then, if $\\mathcal{F} \\to \\mathcal{G}$ is surjective after sheafification, we have $$ H_i(K) = \\left\\{ \\begin{matrix} 0 & \\text{if} & i > 0\\\\ \\mathbf{Z}_\\mathcal{G}^\\# & \\text{if} & i = 0 \\end{matrix} \\right. $$ The isomorphism in degree $0$ is given by the morphism $H_0(K) \\to \\mathbf{Z}_\\mathcal{G}^\\#$ coming from the map $(\\mathbf{Z}_K^\\#)_0 = \\mathbf{Z}_\\mathcal{F}^\\# \\to \\mathbf{Z}_\\mathcal{G}^\\#$."}
+{"_id": "8393", "title": "hypercovering-lemma-acyclicity", "text": "Let $\\mathcal{C}$ be a site. Let $f : L \\to K$ be a morphism of simplicial objects of $\\textit{PSh}(\\mathcal{C})$. Let $n \\geq 0$ be an integer. Assume that \\begin{enumerate} \\item For $i < n$ the morphism $L_i \\to K_i$ is an isomorphism. \\item The morphism $L_n \\to K_n$ is surjective after sheafification. \\item The canonical map $L \\to \\text{cosk}_n \\text{sk}_n L$ is an isomorphism. \\item The canonical map $K \\to \\text{cosk}_n \\text{sk}_n K$ is an isomorphism. \\end{enumerate} Then $H_i(f) : H_i(L) \\to H_i(K)$ is an isomorphism."}
+{"_id": "8394", "title": "hypercovering-lemma-acyclic-hypercover-sheaves", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial presheaf. Let $\\mathcal{G}$ be a presheaf. Let $K \\to \\mathcal{G}$ be an augmentation of $K$ towards $\\mathcal{G}$. Assume that \\begin{enumerate} \\item The morphism of presheaves $K_0 \\to \\mathcal{G}$ becomes a surjection after sheafification. \\item The morphism $$ (d^1_0, d^1_1) : K_1 \\longrightarrow K_0 \\times_\\mathcal{G} K_0 $$ becomes a surjection after sheafification. \\item For every $n \\geq 1$ the morphism $$ K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1} $$ turns into a surjection after sheafification. \\end{enumerate} Then $H_i(K) = 0$ for $i > 0$ and $H_0(K) = \\mathbf{Z}_\\mathcal{G}^\\#$."}
+{"_id": "8395", "title": "hypercovering-lemma-hypercovering-acyclic", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. The homology of the simplicial presheaf $F(K)$ is $0$ in degrees $> 0$ and equal to $\\mathbf{Z}_X^\\#$ in degree $0$."}
+{"_id": "8396", "title": "hypercovering-lemma-h0-cech", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$. Then $\\check{H}^0(K, \\mathcal{F}) = \\mathcal{F}(X)$."}
+{"_id": "8397", "title": "hypercovering-lemma-injective-trivial-cech", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$. Then $$ \\check{H}^p(K, \\mathcal{I}) = \\left\\{ \\begin{matrix} \\mathcal{I}(X) & \\text{if} & p = 0 \\\\ 0 & \\text{if} & p > 0 \\end{matrix} \\right. $$"}
+{"_id": "8398", "title": "hypercovering-lemma-cech-spectral-sequence", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$. There is a map $$ s(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(X, \\mathcal{F}) $$ in $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces natural transformations $$ \\check{H}^i(K, -) \\longrightarrow H^i(X, -) $$ as functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover, there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F})) $$ converging to $H^{p + q}(X, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$ and in the hypercovering $K$."}
+{"_id": "8399", "title": "hypercovering-lemma-h0-cech-variant", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$. Let $K$ be a hypercovering of $\\mathcal{G}$. Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$. Then $\\check{H}^0(K, \\mathcal{F}) = H^0(\\mathcal{G}, \\mathcal{F})$."}
+{"_id": "8400", "title": "hypercovering-lemma-injective-trivial-cech-variant", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$. Let $K$ be a hypercovering of $\\mathcal{G}$. Let $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$. Then $$ \\check{H}^p(K, \\mathcal{I}) = \\left\\{ \\begin{matrix} H^0(\\mathcal{G}, \\mathcal{I}) & \\text{if} & p = 0 \\\\ 0 & \\text{if} & p > 0 \\end{matrix} \\right. $$"}
+{"_id": "8401", "title": "hypercovering-lemma-cech-spectral-sequence-variant", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$. Let $K$ be a hypercovering of $\\mathcal{G}$. Let $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$. There is a map $$ s(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(\\mathcal{G}, \\mathcal{F}) $$ in $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces a natural transformation $$ \\check{H}^i(K, -) \\longrightarrow H^i(\\mathcal{G}, -) $$ of functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover, there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F})) $$ converging to $H^{p + q}(\\mathcal{G}, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$ and in the hypercovering $K$."}
+{"_id": "8403", "title": "hypercovering-lemma-funny-gamma", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K, L, M$ be simplicial objects of $\\text{SR}(\\mathcal{C}, X)$. Let $a : K \\to L$, $b : M \\to L$ be morphisms. Assume \\begin{enumerate} \\item $K$ is a hypercovering of $X$, \\item the morphism $M_0 \\to L_0$ is a covering, and \\item for all $n \\geq 0$ in the diagram $$ \\xymatrix{ M_{n + 1} \\ar[dd] \\ar[rr] \\ar[rd]^\\gamma & & (\\text{cosk}_n \\text{sk}_n M)_{n + 1} \\ar[dd] \\\\ & L_{n + 1} \\times_{(\\text{cosk}_n \\text{sk}_n L)_{n + 1}} (\\text{cosk}_n \\text{sk}_n M)_{n + 1} \\ar[ld] \\ar[ru] & \\\\ L_{n + 1} \\ar[rr] & & (\\text{cosk}_n \\text{sk}_n L)_{n + 1} } $$ the arrow $\\gamma$ is a covering. \\end{enumerate} Then the fibre product $K \\times_L M$ is a hypercovering of $X$."}
+{"_id": "8405", "title": "hypercovering-lemma-covering", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $k \\geq 0$ be an integer. Let $u : Z \\to K_k$ be a covering in $\\text{SR}(\\mathcal{C}, X)$. Then there exists a morphism of hypercoverings $f: L \\to K$ such that $L_k \\to K_k$ factors through $u$."}
+{"_id": "8407", "title": "hypercovering-lemma-one-more-simplex", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \\subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ has finitely many nondegenerate simplices. Suppose $n \\geq 0$ and $x \\in V_n$, $x \\not \\in U_n$ are such that \\begin{enumerate} \\item $V_i = U_i$ for $i < n$, \\item $V_n = U_n \\cup \\{x\\}$, \\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$ is degenerate. \\end{enumerate} Then the morphism $$ \\Hom(V, K)_0 \\longrightarrow \\Hom(U, K)_0 $$ of $\\text{SR}(\\mathcal{C}, X)$ is a covering."}
+{"_id": "8408", "title": "hypercovering-lemma-add-simplices", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \\subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then the morphism $$ \\Hom(V, K)_0 \\longrightarrow \\Hom(U, K)_0 $$ of $\\text{SR}(\\mathcal{C}, X)$ is a covering."}
+{"_id": "8409", "title": "hypercovering-lemma-degeneracy-maps-coverings", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then \\begin{enumerate} \\item $K_n$ is a covering of $X$ for each $n \\geq 0$, \\item $d^n_i : K_n \\to K_{n - 1}$ is a covering for all $n \\geq 1$ and $0 \\leq i \\leq n$. \\end{enumerate}"}
+{"_id": "8410", "title": "hypercovering-lemma-hom-hypercovering", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $L$ be a simplicial object of $\\text{SR}(\\mathcal{C}, X)$. Let $n \\geq 0$. Consider the commutative diagram \\begin{equation} \\label{equation-diagram} \\xymatrix{ \\Hom(\\Delta[1], L)_{n + 1} \\ar[r] \\ar[d] & (\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1} \\ar[d] \\\\ (L \\times L)_{n + 1} \\ar[r] & (\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1} } \\end{equation} coming from the morphism defined above. We can identify the terms in this diagram as follows, where $\\partial \\Delta[n + 1] = i_{n!}\\text{sk}_n \\Delta[n + 1]$ is the $n$-skeleton of the $(n + 1)$-simplex: \\begin{eqnarray*} \\Hom(\\Delta[1], L)_{n + 1} & = & \\Hom(\\Delta[1] \\times \\Delta[n + 1], L)_0 \\\\ (\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1} & = & \\Hom(\\Delta[1] \\times \\partial \\Delta[n + 1], L)_0 \\\\ (L \\times L)_{n + 1} & = & \\Hom( (\\Delta[n + 1] \\amalg \\Delta[n + 1], L)_0 \\\\ (\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1} & = & \\Hom( \\partial \\Delta[n + 1] \\amalg \\partial \\Delta[n + 1], L)_0 \\end{eqnarray*} and the morphism between these objects of $\\text{SR}(\\mathcal{C}, X)$ come from the commutative diagram of simplicial sets \\begin{equation} \\label{equation-dual-diagram} \\xymatrix{ \\Delta[1] \\times \\Delta[n + 1] & \\Delta[1] \\times \\partial\\Delta[n + 1] \\ar[l] \\\\ \\Delta[n + 1] \\amalg \\Delta[n + 1] \\ar[u] & \\partial\\Delta[n + 1] \\amalg \\partial\\Delta[n + 1] \\ar[l] \\ar[u] } \\end{equation} Moreover the fibre product of the bottom arrow and the right arrow in (\\ref{equation-diagram}) is equal to $$ \\Hom(U, L)_0 $$ where $U \\subset \\Delta[1] \\times \\Delta[n + 1]$ is the smallest simplicial subset such that both $\\Delta[n + 1] \\amalg \\Delta[n + 1]$ and $\\Delta[1] \\times \\partial\\Delta[n + 1]$ map into it."}
+{"_id": "8412", "title": "hypercovering-lemma-basis-hypercovering", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology of $X$. There exists a hypercovering $(I, \\{U_i\\})$ of $X$ such that each $U_i$ is an element of $\\mathcal{B}$."}
+{"_id": "8413", "title": "hypercovering-lemma-quasi-separated-quasi-compact-hypercovering", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology of $X$. Assume that \\begin{enumerate} \\item $X$ is quasi-compact, \\item each $U \\in \\mathcal{B}$ is quasi-compact open, and \\item the intersection of any two quasi-compact opens in $X$ is quasi-compact. \\end{enumerate} Then there exists a hypercovering $(I, \\{U_i\\})$ of $X$ with the following properties \\begin{enumerate} \\item each $U_i$ is an element of the basis $\\mathcal{B}$, \\item each of the $I_n$ is a finite set, and in particular \\item each of the coverings  (\\ref{equation-covering-X}), (\\ref{equation-covering-two}), and (\\ref{equation-covering-general}) is finite. \\end{enumerate}"}
+{"_id": "8414", "title": "hypercovering-lemma-split", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be an $r$-truncated simplicial object of $\\text{SR}(\\mathcal{C})$. The following are equivalent \\begin{enumerate} \\item $K$ is split (Simplicial, Definition \\ref{simplicial-definition-split}), \\item $f_{\\varphi, i} : U_{n, i} \\to U_{m, \\alpha(\\varphi)(i)}$ is an isomorphism for $r \\geq n \\geq 0$, $\\varphi : [m] \\to [n]$ surjective, $i \\in I_n$, and \\item $f_{\\sigma^n_j, i} : U_{n, i} \\to U_{n + 1, \\alpha(\\sigma^n_j)(i)}$ is an isomorphism for $0 \\leq j \\leq n < r$, $i \\in I_n$. \\end{enumerate} The same holds for simplicial objects if in (2) and (3) we set $r = \\infty$."}
+{"_id": "8415", "title": "hypercovering-lemma-hypercovering-object", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item any object $U$ of $\\mathcal{C}$ has a covering $\\{U_j \\to U\\}_{j \\in J}$ with $U_j \\in \\mathcal{B}$, and \\item if $\\{U_j \\to U\\}_{j \\in J}$ is a covering with $U_j \\in \\mathcal{B}$ and $\\{U' \\to U\\}$ is a morphism with $U' \\in \\mathcal{B}$, then $\\{U_j \\to U\\}_{j \\in J} \\amalg \\{U' \\to U\\}$ is a covering. \\end{enumerate} Then for any $X$ in $\\mathcal{C}$ there is a hypercovering $K$ of $X$ such that $K_n = \\{U_{n, i}\\}_{i \\in I_n}$ with $U_{n, i} \\in \\mathcal{B}$ for all $i \\in I_n$."}
+{"_id": "8416", "title": "hypercovering-lemma-hypercovering-site", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume that any object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$. Then there is a hypercovering $K$ such that $K_n = \\{U_i\\}_{i \\in I_n}$ with $U_i \\in \\mathcal{B}$ for all $i \\in I_n$."}
+{"_id": "8417", "title": "hypercovering-lemma-hypercovering-morphism-sites", "text": "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given by the functor $u : \\mathcal{D} \\to \\mathcal{C}$. Assume $\\mathcal{D}$ and $\\mathcal{C}$ have equalizers and fibre products and $u$ commutes with them. If a simplicial object $K$ of $\\text{SR}(\\mathcal{D})$ is a hypercovering, then $u(K)$ is a hypercovering."}
+{"_id": "8418", "title": "hypercovering-lemma-hypercovering-continuous-functor", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor. Assume $\\mathcal{D}$ and $\\mathcal{C}$ have fibre products and $u$ commutes with them. Let $Y \\in \\mathcal{D}$ and $K \\in \\text{SR}(\\mathcal{D}, Y)$ a hypercovering of $Y$. Then $u(K)$ is a hypercovering of $u(Y)$."}
+{"_id": "8419", "title": "hypercovering-lemma-w-contractible", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume \\begin{enumerate} \\item $\\mathcal{C}$ has fibre products, \\item for all $X \\in \\Ob(\\mathcal{C})$ there exists a finite covering $\\{U_i \\to X\\}_{i \\in I}$ with $U_i \\in \\mathcal{B}$, \\item if $\\{U_i \\to X\\}_{i \\in I}$ is a finite covering with $U_i \\in \\mathcal{B}$ and $U \\to X$ is a morphism with $U \\in \\mathcal{B}$, then $\\{U_i \\to X\\}_{i \\in I} \\amalg \\{U \\to X\\}$ is a covering. \\end{enumerate} Then for every $X$ there exists a hypercovering $K$ of $X$ such that each $K_n = \\{U_{n, i} \\to X\\}_{i \\in I_n}$ with $I_n$ finite and $U_{n, i} \\in \\mathcal{B}$."}
+{"_id": "8435", "title": "algebraic-theorem-smooth-groupoid-gives-algebraic-stack", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the quotient stack $[U/R]$ is an algebraic stack over $S$."}
+{"_id": "8436", "title": "algebraic-lemma-morphism-schemes-gives-representable-transformation", "text": "Let $S$, $X$, $Y$ be objects of $\\Sch_{fppf}$. Let $f : X \\to Y$ be a morphism of schemes. Then the $1$-morphism induced by $f$ $$ (\\Sch/X)_{fppf} \\longrightarrow (\\Sch/Y)_{fppf} $$ is a representable $1$-morphism."}
+{"_id": "8437", "title": "algebraic-lemma-representable-morphism-equivalent", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Consider a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences. Then $f$ is representable if and only if $f'$ is representable."}
+{"_id": "8438", "title": "algebraic-lemma-composition-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$ Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be representable $1$-morphisms. Then $$ g \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z} $$ is a representable $1$-morphism."}
+{"_id": "8439", "title": "algebraic-lemma-base-change-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$ Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism. Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \\xymatrix{ \\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Z} \\ar[r]^g & \\mathcal{Y} } $$ Then the base change $f'$ is a representable $1$-morphism."}
+{"_id": "8441", "title": "algebraic-lemma-characterize-representable-by-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Then $\\mathcal{X}$ is representable by an algebraic space over $S$ if and only if the following conditions are satisfied: \\begin{enumerate} \\item $\\mathcal{X}$ is fibred in setoids\\footnote{This means that it is fibred in groupoids and objects in the fibre categories have no nontrivial automorphisms, see Categories, Definition \\ref{categories-definition-category-fibred-sets}.}, and \\item the presheaf $U \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong$ is an algebraic space. \\end{enumerate}"}
+{"_id": "8442", "title": "algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are necessary and sufficient conditions for $f$ to be representable by algebraic spaces: \\begin{enumerate} \\item for each scheme $U/S$ the functor $f_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$ between fibre categories is faithful, and \\item for each $U$ and each $y \\in \\Ob(\\mathcal{Y}_U)$ the presheaf $$ (h : V \\to U) \\longmapsto \\{(x, \\phi) \\mid x \\in \\Ob(\\mathcal{X}_V), \\phi : h^*y \\to f(x)\\}/\\cong $$ is an algebraic space over $U$. \\end{enumerate} Here we have made a choice of pullbacks for $\\mathcal{Y}$."}
+{"_id": "8443", "title": "algebraic-lemma-representable-by-spaces-morphism-equivalent", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Consider a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences. Then $f$ is representable by algebraic spaces if and only if $f'$ is representable by algebraic spaces."}
+{"_id": "8444", "title": "algebraic-lemma-morphism-spaces-gives-representable-by-spaces", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $S$. If $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces over $S$, then the $1$-morphism $f$ is representable by algebraic spaces."}
+{"_id": "8445", "title": "algebraic-lemma-map-presheaves-representable-by-algebraic-spaces", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Let $a : F \\to G$ be a map of presheaves of sets on $(\\Sch/S)_{fppf}$. Denote $a' : \\mathcal{S}_F  \\to \\mathcal{S}_G$ the associated map of categories fibred in sets. Then $a$ is representable by algebraic spaces (see Bootstrap, Definition \\ref{bootstrap-definition-morphism-representable-by-spaces}) if and only if $a'$ is representable by algebraic spaces."}
+{"_id": "8446", "title": "algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in setoids over $(\\Sch/S)_{fppf}$. Let $F$, resp.\\ $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$. Let $a : F \\to G$ be the map of presheaves corresponding to $f$. Then $a$ is representable by algebraic spaces (see Bootstrap, Definition \\ref{bootstrap-definition-morphism-representable-by-spaces}) if and only if $f$ is representable by algebraic spaces."}
+{"_id": "8447", "title": "algebraic-lemma-base-change-representable-by-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \\xymatrix{ \\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Z} \\ar[r]^g & \\mathcal{Y} } $$ Then the base change $f'$ is a $1$-morphism representable by algebraic spaces."}
+{"_id": "8448", "title": "algebraic-lemma-base-change-by-space-representable-by-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$ Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms. Assume \\begin{enumerate} \\item $f$ is representable by algebraic spaces, and \\item $\\mathcal{Z}$ is representable by an algebraic space over $S$. \\end{enumerate} Then the $2$-fibre product $\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X}$ is representable by an algebraic space."}
+{"_id": "8449", "title": "algebraic-lemma-composition-representable-by-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ are $1$-morphisms representable by algebraic spaces, then $$ g \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z} $$ is a $1$-morphism representable by algebraic spaces."}
+{"_id": "8450", "title": "algebraic-lemma-product-representable-by-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$ be $1$-morphisms representable by algebraic spaces. Then $$ f_1 \\times f_2 : \\mathcal{X}_1 \\times \\mathcal{X}_2 \\longrightarrow \\mathcal{Y}_1 \\times \\mathcal{Y}_2 $$ is a $1$-morphism representable by algebraic spaces."}
+{"_id": "8452", "title": "algebraic-lemma-property-morphism-equivalent", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Let $\\mathcal{P}$ be as in Definition \\ref{definition-relative-representable-property}. Consider a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences and $f$ (or equivalently $f'$) is representably by algebraic spaces. Then $f$ has $\\mathcal{P}$ if and only if $f'$ has $\\mathcal{P}$."}
+{"_id": "8453", "title": "algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be as in Definition \\ref{definition-relative-representable-property}. Assume $a$ is representable by algebraic spaces. Then $a : F \\to G$ has property $\\mathcal{P}$ (see Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}) if and only if the corresponding morphism $\\mathcal{S}_F \\to \\mathcal{S}_G$ of categories fibred in groupoids has property $\\mathcal{P}$."}
+{"_id": "8454", "title": "algebraic-lemma-map-fibred-setoids-property", "text": "Let $S$ be an object of $\\Sch_{fppf}$. Let $\\mathcal{P}$ be as in Definition \\ref{definition-relative-representable-property}. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in setoids over $(\\Sch/S)_{fppf}$. Let $F$, resp.\\ $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$. Let $a : F \\to G$ be the map of presheaves corresponding to $f$. Then $a$ has $\\mathcal{P}$ if and only if $f$ has $\\mathcal{P}$."}
+{"_id": "8455", "title": "algebraic-lemma-composition-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$, $\\mathcal{Y}$, $\\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property} which is stable under composition. Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms which are representable by algebraic spaces. If $f$ and $g$ have property $\\mathcal{P}$ so does $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$."}
+{"_id": "8456", "title": "algebraic-lemma-base-change-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property}. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism. Consider the $2$-fibre product diagram $$ \\xymatrix{ \\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Z} \\ar[r]^g & \\mathcal{Y} } $$ If $f$ has $\\mathcal{P}$, then the base change $f'$ has $\\mathcal{P}$."}
+{"_id": "8459", "title": "algebraic-lemma-representable-transformations-property-implication", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in Definition \\ref{definition-relative-representable-property}. Suppose that for any morphism of algebraic spaces $a : F \\to G$ we have $\\mathcal{P}(a) \\Rightarrow \\mathcal{P}'(a)$. If $f$ has property $\\mathcal{P}$ then $f$ has property $\\mathcal{P}'$."}
+{"_id": "8461", "title": "algebraic-lemma-representable-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent: \\begin{enumerate} \\item the diagonal $\\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item for every scheme $U$ over $S$, and any $x, y \\in \\Ob(\\mathcal{X}_U)$ the sheaf $\\mathit{Isom}(x, y)$ is an algebraic space over $U$, \\item for every scheme $U$ over $S$, and any $x \\in \\Ob(\\mathcal{X}_U)$ the associated $1$-morphism $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ is representable by algebraic spaces, \\item for every pair of schemes $T_1, T_2$ over $S$, and any $x_i \\in \\Ob(\\mathcal{X}_{T_i})$, $i = 1, 2$ the $2$-fibre product $(\\Sch/T_1)_{fppf} \\times_{x_1, \\mathcal{X}, x_2} (\\Sch/T_2)_{fppf}$ is representable by an algebraic space, \\item for every representable category fibred in groupoids $\\mathcal{U}$ over $(\\Sch/S)_{fppf}$ every $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces, \\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of representable categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and any $1$-morphisms $x_i : \\mathcal{T}_i \\to \\mathcal{X}$, $i = 1, 2$ the $2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$ is representable by an algebraic space, \\item for every category fibred in groupoids $\\mathcal{U}$ over $(\\Sch/S)_{fppf}$ which is representable by an algebraic space every $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces, \\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ which are representable by algebraic spaces, and any $1$-morphisms $x_i : \\mathcal{T}_i \\to \\mathcal{X}$ the $2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$ is representable by an algebraic space. \\end{enumerate}"}
+{"_id": "8462", "title": "algebraic-lemma-equivalent", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are equivalent as categories over $(\\Sch/S)_{fppf}$. Then $\\mathcal{X}$ is an algebraic stack if and only if $\\mathcal{Y}$ is an algebraic stack. Similarly, $\\mathcal{X}$ is a Deligne-Mumford stack if and only if $\\mathcal{Y}$ is a Deligne-Mumford stack."}
+{"_id": "8463", "title": "algebraic-lemma-representable-algebraic", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. \\begin{enumerate} \\item A category fibred in groupoids $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ which is representable by an algebraic space is a Deligne-Mumford stack. \\item If $F$ is an algebraic space over $S$, then the associated category fibred in groupoids $p : \\mathcal{S}_F \\to (\\Sch/S)_{fppf}$ is a Deligne-Mumford stack. \\item If $X \\in \\Ob((\\Sch/S)_{fppf})$, then $(\\Sch/X)_{fppf} \\to (\\Sch/S)_{fppf}$ is a Deligne-Mumford stack. \\end{enumerate}"}
+{"_id": "8464", "title": "algebraic-lemma-algebraic-stack-no-automorphisms", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids, \\item $\\mathcal{X}$ is a Deligne-Mumford stack such that the canonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is an equivalence, and \\item $\\mathcal{X}$ is representable by an algebraic space. \\end{enumerate}"}
+{"_id": "8466", "title": "algebraic-lemma-2-fibre-product-general", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{Z}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$ whose diagonal is representable by algebraic spaces. Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$. Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product $\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack."}
+{"_id": "8467", "title": "algebraic-lemma-2-fibre-product", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be algebraic stacks over $S$. Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of algebraic stacks. Then the $2$-fibre product $\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack. It is also the $2$-fibre product in the $2$-category of algebraic stacks over $(\\Sch/S)_{fppf}$."}
+{"_id": "8468", "title": "algebraic-lemma-lift-morphism-presentations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. Let $V \\in \\Ob((\\Sch/S)_{fppf})$. Let $y : (\\Sch/V)_{fppf} \\to \\mathcal{Y}$ be surjective and smooth. Then there exists an object $U \\in \\Ob((\\Sch/S)_{fppf})$ and a $2$-commutative diagram $$ \\xymatrix{ (\\Sch/U)_{fppf} \\ar[r]_a \\ar[d]_x & (\\Sch/V)_{fppf} \\ar[d]^y \\\\ \\mathcal{X} \\ar[r]^f & \\mathcal{Y} } $$ with $x$ surjective and smooth."}
+{"_id": "8469", "title": "algebraic-lemma-characterize-representable-by-algebraic-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. The following are equivalent: \\begin{enumerate} \\item for $U \\in \\Ob((\\Sch/S)_{fppf})$ the functor $f : \\mathcal{X}_U \\to \\mathcal{Y}_U$ is faithful, \\item the functor $f$ is faithful, and \\item $f$ is representable by algebraic spaces. \\end{enumerate}"}
+{"_id": "8470", "title": "algebraic-lemma-smooth-surjective-morphism-implies-algebraic", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If \\begin{enumerate} \\item $\\mathcal{U}$ is representable by an algebraic space, and \\item $u$ is representable by algebraic spaces, surjective and smooth, \\end{enumerate} then $\\mathcal X$ is an algebraic stack over $S$."}
+{"_id": "8471", "title": "algebraic-lemma-representable-morphism-to-algebraic", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. Assume that \\begin{enumerate} \\item $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic spaces, and \\item $\\mathcal{Y}$ is an algebraic stack over $S$. \\end{enumerate} Then $\\mathcal{X}$ is an algebraic stack over $S$."}
+{"_id": "8472", "title": "algebraic-lemma-open-fibred-category-is-algebraic", "text": "\\begin{reference} Removing the hypothesis that $j$ is a monomorphism was observed in an email from Matthew Emerton dates June 15, 2016 \\end{reference} Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $j : \\mathcal X \\to \\mathcal Y$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $j$ is representable by algebraic spaces. Then, if $\\mathcal{Y}$ is a stack in groupoids (resp.\\ an algebraic stack), so is $\\mathcal{X}$."}
+{"_id": "8473", "title": "algebraic-lemma-map-space-into-stack", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be an algebraic stack over $S$. Let $\\mathcal{U}$ be an algebraic stack over $S$ which is representable by an algebraic space. Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a 1-morphism. Then \\begin{enumerate} \\item the $2$-fibre product $\\mathcal{R} = \\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U}$ is representable by an algebraic space, \\item there is a canonical equivalence $$ \\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U} = \\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}, $$ \\item the projection $\\text{pr}_{02}$ induces via (2) a $1$-morphism $$ \\text{pr}_{02} : \\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R} \\longrightarrow \\mathcal{R} $$ \\item let $U$, $R$ be the algebraic spaces representing $\\mathcal{U}, \\mathcal{R}$ and $t, s : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$ are the morphisms corresponding to the $1$-morphisms $\\text{pr}_0, \\text{pr}_1 : \\mathcal{R} \\to \\mathcal{U}$ and $\\text{pr}_{02} : \\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R} \\to \\mathcal{R}$ above, then the quintuple $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$, \\item the morphism $f$ induces a canonical $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ of stacks in groupoids over $(\\Sch/S)_{fppf}$, and \\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ is fully faithful. \\end{enumerate}"}
+{"_id": "8474", "title": "algebraic-lemma-stack-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be an algebraic stack over $S$. Let $U$ be an algebraic space over $S$. Let $f : \\mathcal{S}_U \\to \\mathcal{X}$ be a surjective smooth morphism. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces and $f_{can} : [U/R] \\to \\mathcal{X}$ be the result of applying Lemma \\ref{lemma-map-space-into-stack} to $U$ and $f$. Then \\begin{enumerate} \\item the morphisms $s$, $t$ are smooth, and \\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ is an equivalence. \\end{enumerate}"}
+{"_id": "8475", "title": "algebraic-lemma-diagonal-quotient-stack", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Then the diagonal of $[U/R]$ is representable by algebraic spaces."}
+{"_id": "8476", "title": "algebraic-lemma-smooth-quotient-smooth-presentation", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the morphism $\\mathcal{S}_U \\to [U/R]$ is smooth and surjective."}
+{"_id": "8477", "title": "algebraic-lemma-change-big-site", "text": "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$. Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$, see Topologies, Section \\ref{topologies-section-change-alpha}. Let $S$ be an object of $\\Sch_{fppf}$. Let $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : (\\Sch/S)_{fppf} \\to (\\Sch'/S)_{fppf}$. Let $\\mathcal{X}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item if $\\mathcal{X}$ is representable by some $X \\in \\Ob((\\Sch/S)_{fppf})$, then $f^{-1}\\mathcal{X}$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$, \\item if $\\mathcal{X}$ is representable by $F \\in \\Sh((\\Sch/S)_{fppf})$ which is an algebraic space, then $f^{-1}\\mathcal{X}$ is representable by the algebraic space $f^{-1}F$, \\item if $\\mathcal{X}$ is an algebraic stack, then $f^{-1}\\mathcal{X}$ is an algebraic stack, and \\item if $\\mathcal{X}$ is a Deligne-Mumford stack, then $f^{-1}\\mathcal{X}$ is a Deligne-Mumford stack too. \\end{enumerate}"}
+{"_id": "8479", "title": "algebraic-lemma-category-of-spaces-over-smaller-base-scheme", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. The constructions A and B of Stacks, Section \\ref{stacks-section-localize} above give isomorphisms of $2$-categories $$ \\left\\{ \\begin{matrix} 2\\text{-category of algebraic}\\\\ \\text{stacks }\\mathcal{X}\\text{ over }S \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} 2\\text{-category of pairs }(\\mathcal{X}', f)\\text{ consisting of an}\\\\ \\text{algebraic stack }\\mathcal{X}'\\text{ over }S'\\text{ and a morphism}\\\\ f : \\mathcal{X}' \\to (\\Sch/S)_{fppf}\\text{ of algebraic stacks over }S' \\end{matrix} \\right\\} $$"}
+{"_id": "8480", "title": "algebraic-proposition-algebraic-stack-no-automorphisms", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a stack in setoids, \\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is an equivalence, and \\item $\\mathcal{X}$ is representable by an algebraic space. \\end{enumerate}"}
+{"_id": "8492", "title": "sites-theorem-plus", "text": "With $\\mathcal{F}$ as above \\begin{enumerate} \\item \\label{item-sep} The presheaf $\\mathcal{F}^+$ is separated. \\item \\label{item-sheaf} If $\\mathcal{F}$ is separated, then $\\mathcal{F}^+$ is a sheaf and the map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$ is injective. \\item \\label{item-plus-iso} If $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to \\mathcal{F}^+$ is an isomorphism. \\item \\label{item-plusplus} The presheaf $\\mathcal{F}^{++}$ is always a sheaf. \\end{enumerate}"}
+{"_id": "8494", "title": "sites-theorem-topology-and-topos", "text": "Let $\\mathcal{C}$ be a category. Let $J$, $J'$ be topologies on $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $J = J'$, \\item sheaves for the topology $J$ are the same as sheaves for the topology $J'$. \\end{enumerate}"}
+{"_id": "8495", "title": "sites-lemma-mono-epi", "text": "The injective (resp.\\ surjective) maps defined above are exactly the monomorphisms (resp.\\ epimorphisms) of $\\textit{PSh}(\\mathcal{C})$. A map is an isomorphism if and only if it is both injective and surjective."}
+{"_id": "8496", "title": "sites-lemma-image", "text": "Let $\\mathcal{C}$ be a category. Suppose that $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves of sets on $\\mathcal{C}$. There exists a unique subpresheaf $\\mathcal{G}' \\subset \\mathcal{G}$ such that $\\varphi$ factors as $\\mathcal{F} \\to \\mathcal{G}' \\to \\mathcal{G}$ and such that the first map is surjective."}
+{"_id": "8497", "title": "sites-lemma-almost-directed", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories. Suppose that $\\mathcal{C}$ has fibre products and equalizers, and that $u$ commutes with them. Then the categories $(\\mathcal{I}_V)^{opp}$ satisfy the hypotheses of Categories, Lemma \\ref{categories-lemma-split-into-directed}."}
+{"_id": "8498", "title": "sites-lemma-directed", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories. Assume \\begin{enumerate} \\item the category $\\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\\mathcal{D}$ , and \\item the category $\\mathcal{C}$ has fibre products and $u$ commutes with them. \\end{enumerate} Then the index categories $(\\mathcal{I}^u_V)^{opp}$ are filtered (see Categories, Definition \\ref{categories-definition-directed})."}
+{"_id": "8499", "title": "sites-lemma-recover", "text": "There is a canonical map $\\mathcal{F}(U) \\to u_p\\mathcal{F}(u(U))$, which is compatible with restriction maps (on $\\mathcal{F}$ and on $u_p\\mathcal{F}$)."}
+{"_id": "8500", "title": "sites-lemma-adjoints-u", "text": "The functor $u_p$ is a left adjoint to the functor $u^p$. In other words the formula $$ \\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G}) = \\Mor_{\\textit{PSh}(\\mathcal{D})}(u_p\\mathcal{F}, \\mathcal{G}) $$ holds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$."}
+{"_id": "8501", "title": "sites-lemma-pullback-representable-presheaf", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories. For any object $U$ of $\\mathcal{C}$ we have $u_ph_U = h_{u(U)}$."}
+{"_id": "8502", "title": "sites-lemma-tautological-combinatorial", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and $\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms with the same fixed target. \\begin{enumerate} \\item If $\\mathcal{U}$ and $\\mathcal{V}$ are combinatorially equivalent then they are tautologically equivalent. \\item If $\\mathcal{U}$ and $\\mathcal{V}$ are tautologically equivalent then $\\mathcal{U}$ is a refinement of $\\mathcal{V}$ and $\\mathcal{V}$ is a refinement of $\\mathcal{U}$. \\item The relation ``being combinatorially equivalent'' is an equivalence relation on all families of morphisms with fixed target. \\item The relation ``being tautologically equivalent'' is an equivalence relation on all families of morphisms with fixed target. \\item The relation ``$\\mathcal{U}$ refines $\\mathcal{V}$ and $\\mathcal{V}$ refines $\\mathcal{U}$'' is an equivalence relation on all families of morphisms with fixed target. \\end{enumerate}"}
+{"_id": "8503", "title": "sites-lemma-tautological-same-sheaf", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and $\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms with the same fixed target. Assume that the fibre products $U_i \\times_U U_{i'}$ and $V_j \\times_U V_{j'}$ exist. If $\\mathcal{U}$ and $\\mathcal{V}$ are tautologically equivalent, then for any presheaf $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf condition for $\\mathcal{F}$ with respect to $\\mathcal{U}$ is equivalent to the sheaf condition for $\\mathcal{F}$ with respect to $\\mathcal{V}$."}
+{"_id": "8504", "title": "sites-lemma-compare-separated-presheaf-condition", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I} \\to \\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ be a morphism of families of maps with fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$. If $\\mathcal{F}(U) \\to \\prod_{j \\in J} \\mathcal{F}(V_j)$ is injective then $\\mathcal{F}(U) \\to \\prod_{i \\in I} \\mathcal{F}(U_i)$ is injective."}
+{"_id": "8505", "title": "sites-lemma-compare-sheaf-condition", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to \\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a morphism of families of maps with fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$. If \\begin{enumerate} \\item the fibre products $U_i \\times_U U_{i'}$, $U_i \\times_U V_j$, $V_j \\times_U V_{j'}$ exist, \\item $\\mathcal{F}$ satisfies the sheaf condition with respect to $\\mathcal{V}$, and \\item for every $i \\in I$ the map $\\mathcal{F}(U_i) \\to \\prod_{j \\in J} \\mathcal{F}(V_j \\times_U U_i)$ is injective. \\end{enumerate} Then $\\mathcal{F}$ satisfies the sheaf condition with respect to $\\mathcal{U}$."}
+{"_id": "8506", "title": "sites-lemma-refine-same-topology", "text": "Let $\\mathcal{C}$ be a category. Let $\\text{Cov}_i$, $i = 1, 2$ be two sets of families of morphisms with fixed target which each define the structure of a site on $\\mathcal{C}$. \\begin{enumerate} \\item If every $\\mathcal{U} \\in \\text{Cov}_1$ is tautologically equivalent to some $\\mathcal{V} \\in \\text{Cov}_2$, then $\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset \\Sh(\\mathcal{C}, \\text{Cov}_1)$. If also, every $\\mathcal{U} \\in \\text{Cov}_2$ is tautologically equivalent to some $\\mathcal{V} \\in \\text{Cov}_1$ then the category of sheaves are equal. \\item Suppose that for each $\\mathcal{U} \\in \\text{Cov}_1$ there exists a $\\mathcal{V} \\in \\text{Cov}_2$ such that $\\mathcal{V}$ refines $\\mathcal{U}$. In this case $\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset \\Sh(\\mathcal{C}, \\text{Cov}_1)$. If also for every $\\mathcal{U} \\in \\text{Cov}_2$ there exists a $\\mathcal{V} \\in \\text{Cov}_1$ such that $\\mathcal{V}$ refines $\\mathcal{U}$, then the categories of sheaves are equal. \\end{enumerate}"}
+{"_id": "8507", "title": "sites-lemma-choice-set-coverings-immaterial", "text": "Let $\\mathcal{C}$ be a category. Let $\\text{Cov}(\\mathcal{C})$ be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition \\ref{definition-site}. Let $\\text{Cov}_1, \\text{Cov}_2 \\subset \\text{Cov}(\\mathcal{C})$ be two subsets of $\\text{Cov}(\\mathcal{C})$ which endow $\\mathcal{C}$ with the structure of a site. If every covering $\\mathcal{U} \\in \\text{Cov}(\\mathcal{C})$ is combinatorially equivalent to a covering in $\\text{Cov}_1$ and combinatorially equivalent to a covering in $\\text{Cov}_2$, then $\\Sh(\\mathcal{C}, \\text{Cov}_1) = \\Sh(\\mathcal{C}, \\text{Cov}_2)$."}
+{"_id": "8508", "title": "sites-lemma-limit-sheaf", "text": "Let $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$ be a diagram. Then $\\lim_\\mathcal{I} \\mathcal{F}$ exists and is equal to the limit in the category of presheaves."}
+{"_id": "8509", "title": "sites-lemma-plus-presheaf", "text": "The constructions above define a presheaf $\\mathcal{F}^+$ together with a canonical map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$."}
+{"_id": "8510", "title": "sites-lemma-plus-functorial", "text": "The association $\\mathcal{F} \\mapsto (\\mathcal{F} \\to \\mathcal{F}^+)$ is a functor."}
+{"_id": "8511", "title": "sites-lemma-common-refinement", "text": "Given a pair of coverings $\\{U_i \\to U\\}$ and $\\{V_j \\to U\\}$ of a given object $U$ of the site $\\mathcal{C}$, there exists a covering which is a common refinement."}
+{"_id": "8512", "title": "sites-lemma-independent-refinement", "text": "Any two morphisms $f, g: \\mathcal{U} \\to \\mathcal{V}$ of coverings inducing the same morphism $U \\to V$ induce the same map $H^0(\\mathcal{V}, \\mathcal{F}) \\to  H^0(\\mathcal{U}, \\mathcal{F})$."}
+{"_id": "8513", "title": "sites-lemma-plus-surjective", "text": "The map $\\theta : \\mathcal{F} \\to \\mathcal{F}^+$ has the following property: For every object $U$ of $\\mathcal{C}$ and every section $s \\in \\mathcal{F}^+(U)$ there exists a covering $\\{U_i \\to U\\}$ such that $s|_{U_i}$ is in the image of $\\theta : \\mathcal{F}(U_i) \\to \\mathcal{F}^{+}(U_i)$."}
+{"_id": "8514", "title": "sites-lemma-colimit-sheaves", "text": "Let $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$ be a diagram. Then $\\colim_\\mathcal{I} \\mathcal{F}$ exists and is the sheafification of the colimit in the category of presheaves."}
+{"_id": "8515", "title": "sites-lemma-sheafification-exact", "text": "The functor $\\textit{PSh}(\\mathcal{C}) \\to \\Sh(\\mathcal{C})$, $\\mathcal{F} \\mapsto \\mathcal{F}^\\#$ is exact."}
+{"_id": "8516", "title": "sites-lemma-sections-sheafification", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. Denote $\\theta^2 : \\mathcal{F} \\to \\mathcal{F}^\\#$ the canonical map of $\\mathcal{F}$ into its sheafification. Let $U$ be an object of $\\mathcal{C}$. Let $s \\in \\mathcal{F}^\\#(U)$. There exists a covering $\\{U_i \\to U\\}$ and sections $s_i \\in \\mathcal{F}(U_i)$ such that \\begin{enumerate} \\item $s|_{U_i} = \\theta^2(s_i)$, and \\item for every $i, j$ there exists a covering $\\{U_{ijk} \\to U_i \\times_U U_j\\}$ of $\\mathcal{C}$ such that the pullback of $s_i$ and $s_j$ to each $U_{ijk}$ agree. \\end{enumerate} Conversely, given any covering $\\{U_i \\to U\\}$, elements $s_i \\in \\mathcal{F}(U_i)$ such that (2) holds, then there exists a unique section $s \\in \\mathcal{F}^\\#(U)$ such that (1) holds."}
+{"_id": "8517", "title": "sites-lemma-mono-epi-sheaves", "text": "The injective (resp.\\ surjective) maps defined above are exactly the monomorphisms (resp.\\ epimorphisms) of the category $\\Sh(\\mathcal{C})$. A map of sheaves is an isomorphism if and only if it is both injective and surjective."}
+{"_id": "8518", "title": "sites-lemma-coequalizer-surjection", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} \\to \\mathcal{G}$ be a surjection of sheaves of sets. Then the diagram $$ \\xymatrix{ \\mathcal{F} \\times_\\mathcal{G} \\mathcal{F} \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\mathcal{F} \\ar[r] & \\mathcal{G}} $$ represents $\\mathcal{G}$ as a coequalizer."}
+{"_id": "8519", "title": "sites-lemma-covering-surjective-after-sheafification", "text": "\\begin{slogan} Coverings become surjective after sheafification. \\end{slogan} Let $\\mathcal{C}$ be a site. If $\\{U_i \\to U\\}_{i \\in I}$ is a covering of the site $\\mathcal{C}$, then the morphism of presheaves of sets $$ \\coprod\\nolimits_{i \\in I} h_{U_i} \\to h_U $$ becomes surjective after sheafification."}
+{"_id": "8520", "title": "sites-lemma-sheaf-coequalizer-representable", "text": "Let $\\mathcal{C}$ be a site. Let $E \\subset \\Ob(\\mathcal{C})$ be a subset such that every object of $\\mathcal{C}$ has a covering by elements of $E$. Let $\\mathcal{F}$ be a sheaf of sets. There exists a diagram of sheaves of sets $$ \\xymatrix{ \\mathcal{F}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\mathcal{F}_0 \\ar[r] & \\mathcal{F} } $$ which represents $\\mathcal{F}$ as a coequalizer, such that $\\mathcal{F}_i$, $i = 0, 1$ are coproducts of sheaves of the form $h_U^\\#$ with $U \\in E$."}
+{"_id": "8521", "title": "sites-lemma-pushforward-sheaf", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor. If $\\mathcal{F}$ is a sheaf on $\\mathcal{D}$ then $u^p\\mathcal{F}$ is a sheaf as well."}
+{"_id": "8522", "title": "sites-lemma-adjoint-sheaves", "text": "In the situation of Lemma \\ref{lemma-pushforward-sheaf}. The functor $u_s : \\mathcal{G} \\mapsto (u_p \\mathcal{G})^\\#$ is a left adjoint to $u^s$."}
+{"_id": "8523", "title": "sites-lemma-technical-up", "text": "In the situation of Lemma \\ref{lemma-pushforward-sheaf}. For any presheaf $\\mathcal{G}$ on $\\mathcal{C}$ we have $(u_p\\mathcal{G})^\\# = (u_p(\\mathcal{G}^\\#))^\\#$."}
+{"_id": "8524", "title": "sites-lemma-pullback-representable-sheaf", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor between sites. For any object $U$ of $\\mathcal{C}$ we have $u_sh_U^\\# = h_{u(U)}^\\#$."}
+{"_id": "8525", "title": "sites-lemma-composition-morphisms-sites", "text": "Let $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let $u : \\mathcal{C}_2 \\to \\mathcal{C}_1$ and $v : \\mathcal{C}_3 \\to \\mathcal{C}_2$ be continuous functors which induce morphisms of sites. Then the functor $u \\circ v : \\mathcal{C}_3 \\to \\mathcal{C}_1$ is continuous and defines a morphism of sites $\\mathcal{C}_1 \\to \\mathcal{C}_3$."}
+{"_id": "8526", "title": "sites-lemma-directed-morphism", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be continuous. Assume all the categories $(\\mathcal{I}_V^u)^{opp}$ of Section \\ref{section-functoriality-PSh} are filtered. Then $u$ defines a morphism of sites $\\mathcal{D} \\to \\mathcal{C}$, in other words $u_s$ is exact."}
+{"_id": "8527", "title": "sites-lemma-morphism-of-sites-covering", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by the functor $u : \\mathcal{C} \\to \\mathcal{D}$. Given any object $V$ of $\\mathcal{D}$ there exists a covering $\\{V_j \\to V\\}$ such that for every $j$ there exists a morphism $V_j \\to u(U_j)$ for some object $U_j$ of $\\mathcal{C}$."}
+{"_id": "8528", "title": "sites-lemma-morphism-sites-topoi", "text": "Given a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$ corresponding to the functor $u : \\mathcal{C} \\to \\mathcal{D}$ the pair of functors $(f^{-1} = u_s, f_* = u^s)$ is a morphism of topoi."}
+{"_id": "8529", "title": "sites-lemma-conclude-quasi-compact", "text": "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$. Consider the following conditions \\begin{enumerate} \\item $U$ is quasi-compact, \\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ there exists a finite covering $\\{V_j \\to U\\}_{j = 1, \\ldots, m}$ of $\\mathcal{C}$ refining $\\mathcal{U}$, and \\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ there exists a finite subset $I' \\subset I$ such that $\\{U_i \\to U\\}_{i \\in I'}$ is a covering in $\\mathcal{C}$. \\end{enumerate} Then we always have (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1) but the reverse implications do not hold in general."}
+{"_id": "8530", "title": "sites-lemma-quasi-compact", "text": "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $U$ is quasi-compact, and \\item for every surjection of sheaves $\\coprod_{i \\in I} \\mathcal{F}_i \\to h_U^\\#$ there is a finite subset $J \\subset I$ such that $\\coprod_{i \\in J} \\mathcal{F}_i \\to h_U^\\#$ is surjective. \\end{enumerate}"}
+{"_id": "8531", "title": "sites-lemma-directed-colimits-sections", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{I} \\to \\Sh(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$ be a filtered diagram of sheaves of sets. Let $U \\in \\Ob(\\mathcal{C})$. Consider the canonical map $$ \\Psi : \\colim_i \\mathcal{F}_i(U) \\longrightarrow \\left(\\colim_i \\mathcal{F}_i\\right)(U) $$ With the terminology introduced above: \\begin{enumerate} \\item If all the transition maps are injective then $\\Psi$ is injective for any $U$. \\item If $U$ is quasi-compact, then $\\Psi$ is injective. \\item If $U$ is quasi-compact and all the transition maps are injective then $\\Psi$ is an isomorphism. \\item If $U$ has a cofinal system of coverings $\\{U_j \\to U\\}_{j \\in J}$ with $J$ finite and $U_j \\times_U U_{j'}$ quasi-compact for all $j, j' \\in J$, then $\\Psi$ is bijective. \\end{enumerate}"}
+{"_id": "8532", "title": "sites-lemma-colimit-sites", "text": "In Situation \\ref{situation-inverse-limit-sites} we can construct a site $(\\mathcal{C}, \\text{Cov}(\\mathcal{C}))$ as follows \\begin{enumerate} \\item as a category $\\mathcal{C} = \\colim \\mathcal{C}_i$, and \\item $\\text{Cov}(\\mathcal{C})$ is the union of the images of $\\text{Cov}(\\mathcal{C}_i)$ by $u_i : \\mathcal{C}_i \\to \\mathcal{C}$. \\end{enumerate}"}
+{"_id": "8533", "title": "sites-lemma-compute-pullback-to-limit", "text": "In Situation \\ref{situation-inverse-limit-sites} let $u_i : \\mathcal{C}_i \\to \\mathcal{C}$ be as constructed in Lemma \\ref{lemma-colimit-sites}. Then $u_i$ defines a morphism of sites $f_i : \\mathcal{C} \\to \\mathcal{C}_i$. For $U_i \\in \\Ob(\\mathcal{C}_i)$ and sheaf $\\mathcal{F}$ on $\\mathcal{C}_i$ we have \\begin{equation} \\label{equation-compute-pullback-to-limit} f_i^{-1}\\mathcal{F}(u_i(U_i)) = \\colim_{a : j \\to i} f_a^{-1}\\mathcal{F}(u_a(U_i)) \\end{equation}"}
+{"_id": "8534", "title": "sites-lemma-colimit", "text": "In Situation \\ref{situation-inverse-limit-sites} assume given \\begin{enumerate} \\item a sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all $i \\in \\Ob(\\mathcal{I})$, \\item for $a : j \\to i$ a map $\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$ of sheaves on $\\mathcal{C}_j$ \\end{enumerate} such that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$ whenever $c = a \\circ b$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$ on the site $\\mathcal{C}$ of Lemma \\ref{lemma-colimit-sites}. Let $i \\in \\Ob(\\mathcal{I})$ and $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then $$ \\colim_{a : j \\to i} \\mathcal{F}_j(u_a(X_i)) = \\mathcal{F}(u_i(X_i)) $$"}
+{"_id": "8535", "title": "sites-lemma-colimit-push-pull", "text": "In Situation \\ref{situation-inverse-limit-sites} assume we have a sheaf $\\mathcal{F}$ on $\\mathcal{C}$. Then $$ \\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F} $$ where the transition maps are $f_j^{-1}\\varphi_a$ for $a : j \\to i$ where $\\varphi_a : f_a^{-1}f_{i, *}\\mathcal{F} \\to f_{j, *}\\mathcal{F}$ is a canonical map satisfying a cocycle condition as in Lemma \\ref{lemma-colimit}."}
+{"_id": "8536", "title": "sites-lemma-recover-pu", "text": "There is a canonical map ${}_pu\\mathcal{F}(u(U)) \\to \\mathcal{F}(U)$, which is compatible with restriction maps."}
+{"_id": "8537", "title": "sites-lemma-adjoints-pu", "text": "The functor ${}_pu$ is a right adjoint to the functor $u^p$. In other words the formula $$ \\Mor_{\\textit{PSh}(\\mathcal{C})}(u^p\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{PSh}(\\mathcal{D})}(\\mathcal{G}, {}_pu\\mathcal{F}) $$ holds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$."}
+{"_id": "8538", "title": "sites-lemma-adjoint-functors", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ and $v : \\mathcal{D} \\to \\mathcal{C}$ be functors of categories. Assume that $v$ is right adjoint to $u$. Then we have \\begin{enumerate} \\item $u^ph_V = h_{v(V)}$ for any $V$ in $\\mathcal{D}$, \\item the category $\\mathcal{I}^v_U$ has an initial object, \\item the category ${}_V^u\\mathcal{I}$ has a final object, \\item ${}_pu = v^p$, and \\item $u^p = v_p$. \\end{enumerate}"}
+{"_id": "8540", "title": "sites-lemma-pu-sheaf", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Then ${}_pu\\mathcal{F}$ is a sheaf on $\\mathcal{D}$, which we will denote ${}_su\\mathcal{F}$."}
+{"_id": "8541", "title": "sites-lemma-exact-cocontinuous", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous. The functor $\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$, $\\mathcal{G} \\mapsto (u^p\\mathcal{G})^\\#$ is a left adjoint to the functor ${}_su$ introduced in Lemma \\ref{lemma-pu-sheaf} above. Moreover, it is exact."}
+{"_id": "8542", "title": "sites-lemma-technical-pu", "text": "In the situation of Lemma \\ref{lemma-exact-cocontinuous}. For any presheaf $\\mathcal{G}$ on $\\mathcal{D}$ we have $(u^p\\mathcal{G})^\\# = (u^p(\\mathcal{G}^\\#))^\\#$."}
+{"_id": "8543", "title": "sites-lemma-cocontinuous-morphism-topoi", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous. The functors $g_* = {}_su$ and $g^{-1} = (u^p\\ )^\\#$ define a morphism of topoi $g$ from  $\\Sh(\\mathcal{C})$ to $\\Sh(\\mathcal{D})$."}
+{"_id": "8544", "title": "sites-lemma-composition-cocontinuous", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{E}$ be cocontinuous functors. Then $v \\circ u$ is cocontinuous and we have $h = g \\circ f$ where $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$, resp.\\ $g : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{E})$, resp.\\ $h : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{E})$ is the morphism of topoi associated to $u$, resp.\\ $v$, resp.\\ $v \\circ u$."}
+{"_id": "8545", "title": "sites-lemma-when-shriek", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, and \\item[(b)] $u$ is continuous. \\end{enumerate} Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the associated morphism of topoi. Then \\begin{enumerate} \\item sheafification in the formula $g^{-1} = (u^p\\ )^\\#$ is unnecessary, in other words $g^{-1}(\\mathcal{G})(U) = \\mathcal{G}(u(U))$, \\item $g^{-1}$ has a left adjoint $g_{!} = (u_p\\ )^\\#$, and \\item $g^{-1}$ commutes with arbitrary limits and colimits. \\end{enumerate}"}
+{"_id": "8546", "title": "sites-lemma-preserve-equalizers", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, \\item[(b)] $u$ is continuous, and \\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and $u$ commutes with them. \\end{enumerate} In this case the functor $g_!$ above commutes with fibre products and equalizers (and more generally with finite connected limits)."}
+{"_id": "8547", "title": "sites-lemma-back-and-forth", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, \\item[(b)] $u$ is continuous, and \\item[(c)] $u$ is fully faithful. \\end{enumerate} For $g_!, g^{-1}, g_*$ as above the canonical maps $\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$ and $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms for all sheaves $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "8548", "title": "sites-lemma-bigger-site", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, \\item[(b)] $u$ is continuous, \\item[(c)] $u$ is fully faithful, \\item[(d)] fibre products exist in $\\mathcal{C}$ and $u$ commutes with them, and \\item[(e)] there exist final objects $e_\\mathcal{C} \\in \\Ob(\\mathcal{C})$, $e_\\mathcal{D} \\in \\Ob(\\mathcal{D})$ such that $u(e_\\mathcal{C}) = e_\\mathcal{D}$. \\end{enumerate} Let $g_!, g^{-1}, g_*$ be as above. Then, $u$ defines a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$ with $f_* = g^{-1}$, $f^{-1} = g_!$. The composition $$ \\xymatrix{ \\Sh(\\mathcal{C}) \\ar[r]^g & \\Sh(\\mathcal{D}) \\ar[r]^f & \\Sh(\\mathcal{C}) } $$ is isomorphic to the identity morphism of the topos $\\Sh(\\mathcal{C})$. Moreover, the functor $f^{-1}$ is fully faithful."}
+{"_id": "8549", "title": "sites-lemma-have-functor-other-way", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{C}$ be functors. Assume that $u$ is cocontinuous, and that $v$ is a right adjoint to $u$. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the morphism of topoi associated to $u$, see Lemma \\ref{lemma-cocontinuous-morphism-topoi}. Then $g_*\\mathcal{F}$ is equal to the presheaf $v^p\\mathcal{F}$, in other words, $(g_*\\mathcal{F})(V) = \\mathcal{F}(v(V))$."}
+{"_id": "8550", "title": "sites-lemma-have-functor-other-way-morphism", "text": "In the situation of Lemma \\ref{lemma-have-functor-other-way}. We have $g_* = v^s = v^p$ and $g^{-1} = v_s = (v_p\\ )^\\#$. If $v$ is continuous then $v$ defines a morphism of sites $f$ from $\\mathcal{C}$ to $\\mathcal{D}$ whose associated morphism of topoi is equal to the morphism $g$ associated to the cocontinuous functor $u$. In other words, a continuous functor which has a cocontinuous left adjoint defines a morphism of sites."}
+{"_id": "8551", "title": "sites-lemma-have-left-adjoint", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{D}$, see Lemmas \\ref{lemma-cocontinuous-morphism-topoi} and \\ref{lemma-when-shriek}. \\begin{enumerate} \\item If $w : \\mathcal{D} \\to \\mathcal{C}$ is a left adjoint to $u$, then \\begin{enumerate} \\item $g_!\\mathcal{F}$ is the sheaf associated to the presheaf $w^p\\mathcal{F}$, and \\item $g_!$ is exact. \\end{enumerate} \\item if $w$ is a continuous left adjoint, then $g_!$ has a left adjoint. \\item If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is the morphism of topoi associated to $w$. \\end{enumerate}"}
+{"_id": "8552", "title": "sites-lemma-existence-lower-shriek", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be two sites. Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. Let $E \\subset \\Ob(\\mathcal{D})$ be a subset such that \\begin{enumerate} \\item for $V \\in E$ there exists a sheaf $\\mathcal{G}$ on $\\mathcal{C}$ such that $f^{-1}\\mathcal{F}(V) =  \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{F})$ functorially for $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$, \\item every object of $\\mathcal{D}$ has a covering by objects of $E$. \\end{enumerate} Then $f^{-1}$ has a left adjoint $f_!$."}
+{"_id": "8553", "title": "sites-lemma-describe-j-shriek", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{G}$ be a presheaf on $\\mathcal{C}/U$. Then $j_{U!}(\\mathcal{G}^\\#)$ is the sheaf associated to the presheaf $$ V \\longmapsto \\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) $$ with obvious restriction mappings."}
+{"_id": "8554", "title": "sites-lemma-describe-j-shriek-representable", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Let $X/U$ be an object of $\\mathcal{C}/U$. Then we have $j_{U!}(h_{X/U}^\\#) = h_X^\\#$."}
+{"_id": "8555", "title": "sites-lemma-essential-image-j-shriek", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories $$ \\Sh(\\mathcal{C}/U) \\longrightarrow \\Sh(\\mathcal{C})/h_U^\\# $$"}
+{"_id": "8556", "title": "sites-lemma-j-shriek-commutes-equalizers-fibre-products", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. The functor $j_{U!}$ commutes with fibre products and equalizers (and more generally finite connected limits). In particular, if $\\mathcal{F} \\subset \\mathcal{F}'$ in $\\Sh(\\mathcal{C}/U)$, then $j_{U!}\\mathcal{F} \\subset j_{U!}\\mathcal{F}'$."}
+{"_id": "8558", "title": "sites-lemma-compute-j-shriek-restrict", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. For any sheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have $j_{U!}j_U^{-1}\\mathcal{F} = \\mathcal{F} \\times h_U^\\#$."}
+{"_id": "8559", "title": "sites-lemma-relocalize", "text": "Let $\\mathcal{C}$ be a site. Let $f : V \\to U$ be a morphism of $\\mathcal{C}$. Then there exists a commutative diagram $$ \\xymatrix{ \\mathcal{C}/V \\ar[rd]_{j_V} \\ar[rr]_j & & \\mathcal{C}/U \\ar[ld]^{j_U} \\\\ & \\mathcal{C} & } $$ of continuous and cocontinuous functors. The functor $j : \\mathcal{C}/V \\to \\mathcal{C}/U$, $(a : W \\to V) \\mapsto (f \\circ a : W \\to U)$ is identified with the functor $j_{V/U} : (\\mathcal{C}/U)/(V/U) \\to \\mathcal{C}/U$ via the identification $(\\mathcal{C}/U)/(V/U) = \\mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \\circ j_!$, $j_V^{-1} = j^{-1} \\circ j_U^{-1}$, and $j_{V*} = j_{U*} \\circ j_*$."}
+{"_id": "8560", "title": "sites-lemma-relocalize-explicit", "text": "Notation $\\mathcal{C}$, $f : V \\to U$, $j_U$, $j_V$, and $j$ as in Lemma \\ref{lemma-relocalize}. Via the identifications $\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$ and $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of Lemma \\ref{lemma-essential-image-j-shriek} we have \\begin{enumerate} \\item the functor $j^{-1}$ has the following description $$ j^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_U^\\#) = (\\mathcal{H} \\times_{\\varphi, h_U^\\#, f} h_V^\\# \\to h_V^\\#). $$ \\item the functor $j_!$ has the following description $$ j_!(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#) = (\\mathcal{H} \\xrightarrow{h_f \\circ \\varphi} h_U^\\#) $$ \\end{enumerate}"}
+{"_id": "8561", "title": "sites-lemma-glue-maps", "text": "\\begin{slogan} Maps of sheaves glue. \\end{slogan} Let $\\mathcal{C}$ be a site. Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves on $\\mathcal{C}$. Given a collection $$ \\varphi_i : \\mathcal{F}|_{\\mathcal{C}/U_i} \\longrightarrow \\mathcal{G}|_{\\mathcal{C}/U_i} $$ of maps of sheaves such that for all $i, j \\in I$ the maps $\\varphi_i, \\varphi_j$ restrict to the same map $\\varphi_{ij} : \\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\to \\mathcal{G}|_{\\mathcal{C}/U_i \\times_U U_j}$ then there exists a unique map of sheaves $$ \\varphi : \\mathcal{F}|_{\\mathcal{C}/U} \\longrightarrow \\mathcal{G}|_{\\mathcal{C}/U} $$ whose restriction to each $\\mathcal{C}/U_i$ agrees with $\\varphi_i$."}
+{"_id": "8562", "title": "sites-lemma-internal-hom-sheaf", "text": "\\begin{slogan} The category of sheaves on a site is cartesian closed \\end{slogan} Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$, $\\mathcal{G}$ and $\\mathcal{H}$ be sheaves on $\\mathcal{C}$. There is a canonical bijection $$ \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}\\times\\mathcal{G},\\mathcal{H}) = \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F},\\SheafHom(\\mathcal{G},\\mathcal{H})) $$ which is functorial in all three entries."}
+{"_id": "8563", "title": "sites-lemma-hom-sheaf-hU", "text": "Let $\\mathcal{C}$ be a site and $U \\in \\Ob(\\mathcal{C})$. Then $\\SheafHom(h_U^\\#, \\mathcal{F}) = j_*(\\mathcal{F}|_{\\mathcal{C}/U})$ for $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$."}
+{"_id": "8564", "title": "sites-lemma-glue-sheaves", "text": "Let $\\mathcal{C}$ be a site. Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. Given any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$ for sheaves of sets with respect to the covering $\\{U_i \\to U\\}_{i \\in I}$ there exists a sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}/U$ together with isomorphisms $$ \\varphi_i : \\mathcal{F}|_{\\mathcal{C}/U_i} \\to \\mathcal{F}_i $$ such that the diagrams $$ \\xymatrix{ \\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[d]_{\\text{id}} \\ar[r]_{\\varphi_i} & \\mathcal{F}_i|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[d]^{\\varphi_{ij}} \\\\ \\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[r]^{\\varphi_j} & \\mathcal{F}_j|_{\\mathcal{C}/U_i \\times_U U_j} } $$ are commutative."}
+{"_id": "8565", "title": "sites-lemma-mapping-property-glue", "text": "Let $\\mathcal{C}$ be a site. Let $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. The category $\\Sh(\\mathcal{C}/U)$ is equivalent to the category of glueing data via the functor that associates to $\\mathcal{F}$ on $\\mathcal{C}/U$ the canonical glueing data."}
+{"_id": "8566", "title": "sites-lemma-describe-j-shriek-good-site", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. If the topology on $\\mathcal{C}$ is subcanonical, see Definition \\ref{definition-weaker-than-canonical}, and if $\\mathcal{G}$ is a sheaf on $\\mathcal{C}/U$, then $$ j_{U!}(\\mathcal{G})(V) = \\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U), $$ in other words sheafification is not necessary in Lemma \\ref{lemma-describe-j-shriek}."}
+{"_id": "8567", "title": "sites-lemma-localize-given-products", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Assume $\\mathcal{C}$ has products of pairs of objects. Then \\begin{enumerate} \\item the functor $j_U$ has a continuous right adjoint, namely the functor $v(X) = X \\times U / U$, \\item the functor $v$ defines a morphism of sites $\\mathcal{C}/U \\to \\mathcal{C}$ whose associated morphism of topoi equals $j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$, and \\item we have $j_{U*}\\mathcal{F}(X) = \\mathcal{F}(X \\times U/U)$. \\end{enumerate}"}
+{"_id": "8568", "title": "sites-lemma-relocalize-given-fibre-products", "text": "Let $\\mathcal{C}$ be a site. Let $U \\to V$ be a morphism of $\\mathcal{C}$. Assume $\\mathcal{C}$ has fibre products. Let $j$ be as in Lemma \\ref{lemma-relocalize}. Then \\begin{enumerate} \\item the functor $j : \\mathcal{C}/U \\to \\mathcal{C}/V$ has a continuous right adjoint, namely the functor $v : (X/V) \\mapsto (X \\times_V U/U)$, \\item the functor $v$ defines a morphism of sites $\\mathcal{C}/U \\to \\mathcal{C}/V$ whose associated morphism of topoi equals $j$, and \\item we have $j_*\\mathcal{F}(X/V) = \\mathcal{F}(X \\times_V U/U)$. \\end{enumerate}"}
+{"_id": "8569", "title": "sites-lemma-restrict-back", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Assume that every $X$ in $\\mathcal{C}$ has at most one morphism to $U$. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}/U$. The canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$ and $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms."}
+{"_id": "8570", "title": "sites-lemma-localize-cartesian-square", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ U' \\ar[d] \\ar[r] & U \\ar[d] \\\\ V' \\ar[r] & V } $$ be a commutative diagram of $\\mathcal{C}$. The morphisms of Lemma \\ref{lemma-relocalize} produce commutative diagrams $$ \\vcenter{ \\xymatrix{ \\mathcal{C}/U' \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} & \\mathcal{C}/U \\ar[d]^{j_{U/V}} \\\\ \\mathcal{C}/V' \\ar[r]^{j_{V'/V}} & \\mathcal{C}/V } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ \\Sh(\\mathcal{C}/U') \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} & \\Sh(\\mathcal{C}/U) \\ar[d]^{j_{U/V}} \\\\ \\Sh(\\mathcal{C}/V') \\ar[r]^{j_{V'/V}} & \\Sh(\\mathcal{C}/V) } } $$ of continuous and cocontinuous functors and of topoi. Moreover, if the initial diagram of $\\mathcal{C}$ is cartesian, then we have $j_{V'/V}^{-1} \\circ j_{U/V, *} = j_{U'/V', *} \\circ j_{U'/U}^{-1}$."}
+{"_id": "8571", "title": "sites-lemma-localize-morphism", "text": "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$ and set $U = u(V)$. Then the functor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$, $V'/V \\mapsto u(V')/U$ determines a morphism of sites $f' : \\mathcal{C}/U \\to \\mathcal{D}/V$. The morphism $f'$ fits into a commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} & \\Sh(\\mathcal{D}). } $$ Using the identifications $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and $\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of Lemma \\ref{lemma-essential-image-j-shriek} the functor $(f')^{-1}$ is described by the rule $$ (f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#) = (f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} h_U^\\#). $$ Finally, we have $f'_*j_U^{-1} = j_V^{-1}f_*$."}
+{"_id": "8572", "title": "sites-lemma-localize-morphism-strong", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{D} \\to \\mathcal{C}$ be a functor. Let $V \\in \\Ob(\\mathcal{D})$. Set $U = u(V)$. Assume that \\begin{enumerate} \\item $\\mathcal{C}$ and $\\mathcal{D}$ have all finite limits, \\item $u$ is continuous, and \\item $u$ commutes with finite limits. \\end{enumerate} There exists a commutative diagram of morphisms of sites $$ \\xymatrix{ \\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{f'} & \\mathcal{C} \\ar[d]^f \\\\ \\mathcal{D}/V \\ar[r]^{j_V} & \\mathcal{D} } $$ where the right vertical arrow corresponds to $u$, the left vertical arrow corresponds to the functor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$, $V'/V \\mapsto u(V')/u(V)$ and the horizontal arrows correspond to the functors $\\mathcal{C} \\to \\mathcal{C}/U$, $X \\mapsto X \\times U$ and $\\mathcal{D} \\to \\mathcal{D}/V$, $Y \\mapsto Y \\times V$ as in Lemma \\ref{lemma-localize-given-products}. Moreover, the associated diagram of morphisms of topoi is equal to the diagram of Lemma \\ref{lemma-localize-morphism}. In particular we have $f'_*j_U^{-1} = j_V^{-1}f_*$."}
+{"_id": "8573", "title": "sites-lemma-relocalize-morphism", "text": "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$ and $c : U \\to u(V)$ a morphism of $\\mathcal{C}$. There exists a commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f_c} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} & \\Sh(\\mathcal{D}). } $$ We have $f_c = f' \\circ j_{U/u(V)}$ where $f' : \\Sh(\\mathcal{C}/u(V)) \\to \\Sh(\\mathcal{D}/V)$ is as in Lemma \\ref{lemma-localize-morphism} and $j_{U/u(V)} : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C}/u(V))$ is as in Lemma \\ref{lemma-relocalize}. Using the identifications $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and $\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of Lemma \\ref{lemma-essential-image-j-shriek} the functor $(f_c)^{-1}$ is described by the rule $$ (f_c)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#) = (f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, h_{u(V)}^\\#, c} h_U^\\# \\rightarrow h_U^\\#). $$ Finally, given any morphisms $b : V' \\to V$, $a : U' \\to U$ and $c' : U' \\to u(V')$ such that $$ \\xymatrix{ U' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\ U \\ar[r]^-c & u(V) } $$ commutes, then the diagram $$ \\xymatrix{ \\Sh(\\mathcal{C}/U') \\ar[r]_{j_{U'/U}} \\ar[d]_{f_{c'}} & \\Sh(\\mathcal{C}/U) \\ar[d]^{f_c} \\\\ \\Sh(\\mathcal{D}/V') \\ar[r]^{j_{V'/V}} & \\Sh(\\mathcal{D}/V). } $$ commutes."}
+{"_id": "8574", "title": "sites-lemma-localize-cocontinuous", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor. Let $U$ be an object of $\\mathcal{C}$, and set $V = u(U)$. We have a commutative diagram $$ \\xymatrix{ \\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{u'} & \\mathcal{C} \\ar[d]^u \\\\ \\mathcal{D}/V \\ar[r]^-{j_V} & \\mathcal{D} } $$ where the left vertical arrow is $u' : \\mathcal{C}/U \\to \\mathcal{D}/V$, $U'/U \\mapsto V'/V$. Then $u'$ is cocontinuous also and we get a commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} & \\Sh(\\mathcal{D}) } $$ where $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$)."}
+{"_id": "8576", "title": "sites-lemma-special-square-cocontinuous", "text": "Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$ and functors $$ \\xymatrix{ \\mathcal{C}' \\ar[r]_{v'} \\ar[d]_{u'} & \\mathcal{C} \\ar[d]^u \\\\ \\mathcal{D}' \\ar[r]^v & \\mathcal{D} } $$ Assume \\begin{enumerate} \\item $u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of topoi $f$, $f'$, $g$, and $g'$ by Lemma \\ref{lemma-cocontinuous-morphism-topoi}, \\item $v \\circ u' = u \\circ v'$, \\item $v$ and $v'$ are continuous as well as cocontinuous, and \\item for any object $V'$ of $\\mathcal{D}'$ the functor ${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$ given by $v$ is cofinal. \\end{enumerate} Then $f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and $g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$."}
+{"_id": "8577", "title": "sites-lemma-special-square-continuous", "text": "Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$ and functors $$ \\xymatrix{ \\mathcal{C}' \\ar[r]_{v'} & \\mathcal{C} \\\\ \\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} & \\mathcal{D} \\ar[u]_u } $$ With notation as in Sections \\ref{section-morphism-sites} and \\ref{section-cocontinuous-morphism-topoi} assume \\begin{enumerate} \\item $u$ and $u'$ are continuous giving rise to morphisms of sites $f$ and $f'$, \\item $v$ and $v'$ are cocontinuous giving rise to morphisms of topoi $g$ and $g'$, \\item $u \\circ v = v' \\circ u'$, and \\item $v$ and $v'$ are continuous as well as cocontinuous. \\end{enumerate} Then\\footnote{In this generality we don't know $f \\circ g'$ is equal to $g \\circ f'$ as morphisms of topoi (there is a canonical $2$-arrow from the first to the second which may not be an isomorphism).} $f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and $g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$."}
+{"_id": "8578", "title": "sites-lemma-equivalence", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item $u$ is cocontinuous, \\item $u$ is continuous, \\item given $a, b : U' \\to U$ in $\\mathcal{C}$ such that $u(a) = u(b)$, then there exists a covering $\\{f_i : U'_i \\to U'\\}$ in $\\mathcal{C}$ such that $a \\circ f_i = b \\circ f_i$, \\item given $U', U \\in \\Ob(\\mathcal{C})$ and a morphism $c : u(U') \\to u(U)$ in $\\mathcal{D}$ there exists a covering $\\{f_i : U_i' \\to U'\\}$ in $\\mathcal{C}$ and morphisms $c_i : U_i' \\to U$ such that $u(c_i) = c \\circ u(f_i)$, and \\item given $V \\in \\Ob(\\mathcal{D})$ there exists a covering of $V$ in $\\mathcal{D}$ of the form $\\{u(U_i) \\to V\\}_{i \\in I}$. \\end{enumerate} Then the morphism of topoi $$ g : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{D}) $$ associated to the cocontinuous functor $u$ by Lemma \\ref{lemma-cocontinuous-morphism-topoi} is an equivalence."}
+{"_id": "8579", "title": "sites-lemma-localize-special-cocontinuous", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a special cocontinuous functor. For every object $U$ of $\\mathcal{C}$ we have a commutative diagram $$ \\xymatrix{ \\mathcal{C}/U \\ar[r]_{j_U} \\ar[d] & \\mathcal{C} \\ar[d]^u \\\\ \\mathcal{D}/u(U) \\ar[r]^-{j_{u(U)}} & \\mathcal{D} } $$ as in Lemma \\ref{lemma-localize-cocontinuous}. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d] & \\Sh(\\mathcal{C}) \\ar[d]^u \\\\ \\Sh(\\mathcal{D}/u(U)) \\ar[r]^-{j_{u(U)}} & \\Sh(\\mathcal{D}) } $$ the vertical arrows are equivalences."}
+{"_id": "8580", "title": "sites-lemma-special-equivalence", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{C}' \\subset \\Sh(\\mathcal{C})$ be a full subcategory (with a set of objects) such that \\begin{enumerate} \\item $h_U^\\# \\in \\Ob(\\mathcal{C}')$ for all $U \\in \\Ob(\\mathcal{C})$, and \\item $\\mathcal{C}'$ is preserved under fibre products in $\\Sh(\\mathcal{C})$. \\end{enumerate} Declare a covering of $\\mathcal{C}'$ to be any family $\\{\\mathcal{F}_i \\to \\mathcal{F}\\}_{i \\in I}$ of maps such that $\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$ is a surjective map of sheaves. Then \\begin{enumerate} \\item $\\mathcal{C}'$ is a site (after choosing a set of coverings, see Sets, Lemma \\ref{sets-lemma-coverings-site}), \\item representable presheaves on $\\mathcal{C}'$ are sheaves (i.e., the topology on $\\mathcal{C}'$ is subcanonical, see Definition \\ref{definition-weaker-than-canonical}), \\item the functor $v : \\mathcal{C} \\to \\mathcal{C}'$, $U \\mapsto h_U^\\#$ is a special cocontinuous functor, hence induces an equivalence $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$, \\item for any $\\mathcal{F} \\in \\Ob(\\mathcal{C}')$ we have $g^{-1}h_\\mathcal{F} = \\mathcal{F}$, and \\item for any $U \\in \\Ob(\\mathcal{C})$ we have $g_*h_U^\\# = h_{v(U)} = h_{h_U^\\#}$. \\end{enumerate}"}
+{"_id": "8581", "title": "sites-lemma-topos-good-site", "text": "Let $\\Sh(\\mathcal{C})$ be a topos. Let $\\{\\mathcal{F}_i\\}_{i \\in I}$ be a set of sheaves on $\\mathcal{C}$. There exists an equivalence of topoi $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ induced by a special cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{C}'$ of sites such that \\begin{enumerate} \\item $\\mathcal{C}'$ has a subcanonical topology, \\item a family $\\{V_j \\to V\\}$ of morphisms of $\\mathcal{C}'$ is (combinatorially equivalent to) a covering of $\\mathcal{C}'$ if and only if $\\coprod h_{V_j} \\to h_V$ is surjective, \\item $\\mathcal{C}'$ has fibre products and a final object (i.e., $\\mathcal{C}'$ has all finite limits), \\item every subsheaf of a representable sheaf on $\\mathcal{C}'$ is representable, and \\item each $g_*\\mathcal{F}_i$ is a representable sheaf. \\end{enumerate}"}
+{"_id": "8582", "title": "sites-lemma-morphism-topoi-comes-from-morphism-sites", "text": "\\begin{reference} This statement is closely related to \\cite[Proposition 4.9.4. Expos\\'e IV]{SGA4}. In order to get the whole result, one should also use \\cite[Remarque 4.7.4, Expos\\'e IV]{SGA4}. \\end{reference} Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Then there exists a site $\\mathcal{C}'$ and a diagram of functors $$ \\xymatrix{ \\mathcal{C} \\ar[r]_v & \\mathcal{C}' & \\mathcal{D} \\ar[l]^u } $$ such that \\begin{enumerate} \\item the functor $v$ is a special cocontinuous functor, \\item the functor $u$ commutes with fibre products, is continuous and defines a morphism of sites $\\mathcal{C}' \\to \\mathcal{D}$, and \\item the morphism of topoi $f$ agrees with the composition of morphisms of topoi $$ \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{C}') \\longrightarrow \\Sh(\\mathcal{D}) $$ where the first arrow comes from $v$ via Lemma \\ref{lemma-equivalence} and the second arrow from $u$ via Lemma \\ref{lemma-morphism-sites-topoi}. \\end{enumerate}"}
+{"_id": "8583", "title": "sites-lemma-localize-topos", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Then the category $\\Sh(\\mathcal{C})/\\mathcal{F}$ is a topos. There is a canonical morphism of topoi $$ j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F} \\longrightarrow \\Sh(\\mathcal{C}) $$ which is a localization as in Section \\ref{section-localize} such that \\begin{enumerate} \\item the functor $j_\\mathcal{F}^{-1}$ is the functor $\\mathcal{H} \\mapsto \\mathcal{H} \\times \\mathcal{F}/\\mathcal{F}$, and \\item the functor $j_{\\mathcal{F}!}$ is the forgetful functor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$. \\end{enumerate}"}
+{"_id": "8585", "title": "sites-lemma-localize-topos-site", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Let $\\mathcal{C}/\\mathcal{F}$ be the category of pairs $(U, s)$ where $U \\in \\Ob(\\mathcal{C})$ and $s \\in \\mathcal{F}(U)$. Let a covering in $\\mathcal{C}/\\mathcal{F}$ be a family $\\{(U_i, s_i) \\to (U, s)\\}$ such that $\\{U_i \\to U\\}$ is a covering of $\\mathcal{C}$. Then $j : \\mathcal{C}/\\mathcal{F} \\to \\mathcal{C}$ is a continuous and cocontinuous functor of sites which induces a morphism of topoi $j : \\Sh(\\mathcal{C}/\\mathcal{F}) \\to \\Sh(\\mathcal{C})$. In fact, there is an equivalence $\\Sh(\\mathcal{C}/\\mathcal{F}) = \\Sh(\\mathcal{C})/\\mathcal{F}$ which turns $j$ into $j_\\mathcal{F}$."}
+{"_id": "8586", "title": "sites-lemma-localize-compare", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} = h_U^\\#$ for some object $U$ of $\\mathcal{C}$. Then $j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$ constructed in Lemma \\ref{lemma-localize-topos} agrees with the morphism of topoi $j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$ constructed in Section \\ref{section-localize} via the identification $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of Lemma \\ref{lemma-essential-image-j-shriek}."}
+{"_id": "8587", "title": "sites-lemma-relocalize-topos", "text": "Let $\\mathcal{C}$ be a site. If $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves on $\\mathcal{C}$ then there exists a natural commutative diagram of morphisms of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C})/\\mathcal{G} \\ar[rd]_{j_\\mathcal{G}} \\ar[rr]_j & & \\Sh(\\mathcal{C})/\\mathcal{F} \\ar[ld]^{j_\\mathcal{F}} \\\\ & \\Sh(\\mathcal{C}) & } $$ where $j = j_{\\mathcal{G}/\\mathcal{F}}$ is the localization of the topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ at the object $\\mathcal{G}/\\mathcal{F}$. In particular we have $$ j^{-1}(\\mathcal{H} \\to \\mathcal{F}) = (\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G}) $$ and $$ j_!(\\mathcal{E} \\xrightarrow{e} \\mathcal{F}) = (\\mathcal{E} \\xrightarrow{s \\circ e} \\mathcal{G}). $$"}
+{"_id": "8588", "title": "sites-lemma-relocalize-compare", "text": "Assume $\\mathcal{C}$ and $s : \\mathcal{G} \\to \\mathcal{F}$ are as in Lemma \\ref{lemma-relocalize-topos}. If $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$ and $s : \\mathcal{G} \\to \\mathcal{F}$ comes from a morphism $V \\to U$ of $\\mathcal{C}$ then the diagram in Lemma \\ref{lemma-relocalize-topos} is identified with diagram (\\ref{equation-relocalize}) via the identifications $\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$ and $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of Lemma \\ref{lemma-essential-image-j-shriek}."}
+{"_id": "8589", "title": "sites-lemma-localize-morphism-topoi", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$. Then there exists a commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f'} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} & \\Sh(\\mathcal{D}). } $$ The morphism $f'$ is characterized by the property that $$ (f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G}) = (f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F}) $$ and we have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$."}
+{"_id": "8590", "title": "sites-lemma-localize-morphism-compare", "text": "Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V$ be an object of $\\mathcal{D}$. Set $U = u(V)$. Set $\\mathcal{G} = h_V^\\#$, and $\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$ (see Lemma \\ref{lemma-pullback-representable-sheaf}). Then the diagram of morphisms of topoi of Lemma \\ref{lemma-localize-morphism-topoi} agrees with the diagram of morphisms of topoi of Lemma \\ref{lemma-localize-morphism} via the identifications $j_\\mathcal{F}= j_U$ and $j_\\mathcal{G} = j_V$ of Lemma \\ref{lemma-localize-compare}."}
+{"_id": "8591", "title": "sites-lemma-relocalize-morphism-topoi", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Let $\\mathcal{G} \\in \\Sh(\\mathcal{D})$, $\\mathcal{F} \\in \\Sh(\\mathcal{C})$ and $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f_s} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} & \\Sh(\\mathcal{D}). } $$ We have $f_s = f' \\circ j_{\\mathcal{F}/f^{-1}\\mathcal{G}}$ where $f' : \\Sh(\\mathcal{C})/f^{-1}\\mathcal{G} \\to \\Sh(\\mathcal{D})/\\mathcal{F}$ is as in Lemma \\ref{lemma-localize-morphism-topoi} and $j_{\\mathcal{F}/f^{-1}\\mathcal{G}} : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}$ is as in Lemma \\ref{lemma-relocalize-topos}. The functor $(f_s)^{-1}$ is described by the rule $$ (f_s)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G}) = (f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, f^{-1}\\mathcal{G}, s} \\mathcal{F} \\rightarrow \\mathcal{F}). $$ Finally, given any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$, $a : \\mathcal{F}' \\to \\mathcal{F}$ and $s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that $$ \\xymatrix{ \\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a & f^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\ \\mathcal{F} \\ar[r]^-s & f^{-1}\\mathcal{G} } $$ commutes, then the diagram $$ \\xymatrix{ \\Sh(\\mathcal{C})/\\mathcal{F}' \\ar[r]_{j_{\\mathcal{F}'/\\mathcal{F}}} \\ar[d]_{f_{s'}} & \\Sh(\\mathcal{C})/\\mathcal{F} \\ar[d]^{f_s} \\\\ \\Sh(\\mathcal{D})/\\mathcal{G}' \\ar[r]^{j_{\\mathcal{G}'/\\mathcal{G}}} & \\Sh(\\mathcal{D})/\\mathcal{G}. } $$ commutes."}
+{"_id": "8593", "title": "sites-lemma-points-recover", "text": "Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor. There are functorial isomorphisms $(h_U)_p = u(U)$ for $U \\in \\Ob(\\mathcal{C})$."}
+{"_id": "8594", "title": "sites-lemma-adjoint-point-push-stalk", "text": "For any functor $u : \\mathcal{C} \\to \\textit{Sets}$. The functor $u^p$ is a right adjoint to the stalk functor on presheaves."}
+{"_id": "8595", "title": "sites-lemma-point-pushforward-sheaf", "text": "Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Suppose that for every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ \\begin{enumerate} \\item the map $\\coprod u(U_i) \\to u(U)$ is surjective, and \\item the maps $u(U_i \\times_U U_j) \\to u(U_i) \\times_{u(U)} u(U_j)$ are surjective. \\end{enumerate} Then we have \\begin{enumerate} \\item the presheaf $u^pE$ is a sheaf for all sets $E$, denote it $u^sE$, \\item the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$ and the functor $u^s: \\textit{Sets} \\to \\Sh(\\mathcal{C})$ are adjoint, and \\item we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$ for every presheaf of sets $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "8596", "title": "sites-lemma-point-site-topos", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item Let $p$ be a point of the site $\\mathcal{C}$. Then the pair of functors $(p_*, p^{-1})$ introduced above define a morphism of topoi $\\Sh(pt) \\to \\Sh(\\mathcal{C})$. \\item Let $p = (p_*, p^{-1})$ be a point of the topos $\\Sh(\\mathcal{C})$. Then the functor $u : U \\mapsto p^{-1}(h_U^\\#)$ gives rise to a point $p'$ of the site $\\mathcal{C}$ whose associated morphism of topoi $(p'_*, (p')^{-1})$ is equal to $p$. \\end{enumerate}"}
+{"_id": "8597", "title": "sites-lemma-site-point-morphism", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by $u : \\mathcal{C} \\to \\textit{Sets}$. Let $S_0$ be an infinite set such that $u(U) \\subset S_0$ for all $U \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{S}$ be the site constructed out of the powerset $S = \\mathcal{P}(S_0)$ in Remark \\ref{remark-pt-topos}. Then \\begin{enumerate} \\item there is an equivalence $i : \\Sh(pt) \\to \\Sh(\\mathcal{S})$, \\item the functor $u : \\mathcal{C} \\to \\mathcal{S}$ induces a morphism of sites $f : \\mathcal{S} \\to \\mathcal{C}$, and \\item the composition $$ \\Sh(pt) \\to \\Sh(\\mathcal{S}) \\to \\Sh(\\mathcal{C}) $$ is the morphism of topoi $(p_*, p^{-1})$ of Lemma \\ref{lemma-point-site-topos}. \\end{enumerate}"}
+{"_id": "8598", "title": "sites-lemma-stalk-skyscraper", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of the topos associated to $\\mathcal{C}$. For any set $E$ there are canonical maps $$ E \\longrightarrow (p_*E)_p \\longrightarrow E $$ whose composition is $\\text{id}_E$."}
+{"_id": "8599", "title": "sites-lemma-skyscraper-functor-exact", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of the topos associated to $\\mathcal{C}$. The functor $p_* : \\textit{Sets} \\to \\Sh(\\mathcal{C})$ has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorphisms."}
+{"_id": "8600", "title": "sites-lemma-neighbourhoods-cofiltered", "text": "Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor. If the category of neighbourhoods of $p$ is cofiltered, then the stalk functor (\\ref{equation-stalk}) is left exact."}
+{"_id": "8601", "title": "sites-lemma-neighbourhoods-directed", "text": "Let $\\mathcal{C}$ be a site. Assume that $\\mathcal{C}$ has a final object $X$ and fibred products. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that \\begin{enumerate} \\item $u(X)$ is a singleton set, and \\item for every pair of morphisms $U \\to W$ and $V \\to W$ with the same target the map $u(U \\times_W V) \\to u(U) \\times_{u(W)} u(V)$ is bijective. \\end{enumerate} Then the the category of neighbourhoods of $p$ is cofiltered and consequently the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\to \\mathcal{F}_p$ commutes with finite limits."}
+{"_id": "8602", "title": "sites-lemma-point-functor", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Let $v : \\mathcal{D} \\to \\textit{Sets}$ be a functor and set $w = v \\circ u$. Denote $q$, resp., $p$ the stalk functor (\\ref{equation-stalk}) associated to $v$, resp.\\ $w$. Then $(u_p\\mathcal{F})_q = \\mathcal{F}_p$ functorially in the presheaf $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "8603", "title": "sites-lemma-point-morphism-sites", "text": "\\begin{slogan} A map of sites defines a map on points, and pullback respects stalks. \\end{slogan} Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. Let $q$ be a point of $\\mathcal{D}$ given by the functor $v : \\mathcal{D} \\to \\textit{Sets}$, see Definition \\ref{definition-point}. Then the functor $v \\circ u : \\mathcal{C} \\to \\textit{Sets}$ defines a point $p$ of $\\mathcal{C}$ and moreover there is a canonical identification $$ (f^{-1}\\mathcal{F})_q = \\mathcal{F}_p $$ for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "8604", "title": "sites-lemma-point-morphism-topoi", "text": "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. Let $q : \\Sh(pt) \\to \\Sh(\\mathcal{D})$ be a point. Then $p = f \\circ q$ is a point of the topos $\\Sh(\\mathcal{C})$ and we have a canonical identification $$ (f^{-1}\\mathcal{F})_q = \\mathcal{F}_p $$ for any sheaf $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "8605", "title": "sites-lemma-point-localize", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by $u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$ and let $x \\in u(U)$. The functor $$ v : \\mathcal{C}/U \\longrightarrow \\textit{Sets}, \\quad (\\varphi : V \\to U) \\longmapsto \\{y \\in u(V) \\mid u(\\varphi)(y) = x\\} $$ defines a point $q$ of the site $\\mathcal{C}/U$ such that the diagram $$ \\xymatrix{ & \\Sh(pt) \\ar[d]^p \\ar[ld]_q \\\\ \\Sh(\\mathcal{C}/U) \\ar[r]^{j_U} & \\Sh(\\mathcal{C}) } $$ commutes. In other words $\\mathcal{F}_p = (j_U^{-1}\\mathcal{F})_q$ for any sheaf on $\\mathcal{C}$."}
+{"_id": "8606", "title": "sites-lemma-points-above-point", "text": "Let $\\mathcal{C}$, $p$, $u$, $U$ be as in Lemma \\ref{lemma-point-localize}. The construction of Lemma \\ref{lemma-point-localize} gives a one to one correspondence between points $q$ of $\\mathcal{C}/U$ lying over $p$ and elements $x$ of $u(U)$."}
+{"_id": "8607", "title": "sites-lemma-stalk-j-shriek", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by $u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$. For any sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$ we have $$ (j_{U!}\\mathcal{G})_p = \\coprod\\nolimits_q \\mathcal{G}_q $$ where the coproduct is over the points $q$ of $\\mathcal{C}/U$ associated to elements $x \\in u(U)$ as in Lemma \\ref{lemma-point-localize}."}
+{"_id": "8608", "title": "sites-lemma-maps-u-points", "text": "Let $\\mathcal{C}$ be a site. Let $u, u' : \\mathcal{C} \\to \\textit{Sets}$ be two functors, and let $t : u' \\to u$ be a transformation of functors. Then we obtain a canonical transformation of stalk functors $t_{stalk} : \\mathcal{F}_{p'} \\to \\mathcal{F}_p$ which agrees with $t$ via the identifications of Lemma \\ref{lemma-points-recover}."}
+{"_id": "8609", "title": "sites-lemma-exactness-stalks", "text": "Let $\\mathcal{C}$ be a site and let $\\{p_i\\}_{i\\in I}$ be a conservative family of points. Then \\begin{enumerate} \\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ we have $\\forall i, \\varphi_{p_i}$ injective implies $\\varphi$ injective. \\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ we have $\\forall i, \\varphi_{p_i}$ surjective implies $\\varphi$ surjective. \\item Given any pair of maps of sheaves $\\varphi_1, \\varphi_2 : \\mathcal{F} \\to \\mathcal{G}$ we have $\\forall i, \\varphi_{1, p_i} = \\varphi_{2, p_i}$ implies $\\varphi_1 = \\varphi_2$. \\item Given a finite diagram $\\mathcal{G} : \\mathcal{J} \\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{F}$ and morphisms $q_j : \\mathcal{F} \\to \\mathcal{G}_j$ then $(\\mathcal{F}, q_j)$ is a limit of the diagram if and only if for each $i$ the stalk $(\\mathcal{F}_{p_i}, (q_j)_{p_i})$ is one. \\item Given a finite diagram $\\mathcal{F} : \\mathcal{J} \\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{G}$ and morphisms $e_j : \\mathcal{F}_j \\to \\mathcal{G}$ then $(\\mathcal{G}, e_j)$ is a colimit of the diagram if and only if for each $i$ the stalk $(\\mathcal{G}_{p_i}, (e_j)_{p_i})$ is one. \\end{enumerate}"}
+{"_id": "8610", "title": "sites-lemma-enough", "text": "Let $\\mathcal{C}$ be a site and let $\\{(p_i, u_i)\\}_{i\\in I}$ be a family of points. The family is conservative if and only if for every sheaf $\\mathcal{F}$ and every $U\\in \\Ob(\\mathcal{C})$ and every pair of distinct sections $s, s' \\in \\mathcal{F}(U)$, $s \\not = s'$ there exists an $i$ and $x\\in u_i(U)$ such that the triples $(U, x, s)$ and $(U, x, s')$ define distinct elements of $\\mathcal{F}_{p_i}$."}
+{"_id": "8612", "title": "sites-lemma-enough-points-local", "text": "Let $\\mathcal{C}$ be a site. Let $\\{U_i\\}_{i \\in I}$ be a family of objects of $\\mathcal{C}$. Assume \\begin{enumerate} \\item $\\coprod h_{U_i}^\\# \\to *$ is a surjective map of sheaves, and \\item each localization $\\mathcal{C}/U_i$ has enough points. \\end{enumerate} Then $\\mathcal{C}$ has enough points."}
+{"_id": "8614", "title": "sites-lemma-refine", "text": "Let $\\mathcal{C}$ be a site. Let $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Let $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements. Let $\\{W_k \\to W\\}$ be a finite covering of $\\mathcal{C}$. Let $f \\in u'(W)$. There exists a refinement $(I, \\geq, U_i, f_{ii'})$ of $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\\mathcal{F}_p$ and that the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$."}
+{"_id": "8615", "title": "sites-lemma-refine-all-at-once", "text": "Let $\\mathcal{C}$ be a site. Let $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Let $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements. There exists a refinement $(I, \\geq, U_i, f_{ii'})$ of $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\\mathcal{F}_p$ and such that for every finite covering $\\{W_k \\to W\\}$ of the site $\\mathcal{C}$, and any $f \\in u'(W)$ the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$."}
+{"_id": "8616", "title": "sites-lemma-criterion-points", "text": "Let $\\mathcal{C}$ be a site. Let $I$ be a set and for $i \\in I$ let $U_i$ be an object of $\\mathcal{C}$ such that \\begin{enumerate} \\item $\\coprod h_{U_i}$ surjects onto the final object of $\\Sh(\\mathcal{C})$, and \\item $\\mathcal{C}/U_i$ satisfies the hypotheses of Proposition \\ref{proposition-criterion-points}. \\end{enumerate} Then $\\mathcal{C}$ has enough points."}
+{"_id": "8617", "title": "sites-lemma-w-contractible", "text": "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$. The following conditions are equivalent \\begin{enumerate} \\item For every covering $\\{U_i \\to U\\}$ there exists a map of sheaves $h_U^\\# \\to \\coprod h_{U_i}^\\#$ right inverse to the sheafification of $\\coprod h_{U_i} \\to h_U$. \\item For every surjection of sheaves of sets $\\mathcal{F} \\to \\mathcal{G}$ the map $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective. \\end{enumerate}"}
+{"_id": "8618", "title": "sites-lemma-exactness-properties", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Consider the following properties (on sheaves of sets): \\begin{enumerate} \\item $f_*$ is faithful, \\item $f_*$ is fully faithful, \\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for all $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$, \\item $f_*$ transforms surjections into surjections, \\item $f_*$ commutes with coequalizers, \\item $f_*$ commutes with pushouts, \\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism for all $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$, \\item $f_*$ reflects injections, \\item $f_*$ reflects surjections, \\item $f_*$ reflects bijections, and \\item for any surjection $\\mathcal{F} \\to f^{-1}\\mathcal{G}$ there exists a surjection $\\mathcal{G}' \\to \\mathcal{G}$ such that $f^{-1}\\mathcal{G}' \\to f^{-1}\\mathcal{G}$ factors through $\\mathcal{F} \\to f^{-1}\\mathcal{G}$. \\end{enumerate} Then we have the following implications \\begin{enumerate} \\item[(a)] (2) $\\Rightarrow$ (1), \\item[(b)] (3) $\\Rightarrow$ (1), \\item[(c)] (7) $\\Rightarrow$ (1), (2), (3), (8), (9), (10). \\item[(d)] (3) $\\Leftrightarrow$ (9), \\item[(e)] (6) $\\Rightarrow$ (4) and (5) $\\Rightarrow$ (4), \\item[(f)] (4) $\\Leftrightarrow$ (11), \\item[(g)] (9) $\\Rightarrow$ (8), (10), and \\item[(h)] (2) $\\Leftrightarrow$ (7). \\end{enumerate} Picture $$ \\xymatrix{ (6) \\ar@{=>}[rd] & & & & & (9) \\ar@{=>}[r] \\ar@{=>}[rd] & (8) \\\\ & (4) \\ar@{<=>}[r] & (11) & (2) \\ar@{<=>}[r] & (7) \\ar@{=>}[ru] \\ar@{=>}[rd] & & (10) \\\\ (5) \\ar@{=>}[ur] & & & & & (3) \\ar@{=>}[r] & (1) } $$"}
+{"_id": "8619", "title": "sites-lemma-weaker", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. Assume that for any object $U$ of $\\mathcal{C}$ and any covering $\\{V_j \\to u(U)\\}$ in $\\mathcal{D}$ there exists a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that the map of sheaves $$ \\coprod h_{u(U_i)}^\\# \\to h_{u(U)}^\\# $$ factors through the map of sheaves $$ \\coprod h_{V_j}^\\# \\to h_{u(U)}^\\#. $$ Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves."}
+{"_id": "8620", "title": "sites-lemma-cover-from-below", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by the functor $u : \\mathcal{C} \\to \\mathcal{D}$. Assume that for every object $V$ of $\\mathcal{D}$ there exist objects $U_i$ of $\\mathcal{C}$ and morphisms $u(U_i) \\to V$ such that $\\{u(U_i) \\to V\\}$ is a covering of $\\mathcal{D}$. In this case the functor $f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ reflects injections and surjections."}
+{"_id": "8622", "title": "sites-lemma-almost-cocontinuous-sheafification", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Let $\\mathcal{G}$ be a presheaf on $\\mathcal{D}$ such that $\\mathcal{G}(V)$ is a singleton whenever $V$ is sheaf theoretically empty. Then $(u^p\\mathcal{G})^\\# = u^p(\\mathcal{G}^\\#)$."}
+{"_id": "8623", "title": "sites-lemma-continuous-almost-cocontinuous", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Then $u^s = u^p : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ commutes with pushouts and coequalizers (and more generally finite connected colimits)."}
+{"_id": "8624", "title": "sites-lemma-morphism-of-sites-almost-cocontinuous", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. If $u$ is almost cocontinuous then $f_*$ commutes with pushouts and coequalizers (and more generally finite connected colimits)."}
+{"_id": "8625", "title": "sites-lemma-open-subtopos", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is a subobject of the final object of $\\Sh(\\mathcal{C})$, and \\item the topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ is a subtopos of $\\Sh(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "8626", "title": "sites-lemma-closed-subtopos", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a subsheaf of the final object $*$ of $\\Sh(\\mathcal{C})$. The full subcategory of sheaves $\\mathcal{G}$ such that $\\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{F}$ is an isomorphism is a subtopos of $\\Sh(\\mathcal{C})$."}
+{"_id": "8627", "title": "sites-lemma-closed-immersion", "text": "Let $i : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a closed immersion of topoi. Then $i_*$ is fully faithful, transforms surjections into surjections, commutes with coequalizers, commutes with pushouts, reflects injections, reflects surjections, and reflects bijections."}
+{"_id": "8629", "title": "sites-lemma-compute-global-sections", "text": "Let $\\mathcal{C}$ be a site. Let $a, b : V \\to U$ be objects of $\\mathcal{C}$ such that $$ \\xymatrix{ h_V^\\# \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_U^\\# \\ar[r] & {*} } $$ is a coequalizer in $\\Sh(\\mathcal{C})$. Then $\\Gamma(\\mathcal{C}, \\mathcal{F})$ is the equalizer of $a^*, b^* : \\mathcal{F}(U) \\to \\mathcal{F}(V)$."}
+{"_id": "8630", "title": "sites-lemma-sieves-set", "text": "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item The collection of sieves on $U$ is a set. \\item Inclusion defines a partial ordering on this set. \\item Unions and intersections of sieves are sieves. \\item \\label{item-sieve-generated} Given a family of morphisms $\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ with target $U$ there exists a unique smallest sieve $S$ on $U$ such that each $U_i \\to U$ belongs to $S(U_i)$. \\item The sieve $S = h_U$ is the maximal sieve. \\item The empty subpresheaf is the minimal sieve. \\end{enumerate}"}
+{"_id": "8631", "title": "sites-lemma-pullback-sieve-section", "text": "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$. Let $S$ be a sieve on $U$. If $f : V \\to U$ is in $S$, then $S \\times_U V = h_V$ is maximal."}
+{"_id": "8632", "title": "sites-lemma-topology-basic", "text": "Let $\\mathcal{C}$ be a category. Let $J$ be a topology on $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item Finite intersections of elements of $J(U)$ are in $J(U)$. \\item If $S \\in J(U)$ and $S' \\supset S$, then $S' \\in J(U)$. \\end{enumerate}"}
+{"_id": "8633", "title": "sites-lemma-play-with-topologies", "text": "Let $\\mathcal{C}$ be a category. Let $\\{J_i\\}_{i\\in I}$ be a set of topologies. \\begin{enumerate} \\item The rule $J(U) = \\bigcap J_i(U)$ defines a topology on $\\mathcal{C}$. \\item There is a coarsest topology finer than all of the topologies $J_i$. \\end{enumerate}"}
+{"_id": "8634", "title": "sites-lemma-topology-presheaves-sheaves", "text": "Let $\\mathcal{C}$ be a category. Let $\\{ \\mathcal{F}_i \\}_{i\\in I}$ be a collection of presheaves of sets on $\\mathcal{C}$. For each $U \\in \\Ob(\\mathcal{C})$ denote $J(U)$ the set of sieves $S$ with the following property: For every morphism $V \\to U$, the maps $$ \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_V, \\mathcal{F}_i) \\longrightarrow \\Mor_{\\textit{PSh}(\\mathcal{C})}(S \\times_U V, \\mathcal{F}_i) $$ are bijective for all $i \\in I$. Then $J$ defines a topology on $\\mathcal{C}$. This topology is the finest topology in which all of the $\\mathcal{F}_i$ are sheaves."}
+{"_id": "8635", "title": "sites-lemma-site-gives-topology", "text": "Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$. For every object $U$ of $\\mathcal{C}$, let $J(U)$ denote the set of sieves $S$ on $U$ with the following property: there exists a covering $\\{f_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ so that the sieve $S'$ generated by the $f_i$ (see Definition \\ref{definition-sieve-generated}) is contained in $S$. \\begin{enumerate} \\item This $J$ is a topology on $\\mathcal{C}$. \\item A presheaf $\\mathcal{F}$ is a sheaf for this topology (see Definition \\ref{definition-sheaf-sets-topology}) if and only if it is a sheaf on the site (see Definition \\ref{definition-sheaf-sets}). \\end{enumerate}"}
+{"_id": "8636", "title": "sites-lemma-L-presheaf", "text": "In the situation above. \\begin{enumerate} \\item The assignment $U \\mapsto L\\mathcal{F}(U)$ combined with the restriction mappings defined above is a presheaf. \\item The maps $\\ell$ glue to give a morphism of presheaves $\\ell : \\mathcal{F} \\to L\\mathcal{F}$. \\item The rule $\\mathcal{F} \\mapsto (\\mathcal{F} \\xrightarrow{\\ell} L\\mathcal{F})$ is a functor. \\item If $\\mathcal{F}$ is a subpresheaf of $\\mathcal{G}$, then $L\\mathcal{F}$ is a subpresheaf of $L\\mathcal{G}$. \\item The map $\\ell : \\mathcal{F} \\to L\\mathcal{F}$ has the following property: For every section $s \\in L\\mathcal{F}(U)$ there exists a covering sieve $S$ on $U$ and an element $\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$ such that $\\ell(\\varphi)$ equals the restriction of $s$ to $S$. \\end{enumerate}"}
+{"_id": "8637", "title": "sites-lemma-sieve-sheafification", "text": "Let $\\mathcal{C}$ be a category endowed with a topology $J$. Let $U$ be an object of $\\mathcal{C}$. Let $S$ be a sieve on $U$. The following are equivalent \\begin{enumerate} \\item The sieve $S$ is a covering sieve. \\item The sheafification $S^\\# \\to h_U^\\#$ of the map $S \\to h_U$ is an isomorphism. \\end{enumerate}"}
+{"_id": "8640", "title": "sites-proposition-sheafification-adjoint", "text": "The canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ has the following universal property: For any map $\\mathcal{F} \\to \\mathcal{G}$, where $\\mathcal{G}$ is a sheaf of sets, there is a unique map $\\mathcal{F}^\\# \\to \\mathcal{G}$ such that $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$ equals the given map."}
+{"_id": "8641", "title": "sites-proposition-get-morphism", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be continuous. Assume furthermore the following: \\begin{enumerate} \\item the category $\\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\\mathcal{D}$ , and \\item the category $\\mathcal{C}$ has fibre products and $u$ commutes with them. \\end{enumerate} Then $u$ defines a morphism of sites $\\mathcal{D} \\to \\mathcal{C}$, in other words $u_s$ is exact."}
+{"_id": "8642", "title": "sites-proposition-point-limits", "text": "Let $\\mathcal{C}$ be a site. Assume that finite limits exist in $\\mathcal{C}$. (I.e., $\\mathcal{C}$ has fibre products, and a final object.) A point $p$ of such a site $\\mathcal{C}$ is given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$ such that \\begin{enumerate} \\item $u$ commutes with finite limits, and \\item if $\\{U_i \\to U\\}$ is a covering, then $\\coprod_i u(U_i) \\to u(U)$ is surjective. \\end{enumerate}"}
+{"_id": "8643", "title": "sites-proposition-criterion-points", "text": "\\begin{reference} \\cite[Expos\\'e VI, Appendix by Deligne, Proposition 9.0]{SGA4} \\end{reference} Let $\\mathcal{C}$ be a site. Assume that \\begin{enumerate} \\item finite limits exist in $\\mathcal{C}$, and \\item every covering $\\{U_i \\to U\\}_{i \\in I}$ has a refinement by a finite covering of $\\mathcal{C}$. \\end{enumerate} Then $\\mathcal{C}$ has enough points."}
+{"_id": "8644", "title": "sites-proposition-functoriality-algebraic-structures-topoi", "text": "\\begin{slogan} Morphisms of topoi preserve algebraic structure. \\end{slogan} Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $f = (f^{-1}, f_*)$ be a morphism of topoi from $\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$. The method introduced above gives rise to an adjoint pair of functors $(f^{-1}, f_*)$ on sheaves of algebraic structures compatible with taking the underlying sheaves of sets for the following types of algebraic structures: \\begin{enumerate} \\item pointed sets, \\item abelian groups, \\item groups, \\item monoids, \\item rings, \\item modules over a fixed ring, and \\item lie algebras over a fixed field. \\end{enumerate} Moreover, in each of these cases the results above labeled ($\\alpha$), ($\\beta$), ($\\gamma$), ($\\delta$), ($\\epsilon$), and ($\\zeta$) hold."}
+{"_id": "8724", "title": "examples-defos-lemma-finite-projective-modules-RS", "text": "Example \\ref{example-finite-projective-modules} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_V$ is a deformation category for any finite dimensional vector space $V$ over $k$."}
+{"_id": "8725", "title": "examples-defos-lemma-finite-projective-modules-TI", "text": "In Example \\ref{example-finite-projective-modules} let $V$ be a finite dimensional $k$-vector space. Then $$ T\\Deformationcategory_V = (0) \\quad\\text{and}\\quad \\text{Inf}(\\Deformationcategory_V) = \\text{End}_k(V) $$ are finite dimensional."}
+{"_id": "8726", "title": "examples-defos-lemma-representations-RS", "text": "Example \\ref{example-representations} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category for any finite dimensional representation $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$."}
+{"_id": "8727", "title": "examples-defos-lemma-representations-TI", "text": "In Example \\ref{example-representations} let  $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation. Then $$ T\\Deformationcategory_{V, \\rho_0} = \\Ext^1_{k[\\Gamma]}(V, V) = H^1(\\Gamma, \\text{End}_k(V)) \\quad\\text{and}\\quad \\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V)) $$ Thus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$ is always finite dimensional and $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional if $\\Gamma$ is finitely generated."}
+{"_id": "8728", "title": "examples-defos-lemma-representations-hull", "text": "In Example \\ref{example-representations} assume $\\Gamma$ finitely generated. Let $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation. Assume $\\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad A \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong $$ of isomorphism classes of objects has a hull. If $H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is prorepresentable."}
+{"_id": "8729", "title": "examples-defos-lemma-continuous-representations-RS", "text": "Example \\ref{example-continuous-representations} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category for any finite dimensional continuous representation $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$."}
+{"_id": "8730", "title": "examples-defos-lemma-continuous-representations-TI", "text": "In Example \\ref{example-continuous-representations} let $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional continuous representation. Then $$ T\\Deformationcategory_{V, \\rho_0} = H^1(\\Gamma, \\text{End}_k(V)) \\quad\\text{and}\\quad \\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V)) $$ Thus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$ is always finite dimensional and $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional if $\\Gamma$ is topologically finitely generated."}
+{"_id": "8732", "title": "examples-defos-lemma-graded-algebras-RS", "text": "Example \\ref{example-graded-algebras} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_P$ is a deformation category for any graded $k$-algebra $P$."}
+{"_id": "8733", "title": "examples-defos-lemma-graded-algebras-TI", "text": "In Example \\ref{example-graded-algebras} let $P$ be a graded $k$-algebra. Then $$ T\\Deformationcategory_P \\quad\\text{and}\\quad \\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P) $$ are finite dimensional if $P$ is finitely generated over $k$."}
+{"_id": "8735", "title": "examples-defos-lemma-rings-RS", "text": "Example \\ref{example-rings} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_P$ is a deformation category for any $k$-algebra $P$."}
+{"_id": "8736", "title": "examples-defos-lemma-rings-TI", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Then $$ T\\Deformationcategory_P = \\text{Ext}^1_P(\\NL_{P/k}, P) \\quad\\text{and}\\quad \\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P) $$"}
+{"_id": "8737", "title": "examples-defos-lemma-smooth", "text": "In Example \\ref{example-rings} let $P$ be a smooth $k$-algebra. Then $T\\Deformationcategory_P = (0)$."}
+{"_id": "8738", "title": "examples-defos-lemma-finite-type-rings-TI", "text": "In Lemma \\ref{lemma-rings-TI} if $P$ is a finite type $k$-algebra, then \\begin{enumerate} \\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional if and only if $\\dim(P) = 0$, and \\item $T\\Deformationcategory_P$ is finite dimensional if $\\Spec(P) \\to \\Spec(k)$ is smooth except at a finite number of points. \\end{enumerate}"}
+{"_id": "8740", "title": "examples-defos-lemma-localization", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Let $S \\subset P$ be a multiplicative subset. There is a natural functor $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P} $$ of deformation categories."}
+{"_id": "8741", "title": "examples-defos-lemma-henselization", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Let $J \\subset P$ be an ideal. Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$. There is a natural functor $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h} $$ of deformation categories."}
+{"_id": "8743", "title": "examples-defos-lemma-completion", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Assume $P$ Noetherian and let $J \\subset P$ be an ideal. Denote $P^\\wedge$ the $J$-adic completion. There is a natural functor $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge} $$ of deformation categories."}
+{"_id": "8745", "title": "examples-defos-lemma-schemes-RS", "text": "Example \\ref{example-schemes} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_X$ is a deformation category for any scheme $X$ over $k$."}
+{"_id": "8746", "title": "examples-defos-lemma-schemes-TI", "text": "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Then $$ \\text{Inf}(\\Deformationcategory_X) = \\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) = \\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) = \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) $$ and $$ T\\Deformationcategory_X = \\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) $$"}
+{"_id": "8747", "title": "examples-defos-lemma-proper-schemes-TI", "text": "In Lemma \\ref{lemma-schemes-TI} if $X$ is proper over $k$, then $\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are finite dimensional."}
+{"_id": "8749", "title": "examples-defos-lemma-open", "text": "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Let $U \\subset X$ be an open subscheme. There is a natural functor $$ \\Deformationcategory_X \\longrightarrow \\Deformationcategory_U $$ of deformation categories."}
+{"_id": "8750", "title": "examples-defos-lemma-affine", "text": "In Example \\ref{example-schemes} let $X = \\Spec(P)$ be an affine scheme over $k$. With $\\Deformationcategory_P$ as in Example \\ref{example-rings} there is a natural equivalence $$ \\Deformationcategory_X \\longrightarrow \\Deformationcategory_P $$ of deformation categories."}
+{"_id": "8752", "title": "examples-defos-lemma-glueing", "text": "In Situation \\ref{situation-glueing} there is an equivalence $$ \\Deformationcategory_X = \\Deformationcategory_{P_1} \\times_{\\Deformationcategory_{P_{12}}} \\Deformationcategory_{P_2} $$ of deformation categories, see Examples \\ref{example-schemes} and \\ref{example-rings}."}
+{"_id": "8753", "title": "examples-defos-lemma-schemes-morphisms-RS", "text": "Example \\ref{example-schemes-morphisms} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_{X \\to Y}$ is a deformation category for any morphism of schemes $X \\to Y$ over $k$."}
+{"_id": "8754", "title": "examples-defos-lemma-schemes-morphisms-TI", "text": "In Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes over $k$. There is a canonical exact sequence of $k$-vector spaces $$ \\xymatrix{ 0 \\ar[r] & \\text{Inf}(\\Deformationcategory_{X \\to Y}) \\ar[r] & \\text{Inf}(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] & \\text{Der}_k(\\mathcal{O}_Y, f_*\\mathcal{O}_X) \\ar[lld] \\\\ & T\\Deformationcategory_{X \\to Y} \\ar[r] & T(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] & \\text{Ext}^1_{\\mathcal{O}_X}(Lf^*\\NL_{Y/k}, \\mathcal{O}_X) } $$"}
+{"_id": "8755", "title": "examples-defos-lemma-proper-schemes-morphisms-TI", "text": "In Lemma \\ref{lemma-schemes-morphisms-TI} if $X$ and $Y$ are both proper over $k$, then $\\text{Inf}(\\Deformationcategory_{X \\to Y})$ and $T\\Deformationcategory_{X \\to Y}$ are finite dimensional."}
+{"_id": "8757", "title": "examples-defos-lemma-schemes-morphisms-smooth-to-base", "text": "\\begin{reference} This is discussed in \\cite[Section 5.3]{Ravi-Murphys-Law} and \\cite[Theorem 3.3]{Ran-deformations}. \\end{reference} In Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes over $k$. If $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $R^1f_*\\mathcal{O}_X = 0$, then the morphism of deformation categories $$ \\Deformationcategory_{X \\to Y} \\to \\Deformationcategory_X $$ is an equivalence."}
+{"_id": "8758", "title": "examples-defos-lemma-spaces-RS", "text": "Example \\ref{example-spaces} satisfies the Rim-Schlessinger condition (RS). In particular, $\\Deformationcategory_X$ is a deformation category for any algebraic space $X$ over $k$."}
+{"_id": "8759", "title": "examples-defos-lemma-spaces-TI", "text": "In Example \\ref{example-spaces} let $X$ be an algebraic space over $k$. Then $$ \\text{Inf}(\\Deformationcategory_X) = \\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) = \\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) = \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) $$ and $$ T\\Deformationcategory_X = \\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) $$"}
+{"_id": "8760", "title": "examples-defos-lemma-proper-spaces-TI", "text": "In Lemma \\ref{lemma-spaces-TI} if $X$ is proper over $k$, then $\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are finite dimensional."}
+{"_id": "8762", "title": "examples-defos-lemma-lift-equivalence-module-derived", "text": "Let $A' \\to A$ be a surjection of rings with nilpotent kernel. Let $A' \\to P'$ be a flat ring map. Set $P = P' \\otimes_{A'} A$. Let $M$ be an $A$-flat $P$-module. Then the following are equivalent \\begin{enumerate} \\item there is an $A'$-flat $P'$-module $M'$ with $M' \\otimes_{P'} P = M$, and \\item there is an object $K' \\in D^-(P')$ with $K' \\otimes_{P'}^\\mathbf{L} P = M$. \\end{enumerate}"}
+{"_id": "8763", "title": "examples-defos-lemma-lift-equivalence-module", "text": "Consider a commutative diagram of Noetherian rings $$ \\xymatrix{ A' \\ar[d] \\ar[r] & P' \\ar[d] \\ar[r] & Q' \\ar[d] \\\\ A \\ar[r] & P \\ar[r] & Q } $$ with cartesian squares, with flat horizontal arrows, and with surjective vertial arrows whose kernels are nilpotent. Let $J' \\subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$. Let $M$ be an $A$-flat $P$-module. Assume for all $g \\in J'$ there exists an $A'$-flat $(P')_g$-module lifting $M_g$. Then the following are equivalent \\begin{enumerate} \\item $M$ has an $A'$-flat lift to a $P'$-module, and \\item $M \\otimes_P Q$ has an $A'$-flat lift to a $Q'$-module. \\end{enumerate}"}
+{"_id": "8764", "title": "examples-defos-lemma-lift-equivalence", "text": "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \\to B$ be a finite type flat ring map. Let $\\mathfrak b \\subset B$ be an ideal such that $\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$. Then $B$ has a flat lift to $A'$ if and only if the $\\mathfrak b$-adic completion $B^\\wedge$ has a flat lift to $A'$."}
+{"_id": "8765", "title": "examples-defos-lemma-first-order-completion", "text": "Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $J \\subset B$ be an ideal such that $\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^\\wedge, N^\\wedge) $$ Here $B^\\wedge$ and $N^\\wedge$ are the $J$-adic completions."}
+{"_id": "8766", "title": "examples-defos-lemma-smooth-completion", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Let $J \\subset P$ be an ideal. Denote $P^\\wedge$ the $J$-adic completion. If \\begin{enumerate} \\item $k \\to P$ is of finite type, and \\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$. \\end{enumerate} then the functor between deformation categories of Lemma \\ref{lemma-completion} $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge} $$ is smooth and induces an isomorphism on tangent spaces."}
+{"_id": "8767", "title": "examples-defos-lemma-lift-equivalence-localization", "text": "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \\to B$ be a finite type flat ring map. Let $S \\subset B$ be a multiplicative subset such that if $\\Spec(B) \\to \\Spec(A)$ is not syntomic at $\\mathfrak q$, then $S \\cap \\mathfrak q = \\emptyset$. Then $B$ has a flat lift to $A'$ if and only if $S^{-1}B$ has a flat lift to $A'$."}
+{"_id": "8768", "title": "examples-defos-lemma-first-order-localization", "text": "Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $S \\subset B$ be a multiplicative subset ideal such that if $\\Spec(B) \\to \\Spec(k)$ is not smooth at $\\mathfrak q$ then $S \\cap \\mathfrak q = \\emptyset$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \\text{Exal}_k(B, N) \\to \\text{Exal}_k(S^{-1}B, S^{-1}N) $$"}
+{"_id": "8769", "title": "examples-defos-lemma-smooth-localization", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Let $S \\subset P$ be a multiplicative subset. If \\begin{enumerate} \\item $k \\to P$ is of finite type, and \\item $\\Spec(P) \\to \\Spec(k)$ is smooth at all points of $V(g)$ for all $g \\in S$. \\end{enumerate} then the functor between deformation categories of Lemma \\ref{lemma-localization} $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P} $$ is smooth and induces an isomorphism on tangent spaces."}
+{"_id": "8770", "title": "examples-defos-lemma-lift-equivalence-henselization", "text": "Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel. Let $A \\to B$ be a finite type flat ring map. Let $\\mathfrak b \\subset B$ be an ideal such that $\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$. Let $(B^h, \\mathfrak b^h)$ be the henselization of the pair $(B, \\mathfrak b)$. Then $B$ has a flat lift to $A'$ if and only if $B^h$ has a flat lift to $A'$."}
+{"_id": "8771", "title": "examples-defos-lemma-first-order-henselization", "text": "Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $J \\subset B$ be an ideal such that $\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$. Let $N$ be a finite $B$-module. Then there is a canonical bijection $$ \\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^h, N^h) $$ Here $(B^h, J^h)$ is the henselization of $(B, J)$ and $N^h = N \\otimes_B B^h$."}
+{"_id": "8772", "title": "examples-defos-lemma-smooth-henselization", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Let $J \\subset P$ be an ideal. Denote $(P^h, J^h)$ the henselization of the pair $(P, J)$. If \\begin{enumerate} \\item $k \\to P$ is of finite type, and \\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$, \\end{enumerate} then the functor between deformation categories of Lemma \\ref{lemma-henselization} $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h} $$ is smooth and induces an isomorphism on tangent spaces."}
+{"_id": "8773", "title": "examples-defos-lemma-isolated", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Assume that $k \\to P$ is of finite type and that $\\Spec(P) \\to \\Spec(k)$ is smooth except at the maximal ideals $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ of $P$. Let $P_{\\mathfrak m_i}$, $P_{\\mathfrak m_i}^h$, $P_{\\mathfrak m_i}^\\wedge$ be the local ring, henselization, completion. Then the maps of deformation categories $$ \\Deformationcategory_P \\to \\prod \\Deformationcategory_{P_{\\mathfrak m_i}} \\to \\prod \\Deformationcategory_{P_{\\mathfrak m_i}^h} \\to \\prod \\Deformationcategory_{P_{\\mathfrak m_i}^\\wedge} $$ are smooth and induce isomorphisms on their finite dimensional tangent spaces."}
+{"_id": "8774", "title": "examples-defos-lemma-lci-unobstructed", "text": "In Example \\ref{example-rings} let $P$ be a local complete intersection over $k$ (Algebra, Definition \\ref{algebra-definition-lci-field}). Then $\\Deformationcategory_P$ is unobstructed."}
+{"_id": "8775", "title": "examples-defos-lemma-glueing-smooth", "text": "In Situation \\ref{situation-glueing} if $U_{12} \\to \\Spec(k)$ is smooth, then the morphism $$ \\Deformationcategory_X \\longrightarrow \\Deformationcategory_{U_1} \\times \\Deformationcategory_{U_2} = \\Deformationcategory_{P_1} \\times \\Deformationcategory_{P_2} $$ is smooth. If in addition $U_1$ is a local complete intersection over $k$, then $$ \\Deformationcategory_X \\longrightarrow \\Deformationcategory_{U_2} = \\Deformationcategory_{P_2} $$ is smooth."}
+{"_id": "8776", "title": "examples-defos-lemma-curve-isolated", "text": "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$, \\item $X \\to \\Spec(k)$ is smooth except at the closed points $p_1, \\ldots, p_n \\in X$. \\end{enumerate} Let $\\mathcal{O}_{X, p_1}$, $\\mathcal{O}_{X, p_1}^h$, $\\mathcal{O}_{X, p_1}^\\wedge$ be the local ring, henselization, completion. Consider the maps of deformation categories $$ \\Deformationcategory_X \\longrightarrow \\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}} \\longrightarrow \\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^h} \\longrightarrow \\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^\\wedge} $$ The first arrow is smooth and the second and third arrows are smooth and induce isomorphisms on tangent spaces."}
+{"_id": "8777", "title": "examples-defos-lemma-curve-isolated-lci", "text": "In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$, \\item $X$ is a local complete intersection over $k$, and \\item $X \\to \\Spec(k)$ is smooth except at finitely many points. \\end{enumerate} Then $\\Deformationcategory_X$ is unobstructed."}
+{"_id": "8778", "title": "examples-defos-lemma-criterion-smoothing", "text": "Let $k$ be a field. Set $S = \\Spec(k[[t]])$ and $S_n = \\Spec(k[t]/(t^n))$. Let $Y \\to S$ be a proper, flat morphism of schemes whose special fibre $X$ is Cohen-Macaulay and equidimensional of dimension $d$. Denote $X_n = Y \\times_S S_n$. If for some $n \\geq 1$ the $d$the Fitting ideal of $\\Omega_{X_n/S_n}$ contains $t^{n - 1}$, then the generic fibre of $Y \\to S$ is smooth."}
+{"_id": "8779", "title": "examples-defos-lemma-jouanolou-type-thing", "text": "Let $k$ be a field. Let $1 \\leq c \\leq n$ be integers. Let $f_1, \\ldots, f_c \\in k[x_1, \\ldots x_n]$ be elements. Let $a_{ij}$, $0 \\leq i \\leq n$, $1 \\leq j \\leq c$ be variables. Consider $$ g_j = f_j + a_{0j} + a_{1j}x_1 + \\ldots + a_{nj}x_n \\in k[a_{ij}][x_1, \\ldots, x_n] $$ Denote $Y \\subset \\mathbf{A}^{n + c(n + 1)}_k$ the closed subscheme cut out by $g_1, \\ldots, g_c$. Denote $\\pi : Y \\to \\mathbf{A}^{c(n + 1)}_k$ the projection onto the affine space with variables $a_{ij}$. Then there is a nonempty Zariski open  of $\\mathbf{A}^{c(n + 1)}_k$ over which $\\pi$ is smooth."}
+{"_id": "8780", "title": "examples-defos-lemma-smoothing-affine-lci", "text": "Let $k$ be a field. Let $A$ be a global complete interesection over $k$. There exists a flat finite type ring map $k[[t]] \\to B$ with $B/tB \\cong A$ such that $B[1/t]$ is smooth over $k((t))$."}
+{"_id": "8781", "title": "examples-defos-lemma-smoothing-artinian-lci", "text": "Let $k$ be a field. Let $A$ be a finite dimensional $k$-algebra which is a local complete intersection over $k$. Then there is a finite flat $k[[t]]$-algebra $B$ with $B/tB \\cong A$ and $B[1/t]$ \\'etale over $k((t))$."}
+{"_id": "8782", "title": "examples-defos-lemma-smoothing-at-lci-point", "text": "Let $k$ be a field. Let $A$ be a $k$-algebra. Assume \\begin{enumerate} \\item $A$ is a local ring essentially of finite type over $k$, \\item $A$ is a complete intersection over $k$ (Algebra, Definition \\ref{algebra-definition-lci-local-ring}). \\end{enumerate} Set $d = \\dim(A) + \\text{trdeg}_k(\\kappa)$ where $\\kappa$ is the residue field of $A$. Then there exists an integer $n$ and a flat, essentially of finite type ring map $k[[t]] \\to B$ with $B/tB \\cong A$ such that $t^n$ is in the $d$th Fitting ideal of $\\Omega_{B/k[[t]]}$."}
+{"_id": "8783", "title": "examples-defos-lemma-smoothing-proper-curve-isolated-lci", "text": "Let $X$ be a scheme over a field $k$. Assume \\begin{enumerate} \\item $X$ is proper over $k$, \\item $X$ is a local complete intersection over $k$, \\item $X$ has dimension $\\leq 1$, and \\item $X \\to \\Spec(k)$ is smooth except at finitely many points. \\end{enumerate} Then there exists a flat projective morphism $Y \\to \\Spec(k[[t]])$ whose generic fibre is smooth and whose special fibre is isomorphic to $X$."}
+{"_id": "8785", "title": "sets-theorem-reflection-principle", "text": "Suppose given $\\phi_1(x_1, \\ldots, x_n), \\ldots, \\phi_m(x_1, \\ldots, x_n)$ a {\\bf finite} collection of formulas of set theory. Let $M_0$ be a set. There exists a set $M$ such that $M_0 \\subset M$ and $\\forall x_1, \\ldots, x_n \\in M$, we have $$ \\forall i = 1, \\ldots, m, \\ \\phi_i^{M}(x_1, \\ldots, x_n) \\Leftrightarrow \\forall i = 1, \\ldots, m, \\ \\phi_i(x_1, \\ldots, x_n). $$ In fact we may take $M = V_\\alpha$ for some limit ordinal $\\alpha$."}
+{"_id": "8786", "title": "sets-lemma-axiom-regularity", "text": "Every set is an element of $V_\\alpha$ for some ordinal $\\alpha$."}
+{"_id": "8787", "title": "sets-lemma-map-from-set-lifts", "text": "Suppose that $T = \\colim_{\\alpha < \\beta} T_\\alpha$ is a colimit of sets indexed by ordinals less than a given ordinal $\\beta$. Suppose that $\\varphi : S \\to T$ is a map of sets. Then $\\varphi$ lifts to a map into $T_\\alpha$ for some $\\alpha < \\beta$ provided that $\\beta$ is not a limit of ordinals indexed by $S$, in other words, if $\\beta$ is an ordinal with $\\text{cf}(\\beta) > |S|$."}
+{"_id": "8788", "title": "sets-lemma-bounded-size", "text": "For every cardinal $\\kappa$, there exists a set $A$ such that every element of $A$ is a scheme and such that for every scheme $S$ with $\\text{size}(S) \\leq \\kappa$, there is an element $X \\in A$ such that $X \\cong S$ (isomorphism of schemes)."}
+{"_id": "8789", "title": "sets-lemma-construct-category", "text": "With notations $\\text{size}$, $Bound$ and $\\Sch_\\alpha$ as above. Let $S_0$ be a set of schemes. There exists a limit ordinal $\\alpha$ with the following properties: \\begin{enumerate} \\item \\label{item-inclusion} We have $S_0 \\subset V_\\alpha$; in other words, $S_0 \\subset \\Ob(\\Sch_\\alpha)$. \\item \\label{item-bounded} For any $S \\in \\Ob(\\Sch_\\alpha)$ and any scheme $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$, there exists a scheme $S' \\in \\Ob(\\Sch_\\alpha)$ such that $T \\cong S'$. \\item \\label{item-limit} For any countable\\footnote{Both the set of objects and the morphism sets are countable. In fact you can prove the lemma with $\\aleph_0$ replaced by any cardinal whatsoever in (3) and (4).} diagram category $\\mathcal{I}$ and any functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the limit $\\lim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and only if it exists in $\\Sch$ and moreover, in this case, the natural morphism between them is an isomorphism. \\item \\label{item-colimit} For any countable diagram category $\\mathcal{I}$ and any functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the colimit $\\colim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and only if it exists in $\\Sch$ and moreover, in this case, the natural morphism between them is an isomorphism. \\end{enumerate}"}
+{"_id": "8790", "title": "sets-lemma-bound-affine", "text": "Let $S$ be an affine scheme. Let $R = \\Gamma(S, \\mathcal{O}_S)$. Then the size of $S$ is equal to $\\max\\{ \\aleph_0, |R|\\}$."}
+{"_id": "8791", "title": "sets-lemma-bound-size", "text": "Let $S$ be a scheme. Let $S = \\bigcup_{i \\in I} S_i$ be an open covering. Then $\\text{size}(S) \\leq \\max\\{|I|, \\sup_i\\{\\text{size}(S_i)\\}\\}$."}
+{"_id": "8792", "title": "sets-lemma-bound-size-fibre-product", "text": "Let $f : X \\to S$, $g : Y \\to S$ be morphisms of schemes. Then we have $\\text{size}(X \\times_S Y) \\leq \\max\\{\\text{size}(X), \\text{size}(Y)\\}$."}
+{"_id": "8793", "title": "sets-lemma-bound-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be locally of finite type with $X$ quasi-compact. Then $\\text{size}(X) \\leq \\text{size}(S)$."}
+{"_id": "8794", "title": "sets-lemma-bound-monomorphism", "text": "Let $f : X \\to Y$ be a monomorphism of schemes. If at least one of the following properties holds, then $\\text{size}(X) \\leq \\text{size}(Y)$: \\begin{enumerate} \\item $f$ is quasi-compact, \\item $f$ is locally of finite presentation, \\item add more here as needed. \\end{enumerate} But the bound does not hold for monomorphisms which are locally of finite type."}
+{"_id": "8795", "title": "sets-lemma-what-is-in-it", "text": "Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-construct-category} above. The category $\\Sch_\\alpha$ satisfies the following properties: \\begin{enumerate} \\item If $X, Y, S \\in \\Ob(\\Sch_\\alpha)$, then for any morphisms $f : X \\to S$, $g : Y \\to S$ the fibre product $X \\times_S Y$ in $\\Sch_\\alpha$ exists and is a fibre product in the category of schemes. \\item Given any at most countable collection $S_1, S_2, \\ldots$ of elements of $\\Ob(\\Sch_\\alpha)$, the coproduct $\\coprod_i S_i$ exists in $\\Ob(\\Sch_\\alpha)$ and is a coproduct in the category of schemes. \\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and any open immersion $U \\to S$, there exists a $V \\in \\Ob(\\Sch_\\alpha)$ with $V \\cong U$. \\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and any closed immersion $T \\to S$, there exists a $S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$. \\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and any finite type morphism $T \\to S$, there exists a $S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$. \\item Suppose $S$ is a scheme which has an open covering $S = \\bigcup_{i \\in I} S_i$ such that there exists a $T \\in \\Ob(\\Sch_\\alpha)$ with (a) $\\text{size}(S_i) \\leq \\text{size}(T)^{\\aleph_0}$ for all $i \\in I$, and (b) $|I| \\leq \\text{size}(T)^{\\aleph_0}$. Then $S$ is isomorphic to an object of $\\Sch_\\alpha$. \\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and any morphism $f : T \\to S$ locally of finite type such that $T$ can be covered by at most $\\text{size}(S)^{\\aleph_0}$ open affines, there exists a $S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$. For example this holds if $T$ can be covered by at most $|\\mathbf{R}| = 2^{\\aleph_0} = \\aleph_0^{\\aleph_0}$ open affines. \\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and any monomorphism $T \\to S$ which is either locally of finite presentation or quasi-compact, there exists a $S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$. \\item Suppose that $T \\in \\Ob(\\Sch_\\alpha)$ is affine. Write $R = \\Gamma(T, \\mathcal{O}_T)$. Then any of the following schemes is isomorphic to a scheme in $\\Sch_\\alpha$: \\begin{enumerate} \\item For any ideal $I \\subset R$ with completion $R^* = \\lim_n R/I^n$, the scheme $\\Spec(R^*)$. \\item For any finite type $R$-algebra $R'$, the scheme $\\Spec(R')$. \\item For any localization $S^{-1}R$, the scheme $\\Spec(S^{-1}R)$. \\item For any prime $\\mathfrak p \\subset R$, the scheme $\\Spec(\\overline{\\kappa(\\mathfrak p)})$. \\item For any subring $R' \\subset R$, the scheme $\\Spec(R')$. \\item Any scheme of finite type over a ring of cardinality at most $|R|^{\\aleph_0}$. \\item And so on. \\end{enumerate} \\end{enumerate}"}
+{"_id": "8796", "title": "sets-lemma-bound-by-covering", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume there exists an fpqc covering $\\{g_j : Y_j \\to Y\\}_{j \\in J}$ such that $g_j$ factors through $f$. Then $\\text{size}(Y) \\leq \\text{size}(X)$."}
+{"_id": "8797", "title": "sets-lemma-bound-fppf-covering", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of a scheme. There exists an fppf covering $\\{W_j \\to X\\}_{j \\in J}$ which is a refinement of $\\{X_i \\to X\\}_{i \\in I}$ such that $\\text{size}(\\coprod W_j) \\leq \\text{size}(X)$."}
+{"_id": "8798", "title": "sets-lemma-sets-with-group-action", "text": "With notations $G$, $G\\textit{-Sets}_\\alpha$, $\\text{size}$, and $Bound$ as above. Let $S_0$ be a set of $G$-sets. There exists a limit ordinal $\\alpha$ with the following properties: \\begin{enumerate} \\item We have $S_0 \\cup \\{{}_GG\\} \\subset \\Ob(G\\textit{-Sets}_\\alpha)$. \\item For any $S \\in \\Ob(G\\textit{-Sets}_\\alpha)$ and any $G$-set $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$, there exists a $S' \\in \\Ob(G\\textit{-Sets}_\\alpha)$ that is isomorphic to $T$. \\item For any countable diagram category $\\mathcal{I}$ and any functor $F : \\mathcal{I} \\to G\\textit{-Sets}_\\alpha$, the limit $\\lim_\\mathcal{I} F$ and colimit $\\colim_\\mathcal{I} F$ exist in $G\\textit{-Sets}_\\alpha$ and are the same as in $G\\textit{-Sets}$. \\end{enumerate}"}
+{"_id": "8799", "title": "sets-lemma-what-is-in-it-G-sets", "text": "Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-sets-with-group-action} above. The category $G\\textit{-Sets}_\\alpha$ satisfies the following properties: \\begin{enumerate} \\item The $G$-set ${}_GG$ is an object of $G\\textit{-Sets}_\\alpha$. \\item (Co)Products, fibre products, and pushouts exist in $G\\textit{-Sets}_\\alpha$ and are the same as their counterparts in $G\\textit{-Sets}$. \\item Given an object $U$ of $G\\textit{-Sets}_\\alpha$, any $G$-stable subset $O \\subset U$  is isomorphic to an object of $G\\textit{-Sets}_\\alpha$. \\end{enumerate}"}
+{"_id": "8800", "title": "sets-lemma-coverings-site", "text": "With notations as above. Let $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})$ be a set contained in $\\text{Cov}(\\mathcal{C})$. There exist a cardinal $\\kappa$ and a limit ordinal $\\alpha$ with the following properties: \\begin{enumerate} \\item We have $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$. \\item The set of coverings $\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$ satisfies (1), (2), and (3) of Sites, Definition \\ref{sites-definition-site} (see above). In other words $(\\mathcal{C}, \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha})$ is a site. \\item Every covering in $\\text{Cov}(\\mathcal{C})$ is combinatorially equivalent to a covering in $\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$. \\end{enumerate}"}
+{"_id": "8801", "title": "sets-lemma-abelian-injectives", "text": "Suppose given a big category $\\mathcal{A}$ (see Categories, Remark \\ref{categories-remark-big-categories}). Assume $\\mathcal{A}$ is abelian and has enough injectives. See Homology, Definitions \\ref{homology-definition-abelian-category} and \\ref{homology-definition-enough-injectives}. Then for any given set of objects $\\{A_s\\}_{s\\in S}$ of $\\mathcal{A}$, there is an abelian subcategory $\\mathcal{A}' \\subset \\mathcal{A}$ with the following properties: \\begin{enumerate} \\item $\\Ob(\\mathcal{A}')$ is a set, \\item $\\Ob(\\mathcal{A}')$ contains $A_s$ for each $s \\in S$, \\item $\\mathcal{A}'$ has enough injectives, and \\item an object of $\\mathcal{A}'$ is injective if and only if it is an injective object of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "8802", "title": "sets-proposition-exist-ordinals-large-cofinality", "text": "Let $\\kappa$ be a cardinal. Then there exists an ordinal whose cofinality is bigger than $\\kappa$."}
+{"_id": "8806", "title": "more-etale-lemma-section-support-in-locally-closed-pre", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $\\varphi : U' \\to U$ be a morphism of $X_\\etale$. Let $Z' \\subset U'$ be a closed subscheme such that $Z' \\to U' \\to U$ is a closed immersion with image $Z \\subset U$. Then there is a canonical bijection $$ \\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} = \\{s' \\in \\mathcal{F}(U') \\mid \\text{Supp}(s') \\subset Z'\\} $$ which is given by restriction if $\\varphi^{-1}(Z) = Z'$."}
+{"_id": "8807", "title": "more-etale-lemma-section-support-in-locally-closed", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a locally closed subscheme. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Given $U, U' \\subset X$ open containing $Z$ as a closed subscheme, there is a canonical bijection $$ \\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} = \\{s \\in \\mathcal{F}(U') \\mid \\text{Supp}(s) \\subset Z\\} $$ which is given by restriction if $U' \\subset U$."}
+{"_id": "8808", "title": "more-etale-lemma-f-shriek-separated", "text": "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. The rule $$ Y_\\etale \\longrightarrow \\textit{Ab},\\quad V \\longmapsto \\{s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V) \\mid \\text{Supp}(s) \\subset X_V \\text{ is proper over }V\\} $$ is an abelian subsheaf of $f_*\\mathcal{F}$."}
+{"_id": "8810", "title": "more-etale-lemma-proper-f-shriek", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Then $f_! = f_*$."}
+{"_id": "8811", "title": "more-etale-lemma-compactify-f-shriek-separated", "text": "Let $Y$ be a scheme. Let $j : X \\to \\overline{X}$ be an open immersion of schemes over $Y$ with $\\overline{X}$ proper over $Y$. Denote $f : X \\to Y$ and $\\overline{f} : \\overline{X} \\to Y$ the structure morphisms. For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical isomorphism (see proof) $$ f_!\\mathcal{F} \\longrightarrow \\overline{f}_!j_!\\mathcal{F} $$ As we have $\\overline{f}_! = \\overline{f}_*$ by Lemma \\ref{lemma-proper-f-shriek} we obtain $\\overline{f}_* \\circ j_! = f_!$ as functors $\\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$."}
+{"_id": "8812", "title": "more-etale-lemma-proper-compact-support", "text": "Let $X$ be a proper scheme over a field $k$. Then $H^0_c(X, \\mathcal{F}) = H^0(X, \\mathcal{F})$."}
+{"_id": "8813", "title": "more-etale-lemma-compactify-compact-support", "text": "Let $k$ be a field. Let $j : X \\to \\overline{X}$ be an open immersion of schemes over $k$ with $\\overline{X}$ proper over $k$. For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical isomorphism (see proof) $$ H^0_c(X, \\mathcal{F}) \\longrightarrow H^0_c(\\overline{X}, j_!\\mathcal{F}) = H^0(\\overline{X}, j_!\\mathcal{F}) $$ where we have the equality on the right by Lemma \\ref{lemma-proper-compact-support}."}
+{"_id": "8814", "title": "more-etale-lemma-stalk-f-shriek-separated", "text": "Let $f : X \\to Y$ be a morphism of schemes which is separated and locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then there is a canonical isomorphism $$ (f_!\\mathcal{F})_{\\overline{y}} \\longrightarrow H^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}}) $$ for any geometric point $\\overline{y} : \\Spec(k) \\to Y$."}
+{"_id": "8815", "title": "more-etale-lemma-base-change-f-shriek-separated", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes with $f$ separated and locally of finite type. For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $f'_!(g')^{-1}\\mathcal{F} = g^{-1}f_!\\mathcal{F}$."}
+{"_id": "8816", "title": "more-etale-lemma-f-shriek-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of schemes which are separated and locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then $g_!f_!\\mathcal{F} = (g \\circ f)_!\\mathcal{F}$ as subsheaves of $(g \\circ f)_*\\mathcal{F}$."}
+{"_id": "8817", "title": "more-etale-lemma-colim-f-shriek-separated", "text": "Let $f : X \\to Y$ be morphism of schemes which is separated and locally of finite type. Let $X = \\bigcup_{i \\in I} X_i$ be an open covering such that for all $i, j \\in I$ there exists a $k$ with $X_i \\cup X_j \\subset X_k$. Denote $f_i : X_i \\to Y$ the restriction of $f$. Then $$ f_!\\mathcal{F} = \\colim_{i \\in I} f_{i, !}(\\mathcal{F}|_{X_i}) $$ functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ where the transition maps are the ones constructed in Remark \\ref{remark-covariance-f-shriek-separated}."}
+{"_id": "8819", "title": "more-etale-lemma-lqf-f-shriek-separated-colimits", "text": "Let $f : X \\to Y$ be a morphism of schemes which is separated and locally quasi-finite. Then \\begin{enumerate} \\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have $$ (f_!\\mathcal{F})_{\\overline{y}} = \\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}} $$ functorially in $\\mathcal{F}$, and \\item the functor $f_!$ is exact. \\end{enumerate}"}
+{"_id": "8820", "title": "more-etale-lemma-finite-support-f-shriek-separated", "text": "Let $f : X \\to Y$ be a separated and locally quasi-finite morphism of schemes. Functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical isomorphism(!) $$ f_{p!}\\mathcal{F} \\longrightarrow f_!\\mathcal{F} $$ of abelian presheaves which identifies the sheaf $f_!\\mathcal{F}$ of Definition \\ref{definition-f-shriek-separated} with the presheaf $f_{p!}\\mathcal{F}$ constructed above."}
+{"_id": "8821", "title": "more-etale-lemma-finite-support-stalk", "text": "Let $f : X \\to Y$ be a morphism of schemes which is locally quasi-finite. Let $\\overline{y} : \\Spec(k) \\to Y$ be a geometric point. Functorially in $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ we have $$ (f_{p!}\\mathcal{F})_{\\overline{y}} = \\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}} $$"}
+{"_id": "8822", "title": "more-etale-lemma-finite-support-etale-shriek", "text": "Let $f = j : U \\to X$ be an \\'etale of schemes. Denote $j_{p!}$ the construction of \\'Etale Cohomology, Equation (\\ref{etale-cohomology-equation-j-p-shriek}) and denote $f_{p!}$ the construction above. Functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical map $$ j_{p!}\\mathcal{F} \\longrightarrow f_{p!}\\mathcal{F} $$ of abelian presheaves which identifies the sheaf $j_!\\mathcal{F} = (j_{p!}\\mathcal{F})^\\#$ of \\'Etale Cohomology, Definition \\ref{etale-cohomology-definition-extension-zero} with $(f_{p!}\\mathcal{F})^\\#$."}
+{"_id": "8823", "title": "more-etale-lemma-lqf-f-shriek-stalk", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. Then \\begin{enumerate} \\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have $$ (f_!\\mathcal{F})_{\\overline{y}} = \\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}} $$ functorially in $\\mathcal{F}$, and \\item the functor $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is exact and commutes with direct sums. \\end{enumerate}"}
+{"_id": "8824", "title": "more-etale-lemma-lqf-colimit-f-shriek", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. Let $X = \\bigcup_{i \\in I} X_i$ be an open covering. Then there exists an exact complex $$ \\ldots \\to \\bigoplus\\nolimits_{i_0, i_1, i_2} f_{i_0i_1i_2, !} \\mathcal{F}|_{X_{i_0i_1i_2}} \\to \\bigoplus\\nolimits_{i_0, i_1} f_{i_0i_1, !} \\mathcal{F}|_{X_{i_0i_1}} \\to \\bigoplus\\nolimits_{i_0} f_{i_0, !} \\mathcal{F}|_{X_{i_0}} \\to f_!\\mathcal{F} \\to 0 $$ functorial in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$, see proof for details."}
+{"_id": "8825", "title": "more-etale-lemma-lqf-base-change-f-shriek", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes with $f$ locally quasi-finite. For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $f'_!(g')^{-1}\\mathcal{F} = g^{-1}f_!\\mathcal{F}$."}
+{"_id": "8826", "title": "more-etale-lemma-lqf-separated-shriek-composition", "text": "Let $f' : X \\to Y'$ and $g : Y' \\to Y$ be composable morphisms of schemes with $f'$ and $f = g \\circ f'$ locally quasi-finite and $g$ separated and locally of finite type. Then there is a canonical isomorphism of functors $g_! \\circ f'_! = f_!$. This isomorphism is compatible with \\begin{enumerate} \\item[(a)] covariance with respect to open embeddings as in Remarks \\ref{remark-covariance-f-shriek-separated} and \\ref{remark-covariance-lqf-f-shriek}, \\item[(b)] the base change isomorphisms of Lemmas \\ref{lemma-lqf-base-change-f-shriek} and \\ref{lemma-base-change-f-shriek-separated}, and \\item[(c)] equal to the isomorphism of Lemma \\ref{lemma-f-shriek-composition} via the identifications of Lemma \\ref{lemma-finite-support-f-shriek-separated} in case $f'$ is separated. \\end{enumerate}"}
+{"_id": "8827", "title": "more-etale-lemma-lqf-shriek-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable locally quasi-finite morphisms of schemes. Then there is a canonical isomorphism of functors $$ (g \\circ f)_! \\longrightarrow g_! \\circ f_! $$ These isomorphisms satisfy the following properties: \\begin{enumerate} \\item If $f$ and $g$ are separated, then the isomorphism agrees with Lemma \\ref{lemma-f-shriek-composition}. \\item If $g$ is separated, then the isomorphism agrees with Lemma \\ref{lemma-lqf-separated-shriek-composition}. \\item For a geometric point $\\overline{z} : \\Spec(k) \\to Z$ the diagram $$ \\xymatrix{ ((g \\circ f)_!\\mathcal{F})_{\\overline{z}} \\ar[d] \\ar[rr] & & \\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}} \\mathcal{F}_{\\overline{x}} \\ar@{=}[d] \\\\ (g_!f_!\\mathcal{F})_{\\overline{z}} \\ar[r] & \\bigoplus\\nolimits_{g(\\overline{y}) = \\overline{z}} (f_!\\mathcal{F})_{\\overline{y}} \\ar[r] & \\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}} \\mathcal{F}_{\\overline{x}} } $$ is commutative where the horizontal arrows are given by Lemma \\ref{lemma-lqf-f-shriek-stalk}. \\item Let $h : Z \\to T$ be a third locally quasi-finite morphism of schemes. Then the diagram $$ \\xymatrix{ (h \\circ g \\circ f)_! \\ar[r] \\ar[d] & (h \\circ g)_! \\circ f_! \\ar[d] \\\\ h_! \\circ (g \\circ f)_! \\ar[r] & h_! \\circ g_! \\circ f_! } $$ commutes. \\item Suppose that we have a diagram of schemes $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_c & X \\ar[d]^f \\\\ Y' \\ar[d]_{g'} \\ar[r]_b & Y \\ar[d]^g \\\\ Z' \\ar[r]^a & Z } $$ with both squares cartesian and $f$ and $g$ locally quasi-finite. Then the diagram $$ \\xymatrix{ a^{-1} \\circ (g \\circ f)_! \\ar[d] \\ar[rr] & & (g' \\circ f')_! \\circ c^{-1} \\ar[d] \\\\ a^{-1} \\circ g_! \\circ f_! \\ar[r] & g'_! \\circ b^{-1} \\circ f_! \\ar[r] & g'_! \\circ f'_! \\circ c^{-1} } $$ commutes where the horizontal arrows are those of Lemma \\ref{lemma-lqf-base-change-f-shriek}. \\end{enumerate}"}
+{"_id": "8828", "title": "more-etale-lemma-lqf-f-upper-shriek", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. The functor $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ has a right adjoint $f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$. Moreover, we have $f^!(\\overline{y}_*A) = \\prod_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*A$."}
+{"_id": "8829", "title": "more-etale-lemma-etale-upper-shriek", "text": "Let $j : U \\to X$ be an \\'etale morphism. Then $j^! = j^{-1}$."}
+{"_id": "8830", "title": "more-etale-lemma-upper-shriek-restriction", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be separated and locally quasi-finite morphisms. There is a canonical isomorphism $(g \\circ f)^! \\to f^! \\circ g^!$. Given a third locally quasi-finite morphism $h : Z \\to T$ the diagram $$ \\xymatrix{ (h \\circ g \\circ f)^! \\ar[r] \\ar[d] & f^! \\circ (h \\circ g)^! \\ar[d] \\\\ (g \\circ f)^! \\circ h^! \\ar[r] & f^! \\circ g^! \\circ h^! } $$ commutes."}
+{"_id": "8831", "title": "more-etale-lemma-upper-shriek-restriction-etale", "text": "Let $j : U \\to X$ and $j' : V \\to U$ be \\'etale morphisms. The isomorphism $(j \\circ j')^{-1} = (j')^{-1} \\circ j^{-1}$ and the isomorphism $(j \\circ j')^! = (j')^! \\circ j^!$ of Lemma \\ref{lemma-upper-shriek-restriction} agree via the isomorphism of Lemma \\ref{lemma-etale-upper-shriek}."}
+{"_id": "8834", "title": "more-etale-lemma-shriek-proper-and-open", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ with $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite. For a torsion ring $\\Lambda$ and $K$ in $D(X'_\\etale, \\Lambda)$ there is a canonical isomorphism $g_!Rf'_*K \\to Rf_*(g'_!K)$ in $D(Y_\\etale, \\Lambda)$."}
+{"_id": "8835", "title": "more-etale-lemma-shriek-proper-and-open-compose", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]_l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z' \\ar[r]^m & Z } $$ with $f$, $f'$, $g$ and $g'$ proper and $k$, $l$, and $m$ separated and locally quasi-finite. Then the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open} for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement)."}
+{"_id": "8836", "title": "more-etale-lemma-shriek-proper-and-open-compose-horizontal", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y } $$ with $f$, $f'$, and $f''$ proper and $g$, $g'$, $h$, and $h'$ separated and locally quasi-finite. Then the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open} for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement)."}
+{"_id": "8837", "title": "more-etale-lemma-shriek-proper-and-open-base-change", "text": "Let $b : Y_1 \\to Y$ be a morphism of schemes. Consider a commutative diagram of schemes $$ \\vcenter{ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } } \\quad\\text{and let}\\quad \\vcenter{ \\xymatrix{ X'_1 \\ar[r]_{g'_1} \\ar[d]_{f'_1} & X_1 \\ar[d]^{f_1} \\\\ Y'_1 \\ar[r]^{g_1} & Y_1 } } $$ be the base change by $b$. Assume $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite. For a torsion ring $\\Lambda$ and $K$ in $D(X'_\\etale, \\Lambda)$ there is commutative diagram $$ \\xymatrix{ b^{-1}g_!Rf'_*K \\ar[d] \\ar[r] & g_{1, !}(b')^{-1}Rf'_*K \\ar[r] & g_{1, !}Rf'_{1, *}(a')^{-1}K \\ar[d] \\\\ b^{-1}Rf_*g'_!K \\ar[r] & Rf_{1, *}a^{-1}g'_!K \\ar[r] & Rf_{1, *}g'_{1, !}(a')^{-1}K } $$ in $D(Y_{1, \\etale}, \\Lambda)$ where $a : X_1 \\to X$, $a' : X'_1 \\to X'$, $b' : Y'_1 \\to Y'$ are the projections, the vertical maps are the arrows of Lemma \\ref{lemma-shriek-proper-and-open} and the horizontal arrows are the base change map (from \\'Etale Cohomology, Section \\ref{etale-cohomology-section-base-change-preliminaries}) and the base change map of Lemma \\ref{lemma-base-change-f-shriek-separated}."}
+{"_id": "8838", "title": "more-etale-lemma-shriek-lqf-and-proper", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X \\ar[r]_f \\ar[rd]_g & Y \\ar[d]^h \\\\ & Z } $$ with $f$ and $g$ locally quasi-finite and $h$ proper. For any torsion ring $\\Lambda$ and $K$ in $D(X_\\etale, \\Lambda)$ there is a canonical isomorphism $g_!K \\to Rh_*(f_!K)$ in $D(Z_\\etale, \\Lambda)$."}
+{"_id": "8839", "title": "more-etale-lemma-shriek-well-defined", "text": "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ is, up to canonical isomorphism, independent of the choice of the compactification."}
+{"_id": "8840", "title": "more-etale-lemma-shriek-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then there is a canonical isomorphism $Rg_! \\circ Rf_! \\to R(g \\circ f)_!$."}
+{"_id": "8843", "title": "more-etale-lemma-derived-lower-shriek-commute-direct-sums", "text": "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ commutes with direct sums."}
+{"_id": "8857", "title": "stacks-properties-lemma-check-representable-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective, locally of finite presentation, and flat. The following are equivalent \\begin{enumerate} \\item $f$ is representable by algebraic spaces, and \\item $W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space. \\end{enumerate}"}
+{"_id": "8858", "title": "stacks-properties-lemma-property-spaces-too", "text": "Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. The following are equivalent: \\begin{enumerate} \\item $f$ has $P$, \\item for every algebraic space $Z$ and morphism $Z \\to \\mathcal{Y}$ the morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$ has $P$. \\end{enumerate}"}
+{"_id": "8859", "title": "stacks-properties-lemma-check-property-covering", "text": "Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective, locally of finite presentation, and flat. Set $V = W \\times_\\mathcal{Y} \\mathcal{X}$. Then $$ (f\\text{ has }P) \\Leftrightarrow (\\text{the projection }V \\to W\\text{ has }P). $$"}
+{"_id": "8860", "title": "stacks-properties-lemma-check-property-weak-covering", "text": "Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Set $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then $$ (f\\text{ has }P) \\Leftrightarrow (\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P). $$"}
+{"_id": "8861", "title": "stacks-properties-lemma-check-property-after-precomposing", "text": "Let $P$ be a property of morphisms of algebraic spaces as above. Let $\\tau \\in \\{\\etale, smooth, syntomic, fppf\\}$. Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks representable by algebraic spaces. Assume \\begin{enumerate} \\item $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and \\'etale, smooth, syntomic, or flat and locally of finite presentation, \\item the composition has $P$, and \\item $P$ is local on the source in the $\\tau$ topology. \\end{enumerate} Then $\\mathcal{Y} \\to \\mathcal{Z}$ has property $P$."}
+{"_id": "8862", "title": "stacks-properties-lemma-representable-in-terms-presentations", "text": "Let $g : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Let $[U/R] \\to \\mathcal{X}$ be a presentation. Set $U' = U \\times_\\mathcal{X} \\mathcal{X}'$, and $R' = R \\times_\\mathcal{X} \\mathcal{X}'$. Then there exists a groupoid in algebraic spaces of the form $(U', R', s', t', c')$, a presentation $[U'/R'] \\to \\mathcal{X}'$, and the diagram $$ \\xymatrix{ [U'/R'] \\ar[d]_{[\\text{pr}]} \\ar[r] & \\mathcal{X}' \\ar[d]^g \\\\ [U/R] \\ar[r] & \\mathcal{X} } $$ is $2$-commutative where the morphism $[\\text{pr}]$ comes from a morphism of groupoids $\\text{pr} : (U', R', s', t', c') \\to (U, R, s, t, c)$."}
+{"_id": "8863", "title": "stacks-properties-lemma-equivalence", "text": "The notion above does indeed define an equivalence relation on morphisms from spectra of fields into the algebraic stack $\\mathcal{X}$."}
+{"_id": "8864", "title": "stacks-properties-lemma-points-cartesian", "text": "Let $$ \\xymatrix{ \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\mathcal{Z} \\ar[r] & \\mathcal{Y} } $$ be a fibre product of algebraic stacks. Then the map of sets of points $$ |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\longrightarrow |\\mathcal{Z}| \\times_{|\\mathcal{Y}|} |\\mathcal{X}| $$ is surjective."}
+{"_id": "8865", "title": "stacks-properties-lemma-characterize-surjective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent: \\begin{enumerate} \\item $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is surjective, and \\item $f$ is surjective (in the sense of Section \\ref{section-properties-morphisms}). \\end{enumerate}"}
+{"_id": "8866", "title": "stacks-properties-lemma-points-presentation", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{X} = [U/R]$ be a presentation of $\\mathcal{X}$, see Algebraic Stacks, Definition \\ref{algebraic-definition-presentation}. Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation and $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation."}
+{"_id": "8867", "title": "stacks-properties-lemma-topology-points", "text": "There exists a unique topology on the sets of points of algebraic stacks with the following properties: \\begin{enumerate} \\item for every morphism of algebraic stacks $\\mathcal{X} \\to \\mathcal{Y}$ the map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is continuous, and \\item for every morphism $U \\to \\mathcal{X}$ which is flat and locally of finite presentation with $U$ an algebraic space the map of topological spaces $|U| \\to |\\mathcal{X}|$ is continuous and open. \\end{enumerate}"}
+{"_id": "8868", "title": "stacks-properties-lemma-space-locally-quasi-compact", "text": "Let $\\mathcal{X}$ be an algebraic stack. Every point of $|\\mathcal{X}|$ has a fundamental system of quasi-compact open neighbourhoods. In particular $|\\mathcal{X}|$ is locally quasi-compact in the sense of Topology, Definition \\ref{topology-definition-locally-quasi-compact}."}
+{"_id": "8869", "title": "stacks-properties-lemma-composition-surjective", "text": "The composition of surjective morphisms is surjective."}
+{"_id": "8870", "title": "stacks-properties-lemma-base-change-surjective", "text": "The base change of a surjective morphism is surjective."}
+{"_id": "8871", "title": "stacks-properties-lemma-descent-surjective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Y}' \\to \\mathcal{Y}$ be a surjective morphism of algebraic stacks. If the base change $f' : \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective."}
+{"_id": "8872", "title": "stacks-properties-lemma-surjective-permanence", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is surjective so is $\\mathcal{Y} \\to \\mathcal{Z}$."}
+{"_id": "8873", "title": "stacks-properties-lemma-quasi-compact-stack", "text": "Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{X}$ is quasi-compact, \\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$ with $U$ a quasi-compact scheme, \\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$ with $U$ a quasi-compact algebraic space, and \\item there exists a surjective morphism $\\mathcal{U} \\to \\mathcal{X}$ of algebraic stacks such that $\\mathcal{U}$ is quasi-compact. \\end{enumerate}"}
+{"_id": "8874", "title": "stacks-properties-lemma-finite-disjoint-quasi-compact", "text": "A finite disjoint union of quasi-compact algebraic stacks is a quasi-compact algebraic stack."}
+{"_id": "8875", "title": "stacks-properties-lemma-type-property", "text": "Let $\\mathcal{P}$ be a property of schemes which is local in the smooth topology, see Descent, Definition \\ref{descent-definition-property-local}. Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item for some scheme $U$ and some surjective smooth morphism $U \\to \\mathcal{X}$ the scheme $U$ has property $\\mathcal{P}$, \\item for every scheme $U$ and every smooth morphism $U \\to \\mathcal{X}$ the scheme $U$ has property $\\mathcal{P}$, \\item for some algebraic space $U$ and some surjective smooth morphism $U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$, and \\item for every algebraic space $U$ and every smooth morphism $U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$. \\end{enumerate} If $\\mathcal{X}$ is a scheme this is equivalent to $\\mathcal{P}(U)$. If $\\mathcal{X}$ is an algebraic space this is equivalent to $X$ having property $\\mathcal{P}$."}
+{"_id": "8876", "title": "stacks-properties-lemma-local-source-target-at-point", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$. Let $\\mathcal{P}$ be a property of germs of schemes which is smooth local, see Descent, Definition \\ref{descent-definition-local-at-point}. The following are equivalent \\begin{enumerate} \\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme and $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$, \\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme and some $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$, \\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ an algebraic space and $u \\in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\\mathcal{P}$ at $u$, and \\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ a an algebraic space and some $u \\in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\\mathcal{P}$ at $u$. \\end{enumerate} If $\\mathcal{X}$ is representable, then this is equivalent to $\\mathcal{P}(\\mathcal{X}, x)$. If $\\mathcal{X}$ is an algebraic space then this is equivalent to $\\mathcal{X}$ having property $\\mathcal{P}$ at $x$."}
+{"_id": "8877", "title": "stacks-properties-lemma-base-change-monomorphism", "text": "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a monomorphism. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$ is a monomorphism."}
+{"_id": "8879", "title": "stacks-properties-lemma-monomorphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \\begin{enumerate} \\item $f$ is a monomorphism, \\item $f$ is fully faithful, \\item the diagonal $\\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is an equivalence, and \\item there exists an algebraic space $W$ and a surjective, flat morphism $W \\to \\mathcal{Y}$ which is locally of finite presentation such that $V = \\mathcal{X} \\times_\\mathcal{Y} W$ is an algebraic space, and the morphism $V \\to W$ is a monomorphism of algebraic spaces. \\end{enumerate}"}
+{"_id": "8880", "title": "stacks-properties-lemma-monomorphism-injective-points", "text": "\\begin{slogan} Monomorphisms of stacks are injective on points. \\end{slogan} A monomorphism of algebraic stacks induces an injective map of sets of points."}
+{"_id": "8881", "title": "stacks-properties-lemma-monomorphism-diagonal", "text": "Let $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{X}'$ is a monomorphism then the canonical diagram $$ \\xymatrix{ \\mathcal{X} \\ar[r] \\ar[d] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\ \\mathcal{X}' \\ar[r] & \\mathcal{X}' \\times_\\mathcal{Y} \\mathcal{X}' } $$ is a fibre product square."}
+{"_id": "8882", "title": "stacks-properties-lemma-base-change-immersion", "text": "Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $\\mathcal{Z} \\to \\mathcal{Y}$ be a (closed, resp.\\ open) immersion. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$ is a (closed, resp.\\ open) immersion."}
+{"_id": "8883", "title": "stacks-properties-lemma-composition-immersion", "text": "Compositions of immersions of algebraic stacks are immersions. Similarly for closed immersions and open immersions."}
+{"_id": "8884", "title": "stacks-properties-lemma-check-immersion-covering", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. let $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be a surjective, flat morphism which is locally of finite presentation. The following are equivalent: \\begin{enumerate} \\item $f$ is an (open, resp.\\ closed) immersion, and \\item $V = W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space, and $V \\to W$ is an (open, resp.\\ closed) immersion. \\end{enumerate}"}
+{"_id": "8885", "title": "stacks-properties-lemma-immersion-monomorphism", "text": "An immersion is a monomorphism."}
+{"_id": "8886", "title": "stacks-properties-lemma-immersion-into-presentation", "text": "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $i : \\mathcal{Z} \\to [U/R]$ be an immersion. Then there exists an $R$-invariant locally closed subspace $Z \\subset U$ and a presentation $[Z/R_Z] \\to \\mathcal{Z}$ where $R_Z$ is the restriction of $R$ to $Z$ such that $$ \\xymatrix{ [Z/R_Z] \\ar[dr] \\ar[rr] & & \\mathcal{Z} \\ar[ld]^i \\\\ & [U/R] } $$ is $2$-commutative. If $i$ is a closed (resp.\\ open) immersion then $Z$ is a closed (resp.\\ open) subspace of $U$."}
+{"_id": "8887", "title": "stacks-properties-lemma-immersion-presentation", "text": "Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $\\mathcal{X} = [U/R]$ be the associated algebraic stack, see Algebraic Stacks, Theorem \\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}. Let $Z \\subset U$ be an $R$-invariant locally closed subspace. Then $$ [Z/R_Z] \\longrightarrow [U/R] $$ is an immersion of algebraic stacks, where $R_Z$ is the restriction of $R$ to $Z$. If $Z \\subset U$ is open (resp.\\ closed) then the morphism is an open (resp.\\ closed) immersion of algebraic stacks."}
+{"_id": "8888", "title": "stacks-properties-lemma-substack-image", "text": "For any immersion $i : \\mathcal{Z} \\to \\mathcal{X}$ there exists a unique locally closed substack $\\mathcal{X}' \\subset \\mathcal{X}$ such that $i$ factors as the composition of an equivalence $i' : \\mathcal{Z} \\to \\mathcal{X}'$ followed by the inclusion morphism $\\mathcal{X}' \\to \\mathcal{X}$. If $i$ is a closed (resp.\\ open) immersion, then $\\mathcal{X}'$ is a closed (resp.\\ open) substack of $\\mathcal{X}$."}
+{"_id": "8889", "title": "stacks-properties-lemma-substacks-presentation", "text": "Let $[U/R] \\to \\mathcal{X}$ be a presentation of an algebraic stack. There is a canonical bijection $$ \\text{locally closed substacks }\\mathcal{Z}\\text{ of }\\mathcal{X} \\longrightarrow R\\text{-invariant locally closed subspaces }Z\\text{ of }U $$ which sends $\\mathcal{Z}$ to $U \\times_\\mathcal{X} \\mathcal{Z}$. Moreover, a morphism of algebraic stacks $f : \\mathcal{Y} \\to \\mathcal{X}$ factors through $\\mathcal{Z}$ if and only if $\\mathcal{Y} \\times_\\mathcal{X} U \\to U$ factors through $Z$. Similarly for closed substacks and open substacks."}
+{"_id": "8890", "title": "stacks-properties-lemma-open-substacks", "text": "Let $\\mathcal{X}$ be an algebraic stack. The rule $\\mathcal{U} \\mapsto |\\mathcal{U}|$ defines an inclusion preserving bijection between open substacks of $\\mathcal{X}$ and open subsets of $|\\mathcal{X}|$."}
+{"_id": "8892", "title": "stacks-properties-lemma-union-open-substacks", "text": "Let $\\mathcal X$ be an algebraic stack and $\\mathcal{X}_i \\subset \\mathcal X$ a collection of open substacks indexed by $i \\in I$. Then there exists an open substack, which we denote $\\bigcup_{i\\in I} \\mathcal{X}_i \\subset \\mathcal X$, such that the $\\mathcal{X}_i$ are open substacks covering it."}
+{"_id": "8894", "title": "stacks-properties-lemma-zariski-open-cover-stack-is-space", "text": "Let $\\mathcal X$ be an algebraic stack. Let $\\mathcal{X}_i$, $i \\in I$ be a set of open substacks of $\\mathcal{X}$. Assume \\begin{enumerate} \\item $\\mathcal{X} = \\bigcup_{i \\in I} \\mathcal{X}_i$, and \\item each $\\mathcal{X}_i$ is an algebraic space. \\end{enumerate} Then $\\mathcal{X}$ is an algebraic space."}
+{"_id": "8896", "title": "stacks-properties-lemma-local-source", "text": "Let $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms of algebraic spaces. Assume \\begin{enumerate} \\item $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ are fppf local on the target and stable under arbitrary base change, \\item $\\text{smooth} \\Rightarrow \\mathcal{R}$, \\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a largest open subspace $W(\\mathcal{P}, f) \\subset X$ such that $f|_{W(\\mathcal{P}, f)}$ has $\\mathcal{P}$, and \\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$, and any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have $Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where $f' : X_{Y'} \\to Y'$ is the base change of $f$. \\end{enumerate} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Assume $f$ has $\\mathcal{Q}$. Then \\begin{enumerate} \\item[(A)] there exists a largest open substack $\\mathcal{X}' \\subset \\mathcal{X}$ such that $f|_{\\mathcal{X}'}$ has $\\mathcal{P}$, and \\item[(B)] if $\\mathcal{Z} \\to \\mathcal{Y}$ is a morphism of algebraic stacks representable by algebraic spaces which has $\\mathcal{R}$ then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}'$ is the largest open substack of $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ over which the base change $\\text{id}_\\mathcal{Z} \\times f$ has property $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "8897", "title": "stacks-properties-lemma-reduced-closed-substack", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $T \\subset |\\mathcal{X}|$ be a closed subset. There exists a unique closed substack $\\mathcal{Z} \\subset \\mathcal{X}$ with the following properties: (a) we have $|\\mathcal{Z}| = T$, and (b) $\\mathcal{Z}$ is reduced."}
+{"_id": "8898", "title": "stacks-properties-lemma-reduced-stack-determined-by-points", "text": "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}' \\subset \\mathcal{X}$ is a closed substack, $\\mathcal{X}$ is reduced and $|\\mathcal{X}'| = |\\mathcal{X}|$, then $\\mathcal{X}' = \\mathcal{X}$."}
+{"_id": "8900", "title": "stacks-properties-lemma-flat-cover-by-field", "text": "Let $\\mathcal{Z}$ be an algebraic stack. Let $k$ be a field and let $\\Spec(k) \\to \\mathcal{Z}$ be surjective and flat. Then any morphism $\\Spec(k') \\to \\mathcal{Z}$ where $k'$ is a field is surjective and flat."}
+{"_id": "8901", "title": "stacks-properties-lemma-unique-point", "text": "Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{Z}$ is reduced and $|\\mathcal{Z}|$ is a singleton, \\item there exists a surjective flat morphism $\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field, and \\item there exists a locally of finite type, surjective, flat morphism $\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "8902", "title": "stacks-properties-lemma-unique-point-better", "text": "Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent \\begin{enumerate} \\item $\\mathcal{Z}$ is reduced, locally Noetherian, and $|\\mathcal{Z}|$ is a singleton, and \\item there exists a locally finitely presented, surjective, flat morphism $\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "8903", "title": "stacks-properties-lemma-monomorphism-into-point", "text": "Let $\\mathcal{Z}' \\to \\mathcal{Z}$ be a monomorphism of algebraic stacks. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\\Spec(k) \\to \\mathcal{Z}$. Then either $\\mathcal{Z}'$ is empty or $\\mathcal{Z}' \\to \\mathcal{Z}$ is an equivalence."}
+{"_id": "8904", "title": "stacks-properties-lemma-improve-unique-point", "text": "Let $\\mathcal{Z}$ be an algebraic stack. Assume $\\mathcal{Z}$ satisfies the equivalent conditions of Lemma \\ref{lemma-unique-point}. Then there exists a unique strictly full subcategory $\\mathcal{Z}' \\subset \\mathcal{Z}$ such that $\\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent conditions of Lemma \\ref{lemma-unique-point-better}. The inclusion morphism $\\mathcal{Z}' \\to \\mathcal{Z}$ is a monomorphism of algebraic stacks."}
+{"_id": "8905", "title": "stacks-properties-lemma-residual-gerbe", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point. The following are equivalent \\begin{enumerate} \\item there exists an algebraic stack $\\mathcal{Z}$ and a monomorphism $\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton and such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$, \\item there exists a reduced algebraic stack $\\mathcal{Z}$ and a monomorphism $\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton and such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$, \\item there exists an algebraic stack $\\mathcal{Z}$, a monomorphism $f : \\mathcal{Z} \\to \\mathcal{X}$, and a surjective flat morphism $z : \\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field such that $x = f(z)$. \\end{enumerate} Moreover, if these conditions hold, then there exists a unique strictly full subcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$ such that $\\mathcal{Z}_x$ is a reduced, locally Noetherian algebraic stack and $|\\mathcal{Z}_x|$ is a singleton which maps to $x$ via the map $|\\mathcal{Z}_x| \\to |\\mathcal{X}|$."}
+{"_id": "8907", "title": "stacks-properties-lemma-residual-gerbe-points", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. Assume that the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ exists. Let $f : \\Spec(K) \\to \\mathcal{X}$ be a morphism where $K$ is a field in the equivalence class of $x$. Then $f$ factors through the inclusion morphism $\\mathcal{Z}_x \\to \\mathcal{X}$."}
+{"_id": "8908", "title": "stacks-properties-lemma-residual-gerbe-unique", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. Let $\\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of Lemma \\ref{lemma-unique-point-better} and let $\\mathcal{Z} \\to \\mathcal{X}$ be a monomorphism such that the image of $|\\mathcal{Z}| \\to |\\mathcal{X}|$ is $x$. Then the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists and $\\mathcal{Z} \\to \\mathcal{X}$ factors as $\\mathcal{Z} \\to \\mathcal{Z}_x \\to \\mathcal{X}$ where the first arrow is an equivalence."}
+{"_id": "8909", "title": "stacks-properties-lemma-residual-gerbe-functorial", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$. If the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$ and $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist, then $f$ induces a commutative diagram $$ \\xymatrix{ \\mathcal{X} \\ar[d]_f & \\mathcal{Z}_x \\ar[l] \\ar[d] \\\\ \\mathcal{Y} & \\mathcal{Z}_y \\ar[l] } $$"}
+{"_id": "8910", "title": "stacks-properties-lemma-residual-gerbe-isomorphic", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$. Assume the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$ and $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist and that there exists a morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence class of $x$ such that $$ \\Spec(k) \\times_\\mathcal{X} \\Spec(k) \\longrightarrow \\Spec(k) \\times_\\mathcal{Y} \\Spec(k) $$ is an isomorphism. Then $\\mathcal{Z}_x \\to \\mathcal{Z}_y$ is an isomorphism."}
+{"_id": "8911", "title": "stacks-properties-lemma-dimension-at-point-well-defined", "text": "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$. Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$. Let $[U/R] \\to \\mathcal{X}$ be a presentation (Algebraic Stacks, Definition \\ref{algebraic-definition-presentation}) where $U$ is a scheme. Let $u \\in U$ be a point that maps to $x$. Let $e : U \\to R$ be the ``identity'' map and let $s : R \\to U$ be the ``source'' map, which is a smooth morphism of algebraic spaces. Let $R_u$ be the fiber of $s : R \\to U$ over $u$. The element $$ \\dim_x(\\mathcal{X}) = \\dim_u(U) - \\dim_{e(u)}(R_u) \\in \\mathbf{Z} \\cup \\infty $$ is independent of the choice of presentation and the point $u$ over $x$."}
+{"_id": "8912", "title": "stacks-properties-lemma-UR-quasi-compact-above-x", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point. The following are equivalent \\begin{enumerate} \\item some morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence class of $x$ is quasi-compact, and \\item any morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence class of $x$ is quasi-compact. \\end{enumerate}"}
+{"_id": "8934", "title": "stacks-lemma-painful", "text": "This actually does give a presheaf."}
+{"_id": "8935", "title": "stacks-lemma-presheaf-mor-map-fibred-categories", "text": "Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism of fibred categories over the category $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$ and $x, y\\in \\Ob((\\mathcal{S}_1)_U)$. Then $F$ defines a canonical morphism of presheaves $$ \\mathit{Mor}_{\\mathcal{S}_1}(x, y) \\longrightarrow \\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y)) $$ on $\\mathcal{C}/U$."}
+{"_id": "8936", "title": "stacks-lemma-isom-as-2-fibre-product", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category, see Categories, Section \\ref{categories-section-fibred-categories}. Let $U \\in \\Ob(\\mathcal{C})$ and let $x, y \\in \\Ob(\\mathcal{S}_U)$. Denote $x, y : \\mathcal{C}/U \\to \\mathcal{S}$ also the corresponding $1$-morphisms, see Categories, Lemma \\ref{categories-lemma-yoneda-2category}. Then \\begin{enumerate} \\item the $2$-fibre product $\\mathcal{S} \\times_{\\mathcal{S} \\times \\mathcal{S}, (x, y)} \\mathcal{C}/U$ is fibred in setoids over $\\mathcal{C}/U$, and \\item $\\mathit{Isom}(x, y)$ is the presheaf of sets corresponding to this category fibred in setoids, see Categories, Lemma \\ref{categories-lemma-2-category-fibred-setoids}. \\end{enumerate}"}
+{"_id": "8937", "title": "stacks-lemma-pullback", "text": "(Pullback of descent data.) Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Make a choice pullbacks as in Categories, Definition \\ref{categories-definition-pullback-functor-fibred-category}. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$, and $\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$ be a families of morphisms of $\\mathcal{C}$ with fixed target. Assume all the fibre products $U_i \\times_U U_{i'}$, $U_i \\times_U U_{i'} \\times_U U_{i''}$, $V_j \\times_V V_{j'}$, and $V_j \\times_V V_{j'} \\times_V V_{j''}$ exist. Let $\\alpha : I \\to J$, $h : U \\to V$ and $g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. \\begin{enumerate} \\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the family $\\{V_j \\to V\\}$. The system $$ \\left( g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')} \\right) $$ is a descent datum relative to $\\mathcal{U}$. \\item This construction defines a functor between descent data relative to $\\mathcal{V}$ and descent data relative to $\\mathcal{U}$. \\item Given a second $\\alpha' : I \\to J$, $h' : U \\to V$ and $g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic. \\end{enumerate}"}
+{"_id": "8938", "title": "stacks-lemma-trivial-cocycle", "text": "In the situation of Definition \\ref{definition-effective-descent-datum} part (2) the maps $can_{ij} : \\text{pr}_0^*f_i^*X \\to \\text{pr}_1^*f_j^*X$ are equal to $(\\alpha_{\\text{pr}_1, f_j})_X \\circ (\\alpha_{\\text{pr}_0, f_i})_X^{-1}$ where $\\alpha_{\\cdot, \\cdot}$ is as in Categories, Lemma \\ref{categories-lemma-fibred} and where we use the equality $f_i \\circ \\text{pr}_0 = f_j \\circ \\text{pr}_1$ as maps $U_i \\times_U U_j \\to U$."}
+{"_id": "8939", "title": "stacks-lemma-compare-descent-condition", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to \\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a morphism of families of maps with fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. If \\begin{enumerate} \\item for $0 \\leq p \\leq 3$ and $0 \\leq q \\leq 3$ with $p + q \\geq 2$ and $i_1, \\ldots, i_p \\in I$ and $j_1, \\ldots, j_q \\in J$ the fibre products $U_{i_1} \\times_U \\ldots \\times_U U_{i_p} \\times_U V_{j_1} \\times_U \\ldots \\times_U V_{j_q}$ exist, \\item the functor $\\mathcal{S}_U \\to DD(\\mathcal{V})$ is an equivalence, \\item for every $i \\in I$ the functor $\\mathcal{S}_{U_i} \\to DD(\\mathcal{V}_i)$ is fully faithful, and \\item for every $i, i' \\in I$ the functor $\\mathcal{S}_{U_i \\times_U U_{i'}} \\to DD(\\mathcal{V}_{ii'})$ is faithful. \\end{enumerate} Here $\\mathcal{V}_i = \\{U_i \\times_U V_j \\to U_i\\}_{j \\in J}$ and $\\mathcal{V}_{ii'} = \\{U_i \\times_U U_{i'} \\times_U V_j \\to U_i \\times_U U_{i'}\\}_{j \\in J}$. Then $\\mathcal{S}_U \\to DD(\\mathcal{U})$ is an equivalence."}
+{"_id": "8940", "title": "stacks-lemma-stack-equivalences", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$, and \\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ of the site $\\mathcal{C}$ the functor $$ \\mathcal{S}_U \\longrightarrow DD(\\mathcal{U}) $$ which associates to an object its canonical descent datum is an equivalence. \\end{enumerate}"}
+{"_id": "8941", "title": "stacks-lemma-substack", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack over the site $\\mathcal{C}$. Let $\\mathcal{S}'$ be a subcategory of $\\mathcal{S}$. Assume \\begin{enumerate} \\item if $\\varphi : y \\to x$ is a strongly cartesian morphism of $\\mathcal{S}$ and $x$ is an object of $\\mathcal{S}'$, then $y$ is isomorphic to an object of $\\mathcal{S}'$, \\item $\\mathcal{S}'$ is a full subcategory of $\\mathcal{S}$, and \\item if $\\{f_i : U_i \\to U\\}$ is a covering of $\\mathcal{C}$, and $x$ an object of $\\mathcal{S}$ over $U$ such that $f_i^*x$ is isomorphic to an object of $\\mathcal{S}'$ for each $i$, then $x$ is isomorphic to an object of $\\mathcal{S}'$. \\end{enumerate} Then $\\mathcal{S}' \\to \\mathcal{C}$ is a stack."}
+{"_id": "8942", "title": "stacks-lemma-stack-equivalent", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$. Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent as categories over $\\mathcal{C}$. Then $\\mathcal{S}_1$ is a stack over $\\mathcal{C}$ if and only if $\\mathcal{S}_2$ is a stack over $\\mathcal{C}$."}
+{"_id": "8943", "title": "stacks-lemma-2-product-stacks", "text": "Let $\\mathcal{C}$ be a site. The $(2, 1)$-category of stacks over $\\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}."}
+{"_id": "8944", "title": "stacks-lemma-characterize-ff", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$. Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism. Then the following are equivalent \\begin{enumerate} \\item $F$ is fully faithful, \\item for every $U \\in \\Ob(\\mathcal{C})$ and for every $x, y \\in \\Ob(\\mathcal{S}_{1, U})$ the map $$ F : \\mathit{Mor}_{\\mathcal{S}_1}(x, y) \\longrightarrow \\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y)) $$ is an isomorphism of sheaves on $\\mathcal{C}/U$. \\end{enumerate}"}
+{"_id": "8945", "title": "stacks-lemma-characterize-essentially-surjective-when-ff", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$. Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism which is fully faithful. Then the following are equivalent \\begin{enumerate} \\item $F$ is an equivalence, \\item for every $U \\in \\Ob(\\mathcal{C})$ and for every $x \\in \\Ob(\\mathcal{S}_{2, U})$ there exists a covering $\\{f_i : U_i \\to U\\}$ such that $f_i^*x$ is in the essential image of the functor $F : \\mathcal{S}_{1, U_i} \\to \\mathcal{S}_{2, U_i}$. \\end{enumerate}"}
+{"_id": "8946", "title": "stacks-lemma-stack-in-groupoids-stack", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$, \\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$ and all fibre categories are groupoids, and \\item $\\mathcal{S}$ is fibred in groupoids over $\\mathcal{C}$ and is a stack over $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "8948", "title": "stacks-lemma-stack-in-groupoids-equivalent", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$. Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent as categories over $\\mathcal{C}$. Then $\\mathcal{S}_1$ is a stack in groupoids over $\\mathcal{C}$ if and only if $\\mathcal{S}_2$ is a stack in groupoids over $\\mathcal{C}$."}
+{"_id": "8949", "title": "stacks-lemma-2-product-stacks-in-groupoids", "text": "Let $\\mathcal{C}$ be a category. The $2$-category of stacks in groupoids over $\\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}."}
+{"_id": "8950", "title": "stacks-lemma-when-stack-in-sets", "text": "Let $\\mathcal{C}$ be a site. Under the equivalence $$ \\left\\{ \\begin{matrix} \\text{the category of presheaves}\\\\ \\text{of sets over }\\mathcal{C} \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{the category of categories}\\\\ \\text{fibred in sets over }\\mathcal{C} \\end{matrix} \\right\\} $$ of Categories, Lemma \\ref{categories-lemma-2-category-fibred-sets} the stacks in sets correspond precisely to the sheaves."}
+{"_id": "8951", "title": "stacks-lemma-stack-in-setoids-characterize", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}$ be a category fibred in setoids over $\\mathcal{C}$. Then $\\mathcal{S}$ is a stack in setoids if and only if the unique equivalent category $\\mathcal{S}'$ fibred in sets (see Categories, Lemma \\ref{categories-lemma-setoid-fibres}) is a stack in sets. In other words, if and only if the presheaf $$ U \\longmapsto \\Ob(\\mathcal{S}_U)/\\!\\!\\cong $$ is a sheaf."}
+{"_id": "8953", "title": "stacks-lemma-2-product-stacks-in-setoids", "text": "Let $\\mathcal{C}$ be a site. The $2$-category of stacks in setoids over $\\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}."}
+{"_id": "8954", "title": "stacks-lemma-2-fibre-product-gives-stack-in-setoids", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}, \\mathcal{T}$ be stacks in groupoids over $\\mathcal{C}$ and let $\\mathcal{R}$ be a stack in setoids over $\\mathcal{C}$. Let $f : \\mathcal{T} \\to \\mathcal{S}$ and $g : \\mathcal{R} \\to \\mathcal{S}$ be $1$-morphisms. If $f$ is faithful, then the $2$-fibre product $$ \\mathcal{T} \\times_{f, \\mathcal{S}, g} \\mathcal{R} $$ is a stack in setoids over $\\mathcal{C}$."}
+{"_id": "8955", "title": "stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}$ be a stack in groupoids over $\\mathcal{C}$ and let $\\mathcal{S}_i$, $i = 1, 2$ be stacks in setoids over $\\mathcal{C}$. Let $f_i : \\mathcal{S}_i \\to \\mathcal{S}$ be $1$-morphisms. Then the $2$-fibre product $$ \\mathcal{S}_1 \\times_{f_1, \\mathcal{S}, f_2} \\mathcal{S}_2 $$ is a stack in setoids over $\\mathcal{C}$."}
+{"_id": "8956", "title": "stacks-lemma-faithful-descent", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ \\mathcal{T}_2 \\ar[r] \\ar[d]_{G'} & \\mathcal{T}_1 \\ar[d]^G \\\\ \\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1 } $$ be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$. Assume \\begin{enumerate} \\item for every $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential image of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and \\item $G'$ is faithful, \\end{enumerate} then $G$ is faithful."}
+{"_id": "8957", "title": "stacks-lemma-stack-in-setoids-descent", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ \\mathcal{T}_2 \\ar[r] \\ar[d] & \\mathcal{T}_1 \\ar[d]^G \\\\ \\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1 } $$ be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$. If \\begin{enumerate} \\item $F : \\mathcal{S}_2 \\to \\mathcal{S}_1$ is fully faithful, \\item for every $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential image of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and \\item $\\mathcal{T}_2$ is a stack in setoids. \\end{enumerate} then $\\mathcal{T}_1$ is a stack in setoids."}
+{"_id": "8958", "title": "stacks-lemma-relative-sheaf-over-stack-is-stack", "text": "Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{S} \\to \\mathcal{T}$ be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$. Assume that \\begin{enumerate} \\item $\\mathcal{T}$ is a stack in groupoids over $\\mathcal{C}$, \\item for every $U \\in \\Ob(\\mathcal{C})$ the functor $\\mathcal{S}_U \\to \\mathcal{T}_U$ of fibre categories is faithful, \\item for each $U$ and each $y \\in \\Ob(\\mathcal{T}_U)$ the presheaf $$ (h : V \\to U) \\longmapsto \\{(x, f) \\mid x \\in \\Ob(\\mathcal{S}_V), f : F(x) \\to f^*y\\text{ over }V\\}/\\cong $$ is a sheaf on $\\mathcal{C}/U$. \\end{enumerate} Then $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$."}
+{"_id": "8960", "title": "stacks-lemma-characterize-stack-in-setoids", "text": "Let $\\mathcal{C}$ be a site. If $\\mathcal{S}$ is a stack in groupoids, then the canonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$ is an equivalence if and only if $\\mathcal{S}$ is a stack in setoids."}
+{"_id": "8961", "title": "stacks-lemma-stackify", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$. There exists a stack $p' : \\mathcal{S}' \\to \\mathcal{C}$ and a $1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$ of fibred categories over $\\mathcal{C}$ (see Categories, Definition \\ref{categories-definition-fibred-categories-over-C}) such that \\begin{enumerate} \\item for every $U \\in \\Ob(\\mathcal{C})$, and any $x, y \\in \\Ob(\\mathcal{S}_U)$ the map $$ \\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y)) $$ induced by $G$ identifies the right hand side with the sheafification of the left hand side, and \\item for every $U \\in \\Ob(\\mathcal{C})$, and any $x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the object $x'|_{U_i}$ is in the essential image of the functor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$. \\end{enumerate} Moreover the stack $\\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions."}
+{"_id": "8962", "title": "stacks-lemma-stackify-universal-property", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$. Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ and $G : \\mathcal{S} \\to \\mathcal{S}'$ the stack and $1$-morphism constructed in Lemma \\ref{lemma-stackify}. This construction has the following universal property: Given a stack $q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism $F : \\mathcal{S} \\to \\mathcal{X}$ of fibred categories over $\\mathcal{C}$ there exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$ such that the diagram $$ \\xymatrix{ \\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\ & \\mathcal{S}' \\ar[ru]_H } $$ is $2$-commutative."}
+{"_id": "8963", "title": "stacks-lemma-stackify-universal-property-more", "text": "Notation and assumptions as in Lemma \\ref{lemma-stackify-universal-property}. There is a canonical equivalence of categories $$ \\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{X}) = \\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}', \\mathcal{X}) $$ given by the constructions in the proof of the aforementioned lemma."}
+{"_id": "8964", "title": "stacks-lemma-stackification-fibre-product-fibred-categories", "text": "Let $\\mathcal{C}$ be a site. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$ be morphisms of fibred categories over $\\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications."}
+{"_id": "8965", "title": "stacks-lemma-stackification-inertia", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{X}$ be a fibred category over $\\mathcal{C}$. The stackification of the inertia fibred category $\\mathcal{I}_\\mathcal{X}$ is inertia of the stackification of $\\mathcal{X}$."}
+{"_id": "8966", "title": "stacks-lemma-stackify-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids over $\\mathcal{C}$. There exists a stack in groupoids $p' : \\mathcal{S}' \\to \\mathcal{C}$ and a $1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$ of categories fibred in groupoids over $\\mathcal{C}$ (see Categories, Definition \\ref{categories-definition-categories-fibred-in-groupoids-over-C}) such that \\begin{enumerate} \\item for every $U \\in \\Ob(\\mathcal{C})$, and any $x, y \\in \\Ob(\\mathcal{S}_U)$ the map $$ \\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y)) $$ induced by $G$ identifies the right hand side with the sheafification of the left hand side, and \\item for every $U \\in \\Ob(\\mathcal{C})$, and any $x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the object $x'|_{U_i}$ is in the essential image of the functor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$. \\end{enumerate} Moreover the stack in groupoids $\\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions."}
+{"_id": "8967", "title": "stacks-lemma-stackify-groupoids-universal-property", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids over $\\mathcal{C}$. Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ and $G : \\mathcal{S} \\to \\mathcal{S}'$ the stack in groupoids and $1$-morphism constructed in Lemma \\ref{lemma-stackify-groupoids}. This construction has the following universal property: Given a stack in groupoids $q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism $F : \\mathcal{S} \\to \\mathcal{X}$ of categories over $\\mathcal{C}$ there exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$ such that the diagram $$ \\xymatrix{ \\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\ & \\mathcal{S}' \\ar[ru]_H } $$ is $2$-commutative."}
+{"_id": "8968", "title": "stacks-lemma-stackification-fibre-product-categories-fibred-in-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$ be morphisms of categories fibred in groupoids over $\\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications."}
+{"_id": "8969", "title": "stacks-lemma-topology-inherited", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Let $\\text{Cov}(\\mathcal{S})$ be the set of families $\\{x_i \\to x\\}_{i \\in I}$ of morphisms in $\\mathcal{S}$ with fixed target such that (a) each $x_i \\to x$ is strongly cartesian, and (b) $\\{p(x_i) \\to p(x)\\}_{i \\in I}$ is a covering of $\\mathcal{C}$. Then $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ is a site."}
+{"_id": "8970", "title": "stacks-lemma-topology-inherited-functorial", "text": "Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of fibred categories over $\\mathcal{C}$. Then $F$ is a continuous and cocontinuous functor between the structure of sites inherited from $\\mathcal{C}$. Hence $F$ induces a morphism of topoi $f : \\Sh(\\mathcal{X}) \\to \\Sh(\\mathcal{Y})$ with $f_* = {}_sF = {}_pF$ and $f^{-1} = F^s = F^p$. In particular $f^{-1}(\\mathcal{G})(x) = \\mathcal{G}(F(x))$ for a sheaf $\\mathcal{G}$ on $\\mathcal{Y}$ and object $x$ of $\\mathcal{X}$."}
+{"_id": "8971", "title": "stacks-lemma-localizing", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$ be a category fibred in groupoids. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\\mathcal{X}/x \\to \\mathcal{C}/U$ where $\\mathcal{X}$ is endowed with the topology inherited from $\\mathcal{C}$."}
+{"_id": "8972", "title": "stacks-lemma-stack-in-groupoids-over-stack-in-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$ and $q : \\mathcal{Y} \\to \\mathcal{C}$ be stacks in groupoids. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories over $\\mathcal{C}$. If $F$ turns $\\mathcal{X}$ into a category fibred in groupoids over $\\mathcal{Y}$, then $\\mathcal{X}$ is a stack in groupoids over $\\mathcal{Y}$ (with topology inherited from $\\mathcal{C}$)."}
+{"_id": "8973", "title": "stacks-lemma-stack-over-stack", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$ be a stack. Endow $\\mathcal{X}$ with the topology inherited from $\\mathcal{C}$ and let $q : \\mathcal{Y} \\to \\mathcal{X}$ be a stack. Then $\\mathcal{Y}$ is a stack over $\\mathcal{C}$. If $p$ and $q$ define stacks in groupoids, then $\\mathcal{Y}$ is a stack in groupoids over $\\mathcal{C}$."}
+{"_id": "8974", "title": "stacks-lemma-gerbe-equivalent", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$. Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent as categories over $\\mathcal{C}$. Then $\\mathcal{S}_1$ is a gerbe over $\\mathcal{C}$ if and only if $\\mathcal{S}_2$ is a gerbe over $\\mathcal{C}$."}
+{"_id": "8975", "title": "stacks-lemma-when-gerbe", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$ and $q : \\mathcal{Y} \\to \\mathcal{C}$ be stacks in groupoids. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories over $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item For some (equivalently any) factorization $F = F' \\circ a$ where $a : \\mathcal{X} \\to \\mathcal{X}'$ is an equivalence of categories over $\\mathcal{C}$ and $F'$ is fibred in groupoids, the map $F' : \\mathcal{X}' \\to \\mathcal{Y}$ is a gerbe (with the topology on $\\mathcal{Y}$ inherited from $\\mathcal{C}$). \\item The following two conditions are satisfied \\begin{enumerate} \\item for $y \\in \\Ob(\\mathcal{Y})$ lying over $U \\in \\Ob(\\mathcal{C})$ there exists a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ and objects $x_i$ of $\\mathcal{X}$ over $U_i$ such that $F(x_i) \\cong y|_{U_i}$ in $\\mathcal{Y}_{U_i}$, and \\item for $U \\in \\Ob(\\mathcal{C})$, $x, x' \\in \\Ob(\\mathcal{X}_U)$, and $b : F(x) \\to F(x')$ in $\\mathcal{Y}_U$ there exists a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ and morphisms $a_i : x|_{U_i} \\to x'|_{U_i}$ in $\\mathcal{X}_{U_i}$ with $F(a_i) = b|_{U_i}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "8976", "title": "stacks-lemma-base-change-gerbe", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\ \\mathcal{Y}' \\ar[r]^G & \\mathcal{Y} } $$ be a $2$-fibre product of stacks in groupoids over $\\mathcal{C}$. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$."}
+{"_id": "8977", "title": "stacks-lemma-composition-gerbe", "text": "Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of stacks in groupoids over $\\mathcal{C}$. If $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ and $\\mathcal{Y}$ is a gerbe over $\\mathcal{Z}$, then $\\mathcal{X}$ is a gerbe over $\\mathcal{Z}$."}
+{"_id": "8978", "title": "stacks-lemma-gerbe-descent", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ \\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\ \\mathcal{Y}' \\ar[r]^G & \\mathcal{Y} } $$ be a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$. If for every $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(\\mathcal{Y}_U)$ there exists a covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential image of $G : \\mathcal{Y}'_{U_i} \\to \\mathcal{Y}_{U_i}$ and $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$, then $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$."}
+{"_id": "8979", "title": "stacks-lemma-gerbe-abelian-auts", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a gerbe over a site $\\mathcal{C}$. Assume that for all $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(\\mathcal{S}_U)$ the sheaf of groups $\\mathit{Aut}(x) = \\mathit{Isom}(x, x)$ on $\\mathcal{C}/U$ is abelian. Then there exist \\begin{enumerate} \\item a sheaf $\\mathcal{G}$ of abelian groups on $\\mathcal{C}$, \\item for every $U \\in \\Ob(\\mathcal{C})$ and every $x \\in \\Ob(\\mathcal{S}_U)$ an isomorphism $\\mathcal{G}|_U \\to \\mathit{Aut}(x)$ \\end{enumerate} such that for every $U$ and every morphism $\\varphi : x \\to y$ in $\\mathcal{S}_U$ the diagram $$ \\xymatrix{ \\mathcal{G}|_U \\ar[d] \\ar@{=}[rr] & & \\mathcal{G}|_U \\ar[d] \\\\ \\mathit{Aut}(x) \\ar[rr]^{\\alpha \\mapsto \\varphi \\circ \\alpha \\circ \\varphi^{-1}} & & \\mathit{Aut}(y) } $$ is commutative."}
+{"_id": "8980", "title": "stacks-lemma-fibred-category-pushforward", "text": "In the situation above, if $\\mathcal{S}$ is a fibred category over $\\mathcal{D}$ then $u^p\\mathcal{S}$ is a fibred category over $\\mathcal{C}$."}
+{"_id": "8981", "title": "stacks-lemma-stack-pushforward", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites. Let $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack over $\\mathcal{D}$. Then $u^p\\mathcal{S}$ is a stack over $\\mathcal{C}$."}
+{"_id": "8983", "title": "stacks-lemma-right-multiplicative-system", "text": "In the situation above assume \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category, \\item $\\mathcal{C}$ has nonempty finite limits, and \\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits. \\end{enumerate} Consider the set $R \\subset \\text{Arrows}(u_{pp}\\mathcal{S})$ of morphisms of the form $$ (a, \\text{id}_V, \\alpha) : (U', \\phi' : V \\to u(U'), x') \\longrightarrow (U, \\phi : V \\to u(U), x) $$ with $\\alpha$ strongly cartesian. Then $R$ is a right multiplicative system."}
+{"_id": "8984", "title": "stacks-lemma-fibred-category-pullback", "text": "With notation and assumptions as in Lemma \\ref{lemma-right-multiplicative-system}. Set $u_p\\mathcal{S} = R^{-1}u_{pp}\\mathcal{S}$, see Categories, Section \\ref{categories-section-localization}. Then $u_p\\mathcal{S}$ is a fibred category over $\\mathcal{D}$."}
+{"_id": "8986", "title": "stacks-lemma-adjointness-pullback-pushforward", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Let $p : \\mathcal{S} \\to \\mathcal{C}$ and $q : \\mathcal{T} \\to \\mathcal{D}$ be categories over $\\mathcal{C}$ and $\\mathcal{D}$. Assume that \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category, \\item $q : \\mathcal{T} \\to \\mathcal{D}$ is a fibred category, \\item $\\mathcal{C}$ has nonempty finite limits, and \\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits. \\end{enumerate} Then we have a canonical equivalence of categories $$ \\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T}) = \\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T}) $$ of morphism categories."}
+{"_id": "8987", "title": "stacks-lemma-adjointness-pullback-pushforward-stacks", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \\ref{sites-proposition-get-morphism}. Let $p : \\mathcal{S} \\to \\mathcal{C}$ and $q : \\mathcal{T} \\to \\mathcal{D}$ be stacks. Then we have a canonical equivalence of categories $$ \\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, f_*\\mathcal{T}) = \\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, \\mathcal{T}) $$ of morphism categories."}
+{"_id": "8988", "title": "stacks-lemma-technical-up", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \\ref{sites-proposition-get-morphism}. Let $\\mathcal{S} \\to \\mathcal{C}$ be a fibred category, and let $\\mathcal{S} \\to \\mathcal{S}'$ be the stackification of $\\mathcal{S}$. Then $f^{-1}\\mathcal{S}'$ is the stackification of $u_p\\mathcal{S}$."}
+{"_id": "8989", "title": "stacks-lemma-bigger-site", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor satisfying the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}. Let $f : \\mathcal{D} \\to \\mathcal{C}$ be the corresponding morphism of sites. Then \\begin{enumerate} \\item for every stack $p : \\mathcal{S} \\to \\mathcal{C}$ the canonical functor $\\mathcal{S} \\to f_*f^{-1}\\mathcal{S}$ is an equivalence of stacks, \\item given stacks $\\mathcal{S}, \\mathcal{S}'$ over $\\mathcal{C}$ the construction $f^{-1}$ induces an equivalence $$ \\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}') \\longrightarrow \\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, f^{-1}\\mathcal{S}') $$ of morphism categories. \\end{enumerate}"}
+{"_id": "8991", "title": "stacks-lemma-localize-stacks", "text": "Assume that $\\mathcal{C}$ is a site, and $U$ is an object of $\\mathcal{C}$ whose associated representable presheaf is a sheaf. Constructions A and B above define mutually inverse (!) functors of $2$-categories $$ \\left\\{ \\begin{matrix} 2\\text{-category of}\\\\ \\text{stacks over }\\mathcal{C}/U \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} 2\\text{-category of pairs }(\\mathcal{T}, p) \\text{ consisting} \\\\ \\text{of a stack }\\mathcal{T}\\text{ over }\\mathcal{C}\\text{ and a morphism} \\\\ p : \\mathcal{T} \\to \\mathcal{C}/U\\text{ of stacks over }\\mathcal{C} \\end{matrix} \\right\\} $$"}
+{"_id": "9010", "title": "spaces-simplicial-lemma-simplicial-site", "text": "Let $X$ be a simplicial space. Then $X_{Zar}$ as defined above is a site."}
+{"_id": "9011", "title": "spaces-simplicial-lemma-describe-sheaves-simplicial-site", "text": "Let $X$ be a simplicial space. There is an equivalence of categories between \\begin{enumerate} \\item $\\Sh(X_{Zar})$, and \\item category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$ described above. \\end{enumerate}"}
+{"_id": "9012", "title": "spaces-simplicial-lemma-simplicial-space-site-functorial", "text": "Let $f : Y \\to X$ be a morphism of simplicial spaces. Then the functor $u : X_{Zar} \\to Y_{Zar}$ which associates to the open $U \\subset X_n$ the open $f_n^{-1}(U) \\subset Y_n$ defines a morphism of sites $f_{Zar} : Y_{Zar} \\to X_{Zar}$."}
+{"_id": "9013", "title": "spaces-simplicial-lemma-describe-functoriality", "text": "Let $f : Y \\to X$ be a morphism of simplicial spaces. In terms of the description of sheaves in Lemma \\ref{lemma-describe-sheaves-simplicial-site} the morphism $f_{Zar}$ of Lemma \\ref{lemma-simplicial-space-site-functorial} can be described as follows. \\begin{enumerate} \\item If $\\mathcal{G}$ is a sheaf on $Y$, then $(f_{Zar, *}\\mathcal{G})_n = f_{n, *}\\mathcal{G}_n$. \\item If $\\mathcal{F}$ is a sheaf on $X$, then $(f_{Zar}^{-1}\\mathcal{F})_n = f_n^{-1}\\mathcal{F}_n$. \\end{enumerate}"}
+{"_id": "9014", "title": "spaces-simplicial-lemma-restriction-to-components", "text": "Let $X$ be a simplicial space. The functor $X_{n, Zar} \\to X_{Zar}$, $U \\mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_n : \\Sh(X_n) \\to \\Sh(X_{Zar})$ satisfies \\begin{enumerate} \\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $X$ the sheaf $\\mathcal{F}_n$ on $X_n$, \\item $g_n^{-1} : \\Sh(X_{Zar}) \\to \\Sh(X_n)$ has a left adjoint $g^{Sh}_{n!}$, \\item $g^{Sh}_{n!}$ commutes with finite connected limits, \\item $g_n^{-1} : \\textit{Ab}(X_{Zar}) \\to \\textit{Ab}(X_n)$ has a left adjoint $g_{n!}$, and \\item $g_{n!}$ is exact. \\end{enumerate}"}
+{"_id": "9015", "title": "spaces-simplicial-lemma-restriction-injective-to-component", "text": "Let $X$ be a simplicial space. If $\\mathcal{I}$ is an injective abelian sheaf on $X_{Zar}$, then $\\mathcal{I}_n$ is an injective abelian sheaf on $X_n$."}
+{"_id": "9017", "title": "spaces-simplicial-lemma-augmentation", "text": "Let $Y$ be a simplicial space and let $a : Y \\to X$ be an augmentation (Simplicial, Definition \\ref{simplicial-definition-augmentation}). Let $a_n : Y_n \\to X$ be the corresponding morphisms of topological spaces. There is a canonical morphism of topoi $$ a : \\Sh(Y_{Zar}) \\to \\Sh(X) $$ with the following properties: \\begin{enumerate} \\item $a^{-1}\\mathcal{F}$ is the sheaf restricting to $a_n^{-1}\\mathcal{F}$ on $Y_n$, \\item $a_m \\circ Y(\\varphi) = a_n$ for all $\\varphi : [m] \\to [n]$, \\item $a \\circ g_n = a_n$ as morphisms of topoi with $g_n$ as in Lemma \\ref{lemma-restriction-to-components}, \\item $a_*\\mathcal{G}$ for $\\mathcal{G} \\in \\Sh(Y_{Zar})$ is the equalizer of the two maps $a_{0, *}\\mathcal{G}_0 \\to a_{1, *}\\mathcal{G}_1$. \\end{enumerate}"}
+{"_id": "9018", "title": "spaces-simplicial-lemma-simplicial-resolution-Z", "text": "Let $X$ be a simplicial topological space. The complex of abelian presheaves on $X_{Zar}$ $$ \\ldots \\to \\mathbf{Z}_{X_2} \\to \\mathbf{Z}_{X_1} \\to \\mathbf{Z}_{X_0} $$ with boundary $\\sum (-1)^i d^n_i$ is a resolution of the constant presheaf $\\mathbf{Z}$."}
+{"_id": "9019", "title": "spaces-simplicial-lemma-simplicial-sheaf-cohomology", "text": "Let $X$ be a simplicial topological space. Let $\\mathcal{F}$ be an abelian sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_1^{p, q} = H^q(X_p, \\mathcal{F}_p) $$ converging to $H^{p + q}(X_{Zar}, \\mathcal{F})$. This spectral sequence is functorial in $\\mathcal{F}$."}
+{"_id": "9022", "title": "spaces-simplicial-lemma-simplicial-site-site", "text": "Let $\\mathcal{C}$ be a simplicial object in the category of sites. With notation as above we construct a site $\\mathcal{C}_{total}$ as follows. \\begin{enumerate} \\item An object of $\\mathcal{C}_{total}$ is an object $U$ of $\\mathcal{C}_n$ for some $n$, \\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$ is given by a map $\\varphi : [m] \\to [n]$ with $U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$ and a morphism $f : U \\to u_\\varphi(V)$ of $\\mathcal{C}_n$, and \\item a covering $\\{(\\text{id}, f_i) :  U_i \\to U\\}$ in $\\mathcal{C}_{total}$ is given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$ of $\\mathcal{C}_n$. \\end{enumerate}"}
+{"_id": "9023", "title": "spaces-simplicial-lemma-simplicial-cocontinuous-site", "text": "Let $\\mathcal{C}$ be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors $u_\\varphi : \\mathcal{C}_n \\to \\mathcal{C}_m$ have property $P$ of Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}. Then we can construct a site $\\mathcal{C}_{total}$ as follows. \\begin{enumerate} \\item An object of $\\mathcal{C}_{total}$ is an object $U$ of $\\mathcal{C}_n$ for some $n$, \\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$ is given by a map $\\varphi : [m] \\to [n]$ with $U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$ and a morphism $f : u_\\varphi(U) \\to V$ of $\\mathcal{C}_m$, and \\item a covering $\\{(\\text{id}, f_i) :  U_i \\to U\\}$ in $\\mathcal{C}_{total}$ is given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$ of $\\mathcal{C}_n$. \\end{enumerate}"}
+{"_id": "9024", "title": "spaces-simplicial-lemma-describe-sheaves-simplicial-site-site", "text": "In Situation \\ref{situation-simplicial-site} there is an equivalence of categories between \\begin{enumerate} \\item $\\Sh(\\mathcal{C}_{total})$, and \\item the category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$ described above. \\end{enumerate} In particular, the topos $\\Sh(\\mathcal{C}_{total})$ only depends on the topoi $\\Sh(\\mathcal{C}_n)$ and the morphisms of topoi $f_\\varphi$."}
+{"_id": "9025", "title": "spaces-simplicial-lemma-restriction-to-components-site", "text": "In Situation \\ref{situation-simplicial-site} the functor $\\mathcal{C}_n \\to \\mathcal{C}_{total}$, $U \\mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$ satisfies \\begin{enumerate} \\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$ the sheaf $\\mathcal{F}_n$ on $\\mathcal{C}_n$, \\item $g_n^{-1} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}_n)$ has a left adjoint $g^{Sh}_{n!}$, \\item for $\\mathcal{G}$ in $\\Sh(\\mathcal{C}_n)$ the restriction of $g_{n!}^{Sh}\\mathcal{G}$ to $\\mathcal{C}_m$ is $\\coprod\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$, \\item $g_{n!}^{Sh}$ commutes with finite connected limits, \\item $g_n^{-1} : \\textit{Ab}(\\mathcal{C}_{total}) \\to \\textit{Ab}(\\mathcal{C}_n)$ has a left adjoint $g_{n!}$, \\item for $\\mathcal{G}$ in $\\textit{Ab}(\\mathcal{C}_n)$ the restriction of $g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is $\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$, and \\item $g_{n!}$ is exact. \\end{enumerate}"}
+{"_id": "9026", "title": "spaces-simplicial-lemma-restriction-injective-to-component-site", "text": "\\begin{slogan} An injective abelian sheaf on a simplicial site is injective on each component \\end{slogan} In Situation \\ref{situation-simplicial-site}. If $\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{C}_{total})$, then $\\mathcal{I}_n$ is injective in $\\textit{Ab}(\\mathcal{C}_n)$. If $\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}(\\mathcal{C}_{total})$, then $\\mathcal{I}_n^\\bullet$ is K-injective in $\\textit{Ab}(\\mathcal{C}_n)$."}
+{"_id": "9027", "title": "spaces-simplicial-lemma-augmentation-site", "text": "In Situation \\ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Remark \\ref{remark-augmentation-site}. Then $a_0$ induces \\begin{enumerate} \\item a morphism of topoi $a_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{D})$ for all $n \\geq 0$, \\item a morphism of topoi $a : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{D})$ \\end{enumerate} such that \\begin{enumerate} \\item for all $\\varphi : [m] \\to [n]$ we have $a_m \\circ f_\\varphi = a_n$, \\item if $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$ is as in Lemma \\ref{lemma-restriction-to-components-site}, then $a \\circ g_n = a_n$, and \\item $a_*\\mathcal{F}$ for $\\mathcal{F} \\in \\Sh(\\mathcal{C}_{total})$ is the equalizer of the two maps $a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1$. \\end{enumerate}"}
+{"_id": "9028", "title": "spaces-simplicial-lemma-morphism-simplicial-sites", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in Situation \\ref{situation-simplicial-site}. Let $h$ be a morphism between simplicial sites as in Remark \\ref{remark-morphism-simplicial-sites}. Then we obtain a morphism of topoi $$ h_{total} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}'_{total}) $$ and commutative diagrams $$ \\xymatrix{ \\Sh(\\mathcal{C}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} & \\Sh(\\mathcal{C}'_n) \\ar[d]^{g'_n} \\\\ \\Sh(\\mathcal{C}_{total}) \\ar[r]^{h_{total}} & \\Sh(\\mathcal{C}'_{total}) } $$ Moreover, we have $(g'_n)^{-1} \\circ h_{total, *} = h_{n, *} \\circ g_n^{-1}$."}
+{"_id": "9029", "title": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites", "text": "With notation and hypotheses as in Lemma \\ref{lemma-morphism-simplicial-sites}. For $K \\in D(\\mathcal{C}_{total})$ we have $(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$."}
+{"_id": "9030", "title": "spaces-simplicial-lemma-morphism-augmentation-simplicial-sites", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi, \\mathcal{D}, a_0$, $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi, \\mathcal{D}', a'_0$, and $h_n$, $n \\geq -1$ be as in Remark \\ref{remark-morphism-augmentation-simplicial-sites}. Then we obtain a commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{C}_{total}) \\ar[d]_a \\ar[r]_{h_{total}} & \\Sh(\\mathcal{C}'_{total}) \\ar[d]^{a'} \\\\ \\Sh(\\mathcal{D}) \\ar[r]^{h_{-1}} & \\Sh(\\mathcal{D}') } $$"}
+{"_id": "9031", "title": "spaces-simplicial-lemma-restriction-module-to-components-site", "text": "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. There is a canonical morphism of ringed topoi $g_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}_{total}), \\mathcal{O})$ agreeing with the morphism $g_n$ of Lemma \\ref{lemma-restriction-to-components-site} on underlying topoi. The functor $g_n^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_n)$ has a left adjoint $g_{n!}$. For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_n)$-modules the restriction of $g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is $$ \\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*\\mathcal{G} $$ where $f_\\varphi : (\\Sh(\\mathcal{C}_m), \\mathcal{O}_m) \\to (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)$ is the morphism of ringed topoi agreeing with the previously defined $f_\\varphi$ on topoi and using the map $\\mathcal{O}(\\varphi) : f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ on sheaves of rings."}
+{"_id": "9032", "title": "spaces-simplicial-lemma-restriction-injective-to-component-limp", "text": "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. If $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then $\\mathcal{I}_n$ is a totally acyclic sheaf on $\\mathcal{C}_n$."}
+{"_id": "9033", "title": "spaces-simplicial-lemma-exactness-g-shriek-modules", "text": "With assumptions as in Lemma \\ref{lemma-restriction-module-to-components-site} the functor $g_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O})$ is exact if the maps $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ are flat for all $\\varphi : [n] \\to [m]$."}
+{"_id": "9035", "title": "spaces-simplicial-lemma-morphism-simplicial-sites-modules", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ and $\\mathcal{O}'$ be a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{C}'_{total}$. Let $(h, h^\\sharp)$ be a morphism between simplicial sites as in Remark \\ref{remark-morphism-simplicial-sites-modules}. Then we obtain a morphism of ringed topoi $$ h_{total} : (\\Sh(\\mathcal{C}_{total}, \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}') $$ and commutative diagrams $$ \\xymatrix{ (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} & (\\Sh(\\mathcal{C}'_n), \\mathcal{O}'_n) \\ar[d]^{g'_n} \\\\ (\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\ar[r]^{h_{total}} & (\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}') } $$ of ringed topoi where $g_n$ and $g'_n$ are as in Lemma \\ref{lemma-restriction-module-to-components-site}. Moreover, we have $(g'_n)^* \\circ h_{total, *} = h_{n, *} \\circ g_n^*$ as functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}'_n)$."}
+{"_id": "9036", "title": "spaces-simplicial-lemma-direct-image-morphism-simplicial-sites-modules", "text": "With notation and hypotheses as in Lemma \\ref{lemma-morphism-simplicial-sites-modules}. For $K \\in D(\\mathcal{O})$ we have $(g'_n)^*Rh_{total, *}K = Rh_{n, *}g_n^*K$."}
+{"_id": "9037", "title": "spaces-simplicial-lemma-simplicial-resolution-Z-site", "text": "In Situation \\ref{situation-simplicial-site} and with notation as above there is a complex $$ \\ldots \\to g_{2!}\\mathbf{Z} \\to g_{1!}\\mathbf{Z} \\to g_{0!}\\mathbf{Z} $$ of abelian sheaves on $\\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\\mathbf{Z}$ on $\\mathcal{C}_{total}$."}
+{"_id": "9038", "title": "spaces-simplicial-lemma-cech-complex", "text": "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}_{total}$ there is a canonical complex $$ 0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to \\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to \\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to \\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots $$ which is exact in degrees $-1, 0$ and exact everywhere if $\\mathcal{F}$ is injective."}
+{"_id": "9039", "title": "spaces-simplicial-lemma-simplicial-sheaf-cohomology-site", "text": "In Situation \\ref{situation-simplicial-site}. For $K$ in $D^+(\\mathcal{C}_{total})$ there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q} $$ converging to $H^{p + q}(\\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$."}
+{"_id": "9040", "title": "spaces-simplicial-lemma-sanity-check", "text": "Let $\\mathcal{C}$ be as in Situation \\ref{situation-simplicial-site}. Let $U \\in \\Ob(\\mathcal{C}_n)$. Let $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{C}_{total})$. Then $H^p(U, \\mathcal{F}) = H^p(U, g_n^{-1}\\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\\mathcal{C}_{total}$."}
+{"_id": "9041", "title": "spaces-simplicial-lemma-simplicial-resolution-augmentation", "text": "In Situation \\ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Remark \\ref{remark-augmentation-site}. For any abelian sheaf $\\mathcal{G}$ on $\\mathcal{D}$  there is an exact complex $$ \\ldots \\to g_{2!}(a_2^{-1}\\mathcal{G}) \\to g_{1!}(a_1^{-1}\\mathcal{G}) \\to g_{0!}(a_0^{-1}\\mathcal{G}) \\to a^{-1}\\mathcal{G} \\to 0 $$ of abelian sheaves on $\\mathcal{C}_{total}$."}
+{"_id": "9042", "title": "spaces-simplicial-lemma-augmentation-cech-complex", "text": "In Situation \\ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Remark \\ref{remark-augmentation-site}. For an abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$ there is a canonical complex $$ 0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to a_{2, *}\\mathcal{F}_2 \\to \\ldots $$ on $\\mathcal{D}$ which is exact in degrees $-1, 0$ and exact everywhere if $\\mathcal{F}$ is injective."}
+{"_id": "9043", "title": "spaces-simplicial-lemma-augmentation-spectral-sequence", "text": "In Situation \\ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Remark \\ref{remark-augmentation-site}. For any $K$ in $D^+(\\mathcal{C}_{total})$ there is a spectral sequence  $(E_r, d_r)_{r \\geq 0}$ with $$ E_1^{p, q} = R^qa_{p, *} K_p,\\quad d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q} $$ converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$."}
+{"_id": "9044", "title": "spaces-simplicial-lemma-simplicial-resolution-ringed", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. There is a complex $$ \\ldots \\to g_{2!}\\mathcal{O}_2 \\to g_{1!}\\mathcal{O}_1 \\to g_{0!}\\mathcal{O}_0 $$ of $\\mathcal{O}$-modules which forms a resolution of $\\mathcal{O}$. Here $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}."}
+{"_id": "9045", "title": "spaces-simplicial-lemma-cech-complex-modules", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. There is a canonical complex $$ 0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to \\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to \\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to \\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots $$ which is exact in degrees $-1, 0$ and exact everywhere if $\\mathcal{F}$ is an injective $\\mathcal{O}$-module."}
+{"_id": "9046", "title": "spaces-simplicial-lemma-simplicial-module-cohomology-site", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings. For $K$ in $D^+(\\mathcal{O})$ there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with $$ E_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad d_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q} $$ converging to $H^{p + q}(\\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$."}
+{"_id": "9047", "title": "spaces-simplicial-lemma-sanity-check-modules", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings. Let $U \\in \\Ob(\\mathcal{C}_n)$. Let $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O})$. Then $H^p(U, \\mathcal{F}) = H^p(U, g_n^*\\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\\mathcal{C}_{total}$."}
+{"_id": "9048", "title": "spaces-simplicial-lemma-flat-augmentation-modules", "text": "With notation as above. The morphism $a : (\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ is flat if and only if $a_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ is flat for $n \\geq 0$."}
+{"_id": "9049", "title": "spaces-simplicial-lemma-simplicial-resolution-augmentation-modules", "text": "With notation as above. For a $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$ there is an exact complex $$ \\ldots \\to g_{2!}(a_2^*\\mathcal{G}) \\to g_{1!}(a_1^*\\mathcal{G}) \\to g_{0!}(a_0^*\\mathcal{G}) \\to a^*\\mathcal{G} \\to 0 $$ of sheaves of $\\mathcal{O}$-modules on $\\mathcal{C}_{total}$. Here $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}."}
+{"_id": "9050", "title": "spaces-simplicial-lemma-augmentation-cech-complex-modules", "text": "With notation as above. For an $\\mathcal{O}$-module $\\mathcal{F}$ on $\\mathcal{C}_{total}$ there is a canonical complex $$ 0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to a_{2, *}\\mathcal{F}_2 \\to \\ldots $$ of $\\mathcal{O}_\\mathcal{D}$-modules which is exact in degrees $-1, 0$. If $\\mathcal{F}$ is an injective $\\mathcal{O}$-module, then the complex is exact in all degrees and remains exact on applying the functor $\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, -)$ for any $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$."}
+{"_id": "9052", "title": "spaces-simplicial-lemma-check-cartesian-module", "text": "In Situation \\ref{situation-simplicial-site}. \\begin{enumerate} \\item A sheaf $\\mathcal{F}$ of sets or abelian groups is cartesian if and only if the maps $(f_{\\delta^n_j})^{-1}\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$ are isomorphisms. \\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only if the maps $(f_{\\delta^n_j})^{-1}K_{n - 1} \\to K_n$ are isomorphisms. \\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$ a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is cartesian if and only if the maps $(f_{\\delta^n_j})^*\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$ are isomorphisms. \\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$ an object $K$ of $D(\\mathcal{O})$ is cartesian if and only if the maps $L(f_{\\delta^n_j})^*K_{n - 1} \\to K_n$ are isomorphisms. \\item Add more here. \\end{enumerate}"}
+{"_id": "9053", "title": "spaces-simplicial-lemma-augmentation-cartesian-module", "text": "In Situation \\ref{situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Remark \\ref{remark-augmentation-site}. \\begin{enumerate} \\item The pullback $a^{-1}\\mathcal{G}$ of a sheaf of sets or abelian groups on $\\mathcal{D}$ is cartesian. \\item The pullback $a^{-1}K$ of an object $K$ of $D(\\mathcal{D})$ is cartesian. \\end{enumerate} Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{O}_\\mathcal{D}$ a sheaf of rings on $\\mathcal{D}$ and $a^\\sharp : \\mathcal{O}_\\mathcal{D} \\to a_*\\mathcal{O}$ a morphism as in Section \\ref{section-cohomology-augmentation-ringed-simplicial-sites}. \\begin{enumerate} \\item[(3)] The pullback $a^*\\mathcal{F}$ of a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules is cartesian. \\item[(4)] The derived pullback $La^*K$ of an object $K$ of $D(\\mathcal{O}_\\mathcal{D})$ is cartesian. \\end{enumerate}"}
+{"_id": "9054", "title": "spaces-simplicial-lemma-characterize-cartesian", "text": "In Situation \\ref{situation-simplicial-site}. The category of cartesian sheaves of sets (resp.\\ abelian groups) is equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$ where $\\mathcal{F}$ is a sheaf of sets (resp.\\ abelian groups) on $\\mathcal{C}_0$ and $$ \\alpha : (f_{\\delta_1^1})^{-1}\\mathcal{F} \\longrightarrow (f_{\\delta_0^1})^{-1}\\mathcal{F} $$ is an isomorphism of sheaves of sets (resp.\\ abelian groups) on $\\mathcal{C}_1$ such that $(f_{\\delta^2_1})^{-1}\\alpha = (f_{\\delta^2_0})^{-1}\\alpha \\circ (f_{\\delta^2_2})^{-1}\\alpha$ as maps of sheaves on $\\mathcal{C}_2$."}
+{"_id": "9055", "title": "spaces-simplicial-lemma-characterize-cartesian-modules", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. The category of cartesian $\\mathcal{O}$-modules is equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$ where $\\mathcal{F}$ is a $\\mathcal{O}_0$-module and $$ \\alpha : (f_{\\delta_1^1})^*\\mathcal{F} \\longrightarrow (f_{\\delta_0^1})^*\\mathcal{F} $$ is an isomorphism of $\\mathcal{O}_1$-modules such that $(f_{\\delta^2_1})^*\\alpha = (f_{\\delta^2_0})^*\\alpha \\circ (f_{\\delta^2_2})^*\\alpha$ as $\\mathcal{O}_2$-module maps."}
+{"_id": "9056", "title": "spaces-simplicial-lemma-Serre-subcat-cartesian-modules", "text": "In Situation \\ref{situation-simplicial-site}. \\begin{enumerate} \\item The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of $\\textit{Ab}(\\mathcal{C}_{total})$. Colimits of systems of cartesian abelian sheaves are cartesian. \\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ such that the morphisms $$ f_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1}) $$ are flat. The full subcategory of cartesian $\\mathcal{O}$-modules forms a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$. Colimits of systems of cartesian $\\mathcal{O}$-modules are cartesian. \\end{enumerate}"}
+{"_id": "9058", "title": "spaces-simplicial-lemma-derived-cartesian-shriek", "text": "In Situation \\ref{situation-simplicial-site}. \\begin{enumerate} \\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only the canonical map $$ g_{n!}K_n \\longrightarrow g_{n!}\\mathbf{Z} \\otimes^\\mathbf{L}_\\mathbf{Z} K $$ is an isomorphism for all $n$. \\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ such that the morphisms $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ are flat for all $\\varphi : [n] \\to [m]$. Then an object $K$ of $D(\\mathcal{O})$ is cartesian if and only if the canonical map $$ g_{n!}K_n \\longrightarrow g_{n!}\\mathcal{O}_n \\otimes^\\mathbf{L}_\\mathcal{O} K $$ is an isomorphism for all $n$. \\end{enumerate}"}
+{"_id": "9059", "title": "spaces-simplicial-lemma-quasi-coherent-sheaf", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Then $\\mathcal{F}$ is quasi-coherent in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local} if and only if $\\mathcal{F}$ is cartesian and $\\mathcal{F}_n$ is a quasi-coherent $\\mathcal{O}_n$-module for all $n$."}
+{"_id": "9060", "title": "spaces-simplicial-lemma-cartesian-objects-derived", "text": "In Situation \\ref{situation-simplicial-site}. If $K \\in D(\\mathcal{C}_{total})$ is an object, then $(K_n, K(\\varphi))$ is a simplicial system of the derived category. If $K$ is cartesian, so is the system."}
+{"_id": "9061", "title": "spaces-simplicial-lemma-abelian-postnikov", "text": "In Situation \\ref{situation-simplicial-site}. Let $K$ be an object of $D(\\mathcal{C}_{total})$. Set $$ X_n = (g_{n!}\\mathbf{Z}) \\otimes^\\mathbf{L}_\\mathbf{Z} K \\quad\\text{and}\\quad Y_n = (g_{n!}\\mathbf{Z} \\to \\ldots \\to g_{0!}\\mathbf{Z})[-n] \\otimes^\\mathbf{L}_\\mathbf{Z} K $$ as objects of $D(\\mathcal{C}_{total})$ where the maps are as in Lemma \\ref{lemma-simplicial-resolution-Z-site}. With the evident canonical maps $Y_n \\to X_n$ and $Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have \\begin{enumerate} \\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$ define a Postnikov system (Derived Categories, Definition \\ref{derived-definition-postnikov-system}) for $\\ldots \\to X_2 \\to X_1 \\to X_0$, \\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{C}_{total})$. \\end{enumerate}"}
+{"_id": "9062", "title": "spaces-simplicial-lemma-nullity-cartesian-objects-derived", "text": "In Situation \\ref{situation-simplicial-site}. If $K, K' \\in D(\\mathcal{C}_{total})$. Assume \\begin{enumerate} \\item $K$ is cartesian, \\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and \\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$. \\end{enumerate} Then any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero."}
+{"_id": "9063", "title": "spaces-simplicial-lemma-hom-cartesian-objects-derived", "text": "In Situation \\ref{situation-simplicial-site}. If $K, K' \\in D(\\mathcal{C}_{total})$. Assume \\begin{enumerate} \\item $K$ is cartesian, \\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$. \\end{enumerate} Then any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems  of $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{C}_{total})$."}
+{"_id": "9064", "title": "spaces-simplicial-lemma-cartesian-object-derived-from-simplicial", "text": "In Situation \\ref{situation-simplicial-site}. Let $(K_n, K_\\varphi)$ be a simplicial system of the derived category. Assume \\begin{enumerate} \\item $(K_n, K_\\varphi)$ is cartesian, \\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$. \\end{enumerate} Then there exists a cartesian object $K$ of $D(\\mathcal{C}_{total})$ whose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$."}
+{"_id": "9065", "title": "spaces-simplicial-lemma-cartesian-objects-derived-modules", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. If $K \\in D(\\mathcal{O})$ is an object, then $(K_n, K(\\varphi))$ is a simplicial system of the derived category of modules. If $K$ is cartesian, so is the system."}
+{"_id": "9066", "title": "spaces-simplicial-lemma-modules-postnikov", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. Let $K$ be an object of $D(\\mathcal{C}_{total})$. Set $$ X_n = (g_{n!}\\mathcal{O}_n) \\otimes^\\mathbf{L}_\\mathcal{O} K \\quad\\text{and}\\quad Y_n = (g_{n!}\\mathcal{O}_n \\to \\ldots \\to g_{0!}\\mathcal{O}_0)[-n] \\otimes^\\mathbf{L}_\\mathcal{O} K $$ as objects of $D(\\mathcal{O})$ where the maps are as in Lemma \\ref{lemma-simplicial-resolution-Z-site}. With the evident canonical maps $Y_n \\to X_n$ and $Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have \\begin{enumerate} \\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$ define a Postnikov system (Derived Categories, Definition \\ref{derived-definition-postnikov-system}) for $\\ldots \\to X_2 \\to X_1 \\to X_0$, \\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{O})$. \\end{enumerate}"}
+{"_id": "9067", "title": "spaces-simplicial-lemma-nullity-cartesian-modules-derived", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. If $K, K' \\in D(\\mathcal{O})$. Assume \\begin{enumerate} \\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for $\\varphi : [m] \\to [n]$, \\item $K$ is cartesian, \\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and \\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$. \\end{enumerate} Then any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero."}
+{"_id": "9068", "title": "spaces-simplicial-lemma-hom-cartesian-modules-derived", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. If $K, K' \\in D(\\mathcal{O})$. Assume \\begin{enumerate} \\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for $\\varphi : [m] \\to [n]$, \\item $K$ is cartesian, \\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$. \\end{enumerate} Then any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems  of $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{O})$."}
+{"_id": "9069", "title": "spaces-simplicial-lemma-cartesian-module-derived-from-simplicial", "text": "In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. Let $(K_n, K_\\varphi)$ be a simplicial system of the derived category of modules. Assume \\begin{enumerate} \\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for $\\varphi : [m] \\to [n]$, \\item $(K_n, K_\\varphi)$ is cartesian, \\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$. \\end{enumerate} Then there exists a cartesian object $K$ of $D(\\mathcal{O})$ whose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$."}
+{"_id": "9071", "title": "spaces-simplicial-lemma-push-pull-localization", "text": "Let $\\mathcal{C}$ be a site and $K$ in $\\text{SR}(\\mathcal{C})$. For $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ we have $$ j_*j^{-1}\\mathcal{F} = \\SheafHom(F(K)^\\#, \\mathcal{F}) $$ where $F$ is as in Hypercoverings, Definition \\ref{hypercovering-definition-SR-F}."}
+{"_id": "9072", "title": "spaces-simplicial-lemma-localize-compare", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor $j_!$ gives an equivalence $\\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})/F(K)^\\#$ where $F$ is as in Hypercoverings, Definition \\ref{hypercovering-definition-SR-F}. \\item The functor $j^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}/K)$ corresponds via the identification of (1) with $\\mathcal{F} \\mapsto (\\mathcal{F} \\times F(K)^\\# \\to F(K)^\\#)$. \\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\\Sh(\\mathcal{C})/F(L)^\\# \\to \\Sh(\\mathcal{C})/F(K)^\\#$, $(\\mathcal{G} \\to F(L)^\\#) \\mapsto (\\mathcal{G} \\times_{F(L)^\\#} F(K)^\\# \\to F(K)^\\#)$. \\end{enumerate}"}
+{"_id": "9073", "title": "spaces-simplicial-lemma-localize-injective", "text": "Let $\\mathcal{C}$ be a site. For $K$ in $\\text{SR}(\\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves."}
+{"_id": "9074", "title": "spaces-simplicial-lemma-augmentation-simplicial-semi-representable", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\\text{SR}(\\mathcal{C})$. The localization functor $j_0 : \\mathcal{C}/K_0 \\to \\mathcal{C}$ defines an augmentation $a_0 : \\Sh(\\mathcal{C}/K_0) \\to \\Sh(\\mathcal{C})$, as in case (B) of Remark \\ref{remark-augmentation-site}. The corresponding morphisms of topoi $$ a_n : \\Sh(\\mathcal{C}/K_n) \\longrightarrow \\Sh(\\mathcal{C}),\\quad a : \\Sh((\\mathcal{C}/K)_{total}) \\longrightarrow \\Sh(\\mathcal{C}) $$ of Lemma \\ref{lemma-augmentation-site} are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors $j_n : \\mathcal{C}/K_n \\to \\mathcal{C}$ and $j_{total} : (\\mathcal{C}/K)_{total} \\to \\mathcal{C}$."}
+{"_id": "9075", "title": "spaces-simplicial-lemma-comparison", "text": "With assumption and notation as in Lemma \\ref{lemma-augmentation-simplicial-semi-representable} we have the following properties: \\begin{enumerate} \\item there is a functor $a^{Sh}_! : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C})$ left adjoint to $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$, \\item there is a functor $a_! : \\textit{Ab}((\\mathcal{C}/K)_{total}) \\to \\textit{Ab}(\\mathcal{C})$ left adjoint to $a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}((\\mathcal{C}/K)_{total})$, \\item the functor $a^{-1}$ associates to $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ the sheaf on $(\\mathcal{C}/K)_{total}$ wich in degree $n$ is equal to $a_n^{-1}\\mathcal{F}$, \\item the functor $a_*$ associates to $\\mathcal{G}$ in $\\textit{Ab}((\\mathcal{C}/K)_{total})$ the equalizer of the two maps $j_{0, *}\\mathcal{G}_0 \\to j_{1, *}\\mathcal{G}_1$, \\end{enumerate}"}
+{"_id": "9076", "title": "spaces-simplicial-lemma-sanity-check-simplicial-semi-representable", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\\text{SR}(\\mathcal{C})$. Let $U/U_{n, i}$ be an object of $\\mathcal{C}/K_n$. Let $\\mathcal{F} \\in \\textit{Ab}((\\mathcal{C}/K)_{total})$. Then $$ H^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}_{n, i}) $$ where \\begin{enumerate} \\item on the left hand side $U$ is viewed as an object of $\\mathcal{C}_{total}$, and \\item on the right hand side $\\mathcal{F}_{n, i}$ is the $i$th component of the sheaf $\\mathcal{F}_n$ on $\\mathcal{C}/K_n$ in the decomposition $\\Sh(\\mathcal{C}/K_n) = \\prod \\Sh(\\mathcal{C}/U_{n, i})$ of Section \\ref{section-semi-representable}. \\end{enumerate}"}
+{"_id": "9077", "title": "spaces-simplicial-lemma-hypercovering-descent-sheaves", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then \\begin{enumerate} \\item $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \\item $a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}((\\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \\end{enumerate} In both cases $a_*$ provides the quasi-inverse functor."}
+{"_id": "9078", "title": "spaces-simplicial-lemma-hypercovering-cech-complex", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. The {\\v C}ech complex of Lemma \\ref{lemma-augmentation-cech-complex} associated to $a^{-1}\\mathcal{F}$ $$ a_{0, *}a_0^{-1}\\mathcal{F} \\to a_{1, *}a_1^{-1}\\mathcal{F} \\to a_{2, *}a_2^{-1}\\mathcal{F} \\to \\ldots $$ is equal to the complex $\\SheafHom(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{F})$. Here $s(\\mathbf{Z}_{F(K)}^\\#)$ is as in Hypercoverings, Definition \\ref{hypercovering-definition-homology}."}
+{"_id": "9079", "title": "spaces-simplicial-lemma-hypercovering-descent-bounded-abelian", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. For $E \\in D(\\mathcal{C})$ the map $$ E \\longrightarrow Ra_*a^{-1}E $$ is an isomorphism."}
+{"_id": "9080", "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then we have a canonical isomorphism $$ R\\Gamma(\\mathcal{C}, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \\in D(\\mathcal{C})$."}
+{"_id": "9081", "title": "spaces-simplicial-lemma-hypercovering-equivalence-bounded", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence $$ D^+(\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9082", "title": "spaces-simplicial-lemma-hypercovering-descent-modules", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. With notation as above $$ a^* : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to \\textit{Mod}(\\mathcal{O}) $$ is fully faithful with essential image the cartesian $\\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse."}
+{"_id": "9083", "title": "spaces-simplicial-lemma-hypercovering-descent-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. For $E \\in D(\\mathcal{O}_\\mathcal{C})$ the map $$ E \\longrightarrow Ra_*La^*E $$ is an isomorphism."}
+{"_id": "9084", "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-modules", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. Then we have a canonical isomorphism $$ R\\Gamma(\\mathcal{C}, E) = R\\Gamma((\\mathcal{C}/K)_{total}, La^*E) $$ for $E \\in D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "9085", "title": "spaces-simplicial-lemma-hypercovering-equivalence-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9086", "title": "spaces-simplicial-lemma-hypercovering-X-descent-sheaves", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then \\begin{enumerate} \\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to \\textit{Ab}((\\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \\end{enumerate} In both cases $a_*$ provides the quasi-inverse functor."}
+{"_id": "9087", "title": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-abelian", "text": "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$. Let $K$ be a hypercovering of $X$. For $E \\in D(\\mathcal{C}/X)$ the map $$ E \\longrightarrow Ra_*a^{-1}E $$ is an isomorphism."}
+{"_id": "9088", "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then we have a canonical isomorphism $$ R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \\in D(\\mathcal{C}/X)$."}
+{"_id": "9089", "title": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $K$ be a hypercovering of $X$. Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence $$ D^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9090", "title": "spaces-simplicial-lemma-hypercovering-X-descent-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. With notation as above $$ a^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}) $$ is fully faithful with essential image the cartesian $\\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse."}
+{"_id": "9091", "title": "spaces-simplicial-lemma-hypercovering-X-descent-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. For $E \\in D(\\mathcal{O}_X)$ the map $$ E \\longrightarrow Ra_*La^*E $$ is an isomorphism."}
+{"_id": "9092", "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. Then we have a canonical isomorphism $$ R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, La^*E) $$ for $E \\in D(\\mathcal{O}_\\mathcal{C})$."}
+{"_id": "9093", "title": "spaces-simplicial-lemma-hypercovering-X-equivalence-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9094", "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-sheaves", "text": "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$. Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above. Then \\begin{enumerate} \\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of sets, \\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to \\textit{Ab}((\\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups. \\end{enumerate} In both cases $a_*$ provides the quasi-inverse functor."}
+{"_id": "9095", "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-abelian", "text": "Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$. Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above. For $E \\in D(\\mathcal{C}/X)$ the map $$ E \\longrightarrow Ra_*a^{-1}E $$ is an isomorphism."}
+{"_id": "9098", "title": "spaces-simplicial-lemma-sr-when-fibre-products", "text": "Let $U$ be a simplicial object of a site $\\mathcal{C}$ with fibre products. \\begin{enumerate} \\item $\\mathcal{C}/U$ has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites, \\item the construction of Lemma \\ref{lemma-simplicial-site-site} applied to the structure in (1) reproduces the site $(\\mathcal{C}/U)_{total}$ above, \\item if $a : U \\to X$ is an augmentation, then $a_0 : \\mathcal{C}/U_0 \\to \\mathcal{C}/X$ is an augmentation as in Remark \\ref{remark-augmentation-site} part (A) and gives the same morphism of topoi $a : \\Sh((\\mathcal{C}/U)_{total}) \\to \\Sh(\\mathcal{C}/X)$ as the one above. \\end{enumerate}"}
+{"_id": "9099", "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$. With notation as above $$ a^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}) $$ is fully faithful with essential image the cartesian $\\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse."}
+{"_id": "9100", "title": "spaces-simplicial-lemma-hypercovering-X-simple-descent-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$. For $E \\in D(\\mathcal{O}_X)$ the map $$ E \\longrightarrow Ra_*La^*E $$ is an isomorphism."}
+{"_id": "9103", "title": "spaces-simplicial-lemma-hypercovering-equivalence-modules", "text": "Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site. Assume given weak Serre subcategories $\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$ satisfying conditions (\\ref{item-restriction}), (\\ref{item-local}), and (\\ref{item-bounded-dimension}) above. Assume $\\mathcal{C}$ has equalizers and fibre products and let $K$ be a hypercovering. Let $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in Remark \\ref{remark-augmentation-ringed}. Let $\\mathcal{A}_{total} \\subset \\textit{Mod}(\\mathcal{O})$ denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules $\\mathcal{F}$ whose restriction $\\mathcal{F}_n$ is in $\\mathcal{A}_{K_n}$ for all $n$ (as defined above). Then the functor $La^*$ defines an equivalence $$ D_\\mathcal{A}(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D_{\\mathcal{A}_{total}}(\\mathcal{O}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9104", "title": "spaces-simplicial-lemma-prepare-bbd-glueing", "text": "In Situation \\ref{situation-locally-given}. Assume negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero. Let $L$ be a simplicial object of $\\text{SR}(\\mathcal{B})$. Consider the simplicial object $K = u(L)$ of $\\text{SR}(\\mathcal{C})$ and let $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in Remark \\ref{remark-augmentation-ringed}. There exists a cartesian object $E$ of $D(\\mathcal{O})$ such that writing $L_n = \\{U_{n, i}\\}_{i \\in I_n}$ the restriction of $E$ to $D(\\mathcal{O}_{\\mathcal{C}/u(U_{n, i})})$ is $E_{U_{n, i}}$ compatibly (see proof for details). Moreover, $E$ is unique up to unique isomorphism."}
+{"_id": "9105", "title": "spaces-simplicial-lemma-bbd-glueing", "text": "In Situation \\ref{situation-locally-given}. Assume \\begin{enumerate} \\item $\\mathcal{C}$ has equalizers and fibre products, \\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$ given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$ such that \\begin{enumerate} \\item $\\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them, \\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$ and $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$, \\item every object of $\\mathcal{D}$ has a covering whose members are objects of $\\mathcal{B}$, \\end{enumerate} \\item all negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero, and \\item there exists a $t \\in \\mathbf{Z}$ such that $H^i(E_U) = 0$ for $i < t$ and $U \\in \\Ob(\\mathcal{B})$. \\end{enumerate} Then there exists a solution unique up to unique isomorphism."}
+{"_id": "9107", "title": "spaces-simplicial-lemma-compare-simplicial-objects", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. There is a commutative diagram $$ \\xymatrix{ \\Sh((\\textit{LC}_{qc}/U)_{total}) \\ar[r]_-h \\ar[d]_{a_{qc}} & \\Sh(U_{Zar}) \\ar[d]^a \\\\ \\Sh(\\textit{LC}_{qc}/X) \\ar[r]^-{h_{-1}} & \\Sh(X) } $$ where the left vertical arrow is defined in Section \\ref{section-hypercovering} and the right vertical arrow is defined in Lemma \\ref{lemma-augmentation}."}
+{"_id": "9108", "title": "spaces-simplicial-lemma-descent-sheaves-for-proper-hypercovering", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$, then $$ a^{-1} : \\Sh(X) \\to \\Sh(U_{Zar}) \\quad\\text{and}\\quad a^{-1} : \\textit{Ab}(X) \\to \\textit{Ab}(U_{Zar}) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in Lemma \\ref{lemma-augmentation}."}
+{"_id": "9109", "title": "spaces-simplicial-lemma-cohomological-descent-for-proper-hypercovering", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$, then for $K \\in D^+(X)$ $$ K \\to Ra_*(a^{-1}K) $$ is an isomorphism where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in Lemma \\ref{lemma-augmentation}."}
+{"_id": "9110", "title": "spaces-simplicial-lemma-compute-via-proper-hypercovering", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. If $U$ is a proper hypercovering of $X$, then $$ R\\Gamma(X, K) = R\\Gamma(U_{Zar}, a^{-1}K) $$ for $K \\in D^+(X)$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in Lemma \\ref{lemma-augmentation}."}
+{"_id": "9113", "title": "spaces-simplicial-lemma-characterize-cartesian-schemes", "text": "Let $X$ be a simplicial scheme. The category of simplicial schemes cartesian over $X$ is equivalent to the category of pairs $(V, \\varphi)$ where $V$ is a scheme over $X_0$ and $$ \\varphi : V \\times_{X_0, d^1_1} X_1 \\longrightarrow X_1 \\times_{d^1_0, X_0} V $$ is an isomorphism over $X_1$ such that $(s_0^0)^*\\varphi = \\text{id}_V$ and such that $$ (d^2_1)^*\\varphi = (d^2_0)^*\\varphi \\circ (d^2_2)^*\\varphi $$ as morphisms of schemes over $X_2$."}
+{"_id": "9114", "title": "spaces-simplicial-lemma-cartesian-over", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism of simplicial schemes. Set $V = Y_0$ considered as a scheme over $X$. The morphisms $d^1_0, d^1_1 : Y_1 \\to Y_0$ and the morphism $\\pi_1 : Y_1 \\to X \\times_S X$ induce isomorphisms $$ \\xymatrix{ V \\times_S X & & Y_1 \\ar[ll]_-{(d^1_1, \\text{pr}_1 \\circ \\pi_1)} \\ar[rr]^-{(\\text{pr}_0 \\circ \\pi_1, d^1_0)} & & X \\times_S V. } $$ Denote $\\varphi : V \\times_S X \\to X \\times_S V$ the resulting isomorphism. Then the pair $(V, \\varphi)$ is a descent datum relative to $X \\to S$."}
+{"_id": "9115", "title": "spaces-simplicial-lemma-cartesian-equivalent-descent-datum", "text": "Let $f : X \\to S$ be a morphism of schemes. The construction $$ \\begin{matrix} \\text{category of cartesian } \\\\ \\text{schemes over } (X/S)_\\bullet \\end{matrix} \\longrightarrow \\begin{matrix} \\text{ category of descent data} \\\\ \\text{ relative to } X/S \\end{matrix} $$ of Lemma \\ref{lemma-cartesian-over} is an equivalence of categories."}
+{"_id": "9116", "title": "spaces-simplicial-lemma-pullback-cartesian-module", "text": "Let $f : V \\to U$ be a morphism of simplicial schemes. Given a quasi-coherent module $\\mathcal{F}$ on $U_{Zar}$ the pullback $f^*\\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$."}
+{"_id": "9117", "title": "spaces-simplicial-lemma-pushforward-cartesian-module", "text": "Let $f : V \\to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are flat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated. For a quasi-coherent module $\\mathcal{G}$ on $V_{Zar}$ the pushforward $f_*\\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$."}
+{"_id": "9120", "title": "spaces-simplicial-lemma-groupoid-simplicial", "text": "Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. There exists a simplicial scheme $X$ over $S$ with the following properties \\begin{enumerate} \\item $X_0 = U$, $X_1 = R$, $X_2 = R \\times_{s, U, t} R$, \\item $s_0^0 = e : X_0 \\to X_1$, \\item $d^1_0 = s : X_1 \\to X_0$, $d^1_1 = t : X_1 \\to X_0$, \\item $s_0^1 = (e \\circ t, 1) : X_1 \\to X_2$, $s_1^1 = (1, e \\circ t) : X_1 \\to X_2$, \\item $d^2_0 = \\text{pr}_1 : X_2 \\to X_1$, $d^2_1 = c : X_2 \\to X_1$, $d^2_2 = \\text{pr}_0$, and \\item $X = \\text{cosk}_2 \\text{sk}_2 X$. \\end{enumerate} For all $n$ we have $X_n = R \\times_{s, U, t} \\ldots \\times_{s, U, t} R$ with $n$ factors. The map $d^n_j : X_n \\to X_{n - 1}$ is given on functors of points by $$ (r_1, \\ldots, r_n) \\longmapsto (r_1, \\ldots, c(r_j, r_{j + 1}), \\ldots, r_n) $$ for $1 \\leq j \\leq n - 1$ whereas $d^n_0(r_1, \\ldots, r_n) = (r_2, \\ldots, r_n)$ and $d^n_n(r_1, \\ldots, r_n) = (r_1, \\ldots, r_{n - 1})$."}
+{"_id": "9123", "title": "spaces-simplicial-lemma-equivalence-relation", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism of simplicial schemes, see Definitions \\ref{definition-cartesian-morphism} and \\ref{definition-fibre-products-simplicial-scheme}. Then the morphism $$ j = (d^1_1, d^1_0) : Y_1 \\to Y_0 \\times_S Y_0 $$ defines an equivalence relation on $Y_0$ over $S$, see Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}."}
+{"_id": "9124", "title": "spaces-simplicial-lemma-equivalence-classes-points", "text": "Let $X \\to S$ be a morphism of schemes. Suppose $Y \\to (X/S)_\\bullet$ is a cartesian morphism of simplicial schemes. For $y \\in Y_0$ a point define $$ T_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\\} $$ as a subset of $Y_0$. Then $y \\in T_y$ and $T_y \\cap T_{y'} \\not = \\emptyset \\Rightarrow T_y = T_{y'}$."}
+{"_id": "9125", "title": "spaces-simplicial-lemma-quasi-compact", "text": "Let $X \\to S$ be a morphism of schemes. Suppose $Y \\to (X/S)_\\bullet$ is a cartesian morphism of simplicial schemes. Let $y \\in Y_0$ be a point. If $X \\to S$ is quasi-compact, then $$ T_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\\} $$ is a quasi-compact subset of $Y_0$."}
+{"_id": "9127", "title": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. There is a commutative diagram $$ \\xymatrix{ \\Sh((\\textit{Spaces}/U)_{fppf, total}) \\ar[r]_-h \\ar[d]_{a_{fppf}} & \\Sh(U_\\etale) \\ar[d]^a \\\\ \\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[r]^-{h_{-1}} & \\Sh(X_\\etale) } $$ where the left vertical arrow is defined in Section \\ref{section-hypercovering} and the right vertical arrow is defined in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9128", "title": "spaces-simplicial-lemma-descent-sheaves-for-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then $$ a^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale) \\quad\\text{and}\\quad a^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9129", "title": "spaces-simplicial-lemma-cohomological-descent-for-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then for $K \\in D^+(X_\\etale)$ $$ K \\to Ra_*(a^{-1}K) $$ is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9130", "title": "spaces-simplicial-lemma-compute-via-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then $$ R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K) $$ for $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9133", "title": "spaces-simplicial-lemma-compare-simplicial-objects-fppf-etale-modules", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. There is a commutative diagram $$ \\xymatrix{ (\\Sh((\\textit{Spaces}/U)_{fppf, total}), \\mathcal{O}_{big, total}) \\ar[r]_-h \\ar[d]_{a_{fppf}} & (\\Sh(U_\\etale), \\mathcal{O}_U) \\ar[d]^a \\\\ (\\Sh((\\textit{Spaces}/X)_{fppf}), \\mathcal{O}_{big}) \\ar[r]^-{h_{-1}} & (\\Sh(X_\\etale), \\mathcal{O}_X) } $$ of ringed topoi where the left vertical arrow is defined in Section \\ref{section-hypercovering-modules} and the right vertical arrow is defined in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9134", "title": "spaces-simplicial-lemma-descent-qcoh-for-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then $$ a^* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U) $$ is an equivalence fully faithful with quasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9135", "title": "spaces-simplicial-lemma-cohomological-descent-qcoh-for-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then for $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module the map $$ \\mathcal{F} \\to Ra_*(a^*\\mathcal{F}) $$ is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9136", "title": "spaces-simplicial-lemma-coh-descent-qcoh-unbounded-for-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. Assume $a : U \\to X$ is an fppf hypercovering of $X$. Then $\\QCoh(\\mathcal{O}_U)$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_U)$ and $$ a^* : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(\\mathcal{O}_U) $$ is an equivalence of categories with quasi-inverse given by $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9137", "title": "spaces-simplicial-lemma-compute-via-fppf-hypercovering-modules", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then $$ R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^*K) $$ for $K \\in D_\\QCoh(\\mathcal{O}_X)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9139", "title": "spaces-simplicial-lemma-fppf-neg-ext-zero-hom", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $K, E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $a : U \\to X$ be an fppf hypercovering. Assume that for all $n \\geq 0$ we have $$ \\Ext_{\\mathcal{O}_{U_n}}^i(La_n^*K, La_n^*E) = 0 \\text{ for } i < 0 $$ Then we have \\begin{enumerate} \\item $\\Ext_{\\mathcal{O}_X}^i(K, E) = 0$ for $i < 0$, and \\item there is an exact sequence $$ 0 \\to \\Hom_{\\mathcal{O}_X}(K, E) \\to \\Hom_{\\mathcal{O}_{U_0}}(La_0^*K, La_0^*E) \\to \\Hom_{\\mathcal{O}_{U_1}}(La_1^*K, La_1^*E) $$ \\end{enumerate}"}
+{"_id": "9140", "title": "spaces-simplicial-lemma-fppf-glue-neg-ext-zero", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $a : U \\to X$ be an fppf hypercovering. Suppose given $K_0 \\in D_\\QCoh(U_0)$ and an isomorphism $$ \\alpha : L(f_{\\delta_1^1})^*K_0 \\longrightarrow L(f_{\\delta_0^1})^*K_0 $$ satisfying the cocycle condition on $U_1$. Set $\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and set $K_n = Lf_{\\tau^n_n}^*K_0$. Assume $\\Ext^i_{\\mathcal{O}_{U_n}}(K_n, K_n) = 0$ for $i < 0$. Then there exists an object $K \\in D_\\QCoh(\\mathcal{O}_X)$ and an isomorphism $La_0^*K \\to K$ compatible with $\\alpha$."}
+{"_id": "9141", "title": "spaces-simplicial-lemma-compare-simplicial-objects-ph-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. There is a commutative diagram $$ \\xymatrix{ \\Sh((\\textit{Spaces}/U)_{ph, total}) \\ar[r]_-h \\ar[d]_{a_{ph}} & \\Sh(U_\\etale) \\ar[d]^a \\\\ \\Sh((\\textit{Spaces}/X)_{ph}) \\ar[r]^-{h_{-1}} & \\Sh(X_\\etale) } $$ where the left vertical arrow is defined in Section \\ref{section-hypercovering} and the right vertical arrow is defined in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9142", "title": "spaces-simplicial-lemma-descent-sheaves-for-ph-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$, then $$ a^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale) \\quad\\text{and}\\quad a^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale) $$ are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9143", "title": "spaces-simplicial-lemma-cohomological-descent-for-ph-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$, then for $K \\in D^+(X_\\etale)$ $$ K \\to Ra_*(a^{-1}K) $$ is an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9144", "title": "spaces-simplicial-lemma-compute-via-ph-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$, then $$ R\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K) $$ for $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9164", "title": "examples-stacks-lemma-quasi-coherent-strongly-cartesian", "text": "A morphism $(f, \\varphi) : (Y, \\mathcal{G}) \\to (X, \\mathcal{F})$ of $\\QCohstack$ is strongly cartesian if and only if the map $\\varphi$ induces an isomorphism $f^*\\mathcal{F} \\to \\mathcal{G}$."}
+{"_id": "9166", "title": "examples-stacks-lemma-stack-of-finite-type-quasi-coherent-sheaves", "text": "The functor $p_{fg} : \\QCohstack_{fg} \\to (\\Sch/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition \\ref{stacks-definition-stack}."}
+{"_id": "9167", "title": "examples-stacks-lemma-finite-type", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item The category of finite type $\\mathcal{O}_X$-modules has a set of isomorphism classes. \\item The category of finite type quasi-coherent $\\mathcal{O}_X$-modules has a set of isomorphism classes. \\end{enumerate}"}
+{"_id": "9170", "title": "examples-stacks-lemma-spaces-strongly-cartesian", "text": "A morphism $(f, g) : X/U \\to Y/V$ of $\\Spacesstack$ is strongly cartesian if and only if the map $f$ induces an isomorphism $X \\to U \\times_{g, V} Y$."}
+{"_id": "9171", "title": "examples-stacks-lemma-pre-stack-of-spaces", "text": "The functor $p : \\Spacesstack \\to (\\Sch/S)_{fppf}$ satisfies conditions (1) and (2) of Stacks, Definition \\ref{stacks-definition-stack}."}
+{"_id": "9172", "title": "examples-stacks-lemma-stack-of-finite-type-spaces", "text": "The functor $p_{ft} : \\Spacesstack_{ft} \\to (\\Sch/S)_{fppf}$ satisfies the conditions (1), (2) and (3) of Stacks, Definition \\ref{stacks-definition-stack}."}
+{"_id": "9174", "title": "examples-stacks-lemma-torsors-sheaf-stack-in-groupoids", "text": "Up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ p : \\mathcal{G}\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9176", "title": "examples-stacks-lemma-torsors-stack-in-groupoids", "text": "Up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ p : G\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9177", "title": "examples-stacks-lemma-compare-torsors", "text": "Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Denote $\\mathcal{G}$, resp.\\ $\\mathcal{B}$ the algebraic space $G$, resp.\\ $B$ seen as a sheaf on $(\\Sch/S)_{fppf}$. The functor $$ G\\textit{-Torsors} \\longrightarrow \\mathcal{G}/\\mathcal{B}\\textit{-Torsors} $$ which associates to a triple $(U, b, X)$ the triple $(U, b, \\mathcal{X})$ where $\\mathcal{X}$ is $X$ viewed as a sheaf is an equivalence of stacks in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9180", "title": "examples-stacks-lemma-picard-stack", "text": "Up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ \\Picardstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9182", "title": "examples-stacks-lemma-faithful-hilbert", "text": "The $1$-morphism $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X})$ is faithful."}
+{"_id": "9183", "title": "examples-stacks-proposition-equal-quotient-stacks", "text": "In Situation \\ref{situation-quotient-stack} there exists a canonical equivalence $$ [X/G] \\longrightarrow [[X/G]] $$ of stacks in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9190", "title": "models-theorem-semistable-reduction", "text": "\\begin{reference} \\cite[Corollary 2.7]{DM} \\end{reference} Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. Then there exists an extension of discrete valuation rings $R \\subset R'$ which induces a finite separable extension of fraction fields $K \\subset K'$ such that $C_{K'}$ has semistable reduction. More precisely, we have the following \\begin{enumerate} \\item If the genus of $C$ is zero, then there exists a degree $2$ separable extension $K'/K$ such that $C_{K'} \\cong \\mathbf{P}^1_{K'}$ and hence $C_{K'}$ is isomorphic to the generic fibre of the smooth projective scheme $\\mathbf{P}^1_{R'}$ over the integral closure $R'$ of $R$ in $K'$. \\item If the genus of $C$ is one, then there exists a finite separable extension $K'/K$ such that $C_{K'}$ has semistable reduction over $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. Moreover, the special fibre of the (unique) minimal model of $C_{K'}$ over $R'_\\mathfrak m$ is either a smooth genus one curve or a cycle of rational curves. \\item If the genus $g$ of $C$ is greater than one, then there exists a finite separable extension $K'/K$ of degree at most $B_g$ (\\ref{equation-bound}) such that $C_{K'}$ has semistable reduction over $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. \\end{enumerate}"}
+{"_id": "9191", "title": "models-lemma-recurring", "text": "\\begin{reference} \\cite[Theorem I]{Taussky} \\end{reference} Let $A = (a_{ij})$ be a complex $n \\times n$ matrix. \\begin{enumerate} \\item If $|a_{ii}| > \\sum_{j \\not = i} |a_{ij}|$ for each $i$, then $\\det(A)$ is nonzero. \\item If there exists a real vector $m = (m_1, \\ldots, m_n)$ with $m_i > 0$ such that $|a_{ii} m_i| > \\sum_{j \\not = i} |a_{ij}m_j|$ for each $i$, then $\\det(A)$ is nonzero. \\end{enumerate}"}
+{"_id": "9192", "title": "models-lemma-recurring-real", "text": "Let $A = (a_{ij})$ be a real $n \\times n$ matrix with $a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be a real vector with $m_i > 0$. For $I \\subset \\{1, \\ldots, n\\}$ let $x_I \\in \\mathbf{R}^n$ be the vector whose $i$th coordinate is $m_i$ if $i \\in I$ and $0$ otherwise. If \\begin{equation} \\label{equation-ineq} -a_{ii}m_i \\geq \\sum\\nolimits_{j \\not = i} a_{ij}m_j \\end{equation} for each $i$, then $\\Ker(A)$ is the vector space spanned by the vectors $x_I$ such that \\begin{enumerate} \\item $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$, and \\item equality holds in (\\ref{equation-ineq}) for $i \\in I$. \\end{enumerate}"}
+{"_id": "9193", "title": "models-lemma-recurring-symmetric-real", "text": "Let $A = (a_{ij})$ be a symmetric real $n \\times n$ matrix with $a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be a real vector with $m_i > 0$. Assume \\begin{enumerate} \\item $Am = 0$, \\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$ such that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$. \\end{enumerate} Then $x^t A x \\leq 0$ with equality if and only if $x = qm$ for some $q \\in \\mathbf{R}$."}
+{"_id": "9194", "title": "models-lemma-orthogonal-direct-sum", "text": "Let $L$ be a finite free $\\mathbf{Z}$-module endowed with an integral symmetric bilinear positive definite form $\\langle\\ ,\\ \\rangle : L \\times L \\to \\mathbf{Z}$. Let $A \\subset L$ be a submodule with $L/A$ torsion free. Set $B = \\{b \\in L \\mid \\langle a, b\\rangle = 0,\\ \\forall a \\in A\\}$. Then we have injective maps $$ A^\\#/A \\leftarrow L/(A \\oplus B) \\rightarrow B^\\#/B $$ whose cokernels are quotients of $L^\\#/L$. Here $A^\\# = \\{a' \\in A \\otimes \\mathbf{Q} \\mid \\langle a, a'\\rangle \\in \\mathbf{Z},\\ \\forall a \\in A\\}$ and similarly for $B$ and $L$."}
+{"_id": "9195", "title": "models-lemma-coker", "text": "Let $L_0$, $L_1$ be a finite free $\\mathbf{Z}$-modules endowed with integral symmetric bilinear positive definite forms $\\langle\\ ,\\ \\rangle : L_i \\times L_i \\to \\mathbf{Z}$. Let $\\text{d} : L_0 \\to L_1$ and $\\text{d}^* : L_1 \\to L_0$ be adjoint. If $\\langle\\ ,\\ \\rangle$ on $L_0$ is unimodular, then there is an isomorphism $$ \\Phi : \\Coker(\\text{d}^*\\text{d})_{torsion} \\longrightarrow \\Im(\\text{d})^\\#/\\Im(\\text{d}) $$ with notation as in Lemma \\ref{lemma-orthogonal-direct-sum}."}
+{"_id": "9196", "title": "models-lemma-recurring-symmetric-integer", "text": "Let $A = (a_{ij})$ be a symmetric $n \\times n$ integer matrix with $a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be an integer vector with $m_i > 0$. Assume \\begin{enumerate} \\item $Am = 0$, \\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$ such that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$. \\end{enumerate} Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. Then for $\\ell$ a prime number coprime with all $a_{ij}$ and $m_i$ we have $$ \\dim_{\\mathbf{F}_\\ell}(\\Coker(A)[\\ell]) \\leq 1 - n + e $$"}
+{"_id": "9197", "title": "models-lemma-genus", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Then the expression $$ g = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii}) $$ is an integer."}
+{"_id": "9198", "title": "models-lemma-irreducible", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. If $n = 1$, then $a_{11} = 0$ and $g = 1 + m_1w_1(g_1 - 1)$. Moreover, we can classify all such numerical types as follows \\begin{enumerate} \\item If $g < 0$, then $g_1 = 0$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1 = 1 - g$. \\item If $g = 0$, then $m_1 = 1$, $w_1 = 1$, $g_1 = 0$ as in Lemma \\ref{lemma-genus-zero}. \\item If $g = 1$, then we conclude $g_1 = 1$ but $m_1, w_1$ can be arbitrary positive integers; this is case (\\ref{item-one}) of Lemma \\ref{lemma-genus-one}. \\item If $g > 1$, then $g_1 > 1$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1(g_1 - 1) = g - 1$. \\end{enumerate}"}
+{"_id": "9199", "title": "models-lemma-diagonal-negative", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. If $n > 1$, then $a_{ii} < 0$ for all $i$."}
+{"_id": "9200", "title": "models-lemma-minus-one", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Assume $n > 1$. If $i$ is such that the contribution $m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ to the genus $g$ is $< 0$, then $g_i = 0$ and $a_{ii} = -w_i$."}
+{"_id": "9201", "title": "models-lemma-contract", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Assume $n$ is a $(-1)$-index. Then there is a numerical type $T'$ given by $n', m'_i, a'_{ij}, w'_i, g'_i$ with \\begin{enumerate} \\item $n' = n - 1$, \\item $m'_i = m_i$, \\item $a'_{ij} = a_{ij} - a_{in}a_{jn}/a_{nn}$, \\item $w'_i = w_i/2$ if $a_{in}/w_n$ even and $a_{in}/w_i$ odd and $w'_i = w_i$ else, \\item $g'_i = \\frac{w_i}{w'_i}(g_i - 1) + 1 + \\frac{a_{in}^2 - w_na_{in}}{2w'_iw_n}$. \\end{enumerate} Moreover, we have $g = g'$."}
+{"_id": "9202", "title": "models-lemma-top-genus", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. Then the expression $g_{top} = 1 - n + e$ is $\\geq 0$."}
+{"_id": "9203", "title": "models-lemma-non-irreducible-minimal-type-genus-at-least-one", "text": "If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type with $n > 1$, then $g \\geq 1$."}
+{"_id": "9204", "title": "models-lemma-genus-nonnegative", "text": "If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type with $n > 1$, then $g \\geq g_{top}$."}
+{"_id": "9205", "title": "models-lemma-minus-two", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Assume $n > 1$. If $i$ is such that the contribution $m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ to the genus $g$ is $0$, then $g_i = 0$ and $a_{ii} = -2w_i$."}
+{"_id": "9207", "title": "models-lemma-picard-T-and-A", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Then $\\Pic(T) \\subset \\Coker(A)$ where $A = (a_{ij})$."}
+{"_id": "9208", "title": "models-lemma-contract-picard-group", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. Assume $n$ is a $(-1)$-index. Let $T'$ be the numerical type constructed in Lemma \\ref{lemma-contract}. There exists an injective map $$ \\Pic(T) \\to \\Pic(T') $$ whose cokernel is an elementary abelian $2$-group."}
+{"_id": "9210", "title": "models-lemma-two-by-two", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet } $$ If $n > 2$, then given a pair $i, j$ of $(-2)$-indices with $a_{ij} > 0$, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-A2} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w \\\\ w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\end{matrix} \\right) $$ with $w$ arbitrary and $2m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or \\item \\label{item-B2} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w \\\\ 2w & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\end{matrix} \\right) $$ with $w$ arbitrary and $m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or \\item \\label{item-G2} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 3w \\\\ 3w & -6w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 3w \\end{matrix} \\right) $$ with $w$ arbitrary and $2m_1 \\geq 3m_2$ and $2m_2 \\geq m_1$. \\end{enumerate}"}
+{"_id": "9211", "title": "models-lemma-three-by-three", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet } $$ If $n > 3$, then given a triple $i, j, k$ of $(-2)$-indices with at least two $a_{ij}, a_{ik}, a_{jk}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-A3} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 \\\\ w & -2w & w \\\\ 0 & w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2$, or \\item \\label{item-C3} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 \\\\ w & -2w & 2w \\\\ 0 & 2w & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ 2w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2$, or \\item \\label{item-B3} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 \\\\ 2w & -4w & 2w \\\\ 0 & 2w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ 2w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $m_3 \\geq m_2$. \\end{enumerate}"}
+{"_id": "9212", "title": "models-lemma-four-by-four", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet } $$ If $n > 4$, then given four $(-2)$-indices $i, j, k, l$ with $a_{ij}, a_{jk}, a_{kl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-A4} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 \\\\ w & -2w & w & 0 \\\\ 0 & w & -2w & w \\\\ 0 & 0 & w & -2w  \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, and $2m_4 \\geq m_3$, or \\item \\label{item-C4} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 \\\\ w & -2w & w & 0 \\\\ 0 & w & -2w & 2w \\\\ 0 & 0 & 2w & -4w  \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ 2w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$, and $2m_4 \\geq m_3$, or \\item \\label{item-B4} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 \\\\ 2w & -4w & 2w & 0 \\\\ 0 & 2w & -4w & 2w \\\\ 0 & 0 & 2w & -2w  \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ 2w \\\\ 2w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, and $m_4 \\geq m_3$, or \\item \\label{item-F4} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 \\\\ w & -2w & 2w & 0 \\\\ 0 & 2w & -4w & 2w \\\\ 0 & 0 & 2w & -4w  \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ 2w \\\\ 2w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2 + m_4$, and $2m_4 \\geq m_3$. \\end{enumerate}"}
+{"_id": "9213", "title": "models-lemma-D4", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\ & \\bullet } $$ If $n > 4$, then given four $(-2)$-indices $i, j, k, l$ with $a_{ij}, a_{ik}, a_{il}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-D4} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & w & w \\\\ w & -2w & 0 & 0 \\\\ w & 0 & -2w & 0 \\\\ w & 0 & 0 & -2w  \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2 + m_3 + m_4$, $2m_2 \\geq m_1$, $2m_3 \\geq m_1$, $2m_4 \\geq m_1$. Observe that this implies $m_1 \\geq \\max(m_2, m_3, m_4)$. \\end{enumerate}"}
+{"_id": "9214", "title": "models-lemma-five-by-five", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet } $$ If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$ with $a_{hi}, a_{ij}, a_{jk}, a_{kl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-A5} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & 0 \\\\ 0 & 0 & w & -2w & w \\\\ 0 & 0 & 0 & w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, $2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or \\item \\label{item-C5} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & 0 \\\\ 0 & 0 & w & -2w & 2w \\\\ 0 & 0 & 0 & 2w & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ 2w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$, $2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or \\item \\label{item-B5} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 & 0 \\\\ 2w & -4w & 2w & 0 & 0 \\\\ 0 & 2w & -4w & 2w & 0 \\\\ 0 & 0 & 2w & -4w & 2w \\\\ 0 & 0 & 0 & 2w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ 2w \\\\ 2w \\\\ 2w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, $2m_4 \\geq m_3 + m_5$, and $m_4 \\geq m_3$. \\end{enumerate}"}
+{"_id": "9215", "title": "models-lemma-fourfold", "text": "Nonexistence of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[ld] \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\ \\bullet & \\bullet } $$ If $n > 5$, there do {\\bf not} exist five $(-2)$-indices $h$, $i$, $j$, $k$ with $a_{hi} > 0$, $a_{hj} > 0$, $a_{hk} > 0$, and $a_{hl} > 0$."}
+{"_id": "9216", "title": "models-lemma-D5", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\ & & \\bullet } $$ If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$ with $a_{hi}, a_{ij}, a_{jk}, a_{jl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s \\begin{enumerate} \\item \\label{item-D5} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & w \\\\ 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & w & 0 & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_5$, $2m_4 \\geq m_3$, and $2m_5 \\geq m_3$. \\end{enumerate}"}
+{"_id": "9217", "title": "models-lemma-long", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet } $$ Let $t > 5$ and $n > t$. Then given $t$ distinct $(-2)$-indices $i_1, \\ldots, i_t$ such that $a_{i_ji_{j + 1}}$ is nonzero for $j = 1, \\ldots, t - 1$, then up to reversing the order of these indices we have the $a$'s and $w$'s \\begin{enumerate} \\item \\label{item-An} are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_t} = w$, $a_{i_ji_{j + 1}} = w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or \\item \\label{item-Cn} are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = w$, $w_{j_t} = 2w$, $a_{i_ji_{j + 1}} = w$ for $j < t - 1$, $a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or \\item \\label{item-Bn} are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = 2w$, $w_{j_t} = w$, $a_{i_ji_{j + 1}} = 2w$, and $a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$. \\end{enumerate}"}
+{"_id": "9218", "title": "models-lemma-Dn", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\ & & & \\bullet } $$ Let $t > 4$ and $n > t + 1$. Then given $t + 1$ distinct $(-2)$-indices $i_1, \\ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}}$ is nonzero for $j = 1, \\ldots, t - 1$ and $a_{i_{t - 1}i_{t + 1}}$ is nonzero, then we have the $a$'s and $w$'s \\begin{enumerate} \\item \\label{item-Dn} are given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t + 1}} = w$, $a_{i_ji_{j + 1}} = w$ for $j = 1, \\ldots, t - 1$, $a_{i_{t - 1}i_{t + 1}} = w$ and $a_{i_ji_k} = 0$ for other pairs $(j, k)$ with $j > k$. \\end{enumerate}"}
+{"_id": "9219", "title": "models-lemma-E6", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & \\bullet } $$ Let $n > 6$. Then given $6$ distinct $(-2)$-indices $i_1, \\ldots, i_6$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{36}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \\begin{enumerate} \\item \\label{item-E6} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\\\ m_6 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 \\\\ 0 & w & -2w & w & 0 & w \\\\ 0 & 0 & w & -2w & w & 0 \\\\ 0 & 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & w & 0 & 0 & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_6$, $2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_3$, and $2m_6 \\geq m_3$. \\end{enumerate}"}
+{"_id": "9220", "title": "models-lemma-double-triple", "text": "Nonexistence of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[d] \\ar@{-}[r] & \\bullet \\\\ & \\bullet & \\bullet } $$ Assume $t \\geq 4$ and $n > t + 2$. There do {\\bf not} exist $t + 2$ distinct $(-2)$-indices $i_0, \\ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}} > 0$ for $j = 1, \\ldots, t - 1$ and $a_{i_0i_2} > 0$ and $a_{i_{t - 1}i_{t + 1}} > 0$."}
+{"_id": "9221", "title": "models-lemma-E6-completed", "text": "Nonexistence of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & \\bullet \\ar@{-}[d] \\\\ & & \\bullet } $$ Assume $n > 7$. There do {\\bf not} exist $7$ distinct $(-2)$-indices $f, g, h, i, j, k, l$ such that $a_{fg}, a_{gh}, a_{ij}, a_{jh}, a_{kl}, a_{lh}$ are nonzero."}
+{"_id": "9222", "title": "models-lemma-E7", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & & \\bullet } $$ Let $n > 7$. Then given $7$ distinct $(-2)$-indices $i_1, \\ldots, i_7$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{47}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \\begin{enumerate} \\item \\label{item-E7} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\\\ m_6 \\\\ m_7 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 & 0 \\\\ 0 & w & -2w & w & 0 & 0 & 0 \\\\ 0 & 0 & w & -2w & w & 0 & w \\\\ 0 & 0 & 0 & w & -2w & w & 0 \\\\ 0 & 0 & 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & 0 & w & 0 & 0 & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, $2m_4 \\geq m_3 + m_5 + m_7$, $2m_5 \\geq m_4 + m_6$, $2m_6 \\geq m_5$, and $2m_7 \\geq m_4$. \\end{enumerate}"}
+{"_id": "9223", "title": "models-lemma-E8", "text": "Classification of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & & & \\bullet } $$ Let $n > 8$. Then given $8$ distinct $(-2)$-indices $i_1, \\ldots, i_8$ such that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{65}, a_{57}$ are nonzero, then we have the $m$'s, $a$'s, and $w$'s \\begin{enumerate} \\item \\label{item-E8} are given by $$ \\left( \\begin{matrix} m_1 \\\\ m_2 \\\\ m_3 \\\\ m_4 \\\\ m_5 \\\\ m_6 \\\\ m_7 \\\\ m_8 \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\ 0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\ 0 & 0 & 0 & w & -2w & w & 0 & w \\\\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right) $$ with $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$, $2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_4 + m_6 + m_8$, $2m_6 \\geq m_5 + m_7$, $2m_7 \\geq m_6$, and $2m_8 \\geq m_5$. \\end{enumerate}"}
+{"_id": "9224", "title": "models-lemma-E7-completed", "text": "Nonexistence of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & & \\bullet } $$ Assume $n > 8$. There do {\\bf not} exist $8$ distinct $(-2)$-indices $e, f, g, h, i, j, k, l$ such that $a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$ are nonzero."}
+{"_id": "9225", "title": "models-lemma-E8-completed", "text": "Nonexistence of proper subgraphs of the form $$ \\xymatrix{ \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\ & & & & & \\bullet } $$ Assume $n > 9$. There do {\\bf not} exist $9$ distinct $(-2)$-indices $d, e, f, g, h, i, j, k, l$ such that $a_{de}, a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$ are nonzero."}
+{"_id": "9226", "title": "models-lemma-genus-zero", "text": "The only minimal numerical type of genus zero is $n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 1$, $g_1 = 0$."}
+{"_id": "9227", "title": "models-lemma-genus-one", "text": "The minimal numerical types of genus one are up to equivalence \\begin{enumerate} \\item \\label{item-one} $n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \\geq 1$ arbitrary, \\item \\label{item-two-cycle} $n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w \\\\ 2w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up4} $n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 4w \\\\ 4w & -8w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-three-cycle} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & w \\\\ w & -2w & w \\\\ w & w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-equal-up3} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 \\\\ w & -2w & 3w \\\\ 0 & 3w & -6w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ 3w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-equal-down3} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -6w & 3w & 0 \\\\ 3w & -6w & 3w \\\\ 0 & 3w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 3w \\\\ 3w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-up} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 \\\\ 2w & -4w & 4w \\\\ 0 & 4w & -8w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-down} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 \\\\ 2w & -4w & 2w \\\\ 0 & 2w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-down-up} $n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 \\\\ 2w & -2w & 2w \\\\ 0 & 2w & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-four-cycle} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & w \\\\ w & -2w & w & 0 \\\\ 0 & w & -2w & w \\\\ w & 0 & w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-equal-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ 2m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 & 0 \\\\ 2w & -4w & 2w & 0 \\\\ 0 & 2w & -4w & 4w \\\\ 0 & 0 & 4w & -8w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 2w \\\\ 4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-equal-down} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 & 0 \\\\ 2w & -4w & 2w & 0 \\\\ 0 & 2w & -4w & 2w \\\\ 0 & 0 & 2w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 2w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-down-equal-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 \\\\ 2w & -2w & w & 0 \\\\ 0 & w & -2w & 2w \\\\ 0 & 0 & 2w & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ w \\\\ w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-triple-with-up} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & w & 2w \\\\ w & -2w & 0 & 0 \\\\ w & 0 & -2w & 0 \\\\ 2w & 0 & 0 & -4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-triple-with-down} $n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ m \\\\ m \\\\ 2m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 2w & 2w \\\\ 2w & -4w & 0 & 0 \\\\ 2w & 0 & -4w & 0 \\\\ 2w & 0 & 0 & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ 2w \\\\ 2w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-five-cycle} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & w \\\\ w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & 0 \\\\ 0 & 0 & w & -2w & w \\\\ w & 0 & 0 & w & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-equal-equal-up-equal} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & 2w & 0 \\\\ 0 & 0 & 2w & -4w & 2w \\\\ 0 & 0 & 0 & 2w & -4w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ 2w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-equal-equal-down-equal} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\\\ 4m \\\\ 2m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 & 0 \\\\ 2w & -4w & 2w & 0 & 0 \\\\ 0 & 2w & -4w & 2w & 0 \\\\ 0 & 0 & 2w & -2w & w \\\\ 0 & 0 & 0 & w & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ 2w \\\\ 2w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-equal-equal-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ 2m \\\\ 2m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 & 0 & 0 \\\\ 2w & -4w & 2w & 0 & 0 \\\\ 0 & 2w & -4w & 2w & 0 \\\\ 0 & 0 & 2w & -4w & 4w \\\\ 0 & 0 & 0 & 4w & -8w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 2w \\\\ 2w \\\\ 4w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-up-equal-equal-down} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 & 0 & 0 \\\\ 2w & -4w & 2w & 0 & 0 \\\\ 0 & 2w & -4w & 2w & 0 \\\\ 0 & 0 & 2w & -4w & 2w \\\\ 0 & 0 & 0 & 2w & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 2w \\\\ 2w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-down-equal-equal-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 2m \\\\ 2m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 & 0 \\\\ 2w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & 0 \\\\ 0 & 0 & w & -2w & 2w \\\\ 0 & 0 & 0 & 2w & -4w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ w \\\\ w \\\\ w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-quadruple} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ m \\\\ m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & w & w & w \\\\ w & -2w & 0 & 0 & 0 \\\\ w & 0 & -2w & 0 & 0 \\\\ w & 0 & 0 & -2w & 0 \\\\ w & 0 & 0 & 0 & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-triple-extended-up} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 2m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -4w & 2w & 0 & 0 & 0 \\\\ 2w & -2w & w & 0 & 0 \\\\ 0 & w & -2w & w & w \\\\ 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & w & 0 & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 2w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-triple-extended-down} $n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} 2m \\\\ 2m \\\\ 2m \\\\ m \\\\ m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & 2w & 0 & 0 & 0 \\\\ 2w & -4w & 2w & 0 & 0 \\\\ 0 & 2w & -4w & 2w & 2w \\\\ 0 & 0 & 2w & -4w & 0 \\\\ 0 & 0 & 2w & 0 & -4w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ 2w \\\\ 2w \\\\ 2w \\\\ 2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-n-cycle} $n \\geq 6$ and we have an $n$-cycle generalizing (\\ref{item-five-cycle}): \\begin{enumerate} \\item $m_1 = \\ldots = m_n = m$, \\item $a_{12} = \\ldots = a_{(n - 1) n} = w$, $a_{1n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = \\ldots = w_n = w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-up-chain-equal-up} $n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-up}): \\begin{enumerate} \\item $m_1 = \\ldots = m_{n - 1} = 2m$, $m_n = m$, \\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = 4w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-up-chain-equal-down} $n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-down}): \\begin{enumerate} \\item $m_1 = \\ldots = m_n = m$, \\item $a_{12} = \\ldots = a_{(n - 1) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-down-chain-equal-up} $n \\geq 6$ and we have a chain generalizing (\\ref{item-down-equal-equal-up}): \\begin{enumerate} \\item $m_1 = w$, $w_2 = \\ldots = m_{n - 1} = 2m$, $m_n = m$, \\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = 2w$, $w_2 = \\ldots = w_{n - 1} = w$, $w_n = 2w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-Dn-extended-up} $n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-up}): \\begin{enumerate} \\item $m_1 = m$, $m_2 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$, \\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = 2w$, $w_2 = \\ldots = w_n = w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-Dn-extended-down} $n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-down}): \\begin{enumerate} \\item $m_1 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$, \\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = w$, $w_2 = \\ldots = w_n = 2w$ \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-double-triple} $n \\geq 6$ and we have a type generalizing (\\ref{item-quadruple}): \\begin{enumerate} \\item $m_1 = m_2 = m$, $m_3 = \\ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_n = m$, \\item $a_{13} = w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$, \\item $w_1 = \\ldots = w_n = w$, \\end{enumerate} with $w$ and $m$ arbitrary, \\item \\label{item-E6-completed} $n = 7$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\\\ m \\\\ 2m \\\\ m \\\\ 2m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 & 0 \\\\ 0 & w & -2w & 0 & w & 0 & w \\\\ 0 & 0 & 0 & -2w & w & 0 & 0 \\\\ 0 & 0 & w & w & -2w & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & -2w & w \\\\ 0 & 0 & w & 0 & 0 & w & -2w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-E7-completed} $n = 8$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\\\ 4m \\\\ 3m \\\\ 2m \\\\ m \\\\ 2m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\ 0 & 0 & w & -2w & w & 0 & 0 & w \\\\ 0 & 0 & 0 & w & -2w & w & 0 & 0 \\\\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary, \\item \\label{item-E8-completed} $n = 9$, and $m_i, a_{ij}, w_i, g_i$ given by $$ \\left( \\begin{matrix} m \\\\ 2m \\\\ 3m \\\\ 4m \\\\ 5m \\\\ 6m \\\\ 4m \\\\ 2m \\\\ 3m \\end{matrix} \\right), \\quad \\left( \\begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ w & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\\\ 0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\ 0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\\\ \\end{matrix} \\right), \\quad \\left( \\begin{matrix} w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\\\ w \\end{matrix} \\right), \\quad \\left( \\begin{matrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{matrix} \\right) $$ with $w$ and $m$ arbitrary. \\end{enumerate}"}
+{"_id": "9228", "title": "models-lemma-bound-neighbours", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Given $i, j$ with $a_{ij} > 0$ we have $m_ia_{ij} \\leq m_j|a_{jj}|$ and $m_iw_i \\leq m_j|a_{jj}|$."}
+{"_id": "9229", "title": "models-lemma-bound-heart", "text": "Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ with $n > 1$ we have \\begin{enumerate} \\item the set $J \\subset \\{1, \\ldots, n\\}$ of non-$(-2)$-indices has at most $2g - 2$ elements, \\item for $j \\in J$ we have $g_j < g$, \\item for $j \\in J$ we have $m_j|a_{jj}| \\leq 6g - 6$, and \\item for $j \\in J$ and $i \\in \\{1, \\ldots, n\\}$ we have $m_ia_{ij} \\leq 6g - 6$. \\end{enumerate}"}
+{"_id": "9230", "title": "models-lemma-bound-wm", "text": "Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ we have $m_i|a_{ij}| \\leq 768g$."}
+{"_id": "9231", "title": "models-lemma-closure-is-model", "text": "Let $V_1 \\to V_2$ be a closed immersion of algebraic schemes over $K$. If $X_2$ is a model for $V_2$, then the scheme theoretic image of $V_1 \\to X_2$ is a model for $V_1$."}
+{"_id": "9232", "title": "models-lemma-normalization", "text": "Let $X$ be a model of a geometrically normal variety $V$ over $K$. Then the normalization $\\nu : X^\\nu \\to X$ is finite and the base change of $X^\\nu$ to the completion $R^\\wedge$ is the normalization of the base change of $X$. Moreover, for each $x \\in X^\\nu$ the completion of $\\mathcal{O}_{X^\\nu, x}$ is normal."}
+{"_id": "9233", "title": "models-lemma-regular", "text": "Let $X$ be a model of a smooth curve $C$ over $K$. Then there exists a resolution of singularities of $X$ and any resolution is a model of $C$."}
+{"_id": "9234", "title": "models-lemma-pre-exists-minimal-model", "text": "\\begin{slogan} A regular proper model of a curve is obtained by successive blowups from a minimal model \\end{slogan} Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. If $X$ is a regular proper model for $C$, then there exists a sequence of morphisms $$ X = X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 $$ of proper regular models of $C$, such that each morphism is a contraction of an exceptional curve of the first kind, and such that $X_0$ is a minimal model."}
+{"_id": "9235", "title": "models-lemma-divisor-special-fiber", "text": "Let $X$ be a regular model of a smooth curve $C$ over $K$. \\begin{enumerate} \\item the special fibre $X_k$ is an effective Cartier divisor on $X$, \\item each irreducible component $C_i$ of $X_k$ is an effective Cartier divisor on $X$, \\item $X_k = \\sum m_i C_i$ (sum of effective Cartier divisors) where $m_i$ is the multiplicity of $C_i$ in $X_k$, \\item $\\mathcal{O}_X(X_k) \\cong \\mathcal{O}_X$. \\end{enumerate}"}
+{"_id": "9236", "title": "models-lemma-gorenstein", "text": "Let $X$ be a regular model of a smooth curve $C$ over $K$. Then \\begin{enumerate} \\item $X \\to \\Spec(R)$ is a Gorenstein morphism of relative dimension $1$, \\item each of the irreducible components $C_i$ of $X_k$ is Gorenstein. \\end{enumerate}"}
+{"_id": "9237", "title": "models-lemma-regular-model-connected", "text": "In Situation \\ref{situation-regular-model} the special fibre $X_k$ is connected."}
+{"_id": "9238", "title": "models-lemma-regular-model-pic", "text": "In Situation \\ref{situation-regular-model} there is an exact sequence $$ 0 \\to \\mathbf{Z} \\to \\mathbf{Z}^{\\oplus n} \\to \\Pic(X) \\to \\Pic(C) \\to 0 $$ where the first map sends $1$ to $(m_1, \\ldots, m_n)$ and the second maps sends the $i$th basis vector to $\\mathcal{O}_X(C_i)$."}
+{"_id": "9239", "title": "models-lemma-intersection-pairing", "text": "In Situation \\ref{situation-regular-model} given $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module and $a = (a_1, \\ldots, a_n) \\in \\mathbf{Z}^{\\oplus n}$ we define $$ \\langle a, \\mathcal{L} \\rangle = \\sum a_i\\deg(\\mathcal{L}|_{C_i}) $$ Then $\\langle , \\rangle$ is bilinear and for $b = (b_1, \\ldots, b_n) \\in \\mathbf{Z}^{\\oplus n}$ we have $$ \\left\\langle a, \\mathcal{O}_X(\\sum b_i C_i) \\right\\rangle = \\left\\langle b, \\mathcal{O}_X(\\sum a_i C_i) \\right\\rangle $$"}
+{"_id": "9240", "title": "models-lemma-properties-form", "text": "In Situation \\ref{situation-regular-model} the symmetric bilinear form (\\ref{equation-form}) has the following properties \\begin{enumerate} \\item $(C_i \\cdot C_j) \\geq 0$ if $i \\not = j$ with equality if and only if $C_i \\cap C_j = \\emptyset$, \\item $(\\sum m_i C_i \\cdot C_j) = 0$, \\item there is no nonempty proper subset $I \\subset \\{1, \\ldots, n\\}$ such that $(C_i \\cdot C_j) = 0$ for $i \\in I$, $j \\not \\in I$. \\item $(\\sum a_i C_i \\cdot \\sum a_i C_i) \\leq 0$ with equality if and only if there exists a $q \\in \\mathbf{Q}$ such that $a_i = qm_i$ for $i = 1, \\ldots, n$, \\end{enumerate}"}
+{"_id": "9241", "title": "models-lemma-multiple-fibre-normal-bundle", "text": "In Situation \\ref{situation-regular-model} set $d = \\gcd(m_1, \\ldots, m_n)$ and let $D = \\sum (m_i/d)C_i$ as an effective Cartier divisor. Then $\\mathcal{O}_X(D)$ has order dividing $d$ in $\\Pic(X)$ and $\\mathcal{C}_{D/X}$ an invertible $\\mathcal{O}_D$-module of order dividing $d$ in $\\Pic(D)$."}
+{"_id": "9242", "title": "models-lemma-regular-model-field", "text": "\\begin{reference} \\cite[Lemma 2.6]{Artin-Winters} \\end{reference} In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$. Let $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor. Then there exists a sequence of effective Cartier divisors $$ (X_k)_{red} = Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_m = D $$ such that $Z_j = Z_{j - 1} + C_{i_j}$ for some $i_j \\in \\{1, \\ldots, n\\}$ for $j = 1, \\ldots, m$ and such that $H^0(Z_j, \\mathcal{O}_{Z_j})$ is a field finite over $k$ for $j = 0, \\ldots m$."}
+{"_id": "9243", "title": "models-lemma-regular-model-genus", "text": "\\begin{reference} \\cite[Lemma 2.6]{Artin-Winters} \\end{reference} In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$. Let $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor on $X$. Then $$ 1 - g_C = d [\\kappa : k] (1 - g_D) $$ where $g_C$ is the genus of $C$, $g_D$ is the genus of $D$, and $\\kappa = H^0(D, \\mathcal{O}_D)$."}
+{"_id": "9244", "title": "models-lemma-exceptional-curves-dont-meet", "text": "In Situation \\ref{situation-regular-model} given a pair of indices $i, j$ such that $C_i$ and $C_j$ are exceptional curves of the first kind and $C_i \\cap C_j \\not = \\emptyset$, then $n = 2$, $m_1 = m_2 = 1$, $C_1 \\cong \\mathbf{P}^1_k$, $C_2 \\cong \\mathbf{P}^1_k$, $C_1$ and $C_2$ meet in a $k$-rational point, and $C$ has genus $0$."}
+{"_id": "9245", "title": "models-lemma-minimal-model-unique", "text": "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$ and genus $> 0$. There is a unique minimal model for $C$."}
+{"_id": "9247", "title": "models-lemma-add-component", "text": "In Situation \\ref{situation-regular-model} suppose we have an effective Cartier divisors $D, D' \\subset X$ such that $D' = D + C_i$ for some $i \\in \\{1, \\ldots, n\\}$ and $D' \\subset X_k$. Then $$ \\chi(X_k, \\mathcal{O}_{D'}) - \\chi(X_k, \\mathcal{O}_D) = \\chi(X_k, \\mathcal{O}_X(-D)|_{C_i}) = -(D \\cdot C_i) + \\chi(C_i, \\mathcal{O}_{C_i}) $$"}
+{"_id": "9248", "title": "models-lemma-genus-formula", "text": "In Situation \\ref{situation-regular-model} we have $$ g_C = 1 + \\sum\\nolimits_{i = 1, \\ldots, n} m_i\\left([\\kappa_i : k] (g_i - 1) - \\frac{1}{2}(C_i \\cdot C_i)\\right) $$ where $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$, $g_i$ is the genus of $C_i$, and $g_C$ is the genus of $C$."}
+{"_id": "9249", "title": "models-lemma-numerical-type-of-model", "text": "In Situation \\ref{situation-regular-model} with $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and $g_i$ the genus of $C_i$ the data $$ n, m_i, (C_i \\cdot C_j), [\\kappa_i : k], g_i $$ is a numerical type of genus equal to the genus of $C$."}
+{"_id": "9250", "title": "models-lemma-numerical-type-minimal-model", "text": "In Situation \\ref{situation-regular-model}. The following are equivalent \\begin{enumerate} \\item $X$ is a minimal model, and \\item the numerical type associated to $X$ is minimal. \\end{enumerate}"}
+{"_id": "9253", "title": "models-lemma-genus-reduction-bigger-than", "text": "In Situation \\ref{situation-regular-model} assume $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k \\to \\Spec(k)$. Then $$ \\dim_k H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}}) \\geq g_{top} + g_{geom}(X_k/k) $$ where $g_{geom}$ is as in Algebraic Curves, Section \\ref{curves-section-genus-geometric-genus} and $g_{top}$ is the topological genus (Definition \\ref{definition-top-genus}) of the numerical type associated to $X_k$ (Definition \\ref{definition-numerical-type-model})."}
+{"_id": "9255", "title": "models-lemma-blowdown-regular-model", "text": "In Situation \\ref{situation-regular-model} assume that $C_n$ is an exceptional curve of the first kind. Let $f : X \\to X'$ be the contraction of $C_n$. Let $C'_i = f(C_i)$. Write $X'_k = \\sum m'_i C'_i$. Then $X'$, $C'_i$, $i = 1, \\ldots, n' = n - 1$, and $m'_i = m_i$ is as in Situation \\ref{situation-regular-model} and we have \\begin{enumerate} \\item for $i, j < n$ we have $(C'_i \\cdot C'_j) = (C_i \\cdot C_j) - (C_i \\cdot C_n) (C_j \\cdot C_n) /(C_n \\cdot C_n)$, \\item for $i < n$ if $C_i \\cap C_n \\not = \\emptyset$, then there are maps $\\kappa_i \\leftarrow \\kappa'_i \\rightarrow \\kappa_n$. \\end{enumerate} Here $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and $\\kappa'_i = H^0(C'_i, \\mathcal{O}_{C'_i})$."}
+{"_id": "9256", "title": "models-lemma-nonuniqueness", "text": "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$ and genus $0$. If there is more than one minimal model for $C$, then the special fibre of every minimal model is isomorphic to $\\mathbf{P}^1_k$."}
+{"_id": "9257", "title": "models-lemma-characterize-trivial", "text": "In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$. If $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module which \\begin{enumerate} \\item restricts to the trivial invertible module on $C$, and \\item has degree $0$ on each $C_i$, \\end{enumerate} then $\\mathcal{L}^{\\otimes d} \\cong \\mathcal{O}_X$."}
+{"_id": "9261", "title": "models-lemma-etale-local-at-worst-nodal", "text": "Let $R$ be a discrete valuation ring. Let $X$ be a scheme which is at-worst-nodal of relative dimension $1$ over $R$. Let $x \\in X$ be a point of the special fibre of $X$ over $R$. Then there exists a commutative diagram $$ \\xymatrix{ X \\ar[d] & U \\ar[r] \\ar[d] \\ar[l] & \\Spec(A) \\ar[dl] \\\\ \\Spec(R) & \\Spec(R') \\ar[l] } $$ where $R \\subset R'$ is an \\'etale extension of discrete valuation rings, the morphism $U \\to X$ is \\'etale, the morphism $U \\to \\Spec(A)$ is \\'etale, there is a point $x' \\in U$ mapping to $x$, and $$ A = R'[u, v]/(uv) \\quad\\text{or}\\quad A = R'[u, v]/(uv - \\pi^n) $$ where $n \\geq 0$ and $\\pi \\in R'$ is a uniformizer."}
+{"_id": "9262", "title": "models-lemma-blowup-at-worst-nodal", "text": "Let $R$ be a discrete valuation ring. Let $X$ be a quasi-compact scheme which is at-worst-nodal of relative dimension $1$ with smooth generic fibre over $R$. Then there exists $m \\geq 0$ and a sequence $$ X_m \\to \\ldots \\to X_1 \\to X_0 = X $$ such that \\begin{enumerate} \\item $X_{i + 1} \\to X_i$ is the blowing up of a closed point $x_i$ where $X_i$ is singular, \\item $X_i \\to \\Spec(R)$ is at-worst-nodal of relative dimension $1$, \\item $X_m$ is regular. \\end{enumerate}"}
+{"_id": "9263", "title": "models-lemma-blowdown-at-worst-nodal", "text": "Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume $X \\to \\Spec(R)$ is at-worst-nodal of relative dimension $1$ over $R$. Let $X \\to X'$ be the contraction of an exceptional curve $E \\subset X$ of the first kind. Then $X'$ is at-worst-nodal of relative dimension $1$ over $R$."}
+{"_id": "9264", "title": "models-lemma-semistable", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. The following are equivalent \\begin{enumerate} \\item there exists a proper model of $C$ which is at-worst-nodal of relative dimension $1$ over $R$, \\item there exists a minimal model of $C$ which is at-worst-nodal of relative dimension $1$ over $R$, and \\item any minimal model of $C$ is at-worst-nodal of relative dimension $1$ over $R$. \\end{enumerate}"}
+{"_id": "9265", "title": "models-lemma-good", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. The following are equivalent \\begin{enumerate} \\item there exists a proper smooth model for $C$, \\item there exists a minimal model for $C$ which is smooth over $R$, \\item any minimal model is smooth over $R$. \\end{enumerate}"}
+{"_id": "9266", "title": "models-proposition-classify-subgraphs", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. Let $I \\subset \\{1, \\ldots, n\\}$ be a proper subset of cardinality $\\geq 2$ consisting of $(-2)$-indices such that there does not exist a nonempty proper subset $I' \\subset I$ with $a_{i'i} = 0$ for $i' \\in I$, $i \\in I \\setminus I'$. Then up to reordering the $m_i$'s, $a_{ij}$'s, $w_i$'s for $i, j \\in I$ are as listed in Lemmas \\ref{lemma-two-by-two}, \\ref{lemma-three-by-three}, \\ref{lemma-four-by-four}, \\ref{lemma-D4}, \\ref{lemma-five-by-five}, \\ref{lemma-D5}, \\ref{lemma-long}, \\ref{lemma-Dn}, \\ref{lemma-E6}, \\ref{lemma-E7}, or \\ref{lemma-E8}."}
+{"_id": "9268", "title": "models-proposition-exists-minimal-model", "text": "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. A minimal model exists."}
+{"_id": "9286", "title": "spaces-groupoids-lemma-restrict-relation", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $U$ be an algebraic space over $B$. Let $j : R \\to U \\times_B U$ be a pre-relation. Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$. Finally, set $$ R' = (U' \\times_B U')\\times_{U \\times_B U} R \\xrightarrow{j'} U' \\times_B U' $$ Then $j'$ is a pre-relation on $U'$ over $B$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation."}
+{"_id": "9287", "title": "spaces-groupoids-lemma-pre-equivalence-equivalence-relation-points", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $j : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$. Consider the relation on $|U|$ defined by the rule $$ x \\sim y \\Leftrightarrow \\exists\\ r \\in |R| : t(r) = x, s(r) = y. $$ If $j$ is a pre-equivalence relation then this is an equivalence relation."}
+{"_id": "9288", "title": "spaces-groupoids-lemma-base-change-group-space", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $B' \\to B$ be a morphism of algebraic spaces. The pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$."}
+{"_id": "9289", "title": "spaces-groupoids-lemma-group-scheme-separated", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then $G \\to B$ is separated (resp.\\ quasi-separated, resp.\\ locally separated) if and only if the identity morphism $e : B \\to G$ is a closed immersion (resp.\\ quasi-compact, resp.\\ an immersion)."}
+{"_id": "9290", "title": "spaces-groupoids-lemma-group-scheme-unramified-or-lqf", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \\to B$ is locally of finite type. Then $G \\to B$ is unramified (resp.\\ locally quasi-finite) if and only if $G \\to B$ is unramified (resp.\\ quasi-finite) at $e(b)$ for all $b \\in |B|$."}
+{"_id": "9291", "title": "spaces-groupoids-lemma-open-over-which-unramified-or-lqf", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Assume $G \\to B$ is locally of finite type. \\begin{enumerate} \\item There exists a maximal open subspace $U \\subset B$ such that $G_U \\to U$ is unramified and formation of $U$ commutes with base change. \\item There exists a maximal open subspace $U \\subset B$ such that $G_U \\to U$ is locally quasi-finite and formation of $U$ commutes with base change. \\end{enumerate}"}
+{"_id": "9292", "title": "spaces-groupoids-lemma-free-action", "text": "Situation as in Definition \\ref{definition-free-action}, The action $a$ is free if and only if $$ G \\times_B X \\to X \\times_B X, \\quad (g, x) \\mapsto (a(g, x), x) $$ is a monomorphism of algebraic spaces."}
+{"_id": "9293", "title": "spaces-groupoids-lemma-characterize-trivial-pseudo-torsors", "text": "In the situation of Definition \\ref{definition-pseudo-torsor}. \\begin{enumerate} \\item The algebraic space $X$ is a pseudo $G$-torsor if and only if for every scheme $T$ over $B$ the set $X(T)$ is either empty or the action of the group $G(T)$ on $X(T)$ is simply transitive. \\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism $X \\to B$ has a section. \\end{enumerate}"}
+{"_id": "9294", "title": "spaces-groupoids-lemma-torsor", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group algebraic space over $S$. Let $X$ be an algebraic space over $S$, and let $a : G \\times_S X \\to X$ be an action of $G$ on $X$. Then $X$ is a $G$-torsor in the $fppf$-topology in the sense of Definition \\ref{definition-principal-homogeneous-space} if and only if $X$ is a $G$-torsor on $(\\Sch/S)_{fppf}$ in the sense of Cohomology on Sites, Definition \\ref{sites-cohomology-definition-torsor}."}
+{"_id": "9295", "title": "spaces-groupoids-lemma-pseudo-torsor-implications", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be a pseudo $G$-torsor over $B$. Assume $G$ and $X$ locally of finite type over $B$. \\begin{enumerate} \\item If $G \\to B$ is unramified, then $X \\to B$ is unramified. \\item If $G \\to B$ is locally quasi-finite, then $X \\to B$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "9296", "title": "spaces-groupoids-lemma-pullback-equivariant", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $G$ be a group algebraic space over $B$. Let $f : X \\to Y$ be a $G$-equivariant morphism between algebraic spaces over $B$ endowed with $G$-actions. Then pullback $f^*$ given by $(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$ defines a functor from the category of $G$-equivariant sheaves on $X$ to the category of quasi-coherent $G$-equivariant sheaves on $Y$."}
+{"_id": "9297", "title": "spaces-groupoids-lemma-groupoid-pre-equivalence", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Given a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ the morphism $j : R \\to U \\times_B U$ is a pre-equivalence relation."}
+{"_id": "9298", "title": "spaces-groupoids-lemma-equivalence-groupoid", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Given an equivalence relation $j : R \\to U \\times_B U$ over $B$ there is a unique way to extend it to a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$."}
+{"_id": "9299", "title": "spaces-groupoids-lemma-diagram", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. In the commutative diagram $$ \\xymatrix{ & U & \\\\ R \\ar[d]_s \\ar[ru]^t & R \\times_{s, U, t} R \\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c & R \\ar[d]^s \\ar[lu]_t \\\\ U & R \\ar[l]_t \\ar[r]^s & U } $$ the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian."}
+{"_id": "9301", "title": "spaces-groupoids-lemma-base-change-groupoid", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $B' \\to B$ be a morphism of algebraic spaces. Then the base changes $U' = B' \\times_B U$, $R' = B' \\times_B R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid in algebraic spaces $(U', R', s', t', c')$ over $B'$ and the projections determine a morphism $(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$."}
+{"_id": "9303", "title": "spaces-groupoids-lemma-pullback", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Consider a morphism $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by $$ (\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$."}
+{"_id": "9305", "title": "spaces-groupoids-lemma-abelian", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian."}
+{"_id": "9306", "title": "spaces-groupoids-lemma-crystals-in-quasi-coherent-modules", "text": "In the situation above, if all the morphisms $f_\\phi$ are flat, then there exists a cardinal $\\kappa$ such that every object $(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$ of $\\textit{CQC}(X)$ is the directed colimit of its $\\kappa$-generated submodules."}
+{"_id": "9307", "title": "spaces-groupoids-lemma-set-generators", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then there exists a set $T$ and a family of objects $(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of $\\QCoh(U, R, s, t, c)$ such that every object $(\\mathcal{F}, \\alpha)$ is the directed colimit of its submodules isomorphic to one of the objects $(\\mathcal{F}_t, \\alpha_t)$."}
+{"_id": "9308", "title": "spaces-groupoids-lemma-groupoid-from-action", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$ with identity $e_G$ and inverse $i_G$. Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$. Then we get a groupoid in algebraic spaces $(U, R, s, t, c, e, i)$ over $B$ in the following manner: \\begin{enumerate} \\item We set $U = X$, and $R = G \\times_B X$. \\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$. \\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$. \\item We set $c : R \\times_{s, U, t} R \\to R$ equal to $((g, x), (g', x')) \\mapsto (m(g, g'), x')$. \\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$. \\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$. \\end{enumerate}"}
+{"_id": "9310", "title": "spaces-groupoids-lemma-groupoid-stabilizer", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The algebraic space $G$ defined by the cartesian square $$ \\xymatrix{ G \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\ U \\ar[r]^-{\\Delta} & U \\times_B U } $$ is a group algebraic space over $U$ with composition law $m$ induced by the composition law $c$."}
+{"_id": "9311", "title": "spaces-groupoids-lemma-groupoid-action-stabilizer", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let $G/U$ be its stabilizer. Denote $R_t/U$ the algebraic space $R$ seen as an algebraic space over $U$ via the morphism $t : R \\to U$. There is a canonical left action $$ a : G \\times_U R_t \\longrightarrow R_t $$ induced by the composition law $c$."}
+{"_id": "9312", "title": "spaces-groupoids-lemma-restrict-groupoid", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \\to U$ be a morphism of algebraic spaces. Consider the following diagram $$ \\xymatrix{ R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}& R \\times_{s, U} U' \\ar[r] \\ar[d] & U' \\ar[d]^g \\\\ U' \\times_{U, t} R \\ar[d] \\ar[r] & R \\ar[r]^s \\ar[d]_t & U \\\\ U' \\ar[r]^g & U } $$ where all the squares are fibre product squares. Then there is a canonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$ such that $(U', R', s', t', c')$ is a groupoid in algebraic spaces over $B$ and such that $U' \\to U$, $R' \\to R$ defines a morphism $(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$. Moreover, for any scheme $T$ over $B$ the functor of groupoids $$ (U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c) $$ is the restriction (see Groupoids, Section \\ref{groupoids-section-restrict-groupoid}) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \\to U(T)$."}
+{"_id": "9313", "title": "spaces-groupoids-lemma-restrict-groupoid-relation", "text": "The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions \\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation} agree via the constructions of Lemmas \\ref{lemma-groupoid-pre-equivalence} and \\ref{lemma-equivalence-groupoid}."}
+{"_id": "9315", "title": "spaces-groupoids-lemma-criterion-quotient-representable", "text": "In the situation of Definition \\ref{definition-quotient-sheaf}. Assume there is an algebraic space $M$ over $S$, and a morphism $U \\to M$ such that \\begin{enumerate} \\item the morphism $U \\to M$ equalizes $s, t$, \\item the map $U \\to M$ is a surjection of sheaves, and \\item the induced map $(t, s) : R \\to U \\times_M U$ is a surjection of sheaves. \\end{enumerate} In this case $M$ represents the quotient sheaf $U/R$."}
+{"_id": "9316", "title": "spaces-groupoids-lemma-quotient-pre-equivalence", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$. For a scheme $S'$ over $S$ and $a, b \\in U(S')$ the following are equivalent: \\begin{enumerate} \\item $a$ and $b$ map to the same element of $(U/R)(S')$, and \\item there exists an fppf covering $\\{f_i : S_i \\to S'\\}$ of $S'$ and morphisms $r_i : S_i \\to R$ such that $a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$. \\end{enumerate} In other words, in this case the map of sheaves $$ R \\longrightarrow U \\times_{U/R} U $$ is surjective."}
+{"_id": "9317", "title": "spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$ and $g : U' \\to U$ a morphism of algebraic spaces over $B$. Let $j' : R' \\to U' \\times_B U'$ be the restriction of $j$ to $U'$. The map of quotient sheaves $$ U'/R' \\longrightarrow U/R $$ is injective. If $U' \\to U$ is surjective as a map of sheaves, for example if $\\{g : U' \\to U\\}$ is an fppf covering (see Topologies on Spaces, Definition \\ref{spaces-topologies-definition-fppf-covering}), then $U'/R' \\to U/R$ is an isomorphism of sheaves."}
+{"_id": "9319", "title": "spaces-groupoids-lemma-quotient-stack-arrows", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ as in Definition \\ref{definition-quotient-stack} (1). There are canonical $1$-morphisms $\\pi : \\mathcal{S}_U \\to [U/R]$, and $[U/R] \\to \\mathcal{S}_B$ of stacks in groupoids over $(\\Sch/S)_{fppf}$. The composition $\\mathcal{S}_U \\to \\mathcal{S}_B$ is the $1$-morphism associated to the structure morphism $U \\to B$."}
+{"_id": "9320", "title": "spaces-groupoids-lemma-quotient-stack-2-arrow", "text": "Assumptions and notation as in Lemma \\ref{lemma-quotient-stack-arrows}. There exists a canonical $2$-morphism $\\alpha : \\pi \\circ s \\to \\pi \\circ t$ making the diagram $$ \\xymatrix{ \\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\ \\mathcal{S}_U \\ar[r]^-\\pi & [U/R] } $$ $2$-commutative."}
+{"_id": "9321", "title": "spaces-groupoids-lemma-quotient-stack-functorial", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ be a morphism of groupoids in algebraic spaces over $B$. Then $f$ induces a canonical $1$-morphism of quotient stacks $$ [f] : [U/R] \\longrightarrow [U'/R']. $$"}
+{"_id": "9322", "title": "spaces-groupoids-lemma-cartesian-square-of-morphism", "text": "Notation and assumption as in Lemma \\ref{lemma-quotient-stack-functorial}. Let $(U'', R'', s'', t'', c'')$ be the groupoid in algebraic spaces over $B$ constructed above. There is a $2$-commutative square $$ \\xymatrix{ [U''/R''] \\ar[d] \\ar[r]_{[g]} & [U/R] \\ar[d]^{[f]} \\\\ \\mathcal{S}_{U'} \\ar[r] & [U'/R'] } $$ which identifies $[U''/R'']$ with the $2$-fibre product."}
+{"_id": "9323", "title": "spaces-groupoids-lemma-quotient-stack-morphisms", "text": "Assume $B \\to S$, $(U, R, s, t, c)$ and $\\pi : \\mathcal{S}_U \\to [U/R]$ are as in Lemma \\ref{lemma-quotient-stack-arrows}. Let $S'$ be a scheme over $S$. Let $x, y \\in \\Ob([U/R]_{S'})$ be objects of the quotient stack over $S'$. If $x = \\pi(x')$ and $y = \\pi(y')$ for some morphisms $x', y' : S' \\to U$, then $$ \\mathit{Isom}(x, y) = S' \\times_{(y', x'), U \\times_S U} R $$ as sheaves over $S'$."}
+{"_id": "9324", "title": "spaces-groupoids-lemma-quotient-stack-2-cartesian", "text": "Assume $B \\to S$, $(U, R, s, t, c)$, and $\\pi : \\mathcal{S}_U \\to [U/R]$ are as in Lemma \\ref{lemma-quotient-stack-arrows}. The $2$-commutative square $$ \\xymatrix{ \\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\ \\mathcal{S}_U \\ar[r]^-\\pi & [U/R] } $$ of Lemma \\ref{lemma-quotient-stack-2-arrow} is a $2$-fibre product of stacks in groupoids of $(\\Sch/S)_{fppf}$."}
+{"_id": "9325", "title": "spaces-groupoids-lemma-quotient-stack-isom", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in Definition \\ref{definition-quotient-stack} (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$ has the following property: There exists a fppf covering $\\{T_i \\to T\\}_{i \\in I}$ such that $\\mathit{Isom}(x, y)|_{(\\Sch/T_i)_{fppf}}$ is representable by an algebraic space."}
+{"_id": "9326", "title": "spaces-groupoids-lemma-quotient-stack-cocycle", "text": "Assumptions and notation as in Lemmas \\ref{lemma-quotient-stack-arrows} and \\ref{lemma-quotient-stack-2-arrow}. The vertical composition of $$ \\xymatrix@C=15pc{ \\mathcal{S}_{R \\times_{s, U, t} R} \\ruppertwocell^{\\pi \\circ s \\circ \\text{pr}_1 = \\pi \\circ s \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_1}} \\ar[r]_(.3){\\pi \\circ t \\circ \\text{pr}_1 = \\pi \\circ s \\circ \\text{pr}_0} \\rlowertwocell_{\\pi \\circ t \\circ \\text{pr}_0 = \\pi \\circ t \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_0}} & [U/R] } $$ is the $2$-morphism $\\alpha \\star \\text{id}_c$. In a formula $\\alpha \\star \\text{id}_c = (\\alpha \\star \\text{id}_{\\text{pr}_0}) \\circ (\\alpha \\star \\text{id}_{\\text{pr}_1}) $."}
+{"_id": "9327", "title": "spaces-groupoids-lemma-quotient-stack-2-coequalizer", "text": "Assumptions and notation as in Lemmas \\ref{lemma-quotient-stack-arrows} and \\ref{lemma-quotient-stack-2-arrow}. The $2$-commutative diagram of Lemma \\ref{lemma-quotient-stack-2-arrow} is a $2$-coequalizer in the following sense: Given \\begin{enumerate} \\item a stack in groupoids $\\mathcal{X}$ over $(\\Sch/S)_{fppf}$, \\item a $1$-morphism $f : \\mathcal{S}_U \\to \\mathcal{X}$, and \\item a $2$-arrow $\\beta : f \\circ s \\to f \\circ t$ \\end{enumerate} such that $$ \\beta \\star \\text{id}_c = (\\beta \\star \\text{id}_{\\text{pr}_0}) \\circ (\\beta \\star \\text{id}_{\\text{pr}_1}) $$ then there exists a $1$-morphism $[U/R] \\to \\mathcal{X}$ which makes the diagram $$ \\xymatrix{ \\mathcal{S}_R \\ar[r]_s \\ar[d]^t & \\mathcal{S}_U \\ar[d] \\ar[ddr]^f \\\\ \\mathcal{S}_U \\ar[r] \\ar[rrd]_f & [U/R] \\ar[rd] \\\\ & & \\mathcal{X} } $$ $2$-commute."}
+{"_id": "9328", "title": "spaces-groupoids-lemma-quotient-stack-objects", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ are as in Definition \\ref{definition-quotient-stack} (1). Let $\\pi : \\mathcal{S}_U \\to [U/R]$ be as in Lemma \\ref{lemma-quotient-stack-arrows}. Let $T$ be a scheme over $S$. \\begin{enumerate} \\item for every object $x$ of the fibre category $[U/R]_T$ there exists an fppf covering $\\{f_i : T_i \\to T\\}_{i \\in I}$ such that $f_i^*x \\cong \\pi(u_i)$ for some $u_i \\in U(T_i)$, \\item the composition of the isomorphisms $$ \\pi(u_i \\circ \\text{pr}_0) = \\text{pr}_0^*\\pi(u_i) \\cong \\text{pr}_0^*f_i^*x \\cong \\text{pr}_1^*f_j^*x \\cong \\text{pr}_1^*\\pi(u_j) = \\pi(u_j \\circ \\text{pr}_1) $$ are of the form $\\pi(r_{ij})$ for certain morphisms $r_{ij} : T_i \\times_T T_j \\to R$, \\item the system $(u_i, r_{ij})$ forms a $[U/R]$-descent datum as defined above, \\item any $[U/R]$-descent datum $(u_i, r_{ij})$ arises in this manner, \\item if $x$ corresponds to $(u_i, r_{ij})$ as above, and $y \\in \\Ob([U/R]_T)$ corresponds to $(u'_i, r'_{ij})$ then there is a canonical bijection $$ \\Mor_{[U/R]_T}(x, y) \\longleftrightarrow \\left\\{ \\begin{matrix} \\text{morphisms }(u_i, r_{ij}) \\to (u'_i, r'_{ij})\\\\ \\text{of }[U/R]\\text{-descent data} \\end{matrix} \\right\\} $$ \\item this correspondence is compatible with refinements of fppf coverings. \\end{enumerate}"}
+{"_id": "9329", "title": "spaces-groupoids-lemma-quotient-stack-restrict", "text": "Notation and assumption as in Lemma \\ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \\longrightarrow [U'/R'] $$ is fully faithful if and only if $R$ is the restriction of $R'$ via the morphism $f : U \\to U'$."}
+{"_id": "9330", "title": "spaces-groupoids-lemma-quotient-stack-restrict-equivalence", "text": "Notation and assumption as in Lemma \\ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \\longrightarrow [U'/R'] $$ is an equivalence if and only if \\begin{enumerate} \\item $(U, R, s, t, c)$ is the restriction of $(U', R', s', t', c')$ via $f : U \\to U'$, and \\item the map $$ \\xymatrix{ U \\times_{f, U', t'} R' \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h & R' \\ar[r]_{s'} & U' } $$ is a surjection of sheaves. \\end{enumerate} Part (2) holds for example if $\\{h : U \\times_{f, U', t'} R' \\to U'\\}$ is an fppf covering, or if $f : U \\to U'$ is a surjection of sheaves, or if $\\{f : U \\to U'\\}$ is an fppf covering."}
+{"_id": "9331", "title": "spaces-groupoids-lemma-criterion-fibre-product", "text": "Notation and assumption as in Lemma \\ref{lemma-quotient-stack-functorial}. Assume that $$ \\xymatrix{ R \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\ U \\ar[r]^f & U' } $$ is cartesian. Then $$ \\xymatrix{ \\mathcal{S}_U \\ar[d] \\ar[r] & [U/R] \\ar[d]^{[f]} \\\\ \\mathcal{S}_{U'} \\ar[r] & [U'/R'] } $$ is a $2$-fibre product square."}
+{"_id": "9332", "title": "spaces-groupoids-lemma-presentation-inertia", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ as in Definition \\ref{definition-quotient-stack} (1). Let $G/U$ be the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Definition \\ref{definition-stabilizer-groupoid}. Set $R' = R \\times_{s, U} G$ and set \\begin{enumerate} \\item $s' : R' \\to G$, $(r, g) \\mapsto g$, \\item $t' : R' \\to G$, $(r, g) \\mapsto c(r, c(g, i(r)))$, \\item $c' : R' \\times_{s', G, t'} R' \\to R'$, $((r_1, g_1), (r_2, g_2) \\mapsto (c(r_1, r_2), g_1)$. \\end{enumerate} Then $(G, R', s', t', c')$ is a groupoid in algebraic spaces over $B$ and $$ \\mathcal{I}_{[U/R]} = [G/ R']. $$ i.e., the associated quotient stack is the inertia stack of $[U/R]$."}
+{"_id": "9333", "title": "spaces-groupoids-lemma-2-cartesian-inertia", "text": "Assume $B \\to S$ and $(U, R, s, t, c)$ as in Definition \\ref{definition-quotient-stack} (1). Let $G/U$ be the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Definition \\ref{definition-stabilizer-groupoid}. There is a canonical $2$-cartesian diagram $$ \\xymatrix{ \\mathcal{S}_G \\ar[r] \\ar[d] & \\mathcal{S}_U \\ar[d] \\\\ \\mathcal{I}_{[U/R]} \\ar[r] & [U/R] } $$ of stacks in groupoids of $(\\Sch/S)_{fppf}$."}
+{"_id": "9334", "title": "spaces-groupoids-lemma-when-gerbe", "text": "Notation and assumption as in Lemma \\ref{lemma-quotient-stack-functorial}. The morphism of quotient stacks $$ [f] : [U/R] \\longrightarrow [U'/R'] $$ turns $[U/R]$ into a gerbe over $[U'/R']$ if $f : U \\to U'$ and $R \\to R'|_U$ are surjective maps of fppf sheaves. Here $R'|_U$ is the restriction of $R'$ to $U$ via $f : U \\to U'$."}
+{"_id": "9335", "title": "spaces-groupoids-lemma-group-quotient-gerbe", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. The morphism $$ [B/G] \\longrightarrow \\mathcal{S}_B $$ (Lemma \\ref{lemma-quotient-stack-arrows}) turns $[B/G]$ into a gerbe over $B$."}
+{"_id": "9336", "title": "spaces-groupoids-lemma-quotient-stack-change-big-site", "text": "Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$. Assume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$, see Topologies, Section \\ref{topologies-section-change-alpha}. Let $S \\in \\Ob(\\Sch_{fppf})$. Let $B, U, R \\in \\Sh((\\Sch/S)_{fppf})$ be algebraic spaces, and let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : \\Sch_{fppf} \\to \\Sch'_{fppf}$. Then we have a canonical equivalence $$ [f^{-1}U/f^{-1}R] \\longrightarrow f^{-1}[U/R] $$ of stacks in groupoids over $(\\Sch'/S)_{fppf}$."}
+{"_id": "9337", "title": "spaces-groupoids-lemma-diagram-diagonal", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \\to U$ be the stabilizer group algebraic space. The commutative diagram $$ \\xymatrix{ R \\ar[d]^{\\Delta_{R/U \\times_B U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & & R \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\ R \\times_{(U \\times_B U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & & R \\times_{s, U} G \\ar[r] & G } $$ the two left horizontal arrows are isomorphisms and the right square is a fibre product square."}
+{"_id": "9338", "title": "spaces-groupoids-lemma-diagonal", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \\to U$ be the stabilizer group algebraic space. \\begin{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $j : R \\to U \\times_B U$ is separated, \\item $G \\to U$ is separated, and \\item $e : U \\to G$ is a closed immersion. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $j : R \\to U \\times_B U$ is locally separated, \\item $G \\to U$ is locally separated, and \\item $e : U \\to G$ is an immersion. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $j : R \\to U \\times_B U$ is quasi-separated, \\item $G \\to U$ is quasi-separated, and \\item $e : U \\to G$ is quasi-compact. \\end{enumerate} \\end{enumerate}"}
+{"_id": "9357", "title": "spaces-descent-lemma-map-families", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and $\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$ be families of morphisms of algebraic spaces over $S$ with fixed targets. Let $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. Let $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\\{V_j \\to V\\}_{j \\in J}$. Then \\begin{enumerate} \\item The system $$ \\left(g_i^*\\mathcal{F}_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right) $$ is a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$. \\item This construction is functorial in the descent datum $(\\mathcal{F}_j, \\varphi_{jj'})$. \\item Given a second morphism $(g', \\alpha' : I \\to J, (g'_i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data $$ (g_i^*\\mathcal{F}_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}) \\cong ((g'_i)^*\\mathcal{F}_{\\alpha'(i)}, (g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}). $$ \\end{enumerate}"}
+{"_id": "9358", "title": "spaces-descent-lemma-zariski-descent-effective", "text": "Let $S$ be a scheme. Let $U$ be an algebraic space over $S$. Let $\\{U_i \\to U\\}$ be a Zariski covering of $U$, see Topologies on Spaces, Definition \\ref{spaces-topologies-definition-zariski-covering}. Any descent datum on quasi-coherent sheaves for the family $\\mathcal{U} = \\{U_i \\to U\\}$ is effective. Moreover, the functor from the category of quasi-coherent $\\mathcal{O}_U$-modules to the category of descent data with respect to $\\{U_i \\to U\\}$ is fully faithful."}
+{"_id": "9360", "title": "spaces-descent-lemma-finite-presentation-descends", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "9363", "title": "spaces-descent-lemma-locally-projective-descends", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module."}
+{"_id": "9365", "title": "spaces-descent-lemma-finite-finitely-presented-module", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ is finite and of finite presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite presentation."}
+{"_id": "9366", "title": "spaces-descent-lemma-open-fpqc-covering", "text": "Let $S$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering of algebraic spaces over $S$. Suppose that for each $i$ we have an open subspace $W_i \\subset T_i$ such that for all $i, j \\in I$ we have $\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open subspaces of $T_i \\times_T T_j$. Then there exists a unique open subspace $W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$."}
+{"_id": "9367", "title": "spaces-descent-lemma-fpqc-universal-effective-epimorphisms", "text": "Let $S$ be a scheme. Let $\\{T_i \\to T\\}$ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition \\ref{spaces-topologies-definition-fpqc-covering}. Then given an algebraic space $B$ over $S$ the sequence $$ \\xymatrix{ \\Mor_S(T, B) \\ar[r] & \\prod\\nolimits_i \\Mor_S(T_i, B) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod\\nolimits_{i, j} \\Mor_S(T_i \\times_T T_j, B) } $$ is an equalizer diagram. In other words, every representable functor on the category of algebraic spaces over $S$ satisfies the sheaf condition for fpqc coverings."}
+{"_id": "9368", "title": "spaces-descent-lemma-curiosity", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphism of algebraic spaces. Let $P$ be one of the following properties of morphisms of algebraic spaces over $S$: flat, locally finite type, locally finite presentation. Assume that $X \\to Z$ has $P$ and that $X \\to Y$ is a surjection of sheaves on $(\\Sch/S)_{fppf}$. Then $Y \\to Z$ is $P$."}
+{"_id": "9369", "title": "spaces-descent-lemma-flat-finitely-presented-permanence", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that $f$ is surjective, flat, and locally of finite presentation and assume that $p$ is locally of finite presentation (resp.\\ locally of finite type). Then $q$ is locally of finite presentation (resp.\\ locally of finite type)."}
+{"_id": "9370", "title": "spaces-descent-lemma-syntomic-smooth-etale-permanence", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale), \\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale). \\end{enumerate} Then $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale)."}
+{"_id": "9371", "title": "spaces-descent-lemma-smooth-permanence", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $f$ is surjective, flat, and locally of finite presentation, \\item $p$ is smooth (resp.\\ \\'etale). \\end{enumerate} Then $q$ is smooth (resp.\\ \\'etale)."}
+{"_id": "9373", "title": "spaces-descent-lemma-descend-unibranch", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$. If $f$ is flat at $x$ and $X$ is geometrically unibranch at $x$, then $Y$ is geometrically unibranch at $f(x)$."}
+{"_id": "9374", "title": "spaces-descent-lemma-descend-reduced", "text": "\\begin{slogan} A flat and surjective morphism of algebraic spaces with a reduced source has a reduced target. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced."}
+{"_id": "9375", "title": "spaces-descent-lemma-descend-locally-Noetherian", "text": "Let $f : X \\to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is locally Noetherian, then $Y$ is locally Noetherian."}
+{"_id": "9376", "title": "spaces-descent-lemma-descend-regular", "text": "Let $f : X \\to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular."}
+{"_id": "9377", "title": "spaces-descent-lemma-reduced-local-smooth", "text": "Let $f : X \\to Y$ be a smooth morphism of algebraic spaces. If $Y$ is reduced, then $X$ is reduced. If $f$ is surjective and $X$ is reduced, then $Y$ is reduced."}
+{"_id": "9378", "title": "spaces-descent-lemma-pullback-property-local-target", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\\tau$ local on the target. Let $f : X \\to Y$ have property $\\mathcal{P}$. For any morphism $Y' \\to Y$ which is flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic, resp.\\ \\'etale, the base change $f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$."}
+{"_id": "9379", "title": "spaces-descent-lemma-largest-open-of-the-base", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\\tau$ local on the target. For any morphism of algebraic spaces $f : X \\to Y$ over $S$ there exists a largest open subspace $W(f) \\subset Y$ such that the restriction $X_{W(f)} \\to W(f)$ has $\\mathcal{P}$. Moreover, \\begin{enumerate} \\item if $g : Y' \\to Y$ is a morphism of algebraic spaces which is flat and locally of finite presentation, syntomic, smooth, or \\'etale and the base change $f' : X_{Y'} \\to Y'$ has $\\mathcal{P}$, then $g$ factors through $W(f)$, \\item if $g : Y' \\to Y$ is flat and locally of finite presentation, syntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and \\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then $g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$ by $Y_i \\to Y$. \\end{enumerate}"}
+{"_id": "9380", "title": "spaces-descent-lemma-descending-properties-morphisms", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item if $X_i \\to Y_i$, $i = 1, 2$ have property $\\mathcal{P}$ so does $X_1 \\amalg X_2 \\to Y_1 \\amalg Y_2$, \\item a morphism of algebraic spaces $f : X \\to Y$ has property $\\mathcal{P}$ if and only if for every affine scheme $Z$ and morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ has property $\\mathcal{P}$, and \\item for any surjective flat morphism of affine schemes $Z' \\to Z$ over $S$ and a morphism $f : X \\to Z$ from an algebraic space to $Z$ we have $$ f' : Z' \\times_Z X \\to Z'\\text{ has }\\mathcal{P} \\Rightarrow f\\text{ has }\\mathcal{P}. $$ \\end{enumerate} Then $\\mathcal{P}$ is fpqc local on the base."}
+{"_id": "9381", "title": "spaces-descent-lemma-descending-property-quasi-compact", "text": "Let $S$ be a scheme. The property $\\mathcal{P}(f) =$``$f$ is quasi-compact'' is fpqc local on the base on algebraic spaces over $S$."}
+{"_id": "9382", "title": "spaces-descent-lemma-descending-property-quasi-separated", "text": "Let $S$ be a scheme. The property $\\mathcal{P}(f) =$``$f$ is quasi-separated'' is fpqc local on the base on algebraic spaces over $S$."}
+{"_id": "9383", "title": "spaces-descent-lemma-descending-property-universally-closed", "text": "Let $S$ be a scheme. The property $\\mathcal{P}(f) =$``$f$ is universally closed'' is fpqc local on the base on algebraic spaces over $S$."}
+{"_id": "9384", "title": "spaces-descent-lemma-descending-property-universally-open", "text": "Let $S$ be a scheme. The property $\\mathcal{P}(f) =$``$f$ is universally open'' is fpqc local on the base on algebraic spaces over $S$."}
+{"_id": "9386", "title": "spaces-descent-lemma-descending-property-surjective", "text": "The property $\\mathcal{P}(f) =$``$f$ is surjective'' is fpqc local on the base."}
+{"_id": "9387", "title": "spaces-descent-lemma-descending-property-universally-injective", "text": "The property $\\mathcal{P}(f) =$``$f$ is universally injective'' is fpqc local on the base."}
+{"_id": "9389", "title": "spaces-descent-lemma-descending-property-locally-finite-type", "text": "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type'' is fpqc local on the base."}
+{"_id": "9390", "title": "spaces-descent-lemma-descending-property-locally-finite-presentation", "text": "The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation'' is fpqc local on the base."}
+{"_id": "9391", "title": "spaces-descent-lemma-descending-property-finite-type", "text": "The property $\\mathcal{P}(f) =$``$f$ is of finite type'' is fpqc local on the base."}
+{"_id": "9392", "title": "spaces-descent-lemma-descending-property-finite-presentation", "text": "The property $\\mathcal{P}(f) =$``$f$ is of finite presentation'' is fpqc local on the base."}
+{"_id": "9393", "title": "spaces-descent-lemma-descending-property-flat", "text": "The property $\\mathcal{P}(f) =$``$f$ is flat'' is fpqc local on the base."}
+{"_id": "9394", "title": "spaces-descent-lemma-descending-property-open-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is an open immersion'' is fpqc local on the base."}
+{"_id": "9395", "title": "spaces-descent-lemma-descending-property-isomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is an isomorphism'' is fpqc local on the base."}
+{"_id": "9396", "title": "spaces-descent-lemma-descending-property-affine", "text": "The property $\\mathcal{P}(f) =$``$f$ is affine'' is fpqc local on the base."}
+{"_id": "9397", "title": "spaces-descent-lemma-descending-property-closed-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is a closed immersion'' is fpqc local on the base."}
+{"_id": "9398", "title": "spaces-descent-lemma-descending-property-separated", "text": "The property $\\mathcal{P}(f) =$``$f$ is separated'' is fpqc local on the base."}
+{"_id": "9399", "title": "spaces-descent-lemma-descending-property-proper", "text": "The property $\\mathcal{P}(f) =$``$f$ is proper'' is fpqc local on the base."}
+{"_id": "9400", "title": "spaces-descent-lemma-descending-property-quasi-affine", "text": "The property $\\mathcal{P}(f) =$``$f$ is quasi-affine'' is fpqc local on the base."}
+{"_id": "9402", "title": "spaces-descent-lemma-descending-property-integral", "text": "The property $\\mathcal{P}(f) =$``$f$ is integral'' is fpqc local on the base."}
+{"_id": "9403", "title": "spaces-descent-lemma-descending-property-finite", "text": "The property $\\mathcal{P}(f) =$``$f$ is finite'' is fpqc local on the base."}
+{"_id": "9404", "title": "spaces-descent-lemma-descending-property-quasi-finite", "text": "The properties $\\mathcal{P}(f) =$``$f$ is locally quasi-finite'' and $\\mathcal{P}(f) =$``$f$ is quasi-finite'' are fpqc local on the base."}
+{"_id": "9405", "title": "spaces-descent-lemma-descending-property-syntomic", "text": "The property $\\mathcal{P}(f) =$``$f$ is syntomic'' is fpqc local on the base."}
+{"_id": "9406", "title": "spaces-descent-lemma-descending-property-smooth", "text": "The property $\\mathcal{P}(f) =$``$f$ is smooth'' is fpqc local on the base."}
+{"_id": "9407", "title": "spaces-descent-lemma-descending-property-unramified", "text": "The property $\\mathcal{P}(f) =$``$f$ is unramified'' is fpqc local on the base."}
+{"_id": "9408", "title": "spaces-descent-lemma-descending-property-etale", "text": "The property $\\mathcal{P}(f) =$``$f$ is \\'etale'' is fpqc local on the base."}
+{"_id": "9410", "title": "spaces-descent-lemma-descending-property-monomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is a monomorphism'' is fpqc local on the base."}
+{"_id": "9411", "title": "spaces-descent-lemma-descending-fppf-property-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is an immersion'' is fppf local on the base."}
+{"_id": "9413", "title": "spaces-descent-lemma-descending-property-ample", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\{g_i : Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering. Let $f_i : X_i \\to Y_i$ be the base change of $f$ and let $\\mathcal{L}_i$ be the pullback of $\\mathcal{L}$ to $X_i$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/Y$, and \\item $\\mathcal{L}_i$ is ample on $X_i/Y_i$ for every $i \\in I$. \\end{enumerate}"}
+{"_id": "9414", "title": "spaces-descent-lemma-ample-in-neighbourhood", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. There exists an open subspace $V \\subset Y$ characterized by the following property: A morphism $Y' \\to Y$ of algebraic spaces factors through $V$ if and only if the pullback $\\mathcal{L}'$ of $\\mathcal{L}$ to $X' = Y' \\times_Y X$ is ample on $X'/Y'$ (as in Divisors on Spaces, Definition \\ref{spaces-divisors-definition-relatively-ample})."}
+{"_id": "9415", "title": "spaces-descent-lemma-precompose-property-local-source", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0] smooth, \\linebreak[0] \\etale\\}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\\tau$ local on the source. Let $f : X \\to Y$ have property $\\mathcal{P}$. For any morphism $a : X' \\to X$ which is flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic, resp.\\ smooth, resp.\\ \\'etale, the composition $f \\circ a : X' \\to Y$ has property $\\mathcal{P}$."}
+{"_id": "9416", "title": "spaces-descent-lemma-transfer-from-schemes", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0] smooth, \\linebreak[0] \\etale\\}$. Suppose that $\\mathcal{P}$ is a property of morphisms of schemes over $S$ which is \\'etale local on the source-and-target. Denote $\\mathcal{P}_{spaces}$ the corresponding property of morphisms of algebraic spaces over $S$, see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-P}. If $\\mathcal{P}$ is local on the source for the $\\tau$-topology, then $\\mathcal{P}_{spaces}$ is local on the source for the $\\tau$-topology."}
+{"_id": "9417", "title": "spaces-descent-lemma-flat-fpqc-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source."}
+{"_id": "9418", "title": "spaces-descent-lemma-locally-finite-presentation-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation'' is fppf local on the source."}
+{"_id": "9419", "title": "spaces-descent-lemma-locally-finite-type-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally of finite type'' is fppf local on the source."}
+{"_id": "9421", "title": "spaces-descent-lemma-universally-open-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is universally open'' is fppf local on the source."}
+{"_id": "9422", "title": "spaces-descent-lemma-syntomic-syntomic-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is syntomic'' is syntomic local on the source."}
+{"_id": "9423", "title": "spaces-descent-lemma-smooth-smooth-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is smooth'' is smooth local on the source."}
+{"_id": "9424", "title": "spaces-descent-lemma-etale-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is \\'etale'' is \\'etale local on the source."}
+{"_id": "9425", "title": "spaces-descent-lemma-locally-quasi-finite-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite'' is \\'etale local on the source."}
+{"_id": "9426", "title": "spaces-descent-lemma-unramified-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is unramified'' is \\'etale local on the source."}
+{"_id": "9427", "title": "spaces-descent-lemma-local-source-target-implies", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Then \\begin{enumerate} \\item $\\mathcal{P}$ is smooth local on the source, \\item $\\mathcal{P}$ is smooth local on the target, \\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is smooth, then $g \\circ f$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "9428", "title": "spaces-descent-lemma-local-source-target-characterize", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item[(a)] $f$ has property $\\mathcal{P}$, \\item[(b)] for every $x \\in |X|$ there exists a smooth morphism of pairs $a : (U, u) \\to (X, x)$, a smooth morphism $b : V \\to Y$, and a morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and $h$ has $\\mathcal{P}$, \\item[(c)] for some commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $a$, $b$ smooth and $a$ surjective the morphism $h$ has $\\mathcal{P}$, \\item[(d)] for any commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $b$ smooth and $U \\to X \\times_Y V$ smooth the morphism $h$ has $\\mathcal{P}$, \\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$, \\item[(f)] there exists a smooth covering $\\{X_i \\to X\\}_{i \\in I}$ such that each composition $X_i \\to Y$ has $\\mathcal{P}$, \\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and for each $i \\in I$ a smooth covering $\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism $X_{ij} \\to Y_i$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "9429", "title": "spaces-descent-lemma-smooth-local-source-target", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $\\mathcal{P}$ is smooth local on the source, \\item $\\mathcal{P}$ is smooth local on the target, and \\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\to Z$ is a smooth morphism then $X \\to Z$ has $\\mathcal{P}$. \\end{enumerate} Then $\\mathcal{P}$ is smooth local on the source-and-target."}
+{"_id": "9430", "title": "spaces-descent-lemma-etale-smooth-local-source-target-implies", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is \\'etale-smooth local on source-and-target. Then \\begin{enumerate} \\item $\\mathcal{P}$ is \\'etale local on the source, \\item $\\mathcal{P}$ is smooth local on the target, \\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms: if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then $g \\circ f$ has $\\mathcal{P}$, and \\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and $g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "9431", "title": "spaces-descent-lemma-etale-smooth-local-source-target-characterize", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is etale-smooth local on source-and-target. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item[(a)] $f$ has property $\\mathcal{P}$, \\item[(b)] for every $x \\in |X|$ there exists a smooth morphism $b : V \\to Y$, an \\'etale morphism $a : U \\to V \\times_Y X$, and a point $u \\in |U|$ mapping to $x$ such that $U \\to V$ has $\\mathcal{P}$, \\item[(c)] for some commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $b$ smooth, $U \\to V \\times_Y X$ \\'etale, and $a$ surjective the morphism $h$ has $\\mathcal{P}$, \\item[(d)] for any commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $b$ smooth and $U \\to X \\times_Y V$ \\'etale, the morphism $h$ has $\\mathcal{P}$, \\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$, \\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such that each composition $X_i \\to Y$ has $\\mathcal{P}$, \\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and for each $i \\in I$ an \\'etale covering $\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism $X_{ij} \\to Y_i$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "9432", "title": "spaces-descent-lemma-etale-smooth-local-source-target", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $\\mathcal{P}$ is \\'etale local on the source, \\item $\\mathcal{P}$ is smooth local on the target, and \\item $\\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open embedding then $X \\to Z$ has $\\mathcal{P}$. \\end{enumerate} Then $\\mathcal{P}$ is \\'etale-smooth local on the source-and-target."}
+{"_id": "9433", "title": "spaces-descent-lemma-family-is-one", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Set $Y = \\coprod_{i \\in I} X_i$. There is a canonical equivalence of categories $$ \\begin{matrix} \\text{category of descent data } \\\\ \\text{relative to the family } \\{X_i \\to X\\}_{i \\in I} \\end{matrix} \\longrightarrow \\begin{matrix} \\text{ category of descent data} \\\\ \\text{ relative to } Y/X \\end{matrix} $$ which maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with $V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$."}
+{"_id": "9434", "title": "spaces-descent-lemma-pullback", "text": "Pullback of descent data. Let $S$ be a scheme. \\begin{enumerate} \\item Let $$ \\xymatrix{ Y' \\ar[r]_f \\ar[d]_{a'} & Y \\ar[d]^a \\\\ X' \\ar[r]^h & X } $$ be a commutative diagram of algebraic spaces over $S$. The construction $$ (V \\to Y, \\varphi) \\longmapsto f^*(V \\to Y, \\varphi) = (V' \\to Y', \\varphi') $$ where $V' = Y' \\times_Y V$ and where $\\varphi'$ is defined as the composition $$ \\xymatrix{ V' \\times_{X'} Y' \\ar@{=}[r] & (Y' \\times_Y V) \\times_{X'} Y' \\ar@{=}[r] & (Y' \\times_{X'} Y') \\times_{Y \\times_X Y} (V \\times_X Y) \\ar[d]^{\\text{id} \\times \\varphi} \\\\ Y' \\times_{X'} V' \\ar@{=}[r] & Y' \\times_{X'} (Y' \\times_Y V) & (Y' \\times_X Y') \\times_{Y \\times_X Y} (Y \\times_X V) \\ar@{=}[l] } $$ defines a functor from the category of descent data relative to $Y \\to X$ to the category of descent data relative to $Y' \\to X'$. \\item Given two morphisms $f_i : Y' \\to Y$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic. \\end{enumerate}"}
+{"_id": "9435", "title": "spaces-descent-lemma-pullback-family", "text": "Let $S$ be a scheme. Let $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$ and $\\mathcal{U} = \\{X_j \\to X\\}_{i \\in I}$ be families of morphisms with fixed target. Let $\\alpha : I' \\to I$, $g : X' \\to X$ and $g_i : X'_i \\to X_{\\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. \\begin{enumerate} \\item Let $(V_i, \\varphi_{ij})$ be a descent datum relative to the family $\\mathcal{U}$. The system $$ \\left( g_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)} \\right) $$ (with notation as in Remark \\ref{remark-easier-family}) is a descent datum relative to $\\mathcal{U}'$. \\item This construction defines a functor between the category of descent data relative to $\\mathcal{U}$ and the category of descent data relative to $\\mathcal{U}'$. \\item Given a second $\\beta : I' \\to I$, $h : X' \\to X$ and $h'_i : X'_i \\to X_{\\beta(i)}$ morphism of families of maps with fixed target, then if $g = h$ the two resulting functors between descent data are canonically isomorphic. \\item These functors agree, via Lemma \\ref{lemma-family-is-one}, with the pullback functors constructed in Lemma \\ref{lemma-pullback}. \\end{enumerate}"}
+{"_id": "9436", "title": "spaces-descent-lemma-descent-data-sheaves", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic spaces over $S$ (Topologies on Spaces, Definition \\ref{spaces-topologies-definition-fppf-covering}). There is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{descent data }(V_i, \\varphi_{ij})\\\\ \\text{relative to }\\{X_i \\to X\\} \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{sheaves }F\\text{ on }(\\Sch/S)_{fppf}\\text{ endowed}\\\\ \\text{with a map }F \\to X\\text{ such that each}\\\\ X_i \\times_X F\\text{ is an algebraic space} \\end{matrix} \\right\\}. $$ Moreover, \\begin{enumerate} \\item the algebraic space $X_i \\times_X F$ on the right hand side corresponds to $V_i$ on the left hand side, and \\item the sheaf $F$ is an algebraic space\\footnote{We will see later that this is always the case if $I$ is not too large, see Bootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space}.} if and only if the corresponding descent datum $(X_i, \\varphi_{ij})$ is effective. \\end{enumerate}"}
+{"_id": "9437", "title": "spaces-descent-proposition-fpqc-descent-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}$ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition \\ref{spaces-topologies-definition-fpqc-covering}. Any descent datum on quasi-coherent sheaves for $\\{X_i \\to X\\}$ is effective. Moreover, the functor from the category of quasi-coherent $\\mathcal{O}_X$-modules to the category of descent data with respect to $\\{X_i \\to X\\}$ is fully faithful."}
+{"_id": "9454", "title": "decent-spaces-theorem-decent-open-dense-scheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is decent, then there exists a dense open subspace of $X$ which is a scheme."}
+{"_id": "9455", "title": "decent-spaces-lemma-composition-universally-bounded", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $V \\to U$ be a morphism of schemes over $S$, and let $U \\to X$ be a morphism from $U$ to $X$. If the fibres of $V \\to U$ and $U \\to X$ are universally bounded, then so are the fibres of $V \\to X$."}
+{"_id": "9456", "title": "decent-spaces-lemma-base-change-universally-bounded", "text": "Let $S$ be a scheme. Let $Y \\to X$ be a representable morphism of algebraic spaces over $S$. Let $U \\to X$ be a morphism from a scheme to $X$. If the fibres of $U \\to X$ are universally bounded, then the fibres of $U \\times_X Y \\to Y$ are universally bounded."}
+{"_id": "9457", "title": "decent-spaces-lemma-descent-universally-bounded", "text": "Let $S$ be a scheme. Let $g : Y \\to X$ be a representable morphism of algebraic spaces over $S$. Let $f : U \\to X$ be a morphism from a scheme towards $X$. Let $f' : U \\times_X Y \\to Y$ be the base change of $f$. If $$ \\Im(|f| : |U| \\to |X|) \\subset \\Im(|g| : |Y| \\to |X|) $$ and $f'$ has universally bounded fibres, then $f$ has universally bounded fibres."}
+{"_id": "9458", "title": "decent-spaces-lemma-universally-bounded-permanence", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider a commutative diagram $$ \\xymatrix{ U \\ar[rd]_g \\ar[rr]_f & & V \\ar[ld]^h \\\\ & X & } $$ where $U$ and $V$ are schemes. If $g$ has universally bounded fibres, and $f$ is surjective and flat, then also $h$ has universally bounded fibres."}
+{"_id": "9459", "title": "decent-spaces-lemma-universally-bounded-finite-fibres", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$. Let $\\varphi : U \\to X$ be a morphism over $S$. If the fibres of $\\varphi$ are universally bounded, then there exists an integer $n$ such that each fibre of $|U| \\to |X|$ has at most $n$ elements."}
+{"_id": "9460", "title": "decent-spaces-lemma-U-finite-above-x", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The following are equivalent: \\begin{enumerate} \\item there exists a family of schemes $U_i$ and \\'etale morphisms $\\varphi_i : U_i \\to X$ such that $\\coprod \\varphi_i : \\coprod U_i \\to X$ is surjective, and such that for each $i$ the fibre of $|U_i| \\to |X|$ over $x$ is finite, and \\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$ the fibre of $|U| \\to |X|$ over $x$ is finite. \\end{enumerate}"}
+{"_id": "9461", "title": "decent-spaces-lemma-R-finite-above-x", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The following are equivalent: \\begin{enumerate} \\item there exists a scheme $U$, an \\'etale morphism $\\varphi : U \\to X$, and points $u, u' \\in U$ mapping to $x$ such that setting $R = U \\times_X U$ the fibre of $$ |R| \\to |U| \\times_{|X|} |U| $$ over $(u, u')$ is finite, \\item for every scheme $U$, \\'etale morphism $\\varphi : U \\to X$ and any points $u, u' \\in U$ mapping to $x$ setting $R = U \\times_X U$ the fibre of $$ |R| \\to |U| \\times_{|X|} |U| $$ over $(u, u')$ is finite, \\item there exists a morphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$ such that the projections $\\Spec(k) \\times_X \\Spec(k) \\to \\Spec(k)$ are \\'etale and quasi-compact, and \\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$. \\end{enumerate}"}
+{"_id": "9462", "title": "decent-spaces-lemma-weak-UR-finite-above-x", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. Let $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism. The following are equivalent: \\begin{enumerate} \\item $x$ is in the image of $|U| \\to |X|$, and setting $R = U \\times_X U$ the fibres of both $$ |U| \\longrightarrow |X| \\quad\\text{and}\\quad |R| \\longrightarrow |X| $$ over $x$ are finite, \\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$, and the fibre product $\\Spec(k) \\times_X U$ is a finite nonempty scheme over $k$. \\end{enumerate}"}
+{"_id": "9463", "title": "decent-spaces-lemma-UR-finite-above-x", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The following are equivalent: \\begin{enumerate} \\item for every affine scheme $U$, any \\'etale morphism $\\varphi : U \\to X$ setting $R = U \\times_X U$ the fibres of both $$ |U| \\longrightarrow |X| \\quad\\text{and}\\quad |R| \\longrightarrow |X| $$ over $x$ are finite, \\item there exist schemes $U_i$ and \\'etale morphisms $U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and for each $i$, setting $R_i = U_i \\times_X U_i$ the fibres of both $$ |U_i| \\longrightarrow |X| \\quad\\text{and}\\quad |R_i| \\longrightarrow |X| $$ over $x$ are finite, \\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$, and for any affine scheme $U$ and \\'etale morphism $U \\to X$ the fibre product $\\Spec(k) \\times_X U$ is a finite scheme over $k$, \\item there exists a quasi-compact monomorphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$, and \\item there exists a quasi-compact morphism $\\Spec(k) \\to X$ with $k$ a field in the equivalence class of $x$. \\end{enumerate}"}
+{"_id": "9464", "title": "decent-spaces-lemma-U-universally-bounded", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent: \\begin{enumerate} \\item there exist schemes $U_i$ and \\'etale morphisms $U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and each $U_i \\to X$ has universally bounded fibres, and \\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$ the fibres of $U \\to X$ are universally bounded. \\end{enumerate}"}
+{"_id": "9465", "title": "decent-spaces-lemma-characterize-very-reasonable", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent: \\begin{enumerate} \\item there exists a Zariski covering $X = \\bigcup X_i$ and for each $i$ a scheme $U_i$ and a quasi-compact surjective \\'etale morphism $U_i \\to X_i$, and \\item there exist schemes $U_i$ and \\'etale morphisms $U_i \\to X$ such that the projections $U_i \\times_X U_i \\to U_i$ are quasi-compact and $\\coprod U_i \\to X$ is surjective. \\end{enumerate}"}
+{"_id": "9466", "title": "decent-spaces-lemma-bounded-fibres", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the following conditions on $X$: \\begin{itemize} \\item[] $(\\alpha)$ For every $x \\in |X|$, the equivalent conditions of Lemma \\ref{lemma-U-finite-above-x} hold. \\item[] $(\\beta)$ For every $x \\in |X|$, the equivalent conditions of Lemma \\ref{lemma-R-finite-above-x} hold. \\item[] $(\\gamma)$ For every $x \\in |X|$, the equivalent conditions of Lemma \\ref{lemma-UR-finite-above-x} hold. \\item[] $(\\delta)$ The equivalent conditions of Lemma \\ref{lemma-U-universally-bounded} hold. \\item[] $(\\epsilon)$ The equivalent conditions of Lemma \\ref{lemma-characterize-very-reasonable} hold. \\item[] $(\\zeta)$ The space $X$ is Zariski locally quasi-separated. \\item[] $(\\eta)$ The space $X$ is quasi-separated \\item[] $(\\theta)$ The space $X$ is representable, i.e., $X$ is a scheme. \\item[] $(\\iota)$ The space $X$ is a quasi-separated scheme. \\end{itemize} We have $$ \\xymatrix{ & (\\theta) \\ar@{=>}[rd] & & & &  \\\\ (\\iota) \\ar@{=>}[ru] \\ar@{=>}[rd] & & (\\zeta) \\ar@{=>}[r] & (\\epsilon) \\ar@{=>}[r] & (\\delta) \\ar@{=>}[r] & (\\gamma) \\ar@{<=>}[r] & (\\alpha) + (\\beta) \\\\ & (\\eta) \\ar@{=>}[ru] & & & & } $$"}
+{"_id": "9467", "title": "decent-spaces-lemma-properties-local", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be one of the properties $(\\alpha)$, $(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, $(\\zeta)$, or $(\\theta)$ of algebraic spaces listed in Lemma \\ref{lemma-bounded-fibres}. Then if $X$ is an algebraic space over $S$, and $X = \\bigcup X_i$ is a Zariski open covering such that each $X_i$ has $\\mathcal{P}$, then $X$ has $\\mathcal{P}$."}
+{"_id": "9468", "title": "decent-spaces-lemma-representable-properties", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be one of the properties $(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, or $(\\theta)$ of algebraic spaces listed in Lemma \\ref{lemma-bounded-fibres}. Let $X$, $Y$ be algebraic spaces over $S$. Let $X \\to Y$ be a representable morphism. If $Y$ has property $\\mathcal{P}$, so does $X$."}
+{"_id": "9470", "title": "decent-spaces-lemma-representable-named-properties", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $X \\to Y$ be a representable morphism. If $Y$ is decent (resp.\\ reasonable), then so is $X$."}
+{"_id": "9471", "title": "decent-spaces-lemma-etale-named-properties", "text": "Let $S$ be a scheme. Let $X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$. If $Y$ is decent, resp.\\ reasonable, then so is $X$."}
+{"_id": "9472", "title": "decent-spaces-lemma-no-specializations-map-to-same-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \\to X$ be an \\'etale morphism from a scheme to $X$. Assume $u, u' \\in |U|$ map to the same point $x$ of $|X|$, and $u' \\leadsto u$. If the pair $(X, x)$ satisfies the equivalent conditions of Lemma \\ref{lemma-U-finite-above-x} then $u = u'$."}
+{"_id": "9473", "title": "decent-spaces-lemma-specialization", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent conditions of Lemma \\ref{lemma-UR-finite-above-x}. Then for every \\'etale morphism $\\varphi : U \\to X$ from a scheme $U$ and any $u \\in U$ with $\\varphi(u) = x$, exists a point $u'\\in U$, $u' \\leadsto u$ with $\\varphi(u') = x'$."}
+{"_id": "9474", "title": "decent-spaces-lemma-generalizations-lift-flat", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a  flat morphism of algebraic spaces over $S$. Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent conditions of Lemma \\ref{lemma-UR-finite-above-x} (for example if $X$ is decent, $X$ is quasi-separated, or $X$ is representable). Then for every $y \\in |Y|$ with $f(y) = x$, there exists a point $y' \\in |Y|$, $y' \\leadsto y$ with $f(y') = x'$."}
+{"_id": "9475", "title": "decent-spaces-lemma-quasi-compact-reasonable-stratification", "text": "Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. There exist reduced closed subspaces $$ \\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset \\ldots \\subset Z_n = X $$ such that with $X_r = Z_r \\setminus Z_{r - 1}$ the stratification $X = \\coprod_{r = 0, \\ldots, n} X_r$ is characterized by the following universal property: Given $g : T \\to X$ the projection $W \\times_X T \\to T$ is finite locally free of degree $r$ if and only if $g(|T|) \\subset |X_r|$."}
+{"_id": "9476", "title": "decent-spaces-lemma-stratify-flat-fp-lqf", "text": "Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, and locally quasi-finite. Then there exist open subspaces $$ X = X_0 \\supset X_1 \\supset X_2 \\supset \\ldots $$ such that a morphism $\\Spec(k) \\to X$ factors through $X_d$ if and only if $W \\times_X \\Spec(k)$ has degree $\\geq d$ over $k$."}
+{"_id": "9477", "title": "decent-spaces-lemma-filter-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$. There exist open subspaces $$ \\ldots \\subset U_4 \\subset U_3 \\subset U_2 \\subset U_1 = X $$ with the following properties: \\begin{enumerate} \\item setting $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$ such that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism, \\item if $x \\in |X|$ can be represented by a quasi-compact morphism $\\Spec(k) \\to X$ from a field, then $x \\in T_p$ for some $p$. \\end{enumerate}"}
+{"_id": "9478", "title": "decent-spaces-lemma-filter-reasonable", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces $$ \\emptyset = U_{n + 1} \\subset U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X $$ with the following property: setting $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$ such that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism."}
+{"_id": "9480", "title": "decent-spaces-lemma-filter-quasi-compact-quasi-separated", "text": "\\begin{reference} This result is almost identical to \\cite[Proposition 5.7.8]{GruRay}. \\end{reference} Let $X$ be a quasi-compact and quasi-separated algebraic space over $\\Spec(\\mathbf{Z})$. There exist an integer $n$ and open subspaces $$ \\emptyset = U_{n + 1} \\subset U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X $$ with the following property: setting $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$ such that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism."}
+{"_id": "9482", "title": "decent-spaces-lemma-extend-integral-morphism", "text": "Let $S$ be a scheme. Let $j : V \\to Y$ be a quasi-compact open immersion of algebraic spaces over $S$. Let $\\pi : Z \\to V$ be an integral morphism. Then there exists an integral morphism $\\nu : Y' \\to Y$ such that $Z$ is $V$-isomorphic to the inverse image of $V$ in $Y'$."}
+{"_id": "9483", "title": "decent-spaces-lemma-there-is-a-scheme-integral-over", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. \\begin{enumerate} \\item There exists a surjective integral morphism $Y \\to X$ where $Y$ is a scheme, \\item given a surjective \\'etale morphism $U \\to X$ we may choose $Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood $V \\subset Y$ such that $V \\to X$ factors through $U$. \\end{enumerate}"}
+{"_id": "9484", "title": "decent-spaces-lemma-when-quotient-scheme-at-point", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a surjective finite locally free morphism of algebraic spaces over $S$. For $y \\in |Y|$ the following are equivalent \\begin{enumerate} \\item $y$ is in the schematic locus of $Y$, and \\item there exists an affine open $U \\subset X$ containing the preimage of $y$. \\end{enumerate}"}
+{"_id": "9486", "title": "decent-spaces-lemma-decent-points-monomorphism", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map $$ \\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\} \\longrightarrow |X| $$ This map is always injective. If $X$ is decent then this map is a bijection."}
+{"_id": "9487", "title": "decent-spaces-lemma-identifies-residue-fields", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \\in |X|$ be a point with image $y = f(x) \\in |Y|$. The following are equivalent \\begin{enumerate} \\item $f$ induces an isomorphism $\\kappa(y) \\to \\kappa(x)$, and \\item the induced morphism $\\Spec(\\kappa(x)) \\to Y$ is a monomorphism. \\end{enumerate}"}
+{"_id": "9488", "title": "decent-spaces-lemma-decent-space-elementary-etale-neighbourhood", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. For every point $x \\in |X|$ there exists an \\'etale morphism $$ (U, u) \\longrightarrow (X, x) $$ where $U$ is an affine scheme, $u$ is the only point of $U$ lying over $x$, and the induced homomorphism $\\kappa(x) \\to \\kappa(u)$ is an isomorphism."}
+{"_id": "9489", "title": "decent-spaces-lemma-elementary-etale-neighbourhoods", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x$ be a point of $X$. The category of elementary \\'etale neighborhoods of $(X, x)$ is cofiltered (see Categories, Definition \\ref{categories-definition-codirected})."}
+{"_id": "9490", "title": "decent-spaces-lemma-describe-henselian-local-ring", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. Let $(U, u) \\to (X, x)$ be an elementary \\'etale neighbourhood. Then $$ \\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h $$ In words: the henselian local ring of $X$ at $x$ is equal to the henselization $\\mathcal{O}_{U, u}^h$ of the local ring $\\mathcal{O}_{U, u}$ of $U$ at $u$."}
+{"_id": "9491", "title": "decent-spaces-lemma-henselian-local-ring-strict", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$ lying over $x \\in |X|$.  The \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ of $X$ at $\\overline{x}$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-etale-local-rings}) is the strict henselization of the henselian local ring $\\mathcal{O}_{X, x}^h$ of $X$ at $x$."}
+{"_id": "9493", "title": "decent-spaces-lemma-decent-no-specializations-map-to-same-point", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $U \\to X$ be an \\'etale morphism from a scheme to $X$. If $u, u' \\in |U|$ map to the same point of $|X|$, and $u' \\leadsto u$, then $u = u'$."}
+{"_id": "9494", "title": "decent-spaces-lemma-decent-specialization", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a specialization of $x'$. Then for every \\'etale morphism $\\varphi : U \\to X$ from a scheme $U$ and any $u \\in U$ with $\\varphi(u) = x$, exists a point $u'\\in U$, $u' \\leadsto u$ with $\\varphi(u') = x'$."}
+{"_id": "9495", "title": "decent-spaces-lemma-kolmogorov", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then $|X|$ is Kolmogorov (see Topology, Definition \\ref{topology-definition-generic-point})."}
+{"_id": "9496", "title": "decent-spaces-lemma-dimension-decent-space", "text": "Let $S$ be a scheme. Dimension as defined in Properties of Spaces, Section \\ref{spaces-properties-section-dimension} behaves well on decent algebraic spaces $X$ over $S$. \\begin{enumerate} \\item If $x \\in |X|$, then $\\dim_x(|X|) = \\dim_x(X)$, and \\item $\\dim(|X|) = \\dim(X)$. \\end{enumerate}"}
+{"_id": "9497", "title": "decent-spaces-lemma-dimension-local-ring-quasi-finite", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism of algebraic spaces over $S$. Let $x \\in |X|$ with image $y \\in |Y|$. Then the dimension of the local ring of $Y$ at $y$ is $\\geq$ to the dimension of the local ring of $X$ at $x$."}
+{"_id": "9498", "title": "decent-spaces-lemma-dimension-quasi-finite", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism of algebraic spaces over $S$. Then $\\dim(X) \\leq \\dim(Y)$."}
+{"_id": "9499", "title": "decent-spaces-lemma-decent-point-like-spaces", "text": "Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space over $S$ and assume that there exists a surjective \\'etale morphism $\\Spec(k) \\to X$. If $X$ is decent, then $X \\cong \\Spec(k')$ where $k' \\subset k$ is a finite separable extension."}
+{"_id": "9500", "title": "decent-spaces-lemma-flat-cover-by-field", "text": "Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. Let $k$ be a field and let $\\Spec(k) \\to Z$ be surjective and flat. Then any morphism $\\Spec(k') \\to Z$ where $k'$ is a field is surjective and flat."}
+{"_id": "9501", "title": "decent-spaces-lemma-unique-point", "text": "Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $Z$ is reduced and $|Z|$ is a singleton, \\item there exists a surjective flat morphism $\\Spec(k) \\to Z$ where $k$ is a field, and \\item there exists a locally of finite type, surjective, flat morphism $\\Spec(k) \\to Z$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "9502", "title": "decent-spaces-lemma-unique-point-better", "text": "Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $Z$ is reduced, locally Noetherian, and $|Z|$ is a singleton, and \\item there exists a locally finitely presented, surjective, flat morphism $\\Spec(k) \\to Z$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "9503", "title": "decent-spaces-lemma-monomorphism-into-point", "text": "Let $S$ be a scheme. Let $Z' \\to Z$ be a monomorphism of algebraic spaces over $S$. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\\Spec(k) \\to Z$. Then either $Z'$ is empty or $Z' = Z$."}
+{"_id": "9504", "title": "decent-spaces-lemma-find-singleton-from-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. Then there exists a unique monomorphism $Z \\to X$ of algebraic spaces over $S$ such that $Z$ is an algebraic space which satisfies the equivalent conditions of Lemma \\ref{lemma-unique-point-better} and such that the image of $|Z| \\to |X|$ is $\\{x\\}$."}
+{"_id": "9506", "title": "decent-spaces-lemma-locally-Noetherian-decent-quasi-separated", "text": "Any locally Noetherian decent algebraic space is quasi-separated."}
+{"_id": "9507", "title": "decent-spaces-lemma-when-field", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. \\begin{enumerate} \\item If $|X|$ is a singleton then $X$ is a scheme. \\item If $|X|$ is a singleton and $X$ is reduced, then $X \\cong \\Spec(k)$ for some field $k$. \\end{enumerate}"}
+{"_id": "9508", "title": "decent-spaces-lemma-algebraic-residue-field-extension-closed-point", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Consider a commutative diagram $$ \\xymatrix{ \\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\ & S } $$ Assume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is a closed point and that $\\kappa(s) \\subset k$ is algebraic. Then the image $x$ of $\\Spec(k) \\to X$ is a closed point of $|X|$."}
+{"_id": "9509", "title": "decent-spaces-lemma-finite-residue-field-extension-finite", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Consider a commutative diagram $$ \\xymatrix{ \\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\ & S } $$ Assume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is a closed point and that $\\kappa(s) \\subset k$ is finite. Then $\\Spec(k) \\to X$ is finite morphism. If $\\kappa(s) = k$ then $\\Spec(k) \\to X$ is closed immersion."}
+{"_id": "9510", "title": "decent-spaces-lemma-decent-space-closed-point", "text": "Let $S$ be a scheme. Suppose $X$ is a decent algebraic space over $S$. Let $x \\in |X|$ be a closed point. Then $x$ can be represented by a closed immersion $i : \\Spec(k) \\to X$ from the spectrum of a field."}
+{"_id": "9511", "title": "decent-spaces-lemma-infinite-number", "text": "Let $A$ be a ring. Let $k$ be a field. Let $\\mathfrak p_n$, $n \\geq 1$ be a sequence of pairwise distinct primes of $A$. Moreover, for each $n$ let $k \\to \\kappa(\\mathfrak p_n)$ be an embedding. Then the closure of the image of $$ \\coprod\\nolimits_{n \\not = m} \\Spec(\\kappa(\\mathfrak p_n) \\otimes_k \\kappa(\\mathfrak p_m)) \\longrightarrow \\Spec(A \\otimes A) $$ meets the diagonal."}
+{"_id": "9512", "title": "decent-spaces-lemma-locally-separated-decent", "text": "A locally separated algebraic space is decent."}
+{"_id": "9513", "title": "decent-spaces-lemma-properties-trivial-implications", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We have the following implications among the conditions on $f$: $$ \\xymatrix{ \\text{representable} \\ar@{=>}[rd] & & & & \\\\ & \\text{very reasonable} \\ar@{=>}[r] & \\text{reasonable} \\ar@{=>}[r] & \\text{decent} \\ar@{=>}[r] & (\\beta) \\\\ \\text{quasi-separated} \\ar@{=>}[ru] & & & & } $$"}
+{"_id": "9514", "title": "decent-spaces-lemma-property-for-morphism-out-of-property", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $X$ is decent (resp.\\ is reasonable, resp.\\ has property $(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then $f$ is decent (resp.\\ reasonable, resp.\\ has property $(\\beta)$)."}
+{"_id": "9515", "title": "decent-spaces-lemma-base-change-relative-conditions", "text": "Having property $(\\beta)$, being decent, or being reasonable is preserved under arbitrary base change."}
+{"_id": "9516", "title": "decent-spaces-lemma-property-over-property", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\omega \\in \\{\\beta, decent, reasonable\\}$. Suppose that $Y$ has property $(\\omega)$ and $f : X \\to Y$ has $(\\omega)$. Then $X$ has $(\\omega)$."}
+{"_id": "9517", "title": "decent-spaces-lemma-composition-relative-conditions", "text": "Having property $(\\beta)$, being decent, or being reasonable is preserved under compositions."}
+{"_id": "9519", "title": "decent-spaces-lemma-descent-conditions", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$. Assume \\begin{enumerate} \\item $f$ is quasi-compact, \\item $f$ is \\'etale, \\item $|f| : |X| \\to |Y|$ is surjective, and \\item the algebraic space $X$ has property $\\mathcal{P}$. \\end{enumerate} Then $Y$ has property $\\mathcal{P}$."}
+{"_id": "9522", "title": "decent-spaces-lemma-surjective-on-fibres", "text": "In the situation of (\\ref{equation-points-fibres}) if $Z' \\to Z$ is a morphism and $z' \\in |Z'|$ maps to $z$, then the induced map $F_{x, z'} \\to F_{x, z}$ is surjective."}
+{"_id": "9523", "title": "decent-spaces-lemma-qf-and-qc-finite-fibre", "text": "In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre}) is finite if $f$ is of finite type and $f$ is quasi-finite at $x$."}
+{"_id": "9524", "title": "decent-spaces-lemma-decent-finite-fibre", "text": "In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre}) is finite if $y$ can be represented by a monomorphism $\\Spec(k) \\to Y$ where $k$ is a field and $g$ is quasi-finite at $z$. (Special case: $Y$ is decent and $g$ is \\'etale.)"}
+{"_id": "9525", "title": "decent-spaces-lemma-topology-fibre", "text": "\\begin{slogan} Fibers of field points of algebraic spaces have the expected Zariski topologies. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$ and assume that $y$ is represented by a quasi-compact monomorphism $\\Spec(k) \\to Y$. Then $|X_k| \\to |X|$ is a homeomorphism onto $f^{-1}(\\{y\\}) \\subset |X|$ with induced topology."}
+{"_id": "9526", "title": "decent-spaces-lemma-conditions-on-point-on-space-over-field", "text": "Let $X$ be an algebraic space locally of finite type over a field $k$. Let $x \\in |X|$. Consider the conditions \\begin{enumerate} \\item $\\dim_x(|X|) = 0$, \\item $x$ is closed in $|X|$ and if $x' \\leadsto x$  in $|X|$ then $x' = x$, \\item $x$ is an isolated point of $|X|$, \\item $\\dim_x(X) = 0$, \\item $X \\to \\Spec(k)$ is quasi-finite at $x$. \\end{enumerate} Then (2), (3), (4), and (5) are equivalent. If $X$ is decent, then (1) is equivalent to the others."}
+{"_id": "9527", "title": "decent-spaces-lemma-conditions-on-space-over-field", "text": "Let $X$ be an algebraic space locally of finite type over a field $k$. Consider the conditions \\begin{enumerate} \\item $|X|$ is a finite set, \\item $|X|$ is a discrete space, \\item $\\dim(|X|) = 0$, \\item $\\dim(X) = 0$, \\item $X \\to \\Spec(k)$ is locally quasi-finite, \\end{enumerate} Then (2), (3), (4), and (5) are equivalent. If $X$ is decent, then (1) implies the others."}
+{"_id": "9528", "title": "decent-spaces-lemma-conditions-on-point-in-fibre-and-qf", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $x \\in |X|$ with image $y \\in |Y|$. Let $F = f^{-1}(\\{y\\})$ with induced topology from $|X|$. Let $k$ be a field and let $\\Spec(k) \\to Y$ be in the equivalence class defining $y$. Set $X_k = \\Spec(k) \\times_Y X$. Let $\\tilde x \\in |X_k|$ map to $x \\in |X|$. Consider the following conditions \\begin{enumerate} \\item \\label{item-fibre-at-x-dim-0} $\\dim_x(F) = 0$, \\item \\label{item-isolated-in-fibre} $x$ is isolated in $F$, \\item \\label{item-no-specializations-in-fibre} $x$ is closed in $F$ and if $x' \\leadsto x$ in $F$, then $x = x'$, \\item \\label{item-dimension-top-k-fibre} $\\dim_{\\tilde x}(|X_k|) = 0$, \\item \\label{item-isolated-in-k-fibre} $\\tilde x$ is isolated in $|X_k|$, \\item \\label{item-no-specializations-in-k-fibre} $\\tilde x$ is closed in $|X_k|$ and if $\\tilde x' \\leadsto \\tilde x$ in $|X_k|$, then $\\tilde x = \\tilde x'$, \\item \\label{item-k-fibre-at-x-dim-0} $\\dim_{\\tilde x}(X_k) = 0$, \\item \\label{item-quasi-finite-at-x} $f$ is quasi-finite at $x$. \\end{enumerate} Then we have $$ \\xymatrix{ (\\ref{item-dimension-top-k-fibre}) \\ar@{=>}[r]_{f\\text{ decent}} & (\\ref{item-isolated-in-k-fibre}) \\ar@{<=>}[r] & (\\ref{item-no-specializations-in-k-fibre}) \\ar@{<=>}[r] & (\\ref{item-k-fibre-at-x-dim-0}) \\ar@{<=>}[r] & (\\ref{item-quasi-finite-at-x}) } $$ If $Y$ is decent, then conditions (\\ref{item-isolated-in-fibre}) and (\\ref{item-no-specializations-in-fibre}) are equivalent to each other and to conditions (\\ref{item-isolated-in-k-fibre}), (\\ref{item-no-specializations-in-k-fibre}), (\\ref{item-k-fibre-at-x-dim-0}), and (\\ref{item-quasi-finite-at-x}). If $Y$ and $X$ are decent, then all conditions are equivalent."}
+{"_id": "9529", "title": "decent-spaces-lemma-conditions-on-fibre-and-qf", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $y \\in |Y|$. Let $k$ be a field and let $\\Spec(k) \\to Y$ be in the equivalence class defining $y$. Set $X_k = \\Spec(k) \\times_Y X$ and let $F = f^{-1}(\\{y\\})$ with the induced topology from $|X|$. Consider the following conditions \\begin{enumerate} \\item \\label{item-fibre-finite} $F$ is finite, \\item \\label{item-fibre-discrete} $F$ is a discrete topological space, \\item \\label{item-fibre-no-specializations} $\\dim(F) = 0$, \\item \\label{item-k-fibre-finite} $|X_k|$ is a finite set, \\item \\label{item-k-fibre-discrete} $|X_k|$ is a discrete space, \\item \\label{item-k-fibre-no-specializations} $\\dim(|X_k|) = 0$, \\item \\label{item-k-fibre-dim-0} $\\dim(X_k) = 0$, \\item \\label{item-quasi-finite-at-points-fibre} $f$ is quasi-finite at all points of $|X|$ lying over $y$. \\end{enumerate} Then we have $$ \\xymatrix{ (\\ref{item-fibre-finite}) & (\\ref{item-k-fibre-finite}) \\ar@{=>}[l] \\ar@{=>}[r]_{f\\text{ decent}} & (\\ref{item-k-fibre-discrete}) \\ar@{<=>}[r] & (\\ref{item-k-fibre-no-specializations}) \\ar@{<=>}[r] & (\\ref{item-k-fibre-dim-0}) \\ar@{<=>}[r] & (\\ref{item-quasi-finite-at-points-fibre}) } $$ If $Y$ is decent, then conditions (\\ref{item-fibre-discrete}) and (\\ref{item-fibre-no-specializations}) are equivalent to each other and to conditions (\\ref{item-k-fibre-discrete}), (\\ref{item-k-fibre-no-specializations}), (\\ref{item-k-fibre-dim-0}), and (\\ref{item-quasi-finite-at-points-fibre}). If $Y$ and $X$ are decent, then (\\ref{item-fibre-finite}) implies all the other conditions."}
+{"_id": "9530", "title": "decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings", "text": "Let $S$ be a scheme. Let $Y$ be a disjoint union of spectra of zero dimensional local rings over $S$. Let $f : X \\to Y$ be a monomorphism of algebraic spaces over $S$. Then $f$ is representable, i.e., $X$ is a scheme."}
+{"_id": "9531", "title": "decent-spaces-lemma-decent-generic-points", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item $x$ is a generic point of an irreducible component of $|X|$, \\item for any \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$, \\item for some \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic spaces, $y$ is a generic point of an irreducible component of $|Y|$, \\item the dimension of the local ring of $X$ at $x$ is zero, and \\item $x$ is a point of codimension $0$ on $X$ \\end{enumerate}"}
+{"_id": "9532", "title": "decent-spaces-lemma-codimension-local-ring", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $T \\subset |X|$ be an irreducible closed subset. Let $\\xi \\in T$ be the generic point (Proposition \\ref{proposition-reasonable-sober}). Then $\\text{codim}(T, |X|)$ (Topology, Definition \\ref{topology-definition-codimension}) is the dimension of the local ring of $X$ at $\\xi$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension-local-ring})."}
+{"_id": "9533", "title": "decent-spaces-lemma-get-reasonable", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \\begin{enumerate} \\item every quasi-compact scheme \\'etale over $X$ has finitely many irreducible components, and \\item every $x \\in |X|$ of codimension $0$ on $X$ can be represented by a monomorphism $\\Spec(k) \\to X$. \\end{enumerate} Then $X$ is a reasonable algebraic space."}
+{"_id": "9534", "title": "decent-spaces-lemma-finitely-many-irreducible-components", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is decent and $|X|$ has finitely many irreducible components, \\item every quasi-compact scheme \\'etale over $X$ has finitely many irreducible components, there are finitely many $x \\in |X|$ of codimension $0$ on $X$, and each of these can be represented by a monomorphism $\\Spec(k) \\to X$, \\item there exists a dense open $X' \\subset X$ which is a scheme, $X'$ has finitely many irreducible components with generic points $\\{x'_1, \\ldots, x'_m\\}$, and the morphism $x'_j \\to X$ is quasi-compact for $j = 1, \\ldots, m$. \\end{enumerate} Moreover, if these conditions hold, then $X$ is reasonable and the points $x'_j \\in |X|$ are the generic points of the irreducible components of $|X|$."}
+{"_id": "9535", "title": "decent-spaces-lemma-generically-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated of finite type. Let $y \\in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent: \\begin{enumerate} \\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$, \\item $X \\to Y$ is quasi-finite at all points of $|X|$ over $y$, \\item there exists an open subspace $Y' \\subset Y$ with $y \\in |Y'|$ such that $Y' \\times_Y X \\to Y'$ is finite. \\end{enumerate} If $Y$ is decent these are also equivalent to \\begin{enumerate} \\item[(4)] the set $f^{-1}(\\{y\\})$ is finite. \\end{enumerate}"}
+{"_id": "9536", "title": "decent-spaces-lemma-generically-finite-reprise", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated and locally of finite type and $Y$ quasi-separated. Let $y \\in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent: \\begin{enumerate} \\item the set $f^{-1}(\\{y\\})$ is finite, \\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$, \\item there exist open subspaces $X' \\subset X$ and $Y' \\subset Y$ with $f(X') \\subset Y'$, $y \\in |Y'|$, and $f^{-1}(\\{y\\}) \\subset |X'|$ such that $f|_{X'} : X' \\to Y'$ is finite. \\end{enumerate}"}
+{"_id": "9537", "title": "decent-spaces-lemma-quasi-finiteness-over-generic-point", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. Let $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of codimension $0$ points of $X$, resp.\\ $Y$. Let $y \\in Y^0$. The following are equivalent \\begin{enumerate} \\item $f^{-1}(\\{y\\}) \\subset X^0$, \\item $f$ is quasi-finite at all points lying over $y$, \\item $f$ is quasi-finite at all $x \\in X^0$ lying over $y$. \\end{enumerate}"}
+{"_id": "9538", "title": "decent-spaces-lemma-finite-over-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. Let $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of codimension $0$ points of $X$, resp.\\ $Y$. Assume \\begin{enumerate} \\item $Y$ is decent, \\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$, \\item either $f$ is quasi-compact or $f$ is separated. \\end{enumerate} Then there exists a dense open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is finite."}
+{"_id": "9542", "title": "decent-spaces-lemma-birational-isomorphism-over-dense-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. Assume \\begin{enumerate} \\item either $f$ is quasi-compact or $f$ is separated, and \\item either $f$ is locally of finite type and $Y$ is reduced or $f$ is locally of finite presentation. \\end{enumerate} Then there exists a dense open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is an isomorphism."}
+{"_id": "9543", "title": "decent-spaces-lemma-birational-etale-localization", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational and $V \\to Y$ is an \\'etale morphism with $V$ affine, then $X \\times_Y V$ is decent with finitely many irreducible components and $X \\times_Y V \\to V$ is birational."}
+{"_id": "9544", "title": "decent-spaces-lemma-birational-induced-morphism-normalizations", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism between algebraic spaces over $S$ which are decent and have finitely many irreducible components. Then the normalizations $X^\\nu \\to X$ and $Y^\\nu \\to Y$ exist and there is a commutative diagram $$ \\xymatrix{ X^\\nu \\ar[r] \\ar[d] &  Y^\\nu \\ar[d] \\\\ X \\ar[r] & Y } $$ of algebraic spaces over $S$. The morphism $X^\\nu \\to Y^\\nu$ is birational."}
+{"_id": "9545", "title": "decent-spaces-lemma-finite-birational-over-normal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X$ and $Y$ are decent and have finitely many irreducible components, \\item $f$ is integral and birational, \\item $Y$ is normal, and \\item $X$ is reduced. \\end{enumerate} Then $f$ is an isomorphism."}
+{"_id": "9547", "title": "decent-spaces-lemma-Jacobson-universally-Jacobson", "text": "Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. Any algebraic space locally of finite type over $X$ is Jacobson."}
+{"_id": "9549", "title": "decent-spaces-lemma-decent-Jacobson-ft-pts", "text": "Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$. Then $X_{\\text{ft-pts}} \\subset |X|$ is the set of closed points."}
+{"_id": "9552", "title": "decent-spaces-lemma-irreducible-local-ring", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$ be a point. The following are equivalent \\begin{enumerate} \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ the local ring $\\mathcal{O}_{U, u}$ has a unique minimal prime, \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there is a unique irreducible component of $U$ through $u$, \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ the local ring $\\mathcal{O}_{U, u}$ is unibranch, \\item the henselian local ring $\\mathcal{O}_{X, x}^h$ has a unique minimal prime. \\end{enumerate}"}
+{"_id": "9553", "title": "decent-spaces-lemma-nr-branches-local-ring", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer. The following are equivalent \\begin{enumerate} \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ the number of minimal primes of the local ring $\\mathcal{O}_{U, u}$ is $\\leq n$ and for at least one choice of $(U, u)$ it is $n$, \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ the number irreducible components of $U$ passing through $u$ is $\\leq n$ and for at least one choice of $(U, u)$ it is $n$, \\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ the number of branches of $U$ at $u$ is $\\leq n$ and for at least one choice of $(U, u)$ it is $n$, \\item the number of minimal prime ideals of $\\mathcal{O}_{X, x}^h$ is $n$. \\end{enumerate}"}
+{"_id": "9554", "title": "decent-spaces-lemma-scheme-with-dimension-function", "text": "Let $S$ be a locally Noetherian and universally catenary scheme. Let $\\delta : S \\to \\mathbf{Z}$ be a dimension function. Let $X$ be a decent algebraic space over $S$ such that the structure morphism $X \\to S$ is locally of finite type. Let $\\delta_X : |X| \\to \\mathbf{Z}$ be the map sending $x$ to $\\delta(f(x))$ plus the transcendence degree of $x/f(x)$. Then $\\delta_X$ is a dimension function on $|X|$."}
+{"_id": "9557", "title": "decent-spaces-lemma-check-dimension-function-finite-cover", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective finite morphism of decent and locally Noetherian algebraic spaces. Let $\\delta : |X| \\to \\mathbf{Z}$ be a function. If $\\delta \\circ |f|$ is a dimension function, then $\\delta$ is a dimension function."}
+{"_id": "9558", "title": "decent-spaces-proposition-reasonable-open-dense-scheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is reasonable, then there exists a dense open subspace of $X$ which is a scheme."}
+{"_id": "9559", "title": "decent-spaces-proposition-reasonable-sober", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then the topological space $|X|$ is sober (see Topology, Definition \\ref{topology-definition-generic-point})."}
+{"_id": "9560", "title": "decent-spaces-proposition-characterize-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact, and $X$ is decent. Then $f$ is universally closed if and only if the existence part of the valuative criterion holds."}
+{"_id": "9578", "title": "groupoids-lemma-restrict-relation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j : R \\to U \\times_S U$ be a pre-relation. Let $g : U' \\to U$ be a morphism of schemes. Finally, set $$ R' = (U' \\times_S U')\\times_{U \\times_S U} R \\xrightarrow{j'} U' \\times_S U' $$ Then $j'$ is a pre-relation on $U'$ over $S$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation."}
+{"_id": "9579", "title": "groupoids-lemma-pre-equivalence-equivalence-relation-points", "text": "Let $j : R \\to U \\times_S U$ be a pre-relation. Consider the relation on points of the scheme $U$ defined by the rule $$ x \\sim y \\Leftrightarrow \\exists\\ r \\in R : t(r) = x, s(r) = y. $$ If $j$ is a pre-equivalence relation then this is an equivalence relation."}
+{"_id": "9580", "title": "groupoids-lemma-etale-equivalence-relation", "text": "Let $j : R \\to U \\times_S U$ be a pre-relation. Assume \\begin{enumerate} \\item $s, t$ are unramified, \\item for any algebraically closed field $k$ over $S$ the map $R(k) \\to U(k) \\times U(k)$ is an equivalence relation, \\item there are morphisms $e : U \\to R$, $i : R \\to R$, $c : R \\times_{s, U, t} R \\to R$ such that $$ \\xymatrix{ U \\ar[r]_e \\ar[d]_\\Delta & R \\ar[d]_j & R \\ar[d]^j \\ar[r]_i & R \\ar[d]^j & R \\times_{s, U, t} R \\ar[d]^{j \\times j} \\ar[r]_c & R \\ar[d]^j \\\\ U \\times_S U \\ar[r] & U \\times_S U & U \\times_S U \\ar[r]^{flip} & U \\times_S U & U \\times_S U \\times_S U \\ar[r]^{\\text{pr}_{02}} & U \\times_S U } $$ are commutative. \\end{enumerate} Then $j$ is an equivalence relation."}
+{"_id": "9581", "title": "groupoids-lemma-base-change-group-scheme", "text": "Let $(G, m)$ be a group scheme over $S$. Let $S' \\to S$ be a morphism of schemes. The pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$."}
+{"_id": "9582", "title": "groupoids-lemma-closed-subgroup-scheme", "text": "Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$. \\begin{enumerate} \\item A closed subscheme $H \\subset G$ is a closed subgroup scheme if and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$, and $i|_H : H \\to G$ factor through $H$. \\item An open subscheme $H \\subset G$ is an open subgroup scheme if and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$, and $i|_H : H \\to G$ factor through $H$. \\end{enumerate}"}
+{"_id": "9583", "title": "groupoids-lemma-group-scheme-separated", "text": "Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Then $G \\to S$ is separated (resp.\\ quasi-separated) if and only if the identity morphism $e : S \\to G$ is a closed immersion (resp.\\ quasi-compact)."}
+{"_id": "9585", "title": "groupoids-lemma-group-scheme-module-differentials", "text": "\\begin{reference} \\cite[Proposition 3.15]{BookAV} \\end{reference} Let $(G, m, e, i)$ be a group scheme over the scheme $S$. Denote $f : G \\to S$ the structure morphism. Then there exist canonical isomorphisms $$ \\Omega_{G/S} \\cong f^*\\mathcal{C}_{S/G} \\cong f^*e^*\\Omega_{G/S} $$ where $\\mathcal{C}_{S/G}$ denotes the conormal sheaf of the immersion $e$. In particular, if $S$ is the spectrum of a field, then $\\Omega_{G/S}$ is a free $\\mathcal{O}_G$-module."}
+{"_id": "9586", "title": "groupoids-lemma-group-scheme-addition-tangent-vectors", "text": "Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $s \\in S$. Then the composition $$ T_{G/S, e(s)} \\oplus T_{G/S, e(s)} = T_{G \\times_S G/S, (e(s), e(s))} \\rightarrow T_{G/S, e(s)} $$ is addition of tangent vectors. Here the $=$ comes from Varieties, Lemma \\ref{varieties-lemma-tangent-space-product} and the right arrow is induced from $m : G \\times_S G \\to G$ via Varieties, Lemma \\ref{varieties-lemma-map-tangent-spaces}."}
+{"_id": "9587", "title": "groupoids-lemma-group-scheme-over-field-open-multiplication", "text": "If $(G, m)$ is a group scheme over a field $k$, then the multiplication map $m : G \\times_k G \\to G$ is open."}
+{"_id": "9588", "title": "groupoids-lemma-group-scheme-over-field-translate-open", "text": "If $(G, m)$ is a group scheme over a field $k$. Let $U \\subset G$ open and $T \\to G$ a morphism of schemes. Then the image of the composition $T \\times_k U \\to G \\times_k G \\to G$ is open."}
+{"_id": "9589", "title": "groupoids-lemma-group-scheme-over-field-separated", "text": "Let $G$ be a group scheme over a field. Then $G$ is a separated scheme."}
+{"_id": "9590", "title": "groupoids-lemma-group-scheme-field-geometrically-irreducible", "text": "Let $G$ be a group scheme over a field $k$. Then \\begin{enumerate} \\item every local ring $\\mathcal{O}_{G, g}$ of $G$ has a unique minimal prime ideal, \\item there is exactly one irreducible component $Z$ of $G$ passing through $e$, and \\item $Z$ is geometrically irreducible over $k$. \\end{enumerate}"}
+{"_id": "9591", "title": "groupoids-lemma-reduced-subgroup-scheme-perfect", "text": "Let $G$ be a group scheme over a perfect field $k$. Then the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$."}
+{"_id": "9592", "title": "groupoids-lemma-open-subgroup-closed-over-field", "text": "Let $k$ be a field. Let $\\psi : G' \\to G$ be a morphism of group schemes over $k$. If $\\psi(G')$ is open in $G$, then $\\psi(G')$ is closed in $G$."}
+{"_id": "9594", "title": "groupoids-lemma-irreducible-group-scheme-over-field-qc", "text": "Let $G$ be a group scheme over a field $k$. If $G$ is irreducible, then $G$ is quasi-compact."}
+{"_id": "9595", "title": "groupoids-lemma-connected-group-scheme-over-field-irreducible", "text": "Let $G$ be a group scheme over a field $k$. If $G$ is connected, then $G$ is irreducible."}
+{"_id": "9596", "title": "groupoids-lemma-profinite-product-over-field", "text": "Let $k$ be a field. Let $T = \\Spec(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any scheme $X$ over $k$ we have $|T \\times_k X| = |T| \\times |X|$ as topological spaces."}
+{"_id": "9597", "title": "groupoids-lemma-compact-set-in-affine", "text": "Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$. Assume that $G$ is Jacobson and that all closed points are $k$-rational. Let $T = \\Spec(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any morphism $f : T \\to G$ there exists an affine open $U \\subset G$ containing $f(T)$."}
+{"_id": "9599", "title": "groupoids-lemma-group-scheme-finite-type-field", "text": "Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then $G$ is equidimensional and $\\dim(G) = \\dim_g(G)$ for all $g \\in G$. For any closed point $g \\in G$ we have $\\dim(G) = \\dim(\\mathcal{O}_{G, g})$."}
+{"_id": "9600", "title": "groupoids-lemma-group-scheme-characteristic-zero-smooth", "text": "Let $k$ be a field of characteristic $0$. Let $G$ be a locally algebraic group scheme over $k$. Then the structure morphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth group scheme."}
+{"_id": "9601", "title": "groupoids-lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth", "text": "Let $k$ be a perfect field of characteristic $p > 0$ (see Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth} for the characteristic zero case). Let $G$ be a locally algebraic group scheme over $k$. If $G$ is reduced then the structure morphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth group scheme."}
+{"_id": "9602", "title": "groupoids-lemma-points-in-affine", "text": "Let $k$ be an algebraically closed field. Let $G$ be a locally algebraic group scheme over $k$. Let $g_1, \\ldots, g_n \\in G(k)$ be $k$-rational points. Then there exists an affine open $U \\subset G$ containing $g_1, \\ldots, g_n$."}
+{"_id": "9603", "title": "groupoids-lemma-algebraic-quasi-projective", "text": "Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$. Then $G$ is quasi-projective over $k$."}
+{"_id": "9605", "title": "groupoids-lemma-abelian-variety-projective", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. Then $A$ is projective."}
+{"_id": "9606", "title": "groupoids-lemma-abelian-variety-change-field", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. For any field extension $K/k$ the base change $A_K$ is an abelian variety over $K$."}
+{"_id": "9607", "title": "groupoids-lemma-abelian-variety-smooth", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. Then $A$ is smooth over $k$."}
+{"_id": "9608", "title": "groupoids-lemma-abelian-variety-abelian", "text": "An abelian variety is an abelian group scheme, i.e., the group law is commutative."}
+{"_id": "9609", "title": "groupoids-lemma-apply-cube", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module. Then there is an isomorphism $$ m_{1, 2, 3}^*\\mathcal{L} \\otimes m_1^*\\mathcal{L} \\otimes m_2^*\\mathcal{L} \\otimes m_3^*\\mathcal{L} \\cong m_{1, 2}^*\\mathcal{L} \\otimes m_{1, 3}^*\\mathcal{L} \\otimes m_{2, 3}^*\\mathcal{L} $$ of invertible modules on $A \\times_k A \\times_k A$ where $m_{i_1, \\ldots, i_t} : A \\times_k A \\times_k A \\to A$ is the morphism $(x_1, x_2, x_3) \\mapsto \\sum x_{i_j}$."}
+{"_id": "9610", "title": "groupoids-lemma-pullbacks-by-n", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module. Then $$ [n]^*\\mathcal{L} \\cong \\mathcal{L}^{\\otimes n(n + 1)/2} \\otimes ([-1]^*\\mathcal{L})^{\\otimes n(n - 1)/2} $$ where $[n] : A \\to A$ sends $x$ to $x + x + \\ldots + x$ with $n$ summands and where $[-1] : A \\to A$ is the inverse of $A$."}
+{"_id": "9611", "title": "groupoids-lemma-degree-multiplication-by-d", "text": "Let $k$ be a field. Let $A$ be an abelian variety over $k$. Let $[d] : A \\to A$ be the multiplication by $d$. Then $[d]$ is finite locally free of degree $d^{2\\dim(A)}$."}
+{"_id": "9612", "title": "groupoids-lemma-abelian-variety-multiplication-by-d-etale", "text": "\\begin{slogan} Multiplication by an integer on an abelian variety is an etale morphism if and only if the integer is invertible in the base field. \\end{slogan} Let $k$ be a field. Let $A$ be a nonzero abelian variety over $k$. Then $[d] : A \\to A$ is \\'etale if and only if $d$ is invertible in $k$."}
+{"_id": "9613", "title": "groupoids-lemma-abelian-variety-multiplication-by-p", "text": "Let $k$ be a field of characteristic $p > 0$. Let $A$ be an abelian variety over $k$. The fibre of $[p] : A \\to A$ over $0$ has at most $p^g$ distinct points."}
+{"_id": "9614", "title": "groupoids-lemma-free-action", "text": "Situation as in Definition \\ref{definition-free-action}, The action $a$ is free if and only if $$ G \\times_S X \\to X \\times_S X, \\quad (g, x) \\mapsto (a(g, x), x) $$ is a monomorphism."}
+{"_id": "9615", "title": "groupoids-lemma-characterize-trivial-pseudo-torsors", "text": "In the situation of Definition \\ref{definition-pseudo-torsor}. \\begin{enumerate} \\item The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme $T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$ on $X(T)$ is simply transitive. \\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism $X \\to S$ has a section. \\end{enumerate}"}
+{"_id": "9616", "title": "groupoids-lemma-torsor", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \\times_S X \\to X$ be an action of $G$ on $X$. Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Then $X$ is a $G$-torsor in the $\\tau$-topology if and only if $\\underline{X}$ is a $\\underline{G}$-torsor on $(\\Sch/S)_\\tau$."}
+{"_id": "9617", "title": "groupoids-lemma-pullback-equivariant", "text": "Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $f : Y \\to X$ be a $G$-equivariant morphism between $S$-schemes endowed with $G$-actions. Then pullback $f^*$ given by $(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$ defines a functor from the category of $G$-equivariant quasi-coherent $\\mathcal{O}_X$-modules  to the category of $G$-equivariant quasi-coherent $\\mathcal{O}_Y$-modules."}
+{"_id": "9618", "title": "groupoids-lemma-complete-reducibility-Gm", "text": "Let $a : \\mathbf{G}_m \\times X \\to X$ be an action on an affine scheme. Then $X$ is the spectrum of a $\\mathbf{Z}$-graded ring and the action is as in Example \\ref{example-Gm-on-affine}."}
+{"_id": "9619", "title": "groupoids-lemma-Gm-equivariant-module", "text": "Let $A$ be a graded ring. Let $X = \\Spec(A)$ with action $a : \\mathbf{G}_m \\times X \\to X$ as in Example \\ref{example-Gm-on-affine}. Let $\\mathcal{F}$ be a $\\mathbf{G}_m$-equivariant quasi-coherent $\\mathcal{O}_X$-module. Then $M = \\Gamma(X, \\mathcal{F})$ has a canonical grading such that it is a graded $A$-module and such that the isomorphism $\\widetilde{M} \\to \\mathcal{F}$ (Schemes, Lemma \\ref{schemes-lemma-quasi-coherent-affine}) is an isomorphism of $\\mathbf{G}_m$-equivariant modules where the $\\mathbf{G}_m$-equivariant structure on $\\widetilde{M}$ is the one from Example \\ref{example-Gm-on-affine}."}
+{"_id": "9620", "title": "groupoids-lemma-groupoid-pre-equivalence", "text": "Given a groupoid scheme $(U, R, s, t, c)$ over $S$ the morphism $j : R \\to U \\times_S U$ is a pre-equivalence relation."}
+{"_id": "9621", "title": "groupoids-lemma-equivalence-groupoid", "text": "Given an equivalence relation $j : R \\to U$ over $S$ there is a unique way to extend it to a groupoid $(U, R, s, t, c)$ over $S$."}
+{"_id": "9622", "title": "groupoids-lemma-diagram", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. In the commutative diagram $$ \\xymatrix{ & U & \\\\ R \\ar[d]_s \\ar[ru]^t & R \\times_{s, U, t} R \\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c & R \\ar[d]^s \\ar[lu]_t \\\\ U & R \\ar[l]_t \\ar[r]^s & U } $$ the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian."}
+{"_id": "9623", "title": "groupoids-lemma-diagram-pull", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid over $S$. The diagram \\begin{equation} \\label{equation-pull} \\xymatrix{ R \\times_{t, U, t} R \\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{(\\text{pr}_0, c \\circ (i, 1))} & R \\ar[r]^t \\ar[d]^{\\text{id}_R} & U \\ar[d]^{\\text{id}_U} \\\\ R \\times_{s, U, t} R \\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} & R \\ar[r]^t \\ar[d]^s & U \\\\ R \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t & U } \\end{equation} is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian."}
+{"_id": "9624", "title": "groupoids-lemma-base-change-groupoid", "text": "Let $(U, R, s, t, c)$ be a groupoid over a scheme $S$. Let $S' \\to S$ be a morphism. Then the base changes $U' = S' \\times_S U$, $R' = S' \\times_S R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid scheme $(U', R', s', t', c')$ over $S'$ and the projections determine a morphism $(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoid schemes over $S$."}
+{"_id": "9626", "title": "groupoids-lemma-pullback", "text": "Let $S$ be a scheme. Consider a morphism $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoid schemes over $S$. Then pullback $f^*$ given by $$ (\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$."}
+{"_id": "9627", "title": "groupoids-lemma-pushforward", "text": "Let $S$ be a scheme. Consider a morphism $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoid schemes over $S$. Assume that \\begin{enumerate} \\item $f : U \\to U'$ is quasi-compact and quasi-separated, \\item the square $$ \\xymatrix{ R \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\ U \\ar[r]^f & U' } $$ is cartesian, and \\item $s'$ and $t'$ are flat. \\end{enumerate} Then pushforward $f_*$ given by $$ (\\mathcal{F}, \\alpha) \\mapsto (f_*\\mathcal{F}, f_*\\alpha) $$ defines a functor from the category of quasi-coherent sheaves on $(U, R, s, t, c)$ to the category of quasi-coherent sheaves on $(U', R', s', t', c')$ which is right adjoint to pullback as defined in Lemma \\ref{lemma-pullback}."}
+{"_id": "9628", "title": "groupoids-lemma-colimits", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits."}
+{"_id": "9629", "title": "groupoids-lemma-abelian", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian."}
+{"_id": "9630", "title": "groupoids-lemma-construct-quasi-coherent", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s, t$ are flat, quasi-compact, and quasi-separated. For any quasi-coherent module $\\mathcal{G}$ on $U$, there exists a canonical isomorphism $\\alpha : t^*t_*s^*\\mathcal{G} \\to s^*t_*s^*\\mathcal{G}$ which turns $(t_*s^*\\mathcal{G}, \\alpha)$ into a quasi-coherent module on $(U, R, s, t, c)$. This construction defines a functor $$ \\QCoh(\\mathcal{O}_U) \\longrightarrow \\QCoh(U, R, s, t, c) $$ which is a right adjoint to the forgetful functor $(\\mathcal{F}, \\beta) \\mapsto \\mathcal{F}$."}
+{"_id": "9631", "title": "groupoids-lemma-push-pull", "text": "Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module, let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module, and let $\\varphi : \\mathcal{G} \\to f^*\\mathcal{F}$ be a module map. Assume \\begin{enumerate} \\item $\\varphi$ is injective, \\item $f$ is quasi-compact, quasi-separated, flat, and surjective, \\item $X$, $Y$ are locally Noetherian, and \\item $\\mathcal{G}$ is a coherent $\\mathcal{O}_Y$-module. \\end{enumerate} Then $\\mathcal{F} \\cap f_*\\mathcal{G}$ defined as the pullback $$ \\xymatrix{ \\mathcal{F} \\ar[r] & f_*f^*\\mathcal{F} \\\\ \\mathcal{F} \\cap f_*\\mathcal{G} \\ar[u] \\ar[r] & f_*\\mathcal{G} \\ar[u] } $$ is a coherent $\\mathcal{O}_X$-module."}
+{"_id": "9632", "title": "groupoids-lemma-colimit-coherent", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that \\begin{enumerate} \\item $U$, $R$ are Noetherian, \\item $s, t$ are flat, quasi-compact, and quasi-separated. \\end{enumerate} Then every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$ is a filtered colimit of coherent modules."}
+{"_id": "9634", "title": "groupoids-lemma-set-of-iso-classes", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $\\kappa$ be a cardinal. There exists a set $T$ and a family $(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of $\\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$ such that every $\\kappa$-generated quasi-coherent module on $(U, R, s, t, c)$ is isomorphic to one of the $(\\mathcal{F}_t, \\alpha_t)$."}
+{"_id": "9635", "title": "groupoids-lemma-colimit-kappa", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that $s, t$ are flat. There exists a cardinal $\\kappa$ such that every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$ is the directed colimit of its $\\kappa$-generated quasi-coherent submodules."}
+{"_id": "9636", "title": "groupoids-lemma-groupoid-from-action", "text": "Let $S$ be a scheme. Let $Y$ be a scheme over $S$. Let $(G, m)$ be a group scheme over $Y$ with identity $e_G$ and inverse $i_G$. Let $X/Y$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$ be an action of $G$ on $X/Y$. Then we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$ in the following manner: \\begin{enumerate} \\item We set $U = X$, and $R = G \\times_Y X$. \\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$. \\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$. \\item We set $c : R \\times_{s, U, t} R \\to R$ equal to $((g, x), (g', x')) \\mapsto (m(g, g'), x')$. \\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$. \\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$. \\end{enumerate}"}
+{"_id": "9638", "title": "groupoids-lemma-groupoid-stabilizer", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The scheme $G$ defined by the cartesian square $$ \\xymatrix{ G \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\ U \\ar[r]^-{\\Delta} & U \\times_S U } $$ is a group scheme over $U$ with composition law $m$ induced by the composition law $c$."}
+{"_id": "9639", "title": "groupoids-lemma-groupoid-action-stabilizer", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer. Denote $R_t/U$ the scheme $R$ seen as a scheme over $U$ via the morphism $t : R \\to U$. There is a canonical left action $$ a : G \\times_U R_t \\longrightarrow R_t $$ induced by the composition law $c$."}
+{"_id": "9640", "title": "groupoids-lemma-groupoid-action-stabilizer-pseudo-torsor", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $G$ be the stabilizer group scheme of $R$. Let $$ G_0 = G \\times_{U, \\text{pr}_0} (U \\times_S U) = G \\times_S U $$ as a group scheme over $U \\times_S U$. The action of $G$ on $R$ of Lemma \\ref{lemma-groupoid-action-stabilizer} induces an action of $G_0$ on $R$ over $U \\times_S U$ which turns $R$ into a pseudo $G_0$-torsor over $U \\times_S U$."}
+{"_id": "9641", "title": "groupoids-lemma-fibres-j", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \\in U \\times_S U$ be a point. Denote $R_p$ the scheme theoretic fibre of $j = (t, s) : R \\to U \\times_S U$. If $R_p \\not = \\emptyset$, then the action $$ G_{0, \\kappa(p)} \\times_{\\kappa(p)} R_p \\longrightarrow R_p $$ (see Lemma \\ref{lemma-groupoid-action-stabilizer-pseudo-torsor}) which turns $R_p$ into a $G_{\\kappa(p)}$-torsor over $\\kappa(p)$."}
+{"_id": "9642", "title": "groupoids-lemma-restrict-groupoid", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism of schemes. Consider the following diagram $$ \\xymatrix{ R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}& R \\times_{s, U} U' \\ar[r] \\ar[d] & U' \\ar[d]^g \\\\ U' \\times_{U, t} R \\ar[d] \\ar[r] & R \\ar[r]^s \\ar[d]_t & U \\\\ U' \\ar[r]^g & U } $$ where all the squares are fibre product squares. Then there is a canonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$ such that $(U', R', s', t', c')$ is a groupoid scheme over $S$ and such that $U' \\to U$, $R' \\to R$ defines a morphism $(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoid schemes over $S$. Moreover, for any scheme $T$ over $S$ the functor of groupoids $$ (U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c) $$ is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \\to U(T)$."}
+{"_id": "9643", "title": "groupoids-lemma-restrict-groupoid-relation", "text": "The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions \\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation} agree via the constructions of Lemmas \\ref{lemma-groupoid-pre-equivalence} and \\ref{lemma-equivalence-groupoid}."}
+{"_id": "9644", "title": "groupoids-lemma-restrict-stabilizer", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let $G'$ be the stabilizer of $(U', R', s', t', c')$. Then $G'$ is the base change of $G$ by $g$, i.e., there is a canonical identification $G' = U' \\times_{g, U} G$."}
+{"_id": "9645", "title": "groupoids-lemma-constructing-invariant-opens", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. \\begin{enumerate} \\item For any subset $W \\subset U$ the subset $t(s^{-1}(W))$ is set-theoretically $R$-invariant. \\item If $s$ and $t$ are open, then for every open $W \\subset U$ the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme. \\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subschemes. \\end{enumerate}"}
+{"_id": "9646", "title": "groupoids-lemma-first-observation", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated. Let $W \\subset U$ be quasi-compact open. Then $t(s^{-1}(W))$ is an intersection of a nonempty family of quasi-compact open subsets of $U$."}
+{"_id": "9647", "title": "groupoids-lemma-second-observation", "text": "Assumptions and notation as in Lemma \\ref{lemma-first-observation}. There exists an $R$-invariant open $V \\subset U$ and a quasi-compact open $W'$ such that $W \\subset V \\subset W' \\subset U$."}
+{"_id": "9648", "title": "groupoids-lemma-criterion-quotient-representable", "text": "In the situation of Definition \\ref{definition-quotient-sheaf}. Assume there is a scheme $M$, and a morphism $U \\to M$ such that \\begin{enumerate} \\item the morphism $U \\to M$ equalizes $s, t$, \\item the morphism $U \\to M$ induces a surjection of sheaves $h_U \\to h_M$ in the $\\tau$-topology, and \\item the induced map $(t, s) : R \\to U \\times_M U$ induces a surjection of sheaves $h_R \\to h_{U \\times_M U}$ in the $\\tau$-topology. \\end{enumerate} In this case $M$ represents the quotient sheaf $U/R$."}
+{"_id": "9649", "title": "groupoids-lemma-quotient-pre-equivalence", "text": "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$. Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$. Assume $U, R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$. For $T \\in \\Ob((\\Sch/S)_\\tau)$ and $a, b \\in U(T)$ the following are equivalent: \\begin{enumerate} \\item $a$ and $b$ map to the same element of $(U/R)(T)$, and \\item there exists a $\\tau$-covering $\\{f_i : T_i \\to T\\}$ of $T$ and morphisms $r_i : T_i \\to R$ such that $a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$. \\end{enumerate} In other words, in this case the map of $\\tau$-sheaves $$ h_R \\longrightarrow h_U \\times_{U/R} h_U $$ is surjective."}
+{"_id": "9650", "title": "groupoids-lemma-quotient-pre-equivalence-relation-restrict", "text": "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$. Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$ and $g : U' \\to U$ a morphism of schemes over $S$. Let $j' : R' \\to U' \\times_S U'$ be the restriction of $j$ to $U'$. Assume  $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$. The map of quotient sheaves $$ U'/R' \\longrightarrow U/R $$ is injective. If $g$ defines a surjection $h_{U'} \\to h_U$ of sheaves in the $\\tau$-topology (for example if $\\{g : U' \\to U\\}$ is a $\\tau$-covering), then $U'/R' \\to U/R$ is an isomorphism."}
+{"_id": "9651", "title": "groupoids-lemma-quotient-groupoid-restrict", "text": "Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ a morphism of schemes over $S$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$. Assume  $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$. The map of quotient sheaves $$ U'/R' \\longrightarrow U/R $$ is injective. If the composition $$ \\xymatrix{ U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h & R \\ar[r]_s & U } $$ defines a surjection of sheaves in the $\\tau$-topology  then the map is bijective. This holds for example if $\\{h : U' \\times_{g, U, t} R \\to U\\}$ is a $\\tau$-covering, or if $U' \\to U$ defines a surjection of sheaves in the $\\tau$-topology, or if $\\{g : U' \\to U\\}$ is a covering in the $\\tau$-topology."}
+{"_id": "9652", "title": "groupoids-lemma-criterion-fibre-product", "text": "Let $S$ be a scheme. Let $f : (U, R, j) \\to (U', R', j')$ be a morphism between equivalence relations over $S$. Assume that $$ \\xymatrix{ R \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\ U \\ar[r]^f & U' } $$ is cartesian. For any $\\tau \\in  \\{Zariski, \\etale, fppf, smooth, syntomic\\}$ the diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & U/R \\ar[d]^f \\\\ U' \\ar[r] & U'/R' } $$ is a fibre product square of $\\tau$-sheaves."}
+{"_id": "9653", "title": "groupoids-lemma-characterize-cartesian-schemes", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of groupoid schemes cartesian over $(U, R, s, t, c)$ is equivalent to the category of pairs $(V, \\varphi)$ where $V$ is a scheme over $U$ and $$ \\varphi : V \\times_{U, t} R \\longrightarrow R \\times_{s, U} V $$ is an isomorphism over $R$ such that $e^*\\varphi = \\text{id}_V$ and such that $$ c^*\\varphi = \\text{pr}_1^*\\varphi \\circ \\text{pr}_0^*\\varphi $$ as morphisms of schemes over $R \\times_{s, U, t} R$."}
+{"_id": "9654", "title": "groupoids-lemma-cartesian-equivalent-descent-datum", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. The construction of Lemma \\ref{lemma-characterize-cartesian-schemes} determines an equivalence $$ \\begin{matrix} \\text{category of groupoid schemes} \\\\ \\text{cartesian over } (X, X \\times_Y X, \\ldots) \\end{matrix} \\longrightarrow \\begin{matrix} \\text{ category of descent data} \\\\ \\text{ relative to } X/Y \\end{matrix} $$"}
+{"_id": "9655", "title": "groupoids-lemma-diagram-diagonal", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \\to U$ be the stabilizer group scheme. The commutative diagram $$ \\xymatrix{ R \\ar[d]^{\\Delta_{R/U \\times_S U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & & R \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\ R \\times_{(U \\times_S U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & & R \\times_{s, U} G \\ar[r] & G } $$ the two left horizontal arrows are isomorphisms and the right square is a fibre product square."}
+{"_id": "9656", "title": "groupoids-lemma-diagonal", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \\to U$ be the stabilizer group scheme. \\begin{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $j : R \\to U \\times_S U$ is separated, \\item $G \\to U$ is separated, and \\item $e : U \\to G$ is a closed immersion. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $j : R \\to U \\times_S U$ is quasi-separated, \\item $G \\to U$ is quasi-separated, and \\item $e : U \\to G$ is quasi-compact. \\end{enumerate} \\end{enumerate}"}
+{"_id": "9657", "title": "groupoids-lemma-determinant-trick", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and $s, t : R \\to U$ finite locally free. Let $C$ be as in (\\ref{equation-invariants}). Let $f \\in A$. Then $\\text{Norm}_{s^\\sharp}(t^\\sharp(f)) \\in C$."}
+{"_id": "9658", "title": "groupoids-lemma-finite-locally-free-disjoint-free", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s, t : R \\to U$ finite locally free. Then $$ U = \\coprod\\nolimits_{r \\geq 1} U_r $$ is a disjoint union of $R$-invariant opens such that the restriction $R_r$ of $R$ to $U_r$ has the property that $s, t : R_r \\to U_r$ are finite locally free of rank $r$."}
+{"_id": "9659", "title": "groupoids-lemma-integral-over-invariants", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and $s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in (\\ref{equation-invariants}). Then $A$ is integral over $C$."}
+{"_id": "9660", "title": "groupoids-lemma-invariants-base-change", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and $s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in (\\ref{equation-invariants}). Let $C \\to C'$ be a ring map, and set $U' = \\Spec(A \\otimes_C C')$, $R' = \\Spec(B \\otimes_C C')$. Then \\begin{enumerate} \\item The maps $s, t, c$ induce maps $s', t', c'$ such that $(U', R', s', t', c')$ is a groupoid scheme. Let $C^1 \\subset A'$ be the $R'$-invariant functions on $U'$. \\item The canonical map $\\varphi : C' \\to C^1$ satisfies \\begin{enumerate} \\item for every $f \\in C^1$ there exists an $n > 0$ and a polynomial $P \\in C'[x]$ whose image in $C^1[x]$ is $(x - f)^n$, and \\item for every $f \\in \\Ker(\\varphi)$ there exists an $n > 0$ such that $f^n = 0$. \\end{enumerate} \\item If $C \\to C'$ is flat then $\\varphi$ is an isomorphism. \\end{enumerate}"}
+{"_id": "9661", "title": "groupoids-lemma-points", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and $s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in (\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$ has the following properties: \\begin{enumerate} \\item the map on points $|U| \\to |M|$ is surjective and $u_0, u_1 \\in |U|$ map to the same point if and only if there exists a $r \\in |R|$ with $t(r) = u_0$ and $s(r) = u_1$, in a formula $$ |M| = |U|/|R| $$ \\item for any algebraically closed field $k$ we have $$ M(k) = U(k)/R(k) $$ \\end{enumerate}"}
+{"_id": "9662", "title": "groupoids-lemma-etale", "text": "Let $S$ be a scheme. Let $f : (U', R', s', t') \\to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. \\begin{enumerate} \\item $U$, $R$, $U'$, $R'$ are affine, \\item $s, t, s', t'$ are finite locally free, \\item the diagrams $$ \\xymatrix{ R' \\ar[d]_{s'} \\ar[r]_f & R \\ar[d]^s \\\\ U' \\ar[r]^f & U } \\quad \\quad \\xymatrix{ R' \\ar[d]_{t'} \\ar[r]_f & R \\ar[d]^t \\\\ U' \\ar[r]^f & U } \\quad \\quad \\xymatrix{ G' \\ar[d] \\ar[r]_f & G \\ar[d] \\\\ U' \\ar[r]^f & U } $$ are cartesian where $G$ and $G'$ are the stabilizer group schemes, and \\item $f : U' \\to U$ is \\'etale. \\end{enumerate} Then the map $C \\to C'$ from the $R$-invariant functions on $U$ to the $R'$-invariant functions on $U'$ is \\'etale and $U' = \\Spec(C') \\times_{\\Spec(C)} U$."}
+{"_id": "9663", "title": "groupoids-lemma-basis", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \\begin{enumerate} \\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine, and \\item there exist elements $x_i \\in A$, $i \\in I$ such that $B = \\bigoplus_{i \\in I} s^\\sharp(A)t^\\sharp(x_i)$. \\end{enumerate} Then $A = \\bigoplus_{i\\in I} Cx_i$, and $B \\cong A \\otimes_C A$ where $C \\subset A$ is the $R$-invariant functions on $U$ as in (\\ref{equation-invariants})."}
+{"_id": "9664", "title": "groupoids-lemma-find-invariant-affine", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s$, $t$ are finite locally free. Let $u \\in U$ be a point such that $t(s^{-1}(\\{u\\}))$ is contained in an affine open of $U$. Then there exists an $R$-invariant affine open neighbourhood of $u$ in $U$."}
+{"_id": "9665", "title": "groupoids-lemma-descend-along-finite", "text": "Let $X \\to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that for all $(y, v_1, \\ldots, v_d)$ where $y \\in Y$ and $v_1, \\ldots, v_d \\in V_y$ there exists an affine open $U \\subset V$ with $v_1, \\ldots, v_d \\in U$. Then any descent datum on $V/X/Y$ is effective."}
+{"_id": "9667", "title": "groupoids-proposition-connected-component", "text": "Let $G$ be a group scheme over a field $k$. There exists a canonical closed subgroup scheme $G^0 \\subset G$ with the following properties \\begin{enumerate} \\item $G^0 \\to G$ is a flat closed immersion, \\item $G^0 \\subset G$ is the connected component of the identity, \\item $G^0$ is geometrically irreducible, and \\item $G^0$ is quasi-compact. \\end{enumerate}"}
+{"_id": "9668", "title": "groupoids-proposition-review-abelian-varieties", "text": "\\begin{reference} Wonderfully explained in \\cite{AVar}. \\end{reference} Let $A$ be an abelian variety over a field $k$. Then \\begin{enumerate} \\item $A$ is projective over $k$, \\item $A$ is a commutative group scheme, \\item the morphism $[n] : A \\to A$ is surjective for all $n \\geq 1$, \\item if $k$ is algebraically closed, then $A(k)$ is a divisible abelian group, \\item $A[n] = \\Ker([n] : A \\to A)$ is a finite group scheme of degree $n^{2\\dim A}$ over $k$, \\item $A[n]$ is \\'etale over $k$ if and only if $n \\in k^*$, \\item if $n \\in k^*$ and $k$ is algebraically closed, then $A(k)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2\\dim(A)}$, \\item if $k$ is algebraically closed of characteristic $p > 0$, then there exists an integer $0 \\leq f \\leq \\dim(A)$ such that $A(k)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$ for all $m \\geq 1$. \\end{enumerate}"}
+{"_id": "9669", "title": "groupoids-proposition-finite-flat-equivalence", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \\begin{enumerate} \\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine, \\item $s, t : R \\to U$ finite locally free, and \\item $j = (t, s)$ is an equivalence. \\end{enumerate} In this case, let $C \\subset A$ be as in (\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$ is finite locally free and $R = U \\times_M U$. Moreover, $M$ represents the quotient sheaf $U/R$ in the fppf topology (see Definition \\ref{definition-quotient-sheaf})."}
+{"_id": "9694", "title": "local-cohomology-theorem-finiteness", "text": "\\begin{reference} This is a special case of \\cite[Satz 2]{Faltings-finiteness}. \\end{reference} Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Set $Z = V(I) \\subset \\Spec(A)$. Let $M$ be a finite $A$-module. Set $s = s_{A, I}(M)$ as in (\\ref{equation-cutoff}). Assume that \\begin{enumerate} \\item $A$ is universally catenary, \\item the formal fibres of the local rings of $A$ are Cohen-Macaulay. \\end{enumerate} Then $H^i_Z(M)$ is finite for $0 \\leq i < s$ and $H^s_Z(M)$ is not finite."}
+{"_id": "9695", "title": "local-cohomology-lemma-local-cohomology-is-local-cohomology", "text": "Let $A$ be a ring and let $I$ be a finitely generated ideal. Set $Z = V(I) \\subset X = \\Spec(A)$. For $K \\in D(A)$ corresponding to $\\widetilde{K} \\in D_\\QCoh(\\mathcal{O}_X)$ via Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded} there is a functorial isomorphism $$ R\\Gamma_Z(K) = R\\Gamma_Z(X, \\widetilde{K}) $$ where on the left we have Dualizing Complexes, Equation (\\ref{dualizing-equation-local-cohomology}) and on the right we have the functor of Cohomology, Section \\ref{cohomology-section-cohomology-support-bis}."}
+{"_id": "9696", "title": "local-cohomology-lemma-local-cohomology", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Set $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$ the inclusion morphism. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_U$-module. Then \\begin{enumerate} \\item there exists an $A$-module $M$ such that $\\mathcal{F}$ is the restriction of $\\widetilde{M}$ to $U$, \\item given $M$ there is an exact sequence $$ 0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0 $$ and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$, \\item we may take $M = H^0(U, \\mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$. \\end{enumerate}"}
+{"_id": "9697", "title": "local-cohomology-lemma-already-torsion", "text": "Let $I, J \\subset A$ be finitely generated ideals of a ring $A$. If $M$ is an $I$-power torsion module, then the canonical map $$ H^i_{V(I) \\cap V(J)}(M) \\to H^i_{V(J)}(M) $$ is an isomorphism for all $i$."}
+{"_id": "9698", "title": "local-cohomology-lemma-multiplicative", "text": "Let $S \\subset A$ be a multiplicative set of a ring $A$. Let $M$ be an $A$-module with $S^{-1}M = 0$. Then $\\colim_{f \\in S} H^0_{V(f)}(M) = M$ and $\\colim_{f \\in S} H^1_{V(f)}(M) = 0$."}
+{"_id": "9699", "title": "local-cohomology-lemma-elements-come-from-bigger", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. Let $\\mathfrak p$ be a prime ideal. Let $M$ be an $A$-module. Let $i \\geq 0$ be an integer and consider the map $$ \\Psi : \\colim_{f \\in A, f \\not \\in \\mathfrak p} H^i_{V((I, f))}(M) \\longrightarrow H^i_{V(I)}(M) $$ Then \\begin{enumerate} \\item $\\Im(\\Psi)$ is the set of elements which map to zero in $H^i_{V(I)}(M)_\\mathfrak p$, \\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = 0$, then $\\Psi$ is injective, \\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = H^i_{V(I)}(M)_\\mathfrak p = 0$, then $\\Psi$ is an isomorphism. \\end{enumerate}"}
+{"_id": "9700", "title": "local-cohomology-lemma-isomorphism", "text": "Let $I \\subset I' \\subset A$ be finitely generated ideals of a Noetherian ring $A$. Let $M$ be an $A$-module. Let $i \\geq 0$ be an integer. Consider the map $$ \\Psi : H^i_{V(I')}(M) \\to H^i_{V(I)}(M) $$ The following are true: \\begin{enumerate} \\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$ for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then $\\Psi$ is surjective, \\item if $H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$ for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then $\\Psi$ is injective, \\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$ for all $\\mathfrak p \\in V(I) \\setminus V(I')$, then $\\Psi$ is an isomorphism. \\end{enumerate}"}
+{"_id": "9701", "title": "local-cohomology-lemma-depth-2-connected-punctured-spectrum", "text": "\\begin{reference} \\cite[Proposition 2.1]{Hartshorne-connectedness} \\end{reference} \\begin{slogan} Hartshorne's connectedness \\end{slogan} Let $A$ be a Noetherian local ring of depth $\\geq 2$. Then the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected."}
+{"_id": "9702", "title": "local-cohomology-lemma-catenary-S2-equidimensional", "text": "\\begin{reference} \\cite[Corollary 5.10.9]{EGA} \\end{reference} Let $A$ be a Noetherian local ring which is catenary and $(S_2)$. Then $\\Spec(A)$ is equidimensional."}
+{"_id": "9703", "title": "local-cohomology-lemma-cd", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. Set $Y = V(I) \\subset X = \\Spec(A)$. Let $d \\geq -1$ be an integer. The following are equivalent \\begin{enumerate} \\item $H^i_Y(A) = 0$ for $i > d$, \\item $H^i_Y(M) = 0$ for $i > d$ for every $A$-module $M$, and \\item if $d = -1$, then $Y = \\emptyset$, if $d = 0$, then $Y$ is open and closed in $X$, and if $d > 0$ then $H^i(X \\setminus Y, \\mathcal{F}) = 0$ for $i \\geq d$ for every quasi-coherent $\\mathcal{O}_{X \\setminus Y}$-module $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "9704", "title": "local-cohomology-lemma-bound-cd", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. Then \\begin{enumerate} \\item $\\text{cd}(A, I)$ is at most equal to the number of generators of $I$, \\item $\\text{cd}(A, I) \\leq r$ if there exist $f_1, \\ldots, f_r \\in A$ such that $V(f_1, \\ldots, f_r) = V(I)$, \\item $\\text{cd}(A, I) \\leq c$ if $\\Spec(A) \\setminus V(I)$ can be covered by $c$ affine opens. \\end{enumerate}"}
+{"_id": "9705", "title": "local-cohomology-lemma-cd-sum", "text": "Let $I, J \\subset A$ be finitely generated ideals of a ring $A$. Then $\\text{cd}(A, I + J) \\leq \\text{cd}(A, I) + \\text{cd}(A, J)$."}
+{"_id": "9706", "title": "local-cohomology-lemma-cd-change-rings", "text": "Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal. Then $\\text{cd}(B, IB) \\leq \\text{cd}(A, I)$. If $A \\to B$ is faithfully flat, then equality holds."}
+{"_id": "9707", "title": "local-cohomology-lemma-cd-local", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. Then $\\text{cd}(A, I) = \\max \\text{cd}(A_\\mathfrak p, I_\\mathfrak p)$."}
+{"_id": "9708", "title": "local-cohomology-lemma-cd-dimension", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. If $M$ is a finite $A$-module, then $H^i_{V(I)}(M) = 0$ for $i > \\dim(\\text{Supp}(M))$. In particular, we have $\\text{cd}(A, I) \\leq \\dim(A)$."}
+{"_id": "9709", "title": "local-cohomology-lemma-cd-is-one", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. If $\\text{cd}(A, I) = 1$ then $\\Spec(A) \\setminus V(I)$ is nonempty affine."}
+{"_id": "9710", "title": "local-cohomology-lemma-cd-maximal", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $d$. Then $H^d_\\mathfrak m(A)$ is nonzero and $\\text{cd}(A, \\mathfrak m) = d$."}
+{"_id": "9711", "title": "local-cohomology-lemma-cd-bound-dim-local", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be a proper ideal. Let $\\mathfrak p \\subset A$ be a prime ideal such that $V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$. Then $\\dim(A/\\mathfrak p) \\leq \\text{cd}(A, I)$."}
+{"_id": "9714", "title": "local-cohomology-lemma-adjoint", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. The functor $RH^0_T$ is the right adjoint to the functor $D(\\text{Mod}_{A, T}) \\to D(A)$."}
+{"_id": "9715", "title": "local-cohomology-lemma-adjoint-ext", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. For any object $K$ of $D(A)$ we have $$ H^i(RH^0_T(K)) = \\colim_{Z \\subset T\\text{ closed}} H^i_Z(K) $$"}
+{"_id": "9716", "title": "local-cohomology-lemma-equal-plus", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. The functor $D^+(\\text{Mod}_{A, T}) \\to D^+_T(A)$ is an equivalence."}
+{"_id": "9717", "title": "local-cohomology-lemma-equal-full", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. If $\\dim(A) < \\infty$, then functor $D(\\text{Mod}_{A, T}) \\to D_T(A)$ is an equivalence."}
+{"_id": "9718", "title": "local-cohomology-lemma-torsion-change-rings", "text": "Let $A \\to B$ be a flat homomorphism of Noetherian rings. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $T' \\subset \\Spec(B)$ be the inverse image of $T$. Then the canonical map $$ R\\Gamma_T(K) \\otimes_A^\\mathbf{L} B \\longrightarrow R\\Gamma_{T'}(K \\otimes_A^\\mathbf{L} B) $$ is an isomorphism for $K \\in D^+(A)$. If $A$ and $B$ have finite dimension, then this is true for $K \\in D(A)$."}
+{"_id": "9719", "title": "local-cohomology-lemma-local-cohomology-ss", "text": "Let $A$ be a ring and let $T, T' \\subset \\Spec(A)$ subsets stable under specialization. For $K \\in D^+(A)$ there is a spectral sequence $$ E_2^{p, q} = H^p_T(H^p_{T'}(K)) \\Rightarrow H^{p + q}_{T \\cap T'}(K) $$ as in Derived Categories, Lemma \\ref{derived-lemma-grothendieck-spectral-sequence}."}
+{"_id": "9721", "title": "local-cohomology-lemma-filter-local-cohomology", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $T' \\subset T$ be the set of nonminimal primes in $T$. Then $T'$ is a subset of $\\Spec(A)$ stable under specialization and for every $A$-module $M$ there is an exact sequence $$ 0 \\to \\colim_{Z, f} H^1_f(H^{i - 1}_Z(M)) \\to H^i_{T'}(M) \\to H^i_T(M) \\to \\bigoplus\\nolimits_{\\mathfrak p \\in T \\setminus T'} H^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p) $$ where the colimit is over closed subsets $Z \\subset T$ and $f \\in A$ with $V(f) \\cap Z \\subset T'$."}
+{"_id": "9722", "title": "local-cohomology-lemma-zero", "text": "Let $A$ be a Noetherian ring of finite dimension. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $\\{M_n\\}_{n \\geq 0}$ be an inverse system of $A$-modules. Let $i \\geq 0$ be an integer. Assume that for every $m$ there exists an integer $m'(m) \\geq m$ such  that for all $\\mathfrak p \\in T$ the induced map $$ H^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p}) \\longrightarrow H^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p}) $$ is zero for $k \\geq m'(m)$. Let $m'' : \\mathbf{N} \\to \\mathbf{N}$ be the $2^{\\dim(T)}$-fold self-composition of $m'$. Then the map $H^i_T(M_k) \\to H^i_T(M_m)$ is zero for all $k \\geq m''(m)$."}
+{"_id": "9723", "title": "local-cohomology-lemma-essential-image", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $\\{M_n\\}_{n \\geq 0}$ be an inverse system of $A$-modules. Let $i \\geq 0$ be an integer. Assume the dimension of $A$ is finite and that for every $m$ there exists an integer $m'(m) \\geq m$ such that for all $\\mathfrak p \\in T$ we have \\begin{enumerate} \\item $H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p}) \\to H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$ is zero for $k \\geq m'(m)$, and \\item $ H^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p}) \\to H^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$ has image $G(\\mathfrak p, m)$ independent of $k \\geq m'(m)$ and moreover $G(\\mathfrak p, m)$ maps injectively into $H^i_{\\mathfrak p A_\\mathfrak p}(M_{0, \\mathfrak p})$. \\end{enumerate} Then there exists an integer $m_0$ such that for every $m \\geq m_0$ there exists an integer $m''(m) \\geq m$ such that for $k \\geq m''(m)$ the image of $H^i_T(M_k) \\to H^i_T(M_m)$ maps injectively into $H^i_T(M_{m_0})$."}
+{"_id": "9724", "title": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator", "text": "\\begin{reference} \\cite[Lemma 3]{Faltings-annulators} \\end{reference} Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $M$ be a finite $A$-module. Let $n \\geq 0$. The following are equivalent \\begin{enumerate} \\item $H^i_T(M)$ is finite for $i \\leq n$, \\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$ such that $J$ annihilates $H^i_T(M)$ for $i \\leq n$. \\end{enumerate} If $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also equivalent to \\begin{enumerate} \\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates $H^i_Z(M)$ for $i \\leq n$. \\end{enumerate}"}
+{"_id": "9725", "title": "local-cohomology-lemma-check-finiteness-local-cohomology-locally", "text": "\\begin{reference} This is a special case of \\cite[Satz 1]{Faltings-finiteness}. \\end{reference} Let $A$ be a Noetherian ring, $I \\subset A$ an ideal, $M$ a finite $A$-module, and $n \\geq 0$ an integer. Let $Z = V(I)$. The following are equivalent \\begin{enumerate} \\item the modules $H^i_Z(M)$ are finite for $i \\leq n$, and \\item for all $\\mathfrak p \\in \\Spec(A)$ the modules $H^i_Z(M)_\\mathfrak p$, $i \\leq n$ are finite $A_\\mathfrak p$-modules. \\end{enumerate}"}
+{"_id": "9726", "title": "local-cohomology-lemma-annihilate-local-cohomology", "text": "Let $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals. Let $i \\geq 0$ be an integer. Set $Z = V(I)$. If $H^i_Z(A)$ is annihilated by $J^n$ for some $n$, then $H^i_Z(M)$ annihilated by $J^m$ for some $m = m(M)$ for every finitely presented $A$-module $M$ such that $M_f$ is a finite locally free $A_f$-module for all $f \\in I$."}
+{"_id": "9727", "title": "local-cohomology-lemma-local-finiteness-for-finite-locally-free", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Set $Z = V(I)$. Let $n \\geq 0$ be an integer. If $H^i_Z(A)$ is finite for $0 \\leq i \\leq n$, then the same is true for $H^i_Z(M)$, $0 \\leq i \\leq n$ for any finite $A$-module $M$ such that $M_f$ is a finite locally free $A_f$-module for all $f \\in I$."}
+{"_id": "9728", "title": "local-cohomology-lemma-check-finiteness-pushforward-on-associated-points", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. For $x \\in U$ let $i_x : W_x \\to U$ be the integral closed subscheme with generic point $x$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. The following are equivalent \\begin{enumerate} \\item for all $x \\in \\text{Ass}(\\mathcal{F})$ the $\\mathcal{O}_X$-module $j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent, \\item $j_*\\mathcal{F}$ is coherent. \\end{enumerate}"}
+{"_id": "9729", "title": "local-cohomology-lemma-finiteness-pushforwards-and-H1-local", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Set $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$ the inclusion morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Then \\begin{enumerate} \\item there exists a finite $A$-module $M$ such that $\\mathcal{F}$ is the restriction of $\\widetilde{M}$ to $U$, \\item given $M$ there is an exact sequence $$ 0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0 $$ and isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$, \\item given $M$ and $p \\geq 0$ the following are equivalent \\begin{enumerate} \\item $R^pj_*\\mathcal{F}$ is coherent, \\item $H^p(U, \\mathcal{F})$ is a finite $A$-module, \\item $H^{p + 1}_Z(M)$ is a finite $A$-module, \\end{enumerate} \\item if the equivalent conditions in (3) hold for $p = 0$, we may take $M = \\Gamma(U, \\mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$. \\end{enumerate}"}
+{"_id": "9730", "title": "local-cohomology-lemma-finiteness-pushforward", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Assume \\begin{enumerate} \\item $X$ is Nagata, \\item $X$ is universally catenary, and \\item for $x \\in \\text{Ass}(\\mathcal{F})$ and $z \\in Z \\cap \\overline{\\{x\\}}$ we have $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$. \\end{enumerate} Then $j_*\\mathcal{F}$ is coherent."}
+{"_id": "9731", "title": "local-cohomology-lemma-sharp-finiteness-pushforward", "text": "Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of a nonempty open subscheme with complement $Z$. Assume that for all $z \\in Z$ and any associated prime $\\mathfrak p$ of the completion $\\mathcal{O}_{X, z}^\\wedge$ we have $\\dim(\\mathcal{O}_{X, z}^\\wedge/\\mathfrak p) \\geq 2$. Then $j_*\\mathcal{O}_U$ is coherent."}
+{"_id": "9733", "title": "local-cohomology-lemma-finiteness-pushforward-general", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Assume \\begin{enumerate} \\item $X$ is universally catenary, \\item for every $z \\in Z$ the formal fibres of $\\mathcal{O}_{X, z}$ are $(S_1)$. \\end{enumerate} In this situation the following are equivalent \\begin{enumerate} \\item[(a)] for $x \\in \\text{Ass}(\\mathcal{F})$ and $z \\in Z \\cap \\overline{\\{x\\}}$ we have $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$, and \\item[(b)] $j_*\\mathcal{F}$ is coherent. \\end{enumerate}"}
+{"_id": "9735", "title": "local-cohomology-lemma-depth-function", "text": "Let $A$ be a Noetherian ring. Let $M$ be a finite $A$-module. Let $\\mathfrak p$ be a prime ideal. Assume $e = \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) < \\infty$. Then there exists a nonempty open $U \\subset V(\\mathfrak p)$ such that $\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) \\geq e$ for all $\\mathfrak q \\in U$ and for all but finitely many $\\mathfrak q \\in U$ we have $\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) > e$."}
+{"_id": "9737", "title": "local-cohomology-lemma-sitting-in-degrees", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite $A$-module. Set $E^i = \\text{Ext}_A^{-i}(M, \\omega_A^\\bullet)$. Then \\begin{enumerate} \\item $E^i$ is a finite $A$-module nonzero only for $0 \\leq i \\leq \\dim(\\text{Supp}(M))$, \\item $\\dim(\\text{Supp}(E^i)) \\leq i$, \\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that $E^\\delta \\not = 0$, \\item $\\mathfrak p \\in \\text{Supp}(E^0 \\oplus \\ldots \\oplus E^i) \\Leftrightarrow \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) \\leq i$, \\item the annihilator of $E^i$ is equal to the annihilator of $H^i_\\mathfrak m(M)$. \\end{enumerate}"}
+{"_id": "9738", "title": "local-cohomology-lemma-kill-local-cohomology-at-prime", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be a finite $A$-module, let $\\mathfrak p \\subset A$ be a prime ideal, and let $s \\geq 0$ be an integer. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $\\mathfrak p \\not \\in V(I)$, and \\item for all primes $\\mathfrak p' \\subset \\mathfrak p$ and $\\mathfrak q \\in V(I)$ with $\\mathfrak p' \\subset \\mathfrak q$ we have $$ \\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) + \\dim((A/\\mathfrak p')_\\mathfrak q) > s $$ \\end{enumerate} Then there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates $H^i_{V(I)}(M)$ for $i \\leq s$."}
+{"_id": "9739", "title": "local-cohomology-lemma-cutoff", "text": "Let $A \\to B$ be a finite homomorphism of Noetherian rings. Let $I \\subset A$ be an ideal and set $J = IB$. Let $M$ be a finite $B$-module. If $A$ is universally catenary, then $s_{B, J}(M) = s_{A, I}(M)$."}
+{"_id": "9740", "title": "local-cohomology-lemma-change-completion", "text": "Let $A$ be a Noetherian ring which has a dualizing complex. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $A', M'$ be the $I$-adic completions of $A, M$. Let $\\mathfrak p' \\subset \\mathfrak q'$ be prime ideals of $A'$ with $\\mathfrak q' \\in V(IA')$ lying over $\\mathfrak p \\subset \\mathfrak q$ in $A$. Then $$ \\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'}) \\geq \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) $$ and $$ \\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'}) + \\dim((A'/\\mathfrak p')_{\\mathfrak q'}) = \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) $$"}
+{"_id": "9741", "title": "local-cohomology-lemma-cutoff-completion", "text": "Let $A$ be a universally catenary Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Then $$ s_{A, I}(M) \\geq s_{A^\\wedge, I^\\wedge}(M^\\wedge) $$ If the formal fibres of $A$ are $(S_n)$, then $\\min(n + 1, s_{A, I}(M)) \\leq s_{A^\\wedge, I^\\wedge}(M^\\wedge)$."}
+{"_id": "9742", "title": "local-cohomology-lemma-local-annihilator", "text": "\\begin{reference} This is a special case of \\cite[Satz 1]{Faltings-annulators}. \\end{reference} Let $A$ be a Gorenstein Noetherian local ring. Let $I \\subset A$ be an ideal and set $Z = V(I) \\subset \\Spec(A)$. Let $M$ be a finite $A$-module. Let $s = s_{A, I}(M)$ as in (\\ref{equation-cutoff}). Then $H^i_Z(M)$ is finite for $i < s$, but $H^s_Z(M)$ is not finite."}
+{"_id": "9743", "title": "local-cohomology-lemma-finiteness-Rjstar", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Let $n \\geq 0$ be an integer. Assume \\begin{enumerate} \\item $X$ is universally catenary, \\item for every $z \\in Z$ the formal fibres of $\\mathcal{O}_{X, z}$ are $(S_n)$. \\end{enumerate} In this situation the following are equivalent \\begin{enumerate} \\item[(a)] for $x \\in \\text{Supp}(\\mathcal{F})$ and $z \\in Z \\cap \\overline{\\{x\\}}$ we have $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > n$, \\item[(b)] $R^pj_*\\mathcal{F}$ is coherent for $0 \\leq p < n$. \\end{enumerate}"}
+{"_id": "9744", "title": "local-cohomology-lemma-finiteness-for-finite-locally-free", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. Let $n \\geq 0$ be an integer. If $R^pj_*\\mathcal{O}_U$ is coherent for $0 \\leq p < n$, then the same is true for $R^pj_*\\mathcal{F}$, $0 \\leq p < n$ for any finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$."}
+{"_id": "9745", "title": "local-cohomology-lemma-annihilate-Hp", "text": "\\begin{reference} \\cite[Lemma 1.9]{Bhatt-local} \\end{reference} Let $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals. Let $p \\geq 0$ be an integer. Set $U = \\Spec(A) \\setminus V(I)$. If $H^p(U, \\mathcal{O}_U)$ is annihilated by $J^n$ for some $n$, then $H^p(U, \\mathcal{F})$ annihilated by $J^m$ for some $m = m(\\mathcal{F})$ for every finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$."}
+{"_id": "9746", "title": "local-cohomology-lemma-check-finiteness-local-cohomology-by-annihilator-complex", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $K$ be an object of $D_{\\textit{Coh}}^+(A)$. Let $n \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $H^i_T(K)$ is finite for $i \\leq n$, \\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$ such that $J$ annihilates $H^i_T(K)$ for $i \\leq n$. \\end{enumerate} If $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also equivalent to \\begin{enumerate} \\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates $H^i_Z(K)$ for $i \\leq n$. \\end{enumerate}"}
+{"_id": "9747", "title": "local-cohomology-lemma-get-depth-1-along-Z", "text": "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a subset stable under specialization. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then there is a unique map $\\mathcal{F} \\to \\mathcal{F}'$ of coherent $\\mathcal{O}_X$-modules such that \\begin{enumerate} \\item $\\mathcal{F} \\to \\mathcal{F}'$ is surjective, \\item $\\mathcal{F}_x \\to \\mathcal{F}'_x$ is an isomorphism for $x \\not \\in T$, \\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 1$ for $x \\in T$. \\end{enumerate} If $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then $f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding quotient for $f^{-1}(T) \\subset Y$ and $f^*\\mathcal{F}$."}
+{"_id": "9748", "title": "local-cohomology-lemma-get-depth-2-along-Z", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume $\\mathcal{F}' = j_*(\\mathcal{F}|_U)$ is coherent. Then $\\mathcal{F} \\to \\mathcal{F}'$ is the unique map of coherent $\\mathcal{O}_X$-modules such that \\begin{enumerate} \\item $\\mathcal{F}|_U \\to \\mathcal{F}'|_U$ is an isomorphism, \\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 2$ for $x \\in X$, $x \\not \\in U$. \\end{enumerate} If $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then $f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding map for $f^{-1}(U) \\subset Y$."}
+{"_id": "9750", "title": "local-cohomology-lemma-make-S2-along-T-simple", "text": "Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \\subset X$ be a subset stable under specialization. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume that if $x \\leadsto x'$ is an immediate specialization of points in $X$ with $x' \\in T'$ and $x \\not \\in T'$, then $\\text{depth}(\\mathcal{F}_x) \\geq 1$. Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$ of coherent $\\mathcal{O}_X$-modules such that \\begin{enumerate} \\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism for $x \\not \\in T'$, \\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$ for $x \\in T'$. \\end{enumerate} If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \\subset Y$."}
+{"_id": "9751", "title": "local-cohomology-lemma-make-S2-along-T", "text": "Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \\subset T \\subset X$ be subsets stable under specialization such that if $x \\leadsto x'$ is an immediate specialization of points in $X$ and $x' \\in T'$, then $x \\in T$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$ of coherent $\\mathcal{O}_X$-modules such that \\begin{enumerate} \\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism for $x \\not \\in T$, \\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective and $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 1$ for $x \\in T$, $x \\not \\in T'$, and \\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$ for $x \\in T'$. \\end{enumerate} If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \\subset f^{-1}(T) \\subset Y$."}
+{"_id": "9752", "title": "local-cohomology-lemma-cd-top-vanishing", "text": "Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset I' \\subset A$ be ideals. If $I'$ is contained in the Jacobson radical of $A$ and $\\text{cd}(A, I') < d$, then $\\text{cd}(A, I) < d$."}
+{"_id": "9753", "title": "local-cohomology-lemma-cd-top-vanishing-some-module", "text": "Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset A$ be an ideal. If $H^d_{V(I)}(M) = 0$ for some finite $A$-module whose support contains all the irreducible components of dimension $d$, then $\\text{cd}(A, I) < d$."}
+{"_id": "9754", "title": "local-cohomology-lemma-top-coh-divisible", "text": "Let $A$ be a Noetherian local ring of dimension $d$. Let $f \\in A$ be an element which is not contained in any minimal prime of dimension $d$. Then $f : H^d_{V(I)}(M) \\to H^d_{V(I)}(M)$ is surjective for any finite $A$-module $M$ and any ideal $I \\subset A$."}
+{"_id": "9755", "title": "local-cohomology-lemma-cd-bound-dualizing", "text": "Let $A$ be a Noetherian local ring with normalized dualizing complex $\\omega_A^\\bullet$. Let $I \\subset A$ be an ideal. If $H^0_{V(I)}(\\omega_A^\\bullet) = 0$, then $\\text{cd}(A, I) < \\dim(A)$."}
+{"_id": "9756", "title": "local-cohomology-lemma-inverse-system-symbolic-powers", "text": "Let $(A, \\mathfrak m)$ be a complete Noetherian local domain. Let $\\mathfrak p \\subset A$ be a prime ideal of dimension $1$. For every $n \\geq 1$ there is an $m \\geq n$ such that $\\mathfrak p^{(m)} \\subset \\mathfrak p^n$."}
+{"_id": "9757", "title": "local-cohomology-lemma-affine-complement", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Assume $A$ is excellent, normal, and $\\dim V(I) \\geq 1$. Then $\\text{cd}(A, I) < \\dim(A)$. In particular, if $\\dim(A) = 2$, then $\\Spec(A) \\setminus V(I)$ is affine."}
+{"_id": "9759", "title": "local-cohomology-lemma-1", "text": "\\begin{reference} See \\cite{Lech-inequalities} and \\cite[Lemma 1 page 299]{MatCA}. \\end{reference} Let $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$ are independent, then $f_1, \\ldots, f_r$ are independent."}
+{"_id": "9760", "title": "local-cohomology-lemma-2", "text": "\\begin{reference} See \\cite{Lech-inequalities} and \\cite[Lemma 2 page 300]{MatCA}. \\end{reference} Let $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$ are independent and if the $A$-module $A/(f_1, \\ldots, f_{r - 1}, f_rg_r)$ has finite length, then \\begin{align*} & \\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_rg_r)) \\\\ & = \\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_r)) + \\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, g_r)) \\end{align*}"}
+{"_id": "9761", "title": "local-cohomology-lemma-3", "text": "\\begin{reference} See \\cite{Lech-inequalities} and \\cite[Lemma 3 page 300]{MatCA}. \\end{reference} Let $(A, \\mathfrak m)$ be a local ring. If $\\mathfrak m = (x_1, \\ldots, x_r)$ and $x_1^{e_1}, \\ldots, x_r^{e_r}$ are independent for some $e_i > 0$, then $\\text{length}_A(A/(x_1^{e_1}, \\ldots, x_r^{e_r})) = e_1\\ldots e_r$."}
+{"_id": "9762", "title": "local-cohomology-lemma-flat-extension-independent", "text": "Let $\\varphi : A \\to B$ be a flat ring map. If $f_1, \\ldots, f_r \\in A$ are independent, then $\\varphi(f_1), \\ldots, \\varphi(f_r) \\in B$ are independent."}
+{"_id": "9763", "title": "local-cohomology-lemma-frobenius-flat-regular", "text": "\\begin{reference} \\cite{Kunz-flat} \\end{reference} Let $p$ be a prime number. Let $A$ be a Noetherian ring with $p = 0$. The following are equivalent \\begin{enumerate} \\item $A$ is regular, and \\item $F : A \\to A$, $a \\mapsto a^p$ is flat. \\end{enumerate}"}
+{"_id": "9764", "title": "local-cohomology-lemma-structure-torsion-D-module-regular", "text": "\\begin{reference} Special case of \\cite[Theorem 2.4]{Lyubeznik} \\end{reference} Let $k$ be a field of characteristic $0$. Let $d \\geq 1$. Let $A = k[[x_1, \\ldots, x_d]]$ with maximal ideal $\\mathfrak m$. Let $M$ be an $\\mathfrak m$-power torsion $A$-module endowed with additive operators $D_1, \\ldots, D_d$ satisfying the leibniz rule $$ D_i(fz) = \\partial_i(f) z + f D_i(z) $$ for $f \\in A$ and $z \\in M$. Here $\\partial_i$ is differentiation with respect to $x_i$. Then $M$ is isomorphic to a direct sum of copies of the injective hull $E$ of $k$."}
+{"_id": "9765", "title": "local-cohomology-lemma-structure-torsion-Frobenius-regular", "text": "\\begin{reference} Follows from \\cite[Corollary 3.6]{Huneke-Sharp} with a little bit of work. Also follows directly from \\cite[Theorem 1.4]{Lyubeznik2}. \\end{reference} Let $p$ be a prime number. Let $(A, \\mathfrak m, k)$ be a regular local ring with $p = 0$. Denote $F : A \\to A$, $a \\mapsto a^p$ be the Frobenius endomorphism. Let $M$ be a $\\mathfrak m$-power torsion module such that $M \\otimes_{A, F} A \\cong M$. Then $M$ is isomorphic to a direct sum of copies of the injective hull $E$ of $k$."}
+{"_id": "9766", "title": "local-cohomology-lemma-derivation", "text": "Let $A$ be a ring. Let $I \\subset A$ be a finitely generated ideal. Set $Z = V(I)$. For each derivation $\\theta : A \\to A$ there exists a canonical additive operator $D$ on the local cohomology modules $H^i_Z(A)$ satisfying the Leibniz rule with respect to $\\theta$."}
+{"_id": "9768", "title": "local-cohomology-lemma-etale-derivation", "text": "Let $A$ be a ring. Let $V \\to \\Spec(A)$ be quasi-compact, quasi-separated, and \\'etale. For each derivation $\\theta : A \\to A$ there exists a canonical additive operator $D$ on $H^i(V, \\mathcal{O}_V)$ satisfying the Leibniz rule with respect to $\\theta$."}
+{"_id": "9770", "title": "local-cohomology-lemma-map-tor-1-zero", "text": "Let $R$ be a ring. Let $M \\to M'$ be a map of $R$-modules with $M$ of finite presentation such that $\\text{Tor}_1^R(M, N) \\to \\text{Tor}_1^R(M', N)$ is zero for all $R$-modules $N$. Then $M \\to M'$ factors through a free $R$-module."}
+{"_id": "9771", "title": "local-cohomology-lemma-characterize-vanishing-tor-ext-above-e", "text": "Let $R$ be a ring. Let $\\alpha : M \\to M'$ be a map of $R$-modules. Let $P_\\bullet \\to M$ and $P'_\\bullet \\to M'$ be resolutions by projective $R$-modules. Let $e \\geq 0$ be an integer. Consider the following conditions \\begin{enumerate} \\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$ inducing $\\alpha$ on cohomology with $a_i = 0$ for $i > e$. \\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$ inducing $\\alpha$ on cohomology with $a_{e + 1} = 0$. \\item The map $\\Ext^i_R(M', N) \\to \\Ext^i_R(M, N)$ is zero for all $R$-modules $N$ and $i > e$. \\item The map $\\Ext^{e + 1}_R(M', N) \\to \\Ext^{e + 1}_R(M, N)$ is zero for all $R$-modules $N$. \\item Let $N = \\Im(P'_{e + 1} \\to P'_e)$ and denote $\\xi \\in \\Ext^{e + 1}_R(M', N)$ the canonical element (see proof). Then $\\xi$ maps to zero in $\\Ext^{e + 1}_R(M, N)$. \\item The map $\\text{Tor}_i^R(M, N) \\to \\text{Tor}_i^R(M', N)$ is zero for all $R$-modules $N$ and $i > e$. \\item The map $\\text{Tor}_{e + 1}^R(M, N) \\to \\text{Tor}_{e + 1}^R(M', N)$ is zero for all $R$-modules $N$. \\end{enumerate} Then we always have the implications $$ (1) \\Leftrightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (4) \\Leftrightarrow (5) \\Rightarrow (6) \\Leftrightarrow (7) $$ If $M$ is $(-e - 1)$-pseudo-coherent (for example if $R$ is Noetherian and $M$ is a finite $R$-module), then all conditions are equivalent."}
+{"_id": "9772", "title": "local-cohomology-lemma-cd-sequence-Koszul", "text": "Let $I$ be an ideal of a Noetherian ring $A$. For all $n \\geq 1$ there exists an $m > n$ such that the map $A/I^m \\to A/I^n$ satisfies the equivalent conditions of Lemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = \\text{cd}(A, I)$."}
+{"_id": "9773", "title": "local-cohomology-lemma-maps-zero-fixed-torsion", "text": "Let $I$ be an ideal of a Noetherian ring $A$. For every $m \\geq 0$ and $i > 0$ there exist a $c = c(A, I, m, i) \\geq 0$ such that for every $A$-module $M$ annihilated by $I^m$ the map $$ \\text{Tor}^A_i(M, A/I^n) \\to \\text{Tor}^A_i(M, A/I^{n - c}) $$ is zero for all $n \\geq c$."}
+{"_id": "9774", "title": "local-cohomology-lemma-annihilates-affine", "text": "Let $I = (a_1, \\ldots, a_t)$ be an ideal of a Noetherian ring $A$. Set $a = a_1$ and denote $B = A[\\frac{I}{a}]$ the affine blowup algebra. There exists a $c > 0$ such that $\\text{Tor}_i^A(B, M)$ is annihilated by $I^c$ for all $A$-modules $M$ and $i \\geq t$."}
+{"_id": "9775", "title": "local-cohomology-lemma-compute-tor-Iq", "text": "In the situation above, for $q \\geq q(A, I)$ and any $A$-module $M$ we have $$ R\\Gamma(X, Lp^*\\widetilde{M}(q)) \\cong M \\otimes_A^\\mathbf{L} I^q $$ in $D(A)$."}
+{"_id": "9776", "title": "local-cohomology-lemma-annihilates", "text": "In the situation above, let $t$ be an upper bound on the number of generators for $I$. There exists an integer $c = c(A, I) \\geq 0$ such that for any $A$-module $M$ the cohomology sheaves $H^j(Lp^*\\widetilde{M})$ are annihilated by $I^c$ for $j \\leq -t$."}
+{"_id": "9777", "title": "local-cohomology-lemma-annihilates-tors", "text": "In the situation above, let $t$ be an upper bound on the number of generators for $I$. There exists an integer $c = c(A, I) \\geq 0$ such that for any $A$-module $M$ the tor modules $\\text{Tor}_i^A(M, A/I^q)$ are annihilated by $I^c$ for $i > t$ and all $q \\geq 0$."}
+{"_id": "9778", "title": "local-cohomology-lemma-tor-maps-vanish", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \\geq 0$ such that the maps $$ \\text{Tor}_{t + 1}^A(M, A/I^n) \\to \\text{Tor}_{t + 1}^A(M, A/I^{n - c}) $$ are zero for any $A$-module $M$ and all $n \\geq N$."}
+{"_id": "9779", "title": "local-cohomology-lemma-bound-q-and-d", "text": "With $q_0 = q(S)$ and $d = d(S)$ as above, we have \\begin{enumerate} \\item for $n \\geq 1$, $q \\geq q_0$, and $i > 0$ we have $H^i(X, \\mathcal{O}_{Y_n}(q)) = 0$, \\item for $n \\geq 1$ and $q \\geq q_0$ we have $H^0(X, \\mathcal{O}_{Y_n}(q)) = I^q/I^{q + n}$, \\item for $q \\geq q_0$ and $i > 0$ we have $H^i(X, \\mathcal{O}_X(q)) = 0$, \\item for $q \\geq q_0$ we have $H^0(X, \\mathcal{O}_X(q)) = I^q$. \\end{enumerate}"}
+{"_id": "9780", "title": "local-cohomology-lemma-almost-exactness", "text": "Let $0 \\to K \\to L \\to M \\to 0$ be a short exact sequence of $A$-modules such that $K$ and $L$ are annihilated by $I^n$ and $M$ is an $(A, n, c)$-module. Then the kernel of $p^*K \\to p^*L$ is scheme theoretically supported on $Y_c$."}
+{"_id": "9781", "title": "local-cohomology-lemma-annihilated", "text": "Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is scheme theoretically supported on $Y_c$ if and only if the canonical map $\\mathcal{F} \\to \\mathcal{F}(c)$ is zero."}
+{"_id": "9782", "title": "local-cohomology-lemma-vanishing-coh-almost-projective", "text": "With $q_0 = q(S)$ and $d = d(S)$ as above, suppose we have integers $n \\geq c \\geq 0$, an $(A, n, c)$-module $M$, an index $i \\in \\{0, 1, \\ldots, d\\}$, and an integer $q$. Then we distinguish the following cases \\begin{enumerate} \\item In the case $i = d \\geq 1$ and $q \\geq q_0$ we have $H^d(X, p^*M(q)) = 0$. \\item In the case $i = d - 1 \\geq 1$ and $q \\geq q_0$ we have $H^{d - 1}(X, p^*M(q)) = 0$. \\item In the case $d - 1 > i > 0$ and $q \\geq q_0 + (d - 1 - i)c$ the map $H^i(X, p^*M(q)) \\to H^i(X, p^*M(q - (d - 1 - i)c))$ is zero. \\item In the case $i = 0$, $d \\in \\{0, 1\\}$, and $q \\geq q_0$, there is a surjection $$ I^qM \\longrightarrow H^0(X, p^*M(q)) $$ \\item In the case $i = 0$, $d > 1$, and $q \\geq q_0 + (d - 1)c$ the map $$ H^0(X, p^*M(q)) \\to H^0(X, p^*M(q - (d - 1)c)) $$ has image contained in the image of the canonical map $I^{q - (d - 1)c}M \\to H^0(X, p^*M(q - (d - 1)c))$. \\end{enumerate}"}
+{"_id": "9783", "title": "local-cohomology-lemma-factor-hom", "text": "With $q_0 = q(S)$ and $d = d(S)$ as above, let $M$ be an $(A, n, c)$-module and let $\\varphi : M \\to I^n/I^{2n}$ be an $A$-linear map. Assume $n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$ and if $d = 0$ assume $n \\geq q_0 + 2c$. Then the composition $$ M \\xrightarrow{\\varphi} I^n/I^{2n} \\to I^{n - (1 + d)c}/I^{2n - (1 + d)c} $$ is of the form $\\sum a_i \\psi_i$ with $a_i \\in I^c$ and $\\psi_i : M \\to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$."}
+{"_id": "9784", "title": "local-cohomology-lemma-bound-two-term-complex", "text": "With $d = d(S)$ and $q_0 = q(S)$ as above. Then \\begin{enumerate} \\item for integers $n \\geq c \\geq 0$ with $n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$, \\item for $K$ of $D(A/I^n)$ with $H^i(K) = 0$ for $i \\not = -1, 0$ and $H^i(K)$ finite for $i = -1, 0$ such that $\\Ext^1_{A/I^c}(K, N)$ is annihilated by $I^c$ for all finite $A/I^c$-modules $N$ \\end{enumerate} the map $$ \\Ext^1_{A/I^n}(K, I^n/I^{2n}) \\longrightarrow \\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c}) $$ is zero."}
+{"_id": "9785", "title": "local-cohomology-proposition-kollar", "text": "\\begin{reference} See \\cite{k-coherent} and see \\cite[IV, Proposition 7.2.2]{EGA} for a special case. \\end{reference} \\begin{slogan} Weak analogue of Hartogs' Theorem: On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent. \\end{slogan} Let $j : U \\to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. The following are equivalent \\begin{enumerate} \\item $j_*\\mathcal{F}$ is coherent, \\item for $x \\in \\text{Ass}(\\mathcal{F})$ and $z \\in Z \\cap \\overline{\\{x\\}}$ and any associated prime $\\mathfrak p$ of the completion $\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge$ we have $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge/\\mathfrak p) \\geq 2$. \\end{enumerate}"}
+{"_id": "9786", "title": "local-cohomology-proposition-annihilator", "text": "\\begin{reference} \\cite{Faltings-annulators}. \\end{reference} Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \\subset T' \\subset \\Spec(A)$ be subsets stable under specialization. Let $s \\geq 0$ an integer. Let $M$ be a finite $A$-module. The following are equivalent \\begin{enumerate} \\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$ such that $J$ annihilates $H^i_T(M)$ for $i \\leq s$, and \\item for all $\\mathfrak p \\not \\in T'$, $\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$ we have $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > s $$ \\end{enumerate}"}
+{"_id": "9787", "title": "local-cohomology-proposition-finiteness", "text": "\\begin{reference} \\cite{Faltings-annulators}. \\end{reference} Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $s \\geq 0$ an integer. Let $M$ be a finite $A$-module. The following are equivalent \\begin{enumerate} \\item $H^i_T(M)$ is a finite $A$-module for $i \\leq s$, and \\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$ we have $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > s $$ \\end{enumerate}"}
+{"_id": "9788", "title": "local-cohomology-proposition-annihilator-complex", "text": "Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \\subset T' \\subset \\Spec(A)$ be subsets stable under specialization. Let $s \\in \\mathbf{Z}$. Let $K$ be an object of $D_{\\textit{Coh}}^+(A)$. The following are equivalent \\begin{enumerate} \\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$ such that $J$ annihilates $H^i_T(K)$ for $i \\leq s$, and \\item for all $\\mathfrak p \\not \\in T'$, $\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$ we have $$ \\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > s $$ \\end{enumerate}"}
+{"_id": "9790", "title": "local-cohomology-proposition-Hartshorne-Lichtenbaum-vanishing", "text": "\\begin{reference} \\cite[Theorem 3.1]{CD} \\end{reference} Let $A$ be a Noetherian local ring with completion $A^\\wedge$. Let $I \\subset A$ be an ideal such that $$ \\dim V(IA^\\wedge + \\mathfrak p) \\geq 1 $$ for every minimal prime $\\mathfrak p \\subset A^\\wedge$ of dimension $\\dim(A)$. Then $\\text{cd}(A, I) < \\dim(A)$."}
+{"_id": "9791", "title": "local-cohomology-proposition-uniform-artin-rees", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \\geq 0$ such that for $n \\geq N$ the maps $$ A/I^n \\to A/I^{n - c} $$ satisfy the equivalent conditions of Lemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = t$."}
+{"_id": "9802", "title": "more-algebra-theorem-regular-fs", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a Noetherian local $k$-algebra. If the characteristic of $k$ is zero then the following are equivalent \\begin{enumerate} \\item $A$ is a regular local ring, and \\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology. \\end{enumerate} If the characteristic of $k$ is $p > 0$ then the following are equivalent \\begin{enumerate} \\item $A$ is geometrically regular over $k$, \\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology. \\item for all $k \\subset k' \\subset k^{1/p}$ finite over $k$ the ring $A \\otimes_k k'$ is regular, \\item $A$ is regular and the canonical map $H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and \\item $A$ is regular and the map $\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$ is injective. \\end{enumerate}"}
+{"_id": "9803", "title": "more-algebra-theorem-formal-glueing", "text": "Let $R$ be a ring, and let $f \\in R$. Let $\\varphi : R \\to S$ be a flat ring map inducing an isomorphism $R/fR \\to S/fS$. Then the functor $$ \\text{Mod}_R \\longrightarrow \\text{Mod}_S \\times_{\\text{Mod}_{S_f}} \\text{Mod}_{R_f}, \\quad M \\longmapsto (M \\otimes_R S, M_f, \\text{can}) $$ is an equivalence."}
+{"_id": "9804", "title": "more-algebra-theorem-BL-glueing", "text": "\\begin{reference} Slight generalization of the main theorem of \\cite{Beauville-Laszlo}. \\end{reference} Let $(R \\to R',f)$ be a glueing pair. The functor $\\text{Can} : \\text{Mod}_R \\longrightarrow \\text{Glue}(R \\to R', f)$ determines an equivalence of the category of $R$-modules glueable for $(R \\to R', f)$ and the category $\\text{Glue}(R \\to R', f)$ of glueing data."}
+{"_id": "9805", "title": "more-algebra-theorem-olivier", "text": "Let $A \\to B$ be a local homomorphism of local rings. If $A$ is strictly henselian and $A \\to B$ is weakly \\'etale, then $A = B$."}
+{"_id": "9806", "title": "more-algebra-theorem-epp", "text": "Let $A \\subset B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. If the characteristic of $\\kappa_A$ is $p > 0$, assume that every element of $$ \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n} $$ is separable algebraic over $\\kappa_A$. Then there exists a finite extension $K \\subset K_1$ which is a weak solution for $A \\to B$ as defined in Definition \\ref{definition-solution}."}
+{"_id": "9807", "title": "more-algebra-lemma-exact-category-stably-free", "text": "Let $R$ be a ring. Let $0 \\to P' \\to P \\to P'' \\to 0$ be a short exact sequence of finite projective $R$-modules. If $2$ out of $3$ of these modules are stably free, then so is the third."}
+{"_id": "9808", "title": "more-algebra-lemma-lift-stably-free", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). For every finite stably free $R/I$-module $E$ there exists a finite stably free $R$-module $M$ such that $M/IM \\cong E$."}
+{"_id": "9809", "title": "more-algebra-lemma-lift-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). Let $M$ be a finite flat $R$-module such that $M/IM$ is a projective $R/I$-module. Then $M$ is a finite projective $R$-module."}
+{"_id": "9810", "title": "more-algebra-lemma-isomorphic-finite-projective-lifts", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the Jacobson radical of $R$). If $P$ and $P'$ are finite projective $R$-modules, then \\begin{enumerate} \\item if $\\varphi : P \\to P'$ is an $R$-module map inducing an isomorphism $\\overline{\\varphi} : P/IP \\to P'/IP'$, then $\\varphi$ is an isomorphism, \\item if $P/IP \\cong P'/IP'$, then $P \\cong P'$. \\end{enumerate}"}
+{"_id": "9811", "title": "more-algebra-lemma-approximate-complex", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal contained in the Jacobson radical of $A$. Let $$ S : L \\xrightarrow{f} M \\xrightarrow{g} N \\quad\\text{and}\\quad S' : L \\xrightarrow{f'} M \\xrightarrow{g'} N $$ be two complexes of finite $A$-modules as shown. Assume that \\begin{enumerate} \\item $c$ works in the Artin-Rees lemma for $f$ and $g$, \\item the complex $S$ is exact, and \\item $f' = f \\bmod I^{c + 1}M$ and $g' = g \\bmod I^{c + 1}N$. \\end{enumerate} Then $c$ works in the Artin-Rees lemma for $g'$ and the complex $S'$ is exact."}
+{"_id": "9812", "title": "more-algebra-lemma-approximate-complex-graded", "text": "Assumptions as in Lemma \\ref{lemma-approximate-complex}. Let $Q = \\Coker(g)$ and $Q' = \\Coker(g')$. Then $\\text{Gr}_I(Q) \\cong \\text{Gr}_I(Q')$ as graded $\\text{Gr}_I(A)$-modules."}
+{"_id": "9813", "title": "more-algebra-lemma-works-flat-extension", "text": "Let $A \\to B$ be a flat map of Noetherian rings. Let $I \\subset A$ be an ideal. Let $f : M \\to N$ be a homomorphism of finite $A$-modules. Assume that $c$ works for $f$ in the Artin-Rees lemma. Then $c$ works for $f \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ in the Artin-Rees lemma for the ideal $IB$."}
+{"_id": "9814", "title": "more-algebra-lemma-fibre-product-finite-type", "text": "Let $R$ be a ring. Let $A \\to B$ and $C \\to B$ be $R$-algebra maps. Assume \\begin{enumerate} \\item $R$ is Noetherian, \\item $A$, $B$, $C$ are of finite type over $R$, \\item $A \\to B$ is surjective, and \\item $B$ is finite over $C$. \\end{enumerate} Then $A \\times_B C$ is of finite type over $R$."}
+{"_id": "9815", "title": "more-algebra-lemma-formal-consequence", "text": "Let $R$ be a Noetherian ring. Let $I$ be a finite set. Suppose given a cartesian diagram $$ \\xymatrix{ \\prod B_i & \\prod A_i \\ar[l]^{\\prod \\varphi_i} \\\\ Q \\ar[u]^{\\prod \\psi_i} & P \\ar[u] \\ar[l] } $$ with $\\psi_i$ and $\\varphi_i$ surjective, and $Q$, $A_i$, $B_i$ of finite type over $R$. Then $P$ is of finite type over $R$."}
+{"_id": "9816", "title": "more-algebra-lemma-diagram-localize", "text": "Suppose given a cartesian diagram of rings $$ \\xymatrix{ R & R' \\ar[l]^t \\\\ B \\ar[u]_s & B'\\ar[u] \\ar[l] } $$ i.e., $B' = B \\times_R R'$. If $h \\in B'$ corresponds to $g \\in B$ and $f \\in R'$ such that $s(g) = t(f)$, then the diagram $$ \\xymatrix{ R_{s(g)} = R_{t(f)} & (R')_f \\ar[l]^-t \\\\ B_g \\ar[u]_s & (B')_h \\ar[u] \\ar[l] } $$ is cartesian too."}
+{"_id": "9817", "title": "more-algebra-lemma-modules", "text": "Given a commutative diagram of rings $$ \\xymatrix{ R & R' \\ar[l] \\\\ B \\ar[u] & B' \\ar[u] \\ar[l] } $$ the functor (\\ref{equation-modules}) has a right adjoint, namely the functor $$ F : (N, M', \\varphi) \\longmapsto N \\times_\\varphi M' $$ (see proof for elucidation)."}
+{"_id": "9818", "title": "more-algebra-lemma-points-of-fibre-product", "text": "In Situation \\ref{situation-module-over-fibre-product} we have $$ \\Spec(B') = \\Spec(B) \\amalg_{\\Spec(A)} \\Spec(A') $$ as topological spaces."}
+{"_id": "9819", "title": "more-algebra-lemma-fibre-product-integral", "text": "In Situation \\ref{situation-module-over-fibre-product} if $B \\to A$ is integral, then $B' \\to A'$ is integral."}
+{"_id": "9820", "title": "more-algebra-lemma-module-over-fibre-product", "text": "In Situation \\ref{situation-module-over-fibre-product} the functor (\\ref{equation-functor}) has a right adjoint, namely the functor $$ F : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M' $$ where $M = M'/IM'$. Moreover, the composition of $F$ with (\\ref{equation-functor}) is the identity functor on $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$. In other words, setting $N' = N \\times_{\\varphi, M} M'$ we have $N' \\otimes_{B'} B = N$ and $N' \\otimes_{B'} A' = M'$."}
+{"_id": "9821", "title": "more-algebra-lemma-module-over-fibre-product-bis", "text": "In the situation of Lemma \\ref{lemma-module-over-fibre-product} for a $B'$-module $L'$ the adjunction map $$ L' \\longrightarrow  (L' \\otimes_{B'} B) \\times_{(L' \\otimes_{B'} A)} (L' \\otimes_{B'} A') $$ is surjective but in general not injective."}
+{"_id": "9822", "title": "more-algebra-lemma-surjection-module-over-fibre-product", "text": "In Situation \\ref{situation-module-over-fibre-product} let $(N_1, M'_1, \\varphi_1) \\to (N_2, M'_2, \\varphi_2)$ be a morphism of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$ with $N_1 \\to N_2$ and $M'_1 \\to M'_2$ surjective. Then $$ N_1 \\times_{\\varphi_1, M_1} M'_1 \\to N_2 \\times_{\\varphi_2, M_2} M'_2 $$ where $M_1 = M'_1/IM'_1$ and $M_2 = M'_2/IM'_2$ is surjective."}
+{"_id": "9823", "title": "more-algebra-lemma-finite-module-over-fibre-product", "text": "Let $A, A', B, B', I, M, M', N, \\varphi$ be as in Lemma \\ref{lemma-module-over-fibre-product}. If $N$ finite over $B$ and $M'$ finite over $A'$, then $N' = N \\times_{\\varphi, M} M'$ is finite over $B'$."}
+{"_id": "9824", "title": "more-algebra-lemma-flat-module-over-fibre-product", "text": "With $A, A', B, B', I$ as in Situation \\ref{situation-module-over-fibre-product}. \\begin{enumerate} \\item Let $(N, M', \\varphi)$ be an object of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \\times_{\\varphi, M} M'$ is flat over $B'$. \\item If $L'$ is a flat $B'$-module, then $L' = (L \\otimes_{B'} B) \\times_{(L \\otimes_{B'} A)} (L \\otimes_{B'} A')$. \\item The category of flat $B'$-modules is equivalent to the full subcategory of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$ consisting of triples $(N, M', \\varphi)$ with $N$ flat over $B$ and $M'$ flat over $A'$. \\end{enumerate}"}
+{"_id": "9825", "title": "more-algebra-lemma-finitely-presented-module-over-fibre-product", "text": "Let $A, A', B, B', I$ be as in Situation \\ref{situation-module-over-fibre-product}. The category of finite projective $B'$-modules is equivalent to the full subcategory of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$ consisting of triples $(N, M', \\varphi)$ with $N$ finite projective over $B$ and $M'$ finite projective over $A'$."}
+{"_id": "9826", "title": "more-algebra-lemma-relative-module-over-fibre-product", "text": "In Situation \\ref{situation-relative-module-over-fibre-product} the functor (\\ref{equation-relative-functor}) has a right adjoint, namely the functor $$ F : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M' $$ where $M = M'/IM'$. Moreover, the composition of $F$ with (\\ref{equation-relative-functor}) is the identity functor on $\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$. In other words, setting $N' = N \\times_{\\varphi, M} M'$ we have $N' \\otimes_{D'} D = N$ and $N' \\otimes_{D'} C' = M'$."}
+{"_id": "9827", "title": "more-algebra-lemma-relative-surjection-ideals", "text": "In Situation \\ref{situation-relative-module-over-fibre-product} the map $JD' \\to IC'$ is surjective where $J = \\Ker(B' \\to B)$."}
+{"_id": "9828", "title": "more-algebra-lemma-relative-finite-module-over-fibre-product", "text": "Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in Lemma \\ref{lemma-relative-module-over-fibre-product}. If $N$ finite over $D$ and $M'$ finite over $C'$, then $N' = N \\times_{\\varphi, M} M'$ is finite over $D'$."}
+{"_id": "9829", "title": "more-algebra-lemma-relative-flat-module-over-fibre-product", "text": "With $A, A', B, B', C, C', D, D', I$ as in Situation \\ref{situation-relative-module-over-fibre-product}. \\begin{enumerate} \\item Let $(N, M', \\varphi)$ be an object of $\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \\times_{\\varphi, M} M'$ is flat over $B'$. \\item If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \\otimes_{D'} D) \\times_{(L \\otimes_{D'} C)} (L \\otimes_{D'} C')$. \\item The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \\varphi)$ of $\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$. \\end{enumerate}"}
+{"_id": "9830", "title": "more-algebra-lemma-relative-finitely-presented-module-over-fibre-product", "text": "Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in Lemma \\ref{lemma-relative-module-over-fibre-product}. If \\begin{enumerate} \\item $N$ is finitely presented over $D$ and flat over $B$, \\item $M'$ finitely presented over $C'$ and flat over $A'$, and \\item the ring map $B' \\to D'$ factors as $B' \\to D'' \\to D'$ with $B' \\to D''$ flat and $D'' \\to D'$ of finite presentation, \\end{enumerate} then $N' = N \\times_M M'$ is finitely presented over $D'$."}
+{"_id": "9831", "title": "more-algebra-lemma-properties-algebras-over-fibre-product", "text": "Let $A, A', B, B', I$ be as in Situation \\ref{situation-module-over-fibre-product}. Let $(D, C', \\varphi)$ be a system consisting of an $B$-algebra $D$, a $A'$-algebra $C'$ and an isomorphism $D \\otimes_B A \\to C'/IC' = C$. Set $D' = D \\times_C C'$ (as in Lemma \\ref{lemma-module-over-fibre-product}). Then \\begin{enumerate} \\item $B' \\to D'$ is finite type if and only if $B \\to D$ and $A' \\to C'$ are finite type, \\item $B' \\to D'$ is flat if and only if $B \\to D$ and $A' \\to C'$ are flat, \\item $B' \\to D'$ is flat and of finite presentation if and only if $B \\to D$ and $A' \\to C'$ are flat and of finite presentation, \\item $B' \\to D'$ is smooth if and only if $B \\to D$ and $A' \\to C'$ are smooth, \\item $B' \\to D'$ is \\'etale if and only if $B \\to D$ and $A' \\to C'$ are \\'etale. \\end{enumerate} Moreover, if $D'$ is a flat $B'$-algebra, then $D' \\to (D' \\otimes_{B'} B) \\times_{(D' \\otimes_{B'} A)} (D' \\otimes_{B'} A')$ is an isomorphism. In this way the category of flat $B'$-algebras is equivalent to the categories of systems $(D, C', \\varphi)$ as above with $D$ flat over $B$ and $C'$ flat over $A'$."}
+{"_id": "9832", "title": "more-algebra-lemma-ideals-generated-by-minors", "text": "Let $R$ be a ring. Let $A$ be an $n \\times m$ matrix with coefficients in $R$. Let $I_r(A)$ be the ideal generated by the $r \\times r$-minors of $A$ with the convention that $I_0(A) = R$ and $I_r(A) = 0$ if $r > \\min(n, m)$. Then \\begin{enumerate} \\item $I_0(A) \\supset I_1(A) \\supset I_2(A) \\supset \\ldots$, \\item if $B$ is an $(n + n') \\times m$ matrix, and $A$ is the first $n$ rows of $B$, then $I_{r + n'}(B) \\subset I_r(A)$, \\item if $C$ is an $n \\times n$ matrix then $I_r(CA) \\subset I_r(A)$. \\item If $A$ is a block matrix $$ \\left( \\begin{matrix} A_1 & 0 \\\\ 0 & A_2  \\end{matrix} \\right) $$ then $I_r(A) = \\sum_{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2)$. \\item Add more here. \\end{enumerate}"}
+{"_id": "9833", "title": "more-algebra-lemma-fitting-ideal", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Choose a presentation $$ \\bigoplus\\nolimits_{j \\in J} R \\longrightarrow R^{\\oplus n} \\longrightarrow M \\longrightarrow 0. $$ of $M$. Let $A = (a_{ij})_{i = 1, \\ldots, n, j \\in J}$ be the matrix of the map $\\bigoplus_{j \\in J} R \\to R^{\\oplus n}$. The ideal $\\text{Fit}_k(M)$ generated by the $(n - k) \\times (n - k)$ minors of $A$ is independent of the choice of the presentation."}
+{"_id": "9834", "title": "more-algebra-lemma-fitting-ideal-basics", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. \\begin{enumerate} \\item If $M$ can be generated by $n$ elements, then $\\text{Fit}_n(M) = R$. \\item Given a second finite $R$-module $M'$ we have $$ \\text{Fit}_l(M \\oplus M') = \\sum\\nolimits_{k + k' = l} \\text{Fit}_k(M)\\text{Fit}_{k'}(M') $$ \\item If $R \\to R'$ is a ring map, then $\\text{Fit}_k(M \\otimes_R R')$ is the ideal of $R'$ generated by the image of $\\text{Fit}_k(M)$. \\item If $M$ is of finite presentation, then $\\text{Fit}_k(M)$ is a finitely generated ideal. \\item If $M \\to M'$ is a surjection, then $\\text{Fit}_k(M) \\subset \\text{Fit}_k(M')$. \\item We have $\\text{Fit}_0(M) \\subset \\text{Ann}_R(M)$. \\item We have $V(\\text{Fit}_0(M)) = \\text{Supp}(M)$. \\item Add more here. \\end{enumerate}"}
+{"_id": "9835", "title": "more-algebra-lemma-fitting-ideal-generate-locally", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$. Let $\\mathfrak p \\subset R$ be a prime ideal. The following are equivalent \\begin{enumerate} \\item $\\text{Fit}_k(M) \\not \\subset \\mathfrak p$, \\item $\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p) \\leq k$, \\item $M_\\mathfrak p$ can be generated by $k$ elements over $R_\\mathfrak p$, and \\item $M_f$ can be generated by $k$ elements over $R_f$ for some $f \\in R$, $f \\not \\in \\mathfrak p$. \\end{enumerate}"}
+{"_id": "9836", "title": "more-algebra-lemma-fitting-ideal-finite-locally-free", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $r \\geq 0$. The following are equivalent \\begin{enumerate} \\item $M$ is finite locally free of rank $r$ (Algebra, Definition \\ref{algebra-definition-locally-free}), \\item $\\text{Fit}_{r - 1}(M) = 0$ and $\\text{Fit}_r(M) = R$, and \\item $\\text{Fit}_k(M) = 0$ for $k < r$ and $\\text{Fit}_k(M) = R$ for $k \\geq r$. \\end{enumerate}"}
+{"_id": "9837", "title": "more-algebra-lemma-principal-fitting-ideal", "text": "Let $R$ be a local ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$. Assume that $\\text{Fit}_k(M) = (f)$ for some $f \\in R$. Let $M'$ be the quotient of $M$ by $\\{x \\in M \\mid fx = 0\\}$. Then $M'$ can be generated by $k$ elements."}
+{"_id": "9838", "title": "more-algebra-lemma-fitting-ideals-and-pd1", "text": "Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $k \\geq 0$. Assume that $\\text{Fit}_k(M) = (f)$ for some nonzerodivisor $f \\in R$ and $\\text{Fit}_{k - 1}(M) = 0$. Then $M$ has projective dimension $\\leq 1$."}
+{"_id": "9839", "title": "more-algebra-lemma-lift-invertible-element", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{u} \\in A/I$ be an invertible element. There exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an invertible element $u' \\in A'$ lifting $\\overline{u}$."}
+{"_id": "9840", "title": "more-algebra-lemma-lift-idempotent", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{e} \\in A/I$ be an idempotent. There exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an idempotent $e' \\in A'$ lifting $\\overline{e}$."}
+{"_id": "9841", "title": "more-algebra-lemma-lift-open-covering", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $\\Spec(A/I) = \\coprod_{j \\in J} \\overline{U}_j$ be a finite disjoint open covering. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a finite disjoint open covering $\\Spec(A') = \\coprod_{j \\in J} U'_j$ lifting the given covering."}
+{"_id": "9842", "title": "more-algebra-lemma-localize-upstairs", "text": "Let $A \\to B$ be a ring map and $J \\subset B$ an ideal. If $A \\to B$ is \\'etale at every prime of $V(J)$, then there exists a $g \\in B$ mapping to an invertible element of $B/J$ such that $A' = B_g$ is \\'etale over $A$."}
+{"_id": "9843", "title": "more-algebra-lemma-lift-factorization-monic", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a monic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a factorization of $f$ in $A/I[x]$ such that $\\overline{g}$ and $\\overline{h}$ are monic and generate the unit ideal in $A/I[x]$. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ with $g'$, $h'$ monic lifting the given factorization over $A/I$."}
+{"_id": "9844", "title": "more-algebra-lemma-lift-factorization-easy", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a monic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a factorization of $f$ in $A/I[x]$ and assume \\begin{enumerate} \\item the leading coefficient of $\\overline{g}$ is an invertible element of $A/I$, and \\item $\\overline{g}$, $\\overline{h}$ generate the unit ideal in $A/I[x]$. \\end{enumerate} Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ lifting the given factorization over $A/I$."}
+{"_id": "9846", "title": "more-algebra-lemma-separate-image-closed-from-closed", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal of $R$ and let $J \\subset S$ be an ideal of $S$. If the closure of the image of $V(J)$ in $\\Spec(R)$ is disjoint from $V(I)$, then there exists an element $f \\in R$ which maps to $1$ in $R/I$ and to an element of $J$ in $S$."}
+{"_id": "9847", "title": "more-algebra-lemma-helper-integral", "text": "Let $I$ be an ideal of a ring $A$. Let $A \\to B$ be an integral ring map. Let $b \\in B$ map to an idempotent in $B/IB$. Then there exists a monic $f \\in A[x]$ with $f(b) = 0$ and $f \\bmod I = x^d(x - 1)^d$ for some $d \\geq 1$."}
+{"_id": "9848", "title": "more-algebra-lemma-lift-idempotent-upstairs", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $A \\to B$ be an integral ring map. Let $\\overline{e} \\in B/IB$ be an idempotent. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an idempotent $e' \\in B \\otimes_A A'$ lifting $\\overline{e}$."}
+{"_id": "9849", "title": "more-algebra-lemma-lift-projective-module", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $\\overline{P}$ be a finite projective $A/I$-module. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a finite projective $A'$-module $P'$ lifting $\\overline{P}$."}
+{"_id": "9850", "title": "more-algebra-lemma-cotangent-complex-symmetric-algebra", "text": "Let $A$ be a ring. Let $0 \\to K \\to A^{\\oplus m} \\to M \\to 0$ be a sequence of $A$-modules. Consider the $A$-algebra $C = \\text{Sym}^*_A(M)$ with its presentation $\\alpha : A[y_1, \\ldots, y_m] \\to C$ coming from the surjection $A^{\\oplus m} \\to M$. Then $$ \\NL(\\alpha) = (K \\otimes_A C \\to \\bigoplus\\nolimits_{j = 1, \\ldots, m} C \\text{d}y_j) $$ (see Algebra, Section \\ref{algebra-section-netherlander}) in particular $\\Omega_{C/A} = M \\otimes_A C$."}
+{"_id": "9851", "title": "more-algebra-lemma-symmetric-algebra-smooth", "text": "Let $A$ be a ring. Let $M$ be an $A$-module. Then $C = \\text{Sym}_A^*(M)$ is smooth over $A$ if and only if $M$ is a finite projective $A$-module."}
+{"_id": "9853", "title": "more-algebra-lemma-idempotents-determined-modulo-radical", "text": "Let $(A, I)$ be a Zariski pair. Then the map from idempotents of $A$ to idempotents of $A/I$ is injective."}
+{"_id": "9854", "title": "more-algebra-lemma-check-isomorphism-zariski", "text": "Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a flat, integral, finitely presented ring map such that $A/I \\to B/IB$ is an isomorphism. Then $A \\to B$ is an isomorphism."}
+{"_id": "9855", "title": "more-algebra-lemma-helper-finite", "text": "Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a finite ring map. Assume \\begin{enumerate} \\item $B/IB = B_1 \\times B_2$ is a product of $A/I$-algebras \\item $A/I \\to B_1/IB_1$ is surjective, \\item $b \\in B$ maps to $(1, 0)$ in the product. \\end{enumerate} Then there exists a monic $f \\in A[x]$ with $f(b) = 0$ and $f \\bmod I = (x - 1)x^d$ for some $d \\geq 1$."}
+{"_id": "9856", "title": "more-algebra-lemma-noetherian-zariski-jacobson-complement", "text": "Let $(A, I)$ be a Zariski pair with $A$ Noetherian. Let $f \\in I$. Then $A_f$ is a Jacobson ring."}
+{"_id": "9857", "title": "more-algebra-lemma-locally-nilpotent-henselian", "text": "Let $(A, I)$ be a pair with $I$ locally nilpotent. Then the functor $B \\mapsto B/IB$ induces an equivalence between the category of \\'etale algebras over $A$ and the category of \\'etale algebras over $A/I$. Moreover, the pair is henselian."}
+{"_id": "9858", "title": "more-algebra-lemma-limit-henselian", "text": "Let $A = \\lim A_n$ where $(A_n)$ is an inverse system of rings whose transition maps are surjective and have locally nilpotent kernels. Then $(A, I_n)$ is a henselian pair, where $I_n = \\Ker(A \\to A_n)$."}
+{"_id": "9859", "title": "more-algebra-lemma-complete-henselian", "text": "Let $(A, I)$ be a pair. If $A$ is $I$-adically complete, then the pair is henselian."}
+{"_id": "9860", "title": "more-algebra-lemma-helper-finite-type", "text": "Let $(A, I)$ be a pair. Let $A \\to B$ be a finite type ring map such that $B/IB = C_1 \\times C_2$ with $A/I \\to C_1$ finite. Let $B'$ be the integral closure of $A$ in $B$. Then we can write $B'/IB' = C_1 \\times C'_2$ such that the map $B'/IB' \\to B/IB$ preserves product decompositions and there exists a $g \\in B'$ mapping to $(1, 0)$ in $C_1 \\times C'_2$ with $B'_g \\to B_g$ an isomorphism."}
+{"_id": "9861", "title": "more-algebra-lemma-characterize-henselian-pair", "text": "\\begin{reference} \\cite[Chapter XI]{Henselian} and \\cite[Proposition 1]{Gabber-henselian} \\end{reference} Let $(A, I)$ be a pair. The following are equivalent \\begin{enumerate} \\item $(A, I)$ is a henselian pair, \\item given an \\'etale ring map $A \\to A'$ and an $A$-algebra map $\\sigma : A' \\to A/I$, there exists an $A$-algebra map $A' \\to A$ lifting $\\sigma$, \\item for any finite $A$-algebra $B$ the map $B \\to B/IB$ induces a bijection on idempotents, \\item for any integral $A$-algebra $B$ the map $B \\to B/IB$ induces a bijection on idempotents, and \\item (Gabber) $I$ is contained in the Jacobson radical of $A$ and every monic polynomial $f(T) \\in A[T]$ of the form $$ f(T) = T^n(T - 1) + a_n T^n + \\ldots + a_1 T + a_0 $$ with $a_n, \\ldots, a_0 \\in I$ and $n \\ge 1$ has a root $\\alpha \\in 1 + I$. \\end{enumerate} Moreover, in part (5) the root is unique."}
+{"_id": "9862", "title": "more-algebra-lemma-change-ideal-henselian-pair", "text": "Let $A$ be a ring. Let $I, J \\subset A$ be ideals with $V(I) = V(J)$. Then $(A, I)$ is henselian if and only if $(A, J)$ is henselian."}
+{"_id": "9863", "title": "more-algebra-lemma-integral-over-henselian-pair", "text": "Let $(A, I)$ be a henselian pair and let $A \\to B$ be an integral ring map. Then $(B, IB)$ is a henselian pair."}
+{"_id": "9864", "title": "more-algebra-lemma-henselian-henselian-pair", "text": "Let $I \\subset J \\subset A$ be ideals of a ring $A$. The following are equivalent \\begin{enumerate} \\item $(A, I)$ and $(A/I, J/I)$ are henselian pairs, and \\item $(A, J)$ is an henselian pair. \\end{enumerate}"}
+{"_id": "9865", "title": "more-algebra-lemma-sum-henselian", "text": "Let $A$ be a ring and let $(A, I)$ and $(A, I')$ be henselian pairs. Then $(A, I + I')$ is an henselian pair."}
+{"_id": "9866", "title": "more-algebra-lemma-product-henselian-pairs", "text": "Let $J$ be a set and let $\\{ (A_j, I_j)\\}_{j \\in J}$ be a collection of pairs. Then $(\\prod_{j \\in J} A_j, \\prod_{j\\in J} I_j)$ is Henselian if and only if so is each $(A_j, I_j)$."}
+{"_id": "9868", "title": "more-algebra-lemma-filtered-colimits-henselian", "text": "The property of being Henselian is preserved under filtered colimits of pairs. More precisely, let $J$ be a directed set and let $(A_j, I_j)$ be a system of henselian pairs over $J$. Then $A = \\colim A_j$ equipped with the ideal $I = \\colim I_j$ is a henselian pair $(A, I)$."}
+{"_id": "9869", "title": "more-algebra-lemma-largest-ideal-henselian", "text": "Let $A$ be a ring. There exists a largest ideal $I \\subset A$ such that $(A, I)$ is a henselian pair."}
+{"_id": "9870", "title": "more-algebra-lemma-irreducible-henselian-pair-connected", "text": "Let $(A, I)$ be a henselian pair. Let $\\mathfrak p \\subset A$ be a prime ideal. Then $V(\\mathfrak p + I)$ is connected."}
+{"_id": "9871", "title": "more-algebra-lemma-henselization", "text": "The inclusion functor $$ \\text{category of henselian pairs} \\longrightarrow \\text{category of pairs} $$ has a left adjoint $(A, I) \\mapsto (A^h, I^h)$."}
+{"_id": "9872", "title": "more-algebra-lemma-henselization-flat", "text": "Let $(A, I)$ be a pair. Let $(A^h, I^h)$ be as in  Lemma \\ref{lemma-henselization}. Then $A \\to A^h$ is flat, $I^h = IA^h$ and $A/I^n \\to A^h/I^nA^h$ is an isomorphism for all $n$."}
+{"_id": "9874", "title": "more-algebra-lemma-henselization-Noetherian-pair", "text": "\\begin{slogan} The henselization of a Noetherian pair is a Noetherian pair with the same completion \\end{slogan} Let $(A, I)$ be a pair with $A$ Noetherian.  Let $(A^h, I^h)$ be as in  Lemma \\ref{lemma-henselization}. Then the map of $I$-adic completions $$ A^\\wedge \\to (A^h)^\\wedge $$ is an isomorphism. Moreover, $A^h$ is Noetherian, the maps $A \\to A^h \\to A^\\wedge$ are flat, and $A^h \\to A^\\wedge$ is faithfully flat."}
+{"_id": "9875", "title": "more-algebra-lemma-henselization-colimit", "text": "Let $(A, I) = \\colim (A_i, I_i)$ be a filtered colimit of pairs. The functor of Lemma \\ref{lemma-henselization} gives $A^h = \\colim A_i^h$ and $I^h = \\colim I_i^h$."}
+{"_id": "9876", "title": "more-algebra-lemma-henselization-change-ideal", "text": "\\begin{slogan} The henselization of a pair only depends on the radical of the ideal \\end{slogan} Let $A$ be a ring with ideals $I$ and $J$. If $V(I) = V(J)$ then the functor of Lemma \\ref{lemma-henselization} produces the same ring for the pair $(A, I)$ as for the pair $(A, J)$."}
+{"_id": "9877", "title": "more-algebra-lemma-henselization-integral", "text": "\\begin{slogan} Henselization commutes with integral base change \\end{slogan} Let $(A, I) \\to (B, J)$ be a map of pairs such that $V(J) = V(IB)$. Let $(A^h , I^h) \\to (B^h, J^h)$ be the induced map on henselizations (Lemma \\ref{lemma-henselization}). If $A \\to B$ is integral, then the induced map $A^h \\otimes_A B \\to B^h$ is an isomorphism."}
+{"_id": "9878", "title": "more-algebra-lemma-lift-finite-projective-module", "text": "Let $(R, I)$ be a henselian pair. Let $\\overline{P}$ be a finite projective $R/I$-module. Then there exists a finite projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$."}
+{"_id": "9879", "title": "more-algebra-lemma-finite-etale-equivalence", "text": "Let $(A, I)$ be a henselian pair. The functor $B \\to B/IB$ determines an equivalence between finite \\'etale $A$-algebras and finite \\'etale $A/I$-algebras."}
+{"_id": "9880", "title": "more-algebra-lemma-lim-finite-projective-gives-finite-projective", "text": "Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $A_n$-modules $M_n$ and $A_{n + 1}$-module maps $M_{n + 1} \\to M_n$. Assume \\begin{enumerate} \\item the transition maps $A_{n + 1} \\to A_n$ are surjective with locally nilpotent kernels, \\item $M_1$ is a finite projective $A_1$-module, \\item $M_n$ is a finite flat $A_n$-module, and \\item the maps induce isomorphisms $M_{n + 1} \\otimes_{A_{n + 1}} A_n \\to M_n$. \\end{enumerate} Then $M = \\lim M_n$ is a finite projective $A$-module and $M \\otimes_A A_n \\to M_n$ is an isomorphism for all $n$."}
+{"_id": "9881", "title": "more-algebra-lemma-absolutely-integrally-closed", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item $A$ is absolutely integrally closed, and \\item any monic $f \\in A[T]$ has a root in $A$. \\end{enumerate}"}
+{"_id": "9882", "title": "more-algebra-lemma-absolutely-integrally-closed-quotient-localization", "text": "Let $A$ be absolutely integrally closed. \\begin{enumerate} \\item Any quotient ring $A/I$ of $A$ is absolutely integrally closed. \\item Any localization $S^{-1}A$ is absolutely integrally closed. \\end{enumerate}"}
+{"_id": "9883", "title": "more-algebra-lemma-integrally-closed-in-absolutely-integrally-closed", "text": "Let $A$ be a ring. Let $S \\subset A$ be a multiplicative subset. If $S^{-1}A$ is absolutely integrally closed and $A \\subset S^{-1}A$ is integrally closed in $S^{-1}A$, then $A$ is absolutely integrally closed."}
+{"_id": "9887", "title": "more-algebra-lemma-absolutely-integrally-closed-henselian-pair", "text": "Let $A$ be absolutely integrally closed. Let $I \\subset A$ be an ideal. Then $(A, I)$ is a henselian pair if (and only if) the following conditions hold \\begin{enumerate} \\item $I$ is contained in the Jacobson radical of $A$, \\item $A \\to A/I$ induces a bijection on idempotents. \\end{enumerate}"}
+{"_id": "9888", "title": "more-algebra-lemma-auto-ass-implies-P", "text": "An auto-associated ring $R$ has the following property: (P) Every proper finitely generated ideal $I \\subset R$ has a nonzero annihilator."}
+{"_id": "9889", "title": "more-algebra-lemma-P-universally-injective", "text": "Let $R$ be a ring having property (P) of Lemma \\ref{lemma-auto-ass-implies-P}. Let $u : N \\to M$ be a homomorphism of projective $R$-modules. Then $u$ is universally injective if and only if $u$ is injective."}
+{"_id": "9890", "title": "more-algebra-lemma-P-fPD-zero", "text": "Let $R$ be a ring. The following are equivalent \\begin{enumerate} \\item $R$ has property (P) of Lemma \\ref{lemma-auto-ass-implies-P}, \\item any injective map of projective $R$-modules is universally injective, \\item if $u : N \\to M$ is injective and $N$, $M$ are finite projective $R$-modules then $\\Coker(u)$ is a finite projective $R$-module, \\item if $N \\subset M$ and $N$, $M$ are finite projective as $R$-modules, then $N$ is a direct summand of $M$, and \\item any injective map $R \\to R^{\\oplus n}$ is a split injection. \\end{enumerate}"}
+{"_id": "9891", "title": "more-algebra-lemma-exact-length-1", "text": "Let $R$ be a ring. Suppose that $\\varphi : R^m \\to R^n$ is a map of finite free modules. The following are equivalent \\begin{enumerate} \\item $\\varphi$ is injective, \\item the rank of $\\varphi$ is $m$ and the annihilator of $I(\\varphi)$ in $R$ is zero. \\end{enumerate} If $R$ is Noetherian these are also equivalent to \\begin{enumerate} \\item[(3)] the rank of $\\varphi$ is $m$ and either $I(\\varphi) = R$ or it contains a nonzerodivisor. \\end{enumerate} Here the rank of $\\varphi$ and $I(\\varphi)$ are defined as in Algebra, Definition \\ref{algebra-definition-rank}."}
+{"_id": "9892", "title": "more-algebra-lemma-coker-injective-free", "text": "Let $R$ be a ring. Suppose that $\\varphi : R^n \\to R^n$ be an injective map of finite free modules of the same rank. Then $\\Hom_R(\\Coker(\\varphi), R) = 0$."}
+{"_id": "9893", "title": "more-algebra-lemma-intersection-flat", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $I_1$, $I_2$ be ideals of $R$. If $M/I_1M$ is flat over $R/I_1$ and $M/I_2M$ is flat over $R/I_2$, then $M/(I_1 \\cap I_2)M$ is flat over $R/(I_1 \\cap I_2)$."}
+{"_id": "9894", "title": "more-algebra-lemma-flattening-artinian", "text": "Let $R$ be an Artinian ring. Let $M$ be an $R$-module. Then there exists a smallest ideal $I \\subset R$ such that $M/IM$ is flat over $R/I$."}
+{"_id": "9895", "title": "more-algebra-lemma-flattening-artinian-universal-property", "text": "Let $R$ be an Artinian ring. Let $M$ be an $R$-module. Let $I \\subset R$ be the smallest ideal $I \\subset R$ such that $M/IM$ is flat over $R/I$. Then $I$ has the following universal property: For every ring map $\\varphi : R \\to R'$ we have $$ R' \\otimes_R M\\text{ is flat over }R' \\Leftrightarrow \\text{we have }\\varphi(I) = 0. $$"}
+{"_id": "9896", "title": "more-algebra-lemma-base-change-flat-at-primes-over", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. Let $M$ be an $S$-module. Let $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal. If (\\ref{equation-flat-at-primes-over}) holds for $(R \\to S, I, M)$, then (\\ref{equation-flat-at-primes-over}) holds for $(R' \\to S \\otimes_R R', I', M \\otimes_R R')$."}
+{"_id": "9898", "title": "more-algebra-lemma-limit-preserving-flat-at-primes-over", "text": "Let $R \\to S$ be a ring map of finite presentation. Let $M$ be an $S$-module of finite presentation. Let $R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ be a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$ be ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all $\\mu \\geq \\lambda$ and set $I' = \\colim_\\lambda I_\\lambda$. If (\\ref{equation-flat-at-primes-over}) holds for $(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then there exists a $\\lambda \\in \\Lambda$ such that (\\ref{equation-flat-at-primes-over}) holds for $(R_\\lambda \\to S \\otimes_R R_\\lambda, I_\\lambda, M \\otimes_R R_\\lambda)$."}
+{"_id": "9900", "title": "more-algebra-lemma-flat-descent-flat-at-primes", "text": "In Situation \\ref{situation-flattening-general} let $R' \\to R''$ be an $R$-algebra map. Let $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals. Assume \\begin{enumerate} \\item the map $V(I'') \\to V(I')$ induced by $\\Spec(R'') \\to \\Spec(R')$ is surjective, and \\item $R''_{\\mathfrak p''}$ is flat over $R'$ for all primes $\\mathfrak p'' \\in V(I'')$. \\end{enumerate} If (\\ref{equation-flat-at-primes}) holds for $(R'', I'')$, then (\\ref{equation-flat-at-primes}) holds for $(R', I')$."}
+{"_id": "9901", "title": "more-algebra-lemma-limit-preserving-flat-at-primes", "text": "In Situation \\ref{situation-flattening-general} assume $R \\to S$ is essentially of finite presentation and $M$ is an $S$-module of finite presentation. Let $R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ be a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$ be ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all $\\mu \\geq \\lambda$ and set $I' = \\colim_\\lambda I_\\lambda$. If (\\ref{equation-flat-at-primes}) holds for $(R', I')$, then there exists a $\\lambda \\in \\Lambda$ such that (\\ref{equation-flat-at-primes}) holds for $(R_\\lambda, I_\\lambda)$."}
+{"_id": "9903", "title": "more-algebra-lemma-flattening-complete-local-noetherian", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $(R, \\mathfrak m)$ is a complete local Noetherian ring, \\item $S$ is a Noetherian ring, and \\item $M$ is finite over $S$. \\end{enumerate} Then there exists an ideal $I \\subset \\mathfrak m$ such that \\begin{enumerate} \\item $(M/IM)_{\\mathfrak q}$ is flat over $R/I$ for all primes $\\mathfrak q$ of $S/IS$ lying over $\\mathfrak m$, and \\item if $J \\subset R$ is an ideal such that $(M/JM)_{\\mathfrak q}$ is flat over $R/J$ for all primes $\\mathfrak q$ lying over $\\mathfrak m$, then $I \\subset J$. \\end{enumerate} In other words, $I$ is the smallest ideal of $R$ such that (\\ref{equation-flat-at-primes-over}) holds for $(\\overline{R} \\to \\overline{S}, \\overline{\\mathfrak m}, \\overline{M})$ where $\\overline{R} = R/I$, $\\overline{S} = S/IS$, $\\overline{\\mathfrak m} = \\mathfrak m/I$ and $\\overline{M} = M/IM$."}
+{"_id": "9904", "title": "more-algebra-lemma-flattening-complete-local-noetherian-property-by-finite-type", "text": "With notation $R \\to S$, $M$, and $I$ and assumptions as in Lemma \\ref{lemma-flattening-complete-local-noetherian}. Consider a local homomorphism of local rings $\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$ such that $R'$ is Noetherian. Then the following are equivalent \\begin{enumerate} \\item condition (\\ref{equation-flat-at-primes-over}) holds for $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and \\item $\\varphi(I) = 0$. \\end{enumerate}"}
+{"_id": "9905", "title": "more-algebra-lemma-flattening-complete-local-universal-property", "text": "With notation $R \\to S$, $M$, and $I$ and assumptions as in Lemma \\ref{lemma-flattening-complete-local-noetherian}. In addition assume that $R \\to S$ is of finite type. Then for any local homomorphism of local rings $\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$ the following are equivalent \\begin{enumerate} \\item condition (\\ref{equation-flat-at-primes-over}) holds for $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and \\item $\\varphi(I) = 0$. \\end{enumerate}"}
+{"_id": "9906", "title": "more-algebra-lemma-have-one-root", "text": "Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. Let $\\alpha \\in R$ be such that $P(\\alpha) = 0$. Then $P(T) = (T - \\alpha)Q(T)$ for some monic polynomial $Q(T) \\in R[T]$."}
+{"_id": "9907", "title": "more-algebra-lemma-adjoin-one-root", "text": "Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients in $R$. There exists a finite free ring map $R \\to R'$ such that $P(T) = (T - \\alpha)Q(T)$ for some $\\alpha \\in R'$ and some monic polynomial $Q(T) \\in R'[T]$."}
+{"_id": "9908", "title": "more-algebra-lemma-finite-split", "text": "Let $R \\to S$ be a finite ring map. There exists a finite free ring extension $R \\subset R'$ such that $S \\otimes_R R'$ is a quotient of a ring of the form $$ R'[T_1, \\ldots, T_n]/(P_1(T_1), \\ldots, P_n(T_n)) $$ with $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some $\\alpha_{ij} \\in R'$."}
+{"_id": "9909", "title": "more-algebra-lemma-split-image", "text": "Let $R$ be a ring. Let $S = R[T_1, \\ldots, T_n]/J$. Assume $J$ contains elements of the form $P_i(T_i)$ with $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some $\\alpha_{ij} \\in R$. For $\\underline{k} = (k_1, \\ldots, k_n)$ with $1 \\leq k_i \\leq d_i$ consider the ring map $$ \\Phi_{\\underline{k}} : R[T_1, \\ldots, T_n] \\to R, \\quad T_i \\longmapsto \\alpha_{ik_i} $$ Set $J_{\\underline{k}} = \\Phi_{\\underline{k}}(J)$. Then the image of $\\Spec(S) \\to \\Spec(R)$ is equal to $V(\\bigcap J_{\\underline{k}})$."}
+{"_id": "9910", "title": "more-algebra-lemma-descent-flatness-injective-finite-Noetherian-rings", "text": "Let $R \\to S$ be a finite injective homomorphism of Noetherian rings. Let $M$ be an $R$-module. If $M \\otimes_R S$ is a flat $S$-module, then $M$ is a flat $R$-module."}
+{"_id": "9912", "title": "more-algebra-lemma-torsion", "text": "Let $R$ be a domain. Let $M$ be an $R$-module. The set of torsion elements of $M$ forms a submodule $M_{tors} \\subset M$. The quotient module $M/M_{tors}$ is torsion free."}
+{"_id": "9913", "title": "more-algebra-lemma-localize-torsion", "text": "Let $R$ be a domain. Let $M$ be a torsion free $R$-module. For any multiplicative set $S \\subset R$ the module $S^{-1}M$ is a torsion free $S^{-1}R$-module."}
+{"_id": "9914", "title": "more-algebra-lemma-flat-pullback-torsion", "text": "Let $R \\to R'$ be a flat homomorphism of domains. If $M$ is a torsion free $R$-module, then $M \\otimes_R R'$ is a torsion free $R'$-module."}
+{"_id": "9915", "title": "more-algebra-lemma-extension-torsion-free", "text": "Let $R$ be a domain. Let $0 \\to M \\to M' \\to M'' \\to 0$ be a short exact sequence of $R$-modules. If $M$ and $M''$ are torsion free, then $M'$ is torsion free."}
+{"_id": "9916", "title": "more-algebra-lemma-check-torsion", "text": "Let $R$ be a domain. Let $M$ be an $R$-module. Then $M$ is torsion free if and only if $M_\\mathfrak m$ is a torsion free $R_\\mathfrak m$-module for all maximal ideals $\\mathfrak m$ of $R$."}
+{"_id": "9917", "title": "more-algebra-lemma-finite-torsion-free-submodule-free", "text": "Let $R$ be a domain. Let $M$ be a finite $R$-module. Then $M$ is torsion free if and only if $M$ is a submodule of a finite free module."}
+{"_id": "9918", "title": "more-algebra-lemma-torsion-free-finite-noetherian-domain", "text": "Let $R$ be a Noetherian domain. Let $M$ be a nonzero finite $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is torsion free, \\item $M$ is a submodule of a finite free module, \\item $(0)$ is the only associated prime of $M$, \\item $(0)$ is in the support of $M$ and $M$ has property $(S_1)$, and \\item $(0)$ is in the support of $M$ and $M$ has no embedded associated prime. \\end{enumerate}"}
+{"_id": "9919", "title": "more-algebra-lemma-flat-torsion-free", "text": "Let $R$ be a domain. Any flat $R$-module is torsion free."}
+{"_id": "9920", "title": "more-algebra-lemma-valuation-ring-torsion-free-flat", "text": "Let $A$ be a valuation ring. An $A$-module $M$ is flat over $A$ if and only if $M$ is torsion free."}
+{"_id": "9921", "title": "more-algebra-lemma-dedekind-torsion-free-flat", "text": "Let $A$ be a Dedekind domain (for example a discrete valuation ring or more generally a PID). \\begin{enumerate} \\item An $A$-module is flat if and only if it is torsion free. \\item A finite torsion free $A$-module is finite locally free. \\item A finite torsion free $A$-module is finite free if $A$ is a PID. \\end{enumerate}"}
+{"_id": "9922", "title": "more-algebra-lemma-hom-into-torsion-free", "text": "Let $R$ be a domain. Let $M$, $N$ be $R$-modules. If $N$ is torsion free, so is $\\Hom_R(M, N)$."}
+{"_id": "9923", "title": "more-algebra-lemma-reflexive-torsion-free", "text": "Let $R$ be a domain and let $M$ be an $R$-module. \\begin{enumerate} \\item If $M$ is reflexive, then $M$ is torsion free. \\item If $M$ is finite, then $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ is injective if and only if $M$ is torsion free \\end{enumerate}"}
+{"_id": "9924", "title": "more-algebra-lemma-cokernel-map-double-dual-dvr", "text": "Let $R$ be a discrete valuation ring and let $M$ be a finite $R$-module. Then the map $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ is surjective."}
+{"_id": "9925", "title": "more-algebra-lemma-check-reflexive", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent: \\begin{enumerate} \\item $M$ is reflexive, \\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module for all primes $\\mathfrak p \\subset R$, and \\item $M_\\mathfrak m$ is a reflexive $R_\\mathfrak m$-module for all maximal ideals $\\mathfrak m$ of $R$. \\end{enumerate}"}
+{"_id": "9926", "title": "more-algebra-lemma-sequence-reflexive", "text": "Let $R$ be a Noetherian domain. Let $0 \\to M \\to M' \\to M''$ an exact sequence of finite $R$-modules. If $M'$ is reflexive and $M''$ is torsion free, then $M$ is reflexive."}
+{"_id": "9927", "title": "more-algebra-lemma-characterize-reflexive", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is reflexive, \\item there exists a short exact sequence $0 \\to M \\to F \\to N \\to 0$ with $F$ finite free and $N$ torsion free. \\end{enumerate}"}
+{"_id": "9928", "title": "more-algebra-lemma-flat-pullback-reflexive", "text": "Let $R \\to R'$ be a flat homomorphism of Noetherian domains. If $M$ is a finite reflexive $R$-module, then $M \\otimes_R R'$ is a finite reflexive $R'$-module."}
+{"_id": "9929", "title": "more-algebra-lemma-dual-reflexive", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. Let $N$ be a finite reflexive $R$-module. Then $\\Hom_R(M, N)$ is reflexive."}
+{"_id": "9930", "title": "more-algebra-lemma-hom-into-depth", "text": "Let $R$ be a Noetherian local ring. Let $M$, $N$ be finite $R$-modules. \\begin{enumerate} \\item If $N$ has depth $\\geq 1$, then $\\Hom_R(M, N)$ has depth $\\geq 1$. \\item If $N$ has depth $\\geq 2$, then $\\Hom_R(M, N)$ has depth $\\geq 2$. \\end{enumerate}"}
+{"_id": "9931", "title": "more-algebra-lemma-hom-into-S2", "text": "Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules. \\begin{enumerate} \\item If $N$ has property $(S_1)$, then $\\Hom_R(M, N)$ has property $(S_1)$. \\item If $N$ has property $(S_2)$, then $\\Hom_R(M, N)$ has property $(S_2)$. \\item If $R$ is a domain, $N$ is torsion free and $(S_2)$, then $\\Hom_R(M, N)$ is torsion free and has property $(S_2)$. \\end{enumerate}"}
+{"_id": "9932", "title": "more-algebra-lemma-check-injective-on-ass", "text": "Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of $R$-modules. Assume that for every prime $\\mathfrak p$ of $R$ at least one of the following happens \\begin{enumerate} \\item $M_\\mathfrak p \\to N_\\mathfrak p$ is injective, or \\item $\\mathfrak p \\not \\in \\text{Ass}(M)$. \\end{enumerate} Then $\\varphi$ is injective."}
+{"_id": "9933", "title": "more-algebra-lemma-check-isomorphism-via-depth-and-ass", "text": "Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of $R$-modules. Assume $M$ is finite and that for every prime $\\mathfrak p$ of $R$ one of the following happens \\begin{enumerate} \\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or \\item $\\text{depth}(M_\\mathfrak p) \\geq 2$ and $\\mathfrak p \\not \\in \\text{Ass}(N)$. \\end{enumerate} Then $\\varphi$ is an isomorphism."}
+{"_id": "9934", "title": "more-algebra-lemma-isom-depth-2-torsion-free", "text": "Let $R$ be a Noetherian domain. Let $\\varphi : M \\to N$ be a map of $R$-modules. Assume $M$ is finite, $N$ is torsion free, and that for every prime $\\mathfrak p$ of $R$ one of the following happens \\begin{enumerate} \\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or \\item $\\text{depth}(M_\\mathfrak p) \\geq 2$. \\end{enumerate} Then $\\varphi$ is an isomorphism."}
+{"_id": "9935", "title": "more-algebra-lemma-reflexive-depth-2", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is reflexive, \\item for every prime $\\mathfrak p$ of $R$ one of the following happens \\begin{enumerate} \\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module, or \\item $\\text{depth}(M_\\mathfrak p) \\geq 2$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "9936", "title": "more-algebra-lemma-reflexive-S2", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite reflexive $R$-module. Let $\\mathfrak p \\subset R$ be a prime ideal. \\begin{enumerate} \\item If $\\text{depth}(R_\\mathfrak p) \\geq 2$, then $\\text{depth}(M_\\mathfrak p) \\geq 2$. \\item If $R$ is $(S_2)$, then $M$ is $(S_2)$. \\end{enumerate}"}
+{"_id": "9937", "title": "more-algebra-lemma-reflexive-over-normal", "text": "Let $R$ be a Noetherian normal domain with fraction field $K$. Let $M$ be a finite $R$-module. The following are equivalent \\begin{enumerate} \\item $M$ is reflexive, \\item $M$ is torsion free and has property $(S_2)$, \\item $M$ is torsion free and $M = \\bigcap_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}$ where the intersection happens in $M_K = M \\otimes_R K$. \\end{enumerate}"}
+{"_id": "9938", "title": "more-algebra-lemma-describe-reflexive-hull", "text": "Let $R$ be a Noetherian normal domain. Let $M$ be a finite $R$-module. Then the reflexive hull of $M$ is the intersection $$ M^{**} = \\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}/(M_\\mathfrak p)_{tors} = \\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1} (M/M_{tors})_\\mathfrak p $$ taken in $M \\otimes_R K$."}
+{"_id": "9939", "title": "more-algebra-lemma-integral-closure-reflexive", "text": "Let $A$ be a Noetherian normal domain with fraction field $K$. Let $L$ be a finite extension of $K$. If the integral closure $B$ of $A$ in $L$ is finite over $A$, then $B$ is reflexive as an $A$-module."}
+{"_id": "9940", "title": "more-algebra-lemma-content-finitely-generated", "text": "Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$. The content ideal of $x$, if it exists, is finitely generated."}
+{"_id": "9941", "title": "more-algebra-lemma-equal-content", "text": "Let $(A, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be a map of flat $A$-modules such that $\\overline{u} : M/\\mathfrak m M \\to N/\\mathfrak m N$ is injective. If $x \\in M$ has content ideal $I$, then $u(x)$ has content ideal $I$ as well."}
+{"_id": "9942", "title": "more-algebra-lemma-content-exists-flat-Mittag-Leffler", "text": "Let $A$ be a ring. Let $M$ be a flat Mittag-Leffler module. Then every element of $M$ has a content ideal."}
+{"_id": "9943", "title": "more-algebra-lemma-flat-finite-type-finite-presentation-local-module", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]$ be a polynomial ring over $R$. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item there exist finitely many primes $\\mathfrak p_1, \\ldots, \\mathfrak p_m$ of $R$ such that the map $R \\to \\prod R_{\\mathfrak p_j}$ is injective, \\item $M$ is a finite $S$-module, \\item $M$ flat over $R$, and \\item for every prime $\\mathfrak p$ of $R$ the module $M_{\\mathfrak p}$ is of finite presentation over $S_{\\mathfrak p}$. \\end{enumerate} Then $M$ is of finite presentation over $S$."}
+{"_id": "9944", "title": "more-algebra-lemma-flat-finite-type-finite-presentation-local", "text": "Let $R \\to S$ be a ring homomorphism. Assume \\begin{enumerate} \\item there exist finitely many primes $\\mathfrak p_1, \\ldots, \\mathfrak p_m$ of $R$ such that the map $R \\to \\prod R_{\\mathfrak p_j}$ is injective, \\item $R \\to S$ is of finite type, \\item $S$ flat over $R$, and \\item for every prime $\\mathfrak p$ of $R$ the ring $S_{\\mathfrak p}$ is of finite presentation over $R_{\\mathfrak p}$. \\end{enumerate} Then $S$ is of finite presentation over $R$."}
+{"_id": "9945", "title": "more-algebra-lemma-flat-graded-finite-type-finite-presentation-module", "text": "Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]$ be a graded polynomial algebra over $R$, i.e., $\\deg(x_i) > 0$ but not necessarily equal to $1$. Let $M$ be a graded $S$-module. Assume \\begin{enumerate} \\item $R$ is a local ring, \\item $M$ is a finite $S$-module, and \\item $M$ is flat over $R$. \\end{enumerate} Then $M$ is finitely presented as an $S$-module."}
+{"_id": "9946", "title": "more-algebra-lemma-flat-graded-finite-type-finite-presentation", "text": "Let $R$ be a ring. Let $S = \\bigoplus_{n \\geq 0} S_n$ be a graded $R$-algebra. Let $M = \\bigoplus_{d \\in \\mathbf{Z}} M_d$ be a graded $S$-module. Assume $S$ is finitely generated as an $R$-algebra, assume $S_0$ is a finite $R$-algebra, and assume there exist finitely many primes $\\mathfrak p_j$, $i = 1, \\ldots, m$ such that $R \\to \\prod R_{\\mathfrak p_j}$ is injective. \\begin{enumerate} \\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra. \\item If $M$ is flat as an $R$-module and finite as an $S$-module, then $M$ is finitely presented as an $S$-module. \\end{enumerate}"}
+{"_id": "9947", "title": "more-algebra-lemma-flat-finite-type-valuation-ring-finite-presentation", "text": "\\begin{reference} \\cite[Theorem 3]{Nagata-Finitely} \\end{reference} Let $A$ be a valuation ring. Let $A \\to B$ be a ring map of finite type. Let $M$ be a finite $B$-module. \\begin{enumerate} \\item If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra. \\item If $M$ is flat as an $A$-module, then $M$ is finitely presented as a $B$-module. \\end{enumerate}"}
+{"_id": "9949", "title": "more-algebra-lemma-blowup-fitting-ideal", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$ and $I = \\text{Fit}_k(M)$. For every $a \\in I$ with $R' = R[\\frac{I}{a}]$ the strict transform $$ M' = (M \\otimes_R R')/a\\text{-power torsion} $$ has $\\text{Fit}_k(M') = R'$."}
+{"_id": "9950", "title": "more-algebra-lemma-blowup-fitting-ideal-locally-free", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$ and $I = \\text{Fit}_k(M)$. Asssume that $M_\\mathfrak p$ is free of rank $k$ for every $\\mathfrak p \\not \\in V(I)$. Then for every $a \\in I$ with $R' = R[\\frac{I}{a}]$ the strict transform $$ M' = (M \\otimes_R R')/a\\text{-power torsion} $$ is locally free of rank $k$."}
+{"_id": "9951", "title": "more-algebra-lemma-blowup-module", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $f \\in R$ be an element such that $M_f$ is finite locally free of rank $r$. Then there exists a finitely generated ideal $I \\subset R$ with $V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$ the strict transform $$ M' = (M \\otimes_R R')/a\\text{-power torsion} $$ is locally free of rank $r$."}
+{"_id": "9952", "title": "more-algebra-lemma-ui-completion-direct-sum-into-product", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian and complete with respect to $I$. There is a canonical map $$ \\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge \\longrightarrow \\prod\\nolimits_{\\alpha \\in A} R $$ from the $I$-adic completion of the direct sum into the product which is universally injective."}
+{"_id": "9953", "title": "more-algebra-lemma-completed-direct-sum-flat", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian. The completion $(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$ is a flat $R$-module."}
+{"_id": "9954", "title": "more-algebra-lemma-tor-strictly-pro-zero", "text": "\\begin{reference} This is \\cite[Lemma 9.9]{quillenhomology}; note that the author forgot the word ``strict'' in the statement although it was clearly intended. \\end{reference} Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $M$ be a finite $A$-module. For every $p > 0$ there exists a $c > 0$ such that $\\text{Tor}_p^A(M, A/I^n) \\to \\text{Tor}_p^A(M, A/I^{n - c})$ is zero for all $n \\geq c$."}
+{"_id": "9955", "title": "more-algebra-lemma-limit-flat", "text": "Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $(M_n)$ be an inverse system of $A$-modules such that \\begin{enumerate} \\item $M_n$ is a flat $A/I^n$-module, \\item $M_{n + 1} \\to M_n$ is surjective. \\end{enumerate} Then $M = \\lim M_n$ is a flat $A$-module and $Q \\otimes_A M = \\lim Q \\otimes_A M_n$ for every finite $A$-module $Q$."}
+{"_id": "9956", "title": "more-algebra-lemma-flat-after-completion", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Assume \\begin{enumerate} \\item $I$ is finitely generated, \\item $R/I$ is Noetherian, \\item $M/IM$ is flat over $R/I$, \\item $\\text{Tor}_1^R(M, R/I) = 0$. \\end{enumerate} Then the $I$-adic completion $R^\\wedge$ is a Noetherian ring and $M^\\wedge$ is flat over $R^\\wedge$."}
+{"_id": "9957", "title": "more-algebra-lemma-functorial", "text": "Let $\\varphi : E \\to R$ and $\\varphi' : E' \\to R$ be $R$-module maps. Let $\\psi : E \\to E'$ be an $R$-module map such that $\\varphi' \\circ \\psi = \\varphi$. Then $\\psi$ induces a homomorphism of differential graded algebras $K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi')$."}
+{"_id": "9958", "title": "more-algebra-lemma-change-basis", "text": "Let $f_1, \\ldots, f_r \\in R$ be a sequence. Let $(x_{ij})$ be an invertible $r \\times r$-matrix with coefficients in $R$. Then the complexes $K_\\bullet(f_\\bullet)$ and $$ K_\\bullet(\\sum x_{1j}f_j, \\sum x_{2j}f_j, \\ldots, \\sum x_{rj}f_j) $$ are isomorphic."}
+{"_id": "9959", "title": "more-algebra-lemma-homotopy-koszul-abstract", "text": "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. Let $e \\in E$ with image $f = \\varphi(e)$ in $R$. Then $$ f = de + ed $$ as endomorphisms of $K_\\bullet(\\varphi)$."}
+{"_id": "9960", "title": "more-algebra-lemma-homotopy-koszul", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be a sequence. Multiplication by $f_i$ on $K_\\bullet(f_\\bullet)$ is homotopic to zero, and in particular the cohomology modules $H_i(K_\\bullet(f_\\bullet))$ are annihilated by the ideal $(f_1, \\ldots, f_r)$."}
+{"_id": "9961", "title": "more-algebra-lemma-cone-koszul-abstract", "text": "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. Let $f \\in R$. Set $E' = E \\oplus R$ and define $\\varphi' : E' \\to R$ by $\\varphi$ on $E$ and multiplication by $f$ on $R$. The complex $K_\\bullet(\\varphi')$ is isomorphic to the cone of the map of complexes $$ f : K_\\bullet(\\varphi) \\longrightarrow K_\\bullet(\\varphi). $$"}
+{"_id": "9962", "title": "more-algebra-lemma-cone-koszul", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r$ be a sequence of elements of $R$. The complex $K_\\bullet(f_1, \\ldots, f_r)$ is isomorphic to the cone of the map of complexes $$ f_r : K_\\bullet(f_1, \\ldots, f_{r - 1}) \\longrightarrow K_\\bullet(f_1, \\ldots, f_{r - 1}). $$"}
+{"_id": "9963", "title": "more-algebra-lemma-cone-squared", "text": "Let $R$ be a ring. Let $A_\\bullet$ be a complex of $R$-modules. Let $f, g \\in R$. Let $C(f)_\\bullet$ be the cone of $f : A_\\bullet \\to A_\\bullet$. Define similarly $C(g)_\\bullet$ and $C(fg)_\\bullet$. Then $C(fg)_\\bullet$ is homotopy equivalent to the cone of a map $$ C(f)_\\bullet[1] \\longrightarrow C(g)_\\bullet $$"}
+{"_id": "9964", "title": "more-algebra-lemma-koszul-mult-abstract", "text": "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. Let $f, g \\in R$. Set $E' = E \\oplus R$ and define $\\varphi'_f, \\varphi'_g, \\varphi'_{fg} : E' \\to R$ by $\\varphi$ on $E$ and multiplication by $f, g, fg$ on $R$. The complex $K_\\bullet(\\varphi'_{fg})$ is isomorphic to the cone of a map of complexes $$ K_\\bullet(\\varphi'_f)[1] \\longrightarrow K_\\bullet(\\varphi'_g). $$"}
+{"_id": "9965", "title": "more-algebra-lemma-koszul-mult", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_{r - 1}$ be a sequence of elements of $R$. Let $f, g \\in R$. The complex $K_\\bullet(f_1, \\ldots, f_{r - 1}, fg)$ is homotopy equivalent to the cone of a map of complexes $$ K_\\bullet(f_1, \\ldots, f_{r - 1}, f)[1] \\longrightarrow K_\\bullet(f_1, \\ldots, f_{r - 1}, g) $$"}
+{"_id": "9966", "title": "more-algebra-lemma-join-sequences-koszul-complex", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r$, $g_1, \\ldots, g_s$ be elements of $R$. Then there is an isomorphism of Koszul complexes $$ K_\\bullet(R, f_1, \\ldots, f_r, g_1, \\ldots, g_s) = \\text{Tot}(K_\\bullet(R, f_1, \\ldots, f_r) \\otimes_R K_\\bullet(R, g_1, \\ldots, g_s)). $$"}
+{"_id": "9967", "title": "more-algebra-lemma-extended-alternating-is-complex", "text": "The extended alternating {\\v C}ech complexes defined above are complexes of $R$-modules."}
+{"_id": "9968", "title": "more-algebra-lemma-extended-alternating-form-module", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module. The extended alternating {\\v C}ech complex of $M$ is the tensor product over $R$ of $M$ with the extended alternating {\\v C}ech complex of $R$."}
+{"_id": "9969", "title": "more-algebra-lemma-extended-alternating-base-change", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module. Let $R \\to S$ be a ring map, denote $g_1, \\ldots, g_r \\in S$ the images of $f_1, \\ldots, f_r$, and set $N = M \\otimes_R S$. The extended alternating {\\v C}ech complex constructed using $S$, $g_1, \\ldots, g_r$, and $N$ is the tensor product of the extended alternating {\\v C}ech complex of $M$ with $S$ over $R$."}
+{"_id": "9970", "title": "more-algebra-lemma-extended-alternating-homotopy-zero", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module. If there exists an $i \\in \\{1, \\ldots, r\\}$ such that $f_i$ is a unit, then the extended alternating {\\v C}ech complex of $M$ is homotopy equivalent to $0$."}
+{"_id": "9971", "title": "more-algebra-lemma-extended-alternating-torsion", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module. Let $H^q$ be the $q$th cohomology module of the extended alternation {\\v C}ech complex of $M$. Then \\begin{enumerate} \\item $H^q = 0$ if $q \\not \\in [0, r]$, \\item for $x \\in H^i$ there exists an $n \\geq 1$ such that $f_i^n x = 0$ for $i = 1, \\ldots, r$, \\item the support of $H^q$ is contained in $V(f_1, \\ldots, f_r)$, \\item if there is an $f \\in (f_1, \\ldots, f_r)$ which acts invertibly on $M$, then $H^q = 0$. \\end{enumerate}"}
+{"_id": "9972", "title": "more-algebra-lemma-extended-alternating-Cech-is-colimit-koszul", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. The extended alternating {\\v C}ech complex $$ R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to \\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to \\ldots \\to R_{f_1\\ldots f_r} $$ is a colimit of the Koszul complexes $K(R, f_1^n, \\ldots, f_r^n)$; see proof for a precise statement."}
+{"_id": "9973", "title": "more-algebra-lemma-regular-koszul-regular", "text": "An $M$-regular sequence is $M$-Koszul-regular. A regular sequence is Koszul-regular."}
+{"_id": "9974", "title": "more-algebra-lemma-koszul-regular-H1-regular", "text": "A $M$-Koszul-regular sequence is $M$-$H_1$-regular. A Koszul-regular sequence is $H_1$-regular."}
+{"_id": "9975", "title": "more-algebra-lemma-mult-koszul-regular", "text": "Let $f_1, \\ldots, f_{r - 1} \\in R$ be a sequence and $f, g \\in R$. Let $M$ be an $R$-module. \\begin{enumerate} \\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$ are $M$-$H_1$-regular then $f_1, \\ldots, f_{r - 1}, fg$ is $M$-$H_1$-regular too. \\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$ are $M$-Koszul-regular then $f_1, \\ldots, f_{r - 1}, fg$ is $M$-Koszul-regular too. \\end{enumerate}"}
+{"_id": "9976", "title": "more-algebra-lemma-koszul-regular-flat-base-change", "text": "Let $\\varphi : R \\to S$ be a flat ring map. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module and set $N = M \\otimes_R S$. \\begin{enumerate} \\item If $f_1, \\ldots, f_r$ in $R$ is an $M$-$H_1$-regular sequence, then $\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-$H_1$-regular sequence in $S$. \\item If $f_1, \\ldots, f_r$ is an $M$-Koszul-regular sequence in $R$, then $\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-Koszul-regular sequence in $S$. \\end{enumerate}"}
+{"_id": "9977", "title": "more-algebra-lemma-H1-regular-quasi-regular", "text": "An $M$-$H_1$-regular sequence is $M$-quasi-regular."}
+{"_id": "9978", "title": "more-algebra-lemma-noetherian-finite-all-equivalent", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \\ldots, f_r \\in \\mathfrak m$. The following are equivalent \\begin{enumerate} \\item $f_1, \\ldots, f_r$ is an $M$-regular sequence, \\item $f_1, \\ldots, f_r$ is a $M$-Koszul-regular sequence, \\item $f_1, \\ldots, f_r$ is an $M$-$H_1$-regular sequence, \\item $f_1, \\ldots, f_r$ is an $M$-quasi-regular sequence. \\end{enumerate} In particular the sequence $f_1, \\ldots, f_r$ is a regular sequence in $R$ if and only if it is a Koszul regular sequence, if and only if it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence."}
+{"_id": "9979", "title": "more-algebra-lemma-H1-regular-in-quotient", "text": "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Let $g_1, \\ldots, g_m$ be a sequence in $A$ whose image in $A/I$ is $H_1$-regular. Then $I \\cap (g_1, \\ldots, g_m) = I(g_1, \\ldots, g_m)$."}
+{"_id": "9980", "title": "more-algebra-lemma-conormal-sequence-H1-regular", "text": "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals. Assume that $J/I \\subset A/I$ is generated by an $H_1$-regular sequence. Then $I \\cap J^2 = IJ$."}
+{"_id": "9981", "title": "more-algebra-lemma-join-quasi-regular-H1-regular", "text": "Let $A$ be a ring. Let $I$ be an ideal generated by a quasi-regular sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form an $H_1$-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is a quasi-regular sequence in $A$."}
+{"_id": "9982", "title": "more-algebra-lemma-join-H1-regular-sequences", "text": "Let $A$ be a ring. Let $I$ be an ideal generated by an $H_1$-regular sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form an $H_1$-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is an $H_1$-regular sequence in $A$."}
+{"_id": "9983", "title": "more-algebra-lemma-truncate-H1-regular", "text": "Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$ be an $H_1$-regular sequence. Then the images $\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$ form an $H_1$-regular sequence."}
+{"_id": "9984", "title": "more-algebra-lemma-join-koszul-regular-sequences", "text": "Let $A$ be a ring. Let $I$ be an ideal generated by a Koszul-regular sequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be elements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a Koszul-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is a Koszul-regular sequence in $A$."}
+{"_id": "9985", "title": "more-algebra-lemma-truncate-koszul-regular", "text": "Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$. If both $f_1, \\ldots, f_n$ and $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ are Koszul-regular sequences in $A$, then $\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$ form a Koszul-regular sequence."}
+{"_id": "9986", "title": "more-algebra-lemma-independence-of-generators", "text": "Let $R$ be a ring. Let $I$ be an ideal generated by $f_1, \\ldots, f_r \\in R$. \\begin{enumerate} \\item If $I$ can be generated by a quasi-regular sequence of length $r$, then $f_1, \\ldots, f_r$ is a quasi-regular sequence. \\item If $I$ can be generated by an $H_1$-regular sequence of length $r$, then $f_1, \\ldots, f_r$ is an $H_1$-regular sequence. \\item If $I$ can be generated by a Koszul-regular sequence of length $r$, then $f_1, \\ldots, f_r$ is a Koszul-regular sequence. \\end{enumerate}"}
+{"_id": "9987", "title": "more-algebra-lemma-make-nonzero-divisor", "text": "\\begin{reference} This is a particular case of \\cite[Corollary]{McCoy} \\end{reference} Let $R$ be a ring. Let $a_1, \\ldots, a_n \\in R$ be elements such that $R \\to R^{\\oplus n}$, $x \\mapsto (xa_1, \\ldots, xa_n)$ is injective. Then the element $\\sum a_i t_i$ of the polynomial ring $R[t_1, \\ldots, t_n]$ is a nonzerodivisor."}
+{"_id": "9988", "title": "more-algebra-lemma-Koszul-regular-flat-locally-regular", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be a Koszul-regular sequence in $R$ such that $(f_1, \\ldots, f_n) \\not = R$. Consider the faithfully flat, smooth ring map $$ R \\longrightarrow S = R[\\{t_{ij}\\}_{i \\leq j}, t_{11}^{-1}, t_{22}^{-1}, \\ldots, t_{nn}^{-1}] $$ For $1 \\leq i \\leq n$ set $$ g_i = \\sum\\nolimits_{i \\leq j} t_{ij} f_j \\in S. $$ Then $g_1, \\ldots, g_n$ is a regular sequence in $S$ and $(f_1, \\ldots, f_n)S = (g_1, \\ldots, g_n)$."}
+{"_id": "9989", "title": "more-algebra-lemma-vanishing-extended-alternating-koszul", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be an Koszul-regular sequence. Then the extended alternating {\\v C}ech complex $R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to \\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to \\ldots \\to R_{f_1\\ldots f_r}$ from Section \\ref{section-alternating-cech} only has cohomology in degree $r$."}
+{"_id": "9990", "title": "more-algebra-lemma-blowup-regular-sequence", "text": "Let $a, a_2, \\ldots, a_r$ be an $H_1$-regular sequence in a ring $R$ (for example a Koszul regular sequence or a regular sequence, see Lemmas \\ref{lemma-regular-koszul-regular} and \\ref{lemma-koszul-regular-H1-regular}). With $I = (a, a_2, \\ldots, a_r)$ the blowup algebra $R' = R[\\frac{I}{a}]$ is isomorphic to $R'' = R[y_2, \\ldots, y_r]/(a y_i - a_i)$."}
+{"_id": "9991", "title": "more-algebra-lemma-base-change-H1-regular", "text": "Let $A \\to B$ be a ring map. Let $f_1, \\ldots, f_r$ be a sequence in $B$ such that $B/(f_1, \\ldots, f_r)$ is $A$-flat. Let $A \\to A'$ be a ring map. Then the canonical map $$ H_1(K_\\bullet(B, f_1, \\ldots, f_r)) \\otimes_A A' \\longrightarrow H_1(K_\\bullet(B', f'_1, \\ldots, f'_r)) $$ is an isomorphism. Here $B' = B \\otimes_A A'$ and $f_i' \\in B'$ is the image of $f_i$."}
+{"_id": "9992", "title": "more-algebra-lemma-relative-regular-immersion-algebra", "text": "Let $A \\to B$ and $A \\to A'$ be ring maps. Set $B' = B \\otimes_A A'$. Let $f_1, \\ldots, f_r \\in B$. Assume $B/(f_1, \\ldots, f_r)B$ is flat over $A$ \\begin{enumerate} \\item If $f_1, \\ldots, f_r$ is a quasi-regular sequence, then the image in $B'$ is a quasi-regular sequence. \\item If $f_1, \\ldots, f_r$ is a $H_1$-regular sequence, then the image in $B'$ is a $H_1$-regular sequence. \\end{enumerate}"}
+{"_id": "9993", "title": "more-algebra-lemma-cut-by-koszul-flat", "text": "Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume \\begin{enumerate} \\item $A' \\to B'$ is flat and of finite presentation, \\item $I$ is locally nilpotent, \\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence, \\item $B/(f_1, \\ldots, f_r)$ is flat over $A$. \\end{enumerate} Then $B'/(f'_1, \\ldots, f'_r)$ is flat over $A'$."}
+{"_id": "9994", "title": "more-algebra-lemma-cut-by-koszul", "text": "Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume \\begin{enumerate} \\item $A' \\to B'$ is flat and of finite presentation (for example smooth), \\item $I$ is locally nilpotent, \\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence, \\item $B/(f_1, \\ldots, f_r)$ is smooth over $A$. \\end{enumerate} Then $B'/(f'_1, \\ldots, f'_r)$ is smooth over $A'$."}
+{"_id": "9995", "title": "more-algebra-lemma-quasi-regular-ideal-finite", "text": "A quasi-regular ideal is finitely generated."}
+{"_id": "9996", "title": "more-algebra-lemma-quasi-regular-ideal-finite-projective", "text": "Let $I \\subset R$ be a quasi-regular ideal of a ring. Then $I/I^2$ is a finite projective $R/I$-module."}
+{"_id": "9997", "title": "more-algebra-lemma-flat-descent-regular-ideal", "text": "Let $A \\to B$ be a faithfully flat ring map. Let $I \\subset A$ be an ideal. If $IB$ is a Koszul-regular (resp.\\ $H_1$-regular, resp.\\ quasi-regular) ideal in $B$, then $I$ is a Koszul-regular (resp.\\ $H_1$-regular, resp.\\ quasi-regular) ideal in $A$."}
+{"_id": "9998", "title": "more-algebra-lemma-conormal-sequence-H1-regular-ideal", "text": "Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals. Assume that $J/I \\subset A/I$ is a $H_1$-regular ideal. Then $I \\cap J^2 = IJ$."}
+{"_id": "10000", "title": "more-algebra-lemma-lci-local", "text": "Let $R \\to S$ be a ring map. Let $g_1, \\ldots, g_m \\in S$ generate the unit ideal. If each $R \\to S_{g_j}$ is a local complete intersection so is $R \\to S$."}
+{"_id": "10001", "title": "more-algebra-lemma-relative-global-complete-intersection-koszul", "text": "Let $R$ be a ring. If $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative global complete intersection, then $f_1, \\ldots, f_c$ is a Koszul regular sequence."}
+{"_id": "10002", "title": "more-algebra-lemma-syntomic-lci", "text": "Let $R \\to S$ be a ring map. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is syntomic (Algebra, Definition \\ref{algebra-definition-lci}), and \\item $R \\to S$ is flat and a local complete intersection. \\end{enumerate}"}
+{"_id": "10003", "title": "more-algebra-lemma-transitive-lci-at-end", "text": "Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is a local complete intersection homomorphism. Choose a presentation $\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation $\\beta : B[y_1, \\ldots, y_m] \\to C$ with kernel $J$. Let $\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel $K$. Then we get a canonical commutative diagram $$ \\xymatrix{ 0 \\ar[r] & \\Omega_{A[x_s]/A} \\otimes C \\ar[r] & \\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] & \\Omega_{B[y_t]/B} \\otimes C \\ar[r] & 0 \\\\ 0 \\ar[r] & I/I^2 \\otimes C \\ar[r] \\ar[u] & K/K^2 \\ar[r] \\ar[u] & J/J^2 \\ar[r] \\ar[u] & 0 } $$ with exact rows. In particular, the six term exact sequence of Algebra, Lemma \\ref{algebra-lemma-exact-sequence-NL} can be completed with a zero on the left, i.e., the sequence $$ 0 \\to H_1(\\NL_{B/A} \\otimes_B C) \\to H_1(L_{C/A}) \\to H_1(L_{C/B}) \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0 $$ is exact."}
+{"_id": "10004", "title": "more-algebra-lemma-transitive-colimit-lci-at-end", "text": "Let $A \\to B \\to C$ be ring maps. If $B \\to C$ is a filtered colimit of local complete intersection homomorphisms then the conclusion of Lemma \\ref{lemma-transitive-lci-at-end} remains valid."}
+{"_id": "10005", "title": "more-algebra-lemma-henselization-NL", "text": "Let $A \\to B$ be a local homomorphism of local rings. Let $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced map on henselizations, resp.\\ strict henselizations (Algebra, Lemma \\ref{algebra-lemma-henselian-functorial}, resp.\\ Lemma \\ref{algebra-lemma-strictly-henselian-functorial}). Then $\\NL_{B/A} \\otimes_B B^h \\to \\NL_{B^h/A^h}$ and $\\NL_{B/A} \\otimes_B B^{sh} \\to \\NL_{B^{sh}/A^{sh}}$ induce isomorphisms on cohomology groups."}
+{"_id": "10006", "title": "more-algebra-lemma-cartier-equality", "text": "Let $K/k$ be a finitely generated field extension. Then $\\Omega_{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and $\\text{trdeg}_k(K) = \\dim_K \\Omega_{K/k} - \\dim_K H_1(L_{K/k})$."}
+{"_id": "10007", "title": "more-algebra-lemma-transitivity-gamma", "text": "Let $K \\subset L \\subset M$ be field extensions. Then the Jacobi-Zariski sequence $$ 0 \\to H_1(L_{L/K}) \\otimes_L M \\to H_1(L_{M/K}) \\to H_1(L_{M/L}) \\to \\Omega_{L/K} \\otimes_L M \\to \\Omega_{M/K} \\to \\Omega_{M/L} \\to 0 $$ is exact."}
+{"_id": "10008", "title": "more-algebra-lemma-gamma-commutative-diagram", "text": "Given a commutative diagram of fields $$ \\xymatrix{ K \\ar[r] & K' \\\\ k \\ar[u] \\ar[r] & k' \\ar[u] } $$ with $k \\subset k'$ and $K \\subset K'$ finitely generated field extensions the kernel and cokernel of the maps $$ \\alpha : \\Omega_{K/k} \\otimes_K K' \\to \\Omega_{K'/k'} \\quad\\text{and}\\quad \\beta : H_1(L_{K/k}) \\otimes_K K' \\to H_1(L_{K'/k'}) $$ are finite dimensional and $$ \\dim \\Ker(\\alpha) - \\dim \\Coker(\\alpha) -\\dim \\Ker(\\beta) + \\dim \\Coker(\\beta) = \\text{trdeg}_k(k') - \\text{trdeg}_K(K') $$"}
+{"_id": "10009", "title": "more-algebra-lemma-geometrically-regular-over-field", "text": "Let $k$ be a field of characteristic $p > 0$. Let $(A, \\mathfrak m, K)$ be a Noetherian local $k$-algebra. Assume $A$ is geometrically regular over $k$. Let $k \\subset F \\subset K$ be a finitely generated subextension. Let $\\varphi : k[y_1, \\ldots, y_m] \\to A$ be a $k$-algebra map such that $y_i$ maps to an element of $F$ in $K$ and such that $\\text{d}y_1, \\ldots, \\text{d}y_m$ map to a basis of $\\Omega_{F/k}$. Set $\\mathfrak p = \\varphi^{-1}(\\mathfrak m)$. Then $$ k[y_1, \\ldots, y_m]_\\mathfrak p \\to A $$ is flat and $A/\\mathfrak pA$ is regular."}
+{"_id": "10010", "title": "more-algebra-lemma-continuous", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ and $J \\subset S$ be ideals and endow $R$ with the $I$-adic topology and $S$ with the $J$-adic topology. Then $\\varphi$ is a homomorphism of topological rings if and only if $\\varphi(I^n) \\subset J$ for some $n \\geq 1$."}
+{"_id": "10011", "title": "more-algebra-lemma-baire-category-complete-module", "text": "Let $M$ be a topological abelian group. Assume $M$ is linearly topologized, complete, and has a countable fundamental system of neighbourhoods of $0$. If $U_n \\subset M$, $n \\geq 1$ are open dense subsets, then $\\bigcap_{n \\geq 1} U_n$ is dense."}
+{"_id": "10012", "title": "more-algebra-lemma-consequence-baire-complete-module", "text": "With same assumptions as Lemma \\ref{lemma-baire-category-complete-module} if $M = \\bigcup_{n \\geq 1} N_n$ for some closed subgroups $N_n$, then $N_n$ is open for some $n$."}
+{"_id": "10013", "title": "more-algebra-lemma-open-mapping", "text": "Let $u : N \\to M$ be a continuous map of linearly topologized abelian groups. Assume that $N$ is complete, $M$ separated, and $N$ has a countable fundamental system of neighbourhoods of $0$. Then exactly one of the following holds \\begin{enumerate} \\item $u$ is open, or \\item for some open subgroup $N' \\subset N$ the image $u(N')$ is nowhere dense in $M$. \\end{enumerate}"}
+{"_id": "10014", "title": "more-algebra-lemma-formally-smooth", "text": "Let $\\varphi : R \\to S$ be a ring map. \\begin{enumerate} \\item If $R \\to S$ is formally smooth in the sense of Algebra, Definition \\ref{algebra-definition-formally-smooth}, then $R \\to S$ is formally smooth for any linear topology on $R$ and any pre-adic topology on $S$ such that $R \\to S$ is continuous. \\item Let $\\mathfrak n \\subset S$ and $\\mathfrak m \\subset R$ ideals such that $\\varphi$ is continuous for the $\\mathfrak m$-adic topology on $R$ and the $\\mathfrak n$-adic topology on $S$. Then the following are equivalent \\begin{enumerate} \\item $\\varphi$ is formally smooth for the $\\mathfrak m$-adic topology on $R$ and the $\\mathfrak n$-adic topology on $S$, and \\item $\\varphi$ is formally smooth for the discrete topology on $R$ and the $\\mathfrak n$-adic topology on $S$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "10015", "title": "more-algebra-lemma-formally-smooth-completion", "text": "Let $(R, \\mathfrak m)$ and $(S, \\mathfrak n)$ be rings endowed with finitely generated ideals. Endow $R$ and $S$ with the $\\mathfrak m$-adic and $\\mathfrak n$-adic topologies. Let $R \\to S$ be a homomorphism of topological rings. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is formally smooth for the $\\mathfrak n$-adic topology, \\item $R \\to S^\\wedge$ is formally smooth for the $\\mathfrak n^\\wedge$-adic topology, \\item $R^\\wedge \\to S^\\wedge$ is formally smooth for the $\\mathfrak n^\\wedge$-adic topology. \\end{enumerate} Here $R^\\wedge$ and $S^\\wedge$ are the $\\mathfrak m$-adic and $\\mathfrak n$-adic completions of $R$ and $S$."}
+{"_id": "10016", "title": "more-algebra-lemma-lift-continuous", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak n$ be an ideal of $S$. Assume that $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology. Consider a solid commutative diagram $$ \\xymatrix{ S \\ar[r]_\\psi \\ar@{-->}[rd] & A/J \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ of homomorphisms of topological rings where $A$ is adic and $A/J$ is the quotient (as topological ring) of $A$ by a closed ideal $J \\subset A$ such that $J^t$ is contained in an ideal of definition of $A$ for some $t \\geq 1$. Then there exists a dotted arrow in the category of topological rings which makes the diagram commute."}
+{"_id": "10017", "title": "more-algebra-lemma-increase-ideal", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset \\mathfrak n' \\subset S$ be ideals. If $R \\to S$ is formally smooth for the $\\mathfrak n$-adic topology, then $R \\to S$ is formally smooth for the $\\mathfrak n'$-adic topology."}
+{"_id": "10018", "title": "more-algebra-lemma-compose-formally-smooth", "text": "A composition of formally smooth continuous homomorphisms of linearly topologized rings is formally smooth."}
+{"_id": "10019", "title": "more-algebra-lemma-base-change-fs", "text": "Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal. Let $R \\to S$ be formally smooth for the $\\mathfrak n$-adic topology. Let $R \\to R'$ be any ring map. Then $R' \\to S' = S \\otimes_R R'$ is formally smooth in the $\\mathfrak n' = \\mathfrak nS'$-adic topology."}
+{"_id": "10020", "title": "more-algebra-lemma-descent-fs", "text": "Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal. Let $R \\to R'$ be a ring map. Set $S' = S \\otimes_R R'$ and $\\mathfrak n' = \\mathfrak nS$. If \\begin{enumerate} \\item the map $R \\to R'$ embeds $R$ as a direct summand of $R'$ as an $R$-module, and \\item $R' \\to S'$ is formally smooth for the $\\mathfrak n'$-adic topology, \\end{enumerate} then $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology."}
+{"_id": "10021", "title": "more-algebra-lemma-fs-local", "text": "Let $(R, \\mathfrak m) \\to (S, \\mathfrak n)$ be a local homomorphism of local rings. The following are equivalent \\begin{enumerate} \\item $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology, \\item for every solid commutative diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/J \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ of local homomorphisms of local rings where $J \\subset A$ is an ideal of square zero, $\\mathfrak m_A^n = 0$ for some $n > 0$, and $S \\to A/J$ induces an isomorphism on residue fields, a dotted arrow exists which makes the diagram commute. \\end{enumerate} If $S$ is Noetherian these conditions are also equivalent to \\begin{enumerate} \\item[(3)] same as in (2) but only for diagrams where in addition $A \\to A/J$ is a small extension (Algebra, Definition \\ref{algebra-definition-small-extension}). \\end{enumerate}"}
+{"_id": "10022", "title": "more-algebra-lemma-fs-implies-regular", "text": "Let $k$ be a field and let $(A, \\mathfrak m, K)$ be a Noetherian local $k$-algebra. If $k \\to A$ is formally smooth for the $\\mathfrak m$-adic topology, then $A$ is a regular local ring."}
+{"_id": "10023", "title": "more-algebra-lemma-lift-residue-field", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete local $k$-algebra. If $\\kappa/k$ is separable, then there exists a $k$-algebra map $\\kappa \\to A$ such that $\\kappa \\to A \\to \\kappa$ is $\\text{id}_\\kappa$."}
+{"_id": "10024", "title": "more-algebra-lemma-power-series-over-residue-field", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete local $k$-algebra. If $\\kappa/k$ is separable and $A$ regular, then there exists an isomorphism of $A \\cong \\kappa[[t_1, \\ldots, t_d]]$ as $k$-algebras."}
+{"_id": "10025", "title": "more-algebra-lemma-regular-implies-fs", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a regular local $k$-algebra such that $K/k$ is separable. Then $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology."}
+{"_id": "10026", "title": "more-algebra-lemma-formally-smooth-finite-type", "text": "Let $A \\to B$ be a finite type ring map with $A$ Noetherian. Let $\\mathfrak q \\subset B$ be a prime ideal lying over $\\mathfrak p \\subset A$. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is smooth at $\\mathfrak q$, and \\item $A_\\mathfrak p \\to B_\\mathfrak q$ is formally smooth in the $\\mathfrak q$-adic topology. \\end{enumerate}"}
+{"_id": "10027", "title": "more-algebra-lemma-power-series-ring-over-Cohen-fs", "text": "Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \\ldots, x_n]]$. Let $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \\ldots, x_n]]$. Let $\\Lambda$ be a Cohen ring. Let $C = \\Lambda[[x_1, \\ldots, x_n]]$. \\begin{enumerate} \\item $\\mathbf{Q} \\to A$ is formally smooth in the  $\\mathfrak m$-adic topology. \\item $\\mathbf{F}_p \\to B$ is formally smooth in the  $\\mathfrak m$-adic topology. \\item $\\mathbf{Z} \\to C$ is formally smooth in the $\\mathfrak m$-adic topology. \\end{enumerate}"}
+{"_id": "10028", "title": "more-algebra-lemma-quotient-power-series-ring-over-Cohen", "text": "Let $K$ be a field and $A = K[[x_1, \\ldots, x_n]]$. Let $\\Lambda$ be a Cohen ring and let $B = \\Lambda[[x_1, \\ldots, x_n]]$. \\begin{enumerate} \\item If $y_1, \\ldots, y_n \\in A$ is a regular system of parameters then $K[[y_1, \\ldots, y_n]] \\to A$ is an isomorphism. \\item If $z_1, \\ldots, z_r \\in A$ form part of a regular system of parameters for $A$, then $r \\leq n$ and $A/(z_1, \\ldots, z_r) \\cong K[[y_1, \\ldots, y_{n - r}]]$. \\item If $p, y_1, \\ldots, y_n \\in B$ is a regular system of parameters then $\\Lambda[[y_1, \\ldots, y_n]] \\to B$ is an isomorphism. \\item If $p, z_1, \\ldots, z_r \\in B$ form part of a regular system of parameters for $B$, then $r \\leq n$ and $B/(z_1, \\ldots, z_r) \\cong \\Lambda[[y_1, \\ldots, y_{n - r}]]$. \\end{enumerate}"}
+{"_id": "10029", "title": "more-algebra-lemma-embed-map-Noetherian-complete-local-rings", "text": "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings. Then there exists a commutative diagram $$ \\xymatrix{ S \\ar[r] & B \\\\ R \\ar[u] \\ar[r] & A \\ar[u] } $$ with the following properties: \\begin{enumerate} \\item the horizontal arrows are surjective, \\item if the characteristic of $A/\\mathfrak m_A$ is zero, then $S$ and $R$ are power series rings over fields, \\item if the characteristic of $A/\\mathfrak m_A$ is $p > 0$, then $S$ and $R$ are power series rings over Cohen rings, and \\item $R \\to S$ maps a regular system of parameters of $R$ to part of a regular system of parameters of $S$. \\end{enumerate} In particular $R \\to S$ is flat (see Algebra, Lemma \\ref{algebra-lemma-flat-over-regular}) with regular fibre $S/\\mathfrak m_R S$ (see Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM})."}
+{"_id": "10030", "title": "more-algebra-lemma-dominate-two-surjections", "text": "Let $S \\to R$ and $S' \\to R$ be surjective maps of complete Noetherian local rings. Then $S \\times_R S'$ is a complete Noetherian local ring."}
+{"_id": "10031", "title": "more-algebra-lemma-formally-smooth-flat", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Assume $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. Then $A \\to B$ is flat."}
+{"_id": "10033", "title": "more-algebra-lemma-lift-fs", "text": "Let $A$ be a Noetherian complete local ring with residue field $k$. Let $B$ be a Noetherian complete local $k$-algebra. Assume $k \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. Then there exists a Noetherian complete local ring $C$ and a local homomorphism $A \\to C$ which is formally smooth in the $\\mathfrak m_C$-adic topology such that $C \\otimes_A k \\cong B$."}
+{"_id": "10034", "title": "more-algebra-lemma-regular-local", "text": "Let $R \\to \\Lambda$ be a ring map with $\\Lambda$ Noetherian. The following are equivalent \\begin{enumerate} \\item $R \\to \\Lambda$ is regular, \\item $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is regular for all $\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and \\item $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ is regular for all maximal ideals $\\mathfrak m' \\subset \\Lambda$ lying over $\\mathfrak m$ in $R$. \\end{enumerate}"}
+{"_id": "10035", "title": "more-algebra-lemma-regular-base-change", "text": "Let $R \\to \\Lambda$ be a regular ring map. For any finite type ring map $R \\to R'$ the base change $R' \\to \\Lambda \\otimes_R R'$ is regular too."}
+{"_id": "10036", "title": "more-algebra-lemma-regular-composition", "text": "Let $A \\to B \\to C$ be regular ring maps. If the fibre rings of $A \\to C$ are Noetherian, then $A \\to C$ is regular."}
+{"_id": "10037", "title": "more-algebra-lemma-colimit-smooth-regular", "text": "Let $R$ be a ring. Let $(A_i, \\varphi_{ii'})$ be a directed system of smooth $R$-algebras. Set $\\Lambda = \\colim A_i$. If the fibre rings $\\Lambda \\otimes_R \\kappa(\\mathfrak p)$ are Noetherian for all $\\mathfrak p \\subset R$, then $R \\to \\Lambda$ is regular."}
+{"_id": "10039", "title": "more-algebra-lemma-regular-permanence", "text": "Let $A \\to B \\to C$ be ring maps. If $A \\to C$ is regular and $B \\to C$ is flat and surjective on spectra, then $A \\to B$ is regular."}
+{"_id": "10040", "title": "more-algebra-lemma-reduced-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is regular, \\item $S$ is Noetherian, and \\item $R$ is Noetherian and reduced. \\end{enumerate} Then $S$ is reduced."}
+{"_id": "10041", "title": "more-algebra-lemma-normal-goes-up", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume \\begin{enumerate} \\item $\\varphi$ is regular, \\item $S$ is Noetherian, and \\item $R$ is Noetherian and normal. \\end{enumerate} Then $S$ is normal."}
+{"_id": "10042", "title": "more-algebra-lemma-completion-dimension", "text": "Let $A$ be a Noetherian local ring. Then $\\dim(A) = \\dim(A^\\wedge)$."}
+{"_id": "10043", "title": "more-algebra-lemma-completion-depth", "text": "Let $A$ be a Noetherian local ring. Then $\\text{depth}(A) = \\text{depth}(A^\\wedge)$."}
+{"_id": "10044", "title": "more-algebra-lemma-completion-CM", "text": "Let $A$ be a Noetherian local ring. Then $A$ is Cohen-Macaulay if and only if $A^\\wedge$ is so."}
+{"_id": "10045", "title": "more-algebra-lemma-completion-regular", "text": "Let $A$ be a Noetherian local ring. Then $A$ is regular if and only if $A^\\wedge$ is so."}
+{"_id": "10046", "title": "more-algebra-lemma-completion-dvr", "text": "Let $A$ be a Noetherian local ring. Then $A$ is a discrete valuation ring if and only if $A^\\wedge$ is so."}
+{"_id": "10047", "title": "more-algebra-lemma-completion-reduced", "text": "Let $A$ be a Noetherian local ring. \\begin{enumerate} \\item If $A^\\wedge$ is reduced, then so is $A$. \\item In general $A$ reduced does not imply $A^\\wedge$ is reduced. \\item If $A$ is Nagata, then $A$ is reduced if and only if $A^\\wedge$ is reduced. \\end{enumerate}"}
+{"_id": "10049", "title": "more-algebra-lemma-flat-completion", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Then the induced map of completions $A^\\wedge \\to B^\\wedge$ is flat if and only if $A \\to B$ is flat."}
+{"_id": "10050", "title": "more-algebra-lemma-flat-unramified", "text": "Let $A \\to B$ be a flat local homomorphism of Noetherian local rings such that $\\mathfrak m_A B = \\mathfrak m_B$ and $\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$. Then $A \\to B$ induces an isomorphism $A^\\wedge \\to B^\\wedge$ of completions."}
+{"_id": "10051", "title": "more-algebra-lemma-Noetherian-etale-extension", "text": "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of $B$ lying over $\\mathfrak p \\subset A$, then $A_{\\mathfrak p}$ is Noetherian if and only if $B_{\\mathfrak q}$ is Noetherian."}
+{"_id": "10052", "title": "more-algebra-lemma-dimension-etale-extension", "text": "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of $B$ lying over $\\mathfrak p \\subset A$, then $\\dim(A_{\\mathfrak p}) = \\dim(B_{\\mathfrak q})$."}
+{"_id": "10053", "title": "more-algebra-lemma-regular-etale-extension", "text": "If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of $B$ lying over $\\mathfrak p \\subset A$, then $A_{\\mathfrak p}$ is regular if and only if $B_{\\mathfrak q}$ is regular."}
+{"_id": "10054", "title": "more-algebra-lemma-Dedekind-etale-extension", "text": "If $A \\to B$ is an \\'etale ring map and $A$ is a Dedekind domain, then $B$ is a finite product of Dedekind domains. In particular, the localizations $B_\\mathfrak q$ for $\\mathfrak q \\subset B$ maximal are discrete valuation rings."}
+{"_id": "10055", "title": "more-algebra-lemma-dumb-properties-henselization", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then we have the following \\begin{enumerate} \\item $R \\to R^h \\to R^{sh}$ are faithfully flat ring maps, \\item $\\mathfrak m R^h = \\mathfrak m^h$ and $\\mathfrak m R^{sh} = \\mathfrak m^h R^{sh} = \\mathfrak m^{sh}$, \\item $R/\\mathfrak m^n = R^h/\\mathfrak m^nR^h$ for all $n$, \\item there exist elements $x_i \\in R^{sh}$ such that $R^{sh}/\\mathfrak m^nR^{sh}$ is a free $R/\\mathfrak m^n$-module on $x_i \\bmod \\mathfrak m^nR^{sh}$. \\end{enumerate}"}
+{"_id": "10056", "title": "more-algebra-lemma-henselization-formally-smooth", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then \\begin{enumerate} \\item $R \\to R^h$, $R^h \\to R^{sh}$, and $R \\to R^{sh}$ are formally \\'etale, \\item $R \\to R^h$, $R^h \\to R^{sh}$, resp.\\ $R \\to R^{sh}$ are formally smooth in the $\\mathfrak m^h$, $\\mathfrak m^{sh}$, resp.\\ $\\mathfrak m^{sh}$-topology. \\end{enumerate}"}
+{"_id": "10057", "title": "more-algebra-lemma-henselization-noetherian", "text": "\\begin{reference} \\cite[IV, Theorem 18.6.6 and Proposition 18.8.8]{EGA} \\end{reference} Let $R$ be a local ring. The following are equivalent \\begin{enumerate} \\item $R$ is Noetherian, \\item $R^h$ is Noetherian, and \\item $R^{sh}$ is Noetherian. \\end{enumerate} In this case we have \\begin{enumerate} \\item[(a)] $(R^h)^\\wedge$ and $(R^{sh})^\\wedge$ are Noetherian complete local rings, \\item[(b)] $R^\\wedge \\to (R^h)^\\wedge$ is an isomorphism, \\item[(c)] $R^h \\to (R^h)^\\wedge$ and $R^{sh} \\to (R^{sh})^\\wedge$ are flat, \\item[(d)] $R^\\wedge \\to (R^{sh})^\\wedge$ is formally smooth in the $\\mathfrak m_{(R^{sh})^\\wedge}$-adic topology, \\item[(e)] $(R^\\wedge)^{sh} = R^\\wedge \\otimes_{R^h} R^{sh}$, and \\item[(f)] $((R^\\wedge)^{sh})^\\wedge = (R^{sh})^\\wedge$. \\end{enumerate}"}
+{"_id": "10058", "title": "more-algebra-lemma-henselization-reduced", "text": "\\begin{slogan} Reducedness passes to the (strict) henselization. \\end{slogan} Let $R$ be a local ring. The following are equivalent: $R$ is reduced, the henselization $R^h$ of $R$ is reduced, and the strict henselization $R^{sh}$ of $R$ is reduced."}
+{"_id": "10059", "title": "more-algebra-lemma-henselization-nil", "text": "Let $R$ be a local ring. Let $nil(R)$ denote the ideal of nilpotent elements of $R$. Then $nil(R)R^h = nil(R^h)$ and $nil(R)R^{sh} = nil(R^{sh})$."}
+{"_id": "10060", "title": "more-algebra-lemma-henselization-normal", "text": "Let $R$ be a local ring. The following are equivalent: $R$ is a normal domain, the henselization $R^h$ of $R$ is a normal domain, and the strict henselization $R^{sh}$ of $R$ is a normal domain."}
+{"_id": "10061", "title": "more-algebra-lemma-henselization-dimension", "text": "Given any local ring $R$ we have $\\dim(R) = \\dim(R^h) = \\dim(R^{sh})$."}
+{"_id": "10062", "title": "more-algebra-lemma-henselization-depth", "text": "Given a Noetherian local ring $R$ we have $\\text{depth}(R) = \\text{depth}(R^h) = \\text{depth}(R^{sh})$."}
+{"_id": "10063", "title": "more-algebra-lemma-henselization-CM", "text": "Let $R$ be a Noetherian local ring. The following are equivalent: $R$ is Cohen-Macaulay, the henselization $R^h$ of $R$ is Cohen-Macaulay, and the strict henselization $R^{sh}$ of $R$ is Cohen-Macaulay."}
+{"_id": "10064", "title": "more-algebra-lemma-henselization-regular", "text": "Let $R$ be a Noetherian local ring. The following are equivalent: $R$ is a regular local ring, the henselization $R^h$ of $R$ is a regular local ring, and the strict henselization $R^{sh}$ of $R$ is a regular local ring."}
+{"_id": "10065", "title": "more-algebra-lemma-henselization-dvr", "text": "Let $R$ be a Noetherian local ring. Then $R$ is a discrete valuation ring if and only if $R^h$ is a discrete valuation ring if and only if $R^{sh}$ is a discrete valuation ring."}
+{"_id": "10066", "title": "more-algebra-lemma-filtered-colimit-etale-noetherian-fibres", "text": "Let $A$ be a ring. Let $B$ be a filtered colimit of \\'etale $A$-algebras. Let $\\mathfrak p$ be a prime of $A$. If $B$ is Noetherian, then there are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ lying over $\\mathfrak p$, we have $B \\otimes_A \\kappa(\\mathfrak p) = \\prod \\kappa(\\mathfrak q_i)$, and each of the field extensions $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q_i)$ is separable algebraic."}
+{"_id": "10067", "title": "more-algebra-lemma-fibres-henselization", "text": "Let $R$ be a Noetherian local ring. Let $\\mathfrak p \\subset R$ be a prime. Then $$ R^h \\otimes_R \\kappa(\\mathfrak p) = \\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i) \\quad\\text{resp.}\\quad R^{sh} \\otimes_R \\kappa(\\mathfrak p) = \\prod\\nolimits_{i = 1, \\ldots, s} \\kappa(\\mathfrak r_i) $$ where $\\mathfrak q_1, \\ldots, \\mathfrak q_t$, resp.\\ $\\mathfrak r_1, \\ldots, \\mathfrak r_s$ are the prime of $R^h$, resp.\\ $R^{sh}$ lying over $\\mathfrak p$. Moreover, the field extensions $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak q_i)$ resp.\\ $\\kappa(\\mathfrak p) \\subset \\kappa(\\mathfrak r_i)$ are separable algebraic."}
+{"_id": "10068", "title": "more-algebra-lemma-p-basis", "text": "Let $k \\subset K$ be a field extension. Assume $k$ has characteristic $p > 0$. Let $\\{x_i\\}$ be a subset of $K$. The following are equivalent \\begin{enumerate} \\item the elements $\\{x_i\\}$ are $p$-independent over $k$, and \\item the elements $\\text{d}x_i$ are $K$-linearly independent in $\\Omega_{K/k}$. \\end{enumerate} Any $p$-independent collection can be extended to a $p$-basis of $K$ over $k$. In particular, the field $K$ has a $p$-basis over $k$. Moreover, the following are equivalent: \\begin{enumerate} \\item[(a)] $\\{x_i\\}$ is a $p$-basis of $K$ over $k$, and \\item[(b)] $\\text{d}x_i$ is a basis of the $K$-vector space $\\Omega_{K/k}$. \\end{enumerate}"}
+{"_id": "10069", "title": "more-algebra-lemma-intersection-subfields-subspace", "text": "Let $k \\subset K$ be a field extension. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$ be a collection of subfields of $K$ with the following properties \\begin{enumerate} \\item $k \\subset K_\\alpha$ for all $\\alpha \\in A$, \\item $k = \\bigcap_{\\alpha \\in A} K_\\alpha$, \\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$ such that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$. \\end{enumerate} Then for $n \\geq 1$ and $V \\subset K^{\\oplus n}$ a $K$-vector space we have $V \\cap k^{\\oplus n} \\not = 0$ if and only if $V \\cap K_\\alpha^{\\oplus n} \\not = 0$ for all $\\alpha \\in A$."}
+{"_id": "10070", "title": "more-algebra-lemma-intersection-subfields", "text": "Let $K$ be a field of characteristic $p$. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$ be a collection of subfields of $K$ with the following properties \\begin{enumerate} \\item $K^p \\subset K_\\alpha$ for all $\\alpha \\in A$, \\item $K^p = \\bigcap_{\\alpha \\in A} K_\\alpha$, \\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$ such that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$. \\end{enumerate} Then \\begin{enumerate} \\item the intersection of the kernels of the maps $\\Omega_{K/\\mathbf{F}_p} \\to \\Omega_{K/K_\\alpha}$ is zero, \\item for any finite extension $K \\subset L$ we have $L^p = \\bigcap_{\\alpha \\in A} L^pK_\\alpha$. \\end{enumerate}"}
+{"_id": "10071", "title": "more-algebra-lemma-power-series-ring-subfields", "text": "Let $k$ be a field of characteristic $p > 0$. Let $n, m \\geq 0$. Let $K$ be the fraction field of $k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$. As $k'$ ranges through all subfields $k/k'/k^p$ with $[k : k'] < \\infty$ the subfields $$ \\text{fraction field of } k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p] \\subset K $$ form a family of subfields as in Lemma \\ref{lemma-intersection-subfields}. Moreover, each of the ring extensions $k'[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p] \\subset k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$ is finite."}
+{"_id": "10072", "title": "more-algebra-lemma-J-1", "text": "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$. The ring $R$ is J-1 if and only if $V(\\mathfrak p) \\cap \\text{Reg}(X)$ contains a nonempty open subset of $V(\\mathfrak p)$ for all $\\mathfrak p \\in \\text{Reg}(X)$."}
+{"_id": "10073", "title": "more-algebra-lemma-intersection-regular-with-closed", "text": "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$. Assume that for all primes $\\mathfrak p \\subset R$ the ring $R/\\mathfrak p$ is J-0. Then $R$ is J-1."}
+{"_id": "10074", "title": "more-algebra-lemma-J-0-goes-down", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R$ is a Noetherian domain, \\item $R \\to S$ is injective and of finite type, and \\item $S$ is a domain and J-0. \\end{enumerate} Then $R$ is J-0."}
+{"_id": "10075", "title": "more-algebra-lemma-J-0-goes-up", "text": "Let $R \\to S$ be a ring map. Assume that \\begin{enumerate} \\item $R$ is a Noetherian domain and J-0, \\item $R \\to S$ is injective and of finite type, and \\item $S$ is a domain, and \\item the induced extension of fraction fields is separable. \\end{enumerate} Then $S$ is J-0."}
+{"_id": "10076", "title": "more-algebra-lemma-J-2", "text": "Let $R$ be a Noetherian ring. The following are equivalent \\begin{enumerate} \\item $R$ is J-2, \\item every finite type $R$-algebra which is a domain is J-0, \\item every finite $R$-algebra is J-1, \\item for every prime $\\mathfrak p$ and every finite purely inseparable extension $\\kappa(\\mathfrak p) \\subset L$ there exists a finite $R$-algebra $R'$ which is a domain, which is J-0, and whose field of fractions is $L$. \\end{enumerate}"}
+{"_id": "10077", "title": "more-algebra-lemma-derivation-extends", "text": "Let $R$ be a ring. Let $D : R \\to R$ be a derivation. \\begin{enumerate} \\item For any ideal $I \\subset R$ the derivation $D$ extends canonically to a derivation $D^\\wedge : R^\\wedge \\to R^\\wedge$ on the $I$-adic completion. \\item For any multiplicative subset $S \\subset R$ the derivation $D$ extends uniquely to the localization $S^{-1}R$ of $R$. \\end{enumerate} If $R \\subset R'$ is a finite type extension of rings such that $R_g \\cong R'_g$ for some $g \\in R$ which is a nonzerodivisor in $R'$, then $g^ND$ extends to $R'$ for some $N \\geq 0$."}
+{"_id": "10078", "title": "more-algebra-lemma-quotient-regular", "text": "\\begin{slogan} The Jacobian criterion for hypersurfaces, done right. \\end{slogan} Let $R$ be a regular ring. Let $f \\in R$. Assume there exists a derivation $D : R \\to R$ such that $D(f)$ is a unit of $R/(f)$. Then $R/(f)$ is regular."}
+{"_id": "10079", "title": "more-algebra-lemma-quotient-sequence-regular", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring. Let $m \\geq 1$. Let $f_1, \\ldots, f_m \\in \\mathfrak m$. Assume there exist derivations $D_1, \\ldots, D_m : R \\to R$ such that $\\det_{1 \\leq i, j \\leq m}(D_i(f_j))$ is a unit of $R$. Then $R/(f_1, \\ldots, f_m)$ is regular and $f_1, \\ldots, f_m$ is a regular sequence."}
+{"_id": "10080", "title": "more-algebra-lemma-degree-p-extension-regular", "text": "Let $R$ be a regular ring. Let $f \\in R$. Assume there exists a derivation $D : R \\to R$ such that $D(f)$ is a unit of $R$. Then $R[z]/(z^n - f)$ is regular for any integer $n \\geq 1$. More generally, $R[z]/(p(z) - f)$ is regular for any $p \\in \\mathbf{Z}[z]$."}
+{"_id": "10081", "title": "more-algebra-lemma-find-D", "text": "Let $p$ be a prime number. Let $B$ be a domain with $p = 0$ in $B$. Let $f \\in B$ be an element which is not a $p$th power in the fraction field of $B$. If $B$ is of finite type over a Noetherian complete local ring, then there exists a derivation $D : B \\to B$ such that $D(f)$ is not zero."}
+{"_id": "10082", "title": "more-algebra-lemma-complete-Noetherian-domain-J-0", "text": "Let $A$ be a Noetherian complete local domain. Then $A$ is J-0."}
+{"_id": "10083", "title": "more-algebra-lemma-lift-derivation-through-fs", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Let $D : A \\to A$ be a derivation. Assume that $B$ is complete and $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. Then there exists an extension $D' : B \\to B$ of $D$."}
+{"_id": "10084", "title": "more-algebra-lemma-check-G-ring-easy", "text": "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$ the algebra $$ (R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q) $$ is geometrically regular over $\\kappa(\\mathfrak q)$."}
+{"_id": "10085", "title": "more-algebra-lemma-G-ring-goes-up-quasi-finite", "text": "Let $R \\to R'$ be a finite type map of Noetherian rings and let $$ \\xymatrix{ \\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\ \\mathfrak q \\ar[r] \\ar@{-}[u] & \\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u] } $$ be primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$. \\begin{enumerate} \\item If the formal fibre $R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$ is geometrically regular over $\\kappa(\\mathfrak q)$, then the formal fibre $R'_{\\mathfrak p'} \\otimes_{R'} \\kappa(\\mathfrak q')$ is geometrically regular over $\\kappa(\\mathfrak q')$. \\item If the formal fibres of $R_\\mathfrak p$ are geometrically regular, then the formal fibres of $R'_{\\mathfrak p'}$ are geometrically regular. \\item If $R \\to R'$ is quasi-finite and $R$ is a G-ring, then $R'$ is a G-ring. \\end{enumerate}"}
+{"_id": "10087", "title": "more-algebra-lemma-helper-G-ring", "text": "Let $k$ be a field of characteristic $p$. Let $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_n]$ and denote $K$ the fraction field of $A$. Let $\\mathfrak p \\subset A$ be a prime. Then $A_\\mathfrak p^\\wedge \\otimes_A K$ is geometrically regular over $K$."}
+{"_id": "10088", "title": "more-algebra-lemma-check-G-ring-maximal-ideals", "text": "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if $R_\\mathfrak m$ has geometrically regular formal fibres for every maximal ideal $\\mathfrak m$ of $R$."}
+{"_id": "10089", "title": "more-algebra-lemma-henselization-G-ring", "text": "Let $R$ be a Noetherian local ring which is a G-ring. Then the henselization $R^h$ and the strict henselization $R^{sh}$ are G-rings."}
+{"_id": "10090", "title": "more-algebra-lemma-another-helper-G-ring", "text": "Let $p$ be a prime number. Let $A$ be a Noetherian complete local domain with fraction field $K$ of characteristic $p$. Let $\\mathfrak q \\subset A[x]$ be a maximal ideal lying over the maximal ideal of $A$ and let $(0) \\not = \\mathfrak r \\subset \\mathfrak q$ be a prime lying over $(0) \\subset A$. Then $A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} \\kappa(\\mathfrak r)$ is geometrically regular over $\\kappa(\\mathfrak r)$."}
+{"_id": "10092", "title": "more-algebra-lemma-map-G-ring-to-completion-regular", "text": "\\begin{reference} \\cite[Theorem 79]{MatCA} \\end{reference} Let $A$ be a G-ring. Let $I \\subset A$ be an ideal and let $A^\\wedge$ be the completion of $A$ with respect to $I$. Then $A \\to A^\\wedge$ is regular."}
+{"_id": "10093", "title": "more-algebra-lemma-henselization-pair-G-ring", "text": "\\begin{slogan} Being a G-ring is stable under Henselizations along ideals \\end{slogan} \\begin{reference} \\cite[Theorem 5.3 i)]{Greco} \\end{reference} Let $A$ be a G-ring. Let $I \\subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see Lemma \\ref{lemma-henselization}. Then $A^h$ is a G-ring."}
+{"_id": "10094", "title": "more-algebra-lemma-check-P-ring-easy", "text": "Let $R$ be a Noetherian ring. Let $P$ be a property as above. Then $R$ is a $P$-ring if and only if for every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$ the $\\kappa(\\mathfrak q)$-algebra $$ (R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q) $$ has property $P$."}
+{"_id": "10095", "title": "more-algebra-lemma-P-local", "text": "Let $R \\to \\Lambda$ be a homomorphism of Noetherian rings. Assume $P$ has property (B). The following are equivalent \\begin{enumerate} \\item the fibres of $R \\to \\Lambda$ have $P$, \\item the fibres of $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ have $P$ for all $\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and \\item the fibres of $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ have $P$ for all maximal ideals $\\mathfrak m' \\subset \\Lambda$ lying over $\\mathfrak m$ in $R$. \\end{enumerate}"}
+{"_id": "10096", "title": "more-algebra-lemma-P-ring-goes-up-quasi-finite", "text": "Let $R \\to R'$ be a finite type map of Noetherian rings and let $$ \\xymatrix{ \\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\ \\mathfrak q \\ar[r] \\ar@{-}[u] & \\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u] } $$ be primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$. Assume $P$ satisfies (A) and (B). \\begin{enumerate} \\item If $\\kappa(\\mathfrak q) \\to R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$ has $P$, then $\\kappa(\\mathfrak q') \\to R'_{\\mathfrak p'} \\otimes_{R'} \\kappa(\\mathfrak q')$ has $P$. \\item If the formal fibres of $R_\\mathfrak p$ have $P$, then the formal fibres of $R'_{\\mathfrak p'}$ have $P$. \\item If $R \\to R'$ is quasi-finite and $R$ is a $P$-ring, then $R'$ is a $P$-ring. \\end{enumerate}"}
+{"_id": "10097", "title": "more-algebra-lemma-check-P-ring-maximal-ideals", "text": "Let $R$ be a Noetherian ring. Assume $P$ satisfies (C) and (D). Then $R$ is a $P$-ring if and only if the formal fibres of $R_\\mathfrak m$ have $P$ for every maximal ideal $\\mathfrak m$ of $R$."}
+{"_id": "10099", "title": "more-algebra-lemma-henselization-pair-P-ring", "text": "\\begin{slogan} Henselization of a ring inherits good properties of formal fibers \\end{slogan} Let $A$ be a $P$-ring where $P$ satisfies (B), (C), (D), and (E). Let $I \\subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see Lemma \\ref{lemma-henselization}. Then $A^h$ is a $P$-ring."}
+{"_id": "10100", "title": "more-algebra-lemma-henselization-P-ring", "text": "Let $R$ be a Noetherian local ring which is a $P$-ring where $P$ satisfies (B), (C), (D), and (E). Then the henselization $R^h$ and the strict henselization $R^{sh}$ are $P$-rings."}
+{"_id": "10101", "title": "more-algebra-lemma-formal-fibres-reduced", "text": "Properties (A), (B), (C), (D), and (E) hold for $P(k \\to R) =$``$R$ is geometrically reduced over $k$''."}
+{"_id": "10102", "title": "more-algebra-lemma-formal-fibres-normal", "text": "Properties (A), (B), (C), (D), and (E) hold for $P(k \\to R) =$``$R$ is geometrically normal over $k$''."}
+{"_id": "10103", "title": "more-algebra-lemma-formal-fibres-Sk", "text": "Fix $n \\geq 1$. Properties (A), (B), (C), (D), and (E) hold for $P(k \\to R) =$``$R$ has $(S_n)$''."}
+{"_id": "10106", "title": "more-algebra-lemma-finite-type-over-excellent", "text": "Any localization of a finite type ring over a (quasi-)excellent ring is (quasi-)excellent."}
+{"_id": "10107", "title": "more-algebra-lemma-Nagata-local-ring", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The following are equivalent \\begin{enumerate} \\item $A$ is Nagata, and \\item the formal fibres of $A$ are geometrically reduced. \\end{enumerate}"}
+{"_id": "10109", "title": "more-algebra-lemma-completion-normal-local-ring", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is normal and the formal fibres of $A$ are normal (for example if $A$ is excellent or quasi-excellent), then $A^\\wedge$ is normal."}
+{"_id": "10110", "title": "more-algebra-lemma-injective-abelian", "text": "An abelian group $J$ is an injective object in the category of abelian groups if and only if $J$ is divisible."}
+{"_id": "10111", "title": "more-algebra-lemma-relation-ext-ext", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be the abelian category of $R$-modules. There is a canonical isomorphism $\\Ext_\\mathcal{A}(M, N) = \\Ext^1_R(M, N)$ compatible with the long exact sequences of Algebra, Lemmas \\ref{algebra-lemma-long-exact-seq-ext} and \\ref{algebra-lemma-reverse-long-exact-seq-ext} and the $6$-term exact sequences of Homology, Lemma \\ref{homology-lemma-six-term-sequence-ext}."}
+{"_id": "10113", "title": "more-algebra-lemma-characterize-injective-bis", "text": "Let $R$ be a ring. Let $J$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $J$ is injective, \\item $\\Ext^1_R(R/I, J) = 0$ for every ideal $I \\subset R$, and \\item for an ideal $I \\subset R$ and module map $I \\to J$ there exists an extension $R \\to J$. \\end{enumerate}"}
+{"_id": "10114", "title": "more-algebra-lemma-vee-exact", "text": "Let $R$ be a ring. The functor $M \\mapsto M^\\vee$ is exact."}
+{"_id": "10115", "title": "more-algebra-lemma-ev-injective", "text": "For any $R$-module $M$ the evaluation map $ev : M \\to (M^\\vee)^\\vee$ is injective."}
+{"_id": "10117", "title": "more-algebra-lemma-injectives-modules", "text": "Let $R$ be a ring. The construction above defines a covariant functor $M \\mapsto (M \\to J(M))$ from the category of $R$-modules to the category of arrows of $R$-modules such that for every module $M$ the output $M \\to J(M)$ is an injective map of $M$ into an injective $R$-module $J(M)$."}
+{"_id": "10118", "title": "more-algebra-lemma-K-injective-flat", "text": "Let $R \\to S$ be a flat ring map. If $I^\\bullet$ is a K-injective complex of $S$-modules, then $I^\\bullet$ is K-injective as a complex of $R$-modules."}
+{"_id": "10120", "title": "more-algebra-lemma-hom-K-injective", "text": "Let $A \\to B$ be a ring map. If $I^\\bullet$ is a K-injective complex of $A$-modules, then $\\Hom_A(B, I^\\bullet)$ is a K-injective complex of $B$-modules."}
+{"_id": "10121", "title": "more-algebra-lemma-derived-tor-homotopy", "text": "Let $R$ be a ring. Let $P^\\bullet$ be a complex of $R$-modules. Let $\\alpha, \\beta : L^\\bullet \\to M^\\bullet$ be homotopy equivalent maps of complexes. Then $\\alpha$ and $\\beta$ induce homotopy equivalent maps $$ \\text{Tot}(\\alpha \\otimes \\text{id}_P), \\text{Tot}(\\beta \\otimes \\text{id}_P) : \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet) \\longrightarrow \\text{Tot}(M^\\bullet \\otimes_R P^\\bullet). $$ In particular the construction $L^\\bullet \\mapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)$ defines an endo-functor of the homotopy category of complexes."}
+{"_id": "10122", "title": "more-algebra-lemma-derived-tor-exact", "text": "Let $R$ be a ring. Let $P^\\bullet$ be a complex of $R$-modules. The functors $$ K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad L^\\bullet \\longmapsto \\text{Tot}(P^\\bullet \\otimes_R L^\\bullet) $$ and $$ K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad L^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet) $$ are exact functors of triangulated categories."}
+{"_id": "10123", "title": "more-algebra-lemma-K-flat-quasi-isomorphism", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a K-flat complex. Then the functor $$ K(\\text{Mod}_R) \\longrightarrow K(\\text{Mod}_R), \\quad L^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R K^\\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms."}
+{"_id": "10124", "title": "more-algebra-lemma-base-change-K-flat", "text": "Let $R \\to R'$ be a ring map. If $K^\\bullet$ is a K-flat complex of $R$-modules, then $K^\\bullet \\otimes_R R'$ is a K-flat complex of $R'$-modules."}
+{"_id": "10125", "title": "more-algebra-lemma-tensor-product-K-flat", "text": "Let $R$ be a ring. If $K^\\bullet$, $L^\\bullet$ are K-flat complexes of $R$-modules, then $\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)$ is a K-flat complex of $R$-modules."}
+{"_id": "10126", "title": "more-algebra-lemma-K-flat-two-out-of-three", "text": "Let $R$ be a ring. Let $(K_1^\\bullet, K_2^\\bullet, K_3^\\bullet)$ be a distinguished triangle in $K(\\text{Mod}_R)$. If two out of three of $K_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "10127", "title": "more-algebra-lemma-K-flat-two-out-of-three-ses", "text": "Let $R$ be a ring. Let $0 \\to K_1^\\bullet \\to K_2^\\bullet \\to K_3^\\bullet \\to 0$ be a short exact sequence of complexes. If $K_3^n$ is flat for all $n \\in \\mathbf{Z}$ and two out of three of $K_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "10128", "title": "more-algebra-lemma-derived-tor-quasi-isomorphism", "text": "Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex of flat $R$-modules. Then $P^\\bullet$ is K-flat."}
+{"_id": "10129", "title": "more-algebra-lemma-colimit-K-flat", "text": "Let $R$ be a ring. Let $K_1^\\bullet \\to K_2^\\bullet \\to \\ldots$ be a system of K-flat complexes. Then $\\colim_i K_i^\\bullet$ is K-flat. More generally any filtered colimit of K-flat complexes is K-flat."}
+{"_id": "10130", "title": "more-algebra-lemma-universally-acyclic-K-flat", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules. If $K^\\bullet \\otimes_R M$ is acyclic for all finitely presented $R$-modules $M$, then $K^\\bullet$ is K-flat."}
+{"_id": "10131", "title": "more-algebra-lemma-K-flat-resolution", "text": "Let $R$ be a ring. For any complex $M^\\bullet$ there exists a K-flat complex $K^\\bullet$ whose terms are flat $R$-modules and a quasi-isomorphism $K^\\bullet \\to M^\\bullet$ which is termwise surjective."}
+{"_id": "10132", "title": "more-algebra-lemma-derived-tor-quasi-isomorphism-other-side", "text": "Let $R$ be a ring. Let $\\alpha : P^\\bullet \\to Q^\\bullet$ be a quasi-isomorphism of K-flat complexes of $R$-modules. For every complex $L^\\bullet$ of $R$-modules the induced map $$ \\text{Tot}(\\text{id}_L \\otimes \\alpha) : \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet) \\longrightarrow \\text{Tot}(L^\\bullet \\otimes_R Q^\\bullet) $$ is a quasi-isomorphism."}
+{"_id": "10133", "title": "more-algebra-lemma-flip-douoble-tensor-product", "text": "Let $R$ be a ring. Let $K^\\bullet, L^\\bullet$ be complexes of $R$-modules. There is a canonical isomorphism $$ K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet \\longrightarrow L^\\bullet \\otimes_R^\\mathbf{L} K^\\bullet $$ functorial in both complexes which uses a sign of $(-1)^{pq}$ for the map $K^p \\otimes_R L^q \\to L^q \\otimes_R K^p$ (see proof for explanation)."}
+{"_id": "10134", "title": "more-algebra-lemma-triple-tensor-product", "text": "Let $R$ be a ring. Let $K^\\bullet, L^\\bullet, M^\\bullet$ be complexes of $R$-modules. There is a canonical isomorphism $$ (K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet) \\otimes_R^\\mathbf{L} M^\\bullet = K^\\bullet \\otimes_R^\\mathbf{L} (L^\\bullet \\otimes_R^\\mathbf{L} M^\\bullet) $$ functorial in all three complexes."}
+{"_id": "10136", "title": "more-algebra-lemma-derived-base-change", "text": "The construction above is independent of choices and defines an exact functor of triangulated categories $- \\otimes_R^\\mathbf{L} N^\\bullet : D(R) \\to D(A)$. There is a functorial isomorphism $$ E^\\bullet \\otimes_R^\\mathbf{L} N^\\bullet = (E^\\bullet \\otimes_R^\\mathbf{L} A) \\otimes_A^\\mathbf{L} N^\\bullet $$ for $E^\\bullet$ in $D(R)$."}
+{"_id": "10137", "title": "more-algebra-lemma-functoriality-derived-base-change", "text": "Let $R \\to A$ be a ring map. Let $f : L^\\bullet \\to N^\\bullet$ be a map of complexes of $A$-modules. Then $f$ induces a transformation of functors $$ 1 \\otimes f : - \\otimes_A^\\mathbf{L} L^\\bullet \\longrightarrow - \\otimes_A^\\mathbf{L} N^\\bullet $$ If $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism of functors."}
+{"_id": "10138", "title": "more-algebra-lemma-double-base-change", "text": "Let $A \\to B \\to C$ be ring maps. Let $N^\\bullet$ be a complex of $B$-modules and $K^\\bullet$ a complex of $C$-modules. The compositions of the functors $$ D(A) \\xrightarrow{- \\otimes_A^\\mathbf{L} N^\\bullet} D(B) \\xrightarrow{- \\otimes_B^\\mathbf{L} K^\\bullet} D(C) $$ is the functor $- \\otimes_A^\\mathbf{L} (N^\\bullet \\otimes_B^\\mathbf{L} K^\\bullet) : D(A) \\to D(C)$. If $M$, $N$, $K$ are modules over $A$, $B$, $C$, then we have $$ (M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K = M \\otimes_A^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K) = (M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K) $$ in $D(C)$. We also have a canonical isomorphism $$ (M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K \\longrightarrow (M \\otimes_A^\\mathbf{L} K) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C) $$ using signs. Similar results holds for complexes."}
+{"_id": "10139", "title": "more-algebra-lemma-base-change-comparison", "text": "The comparison map (\\ref{equation-comparison-map}) is an isomorphism if $A' = A \\otimes_R R'$ and $A$ and $R'$ are Tor independent over $R$."}
+{"_id": "10140", "title": "more-algebra-lemma-tor-independent-flat", "text": "Consider a commutative diagram of rings $$ \\xymatrix{ A' & R' \\ar[r] \\ar[l] & B' \\\\ A \\ar[u] & R \\ar[l] \\ar[u] \\ar[r] & B \\ar[u] } $$ Assume that $R'$ is flat over $R$ and $A'$ is flat over $A \\otimes_R R'$ and $B'$ is flat over $R' \\otimes_R B$. Then $$ \\text{Tor}_i^R(A, B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B') = \\text{Tor}_i^{R'}(A', B') $$"}
+{"_id": "10141", "title": "more-algebra-lemma-flat-base-change-tor-independent", "text": "Let $R \\to A$ and $R \\to B$ be ring maps. Let $R \\to R'$ be a ring map and set $A' = A \\otimes_R R'$ and $B' = B \\otimes_R R'$. If $A$ and $B$ are tor independent over $R$ and $R \\to R'$ is flat, then $A'$ and $B'$ are tor independent over $R'$."}
+{"_id": "10142", "title": "more-algebra-lemma-lemma-tor-independent-flat-compare", "text": "Assumptions as in Lemma \\ref{lemma-tor-independent-flat}. For $M \\in D(A)$ there are canonical isomorphisms $$ H^i((M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B') = H^i(M \\otimes_R^\\mathbf{L} B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B') $$ of $A' \\otimes_{R'} B'$-modules."}
+{"_id": "10143", "title": "more-algebra-lemma-tor-independent", "text": "Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. The following are equivalent \\begin{enumerate} \\item $A$ and $B$ are Tor independent over $R$, \\item for every pair of primes $\\mathfrak p \\subset A$ and $\\mathfrak q \\subset B$ lying over the same prime $\\mathfrak r \\subset R$ the rings $A_\\mathfrak p$ and $B_\\mathfrak q$ are Tor independent over $R_\\mathfrak r$, and \\item For every prime $\\mathfrak s$ of $A \\otimes_R B$ the module $$ \\text{Tor}_i^R(A, B)_\\mathfrak s = \\text{Tor}_i^{R_\\mathfrak r}(A_\\mathfrak p, B_\\mathfrak q)_\\mathfrak s $$ (where $\\mathfrak p = A \\cap \\mathfrak s$, $\\mathfrak q = B \\cap \\mathfrak s$ and $\\mathfrak r = R \\cap \\mathfrak s$) is zero. \\end{enumerate}"}
+{"_id": "10144", "title": "more-algebra-lemma-functoriality-product-tor", "text": "Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \\to C$ be an $R$-algebra map. Then the induced map $$ \\text{Tor}^R_{\\star}(B, A) \\longrightarrow \\text{Tor}^R_{\\star}(C, A) $$ is an $A$-algebra homomorphism."}
+{"_id": "10145", "title": "more-algebra-lemma-cone-pseudo-coherent", "text": "Let $R$ be a ring and $m \\in \\mathbf{Z}$. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. \\begin{enumerate} \\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent and $L^\\bullet$ is $m$-pseudo-coherent then $M^\\bullet$ is $m$-pseudo-coherent. \\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent, then $L^\\bullet$ is $m$-pseudo-coherent. \\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent and $M^\\bullet$ is $m$-pseudo-coherent, then $K^\\bullet$ is $(m + 1)$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "10146", "title": "more-algebra-lemma-finite-cohomology", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$ for $i > m$, then $H^m(K^\\bullet)$ is a finite type $R$-module. \\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$ for $i > m + 1$, then $H^{m + 1}(K^\\bullet)$ is a finitely presented $R$-module. \\end{enumerate}"}
+{"_id": "10147", "title": "more-algebra-lemma-n-pseudo-module", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Then \\begin{enumerate} \\item $M$ is $0$-pseudo-coherent if and only if $M$ is a finite $R$-module, \\item $M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely presented $R$-module, \\item $M$ is $(-d)$-pseudo-coherent if and only if there exists a resolution $$ R^{\\oplus a_d} \\to R^{\\oplus a_{d - 1}} \\to \\ldots \\to R^{\\oplus a_0} \\to M \\to 0 $$ of length $d$, and \\item $M$ is pseudo-coherent if and only if there exists an infinite resolution $$ \\ldots \\to R^{\\oplus a_1} \\to R^{\\oplus a_0} \\to M \\to 0 $$ by finite free $R$-modules. \\end{enumerate}"}
+{"_id": "10148", "title": "more-algebra-lemma-pseudo-coherent", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ is pseudo-coherent, \\item $K^\\bullet$ is $m$-pseudo-coherent for every $m \\in \\mathbf{Z}$, and \\item $K^\\bullet$ is quasi-isomorphic to a bounded above complex of finite projective $R$-modules. \\end{enumerate} If (1), (2), and (3) hold and $H^i(K^\\bullet) = 0$ for $i > b$, then we can find a quasi-isomorphism $F^\\bullet \\to K^\\bullet$ with $F^i$ finite free $R$-modules and $F^i = 0$ for $i > b$."}
+{"_id": "10149", "title": "more-algebra-lemma-two-out-of-three-pseudo-coherent", "text": "Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. If two out of three of $K^\\bullet, L^\\bullet, M^\\bullet$ are pseudo-coherent then the third is also pseudo-coherent."}
+{"_id": "10150", "title": "more-algebra-lemma-recognize-pseudo-coherent", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $H^i(K^\\bullet) = 0$ for all $i \\geq m$, then $K^\\bullet$ is $m$-pseudo-coherent. \\item If $H^i(K^\\bullet) = 0$ for $i > m$ and $H^m(K^\\bullet)$ is a finite $R$-module, then $K^\\bullet$ is $m$-pseudo-coherent. \\item If $H^i(K^\\bullet) = 0$ for $i > m + 1$, the module $H^{m + 1}(K^\\bullet)$ is of finite presentation, and $H^m(K^\\bullet)$ is of finite type, then $K^\\bullet$ is $m$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "10151", "title": "more-algebra-lemma-summands-pseudo-coherent", "text": "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. If $K^\\bullet \\oplus L^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) so are $K^\\bullet$ and $L^\\bullet$."}
+{"_id": "10152", "title": "more-algebra-lemma-complex-pseudo-coherent-modules", "text": "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded above complex of $R$-modules such that $K^i$ is $(m - i)$-pseudo-coherent for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent. In particular, if $K^\\bullet$ is a bounded above complex of pseudo-coherent $R$-modules, then $K^\\bullet$ is pseudo-coherent."}
+{"_id": "10153", "title": "more-algebra-lemma-cohomology-pseudo-coherent", "text": "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet \\in D^{-}(R)$ such that $H^i(K^\\bullet)$ is $(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)."}
+{"_id": "10154", "title": "more-algebra-lemma-finite-push-pseudo-coherent", "text": "Let $A \\to B$ be a ring map. Assume that $B$ is pseudo-coherent as an $A$-module. Let $K^\\bullet$ be a complex of $B$-modules. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ is $m$-pseudo-coherent as a complex of $B$-modules, and \\item $K^\\bullet$ is $m$-pseudo-coherent as a complex of $A$-modules. \\end{enumerate} The same equivalence holds for pseudo-coherence."}
+{"_id": "10155", "title": "more-algebra-lemma-pull-pseudo-coherent", "text": "Let $A \\to B$ be a ring map. Let $K^\\bullet$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is an $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $B$-modules."}
+{"_id": "10156", "title": "more-algebra-lemma-flat-base-change-pseudo-coherent", "text": "Let $A \\to B$ be a flat ring map. Let $M$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent) $A$-module. Then $M \\otimes_A B$ is an $m$-pseudo-coherent (resp.\\ pseudo-coherent) $B$-module."}
+{"_id": "10157", "title": "more-algebra-lemma-glue-pseudo-coherent", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which generate the unit ideal. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $R$-modules. If for each $i$ the complex $K^\\bullet \\otimes_R R_{f_i}$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)."}
+{"_id": "10158", "title": "more-algebra-lemma-flat-descent-pseudo-coherent", "text": "Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat ring map. If the complex $K^\\bullet \\otimes_R R'$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)."}
+{"_id": "10159", "title": "more-algebra-lemma-tensor-pseudo-coherent", "text": "Let $R$ be a ring. Let $K, L$ be objects of $D(R)$. \\begin{enumerate} \\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then $K \\otimes_R^\\mathbf{L} L$ is $t$-pseudo-coherent with $t = \\max(m + a, n + b)$. \\item If $K$ and $L$ are pseudo-coherent, then $K \\otimes_R^\\mathbf{L} L$ is pseudo-coherent. \\end{enumerate}"}
+{"_id": "10160", "title": "more-algebra-lemma-Noetherian-pseudo-coherent", "text": "Let $R$ be a Noetherian ring. Then \\begin{enumerate} \\item A complex of $R$-modules $K^\\bullet$ is $m$-pseudo-coherent if and only if $K^\\bullet \\in D^{-}(R)$ and $H^i(K^\\bullet)$ is a finite $R$-module for $i \\geq m$. \\item A complex of $R$-modules $K^\\bullet$ is pseudo-coherent if and only if $K^\\bullet \\in D^{-}(R)$ and $H^i(K^\\bullet)$ is a finite $R$-module for all $i$. \\item An $R$-module is pseudo-coherent if and only if it is finite. \\end{enumerate}"}
+{"_id": "10161", "title": "more-algebra-lemma-coherent-pseudo-coherent", "text": "Let $R$ be a coherent ring (Algebra, Definition \\ref{algebra-definition-coherent}). Let $K \\in D^-(R)$. The following are equivalent \\begin{enumerate} \\item $K$ is $m$-pseudo-coherent, \\item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is coherent for $i > m$, and \\item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is finitely presented for $i > m$. \\end{enumerate} Thus $K$ is pseudo-coherent if and only if $H^i(K)$ is a coherent module for all $i$."}
+{"_id": "10162", "title": "more-algebra-lemma-pseudo-coherence-colimit-ext", "text": "Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of $R$-modules. Let $K \\in D(R)$ be $m$-pseudo-coherent. Then $\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and $\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective."}
+{"_id": "10164", "title": "more-algebra-lemma-pseudo-coherence-and-ext", "text": "Let $R$ be a ring. Let $L$, $M$, $N$ be $R$-modules. \\begin{enumerate} \\item If $M$ is finitely presented and $L$ is flat, then the canonical map $\\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L)$ is an isomorphism. \\item If $M$ is $(-m)$-pseudo-coherent and $L$ is flat, then the canonical map $\\Ext^i_R(M, N) \\otimes_R L \\to \\Ext^i_R(M, N \\otimes_R L)$ is an isomorphism for $i < m$. \\end{enumerate}"}
+{"_id": "10165", "title": "more-algebra-lemma-pseudo-coherence-and-base-change-ext", "text": "Let $R \\to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules. \\begin{enumerate} \\item If $M$ is a finitely presented $R$-module, then $\\Hom_R(M, N) \\otimes_R R' = \\Hom_{R'}(M \\otimes_R R', N \\otimes_R R')$. \\item If $M$ is $(-m)$-pseudo-coherent, then $\\Ext^i_R(M, N) \\otimes_R R' = \\Ext^i_{R'}(M \\otimes_R R', N \\otimes_R R')$ for $i < m$. \\end{enumerate} In particular if $R$ is Noetherian and $M$ is a finite module this holds for all $i$."}
+{"_id": "10166", "title": "more-algebra-lemma-pseudo-coherent-tensor", "text": "Let $R$ be a ring. Let $K \\in D^-(R)$. The following are equivalent: \\begin{enumerate} \\item $K$ is pseudo-coherent, \\item for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the canonical map $$ \\alpha : K \\otimes_R^\\mathbf{L} \\left( \\prod\\nolimits_\\alpha Q_{\\alpha} \\right) \\longrightarrow \\prod\\nolimits_\\alpha (K \\otimes_R^\\mathbf{L} Q_{\\alpha}) $$ is an isomorphism in $D(A)$, \\item for every $R$-module $Q$ and every set $A$, the canonical map $$ \\beta : K \\otimes_R^\\mathbf{L} Q^A \\longrightarrow (K \\otimes_R^\\mathbf{L} Q)^A $$ is an isomorphism in $D(A)$, and \\item for every set $A$, the canonical map $$ \\gamma : K \\otimes_R^\\mathbf{L} R^A \\longrightarrow K^A $$ is an isomorphism in $D(A)$. \\end{enumerate} Given $m \\in \\mathbf{Z}$ the following are equivalent \\begin{enumerate} \\item[(a)] $K$ is $m$-pseudo-coherent, \\item[(b)] for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, with $\\alpha$ as above $H^i(\\alpha)$ is an isomorphism for $i > m$ and surjective for $i = m$, \\item[(c)] for every $R$-module $Q$ and every set $A$, with $\\beta$ as above $H^i(\\beta)$ is an isomorphism for $i > m$ and surjective for $i = m$, \\item[(d)] for every set $A$, with $\\gamma$ as above $H^i(\\gamma)$ is an isomorphism for $i > m$ and surjective for $i = m$. \\end{enumerate}"}
+{"_id": "10167", "title": "more-algebra-lemma-detect-cohomology-pseudo-coherent", "text": "Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent. Let $i \\in \\mathbf{Z}$. There exists a finitely presented $R$-module $M$ and a map $K \\to M[-i]$ in $D(R)$ which induces an injection $H^i(K) \\to M$."}
+{"_id": "10168", "title": "more-algebra-lemma-detect-cohomology", "text": "Let $A$ be a Noetherian ring. Let $K \\in D(A)$ be pseudo-coherent, i.e., $K \\in D^-(A)$ with finite cohomology modules. Let $\\mathfrak m$ be a maximal ideal of $A$. If $H^i(K)/\\mathfrak m H^i(K) \\not = 0$, then there exists a finite $A$-module $E$ annihilated by a power of $\\mathfrak m$ and a map $K \\to E[-i]$ which is nonzero on $H^i(K)$."}
+{"_id": "10169", "title": "more-algebra-lemma-last-one-flat", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a bounded above complex of flat $R$-modules with tor-amplitude in $[a, b]$. Then $\\Coker(d_K^{a - 1})$ is a flat $R$-module."}
+{"_id": "10170", "title": "more-algebra-lemma-tor-amplitude", "text": "Let $R$ be a ring. Let $K^\\bullet$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ has tor-amplitude in $[a, b]$. \\item $K^\\bullet$ is quasi-isomorphic to a complex $E^\\bullet$ of flat $R$-modules with $E^i = 0$ for $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "10171", "title": "more-algebra-lemma-bounded-below-tor-amplitude", "text": "Let $R$ be a ring. Let $a \\in \\mathbf{Z}$ and let $K$ be an object of $D(R)$. The following are equivalent \\begin{enumerate} \\item $K$ has tor-amplitude in $[a, \\infty]$, and \\item $K$ is quasi-isomorphic to a K-flat complex $E^\\bullet$ whose terms are flat $R$-modules with $E^i = 0$ for $i \\not \\in [a, \\infty]$. \\end{enumerate}"}
+{"_id": "10172", "title": "more-algebra-lemma-cone-tor-amplitude", "text": "Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. Let $a, b \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and $L^\\bullet$ has tor-amplitude in $[a, b]$ then $M^\\bullet$ has tor-amplitude in $[a, b]$. \\item If $K^\\bullet, M^\\bullet$ have tor-amplitude in $[a, b]$, then $L^\\bullet$ has tor-amplitude in $[a, b]$. \\item If $L^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and $M^\\bullet$ has tor-amplitude in $[a, b]$, then $K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$. \\end{enumerate}"}
+{"_id": "10173", "title": "more-algebra-lemma-tor-dimension", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \\geq 0$. The following are equivalent \\begin{enumerate} \\item $M$ has tor dimension $\\leq d$, and \\item there exists a resolution $$ 0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0 $$ with $F_i$ a flat $R$-module. \\end{enumerate} In particular an $R$-module has tor dimension $0$ if and only if it is a flat $R$-module."}
+{"_id": "10174", "title": "more-algebra-lemma-summands-tor-amplitude", "text": "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. If $K^\\bullet \\oplus L^\\bullet$ has tor amplitude in $[a, b]$ so do $K^\\bullet$ and $L^\\bullet$."}
+{"_id": "10175", "title": "more-algebra-lemma-complex-finite-tor-dimension-modules", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of $R$-modules such that $K^i$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\\bullet$ has tor amplitude in $[a, b]$. In particular if $K^\\bullet$ is a finite complex of $R$-modules of finite tor dimension, then $K^\\bullet$ has finite tor dimension."}
+{"_id": "10176", "title": "more-algebra-lemma-cohomology-tor-amplitude", "text": "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet \\in D^b(R)$ such that $H^i(K^\\bullet)$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\\bullet$ has tor amplitude in $[a, b]$. In particular if $K^\\bullet \\in D^b(R)$ and all its cohomology groups have finite tor dimension then $K^\\bullet$ has finite tor dimension."}
+{"_id": "10177", "title": "more-algebra-lemma-push-tor-amplitude", "text": "Let $A \\to B$ be a ring map. Let $K^\\bullet$ and $L^\\bullet$ be complexes of $B$-modules. Let $a, b, c, d \\in \\mathbf{Z}$. If \\begin{enumerate} \\item $K^\\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, \\item $L^\\bullet$ as a complex of $A$-modules has tor amplitude in $[c, d]$, \\end{enumerate} then $K^\\bullet \\otimes^\\mathbf{L}_B L^\\bullet$ as a complex of $A$-modules has tor amplitude in $[a + c, b + d]$."}
+{"_id": "10178", "title": "more-algebra-lemma-flat-push-tor-amplitude", "text": "Let $A \\to B$ be a ring map. Assume that $B$ is flat as an $A$-module. Let $K^\\bullet$ be a complex of $B$-modules. Let $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of $A$-modules has tor amplitude in $[a, b]$."}
+{"_id": "10179", "title": "more-algebra-lemma-finite-tor-dimension-push-tor-amplitude", "text": "Let $A \\to B$ be a ring map. Assume that $B$ has tor dimension $\\leq d$ as an $A$-module. Let $K^\\bullet$ be a complex of $B$-modules. Let $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of $A$-modules has tor amplitude in $[a - d, b]$."}
+{"_id": "10180", "title": "more-algebra-lemma-pull-tor-amplitude", "text": "Let $A \\to B$ be a ring map. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $A$-modules with tor amplitude in $[a, b]$. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ as a complex of $B$-modules has tor amplitude in $[a, b]$."}
+{"_id": "10181", "title": "more-algebra-lemma-flat-base-change-finite-tor-dimension", "text": "Let $A \\to B$ be a flat ring map. Let $d \\geq 0$. Let $M$ be an $A$-module of tor dimension $\\leq d$. Then $M \\otimes_A B$ is a $B$-module of tor dimension $\\leq d$."}
+{"_id": "10182", "title": "more-algebra-lemma-tor-amplitude-localization", "text": "Let $A  \\to B$ be a ring map. Let $K^\\bullet$ be a complex of $B$-modules. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules, \\item $K^\\bullet_\\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $A_\\mathfrak p$-modules for every prime $\\mathfrak q \\subset B$ with $\\mathfrak p = A \\cap \\mathfrak q$, \\item $K^\\bullet_\\mathfrak m$ has tor amplitude in $[a, b]$ as a complex of $A_\\mathfrak p$-modules for every maximal ideal $\\mathfrak m \\subset B$ with $\\mathfrak p = A \\cap \\mathfrak m$. \\end{enumerate}"}
+{"_id": "10183", "title": "more-algebra-lemma-glue-tor-amplitude", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which generate the unit ideal. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $R$-modules. If for each $i$ the complex $K^\\bullet \\otimes_R R_{f_i}$ has tor amplitude in $[a, b]$, then $K^\\bullet$ has tor amplitude in $[a, b]$."}
+{"_id": "10184", "title": "more-algebra-lemma-flat-descent-tor-amplitude", "text": "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat ring map. If the complex $K^\\bullet \\otimes_R R'$ has tor amplitude in $[a, b]$, then $K^\\bullet$ has tor amplitude in $[a, b]$."}
+{"_id": "10185", "title": "more-algebra-lemma-no-change-tor-amplitude", "text": "Given ring maps $R \\to A \\to B$ with $A \\to B$ faithfully flat and $K \\in D(A)$ the tor amplitude of $K$ over $R$ is the same as the tor amplitude of $K \\otimes_A^\\mathbf{L} B$ over $R$."}
+{"_id": "10186", "title": "more-algebra-lemma-finite-gl-dim-tor-dimension", "text": "Let $R$ be a ring of finite global dimension $d$. Then \\begin{enumerate} \\item every module has tor dimension $\\leq d$, \\item a complex of $R$-modules $K^\\bullet$ with $H^i(K^\\bullet) \\not = 0$ only if $i \\in [a, b]$ has tor amplitude in $[a - d, b]$, and \\item a complex of $R$-modules $K^\\bullet$ has finite tor dimension if and only if $K^\\bullet \\in D^b(R)$. \\end{enumerate}"}
+{"_id": "10187", "title": "more-algebra-lemma-projective-amplitude", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $K$ has projective-amplitude in $[a, b]$, \\item $\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i \\not \\in [-b, -a]$, \\item $H^n(K) = 0$ for $n > b$ and $\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i > -a$, and \\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and $\\Ext^{-a + 1}_R(K, N) = 0$ for all $R$-modules $N$. \\end{enumerate}"}
+{"_id": "10188", "title": "more-algebra-lemma-injective-amplitude", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $K$ has injective-amplitude in $[a, b]$, \\item $\\Ext^i_R(N, K) = 0$ for all $R$-modules $N$ and all $i \\not \\in [a, b]$, \\item $\\Ext^i(R/I, K) = 0$ for all ideals $I \\subset R$ and all $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "10189", "title": "more-algebra-lemma-finite-injective-dimension", "text": "Let $R$ be a ring. Let $K \\in D(R)$. \\begin{enumerate} \\item If $K$ is in $D^b(R)$ and $H^i(K)$ has finite injective dimension for all $i$, then $K$ has finite injective dimension. \\item If $K^\\bullet$ represents $K$, is a bounded complex of $R$-modules, and $K^i$ has finite injective dimension for all $i$, then $K$ has finite injective dimension. \\end{enumerate}"}
+{"_id": "10190", "title": "more-algebra-lemma-finite-injective-dimension-Noetherian-radical", "text": "Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal contained in the Jacobson radical of $R$. Let $K \\in D^+(R)$ have finite cohomology modules. Then the following are equivalent \\begin{enumerate} \\item $K$ has finite injective dimension, and \\item there exists a $b$ such that $\\Ext^i_R(R/J, K) = 0$ for $i > b$ and any ideal $J \\supset I$. \\end{enumerate}"}
+{"_id": "10191", "title": "more-algebra-lemma-finite-injective-dimension-Noetherian-local", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local Noetherian ring. Let $K \\in D^+(R)$ have finite cohomology modules. Then the following are equivalent \\begin{enumerate} \\item $K$ has finite injective dimension, and \\item $\\Ext^i_R(\\kappa, K) = 0$ for $i \\gg 0$. \\end{enumerate}"}
+{"_id": "10192", "title": "more-algebra-lemma-ideal-factor-through-projective", "text": "Let $R$ be a ring. Let $M$, $N$ be $R$-modules. \\begin{enumerate} \\item Given an $R$-module map $\\varphi : M \\to N$ the following are equivalent: (a) $\\varphi$ factors through a projective $R$-module, and (b) $\\varphi$ factors through a free $R$-module. \\item The set of $\\varphi : M \\to N$ satisfying the equivalent conditions of (1) is an $R$-submodule of $\\Hom_R(M, N)$. \\item Given maps $\\psi : M' \\to M$ and $\\xi : N \\to N'$, if $\\varphi : M \\to N$ satisfies the equivalent conditions of (1), then $\\xi \\circ \\varphi \\circ \\psi : M' \\to N'$ does too. \\end{enumerate}"}
+{"_id": "10194", "title": "more-algebra-lemma-near-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. The following conditions are equivalent \\begin{enumerate} \\item for every $a \\in I$ the map $a : M \\to M$ factors through a projective $R$-module, \\item for every $a \\in I$ the map $a : M \\to M$ factors through a free $R$-module, and \\item $\\Ext^1_R(M, N)$ is annihilated by $I$ for every $R$-module $N$. \\end{enumerate}"}
+{"_id": "10195", "title": "more-algebra-lemma-torsion-near-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. If $M$ is annihilated by $I$, then $M$ is $I$-projective."}
+{"_id": "10196", "title": "more-algebra-lemma-ses-near-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $$ 0 \\to K \\to P \\to M \\to 0 $$ be a short exact sequence of $R$-modules. If $M$ is $I$-projective and $P$ is projective, then $K$ is $I$-projective."}
+{"_id": "10197", "title": "more-algebra-lemma-dual-near-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. If $M$ is a finite, $I$-projective $R$-module, then $M^\\vee = \\Hom_R(M, R)$ is $I$-projective."}
+{"_id": "10198", "title": "more-algebra-lemma-compose", "text": "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$ of $R$-modules there is a canonical isomorphism $$ \\Hom^\\bullet(K^\\bullet, \\Hom^\\bullet(L^\\bullet, M^\\bullet)) = \\Hom^\\bullet(\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet), M^\\bullet) $$ of complexes of $R$-modules."}
+{"_id": "10199", "title": "more-algebra-lemma-composition", "text": "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$ of $R$-modules there is a canonical morphism $$ \\text{Tot}\\left( \\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R \\Hom^\\bullet(K^\\bullet, L^\\bullet) \\right) \\longrightarrow \\Hom^\\bullet(K^\\bullet, M^\\bullet) $$ of complexes of $R$-modules."}
+{"_id": "10200", "title": "more-algebra-lemma-diagonal-better", "text": "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$ of $R$-modules there is a canonical morphism $$ \\text{Tot}(K^\\bullet \\otimes_R \\Hom^\\bullet(M^\\bullet, L^\\bullet)) \\longrightarrow \\Hom^\\bullet(M^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)) $$ of complexes of $R$-modules functorial in all three complexes."}
+{"_id": "10201", "title": "more-algebra-lemma-diagonal", "text": "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet$ of $R$-modules there is a canonical morphism $$ K^\\bullet \\longrightarrow \\Hom^\\bullet(L^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)) $$ of complexes of $R$-modules functorial in both complexes."}
+{"_id": "10202", "title": "more-algebra-lemma-evaluate-and-more", "text": "Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$ of $R$-modules there is a canonical morphism $$ \\text{Tot}(\\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R K^\\bullet) \\longrightarrow \\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, L^\\bullet), M^\\bullet) $$ of complexes of $R$-modules functorial in all three complexes."}
+{"_id": "10203", "title": "more-algebra-lemma-symmetric-monoidal-cat-complexes", "text": "Let $R$ be a ring. The category of complexes of $R$-modules with tensor product defined by $K^\\bullet \\otimes M^\\bullet = \\text{Tot}(K^\\bullet \\otimes_R M^\\bullet)$ is a symmetric monoidal category with the associativity and commutativity constraints given above."}
+{"_id": "10204", "title": "more-algebra-lemma-left-dual-module", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let  $N, \\eta, \\epsilon$ be a left dual of $M$ in the monoidal category of $R$-modules, see Categories, Definition \\ref{categories-definition-dual}. Then \\begin{enumerate} \\item $M$ and $N$ are finite projective $R$-modules, \\item the map $e : \\Hom_R(M, R) \\to N$, $\\lambda \\mapsto (\\lambda \\otimes 1)(\\eta)$ is an isomorphism, \\item we have $\\epsilon(n, m) = e^{-1}(n)(m)$ for $n \\in N$ and $m \\in M$. \\end{enumerate}"}
+{"_id": "10205", "title": "more-algebra-lemma-left-dual-complex", "text": "Let $R$ be a ring. Let $M^\\bullet$ be a complex of $R$-modules. Let $N^\\bullet, \\eta, \\epsilon$ be a left dual of $M^\\bullet$ in the monoidal category of complexes of $R$-modules. Then \\begin{enumerate} \\item $M^\\bullet$ and $N^\\bullet$ are bounded, \\item $M^n$ and $N^n$ are finite projective $R$-modules, \\item writing $\\epsilon = \\sum \\epsilon_n$ with $\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ and $\\eta = \\sum \\eta_n$ with $\\eta_n : R \\to  M^n \\otimes_R N^{-n}$ then $(N^{-n}, \\eta_n, \\epsilon_n)$ is the left dual of $M^n$ as in Lemma \\ref{lemma-left-dual-module}, \\item the differential $d_N^n : N^n \\to N^{n + 1}$ is equal to $-(-1)^n$ times the map $$ N^n = \\Hom_R(M^{-n}, R) \\xrightarrow{d_M^{-n - 1}} \\Hom_R(M^{-n - 1}, R) = N^{n + 1} $$ where the equality signs are the identifications from Lemma \\ref{lemma-left-dual-module} part (2). \\end{enumerate} Conversely, given a bounded complex $M^\\bullet$ of finite projective $R$-modules, setting $N^n = \\Hom_R(M^{-n}, R)$ with differentials as above, setting $\\epsilon = \\sum \\epsilon_n$ with $\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ given by evaluation, and setting $\\eta = \\sum \\eta_n$ with $\\eta_n : R \\to M^n \\otimes_R N^{-n}$ mapping $1$ to $\\text{id}_{M_n}$ we obtain a left dual of $M^\\bullet$ in the monoidal category of complexes of $R$-modules."}
+{"_id": "10206", "title": "more-algebra-lemma-internal-hom", "text": "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism $$ R\\Hom_R(K, R\\Hom_R(L, M)) = R\\Hom_R(K \\otimes_R^\\mathbf{L} L, M) $$ in $D(R)$ functorial in $K, L, M$ which recovers (\\ref{equation-internal-hom}) by taking $H^0$."}
+{"_id": "10207", "title": "more-algebra-lemma-RHom-out-of-projective", "text": "Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex of projective $R$-modules. Let $L^\\bullet$ be a complex of $R$-modules. Then $R\\Hom_R(P^\\bullet, L^\\bullet)$ is represented by the complex $\\Hom^\\bullet(P^\\bullet, L^\\bullet)$."}
+{"_id": "10208", "title": "more-algebra-lemma-internal-hom-evaluate", "text": "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism $$ R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M) $$ in $D(R)$ functorial in $K, L, M$."}
+{"_id": "10210", "title": "more-algebra-lemma-internal-hom-diagonal-better", "text": "Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism $$ K \\otimes_R^\\mathbf{L} R\\Hom_R(M, L) \\longrightarrow R\\Hom_R(M, K \\otimes_R^\\mathbf{L} L) $$ in $D(R)$ functorial in $K$, $L$, $M$."}
+{"_id": "10211", "title": "more-algebra-lemma-internal-hom-diagonal", "text": "Let $R$ be a ring. Given complexes $K, L$ in $D(R)$ there is a canonical morphism $$ K \\longrightarrow R\\Hom_R(L, K \\otimes_R^\\mathbf{L} L) $$ in $D(R)$ functorial in both $K$ and $L$."}
+{"_id": "10212", "title": "more-algebra-lemma-perfect", "text": "Let $K^\\bullet$ be an object of $D(R)$. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ is perfect, and \\item $K^\\bullet$ is pseudo-coherent and has finite tor dimension. \\end{enumerate} If (1) and (2) hold and $K^\\bullet$ has tor-amplitude in $[a, b]$, then $K^\\bullet$ is quasi-isomorphic to a complex $E^\\bullet$ of finite projective $R$-modules with $E^i = 0$ for $i \\not \\in [a, b]$."}
+{"_id": "10213", "title": "more-algebra-lemma-perfect-module", "text": "Let $M$ be a module over a ring $R$. The following are equivalent \\begin{enumerate} \\item $M$ is a perfect module, and \\item there exists a resolution $$ 0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0 $$ with each $F_i$ a finite projective $R$-module. \\end{enumerate}"}
+{"_id": "10214", "title": "more-algebra-lemma-two-out-of-three-perfect", "text": "Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. If two out of three of $K^\\bullet, L^\\bullet, M^\\bullet$ are perfect then the third is also perfect."}
+{"_id": "10215", "title": "more-algebra-lemma-summands-perfect", "text": "Let $R$ be a ring. If $K^\\bullet \\oplus L^\\bullet$ is perfect, then so are $K^\\bullet$ and $L^\\bullet$."}
+{"_id": "10217", "title": "more-algebra-lemma-cohomology-perfect", "text": "Let $R$ be a ring. If $K^\\bullet \\in D^b(R)$ and all its cohomology modules are perfect, then $K^\\bullet$ is perfect."}
+{"_id": "10219", "title": "more-algebra-lemma-pull-perfect", "text": "Let $A \\to B$ be a ring map. Let $K^\\bullet$ be a perfect complex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is a perfect complex of $B$-modules."}
+{"_id": "10221", "title": "more-algebra-lemma-glue-perfect", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which generate the unit ideal. Let $K^\\bullet$ be a complex of $R$-modules. If for each $i$ the complex $K^\\bullet \\otimes_R R_{f_i}$ is perfect, then $K^\\bullet$ is perfect."}
+{"_id": "10222", "title": "more-algebra-lemma-flat-descent-perfect", "text": "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat ring map. If the complex $K^\\bullet \\otimes_R R'$ is perfect, then $K^\\bullet$ is perfect."}
+{"_id": "10223", "title": "more-algebra-lemma-regular-perfect", "text": "Let $R$ be a regular ring of finite dimension. Then \\begin{enumerate} \\item an $R$-module is perfect if and only if it is a finite $R$-module, and \\item a complex of $R$-modules $K^\\bullet$ is perfect if and only if $K^\\bullet \\in D^b(R)$ and each $H^i(K^\\bullet)$ is a finite $R$-module. \\end{enumerate}"}
+{"_id": "10224", "title": "more-algebra-lemma-dual-perfect-complex", "text": "Let $A$ be a ring. Let $K \\in D(A)$ be perfect. Then $K^\\vee = R\\Hom_A(K, A)$ is a perfect complex and $K \\cong (K^\\vee)^\\vee$. There are functorial isomorphisms $$ L \\otimes_A^\\mathbf{L} K^\\vee = R\\Hom_A(K, L) \\quad\\text{and}\\quad H^0(L \\otimes_A^\\mathbf{L} K^\\vee) = \\Ext_A^0(K, L) $$ for $L \\in D(A)$."}
+{"_id": "10226", "title": "more-algebra-lemma-colimit-perfect-complexes", "text": "Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings. \\begin{enumerate} \\item Given a perfect $K$ in $D(R)$ there exists an $i \\in I$ and a perfect $K_i$ in $D(R_i)$ such that $K \\cong K_i \\otimes_{R_i}^\\mathbf{L} R$ in $D(R)$. \\item Given $0 \\in I$ and $K_0, L_0 \\in D(R_0)$ with $K_0$ perfect, we have $$ \\Hom_{D(R)}(K_0 \\otimes_{R_0}^\\mathbf{L} R, L_0 \\otimes_{R_0}^\\mathbf{L} R) = \\colim_{i \\geq 0} \\Hom_{D(R_i)}(K_0 \\otimes_{R_0}^\\mathbf{L} R_i, L_0 \\otimes_{R_0}^\\mathbf{L} R_i) $$ \\end{enumerate} In other words, the triangulated category of perfect complexes over $R$ is the colimit of the triangulated categories of perfect complexes over $R_i$."}
+{"_id": "10227", "title": "more-algebra-lemma-lift-acyclic-complex", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$ be a class of $R$-modules. Assume \\begin{enumerate} \\item each $P \\in \\mathcal{P}$ is a projective $R$-module, \\item if $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$, then $P_2 \\in \\mathcal{P}$, and \\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective modulo $I$, then $f$ is surjective. \\end{enumerate} Then given any bounded above acyclic complex $E^\\bullet$ whose terms are of the form $P/IP$ for $P \\in \\mathcal{P}$ there exists a bounded above acyclic complex $P^\\bullet$ whose terms are in $\\mathcal{P}$ lifting $E^\\bullet$."}
+{"_id": "10228", "title": "more-algebra-lemma-lift-complex", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$ be a class of $R$-modules. Let $K \\in D(R)$ and let $E^\\bullet$ be a complex of $R/I$-modules representing $K \\otimes_R^\\mathbf{L} R/I$. Assume \\begin{enumerate} \\item each $P \\in \\mathcal{P}$ is a projective $R$-module, \\item $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$ if and only if $P_1, P_2 \\in \\mathcal{P}$, \\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective modulo $I$, then $f$ is surjective, \\item $E^\\bullet$ is bounded above and $E^i$ is of the form $P/IP$ for $P \\in \\mathcal{P}$, and \\item $K$ can be represented by a bounded above complex whose terms are in $\\mathcal{P}$. \\end{enumerate} Then there exists a bounded above complex $P^\\bullet$ whose terms are in $\\mathcal{P}$ with $P^\\bullet/IP^\\bullet$ isomorphic to $E^\\bullet$ and representing $K$ in $D(R)$."}
+{"_id": "10229", "title": "more-algebra-lemma-lift-complex-projectives", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \\begin{enumerate} \\item $E^\\bullet$ is a bounded above complex of projective $R/I$-modules, \\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$, and \\item $I$ is a nilpotent ideal. \\end{enumerate} Then there exists a bounded above complex $P^\\bullet$ of projective $R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$ is isomorphic to $E^\\bullet$."}
+{"_id": "10230", "title": "more-algebra-lemma-lift-complex-stably-frees", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \\begin{enumerate} \\item $E^\\bullet$ is a bounded above complex of finite stably free $R/I$-modules, \\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$, \\item $K^\\bullet$ is pseudo-coherent, and \\item every element of $1 + I$ is invertible. \\end{enumerate} Then there exists a bounded above complex $P^\\bullet$ of finite stably free $R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$ is isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free."}
+{"_id": "10231", "title": "more-algebra-lemma-lift-pseudo-coherent-from-residue-field", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $K \\in D(R)$ be pseudo-coherent. Set $d_i = \\dim_\\kappa H^i(K \\otimes_R^\\mathbf{L} \\kappa)$. Then $d_i < \\infty$ and for some $b \\in \\mathbf{Z}$ we have $d_i = 0$ for $i > b$. Then there exists a complex $$ \\ldots \\to R^{\\oplus d_{b - 2}} \\to R^{\\oplus d_{b - 1}} \\to R^{\\oplus d_b} \\to 0 \\to \\ldots $$ representing $K$ in $D(R)$. Moreover, this complex is unique up to isomorphism(!)."}
+{"_id": "10232", "title": "more-algebra-lemma-lift-perfect-from-residue-field", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $K \\in D(R)$ be perfect. Set $d_i = \\dim_{\\kappa(\\mathfrak p)} H^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$. Then $d_i < \\infty$ and only a finite number are nonzero. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a complex $$ \\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to \\ldots \\to R_f^{\\oplus d_{b - 1}} \\to R_f^{\\oplus d_b} \\to 0 \\to \\ldots $$ representing $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$."}
+{"_id": "10233", "title": "more-algebra-lemma-compare-representatives-perfect", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $M^\\bullet$ and $N^\\bullet$ be bounded complexes of finite projective $R$-modules representing the same object of $D(R)$. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that there is an isomorphism (!) of complexes $$ M^\\bullet_f \\oplus P^\\bullet \\cong N^\\bullet_f \\oplus Q^\\bullet $$ where $P^\\bullet$ and $Q^\\bullet$ are finite direct sums of trivial complexes, i.e., complexes of the form the form $\\ldots \\to 0 \\to R_f \\xrightarrow{1} R_f \\to 0 \\to \\ldots$ (placed in arbitrary degrees)."}
+{"_id": "10234", "title": "more-algebra-lemma-lift-complex-finite-projectives", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that \\begin{enumerate} \\item $E^\\bullet$ is a bounded above complex of finite projective $R/I$-modules, \\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$, \\item $K$ is pseudo-coherent, and \\item $(R, I)$ is a henselian pair. \\end{enumerate} Then there exists a bounded above complex $P^\\bullet$ of finite projective $R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$ is isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free."}
+{"_id": "10235", "title": "more-algebra-lemma-splitting-unique", "text": "Let $R$ be a ring. Let $K$ and $L$ be objects of $D(R)$. Assume $L$ has projective-amplitude in $[a, b]$, for example if $L$ is perfect of tor-amplitude in $[a, b]$. \\begin{enumerate} \\item If $H^i(K) = 0$ for $i \\geq a$, then $\\Hom_{D(R)}(L, K) = 0$. \\item If $H^i(K) = 0$ for $i \\geq a + 1$, then given any distinguished triangle $K \\to M \\to L \\to K[1]$ there is an isomorphism $M \\cong K \\oplus L$ in $D(R)$ compatible with the maps in the distinguished triangle. \\item If $H^i(K) = 0$ for $i \\geq a$, then the isomorphism in (2) exists and is unique. \\end{enumerate}"}
+{"_id": "10236", "title": "more-algebra-lemma-better-cut-complex-in-two", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \\in \\mathbf{Z}$ the map $$ H^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p) \\longrightarrow H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) $$ is surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect object of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$ and a canonical isomorphism $$ K^\\bullet \\otimes_R R_f \\cong \\tau_{\\leq i}(K^\\bullet \\otimes_R R_f) \\oplus \\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f) $$ in $D(R_f)$."}
+{"_id": "10238", "title": "more-algebra-lemma-cut-complex-in-two", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $i \\in \\mathbf{Z}$. Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules such that $H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a canonical direct sum decomposition $$ K^\\bullet \\otimes_R R_f = \\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f) \\oplus \\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f) $$ in $D(R_f)$ with $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ a perfect complex with tor-amplitude in $[i + 1, \\infty]$."}
+{"_id": "10239", "title": "more-algebra-lemma-split-using-ext-injective", "text": "Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$. Assume that for any injective $R$-module map $M \\to M'$ the map $\\Ext^{-a}_R(K, M) \\to \\Ext^{-a}_R(K, M')$ is injective. Then there is a unique direct sum decomposition $K \\cong \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + 1}K$ and $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$."}
+{"_id": "10241", "title": "more-algebra-lemma-lift-bounded-pseudo-coherent-to-perfect", "text": "Let $R$ be a ring and let $\\mathfrak p \\subset R$ be a prime. Let $K$ be pseudo-coherent and bounded below. Set $d_i = \\dim_{\\kappa(\\mathfrak p)} H^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$. If there exists an $a \\in \\mathbf{Z}$ such that $d_i = 0$ for $i < a$, then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a complex $$ \\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to \\ldots \\to R_f^{\\oplus d_{b - 1}} \\to R_f^{\\oplus d_b} \\to 0 \\to \\ldots $$ representing $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$. In particular $K \\otimes_R^\\mathbf{L} R_f$ is perfect."}
+{"_id": "10242", "title": "more-algebra-lemma-check-perfect-pointwise", "text": "Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules. The following are equivalent \\begin{enumerate} \\item $K^\\bullet$ is perfect with tor amplitude in $[a, b]$, \\item for every prime $\\mathfrak p$ we have $H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all $i \\not \\in [a, b]$, and \\item for every maximal ideal $\\mathfrak m$ we have $H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all $i \\not \\in [a, b]$. \\end{enumerate}"}
+{"_id": "10243", "title": "more-algebra-lemma-check-perfect-stalks", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules. Consider the following conditions \\begin{enumerate} \\item $K^\\bullet$ is perfect, \\item for every prime ideal $\\mathfrak p$ the complex $K^\\bullet \\otimes_R R_{\\mathfrak p}$ is perfect, \\item for every maximal ideal $\\mathfrak m$ the complex $K^\\bullet \\otimes_R R_{\\mathfrak m}$ is perfect, \\item for every prime $\\mathfrak p$ we have $H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all $i \\ll 0$, \\item for every maximal ideal $\\mathfrak m$ we have $H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all $i \\ll 0$. \\end{enumerate} We always have the implications $$ (1) \\Rightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (3) \\Leftrightarrow (4) \\Leftrightarrow (5) $$ If $K^\\bullet$ is bounded below, then all conditions are equivalent."}
+{"_id": "10244", "title": "more-algebra-lemma-projective-amplitude-pseudo-coherent", "text": "Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$. Let $a, b \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $K$ has projective-amplitude in $[a, b]$, \\item $K$ is perfect of tor-amplitude in $[a, b]$, \\item $\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i \\not \\in [-b, -a]$, \\item $H^n(K) = 0$ for $n > b$ and $\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all $i > -a$, and \\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and $\\Ext^{-a + 1}_R(K, N) = 0$ for all finitely presented $R$-modules $N$. \\end{enumerate}"}
+{"_id": "10245", "title": "more-algebra-lemma-perfect-over-polynomial-ring", "text": "Let $A \\to B$ be a ring map. Let $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$. Let $K^\\bullet$ be a complex of $B$-modules. Assume \\begin{enumerate} \\item the ring map $A \\to B$ is flat, \\item for every prime $\\mathfrak p \\subset A$ the ring $B \\otimes_A \\kappa(\\mathfrak p)$ has finite global dimension $\\leq d$, \\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and \\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules. \\end{enumerate} Then $K^\\bullet$ is perfect as a complex of $B$-modules with tor amplitude in $[a - d, b]$."}
+{"_id": "10246", "title": "more-algebra-lemma-perfect-over-regular-local-ring", "text": "Let $A \\to B$ be a local ring homomorphism. Let $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$. Let $K^\\bullet$ be a complex of $B$-modules. Assume \\begin{enumerate} \\item the ring map $A \\to B$ is flat, \\item the ring $B/\\mathfrak m_AB$ is regular of dimension $d$, \\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and \\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules, in fact it suffices if $H^i(K^\\bullet \\otimes_A^\\mathbf{L} \\kappa(\\mathfrak m_A))$ is nonzero only for $i \\in [a, b]$. \\end{enumerate} Then $K^\\bullet$ is perfect as a complex of $B$-modules with tor amplitude in $[a - d, b]$."}
+{"_id": "10247", "title": "more-algebra-lemma-perfect-ring-classical-generator", "text": "Let $R$ be a ring. The full subcategory $D_{perf}(R) \\subset D(R)$ of perfect objects is the smallest strictly full, saturated, triangulated subcategory containing $R = R[0]$. In other words $D_{perf}(R) = \\langle R \\rangle$. In particular, $R$ is a classical generator for $D_{perf}(R)$."}
+{"_id": "10248", "title": "more-algebra-lemma-commutes-with-countable-sums", "text": "Let $R$ be a ring. Let $K \\in D(R)$ be an object such that for every countable set of objects $E_n \\in D(R)$ the canonical map $$ \\bigoplus \\Hom_{D(R)}(K, E_n) \\longrightarrow \\Hom_{D(R)}(K, \\bigoplus E_n) $$ is a bijection. Then, given any system $L_n^\\bullet$ of complexes over $\\mathbf{N}$ we have that $$ \\colim \\Hom_{D(R)}(K, L^\\bullet_n) \\longrightarrow \\Hom_{D(R)}(K, L^\\bullet) $$ is a bijection, where $L^\\bullet$ is the termwise colimit, i.e., $L^m = \\colim L_n^m$ for all $m \\in \\mathbf{Z}$."}
+{"_id": "10249", "title": "more-algebra-lemma-perfect-modulo-nilpotent-ideal", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that \\begin{enumerate} \\item $K \\otimes_R^\\mathbf{L} R/I$ is perfect in $D(R/I)$, and \\item $I$ is a nilpotent ideal. \\end{enumerate} Then $K$ is perfect in $D(R)$."}
+{"_id": "10251", "title": "more-algebra-lemma-ghost-lemma", "text": "\\begin{reference} \\cite{Kelly} \\end{reference} Let $R$ be a ring. Let $n \\geq 1$. Let $K \\in \\langle R \\rangle_n$ with notation as in Derived Categories, Section \\ref{derived-section-generators}. Consider maps $$ K \\xrightarrow{f_1} K_1 \\xrightarrow{f_2} K_2 \\xrightarrow{f_3} \\ldots \\xrightarrow{f_n} K_n $$ in $D(R)$. If $H^i(f_j) = 0$ for all $i, j$, then $f_n \\circ \\ldots \\circ f_1 = 0$."}
+{"_id": "10252", "title": "more-algebra-lemma-not-regular-not-strong", "text": "Let $R$ be a Noetherian ring. If $R$ is a strong generator for $D_{perf}(R)$, then $R$ is regular of finite dimension."}
+{"_id": "10253", "title": "more-algebra-lemma-ext-regular", "text": "Let $R$ be a Noetherian regular ring of dimension $d < \\infty$. Let $K, L \\in D^-(R)$. Assume there exists an $k$ such that $H^i(K) = 0$ for $i \\leq k$ and $H^i(L) = 0$ for $i \\geq k - d + 1$. Then $\\Hom_{D(R)}(K, L) = 0$."}
+{"_id": "10254", "title": "more-algebra-lemma-split-complex-regular", "text": "Let $R$ be a Noetherian regular ring of dimension $1 \\leq d < \\infty$. Let $K \\in D(R)$ be perfect and let $k \\in \\mathbf{Z}$ such that $H^i(K) = 0$ for $i = k - d + 2, \\ldots, k$ (empty condition if $d = 1$). Then $K = \\tau_{\\leq k - d + 1}K \\oplus \\tau_{\\geq k + 1}K$."}
+{"_id": "10255", "title": "more-algebra-lemma-regular-strong-generator", "text": "Let $R$ be a Noetherian regular ring of finite dimension. Then $R$ is a strong generator for the full subcategory $D_{perf}(R) \\subset D(R)$ of perfect objects."}
+{"_id": "10256", "title": "more-algebra-lemma-relatively-finitely-presented", "text": "Let $R \\to A$ be a ring map of finite type. Let $M$ be an $A$-module. The following are equivalent \\begin{enumerate} \\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$ the module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module, \\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$ the module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module, and \\item for any surjection $A' \\to A$ where $A'$ is a finitely presented $R$-algebra, the module $M$ is finitely presented as $A'$-module. \\end{enumerate} In this case $M$ is a finitely presented $A$-module."}
+{"_id": "10257", "title": "more-algebra-lemma-finite-extension", "text": "Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras. Let $M$ be a $B$-module. Then $M$ is an $A$-module finitely presented relative to $R$ if and only if $M$ is a $B$-module finitely presented relative to $R$."}
+{"_id": "10258", "title": "more-algebra-lemma-localize-relative-finite-presentation", "text": "Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map, $g \\in A$, and $M$ an $A$-module. If $M$ of finite presentation relative to $R_f$, then $M_g$ is an $A_g$-module of finite presentation relative to $R$."}
+{"_id": "10259", "title": "more-algebra-lemma-base-change-relative-finite-presentation", "text": "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. For any ring map $R \\to R'$ the $A \\otimes_R R'$-module $$ M \\otimes_A A' = M \\otimes_R R' $$ is finitely presented relative to $R'$."}
+{"_id": "10260", "title": "more-algebra-lemma-pull-relative-finite-presentation", "text": "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. Let $A \\to A'$ be a ring map of finite presentation. The $A'$-module $M \\otimes_A A'$ is finitely presented relative to $R$."}
+{"_id": "10261", "title": "more-algebra-lemma-composition-relative-finite-presentation", "text": "Let $R \\to A \\to B$ be finite type ring maps. Let $M$ be a $B$-module. If $M$ is finitely presented relative to $A$ and $A$ is of finite presentation over $R$, then $M$ is finitely presented relative to $R$."}
+{"_id": "10262", "title": "more-algebra-lemma-glue-relative-finite-presentation", "text": "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module. Let $f_1, \\ldots, f_r \\in A$ generate the unit ideal. The following are equivalent \\begin{enumerate} \\item each $M_{f_i}$ is finitely presented relative to $R$, and \\item $M$ is finitely presented relative to $R$. \\end{enumerate}"}
+{"_id": "10263", "title": "more-algebra-lemma-ses-relatively-finite-presentation", "text": "Let $R \\to A$ be a finite type ring map. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of $A$-modules. \\begin{enumerate} \\item If $M', M''$ are finitely presented relative to $R$, then so is $M$. \\item If $M'$ is a finite type $A$-module and $M$ is finitely presented relative to $R$, then $M''$ is finitely presented relative to $R$. \\end{enumerate}"}
+{"_id": "10264", "title": "more-algebra-lemma-sum-relatively-finite-presentation", "text": "Let $R \\to A$ be a finite type ring map. Let $M, M'$ be $A$-modules. If $M \\oplus M'$ is finitely presented relative to $R$, then so are $M$ and $M'$."}
+{"_id": "10265", "title": "more-algebra-lemma-pull-push", "text": "Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules. Consider the $R$-algebra map $R[x] \\to R$ which maps $x$ to zero. Then $$ K^\\bullet \\otimes_{R[x]}^{\\mathbf{L}} R \\cong K^\\bullet \\oplus K^\\bullet[1] $$ in $D(R)$."}
+{"_id": "10266", "title": "more-algebra-lemma-add-variable-pseudo-coherent", "text": "Let $R$ be a ring and $K^\\bullet$ a complex of $R$-modules. Let $m \\in \\mathbf{Z}$. Consider the $R$-algebra map $R[x] \\to R$ which maps $x$ to zero. Then $K^\\bullet$ is $m$-pseudo-coherent as a complex of $R$-modules if and only if $K^\\bullet$ is $m$-pseudo-coherent as a complex of $R[x]$-modules."}
+{"_id": "10267", "title": "more-algebra-lemma-relatively-pseudo-coherent", "text": "Let $R \\to A$ be a ring map of finite type. Let $K^\\bullet$ be a complex of $A$-modules. Let $m \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$ the complex $K^\\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \\ldots, x_n]$-modules, \\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$ the complex $K^\\bullet$ is an $m$-pseudo-coherent complex of $R[x_1, \\ldots, x_n]$-modules. \\end{enumerate} In particular the same equivalence holds for pseudo-coherence."}
+{"_id": "10268", "title": "more-algebra-lemma-finite-extension-pseudo-coherent", "text": "Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules. Then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$ if and only if $K^\\bullet$ seen as a complex of $A$-modules is $m$-pseudo-coherent (pseudo-coherent) relative to $R$."}
+{"_id": "10269", "title": "more-algebra-lemma-cone-relatively-pseudo-coherent", "text": "Let $R$ be a ring. Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished triangle in $D(A)$. \\begin{enumerate} \\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $L^\\bullet$ is $m$-pseudo-coherent relative to $R$ then $M^\\bullet$ is $m$-pseudo-coherent relative to $R$. \\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent relative to $R$, then $L^\\bullet$ is $m$-pseudo-coherent relative to $R$. \\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and $M^\\bullet$ is $m$-pseudo-coherent relative to $R$, then $K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$. \\end{enumerate} Moreover, if two out of three of $K^\\bullet, L^\\bullet, M^\\bullet$ are pseudo-coherent relative to $R$, the so is the third."}
+{"_id": "10270", "title": "more-algebra-lemma-rel-n-pseudo-module", "text": "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module. Then \\begin{enumerate} \\item $M$ is $0$-pseudo-coherent relative to $R$ if and only if $M$ is a finite type $A$-module, \\item $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if $M$ is a finitely presented relative to $R$, \\item $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if for every surjection $R[x_1, \\ldots, x_n] \\to A$ there exists a resolution $$ R[x_1, \\ldots, x_n]^{\\oplus a_d} \\to R[x_1, \\ldots, x_n]^{\\oplus a_{d - 1}} \\to \\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0 $$ of length $d$, and \\item $M$ is pseudo-coherent relative to $R$ if and only if for every presentation $R[x_1, \\ldots, x_n] \\to A$ there exists an infinite resolution $$ \\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_1} \\to R[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0 $$ by finite free $R[x_1, \\ldots, x_n]$-modules. \\end{enumerate}"}
+{"_id": "10271", "title": "more-algebra-lemma-summands-relative-pseudo-coherent", "text": "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet, L^\\bullet \\in D(A)$. If $K^\\bullet \\oplus L^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$ so are $K^\\bullet$ and $L^\\bullet$."}
+{"_id": "10274", "title": "more-algebra-lemma-localize-relative-pseudo-coherent", "text": "Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map, $g \\in A$, and $K^\\bullet$ a complex of $A$-modules. If $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R_f$, then $K^\\bullet \\otimes_A A_g$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$."}
+{"_id": "10275", "title": "more-algebra-lemma-base-change-relative-pseudo-coherent", "text": "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $A$-modules which is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$. Let $R \\to R'$ be a ring map such that $A$ and $R'$ are Tor independent over $R$. Set $A' = A \\otimes_R R'$. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} A'$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R'$."}
+{"_id": "10276", "title": "more-algebra-lemma-pull-relative-pseudo-coherent", "text": "Let $R \\to A \\to B$ be finite type ring maps. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $A$-modules. Assume $B$ as a $B$-module is pseudo-coherent relative to $A$. If $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$, then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$."}
+{"_id": "10278", "title": "more-algebra-lemma-composition-relative-pseudo-coherent", "text": "Let $R$ be a ring. Let $A \\to B$ be a map of finite type $R$-algebras. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules. Assume $A$ is pseudo-coherent relative to $R$. Then the following are equivalent \\begin{enumerate} \\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $A$, and \\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$. \\end{enumerate}"}
+{"_id": "10279", "title": "more-algebra-lemma-glue-relative-pseudo-coherent", "text": "Let $R \\to A$ be a finite type ring map. Let $K^\\bullet$ be a complex of $A$-modules. Let $m \\in \\mathbf{Z}$. Let $f_1, \\ldots, f_r \\in A$ generate the unit ideal. The following are equivalent \\begin{enumerate} \\item each $K^\\bullet \\otimes_A A_{f_i}$ is $m$-pseudo-coherent relative to $R$, and \\item $K^\\bullet$ is $m$-pseudo-coherent relative to $R$. \\end{enumerate} The same equivalence holds for pseudo-coherence relative to $R$."}
+{"_id": "10280", "title": "more-algebra-lemma-Noetherian-relative-pseudo-coherent", "text": "Let $R$ be a Noetherian ring. Let $R \\to A$ be a finite type ring map. Then \\begin{enumerate} \\item A complex of $A$-modules $K^\\bullet$ is $m$-pseudo-coherent relative to $R$ if and only if $K^\\bullet \\in D^{-}(A)$ and $H^i(K^\\bullet)$ is a finite $A$-module for $i \\geq m$. \\item A complex of $A$-modules $K^\\bullet$ is pseudo-coherent relative to $R$ if and only if $K^\\bullet \\in D^{-}(A)$ and $H^i(K^\\bullet)$ is a finite $A$-module for all $i$. \\item An $A$-module is pseudo-coherent relative to $R$ if and only if it is finite. \\end{enumerate}"}
+{"_id": "10281", "title": "more-algebra-lemma-perfect-ring-map", "text": "A ring map $A \\to B$ is perfect if and only if $B = A[x_1, \\ldots, x_n]/I$ and $B$ as an $A[x_1, \\ldots, x_n]$-module has a finite resolution by finite projective $A[x_1, \\ldots, x_n]$-modules."}
+{"_id": "10282", "title": "more-algebra-lemma-Noetherian-pseudo-coherent-ring-map", "text": "A finite type ring map of Noetherian rings is pseudo-coherent."}
+{"_id": "10283", "title": "more-algebra-lemma-flat-finite-presentation-perfect", "text": "A ring map which is flat and of finite presentation is perfect."}
+{"_id": "10284", "title": "more-algebra-lemma-regular-perfect-ring-map", "text": "Let $A \\to B$ be a finite type ring map with $A$ a regular ring of finite dimension. Then $A \\to B$ is perfect."}
+{"_id": "10285", "title": "more-algebra-lemma-lci-perfect", "text": "A local complete intersection homomorphism is perfect."}
+{"_id": "10286", "title": "more-algebra-lemma-relative-pseudo-coherent-is-moot", "text": "Let $R \\to A$ be a pseudo-coherent ring map. Let $K \\in D(A)$. The following are equivalent \\begin{enumerate} \\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$, and \\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) in $D(A)$. \\end{enumerate}"}
+{"_id": "10287", "title": "more-algebra-lemma-more-relative-pseudo-coherent-is-moot", "text": "Let $R \\to B \\to A$ be ring maps with $\\varphi : B \\to A$ surjective and $R \\to B$ and $R \\to A$ flat and of finite presentation. For $K \\in D(A)$ denote $\\varphi_*K \\in D(B)$ the restriction. The following are equivalent \\begin{enumerate} \\item $K$ is pseudo-coherent, \\item $K$ is pseudo-coherent relative to $R$, \\item $K$ is pseudo-coherent relative to $A$, \\item $\\varphi_*K$ is pseudo-coherent, \\item $\\varphi_*K$ is pseudo-coherent relative to $R$. \\end{enumerate} Similar holds for $m$-pseudo-coherence."}
+{"_id": "10288", "title": "more-algebra-lemma-cone-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map of finite presentation. The $R$-perfect objects of $D(A)$ form a saturated\\footnote{Derived Categories, Definition \\ref{derived-definition-saturated}.} triangulated strictly full subcategory."}
+{"_id": "10289", "title": "more-algebra-lemma-perfect-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map of finite presentation. A perfect object of $D(A)$ is $R$-perfect. If $K, M \\in D(A)$ then $K \\otimes_A^\\mathbf{L} M$ is $R$-perfect if $K$ is perfect and $M$ is $R$-perfect."}
+{"_id": "10290", "title": "more-algebra-lemma-structure-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map of finite presentation. Let $K \\in D(A)$. The following are equivalent \\begin{enumerate} \\item $K$ is $R$-perfect, and \\item $K$ is isomorphic to a finite complex of $R$-flat, finitely presented $A$-modules. \\end{enumerate}"}
+{"_id": "10291", "title": "more-algebra-lemma-base-change-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map of finite presentation. Let $R \\to R'$ be a ring map and set $A' = A \\otimes_R R'$. If $K \\in D(A)$ is $R$-perfect, then $K \\otimes_A^\\mathbf{L} A'$ is $R'$-perfect."}
+{"_id": "10292", "title": "more-algebra-lemma-compute-RHom-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map. Let $K, L \\in D(A)$ with $K$ pseudo-coherent and $L$ finite tor dimension over $R$. We may choose \\begin{enumerate} \\item a bounded above complex $P^\\bullet$ of finite free $A$-modules representing $K$, and \\item a bounded complex of $R$-flat $A$-modules $F^\\bullet$ representing $L$. \\end{enumerate} Given these choices we have \\begin{enumerate} \\item[(a)] $E^\\bullet = \\Hom^\\bullet(P^\\bullet, F^\\bullet)$ is a bounded below complex of $R$-flat $A$-modules representing $R\\Hom_A(K, L)$, \\item[(b)] for any ring map $R \\to R'$ with $A' = A \\otimes_R R'$ the complex $E^\\bullet \\otimes_R R'$ represents $R\\Hom_{A'}(K \\otimes_A^\\mathbf{L} A', L \\otimes_A^\\mathbf{L} A')$. \\end{enumerate} If in addition $R \\to A$ is of finite presentation and $L$ is $R$-perfect, then we may choose $F^p$ to be finitely presented $A$-modules and consequently $E^n$ will be finitely presented $A$-modules as well."}
+{"_id": "10293", "title": "more-algebra-lemma-colimit-relatively-perfect", "text": "Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings. Let $0 \\in I$ and $R_0 \\to A_0$ be a flat ring map of finite presentation. For $i \\geq 0$ set $A_i = R_i \\otimes_{R_0} A_0$ and set $A = R \\otimes_{R_0} A_0$. \\begin{enumerate} \\item Given an $R$-perfect $K$ in $D(A)$ there exists an $i \\in I$ and an $R_i$-perfect $K_i$ in $D(A_i)$ such that $K \\cong K_i \\otimes_{A_i}^\\mathbf{L} A$ in $D(A)$. \\item Given $K_0, L_0 \\in D(A_0)$ with $K_0$ pseudo-coherent and $L_0$ finite tor dimension over $R_0$, then we have $$ \\Hom_{D(A)}(K_0 \\otimes_{A_0}^\\mathbf{L} A, L_0 \\otimes_{A_0}^\\mathbf{L} A) = \\colim_{i \\geq 0} \\Hom_{D(A_i)}(K_0 \\otimes_{A_0}^\\mathbf{L} A_i, L_0 \\otimes_{A_0}^\\mathbf{L} A_i) $$ \\end{enumerate} In particular, the triangulated category of $R$-perfect complexes over $A$ is the colimit of the triangulated categories of $R_i$-perfect complexes over $A_i$."}
+{"_id": "10294", "title": "more-algebra-lemma-thickening-relatively-perfect", "text": "Let $R' \\to A'$ be a flat ring map of finite presentation. Let $R' \\to R$ be a surjective ring map whose kernel is a nilpotent ideal. Set $A = A' \\otimes_{R'} R$. Let $K' \\in D(A')$ and set $K = K' \\otimes_{A'}^\\mathbf{L} A$ in $D(A)$. If $K$ is $R$-perfect, then $K'$ is $R'$-perfect."}
+{"_id": "10295", "title": "more-algebra-lemma-lift-from-fibre-relatively-perfect", "text": "Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_d]/I$ be flat and of finite presentation over $R$. Let $\\mathfrak q \\subset A$ be a prime ideal lying over $\\mathfrak p \\subset R$. Let $K \\in D(A)$ be pseudo-coherent. Let $a, b \\in \\mathbf{Z}$. If $H^i(K_\\mathfrak q \\otimes_{R_\\mathfrak p}^\\mathbf{L} \\kappa(\\mathfrak p))$ is nonzero only for $i \\in [a, b]$, then $K_\\mathfrak q$ has tor amplitude in $[a - d, b]$ over $R$."}
+{"_id": "10296", "title": "more-algebra-lemma-bounded-on-fibres-relatively-perfect", "text": "Let $R \\to A$ be a ring map which is flat and of finite presentation. Let $K \\in D(A)$ be pseudo-coherent. The following are equivalent \\begin{enumerate} \\item $K$ is $R$-perfect, and \\item $K$ is bounded below and for every prime ideal $\\mathfrak p \\subset R$ the object $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$ is bounded below. \\end{enumerate}"}
+{"_id": "10297", "title": "more-algebra-lemma-ext-1-zero", "text": "Let $R$ be a ring. Let $K \\in D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. The following are equivalent \\begin{enumerate} \\item $H^{-1}(K) = 0$ and $H^0(K)$ is a projective module and \\item $\\Ext^1_R(K, M) = 0$ for every $R$-module $M$. \\end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \\begin{enumerate} \\item[(3)] $\\Ext^1_R(K, M) = 0$ for every finite $R$-module $M$. \\end{enumerate}"}
+{"_id": "10298", "title": "more-algebra-lemma-represent-two-term-complex", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. Then \\begin{enumerate} \\item $K$ can be represented by a two term complex $K^{-1} \\to K^0$ with $K^0$ a free module, and \\item if $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then $K$ can be represented by a two term complex $K^{-1} \\to K^0$ with $K^0$ a finite free module and $K^{-1}$ finite. \\end{enumerate}"}
+{"_id": "10299", "title": "more-algebra-lemma-map-out-of-almost-free", "text": "Let $R$ be a ring. Let $M^\\bullet$ be a complex of modules over $R$ with $M^i = 0$ for $i > 0$ and $M^0$ a projective $R$-module. Let $K^\\bullet$ be a second complex. \\begin{enumerate} \\item Assume $K^i = 0$ for $i \\leq -2$. Then $\\Hom_{D(R)}(M^\\bullet, K^\\bullet) = \\Hom_{K(R)}(M^\\bullet, K^\\bullet)$. \\item Assume $K^i = 0$ for $i \\not \\in [-1, 0]$ and $K^0$ a projective $R$-module. Then for a map of complexes $a^\\bullet : M^\\bullet \\to K^\\bullet$, the following are equivalent \\begin{enumerate} \\item $a^\\bullet$ induces the zero map $\\Ext^1_R(K^\\bullet, N) \\to \\Ext^1_R(M^\\bullet, N)$ for all $R$-modules $N$, and \\item there is a map $h^0 : M^0 \\to K^{-1}$ such that $a^{-1} + h^0 \\circ d^{-1}_K = 0$. \\end{enumerate} \\item Assume $K^i = 0$ for $i \\leq -3$. Let $\\alpha \\in \\Hom_{D(R)}(M^\\bullet, K^\\bullet)$. If the composition of $\\alpha$ with $K^\\bullet \\to K^{-2}[2]$ comes from an $R$-module map $a : M^{-2} \\to K^{-2}$ with $a \\circ d_M^{-3} = 0$, then $\\alpha$ can be represented by a map of complexes $a^\\bullet : M^\\bullet \\to K^\\bullet$ with $a^{-2} = a$. \\item In (2) for any second map of complexes $(a')^\\bullet : M^\\bullet \\to K^\\bullet$ representing $\\alpha$ with $a = (a')^{-2}$ there exist $h^i : M^i \\to K^{i - 1}$ for $i = 0, -1$ such that $$ h^{-1} \\circ d_M^{-2} = 0, \\quad (a')^{-1} = a^{-1} + d_K^{-2} \\circ h^{-1} + h^0 \\circ d_M^{-1},\\quad (a')^0 = a^0 + d_K^{-1} \\circ h^0 $$ \\end{enumerate}"}
+{"_id": "10300", "title": "more-algebra-lemma-ext-1-annihilated-definite", "text": "Let $R$ be a ring and let $I \\subset R$ be an ideal. Let $K \\in D(R)$. Assume $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. The following are equivalent \\begin{enumerate} \\item $\\Ext^1_R(K, N)$ is annihilated by $I$ for all $R$-modules $N$, \\item $K$ can be represented by a complex $K^{-1} \\to K^0$ with $K^0$ free such that for any $a \\in I$ the map $a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$, \\item whenever $K$ is represented by a two term complex $K^{-1} \\to K^0$ with $K^0$ projective, then for any $a \\in I$ the map $a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$. \\end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \\begin{enumerate} \\item[(4)] $\\Ext^1_R(K, N)$ is annihilated by $I$ for every finite $R$-module $N$, \\item[(5)] $K$ can be represented by a complex $K^{-1} \\to K^0$ with $K^0$ finite free and $K^{-1}$ finite such that for any $a \\in I$ the map $a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$. \\end{enumerate}"}
+{"_id": "10301", "title": "more-algebra-lemma-two-term-base-change", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. Let $K^{-1} \\to K^0$ be a two term complex of $R$-modules representing $K$ such that $K^0$ is a flat $R$-module (for example projective or free). Let $R \\to R'$ be a ring map. Then the complex $K^\\bullet \\otimes_R R'$ represents $\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$."}
+{"_id": "10302", "title": "more-algebra-lemma-base-change-property-ext-1-annihilated", "text": "Let $I$ be an ideal of a ring $R$. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. Let $R \\to R'$ be a ring map. If $K$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R, I)$, then $\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R', IR')$"}
+{"_id": "10303", "title": "more-algebra-lemma-two-term-surjection-map-zero", "text": "Let $R$ be a ring. Let $\\alpha : K \\to K'$ be a morphism of $D(R)$. Assume \\begin{enumerate} \\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$ \\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective. \\end{enumerate} For any $f \\in R$ if $f : K \\to K$ is $0$, then $f : K' \\to K'$ is $0$."}
+{"_id": "10304", "title": "more-algebra-lemma-surjection-property-ext-1-annihilated", "text": "Let $I$ be an ideal of a ring $R$. Let $\\alpha : K \\to K'$ be a morphism of $D(R)$. Assume \\begin{enumerate} \\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$ \\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective. \\end{enumerate} If $K$ satisfies the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-ext-1-annihilated-definite}, then $K'$ does too."}
+{"_id": "10305", "title": "more-algebra-lemma-ext-1-annihilated", "text": "Let $R$ be ring and let $I \\subset R$ be an ideal. Let $K \\in D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. The following are equivalent \\begin{enumerate} \\item there exists a $c \\geq 0$ such that the equivalent conditions (1), (2), (3) of Lemma \\ref{lemma-ext-1-annihilated-definite} hold for $K$ and the ideal $I^c$, \\item there exists a $c \\geq 0$ such that (a) $I^c$ annihilates $H^{-1}(K)$ and (b) $H^0(K)$ is an $I^c$-projective module (see Section \\ref{section-near-projective}). \\end{enumerate} If $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to \\begin{enumerate} \\item[(3)] there exists a $c \\geq 0$ such that the equivalent conditions (4), (5) of Lemma \\ref{lemma-ext-1-annihilated-definite} hold for $K$ and the ideal $I^c$, \\item[(4)] $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \\ldots, f_s \\in R$ with $V(f_1, \\ldots, f_s) \\subset V(I)$ such that the localizations $H^0(K)_{f_i}$ are projective $R_{f_i}$-modules, \\item[(5)] $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \\ldots, f_s \\in I$ with $V(f_1, \\ldots, f_s) = V(I)$ such that the localizations $H^0(K)_{f_i}$ are projective $R_{f_i}$-modules. \\end{enumerate}"}
+{"_id": "10306", "title": "more-algebra-lemma-zero-in-derived", "text": "Let $R$ be a ring. Let $K_j \\in D(R)$, $j = 1, 2, 3$ with $H^i(K_j) = 0$ for $i \\not \\in \\{-1, 0\\}$. Let $\\varphi : K_1 \\to K_2$ and $\\psi : K_2 \\to K_3$ be maps in $D(R)$. If $H^0(\\varphi) = 0$ and $H^{-1}(\\psi) = 0$, then $\\varphi \\circ \\psi = 0$."}
+{"_id": "10307", "title": "more-algebra-lemma-silly", "text": "Let $R$ be a ring. Let $K \\in D(R)$ be given by a two term complex of the form $R^{\\oplus n} \\to R^{\\oplus n}$. Denote $A \\in \\text{Mat}(n \\times n, R)$ the matrix of the differential. Then $\\det(a) : K \\to K$ is zero in $D(R)$."}
+{"_id": "10308", "title": "more-algebra-lemma-tensor-NL", "text": "Let $R \\to S$ and $S \\to S'$ be ring maps. The canonical map $\\NL_{S/R} \\otimes_S^\\mathbf{L} S' \\to \\NL_{S/R} \\otimes_S S'$ induces an isomorphism $\\tau_{\\geq -1}(\\NL_{S/R} \\otimes_S^\\mathbf{L} S') \\to \\NL_{S/R} \\otimes_S S'$ in $D(S')$. Similarly, given a presentation $\\alpha$ of $S$ over $R$ the canonical map $\\NL(\\alpha) \\otimes_S^\\mathbf{L} S' \\to \\NL(\\alpha) \\otimes_S S'$ induces an isomorphism $\\tau_{\\geq -1}(\\NL(\\alpha) \\otimes_S^\\mathbf{L} S') \\to \\NL(\\alpha) \\otimes_S S'$ in $D(S')$."}
+{"_id": "10309", "title": "more-algebra-lemma-base-change-NL", "text": "Let $R \\to S$ and $R \\to R'$ be ring maps. Let $\\alpha : P \\to S$ be a presentation of $S$ over $R$. Then $\\alpha' : P \\otimes_R R' \\to S \\otimes_R R'$ is a presentation of $S' = S \\otimes_R R'$ over $R'$. The canonical map $$ NL(\\alpha) \\otimes_S S' \\to \\NL(\\alpha') $$ is an isomorphism on $H^0$ and surjective on $H^{-1}$. In particular, the canonical map $$ \\NL_{S/R} \\otimes_S S' \\to \\NL_{S'/R'} $$ is an isomorphism on $H^0$ and surjective on $H^{-1}$."}
+{"_id": "10310", "title": "more-algebra-lemma-base-change-NL-flat", "text": "Consider a cocartesian diagram of rings $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[r] \\ar[u] & A' \\ar[u] } $$ If $B$ is flat over $A$, then the canonical map $\\NL_{B/A} \\otimes_B B' \\to \\NL_{B'/A'}$ is a quasi-isomorphism. If in addition $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$ then $\\NL_{B/A} \\otimes_B^\\mathbf{L} B' \\to \\NL_{B'/A'}$ is a quasi-isomorphism too."}
+{"_id": "10311", "title": "more-algebra-lemma-lci-NL", "text": "Let $A \\to B$ be a local complete intersection as in Definition \\ref{definition-local-complete-intersection}. Then $\\NL_{B/A}$ is a perfect object of $D(B)$ with tor amplitude in $[-1, 0]$."}
+{"_id": "10312", "title": "more-algebra-lemma-base-change-lci-bis", "text": "Consider a cocartesian diagram of rings $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[r] \\ar[u] & A' \\ar[u] } $$ If $A \\to B$ and $A' \\to B'$ are local complete intersections as in Definition \\ref{definition-local-complete-intersection}, then the kernel of $H^{-1}(\\NL_{B/A} \\otimes_B B') \\to H^{-1}(\\NL_{B'/A'})$ is a finite projective $B'$-module."}
+{"_id": "10313", "title": "more-algebra-lemma-compute-Rlim", "text": "The functor $\\lim : \\textit{Ab}(\\mathbf{N}) \\to \\textit{Ab}$ has a right derived functor \\begin{equation} \\label{equation-Rlim} R\\lim : D(\\textit{Ab}(\\mathbf{N})) \\longrightarrow D(\\textit{Ab}) \\end{equation} As usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have \\begin{enumerate} \\item for any $(A_n)$ in $\\textit{Ab}(\\mathbf{N})$ we have $R^p\\lim A_n = 0$ for $p > 1$, \\item the object $R\\lim A_n$ of $D(\\textit{Ab})$ is represented by the complex $$ \\prod A_n \\to \\prod A_n,\\quad (x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1})) $$ sitting in degrees $0$ and $1$, \\item if $(A_n)$ is ML, then $R^1\\lim A_n = 0$, i.e., $(A_n)$ is right acyclic for $\\lim$, \\item every $K^\\bullet \\in D(\\textit{Ab}(\\mathbf{N}))$ is quasi-isomorphic to a complex whose terms are right acyclic for $\\lim$, and \\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e., of $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the complex whose term in degree $p$ is $\\lim_n K_n^p$. \\end{enumerate}"}
+{"_id": "10314", "title": "more-algebra-lemma-apply-Mittag-Leffler-again", "text": "Let $$ (A^{-2}_n \\to A^{-1}_n \\to A^0_n \\to A^1_n) $$ be an inverse system of complexes of abelian groups and denote $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote $(H_n^{-1})$, $(H_n^0)$ the inverse systems of cohomologies, and denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$. If \\begin{enumerate} \\item $(A^{-2}_n)$ and $(A^{-1}_n)$ have vanishing $R^1\\lim$, \\item $(H^{-1}_n)$ has vanishing $R^1\\lim$, \\end{enumerate} then $H^0 = \\lim H_n^0$."}
+{"_id": "10317", "title": "more-algebra-lemma-distinguished-triangle-Rlim", "text": "Let $K = (K_n^\\bullet)$ be an object of $D(\\textit{Ab}(\\mathbf{N}))$. There exists a canonical distinguished triangle $$ R\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet \\to R\\lim K[1] $$ in $D(\\textit{Ab})$. In other words, $R\\lim K$ is a derived limit of the inverse system $(K_n^\\bullet)$ of $D(\\textit{Ab})$, see Derived Categories, Definition \\ref{derived-definition-derived-limit}."}
+{"_id": "10318", "title": "more-algebra-lemma-break-long-exact-sequence", "text": "With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim} the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences $$ 0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to H^p(R\\lim K) \\to \\lim_n H^p(K_n^\\bullet) \\to 0 $$"}
+{"_id": "10319", "title": "more-algebra-lemma-lift-to-system-complexes-Ab", "text": "Let $(K_n)$ be an inverse system of objects of $D(\\textit{Ab})$. Then there exists an object $M = (M_n^\\bullet)$ of $D(\\textit{Ab}(\\mathbf{N}))$ and isomorphisms $M_n^\\bullet \\to K_n$ in $D(\\textit{Ab})$ such that the diagrams $$ \\xymatrix{ M_{n + 1}^\\bullet \\ar[d] \\ar[r] & M_n^\\bullet \\ar[d] \\\\ K_{n + 1} \\ar[r] & K_n } $$ commute in $D(\\textit{Ab})$."}
+{"_id": "10320", "title": "more-algebra-lemma-Rlim-pro-equal", "text": "Let $E \\to D$ be a morphism of $D(\\textit{Ab}(\\mathbf{N}))$. Let $(E_n)$, resp.\\ $(D_n)$ be the system of objects of $D(\\textit{Ab})$ associated to $E$, resp.\\ $D$. If $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then $R\\lim E \\to R\\lim D$ is an isomorphism in $D(\\textit{Ab})$."}
+{"_id": "10321", "title": "more-algebra-lemma-emmanouil", "text": "\\begin{reference} Taken from \\cite{Emmanouil}. \\end{reference} Let $(A_n)$ be an inverse system of abelian groups. The following are equivalent \\begin{enumerate} \\item $(A_n)$ is Mittag-Leffler, \\item $R^1\\lim A_n = 0$ and the same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$. \\end{enumerate}"}
+{"_id": "10323", "title": "more-algebra-lemma-Rlim-zero-of-direct-sums", "text": "Let $(A_n)$ be an inverse system of abelian groups. The following are equivalent \\begin{enumerate} \\item $(A_n)$ is zero as a pro-object, \\item $\\lim A_n = 0$ and $R^1\\lim A_n = 0$ and the same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$. \\end{enumerate}"}
+{"_id": "10324", "title": "more-algebra-lemma-compute-Rlim-modules", "text": "In the situation above. The functor $\\lim : \\textit{Mod}(\\mathbf{N}, (A_n)) \\to \\text{Mod}_A$ has a right derived functor $$ R\\lim : D(\\textit{Mod}(\\mathbf{N}, (A_n))) \\longrightarrow D(A) $$ As usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have \\begin{enumerate} \\item for any $(M_n)$ in $\\textit{Mod}(\\mathbf{N}, (A_n))$ we have $R^p\\lim M_n = 0$ for $p > 1$, \\item the object $R\\lim M_n$ of $D(\\text{Mod}_A)$ is represented by the complex $$ \\prod M_n \\to \\prod M_n,\\quad (x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1})) $$ sitting in degrees $0$ and $1$, \\item if $(M_n)$ is ML, then $R^1\\lim M_n = 0$, i.e., $(M_n)$ is right acyclic for $\\lim$, \\item every $K^\\bullet \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ is quasi-isomorphic to a complex whose terms are right acyclic for $\\lim$, and \\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e., of $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the complex whose term in degree $p$ is $\\lim_n K_n^p$. \\end{enumerate}"}
+{"_id": "10325", "title": "more-algebra-lemma-distinguished-triangle-Rlim-modules", "text": "Let $K = (K_n^\\bullet)$ be an object of $D(\\textit{Mod}(\\mathbf{N}, (A_n)))$. There exists a canonical distinguished triangle $$ R\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet \\to R\\lim K[1] $$ in $D(A)$. In other words, $R\\lim K$ is a derived limit of the inverse system $(K_n^\\bullet)$ of $D(A)$, see Derived Categories, Definition \\ref{derived-definition-derived-limit}."}
+{"_id": "10326", "title": "more-algebra-lemma-break-long-exact-sequence-modules", "text": "With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim-modules} the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences $$ 0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to H^p(R\\lim K) \\to \\lim_n H^p(K_n^\\bullet) \\to 0 $$ of $A$-modules."}
+{"_id": "10327", "title": "more-algebra-lemma-lift-to-system-complexes", "text": "Let $(A_n)$ be an inverse system of rings. Suppose that we are given \\begin{enumerate} \\item for every $n$ an object $K_n$ of $D(A_n)$, and \\item for every $n$ a map $\\varphi_n : K_{n + 1} \\to K_n$ of $D(A_{n + 1})$ where we think of $K_n$ as an object of $D(A_{n + 1})$ by restriction via $A_{n + 1} \\to A_n$. \\end{enumerate} There exists an object $M = (M_n^\\bullet) \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ and isomorphisms $\\psi_n : M_n^\\bullet \\to K_n$ in $D(A_n)$ such that the diagrams $$ \\xymatrix{ M_{n + 1}^\\bullet \\ar[d]_{\\psi_{n + 1}} \\ar[r] & M_n^\\bullet \\ar[d]^{\\psi_n} \\\\ K_{n + 1} \\ar[r]^{\\varphi_n} & K_n } $$ commute in $D(A_{n + 1})$."}
+{"_id": "10328", "title": "more-algebra-lemma-get-ML-system", "text": "Let $(A_n)$ be an inverse system of rings. Every $K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ can be represented by a system of complexes $(M_n^\\bullet)$ such that all the transition maps $M_{n + 1}^\\bullet \\to M_n^\\bullet$ are surjective."}
+{"_id": "10329", "title": "more-algebra-lemma-get-K-flat-system", "text": "Let $(A_n)$ be an inverse system of rings. Every $K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ can be represented by a system of complexes $(K_n^\\bullet)$ such that each $K_n^\\bullet$ is K-flat."}
+{"_id": "10330", "title": "more-algebra-lemma-derived-tensor-product-systems", "text": "Let $(A_n)$ be an inverse system of rings. Given $K, L \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ there is a canonical derived tensor product $K \\otimes^\\mathbf{L} L$ in $D(\\mathbf{N}, (A_n))$ compatible with the maps to $D(A_n)$. The construction is symmetric in $K$ and $L$ and an exact functor of triangulated categories in each variable."}
+{"_id": "10331", "title": "more-algebra-lemma-tensor-Rlim-exact", "text": "Let $A$ be a ring. Let $E \\to D \\to F \\to E[1]$ be a distinguished triangle of $D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$, resp.\\ $(F_n)$ be the system of objects of $D(A)$ associated to $E$, resp.\\ $D$, resp.\\ $F$. Then for every $K \\in D(A)$ there is a canonical distinguished triangle $$ R\\lim (K \\otimes^\\mathbf{L}_A E_n) \\to R\\lim (K \\otimes^\\mathbf{L}_A D_n) \\to R\\lim (K \\otimes^\\mathbf{L}_A F_n) \\to R\\lim (K \\otimes^\\mathbf{L}_A E_n)[1] $$ in $D(A)$ with notation as in Remark \\ref{remark-constructing-tensor-with-limits-functorially}."}
+{"_id": "10332", "title": "more-algebra-lemma-tensor-Rlim-pro-equal", "text": "Let $A$ be a ring. Let $E \\to D$ be a morphism of $D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$ be the system of objects of $D(A)$ associated to $E$, resp.\\ $D$. If $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then for every $K \\in D(A)$ the corresponding map $$ R\\lim (K \\otimes^\\mathbf{L}_A E_n) \\longrightarrow R\\lim (K \\otimes^\\mathbf{L}_A D_n) $$ in $D(A)$ is an isomorphism (notation as in Remark \\ref{remark-constructing-tensor-with-limits-functorially})."}
+{"_id": "10333", "title": "more-algebra-lemma-I-power-torsion-presentation", "text": "Let $R$ be a ring. Let $I$ be an ideal of $R$. Let $M$ be an $I$-power torsion module. Then $M$ admits a resolution $$ \\ldots \\to K_2 \\to K_1 \\to K_0 \\to M \\to 0 $$ with each $K_i$ a direct sum of copies of $R/I^n$ for $n$ variable."}
+{"_id": "10334", "title": "more-algebra-lemma-torsion-free", "text": "Let $R$ be a ring. Let $I$ be an ideal of $R$. For any $R$-module $M$ set $M[I^n] = \\{m \\in M \\mid I^nm = 0\\}$. If $I$ is finitely generated then the following are equivalent \\begin{enumerate} \\item $M[I] = 0$, \\item $M[I^n] = 0$ for all $n \\geq 1$, and \\item if $I = (f_1, \\ldots, f_t)$, then the map $M \\to \\bigoplus M_{f_i}$ is injective. \\end{enumerate}"}
+{"_id": "10335", "title": "more-algebra-lemma-divide-by-torsion", "text": "Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$. \\begin{enumerate} \\item For any $R$-module $M$ we have $(M/M[I^\\infty])[I] = 0$. \\item An extension of $I$-power torsion modules is $I$-power torsion. \\end{enumerate}"}
+{"_id": "10336", "title": "more-algebra-lemma-I-power-torsion", "text": "Let $I$ be a finitely generated ideal of a ring $R$. The $I$-power torsion modules form a Serre subcategory of the abelian category $\\text{Mod}_R$, see Homology, Definition \\ref{homology-definition-serre-subcategory}."}
+{"_id": "10337", "title": "more-algebra-lemma-local-cohomology-closed", "text": "Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal. The subcategory $I^\\infty\\text{-torsion} \\subset \\text{Mod}_R$ depends only on the closed subset $Z = V(I) \\subset \\Spec(R)$. In fact, an $R$-module $M$ is $I$-power torsion if and only if its support is contained in $Z$."}
+{"_id": "10338", "title": "more-algebra-lemma-derived-vanishing-mod-I", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $K$ be an object of $D(R)$ such hat $K \\otimes_R^\\mathbf{L} R/I = 0$ in $D(R)$. Then \\begin{enumerate} \\item $K \\otimes_R^\\mathbf{L} R/I^n = 0$ for all $n \\geq 1$, \\item $K \\otimes_R^\\mathbf{L} N = 0$ for any $I$-power torsion $R$-module $N$, \\item $K \\otimes_R^\\mathbf{L} M = 0$ for any $M \\in D^b(R)$ whose cohomology modules are $I$-power torsion. \\end{enumerate}"}
+{"_id": "10339", "title": "more-algebra-lemma-characterize-flatness-on-torsion", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. The following are equivalent \\begin{enumerate} \\item $\\varphi$ is flat and $R/I \\to S/IS$ is faithfully flat, \\item $\\varphi$ is flat, and the map $\\Spec(S/IS) \\to \\Spec(R/I)$ is surjective. \\item $\\varphi$ is flat, and the base change functor $M \\mapsto M \\otimes_R S$ is faithful on modules annihilated by $I$, and \\item $\\varphi$ is flat, and the base change functor $M \\mapsto M \\otimes_R S$ is faithful on $I$-power torsion modules. \\end{enumerate}"}
+{"_id": "10340", "title": "more-algebra-lemma-neighbourhood-isomorphism", "text": "Assume $(\\varphi : R \\to S, I)$ satisfies the equivalent conditions of Lemma \\ref{lemma-characterize-flatness-on-torsion}. The following are equivalent \\begin{enumerate} \\item for any $I$-power torsion module $M$, the natural map $M \\to M \\otimes_R S$ is an isomorphism, and \\item $R/I \\to S/IS$ is an isomorphism. \\end{enumerate}"}
+{"_id": "10341", "title": "more-algebra-lemma-neighbourhood-equivalence", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. Then \\begin{enumerate} \\item for any $R$-module $M$ the map $M \\to M \\otimes_R S$ induces an isomorphism $M[I^\\infty] \\to (M \\otimes_R S)[(IS)^\\infty]$ of $I$-power torsion submodules, \\item the natural map $$ \\Hom_R(M, N) \\longrightarrow \\Hom_S(M \\otimes_R S, N \\otimes_R S) $$ is an isomorphism if either $M$ or $N$ is $I$-power torsion, and \\item the base change functor $M \\mapsto M \\otimes_R S$ defines an equivalence of categories between $I$-power torsion modules and $IS$-power torsion modules. \\end{enumerate}"}
+{"_id": "10342", "title": "more-algebra-lemma-map-identifies-koszul-and-cech-complexes", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. For any $f_1, \\ldots, f_r \\in R$ such that $V(f_1, \\ldots, f_r) = V(I)$ \\begin{enumerate} \\item the map of Koszul complexes $K(R, f_1, \\ldots, f_r) \\to K(S, f_1, \\ldots, f_r)$ is a quasi-isomorphism, and \\item The map of extended alternating {\\v C}ech complexes $$ \\xymatrix{ R \\to \\prod_{i_0} R_{f_{i_0}} \\to \\prod_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to \\ldots \\to R_{f_1\\ldots f_r} \\ar[d] \\\\ S \\to \\prod_{i_0} S_{f_{i_0}} \\to \\prod_{i_0 < i_1} S_{f_{i_0}f_{i_1}} \\to \\ldots \\to S_{f_1\\ldots f_r} } $$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "10343", "title": "more-algebra-lemma-naive-Koszul-complex", "text": "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal of $R$. Let $M$ be the $R$-module generated by elements $e_1, \\ldots, e_n$ subject to the relations $f_i e_j - f_j e_i = 0$. There exists a short exact sequence $$ 0 \\to K \\to M \\to I \\to 0 $$ such that $K$ is annihilated by $I$."}
+{"_id": "10344", "title": "more-algebra-lemma-explicit-ext", "text": "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ set $$ H_1(N, f_\\bullet) = \\frac{\\{(x_1, \\ldots, x_n) \\in N^{\\oplus n} \\mid f_i x_j = f_j x_i \\}} {\\{f_1x, \\ldots, f_nx) \\mid x \\in N\\}} $$ For any $R$-module $N$ there exists a canonical short exact sequence $$ 0 \\to \\Ext_R(R/I, N) \\to H_1(N, f_\\bullet) \\to \\Hom_R(K, N) $$ where $K$ is as in Lemma \\ref{lemma-naive-Koszul-complex}."}
+{"_id": "10345", "title": "more-algebra-lemma-koszul-homology-annihilated", "text": "Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ the Koszul homology group $H_1(N, f_\\bullet)$ defined in Lemma \\ref{lemma-explicit-ext} is annihilated by $I$."}
+{"_id": "10346", "title": "more-algebra-lemma-neighbourhood-extensions", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. Let $M$, $N$ be $R$-modules. Assume $M$ is $I$-power torsion. Given an short exact sequence $$ 0 \\to N \\otimes_R S \\to \\tilde E \\to M \\otimes_R S \\to 0 $$ there exists a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & N \\ar[r] \\ar[d] & E \\ar[r] \\ar[d] & M \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & N \\otimes_R S \\ar[r] & \\tilde E \\ar[r] & M \\otimes_R S \\ar[r] & 0 } $$ with exact rows."}
+{"_id": "10347", "title": "more-algebra-lemma-recover-module-from-glueing-data", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that $R/I \\to S/IS$ is an isomorphism. Let $M$ be an $R$-module. Then the complex (\\ref{equation-glueing-complex}) is exact."}
+{"_id": "10348", "title": "more-algebra-lemma-H0-inverse", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that $R/I \\to S/IS$ is an isomorphism. Then the functor $H^0$ is a left quasi-inverse to the functor $\\text{Can}$ of Remark \\ref{remark-glueing-data}."}
+{"_id": "10349", "title": "more-algebra-lemma-exact", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and let $I = (f_1, \\ldots, f_t) \\subset R$ be an ideal. Then $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ is an abelian category, and the functor $\\text{Can}$ is exact and commutes with arbitrary colimits."}
+{"_id": "10350", "title": "more-algebra-lemma-equivalence-I-unit", "text": "Let $\\varphi : R \\to S$ be a flat ring map and $(f_1, \\ldots, f_t) = R$. Then $\\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories $$ \\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t) $$"}
+{"_id": "10351", "title": "more-algebra-lemma-base-change-glue", "text": "Let $\\varphi : R \\to S$ be a flat ring map and $I = (f_1, \\ldots, f_t)$ and ideal. Let $R \\to R'$ be a flat ring map, and set $S' = S \\otimes_R R'$. Then we obtain a commutative diagram of categories and functors $$ \\xymatrix{ \\text{Mod}_R \\ar[r]_-{\\text{Can}} \\ar[d]_{-\\otimes_R R'} & \\text{Glue}(R \\to S, f_1, \\ldots, f_t) \\ar[r]_-{H^0} \\ar[d]^{-\\otimes_R R'} & \\text{Mod}_R \\ar[d]^{-\\otimes_R R'} \\\\ \\text{Mod}_{R'} \\ar[r]^-{\\text{Can}} & \\text{Glue}(R' \\to S', f_1, \\ldots, f_t) \\ar[r]^-{H^0} & \\text{Mod}_{R'} } $$"}
+{"_id": "10352", "title": "more-algebra-lemma-application-formal-glueing", "text": "Let $\\varphi : R \\to S$ be a flat ring map and let $I \\subset R$ be a finitely generated ideal such that $R/I \\to S/IS$ is an isomorphism.  \\begin{enumerate} \\item Given an $R$-module $N$, an $S$-module $M'$ and an $S$-module map $\\varphi : M' \\to N \\otimes_R S$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\\psi : M \\to N$ and an isomorphism $M \\otimes_R S = M'$ compatible with $\\varphi$ and $\\psi$. \\item Given an $R$-module $M$, an $S$-module $N'$ and an $S$-module map $\\varphi : M \\otimes_R S \\to N'$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\\psi : M \\to N$ and an isomorphism $N \\otimes_R S = N'$ compatible with $\\varphi$ and $\\psi$. \\end{enumerate} In both cases we have $\\Ker(\\varphi) \\cong \\Ker(\\psi)$ and $\\Coker(\\varphi) \\cong \\Coker(\\psi)$."}
+{"_id": "10353", "title": "more-algebra-lemma-same-quotients", "text": "Let $R$ be a ring and let $f \\in R$. For every positive integer $n$ the map $R/f^nR \\to R^\\wedge/f^n R^\\wedge$ is an isomorphism."}
+{"_id": "10354", "title": "more-algebra-lemma-torsion-completion", "text": "\\begin{reference} Slight generalization of \\cite[Lemme~1]{Beauville-Laszlo}. \\end{reference} Let $R$ be a ring, let $f \\in R$ be an element, and let $R \\to R'$ be a ring map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$. For any $f$-power torsion $R$-module $M$ the map $M \\to M \\otimes_R R'$ is an isomorphism. For example, we have $M \\cong M \\otimes_R R^\\wedge$."}
+{"_id": "10355", "title": "more-algebra-lemma-BL-faithful", "text": "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$. The $R$-module $R' \\oplus R_f$ is faithful: for every nonzero $R$-module $M$, the module $M \\otimes_R (R' \\oplus R_f)$ is also nonzero. For example, if $M$ is nonzero, then $M \\otimes_R (R^\\wedge \\oplus R_f)$ is nonzero."}
+{"_id": "10357", "title": "more-algebra-lemma-faithful-descent", "text": "\\begin{reference} Slight generalization of \\cite[Lemme~2(a)]{Beauville-Laszlo}. \\end{reference} Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$. An $R$-module $M$ is finitely generated if and only if the ($R' \\oplus R_f$)-module $M \\otimes_R (R' \\oplus R_f)$ is finitely generated. For example, if $M \\otimes_R (R^\\wedge \\oplus R_f)$ is finitely generated as a module over $R^\\wedge \\oplus R_f$, then $M$ is a finitely generated $R$-module."}
+{"_id": "10358", "title": "more-algebra-lemma-same-f-torsion", "text": "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$. The sequence (\\ref{equation-BL-cech-re}) is \\begin{enumerate} \\item exact on the right, \\item exact on the left if and only if $R[f^\\infty] \\to R'[f^\\infty]$ is injective, and \\item exact in the middle if and only if $R[f^\\infty] \\to R'[f^\\infty]$ is surjective. \\end{enumerate} In particular, $(R \\to R', f)$ is a glueing pair if and only if $R[f^\\infty] \\to R'[f^\\infty]$ is bijective. For example, $(R, f)$ is a glueing pair if and only if $R[f^\\infty] \\to R^\\wedge[f^\\infty]$ is bijective."}
+{"_id": "10359", "title": "more-algebra-lemma-same-f-torsion-module", "text": "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map which induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$. The sequence (\\ref{equation-BL-cech-mod-re}) is \\begin{enumerate} \\item exact on the right, \\item exact on the left if and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is injective, and \\item exact in the middle if and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is surjective. \\end{enumerate} Thus $M$ is glueable for $(R \\to R', f)$ if and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is bijective. If $(R \\to R', f)$ is a glueing pair, then $M$ is glueable for $(R \\to R', f)$ if and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is injective. For example, if $(R, f)$ is a glueing pair, then $M$ is glueable if and only if $M[f^\\infty] \\to (M \\otimes_R R^\\wedge)[f^\\infty]$ is injective."}
+{"_id": "10360", "title": "more-algebra-lemma-first-tor", "text": "Let $(R \\to R', f)$ be a glueing pair. Then $\\text{Tor}^R_1(R', f^n R) = 0$ for each $n > 0$."}
+{"_id": "10361", "title": "more-algebra-lemma-first-tor-total", "text": "Let $(R \\to R',f)$ be a glueing pair. Then $\\text{Tor}^R_1(R', R/R[f^\\infty]) = 0$."}
+{"_id": "10362", "title": "more-algebra-lemma-BL3", "text": "\\begin{reference} Slight generalization of \\cite[Lemme 3(a)]{Beauville-Laszlo} \\end{reference} Let $(R \\to R', f)$ be a glueing pair. For every $R$-module $M$, we have $\\text{Tor}^R_1(R', \\Coker(M \\to M_f)) = 0$."}
+{"_id": "10363", "title": "more-algebra-lemma-BL-flat", "text": "Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module which is not necessarily glueable for $(R \\to R', f)$. Then $M$ is flat over $R$ if and only if $M \\otimes_R R'$ is flat over $R'$ and $M_f$ is flat over $R_f$."}
+{"_id": "10364", "title": "more-algebra-lemma-BL-properties", "text": "Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module which is not necessarily glueable for $(R \\to R', f)$. Then $M$ is a finite projective $R$-module if and only if $M \\otimes_R R'$ is finite projective over $R'$ and $M_f$ is finite projective over $R_f$."}
+{"_id": "10365", "title": "more-algebra-lemma-hom-from-Af", "text": "Let $A$ be a ring. Let $f \\in A$. Let $K \\in D(A)$. The following are equivalent \\begin{enumerate} \\item $\\Ext^n_A(A_f, K) = 0$ for all $n$, \\item $\\Hom_{D(A)}(E, K) = 0$ for all $E$ in $D(A_f)$, \\item $T(K, f) = 0$, \\item for every $p \\in \\mathbf{Z}$ we have $T(H^p(K), f) = 0$, \\item for every $p \\in \\mathbf{Z}$ we have $\\Hom_A(A_f, H^p(K)) = 0$ and $\\Ext^1_A(A_f, H^p(K)) = 0$, \\item $R\\Hom_A(A_f, K) = 0$, \\item the map $\\prod_{n \\geq 0} K \\to \\prod_{n \\geq 0} K$, $(x_0, x_1, \\ldots) \\mapsto (x_0 - fx_1, x_1 - fx_2, \\ldots)$ is an isomorphism in $D(A)$, and \\item add more here. \\end{enumerate}"}
+{"_id": "10366", "title": "more-algebra-lemma-ideal-of-elements-complete-wrt", "text": "Let $A$ be a ring. Let $K \\in D(A)$. The set $I$ of $f \\in A$ such that $T(K, f) = 0$ is a radical ideal of $A$."}
+{"_id": "10367", "title": "more-algebra-lemma-complete-derived-complete", "text": "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Let $M$ be an $A$-module. \\begin{enumerate} \\item If $M$ is $I$-adically complete, then $T(M, f) = 0$ for all $f \\in I$. \\item Conversely, if $T(M, f) = 0$ for all $f \\in I$ and $I$ is finitely generated, then $M \\to \\lim M/I^nM$ is surjective. \\end{enumerate}"}
+{"_id": "10368", "title": "more-algebra-lemma-serre-subcategory", "text": "Let $I$ be an ideal of a ring $A$. \\begin{enumerate} \\item The derived complete $A$-modules form a weak Serre subcategory $\\mathcal{C}$ of $\\text{Mod}_A$. \\item $D_\\mathcal{C}(A) \\subset D(A)$ is the full subcategory of derived complete objects. \\end{enumerate}"}
+{"_id": "10369", "title": "more-algebra-lemma-derived-complete-zero", "text": "Let $I$ be a finitely generated ideal of a ring $A$. Let $M$ be a derived complete $A$-module. If $M/IM = 0$, then $M = 0$."}
+{"_id": "10370", "title": "more-algebra-lemma-pseudo-coherent-is-derived-complete", "text": "Let $A$ be a ring and $I \\subset A$ an ideal. If $A$ is $I$-adically complete then any pseudo-coherent object of $D(A)$ is derived complete."}
+{"_id": "10371", "title": "more-algebra-lemma-double-localize", "text": "Let $A$ be a ring. Let $f, g \\in A$. Then for $K \\in D(A)$ we have $R\\Hom_A(A_f, R\\Hom_A(A_g, K)) = R\\Hom_A(A_{fg}, K)$."}
+{"_id": "10372", "title": "more-algebra-lemma-derived-completion", "text": "\\begin{slogan} Derived completions along finitely generated ideals exist, and can be computed by a {\\v C}ech procedure. \\end{slogan} Let $I$ be a finitely generated ideal of a ring $A$. The inclusion functor $D_{comp}(A, I) \\to D(A)$ has a left adjoint, i.e., given any object $K$ of $D(A)$ there exists a map $K \\to K^\\wedge$ of $K$ into a derived complete object of $D(A)$ such that the map $$ \\Hom_{D(A)}(K^\\wedge, E) \\longrightarrow \\Hom_{D(A)}(K, E) $$ is bijective whenever $E$ is a derived complete object of $D(A)$. In fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have $$ K^\\wedge = R\\Hom\\left((A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to \\ldots \\to A_{f_1\\ldots f_r}), K\\right) $$ functorially in $K$."}
+{"_id": "10373", "title": "more-algebra-lemma-derived-completion-vanishes", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $K^\\bullet$ be a complex of $A$-modules such that $f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some $f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then the derived completion of $K^\\bullet$ is zero."}
+{"_id": "10374", "title": "more-algebra-lemma-completion-RHom", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $K, L \\in D(A)$. Then $$ R\\Hom_A(K, L)^\\wedge = R\\Hom_A(K, L^\\wedge) = R\\Hom_A(K^\\wedge, L^\\wedge) $$"}
+{"_id": "10375", "title": "more-algebra-lemma-naive-derived-completion", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal. Let $(K_n)$ be an inverse system of objects of $D(A)$ such that for all $f \\in I$ and $n$ there exists an $e = e(n, f)$ such that $f^e$ is zero on $K_n$. Then for $K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n)$ is derived complete with respect to $I$."}
+{"_id": "10376", "title": "more-algebra-lemma-koszul-derived-completion-complete", "text": "In Situation \\ref{situation-koszul}. For $K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$ is derived complete with respect to $I$."}
+{"_id": "10377", "title": "more-algebra-lemma-characterize-derived-complete-Koszul", "text": "In Situation \\ref{situation-koszul}. Let $K \\in D(A)$. The following are equivalent \\begin{enumerate} \\item $K$ is derived complete with respect to $I$, and \\item the canonical map $K \\to R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$ is an isomorphism of $D(A)$. \\end{enumerate}"}
+{"_id": "10378", "title": "more-algebra-lemma-derived-completion-koszul", "text": "In Situation \\ref{situation-koszul}. The functor which sends $K \\in D(A)$ to the derived limit $K' = R\\lim( K \\otimes_A^\\mathbf{L} K_n^\\bullet )$ is the left adjoint to the inclusion functor $D_{comp}(A) \\to D(A)$ constructed in Lemma \\ref{lemma-derived-completion}."}
+{"_id": "10380", "title": "more-algebra-lemma-derived-completion-finite-cohomological-dimension", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal which can be generated by $r$ elements. Then derived completion has finite cohomological dimension: \\begin{enumerate} \\item Let $K \\to L$ be a morphism in $D(A)$ such that $H^i(K) \\to H^i(L)$ is an isomorphism for $i \\geq 1$ and surjective for $i = 0$. Then $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\geq 1$ and surjective for $i = 0$. \\item Let $K \\to L$ be a morphism of $D(A)$ such that $H^i(K) \\to H^i(L)$ is an isomorphism for $i \\leq -1$ and injective for $i = 0$. Then $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\leq -r - 1$ and injective for $i = -r$. \\end{enumerate}"}
+{"_id": "10382", "title": "more-algebra-lemma-restriction-derived-complete", "text": "Let $A \\to B$ be a ring map. Let $I \\subset A$ be an ideal. The inverse image of $D_{comp}(A, I)$ under the restriction functor $D(B) \\to D(A)$ is $D_{comp}(B, IB)$."}
+{"_id": "10384", "title": "more-algebra-lemma-lift-universally", "text": "Let $A$ be a ring. Let $f \\in A$. If there exists an integer $c \\geq 1$ such that $A[f^c] = A[f^{c + 1}] = A[f^{c + 2}] = \\ldots$ (for example if $A$ is Noetherian), then for all $n \\geq 1$ there exist maps $$ (A \\xrightarrow{f^n} A) \\longrightarrow A/(f^n), \\quad\\text{and}\\quad A/(f^{n + c}) \\longrightarrow (A \\xrightarrow{f^n} A) $$ in $D(A)$ inducing an isomorphism of the pro-objects $\\{A/(f^n)\\}$ and $\\{(f^n : A \\to A)\\}$ in $D(A)$."}
+{"_id": "10387", "title": "more-algebra-lemma-kernel-to-completion-square-zero", "text": "Let $f \\in A$ be an element of a ring. Set $J = \\bigcap f^nA$. Let $M$ be an $A$-module derived complete with respect to $f$. Then $JM' = 0$ where $M' = \\Ker(M \\to \\lim M/f^nM)$. In particular, if $A$ is derived complete then $J$ is an ideal of square zero."}
+{"_id": "10389", "title": "more-algebra-lemma-kernel-to-completion-nilpotent", "text": "Let $A$ be a ring derived complete with respect to an ideal $I$. Set $J = \\bigcap I^n$. If $I$ can be generated by $r$ elements then $J^N = 0$ where $N = 2^r$."}
+{"_id": "10391", "title": "more-algebra-lemma-sequence-Koszul-complexes", "text": "In Situation \\ref{situation-koszul}. If $A$ is Noetherian, then the pro-objects $\\{K_n^\\bullet\\}$ and $\\{A/(f_1^n, \\ldots, f_r^n)\\}$ of $D(A)$ are isomorphic\\footnote{In particular, for every $n$ there exists an $m \\geq n$ such that $K_m^\\bullet \\to K_n^\\bullet$ factors through the map $K_m^\\bullet \\to A/(f_1^m, \\ldots, f_r^m)$.}."}
+{"_id": "10392", "title": "more-algebra-lemma-noetherian-calculate", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module with derived completion $M^\\wedge$. Then there are short exact sequences $$ 0 \\to R^1\\lim \\text{Tor}_{i + 1}^A(M, A/I^n) \\to H^{-i}(M^\\wedge) \\to \\lim \\text{Tor}_i^A(M, A/I^n) \\to 0 $$ A similar result holds for $M \\in D^-(A)$."}
+{"_id": "10393", "title": "more-algebra-lemma-derived-completion-pseudo-coherent", "text": "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal. Let $K$ be an object of $D(A)$ such that $H^n(K)$ a finite $A$-module for all $n \\in \\mathbf{Z}$. Then the cohomology modules $H^n(K^\\wedge)$ of the derived completion are the $I$-adic completions of the cohomology modules $H^n(K)$."}
+{"_id": "10395", "title": "more-algebra-lemma-when-derived-completion-is-completion", "text": "Let $I$ be an ideal in a Noetherian ring $A$. \\begin{enumerate} \\item If $M$ is a finite $A$-module and $N$ is a flat $A$-module, then the derived $I$-adic completion of $M \\otimes_A N$ is the usual $I$-adic completion of $M \\otimes_A N$. \\item If $M$ is a finite $A$-module and $f \\in A$, then the derived $I$-adic completion of $M_f$ is the usual $I$-adic completion of $M_f$. \\end{enumerate}"}
+{"_id": "10397", "title": "more-algebra-lemma-eta-first-property", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. There is a canonical isomorphism $$ f^i : H^i(M^\\bullet)/H^i(M^\\bullet)[f] \\longrightarrow H^i(\\eta_fM^\\bullet) $$ given by multiplication by $f^i$."}
+{"_id": "10398", "title": "more-algebra-lemma-eta-qis", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. If $M^\\bullet \\to N^\\bullet$ is a quasi-isomorphism of complexes of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$ and $N^i$, then the induced map $\\eta_fM^\\bullet \\to \\eta_fN^\\bullet$ is a quasi-isomorphism too."}
+{"_id": "10400", "title": "more-algebra-lemma-eta-third-property", "text": "Let $A$ be a ring and let $f, g \\in A$ be nonzerodivisors. Let $M^\\bullet$ be a complex of $A$-modules such that $fg$ is a nonzerodivisor on all $M^i$. Then $\\eta_f\\eta_gM^\\bullet = \\eta_{fg}M^\\bullet$."}
+{"_id": "10401", "title": "more-algebra-lemma-eta-locally-free", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. Assume \\begin{enumerate} \\item $M^i$, $M^{i + 1}$ free of ranks $r_i, r_{i + 1}$, \\item the $r_i \\times r_i$ minors of $(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$ generate a principal ideal. \\end{enumerate} Then $(\\eta_fM)^i$ is locally free of rank $r_i$ and the canonical map $(\\eta_fM)^i \\to f^iM^i \\oplus f^{i + 1}M^{i + 1}$ is the inclusion of a direct summand."}
+{"_id": "10402", "title": "more-algebra-lemma-eta-base-change", "text": "Let $A \\to B$ be a ring map. Let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules. Assume \\begin{enumerate} \\item $f$ maps to a nonzerodivisor $g$ in $B$, \\item $f$ is a nonzerodivisor on $M^i$, \\item $M^i$ is finite free of rank $r_i$, \\item the $r_i \\times r_i$ minors of $(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$ generate a principal ideal. \\end{enumerate} for all $i \\in \\mathbf{Z}$. Then there is a canonical isomorphism $\\eta_fM^\\bullet \\otimes_A B = \\eta_g(M^\\bullet \\otimes_A B)$."}
+{"_id": "10403", "title": "more-algebra-lemma-vanishing-beta", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. For $i \\in \\mathbf{Z}$ the following are equivalent \\begin{enumerate} \\item $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$, \\item $\\beta : H^i(M^\\bullet \\otimes_A f^iA/f^{i + 1}A) \\to H^{i + 1}(M^\\bullet \\otimes_A f^{i + 1}A/f^{i + 2}A)$ is zero. \\end{enumerate} These equivalent conditions are implied by the condition $H^{i + 1}(M^\\bullet)[f] = 0$."}
+{"_id": "10404", "title": "more-algebra-lemma-eta-vanishing-beta", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. If $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$, then the canonical map $$ (\\eta_fM)^i / f(\\eta_fM)^i \\longrightarrow f^iM^i/f^{i + 1}M^i \\oplus f^{i + 1}M^{i + 1}/f^{i + 2}M^{i + 1}, x \\longmapsto (x, d^i(x)) $$ identifies the left hand side with a direct sum of submodules of the right hand side."}
+{"_id": "10405", "title": "more-algebra-lemma-ideal-direct-summand", "text": "Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules. Let $s : M \\to N_1 \\oplus N_2$ be a split injection. There exists a finitely generated ideal $I \\subset A$ with the following property: a ring map $A \\to B$ factors through $A/I$ if and only if $s \\otimes \\text{id}_B$ identifies $M \\otimes_A B$ with a direct sum of submodules of $N_1 \\otimes_A B \\oplus N_2 \\otimes_A B$."}
+{"_id": "10406", "title": "more-algebra-lemma-eta-vanishing-beta-plus", "text": "Let $A$ be a ring and let $f \\in \\mathfrak m_A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. Assume \\begin{enumerate} \\item $M^i$ is finite free of rank $r_i$ and $M^i = 0$ for $|i| \\gg 0$, \\item the $r_i \\times r_i$ minors of $(f, d^i) : M^i \\to M^i \\oplus M^{i + 1}$ generate a principal ideal \\end{enumerate} for all $i \\in \\mathbf{Z}$. Consider the set of prime ideals $$ E = \\{f \\in \\mathfrak p \\subset A \\mid \\Ker(d^i \\bmod f^2)_\\mathfrak p \\text{ surjects onto } \\Ker(d^i \\bmod f)_\\mathfrak p \\text{ for all }i \\in \\mathbf{Z}\\} $$ There exists a finitely generated ideal $f \\in I$ such that $I_\\mathfrak p = fA_\\mathfrak p$ for all $\\mathfrak p \\in E$ and such that the cohomology modules of $\\eta_f M^\\bullet \\otimes_A A/I$ are finite free."}
+{"_id": "10407", "title": "more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent", "text": "Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$ in $D(A_{n + 1})$. Assume \\begin{enumerate} \\item the transition maps $A_{n + 1} \\to A_n$ are surjective with locally nilpotent kernels, \\item $K_1$ is pseudo-coherent, and \\item the maps induce isomorphisms $K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$. \\end{enumerate} Then $K = R\\lim K_n$ is a pseudo-coherent object of $D(A)$ and $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$."}
+{"_id": "10408", "title": "more-algebra-lemma-Rlim-pseudo-coherent-gives-complete-pseudo-coherent", "text": "Let $A$ be a ring and $I \\subset A$ an ideal. Suppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$ in $D(A/I^{n + 1})$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, \\item $K_1$ is pseudo-coherent, and \\item the maps induce isomorphisms $K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$. \\end{enumerate} Then $K = R\\lim K_n$ is a pseudo-coherent, derived complete object of $D(A)$ and $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$."}
+{"_id": "10409", "title": "more-algebra-lemma-Rlim-perfect-gives-perfect", "text": "\\begin{reference} \\cite[Lemma 4.2]{Bhatt-Algebraize} \\end{reference} Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings. Suppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$ in $D(A_{n + 1})$. Assume \\begin{enumerate} \\item the transition maps $A_{n + 1} \\to A_n$ are surjective with locally nilpotent kernels, \\item $K_1$ is a perfect object, and \\item the maps induce isomorphisms $K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$. \\end{enumerate} Then $K = R\\lim K_n$ is a perfect object of $D(A)$ and $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$."}
+{"_id": "10410", "title": "more-algebra-lemma-Rlim-perfect-gives-complete", "text": "Let $A$ be a ring and $I \\subset A$ an ideal. Suppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$ in $D(A/I^{n + 1})$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, \\item $K_1$ is a perfect object, and \\item the maps induce isomorphisms $K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$. \\end{enumerate} Then $K = R\\lim K_n$ is a perfect, derived complete object of $D(A)$ and $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$."}
+{"_id": "10412", "title": "more-algebra-lemma-kollar-kovacs", "text": "\\begin{reference} Email from Kovacs of 23/02/2018. \\end{reference} Let $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$. Set $K_n = K \\otimes_A^\\mathbf{L} A/I^n$. Assume for all $i \\in \\mathbf{Z}$ we have \\begin{enumerate} \\item $H^i(K)$ is a finite $A$-module, and \\item the system $H^i(K_n)$ satisfies Mittag-Leffler. \\end{enumerate} Then $\\lim H^i(K)/I^nH^i(K)$ is equal to $\\lim H^i(K_n)$ for all $i \\in \\mathbf{Z}$."}
+{"_id": "10413", "title": "more-algebra-lemma-internal-hom-evaluate-isomorphism", "text": "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. the map $$ R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M) $$ of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism in the following two cases \\begin{enumerate} \\item $K$ perfect, or \\item $K$ is pseudo-coherent, $L \\in D^+(R)$, and $M$ finite injective dimension. \\end{enumerate}"}
+{"_id": "10414", "title": "more-algebra-lemma-internal-hom-evaluate-isomorphism-technical", "text": "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. the map $$ R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M) $$ of Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism if the following three conditions are satisfied \\begin{enumerate} \\item $L, M$ have finite injective dimension, \\item $R\\Hom_R(L, M)$ has finite tor dimension, \\item for every $n \\in \\mathbf{Z}$ the truncation $\\tau_{\\leq n}K$ is pseudo-coherent \\end{enumerate}"}
+{"_id": "10415", "title": "more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism", "text": "Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. The map $$ K \\otimes_R^\\mathbf{L} R\\Hom_R(M, L) \\longrightarrow R\\Hom_R(M, K \\otimes_R^\\mathbf{L} L) $$ of Lemma \\ref{lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \\begin{enumerate} \\item $M$ perfect, or \\item $K$ is perfect, or \\item $M$ is pseudo-coherent, $L \\in D^+(R)$, and $K$ has tor amplitude in $[a, \\infty]$. \\end{enumerate}"}
+{"_id": "10417", "title": "more-algebra-lemma-upgrade-adjoint-tensor-RHom", "text": "Let $R \\to R'$ be a ring map. For $K \\in D(R)$ and $M \\in D(R')$ there is a canonical isomorphism $$ R\\Hom_R(K, M) = R\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M) $$"}
+{"_id": "10418", "title": "more-algebra-lemma-base-change-RHom", "text": "Let $R \\to R'$ be a ring map. Let $K, M \\in D(R)$. The map (\\ref{equation-base-change-RHom}) $$ R\\Hom_R(K, M) \\otimes_R^\\mathbf{L} R' \\longrightarrow R\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M \\otimes_R^\\mathbf{L} R') $$ is an isomorphism in $D(R')$ in the following cases \\begin{enumerate} \\item $K$ is perfect, \\item $R'$ is perfect as an $R$-module, \\item $R \\to R'$ is flat, $K$ is pseudo-coherent, and $M \\in D^{+}(R)$, or \\item $R'$ has finite tor dimension as an $R$-module, $K$ is pseudo-coherent, and $M \\in D^{+}(R)$ \\end{enumerate}"}
+{"_id": "10419", "title": "more-algebra-lemma-consequence-Artin-Rees", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $ K \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M $ be a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$. For $n \\geq 0$ let $$ K/I^nK \\xrightarrow{\\alpha_n} L/I^nL \\xrightarrow{\\beta_n} M/I^nM $$ be the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$. Then there are canonical $A$-module maps giving a commutative diagram $$ \\xymatrix{ & & & H \\ar[lld] \\ar[ld] \\ar[d] \\\\ \\ldots \\ar[r] & H_3 \\ar[r] & H_2 \\ar[r] & H_1 } $$ Moreover, there exists a $c > 0$ and canonical $A$-module maps $H_n \\to H/I^{n - c}H$ for $n \\geq c$ such that the compositions $$ H/I^n H \\to H_n \\to  H/I^{n - c}H \\quad\\text{and}\\quad H_n \\to H/I^{n - c}H \\to H_{n - c} $$ are the canonical ones. Moreover, we have \\begin{enumerate} \\item $(H_n)$ and $(H/I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$, \\item $\\lim H_n = \\lim H/I^n H$, \\item the inverse system $(H_n)$ is Mittag-Leffler, \\item the image of $H_{n + c} \\to H_n$ is equal to the image of $H \\to H_n$, \\item the composition $I^cH_n \\to H_n \\to H/I^{n - c}H \\to H_n/I^{n - c}H_n$ is the inclusion $I^cH_n \\to H_n$ followed by the quotient map $H_n \\to H_n/I^{n - c}H_n$, and \\item the kernel and cokernel of $H/I^nH \\to H_n$ is annihilated by $I^c$. \\end{enumerate}"}
+{"_id": "10420", "title": "more-algebra-lemma-kollar-kovacs-pseudo-coherent", "text": "\\begin{reference} Email from Kovacs of 23/02/2018. \\end{reference} Let $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$ be pseudo-coherent. Set $K_n = K \\otimes_A^\\mathbf{L} A/I^n$. Then for all $i \\in \\mathbf{Z}$ the system $H^i(K_n)$ satisfies Mittag-Leffler and $\\lim H^i(K)/I^nH^i(K)$ is equal to $\\lim H^i(K_n)$."}
+{"_id": "10421", "title": "more-algebra-lemma-derived-completion-plain-completion", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M^\\bullet$ be a bounded complex of finite $A$-modules. The inverse system of maps $$ M^\\bullet \\otimes_A^\\mathbf{L} A/I^n \\longrightarrow M^\\bullet/I^nM^\\bullet $$ defines an isomorphism of pro-objects of $D(A)$."}
+{"_id": "10422", "title": "more-algebra-lemma-hom-systems-ML", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. Then \\begin{enumerate} \\item the systems $(\\Hom_A(M_n, N_n))$ and $(\\text{Isom}_A(M_n, N_n))$ are Mittag-Leffler, \\item there exists a $c \\geq 0$ such that the kernels and cokernels of $$ \\Hom_A(M, N)/I^n\\Hom_A(M, N) \\to \\Hom_A(M_n, N_n) $$ are killed by $I^c$ for all $n$, \\item we have $\\lim \\Hom_A(M_n, N_n) =\\Hom_A(M, N)^\\wedge = \\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge)$ \\item $\\lim \\text{Isom}_A(M_n, N_n) = \\text{Isom}_{A^\\wedge}(M^\\wedge, N^\\wedge)$. \\end{enumerate} Here ${}^\\wedge$ denotes usual $I$-adic completion."}
+{"_id": "10423", "title": "more-algebra-lemma-isomorphic-completions", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be finite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. If $M_n \\cong N_n$ for all $n$, then $M^\\wedge \\cong N^\\wedge$ as $A^\\wedge$-modules."}
+{"_id": "10424", "title": "more-algebra-lemma-iso", "text": "A morphism $(c, \\varphi_n)$ of the category of Remark \\ref{remark-weird-systems} is an isomorphism if and only if there exists a $c' \\geq 0$ such that $\\Ker(\\varphi_n)$ and $\\Coker(\\varphi_n)$ are $I^{c'}$-torsion for all $n \\gg 0$."}
+{"_id": "10425", "title": "more-algebra-lemma-dejong-kollar-kovacs", "text": "\\begin{reference} Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018. \\end{reference} Let $I$ be an ideal of the Noetherian ring $A$. Let $M$ and $N$ be finite $A$-modules. Write $A_n = A/I^n$, $M_n = M/I^nM$, and $N_n = N/I^nN$. For every $i \\geq 0$ the objects $$ \\{\\Ext^i_A(M, N)/I^n\\Ext^i_A(M, N)\\}_{n \\geq 1} \\quad\\text{and}\\quad \\{\\Ext^i_{A_n}(M_n, N_n)\\}_{n \\geq 1} $$ are isomorphic in the category $\\mathcal{C}$ of Remark \\ref{remark-weird-systems}."}
+{"_id": "10427", "title": "more-algebra-lemma-consequence-Artin-Rees-bis", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $ K \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M $ be a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$. For $n \\geq 0$ let $$ I^nK \\xrightarrow{\\alpha_n} I^nL \\xrightarrow{\\beta_n} I^nM $$ be the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$. Then there are canonical $A$-module maps $$ \\ldots \\to H_3 \\to H_2 \\to H_1 \\to H $$ There exists a $c > 0$ such that for $n \\geq c$ the image of $H_n \\to H$ is contained in $I^{n - c}H$ and there is a canonical $A$-module map $I^nH \\to H_{n - c}$ such that the compositions $$ I^n H \\to H_{n - c} \\to  I^{n - 2c}H \\quad\\text{and}\\quad H_n \\to I^{n - c}H \\to H_{n - 2c} $$ are the canonical ones. In particular, the inverse systems $(H_n)$ and $(I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$."}
+{"_id": "10428", "title": "more-algebra-lemma-ext-factors", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be $A$-modules with $M$ finite. For each $p > 0$ there exists a $c \\geq 0$ such that for $n \\geq c$ the map $\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$ factors through $\\Ext^p_A(I^nM, I^{n - c}N) \\to \\Ext_A^p(I^nM, N)$."}
+{"_id": "10429", "title": "more-algebra-lemma-ext-annihilated", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be $A$-modules with $M$ finite and $N$ annihilated by a power of $I$. For each $p > 0$ there exists an $n$ such that the map $\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$ is zero."}
+{"_id": "10430", "title": "more-algebra-lemma-ext-induced-toplogy", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $K \\in D(A)$ be pseudo-coherent and let $M$ be a finite $A$-module. For each $p \\in \\mathbf{Z}$ there exists an $c$ such that the image of $\\Ext_A^p(K, I^nM) \\to \\Ext_A^p(K, M)$ is contained in $I^{n - c}\\Ext_A^p(K, M)$ for $n \\geq c$."}
+{"_id": "10431", "title": "more-algebra-lemma-sequence-powers-pro-bounded", "text": "In Situation \\ref{situation-koszul} assume $A$ is Noetherian. With notation as above, the inverse system $(I^n)$ is pro-isomorphic in $D(A)$ to the inverse system $(I_n^\\bullet)$."}
+{"_id": "10432", "title": "more-algebra-lemma-tensoring-Deligne-system", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M^\\bullet$ be a bounded complex of finite $A$-modules. The inverse system of maps $$ I^n \\otimes_A^\\mathbf{L} M^\\bullet \\longrightarrow I^nM^\\bullet $$ defines an isomorphism of pro-objects of $D(A)$."}
+{"_id": "10433", "title": "more-algebra-lemma-factor-through-derived-tensor-product", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. There exists an integer $n > 0$ such that $I^nM \\to M$ factors through the map $I \\otimes_A^\\mathbf{L} M \\to M$ in $D(A)$."}
+{"_id": "10434", "title": "more-algebra-lemma-ext-annihilated-into", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $K \\in D(A)$ be pseudo-coherent. Let $a \\in \\mathbf{Z}$. Assume that for every finite $A$-module $M$ the modules $\\Ext^i_A(K, M)$ are $I$-power torsion for $i \\geq a$. Then for $i \\geq a$ and $M$ finite the system $\\Ext^i_A(K, M/I^nM)$ is essentially constant with value $$ \\Ext^i_A(K, M) = \\lim \\Ext^i_A(K, M/I^nM) $$"}
+{"_id": "10435", "title": "more-algebra-lemma-tor-annihilated", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$. There exists an integer $n > 0$ such that $\\text{Tor}^A_p(I^nM, N) \\to \\text{Tor}^A_p(M, N)$ is zero for all $p \\geq 0$."}
+{"_id": "10436", "title": "more-algebra-lemma-pseudo-coherent-tensor-limit", "text": "Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent. Let $(M_n)$ be an inverse system of $R$-modules. Then $R\\lim K \\otimes_R^\\mathbf{L} M_n = K \\otimes_R^\\mathbf{L} R\\lim M_n$."}
+{"_id": "10438", "title": "more-algebra-lemma-prod-qis-gives-qis", "text": "Let $$ (A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots) \\longrightarrow (B_0^\\bullet \\to B_1^\\bullet \\to B_2^\\bullet \\to \\ldots) $$ be a map between two complexes of complexes of abelian groups. Set $A^{p, q} = A_p^q$, $B^{p, q} = B_p^q$ to obtain double complexes. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ and $\\text{Tot}_\\pi(B^{\\bullet, \\bullet})$ be the product total complexes associated to the double complexes. If each $A_p^\\bullet \\to B_p^\\bullet$ is a quasi-isomorphism, then $\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to \\text{Tot}_\\pi(B^{\\bullet, \\bullet})$ is a quasi-isomorphism."}
+{"_id": "10439", "title": "more-algebra-lemma-key", "text": "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat. Let $N$ be a $B$-module. If $N$ is flat as an $A$-module, then $N$ is flat as a $B$-module."}
+{"_id": "10440", "title": "more-algebra-lemma-weak-dimension-goes-up", "text": "Let $A \\to B$ be a weakly \\'etale ring map. If $A$ has weak dimension at most $d$, then so does $B$."}
+{"_id": "10441", "title": "more-algebra-lemma-absolutely-flat", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item $A$ has weak dimension $\\leq 0$, \\item $A$ is absolutely flat, and \\item $A$ is reduced and every prime is maximal. \\end{enumerate} In this case every local ring of $A$ is a field."}
+{"_id": "10442", "title": "more-algebra-lemma-product-fields-absolutely-flat", "text": "A product of fields is an absolutely flat ring."}
+{"_id": "10443", "title": "more-algebra-lemma-base-change-weakly-etale", "text": "Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$ be the base change of $B$. \\begin{enumerate} \\item If $B \\otimes_A B \\to B$ is flat, then $B' \\otimes_{A'} B' \\to B'$ is flat. \\item If $A \\to B$ is weakly \\'etale, then $A' \\to B'$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "10444", "title": "more-algebra-lemma-absolutely-flat-over-absolutely-flat", "text": "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat. \\begin{enumerate} \\item If $A$ is an absolutely flat ring, then so is $B$. \\item If $A$ is reduced and $A \\to B$ is weakly \\'etale, then $B$ is reduced. \\end{enumerate}"}
+{"_id": "10445", "title": "more-algebra-lemma-composition-weakly-etale", "text": "Let $A \\to B$ and $B \\to C$ be ring maps. \\begin{enumerate} \\item If $B \\otimes_A B \\to B$ and $C \\otimes_B C \\to C$ are flat, then $C \\otimes_A C \\to C$ is flat. \\item If $A \\to B$ and $B \\to C$ are weakly \\'etale, then $A \\to C$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "10446", "title": "more-algebra-lemma-go-down", "text": "Let $A \\to B \\to C$ be ring maps. \\begin{enumerate} \\item If $B \\to C$ is faithfully flat and $C \\otimes_A C \\to C$ is flat, then $B \\otimes_A B \\to B$ is flat. \\item If $B \\to C$ is faithfully flat and $A \\to C$ is weakly \\'etale, then $A \\to B$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "10447", "title": "more-algebra-lemma-weakly-etale-permanence", "text": "Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of weakly \\'etale $A$-algebras. Then $B \\to C$ is weakly \\'etale."}
+{"_id": "10448", "title": "more-algebra-lemma-formally-unramified", "text": "Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat. Then $\\Omega_{B/A} = 0$, i.e., $B$ is formally unramified over $A$."}
+{"_id": "10450", "title": "more-algebra-lemma-when-weakly-etale", "text": "Let $A \\to B$ be a ring map. Then $A \\to B$ is weakly \\'etale in each of the following cases \\begin{enumerate} \\item $B = S^{-1}A$ is a localization of $A$, \\item $A \\to B$ is \\'etale, \\item $B$ is a filtered colimit of weakly \\'etale $A$-algebras. \\end{enumerate}"}
+{"_id": "10451", "title": "more-algebra-lemma-absolutely-flat-fields", "text": "Let $K \\subset L$ be an extension of fields. If $L \\otimes_K L \\to L$ is flat, then $L$ is an algebraic separable extension of $K$."}
+{"_id": "10452", "title": "more-algebra-lemma-absolutely-flat-over-field", "text": "Let $B$ be an algebra over a field $K$. The following are equivalent \\begin{enumerate} \\item $B \\otimes_K B \\to B$ is flat, \\item $K \\to B$ is weakly \\'etale, and \\item $B$ is a filtered colimit of \\'etale $K$-algebras. \\end{enumerate} Moreover, every finitely generated $K$-subalgebra of $B$ is \\'etale over $K$."}
+{"_id": "10453", "title": "more-algebra-lemma-weakly-etale-residue-field-extensions", "text": "Let $A \\to B$ be a ring map. If $A \\to B$ is weakly \\'etale, then $A \\to B$ induces separable algebraic residue field extensions."}
+{"_id": "10454", "title": "more-algebra-lemma-weak-dimension-at-most-1", "text": "Let $A$ be a ring. The following are equivalent \\begin{enumerate} \\item $A$ has weak dimension $\\leq 1$, \\item every ideal of $A$ is flat, \\item every finitely generated ideal of $A$ is flat, \\item every submodule of a flat $A$-module is flat, and \\item every local ring of $A$ is a valuation ring. \\end{enumerate}"}
+{"_id": "10455", "title": "more-algebra-lemma-product-weak-dimension-at-most-1", "text": "Let $J$ be a set. For each $j \\in J$ let $A_j$ be a valuation ring with fraction field $K_j$. Set $A = \\prod A_j$ and $K = \\prod K_j$. Then $A$ has weak dimension at most $1$ and $A \\to K$ is a localization."}
+{"_id": "10456", "title": "more-algebra-lemma-product-found-valuation-rings", "text": "Let $A$ be a normal domain with fraction field $K$. There exists a cartesian diagram $$ \\xymatrix{ A \\ar[d] \\ar[r] & K \\ar[d] \\\\ V \\ar[r] & L } $$ of rings where $V$ has weak dimension at most $1$ and $V \\to L$ is a flat, injective, epimorphism of rings."}
+{"_id": "10457", "title": "more-algebra-lemma-weak-dimension-at-most-1-integrally-closed", "text": "Let $A$ be a ring of weak dimension at most $1$. If $A \\to B$ is a flat, injective, epimorphism of rings, then $A$ is integrally closed in $B$."}
+{"_id": "10458", "title": "more-algebra-lemma-normality-goes-up", "text": "Let $A$ be a normal domain with fraction field $K$. Let $A \\to B$ be weakly \\'etale. Then $B$ is integrally closed in $B \\otimes_A K$."}
+{"_id": "10459", "title": "more-algebra-lemma-integral-over-henselian", "text": "Let $A \\to B$ be a ring homomorphism. Assume \\begin{enumerate} \\item $A$ is a henselian local ring, \\item $A \\to B$ is integral, \\item $B$ is a domain. \\end{enumerate} Then $B$ is a henselian local ring and $A \\to B$ is a local homomorphism. If $A$ is strictly henselian, then $B$ is a strictly henselian local ring and the extension $\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$ of residue fields is purely inseparable."}
+{"_id": "10460", "title": "more-algebra-lemma-local-tensor-with-integral", "text": "Let $A \\to B$ and $A \\to C$ be local homomorphisms of local rings. If $A \\to C$ is integral and either $\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_C)$ or $\\kappa(\\mathfrak m_A) \\subset \\kappa(\\mathfrak m_B)$ is purely inseparable, then $D = B \\otimes_A C$ is a local ring and $B \\to D$ and $C \\to D$ are local."}
+{"_id": "10461", "title": "more-algebra-lemma-class-weakly-etale-over-field", "text": "Let $K$ be a field. If $B$ is weakly \\'etale over $K$, then \\begin{enumerate} \\item $B$ is reduced, \\item $B$ is integral over $K$, \\item any finitely generated $K$-subalgebra of $B$ is a finite product of finite separable extensions of $K$, \\item $B$ is a field if and only if $B$ does not have nontrivial idempotents and in this case it is a separable algebraic extension of $K$, \\item any sub or quotient $K$-algebra of $B$ is weakly \\'etale over $K$, \\item if $B'$ is weakly \\'etale over $K$, then $B \\otimes_K B'$ is weakly \\'etale over $K$. \\end{enumerate}"}
+{"_id": "10462", "title": "more-algebra-lemma-max-weakly-etale-subalgebra", "text": "Let $K$ be a field. Let $A$ be a $K$-algebra. There exists a maximal weakly \\'etale $K$-subalgebra $B_{max} \\subset A$."}
+{"_id": "10463", "title": "more-algebra-lemma-properties-of-max-weakly-etale-subalgebra", "text": "Let $K$ be a field. For a $K$-algebra $A$ denote $B_{max}(A)$ the maximal weakly \\'etale $K$-subalgebra of $A$ as in Lemma \\ref{lemma-max-weakly-etale-subalgebra}. Then \\begin{enumerate} \\item any $K$-algebra map $A' \\to A$ induces a $K$-algebra map $B_{max}(A') \\to B_{max}(A)$, \\item if $A' \\subset A$, then $B_{max}(A') = B_{max}(A) \\cap A'$, \\item if $A = \\colim A_i$ is a filtered colimit, then $B_{max}(A) = \\colim B_{max}(A_i)$, \\item the map $B_{max}(A) \\to B_{max}(A_{red})$ is an isomorphism, \\item $B_{max}(A_1 \\times \\ldots \\times A_n) = B_{max}(A_1) \\times \\ldots \\times B_{max}(A_n)$, \\item if $A$ has no nontrivial idempotents, then $B_{max}(A)$ is a field and a separable algebraic extension of $K$, \\item add more here. \\end{enumerate}"}
+{"_id": "10464", "title": "more-algebra-lemma-change-fields-max-weakly-etale-subalgebra", "text": "Let $L/K$ be an extension of fields. Let $A$ be a $K$-algebra. Let $B \\subset A$ be the maximal weakly \\'etale $K$-subalgebra of $A$ as in Lemma \\ref{lemma-max-weakly-etale-subalgebra}. Then $B \\otimes_K L$ is the maximal weakly \\'etale $L$-subalgebra of $A \\otimes_K L$."}
+{"_id": "10465", "title": "more-algebra-lemma-branches", "text": "Let $A$ be a local ring. Assume $A$ has finitely many minimal prime ideals. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Let $A^h$ be the henselization of $A$. Consider the maps $$ \\Spec(A') \\leftarrow \\Spec((A')^h) \\rightarrow \\Spec(A^h) $$ where $(A')^h = A' \\otimes_A A^h$. Then \\begin{enumerate} \\item the left arrow is bijective on maximal ideals, \\item the right arrow is bijective on minimal primes, \\item every minimal prime of $(A')^h$ is contained in a unique maximal ideal and every maximal ideal contains exact one minimal prime. \\end{enumerate}"}
+{"_id": "10466", "title": "more-algebra-lemma-unibranch", "text": "\\begin{reference} \\cite[Chapter IV Proposition 18.6.12]{EGA4} \\end{reference} Let $A$ be a local ring. Let $A^h$ be the henselization of $A$. The following are equivalent \\begin{enumerate} \\item $A$ is unibranch, and \\item $A^h$ has a unique minimal prime. \\end{enumerate}"}
+{"_id": "10467", "title": "more-algebra-lemma-geometric-branches", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local ring. Assume $A$ has finitely many minimal prime ideals. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Choose an algebraic closure $\\overline{\\kappa}$ of $\\kappa$ and denote $\\kappa^{sep} \\subset \\overline{\\kappa}$ the separable algebraic closure of $\\kappa$. Let $A^{sh}$ be the strict henselization of $A$ with respect to $\\kappa^{sep}$. Consider the maps $$ \\Spec(A') \\xleftarrow{c} \\Spec((A')^{sh}) \\xrightarrow{e} \\Spec(A^{sh}) $$ where $(A')^{sh} = A' \\otimes_A A^{sh}$. Then \\begin{enumerate} \\item for $\\mathfrak m' \\subset A'$ maximal the residue field $\\kappa'$ is algebraic over $\\kappa$ and the fibre of $c$ over $\\mathfrak m'$ can be canonically identified with $\\Hom_\\kappa(\\kappa', \\overline{\\kappa})$, \\item the right arrow is bijective on minimal primes, \\item every minimal prime of $(A')^{sh}$ is contained in a unique maximal ideal and every maximal ideal contains a unique minimal prime. \\end{enumerate}"}
+{"_id": "10468", "title": "more-algebra-lemma-geometrically-unibranch", "text": "\\begin{reference} \\cite[Lemma 2.2]{Etale-coverings} and \\cite[Chapter IV Proposition 18.8.15]{EGA4} \\end{reference} Let $A$ be a local ring. Let $A^{sh}$ be a strict henselization of $A$. The following are equivalent \\begin{enumerate} \\item $A$ is geometrically unibranch, and \\item $A^{sh}$ has a unique minimal prime. \\end{enumerate}"}
+{"_id": "10469", "title": "more-algebra-lemma-number-of-branches-1", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local ring. \\begin{enumerate} \\item If $A$ has infinitely many minimal prime ideals, then the number of (geometric) branches of $A$ is $\\infty$. \\item The number of branches of $A$ is $1$ if and only if $A$ is unibranch. \\item The number of geometric branches of $A$ is $1$ if and only if $A$ is geometrically unibranch. \\end{enumerate} Assume $A$ has finitely many minimal primes and let $A'$ be the integral closure of $A$ in the total ring of fractions of $A_{red}$. Then \\begin{enumerate} \\item[(4)] the number of branches of $A$ is the number of maximal ideals $\\mathfrak m'$ of $A'$, \\item[(5)] to get the number of geometric branches of $A$ we have to count each maximal ideal $\\mathfrak m'$ of $A'$ with multiplicity given by the separable degree of $\\kappa(\\mathfrak m')/\\kappa$. \\end{enumerate}"}
+{"_id": "10470", "title": "more-algebra-lemma-invariance-number-branches-smooth", "text": "Let $A \\to B$ be a local homomorphism of local rings which is the localization of a smooth ring map. \\begin{enumerate} \\item The number of geometric branches of $A$ is equal to the number of geometric branches of $B$. \\item If $A \\to B$ induces a purely inseparable extension of residue fields, then the number of branches of $A$ is the number of branches of $B$. \\end{enumerate}"}
+{"_id": "10471", "title": "more-algebra-lemma-minimal-primes-tensor-strictly-henselian", "text": "Let $k$ be an algebraically closed field. Let $A$, $B$ be strictly henselian local $k$-algebras with residue field equal to $k$. Let $C$ be the strict henselization of $A \\otimes_k B$ at the maximal ideal $\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B$. Then the minimal primes of $C$ correspond $1$-to-$1$ to pairs of minimal primes of $A$ and $B$."}
+{"_id": "10472", "title": "more-algebra-lemma-nr-branches-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. \\begin{enumerate} \\item The map $A^h \\to A^\\wedge$ defines a surjective map from minimal primes of $A^\\wedge$ to minimal primes of $A^h$. \\item The number of branches of $A$ is at most the number of branches of $A^\\wedge$. \\item The number of geometric branches of $A$ is at most the number of geometric branches of $A^\\wedge$. \\end{enumerate}"}
+{"_id": "10473", "title": "more-algebra-lemma-equal-nr-branches-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The number of branches of $A$ is the same as the number of branches of $A^\\wedge$ if and only if $\\sqrt{\\mathfrak qA^\\wedge}$ is prime for every minimal prime $\\mathfrak q \\subset A^h$ of the henselization."}
+{"_id": "10474", "title": "more-algebra-lemma-glueing-sum-components-open", "text": "Let $A$ be a ring and let $I$ be a finitely generated ideal. Let $A \\to C$ be a ring map such that for all $f \\in I$ the ring map $A_f \\to C_f$ is localization at an idempotent. Then there exists a surjection $A \\to C'$ such that $A_f \\to (C \\times C')_f$ is an isomorphism for all $f \\in I$."}
+{"_id": "10475", "title": "more-algebra-lemma-quotient-by-idempotent", "text": "Let $A$ be a Noetherian ring and $I$ an ideal. Let $B$ be a finite type $A$-algebra. Let $B^\\wedge \\to C$ be a surjective ring map with kernel $J$ where $B^\\wedge$ is the $I$-adic completion. If $J/J^2$ is annihilated by $I^c$ for some $c \\geq 0$, then $C$ is isomorphic to the completion of a finite type $A$-algebra."}
+{"_id": "10476", "title": "more-algebra-lemma-one-dimensional-formal-branch", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring with henselization $A^h$. Let $\\mathfrak q \\subset A^\\wedge$ be a minimal prime with $\\dim(A^\\wedge/\\mathfrak q) = 1$. Then there exists a minimal prime $\\mathfrak q^h$ of $A^h$ such that $\\mathfrak q = \\sqrt{\\mathfrak q^hA^\\wedge}$."}
+{"_id": "10478", "title": "more-algebra-lemma-one-dimensional-number-of-branches", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Then the number of (geometric) branches of $A$ and $A^\\wedge$ is the same."}
+{"_id": "10479", "title": "more-algebra-lemma-geometrically-normal-formal-fibres-number-of-branches", "text": "\\begin{reference} \\cite[Theorem 2.3]{Beddani} \\end{reference} Let $(A, \\mathfrak m)$ be a Noetherian local ring. If the formal fibres of $A$ are geometrically normal (for example if $A$ is excellent or quasi-excellent), then $A$ is Nagata and the number of (geometric) branches of $A$ and $A^\\wedge$ is the same."}
+{"_id": "10480", "title": "more-algebra-lemma-not-formally-catenary", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring which is not formally catenary. Then $A$ is not universally catenary."}
+{"_id": "10481", "title": "more-algebra-lemma-flat-under-catenary-equidimensional", "text": "Let $A \\to B$ be a flat local ring map of local Noetherian rings. Assume $B$ is catenary and is $\\Spec(B)$ equidimensional. Then \\begin{enumerate} \\item $\\Spec(B/\\mathfrak p B)$ is equidimensional for all $\\mathfrak p \\subset A$ and \\item $A$ is catenary and $\\Spec(A)$ is equidimensional. \\end{enumerate}"}
+{"_id": "10482", "title": "more-algebra-lemma-formally-catenary", "text": "Let $A$ be a formally catenary Noetherian local ring. Then $A$ is universally catenary."}
+{"_id": "10484", "title": "more-algebra-lemma-pol-lifting", "text": "Let $\\varphi : A \\to B$ be a surjection of rings. Let $G$ be a finite group of order $n$ acting on $\\varphi : A \\to B$. If $b \\in B^G$, then there exists a monic polynomial $P \\in A^G[T]$ which maps to $(T - b)^n$ in $B^G[T]$."}
+{"_id": "10485", "title": "more-algebra-lemma-invariants-modulo", "text": "Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $I \\subset R$ be an ideal such that $\\sigma(I) \\subset I$ for all $\\sigma \\in G$. Then $R^G/I^G \\subset (R/I)^G$ is an integral extension of rings which induces homeomorphisms on spectra and purely inseparable extensions of residue fields."}
+{"_id": "10486", "title": "more-algebra-lemma-functor-invariants-tensor", "text": "Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$. Let $A$ be an $R^G$-algebra. \\begin{enumerate} \\item for $b \\in (A \\otimes_{R^G} R)^G$ there exists a monic polynomial $P \\in A[T]$ whose image in $(A \\otimes_{R^G} R)^G[T]$ is $(T - b)^n$, \\item for $a \\in A$ mapping to zero in $(A \\otimes_{R^G} R)^G$ we have $(T - a)^{n^2} = T^{n^2}$ in $A[T]$. \\end{enumerate}"}
+{"_id": "10487", "title": "more-algebra-lemma-base-change-invariants", "text": "Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $R^G \\to A$ be a ring map. The map $$ A \\to (A \\otimes_{R^G} R)^G $$ is an isomorphism if $R^G \\to A$ is flat. In general the map is integral, induces a homeomorphism on spectra, and induces purely inseparable residue field extensions."}
+{"_id": "10488", "title": "more-algebra-lemma-one-orbit", "text": "Let $G$ be a finite group acting on a ring $R$. For any two primes $\\mathfrak q, \\mathfrak q' \\subset R$ lying over the same prime in $R^G$ there exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$."}
+{"_id": "10489", "title": "more-algebra-lemma-one-orbit-geometric", "text": "Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$ be a prime lying over $\\mathfrak p \\subset R^G$. Then $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is an algebraic normal extension and the map $$ D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\} \\longrightarrow \\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)) $$ is surjective\\footnote{Recall that we use the notation $\\text{Gal}$ only in the case of Galois extensions.}."}
+{"_id": "10490", "title": "more-algebra-lemma-one-orbit-geometric-galois", "text": "Let $A$ be a normal domain with fraction field $K$. Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \\text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$. \\begin{enumerate} \\item For any two primes $\\mathfrak q, \\mathfrak q' \\subset B$ lying over the same prime in $A$ there exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$. \\item Let $\\mathfrak q \\subset B$ be a prime lying over $\\mathfrak p \\subset A$. Then $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is an algebraic normal extension and the map $$ D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\} \\longrightarrow \\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)) $$ is surjective. \\end{enumerate}"}
+{"_id": "10491", "title": "more-algebra-lemma-one-orbit-geometric-galois-compare", "text": "Let $A$ be a normal domain with fraction field $K$. Let $M/L/K$ be a tower of (possibly infinite) Galois extensions of $K$. Let $H = \\text{Gal}(M/K)$ and $G = \\text{Gal}(L/K)$ and let $C$ and $B$ be the integral closure of $A$ in $M$ and $L$. Let $\\mathfrak r \\subset C$ and $\\mathfrak q = B \\cap \\mathfrak r$. Set $D_\\mathfrak r = \\{\\tau \\in H \\mid \\tau(\\mathfrak r) = \\mathfrak r\\}$ and $I_\\mathfrak r = \\{\\tau \\in D_\\mathfrak r \\mid \\tau \\bmod \\mathfrak r = \\text{id}_{\\kappa(\\mathfrak r)}\\}$ and similarly for $D_\\mathfrak q$ and $I_\\mathfrak q$. Under the map $H \\to G$ the induced maps $D_\\mathfrak r \\to D_\\mathfrak q$ and $I_\\mathfrak r \\to I_\\mathfrak q$ are surjective."}
+{"_id": "10492", "title": "more-algebra-lemma-inequality", "text": "Let $A \\subset B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. If the extension $L/K$ is finite, then the residue field extension is finite and we have $ef \\leq [L : K]$."}
+{"_id": "10493", "title": "more-algebra-lemma-multiplicative-e-f", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings. Then the ramification indices of $B/A$ and $C/B$ multiply to give the ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$. Similarly for the residual degrees in case they are finite."}
+{"_id": "10494", "title": "more-algebra-lemma-ramification-index-a-power-of-p", "text": "Let $A \\subset B$ be an extension of discrete valuation rings inducing the field extension $K \\subset L$. If the characteristic of $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then the ramification index $e$ is a power of $p$."}
+{"_id": "10495", "title": "more-algebra-lemma-extension-dvrs-formally-smooth", "text": "Let $A \\subset B$ be an extension of discrete valuation rings. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology, and \\item $A \\to B$ is weakly unramified and $\\kappa_A \\subset \\kappa_B$ is a separable field extension. \\end{enumerate}"}
+{"_id": "10496", "title": "more-algebra-lemma-permanence-unramified", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. \\begin{enumerate} \\item If $M/L/K$ are finite separable extensions and $M$ is unramified with respect to $A$, then $L$ is unramified with respect to $A$. \\item If $L/K$ is a finite separable extension which is unramified with respect to $A$, then there exists a Galois extension $M/K$ containing $L$ which is unramified with respect to $A$. \\item If $L_1/K$, $L_2/K$ are finite separable extensions which are unramified with respect to $A$, then there exists a a finite separable extension $L/K$ which is unramified with respect to $A$ containing $L_1$ and $L_2$. \\end{enumerate}"}
+{"_id": "10498", "title": "more-algebra-lemma-galois", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Then $G$ acts on the ring $B$ of Remark \\ref{remark-finite-separable-extension} and acts transitively on the set of maximal ideals of $B$."}
+{"_id": "10499", "title": "more-algebra-lemma-galois-conclusion", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Then there are $e \\geq 1$ and $f \\geq 1$ such that $e_i = e$ and $f_i = f$ for all $i$ (notation as in Remark \\ref{remark-finite-separable-extension}). In particular $[L : K] = n e f$."}
+{"_id": "10500", "title": "more-algebra-lemma-galois-galois", "text": "Let $A$ be a discrete valuation ring with fraction field $K$ and residue field $\\kappa$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m$ be a maximal ideal of $B$. Then \\begin{enumerate} \\item the field extension $\\kappa(\\mathfrak m)/\\kappa$ is normal, and \\item $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective. \\end{enumerate} If for some (equivalently all) maximal ideal(s) $\\mathfrak m \\subset B$ the field extension $\\kappa(\\mathfrak m)/\\kappa$ is separable, then \\begin{enumerate} \\item[(3)] $\\kappa(\\mathfrak m)/\\kappa$ is Galois, and \\item[(4)] $D \\to \\text{Gal}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective. \\end{enumerate} Here $D \\subset G$ is the decomposition group of $\\mathfrak m$."}
+{"_id": "10501", "title": "more-algebra-lemma-galois-inertia", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m \\subset B$ be a maximal ideal. The inertia group $I$ of $\\mathfrak m$ sits in a canonical exact sequence $$ 1 \\to P \\to I \\to I_t \\to 1 $$ such that \\begin{enumerate} \\item $P = \\{\\sigma \\in D \\mid \\sigma|_{B/\\mathfrak m^2} = \\text{id}_{B/\\mathfrak m^2}\\}$ where $D$ is the decomposition group, \\item $P$ is a normal subgroup of $D$, \\item $P$ is a $p$-group if the characteristic of $\\kappa_A$ is $p > 0$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$ is zero, \\item $I_t$ is cyclic of order the prime to $p$ part of the integer $e$, and \\item there is a canonical isomorphism $\\theta : I_t \\to \\mu_e(\\kappa(\\mathfrak m))$. \\end{enumerate} Here $e$ is the integer of Lemma \\ref{lemma-galois-conclusion}."}
+{"_id": "10502", "title": "more-algebra-lemma-inertia-character", "text": "With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}. The inertia character $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$ satisfies the following property $$ \\theta(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta(\\sigma)) $$ for $\\tau \\in D$ and $\\sigma \\in I$."}
+{"_id": "10503", "title": "more-algebra-lemma-inertial-invariants-unramified", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Let $\\mathfrak m \\subset B$ be a maximal ideal of the integral closure of $A$ in $L$. Let $I \\subset G$ be the inertia group of $\\mathfrak m$. Then $B^I$ is the integral closure of $A$ in $L^I$ and $A \\to (B^I)_{B^I \\cap \\mathfrak m}$ is \\'etale."}
+{"_id": "10504", "title": "more-algebra-lemma-compare-inertia", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be a tower with $M/K$ and $L/K$ finite Galois. Let $C$, $B$ be the integral closure of $A$ in $M$, $L$. Let $\\mathfrak m' \\subset C$ be a maximal ideal and set $\\mathfrak m = \\mathfrak m' \\cap B$. Let $$ P \\subset I \\subset D \\subset \\text{Gal}(L/K) \\quad\\text{and}\\quad P' \\subset I' \\subset D' \\subset \\text{Gal}(M/K) $$ be the wild inertia, inertia, decomposition group of $\\mathfrak m$ and $\\mathfrak m'$. Then the canonical surjection $\\text{Gal}(M/K) \\to \\text{Gal}(L/K)$ induces surjections $P' \\to P$, $I' \\to I$, and $D' \\to D$. Moreover these fit into commutative diagrams $$ \\vcenter{ \\xymatrix{ D' \\ar[r] \\ar[d] & \\text{Aut}(\\kappa(\\mathfrak m')/\\kappa_A) \\ar[d] \\\\ D \\ar[r] & \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A) } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ I' \\ar[r]_-{\\theta'} \\ar[d] & \\mu_{e'}(\\kappa(\\mathfrak m')) \\ar[d]^{(-)^{e'/e}} \\\\ I \\ar[r]^-\\theta & \\mu_e(\\kappa(\\mathfrak m)) } } $$ where $e'$ and $e$ are the ramification indices of $A \\to C_{\\mathfrak m'}$ and $A \\to B_\\mathfrak m$."}
+{"_id": "10505", "title": "more-algebra-lemma-krasner", "text": "Let $A$ be a complete local domain of dimension $1$. Let $P(t) \\in A[t]$ be a polynomial with coefficients in $A$. Let $\\alpha \\in A$ be a root of $P$ but not a root of the derivative $P' = \\text{d}P/\\text{d}t$. For every $c \\geq 0$ there exists an integer $n$ such that for any $Q \\in A[t]$ whose coefficients are in $\\mathfrak m_A^n$ the polynomial $P + Q$ has a root $\\beta \\in A$ with $\\beta - \\alpha \\in \\mathfrak m_A^c$."}
+{"_id": "10506", "title": "more-algebra-lemma-approximate-separable-extension", "text": "Let $A$ be a discrete valuation ring with field of fractions $K$. Let $A^\\wedge$ be the completion of $A$ with fraction field $K^\\wedge$. If $M/K^\\wedge$ is a finite separable extension, then there exists a finite separable extension $L/K$ such that $M = K^\\wedge \\otimes_K L$."}
+{"_id": "10507", "title": "more-algebra-lemma-pull-root-uniformizer", "text": "Let $A$ be a discrete valuation ring with uniformizer $\\pi$. Let $n \\geq 2$. Then $K_1 = K[\\pi^{1/n}]$ is a degree $n$ extension of $K$ and the integral closure $A_1$ of $A$ in $K_1$ is the ring $A[\\pi^{1/n}]$ which is a discrete valuation ring with ramification index $n$ over $A$."}
+{"_id": "10508", "title": "more-algebra-lemma-formally-smooth-goes-up", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Assume that $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. Then for any finite extension $K \\subset K_1$ we have $L_1 = L \\otimes_K K_1$, $B_1 = B \\otimes_A A_1$, and each extension $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ (see Remark \\ref{remark-construction}) is formally smooth in the $\\mathfrak m_{ij}$-adic topology."}
+{"_id": "10509", "title": "more-algebra-lemma-abhyankar", "text": "Let $A \\subset B$ be an extension of discrete valuation rings. Assume that either the residue characteristic of $A$ is $0$ or it is $p$, the ramification index $e$ is prime to $p$, and $\\kappa_B/\\kappa_A$ is a separable field extension. Let $K_1/K$ be a finite extension. Using the notation of Remark \\ref{remark-construction} assume $e$ divides the ramification index of $A \\subset (A_1)_{\\mathfrak m_i}$ for some $i$. Then $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ is formally smooth $\\mathfrak m_{ij}$-adic topology for all $j = 1, \\ldots, m_i$."}
+{"_id": "10510", "title": "more-algebra-lemma-composition-tame", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be finite separable extensions. Let $B$ be the integral closure of $A$ in $L$. If $L/K$ is tamely ramified with respect to $A$ and $M/L$ is tamely ramified with respect to $B_\\mathfrak m$ for every maximal ideal $\\mathfrak m$ of $B$, then $M/K$ is tamely ramified with respect to $A$."}
+{"_id": "10511", "title": "more-algebra-lemma-subextension-tame", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. If $M/L/K$ are finite separable extensions and $M$ is tamely ramified with respect to $A$, then $L$ is tamely ramified with respect to $A$."}
+{"_id": "10512", "title": "more-algebra-lemma-characterize-tame", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $\\pi \\in A$ be a uniformizer. Let $L/K$ be a finite separable extension. The following are equivalent \\begin{enumerate} \\item $L$ is tamely ramified with respect to $A$, \\item there exists an $e \\geq 1$ invertible in $\\kappa_A$ and an extension $L'/K' = K[\\pi^{1/e}]$ unramified with respect to $A' = A[\\pi^{1/e}]$ such that $L$ is contained in $L'$, and \\item there exists an $e_0 \\geq 1$ invertible in $\\kappa_A$ such that for every $d \\geq 1$ invertible in $\\kappa_A$ (2) holds with $e = de_0$. \\end{enumerate}"}
+{"_id": "10514", "title": "more-algebra-lemma-tame-goes-up", "text": "Let $A \\subset B$ be an extension of discrete valuation rings. Denote $L/K$ the corresponding extension of fraction fields. Let $K'/K$ be a finite separable extension. Then $$ K' \\otimes_K L = \\prod L'_i $$ is a finite product of fields and the following is true \\begin{enumerate} \\item If $K'$ is unramified with respect to $A$, then each $L'_i$ is unramified with respect to $B$. \\item If $K'$ is tamely ramified with respect to $A$, then each $L'_i$ is tamely ramified with respect to $B$. \\end{enumerate}"}
+{"_id": "10515", "title": "more-algebra-lemma-weakly-unramified-goes-up-along-totally-ramified", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Assume that $A \\to B$ is weakly unramified. Then for any finite separable extension $K_1/K$ totally ramified with respect to $A$ we have that $L_1 = L \\otimes_K K_1$ is a field, $A_1$ and $B_1 = B \\otimes_A A_1$ are discrete valuation rings, and the extension $A_1 \\subset B_1$ (see Remark \\ref{remark-construction}) is weakly unramified."}
+{"_id": "10516", "title": "more-algebra-lemma-solutions-go-down", "text": "Let $A \\to B \\to C$ be extensions of discrete valuation rings with fraction fields $K \\subset L \\subset M$. Let $K \\subset K_1$ be a finite extension. \\begin{enumerate} \\item If $K_1$ is a (weak) solution for $A \\to C$, then $K_1$ is a (weak) solution for $A \\to B$. \\item If $K_1$ is a (weak) solution for $A \\to B$ and $L_1 = (L \\otimes_K K_1)_{red}$ is a product of fields which are (weak) solutions for $B \\to C$, then $K_1$ is a weak solution for $A \\to C$. \\end{enumerate}"}
+{"_id": "10517", "title": "more-algebra-lemma-separable-solution-separable-solution", "text": "Let $A \\subset B$ be an extension of discrete valuation rings. Assume \\begin{enumerate} \\item the extension $K \\subset L$ of fraction fields is separable, \\item $B$ is Nagata, and \\item there exists a solution for $A \\subset B$. \\end{enumerate} Then there exists a separable solution for $A \\subset B$."}
+{"_id": "10518", "title": "more-algebra-lemma-solution-after-strict-henselization", "text": "Let $A \\to B$ be an extension of discrete valuation rings. There exists a commutative diagram $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[r] \\ar[u] & A' \\ar[u] } $$ of extensions of discrete valuation rings such that \\begin{enumerate} \\item the extensions $K \\subset K'$ and $L \\subset L'$ of fraction fields are separable algebraic, \\item the residue fields of $A'$ and $B'$ are separable algebraic closures of the residue fields of $A$ and $B$, and \\item if a solution, weak solution, or separable solution exists for $A' \\to B'$, then a solution, weak solution, or separable solution exists for $A \\to B$. \\end{enumerate}"}
+{"_id": "10519", "title": "more-algebra-lemma-galois-relative", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Let $K \\subset K_1$ be a normal extension. Say $G = \\text{Aut}(K_1/K)$. Then $G$ acts on the rings $K_1$, $L_1$, $A_1$ and $B_1$ of Remark \\ref{remark-construction} and acts transitively on the set of maximal ideals of $B_1$."}
+{"_id": "10520", "title": "more-algebra-lemma-make-degree-q-extension", "text": "Let $A$ be a discrete valuation ring with uniformizer $\\pi$. If the residue characteristic of $A$ is $p > 0$, then for every $n > 1$ and $p$-power $q$ there exists a degree $q$ separable extension $L/K$ totally ramified with respect to $A$ such that the integral closure $B$ of $A$ in $L$ has ramification index $q$ and a uniformizer $\\pi_B$ such that $\\pi_B^q = \\pi + \\pi^n b$ and $\\pi_B^q = \\pi + (\\pi_B)^{nq}b'$ for some $b, b' \\in B$."}
+{"_id": "10521", "title": "more-algebra-lemma-pre-purely-inseparable-case", "text": "Let $A$ be a discrete valuation ring. Assume the reside field $\\kappa_A$ has characteristic $p > 0$ and that $a \\in A$ is an element whose residue class in $\\kappa_A$ is not a $p$th power. Then $a$ is not a $p$th power in $K$ and the integral closure of $A$ in $K[a^{1/p}]$ is the ring $A[a^{1/p}]$ which is a discrete valuation ring weakly unramified over $A$."}
+{"_id": "10522", "title": "more-algebra-lemma-purely-inseparable-case", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings with fractions fields $K \\subset L \\subset M$. Let $\\pi \\in A$ be a uniformizer. Assume \\begin{enumerate} \\item $B$ is a Nagata ring, \\item $A \\subset B$ is weakly unramified, \\item $M$ is a degree $p$ purely inseparable extension of $L$. \\end{enumerate} Then either \\begin{enumerate} \\item $A \\to C$ is weakly unramified, or \\item $C = B[\\pi^{1/p}]$, or \\item there exists a degree $p$ separable extension $K_1/K$ totally ramified with respect to $A$ such that $L_1 = L \\otimes_K K_1$ and $M_1 = M \\otimes_K K_1$ are fields and the maps of integral closures $A_1 \\to B_1 \\to C_1$ are weakly unramified extensions of discrete valuation rings. \\end{enumerate}"}
+{"_id": "10523", "title": "more-algebra-lemma-cohen", "text": "Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is nilpotent. There exists a ring map $\\sigma : \\kappa_A \\to A$ which is a section to the residue map $A \\to \\kappa_A$. If $A \\to A'$ is a local homomorphism of local rings, then we can choose a similar ring map $\\sigma' : \\kappa_{A'} \\to A'$ compatible with $\\sigma$ provided that the extension $\\kappa_A \\subset \\kappa_{A'}$ is separable."}
+{"_id": "10524", "title": "more-algebra-lemma-pre-characteristic-p-case", "text": "Let $A$ be a discrete valuation ring with fraction field $K$ of characteristic $p > 0$. Let $\\xi \\in K$. Let $L$ be an extension of $K$ obtained by adjoining a root of $z^p - z = \\xi$. Then $L/K$ is Galois and one of the following happens \\begin{enumerate} \\item $L = K$, \\item $L/K$ is unramified with respect to $A$ of degree $p$, \\item $L/K$ is totally ramified with respect to $A$ with ramification index $p$, and \\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring, $A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable residue field extension of degree $p$. \\end{enumerate} Let $\\pi$ be a uniformizer of $A$. We have the following implications: \\begin{enumerate} \\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2). \\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by $p$ and $a$ is a unit in $A$, then we are in case (3) \\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and the image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4). \\end{enumerate}"}
+{"_id": "10525", "title": "more-algebra-lemma-characteristic-p-case", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings with fractions fields $K \\subset L \\subset M$. Assume \\begin{enumerate} \\item $A \\subset B$ weakly unramified, \\item the characteristic of $K$ is $p$, \\item $M$ is a degree $p$ Galois extension of $L$, and \\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$. \\end{enumerate} Then there exists a finite Galois extension $K_1/K$ totally ramified with respect to $A$ which is a weak solution for $A \\to C$."}
+{"_id": "10526", "title": "more-algebra-lemma-prepare", "text": "Let $A$ be a ring which contains a primitive $p$th root of unity $\\zeta$. Set $w = 1 - \\zeta$. Then $$ P(z) = \\frac{(1 + wz)^p - 1}{w^p} = z^p - z + \\sum\\nolimits_{0 < i < p} a_i z^i $$ is an element of $A[z]$ and in fact $a_i \\in (w)$. Moreover, we have $$ P(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2) $$ in the polynomial ring $A[z_1, z_2]$."}
+{"_id": "10527", "title": "more-algebra-lemma-extension-defined-by-nice-polynial", "text": "Let $A$ be a discrete valuation ring of mixed characteristic $(0, p)$ which contains a primitive $p$th root of $1$. Let $P(t) \\in A[t]$ be the polynomial of Lemma \\ref{lemma-prepare}. Let $\\xi \\in K$. Let $L$ be an extension of $K$ obtained by adjoining a root of $P(z) = \\xi$. Then $L/K$ is Galois and one of the following happens \\begin{enumerate} \\item $L = K$, \\item $L/K$ is unramified with respect to $A$ of degree $p$, \\item $L/K$ is totally ramified with respect to $A$ with ramification index $p$, and \\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring, $A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable residue field extension of degree $p$. \\end{enumerate} Let $\\pi$ be a uniformizer of $A$. We have the following implications: \\begin{enumerate} \\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2). \\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by $p$ and $a$ is a unit in $A$, then we are in case (3) \\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and the image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4). \\end{enumerate}"}
+{"_id": "10528", "title": "more-algebra-lemma-make-finite-level", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings with fractions fields $K \\subset L \\subset M$. Assume that \\begin{enumerate} \\item $A$ has mixed characteristic $(0, p)$, \\item $A \\subset B$ is weakly unramified, \\item $B$ contains a primitive $p$th root of $1$, and \\item $M/L$ is Galois of degree $p$. \\end{enumerate} Then there exists a finite Galois extension $K_1/K$ totally ramified with respect to $A$ which is either a weak solution for $A \\to C$ or is such that $M_1/L_1$ is a degree $p$ extension of finite level."}
+{"_id": "10529", "title": "more-algebra-lemma-lowering-the-level", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings with fractions fields $K \\subset L \\subset M$. Assume \\begin{enumerate} \\item $A$ has mixed characteristic $(0, p)$, \\item $A \\subset B$ weakly unramified, \\item $B$ contains a primitive $p$th root of $1$, \\item $M/L$ is a degree $p$ extension of finite level $l > 0$, \\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$. \\end{enumerate} Then there exists a finite separable extension $K_1$ of $K$ totally ramified with respect to $A$ such that either $K_1$ is a weak solution for $A \\to C$, or the extension $M_1/L_1$ is a degree $p$ extension of finite level $\\leq \\max(0, l - 1, 2l - p)$."}
+{"_id": "10530", "title": "more-algebra-lemma-special-case", "text": "Let $A \\subset B \\subset C$ be extensions of discrete valuation rings with fraction fields $K \\subset L \\subset M$. Assume \\begin{enumerate} \\item the residue field $k$ of $A$ is algebraically closed of characteristic $p > 0$, \\item $A$ and $B$ are complete, \\item $A \\to B$ is weakly unramified, \\item $M$ is a finite extension of $L$, \\item $k = \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}$ \\end{enumerate} Then there exists a finite extension $K \\subset K_1$ which is a weak solution for $A \\to C$."}
+{"_id": "10531", "title": "more-algebra-lemma-big-extension-is-ok", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Assume $B$ is essentially of finite type over $A$. Let $K \\subset K'$ be an algebraic extension of fields such that the integral closure $A'$ of $A$ in $K'$ is Noetherian. Then the integral closure $B'$ of $B$ in $L' = (L \\otimes_K K')_{red}$ is Noetherian as well. Moreover, the map $\\Spec(B') \\to \\Spec(A')$ is surjective and the corresponding residue field extensions are finitely generated field extensions."}
+{"_id": "10532", "title": "more-algebra-lemma-epp-essentially-finite-type-separable", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Assume \\begin{enumerate} \\item $B$ is essentially of finite type over $A$, \\item either $A$ or $B$ is a Nagata ring, and \\item $L/K$ is separable. \\end{enumerate} Then there exists a separable solution for $A \\to B$ (Definition \\ref{definition-solution})."}
+{"_id": "10533", "title": "more-algebra-lemma-invertible", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Equivalent are \\begin{enumerate} \\item $M$ is finite locally free module of rank $1$, \\item $M$ is invertible, and \\item there exists an $R$-module $N$ such that $M \\otimes_R N \\cong R$. \\end{enumerate} Moreover, in this case the module $N$ is (3) is isomorphic to $\\Hom_R(M, R)$."}
+{"_id": "10534", "title": "more-algebra-lemma-UFD-Pic-trivial", "text": "Let $R$ be a UFD. Then $\\Pic(R)$ is trivial."}
+{"_id": "10535", "title": "more-algebra-lemma-det-ses", "text": "Let $R$ be a ring. Let $$ 0 \\to M' \\to M \\to M'' \\to 0 $$ be a short exact sequence of finite projective $R$-modules. Then there is a canonical isomorphism $$ \\gamma : \\det(M') \\otimes \\det(M'') \\longrightarrow \\det(M) $$"}
+{"_id": "10536", "title": "more-algebra-lemma-det-ses-functorial", "text": "Let $R$ be a ring. Let $$ \\xymatrix{ 0 \\ar[r] & M' \\ar[r] \\ar[d]^u & M \\ar[r] \\ar[d]^v & M'' \\ar[r] \\ar[d]^w & 0 \\\\ 0 \\ar[r] & K' \\ar[r] & K \\ar[r] & K'' \\ar[r] & 0 } $$ be a commutative diagram of finite projective $R$-modules whose vertical arrows are isomorphisms. Then we get a commutative diagram of isomorphisms $$ \\xymatrix{ \\det(M') \\otimes \\det(M'') \\ar[r]_-\\gamma \\ar[d]_{\\det(u) \\otimes \\det(w)} & \\det(M) \\ar[d]^{\\det(v)} \\\\ \\det(K') \\otimes \\det(K'') \\ar[r]^-\\gamma & \\det(K) } $$ where the horizontal arrows are the ones constructed in Lemma \\ref{lemma-det-ses}."}
+{"_id": "10537", "title": "more-algebra-lemma-det-filtration", "text": "Let $R$ be a ring. Let $$ K \\subset L \\subset M $$ be $R$-modules such that $K$, $L/K$, and $M/L$ are finite projective $R$-modules. Then the diagram $$ \\xymatrix{ \\det(K) \\otimes \\det(L/K) \\otimes \\det(M/L) \\ar[r] \\ar[d] & \\det(L) \\otimes \\det(M/L) \\ar[d] \\\\ \\det(K) \\otimes \\det(M/K) \\ar[r] & \\det(M) } $$ commutes where the maps are those of Lemma \\ref{lemma-det-ses}."}
+{"_id": "10538", "title": "more-algebra-lemma-det-direct-sum", "text": "Let $R$ be a ring. Let $M'$ and $M''$ be two finite projective $R$-modules. Then the diagram $$ \\xymatrix{ \\det(M') \\otimes \\det(M'') \\ar[r] \\ar[d]_{\\epsilon \\cdot (\\text{switch tensors})} & \\det(M' \\oplus M'') \\ar[d]^{\\det(\\text{swith summands})} \\\\ \\det(M'') \\otimes \\det(M') \\ar[r] & \\det(M'' \\oplus M') } $$ commutes where $\\epsilon = \\det( -\\text{id}_{M' \\otimes M''}) \\in R^*$ and the horizontal arrows are those of Lemma \\ref{lemma-det-ses}."}
+{"_id": "10539", "title": "more-algebra-lemma-det-switch", "text": "Let $R$ be a ring. Let $M$, $N$ be finite projective $R$-modules. Let $a : M \\to N$ and $b : N \\to M$ be $R$-linear maps. Then $$ \\det(\\text{id} + a \\circ b) = \\det(\\text{id} + b \\circ a) $$ as elements of $R$."}
+{"_id": "10540", "title": "more-algebra-lemma-det", "text": "Let $R$ be a ring. There is a map $$ \\det : K_0(R) \\longrightarrow \\Pic(R) $$ which maps $[M]$ to the class of the invertible module $\\wedge^n(M)$ if $M$ is a finite locally free module of rank $n$."}
+{"_id": "10541", "title": "more-algebra-lemma-perfect-to-K-group", "text": "Let $R$ be a ring. There is a map $$ c : \\text{perfect complexes over }R \\longrightarrow K_0(R) $$ with the following properties \\begin{enumerate} \\item $c(K[n]) = (-1)^nc(K)$ for a perfect complex $K$, \\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle of perfect complexes, then $c(L) = c(K) + c(M)$, \\item if $K$ is represented by a finite complex $M^\\bullet$ consisting of finite projective modules, then $c(K) = \\sum (-1)^i[M_i]$. \\end{enumerate}"}
+{"_id": "10542", "title": "more-algebra-lemma-perfect-to-K-group-universal", "text": "Let $R$ be a ring. Let $D_{perf}(R)$ be the derived category of perfect objects, see Lemma \\ref{lemma-perfect-ring-classical-generator}. The map $c$ of Lemma \\ref{lemma-perfect-to-K-group} gives an isomorphism $K_0(D_{perf}(R)) = K_0(R)$."}
+{"_id": "10543", "title": "more-algebra-lemma-regular-local-Pic-zero", "text": "Let $R$ be a regular local ring. Let $f \\in R$. Then $\\Pic(R_f) = 0$."}
+{"_id": "10544", "title": "more-algebra-lemma-regular-local-UFD", "text": "A regular local ring is a UFD."}
+{"_id": "10545", "title": "more-algebra-lemma-picard-group-generic-fibre-regular", "text": "Let $R$ be a valuation ring with fraction field $K$ and residue field $\\kappa$. Let $R \\to A$ be a homomorphism of rings such that \\begin{enumerate} \\item $A$ is local and $R \\to A$ is local, \\item $A$ is flat and essentially of finite type over $R$, \\item $A \\otimes_R \\kappa$ regular. \\end{enumerate} Then $\\Pic(A \\otimes_R K) = 0$."}
+{"_id": "10546", "title": "more-algebra-lemma-canonical-element-well-defined", "text": "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of complexes of $R$-modules satisfying (1), (2), (3) above. If $L^\\bullet$ has rank $0$, then $\\det(a^\\bullet)$ maps the canonical element $\\delta(L^\\bullet)$ to $\\delta(K^\\bullet)$."}
+{"_id": "10547", "title": "more-algebra-lemma-homotopic-surjections", "text": "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of complexes of $R$-modules satisfying (1), (2), (3) above. Let $h : K^0 \\to L^{-1}$ be a map such that $b^0 = a^0 + d \\circ h$ and $b^{-1} = a^{-1} + h \\circ d$ are surjective. Then $\\det(a^\\bullet) = \\det(b^\\bullet)$ as maps $\\det(L^\\bullet) \\to \\det(K^\\bullet)$."}
+{"_id": "10548", "title": "more-algebra-lemma-compose-surjections", "text": "Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ and $b^\\bullet : L^\\bullet \\to M^\\bullet$ be maps of complexes of $R$-modules satisfying (1), (2), (3) above. Then we have $\\det(a^\\bullet) \\circ \\det(b^\\bullet) =  \\det(b^\\bullet \\circ a^\\bullet)$ as maps $\\det(M^\\bullet) \\to \\det(K^\\bullet)$."}
+{"_id": "10549", "title": "more-algebra-lemma-determinant-two-term-complexes", "text": "Let $R$ be a ring. The constructions above determine a functor $$ \\det : \\left\\{ \\begin{matrix} \\text{category of perfect complexes} \\\\ \\text{with tor amplitude in }[-1, 0] \\\\ \\text{morphisms are isomorphisms} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{category of invertible modules} \\\\ \\text{morphisms are isomorphisms} \\end{matrix} \\right\\} $$ Moreover, given a rank $0$ perfect object $L$ of $D(R)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\\delta(L) \\in \\det(L)$ such that for any isomorphism $a : L \\to K$ in $D(R)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$."}
+{"_id": "10550", "title": "more-algebra-lemma-inequality-general", "text": "Let $A \\subset B$ be an extension of valuation rings with fraction fields $K \\subset L$. If the extension $K \\subset L$ is finite, then the residue field extension is finite, the index of $\\Gamma_A$ in $\\Gamma_B$ is finite, and $$ [\\Gamma_B : \\Gamma_A] [\\kappa_B : \\kappa_A] \\leq [L : K]. $$"}
+{"_id": "10551", "title": "more-algebra-lemma-extension-normal-domains-and-roots", "text": "Let $A \\to B$ be a flat local homomorphism of Noetherian local normal domains. Let $f \\in A$ and $h \\in B$ such that $f = w h^n$ for some $n > 1$ and some unit $w$ of $B$. Assume that for every height $1$ prime $\\mathfrak p \\subset A$ there is a height $1$ prime $\\mathfrak q \\subset B$ lying over $\\mathfrak p$ such that the extension $A_\\mathfrak p \\subset B_\\mathfrak q$ is weakly unramified. Then $f = u g^n$ for some $g \\in A$ and unit $u$ of $A$."}
+{"_id": "10552", "title": "more-algebra-lemma-etale-extension-valuation-ring", "text": "Let $A$ be a valuation ring. Let $A \\to B$ be an \\'etale ring map and let $\\mathfrak m \\subset B$ be a prime lying over the maximal ideal of $A$. Then $A \\subset B_\\mathfrak m$ is an extension of valuation rings which is weakly unramified."}
+{"_id": "10553", "title": "more-algebra-lemma-henselization-valuation-ring", "text": "Let $A$ be a valuation ring. Let $A^h$, resp.\\ $A^{sh}$ be its henselization, resp.\\ strict henselization. Then $$ A \\subset A^h \\subset A^{sh} $$ are extensions of valuation rings which induce bijections on value groups, i.e., which are weakly unramified."}
+{"_id": "10554", "title": "more-algebra-lemma-characterize-PD-modules", "text": "\\begin{reference} \\cite[Corollary 1]{Warfield-Purity} \\end{reference} Let $P$ be a module over a ring $R$. The following are equivalent \\begin{enumerate} \\item $P$ is a direct summand of a direct sum of modules of the form $R/fR$, for $f \\in R$ varying. \\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ of $R$-modules such that $fA = A \\cap fB$ for all $f \\in R$ the map $\\Hom_R(P, B) \\to \\Hom_R(P, C)$ is surjective. \\end{enumerate}"}
+{"_id": "10555", "title": "more-algebra-lemma-generalized-valuation-ring", "text": "\\begin{reference} \\cite{Warfield-Decomposition} \\end{reference} Let $R$ be a nonzero ring. The following are equivalent \\begin{enumerate} \\item For $a, b \\in R$ either $a$ divides $b$ or $b$ divides $a$. \\item Every finitely generated ideal is principal and $R$ is local. \\item The set of ideals of $R$ are linearly ordered by inclusion. \\end{enumerate} This holds in particular if $R$ is a valuation ring."}
+{"_id": "10556", "title": "more-algebra-lemma-generalized-valuation-ring-modules", "text": "\\begin{reference} \\cite[Theorem 1]{Warfield-Decomposition} \\end{reference} Let $R$ be a ring satisfying the equivalent conditions of Lemma \\ref{lemma-generalized-valuation-ring}. Then every finitely presented $R$-module is isomorphic to a finite direct sum of modules of the form $R/fR$."}
+{"_id": "10557", "title": "more-algebra-lemma-warfield", "text": "\\begin{reference} \\cite[Theorem 3]{Warfield-Decomposition} \\end{reference} Let $R$ be a ring such that every local ring of $R$ at a maximal ideal satisfies the equivalent conditions of Lemma \\ref{lemma-generalized-valuation-ring}. Then every finitely presented $R$-module is a summand of a  finite direct sum of modules of the form $R/fR$ for $f$ in $R$ varying."}
+{"_id": "10558", "title": "more-algebra-lemma-elementary-divisor-is-bezout", "text": "An elementary divisor domain is B\\'ezout."}
+{"_id": "10559", "title": "more-algebra-lemma-localize-bezout", "text": "The localization of a B\\'ezout domain is B\\'ezout. Every local ring of a B\\'ezout domain is a valuation ring. A local domain is B\\'ezout if and only if it is a valuation ring."}
+{"_id": "10560", "title": "more-algebra-lemma-split-off-free-part", "text": "Let $R$ be a B\\'ezout domain. \\begin{enumerate} \\item Every finite submodule of a free module is finite free. \\item Every finitely presented $R$-module $M$ is a direct sum of a finite free module and a torsion module $M_{tors}$ which is a summand of a module of the form $\\bigoplus_{i = 1, \\ldots, n} R/f_iR$ with $f_1, \\ldots, f_n \\in R$ nonzero. \\end{enumerate}"}
+{"_id": "10561", "title": "more-algebra-lemma-modules-PID", "text": "Let $R$ be a PID. Every finite $R$-module $M$ is of isomorphic to a module of the form $$ R^{\\oplus r} \\oplus \\bigoplus\\nolimits_{i = 1, \\ldots, n} R/f_iR $$ for some $r, n \\geq 0$ and $f_1, \\ldots, f_n \\in R$ nonzero."}
+{"_id": "10563", "title": "more-algebra-lemma-polypoly", "text": "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension one, and let $x\\in\\mathfrak m$ be an element not contained in any minimal prime of $R$. Then \\begin{enumerate} \\item the function $P : n \\mapsto \\text{length}_R(R/x^n R)$ satisfies $P(n) \\leq n P(1)$ for $n \\geq 0$, \\item if $x$ is a nonzerodivisor, then $P(n) = nP(1)$ for $n \\geq 0$. \\end{enumerate}"}
+{"_id": "10564", "title": "more-algebra-lemma-minprimespoly", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $x \\in \\mathfrak m$ be an element not contained in any minimal prime of $R$. Let $t$ be the number of minimal prime ideals of $R$. Then $t \\leq \\text{length}_R(R/xR)$."}
+{"_id": "10565", "title": "more-algebra-lemma-sopexists", "text": "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension $d > 1$, let $f \\in \\mathfrak m$ be an element not contained in any minimal prime ideal of $R$, and let $k\\in\\mathbf{N}$. Then there exist elements $g_1, \\ldots, g_{d - 1} \\in \\mathfrak m^k$ such that $f, g_1, \\ldots, g_{d - 1}$ is a system of parameters."}
+{"_id": "10566", "title": "more-algebra-lemma-syspar", "text": "Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension two, and let $f \\in \\mathfrak m$ be an element not contained in any minimal prime ideal of $R$. Then there exist $g \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that \\begin{enumerate} \\item[(a)] $f,g$ form a system of parameters for $R$. \\item[(b)] If $h \\in \\mathfrak m^N$, then $f + h, g$ is a system of parameters and $\\text{length}_R (R/(f, g)) = \\text{length}_R(R/(f + h, g))$. \\end{enumerate}"}
+{"_id": "10567", "title": "more-algebra-lemma-radical-element", "text": "Let $R$ be a Noetherian local normal domain of dimension $2$. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be pairwise distinct primes of height $1$. There exists a nonzero element $f \\in \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$ such that $R/fR$ is reduced."}
+{"_id": "10568", "title": "more-algebra-lemma-divides-radical", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain of dimension $2$. If $a \\in \\mathfrak m$ is nonzero, then there exists an element $c \\in A$ such that $A/cA$ is reduced and such that $a$ divides $c^n$ for some $n$."}
+{"_id": "10569", "title": "more-algebra-lemma-multiplicity", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let $g_1, \\ldots, g_d$ be a system of parameters, and let $I = (g_1, \\ldots, g_d)$. If $e_I/d!$ is the leading coefficient of the numerical polynomial $n \\mapsto \\text{length}_R(R/I^{n+1})$, then $e_I \\leq \\text{length}_R(R/I)$."}
+{"_id": "10570", "title": "more-algebra-lemma-minprimespolyhigher", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let $t$ be the number of minimal prime ideals of $R$ of dimension $d$, and let $(g_1,\\ldots,g_d)$ be a system of parameters. Then $t \\leq \\text{length}_R(R/(g_1,\\ldots,g_n))$."}
+{"_id": "10571", "title": "more-algebra-lemma-sysparhigher", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, and let $f \\in \\mathfrak m$ be an element not contained in any minimal prime ideal of $R$. Then there exist elements $g_1, \\ldots, g_{d - 1} \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that \\begin{enumerate} \\item $f, g_1, \\ldots, g_{d - 1}$ form a system of parameters for $R$ \\item If $h \\in \\mathfrak m^N$, then $f + h, g_1, \\ldots, g_{d - 1}$ is a system of parameters and we have $\\text{length}_R R/(f, g_1, \\ldots, g_{d-1}) = \\text{length}_R R/(f + h, g_1, \\ldots, g_{d-1})$. \\end{enumerate}"}
+{"_id": "10573", "title": "more-algebra-lemma-complex-bounded-above-free-colim-bounded-finite-free", "text": "Let $R$ be a ring. Let $F^\\bullet$ be a bounded above complex of free $R$-modules. Given pairs $(n_i, f_i)$, $i = 1, \\ldots, N$ with $n_i \\in \\mathbf{Z}$ and $f_i \\in F^{n_i}$ there exists a subcomplex $G^\\bullet \\subset F^\\bullet$ containing all $f_i$ which is bounded and consists of finite free $R$-modules."}
+{"_id": "10574", "title": "more-algebra-lemma-have-dual-derived", "text": "Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following are equivalent \\begin{enumerate} \\item $M$ has a left dual in $D(R)$ as in Categories, Definition \\ref{categories-definition-dual}, \\item $M$ is a perfect object of $D(R)$. \\end{enumerate} Moreover, in this case the left dual of $M$ is the object $M^\\vee$ of Lemma \\ref{lemma-dual-perfect-complex}."}
+{"_id": "10575", "title": "more-algebra-lemma-invertible-derived", "text": "Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following are equivalent \\begin{enumerate} \\item $M$ is invertible in $D(R)$, see Categories, Definition \\ref{categories-definition-invertible}, and \\item for every prime ideal $\\mathfrak p \\subset R$ there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that $M_f \\cong R_f[-n]$ for some $n \\in \\mathbf{Z}$. \\end{enumerate} Moreover, in this case \\begin{enumerate} \\item[(a)] $M$ is a perfect object of $D(R)$, \\item[(b)] $M = \\bigoplus H^n(M)[-n]$ in $D(R)$, \\item[(c)] each $H^n(M)$ is a finite projective $R$-module, \\item[(d)] we can write $R = \\prod_{a \\leq n \\leq b} R_n$ such that $H^n(M)$ corresponds to an invertible $R_n$-module. \\end{enumerate}"}
+{"_id": "10576", "title": "more-algebra-proposition-characterization-geometrically-regular", "text": "Let $k$ be a field of characteristic $p > 0$. Let $(A, \\mathfrak m, K)$ be a Noetherian local $k$-algebra. The following are equivalent \\begin{enumerate} \\item $A$ is geometrically regular over $k$, \\item for all $k \\subset k' \\subset k^{1/p}$ finite over $k$ the ring $A \\otimes_k k'$ is regular, \\item $A$ is regular and the canonical map $H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and \\item $A$ is regular and the map $\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$ is injective. \\end{enumerate}"}
+{"_id": "10577", "title": "more-algebra-proposition-fs-flat-fibre-fs", "text": "Let $A \\to B$ be a local homomorphism of Noetherian local rings. Let $k$ be the residue field of $A$ and $\\overline{B} = B \\otimes_A k$ the special fibre. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular over $k$, \\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth in the $\\mathfrak m_{\\overline{B}}$-adic topology, and \\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. \\end{enumerate}"}
+{"_id": "10578", "title": "more-algebra-proposition-ubiquity-J-2", "text": "The following types of rings are J-2: \\begin{enumerate} \\item fields, \\item Noetherian complete local rings, \\item $\\mathbf{Z}$, \\item Noetherian local rings of dimension $1$, \\item Nagata rings of dimension $1$, \\item Dedekind domains with fraction field of characteristic zero, \\item finite type ring extensions of any of the above. \\end{enumerate}"}
+{"_id": "10579", "title": "more-algebra-proposition-fs-regular", "text": "Let $A \\to B$ be a local homomorphism of Noetherian complete local rings. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is regular, \\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular over $k$, \\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth in the $\\mathfrak m_{\\overline{B}}$-adic topology, and \\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology. \\end{enumerate}"}
+{"_id": "10580", "title": "more-algebra-proposition-Noetherian-complete-G-ring", "text": "A Noetherian complete local ring is a G-ring."}
+{"_id": "10581", "title": "more-algebra-proposition-finite-type-over-G-ring", "text": "Let $R$ be a G-ring. If $R \\to S$ is essentially of finite type then $S$ is a G-ring."}
+{"_id": "10582", "title": "more-algebra-proposition-ubiquity-G-ring", "text": "The following types of rings are G-rings: \\begin{enumerate} \\item fields, \\item Noetherian complete local rings, \\item $\\mathbf{Z}$, \\item Dedekind domains with fraction field of characteristic zero, \\item finite type ring extensions of any of the above. \\end{enumerate}"}
+{"_id": "10583", "title": "more-algebra-proposition-finite-type-over-P-ring", "text": "Let $R$ be a $P$-ring where $P$ satisfies (A), (B), (C), and (D). If $R \\to S$ is essentially of finite type then $S$ is a $P$-ring."}
+{"_id": "10584", "title": "more-algebra-proposition-ubiquity-excellent", "text": "The following types of rings are excellent: \\begin{enumerate} \\item fields, \\item Noetherian complete local rings, \\item $\\mathbf{Z}$, \\item Dedekind domains with fraction field of characteristic zero, \\item finite type ring extensions of any of the above. \\end{enumerate}"}
+{"_id": "10585", "title": "more-algebra-proposition-perfect-is-compact", "text": "Let $R$ be a ring. For an object $K$ of $D(R)$ the following are equivalent \\begin{enumerate} \\item $K$ is perfect, and \\item $K$ is a compact object of $D(R)$. \\end{enumerate}"}
+{"_id": "10587", "title": "more-algebra-proposition-equivalence", "text": "Assume $\\varphi : R \\to S$ is a flat ring map and $I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that $R/I \\to S/IS$ is an isomorphism. Then $\\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories $$ \\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t) $$"}
+{"_id": "10588", "title": "more-algebra-proposition-formal-glueing", "text": "Let $R$ be a Noetherian ring. Let $f \\in R$ be an element. Let $R^\\wedge$ be the $f$-adic completion of $R$. Then the functor $M \\mapsto (M^\\wedge, M_f, \\text{can})$ defines an equivalence $$ \\text{Mod}^{fg}_R \\longrightarrow \\text{Mod}^{fg}_{R^\\wedge} \\times_{\\text{Mod}^{fg}_{(R^\\wedge)_f}} \\text{Mod}^{fg}_{R_f} $$"}
+{"_id": "10589", "title": "more-algebra-proposition-derived-complete-modules", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. Let $M$ be an $A$-module. The following are equivalent \\begin{enumerate} \\item $M$ is $I$-adically complete, and \\item $M$ is derived complete with respect to $I$ and $\\bigcap I^nM = 0$. \\end{enumerate}"}
+{"_id": "10590", "title": "more-algebra-proposition-noetherian-naive-completion-is-completion", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. The functor which sends $K \\in D(A)$ to the derived limit $K' = R\\lim( K \\otimes_A^\\mathbf{L} A/I^n )$ is the left adjoint to the inclusion functor $D_{comp}(A) \\to D(A)$ constructed in Lemma \\ref{lemma-derived-completion}."}
+{"_id": "10591", "title": "more-algebra-proposition-ratliff", "text": "\\begin{reference} \\cite{Ratliff} \\end{reference} A Noetherian local ring is universally catenary if and only if it is formally catenary."}
+{"_id": "10592", "title": "more-algebra-proposition-epp-essentially-finite-type", "text": "\\begin{reference} See \\cite[Lemma 2.13]{alterations} for a special case. \\end{reference} Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. If $B$ is essentially of finite type over $A$, then there exists a finite extension $K \\subset K_1$ which is a solution for $A \\to B$ as defined in Definition \\ref{definition-solution}."}
+{"_id": "10681", "title": "etale-theorem-unramified-equivalence", "text": "Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type. Let $x$ be a point of $X$. The following are equivalent \\begin{enumerate} \\item $f$ is unramified at $x$, \\item the stalk $\\Omega_{X/Y, x}$ of the module of relative differentials at $x$ is trivial, \\item there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a commutative diagram $$ \\xymatrix{ U \\ar[rr]_i \\ar[rd] & & \\mathbf{A}^n_V \\ar[ld] \\\\ & V } $$ where $i$ is a closed immersion defined by a quasi-coherent sheaf of ideals $\\mathcal{I}$ such that the differentials $\\text{d}g$ for $g \\in \\mathcal{I}_{i(x)}$ generate $\\Omega_{\\mathbf{A}^n_V/V, i(x)}$, and \\item the diagonal $\\Delta_{X/Y} : X \\to X \\times_Y X$ is a local isomorphism at $x$. \\end{enumerate}"}
+{"_id": "10683", "title": "etale-theorem-sections-unramified-maps", "text": "Let $Y$ be a connected scheme. Let $f : X \\to Y$ be unramified and separated. Every section of $f$ is an isomorphism onto a connected component. There exists a bijective correspondence $$ \\text{sections of }f \\leftrightarrow \\left\\{ \\begin{matrix} \\text{connected components }X'\\text{ of }X\\text{ such that}\\\\ \\text{the induced map }X' \\to Y\\text{ is an isomorphism} \\end{matrix} \\right\\} $$ In particular, given $x \\in X$ there is at most one section passing through $x$."}
+{"_id": "10692", "title": "etale-theorem-etale-radicial-open", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is an open immersion, \\item $f$ is universally injective and \\'etale, and \\item $f$ is a flat monomorphism, locally of finite presentation. \\end{enumerate}"}
+{"_id": "10693", "title": "etale-theorem-etale-topological", "text": "Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S_0$ be a closed subscheme of $S$ with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). Denote $X_0$ (resp.\\ $Y_0$) the base change $S_0 \\times_S X$ (resp.\\ $S_0 \\times_S Y$). If $X$ is \\'etale over $S$, then the map $$ \\Mor_S(Y, X) \\longrightarrow \\Mor_{S_0}(Y_0, X_0) $$ is bijective."}
+{"_id": "10694", "title": "etale-theorem-remarkable-equivalence", "text": "\\begin{reference} \\cite[IV, Theorem 18.1.2]{EGA} \\end{reference} Let $S$ be a scheme. Let $S_0 \\subset S$ be a closed subscheme with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). The functor $$ X \\longmapsto X_0 = S_0 \\times_S X $$ defines an equivalence of categories $$ \\{ \\text{schemes }X\\text{ \\'etale over }S \\} \\leftrightarrow \\{ \\text{schemes }X_0\\text{ \\'etale over }S_0 \\} $$"}
+{"_id": "10696", "title": "etale-lemma-characterize-unramified-Noetherian", "text": "\\begin{slogan} Unramifiedness is a stalk local condition. \\end{slogan} Let $A \\to B$ be of finite type with $A$ a Noetherian ring. Let $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$. Then $A \\to B$ is unramified at $\\mathfrak q$ if and only if $A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an unramified homomorphism of local rings."}
+{"_id": "10697", "title": "etale-lemma-unramified-completions", "text": "Let $A$, $B$ be Noetherian local rings. Let $A \\to B$ be a local homomorphism. \\begin{enumerate} \\item if $A \\to B$ is an unramified homomorphism of local rings, then $B^\\wedge$ is a finite $A^\\wedge$ module, \\item if $A \\to B$ is an unramified homomorphism of local rings and $\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$, then $A^\\wedge \\to B^\\wedge$ is surjective, \\item if $A \\to B$ is an unramified homomorphism of local rings and $\\kappa(\\mathfrak m_A)$ is separably closed, then $A^\\wedge \\to B^\\wedge$ is surjective, \\item if $A$ and $B$ are complete discrete valuation rings, then $A \\to B$ is an unramified homomorphism of local rings if and only if the uniformizer for $A$ maps to a uniformizer for $B$, and the residue field extension is finite separable (and $B$ is essentially of finite type over $A$). \\end{enumerate}"}
+{"_id": "10698", "title": "etale-lemma-characterize-unramified-completions", "text": "Let $A$, $B$ be Noetherian local rings. Let $A \\to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is an unramified homomorphism of local rings \\item $A^\\wedge \\to B^\\wedge$ is an unramified homomorphism of local rings, and \\item $A^\\wedge \\to B^\\wedge$ is unramified. \\end{enumerate}"}
+{"_id": "10699", "title": "etale-lemma-unramified-definition", "text": "Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$. The morphism $f$ is unramified at $x$ in the sense of Definition \\ref{definition-unramified-schemes} if and only if it is unramified in the sense of Morphisms, Definition \\ref{morphisms-definition-unramified}."}
+{"_id": "10701", "title": "etale-lemma-universally-injective-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is unramified and a monomorphism, \\item $f$ is unramified and universally injective, \\item $f$ is locally of finite type and a monomorphism, \\item $f$ is universally injective, locally of finite type, and formally unramified, \\item $f$ is locally of finite type and $X_s$ is either empty or $X_s \\to s$ is an isomorphism for all $s \\in S$. \\end{enumerate}"}
+{"_id": "10702", "title": "etale-lemma-characterize-closed-immersion", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is a closed immersion, \\item $f$ is a proper monomorphism, \\item $f$ is proper, unramified, and universally injective, \\item $f$ is universally closed, unramified, and a monomorphism, \\item $f$ is universally closed, unramified, and universally injective, \\item $f$ is universally closed, locally of finite type, and a monomorphism, \\item $f$ is universally closed, universally injective, locally of finite type, and formally unramified. \\end{enumerate}"}
+{"_id": "10703", "title": "etale-lemma-finite-unramified-one-point", "text": "Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume that \\begin{enumerate} \\item $\\pi$ is finite, \\item $\\pi$ is unramified, \\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and \\item $\\kappa(s) \\subset \\kappa(x)$ is purely inseparable\\footnote{In view of condition (2) this is equivalent to $\\kappa(s) = \\kappa(x)$.}. \\end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is a closed immersion."}
+{"_id": "10704", "title": "etale-lemma-characterize-etale-Noetherian", "text": "Let $A \\to B$ be of finite type with $A$ a Noetherian ring. Let $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$. Then $A \\to B$ is \\'etale at $\\mathfrak q$ if and only if $A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an \\'etale homomorphism of local rings."}
+{"_id": "10705", "title": "etale-lemma-characterize-etale-completions", "text": "Let $A$, $B$ be Noetherian local rings. Let $A \\to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \\begin{enumerate} \\item $A \\to B$ is an \\'etale homomorphism of local rings \\item $A^\\wedge \\to B^\\wedge$ is an \\'etale homomorphism of local rings, and \\item $A^\\wedge \\to B^\\wedge$ is \\'etale. \\end{enumerate} Moreover, in this case $B^\\wedge \\cong (A^\\wedge)^{\\oplus n}$ as $A^\\wedge$-modules for some $n \\geq 1$."}
+{"_id": "10707", "title": "etale-lemma-finite-etale-one-point", "text": "Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume that \\begin{enumerate} \\item $\\pi$ is finite, \\item $\\pi$ is \\'etale, \\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and \\item $\\kappa(s) \\subset \\kappa(x)$ is purely inseparable\\footnote{In view of condition (2) this is equivalent to $\\kappa(s) = \\kappa(x)$.}. \\end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is an isomorphism."}
+{"_id": "10708", "title": "etale-lemma-relative-frobenius-etale", "text": "Let $U \\to X$ be an \\'etale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \\to U \\times_{X, F_X} X$ is an isomorphism."}
+{"_id": "10709", "title": "etale-lemma-unramified-etale-local", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume $f$ is unramified at each $x_i$. Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$ such that \\begin{enumerate} \\item $V_{i, j} \\to U$ is a closed immersion passing through $u$, \\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \\end{enumerate}"}
+{"_id": "10710", "title": "etale-lemma-unramified-etale-local-technical", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is unramified at each $x_i$. Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and a disjoint union decomposition $$ X_U = W \\amalg \\coprod\\nolimits_{i, j} V_{i, j} $$ such that \\begin{enumerate} \\item $V_{i, j} \\to U$ is a closed immersion passing through $u$, \\item the fibre $W_u$ contains no point mapping to any $x_i$. \\end{enumerate} In particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then the fibre $W_u$ is empty."}
+{"_id": "10711", "title": "etale-lemma-finite-unramified-etale-local", "text": "Let $f : X \\to S$ be a finite unramified morphism of schemes. Let $s \\in S$. There exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and a finite disjoint union decomposition $$ X_U = \\coprod\\nolimits_j V_j $$ such that each $V_j \\to U$ is a closed immersion."}
+{"_id": "10712", "title": "etale-lemma-etale-etale-local", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume $f$ is \\'etale at each $x_i$. Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$ such that \\begin{enumerate} \\item $V_{i, j} \\to U$ is an isomorphism, \\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \\end{enumerate}"}
+{"_id": "10713", "title": "etale-lemma-etale-etale-local-technical", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is \\'etale at each $x_i$. Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and a disjoint union decomposition $$ X_U = W \\amalg \\coprod\\nolimits_{i, j} V_{i, j} $$ such that \\begin{enumerate} \\item $V_{i, j} \\to U$ is an isomorphism, \\item the fibre $W_u$ contains no point mapping to any $x_i$. \\end{enumerate} In particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then the fibre $W_u$ is empty."}
+{"_id": "10714", "title": "etale-lemma-finite-etale-etale-local", "text": "Let $f : X \\to S$ be a finite \\'etale morphism of schemes. Let $s \\in S$. There exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and a disjoint union decomposition $$ X_U = \\coprod\\nolimits_j V_j $$ such that each $V_j \\to U$ is an isomorphism."}
+{"_id": "10715", "title": "etale-lemma-etale-dimension", "text": "Let $A$, $B$ be Noetherian local rings. Let $A \\to B$ be a \\'etale homomorphism of local rings. Then $\\dim(A) = \\dim(B)$."}
+{"_id": "10716", "title": "etale-lemma-faithful", "text": "If $f : X \\to S$ is surjective, then the functor (\\ref{equation-descent-etale}) is faithful."}
+{"_id": "10717", "title": "etale-lemma-fully-faithful", "text": "Assume $f : X \\to S$ is submersive and any \\'etale base change of $f$ is submersive. Then the functor (\\ref{equation-descent-etale}) is fully faithful."}
+{"_id": "10718", "title": "etale-lemma-fully-faithful-cases", "text": "Let $f : X \\to S$ be a morphism of schemes. In the following cases the functor (\\ref{equation-descent-etale}) is fully faithful: \\begin{enumerate} \\item $f$ is surjective and universally closed (e.g., finite, integral, or proper), \\item $f$ is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale), \\item $f$ is surjective, quasi-compact, and flat. \\end{enumerate}"}
+{"_id": "10719", "title": "etale-lemma-reduce-to-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $(V, \\varphi)$ be a descent datum relative to $X/S$ with $V \\to X$ \\'etale. Let $S = \\bigcup S_i$ be an open covering. Assume that \\begin{enumerate} \\item the pullback of the descent datum $(V, \\varphi)$ to $X \\times_S S_i/S_i$ is effective, \\item the functor (\\ref{equation-descent-etale}) for $X \\times_S (S_i \\cap S_j) \\to (S_i \\cap S_j)$ is fully faithful, and \\item the functor (\\ref{equation-descent-etale}) for $X \\times_S (S_i \\cap S_j \\cap S_k) \\to (S_i \\cap S_j \\cap S_k)$ is faithful. \\end{enumerate} Then $(V, \\varphi)$ is effective."}
+{"_id": "10720", "title": "etale-lemma-split-henselian", "text": "Let $(A, I)$ be a henselian pair. Let $U \\to \\Spec(A)$ be a quasi-compact, separated, \\'etale morphism such that $U \\times_{\\Spec(A)} \\Spec(A/I) \\to \\Spec(A/I)$ is finite. Then $$ U = U_{fin} \\amalg U_{away} $$ where $U_{fin} \\to \\Spec(A)$ is finite and $U_{away}$ has no points lying over $Z$."}
+{"_id": "10721", "title": "etale-lemma-strict-normal-crossings", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an effective Cartier divisor. Let $D_i \\subset D$, $i \\in I$ be its irreducible components viewed as reduced closed subschemes of $X$. The following are equivalent \\begin{enumerate} \\item $D$ is a strict normal crossings divisor, and \\item $D$ is reduced, each $D_i$ is an effective Cartier divisor, and for $J \\subset I$ finite the scheme theoretic intersection $D_J = \\bigcap_{j \\in J} D_j$ is a regular scheme each of whose irreducible components has codimension $|J|$ in $X$. \\end{enumerate}"}
+{"_id": "10722", "title": "etale-lemma-smooth-pullback-strict-normal-crossings", "text": "\\begin{slogan} Pullback of a strict normal crossings divisor by a smooth morphism is a strict normal crossings divisor. \\end{slogan} Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a strict normal crossings divisor. If $f : Y \\to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a strict normal crossings divisor on $Y$."}
+{"_id": "10724", "title": "etale-lemma-characterize-normal-crossings-normalization", "text": "Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed subscheme. The following are equivalent \\begin{enumerate} \\item $D$ is a normal crossings divisor in $X$, \\item $D$ is reduced, the normalization $\\nu : D^\\nu \\to D$ is unramified, and for any $n \\geq 1$ the scheme $$ Z_n = D^\\nu \\times_D \\ldots \\times_D D^\\nu \\setminus \\{(p_1, \\ldots, p_n) \\mid p_i = p_j\\text{ for some }i\\not = j\\} $$ is regular, the morphism $Z_n \\to X$ is a local complete intersection morphism whose conormal sheaf is locally free of rank $n$. \\end{enumerate}"}
+{"_id": "10726", "title": "etale-proposition-properties-sections", "text": "Sections of unramified morphisms. \\begin{enumerate} \\item Any section of an unramified morphism is an open immersion. \\item Any section of a separated morphism is a closed immersion. \\item Any section of an unramified separated morphism is open and closed. \\end{enumerate}"}
+{"_id": "10727", "title": "etale-proposition-equality", "text": "Let $S$ is be a scheme. Let $\\pi : X \\to S$ be unramified and separated. Let $Y$ be an $S$-scheme and $y \\in Y$ a point. Let $f, g : Y \\to X$ be two $S$-morphisms. Assume \\begin{enumerate} \\item $Y$ is connected \\item $x = f(y) = g(y)$, and \\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(x) \\to \\kappa(y)$ on residue fields are equal. \\end{enumerate} Then $f = g$."}
+{"_id": "10729", "title": "etale-proposition-etale-CM", "text": "\\begin{slogan} Being Cohen-Macaulay ascends and descends along \\'etale maps. \\end{slogan} Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then $A$ is Cohen-Macaulay if and only if $B$ is so."}
+{"_id": "10733", "title": "etale-proposition-effective", "text": "Let $f : X \\to S$ be a surjective integral morphism. The functor (\\ref{equation-descent-etale}) induces an equivalence $$ \\begin{matrix} \\text{schemes quasi-compact,}\\\\ \\text{separated, \\'etale over }S \\end{matrix} \\longrightarrow \\begin{matrix} \\text{descent data }(V, \\varphi)\\text{ relative to }X/S\\text{ with}\\\\ V\\text{ quasi-compact, separated, \\'etale over }X \\end{matrix} $$"}
+{"_id": "10744", "title": "crystalline-lemma-divided-power-envelope", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal with $IB \\subset J$. There exists a homomorphism of divided power rings $$ (A, I, \\gamma) \\longrightarrow (D, \\bar J, \\bar \\gamma) $$ such that $$ \\Hom_{(A, I, \\gamma)}((D, \\bar J, \\bar \\gamma), (C, K, \\delta)) = \\Hom_{(A, I)}((B, J), (C, K)) $$ functorially in the divided power algebra $(C, K, \\delta)$ over $(A, I, \\gamma)$. Here the LHS is morphisms of divided power rings over $(A, I, \\gamma)$ and the RHS is morphisms of (ring, ideal) pairs over $(A, I)$."}
+{"_id": "10746", "title": "crystalline-lemma-describe-divided-power-envelope", "text": "Let $(B, I, \\gamma)$ be a divided power algebra. Let $I \\subset J \\subset B$ be an ideal. Let $(D, \\bar J, \\bar \\gamma)$ be the divided power envelope of $J$ relative to $\\gamma$. Choose elements $f_t \\in J$, $t \\in T$ such that $J = I + (f_t)$. Then there exists a surjection $$ \\Psi : B\\langle x_t \\rangle \\longrightarrow D $$ of divided power rings mapping $x_t$ to the image of $f_t$ in $D$. The kernel of $\\Psi$ is generated by the elements $x_t - f_t$ and all $$ \\delta_n\\left(\\sum r_t x_t - r_0\\right) $$ whenever $\\sum r_t f_t = r_0$ in $B$ for some $r_t \\in B$, $r_0 \\in I$."}
+{"_id": "10747", "title": "crystalline-lemma-divided-power-envelope-add-variables", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $B$ be an $A$-algebra and $IB \\subset J \\subset B$ an ideal. Let $x_i$ be a set of variables. Then $$ D_{B[x_i], \\gamma}(JB[x_i] + (x_i)) = D_{B, \\gamma}(J) \\langle x_i \\rangle $$"}
+{"_id": "10750", "title": "crystalline-lemma-divided-power-first-order-thickening", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $M$ be an $A$-module. Let $B = A \\oplus M$ as an $A$-algebra where $M$ is an ideal of square zero and set $J = I \\oplus M$. Set $$ \\delta_n(x + z) = \\gamma_n(x) + \\gamma_{n - 1}(x)z $$ for $x \\in I$ and $z \\in M$. Then $\\delta$ is a divided power structure and $A \\to B$ is a homomorphism of divided power rings from $(A, I, \\gamma)$ to $(B, J, \\delta)$."}
+{"_id": "10751", "title": "crystalline-lemma-divided-power-second-order-thickening", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $M$, $N$ be $A$-modules. Let $q : M \\times M \\to N$ be an $A$-bilinear map. Let $B = A \\oplus M \\oplus N$ as an $A$-algebra with multiplication $$ (x, z, w)\\cdot (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z)) $$ and set $J = I \\oplus M \\oplus N$. Set $$ \\delta_n(x, z, w) = (\\gamma_n(x), \\gamma_{n - 1}(x)z, \\gamma_{n - 1}(x)w + \\gamma_{n - 2}(x)q(z, z)) $$ for $(x, z, w) \\in J$. Then $\\delta$ is a divided power structure and $A \\to B$ is a homomorphism of divided power rings from $(A, I, \\gamma)$ to $(B, J, \\delta)$."}
+{"_id": "10752", "title": "crystalline-lemma-affine-thickenings-colimits", "text": "In Situation \\ref{situation-affine}. \\begin{enumerate} \\item $\\text{CRIS}(C/A)$ has products, \\item $\\text{CRIS}(C/A)$ has all finite nonempty colimits and (\\ref{equation-forget-affine}) commutes with these, and \\item $\\text{Cris}(C/A)$ has all finite nonempty colimits and $\\text{Cris}(C/A) \\to \\text{CRIS}(C/A)$ commutes with them. \\end{enumerate}"}
+{"_id": "10753", "title": "crystalline-lemma-list-properties", "text": "In Situation \\ref{situation-affine}. Let $P \\to C$ be a surjection of $A$-algebras with kernel $J$. Write $D_{P, \\gamma}(J) = (D, \\bar J, \\bar\\gamma)$. Let $(D^\\wedge, J^\\wedge, \\bar\\gamma^\\wedge)$ be the $p$-adic completion of $D$, see Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}. For every $e \\geq 1$ set $P_e = P/p^eP$ and $J_e \\subset P_e$ the image of $J$ and write $D_{P_e, \\gamma}(J_e) = (D_e, \\bar J_e, \\bar\\gamma)$. Then for all $e$ large enough we have \\begin{enumerate} \\item $p^eD \\subset \\bar J$ and $p^eD^\\wedge \\subset \\bar J^\\wedge$ are preserved by divided powers, \\item $D^\\wedge/p^eD^\\wedge = D/p^eD = D_e$ as divided power rings, \\item $(D_e, \\bar J_e, \\bar\\gamma)$ is an object of $\\text{Cris}(C/A)$, \\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is equal to $\\lim_e (D_e, \\bar J_e, \\bar\\gamma)$, and \\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is an object of $\\text{Cris}^\\wedge(C/A)$. \\end{enumerate}"}
+{"_id": "10754", "title": "crystalline-lemma-set-generators", "text": "In Situation \\ref{situation-affine}. Let $P$ be a polynomial algebra over $A$ and let $P \\to C$ be a surjection of $A$-algebras with kernel $J$. With $(D_e, \\bar J_e, \\bar\\gamma)$ as in Lemma \\ref{lemma-list-properties}: for every object $(B, J_B, \\delta)$ of $\\text{CRIS}(C/A)$ there exists an $e$ and a morphism $D_e \\to B$ of $\\text{CRIS}(C/A)$."}
+{"_id": "10755", "title": "crystalline-lemma-generator-completion", "text": "In Situation \\ref{situation-affine}. Let $P$ be a polynomial algebra over $A$ and let $P \\to C$ be a surjection of $A$-algebras with kernel $J$. Let $(D, \\bar J, \\bar\\gamma)$ be the $p$-adic completion of $D_{P, \\gamma}(J)$. For every object $(B \\to C, \\delta)$ of $\\text{Cris}^\\wedge(C/A)$ there exists a morphism $D \\to B$ of $\\text{Cris}^\\wedge(C/A)$."}
+{"_id": "10756", "title": "crystalline-lemma-omega", "text": "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring and $A \\to B$ a ring map.  \\begin{enumerate} \\item Consider $B[x]$ with divided power ideal $(JB[x], \\delta')$ where $\\delta'$ is the extension of $\\delta$ to $B[x]$. Then $$ \\Omega_{B[x]/A, \\delta'} = \\Omega_{B/A, \\delta} \\otimes_B B[x] \\oplus B[x]\\text{d}x. $$ \\item Consider $B\\langle x \\rangle$ with divided power ideal $(JB\\langle x \\rangle + B\\langle x \\rangle_{+}, \\delta')$. Then $$ \\Omega_{B\\langle x\\rangle/A, \\delta'} = \\Omega_{B/A, \\delta} \\otimes_B B\\langle x \\rangle \\oplus B\\langle x\\rangle \\text{d}x. $$ \\item Let $K \\subset J$ be an ideal preserved by $\\delta_n$ for all $n > 0$. Set $B' = B/K$ and denote $\\delta'$ the induced divided power on $J/K$. Then $\\Omega_{B'/A, \\delta'}$ is the quotient of $\\Omega_{B/A, \\delta} \\otimes_B B'$ by the $B'$-submodule generated by $\\text{d}k$ for $k \\in K$. \\end{enumerate}"}
+{"_id": "10757", "title": "crystalline-lemma-diagonal-and-differentials", "text": "Let $(A, I, \\gamma) \\to (B, J, \\delta)$ be a homomorphism of divided power rings. Let $(B(1), J(1), \\delta(1))$ be the coproduct of $(B, J, \\delta)$ with itself over $(A, I, \\gamma)$, i.e., such that $$ \\xymatrix{ (B, J, \\delta) \\ar[r] & (B(1), J(1), \\delta(1)) \\\\ (A, I, \\gamma) \\ar[r] \\ar[u] & (B, J, \\delta) \\ar[u] } $$ is cocartesian. Denote $K = \\Ker(B(1) \\to B)$. Then $K \\cap J(1) \\subset J(1)$ is preserved by the divided power structure and $$ \\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right) $$ canonically."}
+{"_id": "10758", "title": "crystalline-lemma-diagonal-and-differentials-affine-site", "text": "In Situation \\ref{situation-affine}. Let $(B, J, \\delta)$ be an object of $\\text{CRIS}(C/A)$. Let $(B(1), J(1), \\delta(1))$ be the coproduct of $(B, J, \\delta)$ with itself in $\\text{CRIS}(C/A)$. Denote $K = \\Ker(B(1) \\to B)$. Then $K \\cap J(1) \\subset J(1)$ is preserved by the divided power structure and $$ \\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right) $$ canonically."}
+{"_id": "10759", "title": "crystalline-lemma-module-differentials-divided-power-envelope", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring map and let $IB \\subset J \\subset B$ be an ideal. Let $D_{B, \\gamma}(J) = (D, \\bar J, \\bar \\gamma)$ be the divided power envelope. Then we have $$ \\Omega_{D/A, \\bar\\gamma} = \\Omega_{B/A} \\otimes_B D $$"}
+{"_id": "10761", "title": "crystalline-lemma-fibre-product", "text": "Let $(U', T', \\delta') \\to (S'_0, S', \\gamma')$ and $(S_0, S, \\gamma) \\to (S'_0, S', \\gamma')$ be morphisms of divided power schemes. If $(U', T', \\delta')$ is a divided power thickening, then there exists a divided power scheme $(T_0, T, \\delta)$ and $$ \\xymatrix{ T \\ar[r] \\ar[d] & T' \\ar[d] \\\\ S \\ar[r] & S' } $$ which is a cartesian diagram in the category of divided power schemes."}
+{"_id": "10762", "title": "crystalline-lemma-divided-power-thickening-fibre-products", "text": "In Situation \\ref{situation-global}. The category $\\text{CRIS}(X/S)$ has all finite nonempty limits, in particular products of pairs and fibre products. The functor (\\ref{equation-forget}) commutes with limits."}
+{"_id": "10764", "title": "crystalline-lemma-compare-big-small", "text": "Assumptions as in Definition \\ref{definition-divided-power-thickening-X}. The inclusion functor $$ \\text{Cris}(X/S) \\to \\text{CRIS}(X/S) $$ commutes with finite nonempty limits, is fully faithful, continuous, and cocontinuous. There are morphisms of topoi $$ (X/S)_{\\text{cris}} \\xrightarrow{i} (X/S)_{\\text{CRIS}} \\xrightarrow{\\pi} (X/S)_{\\text{cris}} $$ whose composition is the identity and of which the first is induced by the inclusion functor. Moreover, $\\pi_* = i^{-1}$."}
+{"_id": "10765", "title": "crystalline-lemma-localize", "text": "In Situation \\ref{situation-global}. Let $X' \\subset X$ and $S' \\subset S$ be open subschemes such that $X'$ maps into $S'$. Then there is a fully faithful functor $\\text{Cris}(X'/S') \\to \\text{Cris}(X/S)$ which gives rise to a morphism of topoi fitting into the commutative diagram $$ \\xymatrix{ (X'/S')_{\\text{cris}} \\ar[r] \\ar[d]_{u_{X'/S'}} & (X/S)_{\\text{cris}} \\ar[d]^{u_{X/S}} \\\\ \\Sh(X'_{Zar}) \\ar[r] & \\Sh(X_{Zar}) } $$ Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma \\ref{sites-lemma-localize-morphism-topoi}."}
+{"_id": "10766", "title": "crystalline-lemma-crystal-quasi-coherent-modules", "text": "With notation $X/S, \\mathcal{I}, \\gamma, \\mathcal{C}, \\mathcal{F}$ as in Definition \\ref{definition-modules}. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is quasi-coherent, and \\item $\\mathcal{F}$ is locally quasi-coherent and a crystal in $\\mathcal{O}_{X/S}$-modules. \\end{enumerate}"}
+{"_id": "10767", "title": "crystalline-lemma-module-differentials-divided-power-scheme", "text": "Let $(T, \\mathcal{J}, \\delta)$ be a divided power scheme. Let $T \\to S$ be a morphism of schemes. The quotient $\\Omega_{T/S} \\to \\Omega_{T/S, \\delta}$ described above is a quasi-coherent $\\mathcal{O}_T$-module. For $W \\subset T$ affine open mapping into $V \\subset S$ affine open we have $$ \\Gamma(W, \\Omega_{T/S, \\delta}) = \\Omega_{\\Gamma(W, \\mathcal{O}_W)/\\Gamma(V, \\mathcal{O}_V), \\delta} $$ where the right hand side is as constructed in Section \\ref{section-differentials}."}
+{"_id": "10768", "title": "crystalline-lemma-module-of-differentials", "text": "In Situation \\ref{situation-global}. For $(U, T, \\delta)$ in $\\text{Cris}(X/S)$ the restriction $(\\Omega_{X/S})_T$ to $T$ is $\\Omega_{T/S, \\delta}$ and the restriction $\\text{d}_{X/S}|_T$ is equal to $\\text{d}_{T/S, \\delta}$."}
+{"_id": "10769", "title": "crystalline-lemma-module-of-differentials-on-affine", "text": "In Situation \\ref{situation-global}. For any affine object $(U, T, \\delta)$ of $\\text{Cris}(X/S)$ mapping into an affine open $V \\subset S$ we have $$ \\Gamma((U, T, \\delta), \\Omega_{X/S}) = \\Omega_{\\Gamma(T, \\mathcal{O}_T)/\\Gamma(V, \\mathcal{O}_V), \\delta} $$ where the right hand side is as constructed in Section \\ref{section-differentials}."}
+{"_id": "10771", "title": "crystalline-lemma-omega-locally-quasi-coherent", "text": "In Situation \\ref{situation-global}. The sheaf of differentials $\\Omega_{X/S}$ has the following two properties: \\begin{enumerate} \\item $\\Omega_{X/S}$ is locally quasi-coherent, and \\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$ of $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion the map $c_f : f^*(\\Omega_{X/S})_{T'} \\to (\\Omega_{X/S})_T$ is surjective. \\end{enumerate}"}
+{"_id": "10772", "title": "crystalline-lemma-automatic-connection", "text": "In Situation \\ref{situation-global}. Let $\\mathcal{F}$ be a crystal in $\\mathcal{O}_{X/S}$-modules on $\\text{Cris}(X/S)$. Then $\\mathcal{F}$ comes equipped with a canonical integrable connection."}
+{"_id": "10773", "title": "crystalline-lemma-homotopy-tensor", "text": "Let $A_*$ be a cosimplicial ring. Let $\\varphi_*, \\psi_* : K_* \\to M_*$ be homomorphisms of cosimplicial $A_*$-modules. \\begin{enumerate} \\item \\label{item-tensor} If $\\varphi_*$ and $\\psi_*$ are homotopic, then $$ \\varphi_* \\otimes 1, \\psi_* \\otimes 1 : K_* \\otimes_{A_*} L_* \\longrightarrow M_* \\otimes_{A_*} L_* $$ are homotopic for any cosimplicial $A_*$-module $L_*$. \\item \\label{item-wedge} If $\\varphi_*$ and $\\psi_*$ are homotopic, then $$ \\wedge^i(\\varphi_*), \\wedge^i(\\psi_*) : \\wedge^i(K_*) \\longrightarrow \\wedge^i(M_*) $$ are homotopic. \\item \\label{item-base-change} If $\\varphi_*$ and $\\psi_*$ are homotopic, and $A_* \\to B_*$ is a homomorphism of cosimplicial rings, then $$ \\varphi_* \\otimes 1, \\psi_* \\otimes 1 : K_* \\otimes_{A_*} B_* \\longrightarrow M_* \\otimes_{A_*} B_* $$ are homotopic as homomorphisms of cosimplicial $B_*$-modules. \\item \\label{item-completion} If $I_* \\subset A_*$ is a cosimplicial ideal, then the induced maps $$ \\varphi^\\wedge_*, \\psi^\\wedge_* : K_*^\\wedge \\longrightarrow M_*^\\wedge $$ between completions are homotopic. \\item Add more here as needed, for example symmetric powers. \\end{enumerate}"}
+{"_id": "10774", "title": "crystalline-lemma-structure-Dn", "text": "Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}). The coprojection $P \\to P \\otimes_A \\ldots \\otimes_A P$, $f \\mapsto f \\otimes 1 \\otimes \\ldots \\otimes 1$ induces an isomorphism \\begin{equation} \\label{equation-structure-Dn} D(n) = \\lim_e D\\langle \\xi_i(j) \\rangle/p^eD\\langle \\xi_i(j) \\rangle \\end{equation} of algebras over $D$ with $$ \\xi_i(j) = x_i \\otimes 1 \\otimes \\ldots \\otimes 1 - 1 \\otimes \\ldots \\otimes 1 \\otimes x_i \\otimes 1 \\otimes \\ldots \\otimes 1 $$ for $j = 1, \\ldots, n$ where the second $x_i$ is placed in the $j + 1$st slot; recall that $D(n)$ is constructed starting with the $n + 1$-fold tensor product of $P$ over $A$."}
+{"_id": "10775", "title": "crystalline-lemma-property-Dn", "text": "Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}). Then $(D, \\bar J, \\bar\\gamma)$ and $(D(n), \\bar J(n), \\bar\\gamma(n))$ are objects of $\\text{Cris}^\\wedge(C/A)$, see Remark \\ref{remark-completed-affine-site}, and $$ D(n) = \\coprod\\nolimits_{j = 0, \\ldots, n} D $$ in $\\text{Cris}^\\wedge(C/A)$."}
+{"_id": "10776", "title": "crystalline-lemma-crystals-on-affine", "text": "In the situation above there is a functor $$ \\begin{matrix} \\text{crystals in quasi-coherent} \\\\ \\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S) \\end{matrix} \\longrightarrow \\begin{matrix} \\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\ \\text{(\\ref{item-complete}), (\\ref{item-connection}), (\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})} \\end{matrix} $$"}
+{"_id": "10777", "title": "crystalline-lemma-crystals-on-affine-smooth", "text": "In Situation \\ref{situation-affine}. Let $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \\gamma}(J')$. There are homomorphisms of divided power $A$-algebras $$ a : D \\longrightarrow D',\\quad b : D' \\longrightarrow D $$ compatible with the maps $D \\to C$ and $D' \\to C$ such that $a \\circ b = \\text{id}_{D'}$. These maps induce an equivalence of categories of pairs $(M, \\nabla)$ satisfying (\\ref{item-complete}), (\\ref{item-connection}), (\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent}) over $D$ and pairs $(M', \\nabla')$  satisfying (\\ref{item-complete}), (\\ref{item-connection}), (\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent}) over $D'$. In particular, the equivalence of categories of Proposition \\ref{proposition-crystals-on-affine} also holds for the corresponding functor towards pairs over $D'$."}
+{"_id": "10778", "title": "crystalline-lemma-vanishing-lqc", "text": "In Situation \\ref{situation-global}. Let $\\mathcal{F}$ be a locally quasi-coherent $\\mathcal{O}_{X/S}$-module on $\\text{Cris}(X/S)$. Then we have $$ H^p((U, T, \\delta), \\mathcal{F}) = 0 $$ for all $p > 0$ and all $(U, T, \\delta)$ with $T$ or $U$ affine."}
+{"_id": "10779", "title": "crystalline-lemma-compare", "text": "In Situation \\ref{situation-global}. Assume moreover $X$ and $S$ are affine schemes. Consider the full subcategory $\\mathcal{C} \\subset \\text{Cris}(X/S)$ consisting of divided power thickenings $(X, T, \\delta)$ endowed with the chaotic topology (see Sites, Example \\ref{sites-example-indiscrete}). For any locally quasi-coherent $\\mathcal{O}_{X/S}$-module $\\mathcal{F}$ we have $$ R\\Gamma(\\mathcal{C}, \\mathcal{F}|_\\mathcal{C}) = R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) $$"}
+{"_id": "10780", "title": "crystalline-lemma-complete", "text": "In Situation \\ref{situation-affine}. Set $\\mathcal{C} = (\\text{Cris}(C/A))^{opp}$ and $\\mathcal{C}^\\wedge = (\\text{Cris}^\\wedge(C/A))^{opp}$ endowed with the chaotic topology, see Remark \\ref{remark-completed-affine-site} for notation. There is a morphism of topoi $$ g : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{C}^\\wedge) $$ such that if $\\mathcal{F}$ is a sheaf of abelian groups on $\\mathcal{C}$, then $$ R^pg_*\\mathcal{F}(B \\to C, \\delta) = \\left\\{ \\begin{matrix} \\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 0 \\\\ R^1\\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 1 \\\\ 0 & \\text{else} \\end{matrix} \\right. $$ where $B_e = B/p^eB$ for $e \\gg 0$."}
+{"_id": "10781", "title": "crystalline-lemma-category-with-covering", "text": "Let $\\mathcal{C}$ be a category endowed with the chaotic topology. Let $X$ be an object of $\\mathcal{C}$ such that every object of $\\mathcal{C}$ has a morphism towards $X$. Assume that $\\mathcal{C}$ has products of pairs. Then for every abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$ the total cohomology $R\\Gamma(\\mathcal{C}, \\mathcal{F})$ is represented by the complex $$ \\mathcal{F}(X) \\to \\mathcal{F}(X \\times X) \\to \\mathcal{F}(X \\times X \\times X) \\to \\ldots $$ associated to the cosimplicial abelian group $[n] \\mapsto \\mathcal{F}(X^n)$."}
+{"_id": "10782", "title": "crystalline-lemma-vanishing-omega-1", "text": "With notation as in (\\ref{equation-omega-Dn}) the complex $$ \\Omega_{D(0)} \\to \\Omega_{D(1)} \\to \\Omega_{D(2)} \\to \\ldots $$ is homotopic to zero as a $D(*)$-cosimplicial module."}
+{"_id": "10783", "title": "crystalline-lemma-vanishing", "text": "With notation as in (\\ref{equation-Dn}) and (\\ref{equation-omega-Dn}), given any cosimplicial module $M_*$ over $D(*)$ and $i > 0$ the cosimplicial module $$ M_0 \\otimes^\\wedge_{D(0)} \\Omega^i_{D(0)} \\to M_1 \\otimes^\\wedge_{D(1)} \\Omega^i_{D(1)} \\to M_2 \\otimes^\\wedge_{D(2)} \\Omega^i_{D(2)} \\to \\ldots $$ is homotopic to zero, where $\\Omega^i_{D(n)}$ is the $p$-adic completion of the $i$th exterior power of $\\Omega_{D(n)}$."}
+{"_id": "10784", "title": "crystalline-lemma-poincare", "text": "Let $A$ be a ring. Let $P = A\\langle x_i \\rangle$ be a divided power polynomial ring over $A$. For any $A$-module $M$ the complex $$ 0 \\to M \\to M \\otimes_A P \\to M \\otimes_A \\Omega^1_{P/A, \\delta} \\to M \\otimes_A \\Omega^2_{P/A, \\delta} \\to \\ldots $$ is exact. Let $D$ be the $p$-adic completion of $P$. Let $\\Omega^i_D$ be the $p$-adic completion of the $i$th exterior power of $\\Omega_{D/A, \\delta}$. For any $p$-adically complete $A$-module $M$ the complex $$ 0 \\to M \\to M \\otimes^\\wedge_A D \\to M \\otimes^\\wedge_A \\Omega^1_D \\to M \\otimes^\\wedge_A \\Omega^2_D \\to \\ldots $$ is exact."}
+{"_id": "10785", "title": "crystalline-lemma-relative-poincare", "text": "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring. Let $P = B\\langle x_i \\rangle$ be a divided power polynomial ring over $B$ with divided power ideal $J = IP + B\\langle x_i \\rangle_{+}$ as usual. Let $M$ be a $B$-module endowed with an integrable connection $\\nabla : M \\to M \\otimes_B \\Omega^1_{B/A, \\delta}$. Then the map of de Rham complexes $$ M \\otimes_B \\Omega^*_{B/A, \\delta} \\longrightarrow M \\otimes_P \\Omega^*_{P/A, \\delta} $$ is a quasi-isomorphism. Let $D$, resp.\\ $D'$ be the $p$-adic completion of $B$, resp.\\ $P$ and let $\\Omega^i_D$, resp.\\ $\\Omega^i_{D'}$ be the $p$-adic completion of $\\Omega^i_{B/A, \\delta}$, resp.\\ $\\Omega^i_{P/A, \\delta}$. Let $M$ be a $p$-adically complete $D$-module endowed with an integral connection $\\nabla : M \\to M \\otimes^\\wedge_D \\Omega^1_D$. Then the map of de Rham complexes $$ M \\otimes^\\wedge_D \\Omega^*_D \\longrightarrow M \\otimes^\\wedge_D \\Omega^*_{D'} $$ is a quasi-isomorphism."}
+{"_id": "10786", "title": "crystalline-lemma-cohomology-is-zero", "text": "Assumptions and notation as in Proposition \\ref{proposition-compute-cohomology}. Then $$ H^j(\\text{Cris}(X/S), \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0 $$ for all $i > 0$ and all $j \\geq 0$."}
+{"_id": "10787", "title": "crystalline-lemma-compute-cohomology-crystal-smooth", "text": "Assumptions as in Proposition \\ref{proposition-compute-cohomology-crystal}. Let $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \\gamma}(J')$. Let $(M', \\nabla')$ be the pair over $D'$ corresponding to $\\mathcal{F}$, see Lemma \\ref{lemma-crystals-on-affine-smooth}. Then the complex $$ M' \\otimes^\\wedge_{D'} \\Omega^*_{D'} $$ computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$."}
+{"_id": "10788", "title": "crystalline-lemma-find-homotopy", "text": "In the situation above there exists a map of complexes $$ e_M^\\bullet : M \\otimes_B (\\Omega')^\\bullet \\longrightarrow M \\otimes_B \\Omega^\\bullet $$ such that $c_M^\\bullet \\circ e_M^\\bullet$ and $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to multiplication by $a$."}
+{"_id": "10789", "title": "crystalline-lemma-computation", "text": "In Situation \\ref{situation-affine}. Assume $D$ and $\\Omega_D$ are as in (\\ref{equation-D}) and (\\ref{equation-omega-D}). Let $\\lambda \\in D$. Let $D'$ be the $p$-adic completion of $$ D[z]\\langle \\xi \\rangle/(\\xi - (z^p - \\lambda)) $$ and let $\\Omega_{D'}$ be the $p$-adic completion of the module of divided power differentials of $D'$ over $A$. For any pair $(M, \\nabla)$ over $D$ satisfying (\\ref{item-complete}), (\\ref{item-connection}), (\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent}) the canonical map of complexes (\\ref{equation-base-change-map-complexes}) $$ c_M^\\bullet : M \\otimes_D^\\wedge \\Omega^\\bullet_D \\longrightarrow M \\otimes_D^\\wedge \\Omega^\\bullet_{D'} $$ has the following property: There exists a map $e_M^\\bullet$ in the opposite direction such that both $c_M^\\bullet \\circ e_M^\\bullet$ and $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to multiplication by $p$."}
+{"_id": "10790", "title": "crystalline-lemma-pullback-along-p-power-cover", "text": "Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma)$ be a divided power scheme over $\\mathbf{Z}_{(p)}$ with $p \\in \\mathcal{I}$. We set $S_0 = V(\\mathcal{I}) \\subset S$. Let $f : X' \\to X$ be an iterated $\\alpha_p$-cover of schemes over $S_0$ with constant degree $q$. Let $\\mathcal{F}$ be any crystal in quasi-coherent sheaves on $X$ and set $\\mathcal{F}' = f_{\\text{cris}}^*\\mathcal{F}$. In the distinguished triangle $$ Ru_{X/S, *}\\mathcal{F} \\longrightarrow f_*Ru_{X'/S, *}\\mathcal{F}' \\longrightarrow E \\longrightarrow Ru_{X/S, *}\\mathcal{F}[1] $$ the object $E$ has cohomology sheaves annihilated by $q$."}
+{"_id": "10791", "title": "crystalline-lemma-pullback-along-p-power-cover-cohomology", "text": "With notations and assumptions as in Lemma \\ref{lemma-pullback-along-p-power-cover} the map $$ f^* : H^i(\\text{Cris}(X/S), \\mathcal{F}) \\longrightarrow H^i(\\text{Cris}(X'/S), \\mathcal{F}') $$ has kernel and cokernel annihilated by $q^{i + 1}$."}
+{"_id": "10792", "title": "crystalline-lemma-pullback-relative-frobenius", "text": "In the situation above, assume that $X \\to S_0$ is smooth of relative dimension $d$. Then $F_{X/S_0}$ is an iterated $\\alpha_p$-cover of degree $p^d$. Hence Lemmas \\ref{lemma-pullback-along-p-power-cover} and \\ref{lemma-pullback-along-p-power-cover-cohomology} apply to this situation. In particular, for any crystal in quasi-coherent modules $\\mathcal{G}$ on $\\text{Cris}(X^{(1)}/S)$ the map $$ F_{X/S_0}^* : H^i(\\text{Cris}(X^{(1)}/S), \\mathcal{G}) \\longrightarrow H^i(\\text{Cris}(X/S), F_{X/S_0, \\text{cris}}^*\\mathcal{G}) $$ has kernel and cokernel annihilated by $p^{d(i + 1)}$."}
+{"_id": "10793", "title": "crystalline-proposition-crystals-on-affine", "text": "The functor $$ \\begin{matrix} \\text{crystals in quasi-coherent} \\\\ \\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S) \\end{matrix} \\longrightarrow \\begin{matrix} \\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\ \\text{(\\ref{item-complete}), (\\ref{item-connection}), (\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})} \\end{matrix} $$ of Lemma \\ref{lemma-crystals-on-affine} is an equivalence of categories."}
+{"_id": "10794", "title": "crystalline-proposition-compute-cohomology", "text": "With notations as above assume that \\begin{enumerate} \\item $\\mathcal{F}$ is locally quasi-coherent, and \\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$ of $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion the map $c_f : f^*\\mathcal{F}_{T'} \\to \\mathcal{F}_T$ is surjective. \\end{enumerate} Then the complex $$ M(0) \\to M(1) \\to M(2) \\to \\ldots $$ computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$."}
+{"_id": "10795", "title": "crystalline-proposition-compute-cohomology-crystal", "text": "Assumptions as in Proposition \\ref{proposition-compute-cohomology} but now assume that $\\mathcal{F}$ is a crystal in quasi-coherent modules. Let $(M, \\nabla)$ be the corresponding module with connection over $D$, see Proposition \\ref{proposition-crystals-on-affine}. Then the complex $$ M \\otimes^\\wedge_D \\Omega^*_D $$ computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$."}
+{"_id": "10796", "title": "crystalline-proposition-compare-with-de-Rham", "text": "In Situation \\ref{situation-global}. Let $\\mathcal{F}$ be a crystal in quasi-coherent modules on $\\text{Cris}(X/S)$. The truncation map of complexes $$ (\\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^1_{X/S} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^2_{X/S} \\to \\ldots) \\longrightarrow \\mathcal{F}[0], $$ while not a quasi-isomorphism, becomes a quasi-isomorphism after applying $Ru_{X/S, *}$. In fact, for any $i > 0$, we have  $$ Ru_{X/S, *}(\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0. $$"}
+{"_id": "10845", "title": "spaces-pushouts-lemma-colimit-agrees", "text": "Let $S$ be a scheme. Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of schemes over $S$ as above. Assume that \\begin{enumerate} \\item $X = \\colim X_i$ exists in the category of schemes, \\item $\\coprod X_i \\to X$ is surjective, \\item if $U \\to X$ is \\'etale and $U_i = X_i \\times_X U$, then $U = \\colim U_i$ in the category of schemes, and \\item every object $(U_i \\to X_i)$ of $\\lim X_{i, \\etale}$ with $U_i \\to X_i$ separated is in the essential image the functor $X_\\etale \\to \\lim X_{i, \\etale}$. \\end{enumerate} Then $X = \\colim X_i$ in the category of algebraic spaces over $S$ also."}
+{"_id": "10846", "title": "spaces-pushouts-lemma-pushout-fpqc-local", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark \\ref{categories-remark-cones-and-cocones}). If there exists a fpqc covering $\\{U_a \\to X\\}_{a \\in A}$ such that \\begin{enumerate} \\item for all $a \\in A$ we have $U_a = \\colim X_i \\times_X U_a$ in the category of algebraic spaces over $B$, and \\item for all $a, b \\in A$ we have $U_a \\times_X U_b = \\colim X_i \\times_X U_a \\times_X U_b$ in the category of algebraic spaces over $B$, \\end{enumerate} then $X = \\colim X_i$ in the category of algebraic spaces over $B$."}
+{"_id": "10847", "title": "spaces-pushouts-lemma-colimit-check-etale-locally", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark \\ref{categories-remark-cones-and-cocones}). Assume that \\begin{enumerate} \\item the base change functor $X_{spaces, \\'etale} \\to \\lim X_{i, spaces, \\etale}$, sending $U$ to $U_i = X_i \\times_X U$ is an equivalence, \\item given \\begin{enumerate} \\item $B'$ affine and \\'etale over $B$, \\item $Z$ an affine scheme over $B'$, \\item $U \\to X \\times_B B'$ an \\'etale morphism of algebraic spaces with $U$ affine, \\item $f_i : U_i \\to Z$ a cocone over $B'$ of the diagram $i \\mapsto U_i = U \\times_X X_i$, \\end{enumerate} there exists a unique morphism $f : U \\to Z$ over $B'$ such that $f_i$ equals the composition $U_i \\to U \\to Z$. \\end{enumerate} Then $X = \\colim X_i$ in the category of all algebraic spaces over $B$."}
+{"_id": "10849", "title": "spaces-pushouts-lemma-glue-etale-sheaf-etale", "text": "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an \\'etale covering of algebraic spaces. The functor $$ \\Sh(X_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\} $$ is an equivalence of categories."}
+{"_id": "10850", "title": "spaces-pushouts-lemma-reduce-to-scheme-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an \\'etale covering of algebraic spaces. If for each $i \\in I$ the functor $$ \\Sh(Y_{i, \\etale}) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Y_i \\to Y_i\\} $$ is an equivalence of categories and for each $i, j \\in I$ the functor $$ \\Sh((Y_i \\times_Y Y_j)_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt } \\{X \\times_Y Y_i \\times_Y Y_j \\to Y_i \\times_Y Y_j\\} $$ is an equivalence of categories, then $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "10851", "title": "spaces-pushouts-lemma-representable-case", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable (by schemes) and $f$ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation Then  $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "10852", "title": "spaces-pushouts-lemma-reduce-to-scheme-source", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\pi : X' \\to X$ be a morphism of algebraic spaces. Assume \\begin{enumerate} \\item $f \\circ \\pi$ is representable (by schemes), \\item $f \\circ \\pi$ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. \\end{enumerate} Then  $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "10853", "title": "spaces-pushouts-lemma-glue-etale-sheaf-proper-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor $$ \\Sh(Y_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\} $$ is an equivalence of categories."}
+{"_id": "10855", "title": "spaces-pushouts-lemma-glue-etale-sheaf-modification", "text": "Let $S$ be a scheme. Let $f : Y' \\to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \\to Y$ be a closed immersion. Set $E = Z \\times_Y Y'$. Picture $$ \\xymatrix{ E \\ar[d]_g \\ar[r]_j & Y' \\ar[d]^f \\\\ Z \\ar[r]^i & Y } $$ If $f$ is an isomorphism over $Y \\setminus Z$, then the functor $$ \\Sh(Y_\\etale) \\longrightarrow \\Sh(Y'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale) $$ is an equivalence of categories."}
+{"_id": "10856", "title": "spaces-pushouts-lemma-descend-etale-proper-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism of algebraic spaces over $S$. Any descent datum $(U/X, \\varphi)$ relative to $f$ (Descent on Spaces, Definition \\ref{spaces-descent-definition-descent-datum}) with $U$ \\'etale over $X$ is effective (Descent on Spaces, Definition \\ref{spaces-descent-definition-effective}). More precisely, there exists an \\'etale morphism $V \\to Y$ of algebraic spaces whose corresponding canonical descent datum is isomorphic to $(U/X, \\varphi)$."}
+{"_id": "10858", "title": "spaces-pushouts-lemma-pushout-along-thickening-schemes", "text": "Let $S$ be a scheme. Let $X \\to X'$ be a thickening of schemes over $S$ and let $X \\to Y$ be an affine morphism of schemes over $S$. Let $Y' = Y \\amalg_X X'$ be the pushout in the category of schemes (see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}). Then $Y'$ is also a pushout in the category of algebraic spaces over $S$."}
+{"_id": "10859", "title": "spaces-pushouts-lemma-pushout-along-thickening", "text": "Let $S$ be a scheme. Let $X \\to X'$ be a thickening of algebraic spaces over $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$. Then there exists a pushout $$ \\xymatrix{ X \\ar[r] \\ar[d]_f & X' \\ar[d]^{f'} \\\\ Y \\ar[r] & Y \\amalg_X X' } $$ in the category of algebraic spaces over $S$. Moreover $Y' = Y \\amalg_X X'$ is a thickening of $Y$ and $$ \\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'} $$ as sheaves on $Y_\\etale = (Y')_\\etale$."}
+{"_id": "10860", "title": "spaces-pushouts-lemma-categories-spaces-over-pushout", "text": "Let $S$ be a base scheme. Let $X \\to X'$ be a thickening of algebraic spaces over $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$. Let $Y' = Y \\amalg_X X'$ be the pushout (see Lemma \\ref{lemma-pushout-along-thickening}). Base change gives a functor $$ F : (\\textit{Spaces}/Y') \\longrightarrow (\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X') $$ given by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$ which sends $(\\Sch/Y')$ into $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$. The functor $F$ has a left adjoint $$ G : (\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X') \\longrightarrow (\\textit{Spaces}/Y') $$ which sends the triple $(V, U', \\varphi)$ to the pushout $V \\amalg_{(V \\times_Y X)} U'$ in the category of algebraic spaces over $S$. The functor $G$ sends $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$ into $(\\Sch/Y')$."}
+{"_id": "10861", "title": "spaces-pushouts-lemma-diagram", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ A \\ar[r] \\ar[d] & C \\ar[d] \\ar[r] & E \\ar[d] \\\\ B \\ar[r] & D \\ar[r] & F } $$ be a commutative diagram of algebraic spaces over $S$. Assume that $A, B, C, D$ and $A, B, E, F$ form cartesian squares and that $B \\to D$ is surjective \\'etale. Then $C, D, E, F$ is a cartesian square."}
+{"_id": "10862", "title": "spaces-pushouts-lemma-equivalence-categories-spaces-over-pushout", "text": "In the situation of Lemma \\ref{lemma-categories-spaces-over-pushout} the functor $F \\circ G$ is isomorphic to the identity functor."}
+{"_id": "10863", "title": "spaces-pushouts-lemma-space-over-pushout-flat-modules", "text": "Let $S$ be a base scheme. Let $X \\to X'$ be a thickening of algebraic spaces over $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$. Let $Y' = Y \\amalg_X X'$ be the pushout (see Lemma \\ref{lemma-pushout-along-thickening}). Let $V' \\to Y'$ be a morphism of algebraic spaces over $S$. Set $V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$. There is an equivalence of categories between \\begin{enumerate} \\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and \\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where \\begin{enumerate} \\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$, \\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat over $X$, and \\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$ is an isomorphism of $\\mathcal{O}_U$-modules. \\end{enumerate} \\end{enumerate} The equivalence maps $\\mathcal{G}'$ to $((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$. Suppose $\\mathcal{G}'$ corresponds to the triple $(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then \\begin{enumerate} \\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type $\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules. \\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then $\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite presentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are $\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation. \\end{enumerate}"}
+{"_id": "10864", "title": "spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat", "text": "In the situation of Lemma \\ref{lemma-equivalence-categories-spaces-over-pushout}. If $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then \\begin{enumerate} \\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and $U' \\to X'$ are locally of finite type, \\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat, \\item $V' \\to Y'$ is flat and locally of finite presentation if and only if $V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation, \\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth, \\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$ are \\'etale, and \\item add more here as needed. \\end{enumerate} If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of spaces flat over $Y'$ and the category of triples $(V, U', \\varphi)$ with $V \\to Y$ and $U' \\to X'$ flat."}
+{"_id": "10866", "title": "spaces-pushouts-lemma-pushout-along-thickening-derived", "text": "Let $S$ be a scheme. Consider a pushout $$ \\xymatrix{ X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\ Y \\ar[r]^j & Y' } $$ in the category of algebraic spaces over $S$ as in Lemma \\ref{lemma-pushout-along-thickening}. Assume $i$ is a thickening. Then the essential image of the functor\\footnote{All functors given by derived pullback.} $$ D(\\mathcal{O}_{Y'}) \\longrightarrow D(\\mathcal{O}_Y) \\times_{D(\\mathcal{O}_X)} D(\\mathcal{O}_{X'}) $$ contains every triple $(M, K', \\alpha)$ where $M \\in D(\\mathcal{O}_Y)$ and $K' \\in D(\\mathcal{O}_{X'})$ are pseudo-coherent."}
+{"_id": "10867", "title": "spaces-pushouts-lemma-elementary-distinguished-square-pushout", "text": "Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be an elementary distinguished square. Then $$ \\xymatrix{ U \\times_W V \\ar[r] \\ar[d] & V \\ar[d]^f \\\\ U \\ar[r] & W } $$ is a pushout in the category of algebraic spaces over $S$."}
+{"_id": "10868", "title": "spaces-pushouts-lemma-construct-elementary-distinguished-square", "text": "Let $S$ be a scheme. Let $V$, $U$ be algebraic spaces over $S$. Let $V' \\subset V$ be an open subspace and let $f' : V' \\to U$ be a separated \\'etale morphism of algebraic spaces over $S$. Then there exists a pushout $$ \\xymatrix{ V' \\ar[r] \\ar[d] & V \\ar[d]^f \\\\ U \\ar[r] & W } $$ in the category of algebraic spaces over $S$ and moreover $(U \\subset W, f : V \\to W)$ is an elementary distinguished square."}
+{"_id": "10869", "title": "spaces-pushouts-lemma-stalk-pushforward-with-support", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is integral and universally injective. Let $\\overline{y}$ be a geometric point of $Y$ and $\\overline{x} = f(\\overline{y})$. We have $$ (Rf_*Q)_{\\overline{x}} = Q_{\\overline{y}} $$ in $D(\\textit{Ab})$ for any object $Q$ of $D(Y_\\etale)$ supported on $|f^{-1}Z|$."}
+{"_id": "10870", "title": "spaces-pushouts-lemma-stalk-formal-glueing", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is integral and universally injective. Let $\\overline{y}$ be a geometric point of $Y$ and $\\overline{x} = f(\\overline{y})$. Let $\\mathcal{G}$ be an abelian sheaf on $Y$. Then the map of two term complexes $$ \\left(f_*\\mathcal{G}_{\\overline{x}} \\to (f \\circ j')_*(\\mathcal{G}|_V)_{\\overline{x}}\\right) \\longrightarrow \\left(\\mathcal{G}_{\\overline{y}} \\to j'_*(\\mathcal{G}|_V)_{\\overline{y}}\\right) $$ induces an isomorphism on kernels and an injection on cokernels. Here $V = Y \\setminus f^{-1}Z$ and $j' : V \\to Y$ is the inclusion."}
+{"_id": "10871", "title": "spaces-pushouts-lemma-stalk-of-pushforward", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism. Let $\\overline{x}$ be a geometric point of $X$ and let $\\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to X$ be the canonical morphism. For a quasi-coherent module $\\mathcal{G}$ on $Y$ we have $$ f_*\\mathcal{G}_{\\overline{x}} = \\Gamma(Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}), p^*\\mathcal{F}) $$ where $p : Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to Y$ is the projection."}
+{"_id": "10872", "title": "spaces-pushouts-lemma-stalk-of-module-with-support", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $i : Z \\to X$ be a closed immersion of finite presentation. Let $Q \\in D_\\QCoh(\\mathcal{O}_X)$ be supported on $|Z|$. Let $\\overline{x}$ be a geometric point of $X$ and let $I_{\\overline{x}} \\subset \\mathcal{O}_{X, \\overline{x}}$ be the stalk of the ideal sheaf of $Z$. Then the cohomology modules $H^n(Q_{\\overline{x}})$ are $I_{\\overline{x}}$-power torsion (see More on Algebra, Definition \\ref{more-algebra-definition-f-power-torsion})."}
+{"_id": "10873", "title": "spaces-pushouts-lemma-formal-glueing-on-closed", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset X$ be a closed subspace. Assume $f^{-1}Z \\to Z$ is an isomorphism and that $f$ is flat in every point of $f^{-1}Z$. For any $Q$ in $D_\\QCoh(\\mathcal{O}_Y)$ supported on $|f^{-1}Z|$ we have $Lf^*Rf_*Q = Q$."}
+{"_id": "10874", "title": "spaces-pushouts-lemma-adjoint", "text": "In Situation \\ref{situation-formal-glueing}. The functor (\\ref{equation-reverse}) is right adjoint to the functor (\\ref{equation-formal-glueing-modules})."}
+{"_id": "10877", "title": "spaces-pushouts-lemma-dominate-by-fpqc-covering", "text": "In Situation \\ref{situation-formal-glueing} there exists an fpqc covering $\\{X_i \\to X\\}_{i \\in I}$ refining the family $\\{U \\to X, Y \\to X\\}$."}
+{"_id": "10878", "title": "spaces-pushouts-lemma-equivalence-on-affine", "text": "In Situation \\ref{situation-formal-glueing} the functor (\\ref{equation-formal-glueing-spaces}) restricts to an equivalence \\begin{enumerate} \\item from the category of algebraic spaces affine over $X$ to the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting of $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$, and $Y' \\to Y$ affine, \\item from the category of closed immersions $X' \\to X$ to the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting of $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$, and $Y' \\to Y$ closed immersions, and \\item same statement as in (2) for finite morphisms. \\end{enumerate}"}
+{"_id": "10879", "title": "spaces-pushouts-lemma-reflects-isomorphisms", "text": "In Situation \\ref{situation-formal-glueing} the functor (\\ref{equation-formal-glueing-spaces}) reflects isomorphisms."}
+{"_id": "10882", "title": "spaces-pushouts-lemma-glueing-affines", "text": "Let $(R \\to R', f)$ be a glueing pair, see above. The functor (\\ref{equation-beauville-laszlo-glueing-spaces}) restricts to an equivalence between the category of affine $Y/X$ which are glueable for $(R \\to R', f)$ and the full subcategory of objects $(V, V', Y')$ of $\\textit{Spaces}(U \\leftarrow U' \\to X')$ with $V$, $V'$, $Y'$ affine."}
+{"_id": "10883", "title": "spaces-pushouts-lemma-glueing-affines-etale", "text": "Let $P$ be one of the following properties of morphisms: ``finite'', ``closed immersion'', ``flat'', ``finite type'', ``flat and finite presentation'', ``\\'etale''. Under the equivalence of Lemma \\ref{lemma-glueing-affines} the morphisms having $P$ correspond to morphisms of triples whose components have $P$."}
+{"_id": "10884", "title": "spaces-pushouts-lemma-glueing-f", "text": "Let $(R \\to R', f)$ be a glueing pair, see above. The functor (\\ref{equation-beauville-laszlo-glueing-spaces}) is faithful on the full subcategory of algebraic spaces $Y/X$ glueable for $(R \\to R', f)$."}
+{"_id": "10887", "title": "spaces-pushouts-lemma-coequalizer", "text": "Let $S$ be a scheme. Let $$ g : Y \\longrightarrow X $$ be a morphism of algebraic spaces over $S$. Assume $X$ is locally Noetherian, and $g$ is proper. Let $R = Y \\times_X Y$ with projection morphisms $t, s : R \\to Y$. There exists a coequalizer $X'$ of $s, t : R \\to Y$ in the category of algebraic spaces over $S$. Moreover \\begin{enumerate} \\item The morphism $X' \\to X$ is finite. \\item The morphism $Y \\to X'$ is proper. \\item The morphism $Y \\to X'$ is surjective. \\item The morphism $X' \\to X$ is universally injective. \\item If $g$ is surjective, the morphism $X' \\to X$ is a universal homeomorphism. \\end{enumerate}"}
+{"_id": "10889", "title": "spaces-pushouts-lemma-essentially-constant", "text": "In Situation \\ref{situation-coequalizer-glue} assume $X$ quasi-compact. In (\\ref{equation-system-coequalizers}) for all $n$ large enough, there exists an $m$ such that $X_n \\to X_{n + m}$ factors through a closed immersion $X \\to X_{n + m}$."}
+{"_id": "10890", "title": "spaces-pushouts-lemma-check-separated", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$. If $(U \\subset X, f : V \\to X)$ is an elementary distinguished square such that $U \\to Y$ and $V \\to Y$ are separated and $U \\times_X V \\to U \\times_Y V$ is closed, then $X \\to Y$ is separated."}
+{"_id": "10891", "title": "spaces-pushouts-lemma-separate-disjoint-locally-closed-by-blowing-up", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \\subset X$ be a quasi-compact open. \\begin{enumerate} \\item If $Z_1, Z_2 \\subset X$ are closed subspaces of finite presentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then there exists a $U$-admissible blowing up $X' \\to X$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. \\item If $T_1, T_2 \\subset |U|$ are disjoint constructible closed subsets, then there is a $U$-admissible blowing up $X' \\to X$ such that the closures of $T_1$ and $T_2$ are disjoint. \\end{enumerate}"}
+{"_id": "10892", "title": "spaces-pushouts-lemma-blowup-iso-along", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $V \\subset Y$ be a quasi-compact open and $U = f^{-1}(V)$. Let $T \\subset |V|$ be a closed subset such that $f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$ in $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that the strict transform $f' : X' \\to Y'$ of $f$ is an isomorphism over an open neighbourhood of the closure of $T$ in $|Y'|$."}
+{"_id": "10893", "title": "spaces-pushouts-lemma-blowup-etale-along", "text": "Let $S$ be a scheme. Consider a diagram $$ \\xymatrix{ X \\ar[d]_f & U \\ar[l] \\ar[d]_{f|_U} & A \\ar[d] \\ar[l] \\\\ Y & V \\ar[l] & B \\ar[l] } $$ of quasi-compact and quasi-separated algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ is proper, \\item $V$ is a quasi-compact open of $Y$, $U = f^{-1}(V)$, \\item $B \\subset V$ and $A \\subset U$ are closed subspaces, \\item $f|_A : A \\to B$ is an isomorphism, and $f$ is \\'etale at every point of $A$. \\end{enumerate} Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that the strict transform $f' : X' \\to Y'$ satisfies: for every geometric point $\\overline{a}$ of the closure of $|A|$ in $|X'|$ there exists a quotient $\\mathcal{O}_{X', \\overline{a}} \\to \\mathcal{O}$ such that $\\mathcal{O}_{Y', f'(\\overline{a})} \\to \\mathcal{O}$ is finite flat."}
+{"_id": "10894", "title": "spaces-pushouts-lemma-replaced-by-strict-transform", "text": "Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of algebraic spaces over $S$. Let $U \\subset X$ be an open subspace. Let $V \\to X \\times_B Y$ be a quasi-compact morphism whose composition with the first projection maps into $U$. Let $Z \\subset X \\times_B Y$ be the scheme theoretic image of $V \\to X \\times_B Y$. Let $X' \\to X$ be a $U$-admissible blowup. Then the scheme theoretic image of $V \\to X' \\times_B Y$ is the strict transform of $Z$ with respect to the blowing up."}
+{"_id": "10895", "title": "spaces-pushouts-lemma-compactification-dominates", "text": "Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U$ be an algebraic space of finite type and separated over $B$. Let $V \\to U$ be an \\'etale morphism. If $V$ has a compactification $V \\subset Y$ over $B$, then there exists a $V$-admissible blowing up $Y' \\to Y$ and an open $V \\subset V' \\subset Y'$ such that $V \\to U$ extends to a proper morphism $V' \\to U$."}
+{"_id": "10896", "title": "spaces-pushouts-lemma-two-compactifications", "text": "Let $B$ be an algebraic space of finite type over $\\mathbf{Z}$. Let $U$ be an algebraic space of finite type and separated over $B$. Let $(U_2 \\subset U, f : U_1 \\to U)$ be an elementary distinguished square. Assume $U_1$ and $U_2$ have compactifications over $B$ and $U_1 \\times_U U_2 \\to U$ has dense image. Then $U$ has a compactification over $B$."}
+{"_id": "10897", "title": "spaces-pushouts-lemma-filter-Noetherian-space", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $U \\subset X$ be a proper dense open subspace. Then there exists an affine scheme $V$ and an \\'etale morphism $V \\to X$ such that \\begin{enumerate} \\item the open subspace $W = U \\cup \\Im(V \\to X)$ is strictly larger than $U$, \\item $(U \\subset W, V \\to W)$ is a distinguished square, and \\item $U \\times_W V \\to U$ has dense image. \\end{enumerate}"}
+{"_id": "10898", "title": "spaces-pushouts-proposition-formal-glueing-modules", "text": "In Situation \\ref{situation-formal-glueing} the functor (\\ref{equation-formal-glueing-modules}) is an equivalence with quasi-inverse given by (\\ref{equation-reverse})."}
+{"_id": "10900", "title": "varieties-theorem-varieties-rational-maps", "text": "Let $k$ be a field. The category of varieties and dominant rational maps is equivalent to the category of finitely generated field extensions $K/k$."}
+{"_id": "10901", "title": "varieties-lemma-product-varieties", "text": "\\begin{slogan} Products of varieties are varieties over algebraically closed fields. \\end{slogan} Let $k$ be an algebraically closed field. Let $X$, $Y$ be varieties over $k$. Then $X \\times_{\\Spec(k)} Y$ is a variety over $k$."}
+{"_id": "10903", "title": "varieties-lemma-change-fields-flat", "text": "Let $K/k$ be an extension of fields. Let $X$ be scheme over $k$ and set $Y = X_K$. If $y \\in Y$ with image $x \\in X$, then \\begin{enumerate} \\item $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ is a faithfully flat local ring homomorphism, \\item with $\\mathfrak p_0 = \\Ker(\\kappa(x) \\otimes_k K \\to \\kappa(y))$ we have $\\kappa(y) = \\kappa(\\mathfrak p_0)$, \\item $\\mathcal{O}_{Y, y} = (\\mathcal{O}_{X, x} \\otimes_k K)_\\mathfrak p$ where $\\mathfrak p \\subset \\mathcal{O}_{X, x} \\otimes_k K$ is the inverse image of $\\mathfrak p_0$. \\item we have $\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y} = (\\kappa(x) \\otimes_k K)_{\\mathfrak p_0}$ \\end{enumerate}"}
+{"_id": "10904", "title": "varieties-lemma-change-fields-algebraic-dim", "text": "Notation as in Lemma \\ref{lemma-change-fields-flat}. Assume $X$ is locally of finite type over $k$. Then $$ \\dim(\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y}) = \\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_K(\\kappa(y)) = \\dim(\\mathcal{O}_{Y, y}) - \\dim(\\mathcal{O}_{X, x}) $$"}
+{"_id": "10905", "title": "varieties-lemma-change-fields-algebraic-unramified", "text": "Notation as in Lemma \\ref{lemma-change-fields-flat}. Assume $X$ is locally of finite type over $k$, that $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$ and that $\\kappa(x) \\otimes_k K$ is reduced (for example if $\\kappa(x)/k$ is separable or $K/k$ is separable). Then $\\mathfrak m_x \\mathcal{O}_{Y, y} = \\mathfrak m_y$."}
+{"_id": "10906", "title": "varieties-lemma-geometrically-reduced-at-point", "text": "\\begin{slogan} Geometric reducedness can be checked on local rings. \\end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \\in X$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced at $x$, and \\item the ring $\\mathcal{O}_{X, x}$ is geometrically reduced over $k$ (see Algebra, Definition \\ref{algebra-definition-geometrically-reduced}). \\end{enumerate}"}
+{"_id": "10907", "title": "varieties-lemma-perfect-reduced", "text": "Let $X$ be a scheme over a perfect field $k$ (e.g.\\ $k$ has characteristic zero). Let $x \\in X$. If $\\mathcal{O}_{X, x}$ is reduced, then $X$ is geometrically reduced at $x$. If $X$ is reduced, then $X$ is geometrically reduced over $k$."}
+{"_id": "10908", "title": "varieties-lemma-geometrically-reduced", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced, \\item $X_{k'}$ is reduced for every field extension $k \\subset k'$, \\item $X_{k'}$ is reduced for every finite purely inseparable field extension $k \\subset k'$, \\item $X_{k^{1/p}}$ is reduced, \\item $X_{k^{perf}}$ is reduced, \\item $X_{\\bar k}$ is reduced, \\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is geometrically reduced (see Algebra, Definition \\ref{algebra-definition-geometrically-reduced}). \\end{enumerate}"}
+{"_id": "10909", "title": "varieties-lemma-check-only-finite-inseparable-extensions", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Let $x \\in X$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced at $x$, \\item $\\mathcal{O}_{X_{k'}, x'}$ is reduced for every finite purely inseparable field extension $k'$ of $k$ and $x' \\in X_{k'}$ the unique point lying over $x$, \\item $\\mathcal{O}_{X_{k^{1/p}}, x'}$ is reduced for $x' \\in X_{k^{1/p}}$ the unique point lying over $x$, and \\item $\\mathcal{O}_{X_{k^{perf}}, x'}$ is reduced for $x' \\in X_{k^{perf}}$ the unique point lying over $x$. \\end{enumerate}"}
+{"_id": "10910", "title": "varieties-lemma-geometrically-reduced-upstairs", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a field extension. Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced at $x$, \\item $X_{k'}$ is geometrically reduced at $x'$. \\end{enumerate} In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$."}
+{"_id": "10911", "title": "varieties-lemma-geometrically-reduced-any-base-change", "text": "Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. \\begin{enumerate} \\item If $X$ is geometrically reduced at $x$, and $Y$ reduced, then $X \\times_k Y$ is reduced at every point lying over $x$. \\item If $X$ geometrically reduced over $k$ and $Y$ reduced. Then $X \\times_k Y$ is reduced. \\end{enumerate}"}
+{"_id": "10912", "title": "varieties-lemma-generic-points-geometrically-reduced", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. \\begin{enumerate} \\item If $x' \\leadsto x$ is a specialization and $X$ is geometrically reduced at $x$, then $X$ is geometrically reduced at $x'$. \\item If $x \\in X$ such that (a) $\\mathcal{O}_{X, x}$ is reduced, and (b) for each specialization $x' \\leadsto x$ where $x'$ is a generic point of an irreducible component of $X$ the scheme $X$ is geometrically reduced at $x'$, then $X$ is geometrically reduced at $x$. \\item If $X$ is reduced and geometrically reduced at all generic points of irreducible components of $X$, then $X$ is geometrically reduced. \\end{enumerate}"}
+{"_id": "10914", "title": "varieties-lemma-finite-extension-geometrically-reduced", "text": "Let $k$ be a field. Let $X \\to \\Spec(k)$ be locally of finite type. Assume $X$ has finitely many irreducible components. Then there exists a finite purely inseparable extension $k \\subset k'$ such that $(X_{k'})_{red}$ is geometrically reduced over $k'$."}
+{"_id": "10915", "title": "varieties-lemma-geometrically-connected-check-after-extension", "text": "Let $X$ be a scheme over the field $k$. Let $k \\subset k'$ be a field extension. Then $X$ is geometrically connected over $k$ if and only if $X_{k'}$ is geometrically connected over $k'$."}
+{"_id": "10916", "title": "varieties-lemma-bijection-connected-components", "text": "Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism $$ p : X \\times_k Y \\longrightarrow Y $$ induces a bijection between connected components."}
+{"_id": "10917", "title": "varieties-lemma-affine-geometrically-connected", "text": "Let $k$ be a field. Let $A$ be a $k$-algebra. Then $X = \\Spec(A)$ is geometrically connected over $k$ if and only if $A$ is geometrically connected over $k$ (see Algebra, Definition \\ref{algebra-definition-geometrically-connected})."}
+{"_id": "10918", "title": "varieties-lemma-separably-closed-field-connected-components", "text": "Let $k \\subset k'$ be an extension of fields. Let $X$ be a scheme over $k$. Assume $k$ separably algebraically closed. Then the morphism $X_{k'} \\to X$ induces a bijection of connected components. In particular, $X$ is geometrically connected over $k$ if and only if $X$ is connected."}
+{"_id": "10919", "title": "varieties-lemma-characterize-geometrically-connected", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $\\overline{k}$ be a separable algebraic closure of $k$. Then $X$ is geometrically connected if and only if the base change $X_{\\overline{k}}$ is connected."}
+{"_id": "10920", "title": "varieties-lemma-descend-open", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $A$ be a $k$-algebra. Let $V \\subset X_A$ be a quasi-compact open. Then there exists a finitely generated $k$-subalgebra $A' \\subset A$ and a quasi-compact open $V' \\subset X_{A'}$ such that $V = V'_A$."}
+{"_id": "10921", "title": "varieties-lemma-Galois-action-quasi-compact-open", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $\\overline{k}$ be a (possibly infinite) Galois extension of $k$. Let $V \\subset X_{\\overline{k}}$ be a quasi-compact open. Then \\begin{enumerate} \\item there exists a finite subextension $k \\subset k' \\subset \\overline{k}$ and a quasi-compact open $V' \\subset X_{k'}$ such that $V = (V')_{\\overline{k}}$, \\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$ such that $\\sigma(V) = V$ for all $\\sigma \\in H$. \\end{enumerate}"}
+{"_id": "10922", "title": "varieties-lemma-closed-fixed-by-Galois", "text": "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite) Galois extension. Let $X$ be a scheme over $k$. Let $\\overline{T} \\subset X_{\\overline{k}}$ have the following properties \\begin{enumerate} \\item $\\overline{T}$ is a closed subset of $X_{\\overline{k}}$, \\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$ we have $\\sigma(\\overline{T}) = \\overline{T}$. \\end{enumerate} Then there exists a closed subset $T \\subset X$ whose inverse image in $X_{\\overline{k}}$ is $\\overline{T}$."}
+{"_id": "10923", "title": "varieties-lemma-characterize-geometrically-disconnected", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically connected, \\item for every finite separable field extension $k \\subset k'$ the scheme $X_{k'}$ is connected. \\end{enumerate}"}
+{"_id": "10924", "title": "varieties-lemma-tricky", "text": "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite) Galois extension. Let $f : T \\to X$ be a morphism of schemes over $k$. Assume $T_{\\overline{k}}$ connected and $X_{\\overline{k}}$ disconnected. Then $X$ is disconnected."}
+{"_id": "10925", "title": "varieties-lemma-geometrically-connected-criterion", "text": "\\begin{reference} \\cite[IV Corollary 4.5.13.1(i)]{EGA} \\end{reference} Let $k$ be a field. Let $T \\to X$ be a morphism of schemes over $k$. Assume $T$ is geometrically connected and $X$ connected. Then $X$ is geometrically connected."}
+{"_id": "10926", "title": "varieties-lemma-geometrically-connected-if-connected-and-point", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Assume $X$ is connected and has a point $x$ such that $k$ is algebraically closed in $\\kappa(x)$. Then $X$ is geometrically connected. In particular, if $X$ has a $k$-rational point and $X$ is connected, then $X$ is geometrically connected."}
+{"_id": "10927", "title": "varieties-lemma-inverse-image-connected-component", "text": "Let $k \\subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every connected component $T$ of $X$ the inverse image $T_K \\subset X_K$ is a union of connected components of $X_K$."}
+{"_id": "10928", "title": "varieties-lemma-image-connected-component-finite-extension", "text": "Let $k \\subset K$ be a finite extension of fields and let $X$ be a scheme over  $k$. Denote by $p : X_K \\to X$ the projection morphism. For every connected  component $T$ of $X_K$ the image $p(T)$ is a connected component of  $X$."}
+{"_id": "10932", "title": "varieties-lemma-geometrically-irreducible-check-after-extension", "text": "Let $X$ be a scheme over the field $k$. Let $k \\subset k'$ be a field extension. Then $X$ is geometrically irreducible over $k$ if and only if $X_{k'}$ is geometrically irreducible over $k'$."}
+{"_id": "10933", "title": "varieties-lemma-separably-closed-irreducible", "text": "Let $X$ be a scheme over a separably closed field $k$. If $X$ is irreducible, then $X_K$ is irreducible for any field extension $k \\subset K$. I.e., $X$ is geometrically irreducible over $k$."}
+{"_id": "10934", "title": "varieties-lemma-bijection-irreducible-components", "text": "Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. Assume $X$ is geometrically irreducible over $k$. Then the projection morphism $$ p : X \\times_k Y \\longrightarrow Y $$ induces a bijection between irreducible components."}
+{"_id": "10935", "title": "varieties-lemma-geometrically-irreducible-local", "text": "\\begin{slogan} Geometric irreductibility is Zariski local modulo connectedness. \\end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically irreducible over $k$, \\item for every nonempty affine open $U$ the $k$-algebra $\\mathcal{O}_X(U)$ is geometrically irreducible over $k$ (see Algebra, Definition \\ref{algebra-definition-geometrically-irreducible}), \\item $X$ is irreducible and there exists an affine open covering $X = \\bigcup U_i$ such that each $k$-algebra $\\mathcal{O}_X(U_i)$ is geometrically irreducible, and \\item there exists an open covering $X = \\bigcup_{i \\in I} X_i$ with $I \\not = \\emptyset$ such that $X_i$ is geometrically irreducible for each $i$ and such that $X_i \\cap X_j \\not = \\emptyset$ for all $i, j \\in I$. \\end{enumerate} Moreover, if $X$ is geometrically irreducible so is every nonempty open subscheme of $X$."}
+{"_id": "10936", "title": "varieties-lemma-geometrically-irreducible-function-field", "text": "Let $X$ be an irreducible scheme over the field $k$. Let $\\xi \\in X$ be its generic point. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically irreducible over $k$, and \\item $\\kappa(\\xi)$ is geometrically irreducible over $k$. \\end{enumerate}"}
+{"_id": "10937", "title": "varieties-lemma-separably-closed-field-irreducible-components", "text": "Let $k \\subset k'$ be an extension of fields. Let $X$ be a scheme over $k$. Set $X' = X_{k'}$. Assume $k$ separably algebraically closed. Then the morphism $X' \\to X$ induces a bijection of irreducible components."}
+{"_id": "10938", "title": "varieties-lemma-characterize-geometrically-irreducible", "text": "\\begin{slogan} Geometric irreducibility can be tested over a separable algebraic closure of the base field. \\end{slogan} Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent: \\begin{enumerate} \\item $X$ is geometrically irreducible over $k$, \\item for every finite separable field extension $k \\subset k'$ the scheme $X_{k'}$ is irreducible, and \\item $X_{\\overline{k}}$ is irreducible, where $k \\subset \\overline{k}$ is a separable algebraic closure of $k$. \\end{enumerate}"}
+{"_id": "10939", "title": "varieties-lemma-inverse-image-irreducible", "text": "Let $k \\subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every irreducible component $T$ of $X$ the inverse image $T_K \\subset X_K$ is a union of irreducible components of $X_K$."}
+{"_id": "10940", "title": "varieties-lemma-image-irreducible", "text": "Let $k \\subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every irreducible component $\\overline{T} \\subset X_K$ the image of $\\overline{T}$ in $X$ is an irreducible component in $X$. This defines a canonical map $$ \\text{IrredComp}(X_K) \\longrightarrow \\text{IrredComp}(X) $$ which is surjective."}
+{"_id": "10942", "title": "varieties-lemma-galois-action-irreducible-components", "text": "Let $k$ be a field, with separable algebraic closure $\\overline{k}$. Let $X$ be a scheme over $k$. There is an action $$ \\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}}) \\longrightarrow \\text{IrredComp}(X_{\\overline{k}}) $$ with the following properties: \\begin{enumerate} \\item An element $\\overline{T} \\in \\text{IrredComp}(X_{\\overline{k}})$ is fixed by the action if and only if there exists an irreducible component $T \\subset X$, which is geometrically irreducible over $k$, such that $T_{\\overline{k}} = \\overline{T}$. \\item For any field extension $k \\subset k'$ with separable algebraic closure $\\overline{k}'$ the diagram $$ \\xymatrix{ \\text{Gal}(\\overline{k}'/k') \\times \\text{IrredComp}(X_{\\overline{k}'}) \\ar[r] \\ar[d] & \\text{IrredComp}(X_{\\overline{k}'}) \\ar[d] \\\\ \\text{Gal}(\\overline{k}/k) \\times \\text{IrredComp}(X_{\\overline{k}}) \\ar[r] & \\text{IrredComp}(X_{\\overline{k}}) } $$ is commutative (where the right vertical arrow is a bijection according to Lemma \\ref{lemma-separably-closed-field-irreducible-components}). \\end{enumerate}"}
+{"_id": "10943", "title": "varieties-lemma-orbit-irreducible-components", "text": "Let $k$ be a field, with separable algebraic closure $\\overline{k}$. Let $X$ be a scheme over $k$. The fibres of the map $$ \\text{IrredComp}(X_{\\overline{k}}) \\longrightarrow \\text{IrredComp}(X) $$ of Lemma \\ref{lemma-image-irreducible} are exactly the orbits of $\\text{Gal}(\\overline{k}/k)$ under the action of Lemma \\ref{lemma-galois-action-irreducible-components}."}
+{"_id": "10944", "title": "varieties-lemma-galois-action-irreducible-components-locally-finite-type", "text": "Let $k$ be a field. Assume $X \\to \\Spec(k)$ locally of finite type. In this case \\begin{enumerate} \\item the action $$ \\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}}) \\longrightarrow \\text{IrredComp}(X_{\\overline{k}}) $$ is continuous if we give $\\text{IrredComp}(X_{\\overline{k}})$ the discrete topology, \\item every irreducible component of $X_{\\overline{k}}$ can be defined over a finite extension of $k$, and \\item given any irreducible component $T \\subset X$ the scheme $T_{\\overline{k}}$ is a finite union of irreducible components of $X_{\\overline{k}}$ which are all in the same $\\text{Gal}(\\overline{k}/k)$-orbit. \\end{enumerate}"}
+{"_id": "10945", "title": "varieties-lemma-finite-extension-geometrically-irreducible-components", "text": "Let $k$ be a field. Let $X \\to \\Spec(k)$ be locally of finite type. Assume $X$ has finitely many irreducible components. Then there exists a finite separable extension $k \\subset k'$ such that every irreducible component of $X_{k'}$ is geometrically irreducible over $k'$."}
+{"_id": "10946", "title": "varieties-lemma-irreducible-components-geometrically-irreducible", "text": "Let $X$ be a scheme over the field $k$. Assume $X$ has finitely many irreducible components which are all geometrically irreducible. Then $X$ has finitely many connected components each of which is geometrically connected."}
+{"_id": "10947", "title": "varieties-lemma-geometrically-integral", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Then $X$ is geometrically integral over $k$ if and only if $X$ is both geometrically reduced and geometrically irreducible over $k$."}
+{"_id": "10948", "title": "varieties-lemma-proper-geometrically-reduced-global-sections", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. \\begin{enumerate} \\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra, \\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian local $k$-algebras, one factor for each connected component of $X$, \\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$ is a product of fields, each a finite extension of $k$, \\item if $X$ is geometrically reduced, then $k_i$ is finite separable over $k$, \\item if $X$ is geometrically connected, then $A$ is geometrically irreducible over $k$, \\item if $X$ is geometrically irreducible, then $A$ is geometrically irreducible over $k$, \\item if $X$ is geometrically reduced and connected, then $A = k$, and \\item if $X$ is geometrically integral, then $A = k$. \\end{enumerate}"}
+{"_id": "10949", "title": "varieties-lemma-baby-stein", "text": "Let $X$ be a proper scheme over a field $k$. Set $A = H^0(X, \\mathcal{O}_X)$. The fibres of the canonical morphism $X \\to \\Spec(A)$ are geometrically connected."}
+{"_id": "10950", "title": "varieties-lemma-geometrically-reduced-stein", "text": "Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$. The following are equivalent \\begin{enumerate} \\item $H^0(X, \\mathcal{O}_X) = k$, and \\item $X$ is geometrically connected. \\end{enumerate}"}
+{"_id": "10951", "title": "varieties-lemma-geometrically-normal-at-point", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \\in X$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically normal at $x$, \\item for every finite purely inseparable field extension $k'$ of $k$ and $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is normal, and \\item the ring $\\mathcal{O}_{X, x}$ is geometrically normal over $k$ (see Algebra, Definition \\ref{algebra-definition-geometrically-normal}). \\end{enumerate}"}
+{"_id": "10952", "title": "varieties-lemma-geometrically-normal", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically normal, \\item $X_{k'}$ is a normal scheme for every field extension $k'/k$, \\item $X_{k'}$ is a normal scheme for every finitely generated field extension $k'/k$, \\item $X_{k'}$ is a normal scheme for every finite purely inseparable field extension $k'/k$, \\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is geometrically normal (see Algebra, Definition \\ref{algebra-definition-geometrically-normal}), and \\item $X_{k^{perf}}$ is a normal scheme. \\end{enumerate}"}
+{"_id": "10953", "title": "varieties-lemma-geometrically-normal-upstairs", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a field extension. Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically normal at $x$, \\item $X_{k'}$ is geometrically normal at $x'$. \\end{enumerate} In particular, $X$ is geometrically normal over $k$ if and only if $X_{k'}$ is geometrically normal over $k'$."}
+{"_id": "10954", "title": "varieties-lemma-fibre-product-normal", "text": "Let $k$ be a field. Let $X$ be a geometrically normal scheme over $k$ and let $Y$ be a normal scheme over $k$. Then $X \\times_k Y$ is a normal scheme."}
+{"_id": "10955", "title": "varieties-lemma-base-change-normal-by-separable", "text": "Let $k$ be a field. Let $X$ be a normal scheme over $k$. Let $K/k$ be a separable field extension. Then $X_K$ is a normal scheme."}
+{"_id": "10956", "title": "varieties-lemma-geometrically-normal-stein", "text": "Let $k$ be a field. Let $X$ be a proper geometrically normal scheme over $k$. The following are equivalent \\begin{enumerate} \\item $H^0(X, \\mathcal{O}_X) = k$, \\item $X$ is geometrically connected, \\item $X$ is geometrically irreducible, and \\item $X$ is geometrically integral. \\end{enumerate}"}
+{"_id": "10957", "title": "varieties-lemma-locally-Noetherian-base-change", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k \\subset k'$ be a finitely generated field extension. Then $X$ is locally Noetherian if and only if $X_{k'}$ is locally Noetherian."}
+{"_id": "10958", "title": "varieties-lemma-geometrically-regular-at-point", "text": "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. Let $x \\in X$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically regular at $x$, \\item for every finite purely inseparable field extension $k'$ of $k$ and $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is regular, and \\item the ring $\\mathcal{O}_{X, x}$ is geometrically regular over $k$ (see Algebra, Definition \\ref{algebra-definition-geometrically-regular}). \\end{enumerate}"}
+{"_id": "10959", "title": "varieties-lemma-geometrically-regular", "text": "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically regular, \\item $X_{k'}$ is a regular scheme for every finitely generated field extension $k \\subset k'$, \\item $X_{k'}$ is a regular scheme for every finite purely inseparable field extension $k \\subset k'$, \\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is geometrically regular (see Algebra, Definition \\ref{algebra-definition-geometrically-regular}), and \\item there exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is geometrically regular over $k$. \\end{enumerate}"}
+{"_id": "10960", "title": "varieties-lemma-geometrically-regular-upstairs", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ be a finitely generated field extension. Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically regular at $x$, \\item $X_{k'}$ is geometrically regular at $x'$. \\end{enumerate} In particular, $X$ is geometrically regular over $k$ if and only if $X_{k'}$ is geometrically regular over $k'$."}
+{"_id": "10961", "title": "varieties-lemma-flat-under-geometrically-regular", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of locally Noetherian schemes over $k$. Let $x \\in X$ be a point and set $y = f(x)$. If $X$ is geometrically regular at $x$ and $f$ is flat at $x$ then $Y$ is geometrically regular at $y$. In particular, if $X$ is geometrically regular over $k$ and $f$ is flat and surjective, then $Y$ is geometrically regular over $k$."}
+{"_id": "10962", "title": "varieties-lemma-geometrically-regular-smooth", "text": "Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $x \\in X$. Then $X$ is geometrically regular at $x$ if and only if $X \\to \\Spec(k)$ is smooth at $x$ (Morphisms, Definition \\ref{morphisms-definition-smooth})."}
+{"_id": "10963", "title": "varieties-lemma-CM-base-change", "text": "Let $X$ be a locally Noetherian scheme over the field $k$. Let $k \\subset k'$ be a finitely generated field extension. Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$. Then we have $$ \\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay} \\Leftrightarrow \\mathcal{O}_{X_{k'}, x'}\\text{ is Cohen-Macaulay} $$ If $X$ is locally of finite type over $k$, the same holds for any field extension $k \\subset k'$."}
+{"_id": "10964", "title": "varieties-lemma-locally-finite-type-Jacobson", "text": "Let $X$ be a scheme which is locally of finite type over $k$. Then \\begin{enumerate} \\item for any closed point $x \\in X$ the extension $k \\subset \\kappa(x)$ is algebraic, and \\item $X$ is a Jacobson scheme (Properties, Definition \\ref{properties-definition-jacobson}). \\end{enumerate}"}
+{"_id": "10965", "title": "varieties-lemma-make-Jacobson", "text": "Let $X$ be a scheme over a field $k$. For any field extension $k \\subset K$ whose cardinality is large enough we have \\begin{enumerate} \\item for any closed point $x \\in X_K$ the extension $K \\subset \\kappa(x)$ is algebraic, and \\item $X_K$ is a Jacobson scheme (Properties, Definition \\ref{properties-definition-jacobson}). \\end{enumerate}"}
+{"_id": "10966", "title": "varieties-lemma-ample-after-field-extension", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. If there exists an ample invertible sheaf on $X_K$ for some field extension $k \\subset K$, then $X$ has an ample invertible sheaf."}
+{"_id": "10968", "title": "varieties-lemma-quasi-projective-after-field-extension", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-projective over $K$ for some field extension $k \\subset K$, then $X$ is quasi-projective over $k$."}
+{"_id": "10969", "title": "varieties-lemma-proper-after-field-extension", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is proper over $K$ for some field extension $k \\subset K$, then $X$ is proper over $k$."}
+{"_id": "10972", "title": "varieties-lemma-tangent-space-cotangent-space", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. There is a canonical isomorphism $$ T_{X/S, x} = \\Hom_{\\mathcal{O}_{X, x}}(\\Omega_{X/S, x}, \\kappa(x)) $$ of vector spaces over $\\kappa(x)$."}
+{"_id": "10973", "title": "varieties-lemma-tangent-space-rational-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point and let $s = f(x) \\in S$. Assume that $\\kappa(x) = \\kappa(s)$. Then there are canonical isomorphisms $$ \\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x}) = \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ and $$ T_{X/S, x} = \\Hom_{\\kappa(x)}( \\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x}), \\kappa(x)) $$ This works more generally if $\\kappa(x)/\\kappa(s)$ is a separable algebraic extension."}
+{"_id": "10974", "title": "varieties-lemma-map-tangent-spaces", "text": "Let $f : X \\to Y$ be a morphism of schemes over a base scheme $S$. Let $x \\in X$ be a point. Set $y = f(x)$. If $\\kappa(y) = \\kappa(x)$, then $f$ induces a natural linear map $$ \\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y} $$ which is dual to the linear map $\\Omega_{Y/S, y} \\otimes \\kappa(y) \\to \\Omega_{X/S, x}$ via the identifications of Lemma \\ref{lemma-tangent-space-cotangent-space}."}
+{"_id": "10975", "title": "varieties-lemma-tangent-space-product", "text": "Let $X$, $Y$ be schemes over a base $S$. Let $x \\in X$ and $y \\in Y$ with the same image point $s \\in S$ such that $\\kappa(s) = \\kappa(x)$ and $\\kappa(s) = \\kappa(y)$. There is a canonical isomorphism $$ T_{X \\times_S Y/S, (x, y)} = T_{X/S, x} \\oplus T_{Y/S, y} $$ The map from left to right is induced by the maps on tangent spaces coming from the projections $X \\times_S Y \\to X$ and $X \\times_S Y \\to Y$. The map from right to left is induced by the maps $1 \\times y : X_s \\to X_s \\times_s Y_s$ and $x \\times 1 : Y_s \\to X_s \\times_s Y_s$ via the identification (\\ref{equation-tangent-space-fibre}) of tangent spaces with tangent spaces of fibres."}
+{"_id": "10976", "title": "varieties-lemma-injective-tangent-spaces-unramified", "text": "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over a base scheme $S$. Let $x \\in X$ be a point. Set $y = f(x)$ and assume that $\\kappa(y) = \\kappa(x)$. Then the following are equivalent \\begin{enumerate} \\item $\\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y}$ is injective, and \\item $f$ is unramified at $x$. \\end{enumerate}"}
+{"_id": "10977", "title": "varieties-lemma-quasi-finite-in-codim-1", "text": "Let $f : X \\to Y$ be locally of finite type. Let $y \\in Y$ be a point such that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$. Assume in addition one of the following conditions is satisfied \\begin{enumerate} \\item for every generic point $\\eta$ of an irreducible component of $X$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic), \\item for every generic point $\\eta$ of an irreducible component of $X$ such that $f(\\eta) \\leadsto y$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic), \\item $f$ is quasi-finite at every generic point of an irreducible component of $X$, \\item $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$, \\item add more here. \\end{enumerate} Then $f$ is quasi-finite at every point of $X$ lying over $y$."}
+{"_id": "10978", "title": "varieties-lemma-finite-in-codim-1", "text": "Let $f : X \\to Y$ be a proper morphism. Let $y \\in Y$ be a point such that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$. Assume in addition one of the following conditions is satisfied \\begin{enumerate} \\item for every generic point $\\eta$ of an irreducible component of $X$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic), \\item for every generic point $\\eta$ of an irreducible component of $X$ such that $f(\\eta) \\leadsto y$ the field extension $\\kappa(\\eta) \\supset \\kappa(f(\\eta))$ is finite (or algebraic), \\item $f$ is quasi-finite at every generic point of $X$, \\item $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$, \\item add more here. \\end{enumerate} Then there exists an open neighbourhood $V \\subset Y$ of $y$ such that $f^{-1}(V) \\to V$ is finite."}
+{"_id": "10979", "title": "varieties-lemma-modification-normal-iso-over-codimension-1", "text": "Let $X$ be a Noetherian scheme. Let $f : Y \\to X$ be a birational proper morphism of schemes with $Y$ reduced. Let $U \\subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains \\begin{enumerate} \\item every point of codimension $0$ in $X$, \\item every $x \\in X$ of codimension $1$ on $X$ such that $\\mathcal{O}_{X, x}$ is a discrete valuation ring, \\item every $x \\in X$ such that the fibre of $Y \\to X$ over $x$ is finite and such that $\\mathcal{O}_{X, x}$ is normal, and \\item every $x \\in X$ such that $f$ is quasi-finite at some $y \\in Y$ lying over $x$ and $\\mathcal{O}_{X, x}$ is normal. \\end{enumerate}"}
+{"_id": "10981", "title": "varieties-lemma-noether-normalization-affine", "text": "Let $f : X \\to S$ be a finite type morphism of affine schemes. Let $s \\in S$. If $\\dim(X_s) = d$, then there exists a factorization $$ X \\xrightarrow{\\pi} \\mathbf{A}^d_S \\to S $$ of $f$ such that the morphism $\\pi_s : X_s \\to \\mathbf{A}^d_{\\kappa(s)}$ of fibres over $s$ is finite."}
+{"_id": "10982", "title": "varieties-lemma-geometric-structure-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Let $V = \\Spec(A)$ be an affine open neighbourhood of $f(x)$ in $S$. If $f$ is unramified at $x$, then there exist exists an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ such that we have a commutative diagram $$ \\xymatrix{ X \\ar[d] & U \\ar[l] \\ar[rd] \\ar[r]^-j & \\Spec(A[t]_{g'}/(g)) \\ar[d] \\ar[r] & \\Spec(A[t]) = \\mathbf{A}^1_V \\ar[ld] \\\\ Y & & V \\ar[ll] } $$ where $j$ is an immersion, $g \\in A[t]$ is a monic polynomial, and $g'$ is the derivative of $g$ with respect to $t$. If $f$ is \\'etale at $x$, then we may choose the diagram such that $j$ is an open immersion."}
+{"_id": "10983", "title": "varieties-lemma-unramfied-over-affine", "text": "Let $f : X \\to S$ be a finite type morphism of affine schemes. Let $x \\in X$ with image $s \\in S$. Let $$ r = \\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) = \\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\dim_{\\kappa(x)} T_{X/S, x} $$ Then there exists a factorization $$ X \\xrightarrow{\\pi} \\mathbf{A}^r_S \\to S $$ of $f$ such that $\\pi$ is unramified at $x$."}
+{"_id": "10985", "title": "varieties-lemma-dimension-fibre-in-codim-1", "text": "Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point with image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$. Let $d \\geq 0$ be an integer such that for every generic point $\\eta$ of an irreducible component of $X$ which contains $x$, we have $\\dim_\\eta(X_{f(\\eta)}) = d$. Then $\\dim_x(X_y) = d$."}
+{"_id": "10987", "title": "varieties-lemma-dimension-fibre-in-higher-codimension", "text": "Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point with image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian. Let $d \\geq 0$ be an integer such that for every generic point $\\eta$ of an irreducible component of $X$ which contains $x$, we have $f(\\eta) \\not = y$ and $\\dim_\\eta(X_{f(\\eta)}) = d$. Then $\\dim_x(X_y) \\leq d + \\dim(\\mathcal{O}_{Y, y}) - 1$."}
+{"_id": "10988", "title": "varieties-lemma-algebraic-scheme-dim-0", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $0$. Then $X$ is a disjoint union of spectra of local Artinian $k$-algebras $A$ with $\\dim_k(A) < \\infty$. If $X$ is an algebraic $k$-scheme of dimension $0$, then in addition $X$ is affine and the morphism $X \\to \\Spec(k)$ is finite."}
+{"_id": "10989", "title": "varieties-lemma-dimension-locally-algebraic", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. \\begin{enumerate} \\item \\label{item-catenary} The topological space of $X$ is catenary (Topology, Definition \\ref{topology-definition-catenary}). \\item \\label{item-dimension-at-closed-point} For $x \\in X$ closed, we have $\\dim_x(X) = \\dim(\\mathcal{O}_{X, x})$. \\item \\label{item-dimension-irreducible} For $X$ irreducible we have $\\dim(X) = \\dim(U)$ for any nonempty open $U \\subset X$ and $\\dim(X) = \\dim_x(X)$ for any $x \\in X$. \\item \\label{item-irreducible-maximal-chains} For $X$ irreducible any chain of irreducible closed subsets can be extended to a maximal chain and all maximal chains of irreducible closed subsets have length equal to $\\dim(X)$. \\item \\label{item-dimension-irreducibles-passing-through} For $x \\in X$ we have $\\dim_x(X) = \\max \\dim(Z) = \\min \\dim(\\mathcal{O}_{X, x'})$ where the maximum is over irreducible components $Z \\subset X$ containing $x$ and the minimum is over specializations $x \\leadsto x'$ with $x'$ closed in $X$. \\item \\label{item-dimension-irreducible-trdeg} If $X$ is irreducible with generic point $x$, then $\\dim(X) = \\text{trdeg}_k(\\kappa(x))$. \\item \\label{item-immediate-specialization} If $x \\leadsto x'$ is an immediate specialization of points of $X$, then we have $\\text{trdeg}_k(\\kappa(x)) = \\text{trdeg}_k(\\kappa(x')) + 1$. \\item \\label{item-dimension-sup-trdeg} The dimension of $X$ is the supremum of the numbers $\\text{trdeg}_k(\\kappa(x))$ where $x$ runs over the generic points of the irreducible components of $X$. \\item \\label{item-specialization} If $x \\leadsto x'$ is a nontrivial specialization of points of $X$, then \\begin{enumerate} \\item $\\dim_x(X) \\leq \\dim_{x'}(X)$, \\item $\\dim(\\mathcal{O}_{X, x}) < \\dim(\\mathcal{O}_{X, x'})$, \\item $\\text{trdeg}_k(\\kappa(x)) > \\text{trdeg}_k(\\kappa(x'))$, and \\item any maximal chain of nontrivial specializations $x = x_0 \\leadsto x_1 \\leadsto \\ldots \\leadsto x_n = x$ has length $n = \\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_k(\\kappa(x'))$. \\end{enumerate} \\item \\label{item-dimension-formula} For $x \\in X$ we have $\\dim_x(X) = \\text{trdeg}_k(\\kappa(x)) + \\dim(\\mathcal{O}_{X, x})$. \\item \\label{item-immediate-specialization-local-ring} If $x \\leadsto x'$ is an immediate specialization of points of $X$ and $X$ is irreducible or equidimensional, then $\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{X, x}) + 1$. \\end{enumerate}"}
+{"_id": "10990", "title": "varieties-lemma-dimension-fibres-locally-algebraic", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of locally algebraic $k$-schemes. \\begin{enumerate} \\item For $y \\in Y$, the fibre $X_y$ is a locally algebraic scheme over $\\kappa(y)$ hence all the results of Lemma \\ref{lemma-dimension-locally-algebraic} apply. \\item Assume $X$ is irreducible. Set $Z = \\overline{f(X)}$ and $d = \\dim(X) - \\dim(Z)$. Then \\begin{enumerate} \\item $\\dim_x(X_{f(x)}) \\geq d$ for all $x \\in X$, \\item the set of $x \\in X$ with $\\dim_x(X_{f(x)}) = d$ is dense open, \\item if $\\dim(\\mathcal{O}_{Z, f(x)}) \\geq 1$, then $\\dim_x(X_{f(x)}) \\leq d + \\dim(\\mathcal{O}_{Z, f(x)}) - 1$, \\item if $\\dim(\\mathcal{O}_{Z, f(x)}) = 1$, then $\\dim_x(X_{f(x)}) = d$, \\end{enumerate} \\item For $x \\in X$ with $y = f(x)$ we have $\\dim_x(X_y) \\geq \\dim_x(X) - \\dim_y(Y)$. \\end{enumerate}"}
+{"_id": "10991", "title": "varieties-lemma-dimension-product-locally-algebraic", "text": "\\begin{slogan} The dimension of the product is the sum of the dimensions. \\end{slogan} Let $k$ be a field. Let $X$, $Y$ be locally algebraic $k$-schemes. \\begin{enumerate} \\item For $z \\in X \\times Y$ lying over $(x, y)$ we have $\\dim_z(X \\times Y) = \\dim_x(X) + \\dim_y(Y)$. \\item We have $\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$. \\end{enumerate}"}
+{"_id": "10992", "title": "varieties-lemma-complete-local-ring-structure-as-algebra", "text": "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. Let $x \\in X$ be a point with residue field $\\kappa$. There is an isomorphism \\begin{equation} \\label{equation-complete-local-ring} \\kappa[[x_1, \\ldots, x_n]]/I \\longrightarrow \\mathcal{O}_{X, x}^\\wedge \\end{equation} inducing the identity on residue fields. In general we cannot choose (\\ref{equation-complete-local-ring}) to be a $k$-algebra isomorphism. However, if the extension $\\kappa/k$ is separable, then we can choose (\\ref{equation-complete-local-ring}) to be an isomorphism of $k$-algebras."}
+{"_id": "10993", "title": "varieties-lemma-base-change-complete-local-ring", "text": "Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic $k$-scheme. Set $Y = X_K$. Let $y \\in Y$ be a point with image $x \\in X$. Assume that $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$ and that $\\kappa(x)/k$ is separable. Choose an isomorphism $$ \\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow \\mathcal{O}_{X, x}^\\wedge $$ of $k$-algebras as in (\\ref{equation-complete-local-ring}). Then we have an isomorphism $$ \\kappa(y)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow \\mathcal{O}_{Y, y}^\\wedge $$ of $K$-algebras as in (\\ref{equation-complete-local-ring}). Here we use $\\kappa(x) \\to \\kappa(y)$ to view $g_j$ as a power series over $\\kappa(y)$."}
+{"_id": "10994", "title": "varieties-lemma-globally-generated-base-change", "text": "Let $X \\to \\Spec(A)$ be a morphism of schemes. Let $A \\subset A'$ be a faithfully flat ring map. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is globally generated if and only if the base change $\\mathcal{F}_{A'}$ is globally generated."}
+{"_id": "10995", "title": "varieties-lemma-very-ample-vanish-at-point", "text": "Let $k$ be an infinite field. Let $X$ be a scheme of finite type over $k$. Let $\\mathcal{L}$ be a very ample invertible sheaf on $X$. Let $n \\geq 0$ and $x, x_1, \\ldots, x_n \\in X$ be points with $x$ a $k$-rational point, i.e., $\\kappa(x) = k$, and $x \\not = x_i$ for $i = 1, \\ldots, n$. Then there exists an $s \\in H^0(X, \\mathcal{L})$ which vanishes at $x$ but not at $x_i$."}
+{"_id": "10997", "title": "varieties-lemma-separate-points-tangent-vectors", "text": "Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $V \\subset H^0(X, \\mathcal{L})$ be a $k$-subvector space. If \\begin{enumerate} \\item for every pair of distinct closed points $x, y \\in X$ there is a section $s \\in V$ which vanishes at $x$ but not at $y$, and \\item for every closed point $x \\in X$ and nonzero tangent vector $\\theta \\in T_{X/k, x}$ there exist a section $s \\in V$ which vanishes at $x$ but whose pullback by $\\theta$ is nonzero, \\end{enumerate} then $\\mathcal{L}$ is very ample and the canonical morphism $\\varphi_{\\mathcal{L}, V} : X \\to \\mathbf{P}(V)$ is a closed immersion."}
+{"_id": "10998", "title": "varieties-lemma-variant-separate-points-tangent-vectors", "text": "Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Suppose that for every closed subscheme $Z \\subset X$ of dimension $0$ and degree $2$ over $k$ the map $$ H^0(X, \\mathcal{L}) \\longrightarrow H^0(Z, \\mathcal{L}|_Z) $$ is surjective. Then $\\mathcal{L}$ is very ample on $X$ over $k$."}
+{"_id": "10999", "title": "varieties-lemma-closure-of-product", "text": "Let $k$ be a field. Let $X$, $Y$ be schemes over $k$, and let $A \\subset X$, $B \\subset Y$ be subsets. Set $$ AB = \\{z \\in X \\times_k Y \\mid \\text{pr}_X(z) \\in A, \\ \\text{pr}_Y(z) \\in B\\} \\subset X \\times_k Y $$ Then set theoretically we have $$ \\overline{A} \\times_k \\overline{B} = \\overline{AB} $$"}
+{"_id": "11000", "title": "varieties-lemma-closure-image-product-map", "text": "Let $k$ be a field. Let $f : A \\to X$, $g : B \\to Y$ be morphisms of schemes over $k$. Then set theoretically we have $$ \\overline{f(A)} \\times_k \\overline{g(B)} = \\overline{(f \\times g)(A \\times_k B)} $$"}
+{"_id": "11001", "title": "varieties-lemma-scheme-theoretic-image-product-map", "text": "Let $k$ be a field. Let $f : A \\to X$, $g : B \\to Y$ be quasi-compact morphisms of schemes over $k$. Let $Z \\subset X$ be the scheme theoretic image of $f$, see Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}. Similarly, let $Z' \\subset Y$ be the scheme theoretic image of $g$. Then $Z \\times_k Z'$ is the scheme theoretic image of $f \\times g$."}
+{"_id": "11002", "title": "varieties-lemma-char-zero-differentials-free-smooth", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is locally of finite type over $k$, \\item $\\Omega_{X/k}$ is locally free, and \\item $k$ has characteristic zero. \\end{enumerate} Then the structure morphism $X \\to \\Spec(k)$ is smooth."}
+{"_id": "11003", "title": "varieties-lemma-char-p-differentials-free-smooth", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is locally of finite type over $k$, \\item $\\Omega_{X/k}$ is locally free, \\item $X$ is reduced, and \\item $k$ is perfect. \\end{enumerate} Then the structure morphism $X \\to \\Spec(k)$ is smooth."}
+{"_id": "11004", "title": "varieties-lemma-smooth-regular", "text": "\\begin{slogan} Smooth over a field implies regular \\end{slogan} Let $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field. Then $X$ is a regular scheme."}
+{"_id": "11005", "title": "varieties-lemma-smooth-geometrically-normal", "text": "Let $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field. Then $X$ is geometrically regular, geometrically normal, and geometrically reduced over $k$."}
+{"_id": "11006", "title": "varieties-lemma-affine-space-over-field", "text": "Let $k$ be a field. Let $d \\geq 0$. Let $W \\subset \\mathbf{A}^d_k$ be nonempty open. Then there exists a closed point $w \\in W$ such that $k \\subset \\kappa(w)$ is finite separable."}
+{"_id": "11007", "title": "varieties-lemma-smooth-separable-closed-points-dense", "text": "Let $k$ be a field. If $X$ is smooth over $\\Spec(k)$ then the set $$ \\{x \\in X\\text{ closed such that }k \\subset \\kappa(x) \\text{ is finite separable}\\} $$ is dense in $X$."}
+{"_id": "11008", "title": "varieties-lemma-geometrically-reduced-dense-smooth-open", "text": "Let $X$ be a scheme over a field $k$. If $X$ is locally of finite type and geometrically reduced over $k$ then $X$ contains a dense open which is smooth over $k$."}
+{"_id": "11009", "title": "varieties-lemma-dense-smooth-open-variety-over-perfect-field", "text": "Let $k$ be a perfect field. Let $X$ be a locally algebraic reduced $k$-scheme, for example a variety over $k$. Then we have $$ \\{x \\in X \\mid X \\to \\Spec(k)\\text{ is smooth at }x\\} = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is regular}\\} $$ and this is a dense open subscheme of $X$."}
+{"_id": "11010", "title": "varieties-lemma-flat-under-smooth", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $k$. Let $x \\in X$ be a point and set $y = f(x)$. If $X \\to \\Spec(k)$ is smooth at $x$ and $f$ is flat at $x$ then $Y \\to \\Spec(k)$ is smooth at $y$. In particular, if $X$ is smooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$."}
+{"_id": "11011", "title": "varieties-lemma-variety-with-smooth-rational-point", "text": "Let $k$ be a field. Let $X$ be a variety over $k$ which has a $k$-rational point $x$ such that $X$ is smooth at $x$. Then $X$ is geometrically integral over $k$."}
+{"_id": "11012", "title": "varieties-lemma-regular-functions-proper-variety", "text": "Let $X$ be a proper variety over $k$. Then \\begin{enumerate} \\item $K = H^0(X, \\mathcal{O}_X)$ is a field which is a finite extension of the field $k$, \\item if $X$ is geometrically reduced, then $K/k$ is separable, \\item if $X$ is geometrically irreducible, then $K/k$ is purely inseparable, \\item if $X$ is geometrically integral, then $K = k$. \\end{enumerate}"}
+{"_id": "11013", "title": "varieties-lemma-normalization-locally-algebraic", "text": "Let $k$ be a field. Let $X$ be a locally algebraic scheme over $k$. Let $\\nu : X^\\nu \\to X$ be the normalization morphism, see Morphisms, Definition \\ref{morphisms-definition-normalization}. Then \\begin{enumerate} \\item $\\nu$ is finite, dominant, and $X^\\nu$ is a disjoint union of normal irreducible locally algebraic schemes over $k$, \\item $\\nu$ factors as $X^\\nu \\to X_{red} \\to X$ and the first morphism is the normalization morphism of $X_{red}$, \\item if $X$ is a reduced algebraic scheme, then $\\nu$ is birational, \\item if $X$ is a variety, then $X^\\nu$ is a variety and $\\nu$ is a finite birational morphism of varieties. \\end{enumerate}"}
+{"_id": "11014", "title": "varieties-lemma-relative-normalization-finite", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a quasi-compact morphism of locally algebraic schemes over $k$. Let $X'$ be the normalization of $X$ in $Y$. If $Y$ is reduced, then $X' \\to X$ is finite."}
+{"_id": "11015", "title": "varieties-lemma-finite-extension-geometrically-normal", "text": "Let $k$ be a field. Let $X$ be an algebraic $k$-scheme. Then there exists a finite purely inseparable extension $k \\subset k'$ such that the normalization $Y$ of $X_{k'}$ is geometrically normal over $k'$."}
+{"_id": "11016", "title": "varieties-lemma-normalization-and-change-of-fields", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $\\nu : X^\\nu \\to X$ be the normalization of $X$ and let $Y^\\nu \\to X_K$ be the normalization of the base change. Then the canonical morphism $$ Y^\\nu \\longrightarrow X^\\nu \\times_{\\Spec(k)} \\Spec(K) $$ is an isomorphism if $K/k$ is separable and a universal homeomorphism in general."}
+{"_id": "11017", "title": "varieties-lemma-geometrically-normal-in-codim-1", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $\\nu : X^\\nu \\to X$ be the normalization of $X$. Let $x \\in X$ be a point such that (a) $\\mathcal{O}_{X, x}$ is reduced, (b) $\\dim(\\mathcal{O}_{X, x}) = 1$, and (c) for every $x' \\in X^\\nu$ with $\\nu(x') = x$ the extension $\\kappa(x')/k$ is separable. Then $X$ is geometrically reduced at $x$ and $X^\\nu$ is geometrically regular at $x'$ with $\\nu(x') = x$."}
+{"_id": "11018", "title": "varieties-lemma-open-in-normal-proper", "text": "Let $k$ be an algebraically closed field. Let $\\overline{X}$ be a proper variety over $k$. Let $X \\subset \\overline{X}$ be an open subscheme. Assume $X$ is normal. Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group."}
+{"_id": "11019", "title": "varieties-lemma-units-integral-finite-type-algebraically-closed", "text": "Let $k$ be an algebraically closed field. Let $X$ be an integral scheme locally of finite type over $k$. Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group."}
+{"_id": "11020", "title": "varieties-lemma-units-general-algebraically-closed", "text": "Let $k$ be an algebraically closed field. Let $X$ be a connected reduced scheme which is locally of finite type over $k$ with finitely many irreducible components. Then $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group."}
+{"_id": "11021", "title": "varieties-lemma-integral-closure-ground-field", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$ which is connected and reduced. Then the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$ is a field."}
+{"_id": "11023", "title": "varieties-lemma-kunneth", "text": "Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and let $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module. Then we have a canonical isomorphism $$ H^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) = \\bigoplus\\nolimits_{p + q = n} H^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G}) $$ provided $X$ and $Y$ are quasi-compact and have affine diagonal\\footnote{The case where $X$ and $Y$ are quasi-separated will be discussed in Lemma \\ref{lemma-kunneth-general} below.} (for example if $X$ and $Y$ are separated)."}
+{"_id": "11024", "title": "varieties-lemma-kunneth-general", "text": "Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and let $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module. Then we have a canonical isomorphism $$ H^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) = \\bigoplus\\nolimits_{p + q = n} H^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G}) $$ provided $X$ and $Y$ are quasi-compact and quasi-separated."}
+{"_id": "11025", "title": "varieties-lemma-change-rings-pic-pre", "text": "Let $A \\to B$ be a faithfully flat ring map. Let $X$ be a quasi-compact and quasi-separated scheme over $A$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module whose pullback to $X_B$ is trivial. Then $H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes -1})$ are invertible $H^0(X, \\mathcal{O}_X)$-modules and the multiplication map induces an isomorphism $$ H^0(X, \\mathcal{L}) \\otimes_{H^0(X, \\mathcal{O}_X)} H^0(X, \\mathcal{L}^{\\otimes -1}) \\longrightarrow H^0(X, \\mathcal{O}_X) $$"}
+{"_id": "11026", "title": "varieties-lemma-change-rings-pic", "text": "Let $A \\to B$ be a faithfully flat ring map. Let $X$ be a scheme over $A$ such that \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated, and \\item $R = H^0(X, \\mathcal{O}_X)$ is a semi-local ring. \\end{enumerate} Then the pullback map $\\Pic(X) \\to \\Pic(X_B)$ is injective."}
+{"_id": "11027", "title": "varieties-lemma-change-fields-pic", "text": "Let $k'/k$ be a field extension. Let $X$ be a scheme over $k$ such that \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated, and \\item $R = H^0(X, \\mathcal{O}_X)$ is semi-local, e.g., if $\\dim_k R < \\infty$. \\end{enumerate} Then the pullback map $\\Pic(X) \\to \\Pic(X_{k'})$ is injective."}
+{"_id": "11028", "title": "varieties-lemma-rational-equivalence-for-Pic", "text": "Let $k$ be a field. Let $X$ be a normal variety over $k$. Let $U \\subset \\mathbf{A}^n_k$ be an open subscheme with $k$-rational points $p, q \\in U(k)$. For every invertible module $\\mathcal{L}$ on $X \\times_{\\Spec(k)} U$ the restrictions $\\mathcal{L}|_{X \\times p}$ and $\\mathcal{L}|_{X \\times q}$ are isomorphic."}
+{"_id": "11029", "title": "varieties-lemma-euler-characteristic-additive", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of coherent modules on $X$. Then $$ \\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3) $$"}
+{"_id": "11030", "title": "varieties-lemma-chi-tensor-finite", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$. Then \\begin{enumerate} \\item $\\mathcal{F}$ is generated by global sections, \\item $H^i(X, \\mathcal{F}) = 0$ for $i > 0$, \\item $\\chi(X, \\mathcal{F}) = \\dim_k\\Gamma(X, \\mathcal{F})$, and \\item $\\chi(X, \\mathcal{F} \\otimes \\mathcal{E}) = n\\chi(X, \\mathcal{F})$ for every locally free module $\\mathcal{E}$ of rank $n$. \\end{enumerate}"}
+{"_id": "11031", "title": "varieties-lemma-euler-characteristic-extend-base-field", "text": "Let $k \\subset k'$ be an extension of fields. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X_{k'}$. Then $\\chi(X, \\mathcal{F}) = \\chi(X', \\mathcal{F}')$."}
+{"_id": "11032", "title": "varieties-lemma-euler-characteristic-morphism", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over $k$. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_Y$-module. Then $$ \\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G}) $$"}
+{"_id": "11033", "title": "varieties-lemma-projective-space-smooth", "text": "\\begin{slogan} Projective space is smooth. \\end{slogan} Let $k$ be a field and $n \\geq 0$. Then $\\mathbf{P}^n_k$ is a smooth projective variety of dimension $n$ over $k$."}
+{"_id": "11034", "title": "varieties-lemma-intersection-in-affine-space", "text": "Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{A}^n_k$ be closed subsets. Assume that $X$ and $Y$ are equidimensional, $\\dim(X) = r$ and $\\dim(Y) = s$. Then every irreducible component of $X \\cap Y$ has dimension $\\geq r + s - n$."}
+{"_id": "11035", "title": "varieties-lemma-intersection-in-projective-space", "text": "Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{P}^n_k$ be nonempty closed subsets. If $\\dim(X) = r$ and $\\dim(Y) = s$ and $r + s \\geq n$, then $X \\cap Y$ is nonempty and $\\dim(X \\cap Y) \\geq r + s - n$."}
+{"_id": "11036", "title": "varieties-lemma-equation-codim-1-in-projective-space", "text": "Let $k$ be a field. Let $Z \\subset \\mathbf{P}^n_k$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has dimension $n - 1$. Then the ideal $I(Z) \\subset k[T_0, \\ldots, T_n]$ corresponding to $Z$ is principal."}
+{"_id": "11037", "title": "varieties-lemma-hyperplane", "text": "Let $k$ be a field. Let $n \\geq 1$. Let $i : H \\to \\mathbf{P}^n_k$ be a hyperplane. Then there exists an isomorphism $$ \\varphi : \\mathbf{P}^{n - 1}_k \\longrightarrow H $$ such that $i^*\\mathcal{O}(1)$ pulls back to $\\mathcal{O}(1)$."}
+{"_id": "11038", "title": "varieties-lemma-exact-sequence-induction", "text": "Let $k$ be an infinite field. Let $n \\geq 1$. Let $\\mathcal{F}$ be a coherent module on $\\mathbf{P}^n_k$. Then there exist a nonzero section $s \\in \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$ and a short exact sequence $$ 0 \\to \\mathcal{F}(-1) \\to \\mathcal{F} \\to i_*\\mathcal{G} \\to 0 $$ where $i : H \\to \\mathbf{P}^n_k$ is the hyperplane $H$ associated to $s$ and $\\mathcal{G} = i^*\\mathcal{F}$."}
+{"_id": "11039", "title": "varieties-lemma-m-regular-extend-base-field", "text": "Let $k \\subset k'$ be an extension of fields. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. Let $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $\\mathbf{P}^n_{k'}$. Then $\\mathcal{F}$ is $m$-regular if and only if $\\mathcal{F}'$ is $m$-regular."}
+{"_id": "11040", "title": "varieties-lemma-m-regular", "text": "In the situation of Lemma \\ref{lemma-exact-sequence-induction}, if $\\mathcal{F}$ is $m$-regular, then $\\mathcal{G}$ is $m$-regular on $H \\cong \\mathbf{P}^{n - 1}_k$."}
+{"_id": "11041", "title": "varieties-lemma-m-regular-up", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. If $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}$ is $(m + 1)$-regular."}
+{"_id": "11042", "title": "varieties-lemma-m-regular-multiply", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. If $\\mathcal{F}$ is $m$-regular, then the multiplication map $$ H^0(\\mathbf{P}^n_k, \\mathcal{F}(m)) \\otimes_k H^0(\\mathbf{P}^n_k, \\mathcal{O}(1)) \\longrightarrow H^0(\\mathbf{P}^n_k, \\mathcal{F}(m + 1)) $$ is surjective."}
+{"_id": "11043", "title": "varieties-lemma-m-regular-globally-generated", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. If $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}(m)$ is globally generated."}
+{"_id": "11044", "title": "varieties-lemma-hilbert-polynomial", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. The function $$ d \\longmapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d)) $$ is a polynomial."}
+{"_id": "11046", "title": "varieties-lemma-bound-quotients-free", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $r \\geq 1$. Let $P \\in \\mathbf{Q}[t]$. There exists an integer $m$ depending on $n$, $r$, and $P$ with the following property: if $$ 0 \\to \\mathcal{K} \\to \\mathcal{O}^{\\oplus r} \\to \\mathcal{F} \\to 0 $$ is a short exact sequence of coherent sheaves on $\\mathbf{P}^n_k$ and $\\mathcal{F}$ has Hilbert polynomial $P$, then $\\mathcal{K}$ is $m$-regular."}
+{"_id": "11047", "title": "varieties-lemma-frobenius-endomorphism-identity", "text": "Let $p > 0$ be a prime number. Let $f : X \\to Y$ be a morphism of schemes in characteristic $p$. Then the diagram $$ \\xymatrix{ X \\ar[d]_f \\ar[r]_{F_X} & X \\ar[d]^f \\\\ Y \\ar[r]^{F_Y} & Y } $$ commutes."}
+{"_id": "11048", "title": "varieties-lemma-frobenius", "text": "Let $p > 0$ be a prime number. Let $X$ be a scheme in characteristic $p$. Then the absolute frobenius $F_X : X \\to X$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions."}
+{"_id": "11049", "title": "varieties-lemma-relative-frobenius-endomorphism-identity", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $f : X \\to Y$ be a morphism of schemes over $S$ . Then the diagram $$ \\xymatrix{ X \\ar[d]_f \\ar[r]_{F_{X/S}} & X^{(p)} \\ar[d]^{f^{(p)}} \\\\ Y \\ar[r]^{F_{Y/S}} & Y^{(p)} } $$ commutes."}
+{"_id": "11050", "title": "varieties-lemma-relative-frobenius", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then the relative frobenius $F_{X/S} : X \\to X^{(p)}$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions."}
+{"_id": "11051", "title": "varieties-lemma-relative-frobenius-omega", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then $\\Omega_{X/S} = \\Omega_{X/X^{(p)}}$."}
+{"_id": "11052", "title": "varieties-lemma-relative-frobenius-finite", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. If $X \\to S$ is locally of finite type, then $F_{X/S}$ is finite."}
+{"_id": "11053", "title": "varieties-lemma-geometrically-reduced-p", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Then $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced."}
+{"_id": "11054", "title": "varieties-lemma-inseparable-deg-p-smooth", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a variety over $k$. The following are equivalent \\begin{enumerate} \\item $X^{(p)}$ is reduced, \\item $X$ is geometrically reduced, \\item there is a nonempty open $U \\subset X$ smooth over $k$. \\end{enumerate} In this case $X^{(p)}$ is a variety over $k$ and $F_{X/k} : X \\to X^{(p)}$ is a finite dominant morphism of degree $p^{\\dim(X)}$."}
+{"_id": "11056", "title": "varieties-lemma-glue-separated", "text": "In Situation \\ref{situation-glue} assume $A$ is a Noetherian ring of dimension $1$. The following are equivalent \\begin{enumerate} \\item $A \\otimes B \\to K$ is not surjective, \\item there exists a discrete valuation ring $\\mathcal{O} \\subset K$ containing both $A$ and $B$. \\end{enumerate}"}
+{"_id": "11057", "title": "varieties-lemma-semi-local", "text": "In Situation \\ref{situation-glue} assume \\begin{enumerate} \\item $A$ is a Noetherian semi-local domain of dimension $1$, \\item $B$ is a discrete valuation ring, \\end{enumerate} Then we have the following two possibilities \\begin{enumerate} \\item[(a)] If $A^*$ is not contained in $R$, then $\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$ are open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$. \\item[(b)] If $A^*$ is contained in $R$, then $B$ dominates one of the local rings of $A$ at a maximal ideal and $A \\otimes B \\to K$ is not surjective. \\end{enumerate}"}
+{"_id": "11058", "title": "varieties-lemma-semi-local-dimension-one-conductor", "text": "Let $B$ be a semi-local Noetherian domain of dimension $1$. Let $B'$ be the integral closure of $B$ in its fraction field. Then $B'$ is a semi-local Dedekind domain. Let $x$ be a nonzero element of the Jacobson radical of $B'$. Then for every $y \\in B'$ there exists an $n$ such that $x^n y \\in B$."}
+{"_id": "11059", "title": "varieties-lemma-semi-local-both-side", "text": "In Situation \\ref{situation-glue} assume \\begin{enumerate} \\item $A$ is a Noetherian semi-local domain of dimension $1$, \\item $B$ is a Noetherian semi-local domain of dimension $1$, \\item $A \\otimes B \\to K$ is surjective. \\end{enumerate} Then $\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$ are open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$."}
+{"_id": "11060", "title": "varieties-lemma-glue-a-bunch-of-local-rings", "text": "Let $K$ be a field. Let $A_1, \\ldots, A_r \\subset K$ be Noetherian semi-local rings of dimension $1$ with fraction field $K$. If $A_i \\otimes A_j \\to K$ is surjective for all $i \\not = j$, then there exists a Noetherian semi-local domain $A \\subset K$ of dimension $1$ contained in $A_1, \\ldots, A_r$ such that \\begin{enumerate} \\item $A \\to A_i$ induces an open immersion $j_i : \\Spec(A_i) \\to \\Spec(A)$, \\item $\\Spec(A)$ is the union of the opens $j_i(\\Spec(A_i))$, \\item each closed point of $\\Spec(A)$ lies in exactly one of these opens. \\end{enumerate}"}
+{"_id": "11061", "title": "varieties-lemma-create-globally-generated", "text": "Let $A$ be a domain with fraction field $K$. Let $B_1, \\ldots, B_r \\subset K$ be Noetherian $1$-dimensional semi-local rings whose fraction fields are $K$. If $A \\otimes B_i \\to K$ are surjective for $i = 1, \\ldots, r$, then there exists an $x \\in A$ such that $x^{-1}$ is in the Jacobson radical of $B_i$ for $i = 1, \\ldots, r$."}
+{"_id": "11062", "title": "varieties-lemma-power-equal", "text": "Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \\prod A_\\mathfrak p$ where the product is over the minimal primes of $A$. Let $a_1, a_2 \\in \\mathfrak m_A$ map to the same element of $L$. Then $a_1^n = a_2^n$ for some $n > 0$."}
+{"_id": "11063", "title": "varieties-lemma-power-works", "text": "Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \\prod A_\\mathfrak p$ and $I = \\bigcap \\mathfrak p$ where the product and intersection are over the minimal primes of $A$. Let $f \\in L$ be an element of the form $f = i + a$ where $a \\in \\mathfrak m_A$ and $i \\in IL$. Then some power of $f$ is in the image of $A \\to L$."}
+{"_id": "11066", "title": "varieties-lemma-affine", "text": "Let $X$ be a scheme all of whose local rings are Noetherian of dimension $\\leq 1$. Let $U \\subset X$ be a retrocompact open. Denote $j : U \\to X$ the inclusion morphism. Then $R^pj_*\\mathcal{F} = 0$, $p > 0$ for every quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}$."}
+{"_id": "11069", "title": "varieties-lemma-find-globally-generated", "text": "Let $X$ be an integral separated scheme. Let $U \\subset X$ be a nonempty affine open such that $X \\setminus U$ is a finite set of points $x_1, \\ldots, x_r$ with $\\mathcal{O}_{X, x_i}$ Noetherian of dimension $1$. Then there exists a globally generated invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and a section $s$ such that $U = X_s$."}
+{"_id": "11070", "title": "varieties-lemma-enough-globally-generated-ample", "text": "Let $X$ be a quasi-compact scheme. If for every $x \\in X$ there exists a pair $(\\mathcal{L}, s)$ consisting of a globally generated invertible sheaf $\\mathcal{L}$ and a global section $s$ such that $x \\in X_s$ and $X_s$ is affine, then $X$ has an ample invertible sheaf."}
+{"_id": "11071", "title": "varieties-lemma-dim-1-noetherian-integral-separated-has-ample", "text": "Let $X$ be a Noetherian integral separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf."}
+{"_id": "11072", "title": "varieties-lemma-surjective-pic-birational-finite", "text": "Let $f : X \\to Y$ be a finite morphism of schemes. Assume there exists an open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is an isomorphism and $Y \\setminus V$ is a discrete space. Then every invertible $\\mathcal{O}_X$-module is the pullback of an invertible $\\mathcal{O}_Y$-module."}
+{"_id": "11073", "title": "varieties-lemma-glue-invertible-sheaves", "text": "Let $X$ be a scheme. Let $Z_1, \\ldots, Z_n \\subset X$ be closed subschemes. Let $\\mathcal{L}_i$ be an invertible sheaf on $Z_i$. Assume that \\begin{enumerate} \\item $X$ is reduced, \\item $X = \\bigcup Z_i$ set theoretically, and \\item $Z_i \\cap Z_j$ is a discrete topological space for $i \\not = j$. \\end{enumerate} Then there exists an invertible sheaf $\\mathcal{L}$ on $X$ whose restriction to $Z_i$ is $\\mathcal{L}_i$. Moreover, if we are given sections $s_i \\in \\Gamma(Z_i, \\mathcal{L}_i)$ which are nonvanishing at the points of $Z_i \\cap Z_j$, then we can choose $\\mathcal{L}$ such that there exists a $s \\in \\Gamma(X, \\mathcal{L})$ with $s|_{Z_i} = s_i$ for all $i$."}
+{"_id": "11074", "title": "varieties-lemma-dim-1-noetherian-reduced-separated-has-ample", "text": "Let $X$ be a Noetherian reduced separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf."}
+{"_id": "11075", "title": "varieties-lemma-lift-line-bundle-from-reduction-dimension-1", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. If the underlying topological space of $X$ is Noetherian and $\\dim(X) \\leq 1$, then $\\Pic(X) \\to \\Pic(Z)$ is surjective."}
+{"_id": "11077", "title": "varieties-lemma-pre-pre-delta-invariant", "text": "Let $(A, \\mathfrak m)$ be a Noetherian $1$-dimensional local ring. Let $f \\in \\mathfrak m$. The following are equivalent \\begin{enumerate} \\item $\\mathfrak m = \\sqrt{(f)}$, \\item $f$ is not contained in any minimal prime of $A$, and \\item $A_f = \\prod_{\\mathfrak p\\text{ minimal}} A_\\mathfrak p$ as $A$-algebras. \\end{enumerate} Such an $f \\in \\mathfrak m$ exists. If $\\text{depth}(A) = 1$ (for example $A$ is reduced), then (1) -- (3) are also equivalent to \\begin{enumerate} \\item[(4)] $f$ is a nonzerodivisor, \\item[(5)] $A_f$ is the total ring of fractions of $A$. \\end{enumerate} If $A$ is reduced, then (1) -- (5) are also equivalent to \\begin{enumerate} \\item[(6)] $A_f$ is the product of the residue fields at the minimal primes of $A$. \\end{enumerate}"}
+{"_id": "11078", "title": "varieties-lemma-pre-delta-invariant", "text": "Let $(A, \\mathfrak m)$ be a reduced Nagata $1$-dimensional local ring. Let $A'$ be the integral closure of $A$ in the total ring of fractions of $A$. Then $A'$ is a normal Nagata ring, $A \\to A'$ is finite, and $A'/A$ has finite length as an $A$-module."}
+{"_id": "11079", "title": "varieties-lemma-delta-invariant-is-zero", "text": "Let $A$ be a reduced Nagata local ring of dimension $1$. The $\\delta$-invariant of $A$ is $0$ if and only if $A$ is a discrete valuation ring."}
+{"_id": "11080", "title": "varieties-lemma-normalization-same-after-completion", "text": "Let $A$ be a reduced Nagata local ring of dimension $1$. Let $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}. Let $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$ be the henselization, strict henselization, reps.\\ completion of $A$. Then $A^h$, $A^{sh}$, resp. $A^\\wedge$ is a reduced Nagata local ring of dimension $1$ and $A' \\otimes_A A^h$, $A' \\otimes_A A^{sh}$, resp. $A' \\otimes_A A^\\wedge$ is the integral closure of $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$ in its total ring of fractions."}
+{"_id": "11081", "title": "varieties-lemma-delta-same-after-completion", "text": "Let $A$ be a reduced Nagata local ring of dimension $1$. The $\\delta$-invariant of $A$ is the same as the $\\delta$-invariant of the henselization, strict henselization, or the completion of $A$."}
+{"_id": "11082", "title": "varieties-lemma-delta-invariant-and-change-of-fields", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be a field extension and set $Y = X_K$. Let $y \\in Y$ with image $x \\in X$. Assume $X$ is geometrically reduced at $x$ and $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) = 1$. Then $$ \\delta\\text{-invariant of }X\\text{ at }x \\leq \\delta\\text{-invariant of }Y\\text{ at }y $$"}
+{"_id": "11083", "title": "varieties-lemma-delta-invariant-and-change-of-fields-better", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be a field extension and set $Y = X_K$. Let $y \\in Y$ with image $x \\in X$. Assume assumptions (a), (b), (c) of Lemma \\ref{lemma-geometrically-normal-in-codim-1} hold for $x \\in X$ and that $\\dim(\\mathcal{O}_{Y, y}) = 1$. Then the $\\delta$-invariant of $X$ at $x$ is $\\delta$-invariant of $Y$ at $y$."}
+{"_id": "11084", "title": "varieties-lemma-number-of-branches", "text": "Let $X$ be a scheme. Assume every quasi-compact open of $X$ has finitely many irreducible components. Let $\\nu : X^\\nu \\to X$ be the normalization of $X$. Let $x \\in X$. \\begin{enumerate} \\item The number of branches of $X$ at $x$ is the number of inverse images of $x$ in $X^\\nu$. \\item The number of geometric branches of $X$ at $x$ is $\\sum_{\\nu(x^\\nu) = x} [\\kappa(x^\\nu) : \\kappa(x)]_s$. \\end{enumerate}"}
+{"_id": "11085", "title": "varieties-lemma-geometric-branches-and-change-of-fields", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \\in X_K$ be a point with image $x$ in $X$. Then the number of geometric branches of $X$ at $x$ is the number of geometric branches of $X_K$ at $y$."}
+{"_id": "11086", "title": "varieties-lemma-geometrically-unibranch-and-change-of-fields", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \\in X_K$ be a point with image $x$ in $X$. Then $X$ is geometrically unibranch at $x$ if and only if $X_K$ is geometrically unibranch at $y$."}
+{"_id": "11087", "title": "varieties-lemma-delta-number-branches-inequality-sh", "text": "Let $(A, \\mathfrak m)$ be a strictly henselian $1$-dimensional reduced Nagata local ring. Then $$ \\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1 $$ If equality holds, then $A$ is a wedge of $n \\geq 1$ strictly henselian discrete valuation rings."}
+{"_id": "11088", "title": "varieties-lemma-delta-number-branches-inequality", "text": "Let $(A, \\mathfrak m)$ be a $1$-dimensional reduced Nagata local ring. Then $$ \\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1 $$"}
+{"_id": "11089", "title": "varieties-lemma-normalize-noetherian-dim-1", "text": "Let $X$ be a locally Noetherian scheme of dimension $1$. Let $\\nu : X^\\nu \\to X$ be the normalization. Then \\begin{enumerate} \\item $\\nu$ is integral, surjective, and induces a bijection on irreducible components, \\item there is a factorization $X^\\nu \\to X_{red} \\to X$ and the morphism $X^\\nu \\to X_{red}$ is the normalization of $X_{red}$, \\item $X^\\nu \\to X_{red}$ is birational, \\item for every closed point $x \\in X$ the stalk $(\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure of $\\mathcal{O}_{X, x}$ in the total ring of fractions of $(\\mathcal{O}_{X, x})_{red} = \\mathcal{O}_{X_{red}, x}$, \\item the fibres of $\\nu$ are finite and the residue field extensions are finite, \\item $X^\\nu$ is a disjoint union of integral normal Noetherian schemes and each affine open is the spectrum of a finite product of Dedekind domains. \\end{enumerate}"}
+{"_id": "11090", "title": "varieties-lemma-prepare-delta-invariant", "text": "Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\\nu : X^\\nu \\to X$ be the normalization. Let $x \\in X$ denote a closed point. Then \\begin{enumerate} \\item $\\nu : X^\\nu \\to X$ is finite, surjective, and birational, \\item $\\mathcal{O}_X \\subset \\nu_*\\mathcal{O}_{X^\\nu}$ and $\\nu_*\\mathcal{O}_{X^\\nu}/\\mathcal{O}_X$ is a direct sum of skyscraper sheaves $\\mathcal{Q}_x$ in the singular points $x$ of $X$, \\item $A' = (\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure of $A = \\mathcal{O}_{X, x}$ in its total ring of fractions, \\item $\\mathcal{Q}_x = A'/A$ has finite length equal to the $\\delta$-invariant of $X$ at $x$, \\item $A'$ is a semi-local ring which is a finite product of Dedekind domains, \\item $A^\\wedge$ is a reduced Noetherian complete local ring of dimension $1$, \\item $(A')^\\wedge$ is the integral closure of $A^\\wedge$ in its total ring of fractions, \\item $(A')^\\wedge$ is a finite product of complete discrete valuation rings, and \\item $A'/A \\cong (A')^\\wedge/A^\\wedge$. \\end{enumerate}"}
+{"_id": "11091", "title": "varieties-lemma-characterize-open-immersion", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set of generic points of irreducible components of $X$. If \\begin{enumerate} \\item $f$ is separated, \\item there is an open covering $X = \\bigcup U_i$ such that $f|_{U_i} : U_i \\to Y$ is an open immersion, and \\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$, \\end{enumerate} then $f$ is an open immersion."}
+{"_id": "11092", "title": "varieties-lemma-local-isomorphism", "text": "Let $X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. If \\begin{enumerate} \\item $\\mathcal{O}_{X, x} = \\mathcal{O}_{S, s}$, \\item $X$ is reduced, \\item $X \\to S$ is of finite type, and \\item $S$ has finitely many irreducible components, \\end{enumerate} then there exists an open neighbourhood $U$ of $x$ such that $f|_U$ is an open immersion."}
+{"_id": "11093", "title": "varieties-lemma-points-in-affine", "text": "Let $f : T \\to X$ be a morphism of schemes. Let $X^0$, resp.\\ $T^0$ denote the sets of generic points of irreducible components. Let $t_1, \\ldots, t_m \\in T$ be a finite set of points with images $x_j = f(t_j)$. If \\begin{enumerate} \\item $T$ is affine, \\item $X$ is quasi-separated, \\item $X^0$ is finite \\item $f(T^0) \\subset X^0$ and $f : T^0 \\to X^0$ is injective, and \\item $\\mathcal{O}_{X, x_j} = \\mathcal{O}_{T, t_j}$, \\end{enumerate} then there exists an affine open of $X$ containing $x_1, \\ldots, x_r$."}
+{"_id": "11094", "title": "varieties-lemma-finite-set-codim-1-points-in-affine", "text": "Let $X$ be an integral separated scheme. Let $x_1, \\ldots, x_r \\in X$ be a finite set of points such that $\\mathcal{O}_{X, x_i}$ is Noetherian of dimension $\\leq 1$. Then there exists an affine open subscheme of $X$ containing all of $x_1, \\ldots, x_r$."}
+{"_id": "11095", "title": "varieties-lemma-extra-silly", "text": "Let $A$ be a ring, $I \\subset A$ an ideal, $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ primes of $A$, and $\\overline{f} \\in A/I$ an element. If $I \\not \\subset \\mathfrak p_i$ for all $i$, then there exists an $f \\in A$, $f \\not \\in \\mathfrak p_i$ which maps to $\\overline{f}$ in $A/I$."}
+{"_id": "11096", "title": "varieties-lemma-finite-set-codim-1-points-in-affine-per-component", "text": "Let $X$ be a scheme. Let $T \\subset X$ be finite set of points. Assume \\begin{enumerate} \\item $X$ has finitely many irreducible components $Z_1, \\ldots, Z_t$, and \\item $Z_i \\cap T$ is contained in an affine open of the reduced induced subscheme corresponding to $Z_i$. \\end{enumerate} Then there exists an affine open subscheme of $X$ containing $T$."}
+{"_id": "11097", "title": "varieties-lemma-proper-minus-point", "text": "Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$. Let $x \\in X$ be a closed point. The open subscheme $X \\setminus \\{x\\}$ is not proper over $k$."}
+{"_id": "11098", "title": "varieties-lemma-dim-1-quasi-projective", "text": "Let $X$ be a separated finite type scheme over a field $k$. If $\\dim(X) \\leq 1$ then $X$ is H-quasi-projective over $k$."}
+{"_id": "11099", "title": "varieties-lemma-dim-1-proper-projective", "text": "Let $X$ be a proper scheme over a field $k$. If $\\dim(X) \\leq 1$ then $X$ is H-projective over $k$."}
+{"_id": "11100", "title": "varieties-lemma-dim-1-projective-completion", "text": "Let $X$ be a separated scheme of finite type over $k$. If $\\dim(X) \\leq 1$, then there exists an open immersion $j : X \\to \\overline{X}$ with the following properties \\begin{enumerate} \\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$ is a closed subscheme of $\\mathbf{P}^d_k$ for some $d$, \\item $j(X) \\subset \\overline{X}$ is dense and scheme theoretically dense, \\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$ for some closed points $x_i \\in \\overline{X}$. \\end{enumerate}"}
+{"_id": "11101", "title": "varieties-lemma-reduced-dim-1-projective-completion", "text": "Let $X$ be a separated scheme of finite type over $k$. If $X$ is reduced and $\\dim(X) \\leq 1$, then there exists an open immersion $j : X \\to \\overline{X}$ such that \\begin{enumerate} \\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$ is a closed subscheme of $\\mathbf{P}^d_k$ for some $d$, \\item $j(X) \\subset \\overline{X}$ is dense and scheme theoretically dense, \\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$ for some closed points $x_i \\in \\overline{X}$, \\item the local rings $\\mathcal{O}_{\\overline{X}, x_i}$ are discrete valuation rings for $i = 1, \\ldots, n$. \\end{enumerate}"}
+{"_id": "11102", "title": "varieties-lemma-curve-affine-projective", "text": "Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$."}
+{"_id": "11104", "title": "varieties-lemma-degree-base-change", "text": "Let $k \\subset k'$ be an extension of fields. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of constant rank $n$. Then the degree of $\\mathcal{E}/X/k$ is equal to the degree of $\\mathcal{E}_{k'}/X_{k'}/k'$."}
+{"_id": "11105", "title": "varieties-lemma-degree-additive", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $0 \\to \\mathcal{E}_1 \\to \\mathcal{E}_2 \\to \\mathcal{E}_3 \\to 0$ be a short exact sequence of locally free $\\mathcal{O}_X$-modules each of finite constant rank. Then $$ \\deg(\\mathcal{E}_2) = \\deg(\\mathcal{E}_1) + \\deg(\\mathcal{E}_3) $$"}
+{"_id": "11106", "title": "varieties-lemma-degree-birational-pullback", "text": "Let $k$ be a field. Let $f : X' \\to X$ be a birational morphism of proper schemes of dimension $\\leq 1$ over $k$. Then $$ \\deg(f^*\\mathcal{E}) = \\deg(\\mathcal{E}) $$ for every finite locally free sheaf of constant rank. More generally it suffices if $f$ induces a bijection between irreducible components of dimension $1$ and isomorphisms of local rings at the corresponding generic points."}
+{"_id": "11107", "title": "varieties-lemma-degree-on-proper-curve", "text": "Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point $\\xi$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$ and let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $$ \\chi(X, \\mathcal{E} \\otimes \\mathcal{F}) = r \\deg(\\mathcal{E}) + n \\chi(X, \\mathcal{F}) $$ where $r = \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi$ is the rank of $\\mathcal{F}$."}
+{"_id": "11108", "title": "varieties-lemma-degree-in-terms-of-components", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$. Then $$ \\deg(\\mathcal{E}) = \\sum m_i \\deg(\\mathcal{E}|_{C_i}) $$ where $C_i \\subset X$, $i = 1, \\ldots, t$ are the irreducible components of dimension $1$ with reduced induced scheme structure and $m_i$ is the multiplicity of $C_i$ in $X$."}
+{"_id": "11109", "title": "varieties-lemma-degree-tensor-product", "text": "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$ over $k$, and let $\\mathcal{E}$, $\\mathcal{V}$ be locally free $\\mathcal{O}_X$-modules of constant finite rank. Then $$ \\deg(\\mathcal{E} \\otimes \\mathcal{V}) = \\text{rank}(\\mathcal{E}) \\deg(\\mathcal{V}) + \\text{rank}(\\mathcal{V}) \\deg(\\mathcal{E}) $$"}
+{"_id": "11111", "title": "varieties-lemma-degree-effective-Cartier-divisor", "text": "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $D$ be an effective Cartier divisor on $X$. Then $D$ is finite over $\\Spec(k)$ of degree $\\deg(D) = \\dim_k \\Gamma(D, \\mathcal{O}_D)$. For a locally free sheaf $\\mathcal{E}$ of rank $n$ we have $$ \\deg(\\mathcal{E}(D)) = n\\deg(D) + \\deg(\\mathcal{E}) $$ where $\\mathcal{E}(D) = \\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D)$."}
+{"_id": "11112", "title": "varieties-lemma-divisible", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced and connected. Let $\\kappa = H^0(X, \\mathcal{O}_X)$. Then $\\kappa/k$ is a finite extension of fields and $w = [\\kappa : k]$ divides \\begin{enumerate} \\item $\\deg(\\mathcal{E})$ for all locally free $\\mathcal{O}_X$-modules $\\mathcal{E}$, \\item $[\\kappa(x) : k]$ for all closed points $x \\in X$, and \\item $\\deg(D)$ for all closed subschemes $D \\subset X$ of dimension zero. \\end{enumerate}"}
+{"_id": "11113", "title": "varieties-lemma-degree-pullback-map-proper-curves", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism of proper curves over $k$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_Y$-module. Then $$ \\deg(f^*\\mathcal{E}) = \\deg(X/Y) \\deg(\\mathcal{E}) $$"}
+{"_id": "11114", "title": "varieties-lemma-check-invertible-sheaf-trivial", "text": "Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $\\mathcal{L}$ has a nonzero section, then $\\deg(\\mathcal{L}) \\geq 0$. \\item If $\\mathcal{L}$ has a nonzero section $s$ which vanishes at a point, then $\\deg(\\mathcal{L}) > 0$. \\item If $\\mathcal{L}$ and $\\mathcal{L}^{-1}$ have nonzero sections, then $\\mathcal{L} \\cong \\mathcal{O}_X$. \\item If $\\deg(\\mathcal{L}) \\leq 0$ and $\\mathcal{L}$ has a nonzero section, then $\\mathcal{L} \\cong \\mathcal{O}_X$. \\item If $\\mathcal{N} \\to \\mathcal{L}$ is a nonzero map of invertible $\\mathcal{O}_X$-modules, then $\\deg(\\mathcal{L}) \\geq \\deg(\\mathcal{N})$ and if equality holds then it is an isomorphism. \\end{enumerate}"}
+{"_id": "11115", "title": "varieties-lemma-no-sections-dual-nef", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and equidimensional of dimension $1$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $\\deg(\\mathcal{L}|_C) \\leq 0$ for all irreducible components $C$ of $X$, then either $H^0(X, \\mathcal{L}) = 0$ or $\\mathcal{L} \\cong \\mathcal{O}_X$."}
+{"_id": "11116", "title": "varieties-lemma-ample-curve", "text": "Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $\\mathcal{L}$ is ample if and only if $\\deg(\\mathcal{L}) > 0$."}
+{"_id": "11117", "title": "varieties-lemma-ampleness-in-terms-of-degrees-components", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $C_i \\subset X$, $i = 1, \\ldots, t$ be the irreducible components of dimension $1$. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{L}$ is ample, and \\item $\\deg(\\mathcal{L}|_{C_i}) > 0$ for $i = 1, \\ldots, t$. \\end{enumerate}"}
+{"_id": "11118", "title": "varieties-lemma-regular-point-on-curve", "text": "Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \\in X$ be a closed point. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf of ideals $\\mathcal{I}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{O}_{X, x}$ is regular, \\item $\\mathcal{O}_{X, x}$ is normal, \\item $\\mathcal{O}_{X, x}$ is a discrete valuation ring, \\item $\\mathcal{I}$ is an invertible $\\mathcal{O}_X$-module, \\item $x$ is an effective Cartier divisor on $X$. \\end{enumerate} If $k$ is perfect, these are also equivalent to \\begin{enumerate} \\item[(6)] $X \\to \\Spec(k)$ is smooth at $x$. \\end{enumerate}"}
+{"_id": "11119", "title": "varieties-lemma-general-degree-g-line-bundle", "text": "Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Then there exist \\begin{enumerate} \\item an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ with $\\dim_k H^0(X, \\mathcal{L}) = 1$ and $H^1(X, \\mathcal{L}) = 0$, and \\item an invertible $\\mathcal{O}_X$-module $\\mathcal{N}$ with $\\dim_k H^0(X, \\mathcal{N}) = 0$ and $H^1(X, \\mathcal{N}) = 0$. \\end{enumerate}"}
+{"_id": "11120", "title": "varieties-lemma-vanishing-degree-2g-and-1-line-bundle", "text": "Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Set $g = \\dim_k H^1(X, \\mathcal{O}_X)$. For every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ with $\\deg(\\mathcal{L}) \\geq 2g - 1$ we have $H^1(X, \\mathcal{L}) = 0$."}
+{"_id": "11121", "title": "varieties-lemma-numerical-polynomial-from-euler", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules. The map $$ (n_1, \\ldots, n_r) \\longmapsto \\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}) $$ is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at most the dimension of the support of $\\mathcal{F}$."}
+{"_id": "11122", "title": "varieties-lemma-numerical-polynomial-leading-term", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules. Let $d = \\dim(\\text{Supp}(\\mathcal{F}))$. Let $Z_i \\subset X$ be the irreducible components of $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\xi_i \\in Z_i$ be the generic point and set $m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{F}_{\\xi_i})$. Then $$ \\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}) - \\sum\\nolimits_i m_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}|_{Z_i}) $$ is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$."}
+{"_id": "11123", "title": "varieties-lemma-intersection-number-integer", "text": "In the situation of Definition \\ref{definition-intersection-number} the intersection number $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is an integer."}
+{"_id": "11124", "title": "varieties-lemma-intersection-number-additive", "text": "In the situation of Definition \\ref{definition-intersection-number} the intersection number $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$, then we have $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) = (\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) + (\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z) $$"}
+{"_id": "11125", "title": "varieties-lemma-intersection-number-in-terms-of-components", "text": "In the situation of Definition \\ref{definition-intersection-number} let $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let $m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{O}_{Z, \\xi_i})$ where $\\xi_i \\in Z_i$ is the generic point. Then $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) = \\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i) $$"}
+{"_id": "11126", "title": "varieties-lemma-intersection-number-and-pullback", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over $k$. Let $Z \\subset Y$ be an integral closed subscheme of dimension $d$ and let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. Then $$ (f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) = \\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z)) $$ where $\\deg(Z \\to f(Z))$ is as in Morphisms, Definition \\ref{morphisms-definition-degree} or $0$ if $\\dim(f(Z)) < d$."}
+{"_id": "11127", "title": "varieties-lemma-numerical-intersection-effective-Cartier-divisor", "text": "Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be a closed subscheme of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. Assume there exists an effective Cartier divisor $D \\subset Z$ such that $\\mathcal{L}_1|_Z \\cong \\mathcal{O}_Z(D)$. Then $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) = (\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D) $$"}
+{"_id": "11128", "title": "varieties-lemma-ample-positive", "text": "Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be a closed subscheme of dimension $d$. If $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ are ample, then $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is positive."}
+{"_id": "11129", "title": "varieties-lemma-degree-finite-morphism-in-terms-degrees", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a finite dominant morphism of proper varieties over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Then $$ \\deg_{f^*\\mathcal{L}}(Y) = \\deg(f) \\deg_\\mathcal{L}(X) $$ where $\\deg(f)$ is as in Morphisms, Definition \\ref{morphisms-definition-degree}."}
+{"_id": "11130", "title": "varieties-lemma-intersection-numbers-and-degrees-on-curves", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \\subset X$ be a closed subscheme of dimension $\\leq 1$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $$ (\\mathcal{L} \\cdot Z) = \\deg(\\mathcal{L}|_Z) $$ where $\\deg(\\mathcal{L}|_Z)$ is as in Definition \\ref{definition-degree-invertible-sheaf}. If $\\mathcal{L}$ is ample, then $\\deg_\\mathcal{L}(Z) = \\deg(\\mathcal{L}|_Z)$."}
+{"_id": "11131", "title": "varieties-lemma-generate-over-complement", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a closed subscheme. Then there exists an integer $n_0$ such that for all $n \\geq n_0$ the kernel $V_n$ of $\\Gamma(X, \\mathcal{L}^{\\otimes n}) \\to \\Gamma(Z, \\mathcal{L}^{\\otimes n}|_Z)$ generates $\\mathcal{L}^{\\otimes n}|_{X \\setminus Z}$ and the canonical morphism $$ X \\setminus Z \\longrightarrow \\mathbf{P}(V_n) $$ is an immersion of schemes over $k$."}
+{"_id": "11132", "title": "varieties-lemma-bertini", "text": "In Situation \\ref{situation-family-divisors} assume \\begin{enumerate} \\item $X$ is smooth over $k$, \\item the image of $\\psi : V \\to \\Gamma(X, \\mathcal{L})$ generates $\\mathcal{L}$, \\item the corresponding morphism $\\varphi_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(V)$ is an immersion. \\end{enumerate} Then for general $v \\in V \\otimes_k k'$ the scheme $H_v$ is smooth over $k'$."}
+{"_id": "11133", "title": "varieties-lemma-vanishin-h0-negative", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. If $\\text{Ass}(\\mathcal{F})$ does not contain any closed points, then $\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$ for $n \\ll 0$."}
+{"_id": "11134", "title": "varieties-lemma-vanishin-h1-negative", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume that for $x \\in X$ closed we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. Then $H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes m}) = 0$ for $m \\ll 0$."}
+{"_id": "11135", "title": "varieties-lemma-connectedness-ample-divisor", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Assume \\begin{enumerate} \\item $s$ is a regular section (Divisors, Definition \\ref{divisors-definition-regular-section}), \\item for every closed point $x \\in X$ we have $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, and \\item $X$ is connected. \\end{enumerate} Then the zero scheme $Z(s)$ of $s$ is connected."}
+{"_id": "11136", "title": "varieties-proposition-units-general", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Assume that $X$ is locally of finite type over $k$, connected, reduced, and has finitely many irreducible components. Then $\\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group if in addition to the conditions above at least one of the following conditions is satisfied: \\begin{enumerate} \\item the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$ is $k$, \\item $X$ has a $k$-rational point, or \\item $X$ is geometrically integral. \\end{enumerate}"}
+{"_id": "11137", "title": "varieties-proposition-unique-base-field", "text": "Let $X$ be a scheme. Let $a : X \\to \\Spec(k_1)$ and $b : X \\to \\Spec(k_2)$ be morphisms from $X$ to spectra of fields. Assume $a, b$ are locally of finite type, and $X$ is reduced, and connected. Then we have $k_1' = k_2'$, where $k_i' \\subset \\Gamma(X, \\mathcal{O}_X)$ is the integral closure of $k_i$ in $\\Gamma(X, \\mathcal{O}_X)$."}
+{"_id": "11138", "title": "varieties-proposition-dim-1-noetherian-separated-has-ample", "text": "Let $X$ be a Noetherian separated scheme of dimension $1$. Then $X$ has an ample invertible sheaf."}
+{"_id": "11139", "title": "varieties-proposition-finite-set-of-points-of-codim-1-in-affine", "text": "Let $X$ be a separated scheme such that every quasi-compact open has a finite number of irreducible components. Let $x_1, \\ldots, x_r \\in X$ be points such that $\\mathcal{O}_{X, x_i}$ is Noetherian of dimension $\\leq 1$. Then there exists an affine open subscheme of $X$ containing all of $x_1, \\ldots, x_r$."}
+{"_id": "11140", "title": "varieties-proposition-asymptotic-riemann-roch", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $d$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Then $$ \\dim_k \\Gamma(X, \\mathcal{L}^{\\otimes n}) \\sim c n^d + l.o.t. $$ where $c = \\deg_\\mathcal{L}(X)/d!$ is a positive constant."}
+{"_id": "11171", "title": "cotangent-theorem-quillen-spectral-sequence", "text": "Let $A \\to B$ be a surjective ring map. Consider the sheaf $\\Omega = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$ of $\\underline{B}$-modules on $\\mathcal{C}_{B/A}$, see Section \\ref{section-compute-L-pi-shriek}. Then there is a spectral sequence with $E_1$-page $$ E_1^{p, q} = H_{- p - q}(\\mathcal{C}_{B/A}, \\text{Sym}^p_{\\underline{B}}(\\Omega)) \\Rightarrow \\text{Tor}^A_{- p - q}(B, B) $$ with $d_r$ of bidegree $(r, -r + 1)$. Moreover, $H_i(\\mathcal{C}_{B/A}, \\text{Sym}^k_{\\underline{B}}(\\Omega)) = 0$ for $i < k$."}
+{"_id": "11172", "title": "cotangent-lemma-colimit-cotangent-complex", "text": "Let $A_i \\to B_i$ be a system of ring maps over a directed index set $I$. Then $\\colim L_{B_i/A_i} = L_{\\colim B_i/\\colim A_i}$."}
+{"_id": "11173", "title": "cotangent-lemma-identify-pi-shriek", "text": "With notation as above let $P_\\bullet$ be a simplicial $A$-algebra endowed with an augmentation $\\epsilon : P_\\bullet \\to B$. Assume each $P_n$ is a polynomial algebra over $A$ and $\\epsilon$ is a trivial Kan fibration on underlying simplicial sets. Then $$ L\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon) $$ in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$."}
+{"_id": "11174", "title": "cotangent-lemma-pi-shriek-standard", "text": "Let $A \\to B$ be a ring map. Let $\\epsilon : P_\\bullet \\to B$ be the standard resolution of $B$ over $A$. Let $\\pi$ be as in (\\ref{equation-pi}). Then $$ L\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon) $$ in $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$."}
+{"_id": "11175", "title": "cotangent-lemma-compute-cotangent-complex", "text": "Let $A \\to B$ be a ring map.  Let $\\pi$ and $i$ be as in (\\ref{equation-pi}). There is a canonical isomorphism $$ L_{B/A} = L\\pi_!(Li^*\\Omega_{\\mathcal{O}/A}) = L\\pi_!(i^*\\Omega_{\\mathcal{O}/A}) = L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}) $$ in $D(B)$."}
+{"_id": "11176", "title": "cotangent-lemma-pi-lower-shriek-constant-sheaf", "text": "If $A \\to B$ is a ring map, then $L\\pi_!(\\pi^{-1}M) = M$ with $\\pi$ as in (\\ref{equation-pi})."}
+{"_id": "11177", "title": "cotangent-lemma-identify-H0", "text": "If $A \\to B$ is a ring map, then $H^0(L_{B/A}) = \\Omega_{B/A}$."}
+{"_id": "11178", "title": "cotangent-lemma-pi-lower-shriek-polynomial-algebra", "text": "If $B$ is a polynomial algebra over the ring $A$, then with $\\pi$ as in (\\ref{equation-pi}) we have that $\\pi_!$ is exact and $\\pi_!\\mathcal{F} = \\mathcal{F}(B \\to B)$."}
+{"_id": "11179", "title": "cotangent-lemma-cotangent-complex-polynomial-algebra", "text": "If $B$ is a polynomial algebra over the ring $A$, then $L_{B/A}$ is quasi-isomorphic to $\\Omega_{B/A}[0]$."}
+{"_id": "11180", "title": "cotangent-lemma-polynomial", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map. Let $\\mathcal{A}$ be the category of $A$-algebra maps $C \\to B$. Let $n \\geq 0$ and let $P_\\bullet$ be a simplicial object of $\\mathcal{A}$ such that \\begin{enumerate} \\item $P_\\bullet \\to B$ is a trivial Kan fibration of simplicial sets, \\item $P_k$ is finite type over $A$ for $k \\leq n$, \\item $P_\\bullet = \\text{cosk}_n \\text{sk}_n P_\\bullet$ as simplicial objects of $\\mathcal{A}$. \\end{enumerate} Then $P_{n + 1}$ is a finite type $A$-algebra."}
+{"_id": "11181", "title": "cotangent-lemma-pi-shriek-finite", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map. Let $\\pi$, $\\underline{B}$ be as in (\\ref{equation-pi}). If $\\mathcal{F}$ is an $\\underline{B}$-module such that $\\mathcal{F}(P, \\alpha)$ is a finite $B$-module for all $\\alpha : P = A[x_1, \\ldots, x_n] \\to B$, then the cohomology modules of $L\\pi_!(\\mathcal{F})$ are finite $B$-modules."}
+{"_id": "11182", "title": "cotangent-lemma-cotangent-finite", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map. Then $H^n(L_{B/A})$ is a finite $B$-module for all $n \\in \\mathbf{Z}$."}
+{"_id": "11183", "title": "cotangent-lemma-O-homology-B-homology", "text": "Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$ be as in (\\ref{equation-pi}). For any $\\mathcal{O}$-module $\\mathcal{F}$ we have $$ L\\pi_!(\\mathcal{F}) = L\\pi_!(Li^*\\mathcal{F}) = L\\pi_!(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B}) $$ in $D(\\textit{Ab})$."}
+{"_id": "11184", "title": "cotangent-lemma-apply-O-B-comparison", "text": "Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$ be as in (\\ref{equation-pi}). We have $$ L\\pi_!(\\mathcal{O}) = L\\pi_!(\\underline{B}) = B \\quad\\text{and}\\quad L_{B/A} = L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}) = L\\pi_!(\\Omega_{\\mathcal{O}/A}) $$ in $D(\\textit{Ab})$."}
+{"_id": "11185", "title": "cotangent-lemma-special-case-triangle", "text": "Let $A \\to B \\to C$ be ring maps. If $B$ is a polynomial algebra over $A$, then there is a distinguished triangle  $L_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to L_{B/A} \\otimes_B^\\mathbf{L} C[1]$ in $D(C)$."}
+{"_id": "11186", "title": "cotangent-lemma-flat-base-change", "text": "Assume (\\ref{equation-commutative-square}) induces a quasi-isomorphism $B \\otimes_A^\\mathbf{L} A' = B'$. Then, with notation as in (\\ref{equation-double-square}) and $\\mathcal{F}' \\in \\textit{Ab}(\\mathcal{C}')$, we have $L\\pi_!(g^{-1}\\mathcal{F}') = L\\pi'_!(\\mathcal{F}')$."}
+{"_id": "11187", "title": "cotangent-lemma-flat-base-change-cotangent-complex", "text": "If (\\ref{equation-commutative-square}) induces a quasi-isomorphism $B \\otimes_A^\\mathbf{L} A' = B'$, then the functoriality map induces an isomorphism $$ L_{B/A} \\otimes_B^\\mathbf{L} B' \\longrightarrow L_{B'/A'} $$"}
+{"_id": "11188", "title": "cotangent-lemma-cotangent-complex-product", "text": "Let $A \\to B$ and $A \\to C$ be ring maps. Then the map $L_{B \\times C/A} \\to L_{B/A} \\oplus L_{C/A}$ is an isomorphism in $D(B \\times C)$."}
+{"_id": "11189", "title": "cotangent-lemma-triangle-ses", "text": "With notation as in (\\ref{equation-three-maps}) set $$ \\begin{matrix} \\Omega_1 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B} \\text{ on }\\mathcal{C}_{B/A} \\\\ \\Omega_2 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{C} \\text{ on }\\mathcal{C}_{C/A} \\\\ \\Omega_3 = \\Omega_{\\mathcal{O}/B} \\otimes_\\mathcal{O} \\underline{C} \\text{ on }\\mathcal{C}_{C/B} \\end{matrix} $$ Then we have a canonical short exact sequence of sheaves of $\\underline{C}$-modules $$ 0 \\to g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C} \\to g_2^{-1}\\Omega_2 \\to g_3^{-1}\\Omega_3 \\to 0 $$ on $\\mathcal{C}_{C/B/A}$."}
+{"_id": "11190", "title": "cotangent-lemma-polynomial-on-top", "text": "With notation as in (\\ref{equation-three-maps}) suppose that $C$ is a polynomial algebra over $B$. Then $L\\pi_!(g_3^{-1}\\mathcal{F}) = L\\pi_{3, !}\\mathcal{F} = \\pi_{3, !}\\mathcal{F}$ for any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{C/B}$"}
+{"_id": "11191", "title": "cotangent-lemma-triangle-compute-lower-shriek", "text": "With notation as in (\\ref{equation-three-maps}) we have $Lg_{i, !} \\circ g_i^{-1} = \\text{id}$ for $i = 1, 2, 3$ and hence also $L\\pi_! \\circ g_i^{-1} = L\\pi_{i, !}$ for $i = 1, 2, 3$."}
+{"_id": "11192", "title": "cotangent-lemma-localize-at-bottom", "text": "Let $A \\to A' \\to B$ be ring maps such that $B = B \\otimes_A^\\mathbf{L} A'$. Then $L_{B/A} = L_{B/A'}$ in $D(B)$."}
+{"_id": "11193", "title": "cotangent-lemma-derived-diagonal", "text": "Let $A \\to B$ be a ring map such that $B = B \\otimes_A^\\mathbf{L} B$. Then $L_{B/A} = 0$ in $D(B)$."}
+{"_id": "11194", "title": "cotangent-lemma-bootstrap", "text": "Let $A \\to B$ be a ring map such that $\\text{Tor}^A_i(B, B) = 0$ for $i > 0$ and such that $L_{B/B \\otimes_A B} = 0$. Then $L_{B/A} = 0$ in $D(B)$."}
+{"_id": "11195", "title": "cotangent-lemma-when-zero", "text": "The cotangent complex $L_{B/A}$ is zero in each of the following cases: \\begin{enumerate} \\item $A \\to B$ and $B \\otimes_A B \\to B$ are flat, i.e., $A \\to B$ is weakly \\'etale (More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale}), \\item $A \\to B$ is a flat epimorphism of rings, \\item $B = S^{-1}A$ for some multiplicative subset $S \\subset A$, \\item $A \\to B$ is unramified and flat, \\item $A \\to B$ is \\'etale, \\item $A \\to B$ is a filtered colimit of ring maps for which the cotangent complex vanishes, \\item $B$ is a henselization of a local ring of $A$, \\item $B$ is a strict henselization of a local ring of $A$, and \\item add more here. \\end{enumerate}"}
+{"_id": "11196", "title": "cotangent-lemma-localize-on-top", "text": "Let $A \\to B \\to C$ be ring maps such that $L_{C/B} = 0$. Then $L_{C/A} = L_{B/A} \\otimes_B^\\mathbf{L} C$."}
+{"_id": "11197", "title": "cotangent-lemma-localize", "text": "Let $A \\to B$ be ring maps and $S \\subset A$, $T \\subset B$ multiplicative subsets such that $S$ maps into $T$. Then $L_{T^{-1}B/S^{-1}A} = L_{B/A} \\otimes_B T^{-1}B$ in $D(T^{-1}B)$."}
+{"_id": "11198", "title": "cotangent-lemma-cotangent-complex-henselization", "text": "Let $A \\to B$ be a local ring homomorphism of local rings. Let $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced maps of henselizations, resp.\\ strict henselizations. Then $$ L_{B^h/A^h} = L_{B^h/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^h \\quad\\text{resp.}\\quad L_{B^{sh}/A^{sh}} = L_{B^{sh}/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^{sh} $$ in $D(B^h)$, resp.\\ $D(B^{sh})$."}
+{"_id": "11199", "title": "cotangent-lemma-when-projective", "text": "If $A \\to B$ is a smooth ring map, then $L_{B/A} = \\Omega_{B/A}[0]$."}
+{"_id": "11200", "title": "cotangent-lemma-frobenius-homotopy", "text": "Let $A \\to B$ be a ring map with $p = 0$ in $A$. Let $P_\\bullet$ be the standard resolution of $B$ over $A$. The map $P_\\bullet \\to P_\\bullet$ induced by the diagram $$ \\xymatrix{ B \\ar[r]_{F_B} & B \\\\ A \\ar[u] \\ar[r]^{F_A} & A \\ar[u] } $$ discussed in Section \\ref{section-functoriality} is homotopic to the Frobenius endomorphism $P_\\bullet \\to P_\\bullet$ given by Frobenius on each $P_n$."}
+{"_id": "11201", "title": "cotangent-lemma-frobenius-acts-as-zero", "text": "Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism and assume that $p = 0$ in $A$. The map $L_{B/A} \\to L_{B/A}$ of Section \\ref{section-functoriality} induced by the Frobenius maps $F_A$ and $F_B$ is homotopic to zero."}
+{"_id": "11202", "title": "cotangent-lemma-perfect-zero", "text": "Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism and assume that $p = 0$ in $A$. If $A$ and $B$ are perfect, then $L_{B/A}$ is zero in $D(B)$."}
+{"_id": "11203", "title": "cotangent-lemma-surjection", "text": "\\begin{slogan} The cohomology of the cotangent complex of a surjective ring map is trivial in degree zero; it is the kernel modulo its square in degree $-1$. \\end{slogan} Let $A \\to B$ be a surjective ring map with kernel $I$. Then $H^0(L_{B/A}) = 0$ and $H^{-1}(L_{B/A}) = I/I^2$. This isomorphism comes from the map (\\ref{equation-comparison-map}) for the object $(A \\to B)$ of $\\mathcal{C}_{B/A}$."}
+{"_id": "11204", "title": "cotangent-lemma-relation-with-naive-cotangent-complex", "text": "Let $A \\to B$ be a ring map. Then $\\tau_{\\geq -1}L_{B/A}$ is canonically quasi-isomorphic to the naive cotangent complex."}
+{"_id": "11205", "title": "cotangent-lemma-vanishing-symmetric-powers", "text": "Notation and assumptions as in Cohomology on Sites, Example \\ref{sites-cohomology-example-category-to-point}. Assume $\\mathcal{C}$ has a cosimplicial object as in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Let $\\mathcal{F}$ be a flat $\\underline{B}$-module such that $H_0(\\mathcal{C}, \\mathcal{F}) = 0$. Then $H_l(\\mathcal{C}, \\text{Sym}_{\\underline{B}}^k(\\mathcal{F})) = 0$ for $l < k$."}
+{"_id": "11206", "title": "cotangent-lemma-map-tors-zero", "text": "Let $A$ be a ring. Let $P = A[E]$ be a polynomial ring. Set $I = (e; e \\in E) \\subset P$. The maps $\\text{Tor}_i^P(A, I^{n + 1}) \\to \\text{Tor}_i^P(A, I^n)$ are zero for all $i$ and $n$."}
+{"_id": "11207", "title": "cotangent-lemma-polynomial-ring-unique", "text": "Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule over $A$. Given $m_s \\in M$ for $s \\in S$, there exists a unique $A$-biderivation $\\lambda : P \\to M$ mapping $s$ to $m_s$ for $s \\in S$."}
+{"_id": "11208", "title": "cotangent-lemma-compare-higher", "text": "In the situation above denote $L$ the complex (\\ref{equation-lichtenbaum-schlessinger}). There is a canonical map $L_{B/A} \\to L$ in $D(A)$ which induces an isomorphism $\\tau_{\\geq -2}L_{B/A} \\to L$ in $D(B)$."}
+{"_id": "11209", "title": "cotangent-lemma-special-case", "text": "Let $A = \\mathbf{Z}[x_1, \\ldots, x_n] \\to B = \\mathbf{Z}$ be the ring map which sends $x_i$ to $0$ for $i = 1, \\ldots, n$. Let $I = (x_1, \\ldots, x_n) \\subset A$. Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$."}
+{"_id": "11211", "title": "cotangent-lemma-mod-Koszul-regular-ideal", "text": "Let $A \\to B$ be a surjective ring map whose kernel $I$ is Koszul. Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$."}
+{"_id": "11212", "title": "cotangent-lemma-tensor-product-tor-independent", "text": "If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$ in (\\ref{equation-tensor-product}) is zero. In this case we have $$ L_{A \\otimes_R B/R} = L_{A/R} \\otimes_A^\\mathbf{L} (A \\otimes_R B) \\oplus L_{B/R} \\otimes_B^\\mathbf{L} (A \\otimes_R B) $$ which is represented by the complex $L_{A/R} \\otimes_R B \\oplus L_{B/R} \\otimes_R A $ of $A \\otimes_R B$-modules."}
+{"_id": "11213", "title": "cotangent-lemma-tensor-product", "text": "Let $R$ be a ring and let $A$, $B$ be $R$-algebras. The object $E$ in (\\ref{equation-tensor-product}) satisfies $$ H^i(E) = \\left\\{ \\begin{matrix} 0 & \\text{if} & i \\geq -1 \\\\ \\text{Tor}_1^R(A, B) & \\text{if} & i = -2 \\end{matrix} \\right. $$"}
+{"_id": "11215", "title": "cotangent-lemma-pullback-cotangent-morphism-topoi", "text": "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. Then $f^{-1}L_{\\mathcal{B}/\\mathcal{A}} = L_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$."}
+{"_id": "11216", "title": "cotangent-lemma-compute-L-morphism-sheaves-rings", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. Then $H^i(L_{\\mathcal{B}/\\mathcal{A}})$ is the sheaf associated to the presheaf $U \\mapsto H^i(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$."}
+{"_id": "11217", "title": "cotangent-lemma-H0-L-morphism-sheaves-rings", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. Then $H^0(L_{\\mathcal{B}/\\mathcal{A}}) = \\Omega_{\\mathcal{B}/\\mathcal{A}}$."}
+{"_id": "11218", "title": "cotangent-lemma-compute-L-product-sheaves-rings", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ and $\\mathcal{A} \\to \\mathcal{B}'$ be homomorphisms of sheaves of rings on $\\mathcal{C}$. Then $$ L_{\\mathcal{B} \\times \\mathcal{B}'/\\mathcal{A}} \\longrightarrow L_{\\mathcal{B}/\\mathcal{A}} \\oplus L_{\\mathcal{B}'/\\mathcal{A}} $$ is an isomorphism in $D(\\mathcal{B} \\times \\mathcal{B}')$."}
+{"_id": "11219", "title": "cotangent-lemma-triangle-sheaves-rings", "text": "Let $\\mathcal{D}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be homomorphisms of sheaves of rings on $\\mathcal{D}$. There is a canonical distinguished triangle $$ L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C} \\to L_{\\mathcal{C}/\\mathcal{A}} \\to L_{\\mathcal{C}/\\mathcal{B}} \\to L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C}[1] $$ in $D(\\mathcal{C})$."}
+{"_id": "11220", "title": "cotangent-lemma-stalk-cotangent-complex", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. If $p$ is a point of $\\mathcal{C}$, then $(L_{\\mathcal{B}/\\mathcal{A}})_p = L_{\\mathcal{B}_p/\\mathcal{A}_p}$."}
+{"_id": "11221", "title": "cotangent-lemma-compare-cotangent-complex-with-naive", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. There is a canonical map $L_{\\mathcal{B}/\\mathcal{A}} \\to \\NL_{\\mathcal{B}/\\mathcal{A}}$ which identifies the naive cotangent complex with the truncation $\\tau_{\\geq -1}L_{\\mathcal{B}/\\mathcal{A}}$."}
+{"_id": "11222", "title": "cotangent-lemma-H0-L-morphism-ringed-spaces", "text": "Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of ringed spaces. Then $H^0(L_{X/S}) = \\Omega_{X/S}$."}
+{"_id": "11223", "title": "cotangent-lemma-triangle-ringed-spaces", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. Then there is a canonical distinguished triangle $$ Lf^* L_{Y/Z} \\to L_{X/Z} \\to L_{X/Y} \\to Lf^*L_{Y/Z}[1] $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "11224", "title": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-spaces", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical map $L_{X/Y} \\to \\NL_{X/Y}$ which identifies the naive cotangent complex with the truncation $\\tau_{\\geq -1}L_{X/Y}$."}
+{"_id": "11226", "title": "cotangent-lemma-H0-L-morphism-ringed-topoi", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of ringed topoi. Then $H^0(L_f) = \\Omega_f$."}
+{"_id": "11227", "title": "cotangent-lemma-triangle-ringed-topoi", "text": "Let $f : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and $g : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to (\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of ringed topoi. Then there is a canonical distinguished triangle $$ Lf^* L_g \\to L_{g \\circ f} \\to L_f \\to Lf^*L_g[1] $$ in $D(\\mathcal{O}_1)$."}
+{"_id": "11228", "title": "cotangent-lemma-compare-cotangent-complex-with-naive-ringed-topoi", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of ringed topoi. There is a canonical map $L_f \\to \\NL_f$ which identifies the naive cotangent complex with the truncation $\\tau_{\\geq -1}L_f$."}
+{"_id": "11230", "title": "cotangent-lemma-morphism-affine-schemes", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $U = \\Spec(A) \\subset X$ and $V = \\Spec(B) \\subset Y$ be affine opens such that $f(U) \\subset V$. There is a canonical map $$ \\widetilde{L_{B/A}} \\longrightarrow L_{X/Y}|_U $$ of complexes which is an isomorphism in $D(\\mathcal{O}_U)$. This map is compatible with restricting to smaller affine opens of $X$ and $Y$."}
+{"_id": "11231", "title": "cotangent-lemma-scheme-over-ring", "text": "Let $\\Lambda$ be a ring. Let $X$ be a scheme over $\\Lambda$. Then $$ L_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}} $$ where $\\underline{\\Lambda}$ is the constant sheaf with value $\\Lambda$ on $X$."}
+{"_id": "11232", "title": "cotangent-lemma-category-fibred", "text": "In the situation above the category $\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_{Zar}$."}
+{"_id": "11235", "title": "cotangent-lemma-compare-spaces-schemes", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$. Then there is a canonical identification $L_{X/Y} = \\epsilon^*L_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$ where $\\epsilon$ is as in Derived Categories of Spaces, Section \\ref{spaces-perfect-section-derived-quasi-coherent-etale} and $L_{X_0/Y_0}$ is as in  Definition \\ref{definition-cotangent-morphism-schemes}."}
+{"_id": "11236", "title": "cotangent-lemma-space-over-ring", "text": "Let $\\Lambda$ be a ring. Let $X$ be an algebraic space over $\\Lambda$. Then $$ L_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}} $$ where $\\underline{\\Lambda}$ is the constant sheaf with value $\\Lambda$ on $X_\\etale$."}
+{"_id": "11237", "title": "cotangent-lemma-category-fibred-space", "text": "In the situation above the category $\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_\\etale$."}
+{"_id": "11240", "title": "cotangent-lemma-fibre-product", "text": "Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of algebraic spaces over $S$. The object $E$ in (\\ref{equation-fibre-product}) satisfies $H^i(E) = 0$ for $i = 0, -1$ and for a geometric point $(\\overline{x}, \\overline{y}) : \\Spec(k) \\to X \\times_B Y$ we have $$ H^{-2}(E)_{(\\overline{x}, \\overline{y})} = \\text{Tor}_1^R(A, B) \\otimes_{A \\otimes_R B} C $$ where $R = \\mathcal{O}_{B, \\overline{b}}$, $A = \\mathcal{O}_{X, \\overline{x}}$, $B = \\mathcal{O}_{Y, \\overline{y}}$, and $C = \\mathcal{O}_{X \\times_B Y, (\\overline{x}, \\overline{y})}$."}
+{"_id": "11241", "title": "cotangent-proposition-polynomial", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map. There exists a simplicial $A$-algebra $P_\\bullet$ with an augmentation $\\epsilon : P_\\bullet \\to B$ such that each $P_n$ is a polynomial algebra of finite type over $A$ and such that $\\epsilon$ is a trivial Kan fibration of simplicial sets."}
+{"_id": "11242", "title": "cotangent-proposition-triangle", "text": "Let $A \\to B \\to C$ be ring maps. There is a canonical distinguished triangle $$ L_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to L_{B/A} \\otimes_B^\\mathbf{L} C[1] $$ in $D(C)$."}
+{"_id": "11270", "title": "spaces-cohomology-theorem-formal-functions", "text": "In Situation \\ref{situation-formal-functions}. Fix $p \\geq 0$. The system of maps $$ H^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F}) \\longrightarrow H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ define an isomorphism of limits $$ H^p(X, \\mathcal{F})^\\wedge \\longrightarrow \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ where the left hand side is the completion of the $A$-module $H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see Algebra, Section \\ref{algebra-section-completion}. Moreover, this is in fact a homeomorphism for the limit topologies."}
+{"_id": "11271", "title": "spaces-cohomology-lemma-higher-direct-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is quasi-compact and quasi-separated, then $R^if_*$ transforms quasi-coherent $\\mathcal{O}_X$-modules into quasi-coherent $\\mathcal{O}_Y$-modules."}
+{"_id": "11272", "title": "spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and any affine object $V$ of $Y_\\etale$ we have $$ H^q(V \\times_Y X, \\mathcal{F}) = H^0(V, R^qf_*\\mathcal{F}) $$ for all $q \\in \\mathbf{Z}$."}
+{"_id": "11273", "title": "spaces-cohomology-lemma-finite-higher-direct-image-zero", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral (for example finite) morphism of algebraic spaces. Then $f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is an exact functor and $R^pf_* = 0$ for $p > 0$."}
+{"_id": "11274", "title": "spaces-cohomology-lemma-stalk-push-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$ with lifts $\\overline{x}_1, \\ldots, \\overline{x}_n$ in $X$. Then $$ (f_*\\mathcal{F})_{\\overline{y}} = \\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{F}_{\\overline{x}_i} $$ for any sheaf $\\mathcal{F}$ on $X_\\etale$."}
+{"_id": "11275", "title": "spaces-cohomology-lemma-finite-rings", "text": "Let $S$ be a scheme. Let $\\pi : X \\to Y$ be a finite morphism of algebraic spaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$. Let $\\mathcal{B}$ be a sheaf of rings on $Y_\\etale$. Let $\\varphi : \\mathcal{B} \\to \\pi_*\\mathcal{A}$ be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi $$ f = (\\pi, \\varphi) : (\\Sh(X_\\etale), \\mathcal{A}) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{B}). $$ For a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a sheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map $$ \\mathcal{G} \\otimes_\\mathcal{B} f_*\\mathcal{F} \\longrightarrow f_*(f^*\\mathcal{G} \\otimes_\\mathcal{A} \\mathcal{F}). $$ is an isomorphism."}
+{"_id": "11276", "title": "spaces-cohomology-lemma-projection-formula-finite", "text": "With $S$, $X$, $Y$, $\\pi$, $\\mathcal{A}$, $\\mathcal{B}$, $\\varphi$, and $f$ as in Lemma \\ref{lemma-finite-rings} we have $$ K \\otimes_\\mathcal{B}^\\mathbf{L} Rf_*M = Rf_*(Lf^*K \\otimes_\\mathcal{A}^\\mathbf{L} M) $$ in $D(\\mathcal{B})$ for any $K \\in D(\\mathcal{B})$ and $M \\in D(\\mathcal{A})$."}
+{"_id": "11277", "title": "spaces-cohomology-lemma-colimits", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is quasi-compact and quasi-separated, then $$ \\colim_i H^p(X, \\mathcal{F}_i) \\longrightarrow H^p(X, \\colim_i \\mathcal{F}_i) $$ is an isomorphism for every filtered diagram of abelian sheaves on $X_\\etale$."}
+{"_id": "11278", "title": "spaces-cohomology-lemma-colimit-cohomology", "text": "\\begin{slogan} Higher direct images of qcqs morphisms commute with filtered colimits of sheaves. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\\mathcal{F} = \\colim \\mathcal{F}_i$ be a filtered colimit of abelian sheaves on $X_\\etale$. Then for any $p \\geq 0$ we have $$ R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i. $$"}
+{"_id": "11279", "title": "spaces-cohomology-lemma-finite-presentation-quasi-compact-colimit", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $I$ be a directed set and let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$ of quasi-coherent $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation. Then we have $$ \\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i) = \\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i). $$"}
+{"_id": "11280", "title": "spaces-cohomology-lemma-product-is-tensor-product", "text": "Let $S$ be a scheme. Let $f_i : U_i \\to X$ be \\'etale morphisms of algebraic spaces over $S$. Then there are isomorphisms $$ f_{1, !}\\underline{\\mathbf{Z}} \\otimes_{\\mathbf{Z}} f_{2, !}\\underline{\\mathbf{Z}} \\longrightarrow f_{12, !}\\underline{\\mathbf{Z}} $$ where $f_{12} : U_1 \\times_X U_2 \\to X$ is the structure morphism and $$ (f_1 \\amalg f_2)_! \\underline{\\mathbf{Z}} \\longrightarrow f_{1, !}\\underline{\\mathbf{Z}} \\oplus f_{2, !}\\underline{\\mathbf{Z}} $$"}
+{"_id": "11281", "title": "spaces-cohomology-lemma-alternating-cech-to-cohomology", "text": "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of $\\textit{Ab}(X_\\etale)$. There exists a canonical map $$ \\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F}) \\longrightarrow R\\Gamma(X, \\mathcal{F}) $$ in $D(\\textit{Ab})$. Moreover, there is a spectral sequence with $E_1$-page $$ E_1^{p, q} = \\Ext_{\\textit{Ab}(X_\\etale)}^q(K^p, \\mathcal{F}) $$ converging to $H^{p + q}(X, \\mathcal{F})$ where $K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$."}
+{"_id": "11282", "title": "spaces-cohomology-lemma-compute", "text": "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and separated morphism of algebraic spaces over $S$. For $p \\geq 0$ set $$ W_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals} $$ where the fibre product has $p + 1$ factors. There is a free action of $S_{p + 1}$ on $W_p$ over $X$ and $$ \\Hom(K^p, \\mathcal{F}) = S_{p + 1}\\text{-anti-invariant elements of } \\mathcal{F}(W_p) $$ functorially in $\\mathcal{F}$ where $K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$."}
+{"_id": "11283", "title": "spaces-cohomology-lemma-twist", "text": "Let $S$ be a scheme. Let $W$ be an algebraic space over $S$. Let $G$ be a finite group acting freely on $W$. Let $U = W/G$, see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}. Let $\\chi : G \\to \\{+1, -1\\}$ be a character. Then there exists a rank 1 locally free sheaf of $\\mathbf{Z}$-modules $\\underline{\\mathbf{Z}}(\\chi)$ on $U_\\etale$ such that for every abelian sheaf $\\mathcal{F}$ on $U_\\etale$ we have $$ H^0(W, \\mathcal{F}|_W)^\\chi = H^0(U, \\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi)) $$"}
+{"_id": "11284", "title": "spaces-cohomology-lemma-alternating-spectral-sequence", "text": "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and separated morphism of algebraic spaces over $S$. For $p \\geq 0$ set $$ W_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals} $$ (with $p + 1$ factors) as in Lemma \\ref{lemma-compute}. Let $\\chi_p : S_{p + 1} \\to \\{+1, -1\\}$ be the sign character. Let $U_p = W_p/S_{p + 1}$ and $\\underline{\\mathbf{Z}}(\\chi_p)$ be as in Lemma \\ref{lemma-twist}. Then the spectral sequence of Lemma \\ref{lemma-alternating-cech-to-cohomology} has $E_1$-page $$ E_1^{p, q} = H^q(U_p, \\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p)) $$ and converges to $H^{p + q}(X, \\mathcal{F})$."}
+{"_id": "11285", "title": "spaces-cohomology-lemma-quasi-coherent-twist", "text": "With $S$, $W$, $G$, $U$, $\\chi$ as in Lemma \\ref{lemma-twist}. If $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module, then so is $\\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi)$."}
+{"_id": "11286", "title": "spaces-cohomology-lemma-vanishing-quasi-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Then we can choose \\begin{enumerate} \\item an affine scheme $U$, \\item a surjective \\'etale morphism $f : U \\to X$, \\item an integer $d$ bounding the degrees of the fibres of $U \\to X$, \\item for every $p = 0, 1, \\ldots, d$ a surjective \\'etale morphism $V_p \\to U_p$ from an affine scheme $V_p$ where $U_p$ is as in Lemma \\ref{lemma-alternating-spectral-sequence}, and \\item an integer $d_p$ bounding the degree of the fibres of $V_p \\to U_p$. \\end{enumerate} Moreover, whenever we have (1) -- (5), then for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^q(X, \\mathcal{F}) = 0$ for $q \\geq \\max(d_p + p)$."}
+{"_id": "11287", "title": "spaces-cohomology-lemma-vanishing-higher-direct-images", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $f$ is quasi-compact and quasi-separated, and \\item $Y$ is quasi-compact. \\end{enumerate} Then there exists an integer $n(X \\to Y)$ such that for any algebraic space $Y'$, any morphism $Y' \\to Y$ and any quasi-coherent sheaf $\\mathcal{F}'$ on $X' = Y' \\times_Y X$ the higher direct images $R^if'_*\\mathcal{F}'$ are zero for $i \\geq n(X \\to Y)$."}
+{"_id": "11288", "title": "spaces-cohomology-lemma-affine-vanishing-higher-direct-images", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. Then $R^if_*\\mathcal{F} = 0$ for $i > 0$ and any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$."}
+{"_id": "11289", "title": "spaces-cohomology-lemma-relative-affine-cohomology", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $H^i(X, \\mathcal{F}) = H^i(Y, f_*\\mathcal{F})$ for all $i \\geq 0$."}
+{"_id": "11292", "title": "spaces-cohomology-lemma-etale-localization-sheaf-with-support", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$. Let $Z \\subset Y$ be a closed subspace such that $f^{-1}(Z) \\to Z$ is an isomorphism of algebraic spaces. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Then $$ \\mathcal{H}^q_Z(\\mathcal{F}) = \\mathcal{H}^q_{f^{-1}(Z)}(f^{-1}\\mathcal{F}) $$ as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^q_Z(Y, \\mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\\mathcal{F})$."}
+{"_id": "11293", "title": "spaces-cohomology-lemma-complexes-with-support-on-closed", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. The map $Ri_* = i_* : D(Z_\\etale) \\to D(X_\\etale)$ induces an equivalence $D(Z_\\etale) \\to D_{|Z|}(X_\\etale)$ with quasi-inverse $$ i^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_{|Z|}(X_\\etale)} $$"}
+{"_id": "11294", "title": "spaces-cohomology-lemma-vanishing-above-dimension", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume $\\dim(X) \\leq d$ for some integer $d$. Let $\\mathcal{F}$ be a quasi-coherent sheaf $\\mathcal{F}$ on $X$. \\begin{enumerate} \\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$, \\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective for any quasi-compact open $U \\subset X$, \\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ for any closed subspace $Z \\subset X$ whose complement is quasi-compact. \\end{enumerate}"}
+{"_id": "11295", "title": "spaces-cohomology-lemma-affine-base-change", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. In this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is a quasi-coherent sheaf, and for every diagram (\\ref{equation-base-change-diagram}) we have $$ g^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}. $$"}
+{"_id": "11296", "title": "spaces-cohomology-lemma-flat-base-change-cohomology", "text": "Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module with pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \\geq 0$ \\begin{enumerate} \\item the base change map of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-base-change-map-flat-case} is an isomorphism $$ g^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}', $$ \\item if $Y = \\Spec(A)$ and $Y' = \\Spec(B)$, then $H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$. \\end{enumerate}"}
+{"_id": "11297", "title": "spaces-cohomology-lemma-coherent-Noetherian", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is coherent, \\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module, \\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module, \\item for any \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme the pullback $\\varphi^*\\mathcal{F}$ is a coherent module on $U$, and \\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme such that the pullback $\\varphi^*\\mathcal{F}$ is a coherent module on $U$. \\end{enumerate} In particular $\\mathcal{O}_X$ is coherent, any invertible $\\mathcal{O}_X$-module is coherent, and more generally any finite locally free $\\mathcal{O}_X$-module is coherent."}
+{"_id": "11298", "title": "spaces-cohomology-lemma-coherent-abelian-Noetherian", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The category of coherent $\\mathcal{O}_X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\\mathcal{O}_X$-modules are coherent. Any extension of coherent sheaves is coherent."}
+{"_id": "11299", "title": "spaces-cohomology-lemma-coherent-Noetherian-quasi-coherent-sub-quotient", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Any quasi-coherent submodule of $\\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\\mathcal{F}$ is coherent."}
+{"_id": "11302", "title": "spaces-cohomology-lemma-coherent-support-closed", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $i : Z \\to X$ be the scheme theoretic support of $\\mathcal{F}$ and $\\mathcal{G}$ the quasi-coherent $\\mathcal{O}_Z$-module such that $i_*\\mathcal{G} = \\mathcal{F}$, see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-scheme-theoretic-support}. Then $\\mathcal{G}$ is a coherent $\\mathcal{O}_Z$-module."}
+{"_id": "11303", "title": "spaces-cohomology-lemma-i-star-equivalence", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of locally Noetherian algebraic spaces over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$ and the category of coherent $\\mathcal{O}_Z$-modules."}
+{"_id": "11304", "title": "spaces-cohomology-lemma-finite-pushforward-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and $f_*\\mathcal{F}$ is coherent."}
+{"_id": "11305", "title": "spaces-cohomology-lemma-acc-coherent", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The ascending chain condition holds for quasi-coherent submodules of $\\mathcal{F}$. In other words, given any sequence $$ \\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F} $$ of quasi-coherent submodules, then $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$."}
+{"_id": "11306", "title": "spaces-cohomology-lemma-power-ideal-kills-sheaf", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals corresponding to a closed subspace $Z \\subset X$. Then there is some $n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$ if and only if $\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically)."}
+{"_id": "11307", "title": "spaces-cohomology-lemma-Artin-Rees", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Let $\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \\geq 0$ such that for all $n \\geq c$ we have $$ \\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G}) = \\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G} $$"}
+{"_id": "11308", "title": "spaces-cohomology-lemma-homs-over-open", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Denote $Z \\subset X$ the corresponding closed subspace and set $U = X \\setminus Z$. There is a canonical isomorphism $$ \\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F}) \\longrightarrow \\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U). $$ In particular we have an isomorphism $$ \\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n, \\mathcal{F}) \\longrightarrow \\Gamma(U, \\mathcal{F}). $$"}
+{"_id": "11309", "title": "spaces-cohomology-lemma-prepare-filter-support", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Suppose that $\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed. Then there exists a short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0 $$ with $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and $\\text{Supp}(\\mathcal{G}) \\subset Z$."}
+{"_id": "11310", "title": "spaces-cohomology-lemma-prepare-filter-irreducible", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Assume that the scheme theoretic support of $\\mathcal{F}$ is a reduced $Z \\subset X$ with $|Z|$ irreducible. Then there exist an integer $r > 0$, a nonzero sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$, and an injective map of coherent sheaves $$ i_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F} $$ whose cokernel is supported on a proper closed subspace of $Z$."}
+{"_id": "11311", "title": "spaces-cohomology-lemma-coherent-filter", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration $$ 0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset \\mathcal{F}_m = \\mathcal{F} $$ by coherent subsheaves such that for each $j = 1, \\ldots, m$ there exists a reduced closed subspace $Z_j \\subset X$ with $|Z_j|$ irreducible and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$ such that $$ \\mathcal{F}_j/\\mathcal{F}_{j - 1} \\cong (Z_j \\to X)_* \\mathcal{I}_j $$"}
+{"_id": "11312", "title": "spaces-cohomology-lemma-property-initial", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \\begin{enumerate} \\item For any short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0 $$ if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$ then so does $\\mathcal{F}$. \\item For every reduced closed subspace $Z \\subset X$ with $|Z|$ irreducible and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$ for $i_*\\mathcal{I}$. \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "11313", "title": "spaces-cohomology-lemma-property-higher-rank-cohomological", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \\begin{enumerate} \\item For any short exact sequence of coherent sheaves $$ 0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0 $$ if $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$ then so does $\\mathcal{F}$. \\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for some $r \\geq 1$, then it holds for $\\mathcal{F}$. \\item For every reduced closed subspace $i : Z \\to X$ with $|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $Z$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z$, \\item for every nonzero quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ there exists a quasi-coherent subsheaf $\\mathcal{G}' \\subset \\mathcal{I}\\mathcal{G}$ such that $\\text{Supp}(\\mathcal{G}/\\mathcal{G}')$ is proper closed in $|Z|$ and such that $\\mathcal{P}$ holds for $i_*\\mathcal{G}'$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "11314", "title": "spaces-cohomology-lemma-property-higher-rank-cohomological-variant", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume \\begin{enumerate} \\item For any short exact sequence of coherent sheaves on $X$ if two out of three have property $\\mathcal{P}$ so does the third. \\item If $\\mathcal{P}$ holds  for $\\mathcal{F}^{\\oplus r}$ for some $r \\geq 1$, then it holds for $\\mathcal{F}$. \\item For every reduced closed subspace $i : Z \\to X$ with $|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $X$ whose scheme theoretic support is $Z$ such that $\\mathcal{P}$ holds for $\\mathcal{G}$. \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
+{"_id": "11315", "title": "spaces-cohomology-lemma-directed-colimit-coherent", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Every quasi-coherent $\\mathcal{O}_X$-module is the filtered colimit of its coherent submodules."}
+{"_id": "11316", "title": "spaces-cohomology-lemma-direct-colimit-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$ with $Y$ Noetherian. Then every quasi-coherent $\\mathcal{O}_X$-module is a filtered colimit of finitely presented $\\mathcal{O}_X$-modules."}
+{"_id": "11317", "title": "spaces-cohomology-lemma-vanishing-compute", "text": "In Situation \\ref{situation-vanishing} for an $A$-module $M$ we have $p_*(M \\otimes_A \\mathcal{O}_X) = \\widetilde{M}$ and $\\Gamma(X, M \\otimes_A \\mathcal{O}_X) = M$."}
+{"_id": "11318", "title": "spaces-cohomology-lemma-vanishing-base-change", "text": "In Situation \\ref{situation-vanishing}. \\begin{enumerate} \\item Given an affine morphism $X' \\to X$ of algebraic spaces, we have $H^1(X', \\mathcal{F}') = 0$ for every quasi-coherent $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$. \\item Given an $A$-algebra $A'$ setting $X' = X \\times_{\\Spec(A)} \\Spec(A')$ the morphism $X' \\to X$ is affine and $\\Gamma(X', \\mathcal{O}_{X'}) = A'$. \\end{enumerate}"}
+{"_id": "11319", "title": "spaces-cohomology-lemma-vanishing-separate-closed", "text": "In Situation \\ref{situation-vanishing}. Let $Z_0, Z_1 \\subset |X|$ be disjoint closed subsets. Then there exists an $a \\in A$ such that $Z_0 \\subset V(a)$ and $Z_1 \\subset V(a - 1)$."}
+{"_id": "11320", "title": "spaces-cohomology-lemma-vanishing-injective", "text": "In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is universally injective."}
+{"_id": "11321", "title": "spaces-cohomology-lemma-vanishing-separated", "text": "In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is separated."}
+{"_id": "11323", "title": "spaces-cohomology-lemma-Noetherian-h1-zero-invertible", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume that for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists an $n \\geq 1$ such that $H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$. Then $X$ is a scheme and $\\mathcal{L}$ is ample on $X$."}
+{"_id": "11324", "title": "spaces-cohomology-lemma-finite-morphism-Noetherian", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $i : Z \\to X$ be a closed immersion. Denote $i' : Z' \\to Y$ the inverse image of $Z$ (Morphisms of Spaces, Section \\ref{spaces-morphisms-section-closed-immersions}) and $f' : Z' \\to Z$ the induced morphism. Then $\\mathcal{G} = f'_*\\mathcal{O}_{Z'}$ is a coherent $\\mathcal{O}_Z$-module whose support is $Z$."}
+{"_id": "11325", "title": "spaces-cohomology-lemma-affine-morphism-projection-ideal", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If $f$ is affine then $\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$ (with notation as explained in the proof)."}
+{"_id": "11326", "title": "spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $f$ finite, \\item $f$ surjective, \\item $Y$ affine, and \\item $X$ Noetherian. \\end{enumerate} Then $X$ is affine."}
+{"_id": "11327", "title": "spaces-cohomology-lemma-weak-chow", "text": "Let $A$ be a ring. Let $X$ be an algebraic space over $\\Spec(A)$ whose structure morphism $X \\to \\Spec(A)$ is separated of finite type. Then there exists a proper surjective morphism $X' \\to X$ where $X'$ is a scheme which is H-quasi-projective over $\\Spec(A)$."}
+{"_id": "11328", "title": "spaces-cohomology-lemma-check-separated-dvr", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $Y$ is locally Noetherian, \\item $f$ is locally of finite type and quasi-separated, \\item for every commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a discrete valuation ring and $K$ its fraction field, there is at most one dotted arrow making the diagram commute. \\end{enumerate} Then $f$ is separated."}
+{"_id": "11329", "title": "spaces-cohomology-lemma-check-proper-dvr", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $Y$ is locally Noetherian, \\item $f$ is of finite type and quasi-separated, \\item for every commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a discrete valuation ring and $K$ its fraction field, there is a unique dotted arrow making the diagram commute. \\end{enumerate} Then $f$ is proper."}
+{"_id": "11330", "title": "spaces-cohomology-lemma-kill-by-twisting", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X \\ar[r]_i \\ar[rd]_f & \\mathbf{P}^n_Y \\ar[d] \\\\ & Y } $$ of algebraic spaces over $S$. Assume $i$ is a closed immersion and $Y$ Noetherian. Set $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_Y}(1)$. Let $\\mathcal{F}$ be a coherent module on $X$. Then there exists an integer $d_0$ such that for all $d \\geq d_0$ we have $R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$ for all $p > 0$."}
+{"_id": "11331", "title": "spaces-cohomology-lemma-proper-pushforward-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_Y$-module for all $i \\geq 0$."}
+{"_id": "11332", "title": "spaces-cohomology-lemma-proper-over-affine-cohomology-finite", "text": "Let $A$ be a Noetherian ring. Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then $H^i(X, \\mathcal{F})$ is finite $A$-module for all $i \\geq 0$."}
+{"_id": "11333", "title": "spaces-cohomology-lemma-graded-finiteness", "text": "Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces. Set $\\mathcal{B} = f^*\\widetilde B$. Let $\\mathcal{F}$ be a quasi-coherent graded $\\mathcal{B}$-module of finite type. For every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$ is a finite $B$-module."}
+{"_id": "11334", "title": "spaces-cohomology-lemma-vanshing-gives-ample", "text": "Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space over $R$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X$, \\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists an $n_0 \\geq 0$ such that $H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$ and $p > 0$, and \\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists an $n \\geq 1$ such that $H^1(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$. \\end{enumerate}"}
+{"_id": "11336", "title": "spaces-cohomology-lemma-cohomology-powers-ideal-times-F", "text": "In Situation \\ref{situation-formal-functions}. Set $B = \\bigoplus_{n \\geq 0} I^n$. Then for every $p \\geq 0$ the graded $B$-module $\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is a finite $B$-module."}
+{"_id": "11337", "title": "spaces-cohomology-lemma-cohomology-powers-ideal-application", "text": "In Situation \\ref{situation-formal-functions}. For every $p \\geq 0$ there exists an integer $c \\geq 0$ such that \\begin{enumerate} \\item the multiplication map $I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$ is surjective for all $n \\geq c$, and \\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$ is contained in the submodule $I^{m - c} H^p(X, I^n\\mathcal{F})$ for all $n \\geq 0$, $m \\geq c$. \\end{enumerate}"}
+{"_id": "11338", "title": "spaces-cohomology-lemma-ML-cohomology-powers-ideal", "text": "In Situation \\ref{situation-formal-functions}. Fix $p \\geq 0$. \\begin{enumerate} \\item There exists a $c_1 \\geq 0$ such that for all $n \\geq c_1$ we have $$ \\Ker( H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}) ) \\subset I^{n - c_1}H^p(X, \\mathcal{F}). $$ \\item The inverse system $$ \\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}} $$ satisfies the Mittag-Leffler condition (see Homology, Definition \\ref{homology-definition-Mittag-Leffler}). \\item In fact for any $p$ and $n$ there exists a $c_2(n) \\geq n$ such that $$ \\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) = \\Im(H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) $$ for all $k \\geq c_2(n)$. \\end{enumerate}"}
+{"_id": "11339", "title": "spaces-cohomology-lemma-spell-out-theorem-formal-functions", "text": "Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces. Let $\\mathcal{F}$ be a coherent sheaf on $X$. Then $$ H^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F}) $$ for all $p \\geq 0$."}
+{"_id": "11340", "title": "spaces-cohomology-lemma-formal-functions-stalk", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ and let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ proper, and \\item $\\mathcal{F}$ coherent. \\end{enumerate} Let $\\overline{y}$ be a geometric point of $Y$. Consider the ``infinitesimal neighbourhoods'' $$ \\xymatrix{ X_n = \\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n) \\times_Y X \\ar[r]_-{i_n} \\ar[d]_{f_n} & X \\ar[d]^f \\\\ \\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n) \\ar[r]^-{c_n} & Y } $$ of the fibre $X_1 = X_{\\overline{y}}$ and set $\\mathcal{F}_n = i_n^*\\mathcal{F}$. Then we have $$ \\left(R^pf_*\\mathcal{F}\\right)_{\\overline{y}}^\\wedge \\cong \\lim_n H^p(X_n, \\mathcal{F}_n) $$ as $\\mathcal{O}_{Y, \\overline{y}}^\\wedge$-modules."}
+{"_id": "11341", "title": "spaces-cohomology-lemma-higher-direct-images-zero-finite-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ is proper, and \\item $X_{\\overline{y}}$ has discrete underlying topological space. \\end{enumerate} Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have $(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > 0$."}
+{"_id": "11342", "title": "spaces-cohomology-lemma-higher-direct-images-zero-above-dimension-fibre", "text": "\\begin{slogan} For proper maps, stalks of higher direct images are trivial in degrees larger than the dimension of the fibre. \\end{slogan} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$. Assume \\begin{enumerate} \\item $Y$ locally Noetherian, \\item $f$ is proper, and \\item $\\dim(X_{\\overline{y}}) = d$. \\end{enumerate} Then for any coherent sheaf $\\mathcal{F}$ on $X$ we have $(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > d$."}
+{"_id": "11343", "title": "spaces-cohomology-lemma-characterize-finite", "text": "(For a more general version see More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-characterize-finite}). Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $Y$ is locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is finite, and \\item $f$ is proper and $|X_k|$ is a discrete space for every morphism $\\Spec(k) \\to Y$ where $k$ is a field. \\end{enumerate}"}
+{"_id": "11344", "title": "spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood", "text": "(For a more general version see More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}). Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$. Assume \\begin{enumerate} \\item $Y$ is locally Noetherian, \\item $f$ is proper, and \\item $|X_{\\overline{y}}|$ is finite. \\end{enumerate} Then there exists an open neighbourhood $V \\subset Y$ of $\\overline{y}$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite."}
+{"_id": "11345", "title": "spaces-cohomology-proposition-vanishing", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and separated. Let $U$ be an affine scheme, and let $f : U \\to X$ be a surjective \\'etale morphism. Let $d$ be an upper bound for the size of the fibres of $|U| \\to |X|$. Then for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^q(X, \\mathcal{F}) = 0$ for $q \\geq d$."}
+{"_id": "11346", "title": "spaces-cohomology-proposition-vanishing-affine", "text": "\\begin{slogan} Serre's criterion for affineness in the setting of algebraic spaces. \\end{slogan} A quasi-compact and quasi-separated algebraic space is affine if and only if all higher cohomology groups of quasi-coherent sheaves vanish. More precisely, any algebraic space as in Situation \\ref{situation-vanishing} is an affine scheme."}
+{"_id": "11350", "title": "artin-theorem-contractions", "text": "\\begin{reference} \\cite[Theorem 3.1]{ArtinII} \\end{reference} Let $S$ be a locally Noetherian scheme such that $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s \\in S$. Let $X'$ be an algebraic space locally of finite type over $S$. Let $T' \\subset |X'|$ be a closed subset. Let $W$ be a locally Noetherian formal algebraic space over $S$ with $W_{red}$ locally of finite type over $S$. Finally, we let $$ g : X'_{/T'} \\longrightarrow W $$ be a formal modification, see Algebraization of Formal Spaces, Definition \\ref{restricted-definition-formal-modification}. If $X'$ and $W$ are separated\\footnote{See Remark \\ref{remark-separated-needed}.} over $S$, then there exists a proper morphism $f : X' \\to X$ of algebraic spaces over $S$, a closed subset $T \\subset |X|$, and an isomorphism $a : X_{/T} \\to W$ of formal algebraic spaces such that \\begin{enumerate} \\item $T'$ is the inverse image of $T$ by $|f| : |X'| \\to |X|$, \\item $f : X' \\to X$ maps $X' \\setminus T'$ isomorphically to $X \\setminus T$, and \\item $g = a \\circ f_{/T}$ where $f_{/T} : X'_{/T'} \\to X_{/T}$ is the induced morphism. \\end{enumerate} In other words, $(f : X' \\to X, T, a)$ is a solution as defined earlier in this section."}
+{"_id": "11352", "title": "artin-lemma-formally-smooth-on-deformation-categories", "text": "Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume either \\begin{enumerate} \\item $F$ is formally smooth on objects (Criteria for Representability, Section \\ref{criteria-section-formally-smooth}), \\item $F$ is representable by algebraic spaces and formally smooth, or \\item $F$ is representable by algebraic spaces and smooth. \\end{enumerate} Then for every finite type field $k$ over $S$ and object $x_0$ of $\\mathcal{X}$ over $k$ the functor (\\ref{equation-functoriality}) is smooth in the sense of Formal Deformation Theory, Definition \\ref{formal-defos-definition-smooth-morphism}."}
+{"_id": "11353", "title": "artin-lemma-fibre-product-deformation-categories", "text": "Let $S$ be a locally Noetherian scheme. Let $$ \\xymatrix{ \\mathcal{W} \\ar[d] \\ar[r] & \\mathcal{Z} \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{Y} } $$ be a $2$-fibre product of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a finite type field over $S$ and $w_0$ an object of $\\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be the images of $w_0$ under the morphisms in the diagram. Then $$ \\xymatrix{ \\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[d] \\ar[r] & \\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[d] \\\\ \\mathcal{F}_{\\mathcal{X}, k, x_0} \\ar[r] & \\mathcal{F}_{\\mathcal{Y}, k, y_0} } $$ is a fibre product of predeformation categories."}
+{"_id": "11354", "title": "artin-lemma-pushout", "text": "\\begin{slogan} Algebraic stacks satisfy the (strong) Rim-Schlessinger condition \\end{slogan} Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[r] \\ar[d] & X' \\ar[d] \\\\ Y \\ar[r] & Y' } $$ be a pushout in the category of schemes over $S$ where $X \\to X'$ is a thickening and $X \\to Y$ is affine, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}. Let $\\mathcal{Z}$ be an algebraic stack over $S$. Then the functor of fibre categories $$ \\mathcal{Z}_{Y'} \\longrightarrow \\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'} $$ is an equivalence of categories."}
+{"_id": "11355", "title": "artin-lemma-algebraic-stack-RS", "text": "Let $\\mathcal{X}$ be an algebraic stack over a locally Noetherian base $S$. Then $\\mathcal{X}$ satisfies (RS)."}
+{"_id": "11356", "title": "artin-lemma-fibre-product-RS", "text": "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$ satisfy (RS), then so does $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$."}
+{"_id": "11357", "title": "artin-lemma-deformation-category", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS). For any field $k$ of finite type over $S$ and any object $x_0$ of $\\mathcal{X}$ lying over $k$ the predeformation category $p : \\mathcal{F}_{\\mathcal{X}, k, x_0} \\to \\mathcal{C}_\\Lambda$ (\\ref{equation-predeformation-category}) is a deformation category, see Formal Deformation Theory, Definition \\ref{formal-defos-definition-deformation-category}."}
+{"_id": "11358", "title": "artin-lemma-change-of-field", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a field of finite type over $S$ and let $l/k$ be a finite extension. Let $x_0$ be an object of $\\mathcal{F}$ lying over $\\Spec(k)$. Denote $x_{l, 0}$ the restriction of $x_0$ to $\\Spec(l)$. Then there is a canonical functor $$ (\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k} \\longrightarrow \\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}} $$ of categories cofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$. If $\\mathcal{X}$ satisfies (RS), then this functor is an equivalence."}
+{"_id": "11359", "title": "artin-lemma-finite-dimension", "text": "Let $S$ be a locally Noetherian scheme. Assume \\begin{enumerate} \\item $\\mathcal{X}$ is an algebraic stack, \\item $U$ is a scheme locally of finite type over $S$, and \\item $(\\Sch/U)_{fppf} \\to \\mathcal{X}$ is a smooth surjective morphism. \\end{enumerate} Then, for any $\\mathcal{F} = \\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in Section \\ref{section-predeformation-categories} the tangent space $T\\mathcal{F}$ and infinitesimal automorphism space $\\text{Inf}(\\mathcal{F})$ have finite dimension over $k$"}
+{"_id": "11360", "title": "artin-lemma-fibre-product-tangent-spaces", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$, $\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS). Let $k$ be a field of finite type over $S$ and let $w_0$ be an object of $\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $k$. Denote $x_0, y_0, z_0$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ you get from $w_0$. Then there is a $6$-term exact sequence $$ \\xymatrix{ 0 \\ar[r] & \\text{Inf}(\\mathcal{F}_{\\mathcal{W}, k, w_0}) \\ar[r] & \\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) \\oplus \\text{Inf}(\\mathcal{F}_{\\mathcal{Z}, k, z_0}) \\ar[r] & \\text{Inf}(\\mathcal{F}_{\\mathcal{Y}, k, y_0}) \\ar[lld] \\\\  & T\\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[r] & T\\mathcal{F}_{\\mathcal{X}, k, x_0} \\oplus T\\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[r] & T\\mathcal{F}_{\\mathcal{Y}, k, y_0} } $$ of $k$-vector spaces."}
+{"_id": "11361", "title": "artin-lemma-smooth-lift-formal", "text": "Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\eta = (R, \\eta_n, g_n)$ be a formal object of $\\mathcal{Y}$ and let $\\xi_1$ be an object of $\\mathcal{X}$ with $F(\\xi_1) \\cong \\eta_1$. If $F$ is formally smooth on objects (see Criteria for Representability, Section \\ref{criteria-section-formally-smooth}), then there exists a formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ such that $F(\\xi) \\cong \\eta$."}
+{"_id": "11362", "title": "artin-lemma-effective", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. The functor (\\ref{equation-approximation}) is an equivalence."}
+{"_id": "11363", "title": "artin-lemma-fibre-product-effective", "text": "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If the functor (\\ref{equation-approximation}) is an equivalence for  $\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$, then it is  an equivalence for $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$."}
+{"_id": "11364", "title": "artin-lemma-approximate", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $x$ be an object of $\\mathcal{X}$ lying over $\\Spec(R)$ where $R$ is a Noetherian complete local ring with residue field $k$ of finite type over $S$. Let $s \\in S$ be the image of $\\Spec(k) \\to S$. Assume that (a) $\\mathcal{O}_{S, s}$ is a G-ring and (b) $p$ is limit preserving on objects. Then for every integer $N \\geq 1$ there exist \\begin{enumerate} \\item a finite type $S$-algebra $A$, \\item a maximal ideal $\\mathfrak m_A \\subset A$, \\item an object $x_A$ of $\\mathcal{X}$ over $\\Spec(A)$, \\item an $S$-isomorphism $R/\\mathfrak m_R^N \\cong A/\\mathfrak m_A^N$, \\item an isomorphism $x|_{\\Spec(R/\\mathfrak m_R^N)} \\cong x_A|_{\\Spec(A/\\mathfrak m_A^N)}$ compatible with (4), and \\item an isomorphism $\\text{Gr}_{\\mathfrak m_R}(R) \\cong \\text{Gr}_{\\mathfrak m_A}(A)$ of graded $k$-algebras. \\end{enumerate}"}
+{"_id": "11365", "title": "artin-lemma-fibre-product-limit-preserving", "text": "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item If $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ and $\\mathcal{Z} \\to (\\Sch/S)_{fppf}$ are limit preserving on objects and $\\mathcal{Y}$ is limit preserving, then $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects. \\item If $\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$ are limit preserving, then so is $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$. \\end{enumerate}"}
+{"_id": "11366", "title": "artin-lemma-limit-preserving-algebraic-space", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{X}$ is a stack in setoids and $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, \\item $\\mathcal{X}$ is a stack in setoids and limit preserving, \\item $\\mathcal{X}$ is representable by an algebraic space locally of finite presentation. \\end{enumerate}"}
+{"_id": "11367", "title": "artin-lemma-diagonal", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces and $\\mathcal{X}$ is limit preserving. Then $\\Delta$ is locally of finite type."}
+{"_id": "11368", "title": "artin-lemma-versality-matches", "text": "With notation as in Definition \\ref{definition-versal}. Let $R = \\mathcal{O}_{U, u_0}^\\wedge$. Let $\\xi$ be the formal object of $\\mathcal{X}$ over $R$ associated to $x|_{\\Spec(R)}$, see (\\ref{equation-approximation}). Then $$ x\\text{ is versal at }u_0 \\Leftrightarrow \\xi\\text{ is versal} $$"}
+{"_id": "11370", "title": "artin-lemma-versal-change-of-field", "text": "Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition \\ref{definition-versal}. Let $l$ be a field and let $u_{l, 0} : \\Spec(l) \\to U$ be a morphism with image $u_0$ such that $l/k = \\kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\\Spec(l)}$. If $\\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then $$ \\mathcal{F}_{(\\Sch/U)_{fppf}, l, u_{l, 0}} \\longrightarrow \\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}} $$ is smooth."}
+{"_id": "11371", "title": "artin-lemma-base-change-versal", "text": "Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition \\ref{definition-versal}. Assume \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\Delta$ is locally of finite type (for example if $\\mathcal{X}$ is limit preserving), and \\item $\\mathcal{X}$ has (RS). \\end{enumerate} Let $V$ be a scheme locally of finite type over $S$ and let $y$ be an object of $\\mathcal{X}$ over $V$. Form the $2$-fibre product $$ \\xymatrix{ \\mathcal{Z} \\ar[r] \\ar[d] & (\\Sch/U)_{fppf} \\ar[d]^x \\\\ (\\Sch/V)_{fppf} \\ar[r]^y & \\mathcal{X} } $$ Let $Z$ be the algebraic space representing $\\mathcal{Z}$ and let $z_0 \\in |Z|$ be a finite type point lying over $u_0$. If $x$ is versal at $u_0$, then the morphism $Z \\to V$ is smooth at $z_0$."}
+{"_id": "11372", "title": "artin-lemma-approximate-versal", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\xi = (R, \\xi_n, f_n)$ be a formal object of $\\mathcal{X}$ with $\\xi_1$ lying over $\\Spec(k) \\to S$ with image $s \\in S$. Assume \\begin{enumerate} \\item $\\xi$ is versal, \\item $\\xi$ is effective, \\item $\\mathcal{O}_{S, s}$ is a G-ring, and \\item $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects. \\end{enumerate} Then there exist a morphism of finite type $U \\to S$, a finite type point $u_0 \\in U$ with residue field $k$, and an object $x$ of $\\mathcal{X}$ over $U$ such that $x$ is versal at $u_0$ and such that $x|_{\\Spec(\\mathcal{O}_{U, u_0}/\\mathfrak m_{u_0}^n)} \\cong \\xi_n$."}
+{"_id": "11373", "title": "artin-lemma-versal-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $U$ be a scheme locally of finite type over $S$. Let $x$ be an object of $\\mathcal{X}$ over $U$. Assume that $x$ is versal at every finite type point of $U$ and that $\\mathcal{X}$ satisfies (RS). Then $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ satisfies (\\ref{equation-smooth})."}
+{"_id": "11374", "title": "artin-lemma-composition-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be composable $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ and $g$ satisfy (\\ref{equation-smooth}) so does $g \\circ f$."}
+{"_id": "11375", "title": "artin-lemma-base-change-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ satisfies (\\ref{equation-smooth}) so does the projection $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$."}
+{"_id": "11376", "title": "artin-lemma-smooth-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ is formally smooth on objects, then $f$ satisfies (\\ref{equation-smooth}). If $f$ is representable by algebraic spaces and smooth, then $f$ satisfies (\\ref{equation-smooth})."}
+{"_id": "11377", "title": "artin-lemma-implies-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $f$ is representable by algebraic spaces, \\item $f$ satisfies (\\ref{equation-smooth}), \\item $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, and \\item $\\mathcal{Y}$ is limit preserving. \\end{enumerate} Then $f$ is smooth."}
+{"_id": "11378", "title": "artin-lemma-get-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $k$ be a finite type field over $S$ and let $x_0$ be an object of $\\mathcal{X}$ over $\\Spec(k)$ with image $s \\in S$. Assume \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ satisfies axioms [1], [2], [3] (see Section \\ref{section-axioms}), \\item every formal object of $\\mathcal{X}$ is effective, \\item openness of versality holds for $\\mathcal{X}$, and \\item $\\mathcal{O}_{S, s}$ is a G-ring. \\end{enumerate} Then there exist a morphism of finite type $U \\to S$ and an object $x$ of $\\mathcal{X}$ over $U$ such that $$ x : (\\Sch/U)_{fppf} \\longrightarrow \\mathcal{X} $$ is smooth and such that there exists a finite type point $u_0 \\in U$ whose residue field is $k$ and such that $x|_{u_0} \\cong x_0$."}
+{"_id": "11380", "title": "artin-lemma-diagonal-representable", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume that \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see Section \\ref{section-axioms}), \\item every formal object of $\\mathcal{X}$ is effective, \\item $\\mathcal{X}$ satisfies openness of versality, and \\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \\end{enumerate} Then $\\mathcal{X}$ is an algebraic stack."}
+{"_id": "11382", "title": "artin-lemma-fibre-product-RS-star", "text": "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$ satisfy (RS*), then so does $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$."}
+{"_id": "11383", "title": "artin-lemma-single-point", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*). Let $x$ be an object of $\\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u \\in U$ be a finite type point such that $x$ is not versal at $u$. Then there exists a morphism $x \\to y$ of $\\mathcal{X}$ lying over $U \\to T$ satisfying \\begin{enumerate} \\item the morphism $U \\to T$ is a first order thickening, \\item we have a short exact sequence $$ 0 \\to \\kappa(u) \\to \\mathcal{O}_T \\to \\mathcal{O}_U \\to 0 $$ \\item there does {\\bf not} exist a pair $(W, \\alpha)$ consisting of an open neighbourhood $W \\subset T$ of $u$ and a morphism $\\beta : y|_W \\to x$ such that the composition $$ x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to y} y|_W \\xrightarrow{\\beta} x $$ is the canonical morphism $x|_{U \\cap W} \\to x$. \\end{enumerate}"}
+{"_id": "11384", "title": "artin-lemma-generalization-versality", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ has (RS*), \\item $\\mathcal{X}$ is limit preserving. \\end{enumerate} Let $x$ be an object of $\\mathcal{X}$ over a scheme $U$ of finite type over $S$. Let $u \\leadsto u_0$ be a specialization of finite type points of $U$ such that $x$ is versal at $u_0$. Then $x$ is versal at $u$."}
+{"_id": "11385", "title": "artin-lemma-infinite-sequence-pre", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*). Let $x$ be an object of $\\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u_n \\in U$, $n \\geq 1$ be finite type points such that (a) there are no specializations $u_n \\leadsto u_m$ for $n \\not = m$, and (b) $x$ is not versal at $u_n$ for all $n$. Then there exist morphisms $$ x \\to x_1 \\to x_2 \\to \\ldots \\quad\\text{in }\\mathcal{X}\\text{ lying over }\\quad U \\to U_1 \\to U_2 \\to \\ldots $$ over $S$ such that \\begin{enumerate} \\item for each $n$ the morphism $U \\to U_n$ is a first order thickening, \\item for each $n$ we have a short exact sequence $$ 0 \\to \\kappa(u_n) \\to \\mathcal{O}_{U_n} \\to \\mathcal{O}_{U_{n - 1}} \\to 0 $$ with $U_0 = U$ for $n = 1$, \\item for each $n$ there does {\\bf not} exist a pair $(W, \\alpha)$ consisting of an open neighbourhood $W \\subset U_n$ of $u_n$ and a morphism $\\alpha : x_n|_W \\to x$ such that the composition $$ x|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to x_n} x_n|_W \\xrightarrow{\\alpha} x $$ is the canonical morphism $x|_{U \\cap W} \\to x$. \\end{enumerate}"}
+{"_id": "11386", "title": "artin-lemma-SGE-implies-openness-versality", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ has (RS*), \\item $\\mathcal{X}$ is limit preserving, \\item systems $(\\xi_n)$ as in Remark \\ref{remark-strong-effectiveness} where $\\Ker(R_m \\to R_n)$ is an ideal of square zero for all $m \\geq n$ are effective. \\end{enumerate} Then $\\mathcal{X}$ satisfies openness of versality."}
+{"_id": "11387", "title": "artin-lemma-functoriality", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $$ \\xymatrix{ B' \\ar[r] & B \\\\ A' \\ar[u] \\ar[r] & A \\ar[u] } $$ be a commutative diagram of $S$-algebras. Let $x$ be an object of $\\mathcal{X}$ over $\\Spec(A)$, let $y$ be an object of $\\mathcal{Y}$ over $\\Spec(B)$, and let $\\phi : f(x)|_{\\Spec(B)} \\to y$ be a morphism of $\\mathcal{Y}$ over $\\Spec(B)$. Then there is a canonical functor $$ \\textit{Lift}(x, A') \\longrightarrow \\textit{Lift}(y, B') $$ of categories of lifts induced by $f$ and $\\phi$. The construction is compatible with compositions of $1$-morphisms of categories fibred in groupoids in an obvious manner."}
+{"_id": "11388", "title": "artin-lemma-properties-lift-RS-star", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra and let $x$ be an object of $\\mathcal{X}$ over $\\Spec(A)$. \\begin{enumerate} \\item There exists an $A$-linear functor $\\text{Inf}_x : \\text{Mod}_A \\to \\text{Mod}_A$ such that given a deformation situation $(x, A' \\to A)$ and a lift $x'$ there is an isomorphism $\\text{Inf}_x(I) \\to \\text{Inf}(x'/x)$ where $I = \\Ker(A' \\to A)$. \\item There exists an $A$-linear functor $T_x : \\text{Mod}_A \\to \\text{Mod}_A$ such that \\begin{enumerate} \\item given $M$ in $\\text{Mod}_A$ there is a bijection $T_x(M) \\to \\text{Lift}(x, A[M])$, \\item given a deformation situation $(x, A' \\to A)$ there is an action $$ T_x(I) \\times \\text{Lift}(x, A') \\to \\text{Lift}(x, A') $$ where $I = \\Ker(A' \\to A)$. It is simply transitive if $\\text{Lift}(x, A') \\not = \\emptyset$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "11389", "title": "artin-lemma-ses-inf-and-T", "text": "Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$, $\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS*). Let $A$ be an $S$-algebra and let $w$ be an object of $\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $A$. Denote $x, y, z$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ you get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence $$ \\xymatrix{ 0 \\ar[r] & \\text{Inf}_w(M) \\ar[r] & \\text{Inf}_x(M) \\oplus \\text{Inf}_z(M) \\ar[r] & \\text{Inf}_y(M) \\ar[lld] \\\\  & T_w(M) \\ar[r] & T_x(M) \\oplus T_z(M) \\ar[r] & T_y(M) } $$ of $A$-modules."}
+{"_id": "11390", "title": "artin-lemma-get-openness-obstruction-theory", "text": "\\begin{reference} This is \\cite[Theorem 4.4]{Hall-coherent} \\end{reference} Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ has (RS*), \\item $\\mathcal{X}$ is limit preserving, \\item there exists an obstruction theory\\footnote{Analyzing the proof the reader sees that in fact it suffices to check the functoriality (ii) of obstruction classes in Definition \\ref{definition-obstruction-theory} for maps $(y, B' \\to B) \\to (x, A' \\to A)$ with $B = A$ and $y = x$.}, \\item for an object $x$ of $\\mathcal{X}$ over $\\Spec(A)$ and $A$-modules $M_n$, $n \\geq 1$ we have \\begin{enumerate} \\item $T_x(\\prod M_n) = \\prod T_x(M_n)$, \\item $\\mathcal{O}_x(\\prod M_n) \\to \\prod \\mathcal{O}_x(M_n)$ is injective. \\end{enumerate} \\end{enumerate} Then $\\mathcal{X}$ satisfies openness of versality."}
+{"_id": "11391", "title": "artin-lemma-compute-ext-into-field", "text": "Let $A \\to k$ be a ring map with $k$ a field. Let $E \\in D^-(A)$. Then $\\Ext^i_A(E, k) = \\Hom_k(H^{-i}(E \\otimes^\\mathbf{L} k), k)$."}
+{"_id": "11392", "title": "artin-lemma-construct-essential-surjection", "text": "Let $\\Lambda \\to A \\to k$ be finite type ring maps of Noetherian rings with $k = \\kappa(\\mathfrak p)$ for some prime $\\mathfrak p$ of $A$. Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be morphism of $D^{-}(A)$ such that $H^{-1}(\\xi \\otimes^{\\mathbf{L}} k)$ is not surjective. Then there exists a surjection $A' \\to A$ of $\\Lambda$-algebras such that \\begin{enumerate} \\item[(a)] $I = \\Ker(A' \\to A)$ has square zero and is isomorphic to $k$ as an $A$-module, \\item[(b)] $\\Omega_{A'/\\Lambda} \\otimes k = \\Omega_{A/\\Lambda} \\otimes k$, and \\item[(c)] $E \\to \\NL_{A/A'}$ is zero. \\end{enumerate}"}
+{"_id": "11393", "title": "artin-lemma-characterize-versal", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*). Let $U = \\Spec(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$ over $U$. Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$. Assume \\begin{enumerate} \\item[(i)] for every deformation situation $(x, A' \\to A)$ we have: $x$ lifts to $\\Spec(A')$ if and only if $E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero, and \\item[(ii)] there is an isomorphism of functors $T_x(-) \\to \\Ext^0_A(E, -)$ such that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$ corresponds to the canonical element (see Remark \\ref{remark-canonical-element}). \\end{enumerate} Let $u_0 \\in U$ be a finite type point with residue field $k = \\kappa(u_0)$. Consider the following statements \\begin{enumerate} \\item $x$ is versal at $u_0$, and \\item $\\xi : E \\to \\NL_{A/\\Lambda}$ induces a surjection $H^{-1}(E \\otimes_A^{\\mathbf{L}} k) \\to H^{-1}(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$ and an injection $H^0(E \\otimes_A^{\\mathbf{L}} k) \\to H^0(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$. \\end{enumerate} Then we always have (2) $\\Rightarrow$ (1) and we have (1) $\\Rightarrow$ (2) if $u_0$ is a closed point."}
+{"_id": "11394", "title": "artin-lemma-openness", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*). Let $U = \\Spec(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$ over $U$. Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$. Assume \\begin{enumerate} \\item[(i)] for every deformation situation $(x, A' \\to A)$ we have: $x$ lifts to $\\Spec(A')$ if and only if $E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero, \\item[(ii)] there is an isomorphism of functors $T_x(-) \\to \\Ext^0_A(E, -)$ such that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$ corresponds to the canonical element (see Remark \\ref{remark-canonical-element}), \\item[(iii)] the cohomology groups of $E$ are finite $A$-modules. \\end{enumerate} If $x$ is versal at a closed point $u_0 \\in U$, then there exists an open neighbourhood $u_0 \\in U' \\subset U$ such that $x$ is versal at every finite type point of $U'$."}
+{"_id": "11397", "title": "artin-lemma-dual-obstruction", "text": "In Situation \\ref{situation-dual}. Assume furthermore that \\begin{enumerate} \\item[(iv)] given a short exact sequence of deformation situations as in Remark \\ref{remark-short-exact-sequence-thickenings} and a lift $x'_2 \\in \\text{Lift}(x, A_2')$ then $o_x(A_3') \\in H^2(K^\\bullet \\otimes_A^\\mathbf{L} I_3)$ equals $\\partial\\theta$ where $\\theta \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} I_1)$ is the element corresponding to $x'_2|_{\\Spec(A_1')}$ via $A_1' = A[I_1]$ and the given map $T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$. \\end{enumerate} In this case there exists an element $\\xi \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\NL_{A/\\Lambda})$ such that \\begin{enumerate} \\item for every deformation situation $(x, A' \\to A)$ we have $\\xi_{A'} = o_x(A')$, and \\item $\\xi_{can}$ matches the canonical element of Remark \\ref{remark-canonical-element} via the given transformation $T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$. \\end{enumerate}"}
+{"_id": "11398", "title": "artin-lemma-dual-openness", "text": "In Situation \\ref{situation-dual} assume that (iv) of Lemma \\ref{lemma-dual-obstruction} holds and that $K^\\bullet$ is a perfect object of $D(A)$. In this case, if $x$ is versal at a closed point $u_0 \\in U$ then there exists an open neighbourhood $u_0 \\in U' \\subset U$ such that $x$ is versal at every finite type point of $U'$."}
+{"_id": "11399", "title": "artin-lemma-canonical-extension", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Restricting along the inclusion functor $(\\textit{Noetherian}/S)_\\tau \\to (\\Sch/S)_\\tau$ defines an equivalence of categories between \\begin{enumerate} \\item the category of limit preserving sheaves on $(\\Sch/S)_\\tau$ and \\item the category of limit preserving sheaves on $(\\textit{Noetherian}/S)_\\tau$ \\end{enumerate}"}
+{"_id": "11400", "title": "artin-lemma-representable-limit-preserving", "text": "Let $X$ be an object of $(\\textit{Noetherian}/S)_\\tau$. If the functor of points $h_X : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$ is limit preserving, then $X$ is locally of finite presentation over $S$."}
+{"_id": "11401", "title": "artin-lemma-representable", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $F', G' : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be limit preserving and sheaves. Let $a' : F' \\to G'$ be a transformation of functors. Denote $a : F \\to G$ the restriction of $a' : F' \\to G'$ to $(\\textit{Noetherian}/S)_\\tau$. The following are equivalent \\begin{enumerate} \\item $a'$ is representable (as a transformation of functors, see Categories, Definition \\ref{categories-definition-representable-morphism}), and \\item for every object $V$ of $(\\textit{Noetherian}/S)_\\tau$ and every map $V \\to G$ the fibre product $F \\times_G V : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$ is a representable functor, and \\item same as in (2) but only for $V$ affine finite type over $S$ mapping into an affine open of $S$. \\end{enumerate}"}
+{"_id": "11402", "title": "artin-lemma-functor", "text": "In Situation \\ref{situation-contractions} the rule $F$ that sends a locally Noetherian scheme $V$ over $S$ to the set of triples $(Z, u', \\hat x)$ satisfying the compatibility condition and which sends a a morphism $\\varphi : V_2 \\to V_1$ of locally Noetherian schemes over $S$ to the map $$ F(\\varphi) : F(V_1) \\longrightarrow F(V_2) $$ sending an element $(Z_1, u'_1, \\hat x_1)$ of $F(V_1)$ to $(Z_2, u'_2, \\hat x_2)$ in $F(V_2)$ given by \\begin{enumerate} \\item $Z_2 \\subset V_2$ is the inverse image of $Z_1$ by $\\varphi$, \\item $u'_2$ is the composition of $u'_1$ and $\\varphi|_{V_2 \\setminus Z_2} : V_2 \\setminus Z_2 \\to V_1 \\setminus Z_1$, \\item $\\hat x_2$ is the composition of $\\hat x_1$ and $\\varphi_{/Z_2} : V_{2, /Z_2} \\to V_{1, /Z_1}$ \\end{enumerate} is a contravariant functor."}
+{"_id": "11403", "title": "artin-lemma-solution", "text": "In Situation \\ref{situation-contractions} if there exists a solution $(f : X' \\to X, T, a)$ then there is a functorial bijection $F(V) = \\Mor_S(V, X)$ on the category of locally Noetherian schemes $V$ over $S$."}
+{"_id": "11404", "title": "artin-lemma-functor-is-solution", "text": "In Situation \\ref{situation-contractions} if there exists an algebraic space $X$ locally of finite type over $S$ and a functorial bijection $F(V) = \\Mor_S(V, X)$ on the category of locally Noetherian schemes $V$ over $S$, then $X$ is a solution."}
+{"_id": "11405", "title": "artin-lemma-closed-immersion", "text": "In Situation \\ref{situation-contractions} assume given a closed subset $Z \\subset S$ such that \\begin{enumerate} \\item the inverse image of $Z$ in $X'$ is $T'$, \\item $U' \\to S \\setminus Z$ is a closed immersion, \\item $W \\to S_{/Z}$ is a closed immersion. \\end{enumerate} Then there exists a solution $(f : X' \\to X, T, a)$ and moreover $X \\to S$ is a closed immersion."}
+{"_id": "11406", "title": "artin-lemma-diagonal-contractions", "text": "In Situation \\ref{situation-contractions} assume $X' \\to S$ and $W \\to S$ are separated. Then the diagonal $\\Delta : F \\to F \\times F$ is representable by closed immersions."}
+{"_id": "11407", "title": "artin-lemma-sheaf", "text": "In Situation \\ref{situation-contractions} the functor $F$ satisfies the sheaf property for all \\'etale coverings of locally Noetherian schemes over $S$."}
+{"_id": "11408", "title": "artin-lemma-limit-preserving", "text": "In Situation \\ref{situation-contractions} the functor $F$ is limit preserving: for any directed limit $V = \\lim V_\\lambda$ of Noetherian affine schemes over $S$ we have $F(V) = \\colim F(V_\\lambda)$."}
+{"_id": "11409", "title": "artin-lemma-rs", "text": "In Situation \\ref{situation-contractions} the functor $F$ satisfies the Rim-Schlessinger condition (RS)."}
+{"_id": "11410", "title": "artin-lemma-finite-dim", "text": "In Situation \\ref{situation-contractions} the tangent spaces of the functor $F$ are finite dimensional."}
+{"_id": "11411", "title": "artin-lemma-formal-object-effective", "text": "In Situation \\ref{situation-contractions} assume $X' \\to S$ is separated. Then every formal object for $F$ is effective."}
+{"_id": "11412", "title": "artin-lemma-openness-smoothness", "text": "Let $S$ be a locally Noetherian scheme. Let $V$ be a scheme locally of finite type over $S$. Let $Z \\subset V$ be closed. Let $W$ be a locally Noetherian formal algebraic space over $S$ such that $W_{red}$ is locally of finite type over $S$. Let $g : V_{/Z} \\to W$ be an adic morphism of formal algebraic spaces over $S$. Let $v \\in V$ be a closed point such that $g$ is versal at $v$ (as in Section \\ref{section-axioms-functors}). Then after replacing $V$ by an open neighbourhood of $v$ the morphism $g$ is smooth (see proof)."}
+{"_id": "11413", "title": "artin-lemma-openness-versality", "text": "In Situation \\ref{situation-contractions} the functor $F$ satisfies openness of versality."}
+{"_id": "11414", "title": "artin-proposition-spaces-diagonal-representable", "text": "Let $S$ be a locally Noetherian scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item $\\Delta : F \\to F \\times F$ is representable by algebraic spaces, \\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see Section \\ref{section-axioms-functors}), and \\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \\end{enumerate} Then $F$ is an algebraic space."}
+{"_id": "11416", "title": "artin-proposition-spaces-diagonal-representable-noetherian", "text": "Let $S$ be a locally Noetherian scheme. Let $F : (\\textit{Noetherian}/S)_\\etale^{opp} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item $\\Delta : F \\to F \\times F$ is representable (as a transformation of functors, see Categories, Definition \\ref{categories-definition-representable-morphism}), \\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and \\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \\end{enumerate} Then there exists a unique algebraic space $F' : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ whose restriction to $(\\textit{Noetherian}/S)_\\etale$ is $F$ (see proof for elucidation)."}
+{"_id": "11446", "title": "obsolete-lemma-lift-elements-ideal", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak p \\subset R$ be a prime. Let $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$. Assume $S_{\\mathfrak q}$ is essentially of finite type over $R_\\mathfrak p$. Assume given \\begin{enumerate} \\item an integer $n \\geq 0$, \\item a prime $\\mathfrak a \\subset \\kappa(\\mathfrak p)[x_1, \\ldots, x_n]$, \\item a surjective $\\kappa(\\mathfrak p)$-homomorphism $$ \\psi : (\\kappa(\\mathfrak p)[x_1, \\ldots, x_n])_{\\mathfrak a} \\longrightarrow S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}, $$ and \\item elements $\\overline{f}_1, \\ldots, \\overline{f}_e$ in $\\Ker(\\psi)$. \\end{enumerate} Then there exist \\begin{enumerate} \\item an integer $m \\geq 0$, \\item and element $g \\in S$, $g \\not\\in \\mathfrak q$, \\item a map $$ \\Psi : R[x_1, \\ldots, x_n, x_{n + 1}, \\ldots, x_{n + m}] \\longrightarrow S_g, $$ and \\item elements $f_1, \\ldots, f_e, f_{e + 1}, \\ldots, f_{e + m}$ of $\\Ker(\\Psi)$ \\end{enumerate} such that \\begin{enumerate} \\item the following diagram commutes $$ \\xymatrix{ R[x_1, \\ldots, x_{n + m}] \\ar[d]_\\Psi \\ar[rr]_-{x_{n + j} \\mapsto 0} & & (\\kappa(\\mathfrak p)[x_1, \\ldots, x_n])_{\\mathfrak a} \\ar[d]^\\psi \\\\ S_g \\ar[rr] & & S_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q} }, $$ \\item the element $f_i$, $i \\leq n$ maps to a unit times $\\overline{f}_i$ in the local ring $$ (\\kappa(\\mathfrak p)[x_1, \\ldots, x_{n + m}])_{ (\\mathfrak a, x_{n + 1}, \\ldots, x_{n + m})}, $$ \\item the element $f_{e + j}$ maps to a unit times $x_{n + j}$ in the same local ring, and \\item the induced map $R[x_1, \\ldots, x_{n + m}]_{\\mathfrak b} \\to S_{\\mathfrak q}$ is surjective, where $\\mathfrak b = \\Psi^{-1}(\\mathfrak qS_g)$. \\end{enumerate}"}
+{"_id": "11450", "title": "obsolete-lemma-relative-effective-cartier-algebra", "text": "Let $A \\to B$ be a ring map. Let $f \\in B$. Assume that \\begin{enumerate} \\item $A \\to B$ is flat, \\item $f$ is a nonzerodivisor, and \\item $A \\to B/fB$ is flat. \\end{enumerate} Then for every ideal $I \\subset A$ the map $f : B/IB \\to B/IB$ is injective."}
+{"_id": "11459", "title": "obsolete-lemma-make-integral-less-trivial", "text": "Let $\\varphi : R \\to S$ be a ring map. Suppose $t \\in S$ satisfies the relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$. Set $u_n = \\varphi(a_n)$, $u_{n-1} = u_n t + \\varphi(a_{n-1})$, and so on till $u_1 = u_2 t + \\varphi(a_1)$. Then all of $u_n, u_{n-1}, \\ldots, u_1$ and $u_nt, u_{n-1}t, \\ldots, u_1t$ are integral over $R$, and the ideals $(\\varphi(a_0), \\ldots, \\varphi(a_n))$ and $(u_n, \\ldots, u_1)$ of $S$ are equal."}
+{"_id": "11461", "title": "obsolete-lemma-P1", "text": "Let $R$ be a ring. Let $F(X, Y) \\in R[X, Y]$ be homogeneous of degree $d$. Assume that for every prime $\\mathfrak p$ of $R$ at least one coefficient of $F$ is not in $\\mathfrak p$. Let $S = R[X, Y]/(F)$ as a graded ring. Then for all $n \\geq d$ the $R$-module $S_n$ is finite locally free of rank $d$."}
+{"_id": "11462", "title": "obsolete-lemma-rel-prime-pols", "text": "Let $k$ be a field. Let $F, G \\in k[X, Y]$ be homogeneous of degrees $d, e$. Assume $F, G$ relatively prime. Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$."}
+{"_id": "11464", "title": "obsolete-lemma-finite-after-localization", "text": "Let $R$ be a ring, let $f \\in R$. Suppose we have $S$, $S'$ and the solid arrows forming the following commutative diagram of rings $$ \\xymatrix{ & S'' \\ar@{-->}[rd] \\ar@{-->}[dd] & \\\\ R \\ar[rr] \\ar@{-->}[ru] \\ar[d] &  & S \\ar[d] \\\\ R_f \\ar[r] & S' \\ar[r] & S_f } $$ Assume that $R_f \\to S'$ is finite. Then we can find a finite ring map $R \\to S''$ and dotted arrows as in the diagram such that $S' = (S'')_f$."}
+{"_id": "11478", "title": "obsolete-lemma-property-irreducible-higher-rank", "text": "Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible closed subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that \\begin{enumerate} \\item For any short exact sequence of coherent sheaves if two out of three of them have property $\\mathcal{P}$ then so does the third. \\item If $\\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$, $Z \\not = Z_0$ and every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$ we have $\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$. \\item There exists some coherent sheaf $\\mathcal{G}$ on $X$ such that \\begin{enumerate} \\item $\\text{Supp}(\\mathcal{G}) = Z_0$, \\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and \\item property $\\mathcal{P}$ holds for $\\mathcal{G}$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf $\\mathcal{F}$ on $X$ whose support is contained in $Z_0$."}
+{"_id": "11480", "title": "obsolete-lemma-section-maps-back-into", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$ be a section. Let $\\mathcal{F}' \\subset \\mathcal{F}$ be quasi-coherent $\\mathcal{O}_X$-modules. Assume that \\begin{enumerate} \\item $X$ is quasi-compact, \\item $\\mathcal{F}$ is of finite type, and \\item $\\mathcal{F}'|_{X_s} = \\mathcal{F}|_{X_s}$. \\end{enumerate} Then there exists an $n \\geq 0$ such that multiplication by $s^n$ on $\\mathcal{F}$ factors through $\\mathcal{F}'$."}
+{"_id": "11485", "title": "obsolete-lemma-sheaf-fpqc-quasi-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated over $S$, then $X$ satisfies the sheaf condition for the fpqc topology."}
+{"_id": "11487", "title": "obsolete-lemma-scheme-very-reasonable", "text": "A scheme is very reasonable."}
+{"_id": "11495", "title": "obsolete-lemma-category-fibred", "text": "Let $X \\to Y$ be a morphism of schemes. \\begin{enumerate} \\item The category $\\mathcal{C}_{X/Y}$ is fibred over $X_{Zar}$. \\item The category $\\mathcal{C}_{X/Y}$ is fibred over $Y_{Zar}$. \\item The category $\\mathcal{C}_{X/Y}$ is fibred over the category of pairs $(U, V)$ where $U \\subset X$, $V \\subset Y$ are open and $f(U) \\subset V$. \\end{enumerate}"}
+{"_id": "11496", "title": "obsolete-lemma-equivalence", "text": "Notation and assumptions as above. Let $p : X_A \\to X$ denote the projection. Given $A'$ denote $p' : X_{A'} \\to X$ the projection. The functor $p'_*$ induces an equivalence of categories between \\begin{enumerate} \\item the category $\\textit{Lift}(\\mathcal{F}, A')$, and \\item the category $\\textit{Lift}(p_*\\mathcal{F}, A')$. \\end{enumerate}"}
+{"_id": "11497", "title": "obsolete-lemma-second-equivalence", "text": "Notation and assumptions as above. The functor $\\pi_!$ induces an equivalence of categories between \\begin{enumerate} \\item the category $\\textit{Lift}_\\mathcal{O}(i_*\\underline{\\mathcal{G}}, A')$, and \\item the category $\\textit{Lift}(\\mathcal{G}, A')$. \\end{enumerate}"}
+{"_id": "11498", "title": "obsolete-lemma-second-equivalence-obs", "text": "Notation and assumptions as in Lemma \\ref{lemma-second-equivalence}. Consider the object $$ L = L(\\Lambda, X, A, \\mathcal{G}) = L\\pi_!(Li^*(i_*(\\underline{\\mathcal{G}}))) $$ of $D(\\mathcal{O}_X \\otimes_\\Lambda A)$. Given a surjection $A' \\to A$ of $\\Lambda$-algebras with square zero kernel $I$ we have \\begin{enumerate} \\item The category $\\textit{Lift}(\\mathcal{G}, A')$ is nonempty if and only if a certain class $\\xi \\in \\Ext^2_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$ is zero. \\item If $\\textit{Lift}(\\mathcal{G}, A')$ is nonempty, then $\\text{Lift}(\\mathcal{G}, A')$ is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$. \\item Given a lift $\\mathcal{G}'$, the set of automorphisms of $\\mathcal{G}'$ which pull back to $\\text{id}_\\mathcal{G}$ is canonically isomorphic to $\\Ext^0_{\\mathcal{O}_X \\otimes A}(L, \\mathcal{G} \\otimes_A I)$. \\end{enumerate}"}
+{"_id": "11499", "title": "obsolete-lemma-pseudo-coherent", "text": "In the situation of Proposition \\ref{proposition-conclusion}, if $X \\to \\Spec(\\Lambda)$ is locally of finite type and $\\Lambda$ is Noetherian, then $L$ is pseudo-coherent."}
+{"_id": "11500", "title": "obsolete-lemma-ob-is-obstruction", "text": "In the situation of Remark \\ref{remark-construction-ob} assume that $\\mathcal{F}$ is flat over $U$. Then the vanishing of the class $\\xi_{U'}$ is a necessary and sufficient condition for the existence of a $\\mathcal{O}_{X \\times_B U'}$-module $\\mathcal{F}'$ flat over $U'$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$."}
+{"_id": "11501", "title": "obsolete-lemma-coherent-defo-thy-general", "text": "In Quot, Situation \\ref{quot-situation-coherent} assume that $S$ is a locally Noetherian scheme and $S = B$. Let $\\mathcal{X} = \\textit{Coh}_{X/B}$. Then we have openness of versality for $\\mathcal{X}$ (see Artin's Axioms, Definition \\ref{artin-definition-openness-versality})."}
+{"_id": "11506", "title": "obsolete-lemma-push-pull-effective-Cartier", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a proper morphism. Let $D \\subset Y$ be an effective Cartier divisor. Assume $X$, $Y$ integral, $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$ and $f$ dominant. Then $$ f_*[f^{-1}(D)]_{n - 1} = [R(X) : R(Y)] [D]_{n - 1}. $$ In particular if $f$ is birational then $f_*[f^{-1}(D)]_{n - 1} = [D]_{n - 1}$."}
+{"_id": "11508", "title": "obsolete-lemma-two-divisors", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z$ be an open and closed subscheme of the scheme $D_1 \\cap D_2$. Assume $\\dim_\\delta(D_1 \\cap D_2 \\setminus Z) \\leq n - 2$. Then there exists a morphism $b : X' \\to X$, and Cartier divisors $D_1', D_2', E$ on $X'$ with the following properties \\begin{enumerate} \\item $X'$ is integral, \\item $b$ is projective, \\item $b$ is the blowup of $X$ in the closed subscheme $Z$, \\item $E = b^{-1}(Z)$, \\item $b^{-1}(D_1) = D'_1 + E$, and $b^{-1}D_2 = D_2' + E$, \\item $\\dim_\\delta(D'_1 \\cap D'_2) \\leq n - 2$, and if $Z = D_1 \\cap D_2$ then $D'_1 \\cap D'_2 = \\emptyset$, \\item for every integral closed subscheme $W'$ with $\\dim_\\delta(W') = n - 1$ we have \\begin{enumerate} \\item if $\\epsilon_{W'}(D'_1, E) > 0$, then setting $W = b(W')$ we have $\\dim_\\delta(W) = n - 1$ and $$ \\epsilon_{W'}(D'_1, E) < \\epsilon_W(D_1, D_2), $$ \\item if $\\epsilon_{W'}(D'_2, E) > 0$, then setting $W = b(W')$ we have $\\dim_\\delta(W) = n - 1$ and $$ \\epsilon_{W'}(D'_2, E) < \\epsilon_W(D_1, D_2), $$ \\end{enumerate} \\end{enumerate}"}
+{"_id": "11509", "title": "obsolete-lemma-sum-divisors-associated-Weil", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given $n_i \\geq 0$ for $i \\in I$. Then $$ [D]_{n - 1} = \\sum\\nolimits_i n_i[D_i]_{n - 1} $$ in $Z_{n - 1}(X)$."}
+{"_id": "11510", "title": "obsolete-lemma-blowing-up-intersections", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = d$. Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection of effective Cartier divisors on $X$. Assume that for all $\\{i, j, k\\} \\subset I$, $\\#\\{i, j, k\\} = 3$ we have $D_i \\cap D_j \\cap D_k = \\emptyset$. Then there exist \\begin{enumerate} \\item an open subscheme $U \\subset X$ with $\\dim_\\delta(X \\setminus U) \\leq d - 3$, \\item a morphism $b : U' \\to U$, and \\item effective Cartier divisors $\\{D'_j\\}_{j \\in J}$ on $U'$ \\end{enumerate} with the following properties: \\begin{enumerate} \\item $b$ is proper morphism $b : U' \\to U$, \\item $U'$ is integral, \\item $b$ is an isomorphism over the complement of the union of the pairwise intersections of the $D_i|_U$, \\item $\\{D'_j\\}_{j \\in J}$ is a locally finite collection of effective Cartier divisors on $U'$, \\item $\\dim_\\delta(D'_j \\cap D'_{j'}) \\leq d - 2$ if $j \\not = j'$, and \\item $b^{-1}(D_i|_U) = \\sum n_{ij} D'_j$ for certain $n_{ij} \\geq 0$. \\end{enumerate} Moreover, if $X$ is quasi-compact, then we may assume $U = X$ in the above."}
+{"_id": "11511", "title": "obsolete-lemma-improved-additivity", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\{i_j : D_j \\to X \\}_{j \\in J}$ be a locally finite collection of effective Cartier divisors on $X$. Let $n_j > 0$, $j\\in J$. Set $D = \\sum_{j \\in J} n_j D_j$, and denote $i : D \\to X$ the inclusion morphism. Let $\\alpha \\in Z_{k + 1}(X)$. Then $$ p : \\coprod\\nolimits_{j \\in J} D_j \\longrightarrow D $$ is proper and $$ i^*\\alpha = p_*\\left(\\sum n_j i_j^*\\alpha\\right) $$ in $\\CH_k(D)$."}
+{"_id": "11512", "title": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $D$, $D'$ be effective Cartier divisors on $X$. Assume $\\dim_\\delta(D \\cap D') = n - 2$. Let $i : D \\to X$, resp.\\ $i' : D' \\to X$ be the corresponding closed immersions. Then \\begin{enumerate} \\item there exists a cycle $\\alpha \\in Z_{n - 2}(D \\cap D')$ whose pushforward to $D$ represents $i^*[D']_{n - 1} \\in \\CH_{n - 2}(D)$ and whose pushforward to $D'$ represents $(i')^*[D]_{n - 1} \\in \\CH_{n - 2}(D')$, and \\item we have $$ D \\cdot [D']_{n - 1} = D' \\cdot [D]_{n - 1} $$ in $\\CH_{n - 2}(X)$. \\end{enumerate}"}
+{"_id": "11513", "title": "obsolete-lemma-commutativity-effective-Cartier-proper-intersection-infinite", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $\\{D_j\\}_{j \\in J}$ be a locally finite collection of effective Cartier divisors on $X$. Let $n_j, m_j \\geq 0$ be collections of nonnegative integers. Set $D = \\sum n_j D_j$ and $D' = \\sum m_j D_j$. Assume that $\\dim_\\delta(D_j \\cap D_{j'}) = n - 2$ for every $j \\not = j'$. Then $D \\cdot [D']_{n - 1} = D' \\cdot [D]_{n - 1}$ in $\\CH_{n - 2}(X)$."}
+{"_id": "11523", "title": "obsolete-proposition-lqf-shriek", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. There exist adjoint functors $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ and $f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$ with the following properties \\begin{enumerate} \\item the functor $f^!$ is the one constructed in More \\'Etale Cohomology, Lemma \\ref{more-etale-lemma-lqf-f-upper-shriek}, \\item for any open $j : U \\to X$ with $f \\circ j$ separated there is a canonical isomorphism $f_! \\circ j_! = (f \\circ j)_!$, and \\item these isomorphisms for $U \\subset U' \\subset X$ are compatible with the isomorphisms in More \\'Etale Cohomology, Lemma \\ref{more-etale-lemma-f-shriek-composition}. \\end{enumerate}"}
+{"_id": "11524", "title": "obsolete-proposition-conclusion", "text": "With $\\Lambda$, $X$, $A$, $\\mathcal{F}$ as above. There exists a canonical object $L = L(\\Lambda, X, A, \\mathcal{F})$ of $D(X_A)$ such that given a surjection $A' \\to A$ of $\\Lambda$-algebras with square zero kernel $I$ we have \\begin{enumerate} \\item The category $\\textit{Lift}(\\mathcal{F}, A')$ is nonempty if and only if a certain class $\\xi \\in \\Ext^2_{X_A}(L, \\mathcal{F} \\otimes_A I)$ is zero. \\item If $\\textit{Lift}(\\mathcal{F}, A')$ is nonempty, then $\\text{Lift}(\\mathcal{F}, A')$ is principal homogeneous under $\\Ext^1_{X_A}(L, \\mathcal{F} \\otimes_A I)$. \\item Given a lift $\\mathcal{F}'$, the set of automorphisms of $\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically isomorphic to $\\Ext^0_{X_A}(L, \\mathcal{F} \\otimes_A I)$. \\end{enumerate}"}
+{"_id": "11565", "title": "stacks-sheaves-lemma-1-morphisms-presheaves", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Then $(g \\circ f)^p = f^p \\circ g^p$ and there is a canonical isomorphism ${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$ compatible with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and $((g \\circ f)^p, {}_p(g \\circ f))$."}
+{"_id": "11567", "title": "stacks-sheaves-lemma-functoriality-sheaves", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. The functors ${}_pf$ and $f^p$ of (\\ref{equation-pushforward-pullback}) transform $\\tau$ sheaves into $\\tau$ sheaves and define a morphism of topoi $f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$."}
+{"_id": "11568", "title": "stacks-sheaves-lemma-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r]^g & \\mathcal{Y} } $$ be a $2$-cartesian diagram of categories fibred in groupoids over $S$. Then we have a canonical isomorphism $$ g^{-1}f_*\\mathcal{F} \\longrightarrow f'_*(g')^{-1}\\mathcal{F} $$ functorial in the presheaf $\\mathcal{F}$ on $\\mathcal{X}$."}
+{"_id": "11569", "title": "stacks-sheaves-lemma-representable", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent \\begin{enumerate} \\item $f$ is representable, and \\item for every $y \\in \\Ob(\\mathcal{Y})$ the functor $\\mathcal{X}^{opp} \\to \\textit{Sets}$, $x \\mapsto \\Mor_\\mathcal{Y}(f(x), y)$ is representable. \\end{enumerate}"}
+{"_id": "11570", "title": "stacks-sheaves-lemma-representable-pushforward", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Then the functor $u : \\mathcal{Y}_\\tau \\to \\mathcal{X}_\\tau$ is continuous and defines a morphism of sites $\\mathcal{X}_\\tau \\to \\mathcal{Y}_\\tau$ which induces the same morphism of topoi $\\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$ as the morphism $f$ constructed in Lemma \\ref{lemma-functoriality-sheaves}. Moreover, $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ for any presheaf $\\mathcal{F}$ on $\\mathcal{X}$."}
+{"_id": "11571", "title": "stacks-sheaves-lemma-functoriality-structure-sheaf", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. There is a canonical identification $f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$ which turns $f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$ into a morphism of ringed topoi."}
+{"_id": "11572", "title": "stacks-sheaves-lemma-compare-with-scheme", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)$. Assume $\\mathcal{X}$ is representable by a scheme $X$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0] smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$ there is a canonical equivalence $$ (\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) = ((\\Sch/X)_\\tau, \\mathcal{O}_X) $$ of ringed sites."}
+{"_id": "11573", "title": "stacks-sheaves-lemma-compare-with-morphism-of-schemes", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories fibred in groupoids over $S$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by schemes $X$, $Y$. Let $f : X \\to Y$ be the morphism of schemes corresponding to $f$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0] smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$ the morphism of ringed topoi $f : (\\Sh(\\mathcal{X}_\\tau), \\mathcal{O}_\\mathcal{X}) \\to (\\Sh(\\mathcal{Y}_\\tau), \\mathcal{O}_\\mathcal{Y})$ agrees with the morphism of ringed topoi $f : (\\Sh((\\Sch/X)_\\tau), \\mathcal{O}_X) \\to  (\\Sh((\\Sch/Y)_\\tau), \\mathcal{O}_Y)$ via the identifications of Lemma \\ref{lemma-compare-with-scheme}."}
+{"_id": "11574", "title": "stacks-sheaves-lemma-localizing", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\\mathcal{X}_\\tau/x \\to (\\Sch/U)_\\tau$."}
+{"_id": "11575", "title": "stacks-sheaves-lemma-comparison", "text": "Let $\\mathcal{F}$ be an \\'etale sheaf on $\\mathcal{X} \\to (\\Sch/S)_{fppf}$. \\begin{enumerate} \\item If $\\varphi : x \\to y$ and $\\psi : y \\to z$ are morphisms of $\\mathcal{X}$ lying over $a : U \\to V$ and $b : V \\to W$, then the composition $$ a_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale})) \\xrightarrow{a_{small}^{-1}c_\\psi} a_{small}^{-1}(\\mathcal{F}|_{V_\\etale}) \\xrightarrow{c_\\varphi} \\mathcal{F}|_{U_\\etale} $$ is equal to $c_{\\psi \\circ \\varphi}$ via the identification $$ (b \\circ a)_{small}^{-1}(\\mathcal{F}|_{W_\\etale}) = a_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale})). $$ \\item If $\\varphi : x \\to y$ lies over an \\'etale morphism of schemes $a : U \\to V$, then (\\ref{equation-comparison}) is an isomorphism. \\item Suppose $f : \\mathcal{Y} \\to \\mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and $y$ is an object of $\\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\\mathcal{F}|_{U_\\etale} = \\mathcal{F}|_{U_\\etale}$. \\item Moreover, given $\\psi : y' \\to y$ in $\\mathcal{Y}$ lying over $a : U' \\to U$ the comparison map $c_\\psi : a_{small}^{-1}(F^{-1}\\mathcal{F}|_{U_\\etale}) \\to F^{-1}\\mathcal{F}|_{U'_\\etale}$ is equal to the comparison map $c_{f(\\psi)} : a_{small}^{-1}\\mathcal{F}|_{U_\\etale} \\to \\mathcal{F}|_{U'_\\etale}$ via the identifications in (3). \\end{enumerate}"}
+{"_id": "11576", "title": "stacks-sheaves-lemma-localizing-structure-sheaf", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$. The equivalence of Lemma \\ref{lemma-localizing} extends to an equivalence of ringed sites $(\\mathcal{X}_\\tau/x, \\mathcal{O}_\\mathcal{X}|_x) \\to ((\\Sch/U)_\\tau, \\mathcal{O})$."}
+{"_id": "11578", "title": "stacks-sheaves-lemma-compare", "text": "Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume $\\mathcal{X}$ is representable by an algebraic space $F$. Then there exists a continuous and cocontinuous functor $ F_\\etale \\to \\mathcal{X}_\\etale $ which induces a morphism of ringed sites $$ \\pi_F : (\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X}) \\longrightarrow (F_\\etale, \\mathcal{O}_F) $$ and a morphism of ringed topoi $$ i_F : (\\Sh(F_\\etale), \\mathcal{O}_F) \\longrightarrow (\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X}) $$ such that $\\pi_F \\circ i_F = \\text{id}$. Moreover $\\pi_{F, *} = i_F^{-1}$."}
+{"_id": "11579", "title": "stacks-sheaves-lemma-compare-morphism", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \\to G$ the induced morphism of algebraic spaces, and $f_{small} : F_\\etale \\to G_\\etale$ the corresponding morphism of ringed topoi. Then $$ \\xymatrix{ (\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X}) \\ar[d]_{\\pi_F} \\ar[rr]_f & & (\\Sh(\\mathcal{Y}_\\etale), \\mathcal{O}_\\mathcal{Y}) \\ar[d]^{\\pi_G} \\\\ (\\Sh(F_\\etale), \\mathcal{O}_F) \\ar[rr]^{f_{small}} & & (\\Sh(G_\\etale), \\mathcal{O}_G) } $$ is a commutative diagram of ringed topoi."}
+{"_id": "11580", "title": "stacks-sheaves-lemma-pullback-quasi-coherent", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \\textit{Mod}(\\mathcal{O}_\\mathcal{Y}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ preserves quasi-coherent sheaves."}
+{"_id": "11581", "title": "stacks-sheaves-lemma-characterize-quasi-coherent", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$ is quasi-coherent if and only if $x^*\\mathcal{F}$ is a quasi-coherent sheaf on $(\\Sch/U)_{fppf}$ for every object $x$ of $\\mathcal{X}$ with $U = p(x)$."}
+{"_id": "11582", "title": "stacks-sheaves-lemma-characterize-quasi-coherent-bis", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\mathcal{F}$ be a presheaf of modules on $\\mathcal{X}$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is an object of $\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$ and $\\mathcal{F}$ is a quasi-coherent module on $(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$ in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local}, \\item $\\mathcal{F}$ is an object of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ and $\\mathcal{F}$ is a quasi-coherent module on $(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local}, and \\item $\\mathcal{F}$ is a quasi-coherent module on $\\mathcal{X}$ in the sense of Definition \\ref{definition-quasi-coherent}. \\end{enumerate}"}
+{"_id": "11583", "title": "stacks-sheaves-lemma-quasi-coherent", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$ is quasi-coherent if and only if the following two conditions hold \\begin{enumerate} \\item $\\mathcal{F}$ is locally quasi-coherent, and \\item for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ lying over $f : U \\to V$ the comparison map $c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to \\mathcal{F}|_{U_\\etale}$ of (\\ref{equation-comparison-modules}) is an isomorphism. \\end{enumerate}"}
+{"_id": "11584", "title": "stacks-sheaves-lemma-pullback-lqc", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y}) \\to \\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ preserves locally quasi-coherent sheaves."}
+{"_id": "11585", "title": "stacks-sheaves-lemma-lqc-colimits", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. \\begin{enumerate} \\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ has colimits and they agree with colimits in the category $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$. \\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ is abelian with kernels and cokernels computed in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$, in other words the inclusion functor is exact. \\item Given a short exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ if two out of three are locally quasi-coherent so is the third. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$ in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ is an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$ with $\\mathcal{F}$ locally of finite presentation on $\\mathcal{X}_\\etale$ the sheaf $\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$ in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ is an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11586", "title": "stacks-sheaves-lemma-qc-colimits", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. \\begin{enumerate} \\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ has colimits and they agree with colimits in the categories $\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$, $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$, $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$, and $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ with $\\mathcal{F}$ locally of finite presentation on $\\mathcal{X}_{fppf}$ the sheaf $\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11587", "title": "stacks-sheaves-lemma-stackification", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then the morphism of topoi $f : \\Sh(\\mathcal{X}_{fppf}) \\to \\Sh(\\mathcal{Y}_{fppf})$ is an equivalence."}
+{"_id": "11588", "title": "stacks-sheaves-lemma-stackification-quasi-coherent", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ induces an equivalence of stackifications, then $f^*$ induces equivalences $\\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$ and $\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\QCoh(\\mathcal{O}_\\mathcal{Y})$."}
+{"_id": "11589", "title": "stacks-sheaves-lemma-map-from-quasi-coherent", "text": "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $\\mathcal{X} = [U/R]$ be the quotient stack. Denote $\\pi : \\mathcal{S}_U \\to \\mathcal{X}$ the quotient map. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module, and let $\\mathcal{H}$ be any object of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. The map $$ \\Hom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{H}) \\longrightarrow \\Hom_{\\mathcal{O}_U}(x^*\\mathcal{F}|_{U_\\etale}, x^*\\mathcal{H}|_{U_\\etale}), \\quad \\phi \\longmapsto x^*\\phi|_{U_\\etale} $$ is injective and its image consists of exactly those $\\varphi : x^*\\mathcal{F}|_{U_\\etale} \\to x^*\\mathcal{H}|_{U_\\etale}$ which give rise to a commutative diagram $$ \\xymatrix{ s_{small}^*(x^*\\mathcal{F}|_{U_\\etale}) \\ar[r] \\ar[d]^{s_{small}^*\\varphi} & (x \\circ s)^*\\mathcal{F}|_{R_\\etale} = (x \\circ t)^*\\mathcal{F}|_{R_\\etale} & t_{small}^*(x^*\\mathcal{F}|_{U_\\etale}) \\ar[l] \\ar[d]_{t_{small}^*\\varphi} \\\\ s_{small}^*(x^*\\mathcal{H}|_{U_\\etale}) \\ar[r] & (x \\circ s)^*\\mathcal{H}|_{R_\\etale} = (x \\circ t)^*\\mathcal{H}|_{R_\\etale} & t_{small}^*(x^*\\mathcal{H}|_{U_\\etale}) \\ar[l] } $$ of modules on $R_\\etale$ where the horizontal arrows are the comparison maps (\\ref{equation-comparison-algebraic-spaces-modules})."}
+{"_id": "11590", "title": "stacks-sheaves-lemma-quasi-coherent-algebraic-stack", "text": "Let $\\mathcal{X}$ be an algebraic stack over $S$. \\begin{enumerate} \\item If $[U/R] \\to \\mathcal{X}$ is a presentation of $\\mathcal{X}$ then there is a canonical equivalence $\\QCoh(\\mathcal{O}_\\mathcal{X}) \\cong \\QCoh(U, R, s, t, c)$. \\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is abelian. \\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ has colimits and they agree with colimits in the category $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ the tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$. \\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ with $\\mathcal{F}$ locally of finite presentation on $\\mathcal{X}_{fppf}$ the sheaf $\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11591", "title": "stacks-sheaves-lemma-cohomology-restriction", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ be an object lying over the scheme $U$. Let $\\mathcal{F}$ be an object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Then $$ H^p_\\tau(x, \\mathcal{F}) = H^p((\\Sch/U)_\\tau, x^{-1}\\mathcal{F}) $$ and if $\\tau = \\etale$, then we also have $$ H^p_\\etale(x, \\mathcal{F}) = H^p(U_\\etale, \\mathcal{F}|_{U_\\etale}). $$"}
+{"_id": "11592", "title": "stacks-sheaves-lemma-pushforward-injective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. \\begin{enumerate} \\item $f_*\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$ for $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$, and \\item $f_*\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ for $\\mathcal{I}$ injective in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11593", "title": "stacks-sheaves-lemma-fibre-products", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. \\begin{enumerate} \\item The category $\\mathcal{X}$ has fibre products. \\item If the $\\mathit{Isom}$-presheaves of $\\mathcal{X}$ are representable by algebraic spaces, then $\\mathcal{X}$ has equalizers. \\item If $\\mathcal{X}$ is an algebraic stack (or more generally a quotient stack), then $\\mathcal{X}$ has equalizers. \\end{enumerate}"}
+{"_id": "11594", "title": "stacks-sheaves-lemma-fibre-products-morphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item The functor $f$ transforms fibre products into fibre products. \\item If $f$ is faithful, then $f$ transforms equalizers into equalizers. \\end{enumerate}"}
+{"_id": "11595", "title": "stacks-sheaves-lemma-fibre-products-preserve-properties", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Z} \\to \\mathcal{Y}$ be faithful $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item the functor $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$ is faithful, and \\item if $\\mathcal{X}, \\mathcal{Z}$ have equalizers, so does $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$. \\end{enumerate}"}
+{"_id": "11596", "title": "stacks-sheaves-lemma-pullback-injective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. The functor $f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$ has a left adjoint $f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$. If $f$ is faithful and $\\mathcal{X}$ has equalizers, then \\begin{enumerate} \\item $f_!$ is exact, and \\item $f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$ for $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$. \\end{enumerate}"}
+{"_id": "11597", "title": "stacks-sheaves-lemma-pullback-injective-modules", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. The functor $f^* : \\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y}) \\to \\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$ has a left adjoint $f_! : \\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) \\to \\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ which agrees with the functor $f_!$ of Lemma \\ref{lemma-pullback-injective} on underlying abelian sheaves. If $f$ is faithful and $\\mathcal{X}$ has equalizers, then \\begin{enumerate} \\item $f_!$ is exact, and \\item $f^{-1}\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$ for $\\mathcal{I}$ injective in $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11598", "title": "stacks-sheaves-lemma-generalities", "text": "Generalities on {\\v C}ech complexes. \\begin{enumerate} \\item If $$ \\xymatrix{ \\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\ \\mathcal{Y} \\ar[r]^e & \\mathcal{X} } $$ is $2$-commutative diagram of categories fibred in groupoids over $(\\Sch/S)_{fppf}$, then there is a morphism of {\\v C}ech complexes $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F}) \\longrightarrow \\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, e^{-1}\\mathcal{F}) $$ \\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then the associated {\\v C}ech complexes are isomorphic, \\end{enumerate}"}
+{"_id": "11599", "title": "stacks-sheaves-lemma-homotopy", "text": "If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$ such that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$ then the extended {\\v C}ech complex is homotopic to zero."}
+{"_id": "11600", "title": "stacks-sheaves-lemma-generalities-sheafified", "text": "Generalities on relative {\\v C}ech complexes. \\begin{enumerate} \\item If $$ \\xymatrix{ \\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\ \\mathcal{Y} \\ar[r]^e & \\mathcal{X} } $$ is $2$-commutative diagram of categories fibred in groupoids over $(\\Sch/S)_{fppf}$, then there is a morphism $e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to \\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$. \\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism, \\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then the associated relative {\\v C}ech complexes are isomorphic, \\end{enumerate}"}
+{"_id": "11601", "title": "stacks-sheaves-lemma-homotopy-sheafified", "text": "If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$ such that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$ then the extended relative {\\v C}ech complex is homotopic to zero."}
+{"_id": "11602", "title": "stacks-sheaves-lemma-base-change-cech-complex", "text": "Let $$ \\xymatrix{ \\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\ \\mathcal{Y} \\ar[r]^e & \\mathcal{X} } $$ be a $2$-fibre product of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and let $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{X}$. Then the map $e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to \\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$ of Lemma \\ref{lemma-generalities-sheafified} is an isomorphism of complexes of abelian presheaves."}
+{"_id": "11603", "title": "stacks-sheaves-lemma-check-exactness-covering", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Let $$ \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} $$ be a complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$. Assume that \\begin{enumerate} \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, and \\item $f^{-1}\\mathcal{F} \\to f^{-1}\\mathcal{G} \\to f^{-1}\\mathcal{H}$ is exact. \\end{enumerate} Then the sequence $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$ is exact."}
+{"_id": "11604", "title": "stacks-sheaves-lemma-cech-to-cohomology", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, \\item the category $\\mathcal{U}$ has equalizers, and \\item the functor $f$ is faithful. \\end{enumerate} Then there is a first quadrant spectral sequence of abelian groups $$ E_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^{-1}\\mathcal{F}) \\Rightarrow H^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F}) $$ converging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology."}
+{"_id": "11606", "title": "stacks-sheaves-lemma-surjective-flat-locally-finite-presentation", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object $y$ of $\\mathcal{Y}$ there exists an fppf covering $\\{y_i \\to y\\}$ and objects $x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$. \\item Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then for any object $y$ of $\\mathcal{Y}$ there exists an \\'etale covering $\\{y_i \\to y\\}$ and objects $x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$. \\end{enumerate}"}
+{"_id": "11607", "title": "stacks-sheaves-lemma-cech-to-cohomology-relative", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, \\item the category $\\mathcal{U}$ has equalizers, and \\item the functor $f$ is faithful. \\end{enumerate} Then there is a first quadrant spectral sequence of abelian sheaves on $\\mathcal{Y}_\\tau$ $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ where all higher direct images are computed in the $\\tau$-topology."}
+{"_id": "11608", "title": "stacks-sheaves-lemma-cech-to-cohomology-relative-modules", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is an object of $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$, \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, \\item the category $\\mathcal{U}$ has equalizers, and \\item the functor $f$ is faithful. \\end{enumerate} Then there is a first quadrant spectral sequence in $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ where all higher direct images are computed in the $\\tau$-topology."}
+{"_id": "11609", "title": "stacks-sheaves-lemma-pushforward-restriction", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic stacks\\footnote{This result should hold for any $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.} over $S$. Let $\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0] smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$. Let $\\mathcal{F}$ be an object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Then the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the presheaf $$ y \\longmapsto H^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}, \\ \\text{pr}^{-1}\\mathcal{F}\\Big) $$ Here $y$ is a typical object of $\\mathcal{Y}$ lying over the scheme $V$."}
+{"_id": "11610", "title": "stacks-sheaves-lemma-base-change-higher-direct-images", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0] smooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$. Let $$ \\xymatrix{ \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r]^g & \\mathcal{Y} } $$ be a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change map is an isomorphism $$ g^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F} $$ functorial for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\tau)$ or $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$."}
+{"_id": "11611", "title": "stacks-sheaves-lemma-compare-injectives", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$ representable by the algebraic space $F$. \\begin{enumerate} \\item If $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\etale)$, then $\\mathcal{I}|_{F_\\etale}$ is injective in $\\textit{Ab}(F_\\etale)$, \\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in $\\textit{Ab}(\\mathcal{X}_\\etale)$, then $\\mathcal{I}^\\bullet|_{F_\\etale}$ is a K-injective complex in $\\textit{Ab}(F_\\etale)$. \\end{enumerate} The same does not hold for modules."}
+{"_id": "11612", "title": "stacks-sheaves-lemma-compare-morphism-cohomology", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks over $S$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \\to G$ the induced morphism of algebraic spaces. \\begin{enumerate} \\item For any $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\etale)$ we have $$ (Rf_*\\mathcal{F})|_{G_\\etale} = Rf_{small, *}(\\mathcal{F}|_{F_\\etale}) $$ in $D(G_\\etale)$. \\item For any object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ we have $$ (Rf_*\\mathcal{F})|_{G_\\etale} = Rf_{small, *}(\\mathcal{F}|_{F_\\etale}) $$ in $D(\\mathcal{O}_G)$. \\end{enumerate}"}
+{"_id": "11613", "title": "stacks-sheaves-lemma-compare-representable-morphism-cohomology", "text": "Let $S$ be a scheme. Consider a $2$-fibre product square $$ \\xymatrix{ \\mathcal{X}' \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Y}' \\ar[r]^g & \\mathcal{Y} } $$ of algebraic stacks over $S$. Assume that $f$ is representable by algebraic spaces and that $\\mathcal{Y}'$ is representable by an algebraic space $G'$. Then $\\mathcal{X}'$ is representable by an algebraic space $F'$ and denoting $f' : F' \\to G'$ the induced morphism of algebraic spaces we have $$ g^{-1}(Rf_*\\mathcal{F})|_{G'_\\etale} = Rf'_{small, *}((g')^{-1}\\mathcal{F}|_{F'_\\etale}) $$ for any $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\etale)$ or in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$"}
+{"_id": "11614", "title": "stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Assume \\begin{enumerate} \\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and \\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies over a morphism of schemes $f : U \\to V$ which is flat and locally of finite presentation the comparison map $c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to \\mathcal{F}|_{U_\\etale}$ of (\\ref{equation-comparison-modules}) is an isomorphism. \\end{enumerate} Then $\\mathcal{F}$ is a sheaf for the fppf topology."}
+{"_id": "11615", "title": "stacks-sheaves-lemma-compare-fppf-etale", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. Let $\\mathcal{F}$ be a presheaf $\\mathcal{O}_\\mathcal{X}$-module such that \\begin{enumerate} \\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and \\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies over a morphism of schemes $f : U \\to V$ which is flat and locally of finite presentation, the comparison map $c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to \\mathcal{F}|_{U_\\etale}$ of (\\ref{equation-comparison-modules}) is an isomorphism. \\end{enumerate} Then $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module and we have the following \\begin{enumerate} \\item If $\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$ is the comparison morphism, then $R\\epsilon_*\\mathcal{F} = \\epsilon_*\\mathcal{F}$. \\item The cohomology groups $H^p_{fppf}(\\mathcal{X}, \\mathcal{F})$ are equal to the cohomology groups computed in the \\'etale topology on $\\mathcal{X}$. Similarly for the cohomology groups $H^p_{fppf}(x, \\mathcal{F})$ and the derived versions $R\\Gamma(\\mathcal{X}, \\mathcal{F})$ and $R\\Gamma(x, \\mathcal{F})$. \\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ then $R^if_*\\mathcal{F}$ is equal to the fppf-sheafification of the higher direct image computed in the \\'etale cohomology. Similarly for derived pullback. \\end{enumerate}"}
+{"_id": "11616", "title": "stacks-sheaves-lemma-cohomology-on-subcategory", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. Let $\\tau = \\etale$ (resp.\\ $\\tau = fppf$). Let $\\mathcal{X}' \\subset \\mathcal{X}$ be a full subcategory with the following properties \\begin{enumerate} \\item if $x \\to x'$ is a morphism of $\\mathcal{X}$ which lies over a smooth (resp.\\ flat and locally finitely presented) morphism of schemes and $x' \\in \\Ob(\\mathcal{X}')$, then $x \\in \\Ob(\\mathcal{X}')$, and \\item there exists an object $x \\in \\Ob(\\mathcal{X}')$ lying over a scheme $U$ such that the associated $1$-morphism $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ is smooth and surjective. \\end{enumerate} We get a site $\\mathcal{X}'_\\tau$ by declaring a covering of $\\mathcal{X}'$ to be any family of morphisms $\\{x_i \\to x\\}$ in $\\mathcal{X}'$ which is a covering in $\\mathcal{X}_\\tau$. Then the inclusion functor $\\mathcal{X}' \\to \\mathcal{X}_\\tau$ is fully faithful, cocontinuous, and continuous, whence defines a morphism of topoi $$ g : \\Sh(\\mathcal{X}'_\\tau) \\longrightarrow \\Sh(\\mathcal{X}_\\tau) $$ and $H^p(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F}) = H^p(\\mathcal{X}_\\tau, \\mathcal{F})$ for all $p \\geq 0$ and all $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\tau)$."}
+{"_id": "11617", "title": "stacks-sheaves-proposition-quasi-coherent", "text": "Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Let $\\mathcal{X} = [U/R]$ be the quotient stack. The category of quasi-coherent modules on $\\mathcal{X}$ is equivalent to the category of quasi-coherent modules on $(U, R, s, t, c)$."}
+{"_id": "11619", "title": "stacks-sheaves-proposition-exactness-cech-complex", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. If \\begin{enumerate} \\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, and \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, \\end{enumerate} then the extended relative {\\v C}ech complex $$ \\ldots \\to 0 \\to \\mathcal{F} \\to f_{0, *}f_0^{-1}\\mathcal{F} \\to f_{1, *}f_1^{-1}\\mathcal{F} \\to f_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots $$ is exact in $\\textit{Ab}(\\mathcal{X}_\\tau)$."}
+{"_id": "11620", "title": "stacks-sheaves-proposition-smooth-covering-compute-cohomology", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of algebraic stacks. \\begin{enumerate} \\item Let $\\mathcal{F}$ be an abelian \\'etale sheaf on $\\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence $$ E_1^{p, q} = H^q_\\etale(\\mathcal{U}_p, f_p^{-1}\\mathcal{F}) \\Rightarrow H^{p + q}_\\etale(\\mathcal{X}, \\mathcal{F}) $$ \\item Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{X}$. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then there is a spectral sequence $$ E_1^{p, q} = H^q_{fppf}(\\mathcal{U}_p, f_p^{-1}\\mathcal{F}) \\Rightarrow H^{p + q}_{fppf}(\\mathcal{X}, \\mathcal{F}) $$ \\end{enumerate}"}
+{"_id": "11621", "title": "stacks-sheaves-proposition-smooth-covering-compute-direct-image", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$ be composable $1$-morphisms of algebraic stacks. \\begin{enumerate} \\item Assume that $f$ is representable by algebraic spaces, surjective and smooth. \\begin{enumerate} \\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X}_\\etale)$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ in $\\textit{Ab}(\\mathcal{Y}_\\etale)$ with higher direct images computed in the \\'etale topology. \\item If $\\mathcal{F}$ is in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ in $\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})$. \\end{enumerate} \\item Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. \\begin{enumerate} \\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ in $\\textit{Ab}(\\mathcal{Y})$ with higher direct images computed in the fppf topology. \\item If $\\mathcal{F}$ is in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ then there is a spectral sequence $$ E_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F} \\Rightarrow R^{p + q}g_*\\mathcal{F} $$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "11635", "title": "resolve-theorem-resolve", "text": "\\begin{reference} \\cite[Theorem on page 151]{Lipman} \\end{reference} Let $Y$ be a two dimensional integral Noetherian scheme. The following are equivalent \\begin{enumerate} \\item there exists an alteration $X \\to Y$ with $X$ regular, \\item there exists a resolution of singularities of $Y$, \\item $Y$ has a resolution of singularities by normalized blowups, \\item the normalization $Y^\\nu \\to Y$ is finite, $Y^\\nu$ has finitely many singular points $y_1, \\ldots, y_m$, and for each $y_i$ the completion of $\\mathcal{O}_{Y^\\nu, y_i}$ is normal. \\end{enumerate}"}
+{"_id": "11636", "title": "resolve-lemma-trace-well-defined", "text": "Let $\\varphi : R[x]/(x^p - a) \\to R[y]/(y^p - b)$ be an $R$-algebra homomorphism. Then $\\text{Tr}_x = \\text{Tr}_y \\circ \\varphi$."}
+{"_id": "11637", "title": "resolve-lemma-trace-higher", "text": "Let $\\mathbf{F}_p \\subset \\Lambda \\subset R \\subset S$ be ring extensions and assume that $S$ is isomorphic to $R[x]/(x^p - a)$ for some $a \\in R$. Then there are canonical $R$-linear maps $$ \\text{Tr} : \\Omega^{t + 1}_{S/\\Lambda} \\longrightarrow \\Omega_{R/\\Lambda}^{t + 1} $$ for $t \\geq 0$ such that $$ \\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge x^i\\text{d}x \\longmapsto \\left\\{ \\begin{matrix} 0 & \\text{if} & 0 \\leq i \\leq p - 2, \\\\ \\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge \\text{d}a & \\text{if} & i = p - 1 \\end{matrix} \\right. $$ for $\\eta_i \\in \\Omega_{R/\\Lambda}$ and such that $\\text{Tr}$ annihilates the image of $S \\otimes_R \\Omega_{R/\\Lambda}^{t + 1} \\to \\Omega_{S/\\Lambda}^{t + 1}$."}
+{"_id": "11638", "title": "resolve-lemma-trace-extends", "text": "Let $S$ be a scheme over $\\mathbf{F}_p$. Let $f : Y \\to X$ be a finite morphism of Noetherian normal integral schemes over $S$. Assume \\begin{enumerate} \\item the extension of function fields is purely inseparable of degree $p$, and \\item $\\Omega_{X/S}$ is a coherent $\\mathcal{O}_X$-module (for example if $X$ is of finite type over $S$). \\end{enumerate} For $i \\geq 1$ there is a canonical map $$ \\text{Tr} : f_*\\Omega^i_{Y/S} \\longrightarrow (\\Omega_{X/S}^i)^{**} $$ whose stalk in the generic point of $X$ recovers the trace map of Lemma \\ref{lemma-trace-higher}."}
+{"_id": "11639", "title": "resolve-lemma-blowup", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$ wotj exceptional divisor $E$. There is a closed immersion $$ r : X \\longrightarrow \\mathbf{P}^1_S $$ over $S$ such that \\begin{enumerate} \\item $r|_E : E \\to \\mathbf{P}^1_\\kappa$ is an isomorphism, \\item $\\mathcal{O}_X(E) = \\mathcal{O}_X(-1) = r^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$, and \\item $\\mathcal{C}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(1)$ and $\\mathcal{N}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$. \\end{enumerate}"}
+{"_id": "11640", "title": "resolve-lemma-blowup-regular", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$. Then $X$ is an irreducible regular scheme."}
+{"_id": "11641", "title": "resolve-lemma-blowup-pic", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$. Then $\\Pic(X) = \\mathbf{Z}$ generated by $\\mathcal{O}_X(E)$."}
+{"_id": "11642", "title": "resolve-lemma-cohomology-of-blowup", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, \\item $H^1(X, \\mathcal{O}_X(n)) = 0$ for $n \\geq -1$, \\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ or $\\mathcal{F}(1)$ is globally generated, \\item $H^0(X, \\mathcal{O}_X(n)) = \\mathfrak m^{\\max(0, n)}$, \\item $\\text{length}_A H^1(X, \\mathcal{O}_X(n)) = -n(-n - 1)/2$ if $n < 0$. \\end{enumerate}"}
+{"_id": "11643", "title": "resolve-lemma-blowup-improve", "text": "Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$. Let $\\mathfrak m^n \\subset I \\subset \\mathfrak m$ be an ideal. Let $d \\geq 0$ be the largest integer such that $$ I \\mathcal{O}_X \\subset \\mathcal{O}_X(-dE) $$ where $E$ is the exceptional divisor. Set $\\mathcal{I}' = I\\mathcal{O}_X(dE) \\subset \\mathcal{O}_X$. Then $d > 0$, the sheaf $\\mathcal{O}_X/\\mathcal{I}'$ is supported in finitely many closed points $x_1, \\ldots, x_r$ of $X$, and \\begin{align*} \\text{length}_A(A/I) & > \\text{length}_A \\Gamma(X, \\mathcal{O}_X/\\mathcal{I}') \\\\ & \\geq \\sum\\nolimits_{i = 1, \\ldots, r} \\text{length}_{\\mathcal{O}_{X, x_i}} (\\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i}) \\end{align*}"}
+{"_id": "11644", "title": "resolve-lemma-differentials-of-blowup", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$. Let $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$. Then $\\Omega_{X/S} = i_*\\Omega_{E/\\kappa}$, where $i : E \\to X$ is the immersion of the exceptional divisor."}
+{"_id": "11645", "title": "resolve-lemma-make-ideal-principal", "text": "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of closed points $x$ such that $\\mathcal{O}_{X, x}$ is regular of dimension $2$ for $x \\in T$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$. Then there exists a sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X $$ where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ such that $\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf."}
+{"_id": "11646", "title": "resolve-lemma-dominate-by-blowing-up-in-points", "text": "Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of closed points $x$ such that $\\mathcal{O}_{X, x}$ is a regular local ring of dimension $2$. Let $f : Y \\to X$ be a proper morphism of schemes which is an isomorphism over $U = X \\setminus T$. Then there exists a sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X $$ where $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ and a factorization $X_n \\to Y \\to X$ of the composition."}
+{"_id": "11647", "title": "resolve-lemma-extend-rational-map-blowing-up", "text": "Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is regular and has dimension $2$. Let $Y$ be a proper scheme over $S$. Given an $S$-rational map $f : U \\to Y$ from $X$ to $Y$ there exists a sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X $$ and an $S$-morphism $f_n : X_n \\to Y$ such that $X_{i + 1} \\to X_i$ blowing up of $X_i$ at a closed point not lying over $U$ and $f_n$ and $f$ agree."}
+{"_id": "11649", "title": "resolve-lemma-dominate-by-normalized-blowing-up", "text": "Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$. Let $f : Y \\to X$ be a proper birational morphism. Then there exists a commutative diagram $$ \\xymatrix{ X_n \\ar[r] \\ar[d] & X_{n - 1} \\ar[r] & \\ldots \\ar[r] & X_1 \\ar[r] & X_0 \\ar[d] \\\\ Y \\ar[rrrr]  & & & & X } $$ where $X_0 \\to X$ is the normalization and where $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed point."}
+{"_id": "11653", "title": "resolve-lemma-equivalence-fibre", "text": "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}). If $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$, then $X_{s_i} \\cong (Y_i)_{s_i}$ as schemes over $\\kappa(s_i)$."}
+{"_id": "11654", "title": "resolve-lemma-equivalence-sequence-blowups", "text": "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}) and assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$. Then there exists a factorization $$ X = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S $$ of $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed point $z_j$ lying over $\\{s_1, \\ldots, s_n\\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i $$ of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$."}
+{"_id": "11655", "title": "resolve-lemma-equivalence-sequence-normalized-blowups", "text": "Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}) and assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$. Assume every quasi-compact open of $S$ has finitely many irreducible components.  Then there exists a factorization $$ X = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S $$ of $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$ at a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i $$ of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$."}
+{"_id": "11656", "title": "resolve-lemma-dominate-by-scheme-modification", "text": "In Situation \\ref{situation-vanishing} there exists a $U$-admissible blowup $X' \\to S$ which dominates $X$."}
+{"_id": "11657", "title": "resolve-lemma-nice-meromorphic-function", "text": "In Situation \\ref{situation-vanishing} there exists a nonzero $f \\in \\mathfrak m$ such that for every $i = 1, \\ldots, r$ there exist \\begin{enumerate} \\item a closed point $x_i \\in C_i$ with $x_i \\not \\in C_j$ for $j \\not = i$, \\item a factorization $f = g_i f_i$ of $f$ in $\\mathcal{O}_{X, x_i}$ such that $g_i \\in \\mathfrak m_{x_i}$ maps to a nonzero element of $\\mathcal{O}_{C_i, x_i}$. \\end{enumerate}"}
+{"_id": "11658", "title": "resolve-lemma-nontrivial-normal-bundle", "text": "In Situation \\ref{situation-vanishing} assume $X$ is normal. Let $Z \\subset X$ be a nonempty effective Cartier divisor such that $Z \\subset X_s$ set theoretically. Then the conormal sheaf of $Z$ is not trivial. More precisely, there exists an $i$ such that $C_i \\subset Z$ and $\\deg(\\mathcal{C}_{Z/X}|_{C_i}) > 0$."}
+{"_id": "11659", "title": "resolve-lemma-H1-injective", "text": "In Situation \\ref{situation-vanishing} assume $X$ is normal and $A$ Nagata. The map $$ H^1(X, \\mathcal{O}_X) \\longrightarrow H^1(f^{-1}(U), \\mathcal{O}_X) $$ is injective."}
+{"_id": "11660", "title": "resolve-lemma-R1-injective", "text": "In Situation \\ref{situation-vanishing} assume $X$ is normal and $A$ Nagata. Then $$ \\Hom_{D(A)}(\\kappa[-1], Rf_*\\mathcal{O}_X) $$ is zero. This uses $D(A) = D_\\QCoh(\\mathcal{O}_S)$ to think of $Rf_*\\mathcal{O}_X$ as an object of $D(A)$."}
+{"_id": "11661", "title": "resolve-lemma-exact-sequence", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain of dimension $2$. Consider a commutative diagram $$ \\xymatrix{ X' \\ar[rd]_{f'} \\ar[rr]_g & & X \\ar[ld]^f \\\\ & \\Spec(A) } $$ where $f$ and $f'$ are modifications as in Situation \\ref{situation-vanishing} and $X$ normal. Then we have a short exact sequence $$ 0 \\to H^1(X, \\mathcal{O}_X) \\to H^1(X', \\mathcal{O}_{X'}) \\to H^0(X, R^1g_*\\mathcal{O}_{X'}) \\to 0 $$ Also $\\dim(\\text{Supp}(R^1g_*\\mathcal{O}_{X'})) = 0$ and $R^1g_*\\mathcal{O}_{X'}$ is generated by global sections."}
+{"_id": "11662", "title": "resolve-lemma-bound-a-torsion", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$. Let $a \\in A$ be nonzero. There exists an integer $N$ such that for every modification $f : X \\to \\Spec(A)$ with $X$ normal the $A$-module $$ M_{X, a} = \\Coker(A \\longrightarrow H^0(Z, \\mathcal{O}_Z)) $$ where $Z \\subset X$ is cut out by $a$ has length bounded by $N$."}
+{"_id": "11663", "title": "resolve-lemma-rational-propagates", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity. Let $A \\subset B$ be a local extension of domains with the same fraction field which is essentially of finite type such that $\\dim(B) = 2$ and $B$ normal. Then $B$ defines a rational singularity."}
+{"_id": "11664", "title": "resolve-lemma-reduce-to-rational", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$. If reduction to rational singularities is possible for $A$, then there exists a finite sequence of normalized blowups $$ X = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A) $$ in closed points such that for any closed point $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ defines a rational singularity. In particular $X \\to \\Spec(A)$ is a modification and $X$ is a normal scheme projective over $A$."}
+{"_id": "11665", "title": "resolve-lemma-go-up-separable", "text": "Let $A \\to B$ be a finite injective local ring map of local normal Nagata domains of dimension $2$. Assume that the induced extension of fraction fields is separable. If reduction to rational singularities is possible for $A$ then it is possible for $B$."}
+{"_id": "11666", "title": "resolve-lemma-regular-rational", "text": "Let $A$ be a Nagata regular local ring of dimension $2$. Then $A$ defines a rational singularity."}
+{"_id": "11667", "title": "resolve-lemma-bound-dualizing-implies-bound", "text": "Let $A$ be a local normal Nagata domain of dimension $2$ which has a dualizing complex $\\omega_A^\\bullet$. If there exists a nonzero $d \\in A$ such that for all normal modifications $X \\to \\Spec(A)$ the cokernel of the trace map $$ \\Gamma(X, \\omega_X) \\to \\omega_A $$ is annihilated by $d$, then reduction to rational singularities is possible for $A$."}
+{"_id": "11668", "title": "resolve-lemma-compare-differentials-dualizing", "text": "Let $p$ be a prime number. Let $A$ be a regular local ring of dimension $2$ and characteristic $p$. Let $A_0 \\subset A$ be a subring such that $\\Omega_{A/A_0}$ is free of rank $r < \\infty$. Set $\\omega_A = \\Omega^r_{A/A_0}$. If $X \\to \\Spec(A)$ is the result of a sequence of blowups in closed points, then there exists a map $$ \\varphi_X : (\\Omega^r_{X/\\Spec(A_0)})^{**} \\longrightarrow \\omega_X $$ extending the given identification in the generic point."}
+{"_id": "11669", "title": "resolve-lemma-go-up-degree-p", "text": "Let $p$ be a prime number. Let $A$ be a complete regular local ring of dimension $2$ and characteristic $p$. Let $L/K$ be a degree $p$ inseparable extension of the fraction field $K$ of $A$. Let $B \\subset L$ be the integral closure of $A$. Then reduction to rational singularities is possible for $B$."}
+{"_id": "11670", "title": "resolve-lemma-globally-generated", "text": "In Situation \\ref{situation-rational}. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and \\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated. \\end{enumerate}"}
+{"_id": "11671", "title": "resolve-lemma-sections-powers-I-rational", "text": "In Situation \\ref{situation-rational} assume $E = X_s$ is an effective Cartier divisor. Let $\\mathcal{I}$ be the ideal sheaf of $E$. Then $H^0(X, \\mathcal{I}^n) = \\mathfrak m^n$ and $H^1(X, \\mathcal{I}^n) = 0$."}
+{"_id": "11672", "title": "resolve-lemma-blow-up-normal-rational", "text": "In Situation \\ref{situation-rational} the blowup of $\\Spec(A)$ in $\\mathfrak m$ is normal."}
+{"_id": "11673", "title": "resolve-lemma-cohomology-blow-up-rational", "text": "In Situation \\ref{situation-rational}. Let $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$. Let $E \\subset X$ be the exceptional divisor. With $\\mathcal{O}_X(1) = \\mathcal{I}$ as usual and $\\mathcal{O}_E(1) = \\mathcal{O}_X(1)|_E$ we have \\begin{enumerate} \\item $E$ is a proper Cohen-Macaulay curve over $\\kappa$. \\item $\\mathcal{O}_E(1)$ is very ample \\item $\\deg(\\mathcal{O}_E(1)) \\geq 1$ and equality holds only if $A$ is a regular local ring, \\item $H^1(E, \\mathcal{O}_E(n)) = 0$ for $n \\geq 0$, and \\item $H^0(E, \\mathcal{O}_E(n)) = \\mathfrak m^n/\\mathfrak m^{n + 1}$ for $n \\geq 0$. \\end{enumerate}"}
+{"_id": "11674", "title": "resolve-lemma-dualizing-rational", "text": "In Situation \\ref{situation-rational} assume $A$ has a dualizing complex $\\omega_A^\\bullet$. With $\\omega_X$ the dualizing module of $X$, the trace map $H^0(X, \\omega_X) \\to \\omega_A$ is an isomorphism and consequently there is a canonical map $f^*\\omega_A \\to \\omega_X$."}
+{"_id": "11675", "title": "resolve-lemma-dualizing-blow-up-rational", "text": "In Situation \\ref{situation-rational} assume $A$ has a dualizing complex $\\omega_A^\\bullet$ and is not regular. Let $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$ with exceptional divisor $E \\subset X$. Let $\\omega_X$ be the dualizing module of $X$. Then \\begin{enumerate} \\item $\\omega_E = \\omega_X|_E \\otimes \\mathcal{O}_E(-1)$, \\item $H^1(X, \\omega_X(n)) = 0$ for $n \\geq 0$, \\item the map $f^*\\omega_A \\to \\omega_X$ of Lemma \\ref{lemma-dualizing-rational} is surjective. \\end{enumerate}"}
+{"_id": "11676", "title": "resolve-lemma-rational-to-gorenstein", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity. Assume $A$ has a dualizing complex. Then there exists a finite sequence of blowups in singular closed points $$ X = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A) $$ such that $X_i$ is normal for each $i$ and such that the dualizing sheaf $\\omega_X$ of $X$ is an invertible $\\mathcal{O}_X$-module."}
+{"_id": "11678", "title": "resolve-lemma-sequence-blowups-along-arc-becomes-nonsingular", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain of dimension $2$. Let $A \\to R$ be a surjection onto a complete discrete valuation ring. This defines a nonsingular arc $a : T = \\Spec(R) \\to \\Spec(A)$. Let $$ \\Spec(A) = X_0 \\leftarrow X_1 \\leftarrow X_2 \\leftarrow X_3 \\leftarrow \\ldots $$ be the sequence of blowing ups constructed from $a$. If $A_\\mathfrak p$ is a regular local ring where $\\mathfrak p = \\Ker(A \\to R)$, then for some $i$ the scheme $X_i$ is regular at $x_i$."}
+{"_id": "11679", "title": "resolve-lemma-iso-completions", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated maximal ideal $\\mathfrak m$. Let $X$ be a scheme over $A$. Let $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ where $A^\\wedge$ is the $\\mathfrak m$-adic completion of $A$. For a point $q \\in Y$ with image $p \\in X$ lying over the closed point of $\\Spec(A)$ the local ring map $\\mathcal{O}_{X, p} \\to \\mathcal{O}_{Y, q}$ induces an isomorphism on completions."}
+{"_id": "11680", "title": "resolve-lemma-port-regularity-to-completion", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type. Set $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in Y$ with image $x \\in X$. Then \\begin{enumerate} \\item if $\\mathcal{O}_{Y, y}$ is regular, then $\\mathcal{O}_{X, x}$ is regular, \\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}$ is regular $\\Leftrightarrow \\mathcal{O}_{X, x}$ is regular, and \\item If $X$ is proper over $A$, then $X$ is regular if and only if $Y$ is regular. \\end{enumerate}"}
+{"_id": "11681", "title": "resolve-lemma-descend-admissible-blowup", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring with completion $A^\\wedge$. Let $U \\subset \\Spec(A)$ and $U^\\wedge \\subset \\Spec(A^\\wedge)$ be the punctured spectra. If $Y \\to \\Spec(A^\\wedge)$ is a $U^\\wedge$-admissible blowup, then there exists a $U$-admissible blowup $X \\to \\Spec(A)$ such that $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$."}
+{"_id": "11683", "title": "resolve-lemma-formally-unramified", "text": "Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let $X$ be a scheme over $A$. Assume \\begin{enumerate} \\item $A$ is analytically unramified (Algebra, Definition \\ref{algebra-definition-analytically-unramified}), \\item $X$ is locally of finite type over $A$, and \\item $X \\to \\Spec(A)$ is \\'etale at the generic points of irreducible components of $X$. \\end{enumerate} Then the normalization of $X$ is finite over $X$."}
+{"_id": "11684", "title": "resolve-lemma-normalization-completion", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type. Set $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. If the complement of the special fibre in $Y$ is normal, then the normalization $X^\\nu \\to X$ is finite and the base change of $X^\\nu$ to $\\Spec(A^\\wedge)$ recovers the normalization of $Y$."}
+{"_id": "11685", "title": "resolve-lemma-normalized-blowup-completion", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain whose completion $A^\\wedge$ is normal. Then given any sequence $$ Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to \\Spec(A^\\wedge) $$ of normalized blowups, there exists a sequence of (proper) normalized blowups $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to \\Spec(A) $$ whose base change to $A^\\wedge$ recovers the given sequence."}
+{"_id": "11687", "title": "resolve-lemma-resolve-rational-double-points", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity, whose completion is normal, and which is Gorenstein. Then there exists a finite sequence of blowups in singular closed points $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A) $$ such that $X_n$ is regular and such that each intervening schemes $X_i$ is normal with finitely many singular points of the same type."}
+{"_id": "11688", "title": "resolve-lemma-regular-alteration-implies", "text": "Let $Y$ be a Noetherian integral scheme. Assume there exists an alteration $f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$ is finite and $Y$ has a dense open which is regular."}
+{"_id": "11689", "title": "resolve-lemma-algebra-helper", "text": "Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let $B \\subset C$ be finite $A$-algebras. Assume that (a) $B$ is a normal ring, and (b) the $\\mathfrak m$-adic completion $C^\\wedge$ is a normal ring. Then $B^\\wedge$ is a normal ring."}
+{"_id": "11690", "title": "resolve-lemma-regular-alteration-implies-local", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain. Assume there exists an alteration $f : X \\to \\Spec(A)$ with $X$ regular. Then \\begin{enumerate} \\item there exists a nonzero $f \\in A$ such that $A_f$ is regular, \\item the integral closure $B$ of $A$ in its fraction field is finite over $A$, \\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the completions of $B$ at its maximal ideals are normal domains, and \\item the generic formal fibre of $A$ is regular. \\end{enumerate}"}
+{"_id": "11691", "title": "resolve-lemma-existence-implies-existence-by-normalized-blowing-ups", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Assume $A$ is normal and has dimension $2$. If $\\Spec(A)$ has a resolution of singularities, then $\\Spec(A)$ has a resolution by normalized blowups."}
+{"_id": "11692", "title": "resolve-lemma-resolve-complete", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian complete local ring. Assume $A$ is a normal domain of dimension $2$. Then $\\Spec(A)$ has a resolution of singularities."}
+{"_id": "11693", "title": "resolve-lemma-resolve-curve", "text": "Let $Y$ be a one dimensional integral Noetherian scheme. The following are equivalent \\begin{enumerate} \\item there exists an alteration $X \\to Y$ with $X$ regular, \\item there exists a resolution of singularities of $Y$, \\item there exists a finite sequence $Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y$ of blowups in closed points with $Y_n$ regular, and \\item the normalization $Y^\\nu \\to Y$ is finite. \\end{enumerate}"}
+{"_id": "11694", "title": "resolve-lemma-blowup-curve", "text": "Let $X$ be a Noetherian scheme. Let $Y \\subset X$ be an integral closed subscheme of dimension $1$ satisfying the equivalent conditions of Lemma \\ref{lemma-resolve-curve}. Then there exists a finite sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X $$ of blowups in closed points such that the strict transform of $Y$ in $X_n$ is a regular curve."}
+{"_id": "11695", "title": "resolve-lemma-blowup-nonsingular-curves-meeting-at-point", "text": "In the situation above let $X' \\to X$ be the blowing up of $X$ in $p$. Let $Y', Z' \\subset X'$ be the strict transforms of $Y, Z$. If $\\mathcal{O}_{Y, p}$ is regular, then \\begin{enumerate} \\item $Y' \\to Y$ is an isomorphism, \\item $Y'$ meets the exceptional fibre $E \\subset X'$ in one point $q$ and $m_q(Y \\cap E) = 1$, \\item if $q \\in Z'$ too, then $m_q(Y \\cap Z') < m_p(Y \\cap Z)$. \\end{enumerate}"}
+{"_id": "11696", "title": "resolve-lemma-blowup-curves", "text": "Let $X$ be a Noetherian scheme. Let $Y_i \\subset X$, $i = 1, \\ldots, n$ be an integral closed subschemes of dimension $1$ each satisfying the equivalent conditions of Lemma \\ref{lemma-resolve-curve}. Then there exists a finite sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X $$ of blowups in closed points such that the strict transform $Y'_i \\subset X_n$ of $Y_i$ in $X_n$ are pairwise disjoint regular curves."}
+{"_id": "11697", "title": "resolve-lemma-turn-into-effective-Cartier", "text": "Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$ be a proper closed subscheme. There exists a sequence $$ X_n \\to \\ldots \\to X_1 \\to X $$ of blowing ups in closed points such that the inverse image $Z_n$ of $Z$ in $X_n$ is an effective Cartier divisor."}
+{"_id": "11699", "title": "resolve-lemma-factor-through-contraction", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an exceptional curve of the first kind. If a contraction $X \\to X'$ of $E$ exists, then it has the following universal property: for every morphism $\\varphi : X \\to Y$ such that $\\varphi(E)$ is a point, there is a unique factorization $X \\to X' \\to Y$ of $\\varphi$."}
+{"_id": "11700", "title": "resolve-lemma-contraction-unique", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an exceptional curve of the first kind. If there exists a contraction of $E$, then it is unique up to unique isomorphism."}
+{"_id": "11701", "title": "resolve-lemma-exceptional-first-kind-local", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an exceptional curve of the first kind. Let $E_n = nE$ and denote $\\mathcal{O}_n$ its structure sheaf. Then $$ A = \\lim H^0(E_n, \\mathcal{O}_n) $$ is a complete local Noetherian regular local ring of dimension $2$ and $\\Ker(A \\to H^0(E_n, \\mathcal{O}_n))$ is the $n$th power of its maximal ideal."}
+{"_id": "11702", "title": "resolve-lemma-contraction", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an exceptional curve of the first kind. If there exists a morphism $f : X \\to Y$ such that \\begin{enumerate} \\item $Y$ is Noetherian, \\item $f$ is proper, \\item $f$ maps $E$ to a point $y$ of $Y$, \\item $f$ is quasi-finite at every point not in $E$, \\end{enumerate} Then there exists a contraction of $E$ and it is the Stein factorization of $f$."}
+{"_id": "11703", "title": "resolve-lemma-pic-blowup", "text": "Let $b : X \\to X'$ be the contraction of an exceptional curve of the first kind $E \\subset X$. Then there is a short exact sequence $$ 0 \\to \\Pic(X') \\to \\Pic(X) \\to \\mathbf{Z} \\to 0 $$ where the first map is pullback by $b$ and the second map sends $\\mathcal{L}$ to the degree of $\\mathcal{L}$ on the exceptional curve $E$. The sequence is split by the map $n \\mapsto \\mathcal{O}_X(-nE)$."}
+{"_id": "11704", "title": "resolve-lemma-lift-sections-and-h1", "text": "Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an exceptional curve of the first kind. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $n$ be the integer such that $\\mathcal{L}|_E$ has degree $n$ viewed as an invertible module on $\\mathbf{P}^1$. Then \\begin{enumerate} \\item If $H^1(X, \\mathcal{L}) = 0$ and $n \\geq 0$, then $H^1(X, \\mathcal{L}(iE)) = 0$ for $0 \\leq i \\leq n + 1$. \\item If $n \\leq 0$, then $H^1(X, \\mathcal{L}) \\subset H^1(X, \\mathcal{L}(E))$. \\end{enumerate}"}
+{"_id": "11705", "title": "resolve-lemma-contract-ample", "text": "Let $S = \\Spec(R)$ be an affine Noetherian scheme. Let $X \\to S$ be a proper morphism. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$. Let $E \\subset X$ be an exceptional curve of the first kind. Then \\begin{enumerate} \\item there exists a contraction $b : X \\to X'$ of $E$, \\item $X'$ is proper over $S$, and \\item the invertible $\\mathcal{O}_{X'}$-module $\\mathcal{L}'$ is ample with $\\mathcal{L}'$ as in Remark \\ref{remark-pic-blowup}. \\end{enumerate}"}
+{"_id": "11706", "title": "resolve-lemma-contract-when-quasi-projective", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of finite type. Let $E \\subset X$ be an exceptional curve of the first kind which is in a fibre of $f$. \\begin{enumerate} \\item If $X$ is projective over $S$, then there exists a contraction $X \\to X'$ of $E$ and $X'$ is projective over $S$. \\item If $X$ is quasi-projective over $S$, then there exists a contraction $X \\to X'$ of $E$ and $X'$ is quasi-projective over $S$. \\end{enumerate}"}
+{"_id": "11707", "title": "resolve-lemma-regular-dim-2-quasi-projective", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a separated morphism of finite type with $X$ regular of dimension $2$. Then $X$ is quasi-projective over $S$."}
+{"_id": "11708", "title": "resolve-lemma-regular-dim-2-projective", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a proper morphism with $X$ regular of dimension $2$. Then $X$ is projective over $S$."}
+{"_id": "11709", "title": "resolve-lemma-proper-birational-regular-surfaces", "text": "Let $f : X \\to Y$ be a proper birational morphism between integral Noetherian schemes regular of dimension $2$. Then $f$ is a sequence of blowups in closed points."}
+{"_id": "11710", "title": "resolve-lemma-birational-regular-surfaces", "text": "Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be proper integral schemes over $S$ which are regular of dimension $2$. Then $X$ and $Y$ are $S$-birational if and only if there exists a diagram of $S$-morphisms $$ X = X_0 \\leftarrow X_1 \\leftarrow \\ldots \\leftarrow X_n = Y_m \\to \\ldots \\to Y_1 \\to Y_0 = Y $$ where each morphism is a blowup in a closed point."}
+{"_id": "11711", "title": "resolve-proposition-Grauert-Riemenschneider", "text": "In Situation \\ref{situation-vanishing} assume \\begin{enumerate} \\item $X$ is a normal scheme, \\item $A$ is Nagata and has a dualizing complex $\\omega_A^\\bullet$. \\end{enumerate} Let $\\omega_X$ be the dualizing module of $X$ (Remark \\ref{remark-dualizing-setup}). Then $R^1f_*\\omega_X = 0$."}
+{"_id": "11783", "title": "spaces-duality-lemma-equivalent-definitions", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item For every \\'etale morphism $U \\to X$ where $U$ is a scheme the restriction $K|_U$ is a dualizing complex for $U$ (as discussed above). \\item There exists a surjective \\'etale morphism $U \\to X$ where $U$ is a scheme such that $K|_U$ is a dualizing complex for $U$. \\end{enumerate}"}
+{"_id": "11784", "title": "spaces-duality-lemma-affine-duality", "text": "Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let $\\mathcal{O}_\\etale$ be the structure sheaf of $X$ on the small \\'etale site of $X$. Let $K, L$ be objects of $D(A)$. If $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective dimension, then $$ \\epsilon^*\\widetilde{R\\Hom_A(K, L)} = R\\SheafHom_{\\mathcal{O}_\\etale}(\\epsilon^*\\widetilde{K}, \\epsilon^*\\widetilde{L}) $$ in $D(\\mathcal{O}_\\etale)$ where $\\epsilon : (X_\\etale, \\mathcal{O}_\\etale) \\to (X, \\mathcal{O}_X)$ is as in Derived Categories of Spaces, Section \\ref{spaces-perfect-section-derived-quasi-coherent-etale}."}
+{"_id": "11788", "title": "spaces-duality-lemma-twisted-inverse-image", "text": "\\begin{reference} This is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}. \\end{reference} Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact algebraic spaces over $S$. The functor $Rf_* : D_\\QCoh(X) \\to D_\\QCoh(Y)$ has a right adjoint."}
+{"_id": "11789", "title": "spaces-duality-lemma-twisted-inverse-image-bounded-below", "text": "Notation and assumptions as in Lemma \\ref{lemma-twisted-inverse-image}. Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ be the right adjoint to $Rf_*$. Then $a$ maps $D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$. In fact, there exists an integer $N$ such that $H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$."}
+{"_id": "11790", "title": "spaces-duality-lemma-iso-on-RSheafHom", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $a$ be the right adjoint to $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$. Let $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$. Then the map (\\ref{equation-sheafy-trace}) $$ Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\longrightarrow R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) $$ becomes an isomorphism after applying the functor $DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$ discussed in Derived Categories of Spaces, Section \\ref{spaces-perfect-section-better-coherator}."}
+{"_id": "11791", "title": "spaces-duality-lemma-iso-global-hom", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact algebraic spaces over $S$. For all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$ (\\ref{equation-sheafy-trace}) induces an isomorphism $R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs."}
+{"_id": "11792", "title": "spaces-duality-lemma-flat-precompose-pus", "text": "In diagram (\\ref{equation-base-change}) the map $a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism."}
+{"_id": "11793", "title": "spaces-duality-lemma-compose-base-change-maps", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z' \\ar[r]^m & Z } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where both diagrams are cartesian and where $f$ and $l$ as well as $g$ and $m$ are Tor independent. Then the maps (\\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement)."}
+{"_id": "11794", "title": "spaces-duality-lemma-compose-base-change-maps-horizontal", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where both diagrams are cartesian and where $f$ and $h$ as well as $f'$ and $h'$ are Tor independent. Then the maps (\\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement)."}
+{"_id": "11796", "title": "spaces-duality-lemma-trace-map-and-base-change", "text": "Suppose we have a diagram (\\ref{equation-base-change}). Then the maps $1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and $\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$ agree via the base change maps $\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$ (Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}) and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$ (\\ref{equation-base-change-map}). More precisely, the diagram $$ \\xymatrix{ Lg^* \\circ Rf_* \\circ a \\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} & Lg^* \\\\ Rf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} & Rf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1} } $$ of transformations of functors commutes."}
+{"_id": "11797", "title": "spaces-duality-lemma-unit-and-base-change", "text": "Suppose we have a diagram (\\ref{equation-base-change}). Then the maps $1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and $\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$ agree via the base change maps $\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$ (Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change}) and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$ (\\ref{equation-base-change-map}). More precisely, the diagram $$ \\xymatrix{ L(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} & L(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\ a' \\circ Rf'_* \\circ L(g')^* & a' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta } $$ of transformations of functors commutes."}
+{"_id": "11798", "title": "spaces-duality-lemma-compare-with-pullback-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. The map $Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to a(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$ defined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$ is an isomorphism if $K$ is perfect. In particular, (\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect."}
+{"_id": "11800", "title": "spaces-duality-lemma-proper-flat", "text": "Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums."}
+{"_id": "11802", "title": "spaces-duality-lemma-properties-relative-dualizing", "text": "Let $Y$ be an affine scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then \\begin{enumerate} \\item $a(\\mathcal{O}_Y)$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$, \\item $Rf_*a(\\mathcal{O}_Y)$ has vanishing cohomology sheaves in positive degrees, \\item $\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))$ is an isomorphism. \\end{enumerate}"}
+{"_id": "11803", "title": "spaces-duality-lemma-relative-dualizing-RHom", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\\omega_{X/Y}^\\bullet, \\tau)$ is a relative dualizing complex, then  $\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$ is an isomorphism and $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology sheaves in positive degrees."}
+{"_id": "11804", "title": "spaces-duality-lemma-uniqueness-relative-dualizing", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\\omega_j^\\bullet, \\tau_j)$, $j = 1, 2$ are two relative dualizing complexes on $X/Y$, then there is a unique isomorphism $(\\omega_1^\\bullet, \\tau_1) \\to (\\omega_2^\\bullet, \\tau_2)$."}
+{"_id": "11805", "title": "spaces-duality-lemma-covering-enough", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. Let $(\\omega^\\bullet, \\tau)$ be a pair consisting of a $Y$-perfect object of $D(\\mathcal{O}_X)$ and a map $\\tau : Rf_*\\omega^\\bullet \\to \\mathcal{O}_Y$. Assume we have cartesian diagrams $$ \\xymatrix{ X_i \\ar[r]_{g_i'} \\ar[d]_{f_i} & X \\ar[d]^f \\\\ Y_i \\ar[r]^{g_i} & Y } $$ with $Y_i$ affine such that $\\{g_i : Y_i \\to Y\\}$ is an \\'etale covering and isomorphisms of pairs $(\\omega^\\bullet|_{X_i}, \\tau|_{Y_i}) \\to (a_i(\\mathcal{O}_{Y_i}), \\text{Tr}_{f_i, \\mathcal{O}_{Y_i}})$ as in Definition \\ref{definition-relative-dualizing-proper-flat}. Then $(\\omega^\\bullet, \\tau)$ is a relative dualizing complex for $X$ over $Y$."}
+{"_id": "11813", "title": "spaces-properties-theorem-exactness-stalks", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective (resp.\\ surjective) if and only if the map on stalks $a_{\\overline{x}} : \\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$ is injective (resp.\\ surjective) for all geometric points of $X$. A sequence of abelian sheaves on $X_\\etale$ is exact if and only if it is exact on all stalks at geometric points of $S$."}
+{"_id": "11814", "title": "spaces-properties-theorem-fully-faithful", "text": "Let $X$, $Y$ be algebraic spaces over $\\Spec(\\mathbf{Z})$. Let $$ (g, g^\\sharp) : (\\Sh(X_\\etale), \\mathcal{O}_X) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{O}_Y) $$ be a morphism of locally ringed topoi. Then there exists a unique morphism of algebraic spaces $f : X \\to Y$ such that $(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$. In other words, the construction $$ \\textit{Spaces}/\\Spec(\\mathbf{Z}) \\longrightarrow \\textit{Locally ringed topoi}, \\quad X \\longrightarrow (X_\\etale, \\mathcal{O}_X) $$ is fully faithful (morphisms up to $2$-isomorphisms on the right hand side)."}
+{"_id": "11815", "title": "spaces-properties-lemma-trivial-implications", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have the following implications among the separation axioms of Definition \\ref{definition-separated}: \\begin{enumerate} \\item separated implies all the others, \\item quasi-separated implies Zariski locally quasi-separated. \\end{enumerate}"}
+{"_id": "11816", "title": "spaces-properties-lemma-characterize-quasi-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is a quasi-separated algebraic space, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \\times_X V$ is quasi-compact, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine the fibre product $U \\times_X V$ is quasi-compact. \\end{enumerate}"}
+{"_id": "11818", "title": "spaces-properties-lemma-scheme-points", "text": "Let $S$ be a scheme. Let $X$ be a scheme over $S$. The points of $X$ as a scheme are in canonical 1-1 correspondence with the points of $X$ as an algebraic space."}
+{"_id": "11819", "title": "spaces-properties-lemma-points-cartesian", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\times_Y X \\ar[r] \\ar[d] & X \\ar[d] \\\\ Z \\ar[r] & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Then the map of sets of points $$ |Z \\times_Y X| \\longrightarrow |Z| \\times_{|Y|} |X| $$ is surjective."}
+{"_id": "11820", "title": "spaces-properties-lemma-characterize-surjective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f : T \\to X$ be a morphism from a scheme to $X$. The following are equivalent \\begin{enumerate} \\item $f : T \\to X$ is surjective (according to Spaces, Definition \\ref{spaces-definition-relative-representable-property}), and \\item $|f| : |T| \\to |X|$ is surjective. \\end{enumerate}"}
+{"_id": "11821", "title": "spaces-properties-lemma-points-presentation", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = U/R$ be a presentation of $X$, see Spaces, Definition \\ref{spaces-definition-presentation}. Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation and $|X|$ is the quotient of $|U|$ by this equivalence relation."}
+{"_id": "11822", "title": "spaces-properties-lemma-topology-points", "text": "Let $S$ be a scheme. There exists a unique topology on the sets of points of algebraic spaces over $S$ with the following properties: \\begin{enumerate} \\item if $X$ is a scheme over $S$, then the topology on $|X|$ is the usual one (via the identification of Lemma \\ref{lemma-scheme-points}), \\item for every morphism of algebraic spaces $X \\to Y$ over $S$ the map $|X| \\to |Y|$ is continuous, and \\item for every \\'etale morphism $U \\to X$ with $U$ a scheme the map of topological spaces $|U| \\to |X|$ is continuous and open. \\end{enumerate}"}
+{"_id": "11823", "title": "spaces-properties-lemma-open-subspaces", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item The rule $X' \\mapsto |X'|$ defines an inclusion preserving bijection between open subspaces $X'$ (see Spaces, Definition \\ref{spaces-definition-immersion}) of $X$, and opens of the topological space $|X|$. \\item A family $\\{X_i \\subset X\\}_{i \\in I}$ of open subspaces of $X$ is a Zariski covering (see Spaces, Definition \\ref{spaces-definition-Zariski-open-covering}) if and only if $|X| = \\bigcup |X_i|$. \\end{enumerate} In other words, the small Zariski site $X_{Zar}$ of $X$ is canonically identified with a site associated to the topological space $|X|$ (see Sites, Example \\ref{sites-example-site-topological})."}
+{"_id": "11824", "title": "spaces-properties-lemma-factor-through-open-subspace", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $X' \\subset X$ be an open subspace. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Then $f$ factors through $X'$ if and only if $|f| : |Y| \\to |X|$ factors through $|X'| \\subset |X|$."}
+{"_id": "11825", "title": "spaces-properties-lemma-etale-image-open", "text": "Let $S$ be a scheme. Let $X$ be an algebraic spaces over $S$. Let $U$ be a scheme and let $f : U \\to X$ be an \\'etale morphism. Let $X' \\subset X$ be the open subspace corresponding to the open $|f|(|U|) \\subset |X|$ via Lemma \\ref{lemma-open-subspaces}. Then $f$ factors through a surjective \\'etale morphism $f' : U \\to X'$. Moreover, if $R = U \\times_X U$, then $R = U \\times_{X'} U$ and $X'$ has the presentation $X' = U/R$."}
+{"_id": "11826", "title": "spaces-properties-lemma-points-monomorphism", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map $$ \\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\} \\longrightarrow |X| $$ This map is injective."}
+{"_id": "11827", "title": "spaces-properties-lemma-quasi-compact-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact."}
+{"_id": "11829", "title": "spaces-properties-lemma-space-locally-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Every point of $|X|$ has a fundamental system of open quasi-compact neighbourhoods. In particular $|X|$ is locally quasi-compact in the sense of Topology, Definition \\ref{topology-definition-locally-quasi-compact}."}
+{"_id": "11830", "title": "spaces-properties-lemma-cover-by-union-affines", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a surjective \\'etale morphism $U \\to X$ where $U$ is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of $S$."}
+{"_id": "11831", "title": "spaces-properties-lemma-union-of-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a Zariski covering $X = \\bigcup X_i$ such that each algebraic space $X_i$ has a surjective \\'etale covering by an affine scheme. We may in addition assume each $X_i$ maps into an affine open of $S$."}
+{"_id": "11832", "title": "spaces-properties-lemma-quasi-compact-affine-cover", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if there exists an \\'etale surjective morphism $U \\to X$ with $U$ an affine scheme."}
+{"_id": "11833", "title": "spaces-properties-lemma-separated-cover", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a separated scheme and $U \\to X$ \\'etale. Then $U \\to X$ is separated, and $R = U \\times_X U$ is a separated scheme."}
+{"_id": "11834", "title": "spaces-properties-lemma-quasi-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a quasi-separated scheme $U$ and a surjective \\'etale morphism $U \\to X$ such that either of the projections $U \\times_X U \\to U$ is quasi-compact, then $X$ is quasi-separated."}
+{"_id": "11835", "title": "spaces-properties-lemma-quasi-separated-quasi-compact-pieces", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is Zariski locally quasi-separated over $S$, \\item $X$ is Zariski locally quasi-separated, \\item there exists a Zariski open covering $X = \\bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ and a quasi-compact surjective \\'etale morphism $U_i \\to X_i$, and \\item there exists a Zariski open covering $X = \\bigcup X_i$ such that for each $i$ there exists an affine scheme $U_i$ which maps into an affine open of $S$ and a quasi-compact surjective \\'etale morphism $U_i \\to X_i$. \\end{enumerate}"}
+{"_id": "11836", "title": "spaces-properties-lemma-finite-fibres-presentation", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a scheme. Let $\\varphi : U \\to X$ be an \\'etale morphism such that the projections $R = U \\times_X U \\to U$ are quasi-compact; for example if $\\varphi$ is quasi-compact. Then the fibres of $$ |U| \\to |X| \\quad\\text{and}\\quad |R| \\to |X| $$ are finite."}
+{"_id": "11837", "title": "spaces-properties-lemma-type-property", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{P}$ be a property of schemes which is local in the \\'etale topology, see Descent, Definition \\ref{descent-definition-property-local}. The following are equivalent \\begin{enumerate} \\item for some scheme $U$ and surjective \\'etale morphism $U \\to X$ the scheme $U$ has property $\\mathcal{P}$, and \\item for every scheme $U$ and every \\'etale morphism $U \\to X$ the scheme $U$ has property $\\mathcal{P}$. \\end{enumerate} If $X$ is representable this is equivalent to $\\mathcal{P}(X)$."}
+{"_id": "11838", "title": "spaces-properties-lemma-local-source-target-at-point", "text": "Let $\\mathcal{P}$ be a property of germs of schemes which is \\'etale local, see Descent, Definition \\ref{descent-definition-local-at-point}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point of $X$. Consider \\'etale morphisms $a : U \\to X$ where $U$ is a scheme. The following are equivalent \\begin{enumerate} \\item for any $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$, and \\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$. \\end{enumerate} If $X$ is representable, then this is equivalent to $\\mathcal{P}(X, x)$."}
+{"_id": "11839", "title": "spaces-properties-lemma-locally-constructible", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\subset |X|$ be a subset. The following are equivalent \\begin{enumerate} \\item for every \\'etale morphism $U \\to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$, \\item for every \\'etale morphism $U \\to X$ where $U$ is an affine scheme the inverse image of $E$ in $U$ is a constructible subset of $U$, \\item for some surjective \\'etale morphism $U \\to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$. \\end{enumerate}"}
+{"_id": "11840", "title": "spaces-properties-lemma-pre-dimension-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are equivalent \\begin{enumerate} \\item for some scheme $U$ and \\'etale morphism $a : U \\to X$ and point $u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$, \\item for any scheme $U$, any \\'etale morphism $a : U \\to X$, and any point $u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$. \\end{enumerate} If $X$ is a scheme, this is equivalent to $\\dim(\\mathcal{O}_{X, x}) = d$."}
+{"_id": "11841", "title": "spaces-properties-lemma-dimension", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following quantities are equal: \\begin{enumerate} \\item The dimension of $X$. \\item The supremum of the dimensions of the local rings of $X$. \\item The supremum of $\\dim_x(X)$ for $x \\in |X|$. \\end{enumerate}"}
+{"_id": "11842", "title": "spaces-properties-lemma-codimension-0-points", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $x \\in |X|$. Consider \\'etale morphisms $a : U \\to X$ where $U$ is a scheme. The following are equivalent \\begin{enumerate} \\item $x$ is a point of codimension $0$ on $X$, \\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$, the point $u$ is the generic point of an irreducible component of $U$, and \\item for any $U \\to X$ as above and any $u \\in U$ mapping to $x$, the point $u$ is the generic point of an irreducible component of $U$. \\end{enumerate} If $X$ is representable, this is equivalent to $x$ being a generic point of an irreducible component of $|X|$."}
+{"_id": "11843", "title": "spaces-properties-lemma-codimension-0-points-dense", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. The set of codimension $0$ points of $X$ is dense in $|X|$."}
+{"_id": "11844", "title": "spaces-properties-lemma-subspace-induced-topology", "text": "Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces. Then $|Z| \\to |X|$ is a homeomorphism of $|Z|$ onto a locally closed subset of $|X|$."}
+{"_id": "11845", "title": "spaces-properties-lemma-subspaces-presentation", "text": "Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be an \\'etale equivalence relation. Let $X = U/R$ be the associated algebraic space (Spaces, Theorem \\ref{spaces-theorem-presentation}). There is a canonical bijection $$ R\\text{-invariant locally closed subschemes }Z'\\text{ of }U \\leftrightarrow \\text{locally closed subspaces }Z\\text{ of }X $$ Moreover, if $Z \\to X$ is closed (resp.\\ open) if and only if $Z' \\to U$ is closed (resp.\\ open)."}
+{"_id": "11846", "title": "spaces-properties-lemma-reduced-closed-subspace", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. There exists a unique closed subspace $Z \\subset X$ with the following properties: (a) we have $|Z| = T$, and (b) $Z$ is reduced."}
+{"_id": "11847", "title": "spaces-properties-lemma-map-into-reduction", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Z \\subset X$ be a closed subspace. Assume $Y$ is reduced. A morphism $f : Y \\to X$ factors through $Z$ if and only if $f(|Y|) \\subset |Z|$."}
+{"_id": "11848", "title": "spaces-properties-lemma-subscheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a largest open subspace $X' \\subset X$ which is a scheme."}
+{"_id": "11849", "title": "spaces-properties-lemma-quasi-separated-finite-etale-cover-dense-open-scheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, \\'etale, surjective morphism $U \\to X$ where $U$ is a quasi-separated scheme, then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \\in |X|$ of codimension $0$ in $X$ is contained in $X'$."}
+{"_id": "11850", "title": "spaces-properties-lemma-quotient-scheme", "text": "Let $S$ be a scheme. Let $G \\to S$ be a group scheme. Let $X \\to S$ be a morphism of schemes. Let $a : G \\times_S X \\to X$ be an action. Assume that \\begin{enumerate} \\item $G \\to S$ is finite locally free, \\item the action $a$ is free, \\item $X \\to S$ is affine, or quasi-affine, or projective, or quasi-projective, or $X$ is isomorphic to an open subscheme of an affine scheme, or $X$ is isomorphic to an open subscheme of $\\text{Proj}(A)$ for some graded ring $A$, or $G \\to S$ is radicial. \\end{enumerate} Then the fppf quotient sheaf $X/G$ is a scheme and $X \\to X/G$ is an fppf $G$-torsor."}
+{"_id": "11851", "title": "spaces-properties-lemma-quotient-separated", "text": "Notation and assumptions as in Proposition \\ref{proposition-finite-flat-equivalence-global}. Then \\begin{enumerate} \\item if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated over $S$, \\item if $U$ is quasi-separated, then $U/R$ is quasi-separated, \\item if $U$ is separated over $S$, then $U/R$ is separated over $S$, \\item if $U$ is separated, then $U/R$ is separated, and \\item add more here. \\end{enumerate} Similar results hold in the setting of Lemma \\ref{lemma-quotient-scheme}."}
+{"_id": "11852", "title": "spaces-properties-lemma-quasi-separated-sober", "text": "Let $S$ be a scheme. Let $X$ be a Zariski locally quasi-separated algebraic space over $S$. Then the topological space $|X|$ is sober (see Topology, Definition \\ref{topology-definition-generic-point})."}
+{"_id": "11853", "title": "spaces-properties-lemma-quasi-compact-quasi-separated-spectral", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The topological space $|X|$ is a spectral space."}
+{"_id": "11854", "title": "spaces-properties-lemma-point-like-spaces", "text": "Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space over $S$ and assume that there exists a surjective \\'etale morphism $\\Spec(k) \\to X$. If $X$ is quasi-separated, then $X \\cong \\Spec(k')$ where $k' \\subset k$ is a finite separable extension."}
+{"_id": "11855", "title": "spaces-properties-lemma-etale-over-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$, $U'$ be schemes over $S$. \\begin{enumerate} \\item If $U \\to U'$ is an \\'etale morphism of schemes, and if $U' \\to X$ is an \\'etale morphism from $U'$ to $X$, then the composition $U \\to X$ is an \\'etale morphism from $U$ to $X$. \\item If $\\varphi : U \\to X$ and $\\varphi' : U' \\to X$ are \\'etale morphisms towards $X$, and if $\\chi : U \\to U'$ is a morphism of schemes such that $\\varphi = \\varphi' \\circ \\chi$, then $\\chi$ is an \\'etale morphism of schemes. \\item If $\\chi : U \\to U'$ is a surjective \\'etale morphism of schemes and $\\varphi' : U' \\to X$ is a morphism such that $\\varphi = \\varphi' \\circ \\chi$ is \\'etale, then $\\varphi'$ is \\'etale. \\end{enumerate}"}
+{"_id": "11856", "title": "spaces-properties-lemma-etale-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent: \\begin{enumerate} \\item $f$ is \\'etale, \\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$, where $U$ is a scheme, such that the composition $f \\circ \\varphi$ is \\'etale (as a morphism of algebraic spaces), \\item there exists a surjective \\'etale morphism $\\psi : V \\to Y$, where $V$ is a scheme, such that the base change $V \\times_X Y \\to V$ is \\'etale (as a morphism of algebraic spaces), \\item there exists a commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are \\'etale, and the left vertical arrow is surjective such that the horizontal arrow is \\'etale. \\end{enumerate}"}
+{"_id": "11857", "title": "spaces-properties-lemma-composition-etale", "text": "The composition of two \\'etale morphisms of algebraic spaces is \\'etale."}
+{"_id": "11858", "title": "spaces-properties-lemma-base-change-etale", "text": "The base change of an \\'etale morphism of algebraic spaces by any morphism of algebraic spaces is \\'etale."}
+{"_id": "11859", "title": "spaces-properties-lemma-etale-permanence", "text": "Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \\to Z$, $h : Y \\to Z$ be \\'etale morphisms and let $f : X \\to Y$ be a morphism such that $h \\circ f = g$. Then $f$ is \\'etale."}
+{"_id": "11860", "title": "spaces-properties-lemma-etale-open", "text": "Let $S$ be a scheme. If $X \\to Y$ is an \\'etale morphism of algebraic spaces over $S$, then the associated map $|X| \\to |Y|$ of topological spaces is open."}
+{"_id": "11861", "title": "spaces-properties-lemma-etale-over-field-scheme", "text": "Let $S$ be a scheme. Let $X \\to \\Spec(k)$ be \\'etale morphism over $S$, where $k$ is a field. Then $X$ is a scheme."}
+{"_id": "11862", "title": "spaces-properties-lemma-compare-etale-sites", "text": "The functor $$ X_\\etale \\longrightarrow X_{spaces, \\etale}, \\quad U/X \\longmapsto U/X $$ is a special cocontinuous functor (Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}) and hence induces an equivalence of topoi $\\Sh(X_\\etale) \\to \\Sh(X_{spaces, \\etale})$."}
+{"_id": "11863", "title": "spaces-properties-lemma-alternative", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X_{affine, \\etale}$ denote the full subcategory of $X_\\etale$ whose objects are those $U/X \\in \\Ob(X_\\etale)$ with $U$ affine. A covering of $X_{affine, \\etale}$ will be a standard \\'etale covering, see Topologies, Definition \\ref{topologies-definition-standard-etale}. Then restriction $$ \\mathcal{F} \\longmapsto \\mathcal{F}|_{X_{affine, \\etale}} $$ defines an equivalence of topoi $\\Sh(X_\\etale) \\cong \\Sh(X_{affine, \\etale})$."}
+{"_id": "11864", "title": "spaces-properties-lemma-functoriality-etale-site", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The continuous functor $$ Y_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale}, \\quad V \\longmapsto X \\times_Y V $$ induces a morphism of sites $$ f_{spaces, \\etale} : X_{spaces, \\etale} \\to Y_{spaces, \\etale}. $$ \\item The rule $f \\mapsto f_{spaces, \\etale}$ is compatible with compositions, in other words $(f \\circ g)_{spaces, \\etale} = f_{spaces, \\etale} \\circ g_{spaces, \\etale}$ (see Sites, Definition \\ref{sites-definition-composition-morphisms-sites}). \\item The morphism of topoi associated to $f_{spaces, \\etale}$ induces, via Lemma \\ref{lemma-compare-etale-sites}, a morphism of topoi $f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ whose construction is compatible with compositions. \\item If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_\\etale \\to Y_\\etale$, corresponding to the continuous functor $V \\mapsto X \\times_Y V$. \\end{enumerate}"}
+{"_id": "11865", "title": "spaces-properties-lemma-f-map", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and let $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$. There are canonical bijections between the following three sets: \\begin{enumerate} \\item The set of maps $\\mathcal{G} \\to f_{small, *}\\mathcal{F}$. \\item The set of maps $f_{small}^{-1}\\mathcal{G} \\to \\mathcal{F}$. \\item The set of $f$-maps $\\varphi : \\mathcal{G} \\to \\mathcal{F}$. \\end{enumerate}"}
+{"_id": "11866", "title": "spaces-properties-lemma-etale-morphism-topoi", "text": "Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is \\'etale. In this case there is a functor $$ j : X_\\etale \\to Y_\\etale, \\quad (\\varphi : U \\to X) \\mapsto (f \\circ \\varphi : U \\to Y) $$ which is cocontinuous. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}. Moreover, $j$ is continuous as well, hence Sites, Lemma \\ref{sites-lemma-when-shriek} applies. In particular $f_{small}^{-1}\\mathcal{G}(U) = \\mathcal{G}(jU)$ for all sheaves $\\mathcal{G}$ on $Y_\\etale$."}
+{"_id": "11867", "title": "spaces-properties-lemma-pushforward-etale-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. If $g$ is \\'etale, then \\begin{enumerate} \\item $f'_{small, *}(\\mathcal{F}|_{X'}) = (f_{small, *}\\mathcal{F})|_{Y'}$ in $\\Sh(Y'_\\etale)$\\footnote{Also $(f')_{small}^{-1}(\\mathcal{G}|_{Y'}) = (f_{small}^{-1}\\mathcal{G})|_{X'}$ because of commutativity of the diagram and (\\ref{equation-restrict})}, and \\item if $\\mathcal{F}$ is an abelian sheaf, then $R^if'_{small, *}(\\mathcal{F}|_{X'}) = (R^if_{small, *}\\mathcal{F})|_{Y'}$. \\end{enumerate}"}
+{"_id": "11868", "title": "spaces-properties-lemma-characterize-sheaf-small-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\\mathcal{F}$ on $X_\\etale$ is given by the following data: \\begin{enumerate} \\item for every $U \\in \\Ob(X_\\etale)$ a sheaf $\\mathcal{F}_U$ on $U_\\etale$, \\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism $c_f : f_{small}^{-1}\\mathcal{F}_U \\to \\mathcal{F}_{U'}$. \\end{enumerate} These data are subject to the condition that given any $f : U' \\to U$ and $g : U'' \\to U'$ in $X_\\etale$ the composition $g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$."}
+{"_id": "11869", "title": "spaces-properties-lemma-descent-sheaf", "text": "With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c, e, i)$ as above. For any sheaf $\\mathcal{F}$ on $X_\\etale$ the sheaf\\footnote{In this lemma and its proof we write simply $\\varphi^{-1}$ instead of $\\varphi_{small}^{-1}$ and similarly for all the other pullbacks.} $\\mathcal{G} = \\varphi^{-1}\\mathcal{F}$ comes equipped with a canonical isomorphism $$ \\alpha : t^{-1}\\mathcal{G} \\longrightarrow s^{-1}\\mathcal{G} $$ such that the diagram $$ \\xymatrix{ & \\text{pr}_1^{-1}t^{-1}\\mathcal{G} \\ar[r]_-{\\text{pr}_1^{-1}\\alpha} & \\text{pr}_1^{-1}s^{-1}\\mathcal{G} \\ar@{=}[rd] & \\\\ \\text{pr}_0^{-1}s^{-1}\\mathcal{G} \\ar@{=}[ru] & & & c^{-1}s^{-1}\\mathcal{G} \\\\ & \\text{pr}_0^{-1}t^{-1}\\mathcal{G} \\ar[lu]^{\\text{pr}_0^{-1}\\alpha} \\ar@{=}[r] & c^{-1}t^{-1}\\mathcal{G} \\ar[ru]_{c^{-1}\\alpha} } $$ is a commutative. The functor $\\mathcal{F} \\mapsto (\\mathcal{G}, \\alpha)$ defines an equivalence of categories between sheaves on $X_\\etale$ and pairs $(\\mathcal{G}, \\alpha)$ as above."}
+{"_id": "11870", "title": "spaces-properties-lemma-cofinal-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. The category of \\'etale neighborhoods is cofiltered. More precisely: \\begin{enumerate} \\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of $\\overline{x}$ in $X$. Then there exists a third \\'etale neighborhood $(U, \\overline{u})$ and morphisms $(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$. \\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two morphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an \\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism $h : (U'', \\overline{u}'') \\to (U, \\overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \\circ h = h_2 \\circ h$. \\end{enumerate} Moreover, given any \\'etale neighbourhood $(U, \\overline{u}) \\to (X, \\overline{x})$ there exists a morphism of \\'etale neighbourhoods $(U', \\overline{u}') \\to (U, \\overline{u})$ where $U'$ is a scheme."}
+{"_id": "11871", "title": "spaces-properties-lemma-geometric-lift-to-usual", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x} : \\Spec(k) \\to X$ be a geometric point of $X$ lying over $x \\in |X|$. Let $\\varphi : U \\to X$ be an \\'etale morphism of algebraic spaces and let $u \\in |U|$ with $\\varphi(u) = x$. Then there exists a geometric point $\\overline{u} : \\Spec(k) \\to U$ lying over $u$ with $\\overline{x} = \\varphi \\circ \\overline{u}$."}
+{"_id": "11872", "title": "spaces-properties-lemma-geometric-lift-to-cover", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. Let $(U, \\overline{u})$ an \\'etale neighborhood of $\\overline{x}$. Let $\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ be an \\'etale covering in $X_{spaces, \\etale}$. Then there exist $i \\in I$ and $\\overline{u}_i : \\overline{x} \\to U_i$ such that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$ is a morphism of \\'etale neighborhoods."}
+{"_id": "11873", "title": "spaces-properties-lemma-stalk-gives-point", "text": "\\begin{slogan} A geometric point of an algebraic space gives a point of its \\'etale topos. \\end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. Consider the functor $$ u : X_\\etale \\longrightarrow \\textit{Sets}, \\quad U \\longmapsto |U_{\\overline{x}}| $$ Then $u$ defines a point $p$ of the site $X_\\etale$ (Sites, Definition \\ref{sites-definition-point}) and its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$ (Sites, Equation \\ref{sites-equation-stalk}) is the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ defined above."}
+{"_id": "11874", "title": "spaces-properties-lemma-stalk-exact", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. \\begin{enumerate} \\item The stalk functor $\\textit{PAb}(X_\\etale) \\to \\textit{Ab}$, $\\mathcal{F}  \\mapsto  \\mathcal{F}_{\\overline{x}}$ is exact. \\item We have $(\\mathcal{F}^\\#)_{\\overline{x}} = \\mathcal{F}_{\\overline{x}}$ for any presheaf of sets $\\mathcal{F}$ on $X_\\etale$. \\item The functor $\\textit{Ab}(X_\\etale) \\to \\textit{Ab}$, $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ is exact. \\item Similarly the functors $\\textit{PSh}(X_\\etale) \\to \\textit{Sets}$ and $\\Sh(X_\\etale) \\to \\textit{Sets}$ given by the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ are exact (see Categories, Definition \\ref{categories-definition-exact}) and commute with arbitrary colimits. \\end{enumerate}"}
+{"_id": "11875", "title": "spaces-properties-lemma-stalk-pullback", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The functor $f_{small}^{-1} : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$ is exact. \\item The functor $f_{small}^{-1} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition \\ref{categories-definition-exact}. \\item For any \\'etale morphism $V \\to Y$ of algebraic spaces we have $f_{small}^{-1}h_V = h_{X \\times_Y V}$. \\item Let $\\overline{x} \\to X$ be a geometric point. Let $\\mathcal{G}$ be a sheaf on $Y_\\etale$. Then there is a canonical identification $$ (f_{small}^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}. $$ where $\\overline{y} = f \\circ \\overline{x}$. \\end{enumerate}"}
+{"_id": "11876", "title": "spaces-properties-lemma-points-small-etale-site", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $p : \\Sh(pt) \\to \\Sh(X_\\etale)$ be a point of the small \\'etale topos of $X$. Then there exists a geometric point $\\overline{x}$ of $X$ such that the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is isomorphic to the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$."}
+{"_id": "11877", "title": "spaces-properties-lemma-support-subsheaf-final", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a subsheaf of the final object of the \\'etale topos of $X$ (see Sites, Example \\ref{sites-example-singleton-sheaf}). Then there exists a unique open $W \\subset X$ such that $\\mathcal{F} = h_W$."}
+{"_id": "11878", "title": "spaces-properties-lemma-zero-over-image", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be an abelian sheaf on $X_{spaces, \\etale}$. Let $\\sigma \\in \\mathcal{F}(U)$ be a local section. There exists an open subspace $W \\subset U$ such that \\begin{enumerate} \\item $W \\subset U$ is the largest open subspace of $U$ such that $\\sigma|_W = 0$, \\item for every $\\varphi : V \\to U$ in $X_\\etale$ we have $$ \\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W, $$ \\item for every geometric point $\\overline{u}$ of $U$ we have $$ (U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{s}} \\Leftrightarrow \\overline{u} \\in W $$ where $\\overline{s} = (U \\to S) \\circ \\overline{u}$. \\end{enumerate}"}
+{"_id": "11880", "title": "spaces-properties-lemma-support-sheaf-rings-closed", "text": "The support of a sheaf of rings on the small \\'etale site of an algebraic space is closed."}
+{"_id": "11881", "title": "spaces-properties-lemma-sheaf-condition-holds", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The rule $U \\mapsto \\Gamma(U, \\mathcal{O}_U)$ defines a sheaf of rings on $X_\\etale$."}
+{"_id": "11882", "title": "spaces-properties-lemma-morphism-ringed-topoi", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then there is a canonical map $f^\\sharp : f_{small}^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$ such that $$ (f_{small}, f^\\sharp) : (\\Sh(X_\\etale), \\mathcal{O}_X) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{O}_Y) $$ is a morphism of ringed topoi. Furthermore, \\begin{enumerate} \\item The construction $f \\mapsto (f_{small}, f^\\sharp)$ is compatible with compositions. \\item If $f$ is a morphism of schemes, then $f^\\sharp$ is the map described in Descent, Remark \\ref{descent-remark-change-topologies-ringed}. \\end{enumerate}"}
+{"_id": "11883", "title": "spaces-properties-lemma-reduced-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is reduced, \\item for every $x \\in |X|$ the local ring of $X$ at $x$ is reduced (Remark \\ref{remark-list-properties-local-ring-local-etale-topology}). \\end{enumerate} In this case $\\Gamma(X, \\mathcal{O}_X)$ is a reduced ring and if $f \\in \\Gamma(X, \\mathcal{O}_X)$ has $X = V(f)$, then $f = 0$."}
+{"_id": "11884", "title": "spaces-properties-lemma-describe-etale-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. Let $(U, \\overline{u})$ be an \\'etale neighbourhood of $\\overline{x}$ where $U$ is a scheme. Then we have $$ \\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, \\overline{u}} = \\mathcal{O}_{U, u}^{sh} $$ where the left hand side is the stalk of the structure sheaf of $X$, and the right hand side is the strict henselization of the local ring of $U$ at the point $u$ at which $\\overline{u}$ is centered."}
+{"_id": "11885", "title": "spaces-properties-lemma-etale-site-locally-ringed", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The small \\'etale site $X_\\etale$ endowed with its structure sheaf $\\mathcal{O}_X$ is a locally ringed site, see Modules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}."}
+{"_id": "11886", "title": "spaces-properties-lemma-dimension-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are equivalent \\begin{enumerate} \\item the dimension of the local ring of $X$ at $x$ (Definition \\ref{definition-dimension-local-ring}) is $d$, \\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for some geometric point $\\overline{x}$ lying over $x$, and \\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for any geometric point $\\overline{x}$ lying over $x$. \\end{enumerate}"}
+{"_id": "11887", "title": "spaces-properties-lemma-dimension-decent-invariant-under-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$. Let $x \\in X$. Then (1) $\\dim_x(X) = \\dim_{f(x)}(Y)$ and (2) the dimension of the local ring of $X$ at $x$ equals the dimension of the local ring of $Y$ at $f(x)$. If $f$ is surjective, then (3) $\\dim(X) = \\dim(Y)$."}
+{"_id": "11889", "title": "spaces-properties-lemma-irreducible-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. The following are equivalent \\begin{enumerate} \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$ has a unique minimal prime, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ there is a unique irreducible component of $U$ through $u$, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$ is unibranch, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$ is geometrically unibranch, \\item $\\mathcal{O}_{X, \\overline{x}}$ has a unique minimal prime for any geometric point $\\overline{x}$ lying over $x$. \\end{enumerate}"}
+{"_id": "11890", "title": "spaces-properties-lemma-nr-branches-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer. The following are equivalent \\begin{enumerate} \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the number of minimal primes of the local ring $\\mathcal{O}_{U, u}$ is $\\leq n$ and for at least one choice of $U, a, u$ it is $n$, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the number irreducible components of $U$ passing through $u$ is $\\leq n$ and for at least one choice of $U, a, u$ it is $n$, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the number of branches of $U$ at $u$ is $\\leq n$ and for at least one choice of $U, a, u$ it is $n$, \\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$ with $a(u) = x$ the number of geometric branches of $U$ at $u$ is $n$, and \\item the number of minimal prime ideals of $\\mathcal{O}_{X, \\overline{x}}$ is $n$. \\end{enumerate}"}
+{"_id": "11891", "title": "spaces-properties-lemma-Noetherian-topology", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X$ is locally Noetherian then $|X|$ is a locally Noetherian topological space. \\item If $X$ is quasi-compact and locally Noetherian, then $|X|$ is a Noetherian topological space. \\end{enumerate}"}
+{"_id": "11892", "title": "spaces-properties-lemma-Noetherian-sober", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space."}
+{"_id": "11893", "title": "spaces-properties-lemma-Noetherian-local-ring-Noetherian", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. Then $\\mathcal{O}_{X, \\overline{x}}$ is a Noetherian local ring."}
+{"_id": "11894", "title": "spaces-properties-lemma-regular", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is regular, and \\item every \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is regular. \\end{enumerate}"}
+{"_id": "11896", "title": "spaces-properties-lemma-regular-normal", "text": "A regular algebraic space is normal."}
+{"_id": "11897", "title": "spaces-properties-lemma-etale-exact-pullback", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$. Then $f^{-1}\\mathcal{O}_Y = \\mathcal{O}_X$, and $f^*\\mathcal{G} = f_{small}^{-1}\\mathcal{G}$ for any sheaf of $\\mathcal{O}_Y$-modules $\\mathcal{G}$. In particular, $f^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$ is exact."}
+{"_id": "11898", "title": "spaces-properties-lemma-pushforward-etale-base-change-modules", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian square of algebraic spaces over $S$. Let $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_X)$. If $g$ is \\'etale, then $f'_*(\\mathcal{F}|_{X'}) = (f_*\\mathcal{F})|_{Y'}$\\footnote{Also $(f')^*(\\mathcal{G}|_{Y'}) = (f^*\\mathcal{G})|_{X'}$ by commutativity of the diagram and (\\ref{equation-restrict-modules})} and $R^if'_*(\\mathcal{F}|_{X'}) = (R^if_*\\mathcal{F})|_{Y'}$ in $\\textit{Mod}(\\mathcal{O}_{Y'})$."}
+{"_id": "11899", "title": "spaces-properties-lemma-characterize-module-small-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is given by the following data: \\begin{enumerate} \\item for every $U \\in \\Ob(X_\\etale)$ a sheaf $\\mathcal{F}_U$ of $\\mathcal{O}_U$-modules on $U_\\etale$, \\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism $c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$. \\end{enumerate} These data are subject to the condition that given any $f : U' \\to U$ and $g : U'' \\to U'$ in $X_\\etale$ the composition $g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$."}
+{"_id": "11900", "title": "spaces-properties-lemma-relocalize-morphism", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ U \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\ X \\ar[r]^f & Y } $$ be a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ \\'etale. Via the identifications (\\ref{equation-localize-ringed}) for $U \\to X$ and $V \\to Y$ the morphism of ringed topoi $$ (g_{spaces, \\etale}, g^\\sharp) : (\\Sh(U_{spaces, \\etale}), \\mathcal{O}_U) \\longrightarrow (\\Sh(V_{spaces, \\etale}), \\mathcal{O}_V) $$ is $2$-isomorphic to the morphism $(f_{spaces, \\etale, c}, f_c^\\sharp)$ constructed in Modules on Sites, Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-sites} starting with the morphism of ringed sites $(f_{spaces, \\etale}, f^\\sharp)$ and the map $c : U \\to V \\times_Y X$ corresponding to $g$."}
+{"_id": "11901", "title": "spaces-properties-lemma-relocalize-morphism-at-schemes", "text": "Same notation and assumptions as in Lemma \\ref{lemma-relocalize-morphism} except that we also assume $U$ and $V$ are schemes. Via the identifications (\\ref{equation-localize-at-scheme-ringed}) for $U \\to X$ and $V \\to Y$ the morphism of ringed topoi $$ (g_{small}, g^\\sharp) : (\\Sh(U_\\etale), \\mathcal{O}_U) \\longrightarrow (\\Sh(V_\\etale), \\mathcal{O}_V) $$ is $2$-isomorphic to the morphism $(f_{small, s}, f_s^\\sharp)$ constructed in Modules on Sites, Lemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-topoi} starting with $(f_{small}, f^\\sharp)$ and the map $s : h_U \\to f_{small}^{-1}h_V$ corresponding to $g$."}
+{"_id": "11902", "title": "spaces-properties-lemma-sheaf-gives-space", "text": "Let $S$ be a scheme and let $Y$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$. Provided a set theoretic condition is satisfied (see proof) the functor $X$ associated to $\\mathcal{F}$ above is an algebraic space and there is an \\'etale morphism $f : X \\to Y$ of algebraic spaces such that $\\mathcal{F} = f_{small, *}*$ where $*$ is the final object of the category $\\Sh(X_\\etale)$ (constant sheaf with value a singleton)."}
+{"_id": "11903", "title": "spaces-properties-lemma-morphism-locally-ringed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The morphism of ringed topoi $(f_{small}, f^\\sharp)$ associated to $f$ is a morphism of locally ringed topoi, see Modules on Sites, Definition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}."}
+{"_id": "11905", "title": "spaces-properties-lemma-faithful", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Any two morphisms $a, b : X \\to Y$ of algebraic spaces over $S$ for which there exists a $2$-isomorphism $(a_{small}, a^\\sharp) \\cong (b_{small}, b^\\sharp)$ in the $2$-category of ringed topoi are equal."}
+{"_id": "11907", "title": "spaces-properties-lemma-pullback-quasi-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The pullback functor $f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$ preserves quasi-coherent sheaves."}
+{"_id": "11908", "title": "spaces-properties-lemma-characterize-quasi-coherent-small-etale", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ is given by the following data: \\begin{enumerate} \\item for every $U \\in \\Ob(X_\\etale)$ a quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}_U$ on $U_\\etale$, \\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism $c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$. \\end{enumerate} These data are subject to the condition that given any $f : U' \\to U$ and $g : U'' \\to U'$ in $X_\\etale$ the composition $g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$."}
+{"_id": "11909", "title": "spaces-properties-lemma-stalk-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$ be a point and let $\\overline{x}$ be a geometric point lying over $x$. Finally, let $\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$ be an \\'etale neighbourhood where $U$ is a scheme. Then $$ (\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}} \\mathcal{O}_{X, \\overline{x}} = \\mathcal{F}_{\\overline{x}} $$ where $u \\in U$ is the image of $\\overline{u}$."}
+{"_id": "11910", "title": "spaces-properties-lemma-stalk-pullback-quasi-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Let $\\overline{x}$ be a geometric point of $X$ and let $\\overline{y} = f \\circ \\overline{x}$ be the image in $Y$. Then there is a canonical isomorphism $$ (f^*\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}} \\otimes_{\\mathcal{O}_{Y, \\overline{y}}} \\mathcal{O}_{X, \\overline{x}} $$ of the stalk of the pullback with the tensor product of the stalk with the local ring of $X$ at $\\overline{x}$."}
+{"_id": "11911", "title": "spaces-properties-lemma-characterize-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module, \\item there exists an \\'etale morphism $f : Y \\to X$ of algebraic spaces over $S$ with $|f| : |Y| \\to |X|$ surjective such that $f^*\\mathcal{F}$ is quasi-coherent on $Y$, \\item there exists a scheme $U$ and a surjective \\'etale morphism $\\varphi : U \\to X$ such that $\\varphi^*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module, and \\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$ the restriction $\\varphi^*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module. \\end{enumerate}"}
+{"_id": "11912", "title": "spaces-properties-lemma-properties-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The category $\\QCoh(\\mathcal{O}_X)$ of quasi-coherent sheaves on $X$ has the following properties: \\begin{enumerate} \\item Any direct sum of quasi-coherent sheaves is quasi-coherent. \\item Any colimit of quasi-coherent sheaves is quasi-coherent. \\item The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent. \\item Given a short exact sequence of $\\mathcal{O}_X$-modules $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ if two out of three are quasi-coherent so is the third. \\item Given two quasi-coherent $\\mathcal{O}_X$-modules the tensor product is quasi-coherent. \\item Given two quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$ is of finite presentation (see Section \\ref{section-properties-modules}), then the internal hom $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is quasi-coherent. \\end{enumerate}"}
+{"_id": "11913", "title": "spaces-properties-lemma-locally-projective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item for some scheme $U$ and surjective \\'etale morphism $U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective on $U$, and \\item for any scheme $U$ and any \\'etale morphism $U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective on $U$. \\end{enumerate}"}
+{"_id": "11914", "title": "spaces-properties-lemma-locally-projective-pullback", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. If $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$ is locally projective on $X$."}
+{"_id": "11915", "title": "spaces-properties-lemma-morphism-to-affine-scheme", "text": "Let $X$ be an algebraic space over $\\mathbf{Z}$. Let $T$ be an affine scheme. The map $$ \\Mor(X, T) \\longrightarrow \\Hom(\\Gamma(T, \\mathcal{O}_T), \\Gamma(X, \\mathcal{O}_X)) $$ which maps $f$ to $f^\\sharp$ (on global sections) is bijective."}
+{"_id": "11916", "title": "spaces-properties-lemma-quotient", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $G$ be an abstract group with a free action on $X$. Then the quotient sheaf $X/G$ is an algebraic space."}
+{"_id": "11917", "title": "spaces-properties-proposition-locally-quasi-separated-open-dense-scheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Zariski locally quasi-separated (for example if $X$ is quasi-separated), then there exists a dense open subspace $X'$ of $X$ which is a scheme. More precisely, every point $x \\in |X|$ of codimension $0$ on $X$ is contained in $X'$."}
+{"_id": "11918", "title": "spaces-properties-proposition-finite-flat-equivalence-global", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \\begin{enumerate} \\item $s, t : R \\to U$ finite locally free, \\item $j = (t, s)$ is an equivalence, and \\item for a dense set of points $u \\in U$ the $R$-equivalence class $t(s^{-1}(\\{u\\}))$ is contained in an affine open of $U$. \\end{enumerate} Then there exists a finite locally free morphism $U \\to M$ of schemes over $S$ such that $R = U \\times_M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology."}
+{"_id": "11919", "title": "spaces-properties-proposition-sheaf-fpqc", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ satisfies the sheaf property for the fpqc topology."}
+{"_id": "11920", "title": "spaces-properties-proposition-quasi-coherent", "text": "With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c)$ as above. For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf $\\varphi^*\\mathcal{F}$ comes equipped with a canonical isomorphism $$ \\alpha : t^*\\varphi^*\\mathcal{F} \\longrightarrow s^*\\varphi^*\\mathcal{F} $$ which satisfies the conditions of Groupoids, Definition \\ref{groupoids-definition-groupoid-module} and therefore defines a quasi-coherent sheaf on $(U, R, s, t, c)$. The functor $\\mathcal{F} \\mapsto (\\varphi^*\\mathcal{F}, \\alpha)$ defines an equivalence of categories $$ \\begin{matrix} \\text{Quasi-coherent} \\\\ \\mathcal{O}_X\\text{-modules} \\end{matrix} \\longleftrightarrow \\begin{matrix} \\text{Quasi-coherent modules}\\\\ \\text{on }(U, R, s, t, c) \\end{matrix} $$"}
+{"_id": "11921", "title": "spaces-properties-proposition-coherator", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$ has enough injectives and all limits. \\item The inclusion functor $\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$ has a right adjoint\\footnote{This functor is sometimes called the {\\it coherator}.} $$ Q : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X) $$ such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "11956", "title": "intersection-theorem-multiplicity-with-koszul", "text": "\\begin{reference} \\cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale} \\end{reference} Let $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r) \\subset A$ be an ideal of definition. Let $M$ be a finite $A$-module. Then $$ e_I(M, r) = \\sum (-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M) $$"}
+{"_id": "11957", "title": "intersection-theorem-well-defined", "text": "Let $X$ be a nonsingular projective variety. Let $\\alpha$, resp.\\ $\\beta$ be an $r$, resp.\\ $s$ cycle on $X$. Assume that $\\alpha$ and $\\beta$ intersect properly so that $\\alpha \\cdot \\beta$ is defined. Finally, assume that $\\alpha \\sim_{rat} 0$. Then $\\alpha \\cdot \\beta \\sim_{rat} 0$."}
+{"_id": "11958", "title": "intersection-lemma-push-coherent", "text": "\\begin{reference} See \\cite[Chapter V]{Serre_algebre_locale}. \\end{reference} Suppose that $f : X \\to Y$ is a proper morphism of varieties. Let $\\mathcal{F}$ be a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq k$, then $f_*[\\mathcal{F}]_k = [f_*\\mathcal{F}]_k$. In particular, if $Z \\subset X$ is a closed subscheme of dimension $\\leq k$, then $f_*[Z]_k = [f_*\\mathcal{O}_Z]_k$."}
+{"_id": "11959", "title": "intersection-lemma-compose-pushforward", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of varieties. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$."}
+{"_id": "11960", "title": "intersection-lemma-pullback", "text": "Let $f : X \\to Y$ be a flat morphism of varieties. Set $r = \\dim(X) - \\dim(Y)$. Then $f^*[\\mathcal{F}]_k = [f^*\\mathcal{F}]_{k + r}$ if $\\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the support of $\\mathcal{F}$ is at most $k$."}
+{"_id": "11963", "title": "intersection-lemma-dimension-product-varieties", "text": "Let $X$ and $Y$ be varieties. Then $X \\times Y$ is a variety and $\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$."}
+{"_id": "11964", "title": "intersection-lemma-pullback-by-regular-immersion", "text": "Let $f : X \\to Y$ be a morphism of varieties. \\begin{enumerate} \\item If $Z \\subset Y$ is a subvariety dimension $d$ and $f$ is a regular immersion of codimension $c$, then every irreducible component of $f^{-1}(Z)$ has dimension $\\geq d - c$. \\item If $Z \\subset Y$ is a subvariety of dimension $d$ and $f$ is a local complete intersection morphism of relative dimension $r$, then every irreducible component of $f^{-1}(Z)$ has dimension $\\geq d + r$. \\end{enumerate}"}
+{"_id": "11965", "title": "intersection-lemma-diagonal-regular-immersion", "text": "Let $X$ be a nonsingular variety. Then the diagonal $\\Delta : X \\to X \\times X$ is a regular immersion of codimension $\\dim(X)$."}
+{"_id": "11966", "title": "intersection-lemma-intersect-in-smooth", "text": "Let $X$ be a nonsingular variety and let $W,V \\subset X$ be closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$. Then every irreducible component $Z$ of $V \\cap W$ has dimension $\\geq r + s - \\dim(X)$."}
+{"_id": "11967", "title": "intersection-lemma-tensor-coherent", "text": "Let $X$ be a locally Noetherian scheme. \\begin{enumerate} \\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules, then $\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is too. \\item If $L$ and $K$ are in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then so is $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$. \\end{enumerate}"}
+{"_id": "11969", "title": "intersection-lemma-transversal", "text": "Let $X$ be a nonsingular variety. Let $V, W \\subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \\cap W$ and assume that the multiplicity (in the sense of Section \\ref{section-cycle-of-closed}) of $Z$ in the closed subscheme $V \\cap W$ is $1$. Then $e(X, V \\cdot W, Z) = 1$ and $V$ and $W$ are smooth in a general point of $Z$."}
+{"_id": "11970", "title": "intersection-lemma-multiplicity-ses", "text": "Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of definition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of finite $A$-modules. Let $d \\geq \\dim(\\text{Supp}(M))$. Then $$ e_I(M, d) = e_I(M', d) + e_I(M'', d) $$"}
+{"_id": "11972", "title": "intersection-lemma-leading-coefficient", "text": "Let $P$ be a polynomial of degree $r$ with leading coefficient $a$. Then $$ r! a = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i{r \\choose i} P(t - i) $$ for any $t$."}
+{"_id": "11973", "title": "intersection-lemma-intersection-multiplicity-CM", "text": "Let $X$ be a nonsingular variety and $W, V \\subset X$ closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \\cap W$ with generic point $\\xi$. Assume that $\\mathcal{O}_{W, \\xi}$ and $\\mathcal{O}_{V, \\xi}$ are Cohen-Macaulay. Then $$ e(X, V \\cdot W, Z) = \\text{length}_{\\mathcal{O}_{X, \\xi}}(\\mathcal{O}_{V \\cap W, \\xi}) $$ where $V \\cap W$ is the scheme theoretic intersection. In particular, if both $V$ and $W$ are Cohen-Macaulay, then $V \\cdot W = [V \\cap W]_{\\dim(V) + \\dim(W) - \\dim(X)}$."}
+{"_id": "11974", "title": "intersection-lemma-one-ideal-ci", "text": "Let $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r)$ be an ideal generated by a regular sequence. Let $M$ be a finite $A$-module. Assume that $\\dim(\\text{Supp}(M/IM)) = 0$. Then $$ e_I(M, r) = \\sum (-1)^i\\text{length}_A(\\text{Tor}_i^A(A/I, M)) $$ Here $e_I(M, r)$ is as in Remark \\ref{remark-trivial-generalization}."}
+{"_id": "11975", "title": "intersection-lemma-multiplicity-with-lci", "text": "Let $X$ be a nonsingular variety. Let $W,V \\subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \\cap W$ with generic point $\\xi$. Suppose the ideal of $V$ in $\\mathcal{O}_{X, \\xi}$ is cut out by a regular sequence $f_1, \\ldots, f_c \\in \\mathcal{O}_{X, \\xi}$. Then $e(X, V\\cdot W, Z)$ is equal to $c!$ times the leading coefficient in the Hilbert polynomial $$ t \\mapsto \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t,\\quad t \\gg 0. $$ In particular, this coefficient is $> 0$."}
+{"_id": "11976", "title": "intersection-lemma-multiplicity-with-effective-Cartier-divisor", "text": "In Lemma \\ref{lemma-multiplicity-with-lci} assume that  $c = 1$, i.e., $V$ is an effective Cartier divisor. Then $$ e(X, V \\cdot W, Z) = \\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi}). $$"}
+{"_id": "11977", "title": "intersection-lemma-multiplicity-lci-CM", "text": "In Lemma \\ref{lemma-multiplicity-with-lci} assume that the local ring $\\mathcal{O}_{W, \\xi}$ is Cohen-Macaulay. Then we have $$ e(X, V \\cdot W, Z) = \\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/ f_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}). $$"}
+{"_id": "11978", "title": "intersection-lemma-rational-equivalence-and-intersection", "text": "Let $X$ be a nonsingular variety. Let $a, b \\in \\mathbf{P}^1$ be distinct closed points. Let $k \\geq 0$. \\begin{enumerate} \\item If $W \\subset X \\times \\mathbf{P}^1$ is a closed subvariety of dimension $k + 1$ which intersects $X \\times a$ properly, then \\begin{enumerate} \\item $[W_a]_k = W \\cdot X \\times a$ as cycles on $X \\times \\mathbf{P}^1$, and \\item $[W_a]_k = \\text{pr}_{X, *}(W \\cdot X \\times a)$ as cycles on $X$. \\end{enumerate} \\item Let $\\alpha$ be a $(k + 1)$-cycle on $X \\times \\mathbf{P}^1$ which intersects $X \\times a$ and $X \\times b$ properly. Then $pr_{X,*}( \\alpha \\cdot X \\times a - \\alpha \\cdot X \\times b)$ is rationally equivalent to zero. \\item Conversely, any $k$-cycle which is rationally equivalent to $0$ is of this form. \\end{enumerate}"}
+{"_id": "11979", "title": "intersection-lemma-transversal-subschemes", "text": "Let $X$ be a nonsingular variety. Let $r, s \\geq 0$ and let $Y, Z \\subset X$ be closed subschemes with $\\dim(Y) \\leq r$ and $\\dim(Z) \\leq s$. Assume $[Y]_r = \\sum n_i[Y_i]$ and $[Z]_s = \\sum m_j[Z_j]$ intersect properly. Let $T$ be an irreducible component of $Y_{i_0} \\cap Z_{j_0}$ for some $i_0$ and $j_0$ and assume that the multiplicity (in the sense of Section \\ref{section-cycle-of-closed}) of $T$ in the closed subscheme $Y \\cap Z$ is $1$. Then \\begin{enumerate} \\item the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$ is $1$, \\item $Y$ and $Z$ are nonsingular at the generic point of $Z$, \\item $n_{i_0} = 1$, $m_{j_0} = 1$, and \\item $T$ is not contained in $Y_i$ or $Z_j$ for $i \\not = i_0$ and $j \\not = j_0$. \\end{enumerate}"}
+{"_id": "11982", "title": "intersection-lemma-tor-and-diagonal", "text": "Let $X$ be a nonsingular variety. \\begin{enumerate} \\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules, then there are canonical isomorphisms $$ \\text{Tor}_i^{\\mathcal{O}_{X \\times X}}(\\mathcal{O}_\\Delta, \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times X}} \\text{pr}_2^*\\mathcal{G}) = \\Delta_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) $$ \\item If $K$ and $M$ are in $D_\\QCoh(\\mathcal{O}_X)$, then there is a canonical isomorphism $$ L\\Delta^* \\left( L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M \\right) = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M $$ in $D_\\QCoh(\\mathcal{O}_X)$ and a canonical isomorphism $$ \\mathcal{O}_\\Delta \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M = \\Delta_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) $$ in $D_\\QCoh(X \\times X)$. \\end{enumerate}"}
+{"_id": "11983", "title": "intersection-lemma-reduction-diagonal", "text": "Let $X$ be a nonsingular variety. Let $\\alpha$, resp.\\ $\\beta$ be an $r$-cycle, resp.\\ $s$-cycle on $X$. Assume $\\alpha$ and $\\beta$ intersect properly. Then \\begin{enumerate} \\item $\\alpha \\times \\beta$ and $[\\Delta]$ intersect properly \\item we have $\\Delta_*(\\alpha \\cdot \\beta) = [\\Delta] \\cdot \\alpha\\times\\beta$ as cycles on $X \\times X$, \\item if $X$ is proper, then $\\text{pr}_{1, *}([\\Delta] \\cdot \\alpha\\times\\beta) = \\alpha\\cdot\\beta$, where $pr_1 : X\\times X \\to X$ is the projection. \\end{enumerate}"}
+{"_id": "11984", "title": "intersection-lemma-tor-sheaf", "text": "\\begin{reference} \\cite[Chapter V]{Serre_algebre_locale} \\end{reference} Let $X$ be a nonsingular variety. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent sheaves on $X$ with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$, $\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and $\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) ) \\leq r + s - \\dim X$. In this case $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$ intersect properly and $$ [\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s = \\sum (-1)^p [\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}. $$"}
+{"_id": "11985", "title": "intersection-lemma-associative", "text": "Let $X$ be a nonsingular variety. Let $U, V, W$ be closed subvarieties. Assume that $U, V, W$ intersect properly pairwise and that $\\dim(U \\cap V \\cap W) \\leq \\dim(U) + \\dim(V) + \\dim(W) - 2\\dim(X)$. Then $$ U \\cdot (V \\cdot W) = (U \\cdot V) \\cdot W $$ as cycles on $X$."}
+{"_id": "11986", "title": "intersection-lemma-flat-pull-back-and-intersections-sheaves", "text": "Let $f : X \\to Y$ be a flat morphism of nonsingular varieties. Set $e = \\dim(X) - \\dim(Y)$. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent sheaves on $Y$ with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$, $\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and $\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) ) \\leq r + s - \\dim(Y)$. In this case the cycles $[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$ intersect properly and $$ f^*([\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s) = [f^*\\mathcal{F}]_{r + e} \\cdot [f^*\\mathcal{G}]_{s + e} $$"}
+{"_id": "11988", "title": "intersection-lemma-projection-formula-flat", "text": "\\begin{reference} See \\cite[Chapter V, C), Section 7, formula (10)]{Serre_algebre_locale} for a more general formula. \\end{reference} Let $f : X \\to Y$ be a flat proper morphism of nonsingular varieties. Set $e = \\dim(X) - \\dim(Y)$. Let $\\alpha$ be an $r$-cycle on $X$ and let $\\beta$ be a $s$-cycle on $Y$. Assume that $\\alpha$ and $f^*(\\beta)$ intersect properly. Then $f_*(\\alpha)$ and $\\beta$ intersect properly and $$ f_*(\\alpha) \\cdot \\beta = f_*( \\alpha \\cdot f^*\\beta) $$"}
+{"_id": "11989", "title": "intersection-lemma-transfer", "text": "Let $X \\to P$ be a closed immersion of nonsingular varieties. Let $C' \\subset P \\times \\mathbf{P}^1$ be a closed subvariety of dimension $r + 1$. Assume \\begin{enumerate} \\item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \\to \\mathbf{P}^1$ is dominant, \\item $C'$ intersects $X \\times \\mathbf{P}^1$ properly, \\item $[C]_r$ intersects $X$ properly. \\end{enumerate} Then setting $\\alpha = [C]_r \\cdot X$ viewed as cycle on $X$ and $\\beta = C' \\cdot X \\times \\mathbf{P}^1$ viewed as cycle on $X \\times \\mathbf{P}^1$, we have $$ \\alpha = \\text{pr}_{X, *}(\\beta \\cdot X \\times 0) $$ as cycles on $X$ where $\\text{pr}_X : X \\times \\mathbf{P}^1 \\to X$ is the projection."}
+{"_id": "11990", "title": "intersection-lemma-projection-generically-finite", "text": "Let $V$ be a vector space of dimension $n + 1$. Let $X \\subset \\mathbf{P}(V)$ be a closed subscheme. If $X \\not = \\mathbf{P}(V)$, then there is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the restriction of the projection $r_p$ defines a finite morphism $r_p|_X : X \\to \\mathbf{P}(W_p)$."}
+{"_id": "11991", "title": "intersection-lemma-projection-generically-immersion", "text": "Let $V$ be a vector space of dimension $n + 1$. Let $X \\subset \\mathbf{P}(V)$ be a closed subvariety. Let $x \\in X$ be a nonsingular point. \\begin{enumerate} \\item If $\\dim(X) < n - 1$, then there is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the morphism $r_p|_X : X \\to r_p(X)$ is an isomorphism over an open neighbourhood of $r_p(x)$. \\item If $\\dim(X) = n - 1$, then there is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the morphism $r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$. \\end{enumerate}"}
+{"_id": "11992", "title": "intersection-lemma-projection-injective", "text": "Let $V$ be a vector space of dimension $n + 1$. Let $Y, Z \\subset \\mathbf{P}(V)$ be closed subvarieties. There is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ we have $$ Y \\cap r_p^{-1}(r_p(Z)) = (Y \\cap Z) \\cup E $$ with $E \\subset Y$ closed and $\\dim(E) \\leq \\dim(Y) + \\dim(Z) + 1 - n$."}
+{"_id": "11993", "title": "intersection-lemma-find-lines", "text": "Let $V$ be a vector space. Let $B \\subset \\mathbf{P}(V)$ be a closed subvariety of codimension $\\geq 2$. Let $p \\in \\mathbf{P}(V)$ be a closed point, $p \\not \\in B$. Then there exists a line $\\ell \\subset \\mathbf{P}(V)$ with $\\ell \\cap B = \\emptyset$. Moreover, these lines sweep out an open subset of $\\mathbf{P}(V)$."}
+{"_id": "11994", "title": "intersection-lemma-doubly-transitive", "text": "Let $V$ be a vector space. Let $G = \\text{PGL}(V)$. Then $G \\times \\mathbf{P}(V) \\to \\mathbf{P}(V)$ is doubly transitive."}
+{"_id": "11995", "title": "intersection-lemma-determinant", "text": "Let $k$ be a field. Let $n \\geq 1$ be an integer and let $x_{ij}, 1 \\leq i, j \\leq n$ be variables. Then $$ \\det \\left( \\begin{matrix} x_{11} & x_{12} & \\ldots & x_{1n} \\\\ x_{21} & \\ldots & \\ldots & \\ldots \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ x_{n1} & \\ldots & \\ldots & x_{nn} \\end{matrix} \\right) $$ is an irreducible element of the polynomial ring $k[x_{ij}]$."}
+{"_id": "11996", "title": "intersection-lemma-make-family", "text": "With notation as above. Let $X, Y$ be closed subvarieties of $\\mathbf{P}(V)$ which intersect properly such that $X \\not = \\mathbf{P}(V)$ and $X \\cap Y \\not = \\emptyset$. For a general line $\\ell \\subset \\mathbf{P}$ with $[\\text{id}_V] \\in \\ell$ we have \\begin{enumerate} \\item $X \\subset U_g$ for all $[g] \\in \\ell$, \\item $g(X)$ intersects $Y$ properly for all $[g] \\in \\ell$. \\end{enumerate}"}
+{"_id": "11997", "title": "intersection-lemma-moving", "text": "\\begin{reference} See \\cite{Roberts}. \\end{reference} Let $X \\subset \\mathbf{P}^N$ be a nonsingular closed subvariety. Let $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let $Z \\subset X$ be a closed subvariety of dimension $d$ and $T_i \\subset X$, $i \\in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \\subset \\mathbf{P}^N$ such that $C$ intersects $X$ properly and such that $$ C \\cdot X = Z + \\sum\\nolimits_{j \\in J} m_j Z_j $$ where $Z_j \\subset X$ are irreducible of dimension $d$, distinct from $Z$, and $$ \\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i) $$ with strict inequality if $Z$ does not intersect $T_i$ properly in $X$."}
+{"_id": "11998", "title": "intersection-lemma-move", "text": "Let $C \\subset \\mathbf{P}^N$ be a closed subvariety. Let $X \\subset \\mathbf{P}^N$ be subvariety and let $T_i \\subset X$ be a finite collection of closed subvarieties. Assume that $C$ and $X$ intersect properly. Then there exists a closed subvariety $C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$ such that \\begin{enumerate} \\item $C' \\to \\mathbf{P}^1$ is dominant, \\item $C'_0 = C$ scheme theoretically, \\item $C'$ and $X \\times \\mathbf{P}^1$ intersect properly, \\item $C'_\\infty$ properly intersects each of the given $T_i$. \\end{enumerate}"}
+{"_id": "11999", "title": "intersection-lemma-moving-move", "text": "Let $X$ be a nonsingular projective variety. Let $\\alpha$ be an $r$-cycle and $\\beta$ be an $s$-cycle on $X$. Then there exists an $r$-cycle $\\alpha'$ such that $\\alpha' \\sim_{rat} \\alpha$ and such that $\\alpha'$ and $\\beta$ intersect properly."}
+{"_id": "12000", "title": "intersection-lemma-well-defined-special-case", "text": "Let $X$ be a nonsingular variety. Let $W \\subset X \\times \\mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety dominating $\\mathbf{P}^1$. Let $W_a$, resp.\\ $W_b$ be the fibre of $W \\to \\mathbf{P}^1$ over $a$, resp.\\ $b$. Let $V$ be a $r$-dimensional subvariety of $X$ such that $V$ intersects both $W_a$ and $W_b$ properly. Then $[V] \\cdot [W_a]_r \\sim_{rat} [V] \\cdot [W_b]_r$."}
+{"_id": "12002", "title": "intersection-proposition-positivity", "text": "\\begin{reference} This is one of the main results of \\cite{Serre_algebre_locale}. \\end{reference} Let $X$ be a nonsingular variety. Let $V \\subset X$ and $W \\subset Y$ be closed subvarieties which intersect properly. Let $Z \\subset V \\cap W$ be an irreducible component. Then $e(X, V \\cdot W, Z) > 0$."}
+{"_id": "12008", "title": "homology-lemma-preadditive-zero", "text": "Let $\\mathcal{A}$ be a preadditive category. Let $x$ be an object of $\\mathcal{A}$. The following are equivalent \\begin{enumerate} \\item $x$ is an initial object, \\item $x$ is a final object, and \\item $\\text{id}_x = 0$ in $\\Mor_\\mathcal{A}(x, x)$. \\end{enumerate} Furthermore, if such an object $0$ exists, then a morphism $\\alpha : x \\to y$ factors through $0$ if and only if $\\alpha = 0$."}
+{"_id": "12009", "title": "homology-lemma-preadditive-direct-sum", "text": "Let $\\mathcal{A}$ be a preadditive category. Let $x, y \\in \\Ob(\\mathcal{A})$. If the product $x \\times y$ exists, then so does the coproduct $x \\amalg y$. If the coproduct $x \\amalg y$ exists, then so does the product $x \\times y$. In this case also $x \\amalg y \\cong x \\times y$."}
+{"_id": "12010", "title": "homology-lemma-additive-additive", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$ be preadditive categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor. Then $F$ transforms direct sums to direct sums and zero to zero."}
+{"_id": "12011", "title": "homology-lemma-additive-cat-biproduct-kernel", "text": "Let $\\mathcal{C}$ be a preadditive category. Let $x \\oplus y$ with morphisms $i, j, p, q$ as in Lemma \\ref{lemma-preadditive-direct-sum} be a direct sum in $\\mathcal{C}$. Then $i : x \\to x \\oplus y$ is a kernel of $q : x \\oplus y \\rightarrow y$. Dually, $p$ is a cokernel for $j$."}
+{"_id": "12012", "title": "homology-lemma-kernel-mono", "text": "Let $\\mathcal{C}$ be a preadditive category. Let $f : x \\to y$ be a morphism in $\\mathcal{C}$. \\begin{enumerate} \\item If a kernel of $f$ exists, then this kernel is a monomorphism. \\item If a cokernel of $f$ exists, then this cokernel is an epimorphism. \\item If a kernel and coimage of $f$ exist, then the coimage is an epimorphism. \\item If a cokernel and image of $f$ exist, then the image is a monomorphism. \\end{enumerate}"}
+{"_id": "12013", "title": "homology-lemma-coim-im-map", "text": "Let $f : x \\to y$ be a morphism in a preadditive category such that the kernel, cokernel, image and coimage all exist. Then $f$ can be factored uniquely as $x \\to \\Coim(f) \\to \\Im(f) \\to y$."}
+{"_id": "12014", "title": "homology-lemma-karoubian", "text": "Let $\\mathcal{C}$ be a preadditive category. The following are equivalent \\begin{enumerate} \\item $\\mathcal{C}$ is Karoubian, \\item every idempotent endomorphism of an object of $\\mathcal{C}$ has a cokernel, and \\item given an idempotent endomorphism $p : z \\to z$ of $\\mathcal{C}$ there exists a direct sum decomposition $z = x \\oplus y$ such that $p$ corresponds to the projection onto $y$. \\end{enumerate}"}
+{"_id": "12015", "title": "homology-lemma-projectors-have-images", "text": "Let $\\mathcal{D}$ be a preadditive category. \\begin{enumerate} \\item If $\\mathcal{D}$ has countable products and kernels of maps which have a right inverse, then $\\mathcal{D}$ is Karoubian. \\item If $\\mathcal{D}$ has countable coproducts and cokernels of maps which have a left inverse, then $\\mathcal{D}$ is Karoubian. \\end{enumerate}"}
+{"_id": "12016", "title": "homology-lemma-abelian-opposite", "text": "Let $\\mathcal{A}$ be a preadditive category. The additions on sets of morphisms make $\\mathcal{A}^{opp}$ into a preadditive category. Furthermore, $\\mathcal{A}$ is additive if and only if $\\mathcal{A}^{opp}$ is additive, and $\\mathcal{A}$ is abelian if and only if $\\mathcal{A}^{opp}$ is abelian."}
+{"_id": "12017", "title": "homology-lemma-characterize-injective", "text": "Let $f : x \\to y$ be a morphism in an abelian category $\\mathcal{A}$. Then \\begin{enumerate} \\item $f$ is injective if and only if $f$ is a monomorphism, and \\item $f$ is surjective if and only if $f$ is an epimorphism. \\end{enumerate}"}
+{"_id": "12018", "title": "homology-lemma-colimit-abelian-category", "text": "Let $\\mathcal{A}$ be an abelian category. All finite limits and finite colimits exist in $\\mathcal{A}$."}
+{"_id": "12019", "title": "homology-lemma-check-exactness", "text": "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a complex of $\\mathcal{A}$. \\begin{enumerate} \\item $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if $$ 0 \\to \\Hom_\\mathcal{A}(M_3, N) \\to \\Hom_\\mathcal{A}(M_2, N) \\to \\Hom_\\mathcal{A}(M_1, N) $$ is an exact sequence of abelian groups for all objects $N$ of $\\mathcal{A}$, and \\item $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if $$ 0 \\to \\Hom_\\mathcal{A}(N, M_1) \\to \\Hom_\\mathcal{A}(N, M_2) \\to \\Hom_\\mathcal{A}(N, M_1) $$ is an exact sequence of abelian groups for all objects $N$ of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "12020", "title": "homology-lemma-ses-split", "text": "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence. \\begin{enumerate} \\item Given a morphism $s : C \\to B$ left inverse to $B \\to C$, there exists a unique $\\pi : B \\to A$ such that $(s, \\pi)$ splits the short exact sequence as in Definition \\ref{definition-ses-split}. \\item Given a morphism $\\pi : B \\to A$ right inverse to $A \\to B$, there exists a unique $s : C \\to B$ such that $(s, \\pi)$ splits the short exact sequence as in Definition \\ref{definition-ses-split}. \\end{enumerate}"}
+{"_id": "12021", "title": "homology-lemma-characterize-cartesian", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ w\\ar[r]^f\\ar[d]_g & y\\ar[d]^h\\\\ x\\ar[r]^k & z } $$ be a commutative diagram.  \\begin{enumerate} \\item The diagram is cartesian if and only if  $$ 0 \\to w \\xrightarrow{(g, f)} x \\oplus y \\xrightarrow{(k, -h)} z $$ is exact. \\item The diagram is cocartesian if and only if  $$ w \\xrightarrow{(g, -f)} x \\oplus y \\xrightarrow{(k, h)} z \\to 0 $$ is exact. \\end{enumerate}"}
+{"_id": "12022", "title": "homology-lemma-cartesian-kernel", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ w\\ar[r]^f\\ar[d]_g & y\\ar[d]^h\\\\ x\\ar[r]^k & z } $$ be a commutative diagram. \\begin{enumerate} \\item If the diagram is cartesian, then the morphism  $\\Ker(f)\\to\\Ker(k)$ induced by $g$ is an isomorphism. \\item If the diagram is cocartesian, then the morphism  $\\Coker(f)\\to\\Coker(k)$ induced by $h$ is an isomorphism. \\end{enumerate}"}
+{"_id": "12023", "title": "homology-lemma-cartesian-cocartesian", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ w\\ar[r]^f\\ar[d]_g & y\\ar[d]^h\\\\ x\\ar[r]^k & z } $$ be a commutative diagram. \\begin{enumerate} \\item If the diagram is cartesian and $k$ is an epimorphism,  then the diagram is cocartesian and $f$ is an epimorphism. \\item If the diagram is cocartesian and $g$ is a monomorphism,  then the diagram is cartesian and $h$ is a monomorphism. \\end{enumerate}"}
+{"_id": "12024", "title": "homology-lemma-epimorphism-universal-abelian-category", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item If $x \\to y$ is surjective, then for every $z \\to y$ the projection $x \\times_y z \\to z$ is surjective. \\item If $x \\to y$ is injective, then for every $x \\to z$ the morphism $z \\to z \\amalg_x y$ is injective. \\end{enumerate}"}
+{"_id": "12025", "title": "homology-lemma-check-exactness-fibre-product", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f:x\\to y$ and $g:y\\to z$  be morphisms with $g\\circ f=0$. Then, the following statements are equivalent: \\begin{enumerate} \\item The sequence $x\\overset{f}\\to y\\overset{g}\\to z$ is exact. \\item For every $h:w\\to y$ with $g\\circ h=0$ there exist an object $v$,  an epimorphism $k:v\\to w$ and a morphism $l:v\\to x$ with $h\\circ k=f\\circ l$. \\end{enumerate}"}
+{"_id": "12026", "title": "homology-lemma-exact-kernel-sequence", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ x \\ar[r]^f \\ar[d]^\\alpha & y \\ar[r]^g \\ar[d]^\\beta & z \\ar[d]^\\gamma\\\\ u \\ar[r]^k & v \\ar[r]^l & w } $$ be a commutative diagram. \\begin{enumerate} \\item If the first row is exact and $k$ is a monomorphism, then the induced  sequence $\\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma)$  is exact. \\item If the second row is exact and $g$ is an epimorphism, then the induced  sequence $\\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma)$  is exact. \\end{enumerate}"}
+{"_id": "12027", "title": "homology-lemma-snake", "text": "Let $\\mathcal{A}$ be an abelian category. Let  $$ \\xymatrix{ & x \\ar[r]^f \\ar[d]^\\alpha & y \\ar[r]^g \\ar[d]^\\beta & z \\ar[r] \\ar[d]^\\gamma & 0 \\\\ 0 \\ar[r] & u \\ar[r]^k & v \\ar[r]^l & w } $$ be a commutative diagram with exact rows. \\begin{enumerate} \\item There exists a unique morphism  $\\delta : \\Ker(\\gamma) \\rightarrow \\Coker(\\alpha)$  such that the diagram $$ \\xymatrix{ y \\ar[d]_\\beta & y \\times_z \\Ker(\\gamma) \\ar[l]_{\\pi'} \\ar[r]^{\\pi} & \\Ker(\\gamma) \\ar[d]^\\delta \\\\ v \\ar[r]^{\\iota'} & \\Coker(\\alpha) \\amalg_u v & \\Coker(\\alpha) \\ar[l]_\\iota } $$ commutes, where $\\pi$ and $\\pi'$ are the canonical projections  and $\\iota$ and $\\iota'$ are the canonical coprojections. \\item The induced sequence  $$ \\Ker(\\alpha)\\overset{f'} \\to \\Ker(\\beta)  \\overset{g'}\\to \\Ker(\\gamma)\\overset{\\delta}\\to \\Coker(\\alpha) \\overset{k'}\\to \\Coker(\\beta)  \\overset{l'}\\to \\Coker(\\gamma) $$ is exact. If $f$ is injective then so is $f'$, and if $l$ is  surjective then so is $l'$. \\end{enumerate}"}
+{"_id": "12028", "title": "homology-lemma-snake-natural", "text": "Let $\\mathcal{A}$ be an abelian category. Let  $$ \\xymatrix{ & & & x\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\alpha & & y\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\beta & & z\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\gamma & & 0\\\\ & & x'\\ar[rr]\\ar[dd]^(.4){\\alpha'} & & y'\\ar[rr]\\ar[dd]^(.4){\\beta'} & & z'\\ar[rr]\\ar[dd]^(.4){\\gamma'} & & 0 & \\\\ & 0\\ar[rr] & & u\\ar[ld]\\ar[rr] & & v\\ar[ld]\\ar[rr] & & w\\ar[ld] & & \\\\ 0\\ar[rr] & & u'\\ar[rr] & & v'\\ar[rr] & & w' & & & } $$ be a commutative diagram with exact rows. Then, the induced diagram $$ \\xymatrix@C=15pt{ \\Ker(\\alpha) \\ar[r] \\ar[d] & \\Ker(\\beta) \\ar[r] \\ar[d] & \\Ker(\\gamma) \\ar[r]^(.45){\\delta} \\ar[d] & \\Coker(\\alpha) \\ar[r] \\ar[d] & \\Coker(\\beta) \\ar[r] \\ar[d] & \\Coker(\\gamma) \\ar[d] \\\\ \\Ker(\\alpha') \\ar[r] & \\Ker(\\beta') \\ar[r] & \\Ker(\\gamma') \\ar[r]^(.45){\\delta'} & \\Coker(\\alpha') \\ar[r] & \\Coker(\\beta') \\ar[r] & \\Coker(\\gamma') } $$ commutes."}
+{"_id": "12029", "title": "homology-lemma-four-lemma", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ w \\ar[r] \\ar[d]^\\alpha & x \\ar[r] \\ar[d]^\\beta & y \\ar[r] \\ar[d]^\\gamma & z \\ar[d]^\\delta \\\\ w' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z' } $$ be a commutative diagram with exact rows. \\begin{enumerate} \\item If $\\alpha, \\gamma$ are surjective and $\\delta$ is injective, then $\\beta$ is surjective. \\item If $\\beta, \\delta$ are injective and $\\alpha$ is surjective, then $\\gamma$ is injective. \\end{enumerate}"}
+{"_id": "12030", "title": "homology-lemma-five-lemma", "text": "\\begin{reference} \\cite[Lemma 4.5 page 16]{Eilenberg-Steenrod} \\end{reference} Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ v \\ar[r] \\ar[d]^\\alpha & w \\ar[r] \\ar[d]^\\beta & x \\ar[r] \\ar[d]^\\gamma & y \\ar[r] \\ar[d]^\\delta & z \\ar[d]^\\epsilon \\\\ v' \\ar[r] & w' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z' } $$ be a commutative diagram with exact rows. If $\\beta, \\delta$ are isomorphisms, $\\epsilon$ is injective, and $\\alpha$ is surjective then $\\gamma$ is an isomorphism."}
+{"_id": "12031", "title": "homology-lemma-baer-sum", "text": "The construction $(E_1, E_2) \\mapsto E_1 + E_2$ above defines a commutative group law on $\\Ext_\\mathcal{A}(B, A)$ which is functorial in both variables."}
+{"_id": "12032", "title": "homology-lemma-six-term-sequence-ext", "text": "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence in $\\mathcal{A}$. \\begin{enumerate} \\item There is a canonical six term exact sequence of abelian groups $$ \\xymatrix{ 0 \\ar[r] & \\Hom_\\mathcal{A}(M_3, N) \\ar[r] & \\Hom_\\mathcal{A}(M_2, N) \\ar[r] & \\Hom_\\mathcal{A}(M_1, N) \\ar[lld] \\\\ & \\Ext_\\mathcal{A}(M_3, N) \\ar[r] & \\Ext_\\mathcal{A}(M_2, N) \\ar[r] & \\Ext_\\mathcal{A}(M_1, N) } $$ for all objects $N$ of $\\mathcal{A}$, and \\item there is a canonical six term exact sequence of abelian groups $$ \\xymatrix{ 0 \\ar[r] & \\Hom_\\mathcal{A}(N, M_1) \\ar[r] & \\Hom_\\mathcal{A}(N, M_2) \\ar[r] & \\Hom_\\mathcal{A}(N, M_3) \\ar[lld] \\\\ & \\Ext_\\mathcal{A}(N, M_1) \\ar[r] & \\Ext_\\mathcal{A}(N, M_2) \\ar[r] & \\Ext_\\mathcal{A}(N, M_3) } $$ for all objects $N$ of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "12033", "title": "homology-lemma-additive-functor", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be additive categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. The following are equivalent \\begin{enumerate} \\item $F$ is additive, \\item $F(A) \\oplus F(B) \\to F(A \\oplus B)$ is an isomorphism for all $A, B \\in \\mathcal{A}$, and \\item $F(A \\oplus B) \\to F(A) \\oplus F(B)$  is an isomorphism for all $A, B \\in \\mathcal{A}$. \\end{enumerate}"}
+{"_id": "12034", "title": "homology-lemma-exact-functor", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. \\begin{enumerate} \\item If $F$ is either left or right exact, then it is additive. \\item $F$ is left exact if and only if for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ the sequence $0 \\to F(A) \\to F(B) \\to F(C)$ is exact. \\item $F$ is right exact if and only if for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ the sequence $F(A) \\to F(B) \\to F(C) \\to 0$ is exact. \\item $F$ is exact if and only if for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ the sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$ is exact. \\end{enumerate}"}
+{"_id": "12035", "title": "homology-lemma-exact-functor-ext", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor. For every pair of objects $A, B$ of $\\mathcal{A}$ the functor $F$ induces an abelian group homomorphism $$ \\Ext_\\mathcal{A}(B, A) \\longrightarrow \\Ext_\\mathcal{B}(F(B), F(A)) $$ which maps the extension $E$ to $F(E)$."}
+{"_id": "12036", "title": "homology-lemma-adjoint-get-abelian", "text": "Let $a : \\mathcal{A} \\to \\mathcal{B}$ and $b : \\mathcal{B} \\to \\mathcal{A}$ be functors. Assume that \\begin{enumerate} \\item $\\mathcal{A}$, $\\mathcal{B}$ are additive categories, $a$, $b$ are additive functors, and $a$ is right adjoint to $b$, \\item $\\mathcal{B}$ is abelian and $b$ is left exact, and \\item $ba \\cong \\text{id}_\\mathcal{A}$. \\end{enumerate} Then $\\mathcal{A}$ is abelian."}
+{"_id": "12037", "title": "homology-lemma-localization-preadditive", "text": "Let $\\mathcal{C}$ be a preadditive category. Let $S$ be a left or right multiplicative system. There exists a canonical preadditive structure on $S^{-1}\\mathcal{C}$ such that the localization functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive."}
+{"_id": "12038", "title": "homology-lemma-localization-additive", "text": "Let $\\mathcal{C}$ be an additive category. Let $S$ be a left or right multiplicative system. Then $S^{-1}\\mathcal{C}$ is an additive category and the localization functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive."}
+{"_id": "12039", "title": "homology-lemma-kernel-localization", "text": "Let $\\mathcal{C}$ be an additive category. Let $S$ be a multiplicative system. Let $X$ be an object of $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item $Q(X) = 0$ in $S^{-1}\\mathcal{C}$, \\item there exists $Y \\in \\Ob(\\mathcal{C})$ such that $0 : X \\to Y$ is an element of $S$, and \\item there exists $Z \\in \\Ob(\\mathcal{C})$ such that $0 : Z \\to X$ is an element of $S$. \\end{enumerate}"}
+{"_id": "12040", "title": "homology-lemma-localization-abelian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item If $S$ is a left multiplicative system, then the category $S^{-1}\\mathcal{A}$ has cokernels and the functor $Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them. \\item If $S$ is a right multiplicative system, then the category $S^{-1}\\mathcal{A}$ has kernels and the functor $Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them. \\item If $S$ is a multiplicative system, then the category $S^{-1}\\mathcal{A}$ is abelian and the functor $Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ is exact. \\end{enumerate}"}
+{"_id": "12041", "title": "homology-lemma-ses-artinian", "text": "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$ be a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Artinian if and only if $A_1$ and $A_3$ are Artinian."}
+{"_id": "12042", "title": "homology-lemma-ses-noetherian", "text": "Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$ be a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Noetherian if and only if $A_1$ and $A_3$ are Noetherian."}
+{"_id": "12043", "title": "homology-lemma-finite-length", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$. The following are equivalent \\begin{enumerate} \\item $A$ is Artinian and Noetherian, and \\item there exists a filtration $0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A$ by subobjects such that $A_i/A_{i - 1}$ is simple for $i = 1, \\ldots, n$. \\end{enumerate}"}
+{"_id": "12044", "title": "homology-lemma-jordan-holder", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ satisfying the equivalent conditions of Lemma \\ref{lemma-finite-length}. Given two filtrations $$ 0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A \\quad\\text{and}\\quad 0 \\subset B_1 \\subset B_2 \\subset \\ldots \\subset B_m = A $$ with $S_i = A_i/A_{i - 1}$ and $T_j = B_j/B_{j - 1}$ simple objects we have $n = m$ and there exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$ such that $S_i \\cong T_{\\sigma(i)}$ for all $i \\in \\{1, \\ldots, n\\}$."}
+{"_id": "12045", "title": "homology-lemma-characterize-serre-subcategory", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$. Then $\\mathcal{C}$ is a Serre subcategory if and only if the following conditions are satisfied: \\begin{enumerate} \\item $0 \\in \\Ob(\\mathcal{C})$, \\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$, \\item any subobject or quotient of an object of $\\mathcal{C}$ is an object of $\\mathcal{C}$, \\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$ then also $A \\in \\Ob(\\mathcal{C})$. \\end{enumerate} Moreover, a Serre subcategory is an abelian category and the inclusion functor is exact."}
+{"_id": "12046", "title": "homology-lemma-characterize-weak-serre-subcategory", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$. Then $\\mathcal{C}$ is a weak Serre subcategory if and only if the following conditions are satisfied: \\begin{enumerate} \\item $0 \\in \\Ob(\\mathcal{C})$, \\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$, \\item kernels and cokernels in $\\mathcal{A}$ of morphisms between objects of $\\mathcal{C}$ are in $\\mathcal{C}$, \\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$ then also $A \\in \\Ob(\\mathcal{C})$. \\end{enumerate} Moreover, a weak Serre subcategory is an abelian category and the inclusion functor is exact."}
+{"_id": "12047", "title": "homology-lemma-kernel-exact-functor", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor. Then the full subcategory of objects $C$ of $\\mathcal{A}$ such that $F(C) = 0$ forms a Serre subcategory of $\\mathcal{A}$."}
+{"_id": "12048", "title": "homology-lemma-serre-subcategory-is-kernel", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory. There exists an abelian category $\\mathcal{A}/\\mathcal{C}$ and an exact functor $$ F : \\mathcal{A} \\longrightarrow \\mathcal{A}/\\mathcal{C} $$ which is essentially surjective and whose kernel is $\\mathcal{C}$. The category $\\mathcal{A}/\\mathcal{C}$ and the functor $F$ are characterized by the following universal property: For any exact functor $G : \\mathcal{A} \\to \\mathcal{B}$ such that $\\mathcal{C} \\subset \\Ker(G)$ there exists a factorization $G = H \\circ F$ for a unique exact functor $H : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$."}
+{"_id": "12049", "title": "homology-lemma-quotient-by-kernel-exact-functor", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor. Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory contained in the kernel of $F$. Then $\\mathcal{C} = \\Ker(F)$ if and only if the induced functor $\\overline{F} : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$ (Lemma \\ref{lemma-serre-subcategory-is-kernel}) is faithful."}
+{"_id": "12050", "title": "homology-lemma-exact-functor-K-groups", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor between abelian categories. Then $F$ induces a homomorphism of $K$-groups $K_0(F) : K_0(\\mathcal{A}) \\to K_0(\\mathcal{B})$ by simply setting $K_0(F)([A]) = [F(A)]$."}
+{"_id": "12051", "title": "homology-lemma-serre-subcategory-K-groups", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory and set $\\mathcal{B} = \\mathcal{A}/\\mathcal{C}$. \\begin{enumerate} \\item The exact functors $\\mathcal{C} \\to \\mathcal{A}$ and $\\mathcal{A} \\to \\mathcal{B}$ induce an exact sequence $$ K_0(\\mathcal{C}) \\to K_0(\\mathcal{A}) \\to K_0(\\mathcal{B}) \\to 0 $$ of $K$-groups, and \\item the kernel of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$ is equal to the collection of elements of the form $$ [H^0(M, \\varphi, \\psi)] - [H^1(M, \\varphi, \\psi)] $$ where $(M, \\varphi, \\psi)$ is a complex as in (\\ref{equation-cyclic-complex}) with the property that it becomes exact in $\\mathcal{B}$; in other words that $H^0(M, \\varphi, \\psi)$ and $H^1(M, \\varphi, \\psi)$ are objects of $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12052", "title": "homology-lemma-efface-implies-universal", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let $F = (F^n, \\delta_F)$ be a $\\delta$-functor from $\\mathcal{A}$ to $\\mathcal{B}$. Suppose that for every $n > 0$ and any $A \\in \\Ob(\\mathcal{A})$ there exists an injective morphism $u : A \\to B$ (depending on $A$ and $n$) such that $F^n(u) : F^n(A) \\to F^n(B)$ is zero. Then $F$ is a universal $\\delta$-functor."}
+{"_id": "12053", "title": "homology-lemma-uniqueness-universal-delta-functor", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. If there exists a universal $\\delta$-functor $(F^n, \\delta_F)$ from $\\mathcal{A}$ to $\\mathcal{B}$ with $F^0 = F$, then it is determined up to unique isomorphism of $\\delta$-functors."}
+{"_id": "12054", "title": "homology-lemma-compose-homotopy", "text": "Let $\\mathcal{A}$ be an additive category. Let $f, g : B_\\bullet \\to C_\\bullet$ be morphisms of chain complexes. Suppose given morphisms of chain complexes $a : A_\\bullet \\to B_\\bullet$, and $c : C_\\bullet \\to D_\\bullet$. If $\\{h_i : B_i \\to C_{i + 1}\\}$ defines a homotopy between $f$ and $g$, then $\\{c_{i + 1} \\circ h_i \\circ a_i\\}$ defines a homotopy between $c \\circ f \\circ a$ and $c \\circ g \\circ a$."}
+{"_id": "12055", "title": "homology-lemma-cat-chain-abelian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item The category of chain complexes in $\\mathcal{A}$ is abelian. \\item A morphism of complexes $f : A_\\bullet \\to B_\\bullet$ is injective if and only if each $f_n : A_n \\to B_n$ is injective. \\item A morphism of complexes $f : A_\\bullet \\to B_\\bullet$ is surjective if and only if each $f_n : A_n \\to B_n$ is surjective. \\item A sequence of chain complexes $$ A_\\bullet \\xrightarrow{f} B_\\bullet \\xrightarrow{g} C_\\bullet $$ is exact at $B_\\bullet$ if and only if each sequence $$ A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i $$ is exact at $B_i$. \\end{enumerate}"}
+{"_id": "12056", "title": "homology-lemma-map-homology-homotopy", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item If the maps $f, g : A_\\bullet \\to B_\\bullet$ are homotopic, then the induced maps $H_i(f)$ and $H_i(g)$ are equal. \\item If the map $f : A_\\bullet \\to B_\\bullet$ is a homotopy equivalence, then $f$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "12057", "title": "homology-lemma-long-exact-sequence-chain", "text": "Let $\\mathcal{A}$ be an abelian category. Suppose that $$ 0 \\to A_\\bullet \\to B_\\bullet \\to C_\\bullet \\to 0 $$ is a short exact sequence of chain complexes of $\\mathcal{A}$. Then there is a canonical long exact homology sequence $$ \\xymatrix{ \\ldots & \\ldots & \\ldots \\ar[lld] \\\\ H_i(A_\\bullet) \\ar[r] & H_i(B_\\bullet) \\ar[r] & H_i(C_\\bullet) \\ar[lld] \\\\ H_{i - 1}(A_\\bullet) \\ar[r] & H_{i - 1}(B_\\bullet) \\ar[r] & H_{i - 1}(C_\\bullet) \\ar[lld] \\\\ \\ldots & \\ldots & \\ldots \\\\ } $$"}
+{"_id": "12058", "title": "homology-lemma-compose-homotopy-cochain", "text": "Let $\\mathcal{A}$ be an additive category. Let $f, g : B^\\bullet \\to C^\\bullet$ be morphisms of cochain complexes. Suppose given morphisms of cochain complexes $a : A^\\bullet \\to B^\\bullet$, and $c : C^\\bullet \\to D^\\bullet$. If $\\{h^i : B^i \\to C^{i - 1}\\}$ defines a homotopy between $f$ and $g$, then $\\{c^{i - 1} \\circ h^i \\circ a^i\\}$ defines a homotopy between $c \\circ f \\circ a$ and $c \\circ g \\circ a$."}
+{"_id": "12059", "title": "homology-lemma-cat-cochain-abelian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item The category of cochain complexes in $\\mathcal{A}$ is abelian. \\item A morphism of cochain complexes $f : A^\\bullet \\to B^\\bullet$ is injective if and only if each $f^n : A^n \\to B^n$ is injective. \\item A morphism of cochain complexes $f : A^\\bullet \\to B^\\bullet$ is surjective if and only if each $f^n : A^n \\to B^n$ is surjective. \\item A sequence of cochain complexes $$ A^\\bullet \\xrightarrow{f} B^\\bullet \\xrightarrow{g} C^\\bullet $$ is exact at $B^\\bullet$ if and only if each sequence $$ A^i \\xrightarrow{f^i} B^i \\xrightarrow{g^i} C^i $$ is exact at $B^i$. \\end{enumerate}"}
+{"_id": "12060", "title": "homology-lemma-map-cohomology-homotopy-cochain", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item If the maps $f, g : A^\\bullet \\to B^\\bullet$ are homotopic, then the induced maps $H^i(f)$ and $H^i(g)$ are equal. \\item If $f : A^\\bullet \\to B^\\bullet$ is a homotopy equivalence, then $f$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "12061", "title": "homology-lemma-long-exact-sequence-cochain", "text": "\\begin{slogan} Short exact sequences of complexes give rise to long exact sequences of (co)homology. \\end{slogan} Let $\\mathcal{A}$ be an abelian category. Suppose that $$ 0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0 $$ is a short exact sequence of chain complexes of $\\mathcal{A}$. Then there is a canonical long exact cohomology sequence $$ \\xymatrix{ \\ldots & \\ldots & \\ldots \\ar[lld] \\\\ H^i(A^\\bullet) \\ar[r] & H^i(B^\\bullet) \\ar[r] & H^i(C^\\bullet) \\ar[lld] \\\\ H^{i + 1}(A^\\bullet) \\ar[r] & H^{i + 1}(B^\\bullet) \\ar[r] & H^{i + 1}(C^\\bullet) \\ar[lld] \\\\ \\ldots & \\ldots & \\ldots \\\\ } $$"}
+{"_id": "12062", "title": "homology-lemma-homotopy-shift", "text": "Let $\\mathcal{A}$ be an additive category. Suppose that $A_\\bullet$ and $B_\\bullet$ are chain complexes. Given any morphism of chain complexes $a : A_\\bullet \\to B_\\bullet$ there is a bijection between the set of homotopies from $a$ to $a$ and $\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$."}
+{"_id": "12063", "title": "homology-lemma-ses-termwise-split", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ 0 \\to A_\\bullet \\to B_\\bullet \\to C_\\bullet \\to 0 $$ be a short exact sequence of complexes. Suppose that $\\{s_n : C_n \\to B_n\\}$ is a family of morphisms which split the short exact sequences $0 \\to A_n \\to B_n \\to C_n \\to 0$. Let $\\pi_n : B_n \\to A_n$ be the associated projections, see Lemma \\ref{lemma-ses-split}. Then the family of morphisms $$ \\pi_{n - 1} \\circ d_{B, n} \\circ s_n : C_n \\to A_{n - 1} $$ define a morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$."}
+{"_id": "12064", "title": "homology-lemma-ses-termwise-split-long", "text": "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above. The morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$ induces the maps $$ H_i(\\delta(s)) : H_i(C_\\bullet) \\longrightarrow H_i(A[-1]_\\bullet) = H_{i - 1}(A_\\bullet) $$ which occur in the long exact homology sequence associated to the short exact sequence of chain complexes by Lemma \\ref{lemma-long-exact-sequence-chain}."}
+{"_id": "12065", "title": "homology-lemma-ses-termwise-split-homotopy", "text": "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above. Suppose $\\{s'_n : C_n \\to B_n\\}$ is a second choice of splittings. Write $s'_n = s_n + i_n \\circ h_n$ for some unique morphisms $h_n : C_n \\to A_n$. The family of maps $\\{h_n : C_n \\to A[-1]_{n + 1}\\}$ is a homotopy between the associated morphisms $\\delta(s), \\delta(s') : C_\\bullet \\to A[-1]_\\bullet$."}
+{"_id": "12066", "title": "homology-lemma-homotopy-shift-cochain", "text": "Let $\\mathcal{A}$ be an additive category. Suppose that $A^\\bullet$ and $B^\\bullet$ are cochain complexes. Given any morphism of cochain complexes $a : A^\\bullet \\to B^\\bullet$ there is a bijection between the set of homotopies from $a$ to $a$ and $\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$."}
+{"_id": "12067", "title": "homology-lemma-ses-termwise-split-cochain", "text": "Let $\\mathcal{A}$ be an additive category. Let $$ 0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0 $$ be a complex (!) of complexes. Suppose that we are given splittings $B^n = A^n \\oplus C^n$ compatible with the maps in the displayed sequence. Let $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$ be the corresponding maps. Then the family of morphisms $$ \\pi^{n + 1} \\circ d_B^n \\circ s^n : C^n \\to A^{n + 1} $$ define a morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$."}
+{"_id": "12068", "title": "homology-lemma-ses-termwise-split-long-cochain", "text": "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split-cochain} above. Assume in addition that $\\mathcal{A}$ is abelian. The morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$ induces the maps $$ H^i(\\delta) : H^i(C^\\bullet) \\longrightarrow H^i(A[1]^\\bullet) = H^{i + 1}(A^\\bullet) $$ which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma \\ref{lemma-long-exact-sequence-cochain}."}
+{"_id": "12069", "title": "homology-lemma-ses-termwise-split-homotopy-cochain", "text": "Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split-cochain}. Let $\\alpha : A^\\bullet \\to B^\\bullet$, $\\beta : B^\\bullet \\to C^\\bullet$ be the given morphisms of complexes. Suppose $(s')^n : C^n \\to B^n$ and $(\\pi')^n : B^n \\to A^n$ is a second choice of splittings. Write $(s')^n = s^n + \\alpha^n \\circ h^n$ and $(\\pi')^n = \\pi^n + g^n \\circ \\beta^n$ for some unique morphisms $h^n : C^n \\to A^n$ and $g^n : C^n \\to A^n$. Then \\begin{enumerate} \\item $g^n = - h^n$, and \\item the family of maps $\\{g^n : C^n \\to A[1]^{n - 1}\\}$ is a homotopy between $\\delta, \\delta' : C^\\bullet \\to A[1]^\\bullet$, more precisely $(\\delta')^n = \\delta^n + g^{n + 1} \\circ d_C^n + d_{A[1]}^{n - 1} \\circ g^n$. \\end{enumerate}"}
+{"_id": "12070", "title": "homology-lemma-graded", "text": "Let $\\mathcal{A}$ be an abelian category. The category of graded objects $\\text{Gr}(\\mathcal{A})$ is abelian."}
+{"_id": "12071", "title": "homology-lemma-additive-dual", "text": "Let $\\mathcal{A}$ be an additive monoidal category. If $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then $Y_1 \\oplus Y_2$ is a left dual of $X_1 \\oplus X_2$."}
+{"_id": "12072", "title": "homology-lemma-Karoubian-dual", "text": "In a Karoubian additive monoidal category every summand of an object which has a left dual has a left dual."}
+{"_id": "12073", "title": "homology-lemma-left-dual-graded-vector-spaces", "text": "Let $F$ be a field. Let $\\mathcal{C}$ be the category of graded $F$-vector spaces viewed as a monoidal category as in Example \\ref{example-graded-vector-spaces}. If $V$ in $\\mathcal{C}$ has a left dual $W$, then $\\sum_n \\dim_F V^n < \\infty$ and the map $\\epsilon$ defines nondegenerate pairings $W^{-n} \\times V^n \\to F$."}
+{"_id": "12074", "title": "homology-lemma-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. The category of filtered objects $\\text{Fil}(\\mathcal{A})$ has the following properties: \\begin{enumerate} \\item It is an additive category. \\item It has a zero object. \\item It has kernels and cokernels, images and coimages. \\item In general it is not an abelian category. \\end{enumerate}"}
+{"_id": "12075", "title": "homology-lemma-characterize-strict-general", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$. The following are equivalent \\begin{enumerate} \\item $f$ is strict, \\item the morphism $\\Coim(f) \\to \\Im(f)$ of Lemma \\ref{lemma-coim-im-map} is an isomorphism. \\end{enumerate}"}
+{"_id": "12076", "title": "homology-lemma-add-summand-strict-monomorphism", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$ be a strict monomorphism of filtered objects. Let $g : A \\to C$ be a morphism of filtered objects. Then $f \\oplus g : A \\to B \\oplus C$ is a strict monomorphism."}
+{"_id": "12077", "title": "homology-lemma-add-summand-strict-epimorphism", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f : B \\to A$ be a strict epimorphism of filtered objects. Let $g : C \\to A$ be a morphism of filtered objects. Then $f \\oplus g : B \\oplus C \\to A$ is a strict epimorphism."}
+{"_id": "12078", "title": "homology-lemma-induced-and-quotient-strict", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A, F)$, $(B, F)$ be filtered objects. Let $u : A \\to B$ be a morphism of filtered objects. If $u$ is injective then $u$ is strict if and only if the filtration on $A$ is the induced filtration. If $u$ is surjective then $u$ is strict if and only if the filtration on $B$ is the quotient filtration."}
+{"_id": "12080", "title": "homology-lemma-filtration-subobject", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A, F)$ be a filtered object of $\\mathcal{A}$. Let $X \\subset Y \\subset A$ be subobjects of $A$. On the object $$ Y/X = \\Ker(A/X \\to A/Y) $$ the quotient filtration coming from the induced filtration on $Y$ and the induced filtration coming from the quotient filtration on $A/X$ agree. Any of the morphisms $X \\to Y$, $X \\to A$, $Y \\to A$, $Y \\to A/X$, $Y \\to Y/X$, $Y/X \\to A/X$ are strict (with induced/quotient filtrations)."}
+{"_id": "12081", "title": "homology-lemma-pushout-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A, B, C \\in \\text{Fil}(\\mathcal{A})$. Let $f : A \\to B$ and $g : A \\to C$ be morphisms. Then there exists a pushout $$ \\xymatrix{ A \\ar[r]_f \\ar[d]_g & B \\ar[d]^{g'} \\\\ C \\ar[r]^{f'} & C \\amalg_A B } $$ in $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$."}
+{"_id": "12083", "title": "homology-lemma-ses-gr", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item Let $A$ be a filtered object and $X \\subset A$. Then for each $p$ the sequence $$ 0 \\to \\text{gr}^p(X) \\to \\text{gr}^p(A) \\to \\text{gr}^p(A/X) \\to 0 $$ is exact (with induced filtration on $X$ and quotient filtration on $A/X$). \\item Let $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$. Then for each $p$ the sequences $$ 0 \\to \\text{gr}^p(\\Ker(f)) \\to \\text{gr}^p(A) \\to \\text{gr}^p(\\Coim(f)) \\to 0 $$ and $$ 0 \\to \\text{gr}^p(\\Im(f)) \\to \\text{gr}^p(B) \\to \\text{gr}^p(\\Coker(f)) \\to 0 $$ are exact. \\end{enumerate}"}
+{"_id": "12084", "title": "homology-lemma-characterize-strict", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$ be a morphism of finite filtered objects of $\\mathcal{A}$. The following are equivalent \\begin{enumerate} \\item $f$ is strict, \\item the morphism $\\Coim(f) \\to \\Im(f)$ is an isomorphism, \\item $\\text{gr}(\\Coim(f)) \\to \\text{gr}(\\Im(f))$ is an isomorphism, \\item the sequence $\\text{gr}(\\Ker(f)) \\to \\text{gr}(A) \\to \\text{gr}(B)$ is exact, \\item the sequence $\\text{gr}(A) \\to \\text{gr}(B) \\to \\text{gr}(\\Coker(f))$ is exact, and \\item the sequence $$ 0 \\to \\text{gr}(\\Ker(f)) \\to \\text{gr}(A) \\to \\text{gr}(B) \\to \\text{gr}(\\Coker(f)) \\to 0 $$ is exact. \\end{enumerate}"}
+{"_id": "12086", "title": "homology-lemma-filtered-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A \\to B \\to C$ be a complex of filtered objects of $\\mathcal{A}$. Assume $A, B, C$ have finite filtrations and that $\\text{gr}(A) \\to \\text{gr}(B) \\to \\text{gr}(C)$ is exact. Then \\begin{enumerate} \\item for each $p \\in \\mathbf{Z}$ the sequence $\\text{gr}^p(A) \\to \\text{gr}^p(B) \\to \\text{gr}^p(C)$ is exact, \\item for each $p \\in \\mathbf{Z}$ the sequence $F^p(A) \\to F^p(B) \\to F^p(C)$ is exact, \\item for each $p \\in \\mathbf{Z}$ the sequence $A/F^p(A) \\to B/F^p(B) \\to C/F^p(C)$ is exact, \\item the maps $A \\to B$ and $B \\to C$ are strict, and \\item $A \\to B \\to C$ is exact (as a sequence in $\\mathcal{A}$). \\end{enumerate}"}
+{"_id": "12087", "title": "homology-lemma-derived-exact-couple", "text": "Let $(A, E, \\alpha, f, g)$ be an exact couple in an abelian category $\\mathcal{A}$. Set \\begin{enumerate} \\item $d = g \\circ f : E \\to E$ so that $d \\circ d = 0$, \\item $E' = \\Ker(d)/\\Im(d)$, \\item $A' = \\Im(\\alpha)$, \\item $\\alpha' : A' \\to A'$ induced by $\\alpha$, \\item $f' : E' \\to A'$ induced by $f$, \\item $g' : A' \\to E'$ induced by ``$g \\circ \\alpha^{-1}$''. \\end{enumerate} Then we have \\begin{enumerate} \\item $\\Ker(d) = f^{-1}(\\Ker(g)) = f^{-1}(\\Im(\\alpha))$, \\item $\\Im(d) = g(\\Im(f)) = g(\\Ker(\\alpha))$, \\item $(A', E', \\alpha', f', g')$ is an exact couple. \\end{enumerate}"}
+{"_id": "12088", "title": "homology-lemma-spectral-sequence-associated-exact-couple", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A, E, \\alpha, f, g)$ be an exact couple. Let $(E_r, d_r)_{r \\geq 1}$ be the spectral sequence associated to the exact couple. In this case we have $$ 0 = B_1 \\subset \\ldots \\subset B_{r + 1} = g(\\Ker(\\alpha^r)) \\subset \\ldots \\subset Z_{r + 1} = f^{-1}(\\Im(\\alpha^r)) \\subset \\ldots \\subset Z_1 = E $$ and the map $d_{r + 1} : E_{r + 1} \\to E_{r + 1}$ is described by the following rule: For any (test) object $T$ of $\\mathcal{A}$ and any elements $x : T \\to Z_{r + 1}$ and $y : T \\to A$ such that $f \\circ x = \\alpha^r \\circ y$ we have $$ d_{r + 1} \\circ \\overline{x} = \\overline{g \\circ y} $$ where $\\overline{x} : T \\to E_{r + 1}$ is the induced morphism."}
+{"_id": "12089", "title": "homology-lemma-differential-objects-abelian", "text": "\\begin{slogan} The category of differential objects of an abelian category is itself an abelian category. \\end{slogan} Let $\\mathcal{A}$ be an abelian category. The category of differential objects of $\\mathcal{A}$ is abelian."}
+{"_id": "12091", "title": "homology-lemma-spectral-sequence-filtered-differential", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ in $\\text{Gr}(\\mathcal{A})$ associated to $(K, F, d)$ such that $d_r : E_r \\to E_r[r]$ for all $r$ and such that the graded pieces $E_r^p$ and maps $d_r^p : E_r^p \\to E_r^{p + r}$ are as given above. Furthermore, $E_0^p = \\text{gr}^p K$, $d_0^p = \\text{gr}^p(d)$, and $E_1^p = H(\\text{gr}^pK, d)$."}
+{"_id": "12093", "title": "homology-lemma-compute-filtered-cohomology", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. If $Z_\\infty^p$ and $B_\\infty^p$ exist (see proof), then \\begin{enumerate} \\item the limit $E_\\infty$ exists and is graded having $E_\\infty^p = Z_\\infty^p/B_\\infty^p$ in degree $p$, and \\item the associated graded $\\text{gr}(H(K))$ of the cohomology of $K$ is a graded subquotient of the graded limit object $E_\\infty$. \\end{enumerate}"}
+{"_id": "12094", "title": "homology-lemma-filtered-differential-ss-converges", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. The associated spectral sequence \\begin{enumerate} \\item weakly converges to $H(K)$ if and only if for every $p \\in \\mathbf{Z}$ we have equality in equations (\\ref{equation-at-bottom}) and (\\ref{equation-on-top}), \\item abuts to $H(K)$ if and only if it weakly converges to $H(K)$ and $\\bigcap_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Im(d)$ and $\\bigcup_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Ker(d)$. \\end{enumerate}"}
+{"_id": "12095", "title": "homology-lemma-spectral-sequence-filtered-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ in the category of bigraded objects of $\\mathcal{A}$ associated to $(K^\\bullet, F)$ such that $d_r$ has bidegree $(r, - r + 1)$ and such that $E_r$ has bigraded pieces $E_r^{p, q}$ and maps $d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ as given above. Furthermore, we have $E_0^{p, q} = \\text{gr}^p(K^{p + q})$, $d_0^{p, q} = \\text{gr}^p(d^{p + q})$, and $E_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet))$."}
+{"_id": "12096", "title": "homology-lemma-spectral-sequence-filtered-complex-d1", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. Assume $\\mathcal{A}$ has countable direct sums. Let $(E_r, d_r)_{r \\geq 0}$ be the spectral sequence associated to $(K^\\bullet, F)$. \\begin{enumerate} \\item The map $$ d_1^{p, q} : E_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet)) \\longrightarrow E_1^{p + 1, q} = H^{p + q + 1}(\\text{gr}^{p + 1}(K^\\bullet)) $$ is equal to the boundary map in cohomology associated to the short exact sequence of complexes $$ 0 \\to \\text{gr}^{p + 1}(K^\\bullet) \\to F^pK^\\bullet/F^{p + 2}K^\\bullet \\to \\text{gr}^p(K^\\bullet) \\to 0. $$ \\item Assume that $d(F^pK) \\subset F^{p + 1}K$ for all $p \\in \\mathbf{Z}$. Then $d$ induces the zero differential on $\\text{gr}^p(K^\\bullet)$ and hence $E_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q}$. Furthermore, in this case $$ d_1^{p, q} : E_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q} \\longrightarrow E_1^{p + 1, q} = \\text{gr}^{p + 1}(K^\\bullet)^{p + q + 1} $$ is the morphism induced by $d$. \\end{enumerate}"}
+{"_id": "12097", "title": "homology-lemma-spectral-sequence-filtered-complex-functorial", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\alpha : (K^\\bullet, F) \\to (L^\\bullet, F)$ be a morphism of filtered complexes of $\\mathcal{A}$. Let $(E_r(K), d_r)_{r \\geq 0}$, resp.\\ $(E_r(L), d_r)_{r \\geq 0}$ be the spectral sequence associated to $(K^\\bullet, F)$, resp.\\ $(L^\\bullet, F)$. The morphism $\\alpha$ induces a canonical morphism of spectral sequences $\\{\\alpha_r : E_r(K) \\to E_r(L)\\}_{r \\geq 0}$ compatible with the bigradings."}
+{"_id": "12098", "title": "homology-lemma-compute-cohomology-filtered-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. If $Z_\\infty^{p, q}$ and $B_\\infty^{p, q}$ exist (see proof), then \\begin{enumerate} \\item the limit $E_\\infty$ exists and is a bigraded object having $E_\\infty^{p, q} = Z_\\infty^{p, q}/B_\\infty^{p, q}$ in bidegree $(p, q)$, \\item the $p$th graded part $\\text{gr}^pH^n(K^\\bullet)$ of the $n$th cohomology object of $K^\\bullet$ is a subquotient of $E_\\infty^{p, n - p}$. \\end{enumerate}"}
+{"_id": "12099", "title": "homology-lemma-relate-boundedness", "text": "In the situation of Definition \\ref{definition-bounded-ss}. Let $Z_r^{p, q}, B_r^{p, q} \\subset E_{r_0}^{p, q}$ be the $(p, q)$-graded parts of $Z_r, B_r$ defined as in Section \\ref{section-spectral-sequence}. \\begin{enumerate} \\item The spectral sequence is regular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $Z_r^{p, q} = Z_{r + 1}^{p, q} = \\ldots$ \\item The spectral sequence is coregular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $B_r^{p, q} = B_{r + 1}^{p, q} = \\ldots$ \\item The spectral sequence is bounded if and only if it is both bounded below and bounded above. \\item If the spectral sequence is bounded below, then it is regular. \\item If the spectral sequence is bounded above, then it is coregular. \\end{enumerate}"}
+{"_id": "12100", "title": "homology-lemma-filtered-complex-ss-converges", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. The associated spectral sequence \\begin{enumerate} \\item weakly converges to $H^*(K^\\bullet)$ if and only if for every $p, q \\in \\mathbf{Z}$ we have equality in equations (\\ref{equation-at-bottom-bigraded}) and (\\ref{equation-on-top-bigraded}), \\item abuts to $H^*(K)$ if and only if it weakly converges to $H^*(K^\\bullet)$ and we have $\\bigcap_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Im(d) \\cap K^n$ and $\\bigcup_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Ker(d) \\cap K^n$. \\end{enumerate}"}
+{"_id": "12101", "title": "homology-lemma-biregular-ss-converges", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$ is finite (see Definition \\ref{definition-filtered}). Then \\begin{enumerate} \\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded, \\item the filtration on each $H^n(K^\\bullet)$ is finite, \\item the spectral sequence associated to $(K^\\bullet, F)$ converges to $H^*(K^\\bullet)$, \\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory and for some $r$ we have $E_r^{p, q} \\in \\mathcal{C}$ for all $p, q \\in \\mathbf{Z}$, then $H^n(K^\\bullet)$ is in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12102", "title": "homology-lemma-biregular-ss-relation-in-K0", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$ is finite (see Definition \\ref{definition-filtered}) and that for some $r$ we have only a finite number of nonzero $E_r^{p, q}$. Then only a finite number of $H^n(K^\\bullet)$ are nonzero and we have $$ \\sum (-1)^n[H^n(K^\\bullet)] = \\sum (-1)^{p + q} [E_r^{p, q}] $$ in $K_0(\\mathcal{A}')$ where $\\mathcal{A}'$ is the smallest weak Serre subcategory of $\\mathcal{A}$ containing the objects $E_r^{p, q}$."}
+{"_id": "12103", "title": "homology-lemma-ss-converges-trivial", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. Assume \\begin{enumerate} \\item for every $n$ there exist $p_0(n)$ such that $H^n(F^pK^\\bullet) = 0$ for $p \\geq p_0(n)$, \\item for every $n$ there exist $p_1(n)$ such that $H^n(F^pK^\\bullet) \\to H^n(K^\\bullet)$ is an isomorphism for $p \\leq p_1(n)$. \\end{enumerate} Then \\begin{enumerate} \\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded, \\item the filtration on each $H^n(K^\\bullet)$ is finite, \\item the spectral sequence associated to $(K^\\bullet, F)$ converges to $H^*(K^\\bullet)$. \\end{enumerate}"}
+{"_id": "12104", "title": "homology-lemma-ss-double-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^{\\bullet, \\bullet}$ be a double complex. The spectral sequences associated to $K^{\\bullet, \\bullet}$ have the following terms: \\begin{enumerate} \\item ${}'E_0^{p, q} = K^{p, q}$ with ${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \\to K^{p, q + 1}$, \\item ${}''E_0^{p, q} = K^{q, p}$ with ${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \\to K^{q + 1, p}$, \\item ${}'E_1^{p, q} = H^q(K^{p, \\bullet})$ with ${}'d_1^{p, q} = H^q(d_1^{p, \\bullet})$, \\item ${}''E_1^{p, q} = H^q(K^{\\bullet, p})$ with ${}''d_1^{p, q} = (-1)^q H^q(d_2^{\\bullet, p})$, \\item ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\\bullet, \\bullet}))$, \\item ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\\bullet, \\bullet}))$. \\end{enumerate}"}
+{"_id": "12105", "title": "homology-lemma-first-quadrant-ss", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^{\\bullet, \\bullet}$ be a double complex. Assume that for every $n \\in \\mathbf{Z}$ there are only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then \\begin{enumerate} \\item the two spectral sequences associated to $K^{\\bullet, \\bullet}$ are bounded, \\item the filtrations $F_I$, $F_{II}$ on each $H^n(K^\\bullet)$ are finite, \\item the spectral sequences $({}'E_r, {}'d_r)_{r \\geq 0}$ and $({}''E_r, {}''d_r)_{r \\geq 0}$ converge to $H^*(\\text{Tot}(K^{\\bullet, \\bullet}))$, \\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory and for some $r$ we have ${}'E_r^{p, q} \\in \\mathcal{C}$ for all $p, q \\in \\mathbf{Z}$, then $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$ is in $\\mathcal{C}$. Similarly for $({}''E_r, {}''d_r)_{r \\geq 0}$. \\end{enumerate}"}
+{"_id": "12106", "title": "homology-lemma-double-complex-gives-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex. Let $A^{\\bullet, \\bullet}$ be a double complex. Let $\\alpha^p : K^p \\to A^{p, 0}$ be morphisms. Assume that \\begin{enumerate} \\item For every $n \\in \\mathbf{Z}$ there are only finitely many nonzero $A^{p, q}$ with $p + q = n$. \\item We have $A^{p, q} = 0$ if $q < 0$. \\item The morphisms $\\alpha^p$ give rise to a morphism of complexes $\\alpha : K^\\bullet \\to A^{\\bullet, 0}$. \\item The complex $A^{p, \\bullet}$ is exact in all degrees $q \\not = 0$ and the morphism $K^p \\to A^{p, 0}$ induces an isomorphism $K^p \\to \\Ker(d_2^{p, 0})$. \\end{enumerate} Then $\\alpha$ induces a quasi-isomorphism $$ K^\\bullet \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet}) $$ of complexes. Moreover, there is a variant of this lemma involving the second variable $q$ instead of $p$."}
+{"_id": "12107", "title": "homology-lemma-homotopy-complex-complexes", "text": "Let $\\mathcal{A}$ be an abelian category. Let $M^\\bullet$ be a complex of $\\mathcal{A}$. Let $$ a : M^\\bullet[0] \\longrightarrow \\left(A^{0, \\bullet} \\to A^{1, \\bullet} \\to A^{2, \\bullet} \\to \\ldots \\right) $$ be a homotopy equivalence in the category of complexes of complexes of $\\mathcal{A}$. Then the map $\\alpha : M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$ induced by $M^\\bullet \\to A^{0, \\bullet}$ is a homotopy equivalence."}
+{"_id": "12109", "title": "homology-lemma-good-resolution-gives-qis", "text": "Let $M^\\bullet$ be a complex of abelian groups. Let $$ \\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0 $$ be an exact complex of complexes of abelian groups such that for all $p \\in \\mathbf{Z}$ the complexes $$ \\ldots \\to \\Ker(d_{A_2^\\bullet}^p) \\to \\Ker(d_{A_1^\\bullet}^p) \\to \\Ker(d_{A_0^\\bullet}^p) \\to \\Ker(d_{M^\\bullet}^p) \\to 0 $$ are exact as well. Set $A^{p, q} = A_{-p}^q$ to obtain a double complex. Then $\\text{Tot}(A^{\\bullet, \\bullet}) \\to M^\\bullet$ induced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism."}
+{"_id": "12110", "title": "homology-lemma-good-right-resolution-gives-qis", "text": "Let $M^\\bullet$ be a complex of abelian groups. Let $$ 0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots $$ be an exact complex of complexes of abelian groups such that for all $p \\in \\mathbf{Z}$ the complexes $$ 0 \\to \\Coker(d_{M^\\bullet}^p) \\to \\Coker(d_{A_0^\\bullet}^p) \\to \\Coker(d_{A_1^\\bullet}^p) \\to \\Coker(d_{A_2^\\bullet}^p) \\to \\ldots $$ are exact as well. Set $A^{p, q} = A_p^q$ to obtain a double complex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ be the product total complex associated to the double complex (see proof). Then the map $M^\\bullet \\to \\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ induced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism."}
+{"_id": "12111", "title": "homology-lemma-resolution-gives-qis", "text": "Let $M^\\bullet$ be a complex of abelian groups. Let $$ \\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0 $$ be an exact complex of complexes of abelian groups. Set $A^{p, q} = A_{-p}^q$ to obtain a double complex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ be the product total complex associated to the double complex (see proof). Then the map $\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to M^\\bullet$ induced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism."}
+{"_id": "12112", "title": "homology-lemma-characterize-injectives", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I$ be an object of $\\mathcal{A}$. The following are equivalent: \\begin{enumerate} \\item The object $I$ is injective. \\item The functor $B \\mapsto \\Hom_\\mathcal{A}(B, I)$ is exact. \\item Any short exact sequence $$ 0 \\to I \\to A \\to B \\to 0 $$ in $\\mathcal{A}$ is split. \\item We have $\\Ext_\\mathcal{A}(B, I) = 0$ for all $B \\in \\Ob(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "12113", "title": "homology-lemma-product-injectives", "text": "Let $\\mathcal{A}$ be an abelian category. Suppose $I_\\omega$, $\\omega \\in \\Omega$ is a set of injective objects of $\\mathcal{A}$. If $\\prod_{\\omega \\in \\Omega} I_\\omega$ exists then it is injective."}
+{"_id": "12114", "title": "homology-lemma-characterize-projectives", "text": "Let $\\mathcal{A}$ be an abelian category. Let $P$ be an object of $\\mathcal{A}$. The following are equivalent: \\begin{enumerate} \\item The object $P$ is projective. \\item The functor $B \\mapsto \\Hom_\\mathcal{A}(P, B)$ is exact. \\item Any short exact sequence $$ 0 \\to A \\to B \\to P \\to 0 $$ in $\\mathcal{A}$ is split. \\item We have $\\Ext_\\mathcal{A}(P, A) = 0$ for all $A \\in \\Ob(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "12115", "title": "homology-lemma-coproduct-projectives", "text": "Let $\\mathcal{A}$ be an abelian category. Suppose $P_\\omega$, $\\omega \\in \\Omega$ is a set of projective objects of $\\mathcal{A}$. If $\\coprod_{\\omega \\in \\Omega} P_\\omega$ exists then it is projective."}
+{"_id": "12116", "title": "homology-lemma-adjoint-preserve-injectives", "text": "\\begin{slogan} A functor with an exact left adjoint preserves injectives \\end{slogan} Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume \\begin{enumerate} \\item $u$ is right adjoint to $v$, and \\item $v$ transforms injective maps into injective maps. \\end{enumerate} Then $u$ transforms injectives into injectives."}
+{"_id": "12117", "title": "homology-lemma-adjoint-enough-injectives", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume \\begin{enumerate} \\item $u$ is right adjoint to $v$, \\item $v$ transforms injective maps into injective maps, \\item $\\mathcal{A}$ has enough injectives, and \\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$. \\end{enumerate} Then $\\mathcal{B}$ has enough injectives."}
+{"_id": "12118", "title": "homology-lemma-adjoint-functorial-injectives", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume \\begin{enumerate} \\item $u$ is right adjoint to $v$, \\item $v$ transforms injective maps into injective maps, \\item $\\mathcal{A}$ has enough injectives, \\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$, and \\item $\\mathcal{A}$ has functorial injective hulls. \\end{enumerate} Then $\\mathcal{B}$ has functorial injective hulls."}
+{"_id": "12119", "title": "homology-lemma-partially-defined-adjoint", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $u : \\mathcal{A} \\to \\mathcal{B}$ be a functor. If there exists a subset $\\mathcal{P} \\subset \\Ob(\\mathcal{B})$ such that \\begin{enumerate} \\item every object of $\\mathcal{B}$ is a quotient of an element of $\\mathcal{P}$, and \\item for every $P \\in \\mathcal{P}$ there exists an object $Q$ of $\\mathcal{A}$ such that $\\Hom_\\mathcal{A}(Q, A) = \\Hom_\\mathcal{B}(P, u(A))$ functorially in $A$, \\end{enumerate} then there exists a left adjoint $v$ of $u$."}
+{"_id": "12121", "title": "homology-lemma-direct-sum-from-product-colimit", "text": "Let $\\mathcal{I}$ be a category. Let $\\mathcal{A}$ be an additive, Karoubian category. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and $G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent \\begin{enumerate} \\item $\\colim_\\mathcal{I} F \\oplus G$ exists, and \\item $\\colim_\\mathcal{I} F$ and $\\colim_\\mathcal{I} G$ exist. \\end{enumerate} In this case $\\colim_\\mathcal{I} F \\oplus G = \\colim_\\mathcal{I} F \\oplus \\colim_\\mathcal{I} G$."}
+{"_id": "12122", "title": "homology-lemma-direct-sum-from-product-essentially-constant", "text": "Let $\\mathcal{I}$ be a filtered category. Let $\\mathcal{A}$ be an additive, Karoubian category. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and $G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent \\begin{enumerate} \\item $F \\oplus G : \\mathcal{I} \\to \\mathcal{A}$ is essentially constant, and \\item $F$ and $G$ are essentially constant. \\end{enumerate}"}
+{"_id": "12123", "title": "homology-lemma-inverse-systems-abelian", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item If $\\mathcal{C}$ is an additive category, then the category of inverse systems with values in $\\mathcal{C}$ is an additive category. \\item If $\\mathcal{C}$ is an abelian category, then the category of inverse systems with values in $\\mathcal{C}$ is an abelian category. A sequence $(K_i) \\to (L_i) \\to (M_i)$ of inverse systems is exact if and only if each $K_i \\to L_i \\to N_i$ is exact. \\end{enumerate}"}
+{"_id": "12124", "title": "homology-lemma-Mittag-Leffler", "text": "Let $$ 0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0 $$ be a short exact sequence of inverse systems of abelian groups. \\begin{enumerate} \\item In any case the sequence $$ 0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i $$ is exact. \\item If $(B_i)$ is ML, then also $(C_i)$ is ML. \\item If $(A_i)$ is ML, then $$ 0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i \\to 0 $$ is exact. \\end{enumerate}"}
+{"_id": "12125", "title": "homology-lemma-apply-Mittag-Leffler", "text": "Let $$ (A_i) \\to (B_i) \\to (C_i) \\to (D_i) $$ be an exact sequence of inverse systems of abelian groups. If the system $(A_i)$ is ML, then the sequence $$ \\lim_i B_i \\to \\lim_i C_i \\to \\lim_i D_i $$ is exact."}
+{"_id": "12126", "title": "homology-lemma-essentially-constant", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A_i)$ be an inverse system in $\\mathcal{A}$ with limit $A = \\lim A_i$. Then $(A_i)$ is essentially constant (see Categories, Definition \\ref{categories-definition-essentially-constant-diagram}) if and only if there exists an $i$ and for all $j \\geq i$ a direct sum decomposition $A_j = A \\oplus Z_j$ such that (a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum decompositions, (b) for all $j$ there exists some $j' \\geq j$ such that $Z_{j'} \\to Z_j$ is zero."}
+{"_id": "12127", "title": "homology-lemma-exact-sequence-ML", "text": "Let $$ 0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0 $$ be an exact sequence of inverse systems of abelian groups. If $(A_i)$ has ML and $(C_i)$ is essentially constant, then $(B_i)$ has ML."}
+{"_id": "12128", "title": "homology-lemma-apply-Mittag-Leffler-again", "text": "Let $$ (A^{-2}_i \\to A^{-1}_i \\to A^0_i \\to A^1_i) $$ be an inverse system of complexes of abelian groups and denote $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote $(H_i^{-1})$, $(H_i^0)$ the inverse systems of cohomologies, and denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$. If $(A^{-2}_i)$ and $(A^{-1}_i)$ are ML and $(H^{-1}_i)$ is essentially constant, then $H^0 = \\lim H_i^0$."}
+{"_id": "12129", "title": "homology-lemma-ML-over-ordinals", "text": "Let $\\alpha$ be an ordinal. Let $K_\\beta^\\bullet$, $\\beta < \\alpha$ be an inverse system of complexes of abelian groups over $\\alpha$. If for all $\\beta < \\alpha$ the complex $K_\\beta^\\bullet$ is acyclic and the map $$ K^n_\\beta \\longrightarrow \\lim_{\\gamma < \\beta} K^n_\\gamma $$ is surjective, then the complex $\\lim_{\\beta < \\alpha} K_\\beta^\\bullet$ is acyclic."}
+{"_id": "12130", "title": "homology-lemma-product-abelian-groups-exact", "text": "Let $I$ be a set. For $i \\in I$ let $L_i \\to M_i \\to N_i$ be a complex of abelian groups. Let $H_i = \\Ker(M_i \\to N_i)/\\Im(L_i \\to M_i)$ be the cohomology. Then $$ \\prod L_i \\to \\prod M_i \\to \\prod N_i $$ is a complex of abelian groups with homology $\\prod H_i$."}
+{"_id": "12200", "title": "categories-theorem-adjoint-functor", "text": "Let $G : \\mathcal{C} \\to \\mathcal{D}$ be a functor of big categories. Assume $\\mathcal{C}$ has limits, $G$ commutes with them, and for every object $y$ of $\\mathcal{D}$ there exists a set of pairs $(x_i, f_i)_{i \\in I}$ with $x_i \\in \\Ob(\\mathcal{C})$, $f_i \\in \\Mor_\\mathcal{D}(y, G(x_i))$ such that for any pair $(x, f)$ with $x \\in \\Ob(\\mathcal{C})$, $f \\in \\Mor_\\mathcal{C}(y, G(x))$ there is an $i$ and a morphism $h : x_i \\to x$ such that $f = G(h) \\circ f_i$. Then $G$ has a left adjoint $F$."}
+{"_id": "12201", "title": "categories-lemma-construct-quasi-inverse", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a fully faithful functor. Suppose for every $X \\in \\Ob(\\mathcal{B})$ given an object $j(X)$ of $\\mathcal{A}$ and an isomorphism $i_X : X \\to F(j(X))$. Then there is a unique functor $j : \\mathcal{B} \\to \\mathcal{A}$ such that $j$ extends the rule on objects, and the isomorphisms $i_X$ define an isomorphism of functors $\\text{id}_\\mathcal{B} \\to F \\circ j$. Moreover, $j$ and $F$ are quasi-inverse equivalences of categories."}
+{"_id": "12202", "title": "categories-lemma-equivalence-categories", "text": "A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective."}
+{"_id": "12203", "title": "categories-lemma-yoneda", "text": "\\begin{reference} Appeared in some form in \\cite{Yoneda-homology}. Used by Grothendieck in a generalized form in \\cite{Gr-II}. \\end{reference} Let $U, V \\in \\Ob(\\mathcal{C})$. Given any morphism of functors $s : h_U \\to h_V$ there is a unique morphism $\\phi : U \\to V$ such that $h(\\phi) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\\mathcal{C}$ we have a natural bijection $$ \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, F) \\longrightarrow F(U), \\quad s \\longmapsto s_U(\\text{id}_U). $$"}
+{"_id": "12204", "title": "categories-lemma-composition-representable", "text": "Let $\\mathcal{C}$ be a category. Let $f : x \\to y$, and $g : y \\to z$ be representable. Then $g \\circ f : x \\to z$ is representable."}
+{"_id": "12205", "title": "categories-lemma-base-change-representable", "text": "Let $\\mathcal{C}$ be a category. Let $f : x \\to y$ be representable. Let $y' \\to y$ be a morphism of $\\mathcal{C}$. Then the morphism $x' := x \\times_y y' \\to y'$ is representable also."}
+{"_id": "12206", "title": "categories-lemma-fibre-product-presheaves", "text": "Let $\\mathcal{C}$ be a category. Let $F, G, H : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be functors. Let $a : F \\to G$ and $b : H \\to G$ be transformations of functors. Then the fibre product $F \\times_{a, G, b} H$ in the category $\\textit{PSh}(\\mathcal{C})$ exists and is given by the formula $$ (F \\times_{a, G, b} H)(X) = F(X) \\times_{a_X, G(X), b_X} H(X) $$ for any object $X$ of $\\mathcal{C}$."}
+{"_id": "12207", "title": "categories-lemma-representable-over-representable", "text": "Let $\\mathcal{C}$ be a category. Let $a : F \\to G$ be a morphism of contravariant functors from $\\mathcal{C}$ to $\\textit{Sets}$. If $a$ is representable, and $G$ is a representable functor, then $F$ is representable."}
+{"_id": "12208", "title": "categories-lemma-representable-diagonal", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor. Assume $\\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent: \\begin{enumerate} \\item the diagonal $\\Delta : F \\to F \\times F$ is representable, \\item for every $U$ in $\\mathcal{C}$, and any $\\xi \\in F(U)$ the map $\\xi : h_U \\to F$ is representable, \\item for every pair $U, V$ in $\\mathcal{C}$ and any $\\xi \\in F(U)$, $\\xi' \\in F(V)$ the fibre product $h_U \\times_{\\xi, F, \\xi'} h_V$ is representable. \\end{enumerate}"}
+{"_id": "12209", "title": "categories-lemma-characterize-mono-epi", "text": "Let $\\mathcal{C}$ be a category, and let $f : X \\to Y$ be a morphism of $\\mathcal{C}$. Then \\begin{enumerate} \\item $f$ is a monomorphism if and only if $X$ is the fibre product $X \\times_Y X$, and \\item $f$ is an epimorphism if and only if $Y$ is the pushout $Y \\amalg_X Y$. \\end{enumerate}"}
+{"_id": "12210", "title": "categories-lemma-functorial-colimit", "text": "Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$, and $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams whose colimits exist. Suppose $H : \\mathcal{I} \\to \\mathcal{J}$ is a functor, and suppose $t : M \\to N \\circ H$ is a transformation of functors. Then there is a unique morphism $$ \\theta : \\colim_\\mathcal{I} M \\longrightarrow \\colim_\\mathcal{J} N $$ such that all the diagrams $$ \\xymatrix{ M_i \\ar[d]_{t_i} \\ar[r] & \\colim_\\mathcal{I} M \\ar[d]^{\\theta} \\\\ N_{H(i)} \\ar[r] & \\colim_\\mathcal{J} N } $$ commute."}
+{"_id": "12211", "title": "categories-lemma-functorial-limit", "text": "Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$, and $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams whose limits exist. Suppose $H : \\mathcal{I} \\to \\mathcal{J}$ is a functor, and suppose $t : N \\circ H \\to M$ is a transformation of functors. Then there is a unique morphism $$ \\theta : \\lim_\\mathcal{J} N \\longrightarrow \\lim_\\mathcal{I} M $$ such that all the diagrams $$ \\xymatrix{ \\lim_\\mathcal{J} N \\ar[d]^{\\theta} \\ar[r] & N_{H(i)} \\ar[d]_{t_i} \\\\ \\lim_\\mathcal{I} M \\ar[r] & M_i } $$ commute."}
+{"_id": "12212", "title": "categories-lemma-colimits-commute", "text": "Let $\\mathcal{I}$, $\\mathcal{J}$ be index categories. Let $M : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$ be a functor. We have $$ \\colim_i \\colim_j M_{i, j} = \\colim_{i, j} M_{i, j} = \\colim_j \\colim_i M_{i, j} $$ provided all the indicated colimits exist. Similar for limits."}
+{"_id": "12213", "title": "categories-lemma-limits-products-equalizers", "text": "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. Write $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$. Denote $s, t : A \\to I$ the source and target maps. Suppose that $\\prod_{i \\in I} M_i$ and $\\prod_{a \\in A} M_{t(a)}$ exist. Suppose that the equalizer of $$ \\xymatrix{ \\prod_{i \\in I} M_i \\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi & \\prod_{a \\in A} M_{t(a)} } $$ exists, where the morphisms are determined by their components as follows: $p_a \\circ \\psi = M(a) \\circ p_{s(a)}$ and $p_a \\circ \\phi = p_{t(a)}$. Then this equalizer is the limit of the diagram."}
+{"_id": "12214", "title": "categories-lemma-colimits-coproducts-coequalizers", "text": "\\begin{slogan} If all coproducts and coequalizers exist, all colimits exist. \\end{slogan} Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. Write $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$. Denote $s, t : A \\to I$ the source and target maps. Suppose that $\\coprod_{i \\in I} M_i$ and $\\coprod_{a \\in A} M_{s(a)}$ exist. Suppose that the coequalizer of $$ \\xymatrix{ \\coprod_{a \\in A} M_{s(a)} \\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi & \\coprod_{i \\in I} M_i } $$ exists, where the morphisms are determined by their components as follows: The component $M_{s(a)}$ maps via $\\psi$ to the component $M_{t(a)}$ via the morphism $a$. The component $M_{s(a)}$ maps via $\\phi$ to the component $M_{s(a)}$ by the identity morphism. Then this coequalizer is the colimit of the diagram."}
+{"_id": "12215", "title": "categories-lemma-connected-limit-over-X", "text": "Let $\\mathcal{C}$ be a category. Let $X$ be an object of $\\mathcal{C}$. Let $M : \\mathcal{I} \\to \\mathcal{C}/X$ be a diagram in the category of objects over $X$. If the index category $\\mathcal{I}$ is connected and the limit of $M$ exists in $\\mathcal{C}/X$, then the limit of the composition $\\mathcal{I} \\to \\mathcal{C}/X \\to \\mathcal{C}$ exists and is the same."}
+{"_id": "12217", "title": "categories-lemma-cofinal", "text": "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories. Assume $\\mathcal{I}$ is cofinal in $\\mathcal{J}$. Then for every diagram $M : \\mathcal{J} \\to \\mathcal{C}$ we have a canonical isomorphism $$ \\colim_\\mathcal{I} M \\circ H = \\colim_\\mathcal{J} M $$ if either side exists."}
+{"_id": "12218", "title": "categories-lemma-initial", "text": "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories. Assume $\\mathcal{I}$ is initial in $\\mathcal{J}$. Then for every diagram $M : \\mathcal{J} \\to \\mathcal{C}$ we have a canonical isomorphism $$ \\lim_\\mathcal{I} M \\circ H = \\lim_\\mathcal{J} M $$ if either side exists."}
+{"_id": "12219", "title": "categories-lemma-colimit-constant-connected-fibers", "text": "Let $F : \\mathcal{I} \\to \\mathcal{I}'$ be a functor. Assume \\begin{enumerate} \\item the fibre categories (see Definition \\ref{definition-fibre-category}) of $\\mathcal{I}$ over $\\mathcal{I}'$ are all connected, and \\item for every morphism $\\alpha' : x' \\to y'$ in $\\mathcal{I}'$ there exist a morphism $\\alpha : x \\to y$ in $\\mathcal{I}$ such that $F(\\alpha) = \\alpha'$. \\end{enumerate} Then for every diagram $M : \\mathcal{I}' \\to \\mathcal{C}$ the colimit $\\colim_\\mathcal{I} M \\circ F$ exists if and only if $\\colim_{\\mathcal{I}'} M$ exists and if so these colimits agree."}
+{"_id": "12221", "title": "categories-lemma-finite-diagram-category", "text": "Let $\\mathcal{I}$ be a category with \\begin{enumerate} \\item $\\Ob(\\mathcal{I})$ is finite, and \\item there exist finitely many morphisms $f_1, \\ldots, f_m \\in \\text{Arrows}(\\mathcal{I})$ such that every morphism of $\\mathcal{I}$ is a composition $f_{j_1} \\circ f_{j_2} \\circ \\ldots \\circ f_{j_k}$. \\end{enumerate} Then there exists a functor $F : \\mathcal{J} \\to \\mathcal{I}$ such that \\begin{enumerate} \\item[(a)] $\\mathcal{J}$ is a finite category, and \\item[(b)] for any diagram $M : \\mathcal{I} \\to \\mathcal{C}$ the (co)limit of $M$ over $\\mathcal{I}$ exists if and only if the (co)limit of $M \\circ F$ over $\\mathcal{J}$ exists and in this case the (co)limits are canonically isomorphic. \\end{enumerate} Moreover, $\\mathcal{J}$ is connected (resp.\\ nonempty) if and only if $\\mathcal{I}$ is so."}
+{"_id": "12222", "title": "categories-lemma-fibre-products-equalizers-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Connected finite limits exist in $\\mathcal{C}$. \\item Equalizers and fibre products exist in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12223", "title": "categories-lemma-almost-finite-limits-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Nonempty finite limits exist in $\\mathcal{C}$. \\item Products of pairs and equalizers exist in $\\mathcal{C}$. \\item Products of pairs and fibre products exist in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12224", "title": "categories-lemma-finite-limits-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Finite limits exist in $\\mathcal{C}$. \\item Finite products and equalizers exist. \\item The category has a final object and fibred products exist. \\end{enumerate}"}
+{"_id": "12226", "title": "categories-lemma-almost-finite-colimits-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Nonempty finite colimits exist in $\\mathcal{C}$. \\item Coproducts of pairs and coequalizers exist in $\\mathcal{C}$. \\item Coproducts of pairs and pushouts exist in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12227", "title": "categories-lemma-colimits-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Finite colimits exist in $\\mathcal{C}$, \\item Finite coproducts and coequalizers exist in $\\mathcal{C}$, and \\item The category has an initial object and pushouts exist. \\end{enumerate}"}
+{"_id": "12228", "title": "categories-lemma-directed-commutes", "text": "Let $\\mathcal{I}$ and $\\mathcal{J}$ be index categories. Assume that $\\mathcal{I}$ is filtered and $\\mathcal{J}$ is finite. Let $M : \\mathcal{I} \\times \\mathcal{J} \\to \\textit{Sets}$, $(i, j) \\mapsto M_{i, j}$ be a diagram of diagrams of sets. In this case $$ \\colim_i \\lim_j M_{i, j} = \\lim_j \\colim_i M_{i, j}. $$ In particular, colimits over $\\mathcal{I}$ commute with finite products, fibre products, and equalizers of sets."}
+{"_id": "12229", "title": "categories-lemma-cofinal-in-filtered", "text": "Let $\\mathcal{I}$ be a category. Let $\\mathcal{J}$ be a full subcategory. Assume that $\\mathcal{I}$ is filtered. Assume also that for any object $i$ of $\\mathcal{I}$, there exists a morphism $i \\to j$ to some object $j$ of $\\mathcal{J}$. Then $\\mathcal{J}$ is filtered and cofinal in $\\mathcal{I}$."}
+{"_id": "12231", "title": "categories-lemma-colimits-abelian-as-sets", "text": "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\\mathcal{I}$ there exists an object $z$ and morphisms $x \\to z$ and $y \\to z$. Let $M : \\mathcal{I} \\to \\textit{Ab}$ be a diagram of abelian groups over $\\mathcal{I}$. Then the colimit of $M$ in the category of sets surjects onto the colimit of $M$ in the category of abelian groups."}
+{"_id": "12232", "title": "categories-lemma-split-into-connected", "text": "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume that for every solid diagram $$ \\xymatrix{ x \\ar[d] \\ar[r] & y \\ar@{..>}[d] \\\\ z \\ar@{..>}[r] & w } $$ in $\\mathcal{I}$ there exists an object $w$ and dotted arrows making the diagram commute. Then $\\mathcal{I}$ is either empty or a nonempty disjoint union of connected categories having the same property."}
+{"_id": "12234", "title": "categories-lemma-split-into-directed", "text": "Let $\\mathcal{I}$ be an index category, i.e., a category. Assume \\begin{enumerate} \\item for every pair of morphisms $a : w \\to x$ and $b : w \\to y$ in $\\mathcal{I}$ there exists an object $z$ and morphisms $c : x \\to z$ and $d : y \\to z$ such that $c \\circ a = d \\circ b$, and \\item for every pair of morphisms $a, b : x \\to y$ there exists a morphism $c : y \\to z$ such that $c \\circ a = c \\circ b$. \\end{enumerate} Then $\\mathcal{I}$ is a (possibly empty) union of disjoint filtered index categories $\\mathcal{I}_j$."}
+{"_id": "12235", "title": "categories-lemma-almost-directed-commutes-equalizers", "text": "Let $\\mathcal{I}$ be an index category satisfying the hypotheses of Lemma \\ref{lemma-split-into-directed} above. Then colimits over $\\mathcal{I}$ commute with fibre products and equalizers in sets (and more generally with finite connected limits)."}
+{"_id": "12236", "title": "categories-lemma-directed-category-system", "text": "Let $\\mathcal{I}$ be a filtered index category. There exists a directed set $I$ and a system $(x_i, \\varphi_{ii'})$ over $I$ in $\\mathcal{I}$ with the following properties: \\begin{enumerate} \\item For every category $\\mathcal{C}$ and every diagram $M : \\mathcal{I} \\to \\mathcal{C}$ with values in $\\mathcal{C}$, denote $(M(x_i), M(\\varphi_{ii'}))$ the corresponding system over $I$. If $\\colim_{i \\in I} M(x_i)$ exists then so does $\\colim_\\mathcal{I} M$ and the transformation $$ \\theta : \\colim_{i \\in I} M(x_i) \\longrightarrow \\colim_\\mathcal{I} M $$ of Lemma \\ref{lemma-functorial-colimit} is an isomorphism. \\item For every category $\\mathcal{C}$ and every diagram $M : \\mathcal{I}^{opp} \\to \\mathcal{C}$ in $\\mathcal{C}$, denote $(M(x_i), M(\\varphi_{ii'}))$ the corresponding inverse system over $I$. If $\\lim_{i \\in I} M(x_i)$ exists then so does $\\lim_\\mathcal{I} M$ and the transformation $$ \\theta : \\lim_{\\mathcal{I}^{opp}} M \\longrightarrow \\lim_{i \\in I} M(x_i) $$ of Lemma \\ref{lemma-functorial-limit} is an isomorphism. \\end{enumerate}"}
+{"_id": "12237", "title": "categories-lemma-nonempty-limit", "text": "If $S : \\mathcal{I} \\to \\textit{Sets}$ is a cofiltered diagram of sets and all the $S_i$ are finite nonempty, then $\\lim_i S_i$ is nonempty. In other words, the limit of a directed inverse system of finite nonempty sets is nonempty."}
+{"_id": "12238", "title": "categories-lemma-essentially-constant-is-limit-colimit", "text": "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. If $\\mathcal{I}$ is filtered and $M$ is essentially constant as an ind-object, then $X = \\colim M_i$ exists and $M$ is essentially constant with value $X$. If $\\mathcal{I}$ is cofiltered and $M$ is essentially constant as a pro-object, then $X = \\lim M_i$ exists and $M$ is essentially constant with value $X$."}
+{"_id": "12239", "title": "categories-lemma-image-essentially-constant", "text": "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram with filtered (resp.\\ cofiltered) index category $\\mathcal{I}$. Let $F : \\mathcal{C} \\to \\mathcal{D}$ be a functor. If $M$ is essentially constant as an ind-object (resp.\\ pro-object), then so is $F \\circ M : \\mathcal{I} \\to \\mathcal{D}$."}
+{"_id": "12240", "title": "categories-lemma-characterize-essentially-constant-ind", "text": "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram with filtered index category $\\mathcal{I}$. The following are equivalent \\begin{enumerate} \\item $M$ is an essentially constant ind-object, and \\item $X = \\colim_i M_i$ exists and for any $W$ in $\\mathcal{C}$ the map $$ \\colim_i \\Mor_\\mathcal{C}(W, M_i) \\longrightarrow \\Mor_\\mathcal{C}(W, X) $$ is bijective. \\end{enumerate}"}
+{"_id": "12241", "title": "categories-lemma-characterize-essentially-constant-pro", "text": "Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram with cofiltered index category $\\mathcal{I}$. The following are equivalent \\begin{enumerate} \\item $M$ is an essentially constant pro-object, and \\item $X = \\lim_i M_i$ exists and for any $W$ in $\\mathcal{C}$ the map $$ \\colim_{i \\in \\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_i, W) \\longrightarrow \\Mor_\\mathcal{C}(X, W) $$ is bijective. \\end{enumerate}"}
+{"_id": "12242", "title": "categories-lemma-cofinal-essentially-constant", "text": "Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of filtered index categories. If $H$ is cofinal, then any diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant if and only if $M \\circ H$ is essentially constant."}
+{"_id": "12243", "title": "categories-lemma-essentially-constant-over-product", "text": "Let $\\mathcal{I}$ and $\\mathcal{J}$ be filtered categories and denote $p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection. Then $\\mathcal{I} \\times \\mathcal{J}$ is filtered and a diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant if and only if $M \\circ p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$ is essentially constant."}
+{"_id": "12245", "title": "categories-lemma-characterize-left-exact", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. Suppose all finite limits exist in $\\mathcal{A}$, see Lemma \\ref{lemma-finite-limits-exist}. The following are equivalent: \\begin{enumerate} \\item $F$ is left exact, \\item $F$ commutes with finite products and equalizers, and \\item $F$ transforms a final object of $\\mathcal{A}$ into a final object of $\\mathcal{B}$, and commutes with fibre products. \\end{enumerate}"}
+{"_id": "12246", "title": "categories-lemma-adjoint-exists", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories. If for each $y \\in \\Ob(\\mathcal{D})$ the functor $x \\mapsto \\Mor_\\mathcal{D}(u(x), y)$ is representable, then $u$ has a right adjoint."}
+{"_id": "12247", "title": "categories-lemma-left-adjoint-composed-fully-faithful", "text": "\\begin{reference} Bhargav Bhatt, private communication. \\end{reference} Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}. \\begin{enumerate} \\item If $v \\circ u$ is fully faithful, then $u$ is fully faithful. \\item If $u \\circ v$ is fully faithful, then $v$ is fully faithful. \\end{enumerate}"}
+{"_id": "12248", "title": "categories-lemma-adjoint-fully-faithful", "text": "Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}. Then \\begin{enumerate} \\item $u$ is fully faithful $\\Leftrightarrow$ $\\text{id} \\cong v \\circ u$ $\\Leftrightarrow$ $\\eta : \\textit{id} \\to v \\circ u$ is an isomorphism, \\item $v$ is fully faithful $\\Leftrightarrow$ $u \\circ v \\cong \\text{id}$ $\\Leftrightarrow$ $\\epsilon : u \\circ v \\to \\text{id}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "12249", "title": "categories-lemma-adjoint-exact", "text": "Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}. \\begin{enumerate} \\item Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$ is a diagram, and suppose that $\\colim_\\mathcal{I} M$ exists in $\\mathcal{C}$. Then $u(\\colim_\\mathcal{I} M) = \\colim_\\mathcal{I} u \\circ M$. In other words, $u$ commutes with (representable) colimits. \\item Suppose that $M : \\mathcal{I} \\to \\mathcal{D}$ is a diagram, and suppose that $\\lim_\\mathcal{I} M$ exists in $\\mathcal{D}$. Then $v(\\lim_\\mathcal{I} M) = \\lim_\\mathcal{I} v \\circ M$. In other words $v$ commutes with representable limits. \\end{enumerate}"}
+{"_id": "12250", "title": "categories-lemma-exact-adjoint", "text": "Let $u$ be a left adjoint of $v$ as in Definition \\ref{definition-adjoint}. \\begin{enumerate} \\item If $\\mathcal{C}$ has finite colimits, then $u$ is right exact. \\item If $\\mathcal{D}$ has finite limits, then $v$ is left exact. \\end{enumerate}"}
+{"_id": "12251", "title": "categories-lemma-transformation-between-functors-and-adjoints", "text": "Let $u_1, u_2 : \\mathcal{C} \\to \\mathcal{D}$ be functors with right adjoints $v_1, v_2 : \\mathcal{D} \\to \\mathcal{C}$. Let $\\beta : u_2 \\to u_1$ be a transformation of functors. Let $\\beta^\\vee : v_1 \\to v_2$ be the corresponding transformation of adjoint functors. Then $$ \\xymatrix{ u_2 \\circ v_1 \\ar[r]_\\beta \\ar[d]_{\\beta^\\vee} & u_1 \\circ v_1 \\ar[d] \\\\ u_2 \\circ v_2 \\ar[r] & \\text{id} } $$ is commutative where the unlabeled arrows are the counit transformations."}
+{"_id": "12252", "title": "categories-lemma-compose-counits", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be categories. Let $v : \\mathcal{A} \\to \\mathcal{B}$ and $v' : \\mathcal{B} \\to \\mathcal{C}$ be functors with left adjoints $u$ and $u'$ respectively. Then \\begin{enumerate} \\item The functor $v'' = v' \\circ v$ has a left adjoint equal to $u'' = u \\circ u'$. \\item Given $X$ in $\\mathcal{A}$ we have \\begin{equation} \\label{equation-compose-counits} \\epsilon_X^v \\circ u(\\epsilon^{v'}_{v(X)}) = \\epsilon^{v''}_X : u''(v''(X)) \\to X \\end{equation} Where $\\epsilon$ is the counit of the adjunctions. \\end{enumerate}"}
+{"_id": "12253", "title": "categories-lemma-a-version-of-brown", "text": "Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.} category which has limits. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item $F$ commutes with limits, \\item there exists a family $\\{x_i\\}_{i \\in I}$ of objects of $\\mathcal{C}$ and for each $i \\in I$ an element $f_i \\in F(x_i)$ such that for $y \\in \\Ob(\\mathcal{C})$ and $g \\in F(y)$ there exists an $i$ and a morphism $\\varphi : x_i \\to y$ with $F(\\varphi)(f_i) = g$. \\end{enumerate} Then $F$ is representable, i.e., there exists an object $x$ of $\\mathcal{C}$ such that $$ F(y) = \\Mor_\\mathcal{C}(x, y) $$ functorially in $y$."}
+{"_id": "12254", "title": "categories-lemma-extend-functor-by-colim", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be big categories having filtered colimits. Let $\\mathcal{C}' \\subset \\mathcal{C}$ be a small full subcategory consisting of categorically compact objects of $\\mathcal{C}$ such that every object of $\\mathcal{C}$ is a filtered colimit of objects of $\\mathcal{C}'$. Then every functor $F' : \\mathcal{C}' \\to \\mathcal{D}$ has a unique extension $F : \\mathcal{C} \\to \\mathcal{D}$ commuting with filtered colimits."}
+{"_id": "12255", "title": "categories-lemma-left-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system. \\begin{enumerate} \\item The relation on pairs defined above is an equivalence relation. \\item The composition rule given above is well defined on equivalence classes. \\item Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category. \\end{enumerate}"}
+{"_id": "12256", "title": "categories-lemma-morphisms-left-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\\mathcal{C}$. Given any finite collection $g_i : X_i \\to Y$ of morphisms of $S^{-1}\\mathcal{C}$ (indexed by $i$), we can find an element $s : Y \\to Y'$ of $S$ and a family of morphisms $f_i : X_i \\to Y'$ of $\\mathcal{C}$ such that each $g_i$ is the equivalence class of the pair $(f_i : X_i \\to Y', s : Y \\to Y')$."}
+{"_id": "12257", "title": "categories-lemma-equality-morphisms-left-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be morphisms of $S^{-1}\\mathcal{C}$ which are the equivalence classes of $(f : X \\to Y', s : Y \\to Y')$ and $(g : X \\to Y', s : Y \\to Y')$. Then $A = B$ if and only if there exists a morphism $a : Y' \\to Y''$ with $a \\circ s \\in S$ and such that $a \\circ f = a \\circ g$."}
+{"_id": "12258", "title": "categories-lemma-properties-left-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\\mathcal{C}$. \\begin{enumerate} \\item The rules $X \\mapsto X$ and $(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_Y : Y \\to Y)$ define a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$. \\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in $S^{-1}\\mathcal{C}$. \\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that $G(s)$ is invertible for every $s \\in S$, then there exists a unique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$ such that $H \\circ Q = G$. \\end{enumerate}"}
+{"_id": "12259", "title": "categories-lemma-left-localization-limits", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\\mathcal{C}$. The localization functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite colimits."}
+{"_id": "12260", "title": "categories-lemma-left-localization-diagram", "text": "Let $\\mathcal{C}$ be a category. Let $S$ be a left multiplicative system. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of $\\mathcal{C}$ and if $$ \\xymatrix{ Q(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\ Q(Y) \\ar[r]^b & Q(Y') } $$ is a commutative diagram in $S^{-1}\\mathcal{C}$, then there exists a morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative diagram $$ \\xymatrix{ X \\ar[d]_f \\ar[r]_g & X'' \\ar[d]^{f''} & X' \\ar[d]^{f'} \\ar[l]^s \\\\ Y \\ar[r]^h & Y'' & Y' \\ar[l]_t } $$ in $\\mathcal{C}$ with $s, t \\in S$ and $a = s^{-1}g$, $b = t^{-1}h$."}
+{"_id": "12261", "title": "categories-lemma-right-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system. \\begin{enumerate} \\item The relation on pairs defined above is an equivalence relation. \\item The composition rule given above is well defined on equivalence classes. \\item Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category. \\end{enumerate}"}
+{"_id": "12262", "title": "categories-lemma-morphisms-right-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\\mathcal{C}$. Given any finite collection $g_i : X \\to Y_i$ of morphisms of $S^{-1}\\mathcal{C}$ (indexed by $i$), we can find an element $s : X' \\to X$ of $S$ and a family of morphisms $f_i : X' \\to Y_i$ of $\\mathcal{C}$ such that $g_i$ is the equivalence class of the pair $(f_i : X' \\to Y_i, s : X' \\to X)$."}
+{"_id": "12263", "title": "categories-lemma-equality-morphisms-right-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be morphisms of $S^{-1}\\mathcal{C}$ which are the equivalence classes of $(f : X' \\to Y, s : X' \\to X)$ and $(g : X' \\to Y, s : X' \\to X)$. Then $A = B$ if and only if there exists a morphism $a : X'' \\to X'$ with $s \\circ a \\in S$ and such that $f \\circ a = g \\circ a$."}
+{"_id": "12264", "title": "categories-lemma-properties-right-localization", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\\mathcal{C}$. \\begin{enumerate} \\item The rules $X \\mapsto X$ and $(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_X : X \\to X)$ define a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$. \\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in $S^{-1}\\mathcal{C}$. \\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that $G(s)$ is invertible for every $s \\in S$, then there exists a unique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$ such that $H \\circ Q = G$. \\end{enumerate}"}
+{"_id": "12265", "title": "categories-lemma-right-localization-limits", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\\mathcal{C}$. The localization functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite limits."}
+{"_id": "12267", "title": "categories-lemma-multiplicative-system", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system. The category of left fractions and the category of right fractions $S^{-1}\\mathcal{C}$ are canonically isomorphic."}
+{"_id": "12268", "title": "categories-lemma-what-gets-inverted", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system. Denote $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ the localization functor. The set $$ \\hat S = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid Q(f) \\text{ is an isomorphism}\\} $$ is equal to $$ S' = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid \\text{there exist }g, h\\text{ such that }gf, fh \\in S\\} $$ and is the smallest saturated multiplicative system containing $S$. In particular, if $S$ is saturated, then $\\hat S = S$."}
+{"_id": "12269", "title": "categories-lemma-properties-2-cat-cats", "text": "The horizontal and vertical compositions have the following properties \\begin{enumerate} \\item $\\circ$ and $\\star$ are associative, \\item the identity transformations $\\text{id}_F$ are units for $\\circ$, \\item the identity transformations of the identity functors $\\text{id}_{\\text{id}_\\mathcal{A}}$ are units for $\\star$ and $\\circ$, and \\item given a diagram $$ \\xymatrix{ \\mathcal{A} \\rruppertwocell^F{t} \\ar[rr]_(.3){F'} \\rrlowertwocell_{F''}{t'} & & \\mathcal{B} \\rruppertwocell^G{s} \\ar[rr]_(.3){G'} \\rrlowertwocell_{G''}{s'} & & \\mathcal{C} } $$ we have $ (s' \\circ s) \\star (t' \\circ t) = (s' \\star t') \\circ (s \\star t)$. \\end{enumerate}"}
+{"_id": "12270", "title": "categories-lemma-2-fibre-product-categories", "text": "In the $(2, 1)$-category of categories $2$-fibre products exist and are given by the construction of Example \\ref{example-2-fibre-product-categories}."}
+{"_id": "12271", "title": "categories-lemma-functoriality-2-fibre-product", "text": "Let $$ \\xymatrix{ & \\mathcal{Y} \\ar[d]_I \\ar[rd]^K & \\\\ \\mathcal{X} \\ar[r]^H \\ar[rd]^L & \\mathcal{Z} \\ar[rd]^M & \\mathcal{B} \\ar[d]^G \\\\ & \\mathcal{A} \\ar[r]^F & \\mathcal{C} } $$ be a $2$-commutative diagram of categories. A choice of isomorphisms $\\alpha : G \\circ K \\to M \\circ I$ and $\\beta : M \\circ H \\to F \\circ L$ determines a morphism $$ \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\longrightarrow \\mathcal{A} \\times_\\mathcal{C} \\mathcal{B} $$ of $2$-fibre products associated to this situation."}
+{"_id": "12272", "title": "categories-lemma-equivalence-2-fibre-product", "text": "Assumptions as in Lemma \\ref{lemma-functoriality-2-fibre-product}. \\begin{enumerate} \\item If $K$ and $L$ are faithful then the morphism $\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$ is faithful. \\item If $K$ and $L$ are fully faithful and $M$ is faithful then the morphism $\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$ is fully faithful. \\item If $K$ and $L$ are equivalences and $M$ is fully faithful then the morphism $\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to \\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$ is an equivalence. \\end{enumerate}"}
+{"_id": "12273", "title": "categories-lemma-associativity-2-fibre-product", "text": "Let $$ \\xymatrix{ \\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\ & \\mathcal{B} & & \\mathcal{D} } $$ be a diagram of categories and functors. Then there is a canonical isomorphism $$ (\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C}) \\times_\\mathcal{D} \\mathcal{E} \\cong \\mathcal{A} \\times_\\mathcal{B} (\\mathcal{C} \\times_\\mathcal{D} \\mathcal{E}) $$ of categories."}
+{"_id": "12274", "title": "categories-lemma-triple-2-fibre-product-pr02", "text": "Let $$ \\xymatrix{ \\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\ & \\mathcal{B} \\ar[rd]_F & & \\mathcal{D} \\ar[ld]^G \\\\ & & \\mathcal{F} & } $$ be a commutative diagram of categories and functors. Then there is a canonical functor $$ \\text{pr}_{02} : \\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{D} \\mathcal{E} \\longrightarrow \\mathcal{A} \\times_\\mathcal{F} \\mathcal{E} $$ of categories."}
+{"_id": "12275", "title": "categories-lemma-2-fibre-product-erase-factor", "text": "Let $$ \\mathcal{A} \\to \\mathcal{B} \\leftarrow \\mathcal{C} \\leftarrow \\mathcal{D} $$ be a diagram of categories and functors. Then there is a canonical isomorphism $$ \\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{C} \\mathcal{D} \\cong \\mathcal{A} \\times_\\mathcal{B} \\mathcal{D} $$ of categories."}
+{"_id": "12276", "title": "categories-lemma-diagonal-1", "text": "Let $$ \\xymatrix{ \\mathcal{C}_3 \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\ \\mathcal{C}_1 \\times \\mathcal{C}_2 \\ar[r]^{G_1 \\times G_2} & \\mathcal{S} \\times \\mathcal{S} } $$ be a $2$-fibre product of categories. Then there is a canonical isomorphism $\\mathcal{C}_3 \\cong \\mathcal{C}_1 \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C}_2$."}
+{"_id": "12277", "title": "categories-lemma-diagonal-2", "text": "Let $$ \\xymatrix{ \\mathcal{C}' \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\ \\mathcal{C} \\ar[r]^{G_1 \\times G_2} & \\mathcal{S} \\times \\mathcal{S} } $$ be a $2$-fibre product of categories. Then there is a canonical isomorphism $$ \\mathcal{C}' \\cong (\\mathcal{C} \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C}) \\times_{(p, q), \\mathcal{C} \\times \\mathcal{C}, \\Delta} \\mathcal{C}. $$"}
+{"_id": "12278", "title": "categories-lemma-fibre-product-after-map", "text": "Let $\\mathcal{A} \\to \\mathcal{C}$, $\\mathcal{B} \\to \\mathcal{C}$ and $\\mathcal{C} \\to \\mathcal{D}$ be functors between categories. Then the diagram $$ \\xymatrix{ \\mathcal{A} \\times_\\mathcal{C} \\mathcal{B} \\ar[d] \\ar[r] & \\mathcal{A} \\times_\\mathcal{D} \\mathcal{B} \\ar[d] \\\\ \\mathcal{C} \\ar[r]^-{\\Delta_{\\mathcal{C}/\\mathcal{D}}} \\ar[r] & \\mathcal{C} \\times_\\mathcal{D} \\mathcal{C} } $$ is a $2$-fibre product diagram."}
+{"_id": "12279", "title": "categories-lemma-base-change-diagonal", "text": "Let $$ \\xymatrix{ \\mathcal{U} \\ar[d] \\ar[r] & \\mathcal{V} \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{Y} } $$ be a $2$-fibre product of categories. Then the diagram $$ \\xymatrix{ \\mathcal{U} \\ar[d] \\ar[r] & \\mathcal{U} \\times_\\mathcal{V} \\mathcal{U} \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} } $$ is $2$-cartesian."}
+{"_id": "12280", "title": "categories-lemma-2-product-categories-over-C", "text": "Let $\\mathcal{C}$ be a category. The $(2, 1)$-category of categories over $\\mathcal{C}$ has 2-fibre products. Suppose that $F : \\mathcal{X} \\to \\mathcal{S}$ and $G : \\mathcal{Y} \\to \\mathcal{S}$ are morphisms of categories over $\\mathcal{C}$. An explicit 2-fibre product $\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}$ is given by the following description \\begin{enumerate} \\item an object of $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$ is a quadruple $(U, x, y, f)$, where $U \\in \\Ob(\\mathcal{C})$, $x\\in \\Ob(\\mathcal{X}_U)$, $y\\in \\Ob(\\mathcal{Y}_U)$, and $f : F(x) \\to G(y)$ is an isomorphism in $\\mathcal{S}_U$, \\item a morphism $(U, x, y, f) \\to (U', x', y', f')$ is given by a pair $(a, b)$, where $a : x \\to x'$ is a morphism in $\\mathcal{X}$, and $b : y \\to y'$ is a morphism in $\\mathcal{Y}$ such that \\begin{enumerate} \\item $a$ and $b$ induce the same morphism $U \\to U'$, and \\item the diagram $$ \\xymatrix{ F(x) \\ar[r]^f \\ar[d]^{F(a)} & G(y) \\ar[d]^{G(b)} \\\\ F(x') \\ar[r]^{f'} & G(y') } $$ is commutative. \\end{enumerate} \\end{enumerate} The functors $p : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{X}$ and $q : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{Y}$ are the forgetful functors in this case. The transformation $\\psi : F \\circ p \\to G \\circ q$ is given on the object $\\xi = (U, x, y, f)$ by $\\psi_\\xi = f : F(p(\\xi)) = F(x) \\to G(y) = G(q(\\xi))$."}
+{"_id": "12281", "title": "categories-lemma-fibre-2-fibre-product-categories-over-C", "text": "Let $\\mathcal{C}$ be a category. Let $f : \\mathcal{X} \\to \\mathcal{S}$ and $g : \\mathcal{Y} \\to \\mathcal{S}$ be morphisms of categories over $\\mathcal{C}$. For any object $U$ of $\\mathcal{C}$ we have the following identity of fibre categories $$ \\left(\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}\\right)_U = \\mathcal{X}_U \\times_{\\mathcal{S}_U} \\mathcal{Y}_U $$"}
+{"_id": "12282", "title": "categories-lemma-composition-cartesian", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. \\begin{enumerate} \\item The composition of two strongly cartesian morphisms is strongly cartesian. \\item Any isomorphism of $\\mathcal{S}$ is strongly cartesian. \\item Any strongly cartesian morphism $\\varphi$ such that $p(\\varphi)$ is an isomorphism, is an isomorphism. \\end{enumerate}"}
+{"_id": "12283", "title": "categories-lemma-cartesian-over-cartesian", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$ be composable functors between categories. Let $x \\to y$ be a morphism of $\\mathcal{A}$. If $x \\to y$ is strongly $\\mathcal{B}$-cartesian and $F(x) \\to F(y)$ is strongly $\\mathcal{C}$-cartesian, then $x \\to y$ is strongly $\\mathcal{C}$-cartesian."}
+{"_id": "12284", "title": "categories-lemma-strongly-cartesian-fibre-product", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$. Assume \\begin{enumerate} \\item $x \\to y$ is strongly cartesian, \\item $p(x) \\times_{p(y)} p(z)$ exists, and \\item there exists a strongly cartesian morphism $a : w \\to z$ in $\\mathcal{S}$ with $p(w) = p(x) \\times_{p(y)} p(z)$ and $p(a) = \\text{pr}_2 : p(x) \\times_{p(y)} p(z) \\to p(z)$. \\end{enumerate} Then the fibre product $x \\times_y z$ exists and is isomorphic to $w$."}
+{"_id": "12285", "title": "categories-lemma-fibred", "text": "Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category. Assume given a choice of pullbacks for $p : \\mathcal{S} \\to \\mathcal{C}$. \\begin{enumerate} \\item For any pair of composable morphisms $f : V \\to U$, $g : W \\to V$ there is a unique isomorphism $$ \\alpha_{g, f} : (f \\circ g)^\\ast \\longrightarrow g^\\ast \\circ f^\\ast $$ as functors $\\mathcal{S}_U \\to \\mathcal{S}_W$ such that for every $y\\in \\Ob(\\mathcal{S}_U)$ the following diagram commutes $$ \\xymatrix{ g^\\ast f^\\ast y \\ar[r] & f^\\ast y \\ar[d] \\\\ (f \\circ g)^\\ast y \\ar[r] \\ar[u]^{(\\alpha_{g, f})_y} & y } $$ \\item If $f = \\text{id}_U$, then there is a canonical isomorphism $\\alpha_U : \\text{id} \\to (\\text{id}_U)^*$ as functors $\\mathcal{S}_U \\to \\mathcal{S}_U$. \\item The quadruple $(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{g, f}, \\alpha_U)$ defines a pseudo functor from $\\mathcal{C}^{opp}$ to the $(2, 1)$-category of categories, see Definition \\ref{definition-functor-into-2-category}. \\end{enumerate}"}
+{"_id": "12286", "title": "categories-lemma-fibred-equivalent", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$. Suppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent as categories over $\\mathcal{C}$. Then $\\mathcal{S}_1$ is fibred over $\\mathcal{C}$ if and only if $\\mathcal{S}_2$ is fibred over $\\mathcal{C}$."}
+{"_id": "12287", "title": "categories-lemma-2-product-fibred-categories-over-C", "text": "Let $\\mathcal{C}$ be a category. The $(2, 1)$-category of fibred categories over $\\mathcal{C}$ has 2-fibre products, and they are described as in Lemma \\ref{lemma-2-product-categories-over-C}."}
+{"_id": "12288", "title": "categories-lemma-cute", "text": "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$. If $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category and $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$ then $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is a fibred category."}
+{"_id": "12289", "title": "categories-lemma-fibred-over-fibred", "text": "Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between categories. If $\\mathcal{A}$ is fibred over $\\mathcal{B}$ and $\\mathcal{B}$ is fibred over $\\mathcal{C}$, then $\\mathcal{A}$ is fibred over $\\mathcal{C}$."}
+{"_id": "12290", "title": "categories-lemma-fibred-category-representable-goes-up", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$ with $x \\to y$ strongly cartesian. If $p(x) \\times_{p(y)} p(z)$ exists, then $x \\times_y z$ exists, $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$, and $x \\times_y z \\to z$ is strongly cartesian."}
+{"_id": "12291", "title": "categories-lemma-ameliorate-morphism-fibred-categories", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of fibred categories over $\\mathcal{C}$. There exist $1$-morphisms of fibred categories over $\\mathcal{C}$ $$ \\xymatrix{ \\mathcal{X} \\ar@<1ex>[r]^u & \\mathcal{X}' \\ar[r]^v \\ar@<1ex>[l]^w & \\mathcal{Y} } $$ such that $F = v \\circ u$ and such that \\begin{enumerate} \\item $u : \\mathcal{X} \\to \\mathcal{X}'$ is fully faithful, \\item $w$ is left adjoint to $u$, and \\item $v : \\mathcal{X}' \\to \\mathcal{Y}$ is a fibred category. \\end{enumerate}"}
+{"_id": "12292", "title": "categories-lemma-inertia-fibred-category", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ and $p' : \\mathcal{S}' \\to \\mathcal{C}$ be fibred categories. Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of fibred categories over $\\mathcal{C}$. Consider the category $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ over $\\mathcal{C}$ whose \\begin{enumerate} \\item objects are pairs $(x, \\alpha)$ where $x \\in \\Ob(\\mathcal{S})$ and $\\alpha : x \\to x$ is an automorphism with $F(\\alpha) = \\text{id}$, \\item morphisms $(x, \\alpha) \\to (y, \\beta)$ are given by morphisms $\\phi : x \\to y$ such that $$ \\xymatrix{ x\\ar[r]_\\phi\\ar[d]_\\alpha & y\\ar[d]^{\\beta} \\\\ x\\ar[r]^\\phi & y \\\\ } $$ commutes, and \\item the functor $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$ is given by $(x, \\alpha) \\mapsto p(x)$. \\end{enumerate} Then \\begin{enumerate} \\item there is an equivalence $$ \\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\longrightarrow \\mathcal{S} \\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta} \\mathcal{S} $$ in the $(2, 1)$-category of categories over $\\mathcal{C}$, and \\item $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ is a fibred category over $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12293", "title": "categories-lemma-relative-inertia-as-fibre-product", "text": "Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of categories fibred over a category $\\mathcal{C}$. Then the diagram $$ \\xymatrix{ \\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\ar[d]_{F \\circ (\\ref{equation-inertia-structure-map})} \\ar[rr]_{(\\ref{equation-comparison})} & & \\mathcal{I}_\\mathcal{S} \\ar[d]^{(\\ref{equation-functorial})} \\\\ \\mathcal{S}' \\ar[rr]^e & & \\mathcal{I}_{\\mathcal{S}'} } $$ is a $2$-fibre product."}
+{"_id": "12294", "title": "categories-lemma-fibred-groupoids", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor. The following are equivalent \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in groupoids, and \\item all fibre categories are groupoids and $\\mathcal{S}$ is a fibred category over $\\mathcal{C}$. \\end{enumerate} Moreover, in this case every morphism of $\\mathcal{S}$ is strongly cartesian. In addition, given $f^\\ast x \\to x$ lying over $f$ for all $f: V \\to U = p(x)$ the data $(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{f, g}, \\alpha_U)$ constructed in Lemma \\ref{lemma-fibred} defines a pseudo functor from $\\mathcal{C}^{opp}$ in to the $(2, 1)$-category of groupoids."}
+{"_id": "12295", "title": "categories-lemma-fibred-gives-fibred-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Let $\\mathcal{S}'$ be the subcategory of $\\mathcal{S}$ defined as follows \\begin{enumerate} \\item $\\Ob(\\mathcal{S}') = \\Ob(\\mathcal{S})$, and \\item for $x, y \\in \\Ob(\\mathcal{S}')$ the set of morphisms between $x$ and $y$ in $\\mathcal{S}'$ is the set of strongly cartesian morphisms between $x$ and $y$ in $\\mathcal{S}$. \\end{enumerate} Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ be the restriction of $p$ to $\\mathcal{S}'$. Then $p' : \\mathcal{S}' \\to \\mathcal{C}$ is fibred in groupoids."}
+{"_id": "12296", "title": "categories-lemma-2-product-fibred-categories", "text": "Let $\\mathcal{C}$ be a category. The $2$-category of categories fibred in groupoids over $\\mathcal{C}$ has 2-fibre products, and they are described as in Lemma \\ref{lemma-2-product-categories-over-C}."}
+{"_id": "12297", "title": "categories-lemma-equivalence-fibred-categories", "text": "Let $p : \\mathcal{S}\\to \\mathcal{C}$ and $p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids, and suppose that $G : \\mathcal{S}\\to \\mathcal {S}'$ is a functor over $\\mathcal{C}$. \\begin{enumerate} \\item Then $G$ is faithful (resp.\\ fully faithful, resp.\\ an equivalence) if and only if for each $U\\in\\Ob(\\mathcal{C})$ the induced functor $G_U : \\mathcal{S}_U\\to \\mathcal{S}'_U$ is faithful (resp.\\ fully faithful, resp.\\ an equivalence). \\item If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12298", "title": "categories-lemma-fully-faithful-diagonal-equivalence", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S}\\to \\mathcal{C}$ and $p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids. Let $G : \\mathcal{S}\\to \\mathcal {S}'$ be a functor over $\\mathcal{C}$. Then $G$ is fully faithful if and only if the diagonal $$ \\Delta_G : \\mathcal{S} \\longrightarrow \\mathcal{S} \\times_{G, \\mathcal{S}', G} \\mathcal{S} $$ is an equivalence."}
+{"_id": "12299", "title": "categories-lemma-morphisms-equivalent-fibred-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}_i$, $i = 1, 2, 3, 4$ be categories fibred in groupoids over $\\mathcal{C}$. Suppose that $\\varphi : \\mathcal{S}_1 \\to \\mathcal{S}_2$ and $\\psi : \\mathcal{S}_3 \\to \\mathcal{S}_4$ are equivalences over $\\mathcal{C}$. Then $$ \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_2, \\mathcal{S}_3) \\longrightarrow \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_4), \\quad \\alpha \\longmapsto \\psi \\circ \\alpha \\circ \\varphi $$ is an equivalence of categories."}
+{"_id": "12301", "title": "categories-lemma-cute-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$. If $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in groupoids and $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$ then $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is fibred in groupoids."}
+{"_id": "12302", "title": "categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids", "text": "Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between categories. If $\\mathcal{A}$ is fibred in groupoids over $\\mathcal{B}$ and $\\mathcal{B}$ is fibred in groupoids over $\\mathcal{C}$, then $\\mathcal{A}$ is fibred in groupoids over $\\mathcal{C}$."}
+{"_id": "12303", "title": "categories-lemma-fibred-groupoids-fibre-product-goes-up", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. Let $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$. If $p(x) \\times_{p(y)} p(z)$ exists, then $x \\times_y z$ exists and $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$."}
+{"_id": "12304", "title": "categories-lemma-ameliorate-morphism-categories-fibred-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$. There exists a factorization $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$ by $1$-morphisms of categories fibred in groupoids over $\\mathcal{C}$ such that $\\mathcal{X} \\to \\mathcal{X}'$ is an equivalence over $\\mathcal{C}$ and such that $\\mathcal{X}'$ is a category fibred in groupoids over $\\mathcal{Y}$."}
+{"_id": "12305", "title": "categories-lemma-amelioration-unique", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$. Assume we have a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{X}' \\ar[rd]_f & \\mathcal{X} \\ar[l]^a \\ar[d]^F \\ar[r]_b & \\mathcal{X}'' \\ar[ld]^g \\\\ & \\mathcal{Y} } $$ where $a$ and $b$ are equivalences of categories over $\\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. Then there exists an equivalence $h : \\mathcal{X}'' \\to \\mathcal{X}'$ of categories over $\\mathcal{Y}$ such that $h \\circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\\mathcal{C}$. If the diagram above actually commutes, then we can arrange it so that $h \\circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\\mathcal{Y}$."}
+{"_id": "12306", "title": "categories-lemma-when-split", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}$ be a fibred category over $\\mathcal{C}$. Then $\\mathcal{S}$ is split if and only if for some choice of pullbacks (see Definition \\ref{definition-pullback-functor-fibred-category}) the pullback functors $(f \\circ g)^*$ and $g^* \\circ f^*$ are equal."}
+{"_id": "12307", "title": "categories-lemma-fibred-strict", "text": "Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. There exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Cat}$ such that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$ in the $2$-category of fibred categories over $\\mathcal{C}$. In other words, every fibred category is equivalent to a split one."}
+{"_id": "12308", "title": "categories-lemma-fibred-groupoids-strict", "text": "Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. There exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Groupoids}$ such that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$ over $\\mathcal{C}$. In other words, every category fibred in groupoids is equivalent to a split one."}
+{"_id": "12309", "title": "categories-lemma-2-product-categories-fibred-sets", "text": "Let $\\mathcal{C}$ be a category. The 2-category of categories fibred in sets over $\\mathcal{C}$ has 2-fibre products. More precisely, the 2-fibre product described in Lemma \\ref{lemma-2-product-categories-over-C} returns a category fibred in sets if one starts out with such."}
+{"_id": "12310", "title": "categories-lemma-2-category-fibred-sets", "text": "\\begin{slogan} Categories fibred in sets are precisely presheaves. \\end{slogan} Let $\\mathcal{C}$ be a category. The only $2$-morphisms between categories fibred in sets are identities. In other words, the $2$-category of categories fibred in sets is a category. Moreover, there is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{the category of presheaves}\\\\ \\text{of sets over }\\mathcal{C} \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{the category of categories}\\\\ \\text{fibred in sets over }\\mathcal{C} \\end{matrix} \\right\\} $$ The functor from left to right is the construction $F \\to \\mathcal{S}_F$ discussed in Example \\ref{example-presheaf}. The functor from right to left assigns to $p : \\mathcal{S} \\to \\mathcal{C}$ the presheaf of objects $U \\mapsto \\Ob(\\mathcal{S}_U)$."}
+{"_id": "12311", "title": "categories-lemma-2-product-categories-fibred-setoids", "text": "Let $\\mathcal{C}$ be a category. The 2-category of categories fibred in setoids over $\\mathcal{C}$ has 2-fibre products. More precisely, the 2-fibre product described in Lemma \\ref{lemma-2-product-categories-over-C} returns a category fibred in setoids if one starts out with such."}
+{"_id": "12312", "title": "categories-lemma-setoid-fibres", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}$ be a category over $\\mathcal{C}$. \\begin{enumerate} \\item If $\\mathcal{S} \\to \\mathcal{S}'$ is an equivalence over $\\mathcal{C}$ with $\\mathcal{S}'$ fibred in sets over $\\mathcal{C}$, then \\begin{enumerate} \\item $\\mathcal{S}$ is fibred in setoids over $\\mathcal{C}$, and \\item for each $U \\in \\Ob(\\mathcal{C})$ the map $\\Ob(\\mathcal{S}_U) \\to \\Ob(\\mathcal{S}'_U)$ identifies the target as the set of isomorphism classes of the source. \\end{enumerate} \\item If $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids, then there exists a category fibred in sets $p' : \\mathcal{S}' \\to \\mathcal{C}$ and an equivalence $\\text{can} : \\mathcal{S} \\to \\mathcal{S}'$ over $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12313", "title": "categories-lemma-2-category-fibred-setoids", "text": "Let $\\mathcal{C}$ be a category. The construction of Lemma \\ref{lemma-setoid-fibres} part (2) gives a functor $$ F : \\left\\{ \\begin{matrix} \\text{the 2-category of categories}\\\\ \\text{fibred in setoids over }\\mathcal{C} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{the category of categories}\\\\ \\text{fibred in sets over }\\mathcal{C} \\end{matrix} \\right\\} $$ (see Definition \\ref{definition-functor-into-2-category}). This functor is an equivalence in the following sense: \\begin{enumerate} \\item for any two 1-morphisms $f, g : \\mathcal{S}_1 \\to \\mathcal{S}_2$ with $F(f) = F(g)$ there exists a unique 2-isomorphism $f \\to g$, \\item for any morphism $h : F(\\mathcal{S}_1) \\to F(\\mathcal{S}_2)$ there exists a 1-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$ with $F(f) = h$, and \\item any category fibred in sets $\\mathcal{S}$ is equal to $F(\\mathcal{S})$. \\end{enumerate} In particular, defining $F_i \\in \\textit{PSh}(\\mathcal{C})$ by the rule $F_i(U) = \\Ob(\\mathcal{S}_{i, U})/\\cong$, we have $$ \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_2) \\Big/ 2\\text{-isomorphism} = \\Mor_{\\textit{PSh}(\\mathcal{C})}(F_1, F_2) $$ More precisely, given any map $\\phi : F_1 \\to F_2$ there exists a $1$-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$ which induces $\\phi$ on isomorphism classes of objects and which is unique up to unique $2$-isomorphism."}
+{"_id": "12314", "title": "categories-lemma-characterize-fibred-setoids-inertia", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. The following are equivalent: \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids, and \\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$, see (\\ref{equation-inertia-structure-map}), is an equivalence (of categories over $\\mathcal{C}$). \\end{enumerate}"}
+{"_id": "12315", "title": "categories-lemma-category-fibred-setoids-presheaves-products", "text": "Let $\\mathcal{C}$ be a category. The construction of Lemma \\ref{lemma-2-category-fibred-setoids} which associates to a category fibred in setoids a presheaf is compatible with products, in the sense that the presheaf associated to a $2$-fibre product $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ is the fibre product of the presheaves associated to $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$."}
+{"_id": "12316", "title": "categories-lemma-characterize-representable-fibred-category", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. \\begin{enumerate} \\item $\\mathcal{S}$ is representable if and only if the following conditions are satisfied: \\begin{enumerate} \\item $\\mathcal{S}$ is fibred in setoids, and \\item the presheaf $U \\mapsto \\Ob(\\mathcal{S}_U)/\\cong$ is representable. \\end{enumerate} \\item If $\\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the equivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$, is uniquely determined up to isomorphism. \\end{enumerate}"}
+{"_id": "12318", "title": "categories-lemma-yoneda-2category", "text": "Let $\\mathcal{S}\\to \\mathcal{C}$ be fibred in groupoids. Let $U \\in \\Ob(\\mathcal{C})$. The functor $$ \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S}) \\longrightarrow \\mathcal{S}_U $$ given by $G \\mapsto G(\\text{id}_U)$ is an equivalence."}
+{"_id": "12319", "title": "categories-lemma-identify-fibre-product", "text": "In the situation above the fibre category of $(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}$ over an object $f : V \\to U$ of $\\mathcal{C}/U$ is the category described as follows: \\begin{enumerate} \\item objects are pairs $(x, \\phi)$, where $x \\in \\Ob(\\mathcal{X}_V)$, and $\\phi : f^*y \\to F(x)$ is a morphism in $\\mathcal{Y}_V$, \\item the set of morphisms between $(x, \\phi)$ and $(x', \\phi')$ is the set of morphisms $\\psi : x \\to x'$ in $\\mathcal{X}_V$ such that $F(\\psi) = \\phi' \\circ \\phi^{-1}$. \\end{enumerate}"}
+{"_id": "12323", "title": "categories-lemma-representable-diagonal-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. Assume $\\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent: \\begin{enumerate} \\item The diagonal $\\mathcal{S} \\to \\mathcal{S} \\times \\mathcal{S}$ is representable. \\item For every $U$ in $\\mathcal{C}$, any $G : \\mathcal{C}/U \\to \\mathcal{S}$ is representable. \\end{enumerate}"}
+{"_id": "12324", "title": "categories-lemma-invertible", "text": "Let $\\mathcal{C}$ be a monoidal category. Let $X$ be an object of $\\mathcal{C}$. The following are equivalent \\begin{enumerate} \\item the functor $L : Y \\mapsto X \\otimes Y$ is an equivalence, \\item the functor $R : Y \\mapsto Y \\otimes X$ is an equivalence, \\item there exists an object $X'$ such that $X \\otimes X' \\cong X' \\otimes X \\cong \\mathbf{1}$. \\end{enumerate}"}
+{"_id": "12325", "title": "categories-lemma-left-dual", "text": "Let $\\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$, then $$ \\Mor(Z' \\otimes X, Z) = \\Mor(Z', Z \\otimes Y) \\quad\\text{and}\\quad \\Mor(Y \\otimes Z', Z) = \\Mor(Z', X \\otimes Z) $$ functorially in $Z$ and $Z'$."}
+{"_id": "12326", "title": "categories-lemma-tensor-dual", "text": "Let $\\mathcal{C}$ be a monoidal category. If $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then $Y_2 \\otimes Y_1$ is a left dual of $X_1 \\otimes X_2$."}
+{"_id": "12327", "title": "categories-lemma-dual-symmetric", "text": "Let $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ be a symmetric monoidal category. Let $X$ be an object of $\\mathcal{C}$ and let $Y$, $\\eta : \\mathbf{1} \\to X \\otimes Y$, and $\\epsilon : Y \\otimes X \\to \\mathbf{1}$ be a left dual of $X$ as in Definition \\ref{definition-dual}. Then $\\eta' = \\psi \\circ \\eta : \\mathbf{1} \\to Y \\otimes X$ and $\\epsilon' = \\epsilon \\circ \\psi : X \\otimes Y \\to \\mathbf{1}$ makes $X$ into a left dual of $Y$."}
+{"_id": "12431", "title": "topologies-lemma-zariski", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a Zariski covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering and for each $i$ we have a Zariski covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a Zariski covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a Zariski covering. \\end{enumerate}"}
+{"_id": "12432", "title": "topologies-lemma-zariski-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a Zariski covering of $T$. Then there exists a Zariski covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is a standard open of $T$, see Schemes, Definition \\ref{schemes-definition-standard-covering}. Moreover, we may choose each $U_j$ to be an open of one of the $T_i$."}
+{"_id": "12433", "title": "topologies-lemma-zariski-induced", "text": "Let $\\Sch_{Zar}$ be a big Zariski site as in Definition \\ref{definition-big-zariski-site}. Let $T \\in \\Ob(\\Sch_{Zar})$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary Zariski covering of $T$. There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{Zar}$ which is tautologically equivalent (see Sites, Definition \\ref{sites-definition-combinatorial-tautological}) to $\\{T_i \\to T\\}_{i \\in I}$."}
+{"_id": "12435", "title": "topologies-lemma-fibre-products-Zariski", "text": "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski site containing $S$. The underlying categories of the sites $\\Sch_{Zar}$, $(\\Sch/S)_{Zar}$, $S_{Zar}$, and $(\\textit{Aff}/S)_{Zar}$ have fibre products. In each case the obvious functor into the category $\\Sch$ of all schemes commutes with taking fibre products. The categories $(\\Sch/S)_{Zar}$, and $S_{Zar}$ both have a final object, namely $S/S$."}
+{"_id": "12436", "title": "topologies-lemma-affine-big-site-Zariski", "text": "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski site containing $S$. The functor $(\\textit{Aff}/S)_{Zar} \\to (\\Sch/S)_{Zar}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_{Zar})$ to $\\Sh((\\Sch/S)_{Zar})$."}
+{"_id": "12437", "title": "topologies-lemma-Zariski-usual", "text": "The category of sheaves on $S_{Zar}$ is equivalent to the category of sheaves on the underlying topological space of $S$."}
+{"_id": "12438", "title": "topologies-lemma-put-in-T", "text": "Let $\\Sch_{Zar}$ be a big Zariski site. Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$. The functor $T_{Zar} \\to (\\Sch/S)_{Zar}$ is cocontinuous and induces a morphism of topoi $$ i_f : \\Sh(T_{Zar}) \\longrightarrow \\Sh((\\Sch/S)_{Zar}) $$ For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_{Zar}$ we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers."}
+{"_id": "12439", "title": "topologies-lemma-at-the-bottom", "text": "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski site containing $S$. The inclusion functor $S_{Zar} \\to (\\Sch/S)_{Zar}$ satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \\pi_S : (\\Sch/S)_{Zar} \\longrightarrow S_{Zar} $$ and a morphism of topoi $$ i_S : \\Sh(S_{Zar}) \\longrightarrow \\Sh((\\Sch/S)_{Zar}) $$ such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$ with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule $i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$."}
+{"_id": "12440", "title": "topologies-lemma-morphism-big", "text": "Let $\\Sch_{Zar}$ be a big Zariski site. Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$. The functor $$ u : (\\Sch/T)_{Zar} \\longrightarrow (\\Sch/S)_{Zar}, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_{Zar} \\longrightarrow (\\Sch/T)_{Zar}, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_{Zar}) \\longrightarrow \\Sh((\\Sch/S)_{Zar}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12441", "title": "topologies-lemma-morphism-big-small", "text": "Let $\\Sch_{Zar}$ be a big Zariski site. Let $f : T \\to S$ be a morphism in $\\Sch_{Zar}$. \\begin{enumerate} \\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in Lemma \\ref{lemma-put-in-T} and $i_T$ as in Lemma \\ref{lemma-at-the-bottom}. \\item The functor $S_{Zar} \\to T_{Zar}$, $(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces a morphism of topoi $$ f_{small} : \\Sh(T_{Zar}) \\longrightarrow \\Sh(S_{Zar}). $$ The functors $f_{small}^{-1}$ and $f_{small, *}$ agree with the usual notions $f^{-1}$ and $f_*$ is we identify sheaves on $T_{Zar}$, resp.\\ $S_{Zar}$ with sheaves on $T$, resp.\\ $S$ via Lemma \\ref{lemma-Zariski-usual}. \\item We have a commutative diagram of morphisms of sites $$ \\xymatrix{ T_{Zar} \\ar[d]_{f_{small}} & (\\Sch/T)_{Zar} \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T} \\\\ S_{Zar} & (\\Sch/S)_{Zar} \\ar[l]_{\\pi_S} } $$ so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi. \\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$. \\end{enumerate}"}
+{"_id": "12445", "title": "topologies-lemma-zariski-etale", "text": "Any Zariski covering is an \\'etale covering."}
+{"_id": "12446", "title": "topologies-lemma-etale", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is an \\'etale covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering and for each $i$ we have an \\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an \\'etale covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an \\'etale covering. \\end{enumerate}"}
+{"_id": "12447", "title": "topologies-lemma-etale-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be an \\'etale covering of $T$. Then there exists an \\'etale covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "12448", "title": "topologies-lemma-etale-induced", "text": "Let $\\Sch_\\etale$ be a big \\'etale site as in Definition \\ref{definition-big-etale-site}. Let $T \\in \\Ob(\\Sch_\\etale)$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary \\'etale covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_\\etale$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard \\'etale covering, then it is tautologically equivalent to a covering in $\\Sch_\\etale$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\\Sch_\\etale$. \\end{enumerate}"}
+{"_id": "12449", "title": "topologies-lemma-verify-site-etale", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale site containing $S$. Both $S_\\etale$ and $(\\textit{Aff}/S)_\\etale$ are sites."}
+{"_id": "12450", "title": "topologies-lemma-fibre-products-etale", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale site containing $S$. The underlying categories of the sites $\\Sch_\\etale$, $(\\Sch/S)_\\etale$, $S_\\etale$, and $(\\textit{Aff}/S)_\\etale$ have fibre products. In each case the obvious functor into the category $\\Sch$ of all schemes commutes with taking fibre products. The categories $(\\Sch/S)_\\etale$, and $S_\\etale$ both have a final object, namely $S/S$."}
+{"_id": "12451", "title": "topologies-lemma-affine-big-site-etale", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale site containing $S$. The functor $(\\textit{Aff}/S)_\\etale \\to (\\Sch/S)_\\etale$ is special cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_\\etale)$ to $\\Sh((\\Sch/S)_\\etale)$."}
+{"_id": "12452", "title": "topologies-lemma-put-in-T-etale", "text": "Let $\\Sch_\\etale$ be a big \\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$. The functor $T_\\etale \\to (\\Sch/S)_\\etale$ is cocontinuous and induces a morphism of topoi $$ i_f : \\Sh(T_\\etale) \\longrightarrow \\Sh((\\Sch/S)_\\etale) $$ For a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\etale$ we have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers."}
+{"_id": "12453", "title": "topologies-lemma-at-the-bottom-etale", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale site containing $S$. The inclusion functor $S_\\etale \\to (\\Sch/S)_\\etale$ satisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \\pi_S : (\\Sch/S)_\\etale \\longrightarrow S_\\etale $$ and a morphism of topoi $$ i_S : \\Sh(S_\\etale) \\longrightarrow \\Sh((\\Sch/S)_\\etale) $$ such that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$ with $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T-etale}. In particular the functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule $i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$."}
+{"_id": "12454", "title": "topologies-lemma-morphism-big-etale", "text": "Let $\\Sch_\\etale$ be a big \\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$. The functor $$ u : (\\Sch/T)_\\etale \\longrightarrow (\\Sch/S)_\\etale, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_\\etale \\longrightarrow (\\Sch/T)_\\etale, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_\\etale) \\longrightarrow \\Sh((\\Sch/S)_\\etale) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12455", "title": "topologies-lemma-morphism-big-small-etale", "text": "Let $\\Sch_\\etale$ be a big \\'etale site. Let $f : T \\to S$ be a morphism in $\\Sch_\\etale$. \\begin{enumerate} \\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in Lemma \\ref{lemma-put-in-T-etale} and $i_T$ as in Lemma \\ref{lemma-at-the-bottom-etale}. \\item The functor $S_\\etale \\to T_\\etale$, $(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces a morphism of sites $$ f_{small} : T_\\etale \\longrightarrow S_\\etale $$ We have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. \\item We have a commutative diagram of morphisms of sites $$ \\xymatrix{ T_\\etale \\ar[d]_{f_{small}} & (\\Sch/T)_\\etale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\ S_\\etale & (\\Sch/S)_\\etale \\ar[l]_{\\pi_S} } $$ so that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi. \\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$. \\end{enumerate}"}
+{"_id": "12457", "title": "topologies-lemma-morphism-big-small-cartesian-diagram-etale", "text": "Let $\\Sch_\\etale$ be a big \\'etale site. Consider a cartesian diagram $$ \\xymatrix{ T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ in $\\Sch_\\etale$. Then $i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$ and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$."}
+{"_id": "12458", "title": "topologies-lemma-characterize-sheaf-big-etale", "text": "Let $S$ be a scheme contained in a big \\'etale site $\\Sch_\\etale$. A sheaf $\\mathcal{F}$ on the big \\'etale site $(\\Sch/S)_\\etale$ is given by the following data: \\begin{enumerate} \\item for every $T/S \\in \\Ob((\\Sch/S)_\\etale)$ a sheaf $\\mathcal{F}_T$ on $T_\\etale$, \\item for every $f : T' \\to T$ in $(\\Sch/S)_\\etale$ a map $c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$. \\end{enumerate} These data are subject to the following conditions: \\begin{enumerate} \\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in $(\\Sch/S)_\\etale$ the composition $c_g \\circ g_{small}^{-1}c_f$ is equal to $c_{f \\circ g}$, and \\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\etale$ is \\'etale then $c_f$ is an isomorphism. \\end{enumerate}"}
+{"_id": "12459", "title": "topologies-lemma-zariski-etale-smooth", "text": "Any \\'etale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering."}
+{"_id": "12460", "title": "topologies-lemma-smooth", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a smooth covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a smooth covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a smooth covering. \\end{enumerate}"}
+{"_id": "12461", "title": "topologies-lemma-smooth-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a smooth covering of $T$. Then there exists a smooth covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme, and such that each morphism $U_j \\to T$ is standard smooth, see Morphisms, Definition \\ref{morphisms-definition-smooth}. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "12462", "title": "topologies-lemma-smooth-induced", "text": "Let $\\Sch_{smooth}$ be a big smooth site as in Definition \\ref{definition-big-smooth-site}. Let $T \\in \\Ob(\\Sch_{smooth})$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary smooth covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{smooth}$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard smooth covering, then it is tautologically equivalent to a covering of $\\Sch_{smooth}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\\Sch_{smooth}$. \\end{enumerate}"}
+{"_id": "12463", "title": "topologies-lemma-affine-big-site-smooth", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big smooth site containing $S$. The functor $(\\textit{Aff}/S)_{smooth} \\to (\\Sch/S)_{smooth}$ is special cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_{smooth})$ to $\\Sh((\\Sch/S)_{smooth})$."}
+{"_id": "12464", "title": "topologies-lemma-morphism-big-smooth", "text": "Let $\\Sch_{smooth}$ be a big smooth site. Let $f : T \\to S$ be a morphism in $\\Sch_{smooth}$. The functor $$ u : (\\Sch/T)_{smooth} \\longrightarrow (\\Sch/S)_{smooth}, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_{smooth} \\longrightarrow (\\Sch/T)_{smooth}, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_{smooth}) \\longrightarrow \\Sh((\\Sch/S)_{smooth}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12465", "title": "topologies-lemma-zariski-etale-smooth-syntomic", "text": "Any smooth covering is a syntomic covering, and a fortiori, any \\'etale or Zariski covering is a syntomic covering."}
+{"_id": "12466", "title": "topologies-lemma-syntomic", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a syntomic covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a syntomic covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a syntomic covering. \\end{enumerate}"}
+{"_id": "12467", "title": "topologies-lemma-syntomic-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a syntomic covering of $T$. Then there exists a syntomic covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme, and such that each morphism $U_j \\to T$ is standard syntomic, see Morphisms, Definition \\ref{morphisms-definition-syntomic}. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "12468", "title": "topologies-lemma-syntomic-induced", "text": "Let $\\Sch_{syntomic}$ be a big syntomic site as in Definition \\ref{definition-big-syntomic-site}. Let $T \\in \\Ob(\\Sch_{syntomic})$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary syntomic covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{syntomic}$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard syntomic covering, then it is tautologically equivalent to a covering in $\\Sch_{syntomic}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\\Sch_{syntomic}$. \\end{enumerate}"}
+{"_id": "12469", "title": "topologies-lemma-affine-big-site-syntomic", "text": "Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic site containing $S$. The functor $(\\textit{Aff}/S)_{syntomic} \\to (\\Sch/S)_{syntomic}$ is special cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_{syntomic})$ to $\\Sh((\\Sch/S)_{syntomic})$."}
+{"_id": "12470", "title": "topologies-lemma-morphism-big-syntomic", "text": "Let $\\Sch_{syntomic}$ be a big syntomic site. Let $f : T \\to S$ be a morphism in $\\Sch_{syntomic}$. The functor $$ u : (\\Sch/T)_{syntomic} \\longrightarrow (\\Sch/S)_{syntomic}, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_{syntomic} \\longrightarrow (\\Sch/T)_{syntomic}, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_{syntomic}) \\longrightarrow \\Sh((\\Sch/S)_{syntomic}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12471", "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf", "text": "Any syntomic covering is an fppf covering, and a fortiori, any smooth, \\'etale, or Zariski covering is an fppf covering."}
+{"_id": "12472", "title": "topologies-lemma-fppf", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is an fppf covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fppf covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fppf covering. \\end{enumerate}"}
+{"_id": "12473", "title": "topologies-lemma-fppf-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be an fppf covering of $T$. Then there exists an fppf covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "12474", "title": "topologies-lemma-fppf-induced", "text": "Let $\\Sch_{fppf}$ be a big fppf site as in Definition \\ref{definition-big-fppf-site}. Let $T \\in \\Ob(\\Sch_{fppf})$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary fppf covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{fppf}$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard fppf covering, then it is tautologically equivalent to a covering of $\\Sch_{fppf}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\\Sch_{fppf}$. \\end{enumerate}"}
+{"_id": "12476", "title": "topologies-lemma-fibre-products-fppf", "text": "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$. The underlying categories of the sites $\\Sch_{fppf}$, $(\\Sch/S)_{fppf}$, and $(\\textit{Aff}/S)_{fppf}$ have fibre products. In each case the obvious functor into the category $\\Sch$ of all schemes commutes with taking fibre products. The category $(\\Sch/S)_{fppf}$ has a final object, namely $S/S$."}
+{"_id": "12477", "title": "topologies-lemma-affine-big-site-fppf", "text": "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$. The functor $(\\textit{Aff}/S)_{fppf} \\to (\\Sch/S)_{fppf}$ is cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_{fppf})$ to $\\Sh((\\Sch/S)_{fppf})$."}
+{"_id": "12478", "title": "topologies-lemma-morphism-big-fppf", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $f : T \\to S$ be a morphism in $\\Sch_{fppf}$. The functor $$ u : (\\Sch/T)_{fppf} \\longrightarrow (\\Sch/S)_{fppf}, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_{fppf} \\longrightarrow (\\Sch/T)_{fppf}, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_{fppf}) \\longrightarrow \\Sh((\\Sch/S)_{fppf}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12480", "title": "topologies-lemma-base-change-standard-ph", "text": "Let $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard ph covering. Let $T' \\to T$ be a morphism of affine schemes. Then $\\{U_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a standard ph covering."}
+{"_id": "12481", "title": "topologies-lemma-refine-by-standard-ph", "text": "Let $T$ be an affine scheme. Each of the following types of families of maps with target $T$ has a refinement by a standard ph covering: \\begin{enumerate} \\item any Zariski open covering of $T$, \\item $\\{W_{ji} \\to T\\}_{j = 1, \\ldots, m, i = 1, \\ldots n_j}$ where $\\{W_{ji} \\to U_j\\}_{i = 1, \\ldots, n_j}$ and $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ are standard ph coverings. \\end{enumerate}"}
+{"_id": "12482", "title": "topologies-lemma-zariski-ph", "text": "A Zariski covering is a ph covering\\footnote{We will see in More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph} that fppf coverings (and hence syntomic, smooth, or \\'etale coverings) are ph coverings as well.}."}
+{"_id": "12483", "title": "topologies-lemma-surjective-proper-ph", "text": "Let $f : Y \\to X$ be a surjective proper morphism of schemes. Then $\\{Y \\to X\\}$ is a ph covering."}
+{"_id": "12484", "title": "topologies-lemma-refine-by-ph", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms such that $f_i$ is locally of finite type for all $i$. The following are equivalent \\begin{enumerate} \\item $\\{T_i \\to T\\}_{i \\in I}$ is a ph covering, \\item there is a ph covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and \\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a ph covering. \\end{enumerate}"}
+{"_id": "12485", "title": "topologies-lemma-ph", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a ph covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering and for each $i$ we have a ph covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a ph covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a ph covering. \\end{enumerate}"}
+{"_id": "12486", "title": "topologies-lemma-ph-induced", "text": "Let $\\Sch_{ph}$ be a big ph site as in Definition \\ref{definition-big-ph-site}. Let $T \\in \\Ob(\\Sch_{ph})$. Let $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary ph covering of $T$. \\begin{enumerate} \\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site $\\Sch_{ph}$ which refines $\\{T_i \\to T\\}_{i \\in I}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard ph covering, then it is tautologically equivalent to a covering of $\\Sch_{ph}$. \\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\\Sch_{ph}$. \\end{enumerate}"}
+{"_id": "12490", "title": "topologies-lemma-characterize-sheaf", "text": "Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_{ph}$. Then $\\mathcal{F}$ is a sheaf if and only if \\begin{enumerate} \\item $\\mathcal{F}$ satisfies the sheaf condition for Zariski coverings, and \\item if $f : V \\to U$ is proper surjective, then $\\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$. \\end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \\begin{enumerate} \\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine. \\end{enumerate}"}
+{"_id": "12491", "title": "topologies-lemma-morphism-big-ph", "text": "Let $\\Sch_{ph}$ be a big ph site. Let $f : T \\to S$ be a morphism in $\\Sch_{ph}$. The functor $$ u : (\\Sch/T)_{ph} \\longrightarrow (\\Sch/S)_{ph}, \\quad V/T \\longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\\Sch/S)_{ph} \\longrightarrow (\\Sch/T)_{ph}, \\quad (U \\to S) \\longmapsto (U \\times_S T \\to T). $$ They induce the same morphism of topoi $$ f_{big} : \\Sh((\\Sch/T)_{ph}) \\longrightarrow \\Sh((\\Sch/S)_{ph}) $$ We have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$. We have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers."}
+{"_id": "12493", "title": "topologies-lemma-recognize-fpqc-covering", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \\begin{enumerate} \\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering, \\item each $f_i$ is flat and for every affine open $U \\subset T$ there exist quasi-compact opens $U_i \\subset T_i$ which are almost all empty, such that $U = \\bigcup f_i(U_i)$, \\item each $f_i$ is flat and there exists an affine open covering $T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$ there exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$ and quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that $U_\\alpha = \\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$. \\end{enumerate} If $T$ is quasi-separated, these are also equivalent to \\begin{enumerate} \\item[(4)] each $f_i$ is flat, and for every $t \\in T$ there exist $i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$ such that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a (not necessarily open) neighbourhood of $t$ in $T$. \\end{enumerate}"}
+{"_id": "12494", "title": "topologies-lemma-disjoint-union-is-fpqc-covering", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \\begin{enumerate} \\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering, and \\item setting $T' = \\coprod_{i \\in I} T_i$, and $f = \\coprod_{i \\in I} f_i$ the family $\\{f : T' \\to T\\}$ is an fpqc covering. \\end{enumerate}"}
+{"_id": "12495", "title": "topologies-lemma-family-flat-dominated-covering", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. Assume that \\begin{enumerate} \\item each $f_i$ is flat, and \\item the family $\\{f_i : T_i \\to T\\}_{i \\in I}$ can be refined by an fpqc covering of $T$. \\end{enumerate} Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$."}
+{"_id": "12496", "title": "topologies-lemma-family-flat-fpqc-local-covering", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. Assume that \\begin{enumerate} \\item each $f_i$ is flat, and \\item there exists an fpqc covering $\\{g_j : S_j \\to T\\}_{j \\in J}$ such that each $\\{S_j \\times_T T_i \\to S_j\\}_{i \\in I}$ is an fpqc covering. \\end{enumerate} Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$."}
+{"_id": "12497", "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "text": "Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, \\'etale or Zariski covering is an fpqc covering."}
+{"_id": "12498", "title": "topologies-lemma-fpqc", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is an fpqc covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering and for each $i$ we have an fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fpqc covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fpqc covering. \\end{enumerate}"}
+{"_id": "12499", "title": "topologies-lemma-fpqc-affine", "text": "Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of $T$. Then there exists an fpqc covering $\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement of $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$."}
+{"_id": "12500", "title": "topologies-lemma-fpqc-affine-axioms", "text": "Let $T$ be an affine scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a standard fpqc covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering and for each $i$ we have a standard fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a standard fpqc covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering and $T' \\to T$ is a morphism of affine schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a standard fpqc covering. \\end{enumerate}"}
+{"_id": "12502", "title": "topologies-lemma-sheaf-property-fpqc", "text": "Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the fpqc topology if and only if it satisfies \\begin{enumerate} \\item the sheaf property for every Zariski covering, and \\item the sheaf property for any standard fpqc covering. \\end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \\begin{enumerate} \\item[(2')] the sheaf property for $\\{V \\to U\\}$ with $V$, $U$ affine and $V \\to U$ faithfully flat. \\end{enumerate}"}
+{"_id": "12503", "title": "topologies-lemma-no-set-of-fpqc-covers-is-initial", "text": "Let $R$ be a nonzero ring. There does not exist a set $A$ of fpqc-coverings of $\\Spec(R)$ such that every fpqc-covering can be refined by an element of $A$."}
+{"_id": "12504", "title": "topologies-lemma-standard-fpqc-standard-V", "text": "A standard fpqc covering is a standard V covering."}
+{"_id": "12505", "title": "topologies-lemma-standard-ph-standard-V", "text": "A standard ph covering is a standard V covering."}
+{"_id": "12506", "title": "topologies-lemma-base-change-standard-V", "text": "Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard V covering. Let $T' \\to T$ be a morphism of affine schemes. Then $\\{T_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a standard V covering."}
+{"_id": "12507", "title": "topologies-lemma-composition-standard-V", "text": "Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard V covering. Let $\\{T_{ji} \\to T_j\\}_{i = 1, \\ldots n_j}$ be a standard V covering. Then $\\{T_{ji} \\to T\\}_{i, j}$ is a standard V covering."}
+{"_id": "12508", "title": "topologies-lemma-refine-standard-V", "text": "Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ be a family of morphisms with $T_j$ affine for all $j$. The following are equivalent \\begin{enumerate} \\item $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ is a standard V covering, \\item there is a standard V covering which refines $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$, and \\item $\\{\\coprod_{j = 1, \\ldots, m} T_j \\to T\\}$ is a standard V covering. \\end{enumerate}"}
+{"_id": "12509", "title": "topologies-lemma-refine-by-V", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms. The following are equivalent \\begin{enumerate} \\item $\\{T_i \\to T\\}_{i \\in I}$ is a V covering, \\item there is a V covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and \\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a V covering. \\end{enumerate}"}
+{"_id": "12510", "title": "topologies-lemma-V", "text": "Let $T$ be a scheme. \\begin{enumerate} \\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$ is a V covering of $T$. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering and for each $i$ we have a V covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then $\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a V covering. \\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering and $T' \\to T$ is a morphism of schemes then $\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a V covering. \\end{enumerate}"}
+{"_id": "12511", "title": "topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V", "text": "Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, \\'etale or Zariski covering is a V covering. Also, a ph covering is a V covering."}
+{"_id": "12512", "title": "topologies-lemma-sheaf-property-V", "text": "Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the V topology if and only if it satisfies \\begin{enumerate} \\item the sheaf property for every Zariski covering, and \\item the sheaf property for any standard V covering. \\end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \\begin{enumerate} \\item[(2')] the sheaf property for a standard V covering of the form $\\{V \\to U\\}$, i.e., consisting of a single arrow. \\end{enumerate}"}
+{"_id": "12513", "title": "topologies-lemma-refine-qcqs-V", "text": "Let $X \\to Y$ be a quasi-compact morphism of schemes. The following are equivalent \\begin{enumerate} \\item $\\{X \\to Y\\}$ is a V covering, \\item for any valuation ring $V$ and morphism $g : \\Spec(V) \\to Y$ there exists an extension of valuation rings $V \\subset W$ and a commutative diagram $$ \\xymatrix{ \\Spec(W) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(V) \\ar[r] & Y } $$ \\item for any morphism $Z \\to Y$ and specialization $z' \\leadsto z$ of points in $Z$, there is a specialization $w' \\leadsto w$ of points in $Z \\times_Y X$ mapping to $z' \\leadsto z$. \\end{enumerate}"}
+{"_id": "12514", "title": "topologies-lemma-V-covering-universally-submersive", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a V covering. Then $$ \\coprod\\nolimits_{i \\in I} f_i : \\coprod\\nolimits_{i \\in I} X_i \\longrightarrow X $$ is a universally submersive morphism of schemes (Morphisms, Definition \\ref{morphisms-definition-submersive})."}
+{"_id": "12515", "title": "topologies-lemma-contained-in", "text": "Any set of big Zariski sites is contained in a common big Zariski site. The same is true, mutatis mutandis, for big fppf and big \\'etale sites."}
+{"_id": "12516", "title": "topologies-lemma-change-alpha", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Suppose given big sites $\\Sch_\\tau$ and $\\Sch'_\\tau$. Assume that $\\Sch_\\tau$ is contained in $\\Sch'_\\tau$. The inclusion functor $\\Sch_\\tau \\to \\Sch'_\\tau$ satisfies the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}. There are morphisms of topoi \\begin{eqnarray*} g : \\Sh(\\Sch_\\tau) & \\longrightarrow & \\Sh(\\Sch'_\\tau) \\\\ f : \\Sh(\\Sch'_\\tau) & \\longrightarrow & \\Sh(\\Sch_\\tau) \\end{eqnarray*} such that $f \\circ g \\cong \\text{id}$. For any object $S$ of $\\Sch_\\tau$ the inclusion functor $(\\Sch/S)_\\tau \\to (\\Sch'/S)_\\tau$ satisfies the assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site} also. Hence similarly we obtain morphisms \\begin{eqnarray*} g : \\Sh((\\Sch/S)_\\tau) & \\longrightarrow & \\Sh((\\Sch'/S)_\\tau) \\\\ f : \\Sh((\\Sch'/S)_\\tau) & \\longrightarrow & \\Sh((\\Sch/S)_\\tau) \\end{eqnarray*} with $f \\circ g \\cong \\text{id}$."}
+{"_id": "12517", "title": "topologies-lemma-extend", "text": "Let $S$ be a scheme. Let $\\mathcal{C}$ be a full subcategory of the category $\\Sch/S$ of all schemes over $S$. Assume \\begin{enumerate} \\item if $X \\to S$ is an object of $\\mathcal{C}$ and $U \\subset X$ is an affine open, then $U \\to S$ is isomorphic to an object of $\\mathcal{C}$, \\item if $V$ is an affine scheme lying over an affine open $U \\subset S$ such that $V \\to U$ is of finite presentation, then $V \\to S$ is isomorphic to an object of $\\mathcal{C}$. \\end{enumerate} Let $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor. Assume \\begin{enumerate} \\item[(a)] for any Zariski covering $\\{f_i : X_i \\to X\\}_{i \\in I}$ with $X, X_i$ objects of $\\mathcal{C}$ we have the sheaf condition for $F$ and this family\\footnote{As we do not know that $X_i \\times_X X_j$ is in $\\mathcal{C}$ this has to be interpreted as follows: by property (1) there exist Zariski coverings $\\{U_{ijk} \\to X_i \\times_X X_j\\}_{k \\in K_{ij}}$ with $U_{ijk}$ an object of $\\mathcal{C}$. Then the sheaf condition says that $F(X)$ is the equalizer of the two maps from $\\prod F(X_i)$ to $\\prod F(U_{ijk})$.}, \\item[(b)] if $X = \\lim X_i$ is a directed limit of affine schemes over $S$ with $X, X_i$ objects of $\\mathcal{C}$, then $F(X) = \\colim F(X_i)$. \\end{enumerate} Then there is a unique way to extend $F$ to a functor $F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$ satisfying the analogues of (a) and (b), i.e., $F'$ satisfies the sheaf condition for any Zariski covering and $F'(X) = \\colim F'(X_i)$ whenever $X = \\lim X_i$ is a directed limit of affine schemes over $S$."}
+{"_id": "12518", "title": "topologies-lemma-limit-fppf-topology", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $T$ be an affine scheme which is written as a limit $T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes. \\begin{enumerate} \\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard $\\tau$-covering of $T$, see Definitions \\ref{definition-standard-Zariski}, \\ref{definition-standard-etale}, \\ref{definition-standard-smooth}, \\ref{definition-standard-syntomic}, and \\ref{definition-standard-fppf}. Then there exists an index $i$ and a standard $\\tau$-covering $\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$ whose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$ is isomorphic to $\\mathcal{V}$. \\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard $\\tau$-coverings of $T_i$. If $f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is a morphism of coverings of $T$, then there exists an index $i' \\geq i$ and a morphism $f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to T_{i'} \\times_{T_i} \\mathcal{V}'_i$ whose base change to $T$ is $f$. \\item If $f, g : \\mathcal{V} \\to \\mathcal{V}'_i$ are morphisms of standard $\\tau$-coverings of $T_i$ whose base changes $f_T, g_T$ to $T$ are equal then there exists an index $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$. \\end{enumerate} In other words, the category of standard $\\tau$-coverings of $T$ is the colimit over $I$ of the categories of standard $\\tau$-coverings of $T_i$."}
+{"_id": "12519", "title": "topologies-lemma-extend-sheaf-general", "text": "Let $S$, $\\mathcal{C}$, $F$ satisfy conditions (1), (2), (a), and (b) of Lemma \\ref{lemma-extend} and denote $F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$ the unique extension constructed in the lemma. Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Assume \\begin{enumerate} \\item[(c)] for any standard $\\tau$-covering $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$ of affines in $\\Sch/S$ such that $V \\to S$ factors through an affine open $U \\subset S$ and $V \\to U$ is of finite presentation, the sheaf condition hold for $F$ and $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$\\footnote{This makes sense as $V$, $V_i$, and $V_i \\times_V V_j$ are isomorphic to objects of $\\mathcal{C}$ by (2).}. \\end{enumerate} Then $F'$ satisfies the sheaf condition for all $\\tau$-coverings."}
+{"_id": "12520", "title": "topologies-lemma-extend-sheaf", "text": "Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $S$ be a scheme contained in a big site $\\Sch_\\tau$. Let $F : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be a $\\tau$-sheaf satisfying property (b) of Lemma \\ref{lemma-extend} with $\\mathcal{C} = (\\Sch/S)_\\tau$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all $\\tau$-coverings."}
+{"_id": "12554", "title": "pic-lemma-hilb-d-sheaf", "text": "Let $X \\to S$ be a morphism of schemes. The functor $\\Hilbfunctor^d_{X/S}$ satisfies the sheaf property for the fpqc topology (Topologies, Definition \\ref{topologies-definition-sheaf-property-fpqc})."}
+{"_id": "12556", "title": "pic-lemma-hilb-d-of-closed", "text": "Let $S$ be a scheme. Let $i : X \\to Y$ be a closed immersion of schemes. If $\\Hilbfunctor^d_{Y/S}$ is representable by a scheme, so is $\\Hilbfunctor^d_{X/S}$ and the corresponding morphism of schemes $\\underline{\\Hilbfunctor}^d_{X/S} \\to \\underline{\\Hilbfunctor}^d_{Y/S}$ is a closed immersion."}
+{"_id": "12557", "title": "pic-lemma-hilb-d-separated", "text": "Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is separated and $\\Hilbfunctor^d_{X/S}$ is representable, then $\\underline{\\Hilbfunctor}^d_{X/S} \\to S$ is separated."}
+{"_id": "12558", "title": "pic-lemma-hilb-d-An", "text": "Let $X \\to S$ be a morphism of affine schemes. Let $d \\geq 0$. Then $\\Hilbfunctor^d_{X/S}$ is representable."}
+{"_id": "12559", "title": "pic-lemma-divisors-on-curves", "text": "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D \\subset X$ be a closed subscheme. Consider the following conditions \\begin{enumerate} \\item $D \\to S$ is finite locally free, \\item $D$ is a relative effective Cartier divisor on $X/S$, \\item $D \\to S$ is locally quasi-finite, flat, and locally of finite presentation, and \\item $D \\to S$ is locally quasi-finite and flat. \\end{enumerate} We always have the implications $$ (1) \\Rightarrow (2) \\Leftrightarrow (3) \\Rightarrow (4) $$ If $S$ is locally Noetherian, then the last arrow is an if and only if. If $X \\to S$ is proper (and $S$ arbitrary), then the first arrow is an if and only if."}
+{"_id": "12560", "title": "pic-lemma-sum-divisors-on-curves", "text": "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \\subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free of degree $d_1 + d_2$ over $S$."}
+{"_id": "12561", "title": "pic-lemma-difference-divisors-on-curves", "text": "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \\subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. If $D_1 \\subset D_2$ (as closed subschemes) then there is a closed subscheme $D \\subset X$ finite locally free of degree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$."}
+{"_id": "12562", "title": "pic-lemma-universal-object", "text": "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The morphism $\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X \\to \\underline{\\Hilbfunctor}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$."}
+{"_id": "12563", "title": "pic-lemma-hilb-d-smooth", "text": "Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$ such that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The schemes $\\underline{\\Hilbfunctor}^d_{X/S}$ are smooth over $S$ of relative dimension $d$."}
+{"_id": "12564", "title": "pic-lemma-flat-geometrically-connected-fibres", "text": "Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}. If $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all $T \\in \\Ob((\\Sch/S)_{fppf})$, then $$ 0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T) $$ is an exact sequence for all $T$."}
+{"_id": "12566", "title": "pic-lemma-criterion", "text": "Let $k$ be a field. Let $G : (\\Sch/k)^{opp} \\to \\textit{Groups}$ be a functor. With terminology as in Schemes, Definition \\ref{schemes-definition-representable-by-open-immersions}, assume that \\begin{enumerate} \\item $G$ satisfies the sheaf property for the Zariski topology, \\item there exists a subfunctor $F \\subset G$ such that \\begin{enumerate} \\item $F$ is representable, \\item $F \\subset G$ is representable by open immersion, \\item for every field extension $K$ of $k$ and $g \\in G(K)$ there exists a $g' \\in G(k)$ such that $g'g \\in F(K)$. \\end{enumerate} \\end{enumerate} Then $G$ is representable by a group scheme over $k$."}
+{"_id": "12567", "title": "pic-lemma-check-conditions", "text": "Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ which has a $k$-rational point. Then the hypotheses of Lemma \\ref{lemma-flat-geometrically-connected-fibres-with-section} are satisfied."}
+{"_id": "12568", "title": "pic-lemma-define-open", "text": "Let $k$ be a field. Let $X$ be a smooth projective curve over $k$ with a $k$-rational point $\\sigma$. For a scheme $T$ over $k$, consider the subset $F(T) \\subset \\Picardfunctor_{X/k, \\sigma}(T)$ consisting of $\\mathcal{L}$ such that $Rf_{T, *}\\mathcal{L}$ is isomorphic to an invertible $\\mathcal{O}_T$-module placed in degree $0$. Then $F \\subset \\Picardfunctor_{X/k, \\sigma}$ is a subfunctor and the inclusion is representable by open immersions."}
+{"_id": "12569", "title": "pic-lemma-open-representable", "text": "Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$ over $k$ with a $k$-rational point $\\sigma$. The open subfunctor $F$ defined in Lemma \\ref{lemma-define-open} is representable by an open subscheme of $\\underline{\\Hilbfunctor}^g_{X/k}$."}
+{"_id": "12570", "title": "pic-lemma-twist-with-general-divisor", "text": "Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. Let $K/k$ be a field extension and let $\\mathcal{L}$ be an invertible sheaf on $X_K$. Then there exists an invertible sheaf $\\mathcal{L}_0$ on $X$ such that $\\dim_K H^0(X_K, \\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 1$ and $\\dim_K H^1(X_K, \\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 0$."}
+{"_id": "12571", "title": "pic-lemma-picard-pieces", "text": "Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. \\begin{enumerate} \\item $\\underline{\\Picardfunctor}_{X/k}$ is a disjoint union of $g$-dimensional smooth proper varieties $\\underline{\\Picardfunctor}^d_{X/k}$, \\item $k$-points of $\\underline{\\Picardfunctor}^d_{X/k}$ correspond to invertible $\\mathcal{O}_X$-modules of degree $d$, \\item $\\underline{\\Picardfunctor}^0_{X/k}$ is an open and closed subgroup scheme, \\item for $d \\geq 0$ there is a canonical morphism $\\gamma_d : \\underline{\\Hilbfunctor}^d_{X/k} \\to \\underline{\\Picardfunctor}^d_{X/k}$ \\item the morphisms $\\gamma_d$ are surjective for $d \\geq g$ and smooth for $d \\geq 2g - 1$, \\item the morphism $\\underline{\\Hilbfunctor}^g_{X/k} \\to \\underline{\\Picardfunctor}^g_{X/k}$ is birational. \\end{enumerate}"}
+{"_id": "12573", "title": "pic-lemma-torsion-descends", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \\mathcal{O}_X) = k$. Let $n$ be an integer prime to $p$. Then the map $$ \\Pic(X)[n] \\longrightarrow \\Pic(X_{k'})[n] $$ is bijective for any purely inseparable extension $k'/k$."}
+{"_id": "12574", "title": "pic-proposition-hilb-d-representable", "text": "Let $X \\to S$ be a morphism of schemes. Let $d \\geq 0$. Assume for all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and $x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$ with $x_1, \\ldots, x_d \\in U$. Then $\\Hilbfunctor^d_{X/S}$ is representable by a scheme."}
+{"_id": "12575", "title": "pic-proposition-hilb-d", "text": "Let $X$ be a geometrically irreducible smooth proper curve over a field $k$. \\begin{enumerate} \\item The functors $\\Hilbfunctor^d_{X/k}$ are representable by smooth proper varieties $\\underline{\\Hilbfunctor}^d_{X/k}$ of dimension $d$ over $k$. \\item For a field extension $k'/k$ the $k'$-rational points of $\\underline{\\Hilbfunctor}^d_{X/k}$ are in $1$-to-$1$ bijection with effective Cartier divisors of degree $d$ on $X_{k'}$. \\item For $d_1, d_2 \\geq 0$ there is a morphism $$ \\underline{\\Hilbfunctor}^{d_1}_{X/k} \\times_k \\underline{\\Hilbfunctor}^{d_2}_{X/k} \\longrightarrow \\underline{\\Hilbfunctor}^{d_1 + d_2}_{X/k} $$ which is finite locally free of degree ${d_1 + d_2 \\choose d_1}$. \\end{enumerate}"}
+{"_id": "12576", "title": "pic-proposition-pic-curve", "text": "Let $k$ be a separably closed field. Let $X$ be a smooth projective curve over $k$. The Picard functor $\\Picardfunctor_{X/k}$ is representable."}
+{"_id": "12581", "title": "constructions-lemma-relative-glueing", "text": "Let $S$ be a scheme. Let $\\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \\begin{enumerate} \\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$. \\item For $U, V \\in \\mathcal{B}$ with $V \\subset U$ a morphism $\\rho^U_V : X_V \\to X_U$ over $U$. \\end{enumerate} Assume that \\begin{enumerate} \\item[(a)] each $\\rho^U_V$ induces an isomorphism $X_V \\to f_U^{-1}(V)$ of schemes over $V$, \\item[(b)] whenever $W, V, U \\in \\mathcal{B}$, with $W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$. \\end{enumerate} Then there exists a morphism $f : X \\to S$ of schemes and isomorphisms $i_U : f^{-1}(U) \\to X_U$ over $U \\in \\mathcal{B}$ such that for $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition $$ \\xymatrix{ X_V \\ar[r]^{i_V^{-1}} & f^{-1}(V) \\ar[rr]^{inclusion} & & f^{-1}(U) \\ar[r]^{i_U} & X_U } $$ is the morphism $\\rho^U_V$. Moreover $X$ is unique up to unique isomorphism over $S$."}
+{"_id": "12582", "title": "constructions-lemma-relative-glueing-sheaves", "text": "Let $S$ be a scheme. Let $\\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \\begin{enumerate} \\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$. \\item For every $U \\in \\mathcal{B}$ a quasi-coherent sheaf $\\mathcal{F}_U$ over $X_U$. \\item For every pair $U, V \\in \\mathcal{B}$ such that $V \\subset U$ a morphism $\\rho^U_V : X_V \\to X_U$. \\item  For every pair $U, V \\in \\mathcal{B}$ such that $V \\subset U$ a morphism $\\theta^U_V : (\\rho^U_V)^*\\mathcal{F}_U \\to \\mathcal{F}_V$. \\end{enumerate} Assume that \\begin{enumerate} \\item[(a)] each $\\rho^U_V$ induces an isomorphism $X_V \\to f_U^{-1}(V)$ of schemes over $V$, \\item[(b)] each $\\theta^U_V$ is an isomorphism, \\item[(c)] whenever $W, V, U \\in \\mathcal{B}$, with $W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$, \\item[(d)] whenever $W, V, U \\in \\mathcal{B}$, with $W \\subset V \\subset U$ we have $\\theta^U_W = \\theta^V_W \\circ (\\rho^V_W)^*\\theta^U_V$. \\end{enumerate} Then there exists a morphism of schemes $f : X \\to S$ together with a quasi-coherent sheaf $\\mathcal{F}$ on $X$ and isomorphisms $i_U : f^{-1}(U) \\to X_U$ and $\\theta_U : i_U^*\\mathcal{F}_U \\to \\mathcal{F}|_{f^{-1}(U)}$ over $U \\in \\mathcal{B}$ such that for $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition $$ \\xymatrix{ X_V \\ar[r]^{i_V^{-1}} & f^{-1}(V) \\ar[rr]^{inclusion} & & f^{-1}(U) \\ar[r]^{i_U} & X_U } $$ is the morphism $\\rho^U_V$, and the composition \\begin{equation} \\label{equation-glue} (\\rho^U_V)^*\\mathcal{F}_U = (i_V^{-1})^*((i_U^*\\mathcal{F}_U)|_{f^{-1}(V)}) \\xrightarrow{\\theta_U|_{f^{-1}(V)}} (i_V^{-1})^*(\\mathcal{F}|_{f^{-1}(V)}) \\xrightarrow{\\theta_V^{-1}} \\mathcal{F}_V \\end{equation} is equal to $\\theta^U_V$. Moreover $(X, \\mathcal{F})$ is unique up to unique isomorphism over $S$."}
+{"_id": "12583", "title": "constructions-lemma-spec-inclusion", "text": "In Situation \\ref{situation-relative-spec}. Suppose $U \\subset U' \\subset S$ are affine opens. Let $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$. The map of rings $A' \\to A$ induces a morphism $\\Spec(A) \\to \\Spec(A')$, and the diagram $$ \\xymatrix{ \\Spec(A) \\ar[r] \\ar[d] & \\Spec(A') \\ar[d] \\\\ U \\ar[r] & U' } $$ is cartesian."}
+{"_id": "12584", "title": "constructions-lemma-transitive-spec", "text": "In Situation \\ref{situation-relative-spec}. Suppose $U \\subset U' \\subset U'' \\subset S$ are affine opens. Let $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$. The composition of the morphisms $\\Spec(A) \\to \\Spec(A')$, and $\\Spec(A') \\to \\Spec(A'')$ of Lemma \\ref{lemma-spec-inclusion} gives the morphism $\\Spec(A) \\to \\Spec(A'')$ of Lemma \\ref{lemma-spec-inclusion}."}
+{"_id": "12585", "title": "constructions-lemma-glue-relative-spec", "text": "In Situation \\ref{situation-relative-spec}. There exists a morphism of schemes $$ \\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\longrightarrow S $$ with the following properties: \\begin{enumerate} \\item for every affine open $U \\subset S$ there exists an isomorphism $i_U : \\pi^{-1}(U) \\to \\Spec(\\mathcal{A}(U))$, and \\item for $U \\subset U' \\subset S$ affine open the composition $$ \\xymatrix{ \\Spec(\\mathcal{A}(U)) \\ar[r]^{i_U^{-1}} & \\pi^{-1}(U) \\ar[rr]^{inclusion} & & \\pi^{-1}(U') \\ar[r]^{i_{U'}} & \\Spec(\\mathcal{A}(U')) } $$ is the open immersion of Lemma \\ref{lemma-spec-inclusion} above. \\end{enumerate}"}
+{"_id": "12586", "title": "constructions-lemma-spec-base-change", "text": "In Situation \\ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \\mathcal{A})$ above. Let $g : S' \\to S$ be a morphism of schemes. Set $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F'$ be the functor associated to $(S', \\mathcal{A}')$ above. Then there is a canonical isomorphism $$ F' \\cong h_{S'} \\times_{h_S} F $$ of functors."}
+{"_id": "12587", "title": "constructions-lemma-spec-affine", "text": "In Situation \\ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \\mathcal{A})$ above. If $S$ is affine, then $F$ is representable by the affine scheme $\\Spec(\\Gamma(S, \\mathcal{A}))$."}
+{"_id": "12589", "title": "constructions-lemma-glueing-gives-functor-spec", "text": "In Situation \\ref{situation-relative-spec}. The scheme $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$ constructed in Lemma \\ref{lemma-glue-relative-spec} and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$."}
+{"_id": "12590", "title": "constructions-lemma-spec-properties", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras. Let $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$ be the relative spectrum of $\\mathcal{A}$ over $S$. \\begin{enumerate} \\item For every affine open $U \\subset S$ the inverse image $\\pi^{-1}(U)$ is affine. \\item For every morphism $g : S' \\to S$ we have $S' \\times_S \\underline{\\Spec}_S(\\mathcal{A}) = \\underline{\\Spec}_{S'}(g^*\\mathcal{A})$. \\item The universal map $$ \\mathcal{A} \\longrightarrow \\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})} $$ is an isomorphism of $\\mathcal{O}_S$-algebras. \\end{enumerate}"}
+{"_id": "12591", "title": "constructions-lemma-canonical-morphism", "text": "Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $f_*\\mathcal{O}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras. There is a canonical morphism $$ can : X \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X) $$ of schemes over $S$. For any affine open $U \\subset S$ the restriction $can|_{f^{-1}(U)}$ is identified with the canonical morphism $$ f^{-1}(U) \\longrightarrow \\Spec(\\Gamma(f^{-1}(U), \\mathcal{O}_X)) $$ coming from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}."}
+{"_id": "12592", "title": "constructions-lemma-category-vector-bundles", "text": "The category of vector bundles over a scheme $S$ is anti-equivalent to the category of quasi-coherent $\\mathcal{O}_S$-modules."}
+{"_id": "12593", "title": "constructions-lemma-standard-open", "text": "Let $S$ be a graded ring. Let $f \\in S$ homogeneous of positive degree. \\begin{enumerate} \\item If $g\\in S$ homogeneous of positive degree and $D_{+}(g) \\subset D_{+}(f)$, then \\begin{enumerate} \\item $f$ is invertible in $S_g$, and $f^{\\deg(g)}/g^{\\deg(f)}$ is invertible in $S_{(g)}$, \\item $g^e = af$ for some $e \\geq 1$ and $a \\in S$ homogeneous, \\item there is a canonical $S$-algebra map $S_f \\to S_g$, \\item there is a canonical $S_0$-algebra map $S_{(f)} \\to S_{(g)}$ compatible with the map $S_f \\to S_g$, \\item the map $S_{(f)} \\to S_{(g)}$ induces an isomorphism $$ (S_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong S_{(g)}, $$ \\item these maps induce a commutative diagram of topological spaces $$ \\xymatrix{ D_{+}(g) \\ar[d] & \\{\\mathbf{Z}\\text{-graded primes of }S_g\\} \\ar[l] \\ar[r] \\ar[d] & \\Spec(S_{(g)}) \\ar[d] \\\\ D_{+}(f) & \\{\\mathbf{Z}\\text{-graded primes of }S_f\\} \\ar[l] \\ar[r] & \\Spec(S_{(f)}) } $$ where the horizontal maps are homeomorphisms and the vertical maps are open immersions, \\item there are compatible canonical $S_f$ and $S_{(f)}$-module maps $M_f \\to M_g$ and $M_{(f)} \\to M_{(g)}$ for any graded $S$-module $M$, and \\item the map $M_{(f)} \\to M_{(g)}$ induces an isomorphism $$ (M_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong M_{(g)}. $$ \\end{enumerate} \\item Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$. \\item Let $g_1, \\ldots, g_n \\in S$ be homogeneous of positive degree. Then $D_{+}(f) \\subset \\bigcup D_{+}(g_i)$ if and only if $g_1^{\\deg(f)}/f^{\\deg(g_1)}, \\ldots, g_n^{\\deg(f)}/f^{\\deg(g_n)}$ generate the unit ideal in $S_{(f)}$. \\end{enumerate}"}
+{"_id": "12594", "title": "constructions-lemma-proj-sheaves", "text": "Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\\widetilde M$ be the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules associated to $M$. \\begin{enumerate} \\item For every $f \\in S$ homogeneous of positive degree we have $$ \\Gamma(D_{+}(f), \\mathcal{O}_{\\text{Proj}(S)}) = S_{(f)}. $$ \\item For every $f\\in S$ homogeneous of positive degree we have $\\Gamma(D_{+}(f), \\widetilde M) = M_{(f)}$ as an $S_{(f)}$-module. \\item Whenever $D_{+}(g) \\subset D_{+}(f)$ the restriction mappings on $\\mathcal{O}_{\\text{Proj}(S)}$ and $\\widetilde M$ are the maps $S_{(f)} \\to S_{(g)}$ and $M_{(f)} \\to M_{(g)}$ from Lemma \\ref{lemma-standard-open}. \\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \\in \\text{Proj}(S)$ be the corresponding point. We have $\\mathcal{O}_{\\text{Proj}(S), x} = S_{(\\mathfrak p)}$. \\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \\in \\text{Proj}(S)$ be the corresponding point. We have $\\mathcal{F}_x = M_{(\\mathfrak p)}$ as an $S_{(\\mathfrak p)}$-module. \\item \\label{item-map} There is a canonical ring map $ S_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde S) $ and a canonical $S_0$-module map $ M_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde M) $ compatible with the descriptions of sections over standard opens and stalks above. \\end{enumerate} Moreover, all these identifications are functorial in the graded $S$-module $M$. In particular, the functor $M \\mapsto \\widetilde M$ is an exact functor from the category of graded $S$-modules to the category of $\\mathcal{O}_{\\text{Proj}(S)}$-modules."}
+{"_id": "12595", "title": "constructions-lemma-standard-open-proj", "text": "Let $S$ be a graded ring. Let $f \\in S$ be homogeneous of positive degree. Suppose that $D(g) \\subset \\Spec(S_{(f)})$ is a standard open. Then there exists a $h \\in S$ homogeneous of positive degree such that $D(g)$ corresponds to $D_{+}(h) \\subset D_{+}(f)$ via the homeomorphism of Algebra, Lemma \\ref{algebra-lemma-topology-proj}. In fact we can take $h$ such that $g = h/f^n$ for some $n$."}
+{"_id": "12596", "title": "constructions-lemma-proj-scheme", "text": "Let $S$ be a graded ring. The locally ringed space $\\text{Proj}(S)$ is a scheme. The standard opens $D_{+}(f)$ are affine opens. For any graded $S$-module $M$ the sheaf $\\widetilde M$ is a quasi-coherent sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules."}
+{"_id": "12597", "title": "constructions-lemma-proj-separated", "text": "Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ is separated."}
+{"_id": "12598", "title": "constructions-lemma-proj-quasi-compact", "text": "Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ is quasi-compact if and only if there exist finitely many homogeneous elements $f_1, \\ldots, f_n \\in S_{+}$ such that $S_{+} \\subset \\sqrt{(f_1, \\ldots, f_n)}$. In this case $\\text{Proj}(S) = D_+(f_1) \\cup \\ldots \\cup D_+(f_n)$."}
+{"_id": "12600", "title": "constructions-lemma-proj-valuative-criterion", "text": "Let $S$ be a graded ring. If $S$ is finitely generated as an algebra over $S_0$, then the morphism $\\text{Proj}(S) \\to \\Spec(S_0)$ satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition \\ref{schemes-definition-valuative-criterion}."}
+{"_id": "12601", "title": "constructions-lemma-widetilde-tensor", "text": "Let $S$ be a graded ring. Let $(X, \\mathcal{O}_X) = (\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$ be the scheme of Lemma \\ref{lemma-proj-scheme}. Let $f \\in S_{+}$ be homogeneous. Let $x \\in X$ be a point corresponding to the homogeneous prime $\\mathfrak p \\subset S$. Let $M$, $N$ be graded $S$-modules. There is a canonical map of $\\mathcal{O}_{\\text{Proj}(S)}$-modules $$ \\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N \\longrightarrow \\widetilde{M \\otimes_S N} $$ which induces the canonical map $ M_{(f)} \\otimes_{S_{(f)}} N_{(f)} \\to (M \\otimes_S N)_{(f)} $ on sections over $D_{+}(f)$ and the canonical map $ M_{(\\mathfrak p)} \\otimes_{S_{(\\mathfrak p)}} N_{(\\mathfrak p)} \\to (M \\otimes_S N)_{(\\mathfrak p)} $ on stalks at $x$. Moreover, the following diagram $$ \\xymatrix{ M_0 \\otimes_{S_0} N_0 \\ar[r] \\ar[d] & (M \\otimes_S N)_0 \\ar[d] \\\\ \\Gamma(X, \\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N) \\ar[r] & \\Gamma(X, \\widetilde{M \\otimes_R N}) } $$ is commutative where the vertical maps are given by (\\ref{equation-map-global-sections})."}
+{"_id": "12602", "title": "constructions-lemma-when-invertible", "text": "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Let $f \\in S$ be homogeneous of degree $d > 0$. The sheaves $\\mathcal{O}_X(nd)|_{D_{+}(f)}$ are invertible, and in fact trivial for all $n \\in \\mathbf{Z}$ (see Modules, Definition \\ref{modules-definition-invertible}). The maps (\\ref{equation-multiply}) restricted to $D_{+}(f)$ $$ \\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}} \\mathcal{O}_X(m)|_{D_{+}(f)} \\longrightarrow \\mathcal{O}_X(nd + m)|_{D_{+}(f)}, $$ the maps (\\ref{equation-multiply-on-sheaf}) restricted to $D_+(f)$ $$ \\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}} \\mathcal{F}(m)|_{D_{+}(f)} \\longrightarrow \\mathcal{F}(nd + m)|_{D_{+}(f)}, $$ and the maps (\\ref{equation-multiply-more-generally}) restricted to $D_{+}(f)$ $$ \\widetilde M(nd)|_{D_{+}(f)} = \\widetilde M|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}} \\mathcal{O}_X(nd)|_{D_{+}(f)} \\longrightarrow \\widetilde{M(nd)}|_{D_{+}(f)} $$ are isomorphisms for all $n, m \\in \\mathbf{Z}$."}
+{"_id": "12603", "title": "constructions-lemma-apply-modules", "text": "Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Set $X = \\text{Proj}(S)$. Assume $X$ is covered by the standard opens $D_+(f)$ with $f \\in S_1$, e.g., if $S$ is generated by $S_1$ over $S_0$. Then the sheaves $\\mathcal{O}_X(n)$ are invertible and the maps (\\ref{equation-multiply}), (\\ref{equation-multiply-on-sheaf}), and (\\ref{equation-multiply-more-generally}) are isomorphisms. In particular, these maps induce isomorphisms $$ \\mathcal{O}_X(1)^{\\otimes n} \\cong \\mathcal{O}_X(n) \\quad \\text{and} \\quad \\widetilde{M} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n) = \\widetilde{M}(n) \\cong \\widetilde{M(n)} $$ Thus (\\ref{equation-map-global-sections-degree-n}) becomes a map \\begin{equation} \\label{equation-map-global-sections-degree-n-simplified} M_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n)) \\end{equation} and (\\ref{equation-global-sections-more-generally}) becomes a map \\begin{equation} \\label{equation-global-sections-more-generally-simplified} M \\longrightarrow \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X, \\widetilde{M}(n)). \\end{equation}"}
+{"_id": "12604", "title": "constructions-lemma-where-invertible", "text": "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an integer. The following open subsets of $X$ are equal: \\begin{enumerate} \\item The largest open subset $W = W_d \\subset X$ such that each $\\mathcal{O}_X(dn)|_W$ is invertible and all the multiplication maps $\\mathcal{O}_X(nd)|_W \\otimes_{\\mathcal{O}_W} \\mathcal{O}_X(md)|_W \\to \\mathcal{O}_X(nd + md)|_W$ (see \\ref{equation-multiply}) are isomorphisms. \\item The union of the open subsets $D_{+}(fg)$ with $f, g \\in S$ homogeneous and $\\deg(f) = \\deg(g) + d$. \\end{enumerate} Moreover, all the maps $\\widetilde M(nd)|_W = \\widetilde M|_W \\otimes_{\\mathcal{O}_W} \\mathcal{O}_X(nd)|_W \\to \\widetilde{M(nd)}|_W$ (see \\ref{equation-multiply-more-generally}) are isomorphisms."}
+{"_id": "12605", "title": "constructions-lemma-principal-open", "text": "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an integer. Let $W = W_d \\subset X$ be the open subscheme defined in Lemma \\ref{lemma-where-invertible}. Let $n \\geq 1$ and $f \\in S_{nd}$. Denote $s \\in \\Gamma(W, \\mathcal{O}_W(nd))$ the section which is the image of $f$ via (\\ref{equation-global-sections}) restricted to $W$. Then $$ W_s = D_{+}(f) \\cap W. $$"}
+{"_id": "12606", "title": "constructions-lemma-ample-on-proj", "text": "Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$. Let $Y \\subset X$ be a quasi-compact open subscheme. Denote $\\mathcal{O}_Y(n)$ the restriction of $\\mathcal{O}_X(n)$ to $Y$. There exists an integer $d \\geq 1$ such that \\begin{enumerate} \\item the subscheme $Y$ is contained in the open $W_d$ defined in Lemma \\ref{lemma-where-invertible}, \\item the sheaf $\\mathcal{O}_Y(dn)$ is invertible for all $n \\in \\mathbf{Z}$, \\item all the maps $\\mathcal{O}_Y(nd) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m) \\longrightarrow \\mathcal{O}_Y(nd + m)$ of Equation (\\ref{equation-multiply}) are isomorphisms, \\item all the maps $\\widetilde M(nd)|_Y = \\widetilde M|_Y \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_X(nd)|_Y \\to \\widetilde{M(nd)}|_Y$ (see \\ref{equation-multiply-more-generally}) are isomorphisms, \\item given $f \\in S_{nd}$ denote $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$ the image of $f$ via (\\ref{equation-global-sections}) restricted to $Y$, then $D_{+}(f) \\cap Y = Y_s$, \\item a basis for the topology on $Y$ is given by the collection of opens $Y_s$, where $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$, $n \\geq 1$, and \\item a basis for the topology of $Y$ is given by those opens $Y_s \\subset Y$, for $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$, $n \\geq 1$ which are affine. \\end{enumerate}"}
+{"_id": "12607", "title": "constructions-lemma-comparison-proj-quasi-coherent", "text": "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Set $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as a graded $S$-module, using (\\ref{equation-global-sections-module}) and (\\ref{equation-global-sections}). Then there is a canonical $\\mathcal{O}_X$-module map $$ \\widetilde{M} \\longrightarrow \\mathcal{F} $$ functorial in $\\mathcal{F}$ such that the induced map $M_0 \\to \\Gamma(X, \\mathcal{F})$ is the identity."}
+{"_id": "12608", "title": "constructions-lemma-morphism-proj", "text": "Let $A$, $B$ be two graded rings. Set $X = \\text{Proj}(A)$ and $Y = \\text{Proj}(B)$. Let $\\psi : A \\to B$ be a graded ring map. Set $$ U(\\psi) = \\bigcup\\nolimits_{f \\in A_{+}\\ \\text{homogeneous}} D_{+}(\\psi(f)) \\subset Y. $$ Then there is a canonical morphism of schemes $$ r_\\psi : U(\\psi) \\longrightarrow X $$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras $$ \\theta = \\theta_\\psi : r_\\psi^*\\left( \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d). $$ The triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the following properties: \\begin{enumerate} \\item For every $d \\geq 0$ the diagram $$ \\xymatrix{ A_d \\ar[d] \\ar[rr]_{\\psi} & & B_d \\ar[d] \\\\ \\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^-\\theta & \\Gamma(U(\\psi), \\mathcal{O}_Y(d)) & \\Gamma(Y, \\mathcal{O}_Y(d)) \\ar[l] } $$ is commutative. \\item For any $f \\in A_{+}$ homogeneous we have $r_\\psi^{-1}(D_{+}(f)) = D_{+}(\\psi(f))$ and the restriction of $r_\\psi$ to $D_{+}(\\psi(f))$ corresponds to the ring map $A_{(f)} \\to B_{(\\psi(f))}$ induced by $\\psi$. \\end{enumerate}"}
+{"_id": "12609", "title": "constructions-lemma-morphism-proj-transitive", "text": "Let $A$, $B$, and $C$ be graded rings. Set $X = \\text{Proj}(A)$, $Y = \\text{Proj}(B)$ and $Z = \\text{Proj}(C)$. Let $\\varphi : A \\to B$, $\\psi : B \\to C$ be graded ring maps. Then we have $$ U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi)) \\quad \\text{and} \\quad r_{\\psi \\circ \\varphi} = r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}. $$ In addition we have $$ \\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi = \\theta_{\\psi \\circ \\varphi} $$ with obvious notation."}
+{"_id": "12610", "title": "constructions-lemma-surjective-graded-rings-map-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above. Assume $A_d \\to B_d$ is surjective for all $d \\gg 0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are surjective but not isomorphisms in general (even if $A \\to B$ is surjective). \\end{enumerate}"}
+{"_id": "12611", "title": "constructions-lemma-eventual-iso-graded-rings-map-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above. Assume $A_d \\to B_d$ is an isomorphism for all $d \\gg 0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is an isomorphism, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12612", "title": "constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above. Assume $A_d \\to B_d$ is surjective for $d \\gg 0$ and that $A$ is generated by $A_1$ over $A_0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12613", "title": "constructions-lemma-base-change-map-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above. Assume there exists a ring map $R \\to A_0$ and a ring map $R \\to R'$ such that $B = R' \\otimes_R A$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item the diagram $$ \\xymatrix{ Y = \\text{Proj}(B) \\ar[r]_{r_\\psi} \\ar[d] & \\text{Proj}(A) = X \\ar[d] \\\\ \\Spec(R') \\ar[r] & \\Spec(R) } $$ is a fibre product square, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12615", "title": "constructions-lemma-d-uple", "text": "Let $S$ be a graded ring. Let $d \\geq 1$. Set $S' = S^{(d)}$ with notation as in Algebra, Section \\ref{algebra-section-graded}. Set $X = \\text{Proj}(S)$ and $X' = \\text{Proj}(S')$. There is a canonical isomorphism $i : X \\to X'$ of schemes such that \\begin{enumerate} \\item for any graded $S$-module $M$ setting $M' = M^{(d)}$, we have a canonical isomorphism $\\widetilde{M} \\to i^*\\widetilde{M'}$, \\item we have canonical isomorphisms $\\mathcal{O}_{X}(nd) \\to i^*\\mathcal{O}_{X'}(n)$ \\end{enumerate} and these isomorphisms are compatible with the multiplication maps of Lemma \\ref{lemma-widetilde-tensor} and hence with the maps (\\ref{equation-multiply}), (\\ref{equation-multiply-on-sheaf}), (\\ref{equation-global-sections}), (\\ref{equation-global-sections-module}), (\\ref{equation-multiply-more-generally}), and (\\ref{equation-global-sections-more-generally}) (see proof for precise statements."}
+{"_id": "12616", "title": "constructions-lemma-converse-construction", "text": "Let $S$ be a graded ring, and $X = \\text{Proj}(S)$. Let $d \\geq 1$ and $U_d \\subset X$ as above. Let $Y$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Let $\\psi : S^{(d)} \\to \\Gamma_*(Y, \\mathcal{L})$ be a graded ring homomorphism such that $\\mathcal{L}$ is generated by the sections in the image of $\\psi|_{S_d} : S_d \\to \\Gamma(Y, \\mathcal{L})$. Then there exists a morphism $\\varphi : Y \\to X$ such that $\\varphi(Y) \\subset U_d$ and an isomorphism $\\alpha : \\varphi^*\\mathcal{O}_{U_d}(d) \\to \\mathcal{L}$ such that $\\psi_\\varphi^d$ agrees with $\\psi$ via $\\alpha$: $$ \\xymatrix{ \\Gamma_*(Y, \\mathcal{L}) & \\Gamma_*(Y, \\varphi^*\\mathcal{O}_{U_d}(d)) \\ar[l]^-\\alpha & \\Gamma_*(U_d, \\mathcal{O}_{U_d}(d)) \\ar[l]^-{\\varphi^*} \\\\ S^{(d)} \\ar[u]^\\psi & & S^{(d)} \\ar[u]^{\\psi^d} \\ar[ul]^{\\psi^d_\\varphi} \\ar[ll]_{\\text{id}} } $$ commutes. Moreover, the pair $(\\varphi, \\alpha)$ is unique."}
+{"_id": "12617", "title": "constructions-lemma-proj-functor-strict", "text": "Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$. The open subscheme $U_d \\subset X$ (\\ref{equation-Ud}) represents the functor $F_d$ and the triple $(d, \\mathcal{O}_{U_d}(d), \\psi^d)$ defined above is the universal family (see Schemes, Section \\ref{schemes-section-representable})."}
+{"_id": "12618", "title": "constructions-lemma-apply", "text": "Let $S$ be a graded ring generated as an $S_0$-algebra by the elements of $S_1$. In this case the scheme $X = \\text{Proj}(S)$ represents the functor which associates to a scheme $Y$ the set of pairs $(\\mathcal{L}, \\psi)$, where \\begin{enumerate} \\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and \\item $\\psi : S \\to \\Gamma_*(Y, \\mathcal{L})$ is a graded ring homomorphism such that $\\mathcal{L}$ is generated by the global sections $\\psi(f)$, with $f \\in S_1$ \\end{enumerate} up to strict equivalence as above."}
+{"_id": "12619", "title": "constructions-lemma-equivalent", "text": "Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Let $T$ be a scheme. Let $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$ be two triples over $T$. The following are equivalent: \\begin{enumerate} \\item Let $n = \\text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists an isomorphism $\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$ with the property that $\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$. \\item The triples $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$ are equivalent. \\item For some positive integer $n = ad = a'd'$ there exists an isomorphism $\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$ with the property that $\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$. \\item The morphisms $\\varphi : T \\to X$ and $\\varphi' : T \\to X$ associated to $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$ are equal. \\end{enumerate}"}
+{"_id": "12620", "title": "constructions-lemma-proj-functor", "text": "Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$. The functor $F$ defined above is representable by the scheme $X$."}
+{"_id": "12621", "title": "constructions-lemma-projective-space", "text": "Let $S = \\mathbf{Z}[T_0, \\ldots, T_n]$ with $\\deg(T_i) = 1$. The scheme $$ \\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(S) $$ represents the functor which associates to a scheme $Y$ the pairs $(\\mathcal{L}, (s_0, \\ldots, s_n))$ where \\begin{enumerate} \\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and \\item $s_0, \\ldots, s_n$ are global sections of $\\mathcal{L}$ which generate $\\mathcal{L}$ \\end{enumerate} up to the following equivalence: $(\\mathcal{L}, (s_0, \\ldots, s_n)) \\sim (\\mathcal{N}, (t_0, \\ldots, t_n))$ $\\Leftrightarrow$ there exists an isomorphism $\\beta : \\mathcal{L} \\to \\mathcal{N}$ with $\\beta(s_i) = t_i$ for $i = 0, \\ldots, n$."}
+{"_id": "12622", "title": "constructions-lemma-standard-covering-projective-space", "text": "Projective $n$-space over $\\mathbf{Z}$ is covered by $n + 1$ standard opens $$ \\mathbf{P}^n_{\\mathbf{Z}} = \\bigcup\\nolimits_{i = 0, \\ldots, n} D_{+}(T_i) $$ where each $D_{+}(T_i)$ is isomorphic to $\\mathbf{A}^n_{\\mathbf{Z}}$ affine $n$-space over $\\mathbf{Z}$."}
+{"_id": "12624", "title": "constructions-lemma-segre-embedding", "text": "Let $S$ be a scheme. There exists a closed immersion $$ \\mathbf{P}^n_S \\times_S \\mathbf{P}^m_S \\longrightarrow \\mathbf{P}^{nm + n + m}_S $$ called the {\\it Segre embedding}."}
+{"_id": "12625", "title": "constructions-lemma-closed-in-projective-space", "text": "Let $R$ be a ring. Let $Z \\subset \\mathbf{P}^n_R$ be a closed subscheme. Let $$ I_d = \\Ker\\left( R[T_0, \\ldots, T_n]_d \\longrightarrow \\Gamma(Z, \\mathcal{O}_{\\mathbf{P}^n_R}(d)|_Z)\\right) $$ Then $I = \\bigoplus I_d \\subset R[T_0, \\ldots, T_n]$ is a graded ideal and $Z = \\text{Proj}(R[T_0, \\ldots, T_n]/I)$."}
+{"_id": "12626", "title": "constructions-lemma-quasi-coherent-projective-space", "text": "Let $R$ be a ring. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathbf{P}^n_R$. For $d \\geq 0$ set $$ M_d = \\Gamma(\\mathbf{P}^n_R, \\mathcal{F} \\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}} \\mathcal{O}_{\\mathbf{P}^n_R}(d)) = \\Gamma(\\mathbf{P}^n_R, \\mathcal{F}(d)) $$ Then $M = \\bigoplus_{d \\geq 0} M_d$ is a graded $R[T_0, \\ldots, R_n]$-module and there is a canonical isomorphism $\\mathcal{F} = \\widetilde{M}$."}
+{"_id": "12627", "title": "constructions-lemma-globally-generated-omega-twist-1", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf and let $s_0, \\ldots, s_n$ be global sections of $\\mathcal{L}$ which generate it. Let $\\mathcal{F}$ be the kernel of the induced map $\\mathcal{O}_X^{\\oplus n + 1} \\to \\mathcal{L}$. Then $\\mathcal{F} \\otimes \\mathcal{L}$ is globally generated."}
+{"_id": "12628", "title": "constructions-lemma-invertible-map-into-proj", "text": "Let $A$ be a graded ring. Set $X = \\text{Proj}(A)$. Let $T$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_T$-module. Let $\\psi : A \\to \\Gamma_*(T, \\mathcal{L})$ be a homomorphism of graded rings. Set $$ U(\\psi) = \\bigcup\\nolimits_{f \\in A_{+}\\text{ homogeneous}} T_{\\psi(f)} $$ The morphism $\\psi$ induces a canonical morphism of schemes $$ r_{\\mathcal{L}, \\psi} : U(\\psi) \\longrightarrow X $$ together with a map of $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras $$ \\theta : r_{\\mathcal{L}, \\psi}^*\\left( \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}. $$ The triple $(U(\\psi), r_{\\mathcal{L}, \\psi}, \\theta)$ is characterized by the following properties: \\begin{enumerate} \\item For $f \\in A_{+}$ homogeneous we have $r_{\\mathcal{L}, \\psi}^{-1}(D_{+}(f)) = T_{\\psi(f)}$. \\item For every $d \\geq 0$ the diagram $$ \\xymatrix{ A_d \\ar[d]_{(\\ref{equation-global-sections})} \\ar[r]_{\\psi} & \\Gamma(T, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\ \\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^{\\theta} & \\Gamma(U(\\psi), \\mathcal{L}^{\\otimes d}) } $$ is commutative. \\end{enumerate} Moreover, for any $d \\geq 1$ and any open subscheme $V \\subset T$ such that the sections in $\\psi(A_d)$ generate $\\mathcal{L}^{\\otimes d}|_V$ the morphism $r_{\\mathcal{L}, \\psi}|_V$ agrees with the morphism $\\varphi : V \\to \\text{Proj}(A)$ and the map $\\theta|_V$ agrees with the map $\\alpha : \\varphi^*\\mathcal{O}_X(d) \\to \\mathcal{L}^{\\otimes d}|_V$ where $(\\varphi, \\alpha)$ is the pair of Lemma \\ref{lemma-converse-construction} associated to $\\psi|_{A^{(d)}} : A^{(d)} \\to \\Gamma_*(V, \\mathcal{L}^{\\otimes d})$."}
+{"_id": "12629", "title": "constructions-lemma-proj-inclusion", "text": "In Situation \\ref{situation-relative-proj}. Suppose $U \\subset U' \\subset S$ are affine opens. Let $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$. The map of graded rings $A' \\to A$ induces a morphism $r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and the diagram $$ \\xymatrix{ \\text{Proj}(A) \\ar[r] \\ar[d] & \\text{Proj}(A') \\ar[d] \\\\ U \\ar[r] & U' } $$ is cartesian. Moreover there are canonical isomorphisms $\\theta : r^*\\mathcal{O}_{\\text{Proj}(A')}(n) \\to \\mathcal{O}_{\\text{Proj}(A)}(n)$ compatible with multiplication maps."}
+{"_id": "12630", "title": "constructions-lemma-transitive-proj", "text": "In Situation \\ref{situation-relative-proj}. Suppose $U \\subset U' \\subset U'' \\subset S$ are affine opens. Let $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$. The composition of the morphisms $r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and $r' : \\text{Proj}(A') \\to \\text{Proj}(A'')$ of Lemma \\ref{lemma-proj-inclusion} gives the morphism $r'' : \\text{Proj}(A) \\to \\text{Proj}(A'')$ of Lemma \\ref{lemma-proj-inclusion}. A similar statement holds for the isomorphisms $\\theta$."}
+{"_id": "12631", "title": "constructions-lemma-glue-relative-proj", "text": "In Situation \\ref{situation-relative-proj}. There exists a morphism of schemes $$ \\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\longrightarrow S $$ with the following properties: \\begin{enumerate} \\item for every affine open $U \\subset S$ there exists an isomorphism $i_U : \\pi^{-1}(U) \\to \\text{Proj}(A)$ with $A = \\mathcal{A}(U)$, and \\item for $U \\subset U' \\subset S$ affine open the composition $$ \\xymatrix{ \\text{Proj}(A) \\ar[r]^{i_U^{-1}} & \\pi^{-1}(U) \\ar[rr]^{inclusion} & & \\pi^{-1}(U') \\ar[r]^{i_{U'}} & \\text{Proj}(A') } $$ with $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ is the open immersion of Lemma \\ref{lemma-proj-inclusion} above. \\end{enumerate}"}
+{"_id": "12632", "title": "constructions-lemma-glue-relative-proj-twists", "text": "In Situation \\ref{situation-relative-proj}. The morphism $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ of Lemma \\ref{lemma-glue-relative-proj} comes with the following additional structure. There exists a quasi-coherent $\\mathbf{Z}$-graded sheaf of $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-algebras $\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)$, and a morphism of graded $\\mathcal{O}_S$-algebras $$ \\psi : \\mathcal{A} \\longrightarrow \\bigoplus\\nolimits_{n \\geq 0} \\pi_*\\left(\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)\\right) $$ uniquely determined by the following property: For every affine open $U \\subset S$ with $A = \\mathcal{A}(U)$ there is an isomorphism $$ \\theta_U : i_U^*\\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\text{Proj}(A)}(n) \\right) \\longrightarrow \\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n) \\right)|_{\\pi^{-1}(U)} $$ of $\\mathbf{Z}$-graded $\\mathcal{O}_{\\pi^{-1}(U)}$-algebras such that $$ \\xymatrix{ A_n \\ar[rr]_\\psi \\ar[dr]_-{(\\ref{equation-global-sections})} & & \\Gamma(\\pi^{-1}(U), \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)) \\\\ & \\Gamma(\\text{Proj}(A), \\mathcal{O}_{\\text{Proj}(A)}(n)) \\ar[ru]_-{\\theta_U} & } $$ is commutative."}
+{"_id": "12633", "title": "constructions-lemma-proj-base-change", "text": "In Situation \\ref{situation-relative-proj}. Let $d \\geq 1$. Let $F_d$ be the functor associated to $(S, \\mathcal{A})$ above. Let $g : S' \\to S$ be a morphism of schemes. Set $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F_d'$ be the functor associated to $(S', \\mathcal{A}')$ above. Then there is a canonical isomorphism $$ F'_d \\cong h_{S'} \\times_{h_S} F_d $$ of functors."}
+{"_id": "12634", "title": "constructions-lemma-relative-proj-affine", "text": "In Situation \\ref{situation-relative-proj}. Let $F_d$ be the functor associated to $(d, S, \\mathcal{A})$ above. If $S$ is affine, then $F_d$ is representable by the open subscheme $U_d$ (\\ref{equation-Ud}) of the scheme $\\text{Proj}(\\Gamma(S, \\mathcal{A}))$."}
+{"_id": "12636", "title": "constructions-lemma-equivalent-relative", "text": "In Situation \\ref{situation-relative-proj}. Let $T$ be a scheme. Let $(d, f, \\mathcal{L}, \\psi)$, $(d', f', \\mathcal{L}', \\psi')$ be two quadruples over $T$. The following are equivalent: \\begin{enumerate} \\item Let $m = \\text{lcm}(d, d')$. Write $m = ad = a'd'$. We have $f = f'$ and there exists an isomorphism $\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$ with the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$ and $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree as graded ring maps $f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$. \\item The quadruples $(d, f, \\mathcal{L}, \\psi)$ and $(d', f', \\mathcal{L}', \\psi')$ are equivalent. \\item We have $f = f'$ and for some positive integer $m = ad = a'd'$ there exists an isomorphism $\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$ with the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$ and $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree as graded ring maps $f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$. \\end{enumerate}"}
+{"_id": "12637", "title": "constructions-lemma-relative-proj", "text": "In Situation \\ref{situation-relative-proj}. The functor $F$ above is representable by a scheme."}
+{"_id": "12638", "title": "constructions-lemma-glueing-gives-functor-proj", "text": "In Situation \\ref{situation-relative-proj}. The scheme $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ constructed in Lemma \\ref{lemma-glue-relative-proj} and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$."}
+{"_id": "12640", "title": "constructions-lemma-relative-proj-separated", "text": "Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-algebras. The morphism $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ is separated."}
+{"_id": "12641", "title": "constructions-lemma-relative-proj-base-change", "text": "Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-algebras. Let $g : S' \\to S$ be any morphism of schemes. Then there is a canonical isomorphism $$ r : \\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A}) \\longrightarrow S' \\times_S \\underline{\\text{Proj}}_S(\\mathcal{A}) $$ as well as a corresponding isomorphism $$ \\theta : r^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)\\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}(d) $$ of $\\mathbf{Z}$-graded $\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}$-algebras."}
+{"_id": "12642", "title": "constructions-lemma-apply-relative", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-modules generated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$. In this case the scheme $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$ represents the functor $F_1$ which associates to a scheme $f : T \\to S$ over $S$ the set of pairs $(\\mathcal{L}, \\psi)$, where \\begin{enumerate} \\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and \\item $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$ is a graded $\\mathcal{O}_T$-algebra homomorphism such that $f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective \\end{enumerate} up to strict equivalence as above. Moreover, in this case all the quasi-coherent sheaves $\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)$ are invertible $\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}$-modules and the multiplication maps induce isomorphisms $ \\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n) \\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}} \\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(m) = \\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n + m)$."}
+{"_id": "12643", "title": "constructions-lemma-relative-proj-modules", "text": "In Situation \\ref{situation-relative-proj}. For any quasi-coherent sheaf of graded $\\mathcal{A}$-modules $\\mathcal{M}$ on $S$, there exists a canonical associated sheaf of $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-modules $\\widetilde{\\mathcal{M}}$ with the following properties: \\begin{enumerate} \\item Given a scheme $T$ and a quadruple $(T \\to S, d, \\mathcal{L}, \\psi)$ over $T$ corresponding to a morphism $h : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ there is a canonical isomorphism $\\widetilde{\\mathcal{M}}_T = h^*\\widetilde{\\mathcal{M}}$ where $\\widetilde{\\mathcal{M}}_T$ is defined by (\\ref{equation-widetilde-M}). \\item The isomorphisms of (1) are compatible with pullbacks. \\item There is a canonical map $$ \\pi^*\\mathcal{M}_0 \\longrightarrow \\widetilde{\\mathcal{M}}. $$ \\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$ is functorial in $\\mathcal{M}$. \\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$ is exact. \\item There are canonical maps $$ \\widetilde{\\mathcal{M}} \\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}} \\widetilde{\\mathcal{N}} \\longrightarrow \\widetilde{\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}} $$ as in Lemma \\ref{lemma-widetilde-tensor}. \\item There exist canonical maps $$ \\pi^*\\mathcal{M} \\longrightarrow \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\widetilde{\\mathcal{M}(n)} $$ generalizing (\\ref{equation-global-sections-more-generally}). \\item The formation of $\\widetilde{\\mathcal{M}}$ commutes with base change. \\end{enumerate}"}
+{"_id": "12644", "title": "constructions-lemma-morphism-relative-proj", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$ be two graded quasi-coherent $\\mathcal{O}_S$-algebras. Set $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ and $q : Y = \\underline{\\text{Proj}}_S(\\mathcal{B}) \\to S$. Let $\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of graded $\\mathcal{O}_S$-algebras. There is a canonical open $U(\\psi) \\subset Y$ and a canonical morphism of schemes $$ r_\\psi : U(\\psi) \\longrightarrow X $$ over $S$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras $$ \\theta = \\theta_\\psi : r_\\psi^*\\left( \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d). $$ The triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property that for any affine open $W \\subset S$ the triple $$ (U(\\psi) \\cap p^{-1}W,\\quad r_\\psi|_{U(\\psi) \\cap p^{-1}W} : U(\\psi) \\cap p^{-1}W \\to q^{-1}W,\\quad \\theta|_{U(\\psi) \\cap p^{-1}W}) $$ is equal to the triple associated to $\\psi : \\mathcal{A}(W) \\to \\mathcal{B}(W)$ in Lemma \\ref{lemma-morphism-proj} via the identifications $p^{-1}W = \\text{Proj}(\\mathcal{A}(W))$ and $q^{-1}W = \\text{Proj}(\\mathcal{B}(W))$ of Section \\ref{section-relative-proj-via-glueing}."}
+{"_id": "12645", "title": "constructions-lemma-morphism-relative-proj-transitive", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be quasi-coherent graded $\\mathcal{O}_S$-algebras. Set $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$, $Y = \\underline{\\text{Proj}}_S(\\mathcal{B})$ and $Z = \\underline{\\text{Proj}}_S(\\mathcal{C})$. Let $\\varphi : \\mathcal{A} \\to \\mathcal{B}$, $\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_S$-algebra maps. Then we have $$ U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi)) \\quad \\text{and} \\quad r_{\\psi \\circ \\varphi} = r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}. $$ In addition we have $$ \\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi = \\theta_{\\psi \\circ \\varphi} $$ with obvious notation."}
+{"_id": "12646", "title": "constructions-lemma-surjective-graded-rings-map-relative-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj} above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are surjective but not isomorphisms in general (even if $\\mathcal{A} \\to \\mathcal{B}$ is surjective). \\end{enumerate}"}
+{"_id": "12647", "title": "constructions-lemma-eventual-iso-graded-rings-map-relative-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj} above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is an isomorphism for all $d \\gg 0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is an isomorphism, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12648", "title": "constructions-lemma-surjective-generated-degree-1-map-relative-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj} above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$ and that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$. Then \\begin{enumerate} \\item $U(\\psi) = Y$, \\item $r_\\psi : Y \\to X$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12649", "title": "constructions-lemma-invertible-map-into-relative-proj", "text": "With assumptions and notation as above. The morphism $\\psi$ induces a canonical morphism of schemes over $S$ $$ r_{\\mathcal{L}, \\psi} : U(\\psi) \\longrightarrow \\underline{\\text{Proj}}_S(\\mathcal{A}) $$ together with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras $$ \\theta : r_{\\mathcal{L}, \\psi}^*\\left( \\bigoplus\\nolimits_{d \\geq 0} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)} $$ characterized by the following properties: \\begin{enumerate} \\item For every open $V \\subset S$ and every $d \\geq 0$ the diagram $$ \\xymatrix{ \\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} & \\Gamma(f^{-1}(V), \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\ \\Gamma(\\pi^{-1}(V), \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)) \\ar[r]^{\\theta} & \\Gamma(f^{-1}(V) \\cap U(\\psi), \\mathcal{L}^{\\otimes d}) } $$ is commutative. \\item For any $d \\geq 1$ and any open subscheme $W \\subset X$ such that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$ is surjective the restriction of the morphism $r_{\\mathcal{L}, \\psi}$ agrees with the morphism $W \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ which exists by the construction of the relative homogeneous spectrum, see Definition \\ref{definition-relative-proj}. \\item For any affine open $V \\subset S$, the restriction $$ (U(\\psi) \\cap f^{-1}(V), r_{\\mathcal{L}, \\psi}|_{U(\\psi) \\cap f^{-1}(V)}, \\theta|_{U(\\psi) \\cap f^{-1}(V)}) $$ agrees via $i_V$ (see Lemma \\ref{lemma-glue-relative-proj}) with the triple $(U(\\psi'), r_{\\mathcal{L}, \\psi'}, \\theta')$ of Lemma \\ref{lemma-invertible-map-into-proj} associated to the map $\\psi' : A = \\mathcal{A}(V) \\to \\Gamma_*(f^{-1}(V), \\mathcal{L}|_{f^{-1}(V)})$ induced by $\\psi$. \\end{enumerate}"}
+{"_id": "12650", "title": "constructions-lemma-twisting-and-proj", "text": "With notation $S$, $\\mathcal{A}$, $\\mathcal{L}$ and $\\mathcal{B}$ as above. There is a canonical isomorphism $$ \\xymatrix{ P = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\ar[rr]_g \\ar[rd]_\\pi & & \\underline{\\text{Proj}}_S(\\mathcal{B}) = P' \\ar[ld]^{\\pi'} \\\\ & S & } $$ with the following properties \\begin{enumerate} \\item There are isomorphisms $\\theta_n : g^*\\mathcal{O}_{P'}(n) \\to \\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n}$ which fit together to give an isomorphism of $\\mathbf{Z}$-graded algebras $$ \\theta : g^*\\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{P'}(n) \\right) \\longrightarrow \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n} $$ \\item For every open $V \\subset S$ the diagrams $$ \\xymatrix{ \\mathcal{A}_n(V) \\otimes \\mathcal{L}^{\\otimes n}(V) \\ar[r]_{multiply} \\ar[d]^{\\psi \\otimes \\pi^*} & \\mathcal{B}_n(V) \\ar[dd]^\\psi \\\\ \\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n)) \\otimes \\Gamma(\\pi^{-1}V, \\pi^*\\mathcal{L}^{\\otimes n}) \\ar[d]^{multiply} \\\\ \\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n}) & \\Gamma(\\pi'^{-1}V, \\mathcal{O}_{P'}(n)) \\ar[l]_-{\\theta_n} } $$ are commutative. \\item Add more here as necessary. \\end{enumerate}"}
+{"_id": "12651", "title": "constructions-lemma-projective-bundle-separated", "text": "Let $S$ be a scheme. The structure morphism $\\mathbf{P}(\\mathcal{E}) \\to S$ of a projective bundle over $S$ is separated."}
+{"_id": "12652", "title": "constructions-lemma-projective-space-bundle", "text": "Let $S$ be a scheme. Let $n \\geq 0$. Then $\\mathbf{P}^n_S$ is a projective bundle over $S$."}
+{"_id": "12673", "title": "algebraization-theorem-final-bootstrap", "text": "In Situation \\ref{situation-bootstrap} the inverse system $\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$ satisfies the Mittag-Leffler condition for $i \\leq s$, the map $$ H^i_T(M) \\longrightarrow \\lim H^i_T(M/I^nM) $$ is an isomorphism for $i \\leq s$, and $H^i_T(M)$ is annihilated by a power of $I$ for $i \\leq s$."}
+{"_id": "12674", "title": "algebraization-theorem-algebraization-formal-sections", "text": "\\begin{reference} The method of proof follows roughly the method of proof of \\cite[Theorem 1]{Faltings-algebraisation} and \\cite[Satz 2]{Faltings-uber}. The result is almost the same as \\cite[Theorem 1.1]{MRaynaud-paper} (affine complement case) and \\cite[Theorem 3.9]{MRaynaud-book} (complement is union of few affines). \\end{reference} Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$. Let $\\mathcal{F}$ be a coherent sheaf on $U$. Assume \\begin{enumerate} \\item $\\text{cd}(A, I) \\leq d$, i.e., $H^i(X \\setminus Y, \\mathcal{G}) = 0$ for $i \\geq d$ and quasi-coherent $\\mathcal{G}$ on $X$, \\item for any $x \\in X \\setminus Y$ whose closure $\\overline{\\{x\\}}$ in $X$ meets $U \\cap Y$ we have $$ \\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) \\geq s \\quad\\text{or}\\quad \\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\dim(\\overline{\\{x\\}}) > d + s $$ \\end{enumerate} Then there exists an open $V_0 \\subset U$ containing $U \\cap Y$ such that for any open $V \\subset V_0$ containing $U \\cap Y$ the map $$ H^i(V, \\mathcal{F}) \\to \\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F}) $$ is an isomorphism for $i < s$. If in addition $ \\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\dim(\\overline{\\{x\\}}) > s $ for all $x \\in U \\cap Y$, then these cohomology groups are finite $A$-modules."}
+{"_id": "12675", "title": "algebraization-lemma-ML-general", "text": "Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. Assume $$ \\bigoplus\\nolimits_{n \\geq 0} H^1(X, I^n\\mathcal{F}_{n + 1}) $$ satisfies the ascending chain condition as a graded $\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module. Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition."}
+{"_id": "12676", "title": "algebraization-lemma-ML-general-better", "text": "Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. Given $n$ define $$ H^1_n = \\bigcap\\nolimits_{m \\geq n} \\Im\\left( H^1(X, I^n\\mathcal{F}_{m + 1}) \\to H^1(X, I^n\\mathcal{F}_{n + 1}) \\right) $$ If $\\bigoplus H^1_n$ satisfies the ascending chain condition as a graded $\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition."}
+{"_id": "12677", "title": "algebraization-lemma-topology-I-adic-general", "text": "Let $I$ be a finitely generated ideal of a ring $A$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. Assume $$ \\bigoplus\\nolimits_{n \\geq 0} H^0(X, I^n\\mathcal{F}_{n + 1}) $$ satisfies the ascending chain condition as a graded $\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module. Then the limit topology on $M = \\lim \\Gamma(X, \\mathcal{F}_n)$ is the $I$-adic topology."}
+{"_id": "12678", "title": "algebraization-lemma-properties-system", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of quasi-coherent $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/\\mathcal{I}^n\\mathcal{F}_{n + 1}$. Set $\\mathcal{F} = \\lim \\mathcal{F}_n$. Then \\begin{enumerate} \\item $\\mathcal{F} = R\\lim \\mathcal{F}_n$, \\item for any affine open $U \\subset X$ we have $H^p(U, \\mathcal{F}) = 0$ for $p > 0$, and \\item for each $p$ there is a short exact sequence $0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}_n) \\to H^p(X, \\mathcal{F}) \\to \\lim H^p(X, \\mathcal{F}_n) \\to 0$. \\end{enumerate} If moreover $\\mathcal{I}$ is of finite type, then \\begin{enumerate} \\item[(4)] $\\mathcal{F}_n = \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$, and \\item[(5)] $\\mathcal{I}^n \\mathcal{F} = \\lim_{m \\geq n} \\mathcal{I}^n\\mathcal{F}_m$. \\end{enumerate}"}
+{"_id": "12679", "title": "algebraization-lemma-equivalent-f-good", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be inverse system of $\\mathcal{O}_X$-modules. The following are equivalent \\begin{enumerate} \\item for all $n \\geq 1$ the map $f : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$ factors through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ to give a short exact sequence $0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$, \\item for all $n \\geq 1$ the map $f^n : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$ factors through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_1$ to give a short exact sequence $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_n \\to 0$ \\item there exists an $\\mathcal{O}_X$-module $\\mathcal{G}$ which is $f$-divisible such that $\\mathcal{F}_n = \\mathcal{G}[f^n]$. \\end{enumerate} If $X$ is a scheme and $\\mathcal{F}_n$ is quasi-coherent, then these are also equivalent to \\begin{enumerate} \\item[(4)] there exists an $\\mathcal{O}_X$-module $\\mathcal{F}$ which is $f$-torsion free such that $\\mathcal{F}_n = \\mathcal{F}/f^n\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "12680", "title": "algebraization-lemma-topology-I-adic-f", "text": "Suppose $X$, $f$, $(\\mathcal{F}_n)$ is as in Lemma \\ref{lemma-equivalent-f-good}. Then the limit topology on $H^p = \\lim H^p(X, \\mathcal{F}_n)$ is the $f$-adic topology."}
+{"_id": "12681", "title": "algebraization-lemma-limit-finite", "text": "Let $A$ be a Noetherian ring complete with respect to a principal ideal $(f)$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item $\\Gamma(X, \\mathcal{F}_1)$ is a finite $A$-module, \\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold. \\end{enumerate} Then $$ M = \\lim \\Gamma(X, \\mathcal{F}_n) $$ is a finite $A$-module, $f$ is a nonzerodivisor on $M$, and $M/fM$ is the image of $M$ in $\\Gamma(X, \\mathcal{F}_1)$."}
+{"_id": "12682", "title": "algebraization-lemma-ML", "text": "Let $A$ be a ring. Let $f \\in A$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item either $H^1(X, \\mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and $H^1(X, \\mathcal{F}_1)$ is a finite $A$-module, \\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold. \\end{enumerate} Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition."}
+{"_id": "12683", "title": "algebraization-lemma-ML-better", "text": "Let $A$ be a ring. Let $f \\in A$. Let $X$ be a scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of $\\mathcal{O}_X$-modules. Assume \\begin{enumerate} \\item either there is an $m \\geq 1$ such that the image of $H^1(X, \\mathcal{F}_m) \\to H^1(X, \\mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and the intersection of the images of $H^1(X, \\mathcal{F}_m) \\to H^1(X, \\mathcal{F}_1)$ is a finite $A$-module, \\item the equivalent conditions of Lemma \\ref{lemma-equivalent-f-good} hold. \\end{enumerate} Then the inverse system $M_n = \\Gamma(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition."}
+{"_id": "12684", "title": "algebraization-lemma-formal-functions-principal", "text": "\\begin{reference} \\cite[Lemma 1.6]{Bhatt-local} \\end{reference} Let $A$ be a ring and $f \\in A$. Let $X$ be a scheme over $A$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume that $\\mathcal{F}[f^n] = \\Ker(f^n : \\mathcal{F} \\to \\mathcal{F})$ stabilizes. Then $$ R\\Gamma(X, \\lim \\mathcal{F}/f^n\\mathcal{F}) = R\\Gamma(X, \\mathcal{F})^\\wedge $$ where the right hand side indicates the derived completion with respect to the ideal $(f) \\subset A$. Let $H^p$ be the $p$th cohomology group of this complex. Then there are short exact sequences $$ 0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}/f^n\\mathcal{F}) \\to H^p \\to \\lim H^p(X, \\mathcal{F}/f^n\\mathcal{F}) \\to 0 $$ and $$ 0 \\to H^0(H^p(X, \\mathcal{F})^\\wedge) \\to H^p \\to T_f(H^{p + 1}(X, \\mathcal{F})) \\to 0 $$ where $T_f(-)$ denote the $f$-adic Tate module as in More on Algebra, Example \\ref{more-algebra-example-spectral-sequence-principal}."}
+{"_id": "12685", "title": "algebraization-lemma-cd-one", "text": "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$. If $\\text{cd}(A, I) = 1$, then there exist $c \\geq 1$ and maps $\\varphi_j : I^c \\to A$ such that $\\sum f_j \\varphi_j : I^c \\to I$ is the inclusion map."}
+{"_id": "12686", "title": "algebraization-lemma-cd-one-extend", "text": "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$, $j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}. Then there is a unique graded $A$-algebra map $$ \\Phi : \\bigoplus\\nolimits_{n \\geq 0} I^{nc} \\to A[T_1, \\ldots, T_r] $$ with $\\Phi(g) = \\sum \\varphi_j(g) T_j$ for $g \\in I^c$. Moreover, the composition of $\\Phi$ with the map $A[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n$, $T_j \\mapsto f_j$ is the inclusion map $\\bigoplus_{n \\geq 0} I^{nc} \\to \\bigoplus_{n \\geq 0} I^n$."}
+{"_id": "12687", "title": "algebraization-lemma-cd-one-extend-to-module", "text": "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$, $j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}. Let $A \\to B$ be a ring map with $B$ Noetherian and let $N$ be a finite $B$-module. Then, after possibly increasing $c$ and adjusting $\\varphi_j$ accordingly, there is a unique unique graded $B$-module map $$ \\Phi_N : \\bigoplus\\nolimits_{n \\geq 0} I^{nc}N \\to N[T_1, \\ldots, T_r] $$ with $\\Phi_N(g x) = \\Phi(g) x$ for $g \\in I^{nc}$ and $x \\in N$ where $\\Phi$ is as in Lemma \\ref{lemma-cd-one-extend}. The composition of $\\Phi_N$ with the map $N[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^nN$, $T_j \\mapsto f_j$ is the inclusion map $\\bigoplus_{n \\geq 0} I^{nc}N \\to \\bigoplus_{n \\geq 0} I^nN$."}
+{"_id": "12688", "title": "algebraization-lemma-cd-is-one-for-system", "text": "Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$ with $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$, $j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}. Let $X$ be a Noetherian scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of coherent $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. Set $\\mathcal{F} = \\lim \\mathcal{F}_n$. Then, after possibly increasing $c$ and adjusting $\\varphi_j$ accordingly, there exists a unique graded $\\mathcal{O}_X$-module map $$ \\Phi_\\mathcal{F} : \\bigoplus\\nolimits_{n \\geq 0} I^{nc}\\mathcal{F} \\longrightarrow \\mathcal{F}[T_1, \\ldots, T_r] $$ with $\\Phi_\\mathcal{F}(g s) = \\Phi(g) s$ for $g \\in I^{nc}$ and $s$ a local section of $\\mathcal{F}$ where $\\Phi$ is as in Lemma \\ref{lemma-cd-one-extend}. The composition of $\\Phi_\\mathcal{F}$ with the map $\\mathcal{F}[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n\\mathcal{F}$, $T_j \\mapsto f_j$ is the canonical inclusion $\\bigoplus_{n \\geq 0} I^{nc}\\mathcal{F} \\to \\bigoplus_{n \\geq 0} I^n\\mathcal{F}$."}
+{"_id": "12689", "title": "algebraization-lemma-topology-I-adic", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian scheme over $\\Spec(A)$. Let $$ \\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 $$ be an inverse system of coherent $\\mathcal{O}_X$-modules such that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$. If $\\text{cd}(A, I) = 1$, then for all $p \\in \\mathbf{Z}$ the limit topology on $\\lim H^p(X, \\mathcal{F}_n)$ is $I$-adic."}
+{"_id": "12690", "title": "algebraization-lemma-descending-chain", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. \\begin{enumerate} \\item Let $M$ be a finite $A$-module. Then the $A$-module $H^i_\\mathfrak m(M)$ satisfies the descending chain condition for any $i$. \\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the punctured spectrum of $A$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Then the $A$-module $H^i(U, \\mathcal{F})$ satisfies the descending chain condition for $i > 0$. \\end{enumerate}"}
+{"_id": "12691", "title": "algebraization-lemma-ML-local", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. \\begin{enumerate} \\item Let $(M_n)$ be an inverse system of finite $A$-modules. Then the inverse system $H^i_\\mathfrak m(M_n)$ satisfies the Mittag-Leffler condition for any $i$. \\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the punctured spectrum of $A$. Let $\\mathcal{F}_n$ be an inverse system of coherent $\\mathcal{O}_U$-modules. Then the inverse system $H^i(U, \\mathcal{F}_n)$ satisfies the Mittag-Leffler condition for $i > 0$. \\end{enumerate}"}
+{"_id": "12692", "title": "algebraization-lemma-terrific", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $(M_n)$ be an inverse system of finite $A$-modules. Let $M \\to \\lim M_n$ be a map where $M$ is a finite $A$-module such that for some $i$ the map $H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M_n)$ is an isomorphism. Then the inverse system $H^i_\\mathfrak m(M_n)$ is essentially constant with value $H^i_\\mathfrak m(M)$."}
+{"_id": "12693", "title": "algebraization-lemma-local-cohomology-derived-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Then $$ H^i(R\\Gamma_\\mathfrak m(M)^\\wedge) = \\lim H^i_\\mathfrak m(M/I^nM) $$ for all $i$ where $R\\Gamma_\\mathfrak m(M)^\\wedge$ denotes the derived $I$-adic completion."}
+{"_id": "12694", "title": "algebraization-lemma-map-twice-localize", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global section of $\\mathcal{O}$. \\begin{enumerate} \\item For $L, N \\in D(\\mathcal{O}_f)$ we have $R\\SheafHom_\\mathcal{O}(L, N) = R\\SheafHom_{\\mathcal{O}_f}(L, N)$. In particular the two $\\mathcal{O}_f$-structures on $R\\SheafHom_\\mathcal{O}(L, N)$ agree. \\item For $K \\in D(\\mathcal{O})$ and $L \\in D(\\mathcal{O}_f)$ we have $$ R\\SheafHom_\\mathcal{O}(L, K) = R\\SheafHom_{\\mathcal{O}_f}(L, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)) $$ In particular $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)) = R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$. \\item If $g$ is a second global section of $\\mathcal{O}$, then $$ R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_g, K)) = R\\SheafHom_\\mathcal{O}(\\mathcal{O}_{gf}, K). $$ \\end{enumerate}"}
+{"_id": "12695", "title": "algebraization-lemma-hom-from-Af", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global section of $\\mathcal{O}$. Let $K \\in D(\\mathcal{O})$. The following are equivalent \\begin{enumerate} \\item $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K) = 0$, \\item $R\\SheafHom_\\mathcal{O}(L, K) = 0$ for all $L$ in $D(\\mathcal{O}_f)$, \\item $T(K, f) = 0$. \\end{enumerate}"}
+{"_id": "12696", "title": "algebraization-lemma-ideal-of-elements-complete-wrt", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K \\in D(\\mathcal{O})$. The rule which associates to $U$ the set $\\mathcal{I}(U)$ of sections $f \\in \\mathcal{O}(U)$ such that $T(K|_U, f) = 0$ is a sheaf of ideals in $\\mathcal{O}$."}
+{"_id": "12697", "title": "algebraization-lemma-derived-complete-internal-hom", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals. If $K \\in D(\\mathcal{O})$ and $L \\in D_{comp}(\\mathcal{O})$, then $R\\SheafHom_\\mathcal{O}(K, L) \\in D_{comp}(\\mathcal{O})$."}
+{"_id": "12699", "title": "algebraization-lemma-pushforward-derived-complete", "text": "Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$ and $\\mathcal{I}' \\subset \\mathcal{O}'$ be sheaves of ideals such that $f^\\sharp$ sends $f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$ into $D_{comp}(\\mathcal{O}, \\mathcal{I})$."}
+{"_id": "12700", "title": "algebraization-lemma-derived-completion", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$ be global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be the ideal sheaf generated by $f_1, \\ldots, f_r$. Then the inclusion functor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$ has a left adjoint, i.e., given any object $K$ of $D(\\mathcal{O})$ there exists a map $K \\to K^\\wedge$ with $K^\\wedge$ in $D_{comp}(\\mathcal{O})$ such that the map $$ \\Hom_{D(\\mathcal{O})}(K^\\wedge, E) \\longrightarrow \\Hom_{D(\\mathcal{O})}(K, E) $$ is bijective whenever $E$ is in $D_{comp}(\\mathcal{O})$. In fact we have $$ K^\\wedge = R\\SheafHom_\\mathcal{O} (\\mathcal{O} \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to \\ldots \\to \\mathcal{O}_{f_1\\ldots f_r}, K) $$ functorially in $K$."}
+{"_id": "12701", "title": "algebraization-lemma-derived-completion-koszul", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$ be global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be the ideal sheaf generated by $f_1, \\ldots, f_r$. Let $K \\in D(\\mathcal{O})$. The derived completion $K^\\wedge$ of Lemma \\ref{lemma-derived-completion} is given by the formula $$ K^\\wedge = R\\lim K \\otimes^\\mathbf{L}_\\mathcal{O} K_n $$ where $K_n = K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)$ is the Koszul complex on $f_1^n, \\ldots, f_r^n$ over $\\mathcal{O}$."}
+{"_id": "12702", "title": "algebraization-lemma-all-rings", "text": "There exist a way to construct \\begin{enumerate} \\item for every pair $(A, I)$ consisting of a ring $A$ and a finitely generated ideal $I \\subset A$ a complex $K(A, I)$ of $A$-modules, \\item a map $K(A, I) \\to A$ of complexes of $A$-modules, \\item for every ring map $A \\to B$ and finitely generated ideal $I \\subset A$ a map of complexes $K(A, I) \\to K(B, IB)$, \\end{enumerate} such that \\begin{enumerate} \\item[(a)] for $A \\to B$ and $I \\subset A$ finitely generated the diagram $$ \\xymatrix{ K(A, I) \\ar[r] \\ar[d] & A \\ar[d] \\\\ K(B, IB) \\ar[r] & B } $$ commutes, \\item[(b)] for $A \\to B \\to C$ and $I \\subset A$ finitely generated the composition of the maps $K(A, I) \\to K(B, IB) \\to K(C, IC)$ is the map $K(A, I) \\to K(C, IC)$. \\item[(c)] for $A \\to B$ and a finitely generated ideal $I \\subset A$ the induced map $K(A, I) \\otimes_A^\\mathbf{L} B \\to K(B, IB)$ is an isomorphism in $D(B)$, and \\item[(d)] if $I = (f_1, \\ldots, f_r) \\subset A$ then there is a commutative diagram $$ \\xymatrix{ (A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to \\ldots \\to A_{f_1\\ldots f_r}) \\ar[r] \\ar[d] &  K(A, I) \\ar[d] \\\\ A \\ar[r]^1 & A } $$ in $D(A)$ whose horizontal arrows are isomorphisms. \\end{enumerate}"}
+{"_id": "12703", "title": "algebraization-lemma-global-extended-cech-complex", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. There exists a map $K \\to \\mathcal{O}$ in $D(\\mathcal{O})$ such that for every $U \\in \\Ob(\\mathcal{C})$ such that $\\mathcal{I}|_U$ is generated by $f_1, \\ldots, f_r \\in \\mathcal{I}(U)$ there is an isomorphism $$ (\\mathcal{O}_U \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{U, f_{i_0}} \\to \\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{U, f_{i_0}f_{i_1}} \\to \\ldots \\to \\mathcal{O}_{U, f_1\\ldots f_r}) \\longrightarrow K|_U $$ compatible with maps to $\\mathcal{O}_U$."}
+{"_id": "12704", "title": "algebraization-lemma-map-identifies-koszul-and-cech-complexes", "text": "Let $\\mathcal{C}$ be a site. Assume $\\varphi : \\mathcal{O} \\to \\mathcal{O}'$ is a flat homomorphism of sheaves of rings. Let $f_1, \\ldots, f_r$ be global sections of $\\mathcal{O}$ such that $\\mathcal{O}/(f_1, \\ldots, f_r) \\cong \\mathcal{O}'/(f_1, \\ldots, f_r)$. Then the map of extended alternating {\\v C}ech complexes $$ \\xymatrix{ \\mathcal{O} \\to \\prod_{i_0} \\mathcal{O}_{f_{i_0}} \\to \\prod_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to \\ldots \\to \\mathcal{O}_{f_1\\ldots f_r} \\ar[d] \\\\ \\mathcal{O}' \\to \\prod_{i_0} \\mathcal{O}'_{f_{i_0}} \\to \\prod_{i_0 < i_1} \\mathcal{O}'_{f_{i_0}f_{i_1}} \\to \\ldots \\to \\mathcal{O}'_{f_1\\ldots f_r} } $$ is a quasi-isomorphism."}
+{"_id": "12706", "title": "algebraization-lemma-pushforward-derived-complete-adjoint", "text": "Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$ and $\\mathcal{I}' \\subset \\mathcal{O}'$  be finite type sheaves of ideals such that $f^\\sharp$ sends $f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$. Then $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$ into $D_{comp}(\\mathcal{O}, \\mathcal{I})$ and has a left adjoint $Lf_{comp}^*$ which is $Lf^*$ followed by derived completion."}
+{"_id": "12707", "title": "algebraization-lemma-pushforward-commutes-with-derived-completion", "text": "\\begin{reference} Generalization of \\cite[Lemma 6.5.9 (2)]{BS}. Compare with \\cite[Theorem 6.5]{HL-P} in the setting of quasi-coherent modules and morphisms of (derived) algebraic stacks. \\end{reference} Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ be a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. Let $\\mathcal{I}' \\subset \\mathcal{O}'$ be the ideal generated by $f^\\sharp(f^{-1}\\mathcal{I})$. Then $Rf_*$ commutes with derived completion, i.e., $Rf_*(K^\\wedge) = (Rf_*K)^\\wedge$."}
+{"_id": "12708", "title": "algebraization-lemma-formal-functions-general", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $\\mathcal{C}$ be a site and let $\\mathcal{O}$ be a sheaf of $A$-algebras. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Then we have $$ R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge = R\\Gamma(\\mathcal{C}, \\mathcal{F}^\\wedge) $$ in $D(A)$ where $\\mathcal{F}^\\wedge$ is the derived completion of $\\mathcal{F}$ with respect to $I\\mathcal{O}$ and on the left hand wide we have the derived completion with respect to $I$. This produces two spectral sequences $$ E_2^{i, j} = H^i(H^j(\\mathcal{C}, \\mathcal{F})^\\wedge) \\quad\\text{and}\\quad E_2^{p, q} = H^p(\\mathcal{C}, H^q(\\mathcal{F}^\\wedge)) $$ both converging to $H^*(R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge) = H^*(\\mathcal{C}, \\mathcal{F}^\\wedge)$"}
+{"_id": "12709", "title": "algebraization-lemma-sections-derived-completion-pseudo-coherent", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $K$ be a pseudo-coherent object of $D(\\mathcal{O}_X)$ with derived completion $K^\\wedge$. Then $$ H^p(U, K^\\wedge) = \\lim H^p(U, K)/I^nH^p(U, K) = H^p(U, K)^\\wedge $$ for any affine open $U \\subset X$ where $I = \\mathcal{I}(U)$ and where on the right we have the derived completion with respect to $I$."}
+{"_id": "12710", "title": "algebraization-lemma-derived-completion-pseudo-coherent", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $K$ be an object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item the derived completion $K^\\wedge$ is equal to $R\\lim (K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}^n)$. \\end{enumerate} Let $K$ is a pseudo-coherent object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item[(2)] the cohomology sheaf $H^q(K^\\wedge)$ is equal to $\\lim H^q(K)/\\mathcal{I}^nH^q(K)$. \\end{enumerate} Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\\footnote{For example $H^q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.}. Then \\begin{enumerate} \\item[(3)] the derived completion $\\mathcal{F}^\\wedge$ is equal to $\\lim \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$, \\item[(4)] $\\lim \\mathcal{F}/I^n \\mathcal{F} = R\\lim \\mathcal{F}/I^n \\mathcal{F}$, \\item[(5)] $H^p(U, \\mathcal{F}^\\wedge) = 0$ for $p \\not = 0$ for all affine opens $U \\subset X$. \\end{enumerate}"}
+{"_id": "12712", "title": "algebraization-lemma-kill-completion-general", "text": "Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $\\mathfrak p \\subset A$ be a prime. Let $s$ and $d$ be integers. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $\\mathfrak p \\not \\in V(J) \\cap V(I)$, \\item $\\text{cd}(A, I) \\leq d$, and \\item for all primes $\\mathfrak p' \\subset \\mathfrak p$ we have $\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) + \\dim((A/\\mathfrak p')_\\mathfrak q) > d + s$ for all $\\mathfrak q \\in V(\\mathfrak p') \\cap V(J) \\cap V(I)$. \\end{enumerate} Then there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates $H^i(R\\Gamma_J(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$ indicates $I$-adic completion."}
+{"_id": "12713", "title": "algebraization-lemma-kill-colimit-weak-general", "text": "Let $I, J$ be ideals of a Noetherian ring. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. With $T$ as in (\\ref{equation-associated-subset}) assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item if $\\mathfrak p \\in V(I)$, then no condition, \\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then $\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq d$ for some $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$, \\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$, then $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s \\quad\\text{or}\\quad \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > d + s $$ for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$. \\end{enumerate} Then there exists an ideal $J_0 \\subset J$ with $V(J_0) \\cap V(I) = V(J) \\cap V(I)$ such that for any $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$ the map $$ R\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_{J_0}(M) $$ induces an isomorphism in cohomology in degrees $\\leq s$ and moreover these modules are annihilated by a power of $J_0I$."}
+{"_id": "12714", "title": "algebraization-lemma-kill-colimit-general", "text": "In Lemma \\ref{lemma-kill-colimit-weak-general} if instead of the empty condition (2) we assume \\begin{enumerate} \\item[(2')] if $\\mathfrak p \\in V(I)$, $\\mathfrak p \\not \\in V(J) \\cap V(I)$, then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > s$ for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$, \\end{enumerate} then the conditions also imply that $H^i_{J_0}(M)$ is a finite $A$-module for $i \\leq s$."}
+{"_id": "12715", "title": "algebraization-lemma-kill-colimit-support-general", "text": "If in Lemma \\ref{lemma-kill-colimit-weak-general} we additionally assume \\begin{enumerate} \\item[(6)] if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$, \\end{enumerate} then $H^i_{J_0}(M) = H^i_J(M) = H^i_{J + I}(M)$ for $i \\leq s$ and these modules are annihilated by a power of $I$."}
+{"_id": "12716", "title": "algebraization-lemma-algebraize-local-cohomology-general", "text": "Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. With $T$ as in (\\ref{equation-associated-subset}) assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $\\mathfrak p \\in V(I)$ no condition, \\item $\\text{cd}(A, I) \\leq d$, \\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$ then $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s \\quad\\text{or}\\quad \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > d + s $$ for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$, \\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$, $V(\\mathfrak p) \\cap V(J) \\cap V(I) \\not = \\emptyset$, and $\\text{depth}(M_\\mathfrak p) < s$, then one of the following holds\\footnote{Our method forces this additional condition. We will return to this (insert future reference).}: \\begin{enumerate} \\item $\\dim(\\text{Supp}(M_\\mathfrak p)) < s + 2$\\footnote{For example if $M$ satisfies Serre's condition $(S_s)$ on the complement of $V(I) \\cup T$.}, or \\item  $\\delta(\\mathfrak p) > d + \\delta_{max} - 1$ where $\\delta$ is a dimension function and $\\delta_{max}$ is the maximum of $\\delta$ on $V(J) \\cap V(I)$, or \\item $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > d + s + \\delta_{max} - \\delta_{min} - 2$ for all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$. \\end{enumerate} \\end{enumerate} Then there exists an ideal $J_0 \\subset J$ with $V(J_0) \\cap V(I) = V(J) \\cap V(I)$ such that for any $J' \\subset J_0$ with $V(J') \\cap V(I) = V(J) \\cap V(I)$ the map $$ R\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_J(M)^\\wedge $$ induces an isomorphism on cohomology in degrees $\\leq s$. Here ${}^\\wedge$ denotes derived $I$-adic completion."}
+{"_id": "12717", "title": "algebraization-lemma-kill-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module and let $\\mathfrak p \\subset A$ be a prime. Let $s$ and $d$ be integers. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $\\text{cd}(A, I) \\leq d$, and \\item $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s$. \\end{enumerate} Then there exists an $f \\in A \\setminus \\mathfrak p$ which annihilates $H^i(R\\Gamma_\\mathfrak m(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$ indicates $I$-adic completion."}
+{"_id": "12718", "title": "algebraization-lemma-kill-colimit-weak", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item if $\\mathfrak p \\in V(I)$, then no condition, \\item if $\\mathfrak p \\not \\in V(I)$ and $V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then $\\dim(A/\\mathfrak p) \\leq d$, \\item if $\\mathfrak p \\not \\in V(I)$ and $V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, then $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s \\quad\\text{or}\\quad \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s $$ \\end{enumerate} Then there exists an ideal $J_0 \\subset A$ with $V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map $$ R\\Gamma_J(M) \\longrightarrow R\\Gamma_{J_0}(M) $$ induces an isomorphism in cohomology in degrees $\\leq s$ and moreover these modules are annihilated by a power of $J_0I$."}
+{"_id": "12719", "title": "algebraization-lemma-kill-colimit", "text": "In Lemma \\ref{lemma-kill-colimit-weak} if instead of the empty condition (2) we assume \\begin{enumerate} \\item[(2')] if $\\mathfrak p \\in V(I)$ and $\\mathfrak p \\not = \\mathfrak m$, then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > s$, \\end{enumerate} then the conditions also imply that $H^i_{J_0}(M)$ is a finite $A$-module for $i \\leq s$."}
+{"_id": "12720", "title": "algebraization-lemma-kill-colimit-support", "text": "If in Lemma \\ref{lemma-kill-colimit-weak} we additionally assume \\begin{enumerate} \\item[(6)] if $\\mathfrak p \\not \\in V(I)$ and $V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$, \\end{enumerate} then $H^i_{J_0}(M) = H^i_J(M) = H^i_\\mathfrak m(M)$ for $i \\leq s$ and these modules are annihilated by a power of $I$."}
+{"_id": "12721", "title": "algebraization-lemma-algebraize-local-cohomology", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $\\mathfrak p \\in V(I)$, no condition, \\item $\\text{cd}(A, I) \\leq d$, \\item if $\\mathfrak p \\not \\in V(I)$ and $V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$ then $$ \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s \\quad\\text{or}\\quad \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s $$ \\end{enumerate} Then there exists an ideal $J_0 \\subset A$ with $V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map $$ R\\Gamma_J(M) \\longrightarrow R\\Gamma_J(M)^\\wedge = R\\Gamma_\\mathfrak m(M)^\\wedge $$ induces an isomorphism in cohomology in degrees $\\leq s$. Here ${}^\\wedge$ denotes derived $I$-adic completion."}
+{"_id": "12722", "title": "algebraization-lemma-helper-bootstrap", "text": "In Situation \\ref{situation-bootstrap} let $\\mathfrak p \\subset \\mathfrak q$ be primes of $A$ with $\\mathfrak p \\not \\in V(I)$ and $\\mathfrak q \\in T$. If there does not exist an $\\mathfrak r \\in V(I) \\setminus T$ with $\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$ then $\\text{depth}(M_\\mathfrak p) > s$."}
+{"_id": "12723", "title": "algebraization-lemma-bootstrap-inherited", "text": "In Situation \\ref{situation-bootstrap} we have \\begin{enumerate} \\item[(E)] if $T' \\subset T$ is a smaller specialization stable subset, then $A, I, T', M$ satisfies the assumptions of Situation \\ref{situation-bootstrap}, \\item[(F)] if $S \\subset A$ is a multiplicative subset, then $S^{-1}A, S^{-1}I, T', S^{-1}M$ satisfies the assumptions of Situation \\ref{situation-bootstrap} where $T' \\subset V(S^{-1}I)$ is the inverse image of $T$, \\item[(G)] the quadruple $A', I', T', M'$ satisfies the assumptions of Situation \\ref{situation-bootstrap} where $A', I', M'$ are the usual $I$-adic completions of $A, I, M$ and $T' \\subset V(I')$ is the inverse image of $T$. \\end{enumerate} Let $I \\subset \\mathfrak a \\subset A$ be an ideal such that $V(\\mathfrak a) \\subset T$. Then \\begin{enumerate} \\item[(A)] if $I$ is contained in the Jacobson radical of $A$, then all hypotheses of Lemmas \\ref{lemma-kill-colimit-weak-general} and \\ref{lemma-kill-colimit-support-general} are satisfied for $A, I, \\mathfrak a, M$, \\item[(B)] if $A$ is complete with respect to $I$, then all hypotheses except for possibly (5) of Lemma \\ref{lemma-algebraize-local-cohomology-general} are satisfied for $A, I, \\mathfrak a, M$, \\item[(C)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$, then all hypotheses of Lemmas \\ref{lemma-kill-colimit-weak} and \\ref{lemma-kill-colimit-support} hold for $A, \\mathfrak m, I, M$, \\item[(D)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$ and $I$-adically complete, then all hypotheses of Lemma \\ref{lemma-algebraize-local-cohomology} hold for $A, \\mathfrak m, I, M$, \\end{enumerate}"}
+{"_id": "12724", "title": "algebraization-lemma-algebraize-local-cohomology-bis", "text": "In Situation \\ref{situation-bootstrap} assume $A$ is local with maximal ideal $\\mathfrak m$ and $T = \\{\\mathfrak m\\}$. Then $H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M/I^nM)$ is an isomorphism for $i \\leq s$ and these modules are annihilated by a power of $I$."}
+{"_id": "12725", "title": "algebraization-lemma-bootstrap-bis-bis", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. If we assume \\begin{enumerate} \\item[(a)] $A$ has a dualizing complex, \\item[(b)] $\\text{cd}(A, I) \\leq d$, \\item[(c)] if $\\mathfrak p \\not \\in V(I)$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ or $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > d + s$. \\end{enumerate} Then $A, I, V(\\mathfrak a), M, s, d$ are as in Situation \\ref{situation-bootstrap}."}
+{"_id": "12726", "title": "algebraization-lemma-bootstrap", "text": "In Situation \\ref{situation-bootstrap} the inverse systems $\\{H^i_T(I^nM)\\}_{n \\geq 0}$ are pro-zero for $i \\leq s$. Moreover, there exists an integer $m_0$ such that for all $m \\geq m_0$ there exists an integer $m'(m) \\geq m$ such that for $k \\geq m'(m)$ the image of $H^{s + 1}_T(I^kM) \\to H^{s + 1}_T(I^mM)$ maps injectively to $H^{s + 1}_T(I^{m_0}M)$."}
+{"_id": "12727", "title": "algebraization-lemma-final-bootstrap", "text": "In Situation \\ref{situation-bootstrap} there exists an integer $m_0 \\geq 0$ such that \\begin{enumerate} \\item $\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$ satisfies the Mittag-Leffler condition for $i < s$. \\item $\\{H^i_T(I^{m_0}M/I^nM)\\}_{n \\geq m_0}$ satisfies the Mittag-Leffler condition for $i \\leq s$, \\item $H^i_T(M) \\to \\lim H^i_T(M/I^nM)$ is an isomorphism for $i < s$, \\item $H^s_T(I^{m_0}M) \\to \\lim H^s_T(I^{m_0}M/I^nM)$ is an isomorphism for $i \\leq s$, \\item $H^s_T(M) \\to \\lim H^s_T(M/I^nM)$ is injective with cokernel killed by $I^{m_0}$, and \\item $R^1\\lim H^s_T(M/I^nM)$ is killed by $I^{m_0}$. \\end{enumerate}"}
+{"_id": "12728", "title": "algebraization-lemma-combine-two", "text": "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$ and let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Suppose that \\begin{enumerate} \\item $A, I, V(\\mathfrak a), M$ satisfy the assumptions of Situation \\ref{situation-bootstrap} for $s$ and $d$, and \\item $A, I, \\mathfrak a, M$ satisfy the conditions of Lemma \\ref{lemma-algebraize-local-cohomology-general} for $s + 1$ and $d$ with $J = \\mathfrak a$. \\end{enumerate} Then there exists an ideal $J_0 \\subset \\mathfrak a$ with $V(J_0) \\cap V(I) = V(\\mathfrak a)$ such that for any $J \\subset J_0$ with $V(J) \\cap V(I) = V(\\mathfrak a)$ the map $$ H^{s + 1}_J(M) \\longrightarrow \\lim H^{s + 1}_\\mathfrak a(M/I^nM) $$ is an isomorphism."}
+{"_id": "12729", "title": "algebraization-lemma-compare-with-derived-completion", "text": "Let $U$ be the punctured spectrum of a Noetherian local ring $A$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. Let $I \\subset A$ be an ideal. Then $$ H^i(R\\Gamma(U, \\mathcal{F})^\\wedge) = \\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F}) $$ for all $i$ where $R\\Gamma(U, \\mathcal{F})^\\wedge$ denotes the derived $I$-adic completion."}
+{"_id": "12730", "title": "algebraization-lemma-application-theorem", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$. Let $\\mathcal{F}$ be a coherent sheaf on $U$. Assume for any associated point $x \\in U$ of $\\mathcal{F}$ we have $\\dim(\\overline{\\{x\\}}) > \\text{cd}(A, I) + 1$ where $\\overline{\\{x\\}}$ is the closure in $X$. Then the map $$ \\colim H^0(V, \\mathcal{F}) \\longrightarrow \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F}) $$ is an isomorphism of finite $A$-modules where the colimit is over opens $V \\subset U$ containing $U \\cap Y$."}
+{"_id": "12731", "title": "algebraization-lemma-application-H0-pre", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $\\mathcal{F}$ be a coherent module on $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$, $\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$ and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$, \\item one of the following holds: \\begin{enumerate} \\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$ is $(S_1)$ \\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In the sense that the difference of the maximal and minimal values on $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}. \\end{enumerate} \\end{enumerate} Then we obtain an isomorphism $$ \\colim H^0(V, \\mathcal{F}) \\longrightarrow \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F}) $$ where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$."}
+{"_id": "12732", "title": "algebraization-lemma-application-H0", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $\\mathcal{F}$ be a coherent module on $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$, $z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$, then $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$, \\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$ we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$, \\end{enumerate} Then we obtain an isomorphism $$ H^0(U, \\mathcal{F}) \\longrightarrow \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F}) $$"}
+{"_id": "12733", "title": "algebraization-lemma-alternative-colim-H0", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor on $M$, \\item $H^1_\\mathfrak a(M/fM)$ is a finite $A$-module. \\end{enumerate} Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map $$ \\colim_V \\Gamma(V, \\widetilde{M}) \\longrightarrow \\lim \\Gamma(U, \\widetilde{M/f^nM}) $$ is an isomorphism where the colimit is over opens $V \\subset U$ containing $U \\cap V(f)$."}
+{"_id": "12734", "title": "algebraization-lemma-alternative-H0", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $H^1_\\mathfrak a(M)$ and $H^2_\\mathfrak a(M)$ are annihilated by a power of $f$. \\end{enumerate} Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map $$ \\Gamma(U, \\widetilde{M}) \\longrightarrow \\lim \\Gamma(U, \\widetilde{M/f^nM}) $$ is an isomorphism."}
+{"_id": "12735", "title": "algebraization-lemma-alternative-higher", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Let $s \\geq 0$. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $H^i_\\mathfrak a(M)$ is annihilated by a power of $f$ for $i \\leq s + 1$. \\end{enumerate} Then with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map $$ H^i(U, \\widetilde{M}) \\longrightarrow \\lim H^i(U, \\widetilde{M/f^nM}) $$ is an isomorphism for $i < s$."}
+{"_id": "12737", "title": "algebraization-lemma-connected", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $\\mathfrak p \\subset A$ is a minimal prime not contained in $V(I)$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$, then $\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$, \\item any nonempty open $V \\subset \\Spec(A)$ which contains $V(I) \\setminus V(\\mathfrak a)$ is connected\\footnote{For example if $A$ is a domain.}. \\end{enumerate} Then $V(I) \\setminus V(\\mathfrak a)$ is either empty or connected."}
+{"_id": "12738", "title": "algebraization-lemma-completion-fully-faithful", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme. Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Consider the following conditions \\begin{enumerate} \\item $X$ is quasi-affine and $\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$ is an isomorphism, \\item $X$ has an ample invertible module $\\mathcal{L}$ and $\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to \\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$ is an isomorphism for all $m \\gg 0$, \\item for every finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ the map $\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$ is an isomorphism, and \\item the completion functor $\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(X, \\mathcal{I})$ is fully faithful on the full subcategory of finite locally free objects. \\end{enumerate} Then (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4) and (4) $\\Rightarrow$ (3)."}
+{"_id": "12739", "title": "algebraization-lemma-completion-fully-faithful-general", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$ ordered by reverse inclusion. \\begin{enumerate} \\item $X$ is quasi-affine and $$ \\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n}) $$ is an isomorphism, \\item $X$ has an ample invertible module $\\mathcal{L}$ and $$ \\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m}) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n}) $$ is an isomorphism for all $m \\gg 0$, \\item for every $V \\in \\mathcal{V}$ and every finite locally free $\\mathcal{O}_V$-module $\\mathcal{E}$ the map $$ \\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'}) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n}) $$ is an isomorphism, and \\item the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(X, \\mathcal{I}), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects (see explanation above). \\end{enumerate} Then (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4) and (4) $\\Rightarrow$ (3)."}
+{"_id": "12740", "title": "algebraization-lemma-recognize-formal-coherent-modules", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. The functor $$ \\textit{Coh}(X, \\mathcal{I}) \\longrightarrow \\text{Pro-}\\QCoh(\\mathcal{O}_X) $$ is fully faithful, see Categories, Remark \\ref{categories-remark-pro-category}."}
+{"_id": "12741", "title": "algebraization-lemma-fully-faithful", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item for any associated prime $\\mathfrak p \\subset A$, $I \\not \\subset \\mathfrak p$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have $\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$. \\item for $\\mathfrak p \\subset A$, $I \\not \\subset \\mathfrak p$ with with $V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$ we have $\\text{depth}(A_\\mathfrak p) \\geq 2$. \\end{enumerate} Then the completion functor $$ \\textit{Coh}(\\mathcal{O}_U) \\longrightarrow \\textit{Coh}(U, I\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12742", "title": "algebraization-lemma-fully-faithful-simple-one", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a$ be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex and is complete with respect to $f$, \\item $A_f$ is $(S_2)$ and for every minimal prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have $\\dim((A/\\mathfrak p)_\\mathfrak q) \\geq 3$. \\end{enumerate} Then the completion functor $$ \\textit{Coh}(\\mathcal{O}_U) \\longrightarrow \\textit{Coh}(U, I\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12743", "title": "algebraization-lemma-fully-faithful-alternative", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$ be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $H^1_\\mathfrak a(A)$ and $H^2_\\mathfrak a(A)$ are annihilated by a power of $f$. \\end{enumerate} Then the completion functor $$ \\textit{Coh}(\\mathcal{O}_U) \\longrightarrow \\textit{Coh}(U, I\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12744", "title": "algebraization-lemma-fully-faithful-simple-two", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a$ be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex and is complete with respect to $f$, \\item for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have $\\text{depth}(A_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > 2$. \\end{enumerate} Then the completion functor $$ \\textit{Coh}(\\mathcal{O}_U) \\longrightarrow \\textit{Coh}(U, I\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12745", "title": "algebraization-lemma-fully-faithful-general", "text": "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be the set of open subschemes of $U$ containing $U \\cap V(I)$ ordered by reverse inclusion. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item for any associated prime $\\mathfrak p \\subset A$ with $I \\not \\subset \\mathfrak p$ and $V(\\mathfrak p) \\cap V(I) \\not \\subset V(\\mathfrak a)$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have $\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$. \\end{enumerate} Then the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(U, I\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12746", "title": "algebraization-lemma-fully-faithful-general-alternative", "text": "Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$ be an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be the set of open subschemes of $U$ containing $U \\cap V(f)$ ordered by reverse inclusion. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak a(A/fA)$ is a finite $A$-module. \\end{enumerate} Then the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(U, f\\mathcal{O}_U), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge $$ is fully faithful on the full subcategory of finite locally free objects."}
+{"_id": "12748", "title": "algebraization-lemma-system-of-modules", "text": "In Situation \\ref{situation-algebraize}. Consider an inverse system $(M_n)$ of $A$-modules such that \\begin{enumerate} \\item $M_n$ is a finite $A$-module, \\item $M_n$ is annihilated by $I^n$, \\item the kernel and cokernel of $M_{n + 1}/I^nM_{n + 1} \\to M_n$ are $\\mathfrak a$-power torsion. \\end{enumerate} Then $(\\widetilde{M}_n|_U)$ is in $\\textit{Coh}(U, I\\mathcal{O}_U)$. Conversely, every object of $\\textit{Coh}(U, I\\mathcal{O}_U)$ arises in this manner."}
+{"_id": "12749", "title": "algebraization-lemma-essential-image-completion", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Consider the following conditions: \\begin{enumerate} \\item $(\\mathcal{F}_n)$ is in the essential image of the functor (\\ref{equation-completion}), \\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module, \\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_V$-module for $U \\cap Y \\subset V \\subset U$ open, \\item $(\\mathcal{F}_n)$ is the completion of the restriction to $U$ of a coherent $\\mathcal{O}_X$-module, \\item $(\\mathcal{F}_n)$ is the restriction to $U$ of the completion of a coherent $\\mathcal{O}_X$-module, \\item there exists an object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction to $U$ is $(\\mathcal{F}_n)$. \\end{enumerate} Then conditions (1), (2), (3), (4), and (5) are equivalent and imply (6). If $A$ is $I$-adically complete then condition (6) implies the others."}
+{"_id": "12750", "title": "algebraization-lemma-algebraizable", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A', I', \\mathfrak a'$ be the $I$-adic completions of $A, I, \\mathfrak a$. Set $X' = \\Spec(A')$ and $U' = X' \\setminus V(\\mathfrak a')$. The following are equivalent \\begin{enumerate} \\item $(\\mathcal{F}_n)$ extends to $X$, and \\item the pullback of $(\\mathcal{F}_n)$ to $U'$ is the completion of a coherent $\\mathcal{O}_{U'}$-module. \\end{enumerate}"}
+{"_id": "12751", "title": "algebraization-lemma-canonically-algebraizable", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $(\\mathcal{F}_n)$ canonically extends to $X$, then \\begin{enumerate} \\item $(\\widetilde{H^0(U, \\mathcal{F}_n)})$ is pro-isomorphic to an object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I \\mathcal{O}_X)$ unique up to unique isomorphism, \\item the restriction of $(\\mathcal{G}_n)$ to $U$ is isomorphic to $(\\mathcal{F}_n)$, i.e., $(\\mathcal{F}_n)$ extends to $X$, \\item the inverse system $\\{H^0(U, \\mathcal{F}_n)\\}$ satisfies the Mittag-Leffler condition, and \\item the module $M$ in (\\ref{equation-guess}) is finite over the $I$-adic completion of $A$ and the limit topology on $M$ is the $I$-adic topology. \\end{enumerate}"}
+{"_id": "12752", "title": "algebraization-lemma-canonically-extend-base-change", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A \\to A'$ be a flat ring map. Set $X' = \\Spec(A')$, let $U' \\subset X'$ be the inverse image of $U$, and denote $g : U' \\to U$ the induced morphism. Set $(\\mathcal{F}'_n) = (g^*\\mathcal{F}_n)$, see Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}. If $(\\mathcal{F}_n)$ canonically extends to $X$, then $(\\mathcal{F}'_n)$ canonically extends to $X'$. Moreover, the extension found in Lemma \\ref{lemma-canonically-algebraizable} for $(\\mathcal{F}_n)$ pulls back to the extension for $(\\mathcal{F}'_n)$."}
+{"_id": "12753", "title": "algebraization-lemma-when-done", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $M$ be as in (\\ref{equation-guess}). Assume \\begin{enumerate} \\item[(a)] the inverse system $H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler, \\item[(b)] the limit topology on $M$ agrees with the $I$-adic topology, and \\item[(c)] the image of $M \\to H^0(U, \\mathcal{F}_n)$ is a finite $A$-module for all $n$. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is $I$-adically complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12754", "title": "algebraization-lemma-algebraization-principal-variant", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak a$, \\item $\\mathcal{F}_n$ is a finite locally free $\\mathcal{O}_U/f^n\\mathcal{O}_U$-module, \\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$ are finite $A$-modules. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12755", "title": "algebraization-lemma-map-kernel-cokernel-on-closed", "text": "In Situation \\ref{situation-algebraize}. Let $(\\mathcal{F}_n) \\to (\\mathcal{F}'_n)$ be a morphism of $\\textit{Coh}(U, I\\mathcal{O}_U)$ whose kernel and cokernel are annihilated by a power of $I$. Then \\begin{enumerate} \\item $(\\mathcal{F}_n)$ extends to $X$ if and only if $(\\mathcal{F}'_n)$ extends to $X$, and \\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module if and only if $(\\mathcal{F}'_n)$ is. \\end{enumerate}"}
+{"_id": "12758", "title": "algebraization-lemma-discussion", "text": "Let $Y$ be a Noetherian scheme and let $Z \\subset Y$ be a closed subset. \\begin{enumerate} \\item For $y \\in Y$ we have $\\delta_Z(y) = 0 \\Leftrightarrow y \\in Z$. \\item The subsets $\\{y \\in Y \\mid \\delta_Z(y) \\leq k\\}$ are stable under specialization. \\item For $y \\in Y$ and $z \\in \\overline{\\{y\\}} \\cap Z$ we have $\\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z}) \\geq \\delta_Z(y)$. \\item If $\\delta$ is a dimension function on $Y$, then $\\delta(y) \\leq \\delta_Z(y) + \\delta_{max}$ where $\\delta_{max}$ is the maximum value of $\\delta$ on $Z$. \\item If $Y = \\Spec(A)$ is the spectrum of a catenary Noetherian local ring with maximal ideal $\\mathfrak m$ and $Z = \\{\\mathfrak m\\}$, then $\\delta_Z(y) = \\dim(\\overline{\\{y\\}})$. \\item Given a pattern of specializations $$ \\xymatrix{ & y'_0 \\ar@{~>}[ld] \\ar@{~>}[rd] & & y'_1 \\ar@{~>}[ld] & \\ldots & y'_{k - 1} \\ar@{~>}[rd] & \\\\ y_0 & & y_1 & & \\ldots & & y_k = y } $$ between points of $Y$ with $y_0 \\in Z$ and $y_i' \\leadsto y_i$ an immediate specialization, then $\\delta_Z(y_k) \\leq k$. \\item If $Y' \\subset Y$ is an open subscheme, then $\\delta^{Y'}_{Y' \\cap Z}(y') \\geq \\delta^Y_Z(y')$ for $y' \\in Y'$. \\end{enumerate}"}
+{"_id": "12759", "title": "algebraization-lemma-change-distance-function", "text": "Let $Y$ be a universally catenary Noetherian scheme. Let $Z \\subset Y$ be a closed subscheme. Let $f : Y' \\to Y$ be a finite type morphism all of whose fibres have dimension $\\leq e$. Set $Z' = f^{-1}(Z)$. Then $$ \\delta_Z(y) \\leq \\delta_{Z'}(y') + e - \\text{trdeg}_{\\kappa(y)}(\\kappa(y')) $$ for $y' \\in Y'$ with image $y \\in Y$."}
+{"_id": "12760", "title": "algebraization-lemma-elementary", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers. \\begin{enumerate} \\item If $(\\mathcal{F}_n)$ is annihilated by a power of $I$, then $(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities for any $a, b$. \\item If $(\\mathcal{F}_n)$ satisfies the $(a + 1, b)$-inequalities, then $(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities. \\end{enumerate} If $\\text{cd}(A, I) \\leq d$ and $A$ has a dualizing complex, then \\begin{enumerate} \\item[(3)] $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities if and only if for all $y \\in U \\cap Y$ the tuple $\\mathcal{O}_{X, y}^\\wedge, I\\mathcal{O}_{X, y}^\\wedge, \\{\\mathfrak m_y^\\wedge\\}, \\mathcal{F}_y^\\wedge, s - \\delta^Y_Z(y), d$ is as in Situation \\ref{situation-bootstrap}. \\item[(4)] If $(\\mathcal{F}_n)$ satisfies the strict $(s, s + d)$-inequalities, then $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities. \\end{enumerate}"}
+{"_id": "12761", "title": "algebraization-lemma-explain-2-3-cd-1", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $\\text{cd}(A, I) = 1$, then $\\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > 3 $$ for all $y \\in U \\cap Y$ and $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$."}
+{"_id": "12762", "title": "algebraization-lemma-sanity", "text": "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, has a dualizing complex, and $\\text{cd}(A, I) \\leq d$, \\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities. \\end{enumerate} Let $x \\in X$ be a point. Let $W = \\overline{\\{x\\}}$. If $W \\cap Y$ has an irreducible component contained in $Z$ and one which is not, then $\\text{depth}(\\mathcal{F}_x) \\geq 1$."}
+{"_id": "12763", "title": "algebraization-lemma-recover", "text": "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, has a dualizing complex, and $\\text{cd}(A, I) \\leq d$, \\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$ satisfies the strict $(1, 1+ d)$-inequalities, and \\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$ we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. \\end{enumerate} Then $H^0(U, \\mathcal{F}) \\to \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$ is an isomorphism."}
+{"_id": "12764", "title": "algebraization-lemma-fully-faithful-inequalities", "text": "In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, has a dualizing complex, and $\\text{cd}(A, I) \\leq d$, \\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities, and \\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$ we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. \\end{enumerate} Then the map $$ \\Hom_U(\\mathcal{G}, \\mathcal{F}) \\longrightarrow \\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge) $$ is bijective for every coherent $\\mathcal{O}_U$-module $\\mathcal{G}$."}
+{"_id": "12766", "title": "algebraization-lemma-algebraization-principal", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal, \\item $A$ has a dualizing complex, \\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak m$, \\item $\\mathcal{F}_n$ is a finite locally free $\\mathcal{O}_U/f^n\\mathcal{O}_U$-module, \\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then $\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12767", "title": "algebraization-lemma-helper-algebraize", "text": "In Situation \\ref{situation-algebraize} let $(M_n)$ be an inverse system of $A$-modules as in Lemma \\ref{lemma-system-of-modules} and let $(\\mathcal{F}_n)$ be the corresponding object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $d \\geq \\text{cd}(A, I)$ and $s \\geq 0$ be integers. With notation as above assume \\begin{enumerate} \\item $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$, \\item $A$ has a dualizing complex, and \\item $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities (Definition \\ref{definition-s-d-inequalities}). \\end{enumerate} Let $E$ be an injective hull of the residue field of $A$. Then for $i \\leq s$ there exists a finite $A$-module $N$ annihilated by a power of $I$ and for $n \\gg 0$ compatible maps $$ H^i_\\mathfrak m(M_n) \\to \\Hom_A(N, E) $$ whose cokernels are finite length $A$-modules and whose kernels $K_n$ form an inverse system such that $\\Im(K_{n''} \\to K_{n'})$ has finite length for $n'' \\gg n' \\gg 0$."}
+{"_id": "12768", "title": "algebraization-lemma-algebraization-principal-bis", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal, \\item $A$ has a dualizing complex, \\item $I = (f)$ is a principal ideal, \\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12769", "title": "algebraization-lemma-unwinding-conditions", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$, \\item $\\text{cd}(A, I) = 1$. \\end{enumerate} Then $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities if and only if for all $y \\in U \\cap Y$ with $\\dim(\\{y\\}) = 1$ and every prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$, $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have $$ \\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 2 $$"}
+{"_id": "12770", "title": "algebraization-lemma-unwinding-conditions-bis", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$, \\item $A$ has a dualizing complex, \\item $\\text{cd}(A, I) = 1$, \\item for $y \\in U \\cap Y$ the module $\\mathcal{F}_y^\\wedge$ is finite locally free outside $V(I\\mathcal{O}_{X, y}^\\wedge)$, for example if $\\mathcal{F}_n$ is a finite locally free $\\mathcal{O}_U/I^n\\mathcal{O}_U$-module, and \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $Y$ has dimension $\\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\end{enumerate} Then $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities."}
+{"_id": "12771", "title": "algebraization-lemma-divide-torsion-formal-coherent-module", "text": "In the situation above assume $X$ locally has a dualizing complex. Let $T \\subset Y$ be a subset stable under specialization. Assume for $y \\in T$ and for a nonmaximal prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $V(\\mathfrak p) \\cap V(\\mathcal{I}^\\wedge_y) = \\{\\mathfrak m_y^\\wedge\\}$ we have $$ \\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p} ((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0 $$ Then there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n')$ of inverse systems of coherent $\\mathcal{O}_X$-modules with the following properties \\begin{enumerate} \\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}'_{n, y}) \\geq 1$, \\item $(\\mathcal{F}'_n)$ is isomorphic as a pro-system to an object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$, \\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$ is surjective with kernel annihilated by a power of $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "12772", "title": "algebraization-lemma-improvement-formal-coherent-module-better", "text": "In the situation above assume $X$ locally has a dualizing complex. Let $T' \\subset T \\subset Y$ be subsets stable under specialization. Let $d \\geq 0$ be an integer. Assume \\begin{enumerate} \\item[(a)] affine locally we have $X = \\Spec(A_0)$ and $Y = V(I_0)$ and $\\text{cd}(A_0, I_0) \\leq d$, \\item[(b)] for $y \\in T$ and a nonmaximal prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$ we have $$ \\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p} ((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0 $$ \\item[(c)] for $y \\in T'$ and for a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(\\mathcal{I}_y^\\wedge)$ and $V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) \\not = \\{\\mathfrak m_y^\\wedge\\}$ we have $$ \\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p} ((\\mathcal{F}^\\wedge_y)_\\mathfrak p) \\geq 1 \\quad\\text{or}\\quad \\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p} ((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 1 + d $$ \\item[(d)] for $y \\in T'$ and a nonmaximal prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$ we have $$ \\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p} ((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 1 $$ \\item[(e)] if $y \\leadsto y'$ is an immediate specialization and $y' \\in T'$, then $y \\in T$. \\end{enumerate} Then there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$ of inverse systems of coherent $\\mathcal{O}_X$-modules with the following properties \\begin{enumerate} \\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}''_{n, y}) \\geq 1$, \\item for $y' \\in T'$ we have $\\text{depth}(\\mathcal{F}''_{n, y'}) \\geq 2$, \\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object $(\\mathcal{H}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$, \\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$ has kernel and cokernel annihilated by a power of $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "12773", "title": "algebraization-lemma-improvement-application", "text": "In Situation \\ref{situation-algebraize} assume that $A$ has a dualizing complex. Let $d \\geq \\text{cd}(A, I)$. Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume $(\\mathcal{F}_n)$ satisfies the $(2, 2 + d)$-inequalities, see Definition \\ref{definition-s-d-inequalities}. Then there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$ of inverse systems of coherent $\\mathcal{O}_U$-modules with the following properties \\begin{enumerate} \\item if $\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$ for all $y \\in U \\cap Y$, \\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object $(\\mathcal{H}_n)$ of $\\textit{Coh}(U, I\\mathcal{O}_U)$, \\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of $\\textit{Coh}(U, I\\mathcal{O}_U)$ has kernel and cokernel annihilated by a power of $I$, \\item the modules $H^0(U, \\mathcal{F}''_n)$ and $H^1(U, \\mathcal{F}''_n)$ are finite $A$-modules for all $n$. \\end{enumerate}"}
+{"_id": "12774", "title": "algebraization-lemma-cd-1-canonical", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$, \\item $(\\mathcal{F}_n)$ is pro-isomorphic to an inverse system $(\\mathcal{F}_n'')$ of coherent $\\mathcal{O}_U$-modules such that $\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$ for all $y \\in U \\cap Y$. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$, see Definition \\ref{definition-canonically-algebraizable}."}
+{"_id": "12775", "title": "algebraization-lemma-blowup", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item all fibres of the blowing up $b : X' \\to X$ of $I$ have dimension $\\leq d - 1$, \\item one of the following is true \\begin{enumerate} \\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities (Definition \\ref{definition-s-d-inequalities}), or \\item for $y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2 $$ \\end{enumerate} \\end{enumerate} Then $(\\mathcal{F}_n)$ extends to $X$."}
+{"_id": "12777", "title": "algebraization-lemma-equivalence-better", "text": "In Situation \\ref{situation-algebraize} assume \\begin{enumerate} \\item $A$ has a dualizing complex and is $I$-adically complete, \\item $I = (f)$ generated by a single element, \\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$, \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and for $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ minimal we have $\\dim(A/\\mathfrak p) \\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\end{enumerate} Then with $U_0 = U \\cap V(f)$ the completion functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{Coh}(\\mathcal{O}_{U'}) \\longrightarrow \\textit{Coh}(U, f\\mathcal{O}_U) $$ is an equivalence on the full subcategories of finite locally free objects."}
+{"_id": "12778", "title": "algebraization-lemma-equivalence", "text": "In Situation \\ref{situation-algebraize} assume \\begin{enumerate} \\item $I = (f)$ is principal, \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$ are finite $A$-modules. \\end{enumerate} Then with $U_0 = U \\cap V(f)$ the completion functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{Coh}(\\mathcal{O}_{U'}) \\longrightarrow \\textit{Coh}(U, f\\mathcal{O}_U) $$ is an equivalence on the full subcategories of finite locally free objects."}
+{"_id": "12779", "title": "algebraization-lemma-prepare-chi-triple", "text": "For any coherent triple $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$ such that $f : \\mathcal{F}' \\to \\mathcal{F}'$ is injective, an isomorphism $\\alpha' : \\mathcal{F}'|_U \\to \\mathcal{F}$, and a map $\\alpha'_0 : \\mathcal{F}'/f\\mathcal{F}' \\to \\mathcal{F}_0$ such that $\\alpha \\circ (\\alpha' \\bmod f) = \\alpha'_0|_{U_0}$."}
+{"_id": "12781", "title": "algebraization-lemma-ses-chi-triple", "text": "We have $\\chi(\\mathcal{G}, \\mathcal{G}_0, \\beta) = \\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha) + \\chi(\\mathcal{H}, \\mathcal{H}_0, \\gamma)$ if $$ 0 \\to (\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\to (\\mathcal{G}, \\mathcal{G}_0, \\beta) \\to (\\mathcal{H}, \\mathcal{H}_0, \\gamma) \\to 0 $$ is a short exact sequence of coherent triples."}
+{"_id": "12782", "title": "algebraization-lemma-nonnegative-chi-triple", "text": "Assume $\\text{depth}(A) \\geq 3$ or equivalently $\\text{depth}(A/fA) \\geq 2$. Let $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$ be an invertible coherent triple. Then $$ \\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) = \\text{length}_A \\Coker(\\Gamma(U, \\mathcal{L}) \\to \\Gamma(U_0, \\mathcal{L}_0)) $$ and in particular this is $\\geq 0$. Moreover,  $\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) = 0$ if and only if $\\mathcal{L} \\cong \\mathcal{O}_U$."}
+{"_id": "12784", "title": "algebraization-lemma-surjective-Pic-first", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite $A$-modules, and \\item $H^3_\\mathfrak m(A/fA) = 0$\\footnote{Observe that (3) and (4) hold if $\\text{depth}(A/fA) \\geq 4$, or equivalently $\\text{depth}(A) \\geq 5$.}. \\end{enumerate} Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. Then $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\Pic(U') \\longrightarrow \\Pic(U_0) $$ is surjective."}
+{"_id": "12785", "title": "algebraization-lemma-surjective-Pic-first-better", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item the conditions of Lemma \\ref{lemma-surjective-Pic-first} hold, and \\item for every maximal ideal $\\mathfrak p \\subset A_f$ the punctured spectrum of $(A_f)_\\mathfrak p$ has trivial Picard group. \\end{enumerate} Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. Then $$ \\Pic(U) \\longrightarrow \\Pic(U_0) $$ is surjective."}
+{"_id": "12786", "title": "algebraization-lemma-local-pic-to-completion", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$. Let $A^\\wedge$ be its completion. Let $U$, resp.\\ $U^\\wedge$ be the punctured spectrum of $A$, resp.\\ $A^\\wedge$. Then $\\Pic(U) \\to \\Pic(U^\\wedge)$ is injective."}
+{"_id": "12787", "title": "algebraization-lemma-trivial-local-pic-regular", "text": "Let $(A, \\mathfrak m)$ be a regular local ring. Then the Picard group of the punctured spectrum of $A$ is trivial."}
+{"_id": "12788", "title": "algebraization-lemma-lefschetz-addendum", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_n = Z(s^n)$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume that for all $x \\in X \\setminus Y$ we have $$ \\text{depth}(\\mathcal{F}_x) + \\dim(\\overline{\\{x\\}}) > 1 $$ Then $\\Gamma(V, \\mathcal{F}) \\to \\lim \\Gamma(Y_n, \\mathcal{F}|_{Y_n})$ is an isomorphism for any open subscheme $V \\subset X$ containing $Y$."}
+{"_id": "12789", "title": "algebraization-lemma-Gm-equivariant-extend-canonically", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ is a graded ring, $\\mathfrak a = A_+$, and $I$ is a homogeneous ideal, \\item $(\\mathcal{F}_n) = (\\widetilde{M_n}|_U)$ where $(M_n)$ is an inverse system of graded $A$-modules, and \\item $(\\mathcal{F}_n)$ extends canonically to $X$. \\end{enumerate} Then there is a finite graded $A$-module $N$ such that \\begin{enumerate} \\item[(a)] the inverse systems $(N/I^nN)$ and $(M_n)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-power torsion modules, and \\item[(b)] $(\\mathcal{F}_n)$ is the completion of of the coherent module associated to $N$. \\end{enumerate}"}
+{"_id": "12790", "title": "algebraization-proposition-derived-completion", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. There exists a left adjoint to the inclusion functor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$."}
+{"_id": "12791", "title": "algebraization-proposition-application-H0", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $\\mathcal{F}$ be a coherent module on $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$, $\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$ and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$. \\end{enumerate} Then we obtain an isomorphism $$ \\colim H^0(V, \\mathcal{F}) \\longrightarrow \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F}) $$ where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$."}
+{"_id": "12793", "title": "algebraization-proposition-cd-1", "text": "\\begin{reference} The local case of this result is \\cite[IV Corollaire 2.9]{MRaynaud-book}. \\end{reference} In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$, \\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities, see Definition \\ref{definition-s-d-inequalities}. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12794", "title": "algebraization-proposition-d-generators", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $V(I) = V(f_1, \\ldots, f_d)$ for some $d \\geq 1$ and $f_1, \\ldots, f_d \\in A$, \\item one of the following is true \\begin{enumerate} \\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities (Definition \\ref{definition-s-d-inequalities}), or \\item for $y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2 $$ \\end{enumerate} \\end{enumerate} Then $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12795", "title": "algebraization-proposition-algebraization-regular-sequence", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume \\begin{enumerate} \\item there exist $f_1, \\ldots, f_d \\in I$ such that for $y \\in U \\cap Y$ the ideal $I\\mathcal{O}_{X, y}$ is generated by $f_1, \\ldots, f_d$ and $f_1, \\ldots, f_d$ form a $\\mathcal{F}_y^\\wedge$-regular sequence, \\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$ are finite $A$-modules. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12796", "title": "algebraization-proposition-algebraization-flat", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume there is Noetherian local ring $(R, \\mathfrak m)$ and a ring map $R \\to A$ such that \\begin{enumerate} \\item $I = \\mathfrak m A$, \\item for $y \\in U \\cap Y$ the stalk $\\mathcal{F}_y^\\wedge$ is $R$-flat, \\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$ are finite $A$-modules. \\end{enumerate} Then $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module."}
+{"_id": "12797", "title": "algebraization-proposition-hilbert-triple", "text": "Let $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ be a coherent triple. Let $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$ be an invertible coherent triple. Then the function $$ \\mathbf{Z} \\longrightarrow \\mathbf{Z},\\quad n \\longmapsto \\chi((\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes (\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n}) $$ is a polynomial of degree $\\leq \\dim(\\text{Supp}(\\mathcal{F}))$."}
+{"_id": "12798", "title": "algebraization-proposition-injective-pic", "text": "\\begin{reference} \\cite[Theorem 1.9]{Kollar-map-pic} \\end{reference} Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $f$ is a nonzerodivisor, \\item $\\text{depth}(A/fA) \\geq 2$, or equivalently $\\text{depth}(A) \\geq 3$, \\item if $f \\in \\mathfrak p \\subset A$ is a prime ideal with $\\dim(A/\\mathfrak p) = 2$, then $\\text{depth}(A_\\mathfrak p) \\geq 2$. \\end{enumerate} Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. The map $$ \\Pic(U) \\to \\Pic(U_0) $$ is injective. Finally, if (1), (2), (3), $A$ is $(S_2)$, and $\\dim(A) \\geq 4$, then (4) holds."}
+{"_id": "12800", "title": "algebraization-proposition-lefschetz", "text": "In the situation above assume for all points $p \\in P \\setminus Q$ we have $$ \\text{depth}(\\mathcal{F}_p) + \\dim(\\overline{\\{p\\}}) > s $$ Then the map $$ H^i(P, \\mathcal{F}) \\longrightarrow \\lim H^i(Q_n, \\mathcal{F}_n) $$ is an isomorphism for $0 \\leq i < s$."}
+{"_id": "12801", "title": "algebraization-proposition-lefschetz-existence", "text": "In the situation above let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(P, \\mathcal{I})$. Assume for all $q \\in Q$ and for all primes $\\mathfrak p \\in \\mathcal{O}_{P, q}^\\wedge$, $\\mathfrak p \\not \\in V(\\mathcal{I}_q^\\wedge)$ we have $$ \\text{depth}((\\mathcal{F}_q^\\wedge)_\\mathfrak p) + \\dim(\\mathcal{O}_{P, q}^\\wedge/\\mathfrak p) + \\dim(\\overline{\\{q\\}}) > 2 $$ Then $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_P$-module."}
+{"_id": "12802", "title": "algebraization-proposition-lefschetz-equivalence", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module and let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the corresponding sheaf of ideals. Let $\\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \\in X \\setminus Y$ we have $$ \\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2 $$ Then the completion functor $$ \\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V) \\longrightarrow \\textit{Coh}(X, \\mathcal{I}) $$ is an equivalence on the full subcategories of finite locally free objects."}
+{"_id": "12821", "title": "spaces-over-fields-lemma-quasi-finite-in-codim-1", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian. Let $y \\in |Y|$ be a point of codimension $\\leq 1$ on $Y$. Let $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied \\begin{enumerate} \\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$, \\item for every $x \\in X^0$ with $f(x) \\leadsto y$ the transcendence degree of $x/f(x)$ is $0$, \\item $f$ is quasi-finite at every $x \\in X^0$, \\item $f$ is quasi-finite at a dense set of points of $|X|$, \\item add more here. \\end{enumerate} Then $f$ is quasi-finite at every point of $X$ lying over $y$."}
+{"_id": "12822", "title": "spaces-over-fields-lemma-finite-in-codim-1", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper and $Y$ is locally Noetherian. Let $y \\in Y$ be a point of codimension $\\leq 1$ in $Y$. Let $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied \\begin{enumerate} \\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$, \\item for every $x \\in X^0$ with $f(x) \\leadsto y$ the transcendence degree of $x/f(x)$ is $0$, \\item $f$ is quasi-finite at every $x \\in X^0$, \\item $f$ is quasi-finite at a dense set of points of $|X|$, \\item add more here. \\end{enumerate} Then there exists an open subspace $Y' \\subset Y$ containing $y$ such that $Y' \\times_Y X \\to Y'$ is finite."}
+{"_id": "12823", "title": "spaces-over-fields-lemma-modification-normal-iso-over-codimension-1", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $f : Y \\to X$ be a birational proper morphism of algebraic spaces with $Y$ reduced. Let $U \\subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains \\begin{enumerate} \\item every point of codimension $0$ in $X$, \\item every $x \\in |X|$ of codimension $1$ on $X$ such that the local ring of $X$ at $x$ is normal (Properties of Spaces, Remark \\ref{spaces-properties-remark-list-properties-local-ring-local-etale-topology}), and \\item every $x \\in |X|$ such that the fibre of $|Y| \\to |X|$ over $x$ is finite and such that the local ring of $X$ at $x$ is normal. \\end{enumerate}"}
+{"_id": "12824", "title": "spaces-over-fields-lemma-integral-algebraic-space-rational-functions", "text": "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. Let $\\eta \\in |X|$ be the generic point of $X$. There are canonical identifications $$ R(X) = \\mathcal{O}_{X, \\eta}^h = \\kappa(\\eta) $$ where $R(X)$ is the ring of rational functions defined in Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-ring-of-rational-functions}, $\\kappa(\\eta)$ is the residue field defined in Decent Spaces, Definition \\ref{decent-spaces-definition-residue-field}, and $\\mathcal{O}_{X, \\eta}^h$ is the henselian local ring defined in Decent Spaces, Definition \\ref{decent-spaces-definition-elemenary-etale-neighbourhood}. In particular, these rings are fields."}
+{"_id": "12825", "title": "spaces-over-fields-lemma-integral-sections", "text": "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. Then $\\Gamma(X, \\mathcal{O}_X)$ is a domain."}
+{"_id": "12826", "title": "spaces-over-fields-lemma-normal-integral-cover-by-affines", "text": "Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. For every $x \\in |X|$ there exists a normal integral affine scheme $U$ and an \\'etale morphism $U \\to X$ such that $x$ is in the image."}
+{"_id": "12827", "title": "spaces-over-fields-lemma-normal-integral-sections", "text": "Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. Then $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain."}
+{"_id": "12828", "title": "spaces-over-fields-lemma-decent-irreducible-closed", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. There are canonical bijections between the following sets: \\begin{enumerate} \\item the set of points of $X$, i.e., $|X|$, \\item the set of irreducible closed subsets of $|X|$, \\item the set of integral closed subspaces of $X$. \\end{enumerate} The bijection from (1) to (2) sends $x$ to $\\overline{\\{x\\}}$. The bijection from (3) to (2) sends $Z$ to $|Z|$."}
+{"_id": "12829", "title": "spaces-over-fields-lemma-finite-degree", "text": "Let $S$ be a scheme. Let $X$, $Y$ be integral algebraic spaces over $S$ Let $x \\in |X|$ and $y \\in |Y|$ be the generic points. Let $f : X \\to Y$ be locally of finite type. Assume $f$ is dominant (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dominant}). The following are equivalent: \\begin{enumerate} \\item the transcendence degree of $x/y$ is $0$, \\item the extension $\\kappa(x) \\supset \\kappa(y)$ (see proof) is finite, \\item there exist nonempty affine opens $U \\subset X$ and $V \\subset Y$ such that $f(U) \\subset V$ and $f|_U : U \\to V$ is finite, \\item $f$ is quasi-finite at $x$, and \\item $x$ is the only point of $|X|$ mapping to $y$. \\end{enumerate} If $f$ is separated or if $f$ is quasi-compact, then these are also equivalent to \\begin{enumerate} \\item[(6)] there exists a nonempty affine open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is finite. \\end{enumerate}"}
+{"_id": "12830", "title": "spaces-over-fields-lemma-degree-composition", "text": "Let $S$ be a scheme. Let $X$, $Y$, $Z$ be integral algebraic spaces over $S$. Let $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally of finite type. Assume any of the equivalent conditions (1) -- (5) of Lemma \\ref{lemma-finite-degree} hold for $f$ and $g$. Then $$ \\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z). $$"}
+{"_id": "12831", "title": "spaces-over-fields-lemma-components-locally-finite", "text": "Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. If $T \\subset |X|$ is a closed subset, then the collection of irreducible components of $T$ is locally finite."}
+{"_id": "12832", "title": "spaces-over-fields-lemma-order-vanishing", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $Z \\subset X$ be a prime divisor and let $\\xi \\in |Z|$ be the generic point. Then the henselian local ring $\\mathcal{O}_{X, \\xi}^h$ is a reduced $1$-dimensional Noetherian local ring and there is a canonical injective map $$ R(X) \\longrightarrow Q(\\mathcal{O}_{X, \\xi}^h) $$ from the function field $R(X)$ of $X$ into the total ring of fractions."}
+{"_id": "12834", "title": "spaces-over-fields-lemma-divisor-locally-finite", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \\in R(X)^*$. Then the collections $$ \\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi \\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\} $$ and $$ \\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\} $$ are locally finite in $X$."}
+{"_id": "12835", "title": "spaces-over-fields-lemma-div-additive", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f, g \\in R(X)^*$. Then $$ \\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g) $$ as Weil divisors on $X$."}
+{"_id": "12836", "title": "spaces-over-fields-lemma-divisor-meromorphic-locally-finite", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets $$ \\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi \\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\} $$ and $$ \\{Z \\subset X \\mid Z \\text{ is a prime divisor and } \\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\} $$ are locally finite in $X$."}
+{"_id": "12837", "title": "spaces-over-fields-lemma-divisor-meromorphic-well-defined", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$ Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero meromorphic sections of $\\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have $$ \\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z] = \\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z] + \\text{div}(f) $$ as Weil divisors."}
+{"_id": "12838", "title": "spaces-over-fields-lemma-c1-additive", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible $\\mathcal{O}_X$-modules. Let $s$, resp.\\ $t$ be a nonzero meromorphic section of $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ and $$ \\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st) = \\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t) $$ in $\\text{Div}(X)$. In particular, the Weil divisor class of $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum of the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$."}
+{"_id": "12841", "title": "spaces-over-fields-lemma-alteration-generically-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper dominant morphism of integral algebraic spaces over $S$. Then $f$ is an alteration if and only if any of the equivalent conditions (1) -- (6) of Lemma \\ref{lemma-finite-degree} hold."}
+{"_id": "12842", "title": "spaces-over-fields-lemma-alteration-contained-in", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism of algebraic spaces over $S$. Assume $Y$ is integral. Then there exists an integral closed subspace $X' \\subset X$ such that $f' = f|_{X'} : X' \\to Y$ is an alteration."}
+{"_id": "12843", "title": "spaces-over-fields-lemma-locally-finite-type-dim-zero", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. In each of the following cases $X$ is a scheme: \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated and $\\dim(X) = 0$, \\item $X$ is locally of finite type over a field $k$ and $\\dim(X) = 0$, \\item $X$ is Noetherian and $\\dim(X) = 0$, and \\item add more here. \\end{enumerate}"}
+{"_id": "12844", "title": "spaces-over-fields-lemma-generic-point-in-schematic-locus", "text": "Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item $x$ is a point of codimension $0$ on $X$, \\item the local ring of $X$ at $x$ has dimension $0$, and \\item $x$ is a generic point of an irreducible component of $|X|$. \\end{enumerate} If true, then there exists an open subspace of $X$ containing $x$ which is a scheme."}
+{"_id": "12845", "title": "spaces-over-fields-lemma-codim-1-point-in-schematic-locus", "text": "\\begin{slogan} Separated algebraic spaces are schemes in codimension 1. \\end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. If $X$ is separated, locally Noetherian, and the dimension of the local ring of $X$ at $x$ is $\\leq 1$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension-local-ring}), then there exists an open subspace of $X$ containing $x$ which is a scheme."}
+{"_id": "12846", "title": "spaces-over-fields-lemma-scheme-after-purely-inseparable-base-change", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. If there exists a purely inseparable field extension $k \\subset k'$ such that $X_{k'}$ is a scheme, then $X$ is a scheme."}
+{"_id": "12847", "title": "spaces-over-fields-lemma-when-scheme-after-base-change", "text": "Let $k$ be a field with algebraic closure $\\overline{k}$. Let $X$ be a quasi-separated algebraic space over $k$. \\begin{enumerate} \\item If there exists a field extension $k \\subset K$ such that $X_K$ is a scheme, then $X_{\\overline{k}}$ is a scheme. \\item If $X$ is quasi-compact and there exists a field extension $k \\subset K$ such that $X_K$ is a scheme, then $X_{k'}$ is a scheme for some finite separable extension $k'$ of $k$. \\end{enumerate}"}
+{"_id": "12848", "title": "spaces-over-fields-lemma-base-change-by-Galois", "text": "Let $k \\subset k'$ be a finite Galois extension with Galois group $G$. Let $X$ be an algebraic space over $k$. Then $G$ acts freely on the algebraic space $X_{k'}$ and $X = X_{k'}/G$ in the sense of Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}."}
+{"_id": "12849", "title": "spaces-over-fields-lemma-when-quotient-scheme-at-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as in Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}. For $y \\in |Y|$ the following are equivalent \\begin{enumerate} \\item $y$ is in the schematic locus of $Y$, and \\item there exists an affine open $U \\subset X$ containing the preimage of $y$. \\end{enumerate}"}
+{"_id": "12850", "title": "spaces-over-fields-lemma-scheme-after-purely-transcendental-base-change", "text": "Let $k$ be a field. Let $X$ be a quasi-separated algebraic space over $k$. If there exists a purely transcendental field extension $k \\subset K$ such that $X_K$ is a scheme, then $X$ is a scheme."}
+{"_id": "12851", "title": "spaces-over-fields-lemma-scheme-over-algebraic-closure-enough-affines", "text": "Let $k$ be a field with algebraic closure $\\overline{k}$. Let $X$ be an algebraic space over $k$ such that \\begin{enumerate} \\item $X$ is decent and locally of finite type over $k$, \\item $X_{\\overline{k}}$ is a scheme, and \\item any finite set of $\\overline{k}$-rational points of $X_{\\overline{k}}$ are contained in an affine. \\end{enumerate} Then $X$ is a scheme."}
+{"_id": "12852", "title": "spaces-over-fields-lemma-locally-quasi-finite-over-field", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is locally quasi-finite over $k$, \\item $X$ is locally of finite type over $k$ and has dimension $0$, \\item $X$ is a scheme and is locally quasi-finite over $k$, \\item $X$ is a scheme and is locally of finite type over $k$ and has dimension $0$, and \\item $X$ is a disjoint union of spectra of Artinian local $k$-algebras $A$ over $k$ with $\\dim_k(A) < \\infty$. \\end{enumerate}"}
+{"_id": "12853", "title": "spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a monomorphism of algebraic spaces over $k$. If $Y$ is locally quasi-finite over $k$ so is $X$."}
+{"_id": "12854", "title": "spaces-over-fields-lemma-geometrically-reduced-at-point", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced at $x$, \\item for some \\'etale neighbourhood $(U, u) \\to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$, \\item for any \\'etale neighbourhood $(U, u) \\to (X, x)$ where $U$ is a scheme, $U$ is geometrically reduced at $u$. \\end{enumerate}"}
+{"_id": "12855", "title": "spaces-over-fields-lemma-geometrically-reduced", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced, \\item for some surjective \\'etale morphism $U \\to X$ where $U$ is a scheme, $U$ is geometrically reduced, \\item for any \\'etale morphism $U \\to X$ where $U$ is a scheme, $U$ is geometrically reduced. \\end{enumerate}"}
+{"_id": "12856", "title": "spaces-over-fields-lemma-perfect-reduced", "text": "Let $X$ be an algebraic space over a perfect field $k$ (for example $k$ has characteristic zero). \\begin{enumerate} \\item For $x \\in |X|$, if $\\mathcal{O}_{X, \\overline{x}}$ is reduced, then $X$ is geometrically reduced at $x$. \\item If $X$ is reduced, then $X$ is geometrically reduced over $k$. \\end{enumerate}"}
+{"_id": "12857", "title": "spaces-over-fields-lemma-geometrically-reduced-positive-characteristic", "text": "Let $k$ be a field of characteristic $p > 0$. Let $X$ be an algebraic space over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced over $k$, \\item $X_{k'}$ is reduced for every field extension $k'/k$, \\item $X_{k'}$ is reduced for every finite purely inseparable field extension $k'/k$, \\item $X_{k^{1/p}}$ is reduced, \\item $X_{k^{perf}}$ is reduced, and \\item $X_{\\bar k}$ is reduced. \\end{enumerate}"}
+{"_id": "12858", "title": "spaces-over-fields-lemma-geometrically-reduced-upstairs", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $k'/k$ be a field extension. Let $x \\in |X|$ be a point and let $x' \\in |X_{k'}|$ be a point lying over $x$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically reduced at $x$, \\item $X_{k'}$ is geometrically reduced at $x'$. \\end{enumerate} In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$."}
+{"_id": "12859", "title": "spaces-over-fields-lemma-geometrically-reduced-etale-local", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a morphism of algebraic spaces over $k$. Let $x \\in |X|$ be a point with image $y \\in |Y|$. \\begin{enumerate} \\item if $f$ is \\'etale at $x$, then $X$ is geometrically reduced at $x$ $\\Leftrightarrow$ $Y$ is geometrically reduced at $y$, \\item if $f$ is surjective \\'etale, then $X$ is geometrically reduced $\\Leftrightarrow$ $Y$ is geometrically reduced. \\end{enumerate}"}
+{"_id": "12860", "title": "spaces-over-fields-lemma-geometrically-connected-check-after-extension", "text": "Let $X$ be an algebraic space over the field $k$. Let $k \\subset k'$ be a field extension. Then $X$ is geometrically connected over $k$ if and only if $X_{k'}$ is geometrically connected over $k'$."}
+{"_id": "12861", "title": "spaces-over-fields-lemma-bijection-connected-components", "text": "Let $k$ be a field. Let $X$, $Y$ be algebraic spaces over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism $$ p : X \\times_k Y \\longrightarrow Y $$ induces a bijection between connected components."}
+{"_id": "12862", "title": "spaces-over-fields-lemma-separably-closed-field-connected-components", "text": "Let $k \\subset k'$ be an extension of fields. Let $X$ be an algebraic space over $k$. Assume $k$ separably algebraically closed. Then the morphism $X_{k'} \\to X$ induces a bijection of connected components. In particular, $X$ is geometrically connected over $k$ if and only if $X$ is connected."}
+{"_id": "12863", "title": "spaces-over-fields-lemma-characterize-geometrically-connected", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $\\overline{k}$ be a separable algebraic closure of $k$. Then $X$ is geometrically connected if and only if the base change $X_{\\overline{k}}$ is connected."}
+{"_id": "12864", "title": "spaces-over-fields-lemma-Galois-action-quasi-compact-open", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. Let $\\overline{k}$ be a (possibly infinite) Galois extension of $k$. Let $V \\subset X_{\\overline{k}}$ be a quasi-compact open. Then \\begin{enumerate} \\item there exists a finite subextension $k \\subset k' \\subset \\overline{k}$ and a quasi-compact open $V' \\subset X_{k'}$ such that $V = (V')_{\\overline{k}}$, \\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$ such that $\\sigma(V) = V$ for all $\\sigma \\in H$. \\end{enumerate}"}
+{"_id": "12865", "title": "spaces-over-fields-lemma-closed-fixed-by-Galois", "text": "Let $k$ be a field. Let $k \\subset \\overline{k}$ be a (possibly infinite) Galois extension. Let $X$ be an algebraic space over $k$. Let $\\overline{T} \\subset |X_{\\overline{k}}|$ have the following properties \\begin{enumerate} \\item $\\overline{T}$ is a closed subset of $|X_{\\overline{k}}|$, \\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$ we have $\\sigma(\\overline{T}) = \\overline{T}$. \\end{enumerate} Then there exists a closed subset $T \\subset |X|$ whose inverse image in $|X_{k'}|$ is $\\overline{T}$."}
+{"_id": "12866", "title": "spaces-over-fields-lemma-characterize-geometrically-disconnected", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically connected, \\item for every finite separable field extension $k \\subset k'$ the algebraic space $X_{k'}$ is connected. \\end{enumerate}"}
+{"_id": "12867", "title": "spaces-over-fields-lemma-geometrically-integral", "text": "Let $k$ be a field. Let $X$ be a decent algebraic space over $k$. Then $X$ is geometrically integral over $k$ if and only if $X$ is both geometrically reduced and geometrically irreducible over $k$."}
+{"_id": "12868", "title": "spaces-over-fields-lemma-proper-geometrically-reduced-global-sections", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. \\begin{enumerate} \\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra, \\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian local $k$-algebras, one factor for each connected component of $|X|$, \\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$ is a product of fields, each a finite extension of $k$, \\item if $X$ is geometrically reduced, then $k_i$ is finite separable over $k$, \\item if $X$ is geometrically connected, then $A$ is geometrically irreducible over $k$, \\item if $X$ is geometrically irreducible, then $A$ is geometrically irreducible over $k$, \\item if $X$ is geometrically reduced and connected, then $A = k$, and \\item if $X$ is geometrically integral, then $A = k$. \\end{enumerate}"}
+{"_id": "12869", "title": "spaces-over-fields-lemma-characterize-trivial-pic-integral", "text": "Let $k$ be a field. Let $X$ be a proper integral algebraic space over $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If $H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes - 1})$ are both nonzero, then $\\mathcal{L} \\cong \\mathcal{O}_X$."}
+{"_id": "12870", "title": "spaces-over-fields-lemma-integral-dimension", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic spaces. Then $\\dim(X) \\leq \\dim(Y)$. If $f$ is surjective then $\\dim(X) = \\dim(Y)$."}
+{"_id": "12871", "title": "spaces-over-fields-lemma-alteration-dimension", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $Y$ is locally Noetherian, \\item $X$ and $Y$ are integral algebraic spaces, \\item $f$ is dominant, and \\item $f$ is locally of finite type. \\end{enumerate} If $x \\in |X|$ and $y \\in |Y|$ are the generic points, then $$ \\dim(X) \\leq \\dim(Y) + \\text{transcendence degree of }x/y. $$ If $f$ is proper, then equality holds."}
+{"_id": "12872", "title": "spaces-over-fields-lemma-smooth-regular", "text": "Let $k$ be a field. Let $X$ be an algebraic space smooth over $k$. Then $X$ is a regular algebraic space."}
+{"_id": "12873", "title": "spaces-over-fields-lemma-smooth-separable-closed-points-dense", "text": "Let $k$ be a field. Let $X$ be an algebraic space smooth over $\\Spec(k)$. The set of $x \\in |X|$ which are image of morphisms $\\Spec(k') \\to X$ with $k' \\supset k$ finite separable is dense in $|X|$."}
+{"_id": "12874", "title": "spaces-over-fields-lemma-euler-characteristic-additive", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of coherent modules on $X$. Then $$ \\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3) $$"}
+{"_id": "12875", "title": "spaces-over-fields-lemma-euler-characteristic-morphism", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of algebraic spaces proper over $k$. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_Y$-module. Then $$ \\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G}) $$"}
+{"_id": "12876", "title": "spaces-over-fields-lemma-numerical-polynomial-from-euler", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules. The map $$ (n_1, \\ldots, n_r) \\longmapsto \\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}) $$ is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at most the dimension of the scheme theoretic support of $\\mathcal{F}$."}
+{"_id": "12877", "title": "spaces-over-fields-lemma-numerical-polynomial-leading-term", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules. Let $d = \\dim(\\text{Supp}(\\mathcal{F}))$. Let $Z_i \\subset X$ be the irreducible components of $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\overline{x}_i$ be a geometric generic point of $Z_i$ and set $m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}} (\\mathcal{F}_{\\overline{x}_i})$. Then $$ \\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}) - \\sum\\nolimits_i m_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_r^{\\otimes n_r}|_{Z_i}) $$ is a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$."}
+{"_id": "12878", "title": "spaces-over-fields-lemma-intersection-number-integer", "text": "In the situation of Definition \\ref{definition-intersection-number} the intersection number $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is an integer."}
+{"_id": "12881", "title": "spaces-over-fields-lemma-intersection-number-and-pullback", "text": "Let $k$ be a field. Let $f : Y \\to X$ be a morphism of algebraic spaces proper over $k$. Let $Z \\subset Y$ be an integral closed subspace of dimension $d$ and let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. Then $$ (f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) = \\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z)) $$ where $\\deg(Z \\to f(Z))$ is as in Definition \\ref{definition-degree} or $0$ if $\\dim(f(Z)) < d$."}
+{"_id": "12902", "title": "spaces-divisors-lemma-associated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$. The following are equivalent \\begin{enumerate} \\item for some \\'etale morphism $f : U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\\mathcal{F}$, \\item for every \\'etale morphism $f : U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\\mathcal{F}$, \\item the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$ is a weakly associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$. \\end{enumerate} If $X$ is locally Noetherian, then these are also equivalent to \\begin{enumerate} \\item[(4)] for some \\'etale morphism $f : U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$, the point $u$ is associated to $f^*\\mathcal{F}$, \\item[(5)] for every \\'etale morphism $f : U \\to X$ with $U$ a scheme and $u \\in U$ mapping to $x$, the point $u$ is associated to $f^*\\mathcal{F}$, \\item[(6)] the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$ is an associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$. \\end{enumerate}"}
+{"_id": "12904", "title": "spaces-divisors-lemma-ses-weakly-ass", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\\text{WeakAss}(\\mathcal{F}_2) \\subset \\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$ and $\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$."}
+{"_id": "12905", "title": "spaces-divisors-lemma-weakly-ass-zero", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $$ \\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset $$"}
+{"_id": "12906", "title": "spaces-divisors-lemma-minimal-support-in-weakly-ass", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$. If \\begin{enumerate} \\item $x \\in \\text{Supp}(\\mathcal{F})$ \\item $x$ is a codimension $0$ point of $X$ (Properties of Spaces, Definition \\ref{spaces-properties-definition-dimension-local-ring}). \\end{enumerate} Then $x \\in \\text{WeakAss}(\\mathcal{F})$. If $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module with scheme theoretic support $Z$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-scheme-theoretic-support}) and $x$ is a codimension $0$ point of $Z$, then $x \\in \\text{WeakAss}(\\mathcal{F})$."}
+{"_id": "12917", "title": "spaces-divisors-lemma-weakly-ass-change-fields", "text": "Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly associated point of the pullback $\\mathcal{F}_K$, then $x$ is a weakly associated point of $\\mathcal{F}$."}
+{"_id": "12919", "title": "spaces-divisors-lemma-etale-weak-assassin-up-down", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of algebraic spaces. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Let $x \\in |X|$ be a point with image $y \\in |Y|$. Then $$ x \\in \\text{WeakAss}(f^*\\mathcal{G}) \\Leftrightarrow y \\in \\text{WeakAss}(\\mathcal{G}) $$"}
+{"_id": "12920", "title": "spaces-divisors-lemma-locally-noetherian-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. The following are equivalent \\begin{enumerate} \\item for some scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, \\item for every scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, and \\item there exists a field $k$ and a morphism $\\Spec(k) \\to Y$ representing $y$ such that $X_k$ is locally Noetherian. \\end{enumerate} If there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$ representing $y$, then these are also equivalent to \\begin{enumerate} \\item[(4)] the algebraic space $X_{k_0}$ is locally Noetherian. \\end{enumerate}"}
+{"_id": "12922", "title": "spaces-divisors-lemma-relative-assassin", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$ and $y = f(x) \\in |Y|$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Consider commutative diagrams $$ \\xymatrix{ X \\ar[d] & X \\times_Y V \\ar[d] \\ar[l] & X_v \\ar[d] \\ar[l] \\\\ Y & V \\ar[l] & v \\ar[l] } \\quad \\xymatrix{ X \\ar[d] & U \\ar[d] \\ar[l] & U_v \\ar[d] \\ar[l] \\\\ Y & V \\ar[l] & v \\ar[l] } \\quad \\xymatrix{ x \\ar@{|->}[d] & x' \\ar@{|->}[d] \\ar@{|->}[l] & u \\ar@{|->}[ld] \\ar@{|->}[l] \\\\ y & v \\ar@{|->}[l] } $$ where $V$ and $U$ are schemes, $V \\to Y$ and $U \\to X \\times_Y V$ are \\'etale, $v \\in V$, $x' \\in |X_v|$, $u \\in U$ are points related as in the last diagram. Denote $\\mathcal{F}|_{X_v}$ and $\\mathcal{F}|_{U_v}$ the pullbacks of $\\mathcal{F}$. The following are equivalent \\begin{enumerate} \\item for some $V, v, x'$ as above $x'$ is a weakly associated point of $\\mathcal{F}|_{X_v}$, \\item for every $V \\to Y, v, x'$ as above $x'$ is a weakly associated point of $\\mathcal{F}|_{X_v}$, \\item for some $U, V, u, v$ as above $u$ is a weakly associated point of $\\mathcal{F}|_{U_v}$, \\item for every $U, V, u, v$ as above $u$ is a weakly associated point of $\\mathcal{F}|_{U_v}$, \\item for some field $k$ and morphism $\\Spec(k) \\to Y$ representing $y$ and some $t \\in |X_k|$ mapping to $x$, the point $t$ is a weakly associated point of $\\mathcal{F}|_{X_k}$. \\end{enumerate} If there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$ representing $y$, then these are also equivalent to \\begin{enumerate} \\item[(6)] $x_0$ is a weakly associated point of $\\mathcal{F}|_{X_{k_0}}$ where $x_0 \\in |X_{k_0}|$ is the unique point mapping to $x$. \\end{enumerate} If the fibre of $f$ over $y$ is locally Noetherian, then in conditions (1), (2), (3), (4), and (6) we may replace ``weakly associated'' with ``associated''."}
+{"_id": "12924", "title": "spaces-divisors-lemma-base-change-relative-assassin", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module and set $\\mathcal{F}' = (g')^*\\mathcal{F}$. If $f$ is locally of finite type, then \\begin{enumerate} \\item $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}') \\Rightarrow g'(x') \\in \\text{Ass}_{X/Y}(\\mathcal{F})$ \\item if $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$, then given $y' \\in |Y'|$ with $f(x) = g(y')$, there exists an $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$ with $g'(x') = x$ and $f'(x') = y'$. \\end{enumerate}"}
+{"_id": "12926", "title": "spaces-divisors-lemma-relative-weak-assassin-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $i : Z \\to X$ be a finite morphism. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Z$-module. Then $\\text{WeakAss}_{X/Y}(i_*\\mathcal{G}) = i(\\text{WeakAss}_{Z/Y}(\\mathcal{G}))$."}
+{"_id": "12927", "title": "spaces-divisors-lemma-relative-assassin-constructible", "text": "Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation. Let $U \\subset X$ be an open subspace such that $U \\to Y$ is quasi-compact. Then the set $$ E = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset |U_y|\\} $$ is locally constructible in $Y$."}
+{"_id": "12929", "title": "spaces-divisors-lemma-fitting-ideal-of-finitely-presented", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Then $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type."}
+{"_id": "12930", "title": "spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. Let $Z_0 \\subset X$ be the closed subspace cut out by $\\text{Fit}_0(\\mathcal{F})$. Let $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$. Then \\begin{enumerate} \\item $Z \\subset Z_0 \\subset X$ as closed subspaces, \\item $|Z| = |Z_0| = \\text{Supp}(\\mathcal{F})$ as closed subsets of $|X|$, \\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module $\\mathcal{G}_0$ with $$ (Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}. $$ \\end{enumerate}"}
+{"_id": "12933", "title": "spaces-divisors-lemma-locally-free-rank-r-pullback", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. The closed subspaces $$ X = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots $$ defined by the Fitting ideals of $\\mathcal{F}$ have the following properties \\begin{enumerate} \\item The intersection $\\bigcap Z_r$ is empty. \\item The functor $(\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule $$ T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by } \\leq r\\text{ sections} \\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ is representable by the open subspace $X \\setminus Z_r$. \\item The functor $F_r : (\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule $$ T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ is representable by the locally closed subspace $Z_{r - 1} \\setminus Z_r$ of $X$. \\end{enumerate} If $\\mathcal{F}$ is of finite presentation, then $Z_r \\to X$, $X \\setminus Z_r \\to X$, and $Z_{r - 1} \\setminus Z_r \\to X$ are of finite presentation."}
+{"_id": "12935", "title": "spaces-divisors-lemma-characterize-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \\subset X$ be a closed subspace. The following are equivalent: \\begin{enumerate} \\item The subspace $D$ is an effective Cartier divisor on $X$. \\item For some scheme $U$ and surjective \\'etale morphism $U \\to X$ the inverse image $D \\times_X U$ is an effective Cartier divisor on $U$. \\item For every scheme $U$ and every \\'etale morphism $U \\to X$ the inverse image $D \\times_X U$ is an effective Cartier divisor on $U$. \\item For every $x \\in |D|$ there exists an \\'etale morphism $(U, u) \\to (X, x)$ of pointed algebraic spaces such that $U = \\Spec(A)$ and $D \\times_X U = \\Spec(A/(f))$ with $f \\in A$ not a zerodivisor. \\end{enumerate}"}
+{"_id": "12936", "title": "spaces-divisors-lemma-complement-locally-principal-closed-subscheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \\subset X$ be a locally principal closed subspace. Let $U = X \\setminus Z$. Then $U \\to X$ is an affine morphism."}
+{"_id": "12937", "title": "spaces-divisors-lemma-complement-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \\subset X$ be an effective Cartier divisor. Let $U = X \\setminus D$. Then $U \\to X$ is an affine morphism and $U$ is scheme theoretically dense in $X$."}
+{"_id": "12939", "title": "spaces-divisors-lemma-sum-effective-Cartier-divisors", "text": "The sum of two effective Cartier divisors is an effective Cartier divisor."}
+{"_id": "12940", "title": "spaces-divisors-lemma-sum-closed-subschemes-effective-Cartier", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y$ be two closed subspaces of $X$ with ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$ defines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$."}
+{"_id": "12941", "title": "spaces-divisors-lemma-pullback-locally-principal", "text": "Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset X$ be a locally principal closed subspace. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subspace of $X'$."}
+{"_id": "12943", "title": "spaces-divisors-lemma-pullback-effective-Cartier-divisors-additive", "text": "Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$."}
+{"_id": "12944", "title": "spaces-divisors-lemma-conormal-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \\subset X$ be an effective Cartier divisor. Then for the conormal sheaf we have $\\mathcal{C}_{D/X} = \\mathcal{I}_D|D = \\mathcal{O}_X(D)^{\\otimes -1}|_D$."}
+{"_id": "12945", "title": "spaces-divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. Let $D = D_1 + D_2$. Then there is a unique isomorphism $$ \\mathcal{O}_X(D_1) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D_2) \\longrightarrow \\mathcal{O}_X(D) $$ which maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$."}
+{"_id": "12946", "title": "spaces-divisors-lemma-regular-section-structure-sheaf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. The following are equivalent: \\begin{enumerate} \\item $f$ is a regular section, and \\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, \\overline{x}}$ is not a zerodivisor. \\item for any affine $U = \\Spec(A)$ \\'etale over $X$ the restriction $f|_U$ is a nonzerodivisor of $A$, and \\item there exists a scheme $U$ and a surjective \\'etale morphism $U \\to X$ such that $f|_U$ is a regular section of $\\mathcal{O}_U$. \\end{enumerate}"}
+{"_id": "12947", "title": "spaces-divisors-lemma-zero-scheme", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. \\begin{enumerate} \\item Consider closed immersions $i : Z \\to X$ such that $i^*s \\in \\Gamma(Z, i^*\\mathcal{L}))$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set. \\item For any morphism of algebraic spaces $f : Y \\to X$ over $S$ we have $f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if $f$ factors through $Z(s)$. \\item The zero scheme $Z(s)$ is a locally principal closed subspace of $X$. \\item The zero scheme $Z(s)$ is an effective Cartier divisor on $X$ if and only if $s$ is a regular section of $\\mathcal{L}$. \\end{enumerate}"}
+{"_id": "12948", "title": "spaces-divisors-lemma-characterize-OD", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $D \\subset X$ is an effective Cartier divisor, then the canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular. \\item Conversely, if $s$ is a regular section of the invertible sheaf $\\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \\subset X$ and a unique isomorphism $\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$. \\end{enumerate} The constructions $D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$ give mutually inverse maps $$ \\left\\{ \\begin{matrix} \\text{effective Cartier divisors on }X \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{pairs }(\\mathcal{L}, s)\\text{ consisting of an invertible}\\\\ \\mathcal{O}_X\\text{-module and a regular global section} \\end{matrix} \\right\\} $$"}
+{"_id": "12952", "title": "spaces-divisors-lemma-Noetherian-regular-separated-pic-effective-Cartier", "text": "Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. Then every invertible $\\mathcal{O}_X$-module is isomorphic to $$ \\mathcal{O}_X(D - D') = \\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1} $$ for some effective Cartier divisors $D, D'$ in $X$."}
+{"_id": "12953", "title": "spaces-divisors-lemma-smooth-over-valuation-ring-effective-Cartier", "text": "Let $R$ be a valuation ring with fraction field $K$. Let $X$ be an algebraic space over $R$ such that $X \\to \\Spec(R)$ is smooth. For every effective Cartier divisor $D \\subset X_K$ there exists an effective Cartier divisor $D' \\subset X$ with $D'_K = D$."}
+{"_id": "12954", "title": "spaces-divisors-lemma-relative-Cartier", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $D \\subset X$ be a closed subspace. Assume \\begin{enumerate} \\item $D$ is an effective Cartier divisor, and \\item $D \\to Y$ is a flat morphism. \\end{enumerate} Then for every morphism of schemes $g : Y' \\to Y$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = Y' \\times_Y X$ where $g' : X' \\to X$ is the projection."}
+{"_id": "12955", "title": "spaces-divisors-lemma-describe-S", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $U$ affine and \\'etale over $X$ the set $\\mathcal{S}_X(U)$ is the set of nonzerodivisors in $\\mathcal{O}_X(U)$."}
+{"_id": "12956", "title": "spaces-divisors-lemma-meromorphic-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \\begin{enumerate} \\item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \\item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}. \\end{enumerate} Then \\begin{enumerate} \\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras, \\item for $U \\in X_\\etale$ affine $\\mathcal{K}_X(U)$ is the total ring of fractions of $\\mathcal{O}_X(U)$, \\item for a geometric point $\\overline{x}$ the set $\\mathcal{S}_{\\overline{x}}$ the set of nonzerodivisors of $\\mathcal{O}_{X, \\overline{x}}$, and \\item for a geometric point $\\overline{x}$ the ring $\\mathcal{K}_{X, \\overline{x}}$ is the total ring of fractions of $\\mathcal{O}_{X, \\overline{x}}$. \\end{enumerate}"}
+{"_id": "12957", "title": "spaces-divisors-lemma-meromorphic-quasi-coherent-agree", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \\begin{enumerate} \\item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \\item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}. \\item[(c)] $X$ is representable by a scheme $X_0$ (awkward but temporary notation). \\end{enumerate} Then the sheaf of meromorphic functions $\\mathcal{K}_X$ is the quasi-coherent sheaf of $\\mathcal{O}_X$-algebras associated to the quasi-coherent sheaf of meromorphic functions $\\mathcal{K}_{X_0}$."}
+{"_id": "12958", "title": "spaces-divisors-lemma-pullback-meromorphic-sections-defined", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Pullbacks of meromorphic sections are defined in each of the following cases \\begin{enumerate} \\item weakly associated points of $X$ are mapped to points of codimension $0$ on $Y$, \\item $f$ is flat, \\item add more here as needed. \\end{enumerate}"}
+{"_id": "12959", "title": "spaces-divisors-lemma-compute-meromorphic", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume \\begin{enumerate} \\item[(a)] every weakly associated point of $X$ is a point of codimension $0$, and \\item[(b)] $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}, \\item[(c)] every codimension $0$ point of $X$ can be represented by a monomorphism $\\Spec(k) \\to X$. \\end{enumerate} Let $X^0 \\subset |X|$ be the set of codimension $0$ points of $X$. Then we have $$ \\mathcal{K}_X = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} $$ where $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map of Schemes, Section \\ref{schemes-section-points}; this makes sense because $X^0$ is contained in the schematic locus of $X$. Similarly, for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we obtain the formula $$ \\mathcal{K}_X(\\mathcal{F}) = \\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta = \\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta $$ for the sheaf of meromorphic sections of $\\mathcal{F}$. Finally, the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$."}
+{"_id": "12960", "title": "spaces-divisors-lemma-meromorphic-sections-pullback", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition \\ref{definition-pullback-meromorphic-sections}). \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules. There is a canonical pullback map $f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to \\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$ for meromorphic sections of $\\mathcal{F}$. \\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. A regular meromorphic section $s$ of $\\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$. \\end{enumerate}"}
+{"_id": "12962", "title": "spaces-divisors-lemma-relative-proj", "text": "In Situation \\ref{situation-relative-proj}. The functor $F$ above is an algebraic space. For any morphism $g : Z \\to X$ where $Z$ is a scheme there is a canonical isomorphism $\\underline{\\text{Proj}}_Z(g^*\\mathcal{A}) = Z \\times_X F$ compatible with further base change."}
+{"_id": "12964", "title": "spaces-divisors-lemma-relative-proj-base-change", "text": "Let $S$ be a scheme. Let $g : X' \\to X$ be a morphism of algebraic spaces over $S$ and let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_X$-algebras. Then there is a canonical isomorphism $$ r : \\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A}) \\longrightarrow X' \\times_X \\underline{\\text{Proj}}_X(\\mathcal{A}) $$ as well as a corresponding isomorphism $$ \\theta : r^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(d)\\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}(d) $$ of $\\mathbf{Z}$-graded $\\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}$-algebras."}
+{"_id": "12965", "title": "spaces-divisors-lemma-relative-proj-separated", "text": "In Situation \\ref{situation-relative-proj} the morphism $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is separated."}
+{"_id": "12970", "title": "spaces-divisors-lemma-relative-proj-generated-in-degree-1", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_X$-modules generated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$. With $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$ we have \\begin{enumerate} \\item $P$ represents the functor $F_1$ which associates to $T$ over $S$ the set of isomorphism classes of triples $(f, \\mathcal{L}, \\psi)$, where $f : T \\to X$ is a morphism over $S$, $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$ is a map of graded $\\mathcal{O}_T$-algebras inducing a surjection $f^*\\mathcal{A}_1 \\to \\mathcal{L}$, \\item the canonical map $\\pi^*\\mathcal{A}_1 \\to \\mathcal{O}_P(1)$ is surjective, and \\item each $\\mathcal{O}_P(n)$ is invertible and the multiplication maps induce isomorphisms $\\mathcal{O}_P(n) \\otimes_{\\mathcal{O}_P} \\mathcal{O}_P(m) = \\mathcal{O}_P(n + m)$. \\end{enumerate}"}
+{"_id": "12971", "title": "spaces-divisors-lemma-morphism-relative-proj", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a map of quasi-coherent graded $\\mathcal{O}_X$-algebras. Set $P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ and $Q = \\underline{\\text{Proj}}_X(\\mathcal{B}) \\to X$. There is a canonical open subspace $U(\\psi) \\subset Q$ and a canonical morphism of algebraic spaces $$ r_\\psi : U(\\psi) \\longrightarrow P $$ over $X$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras $$ \\theta = \\theta_\\psi : r_\\psi^*\\left( \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_P(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d). $$ The triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property that for any scheme $W$ \\'etale over $X$ the triple $$ (U(\\psi) \\times_X W,\\quad r_\\psi|_{U(\\psi) \\times_X W} : U(\\psi) \\times_X W \\to  P \\times_X W,\\quad \\theta|_{U(\\psi) \\times_X W}) $$ is equal to the triple associated to $\\psi : \\mathcal{A}|_W \\to \\mathcal{B}|_W$ of Constructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}."}
+{"_id": "12972", "title": "spaces-divisors-lemma-morphism-relative-proj-transitive", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be quasi-coherent graded $\\mathcal{O}_X$-algebras. Set $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$, $Q = \\underline{\\text{Proj}}_X(\\mathcal{B})$ and $R = \\underline{\\text{Proj}}_X(\\mathcal{C})$. Let $\\varphi : \\mathcal{A} \\to \\mathcal{B}$, $\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_X$-algebra maps. Then we have $$ U(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi)) \\quad \\text{and} \\quad r_{\\psi \\circ \\varphi} = r_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}. $$ In addition we have $$ \\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi = \\theta_{\\psi \\circ \\varphi} $$ with obvious notation."}
+{"_id": "12973", "title": "spaces-divisors-lemma-surjective-graded-rings-map-relative-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj} above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$. Then \\begin{enumerate} \\item $U(\\psi) = Q$, \\item $r_\\psi : Q \\to R$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$ are surjective but not isomorphisms in general (even if $\\mathcal{A} \\to \\mathcal{B}$ is surjective). \\end{enumerate}"}
+{"_id": "12975", "title": "spaces-divisors-lemma-surjective-generated-degree-1-map-relative-proj", "text": "With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj} above. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$ and that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$. Then \\begin{enumerate} \\item $U(\\psi) = Q$, \\item $r_\\psi : Q \\to P$ is a closed immersion, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$ are isomorphisms. \\end{enumerate}"}
+{"_id": "12976", "title": "spaces-divisors-lemma-invertible-map-into-relative-proj", "text": "With assumptions and notation as above. The morphism $\\psi$ induces a canonical morphism of algebraic spaces over $Y$ $$ r_{\\mathcal{L}, \\psi} : U(\\psi) \\longrightarrow \\underline{\\text{Proj}}_Y(\\mathcal{A}) $$ together with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras $$ \\theta : r_{\\mathcal{L}, \\psi}^*\\left( \\bigoplus\\nolimits_{d \\geq 0} \\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d) \\right) \\longrightarrow \\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)} $$ characterized by the following properties: \\begin{enumerate} \\item For $V \\to Y$ \\'etale and $d \\geq 0$ the diagram $$ \\xymatrix{ \\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} & \\Gamma(V \\times_Y X, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\ \\Gamma(V \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A}), \\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d)) \\ar[r]^-\\theta & \\Gamma(V \\times_Y U(\\psi), \\mathcal{L}^{\\otimes d}) } $$ is commutative. \\item For any $d \\geq 1$ and any morphism $W \\to X$ where $W$ is a scheme such that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$ is surjective we have (a) $W \\to X$ factors through $U(\\psi)$ and (b) composition of $W \\to U(\\psi)$ with $r_{\\mathcal{L}, \\psi}$ agrees with the morphism $W \\to \\underline{\\text{Proj}}_Y(\\mathcal{A})$ which exists by the construction of $\\underline{\\text{Proj}}_Y(\\mathcal{A})$, see Definition \\ref{definition-relative-proj}. \\item Consider a commutative diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ where $X'$ and $Y'$ are schemes, set $\\mathcal{A}' = g^*\\mathcal{A}$ and $\\mathcal{L}' = (g')^*\\mathcal{L}$ and denote $\\psi' : (f')^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} (\\mathcal{L}')^{\\otimes d}$ the pullback of $\\psi$. Let $U(\\psi')$, $r_{\\psi', \\mathcal{L}'}$, and $\\theta'$ be the open, morphism, and homomorphism constructed in Constructions, Lemma \\ref{lemma-invertible-map-into-relative-proj}. Then $U(\\psi') = (g')^{-1}(U(\\psi))$ and $r_{\\psi', \\mathcal{L}'}$ agrees with the base change of $r_{\\psi, \\mathcal{L}}$ via the isomorphism $\\underline{\\text{Proj}}_{Y'}(\\mathcal{A}') = Y' \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A})$ of Lemma \\ref{lemma-relative-proj-base-change}. Moreover, $\\theta'$ is the pullback of $\\theta$. \\end{enumerate}"}
+{"_id": "12978", "title": "spaces-divisors-lemma-ample-base-change", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Y' \\to Y$ be a morphism of algebraic spaces over $S$. Let $f' : X' \\to Y'$ be the base change of $f$ and denote $\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$. If $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample."}
+{"_id": "12979", "title": "spaces-divisors-lemma-relatively-ample-properties", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If there exists an $f$-ample invertible sheaf, then $f$ is representable, quasi-compact, and separated."}
+{"_id": "12980", "title": "spaces-divisors-lemma-descend-relatively-ample", "text": "Let $V \\to U$ be a surjective \\'etale morphism of affine schemes. Let $X$ be an algebraic space over $U$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Y = V \\times_U X$ and let $\\mathcal{N}$ be the pullback of $\\mathcal{L}$ to $Y$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/U$, and \\item $\\mathcal{N}$ is ample on $Y/V$. \\end{enumerate}"}
+{"_id": "12981", "title": "spaces-divisors-lemma-relatively-ample-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/Y$, \\item for every scheme $Z$ and every morphism $Z \\to Y$ the algebraic space $X_Z = Z \\times_Y X$ is a scheme and the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the algebraic space $X_Z = Z \\times_Y X$ is a scheme and the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that the algebraic space $X_V = V \\times_Y X$ is a scheme and the pullback $\\mathcal{L}_V$ is ample on $X_V/V$. \\end{enumerate}"}
+{"_id": "12983", "title": "spaces-divisors-lemma-ample-on-fibre", "text": "Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \\to Y$ is proper. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $y \\in Y$ be a point such that $X_y$ is a scheme and $\\mathcal{L}_y$ is ample on $X_y$. Then there exists a $d_0$ such that for all $d \\geq d_0$ we have $$ R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0 \\text{ for }p > 0 $$ and the map $$ f_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y \\longrightarrow H^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d}) $$ is surjective."}
+{"_id": "12984", "title": "spaces-divisors-lemma-ample-in-neighbourhood", "text": "(For a more general version see Descent on Spaces, Lemma \\ref{spaces-descent-lemma-ample-in-neighbourhood}). Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \\to Y$ is proper. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $y \\in Y$ be a point such that $X_y$ is a scheme and $\\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \\subset Y$ of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$ (as in Definition \\ref{definition-relatively-ample})."}
+{"_id": "12985", "title": "spaces-divisors-lemma-closed-subscheme-proj", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subspace. Denote $\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map $$ \\mathcal{A} \\longrightarrow \\bigoplus\\nolimits_{d \\geq 0} \\pi_*\\left((i_*\\mathcal{O}_Z)(d)\\right) $$ If $\\pi$ is quasi-compact, then there is an isomorphism $Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{I})$."}
+{"_id": "12986", "title": "spaces-divisors-lemma-closed-subscheme-proj-finite", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subscheme. If $\\pi$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ such that $Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$."}
+{"_id": "12987", "title": "spaces-divisors-lemma-closed-subscheme-proj-finite-type", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative Proj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subspace. Let $U \\subset X$ be an open. Assume that \\begin{enumerate} \\item $\\pi$ is quasi-compact, \\item $i$ of finite presentation, \\item $|U| \\cap |\\pi|(|i|(|Z|)) = \\emptyset$, \\item $U$ is quasi-compact, \\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_X$-module for all $n$. \\end{enumerate} Then there exists a $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a) $Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$ and (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$."}
+{"_id": "12988", "title": "spaces-divisors-lemma-conormal-sheaf-section-projective-bundle", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_X$-module. There is a bijection $$ \\left\\{ \\begin{matrix} \\text{sections }\\sigma\\text{ of the } \\\\ \\text{morphism } \\mathbf{P}(\\mathcal{E}) \\to X \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{surjections }\\mathcal{E} \\to \\mathcal{L}\\text{ where} \\\\ \\mathcal{L}\\text{ is an invertible }\\mathcal{O}_X\\text{-module} \\end{matrix} \\right\\} $$ In this case $\\sigma$ is a closed immersion and there is a canonical isomorphism $$ \\Ker(\\mathcal{E} \\to \\mathcal{L}) \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1} \\longrightarrow \\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})} $$ Both the bijection and isomorphism are compatible with base change."}
+{"_id": "12989", "title": "spaces-divisors-lemma-blowing-up-affine", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine scheme \\'etale over $X$ and let $I \\subset A$ be the ideal corresponding to $\\mathcal{I}|_U$. If $X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$, then there is a canonical isomorphism $$ U \\times_X X' = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d) $$ of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \\times_X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\\frac{I}{a}]$."}
+{"_id": "12990", "title": "spaces-divisors-lemma-flat-base-change-blowing-up", "text": "Let $S$ be a scheme. Let $X_1 \\to X_2$ be a flat morphism of algebraic spaces over $S$. Let $Z_2 \\subset X_2$ be a closed subspace. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian diagram $$ \\xymatrix{ X_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\ X_1 \\ar[r] & X_2 } $$ of algebraic spaces over $S$."}
+{"_id": "12991", "title": "spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \\subset X$ be a closed subspace. The blowing up $b : X' \\to X$ of $Z$ in $X$ has the following properties: \\begin{enumerate} \\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$ is an isomorphism, \\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$, \\item there is a canonical isomorphism $\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$ \\end{enumerate}"}
+{"_id": "12992", "title": "spaces-divisors-lemma-universal-property-blowing-up", "text": "\\begin{slogan} Blow up a closed subset to make it Cartier. \\end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \\subset X$ be a closed subspace. Let $\\mathcal{C}$ be the full subcategory of $(\\textit{Spaces}/X)$ consisting of $Y \\to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$ is a final object of $\\mathcal{C}$."}
+{"_id": "12993", "title": "spaces-divisors-lemma-blow-up-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \\subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$."}
+{"_id": "12996", "title": "spaces-divisors-lemma-blow-up-pullback-effective-Cartier", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \\to X$ be a blowup of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^{-1}D$ is defined (see Definition \\ref{definition-pullback-effective-Cartier-divisor})."}
+{"_id": "12997", "title": "spaces-divisors-lemma-blowing-up-two-ideals", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ and $\\mathcal{J}$ be quasi-coherent sheaves of ideals. Let $b : X' \\to X$ be the blowing up of $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the blowing up of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$ is canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$."}
+{"_id": "12998", "title": "spaces-divisors-lemma-blowing-up-projective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$ in the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then $b : X' \\to X$ is a proper morphism."}
+{"_id": "12999", "title": "spaces-divisors-lemma-composition-finite-type-blowups", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Let $Z \\subset X$ be a closed subspace of finite presentation. Let $b : X' \\to X$ be the blowing up with center $Z$. Let $Z' \\subset X'$ be a closed subspace of finite presentation. Let $X'' \\to X'$ be the blowing up with center $Z'$. There exists a closed subspace $Y \\subset X$ of finite presentation, such that \\begin{enumerate} \\item $|Y| = |Z| \\cup |b|(|Z'|)$, and \\item the composition $X'' \\to X$ is isomorphic to the blowing up of $X$ in $Y$. \\end{enumerate}"}
+{"_id": "13000", "title": "spaces-divisors-lemma-strict-transform-local", "text": "In the situation of Definition \\ref{definition-strict-transform}. Let $$ \\xymatrix{ U \\ar[r] \\ar[d] & X \\ar[d] \\\\ V \\ar[r] & B } $$ be a commutative diagram of morphisms with $U$ and $V$ schemes and \\'etale horizontal arrows. Let $V' \\to V$ be the blowup of $V$ in $Z \\times_B V$. Then \\begin{enumerate} \\item $V' = V \\times_B B'$ and the maps $V' \\to B'$ and $U \\times_V V' \\to X \\times_B B'$ are \\'etale, \\item the strict transform $U'$ of $U$ relative to $V' \\to V$ is equal to $X' \\times_X U$ where $X'$ is the strict transform of $X$ relative to $B' \\to B$, and \\item for a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the restriction of the strict transform $\\mathcal{F}'$ to $U \\times_V V'$ is the strict transform of $\\mathcal{F}|_U$ relative to $V' \\to V$. \\end{enumerate}"}
+{"_id": "13001", "title": "spaces-divisors-lemma-strict-transform", "text": "In the situation of Definition \\ref{definition-strict-transform}. \\begin{enumerate} \\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subspace $f^{-1}Z$ of $X$. \\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the strict transform $\\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \\to X \\times_B B'$ of the strict transform of $\\mathcal{F}$ relative to the blowing up $X' \\to X$. \\end{enumerate}"}
+{"_id": "13002", "title": "spaces-divisors-lemma-strict-transform-flat", "text": "In the situation of Definition \\ref{definition-strict-transform}. \\begin{enumerate} \\item If $X$ is flat over $B$ at all points lying over $Z$, then the strict transform of $X$ is equal to the base change $X \\times_B B'$. \\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathcal{F}$ is flat over $B$ at all points lying over $Z$, then the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the pullback $\\text{pr}_X^*\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "13003", "title": "spaces-divisors-lemma-strict-transform-affine", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \\subset B$ be a closed subspace. Let $b : B' \\to B$ be the blowing up of $Z$ in $B$. Let $g : X \\to Y$ be an affine morphism of spaces over $B$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \\times_B B' \\to Y \\times_B B'$ be the base change of $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$ relative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform of $g_*\\mathcal{F}$."}
+{"_id": "13005", "title": "spaces-divisors-lemma-strict-transform-composition-blowups", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \\subset B$ be a closed subspace. Let $b : B' \\to B$ be the blowing up with center $Z$. Let $Z' \\subset B'$ be a closed subspace. Let $B'' \\to B'$ be the blowing up with center $Z'$. Let $Y \\subset B$ be a closed subscheme such that $|Y| = |Z| \\cup |b|(|Z'|)$ and the composition $B'' \\to B$ is isomorphic to the blowing up of $B$ in $Y$. In this situation, given any scheme $X$ over $B$ and $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have \\begin{enumerate} \\item the strict transform of $\\mathcal{F}$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \\to B'$ in $Z'$ of the strict transform of $\\mathcal{F}$ with respect to the blowup $B' \\to B$ of $B$ in $Z$, and \\item the strict transform of $X$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \\to B'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $B' \\to B$ of $B$ in $Z$. \\end{enumerate}"}
+{"_id": "13009", "title": "spaces-divisors-lemma-composition-admissible-blowups", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \\subset X$ be a quasi-compact open subspace. Let $b : X' \\to X$ be a $U$-admissible blowup. Let $X'' \\to X'$ be a $U$-admissible blowup. Then the composition $X'' \\to X$ is a $U$-admissible blowup."}
+{"_id": "13010", "title": "spaces-divisors-lemma-extend-admissible-blowups", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U, V \\subset X$ be quasi-compact open subspaces. Let $b : V' \\to V$ be a $U \\cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \\to X$ whose restriction to $V$ is $V'$."}
+{"_id": "13012", "title": "spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U, V$ be quasi-compact disjoint open subspaces of $X$. Then there exist a $U \\cup V$-admissible blowup $b : X' \\to X$ such that $X'$ is a disjoint union of open subspaces $X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and $b^{-1}(V) \\subset X'_2$."}
+{"_id": "13033", "title": "dga-lemma-dgm-abelian", "text": "Let $(A, d)$ be a differential graded algebra. The category $\\text{Mod}_{(A, \\text{d})}$ is abelian and has arbitrary limits and colimits."}
+{"_id": "13035", "title": "dga-lemma-homotopy-direct-sums", "text": "Let $(A, \\text{d})$ be a differential graded algebra. The homotopy category $K(\\text{Mod}_{(A, \\text{d})})$ has direct sums and products."}
+{"_id": "13036", "title": "dga-lemma-functorial-cone", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Suppose that $$ \\xymatrix{ K_1 \\ar[r]_{f_1} \\ar[d]_a & L_1 \\ar[d]^b \\\\ K_2 \\ar[r]^{f_2} & L_2 } $$ is a diagram of homomorphisms of differential graded $A$-modules which is commutative up to homotopy. Then there exists a morphism $c : C(f_1) \\to C(f_2)$ which gives rise to a morphism of triangles $$ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \\to (K_1, L_1, C(f_1), f_2, i_2, p_2) $$ in $K(\\text{Mod}_{(A, \\text{d})})$."}
+{"_id": "13037", "title": "dga-lemma-admissible-ses", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $0 \\to K \\to L \\to M \\to 0$ be an admissible short exact sequence of differential graded $A$-modules. Let $s : M \\to L$ and $\\pi : L \\to K$ be splittings such that $\\Ker(\\pi) = \\Im(s)$. Then we obtain a morphism $$ \\delta = \\pi \\circ \\text{d}_L \\circ s : M \\to K[1] $$ of $\\text{Mod}_{(A, \\text{d})}$ which induces the boundary maps in the long exact sequence of cohomology (\\ref{equation-les})."}
+{"_id": "13038", "title": "dga-lemma-make-commute-map", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $$ \\xymatrix{ K \\ar[r]_f \\ar[d]_a & L \\ar[d]^b \\\\ M \\ar[r]^g & N } $$ be a diagram of homomorphisms of differential graded $A$-modules commuting up to homotopy. \\begin{enumerate} \\item If $f$ is an admissible monomorphism, then $b$ is homotopic to a homomorphism which makes the diagram commute. \\item If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism which makes the diagram commute. \\end{enumerate}"}
+{"_id": "13039", "title": "dga-lemma-make-injective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $\\alpha : K \\to L$ be a homomorphism of differential graded $A$-modules. There exists a factorization $$ \\xymatrix{ K \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha & \\tilde L \\ar[r]^\\pi & L } $$ in $\\text{Mod}_{(A, \\text{d})}$ such that \\begin{enumerate} \\item $\\tilde \\alpha$ is an admissible monomorphism (see Definition \\ref{definition-admissible-ses}), \\item there is a morphism $s : L \\to \\tilde L$ such that $\\pi \\circ s = \\text{id}_L$ and such that $s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L}$. \\end{enumerate}"}
+{"_id": "13040", "title": "dga-lemma-sequence-maps-split", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $L_1 \\to L_2 \\to \\ldots \\to L_n$ be a sequence of composable homomorphisms of differential graded $A$-modules. There exists a commutative diagram $$ \\xymatrix{ L_1 \\ar[r] & L_2 \\ar[r] & \\ldots \\ar[r] & L_n \\\\ M_1 \\ar[r] \\ar[u] & M_2 \\ar[r] \\ar[u] & \\ldots \\ar[r] & M_n \\ar[u] } $$ in $\\text{Mod}_{(A, \\text{d})}$ such that each $M_i \\to M_{i + 1}$ is an admissible monomorphism and each $M_i \\to L_i$ is a homotopy equivalence."}
+{"_id": "13041", "title": "dga-lemma-nilpotent", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $0 \\to K_i \\to L_i \\to M_i \\to 0$, $i = 1, 2, 3$ be admissible short exact sequence of differential graded $A$-modules. Let $b : L_1 \\to L_2$ and $b' : L_2 \\to L_3$ be homomorphisms of differential graded modules such that $$ \\vcenter{ \\xymatrix{ K_1 \\ar[d]_0 \\ar[r] & L_1 \\ar[r] \\ar[d]_b & M_1 \\ar[d]_0 \\\\ K_2 \\ar[r] & L_2 \\ar[r] & M_2 } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ K_2 \\ar[d]^0 \\ar[r] & L_2 \\ar[r] \\ar[d]^{b'} & M_2 \\ar[d]^0 \\\\ K_3 \\ar[r] & L_3 \\ar[r] & M_3 } } $$ commute up to homotopy. Then $b' \\circ b$ is homotopic to $0$."}
+{"_id": "13043", "title": "dga-lemma-rotate-cone", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $f : K \\to L$ be a homomorphism of differential graded modules. The triangle $(L, C(f), K[1], i, p, f[1])$ is the triangle associated to the admissible short exact sequence $$ 0 \\to L \\to C(f) \\to K[1] \\to 0 $$ coming from the definition of the cone of $f$."}
+{"_id": "13044", "title": "dga-lemma-rotate-triangle", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $\\alpha : K \\to L$ and $\\beta : L \\to M$ define an admissible short exact sequence $$ 0 \\to K \\to L \\to M \\to 0 $$ of differential graded $A$-modules. Let $(K, L, M, \\alpha, \\beta, \\delta)$ be the associated triangle. Then the triangles $$ (M[-1], K, L, \\delta[-1], \\alpha, \\beta) \\quad\\text{and}\\quad (M[-1], K, C(\\delta[-1]), \\delta[-1], i, p) $$ are isomorphic."}
+{"_id": "13045", "title": "dga-lemma-third-isomorphism", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $f_1 : K_1 \\to L_1$ and $f_2 : K_2 \\to L_2$ be homomorphisms of differential graded $A$-modules. Let $$ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \\longrightarrow (K_1, L_1, C(f_1), f_2, i_2, p_2) $$ be any morphism of triangles of $K(\\text{Mod}_{(A, \\text{d})})$. If $a$ and $b$ are homotopy equivalences then so is $c$."}
+{"_id": "13046", "title": "dga-lemma-the-same-up-to-isomorphisms", "text": "Let $(A, \\text{d})$ be a differential graded algebra. \\begin{enumerate} \\item Given an admissible short exact sequence $0 \\to K \\xrightarrow{\\alpha} L \\to M \\to 0$ of differential graded $A$-modules there exists a homotopy equivalence $C(\\alpha) \\to M$ such that the diagram $$ \\xymatrix{ K \\ar[r] \\ar[d] & L \\ar[d] \\ar[r] & C(\\alpha) \\ar[r]_{-p} \\ar[d] & K[1] \\ar[d] \\\\ K \\ar[r]^\\alpha & L \\ar[r]^\\beta & M \\ar[r]^\\delta & K[1] } $$ defines an isomorphism of triangles in $K(\\text{Mod}_{(A, \\text{d})})$. \\item Given a morphism of complexes $f : K \\to L$ there exists an isomorphism of triangles $$ \\xymatrix{ K \\ar[r] \\ar[d] & \\tilde L \\ar[d] \\ar[r] & M \\ar[r]_{\\delta} \\ar[d] & K[1] \\ar[d] \\\\ K \\ar[r] & L \\ar[r] & C(f) \\ar[r]^{-p} & K[1] } $$ where the upper triangle is the triangle associated to a admissible short exact sequence $K \\to \\tilde L \\to M$. \\end{enumerate}"}
+{"_id": "13048", "title": "dga-lemma-two-split-injections", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Suppose that $\\alpha : K \\to L$ and $\\beta : L \\to M$ are admissible monomorphisms of differential graded $A$-modules. Then there exist distinguished triangles $(K, L, Q_1, \\alpha, p_1, d_1)$, $(K, M, Q_2, \\beta \\circ \\alpha, p_2, d_2)$ and $(L, M, Q_3, \\beta, p_3, d_3)$ for which TR4 holds."}
+{"_id": "13049", "title": "dga-lemma-left-right", "text": "Let $(A, \\text{d})$ be a differential graded $R$-algebra. The functor $M \\mapsto M^{opp}$ from the category of left differential graded $A$-modules to the category of right differential graded $A^{opp}$-modules is an equivalence."}
+{"_id": "13050", "title": "dga-lemma-left-module-structure", "text": "In the situation above, let $A$ be a differential graded $R$-algebra. To give a left $A$-module structure on $M$ is the same thing as giving a homomorphism $A \\to E$ of differential graded $R$-algebras."}
+{"_id": "13051", "title": "dga-lemma-characterize-hom", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra. Let $M'$ be a right differential graded $A$-module and let $M$ be a left differential graded $A$-module. Let $N^\\bullet$ be a complex of $R$-modules. Then we have $$ \\Hom_{\\text{Mod}_{(A, d)}}(M', \\Hom(M, N^\\bullet)) = \\Hom_{\\text{Comp}(R)}(M' \\otimes_A M, N^\\bullet) $$ where $M \\otimes_A M$ is viewed as a complex of $R$-modules as in Section \\ref{section-tensor-product}."}
+{"_id": "13052", "title": "dga-lemma-right-module-structure", "text": "In the situation above, let $A$ be a differential graded $R$-algebra. To give a right $A$-module structure on $M$ is the same thing as giving a homomorphism $\\tau : A \\to E^{opp}$ of differential graded $R$-algebras."}
+{"_id": "13054", "title": "dga-lemma-target-graded-projective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M \\to P$ be a surjective homomorphism of differential graded $A$-modules. If $P$ is projective as a graded $A$-module, then $M \\to P$ is an admissible epimorphism."}
+{"_id": "13055", "title": "dga-lemma-hom-from-shift-free", "text": "Let $(A, d)$ be a differential graded algebra. Then we have $$ \\Hom_{\\text{Mod}_{(A, \\text{d})}}(A[k], M) = \\Ker(\\text{d} : M^{-k} \\to M^{-k + 1}) $$ and $$ \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(A[k], M) = H^{-k}(M) $$ for any differential graded $A$-module $M$."}
+{"_id": "13056", "title": "dga-lemma-source-graded-injective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $I \\to M$ be an injective homomorphism of differential graded $A$-modules. If $I$ is graded injective, then $I \\to M$ is an admissible monomorphism."}
+{"_id": "13058", "title": "dga-lemma-hom-into-shift-dual-free", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Then we have $$ \\Hom_{\\text{Mod}_{(A, \\text{d})}}(M, A^\\vee[k]) = \\Ker(\\text{d} : (M^\\vee)^k \\to (M^\\vee)^{k + 1}) $$ and $$ \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, A^\\vee[k]) = H^k(M^\\vee) $$ as functors in the differential graded $A$-module $M$."}
+{"_id": "13059", "title": "dga-lemma-property-P-sequence", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module. If $F_\\bullet$ is a filtration as in property (P), then we obtain an admissible short exact sequence $$ 0 \\to \\bigoplus\\nolimits F_iP \\to \\bigoplus\\nolimits F_iP \\to P \\to 0 $$ of differential graded $A$-modules."}
+{"_id": "13060", "title": "dga-lemma-property-P-K-projective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then $$ \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N) = 0 $$ for all acyclic differential graded $A$-modules $N$."}
+{"_id": "13061", "title": "dga-lemma-good-quotient", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $P \\to M$ of differential graded $A$-modules with the following properties \\begin{enumerate} \\item $P \\to M$ is surjective, \\item $\\Ker(\\text{d}_P) \\to \\Ker(\\text{d}_M)$ is surjective, and \\item $P$ sits in an admissible short exact sequence $0 \\to P' \\to P \\to P'' \\to 0$ where $P'$, $P''$ are direct sums of shifts of $A$. \\end{enumerate}"}
+{"_id": "13062", "title": "dga-lemma-resolve", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $P \\to M$ of differential graded $A$-modules such that \\begin{enumerate} \\item $P \\to M$ is a quasi-isomorphism, and \\item $P$ has property (P). \\end{enumerate}"}
+{"_id": "13063", "title": "dga-lemma-property-I-sequence", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $I$ be a differential graded $A$-module. If $F_\\bullet$ is a filtration as in property (I), then we obtain an admissible short exact sequence $$ 0 \\to I \\to \\prod\\nolimits I/F_iI \\to \\prod\\nolimits I/F_iI \\to 0 $$ of differential graded $A$-modules."}
+{"_id": "13064", "title": "dga-lemma-property-I-K-injective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $I$ be a differential graded $A$-module with property (I). Then $$ \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(N, I) = 0 $$ for all acyclic differential graded $A$-modules $N$."}
+{"_id": "13065", "title": "dga-lemma-good-sub", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $M \\to I$ of differential graded $A$-modules with the following properties \\begin{enumerate} \\item $M \\to I$ is injective, \\item $\\Coker(\\text{d}_M) \\to \\Coker(\\text{d}_I)$ is injective, and \\item $I$ sits in an admissible short exact sequence $0 \\to I' \\to I \\to I'' \\to 0$ where $I'$, $I''$ are products of shifts of $A^\\vee$. \\end{enumerate}"}
+{"_id": "13066", "title": "dga-lemma-right-resolution", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $M \\to I$ of differential graded $A$-modules such that \\begin{enumerate} \\item $M \\to I$ is a quasi-isomorphism, and \\item $I$ has property (I). \\end{enumerate}"}
+{"_id": "13067", "title": "dga-lemma-acyclic", "text": "Let $(A, \\text{d})$ be a differential graded algebra. The full subcategory $\\text{Ac}$ of $K(\\text{Mod}_{(A, \\text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\\text{Mod}_{(A, \\text{d})})$. The corresponding saturated multiplicative system (see Derived Categories, Lemma \\ref{derived-lemma-operations}) of $K(\\text{Mod}_{(A, \\text{d})})$ is the class $\\text{Qis}$ of quasi-isomorphisms. In particular, the kernel of the localization functor $$ Q : K(\\text{Mod}_{(A, \\text{d})}) \\to \\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})}) $$ is $\\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$."}
+{"_id": "13068", "title": "dga-lemma-hom-derived", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ and $N$ be differential graded $A$-modules. \\begin{enumerate} \\item Let $P \\to M$ be a P-resolution as in Lemma \\ref{lemma-resolve}. Then $$ \\Hom_{D(A, \\text{d})}(M, N) = \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N) $$ \\item Let $N \\to I$ be an I-resolution as in Lemma \\ref{lemma-right-resolution}. Then $$ \\Hom_{D(A, \\text{d})}(M, N) = \\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, I) $$ \\end{enumerate}"}
+{"_id": "13069", "title": "dga-lemma-derived-products", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Then \\begin{enumerate} \\item $D(A, \\text{d})$ has both direct sums and products, \\item direct sums are obtained by taking direct sums of differential graded modules, \\item products are obtained by taking products of differential graded modules. \\end{enumerate}"}
+{"_id": "13070", "title": "dga-lemma-derived-canonical-delta-functor", "text": "Let $(A, \\text{d})$ be a differential graded algebra. The functor $\\text{Mod}_{(A, \\text{d})} \\to D(A, \\text{d})$ defined has the natural structure of a $\\delta$-functor, with $$ \\delta_{K \\to L \\to M} = - p \\circ q^{-1} $$ with $p$ and $q$ as explained above."}
+{"_id": "13071", "title": "dga-lemma-homotopy-colimit", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M_n$ be a system of differential graded modules. Then the derived colimit $\\text{hocolim} M_n$ in $D(A, \\text{d})$ is represented by the differential graded module $\\colim M_n$."}
+{"_id": "13072", "title": "dga-lemma-functorial", "text": "Let $R$ be a ring. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of differential graded categories over $R$ induces functors $\\text{Comp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{B})$ and $K(\\mathcal{A}) \\to K(\\mathcal{B})$."}
+{"_id": "13073", "title": "dga-lemma-additive-functor-induces-dga-functor", "text": "Let $F : \\mathcal{B} \\to \\mathcal{B}'$ be an additive functor between additive categories. Then $F$ induces a functor of differential graded categories $$ F : \\text{Comp}^{dg}(\\mathcal{B}) \\to \\text{Comp}^{dg}(\\mathcal{B}') $$ of Example \\ref{example-category-complexes} inducing the usual functors on the category of complexes and the homotopy categories."}
+{"_id": "13074", "title": "dga-lemma-homomorphism-induces-dga-functor", "text": "Let $\\varphi : (A, \\text{d}) \\to (E, \\text{d})$ be a homomorphism of differential graded algebras. Then $\\varphi$ induces a functor of differential graded categories $$ F : \\text{Mod}^{dg}_{(E, \\text{d})} \\longrightarrow \\text{Mod}^{dg}_{(A, \\text{d})} $$ of Example \\ref{example-dgm-dg-cat} inducing obvious restriction functors on the categories of differential graded modules and homotopy categories."}
+{"_id": "13075", "title": "dga-lemma-construction", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category over $R$. Let $x$ be an object of $\\mathcal{A}$. Let $$ (E, \\text{d}) = \\Hom_\\mathcal{A}(x, x) $$ be the differential graded $R$-algebra of endomorphisms of $x$. We obtain a functor $$ \\mathcal{A} \\longrightarrow \\text{Mod}^{dg}_{(E, \\text{d})},\\quad y \\longmapsto \\Hom_\\mathcal{A}(x, y) $$ of differential graded categories by letting $E$ act on $\\Hom_\\mathcal{A}(x, y)$ via composition in $\\mathcal{A}$. This functor induces functors $$ \\text{Comp}(\\mathcal{A}) \\to \\text{Mod}_{(A, \\text{d})} \\quad\\text{and}\\quad K(\\mathcal{A}) \\to K(\\text{Mod}_{(A, \\text{d})}) $$ by an application of Lemma \\ref{lemma-functorial}."}
+{"_id": "13076", "title": "dga-lemma-get-triangle", "text": "Let $\\mathcal{A}$ be a differential graded category satisfying axioms (A) and (B). Given an admissible short exact sequence $x \\to y \\to z$ we obtain (see proof) a triangle $$ x \\to y \\to z \\to x[1] $$ in $\\text{Comp}(\\mathcal{A})$ with the property that any two compositions in $z[-1] \\to x \\to y \\to z \\to x[1]$ are zero in $K(\\mathcal{A})$."}
+{"_id": "13077", "title": "dga-lemma-cone", "text": "\\begin{slogan} The homotopy category is a triangulated category. This lemma proves a part of the axioms of a triangulated category. \\end{slogan} In Situation \\ref{situation-ABC} suppose that $$ \\xymatrix{ x_1 \\ar[r]_{f_1} \\ar[d]_a & y_1 \\ar[d]^b \\\\ x_2 \\ar[r]^{f_2} & y_2 } $$ is a diagram of $\\text{Comp}(\\mathcal{A})$ commutative up to homotopy. Then there exists a morphism $c : c(f_1) \\to c(f_2)$ which gives rise to a morphism of triangles $$ (a, b, c) : (x_1, y_1, c(f_1)) \\to (x_1, y_1, c(f_1)) $$ in $K(\\mathcal{A})$."}
+{"_id": "13078", "title": "dga-lemma-id-cone-null", "text": "In Situation \\ref{situation-ABC} given any object $x$ of $\\mathcal{A}$, and the cone $C(1_x)$ of the identity morphism $1_x : x \\to x$, the identity morphism on $C(1_x)$ is homotopic to zero."}
+{"_id": "13079", "title": "dga-lemma-homo-change", "text": "In Situation \\ref{situation-ABC} given a diagram $$ \\xymatrix{x\\ar[r]^f\\ar[d]_a & y\\ar[d]^b\\\\ z\\ar[r]^g & w} $$ in $\\text{Comp}(\\mathcal{A})$ commuting up to homotopy. Then \\begin{enumerate} \\item If $f$ is an admissible monomorphism, then $b$ is homotopic to a morphism $b'$ which makes the diagram commute. \\item If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism $a'$ which makes the diagram commute. \\end{enumerate}"}
+{"_id": "13080", "title": "dga-lemma-factor", "text": "In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$ be a morphism in $\\text{Comp}(\\mathcal{A})$. Then there exists a factorization in $\\text{Comp}(\\mathcal{A})$: $$ \\xymatrix{ x \\ar[r]^{\\tilde{\\alpha}}  & \\tilde{y} \\ar@<0.5ex>[r]^{\\pi} & y\\ar@<0.5ex>[l]^s } $$ such that \\begin{enumerate} \\item $\\tilde{\\alpha}$ is an admissible monomorphism, and $\\pi\\tilde{\\alpha}=\\alpha$. \\item There exists a morphism $s:y\\to\\tilde{y}$ in $\\text{Comp}(\\mathcal{A})$ such that $\\pi s=1_y$ and $s\\pi$ is homotopic to $1_{\\tilde{y}}$.  \\end{enumerate}"}
+{"_id": "13081", "title": "dga-lemma-analogue-sequence-maps-split", "text": "In Situation \\ref{situation-ABC} let $x_1 \\to x_2 \\to \\ldots \\to x_n$ be a sequence of composable morphisms in $\\text{Comp}(\\mathcal{A})$. Then there exists a commutative diagram in $\\text{Comp}(\\mathcal{A})$: $$ \\xymatrix{x_1\\ar[r] & x_2\\ar[r] & \\ldots\\ar[r] & x_n\\\\ y_1\\ar[r]\\ar[u] & y_2\\ar[r]\\ar[u] & \\ldots\\ar[r] & y_n\\ar[u]} $$ such that each $y_i\\to y_{i+1}$ is an admissible monomorphism and each $y_i\\to x_i$ is a homotopy equivalence."}
+{"_id": "13082", "title": "dga-lemma-triseq", "text": "In Situation \\ref{situation-ABC} let $x_i \\to y_i \\to z_i$ be morphisms in $\\mathcal{A}$ ($i=1,2,3$) such that $x_2 \\to y_2\\to z_2$ is an admissible short exact sequence. Let $b : y_1 \\to y_2$ and $b' : y_2\\to y_3$ be morphisms in $\\text{Comp}(\\mathcal{A})$ such that $$ \\vcenter{ \\xymatrix{ x_1 \\ar[d]_0 \\ar[r] & y_1 \\ar[r] \\ar[d]_b & z_1 \\ar[d]_0 \\\\ x_2 \\ar[r] & y_2 \\ar[r] & z_2 } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ x_2 \\ar[d]^0 \\ar[r] & y_2 \\ar[r] \\ar[d]^{b'} & z_2 \\ar[d]^0 \\\\ x_3 \\ar[r] & y_3 \\ar[r] & z_3 } } $$ commute up to homotopy. Then $b'\\circ b$ is homotopic to $0$."}
+{"_id": "13084", "title": "dga-lemma-restate-axiom-c", "text": "In Situation \\ref{situation-ABC} let $f: x \\to y$ be a morphism in $\\text{Comp}(\\mathcal{A})$. The triangle $(y, c(f), x[1], i, p, f[1])$ is the triangle associated to the admissible short exact sequence  $$ \\xymatrix{y\\ar[r] & c(f) \\ar[r] & x[1]} $$ where the cone $c(f)$ is defined as in Lemma \\ref{lemma-get-triangle}."}
+{"_id": "13085", "title": "dga-lemma-cone-rotate-isom", "text": "In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$ and $\\beta : y \\to z$ define an admissible short exact sequence $$ \\xymatrix{ x \\ar[r] & y\\ar[r] & z } $$ in $\\text{Comp}(\\mathcal{A})$. Let $(x, y, z, \\alpha, \\beta, \\delta)$ be the associated triangle in $K(\\mathcal{A})$. Then, the triangles $$ (z[-1], x, y, \\delta[-1], \\alpha, \\beta) \\quad\\text{and}\\quad (z[-1], x, c(\\delta[-1]), \\delta[-1], i, p) $$ are isomorphic."}
+{"_id": "13086", "title": "dga-lemma-analogue-third-isomorphism", "text": "In Situation \\ref{situation-ABC} let $f_1 : x_1 \\to y_1$ and $f_2 : x_2 \\to y_2$ be morphisms in $\\text{Comp}(\\mathcal{A})$. Let  $$ (a,b,c): (x_1,y_1,c(f_1), f_1, i_1, p_1) \\to (x_2,y_2, c(f_2), f_2, i_1, p_1) $$ be any morphism of triangles in $K(\\mathcal{A})$. If $a$ and $b$ are homotopy equivalences, then so is $c$."}
+{"_id": "13087", "title": "dga-lemma-cone-homotopy", "text": "In Situation \\ref{situation-ABC}. \\begin{enumerate} \\item Given an admissible short exact sequence $x\\xrightarrow{\\alpha} y\\xrightarrow{\\beta} z$. Then there exists a homotopy equivalence $e:C(\\alpha)\\to z$ such that the diagram \\begin{equation} \\label{equation-cone-isom-triangle} \\vcenter{ \\xymatrix{ x\\ar[r]^{\\alpha}\\ar[d] & y\\ar[r]^{b}\\ar[d] & C(\\alpha)\\ar[r]^{-c}\\ar@{.>}[d]^{e} & x[1]\\ar[d] \\\\ x\\ar[r]^{\\alpha} & y\\ar[r]^{\\beta} & z\\ar[r]^{\\delta} & x[1] } } \\end{equation} defines an isomorphism of triangles in $K(\\mathcal{A})$. Here $y\\xrightarrow{b}C(\\alpha)\\xrightarrow{c}x[1]$ is the admissible short exact sequence given as in axiom (C). \\item Given a morphism $\\alpha : x \\to y$ in $\\text{Comp}(\\mathcal{A})$, let $x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to y$ be the factorization given as in Lemma \\ref{lemma-factor}, where the admissible monomorphism $x \\xrightarrow{\\tilde{\\alpha}} y$ extends to the admissible short exact sequence $$ \\xymatrix{ x \\ar[r]^{\\tilde{\\alpha}} & \\tilde{y} \\ar[r] & z } $$ Then there exists an isomorphism of triangles $$ \\xymatrix{ x \\ar[r]^{\\tilde{\\alpha}} \\ar[d] & \\tilde{y} \\ar[r] \\ar[d] & z \\ar[r]^{\\delta} \\ar@{.>}[d]^{e} & x[1] \\ar[d] \\\\ x \\ar[r]^{\\alpha} & y \\ar[r] & C(\\alpha) \\ar[r]^{-c} & x[1] } $$ where the upper triangle is the triangle associated to the sequence $x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to z$. \\end{enumerate}"}
+{"_id": "13088", "title": "dga-lemma-analogue-homotopy-category-pre-triangulated", "text": "In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$ with its natural translation functors and distinguished triangles is a pre-triangulated category."}
+{"_id": "13089", "title": "dga-lemma-dgc-analogue-tr4", "text": "In Situation \\ref{situation-ABC} given admissible monomorphisms $x \\xrightarrow{\\alpha} y$, $y \\xrightarrow{\\beta} z$ in $\\mathcal{A}$, there exist distinguished triangles $(x,y,q_1,\\alpha,p_1,\\delta_1)$, $(x,z,q_2,\\beta\\alpha,p_2,\\delta_2)$ and $(y,z,q_3,\\beta,p_3,\\delta_3)$ for which TR4 holds."}
+{"_id": "13090", "title": "dga-lemma-functor-between-ABC", "text": "Let $R$ be a ring. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor between differential graded categories over $R$ satisfying axioms (A), (B), and (C) such that $F(x[1]) = F(x)[1]$. Then $F$ induces an exact functor $K(\\mathcal{A}) \\to K(\\mathcal{B})$ of triangulated categories."}
+{"_id": "13092", "title": "dga-lemma-bimodule-over-tensor", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. The construction above defines an equivalence of categories $$ \\begin{matrix} \\text{differential graded}\\\\ (A, B)\\text{-bimodules} \\end{matrix} \\longleftrightarrow \\begin{matrix} \\text{right differential graded }\\\\ A^{opp} \\otimes_R B\\text{-modules} \\end{matrix} $$"}
+{"_id": "13093", "title": "dga-lemma-bimodule-resolve", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $M$ be a differential graded $(A, B)$-bimodule. There exists a homomorphism $P \\to M$ of differential graded $(A, B)$-bimodules which is a quasi-isomorphism such that $P$ has property (P) as defined above."}
+{"_id": "13094", "title": "dga-lemma-bimodule-property-P-sequence", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $P$ be a differential graded $(A, B)$-bimodule having property (P) with corresponding filtration $F_\\bullet$, then we obtain a short exact sequence $$ 0 \\to \\bigoplus\\nolimits F_iP \\to \\bigoplus\\nolimits F_iP \\to P \\to 0 $$ of differential graded $(A, B)$-bimodules which is split as a sequence of graded $(A, B)$-bimodules."}
+{"_id": "13095", "title": "dga-lemma-tensor", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then $M \\mapsto M \\otimes_A N$ defines a functor $$ - \\otimes_A N : \\text{Mod}^{dg}_{(A, \\text{d})} \\longrightarrow \\text{Mod}^{dg}_{(B, \\text{d})} $$ of differential graded categories. This functor induces functors $$ \\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(B, \\text{d})} \\quad\\text{and}\\quad K(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})}) $$ by an application of Lemma \\ref{lemma-functorial}."}
+{"_id": "13096", "title": "dga-lemma-hom", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. The construction above defines a functor $$ \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, -) : \\text{Mod}^{dg}_{(B, \\text{d})} \\longrightarrow \\text{Mod}^{dg}_{(A, \\text{d})} $$ of differential graded categories. This functor induces functors $$ \\text{Mod}_{(B, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})} \\quad\\text{and}\\quad K(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})}) $$ by an application of Lemma \\ref{lemma-functorial}."}
+{"_id": "13097", "title": "dga-lemma-tensor-hom-adjunction", "text": "Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. Let $M$ be a right $A$-module, $N$ an $(A, B)$-bimodule, and $N'$ a right $B$-module. Then we have a canonical isomorphism $$ \\Hom_B(M \\otimes_A N, N') = \\Hom_A(M, \\Hom_B(N, N')) $$ of $R$-modules. If $A$, $B$, $M$, $N$, $N'$ are compatibly graded, then we have a canonical isomorphism $$ \\Hom_{\\text{Mod}_B^{gr}}(M \\otimes_A N, N') = \\Hom_{\\text{Mod}_A^{gr}}(M, \\Hom_{\\text{Mod}_B^{gr}}(N, N')) $$ of graded $R$-modules If $A$, $B$, $M$, $N$, $N'$ are compatibly differential graded, then we have a canonical isomorphism $$ \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M \\otimes_A N, N') = \\Hom_{\\text{Mod}^{dg}_{(A, \\text{d})}}(M, \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N')) $$ of complexes of $R$-modules."}
+{"_id": "13099", "title": "dga-lemma-derived-restriction", "text": "In the situation above, the right derived functor of $F$ exists. We denote it $R\\Hom(N, -) : D(B, \\text{d}) \\to D(A, \\text{d})$."}
+{"_id": "13100", "title": "dga-lemma-functoriality-derived-restriction", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $f : N \\to N'$ be a homomorphism of differential graded $(A, B)$-bimodules. Then $f$ induces a morphism of functors $$ - \\circ f : R\\Hom(N', -) \\longrightarrow R\\Hom(N, -) $$ If $f$ is a quasi-isomorphism, then $f \\circ -$ is an isomorphism of functors."}
+{"_id": "13101", "title": "dga-lemma-derived-restriction-exts", "text": "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over a ring $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then for every $n \\in \\mathbf{Z}$ there are isomorphisms $$ H^n(R\\Hom(N, M)) = \\Ext^n_{D(B, \\text{d})}(N, M) $$ of $R$-modules functorial in $M$. It is also functorial in $N$ with respect to the operation described in Lemma \\ref{lemma-functoriality-derived-restriction}."}
+{"_id": "13102", "title": "dga-lemma-compute-derived-restriction", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. If $\\Hom_{D(B, \\text{d})}(N, N') = \\Hom_{K(\\text{Mod}_{(B, \\text{d})})}(N, N')$ for all $N' \\in K(B, \\text{d})$, for example if $N$ has property (P) as a differential graded $B$-module, then $$ R\\Hom(N, M) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, M) $$ functorially in $M$ in $D(B, \\text{d})$."}
+{"_id": "13103", "title": "dga-lemma-existence-of-derived", "text": "In the situation above. If the right derived functor $R\\Hom(K^\\bullet, -)$ of $\\Hom(K^\\bullet, -) : K(\\mathcal{A}) \\to D(\\textit{Ab})$ is everywhere defined on $D(\\mathcal{A})$, then we obtain a canonical exact functor $$ R\\Hom(K^\\bullet, -) : D(\\mathcal{A}) \\longrightarrow D(E, \\text{d}) $$ of triangulated categories which reduces to the usual one  on taking associated complexes of abelian groups."}
+{"_id": "13105", "title": "dga-lemma-derived-bc", "text": "In the situation above, the left derived functor of $F$ exists. We denote it $- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d})$."}
+{"_id": "13106", "title": "dga-lemma-functoriality-bc", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $f : N \\to N'$ be a homomorphism of differential graded $(A, B)$-bimodules. Then $f$ induces a morphism of functors $$ 1\\otimes f : - \\otimes_A^\\mathbf{L} N \\longrightarrow - \\otimes_A^\\mathbf{L} N' $$ If $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism of functors."}
+{"_id": "13107", "title": "dga-lemma-tensor-hom-adjoint", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. Then the functor $$ - \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\longrightarrow D(B, \\text{d}) $$ of Lemma \\ref{lemma-derived-bc} is a left adjoint to the functor $$ R\\Hom(N, -) : D(B, \\text{d}) \\longrightarrow D(A, \\text{d}) $$ of Lemma \\ref{lemma-derived-restriction}."}
+{"_id": "13108", "title": "dga-lemma-tensor-with-compact-fully-faithful", "text": "With notation and assumptions as in Lemma \\ref{lemma-tensor-hom-adjoint}. Assume \\begin{enumerate} \\item $N$ defines a compact object of $D(B, \\text{d})$, and \\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an isomorphism for all $k \\in \\mathbf{Z}$. \\end{enumerate} Then the functor $-\\otimes_A^\\mathbf{L} N$ is fully faithful."}
+{"_id": "13110", "title": "dga-lemma-RHom-is-tensor", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $T$ be a differential graded $(A, B)$-bimodule. Assume \\begin{enumerate} \\item $T$ defines a compact object of $D(B, \\text{d})$, and \\item $S = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(T, B)$ represents $R\\Hom(T, B)$ in $D(A, \\text{d})$. \\end{enumerate} Then $S$ has a structure of a differential graded $(B, A)$-bimodule and there is an isomorphism $$ N \\otimes_B^\\mathbf{L} S \\longrightarrow R\\Hom(T, N) $$ functorial in $N$ in $D(B, \\text{d})$."}
+{"_id": "13111", "title": "dga-lemma-compose-tensor-functors-general", "text": "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and $(C, \\text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-module. Assume (\\ref{equation-plain-versus-derived}) is an isomorphism. Then the composition $$ \\xymatrix{ D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & & D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & & D(C, \\text{d}) } $$ is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ with $N'' = N \\otimes_B N'$ viewed as $(A, C)$-bimodule."}
+{"_id": "13112", "title": "dga-lemma-compose-tensor-functors-general-algebra", "text": "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and $(C, \\text{d})$ be differential graded $R$-algebras. Assume that (\\ref{equation-plain-versus-derived-algebras}) is an isomorphism. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-bimodule. Then the composition $$ \\xymatrix{ D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & & D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & & D(C, \\text{d}) } $$ is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for a differential graded $(A, C)$-bimodule $N''$ described in the proof."}
+{"_id": "13113", "title": "dga-lemma-compose-tensor-functors", "text": "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and $(C, \\text{d})$ be differential graded $R$-algebras. If $C$ is K-flat as a complex of $R$-modules, then (\\ref{equation-plain-versus-derived-algebras}) is an isomorphism and the conclusion of Lemma \\ref{lemma-compose-tensor-functors-general-algebra} is valid."}
+{"_id": "13114", "title": "dga-lemma-tensor-with-complex", "text": "In the situation above there is a functor $$ - \\otimes_E K^\\bullet : \\text{Mod}^{dg}_{(E, \\text{d})} \\longrightarrow \\text{Comp}^{dg}(\\mathcal{O}) $$ of differential graded categories. This functor sends $E$ to $K^\\bullet$ and commutes with direct sums."}
+{"_id": "13115", "title": "dga-lemma-tensor-with-complex-homotopy", "text": "The functor of Lemma \\ref{lemma-tensor-with-complex} defines an exact functor of triangulated categories $K(\\text{Mod}_{(E \\text{d})}) \\to K(\\mathcal{O})$."}
+{"_id": "13116", "title": "dga-lemma-tensor-with-complex-derived", "text": "The functor $K(\\text{Mod}_{(E, \\text{d})}) \\to K(\\mathcal{O})$ of Lemma \\ref{lemma-tensor-with-complex-homotopy} has a left derived version defined on all of $D(E, \\text{d})$. We denote it $- \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\to D(\\mathcal{O})$."}
+{"_id": "13117", "title": "dga-lemma-upgrade-tensor-with-complex-derived", "text": "Let $R$ be a ring. Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of commutative $R$-algebras. Let $K^\\bullet$ be a complex of $\\mathcal{O}$-modules. The functor of Lemma \\ref{lemma-tensor-with-complex-derived} has the following property: For every $M$, $N$ in $D(E, \\text{d})$ there is a canonical map $$ R\\Hom(M, N) \\longrightarrow R\\Hom_\\mathcal{O}(M \\otimes_E^\\mathbf{L} K^\\bullet, N \\otimes_E^\\mathbf{L} K^\\bullet) $$ in $D(R)$ which on cohomology modules gives the maps $$ \\Ext^n_{D(E, \\text{d})}(M, N) \\to \\Ext^n_{D(\\mathcal{O})} (M \\otimes_E^\\mathbf{L} K^\\bullet, N \\otimes_E^\\mathbf{L} K^\\bullet) $$ induced by the functor $- \\otimes_E^\\mathbf{L} K^\\bullet$."}
+{"_id": "13118", "title": "dga-lemma-tensor-with-complex-hom-adjoint", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K^\\bullet$ be a complex of $\\mathcal{O}$-modules. Then the functor $$ - \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}) $$ of Lemma \\ref{lemma-tensor-with-complex-derived} is a left adjoint of the functor $$ R\\Hom(K^\\bullet, -) : D(\\mathcal{O}) \\longrightarrow D(E, \\text{d}) $$ of Lemma \\ref{lemma-existence-of-derived}."}
+{"_id": "13119", "title": "dga-lemma-fully-faithful-in-compact-case", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K^\\bullet$ be a complex of $\\mathcal{O}$-modules. Assume \\begin{enumerate} \\item $K^\\bullet$ represents a compact object of $D(\\mathcal{O})$, and \\item $E = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}(K^\\bullet, K^\\bullet)$ computes the ext groups of $K^\\bullet$ in $D(\\mathcal{O})$. \\end{enumerate} Then the functor $$ - \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\longrightarrow D(\\mathcal{O}) $$ of Lemma \\ref{lemma-tensor-with-complex-derived} is fully faithful."}
+{"_id": "13120", "title": "dga-lemma-factor-through-nicer", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be a compact object of $D(A, \\text{d})$. Let $P$ be a differential graded $A$-module which has a finite filtration $$ 0 = F_{-1}P \\subset F_0P \\subset F_1P \\subset \\ldots \\subset F_nP = P $$ by differential graded submodules such that $$ F_{i + 1}P/F_iP \\cong \\bigoplus\\nolimits_{j \\in J_i} A[k_{i, j}] $$ as differential graded $A$-modules for some sets $J_i$ and integers $k_{i, j}$. Let $E \\to P$ be a morphism of $D(A, \\text{d})$. Then there exists a differential graded submodule $P' \\subset P$ such that $F_{i + 1}P \\cap P'/(F_iP \\cap P')$ is equal to $\\bigoplus_{j \\in J'_i} A[k_{i, j}]$ for some finite subsets $J'_i \\subset J_i$ and such that $E \\to P$ factors through $P'$."}
+{"_id": "13121", "title": "dga-lemma-compact-implies-bounded", "text": "Let $(A, \\text{d})$ be a differential graded algebra. For every compact object $E$ of $D(A, \\text{d})$ there exist integers $a \\leq b$ such that $\\Hom_{D(A, \\text{d})}(E, M) = 0$ if $H^i(M) = 0$ for $i \\in [a, b]$."}
+{"_id": "13122", "title": "dga-lemma-compact", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Assume that $A^n = 0$ for $|n| \\gg 0$. Let $E$ be an object of $D(A, \\text{d})$. The following are equivalent \\begin{enumerate} \\item $E$ is a compact object, and \\item $E$ can be represented by a differential graded $A$-module $P$ which is finite projective as a graded $A$-module and satisfies $\\Hom_{K(A, \\text{d})}(P, M) = \\Hom_{D(A, \\text{d})}(P, M)$ for every differential graded $A$-module $M$. \\end{enumerate}"}
+{"_id": "13123", "title": "dga-lemma-qis-equivalence", "text": "Let $R$ be a ring. Let $(A, \\text{d}) \\to (B, \\text{d})$ be a homomorphism of differential graded algebras over $R$, which induces an isomorphism on cohomology algebras. Then $$ - \\otimes_A^\\mathbf{L} B : D(A, \\text{d}) \\to D(B, \\text{d}) $$ gives an $R$-linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \\mapsto N_A$."}
+{"_id": "13124", "title": "dga-lemma-tilting-equivalence", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be  differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Assume that \\begin{enumerate} \\item $N$ defines a compact object of $D(B, \\text{d})$, \\item if $N' \\in D(B, \\text{d})$ and $\\Hom_{D(B, \\text{d})}(N, N'[n]) = 0$ for $n \\in \\mathbf{Z}$, then $N' = 0$, and \\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an isomorphism for all $k \\in \\mathbf{Z}$. \\end{enumerate} Then $$ - \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d}) $$ gives an $R$-linear equivalence of triangulated categories."}
+{"_id": "13125", "title": "dga-lemma-rickard", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Assume that $A = H^0(A)$. The following are equivalent \\begin{enumerate} \\item $D(A, \\text{d})$ and $D(B, \\text{d})$ are equivalent as $R$-linear triangulated categories, and \\item there exists an object $P$ of $D(B, \\text{d})$ such that \\begin{enumerate} \\item $P$ is a compact object of $D(B, \\text{d})$, \\item if $N \\in D(B, \\text{d})$ with $\\Hom_{D(B, \\text{d})}(P, N[i]) = 0$ for $i \\in \\mathbf{Z}$, then $N = 0$, \\item $\\Hom_{D(B, \\text{d})}(P, P[i]) = 0$ for $i \\not = 0$ and equal to $A$ for $i = 0$. \\end{enumerate} \\end{enumerate} The equivalence $D(A, \\text{d}) \\to D(B, \\text{d})$ constructed in (2) sends $A$ to $P$."}
+{"_id": "13127", "title": "dga-lemma-K-flat-resolution", "text": "Let $R$ be a ring. Let $(B, \\text{d})$ be a differential graded $R$-algebra. There exists a quasi-isomorphism $(A, \\text{d}) \\to (B, \\text{d})$ of differential graded $R$-algebras with the following properties \\begin{enumerate} \\item $A$ is K-flat as a complex of $R$-modules, \\item $A$ is a free graded $R$-algebra. \\end{enumerate}"}
+{"_id": "13129", "title": "dga-lemma-countable", "text": "Let $(A, \\text{d})$ be a differential graded algebra with $H^i(A)$ countable for each $i$. Let $M$ be an object of $D(A, \\text{d})$. Then the following are equivalent \\begin{enumerate} \\item $M = \\text{hocolim} E_n$ with $E_n$ compact in $D(A, \\text{d})$, and \\item $H^i(M)$ is countable for each $i$. \\end{enumerate}"}
+{"_id": "13130", "title": "dga-proposition-homotopy-category-triangulated", "text": "Let $(A, \\text{d})$ be a differential graded algebra. The homotopy category $K(\\text{Mod}_{(A, \\text{d})})$ of differential graded $A$-modules with its natural translation functors and distinguished triangles is a triangulated category."}
+{"_id": "13131", "title": "dga-proposition-ABC-homotopy-category-triangulated", "text": "In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$ with its natural translation functors and distinguished triangles is a triangulated category."}
+{"_id": "13132", "title": "dga-proposition-compact", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be an object of $D(A, \\text{d})$. Then the following are equivalent \\begin{enumerate} \\item $E$ is a compact object, \\item $E$ is a direct summand of an object of $D(A, \\text{d})$ which is represented by a differential graded module $P$ which has a finite filtration $F_\\bullet$ by differential graded submodules such that $F_iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. \\end{enumerate}"}
+{"_id": "13133", "title": "dga-proposition-rickard", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Let $F : D(A, \\text{d}) \\to D(B, \\text{d})$ be an $R$-linear equivalence of triangulated categories. Assume that \\begin{enumerate} \\item $A = H^0(A)$, and \\item $B$ is K-flat as a complex of $R$-modules. \\end{enumerate} Then there exists an $(A, B)$-bimodule $N$ as in Lemma \\ref{lemma-tilting-equivalence}."}
+{"_id": "13173", "title": "spaces-more-groupoids-lemma-first-order-structure-c", "text": "The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition $$ I/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2 $$ where the second arrow comes from the equality $J = (I \\otimes B + B \\otimes I)C$. The map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$."}
+{"_id": "13174", "title": "spaces-more-groupoids-lemma-idenity-on-conormal", "text": "In the situation discussed in this section, let $\\delta \\in \\Gamma_0$ and $f = t \\circ \\delta : U \\to U$. If $s, t$ are flat, then the canonical map $\\mathcal{C}_{U_0/U} \\to \\mathcal{C}_{U_0/U}$ induced by $f$ (More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-conormal-functorial}) is the identity map."}
+{"_id": "13175", "title": "spaces-more-groupoids-lemma-composition-is-addition", "text": "The bijection (\\ref{equation-isomorphism}) is an isomorphism of groups."}
+{"_id": "13177", "title": "spaces-more-groupoids-lemma-property-G-invariant", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \\to U$ be its stabilizer group algebraic space. Let $\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\\tau$-local on the target. Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the $\\tau$-topology. Let $W \\subset U$ be the maximal open subspace such that $G_W \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids in Spaces, Definition \\ref{spaces-groupoids-definition-invariant-open})."}
+{"_id": "13178", "title": "spaces-more-groupoids-lemma-two-fibres", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $K$ be a field and let $r, r' : \\Spec(K) \\to R$ be morphisms such that $t \\circ r = t \\circ r' : \\Spec(K) \\to U$. Set $u = s \\circ r$, $u' = s \\circ r'$ and denote $F_u = \\Spec(K) \\times_{u, U, s} R$ and $F_{u'} = \\Spec(K) \\times_{u', U, s} R$ the fibre products. Then $F_u \\cong F_{u'}$ as algebraic spaces over $K$."}
+{"_id": "13179", "title": "spaces-more-groupoids-lemma-restrict-preserves-type", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. \\begin{enumerate} \\item If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type. \\item If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation. \\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat. \\item Add more here. \\end{enumerate}"}
+{"_id": "13180", "title": "spaces-more-groupoids-lemma-groupoid-on-field-open-multiplication", "text": "In Situation \\ref{situation-groupoid-on-field} the composition morphism $c : R \\times_{s, U, t} R \\to R$ is flat and universally open. In Situation \\ref{situation-group-over-field} the group law $m : G \\times_k G \\to G$ is flat and universally open."}
+{"_id": "13181", "title": "spaces-more-groupoids-lemma-group-scheme-over-field-separated", "text": "In Situation \\ref{situation-groupoid-on-field} assume $R$ is a decent space. Then $R$ is a separated algebraic space. In Situation \\ref{situation-group-over-field} assume that $G$ is a decent algebraic space. Then $G$ is separated algebraic space."}
+{"_id": "13182", "title": "spaces-more-groupoids-lemma-restrict-groupoid-on-field", "text": "In Situation \\ref{situation-groupoid-on-field}. Let $k \\subset k'$ be a field extension, $U' = \\Spec(k')$ and let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram $$ \\xymatrix{ R' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] & R \\times_{s, U} U' \\ar[r] \\ar[d] & U' \\ar[d] \\\\ U' \\times_{U, t} R \\ar[d] \\ar[r] & R \\ar[r]^s \\ar[d]_t & U \\\\ U' \\ar[r] & U } $$ all the morphisms are surjective, flat, and universally open. The dotted arrow $R' \\to R$ is in addition affine."}
+{"_id": "13183", "title": "spaces-more-groupoids-lemma-groupoid-on-field-explain-points", "text": "In Situation \\ref{situation-groupoid-on-field}. For any point $r \\in |R|$ there exist \\begin{enumerate} \\item a field extension $k \\subset k'$ with $k'$ algebraically closed, \\item a point $r' : \\Spec(k') \\to R'$ where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $\\Spec(k') \\to \\Spec(k)$ \\end{enumerate} such that \\begin{enumerate} \\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and \\item the maps $s' \\circ r', t' \\circ r' : \\Spec(k') \\to \\Spec(k')$ are automorphisms. \\end{enumerate}"}
+{"_id": "13184", "title": "spaces-more-groupoids-lemma-groupoid-on-field-move-point", "text": "In Situation \\ref{situation-groupoid-on-field}. If $r : \\Spec(k) \\to R$ is a morphism such that $s \\circ r, t \\circ r$ are automorphisms of $\\Spec(k)$, then the map $$ R \\longrightarrow R, \\quad x \\longmapsto c(r, x) $$ is an automorphism $R \\to R$ which maps $e$ to $r$."}
+{"_id": "13186", "title": "spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension", "text": "In Situation \\ref{situation-groupoid-on-field} assume $s, t$ are locally of finite type. For all $r \\in |R|$ \\begin{enumerate} \\item $\\dim(R) = \\dim_r(R)$, \\item the transcendence degree of $r$ over $\\Spec(k)$ via $s$ equals the transcendence degree of $r$ over $\\Spec(k)$ via $t$, and \\item if the transcendence degree mentioned in (2) is $0$, then $\\dim(R) = \\dim(\\mathcal{O}_{R, \\overline{r}})$. \\end{enumerate}"}
+{"_id": "13187", "title": "spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension", "text": "In Situation \\ref{situation-group-over-field} assume $G$ locally of finite type. For all $g \\in |G|$ \\begin{enumerate} \\item $\\dim(G) = \\dim_g(G)$, \\item if the transcendence degree of $g$ over $k$ is $0$, then $\\dim(G) = \\dim(\\mathcal{O}_{G, \\overline{g}})$. \\end{enumerate}"}
+{"_id": "13188", "title": "spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer", "text": "In Situation \\ref{situation-groupoid-on-field} assume $s, t$ are locally of finite type. Let $G = \\Spec(k) \\times_{\\Delta, \\Spec(k) \\times_B \\Spec(k), t \\times s} R$ be the stabilizer group algebraic space. Then we have $\\dim(R) = \\dim(G)$."}
+{"_id": "13189", "title": "spaces-more-groupoids-lemma-group-space-scheme-over-kbar", "text": "Let $k$ be a field with algebraic closure $\\overline{k}$. Let $G$ be a group algebraic space over $k$ which is separated\\footnote{It is enough to assume $G$ is decent, e.g., locally separated or quasi-separated by Lemma \\ref{lemma-group-scheme-over-field-separated}.}. Then $G_{\\overline{k}}$ is a scheme."}
+{"_id": "13190", "title": "spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k", "text": "Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated and locally of finite type over $k$, then $G$ is a scheme."}
+{"_id": "13191", "title": "spaces-more-groupoids-lemma-factor-through-over-open", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \\to Y$ and $g : X \\to Z$ be morphisms of algebraic spaces over $B$. Assume \\begin{enumerate} \\item $Y \\to B$ is separated, \\item $g$ is surjective, flat, and locally of finite presentation, \\item there is a scheme theoretically dense open $V \\subset Z$ such that $f|_{g^{-1}(V)} : g^{-1}(V) \\to Y$ factors through $V$. \\end{enumerate} Then $f$ factors through $g$."}
+{"_id": "13192", "title": "spaces-more-groupoids-lemma-quotient-power-P1", "text": "\\begin{slogan} A morphism from a nonempty product of projective lines over a field to a separated finite type algebraic space over a field factors as a finite morphism after a projection to a product of projective lines. \\end{slogan} Let $k$ be a field. Let $n \\geq 1$ and let $(\\mathbf{P}^1_k)^n$ be the $n$-fold self product over $\\Spec(k)$. Let $f : (\\mathbf{P}^1_k)^n \\to Z$ be a morphism of algebraic spaces over $k$. If $Z$ is separated of finite type over $k$, then $f$ factors as $$ (\\mathbf{P}^1_k)^n \\xrightarrow{projection} (\\mathbf{P}^1_k)^m \\xrightarrow{finite} Z. $$"}
+{"_id": "13193", "title": "spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group", "text": "\\begin{slogan} No complete rational curves on groups. \\end{slogan} Let $k$ be a field. Let $G$ be a separated group algebraic space locally of finite type over $k$. There does not exist a nonconstant morphism $f : \\mathbf{P}^1_k \\to G$ over $\\Spec(k)$."}
+{"_id": "13194", "title": "spaces-more-groupoids-lemma-finite-sheaf", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then we have \\begin{enumerate} \\item The presheaf $(X/Y)_{fin}$ satisfies the sheaf condition for the fppf topology. \\item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, (X/Y)_{fin}) = \\{(a, Z)\\text{ satisfying \\ref{equation-finite-conditions}}\\} $$ \\end{enumerate}"}
+{"_id": "13195", "title": "spaces-more-groupoids-lemma-finite-open", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X' \\ar[rr]_j \\ar[rd] & & X \\ar[ld] \\\\ & Y } $$ of algebraic spaces over $S$. If $j$ is an open immersion, then there is a canonical injective map of sheaves $j : (X'/Y)_{fin} \\to (X/Y)_{fin}$."}
+{"_id": "13196", "title": "spaces-more-groupoids-lemma-finite-lives-on-locally-quasi-finite-part", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $X' \\subset X$ be the maximal open subspace over which $f$ is locally quasi-finite, see Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}. Then $(X/Y)_{fin} = (X'/Y)_{fin}$."}
+{"_id": "13197", "title": "spaces-more-groupoids-lemma-finite-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $T$ be an algebraic space over $S$, and let $(a, Z)$ be a pair as in \\ref{equation-finite-conditions}. If $f$ is separated, then $Z$ is closed in $T \\times_Y X$."}
+{"_id": "13198", "title": "spaces-more-groupoids-lemma-finite-diagonal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The diagonal of $(X/Y)_{fin} \\to Y$ $$ (X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times_Y (X/Y)_{fin} $$ is representable (by schemes) and an open immersion and the ``absolute'' diagonal $$ (X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times (X/Y)_{fin} $$ is representable (by schemes)."}
+{"_id": "13199", "title": "spaces-more-groupoids-lemma-finite-criterion-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Suppose that $U$ is a scheme, $U \\to Y$ is an \\'etale morphism and $Z \\subset U \\times_Y X$ is an open subspace finite over $U$. Then the induced morphism $U \\to (X/Y)_{fin}$ is \\'etale."}
+{"_id": "13200", "title": "spaces-more-groupoids-lemma-finite-pullback", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[d] \\ar[r] & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ be a fibre product square of algebraic spaces over $S$. Then $$ \\xymatrix{ (X'/Y')_{fin} \\ar[d] \\ar[r] & (X/Y)_{fin} \\ar[d] \\\\ Y' \\ar[r] & Y } $$ is a fibre product square of sheaves on $(\\Sch/S)_{fppf}$."}
+{"_id": "13201", "title": "spaces-more-groupoids-lemma-finite-surjective-etale-cover", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated and locally quasi-finite, then there exists a scheme $U$ \\'etale over $Y$ and a surjective \\'etale morphism $U \\to (X/Y)_{fin}$ over $Y$."}
+{"_id": "13202", "title": "spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is separated, flat, and locally of finite presentation. In this case \\begin{enumerate} \\item $(X/Y)_{fin} \\to Y$ is separated, representable, and \\'etale, and \\item if $Y$ is a scheme, then $(X/Y)_{fin}$ is (representable by) a scheme. \\end{enumerate}"}
+{"_id": "13203", "title": "spaces-more-groupoids-lemma-finite-plus-section", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\sigma : Y \\to X$ be a section of $f$. Consider the transformation of functors $$ t : (X/Y, \\sigma)_{fin} \\longrightarrow (X/Y)_{fin}. $$ defined above. Then \\begin{enumerate} \\item $t$ is representable by open immersions, \\item if $f$ is separated, then $t$ is representable by open and closed immersions, \\item if $(X/Y)_{fin}$ is an algebraic space, then $(X/Y, \\sigma)_{fin}$ is an algebraic space and an open subspace of $(X/Y)_{fin}$, and \\item if $(X/Y)_{fin}$ is a scheme, then $(X/Y, \\sigma)_{fin}$ is an open subscheme of it. \\end{enumerate}"}
+{"_id": "13204", "title": "spaces-more-groupoids-lemma-finite-part-groupoid", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. Assume the morphisms $s, t$ are separated and locally of finite type. There exists a canonical morphism $$ (U', Z_{univ}, s', t', c', e', i') \\longrightarrow (U, R, s, t, c, e, i) $$ of groupoids in algebraic spaces over $B$ where \\begin{enumerate} \\item $g : U' \\to U$ is identified with $(R_s/U, e)_{fin} \\to U$, and \\item $Z_{univ} \\subset R \\times_{s, U, g} U'$ is the universal open (and closed) subspace finite over $U'$ which contains the base change of the unit $e$. \\end{enumerate}"}
+{"_id": "13205", "title": "spaces-more-groupoids-lemma-strong-splitting", "text": "In Situation \\ref{situation-strong-splitting} there exists an algebraic space $U'$, an \\'etale morphism $U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$ lying over $u : \\Spec(\\kappa(u)) \\to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$."}
+{"_id": "13206", "title": "spaces-more-groupoids-lemma-splitting", "text": "In Situation \\ref{situation-splitting} there exists an algebraic space $U'$, an \\'etale morphism $U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$ lying over $u : \\Spec(\\kappa(u)) \\to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$."}
+{"_id": "13207", "title": "spaces-more-groupoids-lemma-quasi-splitting", "text": "In Situation \\ref{situation-quasi-splitting} there exists an algebraic space $U'$, an \\'etale morphism $U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$ lying over $u : \\Spec(\\kappa(u)) \\to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$."}
+{"_id": "13208", "title": "spaces-more-groupoids-lemma-strong-splitting-scheme", "text": "In Situation \\ref{situation-strong-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$."}
+{"_id": "13209", "title": "spaces-more-groupoids-lemma-splitting-scheme", "text": "In Situation \\ref{situation-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$."}
+{"_id": "13210", "title": "spaces-more-groupoids-lemma-quasi-splitting-scheme", "text": "In Situation \\ref{situation-quasi-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$."}
+{"_id": "13211", "title": "spaces-more-groupoids-lemma-strong-splitting-affine-scheme", "text": "In Situation \\ref{situation-strong-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$."}
+{"_id": "13212", "title": "spaces-more-groupoids-lemma-splitting-affine-scheme", "text": "In Situation \\ref{situation-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$."}
+{"_id": "13213", "title": "spaces-more-groupoids-lemma-quasi-splitting-affine-scheme", "text": "In Situation \\ref{situation-quasi-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \\'etale morphism $U' \\to U$, and a point $u' \\in U'$ lying over $u$ with $\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$."}
+{"_id": "13214", "title": "spaces-more-groupoids-proposition-group-space-scheme-over-field", "text": "Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated, then $G$ is a scheme."}
+{"_id": "13215", "title": "spaces-more-groupoids-proposition-finite-algebraic-space", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is separated and locally of finite type. Then $(X/Y)_{fin}$ is an algebraic space. Moreover, the morphism $(X/Y)_{fin} \\to Y$ is \\'etale."}
+{"_id": "13221", "title": "modules-lemma-abelian", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The category $\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover a complex $$ \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} $$ is exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the complex $$ \\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x $$ is exact at $\\mathcal{G}_x$."}
+{"_id": "13222", "title": "modules-lemma-limits-colimits", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item All limits exist in $\\textit{Mod}(\\mathcal{O}_X)$. Limits are the same as the corresponding limits of presheaves of $\\mathcal{O}_X$-modules (i.e., commute with taking sections over opens). \\item All colimits exist in $\\textit{Mod}(\\mathcal{O}_X)$. Colimits are the sheafification of the corresponding colimit in the category of presheaves. Taking colimits commutes with taking stalks. \\item Filtered colimits are exact. \\item Finite direct sums are the same as the corresponding finite direct sums of presheaves of $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "13223", "title": "modules-lemma-exactness-pushforward-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. \\begin{enumerate} \\item The functor $f_* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$ is left exact. In fact it commutes with all limits. \\item The functor $f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$ is right exact. In fact it commutes with all colimits. \\item Pullback $f^{-1} : \\textit{Ab}(Y) \\to \\textit{Ab}(X)$ on abelian sheaves is exact. \\end{enumerate}"}
+{"_id": "13224", "title": "modules-lemma-j-shriek-exact", "text": "Let $j : U \\to X$ be an open immersion of topological spaces. The functor $j_! : \\textit{Ab}(U) \\to \\textit{Ab}(X)$ is exact."}
+{"_id": "13226", "title": "modules-lemma-globally-generated", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $I$ be a set. Let $s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$ be global sections. The sections $s_i$ generate $\\mathcal{F}$ if and only if for all $x\\in X$ the elements $s_{i, x} \\in \\mathcal{F}_x$ generate the $\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$."}
+{"_id": "13227", "title": "modules-lemma-tensor-product-globally-generated", "text": "\\begin{slogan} The tensor product of globally generated sheaves of modules is globally generated. \\end{slogan} Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ and $\\mathcal{G}$ are generated by global sections then so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$."}
+{"_id": "13228", "title": "modules-lemma-generated-by-local-sections", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $I$ be a set. Let $s_i$, $i \\in I$ be a collection of local sections of $\\mathcal{F}$, i.e., $s_i \\in \\mathcal{F}(U_i)$ for some opens $U_i \\subset X$. There exists a unique smallest subsheaf of $\\mathcal{O}_X$-modules $\\mathcal{G}$ such that each $s_i$ corresponds to a local section of $\\mathcal{G}$."}
+{"_id": "13229", "title": "modules-lemma-generated-by-local-sections-stalk", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Given a set $I$, and local sections $s_i$, $i \\in I$ of $\\mathcal{F}$. Let $\\mathcal{G}$ be the subsheaf generated by the $s_i$ and let $x\\in X$. Then $\\mathcal{G}_x$ is the $\\mathcal{O}_{X, x}$-submodule of $\\mathcal{F}_x$ generated by the elements $s_{i, x}$ for those $i$ such that $s_i$ is defined at $x$."}
+{"_id": "13230", "title": "modules-lemma-support-section-closed", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $U \\subset X$ open. \\begin{enumerate} \\item The support of $s \\in \\mathcal{F}(U)$ is closed in $U$. \\item The support of $fs$ is contained in the intersections of the supports of $f \\in \\mathcal{O}_X(U)$ and $s \\in \\mathcal{F}(U)$. \\item The support of $s + s'$ is contained in the union of the supports of $s, s' \\in \\mathcal{F}(U)$. \\item The support of $\\mathcal{F}$ is the union of the supports of all local sections of $\\mathcal{F}$. \\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of $\\mathcal{O}_X$-modules, then the support of $\\varphi(s)$ is contained in the support of $s \\in \\mathcal{F}(U)$. \\end{enumerate}"}
+{"_id": "13231", "title": "modules-lemma-support-sheaf-rings-closed", "text": "Let $X$ be a topological space. The support of a sheaf of rings is closed."}
+{"_id": "13232", "title": "modules-lemma-i-star-exact", "text": "Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset. Denote $i : Z \\to X$ the inclusion map. The functor $$ i_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X) $$ is exact, fully faithful, with essential image exactly those abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$ is a left inverse to $i_*$."}
+{"_id": "13233", "title": "modules-lemma-i-star-right-adjoint", "text": "Let $i : Z \\to X$ be the inclusion of a closed subset into the topological space $X$. The functor $\\textit{Ab}(X) \\to \\textit{Ab}(Z)$, $\\mathcal{F} \\mapsto \\mathcal{H}_Z(\\mathcal{F})$ of Remark \\ref{remark-sections-support-in-closed} is a right adjoint to $i_* : \\textit{Ab}(Z) \\to \\textit{Ab}(Z)$. In particular $i_*$ commutes with arbitrary colimits."}
+{"_id": "13234", "title": "modules-lemma-canonical-exact-sequence", "text": "Let $X$ be a topological space. Let $U \\subset X$ be an open subset with complement $Z \\subset X$. Denote $j : U \\to X$ the open immersion and $i : Z \\to X$ the closed immersion. For any sheaf of abelian groups $\\mathcal{F}$ on $X$ the adjunction mappings $j_{!}j^*\\mathcal{F} \\to \\mathcal{F}$ and $\\mathcal{F} \\to i_*i^*\\mathcal{F}$ give a short exact sequence $$ 0 \\to j_{!}j^*\\mathcal{F} \\to \\mathcal{F} \\to i_*i^*\\mathcal{F} \\to 0 $$ of sheaves of abelian groups. For any morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of abelian sheaves on $X$ we obtain a morphism of short exact sequences $$ \\xymatrix{ 0 \\ar[r] & j_{!}j^*\\mathcal{F} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & i_*i^*\\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & j_{!}j^*\\mathcal{G} \\ar[r] & \\mathcal{G} \\ar[r] & i_*i^*\\mathcal{G} \\ar[r] & 0 } $$"}
+{"_id": "13235", "title": "modules-lemma-pullback-locally-generated", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\\mathcal{G}$ is locally generated by sections if $\\mathcal{G}$ is locally generated by sections."}
+{"_id": "13236", "title": "modules-lemma-pullback-finite-type", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\\mathcal{G}$ of a finite type $\\mathcal{O}_Y$-module is a finite type $\\mathcal{O}_X$-module."}
+{"_id": "13237", "title": "modules-lemma-extension-finite-type", "text": "Let $X$ be a ringed space. The image of a morphism of $\\mathcal{O}_X$-modules of finite type is of finite type. Let $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ be a short exact sequence of $\\mathcal{O}_X$-modules. If $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are of finite type, so is $\\mathcal{F}_2$."}
+{"_id": "13238", "title": "modules-lemma-finite-type-surjective-on-stalk", "text": "Let $X$ be a ringed space. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism of $\\mathcal{O}_X$-modules. Let $x \\in X$. Assume $\\mathcal{F}$ of finite type and the map on stalks $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ surjective. Then there exists an open neighbourhood $x \\in U \\subset X$ such that $\\varphi|_U$ is surjective."}
+{"_id": "13239", "title": "modules-lemma-finite-type-stalk-zero", "text": "Let $X$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $x \\in X$. Assume $\\mathcal{F}$ of finite type and $\\mathcal{F}_x = 0$. Then there exists an open neighbourhood $x \\in U \\subset X$ such that $\\mathcal{F}|_U$ is zero."}
+{"_id": "13240", "title": "modules-lemma-support-finite-type-closed", "text": "\\begin{slogan} Over any ringed space, sheaves of modules of finite type have closed support. \\end{slogan} Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is of finite type then support of $\\mathcal{F}$ is closed."}
+{"_id": "13241", "title": "modules-lemma-finite-type-quasi-compact-colimit", "text": "Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\\mathcal{F}_i, f_{ii'})$ be a system over $I$ consisting of sheaves of $\\mathcal{O}_X$-modules (see Categories, Section \\ref{categories-section-posets-limits}). Let $\\mathcal{F} = \\colim \\mathcal{F}_i$ be the colimit. Assume (a) $I$ is directed, (b) $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and (c) $X$ is quasi-compact. Then there exists an $i$ such that $\\mathcal{F}_i \\to \\mathcal{F}$ is surjective. If the transition maps $f_{ii'}$ are injective then we conclude that $\\mathcal{F} = \\mathcal{F}_i$ for some $i \\in I$."}
+{"_id": "13242", "title": "modules-lemma-set-isomorphism-classes-finite-type-modules", "text": "Let $X$ be a ringed space. There exists a set of $\\mathcal{O}_X$-modules $\\{\\mathcal{F}_i\\}_{i \\in I}$ of finite type such that each finite type $\\mathcal{O}_X$-module on $X$ is isomorphic to exactly one of the $\\mathcal{F}_i$."}
+{"_id": "13243", "title": "modules-lemma-direct-sum-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The direct sum of two quasi-coherent $\\mathcal{O}_X$-modules is a quasi-coherent $\\mathcal{O}_X$-module."}
+{"_id": "13244", "title": "modules-lemma-pullback-quasi-coherent", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\\mathcal{G}$ of a quasi-coherent $\\mathcal{O}_Y$-module is quasi-coherent."}
+{"_id": "13245", "title": "modules-lemma-construct-quasi-coherent-sheaves", "text": "Let $(X, \\mathcal{O}_X)$ be ringed space. Let $\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$ be a ring homomorphism from a ring $R$ into the ring of global sections on $X$. Let $M$ be an $R$-module. The following three constructions give canonically isomorphic sheaves of $\\mathcal{O}_X$-modules: \\begin{enumerate} \\item Let $\\pi : (X, \\mathcal{O}_X) \\longrightarrow (\\{*\\}, R)$ be the morphism of ringed spaces with $\\pi : X \\to \\{*\\}$ the unique map and with $\\pi$-map $\\pi^\\sharp$ the given map $\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$. Set $\\mathcal{F}_1 = \\pi^*M$. \\item Choose a presentation $\\bigoplus_{j \\in J} R \\to \\bigoplus_{i \\in I} R \\to M \\to 0$. Set $$ \\mathcal{F}_2 = \\Coker\\left( \\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X \\to \\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X \\right). $$ Here the map on the component $\\mathcal{O}_X$ corresponding to $j \\in J$ given by the section $\\sum_i \\alpha(r_{ij})$ where the $r_{ij}$ are the matrix coefficients of the map in the presentation of $M$. \\item Set $\\mathcal{F}_3$ equal to the sheaf associated to the presheaf $U \\mapsto \\mathcal{O}_X(U) \\otimes_R M$, where the map $R \\to \\mathcal{O}_X(U)$ is the composition of $\\alpha$ and the restriction map $\\mathcal{O}_X(X) \\to \\mathcal{O}_X(U)$. \\end{enumerate} This construction has the following properties: \\begin{enumerate} \\item The resulting sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}_M = \\mathcal{F}_1 = \\mathcal{F}_2 = \\mathcal{F}_3$ is quasi-coherent. \\item The construction gives a functor from the category of $R$-modules to the category of quasi-coherent sheaves on $X$ which commutes with arbitrary colimits. \\item For any $x \\in X$ we have $\\mathcal{F}_{M, x} = \\mathcal{O}_{X, x} \\otimes_R M$ functorial in $M$. \\item Given any $\\mathcal{O}_X$-module $\\mathcal{G}$ we have $$ \\Mor_{\\mathcal{O}_X}(\\mathcal{F}_M, \\mathcal{G}) = \\Hom_R(M, \\Gamma(X, \\mathcal{G})) $$ where the $R$-module structure on $\\Gamma(X, \\mathcal{G})$ comes from the $\\Gamma(X, \\mathcal{O}_X)$-module structure via $\\alpha$. \\end{enumerate}"}
+{"_id": "13246", "title": "modules-lemma-restrict-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $R = \\Gamma(X, \\mathcal{O}_X)$. Let $M$ be an $R$-module. Let $\\mathcal{F}_M$ be the quasi-coherent sheaf of $\\mathcal{O}_X$-modules associated to $M$. If $g : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$ is a morphism of ringed spaces, then $g^*\\mathcal{F}_M$ is the sheaf associated to the $\\Gamma(Y, \\mathcal{O}_Y)$-module $\\Gamma(Y, \\mathcal{O}_Y) \\otimes_R M$."}
+{"_id": "13247", "title": "modules-lemma-quasi-coherent-module", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $x \\in X$ be a point. Assume that $x$ has a fundamental system of quasi-compact neighbourhoods. Consider any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$. Then there exists an open neighbourhood $U$ of $x$ such that $\\mathcal{F}|_U$ is isomorphic to the sheaf of modules $\\mathcal{F}_M$ on $(U, \\mathcal{O}_U)$ associated to some $\\Gamma(U, \\mathcal{O}_U)$-module $M$."}
+{"_id": "13248", "title": "modules-lemma-finite-presentation-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any $\\mathcal{O}_X$-module of finite presentation is quasi-coherent."}
+{"_id": "13249", "title": "modules-lemma-kernel-surjection-finite-free-onto-finite-presentation", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a $\\mathcal{O}_X$-module of finite presentation. \\begin{enumerate} \\item If $\\psi : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{F}$ is a surjection, then $\\Ker(\\psi)$ is of finite type. \\item If $\\theta : \\mathcal{G} \\to \\mathcal{F}$ is surjective with $\\mathcal{G}$ of finite type, then $\\Ker(\\theta)$ is of finite type. \\end{enumerate}"}
+{"_id": "13250", "title": "modules-lemma-pullback-finite-presentation", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\\mathcal{G}$ of a module of finite presentation is of finite presentation."}
+{"_id": "13252", "title": "modules-lemma-finite-presentation-quasi-compact-colimit", "text": "Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$ consisting of sheaves of $\\mathcal{O}_X$-modules (see Categories, Section \\ref{categories-section-posets-limits}). Assume \\begin{enumerate} \\item $I$ is directed, \\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation, and \\item $X$ has a cofinal system of open coverings $\\mathcal{U} : X = \\bigcup_{j\\in J} U_j$ with $J$ finite and $U_j \\cap U_{j'}$ quasi-compact for all $j, j' \\in J$. \\end{enumerate} Then we have $$ \\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i) = \\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i). $$"}
+{"_id": "13253", "title": "modules-lemma-finite-presentation-stalk-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $x \\in X$ such that $\\mathcal{F}_x \\cong \\mathcal{O}_{X, x}^{\\oplus r}$. Then there exists an open neighbourhood $U$ of $x$ such that $\\mathcal{F}|_U \\cong \\mathcal{O}_U^{\\oplus r}$."}
+{"_id": "13254", "title": "modules-lemma-coherent-finite-presentation", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any coherent $\\mathcal{O}_X$-module is of finite presentation and hence quasi-coherent."}
+{"_id": "13255", "title": "modules-lemma-coherent-abelian", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item Any finite type subsheaf of a coherent sheaf is coherent. \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism from a finite type sheaf $\\mathcal{F}$ to a coherent sheaf $\\mathcal{G}$. Then $\\Ker(\\varphi)$ is finite type. \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of coherent $\\mathcal{O}_X$-modules. Then $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are coherent. \\item Given a short exact sequence of $\\mathcal{O}_X$-modules $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ if two out of three are coherent so is the third. \\item The category $\\textit{Coh}(\\mathcal{O}_X)$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_X)$. In particular, the category of coherent modules is abelian and the inclusion functor $\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$ is exact. \\end{enumerate}"}
+{"_id": "13256", "title": "modules-lemma-coherent-structure-sheaf", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume $\\mathcal{O}_X$ is a coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is coherent if and only if it is of finite presentation."}
+{"_id": "13257", "title": "modules-lemma-finite-type-to-coherent-injective-on-stalk", "text": "Let $X$ be a ringed space. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism of $\\mathcal{O}_X$-modules. Let $x \\in X$. Assume $\\mathcal{G}$ of finite type, $\\mathcal{F}$ coherent and the map on stalks $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ injective. Then there exists an open neighbourhood $x \\in U \\subset X$ such that $\\varphi|_U$ is injective."}
+{"_id": "13259", "title": "modules-lemma-i-star-reflects-finite-type", "text": "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism of ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and that $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective. Let $\\mathcal{F}$ be an $\\mathcal{O}_Z$-module. Then $i_*\\mathcal{F}$ is of finite type if and only if $\\mathcal{F}$ is of finite type."}
+{"_id": "13260", "title": "modules-lemma-i-star-equivalence", "text": "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism of ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective. Denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the kernel of $i^\\sharp$. The functor $$ i_* : \\textit{Mod}(\\mathcal{O}_Z) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X) $$ is exact, fully faithful, with essential image those $\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$."}
+{"_id": "13261", "title": "modules-lemma-adjoint-section-with-support", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. The functor $\\mathcal{H}_Z : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X|_Z)$ of Remark \\ref{remark-sections-support-in-closed-modules} is right adjoint to $i_* : \\textit{Mod}(\\mathcal{O}_X|_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$."}
+{"_id": "13262", "title": "modules-lemma-locally-free-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is locally free then it is quasi-coherent."}
+{"_id": "13263", "title": "modules-lemma-pullback-locally-free", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. If $\\mathcal{G}$ is a locally free $\\mathcal{O}_Y$-module, then $f^*\\mathcal{G}$ is a locally free $\\mathcal{O}_X$-module."}
+{"_id": "13264", "title": "modules-lemma-rank", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Suppose that the support of $\\mathcal{O}_X$ is $X$, i.e., all stalk of $\\mathcal{O}_X$ are nonzero rings. Let $\\mathcal{F}$ be a locally free sheaf of $\\mathcal{O}_X$-modules. There exists a locally constant function $$ \\text{rank}_\\mathcal{F} : X \\longrightarrow \\{0, 1, 2, \\ldots\\}\\cup\\{\\infty\\} $$ such that for any point $x \\in X$ the cardinality of any set $I$ such that $\\mathcal{F}$ is isomorphic to $\\bigoplus_{i\\in I} \\mathcal{O}_X$ in a neighbourhood of $x$ is $\\text{rank}_\\mathcal{F}(x)$."}
+{"_id": "13265", "title": "modules-lemma-map-finite-locally-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $r \\geq 0$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of finite locally free $\\mathcal{O}_X$-modules of rank $r$. Then $\\varphi$ is an isomorphism if and only if $\\varphi$ is surjective."}
+{"_id": "13266", "title": "modules-lemma-direct-summand-of-locally-free-is-locally-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. If all stalks $\\mathcal{O}_{X, x}$ are local rings, then any direct summand of a finite locally free $\\mathcal{O}_X$-module is finite locally free."}
+{"_id": "13267", "title": "modules-lemma-stalk-tensor-product", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. Let $x \\in X$. There is a canonical isomorphism of $\\mathcal{O}_{X, x}$-modules $$ (\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})_x = \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{G}_x $$ functorial in $\\mathcal{F}$ and $\\mathcal{G}$."}
+{"_id": "13268", "title": "modules-lemma-tensor-product-sheafification", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be presheaves of $\\mathcal{O}_X$-modules with sheafifications $\\mathcal{F}$, $\\mathcal{G}$. Then $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} = (\\mathcal{F}' \\otimes_{p, \\mathcal{O}_X} \\mathcal{G}')^\\#$."}
+{"_id": "13269", "title": "modules-lemma-tensor-product-exact", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module. If $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ is an exact sequence of $\\mathcal{O}_X$-modules then the induced sequence $$ \\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{F}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to 0 $$ is exact."}
+{"_id": "13270", "title": "modules-lemma-tensor-product-pullback", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_Y$-modules. Then $f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_Y} \\mathcal{G}) = f^*\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}$ functorially in $\\mathcal{F}$, $\\mathcal{G}$."}
+{"_id": "13271", "title": "modules-lemma-tensor-product-permanence", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated by sections, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent, then $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$ is coherent. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$. \\end{enumerate}"}
+{"_id": "13272", "title": "modules-lemma-tensor-commute-colimits", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For any $\\mathcal{O}_X$-module $\\mathcal{F}$ the functor $$ \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X) , \\quad \\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} $$ commutes with arbitrary colimits."}
+{"_id": "13273", "title": "modules-lemma-flat-stalks-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. An $\\mathcal{O}_X$-module $\\mathcal{F}$ is flat if and only if the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{X, x}$-module for all $x \\in X$."}
+{"_id": "13274", "title": "modules-lemma-colimits-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. A filtered colimit of flat $\\mathcal{O}_X$-modules is flat. A direct sum of flat $\\mathcal{O}_X$-modules is flat."}
+{"_id": "13275", "title": "modules-lemma-j-shriek-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $U \\subset X$ be open. The sheaf $j_{U!}\\mathcal{O}_U$ is a flat sheaf of $\\mathcal{O}_X$-modules."}
+{"_id": "13276", "title": "modules-lemma-module-quotient-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item Any sheaf of $\\mathcal{O}_X$-modules is a quotient of a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$. \\item Any $\\mathcal{O}_X$-module is a quotient of a flat $\\mathcal{O}_X$-module. \\end{enumerate}"}
+{"_id": "13277", "title": "modules-lemma-flat-tor-zero", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $$ 0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 $$ be a short exact sequence of $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is flat. Then for any $\\mathcal{O}_X$-module $\\mathcal{G}$ the sequence $$ 0 \\to \\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to \\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0 $$ is exact."}
+{"_id": "13278", "title": "modules-lemma-flat-ses", "text": "\\begin{slogan} Kernels of epimorphisms and extensions of flat sheaves of modules over a ringed space are again flat. \\end{slogan} Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $$ 0 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F}_0 \\to 0 $$ be a short exact sequence of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is $\\mathcal{F}_1$. \\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is $\\mathcal{F}_2$. \\end{enumerate}"}
+{"_id": "13280", "title": "modules-lemma-flat-eq", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a flat $\\mathcal{O}_X$-module. Let $U \\subset X$ be open and let $$ \\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)} \\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)} \\mathcal{F}|_U $$ be a complex of $\\mathcal{O}_U$-modules. For every $x \\in U$ there exists an open neighbourhood $V \\subset U$ of $x$ and a factorization $$ \\mathcal{O}_V^{\\oplus n} \\xrightarrow{A} \\mathcal{O}_V^{\\oplus m} \\xrightarrow{(t_1, \\ldots, t_m)} \\mathcal{F}|_V $$ of $(s_1, \\ldots, s_n)|_V$ such that $A \\circ (f_1, \\ldots, f_n)|_V = 0$."}
+{"_id": "13281", "title": "modules-lemma-left-dual-module", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $\\mathcal{G}, \\eta, \\epsilon$ be a left dual of $\\mathcal{F}$ in the monoidal category of $\\mathcal{O}_X$-modules, see Categories, Definition \\ref{categories-definition-dual}. Then \\begin{enumerate} \\item $\\mathcal{F}$ is locally a direct summand of a finite free $\\mathcal{O}_X$-module, \\item the map $e : \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\to \\mathcal{G}$ sending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$ is an isomorphism, \\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections $f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "13282", "title": "modules-lemma-flat-locally-finite-presentation", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a flat $\\mathcal{O}_X$-module of finite presentation. Then $\\mathcal{F}$ is locally a direct summand of a finite free $\\mathcal{O}_X$-module."}
+{"_id": "13283", "title": "modules-lemma-surjection", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{F}$ be a sheaf of sets on $X$. There exists a set $I$ and for each $i \\in I$ an element $U_i \\in \\mathcal{B}$ and a finite set $S_i$ such that there exists a surjection $\\coprod_{i \\in I} j_{U_i!}\\underline{S_i} \\to \\mathcal{F}$."}
+{"_id": "13284", "title": "modules-lemma-filtered-colimit-constructibles", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology of $X$ and assume that each $U \\in \\mathcal{B}$ is quasi-compact. Then every sheaf of sets on $X$ is a filtered colimit of sheaves of the form \\begin{equation} \\label{equation-towards-constructible-sets} \\text{Coequalizer}\\left( \\xymatrix{ \\coprod\\nolimits_{b = 1, \\ldots, m} j_{V_b!}\\underline{S_b} \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\coprod\\nolimits_{a = 1, \\ldots, n} j_{U_a!}\\underline{S_a} } \\right) \\end{equation} with $U_a$ and $V_b$ in $\\mathcal{B}$ and $S_a$ and $S_b$ finite sets."}
+{"_id": "13285", "title": "modules-lemma-constructible-comes-from-finite", "text": "Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\\mathcal{F}$ be a sheaf of sets as in Equation (\\ref{equation-towards-constructible-sets}). Then there exists a continuous spectral map $f : X \\to Y$ to a finite sober topological space $Y$ and a sheaf of sets $\\mathcal{G}$ on $Y$ with finite stalks such that $f^{-1}\\mathcal{G} \\cong \\mathcal{F}$."}
+{"_id": "13286", "title": "modules-lemma-constructible-in-constant", "text": "Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\\mathcal{F}$ be a sheaf of sets as in Equation (\\ref{equation-towards-constructible-sets}). Then there exist finitely many constructible closed subsets $Z_1, \\ldots, Z_n \\subset X$ and finite sets $S_i$ such that $\\mathcal{F}$ is isomorphic to a subsheaf of $\\prod (Z_i \\to X)_*\\underline{S_i}$."}
+{"_id": "13287", "title": "modules-lemma-pullback-flat", "text": "Let $f : X \\to Y$ be a flat morphism of ringed spaces. Then the pullback functor $f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$ is exact."}
+{"_id": "13288", "title": "modules-lemma-local-tensor-algebra", "text": "In the situation described above. The sheaf $\\wedge^n\\mathcal{F}$ is the sheafification of the presheaf $$ U \\longmapsto \\wedge^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)). $$ See Algebra, Section \\ref{algebra-section-tensor-algebra}. Similarly, the sheaf $\\text{Sym}^n\\mathcal{F}$ is the sheafification of the presheaf $$ U \\longmapsto \\text{Sym}^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)). $$"}
+{"_id": "13290", "title": "modules-lemma-pullback-tensor-algebra", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules. Then $f^*\\text{T}(\\mathcal{F}) = \\text{T}(f^*\\mathcal{F})$, and similarly for the exterior and symmetric algebras associated to $\\mathcal{F}$."}
+{"_id": "13291", "title": "modules-lemma-presentation-sym-exterior", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$ be an exact sequence of sheaves of $\\mathcal{O}_X$-modules. For each $n \\geq 1$ there is an exact sequence $$ \\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\text{Sym}^{n - 1}(\\mathcal{F}_1) \\to \\text{Sym}^n(\\mathcal{F}_1) \\to \\text{Sym}^n(\\mathcal{F}) \\to 0 $$ and similarly an exact sequence $$ \\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\wedge^{n - 1}(\\mathcal{F}_1) \\to \\wedge^n(\\mathcal{F}_1) \\to \\wedge^n(\\mathcal{F}) \\to 0 $$"}
+{"_id": "13292", "title": "modules-lemma-tensor-algebra-permanence", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{F}$ is locally generated by sections, then so is each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$. \\item If $\\mathcal{F}$ is of finite type, then so is each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$. \\item If $\\mathcal{F}$ is of finite presentation, then so is each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$. \\item If $\\mathcal{F}$ is coherent, then for $n > 0$ each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$ is coherent. \\item If $\\mathcal{F}$ is quasi-coherent, then so is each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$. \\item If $\\mathcal{F}$ is locally free, then so is each $\\text{T}^n(\\mathcal{F})$, $\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$. \\end{enumerate}"}
+{"_id": "13294", "title": "modules-lemma-internal-hom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be $\\mathcal{O}_X$-modules. There is a canonical isomorphism $$ \\SheafHom_{\\mathcal{O}_X} (\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}, \\mathcal{H}) \\longrightarrow \\SheafHom_{\\mathcal{O}_X} (\\mathcal{F}, \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, to give a morphism $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{H}$ is the same as giving a morphism $\\mathcal{F} \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H})$."}
+{"_id": "13295", "title": "modules-lemma-internal-hom-exact", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$ is an exact sequence of $\\mathcal{O}_X$-modules, then $$ 0 \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_1, \\mathcal{G}) \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_2, \\mathcal{G}) $$ is exact. \\item If $0 \\to \\mathcal{G} \\to \\mathcal{G}_1 \\to \\mathcal{G}_2$ is an exact sequence of $\\mathcal{O}_X$-modules, then $$ 0 \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_1) \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_2) $$ is exact. \\end{enumerate}"}
+{"_id": "13296", "title": "modules-lemma-stalk-internal-hom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is finitely presented then the canonical map $$ \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x \\to \\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x) $$ is an isomorphism."}
+{"_id": "13297", "title": "modules-lemma-pullback-internal-hom", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_Y$-modules. If $\\mathcal{F}$ is finitely presented and $f$ is flat, then the canonical map $$ f^*\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{F}, \\mathcal{G}) \\longrightarrow \\SheafHom_{\\mathcal{O}_X}(f^*\\mathcal{F}, f^*\\mathcal{G}) $$ is an isomorphism."}
+{"_id": "13298", "title": "modules-lemma-internal-hom-locally-kernel-direct-sum", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is finitely presented then the sheaf $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is locally a kernel of a map between finite direct sums of copies of $\\mathcal{G}$. In particular, if $\\mathcal{G}$ is coherent then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is coherent too."}
+{"_id": "13299", "title": "modules-lemma-adjoint-tensor-restrict", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Then we have $$ \\Hom_{\\mathcal{O}_1}(\\mathcal{F}_{\\mathcal{O}_1}, \\mathcal{G}) = \\Hom_{\\mathcal{O}_2}(\\mathcal{F}, \\SheafHom_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{G})) $$ bifunctorially in $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_2)$ and $\\mathcal{G} \\in \\textit{Mod}(\\mathcal{O}_1)$."}
+{"_id": "13300", "title": "modules-lemma-invertible", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{L}$ be an $\\mathcal{O}_X$-module. Equivalent are \\begin{enumerate} \\item $\\mathcal{L}$ is invertible, and \\item there exists an $\\mathcal{O}_X$-module $\\mathcal{N}$ such that $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N} \\cong \\mathcal{O}_X$. \\end{enumerate} In this case $\\mathcal{L}$ is locally a direct summand of a finite free $\\mathcal{O}_X$-module and the module $\\mathcal{N}$ in (2) is isomorphic to $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$."}
+{"_id": "13302", "title": "modules-lemma-invertible-is-locally-free-rank-1", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Any locally free $\\mathcal{O}_X$-module of rank $1$ is invertible. If all stalks $\\mathcal{O}_{X, x}$ are local rings, then the converse holds as well (but in general this is not the case)."}
+{"_id": "13303", "title": "modules-lemma-constructions-invertible", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. \\begin{enumerate} \\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible $\\mathcal{O}_X$-modules, then so is $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$. \\item If $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, then so is $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$ and the evaluation map $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X) \\to \\mathcal{O}_X$ is an isomorphism. \\end{enumerate}"}
+{"_id": "13304", "title": "modules-lemma-pic-set", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. There exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$ such that each invertible module on $X$ is isomorphic to exactly one of the $\\mathcal{L}_i$."}
+{"_id": "13305", "title": "modules-lemma-s-open", "text": "\\begin{slogan} A (local) trivialisation of a linebundle is the same as a (local) nonvanishing section. \\end{slogan} Let $X$ be a ringed space. Assume that each stalk $\\mathcal{O}_{X, x}$ is a local ring with maximal ideal $\\mathfrak m_x$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. For any section $s \\in \\Gamma(X, \\mathcal{L})$ the set $$ X_s = \\{x \\in X \\mid \\text{image }s \\not\\in \\mathfrak m_x\\mathcal{L}_x\\} $$ is open in $X$. The map $s : \\mathcal{O}_{X_s} \\to \\mathcal{L}|_{X_s}$ is an isomorphism, and there exists a section $s'$ of $\\mathcal{L}^{\\otimes -1}$ over $X_s$ such that $s' (s|_{X_s}) = 1$."}
+{"_id": "13306", "title": "modules-lemma-det-ses", "text": "Let $X$ be a ringed space. Let $0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{E}'' \\to 0$ be a short exact sequence of finite locally free $\\mathcal{O}_X$-modules, Then there is a canonical isomorphism $$ \\det(\\mathcal{E}') \\otimes_{\\mathcal{O}_X}\\det(\\mathcal{E}'') \\longrightarrow \\det(\\mathcal{E}) $$ of $\\mathcal{O}_X$-modules."}
+{"_id": "13308", "title": "modules-lemma-simple-invert", "text": "Let $X$ be a topological space and let $\\mathcal{O}_X$ be a presheaf of rings. Let $\\mathcal{S} \\subset \\mathcal{O}_X$ be a pre-sheaf of sets contained in $\\mathcal{O}_X$. Suppose that for every open $U \\subset X$ the set $\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset. \\begin{enumerate} \\item There is a map of presheaves of rings $\\mathcal{O}_X \\to \\mathcal{S}^{-1}\\mathcal{O}_X$ such that every local section of $\\mathcal{S}$ maps to an invertible section of $\\mathcal{O}_X$. \\item For any homomorphism of presheaves of rings $\\mathcal{O}_X \\to \\mathcal{A}$ such that each local section of $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$ there exists a unique factorization $\\mathcal{S}^{-1}\\mathcal{O}_X \\to \\mathcal{A}$. \\item For any $x \\in X$ we have $$ (\\mathcal{S}^{-1}\\mathcal{O}_X)_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}. $$ \\item The sheafification $(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$ is a sheaf of rings with a map of sheaves of rings $(\\mathcal{O}_X)^\\# \\to (\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$ which is universal for maps of $(\\mathcal{O}_X)^\\#$ into sheaves of rings such that each local section of $\\mathcal{S}$ maps to an invertible section. \\item For any $x \\in X$ we have $$ (\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}. $$ \\end{enumerate}"}
+{"_id": "13309", "title": "modules-lemma-simple-invert-module", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_X$ be a presheaf of rings. Let $\\mathcal{S} \\subset \\mathcal{O}_X$ be a pre-sheaf of sets contained in $\\mathcal{O}_X$. Suppose that for every open $U \\subset X$ the set $\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset. For any presheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have $$ \\mathcal{S}^{-1}\\mathcal{F} = \\mathcal{S}^{-1}\\mathcal{O}_X \\otimes_{p, \\mathcal{O}_X} \\mathcal{F} $$ (see Sheaves, Section \\ref{sheaves-section-presheaves-modules} for notation) and if $\\mathcal{F}$ and $\\mathcal{O}_X$ are sheaves then $$ (\\mathcal{S}^{-1}\\mathcal{F})^\\# = (\\mathcal{S}^{-1}\\mathcal{O}_X)^\\# \\otimes_{\\mathcal{O}_X} \\mathcal{F} $$ (see Sheaves, Section \\ref{sheaves-section-sheafification-presheaves-modules} for notation)."}
+{"_id": "13310", "title": "modules-lemma-universal-module", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor $$ \\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad \\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F}) $$ is representable."}
+{"_id": "13311", "title": "modules-lemma-differentials-sheafify", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$ is the sheaf associated to the presheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$."}
+{"_id": "13312", "title": "modules-lemma-localize-differentials", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. For $U \\subset X$ open there is a canonical isomorphism $$ \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U = \\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)} $$ compatible with universal derivations."}
+{"_id": "13313", "title": "modules-lemma-pullback-differentials", "text": "Let $f : Y \\to X$ be a continuous map of topological spaces. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} = \\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$ compatible with universal derivations."}
+{"_id": "13314", "title": "modules-lemma-stalk-module-differentials", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $x \\in X$. Then we have $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1, x} = \\Omega_{\\mathcal{O}_{2, x}/\\mathcal{O}_{1, x}}$."}
+{"_id": "13315", "title": "modules-lemma-functoriality-differentials", "text": "Let $X$ be a topological space. Let $$ \\xymatrix{ \\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\ \\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u] } $$ be a commutative diagram of sheaves of rings on $X$. The map $\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map $\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$ is a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_2$-modules $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$. It is uniquely characterized by the property that $\\text{d}(f) \\mapsto \\text{d}(\\varphi(f))$ for any local section $f$ of $\\mathcal{O}_2$. In this way $\\Omega_{-/-}$ becomes a functor on the category of arrows of sheaves of rings."}
+{"_id": "13317", "title": "modules-lemma-double-structure-gives-derivation", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of ringed spaces. Consider a short exact sequence $$ 0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0 $$ Here $\\mathcal{A}$ is a sheaf of $f^{-1}\\mathcal{O}_S$-algebras, $\\pi : \\mathcal{A} \\to \\mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and $\\mathcal{I} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{I}$ an ideal sheaf with square zero in $\\mathcal{A}$. So $\\mathcal{I}$ has a natural structure of an $\\mathcal{O}_X$-module. A section $s : \\mathcal{O}_X \\to \\mathcal{A}$ of $\\pi$ is a $f^{-1}\\mathcal{O}_S$-algebra map such that $\\pi \\circ s = \\text{id}$. Given any section $s : \\mathcal{O}_X \\to \\mathcal{A}$ of $\\pi$ and any $S$-derivation $D : \\mathcal{O}_X \\to \\mathcal{I}$ the map $$ s + D : \\mathcal{O}_X \\to \\mathcal{A} $$ is a section of $\\pi$ and every section $s'$ is of the form $s + D$ for a unique $S$-derivation $D$."}
+{"_id": "13318", "title": "modules-lemma-functoriality-differentials-ringed-spaces", "text": "Let $$ \\xymatrix{ X' \\ar[d]_{h'} \\ar[r]_f & X \\ar[d]^h \\\\ S' \\ar[r]^g & S } $$ be a commutative diagram of ringed spaces. \\begin{enumerate} \\item The canonical map $\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with $f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a $S$-derivation and we obtain a canonical map of $\\mathcal{O}_X$-modules $\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$. \\item The commutative diagram $$ \\xymatrix{ f^{-1}\\mathcal{O}_X \\ar[r] & \\mathcal{O}_{X'} \\\\ f^{-1}h^{-1}\\mathcal{O}_S \\ar[u] \\ar[r] & (h')^{-1}\\mathcal{O}_{S'} \\ar[u] } $$ induces by Lemmas \\ref{lemma-pullback-differentials} and \\ref{lemma-functoriality-differentials} a canonical map $f^{-1}\\Omega_{X/S} \\to \\Omega_{X'/S'}$. \\end{enumerate} These two maps correspond (via adjointness of $f_*$ and $f^*$ and via $f^*\\Omega_{X/S} = f^{-1}\\Omega_{X/S} \\otimes_{f^{-1}\\mathcal{O}_X} \\mathcal{O}_{X'}$ and Sheaves, Lemma \\ref{sheaves-lemma-adjointness-tensor-restrict}) to the same $\\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'} $$ which is uniquely characterized by the property that $f^*\\text{d}_{X/S}(a)$ maps to $\\text{d}_{X'/S'}(f^*a)$ for any local section $a$ of $\\mathcal{O}_X$."}
+{"_id": "13319", "title": "modules-lemma-check-functoriality-differentials", "text": "Let $$ \\xymatrix{ X'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\ S'' \\ar[r] & S' \\ar[r] & S } $$ be a commutative diagram of ringed spaces. With notation as in Lemma \\ref{lemma-functoriality-differentials-ringed-spaces} we have $$ c_{f \\circ g} = c_g \\circ g^* c_f $$ as maps $(f \\circ g)^*\\Omega_{X/S} \\to \\Omega_{X''/S''}$."}
+{"_id": "13320", "title": "modules-lemma-composition-differential-operators", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules. If $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$ are differential operators of order $k$ and $k'$, then $D' \\circ D$ is a differential operator of order $k + k'$."}
+{"_id": "13321", "title": "modules-lemma-module-principal-parts", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. Let $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$ and a canonical isomorphism $$ \\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G}) $$ functorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$."}
+{"_id": "13322", "title": "modules-lemma-differential-operators-sheafify", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of presheaves of rings on $X$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_2$-modules. Then $\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$ is the sheaf associated to the presheaf $U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$."}
+{"_id": "13323", "title": "modules-lemma-sequence-of-principal-parts", "text": "Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. There is a canonical short exact sequence $$ 0 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to \\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to \\mathcal{F} \\to 0 $$ functorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}."}
+{"_id": "13325", "title": "modules-lemma-differentials-de-rham-complex-order-1", "text": "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $X$. The differentials $\\text{d} : \\Omega^i_{\\mathcal{B}/\\mathcal{A}}  \\to \\Omega^{i + 1}_{\\mathcal{B}/\\mathcal{A}}$ are differential operators of order $1$."}
+{"_id": "13327", "title": "modules-lemma-NL-up-to-qis", "text": "In the situation above there is a canonical isomorphism $\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$."}
+{"_id": "13328", "title": "modules-lemma-pullback-NL", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $Y$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} = \\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$."}
+{"_id": "13329", "title": "modules-lemma-stalk-NL", "text": "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Let $x \\in X$. Then we have $\\NL_{\\mathcal{B}/\\mathcal{A}, x} = \\NL_{\\mathcal{B}_x/\\mathcal{A}_x}$."}
+{"_id": "13330", "title": "modules-lemma-exact-sequence-NL", "text": "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be maps of sheaves of rings. Let $C$ be the cone (Derived Categories, Definition \\ref{derived-definition-cone}) of the map of complexes $\\NL_{\\mathcal{C}/\\mathcal{A}} \\to \\NL_{\\mathcal{C}/\\mathcal{B}}$. There is a canonical map $$ c : \\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C} \\longrightarrow C[-1] $$ of complexes of $\\mathcal{C}$-modules which produces a canonical six term exact sequence $$ \\xymatrix{ H^0(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] & H^0(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] & H^0(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[r] & 0 \\\\ H^{-1}(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] & H^{-1}(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] & H^{-1}(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[llu] } $$ of cohomology sheaves."}
+{"_id": "13331", "title": "modules-lemma-exact-sequence-NL-ringed-topoi", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. Let $C$ be the cone of the map $\\NL_{X/Z} \\to \\NL_{X/Y}$ of complexes of $\\mathcal{O}_X$-modules. There is a canonical map $$ f^*\\NL_{Y/Z} \\to C[-1] $$ which produces a canonical six term exact sequence $$ \\xymatrix{ H^0(f^*\\NL_{Y/Z}) \\ar[r] & H^0(\\NL_{X/Z}) \\ar[r] & H^0(\\NL_{X/Y}) \\ar[r] & 0 \\\\ H^{-1}(f^*\\NL_{Y/Z}) \\ar[r] & H^{-1}(\\NL_{X/Z}) \\ar[r] & H^{-1}(\\NL_{X/Y}) \\ar[llu] } $$ of cohomology sheaves."}
+{"_id": "13369", "title": "defos-lemma-huge-diagram", "text": "Given a commutative diagram $$ \\xymatrix{ & 0 \\ar[r] & N_2 \\ar[r] & B'_2 \\ar[r] & B_2 \\ar[r] & 0 \\\\ & 0 \\ar[r]|\\hole & I_2 \\ar[u]_{c_2} \\ar[r] & A'_2 \\ar[u] \\ar[r]|\\hole & A_2 \\ar[u] \\ar[r] & 0 \\\\ 0 \\ar[r] & N_1 \\ar[ruu] \\ar[r] & B'_1 \\ar[r] & B_1 \\ar[ruu] \\ar[r] & 0 \\\\ 0 \\ar[r] & I_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] & A'_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & A_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0 } $$ with front and back solutions to (\\ref{equation-to-solve}) we have \\begin{enumerate} \\item There exist a canonical element in $\\Ext^1_{B_1}(\\NL_{B_1/A_1}, N_2)$ whose vanishing is a necessary and sufficient condition for the existence of a ring map $B'_1 \\to B'_2$ fitting into the diagram. \\item If there exists a map $B'_1 \\to B'_2$ fitting into the diagram the set of all such maps is a principal homogeneous space under $\\Hom_{B_1}(\\Omega_{B_1/A_1}, N_2)$. \\end{enumerate}"}
+{"_id": "13371", "title": "defos-lemma-choices", "text": "If there exists a solution to (\\ref{equation-to-solve}), then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_B(\\NL_{B/A}, N)$."}
+{"_id": "13372", "title": "defos-lemma-extensions-of-rings", "text": "Let $A$ be a ring and let $I$ be an $A$-module. \\begin{enumerate} \\item The set of extensions of rings $0 \\to I \\to A' \\to A \\to 0$ where $I$ is an ideal of square zero is canonically bijective to $\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, I)$. \\item Given a ring map $A \\to B$, a $B$-module $N$, an $A$-module map $c : I \\to N$, and given extensions of rings with square zero kernels: \\begin{enumerate} \\item[(a)] $0 \\to I \\to A' \\to A \\to 0$ corresponding to $\\alpha \\in \\Ext^1_A(\\NL_{A/\\mathbf{Z}}, I)$, and \\item[(b)] $0 \\to N \\to B' \\to B \\to 0$ corresponding to $\\beta \\in \\Ext^1_B(\\NL_{B/\\mathbf{Z}}, N)$ \\end{enumerate} then there is a map $A' \\to B'$ fitting into a diagram (\\ref{equation-to-solve}) if and only if $\\beta$ and $\\alpha$ map to the same element of $\\Ext^1_A(\\NL_{A/\\mathbf{Z}}, N)$. \\end{enumerate}"}
+{"_id": "13373", "title": "defos-lemma-strict-morphism-thickenings", "text": "In Situation \\ref{situation-morphism-thickenings} the morphism $(f, f')$ is a strict morphism of thickenings if and only if (\\ref{equation-morphism-thickenings}) is cartesian in the category of ringed spaces."}
+{"_id": "13374", "title": "defos-lemma-inf-map", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given extensions $$ 0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0 $$ as in (\\ref{equation-extension}) and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{K} \\to \\mathcal{L}$. \\begin{enumerate} \\item If there exists an $\\mathcal{O}_{X'}$-module map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$, then the diagram $$ \\xymatrix{ \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} & \\mathcal{K} \\ar[d]^\\psi \\\\ \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\ar[r]^-{c_{\\mathcal{G}'}} & \\mathcal{L} } $$ is commutative. \\item The set of $\\mathcal{O}_{X'}$-module maps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$ is, if nonempty, a principal homogeneous space under $\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})$. \\end{enumerate}"}
+{"_id": "13375", "title": "defos-lemma-inf-obs-map", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given extensions $$ 0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0 $$ as in (\\ref{equation-extension}) and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram $$ \\xymatrix{ \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} & \\mathcal{K} \\ar[d]^\\psi \\\\ \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\ar[r]^-{c_{\\mathcal{G}'}} & \\mathcal{L} } $$ is commutative. Then there exists an element $$ o(\\varphi, \\psi) \\in \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L}) $$ whose vanishing is a necessary and sufficient condition for the existence of a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$."}
+{"_id": "13376", "title": "defos-lemma-inf-ext", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given $\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$ and an $\\mathcal{O}_X$-linear map $c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$. If there exists a sequence (\\ref{equation-extension}) with $c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these extensions is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})$."}
+{"_id": "13377", "title": "defos-lemma-inf-obs-ext", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Assume given $\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$ and an $\\mathcal{O}_X$-linear map $c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$. Then there exists an element $$ o(\\mathcal{F}, \\mathcal{K}, c) \\in \\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}) $$ whose vanishing is a necessary and sufficient condition for the existence of a sequence (\\ref{equation-extension}) with $c_{\\mathcal{F}'} = c$."}
+{"_id": "13378", "title": "defos-lemma-inf-map-special", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules. Set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map. The set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal homogeneous space under $\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$."}
+{"_id": "13379", "title": "defos-lemma-deform-module", "text": "Let $(f, f')$ be a morphism of first order thickenings of ringed spaces as in Situation \\ref{situation-morphism-thickenings}. Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module and set $\\mathcal{F} = i^*\\mathcal{F}'$. Assume that $\\mathcal{F}$ is flat over $S$ and that $(f, f')$ is a strict morphism of thickenings (Definition \\ref{definition-strict-morphism-thickenings}). Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}'$ is flat over $S'$, and \\item the canonical map $f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}'$ is an isomorphism. \\end{enumerate} Moreover, in this case the maps $$ f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}' $$ are isomorphisms."}
+{"_id": "13380", "title": "defos-lemma-inf-map-rel", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings}. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map. Assume that $\\mathcal{G}'$ is flat over $S'$ and that $(f, f')$ is a strict morphism of thickenings. The set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal homogeneous space under $$ \\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J}) $$"}
+{"_id": "13381", "title": "defos-lemma-inf-obs-map-special", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order thickening of ringed spaces. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map. There exists an element $$ o(\\varphi) \\in \\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}') $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$."}
+{"_id": "13383", "title": "defos-lemma-inf-ext-rel", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings}. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\\mathcal{F}$ flat over $S$. If there exists a pair $(\\mathcal{F}', \\alpha)$ consisting of an $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ and an isomorphism $\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of isomorphism classes of such pairs is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}( \\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$."}
+{"_id": "13384", "title": "defos-lemma-inf-obs-ext-rel", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings}. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\\mathcal{F}$ flat over $S$. There exists an $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if \\begin{enumerate} \\item the canonical map $ f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$ is an isomorphism, and \\item the class $o(\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1) \\in \\Ext^2_{\\mathcal{O}_X}( \\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$ of Lemma \\ref{lemma-inf-obs-ext} is zero. \\end{enumerate}"}
+{"_id": "13385", "title": "defos-lemma-flat", "text": "In the situation above. \\begin{enumerate} \\item There exists an $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if the class $o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1) \\in \\Ext^2_{\\mathcal{O}_X}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$ of Lemma \\ref{lemma-inf-obs-ext} is zero. \\item If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$. \\item Given a lift $\\mathcal{F}'$, the set of automorphisms of $\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically isomorphic to $\\Ext^0_{\\mathcal{O}_X}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$. \\end{enumerate}"}
+{"_id": "13387", "title": "defos-lemma-huge-diagram-ringed-spaces", "text": "Assume given a commutative diagram of morphisms of ringed spaces \\begin{equation} \\label{equation-huge-1} \\vcenter{ \\xymatrix{ & (X_2, \\mathcal{O}_{X_2}) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g & (X'_2, \\mathcal{O}_{X'_2}) \\ar[d]^{f'_2} \\\\ & (S_2, \\mathcal{O}_{S_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole & (S'_2, \\mathcal{O}_{S'_2}) \\ar[ddl] \\\\ (X_1, \\mathcal{O}_{X_1}) \\ar[r]_{i_1} \\ar[d]_{f_1} & (X'_1, \\mathcal{O}_{X'_1}) \\ar[d]^{f'_1} \\\\ (S_1, \\mathcal{O}_{S_1}) \\ar[r]^{t_1} & (S'_1, \\mathcal{O}_{S'_1}) } } \\end{equation} whose horizontal arrows are first order thickenings. Set $\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given a $g$-map $\\nu : \\mathcal{G}_1 \\to \\mathcal{G}_2$ of modules giving rise to the commutative diagram \\begin{equation} \\label{equation-huge-2} \\vcenter{ \\xymatrix{ & 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] & \\mathcal{O}_{X'_2} \\ar[r] & \\mathcal{O}_{X_2} \\ar[r] & 0 \\\\ & 0 \\ar[r]|\\hole & \\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] & \\mathcal{O}_{S'_2} \\ar[u] \\ar[r]|\\hole & \\mathcal{O}_{S_2} \\ar[u] \\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathcal{G}_1 \\ar[ruu] \\ar[r] & \\mathcal{O}_{X'_1} \\ar[r] & \\mathcal{O}_{X_1} \\ar[ruu] \\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] & \\mathcal{O}_{S'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & \\mathcal{O}_{S_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0 } } \\end{equation} with front and back solutions to (\\ref{equation-to-solve-ringed-spaces}). \\begin{enumerate} \\item There exist a canonical element in $\\Ext^1_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2)$ whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed spaces $X'_2 \\to X'_1$ fitting into (\\ref{equation-huge-1}) compatibly with $\\nu$. \\item If there exists a morphism $X'_2 \\to X'_1$ fitting into (\\ref{equation-huge-1}) compatibly with $\\nu$ the set of all such morphisms is a principal homogeneous space under $$ \\Hom_{\\mathcal{O}_{X_1}}(\\Omega_{X_1/S_1}, g_*\\mathcal{G}_2) = \\Hom_{\\mathcal{O}_{X_2}}(g^*\\Omega_{X_1/S_1}, \\mathcal{G}_2) = \\Ext^0_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2). $$ \\end{enumerate}"}
+{"_id": "13388", "title": "defos-lemma-NL-represent-ext-class", "text": "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings. Let $\\mathcal{G}$ be a $\\mathcal{B}$-module. Let $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$.  There exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$ such that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$ is the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$ (see proof for notation)."}
+{"_id": "13389", "title": "defos-lemma-choices-ringed-spaces", "text": "If there exists a solution to (\\ref{equation-to-solve-ringed-spaces}), then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$."}
+{"_id": "13390", "title": "defos-lemma-extensions-of-ringed-spaces", "text": "Let $(S, \\mathcal{O}_S)$ be a ringed space and let $\\mathcal{J}$ be an $\\mathcal{O}_S$-module. \\begin{enumerate} \\item The set of extensions of sheaves of rings $0 \\to \\mathcal{J} \\to \\mathcal{O}_{S'} \\to \\mathcal{O}_S \\to 0$ where $\\mathcal{J}$ is an ideal of square zero is canonically bijective to $\\Ext^1_{\\mathcal{O}_S}(\\NL_{S/\\mathbf{Z}}, \\mathcal{J})$. \\item Given a morphism of ringed spaces $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$, an $\\mathcal{O}_X$-module $\\mathcal{G}$, an $f$-map $c : \\mathcal{J} \\to \\mathcal{G}$, and given extensions of sheaves of rings with square zero kernels: \\begin{enumerate} \\item[(a)] $0 \\to \\mathcal{J} \\to \\mathcal{O}_{S'} \\to \\mathcal{O}_S \\to 0$ corresponding to $\\alpha \\in \\Ext^1_{\\mathcal{O}_S}(\\NL_{S/\\mathbf{Z}}, \\mathcal{J})$, \\item[(b)] $0 \\to \\mathcal{G} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0$ corresponding to $\\beta \\in \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/\\mathbf{Z}}, \\mathcal{G})$ \\end{enumerate} then there is a morphism $X' \\to S'$ fitting into a diagram (\\ref{equation-to-solve-ringed-spaces}) if and only if $\\beta$ and $\\alpha$ map to the same element of $\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_{S/\\mathbf{Z}}, \\mathcal{G})$. \\end{enumerate}"}
+{"_id": "13391", "title": "defos-lemma-deform", "text": "Let $S \\subset S'$ be a first order thickening of schemes. Let $f : X \\to S$ be a flat morphism of schemes. If there exists a flat morphism $f' : X' \\to S'$ of schemes and an isomorphism $a : X \\to X' \\times_{S'} S$ over $S$, then \\begin{enumerate} \\item the set of isomorphism classes of pairs $(f' : X' \\to S', a)$ is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$, and \\item the set of automorphisms of $\\varphi : X' \\to X'$ over $S'$ which reduce to the identity on $X' \\times_{S'} S$ is $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$. \\end{enumerate}"}
+{"_id": "13392", "title": "defos-lemma-inf-map-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions $$ 0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0 $$ as in (\\ref{equation-extension-ringed-topoi}) and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{K} \\to \\mathcal{L}$. \\begin{enumerate} \\item If there exists an $\\mathcal{O}'$-module map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$, then the diagram $$ \\xymatrix{ \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} & \\mathcal{K} \\ar[d]^\\psi \\\\ \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G} \\ar[r]^-{c_{\\mathcal{G}'}} & \\mathcal{L} } $$ is commutative. \\item The set of $\\mathcal{O}'$-module maps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$ is, if nonempty, a principal homogeneous space under $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})$. \\end{enumerate}"}
+{"_id": "13393", "title": "defos-lemma-inf-obs-map-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Assume given extensions $$ 0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0 $$ as in (\\ref{equation-extension-ringed-topoi}) and maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram $$ \\xymatrix{ \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} & \\mathcal{K} \\ar[d]^\\psi \\\\ \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G} \\ar[r]^-{c_{\\mathcal{G}'}} & \\mathcal{L} } $$ is commutative. Then there exists an element $$ o(\\varphi, \\psi) \\in \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{L}) $$ whose vanishing is a necessary and sufficient condition for the existence of a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$ and $\\psi$."}
+{"_id": "13394", "title": "defos-lemma-inf-ext-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$ and an $\\mathcal{O}$-linear map $c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$. If there exists a sequence (\\ref{equation-extension-ringed-topoi}) with $c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these extensions is principal homogeneous under $\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})$."}
+{"_id": "13395", "title": "defos-lemma-inf-obs-ext-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Assume given $\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$ and an $\\mathcal{O}$-linear map $c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$. Then there exists an element $$ o(\\mathcal{F}, \\mathcal{K}, c) \\in \\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}) $$ whose vanishing is a necessary and sufficient condition for the existence of a sequence (\\ref{equation-extension-ringed-topoi}) with $c_{\\mathcal{F}'} = c$."}
+{"_id": "13396", "title": "defos-lemma-inf-map-special-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules. Set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map. The set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal homogeneous space under $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$."}
+{"_id": "13397", "title": "defos-lemma-deform-module-ringed-topoi", "text": "Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}. Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module and set $\\mathcal{F} = i^*\\mathcal{F}'$. Assume that $\\mathcal{F}$ is flat over $\\mathcal{O}_\\mathcal{B}$ and that $(f, f')$ is a strict morphism of thickenings (Definition \\ref{definition-strict-morphism-thickenings-ringed-topoi}). Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$, and \\item the canonical map $f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}'$ is an isomorphism. \\end{enumerate} Moreover, in this case the maps $$ f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{I}\\mathcal{F}' $$ are isomorphisms."}
+{"_id": "13398", "title": "defos-lemma-deform-fp-module-ringed-topoi", "text": "Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}. Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module and set $\\mathcal{F} = i^*\\mathcal{F}'$. Assume that $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}'$ is an $\\mathcal{O}'$-module of finite presentation, and \\item $\\mathcal{F}$ is an $\\mathcal{O}$-module of finite presentation. \\end{enumerate}"}
+{"_id": "13399", "title": "defos-lemma-inf-map-rel-ringed-topoi", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map. Assume that $\\mathcal{G}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. The set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal homogeneous space under $$ \\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J}) $$"}
+{"_id": "13400", "title": "defos-lemma-inf-obs-map-special-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a first order thickening of ringed topoi. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map. There exists an element $$ o(\\varphi) \\in \\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}') $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\\varphi$ to an $\\mathcal{O}'$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$."}
+{"_id": "13402", "title": "defos-lemma-inf-ext-rel-ringed-topoi", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. If there exists a pair $(\\mathcal{F}', \\alpha)$ consisting of an $\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$ and an isomorphism $\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of isomorphism classes of such pairs is principal homogeneous under $\\Ext^1_\\mathcal{O}( \\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$."}
+{"_id": "13403", "title": "defos-lemma-inf-obs-ext-rel-ringed-topoi", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings-ringed-topoi}. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Assume $(f, f')$ is a strict morphism of thickenings and $\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. There exists an $\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if \\begin{enumerate} \\item the canonical map $f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}$ is an isomorphism, and \\item the class $o(\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}, 1) \\in \\Ext^2_\\mathcal{O}( \\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$ of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero. \\end{enumerate}"}
+{"_id": "13404", "title": "defos-lemma-flat-ringed-topoi", "text": "In the situation above. \\begin{enumerate} \\item There exists an $\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if the class $o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F}, 1) \\in \\Ext^2_\\mathcal{O}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$ of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero. \\item If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under $\\Ext^1_\\mathcal{O}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$. \\item Given a lift $\\mathcal{F}'$, the set of automorphisms of $\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically isomorphic to $\\Ext^0_\\mathcal{O}( \\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$. \\end{enumerate}"}
+{"_id": "13405", "title": "defos-lemma-functorial-ringed-topoi", "text": "In Situation \\ref{situation-morphism-flat-thickenings-ringed-topoi} the obstruction class $o(\\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$ maps to the obstruction class $o(\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$ under the canonical map $$ \\Ext^2_\\mathcal{O}( \\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F}) \\to \\Ext^2_\\mathcal{O}( \\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F}) $$"}
+{"_id": "13406", "title": "defos-lemma-verify-iv-ringed-topoi", "text": "In Situation \\ref{situation-ses-flat-thickenings-ringed-topoi} the modules $\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules flat over $\\mathcal{O}_{\\mathcal{B}'_1}$ restricting to $\\mathcal{F}$ on $(\\Sh(\\mathcal{C}), \\mathcal{O})$. Their difference (Lemma \\ref{lemma-flat-ringed-topoi}) is an element $\\theta$ of $\\Ext^1_\\mathcal{O}(\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F})$ whose boundary in $\\Ext^2_\\mathcal{O}(\\mathcal{F}, f^*\\mathcal{J}_3 \\otimes_\\mathcal{O} \\mathcal{F})$ equals the obstruction (Lemma \\ref{lemma-flat-ringed-topoi}) to lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over $\\mathcal{O}_{\\mathcal{B}'_3}$."}
+{"_id": "13407", "title": "defos-lemma-huge-diagram-ringed-topoi", "text": "Assume given a commutative diagram of morphisms ringed topoi \\begin{equation} \\label{equation-huge-1-ringed-topoi} \\vcenter{ \\xymatrix{ & (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g & (\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\ar[d]^{f'_2} \\\\ & (\\Sh(\\mathcal{B}_2), \\mathcal{O}_{\\mathcal{B}_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole & (\\Sh(\\mathcal{B}'_2), \\mathcal{O}_{\\mathcal{B}'_2}) \\ar[ddl] \\\\ (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\ar[r]_{i_1} \\ar[d]_{f_1} & (\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1) \\ar[d]^{f'_1} \\\\ (\\Sh(\\mathcal{B}_1), \\mathcal{O}_{\\mathcal{B}_1}) \\ar[r]^{t_1} & (\\Sh(\\mathcal{B}'_1), \\mathcal{O}_{\\mathcal{B}'_1}) } } \\end{equation} whose horizontal arrows are first order thickenings. Set $\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given a map of $g^{-1}\\mathcal{O}_1$-modules $\\nu : g^{-1}\\mathcal{G}_1 \\to \\mathcal{G}_2$ giving rise to the commutative diagram \\begin{equation} \\label{equation-huge-2-ringed-topoi} \\vcenter{ \\xymatrix{ & 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] & \\mathcal{O}'_2 \\ar[r] & \\mathcal{O}_2 \\ar[r] & 0 \\\\ & 0 \\ar[r]|\\hole & f_2^{-1}\\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] & f_2^{-1}\\mathcal{O}_{\\mathcal{B}'_2} \\ar[u] \\ar[r]|\\hole & f_2^{-1}\\mathcal{O}_{\\mathcal{B}_2} \\ar[u] \\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathcal{G}_1 \\ar[ruu] \\ar[r] & \\mathcal{O}'_1 \\ar[r] & \\mathcal{O}_1 \\ar[ruu] \\ar[r] & 0 \\\\ 0 \\ar[r] & f_1^{-1}\\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] & f_1^{-1}\\mathcal{O}_{\\mathcal{B}'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & f_1^{-1}\\mathcal{O}_{\\mathcal{B}_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0 } } \\end{equation} with front and back solutions to (\\ref{equation-to-solve-ringed-topoi}). (The north-north-west arrows are maps on $\\mathcal{C}_2$ after applying $g^{-1}$ to the source.) \\begin{enumerate} \\item There exist a canonical element in $\\Ext^1_{\\mathcal{O}_2}( Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2)$ whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed topoi $(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to (\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$ fitting into (\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$. \\item If there exists a morphism $(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to (\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$ fitting into (\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$ the set of all such morphisms is a principal homogeneous space under $$ \\Hom_{\\mathcal{O}_1}( \\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, g_*\\mathcal{G}_2) = \\Hom_{\\mathcal{O}_2}( g^*\\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2) = \\Ext^0_{\\mathcal{O}_2}( Lg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2). $$ \\end{enumerate}"}
+{"_id": "13408", "title": "defos-lemma-NL-represent-ext-class-ringed-topoi", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. Let $\\mathcal{G}$ be a $\\mathcal{B}$-module. Let $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$.  There exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$ such that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$ is the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$ (see proof for notation)."}
+{"_id": "13409", "title": "defos-lemma-choices-ringed-topoi", "text": "If there exists a solution to (\\ref{equation-to-solve-ringed-topoi}), then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_\\mathcal{O}( \\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$."}
+{"_id": "13410", "title": "defos-lemma-extensions-of-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a ringed topos and let $\\mathcal{J}$ be an $\\mathcal{O}_\\mathcal{B}$-module. \\begin{enumerate} \\item The set of extensions of sheaves of rings $0 \\to \\mathcal{J} \\to \\mathcal{O}_{\\mathcal{B}'} \\to \\mathcal{O}_\\mathcal{B} \\to 0$ where $\\mathcal{J}$ is an ideal of square zero is canonically bijective to $\\Ext^1_{\\mathcal{O}_\\mathcal{B}}( \\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{J})$. \\item Given a morphism of ringed topoi $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$, an $\\mathcal{O}$-module $\\mathcal{G}$, an $f^{-1}\\mathcal{O}_\\mathcal{B}$-module map $c : f^{-1}\\mathcal{J} \\to \\mathcal{G}$, and given extensions of sheaves of rings with square zero kernels: \\begin{enumerate} \\item[(a)] $0 \\to \\mathcal{J} \\to \\mathcal{O}_{\\mathcal{B}'} \\to \\mathcal{O}_\\mathcal{B} \\to 0$ corresponding to $\\alpha \\in \\Ext^1_{\\mathcal{O}_\\mathcal{B}}( \\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{J})$, \\item[(b)] $0 \\to \\mathcal{G} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0$ corresponding to $\\beta \\in \\Ext^1_\\mathcal{O}(\\NL_{\\mathcal{O}/\\mathbf{Z}}, \\mathcal{G})$ \\end{enumerate} then there is a morphism $(\\Sh(\\mathcal{C}), \\mathcal{O}') \\to (\\Sh(\\mathcal{B}, \\mathcal{O}_{\\mathcal{B}'})$ fitting into a diagram (\\ref{equation-to-solve-ringed-topoi}) if and only if $\\beta$ and $\\alpha$ map to the same element of $\\Ext^1_\\mathcal{O}( Lf^*\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathbf{Z}}, \\mathcal{G})$. \\end{enumerate}"}
+{"_id": "13411", "title": "defos-lemma-match-thickenings", "text": "Let $S$ be a scheme. Let $i : Z \\to Z'$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $i$ is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section \\ref{spaces-more-morphisms-section-thickenings}, and \\item the associated morphism $i_{small} : (\\Sh(Z_\\etale), \\mathcal{O}_Z) \\to (\\Sh(Z'_\\etale), \\mathcal{O}_{Z'})$ of ringed topoi (Properties of Spaces, Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}) is a thickening in the sense of Section \\ref{section-thickenings-ringed-topoi}. \\end{enumerate}"}
+{"_id": "13412", "title": "defos-lemma-deform-spaces", "text": "Let $S$ be a scheme. Let $Y \\subset Y'$ be a first order thickening of algebraic spaces over $S$. Let $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$. If there exists a flat morphism $f' : X' \\to Y'$ of algebraic spaces over $S$ and an isomorphsm $a : X \\to X' \\times_{Y'} Y$ over $Y$, then \\begin{enumerate} \\item the set of isomorphism classes of pairs $(f' : X' \\to Y', a)$ is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$, and \\item the set of automorphisms of $\\varphi : X' \\to X'$ over $Y'$ which reduce to the identity on $X' \\times_{Y'} Y$ is $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$. \\end{enumerate}"}
+{"_id": "13413", "title": "defos-lemma-thickening-space-quasi-coherent", "text": "In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$ (Derived Categories of Spaces, Section \\ref{spaces-perfect-section-better-coherator}) is an isomorphism. Then the functor $F$ is an equivalence of categories."}
+{"_id": "13414", "title": "defos-lemma-thickening-over-thickening-space-quasi-coherent", "text": "In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$ (Derived Categories of Spaces, Section \\ref{spaces-perfect-section-better-coherator}) is an isomorphism. Then the functor $FT$ is an equivalence of categories."}
+{"_id": "13415", "title": "defos-lemma-canonical-class-algebra", "text": "Let $R' \\to R$ be a surjection of rings whose kernel is an ideal $I$ of square zero. For every $K \\in D^-(R)$ there is a canonical map $$ \\omega(K) : K \\longrightarrow K \\otimes_R^\\mathbf{L} I[2] $$ in $D(R)$ with the following properties \\begin{enumerate} \\item $\\omega(K) = 0$ if and only if there exists $K' \\in D(R')$ with $K' \\otimes_{R'}^\\mathbf{L} R = K$, \\item given $K \\to L$ in $D^-(R)$ the diagram $$ \\xymatrix{ K \\ar[d] \\ar[rr]_-{\\omega(K)} & & K \\otimes^\\mathbf{L}_R I[2] \\ar[d] \\\\ L \\ar[rr]^-{\\omega(L)} & & L \\otimes^\\mathbf{L}_R I[2] } $$ commutes, and \\item formation of $\\omega(K)$ is compatible with ring maps $R' \\to S'$ (see proof for a precise statement). \\end{enumerate}"}
+{"_id": "13416", "title": "defos-lemma-lift-complex", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data \\begin{enumerate} \\item flat $\\mathcal{O}$-modules $\\mathcal{G}^n$, \\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{G}^{n + 1}$, \\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules, \\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{K}_0^n$ \\end{enumerate} such that \\begin{enumerate} \\item[(a)] $H^n(\\mathcal{K}_0^\\bullet) = 0$ for $n \\gg 0$, \\item[(b)] $\\mathcal{G}^n = 0$ for $n \\gg 0$, \\item[(c)] with $\\mathcal{G}^n_0 = \\mathcal{G}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$ the induced maps determine a complex $\\mathcal{G}_0^\\bullet$ and a map of complexes $\\mathcal{G}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$. \\end{enumerate} Then there exist \\begin{enumerate} \\item[(\\romannumeral1)] flat $\\mathcal{O}$-modules $\\mathcal{F}^n$, \\item[(\\romannumeral2)] maps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$, \\item[(\\romannumeral3)] maps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{K}_0^n$, \\item[(\\romannumeral4)] maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{F}^n$, \\end{enumerate} such that $\\mathcal{F}^n = 0$ for $n \\gg 0$, such that the diagrams $$ \\xymatrix{ \\mathcal{G}^n \\ar[r] \\ar[d] & \\mathcal{G}^{n + 1} \\ar[d] \\\\ \\mathcal{F}^n \\ar[r] & \\mathcal{F}^{n + 1} } $$ commute for all $n$, such that the composition $\\mathcal{G}^n \\to \\mathcal{F}^n \\to \\mathcal{K}_0^n$ is the given map $\\mathcal{G}^n \\to \\mathcal{K}_0^n$, and such that with $\\mathcal{F}^n_0 = \\mathcal{F}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$ we obtain a complex $\\mathcal{F}_0^\\bullet$ and map of complexes $\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$ which is a quasi-isomorphism."}
+{"_id": "13417", "title": "defos-lemma-canonical-class", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\\mathcal{I}$ of square zero. For every object $K_0$ in $D^-(\\mathcal{O}_0)$ there is a canonical map $$ \\omega(K_0) : K_0 \\longrightarrow K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}[2] $$ in $D(\\mathcal{O}_0)$ such that for any map $K_0 \\to L_0$ in $D^-(\\mathcal{O}_0)$ the diagram $$ \\xymatrix{ K_0 \\ar[d] \\ar[rr]_-{\\omega(K_0)} & & (K_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2] \\ar[d] \\\\ L_0 \\ar[rr]^-{\\omega(L_0)} & & (L_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2] } $$ commutes."}
+{"_id": "13418", "title": "defos-lemma-induced-map", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\alpha : K \\to L$ be a map of $D^-(\\mathcal{O})$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Let $n \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $H^i(\\alpha)$ is an isomorphism for $i \\geq n$, then $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$ is an isomorphism for $i \\geq n$. \\item If $H^i(\\alpha)$ is an isomorphism for $i > n$  and surjective for $i = n$, then $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$ is an isomorphism for $i > n$ and surjective for $i = n$. \\end{enumerate}"}
+{"_id": "13419", "title": "defos-lemma-canonical-class-obstruction", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\\mathcal{I}$ of square zero. For every object $K_0$ in $D^-(\\mathcal{O}_0)$ the following are equivalent \\begin{enumerate} \\item  the class $\\omega(K_0) \\in \\Ext^2_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0} \\mathcal{I})$ constructed in Lemma \\ref{lemma-canonical-class} is zero, \\item there exists $K \\in D^-(\\mathcal{O})$ with $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 = K_0$ in $D(\\mathcal{O}_0)$. \\end{enumerate}"}
+{"_id": "13420", "title": "defos-lemma-lift-map-complexes", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data \\begin{enumerate} \\item a complex of $\\mathcal{O}$-modules $\\mathcal{F}^\\bullet$, \\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules, \\item a quasi-isomorphism $\\mathcal{K}_0^\\bullet \\to \\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$, \\end{enumerate} Then there exist a quasi-isomorphism $\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet$ such that the map of complexes $\\mathcal{G}^\\bullet  \\otimes_\\mathcal{O} \\mathcal{O}_0 \\to \\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$ factors through $\\mathcal{K}_0^\\bullet$ in the homotopy category of complexes of $\\mathcal{O}_0$-modules."}
+{"_id": "13421", "title": "defos-lemma-inf-obs-map-defo-complex", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\\mathcal{I}$ of square zero. Let $K, L \\in D^-(\\mathcal{O})$. Set $K_0 = K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$ and $L_0 = L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$ in $D^-(\\mathcal{O}_0)$. Given $\\alpha_0 : K_0 \\to L_0$ in $D(\\mathcal{O}_0)$ there is a canonical element $$ o(\\alpha_0) \\in \\Ext^1_{\\mathcal{O}_0}(K_0, L_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}) $$ whose vanishing is necessary and sufficient for the existence of a map $\\alpha : K \\to L$ in $D(\\mathcal{O})$ with $\\alpha_0 = \\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}$."}
+{"_id": "13422", "title": "defos-lemma-first-order-defos-complex", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\\mathcal{I}$ of square zero. Let $K_0 \\in D^-(\\mathcal{O})$. A lift of $K_0$ is a pair $(K, \\alpha_0)$ consisting of an object $K$ in $D^-(\\mathcal{O})$ and an isomorphism $\\alpha_0 : K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 \\to K_0$ in $D(\\mathcal{O}_0)$. \\begin{enumerate} \\item Given a lift $(K, \\alpha)$ the group of automorphism of the pair is canonically the cokernel of a map $$ \\Ext^{-1}_{\\mathcal{O}_0}(K_0, K_0) \\longrightarrow \\Hom_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}) $$ \\item If there is a lift, then the set of isomorphism classes of lifts is principal homogenenous under $\\Ext^1_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})$. \\end{enumerate}"}
+{"_id": "13438", "title": "groupoids-quotients-lemma-invariant", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$. A morphism of algebraic spaces $\\phi : U \\to X$ is $R$-invariant if and only if it factors as $U \\to U/R \\to X$."}
+{"_id": "13439", "title": "groupoids-quotients-lemma-base-change-on-invariant", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$. Let $U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Let $X' \\to X$ be any morphism of algebraic spaces. \\begin{enumerate} \\item Setting $U' = X' \\times_X U$, $R' = X' \\times_X R$ we obtain a pre-relation $j' : R' \\to U' \\times_B U'$. \\item If $j$ is a relation, then $j'$ is a relation. \\item If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. \\item If $j$ is an equivalence relation, then $j'$ is an equivalence relation. \\item If $j$ comes from a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$, then \\begin{enumerate} \\item $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $X$, and \\item $j'$ comes from the base change $(U', R', s', t', c')$ of this groupoid to $X'$, see Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-base-change-groupoid}. \\end{enumerate} \\item If $j$ comes from the action of a group algebraic space $G/B$ on $U$ as in Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-groupoid-from-action} then $j'$ comes from the induced action of $G$ on $U'$. \\end{enumerate}"}
+{"_id": "13440", "title": "groupoids-quotients-lemma-base-change-quotient-sheaf", "text": "In the situation of Lemma \\ref{lemma-base-change-on-invariant} there is an isomorphism of sheaves $$ U'/R' = X' \\times_X U/R $$ For the construction of quotient sheaves, see Groupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves}."}
+{"_id": "13441", "title": "groupoids-quotients-lemma-base-change-quotient-stack", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Let $g : X' \\to X$ be a morphism of algebraic spaces over $B$ and let $(U', R', s', t', c')$ be the base change as in Lemma \\ref{lemma-base-change-on-invariant}. Then $$ \\xymatrix{ [U'/R'] \\ar[r] \\ar[d] & [U/R] \\ar[d] \\\\ \\mathcal{S}_{X'} \\ar[r] & \\mathcal{S}_X } $$ is a $2$-fibre product of stacks in groupoids over $(\\Sch/S)_{fppf}$. For the construction of quotient stacks and the morphisms in this diagram, see Groupoids in Spaces, Section \\ref{spaces-groupoids-section-stacks}."}
+{"_id": "13442", "title": "groupoids-quotients-lemma-categorical-unique", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation in algebraic spaces over $B$. If a categorical quotient in the category of algebraic spaces over $B$ exists, then it is unique up to unique isomorphism. Similarly for categorical quotients in full subcategories of $\\textit{Spaces}/B$."}
+{"_id": "13443", "title": "groupoids-quotients-lemma-categorical-reduced", "text": "In the situation of Definition \\ref{definition-categorical}. If $\\phi : U \\to X$ is a categorical quotient and $U$ is reduced, then $X$ is reduced. The same holds for categorical quotients in a category of spaces $\\mathcal{C}$ listed in Example \\ref{example-categories}."}
+{"_id": "13445", "title": "groupoids-quotients-lemma-invariant-map-constant-on-orbit", "text": "In the situation of Definition \\ref{definition-orbit}. Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Then $|\\phi| : |U| \\to |X|$ is constant on the orbits."}
+{"_id": "13446", "title": "groupoids-quotients-lemma-weak-orbit-pre-equivalence", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\\Spec(k) \\to B$ be a geometric point of $B$. Let $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$. In this case the weak orbit of $\\overline{u} \\in U(k)$ is simply $$ \\{ \\overline{u}' \\in U(k) \\text{ such that } \\exists \\overline{r} \\in R(k), \\ s(\\overline{r}) = \\overline{u}, \\ t(\\overline{r}) = \\overline{u}' \\} $$ and the orbit of $\\overline{u} \\in U(k)$ is $$ \\{ \\overline{u}' \\in U(k) : \\exists\\text{ field extension }k \\subset K, \\ \\exists\\ r \\in R(K), \\ s(r) = \\overline{u}, \\ t(r) = \\overline{u}'\\} $$"}
+{"_id": "13447", "title": "groupoids-quotients-lemma-make-pre-equivalence", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. Then $j_\\infty : R_\\infty \\to U \\times_B U$ is a pre-equivalence relation over $B$. Moreover \\begin{enumerate} \\item $\\phi : U \\to X$ is $R$-invariant if and only if it is $R_\\infty$-invariant, \\item the canonical map of quotient sheaves $U/R \\to U/R_\\infty$ (see Groupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves}) is an isomorphism, \\item weak $R$-orbits agree with weak $R_\\infty$-orbits, \\item $R$-orbits agree with $R_\\infty$-orbits, \\item if $s, t$ are locally of finite type, then $s_\\infty$, $t_\\infty$ are locally of finite type, \\item add more here as needed. \\end{enumerate}"}
+{"_id": "13448", "title": "groupoids-quotients-lemma-geometric-orbits", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. Let $\\Spec(k) \\to B$ be a geometric point of $B$. \\begin{enumerate} \\item If $s, t : R \\to U$ are locally of finite type then weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree with $R$-orbits on $U(k)$. \\item If $k$ has sufficiently large cardinality then weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree with $R$-orbits on $U(k)$. \\end{enumerate}"}
+{"_id": "13449", "title": "groupoids-quotients-lemma-set-theoretic-invariant", "text": "In the situation of Definition \\ref{definition-set-theoretically-invariant}. A morphism $\\phi : U \\to X$ is set-theoretically $R$-invariant if and only if for any algebraically closed field $k$ over $B$ the map $U(k) \\to X(k)$ is constant on orbits."}
+{"_id": "13450", "title": "groupoids-quotients-lemma-invariant-set-theoretically-invariant", "text": "In the situation of Definition \\ref{definition-set-theoretically-invariant}. An invariant morphism is set-theoretically invariant."}
+{"_id": "13452", "title": "groupoids-quotients-lemma-set-theoretic-pre-equivalence-geometric", "text": "In the situation of Definition \\ref{definition-set-theoretic-equivalence}. The following are equivalent: \\begin{enumerate} \\item The morphism $j$ is a set-theoretic pre-equivalence relation. \\item The subset $j(|R|) \\subset |U \\times_B U|$ contains the image of $|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}). \\item For every algebraically closed field $k$ over $B$ of sufficiently large cardinality the subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence relation. \\end{enumerate} If $s, t$ are locally of finite type these are also equivalent to \\begin{enumerate} \\item[(4)] For every algebraically closed field $k$ over $B$ the subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence relation. \\end{enumerate}"}
+{"_id": "13454", "title": "groupoids-quotients-lemma-set-theoretic-equivalence", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. \\begin{enumerate} \\item If $j$ is a pre-equivalence relation, then $j$ is a set-theoretic pre-equivalence relation. This holds in particular when $j$ comes from a groupoid in algebraic spaces, or from an action of a group algebraic space on $U$. \\item If $j$ is an equivalence relation, then $j$ is a set-theoretic equivalence relation. \\end{enumerate}"}
+{"_id": "13455", "title": "groupoids-quotients-lemma-separates-orbits", "text": "Let $B \\to S$ be as in Section \\ref{section-conventions-notation}. Let $j : R \\to U \\times_B U$ be a pre-relation. Let $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$. Consider the diagram $$ \\xymatrix{ (U \\times_X U) \\times_{(U \\times_B U)} R \\ar[d]^q \\ar[r]_-p & R \\ar[d]^j \\\\ U \\times_X U \\ar[r]^c & U \\times_B U } $$ Then we have: \\begin{enumerate} \\item The morphism $\\phi$ is set-theoretically invariant if and only if $p$ is surjective. \\item If $j$ is a set-theoretic pre-equivalence relation then $\\phi$ separates orbits if and only if $p$ and $q$ are surjective. \\item If $p$ and $q$ are surjective, then $j$ is a set-theoretic pre-equivalence relation (and $\\phi$ separates orbits). \\item If $\\phi$ is $R$-invariant and $j$ is a set-theoretic pre-equivalence relation, then $\\phi$ separates orbits if and only if the induced morphism $R \\to U \\times_X U$ is surjective. \\end{enumerate}"}
+{"_id": "13475", "title": "spaces-resolve-lemma-dimension-special-fibre", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain of dimension $\\geq 1$. Let $f : X \\to \\Spec(A)$ be a morphism of algebraic spaces. Assume at least one of the following conditions is satisfied \\begin{enumerate} \\item $f$ is a modification (Spaces over Fields, Definition \\ref{spaces-over-fields-definition-modification}), \\item $f$ is an alteration (Spaces over Fields, Definition \\ref{spaces-over-fields-definition-alteration}), \\item $f$ is locally of finite type, quasi-separated, $X$ is integral, and there is exactly one point of $|X|$ mapping to the generic point of $\\Spec(A)$, \\item $f$ is locally of finite type, $X$ is decent, and the points of $|X|$ mapping to the generic point of $\\Spec(A)$ are the generic points of irreducible components of $|X|$, \\item add more here. \\end{enumerate} Then $\\dim(X_\\kappa) \\leq \\dim(A) - 1$."}
+{"_id": "13477", "title": "spaces-resolve-lemma-equivalence", "text": "The functor $F$ (\\ref{equation-equivalence}) is an equivalence."}
+{"_id": "13479", "title": "spaces-resolve-lemma-equivalence-fibre", "text": "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}). If $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$, then $Y_{x_i} \\cong (Y_i)_{u_i}$ as algebraic spaces."}
+{"_id": "13480", "title": "spaces-resolve-lemma-equivalence-sequence-blowups", "text": "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}) and assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$. Then there exists a factorization $$ Y = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X $$ of $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i $$ of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$."}
+{"_id": "13481", "title": "spaces-resolve-lemma-make-ideal-principal", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \\subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$. Then there exists a sequence $$ X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 = X $$ where $X_{j + 1} \\to X_j$ is the blowing up of $X_j$ at a closed point $x_j$ lying above a point of $T$ such that $\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf."}
+{"_id": "13484", "title": "spaces-resolve-lemma-equivalence-sequence-normalized-blowups", "text": "Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}) and assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$. Assume $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}. Then there exists a factorization $$ Y = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X $$ of $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$ at a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if for each $i$ there exists a factorization $$ Y_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i $$ of $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$."}
+{"_id": "13487", "title": "spaces-resolve-lemma-port-regularity-to-completion", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $X \\to \\Spec(A)$ be a morphism which is locally of finite type with $X$ a decent algebraic space. Set $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in |Y|$ with image $x \\in |X|$. Then \\begin{enumerate} \\item if $\\mathcal{O}_{Y, y}^h$ is regular, then $\\mathcal{O}_{X, x}^h$ is regular, \\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}^h$ is regular $\\Leftrightarrow \\mathcal{O}_{X, x}^h$ is regular, and \\item If $X$ is proper over $A$, then $X$ is regular if and only if $Y$ is regular. \\end{enumerate}"}
+{"_id": "13489", "title": "spaces-resolve-lemma-regular-alteration-implies", "text": "Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over $S$. Assume there exists an alteration $f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$ is finite and $Y$ has a dense open which is regular."}
+{"_id": "13490", "title": "spaces-resolve-lemma-regular-alteration-implies-local", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain. Assume there exists an alteration $f : X \\to \\Spec(A)$ with $X$ regular. Then \\begin{enumerate} \\item there exists a nonzero $f \\in A$ such that $A_f$ is regular, \\item the integral closure $B$ of $A$ in its fraction field is finite over $A$, \\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the completions of $B$ at its maximal ideals are normal domains, and \\item the generic formal fibre of $A$ is regular. \\end{enumerate}"}
+{"_id": "13496", "title": "duality-lemma-equivalent-definitions", "text": "Let $X$ be a locally Noetherian scheme. Let $K$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent \\begin{enumerate} \\item For every affine open $U = \\Spec(A) \\subset X$ there exists a dualizing complex $\\omega_A^\\bullet$ for $A$ such that $K|_U$ is isomorphic to the image of $\\omega_A^\\bullet$ by the functor $\\widetilde{} : D(A) \\to D(\\mathcal{O}_U)$. \\item There is an affine open covering $X = \\bigcup U_i$, $U_i = \\Spec(A_i)$ such that for each $i$ there exists a dualizing complex $\\omega_i^\\bullet$ for $A_i$ such that $K|_{U_i}$ is isomorphic to the image of $\\omega_i^\\bullet$ by the functor $\\widetilde{} : D(A_i) \\to D(\\mathcal{O}_{U_i})$. \\end{enumerate}"}
+{"_id": "13497", "title": "duality-lemma-affine-duality", "text": "Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let $K, L$ be objects of $D(A)$. If $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective dimension, then $$ R\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L}) = \\widetilde{R\\Hom_A(K, L)} $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "13498", "title": "duality-lemma-internal-hom-evaluate-isom", "text": "Let $X$ be a Noetherian scheme. Let $K, L, M \\in D_\\QCoh(\\mathcal{O}_X)$.  Then the map $$ R\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\longrightarrow R\\SheafHom(R\\SheafHom(K, L), M) $$ of Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-evaluate} is an isomorphism in the following two cases \\begin{enumerate} \\item $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, $L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$, and $M$ affine locally has finite injective dimension (see proof), or \\item $K$ and $L$ are in $D_{\\textit{Coh}}(\\mathcal{O}_X)$, the object $R\\SheafHom(L, M)$ has finite tor dimension, and $L$ and $M$ affine locally have finite injective dimension (in particular $L$ and $M$ are bounded). \\end{enumerate}"}
+{"_id": "13499", "title": "duality-lemma-dualizing-schemes", "text": "Let $K$ be a dualizing complex on a locally Noetherian scheme $X$. Then $K$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_X)$ and $D = R\\SheafHom_{\\mathcal{O}_X}(-, K)$ induces an anti-equivalence $$ D : D_{\\textit{Coh}}(\\mathcal{O}_X) \\longrightarrow D_{\\textit{Coh}}(\\mathcal{O}_X) $$ which comes equipped with a canonical isomorphism $\\text{id} \\to D \\circ D$. If $X$ is quasi-compact, then $D$ exchanges $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and induces an equivalence $D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "13500", "title": "duality-lemma-dualizing-unique-schemes", "text": "Let $X$ be a locally Noetherian scheme. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ for some invertible object $L$ of $D(\\mathcal{O}_X)$."}
+{"_id": "13501", "title": "duality-lemma-dimension-function-scheme", "text": "Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$ be a dualizing complex on $X$. Then $X$ is universally catenary and the function $X \\to \\mathbf{Z}$ defined by $$ x \\longmapsto \\delta(x)\\text{ such that } \\omega_{X, x}^\\bullet[-\\delta(x)] \\text{ is a normalized dualizing complex over } \\mathcal{O}_{X, x} $$ is a dimension function."}
+{"_id": "13502", "title": "duality-lemma-sitting-in-degrees", "text": "Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$ be a dualizing complex on $X$ with associated dimension function $\\delta$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Set $\\mathcal{E}^i = \\SheafExt^{-i}_{\\mathcal{O}_X}(\\mathcal{F}, \\omega_X^\\bullet)$. Then $\\mathcal{E}^i$ is a coherent $\\mathcal{O}_X$-module and for $x \\in X$ we have \\begin{enumerate} \\item $\\mathcal{E}^i_x$ is nonzero only for $\\delta(x) \\leq i \\leq \\delta(x) + \\dim(\\text{Supp}(\\mathcal{F}_x))$, \\item $\\dim(\\text{Supp}(\\mathcal{E}^{i + \\delta(x)}_x)) \\leq i$, \\item $\\text{depth}(\\mathcal{F}_x)$ is the smallest integer $i \\geq 0$ such that $\\mathcal{E}^{i + \\delta(x)} \\not = 0$, and \\item we have $x \\in \\text{Supp}(\\bigoplus_{j \\leq i} \\mathcal{E}^j) \\Leftrightarrow \\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\delta(x) \\leq i$. \\end{enumerate}"}
+{"_id": "13503", "title": "duality-lemma-twisted-inverse-image", "text": "\\begin{reference} This is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}. \\end{reference} Let $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact schemes. The functor $Rf_* : D_\\QCoh(X) \\to D_\\QCoh(Y)$ has a right adjoint."}
+{"_id": "13504", "title": "duality-lemma-twisted-inverse-image-bounded-below", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ be the right adjoint to $Rf_*$ of Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ maps $D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$. In fact, there exists an integer $N$ such that $H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$."}
+{"_id": "13505", "title": "duality-lemma-iso-on-RSheafHom", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$. Let $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$. Then the map (\\ref{equation-sheafy-trace}) $$ Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\longrightarrow R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) $$ becomes an isomorphism after applying the functor $DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$ discussed in Derived Categories of Schemes, Section \\ref{perfect-section-better-coherator}."}
+{"_id": "13506", "title": "duality-lemma-iso-global-hom", "text": "Let $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact schemes. For all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$ (\\ref{equation-sheafy-trace}) induces an isomorphism $R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs."}
+{"_id": "13507", "title": "duality-lemma-flat-precompose-pus", "text": "In diagram (\\ref{equation-base-change}) assume that $g$ is flat or more generally that $f$ and $g$ are Tor independent. Then $a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism."}
+{"_id": "13508", "title": "duality-lemma-when-sheafy", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$. Let $V \\subset Y$ be quasi-compact open with inverse image $U \\subset X$. \\begin{enumerate} \\item For every $Q \\in D_\\QCoh^+(\\mathcal{O}_Y)$ supported on $Y \\setminus V$ the image $a(Q)$ is supported on $X \\setminus U$ if and only if (\\ref{equation-sheafy}) is an isomorphism on all $K$ in $D_\\QCoh^+(\\mathcal{O}_Y)$. \\item For every $Q \\in D_\\QCoh(\\mathcal{O}_Y)$ supported on $Y \\setminus V$ the image $a(Q)$ is supported on $X \\setminus U$ if and only if (\\ref{equation-sheafy}) is an isomorphism on all $K$ in $D_\\QCoh(\\mathcal{O}_Y)$. \\item If $a$ commutes with direct sums, then the equivalent conditions of (1) imply the equivalent conditions of (2). \\end{enumerate}"}
+{"_id": "13509", "title": "duality-lemma-proper-noetherian", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a proper morphism. If\\footnote{This proof works for those morphisms of quasi-compact and quasi-separated schemes such that $Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows easily from a theorem of Kiehl \\cite{Kiehl} that this holds if $f$ is proper and pseudo-coherent. This is the correct generality for this lemma and some of the other results in this chapter.} \\begin{enumerate} \\item $f$ is flat and of finite presentation, or \\item $Y$ is Noetherian \\end{enumerate} then the equivalent conditions of Lemma \\ref{lemma-when-sheafy} part (1) hold for all quasi-compact opens $V$ of $Y$."}
+{"_id": "13510", "title": "duality-lemma-compose-base-change-maps", "text": "Consider a commutative diagram $$ \\xymatrix{ X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z' \\ar[r]^m & Z } $$ of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where $f$ and $l$ as well as $g$ and $m$ are Tor independent. Then the maps (\\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement)."}
+{"_id": "13511", "title": "duality-lemma-compose-base-change-maps-horizontal", "text": "Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y } $$ of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where $f$ and $h$ as well as $f'$ and $h'$ are Tor independent. Then the maps (\\ref{equation-base-change-map}) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement)."}
+{"_id": "13512", "title": "duality-lemma-more-base-change", "text": "In diagram (\\ref{equation-base-change}) assume \\begin{enumerate} \\item $g : Y' \\to Y$ is a morphism of affine schemes, \\item $f : X \\to Y$ is proper, and \\item $f$ and $g$ are Tor independent. \\end{enumerate} Then the base change map (\\ref{equation-base-change-map}) induces an isomorphism $$ L(g')^*a(K) \\longrightarrow a'(Lg^*K) $$ in the following cases \\begin{enumerate} \\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$ is flat of finite presentation, \\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$ is perfect and $Y$ Noetherian, \\item for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$ if $g$ has finite Tor dimension and $Y$ Noetherian. \\end{enumerate}"}
+{"_id": "13513", "title": "duality-lemma-trace-map-and-base-change", "text": "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$ are tor independent. Then the maps $1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and $\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$ agree via the base change maps $\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$ (Cohomology, Remark \\ref{cohomology-remark-base-change}) and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$ (\\ref{equation-base-change-map}). More precisely, the diagram $$ \\xymatrix{ Lg^* \\circ Rf_* \\circ a \\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} & Lg^* \\\\ Rf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} & Rf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1} } $$ of transformations of functors commutes."}
+{"_id": "13514", "title": "duality-lemma-unit-and-base-change", "text": "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$ are tor independent. Then the maps $1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and $\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$ agree via the base change maps $\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$ (Cohomology, Remark \\ref{cohomology-remark-base-change}) and $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$ (\\ref{equation-base-change-map}). More precisely, the diagram $$ \\xymatrix{ L(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} & L(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\ a' \\circ Rf'_* \\circ L(g')^* & a' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta } $$ of transformations of functors commutes."}
+{"_id": "13515", "title": "duality-lemma-compare-with-pullback-perfect", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. The map $Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to a(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$ defined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$ is an isomorphism if $K$ is perfect. In particular, (\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect."}
+{"_id": "13516", "title": "duality-lemma-restriction-compare-with-pullback", "text": "Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$ are tor independent. Let $K \\in D_\\QCoh(\\mathcal{O}_Y)$. The diagram $$ \\xymatrix{ L(g')^*(Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(\\mathcal{O}_Y)) \\ar[r] \\ar[d] & L(g')^*a(K) \\ar[d] \\\\ L(f')^*Lg^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} a'(\\mathcal{O}_{Y'}) \\ar[r] & a'(Lg^*K) } $$ commutes where the horizontal arrows are the maps (\\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$ and the vertical maps are constructed using Cohomology, Remark \\ref{cohomology-remark-base-change} and (\\ref{equation-base-change-map})."}
+{"_id": "13517", "title": "duality-lemma-compare-on-open", "text": "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes. Let $V \\subset Y$ be an open such that $f^{-1}(V) \\to V$ is an isomorphism. Then for $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$ the map (\\ref{equation-compare-with-pullback}) restricts to an isomorphism over $f^{-1}(V)$."}
+{"_id": "13518", "title": "duality-lemma-transitivity-compare-with-pullback", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of quasi-compact and quasi-separated schemes and set $h = g \\circ f$. Let $a, b, c$ be the adjoints of Lemma \\ref{lemma-twisted-inverse-image} for $f, g, h$. For any $K \\in D_\\QCoh(\\mathcal{O}_Z)$ the diagram $$ \\xymatrix{ Lf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} b(\\mathcal{O}_Z)) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y) \\ar@{=}[d] \\ar[r] & a(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} b(\\mathcal{O}_Z)) \\ar[r] & a(b(K)) \\ar@{=}[d] \\\\ Lh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*b(\\mathcal{O}_Z) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y) \\ar[r] & Lh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} c(\\mathcal{O}_Z) \\ar[r] & c(K) } $$ is commutative where the arrows are (\\ref{equation-compare-with-pullback}) and we have used $Lh^* = Lf^* \\circ Lg^*$ and $c = a \\circ b$."}
+{"_id": "13519", "title": "duality-lemma-compute-sheaf-with-exact-support", "text": "With notation as above. The functor $\\SheafHom(\\mathcal{O}_Z, -)$ is a right adjoint to the functor $i_* : \\textit{Mod}(\\mathcal{O}_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$. For $V \\subset Z$ open we have $$ \\Gamma(V, \\SheafHom(\\mathcal{O}_Z, \\mathcal{F})) = \\{s \\in \\Gamma(U, \\mathcal{F}) \\mid \\mathcal{I}s = 0\\} $$ where $U \\subset X$ is an open whose intersection with $Z$ is $V$."}
+{"_id": "13520", "title": "duality-lemma-sheaf-with-exact-support-adjoint", "text": "With notation as above. The functor $R\\SheafHom(\\mathcal{O}_Z, -)$ is the right adjoint of the functor $Ri_* : D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$."}
+{"_id": "13521", "title": "duality-lemma-sheaf-with-exact-support-ext", "text": "With notation as above. We have $$ Ri_*R\\SheafHom(\\mathcal{O}_Z, K) = R\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Z, K) $$ in $D(\\mathcal{O}_X)$ for all $K$ in $D(\\mathcal{O}_X)$."}
+{"_id": "13522", "title": "duality-lemma-sheaf-with-exact-support-internal-home", "text": "With notation as above. For $M \\in D(\\mathcal{O}_Z)$ we have $$ R\\SheafHom_{\\mathcal{O}_X}(Ri_*M, K) = Ri_*R\\SheafHom_{\\mathcal{O}_Z}(M, R\\SheafHom(\\mathcal{O}_Z, K)) $$ in $D(\\mathcal{O}_Z)$ for all $K$ in $D(\\mathcal{O}_X)$."}
+{"_id": "13523", "title": "duality-lemma-sheaf-with-exact-support-quasi-coherent", "text": "Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes (any closed immersion if $X$ is locally Noetherian). Then \\begin{enumerate} \\item $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_\\QCoh(\\mathcal{O}_X)$ into $D^+_\\QCoh(\\mathcal{O}_Z)$, and \\item if $X = \\Spec(A)$ and $Z = \\Spec(B)$, then the diagram $$ \\xymatrix{ D^+(B) \\ar[r] & D_\\QCoh^+(\\mathcal{O}_Z) \\\\ D^+(A) \\ar[r] \\ar[u]^{R\\Hom(B, -)} & D_\\QCoh^+(\\mathcal{O}_X) \\ar[u]_{R\\SheafHom(\\mathcal{O}_Z, -)} } $$ is commutative. \\end{enumerate}"}
+{"_id": "13524", "title": "duality-lemma-sheaf-with-exact-support-coherent", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Assume $X$ is a locally Noetherian. Then $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^+_{\\textit{Coh}}(\\mathcal{O}_Z)$."}
+{"_id": "13525", "title": "duality-lemma-twisted-inverse-image-closed", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \\to X$ be a pseudo-coherent closed immersion (if $X$ is Noetherian, then any closed immersion is pseudo-coherent). Let $a : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Z)$ be the right adjoint to $Ri_*$. Then there is a functorial isomorphism $$ a(K) = R\\SheafHom(\\mathcal{O}_Z, K) $$ for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$."}
+{"_id": "13526", "title": "duality-lemma-check-base-change-is-iso", "text": "In the situation above, the map (\\ref{equation-base-change-exact-support}) is an isomorphism if and only if the base change map $$ Lf^*R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_Z, K) \\longrightarrow R\\SheafHom_{\\mathcal{O}_{X'}}(\\mathcal{O}_{Z'}, Lf^*K) $$ of Cohomology, Remark \\ref{cohomology-remark-prepare-fancy-base-change} is an isomorphism."}
+{"_id": "13527", "title": "duality-lemma-flat-bc-sheaf-with-exact-support", "text": "In the situation above, assume $f$ is flat and $i$ pseudo-coherent. Then (\\ref{equation-base-change-exact-support}) is an isomorphism for $K$ in $D^+_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "13528", "title": "duality-lemma-sheaf-with-exact-support-tensor", "text": "Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes. Let $M \\in D_\\QCoh(\\mathcal{O}_X)$ locally have tor-amplitude in $[a, \\infty)$. Let $K \\in D_\\QCoh^+(\\mathcal{O}_X)$. Then there is a canonical isomorphism $$ R\\SheafHom(\\mathcal{O}_Z, K) \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Li^*M = R\\SheafHom(\\mathcal{O}_Z, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) $$ in $D(\\mathcal{O}_Z)$."}
+{"_id": "13529", "title": "duality-lemma-compute-sheafhom-affine", "text": "With notation as above. The functor $\\SheafHom(f_*\\mathcal{O}_Y, -)$ is a right adjoint to the restriction functor $\\textit{Mod}(f_*\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$. For an affine open $U \\subset X$ we have $$ \\Gamma(U, \\SheafHom(f_*\\mathcal{O}_Y, \\mathcal{F})) = \\Hom_A(B, \\mathcal{F}(U)) $$ where $A = \\mathcal{O}_X(U)$ and $B = \\mathcal{O}_Y(f^{-1}(U))$."}
+{"_id": "13530", "title": "duality-lemma-sheafhom-affine-adjoint", "text": "With notation as above. The functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$ is the right adjoint of the functor $D(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$."}
+{"_id": "13531", "title": "duality-lemma-sheafhom-affine-ext", "text": "With notation as above. The composition $$ D(\\mathcal{O}_X) \\xrightarrow{R\\SheafHom(f_*\\mathcal{O}_Y, -)} D(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X) $$ is the functor $K \\mapsto R\\SheafHom_{\\mathcal{O}_X}(f_*\\mathcal{O}_Y, K)$."}
+{"_id": "13532", "title": "duality-lemma-finite-twisted", "text": "Let $f : Y \\to X$ be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$ maps $D_\\QCoh^+(\\mathcal{O}_X)$ into $D_\\QCoh^+(f_*\\mathcal{O}_Y)$. If $X$ is quasi-compact and quasi-separated, then the diagram $$ \\xymatrix{ D_\\QCoh^+(\\mathcal{O}_X) \\ar[rr]_a \\ar[rd]_{R\\SheafHom(f_*\\mathcal{O}_Y, -)} & & D_\\QCoh^+(\\mathcal{O}_Y) \\ar[ld]^\\Phi \\\\ & D_\\QCoh^+(f_*\\mathcal{O}_Y) } $$ is commutative, where $a$ is the right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for $f$ and $\\Phi$ is the equivalence of Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-morphism-equivalence}."}
+{"_id": "13533", "title": "duality-lemma-proper-flat", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums."}
+{"_id": "13534", "title": "duality-lemma-proper-flat-relative", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then \\begin{enumerate} \\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$, \\item for every open $V \\subset Y$ and any $K \\in D_\\QCoh(\\mathcal{O}_Y)$ the map (\\ref{equation-sheafy}) is an isomorphism, and \\end{enumerate}"}
+{"_id": "13535", "title": "duality-lemma-compare-with-pullback-flat-proper", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. The map (\\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "13536", "title": "duality-lemma-proper-flat-base-change", "text": "Let $g : Y' \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $f : X \\to Y$ be a proper, flat morphism of finite presentation. Then the base change map (\\ref{equation-base-change-map}) is an isomorphism for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "13537", "title": "duality-lemma-properties-relative-dualizing", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a morphism of schemes which is proper, flat, and of finite presentation with relative dualizing complex $\\omega_{X/Y}^\\bullet$ (Remark \\ref{remark-relative-dualizing-complex}). Then \\begin{enumerate} \\item $\\omega_{X/Y}^\\bullet$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$, \\item $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology sheaves in positive degrees, \\item $\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$ is an isomorphism. \\end{enumerate}"}
+{"_id": "13538", "title": "duality-lemma-van-den-bergh", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a proper, flat morphism of finite presentation with relative dualizing complex $\\omega_{X/Y}^\\bullet$ (Remark \\ref{remark-relative-dualizing-complex}). There is a canonical isomorphism \\begin{equation} \\label{equation-pre-rigid} \\mathcal{O}_X = c(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet) = c(L\\text{pr}_2^*\\omega_{X/Y}^\\bullet) \\end{equation} and a canonical isomorphism \\begin{equation} \\label{equation-rigid} \\omega_{X/Y}^\\bullet = c\\left(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet \\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L} L\\text{pr}_2^*\\omega_{X/Y}^\\bullet\\right) \\end{equation} where $c$ is the right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for the diagonal $\\Delta : X \\to X \\times_Y X$."}
+{"_id": "13539", "title": "duality-lemma-proper-flat-noetherian", "text": "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums."}
+{"_id": "13540", "title": "duality-lemma-proper-flat-noetherian-relative", "text": "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then \\begin{enumerate} \\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$, then $a(Q)$ is supported on $f^{-1}(T)$, \\item for every open $V \\subset Y$ and any $K \\in D_\\QCoh(\\mathcal{O}_Y)$ the map (\\ref{equation-sheafy}) is an isomorphism, and \\end{enumerate}"}
+{"_id": "13541", "title": "duality-lemma-compare-with-pullback-flat-proper-noetherian", "text": "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes. The map (\\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "13543", "title": "duality-lemma-compute-for-effective-Cartier", "text": "As above, let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor. There is a canonical isomorphism $R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X) = \\mathcal{N}[-1]$ in $D(\\mathcal{O}_D)$."}
+{"_id": "13544", "title": "duality-lemma-sheaf-with-exact-support-effective-Cartier", "text": "As above, let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor. Then (\\ref{equation-map-effective-Cartier}) combined with Lemma \\ref{lemma-compute-for-effective-Cartier} defines an isomorphism $$ Li^*K \\otimes_{\\mathcal{O}_D}^\\mathbf{L} \\mathcal{N}[-1] \\longrightarrow R\\SheafHom(\\mathcal{O}_D, K) $$ functorial in $K$ in $D(\\mathcal{O}_X)$."}
+{"_id": "13545", "title": "duality-lemma-upper-shriek-P1", "text": "Let $Y$ be a Noetherian scheme. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_Y$-module of rank $n + 1$ with determinant $\\mathcal{L} = \\wedge^{n + 1}(\\mathcal{E})$. Let $f : X = \\mathbf{P}(\\mathcal{E}) \\to Y$ be the projection. Let $a$ be the right adjoint for $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ c : f^*\\mathcal{L}(-n - 1)[n] \\longrightarrow a(\\mathcal{O}_Y) $$ In particular, if $\\mathcal{E} = \\mathcal{O}_Y^{\\oplus n + 1}$, then $X = \\mathbf{P}^n_Y$ and we obtain $a(\\mathcal{O}_Y) = \\mathcal{O}_X(-n - 1)[n]$."}
+{"_id": "13546", "title": "duality-lemma-ext", "text": "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and $\\mathcal{N} = \\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$. There is a canonical isomorphism $c : \\mathcal{N} \\to \\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X) $."}
+{"_id": "13547", "title": "duality-lemma-regular-ideal-ext", "text": "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$. If $\\mathcal{I}$ is Koszul-regular (Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf}) then composition on $R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)$ defines isomorphisms $$ \\wedge^i(\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)) \\longrightarrow \\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X) $$ for all $i$."}
+{"_id": "13548", "title": "duality-lemma-regular-immersion-ext", "text": "Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and $\\mathcal{N} = \\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$. If $\\mathcal{I}$ is Koszul-regular (Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf}) then $$ R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y) = \\wedge^r \\mathcal{N}[r] $$ where $r : Y \\to \\{1, 2, 3, \\ldots \\}$ sends $y$ to the minimal number of generators of $\\mathcal{I}$ needed in a neighbourhood of $y$."}
+{"_id": "13549", "title": "duality-lemma-regular-immersion", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $i : X \\to Y$ be a Koszul-regular closed immersion. Let $a$ be the right adjoint of $Ri_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ \\wedge^r\\mathcal{N}[-r] \\longrightarrow a(\\mathcal{O}_Y) $$ where $\\mathcal{N} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{C}_{X/Y}, \\mathcal{O}_X)$ is the normal sheaf of $i$ (Morphisms, Section \\ref{morphisms-section-conormal-sheaf}) and $r$ is its rank viewed as a locally constant function on $X$."}
+{"_id": "13550", "title": "duality-lemma-smooth-proper", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a smooth proper morphism of relative dimension $d$. Let $a$ be the right adjoint of $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$ as in Lemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism $$ \\wedge^d \\Omega_{X/S}[d] \\longrightarrow a(\\mathcal{O}_S) $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "13551", "title": "duality-lemma-shriek-well-defined", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. The functor $f^!$ is, up to canonical isomorphism, independent of the choice of the compactification."}
+{"_id": "13552", "title": "duality-lemma-upper-shriek-composition", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of $\\textit{FTS}_S$. Then there is a canonical isomorphism $(g \\circ f)^! \\to f^! \\circ g^!$."}
+{"_id": "13554", "title": "duality-lemma-map-pullback-to-shriek-well-defined", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. There are canonical maps $$ \\mu_{f, K} : Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\longrightarrow f^!K $$ functorial in $K$ in $D^+_\\QCoh(\\mathcal{O}_Y)$. If $g : Y \\to Z$ is another morphism of $\\textit{FTS}_S$, then the diagram $$ \\xymatrix{ Lf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\ar@{=}[d] \\ar[r]_-{\\mu_f} & f^!(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z) \\ar[r]_-{f^!\\mu_g} & f^!g^!K \\ar@{=}[d] \\\\ Lf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^* g^!\\mathcal{O}_Z \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\ar[r]^-{\\mu_f} & Lf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!g^!\\mathcal{O}_Z \\ar[r]^-{\\mu_{g \\circ f}} & f^!g^!K } $$ commutes for all $K \\in D^+_\\QCoh(\\mathcal{O}_Z)$."}
+{"_id": "13555", "title": "duality-lemma-shriek-open-immersion", "text": "In Situation \\ref{situation-shriek} let $Y$ be an object of $\\textit{FTS}_S$ and let $j : X \\to Y$ be an open immersion. Then there is a canonical isomorphism $j^! = j^*$ of functors."}
+{"_id": "13557", "title": "duality-lemma-shriek-affine-line", "text": "In Situation \\ref{situation-shriek} let $Y$ be an object of $\\textit{FTS}_S$ and let $f : X = \\mathbf{A}^1_Y \\to Y$ be the projection. Then there is a (noncanonical) isomorphism $f^!(-) \\cong Lf^*(-) [1]$ of functors."}
+{"_id": "13558", "title": "duality-lemma-shriek-closed-immersion", "text": "In Situation \\ref{situation-shriek} let $Y$ be an object of $\\textit{FTS}_S$ and let $i : X \\to Y$ be a closed immersion. Then there is a canonical isomorphism $i^!(-) = R\\SheafHom(\\mathcal{O}_X, -)$ of functors."}
+{"_id": "13559", "title": "duality-lemma-shriek-coherent", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Then $f^!$ maps $D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ into $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$."}
+{"_id": "13560", "title": "duality-lemma-shriek-dualizing", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. If $K$ is a dualizing complex for $Y$, then $f^!K$ is a dualizing complex for $X$."}
+{"_id": "13561", "title": "duality-lemma-shriek-via-duality", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $K$ be a dualizing complex on $Y$. Set $D_Y(M) = R\\SheafHom_{\\mathcal{O}_Y}(M, K)$ for $M \\in D_{\\textit{Coh}}(\\mathcal{O}_Y)$ and $D_X(E) = R\\SheafHom_{\\mathcal{O}_X}(E, f^!K)$ for $E \\in D_{\\textit{Coh}}(\\mathcal{O}_X)$. Then there is a canonical isomorphism $$ f^!M \\longrightarrow D_X(Lf^*D_Y(M)) $$ for $M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$."}
+{"_id": "13562", "title": "duality-lemma-perfect-comparison-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Assume $f$ is perfect (e.g., flat). Then \\begin{enumerate} \\item[(a)] $f^!$ maps $D_{\\textit{Coh}}^b(\\mathcal{O}_Y)$ into $D_{\\textit{Coh}}^b(\\mathcal{O}_X)$, \\item[(b)] the map $\\mu_{f,  K} : Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\to f^!K$ of Lemma \\ref{lemma-map-pullback-to-shriek-well-defined} is an isomorphism for all $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$. \\end{enumerate}"}
+{"_id": "13563", "title": "duality-lemma-flat-shriek-relatively-perfect", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. If $f$ is flat, then $f^!\\mathcal{O}_Y$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$ and $\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(f^!\\mathcal{O}_Y, f^!\\mathcal{O}_Y)$ is an isomorphism."}
+{"_id": "13564", "title": "duality-lemma-lci-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Assume $f : X \\to Y$ is a local complete intersection morphism. Then \\begin{enumerate} \\item $f^!\\mathcal{O}_Y$ is an invertible object of $D(\\mathcal{O}_X)$, and \\item $f^!$ maps perfect complexes to perfect complexes. \\end{enumerate}"}
+{"_id": "13565", "title": "duality-lemma-base-change-shriek-flat", "text": "In Situation \\ref{situation-shriek} let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of $\\textit{FTS}_S$ with $g$ flat. Then there is an isomorphism $L(g')^* \\circ f^! \\to (f')^! \\circ Lg^*$ on $D_\\QCoh^+(\\mathcal{O}_Y)$."}
+{"_id": "13566", "title": "duality-lemma-shriek-etale", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be an \\'etale morphism of $\\textit{FTS}_S$. Then $f^! \\cong f^*$ as functors on $D^+_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "13567", "title": "duality-lemma-base-change-locally", "text": "In Situation \\ref{situation-shriek} let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ be a cartesian diagram of $\\textit{FTS}_S$. Let $E \\in D^+_\\QCoh(\\mathcal{O}_Y)$ be an object such that $Lg^*E$ is in $D^+(\\mathcal{O}_Y)$. If $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$ restrict to isomorphic objects of $D(\\mathcal{O}_{U'})$ for $U' \\subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$."}
+{"_id": "13568", "title": "duality-lemma-relative-dualizing-fibres", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Assume $f$ is flat. Set $\\omega_{X/Y}^\\bullet = f^!\\mathcal{O}_Y$ in $D^b_{\\textit{Coh}}(X)$. Let $y \\in Y$ and $h : X_y \\to X$ the projection. Then $Lh^*\\omega_{X/Y}^\\bullet$ is a dualizing complex on $X_y$."}
+{"_id": "13569", "title": "duality-lemma-good-dualizing-unique", "text": "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\\mathcal{U}$ be a finite open covering of $X$ by schemes separated over $S$. If there exists a dualizing complex normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then it is unique up to unique isomorphism."}
+{"_id": "13572", "title": "duality-lemma-good-over-both", "text": "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}. Let $f : X \\to Y$ be a morphism of finite type schemes over $S$. Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes normalized relative to $\\omega_S^\\bullet$. Then $\\omega_X^\\bullet$ is a dualizing complex normalized relative to $\\omega_Y^\\bullet$."}
+{"_id": "13574", "title": "duality-lemma-proper-map-good-dualizing-complex", "text": "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}. Let $f : X \\to Y$ be a proper morphism of schemes of finite type over $S$. Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes normalized relative to $\\omega_S^\\bullet$. Let $a$ be the right adjoint of Lemma \\ref{lemma-twisted-inverse-image} for $f$. Then there is a canonical isomorphism $a(\\omega_Y^\\bullet) = \\omega_X^\\bullet$."}
+{"_id": "13575", "title": "duality-lemma-duality-bootstrap", "text": "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}. With $f^!_{new}$ and $\\omega_X^\\bullet$ defined for all (morphisms of) schemes of finite type over $S$ as above: \\begin{enumerate} \\item the functors $f^!_{new}$ and the arrows $(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$ turn $D_{\\textit{Coh}}^+$ into a pseudo functor from the category of schemes of finite type over $S$ into the $2$-category of categories, \\item $\\omega_X^\\bullet = (X \\to S)^!_{new} \\omega_S^\\bullet$, \\item the functor $D_X$ defines an involution of $D_{\\textit{Coh}}(\\mathcal{O}_X)$ switching $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$ and $D_{\\textit{Coh}}^-(\\mathcal{O}_X)$ and fixing $D_{\\textit{Coh}}^b(\\mathcal{O}_X)$, \\item $\\omega_X^\\bullet = f^!_{new}\\omega_Y^\\bullet$ for $f : X \\to Y$ a morphism of finite type schemes over $S$, \\item $f^!_{new}M = D_X(Lf^*D_Y(M))$ for $M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and \\item if in addition $f$ is proper, then $f^!_{new}$ is isomorphic to the restriction of the right adjoint of $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ to $D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ and there is a canonical isomorphism $$ Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, f^!_{new}M) \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, M) $$ for $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and $M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and $$ Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, \\omega_X^\\bullet) = R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, \\omega_Y^\\bullet) $$ for $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and \\end{enumerate} If $X$ is separated over $S$, then $\\omega_X^\\bullet$ is canonically isomorphic to $(X \\to S)^!\\omega_S^\\bullet$ and if $f$ is a morphism between schemes separated over $S$, then there is a canonical isomorphism\\footnote{We haven't checked that these are compatible with the isomorphisms $(g \\circ f)^! \\to f^! \\circ g^!$ and $(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$. We will do this here if we need this later.} $f_{new}^!K = f^!K$ for $K$ in $D_{\\textit{Coh}}^+$."}
+{"_id": "13576", "title": "duality-lemma-good-dualizing-normalized", "text": "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing complex. Let $X$ be a scheme of finite type over $S$ and let $\\omega_X^\\bullet$ be the dualizing complex normalized relative to $\\omega_S^\\bullet$. If $x \\in X$ is a closed point lying over a closed point $s$ of $S$, then $\\omega_{X, x}^\\bullet$ is a normalized dualizing complex over $\\mathcal{O}_{X, x}$ provided that $\\omega_{S, s}^\\bullet$ is a normalized dualizing complex over $\\mathcal{O}_{S, s}$."}
+{"_id": "13577", "title": "duality-lemma-good-dualizing-dimension-function", "text": "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing complex. Let $f : X \\to S$ be of finite type and let $\\omega_X^\\bullet$ be the dualizing complex normalized relative to $\\omega_S^\\bullet$. For all $x \\in X$ we have $$ \\delta_X(x) - \\delta_S(f(x)) = \\text{trdeg}_{\\kappa(f(x))}(\\kappa(x)) $$ where $\\delta_S$, resp.\\ $\\delta_X$ is the dimension function of $\\omega_S^\\bullet$, resp.\\ $\\omega_X^\\bullet$, see Lemma \\ref{lemma-dimension-function-scheme}."}
+{"_id": "13578", "title": "duality-lemma-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Then $$ H^i(f^!\\mathcal{O}_Y)_x \\not = 0 \\Rightarrow - \\dim_x(X_y) \\leq i. $$"}
+{"_id": "13579", "title": "duality-lemma-flat-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. If $f$ is flat, then $$ H^i(f^!\\mathcal{O}_Y)_x \\not = 0 \\Rightarrow - \\dim_x(X_y) \\leq i \\leq 0. $$ In fact, if all fibres of $f$ have dimension $\\leq d$, then $f^!\\mathcal{O}_Y$ has tor-amplitude in $[-d, 0]$ as an object of $D(X, f^{-1}\\mathcal{O}_Y)$."}
+{"_id": "13580", "title": "duality-lemma-shriek-over-CM", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Assume \\begin{enumerate} \\item $\\mathcal{O}_{Y, y}$ is Cohen-Macaulay, and \\item $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi)) \\leq r$ for any generic point $\\xi$ of an irreducible component of $X$ containing $x$. \\end{enumerate} Then $$ H^i(f^!\\mathcal{O}_Y)_x \\not = 0 \\Rightarrow - r \\leq i $$ and the stalk $H^{-r}(f^!\\mathcal{O}_Y)_x$ is $(S_2)$ as an $\\mathcal{O}_{X, x}$-module."}
+{"_id": "13581", "title": "duality-lemma-flat-quasi-finite-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. If $f$ is flat and quasi-finite, then $$ f^!\\mathcal{O}_Y = \\omega_{X/Y}[0] $$ for some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$."}
+{"_id": "13582", "title": "duality-lemma-CM-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. If $f$ is Cohen-Macaulay (More on Morphisms, Definition \\ref{more-morphisms-definition-CM}), then $$ f^!\\mathcal{O}_Y = \\omega_{X/Y}[d] $$ for some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$ where $d$ is the locally constant function on $X$ which gives the relative dimension of $X$ over $Y$."}
+{"_id": "13583", "title": "duality-lemma-dualizing-module", "text": "Let $X$ be a connected Noetherian scheme and let $\\omega_X$ be a dualizing module on $X$. The support of $\\omega_X$ is the union of the irreducible components of maximal dimension with respect to any dimension function and $\\omega_X$ is a coherent $\\mathcal{O}_X$-module having property $(S_2)$."}
+{"_id": "13584", "title": "duality-lemma-vanishing-good-dualizing", "text": "Let $X/A$ with $\\omega_X^\\bullet$ and $\\omega_X$ be as in Example \\ref{example-proper-over-local}. Then \\begin{enumerate} \\item $H^i(\\omega_X^\\bullet) \\not = 0 \\Rightarrow i \\in \\{-\\dim(X), \\ldots, 0\\}$, \\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$, \\item $\\text{Supp}(\\omega_X)$ is the union of the components of dimension $\\dim(X)$, and \\item $\\omega_X$ has property $(S_2)$. \\end{enumerate}"}
+{"_id": "13585", "title": "duality-lemma-dualizing-module-proper-over-A", "text": "Let $X/A$ with dualizing module $\\omega_X$ be as in Example \\ref{example-proper-over-local}. Let $d = \\dim(X_s)$ be the dimension of the closed fibre. If $\\dim(X) = d + \\dim(A)$, then the dualizing module $\\omega_X$ represents the functor $$ \\mathcal{F} \\longmapsto \\Hom_A(H^d(X, \\mathcal{F}), \\omega_A) $$ on the category of coherent $\\mathcal{O}_X$-modules."}
+{"_id": "13586", "title": "duality-lemma-dualizing-module-CM-scheme", "text": "Let $X$ be a locally Noetherian scheme with dualizing complex $\\omega_X^\\bullet$. \\begin{enumerate} \\item $X$ is Cohen-Macaulay $\\Leftrightarrow$ $\\omega_X^\\bullet$ locally has a unique nonzero cohomology sheaf, \\item $\\mathcal{O}_{X, x}$ is Cohen-Macaulay $\\Leftrightarrow$ $\\omega_{X, x}^\\bullet$ has a unique nonzero cohomology, \\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay}\\}$ is open and Cohen-Macaulay. \\end{enumerate} If $X$ is connected and Cohen-Macaulay, then there is an integer $n$ and a coherent Cohen-Macaulay $\\mathcal{O}_X$-module $\\omega_X$ such that $\\omega_X^\\bullet = \\omega_X[-n]$."}
+{"_id": "13588", "title": "duality-lemma-affine-flat-Noetherian-CM", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then the following are equivalent \\begin{enumerate} \\item $f$ is Cohen-Macaulay at $x$, \\item $f^!\\mathcal{O}_Y$ has a unique nonzero cohomology sheaf in a neighbourhood of $x$. \\end{enumerate}"}
+{"_id": "13589", "title": "duality-lemma-gorenstein-CM", "text": "A Gorenstein scheme is Cohen-Macaulay."}
+{"_id": "13590", "title": "duality-lemma-regular-gorenstein", "text": "A regular scheme is Gorenstein."}
+{"_id": "13591", "title": "duality-lemma-gorenstein", "text": "Let $X$ be a locally Noetherian scheme. \\begin{enumerate} \\item If $X$ has a dualizing complex $\\omega_X^\\bullet$, then \\begin{enumerate} \\item $X$ is Gorenstein $\\Leftrightarrow$ $\\omega_X^\\bullet$ is an invertible object of $D(\\mathcal{O}_X)$, \\item $\\mathcal{O}_{X, x}$ is Gorenstein $\\Leftrightarrow$ $\\omega_{X, x}^\\bullet$ is an invertible object of $D(\\mathcal{O}_{X, x})$, \\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Gorenstein}\\}$ is an open Gorenstein subscheme. \\end{enumerate} \\item If $X$ is Gorenstein, then $X$ has a dualizing complex if and only if $\\mathcal{O}_X[0]$ is a dualizing complex. \\end{enumerate}"}
+{"_id": "13592", "title": "duality-lemma-gorenstein-lci", "text": "If $f : Y \\to X$ is a local complete intersection morphism with $X$ a Gorenstein scheme, then $Y$ is Gorenstein."}
+{"_id": "13593", "title": "duality-lemma-gorenstein-local-syntomic", "text": "The property $\\mathcal{P}(S) =$``$S$ is Gorenstein'' is local in the syntomic topology."}
+{"_id": "13594", "title": "duality-lemma-gorenstein-base-change", "text": "Let $X$ be a locally Noetherian scheme over the field $k$. Let $k \\subset k'$ be a finitely generated field extension. Let $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$. Then we have $$ \\mathcal{O}_{X, x}\\text{ is Gorenstein} \\Leftrightarrow \\mathcal{O}_{X_{k'}, x'}\\text{ is Gorenstein} $$ If $X$ is locally of finite type over $k$, the same holds for any field extension $k \\subset k'$."}
+{"_id": "13595", "title": "duality-lemma-gorenstein-morphism", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is Gorenstein, and \\item $f$ is flat and its fibres are Gorenstein schemes. \\end{enumerate}"}
+{"_id": "13596", "title": "duality-lemma-gorenstein-CM-morphism", "text": "A Gorenstein morphism is Cohen-Macaulay."}
+{"_id": "13597", "title": "duality-lemma-lci-gorenstein", "text": "A syntomic morphism is Gorenstein. Equivalently a flat local complete intersection morphism is Gorenstein."}
+{"_id": "13598", "title": "duality-lemma-composition-gorenstein", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms. Assume that the fibres $X_y$, $Y_z$ and $X_z$ of $f$, $g$, and $g \\circ f$ are locally Noetherian. \\begin{enumerate} \\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$, then $g \\circ f$ is Gorenstein at $x$. \\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein. \\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$, then $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$. \\item If $f \\circ g$ is Gorenstein and $f$ is flat, then $f$ is Gorenstein and $g$ is Gorenstein at every point in the image of $f$. \\end{enumerate}"}
+{"_id": "13599", "title": "duality-lemma-flat-morphism-from-gorenstein-scheme", "text": "\\begin{slogan} Gorensteinnes of the total space of a flat fibration implies same for base and fibres \\end{slogan} Let $f : X \\to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Gorenstein, then $f$ is Gorenstein and $\\mathcal{O}_{Y, f(x)}$ is Gorenstein for all $x \\in X$."}
+{"_id": "13600", "title": "duality-lemma-base-change-gorenstein", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$ be the base change of $f$. Let $x' \\in X'$ be a point with image $x \\in X$. \\begin{enumerate} \\item If $f$ is Gorenstein at $x$, then $f' : X' \\to Y'$ is Gorenstein at $x'$. \\item If $f$ is flat and $x$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \\end{enumerate}"}
+{"_id": "13601", "title": "duality-lemma-flat-lft-base-change-gorenstein", "text": "Let $f : X \\to Y$ be a morphism of schemes which is flat and locally of finite type. Then formation of the set $\\{x \\in X \\mid f\\text{ is Gorenstein at }x\\}$ commutes with arbitrary base change."}
+{"_id": "13602", "title": "duality-lemma-affine-flat-Noetherian-gorenstein", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then the following are equivalent \\begin{enumerate} \\item $f$ is Gorenstein at $x$, \\item $f^!\\mathcal{O}_X$ is isomorphic to an invertible object in a neighbourhood of $x$. \\end{enumerate} In particular, the set of points where $f$ is Gorenstein is open in $X$."}
+{"_id": "13603", "title": "duality-lemma-flat-finite-presentation-characterize-gorenstein", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \\in X$ with image $s \\in S$. Set $d = \\dim_x(X_s)$. The following are equivalent \\begin{enumerate} \\item $f$ is Gorenstein at $x$, \\item there exists an open neighbourhood $U \\subset X$ of $x$ and a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$ which is Gorenstein at $x$, \\item there exists an open neighbourhood $U \\subset X$ of $x$ and a locally quasi-finite Gorenstein morphism $U \\to \\mathbf{A}^d_S$ over $S$, \\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$ of an open neighbourhood $U \\subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is Gorenstein at $x$. \\end{enumerate} In particular, the set of points where $f$ is Gorenstein is open in $X$."}
+{"_id": "13605", "title": "duality-lemma-descent-ascent", "text": "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Assume \\begin{enumerate} \\item $f$ is syntomic and surjective, or \\item $f$ is a surjective flat local complete intersection morphism, or \\item $f$ is a surjective Gorenstein morphism of finite type. \\end{enumerate} Then $K \\in D_\\QCoh(\\mathcal{O}_Y)$ is a dualizing complex on $Y$ if and only if $Lf^*K$ is a dualizing complex on $X$."}
+{"_id": "13606", "title": "duality-lemma-duality-proper-over-field", "text": "Let $X$ be a proper scheme over a field $k$. There exists a dualizing complex $\\omega_X^\\bullet$ with the following properties \\begin{enumerate} \\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i \\in [-\\dim(X), 0]$, \\item $\\omega_X = H^{-\\dim(X)}(\\omega_X^\\bullet)$ is a coherent $(S_2)$-module whose support is the irreducible components of dimension $d$, \\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$, \\item for $x \\in X$ closed the module $H^i(\\omega_{X, x}^\\bullet) \\oplus \\ldots \\oplus H^0(\\omega_{X, x}^\\bullet)$ is nonzero if and only if $\\text{depth}(\\mathcal{O}_{X, x}) \\leq -i$, \\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial isomorphisms\\footnote{This property characterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$ up to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.} $$ \\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k) $$ compatible with shifts and distinguished triangles, \\item there are functorial isomorphisms $\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{\\dim(X)}(X, \\mathcal{F}), k)$ for $\\mathcal{F}$ quasi-coherent on $X$, and \\item if $X \\to \\Spec(k)$ is smooth of relative dimension $d$, then $\\omega_X^\\bullet \\cong \\wedge^d\\Omega_{X/k}[d]$ and $\\omega_X \\cong \\wedge^d\\Omega_{X/k}$. \\end{enumerate}"}
+{"_id": "13607", "title": "duality-lemma-duality-proper-over-field-perfect", "text": "Let $k$, $X$, and $\\omega_X^\\bullet$ be as in Lemma \\ref{lemma-duality-proper-over-field}. Let $t : H^0(X, \\omega_X^\\bullet) \\to k$ be as in Remark \\ref{remark-duality-proper-over-field}. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then the pairings $$ H^i(X, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee) \\times H^{-i}(X, E) \\longrightarrow k, \\quad (\\xi, \\eta) \\longmapsto t((1_{\\omega_X^\\bullet} \\otimes \\epsilon)(\\xi \\cup \\eta)) $$ are perfect for all $i$. Here $\\cup$ denotes the cupproduct of Cohomology, Section \\ref{cohomology-section-cup-product} and $\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$ is as in Cohomology, Example \\ref{cohomology-example-dual-derived}."}
+{"_id": "13608", "title": "duality-lemma-duality-proper-over-field-CM", "text": "Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $d$. The module $\\omega_X$ of Lemma \\ref{lemma-duality-proper-over-field} has the following properties \\begin{enumerate} \\item $\\omega_X$ is a dualizing module on $X$ (Section \\ref{section-dualizing-module}), \\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$, \\item there are functorial isomorphisms $\\Ext^i_X(K, \\omega_X[d]) = \\Hom_k(H^{-i}(X, K), k)$ compatible with shifts and distinguished triangles for $K \\in D_\\QCoh(X)$, \\item there are functorial isomorphisms $\\Ext^{d - i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^i(X, \\mathcal{F}), k)$ for $\\mathcal{F}$ quasi-coherent on $X$. \\end{enumerate}"}
+{"_id": "13610", "title": "duality-lemma-relative-dualizing-complex-algebra", "text": "Let $X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \\xi)$ be a relative dualizing complex. Then for any commutative diagram $$ \\xymatrix{ \\Spec(A) \\ar[d] \\ar[r] & X \\ar[d] \\\\ \\Spec(R) \\ar[r] & S } $$ whose horizontal arrows are open immersions, the restriction of $K$ to $\\Spec(A)$ corresponds via Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded} to a relative dualizing complex for $R \\to A$ in the sense of Dualizing Complexes, Definition \\ref{dualizing-definition-relative-dualizing-complex}."}
+{"_id": "13611", "title": "duality-lemma-relative-dualizing-RHom", "text": "Let $X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $(K, \\xi)$ be a relative dualizing complex. Then $\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(K, K)$ is an isomorphism."}
+{"_id": "13612", "title": "duality-lemma-uniqueness-relative-dualizing", "text": "Let $X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. If $(K, \\xi)$ and $(L, \\eta)$ are two relative dualizing complexes on $X/S$, then there is a unique isomorphism $K \\to L$ sending $\\xi$ to $\\eta$."}
+{"_id": "13613", "title": "duality-lemma-existence-relative-dualizing", "text": "Let $X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. There exists a relative dualizing complex $(K, \\xi)$."}
+{"_id": "13614", "title": "duality-lemma-base-change-relative-dualizing", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ of schemes. Assume $X \\to S$ is flat and locally of finite presentation. Let $(K, \\xi)$ be a relative dualizing complex for $f$. Set $K' = L(g')^*K$. Let $\\xi'$ be the derived base change of $\\xi$ (see proof). Then $(K', \\xi')$ is a relative dualizing complex for $f'$."}
+{"_id": "13615", "title": "duality-lemma-flat-proper-relative-dualizing", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a proper, flat morphism of finite presentation. The relative dualizing complex $\\omega_{X/S}^\\bullet$ of Remark \\ref{remark-relative-dualizing-complex} together with (\\ref{equation-pre-rigid}) is a relative dualizing complex in the sense of Definition \\ref{definition-relative-dualizing-complex}."}
+{"_id": "13618", "title": "duality-lemma-determinant", "text": "Let $X$ be a locally ringed space. Let $$ \\mathcal{E}_1 \\xrightarrow{\\alpha} \\mathcal{E}_0 \\to \\mathcal{F} \\to 0 $$ be a short exact sequence of $\\mathcal{O}_X$-modules. Assume $\\mathcal{E}_1$ and $\\mathcal{E}_0$ are locally free of ranks $r_1, r_0$. Then there is a canonical map $$ \\wedge^{r_0 - r_1}\\mathcal{F} \\longrightarrow \\wedge^{r_1}(\\mathcal{E}_1^\\vee) \\otimes \\wedge^{r_0}\\mathcal{E}_0 $$ which is an isomorphism on the stalk at $x \\in X$ if and only if $\\mathcal{F}$ is locally free of rank $r_0 - r_1$ in an open neighbourhood of $x$."}
+{"_id": "13619", "title": "duality-lemma-fundamental-class-lci", "text": "Let $Y$ be a Noetherian scheme. Let $f : X \\to Y$ be a local complete intersection morphism which factors as an immersion $X \\to P$ followed by a proper smooth morphism $P \\to Y$. Let $r$ be the locally constant function on $X$ such that $\\omega_{Y/X} = H^{-r}(f^!\\mathcal{O}_Y)$ is the unique nonzero cohomology sheaf of $f^!\\mathcal{O}_Y$, see Lemma \\ref{lemma-lci-shriek}. Then there is a map $$ \\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{Y/X} $$ which is an isomorphism on the stalk at a point $x$ if and only if $f$ is smooth at $x$."}
+{"_id": "13620", "title": "duality-lemma-fundamental-class-almost-lci", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $r \\geq 0$. Assume \\begin{enumerate} \\item $Y$ is Cohen-Macaulay (Properties, Definition \\ref{properties-definition-Cohen-Macaulay}), \\item $f$ factors as $X \\to P \\to Y$ where the first morphism is an immersion and the second is smooth and proper, \\item if $x \\in X$ and $\\dim(\\mathcal{O}_{X, x}) \\leq 1$, then $f$ is Koszul at $x$ (More on Morphisms, Definition \\ref{more-morphisms-definition-lci}), and \\item if $\\xi$ is a generic point of an irreducible component of $X$, then we have $\\text{trdeg}_{\\kappa(f(\\xi))} \\kappa(\\xi) = r$. \\end{enumerate} Then with $\\omega_{Y/X} = H^{-r}(f^!\\mathcal{O}_Y)$ there is a map $$ \\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{Y/X} $$ which is an isomorphism on the locus where $f$ is smooth."}
+{"_id": "13621", "title": "duality-lemma-lift-map", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $(K_n)$ be a Deligne system and denote $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ the value of the constant system $(K_n|_U)$. Let $L$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Then $\\colim \\Hom_X(K_n, L) = \\Hom_U(K, L|_U)$."}
+{"_id": "13622", "title": "duality-lemma-lift-map-plus", "text": "The result of Lemma \\ref{lemma-lift-map} holds even for $L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "13623", "title": "duality-lemma-extension-by-zero", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. \\begin{enumerate} \\item Let $(K_n)$ and $(L_n)$ be Deligne systems. Let $K$ and $L$ be the values of the constant systems $(K_n|_U)$ and $(L_n|_U)$. Given a morphism $\\alpha : K \\to L$ of $D(\\mathcal{O}_U)$ there is a unique morphism of pro-systems $(K_n) \\to (L_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ whose restriction to $U$ is $\\alpha$. \\item Given $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ there exists a Deligne system $(K_n)$ such that $(K_n|_U)$ is constant with value $K$. \\item The pro-object $(K_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ of (2) is unique up to unique isomorphism (as a pro-object). \\end{enumerate}"}
+{"_id": "13624", "title": "duality-lemma-extension-by-zero-triangle", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $$ K \\to L \\to M \\to K[1] $$ be a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$. Then there exists an inverse system of distinguished triangles $$ K_n \\to L_n \\to M_n \\to K_n[1] $$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that $(K_n)$, $(L_n)$, $(M_n)$ are Deligne systems and such that the restriction of these distinguished triangles to $U$ is isomorphic to the distinguished triangle we started out with."}
+{"_id": "13625", "title": "duality-lemma-deligne-system-2-out-of-3", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $$ K_n \\to L_n \\to M_n \\to K_n[1] $$ be an inverse system of distinguished triangles in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. If $(K_n)$ and $(M_n)$ are pro-isomorphic to Deligne systems, then so is $(L_n)$."}
+{"_id": "13626", "title": "duality-lemma-consequence-Artin-Rees-bis", "text": "Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $\\mathcal{F}^\\bullet$ be a complex of coherent $\\mathcal{O}_X$-modules. Let $p \\in \\mathbf{Z}$. Set $\\mathcal{H} = H^p(\\mathcal{F}^\\bullet)$ and $\\mathcal{H}_n = H^p(\\mathcal{I}^n\\mathcal{F}^\\bullet)$. Then there are canonical $\\mathcal{O}_X$-module maps $$ \\ldots \\to \\mathcal{H}_3 \\to \\mathcal{H}_2 \\to \\mathcal{H}_1 \\to \\mathcal{H} $$ There exists a $c > 0$ such that for $n \\geq c$ the image of $\\mathcal{H}_n \\to \\mathcal{H}$ is contained in $\\mathcal{I}^{n - c}\\mathcal{H}$ and there is a canonical $\\mathcal{O}_X$-module map $\\mathcal{I}^n\\mathcal{H} \\to \\mathcal{H}_{n - c}$ such that the compositions $$ \\mathcal{I}^n \\mathcal{H} \\to \\mathcal{H}_{n - c} \\to \\mathcal{I}^{n - 2c}\\mathcal{H} \\quad\\text{and}\\quad \\mathcal{H}_n \\to \\mathcal{I}^{n - c}\\mathcal{H} \\to \\mathcal{H}_{n - 2c} $$ are the canonical ones. In particular, the inverse systems $(\\mathcal{H}_n)$ and $(\\mathcal{I}^n\\mathcal{H})$ are isomorphic as pro-objects of $\\textit{Mod}(\\mathcal{O}_X)$."}
+{"_id": "13627", "title": "duality-lemma-characterize-extension-by-zero-algebra", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $a \\leq b$ be integers. Let $(K_n)$ be an inverse system of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that $H^i(K_n) = 0$ for $i \\not \\in [a, b]$. The following are equivalent \\begin{enumerate} \\item $(K_n)$ is pro-isomorphic to a Deligne system, \\item for every $p \\in \\mathbf{Z}$ there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ such that the pro-systems $(H^p(K_n))$ and $(\\mathcal{I}^n\\mathcal{F})$ are pro-isomorphic. \\end{enumerate}"}
+{"_id": "13628", "title": "duality-lemma-characterize-extension-by-zero", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $(K_n)$ be an inverse system in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Let $X = W_1 \\cup \\ldots \\cup W_r$ be an open covering. The following are equivalent \\begin{enumerate} \\item $(K_n)$ is pro-isomorphic to a Deligne system, \\item for each $i$ the restriction $(K_n|_{W_i})$ is pro-isomorphic to a Deligne system with respect to the open immersion $U \\cap W_i \\to W_i$. \\end{enumerate}"}
+{"_id": "13629", "title": "duality-lemma-tensoring-Deligne-system", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals with $V(\\mathcal{I}) = X \\setminus U$. Let $K$ be in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Then $$ K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I}^n $$ is pro-isomorphic to a Deligne system with constant value $K|_U$ over $U$."}
+{"_id": "13630", "title": "duality-lemma-well-defined-pre", "text": "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes. Let $V \\subset Y$ be an open subscheme and set $U = f^{-1}(V)$. Picture $$ \\xymatrix{ U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\ V \\ar[r]^{j'} & Y } $$ Then we have a canonical isomorphism $Rj'_! \\circ Rg_* \\to Rf_* \\circ Rj_!$ of functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to \\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ where $Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}."}
+{"_id": "13631", "title": "duality-lemma-well-defined", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let $j' : U \\to X'$ be a compactification of $U$ over $X$ (see proof) and denote $f : X' \\to X$ the structure morphism. Then we have a canonical isomorphism $Rj_! \\to Rf_* \\circ R(j')_!$ of functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to \\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ where $Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}."}
+{"_id": "13634", "title": "duality-lemma-composition-lower-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of $\\textit{FTS}_S$. With notation as in Remark \\ref{remark-composition-lower-shriek} we have $Rg_! \\circ Rf_! = R(g \\circ f)_!$."}
+{"_id": "13635", "title": "duality-lemma-duality-compact-support", "text": "Let $p : U \\to \\Spec(k)$ be separated of finite type where $k$ is a field. Let $\\omega_{U/k}^\\bullet = p^!\\mathcal{O}_{\\Spec(k)}$. There are canonical isomorphisms $$ \\Hom_k(H^i(U, K), k) = H^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet)) $$ of topological $k$-vector spaces functorial for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$."}
+{"_id": "13637", "title": "duality-lemma-h0-compactly-supported", "text": "Let $X$ be a proper scheme over a field $k$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ with $H^i(K) = 0$ for $i < 0$. Set $\\mathcal{F} = H^0(K)$. Let $Z \\subset X$ be closed with complement $U = X \\setminus U$. Then $$ H^0_c(U, K|_U) \\subset H^0(X, \\mathcal{F}) $$ is given by those global sections of $\\mathcal{F}$ which vanish in an open neighbourhood of $Z$."}
+{"_id": "13638", "title": "duality-lemma-lichtenbaum", "text": "Let $U$ be a variety. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. If $H^d(U, \\mathcal{F})$ is nonzero, then $\\dim(U) \\geq d$ and if equality holds, then $U$ is proper."}
+{"_id": "13639", "title": "duality-proposition-duality-compactly-supported", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Then the functors $Rf_!$ and $f^!$ are adjoint in the following sense: for all $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$ we have $$ \\Hom_X(K, f^!L) = \\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L) $$ bifunctorially in $K$ and $L$."}
+{"_id": "13670", "title": "more-morphisms-theorem-openness-flatness", "text": "\\begin{reference} \\cite[IV Theorem 11.3.1]{EGA} \\end{reference} Let $S$ be a scheme. Let $f : X \\to S$ be a morphism which is locally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is locally of finite presentation. Then $$ U = \\{x \\in X \\mid \\mathcal{F}\\text{ is flat over }S\\text{ at }x\\} $$ is open in $X$."}
+{"_id": "13671", "title": "more-morphisms-theorem-criterion-flatness-fibre-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$. Set $y = f(x)$ and $s \\in S$ the image of $x$ in $S$. Assume $S$, $X$, $Y$ locally Noetherian, $\\mathcal{F}$ coherent, and $\\mathcal{F}_x \\not = 0$. Then the following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $S$ at $x$, and $\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and \\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is flat over $Y$ at $x$. \\end{enumerate}"}
+{"_id": "13672", "title": "more-morphisms-theorem-criterion-flatness-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $S$, \\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and \\item $Y$ is locally of finite type over $S$. \\end{enumerate} Let $x \\in X$. Set $y = f(x)$ and let $s \\in S$ be the image of $x$ in $S$. If $\\mathcal{F}_x \\not = 0$, then the following are equivalent: \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $S$ at $x$, and $\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and \\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is flat over $Y$ at $x$. \\end{enumerate} Moreover, the set of points $x$ where (1) and (2) hold is open in $\\text{Supp}(\\mathcal{F})$."}
+{"_id": "13673", "title": "more-morphisms-theorem-of-the-cube", "text": "Let $S$ be a scheme. Let $X$, $Y$, and $Z$ be schemes over $S$. Let $x : S \\to X$ and $y : S \\to Y$ be sections of the structure morphisms. Let $\\mathcal{L}$ be an invertible module on $X \\times_S Y \\times_S Z$. If \\begin{enumerate} \\item $X \\to S$ and $Y \\to S$ are flat, proper morphisms of finite presentation with geometrically integral fibres, \\item the pullbacks of $\\mathcal{L}$ by $x \\times \\text{id}_Y \\times \\text{id}_Z$ and $\\text{id}_X \\times y \\times \\text{id}_Z$ are trivial over $Y \\times_S Z$ and $X \\times_S Z$, \\item there is a point $z \\in Z$ such that $\\mathcal{L}$ restricted to $X \\times_S Y \\times_S z$ is trivial, and \\item $Z$ is connected, \\end{enumerate} then $\\mathcal{L}$ is trivial."}
+{"_id": "13674", "title": "more-morphisms-theorem-stein-factorization-Noetherian", "text": "Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a proper morphism. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\ & S & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is proper with geometrically connected fibres, \\item the morphism $\\pi : S' \\to S$ is finite, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$, \\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and \\item $S'$ is the normalization of $S$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. \\end{enumerate}"}
+{"_id": "13675", "title": "more-morphisms-theorem-stein-factorization-general", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a proper morphism. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\ & S & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is proper with geometrically connected fibres, \\item the morphism $\\pi : S' \\to S$ is integral, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$, \\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and \\item $S'$ is the normalization of $S$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. \\end{enumerate}"}
+{"_id": "13676", "title": "more-morphisms-theorem-normalized-base-change-with-reduced-fibre", "text": "Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring. There exists a finite extension $K \\subset L$ such that the normalized base change $Y$ is smooth over $\\Spec(B)$ at all generic points of all fibres."}
+{"_id": "13677", "title": "more-morphisms-lemma-first-order-thickening", "text": "Let $X$ be a scheme over a base $S$. Consider a short exact sequence $$ 0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0 $$ of sheaves on $X$ where $\\mathcal{A}$ is a sheaf of $f^{-1}\\mathcal{O}_S$-algebras, $\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection of sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and $\\mathcal{I}$ is its kernel. If \\begin{enumerate} \\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and \\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module \\end{enumerate} then $X' = (X, \\mathcal{A})$ is a scheme and $X \\to X'$ is a first order thickening over $S$. Moreover, any first order thickening over $S$ is of this form."}
+{"_id": "13678", "title": "more-morphisms-lemma-thickening-affine-scheme", "text": "\\begin{slogan} Affineness is insensitive to thickenings \\end{slogan} Any thickening of an affine scheme is affine."}
+{"_id": "13679", "title": "more-morphisms-lemma-base-change-thickening", "text": "Let $S \\subset S'$ be a thickening of schemes. Let $X' \\to S'$ be a morphism and set $X = S \\times_{S'} X'$. Then $(X \\subset X') \\to (S \\subset S')$ is a morphism of thickenings. If $S \\subset S'$ is a first (resp.\\ finite order) thickening, then $X \\subset X'$ is a first (resp.\\ finite order) thickening."}
+{"_id": "13680", "title": "more-morphisms-lemma-composition-thickening", "text": "\\begin{slogan} Compositions of thickenings are thickenings \\end{slogan} If $S \\subset S'$ and $S' \\subset S''$ are thickenings, then so is $S \\subset S''$."}
+{"_id": "13682", "title": "more-morphisms-lemma-thicken-property-morphisms", "text": "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism of thickenings. Then \\begin{enumerate} \\item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \\item $f$ is quasi-compact if and only if $f'$ quasi-compact, \\item $f$ is universally closed if and only if $f'$ is universally closed, \\item $f$ is integral if and only if $f'$ is integral, \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\item $f$ is universally injective if and only if $f'$ is universally injective, \\item $f$ is universally open if and only if $f'$ is universally open, \\item $f$ is quasi-affine if and only if $f'$ is quasi-affine, and \\item add more here. \\end{enumerate}"}
+{"_id": "13683", "title": "more-morphisms-lemma-thicken-property-relatively-ample", "text": "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism of thickenings. Let $\\mathcal{L}'$ be an invertible sheaf on $X'$ and denote $\\mathcal{L}$ the restriction to $X$. Then $\\mathcal{L}'$ is $f'$-ample if and only if $\\mathcal{L}$ is $f$-ample."}
+{"_id": "13684", "title": "more-morphisms-lemma-thicken-property-morphisms-cartesian", "text": "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism of thickenings such that $X = S \\times_{S'} X'$. If $S \\subset S'$ is a finite order thickening, then \\begin{enumerate} \\item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \\item $f$ is locally of finite type if and only if $f'$ is locally of finite type, \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\item $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $\\Omega_{X/S} = 0$ if and only if $\\Omega_{X'/S'} = 0$, \\item $f$ is unramified if and only if $f'$ is unramified, \\item $f$ is proper if and only if $f'$ is proper, \\item $f$ is finite if and only if $f'$ is finite, \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\item $f$ is an immersion if and only if $f'$ is an immersion, and \\item add more here. \\end{enumerate}"}
+{"_id": "13685", "title": "more-morphisms-lemma-properties-that-extend-over-thickenings", "text": "Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a morphism of thickenings. Assume $f$ and $f'$ are locally of finite type and $X = Y \\times_{Y'} X'$. Then \\begin{enumerate} \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\item $f$ is finite if and only if $f'$ is finite, \\item $f$ is a closed immersion if and only if $f'$ is a closed immersion, \\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$, \\item $f$ is unramified if and only if $f'$ is unramified, \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\item $f$ is an immersion if and only if $f'$ is an immersion, \\item $f$ is proper if and only if $f'$ is proper, and \\item add more here. \\end{enumerate}"}
+{"_id": "13686", "title": "more-morphisms-lemma-picard-group-first-order-thickening", "text": "Let $X \\subset X'$ be a first order thickening with ideal sheaf $\\mathcal{I}$. Then there is a canonical exact sequence $$ \\xymatrix{ 0 \\ar[r] & H^0(X, \\mathcal{I}) \\ar[r] & H^0(X', \\mathcal{O}_{X'}^*) \\ar[r] & H^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\ & H^1(X, \\mathcal{I}) \\ar[r] & \\Pic(X') \\ar[r] & \\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\ & H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots } $$ of abelian groups."}
+{"_id": "13687", "title": "more-morphisms-lemma-torsion-pic-thickening", "text": "Let $X \\subset X'$ be a thickening. Let $n$ be an integer invertible in $\\mathcal{O}_X$. Then the map $\\Pic(X')[n] \\to \\Pic(X)[n]$ is bijective."}
+{"_id": "13688", "title": "more-morphisms-lemma-first-order-infinitesimal-neighbourhood", "text": "Let $i : Z \\to X$ be an immersion of schemes. The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$ has the following universal property: Given any commutative diagram $$ \\xymatrix{ Z \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\ X & T' \\ar[l]_b } $$ where $T \\subset T'$ is a first order thickening over $X$, there exists a unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of thickenings over $X$."}
+{"_id": "13689", "title": "more-morphisms-lemma-infinitesimal-neighbourhood-conormal", "text": "Let $i : Z \\to X$ be an immersion of schemes. Let $Z \\subset Z'$ be the first order infinitesimal neighbourhood of $Z$ in $X$. Then the diagram $$ \\xymatrix{ Z \\ar[r] \\ar[d] & Z' \\ar[d] \\\\ Z \\ar[r] & X } $$ induces a map of conormal sheaves $\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}. This map is an isomorphism."}
+{"_id": "13690", "title": "more-morphisms-lemma-formally-unramified-not-affine", "text": "If $f : X \\to S$ is a formally unramified morphism, then given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of schemes over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition \\ref{definition-formally-unramified} the condition that $T$ be affine may be dropped."}
+{"_id": "13691", "title": "more-morphisms-lemma-composition-formally-unramified", "text": "A composition of formally unramified morphisms is formally unramified."}
+{"_id": "13692", "title": "more-morphisms-lemma-base-change-formally-unramified", "text": "A base change of a formally unramified morphism is formally unramified."}
+{"_id": "13693", "title": "more-morphisms-lemma-formally-unramified-on-opens", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$ and $V \\subset S$ be open such that $f(U) \\subset V$. If $f$ is formally unramified, so is $f|_U : U \\to V$."}
+{"_id": "13695", "title": "more-morphisms-lemma-formally-unramified-differentials", "text": "Let $f : X \\to S$ be a morphism of schemes. Then $f$ is formally unramified if and only if $\\Omega_{X/S} = 0$."}
+{"_id": "13696", "title": "more-morphisms-lemma-unramified-formally-unramified", "text": "\\begin{slogan} Unramified morphisms are the same as formally unramified morphism that are locally of finite type. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is unramified (resp.\\ G-unramified), and \\item the morphism $f$ is locally of finite type (resp.\\ locally of finite presentation) and formally unramified. \\end{enumerate}"}
+{"_id": "13697", "title": "more-morphisms-lemma-universal-thickening", "text": "Let $h : Z \\to X$ be a formally unramified morphism of schemes. There exists a universal first order thickening $Z \\subset Z'$ of $Z$ over $X$."}
+{"_id": "13699", "title": "more-morphisms-lemma-universal-thickening-unramified", "text": "Let $Z \\to X$ be a formally unramified morphism of schemes. Then the universal first order thickening $Z'$ is formally unramified over $X$."}
+{"_id": "13700", "title": "more-morphisms-lemma-universal-thickening-functorial", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\ W \\ar[r]^{h'} & Y } $$ with $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal first order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal first order thickening of $W$ over $Y$. There exists a canonical morphism $(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into the following commutative diagram $$ \\xymatrix{ & & & Z' \\ar[ld] \\ar[d]^{f'} \\\\ Z \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\ W \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y } $$ In particular the morphism $(f, f')$ of thickenings induces a morphism of conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$."}
+{"_id": "13701", "title": "more-morphisms-lemma-universal-thickening-fibre-product", "text": "Let $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\ W \\ar[r]^{h'} & Y } $$ be a fibre product diagram in the category of schemes with $h'$ formally unramified. Then $h$ is formally unramified and if $W \\subset W'$ is the universal first order thickening of $W$ over $Y$, then $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal first order thickening of $Z$ over $X$. In particular the canonical map $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of Lemma \\ref{lemma-universal-thickening-functorial} is surjective."}
+{"_id": "13702", "title": "more-morphisms-lemma-universal-thickening-fibre-product-flat", "text": "Let $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\ W \\ar[r]^{h'} & Y } $$ be a fibre product diagram in the category of schemes with $h'$ formally unramified and $g$ flat. In this case the corresponding map $Z' \\to W'$ of universal first order thickenings is flat, and $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ is an isomorphism."}
+{"_id": "13703", "title": "more-morphisms-lemma-universal-thickening-localize", "text": "Taking the universal first order thickenings commutes with taking opens. More precisely, let $h : Z \\to X$ be a formally unramified morphism of schemes. Let $V \\subset Z$, $U \\subset X$ be opens such that $h(V) \\subset U$. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $h|_V : V \\to U$ is formally unramified and the universal first order thickening of $V$ over $U$ is the open subscheme $V' \\subset Z'$ such that $V = Z \\cap V'$. In particular, $\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$."}
+{"_id": "13704", "title": "more-morphisms-lemma-differentials-universally-unramified", "text": "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$. Let $Z \\subset Z'$ be the universal first order thickening of $Z$ over $X$ with structure morphism $h' : Z' \\to X$. The canonical map $$ c_{h'} : (h')^*\\Omega_{X/S} \\longrightarrow \\Omega_{Z'/S} $$ induces an isomorphism $h^*\\Omega_{X/S} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z$."}
+{"_id": "13705", "title": "more-morphisms-lemma-universally-unramified-differentials-sequence", "text": "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$. There is a canonical exact sequence $$ \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0. $$ The first arrow is induced by $\\text{d}_{Z'/S}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$."}
+{"_id": "13706", "title": "more-morphisms-lemma-two-unramified-morphisms", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are formally unramified. Then there is a canonical exact sequence $$ \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0 $$ where the first arrow comes from Lemma \\ref{lemma-universal-thickening-functorial} and the second from Lemma \\ref{lemma-universally-unramified-differentials-sequence}."}
+{"_id": "13707", "title": "more-morphisms-lemma-transitivity-conormal", "text": "Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes. \\begin{enumerate} \\item If $Z \\subset Z'$ is the universal first order thickening of $Z$ over $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$ over $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is the universal first order thickening of $Z$ over $Y$. \\item There is a canonical exact sequence $$ i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ where the maps come from Lemma \\ref{lemma-universal-thickening-functorial} and $i : Z \\to Y$ is the first morphism. \\end{enumerate}"}
+{"_id": "13709", "title": "more-morphisms-lemma-composition-formally-etale", "text": "A composition of formally \\'etale morphisms is formally \\'etale."}
+{"_id": "13710", "title": "more-morphisms-lemma-base-change-formally-etale", "text": "A base change of a formally \\'etale morphism is formally \\'etale."}
+{"_id": "13711", "title": "more-morphisms-lemma-formally-etale-on-opens", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$ and $V \\subset S$ be open subschemes such that $f(U) \\subset V$. If $f$ is formally \\'etale, so is $f|_U : U \\to V$."}
+{"_id": "13712", "title": "more-morphisms-lemma-characterize-formally-etale", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is formally \\'etale, \\item $f$ is formally unramified and the universal first order thickening of $X$ over $S$ is equal to $X$, \\item $f$ is formally unramified and $\\mathcal{C}_{X/S} = 0$, and \\item $\\Omega_{X/S} = 0$ and $\\mathcal{C}_{X/S} = 0$. \\end{enumerate}"}
+{"_id": "13713", "title": "more-morphisms-lemma-unramified-flat-formally-etale", "text": "An unramified flat morphism is formally \\'etale."}
+{"_id": "13714", "title": "more-morphisms-lemma-affine-formally-etale", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally \\'etale if and only if $\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally \\'etale ring map."}
+{"_id": "13715", "title": "more-morphisms-lemma-etale-formally-etale", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is \\'etale, and \\item the morphism $f$ is locally of finite presentation and formally \\'etale. \\end{enumerate}"}
+{"_id": "13716", "title": "more-morphisms-lemma-difference-derivation", "text": "Let $S$ be a scheme. Let $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings over $S$. Let $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$ be two morphisms of thickenings over $S$. Assume that \\begin{enumerate} \\item $a = b$, and \\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ (Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}) are equal. \\end{enumerate} Then the map $(a')^\\sharp - (b')^\\sharp$ factors as $$ \\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D} a_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'} $$ where $D$ is an $\\mathcal{O}_S$-derivation."}
+{"_id": "13717", "title": "more-morphisms-lemma-action-by-derivations", "text": "Let $S$ be a scheme. Let $(a, a') : (X \\subset X') \\to (Y \\subset Y')$ be a morphism of first order thickenings over $S$. Let $$ \\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'} $$ be an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs $(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that (1) and (2) of Lemma \\ref{lemma-difference-derivation} hold and the derivation $D$ and $\\theta$ are related by Equation (\\ref{equation-D})."}
+{"_id": "13718", "title": "more-morphisms-lemma-sheaf", "text": "Let $S$ be a scheme. Let $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings over $S$. Assume given a morphism $a : X \\to Y$ and a map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of $\\mathcal{O}_X$-modules. For an open subscheme $U' \\subset X'$ consider morphisms $a' : U' \\to Y'$ such that \\begin{enumerate} \\item $a'$ is a morphism over $S$, \\item $a'|_U = a|_U$, and \\item the induced map $a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$ is the restriction of $A$ to $U$. \\end{enumerate} Here $U = X \\cap U'$. Then the rule \\begin{equation} \\label{equation-sheaf} U' \\mapsto \\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\} \\end{equation} defines a sheaf of sets on $X'$."}
+{"_id": "13719", "title": "more-morphisms-lemma-action-sheaf", "text": "Same notation and assumptions as in Lemma \\ref{lemma-sheaf}. There is an action of the sheaf $$ \\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/S}, \\mathcal{C}_{X/X'}) $$ on the sheaf (\\ref{equation-sheaf}). Moreover, the action is simply transitive for any open $U' \\subset X'$ over which the sheaf (\\ref{equation-sheaf}) has a section."}
+{"_id": "13720", "title": "more-morphisms-lemma-omega-deformation", "text": "Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening over $S$. Let $Y$ be a scheme over $S$. Let $a', b' : X' \\to Y$ be two morphisms over $S$ with $a = a'|_X = b'|_X$. This gives rise to a commutative diagram $$ \\xymatrix{ X \\ar[r] \\ar[d]_a & X' \\ar[d]^{(b', a')} \\\\ Y \\ar[r]^-{\\Delta_{Y/S}} & Y \\times_S Y } $$ Since the horizontal arrows are immersions with conormal sheaves $\\mathcal{C}_{X/X'}$ and $\\Omega_{Y/S}$, by Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}, we obtain a map $\\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'}$. Then this $\\theta$ and the derivation $D$ of Lemma \\ref{lemma-difference-derivation} are related by Equation (\\ref{equation-D})."}
+{"_id": "13721", "title": "more-morphisms-lemma-sheaf-differentials-etale-localization", "text": "Let $$ \\xymatrix{ X_1 \\ar[d] & X_2 \\ar[l]^f \\ar[d] \\\\ S_1 & S_2 \\ar[l] } $$ be a commutative diagram of schemes with $X_2 \\to X_1$ and $S_2 \\to S_1$ \\'etale. Then the map $c_f : f^*\\Omega_{X_1/S_1} \\to \\Omega_{X_2/S_2}$ of Morphisms, Lemma \\ref{morphisms-lemma-functoriality-differentials} is an isomorphism."}
+{"_id": "13722", "title": "more-morphisms-lemma-action-by-derivations-etale-localization", "text": "Consider a commutative diagram of first order thickenings $$ \\vcenter{ \\xymatrix{ (T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & & (X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\ (T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & & (X_1 \\subset X_1') } } \\quad \\begin{matrix} \\text{and a commutative} \\\\ \\text{diagram of schemes} \\end{matrix} \\quad \\vcenter{ \\xymatrix{ X_2' \\ar[r] \\ar[d] & S_2 \\ar[d] \\\\ X_1' \\ar[r] & S_1 } } $$ with $X_2 \\to X_1$ and $S_2 \\to S_1$ \\'etale. For any $\\mathcal{O}_{T_1}$-linear map $\\theta_1 : a_1^*\\Omega_{X_1/S_1} \\to \\mathcal{C}_{T_1/T'_1}$ let $\\theta_2$ be the composition $$ \\xymatrix{ a_2^*\\Omega_{X_2/S_2} \\ar@{=}[r] & h^*a_1^*\\Omega_{X_1/S_1} \\ar[r]^-{h^*\\theta_1} & h^*\\mathcal{C}_{T_1/T'_1} \\ar[r] & \\mathcal{C}_{T_2/T'_2} } $$ (equality sign is explained in the proof). Then the diagram $$ \\xymatrix{ T_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\ T_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1 } $$ commutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$ are as in Remark \\ref{remark-action-by-derivations}."}
+{"_id": "13723", "title": "more-morphisms-lemma-deform", "text": "Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism of first order thickenings. Assume that $f$ is flat. Then the following are equivalent \\begin{enumerate} \\item $f'$ is flat and $X = S \\times_{S'} X'$, and \\item the canonical map $f^*\\mathcal{C}_{S/S'} \\to \\mathcal{C}_{X/X'}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "13724", "title": "more-morphisms-lemma-flatness-morphism-thickenings", "text": "Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (S \\subset S') } $$ of thickenings. Assume \\begin{enumerate} \\item $X'$ is flat over $S'$, \\item $f$ is flat, \\item $S \\subset S'$ is a finite order thickening, and \\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$. \\end{enumerate} Then $f'$ is flat and $Y'$ is flat over $S'$ at all points in the image of $f'$."}
+{"_id": "13725", "title": "more-morphisms-lemma-deform-property", "text": "Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (S \\subset S') } $$ of thickenings. Assume $S \\subset S'$ is a finite order thickening, $X'$ flat over $S'$, $X = S \\times_{S'} X'$, and $Y = S \\times_{S'} Y'$. Then \\begin{enumerate} \\item $f$ is flat if and only if $f'$ is flat, \\label{item-flat} \\item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \\label{item-isomorphism} \\item $f$ is an open immersion if and only if $f'$ is an open immersion, \\label{item-open-immersion} \\item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \\label{item-quasi-compact} \\item $f$ is universally closed if and only if $f'$ is universally closed, \\label{item-universally-closed} \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\label{item-separated} \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\label{item-monomorphism} \\item $f$ is surjective if and only if $f'$ is surjective, \\label{item-surjective} \\item $f$ is universally injective if and only if $f'$ is universally injective, \\label{item-universally-injective} \\item $f$ is affine if and only if $f'$ is affine, \\label{item-affine} \\item \\label{item-finite-type} $f$ is locally of finite type if and only if $f'$ is locally of finite type, \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\label{item-quasi-finite} \\item \\label{item-finite-presentation} $f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation, \\item \\label{item-relative-dimension-d} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $f$ is universally open if and only if $f'$ is universally open, \\label{item-universally-open} \\item $f$ is syntomic if and only if $f'$ is syntomic, \\label{item-syntomic} \\item $f$ is smooth if and only if $f'$ is smooth, \\label{item-smooth} \\item $f$ is unramified if and only if $f'$ is unramified, \\label{item-unramified} \\item $f$ is \\'etale if and only if $f'$ is \\'etale, \\label{item-etale} \\item $f$ is proper if and only if $f'$ is proper, \\label{item-proper} \\item $f$ is integral if and only if $f'$ is integral, \\label{item-integral} \\item $f$ is finite if and only if $f'$ is finite, \\label{item-finite} \\item \\label{item-finite-locally-free} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \\item add more here. \\end{enumerate}"}
+{"_id": "13726", "title": "more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft", "text": "Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (S \\subset S') } $$ of thickenings. Assume \\begin{enumerate} \\item $Y' \\to S'$ is locally of finite type, \\item $X' \\to S'$ is flat and locally of finite presentation, \\item $f$ is flat, and \\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$. \\end{enumerate} Then $f'$ is flat and for all $y' \\in Y'$ in the image of $f'$ the local ring $\\mathcal{O}_{Y', y'}$ is flat and essentially of finite presentation over $\\mathcal{O}_{S', s'}$."}
+{"_id": "13727", "title": "more-morphisms-lemma-deform-property-fp-over-ft", "text": "Consider a commutative diagram $$ \\xymatrix{ (X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\ & (S \\subset S') } $$ of thickenings. Assume $Y' \\to S'$ locally of finite type, $X' \\to S'$ flat and locally of finite presentation, $X = S \\times_{S'} X'$, and $Y = S \\times_{S'} Y'$. Then \\begin{enumerate} \\item $f$ is flat if and only if $f'$ is flat, \\label{item-flat-fp-over-ft} \\item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \\label{item-isomorphism-fp-over-ft} \\item $f$ is an open immersion if and only if $f'$ is an open immersion, \\label{item-open-immersion-fp-over-ft} \\item $f$ is quasi-compact if and only if $f'$ is quasi-compact, \\label{item-quasi-compact-fp-over-ft} \\item $f$ is universally closed if and only if $f'$ is universally closed, \\label{item-universally-closed-fp-over-ft} \\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \\label{item-separated-fp-over-ft} \\item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \\label{item-monomorphism-fp-over-ft} \\item $f$ is surjective if and only if $f'$ is surjective, \\label{item-surjective-fp-over-ft} \\item $f$ is universally injective if and only if $f'$ is universally injective, \\label{item-universally-injective-fp-over-ft} \\item $f$ is affine if and only if $f'$ is affine, \\label{item-affine-fp-over-ft} \\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \\label{item-quasi-finite-fp-over-ft} \\item \\label{item-relative-dimension-d-fp-over-ft} $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$, \\item $f$ is universally open if and only if $f'$ is universally open, \\label{item-universally-open-fp-over-ft} \\item $f$ is syntomic if and only if $f'$ is syntomic, \\label{item-syntomic-fp-over-ft} \\item $f$ is smooth if and only if $f'$ is smooth, \\label{item-smooth-fp-over-ft} \\item $f$ is unramified if and only if $f'$ is unramified, \\label{item-unramified-fp-over-ft} \\item $f$ is \\'etale if and only if $f'$ is \\'etale, \\label{item-etale-fp-over-ft} \\item $f$ is proper if and only if $f'$ is proper, \\label{item-proper-fp-over-ft} \\item $f$ is finite if and only if $f'$ is finite, \\label{item-finite-fp-over-ft} \\item \\label{item-finite-locally-free-fp-over-ft} $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and \\item add more here. \\end{enumerate}"}
+{"_id": "13728", "title": "more-morphisms-lemma-deform-projective", "text": "Let $f : X \\to S$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $\\mathcal{L}$ be $f$-ample. Assume $S$ is quasi-compact. There exists a $d_0 \\geq 0$ such that for every cartesian diagram $$ \\vcenter{ \\xymatrix{ X \\ar[r]_{i'} \\ar[d]_f & X' \\ar[d]^{f'} \\\\ S \\ar[r]^i & S' } } \\quad\\text{and}\\quad \\begin{matrix} \\text{invertible }\\mathcal{O}_{X'}\\text{-module}\\\\ \\mathcal{L}'\\text{ with }\\mathcal{L} \\cong (i')^*\\mathcal{L}' \\end{matrix} $$ where $S \\subset S'$ is a thickening and $f'$ is proper, flat, of finite presentation we have \\begin{enumerate} \\item $R^p(f')_*(\\mathcal{L}')^{\\otimes d} = 0$ for all $p > 0$ and $d \\geq d_0$, \\item $\\mathcal{A}'_d = (f')_*(\\mathcal{L}')^{\\otimes d}$ is finite locally free for $d \\geq d_0$, \\item $\\mathcal{A}' = \\mathcal{O}_{S'} \\oplus \\bigoplus_{d \\geq d_0} \\mathcal{A}'_d$ is a quasi-coherent $\\mathcal{O}_{S'}$-algebra of finite presentation, \\item there is a canonical isomorphism $r' : X' \\to \\underline{\\text{Proj}}_{S'}(\\mathcal{A}')$, and \\item there is a canonical isomorphism $\\theta' : (r')^*\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(\\mathcal{A}')}(1) \\to \\mathcal{L}'$. \\end{enumerate} The construction of $\\mathcal{A}'$, $r'$, $\\theta'$ is functorial in the data $(X', S', i, i', f', \\mathcal{L}')$."}
+{"_id": "13729", "title": "more-morphisms-lemma-composition-formally-smooth", "text": "A composition of formally smooth morphisms is formally smooth."}
+{"_id": "13731", "title": "more-morphisms-lemma-formally-etale-unramified-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. Then $f$ is formally \\'etale if and only if $f$ is formally smooth and formally unramified."}
+{"_id": "13732", "title": "more-morphisms-lemma-formally-smooth-on-opens", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$ and $V \\subset S$ be open subschemes such that $f(U) \\subset V$. If $f$ is formally smooth, so is $f|_U : U \\to V$."}
+{"_id": "13733", "title": "more-morphisms-lemma-affine-formally-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally smooth if and only if $\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally smooth ring map."}
+{"_id": "13734", "title": "more-morphisms-lemma-smooth-formally-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is smooth, and \\item the morphism $f$ is locally of finite presentation and formally smooth. \\end{enumerate}"}
+{"_id": "13735", "title": "more-morphisms-lemma-formally-smooth-sheaf-differentials", "text": "Let $f : X \\to Y$ be a formally smooth morphism of schemes. Then $\\Omega_{X/Y}$ is locally projective on $X$."}
+{"_id": "13736", "title": "more-morphisms-lemma-h1-is-zero", "text": "Let $T$ be an affine scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_T$-modules. Consider $\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$. If $\\mathcal{F}$ is locally projective, then $H^1(T, \\mathcal{H}) = 0$."}
+{"_id": "13737", "title": "more-morphisms-lemma-formally-smooth", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is formally smooth, \\item for every $x \\in X$ there exist opens $x \\in U \\subset X$ and $f(x) \\in V \\subset Y$ with $f(U) \\subset V$ such that $f|_U : U \\to V$ is formally smooth, \\item for every pair of affine opens $U \\subset X$ and $V \\subset Y$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is formally smooth, and \\item there exists an affine open covering $Y = \\bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is a formally smooth ring map for all $j$ and $i$. \\end{enumerate}"}
+{"_id": "13738", "title": "more-morphisms-lemma-triangle-differentials-formally-smooth", "text": "Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes. Assume $f$ is formally smooth. Then $$ 0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0 $$ (see Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials}) is short exact."}
+{"_id": "13740", "title": "more-morphisms-lemma-two-unramified-morphisms-formally-smooth", "text": "Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are formally unramified and $f$ is formally smooth. Then the canonical exact sequence $$ 0 \\to \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0 $$ of Lemma \\ref{lemma-two-unramified-morphisms} is exact and locally split."}
+{"_id": "13741", "title": "more-morphisms-lemma-lifting-along-artinian-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent: \\begin{enumerate} \\item $f$ is smooth at $x$, \\item for every solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\ S & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu] } $$ where $B' \\to B$ is a surjection of local rings with $\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the closed point of $\\Spec(B)$ to $x$ there exists a dotted arrow making the diagram commute, \\item same as in (2) but with $B' \\to B$ ranging over small extensions (see Algebra, Definition \\ref{algebra-definition-small-extension}), and \\item same as in (2) but with $B' \\to B$ ranging over small extensions such that $\\alpha$ induces an isomorphism $\\kappa(x) \\to \\kappa(\\mathfrak m)$ where $\\mathfrak m \\subset B$ is the maximal ideal. \\end{enumerate}"}
+{"_id": "13743", "title": "more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds", "text": "Let $f : X \\to S$ be a finite type morphism of locally Noetherian schemes. Let $Z \\subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$. \\begin{enumerate} \\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$. \\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$ is \\'etale at every point of $f^{-1}(Z)$. \\end{enumerate}"}
+{"_id": "13744", "title": "more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds", "text": "Let $f : X \\to S$ be a morphism of locally Noetherian schemes. Let $Z \\subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$. If $X_n \\to Z_n$ is flat for all $n$, then $f$ is flat at every point of $f^{-1}(Z)$."}
+{"_id": "13745", "title": "more-morphisms-lemma-NL-affine", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$ be affine opens with $f(U) \\subset V$. There is a canonical map $$ \\widetilde{\\NL_{A/R}} \\longrightarrow \\NL_{X/Y}|_U $$ of complexes which is an isomorphism in $D(\\mathcal{O}_U)$."}
+{"_id": "13746", "title": "more-morphisms-lemma-netherlander-quasi-coherent", "text": "Let $f : X \\to Y$ be a morphism of schemes. The cohomology sheaves of the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$."}
+{"_id": "13747", "title": "more-morphisms-lemma-netherlander-fp", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\\NL_{X/Y}$ is locally on $X$ quasi-isomorphic to a complex $$ \\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots $$ of quasi-coherent $\\mathcal{O}_X$-modules with $\\mathcal{F}^0$ of finite presentation and $\\mathcal{F}^{-1}$ of finite type."}
+{"_id": "13748", "title": "more-morphisms-lemma-NL-formally-smooth", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is formally smooth, \\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$ is locally projective. \\end{enumerate}"}
+{"_id": "13750", "title": "more-morphisms-lemma-NL-smooth", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is smooth, and \\item $f$ is locally of finite presentation, $H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$ is finite locally free. \\end{enumerate}"}
+{"_id": "13751", "title": "more-morphisms-lemma-NL-etale", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is \\'etale, and \\item $f$ is locally of finite presentation and $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$. \\end{enumerate}"}
+{"_id": "13752", "title": "more-morphisms-lemma-NL-immersion", "text": "Let $i : Z \\to X$ be an immersion of schemes. Then $\\NL_{Z/X}$ is isomorphic to $\\mathcal{C}_{Z/X}[1]$ in $D(\\mathcal{O}_Z)$ where $\\mathcal{C}_{Z/X}$ is the conormal sheaf of $Z$ in $X$."}
+{"_id": "13753", "title": "more-morphisms-lemma-exact-sequence-NL", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. There is a canonical six term exact sequence $$ H^{-1}(f^*\\NL_{Y/Z}) \\to H^{-1}(\\NL_{X/Z}) \\to H^{-1}(\\NL_{X/Y}) \\to f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0 $$ of cohomology sheaves."}
+{"_id": "13754", "title": "more-morphisms-lemma-get-triangle-NL", "text": "Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of schemes. Assume $X \\to Y$ is a complete intersection morphism. Then there is a canonical distinguished triangle $$ f^*\\NL_{Y/Z} \\to \\NL_{X/Z} \\to \\NL_{X/Y} \\to f^*\\NL_{Y/Z}[1] $$ in $D(\\mathcal{O}_X)$ which recovers the $6$-term exact sequence of Lemma \\ref{lemma-exact-sequence-NL}."}
+{"_id": "13756", "title": "more-morphisms-lemma-get-NL", "text": "Let $f : X \\to Y$ be a morphism of schemes which factors as $f = g \\circ i$ with $i$ an immersion and $g : P \\to Y$ formally smooth (for example smooth). Then there is a canonical isomorphism $$ \\NL_{X/Y} \\cong \\left(\\mathcal{C}_{X/P} \\to i^*\\Omega_{P/Y}\\right) $$ in $D(\\mathcal{O}_X)$ where the conormal sheaf $\\mathcal{C}_{X/P}$ is placed in degree $-1$."}
+{"_id": "13761", "title": "more-morphisms-lemma-pushout-fpqc-local", "text": "Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of schemes. Let $(W, X_i \\to W)$ be a cocone for the diagram in the category of schemes (Categories, Remark \\ref{categories-remark-cones-and-cocones}). If there exists a fpqc covering $\\{W_a \\to W\\}_{a \\in A}$ of schemes such that \\begin{enumerate} \\item for all $a \\in A$ we have $W_a = \\colim X_i \\times_W W_a$ in the category of schemes, and \\item for all $a, b \\in A$ we have $W_a \\times_W W_b = \\colim X_i \\times_W W_a \\times_W W_b$ in the category of schemes, \\end{enumerate} then $W = \\colim X_i$ in the category of schemes."}
+{"_id": "13762", "title": "more-morphisms-lemma-pushout-along-thickening", "text": "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an affine morphism of schemes. Then there exists a pushout $$ \\xymatrix{ X \\ar[r] \\ar[d]_f & X' \\ar[d]^{f'} \\\\ Y \\ar[r] & Y' } $$ in the category of schemes. Moreover, $Y \\subset Y'$ is a thickening, $X = Y \\times_{Y'} X'$, and $$ \\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'} $$ as sheaves on $|Y| = |Y'|$."}
+{"_id": "13763", "title": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout", "text": "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an affine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout (see Lemma \\ref{lemma-pushout-along-thickening}). Base change gives a functor $$ F : (\\Sch/Y') \\longrightarrow (\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X') $$ given by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$ which has a left adjoint $$ G : (\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X') \\longrightarrow (\\Sch/Y') $$ which sends the triple $(V, U', \\varphi)$ to the pushout $V \\amalg_{(V \\times_Y X)} U'$. Finally, $F \\circ G$ is isomorphic to the identity functor."}
+{"_id": "13764", "title": "more-morphisms-lemma-scheme-over-pushout-flat-modules", "text": "Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an affine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout (see Lemma \\ref{lemma-pushout-along-thickening}). Let $V' \\to Y'$ be a morphism of schemes. Set $V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$. There is an equivalence of categories between \\begin{enumerate} \\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and \\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where \\begin{enumerate} \\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$, \\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat over $X'$, and \\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$ is an isomorphism of $\\mathcal{O}_U$-modules. \\end{enumerate} \\end{enumerate} The equivalence maps $\\mathcal{G}'$ to $((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$. Suppose $\\mathcal{G}'$ corresponds to the triple $(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then \\begin{enumerate} \\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type $\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules. \\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then $\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite presentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are $\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation. \\end{enumerate}"}
+{"_id": "13765", "title": "more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat", "text": "In the situation of Lemma \\ref{lemma-equivalence-categories-schemes-over-pushout}. If $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then \\begin{enumerate} \\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and $U' \\to X'$ are locally of finite type, \\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat, \\item $V' \\to Y'$ is flat and locally of finite presentation if and only if $V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation, \\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth, \\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$ are \\'etale, and \\item add more here as needed. \\end{enumerate} If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of schemes flat over $Y'$ and the category of triples $(V, U', \\varphi)$ with $V \\to Y$ and $U' \\to X'$ flat."}
+{"_id": "13766", "title": "more-morphisms-lemma-flat-locus-base-change", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x' \\in X'$ with images $x = g'(x')$ and $s' = g'(x')$. \\begin{enumerate} \\item If $\\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$. \\item If $g$ is flat at $s'$ and $(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$, then $\\mathcal{F}$ is flat over $S$ at $x$. \\end{enumerate} In particular, if $g$ is flat, $f$ is locally of finite presentation, and $\\mathcal{F}$ is locally of finite presentation, then formation of the open subset of Theorem \\ref{theorem-openness-flatness} commutes with base change."}
+{"_id": "13767", "title": "more-morphisms-lemma-morphism-between-flat-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Assume \\begin{enumerate} \\item $S$, $X$, $Y$ are locally Noetherian, \\item $X$ is flat over $S$, \\item for every $s \\in S$ the morphism $f_s : X_s \\to Y_s$ is flat. \\end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$."}
+{"_id": "13768", "title": "more-morphisms-lemma-morphism-between-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $S$, \\item $X$ is flat over $S$, \\item for every $s \\in S$ the morphism $f_s : X_s \\to Y_s$ is flat, and \\item $Y$ is locally of finite type over $S$. \\end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$."}
+{"_id": "13769", "title": "more-morphisms-lemma-base-change-criterion-flatness-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $S$, \\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, \\item $\\mathcal{F}$ is flat over $S$, and \\item $Y$ is locally of finite type over $S$. \\end{enumerate} Then the set $$ U = \\{x \\in X \\mid \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}. $$ is open in $X$ and its formation commutes with arbitrary base change: If $S' \\to S$ is a morphism of schemes, and $U'$ is the set of points of $X' = X \\times_S S'$ where $\\mathcal{F}' = \\mathcal{F} \\times_S S'$ is flat over $Y' = Y \\times_S S'$, then $U' = U \\times_S S'$."}
+{"_id": "13771", "title": "more-morphisms-lemma-flat-and-free-at-point-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes of finite presentation. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module. Let $x \\in X$ with image $s \\in S$. If $\\mathcal{F}$ is flat at $x$ over $S$ and $(\\mathcal{F}_s)_x$ is a flat $\\mathcal{O}_{X_s, x}$-module, then $\\mathcal{F}$ is finite free in a neighbourhood of $x$."}
+{"_id": "13772", "title": "more-morphisms-lemma-finite-free-open", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite presentation. Let $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module flat over $S$. Then the set $$ \\{x \\in X : \\mathcal{F}\\text{ free in a neighbourhood of }x\\} $$ is open in $X$ and its formation commutes with arbitrary base change $S' \\to S$."}
+{"_id": "13773", "title": "more-morphisms-lemma-integral-closure-smooth-pullback", "text": "Let $f : Y \\to X$ be a smooth morphism of schemes. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The integral closure of $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$ where $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$."}
+{"_id": "13774", "title": "more-morphisms-lemma-normalization-smooth-localization", "text": "Let $$ \\xymatrix{ Y_2 \\ar[r] \\ar[d]_{f_2} & Y_1 \\ar[d]^{f_1} \\\\ X_2 \\ar[r]^\\varphi & X_1 } $$ be a fibre square in the category of schemes. Assume $f_1$ is quasi-compact and quasi-separated, and $\\varphi$ is smooth. Let $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$. Then $X_2' \\cong X_2 \\times_{X_1} X_1'$."}
+{"_id": "13775", "title": "more-morphisms-lemma-normalization-and-smooth", "text": "Let $X \\to Y$ be a smooth morphism of schemes. Assume every quasi-compact open of $Y$ has finitely many irreducible components. Then the same is true for $X$ and there is a canonical isomorphism $X^\\nu = X \\times_Y Y^\\nu$ where $X^\\nu$, $Y^\\nu$ are the normalizations of $X$, $Y$."}
+{"_id": "13776", "title": "more-morphisms-lemma-normalization-and-henselization", "text": "Let $X$ be a locally Noetherian scheme. Let $\\nu : X^\\nu \\to X$ be the normalization morphism. Then for any point $x \\in X$ the base change $$ X^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^h) \\to \\Spec(\\mathcal{O}_{X, x}^h), \\quad\\text{resp.}\\quad X^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh}) \\to \\Spec(\\mathcal{O}_{X, x}^{sh}) $$ is the normalization of $\\Spec(\\mathcal{O}_{X, x}^h)$, resp.\\ $\\Spec(\\mathcal{O}_{X, x}^{sh})$."}
+{"_id": "13777", "title": "more-morphisms-lemma-normal", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is normal, and \\item $f$ is flat and its fibres are geometrically normal schemes. \\end{enumerate}"}
+{"_id": "13778", "title": "more-morphisms-lemma-smooth-normal", "text": "A smooth morphism is normal."}
+{"_id": "13779", "title": "more-morphisms-lemma-locally-Noetherian-fibres-fppf-local-source-and-target", "text": "The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian'' is local in the fppf topology on the source and the target."}
+{"_id": "13781", "title": "more-morphisms-lemma-regular", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is regular, \\item $f$ is flat and its fibres are geometrically regular schemes, \\item for every pair of affine opens $U \\subset X$, $V \\subset Y$ with $f(U) \\subset V$ the ring map $\\mathcal{O}(V) \\to \\mathcal{O}(U)$ is regular, \\item there exists an open covering $Y = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$ is regular, and \\item there exists an affine open covering $Y = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring maps $\\mathcal{O}(V_j) \\to \\mathcal{O}(U_i)$ are regular. \\end{enumerate}"}
+{"_id": "13784", "title": "more-morphisms-lemma-CM", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent \\begin{enumerate} \\item $f$ is Cohen-Macaulay, and \\item $f$ is flat and its fibres are Cohen-Macaulay schemes. \\end{enumerate}"}
+{"_id": "13786", "title": "more-morphisms-lemma-composition-CM", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. Assume that the fibres of $f$, $g$, and $g \\circ f$ are locally Noetherian. Let $x \\in X$ with images $y \\in Y$ and $z \\in Z$. \\begin{enumerate} \\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$. \\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay. \\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$, then $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$. \\item If $g \\circ f$ is Cohen-Macaulay and $f$ is flat, then $f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in the image of $f$. \\end{enumerate}"}
+{"_id": "13788", "title": "more-morphisms-lemma-base-change-CM", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$ be the base change of $f$. Let $x' \\in X'$ be a point with image $x \\in X$. \\begin{enumerate} \\item If $f$ is Cohen-Macaulay at $x$, then $f' : X' \\to Y'$ is Cohen-Macaulay at $x'$. \\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \\end{enumerate}"}
+{"_id": "13789", "title": "more-morphisms-lemma-flat-finite-presentation-CM-open", "text": "\\begin{reference} \\cite[IV Corollary 12.1.7(iii)]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $$ W = \\{x \\in X \\mid f\\text{ is Cohen-Macaulay at }x\\} $$ Then \\begin{enumerate} \\item $W = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x}\\text{ is Cohen-Macaulay}\\}$, \\item $W$ is open in $X$, \\item $W$ dense in every fibre of $X \\to S$, \\item the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : S' \\to S$, consider the base change $f' : X' \\to S'$ of $f$ and the projection $g' : X' \\to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$. \\end{enumerate}"}
+{"_id": "13790", "title": "more-morphisms-lemma-flat-finite-presentation-characterize-CM", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $x \\in X$ with image $s \\in S$. Set $d = \\dim_x(X_s)$. The following are equivalent \\begin{enumerate} \\item $f$ is Cohen-Macaulay at $x$, \\item there exists an open neighbourhood $U \\subset X$ of $x$ and a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$ which is flat at $x$, \\item there exists an open neighbourhood $U \\subset X$ of $x$ and a locally quasi-finite flat morphism $U \\to \\mathbf{A}^d_S$ over $S$, \\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$ of an open neighbourhood $U \\subset X$ of $x$ we have: $g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is flat at $x$. \\end{enumerate}"}
+{"_id": "13791", "title": "more-morphisms-lemma-flat-finite-presentation-CM-pieces", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. For $d \\geq 0$ there exist opens $U_d \\subset X$ with the following properties \\begin{enumerate} \\item $W = \\bigcup_{d \\geq 0} U_d$ is dense in every fibre of $f$, and \\item $U_d \\to S$ is of relative dimension $d$ (see Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}). \\end{enumerate}"}
+{"_id": "13792", "title": "more-morphisms-lemma-flat-finite-presentation-specialization-dimension", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Suppose $x' \\leadsto x$ is a specialization of points of $X$ with image $s' \\leadsto s$ in $S$. If $x$ is a generic point of an irreducible component of $X_s$ then $\\dim_{x'}(X_{s'}) = \\dim_x(X_s)$."}
+{"_id": "13793", "title": "more-morphisms-lemma-CM-local-source-and-target", "text": "The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is Cohen-Macaulay'' is local in the fppf topology on the target and local in the syntomic topology on the source."}
+{"_id": "13794", "title": "more-morphisms-lemma-slice-given-element", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Let $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is flat at $x$, and \\item the image $\\overline{h}$ of $h$ in $\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$ is a nonzerodivisor. \\end{enumerate} Then there exists an affine open neighbourhood $U \\subset X$ of $x$ such that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and $D \\to S$ flat and locally of finite presentation."}
+{"_id": "13795", "title": "more-morphisms-lemma-slice-given-elements", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Let $h_1, \\ldots, h_r \\in \\mathcal{O}_{X, x}$. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is flat at $x$, and \\item the images of $h_1, \\ldots, h_r$ in $\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$ form a regular sequence. \\end{enumerate} Then there exists an affine open neighbourhood $U \\subset X$ of $x$ such that $h_1, \\ldots, h_r$ come from $h_1, \\ldots, h_r \\in \\Gamma(U, \\mathcal{O}_U)$ and such that $Z = V(h_1, \\ldots, h_r) \\to U$ is a regular immersion with $x \\in Z$ and $Z \\to S$ flat and locally of finite presentation. Moreover, the base change $Z_{S'} \\to U_{S'}$ is a regular immersion for any scheme $S'$ over $S$."}
+{"_id": "13796", "title": "more-morphisms-lemma-slice-once", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is flat at $x$, and \\item $\\mathcal{O}_{X_s, x}$ has $\\text{depth} \\geq 1$. \\end{enumerate} Then there exists an affine open neighbourhood $U \\subset X$ of $x$ and an effective Cartier divisor $D \\subset U$ containing $x$ such that $D \\to S$ is flat and of finite presentation."}
+{"_id": "13797", "title": "more-morphisms-lemma-slice-CM", "text": "\\begin{reference} \\cite[IV Proposition 17.16.1]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is Cohen-Macaulay at $x$, and \\item $x$ is a closed point of $X_s$. \\end{enumerate} Then there exists a regular immersion $Z \\to X$ containing $x$ such that \\begin{enumerate} \\item[(a)] $Z \\to S$ is flat and locally of finite presentation, \\item[(b)] $Z \\to S$ is locally quasi-finite, and \\item[(c)] $Z_s = \\{x\\}$ set theoretically. \\end{enumerate}"}
+{"_id": "13798", "title": "more-morphisms-lemma-qf-fp-flat-neighbourhood-dominates-fppf", "text": "Let $f : X \\to S$ be a flat morphism of schemes which is locally of finite presentation. Let $s \\in S$ be a point in the image of $f$. Then there exists a commutative diagram $$ \\xymatrix{ S' \\ar[rr] \\ar[rd]_g & & X \\ar[ld]^f \\\\ & S } $$ where $g : S' \\to S$ is flat, locally of finite presentation, locally quasi-finite, and $s \\in g(S')$."}
+{"_id": "13799", "title": "more-morphisms-lemma-qf-fp-flat-dominates-fppf", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be an fppf covering of $S$, see Topologies, Definition \\ref{topologies-definition-fppf-covering}. Then there exists an fppf covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$ which refines (see Sites, Definition \\ref{sites-definition-morphism-coverings}) $\\mathcal{U}$ such that each $T_j \\to S$ is locally quasi-finite."}
+{"_id": "13800", "title": "more-morphisms-lemma-empty-generic-fibre", "text": "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$. If $X_\\eta = \\emptyset$ then there exists a nonempty open $V \\subset Y$ such that $X_V = V \\times_Y X = \\emptyset$."}
+{"_id": "13801", "title": "more-morphisms-lemma-nonempty-generic-fibre", "text": "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$. If $X_\\eta \\not = \\emptyset$ then there exists a nonempty open $V \\subset Y$ such that $X_V = V \\times_Y X \\to V$ is surjective."}
+{"_id": "13802", "title": "more-morphisms-lemma-nowhere-dense-generic-fibre", "text": "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$. If $Z \\subset X$ is a closed subset with $Z_\\eta$ nowhere dense in $X_\\eta$, then there exists a nonempty open $V \\subset Y$ such that $Z_y$ is nowhere dense in $X_y$ for all $y \\in V$."}
+{"_id": "13803", "title": "more-morphisms-lemma-scheme-theoretically-dense-generic-fibre", "text": "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$. Let $U \\subset X$ be an open subscheme such that $U_\\eta$ is scheme theoretically dense in $X_\\eta$. Then there exists a nonempty open $V \\subset Y$ such that $U_y$ is scheme theoretically dense in $X_y$ for all $y \\in V$."}
+{"_id": "13804", "title": "more-morphisms-lemma-cover-generic-fibre-neighbourhood", "text": "Let $f : X \\to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$. Let $X_\\eta = Z_{1, \\eta} \\cup \\ldots \\cup Z_{n, \\eta}$ be a covering of the generic fibre by closed subsets of $X_\\eta$. Let $Z_i$ be the closure of $Z_{i, \\eta}$ in $X$ (see discussion above). Then there exists a nonempty open $V \\subset Y$ such that $X_y = Z_{1, y} \\cup \\ldots \\cup Z_{n, y}$ for all $y \\in V$."}
+{"_id": "13805", "title": "more-morphisms-lemma-reduction-generic-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\eta \\in Y$ be a generic point of an irreducible component of $Y$. Then $(X_\\eta)_{red} = (X_{red})_\\eta$."}
+{"_id": "13806", "title": "more-morphisms-lemma-make-generic-fibre-geometrically-reduced", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that $Y$ is irreducible and $f$ is of finite type. There exists a diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\ Y' \\ar[r]^g & V \\ar[r] & Y } $$ where \\begin{enumerate} \\item $V$ is a nonempty open of $Y$, \\item $X_V = V \\times_Y X$, \\item $g : Y' \\to V$ is a finite universal homeomorphism, \\item $X' = (Y' \\times_Y X)_{red} = (Y' \\times_V X_V)_{red}$, \\item $g'$ is a finite universal homeomorphism, \\item $Y'$ is an integral affine scheme, \\item $f'$ is flat and of finite presentation, and \\item the generic fibre of $f'$ is geometrically reduced. \\end{enumerate}"}
+{"_id": "13807", "title": "more-morphisms-lemma-make-components-generic-fibre-geometrically-irreducible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that $Y$ is irreducible and $f$ is of finite type. There exists a diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\ Y' \\ar[r]^g & V \\ar[r] & Y } $$ where \\begin{enumerate} \\item $V$ is a nonempty open of $Y$, \\item $X_V = V \\times_Y X$, \\item $g : Y' \\to V$ is surjective finite \\'etale, \\item $X' = Y' \\times_Y X = Y' \\times_V X_V$, \\item $g'$ is surjective finite \\'etale, \\item $Y'$ is an irreducible affine scheme, and \\item all irreducible components of the generic fibre of $f'$ are geometrically irreducible. \\end{enumerate}"}
+{"_id": "13808", "title": "more-morphisms-lemma-relative-assassin-in-neighbourhood", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\xi \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and set $Z = \\overline{\\{\\xi\\}} \\subset X$. If $f$ is locally of finite type and $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, then there exists a nonempty open $V \\subset Z$ such that for every $s \\in f(V)$ the generic points of $V_s$ are elements of $\\text{Ass}_{X/S}(\\mathcal{F})$."}
+{"_id": "13809", "title": "more-morphisms-lemma-bad-case", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open subscheme. Assume \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is of finite type, \\item $Y$ is irreducible with generic point $\\eta$, and \\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta)$ is not contained in $U_\\eta$. \\end{enumerate} Then there exists a nonempty open subscheme $V \\subset Y$ such that for all $y \\in V$ the set $\\text{Ass}_{X_y}(\\mathcal{F}_y)$ is not contained in $U_y$."}
+{"_id": "13810", "title": "more-morphisms-lemma-good-case", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open subscheme. Assume \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is of finite type, \\item $Y$ is irreducible with generic point $\\eta$, and \\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta) \\subset U_\\eta$. \\end{enumerate} Then there exists a nonempty open subscheme $V \\subset Y$ such that for all $y \\in V$ we have $\\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y$."}
+{"_id": "13811", "title": "more-morphisms-lemma-base-change-assassin-in-U", "text": "Let $f : X \\to S$ be a morphism which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $U \\subset X$ be an open subscheme. Let $g : S' \\to S$ be a morphism of schemes, let $f' : X' = X_{S'} \\to S'$ be the base change of $f$, let $g' : X' \\to X$ be the projection, set $\\mathcal{F}' = (g')^*\\mathcal{F}$, and set $U' = (g')^{-1}(U)$. Finally, let $s' \\in S'$ with image $s = g(s')$. In this case $$ \\text{Ass}_{X_s}(\\mathcal{F}_s) \\subset U_s \\Leftrightarrow \\text{Ass}_{X'_{s'}}(\\mathcal{F}'_{s'}) \\subset U'_{s'}. $$"}
+{"_id": "13812", "title": "more-morphisms-lemma-relative-assassin-constructible", "text": "Let $f : X \\to Y$ be a morphism of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation. Let $U \\subset X$ be an open subscheme such that $U \\to Y$ is quasi-compact. Then the set $$ E = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y\\} $$ is locally constructible in $Y$."}
+{"_id": "13813", "title": "more-morphisms-lemma-nonreduced-in-neighbourhood", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ is nonreduced, then there exists a nonempty open $V \\subset Y$ such that for all $y \\in V$ the fibre $X_y$ is nonreduced."}
+{"_id": "13814", "title": "more-morphisms-lemma-base-change-fibres-geometrically-reduced", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $g : Y' \\to Y$ be any morphism, and denote $f' : X' \\to Y'$ the base change of $f$. Then \\begin{align*} \\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically reduced}\\} \\\\ = g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}). \\end{align*}"}
+{"_id": "13815", "title": "more-morphisms-lemma-not-geometrically-reduced-in-neighbourhood", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ is not geometrically reduced, then there exists a nonempty open $V \\subset Y$ such that for all $y \\in V$ the fibre $X_y$ is not geometrically reduced."}
+{"_id": "13816", "title": "more-morphisms-lemma-geometrically-reduced-generic-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $Y$ is irreducible with generic point $\\eta$, \\item $X_\\eta$ is geometrically reduced, and \\item $f$ is of finite type. \\end{enumerate} Then there exists a nonempty open subscheme $V \\subset Y$ such that $X_V \\to V$ has geometrically reduced fibres."}
+{"_id": "13817", "title": "more-morphisms-lemma-geometrically-reduced-constructible", "text": "Let $f : X \\to Y$ be a morphism which is quasi-compact and locally of finite presentation. Then the set $$ E = \\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\} $$ is locally constructible in $Y$."}
+{"_id": "13818", "title": "more-morphisms-lemma-proper-flat-over-dvr-reduced-fibre", "text": "Let $X \\to \\Spec(R)$ be a proper flat morphism where $R$ is a discrete valuation ring. If the special fibre is reduced, then both $X$ and the generic fibre $X_\\eta$ are reduced."}
+{"_id": "13819", "title": "more-morphisms-lemma-geometrically-reduced-open", "text": "Let $f : X \\to Y$ be a flat proper morphism of finite presentation. Then the set $\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}$ is open in $Y$."}
+{"_id": "13820", "title": "more-morphisms-lemma-irreducible-components-in-neighbourhood", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$ irreducible components, then there exists a nonempty open $V \\subset Y$ such that for all $y \\in V$ the fibre $X_y$ has at least $n$ irreducible components."}
+{"_id": "13821", "title": "more-morphisms-lemma-base-change-fibres-geometrically-irreducible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $g : Y' \\to Y$ be any morphism, and denote $f' : X' \\to Y'$ the base change of $f$. Then \\begin{align*} \\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically irreducible}\\} \\\\ = g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically irreducible}\\}). \\end{align*}"}
+{"_id": "13822", "title": "more-morphisms-lemma-base-change-fibres-nr-geometrically-irreducible-components", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $$ n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\} $$ be the function which associates to $y \\in Y$ the number of irreducible components of $(X_y)_K$ where $K$ is a separably closed extension of $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \\circ g $$ where $X' \\to Y'$ is the base change of $f$."}
+{"_id": "13823", "title": "more-morphisms-lemma-irreducible-polynomial-over-domain", "text": "Let $A$ be a domain with fraction field $K$. Let $P \\in A[x_1, \\ldots, x_n]$. Denote $\\overline{K}$ the algebraic closure of $K$. Assume $P$ is irreducible in $\\overline{K}[x_1, \\ldots, x_n]$. Then there exists a $f \\in A$ such that $P^\\varphi \\in \\kappa[x_1, \\ldots, x_n]$ is irreducible for all homomorphisms $\\varphi : A_f \\to \\kappa$ into fields."}
+{"_id": "13824", "title": "more-morphisms-lemma-geom-irreducible-generic-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $Y$ is irreducible with generic point $\\eta$, \\item $X_\\eta$ is geometrically irreducible, and \\item $f$ is of finite type. \\end{enumerate} Then there exists a nonempty open subscheme $V \\subset Y$ such that $X_V \\to V$ has geometrically irreducible fibres."}
+{"_id": "13825", "title": "more-morphisms-lemma-nr-geom-irreducible-components-good", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically irreducible components of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}. Assume $f$ of finite type. Let $y \\in Y$ be a point. Then there exists a nonempty open $V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant."}
+{"_id": "13826", "title": "more-morphisms-lemma-nr-geom-irreducible-components-constructible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically irreducible components of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\} $$ of $n_{X/Y}$ are locally constructible in $Y$."}
+{"_id": "13827", "title": "more-morphisms-lemma-connected-components-in-neighbourhood", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$ connected components, then there exists a nonempty open $V \\subset Y$ such that for all $y \\in V$ the fibre $X_y$ has at least $n$ connected components."}
+{"_id": "13828", "title": "more-morphisms-lemma-base-change-fibres-geometrically-connected", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $g : Y' \\to Y$ be any morphism, and denote $f' : X' \\to Y'$ the base change of $f$. Then \\begin{align*} \\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically connected}\\} \\\\ = g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically connected}\\}). \\end{align*}"}
+{"_id": "13829", "title": "more-morphisms-lemma-base-change-fibres-nr-geometrically-connected-components", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $$ n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\} $$ be the function which associates to $y \\in Y$ the number of connected components of $(X_y)_K$ where $K$ is a separably closed extension of $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \\circ g $$ where $X' \\to Y'$ is the base change of $f$."}
+{"_id": "13830", "title": "more-morphisms-lemma-geometrically-connected-generic-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $Y$ is irreducible with generic point $\\eta$, \\item $X_\\eta$ is geometrically connected, and \\item $f$ is of finite type. \\end{enumerate} Then there exists a nonempty open subscheme $V \\subset Y$ such that $X_V \\to V$ has geometrically connected fibres."}
+{"_id": "13831", "title": "more-morphisms-lemma-nr-geom-connected-components-good", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically connected components of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Assume $f$ of finite type. Let $y \\in Y$ be a point. Then there exists a nonempty open $V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant."}
+{"_id": "13832", "title": "more-morphisms-lemma-nr-geom-connected-components-constructible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ counting the numbers of geometric connected components of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\} $$ of $n_{X/Y}$ are locally constructible in $Y$."}
+{"_id": "13833", "title": "more-morphisms-lemma-connected-flat-over-dvr", "text": "\\begin{slogan} A flat degeneration of a disconnected scheme is either disconnected or nonreduced. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume that \\begin{enumerate} \\item $S$ is the spectrum of a discrete valuation ring, \\item $f$ is flat, \\item $X$ is connected, \\item the closed fibre $X_s$ is reduced. \\end{enumerate} Then the generic fibre $X_\\eta$ is connected."}
+{"_id": "13838", "title": "more-morphisms-lemma-connected-along-section-open", "text": "Let $f : X \\to Y$, $s : Y \\to X$ be as in Situation \\ref{situation-connected-along-section}. Assume \\begin{enumerate} \\item $f$ is of finite presentation and flat, and \\item all fibres of $f$ are geometrically reduced. \\end{enumerate} Then $X^0$ is open in $X$."}
+{"_id": "13839", "title": "more-morphisms-lemma-dimension-in-neighbourhood", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with generic point $\\eta$ and $f$ of finite type. If $X_\\eta$ has dimension $n$, then there exists a nonempty open $V \\subset Y$ such that for all $y \\in V$ the fibre $X_y$ has dimension $n$."}
+{"_id": "13840", "title": "more-morphisms-lemma-base-change-dimension-fibres", "text": "Let $f : X \\to Y$ be a morphism of finite type. Let $$ n_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\} $$ be the function which associates to $y \\in Y$ the dimension of $X_y$. If $g : Y' \\to Y$ is a morphism then $$ n_{X'/Y'} = n_{X/Y} \\circ g $$ where $X' \\to Y'$ is the base change of $f$."}
+{"_id": "13841", "title": "more-morphisms-lemma-dimension-fibres-constructible", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Assume $f$ of finite presentation. Then the level sets $$ E_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\} $$ of $n_{X/Y}$ are locally constructible in $Y$."}
+{"_id": "13842", "title": "more-morphisms-lemma-dimension-fibres-flat", "text": "Let $f : X \\to Y$ be a flat morphism of schemes of finite presentation. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is lower semi-continuous."}
+{"_id": "13843", "title": "more-morphisms-lemma-dimension-fibres-proper", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is upper semi-continuous."}
+{"_id": "13844", "title": "more-morphisms-lemma-dimension-fibres-proper-flat", "text": "Let $f : X \\to Y$ be a proper, flat morphism of schemes of finite presentation. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-dimension-fibres}. Then $n_{X/Y}$ is locally constant."}
+{"_id": "13845", "title": "more-morphisms-lemma-amazing-bertini-lemma", "text": "\\begin{reference} See pages 71 and 72 of \\cite{Jou} \\end{reference} Let $K/k$ be a geometrically irreducible and finitely generated field extension. Let $n \\geq 1$. Let $g_1, \\ldots, g_n \\in K$ be elements such that there exist $c_1, \\ldots, c_n \\in k$ such that the elements $$ x_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in K(x_1, \\ldots, x_n) $$ are algebraically independent over $k$. Then $K(x_1, \\ldots, x_n)$ is geometrically irreducible over $k(x_1, \\ldots, x_n, \\sum g_ix_i)$."}
+{"_id": "13846", "title": "more-morphisms-lemma-algebraically-independent", "text": "Let $A$ be a domain of finite type over a field $k$. Let $n \\geq 2$. Let $g_1, \\ldots, g_n \\in A$ be elements such that $V(g_1, g_2)$ has an irreducible component of dimension $\\dim(A) - 2$. Then there exist $c_1, \\ldots, c_n \\in k$ such that the elements $$ x_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in \\text{Frac}(A)(x_1, \\ldots, x_n) $$ are algebraically independent over $k$."}
+{"_id": "13847", "title": "more-morphisms-lemma-bertini-irreducible", "text": "\\begin{reference} \\cite[Theorem 6.3 part 4)]{Jou} \\end{reference} In Varieties, Situation \\ref{varieties-situation-family-divisors} assume \\begin{enumerate} \\item $X$ is of finite type over $k$, \\item $X$ is geometrically irreducible over $k$, \\item there exist $v_1, v_2, v_3 \\in V$ and an irreducible component $Z$ of $H_{v_2} \\cap H_{v_3}$ such that $Z \\not \\subset H_{v_1}$ and $\\text{codim}(Z, X) = 2$, and \\item every irreducible component $Y$ of $\\bigcap_{v \\in V} H_v$ has $\\text{codim}(Y, X) \\geq 2$. \\end{enumerate} Then for general $v \\in V \\otimes_k k'$ the scheme $H_v$ is geometrically irreducible over $k'$."}
+{"_id": "13848", "title": "more-morphisms-lemma-diagonal-picard-flat-proper", "text": "Let $f : X \\to S$ be a flat, proper morphism of finite presentation. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. For a morphism $g : T \\to S$ consider the base change diagram $$ \\xymatrix{ X_T \\ar[d]_p \\ar[r]_q & X \\ar[d]^f \\\\ T \\ar[r]^g & S } $$ Assume $\\mathcal{O}_T \\to p_*\\mathcal{O}_{X_T}$ is an isomorphism for all $g : T \\to S$. Then there exists an immersion $j : Z \\to S$ of finite presentation such that a morphism $g : T \\to S$ factors through $Z$ if and only if there exists a finite locally free $\\mathcal{O}_T$-module $\\mathcal{N}$ with $p^*\\mathcal{N} \\cong q^*\\mathcal{E}$."}
+{"_id": "13850", "title": "more-morphisms-lemma-triviality-generic-fibre-valuation-ring", "text": "Let $f : X \\to S$ be a proper flat morphism of finite presentation. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $S$ is the spectrum of a valuation ring, \\item $\\mathcal{L}$ is trivial on the generic fibre $X_\\eta$ of $f$, \\item the closed fibre $X_0$ of $f$ is integral, \\item $H^0(X_\\eta, \\mathcal{O}_{X_\\eta})$ is equal to the function field of $S$. \\end{enumerate} Then $\\mathcal{L}$ is trivial."}
+{"_id": "13851", "title": "more-morphisms-lemma-get-a-closed", "text": "Let $f : X \\to S$ and $\\mathcal{E}$ be as in Lemma \\ref{lemma-diagonal-picard-flat-proper} and in addition assume $\\mathcal{E}$ is an invertible $\\mathcal{O}_X$-module. If moreover the geometric fibres of $f$ are integral, then $Z$ is closed in $S$."}
+{"_id": "13852", "title": "more-morphisms-lemma-H1-O-picard-flat-proper", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X' \\ar[rr] \\ar[dr]_{f'} & & X \\ar[dl]^f \\\\ & S } $$ with $f' : X' \\to S$ and $f : X \\to S$ satisfying the hypotheses of Lemma \\ref{lemma-diagonal-picard-flat-proper}. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module and let $\\mathcal{L}'$ be the pullback to $X'$. Let $Z \\subset S$, resp.\\ $Z' \\subset S$ be the locally closed subscheme constructed in Lemma \\ref{lemma-diagonal-picard-flat-proper} for $(f, \\mathcal{L})$, resp.\\ $(f', \\mathcal{L}')$ so that $Z \\subset Z'$. If $s \\in Z$ and $$ H^1(X_s, \\mathcal{O}) \\longrightarrow H^1(X'_s, \\mathcal{O}) $$ is injective, then $Z \\cap U = Z' \\cap U$ for some open neighbourhood $U$ of $s$."}
+{"_id": "13853", "title": "more-morphisms-lemma-H1-O-multiple-picard-flat-proper", "text": "Consider $n$ commutative diagrams of schemes $$ \\xymatrix{ X_i \\ar[rr] \\ar[dr]_{f_i} & & X \\ar[dl]^f \\\\ & S } $$ with $f_i : X_i \\to S$ and $f : X \\to S$ satisfying the hypotheses of Lemma \\ref{lemma-diagonal-picard-flat-proper}. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module and let $\\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \\subset S$, resp.\\ $Z_i \\subset S$ be the locally closed subscheme constructed in Lemma \\ref{lemma-diagonal-picard-flat-proper} for $(f, \\mathcal{L})$, resp.\\ $(f_i, \\mathcal{L}_i)$ so that $Z \\subset \\bigcap_{i = 1, \\ldots, n} Z_i$. If $s \\in Z$ and $$ H^1(X_s, \\mathcal{O}) \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} H^1(X_{i, s}, \\mathcal{O}) $$ is injective, then $Z \\cap U = (\\bigcap_{i = 1, \\ldots, n} Z_i) \\cap U$ (scheme theoretic intersection) for some open neighbourhood $U$ of $s$."}
+{"_id": "13854", "title": "more-morphisms-lemma-pic-of-product", "text": "Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes satisfying the hypotheses of Lemma \\ref{lemma-diagonal-picard-flat-proper}. Let $\\sigma : S \\to X$ and $\\tau : S \\to Y$ be sections of $f$ and $g$. Let $s \\in S$. Let $\\mathcal{L}$ be an invertible sheaf on $X \\times_S Y$. If $(1 \\times \\tau)^*\\mathcal{L}$ on $X$, $(\\sigma \\times 1)^*\\mathcal{L}$ on $Y$, and $\\mathcal{L}|_{(X \\times_S Y)_s}$ are trivial, then there is an open neighbourhood $U$ of $s$ such that $\\mathcal{L}$ is trivial over $(X \\times_S Y)_U$."}
+{"_id": "13855", "title": "more-morphisms-lemma-Noetherian-approximation", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram $$ \\xymatrix{ X_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\ S_0 & S \\ar[l] } $$ such that \\begin{enumerate} \\item $X_0$, $S_0$ are affine schemes, \\item $S_0$ of finite type over $\\mathbf{Z}$, \\item $f_0$ is finite of finite type. \\end{enumerate}"}
+{"_id": "13856", "title": "more-morphisms-lemma-Noetherian-approximation-module", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that there exists a coherent $\\mathcal{O}_{X_0}$-module $\\mathcal{F}_0$ with $g^*\\mathcal{F}_0 = \\mathcal{F}$."}
+{"_id": "13857", "title": "more-morphisms-lemma-Noetherian-approximation-flat-module", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite presentation and flat over $S$. Then we may choose a diagram as in Lemma \\ref{lemma-Noetherian-approximation-module} and sheaf $\\mathcal{F}_0$ such that in addition $\\mathcal{F}_0$ is flat over $S_0$."}
+{"_id": "13858", "title": "more-morphisms-lemma-Noetherian-approximation-flat", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation and flat. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ is flat."}
+{"_id": "13860", "title": "more-morphisms-lemma-Noetherian-approximation-geometrically-reduced", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation with geometrically reduced fibres. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically reduced fibres."}
+{"_id": "13861", "title": "more-morphisms-lemma-Noetherian-approximation-geometrically-irreducible", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically irreducible fibres."}
+{"_id": "13863", "title": "more-morphisms-lemma-Noetherian-approximation-dimension-d", "text": "Let $d \\geq 0$ be an integer. Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension $d$. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition all fibres of $f_0$ have dimension $d$."}
+{"_id": "13865", "title": "more-morphisms-lemma-Noetherian-approximation-combine", "text": "(Noetherian approximation and combining properties.) Let $P$, $Q$ be properties of morphisms of schemes which are stable under base change. Let $f : X \\to S$ be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams $$ \\vcenter{ \\xymatrix{ X_1 \\ar[d]_{f_1} & X \\ar[l] \\ar[d]^f \\\\ S_1 & S \\ar[l] } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ X_2 \\ar[d]_{f_2} & X \\ar[l] \\ar[d]^f \\\\ S_2 & S \\ar[l] } } $$ of affine schemes, with $S_1$, $S_2$ of finite type over $\\mathbf{Z}$ and $f_1$, $f_2$ of finite type such that $f_1$ has property $P$ and $f_2$ has property $Q$. Then we can find a cartesian diagram $$ \\xymatrix{ X_0 \\ar[d]_{f_0} & X \\ar[l] \\ar[d]^f \\\\ S_0 & S \\ar[l] } $$ of affine schemes with $S_0$ of finite type over $\\mathbf{Z}$ and $f_0$ of finite type such that $f_0$ has both property $P$ and property $Q$."}
+{"_id": "13866", "title": "more-morphisms-lemma-realize-prescribed-residue-field-extension-etale", "text": "Let $S$ be a scheme. Let $s \\in S$. Let $\\kappa(s) \\subset k$ be a finite separable field extension. Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ such that the field extension $\\kappa(s) \\subset \\kappa(u)$ is isomorphic to $\\kappa(s) \\subset k$."}
+{"_id": "13867", "title": "more-morphisms-lemma-etale-neighbourhoods-not-quite-filtered", "text": "Let $S$ be a scheme, and let $s$ be a point of $S$. The category of \\'etale neighborhoods has the following properties: \\begin{enumerate} \\item Let $(U_i, u_i)_{i=1, 2}$ be two \\'etale neighborhoods of $s$ in $S$. Then there exists a third \\'etale neighborhood $(U, u)$ and morphisms $(U, u) \\to (U_i, u_i)$, $i = 1, 2$. \\item Let $h_1, h_2: (U, u) \\to (U', u')$ be two morphisms between \\'etale neighborhoods of $s$. Assume $h_1$, $h_2$ induce the same map $\\kappa(u') \\to \\kappa(u)$ of residue fields. Then there exist an \\'etale neighborhood $(U'', u'')$ and a morphism $h : (U'', u'') \\to (U, u)$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \\circ h = h_2 \\circ h$. \\end{enumerate}"}
+{"_id": "13868", "title": "more-morphisms-lemma-elementary-etale-neighbourhoods", "text": "Let $S$ be a scheme, and let $s$ be a point of $S$. The category of elementary \\'etale neighborhoods of $(S, s)$ is cofiltered (see Categories, Definition \\ref{categories-definition-codirected})."}
+{"_id": "13869", "title": "more-morphisms-lemma-describe-henselization", "text": "Let $S$ be a scheme. Let $s \\in S$. Then we have $$ \\mathcal{O}_{S, s}^h = \\colim_{(U, u)} \\mathcal{O}(U) $$ where the colimit is over the filtered category which is opposite to the category of elementary \\'etale neighbourhoods $(U, u)$ of $(S, s)$."}
+{"_id": "13870", "title": "more-morphisms-lemma-lift-etale-neighbourhood-fibre", "text": "\\begin{slogan} Lift \\'etale neighbourhood of point on fibre to total space. \\end{slogan} Let $X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$. Let $(V, v) \\to (X_s, x)$ be an \\'etale neighbourhood. Then there exists an \\'etale neighbourhood $(U, u) \\to (X, x)$ such that there exists a morphism $(U_s, u) \\to (V, v)$ of \\'etale neighbourhoods of $(X_s, x)$ which is an open immersion."}
+{"_id": "13871", "title": "more-morphisms-lemma-nr-minimal-primes", "text": "Let $R = \\colim R_i$ be colimit of a directed system of rings whose transition maps are faithfully flat. Then the number of minimal primes of $R$ taken as an element of $\\{0, 1, 2, \\ldots, \\infty\\}$ is the supremum of the numbers of minimal primes of the $R_i$."}
+{"_id": "13872", "title": "more-morphisms-lemma-nr-branches", "text": "Let $X$ be a scheme and $x \\in X$ a point. Then \\begin{enumerate} \\item the number of branches of $X$ at $x$ is equal to the supremum of the number of irreducible components of $U$ passing through $u$ taken over elementary \\'etale neighbourhoods $(U, u) \\to (X, x)$, \\item the number of geometric branches of $X$ at $x$ is equal to the supremum of the number of irreducible components of $U$ passing through $u$ taken over \\'etale neighbourhoods $(U, u) \\to (X, x)$, \\item $X$ is unibranch at $x$ if and only if for every elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there is exactly one irreducible component of $U$ passing through $u$, and \\item $X$ is geometrically unibranch at $x$ if and only if for every \\'etale neighbourhood $(U, u) \\to (X, x)$ there is exactly one irreducible component of $U$ passing through $u$. \\end{enumerate}"}
+{"_id": "13874", "title": "more-morphisms-lemma-number-of-branches-and-smooth", "text": "Let $X \\to S$ be a smooth morphism of schemes. Let $x \\in X$ with image $s \\in S$. Then \\begin{enumerate} \\item The number of geometric branches of $X$ at $x$ is equal to the number of geometric branches of $S$ at $s$. \\item If $\\kappa(x)/\\kappa(s)$ is a purely inseparable\\footnote{In fact, it would suffice if $\\kappa(x)$ is geometrically irreducible over $\\kappa(s)$. If we ever need this we will add a detailed proof.} extension of fields, then number of branches of $X$ at $x$ is equal to the number of branches of $S$ at $s$. \\end{enumerate}"}
+{"_id": "13875", "title": "more-morphisms-lemma-slice-smooth-given-element", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Let $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$. Assume \\begin{enumerate} \\item $f$ is smooth at $x$, and \\item the image $\\text{d}\\overline{h}$ of $\\text{d}h$ in $$ \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) $$ is nonzero. \\end{enumerate} Then there exists an affine open neighbourhood $U \\subset X$ of $x$ such that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and $D \\to S$ smooth."}
+{"_id": "13876", "title": "more-morphisms-lemma-slice-smooth-once", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Assume \\begin{enumerate} \\item $f$ is smooth at $x$, and \\item the map $$ \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) \\longrightarrow \\Omega_{\\kappa(x)/\\kappa(s)} $$ has a nonzero kernel. \\end{enumerate} Then there exists an affine open neighbourhood $U \\subset X$ of $x$ and an effective Cartier divisor $D \\subset U$ containing $x$ such that $D \\to S$ is smooth."}
+{"_id": "13878", "title": "more-morphisms-lemma-slice-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with image $s \\in S$. Assume \\begin{enumerate} \\item $f$ is smooth at $x$, and \\item $x$ is a closed point of $X_s$ and $\\kappa(s) \\subset \\kappa(x)$ is separable. \\end{enumerate} Then there exists an immersion $Z \\to X$ containing $x$ such that \\begin{enumerate} \\item $Z \\to S$ is \\'etale, and \\item $Z_s = \\{x\\}$ set theoretically. \\end{enumerate}"}
+{"_id": "13879", "title": "more-morphisms-lemma-etale-nbhd-dominates-smooth", "text": "Let $f : X \\to S$ be a smooth morphism of schemes. Let $s \\in S$ be a point in the image of $f$. Then there exists an \\'etale neighbourhood $(S', s') \\to (S, s)$ and a $S$-morphism $S' \\to X$."}
+{"_id": "13880", "title": "more-morphisms-lemma-etale-dominates-smooth", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be a smooth covering of $S$, see Topologies, Definition \\ref{topologies-definition-smooth-covering}. Then there exists an \\'etale covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$ (see Topologies, Definition \\ref{topologies-definition-etale-covering}) which refines (see Sites, Definition \\ref{sites-definition-morphism-coverings}) $\\mathcal{U}$."}
+{"_id": "13881", "title": "more-morphisms-lemma-cover-smooth-by-special", "text": "Let $f : X \\to S$ be a smooth morphism of schemes. Then there exists an \\'etale covering $\\{U_i \\to X\\}_{i \\in I}$ such that $U_i \\to S$ factors as $U_i \\to V_i \\to S$ where $V_i \\to S$ is \\'etale and $U_i \\to V_i$ is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres."}
+{"_id": "13882", "title": "more-morphisms-lemma-etale-local-structure", "text": "Let $S$ be a scheme. Let $Y \\to X$ be a closed immersion of schemes smooth over $S$. For every $y \\in Y$ there exist integers $0 \\leq m, n$ and a commutative diagram $$ \\xymatrix{ Y \\ar[d] & V \\ar[l] \\ar[d] \\ar[r] & \\mathbf{A}^m_S \\ar[d]^{(a_1, \\ldots, a_m) \\mapsto (a_1, \\ldots, a_m, 0 \\ldots, 0)} \\\\ X & U \\ar[l] \\ar[r]^-\\pi & \\mathbf{A}^{m + n}_S } $$ where $U \\subset X$ is open, $V = Y \\cap U$, $\\pi$ is \\'etale, $V = \\pi^{-1}(\\mathbf{A}^m_S)$, and $y \\in V$."}
+{"_id": "13883", "title": "more-morphisms-lemma-map-approximation", "text": "Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the same point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring. Assume further we are given a local $\\mathcal{O}_{S, s}$-algebra map $$ \\varphi : \\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}^\\wedge $$ For every $N \\geq 1$ there exists an elementary \\'etale neighbourhood $(U, u) \\to (X, x)$ and an $S$-morphism $f : U \\to Y$ mapping $u$ to $y$ such that the diagram $$ \\xymatrix{ \\mathcal{O}_{X, x}^\\wedge \\ar[r] & \\mathcal{O}_{U, u}^\\wedge \\\\ \\mathcal{O}_{Y, y} \\ar[r]^{f^\\sharp_u} \\ar[u]^\\varphi & \\mathcal{O}_{U, u} \\ar[u] } $$ commutes modulo $\\mathfrak m_u^N$."}
+{"_id": "13884", "title": "more-morphisms-lemma-isomorphism-approximation", "text": "Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the same point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring. Assume we have an $\\mathcal{O}_{S, s}$-algebra isomorphism $$ \\varphi : \\mathcal{O}_{Y, y}^\\wedge \\longrightarrow \\mathcal{O}_{X, x}^\\wedge $$ between the complete local rings. Then for every $N \\geq 1$ there exists morphisms $$ (X, x) \\leftarrow (U, u) \\rightarrow (Y, y) $$ of pointed schemes over $S$ such that both arrows define elementary \\'etale neighbourhoods and such that the diagram $$ \\xymatrix{ & \\mathcal{O}_{U, u}^\\wedge \\\\ \\mathcal{O}_{Y, y}^\\wedge \\ar[rr]^\\varphi \\ar[ru] & & \\mathcal{O}_{X, x}^\\wedge \\ar[lu] } $$ commutes modulo $\\mathfrak m_u^N$."}
+{"_id": "13885", "title": "more-morphisms-lemma-relative-map-approximation-pre", "text": "Let $X \\to S$, $Y \\to S$, $x$, $s$, $y$, $t$, $\\sigma$, $y_\\sigma$, and $\\varphi$ be given as follows: we have morphisms of schemes $$ \\vcenter{ \\xymatrix{ X \\ar[d] & Y \\ar[d] \\\\ S & T } } \\quad\\text{with points}\\quad \\vcenter{ \\xymatrix{ x \\ar[d] & y \\ar[d] \\\\ s & t } } $$ Here $S$ is locally Noetherian and $T$ is of finite type over $\\mathbf{Z}$. The morphisms $X \\to S$ and $Y \\to T$ are locally of finite type. The local ring $\\mathcal{O}_{S, s}$ is a G-ring. The map $$ \\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge $$ is a local homomorphism. Set $Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$. Next, $y_\\sigma$ is a point of $Y_\\sigma$ mapping to $y$ and the closed point of $\\Spec(\\mathcal{O}_{S, s}^\\wedge)$. Finally $$ \\varphi : \\mathcal{O}_{X, x}^\\wedge \\longrightarrow \\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge $$ is an isomorphism of $\\mathcal{O}_{S, s}^\\wedge$-algebras. In this situation there exists a commutative diagram $$ \\xymatrix{ X \\ar[d] & W \\ar[l] \\ar[rd] \\ar[rr] & & Y \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\ S & & V \\ar[ll] \\ar[rr]^\\tau & & T } $$ of schemes and points $w \\in W$, $v \\in V$ such that \\begin{enumerate} \\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood, \\item $(W, w) \\to (X, x)$ is an elementary \\'etale neighbourhood, and \\item $\\tau(v) = t$. \\end{enumerate} Let $y_\\tau \\in Y \\times_T V$ correspond to $y_\\sigma$ via the identification $(Y_\\sigma)_s = (Y \\times_T V)_v$. Then \\begin{enumerate} \\item[(4)] $(W, w) \\to (Y \\times_{T, \\tau} V, y_\\tau)$ is an elementary \\'etale neighbourhood. \\end{enumerate}"}
+{"_id": "13886", "title": "more-morphisms-lemma-relative-map-approximation", "text": "Consider a diagram $$ \\vcenter{ \\xymatrix{ X \\ar[d] & Y \\ar[d] \\\\ S & T \\ar[l] } } \\quad\\text{with points}\\quad \\vcenter{ \\xymatrix{ x \\ar[d] & y \\ar[d] \\\\ s & t \\ar[l] } } $$ where $S$ be a locally Noetherian scheme and the morphisms are locally of finite type. Assume $\\mathcal{O}_{S, s}$ is a G-ring. Assume further we are given a local $\\mathcal{O}_{S, s}$-algebra map $$ \\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge $$ and a local $\\mathcal{O}_{S, s}$-algebra map $$ \\varphi : \\mathcal{O}_{X, x} \\longrightarrow \\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge $$ where $Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$ and $y_\\sigma$ is the unique point of $Y_\\sigma$ lying over $y$. For every $N \\geq 1$ there exists a commutative diagram $$ \\xymatrix{ X \\ar[d] & X \\times_S V \\ar[l] \\ar[rd] & W \\ar[l]^-f \\ar[r] \\ar[d] & Y \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\ S & & V \\ar[ll] \\ar[rr]^\\tau & & T } $$ of schemes over $S$ and points $w \\in W$, $v \\in V$ such that \\begin{enumerate} \\item $v \\mapsto s$, $\\tau(v) = t$, $f(w) = (x, v)$, and $w \\mapsto (y, v)$, \\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood, \\item the diagram $$ \\xymatrix{ \\mathcal{O}_{S, s}^\\wedge \\ar[r] & \\mathcal{O}_{V, v}^\\wedge \\\\ \\mathcal{O}_{T, t} \\ar[r]^{\\tau^\\sharp_v} \\ar[u]_\\sigma & \\mathcal{O}_{V, v} \\ar[u] } $$ commutes module $\\mathfrak m_v^N$, \\item $(W, w) \\to (Y \\times_{T, \\tau} V, (y, v))$ is an elementary \\'etale neighbourhood, \\item the diagram $$ \\xymatrix{ \\mathcal{O}_{X, x} \\ar[r]_\\varphi & \\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge \\ar[r] & \\mathcal{O}_{Y_\\sigma, y_\\sigma}/\\mathfrak m_{y_\\sigma}^N \\ar@{=}[r] & \\mathcal{O}_{Y \\times_{T, \\tau} V, (y, v)}/\\mathfrak m_{(y, v)}^N \\ar[d]_{\\cong} \\\\ \\mathcal{O}_{X, x} \\ar[r] \\ar@{=}[u] & \\mathcal{O}_{X \\times_S V, (x, v)} \\ar[r]^{f^\\sharp_w} & \\mathcal{O}_{W, w} \\ar[r] & \\mathcal{O}_{W, w}/\\mathfrak m_w^N } $$ commutes. The equality comes from the fact that $Y_\\sigma$ and $Y \\times_{T, \\tau} V$ are canonically isomorphic over $\\mathcal{O}_{V, v}/\\mathfrak m_v^N = \\mathcal{O}_{S, s}/\\mathfrak m_s^N$ by parts (2) and (3). \\end{enumerate}"}
+{"_id": "13887", "title": "more-morphisms-lemma-control-agreement", "text": "Let $T \\to S$ be finite type morphisms of Noetherian schemes. Let $t \\in T$ map to $s \\in S$ and let $\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$ be a local $\\mathcal{O}_{S, s}$-algebra map. For every $N \\geq 1$ there exists a finite type morphism $(T', t') \\to (T, t)$ such that $\\sigma$ factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$ and such that for every local $\\mathcal{O}_{S, s}$-algebra map $\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$ which factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$ the maps $\\sigma$ and $\\sigma'$ agree modulo $\\mathfrak m_s^N$."}
+{"_id": "13888", "title": "more-morphisms-lemma-control-graded", "text": "Let $Y \\to T \\to S$ be finite type morphisms of Noetherian schemes. Let $t \\in T$ map to $s \\in S$ and let $\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$ be a local $\\mathcal{O}_{S, s}$-algebra map. There exists a finite type morphism $(T', t') \\to (T, t)$ such that $\\sigma$ factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$ and such that for every local $\\mathcal{O}_{S, s}$-algebra map $\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$ which factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$ the closed immersions $$ Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge) = Y_\\sigma \\longleftarrow Y_t \\longrightarrow Y_{\\sigma'} = Y \\times_{T, \\sigma'} \\Spec(\\mathcal{O}_{S, s}^\\wedge) $$ have isomorphic conormal algebras."}
+{"_id": "13890", "title": "more-morphisms-lemma-dominate-etale-neighbourhood-finite-flat", "text": "Let $S$ be a scheme. Let $s \\in S$. Let $f : (U, u) \\to (S, s)$ be an \\'etale neighbourhood. There exists an affine open neighbourhood $s \\in V \\subset S$ and a surjective, finite locally free morphism $\\pi : T \\to V$ such that for every $t \\in \\pi^{-1}(s)$ there exists an open neighbourhood $t \\in W_t \\subset T$ and a commutative diagram $$ \\xymatrix{ T \\ar[d]^\\pi & W_t \\ar[l] \\ar[rr]_{h_t} \\ar[rd] & & U \\ar[dl] \\\\ V \\ar[rr] & & S } $$ with $h_t(t) = u$."}
+{"_id": "13891", "title": "more-morphisms-lemma-dominate-etale-affine-finite-flat", "text": "Let $f : U \\to S$ be a surjective \\'etale morphism of affine schemes. There exists a surjective, finite locally free morphism $\\pi : T \\to S$ and a finite open covering $T = T_1 \\cup \\ldots \\cup T_n$ such that each $T_i \\to S$ factors through $U \\to S$. Diagram: $$ \\xymatrix{ & \\coprod T_i  \\ar[rd] \\ar[ld] & \\\\ T \\ar[rd]^\\pi & & U \\ar[ld]_f \\\\ & S & } $$ where the south-west arrow is a Zariski-covering."}
+{"_id": "13892", "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. Set $s = f(x)$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, and \\item $x \\in X_s$ is isolated\\footnote{In the presence of (1) this means that $f$ is quasi-finite at $x$, see Morphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.}. \\end{enumerate} Then there exist \\begin{enumerate} \\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$, \\item[(b)] an open subscheme $V \\subset X_U$ (see \\ref{equation-basic-diagram}) \\end{enumerate} such that \\begin{enumerate} \\item[(\\romannumeral1)] $V \\to U$ is a finite morphism, \\item[(\\romannumeral2)] there is a unique point $v$ of $V$ mapping to $u$ in $U$, and \\item[(\\romannumeral3)] the point $v$ maps to $x$ under the morphism $X_U \\to X$, inducing $\\kappa(x) = \\kappa(v)$. \\end{enumerate} Moreover, for any elementary \\'etale neighbourhood $(U', u') \\to (U, u)$ setting $V' = U' \\times_U V \\subset X_{U'}$ the triple $(U', u', V')$ satisfies the properties (\\romannumeral1), (\\romannumeral2), and (\\romannumeral3) as well."}
+{"_id": "13893", "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, and \\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$. \\end{enumerate} Then there exist \\begin{enumerate} \\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$, \\item[(b)] for each $i$ an open subscheme $V_i \\subset X_U$, \\end{enumerate} such that for each $i$ we have \\begin{enumerate} \\item[(\\romannumeral1)] $V_i \\to U$ is a finite morphism, \\item[(\\romannumeral2)] there is a unique point $v_i$ of $V_i$ mapping to $u$ in $U$, and \\item[(\\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and $\\kappa(x_i) = \\kappa(v_i)$. \\end{enumerate}"}
+{"_id": "13894", "title": "more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, and \\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$. \\end{enumerate} Then there exist \\begin{enumerate} \\item[(a)] an \\'etale neighbourhood $(U, u) \\to (S, s)$, \\item[(b)] for each $i$ an integer $m_i$ and open subschemes $V_{i, j} \\subset X_U$, $j = 1, \\ldots, m_i$ \\end{enumerate} such that we have \\begin{enumerate} \\item[(\\romannumeral1)] each $V_{i, j} \\to U$ is a finite morphism, \\item[(\\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\\kappa(u) \\subset \\kappa(v_{i, j})$ finite purely inseparable, \\item[(\\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and \\item[(\\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and no other points of $(X_U)_u$ map to $x_i$. \\end{enumerate}"}
+{"_id": "13895", "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is separated, and \\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$ and a decomposition $$ U \\times_S X = W \\amalg V_1 \\amalg \\ldots \\amalg V_n $$ into open and closed subschemes such that the morphisms $V_i \\to U$ are finite, the fibres of $V_i \\to U$ over $u$ are singletons $\\{v_i\\}$, each $v_i$ maps to $x_i$ with $\\kappa(x_i) = \\kappa(v_i)$, and the fibre of $W \\to U$ over $u$ contains no points mapping to any of the $x_i$."}
+{"_id": "13896", "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is separated, and \\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$. \\end{enumerate} Then there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$ and a decomposition $$ U \\times_S X = W \\amalg \\ \\coprod\\nolimits_{i = 1, \\ldots, n} \\ \\coprod\\nolimits_{j = 1, \\ldots, m_i} V_{i, j} $$ into open and closed subschemes such that the morphisms $V_{i, j} \\to U$ are finite, the fibres of $V_{i, j} \\to U$ over $u$ are singletons $\\{v_{i, j}\\}$, each $v_{i, j}$ maps to $x_i$, $\\kappa(u) \\subset \\kappa(v_{i, j})$ is purely inseparable, and the fibre of $W \\to U$ over $u$ contains no points mapping to any of the $x_i$."}
+{"_id": "13897", "title": "more-morphisms-lemma-etale-splits-off-quasi-finite-part", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is separated, and \\item $X_s$ has at most finitely many isolated points. \\end{enumerate} Then there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$ and a decomposition $$ U \\times_S X = W \\amalg V $$ into open and closed subschemes such that the morphism $V \\to U$ is finite, and the fibre $W_u$ of the morphism $W \\to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_u = \\emptyset$."}
+{"_id": "13898", "title": "more-morphisms-lemma-etale-makes-integral-split", "text": "Let $R \\to S$ be an integral ring map. Let $\\mathfrak p \\subset R$ be a prime ideal. Assume \\begin{enumerate} \\item there are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_n$ lying over $\\mathfrak p$, and \\item for each $i$ the maximal separable subextension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak q_i)_{sep}/\\kappa(\\mathfrak p)$ (Fields, Lemma \\ref{fields-lemma-separable-first}) is finite over $\\kappa(\\mathfrak p)$. \\end{enumerate} Then there exists an \\'etale ring map $R \\to R'$ and a prime $\\mathfrak p'$ lying over $\\mathfrak p$ such that $$ S \\otimes_R R' = A_1 \\times \\ldots \\times A_m $$ with $R' \\to A_j$ integral having a unique prime $\\mathfrak r_j$ over $\\mathfrak p'$ such that $\\kappa(\\mathfrak r_j)/\\kappa(\\mathfrak p')$ is purely inseparable."}
+{"_id": "13899", "title": "more-morphisms-lemma-finite-type-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is of finite type and separated. Let $S'$ be the normalization of $S$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\ & S & } $$ Then there exists an open subscheme $U' \\subset S'$ such that \\begin{enumerate} \\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and \\item $(f')^{-1}(U') \\subset X$ is the set of points at which $f$ is quasi-finite. \\end{enumerate}"}
+{"_id": "13900", "title": "more-morphisms-lemma-quasi-finite-separated-quasi-affine", "text": "\\begin{slogan} Quasi-finite, separated morphisms are quasi-affine \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated. Let $S'$ be the normalization of $S$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. Picture: $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\ & S & } $$ Then $f'$ is a quasi-compact open immersion and $\\nu$ is integral. In particular $f$ is quasi-affine."}
+{"_id": "13901", "title": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite", "text": "\\begin{reference} \\cite[IV Corollary 18.12.13]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated and assume that $S$ is quasi-compact and quasi-separated. Then there exists a factorization $$ \\xymatrix{ X \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\ & S & } $$ where $j$ is a quasi-compact open immersion and $\\pi$ is finite."}
+{"_id": "13903", "title": "more-morphisms-lemma-characterize-finite", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ is finite, \\item $f$ is proper with finite fibres, \\item $f$ is proper and locally quasi-finite, \\item $f$ is universally closed, separated, locally of finite type and has finite fibres. \\end{enumerate}"}
+{"_id": "13904", "title": "more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume that $f$ is proper and $f^{-1}(\\{s\\})$ is a finite set. Then there exists an open neighbourhood $V \\subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite."}
+{"_id": "13905", "title": "more-morphisms-lemma-flat-proper-family-cannot-collapse-fibre", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\ & S } $$ Let $s \\in S$. Assume \\begin{enumerate} \\item $X \\to S$ is a proper morphism, \\item $Y \\to S$ is separated and locally of finite type, and \\item the image of $X_s \\to Y_s$ is finite. \\end{enumerate} Then there is an open subspace $U \\subset S$ containing $s$ such that $X_U \\to Y_U$ factors through a closed subscheme $Z \\subset Y_U$ finite over $U$."}
+{"_id": "13906", "title": "more-morphisms-lemma-separated-locally-quasi-finite-over-affine", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism with $Y$ affine. Then every finite set of points of $X$ is contained in an open affine of $X$."}
+{"_id": "13907", "title": "more-morphisms-lemma-quasi-finite-finite-over-dense-open", "text": "Let $f : Y \\to X$ be a quasi-finite morphism. There exists a dense open $U \\subset X$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is finite."}
+{"_id": "13908", "title": "more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded", "text": "Let $f : X \\to S$ be flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. Then there exist closed subsets $$ \\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset \\ldots \\subset Z_n = S $$ such that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification $S = \\coprod_{r = 0, \\ldots, n} S_r$ is characterized by the following universal property: Given $g : T \\to S$ the projection $X \\times_S T \\to T$ is finite locally free of degree $r$ if and only if $g(T) \\subset S_r$ (set theoretically)."}
+{"_id": "13909", "title": "more-morphisms-lemma-stratify-flat-fp-qf", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat, locally of finite presentation, separated, and quasi-finite. Then there exist closed subsets $$ \\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset \\ldots \\subset S $$ such that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification $S = \\coprod S_r$ is characterized by the following universal property: Given a morphism $g : T \\to S$ the projection $X \\times_S T \\to T$ is finite locally free of degree $r$ if and only if $g(T) \\subset S_r$ (set theoretically). Moreover, the inclusion maps $S_r \\to S$ are quasi-compact."}
+{"_id": "13910", "title": "more-morphisms-lemma-stratify-flat-fp-lqf", "text": "Let $f : X \\to S$ be a flat, locally of finite presentation, separated, and locally quasi-finite morphism of schemes. Then there exist open subschemes $$ S = U_0 \\supset U_1 \\supset U_2 \\supset \\ldots $$ such that a morphism $\\Spec(k) \\to S$ factors through $U_d$ if and only if $X \\times_S \\Spec(k)$ has degree $\\geq d$ over $k$."}
+{"_id": "13911", "title": "more-morphisms-lemma-go-down-with-annihilators", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Let $g \\in \\Gamma(X, \\mathcal{O}_X)$ nonzero. Then there exist an open $V \\subset X$ such that $g|_V \\not = 0$, an open $U \\subset S$ fitting into a commutative diagram $$ \\xymatrix{ V \\ar[r] \\ar[d]_\\pi & X \\ar[d]^f \\\\ U \\ar[r] & S, } $$ a quasi-coherent subsheaf $\\mathcal{F} \\subset \\mathcal{O}_U$, an integer $r > 0$, and an injective $\\mathcal{O}_U$-module map $\\mathcal{F}^{\\oplus r} \\to \\pi_*\\mathcal{O}_V$ whose image contains $g|_V$."}
+{"_id": "13912", "title": "more-morphisms-lemma-there-is-a-scheme-integral-over", "text": "Let $U \\to X$ be a surjective \\'etale morphism of schemes. Assume $X$ is quasi-compact and quasi-separated. Then there exists a surjective integral morphism $Y \\to X$, such that for every $y \\in Y$ there is an open neighbourhood $V \\subset Y$ such that $V \\to X$ factors through $U$. In fact, we may assume $Y \\to X$ is finite and of finite presentation."}
+{"_id": "13913", "title": "more-morphisms-lemma-descent-connected-fibres", "text": "Consider a diagram of morphisms of schemes $$ \\xymatrix{ Z \\ar[r]_{\\sigma} \\ar[rd] & X \\ar[d] \\\\ & Y } $$ an a point $y \\in Y$. Assume \\begin{enumerate} \\item $X \\to Y$ is of finite presentation and flat, \\item $Z \\to Y$ is finite locally free, \\item $Z_y \\not = \\emptyset$, \\item all fibres of $X \\to Y$ are geometrically reduced, and \\item $X_y$ is geometrically connected over $\\kappa(y)$. \\end{enumerate} Then there exists a quasi-compact open $X^0 \\subset X$ such that $X^0_y = X_y$ and such that all nonempty fibres of $X^0 \\to Y$ are geometrically connected."}
+{"_id": "13914", "title": "more-morphisms-lemma-fibre-geometrically-connected-reduced", "text": "Let $h : Y \\to S$ be a morphism of schemes. Let $s \\in S$ be a point. Let $T \\subset Y_s$ be an open subscheme. Assume \\begin{enumerate} \\item $h$ is flat and of finite presentation, \\item all fibres of $h$ are geometrically reduced, and \\item $T$ is geometrically connected over $\\kappa(s)$. \\end{enumerate} Then we can find an affine elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a quasi-compact open $V \\subset Y_{S'}$ such that \\begin{enumerate} \\item[(a)] all fibres of $V \\to S'$ are geometrically connected, \\item[(b)] $V_{s'} = T \\times_s s'$. \\end{enumerate}"}
+{"_id": "13915", "title": "more-morphisms-lemma-cover-by-geometrically-connected", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite presentation and flat with geometrically reduced fibres. Then there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such that $X_i \\to S$ factors as $X_i \\to S_i \\to S$ where $S_i \\to S$ is \\'etale and $X_i \\to S_i$ is flat of finite presentation with geometrically connected and geometrically reduced fibres."}
+{"_id": "13916", "title": "more-morphisms-lemma-normal-morphism-irreducible", "text": "Let $h : Y \\to S$ be a morphism of schemes. Let $s \\in S$ be a point. Let $T \\subset Y_s$ be an open subscheme. Assume \\begin{enumerate} \\item $h$ is of finite presentation, \\item $h$ is normal, and \\item $T$ is geometrically irreducible over $\\kappa(s)$. \\end{enumerate} Then we can find an affine elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a quasi-compact open $V \\subset Y_{S'}$ such that \\begin{enumerate} \\item[(a)] all fibres of $V \\to S'$ are geometrically integral, \\item[(b)] $V_{s'} = T \\times_s s'$. \\end{enumerate}"}
+{"_id": "13917", "title": "more-morphisms-lemma-local-structure-finite-type", "text": "Let $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$. Assume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$. Then there exists a commutative diagram $$ \\xymatrix{ X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] & x' \\ar@{|->}[l]  \\ar@{|->}[d] \\\\ & Y \\ar[d]^h & & y \\ar@{|->}[d] \\\\ S \\ar@{=}[r] & S & s & s \\ar@{=}[l] } $$ and a point $x' \\in X'$ with $g(x') = x$ such that with $y = \\pi(x')$ we have \\begin{enumerate} \\item $h : Y \\to S$ is smooth of relative dimension $n$, \\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood, \\item $\\pi$ is finite, and $\\pi^{-1}(\\{y\\}) = \\{x'\\}$, and \\item $\\kappa(y)$ is a purely transcendental extension of $\\kappa(s)$. \\end{enumerate} Moreover, if $f$ is locally of finite presentation then $\\pi$ is of finite presentation."}
+{"_id": "13918", "title": "more-morphisms-lemma-local-local-structure-finite-type", "text": "\\begin{slogan} A morphism of finite type is, in \\'etale neighbourhoods, finite over a smooth morphism. \\end{slogan} Let $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$. Assume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$. Then there exists a commutative diagram $$ \\xymatrix{ X \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] & x' \\ar@{|->}[l] \\ar@{|->}[d] \\\\ & Y' \\ar[d]^h & & y' \\ar@{|->}[d] \\\\ S & S' \\ar[l]_e & s & s' \\ar@{|->}[l] } $$ and a point $x' \\in X'$ with $g(x') = x$ such that with $y' = \\pi(x')$, $s' = h(y')$ we have \\begin{enumerate} \\item $h : Y' \\to S'$ is smooth of relative dimension $n$, \\item all fibres of $Y' \\to S'$ are geometrically integral, \\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood, \\item $\\pi$ is finite, and $\\pi^{-1}(\\{y'\\}) = \\{x'\\}$, \\item $\\kappa(y')$ is a purely transcendental extension of $\\kappa(s')$, and \\item $e : (S', s') \\to (S, s)$ is an elementary \\'etale neighbourhood. \\end{enumerate} Moreover, if $f$ is locally of finite presentation, then $\\pi$ is of finite presentation."}
+{"_id": "13919", "title": "more-morphisms-lemma-local-local-structure-finite-type-affine", "text": "Assumption and notation as in Lemma \\ref{lemma-local-local-structure-finite-type}. In addition to properties (1) -- (6) we may also arrange it so that \\begin{enumerate} \\item[(7)] $S'$, $Y'$, $X'$ are affine. \\end{enumerate}"}
+{"_id": "13920", "title": "more-morphisms-lemma-finite-morphism-single-point-in-fibre", "text": "Let $\\pi : X \\to Y$ be a finite morphism. Let $x \\in X$ with $y = \\pi(x)$ such that $\\pi^{-1}(\\{y\\}) = \\{x\\}$. Then \\begin{enumerate} \\item For every neighbourhood $U \\subset X$ of $x$ in $X$, there exists a neighbourhood $V \\subset Y$ of $y$ such that $\\pi^{-1}(V) \\subset U$. \\item The ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is finite. \\item If $\\pi$ is of finite presentation, then $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is of finite presentation. \\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $\\mathcal{F}_x = \\pi_*\\mathcal{F}_y$ as $\\mathcal{O}_{Y, y}$-modules. \\end{enumerate}"}
+{"_id": "13921", "title": "more-morphisms-lemma-dominate-fppf-etale-locally", "text": "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering. Then there exist \\begin{enumerate} \\item an \\'etale covering $\\{S'_a \\to S\\}$, \\item surjective finite locally free morphisms $V_a \\to S'_a$, \\end{enumerate} such that the fppf covering $\\{V_a \\to S\\}$ refines the given covering $\\{S_i \\to S\\}$."}
+{"_id": "13922", "title": "more-morphisms-lemma-dominate-fppf", "text": "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering. Then there exist \\begin{enumerate} \\item a Zariski open covering $S = \\bigcup U_j$, \\item surjective finite locally free morphisms $W_j \\to U_j$, \\item Zariski open coverings $W_j = \\bigcup_k W_{j, k}$, \\item surjective finite locally free morphisms $T_{j, k} \\to W_{j, k}$ \\end{enumerate} such that the fppf covering $\\{T_{j, k} \\to S\\}$ refines the given covering $\\{S_i \\to S\\}$."}
+{"_id": "13923", "title": "more-morphisms-lemma-extend-integral-surjective-morphisms", "text": "Let $S$ be a scheme. If $U \\subset S$ is open and $V \\to U$ is a surjective integral morphism, then there exists a surjective integral morphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$ isomorphic to $V$ as schemes over $U$."}
+{"_id": "13924", "title": "more-morphisms-lemma-extend-finite-surjective-morphisms", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. If $U \\subset S$ is a quasi-compact open and $V \\to U$ is a surjective finite morphism, then there exists a surjective finite morphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$ isomorphic to $V$ as schemes over $U$."}
+{"_id": "13925", "title": "more-morphisms-lemma-dominate-fppf-integral", "text": "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering. Then there exists a surjective integral morphism $S' \\to S$ and an open covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the morphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$."}
+{"_id": "13926", "title": "more-morphisms-lemma-dominate-fppf-finite", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering. Then there exists a surjective finite morphism $S' \\to S$ of finite presentation and an open covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the morphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$."}
+{"_id": "13927", "title": "more-morphisms-lemma-fppf-ph", "text": "An fppf covering of schemes is a ph covering."}
+{"_id": "13928", "title": "more-morphisms-lemma-quasi-projective", "text": "Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $X \\to S$ is quasi-projective, \\item $X \\to S$ is H-quasi-projective, \\item there exists a quasi-compact open immersion $X \\to X'$ of schemes over $S$ with $X' \\to S$ projective, \\item $X \\to S$ is of finite type and $X$ has an ample invertible sheaf, and \\item $X \\to S$ is of finite type and there exists an $f$-very ample invertible sheaf. \\end{enumerate}"}
+{"_id": "13931", "title": "more-morphisms-lemma-projective", "text": "Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $X \\to S$ is projective, \\item $X \\to S$ is H-projective, \\item $X \\to S$ is quasi-projective and proper, \\item $X \\to S$ is H-quasi-projective and proper, \\item $X \\to S$ is proper and $X$ has an ample invertible sheaf, \\item $X \\to S$ is proper and there exists an $f$-ample invertible sheaf, \\item $X \\to S$ is proper and there exists an $f$-very ample invertible sheaf, \\item there is a quasi-coherent graded $\\mathcal{O}_S$-algebra $\\mathcal{A}$ generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$ with $\\mathcal{A}_1$ a finite type $\\mathcal{O}_S$-module such that $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$.  \\end{enumerate}"}
+{"_id": "13932", "title": "more-morphisms-lemma-category-projective", "text": "Let $S$ be a scheme which has an ample invertible sheaf. Let $\\text{P}_S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-projective}. \\begin{enumerate} \\item if $S' \\to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\\text{P}_S \\to \\text{P}_{S'}$, \\item if $X \\in \\text{P}_S$ and $Y \\in \\text{P}_X$, then $Y \\in \\text{P}_S$, \\item the category $\\text{P}_S$ is closed under fibre products, \\item the category $\\text{P}_S$ is closed under finite disjoint unions, \\item if $X \\to S$ is finite, then $X$ is in $\\text{P}_S$, \\item add more here. \\end{enumerate}"}
+{"_id": "13933", "title": "more-morphisms-lemma-ample-in-neighbourhood", "text": "\\begin{reference} \\cite[IV Corollary 9.6.4]{EGA} \\end{reference} Let $f : X \\to Y$ be a proper morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \\subset Y$ of $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$."}
+{"_id": "13934", "title": "more-morphisms-lemma-apply-proj-spec", "text": "Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_P$-module. Set $A = \\bigoplus_{m \\geq 0} \\Gamma(P, \\mathcal{L}^{\\otimes m})$. Then $P = \\text{Proj}(A)$ and diagram (\\ref{equation-proj-and-spec}) becomes the diagram $$ \\xymatrix{ \\underline{\\Spec}_P \\left( \\bigoplus\\nolimits_{m \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes m} \\right) \\ar[r] \\ar@{=}[d] & L = \\underline{\\Spec}_P \\left( \\bigoplus\\nolimits_{m \\geq 0} \\mathcal{L}^{\\otimes m} \\right) \\ar[d]^\\sigma \\ar[r]_-\\pi & P \\ar[d] \\\\ U \\ar[r] & X \\ar[r] & Z } $$ having the properties explained above."}
+{"_id": "13935", "title": "more-morphisms-lemma-locally-principal-vertical", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $Z \\subset X$ be a closed subscheme. Let $s \\in S$. Assume \\begin{enumerate} \\item $S$ is irreducible with generic point $\\eta$, \\item $X$ is irreducible, \\item $f$ is dominant, \\item $f$ is locally of finite type, \\item $\\dim(X_s) \\leq \\dim(X_\\eta)$, \\item $Z$ is locally principal in $X$, and \\item $Z_\\eta = \\emptyset$. \\end{enumerate} Then the fibre $Z_s$ is (set theoretically) a union of irreducible components of $X_s$."}
+{"_id": "13938", "title": "more-morphisms-lemma-closed-point-nearby-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\leadsto x'$ be a specialization of points in $X$. Set $s = f(x)$ and $s' = f(x')$. Assume \\begin{enumerate} \\item $x'$ is a closed point of $X_{s'}$, and \\item $f$ is locally of finite type. \\end{enumerate} Then the set $$ \\{x_1 \\in X \\text{ such that } f(x_1) = s \\text{ and } x_1\\text{ is closed in }X_s \\text{ and } x \\leadsto x_1 \\leadsto x' \\} $$ is dense in the closure of $x$ in $X_s$."}
+{"_id": "13940", "title": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$. Let $U \\subset X$ be an open subscheme. Assume $f$ locally of finite type, $S$ locally Noetherian, $x$ a closed point of $X_s$, and assume there exists a point $x' \\in U$ with $x' \\leadsto x$ and $f(x') \\not = s$. Then there exists a closed subscheme $Z \\subset X$ such that (a) $x \\in Z$, (b) $f|_Z : Z \\to S$ is quasi-finite at $x$, and (c) there exists a $z \\in Z$, $z \\in U$, $z \\leadsto x$ and $f(z) \\not = s$."}
+{"_id": "13942", "title": "more-morphisms-lemma-stein-universally-closed", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a universally closed and quasi-separated morphism. There exists a factorization $$ \\xymatrix{ X \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\ & S & } $$ with the following properties: \\begin{enumerate} \\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated, and surjective, \\item the morphism $\\pi : S' \\to S$ is integral, \\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$, \\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and \\item $S'$ is the normalization of $S$ in $X$, see Morphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}. \\end{enumerate} Formation of the factorization $f = \\pi \\circ f'$ commutes with flat base change."}
+{"_id": "13943", "title": "more-morphisms-lemma-stein-universally-closed-residue-fields", "text": "In Lemma \\ref{lemma-stein-universally-closed} assume in addition that $f$ is locally of finite type. Then for $s \\in S$ the fibre $\\pi^{-1}(\\{s\\}) = \\{s_1, \\ldots, s_n\\}$ is finite and the field extensions $\\kappa(s_i)/\\kappa(s)$ are finite."}
+{"_id": "13944", "title": "more-morphisms-lemma-characterize-geometrically-connected-fibres", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Then $X_s$ is geometrically connected, if and only if for every \\'etale neighbourhood $(U, u) \\to (S, s)$ the base change $X_U \\to U$ has connected fibre $X_u$."}
+{"_id": "13945", "title": "more-morphisms-lemma-geometrically-connected-fibres-towards-normal", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is proper, \\item $S$ is integral with generic point $\\xi$, \\item $S$ is normal, \\item $X$ is reduced, \\item every generic point of an irreducible component of $X$ maps to $\\xi$, \\item we have $H^0(X_\\xi, \\mathcal{O}) = \\kappa(\\xi)$. \\end{enumerate} Then $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and $f$ has geometrically connected fibres."}
+{"_id": "13946", "title": "more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous", "text": "Let $X \\to S$ be a flat proper morphism of finite presentation. Let $n_{X/S}$ be the function on $S$ counting the numbers of geometric connected components of fibres of $f$ introduced in Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}. Then $n_{X/S}$ is lower semi-continuous."}
+{"_id": "13948", "title": "more-morphisms-lemma-split-off-proper-part-henselian", "text": "\\begin{reference} A reference for the case of an adic Noetherian base is \\cite[III, Proposition 5.5.1]{EGA} \\end{reference} Let $(A, I)$ be a henselian pair. Let $X \\to \\Spec(A)$ be separated and of finite type. Set $X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$. Let $Y \\subset X_0$ be an open and closed subscheme such that $Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed subscheme $W \\subset X$ which is proper over $A$ with $W \\times_{\\Spec(A)} \\Spec(A/I) = Y$."}
+{"_id": "13949", "title": "more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fppf covering, see Topologies, Definition \\ref{topologies-definition-fppf-covering}. Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$. If each morphism $V_i \\to X_i$ is separated and locally quasi-finite, then the descent datum is effective."}
+{"_id": "13958", "title": "more-morphisms-lemma-relatively-pseudo-coherent", "text": "Let $X \\to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item for some closed immersion $i : X \\to \\mathbf{A}^n_S$ the object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$ is $m$-pseudo-coherent, and \\item for all closed immersions $i : X \\to \\mathbf{A}^n_S$ the object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$ is $m$-pseudo-coherent. \\end{enumerate}"}
+{"_id": "13960", "title": "more-morphisms-lemma-localize-relative-pseudo-coherent", "text": "Let $S$ be an affine scheme. Let $V \\subset S$ be a standard open. Let $X \\to V$ be a finite type morphism of affine schemes. Let $U \\subset X$ be an affine open. Let $E$ be an object of $D(\\mathcal{O}_X)$. If the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(X \\to V, E)$, then the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U \\to S, E|_U)$."}
+{"_id": "13961", "title": "more-morphisms-lemma-glue-relative-pseudo-coherent", "text": "Let $X \\to S$ be a finite type morphism of affine schemes. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Let $X = \\bigcup U_i$ be a standard affine open covering. The following are equivalent \\begin{enumerate} \\item the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} hold for the pairs $(U_i \\to S, E|_{U_i})$, \\item the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} hold for the pair $(X \\to S, E)$. \\end{enumerate}"}
+{"_id": "13962", "title": "more-morphisms-lemma-relative-pseudo-coherence-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\\mathcal{O}_X)$. Fix $m \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent relative to $S$, \\item for every affine opens $U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$ the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U \\to V, E|_U)$. \\end{enumerate} Moreover, if this is true, then for every open subschemes $U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$ the restriction $E|_U$ is $m$-pseudo-coherent relative to $V$."}
+{"_id": "13963", "title": "more-morphisms-lemma-qcoh-relative-pseudo-coherence-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Fix $m \\in \\mathbf{Z}$. The following are equivalent \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent relative to $S$, \\item there exists an affine open covering $S = \\bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$ such that the complex of $\\mathcal{O}_X(U_{ij})$-modules $R\\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to $\\mathcal{O}_S(V_i)$, and \\item for every affine opens $U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$ the complex of $\\mathcal{O}_X(U)$-modules $R\\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\\mathcal{O}_S(V)$. \\end{enumerate}"}
+{"_id": "13965", "title": "more-morphisms-lemma-finite-morphism-relative-pseudo-coherence", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $R\\pi_*E$ is $m$-pseudo-coherent relative to $S$."}
+{"_id": "13966", "title": "more-morphisms-lemma-cone-relatively-pseudo-coherent", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $(E, E', E'')$ be a distinguished triangle of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is $m$-pseudo-coherent relative to $S$. \\item If $E, E''$ are $m$-pseudo-coherent relative to $S$, then $E'$ is $m$-pseudo-coherent relative to $S$. \\item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E''$ is $m$-pseudo-coherent relative to $S$, then $E$ is $(m + 1)$-pseudo-coherent relative to $S$. \\end{enumerate} Moreover, if two out of three of $E, E', E''$ are pseudo-coherent relative to $S$, the so is the third."}
+{"_id": "13968", "title": "more-morphisms-lemma-summands-relative-pseudo-coherent", "text": "Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $m \\in \\mathbf{Z}$. Let $E, K$ be objects of $D(\\mathcal{O}_X)$. If $E \\oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$."}
+{"_id": "13971", "title": "more-morphisms-lemma-base-change-relative-pseudo-coherent", "text": "Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$ which is $m$-pseudo-coherent relative to $S$. Let $S' \\to S$ be a morphism of schemes. Set $X' = X \\times_S S'$ and denote $E'$ the derived pullback of $E$ to $X'$. If $S'$ and $X$ are Tor independent over $S$, then $E'$ is $m$-pseudo-coherent relative to $S'$."}
+{"_id": "13972", "title": "more-morphisms-lemma-pull-relative-pseudo-coherent", "text": "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_Y)$. Assume \\begin{enumerate} \\item $\\mathcal{O}_X$ is pseudo-coherent relative to $Y$\\footnote{This means $f$ is pseudo-coherent, see Definition \\ref{definition-pseudo-coherent}.}, and \\item $E$ is $m$-pseudo-coherent relative to $S$. \\end{enumerate} Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$."}
+{"_id": "13973", "title": "more-morphisms-lemma-composition-relative-pseudo-coherent", "text": "Let $f : X \\to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$. Assume $\\mathcal{O}_Y$ is pseudo-coherent relative to $S$\\footnote{This means $Y \\to S$ is pseudo-coherent, see Definition \\ref{definition-pseudo-coherent}.}. Then the following are equivalent \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent relative to $Y$, and \\item $E$ is $m$-pseudo-coherent relative to $S$. \\end{enumerate}"}
+{"_id": "13974", "title": "more-morphisms-lemma-check-relative-pseudo-coherence-on-charts", "text": "Let $$ \\xymatrix{ X \\ar[rd] \\ar[rr]_i & & P \\ar[ld] \\\\ & S } $$ be a commutative diagram of schemes. Assume $i$ is a closed immersion and $P \\to S$ flat and locally of finite presentation. Let $E$ be an object of $D(\\mathcal{O}_X)$. Then the following are equivalent \\begin{enumerate} \\item $E$ is $m$-pseudo-coherent relative to $S$, \\item $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and \\item $Ri_*E$ is $m$-pseudo-coherent on $P$. \\end{enumerate}"}
+{"_id": "13975", "title": "more-morphisms-lemma-pseudo-coherent", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a pseudo-coherent ring map, \\item for every pair of affine opens $U \\subset X$, $V \\subset S$ such that $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is pseudo-coherent, and \\item $f$ is locally of finite type and $\\mathcal{O}_X$ is pseudo-coherent relative to $S$. \\end{enumerate}"}
+{"_id": "13976", "title": "more-morphisms-lemma-flat-base-change-pseudo-coherent", "text": "A flat base change of a pseudo-coherent morphism is pseudo-coherent."}
+{"_id": "13977", "title": "more-morphisms-lemma-composition-pseudo-coherent", "text": "A composition of pseudo-coherent morphisms of schemes is pseudo-coherent."}
+{"_id": "13978", "title": "more-morphisms-lemma-pseudo-coherent-finite-presentation", "text": "A pseudo-coherent morphism is locally of finite presentation."}
+{"_id": "13979", "title": "more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "text": "A flat morphism which is locally of finite presentation is pseudo-coherent."}
+{"_id": "13980", "title": "more-morphisms-lemma-permanence-pseudo-coherent", "text": "Let $f : X \\to Y$ be a morphism of schemes pseudo-coherent over a base scheme $S$. Then $f$ is pseudo-coherent."}
+{"_id": "13981", "title": "more-morphisms-lemma-finite-pseudo-coherent", "text": "Let $f : X \\to S$ be a finite morphism of schemes. Then $f$ is pseudo-coherent if and only if $f_*\\mathcal{O}_X$ is pseudo-coherent as an $\\mathcal{O}_S$-module."}
+{"_id": "13982", "title": "more-morphisms-lemma-Noetherian-pseudo-coherent", "text": "Let $f : X \\to S$ be a morphism of schemes. If $S$ is locally Noetherian, then $f$ is pseudo-coherent if and only if $f$ is locally of finite type."}
+{"_id": "13984", "title": "more-morphisms-lemma-quotient-of-flat-finitely-presented", "text": "Let $A \\to B$ be a flat ring map of finite presentation. Let $I \\subset B$ be an ideal. Then $A \\to B/I$ is pseudo-coherent if and only if $I$ is pseudo-coherent as a $B$-module."}
+{"_id": "13986", "title": "more-morphisms-lemma-pseudo-coherent-fppf-local-source", "text": "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is fppf local on the source."}
+{"_id": "13987", "title": "more-morphisms-lemma-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. The following are equivalent \\begin{enumerate} \\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a perfect ring map, and \\item for every pair of affine opens $U \\subset X$, $V \\subset S$ such that $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is perfect. \\end{enumerate}"}
+{"_id": "13988", "title": "more-morphisms-lemma-flat-base-change-perfect", "text": "A flat base change of a perfect morphism is perfect."}
+{"_id": "13989", "title": "more-morphisms-lemma-composition-perfect", "text": "A composition of perfect morphisms of schemes is perfect."}
+{"_id": "13990", "title": "more-morphisms-lemma-flat-finite-presentation-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is flat and perfect, and \\item $f$ is flat and locally of finite presentation. \\end{enumerate}"}
+{"_id": "13991", "title": "more-morphisms-lemma-regular-target-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is regular and $f$ is locally of finite type. Then $f$ is perfect."}
+{"_id": "13992", "title": "more-morphisms-lemma-regular-immersion-perfect", "text": "A regular immersion of schemes is perfect. A Koszul-regular immersion of schemes is perfect."}
+{"_id": "13993", "title": "more-morphisms-lemma-perfect-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $Y \\to S$ smooth and $X \\to S$ perfect. Then $f : X \\to Y$ is perfect."}
+{"_id": "13995", "title": "more-morphisms-lemma-check-perfect-stalks", "text": "Let $f : X \\to S$ be a pseudo-coherent morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is perfect, \\item $\\mathcal{O}_X$ locally has finite tor dimension as a sheaf of $f^{-1}\\mathcal{O}_S$-modules, and \\item for all $x \\in X$ the ring $\\mathcal{O}_{X, x}$ has finite tor dimension as an $\\mathcal{O}_{S, f(x)}$-module. \\end{enumerate}"}
+{"_id": "13997", "title": "more-morphisms-lemma-perfect-proper-perfect-direct-image", "text": "Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a perfect proper morphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$."}
+{"_id": "14001", "title": "more-morphisms-lemma-lci", "text": "Let $f : X \\to S$ be a local complete intersection morphism. Let $P$ be a scheme smooth over $S$. Let $U \\subset X$ be an open subscheme and $i : U \\to P$ an immersion of schemes over $S$. Then $i$ is a Koszul-regular immersion."}
+{"_id": "14002", "title": "more-morphisms-lemma-lci-properties", "text": "Let $f : X \\to S$ be a local complete intersection morphism. Then \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is pseudo-coherent, and \\item $f$ is perfect. \\end{enumerate}"}
+{"_id": "14003", "title": "more-morphisms-lemma-affine-lci", "text": "Let $f : X = \\Spec(B) \\to S = \\Spec(A)$ be a morphism of affine schemes. Then $f$ is a local complete intersection morphism if and only if $A \\to B$ is a local complete intersection homomorphism, see More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}."}
+{"_id": "14004", "title": "more-morphisms-lemma-flat-base-change-lci", "text": "A flat base change of a local complete intersection morphism is a local complete intersection morphism."}
+{"_id": "14005", "title": "more-morphisms-lemma-composition-lci", "text": "A composition of local complete intersection morphisms is a local complete intersection morphism."}
+{"_id": "14006", "title": "more-morphisms-lemma-flat-lci", "text": "\\begin{slogan} A morphism is flat and lci if and only if it is syntomic. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is flat and a local complete intersection morphism, and \\item $f$ is syntomic. \\end{enumerate}"}
+{"_id": "14007", "title": "more-morphisms-lemma-regular-immersion-lci", "text": "A regular immersion of schemes is a local complete intersection morphism. A Koszul-regular immersion of schemes is a local complete intersection morphism."}
+{"_id": "14008", "title": "more-morphisms-lemma-lci-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $Y \\to S$ smooth and $X \\to S$ is a local complete intersection morphism. Then $f : X \\to Y$ is a local complete intersection morphism."}
+{"_id": "14009", "title": "more-morphisms-lemma-morphism-regular-schemes-is-lci", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally of finite type and $X$ and $Y$ are regular, then $f$ is a local complete intersection morphism."}
+{"_id": "14010", "title": "more-morphisms-lemma-lci-avramov", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume \\begin{enumerate} \\item $S$ is locally Noetherian, \\item $Y \\to S$ is locally of finite type, \\item $f : X \\to Y$ is perfect, \\item $X \\to S$ is a local complete intersection morphism. \\end{enumerate} Then $X \\to Y$ is a local complete intersection morphism and $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$."}
+{"_id": "14015", "title": "more-morphisms-lemma-flat-fp-NL-lci", "text": "Let $f : X \\to Y$ be a flat morphism of finite presentation. The following are equivalent \\begin{enumerate} \\item $f$ is a local complete intersection morphism, \\item $f$ is syntomic, \\item $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \\item $\\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$. \\end{enumerate}"}
+{"_id": "14016", "title": "more-morphisms-lemma-smooth-diagonal-perfect", "text": "Let $f : X \\to Y$ be a finite type morphism of locally Noetherian schemes. Denote $\\Delta : X \\to X \\times_Y X$ the diagonal morphism. The following are equivalent \\begin{enumerate} \\item $f$ is smooth, \\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a regular immersion, \\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a local complete intersection morphism, \\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is perfect. \\end{enumerate}"}
+{"_id": "14017", "title": "more-morphisms-lemma-descending-property-lci", "text": "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is fpqc local on the base."}
+{"_id": "14018", "title": "more-morphisms-lemma-lci-syntomic-local-source", "text": "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is syntomic local on the source."}
+{"_id": "14019", "title": "more-morphisms-lemma-base-change-lci-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Assume both $X$ and $Y$ are flat and locally of finite presentation over $S$. Then the set $$ \\{x \\in X \\mid f\\text{ Koszul at }x\\}. $$ is open in $X$ and its formation commutes with arbitrary base change $S' \\to S$."}
+{"_id": "14020", "title": "more-morphisms-lemma-unramified-lci", "text": "Let $f : X \\to Y$ be a local complete intersection morphism of schemes. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free on $X$."}
+{"_id": "14021", "title": "more-morphisms-lemma-transitivity-conormal-lci", "text": "Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes. Assume that $Z \\to Y$ is a local complete intersection morphism. The exact sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ of Lemma \\ref{lemma-transitivity-conormal} is short exact."}
+{"_id": "14022", "title": "more-morphisms-lemma-check-weakly-etale-stalks", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $X \\to Y$ is weakly \\'etale, and \\item for every $x \\in X$ the ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "14023", "title": "more-morphisms-lemma-key", "text": "Let $X \\to Y$ be a morphism of schemes such that $X \\to X \\times_Y X$ is flat. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. If $\\mathcal{F}$ is flat over $Y$, then $\\mathcal{F}$ is flat over $X$."}
+{"_id": "14024", "title": "more-morphisms-lemma-weakly-etale-characterize", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is weakly \\'etale. \\item For every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is weakly \\'etale. \\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$ and open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$ is weakly \\'etale. \\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$ and affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such that the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is of weakly \\'etale, for all $j\\in J, i\\in I_j$. \\end{enumerate} Moreover, if $f$ is weakly \\'etale then for any open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the restriction $f|_U : U \\to V$ is weakly-\\'etale."}
+{"_id": "14025", "title": "more-morphisms-lemma-composition-weakly-etale", "text": "Let $X \\to Y \\to Z$ be morphisms of schemes. \\begin{enumerate} \\item If $X \\to X \\times_Y X$ and $Y \\to Y \\times_Z Y$ are flat, then $X \\to X \\times_Z X$ is flat. \\item If $X \\to Y$ and $Y \\to Z$ are weakly \\'etale, then $X \\to Z$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "14026", "title": "more-morphisms-lemma-base-change-weakly-etale", "text": "Let $X \\to Y$ and $Y' \\to Y$ be morphisms of schemes and let $X' = Y' \\times_Y X$ be the base change of $X$. \\begin{enumerate} \\item If $X \\to X \\times_Y X$ is flat, then $X' \\to X' \\times_{Y'} X'$ is flat. \\item If $X \\to Y$ is weakly \\'etale, then $X' \\to Y'$ is weakly \\'etale. \\end{enumerate}"}
+{"_id": "14029", "title": "more-morphisms-lemma-when-weakly-etale", "text": "Let $f : X \\to Y$ be a morphism of schemes. Then $X \\to Y$ is weakly \\'etale in each of the following cases \\begin{enumerate} \\item $X \\to Y$ is a flat monomorphism, \\item $X \\to Y$ is an open immersion, \\item $X \\to Y$ is flat and unramified, \\item $X \\to Y$ is \\'etale. \\end{enumerate}"}
+{"_id": "14031", "title": "more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is weakly \\'etale, and \\item for $x \\in X$ the local ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ induces an isomorphism on strict henselizations. \\end{enumerate}"}
+{"_id": "14033", "title": "more-morphisms-lemma-weakly-etale-permanence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. If $X$, $Y$ are weakly \\'etale over $S$, then $f$ is weakly \\'etale."}
+{"_id": "14034", "title": "more-morphisms-lemma-weakly-etale-universal-homeomorphism", "text": "Let $f : X \\to Y$ be a morphism of schemes. If $f$ is weakly \\'etale and a universal homeomorphism, it is an isomorphism."}
+{"_id": "14036", "title": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve", "text": "Let $f : X \\to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Then there exists a finite extension $L/K$ such that in the diagram $$ \\xymatrix{ Y \\ar[rd]_g \\ar[r]_-\\nu & X \\times_S T \\ar[d] \\ar[r] & X \\ar[d]_f \\\\ & T \\ar[r] & S } $$ the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\\Spec(L)$ and $\\nu : Y \\to X \\times_S T$ is the normalization."}
+{"_id": "14037", "title": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-separable", "text": "Let $A$ be a Dedekind ring with fraction field $K$. Let $X$ be a scheme flat and of finite type over $A$. Assume $A$ is a Nagata ring and that for every generic point $\\eta$ of an irreducible component of $X$ the field extension $K \\subset \\kappa(\\eta)$ is separable. Then there exists a finite separable extension $K \\subset L$ such that the normalized base change $Y$ is smooth over $\\Spec(B)$ at all generic points of all fibres."}
+{"_id": "14039", "title": "more-morphisms-lemma-ind-quasi-affine-alternative-definition", "text": "For a morphism of schemes $f : X \\to Y$, the following are equivalent: \\begin{enumerate} \\item $f$ is ind-quasi-affine, \\item for every affine open subscheme $V \\subset Y$ and every quasi-compact open subscheme $U \\subset f^{-1}(V)$, the induced morphism $U \\to V$ is quasi-affine.  \\item for some cover $\\{ V_j \\}_{j \\in J}$ of $Y$ by quasi-compact and quasi-separated open subschemes $V_j \\subset Y$, every $j \\in J$, and every quasi-compact open subscheme $U \\subset f^{-1}(V_j)$, the induced morphism $U \\to V_j$ is quasi-affine. \\item for every quasi-compact and quasi-separated open subscheme $V \\subset Y$ and every quasi-compact open subscheme $U \\subset f^{-1}(V)$, the induced morphism $U \\to V$ is quasi-affine. \\end{enumerate} In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base."}
+{"_id": "14040", "title": "more-morphisms-lemma-ind-quasi-affine-composition", "text": "The property of being an ind-quasi-affine morphism is stable under composition."}
+{"_id": "14041", "title": "more-morphisms-lemma-ind-quasi-affine-examples", "text": "Any quasi-affine morphism is ind-quasi-affine. Any immersion is ind-quasi-affine."}
+{"_id": "14042", "title": "more-morphisms-lemma-ind-quasi-affine-permanence", "text": "If $f : X \\to Y$ and $g : Y \\to Z$ are morphisms of schemes such that $g \\circ f$ is ind-quasi-affine, then $f$ is ind-quasi-affine."}
+{"_id": "14043", "title": "more-morphisms-lemma-base-change-ind-quasi-affine", "text": "The property of being ind-quasi-affine is stable under base change."}
+{"_id": "14045", "title": "more-morphisms-lemma-etale-separated-ind-quasi-affine", "text": "A separated locally quasi-finite morphism of schemes is ind-quasi-affine."}
+{"_id": "14046", "title": "more-morphisms-lemma-prepare-pushout-along-closed-immersion-and-integral", "text": "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral} then for $y \\in Y$ there exist affine opens $U \\subset X$ and $V \\subset Y$ with $i^{-1}(U) = j^{-1}(V)$ and $y \\in V$."}
+{"_id": "14049", "title": "more-morphisms-lemma-pushout-functor", "text": "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral} suppose given a commutative diagram $$ \\xymatrix{ Y' \\ar[d]^g & Z' \\ar[l]^{j'} \\ar[r]_{i'} \\ar[d]^h & X' \\ar[d]^f \\\\ Y & Z \\ar[l] \\ar[r] & X } $$ with cartesian squares and $f, g, h$ separated and locally quasi-finite. Then \\begin{enumerate} \\item the pushouts $Y \\amalg_Z X$ and $Y' \\amalg_{Z'} X'$ exist, \\item $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is separated and locally quasi-finite, and \\item the squares $$ \\xymatrix{ Y' \\ar[r] \\ar[d] & Y' \\amalg_{Z'} X' \\ar[d] & X' \\ar[l] \\ar[d] \\\\ Y \\ar[r] & Y \\amalg_Z X & X \\ar[l] } $$ are cartesian. \\end{enumerate}"}
+{"_id": "14050", "title": "more-morphisms-lemma-pushout-functor-equivalence-flat", "text": "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral} the category of schemes flat, separated, and locally quasi-finite over the pushout $Y \\amalg_Z X$ is equivalent to the category of $(X', Y', Z', i', j', f, g, h)$ as in Lemma \\ref{lemma-pushout-functor} with $f, g, h$ flat. Similarly with ``flat'' replaced with ``\\'etale''."}
+{"_id": "14051", "title": "more-morphisms-lemma-pushout-along-closed-immersions", "text": "Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes. Then the pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\ Y \\ar[r]^-b & Y \\amalg_Z X } $$ The diagram is a fibre square, the morphisms $a$ and $b$ are closed immersions, and there is a short exact sequence $$ 0 \\to \\mathcal{O}_{Y \\amalg_Z X} \\to a_*\\mathcal{O}_X \\oplus b_*\\mathcal{O}_Y \\to c_*\\mathcal{O}_Z \\to 0 $$ where $c = a \\circ i = b \\circ j$."}
+{"_id": "14052", "title": "more-morphisms-lemma-pushout-along-closed-immersions-properties-above", "text": "Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes. Let $f : X' \\to X$ and $g : Y' \\to Y$ be morphisms of schemes and let $\\varphi : X' \\times_{X, i} Z \\to Y' \\times_{Y, j} Z$ be an isomorphism of schemes over $Z$. Consider the morphism $$ h : X' \\amalg_{X' \\times_{X, i} Z, \\varphi} Y' \\longrightarrow X \\amalg_Z Y $$ Then we have \\begin{enumerate} \\item $h$ is locally of finite type if and only if $f$ and $g$ are locally of finite type, \\item $h$ is flat if and only if $f$ and $g$ are flat, \\item $h$ is flat and locally of finite presentation if and only if $f$ and $g$ are flat and locally of finite presentation, \\item $h$ is smooth if and only if $f$ and $g$ are smooth, \\item $h$ is \\'etale if and only if $f$ and $g$ are \\'etale, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "14053", "title": "more-morphisms-lemma-hom-from-finite-free-into-affine", "text": "Let $Z \\to S$ and $X \\to S$ be morphisms of affine schemes. Assume $\\Gamma(Z, \\mathcal{O}_Z)$ is a finite free $\\Gamma(S, \\mathcal{O}_S)$-module. Then $\\mathit{Mor}_S(Z, X)$ is representable by an affine scheme over $S$."}
+{"_id": "14054", "title": "more-morphisms-lemma-hom-from-finite-locally-free-into-affine", "text": "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes. If $Z \\to S$ is finite locally free and $X \\to S$ is affine, then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme affine over $S$."}
+{"_id": "14055", "title": "more-morphisms-lemma-hom-from-finite-locally-free-representable", "text": "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes. Assume \\begin{enumerate} \\item $Z \\to S$ is finite locally free, and \\item for all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and $x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$ with $x_1, \\ldots, x_d \\in U$. \\end{enumerate} Then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme."}
+{"_id": "14056", "title": "more-morphisms-lemma-hom-from-finite-locally-free-separated-lqf", "text": "Let $Z \\to S$ and $X \\to S$ be morphisms of schemes. Assume $Z \\to S$ is finite locally free and $X \\to S$ is separated and locally quasi-finite. Then $\\mathit{Mor}_S(Z, X)$ is representable by a scheme."}
+{"_id": "14057", "title": "more-morphisms-lemma-case-of-tor-independence", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ Z' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\ X' \\ar[r] & S' } $$ Let $S \\to S'$ be a morphism. Denote by $X$ and $Y$ the base changes of $X'$ and $Y'$ to $S$. Assume $Y' \\to S'$ and $Z' \\to X'$ are flat. Then $X \\times_S Y$ and $Z'$ are Tor independent over $X' \\times_{S'} Y'$."}
+{"_id": "14058", "title": "more-morphisms-lemma-derived-chow", "text": "Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation over $A$. Let $x \\in X$. Then there exist an open neighbourhood $U \\subset X$ of $x$, an $n \\geq 0$, an open $V \\subset \\mathbf{P}^n_A$, a closed subscheme $Z \\subset X \\times_A \\mathbf{P}^n_A$, a point $z \\in Z$, and an object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that \\begin{enumerate} \\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation, \\item $b : Z \\to X$ is an isomorphism over $U$ and $b(z) = x$, \\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$, \\item $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \\in V$, \\item $E|_{X \\times_A V} \\cong (b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$, \\item $E$ is pseudo-coherent and supported on $Z$. \\end{enumerate}"}
+{"_id": "14059", "title": "more-morphisms-lemma-compute-Fourier-Mukai-for-derived-chow", "text": "Let $A$, $x \\in X$, and $U, n, V, Z, z, E$ be as in Lemma \\ref{lemma-derived-chow}. For any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have $$ Rq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(U \\to V)_*K|_U $$ where $p : X \\times_A \\mathbf{P}^n_A \\to X$ and $q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are the projections and where the morphism $U \\to V$ is the finitely presented closed immersion $c \\circ (b|_U)^{-1}$."}
+{"_id": "14060", "title": "more-morphisms-lemma-characterize-pseudo-coherent", "text": "Let $A$ be a ring. Let $X$ be a scheme separated and of finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$."}
+{"_id": "14061", "title": "more-morphisms-lemma-characterize-pseudo-coh-improved", "text": "Let $A$ be a ring. Let $X$ be a scheme separated and of finite presentation over $A$. Let  $K \\in D_\\QCoh(\\mathcal{O}_X).$ If  $R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every perfect  $E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent relative to $A$."}
+{"_id": "14062", "title": "more-morphisms-lemma-characterize-relatively-perfect", "text": "Let $A$ be a ring. Let $X$ be a scheme separated, of finite presentation, and flat over $A$. Let  $K \\in D_\\QCoh(\\mathcal{O}_X).$ If  $R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in $D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is $\\Spec(A)$-perfect."}
+{"_id": "14063", "title": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf", "text": "Let $X \\to S$ be locally of finite type. Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$."}
+{"_id": "14064", "title": "more-morphisms-lemma-relative-pseudo-coherent-post-compose", "text": "Let $X \\to T \\to S$ be morphisms of schemes. Assume $T \\to S$ is flat and locally of finite presentation and $X \\to T$ locally of finite type. Let $E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $E$ is $m$-pseudo-coherent relative to $T$."}
+{"_id": "14066", "title": "more-morphisms-lemma-thickening-pseudo-coherent", "text": "Let $i : X \\to X'$ be a finite order thickening of schemes. Let $K' \\in D(\\mathcal{O}_{X'})$ be an object such that $K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent."}
+{"_id": "14067", "title": "more-morphisms-lemma-thickening-relatively-perfect", "text": "Consider a cartesian diagram $$ \\xymatrix{ X \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\ Y \\ar[r]^j & Y' } $$ of schemes. Assume $X' \\to Y'$ is flat and locally of finite presentation and $Y \\to Y'$ is a finite order thickening. Let $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect."}
+{"_id": "14069", "title": "more-morphisms-lemma-check-h1-fibre-zero", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$ be a point with $\\dim(X_y) \\leq 1$. If \\begin{enumerate} \\item $R^1f_*\\mathcal{O}_X = 0$, or more generally \\item there is a morphism $g : Y' \\to Y$ such that $y$ is in the image of $g$ and such that $R'f'_*\\mathcal{O}_{X'} = 0$ where $f' : X' \\to Y'$ is the base change of $f$ by $g$. \\end{enumerate} Then $H^1(X_y, \\mathcal{O}_{X_y}) = 0$."}
+{"_id": "14070", "title": "more-morphisms-lemma-h1-fibre-zero", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$ be a point with $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$. Then there is an open neighbourhood $V \\subset Y$ of $y$ such that $R^1f_*\\mathcal{O}_X|_V = 0$ and the same is true after base change by any $Y' \\to V$."}
+{"_id": "14071", "title": "more-morphisms-lemma-globally-generated-vanishing", "text": "Let $f : X \\to Y$ be a proper morphism of schemes such that $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$ for all $y \\in Y$. Let $\\mathcal{F}$ be quasi-coherent on $X$. Then \\begin{enumerate} \\item $R^pf_*\\mathcal{F} = 0$ for $p > 1$, and \\item $R^1f_*\\mathcal{F} = 0$ if there is a surjection $f^*\\mathcal{G} \\to \\mathcal{F}$ with $\\mathcal{G}$ quasi-coherent on $Y$. \\end{enumerate} If $Y$ is affine, then we also have \\begin{enumerate} \\item[(3)] $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and \\item[(4)] $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated. \\end{enumerate}"}
+{"_id": "14072", "title": "more-morphisms-lemma-h1-fibre-zero-check-h0-kappa", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Assume \\begin{enumerate} \\item for all $y \\in Y$ we have $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$, and \\item $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective. \\end{enumerate} Then $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is surjective for any base change $f' : X' \\to Y'$ of $f$."}
+{"_id": "14073", "title": "more-morphisms-lemma-h1-fibre-zero-h0-kappa", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ of morphisms of schemes. Let $s \\in S$ be a point. Assume \\begin{enumerate} \\item $X \\to S$ is locally of finite presentation and flat at points of $X_s$, \\item $f$ is proper, \\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$ and $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$, \\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is surjective. \\end{enumerate} Then there is an open $Y_s \\subset V \\subset Y$ such that (a) $f^{-1}(V)$ is flat over $S$, (b) $\\dim(X_y) \\leq 1$ for $y \\in V$, (c) $R^1f_*\\mathcal{O}_X|_V = 0$, (d) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$ is surjective, and (b), (c), and (d) remain true after base change by any $Y' \\to V$."}
+{"_id": "14074", "title": "more-morphisms-lemma-h1-fibre-zero-h0-flat", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ of morphisms of schemes. Assume $X \\to S$ is flat, $f$ is proper, $\\dim(X_y) \\leq 1$ for $y \\in Y$, and $R^1f_*\\mathcal{O}_X = 0$. Then $f_*\\mathcal{O}_X$ is $S$-flat and formation of $f_*\\mathcal{O}_X$ commutes with arbitrary base change $S' \\to S$."}
+{"_id": "14075", "title": "more-morphisms-lemma-h1-fibre-zero-isom", "text": "Consider a commutative diagram $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ of morphisms of schemes. Let $s \\in S$ be a point. Assume \\begin{enumerate} \\item $X \\to S$ is locally of finite presentation and flat at points of $X_s$, \\item $Y \\to S$ is locally of finite presentation, \\item $f$ is proper, \\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$ and $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$, \\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is an isomorphism. \\end{enumerate} Then there is an open $Y_s \\subset V \\subset Y$ such that (a) $V$ is flat over $S$, (b) $f^{-1}(V)$ is flat over $S$, (c) $\\dim(X_y) \\leq 1$ for $y \\in V$, (d) $R^1f_*\\mathcal{O}_X|_V = 0$, (e) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$ is an isomorphism, and (a) -- (e) remain true after base change of $f^{-1}(V) \\to V$ by any $S' \\to S$."}
+{"_id": "14076", "title": "more-morphisms-lemma-bijection-on-Pic", "text": "Let $f : X \\to Y$ be a proper morphism of Noetherian schemes such that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$, such that the fibres of $f$ have dimension $\\leq 1$, and such that $H^1(X_y, \\mathcal{O}_{X_y}) = 0$ for $y \\in Y$. Then $f^* : \\Pic(Y) \\to \\Pic(X)$ is a bijection onto the subgroup of $\\mathcal{L} \\in \\Pic(X)$ with $\\mathcal{L}|_{X_y} \\cong \\mathcal{O}_{X_y}$ for all $y \\in Y$."}
+{"_id": "14081", "title": "more-morphisms-lemma-test-universally-open", "text": "Let $f : X \\to S$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is universally open, \\item for every morphism $S' \\to S$ which is locally of finite presentation the base change $X_{S'} \\to S'$ is open, and \\item for every $n$ the morphism $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is open. \\end{enumerate}"}
+{"_id": "14083", "title": "more-morphisms-lemma-characterize-universally-open-finite", "text": "Let $A \\to B$ be a ring map. Say $B$ is generated as an $A$-module by $b_1, \\ldots, b_d \\in B$. Set $h = \\sum x_ib_i \\in B[x_1, \\ldots, x_d]$. Then $\\Spec(B) \\to \\Spec(A)$ is universally open if and only if the image of $D(h)$ in $\\Spec(A[x_1, \\ldots, x_d])$ is open."}
+{"_id": "14085", "title": "more-morphisms-lemma-count-geometric-fibres", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. Then \\begin{enumerate} \\item the functions $n_{X/Y}$ of Lemmas \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components} and \\ref{lemma-base-change-fibres-nr-geometrically-connected-components} agree, \\item if $X$ is quasi-compact, then $n_{X/Y}$ attains a maximum $d < \\infty$. \\end{enumerate}"}
+{"_id": "14086", "title": "more-morphisms-lemma-count-geometric-fibres-universally-open", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open morphism of schemes. Let $n_{X/Y}$ be as in Lemma \\ref{lemma-count-geometric-fibres}. If $n_{X/Y}(y) \\geq d$ for some $y \\in Y$ and $d \\geq 0$, then $n_{X/Y} \\geq d$ in an open neighbourhood of $y$."}
+{"_id": "14088", "title": "more-morphisms-lemma-weighting-check-after-etale-base-change", "text": "Given a cartesian square $$ \\xymatrix{ U \\ar[d]_\\pi & U' \\ar[l]^h \\ar[d]^{\\pi'} \\\\ V & V' \\ar[l]_g } $$ with $\\pi$ locally quasi-finite with finite fibres and a function $w : U \\to \\mathbf{Z}$ we have $(\\int_\\pi w) \\circ g = \\int_{\\pi'} (w \\circ h)$."}
+{"_id": "14089", "title": "more-morphisms-lemma-weighting-base-change", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. Let $w : X \\to \\mathbf{Z}$ be a weighting. Let $f' : X' \\to Y'$ be the base change of $f$ by a morphism $Y' \\to Y$. Then the composition $w' : X' \\to \\mathbf{Z}$ of $w$ and the projection $X' \\to X$ is a weighting of $f'$."}
+{"_id": "14090", "title": "more-morphisms-lemma-weighting-universally-open", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. Let $w : X \\to \\mathbf{Z}$ be a weighting. If $w(x) > 0$ for all $x \\in X$, then $f$ is universally open."}
+{"_id": "14093", "title": "more-morphisms-lemma-jumps-w", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $w$ are locally constructible in $X$."}
+{"_id": "14094", "title": "more-morphisms-lemma-jumps-int-w", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of finite presentation. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $\\int_f w$ are locally constructible in $Y$."}
+{"_id": "14096", "title": "more-morphisms-lemma-semicontinuous-int-w", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism with finite fibres. Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $\\int_f w$ is lower semi-continuous."}
+{"_id": "14097", "title": "more-morphisms-lemma-max-int-w", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism with $X$ quasi-compact. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then $\\int_f w$ attains its maximum."}
+{"_id": "14098", "title": "more-morphisms-lemma-max-int-finite", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism. Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $\\int_w f$ attains its maximum $d$ and let $Y_d \\subset Y$ be the open set of points $y$ with $(\\int_f w)(y) = d$. Then the morphism $f^{-1}(Y_d) \\to Y_d$ is finite."}
+{"_id": "14099", "title": "more-morphisms-lemma-open-and-closed-in-finite", "text": "Let $A \\to B$ be a ring map which is finite and of finite presentation. There exists a finitely presented ring map $A \\to A_{univ}$ and an idempotent $e_{univ} \\in B \\otimes_A A_{univ}$ such that for any ring map $A \\to A'$ and idempotent $e \\in B \\otimes_A A'$ there is a ring map $A_{univ} \\to A'$ mapping $e_{univ}$ to $e$."}
+{"_id": "14100", "title": "more-morphisms-lemma-open-and-closed-in-quasi-finite", "text": "Let $X \\to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. There exists a morphism $Y_{univ} \\to Y$ of finite presentation and an open subscheme $U_{univ} \\subset Y_{univ} \\times_Y X$ such that $U_{univ} \\to Y_{univ}$ is finite with the following property: given any morphism $Y' \\to Y$ of affine schemes and an open subscheme $U' \\subset Y' \\times_Y X$ such that $U' \\to Y'$ is finite, there exists a morphism $Y' \\to Y_{univ}$ such that the inverse image of $U_{univ}$ is $U'$."}
+{"_id": "14101", "title": "more-morphisms-lemma-descend-weighting", "text": "Let $Y = \\lim Y_i$ be a directed limit of affine schemes. Let $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $f : X  \\to Y$ and $f_i : X_i \\to Y_i$ for $i \\geq 0$ be the base changes of $f_0$. If $w : X \\to \\mathbf{Z}$ is a weighting of $f$, then for sufficiently large $i$ there exists a weighting $w_i : X_i \\to \\mathbf{Z}$ of $f_i$ whose pullback to $X$ is $w$."}
+{"_id": "14102", "title": "more-morphisms-lemma-affineness-of-large-open", "text": "Let $f : X \\to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $w : X \\to \\mathbf{Z}_{> 0}$ be a postive weighting of $f$. Let $d < \\infty$ be the maximum value of $\\int_f w$. The open $$ Y_d = \\{y \\in Y \\mid (\\textstyle{\\int}_f w)(y) = d \\} $$ of $Y$ is affine."}
+{"_id": "14103", "title": "more-morphisms-proposition-pushout-along-closed-immersion-and-integral", "text": "\\begin{reference} \\cite[Theorem 7.1 part iii]{Ferrand-Conducteur} \\end{reference} In Situation \\ref{situation-pushout-along-closed-immersion-and-integral} the pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture $$ \\xymatrix{ Z \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\ Y \\ar[r]^-b & Y \\amalg_Z X } $$ The diagram is a fibre square, the morphism $a$ is integral, the morphism $b$ is a closed immersion, and $$ \\mathcal{O}_{Y \\amalg_Z X} = b_*\\mathcal{O}_Y \\times_{c_*\\mathcal{O}_Z} a_*\\mathcal{O}_X $$ as sheaves of rings where $c = a \\circ i = b \\circ j$."}
+{"_id": "14138", "title": "sites-modules-lemma-limits-colimits-abelian-presheaves", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item All limits and colimits exist in $\\textit{PAb}(\\mathcal{C})$. \\item All limits and colimits commute with taking sections over objects of $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "14139", "title": "sites-modules-lemma-abelian-abelian", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of abelian sheaves on $\\mathcal{C}$. \\begin{enumerate} \\item The category $\\textit{Ab}(\\mathcal{C})$ is an abelian category. \\item The kernel $\\Ker(\\varphi)$ of $\\varphi$ is the same as the kernel of $\\varphi$ as a morphism of presheaves. \\item The morphism $\\varphi$ is injective (Homology, Definition \\ref{homology-definition-injective-surjective}) if and only if $\\varphi$ is injective as a map of presheaves (Sites, Definition \\ref{sites-definition-presheaves-injective-surjective}), if and only if $\\varphi$ is injective as a map of sheaves (Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}). \\item The cokernel $\\Coker(\\varphi)$ of $\\varphi$ is the sheafification of the cokernel of $\\varphi$ as a morphism of presheaves. \\item The morphism $\\varphi$ is surjective (Homology, Definition \\ref{homology-definition-injective-surjective}) if and only if $\\varphi$ is surjective as a map of sheaves (Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}). \\item A complex of abelian sheaves $$ \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} $$ is exact at $\\mathcal{G}$ if and only if for all $U \\in \\Ob(\\mathcal{C})$ and all $s \\in \\mathcal{G}(U)$ mapping to zero in $\\mathcal{H}(U)$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that each $s|_{U_i}$ is in the image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$. \\end{enumerate}"}
+{"_id": "14140", "title": "sites-modules-lemma-limits-colimits-abelian-sheaves", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item All limits and colimits exist in $\\textit{Ab}(\\mathcal{C})$. \\item Limits are the same as the corresponding limits of abelian presheaves over $\\mathcal{C}$ (i.e., commute with taking sections over objects of $\\mathcal{C}$). \\item Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over $\\mathcal{C}$. \\item A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves. \\item Filtered colimits are exact. \\end{enumerate}"}
+{"_id": "14141", "title": "sites-modules-lemma-obvious-adjointness", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{G}$, $\\mathcal{F}$ be a presheaves of sets. Let $\\mathcal{A}$ be an abelian presheaf. Let $U$ be an object of $\\mathcal{C}$. Then we have \\begin{align*} \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F}) & = \\mathcal{F}(U), \\\\ \\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}, \\mathcal{A}) & = \\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\ \\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_U, \\mathcal{A}) & = \\mathcal{A}(U). \\end{align*} All of these equalities are functorial."}
+{"_id": "14142", "title": "sites-modules-lemma-coproduct-sum-free-abelian-presheaf", "text": "Let $\\mathcal{C}$ be a category. Let $I$ be a set. For each $i \\in I$ let $\\mathcal{G}_i$ be a presheaf of sets. Then $$ \\mathbf{Z}_{\\coprod_i \\mathcal{G}_i} = \\bigoplus\\nolimits_{i \\in I} \\mathbf{Z}_{\\mathcal{G}_i} $$ in $\\textit{PAb}(\\mathcal{C})$."}
+{"_id": "14143", "title": "sites-modules-lemma-obvious-adjointness-sheaves", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$, $\\mathcal{F}$ be a sheaves of sets. Let $\\mathcal{A}$ be an abelian sheaf. Let $U$ be an object of $\\mathcal{C}$. Then we have \\begin{align*} \\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F}) & = \\mathcal{F}(U), \\\\ \\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}^\\#, \\mathcal{A}) & = \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\ \\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_U^\\#, \\mathcal{A}) & = \\mathcal{A}(U). \\end{align*} All of these equalities are functorial."}
+{"_id": "14144", "title": "sites-modules-lemma-may-sheafify-before-abelianize", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets. Then $\\mathbf{Z}_\\mathcal{G}^\\# = (\\mathbf{Z}_{\\mathcal{G}^\\#})^\\#$."}
+{"_id": "14145", "title": "sites-modules-lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. There exists a factorization $$ \\xymatrix{ (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[rr]_{(f, f^\\sharp)} \\ar[d]_{(g, g^\\sharp)} & & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\ar[d]^{(e, e^\\sharp)} \\\\ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[rr]^{(h, h^\\sharp)} & & (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) } $$ where \\begin{enumerate} \\item $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ is an equivalence of topoi induced by a special cocontinuous functor $\\mathcal{C} \\to \\mathcal{C}'$ (see Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}), \\item $e : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{D}')$ is an equivalence of topoi induced by a special cocontinuous functor $\\mathcal{D} \\to \\mathcal{D}'$ (see Sites, Definition \\ref{sites-definition-special-cocontinuous-functor}), \\item $\\mathcal{O}_{\\mathcal{C}'} = g_*\\mathcal{O}_\\mathcal{C}$ and $g^\\sharp$ is the obvious map, \\item $\\mathcal{O}_{\\mathcal{D}'} = e_*\\mathcal{O}_\\mathcal{D}$ and $e^\\sharp$ is the obvious map, \\item the sites $\\mathcal{C}'$ and $\\mathcal{D}'$ have final objects and fibre products (i.e., all finite limits), \\item $h$ is a morphism of sites induced by a continuous functor $u : \\mathcal{D}' \\to \\mathcal{C}'$ which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition \\ref{sites-proposition-get-morphism}), and \\item given any set of sheaves $\\mathcal{F}_i$ (resp.\\ $\\mathcal{G}_j$) on $\\mathcal{C}$ (resp.\\ $\\mathcal{D}$) we may assume each of these is a representable sheaf on $\\mathcal{C}'$ (resp.\\ $\\mathcal{D}'$). \\end{enumerate} Moreover, if $(f, f^\\sharp)$ is an equivalence of ringed topoi, then we can choose the diagram such that $\\mathcal{C}' = \\mathcal{D}'$, $\\mathcal{O}_{\\mathcal{C}'} = \\mathcal{O}_{\\mathcal{D}'}$ and $(h, h^\\sharp)$ is the identity."}
+{"_id": "14146", "title": "sites-modules-lemma-adjointness-tensor-restrict-presheaves", "text": "With $\\mathcal{C}$, $\\mathcal{O}_1 \\to \\mathcal{O}_2$, $\\mathcal{F}$ and $\\mathcal{G}$ as above there exists a canonical bijection $$ \\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G}, \\mathcal{F} ) $$ In other words, the restriction and change of rings functors defined above are adjoint to each other."}
+{"_id": "14147", "title": "sites-modules-lemma-sheafification-presheaf-modules", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. Let $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$ as a presheaf of rings, see Sites, Section \\ref{sites-section-sheaves-algebraic-structures}. Let $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$ as a presheaf of abelian groups. There exists a unique map of sheaves of sets $$ \\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\longrightarrow \\mathcal{F}^\\# $$ which makes the diagram $$ \\xymatrix{ \\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[d] \\\\ \\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] & \\mathcal{F}^\\# } $$ commute and which makes $\\mathcal{F}^\\#$ into a sheaf of $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$ is a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism of presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$ (into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module) factors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$ where $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of $\\mathcal{O}^\\#$-modules."}
+{"_id": "14148", "title": "sites-modules-lemma-sheafification-exact", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$ The sheafification functor $$ \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}^\\#), \\quad \\mathcal{F} \\longmapsto \\mathcal{F}^\\# $$ is exact."}
+{"_id": "14149", "title": "sites-modules-lemma-adjointness-tensor-restrict", "text": "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and $\\mathcal{G}$ as above there exists a canonical bijection $$ \\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G}, \\mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other."}
+{"_id": "14150", "title": "sites-modules-lemma-epimorphism-modules", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be an epimorphism of sheaves of rings. Let $\\mathcal{G}_1, \\mathcal{G}_2$ be $\\mathcal{O}'$-modules. Then $$ \\Hom_{\\mathcal{O}'}(\\mathcal{G}_1, \\mathcal{G}_2) = \\Hom_\\mathcal{O}(\\mathcal{G}_1, \\mathcal{G}_2). $$ In other words, the restriction functor $\\textit{Mod}(\\mathcal{O}') \\to \\textit{Mod}(\\mathcal{O})$ is fully faithful."}
+{"_id": "14151", "title": "sites-modules-lemma-pushforward-module", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. There is a natural map of sheaves of sets $$ f_*\\mathcal{O} \\times f_*\\mathcal{F} \\longrightarrow f_*\\mathcal{F} $$ which turns $f_*\\mathcal{F}$ into a sheaf of $f_*\\mathcal{O}$-modules. This construction is functorial in $\\mathcal{F}$."}
+{"_id": "14152", "title": "sites-modules-lemma-pullback-module", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules. There is a natural map of sheaves of sets $$ f^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G} \\longrightarrow f^{-1}\\mathcal{G} $$ which turns $f^{-1}\\mathcal{G}$ into a sheaf of $f^{-1}\\mathcal{O}$-modules. This construction is functorial in $\\mathcal{G}$."}
+{"_id": "14153", "title": "sites-modules-lemma-adjoint-push-pull-modules", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules. Let $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules. Then $$ \\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}). $$ Here we use Lemmas \\ref{lemma-pullback-module} and \\ref{lemma-pushforward-module}, and we think of $f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via $\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$."}
+{"_id": "14155", "title": "sites-modules-lemma-adjoint-pullback-pushforward-modules", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules. There is a canonical bijection $$ \\Hom_{\\mathcal{O}_\\mathcal{C}}(f^*\\mathcal{G}, \\mathcal{F}) = \\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, f_*\\mathcal{F}). $$ In other words: the functor $f^*$ is the left adjoint to $f_*$."}
+{"_id": "14156", "title": "sites-modules-lemma-push-pull-composition-modules", "text": "$(f, f^\\sharp) : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and $(g, g^\\sharp) : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to (\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of ringed topoi. There are canonical isomorphisms of functors $(g \\circ f)_* \\cong g_* \\circ f_*$ and $(g \\circ f)^* \\cong f^* \\circ g^*$."}
+{"_id": "14157", "title": "sites-modules-lemma-abelian", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. The category $\\textit{Mod}(\\mathcal{O})$ is an abelian category. The forgetful functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$ is exact, hence kernels, cokernels and exactness of $\\mathcal{O}$-modules, correspond to the corresponding notions for abelian sheaves."}
+{"_id": "14158", "title": "sites-modules-lemma-limits-colimits", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. All limits and colimits exist in $\\textit{Mod}(\\mathcal{O})$ and the forgetful functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$ commutes with them. Moreover, filtered colimits are exact."}
+{"_id": "14159", "title": "sites-modules-lemma-exactness-pushforward-pullback", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. \\begin{enumerate} \\item The functor $f_*$ is left exact. In fact it commutes with all limits. \\item The functor $f^*$ is right exact. In fact it commutes with all colimits. \\end{enumerate}"}
+{"_id": "14160", "title": "sites-modules-lemma-check-exactness-stalks", "text": "Let $\\mathcal{C}$ be a site. If $\\{p_i\\}_{i \\in I}$ is a conservative family of points, then we may check exactness of a sequence of abelian sheaves on the stalks at the points $p_i$, $i \\in I$. If $\\mathcal{C}$ has enough points, then exactness of a sequence of abelian sheaves may be checked on stalks."}
+{"_id": "14161", "title": "sites-modules-lemma-reflect-surjections", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. The following are equivalent: \\begin{enumerate} \\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for all $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, and \\item $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$ reflects surjections. \\end{enumerate} In this case the functor $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$ is faithful."}
+{"_id": "14162", "title": "sites-modules-lemma-exactness", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. Assume at least one of the following properties holds \\begin{enumerate} \\item $f_*$ transforms surjections of sheaves of sets into surjections, \\item $f_*$ transforms surjections of abelian sheaves into surjections, \\item $f_*$ commutes with coequalizers on sheaves of sets, \\item $f_*$ commutes with pushouts on sheaves of sets, \\end{enumerate} Then $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$ is exact."}
+{"_id": "14163", "title": "sites-modules-lemma-morphism-ringed-sites-almost-cocontinuous", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. Assume $u$ is almost cocontinuous. Then \\begin{enumerate} \\item $f_* : \\textit{Ab}(\\mathcal{D}) \\to \\textit{Ab}(\\mathcal{C})$ is exact. \\item if $f^\\sharp : f^{-1}\\mathcal{O}_\\mathcal{C} \\to \\mathcal{O}_\\mathcal{D}$ is given so that $f$ becomes a morphism of ringed sites, then $f_* : \\textit{Mod}(\\mathcal{O}_\\mathcal{D}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ is exact. \\end{enumerate}"}
+{"_id": "14164", "title": "sites-modules-lemma-g-shriek-adjoint", "text": "The functor $g_{p!}$ is a left adjoint to the functor $u^p$. The functor $g_!$ is a left adjoint to the functor $g^{-1}$. In other words the formulas \\begin{align*} \\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G}) & = \\Mor_{\\textit{PAb}(\\mathcal{D})}(g_{p!}\\mathcal{F}, \\mathcal{G}), \\\\ \\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathcal{F}, g^{-1}\\mathcal{G}) & = \\Mor_{\\textit{Ab}(\\mathcal{D})}(g_!\\mathcal{F}, \\mathcal{G}) \\end{align*} hold bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$."}
+{"_id": "14165", "title": "sites-modules-lemma-exactness-lower-shriek", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, \\item[(b)] $u$ is continuous, and \\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and $u$ commutes with them. \\end{enumerate} In this case the functor $g_! : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$ is exact."}
+{"_id": "14166", "title": "sites-modules-lemma-back-and-forth", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Assume that \\begin{enumerate} \\item[(a)] $u$ is cocontinuous, \\item[(b)] $u$ is continuous, and \\item[(c)] $u$ is fully faithful. \\end{enumerate} For $g_!, g^{-1}, g_*$ as above the canonical maps $\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$ and $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms for all abelian sheaves $\\mathcal{F}$ on $\\mathcal{C}$."}
+{"_id": "14167", "title": "sites-modules-lemma-have-left-adjoint", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. \\begin{enumerate} \\item If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\\Sh}$ on underlying sheaves of sets and $g_!$ is exact. \\item If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is the morphism of topoi associated to $w$. \\end{enumerate}"}
+{"_id": "14168", "title": "sites-modules-lemma-global-pullback", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module. \\begin{enumerate} \\item If $\\mathcal{F}$ is free then $f^*\\mathcal{F}$ is free. \\item If $\\mathcal{F}$ is finite free then $f^*\\mathcal{F}$ is finite free. \\item If $\\mathcal{F}$ is generated by global sections then $f^*\\mathcal{F}$ is generated by global sections. \\item Given $r \\geq 0$ if $\\mathcal{F}$ is generated by $r$ global sections, then $f^*\\mathcal{F}$ is generated by $r$ global sections. \\item If $\\mathcal{F}$ is generated by finitely many global sections then $f^*\\mathcal{F}$ is generated by finitely many global sections. \\item If $\\mathcal{F}$ has a global presentation then $f^*\\mathcal{F}$ has a global presentation. \\item If $\\mathcal{F}$ has a finite global presentation then $f^*\\mathcal{F}$ has a finite global presentation. \\end{enumerate}"}
+{"_id": "14169", "title": "sites-modules-lemma-extension-by-zero", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. The restriction functor $j_U^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_U)$ has a left adjoint $j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$. So $$ \\Mor_{\\textit{Mod}(\\mathcal{O}_U)}(\\mathcal{G}, j_U^*\\mathcal{F}) = \\Mor_{\\textit{Mod}(\\mathcal{O})}(j_{U!}\\mathcal{G}, \\mathcal{F}) $$ for $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}))$ and $\\mathcal{G} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_U))$. Moreover, the extension by zero $j_{U!}\\mathcal{G}$ of $\\mathcal{G}$ is the sheaf associated to the presheaf $$ V \\longmapsto \\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) $$ with obvious restriction mappings and an obvious $\\mathcal{O}$-module structure."}
+{"_id": "14170", "title": "sites-modules-lemma-extension-by-zero-exact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. The functor $j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$ is exact."}
+{"_id": "14171", "title": "sites-modules-lemma-j-shriek-reflects-exactness", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. A complex of $\\mathcal{O}_U$-modules $\\mathcal{G}_1 \\to \\mathcal{G}_2 \\to \\mathcal{G}_3$ is exact if and only if $j_{U!}\\mathcal{G}_1 \\to j_{U!}\\mathcal{G}_2 \\to j_{U!}\\mathcal{G}_3$ is exact as a sequence of $\\mathcal{O}$-modules."}
+{"_id": "14172", "title": "sites-modules-lemma-relocalize", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f : V \\to U$ be a morphism of $\\mathcal{C}$. Then there exists a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}/V), \\mathcal{O}_V) \\ar[rd]_{(j_V, j_V^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & & (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\ar[ld]^{(j_U, j_U^\\sharp)} \\\\ & (\\Sh(\\mathcal{C}), \\mathcal{O}) & } $$ of ringed topoi. Here $(j, j^\\sharp)$ is the localization morphism associated to the object $V/U$ of the ringed site $(\\mathcal{C}/V, \\mathcal{O}_V)$."}
+{"_id": "14173", "title": "sites-modules-lemma-restrict-back", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Assume that every $X$ in $\\mathcal{C}$ has at most one morphism to $U$. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}/U$. The canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$ and $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms."}
+{"_id": "14174", "title": "sites-modules-lemma-localize-morphism-ringed-sites", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}) \\longrightarrow (\\mathcal{D}, \\mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V$ be an object of $\\mathcal{D}$ and set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\\sharp$ such that the diagram of Sites, Lemma \\ref{sites-lemma-localize-morphism} is turned into a commutative diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & & (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V) \\ar[rr]^{(j_V, j_V^\\sharp)} & & (\\Sh(\\mathcal{D}), \\mathcal{O}'). } $$ Moreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$ and $f'_*j_U^* = j_V^*f_*$."}
+{"_id": "14175", "title": "sites-modules-lemma-relocalize-morphism-ringed-sites", "text": "Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}) \\longrightarrow (\\mathcal{D}, \\mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$ and $c : U \\to u(V)$ a morphism of $\\mathcal{C}$. There exists a commutative diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f_c, f_c^\\sharp)} & & (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V) \\ar[rr]^{(j_V, j_V^\\sharp)} & & (\\Sh(\\mathcal{D}), \\mathcal{O}'). } $$ The morphism $(f_c, f_c^\\sharp)$ is equal to the composition of the morphism $$ (f', (f')^\\sharp) : (\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)}) \\longrightarrow (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V) $$ of Lemma \\ref{lemma-localize-morphism-ringed-sites} and the morphism $$ (j, j^\\sharp) : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)}) $$ of Lemma \\ref{lemma-relocalize}. Given any morphisms $b : V' \\to V$, $a : U' \\to U$ and $c' : U' \\to u(V')$ such that $$ \\xymatrix{ U' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\ U \\ar[r]^-c & u(V) } $$ commutes, then the following diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C}/U'), \\mathcal{O}_{U'}) \\ar[rr]_{(j_{U'/U}, j_{U'/U}^\\sharp)} \\ar[d]_{(f_{c'}, f_{c'}^\\sharp)} & & (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\ar[d]^{(f_c, f_c^\\sharp)} \\\\ (\\Sh(\\mathcal{D}/V'), \\mathcal{O}'_{V'}) \\ar[rr]^{(j_{V'/V}, j_{V'/V}^\\sharp)} & & (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_{V'}) } $$ commutes."}
+{"_id": "14176", "title": "sites-modules-lemma-localize-ringed-topos", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $\\mathcal{F} \\in \\Sh(\\mathcal{C})$ be a sheaf. For a sheaf $\\mathcal{H}$ on $\\mathcal{C}$ denote $\\mathcal{H}_\\mathcal{F}$ the sheaf $\\mathcal{H} \\times \\mathcal{F}$ seen as an object of the category $\\Sh(\\mathcal{C})/\\mathcal{F}$. The pair $(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$ is a ringed topos and there is a canonical morphism of ringed topoi $$ (j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\longrightarrow (\\Sh(\\mathcal{C}), \\mathcal{O}) $$ which is a localization as in Section \\ref{section-localize} such that \\begin{enumerate} \\item the functor $j_\\mathcal{F}^{-1}$ is the functor $\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$, \\item the functor $j_\\mathcal{F}^*$ is the functor $\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$, \\item the functor $j_{\\mathcal{F}!}$ on sheaves of sets is the forgetful functor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$, \\item the functor $j_{\\mathcal{F}!}$ on sheaves of modules associates to the $\\mathcal{O}_\\mathcal{F}$-module $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the $\\mathcal{O}$-module which is the sheafification of the presheaf $$ V \\longmapsto \\bigoplus\\nolimits_{s \\in \\mathcal{F}(V)} \\{\\sigma \\in \\mathcal{G}(V) \\mid \\varphi(\\sigma) = s \\} $$ for $V \\in \\Ob(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "14177", "title": "sites-modules-lemma-localize-compare", "text": "With $(\\Sh(\\mathcal{C}), \\mathcal{O})$ and $\\mathcal{F} \\in \\Sh(\\mathcal{C})$ as in Lemma \\ref{lemma-localize-ringed-topos}. If $\\mathcal{F} = h_U^\\#$ for some object $U$ of $\\mathcal{C}$ then via the identification $\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of Sites, Lemma \\ref{sites-lemma-essential-image-j-shriek} we have \\begin{enumerate} \\item canonically $\\mathcal{O}_U = \\mathcal{O}_\\mathcal{F}$, and \\item with these identifications we have $(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$. \\end{enumerate}"}
+{"_id": "14178", "title": "sites-modules-lemma-relocalize-ringed-topos", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. If $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves on $\\mathcal{C}$ then there exists a natural commutative diagram of morphisms of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G}) \\ar[rd]_{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & & (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\ar[ld]^{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)} \\\\ & (\\Sh(\\mathcal{C}), \\mathcal{O}) & } $$ where $(j, j^\\sharp)$ is the localization morphism of the ringed topos $(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$ at the object $\\mathcal{G}/\\mathcal{F}$."}
+{"_id": "14179", "title": "sites-modules-lemma-relocalize-compare", "text": "With $(\\Sh(\\mathcal{C}), \\mathcal{O})$, $s : \\mathcal{G} \\to \\mathcal{F}$ as in Lemma \\ref{lemma-relocalize-ringed-topos}. If there exist a morphism $f : V \\to U$ of $\\mathcal{C}$ such that $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$ and $s$ is induced by $f$, then the diagrams of Lemma \\ref{lemma-relocalize} and Lemma \\ref{lemma-relocalize-ringed-topos} agree via the identifications $(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$ and $(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp) = (j_V, j_V^\\sharp)$ of Lemma \\ref{lemma-localize-compare}."}
+{"_id": "14180", "title": "sites-modules-lemma-localize-morphism-ringed-topoi", "text": "Let $$ f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{D}), \\mathcal{O}') $$ be a morphism of ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$. Then there exists a commutative diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & & (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G}) \\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & & (\\Sh(\\mathcal{D}), \\mathcal{O}') } $$ We have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$ and $f'_*j_\\mathcal{F}^* = j_\\mathcal{G}^*f_*$. Moreover, the morphism $f'$ is characterized by the rule $$ (f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G}) = (f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F}). $$"}
+{"_id": "14181", "title": "sites-modules-lemma-localize-morphism-compare", "text": "Let $$ f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{D}), \\mathcal{O}') $$ be a morphism of ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$. If $f$ is given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$ and $\\mathcal{G} = h_V^\\#$, then the commutative diagrams of Lemma \\ref{lemma-localize-morphism-ringed-sites} and Lemma \\ref{lemma-localize-morphism-ringed-topoi} agree via the identifications of Lemma \\ref{lemma-localize-compare}."}
+{"_id": "14182", "title": "sites-modules-lemma-relocalize-morphism-ringed-topoi", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$, let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$, and let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)} \\ar[d]_{(f_c, f_c^\\sharp)} & & (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G}) \\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & & (\\Sh(\\mathcal{D}), \\mathcal{O}'). } $$ The morphism $(f_s, f_s^\\sharp)$ is equal to the composition of the morphism $$ (f', (f')^\\sharp) : (\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}}) \\longrightarrow (\\Sh(\\mathcal{D})/{\\mathcal{G}}, \\mathcal{O}'_\\mathcal{G}) $$ of Lemma \\ref{lemma-localize-morphism-ringed-topoi} and the morphism $$ (j, j^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\to (\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}}) $$ of Lemma \\ref{lemma-relocalize-ringed-topos}. Given any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$, $a : \\mathcal{F}' \\to \\mathcal{F}$, and $s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that $$ \\xymatrix{ \\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a & f^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\ \\mathcal{F} \\ar[r]^-s & f^{-1}\\mathcal{G} } $$ commutes, then the following diagram of ringed topoi $$ \\xymatrix{ (\\Sh(\\mathcal{C})/\\mathcal{F}', \\mathcal{O}_{\\mathcal{F}'}) \\ar[rr]_{(j_{\\mathcal{F}'/\\mathcal{F}}, j_{\\mathcal{F}'/\\mathcal{F}}^\\sharp)} \\ar[d]_{(f_{s'}, f_{s'}^\\sharp)} & & (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\ar[d]^{(f_s, f_s^\\sharp)} \\\\ (\\Sh(\\mathcal{D})/\\mathcal{G}', \\mathcal{O}'_{\\mathcal{G}'}) \\ar[rr]^{(j_{\\mathcal{G}'/\\mathcal{G}}, j_{\\mathcal{G}'/\\mathcal{G}}^\\sharp)} & & (\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_{\\mathcal{G}'}) } $$ commutes."}
+{"_id": "14183", "title": "sites-modules-lemma-relocalize-morphism-compare", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$, $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be as in Lemma \\ref{lemma-relocalize-morphism-ringed-topoi}. If $f$ is given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$ and $\\mathcal{G} = h_V^\\#$, $\\mathcal{F} = h_U^\\#$ and $s$ comes from a morphism $c : U \\to u(V)$, then the commutative diagrams of Lemma \\ref{lemma-relocalize-morphism-ringed-sites} and Lemma \\ref{lemma-relocalize-morphism-ringed-topoi} agree via the identifications of Lemma \\ref{lemma-localize-compare}."}
+{"_id": "14185", "title": "sites-modules-lemma-local-final-object", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Assume that the site $\\mathcal{C}$ has a final object $X$. Then \\begin{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is locally free, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a locally free $\\mathcal{O}_{X_i}$-module, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a free $\\mathcal{O}_{X_i}$-module. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is finite locally free, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a finite locally free $\\mathcal{O}_{X_i}$-module, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a finite free $\\mathcal{O}_{X_i}$-module. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is locally generated by sections, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module locally generated by sections, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module globally generated by sections. \\end{enumerate} \\item Given $r \\geq 0$, the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is locally generated by $r$ sections, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module locally generated by $r$ sections, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module globally generated by $r$ sections. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is of finite type, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module of finite type, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module globally generated by finitely many sections. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is quasi-coherent, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a quasi-coherent $\\mathcal{O}_{X_i}$-module, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module which has a global presentation. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is of finite presentation, \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module of finite presentation, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is an $\\mathcal{O}_{X_i}$-module has a finite global presentation. \\end{enumerate} \\item The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is coherent, and \\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a coherent $\\mathcal{O}_{X_i}$-module. \\end{enumerate} \\end{enumerate}"}
+{"_id": "14186", "title": "sites-modules-lemma-local-pullback", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module. \\begin{enumerate} \\item If $\\mathcal{F}$ is locally free then $f^*\\mathcal{F}$ is locally free. \\item If $\\mathcal{F}$ is finite locally free then $f^*\\mathcal{F}$ is finite locally free. \\item If $\\mathcal{F}$ is locally generated by sections then $f^*\\mathcal{F}$ is locally generated by sections. \\item If $\\mathcal{F}$ is locally generated by $r$ sections then $f^*\\mathcal{F}$ is locally generated by $r$ sections. \\item If $\\mathcal{F}$ is of finite type then $f^*\\mathcal{F}$ is of finite type. \\item If $\\mathcal{F}$ is quasi-coherent then $f^*\\mathcal{F}$ is quasi-coherent. \\item If $\\mathcal{F}$ is of finite presentation then $f^*\\mathcal{F}$ is of finite presentation. \\end{enumerate}"}
+{"_id": "14187", "title": "sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\theta : \\mathcal{G} \\to \\mathcal{F}$ be a surjective $\\mathcal{O}$-module map with $\\mathcal{F}$ of finite presentation and $\\mathcal{G}$ of finite type. Then $\\Ker(\\theta)$ is of finite type."}
+{"_id": "14188", "title": "sites-modules-lemma-i-star-equivalence", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of ringed topoi. Assume $i$ is a closed immersion of topoi and $i^\\sharp : \\mathcal{O}' \\to i_*\\mathcal{O}$ is surjective. Denote $\\mathcal{I} \\subset \\mathcal{O}'$ the kernel of $i^\\sharp$. The functor $$ i_* : \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}') $$ is exact, fully faithful, with essential image those $\\mathcal{O}'$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$."}
+{"_id": "14189", "title": "sites-modules-lemma-tensor-product-pullback", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_\\mathcal{D}$-modules. Then $f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{G}) = f^*\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{C}} f^*\\mathcal{G}$ functorially in $\\mathcal{F}$, $\\mathcal{G}$."}
+{"_id": "14191", "title": "sites-modules-lemma-internal-hom", "text": "If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings, $\\mathcal{F}$ is a presheaf of $\\mathcal{O}$-modules, and $\\mathcal{G}$ is a sheaf of $\\mathcal{O}$-modules, then $\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$ is a sheaf of $\\mathcal{O}$-modules."}
+{"_id": "14192", "title": "sites-modules-lemma-internal-hom-restriction", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}, \\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules. Then formation of $\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$ commutes with restriction to $U$ for $U \\in \\Ob(\\mathcal{C})$."}
+{"_id": "14193", "title": "sites-modules-lemma-internal-hom-commute-limits", "text": "Internal hom and (co)limits. Let $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings. \\begin{enumerate} \\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor $$ \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O}) , \\quad \\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) $$ commutes with arbitrary limits. \\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor $$ \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})^{opp} , \\quad \\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) $$ commutes with arbitrary colimits, in a formula $$ \\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G}) = \\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}). $$ \\end{enumerate} Suppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings. \\begin{enumerate} \\item[(3)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}) , \\quad \\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) $$ commutes with arbitrary limits. \\item[(4)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})^{opp} , \\quad \\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) $$ commutes with arbitrary colimits, in a formula $$ \\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G}) = \\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}). $$ \\end{enumerate}"}
+{"_id": "14194", "title": "sites-modules-lemma-internal-hom-adjoint-tensor", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. \\begin{enumerate} \\item Let $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be presheaves of $\\mathcal{O}$-modules. There is a canonical isomorphism $$ \\SheafHom_\\mathcal{O} (\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H}) \\longrightarrow \\SheafHom_\\mathcal{O} (\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, $$ \\Mor_{\\textit{PMod}(\\mathcal{O})}( \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H}) = \\Mor_{\\textit{PMod}(\\mathcal{O})}( \\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H})) $$ \\item Suppose that $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings, and $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ are sheaves of $\\mathcal{O}$-modules. There is a canonical isomorphism $$ \\SheafHom_\\mathcal{O} (\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H}) \\longrightarrow \\SheafHom_\\mathcal{O} (\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, $$ \\Mor_{\\textit{Mod}(\\mathcal{O})}( \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H}) = \\Mor_{\\textit{Mod}(\\mathcal{O})}( \\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H})) $$ \\end{enumerate}"}
+{"_id": "14195", "title": "sites-modules-lemma-tensor-commute-colimits", "text": "Tensor product and colimits. Let $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings. \\begin{enumerate} \\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor $$ \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O}) , \\quad \\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G} $$ commutes with arbitrary colimits. \\item Suppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings. For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}) , \\quad \\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} $$ commutes with arbitrary colimits. \\end{enumerate}"}
+{"_id": "14196", "title": "sites-modules-lemma-adjoint-hom-restrict", "text": "Let $\\mathcal{C}$ be a category, resp.\\ a site Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map of presheaves, resp.\\ sheaves of rings. Then $$ \\Hom_\\mathcal{O}(\\mathcal{G}, \\mathcal{F}) = \\Hom_{\\mathcal{O}'}(\\mathcal{G}, \\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{F})) $$ for any $\\mathcal{O}'$-module $\\mathcal{G}$ and $\\mathcal{O}$-module $\\mathcal{F}$."}
+{"_id": "14197", "title": "sites-modules-lemma-j-shriek-and-tensor", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$ and $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{O})$ we have $j_{U!}\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} = j_{U!}(\\mathcal{G} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U)$."}
+{"_id": "14198", "title": "sites-modules-lemma-flatness-presheaves", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. If each $\\mathcal{F}(U)$ is a flat $\\mathcal{O}(U)$-module, then $\\mathcal{F}$ is flat."}
+{"_id": "14199", "title": "sites-modules-lemma-flatness-sheafification", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then $\\mathcal{F}^\\#$ is a flat $\\mathcal{O}^\\#$-module."}
+{"_id": "14200", "title": "sites-modules-lemma-colimits-flat", "text": "Colimits and tensor product. \\begin{enumerate} \\item A filtered colimit of flat presheaves of modules is flat. A direct sum of flat presheaves of modules is flat. \\item A filtered colimit of flat sheaves of modules is flat. A direct sum of flat sheaves of modules is flat. \\end{enumerate}"}
+{"_id": "14201", "title": "sites-modules-lemma-restriction-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of $\\mathcal{C}$. If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then $\\mathcal{F}|_U$ is a flat $\\mathcal{O}_U$-module."}
+{"_id": "14202", "title": "sites-modules-lemma-j-shriek-flat", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\\mathcal{C}$. Consider the functor $j_U : \\mathcal{C}/U \\to \\mathcal{C}$. \\begin{enumerate} \\item The presheaf of $\\mathcal{O}$-modules $j_{U!}\\mathcal{O}_U$ (see Remark \\ref{remark-localize-presheaves}) is flat. \\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings, $j_{U!}\\mathcal{O}_U$ is a flat sheaf of $\\mathcal{O}$-modules. \\end{enumerate}"}
+{"_id": "14203", "title": "sites-modules-lemma-module-quotient-flat", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. \\begin{enumerate} \\item Any presheaf of $\\mathcal{O}$-modules is a quotient of a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$. \\item Any presheaf of $\\mathcal{O}$-modules is a quotient of a flat presheaf of $\\mathcal{O}$-modules. \\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings, then any sheaf of $\\mathcal{O}$-modules is a quotient of a direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$. \\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings, then any sheaf of $\\mathcal{O}$-modules is a quotient of a flat sheaf of $\\mathcal{O}$-modules. \\end{enumerate}"}
+{"_id": "14204", "title": "sites-modules-lemma-flat-tor-zero", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $$ 0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0 $$ be a short exact sequence of presheaves of $\\mathcal{O}$-modules. Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules. \\begin{enumerate} \\item If $\\mathcal{F}$ is a flat presheaf of modules, then the sequence $$ 0 \\to \\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to \\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0 $$ is exact. \\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$, $\\mathcal{F}$, $\\mathcal{F}'$, $\\mathcal{F}''$, and $\\mathcal{G}$ are sheaves, and $\\mathcal{F}$ is flat as a sheaf of modules, then the sequence $$ 0 \\to \\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to \\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0 $$ is exact. \\end{enumerate}"}
+{"_id": "14205", "title": "sites-modules-lemma-flat-ses", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $$ 0 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F}_0 \\to 0 $$ be a short exact sequence of presheaves of $\\mathcal{O}$-modules. \\begin{enumerate} \\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is $\\mathcal{F}_1$. \\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is $\\mathcal{F}_2$. \\end{enumerate} If $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings then the same result holds in $\\textit{Mod}(\\mathcal{O})$."}
+{"_id": "14206", "title": "sites-modules-lemma-flat-resolution-of-flat", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $$ \\ldots \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F}_0 \\to \\mathcal{Q} \\to 0 $$ be an exact complex of presheaves of $\\mathcal{O}$-modules. If $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}$-modules, then for any presheaf $\\mathcal{G}$ of $\\mathcal{O}$-modules the complex $$ \\ldots \\to \\mathcal{F}_2 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to \\mathcal{F}_1 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to \\mathcal{F}_0 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to \\mathcal{Q} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0 $$ is exact also. If $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings then the same result holds $\\textit{Mod}(\\mathcal{O})$."}
+{"_id": "14207", "title": "sites-modules-lemma-tensor-flats", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $\\mathcal{G}$ and $\\mathcal{F}$ are flat $\\mathcal{O}$-modules, then $\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}$ is a flat $\\mathcal{O}$-module."}
+{"_id": "14208", "title": "sites-modules-lemma-flat-change-of-rings", "text": "Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on a site $\\mathcal{C}$. If $\\mathcal{G}$ is a flat $\\mathcal{O}_1$-module, then $\\mathcal{G} \\otimes_{\\mathcal{O}_1} \\mathcal{O}_2$ is a flat $\\mathcal{O}_2$-module."}
+{"_id": "14209", "title": "sites-modules-lemma-flat-eq", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module. \\item Let $U$ be an object of $\\mathcal{C}$ and let $$ \\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)} \\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)} \\mathcal{F}|_U $$ be a complex of $\\mathcal{O}_U$-modules. Then there exists a covering $\\{U_i \\to U\\}$ and for each $i$ a factorization $$ \\mathcal{O}_{U_i}^{\\oplus n} \\xrightarrow{B_i} \\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})} \\mathcal{F}|_{U_i} $$ of $(s_1, \\ldots, s_n)|_{U_i}$ such that $B_i \\circ (f_1, \\ldots, f_n)|_{U_i} = 0$. \\item Let $U$ be an object of $\\mathcal{C}$ and let $$ \\mathcal{O}_U^{\\oplus m} \\xrightarrow{A} \\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)} \\mathcal{F}|_U $$ be a complex of $\\mathcal{O}_U$-modules. Then there exists a covering $\\{U_i \\to U\\}$ and for each $i$ a factorization $$ \\mathcal{O}_{U_i}^{\\oplus n} \\xrightarrow{B_i} \\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})} \\mathcal{F}|_{U_i} $$ of $(s_1, \\ldots, s_n)|_{U_i}$ such that $B_i \\circ A|_{U_i} = 0$. \\end{enumerate}"}
+{"_id": "14210", "title": "sites-modules-lemma-flat-over-thickening", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}' \\to \\mathcal{O}$ be a surjection of sheaves of rings whose kernel $\\mathcal{I}$ is an ideal of square zero. Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module and set $\\mathcal{F} = \\mathcal{F}'/\\mathcal{I}\\mathcal{F}'$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}'$ is a flat $\\mathcal{O}'$-module, and \\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module and $\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{F}'$ is injective. \\end{enumerate}"}
+{"_id": "14211", "title": "sites-modules-lemma-left-dual-module", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a $\\mathcal{O}$-module. Let $\\mathcal{G}, \\eta, \\epsilon$ be a left dual of $\\mathcal{F}$ in the monoidal category of $\\mathcal{O}$-modules, see Categories, Definition \\ref{categories-definition-dual}. Then \\begin{enumerate} \\item for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$ is a direct summand of a finite free $\\mathcal{O}|_{U_i}$-module, \\item the map $e : \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{O}) \\to \\mathcal{G}$ sending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$ is an isomorphism, \\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections $f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "14212", "title": "sites-modules-lemma-flat-locally-finite-presentation", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be locally of finite presentation and flat. Then given an object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$ is a direct summand of a finite free $\\mathcal{O}_{U_i}$-module."}
+{"_id": "14213", "title": "sites-modules-lemma-covering-gives-surjection", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$. Then the sequence $$ \\bigoplus j_{U_i \\times_U U_j!}\\mathcal{O}_{U_i \\times_U U_j} \\to \\bigoplus j_{U_i!}\\mathcal{O}_{U_i} \\to j_!\\mathcal{O}_U \\to 0 $$ is exact."}
+{"_id": "14214", "title": "sites-modules-lemma-silly-quasi-compact", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be covering of $\\mathcal{C}$. If $U$ is quasi-compact, then there exist a finite subset $I' \\subset I$ such that the sequence $$ \\bigoplus\\nolimits_{i, i' \\in I'} j_{U_i \\times_U U_{i'}!}\\mathcal{O}_{U_i \\times_U U_{i'}} \\to \\bigoplus\\nolimits_{i \\in I'} j_{U_i!}\\mathcal{O}_{U_i} \\to j_!\\mathcal{O}_U \\to 0 $$ is exact."}
+{"_id": "14215", "title": "sites-modules-lemma-sections-over-quasi-compact", "text": "Let $\\mathcal{C}$ be a site. Let $W$ be a quasi-compact object of $\\mathcal{C}$. \\begin{enumerate} \\item The functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\mapsto \\mathcal{F}(W)$ commutes with coproducts. \\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. The functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}$, $\\mathcal{F} \\mapsto \\mathcal{F}(W)$ commutes with direct sums. \\end{enumerate}"}
+{"_id": "14216", "title": "sites-modules-lemma-quasi-compact-hom-from", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact object of $\\mathcal{C}$. Then the functor $\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, -)$ commutes with direct sums."}
+{"_id": "14217", "title": "sites-modules-lemma-module-quotient-direct-sum", "text": "In Situation \\ref{situation-quasi-compact-objects} assume (\\ref{item-enough}) holds. \\begin{enumerate} \\item Every sheaf of sets is the target of a surjective map whose source is a coproduct $\\coprod h_{U_i}^\\#$ with $U_i$ in $\\mathcal{B}$. \\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module is a quotient of a direct sum $\\bigoplus\\nolimits j_{U_i!}\\mathcal{O}_{U_i}$ with $U_i$ in $\\mathcal{B}$. \\end{enumerate}"}
+{"_id": "14218", "title": "sites-modules-lemma-module-filtered-colimit-constructibles", "text": "In Situation \\ref{situation-quasi-compact-objects} assume (\\ref{item-enough}) and (\\ref{item-enough-qc}) hold. \\begin{enumerate} \\item Every sheaf of sets is a filtered colimit of sheaves of the form \\begin{equation} \\label{equation-towards-constructible-sets} \\text{Coequalizer}\\left( \\xymatrix{ \\coprod\\nolimits_{j = 1, \\ldots, m} h_{V_j}^\\# \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\coprod\\nolimits_{i = 1, \\ldots, n} h_{U_i}^\\# } \\right) \\end{equation} with $U_i$ and $V_j$ in $\\mathcal{B}$. \\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module is a filtered colimit of sheaves of the form \\begin{equation} \\label{equation-towards-constructible} \\Coker\\left( \\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i} \\right) \\end{equation} with $U_i$ and $V_j$ in $\\mathcal{B}$. \\end{enumerate}"}
+{"_id": "14219", "title": "sites-modules-lemma-cokernel-map-towards-constructibles", "text": "In Situation \\ref{situation-quasi-compact-objects} assume (\\ref{item-enough}) and (\\ref{item-enough-qc}) hold. Let $\\mathcal{O}$ be a sheaf of rings. Then a cokernel of a map between modules as in (\\ref{equation-towards-constructible}) is another module as in (\\ref{equation-towards-constructible})."}
+{"_id": "14220", "title": "sites-modules-lemma-change-presentation-towards-constructibles", "text": "In Situation \\ref{situation-quasi-compact-objects} assume (\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs}) hold. Let $\\mathcal{O}$ be a sheaf of rings. Assume given a map $$ \\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i} $$ with $U_i$ and $V_j$ in $\\mathcal{B}$, and coverings $\\{U_{ik} \\to U_i\\}_{k \\in K_i}$ with $U_{ik} \\in \\mathcal{B}$. Then there exist finite subsets $K'_i \\subset K_i$ and a finite set $L$ of $W_l \\in \\mathcal{B}$ and a commutative diagram $$ \\xymatrix{ \\bigoplus_{l \\in L} j_{W_l!}\\mathcal{O}_{W_l} \\ar[d] \\ar[r] & \\bigoplus_{i = 1, \\ldots, n} \\bigoplus_{k \\in K'_i} j_{U_{ik}!}\\mathcal{O}_{U_{ik}} \\ar[d] \\\\ \\bigoplus_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\ar[r] & \\bigoplus_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i} } $$ inducing an isomorphism on cokernels of the horizontal maps."}
+{"_id": "14221", "title": "sites-modules-lemma-extension-towards-constructibles", "text": "In Situation \\ref{situation-quasi-compact-objects} assume (\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs}) hold. Let $\\mathcal{O}$ be a sheaf of rings. Then an extension of modules as in (\\ref{equation-towards-constructible}) is another module as in (\\ref{equation-towards-constructible})."}
+{"_id": "14223", "title": "sites-modules-lemma-flat-pullback-exact", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ be a morphism of ringed topoi. Then $$ f^{-1} : \\textit{Ab}(\\mathcal{C}') \\longrightarrow \\textit{Ab}(\\mathcal{C}), \\quad \\mathcal{F} \\longmapsto f^{-1}\\mathcal{F} $$ is exact. If $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ is a flat morphism of ringed topoi then $$ f^* : \\textit{Mod}(\\mathcal{O}') \\longrightarrow \\textit{Mod}(\\mathcal{O}), \\quad \\mathcal{F} \\longmapsto f^*\\mathcal{F} $$ is exact."}
+{"_id": "14224", "title": "sites-modules-lemma-invertible", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{L}$ be an $\\mathcal{O}$-module. The following are equivalent: \\begin{enumerate} \\item $\\mathcal{L}$ is invertible, and \\item there exists an $\\mathcal{O}$-module $\\mathcal{N}$ such that $\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N} \\cong \\mathcal{O}$. \\end{enumerate} In this case we have \\begin{enumerate} \\item[(a)] $\\mathcal{L}$ is a flat $\\mathcal{O}$-module of finite presentation, \\item[(b)] for every object $U$ of $\\mathcal{C}$ there exists a covering $U\\{U_i \\to U\\}$ such that $\\mathcal{L}|_{U_i}$ is a direct summand of a finite free module, and \\item[(c)] the module $\\mathcal{N}$ in (2) is isomorphic to $\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$. \\end{enumerate}"}
+{"_id": "14228", "title": "sites-modules-lemma-universal-module", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor $$ \\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad \\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F}) $$ is representable."}
+{"_id": "14229", "title": "sites-modules-lemma-differentials-sheafify", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of presheaves of rings. Then $\\Omega_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}$ is the sheaf associated to the presheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$."}
+{"_id": "14230", "title": "sites-modules-lemma-pullback-differentials", "text": "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. Then there is a canonical identification $f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} = \\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$ compatible with universal derivations."}
+{"_id": "14231", "title": "sites-modules-lemma-localize-differentials", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. For any object $U$ of $\\mathcal{C}$ there is a canonical isomorphism $$ \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U = \\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)} $$ compatible with universal derivations."}
+{"_id": "14232", "title": "sites-modules-lemma-functoriality-differentials", "text": "Let $\\mathcal{C}$ be a site. Let $$ \\xymatrix{ \\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\ \\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u] } $$ be a commutative diagram of sheaves of rings on $\\mathcal{C}$. The map $\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map $\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$ is a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_2$-modules $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$. It is uniquely characterized by the property that $\\text{d}(f)$ mapsto $\\text{d}(\\varphi(f))$ for any local section $f$ of $\\mathcal{O}_2$. In this way $\\Omega_{-/-}$ becomes a functor on the category of arrows of sheaves of rings."}
+{"_id": "14233", "title": "sites-modules-lemma-differential-seq", "text": "In Lemma \\ref{lemma-functoriality-differentials} suppose that $\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel $\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that $\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact sequence of $\\mathcal{O}'_2$-modules $$ \\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2 \\longrightarrow \\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1} \\longrightarrow 0 $$ The leftmost map is characterized by the rule that a local section $f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$."}
+{"_id": "14234", "title": "sites-modules-lemma-double-structure-gives-derivation", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Consider a short exact sequence $$ 0 \\to \\mathcal{F} \\to \\mathcal{A} \\to \\mathcal{O}_2 \\to 0 $$ Here $\\mathcal{A}$ is a sheaf of $\\mathcal{O}_1$-algebras, $\\pi : \\mathcal{A} \\to \\mathcal{O}_2$ is a surjection of sheaves of $\\mathcal{O}_1$-algebras, and $\\mathcal{F} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{F}$ an ideal sheaf with square zero in $\\mathcal{A}$. So $\\mathcal{F}$ has a natural structure of an $\\mathcal{O}_2$-module. A section $s : \\mathcal{O}_2 \\to \\mathcal{A}$ of $\\pi$ is a $\\mathcal{O}_1$-algebra map such that $\\pi \\circ s = \\text{id}$. Given any section $s : \\mathcal{O}_2 \\to \\mathcal{F}$ of $\\pi$ and any $\\varphi$-derivation $D : \\mathcal{O}_1 \\to \\mathcal{F}$ the map $$ s + D : \\mathcal{O}_1 \\to \\mathcal{A} $$ is a section of $\\pi$ and every section $s'$ is of the form $s + D$ for a unique $\\varphi$-derivation $D$."}
+{"_id": "14235", "title": "sites-modules-lemma-functoriality-differentials-ringed-topoi", "text": "Let $X = (\\Sh(\\mathcal{C}_X), \\mathcal{O}_X)$, $Y = (\\Sh(\\mathcal{C}_Y), \\mathcal{O}_Y)$, $X' = (\\Sh(\\mathcal{C}_{X'}), \\mathcal{O}_{X'})$, and $Y' = (\\Sh(\\mathcal{C}_{Y'}), \\mathcal{O}_{Y'})$ be ringed topoi. Let $$ \\xymatrix{ X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ be a commutative diagram of morphisms of ringed topoi. The map $f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map $f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules $\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\\mathcal{O}_{X'}$-module homomorphism $$ c_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}. $$ It is uniquely characterized by the property that $f^*\\text{d}_{X/Y}(t)$ mapsto $\\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\\mathcal{O}_X$."}
+{"_id": "14236", "title": "sites-modules-lemma-composition-differential-operators", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings. Let $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules. If $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$ are differential operators of order $k$ and $k'$, then $D' \\circ D$ is a differential operator of order $k + k'$."}
+{"_id": "14237", "title": "sites-modules-lemma-module-principal-parts", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. Let $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$ and a canonical isomorphism $$ \\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G}) $$ functorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$."}
+{"_id": "14238", "title": "sites-modules-lemma-differential-operators-sheafify", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of presheaves of rings. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_2$-modules. Then $\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$ is the sheaf associated to the presheaf $U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$."}
+{"_id": "14239", "title": "sites-modules-lemma-sequence-of-principal-parts", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. There is a canonical short exact sequence $$ 0 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to \\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to \\mathcal{F} \\to 0 $$ functorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}."}
+{"_id": "14240", "title": "sites-modules-lemma-NL-up-to-qis", "text": "In the situation above there is a canonical isomorphism $\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$."}
+{"_id": "14241", "title": "sites-modules-lemma-pullback-NL", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be morphism of topoi. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{D}$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} = \\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$."}
+{"_id": "14242", "title": "sites-modules-lemma-stalk-exact", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$. \\begin{enumerate} \\item We have $(\\mathcal{F}^\\#)_p = \\mathcal{F}_p$ for any presheaf of sets on $\\mathcal{C}$. \\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see Categories, Definition \\ref{categories-definition-exact}) and commutes with arbitrary colimits. \\item The stalk functor $\\textit{PSh}(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see Categories, Definition \\ref{categories-definition-exact}) and commutes with arbitrary colimits. \\end{enumerate}"}
+{"_id": "14243", "title": "sites-modules-lemma-stalk-exact-abelian", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$. \\begin{enumerate} \\item The functor $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact. \\item The stalk functor $\\textit{PAb}(\\mathcal{C}) \\to \\textit{Ab}$, $\\mathcal{F}  \\mapsto  \\mathcal{F}_p$ is exact. \\item For $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$ we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$. \\end{enumerate}"}
+{"_id": "14244", "title": "sites-modules-lemma-stalk-exact-modules", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $p$ be a point of $\\mathcal{C}$. \\begin{enumerate} \\item The functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact. \\item The stalk functor $\\textit{PMod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact. \\item For $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}))$ we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$. \\end{enumerate}"}
+{"_id": "14245", "title": "sites-modules-lemma-pullback-stalk", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Let $p$ be a point of $\\mathcal{C}$ or $\\Sh(\\mathcal{C})$ and set $q = f \\circ p$. Then $$ (f^*\\mathcal{F})_p = \\mathcal{F}_q \\otimes_{\\mathcal{O}_{\\mathcal{D}, q}} \\mathcal{O}_{\\mathcal{C}, p} $$ for any $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{F}$."}
+{"_id": "14246", "title": "sites-modules-lemma-skyscraper-exact", "text": "Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ or of its associated topos. \\begin{enumerate} \\item The functor $p_* : \\textit{Ab} \\to \\textit{Ab}(\\mathcal{C})$, $A \\mapsto p_*A$ is exact. \\item There is a functorial direct sum decomposition $$ p^{-1}p_*A = A \\oplus I(A) $$ for $A \\in \\Ob(\\textit{Ab})$. \\end{enumerate}"}
+{"_id": "14248", "title": "sites-modules-lemma-stalk-j-shriek", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $p$ be a point of $\\mathcal{C}$. Let $U$ be an object of $\\mathcal{C}$. For $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$ we have $$ (j_{U!}\\mathcal{G})_p = \\bigoplus\\nolimits_q \\mathcal{G}_q $$ where the coproduct is over the points $q$ of $\\mathcal{C}/U$ lying over $p$, see Sites, Lemma \\ref{sites-lemma-points-above-point}."}
+{"_id": "14249", "title": "sites-modules-lemma-pullback-flat", "text": "\\begin{reference} \\cite[Expos\\'e V, Corollary 1.7.1]{SGA4} \\end{reference} Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Then $f^*\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{C}$-module whenever $\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{D}$-module."}
+{"_id": "14250", "title": "sites-modules-lemma-stalk-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $p$ be a point of $\\mathcal{C}$. If $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then $\\mathcal{F}_p$ is a flat $\\mathcal{O}_p$-module."}
+{"_id": "14251", "title": "sites-modules-lemma-check-flat-stalks", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Let $\\{p_i\\}_{i \\in I}$ be a conservative family of points of $\\mathcal{C}$. Then $\\mathcal{F}$ is flat if and only if $\\mathcal{F}_{p_i}$ is a flat $\\mathcal{O}_{p_i}$-module for all $i \\in I$."}
+{"_id": "14252", "title": "sites-modules-lemma-pullback-ses", "text": "Let $f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}'), \\mathcal{O})$ be a morphism of ringed topoi. Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$ be a short exact sequence of $\\mathcal{O}$-modules with $\\mathcal{H}$ a flat $\\mathcal{O}$-module. Then the sequence $0 \\to f^*\\mathcal{F} \\to f^*\\mathcal{G} \\to f^*\\mathcal{H} \\to 0$ is exact as well."}
+{"_id": "14253", "title": "sites-modules-lemma-locally-ringed", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The following are equivalent \\begin{enumerate} \\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$ there exists a covering $\\{U_j \\to U\\}$ such that for each $j$ either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible. \\item For $U \\in \\Ob(\\mathcal{C})$, $n \\geq 1$, and $f_1, \\ldots, f_n \\in \\mathcal{O}(U)$ which generate the unit ideal in $\\mathcal{O}(U)$ there exists a covering $\\{U_j \\to U\\}$ such that for each $j$ there exists an $i$ such that $f_i|_{U_j}$ is invertible. \\item The map of sheaves of sets $$ (\\mathcal{O} \\times \\mathcal{O}) \\amalg (\\mathcal{O} \\times \\mathcal{O}) \\longrightarrow \\mathcal{O} \\times \\mathcal{O} $$ which maps $(f, a)$ in the first component to $(f, af)$ and $(f, b)$ in the second component to $(f, b(1 - f))$ is surjective. \\end{enumerate}"}
+{"_id": "14254", "title": "sites-modules-lemma-locally-ringed-stalk", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Consider the following conditions \\begin{enumerate} \\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$ there exists a covering $\\{U_j \\to U\\}$ such that for each $j$ either $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible. \\item For every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$ is either the zero ring or a local ring. \\end{enumerate} We always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points then (1) and (2) are equivalent."}
+{"_id": "14255", "title": "sites-modules-lemma-ringed-stalk-not-zero", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Consider the statements \\begin{enumerate} \\item (\\ref{equation-one-is-never-zero}) is an isomorphism, and \\item for every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$ is not the zero ring. \\end{enumerate} We always have (1) $\\Rightarrow$ (2) and if $\\mathcal{C}$ has enough points then (1) $\\Leftrightarrow$ (2)."}
+{"_id": "14256", "title": "sites-modules-lemma-locally-ringed-intrinsic", "text": "Being a locally ringed site is an intrinsic property. More precisely, \\begin{enumerate} \\item if $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$ is a morphism of topoi and $(\\mathcal{C}, \\mathcal{O})$ is a locally ringed site, then $(\\mathcal{C}', f^{-1}\\mathcal{O})$ is a locally ringed site, and \\item if $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ is an equivalence of ringed topoi, then $(\\mathcal{C}, \\mathcal{O})$ is locally ringed if and only if $(\\mathcal{C}', \\mathcal{O}')$ is locally ringed. \\end{enumerate}"}
+{"_id": "14257", "title": "sites-modules-lemma-invertible-is-locally-free-rank-1", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Any locally free $\\mathcal{O}$-module of rank $1$ is invertible. If $(\\mathcal{C}, \\mathcal{O})$ is locally ringed, then the converse holds as well (but in general this is not the case)."}
+{"_id": "14258", "title": "sites-modules-lemma-locally-ringed-morphism", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Consider the following conditions \\begin{enumerate} \\item The diagram of sheaves $$ \\xymatrix{ f^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] & \\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\ f^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} & \\mathcal{O}_\\mathcal{C} } $$ is cartesian. \\item For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$, the diagram $$ \\xymatrix{ \\mathcal{O}^*_{\\mathcal{D}, q} \\ar[r] \\ar[d] & \\mathcal{O}^*_{\\mathcal{C}, p} \\ar[d] \\\\ \\mathcal{O}_{\\mathcal{D}, q} \\ar[r] & \\mathcal{O}_{\\mathcal{C}, p} } $$ of sets is cartesian. \\end{enumerate} We always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points then (1) and (2) are equivalent. If $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ and $(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ are locally ringed topoi then (2) is equivalent to \\begin{enumerate} \\item[(3)] For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$, the ring map $\\mathcal{O}_{\\mathcal{D}, q} \\to \\mathcal{O}_{\\mathcal{C}, p}$ is a local ring map. \\end{enumerate} In fact, properties (2), or (3) for a conservative family of points implies (1)."}
+{"_id": "14259", "title": "sites-modules-lemma-composition-morphisms-locally-ringed-topoi", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and $(g, g^\\sharp) : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to (\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of locally ringed topoi. Then the composition $(g, g^\\sharp) \\circ (f, f^\\sharp)$ (see Definition \\ref{definition-ringed-topos}) is also a morphism of locally ringed topoi."}
+{"_id": "14260", "title": "sites-modules-lemma-locally-ringed-intrinsic-morphism", "text": "If $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$ is a morphism of topoi. If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}$, then $$ f^{-1}(\\mathcal{O}^*) = (f^{-1}\\mathcal{O})^*. $$ In particular, if $\\mathcal{O}$ turns $\\mathcal{C}$ into a locally ringed site, then setting $f^\\sharp = \\text{id}$ the morphism of ringed topoi $$ (f, f^\\sharp) : (\\Sh(\\mathcal{C}'), f^{-1}\\mathcal{O}) \\to (\\Sh(\\mathcal{C}, \\mathcal{O}) $$ is a morphism of locally ringed topoi."}
+{"_id": "14261", "title": "sites-modules-lemma-localize-morphism-locally-ringed-topoi", "text": "Localization of locally ringed sites and topoi. \\begin{enumerate} \\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site. Let $U$ be an object of $\\mathcal{C}$. Then the localization $(\\mathcal{C}/U, \\mathcal{O}_U)$ is a locally ringed site, and the localization morphism $$ (j_U, j_U^\\sharp) : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}) $$ is a morphism of locally ringed topoi. \\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site. Let $f : V \\to U$ be a morphism of $\\mathcal{C}$. Then the morphism $$ (j, j^\\sharp) : (\\Sh(\\mathcal{C}/V), \\mathcal{O}_V) \\to (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) $$ of Lemma \\ref{lemma-relocalize} is a morphism of locally ringed topoi. \\item Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}) \\longrightarrow (\\mathcal{D}, \\mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V$ be an object of $\\mathcal{D}$ and let $U = u(V)$. Then the morphism $$ (f', (f')^\\sharp) : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V) $$ of Lemma \\ref{lemma-localize-morphism-ringed-sites} is a morphism of locally ringed sites. \\item Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}) \\longrightarrow (\\mathcal{D}, \\mathcal{O}')$ be a morphism of locally ringed sites where $f$ is given by the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$, and $c : U \\to u(V)$. Then the morphism $$ (f_c, (f_c)^\\sharp) : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V) $$ of Lemma \\ref{lemma-relocalize-morphism-ringed-sites} is a morphism of locally ringed topoi. \\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally ringed topos. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. Then the localization $(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$ is a locally ringed topos and the localization morphism $$ (j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}) $$ is a morphism of locally ringed topoi. \\item  Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally ringed topos. Let $s : \\mathcal{G} \\to \\mathcal{F}$ be a map of sheaves on $\\mathcal{C}$. Then the morphism $$ (j, j^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G}) \\longrightarrow (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) $$ of Lemma \\ref{lemma-relocalize-ringed-topos} is a morphism of locally ringed topoi. \\item Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$. Then the morphism $$ (f', (f')^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\longrightarrow (\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G}) $$ of Lemma \\ref{lemma-localize-morphism-ringed-topoi} is a morphism of locally ringed topoi. \\item  Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$, let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$, and let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be a morphism of sheaves. Then the morphism $$ (f_s, (f_s)^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\longrightarrow (\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G}) $$ of Lemma \\ref{lemma-relocalize-morphism-ringed-topoi} is a morphism of locally ringed topoi. \\end{enumerate}"}
+{"_id": "14262", "title": "sites-modules-lemma-lower-shriek-modules", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor between sites. Denote $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ the associated morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings on $\\mathcal{D}$. Set $\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$. Hence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$. In this case there exists a functor $$ g_! : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\longrightarrow \\textit{Mod}(\\mathcal{O}_\\mathcal{D}) $$ which is left adjoint to $g^*$."}
+{"_id": "14263", "title": "sites-modules-lemma-special-square-cocontinuous", "text": "Assume given a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ of ringed topoi. Assume \\begin{enumerate} \\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors $u$, $u'$, $v$, and $v'$ as in Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}, \\item $v \\circ u' = u \\circ v'$, \\item $v$ and $v'$ are continuous as well as cocontinuous, \\item for any object $V'$ of $\\mathcal{D}'$ the functor ${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$ given by $v$ is cofinal, and \\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$ and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$. \\end{enumerate} Then we have $f'_* \\circ (g')^* = g^* \\circ f_*$ and $g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules."}
+{"_id": "14264", "title": "sites-modules-lemma-special-square-continuous", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ of ringed topoi and suppose we have functors $$ \\xymatrix{ \\mathcal{C}' \\ar[r]_{v'} & \\mathcal{C} \\\\ \\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} & \\mathcal{D} \\ar[u]_u } $$ such that (with notation as in Sites, Sections \\ref{sites-section-morphism-sites} and \\ref{sites-section-cocontinuous-morphism-topoi}) we have \\begin{enumerate} \\item $u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$, \\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$, \\item $u \\circ v = v' \\circ u'$, \\item $v$ and $v'$ are continuous as well as cocontinuous, and \\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$ and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$. \\end{enumerate} Then $f'_* \\circ (g')^* = g^* \\circ f_*$ and $g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules."}
+{"_id": "14265", "title": "sites-modules-lemma-constant-exact", "text": "Let $\\mathcal{C}$ be a site. If $0 \\to A \\to B \\to C \\to 0$ is a short exact sequence of abelian groups, then $0 \\to \\underline{A} \\to \\underline{B} \\to \\underline{C} \\to 0$ is an exact sequence of abelian sheaves and in fact it is even exact as a sequence of abelian presheaves."}
+{"_id": "14266", "title": "sites-modules-lemma-tensor-with-finitely-presented", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$ and $Q$ be $\\Lambda$-modules. If $Q$ is a finitely presented $\\Lambda$-module, then we have $\\underline{M \\otimes_\\Lambda Q}(U) = \\underline{M}(U) \\otimes_\\Lambda Q$ for all $U \\in \\Ob(\\mathcal{C})$."}
+{"_id": "14267", "title": "sites-modules-lemma-flat-sections", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a coherent ring. Let $M$ be a flat $\\Lambda$-module. For $U \\in \\Ob(\\mathcal{C})$ the module $\\underline{M}(U)$ is a flat $\\Lambda$-module."}
+{"_id": "14268", "title": "sites-modules-lemma-completion-flat", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal. The sheaf $\\underline{\\Lambda}^\\wedge = \\lim \\underline{\\Lambda/I^n}$ is a flat $\\underline{\\Lambda}$-algebra. Moreover we have canonical identifications $$ \\underline{\\Lambda}/I\\underline{\\Lambda} = \\underline{\\Lambda}/\\underline{I} = \\underline{\\Lambda}^\\wedge/I\\underline{\\Lambda}^\\wedge = \\underline{\\Lambda}^\\wedge/\\underline{I} \\cdot \\underline{\\Lambda}^\\wedge = \\underline{\\Lambda}^\\wedge/\\underline{I}^\\wedge = \\underline{\\Lambda/I} $$ where $\\underline{I}^\\wedge = \\lim \\underline{I/I^n}$."}
+{"_id": "14269", "title": "sites-modules-lemma-locally-constant-finite-type", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$ be a $\\Lambda$-module. Assume $\\Sh(\\mathcal{C})$ is not the empty topos. Then \\begin{enumerate} \\item $\\underline{M}$ is a finite type sheaf of $\\underline{\\Lambda}$-modules if and only if $M$ is a finite $\\Lambda$-module, and \\item $\\underline{M}$ is a finitely presented sheaf of $\\underline{\\Lambda}$-modules if and only if $M$ is a  finitely presented $\\Lambda$-module. \\end{enumerate}"}
+{"_id": "14270", "title": "sites-modules-lemma-pullback-locally-constant", "text": "Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi. If $\\mathcal{G}$ is a locally constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc on $\\mathcal{D}$, the same is true for $f^{-1}\\mathcal{G}$ on $\\mathcal{C}$."}
+{"_id": "14271", "title": "sites-modules-lemma-morphism-locally-constant", "text": "Let $\\mathcal{C}$ be a site with a final object $X$. \\begin{enumerate} \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of sets on $\\mathcal{C}$. If $\\mathcal{F}$ is finite locally constant, there exists a covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant sheaves associated to a map of sets. \\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of abelian groups on $\\mathcal{C}$. If $\\mathcal{F}$ is finite locally constant, there exists a covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant abelian sheaves associated to a map of abelian groups. \\item Let $\\Lambda$ be a ring. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of locally constant sheaves of $\\Lambda$-modules on $\\mathcal{C}$. If $\\mathcal{F}$ is of finite type, then there exists a covering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant sheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules. \\end{enumerate}"}
+{"_id": "14272", "title": "sites-modules-lemma-locally-constant", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring. Let $M$, $N$ be $\\Lambda$-modules. Let $\\mathcal{F}, \\mathcal{G}$ be a locally constant sheaves of $\\Lambda$-modules. \\begin{enumerate} \\item If $M$ is of finite presentation, then $$ \\underline{\\Hom_\\Lambda(M, N)} = \\SheafHom_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N}) $$ \\item If $M$ and $N$ are both of finite presentation, then $$ \\underline{\\text{Isom}_\\Lambda(M, N)} = \\mathit{Isom}_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N}) $$ \\item If $\\mathcal{F}$ is of finite presentation, then $\\SheafHom_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$ is a locally constant sheaf of $\\Lambda$-modules. \\item If $\\mathcal{F}$ and $\\mathcal{G}$ are both of finite presentation, then $\\mathit{Isom}_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$ is a locally constant sheaf of sets. \\end{enumerate}"}
+{"_id": "14273", "title": "sites-modules-lemma-kernel-finite-locally-constant", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside $\\Sh(\\mathcal{C})$. \\item The category of finite locally constant abelian sheaves is a weak Serre subcategory of $\\textit{Ab}(\\mathcal{C})$. \\item Let $\\Lambda$ be a Noetherian ring. The category of finite type, locally constant sheaves of $\\Lambda$-modules on $\\mathcal{C}$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{C}, \\Lambda)$. \\end{enumerate}"}
+{"_id": "14274", "title": "sites-modules-lemma-tensor-product-locally-constant", "text": "Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\\Lambda$-modules on $\\mathcal{C}$ is a locally constant sheaf of $\\Lambda$-modules."}
+{"_id": "14275", "title": "sites-modules-lemma-simple-invert", "text": "In the situation above the map to the sheafification $$ \\mathcal{O} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{O})^\\# $$ is a homomorphism of sheaves of rings with the following universal property: for any homomorphism of sheaves of rings $\\mathcal{O} \\to \\mathcal{A}$ such that each local section of $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$ there exists a unique factorization $(\\mathcal{S}^{-1}\\mathcal{O})^\\# \\to \\mathcal{A}$."}
+{"_id": "14276", "title": "sites-modules-lemma-simple-invert-module", "text": "In the situation above the map to the sheafification $$ \\mathcal{F} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{F})^\\# $$ has the following universal property: for any homomorphism of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$ such that each local section of $\\mathcal{S}$ acts invertibly on $\\mathcal{G}$ there exists a unique factorization $(\\mathcal{S}^{-1}\\mathcal{F})^\\# \\to \\mathcal{G}$. Moreover we have $$ (\\mathcal{S}^{-1}\\mathcal{F})^\\# = (\\mathcal{S}^{-1}\\mathcal{O})^\\# \\otimes_\\mathcal{O} \\mathcal{F} $$ as sheaves of $(\\mathcal{S}^{-1}\\mathcal{O})^\\#$-modules."}
+{"_id": "14313", "title": "derham-lemma-base-change-de-rham", "text": "Let $$ \\xymatrix{ X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\ S' \\ar[r] & S } $$ be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms $f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$."}
+{"_id": "14314", "title": "derham-lemma-etale", "text": "Consider a commutative diagram of schemes $$ \\xymatrix{ X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\ S' \\ar[r] & S } $$ If $X' \\to X$ and $S' \\to S$ are \\'etale, then the maps discussed above induce isomorphisms $f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$."}
+{"_id": "14315", "title": "derham-lemma-de-rham-affine", "text": "Let $X \\to S$ be a morphism of affine schemes given by the ring map $R \\to A$. Then $R\\Gamma(X, \\Omega^\\bullet_{X/S}) = \\Omega^\\bullet_{A/R}$ in $D(R)$ and $H^i_{dR}(X/S) = H^i(\\Omega^\\bullet_{A/R})$."}
+{"_id": "14316", "title": "derham-lemma-quasi-coherence-relative", "text": "Let $p : X \\to S$ be a morphism of schemes. If $p$ is quasi-compact and quasi-separated, then $Rp_*\\Omega^\\bullet_{X/S}$ is an object of $D_\\QCoh(\\mathcal{O}_S)$."}
+{"_id": "14317", "title": "derham-lemma-coherence-relative", "text": "Let $p : X \\to S$ be a proper morphism of schemes with $S$ locally Noetherian. Then $Rp_*\\Omega^\\bullet_{X/S}$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_S)$."}
+{"_id": "14319", "title": "derham-lemma-proper-smooth-de-Rham", "text": "Let $f : X \\to S$ be a proper smooth morphism of schemes. Then $Rf_*\\Omega^p_{X/S}$, $p \\geq 0$ and $Rf_*\\Omega^\\bullet_{X/S}$ are perfect objects of $D(\\mathcal{O}_S)$ whose formation commutes with arbitrary change of base."}
+{"_id": "14320", "title": "derham-lemma-cup-product-graded-commutative", "text": "Let $p : X \\to S$ be a morphism of schemes. The cup product on $H^*_{dR}(X/S)$ is associative and graded commutative."}
+{"_id": "14322", "title": "derham-lemma-de-rham-complex-product", "text": "In the situation above there is a canonical isomorphism $$ \\text{Tot}(\\Omega^\\bullet_{X/S} \\boxtimes \\Omega^\\bullet_{Y/S}) \\longrightarrow \\Omega^\\bullet_{X \\times_S Y/S} $$ of complexes of $f^{-1}\\mathcal{O}_S$-modules."}
+{"_id": "14323", "title": "derham-lemma-kunneth-de-rham", "text": "Assume $X$ and $Y$ are smooth, quasi-compact, with affine diagonal over $S = \\Spec(A)$. Then the map $$ R\\Gamma(X, \\Omega^\\bullet_{X/S}) \\otimes_A^\\mathbf{L} R\\Gamma(Y, \\Omega^\\bullet_{Y/S}) \\longrightarrow R\\Gamma(X \\times_S Y, \\Omega^\\bullet_{X \\times_S Y/S}) $$ is an isomorphism in $D(A)$."}
+{"_id": "14325", "title": "derham-lemma-pullback-c1", "text": "Given a commutative diagram $$ \\xymatrix{ X' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\ S' \\ar[r] & S } $$ of schemes the diagrams $$ \\xymatrix{ \\Pic(X') \\ar[d]_{c_1^{dR}} & \\Pic(X) \\ar[d]^{c_1^{dR}} \\ar[l]^{f^*} \\\\ H^2_{dR}(X'/S') & H^2_{dR}(X/S) \\ar[l]_{f^*} } \\quad \\xymatrix{ \\Pic(X') \\ar[d]_{c_1^{Hodge}} & \\Pic(X) \\ar[d]^{c_1^{Hodge}} \\ar[l]^{f^*} \\\\ H^1(X', \\Omega^1_{X'/S'}) & H^1(X, \\Omega^1_{X/S}) \\ar[l]_{f^*} } $$ commute."}
+{"_id": "14326", "title": "derham-lemma-the-complex-for-L-star", "text": "With notation as above, there is a short exact sequence of complexes $$ 0 \\to \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{L^\\star/S, 0} \\to \\Omega^\\bullet_{X/S}[-1] \\to 0 $$"}
+{"_id": "14327", "title": "derham-lemma-the-complex-for-L-star-gives-chern-class", "text": "The ``boundary'' map $\\delta : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]$ in $D(X, f^{-1}\\mathcal{O}_S)$ coming from the short exact sequence in Lemma \\ref{lemma-the-complex-for-L-star} is the map of Remark \\ref{remark-cup-product-as-a-map} for $\\xi = c_1^{dR}(\\mathcal{L})$."}
+{"_id": "14328", "title": "derham-lemma-push-omega-a", "text": "With notation as above we have \\begin{enumerate} \\item $\\Omega^p_{L^\\star/S, n} = \\Omega^p_{L^\\star/S, 0} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$ for all $n \\in \\mathbf{Z}$ as quasi-coherent $\\mathcal{O}_X$-modules, \\item $\\Omega^\\bullet_{X/S} = \\Omega^\\bullet_{L/X, 0}$ as complexes, and \\item for $n > 0$ and $p \\geq 0$ we have $\\Omega^p_{L/X, n} = \\Omega^p_{L^\\star/S, n}$. \\end{enumerate}"}
+{"_id": "14329", "title": "derham-lemma-line-bundle-characteristic-zero", "text": "In the situation above, assume there is a morphism $S \\to \\Spec(\\mathbf{Q})$. Then $\\Omega^\\bullet_{X/S} \\to \\pi_*\\Omega^\\bullet_{L/S}$ is a quasi-isomorphism and $H_{dR}^*(X/S) = H_{dR}^*(L/S)$."}
+{"_id": "14330", "title": "derham-lemma-euler-sequence", "text": "There exists a short exact sequence $$ 0 \\to \\Omega \\to \\mathcal{O}(-1)^{\\oplus n + 1} \\to \\mathcal{O} \\to 0 $$"}
+{"_id": "14331", "title": "derham-lemma-twisted-hodge-cohomology-projective-space", "text": "In the situation above we have the following cohomology groups \\begin{enumerate} \\item $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$ unless $0 \\leq p = q \\leq n$, \\item for $0 \\leq p \\leq n$ the $A$-module $H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$. \\item for $q > 0$, $k > 0$, and $p$ arbitrary we have $H^q(\\mathbf{P}^n_A, \\Omega^p(k)) = 0$, and \\item add more here. \\end{enumerate}"}
+{"_id": "14332", "title": "derham-lemma-hodge-cohomology-projective-space", "text": "We have $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$ unless $0 \\leq p = q \\leq n$. For $0 \\leq p \\leq n$ the $A$-module $H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$ with basis element $c_1^{Hodge}(\\mathcal{O}(1))^p$."}
+{"_id": "14333", "title": "derham-lemma-de-rham-cohomology-projective-space", "text": "For $0 \\leq i \\leq n$ the de Rham cohomology $H^{2i}_{dR}(\\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$ with basis element $c_1^{dR}(\\mathcal{O}(1))^i$. In all other degrees the de Rham cohomology of $\\mathbf{P}^n_A$ over $A$ is zero."}
+{"_id": "14334", "title": "derham-lemma-spectral-sequence-smooth", "text": "Let $f : X \\to Y$ be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme $S$. There is a bounded spectral sequence with first page $$ E_1^{p, q} = H^q(\\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/Y}) $$ converging to $R^{p + q}f_*\\Omega^\\bullet_{X/S}$."}
+{"_id": "14335", "title": "derham-lemma-relative-global-generation-on-fibres", "text": "Let $f : X \\to Y$ be a smooth proper morphism of schemes. Let $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let $\\xi_i \\in H^{n_i}_{dR}(X/Y)$, $1 \\leq i \\leq N$. Assume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. Then the map $$ \\bigoplus\\nolimits_{i = 1}^N \\mathcal{O}_Y[-n_i] \\longrightarrow Rf_*\\Omega^\\bullet_{X/Y} $$ associated to $\\xi_1, \\ldots, \\xi_N$ is an isomorphism."}
+{"_id": "14336", "title": "derham-lemma-global-generation-on-fibres", "text": "Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$. Assume \\begin{enumerate} \\item $Y$ and $S$ are affine, and \\item there exist integers $N$ and $n_1, \\ldots, n_N \\geq 0$ and $\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$ such that for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. \\end{enumerate} Then the map $$ \\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow H^*_{dR}(X/S), \\quad (a_1, \\ldots, a_N) \\longmapsto  \\sum \\xi_i \\cup f^*a_i $$ is an isomorphism."}
+{"_id": "14337", "title": "derham-lemma-log-complex", "text": "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. There is a canonical short exact sequence of complexes $$ 0 \\to \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}(\\log Y) \\to \\Omega^\\bullet_{Y/S}[-1] \\to 0 $$"}
+{"_id": "14339", "title": "derham-lemma-gysin-via-log-complex", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. Denote $$ \\delta : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}[2] $$ in $D(X, f^{-1}\\mathcal{O}_S)$ the ``boundary'' map coming from the short exact sequence in Lemma \\ref{lemma-log-complex}. Denote $$ \\xi' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2] $$ in $D(X, f^{-1}\\mathcal{O}_S)$ the map of Remark \\ref{remark-cup-product-as-a-map} corresponding to $\\xi = c_1^{dR}(\\mathcal{O}_X(-Y))$. Denote $$ \\zeta' : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{Y/S}[2] $$ in $D(Y, f|_Y^{-1}\\mathcal{O}_S)$ the map of Remark \\ref{remark-cup-product-as-a-map} corresponding to $\\zeta = c_1^{dR}(\\mathcal{O}_X(-Y)|_Y)$. Then the diagram $$ \\xymatrix{ \\Omega^\\bullet_{X/S} \\ar[d]_{\\xi'} \\ar[r] & \\Omega^\\bullet_{Y/S} \\ar[d]^{\\zeta'} \\ar[ld]_\\delta \\\\ \\Omega^\\bullet_{X/S}[2] \\ar[r] & \\Omega^\\bullet_{Y/S}[2] } $$ is commutative in $D(X, f^{-1}\\mathcal{O}_S)$."}
+{"_id": "14340", "title": "derham-lemma-log-complex-consequence", "text": "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. Let $b \\in H^m_{dR}(X/S)$ be a de Rham cohomology class whose restriction to $Y$ is zero. Then $c_1^{dR}(\\mathcal{O}_X(Y)) \\cup b = 0$ in $H^{m + 2}_{dR}(X/S)$."}
+{"_id": "14341", "title": "derham-lemma-check-log-smooth", "text": "Let $X \\to T \\to S$ be morphisms of schemes. Let $Y \\subset X$ be an effective Cartier divisor. If both $X \\to T$ and $Y \\to T$ are smooth, then the de Rham complex of log poles is defined for $Y \\subset X$ over $S$."}
+{"_id": "14342", "title": "derham-lemma-comparison", "text": "For $a \\geq 0$ we have \\begin{enumerate} \\item the map $\\Omega^a_{X/S} \\to b_*\\Omega^a_{L/S}$ is an isomorphism, \\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{P/S}$ is an isomorphism, and \\item the map $Rb_*\\Omega^a_{L/S} \\to i_*Rp_*\\Omega^a_{P/S}$ is an isomorphism on cohomology sheaves in degree $\\geq 1$. \\end{enumerate}"}
+{"_id": "14343", "title": "derham-lemma-comparison-bis", "text": "For $a \\geq 0$ there are canonical maps $$ b^*\\Omega^a_{X/S} \\longrightarrow \\Omega^a_{L/S} \\longrightarrow b^*\\Omega^a_{X/S} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L((n - 1)E) $$ whose composition is induced by the inclusion $\\mathcal{O}_L \\subset \\mathcal{O}_L((n - 1)E)$."}
+{"_id": "14344", "title": "derham-lemma-blowup-twist-same-cohomology", "text": "Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$. For any locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ and $0 \\leq i \\leq n - 1$ the map $$ \\mathcal{E} \\longrightarrow Rb_*(b^*\\mathcal{E} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L(iE)) $$ is an isomorphism in $D(\\mathcal{O}_X)$."}
+{"_id": "14345", "title": "derham-lemma-blowup", "text": "Let $S$ be a scheme. Let $Z \\to X$ be a closed immersion of schemes smooth over $S$. Let $b : X' \\to X$ be the blowing up of $Z$ with exceptional divisor $E \\subset X'$. Then $X'$ and $E$ are smooth over $S$. The morphism $p : E \\to Z$ is canonically isomorphic to the projective space bundle $$ \\mathbf{P}(\\mathcal{I}/\\mathcal{I}^2) \\longrightarrow Z $$ where $\\mathcal{I} \\subset \\mathcal{O}_X$ is the ideal sheaf of $Z$. The relative $\\mathcal{O}_E(1)$ coming from the projective space bundle structure is isomorphic to the restriction of $\\mathcal{O}_{X'}(-E)$ to $E$."}
+{"_id": "14346", "title": "derham-lemma-comparison-general", "text": "With notation as in Lemma \\ref{lemma-blowup} for $a \\geq 0$ we have \\begin{enumerate} \\item the map $\\Omega^a_{X/S} \\to b_*\\Omega^a_{X'/S}$ is an isomorphism, \\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{E/S}$ is an isomorphism, \\item the map $Rb_*\\Omega^a_{X'/S} \\to i_*Rp_*\\Omega^a_{E/S}$ is an isomorphism on cohomology sheaves in degree $\\geq 1$. \\end{enumerate}"}
+{"_id": "14347", "title": "derham-lemma-distinguished-triangle-blowup", "text": "With notation as in Lemma \\ref{lemma-blowup} and denoting $f : X \\to S$ the structure morphism there is a canonical distinguished triangle $$ \\Omega^\\bullet_{X/S} \\to Rb_*(\\Omega^\\bullet_{X'/S}) \\oplus i_*\\Omega^\\bullet_{Z/S} \\to i_*Rp_*(\\Omega^\\bullet_{E/S}) \\to \\Omega^\\bullet_{X/S}[1] $$ in $D(X, f^{-1}\\mathcal{O}_S)$ where the four maps $$ \\begin{matrix} \\Omega^\\bullet_{X/S} & \\to & Rb_*(\\Omega^\\bullet_{X'/S}), \\\\ \\Omega^\\bullet_{X/S} & \\to & i_*\\Omega^\\bullet_{Z/S}, \\\\ Rb_*(\\Omega^\\bullet_{X'/S}) & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S}), \\\\ i_*\\Omega^\\bullet_{Z/S} & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S}) \\end{matrix} $$ are the canonical ones (Section \\ref{section-de-rham-complex}), except with sign reversed for one of them."}
+{"_id": "14348", "title": "derham-lemma-ext-zero", "text": "Let $i : Z \\to X$ be a closed immersion of schemes which is regular of codimension $c$. Then $\\Ext^q_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E}) = 0$ for $q < c$ for $\\mathcal{E}$ locally free on $X$ and $\\mathcal{F}$ any $\\mathcal{O}_Z$-module."}
+{"_id": "14350", "title": "derham-lemma-funny-map", "text": "Let $R$ be a ring and consider a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & K^0 \\ar[r] & L^0 \\ar[r] & M^0 \\ar[r] & 0 \\\\ & & L^{-1} \\ar[u]_\\partial \\ar@{=}[r] & M^{-1} \\ar[u] } $$ of $R$-modules with exact top row and $M^0$ and $M^{-1}$ finite free of the same rank. Then there are canonical maps $$ \\wedge^i(H^0(L^\\bullet)) \\longrightarrow \\wedge^i(K^0) \\otimes_R \\det(M^\\bullet) $$ whose composition with $\\wedge^i(K^0) \\to \\wedge^i(H^0(L^\\bullet))$ is equal to multiplication with $\\delta(M^\\bullet)$."}
+{"_id": "14351", "title": "derham-lemma-Garel-upstairs", "text": "There exists a unique rule that to every locally quasi-finite syntomic morphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_Y$-module maps $$ c^p_{Y/X} : \\Omega^p_{Y/\\mathbf{Z}} \\longrightarrow f^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X}) $$ satisfying the following two properties \\begin{enumerate} \\item the composition with $f^*\\Omega^p_{X/\\mathbf{Z}} \\to \\Omega^p_{Y/\\mathbf{Z}}$ is multiplication by $\\delta(\\NL_{Y/X})$, and \\item the rule is compatible with restriction to opens and with base change. \\end{enumerate}"}
+{"_id": "14352", "title": "derham-lemma-Garel", "text": "There exists a unique rule that to every finite syntomic morphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_X$-module maps $$ \\Theta^p_{Y/X} : f_*\\Omega^p_{Y/\\mathbf{Z}} \\longrightarrow \\Omega^p_{X/\\mathbf{Z}} $$ satisfying the following properties \\begin{enumerate} \\item the composition with $\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_Y \\to f_*\\Omega^p_{Y/\\mathbf{Z}}$ is equal to $\\text{id} \\otimes \\text{Trace}_f$ where $\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$ is the map from Discriminants, Section \\ref{discriminant-section-discriminant}, \\item the rule is compatible with base change. \\end{enumerate}"}
+{"_id": "14353", "title": "derham-lemma-Garel-map-frobenius-smooth-char-p", "text": "Let $p$ be a prime number. Let $X \\to S$ be a smooth morphism of relative dimension $d$ of schemes in characteristic $p$. The relative Frobenius $F_{X/S} : X \\to X^{(p)}$ of $X/S$ (Varieties, Definition \\ref{varieties-definition-relative-frobenius}) is finite syntomic and the corresponding map $$ \\Theta_{X/X^{(p)}} : F_{X/S, *}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X^{(p)}/S} $$ is zero in all degrees except in degree $d$ where it defines a surjection."}
+{"_id": "14354", "title": "derham-lemma-duality-hodge", "text": "Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$ equidimensional of dimension $d$. There exists a $k$-linear map $$ t : H^d(X, \\Omega^d_{X/k}) \\longrightarrow k $$ unique up to precomposing by multiplication by a unit of $H^0(X, \\mathcal{O}_X)$ with the following property: for all $p, q$ the pairing $$ H^q(X, \\Omega^p_{X/k}) \\times H^{d - q}(X, \\Omega^{d - p}_{X/k}) \\longrightarrow k, \\quad (\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi') $$ is perfect."}
+{"_id": "14355", "title": "derham-lemma-bottom-part-degenerates", "text": "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. The map $$ \\text{d} : H^0(X, \\mathcal{O}_X) \\to H^0(X, \\Omega^1_{X/k}) $$ is zero."}
+{"_id": "14356", "title": "derham-lemma-top-part-degenerates", "text": "Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$ equidimensional of dimension $d$. The map $$ \\text{d} : H^d(X, \\Omega^{d - 1}_{X/k}) \\to H^d(X, \\Omega^d_{X/k}) $$ is zero."}
+{"_id": "14357", "title": "derham-lemma-chern-classes", "text": "There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ a total Chern class $$ c^{dR} : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} H^{2i}_{dR}(X/\\mathbf{Z}) $$ with the following properties \\begin{enumerate} \\item we have $c^{dR}(\\alpha + \\beta) = c^{dR}(\\alpha) c^{dR}(\\beta)$ for $\\alpha, \\beta \\in K_0(\\textit{Vect}(X))$, \\item if $f : X \\to X'$ is a morphism of quasi-compact and quasi-separated schemes, then $c^{dR}(f^*\\alpha) = f^*c^{dR}(\\alpha)$, \\item given $\\mathcal{L} \\in \\Pic(X)$ we have $c^{dR}([\\mathcal{L}]) = 1 + c_1^{dR}(\\mathcal{L})$ \\end{enumerate}"}
+{"_id": "14359", "title": "derham-lemma-gysin-differential", "text": "The gysin map (\\ref{equation-gysin}) is compatible with the de Rham differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$."}
+{"_id": "14360", "title": "derham-lemma-gysin-global", "text": "Let $X \\to S$ be a morphism of schemes. Let $Z \\to X$ be a closed immersion of finite presentation whose conormal sheaf $\\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map $$ \\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}) $$ which is locally given by the maps $\\gamma^p_{f_1, \\ldots, f_c}$ of Remark \\ref{remark-gysin-equations}."}
+{"_id": "14361", "title": "derham-lemma-gysin-differential-global", "text": "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}. The gysin map $\\gamma^p$ is compatible with the de Rham differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$."}
+{"_id": "14362", "title": "derham-lemma-gysin-projection", "text": "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}. Given $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have $\\gamma^p(\\alpha|_Z) = i^{-1}\\alpha \\wedge \\gamma^0(1)$ in $H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}))$. Please see proof for notation."}
+{"_id": "14363", "title": "derham-lemma-gysin-transverse", "text": "Let $c \\geq 0$ be a integer. Let $$ \\xymatrix{ Z' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\ Z \\ar[r] & X \\ar[r] & S } $$ be a commutative diagram of schemes. Assume \\begin{enumerate} \\item $Z \\to X$ and $Z' \\to X'$ satisfy the assumptions of Lemma \\ref{lemma-gysin-global}, \\item the left square in the diagram is cartesian, and \\item $h^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z'/X'}$ (Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}) is an isomorphism. \\end{enumerate} Then the diagram $$ \\xymatrix{ h^*\\Omega^p_{Z/S} \\ar[rr]_-{h^{-1}\\gamma^p} \\ar[d] & & \\mathcal{O}_{X'}|_{Z'} \\otimes_{h^{-1}\\mathcal{O}_X|_Z} h^{-1}\\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}) \\ar[d] \\\\ \\Omega^p_{Z'/S'} \\ar[rr]^{\\gamma^p} & & \\mathcal{H}^c_{Z'}(\\Omega^{p + c}_{X'/S'}) } $$ is commutative. The left vertical arrow is functoriality of modules of differentials and the right vertical arrow uses Cohomology, Remark \\ref{cohomology-remark-support-functorial}."}
+{"_id": "14368", "title": "derham-lemma-relative-bottom-part-degenerates", "text": "In Situation \\ref{situation-relative-duality} the psuhforward $f_*\\mathcal{O}_X$ is a finite \\'etale $\\mathcal{O}_S$-algebra and locally on $S$ we have $Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$ in $D(\\mathcal{O}_S)$ with $P$ perfect of tor amplitude in $[1, \\infty)$. The map $\\text{d} : f_*\\mathcal{O}_X \\to f_*\\Omega_{X/S}$ is zero."}
+{"_id": "14369", "title": "derham-lemma-relative-duality-hodge", "text": "In Situation \\ref{situation-relative-duality} there exists an $\\mathcal{O}_S$-module map $$ t : Rf_*\\Omega^d_{X/S}[d] \\longrightarrow \\mathcal{O}_S $$ unique up to precomposing by multiplication by a unit of $H^0(X, \\mathcal{O}_X)$ with the following property: for all $p$ the pairing $$ Rf_*\\Omega^p_{X/S} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Rf_*\\Omega^{d - p}_{X/S}[d] \\longrightarrow \\mathcal{O}_S $$ given by the relative cup product composed with $t$ is a perfect pairing of perfect complexes on $S$."}
+{"_id": "14370", "title": "derham-lemma-relative-top-part-degenerates", "text": "In Situation \\ref{situation-relative-duality} the map $\\text{d} : R^df_*\\Omega^{d - 1}_{X/S} \\to R^df_*\\Omega^d_{X/S}$ is zero."}
+{"_id": "14371", "title": "derham-proposition-global-generation-on-fibres", "text": "Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$. Let $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let $\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$. Assume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$ in $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. The map $$ \\tilde \\xi = \\bigoplus \\tilde \\xi_i[-n_i] : \\bigoplus \\Omega^\\bullet_{Y/S}[-n_i] \\longrightarrow Rf_*\\Omega^\\bullet_{X/S} $$ (see proof) is an isomorphism in $D(Y, (Y \\to S)^{-1}\\mathcal{O}_S)$ and correspondingly the map $$ \\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow H^*_{dR}(X/S), \\quad (a_1, \\ldots, a_N) \\longmapsto  \\sum \\xi_i \\cup f^*a_i $$ is an isomorphism."}
+{"_id": "14372", "title": "derham-proposition-projective-space-bundle-formula", "text": "Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of constant rank $r$. Consider the morphism $p : P = \\mathbf{P}(\\mathcal{E}) \\to X$. Then the map $$ \\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} H^*_{dR}(X/S) \\longrightarrow H^*_{dR}(P/S) $$ given by the rule $$ (a_0, \\ldots, a_{r - 1}) \\longmapsto \\sum\\nolimits_{i = 0, \\ldots, r - 1} c_1^{dR}(\\mathcal{O}_P(1))^i \\cup p^*(a_i) $$ is an isomorphism."}
+{"_id": "14373", "title": "derham-proposition-blowup-split", "text": "With notation as in Lemma \\ref{lemma-blowup} the map $\\Omega^\\bullet_{X/S} \\to Rb_*\\Omega^\\bullet_{X'/S}$ has a splitting in $D(X, (X \\to S)^{-1}\\mathcal{O}_S)$."}
+{"_id": "14374", "title": "derham-proposition-Garel", "text": "\\begin{reference} \\cite{Garel} \\end{reference} Let $f : Y \\to X$ be a finite syntomic morphism of schemes. The maps $\\Theta^p_{Y/X}$ of Lemma \\ref{lemma-Garel} define a map of complexes $$ \\Theta_{Y/X} : f_*\\Omega^\\bullet_{Y/\\mathbf{Z}} \\longrightarrow \\Omega^\\bullet_{X/\\mathbf{Z}} $$ with the following properties \\begin{enumerate} \\item in degree $0$ we get $\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$, see Discriminants, Section \\ref{discriminant-section-discriminant}, \\item we have $\\Theta_{Y/X}(\\omega \\wedge \\eta) = \\omega \\wedge \\Theta_{Y/X}(\\eta)$ for $\\omega$ in $\\Omega^\\bullet_{X/\\mathbf{Z}}$ and $\\eta$ in $f_*\\Omega^\\bullet_{Y/\\mathbf{Z}}$, \\item if $f$ is a morphism over a base scheme $S$, then $\\Theta_{Y/X}$ induces a map of complexes $f_*\\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}$. \\end{enumerate}"}
+{"_id": "14375", "title": "derham-proposition-poincare-duality", "text": "Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$ equidimensional of dimension $d$. There exists a $k$-linear map $$ t : H^{2d}_{dR}(X/k) \\longrightarrow k $$ unique up to precomposing by multiplication by a unit of $H^0(X, \\mathcal{O}_X)$ with the following property: for all $i$ the pairing $$ H^i_{dR}(X/k) \\times H_{dR}^{2d - i}(X/k) \\longrightarrow k, \\quad (\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi') $$ is perfect."}
+{"_id": "14395", "title": "trace-theorem-baffling", "text": "Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers $j\\geq 0$, $$ F_X^* : H^j (X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\longrightarrow H^j (X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) $$ is the identity."}
+{"_id": "14396", "title": "trace-theorem-geometric-arithmetic-inverse", "text": "Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then for all $j\\geq 0$, $\\text{frob}_k$ acts on the cohomology group $H^j(X_{\\bar k}, \\mathcal{F}|_{X_{\\bar k}})$ as the inverse of the map $\\pi_X^*$."}
+{"_id": "14397", "title": "trace-theorem-trace", "text": "Let $X$ be a projective curve over a finite field $k$, $\\Lambda$ a finite ring and $K \\in D_{ctf}(X, \\Lambda)$. Then the global and local Lefschetz numbers of $K$ are equal, i.e., \\begin{equation} \\label{equation-trace-formula} \\text{Tr}(\\pi^*_X |_{R\\Gamma(X_{\\bar k}, K)}) = \\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_X |_{K_{\\bar x}}) \\end{equation} in $\\Lambda^\\natural$."}
+{"_id": "14398", "title": "trace-theorem-weil-trace-formula", "text": "Let $C$ be a nonsingular projective curve over an algebraically closed field $k$, and $\\varphi : C \\to C$ a $k$-endomorphism of $C$ distinct from the identity. Let $V(\\varphi) = \\Delta_C \\cdot \\Gamma_\\varphi$, where $\\Delta_C$ is the diagonal, $\\Gamma_\\varphi$ is the graph of $\\varphi$, and the intersection number is taken on $C \\times C$. Let $J = \\underline{\\Picardfunctor}^0_{C/k}$ be the jacobian of $C$ and denote $\\varphi^* : J \\to J$ the action induced by $\\varphi$ by taking pullbacks. Then $$ V(\\varphi) = 1 - \\text{Tr}_J(\\varphi^*) + \\deg \\varphi. $$"}
+{"_id": "14399", "title": "trace-theorem-trace-formula-again", "text": "Let $k$ be a finite field and $X$ a finite type, separated scheme of dimension at most 1 over $k$. Let $\\Lambda$ be a finite ring whose cardinality is prime to that of $k$, and $K\\in D_{ctf}(X, \\Lambda)$. Then \\begin{equation} \\label{equation-trace-formula-again} \\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K)}) = \\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_x |_{K_{\\bar x}}) \\end{equation} in $\\Lambda^{\\natural}$."}
+{"_id": "14400", "title": "trace-theorem-A", "text": "Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a constructible flat $\\Lambda$-module on $X_\\etale$. Then $$ L(X, \\mathcal{F}) = \\det(1 - \\pi_X^*\\ T |_{R\\Gamma_c(X_{\\bar k}, \\mathcal{F})})^{-1} \\in \\Lambda[[T]]. $$"}
+{"_id": "14401", "title": "trace-theorem-B", "text": "Let $X$ be a scheme of finite type over a finite field $k$, and $\\mathcal{F}$ a $\\mathbf{Q}_\\ell$-sheaf on $X$. Then $$ L(X, \\mathcal{F}) = \\prod\\nolimits_i \\det(1 - \\pi_X^*T |_{H_c^i(X_{\\bar k} , \\mathcal{F})})^{(-1)^{i + 1}} \\in \\mathbf{Q}_\\ell[[T]]. $$"}
+{"_id": "14402", "title": "trace-theorem-D", "text": "Let $X/k$ be as above, let $\\Lambda$ be a finite ring with $\\#\\Lambda \\in k^*$ and $K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma_c(X_{\\bar k}, K)\\in D_{perf}(\\Lambda)$ and $$ \\sum_{x\\in X(k)}\\text{Tr}\\left(\\pi_x |_{K_{\\bar x}}\\right) = \\text{Tr}\\left(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K )}\\right). $$"}
+{"_id": "14403", "title": "trace-theorem-C", "text": "Let $X$ be a separated scheme of finite type over a finite field $k$ and $\\mathcal{F}$ be a $\\mathbf{Q}_\\ell$-sheaf on $X$. Then $\\dim_{\\mathbf{Q}_\\ell}H_c^i(X_{\\bar k}, \\mathcal{F})$ is finite for all $i$, and is nonzero for $0\\leq i \\leq 2 \\dim X$ only. Furthermore, we have $$ \\sum_{x\\in X(k)} \\text{Tr}\\left(\\pi_x |_{\\mathcal{F}_{\\bar x}}\\right) = \\sum_i (-1)^i\\text{Tr}\\left(\\pi_X^* |_{H_c^i(X_{\\bar k}, \\mathcal{F})}\\right). $$"}
+{"_id": "14404", "title": "trace-theorem-fundamental-group", "text": "Let $X$ be a connected scheme. \\begin{enumerate} \\item There is a topology on $\\pi_1(X, \\overline{x})$ such that the open subgroups form a fundamental system of open nbhds of $e\\in \\pi_1(X, \\overline x)$. \\item With topology of (1) the group $\\pi_1(X, \\overline{x})$ is a profinite group. \\item The functor $$ \\begin{matrix} \\text{ schemes finite } \\atop \\text{ \\'etale over }X & \\to & \\text{ finite discrete continuous } \\atop \\pi_1(X, \\overline{x})\\text{-sets}\\\\ Y / X& \\mapsto & F_{\\overline{x}}(Y) \\text{ with its natural action} \\end{matrix} $$ is an equivalence of categories. \\end{enumerate}"}
+{"_id": "14405", "title": "trace-theorem-weil-II", "text": "For a sheaf $\\mathcal{F}_\\rho$ with $\\rho$ satisfying the conclusions of the conjecture above then the eigenvalues of $\\pi_X^*$ on $H_c^i(X_{\\overline{k}}, \\mathcal{F}_{\\rho})$ are algebraic numbers $\\alpha$ with absolute values $$ |\\alpha|=q^{w/2}, \\text{ for }w\\in \\mathbf{Z},\\ w\\leq i $$ Moreover, if $X$ smooth and proj. then $w = i$."}
+{"_id": "14406", "title": "trace-theorem-drinfeld-make-rho", "text": "Given an eigenform $f$ with values in $\\overline{\\mathbf{Q}}_l$ and eigenvalues $u_v\\in \\overline{\\mathbf{Z}}_l^*$ then there exists $$ \\rho : \\pi_1(X)\\to \\text{GL}_2(E) $$ continuous, absolutely irreducible where $E$ is a finite extension of $\\mathbf{Q}_\\ell$ contained in $\\overline{\\mathbf{Q}}_l$ such that $t_v = \\text{Tr}(\\rho(F_v))$, and $u_v = q_v^{-1}\\det\\left(\\rho(F_v)\\right)$ for all places $v$."}
+{"_id": "14407", "title": "trace-theorem-drinfeld-make-f", "text": "Suppose $\\mathbf{Q}_l \\subset E$ finite, and $$ \\rho : \\pi_1(X)\\to \\text{GL}_2(E) $$ absolutely irreducible, continuous. Then there exists an eigenform $f$ with values in $\\overline{\\mathbf{Q}}_l$ whose eigenvalues $t_v$, $u_v$ satisfy the equalities $t_v = \\text{Tr}(\\rho(F_v))$ and $u_v = q_v^{-1}\\det(\\rho(F_v))$."}
+{"_id": "14408", "title": "trace-theorem-conjecture-n-2", "text": "The Conjecture holds if $n\\leq 2$."}
+{"_id": "14409", "title": "trace-theorem-conjecture-l-bigger-2n", "text": "Conjecture holds if $l > 2n$ modulo some unproven things."}
+{"_id": "14410", "title": "trace-theorem-deformation-rings", "text": "(See \\cite[Theorem 3.5]{dJ-conjecture}) Suppose $$ \\rho_0: \\pi_1(X)\\to \\text{GL}_n(\\mathbf{F}_l) $$ is a continuous, $l\\neq p$. Assume \\begin{enumerate} \\item Conj. holds for $X$, \\item $\\rho_0 |_{\\pi_1(X_{\\overline{k}})}$ abs. irred., and \\item $l$ does not divide $n$. \\end{enumerate} Then the universal deformation ring $R_{\\text{univ}}$ of $\\rho_0$ is finite flat over $\\mathbf{Z}_l$."}
+{"_id": "14411", "title": "trace-lemma-baffling", "text": "Let $X$ be a scheme and $g : X \\to X$ a morphism. Assume that for all $\\varphi : U \\to X$ \\'etale, there is an isomorphism $$ \\xymatrix{ U \\ar[rd]_\\varphi \\ar[rr]^-\\sim & & {U \\times_{\\varphi, X, g} X} \\ar[ld]^{\\text{pr}_2} \\\\ & X } $$ functorial in $U$. Then $g$ induces the identity on cohomology (for any sheaf)."}
+{"_id": "14413", "title": "trace-lemma-two-actions-agree", "text": "In the situation above denote $\\alpha : X \\to \\Spec(k)$ the structure morphism. Consider the stalk $(R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)}$ endowed with its natural Galois action as in \\'Etale Cohomology, Section \\ref{etale-cohomology-section-galois-action-stalks}. Then the identification $$ (R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)} \\cong H^j (X_{\\bar k}, \\mathcal{F}|_{X_{\\bar k}}) $$ from \\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-higher-direct-images} is an isomorphism of $G_k$-modules."}
+{"_id": "14415", "title": "trace-lemma-derived-categories", "text": "Morphisms between objects in the derived category. \\begin{enumerate} \\item Let $I^\\bullet \\in \\text{Comp}^+(\\mathcal{A})$ with $I^n$ injective for all $n \\in \\mathbf{Z}$. Then $$ \\Hom_{D(\\mathcal{A})}(K^\\bullet, I^\\bullet) = \\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet). $$ \\item Let $P^\\bullet \\in \\text{Comp}^-(\\mathcal{A})$ with $P^n$ is projective for all $n \\in \\mathbf{Z}$. Then $$ \\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet) = \\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet). $$ \\item If $\\mathcal{A}$ has enough injectives and $\\mathcal{I} \\subset \\mathcal{A}$ is the additive subcategory of injectives, then $ D^+(\\mathcal{A})\\cong K^+(\\mathcal{I}) $ (as triangulated categories). \\item If $\\mathcal{A}$ has enough projectives and $\\mathcal{P} \\subset \\mathcal{A}$ is the additive subcategory of projectives, then $ D^-(\\mathcal{A}) \\cong K^-(\\mathcal{P}). $ \\end{enumerate}"}
+{"_id": "14416", "title": "trace-lemma-filtered-derived-category", "text": "If $\\mathcal{A}$ has enough injectives, then $DF^+(\\mathcal{A}) \\cong K^+(\\mathcal{I})$, where $\\mathcal{I}$ is the full additive subcategory of $\\text{Fil}^f(\\mathcal{A})$ consisting of filtered injective objects. Similarly, if $\\mathcal{A}$ has enough projectives, then $DF^-(\\mathcal{A}) \\cong K^+(\\mathcal{P})$, where $\\mathcal P$ is the full additive subcategory of $\\text{Fil}^f(\\mathcal{A})$ consisting of filtered projective objects."}
+{"_id": "14418", "title": "trace-lemma-additive-filtered-finite-projective", "text": "Let $P \\in \\text{Fil}^f(\\text{Mod}_\\Lambda)$ be filtered finite projective, and $f : P \\to P$ an endomorphism in $\\text{Fil}^f(\\text{Mod}_\\Lambda)$. Then $$ \\text{Tr}(f|_P) = \\sum\\nolimits_p \\text{Tr}(f|_{\\text{gr}^p(P)}). $$"}
+{"_id": "14419", "title": "trace-lemma-characterize-perfect", "text": "Let $\\Lambda$ be a left Noetherian ring and $K\\in D(\\Lambda)$. Then $K$ is perfect if and only if the two following conditions hold: \\begin{enumerate} \\item $K$ has finite $\\text{Tor}$-dimension, and \\item for all $i \\in \\mathbf{Z}$, $H^i(K)$ is a finite $\\Lambda$-module. \\end{enumerate}"}
+{"_id": "14421", "title": "trace-lemma-epsilon", "text": "Let $e\\in G$ denote the neutral element. The map $$ \\begin{matrix} \\Lambda[G] & \\longrightarrow & \\Lambda^{\\natural}\\\\ \\sum \\lambda_g\\cdot g & \\longmapsto & \\lambda_e \\end{matrix} $$ factors through $\\Lambda[G]^\\natural$. We denote $\\varepsilon : \\Lambda[G]^\\natural\\to \\Lambda^\\natural$ the induced map."}
+{"_id": "14422", "title": "trace-lemma-lambda-trace", "text": "Let $f : P\\to P$ be an endomorphism of the finite projective $\\Lambda[G]$-module $P$. Then $$ \\text{Tr}_{\\Lambda}(f; P) = \\# G \\cdot \\text{Tr}_\\Lambda^G(f; P). $$"}
+{"_id": "14423", "title": "trace-lemma-A-module-structure", "text": "The map $A\\to \\Lambda$ defines an $A$-module structure on $\\Lambda^\\natural$."}
+{"_id": "14424", "title": "trace-lemma-diagonal-action-projective-module", "text": "Let $P$ be a finite projective $A[G]$-module and $M$ a $\\Lambda[G]$-module, finite projective as a $\\Lambda$-module. Then $P \\otimes_A M$ is a finite projective $\\Lambda[G]$-module, for the structure induced by the diagonal action of $G$."}
+{"_id": "14425", "title": "trace-lemma-multiplicative-trace", "text": "With assumptions as in Lemma \\ref{lemma-diagonal-action-projective-module}, let $u\\in \\text{End}_{A[G]}(P)$ and $v\\in \\text{End}_{\\Lambda[G]}(M)$. Then $$ \\text{Tr}_\\Lambda^G \\left(u \\otimes v; P \\otimes_A M\\right) = \\text{Tr}_A^G(u; P)\\cdot \\text{Tr}_\\Lambda(v;M). $$"}
+{"_id": "14426", "title": "trace-lemma-gamma-z-gamma-trace", "text": "Let $P$ be a $\\Lambda[\\Gamma]$-module, finite and projective as a $\\Lambda[G]$-module, and $\\gamma \\in \\Gamma$. Then $$ \\text{Tr}_{\\Lambda}(\\gamma, P) = \\# Z_\\gamma \\cdot \\text{Tr}_\\Lambda^{Z_\\gamma}\\left(\\gamma, P\\right). $$"}
+{"_id": "14427", "title": "trace-lemma-weak-trace", "text": "Let $P$ be an $A[\\Gamma]$-module, finite projective as $A[G]$-module. Let $M$ be a $\\Lambda[\\Gamma]$-module, finite projective as a $\\Lambda$-module. Then $$ \\text{Tr}_{\\Lambda}^{Z_\\gamma}(\\gamma, P \\otimes_A M) = \\text{Tr}_A^{Z_\\gamma}(\\gamma, P)\\cdot \\text{Tr}_\\Lambda(\\gamma, M). $$"}
+{"_id": "14428", "title": "trace-lemma-trivial-trace", "text": "Let $P$ be a $\\Lambda[\\Gamma]$-module, finite projective as $\\Lambda[G]$-module. Then the coinvariants $P_G = \\Lambda \\otimes_{\\Lambda[G]} P$ form a finite projective $\\Lambda$-module, endowed with an action of $\\Gamma/G = \\mathbf{N}$. Moreover, we have $$ \\text{Tr}_\\Lambda(1; P_G) = \\sum\\nolimits'_{\\gamma \\mapsto 1} \\text{Tr}_\\Lambda^{Z_\\gamma}(\\gamma, P) $$ where $\\sum_{\\gamma\\mapsto 1}'$ means taking the sum over the $G$-conjugacy classes in $\\Gamma$."}
+{"_id": "14429", "title": "trace-lemma-eventually-constant", "text": "Let $\\{\\mathcal{G}_n\\}_{n\\geq 1}$ be an inverse system of constructible $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-modules. Suppose that for all $k\\geq 1$, the maps $$ \\mathcal{G}_{n+1}/\\ell^k \\mathcal{G}_{n+1}\\to \\mathcal{G}_n /\\ell^k \\mathcal{G}_n $$ are isomorphisms for all $n\\gg 0$ (where the bound possibly depends on $k$). In other words, assume that the system $\\{\\mathcal{G}_n/\\ell^k\\mathcal{G}_n\\}_{n\\geq 1}$ is eventually constant, and call $\\mathcal{F}_k$ the corresponding sheaf. Then the system $\\left\\{\\mathcal{F}_k\\right\\}_{k\\geq 1}$ forms a $\\mathbf{Z}_\\ell$-sheaf on $X$."}
+{"_id": "14433", "title": "trace-lemma-identify-h2c", "text": "There is a canonical isomorphism $$ H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)=(M)_{\\pi_1(X_{\\overline{k}}, \\overline\\eta)}(-1) $$ as $\\text{Gal}(k^{^{sep}}/k)$-modules."}
+{"_id": "14436", "title": "trace-proposition-compare-filtered-graded", "text": "In the situation above, we have $$ \\text{gr}^p \\circ RT = RT \\circ \\text{gr}^p $$ where the $RT$ on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram $$ \\xymatrix{ DF^+(\\mathcal{A}) \\ar[r]^{RT} \\ar[d]_{\\text{gr}^p} & DF^+(\\mathcal{B}) \\ar[d]^{\\text{gr}^p}\\\\ D^+(\\mathcal{A}) \\ar[r]^{RT} & D^+(\\mathcal{B}).} $$"}
+{"_id": "14437", "title": "trace-proposition-trace-well-defined", "text": "Let $K\\in D_{perf}(\\Lambda)$ and $f\\in \\text{End}_{D(\\Lambda)}(K)$. Then the trace $\\text{Tr}(f)\\in \\Lambda^\\natural$ is well defined."}
+{"_id": "14438", "title": "trace-proposition-projective-curve-constructible-cohomology", "text": "Let $X$ be a projective curve over a field $k$, $\\Lambda$ a finite ring and $K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma(X_{\\bar k}, K)\\in D_{perf}(\\Lambda)$."}
+{"_id": "14439", "title": "trace-proposition-integral-normal-fundamental-group", "text": "Let $X$ be an integral normal Noetherian scheme. Let $\\overline y\\to X$ be an algebraic geometric point lying over the generic point $\\eta\\in X$. Then $$ \\pi_x(X, \\overline \\eta) = Gal(M/\\kappa(\\eta)) $$ ($\\kappa(\\eta)$, function field of $X$) where $$ \\kappa(\\overline \\eta)\\supset M\\supset \\kappa(\\eta) = k(X) $$ is the max sub-extension such that for every finite sub extension $M\\supset L\\supset \\kappa(\\eta)$ the normalization of $X$ in $L$ is finite \\'etale over $X$."}
+{"_id": "14440", "title": "trace-proposition-curve-kpi1", "text": "Let $X/k$ as before but $X_{\\overline{k}}\\neq \\mathbf{P}^1_{\\overline{k}}$ The functors $ (M, \\rho)\\mapsto H_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho) $ are the left derived functor of $(M, \\rho)\\mapsto H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)$ so $$ H_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho) = H_i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M)(-1) $$ Moreover, there is a derived version, namely $$ R\\Gamma_c(X_{\\overline{k}}, \\mathcal{F}_\\rho) = LH_0(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M(-1)) = M(-1) \\otimes_{\\Lambda[[\\pi_1(X_{\\overline{k}}, \\overline \\eta)]]}^\\mathbf{L} \\Lambda $$ in $D(\\Lambda[[\\widehat{\\mathbf{Z}}]])$. Similarly, the functors $(M, \\rho)\\mapsto H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho)$ are the right derived functor of $(M, \\rho)\\mapsto M^{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}$ so $$ H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho) = H^i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M) $$ Moreover, in this case there is a derived version too."}
+{"_id": "14441", "title": "trace-proposition-cusp-forms-finite", "text": "If $\\Lambda$ is Noetherian then $C(\\Lambda)$ is a finitely generated $\\Lambda$-module. Moreover, if $\\Lambda$ is a field with prime subfield $\\mathbf{F} \\subset \\Lambda$ then $$ C(\\Lambda)=(C(\\mathbf{F}))\\otimes_{\\mathbf{F}}\\Lambda $$ compatibly with $T_v$ acting."}
+{"_id": "14442", "title": "trace-proposition-finite-set-frobenii-generate-topologically", "text": "There exists a finite set $x_1, \\ldots, x_n$ of closed points of $X$ such that set of {\\bf all} frobenius elements corresponding to these points topologically generate $\\pi_1(X)$."}
+{"_id": "14478", "title": "sheaves-lemma-product-presheaves", "text": "Let $X$ be a topological space. The category of presheaves of sets on $X$ has products (see Categories, Definition \\ref{categories-definition-product}). Moreover, the set of sections of the product $\\mathcal{F} \\times \\mathcal{G}$ over an open $U$ is the product of the sets of sections of $\\mathcal{F}$ and $\\mathcal{G}$ over $U$."}
+{"_id": "14479", "title": "sheaves-lemma-abelian-presheaves", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf of sets. Consider the following types of structure on $\\mathcal{F}$: \\begin{enumerate} \\item For every open $U$ the structure of an abelian group on $\\mathcal{F}(U)$ such that all restriction maps are abelian group homomorphisms. \\item A map of presheaves $+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$, a map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$ and a map $0 : * \\to \\mathcal{F}$ (see Example \\ref{example-singleton-presheaf}) satisfying all the axioms of $+, -, 0$ in a usual abelian group. \\item A map of presheaves $+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$, a map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$ and a map $0 : * \\to \\mathcal{F}$ such that for each open $U \\subset X$ the quadruple $(\\mathcal{F}(U), +, -, 0)$ is an abelian group, \\item A map of presheaves $+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$ such that for every open $U \\subset X$ the map $+ : \\mathcal{F}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ defines the structure of an abelian group. \\end{enumerate} There are natural bijections between the collections of types of data (1) - (4) above."}
+{"_id": "14480", "title": "sheaves-lemma-adjointness-tensor-restrict-presheaves", "text": "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and $\\mathcal{G}$ as above there exists a canonical bijection $$ \\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G}, \\mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other."}
+{"_id": "14482", "title": "sheaves-lemma-sheaf-subset-stalks", "text": "Let $\\mathcal{F}$ be a sheaf of sets on the topological space $X$. For every open $U \\subset X$ the map $$ \\mathcal{F}(U) \\longrightarrow \\prod\\nolimits_{x \\in U} \\mathcal{F}_x $$ is injective."}
+{"_id": "14483", "title": "sheaves-lemma-stalk-abelian-presheaf", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf of abelian groups on $X$. There exists a unique structure of an abelian group on $\\mathcal{F}_x$ such that for every $U \\subset X$ open, $x\\in U$ the map $\\mathcal{F}(U) \\to \\mathcal{F}_x$ is a group homomorphism. Moreover, $$ \\mathcal{F}_x = \\colim_{x\\in U} \\mathcal{F}(U) $$ holds in the category of abelian groups."}
+{"_id": "14484", "title": "sheaves-lemma-stalk-presheaf-values-in-category", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor. Assume that \\begin{enumerate} \\item $F$ is faithful, and \\item directed colimits exist in $\\mathcal{C}$ and $F$ commutes with them. \\end{enumerate} Let $X$ be a topological space. Let $x \\in X$. Let $\\mathcal{F}$ be a presheaf with values in $\\mathcal{C}$. Then $$ \\mathcal{F}_x = \\colim_{x\\in U} \\mathcal{F}(U) $$ exists in $\\mathcal{C}$. Its underlying set is equal to the stalk of the underlying presheaf of sets of $\\mathcal{F}$. Furthermore, the construction $\\mathcal{F} \\mapsto \\mathcal{F}_x$ is a functor from the category of presheaves with values in $\\mathcal{C}$ to $\\mathcal{C}$."}
+{"_id": "14485", "title": "sheaves-lemma-stalk-module", "text": "Let $X$ be a topological space. Let $\\mathcal{O}$ be a presheaf of rings on $X$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. Let $x \\in X$. The canonical map $\\mathcal{O}_x \\times \\mathcal{F}_x \\to \\mathcal{F}_x$ coming from the multiplication map $\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$ defines a $\\mathcal{O}_x$-module structure on the abelian group $\\mathcal{F}_x$."}
+{"_id": "14486", "title": "sheaves-lemma-stalk-tensor-presheaf-modules", "text": "Let $X$ be a topological space. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of presheaves of rings on $X$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. Let $x \\in X$. We have $$ \\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x = (\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{O}')_x $$ as $\\mathcal{O}'_x$-modules."}
+{"_id": "14487", "title": "sheaves-lemma-list-algebraic-structures", "text": "The following categories, endowed with the obvious forgetful functor, define types of algebraic structures: \\begin{enumerate} \\item The category of pointed sets. \\item The category of abelian groups. \\item The category of groups. \\item The category of monoids. \\item The category of rings. \\item The category of $R$-modules for a fixed ring $R$. \\item The category of Lie algebras over a fixed field. \\end{enumerate}"}
+{"_id": "14488", "title": "sheaves-lemma-properties-algebraic-structures", "text": "Let $(\\mathcal{C}, F)$ be a type of algebraic structure. \\begin{enumerate} \\item $\\mathcal{C}$ has a final object $0$ and $F(0) = \\{ * \\}$. \\item $\\mathcal{C}$ has products and $F(\\prod A_i) = \\prod F(A_i)$. \\item $\\mathcal{C}$ has fibre products and $F(A \\times_B C) = F(A)\\times_{F(B)}F(C)$. \\item $\\mathcal{C}$ has equalizers, and if $E \\to A$ is the equalizer of $a, b : A \\to B$, then $F(E) \\to F(A)$ is the equalizer of $F(a), F(b) : F(A) \\to F(B)$. \\item $A \\to B$ is a monomorphism if and only if $F(A) \\to F(B)$ is injective. \\item if $F(a) : F(A) \\to F(B)$ is surjective, then $a$ is an epimorphism. \\item given $A_1 \\to A_2 \\to A_3 \\to \\ldots$, then $\\colim A_i$ exists and $F(\\colim A_i) = \\colim F(A_i)$, and more generally for any filtered colimit. \\end{enumerate}"}
+{"_id": "14489", "title": "sheaves-lemma-image-contained-in", "text": "Let $(\\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $A, B, C \\in \\Ob(\\mathcal{C})$. Let $f : A \\to B$ and $g : C \\to B$ be morphisms of $\\mathcal{C}$. If $F(g)$ is injective, and $\\Im(F(f)) \\subset \\Im(F(g))$, then $f$ factors as $f = g \\circ t$ for some morphism $t : A \\to C$."}
+{"_id": "14490", "title": "sheaves-lemma-points-exactness", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of sheaves of sets on $X$. \\begin{enumerate} \\item The map $\\varphi$ is a monomorphism in the category of sheaves if and only if for all $x \\in X$ the map $\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$ is injective. \\item The map $\\varphi$ is an epimorphism in the category of sheaves if and only if for all $x \\in X$ the map $\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$ is surjective. \\item The map $\\varphi$ is an isomorphism in the category of sheaves if and only if for all $x \\in X$ the map $\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$ is bijective. \\end{enumerate}"}
+{"_id": "14491", "title": "sheaves-lemma-characterize-epi-mono", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item Epimorphisms (resp.\\ monomorphisms) in the category of presheaves are exactly the surjective (resp.\\ injective) maps of presheaves. \\item Epimorphisms (resp.\\ monomorphisms) in the category of sheaves are exactly the surjective (resp.\\ injective) maps of sheaves, and are exactly those maps with are surjective (resp.\\ injective) on all the stalks. \\item The sheafification of a surjective (resp.\\ injective) morphism of presheaves of sets is surjective (resp.\\ injective). \\end{enumerate}"}
+{"_id": "14492", "title": "sheaves-lemma-check-homomorphism-stalks", "text": "let $X$ be a topological space. Let $(\\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $\\mathcal{F}$, $\\mathcal{G}$ are sheaves on $X$ with values in $\\mathcal{C}$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of the underlying sheaves of sets. If for all points $x \\in X$ the map $\\mathcal{F}_x \\to \\mathcal{G}_x$ is a morphism of algebraic structures, then $\\varphi$ is a morphism of sheaves of algebraic structures."}
+{"_id": "14493", "title": "sheaves-lemma-sheafification-sheaf", "text": "The presheaf $\\mathcal{F}^{\\#}$ is a sheaf."}
+{"_id": "14494", "title": "sheaves-lemma-stalk-sheafification", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf of sets on $X$. Let $x \\in X$. Then $\\mathcal{F}_x = \\mathcal{F}^\\#_x$."}
+{"_id": "14496", "title": "sheaves-lemma-separated-presheaf-into-sheaf", "text": "Let $X$ be a topological space. A presheaf $\\mathcal{F}$ is separated (see Definition \\ref{definition-separated}) if and only if the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ is injective."}
+{"_id": "14497", "title": "sheaves-lemma-diagram-fibre-product", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf of sets on $X$. Let $U \\subset X$ be open. There is a canonical fibre product diagram $$ \\xymatrix{ \\mathcal{F}^\\#(U) \\ar[d] \\ar[r] & \\Pi(\\mathcal{F})(U) \\ar[d] \\\\ \\prod_{x \\in U} \\mathcal{F}_x \\ar[r] & \\prod_{x \\in U} \\Pi(\\mathcal{F})_x } $$ where the maps are the following: \\begin{enumerate} \\item The left vertical map has components $\\mathcal{F}^\\#(U) \\to \\mathcal{F}^\\#_x = \\mathcal{F}_x$ where the equality is Lemma \\ref{lemma-stalk-sheafification}. \\item The top horizontal map comes from the map of presheaves $\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described in Section \\ref{section-sheafification}. \\item The right vertical map has obvious component maps $\\Pi(\\mathcal{F})(U) \\to \\Pi(\\mathcal{F})_x$. \\item The bottom horizontal map has components $\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$ which come from the map of presheaves $\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described in Section \\ref{section-sheafification}. \\end{enumerate}"}
+{"_id": "14498", "title": "sheaves-lemma-sheafify-abelian-presheaf", "text": "Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$. Then there exists a unique structure of abelian sheaf on $\\mathcal{F}^\\#$ such that $\\mathcal{F} \\to \\mathcal{F}^\\#$ is a morphism of abelian presheaves. Moreover, the following adjointness property holds $$ \\Mor_{\\textit{PAb}(X)}(\\mathcal{F}, i(\\mathcal{G})) = \\Mor_{\\textit{Ab}(X)}(\\mathcal{F}^\\#, \\mathcal{G}). $$"}
+{"_id": "14500", "title": "sheaves-lemma-sheafification-presheaf-modules", "text": "Let $X$ be a topological space. Let $\\mathcal{O}$ be a presheaf of rings on $X$. Let $\\mathcal{F}$ be a presheaf $\\mathcal{O}$-modules. Let $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$. Let $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$ as a presheaf of abelian groups. There exists a map of sheaves of sets $$ \\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\longrightarrow \\mathcal{F}^\\# $$ which makes the diagram $$ \\xymatrix{ \\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[d] \\\\ \\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] & \\mathcal{F}^\\# } $$ commute and which makes $\\mathcal{F}^\\#$ into a sheaf of $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$ is a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism of presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$ (into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module) factors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$ where $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of $\\mathcal{O}^\\#$-modules."}
+{"_id": "14501", "title": "sheaves-lemma-adjointness-tensor-restrict", "text": "With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and $\\mathcal{G}$ as above there exists a canonical bijection $$ \\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1}) = \\Hom_{\\mathcal{O}_2}( \\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G}, \\mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other."}
+{"_id": "14502", "title": "sheaves-lemma-stalk-tensor-sheaf-modules", "text": "Let $X$ be a topological space. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of sheaves of rings on $X$. Let $\\mathcal{F}$ be a sheaf $\\mathcal{O}$-modules. Let $x \\in X$. We have $$ \\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x = (\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{O}')_x $$ as $\\mathcal{O}'_x$-modules."}
+{"_id": "14503", "title": "sheaves-lemma-pushforward-sheaf", "text": "Let $f : X \\to Y$ be a continuous map. Let $\\mathcal{F}$ be a sheaf of sets on $X$. Then $f_*\\mathcal{F}$ is a sheaf on $Y$."}
+{"_id": "14504", "title": "sheaves-lemma-pushforward-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps of topological spaces. The functors $(g \\circ f)_*$ and $g_* \\circ f_*$ are equal (on both presheaves and sheaves of sets)."}
+{"_id": "14505", "title": "sheaves-lemma-pullback-presheaves", "text": "Let $f : X \\to Y$ be a continuous map. There exists a functor $f_p : \\textit{PSh}(Y) \\to \\textit{PSh}(X)$ which is left adjoint to $f_*$. For a presheaf $\\mathcal{G}$ it is determined by the rule $$ f_p\\mathcal{G}(U) = \\colim_{f(U) \\subset V} \\mathcal{G}(V) $$ where the colimit is over the collection of open neighbourhoods $V$ of $f(U)$ in $Y$. The colimits are over directed partially ordered sets. (The restriction mappings of $f_p\\mathcal{G}$ are explained in the proof.)"}
+{"_id": "14506", "title": "sheaves-lemma-stalk-pullback-presheaf", "text": "Let $f : X \\to Y$ be a continuous map. Let $x \\in X$. Let $\\mathcal{G}$ be a presheaf of sets on $Y$. There is a canonical bijection of stalks $(f_p\\mathcal{G})_x = \\mathcal{G}_{f(x)}$."}
+{"_id": "14507", "title": "sheaves-lemma-stalk-pullback", "text": "Let $x \\in X$. Let $\\mathcal{G}$ be a sheaf of sets on $Y$. There is a canonical bijection of stalks $(f^{-1}\\mathcal{G})_x = \\mathcal{G}_{f(x)}$."}
+{"_id": "14508", "title": "sheaves-lemma-pullback-composition", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps of topological spaces. The functors $(g \\circ f)^{-1}$ and $f^{-1} \\circ g^{-1}$ are canonically isomorphic. Similarly $(g \\circ f)_p \\cong f_p \\circ g_p$ on presheaves."}
+{"_id": "14509", "title": "sheaves-lemma-f-map", "text": "Let $f : X \\to Y$ be a continuous map. Let $\\mathcal{F}$ be a sheaf of sets on $X$ and let $\\mathcal{G}$ be a sheaf of sets on $Y$. There are canonical bijections between the following three sets: \\begin{enumerate} \\item The set of maps $\\mathcal{G} \\to f_*\\mathcal{F}$. \\item The set of maps $f^{-1}\\mathcal{G} \\to \\mathcal{F}$. \\item The set of $f$-maps $\\xi : \\mathcal{G} \\to \\mathcal{F}$. \\end{enumerate}"}
+{"_id": "14511", "title": "sheaves-lemma-pullback-abelian-stalk", "text": "Let $f : X \\to Y$ be a continuous map. \\begin{enumerate} \\item Let $\\mathcal{G}$ be an abelian presheaf on $Y$. Let $x \\in X$. The bijection $\\mathcal{G}_{f(x)} \\to (f_p\\mathcal{G})_x$ of Lemma \\ref{lemma-stalk-pullback-presheaf} is an isomorphism of abelian groups. \\item Let $\\mathcal{G}$ be an abelian sheaf on $Y$. Let $x \\in X$. The bijection $\\mathcal{G}_{f(x)} \\to (f^{-1}\\mathcal{G})_x$ of Lemma \\ref{lemma-stalk-pullback} is an isomorphism of abelian groups. \\end{enumerate}"}
+{"_id": "14512", "title": "sheaves-lemma-f-map-sets-algebraic-structures", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Suppose given sheaves of algebraic structures $\\mathcal{F}$ on $X$, $\\mathcal{G}$ on $Y$. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map of underlying sheaves of sets. If for every $V \\subset Y$ open the map of sets $\\varphi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}V)$ is the effect of a morphism in $\\mathcal{C}$ on underlying sets, then $\\varphi$ comes from a unique $f$-morphism between sheaves of algebraic structures."}
+{"_id": "14513", "title": "sheaves-lemma-pushforward-presheaf-module", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a presheaf of rings on $X$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f_*\\mathcal{O} \\times f_*\\mathcal{F} \\longrightarrow f_*\\mathcal{F} $$ which turns $f_*\\mathcal{F}$ into a presheaf of $f_*\\mathcal{O}$-modules. This construction is functorial in $\\mathcal{F}$."}
+{"_id": "14514", "title": "sheaves-lemma-pullback-presheaf-module", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a presheaf of rings on $Y$. Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f_p\\mathcal{O} \\times f_p\\mathcal{G} \\longrightarrow f_p\\mathcal{G} $$ which turns $f_p\\mathcal{G}$ into a presheaf of $f_p\\mathcal{O}$-modules. This construction is functorial in $\\mathcal{G}$."}
+{"_id": "14515", "title": "sheaves-lemma-adjoint-push-pull-presheaves-modules", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a presheaf of rings on $Y$. Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules. Let $\\mathcal{F}$ be a presheaf of $f_p\\mathcal{O}$-modules. Then $$ \\Mor_{\\textit{PMod}(f_p\\mathcal{O})}(f_p\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{PMod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}). $$ Here we use Lemmas \\ref{lemma-pullback-presheaf-module} and \\ref{lemma-pushforward-presheaf-module}, and we think of $f_*\\mathcal{F}$ as an $\\mathcal{O}$-module via the map $i_\\mathcal{O} : \\mathcal{O} \\to f_*f_p\\mathcal{O}$ (defined first in the proof of Lemma \\ref{lemma-pullback-presheaves})."}
+{"_id": "14517", "title": "sheaves-lemma-pushforward-module", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a sheaf of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. The pushforward $f_*\\mathcal{F}$, as defined in Lemma \\ref{lemma-pushforward-presheaf-module} is a sheaf of $f_*\\mathcal{O}$-modules."}
+{"_id": "14518", "title": "sheaves-lemma-pullback-module", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a sheaf of rings on $Y$. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G} \\longrightarrow f^{-1}\\mathcal{G} $$ which turns $f^{-1}\\mathcal{G}$ into a sheaf of $f^{-1}\\mathcal{O}$-modules."}
+{"_id": "14519", "title": "sheaves-lemma-adjoint-push-pull-modules", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a sheaf of rings on $Y$. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules. Let $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules. Then $$ \\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}). $$ Here we use Lemmas \\ref{lemma-pullback-module} and \\ref{lemma-pushforward-module}, and we think of $f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via $\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$."}
+{"_id": "14521", "title": "sheaves-lemma-adjoint-pullback-pushforward-modules", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules. There is a canonical bijection $$ \\Hom_{\\mathcal{O}_X}(f^*\\mathcal{G}, \\mathcal{F}) = \\Hom_{\\mathcal{O}_Y}(\\mathcal{G}, f_*\\mathcal{F}). $$ In other words: the functor $f^*$ is the left adjoint to $f_*$."}
+{"_id": "14522", "title": "sheaves-lemma-push-pull-composition-modules", "text": "Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces. The functors $(g \\circ f)_*$ and $g_* \\circ f_*$ are equal. There is a canonical isomorphism of functors $(g \\circ f)^* \\cong f^* \\circ g^*$."}
+{"_id": "14523", "title": "sheaves-lemma-stalk-pullback-modules", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules. Let $x \\in X$. Then $$ (f^*\\mathcal{G})_x = \\mathcal{G}_{f(x)} \\otimes_{\\mathcal{O}_{Y, f(x)}} \\mathcal{O}_{X, x} $$ as $\\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\\sharp_x : \\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$."}
+{"_id": "14524", "title": "sheaves-lemma-skyscraper-stalks", "text": "Let $X$ be a topological space, $x \\in X$ a point, and $A$ a set. For any point $x' \\in X$ the stalk of the skyscraper sheaf at $x$ with value $A$ at $x'$ is $$ (i_{x, *}A)_{x'} = \\left\\{ \\begin{matrix} A & \\text{if} & x' \\in \\overline{\\{x\\}} \\\\ \\{*\\} & \\text{if} & x' \\not\\in \\overline{\\{x\\}} \\end{matrix} \\right. $$ A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules."}
+{"_id": "14525", "title": "sheaves-lemma-stalk-skyscraper-adjoint", "text": "Let $X$ be a topological space, and let $x \\in X$ a point. The functors $\\mathcal{F} \\mapsto \\mathcal{F}_x$ and $A \\mapsto i_{x, *}A$ are adjoint. In a formula $$ \\Mor_{\\textit{Sets}}(\\mathcal{F}_x, A) = \\Mor_{\\Sh(X)}(\\mathcal{F}, i_{x, *}A). $$ A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have $$ \\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, A) = \\Hom_{\\mathcal{O}_X}(\\mathcal{F}, i_{x, *}A). $$"}
+{"_id": "14526", "title": "sheaves-lemma-directed-colimits-sections", "text": "Let $X$ be a topological space. Let $I$ be a directed set. Let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system of sheaves of sets over $I$, see Categories, Section \\ref{categories-section-posets-limits}. Let $U \\subset X$ be an open subset. Consider the canonical map $$ \\Psi : \\colim_i \\mathcal{F}_i(U) \\longrightarrow \\left(\\colim_i \\mathcal{F}_i\\right)(U) $$ \\begin{enumerate} \\item If all the transition maps are injective then $\\Psi$ is injective for any open $U$. \\item If $U$ is quasi-compact, then $\\Psi$ is injective. \\item If $U$ is quasi-compact and all the transition maps are injective then $\\Psi$ is an isomorphism. \\item If $U$ has a cofinal system of open coverings $\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with $J$ finite and $U_j \\cap U_{j'}$ quasi-compact for all $j, j' \\in J$, then $\\Psi$ is bijective. \\end{enumerate}"}
+{"_id": "14527", "title": "sheaves-lemma-compute-pullback-to-limit", "text": "In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let $\\mathcal{G}$ be a sheaf on $X_i$. For $U_i \\subset X_i$ quasi-compact open we have $$ p_i^{-1}\\mathcal{G}(p_i^{-1}(U_i)) = \\colim_{a : j \\to i} f_a^{-1}\\mathcal{G}(f_a^{-1}(U_i)) $$"}
+{"_id": "14528", "title": "sheaves-lemma-descend-opens", "text": "In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let $U_i \\subset X_i$ be a quasi-compact open. Then $$ \\colim_{a : j \\to i} \\mathcal{F}_j(f_a^{-1}(U_i)) = \\mathcal{F}(p_i^{-1}(U_i)) $$"}
+{"_id": "14529", "title": "sheaves-lemma-cofinal-systems-coverings", "text": "With notation as above. For each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$ be a cofinal system. For each $U \\in \\mathcal{B}$, and each $\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$, let coverings $\\mathcal{U}_{ij} : U_i \\cap U_j = \\bigcup U_{ijk}$, $U_{ijk} \\in \\mathcal{B}$ be given. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$. The following are equivalent \\begin{enumerate} \\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ and every covering $\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ the sheaf condition $(**)$ holds (for the given coverings $\\mathcal{U}_{ij}$). \\end{enumerate}"}
+{"_id": "14530", "title": "sheaves-lemma-cofinal-systems-coverings-standard-case", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Assume that for every triple $U, U', U'' \\in \\mathcal{B}$ with $U' \\subset U$ and $U'' \\subset U$ we have $U' \\cap U'' \\in \\mathcal{B}$. For each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$ be a cofinal system. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$. The following are equivalent \\begin{enumerate} \\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$. \\item For every $U \\in \\mathcal{B}$ and every covering $\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ and for every family of sections $s_i \\in \\mathcal{F}(U_i)$ such that $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$ there exists a unique section $s \\in \\mathcal{F}(U)$ which restricts to $s_i$ on $U_i$. \\end{enumerate}"}
+{"_id": "14531", "title": "sheaves-lemma-condition-star-sections", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $U \\in \\mathcal{B}$. Let $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$. The map $$ \\mathcal{F}(U) \\to \\prod\\nolimits_{x \\in U} \\mathcal{F}_x $$ identifies $\\mathcal{F}(U)$ with the elements $(s_x)_{x\\in U}$ with the property \\begin{itemize} \\item[(*)] For any $x \\in U$ there exists a $V \\in \\mathcal{B}$, with $x \\in V \\subset U$ and a section $\\sigma \\in \\mathcal{F}(V)$ such that for all $y \\in V$ we have $s_y = (V, \\sigma)$ in $\\mathcal{F}_y$. \\end{itemize}"}
+{"_id": "14532", "title": "sheaves-lemma-extend-off-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$. There exists a unique sheaf of sets $\\mathcal{F}^{ext}$ on $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$ for all $U \\in \\mathcal{B}$ compatibly with the restriction mappings."}
+{"_id": "14533", "title": "sheaves-lemma-restrict-basis-equivalence", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Denote $\\Sh(\\mathcal{B})$ the category of sheaves on $\\mathcal{B}$. There is an equivalence of categories $$ \\Sh(X) \\longrightarrow \\Sh(\\mathcal{B}) $$ which assigns to a sheaf on $X$ its restriction to the members of $\\mathcal{B}$."}
+{"_id": "14534", "title": "sheaves-lemma-extend-off-basis-structures", "text": "Let $X$ be a topological space. Let $(\\mathcal{C}, F)$ be a type of algebraic structure. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $\\mathcal{B}$. There exists a unique sheaf $\\mathcal{F}^{ext}$ with values in $\\mathcal{C}$ on $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$ for all $U \\in \\mathcal{B}$ compatibly with the restriction mappings."}
+{"_id": "14535", "title": "sheaves-lemma-restrict-basis-equivalence-structures", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $(\\mathcal{C}, F)$ be a type of algebraic structure. Denote $\\Sh(\\mathcal{B}, \\mathcal{C})$ the category of sheaves with values in $\\mathcal{C}$ on $\\mathcal{B}$. There is an equivalence of categories $$ \\Sh(X, \\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{B}, \\mathcal{C}) $$ which assigns to a sheaf on $X$ its restriction to the members of $\\mathcal{B}$."}
+{"_id": "14536", "title": "sheaves-lemma-extend-off-basis-module", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{B}$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{B}$. Let $\\mathcal{O}^{ext}$ be the sheaf of rings on $X$ extending $\\mathcal{O}$ and let $\\mathcal{F}^{ext}$ be the abelian sheaf on $X$ extending $\\mathcal{F}$, see Lemma \\ref{lemma-extend-off-basis-structures}. There exists a canonical map $$ \\mathcal{O}^{ext} \\times \\mathcal{F}^{ext} \\longrightarrow \\mathcal{F}^{ext} $$ which agrees with the given map over elements of $\\mathcal{B}$ and which endows $\\mathcal{F}^{ext}$ with the structure of an $\\mathcal{O}^{ext}$-module."}
+{"_id": "14537", "title": "sheaves-lemma-restrict-basis-equivalence-modules", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{O}$ be a sheaf of rings on $X$. Denote $\\textit{Mod}(\\mathcal{O}|_\\mathcal{B})$ the category of sheaves of $\\mathcal{O}|_\\mathcal{B}$-modules on $\\mathcal{B}$. There is an equivalence of categories $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}|_\\mathcal{B}) $$ which assigns to a sheaf of $\\mathcal{O}$-modules on $X$ its restriction to the members of $\\mathcal{B}$."}
+{"_id": "14538", "title": "sheaves-lemma-f-map-basis-below-structures", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $(\\mathcal{C}, F)$ be a type of algebraic structures. Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$. Let $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$. Let $\\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \\in \\mathcal{B}$ a morphism $$ \\varphi_V : \\mathcal{G}(V) \\longrightarrow \\mathcal{F}(f^{-1}V) $$ of $\\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition \\ref{definition-f-map} and discussion of $f$-maps in Section \\ref{section-presheaves-structures-functorial}) $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ recovering $\\varphi_V$ for $V \\in \\mathcal{B}$."}
+{"_id": "14539", "title": "sheaves-lemma-f-map-basis-below-modules", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules. Let $\\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \\in \\mathcal{B}$ a $\\mathcal{O}_Y(V)$-module map $$ \\varphi_V : \\mathcal{G}(V) \\longrightarrow \\mathcal{F}(f^{-1}V) $$ (where $\\mathcal{F}(f^{-1}V)$ has a module structure using $f^\\sharp_V : \\mathcal{O}_Y(V) \\to \\mathcal{O}_X(f^{-1}V)$) compatible with restriction mappings. Then there is a unique $f$-map (see discussion of $f$-maps in Section \\ref{section-ringed-spaces-functoriality-modules}) $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ recovering $\\varphi_V$ for $V \\in \\mathcal{B}$."}
+{"_id": "14540", "title": "sheaves-lemma-f-map-basis-above-and-below-structures", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $(\\mathcal{C}, F)$ be a type of algebraic structures. Let $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$. Let $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$. Let $\\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \\in \\mathcal{B}_Y$, and $U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a morphism $$ \\varphi_V^U : \\mathcal{G}(V) \\longrightarrow \\mathcal{F}(U) $$ of $\\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition \\ref{definition-f-map} and the discussion of $f$-maps in Section \\ref{section-presheaves-structures-functorial}) $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ recovering $\\varphi_V^U$ as the composition $$ \\mathcal{G}(V) \\xrightarrow{\\varphi_V} \\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}} \\mathcal{F}(U) $$ for every pair $(U, V)$ as above."}
+{"_id": "14541", "title": "sheaves-lemma-f-map-basis-above-and-below-modules", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules. Let $\\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \\in \\mathcal{B}_Y$, and $U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a $\\mathcal{O}_Y(V)$-module map $$ \\varphi_V^U : \\mathcal{G}(V) \\longrightarrow \\mathcal{F}(U) $$ compatible with restriction mappings. Here the $\\mathcal{O}_Y(V)$-module structure on $\\mathcal{F}(U)$ comes from the $\\mathcal{O}_X(U)$-module structure via the map $f^\\sharp_V : \\mathcal{O}_Y(V) \\to \\mathcal{O}_X(f^{-1}V) \\to \\mathcal{O}_X(U)$. Then there is a unique $f$-map of sheaves of modules (see Definition \\ref{definition-f-map} and the discussion of $f$-maps in Section \\ref{section-ringed-spaces-functoriality-modules}) $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ recovering $\\varphi_V^U$ as the composition $$ \\mathcal{G}(V) \\xrightarrow{\\varphi_V} \\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}} \\mathcal{F}(U) $$ for every pair $(U, V)$ as above."}
+{"_id": "14542", "title": "sheaves-lemma-j-pullback", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset $U$ into $X$. \\begin{enumerate} \\item Let $\\mathcal{G}$ be a presheaf of sets on $X$. The presheaf $j_p\\mathcal{G}$ (see Section \\ref{section-presheaves-functorial}) is given by the rule $V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open. \\item Let $\\mathcal{G}$ be a sheaf of sets on $X$. The sheaf $j^{-1}\\mathcal{G}$ is given by the rule $V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open. \\item For any point $u \\in U$ and any sheaf $\\mathcal{G}$ on $X$ we have a canonical identification of stalks $$ j^{-1}\\mathcal{G}_u = (\\mathcal{G}|_U)_u = \\mathcal{G}_u. $$ \\item On the category of presheaves of $U$ we have $j_pj_* = \\text{id}$. \\item On the category of sheaves of $U$ we have $j^{-1}j_* = \\text{id}$. \\end{enumerate} The same description holds for (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures, and (pre)sheaves of modules."}
+{"_id": "14543", "title": "sheaves-lemma-j-shriek", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. \\begin{enumerate} \\item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}). \\item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \\Mor_{\\Sh(X)}(j_!\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\Sh(U)}(\\mathcal{F}, j^{-1}\\mathcal{G}) = \\Mor_{\\Sh(U)}(\\mathcal{F}, \\mathcal{G}|_U) $$ bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$. \\item Let $\\mathcal{F}$ be a sheaf of sets on $U$. The stalks of the sheaf $j_!\\mathcal{F}$ are described as follows $$ j_{!}\\mathcal{F}_x = \\left\\{ \\begin{matrix} \\emptyset & \\text{if} & x \\not \\in U \\\\ \\mathcal{F}_x & \\text{if} & x \\in U \\end{matrix} \\right. $$ \\item On the category of presheaves of $U$ we have $j_pj_{p!} = \\text{id}$. \\item On the category of sheaves of $U$ we have $j^{-1}j_! = \\text{id}$. \\end{enumerate}"}
+{"_id": "14544", "title": "sheaves-lemma-j-shriek-abelian", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. Consider the functors of restriction and extension by $0$ for abelian (pre)sheaves. \\begin{enumerate} \\item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}). \\item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \\Mor_{\\textit{Ab}(X)}(j_!\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, j^{-1}\\mathcal{G}) = \\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, \\mathcal{G}|_U) $$ bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$. \\item Let $\\mathcal{F}$ be an abelian sheaf on $U$. The stalks of the sheaf $j_!\\mathcal{F}$ are described as follows $$ j_{!}\\mathcal{F}_x = \\left\\{ \\begin{matrix} 0 & \\text{if} & x \\not \\in U \\\\ \\mathcal{F}_x & \\text{if} & x \\in U \\end{matrix} \\right. $$ \\item On the category of abelian presheaves of $U$ we have $j_pj_{p!} = \\text{id}$. \\item On the category of abelian sheaves of $U$ we have $j^{-1}j_! = \\text{id}$. \\end{enumerate}"}
+{"_id": "14545", "title": "sheaves-lemma-j-shriek-structures", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. Let $(\\mathcal{C}, F)$ be a type of algebraic structure such that $\\mathcal{C}$ has an initial object $e$. Consider the functors of restriction and extension by $e$ for (pre)sheaves of algebraic structure defined above. \\begin{enumerate} \\item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}). \\item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \\Mor_{\\Sh(X, \\mathcal{C})}(j_!\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, j^{-1}\\mathcal{G}) = \\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, \\mathcal{G}|_U) $$ bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$. \\item Let $\\mathcal{F}$ be a sheaf on $U$. The stalks of the sheaf $j_!\\mathcal{F}$ are described as follows $$ j_{!}\\mathcal{F}_x = \\left\\{ \\begin{matrix} e & \\text{if} & x \\not \\in U \\\\ \\mathcal{F}_x & \\text{if} & x \\in U \\end{matrix} \\right. $$ \\item On the category of presheaves of algebraic structures on $U$ we have $j_pj_{p!} = \\text{id}$. \\item On the category of sheaves of algebraic structures on $U$ we have $j^{-1}j_! = \\text{id}$. \\end{enumerate}"}
+{"_id": "14546", "title": "sheaves-lemma-j-shriek-modules", "text": "Let $(X, \\mathcal{O})$ be a ringed space. Let $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$ be an open subspace. Consider the functors of restriction and extension by $0$ for (pre)sheaves of modules defined above. \\begin{enumerate} \\item The functor $j_{p!}$ is a left adjoint to restriction, in a formula $$ \\Mor_{\\textit{PMod}(\\mathcal{O})}(j_{p!}\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\textit{PMod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U) $$ bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$. \\item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \\Mor_{\\textit{Mod}(\\mathcal{O})}(j_!\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\textit{Mod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U) $$ bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$. \\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $U$. The stalks of the sheaf $j_!\\mathcal{F}$ are described as follows $$ j_{!}\\mathcal{F}_x = \\left\\{ \\begin{matrix} 0 & \\text{if} & x \\not \\in U \\\\ \\mathcal{F}_x & \\text{if} & x \\in U \\end{matrix} \\right. $$ \\item On the category of sheaves of $\\mathcal{O}|_U$-modules on $U$ we have $j^{-1}j_! = \\text{id}$. \\end{enumerate}"}
+{"_id": "14547", "title": "sheaves-lemma-equivalence-categories-open", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. The functor $$ j_! : \\Sh(U) \\longrightarrow \\Sh(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = \\emptyset$ for all $x \\in X \\setminus U$."}
+{"_id": "14548", "title": "sheaves-lemma-equivalence-categories-open-abelian", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. The functor $$ j_! : \\textit{Ab}(U) \\longrightarrow \\textit{Ab}(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$."}
+{"_id": "14549", "title": "sheaves-lemma-equivalence-categories-open-structures", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. Let $(\\mathcal{C}, F)$ be a type of algebraic structure such that $\\mathcal{C}$ has an initial object $e$. The functor $$ j_! : \\Sh(U, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = e$ for all $x \\in X \\setminus U$."}
+{"_id": "14550", "title": "sheaves-lemma-equivalence-categories-open-modules", "text": "Let $(X, \\mathcal{O})$ be a ringed space. Let $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$ be an open subspace. The functor $$ j_! : \\textit{Mod}(\\mathcal{O}|_U) \\longrightarrow \\textit{Mod}(\\mathcal{O}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$."}
+{"_id": "14551", "title": "sheaves-lemma-stalks-closed-pushforward", "text": "Let $X$ be a topological space. Let $i : Z \\to X$ be the inclusion of a closed subset $Z$ into $X$. Let $\\mathcal{F}$ be a sheaf of sets on $Z$. The stalks of $i_*\\mathcal{F}$ are described as follows $$ i_*\\mathcal{F}_x = \\left\\{ \\begin{matrix} \\{*\\} & \\text{if} & x \\not \\in Z \\\\ \\mathcal{F}_x & \\text{if} & x \\in Z \\end{matrix} \\right. $$ where $\\{*\\}$ denotes a singleton set. Moreover, $i^{-1}i_* = \\text{id}$ on the category of sheaves of sets on $Z$. Moreover, the same holds for abelian sheaves on $Z$, resp.\\ sheaves of algebraic structures on $Z$ where $\\{*\\}$ has to be replaced by $0$, resp.\\ a final object of the category of algebraic structures."}
+{"_id": "14552", "title": "sheaves-lemma-equivalence-categories-closed", "text": "Let $X$ be a topological space. Let $i : Z \\to X$ be the inclusion of a closed subset. The functor $$ i_* : \\Sh(Z) \\longrightarrow \\Sh(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = \\{*\\}$ for all $x \\in X \\setminus Z$."}
+{"_id": "14553", "title": "sheaves-lemma-equivalence-categories-closed-abelian", "text": "Let $X$ be a topological space. Let $i : Z \\to X$ be the inclusion of a closed subset. The functor $$ i_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$."}
+{"_id": "14554", "title": "sheaves-lemma-equivalence-categories-closed-structures", "text": "Let $X$ be a topological space. Let $i : Z \\to X$ be the inclusion of a closed subset. Let $(\\mathcal{C}, F)$ be a type of algebraic structure with final object $0$. The functor $$ i_* : \\Sh(Z, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\\mathcal{G}$ such that $\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$."}
+{"_id": "14555", "title": "sheaves-lemma-glue-maps", "text": "Let $X$ be a topological space. Let $X = \\bigcup U_i$ be an open covering. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of sets on $X$. Given a collection $$ \\varphi_i : \\mathcal{F}|_{U_i} \\longrightarrow \\mathcal{G}|_{U_i} $$ of maps of sheaves such that for all $i, j \\in I$ the maps $\\varphi_i, \\varphi_j$ restrict to the same map $\\mathcal{F}|_{U_i \\cap U_j} \\to \\mathcal{G}|_{U_i \\cap U_j}$ then there exists a unique map of sheaves $$ \\varphi : \\mathcal{F} \\longrightarrow \\mathcal{G} $$ whose restriction to each $U_i$ agrees with $\\varphi_i$."}
+{"_id": "14556", "title": "sheaves-lemma-glue-sheaves", "text": "Let $X$ be a topological space. Let $X = \\bigcup_{i\\in I} U_i$ be an open covering. Given any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$ for sheaves of sets with respect to the covering $X = \\bigcup U_i$ there exists a sheaf of sets $\\mathcal{F}$ on $X$ together with isomorphisms $$ \\varphi_i : \\mathcal{F}|_{U_i} \\to \\mathcal{F}_i $$ such that the diagrams $$ \\xymatrix{ \\mathcal{F}|_{U_i \\cap U_j} \\ar[r]_{\\varphi_i} \\ar[d]_{\\text{id}} & \\mathcal{F}_i|_{U_i \\cap U_j} \\ar[d]^{\\varphi_{ij}} \\\\ \\mathcal{F}|_{U_i \\cap U_j} \\ar[r]^{\\varphi_j} & \\mathcal{F}_j|_{U_i \\cap U_j} } $$ are commutative."}
+{"_id": "14557", "title": "sheaves-lemma-glue-sheaves-structures", "text": "Let $X$ be a topological space. Let $X = \\bigcup U_i$ be an open covering. Let $(\\mathcal{F}_i, \\varphi_{ij})$ be a glueing data of sheaves of abelian groups, resp.\\ sheaves of algebraic structures, resp.\\ sheaves of $\\mathcal{O}$-modules for some sheaf of rings $\\mathcal{O}$ on $X$. Then the construction in the proof of Lemma \\ref{lemma-glue-sheaves} above leads to a sheaf of abelian groups, resp.\\ sheaf of algebraic structures, resp.\\ sheaf of $\\mathcal{O}$-modules."}
+{"_id": "14558", "title": "sheaves-lemma-mapping-property-glue", "text": "Let $X$ be a topological space. Let $X = \\bigcup_{i\\in I} U_i$ be an open covering. The functor which associates to a sheaf of sets $\\mathcal{F}$ the following collection of glueing data $$ (\\mathcal{F}|_{U_i}, (\\mathcal{F}|_{U_i})|_{U_i \\cap U_j} \\to (\\mathcal{F}|_{U_j})|_{U_i \\cap U_j} ) $$ with respect to the covering $X = \\bigcup U_i$ defines an equivalence of categories between $\\Sh(X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp.\\ sheaves of algebraic structures, resp.\\ sheaves of $\\mathcal{O}$-modules."}
+{"_id": "14590", "title": "descent-theorem-descent", "text": "The following conditions are equivalent. \\begin{enumerate} \\item[(a)] The morphism $f$ is a descent morphism for modules. \\item[(b)] The morphism $f$ is an effective descent morphism for modules. \\item[(c)] The morphism $f$ is universally injective. \\end{enumerate}"}
+{"_id": "14593", "title": "descent-lemma-refine-descent-datum", "text": "Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and $\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$ be families of morphisms of schemes with fixed target. Let $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. Let $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\\{V_j \\to V\\}_{j \\in J}$. Then \\begin{enumerate} \\item The system $$ \\left(g_i^*\\mathcal{F}_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right) $$ is a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$. \\item This construction is functorial in the descent datum $(\\mathcal{F}_j, \\varphi_{jj'})$. \\item Given a second morphism $(g', \\alpha' : I \\to J, (g'_i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data $$ (g_i^*\\mathcal{F}_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}) \\cong ((g'_i)^*\\mathcal{F}_{\\alpha'(i)}, (g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}). $$ \\end{enumerate}"}
+{"_id": "14594", "title": "descent-lemma-zariski-descent-effective", "text": "Let $S$ be a scheme. Let $S = \\bigcup U_i$ be an open covering. Any descent datum on quasi-coherent sheaves for the family $\\mathcal{U} = \\{U_i \\to S\\}$ is effective. Moreover, the functor from the category of quasi-coherent $\\mathcal{O}_S$-modules to the category of descent data with respect to $\\mathcal{U}$ is fully faithful."}
+{"_id": "14595", "title": "descent-lemma-descent-datum-cosimplicial", "text": "Let $R \\to A$ be a ring map. Given a descent datum $(N, \\varphi)$ we can associate to it a cosimplicial $(A/R)_\\bullet$-module $N_\\bullet$\\footnote{We should really write $(N, \\varphi)_\\bullet$.} by the rules $N_n = N_{n, n}$ and given $\\beta : [n] \\to [m]$ setting we define $$ N_\\bullet(\\beta) = (\\varphi^m_{\\beta(n)m}) \\circ N_{\\beta, n} : N_{n, n} \\longrightarrow N_{m, m}. $$ This procedure is functorial in the descent datum."}
+{"_id": "14596", "title": "descent-lemma-canonical-descent-datum-cosimplicial", "text": "Let $R \\to A$ be a ring map. Let $M$ be an $R$-module. The cosimplicial $(A/R)_\\bullet$-module associated to the canonical descent datum is isomorphic to the cosimplicial module $(A/R)_\\bullet \\otimes_R M$."}
+{"_id": "14598", "title": "descent-lemma-ff-exact", "text": "Suppose that $R \\to A$ is faithfully flat, see Algebra, Definition \\ref{algebra-definition-flat}. Then for any $R$-module $M$ the extended cochain complex (\\ref{equation-extended-complex}) is exact."}
+{"_id": "14599", "title": "descent-lemma-recognize-effective", "text": "Let $R \\to A$ be a faithfully flat ring map. Let $(N, \\varphi)$ be a descent datum. Then $(N, \\varphi)$ is effective if and only if the canonical map $$ A \\otimes_R H^0(s(N_\\bullet)) \\longrightarrow N $$ is an isomorphism."}
+{"_id": "14600", "title": "descent-lemma-descent-descends", "text": "Let $R \\to A$ be a faithfully flat ring map, and let $R \\to R'$ be faithfully flat. Set $A' = R' \\otimes_R A$. If all descent data for $R' \\to A'$ are effective, then so are all descent data for $R \\to A$."}
+{"_id": "14602", "title": "descent-lemma-C-is-faithful", "text": "For a ring $R$, the functor $C : \\text{Mod}_R \\to \\text{Mod}_R$ is exact and reflects injections and surjections."}
+{"_id": "14603", "title": "descent-lemma-split-surjection", "text": "Let $R$ be a ring. A morphism $f: M \\to N$ in $\\text{Mod}_R$ is universally injective if and only if $C(f): C(N) \\to C(M)$ is a split surjection."}
+{"_id": "14604", "title": "descent-lemma-equalizer-M", "text": "For $(M,\\theta) \\in DD_{S/R}$, the diagram \\begin{equation} \\label{equation-equalizer-M} \\xymatrix@C=8pc{ M \\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} & M \\otimes_{S, \\delta_1^1} S_2 \\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)} \\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} &  M \\otimes_{S, \\delta_{12}^1} S_3 } \\end{equation} is a split equalizer."}
+{"_id": "14605", "title": "descent-lemma-equalizer-CM", "text": "For $(M, \\theta) \\in DD_{S/R}$, the diagram \\begin{equation} \\label{equation-coequalizer-CM} \\xymatrix@C=8pc{ C(M \\otimes_{S, \\delta_{12}^1} S_3) \\ar@<1ex>[r]^{C((\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0))} \\ar@<-1ex>[r]_{C(1_{M \\otimes S_2} \\otimes \\delta^2_1)} & C(M \\otimes_{S, \\delta_1^1} S_2 ) \\ar[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))} & C(M). } \\end{equation} obtained by applying $C$ to (\\ref{equation-equalizer-M}) is a split coequalizer."}
+{"_id": "14607", "title": "descent-lemma-descent-lemma", "text": "If $f$ is universally injective, then the diagram \\begin{equation} \\label{equation-equalizer-f2} \\xymatrix@C=8pc{ f_*(M, \\theta) \\otimes_R S \\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} & M \\otimes_{S, \\delta_1^1} S_2  \\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)} \\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} & M \\otimes_{S, \\delta_{12}^1} S_3 } \\end{equation} obtained by tensoring (\\ref{equation-equalizer-f}) over $R$ with $S$ is an  equalizer."}
+{"_id": "14608", "title": "descent-lemma-flat-to-injective", "text": "If $M \\in \\text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module."}
+{"_id": "14609", "title": "descent-lemma-standard-fpqc-covering", "text": "Let $S$ be an affine scheme. Let $\\mathcal{U} = \\{f_i : U_i \\to S\\}_{i = 1, \\ldots, n}$ be a standard fpqc covering of $S$, see Topologies, Definition \\ref{topologies-definition-standard-fpqc}. Any descent datum on quasi-coherent sheaves for $\\mathcal{U} = \\{U_i \\to S\\}$ is effective. Moreover, the functor from the category of quasi-coherent $\\mathcal{O}_S$-modules to the category of descent data with respect to $\\mathcal{U}$ is fully faithful."}
+{"_id": "14610", "title": "descent-lemma-galois-descent", "text": "Let $k'/k$ be a (finite) Galois extension with Galois group $G$. Let $X$ be a scheme over $k$. The category of quasi-coherent $\\mathcal{O}_X$-modules is equivalent to the category of systems $(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where \\begin{enumerate} \\item $\\mathcal{F}$ is a quasi-coherent module on $X_{k'}$, \\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$ is an isomorphism of modules, \\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$ for all $\\sigma, \\tau \\in G$. \\end{enumerate} Here $f_\\sigma = \\text{id}_X \\times \\Spec(\\sigma) : X_{k'} \\to X_{k'}$."}
+{"_id": "14611", "title": "descent-lemma-galois-descent-more-general", "text": "Let $X \\to Y$, $G$, and $f_\\sigma : X \\to X$ be as above. The category of quasi-coherent $\\mathcal{O}_Y$-modules is equivalent to the category of systems $(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where \\begin{enumerate} \\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module, \\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$ is an isomorphism of modules, \\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$ for all $\\sigma, \\tau \\in G$. \\end{enumerate}"}
+{"_id": "14612", "title": "descent-lemma-finite-type-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module."}
+{"_id": "14613", "title": "descent-lemma-finite-type-descends-fppf", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of locally ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules. If \\begin{enumerate} \\item $f$ is open as a map of topological spaces, \\item $f$ is surjective and flat, and \\item $f^*\\mathcal{F}$ is of finite type, \\end{enumerate} then $\\mathcal{F}$ is of finite type."}
+{"_id": "14614", "title": "descent-lemma-finite-presentation-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "14615", "title": "descent-lemma-locally-generated-by-r-sections-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is locally generated by $r$ sections as an $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is locally generated by $r$ sections as an $\\mathcal{O}_X$-module."}
+{"_id": "14616", "title": "descent-lemma-flat-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module."}
+{"_id": "14617", "title": "descent-lemma-finite-locally-free-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a finite locally free $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module."}
+{"_id": "14618", "title": "descent-lemma-locally-projective-descends", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module."}
+{"_id": "14619", "title": "descent-lemma-finite-over-finite-module", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ is a finite morphism. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite type."}
+{"_id": "14620", "title": "descent-lemma-finite-finitely-presented-module", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ is finite and of finite presentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation if and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite presentation."}
+{"_id": "14621", "title": "descent-lemma-sheaf-condition-holds", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module. Let $\\tau \\in \\{Zariski, \\linebreak[0] fpqc, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. The functor defined in (\\ref{equation-quasi-coherent-presheaf}) satisfies the sheaf condition with respect to any $\\tau$-covering $\\{T_i \\to T\\}_{i \\in I}$ of any scheme $T$ over $S$."}
+{"_id": "14622", "title": "descent-lemma-compare-sites", "text": "Let $S$ be a scheme. Denote $$ \\begin{matrix} \\text{id}_{\\tau, Zar} & : & (\\Sch/S)_\\tau \\to S_{Zar}, & \\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\} \\\\ \\text{id}_{\\tau, \\etale} & : & (\\Sch/S)_\\tau \\to S_\\etale, & \\tau \\in \\{\\etale, smooth, syntomic, fppf\\} \\\\ \\text{id}_{small, \\etale, Zar} & : & S_\\etale \\to S_{Zar}, \\end{matrix} $$ the morphisms of ringed sites of Remark \\ref{remark-change-topologies-ringed}. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_S$-modules which we view a sheaf of $\\mathcal{O}$-modules on $S_{Zar}$. Then \\begin{enumerate} \\item $(\\text{id}_{\\tau, Zar})^*\\mathcal{F}$ is the $\\tau$-sheafification of the Zariski sheaf $$ (f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F}) $$ on $(\\Sch/S)_\\tau$, and \\item $(\\text{id}_{small, \\etale, Zar})^*\\mathcal{F}$ is the \\'etale sheafification of the Zariski sheaf $$ (f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F}) $$ on $S_\\etale$. \\end{enumerate} Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules on $S_\\etale$. Then \\begin{enumerate} \\item[(3)] $(\\text{id}_{\\tau, \\etale})^*\\mathcal{G}$ is the $\\tau$-sheafification of the \\'etale sheaf $$ (f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{G}) $$ where $f_{small} : T_\\etale \\to S_\\etale$ is the morphism of ringed small \\'etale sites of Remark \\ref{remark-change-topologies-ringed}. \\end{enumerate}"}
+{"_id": "14623", "title": "descent-lemma-quasi-coherent-gives-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. \\begin{enumerate} \\item The sheaf $\\mathcal{F}^a$ is a quasi-coherent $\\mathcal{O}$-module on $(\\Sch/S)_\\tau$, as defined in Modules on Sites, Definition \\ref{sites-modules-definition-site-local}. \\item If $\\tau = \\etale$ (resp.\\ $\\tau = Zariski$), then the sheaf $\\mathcal{F}^a$ is a quasi-coherent $\\mathcal{O}$-module on $S_\\etale$ (resp.\\ $S_{Zar}$) as defined in Modules on Sites, Definition \\ref{sites-modules-definition-site-local}. \\end{enumerate}"}
+{"_id": "14624", "title": "descent-lemma-standard-covering-Cech", "text": "Let $S$ be a scheme. Let \\begin{enumerate} \\item[(a)] $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$ and $\\mathcal{C} = (\\Sch/S)_\\tau$, or \\item[(b)] let $\\tau = \\etale$ and $\\mathcal{C} = S_\\etale$, or \\item[(c)] let $\\tau = Zariski$ and $\\mathcal{C} = S_{Zar}$. \\end{enumerate} Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$ be affine. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$ be a standard affine $\\tau$-covering in $\\mathcal{C}$. Then \\begin{enumerate} \\item $\\mathcal{V} = \\{\\coprod_{i = 1, \\ldots, n} U_i \\to U\\}$ is a $\\tau$-covering of $U$, \\item $\\mathcal{U}$ is a refinement of $\\mathcal{V}$, and \\item the induced map on {\\v C}ech complexes (Cohomology on Sites, Equation (\\ref{sites-cohomology-equation-map-cech-complexes})) $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F}) \\longrightarrow \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) $$ is an isomorphism of complexes. \\end{enumerate}"}
+{"_id": "14625", "title": "descent-lemma-standard-covering-Cech-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\\tau$, $\\mathcal{C}$, $U$, $\\mathcal{U}$ be as in Lemma \\ref{lemma-standard-covering-Cech}. Then there is an isomorphism of complexes $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^a) \\cong s((A/R)_\\bullet \\otimes_R M) $$ (see Section \\ref{section-descent-modules}) where $R = \\Gamma(U, \\mathcal{O}_U)$, $M = \\Gamma(U, \\mathcal{F}^a)$ and $R \\to A$ is a faithfully flat ring map. In particular $$ \\check{H}^p(\\mathcal{U}, \\mathcal{F}^a) = 0 $$ for all $p \\geq 1$."}
+{"_id": "14627", "title": "descent-lemma-equivalence-quasi-coherent-limits", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. The functors $$ \\QCoh(\\mathcal{O}_S) \\longrightarrow \\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O}) \\quad\\text{and}\\quad \\QCoh(\\mathcal{O}_S) \\longrightarrow \\textit{Mod}(S_\\tau, \\mathcal{O}) $$ defined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in Proposition \\ref{proposition-equivalence-quasi-coherent} are \\begin{enumerate} \\item fully faithful, \\item compatible with direct sums, \\item compatible with colimits, \\item right exact, \\item exact as a functor $\\QCoh(\\mathcal{O}_S) \\to \\textit{Mod}(S_\\etale, \\mathcal{O})$, \\item {\\bf not} exact as a functor $\\QCoh(\\mathcal{O}_S) \\to \\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$ in general, \\item given two quasi-coherent $\\mathcal{O}_S$-modules $\\mathcal{F}$, $\\mathcal{G}$ we have $(\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{G})^a = \\mathcal{F}^a \\otimes_\\mathcal{O} \\mathcal{G}^a$, \\item given two quasi-coherent $\\mathcal{O}_S$-modules $\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$ is of finite presentation we have $(\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{G}))^a = \\SheafHom_\\mathcal{O}(\\mathcal{F}^a, \\mathcal{G}^a)$, and \\item given a short exact sequence $0 \\to \\mathcal{F}_1^a \\to \\mathcal{E} \\to \\mathcal{F}_2^a \\to 0$ of $\\mathcal{O}$-modules then $\\mathcal{E}$ is quasi-coherent\\footnote{Warning: This is misleading. See part (6).}, i.e., $\\mathcal{E}$ is in the essential image of the functor. \\end{enumerate}"}
+{"_id": "14628", "title": "descent-lemma-higher-direct-images-small-etale", "text": "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \\'etale or Zariski topology, there are canonical isomorphisms $R^if_{small, *}(\\mathcal{F}^a) = (R^if_*\\mathcal{F})^a$."}
+{"_id": "14629", "title": "descent-lemma-cohomology-parasitic", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Let $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$. \\begin{enumerate} \\item If $\\mathcal{G}$ is parasitic for the $\\tau$-topology, then $H^p_\\tau(U, \\mathcal{G}) = 0$ for every $U$ open in $S$, resp.\\ \\'etale over $S$, resp.\\ smooth over $S$, resp.\\ syntomic over $S$, resp.\\ flat and locally of finite presentation over $S$. \\item If $\\mathcal{G}$ is parasitic then $H^p_\\tau(U, \\mathcal{G}) = 0$ for every $U$ flat over $S$. \\end{enumerate}"}
+{"_id": "14630", "title": "descent-lemma-direct-image-parasitic", "text": "Let $f : T \\to S$ be a morphism of schemes. For any parasitic $\\mathcal{O}$-module on $(\\Sch/T)_\\tau$ the pushforward $f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$ are parasitic $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$."}
+{"_id": "14631", "title": "descent-lemma-quasi-coherent-and-flat-base-change", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale\\}$. Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules on $(\\Sch/S)_{fppf}$ such that \\begin{enumerate} \\item $\\mathcal{G}|_{S_\\tau}$ is quasi-coherent, and \\item for every flat, locally finitely presented morphism $g : U \\to S$ the canonical map $g_{\\tau, small}^*(\\mathcal{G}|_{S_\\tau}) \\to \\mathcal{G}|_{U_\\tau}$ is an isomorphism. \\end{enumerate} Then $H^p(U, \\mathcal{G}) = H^p(U, \\mathcal{G}|_{U_\\tau})$ for every $U$ flat and locally of finite presentation over $S$."}
+{"_id": "14632", "title": "descent-lemma-equiv-fibre-product", "text": "For a scheme $X$ denote $|X|$ the underlying set. Let $f : X \\to S$ be a morphism of schemes. Then $$ |X \\times_S X| \\to |X| \\times_{|S|} |X| $$ is surjective."}
+{"_id": "14633", "title": "descent-lemma-universal-effective-epimorphism-affine", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine schemes. The following are equivalent \\begin{enumerate} \\item for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $$ \\Gamma(X, \\mathcal{F}) = \\text{Equalizer}\\left( \\xymatrix{ \\prod\\nolimits_{i \\in I} \\Gamma(X_i, f_i^*\\mathcal{F}) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod\\nolimits_{i, j \\in I} \\Gamma(X_i \\times_X X_j, (f_i \\times f_j)^*\\mathcal{F}) } \\right) $$ \\item $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a universal effective epimorphism (Sites, Definition \\ref{sites-definition-universal-effective-epimorphisms}) in the category of affine schemes. \\end{enumerate}"}
+{"_id": "14634", "title": "descent-lemma-universal-effective-epimorphism-surjective", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes. \\begin{enumerate} \\item If the family is universal effective epimorphism in the category of schemes, then $\\coprod f_i$ is surjective. \\item If $X$ and $X_i$ are affine and the family is a universal effective epimorphism in the category of affine schemes, then $\\coprod f_i$ is surjective. \\end{enumerate}"}
+{"_id": "14635", "title": "descent-lemma-check-universal-effective-epimorphism-affine", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes. If for every morphism $Y \\to X$ with $Y$ affine the family of base changes $g_i : Y_i \\to Y$ forms an effective epimorphism, then the family of $f_i$ forms a universally effective epimorphism in the category of schemes."}
+{"_id": "14636", "title": "descent-lemma-universal-effective-epimorphism", "text": "Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine schemes. Assume the equivalent assumption of Lemma \\ref{lemma-universal-effective-epimorphism-affine} hold and that moreover for any morphism of affines $Y \\to X$ the map $$ \\coprod X_i \\times_X Y \\longrightarrow Y $$ is a submersive map of topological spaces (Topology, Definition \\ref{topology-definition-submersive}). Then our family of morphisms is a universal effective epimorphism in the category of schemes."}
+{"_id": "14637", "title": "descent-lemma-open-fpqc-covering", "text": "Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a fpqc covering. Suppose that for each $i$ we have an open subset $W_i \\subset T_i$ such that for all $i, j \\in I$ we have $\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open subsets of $T_i \\times_T T_j$. Then there exists a unique open subset $W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$."}
+{"_id": "14638", "title": "descent-lemma-fpqc-universal-effective-epimorphisms", "text": "Let $\\{T_i \\to T\\}$ be an fpqc covering, see Topologies, Definition \\ref{topologies-definition-fpqc-covering}. Then $\\{T_i \\to T\\}$ is a universal effective epimorphism in the category of schemes, see Sites, Definition \\ref{sites-definition-universal-effective-epimorphisms}. In other words, every representable functor on the category of schemes satisfies the sheaf condition for the fpqc topology, see Topologies, Definition \\ref{topologies-definition-sheaf-property-fpqc}."}
+{"_id": "14639", "title": "descent-lemma-coequalizer-fpqc-local", "text": "Consider schemes $X, Y, Z$ and morphisms $a, b : X \\to Y$ and a morphism $c : Y \\to Z$ with $c \\circ a = c \\circ b$. Set $d = c \\circ a = c \\circ b$. If there exists an fpqc covering $\\{Z_i \\to Z\\}$ such that \\begin{enumerate} \\item for all $i$ the morphism $Y \\times_{c, Z} Z_i \\to Z_i$ is the coequalizer of $(a, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$ and $(b, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$, and \\item for all $i$ and $i'$ the morphism $Y \\times_{c, Z} (Z_i \\times_Z Z_{i'}) \\to (Z_i \\times_Z Z_{i'})$ is the coequalizer of $(a, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to Y \\times_{c, Z} (Z_i \\times_Z Z_{i'})$ and $(b, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to Y \\times_{c, Z} (Z_i \\times_Z Z_{i'})$ \\end{enumerate} then $c$ is the coequalizer of $a$ and $b$."}
+{"_id": "14640", "title": "descent-lemma-flat-finitely-presented-permanence-algebra", "text": "Let $R \\to A \\to B$ be ring maps. Assume $R \\to B$ is of finite presentation and $A \\to B$ faithfully flat and of finite presentation. Then $R \\to A$ is of finite presentation."}
+{"_id": "14641", "title": "descent-lemma-finite-type-local-source-fppf-algebra", "text": "Let $R \\to A \\to B$ be ring maps. Assume $R \\to B$ is of finite type and $A \\to B$ faithfully flat and of finite presentation. Then $R \\to A$ is of finite type."}
+{"_id": "14642", "title": "descent-lemma-flat-finitely-presented-permanence", "text": "\\begin{reference} \\cite[IV, 17.7.5 (i) and (ii)]{EGA}. \\end{reference} Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $f$ is surjective, flat and locally of finite presentation and assume that $p$ is locally of finite presentation (resp.\\ locally of finite type). Then $q$ is locally of finite presentation (resp.\\ locally of finite type)."}
+{"_id": "14643", "title": "descent-lemma-syntomic-smooth-etale-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale), \\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale). \\end{enumerate} Then $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale)."}
+{"_id": "14644", "title": "descent-lemma-smooth-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective, flat, and locally of finite presentation, \\item $p$ is smooth (resp.\\ \\'etale). \\end{enumerate} Then $q$ is smooth (resp.\\ \\'etale)."}
+{"_id": "14645", "title": "descent-lemma-syntomic-permanence", "text": "Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \\begin{enumerate} \\item $f$ is surjective, flat, and locally of finite presentation, \\item $p$ is syntomic. \\end{enumerate} Then both $q$ and $f$ are syntomic."}
+{"_id": "14646", "title": "descent-lemma-curiosity", "text": "Let $X \\to Y \\to Z$ be morphism of schemes. Let $P$ be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that $X \\to Z$ has $P$ and that $\\{X \\to Y\\}$ can be refined by an fppf covering of $Y$. Then $Y \\to Z$ is $P$."}
+{"_id": "14647", "title": "descent-lemma-descending-properties", "text": "Let $\\mathcal{P}$ be a property of schemes. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Assume that \\begin{enumerate} \\item the property is local in the Zariski topology, \\item for any morphism of affine schemes $S' \\to S$ which is flat, flat of finite presentation, \\'etale, smooth or syntomic depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, property $\\mathcal{P}$ holds for $S'$ if property $\\mathcal{P}$ holds for $S$, and \\item for any surjective morphism of affine schemes $S' \\to S$ which is flat, flat of finite presentation, \\'etale, smooth or syntomic depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, property $\\mathcal{P}$ holds for $S$ if property $\\mathcal{P}$ holds for $S'$. \\end{enumerate} Then $\\mathcal{P}$ is $\\tau$ local on the base."}
+{"_id": "14648", "title": "descent-lemma-Noetherian-local-fppf", "text": "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian'' is local in the fppf topology."}
+{"_id": "14649", "title": "descent-lemma-Jacobson-local-fppf", "text": "The property $\\mathcal{P}(S) =$``$S$ is Jacobson'' is local in the fppf topology."}
+{"_id": "14650", "title": "descent-lemma-locally-finite-nr-irred-local-fppf", "text": "The property $\\mathcal{P}(S) =$``every quasi-compact open of $S$ has a finite number of irreducible components'' is local in the fppf topology."}
+{"_id": "14651", "title": "descent-lemma-Sk-local-syntomic", "text": "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(S_k)$'' is local in the syntomic topology."}
+{"_id": "14652", "title": "descent-lemma-CM-local-syntomic", "text": "The property $\\mathcal{P}(S) =$``$S$ is Cohen-Macaulay'' is local in the syntomic topology."}
+{"_id": "14653", "title": "descent-lemma-reduced-local-smooth", "text": "The property $\\mathcal{P}(S) =$``$S$ is reduced'' is local in the smooth topology."}
+{"_id": "14654", "title": "descent-lemma-normal-local-smooth", "text": "\\begin{slogan} Normality is local in the smooth topology. \\end{slogan} The property $\\mathcal{P}(S) =$``$S$ is normal'' is local in the smooth topology."}
+{"_id": "14655", "title": "descent-lemma-Rk-local-smooth", "text": "The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(R_k)$'' is local in the smooth topology."}
+{"_id": "14656", "title": "descent-lemma-regular-local-smooth", "text": "The property $\\mathcal{P}(S) =$``$S$ is regular'' is local in the smooth topology."}
+{"_id": "14657", "title": "descent-lemma-Nagata-local-smooth", "text": "The property $\\mathcal{P}(S) =$``$S$ is Nagata'' is local in the smooth topology."}
+{"_id": "14658", "title": "descent-lemma-descend-reduced", "text": "If $f : X \\to Y$ is a flat and surjective morphism of schemes and $X$ is reduced, then $Y$ is reduced."}
+{"_id": "14659", "title": "descent-lemma-descend-regular", "text": "Let $f : X \\to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular."}
+{"_id": "14660", "title": "descent-lemma-dimension-at-point-local", "text": "Let $f : U \\to V$ be an \\'etale morphism of schemes. Let $u \\in U$ and $v = f(u)$. Then $\\dim_u(U) = \\dim_v(V)$."}
+{"_id": "14661", "title": "descent-lemma-dimension-local-ring-local", "text": "Let $f : U \\to V$ be an \\'etale morphism of schemes. Let $u \\in U$ and $v = f(u)$. Then $\\dim(\\mathcal{O}_{U, u}) = \\dim(\\mathcal{O}_{V, v})$."}
+{"_id": "14663", "title": "descent-lemma-pullback-property-local-target", "text": "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$. Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local on the target. Let $f : X \\to Y$ have property $\\mathcal{P}$. For any morphism $Y' \\to Y$ which is flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic, resp.\\ \\'etale, resp.\\ an open immersion, the base change $f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$."}
+{"_id": "14664", "title": "descent-lemma-largest-open-of-the-base", "text": "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$. Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local on the target. For any morphism of schemes $f : X \\to Y$ there exists a largest open $W(f) \\subset Y$ such that the restriction $X_{W(f)} \\to W(f)$ has $\\mathcal{P}$. Moreover, \\begin{enumerate} \\item if $g : Y' \\to Y$ is flat and locally of finite presentation, syntomic, smooth, or \\'etale and the base change $f' : X_{Y'} \\to Y'$ has $\\mathcal{P}$, then $g(Y') \\subset W(f)$, \\item if $g : Y' \\to Y$ is flat and locally of finite presentation, syntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and \\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then $g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$ by $Y_i \\to Y$. \\end{enumerate}"}
+{"_id": "14665", "title": "descent-lemma-descending-properties-morphisms", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$. Assume that \\begin{enumerate} \\item the property is preserved under flat, flat and locally of finite presentation, \\'etale, smooth, or syntomic base change depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic (compare with Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}), \\item the property is Zariski local on the base. \\item for any surjective morphism of affine schemes $S' \\to S$ which is flat, flat of finite presentation, \\'etale, smooth or syntomic depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, and any morphism of schemes $f : X \\to S$ property $\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$ holds for the base change $f' : X' = S' \\times_S X \\to S'$. \\end{enumerate} Then $\\mathcal{P}$ is $\\tau$ local on the base."}
+{"_id": "14666", "title": "descent-lemma-descending-property-quasi-compact", "text": "The property $\\mathcal{P}(f) =$``$f$ is quasi-compact'' is fpqc local on the base."}
+{"_id": "14667", "title": "descent-lemma-descending-property-quasi-separated", "text": "The property $\\mathcal{P}(f) =$``$f$ is quasi-separated'' is fpqc local on the base."}
+{"_id": "14668", "title": "descent-lemma-descending-property-universally-closed", "text": "The property $\\mathcal{P}(f) =$``$f$ is universally closed'' is fpqc local on the base."}
+{"_id": "14669", "title": "descent-lemma-descending-property-universally-open", "text": "The property $\\mathcal{P}(f) =$``$f$ is universally open'' is fpqc local on the base."}
+{"_id": "14670", "title": "descent-lemma-descending-property-universally-submersive", "text": "The property $\\mathcal{P}(f) =$``$f$ is universally submersive'' is fpqc local on the base."}
+{"_id": "14671", "title": "descent-lemma-descending-property-separated", "text": "The property $\\mathcal{P}(f) =$``$f$ is separated'' is fpqc local on the base."}
+{"_id": "14672", "title": "descent-lemma-descending-property-surjective", "text": "The property $\\mathcal{P}(f) =$``$f$ is surjective'' is fpqc local on the base."}
+{"_id": "14673", "title": "descent-lemma-descending-property-universally-injective", "text": "The property $\\mathcal{P}(f) =$``$f$ is universally injective'' is fpqc local on the base."}
+{"_id": "14674", "title": "descent-lemma-descending-property-universal-homeomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism'' is fpqc local on the base."}
+{"_id": "14675", "title": "descent-lemma-descending-property-locally-finite-type", "text": "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type'' is fpqc local on the base."}
+{"_id": "14676", "title": "descent-lemma-descending-property-locally-finite-presentation", "text": "The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation'' is fpqc local on the base."}
+{"_id": "14677", "title": "descent-lemma-descending-property-finite-type", "text": "The property $\\mathcal{P}(f) =$``$f$ is of finite type'' is fpqc local on the base."}
+{"_id": "14678", "title": "descent-lemma-descending-property-finite-presentation", "text": "The property $\\mathcal{P}(f) =$``$f$ is of finite presentation'' is fpqc local on the base."}
+{"_id": "14679", "title": "descent-lemma-descending-property-proper", "text": "The property $\\mathcal{P}(f) =$``$f$ is proper'' is fpqc local on the base."}
+{"_id": "14680", "title": "descent-lemma-descending-property-flat", "text": "The property $\\mathcal{P}(f) =$``$f$ is flat'' is fpqc local on the base."}
+{"_id": "14681", "title": "descent-lemma-descending-property-open-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is an open immersion'' is fpqc local on the base."}
+{"_id": "14682", "title": "descent-lemma-descending-property-isomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is an isomorphism'' is fpqc local on the base."}
+{"_id": "14683", "title": "descent-lemma-descending-property-affine", "text": "The property $\\mathcal{P}(f) =$``$f$ is affine'' is fpqc local on the base."}
+{"_id": "14684", "title": "descent-lemma-descending-property-closed-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is a closed immersion'' is fpqc local on the base."}
+{"_id": "14685", "title": "descent-lemma-descending-property-quasi-affine", "text": "The property $\\mathcal{P}(f) =$``$f$ is quasi-affine'' is fpqc local on the base."}
+{"_id": "14686", "title": "descent-lemma-descending-property-quasi-compact-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion'' is fpqc local on the base."}
+{"_id": "14687", "title": "descent-lemma-descending-property-integral", "text": "The property $\\mathcal{P}(f) =$``$f$ is integral'' is fpqc local on the base."}
+{"_id": "14688", "title": "descent-lemma-descending-property-finite", "text": "The property $\\mathcal{P}(f) =$``$f$ is finite'' is fpqc local on the base."}
+{"_id": "14689", "title": "descent-lemma-descending-property-quasi-finite", "text": "The properties $\\mathcal{P}(f) =$``$f$ is locally quasi-finite'' and $\\mathcal{P}(f) =$``$f$ is quasi-finite'' are fpqc local on the base."}
+{"_id": "14690", "title": "descent-lemma-descending-property-relative-dimension-d", "text": "The property $\\mathcal{P}(f) =$``$f$ is locally of finite type of relative dimension $d$'' is fpqc local on the base."}
+{"_id": "14691", "title": "descent-lemma-descending-property-syntomic", "text": "The property $\\mathcal{P}(f) =$``$f$ is syntomic'' is fpqc local on the base."}
+{"_id": "14692", "title": "descent-lemma-descending-property-smooth", "text": "The property $\\mathcal{P}(f) =$``$f$ is smooth'' is fpqc local on the base."}
+{"_id": "14693", "title": "descent-lemma-descending-property-unramified", "text": "The property $\\mathcal{P}(f) =$``$f$ is unramified'' is fpqc local on the base. The property $\\mathcal{P}(f) =$``$f$ is G-unramified'' is fpqc local on the base."}
+{"_id": "14694", "title": "descent-lemma-descending-property-etale", "text": "The property $\\mathcal{P}(f) =$``$f$ is \\'etale'' is fpqc local on the base."}
+{"_id": "14695", "title": "descent-lemma-descending-property-finite-locally-free", "text": "The property $\\mathcal{P}(f) =$``$f$ is finite locally free'' is fpqc local on the base. Let $d \\geq 0$. The property $\\mathcal{P}(f) =$``$f$ is finite locally free of degree $d$'' is fpqc local on the base."}
+{"_id": "14696", "title": "descent-lemma-descending-property-monomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is a monomorphism'' is fpqc local on the base."}
+{"_id": "14697", "title": "descent-lemma-descending-property-regular-immersion", "text": "The properties \\begin{enumerate} \\item[] $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion'', \\item[] $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion'', and \\item[] $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion'' \\end{enumerate} are fpqc local on the base."}
+{"_id": "14698", "title": "descent-lemma-descending-fppf-property-immersion", "text": "The property $\\mathcal{P}(f) =$``$f$ is an immersion'' is fppf local on the base."}
+{"_id": "14699", "title": "descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism", "text": "Let $f : X \\to Y$ be a flat, quasi-compact, surjective monomorphism. Then f is an isomorphism."}
+{"_id": "14700", "title": "descent-lemma-universally-injective-etale-open-immersion", "text": "A universally injective \\'etale morphism is an open immersion."}
+{"_id": "14701", "title": "descent-lemma-flat-universally-injective", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set of generic points of irreducible components of $X$. If \\begin{enumerate} \\item $f$ is flat and separated, \\item for $\\xi \\in X^0$ we have $\\kappa(f(\\xi)) = \\kappa(\\xi)$, and \\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$, \\end{enumerate} then $f$ is universally injective."}
+{"_id": "14703", "title": "descent-lemma-descending-property-proper-over-base", "text": "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type. Let $Z$ be a closed subset of $X$. If there exists an fpqc covering $\\{Y_i \\to Y\\}$ such that the inverse image $Z_i \\subset Y_i \\times_Y X$ is proper over $Y_i$ (Cohomology of Schemes, Definition \\ref{coherent-definition-proper-over-base}) then $Z$ is proper over $Y$."}
+{"_id": "14704", "title": "descent-lemma-descending-property-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\{g_i : S_i \\to S\\}_{i \\in I}$ be an fpqc covering. Let $f_i : X_i \\to S_i$ be the base change of $f$ and let $\\mathcal{L}_i$ be the pullback of $\\mathcal{L}$ to $X_i$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/S$, and \\item $\\mathcal{L}_i$ is ample on $X_i/S_i$ for every $i \\in I$. \\end{enumerate}"}
+{"_id": "14705", "title": "descent-lemma-precompose-property-local-source", "text": "Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$. Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local on the source. Let $f : X \\to Y$ have property $\\mathcal{P}$. For any morphism $a : X' \\to X$ which is flat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic, resp.\\ \\'etale, resp.\\ an open immersion, the composition $f \\circ a : X' \\to Y$ has property $\\mathcal{P}$."}
+{"_id": "14706", "title": "descent-lemma-largest-open-of-the-source", "text": "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$. Let $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local on the source. For any morphism of schemes $f : X \\to Y$ there exists a largest open $W(f) \\subset X$ such that the restriction $f|_{W(f)} : W(f) \\to Y$ has $\\mathcal{P}$. Moreover, if $g : X' \\to X$ is flat and locally of finite presentation, syntomic, smooth, or \\'etale and $f' = f \\circ g : X' \\to Y$, then $g^{-1}(W(f)) = W(f')$."}
+{"_id": "14707", "title": "descent-lemma-properties-morphisms-local-source", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Assume that \\begin{enumerate} \\item the property is preserved under precomposing with flat, flat locally of finite presentation, \\'etale, smooth or syntomic morphisms depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, \\item the property is Zariski local on the source, \\item the property is Zariski local on the target, \\item for any morphism of affine schemes $X \\to Y$, and any surjective morphism of affine schemes $X' \\to X$ which is flat, flat of finite presentation, \\'etale, smooth or syntomic depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, property $\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$ holds for the composition $f' : X' \\to Y$. \\end{enumerate} Then $\\mathcal{P}$ is $\\tau$ local on the source."}
+{"_id": "14708", "title": "descent-lemma-flat-fpqc-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source."}
+{"_id": "14709", "title": "descent-lemma-injective-local-rings-fpqc-local-source", "text": "Then property $\\mathcal{P}(f : X \\to Y)=$``for every $x \\in X$ the map of local rings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is injective'' is fpqc local on the source."}
+{"_id": "14710", "title": "descent-lemma-locally-finite-presentation-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation'' is fppf local on the source."}
+{"_id": "14711", "title": "descent-lemma-locally-finite-type-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally of finite type'' is fppf local on the source."}
+{"_id": "14712", "title": "descent-lemma-open-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is open'' is fppf local on the source."}
+{"_id": "14713", "title": "descent-lemma-universally-open-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is universally open'' is fppf local on the source."}
+{"_id": "14714", "title": "descent-lemma-syntomic-syntomic-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is syntomic'' is syntomic local on the source."}
+{"_id": "14715", "title": "descent-lemma-smooth-smooth-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is smooth'' is smooth local on the source."}
+{"_id": "14716", "title": "descent-lemma-etale-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is \\'etale'' is \\'etale local on the source."}
+{"_id": "14717", "title": "descent-lemma-locally-quasi-finite-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite'' is \\'etale local on the source."}
+{"_id": "14718", "title": "descent-lemma-unramified-etale-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is unramified'' is \\'etale local on the source. The property $\\mathcal{P}(f)=$``$f$ is G-unramified'' is \\'etale local on the source."}
+{"_id": "14719", "title": "descent-lemma-local-source-target-implies", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on source-and-target. Then \\begin{enumerate} \\item $\\mathcal{P}$ is \\'etale local on the source, \\item $\\mathcal{P}$ is \\'etale local on the target, \\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms: if $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then $g \\circ f$ has $\\mathcal{P}$, and \\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and $g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "14720", "title": "descent-lemma-local-source-target-characterize", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on source-and-target. Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item[(a)] $f$ has property $\\mathcal{P}$, \\item[(b)] for every $x \\in X$ there exists an \\'etale morphism of germs $a : (U, u) \\to (X, x)$, an \\'etale morphism $b : V \\to Y$, and a morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and $h$ has $\\mathcal{P}$, \\item[(c)] for any commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $a$, $b$ \\'etale the morphism $h$ has $\\mathcal{P}$, \\item[(d)] for some diagram as in (c) with $a : U \\to X$ surjective $h$ has $\\mathcal{P}$, \\item[(e)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ such that each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$, \\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such that each composition $X_i \\to Y$ has $\\mathcal{P}$, \\item[(g)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ and for each $i \\in I$ an \\'etale covering $\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism $X_{ij} \\to Y_i$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "14721", "title": "descent-lemma-etale-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes. Assume \\begin{enumerate} \\item $\\mathcal{P}$ is \\'etale local on the source, \\item $\\mathcal{P}$ is \\'etale local on the target, and \\item $\\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open subscheme then $X \\to Z$ has $\\mathcal{P}$. \\end{enumerate} Then $\\mathcal{P}$ is \\'etale local on the source-and-target."}
+{"_id": "14722", "title": "descent-lemma-etale-etale-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Given a commutative diagram of schemes $$ \\vcenter{ \\xymatrix{ X' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\ X \\ar[r]^f & Y } } \\quad\\text{with points}\\quad \\vcenter{ \\xymatrix{ x' \\ar[d] \\ar[r] & y' \\ar[d] \\\\ x \\ar[r] & y } } $$ such that $g'$ is \\'etale at $x'$ and $g$ is \\'etale at $y'$, then $x \\in W(f) \\Leftrightarrow x' \\in W(f')$ where $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}."}
+{"_id": "14723", "title": "descent-lemma-orbits", "text": "Let $k$ be a field. Let $n \\geq 2$. For $1 \\leq i, j \\leq n$ with $i \\not = j$ and $d \\geq 0$ denote $T_{i, j, d}$ the automorphism of $\\mathbf{A}^n_k$ given in coordinates by $$ (x_1, \\ldots, x_n) \\longmapsto (x_1, \\ldots, x_{i - 1}, x_i + x_j^d, x_{i + 1}, \\ldots, x_n) $$ Let $W \\subset \\mathbf{A}^n_k$ be a nonempty open subscheme such that $T_{i, j, d}(W) = W$ for all $i, j, d$ as above. Then either $W = \\mathbf{A}^n_k$ or the characteristic of $k$ is $p > 0$ and $\\mathbf{A}^n_k \\setminus W$ is a finite set of closed points whose coordinates are algebraic over $\\mathbf{F}_p$."}
+{"_id": "14724", "title": "descent-lemma-etale-tau-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes. Assume \\begin{enumerate} \\item $\\mathcal{P}$ is \\'etale local on the source, \\item $\\mathcal{P}$ is smooth local on the target, \\item $\\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open subscheme then $X \\to Z$ has $\\mathcal{P}$. \\end{enumerate} Given a commutative diagram of schemes $$ \\vcenter{ \\xymatrix{ X' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\ X \\ar[r]^f & Y } } \\quad\\text{with points}\\quad \\vcenter{ \\xymatrix{ x' \\ar[d] \\ar[r] & y' \\ar[d] \\\\ x \\ar[r] & y } } $$ such that $g$ is smooth $y'$ and $X' \\to X \\times_Y Y'$ is \\'etale at $x'$, then $x \\in W(f) \\Leftrightarrow x' \\in W(f')$ where $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}."}
+{"_id": "14725", "title": "descent-lemma-local-source-target-global-implies-local", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Consider the property $\\mathcal{Q}$ of morphisms of germs defined by the rule $$ \\mathcal{Q}((X, x) \\to (S, s)) \\Leftrightarrow \\text{there exists a representative }U \\to S \\text{ which has }\\mathcal{P} $$ Then $\\mathcal{Q}$ is \\'etale local on the source-and-target as in Definition \\ref{definition-local-source-target-at-point}."}
+{"_id": "14726", "title": "descent-lemma-local-source-target-local-implies-global", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on source-and-target. Let $Q$ be the associated property of morphisms of germs, see Lemma \\ref{lemma-local-source-target-global-implies-local}. Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item $f$ has property $\\mathcal{P}$, and \\item for every $x \\in X$ the morphism of germs $(X, x) \\to (Y, f(x))$ has property $\\mathcal{Q}$. \\end{enumerate}"}
+{"_id": "14727", "title": "descent-lemma-flat-at-point", "text": "The property of morphisms of germs $$ \\mathcal{P}((X, x) \\to (S, s)) = \\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}\\text{ is flat} $$ is \\'etale local on the source-and-target."}
+{"_id": "14728", "title": "descent-lemma-etale-on-fiber", "text": "Consider a commutative diagram of morphisms of schemes $$ \\xymatrix{ U' \\ar[r] \\ar[d] & V' \\ar[d] \\\\ U \\ar[r] & V } $$ with \\'etale vertical arrows and a point $v' \\in V'$ mapping to $v \\in V$. Then the morphism of fibres $U'_{v'} \\to U_v$ is \\'etale."}
+{"_id": "14729", "title": "descent-lemma-dimension-local-ring-fibre", "text": "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The property of morphisms of germs $$ \\mathcal{P}_d((X, x) \\to (S, s)) = \\text{the local ring } \\mathcal{O}_{X_s, x} \\text{ of the fibre has dimension }d $$ is \\'etale local on the source-and-target."}
+{"_id": "14730", "title": "descent-lemma-transcendence-degree-at-point", "text": "Let $r \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The property of morphisms of germs $$ \\mathcal{P}_r((X, x) \\to (S, s)) \\Leftrightarrow \\text{trdeg}_{\\kappa(s)} \\kappa(x) = r $$ is \\'etale local on the source-and-target."}
+{"_id": "14731", "title": "descent-lemma-dimension-at-point", "text": "Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The property of morphisms of germs $$ \\mathcal{P}_d((X, x) \\to (S, s)) \\Leftrightarrow \\dim_x (X_s) = d $$ is \\'etale local on the source-and-target."}
+{"_id": "14732", "title": "descent-lemma-family-is-one", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$. Set $X = \\coprod_{i \\in I} X_i$, and consider it as an $S$-scheme. There is a canonical equivalence of categories $$ \\begin{matrix} \\text{category of descent data } \\\\ \\text{relative to the family } \\{X_i \\to S\\}_{i \\in I} \\end{matrix} \\longrightarrow \\begin{matrix} \\text{ category of descent data} \\\\ \\text{ relative to } X/S \\end{matrix} $$ which maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with $V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$."}
+{"_id": "14733", "title": "descent-lemma-pullback", "text": "Pullback of descent data for schemes over schemes. \\begin{enumerate} \\item Let $$ \\xymatrix{ X' \\ar[r]_f \\ar[d]_{a'} & X \\ar[d]^a \\\\ S' \\ar[r]^h & S } $$ be a commutative diagram of morphisms of schemes. The construction $$ (V \\to X, \\varphi) \\longmapsto f^*(V \\to X, \\varphi) = (V' \\to X', \\varphi') $$ where $V' = X' \\times_X V$ and where $\\varphi'$ is defined as the composition $$ \\xymatrix{ V' \\times_{S'} X' \\ar@{=}[r] & (X' \\times_X V) \\times_{S'} X' \\ar@{=}[r] & (X' \\times_{S'} X') \\times_{X \\times_S X} (V \\times_S X) \\ar[d]^{\\text{id} \\times \\varphi} \\\\ X' \\times_{S'} V' \\ar@{=}[r] & X' \\times_{S'} (X' \\times_X V) & (X' \\times_{S'} X') \\times_{X \\times_S X} (X \\times_S V) \\ar@{=}[l] } $$ defines a functor from the category of descent data relative to $X \\to S$ to the category of descent data relative to $X' \\to S'$. \\item Given two morphisms $f_i : X' \\to X$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic. \\end{enumerate}"}
+{"_id": "14734", "title": "descent-lemma-pullback-family", "text": "Let $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$ and $\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$ be families of morphisms with fixed target. Let $\\alpha : I \\to J$, $h : S' \\to S$ and $g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. \\begin{enumerate} \\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the family $\\{V_j \\to S'\\}$. The system $$ \\left( g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')} \\right) $$ (with notation as in Remark \\ref{remark-easier-family}) is a descent datum relative to $\\mathcal{V}$. \\item This construction defines a functor between descent data relative to $\\mathcal{U}$ and descent data relative to $\\mathcal{V}$. \\item Given a second $\\alpha' : I \\to J$, $h' : S' \\to S$ and $g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic. \\item These functors agree, via Lemma \\ref{lemma-family-is-one}, with the pullback functors constructed in Lemma \\ref{lemma-pullback}. \\end{enumerate}"}
+{"_id": "14735", "title": "descent-lemma-surjective-flat-epi", "text": "A surjective and flat morphism is an epimorphism in the category of schemes."}
+{"_id": "14736", "title": "descent-lemma-ff-base-change-faithful", "text": "Let $h : S' \\to S$ be a surjective, flat morphism of schemes. The base change functor $$ \\Sch/S \\longrightarrow \\Sch/S', \\quad X \\longmapsto S' \\times_S X $$ is faithful."}
+{"_id": "14737", "title": "descent-lemma-faithful", "text": "In the situation of Lemma \\ref{lemma-pullback} assume that $f : X' \\to X$ is surjective and flat. Then the pullback functor is faithful."}
+{"_id": "14738", "title": "descent-lemma-fully-faithful", "text": "In the situation of Lemma \\ref{lemma-pullback} assume \\begin{enumerate} \\item $\\{f : X' \\to X\\}$ is an fpqc covering (for example if $f$ is surjective, flat, and quasi-compact), and \\item $S = S'$. \\end{enumerate} Then the pullback functor is fully faithful."}
+{"_id": "14739", "title": "descent-lemma-pullback-selfmap", "text": "Let $X \\to S$ be a morphism of schemes. Let $f : X \\to X$ be a selfmap of $X$ over $S$. In this case pullback by $f$ is isomorphic to the identity functor on the category of descent data relative to $X \\to S$."}
+{"_id": "14743", "title": "descent-lemma-fpqc-refinement-coverings-fully-faithful", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and $\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, be families of morphisms with target $S$. Let $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and $g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. Assume that for each $j \\in J$ the family $\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is an fpqc covering of $V_j$. Then the pullback functor $$ \\text{descent data relative to } \\mathcal{V} \\longrightarrow \\text{descent data relative to } \\mathcal{U} $$ of Lemma \\ref{lemma-pullback-family} is fully faithful."}
+{"_id": "14744", "title": "descent-lemma-Zariski-refinement-coverings-equivalence", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and $\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, be families of morphisms with target $S$. Let $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and $g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \\ref{sites-definition-morphism-coverings}. Assume that for each $j \\in J$ the family $\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is a Zariski covering (see Topologies, Definition \\ref{topologies-definition-zariski-covering}) of $V_j$. Then the pullback functor $$ \\text{descent data relative to } \\mathcal{V} \\longrightarrow \\text{descent data relative to } \\mathcal{U} $$ of Lemma \\ref{lemma-pullback-family} is an equivalence of categories. In particular, the category of schemes over $S$ is equivalent to the category of descent data relative to any Zariski covering of $S$."}
+{"_id": "14745", "title": "descent-lemma-refine-coverings-fully-faithful", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and $\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, be fpqc-coverings of $S$. If $\\mathcal{U}$ is a refinement of $\\mathcal{V}$, then the pullback functor $$ \\text{descent data relative to } \\mathcal{V} \\longrightarrow \\text{descent data relative to } \\mathcal{U} $$ is fully faithful. In particular, the category of schemes over $S$ is identified with a full subcategory of the category of descent data relative to any fpqc-covering of $S$."}
+{"_id": "14746", "title": "descent-lemma-effective-for-fpqc-is-local-upstairs", "text": "Let $X \\to S$ be a surjective, quasi-compact, flat morphism of schemes. Let $(V, \\varphi)$ be a descent datum relative to $X/S$. Suppose that for all $v \\in V$ there exists an open subscheme $v \\in W \\subset V$ such that $\\varphi(W \\times_S X) \\subset X \\times_S W$ and such that the descent datum $(W, \\varphi|_{W \\times_S X})$ is effective. Then $(V, \\varphi)$ is effective."}
+{"_id": "14747", "title": "descent-lemma-descending-types-morphisms", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$. Suppose that \\begin{enumerate} \\item $\\mathcal{P}$ is stable under any base change (see Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}), \\item if $Y_j \\to V_j$, $j = 1, \\ldots, m$ have $\\mathcal{P}$, then so does $\\coprod Y_j \\to \\coprod V_j$, and \\item for any surjective morphism of affines $X \\to S$ which is flat, flat of finite presentation, \\'etale, smooth or syntomic depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic, any descent datum $(V, \\varphi)$ relative to $X$ over $S$ such that $\\mathcal{P}$ holds for $V \\to X$ is effective. \\end{enumerate} Then morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings."}
+{"_id": "14748", "title": "descent-lemma-affine", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see Topologies, Definition \\ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$. If each morphism $V_i \\to X_i$ is affine, then the descent datum is effective."}
+{"_id": "14749", "title": "descent-lemma-closed-immersion", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see Topologies, Definition \\ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$. If each morphism $V_i \\to X_i$ is a closed immersion, then the descent datum is effective."}
+{"_id": "14750", "title": "descent-lemma-quasi-affine", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see Topologies, Definition \\ref{topologies-definition-fpqc-covering}. Let $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$. If each morphism $V_i \\to X_i$ is quasi-affine, then the descent datum is effective."}
+{"_id": "14751", "title": "descent-lemma-descent-data-sheaves", "text": "Let $\\tau \\in \\{Zariski, fppf, \\etale, smooth, syntomic\\}$\\footnote{The fact that fpqc is missing is not a typo. See discussion in Topologies, Section \\ref{topologies-section-fpqc}.}. Let $\\Sch_\\tau$ be a big $\\tau$-site. Let $S \\in \\Ob(\\Sch_\\tau)$. Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering in the site $(\\Sch/S)_\\tau$. There is an equivalence of categories $$ \\left\\{ \\begin{matrix} \\text{descent data }(X_i, \\varphi_{ii'})\\text{ such that}\\\\ \\text{each }X_i \\in \\Ob((\\Sch/S)_\\tau) \\end{matrix} \\right\\} \\leftrightarrow \\left\\{ \\begin{matrix} \\text{sheaves }F\\text{ on }(\\Sch/S)_\\tau\\text{ such that}\\\\ \\text{each }h_{S_i} \\times F\\text{ is representable} \\end{matrix} \\right\\}. $$ Moreover, \\begin{enumerate} \\item the objects representing $h_{S_i} \\times F$ on the right hand side correspond to the schemes $X_i$ on the left hand side, and \\item the sheaf $F$ is representable if and only if the corresponding descent datum $(X_i, \\varphi_{ii'})$ is effective. \\end{enumerate}"}
+{"_id": "14752", "title": "descent-proposition-descent-module", "text": "\\begin{slogan} Effective descent for modules along faithfully flat ring maps. \\end{slogan} Let $R \\to A$ be a faithfully flat ring map. Then \\begin{enumerate} \\item any descent datum on modules with respect to $R \\to A$ is effective, \\item the functor $M \\mapsto (A \\otimes_R M, can)$ from $R$-modules to the category of descent data is an equivalence, and \\item the inverse functor is given by $(N, \\varphi) \\mapsto H^0(s(N_\\bullet))$. \\end{enumerate}"}
+{"_id": "14753", "title": "descent-proposition-fpqc-descent-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to S\\}$ be an fpqc covering, see Topologies, Definition \\ref{topologies-definition-fpqc-covering}. Any descent datum on quasi-coherent sheaves for $\\mathcal{U} = \\{U_i \\to S\\}$ is effective. Moreover, the functor from the category of quasi-coherent $\\mathcal{O}_S$-modules to the category of descent data with respect to $\\mathcal{U}$ is fully faithful."}
+{"_id": "14754", "title": "descent-proposition-same-cohomology-quasi-coherent", "text": "\\begin{slogan} Cohomology of quasi-coherent sheaves is the same no matter which topology you use. \\end{slogan} Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. \\begin{enumerate} \\item There is a canonical isomorphism $$ H^q(S, \\mathcal{F}) = H^q((\\Sch/S)_\\tau, \\mathcal{F}^a). $$ \\item There are canonical isomorphisms $$ H^q(S, \\mathcal{F}) = H^q(S_{Zar}, \\mathcal{F}^a) = H^q(S_\\etale, \\mathcal{F}^a). $$ \\end{enumerate}"}
+{"_id": "14755", "title": "descent-proposition-equivalence-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. \\begin{enumerate} \\item The functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence of categories $$ \\QCoh(\\mathcal{O}_S) \\longrightarrow \\QCoh((\\Sch/S)_\\tau, \\mathcal{O}) $$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\\mathcal{O}$-modules on the big $\\tau$ site of $S$. \\item Let $\\tau = \\etale$, or $\\tau = Zariski$. The functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines an equivalence of categories $$ \\QCoh(\\mathcal{O}_S) \\longrightarrow \\QCoh(S_\\tau, \\mathcal{O}) $$ between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\\mathcal{O}$-modules on the small $\\tau$ site of $S$. \\end{enumerate}"}
+{"_id": "14756", "title": "descent-proposition-equivalence-quasi-coherent-functorial", "text": "Let $f : T \\to S$ be a morphism of schemes. \\begin{enumerate} \\item The equivalences of categories of Proposition \\ref{proposition-equivalence-quasi-coherent} are compatible with pullback. More precisely, we have $f^*(\\mathcal{G}^a) = (f^*\\mathcal{G})^a$ for any quasi-coherent sheaf $\\mathcal{G}$ on $S$. \\item The equivalences of categories of Proposition \\ref{proposition-equivalence-quasi-coherent} part (1) are {\\bf not} compatible with pushforward in general. \\item If $f$ is quasi-compact and quasi-separated, and $\\tau \\in \\{Zariski, \\etale\\}$ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram $$ \\xymatrix{ \\QCoh(\\mathcal{O}_T) \\ar[rr]_{f_*} \\ar[d]_{\\mathcal{F} \\mapsto \\mathcal{F}^a} & & \\QCoh(\\mathcal{O}_S) \\ar[d]^{\\mathcal{G} \\mapsto \\mathcal{G}^a} \\\\ \\QCoh(T_\\tau, \\mathcal{O}) \\ar[rr]^{f_{small, *}} & & \\QCoh(S_\\tau, \\mathcal{O}) } $$ is commutative, i.e., $f_{small, *}(\\mathcal{F}^a) = (f_*\\mathcal{F})^a$. \\end{enumerate}"}
+{"_id": "14804", "title": "simplicial-theorem-dold-kan", "text": "Let $\\mathcal{A}$ be an abelian category. The functor $N$ induces an equivalence of categories $$ N : \\text{Simp}(\\mathcal{A}) \\longrightarrow \\text{Ch}_{\\geq 0}(\\mathcal{A}) $$"}
+{"_id": "14805", "title": "simplicial-lemma-face-degeneracy", "text": "Any morphism in $\\Delta$ can be written as a composition of the morphisms $\\delta^n_j$ and $\\sigma^n_j$."}
+{"_id": "14806", "title": "simplicial-lemma-relations-face-degeneracy", "text": "The morphisms $\\delta^n_j$ and $\\sigma^n_j$ satisfy the following relations. \\begin{enumerate} \\item If $0 \\leq i < j \\leq n + 1$, then $\\delta^{n + 1}_j \\circ \\delta^n_i = \\delta^{n + 1}_i \\circ \\delta^n_{j - 1}$. In other words the diagram $$ \\xymatrix{ & [n] \\ar[rd]^{\\delta^{n + 1}_j} & \\\\ [n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\delta^n_{j - 1}} & & [n + 1] \\\\ & [n] \\ar[ru]_{\\delta^{n + 1}_i} & } $$ commutes. \\item If $0 \\leq i < j \\leq n - 1$, then $\\sigma^{n - 1}_j \\circ \\delta^n_i = \\delta^{n - 1}_i \\circ \\sigma^{n - 2}_{j - 1}$. In other words the diagram $$ \\xymatrix{ & [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\ [n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_{j - 1}} & & [n - 1] \\\\ & [n - 2] \\ar[ru]_{\\delta^{n - 1}_i} & } $$ commutes. \\item If $0 \\leq j \\leq n - 1$, then $\\sigma^{n - 1}_j \\circ \\delta^n_j = \\text{id}_{[n - 1]}$ and $\\sigma^{n - 1}_j \\circ \\delta^n_{j + 1} = \\text{id}_{[n - 1]}$. In other words the diagram $$ \\xymatrix{ & [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\ [n - 1] \\ar[ru]^{\\delta^n_j} \\ar[rd]_{\\delta^n_{j + 1}} \\ar[rr]^{\\text{id}_{[n - 1]}} & & [n - 1] \\\\ & [n] \\ar[ru]_{\\sigma^{n - 1}_j} & } $$ commutes. \\item If $0 < j + 1 < i \\leq n$, then $\\sigma^{n - 1}_j \\circ \\delta^n_i = \\delta^{n - 1}_{i - 1} \\circ \\sigma^{n - 2}_j$. In other words the diagram $$ \\xymatrix{ & [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\ [n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_j} & & [n - 1] \\\\ & [n - 2] \\ar[ru]_{\\delta^{n - 1}_{i - 1}} & } $$ commutes. \\item If $0 \\leq i \\leq j \\leq n - 1$, then $\\sigma^{n - 1}_j \\circ \\sigma^n_i = \\sigma^{n - 1}_i \\circ \\sigma^n_{j + 1}$. In other words the diagram $$ \\xymatrix{ & [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\ [n + 1] \\ar[ru]^{\\sigma^n_i} \\ar[rd]_{\\sigma^n_{j + 1}} & & [n - 1] \\\\ & [n] \\ar[ru]_{\\sigma^{n - 1}_i} & } $$ commutes. \\end{enumerate}"}
+{"_id": "14807", "title": "simplicial-lemma-face-degeneracy-category", "text": "The category $\\Delta$ is the universal category with objects $[n]$, $n \\geq 0$ and morphisms $\\delta^n_j$ and $\\sigma^n_j$ such that (a) every morphism is a composition of these morphisms, (b) the relations listed in Lemma \\ref{lemma-relations-face-degeneracy} are satisfied, and (c) any relation among the morphisms is a consequence of those relations."}
+{"_id": "14808", "title": "simplicial-lemma-characterize-simplicial-object", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item Given a simplicial object $U$ in $\\mathcal{C}$ we obtain a sequence of objects $U_n = U([n])$ endowed with the morphisms $d^n_j = U(\\delta^n_j) : U_n \\to U_{n-1}$ and $s^n_j = U(\\sigma^n_j) : U_n \\to U_{n + 1}$. These morphisms satisfy the opposites of the relations displayed in Lemma \\ref{lemma-relations-face-degeneracy}, namely \\begin{enumerate} \\item If $0 \\leq i < j \\leq n + 1$, then $d^n_i \\circ d^{n + 1}_j = d^n_{j - 1} \\circ d^{n + 1}_i$. \\item If $0 \\leq i < j \\leq n - 1$, then $d^n_i \\circ s^{n - 1}_j = s^{n - 2}_{j - 1} \\circ d^{n - 1}_i$. \\item If $0 \\leq j \\leq n - 1$, then $\\text{id} = d^n_j \\circ s^{n - 1}_j = d^n_{j + 1} \\circ s^{n - 1}_j$. \\item If $0 < j + 1 < i \\leq n$, then $d^n_i \\circ s^{n - 1}_j = s^{n - 2}_j \\circ d^{n - 1}_{i - 1}$. \\item If $0 \\leq i \\leq j \\leq n - 1$, then $s^n_i \\circ s^{n - 1}_j = s^n_{j + 1} \\circ s^{n - 1}_i$. \\end{enumerate} \\item Conversely, given a sequence of objects $U_n$ and morphisms $d^n_j$, $s^n_j$ satisfying (1)(a) -- (e) there exists a unique simplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$, $d^n_j = U(\\delta^n_j)$, and $s^n_j = U(\\sigma^n_j)$. \\item A morphism between simplicial objects $U$ and $U'$ is given by a family of morphisms $U_n \\to U'_n$ commuting with the morphisms $d^n_j$ and $s^n_j$. \\end{enumerate}"}
+{"_id": "14809", "title": "simplicial-lemma-si-injective", "text": "Let $\\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\\mathcal{C}$. Each of the morphisms $s^n_i : U_n \\to U_{n + 1}$ has a left inverse. In particular $s^n_i$ is a monomorphism."}
+{"_id": "14811", "title": "simplicial-lemma-di-injective", "text": "Let $\\mathcal{C}$ be a category. Let $U$ be a cosimplicial object of $\\mathcal{C}$. Each of the morphisms $\\delta^n_i : U_{n - 1} \\to U_n$ has a left inverse. In particular $\\delta^n_i$ is a monomorphism."}
+{"_id": "14812", "title": "simplicial-lemma-product", "text": "If $U$ and $V$ are simplicial objects in the category $\\mathcal{C}$, and if $U \\times V$ exists, then we have $$ \\Mor(W, U \\times V) = \\Mor(W, U) \\times \\Mor(W, V) $$ for any third simplicial object $W$ of $\\mathcal{C}$."}
+{"_id": "14813", "title": "simplicial-lemma-fibre-product", "text": "If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$, and if $a : V \\to U$, $b : W \\to U$ are morphisms and if $V \\times_U W$ exists, then we have $$ \\Mor(T, V \\times_U W) = \\Mor(T, V) \\times_{\\Mor(T, U)} \\Mor(T, W) $$ for any fourth simplicial object $T$ of $\\mathcal{C}$."}
+{"_id": "14814", "title": "simplicial-lemma-push-out", "text": "If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$, and if $a : U \\to V$, $b : U \\to W$ are morphisms and if $V\\amalg_U W$ exists, then we have $$ \\Mor(V\\amalg_U W, T) = \\Mor(V, T) \\times_{\\Mor(U, T)} \\Mor(W, T) $$ for any fourth simplicial object $T$ of $\\mathcal{C}$."}
+{"_id": "14815", "title": "simplicial-lemma-product-cosimplicial-objects", "text": "If $U$ and $V$ are cosimplicial objects in the category $\\mathcal{C}$, and if $U \\times V$ exists, then we have $$ \\Mor(W, U \\times V) = \\Mor(W, U) \\times \\Mor(W, V) $$ for any third cosimplicial object $W$ of $\\mathcal{C}$."}
+{"_id": "14816", "title": "simplicial-lemma-fibre-product-cosimplicial-objects", "text": "If $U, V, W$ are cosimplicial objects in the category $\\mathcal{C}$, and if $a : V \\to U$, $b : W \\to U$ are morphisms and if $V \\times_U W$ exists, then we have $$ \\Mor(T, V \\times_U W) = \\Mor(T, V) \\times_{\\Mor(T, U)} \\Mor(T, W) $$ for any fourth cosimplicial object $T$ of $\\mathcal{C}$."}
+{"_id": "14817", "title": "simplicial-lemma-simplex-map", "text": "Let $U$ be a simplicial set. Let $n \\geq 0$ be an integer. There is a canonical bijection $$ \\Mor(\\Delta[n], U) \\longrightarrow U_n $$ which maps a morphism $\\varphi$ to the value of $\\varphi$ on the unique nondegenerate $n$-simplex of $\\Delta[n]$."}
+{"_id": "14818", "title": "simplicial-lemma-product-degenerate", "text": "Let $U$, $V$ be simplicial sets. Let $a, b \\geq 0$ be integers. Assume every $n$-simplex of $U$ is degenerate if $n > a$. Assume every $n$-simplex of $V$ is degenerate if $n > b$. Then every $n$-simplex of $U \\times V$ is degenerate if $n > a + b$."}
+{"_id": "14819", "title": "simplicial-lemma-check-product-with-simplicial-set", "text": "Let $\\mathcal{C}$ be a category such that the coproduct of any two objects of $\\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\\mathcal{C}$. Assume that each $U_n$ is finite nonempty. The functor $W \\mapsto \\Mor_{\\text{Simp}(\\mathcal{C})}(U \\times V, W)$ is canonically isomorphic to the functor which maps $W$ to the set in Equation (\\ref{equation-functor-product-with-simplicial-set})."}
+{"_id": "14820", "title": "simplicial-lemma-back-to-U", "text": "Let $\\mathcal{C}$ be a category such that the coproduct of any two objects of $\\mathcal{C}$ exists. Let us temporarily denote $\\textit{FSSets}$ the category of simplicial sets all of whose components are finite nonempty. \\begin{enumerate} \\item The rule $(U, V) \\mapsto U \\times V$ defines a functor $\\textit{FSSets} \\times \\text{Simp}(\\mathcal{C}) \\to \\text{Simp}(\\mathcal{C})$. \\item For every $U$, $V$ as above there is a canonical map of simplicial objects $$ U \\times V \\longrightarrow V $$ defined by taking the identity on each component of $(U \\times V)_n = \\coprod_u V_n$. \\end{enumerate}"}
+{"_id": "14821", "title": "simplicial-lemma-morphism-from-coproduct", "text": "With $X$ and $k$ as above. For any simplicial object $V$ of $\\mathcal{C}$ we have the following canonical bijection $$ \\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times \\Delta[k], V) \\longrightarrow \\Mor_\\mathcal{C}(X, V_k). $$ which maps $\\gamma$ to the restriction of the morphism $\\gamma_k$ to the component corresponding to $\\text{id}_{[k]}$. Similarly, for any $n \\geq k$, if $W$ is an $n$-truncated simplicial object of $\\mathcal{C}$, then we have $$ \\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_n(X \\times \\Delta[k]), W) = \\Mor_\\mathcal{C}(X, W_k). $$"}
+{"_id": "14822", "title": "simplicial-lemma-morphism-into-product", "text": "With $X$, $k$ and $U$ as above. \\begin{enumerate} \\item For any simplicial object $V$ of $\\mathcal{C}$ we have the following canonical bijection $$ \\Mor_{\\text{Simp}(\\mathcal{C})}(V, U) \\longrightarrow \\Mor_\\mathcal{C}(V_k, X). $$ wich maps $\\gamma$ to the morphism $\\gamma_k$ composed with the projection onto the factor corresponding to $\\text{id}_{[k]}$. \\item Similarly, if $W$ is an $k$-truncated simplicial object of $\\mathcal{C}$, then we have $$ \\Mor_{\\text{Simp}_k(\\mathcal{C})}(W, \\text{sk}_k U) = \\Mor_\\mathcal{C}(W_k, X). $$ \\item The object $U$ constructed above is an incarnation of $\\Hom(C[k], X)$ where $C[k]$ is the cosimplicial set from Example \\ref{example-simplex-cosimplicial-set}. \\end{enumerate}"}
+{"_id": "14823", "title": "simplicial-lemma-exists-hom-0-from-simplicial-set", "text": "Assume the category $\\mathcal{C}$ has coproducts of any two objects and countable limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \\geq 0$. Let $V$ be a simplicial object of $\\mathcal{C}$. Then the functor \\begin{eqnarray*} \\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\ X & \\longmapsto & \\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V) \\end{eqnarray*} is representable."}
+{"_id": "14824", "title": "simplicial-lemma-exists-hom-0-from-simplicial-set-finite", "text": "Assume the category $\\mathcal{C}$ has coproducts of any two objects and finite limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \\geq 0$. Assume that all $n$-simplices of $U$ are degenerate for all $n \\gg 0$. Let $V$ be a simplicial object of $\\mathcal{C}$. Then the functor \\begin{eqnarray*} \\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\ X & \\longmapsto & \\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V) \\end{eqnarray*} is representable."}
+{"_id": "14825", "title": "simplicial-lemma-exists-hom-from-simplicial-set-finite", "text": "Assume the category $\\mathcal{C}$ has coproducts of any two objects and finite limits. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \\geq 0$. Assume that all $n$-simplices of $U$ are degenerate for all $n \\gg 0$. Let $V$ be a simplicial object of $\\mathcal{C}$. Then $\\Hom(U, V)$ exists, moreover we have the expected equalities $$ \\Hom(U, V)_n = \\Hom(U \\times \\Delta[n], V)_0. $$"}
+{"_id": "14826", "title": "simplicial-lemma-hom-from-coprod", "text": "Assume the category $\\mathcal{C}$ has coproducts of any two objects and finite limits. Let $a : U \\to V$, $b : U \\to W$ be morphisms of simplicial sets. Assume $U_n, V_n, W_n$ finite nonempty for all $n \\geq 0$. Assume that all $n$-simplices of $U, V, W$ are degenerate for all $n \\gg 0$. Let $T$ be a simplicial object of $\\mathcal{C}$. Then $$ \\Hom(V, T) \\times_{\\Hom(U, T)} \\Hom(W, T) = \\Hom(V \\amalg_U W, T) $$ In other words, the fibre product on the left hand side is represented by the Hom object on the right hand side."}
+{"_id": "14827", "title": "simplicial-lemma-splitting-simplicial-sets", "text": "Let $U$ be a simplicial set. Then $U$ has a unique splitting with $N(U_m)$ equal to the set of nondegenerate $m$-simplices."}
+{"_id": "14828", "title": "simplicial-lemma-injective-map-simplicial-sets", "text": "Let $f : U \\to V$ be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of $U$ is a nondegenerate simplex of $V$ and (b) the restriction of $f$ to a map from the set of nondegenerate simplices of $U$ to the set of nondegenerate simplices of $V$ is injective. Then $f_n$ is injective for all $n$. Same holds with ``injective'' replaced by ``surjective'' or ``bijective''."}
+{"_id": "14829", "title": "simplicial-lemma-simplicial-set-n-skel-sub", "text": "Let $U$ be a simplicial set. Let $n \\geq 0$ be an integer. The rule $$ U'_m = \\bigcup\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi)) $$ defines a sub simplicial set $U' \\subset U$ with $U'_i = U_i$ for $i \\leq n$. Moreover, all $m$-simplices of $U'$ are degenerate for all $m > n$."}
+{"_id": "14830", "title": "simplicial-lemma-splitting-simplicial-groups", "text": "Let $U$ be a simplicial abelian group. Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and for $m \\geq 1$ taking $$ N(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i). $$ Moreover, this splitting is functorial on the category of simplicial abelian groups."}
+{"_id": "14831", "title": "simplicial-lemma-splitting-abelian-category", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object in $\\mathcal{A}$. Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and for $m \\geq 1$ taking $$ N(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i). $$ Moreover, this splitting is functorial on the category of simplicial objects of $\\mathcal{A}$."}
+{"_id": "14832", "title": "simplicial-lemma-injective-map-simplicial-abelian", "text": "\\begin{slogan} The Dold-Kan normalization functor reflects injectivity, surjectivity, and isomorphy. \\end{slogan} Let $\\mathcal{A}$ be an abelian category. Let $f : U \\to V$ be a morphism of simplicial objects of $\\mathcal{A}$. If the induced morphisms $N(f)_i : N(U)_i \\to N(V)_i$ are injective for all $i$, then $f_i$ is injective for all $i$. Same holds with ``injective'' replaced with ``surjective'', or ``isomorphism''."}
+{"_id": "14834", "title": "simplicial-lemma-simplicial-abelian-n-skel-sub", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\\mathcal{A}$. Let $n \\geq 0$ be an integer. The rule $$ U'_m = \\sum\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi)) $$ defines a sub simplicial object $U' \\subset U$ with $U'_i = U_i$ for $i \\leq n$. Moreover, $N(U'_m) = 0$ for all $m > n$."}
+{"_id": "14835", "title": "simplicial-lemma-existence-cosk", "text": "If the category $\\mathcal{C}$ has finite limits, then $\\text{cosk}_m$ functors exist for all $m$. Moreover, for any $m$-truncated simplicial object $U$ the simplicial object $\\text{cosk}_mU$ is described by the formula $$ (\\text{cosk}_mU)_n = \\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n) $$ and for $\\varphi : [n] \\to [n']$ the map $\\text{cosk}_mU(\\varphi)$ comes from the identification $U(n') \\circ \\overline{\\varphi} = U(n)$ above via Categories, Lemma \\ref{categories-lemma-functorial-limit}."}
+{"_id": "14836", "title": "simplicial-lemma-trivial-cosk", "text": "Let $\\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$. For $n \\leq m$ the limit $\\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)$ exists and is canonically isomorphic to $U_n$."}
+{"_id": "14837", "title": "simplicial-lemma-recover-cosk", "text": "Let $\\mathcal{C}$ be a category with finite limits. Let $U$ be an $n$-truncated simplicial object of $\\mathcal{C}$. The morphism $\\text{sk}_n \\text{cosk}_n U \\to U$ is an isomorphism."}
+{"_id": "14838", "title": "simplicial-lemma-formula-limit", "text": "Let $n$ be an integer $\\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\\mathcal{C}$. Consider the contravariant functor from $\\mathcal{C}$ to $\\textit{Sets}$ which associates to an object $T$ the set $$ \\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n) \\mid d^n_{j - 1} \\circ f_i = d^n_i \\circ f_j \\ \\forall\\ 0\\leq i < j\\leq n + 1\\} $$ If this functor is representable by some object $U_{n + 1}$ of $\\mathcal{C}$, then $$ U_{n + 1} = \\lim_{(\\Delta/[n + 1])_{\\leq n}^{opp}} U(n) $$"}
+{"_id": "14839", "title": "simplicial-lemma-work-out", "text": "Let $n$ be an integer $\\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\\mathcal{C}$. Consider the contravariant functor from $\\mathcal{C}$ to $\\textit{Sets}$ which associates to an object $T$ the set $$ \\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n) \\mid d^n_{j - 1} \\circ f_i = d^n_i \\circ f_j \\ \\forall\\ 0\\leq i < j\\leq n + 1\\} $$ If this functor is representable by some object $U_{n + 1}$ of $\\mathcal{C}$, then there exists an $(n + 1)$-truncated simplicial object $\\tilde U$, with $\\text{sk}_n \\tilde U = U$ and $\\tilde U_{n + 1} = U_{n + 1}$ such that the following adjointness holds $$ \\Mor_{\\text{Simp}_{n + 1}(\\mathcal{C})}(V, \\tilde U) = \\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_nV, U) $$"}
+{"_id": "14841", "title": "simplicial-lemma-cosk-product", "text": "Let $U$, $V$ be $n$-truncated simplicial objects of a category $\\mathcal{C}$. Then $$ \\text{cosk}_n (U \\times V) = \\text{cosk}_nU \\times \\text{cosk}_nV $$ whenever the left and right hand sides exist."}
+{"_id": "14842", "title": "simplicial-lemma-cosk-fibre-product", "text": "Assume $\\mathcal{C}$ has fibre products. Let $U \\to V$ and $W \\to V$ be morphisms of $n$-truncated simplicial objects of the category $\\mathcal{C}$. Then $$ \\text{cosk}_n (U \\times_V W) = \\text{cosk}_nU \\times_{\\text{cosk}_n V} \\text{cosk}_nW $$ whenever the left and right hand side exist."}
+{"_id": "14843", "title": "simplicial-lemma-cosk-above-object", "text": "Let $\\mathcal{C}$ be a category with finite limits. Let $X \\in \\Ob(\\mathcal{C})$. The functor $\\mathcal{C}/X \\to \\mathcal{C}$ commutes with the coskeleton functors $\\text{cosk}_k$ for $k \\geq 1$."}
+{"_id": "14844", "title": "simplicial-lemma-simplex-cosk", "text": "The canonical map $\\Delta[n] \\to \\text{cosk}_1 \\text{sk}_1 \\Delta[n]$ is an isomorphism."}
+{"_id": "14845", "title": "simplicial-lemma-augmentation-howto", "text": "Let $\\mathcal{C}$ be a category. Let $X \\in \\Ob(\\mathcal{C})$. Let $U$ be a simplicial object of $\\mathcal{C}$. To give an augmentation of $U$ towards $X$ is the same as giving a morphism $\\epsilon_0 : U_0 \\to X$ such that $\\epsilon_0 \\circ d^1_0 = \\epsilon_0 \\circ d^1_1$."}
+{"_id": "14846", "title": "simplicial-lemma-cosk-minus-one", "text": "Let $\\mathcal{C}$ be a category with fibred products. Let $f : Y\\to X$ be a morphism of $\\mathcal{C}$. Let $U$ be the simplicial object of $\\mathcal{C}$ whose $n$th term is the $(n + 1)$fold fibred product $Y \\times_X Y \\times_X \\ldots \\times_X Y$. See Example \\ref{example-fibre-products-simplicial-object}. For any simplicial object $V$ of $\\mathcal{C}$ we have \\begin{align*} \\Mor_{\\text{Simp}(\\mathcal{C})}(V, U) & = \\Mor_{\\text{Simp}_1(\\mathcal{C})}(\\text{sk}_1 V, \\text{sk}_1 U) \\\\ & = \\{g_0 : V_0 \\to Y \\mid f \\circ g_0 \\circ d^1_0 = f \\circ g_0 \\circ d^1_1\\} \\end{align*} In particular we have $U = \\text{cosk}_1 \\text{sk}_1 U$."}
+{"_id": "14847", "title": "simplicial-lemma-left-adjoint-exists", "text": "Let $\\mathcal{C}$ be a category which has finite colimits. The functors $i_{m!}$ exist for all $m$. Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$. The simplicial object $i_{m!}U$ is described by the formula $$ (i_{m!}U)_n = \\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n) $$ and for $\\varphi : [n] \\to [n']$ the map $i_{m!}U(\\varphi)$ comes from the identification $U(n) \\circ \\underline{\\varphi} = U(n')$ above via Categories, Lemma \\ref{categories-lemma-functorial-colimit}."}
+{"_id": "14848", "title": "simplicial-lemma-recovering-U", "text": "Let $\\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$. For any $n \\leq m$ the colimit $$ \\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n) $$ exists and is equal to $U_n$."}
+{"_id": "14849", "title": "simplicial-lemma-recovering-U-for-real", "text": "Let $\\mathcal{C}$ be a category which has finite colimits. Let $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$. The map $U \\to \\text{sk}_m i_{m!}U$ is an isomorphism."}
+{"_id": "14850", "title": "simplicial-lemma-imshriek-sets", "text": "If $U$ is an $m$-truncated simplicial set and $n > m$ then all $n$-simplices of $i_{m!}U$ are degenerate."}
+{"_id": "14851", "title": "simplicial-lemma-n-skeleton-sets", "text": "Let $U$ be a simplicial set. Let $n \\geq 0$ be an integer. The morphism $i_{n!} \\text{sk}_n U \\to U$ identifies $i_{n!} \\text{sk}_n U$ with the simplicial set $U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-set-n-skel-sub}."}
+{"_id": "14852", "title": "simplicial-lemma-glue-simplex", "text": "Let $U \\subset V$ be simplicial sets. Suppose $n \\geq 0$ and $x \\in V_n$, $x \\not \\in U_n$ are such that \\begin{enumerate} \\item $V_i = U_i$ for $i < n$, \\item $V_n = U_n \\cup \\{x\\}$, \\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$ is degenerate. \\end{enumerate} Let $\\Delta[n] \\to V$ be the unique morphism mapping the nondegenerate $n$-simplex of $\\Delta[n]$ to $x$. In this case the diagram $$ \\xymatrix{ \\Delta[n] \\ar[r] & V \\\\ i_{(n - 1)!} \\text{sk}_{n - 1} \\Delta[n] \\ar[r] \\ar[u] & U \\ar[u] } $$ is a pushout diagram."}
+{"_id": "14853", "title": "simplicial-lemma-add-simplices", "text": "Let $U \\subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then there exists a sequence of sub simplicial sets $$ U = W^0 \\subset W^1 \\subset W^2 \\subset \\ldots W^r = V $$ such that Lemma \\ref{lemma-glue-simplex} applies to each of the inclusions $W^i \\subset W^{i + 1}$."}
+{"_id": "14854", "title": "simplicial-lemma-imshriek-abelian", "text": "Let $\\mathcal{A}$ be an abelian category Let $U$ be an $m$-truncated simplicial object of $\\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_n = 0$."}
+{"_id": "14855", "title": "simplicial-lemma-n-skeleton-abelian", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\\mathcal{A}$. Let $n \\geq 0$ be an integer. The morphism $i_{n!} \\text{sk}_n U \\to U$ identifies $i_{n!} \\text{sk}_n U$ with the simplicial subobject $U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-abelian-n-skel-sub}."}
+{"_id": "14856", "title": "simplicial-lemma-cosk-shriek", "text": "Let $\\mathcal{C}$ be a category with finite coproducts and finite limits. Let $V$ be a simplicial object of $\\mathcal{C}$. In this case $$ (\\text{cosk}_n \\text{sk}_n V)_{n + 1} = \\Hom(i_{n !}\\text{sk}_n \\Delta[n + 1], V)_0. $$"}
+{"_id": "14857", "title": "simplicial-lemma-abelian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item The categories $\\text{Simp}(\\mathcal{A})$ and $\\text{CoSimp}(\\mathcal{A})$ are abelian. \\item A morphism of (co)simplicial objects $f : A \\to B$ is injective if and only if each $f_n : A_n \\to B_n$ is injective. \\item A morphism of (co)simplicial objects $f : A \\to B$ is surjective if and only if each $f_n : A_n \\to B_n$ is surjective. \\item A sequence of (co)simplicial objects $$ A \\xrightarrow{f} B \\xrightarrow{g} C $$ is exact at $B$ if and only if each sequence $$ A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i $$ is exact at $B_i$. \\end{enumerate}"}
+{"_id": "14858", "title": "simplicial-lemma-eilenberg-maclane-object", "text": "With $A$, $k$ and $U$ as above, so $U_i = 0$, $i < k$ and $U_k = A$. \\begin{enumerate} \\item Given a $k$-truncated simplicial object $V$ we have $$ \\Mor(U, V) = \\{ f : A \\to V_k \\mid d^k_i \\circ f = 0, \\ i = 0, \\ldots, k \\} $$ and $$ \\Mor(V, U) = \\{ f : V_k \\to A \\mid f \\circ s^{k - 1}_i = 0, \\ i = 0, \\ldots, k - 1 \\}. $$ \\item The object $i_{k!} U$ has $n$th term equal to $\\bigoplus_\\alpha A$ where $\\alpha$ runs over all surjective morphisms $\\alpha : [n] \\to [k]$. \\item For any $\\varphi : [m] \\to [n]$ the map $i_{k!} U(\\varphi)$ is described as the mapping $\\bigoplus_\\alpha A \\to \\bigoplus_{\\alpha'} A$ which maps to component corresponding to $\\alpha : [n] \\to [k]$ to zero if $\\alpha \\circ \\varphi$ is not surjective and by the identity to the component corresponding to $\\alpha \\circ \\varphi$ if it is surjective. \\item The object $\\text{cosk}_k U$ has $n$th term equal to $\\bigoplus_\\beta A$, where $\\beta$ runs over all injective morphisms $\\beta : [k] \\to [n]$. \\item For any $\\varphi : [m] \\to [n]$ the map $\\text{cosk}_k U(\\varphi)$ is described as the mapping $\\bigoplus_\\beta A \\to \\bigoplus_{\\beta'} A$ which maps to component corresponding to $\\beta : [k] \\to [n]$ to zero if $\\beta$ does not factor through $\\varphi$ and by the identity to each of the components corresponding to $\\beta'$ such that $\\beta = \\varphi \\circ \\beta'$ if it does. \\item The canonical map $ c : i_{k !} U \\to \\text{cosk}_k U $ in degree $n$ has $(\\alpha, \\beta)$ coefficient $A \\to A$ equal to zero if $\\alpha \\circ \\beta$ is not the identity and equal to $\\text{id}_A$ if it is. \\item The canonical map $ c : i_{k !} U \\to \\text{cosk}_k U $ is injective. \\end{enumerate}"}
+{"_id": "14859", "title": "simplicial-lemma-extension", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ and let $k$ be an integer $\\geq 0$. Consider the simplicial object $E$ defined by the following rules \\begin{enumerate} \\item $E_n = \\bigoplus_\\alpha A$, where the sum is over $\\alpha : [n] \\to [k + 1]$ whose image is either $[k]$ or $[k + 1]$. \\item Given $\\varphi : [m] \\to [n]$ the map $E_n \\to E_m$ maps the summand corresponding to $\\alpha$ via $\\text{id}_A$ to the summand corresponding to $\\alpha \\circ \\varphi$, provided $\\Im(\\alpha \\circ \\varphi)$ is equal to $[k]$ or $[k + 1]$. \\end{enumerate} Then there exists a short exact sequence $$ 0 \\to K(A, k) \\to E \\to K(A, k + 1) \\to 0 $$ which is term by term split exact."}
+{"_id": "14861", "title": "simplicial-lemma-s-exact", "text": "The functor $s$ is exact."}
+{"_id": "14862", "title": "simplicial-lemma-homology-extension", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ and let $k$ be an integer. Let $E$ be the object described in Lemma \\ref{lemma-extension}. Then the complex $s(E)$ is acyclic."}
+{"_id": "14863", "title": "simplicial-lemma-homology-eilenberg-maclane", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ and let $k$ be an integer. We have $H_i(s(K(A, k))) = A$ if $i = k$ and $0$ else."}
+{"_id": "14864", "title": "simplicial-lemma-map-associated-complexes", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\\mathcal{A}$. The canonical map $N(U_n) \\to U_n$ gives rise to a morphism of complexes $N(U) \\to s(U)$."}
+{"_id": "14865", "title": "simplicial-lemma-N-K", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ and let $k$ be an integer. We have $N(K(A, k))_i = A$ if $i = k$ and $0$ else."}
+{"_id": "14866", "title": "simplicial-lemma-decompose-associated-complexes", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\\mathcal{A}$. The canonical morphism of chain complexes $N(U) \\to s(U)$ is split. In fact, $$ s(U) = N(U) \\oplus D(U) $$ for some complex $D(U)$. The construction $U \\mapsto D(U)$ is functorial."}
+{"_id": "14867", "title": "simplicial-lemma-N-exact", "text": "The functor $N$ is exact."}
+{"_id": "14868", "title": "simplicial-lemma-quasi-isomorphism", "text": "Let $\\mathcal{A}$ be an abelian category. Let $V$ be a simplicial object of $\\mathcal{A}$. The canonical morphism of chain complexes $N(V) \\to s(V)$ is a quasi-isomorphism. In other words, the complex $D(V)$ of Lemma \\ref{lemma-decompose-associated-complexes} is acyclic."}
+{"_id": "14870", "title": "simplicial-lemma-S-N", "text": "Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories. Let $N : \\mathcal{A} \\to \\mathcal{B}$, and $S : \\mathcal{B} \\to \\mathcal{A}$ be functors. Suppose that \\begin{enumerate} \\item the functors $S$ and $N$ are exact, \\item there is an isomorphism $g : N \\circ S \\to \\text{id}_\\mathcal{B}$ to the identity functor of $\\mathcal{B}$, \\item $N$ is faithful, and \\item $S$ is essentially surjective. \\end{enumerate} Then $S$ and $N$ are quasi-inverse equivalences of categories."}
+{"_id": "14872", "title": "simplicial-lemma-relations-homotopy", "text": "In the situation above, we have the following relations: \\begin{enumerate} \\item We have $h_{n, 0} = b_n$ and $h_{n, n + 1} = a_n$. \\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i - 1} \\circ d^n_j$ for $i > j$. \\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i} \\circ d^n_j$ for $i \\leq j$. \\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i + 1} \\circ s^n_j$ for $i > j$. \\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i} \\circ s^n_j$ for $i \\leq j$. \\end{enumerate} Conversely, given a system of maps $h_{n, i}$ satisfying the properties listed above, then these define a morphism $h$ which is a homotopy from $a$ to $b$."}
+{"_id": "14875", "title": "simplicial-lemma-products-homotopy", "text": "Let $\\mathcal{C}$ be a category. Let $T$ be a set. For $t \\in T$ let $X_t$, $Y_t$ be simplicial objects of $\\mathcal{C}$. Assume $X = \\prod_{t \\in T} X_t$ and $Y = \\prod_{t \\in T} Y_t$ exist. \\begin{enumerate} \\item If $X_t$ and $Y_t$ are homotopy equivalent for all $t \\in T$ and $T$ is finite, then $X$ and $Y$ are homotopy equivalent. \\end{enumerate} For $t \\in T$ let $a_t, b_t : X_t \\to Y_t$ be morphisms. Set $a = \\prod a_t : X \\to Y$ and $b = \\prod b_t : X \\to Y$. \\begin{enumerate} \\item[(2)] If there exists a homotopy from $a_t$ to $b_t$ for all $t \\in T$, then there exists a homotopy from $a$ to $b$. \\item[(3)] If $T$ is finite and $a_t, b_t : X_t \\to Y_t$ for $t \\in T$ are homotopic, then $a$ and $b$ are homotopic. \\end{enumerate}"}
+{"_id": "14876", "title": "simplicial-lemma-homotopy-s-N", "text": "Let $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of chain complexes. If $\\mathcal{A}$ is abelian, then also $N(a), N(b) : N(U) \\to N(V)$ are homotopic maps of chain complexes."}
+{"_id": "14877", "title": "simplicial-lemma-homotopy-equivalence-s-N", "text": "Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism of simplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence, then $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes. If in addition $\\mathcal{A}$ is abelian, then also $N(a) : N(U) \\to N(V)$ is a homotopy equivalence of chain complexes."}
+{"_id": "14878", "title": "simplicial-lemma-compare-homotopies", "text": "Let $\\mathcal{C}$ be a category. Suppose that $U$ and $V$ are two cosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be morphisms of cosimplicial objects. Recall that $U$, $V$ correspond to simplicial objects $U'$, $V'$ of $\\mathcal{C}^{opp}$. Moreover $a, b$ correspond to morphisms $a', b' : V' \\to U'$. The following are equivalent \\begin{enumerate} \\item There exists a homotopy $h = \\{h_{n, \\alpha}\\}$ from $a$ to $b$ as in Remark \\ref{remark-homotopy-cosimplicial-better}. \\item There exists a homotopy $h = \\{h_{n, i}\\}$ from $a'$ to $b'$ as in Remark \\ref{remark-homotopy-better}. \\end{enumerate} Thus $a$ is homotopic to $b$ as in Remark \\ref{remark-homotopy-cosimplicial-better} if and only if $a'$ is homotopic to $b'$ as in Remark \\ref{remark-homotopy-better}."}
+{"_id": "14879", "title": "simplicial-lemma-functorial-homotopy", "text": "\\begin{slogan} Functors preserve homotopic morphisms of (co)simplicial objects. \\end{slogan} Let $\\mathcal{C}, \\mathcal{C}', \\mathcal{D}, \\mathcal{D}'$ be categories. With terminology as in Remarks \\ref{remark-homotopy-cosimplicial-better} and \\ref{remark-homotopy-better}. \\begin{enumerate} \\item Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{D}$. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \\to F(V)$ of simplicial objects. \\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects of $\\mathcal{C}$. Let $F : \\mathcal{C} \\to \\mathcal{C}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \\to F(V)$ of cosimplicial objects. \\item Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{D}$. Let $F : \\mathcal{D} \\to \\mathcal{C}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \\to F(U)$ of cosimplicial objects. \\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects of $\\mathcal{C}$. Let $F : \\mathcal{C} \\to \\mathcal{D}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \\to F(U)$ of simplicial objects. \\end{enumerate}"}
+{"_id": "14881", "title": "simplicial-lemma-homotopy-s-Q", "text": "\\begin{slogan} The (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps. \\end{slogan} Let $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms of cosimplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of cochain complexes. If in addition $\\mathcal{A}$ is abelian, then $Q(a), Q(b) : Q(U) \\to Q(V)$ are homotopic maps of cochain complexes."}
+{"_id": "14882", "title": "simplicial-lemma-homotopy-equivalence-s-Q", "text": "Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism of cosimplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence, then $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes. If in addition $\\mathcal{A}$ is abelian, then also $Q(a) : Q(U) \\to Q(V)$ is a homotopy equivalence of chain complexes."}
+{"_id": "14883", "title": "simplicial-lemma-represent-homotopy", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be a chain complex. Consider the covariant functor $$ B \\longmapsto \\{ (a, b, h) \\mid a, b : A \\to B\\text{ and }h\\text{ a homotopy between }a, b \\} $$ There exists a chain complex $\\diamond A$ such that $\\Mor_{\\text{Ch}(\\mathcal{A})}(\\diamond A, -)$ is isomorphic to the displayed functor. The construction $A \\mapsto \\diamond A$ is functorial."}
+{"_id": "14884", "title": "simplicial-lemma-map-into-diamond", "text": "Let $\\mathcal{A}$ be an abelian category. Let $$ 0 \\to A \\oplus A \\to B \\to C \\to 0 $$ be a short exact sequence of chain complexes of $\\mathcal{A}$. Suppose given in addition morphisms $s_n : C_n \\to B_n$ splitting the associated short exact sequence in degree $n$. Let $\\delta(s) : C \\to (A \\oplus A)[-1] = A[-1] \\oplus A[-1]$ be the associated morphism of complexes, see Homology, Lemma \\ref{homology-lemma-ses-termwise-split}. If $\\delta(s)$ factors through the morphism $(1, -1) : A[-1] \\to A[-1] \\oplus A[-1]$, then there is a unique morphism $B \\to \\diamond A$ fitting into a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & A \\oplus A \\ar[d] \\ar[r] & B \\ar[r] \\ar[d] & C \\ar[d] \\ar[r] & 0 \\\\ 0 \\ar[r] & A \\oplus A \\ar[r] & \\diamond A \\ar[r] & A[-1] \\ar[r] & 0 } $$ where the vertical maps are compatible with the splittings $s_n$ and the splittings of $\\diamond A_n \\to A[-1]_n$ as well."}
+{"_id": "14886", "title": "simplicial-lemma-trivial-kan", "text": "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets. For any solid commutative diagram $$ \\xymatrix{ Z \\ar[r]_b \\ar[d] & X \\ar[d] \\\\ W \\ar[r]^a \\ar@{-->}[ru] & Y } $$ of simplicial sets with $Z \\to W$ (termwise) injective a dotted arrow exists making the diagram commute."}
+{"_id": "14887", "title": "simplicial-lemma-trivial-kan-base-change", "text": "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets. Let $Y' \\to Y$ be a morphism of simplicial sets. Then $X \\times_Y Y' \\to Y'$ is a trivial Kan fibration."}
+{"_id": "14888", "title": "simplicial-lemma-trivial-kan-composition", "text": "The composition of two trivial Kan fibrations is a trivial Kan fibration."}
+{"_id": "14889", "title": "simplicial-lemma-limit-trivial-kan", "text": "Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of trivial Kan fibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$. Then $U \\to U^0$ is a trivial Kan fibration."}
+{"_id": "14890", "title": "simplicial-lemma-product-trivial-kan", "text": "\\begin{slogan} Products of trivial Kan fibrations are trivial Kan fibrations. \\end{slogan} Let $X_i \\to Y_i$ be a set of trivial Kan fibrations. Then $\\prod X_i \\to \\prod Y_i$ is a trivial Kan fibration."}
+{"_id": "14891", "title": "simplicial-lemma-filtered-colimit-trivial-kan", "text": "A filtered colimit of trivial Kan fibrations is a trivial Kan fibration."}
+{"_id": "14892", "title": "simplicial-lemma-trivial-kan-homotopy", "text": "Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets. Then $f$ is a homotopy equivalence."}
+{"_id": "14894", "title": "simplicial-lemma-kan-composition", "text": "The composition of two Kan fibrations is a Kan fibration."}
+{"_id": "14895", "title": "simplicial-lemma-limit-kan", "text": "Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of Kan fibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$. Then $U \\to U^0$ is a Kan fibration."}
+{"_id": "14896", "title": "simplicial-lemma-product-kan", "text": "Let $X_i \\to Y_i$ be a set of Kan fibrations. Then $\\prod X_i \\to \\prod Y_i$ is a Kan fibration."}
+{"_id": "14897", "title": "simplicial-lemma-simplicial-group-kan", "text": "Let $X$ be a simplicial group. Then $X$ is a Kan complex."}
+{"_id": "14899", "title": "simplicial-lemma-qis-simplicial-abelian-groups", "text": "Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups which is termwise surjective and induces a quasi-isomorphism on associated chain complexes. Then $f$ is a trivial Kan fibration of simplicial sets."}
+{"_id": "14900", "title": "simplicial-lemma-homotopy-equivalence", "text": "Let $f : X \\to Y$ be a map of simplicial abelian groups. If $f$ is a homotopy equivalence of simplicial sets, then $f$ induces a quasi-isomorphism of associated chain complexes."}
+{"_id": "14903", "title": "simplicial-lemma-cosk-minus-one-equivalence", "text": "Let $A$, $B$ be sets, and that $f : A \\to B$ is a map. Consider the simplicial set $U$ with $n$-simplices $$ A \\times_B A \\times_B \\ldots \\times_B A\\ (n + 1 \\text{ factors)}. $$ see Example \\ref{example-fibre-products-simplicial-object}. If $f$ is surjective, the morphism $U \\to B$ where $B$ indicates the constant simplicial set with value $B$ is a trivial Kan fibration."}
+{"_id": "14904", "title": "simplicial-lemma-godement", "text": "In Example \\ref{example-godement} if $$ 1_Y = (d \\star 1_Y) \\circ s = (1_Y \\star d) \\circ s \\quad\\text{and}\\quad (s \\star 1) \\circ s = (1 \\star s) \\circ s $$ then $X = (X_n, d^n_j, s^n_j)$ is a simplicial object in the category of endofunctors of $\\mathcal{C}$ and $d : X_0 = Y \\to \\text{id}_\\mathcal{C}$ defines an augmentation."}
+{"_id": "14905", "title": "simplicial-lemma-godement-section", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$ be as in Example \\ref{example-godement-functorial}. Given a transformation of functors $h_0 : G \\circ F \\to G \\circ Y \\circ F$ such that $$ 1_{G \\circ F} = (1_G \\star d \\star 1_F) \\circ h_0 $$ Then there is a morphism $h : G \\circ F \\to G \\circ X \\circ F$ of simplicial objects such that $\\epsilon \\circ h = \\text{id}$ where $\\epsilon : G \\circ X \\circ F \\to G \\circ F$ is the augmentation."}
+{"_id": "14906", "title": "simplicial-lemma-godement-two-maps", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$ be as in Example \\ref{example-godement-functorial}. Let $F' : \\mathcal{A} \\to \\mathcal{C}$ and $G' : \\mathcal{C} \\to \\mathcal{B}$ be two functors. Let $(a_n) : G \\circ X \\to G' \\circ X$ be a morphism of simplicial objects compatible via augmentations with $a : G \\to G'$. Let $(b_n) : X \\circ F \\to X \\circ F'$ be a morphism of simplicial objects compatible via augmentations with $b : F \\to F'$. Then the two maps $$ a \\star (b_n), (a_n) \\star b : G \\circ X \\circ F \\to G' \\circ X \\circ F' $$ are homotopic."}
+{"_id": "14907", "title": "simplicial-lemma-godement-before-after", "text": "Let $\\mathcal{C}$, $Y$, $d$, $s$ be as in Example \\ref{example-godement} satisfying the equations of Lemma \\ref{lemma-godement}. Let $f : \\text{id}_\\mathcal{C} \\to \\text{id}_\\mathcal{C}$ be an endomorphism of the identity functor. Then $f \\star 1_X, 1_X \\star f : X \\to X$ are maps of simplicial objects compatible with $f$ via the augmentation $\\epsilon : X \\to \\text{id}_\\mathcal{C}$. Moreover, $f \\star 1_X$ and $1_X \\star f$ are homotopic."}
+{"_id": "14908", "title": "simplicial-lemma-standard-simplicial", "text": "In Situation \\ref{situation-adjoint-functors} the system $X = (X_n, d^n_j, s^n_j)$ is a simplicial object of $\\text{Fun}(\\mathcal{A}, \\mathcal{A})$ and $\\epsilon_0$ defines an augmentation $\\epsilon$ from $X$ to the constant simplicial object with value $X_{-1} = \\text{id}_\\mathcal{A}$."}
+{"_id": "14909", "title": "simplicial-lemma-standard-simplicial-homotopy", "text": "In Situation \\ref{situation-adjoint-functors} the maps $$ 1_V \\star \\epsilon : V \\circ X \\to V, \\quad\\text{and}\\quad \\epsilon \\star 1_U : X \\circ U \\to U $$ are homotopy equivalences."}
+{"_id": "14944", "title": "discriminant-lemma-dominate-factorizations", "text": "Let $A \\to B$ be a quasi-finite ring map. Given two factorizations $A \\to B' \\to B$ and $A \\to B'' \\to B$ with $A \\to B'$ and $A \\to B''$ finite and $\\Spec(B) \\to \\Spec(B')$ and $\\Spec(B) \\to \\Spec(B'')$ open immersions, there exists an $A$-subalgebra $B''' \\subset B$ finite over $A$ such that $\\Spec(B) \\to \\Spec(B''')$ an open immersion and $B' \\to B$ and $B'' \\to B$ factor through $B'''$."}
+{"_id": "14945", "title": "discriminant-lemma-dualizing-well-defined", "text": "The module (\\ref{equation-dualizing}) is well defined, i.e., independent of the choice of the factorization."}
+{"_id": "14946", "title": "discriminant-lemma-localize-dualizing", "text": "Let $A \\to B$ be a quasi-finite map of Noetherian rings. \\begin{enumerate} \\item If $A \\to B$ factors as $A \\to A_f \\to B$ for some $f \\in A$, then $\\omega_{B/A} = \\omega_{B/A_f}$. \\item If $g \\in B$, then $(\\omega_{B/A})_g = \\omega_{B_g/A}$. \\item If $f \\in A$, then $\\omega_{B_f/A_f} = (\\omega_{B/A})_f$. \\end{enumerate}"}
+{"_id": "14947", "title": "discriminant-lemma-bc-map-dualizing", "text": "The base change map (\\ref{equation-bc-dualizing}) is independent of the choice of the factorization $A \\to B' \\to B$. Given ring maps $A \\to A_1 \\to A_2$ the composition of the base change maps for $A \\to A_1$ and $A_1 \\to A_2$ is the base change map for $A \\to A_2$."}
+{"_id": "14948", "title": "discriminant-lemma-dualizing-flat-base-change", "text": "If $A \\to A_1$ is flat, then the base change map (\\ref{equation-bc-dualizing}) induces an isomorphism $\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$."}
+{"_id": "14949", "title": "discriminant-lemma-dualizing-composition", "text": "Let $A \\to B \\to C$ be quasi-finite homomorphisms of Noetherian rings. There is a canonical map $\\omega_{B/A} \\otimes_B \\omega_{C/B} \\to \\omega_{C/A}$."}
+{"_id": "14950", "title": "discriminant-lemma-dualizing-product", "text": "Let $A \\to B$ and $A \\to C$ be quasi-finite maps of Noetherian rings. Then $\\omega_{B \\times C/A} = \\omega_{B/A} \\times \\omega_{C/A}$ as modules over $B \\times C$."}
+{"_id": "14951", "title": "discriminant-lemma-dualizing-associated-primes", "text": "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings. Then $\\text{Ass}_B(\\omega_{B/A})$ is the set of primes of $B$ lying over associated primes of $A$."}
+{"_id": "14952", "title": "discriminant-lemma-dualizing-base-flat-flat", "text": "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings. Then $\\omega_{B/A}$ is a flat $A$-module."}
+{"_id": "14953", "title": "discriminant-lemma-dualizing-base-change-of-flat", "text": "If $A \\to B$ is flat, then the base change map (\\ref{equation-bc-dualizing}) induces an isomorphism $\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$."}
+{"_id": "14955", "title": "discriminant-lemma-discriminant", "text": "Let $\\pi : X \\to Y$ be a morphism of schemes which is finite locally free. Then $\\pi$ is \\'etale if and only if its discriminant is empty."}
+{"_id": "14956", "title": "discriminant-lemma-trace-unique", "text": "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings. Then there is at most one trace element in $\\omega_{B/A}$."}
+{"_id": "14957", "title": "discriminant-lemma-finite-flat-trace", "text": "Let $A \\to B$ be a finite flat map of Noetherian rings. Then $\\text{Trace}_{B/A} \\in \\omega_{B/A}$ is the trace element."}
+{"_id": "14958", "title": "discriminant-lemma-trace-base-change", "text": "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings. Let $\\tau \\in \\omega_{B/A}$ be a trace element. \\begin{enumerate} \\item If $A \\to A_1$ is a map with $A_1$ Noetherian, then with $B_1 = A_1 \\otimes_A B$ the image of $\\tau$ in $\\omega_{B_1/A_1}$ is a trace element. \\item If $A = R_f$, then $\\tau$ is a trace element in $\\omega_{B/R}$. \\item If $g \\in B$, then the image of $\\tau$ in $\\omega_{B_g/A}$ is a trace element. \\item If $B = B_1 \\times B_2$, then $\\tau$ maps to a trace element in both $\\omega_{B_1/A}$ and $\\omega_{B_2/A}$. \\end{enumerate}"}
+{"_id": "14959", "title": "discriminant-lemma-glue-trace", "text": "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings. Let $g_1, \\ldots, g_m \\in B$ be elements generating the unit ideal. Let $\\tau \\in \\omega_{B/A}$ be an element whose image in $\\omega_{B_{g_i}/A}$ is a trace element for $A \\to B_{g_i}$. Then $\\tau$ is a trace element."}
+{"_id": "14960", "title": "discriminant-lemma-dualizing-tau", "text": "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings. There exists a trace element $\\tau \\in \\omega_{B/A}$."}
+{"_id": "14961", "title": "discriminant-lemma-tau-nonzero", "text": "Let $k$ be a field and let $A$ be a finite $k$-algebra. Assume $A$ is local with residue field $k'$. The following are equivalent \\begin{enumerate} \\item $\\text{Trace}_{A/k}$ is nonzero, \\item $\\tau_{A/k} \\in \\omega_{A/k}$ is nonzero, and \\item $k'/k$ is separable and $\\text{length}_A(A)$ is prime to the characteristic of $k$. \\end{enumerate}"}
+{"_id": "14962", "title": "discriminant-lemma-noether-different-product", "text": "Let $A \\to B_i$, $i = 1, 2$ be ring maps. Set $B = B_1 \\times B_2$. \\begin{enumerate} \\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is $J_1 \\times J_2$ where $J_i$ is the annihilator of $\\Ker(B_i \\otimes_A B_i \\to B_i)$. \\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is $\\mathfrak{D}_1 \\times \\mathfrak{D}_2$, where $\\mathfrak{D}_i$ is the Noether different of $B_i$ over $A$. \\end{enumerate}"}
+{"_id": "14963", "title": "discriminant-lemma-noether-different-base-change", "text": "Let $A \\to B$ be a finite type ring map. Let $A \\to A'$ be a flat ring map. Set $B' = B \\otimes_A A'$. \\begin{enumerate} \\item The annihilator $J'$ of $\\Ker(B' \\otimes_{A'} B' \\to B')$ is $J \\otimes_A A'$ where $J$ is the annihilator of $\\Ker(B \\otimes_A B \\to B)$. \\item The Noether different $\\mathfrak{D}'$ of $B'$ over $A'$ is $\\mathfrak{D}B'$, where $\\mathfrak{D}$ is the Noether different of $B$ over $A$. \\end{enumerate}"}
+{"_id": "14964", "title": "discriminant-lemma-noether-different-localization", "text": "Let $A \\to B' \\to B$ be ring maps with $A \\to B'$ of finite type and $B' \\to B$ inducing an open immersion of spectra. \\begin{enumerate} \\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is $J' \\otimes_{B'} B$ where $J'$ is the annihilator of $\\Ker(B' \\otimes_A B' \\to B')$. \\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is $\\mathfrak{D}'B$, where $\\mathfrak{D}'$ is the Noether different of $B'$ over $A$. \\end{enumerate}"}
+{"_id": "14965", "title": "discriminant-lemma-noether-pairing-compatibilities", "text": "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings. \\begin{enumerate} \\item If $A \\to A'$ is a flat map of Noetherian rings, then $$ \\xymatrix{ \\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d]  \\\\ \\omega_{B'/A'} \\times J' \\ar[r] & B' } $$ is commutative where notation as in Lemma \\ref{lemma-noether-different-base-change} and horizontal arrows are given by (\\ref{equation-pairing-noether}). \\item If $B = B_1 \\times B_2$, then $$ \\xymatrix{ \\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d]  \\\\ \\omega_{B_i/A} \\times J_i \\ar[r] & B_i } $$ is commutative for $i = 1, 2$ where notation as in Lemma \\ref{lemma-noether-different-product} and horizontal arrows are given by (\\ref{equation-pairing-noether}). \\end{enumerate}"}
+{"_id": "14966", "title": "discriminant-lemma-noether-pairing-flat-quasi-finite", "text": "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings. The pairing of Remark \\ref{remark-construction-pairing} induces an isomorphism $J \\to \\Hom_B(\\omega_{B/A}, B)$."}
+{"_id": "14967", "title": "discriminant-lemma-noether-different-flat-quasi-finite", "text": "Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings. The diagram $$ \\xymatrix{ J \\ar[rr] \\ar[rd]_\\mu & & \\Hom_B(\\omega_{B/A}, B) \\ar[ld]^{\\varphi \\mapsto \\varphi(\\tau_{B/A})} \\\\ & B } $$ commutes where the horizontal arrow is the isomorphism of Lemma \\ref{lemma-noether-pairing-flat-quasi-finite}. Hence the Noether different of $B$ over $A$ is the image of the map $\\Hom_B(\\omega_{B/A}, B) \\to B$."}
+{"_id": "14971", "title": "discriminant-lemma-kahler-different-complete-intersection", "text": "Let $A$ be a ring. Let $n \\geq 1$ and $f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$. Set $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$. The K\\\"ahler different of $B$ over $A$ is the ideal of $B$ generated by $\\det(\\partial f_i/\\partial x_j)$."}
+{"_id": "14972", "title": "discriminant-lemma-dedekind-different-ideal", "text": "Assume the Dedekind different of $A \\to B$ is defined. Consider the statements \\begin{enumerate} \\item $A \\to B$ is flat, \\item $A$ is a normal ring, \\item $\\text{Trace}_{L/K}(B) \\subset A$, \\item $1 \\in \\mathcal{L}_{B/A}$, and \\item the Dedekind different $\\mathfrak{D}_{B/A}$ is an ideal of $B$. \\end{enumerate} Then we have (1) $\\Rightarrow$ (3), (2) $\\Rightarrow$ (3), (3) $\\Leftrightarrow$ (4), and (4) $\\Rightarrow$ (5)."}
+{"_id": "14973", "title": "discriminant-lemma-dedekind-complementary-module", "text": "If the Dedekind different of $A \\to B$ is defined, then there is a canonical isomorphism $\\mathcal{L}_{B/A} \\to \\omega_{B/A}$."}
+{"_id": "14974", "title": "discriminant-lemma-flat-dedekind-complementary-module-trace", "text": "If the Dedekind different of $A \\to B$ is defined and $A \\to B$ is flat, then \\begin{enumerate} \\item the canonical isomorphism $\\mathcal{L}_{B/A} \\to \\omega_{B/A}$ sends $1 \\in \\mathcal{L}_{B/A}$ to the trace element $\\tau_{B/A} \\in \\omega_{B/A}$, and \\item the Dedekind different is $\\mathfrak{D}_{B/A} = \\{b \\in B \\mid b\\omega_{B/A} \\subset B\\tau_{B/A}\\}$. \\end{enumerate}"}
+{"_id": "14976", "title": "discriminant-lemma-flat-gorenstein-agree-noether", "text": "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ be affine open subschemes with $f(V) \\subset U$. If $\\omega_{Y/X}|_V$ is invertible, i.e., if $\\omega_{B/A}$ is an invertible $B$-module, then $$ \\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}} $$ as coherent ideal sheaves on $V$ where $\\mathfrak{D} \\subset B$ is the Noether different of $B$ over $A$."}
+{"_id": "14977", "title": "discriminant-lemma-base-change-different", "text": "Consider a cartesian diagram of Noetherian schemes $$ \\xymatrix{ Y' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\ X' \\ar[r]^g & X } $$ with $f$ flat and quasi-finite. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$ be the closed subscheme cut out by the different $\\mathfrak{D}_f$, resp.\\ $\\mathfrak{D}_{f'}$. Then $Y' \\to Y$ induces a bijective closed immersion $R' \\to R \\times_Y Y'$. If $g$ is flat or if $\\omega_{Y/X}$ is invertible, then $R' = R \\times_Y Y'$."}
+{"_id": "14978", "title": "discriminant-lemma-norm-different-in-discriminant", "text": "Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes. Then $\\text{Norm}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$ maps $f_*\\mathfrak{D}_f$ into the ideal sheaf of the discriminant $D_f$."}
+{"_id": "14979", "title": "discriminant-lemma-different-ramification", "text": "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes. The closed subscheme $R \\subset Y$ defined by the different $\\mathfrak{D}_f$ is exactly the set of points where $f$ is not \\'etale (equivalently not unramified)."}
+{"_id": "14980", "title": "discriminant-lemma-norm-different-is-discriminant", "text": "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $R \\subset Y$ be the closed subscheme defined by $\\mathfrak{D}_f$. \\begin{enumerate} \\item If $\\omega_{Y/X}$ is invertible, then $R$ is a locally principal closed subscheme of $Y$. \\item If $\\omega_{Y/X}$ is invertible and $f$ is finite, then the norm of $R$ is the discriminant $D_f$ of $f$. \\item If $\\omega_{Y/X}$ is invertible and $f$ is \\'etale at the associated points of $Y$, then $R$ is an effective Cartier divisor and there is an isomorphism $\\mathcal{O}_Y(R) = \\omega_{Y/X}$. \\end{enumerate}"}
+{"_id": "14981", "title": "discriminant-lemma-syntomic-quasi-finite", "text": "Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is locally quasi-finite and syntomic, \\item $f$ is locally quasi-finite, flat, and a local complete intersection morphism, \\item $f$ is locally quasi-finite, flat, locally of finite presentation, and the fibres of $f$ are local complete intersections, \\item $f$ is locally quasi-finite and for every $y \\in Y$ there are affine opens $y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ with $f(V) \\subset U$ an integer $n$ and $h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$, \\item for every $y \\in Y$ there are affine opens $y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ with $f(V) \\subset U$ such that $A \\to B$ is a relative global complete intersection of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$, \\item $f$ is locally quasi-finite, flat, locally of finite presentation, and $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \\item $f$ is flat, locally of finite presentation, $\\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$, \\end{enumerate}"}
+{"_id": "14982", "title": "discriminant-lemma-characterize-invertible", "text": "Invertibility of the relative dualizing module. \\begin{enumerate} \\item If $A \\to B$ is a quasi-finite flat homomorphism of Noetherian rings, then $\\omega_{B/A}$ is an invertible $B$-module if and only if $\\omega_{B \\otimes_A \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)}$ is an invertible $B \\otimes_A \\kappa(\\mathfrak p)$-module for all primes $\\mathfrak p \\subset A$. \\item If $Y \\to X$ is a quasi-finite flat morphism of Noetherian schemes, then $\\omega_{Y/X}$ is invertible if and only if $\\omega_{Y_x/x}$ is invertible for all $x \\in X$. \\end{enumerate}"}
+{"_id": "14983", "title": "discriminant-lemma-dim-zero-global-complete-intersection-over-field", "text": "Let $k$ be a field. Let $B = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ be a global complete intersection over $k$ of dimension $0$. Then $\\omega_{B/k}$ is invertible."}
+{"_id": "14984", "title": "discriminant-lemma-dualizing-syntomic-quasi-finite", "text": "Let $f : Y \\to X$ be a morphism of locally Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite} then $\\omega_{Y/X}$ is an invertible $\\mathcal{O}_Y$-module."}
+{"_id": "14985", "title": "discriminant-lemma-universal-quasi-finite-syntomic-etale", "text": "With notation as in Example \\ref{example-universal-quasi-finite-syntomic} the schemes $X_{n, d}$ and $Y_{n, d}$ are regular and irreducible, the morphism $Y_{n, d} \\to X_{n, d}$ is locally quasi-finite and syntomic, and there is a dense open subscheme $V \\subset Y_{n, d}$ such that $Y_{n, d} \\to X_{n, d}$ restricts to an \\'etale morphism $V \\to X_{n, d}$."}
+{"_id": "14986", "title": "discriminant-lemma-locally-comes-from-universal", "text": "Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite} then for every $y \\in Y$ there exist $n, d$ and a commutative diagram $$ \\xymatrix{ Y \\ar[d] & V \\ar[d] \\ar[l] \\ar[r] & Y_{n, d} \\ar[d] \\\\ X & U \\ar[l] \\ar[r] & X_{n, d} } $$ where $U \\subset X$ and $V \\subset Y$ are open, where $Y_{n, d} \\to X_{n, d}$ is as in Example \\ref{example-universal-quasi-finite-syntomic}, and where the square on the right hand side is cartesian."}
+{"_id": "14987", "title": "discriminant-lemma-syntomic-finite", "text": "Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is finite and syntomic, \\item $f$ is finite, flat, and a local complete intersection morphism, \\item $f$ is finite, flat, locally of finite presentation, and the fibres of $f$ are local complete intersections, \\item $f$ is finite and for every $x \\in X$ there is an affine open $x \\in U = \\Spec(A) \\subset X$ an integer $n$ and $f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that $f^{-1}(U)$ is isomorphic to the spectrum of $A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$, \\item $f$ is finite, flat, locally of finite presentation, and $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and \\item $f$ is finite, flat, locally of finite presentation, and $\\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$, \\end{enumerate}"}
+{"_id": "14988", "title": "discriminant-lemma-universal-finite-syntomic", "text": "With notation as in Example \\ref{example-universal-finite-syntomic} there is an open subscheme $U_d \\subset X_d$ with the following property: a morphism of schemes $X \\to X_d$ factors through $U_d$ if and only if $Y_d \\times_{X_d} X \\to X$ is syntomic."}
+{"_id": "14989", "title": "discriminant-lemma-universal-finite-syntomic-smooth", "text": "With notation as in Example \\ref{example-universal-finite-syntomic} and $U_d$ as in Lemma \\ref{lemma-universal-finite-syntomic} then $U_d$ is smooth over $\\Spec(\\mathbf{Z})$."}
+{"_id": "14990", "title": "discriminant-lemma-universal-finite-syntomic-etale", "text": "With notation as in Example \\ref{example-universal-finite-syntomic} consider the open subscheme $U'_d \\subset X_d$ over which $\\pi_d$ is \\'etale. Then $U'_d$ is a dense subset of the open $U_d$ of Lemma \\ref{lemma-universal-finite-syntomic}"}
+{"_id": "14991", "title": "discriminant-lemma-locally-comes-from-universal-finite", "text": "Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-finite} then for every $x \\in X$ there exist a $d$ and a commutative diagram $$ \\xymatrix{ Y \\ar[d] & V \\ar[d] \\ar[l] \\ar[r] & V_d \\ar[d] \\ar[r] & Y_d \\ar[d]^{\\pi_d}\\\\ X & U \\ar[l] \\ar[r] & U_d \\ar[r] & X_d } $$ with the following properties \\begin{enumerate} \\item $U \\subset X$ is open and $V = f^{-1}(U)$, \\item $\\pi_d : Y_d \\to X_d$ is as in Example \\ref{example-universal-finite-syntomic}, \\item $U_d \\subset X_d$ is as in Lemma \\ref{lemma-universal-finite-syntomic} and $V_d = \\pi_d^{-1}(U_d) \\subset Y_d$, \\item where the middle square is cartesian. \\end{enumerate}"}
+{"_id": "14992", "title": "discriminant-lemma-tate", "text": "\\begin{reference} \\cite[Appendix]{Mazur-Roberts} \\end{reference} Let $A \\to P$ be a ring map. Let $f_1, \\ldots, f_n \\in P$ be a Koszul regular sequence. Assume $B = P/(f_1, \\ldots, f_n)$ is flat over $A$. Let $g_1, \\ldots, g_n \\in P \\otimes_A B$ be a Koszul regular sequence generating the kernel of the multiplication map $P \\otimes_A B \\to B$. Write $f_i \\otimes 1 = \\sum g_{ij} g_j$. Then the annihilator of $\\Ker(B \\otimes_A B \\to B)$ is a principal ideal generated by the image of $\\det(g_{ij})$."}
+{"_id": "14993", "title": "discriminant-lemma-quasi-finite-complete-intersection", "text": "Let $A$ be a ring. Let $n \\geq 1$ and $h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$. Set $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$. Assume that $B$ is quasi-finite over $A$. Then \\begin{enumerate} \\item $B$ is flat over $A$ and $A \\to B$ is a relative local complete intersection, \\item the annihilator $J$ of $I = \\Ker(B \\otimes_A B \\to B)$ is free of rank $1$ over $B$, \\item the Noether different of $B$ over $A$ is generated by $\\det(\\partial f_i/\\partial x_j)$ in $B$. \\end{enumerate}"}
+{"_id": "14994", "title": "discriminant-lemma-different-syntomic-quasi-finite", "text": "Let $f : Y \\to X$ be a morphism of Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite} then the different $\\mathfrak{D}_f$ of $f$ is the K\\\"ahler different of $f$."}
+{"_id": "14995", "title": "discriminant-lemma-different-quasi-finite-complete-intersection", "text": "Let $A$ be a ring. Let $n \\geq 1$ and $h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$. Set $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$. Assume that $B$ is quasi-finite over $A$. Then there is an isomorphism $B \\to \\omega_{B/A}$ mapping $\\det(\\partial f_i/\\partial x_j)$ to $\\tau_{B/A}$."}
+{"_id": "14996", "title": "discriminant-lemma-discriminant-quasi-finite-morphism-smooth", "text": "Let $S$ be a Noetherian scheme. Let $X$, $Y$ be smooth schemes of relative dimension $n$ over $S$. Let $f : Y \\to X$ be a quasi-finite morphism over $S$. Then $f$ is flat and the closed subscheme $R \\subset Y$ cut out by the different of $f$ is the locally principal closed subscheme cut out by $$ \\wedge^n(\\text{d}f) \\in \\Gamma(Y, (f^*\\Omega^n_{X/S})^{\\otimes -1} \\otimes_{\\mathcal{O}_Y} \\Omega^n_{Y/S}) $$ If $f$ is \\'etale at the associated points of $Y$, then $R$ is an effective Cartier divisor and $$ f^*\\Omega^n_{X/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{O}(R) = \\Omega^n_{Y/S} $$ as invertible sheaves on $Y$."}
+{"_id": "14997", "title": "discriminant-lemma-explain-condition", "text": "Let $A \\to B$ be a map of Noetherian rings. Consider the conditions \\begin{enumerate} \\item nonzerodivisors of $A$ map to nonzerodivisors of $B$, \\item (1) holds and $Q(A) \\to Q(A) \\otimes_A B$ is flat, \\item $A \\to B_\\mathfrak q$ is flat for every $\\mathfrak q \\in \\text{Ass}(B)$, \\item (3) holds and $A \\to B_\\mathfrak q$ is flat for every $\\mathfrak q$ lying over an element in $\\text{Ass}(A)$. \\end{enumerate} Then we have the following implications $$ \\xymatrix{ (1) & (2) \\ar@{=>}[l] \\ar@{=>}[d] \\\\ (3) \\ar@{=>}[u] & (4) \\ar@{=>}[l] } $$ If going up holds for $A \\to B$ then (2) and (4) are equivalent."}
+{"_id": "14998", "title": "discriminant-lemma-agree-dedekind", "text": "Assume the Dedekind different is defined for $A \\to B$. Set $X = \\Spec(A)$ and $Y = \\Spec(B)$. The generalization of Remark \\ref{remark-different-generalization} applies to the morphism $f : Y \\to X$ if and only if $1 \\in \\mathcal{L}_{B/A}$ (e.g., if $A$ is normal, see Lemma \\ref{lemma-dedekind-different-ideal}). In this case $\\mathfrak{D}_{B/A}$ is an ideal of $B$ and we have $$ \\mathfrak{D}_f = \\widetilde{\\mathfrak{D}_{B/A}} $$ as coherent ideal sheaves on $Y$."}
+{"_id": "14999", "title": "discriminant-lemma-compare-dualizing", "text": "Let $f : Y \\to X$ be a quasi-finite separated morphism of Noetherian schemes. For every pair of affine opens $\\Spec(B) = V \\subset Y$, $\\Spec(A) = U \\subset X$ with $f(V) \\subset U$ there is an isomorphism $$ H^0(V, f^!\\mathcal{O}_Y) = \\omega_{B/A} $$ where $f^!$ is as in Duality for Schemes, Section \\ref{duality-section-upper-shriek}. These isomorphisms are compatible with restriction maps and define a canonical isomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{Y/X}$ with $\\omega_{Y/X}$ as in Remark \\ref{remark-relative-dualizing-for-quasi-finite}. Similarly, if $f : Y \\to X$ is a quasi-finite morphism of schemes of finite type over a Noetherian base $S$ endowed with a dualizing complex $\\omega_S^\\bullet$, then $H^0(f_{new}^!\\mathcal{O}_X) = \\omega_{Y/X}$."}
+{"_id": "15000", "title": "discriminant-lemma-compare-trace", "text": "Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes. The map $$ \\text{Trace}_f : f_*\\mathcal{O}_Y \\longrightarrow \\mathcal{O}_X $$ of Section \\ref{section-discriminant} corresponds to a map $\\mathcal{O}_Y \\to f^!\\mathcal{O}_X$. Denote $\\tau_{Y/X} \\in H^0(Y, f^!\\mathcal{O}_X)$ the image of $1$. Via the isomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{X/Y}$ of Lemma \\ref{lemma-compare-dualizing} this agrees with the construction in Remark \\ref{remark-relative-dualizing-for-flat-quasi-finite}."}
+{"_id": "15001", "title": "discriminant-lemma-gorenstein-quasi-finite", "text": "Let $f : Y \\to X$ be a quasi-finite morphism of Noetherian schemes. The following are equivalent \\begin{enumerate} \\item $f$ is Gorenstein, \\item $f$ is flat and the fibres of $f$ are Gorenstein, \\item $f$ is flat and $\\omega_{Y/X}$ is invertible (Remark \\ref{remark-relative-dualizing-for-quasi-finite}), \\item for every $y \\in Y$ there are affine opens $y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ with $f(V) \\subset U$ such that $A \\to B$ is flat and $\\omega_{B/A}$ is an invertible $B$-module. \\end{enumerate}"}
+{"_id": "15002", "title": "discriminant-proposition-tate-map", "text": "There exists a unique rule that to every locally quasi-finite syntomic morphism of locally Noetherian schemes $Y \\to X$ assigns an isomorphism $$ c_{Y/X} : \\det(\\NL_{Y/X}) \\longrightarrow \\omega_{Y/X} $$ satisfying the following two properties \\begin{enumerate} \\item the section $\\delta(\\NL_{Y/X})$ is mapped to $\\tau_{Y/X}$, and \\item the rule is compatible with restriction to opens and with base change. \\end{enumerate}"}
+{"_id": "15014", "title": "stacks-limits-lemma-limit-preserving-objects", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ is limit preserving (Definition \\ref{definition-limit-preserving}), then $f$ is limit preserving on objects (Criteria for Representability, Section \\ref{criteria-section-limit-preserving})."}
+{"_id": "15015", "title": "stacks-limits-lemma-base-change-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving, then so is the base change $p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$ of $p$ by $q$."}
+{"_id": "15016", "title": "stacks-limits-lemma-composition-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ and $q$ are limit preserving, then so is the composition $q \\circ p$."}
+{"_id": "15017", "title": "stacks-limits-lemma-representable-by-spaces-limit-preserving", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent: \\begin{enumerate} \\item $p$ is limit preserving, \\item $p$ is limit preserving on objects, and \\item $p$ is locally of finite presentation (see Algebraic Stacks, Definition \\ref{algebraic-definition-relative-representable-property}). \\end{enumerate}"}
+{"_id": "15018", "title": "stacks-limits-lemma-limit-preserving-diagonal", "text": "Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent \\begin{enumerate} \\item the diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ is limit preserving, and \\item for every directed limit $U = \\lim U_i$ of affine schemes over $S$ the functor $$ \\colim \\mathcal{X}_{U_i} \\longrightarrow \\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i} $$ is fully faithful. \\end{enumerate} In particular, if $p$ is limit preserving, then $\\Delta$ is too."}
+{"_id": "15019", "title": "stacks-limits-lemma-locally-finite-presentation-limit-preserving", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$. If $\\mathcal{X} \\to S$ is locally of finite presentation, then $\\mathcal{X}$ is limit preserving in the sense of Artin's Axioms, Definition \\ref{artin-definition-limit-preserving} (equivalently: the morphism $\\mathcal{X} \\to S$ is limit preserving)."}
+{"_id": "15020", "title": "stacks-limits-lemma-eventually-separated", "text": "In Situation \\ref{situation-descent} assume that $\\mathcal{X}_0 \\to Y_0$ is a morphism from algebraic stack to $Y_0$. Assume $\\mathcal{X}_0$ is quasi-compact and quasi-separated. If $Y \\times_{Y_0} \\mathcal{X}_0 \\to Y$ is separated, then $Y_i \\times_{Y_0} \\mathcal{X}_0 \\to Y_i$ is separated for all sufficiently large $i \\in I$."}
+{"_id": "15021", "title": "stacks-limits-lemma-descend-a-stack-down", "text": "Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system of algebraic spaces over $I$. Assume \\begin{enumerate} \\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine, \\item the spaces $X_i$ are quasi-compact and quasi-separated. \\end{enumerate} Let $X = \\lim X_i$. If $\\mathcal{X}$ is an algebraic stack of finite presentation over $X$, then there exists an $i \\in I$ and an algebraic stack $\\mathcal{X}_i$ of finite presentation over $X_i$ with $\\mathcal{X} \\cong \\mathcal{X}_i \\times_{X_i} X$ as algebraic stacks over $X$."}
+{"_id": "15022", "title": "stacks-limits-lemma-finite-type-closed-in-finite-presentation", "text": "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. Assume: \\begin{enumerate} \\item $f$ is of finite type and quasi-separated, \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a morphism of finite presentation $f' : \\mathcal{X}' \\to Y$ and a closed immersion $\\mathcal{X} \\to \\mathcal{X}'$ of algebraic stacks over $Y$."}
+{"_id": "15023", "title": "stacks-limits-lemma-separated-closed-in-finite-presentation", "text": "Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. Assume: \\begin{enumerate} \\item $f$ is of finite type and separated, \\item $Y$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a separated morphism of finite presentation $f' : \\mathcal{X}' \\to Y$ and a closed immersion $\\mathcal{X} \\to \\mathcal{X}'$ of algebraic stacks over $Y$."}
+{"_id": "15024", "title": "stacks-limits-proposition-characterize-locally-finite-presentation", "text": "\\begin{reference} This is a special case of \\cite[Lemma 2.3.15]{Emerton-Gee} \\end{reference} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is limit preserving, \\item $f$ is limit preserving on objects, and \\item $f$ is locally of finite presentation. \\end{enumerate}"}
+{"_id": "15026", "title": "limits-lemma-directed-inverse-system-affine-schemes-has-limit", "text": "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.  If all the schemes $S_i$ are affine, then the limit $S = \\lim_i S_i$ exists in the category of schemes. In fact $S$ is affine and $S = \\Spec(\\colim_i R_i)$ with $R_i = \\Gamma(S_i, \\mathcal{O})$."}
+{"_id": "15027", "title": "limits-lemma-directed-inverse-system-has-limit", "text": "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. If all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine, then the limit $S = \\lim_i S_i$ exists in the category of schemes. Moreover, \\begin{enumerate} \\item each of the morphisms $f_i : S \\to S_i$ is affine, \\item for an element $0 \\in I$ and any open subscheme $U_0 \\subset S_0$ we have $$ f_0^{-1}(U_0) = \\lim_{i \\geq 0} f_{i0}^{-1}(U_0) $$ in the category of schemes. \\end{enumerate}"}
+{"_id": "15028", "title": "limits-lemma-scheme-over-limit", "text": "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine, Let $S = \\lim_i S_i$. Let $0 \\in I$. Suppose that $T$ is a scheme over $S_0$. Then $$ T \\times_{S_0} S = \\lim_{i \\geq 0} T \\times_{S_0} S_i $$"}
+{"_id": "15029", "title": "limits-lemma-infinite-product", "text": "\\begin{slogan} Infinite products of affine schemes exist and are affine. \\end{slogan} Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$ let $f_i : T_i \\to S$ be an affine morphism. Then the product $T = \\prod T_i$ exists in the category of schemes over $S$. In fact, we have $$ T = \\lim_{\\{i_1, \\ldots, i_n\\} \\subset I} T_{i_1} \\times_S \\ldots \\times_S T_{i_n} $$ and the projection morphisms $T \\to T_{i_1} \\times_S \\ldots \\times_S T_{i_n}$ are affine."}
+{"_id": "15030", "title": "limits-lemma-infinite-product-surjective", "text": "Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$ let $f_i : T_i \\to S$ be a surjective affine morphism. Then the product $T = \\prod T_i$ in the category of schemes over $S$ (Lemma \\ref{lemma-infinite-product}) maps surjectively to $S$."}
+{"_id": "15031", "title": "limits-lemma-infinite-product-integral", "text": "Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$ let $f_i : T_i \\to S$ be an integral morphism. Then the product $T = \\prod T_i$ in the category of schemes over $S$ (Lemma \\ref{lemma-infinite-product}) is integral over $S$."}
+{"_id": "15032", "title": "limits-lemma-inverse-limit-sets", "text": "Let $S = \\lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then $S_{set} = \\lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$."}
+{"_id": "15033", "title": "limits-lemma-inverse-limit-top", "text": "\\begin{reference} \\cite[IV, Proposition 8.2.9]{EGA} \\end{reference} Let $S = \\lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then $S_{top} = \\lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$."}
+{"_id": "15034", "title": "limits-lemma-limit-nonempty", "text": "Let $S = \\lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). If all the schemes $S_i$ are nonempty and quasi-compact, then the limit $S = \\lim_i S_i$ is nonempty."}
+{"_id": "15035", "title": "limits-lemma-inverse-limit-irreducibles", "text": "Let $S = \\lim S_i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma \\ref{lemma-directed-inverse-system-has-limit}). Let $s \\in S$ with images $s_i \\in S_i$. Then \\begin{enumerate} \\item $s = \\lim s_i$ as schemes, i.e., $\\kappa(s) = \\colim \\kappa(s_i)$, \\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as sets, and \\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as schemes where $\\overline{\\{s\\}}$ and $\\overline{\\{s_i\\}}$ are endowed with the reduced induced scheme structure. \\end{enumerate}"}
+{"_id": "15036", "title": "limits-lemma-topology-limit", "text": "In Situation \\ref{situation-descent}. \\begin{enumerate} \\item We have $S_{set} = \\lim_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$. \\item We have $S_{top} = \\lim_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$. \\item If $s, s' \\in S$ and $s'$ is not a specialization of $s$ then for some $i \\in I$ the image $s'_i \\in S_i$ of $s'$ is not a specialization of the image $s_i \\in S_i$ of $s$. \\item Add more easy facts on topology of $S$ here. (Requirement: whatever is added should be easy in the affine case.) \\end{enumerate}"}
+{"_id": "15038", "title": "limits-lemma-limit-closed-nonempty", "text": "In Situation \\ref{situation-descent}. Suppose for each $i$ we are given a nonempty closed subset $Z_i \\subset S_i$ with $f_{ii'}(Z_i) \\subset Z_{i'}$. Then there exists a point $s \\in S$ with $f_i(s) \\in Z_i$ for all $i$."}
+{"_id": "15039", "title": "limits-lemma-limit-fibre-product-empty", "text": "In Situation \\ref{situation-descent}. Suppose we are given an $i$ and a morphism $T \\to S_i$ such that \\begin{enumerate} \\item $T \\times_{S_i} S = \\emptyset$, and \\item $T$ is quasi-compact. \\end{enumerate} Then $T \\times_{S_i} S_{i'} = \\emptyset$ for all sufficiently large $i'$."}
+{"_id": "15040", "title": "limits-lemma-limit-contained-in-constructible", "text": "In Situation \\ref{situation-descent}. Suppose we are given an $i$ and a locally constructible subset $E \\subset S_i$ such that $f_i(S) \\subset E$. Then $f_{i'i}(S_{i'}) \\subset E$ for all sufficiently large $i'$."}
+{"_id": "15041", "title": "limits-lemma-descend-opens", "text": "In Situation \\ref{situation-descent} we have the following: \\begin{enumerate} \\item Given any quasi-compact open $V \\subset S = \\lim_i S_i$ there exists an $i \\in I$ and a quasi-compact open $V_i \\subset S_i$ such that $f_i^{-1}(V_i) = V$. \\item Given $V_i \\subset S_i$ and $V_{i'} \\subset S_{i'}$ quasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$ there exists an index $i'' \\geq i, i'$ such that $f_{i''i}^{-1}(V_i) = f_{i''i'}^{-1}(V_{i'})$. \\item If $V_{1, i}, \\ldots, V_{n, i} \\subset S_i$ are quasi-compact opens and $S = f_i^{-1}(V_{1, i}) \\cup \\ldots \\cup f_i^{-1}(V_{n, i})$ then $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \\cup \\ldots \\cup f_{i'i}^{-1}(V_{n, i})$ for some $i' \\geq i$. \\end{enumerate}"}
+{"_id": "15042", "title": "limits-lemma-limit-quasi-affine", "text": "In Situation \\ref{situation-descent} if $S$ is quasi-affine, then for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$ are quasi-affine."}
+{"_id": "15043", "title": "limits-lemma-limit-affine", "text": "In Situation \\ref{situation-descent} if $S$ is affine, then for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$ are affine."}
+{"_id": "15044", "title": "limits-lemma-limit-separated", "text": "In Situation \\ref{situation-descent} if $S$ is separated, then for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$ are separated."}
+{"_id": "15045", "title": "limits-lemma-limit-ample", "text": "In Situation \\ref{situation-descent} let $\\mathcal{L}_0$ be an invertible sheaf of modules on $S_0$. If the pullback $\\mathcal{L}$ to $S$ is ample, then for some $i \\in I$ the pullback $\\mathcal{L}_i$ to $S_i$ is ample."}
+{"_id": "15046", "title": "limits-lemma-finite-type-eventually-closed", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \\to X$ be a morphism of schemes over $S$. \\begin{enumerate} \\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and $Y$ locally of finite type over $S$, then $Y \\to X_i$ is a closed immersion for $i$ large enough. \\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to S$ locally of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an immersion for $i$ large enough. \\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact, $X_i \\to S$ locally of finite type, the transition morphisms $X_{i'} \\to X_i$ are closed immersions, and $Y \\to S$ is locally of finite presentation, then $Y \\to X_i$ is an isomorphism for $i$ large enough. \\end{enumerate}"}
+{"_id": "15047", "title": "limits-lemma-eventually-separated", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \\begin{enumerate} \\item $S$ quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item $X \\to S$ separated. \\end{enumerate} Then $X_i \\to S$ is separated for all $i$ large enough."}
+{"_id": "15048", "title": "limits-lemma-eventually-affine", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \\begin{enumerate} \\item $S$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item $X \\to S$ affine. \\end{enumerate} Then $X_i \\to S$ is affine for $i$ large enough."}
+{"_id": "15049", "title": "limits-lemma-eventually-finite", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \\begin{enumerate} \\item $S$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item the transition morphisms $X_{i'} \\to X_i$ are finite, \\item $X_i \\to S$ locally of finite type \\item $X \\to S$ integral. \\end{enumerate} Then $X_i \\to S$ is finite for $i$ large enough."}
+{"_id": "15050", "title": "limits-lemma-eventually-closed-immersion", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume \\begin{enumerate} \\item $S$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions, \\item $X_i \\to S$ locally of finite type \\item $X \\to S$ a closed immersion. \\end{enumerate} Then $X_i \\to S$ is a closed immersion for $i$ large enough."}
+{"_id": "15051", "title": "limits-lemma-quasi-affine-finite-type-over-Z", "text": "Let $W$ be a quasi-affine scheme of finite type over $\\mathbf{Z}$. Suppose $W \\to \\Spec(R)$ is an open immersion into an affine scheme. There exists a finite type $\\mathbf{Z}$-algebra $A \\subset R$ which induces an open immersion $W \\to \\Spec(A)$. Moreover, $R$ is the directed colimit of such subalgebras."}
+{"_id": "15052", "title": "limits-lemma-diagram", "text": "Suppose given a cartesian diagram of rings $$ \\xymatrix{ B \\ar[r]_s & R \\\\ B'\\ar[u] \\ar[r] & R' \\ar[u]_t } $$ Let $W' \\subset \\Spec(R')$ be an open of the form $W' = D(f_1) \\cup \\ldots \\cup D(f_n)$ such that $t(f_i) = s(g_i)$ for some $g_i \\in B$ and $B_{g_i} \\cong R_{s(g_i)}$. Then $B' \\to R'$ induces an open immersion of $W'$ into $\\Spec(B')$."}
+{"_id": "15053", "title": "limits-lemma-approximate", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \\subset S$ be a quasi-compact open. Let $I$ be a directed set and let $(V_i, f_{ii'})$ be an inverse system of schemes over $I$ with affine transition maps, with each $V_i$ of finite type over $\\mathbf{Z}$, and with $V = \\lim V_i$. Then there exist \\begin{enumerate} \\item a directed set $J$, \\item an inverse system of schemes $(S_j, g_{jj'})$ over $J$, \\item an order preserving map $\\alpha : J \\to I$, \\item open subschemes $V'_j \\subset S_j$, and \\item isomorphisms $V'_j \\to V_{\\alpha(j)}$ \\end{enumerate} such that \\begin{enumerate} \\item the transition morphisms $g_{jj'} : S_j \\to S_{j'}$ are affine, \\item each $S_j$ is of finite type over $\\mathbf{Z}$, \\item $g_{jj'}^{-1}(V'_{j'}) = V'_j$, \\item $S = \\lim S_j$ and $V = \\lim V'_j$, and \\item the diagrams $$ \\vcenter{ \\xymatrix{ V \\ar[d] \\ar[rd] \\\\ V'_j \\ar[r] & V_{\\alpha(j)} } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ V'_j \\ar[r] \\ar[d] & V_{\\alpha(j)} \\ar[d] \\\\ V'_{j'} \\ar[r] & V_{\\alpha(j')} } } $$ are commutative. \\end{enumerate}"}
+{"_id": "15054", "title": "limits-lemma-surjection-is-enough", "text": "Let $f : X \\to S$ be a morphism of schemes. If for every directed limit $T = \\lim_{i \\in I} T_i$ of affine schemes over $S$ the map $$ \\colim \\Mor_S(T_i, X) \\longrightarrow \\Mor_S(T, X) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition \\ref{proposition-characterize-locally-finite-presentation} parts (2) and (3) it suffices to check surjectivity of the map."}
+{"_id": "15055", "title": "limits-lemma-relative-approximation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume that \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated, and \\item $S$ is quasi-separated. \\end{enumerate} Then $X = \\lim X_i$ is a limit of a directed system of schemes $X_i$ of finite presentation over $S$ with affine transition morphisms over $S$."}
+{"_id": "15056", "title": "limits-lemma-integral-limit-finite-and-finite-presentation", "text": "Let $X \\to S$ be an integral morphism with $S$ quasi-compact and quasi-separated. Then $X = \\lim X_i$ with $X_i \\to S$ finite and of finite presentation."}
+{"_id": "15057", "title": "limits-lemma-descend-affine-finite-presentation", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If $f$ is affine, then there exists an index $i \\geq 0$ such that $f_i$ is affine."}
+{"_id": "15058", "title": "limits-lemma-descend-finite-finite-presentation", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a finite morphism, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is finite."}
+{"_id": "15059", "title": "limits-lemma-descend-unramified", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is unramified, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is unramified."}
+{"_id": "15060", "title": "limits-lemma-descend-closed-immersion-finite-presentation", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a closed immersion, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is a closed immersion."}
+{"_id": "15061", "title": "limits-lemma-descend-separated-finite-presentation", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If $f$ is separated, then $f_i$ is separated for some $i \\geq 0$."}
+{"_id": "15062", "title": "limits-lemma-descend-flat-finite-presentation", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is flat, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is flat for some $i \\geq 0$."}
+{"_id": "15063", "title": "limits-lemma-descend-finite-locally-free", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is finite locally free (of degree $d$), \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is finite locally free (of degree $d$) for some $i \\geq 0$."}
+{"_id": "15064", "title": "limits-lemma-descend-smooth", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is smooth, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is smooth for some $i \\geq 0$."}
+{"_id": "15065", "title": "limits-lemma-descend-etale", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is \\'etale, \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is \\'etale for some $i \\geq 0$."}
+{"_id": "15066", "title": "limits-lemma-descend-isomorphism", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is an isomorphism, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is an isomorphism for some $i \\geq 0$."}
+{"_id": "15067", "title": "limits-lemma-descend-open-immersion", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is an open immersion, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then $f_i$ is an open immersion for some $i \\geq 0$."}
+{"_id": "15068", "title": "limits-lemma-descend-monomorphism", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a monomorphism, and \\item $f_0$ is locally of finite type, \\end{enumerate} then $f_i$ is a monomorphism for some $i \\geq 0$."}
+{"_id": "15069", "title": "limits-lemma-descend-surjective", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is surjective, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is surjective."}
+{"_id": "15070", "title": "limits-lemma-descend-syntomic", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is syntomic, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is syntomic."}
+{"_id": "15071", "title": "limits-lemma-locally-finite-type-in-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume: \\begin{enumerate} \\item The morphism $f$ is locally of finite type. \\item The scheme $X$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a morphism of finite presentation $f' : X' \\to S$ and an immersion $X \\to X'$ of schemes over $S$."}
+{"_id": "15072", "title": "limits-lemma-finite-type-closed-in-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume: \\begin{enumerate} \\item The morphism $f$ is of locally of finite type. \\item The scheme $X$ is quasi-compact and quasi-separated, and \\item The scheme $S$ is quasi-separated. \\end{enumerate} Then there exists a morphism of finite presentation $f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$."}
+{"_id": "15073", "title": "limits-lemma-closed-is-limit-closed-and-finite-presentation", "text": "Let $X \\to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$ of schemes over $Y$ where $X_i \\to Y$ is a closed immersion of finite presentation."}
+{"_id": "15074", "title": "limits-lemma-finite-type-is-limit-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item The morphism $f$ is of locally of finite type. \\item The scheme $X$ is quasi-compact and quasi-separated, and \\item The scheme $S$ is quasi-separated. \\end{enumerate} Then $X = \\lim X_i$ where the $X_i \\to S$ are of finite presentation, the $X_i$ are quasi-compact and quasi-separated, and the transition morphisms $X_{i'} \\to X_i$ are closed immersions (which implies that $X \\to X_i$ are closed immersions for all $i$)."}
+{"_id": "15076", "title": "limits-lemma-finite-in-finite-and-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is finite, and \\item $S$ quasi-compact and quasi-separated. \\end{enumerate} Then $X$ is a directed limit $X = \\lim X_i$ where the transition maps are closed immersions and the objects $X_i$ are finite and of finite presentation over $S$."}
+{"_id": "15077", "title": "limits-lemma-descend-finite-presentation", "text": "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \\begin{enumerate} \\item the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine, \\item the schemes $S_i$ are quasi-compact and quasi-separated. \\end{enumerate} Let $S = \\lim_i S_i$. Then we have the following: \\begin{enumerate} \\item For any morphism of finite presentation $X \\to S$ there exists an index $i \\in I$ and a morphism of finite presentation $X_i \\to S_i$ such that $X \\cong X_{i, S}$ as schemes over $S$. \\item Given an index $i \\in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$, and a morphism $\\varphi : X_{i, S} \\to Y_{i, S}$ over $S$, there exists an index $i' \\geq i$ and a morphism $\\varphi_{i'} : X_{i, S_{i'}} \\to Y_{i, S_{i'}}$ whose base change to $S$ is $\\varphi$. \\item Given an index $i \\in I$, schemes $X_i$, $Y_i$ of finite presentation over $S_i$ and a pair of morphisms $\\varphi_i, \\psi_i : X_i \\to Y_i$ whose base changes $\\varphi_{i, S} = \\psi_{i, S}$ are equal, there exists an index $i' \\geq i$ such that $\\varphi_{i, S_{i'}} = \\psi_{i, S_{i'}}$. \\end{enumerate} In other words, the category of schemes of finite presentation over $S$ is the colimit over $I$ of the categories of schemes of finite presentation over $S_i$."}
+{"_id": "15078", "title": "limits-lemma-descend-modules-finite-presentation", "text": "Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of schemes over $I$. Assume \\begin{enumerate} \\item all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine, \\item all the schemes $S_i$ are quasi-compact and quasi-separated. \\end{enumerate} Let $S = \\lim_i S_i$. Then we have the following: \\begin{enumerate} \\item For any sheaf of $\\mathcal{O}_S$-modules $\\mathcal{F}$ of finite presentation there exists an index $i \\in I$ and a sheaf of $\\mathcal{O}_{S_i}$-modules of finite presentation $\\mathcal{F}_i$ such that $\\mathcal{F} \\cong f_i^*\\mathcal{F}_i$. \\item Suppose given an index $i \\in I$, sheaves of $\\mathcal{O}_{S_i}$-modules $\\mathcal{F}_i$, $\\mathcal{G}_i$ of finite presentation and a morphism $\\varphi : f_i^*\\mathcal{F}_i \\to f_i^*\\mathcal{G}_i$ over $S$. Then there exists an index $i' \\geq i$ and a morphism $\\varphi_{i'} : f_{i'i}^*\\mathcal{F}_i \\to f_{i'i}^*\\mathcal{G}_i$ whose base change to $S$ is $\\varphi$. \\item Suppose given an index $i \\in I$, sheaves of $\\mathcal{O}_{S_i}$-modules $\\mathcal{F}_i$, $\\mathcal{G}_i$ of finite presentation and a pair of morphisms $\\varphi_i, \\psi_i : \\mathcal{F}_i \\to \\mathcal{G}_i$. Assume that the base changes are equal: $f_i^*\\varphi_i = f_i^*\\psi_i$. Then there exists an index $i' \\geq i$ such that $f_{i'i}^*\\varphi_i = f_{i'i}^*\\psi_i$. \\end{enumerate} In other words, the category of modules of finite presentation over $S$ is the colimit over $I$ of the categories modules of finite presentation over $S_i$."}
+{"_id": "15079", "title": "limits-lemma-descend-invertible-modules", "text": "Let $S = \\lim S_i$ be the limit of a directed system of quasi-compact and quasi-separated schemes $S_i$ with affine transition morphisms. Then \\begin{enumerate} \\item any finite locally free $\\mathcal{O}_S$-module is the pullback of a finite locally free $\\mathcal{O}_{S_i}$-module for some $i$, \\item any invertible $\\mathcal{O}_S$-module is the pullback of an invertible $\\mathcal{O}_{S_i}$-module for some $i$, and \\item any finite type quasi-coherent ideal $\\mathcal{I} \\subset \\mathcal{O}_S$ is of the form $\\mathcal{I}_i \\cdot \\mathcal{O}_S$ for some $i$ and some finite type quasi-coherent ideal $\\mathcal{I}_i \\subset \\mathcal{O}_{S_i}$. \\end{enumerate}"}
+{"_id": "15080", "title": "limits-lemma-descend-module-flat-finite-presentation", "text": "With notation and assumptions as in Lemma \\ref{lemma-descend-finite-presentation}. Let $i \\in I$. Suppose that $\\varphi_i : X_i \\to Y_i$ is a morphism of schemes of finite presentation over $S_i$ and that $\\mathcal{F}_i$ is a quasi-coherent $\\mathcal{O}_{X_i}$-module of finite presentation. If the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S$ is flat over $Y_i \\times_{S_i} S$, then there exists an index $i' \\geq i$ such that the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S_{i'}$ is flat over $Y_i \\times_{S_i} S_{i'}$."}
+{"_id": "15081", "title": "limits-lemma-descend-finite-presentation-variant", "text": "For a scheme $T$ denote $\\mathcal{C}_T$ the full subcategory of schemes $W$ over $T$ such that $W$ is quasi-compact and quasi-separated and such that the structure morphism $W \\to T$ is locally of finite presentation. Let $S = \\lim S_i$ be a directed limit of schemes with affine transition morphisms. Then there is an equivalence of categories $$ \\colim \\mathcal{C}_{S_i} \\longrightarrow \\mathcal{C}_S $$ given by the base change functors."}
+{"_id": "15082", "title": "limits-lemma-affine", "text": "\\begin{slogan} A scheme, admitting a finite surjective map from an affine scheme, is affine. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $S$ is affine."}
+{"_id": "15083", "title": "limits-lemma-affines-glued-in-closed-affine", "text": "Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine."}
+{"_id": "15084", "title": "limits-lemma-ample-on-reduction", "text": "Let $i : Z \\to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Let $\\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\\mathcal{L}$ is ample on $Z$, if and only if $\\mathcal{L}$ is ample on $X$."}
+{"_id": "15085", "title": "limits-lemma-thickening-quasi-affine", "text": "Let $i : Z \\to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine."}
+{"_id": "15086", "title": "limits-lemma-ample-profinite-set-in-principal-affine", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$. Assume we have morphisms of schemes $$ \\Spec(k) \\leftarrow \\Spec(A) \\to W \\subset X $$ where $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$. Then there exists an $n > 0$ and a section $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that $X_s$ is affine, $X_s \\subset W$, and $\\Spec(A) \\to W$ factors through $X_s$"}
+{"_id": "15087", "title": "limits-lemma-chow-finite-type", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a separated morphism of finite type. Then there exists an $n \\geq 0$ and a diagram $$ \\xymatrix{ X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\ & S & } $$ where $X' \\to \\mathbf{P}^n_S$ is an immersion, and $\\pi : X' \\to X$ is proper and surjective."}
+{"_id": "15089", "title": "limits-lemma-eventually-proper", "text": "\\begin{slogan} If the base change of a scheme to a limit is proper, then already the base change is proper at a finite level. \\end{slogan} Assumptions and notation as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is proper, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i$ such that $f_i$ is proper."}
+{"_id": "15090", "title": "limits-lemma-proper-limit-of-proper-finite-presentation", "text": "Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then $X = \\lim X_i$ is a directed limit of schemes $X_i$ proper and of finite presentation over $S$ such that all transition morphisms and the morphisms $X \\to X_i$ are closed immersions."}
+{"_id": "15091", "title": "limits-lemma-proper-limit-of-proper-finite-presentation-noetherian", "text": "Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_i : X_i \\to S_i)$ of morphisms of schemes over $I$, such that the transition morphisms $X_i \\to X_{i'}$ and $S_i \\to S_{i'}$ are affine, such that $f_i$ is proper, such that $S_i$ is of finite type over $\\mathbf{Z}$, and such that $(X \\to S) = \\lim (X_i \\to S_i)$."}
+{"_id": "15092", "title": "limits-lemma-finite-type-eventually-proper", "text": "Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \\to X$ be a morphism of schemes over $S$. If $Y \\to X$ is proper, $X_i$ quasi-compact and quasi-separated, and $Y$ locally of finite type over $S$, then $Y \\to X_i$ is proper for $i$ large enough."}
+{"_id": "15093", "title": "limits-lemma-eventually-proper-support", "text": "Assumptions and notation as in Situation \\ref{situation-descent-property}. Let $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module. Denote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of $\\mathcal{F}_0$ to $X$ and $X_i$. Assume \\begin{enumerate} \\item $f_0$ is locally of finite type, \\item $\\mathcal{F}_0$ is of finite type, \\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$. \\end{enumerate} Then the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$ for some $i$."}
+{"_id": "15094", "title": "limits-lemma-separate", "text": "Let $f : X \\to S$ be a quasi-compact morphism of schemes. Let $g : T \\to S$ be a morphism of schemes. Let $t \\in T$ be a point and $Z \\subset X_T$ be a closed subscheme such that $Z \\cap X_t = \\emptyset$. Then there exists an open neighbourhood $V \\subset T$ of $t$, a commutative diagram $$ \\xymatrix{ V \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\ T \\ar[r]^g & S, } $$ and a closed subscheme $Z' \\subset X_{T'}$ such that \\begin{enumerate} \\item the morphism $b : T' \\to S$ is locally of finite presentation, \\item with $t' = a(t)$ we have $Z' \\cap X_{t'} = \\emptyset$, and \\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$. \\end{enumerate} Moreover, we may assume $V$ and $T'$ are affine."}
+{"_id": "15095", "title": "limits-lemma-test-universally-closed", "text": "Let $f : X \\to S$ be a quasi-compact morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item for every morphism $S' \\to S$ which is locally of finite presentation the base change $X_{S'} \\to S'$ is closed, and \\item for every $n$ the morphism $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed. \\end{enumerate}"}
+{"_id": "15096", "title": "limits-lemma-limited-base-change", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a separated morphism of finite type. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is proper. \\item For any morphism $S' \\to S$ which is locally of finite type the base change $X_{S'} \\to S'$ is closed. \\item For every $n \\geq 0$ the morphism $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed. \\end{enumerate}"}
+{"_id": "15097", "title": "limits-lemma-reach-point-closure-Noetherian", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \\in Y$ be a point in the closure of the image of $f$. Then there exists a commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\ \\Spec(A) \\ar[r] & Y } $$ where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume that the image point of $\\Spec(K) \\to X$ is a generic point $\\eta$ of an irreducible component of $X$ and that $K = \\kappa(\\eta)$."}
+{"_id": "15098", "title": "limits-lemma-Noetherian-dvr-valuative-separation", "text": "Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is locally of finite type. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is separated. \\item For any diagram (\\ref{equation-valuative}) there is at most one dotted arrow. \\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete valuation ring there is at most one dotted arrow. \\item For any irreducible component $X_0$ of $X$ with generic point $\\eta \\in X_0$, for any discrete valuation ring $A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any diagram (\\ref{equation-valuative}) such that the morphism $\\Spec(K) \\to X$ is the canonical one (see Schemes, Section \\ref{schemes-section-points}) there is at most one dotted arrow. \\end{enumerate}"}
+{"_id": "15099", "title": "limits-lemma-Noetherian-dvr-valuative-proper", "text": "Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a morphism of finite type. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is proper. \\item For any diagram (\\ref{equation-valuative}) there exists exactly one dotted arrow. \\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete valuation ring there exists exactly one dotted arrow. \\item For any irreducible component $X_0$ of $X$ with generic point $\\eta \\in X_0$, for any discrete valuation ring $A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any diagram (\\ref{equation-valuative}) such that the morphism $\\Spec(K) \\to X$ is the canonical one (see Schemes, Section \\ref{schemes-section-points}) there exists exactly one dotted arrow. \\end{enumerate}"}
+{"_id": "15100", "title": "limits-lemma-check-universally-closed-Noetherian", "text": "Let $f : X \\to S$ be a finite type morphism of schemes. Assume $S$ is locally Noetherian. Then the following are equivalent \\begin{enumerate} \\item $f$ is universally closed, \\item for every $n$ the morphism $\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed, \\item for any diagram (\\ref{equation-valuative}) there exists some dotted arrow, \\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete valuation ring there exists some dotted arrow. \\end{enumerate}"}
+{"_id": "15101", "title": "limits-lemma-refined-valuative-criterion-proper", "text": "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper."}
+{"_id": "15102", "title": "limits-lemma-refined-valuative-criterion-separated", "text": "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ is locally of finite type, that $h$ is of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists at most one dotted arrow making the diagram commute, then $f$ is separated."}
+{"_id": "15104", "title": "limits-lemma-limit-dimension", "text": "Let $I$ be a directed set. Let $(f_i : X_i \\to S_i)$ be an inverse system of morphisms of schemes over $I$. Assume \\begin{enumerate} \\item all the morphisms $S_{i'} \\to S_i$ are affine, \\item all the schemes $S_i$ are quasi-compact and quasi-separated, \\item the morphisms $f_i$ are of finite type, and \\item the morphisms $X_{i'} \\to X_i \\times_{S_i} S_{i'}$ are closed immersions. \\end{enumerate} Let $f : X = \\lim_i X_i \\to S = \\lim_i S_i$ be the limit. Let $d \\geq 0$. If every fibre of $f$ has dimension $\\leq d$, then for some $i$ every fibre of $f_i$ has dimension $\\leq d$."}
+{"_id": "15105", "title": "limits-lemma-descend-quasi-finite", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a quasi-finite morphism, and \\item $f_0$ is locally of finite type, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is quasi-finite."}
+{"_id": "15106", "title": "limits-lemma-descend-dimension-d", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ has relative dimension $d$, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ has relative dimension $d$."}
+{"_id": "15107", "title": "limits-lemma-approximate-given-relative-dimension", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a morphism of finite presentation. Let $d \\geq 0$ be an integer. If $Z \\subset X$ be a closed subscheme such that $\\dim(Z_s) \\leq d$ for all $s \\in S$, then there exists a closed subscheme $Z' \\subset X$ such that \\begin{enumerate} \\item $Z \\subset Z'$, \\item $Z' \\to X$ is of finite presentation, and \\item $\\dim(Z'_s) \\leq d$ for all $s \\in S$. \\end{enumerate}"}
+{"_id": "15108", "title": "limits-lemma-top-cohomology-functor", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Assume \\begin{enumerate} \\item $X$ and $Y$ are quasi-compact and quasi-separated, and \\item $R^if_*\\mathcal{F} = 0$ for $i > d$ and every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$. \\end{enumerate} Then we have \\begin{enumerate} \\item[(a)] for any base change diagram $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ we have $R^if'_*\\mathcal{F}' = 0$ for $i > d$ and any quasi-coherent $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$, \\item[(b)] $R^df'_*(\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{G}') = R^df'_*\\mathcal{F}' \\otimes_{\\mathcal{O}_{Y'}} \\mathcal{G}'$ for any quasi-coherent $\\mathcal{O}_{Y'}$-module $\\mathcal{G}'$, \\item[(c)] formation of $R^df'_*\\mathcal{F}'$ commutes with arbitrary further base change (see proof for explanation). \\end{enumerate}"}
+{"_id": "15109", "title": "limits-lemma-higher-direct-images-zero-above-dimension-fibre", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $y \\in Y$. Assume $f$ is proper and $\\dim(X_y) = d$. Then \\begin{enumerate} \\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have $(R^if_*\\mathcal{F})_y = 0$ for all $i > d$, \\item there is an affine open neighbourhood $V \\subset Y$ of $y$ such that $f^{-1}(V) \\to V$ and $d$ satisfy the assumptions and conclusions of Lemma \\ref{lemma-top-cohomology-functor}. \\end{enumerate}"}
+{"_id": "15110", "title": "limits-lemma-proper-top-cohomology-finite-type", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $f$ is a proper morphism all of whose fibres have dimension $\\leq d$, \\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite type. \\end{enumerate} Then $R^df_*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite type."}
+{"_id": "15112", "title": "limits-lemma-glueing-near-closed-point", "text": "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that $U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With $V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is an equivalence of categories $$ \\left\\{ \\begin{matrix} X \\to S\\text{ of finite presentation} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\vcenter{ \\xymatrix{ X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\ U & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s}) } } \\right\\} $$ where on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation."}
+{"_id": "15113", "title": "limits-lemma-glueing-near-closed-point-modules", "text": "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that $U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With $V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is an equivalence of categories $$ \\left\\{ \\mathcal{O}_S\\text{-modules }\\mathcal{F}\\text{ of finite presentation} \\right\\} \\longrightarrow \\left\\{ (\\mathcal{G}, \\mathcal{H}, \\alpha) \\right\\} $$ where on the right hand side we consider triples consisting of a $\\mathcal{O}_U$-module $\\mathcal{G}$ of finite presentation, a $\\mathcal{O}_{\\Spec(\\mathcal{O}_{S, s})}$-module $\\mathcal{H}$ of finite presentation, and an isomorphism $\\alpha : \\mathcal{G}|_V \\to \\mathcal{H}|_V$ of $\\mathcal{O}_V$-modules."}
+{"_id": "15114", "title": "limits-lemma-glueing-near-point", "text": "Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open. Let $s \\in S$ be a point in the complement of $U$. With $V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is an equivalence of categories $$ \\colim_{s \\in U' \\supset U\\text{ open}} \\left\\{ \\vcenter{ \\xymatrix{ X \\ar[d] \\\\ U' } } \\right\\} \\longrightarrow \\left\\{ \\vcenter{ \\xymatrix{ X' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\ U & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s}) } } \\right\\} $$ where on the left hand side the vertical arrow is of finite presentation and on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation."}
+{"_id": "15115", "title": "limits-lemma-glueing-near-point-properties", "text": "Notation and assumptions as in Lemma \\ref{lemma-glueing-near-point}. Let $U \\subset U' \\subset X$ be an open containing $s$. \\begin{enumerate} \\item  Let $f' : X \\to U'$ correspond to $f : X' \\to U$ and $g : Y \\to \\Spec(\\mathcal{O}_{S, s})$ via the equivalence. If $f$ and $g$ are separated, proper, finite, \\'etale, then after possibly shrinking $U'$ the morphism $f'$ has the same property. \\item Let $a : X_1 \\to X_2$ be a morphism of schemes of finite presentation over $U'$ with base change $a' : X'_1 \\to X'_2$ over $U$ and $b : Y_1 \\to Y_2$ over $\\Spec(\\mathcal{O}_{S, s})$. If $a'$ and $b$ are separated, proper, finite, \\'etale, then after possibly shrinking $U'$ the morphism $a$ has the same property. \\end{enumerate}"}
+{"_id": "15117", "title": "limits-lemma-modifications", "text": "Let $S$ be a scheme. Let $s \\in S$ be a closed point such that $U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With $V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ the base change functor $$ \\left\\{ \\begin{matrix} f : X \\to S\\text{ of finite presentation} \\\\ f^{-1}(U) \\to U\\text{ is an isomorphism} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} g : Y \\to \\Spec(\\mathcal{O}_{S, s})\\text{ of finite presentation} \\\\ g^{-1}(V) \\to V\\text{ is an isomorphism} \\end{matrix} \\right\\} $$ is an equivalence of categories."}
+{"_id": "15118", "title": "limits-lemma-modifications-properties", "text": "Notation and assumptions as in Lemma \\ref{lemma-modifications}. Let $f : X \\to S$ correspond to $g : Y \\to \\Spec(\\mathcal{O}_{S, s})$ via the equivalence. Then $f$ is separated, proper, finite, \\'etale and add more here if and only if $g$ is so."}
+{"_id": "15119", "title": "limits-lemma-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $X \\to S$ be quasi-separated and of finite type. Then there exists an $i \\in I$ and a diagram \\begin{equation} \\label{equation-good-diagram} \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ S \\ar[r] & S_i } } \\end{equation} such that $W \\to S_i$ is of finite type and such that the induced morphism $X \\to S \\times_{S_i} W$ is a closed immersion."}
+{"_id": "15120", "title": "limits-lemma-limit-from-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $X \\to S$ be quasi-separated and of finite type. Given $i \\in I$ and a diagram $$ \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ S \\ar[r] & S_i } } $$ as in (\\ref{equation-good-diagram}) for $i' \\geq i$ let $X_{i'}$ be the scheme theoretic image of $X \\to S_{i'} \\times_{S_i} W$. Then $X = \\lim_{i' \\geq i} X_{i'}$."}
+{"_id": "15121", "title": "limits-lemma-morphism-good-diagram", "text": "In Situation \\ref{situation-limit-noetherian}. Let $f : X \\to Y$ be a morphism of schemes quasi-separated and of finite type over $S$. Let $$ \\vcenter{ \\xymatrix{ X \\ar[r] \\ar[d] & W \\ar[d] \\\\ S \\ar[r] & S_{i_1} } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ Y \\ar[r] \\ar[d] & V \\ar[d] \\\\ S \\ar[r] & S_{i_2} } } $$ be diagrams as in (\\ref{equation-good-diagram}). Let $X = \\lim_{i \\geq i_1} X_i$ and $Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding limit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}. Then there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism $$ (f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0} $$ of inverse systems over $(S_i)_{i \\geq i_0}$ such that such that $f = \\lim_{i \\geq i_0} f_i$. If $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$ is a second morphism of inverse systems over $(S_i)_{i \\geq i_0}$ such that such that $f = \\lim_{i \\geq i_0} g_i$ then $f_i = g_i$ for all $i \\gg i_0$."}
+{"_id": "15122", "title": "limits-lemma-morphism-good-diagram-flat", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is flat and of finite presentation, then there exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have $f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and $X = Y \\times_{Y_{i_3}} X_{i_3}$."}
+{"_id": "15126", "title": "limits-proposition-approximate", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. There exist a directed set $I$ and an inverse system of schemes $(S_i, f_{ii'})$ over $I$ such that \\begin{enumerate} \\item the transition morphisms $f_{ii'}$ are affine \\item each $S_i$ is of finite type over $\\mathbf{Z}$, and \\item $S = \\lim_i S_i$. \\end{enumerate}"}
+{"_id": "15127", "title": "limits-proposition-characterize-locally-finite-presentation", "text": "\\begin{reference} \\cite[IV, Proposition 8.14.2]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is locally of finite presentation. \\item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $T_i$ affine, we have $$ \\Mor_S(\\lim_i T_i, X) = \\colim_i \\Mor_S(T_i, X) $$ \\item For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $f_{ii'}$ affine and every $T_i$ quasi-compact and quasi-separated as a scheme, we have $$ \\Mor_S(\\lim_i T_i, X) = \\colim_i \\Mor_S(T_i, X) $$ \\end{enumerate}"}
+{"_id": "15128", "title": "limits-proposition-separated-closed-in-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is of finite type and separated, and \\item $S$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a separated morphism of finite presentation $f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$."}
+{"_id": "15129", "title": "limits-proposition-affine", "text": "\\begin{slogan} A scheme admitting a surjective integral map from an affine scheme is affine. \\end{slogan} Let $f : X \\to S$ be a morphism of schemes. Assume that $f$ is surjective and integral, and assume that $X$ is affine. Then $S$ is affine."}
+{"_id": "8", "title": "stacks-perfect-definition-derived", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{M}_\\mathcal{X} \\subset \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ denote the category of locally quasi-coherent $\\mathcal{O}_\\mathcal{X}$-modules with the flat base change property. Let $\\mathcal{P}_\\mathcal{X} \\subset \\mathcal{M}_\\mathcal{X}$ be the full subcategory consisting of parasitic objects. We define the {\\it derived category of $\\mathcal{O}_\\mathcal{X}$-modules with quasi-coherent cohomology sheaves} as the Verdier quotient\\footnote{This definition is different from the one in the literature, see \\cite[6.3]{olsson_sheaves}, but it agrees with that definition by Lemma \\ref{lemma-derived-quasi-coherent}.} $$ D_\\QCoh(\\mathcal{O}_\\mathcal{X}) = D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})/ D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) $$"}
+{"_id": "278", "title": "spaces-more-morphisms-definition-radicial", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it radicial} if for any morphism $\\Spec(K) \\to Y$ where $K$ is a field the reduction $(\\Spec(K) \\times_Y X)_{red}$ is either empty or representable by the spectrum of a purely inseparable field extension of $K$."}
+{"_id": "279", "title": "spaces-more-morphisms-definition-conormal-sheaf", "text": "Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf $\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$} is the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$ described above."}
+{"_id": "280", "title": "spaces-more-morphisms-definition-conormal-algebra", "text": "Let $i : Z \\to X$ be an immersion. The {\\it conormal algebra $\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $i$} is the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras $\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above."}
+{"_id": "281", "title": "spaces-more-morphisms-definition-normal-cone", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. The {\\it normal cone $C_ZX$} of $Z$ in $X$ is $$ C_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *}) $$ see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-relative-spec}. The {\\it normal bundle} of $Z$ in $X$ is the vector bundle $$ N_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X})) $$"}
+{"_id": "282", "title": "spaces-more-morphisms-definition-sheaf-differentials", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$} is sheaf of differentials (Modules on Sites, Definition \\ref{sites-modules-definition-sheaf-differentials}) for the morphism of ringed topoi $$ (f_{small}, f^\\sharp) : (X_\\etale, \\mathcal{O}_X) \\to (Y_\\etale, \\mathcal{O}_Y) $$ of Properties of Spaces, Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}. The {\\it universal $Y$-derivation} will be denoted $\\text{d}_{X/Y} : \\mathcal{O}_X \\to \\Omega_{X/Y}$."}
+{"_id": "283", "title": "spaces-more-morphisms-definition-thickening", "text": "Thickenings. Let $S$ be a scheme. \\begin{enumerate} \\item We say an algebraic space $X'$ is a {\\it thickening} of an algebraic space $X$ if $X$ is a closed subspace of $X'$ and the associated topological spaces are equal. \\item We say $X'$ is a {\\it first order thickening} of $X$ if $X$ is a closed subspace of $X'$ and the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero. \\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a {\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that $f(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed subspace $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say that $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of thickenings. \\item Let $B$ be an algebraic space. We similarly define {\\it thickenings over $B$}, and {\\it morphisms of thickenings over $B$}. This means that the spaces $X, X', Y, Y'$ above are algebraic spaces endowed with a structure morphism to $B$, and that the morphisms $X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $B$. \\end{enumerate}"}
+{"_id": "284", "title": "spaces-more-morphisms-definition-first-order-infinitesimal-neighbourhood", "text": "Let $i : Z \\to X$ be an immersion of algebraic spaces. The {\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is the first order thickening $Z \\subset Z'$ over $X$ described above."}
+{"_id": "285", "title": "spaces-more-morphisms-definition-formally-smooth-etale-unramified", "text": "Let $S$ be a scheme. Let $a : F \\to G$ be a transformation of functors $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Consider commutative solid diagrams of the form $$ \\xymatrix{ F \\ar[d]_a & T \\ar[d]^i \\ar[l] \\\\ G & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T$ and $T'$ are affine schemes and $i$ is a closed immersion defined by an ideal of square zero. \\begin{enumerate} \\item We say $a$ is {\\it formally smooth} if given any solid diagram as above there exists a dotted arrow making the diagram commute\\footnote{This is just one possible definition that one can make here. Another slightly weaker condition would be to require that the dotted arrow exists fppf locally on $T'$. This weaker notion has in some sense better formal properties.}. \\item We say $a$ is {\\it formally \\'etale} if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute. \\item We say $a$ is {\\it formally unramified} if given any solid diagram as above there exists at most one dotted arrow making the diagram commute. \\end{enumerate}"}
+{"_id": "286", "title": "spaces-more-morphisms-definition-formally-unramified", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is said to be {\\it formally unramified} if it is formally unramified as a transformation of functors as in Definition \\ref{definition-formally-smooth-etale-unramified}."}
+{"_id": "287", "title": "spaces-more-morphisms-definition-universal-thickening", "text": "Let $S$ be a scheme. Let $h : Z \\to X$ be a formally unramified morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The {\\it universal first order thickening} of $Z$ over $X$ is the thickening $Z \\subset Z'$ constructed in Lemma \\ref{lemma-universal-thickening}. \\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf of $Z$ in its universal first order thickening $Z'$ over $X$. \\end{enumerate} We often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation."}
+{"_id": "288", "title": "spaces-more-morphisms-definition-formally-etale", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is said to be {\\it formally \\'etale} if it is formally \\'etale as a transformation of functors as in Definition \\ref{definition-formally-smooth-etale-unramified}."}
+{"_id": "289", "title": "spaces-more-morphisms-definition-formally-smooth", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is said to be {\\it formally smooth} if it is formally smooth as a transformation of functors as in Definition \\ref{definition-formally-smooth-etale-unramified}."}
+{"_id": "290", "title": "spaces-more-morphisms-definition-netherlander", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The {\\it naive cotangent complex of $f$} is the complex defined in Modules on Sites, Definition \\ref{sites-modules-definition-cotangent-complex-morphism-ringed-topoi} for the morphism of ringed topoi $f_{small}$ between the small \\'etale sites of $X$ and $Y$, see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}. Notation: $\\NL_f$ or $\\NL_{X/Y}$."}
+{"_id": "291", "title": "spaces-more-morphisms-definition-module-flat-on-fibre", "text": "Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$ be a point and denote $z \\in |Z|$ its image. \\begin{enumerate} \\item We say {\\it the restriction of $\\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of Lemma \\ref{lemma-flat-on-fibres-at-point} are satisfied. \\item We say {\\it the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of Lemma \\ref{lemma-flat-on-fibres-at-point} hold with $\\mathcal{F} = \\mathcal{O}_X$. \\item We say {\\it the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$} if for all $x \\in |X|$ lying over $z$ the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$ \\end{enumerate}"}
+{"_id": "292", "title": "spaces-more-morphisms-definition-CM", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian (Divisors on Spaces, Definition \\ref{spaces-divisors-definition-locally-Noetherian-fibre}). \\begin{enumerate} \\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is {\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$ and the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target-at-point} hold for the property $\\mathcal{P}$ described in Lemma \\ref{lemma-CM-local-ring-fibre}. \\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is Cohen-Macaulay at every point of $X$. \\end{enumerate}"}
+{"_id": "293", "title": "spaces-more-morphisms-definition-gorenstein", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian (Divisors on Spaces, Definition \\ref{spaces-divisors-definition-locally-Noetherian-fibre}). \\begin{enumerate} \\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is {\\it Gorenstein at $x$} if $f$ is flat at $x$ and the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target-at-point} hold for the property $\\mathcal{P}$ described in Lemma \\ref{lemma-gorenstein-local-ring-fibre}. \\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is Gorenstein at every point of $X$. \\end{enumerate}"}
+{"_id": "294", "title": "spaces-more-morphisms-definition-geometrically-reduced-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. We say {\\it the fibre of $f : X \\to Y$ at $y$ is geometrically reduced} if the equivalent conditions of Lemma \\ref{lemma-geometrically-reduced-fibre} hold."}
+{"_id": "295", "title": "spaces-more-morphisms-definition-regular-immersion", "text": "Let $S$ be a scheme. Let $i : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $i$ is a {\\it Koszul-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \\ref{lemma-representable-etale-local-target} hold with $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion''. \\item We say $i$ is an {\\it $H_1$-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \\ref{lemma-representable-etale-local-target} hold with $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion''. \\item We say $i$ is a {\\it quasi-regular immersion} if $i$ is representable and the equivalent conditions of Lemma \\ref{lemma-representable-etale-local-target} hold with $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''. \\end{enumerate}"}
+{"_id": "296", "title": "spaces-more-morphisms-definition-relative-pseudo-coherence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item We say $E$ is {\\it $m$-pseudo-coherent relative to $Y$} if the equivalent conditions of Lemma \\ref{lemma-qcoh-relative-pseudo-coherence-characterize} are satisfied. \\item We say $E$ is {\\it pseudo-coherent relative to $Y$} if $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \\in \\mathbf{Z}$. \\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $Y$} if $\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $Y$. \\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $Y$} if $\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is pseudo-coherent relative to $Y$. \\end{enumerate}"}
+{"_id": "297", "title": "spaces-more-morphisms-definition-pseudo-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it pseudo-coherent} if the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target} hold with $\\mathcal{P} =$``pseudo-coherent''. \\item Let $x \\in |X|$. We say $f$ is {\\it pseudo-coherent at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is pseudo-coherent. \\end{enumerate}"}
+{"_id": "298", "title": "spaces-more-morphisms-definition-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it perfect} if the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target} hold with $\\mathcal{P} =$``perfect''. \\item Let $x \\in |X|$. We say $f$ is {\\it perfect at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is perfect. \\end{enumerate}"}
+{"_id": "299", "title": "spaces-more-morphisms-definition-lci", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is a {\\it Koszul morphism}, or that $f$ is a {\\it local complete intersection morphism} if the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target} hold with $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism''. \\item Let $x \\in |X|$. We say $f$ is {\\it Koszul at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is a local complete intersection morphism. \\end{enumerate}"}
+{"_id": "300", "title": "spaces-more-morphisms-definition-relatively-perfect", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. An object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect relative to $Y$} or {\\it $Y$-perfect} if $E$ is pseudo-coherent (Cohomology on Sites, Definition \\ref{sites-cohomology-definition-pseudo-coherent}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_Y)$ (Cohomology on Sites, Definition \\ref{sites-cohomology-definition-tor-amplitude})."}
+{"_id": "301", "title": "spaces-more-morphisms-definition-nodal-family", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it at-worst-nodal of relative dimension $1$} if the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target} hold with $\\mathcal{P} =$``at-worst-nodal of relative dimension $1$''."}
+{"_id": "1432", "title": "algebra-definition-module-finite-type", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item We say $M$ is a {\\it finite $R$-module}, or a {\\it finitely generated $R$-module} if there exist $n \\in \\mathbf{N}$ and $x_1, \\ldots, x_n \\in M$ such that every element of $M$ is a $R$-linear combination of the $x_i$. Equivalently, this means there exists a surjection $R^{\\oplus n} \\to M$ for some $n \\in \\mathbf{N}$. \\item We say $M$ is a {\\it finitely presented $R$-module} or an {\\it $R$-module of finite presentation} if there exist integers $n, m \\in \\mathbf{N}$ and an exact sequence $$ R^{\\oplus m} \\longrightarrow R^{\\oplus n} \\longrightarrow M \\longrightarrow 0 $$ \\end{enumerate}"}
+{"_id": "1433", "title": "algebra-definition-finite-type", "text": "Let $R \\to S$ be a ring map. \\begin{enumerate} \\item We say $R \\to S$ is of {\\it finite type}, or that {\\it $S$ is a finite type $R$-algebra} if there exists an $n \\in \\mathbf{N}$ and an surjection of $R$-algebras $R[x_1, \\ldots, x_n] \\to S$. \\item We say $R \\to S$ is of {\\it finite presentation} if there exist integers $n, m \\in \\mathbf{N}$ and polynomials $f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$ and an isomorphism of $R$-algebras $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m) \\cong S$. \\end{enumerate}"}
+{"_id": "1434", "title": "algebra-definition-finite-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. We say $\\varphi : R \\to S$ is {\\it finite} if $S$ is finite as an $R$-module."}
+{"_id": "1435", "title": "algebra-definition-directed-system", "text": "Let $(I, \\leq)$ be a preordered set. A {\\it system $(M_i, \\mu_{ij})$ of $R$-modules over $I$} consists of a family of $R$-modules $\\{M_i\\}_{i\\in I}$ indexed by $I$ and a family of $R$-module maps $\\{\\mu_{ij} : M_i \\to M_j\\}_{i \\leq j}$ such that for all $i \\leq j \\leq k$ $$ \\mu_{ii} = \\text{id}_{M_i}\\quad \\mu_{ik} = \\mu_{jk}\\circ \\mu_{ij} $$ We say $(M_i, \\mu_{ij})$ is a {\\it directed system} if $I$ is a directed set."}
+{"_id": "1436", "title": "algebra-definition-homomorphism-directed-systems", "text": "Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be systems of $R$-modules over the same preordered set $I$. A {\\it homomorphism of systems} $\\Phi$ from $(M_i, \\mu_{ij})$ to $(N_i, \\nu_{ij})$ is by definition a family of $R$-module homomorphisms $\\phi_i : M_i \\to N_i$ such that $\\phi_j \\circ \\mu_{ij} = \\nu_{ij} \\circ \\phi_i$ for all $i \\leq j$."}
+{"_id": "1437", "title": "algebra-definition-multiplicative-subset", "text": "Let $R$ be a ring, $S$ a subset of $R$. We say $S$ is a {\\it multiplicative subset of $R$} if $1\\in S$ and $S$ is closed under multiplication, i.e., $s, s' \\in S \\Rightarrow ss' \\in S$."}
+{"_id": "1438", "title": "algebra-definition-localization", "text": "This ring is called the {\\it localization of $A$ with respect to $S$}."}
+{"_id": "1439", "title": "algebra-definition-localization-module", "text": "The $S^{-1}A$-module $S^{-1}M$ is called the {\\it localization} of $M$ at $S$."}
+{"_id": "1440", "title": "algebra-definition-relation", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. Let $n \\geq 0$ and $x_i \\in M$ for $i = 1, \\ldots, n$. A {\\it relation} between $x_1, \\ldots, x_n$ in $M$ is a sequence of elements $f_1, \\ldots, f_n \\in R$ such that $\\sum_{i = 1, \\ldots, n} f_i x_i = 0$."}
+{"_id": "1441", "title": "algebra-definition-bilinear", "text": "Let $R$ be a ring, $M, N, P$ be three $R$-modules. A mapping $f : M \\times N \\to P$ (where $M \\times N$ is viewed only as Cartesian product of two $R$-modules) is said to be {\\it $R$-bilinear} if for each $x \\in M$ the mapping $y\\mapsto f(x, y)$ of $N$ into $P$ is $R$-linear, and for each $y\\in N$ the mapping $x\\mapsto f(x, y)$ is also $R$-linear."}
+{"_id": "1442", "title": "algebra-definition-bimodule", "text": "An abelian group $N$ is called an {\\it $(A, B)$-bimodule} if it is both an $A$-module and a $B$-module, and the actions $A \\to End(M)$ and $B \\to End(M)$ are compatible in the sense that $(ax)b = a(xb)$ for all $a\\in A, b\\in B, x\\in N$. Usually we denote it as $_AN_B$."}
+{"_id": "1443", "title": "algebra-definition-base-change", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Let $R \\to R'$ be any ring map. The {\\it base change} of $\\varphi$ by $R \\to R'$ is the ring map $R' \\to S \\otimes_R R'$. In this situation we often write $S' = S \\otimes_R R'$. The {\\it base change} of the $S$-module $M$ is the $S'$-module $M \\otimes_R R'$."}
+{"_id": "1444", "title": "algebra-definition-spectrum-ring", "text": "Let $R$ be a ring. \\begin{enumerate} \\item The {\\it spectrum} of $R$ is the set of prime ideals of $R$. It is usually denoted $\\Spec(R)$. \\item Given a subset $T \\subset R$ we let $V(T) \\subset \\Spec(R)$ be the set of primes containing $T$, i.e., $V(T) = \\{ \\mathfrak p \\in \\Spec(R) \\mid \\forall f\\in T, f\\in \\mathfrak p\\}$. \\item Given an element $f \\in R$ we let $D(f) \\subset \\Spec(R)$ be the set of primes not containing $f$. \\end{enumerate}"}
+{"_id": "1445", "title": "algebra-definition-Zariski-topology", "text": "Let $R$ be a ring. The topology on $\\Spec(R)$ whose closed sets are the sets $V(T)$ is called the {\\it Zariski} topology. The open subsets $D(f)$ are called the {\\it standard opens} of $\\Spec(R)$."}
+{"_id": "1446", "title": "algebra-definition-local-ring", "text": "A {\\it local ring} is a ring with exactly one maximal ideal. The maximal ideal is often denoted $\\mathfrak m_R$ in this case. We often say ``let $(R, \\mathfrak m, \\kappa)$ be a local ring'' to indicate that $R$ is local, $\\mathfrak m$ is its unique maximal ideal and $\\kappa = R/\\mathfrak m$ is its residue field. A {\\it local homomorphism of local rings} is a ring map $\\varphi : R \\to S$ such that $R$ and $S$ are local rings and such that $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$. If it is given that $R$ and $S$ are local rings, then the phrase ``{\\it local ring map $\\varphi : R \\to S$}'' means that $\\varphi$ is a local homomorphism of local rings."}
+{"_id": "1447", "title": "algebra-definition-oka-family", "text": "Let $R$ be a ring. Let $\\mathcal{F}$ be a set of ideals of $R$. We say $\\mathcal{F}$ is an {\\it Oka family} if $R \\in \\mathcal{F}$ and whenever $I \\subset R$ is an ideal and $(I : a), (I, a) \\in \\mathcal{F}$ for some $a \\in R$, then $I \\in \\mathcal{F}$."}
+{"_id": "1448", "title": "algebra-definition-locally-nilpotent-ideal", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. We say $I$ is {\\it locally nilpotent} if for every $x \\in I$ there exists an $n \\in \\mathbf{N}$ such that $x^n = 0$. We say $I$ is {\\it nilpotent} if there exists an $n \\in \\mathbf{N}$ such that $I^n = 0$."}
+{"_id": "1449", "title": "algebra-definition-ring-jacobson", "text": "Let $R$ be a ring. We say that $R$ is a {\\it Jacobson ring} if every radical ideal $I$ is the intersection of the maximal ideals containing it."}
+{"_id": "1450", "title": "algebra-definition-integral-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. \\begin{enumerate} \\item An element $s \\in S$ is {\\it integral over $R$} if there exists a monic polynomial $P(x) \\in R[x]$ such that $P^\\varphi(s) = 0$, where $P^\\varphi(x) \\in S[x]$ is the image of $P$ under $\\varphi : R[x] \\to S[x]$. \\item  The ring map $\\varphi$ is {\\it integral} if every $s \\in S$ is integral over $R$. \\end{enumerate}"}
+{"_id": "1451", "title": "algebra-definition-integral-closure", "text": "Let $R \\to S$ be a ring map. The ring $S' \\subset S$ of elements integral over $R$, see Lemma \\ref{lemma-integral-closure-is-ring}, is called the {\\it integral closure} of $R$ in $S$. If $R \\subset S$ we say that $R$ is {\\it integrally closed} in $S$ if $R = S'$."}
+{"_id": "1452", "title": "algebra-definition-domain-normal", "text": "A domain $R$ is called {\\it normal} if it is integrally closed in its field of fractions."}
+{"_id": "1453", "title": "algebra-definition-almost-integral", "text": "Let $R$ be a domain. \\begin{enumerate} \\item An element $g$ of the fraction field of $R$ is called {\\it almost integral over $R$} if there exists an element $r \\in R$, $r\\not = 0$ such that $rg^n \\in R$ for all $n \\geq 0$. \\item The domain $R$ is called {\\it completely normal} if every almost integral element of the fraction field of $R$ is contained in $R$. \\end{enumerate}"}
+{"_id": "1454", "title": "algebra-definition-ring-normal", "text": "A ring $R$ is called {\\it normal} if for every prime $\\mathfrak p \\subset R$ the localization $R_{\\mathfrak p}$ is a normal domain (see Definition \\ref{definition-domain-normal})."}
+{"_id": "1455", "title": "algebra-definition-integral-over-ideal", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. We say an element $g \\in S$ is {\\it integral over $I$} if there exists a monic polynomial $P = x^d + \\sum_{j < d} a_j x^j$ with coefficients $a_j \\in I^{d-j}$ such that $P^\\varphi(g) = 0$ in $S$."}
+{"_id": "1456", "title": "algebra-definition-flat", "text": "Let $R$ be a ring. \\begin{enumerate} \\item An $R$-module $M$ is called {\\it flat} if whenever $N_1 \\to N_2 \\to N_3$ is an exact sequence of $R$-modules the sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$ is exact as well. \\item An $R$-module $M$ is called {\\it faithfully flat} if the complex of $R$-modules $N_1 \\to N_2 \\to N_3$ is exact if and only if the sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$ is exact. \\item A ring map $R \\to S$ is called {\\it flat} if $S$ is flat as an $R$-module. \\item A ring map $R \\to S$ is called {\\it faithfully flat} if $S$ is faithfully flat as an $R$-module. \\end{enumerate}"}
+{"_id": "1457", "title": "algebra-definition-support-module", "text": "Let $R$ be a ring and let $M$ be an $R$-module. The {\\it support of $M$} is the set $$ \\text{Supp}(M) = \\{ \\mathfrak p \\in \\Spec(R) \\mid M_{\\mathfrak p} \\not = 0 \\} $$"}
+{"_id": "1458", "title": "algebra-definition-annihilator", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item Given an element $m \\in M$ the {\\it annihilator of $m$} is the ideal $$ \\text{Ann}_R(m) = \\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}. $$ \\item The {\\it annihilator of $M$} is the ideal $$ \\text{Ann}_R(M) = \\text{Ann}(M) = \\{f \\in R \\mid fm = 0\\ \\forall m \\in M\\}. $$ \\end{enumerate}"}
+{"_id": "1459", "title": "algebra-definition-going-up-down", "text": "Let $\\varphi : R \\to S$ be a ring map. \\begin{enumerate} \\item We say a $\\varphi : R \\to S$ satisfies {\\it going up} if given primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$ and a prime $\\mathfrak q$ in $S$ lying over $\\mathfrak p$ there exists a prime $\\mathfrak q'$ of $S$ such that (a) $\\mathfrak q \\subset \\mathfrak q'$, and (b) $\\mathfrak q'$ lies over $\\mathfrak p'$. \\item We say a $\\varphi : R \\to S$ satisfies {\\it going down} if given primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$ and a prime $\\mathfrak q'$ in $S$ lying over $\\mathfrak p'$ there exists a prime $\\mathfrak q$ of $S$ such that (a) $\\mathfrak q \\subset \\mathfrak q'$, and (b) $\\mathfrak q$ lies over $\\mathfrak p$. \\end{enumerate}"}
+{"_id": "1460", "title": "algebra-definition-separable-field-extension", "text": "Let $k \\subset K$ be a field extension. \\begin{enumerate} \\item We say $K$ is {\\it separably generated over $k$} if there exists a transcendence basis $\\{x_i; i \\in I\\}$ of $K/k$ such that the extension $k(x_i; i\\in I) \\subset K$ is a separable algebraic extension. \\item We say $K$ is {\\it separable over $k$} if for every subextension $k \\subset K' \\subset K$ with $K'$ finitely generated over $k$, the extension $k \\subset K'$ is separably generated. \\end{enumerate}"}
+{"_id": "1461", "title": "algebra-definition-geometrically-reduced", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\\it geometrically reduced over $k$} if for every field extension $k \\subset K$ the $K$-algebra $K \\otimes_k S$ is reduced."}
+{"_id": "1462", "title": "algebra-definition-perfect", "text": "Let $k$ be a field. We say $k$ is {\\it perfect} if every field extension of $k$ is separable over $k$."}
+{"_id": "1463", "title": "algebra-definition-perfection", "text": "Let $k$ be a field. The field extension $k \\subset k'$ of Lemma \\ref{lemma-perfection} is called the {\\it perfect closure} of $k$. Notation $k \\subset k^{perf}$."}
+{"_id": "1464", "title": "algebra-definition-geometrically-irreducible", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\\it geometrically irreducible over $k$} if for every field extension $k \\subset k'$ the spectrum of $S \\otimes_k k'$ is irreducible\\footnote{An irreducible space is nonempty.}."}
+{"_id": "1465", "title": "algebra-definition-geometrically-connected", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\\it geometrically connected over $k$} if for every field extension $k \\subset k'$ the spectrum of $S \\otimes_k k'$ is connected."}
+{"_id": "1466", "title": "algebra-definition-geometrically-integral", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. We say $S$ is {\\it geometrically integral over $k$} if for every field extension $k \\subset k'$ the ring of $S \\otimes_k k'$ is a domain."}
+{"_id": "1467", "title": "algebra-definition-valuation-ring", "text": "Valuation rings. \\begin{enumerate} \\item Let $K$ be a field. Let $A$, $B$ be local rings contained in $K$. We say that $B$ {\\it dominates} $A$ if $A \\subset B$ and $\\mathfrak m_A = A \\cap \\mathfrak m_B$. \\item Let $A$ be a ring. We say $A$ is a {\\it valuation ring} if $A$ is a local domain and if $A$ is maximal for the relation of domination among local rings contained in the fraction field of $A$. \\item Let $A$ be a valuation ring with fraction field $K$. If $R \\subset K$ is a subring of $K$, then we say $A$ is {\\it centered} on $R$ if $R \\subset A$. \\end{enumerate}"}
+{"_id": "1468", "title": "algebra-definition-value-group", "text": "Let $A$ be a valuation ring. \\begin{enumerate} \\item The totally ordered abelian group $(\\Gamma, \\geq)$ of Lemma \\ref{lemma-valuation-group} is called the {\\it value group} of the valuation ring $A$. \\item The map $v : A - \\{0\\} \\to \\Gamma$ and also $v : K^* \\to \\Gamma$ is called the {\\it valuation} associated to $A$. \\item The valuation ring $A$ is called a {\\it discrete valuation ring} if $\\Gamma \\cong \\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "1469", "title": "algebra-definition-length", "text": "Let $R$ be a ring. For any $R$-module $M$ we define the {\\it length} of $M$ over $R$ by the formula $$ \\text{length}_R(M) = \\sup \\{ n \\mid \\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M, \\text{ }M_i \\not = M_{i + 1} \\}. $$"}
+{"_id": "1470", "title": "algebra-definition-simple-module", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ is {\\it simple} if $M \\not = 0$ and every submodule of $M$ is either equal to $M$ or to $0$."}
+{"_id": "1471", "title": "algebra-definition-artinian", "text": "A ring $R$ is {\\it Artinian} if it satisfies the descending chain condition for ideals."}
+{"_id": "1472", "title": "algebra-definition-essentially-finite-p-t", "text": "Let $R \\to S$ be a ring map. \\begin{enumerate} \\item We say that $R \\to S$ is {\\it essentially of finite type} if $S$ is the localization of an $R$-algebra of finite type. \\item We say that $R \\to S$ is {\\it essentially of finite presentation} if $S$ is the localization of an $R$-algebra of finite presentation. \\end{enumerate}"}
+{"_id": "1473", "title": "algebra-definition-proj", "text": "Let $S$ be a graded ring. We define $\\text{Proj}(S)$ to be the set of homogeneous prime ideals $\\mathfrak p$ of $S$ such that $S_{+} \\not \\subset \\mathfrak p$. The set $\\text{Proj}(S)$ is a subset of $\\Spec(S)$ and we endow it with the induced topology. The topological space $\\text{Proj}(S)$ is called the {\\it homogeneous spectrum} of the graded ring $S$."}
+{"_id": "1474", "title": "algebra-definition-numerical-polynomial", "text": "Let $A$ be an abelian group. We say that a function $f : n \\mapsto f(n) \\in A$ defined for all sufficient large integers $n$ is a {\\it numerical polynomial} if there exists $r \\geq 0$, elements $a_0, \\ldots, a_r\\in A$ such that $$ f(n) = \\sum\\nolimits_{i = 0}^r \\binom{n}{i} a_i $$ for all $n \\gg 0$."}
+{"_id": "1475", "title": "algebra-definition-ideal-definition", "text": "Let $(R, \\mathfrak m)$ be a local Noetherian ring. An ideal $I \\subset R$ such that $\\sqrt{I} = \\mathfrak m$ is called {\\it an ideal of definition of $R$}."}
+{"_id": "1476", "title": "algebra-definition-hilbert-polynomial", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. The {\\it Hilbert polynomial} of $M$ over $R$ is the element $P(t) \\in \\mathbf{Q}[t]$ such that $P(n) = \\varphi_M(n)$ for $n \\gg 0$."}
+{"_id": "1477", "title": "algebra-definition-d", "text": "Let $R$ be a local Noetherian ring and $M$ a finite $R$-module. We denote {\\it $d(M)$} the element of $\\{-\\infty, 0, 1, 2, \\ldots \\}$ defined as follows: \\begin{enumerate} \\item If $M = 0$ we set $d(M) = -\\infty$, \\item if $M \\not = 0$ then $d(M)$ is the degree of the numerical polynomial $\\chi_M$. \\end{enumerate}"}
+{"_id": "1478", "title": "algebra-definition-Krull", "text": "The {\\it Krull dimension} of the ring $R$ is the Krull dimension of the topological space $\\Spec(R)$, see Topology, Definition \\ref{topology-definition-Krull}. In other words it is the supremum of the integers $n\\geq 0$ such that there exists a chain of prime ideals of length $n$: $$ \\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n, \\quad \\mathfrak p_i \\not = \\mathfrak p_{i + 1}. $$"}
+{"_id": "1479", "title": "algebra-definition-height", "text": "The {\\it height} of a prime ideal $\\mathfrak p$ of a ring $R$ is the dimension of the local ring $R_{\\mathfrak p}$."}
+{"_id": "1480", "title": "algebra-definition-regular-local", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$. \\begin{enumerate} \\item A {\\it system of parameters of $R$} is a sequence of elements $x_1, \\ldots, x_d \\in \\mathfrak m$ which generates an ideal of definition of $R$, \\item if there exist $x_1, \\ldots, x_d \\in \\mathfrak m$ such that $\\mathfrak m = (x_1, \\ldots, x_d)$ then we call $R$ a {\\it regular local ring} and $x_1, \\ldots, x_d$ a {\\it regular system of parameters}. \\end{enumerate}"}
+{"_id": "1481", "title": "algebra-definition-associated", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. A prime $\\mathfrak p$ of $R$ is {\\it associated} to $M$ if there exists an element $m \\in M$ whose annihilator is $\\mathfrak p$. The set of all such primes is denoted $\\text{Ass}_R(M)$ or $\\text{Ass}(M)$."}
+{"_id": "1482", "title": "algebra-definition-symbolic-power", "text": "Let $R$ be a ring. Let $\\mathfrak p$ be a prime ideal. For $n \\geq 0$ the $n$th {\\it symbolic power} of $\\mathfrak p$ is the ideal $\\mathfrak p^{(n)} = \\Ker(R \\to R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p)$."}
+{"_id": "1483", "title": "algebra-definition-relative-assassin", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. The {\\it relative assassin of $N$ over $S/R$} is the set $$ \\text{Ass}_{S/R}(N) = \\{ \\mathfrak q \\subset S \\mid \\mathfrak q \\in \\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p)) \\text{ with }\\mathfrak p = R \\cap \\mathfrak q\\}. $$ This is the set named $A$ in Lemma \\ref{lemma-compare-relative-assassins}."}
+{"_id": "1484", "title": "algebra-definition-weakly-associated", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. A prime $\\mathfrak p$ of $R$ is {\\it weakly associated} to $M$ if there exists an element $m \\in M$ such that $\\mathfrak p$ is minimal among the prime ideals containing the annihilator $\\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}$. The set of all such primes is denoted $\\text{WeakAss}_R(M)$ or $\\text{WeakAss}(M)$."}
+{"_id": "1485", "title": "algebra-definition-embedded-primes", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item  The associated primes of $M$ which are not minimal among the associated primes of $M$ are called the {\\it embedded associated primes} of $M$. \\item The {\\it embedded primes of $R$} are the embedded associated primes of $R$ as an $R$-module. \\end{enumerate}"}
+{"_id": "1486", "title": "algebra-definition-regular-sequence", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements $f_1, \\ldots, f_r$ of $R$ is called an {\\it $M$-regular sequence} if the following conditions hold: \\begin{enumerate} \\item $f_i$ is a nonzerodivisor on $M/(f_1, \\ldots, f_{i - 1})M$ for each $i = 1, \\ldots, r$, and \\item the module $M/(f_1, \\ldots, f_r)M$ is not zero. \\end{enumerate} If $I$ is an ideal of $R$ and $f_1, \\ldots, f_r \\in I$ then we call $f_1, \\ldots, f_r$ a {\\it $M$-regular sequence in $I$}. If $M = R$, we call $f_1, \\ldots, f_r$ simply a {\\it regular sequence} (in $I$)."}
+{"_id": "1487", "title": "algebra-definition-quasi-regular-sequence", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements $f_1, \\ldots, f_c$ of $R$ is called {\\it $M$-quasi-regular} if (\\ref{equation-quasi-regular}) is an isomorphism. If $M = R$, we call $f_1, \\ldots, f_c$ simply a {\\it quasi-regular sequence}."}
+{"_id": "1488", "title": "algebra-definition-blow-up", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. \\begin{enumerate} \\item The {\\it blowup algebra}, or the {\\it Rees algebra}, associated to the pair $(R, I)$ is the graded $R$-algebra $$ \\text{Bl}_I(R) = \\bigoplus\\nolimits_{n \\geq 0} I^n = R \\oplus I \\oplus I^2 \\oplus \\ldots $$ where the summand $I^n$ is placed in degree $n$. \\item Let $a \\in I$ be an element. Denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra. Then the {\\it affine blowup algebra} $R[\\frac{I}{a}]$ is the algebra $(\\text{Bl}_I(R))_{(a^{(1)})}$ constructed in Section \\ref{section-proj}. \\end{enumerate}"}
+{"_id": "1489", "title": "algebra-definition-finite-free-resolution", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item A (left) {\\it resolution} $F_\\bullet \\to M$ of $M$ is an exact complex $$ \\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0 $$ of $R$-modules. \\item A {\\it resolution of $M$ by free $R$-modules} is a resolution $F_\\bullet \\to M$ where each $F_i$ is a free $R$-module. \\item A {\\it resolution of $M$ by finite free $R$-modules} is a resolution $F_\\bullet \\to M$ where each $F_i$ is a finite free $R$-module. \\end{enumerate}"}
+{"_id": "1490", "title": "algebra-definition-depth", "text": "Let $R$ be a ring, and $I \\subset R$ an ideal. Let $M$ be a finite $R$-module. The {\\it $I$-depth} of $M$, denoted $\\text{depth}_I(M)$, is defined as follows: \\begin{enumerate} \\item if $IM \\not = M$, then $\\text{depth}_I(M)$ is the supremum in $\\{0, 1, 2, \\ldots, \\infty\\}$ of the lengths of $M$-regular sequences in $I$, \\item if $IM = M$ we set $\\text{depth}_I(M) = \\infty$. \\end{enumerate} If $(R, \\mathfrak m)$ is local we call $\\text{depth}_{\\mathfrak m}(M)$ simply the {\\it depth} of $M$."}
+{"_id": "1491", "title": "algebra-definition-projective", "text": "Let $R$ be a ring. An $R$-module $P$ is {\\it projective} if and only if the functor $\\Hom_R(P, -) : \\text{Mod}_R \\to \\text{Mod}_R$ is an exact functor."}
+{"_id": "1492", "title": "algebra-definition-locally-free", "text": "Let $R$ be a ring and $M$ an $R$-module. \\begin{enumerate} \\item We say that $M$ is {\\it locally free} if we can cover $\\Spec(R)$ by standard opens $D(f_i)$, $i \\in I$ such that $M_{f_i}$ is a free $R_{f_i}$-module for all $i \\in I$. \\item We say that $M$ is {\\it finite locally free} if we can choose the covering such that each $M_{f_i}$ is finite free. \\item We say that $M$ is {\\it finite locally free of rank $r$} if we can choose the covering such that each $M_{f_i}$ is isomorphic to $R_{f_i}^{\\oplus r}$. \\end{enumerate}"}
+{"_id": "1493", "title": "algebra-definition-universally-injective", "text": "Let $f: M \\to N$ be a map of $R$-modules.  Then $f$ is called {\\it universally injective} if for every $R$-module $Q$, the map $f \\otimes_R \\text{id}_Q: M \\otimes_R Q \\to N \\otimes_R Q$ is injective.  A sequence $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ of $R$-modules is called {\\it universally exact} if it is exact and $M_1 \\to M_2$ is universally injective."}
+{"_id": "1494", "title": "algebra-definition-devissage", "text": "Let $M$ be an $R$-module.  A {\\it direct sum d\\'evissage} of $M$ is a family of submodules $(M_{\\alpha})_{\\alpha \\in S}$, indexed by an ordinal $S$ and increasing (with respect to inclusion), such that: \\begin{enumerate} \\item[(0)] $M_0 = 0$; \\item[(1)] $M = \\bigcup_{\\alpha} M_{\\alpha}$; \\item[(2)] if $\\alpha \\in S$ is a limit ordinal, then $M_{\\alpha} = \\bigcup_{\\beta < \\alpha} M_{\\beta}$; \\item[(3)] if $\\alpha + 1 \\in S$, then $M_{\\alpha}$ is a direct summand of $M_{\\alpha + 1}$. \\end{enumerate} If moreover \\begin{enumerate} \\item[(4)] $M_{\\alpha + 1}/M_{\\alpha}$ is countably generated for $\\alpha + 1 \\in S$, \\end{enumerate} then $(M_{\\alpha})_{\\alpha \\in S}$ is called a {\\it Kaplansky d\\'evissage} of $M$."}
+{"_id": "1495", "title": "algebra-definition-ML-system", "text": "Let $(A_i, \\varphi_{ji})$ be a directed inverse system of sets over $I$.  Then we say  $(A_i, \\varphi_{ji})$ is {\\it Mittag-Leffler} if for each $i \\in I$, the family $\\varphi_{ji}(A_j) \\subset A_i$ for $j \\geq i$ stabilizes.  Explicitly, this means that for each $i \\in I$, there exists $j \\geq i$ such that for $k \\geq j$ we have $\\varphi_{ki}(A_k) = \\varphi_{ji}( A_j)$.  If $(A_i, \\varphi_{ji})$ is a directed inverse system of modules over a ring $R$, we say that it is Mittag-Leffler if the underlying inverse system of sets is Mittag-Leffler."}
+{"_id": "1496", "title": "algebra-definition-ML-inductive-system", "text": "Let $(M_i, f_{ij})$ be a directed system of $R$-modules.  We say that $(M_i, f_{ij})$ is a {\\it Mittag-Leffler directed system of modules} if each $M_i$ is an $R$-module of finite presentation and if for every $R$-module $N$, the inverse system $$ (\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N)) $$ is Mittag-Leffler."}
+{"_id": "1497", "title": "algebra-definition-domination", "text": "Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules. Then we say $g$ {\\it dominates} $f$ if for any $R$-module $Q$, we have $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R \\text{id}_Q)$."}
+{"_id": "1498", "title": "algebra-definition-mittag-leffler-module", "text": "Let $M$ be an $R$-module.  We say that $M$ is {\\it Mittag-Leffler} if the equivalent conditions of Proposition \\ref{proposition-ML-characterization} hold."}
+{"_id": "1499", "title": "algebra-definition-coherent", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item We say $M$ is a {\\it coherent module} if it is finitely generated and every finitely generated submodule of $M$ is finitely presented over $R$. \\item We say $R$ is a {\\it coherent ring} if it is coherent as a module over itself. \\end{enumerate}"}
+{"_id": "1500", "title": "algebra-definition-complete", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is {\\it $I$-adically complete} if the map $$ M \\longrightarrow M^\\wedge = \\lim_n M/I^nM $$ is an isomorphism\\footnote{This includes the condition that $\\bigcap I^nM = (0)$.}. We say $R$ is {\\it $I$-adically complete} if $R$ is $I$-adically complete as an $R$-module."}
+{"_id": "1501", "title": "algebra-definition-rank", "text": "Let $R$ be a ring. Suppose that $\\varphi : R^m \\to R^n$ is a map of finite free modules. \\begin{enumerate} \\item The {\\it rank} of $\\varphi$ is the maximal $r$ such that $\\wedge^r \\varphi : \\wedge^r R^m \\to \\wedge^r R^n$ is nonzero. \\item We let $I(\\varphi) \\subset R$ be the ideal generated by the $r \\times r$ minors of the matrix of $\\varphi$, where $r$ is the rank as defined above. \\end{enumerate}"}
+{"_id": "1502", "title": "algebra-definition-CM", "text": "Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. We say $M$ is {\\it Cohen-Macaulay} if $\\dim(\\text{Supp}(M)) = \\text{depth}(M)$."}
+{"_id": "1503", "title": "algebra-definition-maximal-CM", "text": "Let $R$ be a Noetherian local ring. A finite module $M$ over $R$ is called a {\\it maximal Cohen-Macaulay} module if $\\text{depth}(M) = \\dim(R)$."}
+{"_id": "1504", "title": "algebra-definition-module-CM", "text": "Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. We say $M$ is {\\it Cohen-Macaulay} if $M_\\mathfrak p$ is a Cohen-Macaulay module over $R_\\mathfrak p$ for all primes $\\mathfrak p$ of $R$."}
+{"_id": "1505", "title": "algebra-definition-local-ring-CM", "text": "A Noetherian local ring $R$ is called {\\it Cohen-Macaulay} if it is Cohen-Macaulay as a module over itself."}
+{"_id": "1506", "title": "algebra-definition-ring-CM", "text": "A Noetherian ring $R$ is called {\\it Cohen-Macaulay} if all its local rings are Cohen-Macaulay."}
+{"_id": "1507", "title": "algebra-definition-catenary", "text": "A ring $R$ is said to be {\\it catenary} if for any pair of prime ideals $\\mathfrak p \\subset \\mathfrak q$, all maximal chains of primes $\\mathfrak p = \\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_e = \\mathfrak q$ have the same (finite) length."}
+{"_id": "1508", "title": "algebra-definition-universally-catenary", "text": "A Noetherian ring $R$ is said to be {\\it universally catenary} if every $R$ algebra of finite type is catenary."}
+{"_id": "1509", "title": "algebra-definition-pure-ideal", "text": "Let $R$ be a ring. We say that $I \\subset R$ is {\\it pure} if the quotient ring $R/I$ is flat over $R$."}
+{"_id": "1510", "title": "algebra-definition-finite-proj-dim", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ has {\\it finite projective dimension} if it has a finite length resolution by projective $R$-modules. The minimal length of such a resolution is called the {\\it projective dimension} of $M$."}
+{"_id": "1511", "title": "algebra-definition-finite-gl-dim", "text": "Let $R$ be a ring. The ring $R$ is said to have {\\it finite global dimension} if there exists an integer $n$ such that every $R$-module has a resolution by projective $R$-modules of length at most $n$. The minimal such $n$ is then called the {\\it global dimension} of $R$."}
+{"_id": "1512", "title": "algebra-definition-regular", "text": "A Noetherian ring $R$ is said to be {\\it regular} if all the localizations $R_{\\mathfrak p}$ at primes are regular local rings."}
+{"_id": "1513", "title": "algebra-definition-fibre", "text": "Suppose that $R \\to S$ is a ring map. Let $\\mathfrak q \\subset S$ be a prime lying over the prime $\\mathfrak p$ of $R$. The {\\it local ring of the fibre at $\\mathfrak q$} is the local ring $$ S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q} = (S/\\mathfrak pS)_{\\mathfrak q} = (S \\otimes_R \\kappa(\\mathfrak p))_{\\mathfrak q} $$"}
+{"_id": "1514", "title": "algebra-definition-uniformizer", "text": "Let $A$ be a discrete valuation ring. A {\\it uniformizer} is an element $\\pi \\in A$ which generates the maximal ideal of $A$."}
+{"_id": "1515", "title": "algebra-definition-irreducible-prime-element", "text": "Let $R$ be a domain. \\begin{enumerate} \\item Elements $x, y \\in R$ are called {\\it associates} if there exists a unit $u \\in R^*$ such that $x = uy$. \\item An element $x \\in R$ is called {\\it irreducible} if it is nonzero, not a unit and whenever $x = yz$, $y, z \\in R$, then $y$ is either a unit or an associate of $x$. \\item An element $x \\in R$ is called {\\it prime} if the ideal generated by $x$ is a prime ideal. \\end{enumerate}"}
+{"_id": "1516", "title": "algebra-definition-UFD", "text": "A {\\it unique factorization domain}, abbreviated {\\it UFD}, is a domain $R$ such that if $x \\in R$ is a nonzero, nonunit, then $x$ has a factorization into irreducibles, and if $$ x = a_1 \\ldots a_m = b_1 \\ldots b_n $$ are factorizations into irreducibles then $n = m$ and there exists a permutation $\\sigma : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$ such that $a_i$ and $b_{\\sigma(i)}$ are associates."}
+{"_id": "1517", "title": "algebra-definition-PID", "text": "A {\\it principal ideal domain}, abbreviated {\\it PID}, is a domain $R$ such that every ideal is a principal ideal."}
+{"_id": "1518", "title": "algebra-definition-dedekind-domain", "text": "A {\\it Dedekind domain} is a domain $R$ such that every nonzero ideal $I \\subset R$ can be written as a product $$ I = \\mathfrak p_1 \\ldots \\mathfrak p_r $$ of nonzero prime ideals uniquely up to permutation of the $\\mathfrak p_i$."}
+{"_id": "1519", "title": "algebra-definition-ord", "text": "Suppose that $K$ is a field, and $R \\subset K$ is a local\\footnote{We could also define this when $R$ is only semi-local but this is probably never really what you want!} Noetherian subring of dimension $1$ with fraction field $K$. In this case we define the {\\it order of vanishing along $R$} $$ \\text{ord}_R : K^* \\longrightarrow \\mathbf{Z} $$ by the rule $$ \\text{ord}_R(x) = \\text{length}_R(R/(x)) $$ if $x \\in R$ and we set $\\text{ord}_R(x/y) = \\text{ord}_R(x) - \\text{ord}_R(y)$ for $x, y \\in R$ both nonzero."}
+{"_id": "1520", "title": "algebra-definition-lattice", "text": "Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. A {\\it lattice in $V$} is a finite $R$-submodule $M \\subset V$ such that $V = K \\otimes_R M$."}
+{"_id": "1521", "title": "algebra-definition-distance", "text": "Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. Let $M$, $M'$ be two lattices in $V$. The {\\it distance between $M$ and $M'$} is the integer $$ d(M, M') = \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M') $$ of Lemma \\ref{lemma-compare-lattices} part (5)."}
+{"_id": "1522", "title": "algebra-definition-quasi-finite", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$ be a prime. \\begin{enumerate} \\item If the equivalent conditions of Lemma \\ref{lemma-isolated-point-fibre} are satisfied then we say $R \\to S$ is {\\it quasi-finite at $\\mathfrak q$}. \\item We say a ring map $A \\to B$ is {\\it quasi-finite} if it is of finite type and quasi-finite at all primes of $B$. \\end{enumerate}"}
+{"_id": "1523", "title": "algebra-definition-strongly-transcendental", "text": "Given an inclusion of rings $R \\subset S$ and an element $x \\in S$ we say that $x$ is {\\it strongly transcendental over $R$} if whenever $u(a_0 + a_1 x + \\ldots + a_k x^k) = 0$ with $u \\in S$ and $a_i \\in R$, then we have $ua_i = 0$ for all $i$."}
+{"_id": "1524", "title": "algebra-definition-relative-dimension", "text": "Suppose that $R \\to S$ is of finite type, and let $\\mathfrak q \\subset S$ be a prime lying over a prime $\\mathfrak p$ of $R$. We define the {\\it relative dimension of $S/R$ at $\\mathfrak q$}, denoted $\\dim_{\\mathfrak q}(S/R)$, to be the dimension of $\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$ at the point corresponding to $\\mathfrak q$. We let $\\dim(S/R)$ be the supremum of $\\dim_{\\mathfrak q}(S/R)$ over all $\\mathfrak q$. This is called the {\\it relative dimension of} $S/R$."}
+{"_id": "1525", "title": "algebra-definition-derivation", "text": "Let $\\varphi : R \\to S$ be a ring map and let $M$ be an $S$-module. A {\\it derivation}, or more precisely an {\\it $R$-derivation} into $M$ is a map $D : S \\to M$ which is additive, annihilates elements of $\\varphi(R)$, and satisfies the {\\it Leibniz rule}: $D(ab) = aD(b) + bD(a)$."}
+{"_id": "1526", "title": "algebra-definition-differentials", "text": "The pair $(\\Omega_{S/R}, \\text{d})$ is called the {\\it module of K\\\"ahler differentials} or the {\\it module of differentials} of $S$ over $R$."}
+{"_id": "1527", "title": "algebra-definition-differential-operators", "text": "Let $R \\to S$ be a ring map. Let $M$, $N$ be $S$-modules. Let $k \\geq 0$ be an integer. We inductively define a {\\it differential operator $D : M \\to N$ of order $k$} to be an $R$-linear map such that for all $g \\in S$ the map $m \\mapsto D(gm) - gD(m)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $S$-linear map."}
+{"_id": "1528", "title": "algebra-definition-module-principal-parts", "text": "Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. The module $P^k_{S/R}(M)$ constructed in Lemma \\ref{lemma-module-principal-parts} is called the {\\it module of principal parts of order $k$} of $M$."}
+{"_id": "1529", "title": "algebra-definition-naive-cotangent-complex", "text": "Let $R \\to S$ be a ring map. The {\\it naive cotangent complex} $\\NL_{S/R}$ is the chain complex (\\ref{equation-naive-cotangent-complex}) $$ \\NL_{S/R} = \\left(I/I^2 \\longrightarrow \\Omega_{R[S]/R} \\otimes_{R[S]} S\\right) $$ with $I/I^2$ placed in (homological) degree $1$ and $\\Omega_{R[S]/R} \\otimes_{R[S]} S$ placed in degree $0$. We will denote $H_1(L_{S/R}) = H_1(\\NL_{S/R})$\\footnote{This module is sometimes denoted $\\Gamma_{S/R}$ in the literature.} the homology in degree $1$."}
+{"_id": "1530", "title": "algebra-definition-lci-field", "text": "Let $k$ be a field. Let $S$ be a finite type $k$-algebra. \\begin{enumerate} \\item We say that $S$ is a {\\it global complete intersection over $k$} if there exists a presentation $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ such that $\\dim(S) = n - c$. \\item We say that $S$ is a {\\it local complete intersection over $k$} if there exists a covering $\\Spec(S) = \\bigcup D(g_i)$ such that each of the rings $S_{g_i}$ is a global complete intersection over $k$. \\end{enumerate} We will also use the convention that the zero ring is a global complete intersection over $k$."}
+{"_id": "1531", "title": "algebra-definition-lci-local-ring", "text": "Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type over $k$. We say $S$ is a {\\it complete intersection (over $k$)} if there exists a local $k$-algebra $R$ and elements $f_1, \\ldots, f_c \\in \\mathfrak m_R$ such that \\begin{enumerate} \\item $R$ is essentially of finite type over $k$, \\item $R$ is a regular local ring, \\item $f_1, \\ldots, f_c$ form a regular sequence in $R$, and \\item $S \\cong R/(f_1, \\ldots, f_c)$ as $k$-algebras. \\end{enumerate}"}
+{"_id": "1532", "title": "algebra-definition-lci", "text": "A ring map $R \\to S$ is called {\\it syntomic}, or we say $S$ is a {\\it flat local complete intersection over $R$} if it is flat, of finite presentation, and if all of its fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are local complete intersections, see Definition \\ref{definition-lci-field}."}
+{"_id": "1533", "title": "algebra-definition-relative-global-complete-intersection", "text": "Let $R \\to S$ be a ring map. We say that $R \\to S$ is a {\\it relative global complete intersection} if there exists a presentation $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ and every nonempty fibre of $\\Spec(S) \\to \\Spec(R)$ has dimension $n - c$. We will say ``let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative global complete intersection'' to indicate this situation."}
+{"_id": "1534", "title": "algebra-definition-smooth", "text": "A ring map $R \\to S$ is {\\it smooth} if it is of finite presentation and the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a finite projective $S$-module placed in degree $0$."}
+{"_id": "1535", "title": "algebra-definition-standard-smooth", "text": "Let $R$ be a ring. Given integers $n \\geq c \\geq 0$ and $f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$ we say $$ S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) $$ is a {\\it standard smooth algebra over $R$} if the polynomial $$ g = \\det \\left( \\begin{matrix} \\partial f_1/\\partial x_1 & \\partial f_2/\\partial x_1 & \\ldots & \\partial f_c/\\partial x_1 \\\\ \\partial f_1/\\partial x_2 & \\partial f_2/\\partial x_2 & \\ldots & \\partial f_c/\\partial x_2 \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ \\partial f_1/\\partial x_c & \\partial f_2/\\partial x_c & \\ldots & \\partial f_c/\\partial x_c \\end{matrix} \\right) $$ maps to an invertible element in $S$."}
+{"_id": "1536", "title": "algebra-definition-smooth-at-prime", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$. We say $R \\to S$ is {\\it smooth at $\\mathfrak q$} if there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is smooth."}
+{"_id": "1537", "title": "algebra-definition-formally-smooth", "text": "Let $R \\to S$ be a ring map. We say $S$ is {\\it formally smooth over $R$} if for every commutative solid diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/I \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ where $I \\subset A$ is an ideal of square zero, a dotted arrow exists which makes the diagram commute."}
+{"_id": "1538", "title": "algebra-definition-small-extension", "text": "Let $\\varphi : B' \\to B$ be a ring map. We say $\\varphi$ is a {\\it small extension} if $B'$ and $B$ are local Artinian rings, $\\varphi$ is surjective and $I = \\Ker(\\varphi)$ has length $1$ as a $B'$-module."}
+{"_id": "1539", "title": "algebra-definition-etale", "text": "Let $R \\to S$ be a ring map. We say $R \\to S$ is {\\it \\'etale} if it is of finite presentation and the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to zero. Given a prime $\\mathfrak q$ of $S$ we say that $R \\to S$ is {\\it \\'etale at $\\mathfrak q$} if there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is \\'etale."}
+{"_id": "1540", "title": "algebra-definition-standard-etale", "text": "Let $R$ be a ring. Let $g , f  \\in R[x]$. Assume that $f$ is monic and the derivative $f'$ is invertible in the localization $R[x]_g/(f)$. In this case the ring map $R \\to R[x]_g/(f)$ is said to be {\\it standard \\'etale}."}
+{"_id": "1541", "title": "algebra-definition-formally-unramified", "text": "Let $R \\to S$ be a ring map. We say $S$ is {\\it formally unramified over $R$} if for every commutative solid diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/I \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ where $I \\subset A$ is an ideal of square zero, there exists at most one dotted arrow making the diagram commute."}
+{"_id": "1542", "title": "algebra-definition-universal-thickening", "text": "Let $R \\to S$ be a formally unramified ring map. \\begin{enumerate} \\item The {\\it universal first order thickening} of $S$ over $R$ is the surjection of $R$-algebras $S' \\to S$ of Lemma \\ref{lemma-universal-thickening}. \\item The {\\it conormal module} of $R \\to S$ is the kernel $I$ of the universal first order thickening $S' \\to S$, seen as an $S$-module. \\end{enumerate} We often denote the conormal module {\\it $C_{S/R}$} in this situation."}
+{"_id": "1543", "title": "algebra-definition-formally-etale", "text": "Let $R \\to S$ be a ring map. We say $S$ is {\\it formally \\'etale over $R$} if for every commutative solid diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/I \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ where $I \\subset A$ is an ideal of square zero, there exists a unique dotted arrow making the diagram commute."}
+{"_id": "1544", "title": "algebra-definition-unramified", "text": "Let $R \\to S$ be a ring map. \\begin{enumerate} \\item We say $R \\to S$ is {\\it unramified} if $R \\to S$ is of finite type and $\\Omega_{S/R} = 0$. \\item We say $R \\to S$ is {\\it G-unramified} if $R \\to S$ is of finite presentation and $\\Omega_{S/R} = 0$. \\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is {\\it unramified at $\\mathfrak q$} if there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is unramified. \\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is {\\it G-unramified at $\\mathfrak q$} if there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is G-unramified. \\end{enumerate}"}
+{"_id": "1545", "title": "algebra-definition-henselian", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. \\begin{enumerate} \\item We say $R$ is {\\it henselian} if for every monic $f \\in R[T]$ and every root $a_0 \\in \\kappa$ of $\\overline{f}$ such that $\\overline{f'}(a_0) \\not = 0$ there exists an $a \\in R$ such that $f(a) = 0$ and $a_0 = \\overline{a}$. \\item We say $R$ is {\\it strictly henselian} if $R$ is henselian and its residue field is separably algebraically closed. \\end{enumerate}"}
+{"_id": "1546", "title": "algebra-definition-henselization", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. \\begin{enumerate} \\item The local ring map $R \\to R^h$ constructed in Lemma \\ref{lemma-henselization} is called the {\\it henselization} of $R$. \\item Given a separable algebraic closure $\\kappa \\subset \\kappa^{sep}$ the local ring map $R \\to R^{sh}$ constructed in Lemma \\ref{lemma-strict-henselization} is called the {\\it strict henselization of $R$ with respect to $\\kappa \\subset \\kappa^{sep}$}. \\item A local ring map $R \\to R^{sh}$ is called a {\\it strict henselization} of $R$ if it is isomorphic to one of the local ring maps constructed in Lemma \\ref{lemma-strict-henselization} \\end{enumerate}"}
+{"_id": "1547", "title": "algebra-definition-conditions", "text": "Let $R$ be a Noetherian ring. Let $k \\geq 0$ be an integer. \\begin{enumerate} \\item We say $R$ has property {\\it $(R_k)$} if for every prime $\\mathfrak p$ of height $\\leq k$ the local ring $R_{\\mathfrak p}$ is regular. We also say that $R$ is {\\it regular in codimension $\\leq k$}. \\item We say $R$ has property {\\it $(S_k)$} if for every prime $\\mathfrak p$ the local ring $R_{\\mathfrak p}$ has depth at least $\\min\\{k, \\dim(R_{\\mathfrak p})\\}$. \\item Let $M$ be a finite $R$-module. We say $M$ has property $(S_k)$ if for every prime $\\mathfrak p$ the module $M_{\\mathfrak p}$ has depth at least $\\min\\{k, \\dim(\\text{Supp}(M_{\\mathfrak p}))\\}$. \\end{enumerate}"}
+{"_id": "1548", "title": "algebra-definition-complete-local-ring", "text": "Let $(R, \\mathfrak m)$ be a local ring. We say $R$ is a {\\it complete local ring} if the canonical map $$ R \\longrightarrow \\lim_n R/\\mathfrak m^n $$ to the completion of $R$ with respect to $\\mathfrak m$ is an isomorphism\\footnote{This includes the condition that $\\bigcap \\mathfrak m^n = (0)$; in some texts this may be indicated by saying that $R$ is complete and separated. Warning: It can happen that the completion $\\lim_n R/\\mathfrak m^n$ of a local ring is non-complete, see Examples, Lemma \\ref{examples-lemma-noncomplete-completion}. This does not happen when $\\mathfrak m$ is finitely generated, see Lemma \\ref{lemma-hathat-finitely-generated} in which case the completion is Noetherian, see Lemma \\ref{lemma-completion-Noetherian}.}."}
+{"_id": "1549", "title": "algebra-definition-coefficient-ring", "text": "Let $(R, \\mathfrak m)$ be a complete local ring. A subring $\\Lambda \\subset R$ is called a {\\it coefficient ring} if the following conditions hold: \\begin{enumerate} \\item $\\Lambda$ is a complete local ring with maximal ideal $\\Lambda \\cap \\mathfrak m$, \\item the residue field of $\\Lambda$ maps isomorphically to the residue field of $R$, and \\item $\\Lambda \\cap \\mathfrak m = p\\Lambda$, where $p$ is the characteristic of the residue field of $R$. \\end{enumerate}"}
+{"_id": "1550", "title": "algebra-definition-cohen-ring", "text": "A {\\it Cohen ring} is a complete discrete valuation ring with uniformizer $p$ a prime number."}
+{"_id": "1551", "title": "algebra-definition-N", "text": "\\begin{reference} \\cite[Chapter 0, Definition 23.1.1]{EGA} \\end{reference} Let $R$ be a domain with field of fractions $K$. \\begin{enumerate} \\item We say $R$ is {\\it N-1} if the integral closure of $R$ in $K$ is a finite $R$-module. \\item We say $R$ is {\\it N-2} or {\\it Japanese} if for any finite extension $K \\subset L$ of fields the integral closure of $R$ in $L$ is finite over $R$. \\end{enumerate}"}
+{"_id": "1552", "title": "algebra-definition-nagata", "text": "Let $R$ be a ring. \\begin{enumerate} \\item We say $R$ is {\\it universally Japanese} if for any finite type ring map $R \\to S$ with $S$ a domain we have that $S$ is N-2 (i.e., Japanese). \\item We say that $R$ is a {\\it Nagata ring} if $R$ is Noetherian and for every prime ideal $\\mathfrak p$ the ring $R/\\mathfrak p$ is N-2. \\end{enumerate}"}
+{"_id": "1553", "title": "algebra-definition-analytically-unramified", "text": "Let $(R, \\mathfrak m)$ be a Noetherian local ring. We say $R$ is {\\it analytically unramified} if its completion $R^\\wedge = \\lim_n R/\\mathfrak m^n$ is reduced. A prime ideal $\\mathfrak p \\subset R$ is said to be {\\it analytically unramified} if $R/\\mathfrak p$ is analytically unramified."}
+{"_id": "1554", "title": "algebra-definition-geometrically-normal", "text": "Let $k$ be a field. A $k$-algebra $R$ is called {\\it geometrically normal} over $k$ if the equivalent conditions of Lemma \\ref{lemma-geometrically-normal} hold."}
+{"_id": "1555", "title": "algebra-definition-geometrically-regular", "text": "Let $k$ be a field. Let $R$ be a Noetherian $k$-algebra. The $k$-algebra $R$ is called {\\it geometrically regular} over $k$ if the equivalent conditions of Lemma \\ref{lemma-geometrically-regular} hold."}
+{"_id": "1643", "title": "moduli-curves-definition-deligne-mumford-smooth", "text": "\\begin{reference} \\cite{DM} \\end{reference} We denote $\\mathcal{M}$ and we name it the {\\it moduli stack of smooth proper curves} the algebraic stack $\\Curvesstack^{smooth, h0}$ parametrizing families of curves introduced in Lemma \\ref{lemma-smooth-curves-h0}. For $g \\geq 0$ we denote $\\mathcal{M}_g$ and we name it the {\\it moduli stack of smooth proper curves of genus $g$} the algebraic stack introduced in Lemma \\ref{lemma-smooth-one-piece-per-genus}."}
+{"_id": "1644", "title": "moduli-curves-definition-relative-dualizing-sheaf", "text": "Let $f : X \\to S$ be a family of curves with Cohen-Macaulay fibres equidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}). Then the $\\mathcal{O}_X$-module $$ \\omega_{X/S} = H^{-1}(\\omega_{X/S}^\\bullet) $$ studied in Lemma \\ref{lemma-CM-dualizing} is called the {\\it relative dualizing sheaf} of $f$."}
+{"_id": "1645", "title": "moduli-curves-definition-prestable", "text": "Let $f : X \\to S$ be a family of curves. We say $f$ is a {\\it prestable family of curves} if \\begin{enumerate} \\item $f$ is at-worst-nodal of relative dimension $1$, and \\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after any base change\\footnote{In fact, it suffices to require $f_*\\mathcal{O}_X = \\mathcal{O}_S$ because the Stein factorization of $f$ is \\'etale in this case, see More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-stein-factorization-etale}. The condition may also be replaced by asking the geometric fibres to be connected, see Lemma \\ref{lemma-geomredcon-in-h0-1}.}. \\end{enumerate}"}
+{"_id": "1646", "title": "moduli-curves-definition-semistable", "text": "Let $f : X \\to S$ be a family of curves. We say $f$ is a {\\it semistable family of curves} if \\begin{enumerate} \\item $X \\to S$ is a prestable family of curves, and \\item $X_s$ has genus $\\geq 1$ and does not have a rational tail for all $s \\in S$. \\end{enumerate}"}
+{"_id": "1647", "title": "moduli-curves-definition-stable", "text": "Let $f : X \\to S$ be a family of curves. We say $f$ is a {\\it stable family of curves} if \\begin{enumerate} \\item $X \\to S$ is a prestable family of curves, and \\item $X_s$ has genus $\\geq 2$ and does not have a rational tails or bridges for all $s \\in S$. \\end{enumerate}"}
+{"_id": "1648", "title": "moduli-curves-definition-deligne-mumford", "text": "\\begin{reference} \\cite{DM} \\end{reference} We denote $\\overline{\\mathcal{M}}$ and we name the {\\it moduli stack of stable curves} the algebraic stack $\\Curvesstack^{stable}$ parametrizing stable families of curves introduced in Lemma \\ref{lemma-stable-curves}. For $g \\geq 2$ we denote $\\overline{\\mathcal{M}}_g$ and we name the {\\it moduli stack of stable curves of genus $g$} the algebraic stack introduced in Lemma \\ref{lemma-stable-one-piece-per-genus}."}
+{"_id": "1696", "title": "dpa-definition-divided-powers", "text": "Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps $\\gamma_n : I \\to I$, $n > 0$ is called a {\\it divided power structure} on $I$ if for all $n \\geq 0$, $m > 0$, $x, y \\in I$, and $a \\in A$ we have \\begin{enumerate} \\item $\\gamma_1(x) = x$, we also set $\\gamma_0(x) = 1$, \\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$, \\item $\\gamma_n(ax) = a^n \\gamma_n(x)$, \\item $\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$, \\item $\\gamma_n(\\gamma_m(x)) = \\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$. \\end{enumerate}"}
+{"_id": "1697", "title": "dpa-definition-divided-power-ring", "text": "A {\\it divided power ring} is a triple $(A, I, \\gamma)$ where $A$ is a ring, $I \\subset A$ is an ideal, and $\\gamma = (\\gamma_n)_{n \\geq 1}$ is a divided power structure on $I$. A {\\it homomorphism of divided power rings} $\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a ring homomorphism $\\varphi : A \\to B$ such that $\\varphi(I) \\subset J$ and such that $\\delta_n(\\varphi(x)) = \\varphi(\\gamma_n(x))$ for all $x \\in I$ and $n \\geq 1$."}
+{"_id": "1698", "title": "dpa-definition-extends", "text": "Given a divided power ring $(A, I, \\gamma)$ and a ring map $A \\to B$ we say $\\gamma$ {\\it extends} to $B$ if there exists a divided power structure $\\bar \\gamma$ on $IB$ such that $(A, I, \\gamma) \\to (B, IB, \\bar\\gamma)$ is a homomorphism of divided power rings."}
+{"_id": "1699", "title": "dpa-definition-divided-powers-graded", "text": "Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a graded $R$-algebra which is strictly graded commutative. A collection of maps $\\gamma_n : A_{even, +} \\to A_{even, +}$ defined for all $n > 0$ is called a {\\it divided power structure} on $A$ if we have \\begin{enumerate} \\item $\\gamma_n(x) \\in A_{2nd}$ if $x \\in A_{2d}$, \\item $\\gamma_1(x) = x$ for any $x$, we also set $\\gamma_0(x) = 1$, \\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$, \\item $\\gamma_n(xy) = x^n \\gamma_n(y)$ for all $x \\in A_{even}$ and $y \\in A_{even, +}$, \\item $\\gamma_n(xy) = 0$ if $x, y \\in A_{odd}$ homogeneous and $n > 1$ \\item if $x, y \\in A_{even, +}$ then $\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$, \\item $\\gamma_n(\\gamma_m(x)) = \\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$ for $x \\in A_{even, +}$. \\end{enumerate}"}
+{"_id": "1700", "title": "dpa-definition-divided-powers-dga", "text": "Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a differential graded $R$-algebra which is strictly graded commutative. A divided power structure $\\gamma$ on $A$ is {\\it compatible with the differential graded structure} if $\\text{d}(\\gamma_n(x)) = \\text{d}(x) \\gamma_{n - 1}(x)$ for all $x \\in A_{even, +}$."}
+{"_id": "1701", "title": "dpa-definition-lci", "text": "Let $A$ be a Noetherian ring. \\begin{enumerate} \\item If $A$ is local, then we say $A$ is a {\\it complete intersection} if its completion is a complete intersection in the sense above. \\item In general we say $A$ is a {\\it local complete intersection} if all of its local rings are complete intersections. \\end{enumerate}"}
+{"_id": "1967", "title": "derived-definition-triangle", "text": "Let $\\mathcal{D}$ be an additive category. Let $[n] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[n]$ be a collection of additive functors indexed by $n \\in \\mathbf{Z}$ such that $[n] \\circ [m] = [n + m]$  and $[0] = \\text{id}$ (equality as functors). In this situation we define a {\\it triangle} to be a sextuple $(X, Y, Z, f, g, h)$ where $X, Y, Z \\in \\Ob(\\mathcal{D})$ and $f : X \\to Y$, $g : Y \\to Z$ and $h : Z \\to X[1]$ are morphisms of $\\mathcal{D}$. A {\\it morphism of triangles} $(X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')$ is given by morphisms $a : X \\to X'$, $b : Y \\to Y'$ and $c : Z \\to Z'$ of $\\mathcal{D}$ such that $b \\circ f = f' \\circ a$, $c  \\circ g = g' \\circ b$ and $a[1] \\circ h = h' \\circ c$."}
+{"_id": "1968", "title": "derived-definition-triangulated-category", "text": "A {\\it triangulated category} consists of a triple $(\\mathcal{D}, \\{[n]\\}_{n\\in \\mathbf{Z}}, \\mathcal{T})$ where \\begin{enumerate} \\item $\\mathcal{D}$ is an additive category, \\item $[n] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[n]$ is a collection of additive functors indexed by $n \\in \\mathbf{Z}$ such that $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}$ (equality as functors), and \\item $\\mathcal{T}$ is a set of triangles called the {\\it distinguished triangles} \\end{enumerate} subject to the following conditions \\begin{enumerate} \\item[TR1] Any triangle isomorphic to a distinguished triangle is a distinguished triangle. Any triangle of the form $(X, X, 0, \\text{id}, 0, 0)$ is distinguished. For any morphism $f : X \\to Y$ of $\\mathcal{D}$ there exists a distinguished triangle of the form $(X, Y, Z, f, g, h)$. \\item[TR2] The triangle $(X, Y, Z, f, g, h)$ is distinguished if and only if the triangle $(Y, Z, X[1], g, h, -f[1])$ is. \\item[TR3] Given a solid diagram $$ \\xymatrix{ X \\ar[r]^f \\ar[d]^a & Y \\ar[r]^g \\ar[d]^b & Z \\ar[r]^h \\ar@{-->}[d] & X[1] \\ar[d]^{a[1]} \\\\ X' \\ar[r]^{f'} & Y' \\ar[r]^{g'} & Z' \\ar[r]^{h'} & X'[1] } $$ whose rows are distinguished triangles and which satisfies $b \\circ f = f' \\circ a$, there exists a morphism $c : Z \\to Z'$ such that $(a, b, c)$ is a morphism of triangles. \\item[TR4] Given objects $X$, $Y$, $Z$ of $\\mathcal{D}$, and morphisms $f : X \\to Y$, $g : Y \\to Z$, and distinguished triangles $(X, Y, Q_1, f, p_1, d_1)$, $(X, Z, Q_2, g \\circ f, p_2, d_2)$, and $(Y, Z, Q_3, g, p_3, d_3)$, there exist morphisms $a : Q_1 \\to Q_2$ and $b : Q_2 \\to Q_3$ such that \\begin{enumerate} \\item $(Q_1, Q_2, Q_3, a, b, p_1[1] \\circ d_3)$ is a distinguished triangle, \\item the triple $(\\text{id}_X, g, a)$ is a morphism of triangles $(X, Y, Q_1, f, p_1, d_1) \\to (X, Z, Q_2, g \\circ f, p_2, d_2)$, and \\item the triple $(f, \\text{id}_Z, b)$ is a morphism of triangles $(X, Z, Q_2, g \\circ f, p_2, d_2) \\to (Y, Z, Q_3, g, p_3, d_3)$. \\end{enumerate} \\end{enumerate} We will call $(\\mathcal{D}, [\\ ], \\mathcal{T})$ a {\\it pre-triangulated category} if TR1, TR2 and TR3 hold.\\footnote{We use $[\\ ]$ as an abbreviation for the family $\\{[n]\\}_{n\\in \\mathbf{Z}}$.}"}
+{"_id": "1969", "title": "derived-definition-exact-functor-triangulated-categories", "text": "Let $\\mathcal{D}$, $\\mathcal{D}'$ be pre-triangulated categories. An {\\it exact functor}, or a {\\it triangulated functor} from $\\mathcal{D}$ to $\\mathcal{D}'$ is a functor $F : \\mathcal{D} \\to \\mathcal{D}'$ together with given functorial isomorphisms $\\xi_X : F(X[1]) \\to F(X)[1]$ such that for every distinguished triangle $(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ the triangle $(F(X), F(Y), F(Z), F(f), F(g), \\xi_X \\circ F(h))$ is a distinguished triangle of $\\mathcal{D}'$."}
+{"_id": "1970", "title": "derived-definition-triangulated-subcategory", "text": "Let $(\\mathcal{D}, [\\ ], \\mathcal{T})$ be a pre-triangulated category. A {\\it pre-triangulated subcategory}\\footnote{This definition may be nonstandard. If $\\mathcal{D}'$ is a full subcategory then $\\mathcal{T}'$ is the intersection of the set of triangles in $\\mathcal{D}'$ with $\\mathcal{T}$, see Lemma \\ref{lemma-triangulated-subcategory}. In this case we drop $\\mathcal{T}'$ from the notation.} is a pair $(\\mathcal{D}', \\mathcal{T}')$ such that \\begin{enumerate} \\item $\\mathcal{D}'$ is an additive subcategory of $\\mathcal{D}$ which is preserved under $[1]$ and $[-1]$, \\item $\\mathcal{T}' \\subset \\mathcal{T}$ is a subset such that for every $(X, Y, Z, f, g, h) \\in \\mathcal{T}'$ we have $X, Y, Z \\in \\Ob(\\mathcal{D}')$ and $f, g, h \\in \\text{Arrows}(\\mathcal{D}')$, and \\item $(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a pre-triangulated category. \\end{enumerate} If $\\mathcal{D}$ is a triangulated category, then we say $(\\mathcal{D}', \\mathcal{T}')$ is a {\\it triangulated subcategory} if it is a pre-triangulated subcategory and $(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a triangulated category."}
+{"_id": "1971", "title": "derived-definition-homological", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $\\mathcal{A}$ be an abelian category. An additive functor $H : \\mathcal{D} \\to \\mathcal{A}$ is called {\\it homological} if for every distinguished triangle $(X, Y, Z, f, g, h)$ the sequence $$ H(X) \\to H(Y) \\to H(Z) $$ is exact in the abelian category $\\mathcal{A}$. An additive functor $H : \\mathcal{D}^{opp} \\to \\mathcal{A}$ is called {\\it cohomological} if the corresponding functor $\\mathcal{D} \\to \\mathcal{A}^{opp}$ is homological."}
+{"_id": "1972", "title": "derived-definition-delta-functor", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a triangulated category. A {\\it $\\delta$-functor from $\\mathcal{A}$ to $\\mathcal{D}$} is given by a functor $G : \\mathcal{A} \\to \\mathcal{D}$ and a rule which assigns to every short exact sequence $$ 0 \\to A \\xrightarrow{a} B \\xrightarrow{b} C \\to 0 $$ a morphism $\\delta = \\delta_{A \\to B \\to C} : G(C) \\to G(A)[1]$ such that \\begin{enumerate} \\item the triangle $(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$ is a distinguished triangle of $\\mathcal{D}$ for any short exact sequence as above, and \\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$ of short exact sequences the diagram $$ \\xymatrix{ G(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & & G(A)[1] \\ar[d] \\\\ G(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & & G(A')[1] } $$ is commutative. \\end{enumerate} In this situation we call $(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$ the {\\it image of the short exact sequence under the given $\\delta$-functor}."}
+{"_id": "1973", "title": "derived-definition-localization", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. We say a multiplicative system $S$ is {\\it compatible with the triangulated structure} if the following two conditions hold: \\begin{enumerate} \\item[MS5] For $s \\in S$ we have $s[n] \\in S$ for all $n \\in \\mathbf{Z}$. \\item[MS6] Given a solid commutative square $$ \\xymatrix{ X \\ar[r] \\ar[d]^s & Y \\ar[r] \\ar[d]^{s'} & Z \\ar[r] \\ar@{-->}[d] & X[1] \\ar[d]^{s[1]} \\\\ X' \\ar[r] & Y' \\ar[r] & Z' \\ar[r] & X'[1] } $$ whose rows are distinguished triangles with $s, s' \\in S$ there exists a morphism $s'' : Z \\to Z'$ in $S$ such that $(s, s', s'')$ is a morphism of triangles. \\end{enumerate}"}
+{"_id": "1974", "title": "derived-definition-saturated", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. We say a full pre-triangulated subcategory $\\mathcal{D}'$ of $\\mathcal{D}$ is {\\it saturated} if whenever $X \\oplus Y$ is isomorphic to an object of $\\mathcal{D}'$ then both $X$ and $Y$ are isomorphic to objects of $\\mathcal{D}'$."}
+{"_id": "1975", "title": "derived-definition-kernel-category", "text": "Let $\\mathcal{D}$ be a (pre-)triangulated category. \\begin{enumerate} \\item Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor. The {\\it kernel of $F$} is the strictly full saturated (pre-)triangulated subcategory described in Lemma \\ref{lemma-triangle-functor-kernel}. \\item Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor. The {\\it kernel of $H$} is the strictly full saturated (pre-)triangulated subcategory described in Lemma \\ref{lemma-homological-functor-kernel}. \\end{enumerate} These are sometimes denoted $\\Ker(F)$ or $\\Ker(H)$."}
+{"_id": "1976", "title": "derived-definition-quotient-category", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$ be a full triangulated subcategory. We define the {\\it quotient category $\\mathcal{D}/\\mathcal{B}$} by the formula $\\mathcal{D}/\\mathcal{B} = S^{-1}\\mathcal{D}$, where $S$ is the multiplicative system of $\\mathcal{D}$ associated to $\\mathcal{B}$ via Lemma \\ref{lemma-construct-multiplicative-system}. The localization functor $Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$ is called the {\\it quotient functor} in this case."}
+{"_id": "1977", "title": "derived-definition-complexes-notation", "text": "Let $\\mathcal{A}$ be an additive category. \\begin{enumerate} \\item We set $\\text{Comp}(\\mathcal{A}) = \\text{CoCh}(\\mathcal{A})$ be the {\\it category of (cochain) complexes}. \\item A complex $K^\\bullet$ is said to be {\\it bounded below} if $K^n = 0$ for all $n \\ll 0$. \\item A complex $K^\\bullet$ is said to be {\\it bounded above} if $K^n = 0$ for all $n \\gg 0$. \\item A complex $K^\\bullet$ is said to be {\\it bounded} if $K^n = 0$ for all $|n| \\gg 0$. \\item We let $\\text{Comp}^{+}(\\mathcal{A})$, $\\text{Comp}^{-}(\\mathcal{A})$, resp.\\ $\\text{Comp}^b(\\mathcal{A})$ be the full subcategory of $\\text{Comp}(\\mathcal{A})$ whose objects are the complexes which are bounded below, bounded above, resp.\\ bounded. \\item We let $K(\\mathcal{A})$ be the category with the same objects as $\\text{Comp}(\\mathcal{A})$ but as morphisms homotopy classes of maps of complexes (see Homology, Lemma \\ref{homology-lemma-compose-homotopy-cochain}). \\item We let $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, resp.\\ $K^b(\\mathcal{A})$ be the full subcategory of $K(\\mathcal{A})$ whose objects are bounded below, bounded above, resp.\\ bounded complexes of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "1978", "title": "derived-definition-cone", "text": "Let $\\mathcal{A}$ be an additive category. Let $f : K^\\bullet \\to L^\\bullet$ be a morphism of complexes of $\\mathcal{A}$. The {\\it cone} of $f$ is the complex $C(f)^\\bullet$ given by $C(f)^n = L^n \\oplus K^{n + 1}$ and differential $$ d_{C(f)}^n = \\left( \\begin{matrix} d^n_L & f^{n + 1} \\\\ 0 & -d_K^{n + 1} \\end{matrix} \\right) $$ It comes equipped with canonical morphisms of complexes $i : L^\\bullet \\to C(f)^\\bullet$ and $p : C(f)^\\bullet \\to K^\\bullet[1]$ induced by the obvious maps $L^n \\to C(f)^n \\to K^{n + 1}$."}
+{"_id": "1979", "title": "derived-definition-termwise-split-map", "text": "Let $\\mathcal{A}$ be an additive category. A {\\it termwise split injection $\\alpha : A^\\bullet \\to B^\\bullet$} is a morphism of complexes such that each $A^n \\to B^n$ is isomorphic to the inclusion of a direct summand. A {\\it termwise split surjection $\\beta : B^\\bullet \\to C^\\bullet$} is a morphism of complexes such that each $B^n \\to C^n$ is isomorphic to the projection onto a direct summand."}
+{"_id": "1980", "title": "derived-definition-split-ses", "text": "Let $\\mathcal{A}$ be an additive category. A {\\it termwise split exact sequence of complexes of $\\mathcal{A}$} is a complex of complexes $$ 0 \\to A^\\bullet \\xrightarrow{\\alpha} B^\\bullet \\xrightarrow{\\beta} C^\\bullet \\to 0 $$ together with given direct sum decompositions $B^n = A^n \\oplus C^n$ compatible with $\\alpha^n$ and $\\beta^n$. We often write $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$ for the maps induced by the direct sum decompositions. According to Homology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain} we get an associated morphism of complexes $$ \\delta : C^\\bullet \\longrightarrow A^\\bullet[1] $$ which in degree $n$ is the map $\\pi^{n + 1} \\circ d_B^n \\circ s^n$. In other words $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$ forms a triangle $$ A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to A^\\bullet[1] $$ This will be the {\\it triangle associated to the termwise split sequence of complexes}."}
+{"_id": "1981", "title": "derived-definition-distinguished-triangle", "text": "Let $\\mathcal{A}$ be an additive category. A triangle $(X, Y, Z, f, g, h)$ of $K(\\mathcal{A})$ is called a {\\it distinguished triangle of $K(\\mathcal{A})$} if it is isomorphic to the triangle associated to a termwise split exact sequence of complexes, see Definition \\ref{definition-split-ses}. Same definition for $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$."}
+{"_id": "1982", "title": "derived-definition-unbounded-derived-category", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\text{Ac}(\\mathcal{A})$ and $\\text{Qis}(\\mathcal{A})$ be as in Lemma \\ref{lemma-acyclic}. The {\\it derived category of $\\mathcal{A}$} is the triangulated category $$ D(\\mathcal{A}) = K(\\mathcal{A})/\\text{Ac}(\\mathcal{A}) = \\text{Qis}(\\mathcal{A})^{-1} K(\\mathcal{A}). $$ We denote $H^0 : D(\\mathcal{A}) \\to \\mathcal{A}$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above. Using Lemma \\ref{lemma-homological-functor-bounded} we introduce the strictly full saturated triangulated subcategories $D^{+}(\\mathcal{A}), D^{-}(\\mathcal{A}), D^b(\\mathcal{A})$ whose sets of objects are $$ \\begin{matrix} \\Ob(D^{+}(\\mathcal{A})) = \\{X \\in \\Ob(D(\\mathcal{A})) \\mid H^n(X) = 0\\text{ for all }n \\ll 0\\} \\\\ \\Ob(D^{-}(\\mathcal{A})) = \\{X \\in \\Ob(D(\\mathcal{A})) \\mid H^n(X) = 0\\text{ for all }n \\gg 0\\} \\\\ \\Ob(D^b(\\mathcal{A})) = \\{X \\in \\Ob(D(\\mathcal{A})) \\mid H^n(X) = 0\\text{ for all }|n| \\gg 0\\} \\end{matrix} $$ The category $D^b(\\mathcal{A})$ is called the {\\it bounded derived category} of $\\mathcal{A}$."}
+{"_id": "1983", "title": "derived-definition-finite-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. The {\\it category of finite filtered objects of $\\mathcal{A}$} is the category of filtered objects $(A, F)$ of $\\mathcal{A}$ whose filtration $F$ is finite. We denote it $\\text{Fil}^f(\\mathcal{A})$."}
+{"_id": "1984", "title": "derived-definition-filtered-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of $K(\\text{Fil}^f(\\mathcal{A}))$. We say that $\\alpha$ is a {\\it filtered quasi-isomorphism} if the morphism $\\text{gr}(\\alpha)$ is a quasi-isomorphism. \\item Let $K^\\bullet$ be an object of $K(\\text{Fil}^f(\\mathcal{A}))$. We say that $K^\\bullet$ is {\\it filtered acyclic} if the complex $\\text{gr}(K^\\bullet)$ is acyclic. \\end{enumerate}"}
+{"_id": "1985", "title": "derived-definition-filtered-derived", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\text{FAc}(\\mathcal{A})$ and $\\text{FQis}(\\mathcal{A})$ be as in Lemma \\ref{lemma-filtered-acyclic}. The {\\it filtered derived category of $\\mathcal{A}$} is the triangulated category $$ DF(\\mathcal{A}) = K(\\text{Fil}^f(\\mathcal{A}))/\\text{FAc}(\\mathcal{A}) = \\text{FQis}(\\mathcal{A})^{-1} K(\\text{Fil}^f(\\mathcal{A})). $$"}
+{"_id": "1986", "title": "derived-definition-filtered-derived-bounded", "text": "Let $\\mathcal{A}$ be an abelian category. The {\\it bounded filtered derived category} $DF^b(\\mathcal{A})$ is the full subcategory of $DF(\\mathcal{A})$ with objects those $X$ such that $\\text{gr}(X) \\in D^b(\\mathcal{A})$. Similarly for the bounded below filtered derived category $DF^{+}(\\mathcal{A})$ and the bounded above filtered derived category $DF^{-}(\\mathcal{A})$."}
+{"_id": "1987", "title": "derived-definition-right-derived-functor-defined", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $X \\in \\Ob(\\mathcal{D})$. \\begin{enumerate} \\item we say the {\\it right derived functor $RF$ is defined at} $X$ if the ind-object $$ (X/S) \\longrightarrow \\mathcal{D}', \\quad (s : X \\to X') \\longmapsto F(X') $$ is essentially constant\\footnote{For a discussion of when an ind-object or pro-object of a category is essentially constant we refer to Categories, Section \\ref{categories-section-essentially-constant}.}; in this case the value $Y$ in $\\mathcal{D}'$ is called the {\\it value of $RF$ at $X$}. \\item we say the {\\it left derived functor $LF$ is defined at} $X$ if the pro-object $$ (S/X) \\longrightarrow \\mathcal{D}', \\quad (s: X' \\to X) \\longmapsto F(X') $$ is essentially constant; in this case the value $Y$ in $\\mathcal{D}'$ is called the {\\it value of $LF$ at $X$}. \\end{enumerate} By abuse of notation we often denote the values simply $RF(X)$ or $LF(X)$."}
+{"_id": "1988", "title": "derived-definition-everywhere-defined", "text": "In Situation \\ref{situation-derived-functor}. We say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined} if $RF$ is defined at every object of $\\mathcal{D}$. We say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined} if $LF$ is defined at every object of $\\mathcal{D}$."}
+{"_id": "1989", "title": "derived-definition-computes", "text": "In Situation \\ref{situation-derived-functor}. \\begin{enumerate} \\item An object $X$ of $\\mathcal{D}$ {\\it computes} $RF$ if $RF$ is defined at $X$ and the canonical map $F(X) \\to RF(X)$ is an isomorphism. \\item An object $X$ of $\\mathcal{D}$ {\\it computes} $LF$ if $LF$ is defined at $X$ and the canonical map $LF(X) \\to F(X)$ is an isomorphism. \\end{enumerate}"}
+{"_id": "1990", "title": "derived-definition-derived-functor", "text": "In Situation \\ref{situation-classical}. \\begin{enumerate} \\item The {\\it right derived functors of $F$} are the partial functors $RF$ associated to cases (1) and (2) of Situation \\ref{situation-classical}. \\item The {\\it left derived functors of $F$} are the partial functors $LF$ associated to cases (3) and (4) of Situation \\ref{situation-classical}. \\item An object $A$ of $\\mathcal{A}$ is said to be {\\it right acyclic for $F$}, or {\\it acyclic for $RF$} if $A[0]$ computes $RF$. \\item An object $A$ of $\\mathcal{A}$ is said to be {\\it left acyclic for $F$}, or {\\it acyclic for $LF$} if $A[0]$ computes $LF$. \\end{enumerate}"}
+{"_id": "1991", "title": "derived-definition-higher-derived-functors", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor between abelian categories. Assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. Let $i \\in \\mathbf{Z}$. The {\\it $i$th right derived functor $R^iF$ of $F$} is the functor $$ R^iF = H^i \\circ RF : \\mathcal{A} \\longrightarrow \\mathcal{B} $$"}
+{"_id": "1992", "title": "derived-definition-injective-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A \\in \\Ob(\\mathcal{A})$. An {\\it injective resolution of $A$} is a complex $I^\\bullet$ together with a map $A \\to I^0$ such that: \\begin{enumerate} \\item We have $I^n = 0$ for $n < 0$. \\item Each $I^n$ is an injective object of $\\mathcal{A}$. \\item The map $A \\to I^0$ is an isomorphism onto $\\Ker(d^0)$. \\item We have $H^i(I^\\bullet) = 0$ for $i > 0$. \\end{enumerate} Hence $A[0] \\to I^\\bullet$ is a quasi-isomorphism. In other words the complex $$ \\ldots \\to 0 \\to A \\to I^0 \\to I^1 \\to \\ldots $$ is acyclic. Let $K^\\bullet$ be a complex in $\\mathcal{A}$. An {\\it injective resolution of $K^\\bullet$} is a complex $I^\\bullet$ together with a map $\\alpha : K^\\bullet \\to I^\\bullet$ of complexes such that \\begin{enumerate} \\item We have $I^n = 0$ for $n \\ll 0$, i.e., $I^\\bullet$ is bounded below. \\item Each $I^n$ is an injective object of $\\mathcal{A}$. \\item The map $\\alpha : K^\\bullet \\to I^\\bullet$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "1993", "title": "derived-definition-projective-resolution", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A \\in \\Ob(\\mathcal{A})$. An {\\it projective resolution of $A$} is a complex $P^\\bullet$ together with a map $P^0 \\to A$ such that: \\begin{enumerate} \\item We have $P^n = 0$ for $n > 0$. \\item Each $P^n$ is an projective object of $\\mathcal{A}$. \\item The map $P^0 \\to A$ induces an isomorphism $\\Coker(d^{-1}) \\to A$. \\item We have $H^i(P^\\bullet) = 0$ for $i < 0$. \\end{enumerate} Hence $P^\\bullet \\to A[0]$ is a quasi-isomorphism. In other words the complex $$ \\ldots \\to P^{-1} \\to P^0 \\to A \\to 0 \\to \\ldots $$ is acyclic. Let $K^\\bullet$ be a complex in $\\mathcal{A}$. An {\\it projective resolution of $K^\\bullet$} is a complex $P^\\bullet$ together with a map $\\alpha : P^\\bullet \\to K^\\bullet$ of complexes such that \\begin{enumerate} \\item We have $P^n = 0$ for $n \\gg 0$, i.e., $P^\\bullet$ is bounded above. \\item Each $P^n$ is an projective object of $\\mathcal{A}$. \\item The map $\\alpha : P^\\bullet \\to K^\\bullet$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "1994", "title": "derived-definition-cartan-eilenberg", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a bounded below complex. A {\\it Cartan-Eilenberg resolution} of $K^\\bullet$ is given by a double complex $I^{\\bullet, \\bullet}$ and a morphism of complexes $\\epsilon : K^\\bullet \\to I^{\\bullet, 0}$ with the following properties: \\begin{enumerate} \\item There exists a $i \\ll 0$ such that $I^{p, q} = 0$ for all $p < i$ and all $q$. \\item We have $I^{p, q} = 0$ if $q < 0$. \\item The complex $I^{p, \\bullet}$ is an injective resolution of $K^p$. \\item The complex $\\Ker(d_1^{p, \\bullet})$ is an injective resolution of $\\Ker(d_K^p)$. \\item The complex $\\Im(d_1^{p, \\bullet})$ is an injective resolution of $\\Im(d_K^p)$. \\item The complex $H^p_I(I^{\\bullet, \\bullet})$ is an injective resolution of $H^p(K^\\bullet)$. \\end{enumerate}"}
+{"_id": "1995", "title": "derived-definition-localization-functor", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. A {\\it resolution functor}\\footnote{This is likely nonstandard terminology.} for $\\mathcal{A}$ is given by the following data: \\begin{enumerate} \\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a bounded below complex of injectives $j(K^\\bullet)$, and \\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a quasi-isomorphism $i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet)$. \\end{enumerate}"}
+{"_id": "1996", "title": "derived-definition-filtered-complexes-notation", "text": "Let $\\mathcal{A}$ be an abelian category. We say an object $I$ of $\\text{Fil}^f(\\mathcal{A})$ is {\\it filtered injective} if each $\\text{gr}^p(I)$ is an injective object of $\\mathcal{A}$."}
+{"_id": "1997", "title": "derived-definition-ext", "text": "Let $\\mathcal{A}$ be an abelian category. Let $i \\in \\mathbf{Z}$. Let $X, Y$ be objects of $D(\\mathcal{A})$. The {\\it $i$th extension group} of $X$ by $Y$ is the group $$ \\Ext^i_\\mathcal{A}(X, Y) = \\Hom_{D(\\mathcal{A})}(X, Y[i]) = \\Hom_{D(\\mathcal{A})}(X[-i], Y). $$ If $A, B \\in \\Ob(\\mathcal{A})$ we set $\\Ext^i_\\mathcal{A}(A, B) = \\text{Ext}^i_\\mathcal{A}(A[0], B[0])$."}
+{"_id": "1998", "title": "derived-definition-yoneda-extension", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A, B \\in \\Ob(\\mathcal{A})$. A degree $i$ {\\it Yoneda extension} of $B$ by $A$ is an exact sequence $$ E : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_0 \\to B \\to 0 $$ in $\\mathcal{A}$. We say two Yoneda extensions $E$ and $E'$ of the same degree are {\\it equivalent} if there exists a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & A \\ar[r] & Z_{i - 1} \\ar[r] & \\ldots \\ar[r] & Z_0 \\ar[r] & B \\ar[r] & 0 \\\\ 0 \\ar[r] & A \\ar[r] \\ar[u]^{\\text{id}} \\ar[d]_{\\text{id}} & Z''_{i - 1} \\ar[r] \\ar[u] \\ar[d] & \\ldots \\ar[r] & Z''_0 \\ar[r] \\ar[u] \\ar[d] & B \\ar[r] \\ar[u]_{\\text{id}} \\ar[d]^{\\text{id}} & 0 \\\\ 0 \\ar[r] & A \\ar[r] & Z'_{i - 1} \\ar[r] & \\ldots \\ar[r] & Z'_0 \\ar[r] & B \\ar[r] & 0 } $$ where the middle row is a Yoneda extension as well."}
+{"_id": "1999", "title": "derived-definition-K-zero", "text": "Let $\\mathcal{D}$ be a triangulated category. We denote $K_0(\\mathcal{D})$ the {\\it zeroth $K$-group of $\\mathcal{D}$}. It is the abelian group constructed as follows. Take the free abelian group on the objects on $\\mathcal{D}$ and for every distinguished triangle $X \\to Y \\to Z$ impose the relation $[Y] - [X] - [Z] = 0$."}
+{"_id": "2000", "title": "derived-definition-K-injective", "text": "Let $\\mathcal{A}$ be an abelian category. A complex $I^\\bullet$ is {\\it K-injective} if for every acyclic complex $M^\\bullet$ we have $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$."}
+{"_id": "2001", "title": "derived-definition-derived-colimit", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$. We say an object $K$ is a {\\it derived colimit}, or a {\\it homotopy colimit} of the system $(K_n)$ if the direct sum $\\bigoplus K_n$ exists and there is a distinguished triangle $$ \\bigoplus K_n \\to \\bigoplus K_n \\to K \\to \\bigoplus K_n[1] $$ where the map $\\bigoplus K_n \\to \\bigoplus K_n$ is given by $1 - f_n$ in degree $n$. If this is the case, then we sometimes indicate this by the notation $K = \\text{hocolim} K_n$."}
+{"_id": "2002", "title": "derived-definition-derived-limit", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be an inverse system of objects of $\\mathcal{D}$. We say an object $K$ is a {\\it derived limit}, or a {\\it homotopy limit} of the system $(K_n)$ if the product $\\prod K_n$ exists and there is a distinguished triangle $$ K \\to \\prod K_n \\to \\prod K_n \\to K[1] $$ where the map $\\prod K_n \\to \\prod K_n$ is given by $(k_n) \\mapsto (k_n - f_{n+1}(k_{n + 1}))$. If this is the case, then we sometimes indicate this by the notation $K = R\\lim K_n$."}
+{"_id": "2003", "title": "derived-definition-generators", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\\mathcal{D}$. \\begin{enumerate} \\item We say $E$ is a {\\it classical generator} of $\\mathcal{D}$ if the smallest strictly full, saturated, triangulated subcategory of $\\mathcal{D}$ containing $E$ is equal to $\\mathcal{D}$, in other words, if $\\langle E \\rangle = \\mathcal{D}$. \\item We say $E$ is a {\\it strong generator} of $\\mathcal{D}$ if $\\langle E \\rangle_n = \\mathcal{D}$ for some $n \\geq 1$. \\item We say $E$ is a {\\it weak generator} or a {\\it generator} of $\\mathcal{D}$ if for any nonzero object $K$ of $\\mathcal{D}$ there exists an integer $n$ and a nonzero map $E \\to K[n]$. \\end{enumerate}"}
+{"_id": "2004", "title": "derived-definition-compact-object", "text": "Let $\\mathcal{D}$ be an additive category with arbitrary direct sums. A {\\it compact object} of $\\mathcal{D}$ is an object $K$ such that the map $$ \\bigoplus\\nolimits_{i \\in I} \\Hom_{\\mathcal{D}}(K, E_i) \\longrightarrow \\Hom_{\\mathcal{D}}(K, \\bigoplus\\nolimits_{i \\in I} E_i) $$ is bijective for any set $I$ and objects $E_i \\in \\Ob(\\mathcal{D})$ parametrized by $i \\in I$."}
+{"_id": "2005", "title": "derived-definition-compactly-generated", "text": "Let $\\mathcal{D}$ be a triangulated category with arbitrary direct sums. We say $\\mathcal{D}$ is {\\it compactly generated} if there exists a set $E_i$, $i \\in I$ of compact objects such that $\\bigoplus E_i$ generates $\\mathcal{D}$."}
+{"_id": "2006", "title": "derived-definition-orthogonal", "text": "Let $\\mathcal{D}$ be an additive category. Let $\\mathcal{A} \\subset \\mathcal{D}$ be a full subcategory. The {\\it right orthogonal} $\\mathcal{A}^\\perp$ of $\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\\mathcal{D}$ such that $\\Hom(A, X) = 0$ for all $A \\in \\Ob(\\mathcal{A})$. The {\\it left orthogonal} ${}^\\perp\\mathcal{A}$ of $\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\\mathcal{D}$ such that $\\Hom(X, A) = 0$ for all $A \\in \\Ob(\\mathcal{A})$."}
+{"_id": "2007", "title": "derived-definition-admissible", "text": "Let $\\mathcal{D}$ be a triangulated category. A {\\it right admissible} subcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma \\ref{lemma-right-adjoint}. A {\\it left admissible} subcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma \\ref{lemma-left-adjoint}. A {\\it two-sided admissible} subcategory is one which is both right and left admissible."}
+{"_id": "2008", "title": "derived-definition-postnikov-system", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_0 $$ be a complex in $\\mathcal{D}$. A {\\it Postnikov system} is defined inductively as follows. \\begin{enumerate} \\item If $n = 0$, then it is an isomorphism $Y_0 \\to X_0$. \\item If $n = 1$, then it is a choice of a distinguished triangle $$ Y_1 \\to X_1 \\to Y_0 \\to Y_1[1] $$ where $X_1 \\to Y_0$ composed with $Y_0 \\to X_0$ is the given morphism $X_1 \\to X_0$. \\item If $n > 1$, then it is a choice of a Postnikov system for $X_{n - 1} \\to \\ldots \\to X_0$ and a choice of a distinguished triangle $$ Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1] $$ where the morphism $X_n \\to Y_{n - 1}$ composed with $Y_{n - 1} \\to X_{n - 1}$ is the given morphism $X_n \\to X_{n - 1}$. \\end{enumerate} Given a morphism \\begin{equation} \\label{equation-map-complexes} \\vcenter{ \\xymatrix{ X_n \\ar[r] \\ar[d] & X_{n - 1} \\ar[r] \\ar[d] & \\ldots \\ar[r] & X_0 \\ar[d] \\\\ X'_n \\ar[r] & X'_{n - 1} \\ar[r] & \\ldots \\ar[r] & X'_0 } } \\end{equation} between complexes of the same length in $\\mathcal{D}$ there is an obvious notion of a {\\it morphism of Postnikov systems}."}
+{"_id": "2248", "title": "cohomology-definition-torsor", "text": "Let $X$ be a topological space. Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A {\\it torsor}, or more precisely a {\\it $\\mathcal{G}$-torsor}, is a sheaf of sets $\\mathcal{F}$ on $X$ endowed with an action $\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that \\begin{enumerate} \\item whenever $\\mathcal{F}(U)$ is nonempty the action $\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ is simply transitive, and \\item for every $x \\in X$ the stalk $\\mathcal{F}_x$ is nonempty. \\end{enumerate} A {\\it morphism of $\\mathcal{G}$-torsors} $\\mathcal{F} \\to \\mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\\mathcal{G}$-actions. The {\\it trivial $\\mathcal{G}$-torsor} is the sheaf $\\mathcal{G}$ endowed with the obvious left $\\mathcal{G}$-action."}
+{"_id": "2249", "title": "cohomology-definition-cech-complex", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Let $\\mathcal{F}$ be an abelian presheaf on $X$. The complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the open covering $\\mathcal{U}$. Its cohomology groups $H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are called the {\\it {\\v C}ech cohomology groups} associated to $\\mathcal{F}$ and the covering $\\mathcal{U}$. They are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$."}
+{"_id": "2250", "title": "cohomology-definition-flasque", "text": "Let $X$ be a topological space. We say a presheaf of sets $\\mathcal{F}$ is {\\it flasque} or {\\it flabby} if for every $U \\subset V$ open in $X$ the restriction map $\\mathcal{F}(V) \\to \\mathcal{F}(U)$ is surjective."}
+{"_id": "2251", "title": "cohomology-definition-alternating-cech-complex", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Let $\\mathcal{F}$ be an abelian presheaf on $X$. The complex $\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the {\\it alternating {\\v C}ech complex} associated to $\\mathcal{F}$ and the open covering $\\mathcal{U}$."}
+{"_id": "2252", "title": "cohomology-definition-ordered-cech-complex", "text": "Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. Assume given a total ordering on $I$. Let $\\mathcal{F}$ be an abelian presheaf on $X$. The complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the open covering $\\mathcal{U}$ and the given total ordering on $I$."}
+{"_id": "2253", "title": "cohomology-definition-covering-locally-finite", "text": "Let $X$ be a topological space. An open covering $X = \\bigcup_{i \\in I} U_i$ is said to be {\\it locally finite} if for every $x \\in X$ there exists an open neighbourhood $W$ of $x$ such that $\\{i \\in I \\mid W \\cap U_i \\not = \\emptyset\\}$ is finite."}
+{"_id": "2254", "title": "cohomology-definition-K-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. A complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}_X$-modules is called {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules the complex $$ \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet) $$ is acyclic."}
+{"_id": "2255", "title": "cohomology-definition-derived-tor", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O}_X)$. The {\\it derived tensor product} $$ - \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} \\mathcal{F}^\\bullet : D(\\mathcal{O}_X) \\longrightarrow D(\\mathcal{O}_X) $$ is the exact functor of triangulated categories described above."}
+{"_id": "2256", "title": "cohomology-definition-tor", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. The {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are define by the formula $$ \\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) = H^{-p}(\\mathcal{F} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}) $$ with derived tensor product as defined above."}
+{"_id": "2257", "title": "cohomology-definition-strictly-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. We say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect} if $\\mathcal{E}^i$ is zero for all but finitely many $i$ and $\\mathcal{E}^i$ is a direct summand of a finite free $\\mathcal{O}_X$-module for all $i$."}
+{"_id": "2258", "title": "cohomology-definition-pseudo-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent} if there exists an open covering $X = \\bigcup U_i$ and for each $i$ a morphism of complexes $\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$ where $\\mathcal{E}_i$ is strictly perfect on $U_i$ and $H^j(\\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\\alpha_i)$ is surjective. \\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent} if it is $m$-pseudo-coherent for all $m$. \\item We say an object $E$ of $D(\\mathcal{O}_X)$ is {\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent}) if and only if it can be represented by a $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "2259", "title": "cohomology-definition-tor-amplitude", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. \\begin{enumerate} \\item We say $E$ has {\\it tor-amplitude in $[a, b]$} if $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$ for all $\\mathcal{O}_X$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$. \\item We say $E$ has {\\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \\item We say $E$ {\\it locally has finite tor dimension} if there exists an open covering $X = \\bigcup U_i$ such that $E|_{U_i}$ has finite tor dimension for all $i$. \\end{enumerate} An $\\mathcal{O}_X$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$} if $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O}_X)$ has tor-amplitude in $[-d, 0]$."}
+{"_id": "2260", "title": "cohomology-definition-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. We say $\\mathcal{E}^\\bullet$ is {\\it perfect} if there exists an open covering $X = \\bigcup U_i$ such that for each $i$ there exists a morphism of complexes $\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$ which is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$ a strictly perfect complex of $\\mathcal{O}_{U_i}$-modules. An object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect} if it can be represented by a perfect complex of $\\mathcal{O}_X$-modules."}
+{"_id": "2288", "title": "stacks-introduction-definition-smooth", "text": "We say a morphism $S \\to \\mathcal{M}_{1, 1}$ is {\\it smooth} if for every morphism $S' \\to \\mathcal{M}_{1, 1}$ the projection morphism $$ S \\times_{\\mathcal{M}_{1, 1}} S' \\longrightarrow S' $$ is smooth."}
+{"_id": "2289", "title": "stacks-introduction-definition-algebraic-stack", "text": "We say $\\mathcal{M}_{1, 1}$ is an {\\it algebraic stack} if and only if \\begin{enumerate} \\item We have descent for objects for the \\'etale topology on $\\Sch$. \\item The key fact holds. \\item there exists a surjective and smooth morphism $S \\to \\mathcal{M}_{1, 1}$. \\end{enumerate}"}
+{"_id": "2434", "title": "restricted-definition-rig-smooth-homomorphism", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). We say $B$ is {\\it rig-smooth over $(A, I)$} if there exists an integer $c \\geq 0$ such that $I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ for every $B$-module $N$."}
+{"_id": "2435", "title": "restricted-definition-rig-etale-homomorphism", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). We say $B$ is {\\it rig-\\'etale over $(A, I)$} if there exists an integer $c \\geq 0$ such that for all $a \\in I^c$ multiplication by $a$ on $\\NL_{B/A}^\\wedge$ is zero in $D(B)$."}
+{"_id": "2436", "title": "restricted-definition-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\\it flat} if for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a flat map of adic Noetherian topological rings."}
+{"_id": "2437", "title": "restricted-definition-rig-closed", "text": "Let $A$ be a Noetherian adic topological ring. Let $\\mathfrak q \\subset A$ be a prime ideal. We say $\\mathfrak q$ is {\\it rig-closed} if the equivalent conditions of Lemma \\ref{lemma-rig-point} are satisfied."}
+{"_id": "2438", "title": "restricted-definition-completed-principal-localization", "text": "Let $A$ be an adic topological ring which has a finitely generated ideal of definition. Let $f \\in A$. The {\\it completed principal localization} $A_{\\{f\\}}$ of $A$ is the completion of $A_f = A[1/f]$ of the principal localization of $A$ at $f$ with respect to any ideal of definition of $A$."}
+{"_id": "2439", "title": "restricted-definition-naively-rig-flat", "text": "Let $\\varphi : A \\to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\\varphi$ is an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it naively rig-flat} if $\\varphi$ is adic, topologically of finite type, and satisfies the equivalent conditions of Lemma \\ref{lemma-naively-rig-flat-continuous}."}
+{"_id": "2440", "title": "restricted-definition-rig-flat-continuous-homomorphism", "text": "Let $\\varphi : A \\to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\\varphi$ is an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it rig-flat} if $\\varphi$ is adic, topologically of finite type, and for all $f \\in A$ the induced map $$ A_{\\{f\\}} \\longrightarrow B_{\\{f\\}} $$ is naively rig-flat (Definition \\ref{definition-naively-rig-flat})."}
+{"_id": "2441", "title": "restricted-definition-rig-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\\it rig-flat} if for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-flat map of adic Noetherian topological rings."}
+{"_id": "2442", "title": "restricted-definition-rig-smooth-continuous-homomorphism", "text": "Let $\\varphi : A \\to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\\varphi$ is an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it rig-smooth} if the equivalent conditions of Lemma \\ref{lemma-rig-smooth-continuous} hold."}
+{"_id": "2443", "title": "restricted-definition-rig-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\\it rig-smooth} if for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-smooth map of adic Noetherian topological rings."}
+{"_id": "2444", "title": "restricted-definition-rig-etale-continuous-homomorphism", "text": "Let $\\varphi : A \\to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\\varphi$ is an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it rig-etale} if the equivalent conditions of Lemma \\ref{lemma-rig-etale-continuous} hold."}
+{"_id": "2445", "title": "restricted-definition-rig-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is {\\it rig-\\'etale} if for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-\\'etale map of adic Noetherian topological rings."}
+{"_id": "2446", "title": "restricted-definition-rig-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. Assume that $X$ and $Y$ are locally Noetherian and that $f$ is locally of finite type. We say $f$ is {\\it rig-surjective} if for every solid diagram $$ \\xymatrix{ \\text{Spf}(R') \\ar@{..>}[r] \\ar@{..>}[d] & X \\ar[d]^f \\\\ \\text{Spf}(R) \\ar[r]^-p & Y } $$ where $R$ is a complete discrete valuation ring and where $p$ is an adic morphism there exists an extension of complete discrete valuation rings $R \\subset R'$ and a morphism $\\text{Spf}(R') \\to X$ making the displayed diagram commute."}
+{"_id": "2447", "title": "restricted-definition-formal-modification", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is a {\\it formal modification} if \\begin{enumerate} \\item $f$ is a proper morphism (Formal Spaces, Definition \\ref{formal-spaces-definition-proper}), \\item $f$ is rig-\\'etale, \\item $f$ is rig-surjective, \\item $\\Delta_f : X \\to X \\times_Y X$ is rig-surjective. \\end{enumerate}"}
+{"_id": "2637", "title": "bootstrap-definition-morphism-representable-by-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$, $G$ be presheaves on $\\Sch_{fppf}/S$. We say a morphism $a : F \\to G$ is {\\it representable by algebraic spaces} if for every $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $\\xi : U \\to G$ the fiber product $U \\times_{\\xi, G} F$ is an algebraic space."}
+{"_id": "2638", "title": "bootstrap-definition-property-transformation", "text": "Let $S$ be a scheme. Let $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$ which is representable by algebraic spaces. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which \\begin{enumerate} \\item is preserved under any base change, and \\item is fppf local on the base, see Descent on Spaces, Definition \\ref{spaces-descent-definition-property-morphisms-local}. \\end{enumerate} In this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every scheme $U$ and $\\xi : U \\to G$ the resulting morphism of algebraic spaces $U \\times_G F \\to U$ has property $\\mathcal{P}$."}
+{"_id": "2761", "title": "spaces-perfect-definition-supported-on", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $T \\subset |X|$ be a closed subset. We say $E$ is {\\it supported on $T$} if the cohomology sheaves $H^i(E)$ are supported on $T$."}
+{"_id": "2762", "title": "spaces-perfect-definition-derived-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it derived category of $\\mathcal{O}_X$-modules with quasi-coherent cohomology sheaves} is denoted $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "2763", "title": "spaces-perfect-definition-proper-over-base", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T \\subset |X|$ be a closed subset. We say {\\it $T$ is proper over $Y$} if the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base} are satisfied."}
+{"_id": "2764", "title": "spaces-perfect-definition-elementary-distinguished-square", "text": "Let $S$ be a scheme. A commutative diagram $$ \\xymatrix{ U \\times_W V \\ar[r] \\ar[d] & V \\ar[d]^f \\\\ U \\ar[r]^j & W } $$ of algebraic spaces over $S$ is called an {\\it elementary distinguished square} if \\begin{enumerate} \\item $U$ is an open subspace of $W$ and $j$ is the inclusion morphism, \\item $f$ is \\'etale, and \\item setting $T = W \\setminus U$ (with reduced induced subspace structure) the morphism $f^{-1}(T) \\to T$ is an isomorphism. \\end{enumerate} We will indicate this by saying: ``Let $(U \\subset W, f : V \\to W)$ be an elementary distinguished square.''"}
+{"_id": "2765", "title": "spaces-perfect-definition-approximation-holds", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider triples $(T, E, m)$ where \\begin{enumerate} \\item $T \\subset |X|$ is a closed subset, \\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and \\item $m \\in \\mathbf{Z}$. \\end{enumerate} We say {\\it approximation holds for the triple} $(T, E, m)$ if there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$ and a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$ for $i > m$ and a surjection $H^m(P) \\to H^m(E)$."}
+{"_id": "2766", "title": "spaces-perfect-definition-approximation", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say {\\it approximation by perfect complexes holds} on $X$ if for any closed subset $T \\subset |X|$ such that the morphism $X \\setminus T \\to X$ is quasi-compact there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds} with \\begin{enumerate} \\item $E$ is $(m - r)$-pseudo-coherent, and \\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$ \\end{enumerate} approximation holds."}
+{"_id": "2767", "title": "spaces-perfect-definition-tor-independent", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. We say $X$ and $Y$ are {\\it Tor independent over $B$} if and only if for every commutative diagram $$ \\xymatrix{ \\Spec(k) \\ar[d]_{\\overline{y}} \\ar[dr]_{\\overline{b}} \\ar[r]_-{\\overline{x}} & X \\ar[d] \\\\ Y \\ar[r] & B } $$ of geometric points the rings $\\mathcal{O}_{X, \\overline{x}}$ and $\\mathcal{O}_{Y, \\overline{y}}$ are Tor independent over $\\mathcal{O}_{B, \\overline{b}}$ (see More on Algebra, Definition \\ref{more-algebra-definition-tor-independent})."}
+{"_id": "2927", "title": "dualizing-definition-essential", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item An injection $A \\subset B$ of $\\mathcal{A}$ is {\\it essential}, or we say that $B$ is an {\\it essential extension of} $A$, if every nonzero subobject $B' \\subset B$ has nonzero intersection with $A$. \\item A surjection $f : A \\to B$ of $\\mathcal{A}$ is {\\it essential} if for every proper subobject $A' \\subset A$ we have $f(A') \\not = B$. \\end{enumerate}"}
+{"_id": "2928", "title": "dualizing-definition-projective-cover", "text": "Let $R$ be a ring. A surjection $P \\to M$ of $R$-modules is said to be a {\\it projective cover}, or sometimes a {\\it projective envelope}, if $P$ is a projective $R$-module and $P \\to M$ is an essential surjection."}
+{"_id": "2929", "title": "dualizing-definition-injective-hull", "text": "Let $R$ be a ring. A injection $M \\to I$ of $R$-modules is said to be an {\\it injective hull} if $I$ is a injective $R$-module and $M \\to I$ is an essential injection."}
+{"_id": "2930", "title": "dualizing-definition-indecomposable", "text": "An object $X$ of an additive category is called {\\it indecomposable} if it is nonzero and if $X = Y \\oplus Z$, then either $Y = 0$ or $Z = 0$."}
+{"_id": "2931", "title": "dualizing-definition-dualizing", "text": "Let $A$ be a Noetherian ring. A {\\it dualizing complex} is a complex of $A$-modules $\\omega_A^\\bullet$ such that \\begin{enumerate} \\item $\\omega_A^\\bullet$ has finite injective dimension, \\item $H^i(\\omega_A^\\bullet)$ is a finite $A$-module for all $i$, and \\item $A \\to R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet)$ is a quasi-isomorphism. \\end{enumerate}"}
+{"_id": "2932", "title": "dualizing-definition-gorenstein", "text": "Gorenstein rings. \\begin{enumerate} \\item Let $A$ be a Noetherian local ring. We say $A$ is {\\it Gorenstein} if $A[0]$ is a dualizing complex for $A$. \\item Let $A$ be a Noetherian ring. We say $A$ is {\\it Gorenstein} if $A_\\mathfrak p$ is Gorenstein for every prime $\\mathfrak p$ of $A$. \\end{enumerate}"}
+{"_id": "2933", "title": "dualizing-definition-relative-dualizing-complex", "text": "Let $R \\to A$ be a flat ring map of finite presentation. A {\\it relative dualizing complex} is an object $K \\in D(A)$ such that \\begin{enumerate} \\item $K$ is $R$-perfect (More on Algebra, Definition \\ref{more-algebra-definition-relatively-perfect}), and \\item $R\\Hom_{A \\otimes_R A}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$ is isomorphic to $A$. \\end{enumerate}"}
+{"_id": "3068", "title": "properties-definition-integral", "text": "Let $X$ be a scheme. We say $X$ is {\\it integral} if it is nonempty and for every nonempty affine open $\\Spec(R) = U \\subset X$ the ring $R$ is an integral domain."}
+{"_id": "3069", "title": "properties-definition-property-local", "text": "Let $P$ be a property of rings. We say that $P$ is {\\it local} if the following hold: \\begin{enumerate} \\item For any ring $R$, and any $f \\in R$ we have $P(R) \\Rightarrow P(R_f)$. \\item For any ring $R$, and $f_i \\in R$ such that $(f_1, \\ldots, f_n) = R$ then $\\forall i, P(R_{f_i}) \\Rightarrow P(R)$. \\end{enumerate}"}
+{"_id": "3070", "title": "properties-definition-locally-P", "text": "Let $P$ be a property of rings. Let $X$ be a scheme. We say $X$ is {\\it locally $P$} if for any $x \\in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ such that $\\mathcal{O}_X(U)$ has property $P$."}
+{"_id": "3071", "title": "properties-definition-noetherian", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item We say $X$ is {\\it locally Noetherian} if every $x \\in X$ has an affine open neighbourhood $\\Spec(R) = U \\subset X$ such that the ring $R$ is Noetherian. \\item We say $X$ is {\\it Noetherian} if $X$ is locally Noetherian and quasi-compact. \\end{enumerate}"}
+{"_id": "3072", "title": "properties-definition-jacobson", "text": "A scheme $S$ is said to be {\\it Jacobson} if its underlying topological space is Jacobson."}
+{"_id": "3073", "title": "properties-definition-normal", "text": "A scheme $X$ is {\\it normal} if and only if for all $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is a normal domain."}
+{"_id": "3074", "title": "properties-definition-Cohen-Macaulay", "text": "Let $X$ be a scheme. We say $X$ is {\\it Cohen-Macaulay} if for every $x \\in X$ there exists an affine open neighbourhood $U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay."}
+{"_id": "3075", "title": "properties-definition-regular", "text": "Let $X$ be a scheme. We say $X$ is {\\it regular}, or {\\it nonsingular} if for every $x \\in X$ there exists an affine open neighbourhood $U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is Noetherian and regular."}
+{"_id": "3076", "title": "properties-definition-dimension", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The {\\it dimension} of $X$ is just the dimension of $X$ as a topological spaces, see Topology, Definition \\ref{topology-definition-Krull}. \\item For $x \\in X$ we denote $\\dim_x(X)$ the dimension of the underlying topological space of $X$ at $x$ as in Topology, Definition \\ref{topology-definition-Krull}. We say $\\dim_x(X)$ is the {\\it dimension of $X$ at $x$}. \\end{enumerate}"}
+{"_id": "3077", "title": "properties-definition-catenary", "text": "Let $S$ be a scheme. We say $S$ is {\\it catenary} if the underlying topological space of $S$ is catenary."}
+{"_id": "3078", "title": "properties-definition-Rk", "text": "Let $X$ be a locally Noetherian scheme. Let $k \\geq 0$. \\begin{enumerate} \\item We say $X$ is {\\it regular in codimension $k$}, or we say $X$ has property {\\it $(R_k)$} if for every $x \\in X$ we have $$ \\dim(\\mathcal{O}_{X, x}) \\leq k \\Rightarrow \\mathcal{O}_{X, x}\\text{ is regular} $$ \\item We say $X$ has property {\\it $(S_k)$} if for every $x \\in X$ we have $\\text{depth}(\\mathcal{O}_{X, x}) \\geq \\min(k, \\dim(\\mathcal{O}_{X, x}))$. \\end{enumerate}"}
+{"_id": "3079", "title": "properties-definition-nagata", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item Assume $X$ integral. We say $X$ is {\\it Japanese} if for every $x \\in X$ there exists an affine open neighbourhood $x \\in U \\subset X$ such that the ring $\\mathcal{O}_X(U)$ is Japanese (see Algebra, Definition \\ref{algebra-definition-N}). \\item We say $X$ is {\\it universally Japanese} if for every $x \\in X$ there exists an affine open neighbourhood $x \\in U \\subset X$ such that the ring $\\mathcal{O}_X(U)$ is universally Japanese (see Algebra, Definition \\ref{algebra-definition-nagata}). \\item We say $X$ is {\\it Nagata} if for every $x \\in X$ there exists an affine open neighbourhood $x \\in U \\subset X$ such that the ring $\\mathcal{O}_X(U)$ is Nagata (see Algebra, Definition \\ref{algebra-definition-nagata}). \\end{enumerate}"}
+{"_id": "3080", "title": "properties-definition-singular-locus", "text": "Let $X$ be a locally Noetherian scheme. The {\\it regular locus} $\\text{Reg}(X)$ of $X$ is the set of $x \\in X$ such that $\\mathcal{O}_{X, x}$ is a regular local ring. The {\\it singular locus} $\\text{Sing}(X)$ is the complement $X \\setminus \\text{Reg}(X)$, i.e., the set of points $x \\in X$ such that $\\mathcal{O}_{X, x}$ is not a regular local ring."}
+{"_id": "3081", "title": "properties-definition-unibranch", "text": "\\begin{reference} \\cite[Chapter IV (6.15.1)]{EGA4} \\end{reference} Let $X$ be a scheme. Let $x \\in X$. We say $X$ is {\\it unibranch at $x$} if the local ring $\\mathcal{O}_{X, x}$ is unibranch. We say $X$ is {\\it geometrically unibranch at $x$} if the local ring $\\mathcal{O}_{X, x}$ is geometrically unibranch. We say $X$ is {\\it unibranch} if $X$ is unibranch at all of its points. We say $X$ is {\\it geometrically unibranch} if $X$ is geometrically unibranch at all of its points."}
+{"_id": "3082", "title": "properties-definition-number-of-branches", "text": "Let $X$ be a scheme. Let $x \\in X$. The {\\it number of branches of $X$ at $x$} is the number of branches of the local ring $\\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}. The {\\it number of geometric branches of $X$ at $x$} is the number of geometric branches of the local ring $\\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}."}
+{"_id": "3083", "title": "properties-definition-quasi-affine", "text": "A scheme $X$ is called {\\it quasi-affine} if it is quasi-compact and isomorphic to an open subscheme of an affine scheme."}
+{"_id": "3084", "title": "properties-definition-locally-projective", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it locally projective} if for every affine open $U \\subset X$ the $\\mathcal{O}_X(U)$-module $\\mathcal{F}(U)$ is projective."}
+{"_id": "3085", "title": "properties-definition-kappa-generated", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be an infinite cardinal. We say a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ is {\\it $\\kappa$-generated} if there exists an open covering $X = \\bigcup U_i$ such that $\\mathcal{F}|_{U_i}$ is generated by a subset $R_i \\subset \\mathcal{F}(U_i)$ whose cardinality is at most $\\kappa$."}
+{"_id": "3086", "title": "properties-definition-subsheaf-sections-annihilated-by-ideal", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in Lemma \\ref{lemma-sections-annihilated-by-ideal} above is called the {\\it subsheaf of sections annihilated by $\\mathcal{I}$}."}
+{"_id": "3087", "title": "properties-definition-subsheaf-sections-supported-on-closed", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset whose complement is retrocompact in $X$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in Lemma \\ref{lemma-sections-supported-on-closed-subset} is called the {\\it subsheaf of sections supported on $T$}."}
+{"_id": "3088", "title": "properties-definition-ample", "text": "\\begin{reference} \\cite[II Definition 4.5.3]{EGA} \\end{reference} Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We say $\\mathcal{L}$ is {\\it ample} if \\begin{enumerate} \\item $X$ is quasi-compact, and \\item for every $x \\in X$ there exists an $n \\geq 1$ and $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that $x \\in X_s$ and $X_s$ is affine. \\end{enumerate}"}
+{"_id": "3145", "title": "criteria-definition-algebraic", "text": "Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. We say that $F$ is {\\it algebraic} if for every scheme $T$ and every object $\\xi$ of $\\mathcal{Y}$ over $T$ the $2$-fibre product $$ (\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{X} $$ is an algebraic stack over $S$."}
+{"_id": "3403", "title": "coherent-definition-depth", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $k \\geq 0$ be an integer. \\begin{enumerate} \\item We say $\\mathcal{F}$ has {\\it depth $k$ at a point} $x$ of $X$ if $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) = k$. \\item We say $X$ has {\\it depth $k$ at a point} $x$ of $X$ if $\\text{depth}(\\mathcal{O}_{X, x}) = k$. \\item We say $\\mathcal{F}$ has property {\\it $(S_k)$} if $$ \\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) \\geq \\min(k, \\dim(\\text{Supp}(\\mathcal{F}_x))) $$ for all $x \\in X$. \\item We say $X$ has property {\\it $(S_k)$} if $\\mathcal{O}_X$ has property $(S_k)$. \\end{enumerate}"}
+{"_id": "3404", "title": "coherent-definition-Cohen-Macaulay", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it Cohen-Macaulay} if and only if $(S_k)$ holds for all $k \\geq 0$."}
+{"_id": "3405", "title": "coherent-definition-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $Z \\subset X$ be a closed subset. We say {\\it $Z$ is proper over $S$} if the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base} are satisfied."}
+{"_id": "3510", "title": "formal-defos-definition-CLambda", "text": "Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite ring map where $k$ is a field. We define {\\it $\\mathcal{C}_\\Lambda$} to be the category with \\begin{enumerate} \\item objects are pairs $(A, \\varphi)$ where $A$ is an Artinian local $\\Lambda$-algebra and where $\\varphi : A/\\mathfrak m_A \\to k$ is a $\\Lambda$-algebra isomorphism, and \\item morphisms $f : (B, \\psi) \\to (A, \\varphi)$ are local $\\Lambda$-algebra homomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$. \\end{enumerate} We say we are in the {\\it classical case} if $\\Lambda$ is a Noetherian complete local ring and $k$ is its residue field."}
+{"_id": "3511", "title": "formal-defos-definition-small-extension", "text": "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.  We say $f$ is a {\\it small extension} if it is surjective and $\\Ker(f)$ is a nonzero principal ideal which is annihilated by $\\mathfrak{m}_B$."}
+{"_id": "3512", "title": "formal-defos-definition-tangent-space-ring", "text": "Let $R \\to S$ be a local homomorphism of local rings. The {\\it relative cotangent space}\\footnote{Caution: We will see later that in our general setting the tangent space of an object $A \\in \\mathcal{C}_\\Lambda$ over $\\Lambda$ should not be defined simply as the $k$-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is $\\Omega_{S/R} \\otimes_S S/\\mathfrak m_S$.} of $R$ over $S$ is the $S/\\mathfrak m_S$-vector space $\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2)$."}
+{"_id": "3513", "title": "formal-defos-definition-essential-surjection", "text": "Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.  We say $f$ is an {\\it essential surjection} if it has the following properties: \\begin{enumerate} \\item $f$ is surjective. \\item If $g: C \\to B$ is a ring map in $\\mathcal{C}_\\Lambda$ such that $f \\circ g$ is surjective, then $g$ is surjective. \\end{enumerate}"}
+{"_id": "3514", "title": "formal-defos-definition-completion-CLambda", "text": "Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite ring map where $k$ is a field. We define {\\it $\\widehat{\\mathcal{C}}_\\Lambda$} to be the category with \\begin{enumerate} \\item objects are pairs $(R, \\varphi)$ where $R$ is a Noetherian complete local $\\Lambda$-algebra and where $\\varphi : R/\\mathfrak m_R \\to k$ is a $\\Lambda$-algebra isomorphism, and \\item morphisms $f : (S, \\psi) \\to (R, \\varphi)$ are local $\\Lambda$-algebra homomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$. \\end{enumerate}"}
+{"_id": "3515", "title": "formal-defos-definition-category-cofibred-groupoids", "text": "Let $\\mathcal{C}$ be a category.  A {\\it category cofibered in groupoids over $\\mathcal{C}$} is a category $\\mathcal{F}$ equipped with a functor $p: \\mathcal{F} \\to \\mathcal{C}$ such that $\\mathcal{F}^{opp}$ is a category fibered in groupoids over $\\mathcal{C}^{opp}$ via $p^{opp}: \\mathcal{F}^{opp} \\to \\mathcal{C}^{opp}$."}
+{"_id": "3516", "title": "formal-defos-definition-prorepresentable", "text": "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor. We say $F$ is {\\it prorepresentable} if there exists an isomorphism $F \\cong \\underline{R}|_{\\mathcal{C}_\\Lambda}$ of functors for some $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$."}
+{"_id": "3517", "title": "formal-defos-definition-predeformation-category", "text": "A {\\it predeformation category} $\\mathcal{F}$ is a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ such that $\\mathcal{F}(k)$ is equivalent to a category with a single object and a single morphism, i.e., $\\mathcal{F}(k)$ contains at least one object and there is a unique morphism between any two objects. A {\\it morphism of predeformation categories} is a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$."}
+{"_id": "3518", "title": "formal-defos-definition-formal-objects", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. The {\\it category $\\widehat{\\mathcal{F}}$ of formal objects of  $\\mathcal{F}$} is the category with the following objects and morphisms. \\begin{enumerate} \\item A {\\it formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{F}$} consists of an object $R$ of $\\widehat{\\mathcal{C}}_\\Lambda$, and a collection indexed by $n \\in \\mathbf{N}$ of objects $\\xi_n$ of $\\mathcal{F}(R/\\mathfrak m_R^n)$ and morphisms $f_n : \\xi_{n + 1} \\to \\xi_n$ lying over the projection $R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$. \\item Let $\\xi = (R, \\xi_n, f_n)$ and $\\eta = (S, \\eta_n, g_n)$ be formal objects of $\\mathcal{F}$.  A {\\it morphism $a : \\xi \\to \\eta$ of formal objects} consists of a map $a_0 : R \\to S$ in $\\widehat{\\mathcal{C}}_\\Lambda$ and a collection $a_n : \\xi_n \\to \\eta_n$ of morphisms of $\\mathcal{F}$ lying over $R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$, such that for every $n$ the diagram $$ \\xymatrix{ \\xi_{n + 1} \\ar[r]_{f_n} \\ar[d]_{a_{n + 1}} & \\xi_n \\ar[d]^{a_n} \\\\ \\eta_{n + 1} \\ar[r]^{g_n} & \\eta_n } $$ commutes. \\end{enumerate}"}
+{"_id": "3519", "title": "formal-defos-definition-completion", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids. The category cofibered in groupoids $\\widehat{p} : \\widehat{\\mathcal  F} \\to \\widehat{\\mathcal{C}}_\\Lambda$ is called the {\\it completion of $\\mathcal{F}$}."}
+{"_id": "3520", "title": "formal-defos-definition-smooth-morphism", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$.  We say  $\\varphi$ is {\\it smooth} if it satisfies the following condition: Let $B \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$.  Let $y \\in \\Ob(\\mathcal{G}(B)), x \\in \\Ob(\\mathcal{F}(A))$, and $y \\to \\varphi(x)$ be a morphism lying over $B \\to A$.  Then there exists $x' \\in \\Ob(\\mathcal{F}(B))$, a morphism $x' \\to x$ lying over $B \\to A$, and a morphism $\\varphi(x') \\to y$ lying over $\\text{id}: B \\to B$, such that the diagram $$ \\xymatrix{ \\varphi(x') \\ar[r] \\ar[dr] & y \\ar[d] \\\\ & \\varphi(x) } $$ commutes."}
+{"_id": "3521", "title": "formal-defos-definition-versal", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids.  Let $\\xi$ be a formal object of $\\mathcal{F}$ lying over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. We say $\\xi$ is {\\it versal} if the corresponding morphism $\\underline{\\xi}: \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ of Remark \\ref{remark-formal-objects-yoneda} is smooth."}
+{"_id": "3522", "title": "formal-defos-definition-cofibered-groupoid-projection-smooth", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids. We say $\\mathcal{F}$ is {\\it smooth} or {\\it unobstructed} if its structure morphism $p$ is smooth in the sense of Definition \\ref{definition-smooth-morphism}."}
+{"_id": "3523", "title": "formal-defos-definition-S1-S2", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$. We define {\\it conditions (S1) and (S2)} on $\\mathcal{F}$ as follows: \\begin{enumerate} \\item[(S1)] Every diagram in $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{            & x_2 \\ar[d] \\\\ x_1 \\ar[r] & x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{            & A_2 \\ar[d] \\\\ A_1 \\ar[r] & A } } $$ in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective can be completed to a commutative diagram $$ \\vcenter{ \\xymatrix{ y \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\ x_1 \\ar[r]      & x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{ A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\ A_1 \\ar[r]      & A. } } $$ \\item[(S2)] The condition of (S1) holds for diagrams in $\\mathcal{F}$ lying over a diagram in $\\mathcal{C}_\\Lambda$ of the form $$ \\xymatrix{           & k[\\epsilon] \\ar[d] \\\\ A  \\ar[r] & k. } $$ Moreover, if we have two commutative diagrams in $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{ y \\ar[r]_c \\ar[d]_a & x_\\epsilon \\ar[d]^e \\\\ x \\ar[r]^d          & x_0 } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ y' \\ar[r]_{c'} \\ar[d]_{a'} & x_\\epsilon \\ar[d]^e \\\\ x \\ar[r]^d                 & x_0 } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{ A \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\ A  \\ar[r] & k } } $$ then there exists a morphism $b : y \\to y'$ in $\\mathcal{F}(A \\times_k k[\\epsilon])$ such that $a = a' \\circ b$. \\end{enumerate}"}
+{"_id": "3524", "title": "formal-defos-definition-linear", "text": "Let $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$, resp.\\ $L: \\text{Mod}_R \\to \\text{Mod}_R$ be a functor.  We say that $L$ is {\\it $R$-linear} if for every pair of objects $M, N$ of $\\text{Mod}^{fg}_R$, resp.\\ $\\text{Mod}_R$ the map $$ L : \\Hom_R(M, N) \\longrightarrow \\Hom_R(L(M), L(N)) $$ is a map of $R$-modules."}
+{"_id": "3525", "title": "formal-defos-definition-tangent-space-over-R", "text": "Let $\\mathcal{C}$ be a category as in Lemma \\ref{lemma-tangent-space-functor}. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that $F(R)$ is a one element set. The {\\it tangent space $TF$ of $F$} is $F(R[\\epsilon])$."}
+{"_id": "3526", "title": "formal-defos-definition-tangent-space", "text": "Let $\\mathcal{F}$ be a predeformation category. The {\\it tangent space $T \\mathcal{F}$ of $\\mathcal{F}$} is the set $\\overline{\\mathcal{F}}(k[\\epsilon])$ of isomorphism classes of objects in the fiber category $\\mathcal F(k[\\epsilon])$."}
+{"_id": "3527", "title": "formal-defos-definition-differential", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism predeformation categories. The {\\it differential $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ of $\\varphi$} is the map obtained by evaluating the morphism of functors $\\overline{\\varphi}: \\overline{\\mathcal{F}} \\to \\overline{\\mathcal{G}}$ at $A = k[\\epsilon]$."}
+{"_id": "3528", "title": "formal-defos-definition-minimal-versal", "text": "Let $\\mathcal{F}$ be a predeformation category. We say a versal formal object $\\xi$ of $\\mathcal{F}$ is {\\it minimal}\\footnote{This may be nonstandard terminology. Many authors tie this notion in with properties of tangent spaces. We will make the link in Section \\ref{section-miniversal-objects-existence}.} if for any morphism of formal objects $\\xi' \\to \\xi$ the underlying map on rings is surjective. Sometimes a minimal versal formal object is called {\\it miniversal}."}
+{"_id": "3529", "title": "formal-defos-definition-RS", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$.  We say that $\\mathcal{F}$ satisfies {\\it condition (RS)} if for every diagram in $\\mathcal{F}$ $$ \\vcenter{ \\xymatrix{            & x_2 \\ar[d] \\\\ x_1 \\ar[r] & x } } \\quad\\text{lying over}\\quad \\vcenter{ \\xymatrix{            & A_2 \\ar[d] \\\\ A_1 \\ar[r] & A } } $$ in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, there exists a fiber product $x_1 \\times_x x_2$ in $\\mathcal{F}$ such that the diagram $$ \\vcenter{ \\xymatrix{ x_1 \\times_x x_2 \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\ x_1 \\ar[r]      & x } } \\quad\\text{lies over}\\quad \\vcenter{ \\xymatrix{ A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\ A_1 \\ar[r]      & A. } } $$"}
+{"_id": "3530", "title": "formal-defos-definition-deformation-category", "text": "A {\\it deformation category} is a predeformation category $\\mathcal{F}$ satisfying (RS). A morphism of deformation categories is a morphism of categories over $\\mathcal{C}_\\Lambda$."}
+{"_id": "3531", "title": "formal-defos-definition-lifts", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $f: A' \\to A$ be a map in $\\mathcal{C}_\\Lambda$. Let $x \\in \\mathcal{F}(A)$. The category $\\textit{Lift}(x, f)$ of lifts of $x$ along $f$ is the category with the following objects and morphisms. \\begin{enumerate} \\item Objects: A {\\it lift of $x$ along $f$} is a morphism $x' \\to x$ lying over $f$. \\item Morphisms: A {\\it morphism of lifts} from $a_1 : x'_1 \\to x$ to $a_2 : x'_2 \\to x$ is a morphism $b : x'_1 \\to x'_2$ in $\\mathcal{F}(A')$ such that $a_2 = a_1 \\circ b$. \\end{enumerate} The set $\\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of isomorphism classes of $\\textit{Lift}(x, f)$."}
+{"_id": "3532", "title": "formal-defos-definition-relative-infinitesimal-auts", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$. Let $x' \\to x$ be a morphism in $\\mathcal{F}$ lying over $A' \\to A$. The kernel $$ \\text{Inf}(x'/x) = \\Ker(\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)) $$ is the {\\it group of infinitesimal automorphisms of $x'$ over $x$}."}
+{"_id": "3533", "title": "formal-defos-definition-infinitesimal-auts", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Assume a choice of pushforward $x_0 \\to x_0'$ of $x_0$ along the map $k \\to k[\\epsilon], a \\mapsto a$ has been made. Then there is a unique map $x'_0 \\to x_0$ such that $x_0 \\to x_0' \\to x_0$ is the identity on $x_0$. Then $$ \\text{Inf}_{x_0}(\\mathcal F) = \\text{Inf}(x'_0/x_0) $$ is the {\\it group of infinitesimal automorphisms of $x_0$}"}
+{"_id": "3534", "title": "formal-defos-definition-automorphism-functor", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}$ be a category cofibered in groupoids over an arbitrary base category $\\mathcal{C}$. Assume a choice of pushforwards has been made. Let $x \\in \\Ob(\\mathcal{F})$ and let $U = p(x)$. Let $U/\\mathcal{C}$ denote the category of objects under $U$. The {\\it automorphism functor of $x$} is the functor $\\mathit{Aut}(x) : U/\\mathcal{C} \\to \\textit{Sets}$ sending an object $f : U \\to V$ to $\\text{Aut}_V(f_*x)$ and sending a morphism $$ \\xymatrix{ V' \\ar[rr] &                    & V\\\\           & U \\ar[ul]^{f'}  \\ar[ur]_f & } $$ to the homomorphism $\\text{Aut}_{V'}(f'_*x) \\to \\text{Aut}_V(f_*x)$ coming from the unique morphism $f'_*x \\to f_*x$ lying over $V' \\to V$ and compatible with $x \\to f'_*x$ and $x \\to f_*x$."}
+{"_id": "3535", "title": "formal-defos-definition-groupoid-in-functors", "text": "Let $\\mathcal{C}$ be a category. The {\\it category of groupoids in functors on $\\mathcal{C}$} is the category with the following objects and morphisms. \\begin{enumerate} \\item Objects: A {\\it groupoid in functors on $\\mathcal{C}$} is a quintuple $(U, R, s, t, c)$ where $U, R : \\mathcal{C} \\to \\textit{Sets}$ are functors and $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$ are morphisms with the following property: For any object $T$ of $\\mathcal{C}$, the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category. \\item Morphisms: A {\\it morphism $(U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoids in functors on $\\mathcal{C}$} consists of morphisms $U \\to U'$ and $R \\to R'$ with the following property: For any object $T$ of $\\mathcal{C}$, the induced maps $U(T) \\to U'(T)$ and $R(T) \\to R'(T)$ define a functor between groupoid categories $$ (U(T), R(T), s, t, c) \\to (U'(T), R'(T), s', t', c'). $$ \\end{enumerate}"}
+{"_id": "3536", "title": "formal-defos-definition-representable", "text": "Let $\\mathcal{C}$ be a category. A groupoid in functors on $\\mathcal{C}$ is {\\it representable} if it is isomorphic to one of the form $(\\underline{U}, \\underline{R}, s, t, c)$ where $U$ and $R$ are objects of $\\mathcal{C}$ and the pushout $R \\amalg_{s, U, t} R$ exists."}
+{"_id": "3537", "title": "formal-defos-definition-restricting-groupoids-in-functors", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$. Let $\\mathcal{C}'$ be a subcategory of $\\mathcal{C}$. The {\\it restriction $(U, R, s, t, c)|_{\\mathcal{C}'}$ of $(U, R, s, t, c)$ to $\\mathcal{C}'$} is the groupoid in functors on $\\mathcal{C}'$ given by $(U|_{\\mathcal{C}'}, R|_{\\mathcal C'}, s|_{\\mathcal{C}'}, t|_{\\mathcal{C}'}, c|_{\\mathcal{C}'})$."}
+{"_id": "3538", "title": "formal-defos-definition-quotient", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$. \\begin{enumerate} \\item The assignment $T \\mapsto  (U(T), R(T), s, t, c)$ determines a functor $\\mathcal{C} \\to \\textit{Groupoids}$. The {\\it quotient category cofibered in groupoids $[U/R] \\to \\mathcal{C}$} is the category cofibered in groupoids over $\\mathcal{C}$ associated to this functor (as in Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-construction-associated-cofibered-groupoid})). \\item The {\\it quotient morphism $U \\to [U/R]$} is the morphism of categories cofibered in groupoids over $\\mathcal{C}$ defined by the rules \\begin{enumerate} \\item $x \\in U(T)$ maps to the object $(T, x) \\in \\Ob([U/R](T))$, and \\item $x \\in U(T)$ and $f : T \\to T'$ give rise to the morphism $(f, \\text{id}_{U(f)(x)}): (T, x) \\to (T, U(f)(x))$ lying over $f : T \\to T'$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "3539", "title": "formal-defos-definition-prorepresentable-groupoid-in-functors", "text": "A groupoid in functors on $\\mathcal{C}_\\Lambda$ is {\\it prorepresentable} if it is isomorphic to $(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$ for some representable groupoid in functors $(\\underline{R_0}, \\underline{R_1}, s, t, c)$ on the category $\\widehat{\\mathcal{C}}_\\Lambda$."}
+{"_id": "3540", "title": "formal-defos-definition-completion-groupoid-in-functors", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. The {\\it completion $(U, R, s, t, c)^{\\wedge}$ of $(U, R, s, t, c)$} is the groupoid in functors $(\\widehat{U}, \\widehat{R}, \\widehat{s}, \\widehat{t}, \\widehat{c})$ on $\\widehat{\\mathcal{C}}_\\Lambda$ described above."}
+{"_id": "3541", "title": "formal-defos-definition-smooth-groupoid-in-functors", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. We say $(U, R, s, t, c)$ is {\\it smooth} if $s, t: R \\to U$ are smooth."}
+{"_id": "3542", "title": "formal-defos-definition-presentation", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over a category $\\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}$. A {\\it presentation of $\\mathcal{F}$ by $(U, R, s, t, c)$} is an equivalence $\\varphi : [U/R] \\to \\mathcal{F}$ of categories cofibered in groupoids over $\\mathcal{C}$."}
+{"_id": "3543", "title": "formal-defos-definition-minimal-groupoid-in-functors", "text": "Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$. \\begin{enumerate} \\item We say $(U, R, s, t, c)$ is {\\it normalized} if the groupoid $(U(k[\\epsilon]), R(k[\\epsilon]), s, t, c)$ is totally disconnected, i.e., there are no morphisms between distinct objects. \\item We say $(U, R, s, t, c)$ is {\\it minimal} if the $U \\to [U/R]$ is given by a minimal versal formal object of $[U/R]$. \\end{enumerate}"}
+{"_id": "3637", "title": "adequate-definition-module-valued-functor", "text": "Let $A$ be a ring. A {\\it module-valued functor} is a functor $F : \\textit{Alg}_A \\to \\textit{Ab}$ such that \\begin{enumerate} \\item for every object $B$ of $\\textit{Alg}_A$ the group $F(B)$ is endowed with the structure of a $B$-module, and \\item for any morphism $B \\to B'$ of $\\textit{Alg}_A$ the map $F(B) \\to F(B')$ is $B$-linear. \\end{enumerate} A {\\it morphism of module-valued functors} is a transformation of functors $\\varphi : F \\to G$ such that $F(B) \\to G(B)$ is $B$-linear for all $B \\in \\Ob(\\textit{Alg}_A)$."}
+{"_id": "3638", "title": "adequate-definition-adequate-functor", "text": "Let $A$ be a ring. A module-valued functor $F$ on $\\textit{Alg}_A$ is called \\begin{enumerate} \\item {\\it adequate} if there exists a map of $A$-modules $M \\to N$ such that $F$ is isomorphic to $\\Ker(\\underline{M} \\to \\underline{N})$. \\item {\\it linearly adequate} if $F$ is isomorphic to the kernel of a map $\\underline{A^{\\oplus n}} \\to \\underline{A^{\\oplus m}}$. \\end{enumerate}"}
+{"_id": "3639", "title": "adequate-definition-adequate", "text": "A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is {\\it adequate} if there exists a $\\tau$-covering $\\{\\Spec(A_i) \\to S\\}_{i \\in I}$ such that $F_{\\mathcal{F}, A_i}$ is adequate for all $i \\in I$."}
+{"_id": "3640", "title": "adequate-definition-category-adequate-modules", "text": "Let $S$ be a scheme. The category of adequate $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$ is denoted {\\it $\\textit{Adeq}(\\mathcal{O})$} or {\\it $\\textit{Adeq}((\\Sch/S)_\\tau, \\mathcal{O})$}. If we want to think just about the abelian category of adequate modules without choosing a topology we simply write {\\it $\\textit{Adeq}(S)$}."}
+{"_id": "3641", "title": "adequate-definition-pure", "text": "Let $A$ be a ring. \\begin{enumerate} \\item An $A$-module $P$ is said to be {\\it pure projective} if for every universally exact sequence $0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence $0 \\to \\Hom_A(P, K) \\to \\Hom_A(P, M) \\to \\Hom_A(P, N) \\to 0$ is exact. \\item An $A$-module $I$ is said to be {\\it pure injective} if for every universally exact sequence $0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence $0 \\to \\Hom_A(N, I) \\to \\Hom_A(M, I) \\to \\Hom_A(K, I) \\to 0$ is exact. \\end{enumerate}"}
+{"_id": "3642", "title": "adequate-definition-pure-resolution", "text": "Let $A$ be a ring. Let $M$ be an $A$-module. \\begin{enumerate} \\item A {\\it pure projective resolution} $P_\\bullet \\to M$ is a universally exact sequence $$ \\ldots \\to P_1 \\to P_0 \\to M \\to 0 $$ with each $P_i$ pure projective. \\item A {\\it pure injective resolution} $M \\to I^\\bullet$ is a universally exact sequence $$ 0 \\to M \\to I^0 \\to I^1 \\to \\ldots $$ with each $I^i$ pure injective. \\end{enumerate}"}
+{"_id": "3643", "title": "adequate-definition-pure-ext", "text": "Let $A$ be a ring and let $M$, $N$ be $A$-modules. The $i$th {\\it pure extension module} $\\text{Pext}^i_A(M, N)$ is the $i$th cohomology module of the complex $\\Hom_A(M, I^\\bullet)$ where $I^\\bullet$ is a pure injective resolution of $N$."}
+{"_id": "3681", "title": "spaces-topologies-definition-zariski-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\\it Zariski covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that each $f_i$ is an open immersion and such that $$ |X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective."}
+{"_id": "3682", "title": "spaces-topologies-definition-etale-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\\it \\'etale covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that each $f_i$ is \\'etale and such that $$ |X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective."}
+{"_id": "3683", "title": "spaces-topologies-definition-big-etale-site", "text": "Let $S$ be a scheme. A big \\'etale site {\\it $(\\textit{Spaces}/S)_\\etale$} is any site constructed as follows: \\begin{enumerate} \\item Choose a big \\'etale site $(\\Sch/S)_\\etale$ as in Topologies, Section \\ref{topologies-section-etale}. \\item As underlying category take the category $\\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \\ref{section-procedure} why this is a set). \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\textit{Spaces}/S$ and the class of \\'etale coverings of Definition \\ref{definition-etale-covering}. \\end{enumerate}"}
+{"_id": "3684", "title": "spaces-topologies-definition-big-small-etale", "text": "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_\\etale$ be as in Definition \\ref{definition-big-etale-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\\textit{Spaces}/S)_\\etale$. Then the big \\'etale site {\\it $(\\textit{Spaces}/X)_\\etale$} of $X$ is the localization of the site $(\\textit{Spaces}/S)_\\etale$ at $X$ introduced in Sites, Section \\ref{sites-section-localize}."}
+{"_id": "3685", "title": "spaces-topologies-definition-restriction-small-etale", "text": "In the situation of Lemma \\ref{lemma-at-the-bottom-etale} the functor $i_X^{-1} = \\pi_{X, *}$ is often called the {\\it restriction to the small \\'etale site}, and for a sheaf $\\mathcal{F}$ on the big \\'etale site we often denote $\\mathcal{F}|_{X_\\etale}$ this restriction."}
+{"_id": "3686", "title": "spaces-topologies-definition-smooth-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\\it smooth covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that each $f_i$ is smooth and such that $$ |X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective."}
+{"_id": "3687", "title": "spaces-topologies-definition-syntomic-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A {\\it syntomic covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that each $f_i$ is syntomic and such that $$ |X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective."}
+{"_id": "3688", "title": "spaces-topologies-definition-fppf-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\\it fppf covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that each $f_i$ is flat and locally of finite presentation and such that $$ |X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|), $$ i.e., the morphisms are jointly surjective."}
+{"_id": "3689", "title": "spaces-topologies-definition-big-fppf-site", "text": "Let $S$ be a scheme. A big fppf site {\\it $(\\textit{Spaces}/S)_{fppf}$} is any site constructed as follows: \\begin{enumerate} \\item Choose a big fppf site $(\\Sch/S)_{fppf}$ as in Topologies, Section \\ref{topologies-section-fppf}. \\item As underlying category take the category $\\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \\ref{section-procedure} why this is a set). \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\textit{Spaces}/S$ and the class of fppf coverings of Definition \\ref{definition-fppf-covering}. \\end{enumerate}"}
+{"_id": "3690", "title": "spaces-topologies-definition-big-small-fppf", "text": "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{fppf}$ be as in Definition \\ref{definition-big-fppf-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\\textit{Spaces}/S)_{fppf}$. Then the big fppf site {\\it $(\\textit{Spaces}/X)_{fppf}$} of $X$ is the localization of the site $(\\textit{Spaces}/S)_{fppf}$ at $X$ introduced in Sites, Section \\ref{sites-section-localize}."}
+{"_id": "3691", "title": "spaces-topologies-definition-ph-covering", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. A {\\it ph covering of $X$} is a family of morphisms $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$ such that $f_i$ is locally of finite type and such that for every $U \\to X$ with $U$ affine there exists a standard ph covering $\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family $\\{X_i \\times_X U \\to U\\}_{i \\in I}$."}
+{"_id": "3692", "title": "spaces-topologies-definition-big-ph-site", "text": "Let $S$ be a scheme. A big ph site {\\it $(\\textit{Spaces}/S)_{ph}$} is any site constructed as follows: \\begin{enumerate} \\item Choose a big ph site $(\\Sch/S)_{ph}$ as in Topologies, Section \\ref{topologies-section-ph}. \\item As underlying category take the category $\\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section \\ref{section-procedure} why this is a set). \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\textit{Spaces}/S$ and the class of ph coverings of Definition \\ref{definition-ph-covering}. \\end{enumerate}"}
+{"_id": "3693", "title": "spaces-topologies-definition-big-small-ph", "text": "Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{ph}$ be as in Definition \\ref{definition-big-ph-site}. Let $X$ be an algebraic space over $S$, i.e., an object of $(\\textit{Spaces}/S)_{ph}$. Then the big ph site {\\it $(\\textit{Spaces}/X)_{ph}$} of $X$ is the localization of the site $(\\textit{Spaces}/S)_{ph}$ at $X$ introduced in Sites, Section \\ref{sites-section-localize}."}
+{"_id": "3694", "title": "spaces-topologies-definition-fpqc-covering", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An {\\it fpqc covering of $X$} is a family of morphisms $\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces such that each $f_i$ is flat and such that for every affine scheme $Z$ and morphism $h : Z \\to X$ there exists a standard fpqc covering $\\{g_j : Z_j \\to Z\\}_{j = 1, \\ldots, m}$ which refines the family $\\{X_i \\times_X Z \\to Z\\}_{i \\in I}$."}
+{"_id": "3830", "title": "proetale-definition-w-local", "text": "A spectral space $X$ is {\\it w-local} if the set of closed points $X_0$ is closed and every point of $X$ specializes to a unique closed point. A continuous map $f : X \\to Y$ of w-local spaces is {\\it w-local} if it is spectral and maps any closed point of $X$ to a closed point of $Y$."}
+{"_id": "3831", "title": "proetale-definition-local-isomorphism", "text": "Let $\\varphi : A \\to B$ be a ring map. \\begin{enumerate} \\item We say $A \\to B$ is a {\\it local isomorphism} if for every prime $\\mathfrak q \\subset B$ there exists a $g \\in B$, $g \\not \\in \\mathfrak q$ such that $A \\to B_g$ induces an open immersion $\\Spec(B_g) \\to \\Spec(A)$. \\item We say $A \\to B$ {\\it identifies local rings} if for every prime $\\mathfrak q \\subset B$ the canonical map $A_{\\varphi^{-1}(\\mathfrak q)} \\to B_\\mathfrak q$ is an isomorphism. \\end{enumerate}"}
+{"_id": "3832", "title": "proetale-definition-ind-zariski", "text": "A ring map $A \\to B$ is said to be {\\it ind-Zariski} if $B$ can be written as a filtered colimit $B = \\colim B_i$ with each $A \\to B_i$ a local isomorphism."}
+{"_id": "3833", "title": "proetale-definition-ind-etale", "text": "A ring map $A \\to B$ is said to be {\\it ind-\\'etale} if $B$ can be written as a filtered colimit of \\'etale $A$-algebras."}
+{"_id": "3834", "title": "proetale-definition-w-contractible", "text": "Let $A$ be a ring. We say $A$ is {\\it w-contractible} if every faithfully flat weakly \\'etale ring map $A \\to B$ has a section."}
+{"_id": "3835", "title": "proetale-definition-fpqc-covering", "text": "Let $T$ be a scheme. A {\\it pro-\\'etale covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is weakly-\\'etale and such that for every affine open $U \\subset T$ there exists $n \\geq 0$, a map $a : \\{1, \\ldots, n\\} \\to I$ and affine opens $V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$ with $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$."}
+{"_id": "3836", "title": "proetale-definition-standard-proetale", "text": "Let $T$ be an affine scheme. A {\\it standard pro-\\'etale covering} of $T$ is a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$ where each $T_j$ is affine, each $f_i$ is weakly \\'etale, and $T = \\bigcup f_i(T_i)$."}
+{"_id": "3837", "title": "proetale-definition-big-proetale-site", "text": "A {\\it big pro-\\'etale site} is any site $\\Sch_\\proetale$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of pro-\\'etale coverings $\\text{Cov}_0$ among these schemes. \\item Change the function $Bound$ of Sets, Equation (\\ref{sets-equation-bound}) into $$ Bound(\\kappa) = \\max\\{\\kappa^{2^{2^{2^\\kappa}}}, \\kappa^{\\aleph_0}, \\kappa^+\\}. $$ \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$ and the function $Bound$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of pro-\\'etale coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "3838", "title": "proetale-definition-big-small-proetale", "text": "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site containing $S$. \\begin{enumerate} \\item The {\\it big pro-\\'etale site of $S$}, denoted $(\\Sch/S)_\\proetale$, is the site $\\Sch_\\proetale/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it small pro-\\'etale site of $S$}, which we denote $S_\\proetale$, is the full subcategory of $(\\Sch/S)_\\proetale$ whose objects are those $U/S$ such that $U \\to S$ is weakly \\'etale. A covering of $S_\\proetale$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\proetale$ with $U \\in \\Ob(S_\\proetale)$. \\item The {\\it big affine pro-\\'etale site of $S$}, denoted $(\\textit{Aff}/S)_\\proetale$, is the full subcategory of $(\\Sch/S)_\\proetale$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_\\proetale$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\proetale$ which is a standard pro-\\'etale covering. \\end{enumerate}"}
+{"_id": "3839", "title": "proetale-definition-restriction-small-proetale", "text": "In the situation of Lemma \\ref{lemma-at-the-bottom} the functor $i_S^{-1} = \\pi_{S, *}$ is often called the {\\it restriction to the small pro-\\'etale site}, and for a sheaf $\\mathcal{F}$ on the big pro-\\'etale site we denote $\\mathcal{F}|_{S_\\proetale}$ this restriction."}
+{"_id": "3840", "title": "proetale-definition-extension-zero", "text": "Let $j : U \\to X$ be a weakly \\'etale morphism of schemes. \\begin{enumerate} \\item The restriction functor $j^{-1} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$ has a left adjoint $j_!^{Sh} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$. \\item The restriction functor $j^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(U_\\proetale)$ has a left adjoint which is denoted $j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$ and called {\\it extension by zero}. \\item Let $\\Lambda$ be a ring. The functor $j^{-1} : \\textit{Mod}(X_\\proetale, \\Lambda) \\to \\textit{Mod}(U_\\proetale, \\Lambda)$ has a left adjoint $j_! : \\textit{Mod}(U_\\proetale, \\Lambda) \\to \\textit{Mod}(X_\\proetale, \\Lambda)$ and called {\\it extension by zero}. \\end{enumerate}"}
+{"_id": "3841", "title": "proetale-definition-constructible", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules on $X_\\proetale$ is {\\it constructible} if for every affine open $U \\subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$ such that $\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$."}
+{"_id": "3842", "title": "proetale-definition-adic", "text": "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $X$ be a scheme. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\proetale$. \\begin{enumerate} \\item We say $\\mathcal{F}$ is a {\\it constructible $\\Lambda$-sheaf} if $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ and each $\\mathcal{F}/I^n\\mathcal{F}$ is a constructible sheaf of $\\Lambda/I^n$-modules. \\item If $\\mathcal{F}$ is a constructible $\\Lambda$-sheaf, then we say $\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}/I^n\\mathcal{F}$ is locally constant. \\item We say $\\mathcal{F}$ is {\\it adic lisse}\\footnote{This may be nonstandard notation.} if there exists a $I$-adically complete $\\Lambda$-module $M$ with $M/IM$ finite such that $\\mathcal{F}$ is locally isomorphic to $$ \\underline{M}^\\wedge = \\lim \\underline{M/I^nM}. $$ \\item We say $\\mathcal{F}$ is {\\it adic constructible}\\footnote{This may be nonstandard notation.} if for every affine open $U \\subset X$ there exists a decomposition $U = \\coprod U_i$ into constructible locally closed subschemes such that $\\mathcal{F}|_{U_i}$ is adic lisse. \\end{enumerate}"}
+{"_id": "3843", "title": "proetale-definition-Dbc", "text": "Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $X$ be a scheme. An object $K$ of $D(X_\\proetale, \\Lambda)$ is called {\\it constructible} if \\begin{enumerate} \\item $K$ is derived complete with respect to $I$, \\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ has constructible cohomology sheaves and locally has finite tor dimension. \\end{enumerate} We denote $D_{cons}(X, \\Lambda)$ the full subcategory of constructible $K$ in $D(X_\\proetale, \\Lambda)$."}
+{"_id": "3844", "title": "proetale-definition-adic-constructible", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $K \\in D(X_\\proetale, \\Lambda)$. \\begin{enumerate} \\item We say $K$ is {\\it adic lisse}\\footnote{This may be nonstandard notation} if there exists a finite complex of finite projective $\\Lambda^\\wedge$-modules $M^\\bullet$ such that $K$ is locally isomorphic to $$ \\underline{M^a}^\\wedge \\to \\ldots \\to \\underline{M^b}^\\wedge $$ \\item We say $K$ is {\\it adic constructible}\\footnote{This may be nonstandard notation.} if for every affine open $U \\subset X$ there exists a decomposition $U = \\coprod U_i$ into constructible locally closed subschemes such that $K|_{U_i}$ is adic lisse. \\end{enumerate}"}
+{"_id": "3974", "title": "formal-spaces-definition-toplogy-tensor-product", "text": "Let $R$ be a topological ring. Let $M$ and $N$ be linearly topologized $R$-modules. The {\\it tensor product} of $M$ and $N$ is the (usual) tensor product $M \\otimes_R N$ endowed with the linear topology defined by declaring $$ \\Im(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu \\longrightarrow M \\otimes_R N) $$ to be a fundamental system of open submodules, where $M_\\mu \\subset M$ and $N_\\nu \\subset N$ run through fundamental systems of open submodules in $M$ and $N$. The {\\it completed tensor product} $$ M \\widehat{\\otimes}_R N = \\lim M \\otimes_R N/(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu) = \\lim M/M_\\mu \\otimes_R N/N_\\nu $$ is the completion of the tensor product."}
+{"_id": "3975", "title": "formal-spaces-definition-weakly-admissible", "text": "Let $A$ be a linearly topologized ring. \\begin{enumerate} \\item An element $f \\in A$ is called {\\it topologically nilpotent} if $f^n \\to 0$ as $n \\to \\infty$. \\item A {\\it weak ideal of definition} for $A$ is an open ideal $I \\subset A$ consisting entirely of topologically nilpotent elements. \\item We say $A$ is {\\it weakly pre-admissible} if $A$ has a weak ideal of definition. \\item We say $A$ is {\\it weakly admissible} if $A$ is weakly pre-admissible and complete\\footnote{By our conventions this includes separated.}. \\end{enumerate}"}
+{"_id": "3976", "title": "formal-spaces-definition-taut", "text": "Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings. We say $\\varphi$ is {\\it taut}\\footnote{This is nonstandard notation. The definition generalizes to modules, by saying a linearly topologized $A$-module $M$ is $A$-taut if for every open ideal $I \\subset A$ the closure of $IM$ in $M$ is open and these closures form a fundamental system of neighbourhoods of $0$ in $M$.} if for every open ideal $I \\subset A$ the closure of the ideal $\\varphi(I)B$ is open and these closures form a fundamental system of open ideals."}
+{"_id": "3977", "title": "formal-spaces-definition-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is an {\\it affine formal algebraic space} if there exist \\begin{enumerate} \\item a directed set $\\Lambda$, \\item a system $(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$ in $(\\Sch/S)_{fppf}$ where \\begin{enumerate} \\item each $X_\\lambda$ is affine, \\item each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening, \\end{enumerate} \\end{enumerate} such that $$ X \\cong \\colim_{\\lambda \\in \\Lambda} X_\\lambda $$ as fppf sheaves and $X$ satisfies a set theoretic condition (see Remark \\ref{remark-set-theoretic}). A {\\it morphism of affine formal algebraic spaces} over $S$ is a map of sheaves."}
+{"_id": "3978", "title": "formal-spaces-definition-types-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\\it McQuillan} if $X$ satisfies the equivalent conditions of Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}. Let $A$ be the weakly admissible topological ring associated to $X$. We say \\begin{enumerate} \\item $X$ is {\\it classical} if $X$ is McQuillan and $A$ is admissible, \\item $X$ is {\\it adic} if $X$ is McQuillan and $A$ is adic, \\item $X$ is {\\it adic*} if $X$ is McQuillan, $A$ is adic, and $A$ has a finitely generated ideal of definition, and \\item $X$ is {\\it Noetherian} if $X$ is McQuillan and $A$ is both Noetherian and adic. \\end{enumerate}"}
+{"_id": "3979", "title": "formal-spaces-definition-affine-formal-spectrum", "text": "Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$, see Definition \\ref{definition-weakly-admissible}\\footnote{See More on Algebra, Definition \\ref{more-algebra-definition-topological-ring} for the classical case and see Remark \\ref{remark-mcquillan} for a discussion of differences.}. The {\\it formal spectrum} of $A$ is the affine formal algebraic space $$ \\text{Spf}(A) = \\colim \\Spec(A/I) $$ where the colimit is over the set of weak ideals of definition of $A$ and taken in the category $\\Sh((\\Sch/S)_{fppf})$."}
+{"_id": "3980", "title": "formal-spaces-definition-countable", "text": "Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\\it countably indexed} if the equivalent conditions of Lemma \\ref{lemma-countable-affine-formal-algebraic-space} are satisfied."}
+{"_id": "3981", "title": "formal-spaces-definition-formal-algebraic-space", "text": "Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is a {\\it formal algebraic space} if there exist a family of maps $\\{X_i \\to X\\}_{i \\in I}$ of sheaves such that \\begin{enumerate} \\item $X_i$ is an affine formal algebraic space, \\item $X_i \\to X$ is representable by algebraic spaces and \\'etale, \\item $\\coprod X_i \\to X$ is surjective as a map of sheaves \\end{enumerate} and $X$ satisfies a set theoretic condition (see Remark \\ref{remark-set-theoretic}). A {\\it morphism of formal algebraic spaces} over $S$ is a map of sheaves."}
+{"_id": "3982", "title": "formal-spaces-definition-completion", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset. The formal algebraic space of Lemma \\ref{lemma-completion-is-formal-algebraic-space} is called the {\\it completion of $X$ along $T$}."}
+{"_id": "3983", "title": "formal-spaces-definition-separated", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say \\begin{enumerate} \\item $X$ is {\\it quasi-separated} if the equivalent conditions of Lemma \\ref{lemma-characterize-quasi-separated} are satisfied. \\item $X$ is {\\it separated} if the equivalent conditions of Lemma \\ref{lemma-characterize-separated} are satisfied. \\end{enumerate}"}
+{"_id": "3984", "title": "formal-spaces-definition-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is {\\it quasi-compact} if the equivalent conditions of Lemma \\ref{lemma-characterize-quasi-compact} are satisfied."}
+{"_id": "3985", "title": "formal-spaces-definition-quasi-compact-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. We say $f$ is {\\it quasi-compact} if the equivalent conditions of Lemma \\ref{lemma-characterize-quasi-compact-morphism} are satisfied."}
+{"_id": "3986", "title": "formal-spaces-definition-types-formal-algebraic-spaces", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is {\\it locally countably indexed}, {\\it locally adic*}, or {\\it locally Noetherian} if the equivalent conditions of Lemma \\ref{lemma-type-local} hold for the corresponding property."}
+{"_id": "3987", "title": "formal-spaces-definition-adic-homomorphism", "text": "Let $A$ and $B$ be pre-adic topological rings. A ring homomorphism $\\varphi : A \\to B$ is {\\it adic}\\footnote{This may be nonstandard terminology.} if there exists an ideal of definition $I \\subset A$ such that the topology on $B$ is the $I$-adic topology."}
+{"_id": "3988", "title": "formal-spaces-definition-adic-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. Assume $X$ and $Y$ are locally adic*. We say $f$ is an {\\it adic morphism} if $f$ is representable by algebraic spaces. See discussion above."}
+{"_id": "3989", "title": "formal-spaces-definition-finite-type", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it locally of finite type} if $f$ is representable by algebraic spaces and is locally of finite type in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}. \\item We say $f$ is of {\\it finite type} if $f$ is locally of finite type and quasi-compact (Definition \\ref{definition-quasi-compact-morphism}). \\end{enumerate}"}
+{"_id": "3990", "title": "formal-spaces-definition-surjective", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of formal algebraic spaces over $S$ is said to be {\\it surjective} if it induces a surjective morphism $X_{red} \\to Y_{red}$ on underlying reduced algebraic spaces."}
+{"_id": "3991", "title": "formal-spaces-definition-monomorphism", "text": "Let $S$ be a scheme. A morphism of formal algebraic spaces over $S$ is called a {\\it monomorphism} if it is an injective map of sheaves."}
+{"_id": "3992", "title": "formal-spaces-definition-closed-immersion", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic spaces over $S$. We say $f$ is a {\\it closed immersion} if $f$ is representable by algebraic spaces and a closed immersion in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}."}
+{"_id": "3993", "title": "formal-spaces-definition-topologically-finite-type", "text": "Let $A \\to B$ be a continuous map of topological rings (More on Algebra, Definition \\ref{more-algebra-definition-topological-ring}). We say $B$ is {\\it topologically of finite type over} $A$ if there exists an $A$-algebra map $A[x_1, \\ldots, x_n] \\to B$ whose image is dense in $B$."}
+{"_id": "3994", "title": "formal-spaces-definition-separated-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism. \\begin{enumerate} \\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion. \\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact. \\end{enumerate}"}
+{"_id": "3995", "title": "formal-spaces-definition-proper", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic spaces over $S$. We say $f$ is {\\it proper} if $f$ is representable by algebraic spaces and is proper in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}."}
+{"_id": "3996", "title": "formal-spaces-definition-etale-sites", "text": "Let $S$ be a scheme. Let $X$ be a formal algebraic space with reduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}). \\begin{enumerate} \\item The {\\it small \\'etale site} $X_\\etale$ of $X$ is the site $X_{red, \\etale}$ of Properties of Spaces, Definition \\ref{spaces-properties-definition-etale-site}. \\item The site $X_{spaces, \\etale}$ is the site $X_{red, spaces, \\etale}$ of Properties of Spaces, Definition \\ref{spaces-properties-definition-spaces-etale-site}. \\item The site $X_{affine, \\etale}$ is the site $X_{red, affine, \\etale}$ of Properties of Spaces, Lemma \\ref{spaces-properties-lemma-alternative}. \\end{enumerate}"}
+{"_id": "4141", "title": "pione-definition-G-set-continuous", "text": "Let $G$ be a topological group. A {\\it $G$-set}, sometimes called a {\\it discrete $G$-set}, is a set $X$ endowed with a left action $a : G \\times X \\to X$ such that $a$ is continuous when $X$ is given the discrete topology and $G \\times X$ the product topology. A {\\it morphism of $G$-sets} $f : X \\to Y$ is simply any $G$-equivariant map from $X$ to $Y$. The category of $G$-sets is denoted {\\it $G\\textit{-Sets}$}."}
+{"_id": "4142", "title": "pione-definition-galois-category", "text": "\\begin{reference} Different from the definition in \\cite[Expos\\'e V, Definition 5.1]{SGA1}. Compare with \\cite[Definition 7.2.1]{BS}. \\end{reference} Let $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor. The pair $(\\mathcal{C}, F)$ is a {\\it Galois category} if \\begin{enumerate} \\item $\\mathcal{C}$ has finite limits and finite colimits, \\item \\label{item-connected-components} every object of $\\mathcal{C}$ is a finite (possibly empty) coproduct of connected objects, \\item $F(X)$ is finite for all $X \\in \\Ob(\\mathcal{C})$, and \\item $F$ reflects isomorphisms and is exact. \\end{enumerate} Here we say $X \\in \\Ob(\\mathcal{C})$ is connected if it is not initial and for any monomorphism $Y \\to X$ either $Y$ is initial or $Y \\to X$ is an isomorphism."}
+{"_id": "4143", "title": "pione-definition-fundamental-group", "text": "Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point of $X$. The {\\it fundamental group} of $X$ with {\\it base point} $\\overline{x}$ is the group $$ \\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}}) $$ of automorphisms of the fibre functor $F_{\\overline{x}} : \\textit{F\\'Et}_X \\to \\textit{Sets}$ endowed with its canonical profinite topology from Lemma \\ref{lemma-aut-inverse-limit}."}
+{"_id": "4177", "title": "stacks-cohomology-definition-flat-base-change", "text": "Let $\\mathcal{X}$ be an algebraic stack and let $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$. We say $\\mathcal{F}$ has the {\\it flat base change property}\\footnote{This may be nonstandard notation.} if and only if $c_\\varphi$ is an isomorphism whenever $f$ is flat."}
+{"_id": "4178", "title": "stacks-cohomology-definition-parasitic", "text": "Let $\\mathcal{X}$ be an algebraic stack. A presheaf of $\\mathcal{O}_\\mathcal{X}$-modules $\\mathcal{F}$ is {\\it parasitic} if we have $\\mathcal{F}(x) = 0$ for any object $x$ of $\\mathcal{X}$ which lies over a scheme $U$ such that the corresponding morphism $x : U \\to \\mathcal{X}$ is flat."}
+{"_id": "4179", "title": "stacks-cohomology-definition-lisse-etale", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item The {\\it lisse-\\'etale site} of $\\mathcal{X}$ is the full subcategory $\\mathcal{X}_{lisse,\\etale}$\\footnote{In the literature the site is denoted $\\text{Lis-\\'et}(\\mathcal{X})$ or $\\text{Lis-Et}(\\mathcal{X})$ and the associated topos is denoted $\\mathcal{X}_{\\text{lis-\\'e}t}$ or $\\mathcal{X}_{\\text{lis-et}}$. In the Stacks project our convention is to name the site and denote the corresponding topos by $\\Sh(\\mathcal{C})$.} of $\\mathcal{X}$ whose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$ such that $x : U \\to \\mathcal{X}$ is smooth. A covering of $\\mathcal{X}_{lisse,\\etale}$ is a family of morphisms $\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{lisse,\\etale}$ which forms a covering of $\\mathcal{X}_\\etale$. \\item The {\\it flat-fppf site} of $\\mathcal{X}$ is the full subcategory $\\mathcal{X}_{flat,fppf}$ of $\\mathcal{X}$ whose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$ such that $x : U \\to \\mathcal{X}$ is flat. A covering of $\\mathcal{X}_{flat,fppf}$ is a family of morphisms $\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{flat,fppf}$ which forms a covering of $\\mathcal{X}_{fppf}$. \\end{enumerate}"}
+{"_id": "4411", "title": "sites-cohomology-definition-torsor", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $\\mathcal{C}$. A {\\it pseudo torsor}, or more precisely a {\\it pseudo $\\mathcal{G}$-torsor}, is a sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}$ endowed with an action $\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that \\begin{enumerate} \\item whenever $\\mathcal{F}(U)$ is nonempty the action $\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ is simply transitive. \\end{enumerate} A {\\it morphism of pseudo $\\mathcal{G}$-torsors} $\\mathcal{F} \\to \\mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\\mathcal{G}$-actions. A {\\it torsor}, or more precisely a {\\it $\\mathcal{G}$-torsor}, is a pseudo $\\mathcal{G}$-torsor such that in addition \\begin{enumerate} \\item[(2)] for every $U \\in \\Ob(\\mathcal{C})$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $U$ such that $\\mathcal{F}(U_i)$ is nonempty for all $i \\in I$. \\end{enumerate} A {\\it morphism of $\\mathcal{G}$-torsors} is simply a morphism of pseudo $\\mathcal{G}$-torsors. The {\\it trivial $\\mathcal{G}$-torsor} is the sheaf $\\mathcal{G}$ endowed with the obvious left $\\mathcal{G}$-action."}
+{"_id": "4412", "title": "sites-cohomology-definition-cech-complex", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$. Let $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$. The complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ is the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the family $\\mathcal{U}$. Its cohomology groups $H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are called the {\\it {\\v C}ech cohomology groups} of $\\mathcal{F}$ with respect to $\\mathcal{U}$. They are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$."}
+{"_id": "4413", "title": "sites-cohomology-definition-limp", "text": "Let $\\mathcal{C}$ be a site. We say an abelian sheaf $\\mathcal{F}$ is {\\it totally acyclic}\\footnote{Although this terminology is is used in \\cite[Vbis, Proposition 1.3.10]{SGA4} this is probably nonstandard notation. In \\cite[V, Definition 4.1]{SGA4} this property is dubbed ``flasque'', but we cannot use this because it would clash with our definition of flasque sheaves on topological spaces. Please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com} if you have a better suggestion.} if for every sheaf of sets $K$ we have $H^p(K, \\mathcal{F}) = 0$ for all $p \\geq 1$."}
+{"_id": "4414", "title": "sites-cohomology-definition-K-flat", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}$-modules is called {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}$-modules the complex $$ \\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet) $$ is acyclic."}
+{"_id": "4415", "title": "sites-cohomology-definition-derived-tor", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O})$. The {\\it derived tensor product} $$ - \\otimes_\\mathcal{O}^{\\mathbf{L}} \\mathcal{F}^\\bullet : D(\\mathcal{O}) \\longrightarrow D(\\mathcal{O}) $$ is the exact functor of triangulated categories described above."}
+{"_id": "4416", "title": "sites-cohomology-definition-tor", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}$-modules. The {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are defined by the formula $$ \\text{Tor}_p^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) = H^{-p}(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{G}) $$ with derived tensor product as defined above."}
+{"_id": "4417", "title": "sites-cohomology-definition-covering-LC", "text": "Let $\\{f_i : X_i \\to X\\}$ be a family of morphisms with fixed target in the category $\\textit{LC}$. We say this family is a {\\it qc covering}\\footnote{This is nonstandard notation. We chose it to remind the reader of fpqc coverings of schemes.} if for every $x \\in X$ there exist $i_1, \\ldots, i_n \\in I$ and quasi-compact subsets $E_j \\subset X_{i_j}$ such that $\\bigcup f_{i_j}(E_j)$ is a neighbourhood of $x$."}
+{"_id": "4418", "title": "sites-cohomology-definition-simplicial-module", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet$ be a simplicial sheaf of rings on $\\mathcal{C}$. A {\\it simplicial $\\mathcal{A}_\\bullet$-module} $\\mathcal{F}_\\bullet$ (sometimes called a {\\it simplicial sheaf of $\\mathcal{A}_\\bullet$-modules}) is a sheaf of modules over the sheaf of rings on $\\Delta \\times \\mathcal{C}$ associated to $\\mathcal{A}_\\bullet$."}
+{"_id": "4419", "title": "sites-cohomology-definition-strictly-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules. We say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect} if $\\mathcal{E}^i$ is zero for all but finitely many $i$ and $\\mathcal{E}^i$ is a direct summand of a finite free $\\mathcal{O}$-module for all $i$."}
+{"_id": "4420", "title": "sites-cohomology-definition-pseudo-coherent", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ and for each $i$ a morphism of complexes $\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$ where $\\mathcal{E}_i$ is a strictly perfect complex of $\\mathcal{O}_{U_i}$-modules and $H^j(\\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\\alpha_i)$ is surjective. \\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent} if it is $m$-pseudo-coherent for all $m$. \\item We say an object $E$ of $D(\\mathcal{O})$ is {\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent}) if and only if it can be represented by a $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $\\mathcal{O}$-modules. \\end{enumerate}"}
+{"_id": "4421", "title": "sites-cohomology-definition-tor-amplitude", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. \\begin{enumerate} \\item We say $E$ has {\\it tor-amplitude in $[a, b]$} if $H^i(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}) = 0$ for all $\\mathcal{O}$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$. \\item We say $E$ has {\\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \\item We say $E$ {\\it locally has finite tor dimension} if for any object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $E|_{U_i}$ has finite tor dimension for all $i$. \\end{enumerate} An $\\mathcal{O}$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$} if $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O})$ has tor-amplitude in $[-d, 0]$."}
+{"_id": "4422", "title": "sites-cohomology-definition-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules. We say $\\mathcal{E}^\\bullet$ is {\\it perfect} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that for each $i$ there exists a morphism of complexes $\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$ which is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$ strictly perfect. An object $E$ of $D(\\mathcal{O})$ is {\\it perfect} if it can be represented by a perfect complex of $\\mathcal{O}$-modules."}
+{"_id": "4524", "title": "fields-definition-field", "text": "A {\\it field} is a nonzero ring where every nonzero element is invertible. Given a field a {\\it subfield} is a subring that is itself a field."}
+{"_id": "4525", "title": "fields-definition-domain", "text": "A {\\it domain} or an {\\it integral domain} is a nonzero ring where $0$ is the only zerodivisor."}
+{"_id": "4526", "title": "fields-definition-characteristic", "text": "The {\\it characteristic} of a field $F$ is $0$ if $\\mathbf{Z} \\subset F$, or is a prime $p$ if $p = 0$ in $F$. The {\\it prime subfield of $F$} is the smallest subfield of $F$ which is either $\\mathbf{Q} \\subset F$ if the characteristic is zero, or $\\mathbf{F}_p \\subset F$ if the characteristic is $p > 0$."}
+{"_id": "4527", "title": "fields-definition-extension", "text": "If $F$ is a field contained in a field $E$, then $E$ is said to be a {\\it field extension} of $F$. We shall write $E/F$ to indicate that $E$ is an extension of $F$."}
+{"_id": "4528", "title": "fields-definition-tower", "text": "A {\\it tower} of fields $E_n/E_{n - 1}/\\ldots/E_0$ consists of a sequence of extensions of fields $E_n/E_{n - 1}$, $E_{n - 1}/E_{n - 2}$, $\\ldots$, $E_1/E_0$."}
+{"_id": "4529", "title": "fields-definition-generated-by", "text": "Let $k$ be a field. If $F/k$ is an extension of fields and $S \\subset F$, we write $k(S)$ for the smallest subfield of $F$ containing $k$ and $S$. We will say that $S$ {\\it generates the field extension} $k(S)/k$. If $S = \\{\\alpha\\}$ is a singleton, then we write $k(\\alpha)$ instead of $k(\\{\\alpha\\})$. We say $F/k$ is a {\\it finitely generated field extension} if there exists a finite subset $S \\subset F$ with $F = k(S)$."}
+{"_id": "4530", "title": "fields-definition-degree", "text": "Let $F/E$ be an extension of fields. The dimension of $F$ considered as an $E$-vector space is called the {\\it degree} of the extension and is denoted $[F : E]$. If $[F : E] < \\infty$ then $F$ is said to be a {\\it finite} extension of $E$."}
+{"_id": "4531", "title": "fields-definition-number-field", "text": "A field $K$ is said to be a {\\it number field} if it has characteristic $0$ and the extension $\\mathbf{Q} \\subset K$ is finite."}
+{"_id": "4532", "title": "fields-definition-algebraic", "text": "Consider a field extension $F/E$. An element $\\alpha \\in F$ is said to be {\\it algebraic} over $E$ if $\\alpha$ is the root of some nonzero polynomial with coefficients in $E$. If all elements of $F$ are algebraic then $F$ is said to be an {\\it algebraic extension} of $E$."}
+{"_id": "4533", "title": "fields-definition-minimal-polynomial", "text": "The polynomial $P$ above is called the {\\it minimal polynomial} of $\\alpha$ over $k$."}
+{"_id": "4534", "title": "fields-definition-algebraically-closed", "text": "A field $F$ is said to be {\\it algebraically closed} if every algebraic extension $E/F$ is trivial, i.e., $E = F$."}
+{"_id": "4535", "title": "fields-definition-algebraic-closure", "text": "Let $F$ be a field. We say $F$ is {\\it algebraically closed} if every algebraic extension $E/F$ is trivial, i.e., $E = F$. An {\\it algebraic closure} of $F$ is a field $\\overline{F}$ containing $F$ such that: \\begin{enumerate} \\item $\\overline{F}$ is algebraic over $F$. \\item $\\overline{F}$ is algebraically closed. \\end{enumerate}"}
+{"_id": "4536", "title": "fields-definition-relatively-prime", "text": "If $k$ is any field, we say that two polynomials in $k[x]$ are {\\it relatively prime} if they generate the unit ideal in $k[x]$."}
+{"_id": "4537", "title": "fields-definition-separable", "text": "Let $F$ be a field. Let $K/F$ be an extension of fields. \\begin{enumerate} \\item We say an irreducible polynomial $P$ over $F$ is {\\it separable} if it is relatively prime to its derivative. \\item Given $\\alpha \\in K$ algebraic over $F$ we say $\\alpha$ is {\\it separable} over $F$ if its minimal polynomial is separable over $F$. \\item If $K$ is an algebraic extension of $F$, we say $K$ is {\\it separable}\\footnote{For nonalgebraic extensions this definition does not make sense and is not the correct one. We refer the reader to Algebra, Sections \\ref{algebra-section-separability} and \\ref{algebra-section-separability-continued}.} over $F$ if every element of $K$ is separable over $F$. \\end{enumerate}"}
+{"_id": "4538", "title": "fields-definition-separable-degree", "text": "Let $F$ be a field. Let $P$ be an irreducible polynomial over $F$. The {\\it separable degree} of $P$ is the cardinality of the set of roots of $P$ in any algebraic closure of $F$ (see discussion above). Notation $\\deg_s(P)$."}
+{"_id": "4539", "title": "fields-definition-purely-inseparable", "text": "Let $F$ be a field of characteristic $p > 0$. Let $K/F$ be an extension. \\begin{enumerate} \\item An element $\\alpha \\in K$ is {\\it purely inseparable} over $F$ if there exists a power $q$ of $p$ such that $\\alpha^q \\in F$. \\item The extension $K/F$ is said to be {\\it purely inseparable} if and only if every element of $K$ is purely inseparable over $F$. \\end{enumerate}"}
+{"_id": "4540", "title": "fields-definition-insep-degree", "text": "Let $E/F$ be an algebraic field extension. Let $E_{sep}$ be the subextension found in Lemma \\ref{lemma-separable-first}. \\begin{enumerate} \\item The integer $[E_{sep} : F]$ is called the {\\it separable degree} of the extension. Notation $[E : F]_s$. \\item The integer $[E : E_{sep}]$ is called the {\\it inseparable degree}, or the {\\it degree of inseparability} of the extension. Notation $[E : F]_i$. \\end{enumerate}"}
+{"_id": "4541", "title": "fields-definition-normal", "text": "Let $E/F$ be an algebraic field extension. We say $E$ is {\\it normal} over $F$ if for all $\\alpha \\in E$ the minimal polynomial $P$ of $\\alpha$ over $F$ splits completely into linear factors over $E$."}
+{"_id": "4542", "title": "fields-definition-automorphisms", "text": "Let $E/F$ be an extension of fields. Then $\\text{Aut}(E/F)$ or $\\text{Aut}_F(E)$ denotes the automorphism group of $E$ as an object of the category of $F$-extensions. Elements of $\\text{Aut}(E/F)$ are called {\\it automorphisms of $E$ over $F$} or {\\it automorphisms of $E/F$}."}
+{"_id": "4543", "title": "fields-definition-splitting-field", "text": "Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial. The field extension $E/F$ constructed in Lemma \\ref{lemma-splitting-field} is called the {\\it splitting field of $P$ over $F$}."}
+{"_id": "4544", "title": "fields-definition-normal-closure", "text": "Let $E/F$ be a finite extension of fields. The field extension $K/E$ constructed in Lemma \\ref{lemma-normal-closure} is called the {\\it normal closure $E$ over $F$}."}
+{"_id": "4545", "title": "fields-definition-trace-norm", "text": "Let $L/K$ be a finite extension of fields. For $\\alpha \\in L$ we define the {\\it trace} $\\text{Trace}_{L/K}(\\alpha) = \\text{Trace}_K(\\alpha : L \\to L)$ and the {\\it norm} $\\text{Norm}_{L/K}(\\alpha) = \\det_K(\\alpha : L \\to L)$."}
+{"_id": "4546", "title": "fields-definition-trace-pairing", "text": "Let $L/K$ be a finite extension of fields. The {\\it trace pairing} for $L/K$ is the symmetric $K$-bilinear form $$ Q_{L/K} : L \\times L \\longrightarrow K,\\quad (\\alpha, \\beta) \\longmapsto \\text{Trace}_{L/K}(\\alpha\\beta) $$"}
+{"_id": "4547", "title": "fields-definition-discriminant", "text": "Let $L/K$ be a finite extension of fields. The {\\it discriminant of $L/K$} is the discriminant of the trace pairing $Q_{L/K}$."}
+{"_id": "4548", "title": "fields-definition-galois", "text": "A field extension $E/F$ is called {\\it Galois} if it is algebraic, separable, and normal."}
+{"_id": "4549", "title": "fields-definition-galois-group", "text": "If $E/F$ is a Galois extension, then the group $\\text{Aut}(E/F)$ is called the {\\it Galois group} and it is denoted $\\text{Gal}(E/F)$."}
+{"_id": "4550", "title": "fields-definition-transcendence", "text": "Let $k \\subset K$ be a field extension. \\begin{enumerate} \\item A collection of elements $\\{x_i\\}_{i \\in I}$ of $K$ is called {\\it algebraically independent} over $k$ if the map $$ k[X_i; i\\in I] \\longrightarrow K $$ which maps $X_i$ to $x_i$ is injective. \\item The field of fractions of a polynomial ring $k[x_i; i \\in I]$ is denoted $k(x_i; i\\in I)$. \\item A {\\it purely transcendental extension} of $k$ is any field extension $k \\subset K$ isomorphic to the field of fractions of a polynomial ring over $k$. \\item A {\\it transcendence basis} of $K/k$ is a collection of elements $\\{x_i\\}_{i \\in I}$ which are algebraically independent over $k$ and such that the extension $k(x_i; i\\in I) \\subset K$ is algebraic. \\end{enumerate}"}
+{"_id": "4551", "title": "fields-definition-transcendence-degree", "text": "Let $k \\subset K$ be a field extension. The {\\it transcendence degree} of $K$ over $k$ is the cardinality of a transcendence basis of $K$ over $k$. It is denoted $\\text{trdeg}_k(K)$."}
+{"_id": "4552", "title": "fields-definition-algebraically-closed-in", "text": "Let $k \\subset K$ be a field extension. \\begin{enumerate} \\item The {\\it algebraic closure of $k$ in $K$} is the subfield $k'$ of $K$ consisting of elements of $K$ which are algebraic over $k$. \\item We say $k$ is {\\it algebraically closed in $K$} if every element of $K$ which is algebraic over $k$ is contained in $k$. \\end{enumerate}"}
+{"_id": "4553", "title": "fields-definition-compositum", "text": "Consider a diagram \\begin{equation} \\label{equation-inside-omega} \\vcenter{ \\xymatrix{ L \\ar[r] & \\Omega \\\\ k \\ar[r] \\ar[u] & K \\ar[u] } } \\end{equation} of field extensions. The {\\it compositum of $K$ and $L$ in $\\Omega$} written $KL$ is the smallest subfield of $\\Omega$ containing both $L$ and $K$."}
+{"_id": "4554", "title": "fields-definition-linearly-disjoint", "text": "Consider a diagram of fields as in (\\ref{equation-inside-omega}). We say that $K$ and $L$ are {\\it linearly disjoint over $k$ in $\\Omega$} if the map $$ K \\otimes_k L \\longrightarrow KL,\\quad \\sum x_i \\otimes y_i \\longmapsto \\sum x_i y_i $$ is injective."}
+{"_id": "4555", "title": "fields-definition-separable-algebraic", "text": "Algebraic field extensions. \\begin{enumerate} \\item A field extension $k \\subset K$ is called {\\it algebraic} if every element of $K$ is algebraic over $k$. \\item An algebraic extension $k \\subset k'$ is called {\\it separable} if every $\\alpha \\in k'$ is separable over $k$. \\item An algebraic extension $k \\subset k'$ is called {\\it purely inseparable} if the characteristic of $k$ is $p > 0$ and for every element $\\alpha \\in k'$ there exists a power $q$ of $p$ such that $\\alpha^q \\in k$. \\item An algebraic extension $k \\subset k'$ is called {\\it normal} if for every $\\alpha \\in k'$ the minimal polynomial $P(T) \\in k[T]$ of $\\alpha$ over $k$ splits completely into linear factors over $k'$. \\item An algebraic extension $k \\subset k'$ is called {\\it Galois} if it is separable and normal. \\end{enumerate}"}
+{"_id": "4660", "title": "spaces-limits-definition-locally-finite-presentation", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item A functor $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ is said to be {\\it limit preserving} or {\\it locally of finite presentation} if for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have $$ F(T) = \\colim F(T_i). $$ We sometimes say that $F$ is {\\it locally of finite presentation over $S$}. \\item Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. A transformation of functors $a : F \\to G$ is {\\it limit preserving} or {\\it locally of finite presentation} if for every scheme $T$ over $S$ and every $y \\in G(T)$ the functor $$ F_y : (\\Sch/T)_{fppf}^{opp} \\longrightarrow \\textit{Sets}, \\quad T'/T \\longmapsto \\{x \\in F(T') \\mid a(x) = y|_{T'}\\} $$ is locally of finite presentation over $T$\\footnote{The characterization (2) in Lemma \\ref{lemma-characterize-relative-limit-preserving} may be easier to parse.}. We sometimes say that $F$ is {\\it relatively limit preserving} over $G$. \\end{enumerate}"}
+{"_id": "4661", "title": "spaces-limits-definition-subsheaf-sections-annihilated-by-ideal", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in Lemma \\ref{lemma-sections-annihilated-by-ideal} above is called the {\\it subsheaf of sections annihilated by $\\mathcal{I}$}."}
+{"_id": "4662", "title": "spaces-limits-definition-subsheaf-sections-supported-on-closed", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a closed subset whose complement corresponds to an open subspace $U \\subset X$ with quasi-compact inclusion morphism $U \\to X$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in Lemma \\ref{lemma-sections-supported-on-closed-subset} above is called the {\\it subsheaf of sections supported on $T$}."}
+{"_id": "4696", "title": "stacks-geometry-definition-versal-ring-at-x", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. A {\\it versal ring to $\\mathcal{X}$ at $x_0$} is a complete Noetherian local $S$-algebra $A$ with residue field $k$ such that there exists a versal formal object $(A, \\xi_n, f_n)$ as in Artin's Axioms, Definition \\ref{artin-definition-versal-formal-object} with $\\xi_1 \\cong x_0$ (a $2$-isomorphism)."}
+{"_id": "4697", "title": "stacks-geometry-definition-multiplicity", "text": "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let $T \\subset |\\mathcal{X}|$ be an irreducible component. The {\\it multiplicity} of $T$ in $\\mathcal{X}$ is defined as $m_{T, \\mathcal{X}} = m_{T', U}$ where $f : U \\to \\mathcal{X}$ is a smooth morphism from a scheme and $T' \\subset |U|$ is an irreducible component with $f(T') \\subset T$."}
+{"_id": "4698", "title": "stacks-geometry-definition-formal-branches", "text": "Let $\\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$. The {\\it formal branches of $\\mathcal{X}$ through $x_0$} is the set of irreducible components of $\\Spec(A)$ for any choice of versal ring to $\\mathcal{X}$ at $x_0$ identified for different choices of $A$ by the procedure described above."}
+{"_id": "4699", "title": "stacks-geometry-definition-multiplicity-formal-branches", "text": "Let $\\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$. The {\\it multiplicity of a formal branch of $\\mathcal{X}$ through $x_0$} is the multiplicity of the corresponding irreducible component of $\\Spec(A)$ for any choice of versal ring to $\\mathcal{X}$ at $x_0$ (see discussion above)."}
+{"_id": "4700", "title": "stacks-geometry-definition-relative-dimension", "text": "If $f : T \\to \\mathcal{X}$ is a locally of finite type morphism from an algebraic space to an algebraic stack, and if $t \\in |T|$ is a point with image $x \\in | \\mathcal{X}|$, then we define {\\it the relative dimension} of $f$ at $t$, denoted $\\dim_t(T_x),$ as follows: choose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of a field, which represents $x$, and choose a point $t' \\in |T \\times_{\\mathcal{X}} \\Spec k|$ mapping to $t$ under the projection to $|T|$ (such a point $t'$ exists, by Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}); then $$ \\dim_t(T_x) = \\dim_{t'}(T \\times_{\\mathcal{X}} \\Spec k ). $$"}
+{"_id": "4701", "title": "stacks-geometry-definition-relative-dimension-for-stacks", "text": "If $f : \\mathcal{T} \\to \\mathcal{X}$ is a locally of finite type morphism between locally Noetherian algebraic stacks, and if $t \\in |\\mathcal{T}|$ is a point with image $x \\in |\\mathcal{X}|$, then we define the {\\it relative dimension} of $f$ at $t$, denoted $\\dim_t(\\mathcal{T}_x),$ as follows: choose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of a field, which represents $x$, and choose a point $t' \\in |\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k|$ mapping to $t$ under the projection to $|\\mathcal{T}|$ (such a point $t'$ exists, by Properties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian}; then $$ \\dim_t(\\mathcal{T}_x) = \\dim_{t'}(\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k ). $$"}
+{"_id": "4702", "title": "stacks-geometry-definition-pseudo-catenary", "text": "We say that a locally Noetherian algebraic stack $\\mathcal{X}$ is {\\it pseudo-catenary} if there exists a smooth and surjective morphism $U \\to \\mathcal{X}$ whose source is a universally catenary scheme."}
+{"_id": "4703", "title": "stacks-geometry-definition-dimension-local-ring", "text": "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let $x \\in |\\mathcal{X}|$ be a finite type point. The {\\it dimension of the local ring of $\\mathcal{X}$ at $x$} is $d \\in \\mathbf{Z}$ if the equivalent conditions of Lemma \\ref{lemma-dimension-local-ring} are satisfied."}
+{"_id": "4984", "title": "spaces-morphisms-definition-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism. \\begin{enumerate} \\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion. \\item We say $f$ is {\\it locally separated}\\footnote{In the literature this term often refers to quasi-separated and locally separated morphisms.} if $\\Delta_{X/Y}$ is an immersion. \\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact. \\end{enumerate}"}
+{"_id": "4985", "title": "spaces-morphisms-definition-surjective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it surjective} if the map $|f| : |X| \\to |Y|$ of associated topological spaces is surjective."}
+{"_id": "4986", "title": "spaces-morphisms-definition-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it open} if the map of topological spaces $|f| : |X| \\to |Y|$ is open. \\item We say $f$ is {\\it universally open} if for every morphism of algebraic spaces $Z \\to Y$ the morphism of topological spaces $$ |Z \\times_Y X| \\to |Z| $$ is open, i.e., the base change $Z \\times_Y X \\to Z$ is open. \\end{enumerate}"}
+{"_id": "4987", "title": "spaces-morphisms-definition-submersive", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it submersive}\\footnote{This is very different from the notion of a submersion of differential manifolds.} if the continuous map $|X| \\to |Y|$ is submersive, see Topology, Definition \\ref{topology-definition-submersive}. \\item We say $f$ is {\\it universally submersive} if for every morphism of algebraic spaces $Y' \\to Y$ the base change $Y' \\times_Y X \\to Y'$ is submersive. \\end{enumerate}"}
+{"_id": "4988", "title": "spaces-morphisms-definition-quasi-compact", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it quasi-compact} if for every quasi-compact algebraic space $Z$ and morphism $Z \\to Y$ the fibre product $Z \\times_Y X$ is quasi-compact."}
+{"_id": "4989", "title": "spaces-morphisms-definition-closed", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it closed} if the map of topological spaces $|X| \\to |Y|$ is closed. \\item We say $f$ is {\\it universally closed} if for every morphism of algebraic spaces $Z \\to Y$ the morphism of topological spaces $$ |Z \\times_Y X| \\to |Z| $$ is closed, i.e., the base change $Z \\times_Y X \\to Z$ is closed. \\end{enumerate}"}
+{"_id": "4990", "title": "spaces-morphisms-definition-monomorphism", "text": "Let $S$ be a scheme. A morphism of algebraic spaces over $S$ is called a {\\it monomorphism} if it is an injective map of sheaves, i.e., a monomorphism in the category of sheaves on $(\\Sch/S)_{fppf}$."}
+{"_id": "4991", "title": "spaces-morphisms-definition-inverse-image-closed-subspace", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $Z \\subset X$ be a closed subspace. The {\\it inverse image $f^{-1}(Z)$ of the closed subspace $Z$} is the closed subspace $Z \\times_X Y$ of $Y$."}
+{"_id": "4992", "title": "spaces-morphisms-definition-scheme-theoretic-intersection-union", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \\subset X$ be closed subspaces corresponding to quasi-coherent ideal sheaves $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$. The {\\it scheme theoretic intersection} of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$. Then {\\it scheme theoretic union} of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\\mathcal{I} \\cap \\mathcal{J}$."}
+{"_id": "4993", "title": "spaces-morphisms-definition-scheme-theoretic-support", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. The {\\it scheme theoretic support of $\\mathcal{F}$} is the closed subspace $Z \\subset X$ constructed in Lemma \\ref{lemma-scheme-theoretic-support}."}
+{"_id": "4994", "title": "spaces-morphisms-definition-scheme-theoretic-image", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The {\\it scheme theoretic image} of $f$ is the smallest closed subspace $Z \\subset Y$ through which $f$ factors, see Lemma \\ref{lemma-scheme-theoretic-image} above."}
+{"_id": "4995", "title": "spaces-morphisms-definition-scheme-theoretically-dense", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \\subset X$ be an open subspace. \\begin{enumerate} \\item The scheme theoretic image of the morphism $U \\to X$ is called the {\\it scheme theoretic closure of $U$ in $X$}. \\item We say $U$ is {\\it scheme theoretically dense in $X$} if the equivalent conditions of Lemma \\ref{lemma-scheme-theoretically-dense} are satisfied. \\end{enumerate}"}
+{"_id": "4996", "title": "spaces-morphisms-definition-dominant", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is called {\\it dominant} if the image of $|f| : |X| \\to |Y|$ is dense in $|Y|$."}
+{"_id": "4997", "title": "spaces-morphisms-definition-universally-injective", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it universally injective} if for every morphism $Y' \\to Y$ the induced map $|Y' \\times_Y X| \\to |Y'|$ is injective."}
+{"_id": "4998", "title": "spaces-morphisms-definition-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it affine} if for every affine scheme $Z$ and morphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable by an affine scheme."}
+{"_id": "4999", "title": "spaces-morphisms-definition-relative-spec", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over $X$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $X$} is the affine morphism $\\underline{\\Spec}(\\mathcal{A}) \\to X$ corresponding to $\\mathcal{A}$ under the equivalence of categories of Lemma \\ref{lemma-affine-equivalence-algebras}."}
+{"_id": "5000", "title": "spaces-morphisms-definition-quasi-affine", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it quasi-affine} if for every affine scheme $Z$ and morphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable by a quasi-affine scheme."}
+{"_id": "5001", "title": "spaces-morphisms-definition-P", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. We say a morphism $f : X \\to Y$ of algebraic spaces over $S$ {\\it has property $\\mathcal{P}$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold."}
+{"_id": "5002", "title": "spaces-morphisms-definition-P-at-point", "text": "Let $\\mathcal{Q}$ be a property of morphisms of germs of schemes which is \\'etale local on the source-and-target. Let $S$ be a scheme. Given a morphism $f : X \\to Y$ of algebraic spaces over $S$ and a point $x \\in |X|$ we say that $f$ {\\it has property $\\mathcal{Q}$ at $x$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold."}
+{"_id": "5003", "title": "spaces-morphisms-definition-locally-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ {\\it locally of finite type} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{locally of finite type}$. \\item Let $x \\in |X|$. We say $f$ is of {\\it finite type at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is locally of finite type. \\item We say $f$ is {\\it of finite type} if it is locally of finite type and quasi-compact. \\end{enumerate}"}
+{"_id": "5004", "title": "spaces-morphisms-definition-finite-type-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say a point $x \\in |X|$ is a {\\it finite type point}\\footnote{This is a slight abuse of language as it would perhaps be more correct to say ``locally finite type point''.} if the equivalent conditions of Lemma \\ref{lemma-point-finite-type} are satisfied. We denote $X_{\\text{ft-pts}}$ the set of finite type points of $X$."}
+{"_id": "5005", "title": "spaces-morphisms-definition-locally-quasi-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it locally quasi-finite} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{locally quasi-finite}$. \\item Let $x \\in |X|$. We say $f$ is {\\it quasi-finite at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is locally quasi-finite. \\item A morphism of algebraic spaces $f : X \\to Y$ is {\\it quasi-finite} if it is locally quasi-finite and quasi-compact. \\end{enumerate}"}
+{"_id": "5006", "title": "spaces-morphisms-definition-locally-finite-presentation", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it locally of finite presentation} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} =$``locally of finite presentation''. \\item Let $x \\in |X|$. We say $f$ is of {\\it finite presentation at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is locally of finite presentation\\footnote{It seems awkward to use ``locally of finite presentation at $x$'', but the current terminology may be misleading in the sense that ``of finite presentation at $x$'' does {\\bf not} mean that there is an open neighbourhood $X' \\subset X$ such that $f|_{X'}$ is of finite presentation.}. \\item A morphism of algebraic spaces $f : X \\to Y$ is {\\it of finite presentation} if it is locally of finite presentation, quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "5007", "title": "spaces-morphisms-definition-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it flat} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} with $\\mathcal{P} =$``flat''. \\item Let $x \\in |X|$. We say $f$ is {\\it flat at $x$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} holds with $\\mathcal{Q} =$``induced map local rings is flat''. \\end{enumerate} Note that the second part makes sense by Descent, Lemma \\ref{descent-lemma-flat-at-point}."}
+{"_id": "5008", "title": "spaces-morphisms-definition-flat-module", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. \\begin{enumerate} \\item Let $x \\in |X|$. We say $\\mathcal{F}$ is {\\it flat at $x$ over $Y$} if the equivalent conditions of Lemma \\ref{lemma-flat-at-point} hold. \\item We say $\\mathcal{F}$ is {\\it flat over $Y$} if $\\mathcal{F}$ is flat over $Y$ at all $x \\in |X|$. \\end{enumerate}"}
+{"_id": "5009", "title": "spaces-morphisms-definition-dimension-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$. Let $d, r \\in \\{0, 1, 2, \\ldots, \\infty\\}$. \\begin{enumerate} \\item We say the {\\it dimension of the local ring of the fibre of $f$ at $x$} is $d$ if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold for the property $\\mathcal{P}_d$ described in Descent, Lemma \\ref{descent-lemma-dimension-local-ring-fibre}. \\item We say the {\\it transcendence degree of $x/f(x)$} is $r$ if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold for the property $\\mathcal{P}_r$ described in Descent, Lemma \\ref{descent-lemma-transcendence-degree-at-point}. \\item We say {\\it $f$ has relative dimension $d$ at $x$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold for the property $\\mathcal{P}_d$ described in Descent, Lemma \\ref{descent-lemma-dimension-at-point}. \\end{enumerate}"}
+{"_id": "5010", "title": "spaces-morphisms-definition-relative-dimension", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $d \\in \\{0, 1, 2, \\ldots\\}$. \\begin{enumerate} \\item We say $f$ has {\\it relative dimension $\\leq d$} if $f$ has relative dimension $\\leq d$ at all $x \\in |X|$. \\item We say $f$ has {\\it relative dimension $d$} if $f$ has relative dimension $d$ at all $x \\in |X|$. \\end{enumerate}"}
+{"_id": "5011", "title": "spaces-morphisms-definition-syntomic", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it syntomic} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} =$``syntomic''. \\item Let $x \\in |X|$. We say $f$ is {\\it syntomic at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is syntomic. \\end{enumerate}"}
+{"_id": "5012", "title": "spaces-morphisms-definition-smooth", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it smooth} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} =$``smooth''. \\item Let $x \\in |X|$. We say $f$ is {\\it smooth at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is smooth. \\end{enumerate}"}
+{"_id": "5013", "title": "spaces-morphisms-definition-unramified", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ is {\\it unramified} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{unramified}$. \\item Let $x \\in |X|$. We say $f$ is {\\it unramified at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is unramified. \\item We say $f$ is {\\it G-unramified} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{G-unramified}$. \\item Let $x \\in |X|$. We say $f$ is {\\it G-unramified at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is G-unramified. \\end{enumerate}"}
+{"_id": "5014", "title": "spaces-morphisms-definition-etale", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$. We say $f$ is {\\it \\'etale at $x$} if there exists an open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$ is \\'etale."}
+{"_id": "5015", "title": "spaces-morphisms-definition-proper", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is {\\it proper} if $f$ is separated, finite type, and universally closed."}
+{"_id": "5016", "title": "spaces-morphisms-definition-valuative-criterion", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say $f$ {\\it satisfies the uniqueness part of the valuative criterion} if given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists at most one dotted arrow (without requiring existence). We say $f$ {\\it satisfies the existence part of the valuative criterion} if given any solid diagram as above there exists an extension $K \\subset K'$ of fields, a valuation ring $A' \\subset K'$ dominating $A$ and a morphism $\\Spec(A') \\to X$ such that the following diagram commutes $$ \\xymatrix{ \\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\ \\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y } $$ We say $f$ {\\it satisfies the valuative criterion} if $f$ satisfies both the existence and uniqueness part."}
+{"_id": "5017", "title": "spaces-morphisms-definition-integral", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say that $f$ is {\\it integral} if for every affine scheme $Z$ and morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable by an affine scheme integral over $Z$. \\item We say that $f$ is {\\it finite} if for every affine scheme $Z$ and morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable by an affine scheme finite over $Z$. \\end{enumerate}"}
+{"_id": "5018", "title": "spaces-morphisms-definition-finite-locally-free", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say that $f$ is {\\it finite locally free} if $f$ is affine and $f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_Y$-module. In this case we say $f$ is has {\\it rank} or {\\it degree} $d$ if the sheaf $f_*\\mathcal{O}_X$ is finite locally free of rank $d$."}
+{"_id": "5019", "title": "spaces-morphisms-definition-rational-map", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. \\begin{enumerate} \\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of algebraic spaces over $S$ defined on dense open subspaces $U$, $V$ of $X$. We say that $f$ is {\\it equivalent} to $g$ if $f|_W = g|_W$ for some dense open subspace $W \\subset U \\cap V$. \\item A {\\it rational map from $X$ to $Y$} is an equivalence class for the equivalence relation defined in (1). \\item Given morphisms $X \\to B$ and $Y \\to B$ of algebraic spaces over $S$ we say that a rational map from $X$ to $Y$ is a {\\it $B$-rational map from $X$ to $Y$} if there exists a representative $f : U \\to Y$ of the equivalence class which is a morphism over $B$. \\end{enumerate}"}
+{"_id": "5020", "title": "spaces-morphisms-definition-rational-function", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\\it rational function on $X$} is a rational map from $X$ to $\\mathbf{A}^1_S$."}
+{"_id": "5021", "title": "spaces-morphisms-definition-ring-of-rational-functions", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it ring of rational functions on $X$} is the ring $R(X)$ whose elements are rational functions with addition and multiplication as just described."}
+{"_id": "5022", "title": "spaces-morphisms-definition-domain-of-definition", "text": "Let $S$ be a scheme. Let $\\varphi$ be a rational map between two algebraic spaces $X$ and $Y$ over $S$. We say $\\varphi$ is {\\it defined in a point $x \\in |X|$} if there exists a representative $(U, f)$ of $\\varphi$ with $x \\in |U|$. The {\\it domain of definition} of $\\varphi$ is the set of all points where $\\varphi$ is defined."}
+{"_id": "5023", "title": "spaces-morphisms-definition-dominant-rational", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$. Assume $|X|$ and $|Y|$ are irreducible. A rational map from $X$ to $Y$ is called {\\it dominant} if any representative $f : U \\to Y$ is a dominant morphism in the sense of Definition \\ref{definition-dominant}."}
+{"_id": "5024", "title": "spaces-morphisms-definition-birational-spaces", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$ with $|X|$ and $|Y|$ irreducible. We say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic in the category of irreducible algebraic spaces over $S$ and dominant rational maps."}
+{"_id": "5025", "title": "spaces-morphisms-definition-integral-closure", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The {\\it integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$} is the quasi-coherent $\\mathcal{O}_X$-subalgebra $\\mathcal{A}' \\subset \\mathcal{A}$ constructed in Lemma \\ref{lemma-integral-closure} above."}
+{"_id": "5026", "title": "spaces-morphisms-definition-normalization-X-in-Y", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\\mathcal{O}'$ be the integral closure of $\\mathcal{O}_X$ in $f_*\\mathcal{O}_Y$. The {\\it normalization of $X$ in $Y$} is the morphism of algebraic spaces $$ \\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X $$ over $S$. It comes equipped with a natural factorization $$ Y \\xrightarrow{f'} X' \\xrightarrow{\\nu} X $$ of the initial morphism $f$."}
+{"_id": "5027", "title": "spaces-morphisms-definition-normalization", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-prepare-normalization}. We define the {\\it normalization} of $X$ as the morphism $$ \\nu : X^\\nu \\longrightarrow X $$ constructed in Lemma \\ref{lemma-normalization}."}
+{"_id": "5028", "title": "spaces-morphisms-definition-universal-homeomorphism", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is called a {\\it universal homeomorphism} if and only if for every morphism of algebraic spaces $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ induces a homeomorphism $|Z \\times_Y X| \\to |Z|$."}
+{"_id": "5114", "title": "weil-definition-chow-group-motives", "text": "Let $k$ be a base field. Let $M = (X, p, m)$ be a Chow motive over $k$. For $i \\in \\mathbf{Z}$ we define the {\\it $i$th Chow group of $M$} by the formula $$ \\CH^i(M) = p\\left(\\CH^{i + m}(X) \\otimes \\mathbf{Q}\\right) $$"}
+{"_id": "5115", "title": "weil-definition-weil-cohomology-theory-classical", "text": "Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A {\\it classical Weil cohomology theory} over $k$ with coefficients in $F$ is given by data (D1), (D2), and (D3) satisfying Poincar\\'e duality, the K\\\"unneth formula, and compatibility with cycle classes, more precisely, satisfying (A), (B), and (C)."}
+{"_id": "5116", "title": "weil-definition-weil-cohomology-theory", "text": "Let $k$ be a field. Let $F$ be a field of characteristic $0$. A {\\it Weil cohomology theory} over $k$ with coefficients in $F$ is given by data (D0), (D1), (D2), and (D3) satisfying Poincar\\'e duality, the K\\\"unneth formula, and compatibility with cycle classes, more precisely, satisfying axioms (A), (B), and (C) of Section \\ref{section-axioms} and in addition such that the equivalent conditions (1) and (2) of Lemma \\ref{lemma-H-0-separable} hold for every smooth projective $X$ over $k$."}
+{"_id": "5537", "title": "morphisms-definition-scheme-theoretic-intersection-union", "text": "Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$. The {\\it scheme theoretic intersection} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$. The {\\it scheme theoretic union} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\\mathcal{I} \\cap \\mathcal{J}$."}
+{"_id": "5538", "title": "morphisms-definition-scheme-theoretic-support", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. The {\\it scheme theoretic support of $\\mathcal{F}$} is the closed subscheme $Z \\subset X$ constructed in Lemma \\ref{lemma-scheme-theoretic-support}."}
+{"_id": "5539", "title": "morphisms-definition-scheme-theoretic-image", "text": "Let $f : X \\to Y$ be a morphism of schemes. The {\\it scheme theoretic image} of $f$ is the smallest closed subscheme $Z \\subset Y$ through which $f$ factors, see Lemma \\ref{lemma-scheme-theoretic-image} above."}
+{"_id": "5540", "title": "morphisms-definition-scheme-theoretically-dense", "text": "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. \\begin{enumerate} \\item The scheme theoretic image of the morphism $U \\to X$ is called the {\\it scheme theoretic closure of $U$ in $X$}. \\item We say $U$ is {\\it scheme theoretically dense in $X$} if for every open $V \\subset X$ the scheme theoretic closure of $U \\cap V$ in $V$ is equal to $V$. \\end{enumerate}"}
+{"_id": "5541", "title": "morphisms-definition-dominant", "text": "A morphism $f : X \\to S$ of schemes is called {\\it dominant} if the image of $f$ is a dense subset of $S$."}
+{"_id": "5542", "title": "morphisms-definition-surjective", "text": "A morphism of schemes is said to be {\\it surjective} if it is surjective on underlying topological spaces."}
+{"_id": "5543", "title": "morphisms-definition-universally-injective", "text": "Let $f : X \\to S$ be a morphism. \\begin{enumerate} \\item We say that $f$ is {\\it universally injective} if and only if for any morphism of schemes $S' \\to S$ the base change $f' : X_{S'} \\to S'$ is injective (on underlying topological spaces). \\item We say $f$ is {\\it radicial} if $f$ is injective as a map of topological spaces, and for every $x \\in X$ the field extension $\\kappa(x) \\supset \\kappa(f(x))$ is purely inseparable. \\end{enumerate}"}
+{"_id": "5544", "title": "morphisms-definition-affine", "text": "A morphism of schemes $f : X \\to S$ is called {\\it affine} if the inverse image of every affine open of $S$ is an affine open of $X$."}
+{"_id": "5545", "title": "morphisms-definition-family-ample-invertible-modules", "text": "\\begin{reference} \\cite[II Definition 2.2.4]{SGA6} \\end{reference} Let $X$ be a scheme. Let $\\{\\mathcal{L}_i\\}_{i \\in I}$ be a family of invertible $\\mathcal{O}_X$-modules. We say $\\{\\mathcal{L}_i\\}_{i \\in I}$ is an {\\it ample family of invertible modules on $X$} if \\begin{enumerate} \\item $X$ is quasi-compact, and \\item for every $x \\in X$ there exists an $i \\in I$, an $n \\geq 1$, and $s \\in \\Gamma(X, \\mathcal{L}_i^{\\otimes n})$ such that $x \\in X_s$ and $X_s$ is affine. \\end{enumerate}"}
+{"_id": "5546", "title": "morphisms-definition-quasi-affine", "text": "A morphism of schemes $f : X \\to S$ is called {\\it quasi-affine} if the inverse image of every affine open of $S$ is a quasi-affine scheme."}
+{"_id": "5547", "title": "morphisms-definition-property-local", "text": "Let $P$ be a property of ring maps. \\begin{enumerate} \\item We say that $P$ is {\\it local} if the following hold: \\begin{enumerate} \\item For any ring map $R \\to A$, and any $f \\in R$ we have $P(R \\to A) \\Rightarrow P(R_f \\to A_f)$. \\item For any rings $R$, $A$, any $f \\in R$, $a\\in A$, and any ring map $R_f \\to A$ we have $P(R_f \\to A) \\Rightarrow P(R \\to A_a)$. \\item For any ring map $R \\to A$, and $a_i \\in A$ such that $(a_1, \\ldots, a_n) = A$ then $\\forall i, P(R \\to A_{a_i}) \\Rightarrow P(R \\to A)$. \\end{enumerate} \\item We say that $P$ is {\\it stable under base change} if for any ring maps $R \\to A$, $R \\to R'$ we have $P(R \\to A) \\Rightarrow P(R' \\to R' \\otimes_R A)$. \\item We say that $P$ is {\\it stable under composition} if for any ring maps $A \\to B$, $B \\to C$ we have $P(A \\to B) \\wedge P(B \\to C) \\Rightarrow P(A \\to C)$. \\end{enumerate}"}
+{"_id": "5548", "title": "morphisms-definition-locally-P", "text": "Let $P$ be a property of ring maps. Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it locally of type $P$} if for any $x \\in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ which maps into an affine open $V \\subset S$ such that the induced ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ has property $P$."}
+{"_id": "5549", "title": "morphisms-definition-finite-type", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is of {\\it finite type at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and an affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is of finite type. \\item We say that $f$ is {\\it locally of finite type} if it is of finite type at every point of $X$. \\item We say that $f$ is of {\\it finite type} if it is locally of finite type and quasi-compact. \\end{enumerate}"}
+{"_id": "5550", "title": "morphisms-definition-finite-type-point", "text": "Let $S$ be a scheme. Let us say that a point $s$ of $S$ is a {\\it finite type point} if the canonical morphism $\\Spec(\\kappa(s)) \\to S$ is of finite type. We denote $S_{\\text{ft-pts}}$ the set of finite type points of $S$."}
+{"_id": "5551", "title": "morphisms-definition-universally-catenary", "text": "Let $S$ be a scheme. Assume $S$ is locally Noetherian. We say $S$ is {\\it universally catenary} if for every morphism $X \\to S$ locally of finite type the scheme $X$ is catenary."}
+{"_id": "5552", "title": "morphisms-definition-J", "text": "Let $X$ be a locally Noetherian scheme. We say $X$ is {\\it J-2} if for every morphism $Y \\to X$ which is locally of finite type the regular locus $\\text{Reg}(Y)$ is open in $Y$."}
+{"_id": "5553", "title": "morphisms-definition-quasi-finite", "text": "\\begin{reference} \\cite[II Definition 6.2.3]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it quasi-finite at a point $x \\in X$} if there exist an affine neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and an affine open $\\Spec(R) = V \\subset S$ such that $f(U) \\subset V$, the ring map $R \\to A$ is of finite type, and $R \\to A$ is quasi-finite at the prime of $A$ corresponding to $x$ (see above). \\item We say $f$ is {\\it locally quasi-finite} if $f$ is quasi-finite at every point $x$ of $X$. \\item We say that $f$ is {\\it quasi-finite} if $f$ is of finite type and every point $x$ is an isolated point of its fibre. \\end{enumerate}"}
+{"_id": "5554", "title": "morphisms-definition-finite-presentation", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is of {\\it finite presentation at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is of finite presentation. \\item We say that $f$ is {\\it locally of finite presentation} if it is of finite presentation at every point of $X$. \\item We say that $f$ is of {\\it finite presentation} if it is locally of finite presentation, quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "5555", "title": "morphisms-definition-open", "text": "Let $f : X \\to S$ be a morphism. \\begin{enumerate} \\item We say $f$ is {\\it open} if the map on underlying topological spaces is open. \\item We say $f$ is {\\it universally open} if for any morphism of schemes $S' \\to S$ the base change $f' : X_{S'} \\to S'$ is open. \\end{enumerate}"}
+{"_id": "5556", "title": "morphisms-definition-submersive", "text": "Let $f : X \\to Y$ be a morphism of schemes. \\begin{enumerate} \\item We say $f$ is {\\it submersive}\\footnote{This is very different from the notion of a submersion of differential manifolds.} if the continuous map of underlying topological spaces is submersive, see Topology, Definition \\ref{topology-definition-submersive}. \\item We say $f$ is {\\it universally submersive} if for every morphism of schemes $Y' \\to Y$ the base change $Y' \\times_Y X \\to Y'$ is submersive. \\end{enumerate}"}
+{"_id": "5557", "title": "morphisms-definition-flat", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item We say $f$ is {\\it flat at a point $x \\in X$} if the local ring $\\mathcal{O}_{X, x}$ is flat over the local ring $\\mathcal{O}_{S, f(x)}$. \\item We say that $\\mathcal{F}$ is {\\it flat over $S$ at a point $x \\in X$} if the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{S, f(x)}$-module. \\item We say $f$ is {\\it flat} if $f$ is flat at every point of $X$. \\item We say that $\\mathcal{F}$ is {\\it flat over $S$} if $\\mathcal{F}$ is flat over $S$ at every point $x$ of $X$. \\end{enumerate}"}
+{"_id": "5558", "title": "morphisms-definition-scheme-structure-connected-component", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a connected component. The {\\it canonical scheme structure on $T$} is the unique scheme structure on $T$ such that the closed immersion $T \\to X$ is flat, see Lemma \\ref{lemma-characterize-flat-closed-immersions}."}
+{"_id": "5559", "title": "morphisms-definition-relative-dimension-d", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $f$ is locally of finite type. \\begin{enumerate} \\item We say $f$ is of {\\it relative dimension $\\leq d$ at $x$} if $\\dim_x(X_{f(x)}) \\leq d$. \\item We say $f$ is of {\\it relative dimension $\\leq d$} if $\\dim_x(X_{f(x)}) \\leq d$ for all $x \\in X$. \\item We say $f$ is of {\\it relative dimension $d$} if all nonempty fibres $X_s$ are equidimensional of dimension $d$. \\end{enumerate}"}
+{"_id": "5560", "title": "morphisms-definition-syntomic", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it syntomic at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is syntomic. \\item We say that $f$ is {\\it syntomic} if it is syntomic at every point of $X$. \\item If $S = \\Spec(k)$ and $f$ is syntomic, then we say that $X$ is a {\\it local complete intersection over $k$}. \\item A morphism of affine schemes $f : X \\to S$ is called {\\it standard syntomic} if there exists a global relative complete intersection $R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see Algebra, Definition \\ref{algebra-definition-relative-global-complete-intersection}) such that $X \\to S$ is isomorphic to $$ \\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R). $$ \\end{enumerate}"}
+{"_id": "5561", "title": "morphisms-definition-syntomic-relative-dimension", "text": "Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$ is {\\it syntomic of relative dimension $d$} if $f$ is syntomic and the function $\\dim_x(X_{f(x)}) = d$ for all $x \\in X$."}
+{"_id": "5562", "title": "morphisms-definition-conormal-sheaf", "text": "Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf $\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$} is the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$ described above."}
+{"_id": "5563", "title": "morphisms-definition-sheaf-differentials", "text": "Let $f : X \\to S$ be a morphism of schemes. The {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$} is the sheaf of differentials of $f$ viewed as a morphism of ringed spaces (Modules, Definition \\ref{modules-definition-differentials}) equipped with its {\\it universal $S$-derivation} $$ \\text{d}_{X/S} : \\mathcal{O}_X \\longrightarrow \\Omega_{X/S}. $$"}
+{"_id": "5564", "title": "morphisms-definition-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it smooth at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is smooth. \\item We say that $f$ is {\\it smooth} if it is smooth at every point of $X$. \\item A morphism of affine schemes $f : X \\to S$ is called {\\it standard smooth} if there exists a standard smooth ring map $R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see Algebra, Definition \\ref{algebra-definition-standard-smooth}) such that $X \\to S$ is isomorphic to $$ \\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R). $$ \\end{enumerate}"}
+{"_id": "5565", "title": "morphisms-definition-smooth-relative-dimension", "text": "Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$ is {\\it smooth of relative dimension $d$} if $f$ is smooth and $\\Omega_{X/S}$ is finite locally free of constant rank $d$."}
+{"_id": "5566", "title": "morphisms-definition-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it unramified at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is unramified. \\item We say that $f$ is {\\it G-unramified at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is G-unramified. \\item We say that $f$ is {\\it unramified} if it is unramified at every point of $X$. \\item We say that $f$ is {\\it G-unramified} if it is G-unramified at every point of $X$. \\end{enumerate}"}
+{"_id": "5567", "title": "morphisms-definition-etale", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it \\'etale at $x \\in X$} if there exists an affine open neighbourhood $\\Spec(A) = U \\subset X$ of $x$ and affine open $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$ such that the induced ring map $R \\to A$ is \\'etale. \\item We say that $f$ is {\\it \\'etale} if it is \\'etale at every point of $X$. \\item A morphism of affine schemes $f : X \\to S$ is called {\\it standard \\'etale} if $X \\to S$ is isomorphic to $$ \\Spec(R[x]_h/(g)) \\to \\Spec(R) $$ where $R \\to R[x]_h/(g)$ is a standard \\'etale ring map, see Algebra, Definition \\ref{algebra-definition-standard-etale}, i.e., $g$ is monic and $g'$ invertible in $R[x]_h/(g)$. \\end{enumerate}"}
+{"_id": "5568", "title": "morphisms-definition-relatively-ample", "text": "\\begin{reference} \\cite[II Definition 4.6.1]{EGA} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample}, or {\\it ample on $X/S$}, or {\\it $f$-ample} if $f : X \\to S$ is quasi-compact, and if for every affine open $V \\subset S$ the restriction of $\\mathcal{L}$ to the open subscheme $f^{-1}(V)$ of $X$ is ample."}
+{"_id": "5569", "title": "morphisms-definition-very-ample", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We say $\\mathcal{L}$ is {\\it relatively very ample} or more precisely {\\it $f$-relatively very ample}, or {\\it very ample on $X/S$}, or {\\it $f$-very ample} if there exist a quasi-coherent $\\mathcal{O}_S$-module $\\mathcal{E}$ and an immersion $i : X \\to \\mathbf{P}(\\mathcal{E})$ over $S$ such that $\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$."}
+{"_id": "5570", "title": "morphisms-definition-quasi-projective", "text": "\\begin{reference} \\cite[II, Definition 5.3.1]{EGA} and \\cite[page 103]{H} \\end{reference} Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say $f$ is {\\it quasi-projective} if $f$ is of finite type and there exists an $f$-relatively ample invertible $\\mathcal{O}_X$-module. \\item We say $f$ is {\\it H-quasi-projective} if there exists a quasi-compact immersion $X \\to \\mathbf{P}^n_S$ over $S$ for some $n$.\\footnote{This is not exactly the same as the definition in Hartshorne. Namely, the definition in Hartshorne (8th corrected printing, 1997) is that $f$ should be the composition of an open immersion followed by a H-projective morphism (see Definition \\ref{definition-projective}), which does not imply $f$ is quasi-compact. See Lemma \\ref{lemma-H-quasi-projective-open-H-projective} for the implication in the other direction.} \\item We say $f$ is {\\it locally quasi-projective} if there exists an open covering $S = \\bigcup V_j$ such that each $f^{-1}(V_j) \\to V_j$ is quasi-projective. \\end{enumerate}"}
+{"_id": "5571", "title": "morphisms-definition-proper", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it proper} if $f$ is separated, finite type, and universally closed."}
+{"_id": "5572", "title": "morphisms-definition-projective", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say $f$ is {\\it projective} if $X$ is isomorphic as an $S$-scheme to a closed subscheme of a projective bundle $\\mathbf{P}(\\mathcal{E})$ for some quasi-coherent, finite type $\\mathcal{O}_S$-module $\\mathcal{E}$. \\item We say $f$ is {\\it H-projective} if there exists an integer $n$ and a closed immersion $X \\to \\mathbf{P}^n_S$ over $S$. \\item We say $f$ is {\\it locally projective} if there exists an open covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is projective. \\end{enumerate}"}
+{"_id": "5573", "title": "morphisms-definition-integral", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say that $f$ is {\\it integral} if $f$ is affine and if for every affine open $\\Spec(R) = V \\subset S$ with inverse image $\\Spec(A) = f^{-1}(V) \\subset X$ the associated ring map $R \\to A$ is integral. \\item We say that $f$ is {\\it finite} if $f$ is affine and if for every affine open $\\Spec(R) = V \\subset S$ with inverse image $\\Spec(A) = f^{-1}(V) \\subset X$ the associated ring map $R \\to A$ is finite. \\end{enumerate}"}
+{"_id": "5574", "title": "morphisms-definition-universal-homeomorphism", "text": "A morphisms $f : X \\to Y$ of schemes is called a {\\it universal homeomorphism} if the base change $f' : Y' \\times_Y X \\to Y'$ is a homeomorphism for every morphism $Y' \\to Y$."}
+{"_id": "5575", "title": "morphisms-definition-seminormal-ring", "text": "Let $A$ be a ring. \\begin{enumerate} \\item We say $A$ is {\\it seminormal} if for all $x, y \\in A$ with $x^3 = y^2$ there is a unique $a \\in A$ with $x = a^2$ and $y = a^3$. \\item We say $A$ is {\\it absolutely weakly normal} if (a) $A$ is seminormal and (b) for any prime number $p$ and $x, y \\in A$ with $p^px = y^p$ there is a unique $a \\in A$ with $x = a^p$ and $y = pa$. \\end{enumerate}"}
+{"_id": "5576", "title": "morphisms-definition-seminormal", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item We say $X$ is {\\it seminormal} if every $x \\in X$ has an affine open neighbourhood $\\Spec(R) = U \\subset X$ such that the ring $R$ is seminormal. \\item We say $X$ is {\\it absolutely weakly normal} if every $x \\in X$ has an affine open neighbourhood $\\Spec(R) = U \\subset X$ such that the ring $R$ is absolutely weakly normal. \\end{enumerate}"}
+{"_id": "5577", "title": "morphisms-definition-seminormalization", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item The morphism $X^{sn} \\to X$ constructed in Lemma \\ref{lemma-seminormalization} is the {\\it seminormalization} of $X$. \\item The morphism $X^{awn} \\to X$ constructed in Lemma \\ref{lemma-seminormalization} is the {\\it absolute weak normalization} of $X$. \\end{enumerate}"}
+{"_id": "5578", "title": "morphisms-definition-finite-locally-free", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it finite locally free} if $f$ is affine and $f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module. In this case we say $f$ is has {\\it rank} or {\\it degree} $d$ if the sheaf $f_*\\mathcal{O}_X$ is finite locally free of degree $d$."}
+{"_id": "5579", "title": "morphisms-definition-rational-map", "text": "Let $X$, $Y$ be schemes. \\begin{enumerate} \\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of schemes defined on dense open subsets $U$, $V$ of $X$. We say that $f$ is {\\it equivalent} to $g$ if $f|_W = g|_W$ for some $W \\subset U \\cap V$ dense open in $X$. \\item A {\\it rational map from $X$ to $Y$} is an equivalence class for the equivalence relation defined in (1). \\item If $X$, $Y$ are schemes over a base scheme $S$ we say that a rational map from $X$ to $Y$ is an {\\it $S$-rational map from $X$ to $Y$} if there exists a representative $f : U \\to Y$ of the equivalence class which is an $S$-morphism. \\end{enumerate}"}
+{"_id": "5580", "title": "morphisms-definition-rational-function", "text": "Let $X$ be a scheme. A {\\it rational function on $X$} is a rational map from $X$ to $\\mathbf{A}^1_{\\mathbf{Z}}$."}
+{"_id": "5581", "title": "morphisms-definition-ring-of-rational-functions", "text": "Let $X$ be a scheme. The {\\it ring of rational functions on $X$} is the ring $R(X)$ whose elements are rational functions with addition and multiplication as just described."}
+{"_id": "5582", "title": "morphisms-definition-function-field", "text": "Let $X$ be an integral scheme. The {\\it function field}, or the {\\it field of rational functions} of $X$ is the field $R(X)$."}
+{"_id": "5583", "title": "morphisms-definition-domain-of-definition", "text": "Let $\\varphi$ be a rational map between two schemes $X$ and $Y$. We say $\\varphi$ is {\\it defined in a point $x \\in X$} if there exists a representative $(U, f)$ of $\\varphi$ with $x \\in U$. The {\\it domain of definition} of $\\varphi$ is the set of all points where $\\varphi$ is defined."}
+{"_id": "5584", "title": "morphisms-definition-dominant-rational", "text": "Let $X$ and $Y$ be irreducible schemes. A rational map from $X$ to $Y$ is called {\\it dominant} if any representative $f : U \\to Y$ is a dominant morphism of schemes."}
+{"_id": "5585", "title": "morphisms-definition-birational-schemes", "text": "Let $X$ and $Y$ be irreducible schemes. \\begin{enumerate} \\item We say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic in the category of irreducible schemes and dominant rational maps. \\item Assume $X$ and $Y$ are schemes over a base scheme $S$. We say $X$ and $Y$ are {\\it $S$-birational} if $X$ and $Y$ are isomorphic in the category of irreducible schemes over $S$ and dominant $S$-rational maps. \\end{enumerate}"}
+{"_id": "5586", "title": "morphisms-definition-birational", "text": "\\begin{reference} \\cite[(2.2.9)]{EGA1} \\end{reference} Let $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many irreducible components. We say a morphism $f : X \\to Y$ is {\\it birational} if \\begin{enumerate} \\item $f$ induces a bijection between the set of generic points of irreducible components of $X$ and the set of generic points of the irreducible components of $Y$, and \\item for every generic point $\\eta \\in X$ of an irreducible component of $X$ the local ring map $\\mathcal{O}_{Y, f(\\eta)} \\to \\mathcal{O}_{X, \\eta}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "5587", "title": "morphisms-definition-degree", "text": "Let $X$ and $Y$ be integral schemes. Let $f : X \\to Y$ be locally of finite type and dominant. Assume $[R(X) : R(Y)] < \\infty$, or any other of the equivalent conditions (1) -- (4) of Lemma \\ref{lemma-finite-degree}. Then the positive integer $$ \\text{deg}(X/Y) = [R(X) : R(Y)] $$ is called the {\\it degree of $X$ over $Y$}."}
+{"_id": "5588", "title": "morphisms-definition-modification", "text": "Let $X$ be an integral scheme. A {\\it modification of $X$} is a birational proper morphism $f : X' \\to X$ with $X'$ integral."}
+{"_id": "5589", "title": "morphisms-definition-alteration", "text": "\\begin{reference} \\cite[Definition 2.20]{alterations} \\end{reference} Let $X$ be an integral scheme. An {\\it alteration of $X$} is a proper dominant morphism $f : Y \\to X$ with $Y$ integral such that $f^{-1}(U) \\to U$ is finite for some nonempty open $U \\subset X$."}
+{"_id": "5590", "title": "morphisms-definition-integral-closure", "text": "Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The {\\it integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$} is the quasi-coherent $\\mathcal{O}_X$-subalgebra $\\mathcal{A}' \\subset \\mathcal{A}$ constructed in Lemma \\ref{lemma-integral-closure} above."}
+{"_id": "5591", "title": "morphisms-definition-normalization-X-in-Y", "text": "Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $\\mathcal{O}'$ be the integral closure of $\\mathcal{O}_X$ in $f_*\\mathcal{O}_Y$. The {\\it normalization of $X$ in $Y$} is the scheme\\footnote{The scheme $X'$ need not be normal, for example if $Y = X$ and $f = \\text{id}_X$, then $X' = X$.} $$ \\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X $$ over $X$. It comes equipped with a natural factorization $$ Y \\xrightarrow{f'} X' \\xrightarrow{\\nu} X $$ of the initial morphism $f$."}
+{"_id": "5592", "title": "morphisms-definition-normalization", "text": "Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. We define the {\\it normalization} of $X$ as the morphism $$ \\nu : X^\\nu \\longrightarrow X $$ which is the normalization of $X$ in the morphism $f : Y \\to X$ (\\ref{equation-generic-points}) constructed above."}
+{"_id": "5593", "title": "morphisms-definition-universally-bounded", "text": "Let $f : X \\to Y$ be a morphism of schemes. \\begin{enumerate} \\item We say the integer $n$ {\\it bounds the degrees of the fibres of $f$} if for all $y \\in Y$ the fibre $X_y$ is a finite scheme over $\\kappa(y)$ whose degree over $\\kappa(y)$ is $\\leq n$. \\item We say the {\\it fibres of $f$ are universally bounded}\\footnote{This is probably nonstandard notation.} if there exists an integer $n$ which bounds the degrees of the fibres of $f$. \\end{enumerate}"}
+{"_id": "5647", "title": "smoothing-definition-singular-ideal", "text": "Let $R \\to A$ be a ring map. The {\\it singular ideal of $A$ over $R$}, denoted $H_{A/R}$ is the unique radical ideal $H_{A/R} \\subset A$ with $$ V(H_{A/R}) = \\{\\mathfrak q \\in \\Spec(A) \\mid R \\to A \\text{ not smooth at }\\mathfrak q\\} $$"}
+{"_id": "5648", "title": "smoothing-definition-strictly-standard", "text": "Let $R \\to A$ be a ring map of finite presentation. We say an element $a \\in A$ is {\\it elementary standard in $A$ over $R$} if there exists a presentation $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and $0 \\leq c \\leq \\min(n, m)$ such that \\begin{equation} \\label{equation-elementary-standard-one} a = a' \\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c} \\end{equation} for some $a' \\in A$ and \\begin{equation} \\label{equation-elementary-standard-two} a f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2 \\end{equation} for $j = 1, \\ldots, m - c$. We say $a \\in A$ is {\\it strictly standard in $A$ over $R$} if there exists a presentation $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and $0 \\leq c \\leq \\min(n, m)$ such that \\begin{equation} \\label{equation-strictly-standard-one} a = \\sum\\nolimits_{I \\subset \\{1, \\ldots, n\\},\\ |I| = c} a_I \\det(\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c,\\ i \\in I} \\end{equation} for some $a_I \\in A$ and \\begin{equation} \\label{equation-strictly-standard-two} a f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2 \\end{equation} for $j = 1, \\ldots, m - c$."}
+{"_id": "5903", "title": "chow-definition-periodic-complex", "text": "Let $R$ be a ring. \\begin{enumerate} \\item A {\\it $2$-periodic complex} over $R$ is given by a quadruple $(M, N, \\varphi, \\psi)$ consisting of $R$-modules $M$, $N$ and $R$-module maps $\\varphi : M \\to N$, $\\psi : N \\to M$ such that $$ \\xymatrix{ \\ldots \\ar[r] & M \\ar[r]^\\varphi & N \\ar[r]^\\psi & M \\ar[r]^\\varphi & N \\ar[r] & \\ldots } $$ is a complex. In this setting we define the {\\it cohomology modules} of the complex to be the $R$-modules $$ H^0(M, N, \\varphi, \\psi) = \\Ker(\\varphi)/\\Im(\\psi) \\quad\\text{and}\\quad H^1(M, N, \\varphi, \\psi) = \\Ker(\\psi)/\\Im(\\varphi). $$ We say the $2$-periodic complex is {\\it exact} if the cohomology groups are zero. \\item A {\\it $(2, 1)$-periodic complex} over $R$ is given by a triple $(M, \\varphi, \\psi)$ consisting of an $R$-module $M$ and $R$-module maps $\\varphi : M \\to M$, $\\psi : M \\to M$ such that $$ \\xymatrix{ \\ldots \\ar[r] & M \\ar[r]^\\varphi & M \\ar[r]^\\psi & M \\ar[r]^\\varphi & M \\ar[r] & \\ldots } $$ is a complex. Since this is a special case of a $2$-periodic complex we have its {\\it cohomology modules} $H^0(M, \\varphi, \\psi)$, $H^1(M, \\varphi, \\psi)$ and a notion of exactness. \\end{enumerate}"}
+{"_id": "5904", "title": "chow-definition-periodic-length", "text": "Let $(M, N, \\varphi, \\psi)$ be a $2$-periodic complex over a ring $R$ whose cohomology modules have finite length. In this case we define the {\\it multiplicity} of $(M, N, \\varphi, \\psi)$ to be the integer $$ e_R(M, N, \\varphi, \\psi) = \\text{length}_R(H^0(M, N, \\varphi, \\psi)) - \\text{length}_R(H^1(M, N, \\varphi, \\psi)) $$ In the case of a $(2, 1)$-periodic complex $(M, \\varphi, \\psi)$, we denote this by $e_R(M, \\varphi, \\psi)$ and we will sometimes call this the {\\it (additive) Herbrand quotient}."}
+{"_id": "5905", "title": "chow-definition-delta-dimension", "text": "Let $(S, \\delta)$ as in Situation \\ref{situation-setup}. For any scheme $X$ locally of finite type over $S$ and any irreducible closed subset $Z \\subset X$ we define $$ \\dim_\\delta(Z) = \\delta(\\xi) $$ where $\\xi \\in Z$ is the generic point of $Z$. We will call this the {\\it $\\delta$-dimension of $Z$}. If $Z$ is a closed subscheme of $X$, then we define $\\dim_\\delta(Z)$ as the supremum of the $\\delta$-dimensions of its irreducible components."}
+{"_id": "5906", "title": "chow-definition-cycles", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $k \\in \\mathbf{Z}$. \\begin{enumerate} \\item A {\\it cycle on $X$} is a formal sum $$ \\alpha = \\sum n_Z [Z] $$ where the sum is over integral closed subschemes $Z \\subset X$, each $n_Z \\in \\mathbf{Z}$, and the collection $\\{Z; n_Z \\not = 0\\}$ is locally finite (Topology, Definition \\ref{topology-definition-locally-finite}). \\item A {\\it $k$-cycle} on $X$ is a cycle $$ \\alpha = \\sum n_Z [Z] $$ where $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$. \\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$. \\end{enumerate}"}
+{"_id": "5907", "title": "chow-definition-cycle-associated-to-closed-subscheme", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme. \\begin{enumerate} \\item For any irreducible component $Z' \\subset Z$ with generic point $\\xi$ the integer $m_{Z', Z} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi}$ (Lemma \\ref{lemma-multiplicity-finite}) is called the {\\it multiplicity of $Z'$ in $Z$}. \\item Assume $\\dim_\\delta(Z) \\leq k$. The {\\it $k$-cycle associated to $Z$} is $$ [Z]_k = \\sum m_{Z', Z}[Z'] $$ where the sum is over the irreducible components of $Z$ of $\\delta$-dimension $k$. (This is a $k$-cycle by Divisors, Lemma \\ref{divisors-lemma-components-locally-finite}.) \\end{enumerate}"}
+{"_id": "5908", "title": "chow-definition-cycle-associated-to-coherent-sheaf", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For any irreducible component $Z' \\subset \\text{Supp}(\\mathcal{F})$ with generic point $\\xi$ the integer $m_{Z', \\mathcal{F}} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi$ (Lemma \\ref{lemma-length-finite}) is called the {\\it multiplicity of $Z'$ in $\\mathcal{F}$}. \\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. The {\\it $k$-cycle associated to $\\mathcal{F}$} is $$ [\\mathcal{F}]_k = \\sum m_{Z', \\mathcal{F}}[Z'] $$ where the sum is over the irreducible components of $\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$. (This is a $k$-cycle by Lemma \\ref{lemma-length-finite}.) \\end{enumerate}"}
+{"_id": "5909", "title": "chow-definition-proper-pushforward", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $f$ is proper. \\begin{enumerate} \\item Let $Z \\subset X$ be an integral closed subscheme with $\\dim_\\delta(Z) = k$. We define $$ f_*[Z] = \\left\\{ \\begin{matrix} 0 & \\text{if} & \\dim_\\delta(f(Z))< k, \\\\ \\deg(Z/f(Z)) [f(Z)] & \\text{if} & \\dim_\\delta(f(Z)) = k. \\end{matrix} \\right. $$ Here we think of $f(Z) \\subset Y$ as an integral closed subscheme. The degree of $Z$ over $f(Z)$ is finite if $\\dim_\\delta(f(Z)) = \\dim_\\delta(Z)$ by Lemma \\ref{lemma-equal-dimension}. \\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The {\\it pushforward} of $\\alpha$ as the sum $$ f_* \\alpha = \\sum n_Z f_*[Z] $$ where each $f_*[Z]$ is defined as above. The sum is locally finite by Lemma \\ref{lemma-quasi-compact-locally-finite} above. \\end{enumerate}"}
+{"_id": "5910", "title": "chow-definition-flat-pullback", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \\to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. \\begin{enumerate} \\item Let $Z \\subset Y$ be an integral closed subscheme of $\\delta$-dimension $k$. We define $f^*[Z]$ to be the $(k+r)$-cycle on $X$ to the scheme theoretic inverse image $$ f^*[Z] = [f^{-1}(Z)]_{k+r}. $$ This makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$ by Lemma \\ref{lemma-flat-inverse-image-dimension}. \\item Let $\\alpha = \\sum n_i [Z_i]$ be a $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$} is the sum $$ f^* \\alpha = \\sum n_i f^*[Z_i] $$ where each $f^*[Z_i]$ is defined as above. The sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}. \\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian groups so obtained. \\end{enumerate}"}
+{"_id": "5911", "title": "chow-definition-principal-divisor", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral with $\\dim_\\delta(X) = n$. Let $f \\in R(X)^*$. The {\\it principal divisor associated to $f$} is the $(n - 1)$-cycle $$ \\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z] $$ defined in Divisors, Definition \\ref{divisors-definition-principal-divisor}. This makes sense because prime divisors have $\\delta$-dimension $n - 1$ by Lemma \\ref{lemma-divisor-delta-dimension}."}
+{"_id": "5912", "title": "chow-definition-rational-equivalence", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $k \\in \\mathbf{Z}$. \\begin{enumerate} \\item Given any locally finite collection $\\{W_j \\subset X\\}$ of integral closed subschemes with $\\dim_\\delta(W_j) = k + 1$, and any $f_j \\in R(W_j)^*$ we may consider $$ \\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X) $$ where $i_j : W_j \\to X$ is the inclusion morphism. This makes sense as the morphism $\\coprod i_j : \\coprod W_j \\to X$ is proper. \\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero} if $\\alpha$ is a cycle of the form displayed above. \\item We say $\\alpha, \\beta \\in Z_k(X)$ are {\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$ if $\\alpha - \\beta$ is rationally equivalent to zero. \\item We define $$ \\CH_k(X) = Z_k(X) / \\sim_{rat} $$ to be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called the {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}. \\end{enumerate}"}
+{"_id": "5913", "title": "chow-definition-divisor-invertible-sheaf", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$ we define the {\\it Weil divisor associated to $s$} is the $(n - 1)$-cycle $$ \\text{div}_\\mathcal{L}(s) = \\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] $$ defined in Divisors, Definition \\ref{divisors-definition-divisor-invertible-sheaf}. This makes sense because Weil divisors have $\\delta$-dimension $n - 1$ by Lemma \\ref{lemma-divisor-delta-dimension}. \\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as $$ c_1(\\mathcal{L}) \\cap [X] = \\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X) $$ where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$. This is well defined by Divisors, Lemma \\ref{divisors-lemma-divisor-meromorphic-well-defined}. \\end{enumerate}"}
+{"_id": "5914", "title": "chow-definition-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We define, for every integer $k$, an operation $$ c_1(\\mathcal{L}) \\cap - : Z_{k + 1}(X) \\to \\CH_k(X) $$ called {\\it intersection with the first Chern class of $\\mathcal{L}$}. \\begin{enumerate} \\item Given an integral closed subscheme $i : W \\to X$ with $\\dim_\\delta(W) = k + 1$ we define $$ c_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W]) $$ where the right hand side is defined in Definition \\ref{definition-divisor-invertible-sheaf}. \\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set $$ c_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i] $$ \\end{enumerate}"}
+{"_id": "5915", "title": "chow-definition-gysin-homomorphism", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $(\\mathcal{L}, s)$ be a pair consisting of an invertible sheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$. Let $D = Z(s)$ be the zero scheme of $s$, and denote $i : D \\to X$ the closed immersion. We define, for every integer $k$, a {\\it Gysin homomorphism} $$ i^* : Z_{k + 1}(X) \\to \\CH_k(D). $$ by the following rules: \\begin{enumerate} \\item Given a integral closed subscheme $W \\subset X$ with $\\dim_\\delta(W) = k + 1$ we define \\begin{enumerate} \\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a $k$-cycle on $D$, and \\item if $W \\subset D$, then $i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$, where $i' : W \\to D$ is the induced closed immersion. \\end{enumerate} \\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$ we set $$ i^*\\alpha = \\sum n_j i^*[W_j] $$ \\item If $D$ is an effective Cartier divisor, then we denote $D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of the class $i^*\\alpha$ to a class on $X$. \\end{enumerate}"}
+{"_id": "5916", "title": "chow-definition-bivariant-class", "text": "\\begin{reference} Similar to \\cite[Definition 17.1]{F} \\end{reference} Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \\in \\mathbf{Z}$. A {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule which assigns to every locally of finite type morphism $Y' \\to Y$ and every $k$ a map $$ c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X') $$ where $X' = Y' \\times_Y X$, satisfying the following conditions \\begin{enumerate} \\item if $Y'' \\to Y'$ is a proper, then $c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$ for all $\\alpha''$ on $Y''$ where $X'' = Y'' \\times_Y X$, \\item if $Y'' \\to Y'$ is flat locally of finite type of fixed relative dimension, then $c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$ for all $\\alpha'$ on $Y'$, and \\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in Definition \\ref{definition-gysin-homomorphism} with pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$, then we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$ for all $\\alpha'$ on $Y'$. \\end{enumerate} The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^p(X \\to Y)$."}
+{"_id": "5917", "title": "chow-definition-chow-cohomology", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. The {\\it Chow cohomology} of $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree $p$ component is $A^p(X \\to X)$."}
+{"_id": "5918", "title": "chow-definition-first-chern-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The {\\it first Chern class} $c_1(\\mathcal{L}) \\in A^1(X)$ of $\\mathcal{L}$ is the bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}."}
+{"_id": "5919", "title": "chow-definition-chern-classes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space bundle associated to $\\mathcal{E}$. \\begin{enumerate} \\item By Lemma \\ref{lemma-chow-ring-projective-bundle} there are elements $c_i \\in \\CH_{n - i}(X)$, $i = 0, \\ldots, r$ such that $c_0 = [X]$, and \\begin{equation} \\label{equation-chern-classes} \\sum\\nolimits_{i = 0}^r (-1)^i c_1(\\mathcal{O}_P(1))^i \\cap \\pi^*c_{r - i} = 0. \\end{equation} \\item With notation as above we set $c_i(\\mathcal{E}) \\cap [X] = c_i$ as an element of $\\CH_{n - i}(X)$. We call these the {\\it Chern classes of $\\mathcal{E}$ on $X$}. \\item The {\\it total Chern class of $\\mathcal{E}$ on $X$} is the combination $$ c({\\mathcal E}) \\cap [X] = c_0({\\mathcal E}) \\cap [X] + c_1({\\mathcal E}) \\cap [X] + \\ldots + c_r({\\mathcal E}) \\cap [X] $$ which is an element of $\\CH_*(X) = \\bigoplus_{k \\in \\mathbf{Z}} \\CH_k(X)$. \\end{enumerate}"}
+{"_id": "5920", "title": "chow-definition-cap-chern-classes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. We define, for every integer $k$ and any $0 \\leq j \\leq r$, an operation $$ c_j(\\mathcal{E}) \\cap - : Z_k(X) \\to \\CH_{k - j}(X) $$ called {\\it intersection with the $j$th Chern class of $\\mathcal{E}$}. \\begin{enumerate} \\item Given an integral closed subscheme $i : W \\to X$ of $\\delta$-dimension $k$ we define $$ c_j(\\mathcal{E}) \\cap [W] = i_*(c_j({i^*\\mathcal{E}}) \\cap [W]) \\in \\CH_{k - j}(X) $$ where $c_j({i^*\\mathcal{E}}) \\cap [W]$ is as defined in Definition \\ref{definition-chern-classes}. \\item For a general $k$-cycle $\\alpha = \\sum n_i [W_i]$ we set $$ c_j(\\mathcal{E}) \\cap \\alpha = \\sum n_i c_j(\\mathcal{E}) \\cap [W_i] $$ \\end{enumerate}"}
+{"_id": "5921", "title": "chow-definition-chern-classes-final", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. For $i = 0, \\ldots, r$ the {\\it $i$th Chern class} of $\\mathcal{E}$ is the bivariant class $c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$ constructed in Lemma \\ref{lemma-cap-cp-bivariant}. The {\\it total Chern class} of $\\mathcal{E}$ is the formal sum $$ c(\\mathcal{E}) =  c_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E}) $$ which is viewed as a nonhomogeneous bivariant class on $X$."}
+{"_id": "5922", "title": "chow-definition-degree-zero-cycle", "text": "Let $k$ be a field (Example \\ref{example-field}). Let $p : X \\to \\Spec(k)$ be proper. The {\\it degree of a zero cycle} on $X$ is given by proper pushforward $$ p_* : \\CH_0(X) \\to \\CH_0(\\Spec(k)) $$ (Lemma \\ref{lemma-proper-pushforward-rational-equivalence}) combined with the natural isomorphism $\\CH_0(\\Spec(k)) = \\mathbf{Z}$ which maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$."}
+{"_id": "5923", "title": "chow-definition-defined-on-perfect", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object. If $E$ is isomorphic in $D(\\mathcal{O}_X)$ to a finite complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules, then we say {\\it Chern classes of $E$ are defined}. If this is the case, then we define $c(E) = c(\\mathcal{E}^\\bullet) \\in A^*(X)$, $ch(E) = ch(\\mathcal{E}^\\bullet) \\in A^*(X) \\otimes \\mathbf{Q}$, and $P_p(E) = P_p(\\mathcal{E}^\\bullet) \\in A^p(X)$."}
+{"_id": "5924", "title": "chow-definition-localized-chern", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \\to X$ be a closed immersion. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes are defined. \\begin{enumerate} \\item If the restriction $E|_{X \\setminus Z}$ is zero, then for all $p \\geq 0$ we define $$ P_p(Z \\to X, E) \\in A^p(Z \\to X) $$ by the construction given above and we define the {\\it localized Chern character} by the formula $$ ch(Z \\to X, E) = \\sum\\nolimits_{p = 0, 1, 2, \\ldots} \\frac{P_p(Z \\to X, E)}{p!} \\quad\\text{in}\\quad A^*(Z \\to X) \\otimes \\mathbf{Q} $$ \\item If the restriction $E|_{X \\setminus Z}$ is isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, then we define the {\\it localized $p$th Chern class} $c_p(Z \\to X, E)$ by the construction above. \\end{enumerate}"}
+{"_id": "5925", "title": "chow-definition-lci-gysin", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. We say {\\it the gysin map for $f$ exists} if we can write $f = g \\circ i$ with $g$ smooth and $i$ an immersion. In this case we define the {\\it gysin map} $f^! = i^! \\circ g^* \\in A^*(X \\to Y)$ as above."}
+{"_id": "5926", "title": "chow-definition-determinant", "text": "Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $\\kappa$. Let $M$ be a finite length $R$-module. Say $l = \\text{length}_R(M)$. \\begin{enumerate} \\item Given elements $x_1, \\ldots, x_r \\in M$ we denote $\\langle x_1, \\ldots, x_r \\rangle = Rx_1 + \\ldots + Rx_r$ the $R$-submodule of $M$ generated by $x_1, \\ldots, x_r$. \\item We will say an $l$-tuple of elements $(e_1, \\ldots, e_l)$ of $M$ is {\\it admissible} if $\\mathfrak m e_i \\subset \\langle e_1, \\ldots, e_{i - 1} \\rangle$ for $i = 1, \\ldots, l$. \\item A {\\it symbol} $[e_1, \\ldots, e_l]$ will mean $(e_1, \\ldots, e_l)$ is an admissible $l$-tuple. \\item An {\\it admissible relation} between symbols is one of the following: \\begin{enumerate} \\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and for some $1 \\leq a \\leq l$ we have $e_a \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then $[e_1, \\ldots, e_l] = 0$, \\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and for some $1 \\leq a \\leq l$ we have $e_a = \\lambda e'_a + x$ with $\\lambda \\in R^*$, and $x \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then $$ [e_1, \\ldots, e_l] = \\overline{\\lambda} [e_1, \\ldots, e_{a - 1}, e'_a, e_{a + 1}, \\ldots, e_l] $$ where $\\overline{\\lambda} \\in \\kappa^*$ is the image of $\\lambda$ in the residue field, and \\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and $\\mathfrak m e_a \\subset \\langle e_1, \\ldots, e_{a - 2}\\rangle$ then $$ [e_1, \\ldots, e_l] = - [e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l]. $$ \\end{enumerate} \\item We define the {\\it determinant of the finite length $R$-module $M$} to be $$ \\det\\nolimits_\\kappa(M) = \\left\\{ \\frac{\\kappa\\text{-vector space generated by symbols}} {\\kappa\\text{-linear combinations of admissible relations}} \\right\\} $$ \\end{enumerate}"}
+{"_id": "5927", "title": "chow-definition-periodic-determinant", "text": "Let $R$ be a local ring with residue field $\\kappa$. Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is exact. The {\\it determinant of $(M, \\varphi, \\psi)$} is the element $$ \\det\\nolimits_\\kappa(M, \\varphi, \\psi) \\in \\kappa^* $$ such that the composition $$ \\det\\nolimits_\\kappa(M) \\xrightarrow{\\gamma_\\psi \\circ \\sigma \\circ \\gamma_\\varphi^{-1}} \\det\\nolimits_\\kappa(M) $$ is multiplication by $(-1)^{\\text{length}_R(I_\\varphi)\\text{length}_R(I_\\psi)} \\det\\nolimits_\\kappa(M, \\varphi, \\psi)$."}
+{"_id": "5928", "title": "chow-definition-symbol-M", "text": "Let $A$ be a Noetherian local ring with residue field $\\kappa$. Let $a, b \\in A$. Let $M$ be a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$. We define the {\\it symbol associated to $M, a, b$} to be the element $$ d_M(a, b) = \\det\\nolimits_\\kappa(M/abM, a, b) \\in \\kappa^* $$"}
+{"_id": "5929", "title": "chow-definition-tame-symbol", "text": "Let $A$ be a Noetherian local domain of dimension $1$ with residue field $\\kappa$. Let $K$ be the fraction field of $A$. We define the {\\it tame symbol} of $A$ to be the map $$ K^* \\times K^* \\longrightarrow \\kappa^*, \\quad (x, y) \\longmapsto d_A(x, y) $$ where $d_A(x, y)$ is extended to $K^* \\times K^*$ by the multiplicativity of Lemma \\ref{lemma-multiplicativity-symbol}."}
+{"_id": "6206", "title": "flat-definition-one-step-devissage", "text": "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $s \\in S$ be a point. A {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$} is given by morphisms of schemes over $S$ $$ \\xymatrix{ X & Z \\ar[l]_i \\ar[r]^\\pi & Y } $$ and a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ of finite type such that \\begin{enumerate} \\item $X$, $S$, $Z$ and $Y$ are affine, \\item $i$ is a closed immersion of finite presentation, \\item $\\mathcal{F} \\cong i_*\\mathcal{G}$, \\item $\\pi$ is finite, and \\item the structure morphism $Y \\to S$ is smooth with geometrically irreducible fibres of dimension $\\dim(\\text{Supp}(\\mathcal{F}_s))$. \\end{enumerate} In this case we say $(Z, Y, i, \\pi, \\mathcal{G})$ is a one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$."}
+{"_id": "6207", "title": "flat-definition-one-step-devissage-at-x", "text": "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $x \\in X$ be a point with image $s$ in $S$. A {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$} is a system $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$, where $(Z, Y, i, \\pi, \\mathcal{G})$ is a one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$ and \\begin{enumerate} \\item $\\dim_x(\\text{Supp}(\\mathcal{F}_s)) = \\dim(\\text{Supp}(\\mathcal{F}_s))$, \\item $z \\in Z$ is a point with $i(z) = x$ and $\\pi(z) = y$, \\item we have $\\pi^{-1}(\\{y\\}) = \\{z\\}$, \\item the extension $\\kappa(s) \\subset \\kappa(y)$ is purely transcendental. \\end{enumerate}"}
+{"_id": "6208", "title": "flat-definition-shrink", "text": "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in Definition \\ref{definition-one-step-devissage-at-x}. Let $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$. Let us define a {\\it standard shrinking} of this situation to be given by standard opens $S' \\subset S$, $X' \\subset X$, $Z' \\subset Z$, and $Y' \\subset Y$ such that $s \\in S'$, $x \\in X'$, $z \\in Z'$, and $y \\in Y'$ and such that $$ (Z', Y', i|_{Z'}, \\pi|_{Z'}, \\mathcal{G}|_{Z'}, z, y) $$ is a one step d\\'evissage of $\\mathcal{F}|_{X'}/X'/S'$ at $x$."}
+{"_id": "6209", "title": "flat-definition-complete-devissage", "text": "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $s \\in S$ be a point. A {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$} is given by a diagram $$ \\xymatrix{ X & Z_1 \\ar[l]^{i_1} \\ar[d]^{\\pi_1} \\\\ & Y_1 & Z_2 \\ar[l]^{i_2} \\ar[d]^{\\pi_2} \\\\ & & Y_2 & Z_3 \\ar[l] \\ar[d] \\\\ & & & ... & ... \\ar[l] \\ar[d] \\\\ & & & & Y_n } $$ of schemes over $S$, finite type quasi-coherent $\\mathcal{O}_{Z_k}$-modules $\\mathcal{G}_k$, and $\\mathcal{O}_{Y_k}$-module maps $$ \\alpha_k : \\mathcal{O}_{Y_k}^{\\oplus r_k} \\longrightarrow \\pi_{k, *}\\mathcal{G}_k, \\quad k = 1, \\ldots, n $$ satisfying the following properties: \\begin{enumerate} \\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ is a one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$, \\item the map $\\alpha_k$ induces an isomorphism $$ \\kappa(\\xi_k)^{\\oplus r_k} \\longrightarrow (\\pi_{k, *}\\mathcal{G}_k)_{\\xi_k} \\otimes_{\\mathcal{O}_{Y_k, \\xi_k}} \\kappa(\\xi_k) $$ where $\\xi_k \\in (Y_k)_s$ is the unique generic point, \\item for $k = 2, \\ldots, n$ the system $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k)$ is a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$ over $s$, \\item $\\Coker(\\alpha_n) = 0$. \\end{enumerate} In this case we say that $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$ is a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$."}
+{"_id": "6210", "title": "flat-definition-complete-devissage-at-x", "text": "Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $x \\in X$ be a point with image $s \\in S$. A {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$} is given by a system $$ (Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n} $$ such that $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)$ is a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$, and such that \\begin{enumerate} \\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1, z_1, y_1)$ is a one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$, \\item for $k = 2, \\ldots, n$ the system $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, z_k, y_k)$ is a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$ at $y_{k - 1}$. \\end{enumerate}"}
+{"_id": "6211", "title": "flat-definition-shrink-complete", "text": "Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in Definition \\ref{definition-complete-devissage-at-x}. Consider a complete d\\'evissage $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$ of $\\mathcal{F}/X/S$ at $x$. Let us define a {\\it standard shrinking} of this situation to be given by standard opens $S' \\subset S$, $X' \\subset X$, $Z'_k \\subset Z_k$, and $Y'_k \\subset Y_k$ such that $s_k \\in S'$, $x_k \\in X'$, $z_k \\in Z'$, and $y_k \\in Y'$ and such that $$ (Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k, z_k, y_k)_{k = 1, \\ldots, n} $$ is a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x$ where $\\mathcal{G}'_k = \\mathcal{G}_k|_{Z'_k}$ and $\\mathcal{F}' = \\mathcal{F}|_{X'}$."}
+{"_id": "6212", "title": "flat-definition-elementary-etale-neighbourhood", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak p$ of $R$. A {\\it elementary \\'etale localization of the ring map $R \\to S$ at $\\mathfrak q$} is given by a commutative diagram of rings and accompanying primes $$ \\xymatrix{ S \\ar[r] & S' \\\\ R \\ar[u] \\ar[r] & R' \\ar[u] } \\quad\\quad \\xymatrix{ \\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\ \\mathfrak p \\ar@{-}[u] \\ar@{-}[r] & \\mathfrak p' \\ar@{-}[u] } $$ such that $R \\to R'$ and $S \\to S'$ are \\'etale ring maps and $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and $\\kappa(\\mathfrak q) = \\kappa(\\mathfrak q')$."}
+{"_id": "6213", "title": "flat-definition-complete-devissage-algebra", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak r$ be a prime of $R$. Let $N$ be a finite $S$-module. A {\\it complete d\\'evissage of $N/S/R$ over $\\mathfrak r$} is given by $R$-algebra maps $$ \\xymatrix{ & A_1 & & A_2 & & ... & & A_n \\\\ S \\ar[ru] & & B_1 \\ar[lu] \\ar[ru] & & ... \\ar[lu] \\ar[ru] & & ... \\ar[lu] \\ar[ru] & & B_n \\ar[lu] } $$ finite $A_i$-modules $M_i$ and $B_i$-module maps $\\alpha_i : B_i^{\\oplus r_i} \\to M_i$ such that \\begin{enumerate} \\item $S \\to A_1$ is surjective and of finite presentation, \\item $B_i \\to A_{i + 1}$ is surjective and of finite presentation, \\item $B_i \\to A_i$ is finite, \\item $R \\to B_i$ is smooth with geometrically irreducible fibres, \\item $N \\cong M_1$ as $S$-modules, \\item $\\Coker(\\alpha_i) \\cong M_{i + 1}$ as $B_i$-modules, \\item $\\alpha_i : \\kappa(\\mathfrak p_i)^{\\oplus r_i} \\to M_i \\otimes_{B_i} \\kappa(\\mathfrak p_i)$ is an isomorphism where $\\mathfrak p_i = \\mathfrak rB_i$, and \\item $\\Coker(\\alpha_n) = 0$. \\end{enumerate} In this situation we say that $(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$ is a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$."}
+{"_id": "6214", "title": "flat-definition-complete-devissage-at-x-algebra", "text": "Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak r$ of $R$. Let $N$ be a finite $S$-module. A {\\it complete d\\'evissage of $N/S/R$ at $\\mathfrak q$} is given by a complete d\\'evissage $(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$ of $N/S/R$ over $\\mathfrak r$ and prime ideals $\\mathfrak q_i \\subset B_i$ lying over $\\mathfrak r$ such that \\begin{enumerate} \\item $\\kappa(\\mathfrak r) \\subset \\kappa(\\mathfrak q_i)$ is purely transcendental, \\item there is a unique prime $\\mathfrak q'_i \\subset A_i$ lying over $\\mathfrak q_i \\subset B_i$, \\item $\\mathfrak q = \\mathfrak q'_1 \\cap S$ and $\\mathfrak q_i = \\mathfrak q'_{i + 1} \\cap A_i$, \\item $R \\to B_i$ has relative dimension $\\dim_{\\mathfrak q_i}(\\text{Supp}(M_i \\otimes_R \\kappa(\\mathfrak r)))$. \\end{enumerate}"}
+{"_id": "6215", "title": "flat-definition-impurity", "text": "In Situation \\ref{situation-pre-pure} we say a diagram (\\ref{equation-impurity}) defines an {\\it impurity of $\\mathcal{F}$ above $s$} if $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and $\\overline{\\{\\xi\\}} \\cap X_t = \\emptyset$. We will indicate this by saying ``let $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$ above $s$''."}
+{"_id": "6216", "title": "flat-definition-pure", "text": "Let $f : X \\to S$ be a morphism of schemes which is of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item Let $s \\in S$. We say $\\mathcal{F}$ is {\\it pure along $X_s$} if there is no impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ with $(T, t) \\to (S, s)$ an elementary \\'etale neighbourhood. \\item We say $\\mathcal{F}$ is {\\it universally pure along $X_s$} if there does not exist any impurity of $\\mathcal{F}$ above $s$. \\item We say that $X$ is {\\it pure along $X_s$} if $\\mathcal{O}_X$ is pure along $X_s$. \\item We say $\\mathcal{F}$ is {\\it universally $S$-pure}, or {\\it universally pure relative to $S$} if $\\mathcal{F}$ is universally pure along $X_s$ for every $s \\in S$. \\item We say $\\mathcal{F}$ is {\\it $S$-pure}, or {\\it pure relative to $S$} if $\\mathcal{F}$ is pure along $X_s$ for every $s \\in S$. \\item We say that $X$ is {\\it $S$-pure} or {\\it pure relative to $S$} if $\\mathcal{O}_X$ is pure relative to $S$. \\end{enumerate}"}
+{"_id": "6217", "title": "flat-definition-flat-dimension-n", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $n \\geq 0$. We say {\\it $\\mathcal{F}$ is flat over $S$ in dimensions $\\geq n$} if the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n} are satisfied."}
+{"_id": "6218", "title": "flat-definition-flattening", "text": "Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say that the {\\it universal flattening of $\\mathcal{F}$ exists} if the functor $F_{flat}$ defined in Situation \\ref{situation-flat} is representable by a scheme $S'$ over $S$. We say that the {\\it universal flattening of $X$ exists} if the universal flattening of $\\mathcal{O}_X$ exists."}
+{"_id": "6219", "title": "flat-definition-flattening-stratification", "text": "Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say that $\\mathcal{F}$ has a {\\it flattening stratification} if the functor $F_{flat}$ defined in Situation \\ref{situation-flat} is representable by a monomorphism $S' \\to S$ associated to a stratification of $S$ by locally closed subschemes. We say that $X$ has a {\\it flattening stratification} if $\\mathcal{O}_X$ has a flattening stratification."}
+{"_id": "6220", "title": "flat-definition-h-covering", "text": "Let $T$ be a scheme. A {\\it h covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ such that each $f_i$ is locally of finite presentation and one of the equivalent conditions of Lemma \\ref{lemma-equivalence-h-v-locally-finite-presentation} is satisfied."}
+{"_id": "6221", "title": "flat-definition-big-h-site", "text": "A {\\it big h site} is any site $\\Sch_h$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of h coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of h coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "6222", "title": "flat-definition-standard-h", "text": "Let $T$ be an affine scheme. A {\\it standard h covering} of $T$ is a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$ with each $T_i$ affine, with $f_i$ of finite presentation satisfying either of the following equivalent conditions: (1) $\\{U_i \\to U\\}$ can be refined by a standard ph covering or (2) $\\{U_i \\to U\\}$ is a V covering."}
+{"_id": "6223", "title": "flat-definition-big-small-h", "text": "Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$. \\begin{enumerate} \\item The {\\it big h site of $S$}, denoted $(\\Sch/S)_h$, is the site $\\Sch_h/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it big affine h site of $S$}, denoted $(\\textit{Aff}/S)_h$, is the full subcategory of $(\\Sch/S)_h$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_h$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_h$ which is a standard h covering. \\end{enumerate}"}
+{"_id": "6353", "title": "curves-definition-normal-projective-model", "text": "Let $k$ be a field. Let $X$ be a curve. A {\\it nonsingular projective model of $X$} is a pair $(Y, \\varphi)$ where $Y$ is a nonsingular projective curve and $\\varphi : k(X) \\to k(Y)$ is an isomorphism of function fields."}
+{"_id": "6354", "title": "curves-definition-linear-series", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $d \\geq 0$ and $r \\geq 0$. A {\\it linear series of degree $d$ and dimension $r$} is a pair $(\\mathcal{L}, V)$ where $\\mathcal{L}$ in an invertible $\\mathcal{O}_X$-module of degree $d$ (Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}) and $V \\subset H^0(X, \\mathcal{L})$ is a $k$-subvector space of dimension $r + 1$. We will abbreviate this by saying $(\\mathcal{L}, V)$ is a {\\it $\\mathfrak g^r_d$} on $X$."}
+{"_id": "6355", "title": "curves-definition-genus", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Then the {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$."}
+{"_id": "6356", "title": "curves-definition-geometric-genus", "text": "Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$. The {\\it geometric genus} of $X$ is the genus of a smooth projective model of $X$ possibly defined over an extension field of $k$ as in Lemma \\ref{lemma-smooth-models}."}
+{"_id": "6357", "title": "curves-definition-multicross", "text": "Let $k$ be an algebraically closed field. Let $X$ be an algebraic $1$-dimensional $k$-scheme. Let $x \\in X$ be a closed point. We say $x$ defines a {\\it multicross singularity} if the completion $\\mathcal{O}_{X, x}^\\wedge$ is isomorphic to (\\ref{equation-multicross}) for some $n \\geq 2$. We say $x$ is a {\\it node}, or an {\\it ordinary double point}, or {\\it defines a nodal singularity} if $n = 2$."}
+{"_id": "6358", "title": "curves-definition-nodal", "text": "Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme. \\begin{enumerate} \\item We say a closed point $x \\in X$ is a {\\it node}, or an {\\it ordinary double point}, or {\\it defines a nodal singularity} if there exists an ordinary double point $\\overline{x} \\in X_{\\overline{k}}$ mapping to $x$. \\item We say the {\\it singularities of $X$ are at-worst-nodal} if all closed points of $X$ are either in the smooth locus of the structure morphism $X \\to \\Spec(k)$ or are ordinary double points. \\end{enumerate}"}
+{"_id": "6359", "title": "curves-definition-split-node", "text": "Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \\in X$ be a closed point. We say $x$ is a {\\it split node} if $x$ is a node, $\\kappa(x) = k$, and the equivalent assertions of Remark \\ref{remark-trivial-quadratic-extension} hold for $A = \\mathcal{O}_{X, x}$."}
+{"_id": "6360", "title": "curves-definition-nodal-family", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it at-worst-nodal of relative dimension $1$} if $f$ satisfies the equivalent conditions of Lemma \\ref{lemma-nodal-family}."}
+{"_id": "6710", "title": "etale-cohomology-definition-etale-covering-initial", "text": "A family of morphisms $\\{ \\varphi_i : U_i \\to X\\}_{i \\in I}$ is called an {\\it \\'etale covering} if each $\\varphi_i$ is an \\'etale morphism and their images cover $X$, i.e., $X = \\bigcup_{i \\in I} \\varphi_i(U_i)$."}
+{"_id": "6711", "title": "etale-cohomology-definition-presheaf", "text": "Let $\\mathcal{C}$ be a category. A {\\it presheaf of sets} (respectively, an {\\it abelian presheaf}) on $\\mathcal{C}$ is a functor $\\mathcal{C}^{opp} \\to \\textit{Sets}$ (resp.\\ $\\textit{Ab}$)."}
+{"_id": "6712", "title": "etale-cohomology-definition-family-morphisms-fixed-target", "text": "Let $\\mathcal{C}$ be a category. A {\\it family of morphisms with fixed target} $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is the data of \\begin{enumerate} \\item an object $U \\in \\mathcal{C}$, \\item a set $I$ (possibly empty), and \\item for all $i\\in I$, a morphism $\\varphi_i : U_i \\to U$ of $\\mathcal{C}$ with target $U$. \\end{enumerate}"}
+{"_id": "6713", "title": "etale-cohomology-definition-site", "text": "A {\\it site}\\footnote{What we call a site is a called a category endowed with a pretopology in \\cite[Expos\\'e II, D\\'efinition 1.3]{SGA4}. In \\cite{ArtinTopologies} it is called a category with a Grothendieck topology.} consists of a category $\\mathcal{C}$ and a set $\\text{Cov}(\\mathcal{C})$ consisting of families of morphisms with fixed target called {\\it coverings}, such that \\begin{enumerate} \\item (isomorphism) if $\\varphi : V \\to U$ is an isomorphism in $\\mathcal{C}$, then $\\{\\varphi : V \\to U\\}$ is a covering, \\item (locality) if $\\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is a covering and for all $i \\in I$ we are given a covering $\\{\\psi_{ij} : U_{ij} \\to U_i \\}_{j\\in I_i}$, then $$ \\{ \\varphi_i \\circ \\psi_{ij} : U_{ij} \\to U \\}_{(i, j)\\in \\prod_{i\\in I} \\{i\\} \\times I_i} $$ is also a covering, and \\item (base change) if $\\{U_i \\to U\\}_{i\\in I}$ is a covering and $V \\to U$ is a morphism in $\\mathcal{C}$, then \\begin{enumerate} \\item for all $i \\in I$ the fibre product $U_i \\times_U V$ exists in $\\mathcal{C}$, and \\item $\\{U_i \\times_U V \\to V\\}_{i\\in I}$ is a covering. \\end{enumerate} \\end{enumerate}"}
+{"_id": "6714", "title": "etale-cohomology-definition-sheaf", "text": "A presheaf $\\mathcal{F}$ of sets (resp. abelian presheaf) on a site $\\mathcal{C}$ is said to be a {\\it separated presheaf} if for all coverings $\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$ the map $$ \\mathcal{F}(U) \\longrightarrow \\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i) $$ is injective. Here the map is $s \\mapsto (s|_{U_i})_{i\\in I}$. The presheaf $\\mathcal{F}$ is a {\\it sheaf} if for all coverings $\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$, the diagram \\begin{equation} \\label{equation-sheaf-axiom} \\xymatrix{ \\mathcal{F}(U) \\ar[r] & \\prod_{i\\in I} \\mathcal{F}(U_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod_{i, j \\in I} \\mathcal{F}(U_i \\times_U U_j), } \\end{equation} where the first map is $s \\mapsto (s|_{U_i})_{i\\in I}$ and the two maps on the right are $(s_i)_{i\\in I} \\mapsto (s_i |_{U_i \\times_U U_j})$ and $(s_i)_{i\\in I} \\mapsto (s_j |_{U_i \\times_U U_j})$, is an equalizer diagram in the category of sets (resp.\\ abelian groups)."}
+{"_id": "6715", "title": "etale-cohomology-definition-category-sheaves", "text": "We denote $\\Sh(\\mathcal{C})$ (resp.\\ $\\textit{Ab}(\\mathcal{C})$) the full subcategory of $\\textit{PSh}(\\mathcal{C})$ (resp.\\ $\\textit{PAb}(\\mathcal{C})$) whose objects are sheaves. This is the {\\it category of sheaves of sets} (resp.\\ {\\it abelian sheaves}) on $\\mathcal{C}$."}
+{"_id": "6716", "title": "etale-cohomology-definition-0-cech", "text": "Let $\\mathcal{F}$ be a presheaf on the site $\\mathcal{C}$ and $\\mathcal{U} = \\{U_i \\to U\\} \\in \\text{Cov} (\\mathcal{C})$. We define the {\\it zeroth {\\v C}ech cohomology group} of $\\mathcal{F}$ with respect to $\\mathcal{U}$ by $$ \\check H^0 (\\mathcal{U}, \\mathcal{F}) = \\left\\{ (s_i)_{i\\in I} \\in \\prod\\nolimits_{i\\in I }\\mathcal{F}(U_i) \\text{ such that } s_i|_{U_i \\times_U U_j} = s_j |_{U_i \\times_U U_j} \\right\\}. $$"}
+{"_id": "6717", "title": "etale-cohomology-definition-fpqc-covering", "text": "Let $T$ be a scheme. An {\\it fpqc covering} of $T$ is a family $\\{ \\varphi_i : T_i \\to T\\}_{i \\in I}$ such that \\begin{enumerate} \\item each $\\varphi_i$ is a flat morphism and $\\bigcup_{i\\in I} \\varphi_i(T_i) = T$, and \\item for each affine open $U \\subset T$ there exists a finite set $K$, a map $\\mathbf{i} : K \\to I$ and affine opens $U_{\\mathbf{i}(k)} \\subset T_{\\mathbf{i}(k)}$ such that $U = \\bigcup_{k \\in K} \\varphi_{\\mathbf{i}(k)}(U_{\\mathbf{i}(k)})$. \\end{enumerate}"}
+{"_id": "6718", "title": "etale-cohomology-definition-sheaf-property-fpqc", "text": "Let $S$ be a scheme. The category of schemes over $S$ is denoted $\\Sch/S$. Consider a functor $\\mathcal{F} : (\\Sch/S)^{opp} \\to \\textit{Sets}$, in other words a presheaf of sets. We say $\\mathcal{F}$ {\\it satisfies the sheaf property for the fpqc topology} if for every fpqc covering $\\{U_i \\to U\\}_{i \\in I}$ of schemes over $S$ the diagram (\\ref{equation-sheaf-axiom}) is an equalizer diagram."}
+{"_id": "6719", "title": "etale-cohomology-definition-descent-datum", "text": "Let $\\mathcal{U} = \\{ t_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with fixed target. A {\\it descent datum} for quasi-coherent sheaves with respect to $\\mathcal{U}$ is a collection $((\\mathcal{F}_i)_{i \\in I}, (\\varphi_{ij})_{i, j \\in I})$ where \\begin{enumerate} \\item $\\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and \\item $\\varphi_{ij} : \\text{pr}_0^* \\mathcal{F}_i \\to \\text{pr}_1^* \\mathcal{F}_j$ is an isomorphism of modules on $T_i \\times_T T_j$, \\end{enumerate} such that the {\\it cocycle condition} holds: the diagrams $$ \\xymatrix{ \\text{pr}_0^*\\mathcal{F}_i \\ar[dr]_{\\text{pr}_{02}^*\\varphi_{ik}} \\ar[rr]^{\\text{pr}_{01}^*\\varphi_{ij}} & & \\text{pr}_1^*\\mathcal{F}_j \\ar[dl]^{\\text{pr}_{12}^*\\varphi_{jk}} \\\\ & \\text{pr}_2^*\\mathcal{F}_k } $$ commute on $T_i \\times_T T_j \\times_T T_k$. This descent datum is called {\\it effective} if there exist a quasi-coherent sheaf $\\mathcal{F}$ over $T$ and $\\mathcal{O}_{T_i}$-module isomorphisms $\\varphi_i : t_i^* \\mathcal{F} \\cong \\mathcal{F}_i$ compatible with the maps $\\varphi_{ij}$, namely $$ \\varphi_{ij} = \\text{pr}_1^* (\\varphi_j) \\circ \\text{pr}_0^* (\\varphi_i)^{-1}. $$"}
+{"_id": "6720", "title": "etale-cohomology-definition-descent-datum-modules", "text": "Let $A \\to B$ be a ring map and $N$ a $B$-module. A {\\it descent datum} for $N$ with respect to $A \\to B$ is an isomorphism $\\varphi : N \\otimes_A B \\cong B \\otimes_A N$ of $B \\otimes_A B$-modules such that the diagram of $B \\otimes_A B \\otimes_A B$-modules $$ \\xymatrix{ {N \\otimes_A B \\otimes_A B} \\ar[dr]_{\\varphi_{02}} \\ar[rr]^{\\varphi_{01}} & & {B \\otimes_A N \\otimes_A B} \\ar[dl]^{\\varphi_{12}} \\\\ & {B \\otimes_A B \\otimes_A N} } $$ commutes where $\\varphi_{01} = \\varphi \\otimes \\text{id}_B$ and similarly for $\\varphi_{12}$ and $\\varphi_{02}$."}
+{"_id": "6721", "title": "etale-cohomology-definition-effective-modules", "text": "A descent datum $(N, \\varphi)$ is called {\\it effective} if there exists an $A$-module $M$ such that $(N, \\varphi) \\cong (B \\otimes_A M, \\varphi_\\text{can})$, with the obvious notion of isomorphism of descent data."}
+{"_id": "6722", "title": "etale-cohomology-definition-ringed-site", "text": "Let $\\mathcal{C}$ be a {\\it ringed site}, i.e., a site endowed with a sheaf of rings $\\mathcal{O}$. A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $\\mathcal{C}$ is called {\\it quasi-coherent} if for all $U \\in \\Ob(\\mathcal{C})$ there exists a covering $\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ such that the restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is isomorphic to the cokernel of an $\\mathcal{O}$-linear map of free $\\mathcal{O}$-modules $$ \\bigoplus\\nolimits_{k \\in K} \\mathcal{O}|_{\\mathcal{C}/U_i} \\longrightarrow \\bigoplus\\nolimits_{l \\in L} \\mathcal{O}|_{\\mathcal{C}/U_i}. $$ The direct sum over $K$ is the sheaf associated to the presheaf $V \\mapsto \\bigoplus_{k \\in K} \\mathcal{O}(V)$ and similarly for the other."}
+{"_id": "6723", "title": "etale-cohomology-definition-cech-complex", "text": "Let $\\mathcal{C}$ be a category, $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ a family of morphisms of $\\mathcal{C}$ with fixed target, and $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ an abelian presheaf. We define the {\\it {\\v C}ech complex} $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ by $$ \\prod_{i_0\\in I} \\mathcal{F}(U_{i_0}) \\to \\prod_{i_0, i_1\\in I} \\mathcal{F}(U_{i_0} \\times_U U_{i_1}) \\to \\prod_{i_0, i_1, i_2 \\in I} \\mathcal{F}(U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}) \\to \\ldots $$ where the first term is in degree 0, and the maps are the usual ones. Again, it is essential to allow the case $i_0 = i_1$ etc. The {\\it {\\v C}ech cohomology groups} are defined by $$ \\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})). $$"}
+{"_id": "6724", "title": "etale-cohomology-definition-free-abelian-presheaf", "text": "Let $\\mathcal{C}$ be a category. Given a presheaf of sets $\\mathcal{G}$, we define the {\\it free abelian presheaf on $\\mathcal{G}$}, denoted $\\mathbf{Z}_\\mathcal{G}$, by the rule $$ \\mathbf{Z}_\\mathcal{G}(U) = \\mathbf{Z}[\\mathcal{G}(U)] $$ for $U \\in \\Ob(\\mathcal{C})$ with restriction maps induced by the restriction maps of $\\mathcal{G}$. In the special case $\\mathcal{G} = h_U$ we write simply $\\mathbf{Z}_U = \\mathbf{Z}_{h_U}$."}
+{"_id": "6725", "title": "etale-cohomology-definition-tau-covering", "text": "(See Topologies, Definitions \\ref{topologies-definition-fppf-covering}, \\ref{topologies-definition-syntomic-covering}, \\ref{topologies-definition-smooth-covering}, \\ref{topologies-definition-etale-covering}, and \\ref{topologies-definition-zariski-covering}.) Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$. A family of morphisms of schemes $\\{f_i : T_i \\to T\\}_{i \\in I}$ with fixed target is called a {\\it $\\tau$-covering} if and only if each $f_i$ is flat of finite presentation, syntomic, smooth, \\'etale, resp.\\ an open immersion, and we have $\\bigcup f_i(T_i) = T$."}
+{"_id": "6726", "title": "etale-cohomology-definition-standard-tau", "text": "(See Topologies, Definitions \\ref{topologies-definition-standard-fppf}, \\ref{topologies-definition-standard-syntomic}, \\ref{topologies-definition-standard-smooth}, \\ref{topologies-definition-standard-etale}, and \\ref{topologies-definition-standard-Zariski}.) Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$. Let $T$ be an affine scheme. A {\\it standard $\\tau$-covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine, and each $f_j$ flat and of finite presentation, standard syntomic, standard smooth, \\'etale, resp.\\ the immersion of a standard principal open in $T$ and $T = \\bigcup f_j(U_j)$."}
+{"_id": "6727", "title": "etale-cohomology-definition-tau-site", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, \\linebreak[0] Zariski\\}$. \\begin{enumerate} \\item A {\\it big $\\tau$-site of $S$} is any of the sites $(\\Sch/S)_\\tau$ constructed as explained above and in more detail in Topologies, Definitions \\ref{topologies-definition-big-small-fppf}, \\ref{topologies-definition-big-small-syntomic}, \\ref{topologies-definition-big-small-smooth}, \\ref{topologies-definition-big-small-etale}, and \\ref{topologies-definition-big-small-Zariski}. \\item If $\\tau \\in \\{\\etale, Zariski\\}$, then the {\\it small $\\tau$-site of $S$} is the full subcategory $S_\\tau$ of $(\\Sch/S)_\\tau$ whose objects are schemes $T$ over $S$ whose structure morphism $T \\to S$ is \\'etale, resp.\\ an open immersion. A covering in $S_\\tau$ is a covering $\\{U_i \\to U\\}$ in $(\\Sch/S)_\\tau$ such that $U$ is an object of $S_\\tau$. \\end{enumerate}"}
+{"_id": "6728", "title": "etale-cohomology-definition-etale-topos", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item The {\\it \\'etale topos}, or the {\\it small \\'etale topos} of $S$ is the category $\\Sh(S_\\etale)$ of sheaves of sets on the small \\'etale site of $S$. \\item The {\\it Zariski topos}, or the {\\it small Zariski topos} of $S$ is the category $\\Sh(S_{Zar})$ of sheaves of sets on the small Zariski site of $S$. \\item For $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ a {\\it big $\\tau$-topos} is the category of sheaves of set on a big $\\tau$-topos of $S$. \\end{enumerate}"}
+{"_id": "6729", "title": "etale-cohomology-definition-additive-sheaf", "text": "On any of the sites $(\\Sch/S)_\\tau$ or $S_\\tau$ of Section \\ref{section-big-small}. \\begin{enumerate} \\item The sheaf $T \\mapsto \\Gamma(T, \\mathcal{O}_T)$ is denoted $\\mathcal{O}_S$, or $\\mathbf{G}_a$, or $\\mathbf{G}_{a, S}$ if we want to indicate the base scheme. \\item Similarly, the sheaf $T \\mapsto \\Gamma(T, \\mathcal{O}^*_T)$ is denoted $\\mathcal{O}_S^*$, or $\\mathbf{G}_m$, or $\\mathbf{G}_{m, S}$ if we want to indicate the base scheme. \\item The {\\it constant sheaf} $\\underline{\\mathbf{Z}/n\\mathbf{Z}}$ on any site is the sheafification of the constant presheaf $U \\mapsto \\mathbf{Z}/n\\mathbf{Z}$. \\end{enumerate}"}
+{"_id": "6730", "title": "etale-cohomology-definition-structure-sheaf", "text": "Let $S$ be a scheme. The {\\it structure sheaf} of $S$ is the sheaf of rings $\\mathcal{O}_S$ on any of the sites $S_{Zar}$, $S_\\etale$, or $(\\Sch/S)_\\tau$ discussed above."}
+{"_id": "6731", "title": "etale-cohomology-definition-etale-morphism", "text": "A morphism of schemes is {\\it \\'etale} if it is smooth of relative dimension 0."}
+{"_id": "6732", "title": "etale-cohomology-definition-standard-etale", "text": "A ring map $A \\to B$ is called {\\it standard \\'etale} if $B \\cong \\left(A[t]/(f)\\right)_g$ with $f, g \\in A[t]$, with $f$ monic, and $\\text{d}f/\\text{d}t$ invertible in $B$."}
+{"_id": "6733", "title": "etale-cohomology-definition-etale-covering", "text": "An {\\it \\'etale covering} of a scheme $U$ is a family of morphisms of schemes $\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ such that \\begin{enumerate} \\item each $\\varphi_i$ is an \\'etale morphism, \\item the $U_i$ cover $U$, i.e., $U = \\bigcup_{i\\in I}\\varphi_i(U_i)$. \\end{enumerate}"}
+{"_id": "6734", "title": "etale-cohomology-definition-big-etale-site", "text": "(For more details see Section \\ref{section-big-small}, or Topologies, Section \\ref{topologies-section-etale}.) Let $S$ be a scheme. The {\\it big \\'etale site over $S$} is the site $(\\Sch/S)_\\etale$, see Definition \\ref{definition-tau-site}. The {\\it small \\'etale site over $S$} is the site $S_\\etale$, see Definition \\ref{definition-tau-site}. We define similarly the {\\it big} and {\\it small Zariski sites} on $S$, denoted $(\\Sch/S)_{Zar}$ and $S_{Zar}$."}
+{"_id": "6735", "title": "etale-cohomology-definition-geometric-point", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item A {\\it geometric point} of $S$ is a morphism $\\Spec(k) \\to S$ where $k$ is algebraically closed. Such a point is usually denoted $\\overline{s}$, i.e., by an overlined small case letter. We often use $\\overline{s}$ to denote the scheme $\\Spec(k)$ as well as the morphism, and we use $\\kappa(\\overline{s})$ to denote $k$. \\item We say $\\overline{s}$ {\\it lies over} $s$ to indicate that $s \\in S$ is the image of $\\overline{s}$. \\item An {\\it \\'etale neighborhood} of a geometric point $\\overline{s}$ of $S$ is a commutative diagram $$ \\xymatrix{ & U \\ar[d]^\\varphi \\\\ {\\overline{s}} \\ar[r]^{\\overline{s}} \\ar[ur]^{\\bar u} & S } $$ where $\\varphi$ is an \\'etale morphism of schemes. We write $(U, \\overline{u}) \\to (S, \\overline{s})$. \\item A {\\it morphism of \\'etale neighborhoods} $(U, \\overline{u}) \\to (U', \\overline{u}')$ is an $S$-morphism $h: U \\to U'$ such that $\\overline{u}' = h \\circ \\overline{u}$. \\end{enumerate}"}
+{"_id": "6736", "title": "etale-cohomology-definition-stalk", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be a presheaf on $S_\\etale$. Let $\\overline{s}$ be a geometric point of $S$. The {\\it stalk} of $\\mathcal{F}$ at $\\overline{s}$ is $$ \\mathcal{F}_{\\overline{s}} = \\colim_{(U, \\overline{u})} \\mathcal{F}(U) $$ where $(U, \\overline{u})$ runs over all \\'etale neighborhoods of $\\overline{s}$ in $S$."}
+{"_id": "6737", "title": "etale-cohomology-definition-support", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$. \\begin{enumerate} \\item The {\\it support of $\\mathcal{F}$} is the set of points $s \\in S$ such that $\\mathcal{F}_{\\overline{s}} \\not = 0$ for any (some) geometric point $\\overline{s}$ lying over $s$. \\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section. The {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where $W \\subset U$ is the largest open subset of $U$ on which $\\sigma$ restricts to zero (see Lemma \\ref{lemma-zero-over-image}). \\end{enumerate}"}
+{"_id": "6738", "title": "etale-cohomology-definition-henselian", "text": "(See Algebra, Definition \\ref{algebra-definition-henselian}.) A local ring $(R, \\mathfrak m, \\kappa)$ is called {\\it henselian} if for all $f \\in R[T]$ monic, for all $a_0 \\in \\kappa$ such that $\\bar f(a_0) = 0$ and $\\bar f'(a_0) \\neq 0$, there exists an $a \\in R$ such that $f(a) = 0$ and $a \\bmod \\mathfrak m = a_0$."}
+{"_id": "6739", "title": "etale-cohomology-definition-strictly-henselian", "text": "A local ring $R$ is called {\\it strictly henselian} if it is henselian and its residue field is separably closed."}
+{"_id": "6740", "title": "etale-cohomology-definition-etale-local-rings", "text": "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$ lying over the point $s \\in S$. \\begin{enumerate} \\item The {\\it \\'etale local ring of $S$ at $\\overline{s}$} is the stalk of the structure sheaf $\\mathcal{O}_S$ on $S_\\etale$ at $\\overline{s}$. We sometimes call this the {\\it strict henselization of $\\mathcal{O}_{S, s}$} relative to the geometric point $\\overline{s}$. Notation used: $\\mathcal{O}_{S, \\overline{s}} = \\mathcal{O}_{S, s}^{sh}$. \\item The {\\it henselization of $\\mathcal{O}_{S, s}$} is the henselization of the local ring of $S$ at $s$. See Algebra, Definition \\ref{algebra-definition-henselization}, and Theorem \\ref{theorem-henselization}. Notation: $\\mathcal{O}_{S, s}^h$. \\item The {\\it strict henselization of $S$ at $\\overline{s}$} is the scheme $\\Spec(\\mathcal{O}_{S, s}^{sh})$. \\item The {\\it henselization of $S$ at $s$} is the scheme $\\Spec(\\mathcal{O}_{S, s}^h)$. \\end{enumerate}"}
+{"_id": "6741", "title": "etale-cohomology-definition-direct-image-presheaf", "text": "Let $f: X\\to Y$ be a morphism of schemes. Let $\\mathcal{F} $ a presheaf of sets on $X_\\etale$. The {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$ (under $f$) is $$ f_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad (V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X). $$ We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$)."}
+{"_id": "6742", "title": "etale-cohomology-definition-direct-image-sheaf", "text": "Let $f: X\\to Y$ be a morphism of schemes. Let $\\mathcal{F} $ a sheaf of sets on $X_\\etale$. The {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$ (under $f$) is $$ f_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad (V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X) $$ which is a sheaf by Remark \\ref{remark-direct-image-sheaf}. We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$)."}
+{"_id": "6743", "title": "etale-cohomology-definition-higher-direct-images", "text": "Let $f: X \\to Y$ be a morphism of schemes. The right derived functors $\\{R^pf_*\\}_{p \\geq 1}$ of $f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ are called {\\it higher direct images}."}
+{"_id": "6744", "title": "etale-cohomology-definition-inverse-image", "text": "Let $f: X\\to Y$ be a morphism of schemes. The {\\it inverse image}, or {\\it pullback}\\footnote{We use the notation $f^{-1}$ for pullbacks of sheaves of sets or sheaves of abelian groups, and we reserve $f^*$ for pullbacks of sheaves of modules via a morphism of ringed sites/topoi.} functors are the functors $$ f^{-1} = f_{small}^{-1} : \\Sh(Y_\\etale) \\longrightarrow \\Sh(X_\\etale) $$ and $$ f^{-1} = f_{small}^{-1} : \\textit{Ab}(Y_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale) $$ which are left adjoint to $f_* = f_{small, *}$. Thus $f^{-1}$ thus characterized by the fact that $$ \\Hom_{{\\Sh(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Hom_{\\Sh(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F}) $$ functorially, for any $\\mathcal{F} \\in \\Sh(X_\\etale)$ and $\\mathcal{G} \\in \\Sh(Y_\\etale)$. We similarly have $$ \\Hom_{{\\textit{Ab}(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Hom_{\\textit{Ab}(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F}) $$ for $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ and $\\mathcal{G} \\in \\textit{Ab}(Y_\\etale)$."}
+{"_id": "6745", "title": "etale-cohomology-definition-inverse-system-sheaves", "text": "Let $I$ be a preordered set. Let $(X_i, f_{i'i})$ be an inverse system of schemes over $I$. A {\\it system $(\\mathcal{F}_i, \\varphi_{i'i})$ of sheaves on $(X_i, f_{i'i})$} is given by \\begin{enumerate} \\item a sheaf $\\mathcal{F}_i$ on $(X_i)_\\etale$ for all $i \\in I$, \\item for $i' \\geq i$ a map $\\varphi_{i'i} : f_{i'i}^{-1}\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ of sheaves on $(X_{i'})_\\etale$ \\end{enumerate} such that $\\varphi_{i''i} = \\varphi_{i''i'} \\circ f_{i'' i'}^{-1}\\varphi_{i'i}$ whenever $i'' \\geq i' \\geq i$."}
+{"_id": "6746", "title": "etale-cohomology-definition-algebraic-geometric-point", "text": "Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point lying over the point $s$ of $S$. Let $\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$ denote the separable algebraic closure of $\\kappa(s)$ in the algebraically closed field $\\kappa(\\overline{s})$. \\begin{enumerate} \\item In this situation the {\\it absolute Galois group} of $\\kappa(s)$ is $\\text{Gal}(\\kappa(s)^{sep}/\\kappa(s))$. It is sometimes denoted $\\text{Gal}_{\\kappa(s)}$. \\item The geometric point $\\overline{s}$ is called {\\it algebraic} if $\\kappa(s) \\subset \\kappa(\\overline{s})$ is an algebraic closure of $\\kappa(s)$. \\end{enumerate}"}
+{"_id": "6747", "title": "etale-cohomology-definition-G-module-continuous", "text": "Let $G$ be a topological group. \\begin{enumerate} \\item A {\\it $G$-module}, sometimes called a {\\it discrete $G$-module}, is an abelian group $M$ endowed with a left action $a : G \\times M \\to M$ by group homomorphisms such that $a$ is continuous when $M$ is given the discrete topology. \\item A {\\it morphism of $G$-modules} $f : M \\to N$ is a $G$-equivariant homomorphism from $M$ to $N$. \\item The category of $G$-modules is denoted $\\text{Mod}_G$. \\end{enumerate} Let $R$ be a ring. \\begin{enumerate} \\item An {\\it $R\\text{-}G$-module} is an $R$-module $M$ endowed with a left action $a : G \\times M \\to M$ by $R$-linear maps such that $a$ is continuous when $M$ is given the discrete topology. \\item A {\\it morphism of $R\\text{-}G$-modules} $f : M \\to N$ is a $G$-equivariant $R$-module map from $M$ to $N$. \\item The category of $R\\text{-}G$-modules is denoted $\\text{Mod}_{R, G}$. \\end{enumerate}"}
+{"_id": "6748", "title": "etale-cohomology-definition-galois-cohomology", "text": "Let $G$ be a topological group. Let $M$ be a discrete $G$-module with continuous $G$-action. In other words, $M$ is an object of the category $\\text{Mod}_G$ introduced in Definition \\ref{definition-G-module-continuous}. \\begin{enumerate} \\item The right derived functors $H^i(G, M)$ of $H^0(G, M)$ on the category $\\text{Mod}_G$ are called the {\\it continuous group cohomology groups} of $M$. \\item If $G$ is an abstract group endowed with the discrete topology then the $H^i(G, M)$ are called the {\\it group cohomology groups} of $M$. \\item If $G$ is a Galois group, then the groups $H^i(G, M)$ are called the {\\it Galois cohomology groups} of $M$. \\item If $G$ is the absolute Galois group of a field $K$, then the groups $H^i(G, M)$ are sometimes called the {\\it Galois cohomology groups of $K$ with coefficients in $M$}. In this case we sometimes write $H^i(K, M)$ instead of $H^i(G, M)$. \\end{enumerate}"}
+{"_id": "6749", "title": "etale-cohomology-definition-brauer-equivalent", "text": "Two finite central simple algebras $A_1$ and $A_2$ over $K$ are called {\\it similar}, or {\\it equivalent} if there exist $m, n \\geq 1$ such that $\\text{Mat}(n \\times n, A_1) \\cong \\text{Mat}(m \\times m, A_2)$. We write $A_1 \\sim A_2$."}
+{"_id": "6750", "title": "etale-cohomology-definition-brauer-group", "text": "Let $K$ be a field. The {\\it Brauer group} of $K$ is the set $\\text{Br} (K)$ of similarity classes of finite central simple algebras over $K$, endowed with the group law induced by tensor product (over $K$). The class of $A$ in $\\text{Br}(K)$ is denoted by $[A]$. The neutral element is $[K] = [\\text{Mat}(d \\times d, K)]$ for any $d \\geq 1$."}
+{"_id": "6751", "title": "etale-cohomology-definition-finite-locally-constant", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. \\begin{enumerate} \\item Let $E$ be a set. We say $\\mathcal{F}$ is the {\\it constant sheaf with value $E$} if $\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto E$. Notation: $\\underline{E}_X$ or $\\underline{E}$. \\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic to a sheaf as in (1). \\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf. \\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it is locally constant and the values are finite sets. \\end{enumerate} Let $\\mathcal{F}$ be a sheaf of abelian groups on $X_\\etale$. \\begin{enumerate} \\item Let $A$ be an abelian group. We say $\\mathcal{F}$ is the {\\it constant sheaf with value $A$} if $\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto A$. Notation: $\\underline{A}_X$ or $\\underline{A}$. \\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic as an abelian sheaf to a sheaf as in (1). \\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf. \\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it is locally constant and the values are finite abelian groups. \\end{enumerate} Let $\\Lambda$ be a ring. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\etale$. \\begin{enumerate} \\item Let $M$ be a $\\Lambda$-module. We say $\\mathcal{F}$ is the {\\it constant sheaf with value $M$} if $\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto M$. Notation: $\\underline{M}_X$ or $\\underline{M}$. \\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic as a sheaf of $\\Lambda$-modules to a sheaf as in (1). \\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a covering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf. \\end{enumerate}"}
+{"_id": "6752", "title": "etale-cohomology-definition-trace-map", "text": "Let $f : Y \\to X$ be a finite \\'etale morphism of schemes. The map $f_* f^{-1} \\to \\text{id}$ described above and explicitly below is called the {\\it trace}."}
+{"_id": "6753", "title": "etale-cohomology-definition-Cr", "text": "A field $K$ is called {\\it $C_r$} if for every $0 < d^r < n$ and every $f \\in K[T_1, \\ldots, T_n]$ homogeneous of degree $d$, there exist $\\alpha = (\\alpha_1, \\ldots, \\alpha_n)$, $\\alpha_i \\in K$ not all zero, such that $f(\\alpha) = 0$. Such an $\\alpha$ is called a {\\it nontrivial solution} of $f$."}
+{"_id": "6754", "title": "etale-cohomology-definition-variety", "text": "Let $k$ be a field. A {\\it variety} is separated, integral scheme of finite type over $k$. A {\\it curve} is a variety of dimension $1$."}
+{"_id": "6755", "title": "etale-cohomology-definition-extension-zero", "text": "Let $j : U \\to X$ be an \\'etale morphism of schemes. \\begin{enumerate} \\item The restriction functor $j^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)$ has a left adjoint $j_!^{Sh} : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$. \\item The restriction functor $j^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)$ has a left adjoint which is denoted $j_! : \\textit{Ab}(U_\\etale) \\to \\textit{Ab}(X_\\etale)$ and called {\\it extension by zero}. \\item Let $\\Lambda$ be a ring. The restriction functor $j^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to \\textit{Mod}(U_\\etale, \\Lambda)$ has a left adjoint which is denoted $j_! : \\textit{Mod}(U_\\etale, \\Lambda) \\to \\textit{Mod}(X_\\etale, \\Lambda)$ and called {\\it extension by zero}. \\end{enumerate}"}
+{"_id": "6756", "title": "etale-cohomology-definition-constructible", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item A sheaf of sets on $X_\\etale$ is {\\it constructible} if for every affine open $U \\subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$ such that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$. \\item A sheaf of abelian groups on $X_\\etale$ is {\\it constructible} if for every affine open $U \\subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$ such that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$. \\item Let $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules on $X_\\etale$ is {\\it constructible} if for every affine open $U \\subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$ such that $\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$. \\end{enumerate}"}
+{"_id": "6757", "title": "etale-cohomology-definition-c", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. We denote {\\it $D_c(X_\\etale, \\Lambda)$} the full subcategory of $D(X_\\etale, \\Lambda)$ of complexes whose cohomology sheaves are constructible sheaves of $\\Lambda$-modules."}
+{"_id": "6758", "title": "etale-cohomology-definition-ctf", "text": "Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. We denote {\\it $D_{ctf}(X_\\etale, \\Lambda)$} the full subcategory of $D_c(X_\\etale, \\Lambda)$ consisting of objects having locally finite tor dimension."}
+{"_id": "6759", "title": "etale-cohomology-definition-cd", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. The {\\it cohomological dimension of $X$} is the smallest element $$ \\text{cd}(X) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\} $$ such that for any abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$ we have $H^i_\\etale(X, \\mathcal{F}) = 0$ for $i > \\text{cd}(X)$. If $X = \\Spec(A)$ we sometimes call this the cohomological dimension of $A$."}
+{"_id": "6760", "title": "etale-cohomology-definition-cd-f", "text": "Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. The {\\it cohomological dimension of $f$} is the smallest element $$ \\text{cd}(f) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\} $$ such that for any abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$ we have $R^if_*\\mathcal{F} = 0$ for $i > \\text{cd}(f)$."}
+{"_id": "6874", "title": "equiv-definition-Serre-functor", "text": "Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear triangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$ for all $X, Y \\in \\Ob(\\mathcal{T})$. We say {\\it a Serre functor exists} if the equivalent conditions of Lemma \\ref{lemma-Serre-functor-exists} are satisfied. In this case a {\\it Serre functor} is a $k$-linear equivalence $S : \\mathcal{T} \\to \\mathcal{T}$ endowed with $k$-linear isomorphisms $c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$ functorial in $X, Y \\in \\Ob(\\mathcal{T})$."}
+{"_id": "6875", "title": "equiv-definition-fourier-mukai-functor", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \\in D(\\mathcal{O}_{X \\times_S Y})$. The exact functor $$ \\Phi_K : D(\\mathcal{O}_X) \\longrightarrow D(\\mathcal{O}_Y),\\quad M \\longmapsto R\\text{pr}_{2, *}( L\\text{pr}_1^*M \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K) $$ of triangulated categories is called a {\\it Fourier-Mukai functor} and $K$ is called a {\\it Fourier-Mukai kernel} for this functor. Moreover, \\begin{enumerate} \\item if $\\Phi_K$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$ then the resulting exact functor $\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is called a Fourier-Mukai functor, \\item if $\\Phi_K$ sends $D_{perf}(\\mathcal{O}_X)$ into $D_{perf}(\\mathcal{O}_Y)$ then the resulting exact functor $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ is called a Fourier-Mukai functor, and \\item if $X$ and $Y$ are Noetherian and $\\Phi_K$ sends $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ then the resulting exact functor $\\Phi_K : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ is called a Fourier-Mukai functor. Similarly for $D_{\\textit{Coh}}$, $D^+_{\\textit{Coh}}$, $D^-_{\\textit{Coh}}$. \\end{enumerate}"}
+{"_id": "6876", "title": "equiv-definition-siblings", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a triangulated category. We say two exact functors of triangulated categories $$ F, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D} $$ are {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$, if the following two conditions are satisfied \\begin{enumerate} \\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic where $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and \\item $F(K) \\cong F'(K)$ for any $K$ in $D^b(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "6877", "title": "equiv-definition-siblings-geometric", "text": "Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Recall that $D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))$ by Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}. We say two $k$-linear exact functors $$ F, F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X)) \\longrightarrow D^b_{\\textit{Coh}}(\\mathcal{O}_Y) $$ are {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$ if $F$ and $F'$ are siblings in the sense of Definition \\ref{definition-siblings} with abelian category being $\\textit{Coh}(\\mathcal{O}_X)$. If $X$ is regular then $D_{perf}(\\mathcal{O}_X) = D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-noetherian} and we use the same terminology for $k$-linear exact functors $F, F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$."}
+{"_id": "6878", "title": "equiv-definition-relative-equivalence-kernel", "text": "Let $S$ be a scheme. Let $X \\to S$ and $Y \\to S$ be smooth proper morphisms. An object $K \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$ is said to be {\\it the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$} if there exist an object $K' \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$ such that $$ \\Delta_{X/S, *}\\mathcal{O}_X \\cong R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K \\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S X}}^\\mathbf{L} L\\text{pr}_{23}^*K') $$ in $D(\\mathcal{O}_{X \\times_S X})$ and $$ \\Delta_{Y/S, *}\\mathcal{O}_Y \\cong R\\text{pr}_{13, *}(L\\text{pr}_{12}^*K' \\otimes_{\\mathcal{O}_{Y \\times_S X \\times_S Y}}^\\mathbf{L} L\\text{pr}_{23}^*K) $$ in $D(\\mathcal{O}_{Y \\times_S Y})$. In other words, the isomorphism class of $K$ defines an invertible arrow in the category defined in Section \\ref{section-category-Fourier-Mukai-kernels}."}
+{"_id": "6879", "title": "equiv-definition-derived-equivalent", "text": "Let $k$ be a field. Let $X$ and $Y$ be smooth projective schemes over $k$. We say $X$ and $Y$ are {\\it derived equivalent} if there exists a $k$-linear exact equivalence $D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$."}
+{"_id": "6928", "title": "stacks-more-morphisms-definition-thickening", "text": "Thickenings. \\begin{enumerate} \\item We say an algebraic stack $\\mathcal{X}'$ is a {\\it thickening} of an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack of $\\mathcal{X}'$ and the associated topological spaces are equal. \\item Given two thickenings $\\mathcal{X} \\subset \\mathcal{X}'$ and $\\mathcal{Y} \\subset \\mathcal{Y}'$ a {\\it morphism of thickenings} is a morphism $f' : \\mathcal{X}' \\to \\mathcal{Y}'$ of algebraic stacks such that $f'|_\\mathcal{X}$ factors through the closed substack $\\mathcal{Y}$. In this situation we set $f = f'|_\\mathcal{X} : \\mathcal{X} \\to \\mathcal{Y}$ and we say that $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$ is a morphism of thickenings. \\item Let $\\mathcal{Z}$ be an algebraic stack. We similarly define {\\it thickenings over $\\mathcal{Z}$} and {\\it morphisms of thickenings over $\\mathcal{Z}$}. This means that the algebraic stacks $\\mathcal{X}'$ and $\\mathcal{Y}'$ are algebraic stack endowed with a structure morphism to $\\mathcal{Z}$ and that $f'$ fits into a suitable $2$-commutative diagram of algebraic stacks. \\end{enumerate}"}
+{"_id": "6929", "title": "stacks-more-morphisms-definition-first-order-thickening", "text": "We say an algebraic stack $\\mathcal{X}'$ is a {\\it first order thickening} of an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack of $\\mathcal{X}'$ and $\\mathcal{X} \\to \\mathcal{X}'$ is a first order thickening in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}."}
+{"_id": "6930", "title": "stacks-more-morphisms-definition-formally-smooth", "text": "A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is said to be {\\it formally smooth} if it is formally smooth on objects as a $1$-morphism in categories fibred in groupoids as explained in Criteria for Representability, Section \\ref{criteria-section-formally-smooth}."}
+{"_id": "6931", "title": "stacks-more-morphisms-definition-categorical-quotient", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $f : \\mathcal{X} \\to Y$ be a morphism to an algebraic space $Y$. \\begin{enumerate} \\item We say $f$ is a {\\it categorical moduli space} if any morphism $\\mathcal{X} \\to W$ to an algebraic space $W$ factors uniquely through $f$. \\item We say $f$ is a {\\it uniform categorical moduli space} if for any flat morphism $Y' \\to Y$ of algebraic spaces the base change $f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space. \\end{enumerate} Let $\\mathcal{C}$ be a full subcategory of the category of algebraic spaces. \\begin{enumerate} \\item[(3)] We say $f$ is a {\\it categorical moduli space in $\\mathcal{C}$} if $Y \\in \\Ob(\\mathcal{C})$ and any morphism $\\mathcal{X} \\to W$ with $W \\in \\Ob(\\mathcal{C})$ factors uniquely through $f$. \\item[(4)] We say is a {\\it uniform categorical moduli space in $\\mathcal{C}$} if $Y \\in \\Ob(\\mathcal{C})$ and for every flat morphism $Y' \\to Y$ in $\\mathcal{C}$ the base change $f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "6932", "title": "stacks-more-morphisms-definition-well-nigh-affine", "text": "Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is {\\it well-nigh affine} if there exists an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \\to \\mathcal{X}$."}
+{"_id": "7115", "title": "perfect-definition-supported-on", "text": "Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $T \\subset X$ be a closed subset. We say $E$ is {\\it supported on $T$} if the cohomology sheaves $H^i(E)$ are supported on $T$."}
+{"_id": "7116", "title": "perfect-definition-approximation-holds", "text": "Let $X$ be a scheme. Consider triples $(T, E, m)$ where \\begin{enumerate} \\item $T \\subset X$ is a closed subset, \\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and \\item $m \\in \\mathbf{Z}$. \\end{enumerate} We say {\\it approximation holds for the triple} $(T, E, m)$ if there exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$ and a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$ for $i > m$ and a surjection $H^m(P) \\to H^m(E)$."}
+{"_id": "7117", "title": "perfect-definition-approximation", "text": "Let $X$ be a scheme. We say {\\it approximation by perfect complexes holds} on $X$ if for any closed subset $T \\subset X$ with $X \\setminus T$ retro-compact in $X$ there exists an integer $r$ such that for every triple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds} with \\begin{enumerate} \\item $E$ is $(m - r)$-pseudo-coherent, and \\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$ \\end{enumerate} approximation holds."}
+{"_id": "7118", "title": "perfect-definition-tor-independent", "text": "Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. We say $X$ and $Y$ are {\\it Tor independent over $S$} if for every $x \\in X$ and $y \\in Y$ mapping to the same point $s \\in S$ the rings $\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are Tor independent over $\\mathcal{O}_{S, s}$ (see More on Algebra, Definition \\ref{more-algebra-definition-tor-independent})."}
+{"_id": "7119", "title": "perfect-definition-relatively-perfect", "text": "Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. An object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect relative to $S$} or {\\it $S$-perfect} if $E$ is pseudo-coherent (Cohomology, Definition \\ref{cohomology-definition-pseudo-coherent}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$ (Cohomology, Definition \\ref{cohomology-definition-tor-amplitude})."}
+{"_id": "7120", "title": "perfect-definition-resolution-property", "text": "Let $X$ be a scheme. We say $X$ has the {\\it resolution property} if every quasi-coherent $\\mathcal{O}_X$-module of finite type is the quotient of a finite locally free $\\mathcal{O}_X$-module."}
+{"_id": "7121", "title": "perfect-definition-K-group", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item We denote $K_0(X)$ the {\\it Grothendieck group of $X$}. It is the zeroth K-group of the strictly full, saturated, triangulated subcategory $D_{perf}(\\mathcal{O}_X)$ of $D(\\mathcal{O}_X)$ consisting of perfect objects. In a formula $$ K_0(X) = K_0(D_{perf}(\\mathcal{O}_X)) $$ \\item If $X$ is locally Noetherian, then we denote $K'_0(X)$ the {\\it Grothendieck group of coherent sheaves on $X$}. It is the is the zeroth $K$-group of the abelian category of coherent $\\mathcal{O}_X$-modules. In a formula $$ K'_0(X) = K_0(\\textit{Coh}(\\mathcal{O}_X)) $$ \\end{enumerate}"}
+{"_id": "7201", "title": "spaces-flat-definition-impurity", "text": "In Situation \\ref{situation-pre-pure} we say a diagram (\\ref{equation-impurity}) defines an {\\it impurity of $\\mathcal{F}$ above $y$} if $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and $t \\not \\in f_T(\\overline{\\{\\xi\\}})$. We will indicate this by saying ``let $(g : T \\to Y, t' \\leadsto t, \\xi)$ be an impurity of $\\mathcal{F}$ above $y$''."}
+{"_id": "7202", "title": "spaces-flat-definition-pure", "text": "In Situation \\ref{situation-pre-pure}. \\begin{enumerate} \\item We say $\\mathcal{F}$ is {\\it pure above $y$} if {\\bf none} of the equivalent conditions of Lemma \\ref{lemma-pure-along-X-y} hold. \\item We say $\\mathcal{F}$ is {\\it universally pure above $y$} if there does not exist any impurity of $\\mathcal{F}$ above $y$. \\item We say that $X$ is {\\it pure above $y$} if $\\mathcal{O}_X$ is pure above $y$. \\item We say $\\mathcal{F}$ is {\\it universally $Y$-pure}, or {\\it universally pure relative to $Y$} if $\\mathcal{F}$ is universally pure above $y$ for every $y \\in |Y|$. \\item We say $\\mathcal{F}$ is {\\it $Y$-pure}, or {\\it pure relative to $Y$} if $\\mathcal{F}$ is pure above $y$ for every $y \\in |Y|$. \\item We say that $X$ is {\\it $Y$-pure} or {\\it pure relative to $Y$} if $\\mathcal{O}_X$ is pure relative to $Y$. \\end{enumerate}"}
+{"_id": "7203", "title": "spaces-flat-definition-flattening", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say that the {\\it universal flattening of $\\mathcal{F}$ exists} if the functor $F_{flat}$ defined in Situation \\ref{situation-flat} is an algebraic space. We say that the {\\it universal flattening of $X$ exists} if the universal flattening of $\\mathcal{O}_X$ exists."}
+{"_id": "7204", "title": "spaces-flat-definition-flat-dimension-n", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $n \\geq 0$. We say {\\it $\\mathcal{F}$ is flat over $Y$ in dimensions $\\geq n$} if the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n} are satisfied."}
+{"_id": "7281", "title": "spaces-chow-definition-delta-dimension", "text": "In Situation \\ref{situation-setup} for any good $X/B$ and any irreducible closed subset $T \\subset |X|$ we define $$ \\dim_\\delta(T) = \\delta(\\xi) $$ where $\\xi \\in T$ is the generic point of $T$. We will call this the {\\it $\\delta$-dimension of $T$}. If $T \\subset |X|$ is any closed subset, then we define $\\dim_\\delta(T)$ as the supremum of the $\\delta$-dimensions of the irreducible components of $T$. If $Z$ is a closed subspace of $X$, then we set $\\dim_\\delta(Z) = \\dim_\\delta(|Z|)$."}
+{"_id": "7282", "title": "spaces-chow-definition-cycles", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $k \\in \\mathbf{Z}$. \\begin{enumerate} \\item A {\\it cycle on $X$} is a formal sum $$ \\alpha = \\sum n_Z [Z] $$ where the sum is over integral closed subspaces $Z \\subset X$, each $n_Z \\in \\mathbf{Z}$, and $\\{|Z|; n_Z \\not = 0\\}$ is a locally finite collection of subsets of $|X|$ (Topology, Definition \\ref{topology-definition-locally-finite}). \\item A {\\it $k$-cycle} on $X$ is a cycle $$ \\alpha = \\sum n_Z [Z] $$ where $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$. \\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$. \\end{enumerate}"}
+{"_id": "7283", "title": "spaces-chow-definition-length-at-x", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. We say {\\it $\\mathcal{F}$ has length $d$ at $x$} if the equivalent conditions of Lemma \\ref{lemma-length} are satisfied."}
+{"_id": "7284", "title": "spaces-chow-definition-cycle-associated-to-closed-subscheme", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $Y \\subset X$ be a closed subspace. \\begin{enumerate} \\item For an irreducible component $Z \\subset Y$ with generic point $\\xi$ the length of $\\mathcal{O}_Y$ at $\\xi$ (Definition \\ref{definition-length-at-x}) is called the {\\it multiplicity of $Z$ in $Y$}. By Lemma \\ref{lemma-length-finite} applied to $\\mathcal{O}_Y$ on $Y$ this is a positive integer. \\item Assume $\\dim_\\delta(Y) \\leq k$. The {\\it $k$-cycle associated to $Y$} is $$ [Y]_k = \\sum m_{Z, Y}[Z] $$ where the sum is over the irreducible components $Z$ of $Y$ of $\\delta$-dimension $k$ and $m_{Z, Y}$ is the multiplicity of $Z$ in $Y$. This is a $k$-cycle by Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-components-locally-finite}. \\end{enumerate}"}
+{"_id": "7285", "title": "spaces-chow-definition-cycle-associated-to-coherent-sheaf", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For an integral closed subspace $Z \\subset X$ with generic point $\\xi$ such that $|Z|$ is an irreducible component of $\\text{Supp}(\\mathcal{F})$ the length of $\\mathcal{F}$ at $\\xi$ (Definition \\ref{definition-length-at-x}) is called the {\\it multiplicity of $Z$ in $\\mathcal{F}$}. By Lemma \\ref{lemma-length-finite} this is a positive integer. \\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. The {\\it $k$-cycle associated to $\\mathcal{F}$} is $$ [\\mathcal{F}]_k = \\sum m_{Z, \\mathcal{F}}[Z] $$ where the sum is over the integral closed subspaces $Z \\subset X$ corresponding to irreducible components of $\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$ and $m_{Z, \\mathcal{F}}$ is the multiplicity of $Z$ in $\\mathcal{F}$. This is a $k$-cycle by Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-components-locally-finite}. \\end{enumerate}"}
+{"_id": "7286", "title": "spaces-chow-definition-proper-pushforward", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a morphism over $B$. Assume $f$ is proper. \\begin{enumerate} \\item Let $Z \\subset X$ be an integral closed subspace with $\\dim_\\delta(Z) = k$. Let $Z' \\subset Y$ be the image of $Z$ as in Lemma \\ref{lemma-proper-image}. We define $$ f_*[Z] = \\left\\{ \\begin{matrix} 0 & \\text{if} & \\dim_\\delta(Z')< k, \\\\ \\deg(Z/Z') [Z'] & \\text{if} & \\dim_\\delta(Z') = k. \\end{matrix} \\right. $$ The degree of $Z$ over $Z'$ is defined and finite if $\\dim_\\delta(Z') = \\dim_\\delta(Z)$ by Lemma \\ref{lemma-equal-dimension} and Spaces over Fields, Definition \\ref{spaces-over-fields-definition-degree}. \\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The {\\it pushforward} of $\\alpha$ as the sum $$ f_* \\alpha = \\sum n_Z f_*[Z] $$ where each $f_*[Z]$ is defined as above. The sum is locally finite by Lemma \\ref{lemma-quasi-compact-locally-finite} above. \\end{enumerate}"}
+{"_id": "7287", "title": "spaces-chow-definition-flat-pullback", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a morphism over $B$. Assume $f$ is flat of relative dimension $r$. \\begin{enumerate} \\item Let $Z \\subset Y$ be an integral closed subspace of $\\delta$-dimension $k$. We define $f^*[Z]$ to be the $(k+r)$-cycle on $X$ associated to the scheme theoretic inverse image $$ f^*[Z] = [f^{-1}(Z)]_{k+r}. $$ This makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$ by Lemma \\ref{lemma-flat-inverse-image-dimension}. \\item Let $\\alpha = \\sum n_i [Z_i]$ be a $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$} is the sum $$ f^* \\alpha = \\sum n_i f^*[Z_i] $$ where each $f^*[Z_i]$ is defined as above. The sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}. \\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian groups so obtained. \\end{enumerate}"}
+{"_id": "7288", "title": "spaces-chow-definition-principal-divisor", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral with $\\dim_\\delta(X) = n$. Let $f \\in R(X)^*$. The {\\it principal divisor associated to $f$} is the $(n - 1)$-cycle $$ \\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z] $$ defined in Spaces over Fields, Definition \\ref{spaces-over-fields-definition-principal-divisor}. This makes sense because prime divisors have $\\delta$-dimension $n - 1$ by Lemma \\ref{lemma-divisor-delta-dimension}."}
+{"_id": "7289", "title": "spaces-chow-definition-rational-equivalence", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $k \\in \\mathbf{Z}$. \\begin{enumerate} \\item Given any locally finite collection $\\{W_j \\subset X\\}$ of integral closed subspaces with $\\dim_\\delta(W_j) = k + 1$, and any $f_j \\in R(W_j)^*$ we may consider $$ \\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X) $$ where $i_j : W_j \\to X$ is the inclusion morphism. This makes sense as the morphism $\\coprod i_j : \\coprod W_j \\to X$ is proper. \\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero} if $\\alpha$ is a cycle of the form displayed above. \\item We say $\\alpha, \\beta \\in Z_k(X)$ are {\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$ if $\\alpha - \\beta$ is rationally equivalent to zero. \\item We define $$ \\CH_k(X) = Z_k(X) / \\sim_{rat} $$ to be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called the {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}. \\end{enumerate}"}
+{"_id": "7290", "title": "spaces-chow-definition-divisor-invertible-sheaf", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$ we define the {\\it Weil divisor associated to $s$} is the $(n - 1)$-cycle $$ \\text{div}_\\mathcal{L}(s) = \\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] $$ defined in Spaces over Fields, Definition \\ref{spaces-over-fields-definition-divisor-invertible-sheaf}. This makes sense because Weil divisors have $\\delta$-dimension $n - 1$ by Lemma \\ref{lemma-divisor-delta-dimension}. \\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as $$ c_1(\\mathcal{L}) \\cap [X] = \\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X) $$ where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$. This is well defined by Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-divisor-meromorphic-well-defined}. \\end{enumerate}"}
+{"_id": "7291", "title": "spaces-chow-definition-cap-c1", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We define, for every integer $k$, an operation $$ c_1(\\mathcal{L}) \\cap - : Z_{k + 1}(X) \\to \\CH_k(X) $$ called {\\it intersection with the first Chern class of $\\mathcal{L}$}. \\begin{enumerate} \\item Given an integral closed subspace $i : W \\to X$ with $\\dim_\\delta(W) = k + 1$ we define $$ c_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W]) $$ where the right hand side is defined in Definition \\ref{definition-divisor-invertible-sheaf}. \\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set $$ c_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i] $$ \\end{enumerate}"}
+{"_id": "7292", "title": "spaces-chow-definition-gysin-homomorphism", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $(\\mathcal{L}, s)$ be a pair consisting of an invertible sheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$. Let $D = Z(s)$ be the vanishing locus of $s$, and denote $i : D \\to X$ the closed immersion. We define, for every integer $k$, a (refined) {\\it Gysin homomorphism} $$ i^* : Z_{k + 1}(X) \\to \\CH_k(D). $$ by the following rules: \\begin{enumerate} \\item Given a integral closed subspace $W \\subset X$ with $\\dim_\\delta(W) = k + 1$ we define \\begin{enumerate} \\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a $k$-cycle on $D$, and \\item if $W \\subset D$, then $i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$, where $i' : W \\to D$ is the induced closed immersion. \\end{enumerate} \\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$ we set $$ i^*\\alpha = \\sum n_j i^*[W_j] $$ \\item If $D$ is an effective Cartier divisor, then we denote $D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of the class to a class on $X$. \\end{enumerate}"}
+{"_id": "7293", "title": "spaces-chow-definition-bivariant-class", "text": "\\begin{reference} Similar to \\cite[Definition 17.1]{F} \\end{reference} In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of good algebraic spaces over $B$. Let $p \\in \\mathbf{Z}$. A {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule which assigns to every morphism $Y' \\to Y$ of good algebraic spaces over $B$ and every $k$ a map $$ c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X') $$ where $X' = Y' \\times_Y X$, satisfying the following conditions \\begin{enumerate} \\item if $Y'' \\to Y'$ is a proper morphism, then $c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$ for all $\\alpha''$ on $Y''$, \\item if $Y'' \\to Y'$ a morphism of good algebraic spaces over $B$ which is flat of relative dimension $r$, then $c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$ for all $\\alpha'$ on $Y'$, \\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in Definition \\ref{definition-gysin-homomorphism} with pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$, then we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$ for all $\\alpha'$ on $Y'$. \\end{enumerate} The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^p(X \\to Y)$."}
+{"_id": "7294", "title": "spaces-chow-definition-chow-cohomology", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. The {\\it Chow cohomology} of $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree $p$ component is $A^p(X \\to X)$."}
+{"_id": "7295", "title": "spaces-chow-definition-chern-classes", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. For $i = 0, \\ldots, r$ the {\\it $i$th Chern class of $\\mathcal{E}$} is the bivariant class $c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$ constructed in Lemma \\ref{lemma-segre-classes}. The {\\it total Chern class of $\\mathcal{E}$} is the formal sum $$ c(\\mathcal{E}) = c_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E}) $$ which is viewed as a nonhomogeneous bivariant class on $X$."}
+{"_id": "7296", "title": "spaces-chow-definition-degree-zero-cycle", "text": "Let $k$ be a field. Let $p : X \\to \\Spec(k)$ be a proper morphism of algebraic spaces. The {\\it degree of a zero cycle} on $X$ is given by proper pushforward $$ p_* : \\CH_0(X) \\longrightarrow \\CH_0(\\Spec(k)) \\longrightarrow \\mathbf{Z} $$ (Lemma \\ref{lemma-proper-pushforward-rational-equivalence}) composed with the natural isomorphism $\\CH_0(\\Spec(k)) \\to \\mathbf{Z}$ which maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$."}
+{"_id": "7369", "title": "sdga-definition-ga", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A {\\it sheaf of graded $\\mathcal{O}$-algebras} or a {\\it sheaf of graded algebras} on $(\\mathcal{C}, \\mathcal{O})$ is given by a family $\\mathcal{A}^n$ indexed by $n \\in \\mathbf{Z}$ of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad (a, b) \\longmapsto ab $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication is associative, and \\item there is a global section $1$ of $\\mathcal{A}^0$ which is a two-sided identity for multiplication. \\end{enumerate} We often denote such a structure $\\mathcal{A}$. A {\\it homomorphism of graded $\\mathcal{O}$-algebras} $f : \\mathcal{A} \\to \\mathcal{B}$ is a family of maps $f^n : \\mathcal{A}^n \\to \\mathcal{B}^n$ of $\\mathcal{O}$-modules compatible with the multiplication maps."}
+{"_id": "7370", "title": "sdga-definition-gm", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A (right) {\\it graded $\\mathcal{A}$-module} or (right) {\\it graded module} over $\\mathcal{A}$ is given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$ of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad (x, a) \\longmapsto xa $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication satisfies $(xa)a' = x(aa')$, \\item the identity section $1$ of $\\mathcal{A}^0$ acts as the identity on $\\mathcal{M}^n$ for all $n$. \\end{enumerate} We often say ``let $\\mathcal{M}$ be a graded $\\mathcal{A}$-module'' to indicate this situation. A {\\it homomorphism of graded $\\mathcal{A}$-modules} $f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps $f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$ of $\\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) graded $\\mathcal{A}$-modules is denoted $\\text{Mod}_\\mathcal{A}$."}
+{"_id": "7371", "title": "sdga-definition-bimodule", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ and $\\mathcal{B}$ be a sheaves of graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A {\\it graded $(\\mathcal{A}, \\mathcal{B})$-bimodule} is given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$ of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad (x, b) \\longmapsto xb $$ and $$ \\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad (a, x) \\longmapsto ax $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$, \\item $(ax)b = a(xb)$, \\item the identity section $1$ of $\\mathcal{A}^0$ acts as the identity by multiplication, and \\item the identity section $1$ of $\\mathcal{B}^0$ acts as the identity by multiplication. \\end{enumerate} We often denote such a structure $\\mathcal{M}$. A {\\it homomorphism of graded $(\\mathcal{A}, \\mathcal{B})$-bimodules} $f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps $f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$ of $\\mathcal{O}$-modules compatible with the multiplication maps."}
+{"_id": "7372", "title": "sdga-definition-dga", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A {\\it sheaf of differential graded $\\mathcal{O}$-algebras} or a {\\it sheaf of differential graded algebras} on $(\\mathcal{C}, \\mathcal{O})$ is a cochain complex $\\mathcal{A}^\\bullet$ of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad (a, b) \\longmapsto ab $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication is associative, \\item there is a global section $1$ of $\\mathcal{A}^0$ which is a two-sided identity for multiplication, \\item for $U \\in \\Ob(\\mathcal{C})$, $a \\in \\mathcal{A}^n(U)$, and $b \\in \\mathcal{A}^m(U)$ we have $$ \\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b) $$ \\end{enumerate} We often denote such a structure $(\\mathcal{A}, \\text{d})$. A {\\it homomorphism of differential graded $\\mathcal{O}$-algebras} from $(\\mathcal{A}, \\text{d})$ to $(\\mathcal{B}, \\text{d})$ is a map $f : \\mathcal{A}^\\bullet \\to \\mathcal{B}^\\bullet$ of complexes of $\\mathcal{O}$-modules compatible with the multiplication maps."}
+{"_id": "7373", "title": "sdga-definition-dgm", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A (right) {\\it differential graded $\\mathcal{A}$-module} or (right) {\\it differential graded module} over $\\mathcal{A}$ is a cochain complex $\\mathcal{M}^\\bullet$ endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad (x, a) \\longmapsto xa $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication satisfies $(xa)a' = x(aa')$, \\item the identity section $1$ of $\\mathcal{A}^0$ acts as the identity on $\\mathcal{M}^n$ for all $n$, \\item for $U \\in \\Ob(\\mathcal{C})$, $x \\in \\mathcal{M}^n(U)$, and $a \\in \\mathcal{A}^m(U)$ we have $$ \\text{d}^{n + m}(xa) = \\text{d}^n(x)a + (-1)^n x\\text{d}^m(a) $$ \\end{enumerate} We often say ``let $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module'' to indicate this situation. A {\\it homomorphism of differential graded $\\mathcal{A}$-modules} from $\\mathcal{M}$ to $\\mathcal{N}$ is a map $f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$ of complexes of $\\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) differential graded $\\mathcal{A}$-modules is denoted $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$."}
+{"_id": "7374", "title": "sdga-definition-dg-bimodule", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ and $\\mathcal{B}$ be a sheaves of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A {\\it differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule} is given by a complex $\\mathcal{M}^\\bullet$ of $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps $$ \\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad (x, b) \\longmapsto xb $$ and $$ \\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad (a, x) \\longmapsto ax $$ called the multiplication maps with the following properties \\begin{enumerate} \\item multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$, \\item $(ax)b = a(xb)$, \\item $\\text{d}(ax) = \\text{d}(a) x + (-1)^{\\deg(a)}a \\text{d}(x)$ and $\\text{d}(xb) = \\text{d}(x) b + (-1)^{\\deg(x)}x \\text{d}(b)$, \\item the identity section $1$ of $\\mathcal{A}^0$ acts as the identity by multiplication, and \\item the identity section $1$ of $\\mathcal{B}^0$ acts as the identity by multiplication. \\end{enumerate} We often denote such a structure $\\mathcal{M}$ and sometimes we write ${}_\\mathcal{A}\\mathcal{M}_\\mathcal{B}$. A {\\it homomorphism of differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodules} $f : \\mathcal{M} \\to \\mathcal{N}$ is a map of complexes $f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$ of $\\mathcal{O}$-modules compatible with the multiplication maps."}
+{"_id": "7375", "title": "sdga-definition-homotopy", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $f, g : \\mathcal{M} \\to \\mathcal{N}$ be homomorphisms of differential graded $\\mathcal{A}$-modules. A {\\it homotopy between $f$ and $g$} is a graded $\\mathcal{A}$-module map $h : \\mathcal{M} \\to \\mathcal{N}$ homogeneous of degree $-1$ such that $$ f - g = \\text{d}_\\mathcal{N} \\circ h + h \\circ \\text{d}_\\mathcal{M} $$ If a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}."}
+{"_id": "7376", "title": "sdga-definition-complexes-notation", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The {\\it homotopy category}, denoted $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$, is the category whose objects are the objects of $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ and whose morphisms are homotopy classes of homomorphisms of differential graded $\\mathcal{A}$-modules."}
+{"_id": "7377", "title": "sdga-definition-cone", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $f : \\mathcal{K} \\to \\mathcal{L}$ be a homomorphism of differential graded $\\mathcal{A}$-modules. The {\\it cone} of $f$ is the differential graded $\\mathcal{A}$-module $C(f)$ defined as follows: \\begin{enumerate} \\item the underlying complex of $\\mathcal{O}$-modules is the cone of the corresponding map $f : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ of complexes of $\\mathcal{A}$-modules, i.e., we have $C(f)^n = \\mathcal{L}^n \\oplus \\mathcal{K}^{n + 1}$ and differential $$ d_{C(f)} = \\left( \\begin{matrix} \\text{d}_\\mathcal{L} & f \\\\ 0 & -\\text{d}_\\mathcal{K} \\end{matrix} \\right) $$ \\item the multiplication map $$ C(f)^n \\times \\mathcal{A}^m \\to C(f)^{n + m} $$ is the direct sum of the multiplication map $\\mathcal{L}^n \\times \\mathcal{A}^m \\to \\mathcal{L}^{n + m}$ and the multiplication map $\\mathcal{K}^{n + 1} \\times \\mathcal{A}^m \\to \\mathcal{K}^{n + 1 + m}$. \\end{enumerate} It comes equipped with canonical hommorphisms of differential graded $\\mathcal{A}$-modules $i : \\mathcal{L} \\to C(f)$ and $p : C(f) \\to \\mathcal{K}[1]$ induced by the obvious maps."}
+{"_id": "7378", "title": "sdga-definition-graded-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module $\\mathcal{I}$ is said to be {\\it graded injective}\\footnote{This may be nonstandard terminology.} if $\\mathcal{M}$ viewed as a graded $\\mathcal{A}$-module is an injective object of the category $\\text{Mod}_\\mathcal{A}$ of graded $\\mathcal{A}$-modules."}
+{"_id": "7379", "title": "sdga-definition-K-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module $\\mathcal{I}$ is {\\it K-injective} if for every acyclic differential graded $\\mathcal{M}$ we have  $$ \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}, \\mathcal{I}) = 0 $$"}
+{"_id": "7380", "title": "sdga-definition-derived-category", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\text{Qis}$ be as in Lemma \\ref{lemma-qis}. The {\\it derived category of $(\\mathcal{A}, \\text{d})$} is the triangulated category $$ D(\\mathcal{A}, \\text{d}) = \\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})}) $$ discussed in more detail above."}
+{"_id": "7381", "title": "sdga-definition-pullback", "text": "Derived tensor product and derived pullback. \\begin{enumerate} \\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras. Let $\\mathcal{N}$ be a  differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. The functor $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{B}, \\text{d})$ constructed in Lemma \\ref{lemma-derived-tensor-product} is called the {\\it derived tensor product} and denoted $- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$. \\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let $\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras. The functor $D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}, \\text{d})$ constructed in Lemma \\ref{lemma-derived-tensor-product} is called {\\it derived pullback} and denote $Lf^*$. \\end{enumerate}"}
+{"_id": "7382", "title": "sdga-definition-pushforward", "text": "Derived internal hom and derived pushforward. \\begin{enumerate} \\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras. Let $\\mathcal{N}$ be a  differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. The right derived extension $$ R\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) : D(\\mathcal{B}, \\text{d}) \\longrightarrow D(\\mathcal{A}, \\text{d}) $$ of the internal hom functor $\\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, -)$ is called {\\it derived internal hom}. \\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let $\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras. The right derived extension $$ Rf_* : D(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}) $$ of the pushforward $f_*$ is called {\\it derived pushforward}. \\end{enumerate}"}
+{"_id": "7601", "title": "stacks-morphisms-definition-separated", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ is {\\it DM} if $\\Delta_f$ is unramified\\footnote{The letters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme $T$ and any morphism $T \\to \\mathcal{Y}$ the fibre product $\\mathcal{X}_T = \\mathcal{X} \\times_\\mathcal{Y} T$ is an algebraic stack over $T$ whose diagonal is unramified, i.e., $\\mathcal{X}_T$ is DM. This implies $\\mathcal{X}_T$ is a Deligne-Mumford stack, see Theorem \\ref{theorem-DM}. In other words a DM morphism is one whose ``fibres'' are Deligne-Mumford stacks. This hopefully at least motivates the terminology.}. \\item We say $f$ is {\\it quasi-DM} if $\\Delta_f$ is locally quasi-finite\\footnote{If $f$ is quasi-DM, then the ``fibres'' $\\mathcal{X}_T$ of $\\mathcal{X} \\to \\mathcal{Y}$ are quasi-DM. An algebraic stack $\\mathcal{X}$ is quasi-DM exactly if there exists a scheme $U$ and a surjective flat morphism $U \\to \\mathcal{X}$ of finite presentation which is locally quasi-finite, see Theorem \\ref{theorem-quasi-DM}. Note the similarity to being Deligne-Mumford, which is defined in terms of having an \\'etale covering by a scheme.}. \\item We say $f$ is {\\it separated} if $\\Delta_f$ is proper. \\item We say $f$ is {\\it quasi-separated} if $\\Delta_f$ is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "7602", "title": "stacks-morphisms-definition-absolute-separated", "text": "Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$. Denote $p : \\mathcal{X} \\to S$ the structure morphism. \\begin{enumerate} \\item We say $\\mathcal{X}$ is {\\it DM over $S$} if $p : \\mathcal{X} \\to S$ is DM. \\item We say $\\mathcal{X}$ is {\\it quasi-DM over $S$} if $p : \\mathcal{X} \\to S$ is quasi-DM. \\item We say $\\mathcal{X}$ is {\\it separated over $S$} if $p : \\mathcal{X} \\to S$ is separated. \\item We say $\\mathcal{X}$ is {\\it quasi-separated over $S$} if $p : \\mathcal{X} \\to S$ is quasi-separated. \\item We say $\\mathcal{X}$ is {\\it DM} if $\\mathcal{X}$ is DM\\footnote{Theorem \\ref{theorem-DM} shows that this is equivalent to $\\mathcal{X}$ being a Deligne-Mumford stack.} over $\\Spec(\\mathbf{Z})$. \\item We say $\\mathcal{X}$ is {\\it quasi-DM} if $\\mathcal{X}$ is quasi-DM over $\\Spec(\\mathbf{Z})$. \\item We say $\\mathcal{X}$ is {\\it separated} if $\\mathcal{X}$ is separated over $\\Spec(\\mathbf{Z})$. \\item We say $\\mathcal{X}$ is {\\it quasi-separated} if $\\mathcal{X}$ is quasi-separated over $\\Spec(\\mathbf{Z})$. \\end{enumerate} In the last 4 definitions we view $\\mathcal{X}$ as an algebraic stack over $\\Spec(\\mathbf{Z})$ via Algebraic Stacks, Definition \\ref{algebraic-definition-viewed-as}."}
+{"_id": "7603", "title": "stacks-morphisms-definition-isom", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space. \\begin{enumerate} \\item Let $x : Z \\to \\mathcal{X}$ be a morphism. We set $$ \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x) = Z \\times_{x, \\mathcal{X}} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} $$ We endow it with the structure of a group algebraic space over $Z$ by pulling back the composition law discussed in Remark \\ref{remark-inertia-is-group-in-spaces}. We will sometimes refer to $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x)$ as the {\\it relative sheaf of automorphisms of $x$}. \\item Let $x_1, x_2 : Z \\to \\mathcal{X}$ be morphisms. Set $y_i = f \\circ x_i$. Let $\\alpha : y_1 \\to y_2$ be a $2$-morphism. Then $\\alpha$ determines a morphism $\\Delta^\\alpha : Z \\to Z \\times_{y_1, \\mathcal{Y}, y_2} Z$ and we set $$ \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) = (Z \\times_{x_1, \\mathcal{X}, x_2} Z) \\times_{Z \\times_{y_1, \\mathcal{Y}, y_2} Z, \\Delta^\\alpha} Z. $$ We will sometimes refer to $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$ as the {\\it relative sheaf of isomorphisms from $x_1$ to $x_2$}. \\end{enumerate} If $\\mathcal{Y} = \\Spec(\\mathbf{Z})$ or more generally when $\\mathcal{Y}$ is an algebraic space, then we use the notation $\\mathit{Isom}_\\mathcal{X}(x, x)$ and $\\mathit{Isom}_\\mathcal{X}(x_1, x_2)$ and we use the terminology {\\it sheaf of automorphisms of $x$} and {\\it sheaf of isomorphisms from $x_1$ to $x_2$}."}
+{"_id": "7604", "title": "stacks-morphisms-definition-quasi-compact", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it quasi-compact} if for every quasi-compact algebraic stack $\\mathcal{Z}$ and morphism $\\mathcal{Z} \\to \\mathcal{Y}$ the fibre product $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-compact."}
+{"_id": "7605", "title": "stacks-morphisms-definition-noetherian", "text": "Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is {\\it Noetherian} if $\\mathcal{X}$ is quasi-compact, quasi-separated and locally Noetherian."}
+{"_id": "7606", "title": "stacks-morphisms-definition-affine", "text": "A morphism of algebraic stacks is said to be {\\it affine} if it is representable and affine in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}."}
+{"_id": "7607", "title": "stacks-morphisms-definition-integral", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ is {\\it integral} if $f$ is representable and integral in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}. \\item We say $f$ is {\\it finite} if $f$ is representable and finite in the sense of Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}. \\end{enumerate}"}
+{"_id": "7608", "title": "stacks-morphisms-definition-open", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ is {\\it open} if the map of topological spaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is open. \\item We say $f$ is {\\it universally open} if for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the morphism of topological spaces $$ |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}| $$ is open, i.e., the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is open. \\end{enumerate}"}
+{"_id": "7609", "title": "stacks-morphisms-definition-submersive", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ is {\\it submersive}\\footnote{This is very different from the notion of a submersion of differential manifolds.} if the continuous map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is submersive, see Topology, Definition \\ref{topology-definition-submersive}. \\item We say $f$ is {\\it universally submersive} if for every morphism of algebraic stacks $\\mathcal{Y}' \\to \\mathcal{Y}$ the base change $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ is submersive. \\end{enumerate}"}
+{"_id": "7610", "title": "stacks-morphisms-definition-closed", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ is {\\it closed} if the map of topological spaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is closed. \\item We say $f$ is {\\it universally closed} if for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the morphism of topological spaces $$ |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}| $$ is closed, i.e., the base change $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is closed. \\end{enumerate}"}
+{"_id": "7611", "title": "stacks-morphisms-definition-universally-injective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it universally injective} if for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map $$ |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}| $$ is injective."}
+{"_id": "7612", "title": "stacks-morphisms-definition-universal-homeomorphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\\it universal homeomorphism} if for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map of topological spaces $$ |\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}| $$ is a homeomorphism."}
+{"_id": "7613", "title": "stacks-morphisms-definition-P", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. We say a morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks {\\it has property $\\mathcal{P}$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold."}
+{"_id": "7614", "title": "stacks-morphisms-definition-locally-finite-type", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ {\\it locally of finite type} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{locally of finite type}$. \\item We say $f$ is {\\it of finite type} if it is locally of finite type and quasi-compact. \\end{enumerate}"}
+{"_id": "7615", "title": "stacks-morphisms-definition-finite-type-point", "text": "Let $\\mathcal{X}$ be an algebraic stack. We say a point $x \\in |\\mathcal{X}|$ is a {\\it finite type point}\\footnote{This is a slight abuse of language as it would perhaps be more correct to say ``locally finite type point''.} if the equivalent conditions of Lemma \\ref{lemma-point-finite-type} are satisfied. We denote $\\mathcal{X}_{\\text{ft-pts}}$ the set of finite type points of $\\mathcal{X}$."}
+{"_id": "7616", "title": "stacks-morphisms-definition-locally-quasi-finite", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it locally quasi-finite} if $f$ is quasi-DM, locally of finite type, and for every morphism $\\Spec(k) \\to \\mathcal{Y}$ where $k$ is a field the space $|\\mathcal{X}_k|$ is discrete."}
+{"_id": "7617", "title": "stacks-morphisms-definition-quasi-finite", "text": "\\begin{reference} \\cite{rydh_approx} \\end{reference} Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it quasi-finite} if $f$ is locally quasi-finite (Definition \\ref{definition-locally-quasi-finite}) and quasi-compact (Definition \\ref{definition-quasi-compact})."}
+{"_id": "7618", "title": "stacks-morphisms-definition-flat", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it flat} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{flat}$."}
+{"_id": "7619", "title": "stacks-morphisms-definition-flat-at-point", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$. We say $f$ is {\\it flat at $x$} if the equivalent conditions of Lemma \\ref{lemma-flat-at-point} hold."}
+{"_id": "7620", "title": "stacks-morphisms-definition-locally-finite-presentation", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. \\begin{enumerate} \\item We say $f$ {\\it locally of finite presentation} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{locally of finite presentation}$. \\item We say $f$ is {\\it of finite presentation} if it is locally of finite presentation, quasi-compact, and quasi-separated. \\end{enumerate}"}
+{"_id": "7621", "title": "stacks-morphisms-definition-gerbe", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$ if $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ as stacks in groupoids over $(\\Sch/S)_{fppf}$, see Stacks, Definition \\ref{stacks-definition-gerbe-over-stack-in-groupoids}. We say an algebraic stack $\\mathcal{X}$ is a {\\it gerbe} if there exists a morphism $\\mathcal{X} \\to X$ where $X$ is an algebraic space which turns $\\mathcal{X}$ into a gerbe over $X$."}
+{"_id": "7622", "title": "stacks-morphisms-definition-smooth", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it smooth} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{smooth}$."}
+{"_id": "7623", "title": "stacks-morphisms-definition-etale-smooth-P", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is \\'etale-smooth local on the source-and-target. We say a DM morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks {\\it has property $\\mathcal{P}$} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold."}
+{"_id": "7624", "title": "stacks-morphisms-definition-etale", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it \\'etale} if $f$ is DM and the equivalent conditions of Lemma \\ref{lemma-etale-smooth-local-source-target} hold with $\\mathcal{P} = \\etale$."}
+{"_id": "7625", "title": "stacks-morphisms-definition-unramified", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it unramified} if $f$ is DM and the equivalent conditions of Lemma \\ref{lemma-etale-smooth-local-source-target} hold with $\\mathcal{P} =$``unramified''."}
+{"_id": "7626", "title": "stacks-morphisms-definition-proper", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it proper} if $f$ is separated, finite type, and universally closed."}
+{"_id": "7627", "title": "stacks-morphisms-definition-scheme-theoretic-image", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The {\\it scheme theoretic image} of $f$ is the smallest closed substack $\\mathcal{Z} \\subset \\mathcal{Y}$ through which $f$ factors\\footnote{We will see in Lemma \\ref{lemma-scheme-theoretic-image-existence} that the scheme theoretic image always exists.}."}
+{"_id": "7628", "title": "stacks-morphisms-definition-fill-in-diagram", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram \\begin{equation} \\label{equation-diagram} \\vcenter{ \\xymatrix{ \\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^-y \\ar@{..>}[ru] & \\mathcal{Y} } } \\end{equation} where $A$ is a valuation ring with field of fractions $K$. Let $$ \\gamma : y \\circ j \\longrightarrow f \\circ x $$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. (Notation as in Categories, Sections \\ref{categories-section-formal-cat-cat} and \\ref{categories-section-2-categories}.) Given (\\ref{equation-diagram}) and $\\gamma$ a {\\it dotted arrow} is a triple $(a, \\alpha, \\beta)$ consisting of a morphism $a : \\Spec(A) \\to \\mathcal{X}$ and $2$-arrows $\\alpha : a \\circ j \\to x$, $\\beta : y \\to f \\circ a$ such that $\\gamma = (\\text{id}_f \\star \\alpha) \\circ (\\beta \\star \\text{id}_j)$, in other words such that $$ \\xymatrix{ & f \\circ a \\circ j \\ar[rd]^{\\text{id}_f \\star \\alpha} \\\\ y \\circ j \\ar[ru]^{\\beta \\star \\text{id}_j} \\ar[rr]^\\gamma & & f \\circ x } $$ is commutative. A {\\it morphism of dotted arrows} $(a, \\alpha, \\beta) \\to (a', \\alpha', \\beta')$ is a $2$-arrow $\\theta : a \\to a'$ such that $\\alpha = \\alpha' \\circ (\\theta \\star \\text{id}_j)$ and $\\beta' = (\\text{id}_f \\star \\theta) \\circ \\beta$."}
+{"_id": "7629", "title": "stacks-morphisms-definition-uniqueness", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the {\\it uniqueness part of the valuative criterion} if for every diagram (\\ref{equation-diagram}) and $\\gamma$ as in Definition \\ref{definition-fill-in-diagram} the category of dotted arrows is either empty or a setoid with exactly one isomorphism class."}
+{"_id": "7630", "title": "stacks-morphisms-definition-existence", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the {\\it existence part of the valuative criterion} if for every diagram (\\ref{equation-diagram}) and $\\gamma$ as in Definition \\ref{definition-fill-in-diagram} there exists an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$ dominating $A$ such that the category of dotted arrows for the outer rectangle of the diagram $$ \\xymatrix{ \\Spec(K') \\ar[r] \\ar@/^2em/[rr]_{x'} \\ar[d]_{j'} & \\Spec(K) \\ar[d]_j \\ar[r]_-x & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A') \\ar[r] \\ar@/_2em/[rr]^{y'} & \\Spec(A) \\ar[r]^-y & \\mathcal{Y} } $$ with induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$ is nonempty."}
+{"_id": "7631", "title": "stacks-morphisms-definition-lci", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\\it local complete intersection morphism} or {\\it Koszul} if the equivalent conditions of Lemma \\ref{lemma-local-source-target} hold with $\\mathcal{P} = \\text{local complete intersection}$."}
+{"_id": "7734", "title": "schemes-definition-locally-ringed-space", "text": "Locally ringed spaces. \\begin{enumerate} \\item A {\\it locally ringed space $(X, \\mathcal{O}_X)$} is a pair consisting of a topological space $X$ and a sheaf of rings $\\mathcal{O}_X$ all of whose stalks are local rings. \\item Given a locally ringed space $(X, \\mathcal{O}_X)$ we say that $\\mathcal{O}_{X, x}$ is the {\\it local ring of $X$ at $x$}. We denote $\\mathfrak{m}_{X, x}$ or simply $\\mathfrak{m}_x$ the maximal ideal of $\\mathcal{O}_{X, x}$. Moreover, the {\\it residue field of $X$ at $x$} is the residue field $\\kappa(x) = \\mathcal{O}_{X, x}/\\mathfrak{m}_x$. \\item A {\\it morphism of locally ringed spaces} $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is a morphism of ringed spaces such that for all $x \\in X$ the induced ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is a local ring map. \\end{enumerate}"}
+{"_id": "7735", "title": "schemes-definition-immersion-locally-ringed-spaces", "text": "Let $f : X \\to Y$ be a morphism of locally ringed spaces. We say that $f$ is an {\\it open immersion} if $f$ is a homeomorphism of $X$ onto an open subset of $Y$, and the map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$ is an isomorphism."}
+{"_id": "7736", "title": "schemes-definition-open-subspace", "text": "Let $X$ be a locally ringed space. Let $U \\subset X$ be an open subset. The locally ringed space $(U, \\mathcal{O}_U)$ of Example \\ref{example-open-subspace} above is the {\\it open subspace of $X$ associated to $U$}."}
+{"_id": "7737", "title": "schemes-definition-closed-immersion-locally-ringed-spaces", "text": "Let $i : Z \\to X$ be a morphism of locally ringed spaces. We say that $i$ is a {\\it closed immersion} if: \\begin{enumerate} \\item The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$. \\item The map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective; let $\\mathcal{I}$ denote the kernel. \\item The $\\mathcal{O}_X$-module $\\mathcal{I}$ is locally generated by sections. \\end{enumerate}"}
+{"_id": "7738", "title": "schemes-definition-closed-subspace", "text": "Let $X$ be a locally ringed space. Let $\\mathcal{I}$ be a sheaf of ideals on $X$ which is locally generated by sections. The locally ringed space $(Z, \\mathcal{O}_Z)$ of Example \\ref{example-closed-subspace} above is the {\\it closed subspace of $X$ associated to the sheaf of ideals $\\mathcal{I}$}."}
+{"_id": "7739", "title": "schemes-definition-standard-covering", "text": "Let $R$ be a ring. \\begin{enumerate} \\item A {\\it standard open covering} of $\\Spec(R)$ is a covering $\\Spec(R) = \\bigcup_{i = 1}^n D(f_i)$, where $f_1, \\ldots, f_n \\in R$. \\item Suppose that $D(f) \\subset \\Spec(R)$ is a standard open. A {\\it standard open covering} of $D(f)$ is a covering $D(f) = \\bigcup_{i = 1}^n D(g_i)$, where $g_1, \\ldots, g_n \\in R$. \\end{enumerate}"}
+{"_id": "7740", "title": "schemes-definition-structure-sheaf", "text": "Let $R$ be a ring. \\begin{enumerate} \\item The {\\it structure sheaf $\\mathcal{O}_{\\Spec(R)}$ of the spectrum of $R$} is the unique sheaf of rings $\\mathcal{O}_{\\Spec(R)}$ which agrees with $\\widetilde R$ on the basis of standard opens. \\item The locally ringed space $(\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ is called the {\\it spectrum} of $R$ and denoted $\\Spec(R)$. \\item The sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules extending $\\widetilde M$ to all opens of $\\Spec(R)$ is called the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules associated to $M$. This sheaf is denoted $\\widetilde M$ as well. \\end{enumerate}"}
+{"_id": "7741", "title": "schemes-definition-affine-scheme", "text": "An {\\it affine scheme} is a locally ringed space isomorphic as a locally ringed space to $\\Spec(R)$ for some ring $R$. A {\\it morphism of affine schemes} is a morphism in the category of locally ringed spaces."}
+{"_id": "7742", "title": "schemes-definition-scheme", "text": "\\begin{history} In \\cite{EGA1} what we call a scheme was called a ``pre-sch\\'ema'' and the name ``sch\\'ema'' was reserved for what is a separated scheme in the Stacks project. In the second edition \\cite{EGA1-second} the terminology was changed to the terminology that is now standard. However, one may occasionally encounter the terminology ``prescheme'', for example in \\cite{Murre-lectures}. \\end{history} A {\\it scheme} is a locally ringed space with the property that every point has an open neighbourhood which is an affine scheme. A {\\it morphism of schemes} is a morphism of locally ringed spaces. The category of schemes will be denoted $\\Sch$."}
+{"_id": "7743", "title": "schemes-definition-immersion", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item A morphism of schemes is called an {\\it open immersion} if it is an open immersion of locally ringed spaces (see Definition \\ref{definition-immersion-locally-ringed-spaces}). \\item An {\\it open subscheme} of $X$ is an open subspace of $X$ in the sense of Definition \\ref{definition-open-subspace}; an open subscheme of $X$ is a scheme by Lemma \\ref{lemma-open-subspace-scheme}. \\item A morphism of schemes is called a {\\it closed immersion} if it is a closed immersion of locally ringed spaces (see Definition \\ref{definition-closed-immersion-locally-ringed-spaces}). \\item A {\\it closed subscheme} of $X$ is a closed subspace of $X$ in the sense of Definition \\ref{definition-closed-subspace}; a closed subscheme is a scheme by Lemma \\ref{lemma-closed-subspace-scheme}. \\item A morphism of schemes $f : X \\to Y$ is called an {\\it immersion}, or a {\\it locally closed immersion} if it can be factored as $j \\circ i$ where $i$ is a closed immersion and $j$ is an open immersion. \\end{enumerate}"}
+{"_id": "7744", "title": "schemes-definition-reduced", "text": "Let $X$ be a scheme. We say $X$ is {\\it reduced} if every local ring $\\mathcal{O}_{X, x}$ is reduced."}
+{"_id": "7745", "title": "schemes-definition-reduced-induced-scheme", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset. A {\\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of $X$ whose underlying set is equal to $Z$. We often say ``let $(Z, \\mathcal{O}_Z)$ be a scheme structure on $Z$'' to indicate this. The {\\it reduced induced scheme structure} on $Z$ is the one constructed in Lemma \\ref{lemma-reduced-closed-subscheme}. The {\\it reduction $X_{red}$ of $X$} is the reduced induced scheme structure on $X$ itself."}
+{"_id": "7746", "title": "schemes-definition-representable-functor", "text": "(See Categories, Definition \\ref{categories-definition-representable-functor}.) Let $F$ be a contravariant functor from the category of schemes to the category of sets (as above). We say that $F$ is {\\it representable by a scheme} or {\\it representable} if there exists a scheme $X$ such that $h_X \\cong F$."}
+{"_id": "7747", "title": "schemes-definition-representable-by-open-immersions", "text": "Let $F$ be a contravariant functor on the category of schemes with values in sets. \\begin{enumerate} \\item We say that $F$ {\\it satisfies the sheaf property for the Zariski topology} if for every scheme $T$ and every open covering $T = \\bigcup_{i \\in I} U_i$, and for any collection of elements $\\xi_i \\in F(U_i)$ such that $\\xi_i|_{U_i \\cap U_j} = \\xi_j|_{U_i \\cap U_j}$ there exists a unique element $\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$. \\item A {\\it subfunctor $H \\subset F$} is a rule that associates to every scheme $T$ a subset $H(T) \\subset F(T)$ such that the maps $F(f) : F(T) \\to F(T')$ maps $H(T)$ into $H(T')$ for all morphisms of schemes $f : T' \\to T$. \\item Let $H \\subset F$ be a subfunctor. We say that $H \\subset F$ is {\\it representable by open immersions} if for all pairs $(T, \\xi)$, where $T$ is a scheme and $\\xi \\in F(T)$ there exists an open subscheme $U_\\xi \\subset T$ with the following property: \\begin{itemize} \\item[(*)] A morphism $f : T' \\to T$ factors through $U_\\xi$ if and only if $f^*\\xi \\in H(T')$. \\end{itemize} \\item Let $I$ be a set. For each $i \\in I$ let $H_i \\subset F$ be a subfunctor. We say that the collection $(H_i)_{i \\in I}$ {\\it covers $F$} if and only if for every $\\xi \\in F(T)$ there exists an open covering $T = \\bigcup U_i$ such that $\\xi|_{U_i} \\in H_i(U_i)$. \\end{enumerate}"}
+{"_id": "7748", "title": "schemes-definition-fibre-product", "text": "Given morphisms of schemes $f : X \\to S$ and $g : Y \\to S$ the {\\it fibre product} is a scheme $X \\times_S Y$ together with projection morphisms $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ sitting into the following commutative diagram $$ \\xymatrix{ X \\times_S Y \\ar[r]_q \\ar[d]_p & Y \\ar[d]^g \\\\ X \\ar[r]^f & S } $$ which is universal among all diagrams of this sort, see Categories, Definition \\ref{categories-definition-fibre-products}."}
+{"_id": "7749", "title": "schemes-definition-inverse-image-closed-subscheme", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\subset Y$ be a closed subscheme of $Y$. The {\\it inverse image $f^{-1}(Z)$ of the closed subscheme $Z$} is the closed subscheme $Z \\times_Y X$ of $X$. See Lemma \\ref{lemma-fibre-product-immersion} above."}
+{"_id": "7750", "title": "schemes-definition-base-change", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item We say $X$ is a {\\it scheme over $S$} to mean that $X$ comes equipped with a morphism of schemes $X \\to S$. The morphism $X \\to S$ is sometimes called the {\\it structure morphism}. \\item If $R$ is a ring we say $X$ is a {\\it scheme over $R$} instead of $X$ is a scheme over $\\Spec(R)$. \\item A {\\it morphism $f : X \\to Y$ of schemes over $S$} is a morphism of schemes such that the composition $X \\to Y \\to S$ of $f$ with the structure morphism of $Y$ is equal to the structure morphism of $X$. \\item We denote $\\Mor_S(X, Y)$ the set of all morphisms from $X$ to $Y$ over $S$. \\item Let $X$ be a scheme over $S$. Let $S' \\to S$ be a morphism of schemes. The {\\it base change} of $X$ is the scheme $X_{S'} = S' \\times_S X$ over $S'$. \\item Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $S' \\to S$ be a morphism of schemes. The {\\it base change} of $f$ is the induced morphism $f' : X_{S'} \\to Y_{S'}$ (namely the morphism $\\text{id}_{S'} \\times_{\\text{id}_S} f$). \\item Let $R$ be a ring. Let $X$ be a scheme over $R$. Let $R \\to R'$ be a ring map. The {\\it base change} $X_{R'}$ is the scheme $\\Spec(R') \\times_{\\Spec(R)} X$ over $R'$. \\end{enumerate}"}
+{"_id": "7751", "title": "schemes-definition-preserved-by-base-change", "text": "Properties and base change. \\begin{enumerate} \\item Let $\\mathcal{P}$ be a property of schemes over a base. We say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change}, or simply that $\\mathcal{P}$ is {\\it preserved under base change} if whenever $X/S$ has $\\mathcal{P}$, any base change $X_{S'}/S'$ has $\\mathcal{P}$. \\item Let $\\mathcal{P}$ be a property of morphisms of schemes over a base. We say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change}, or simply that {\\it preserved under base change} if whenever $f : X \\to Y$ over $S$ has $\\mathcal{P}$, any base change $f' : X_{S'} \\to Y_{S'}$ over $S'$ has $\\mathcal{P}$. \\end{enumerate}"}
+{"_id": "7752", "title": "schemes-definition-fibre", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$ be a point. The {\\it scheme theoretic fibre $X_s$ of $f$ over $s$}, or simply the {\\it fibre of $f$ over $s$}, is the scheme fitting in the following fibre product diagram $$ \\xymatrix{ X_s = \\Spec(\\kappa(s)) \\times_S X \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(\\kappa(s)) \\ar[r] & S } $$ We think of the fibre $X_s$ always as a scheme over $\\kappa(s)$."}
+{"_id": "7753", "title": "schemes-definition-quasi-compact", "text": "A morphism of schemes is called {\\it quasi-compact} if the underlying map of topological spaces is quasi-compact, see Topology, Definition \\ref{topology-definition-quasi-compact}."}
+{"_id": "7754", "title": "schemes-definition-universally-closed", "text": "A morphism of schemes $f : X \\to S$ is said to be {\\it universally closed} if every base change $f' : X_{S'} \\to S'$ is closed."}
+{"_id": "7755", "title": "schemes-definition-valuative-criterion", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ {\\it satisfies the existence part of the valuative criterion} if given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & S } $$ where $A$ is a valuation ring with field of fractions $K$, the dotted arrow exists. We say $f$ {\\it satisfies the uniqueness part of the valuative criterion} if there is at most one dotted arrow given any diagram as above (without requiring existence of course)."}
+{"_id": "7756", "title": "schemes-definition-separated", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item We say $f$ is {\\it separated} if the diagonal morphism $\\Delta_{X/S}$ is a closed immersion. \\item We say $f$ is {\\it quasi-separated} if the diagonal morphism $\\Delta_{X/S}$ is a quasi-compact morphism. \\item We say a scheme $Y$ is {\\it separated} if the morphism $Y \\to \\Spec(\\mathbf{Z})$ is separated. \\item We say a scheme $Y$ is {\\it quasi-separated} if the morphism $Y \\to \\Spec(\\mathbf{Z})$ is quasi-separated. \\end{enumerate}"}
+{"_id": "7757", "title": "schemes-definition-monomorphism", "text": "A morphism of schemes is called a {\\it monomorphism} if it is a monomorphism in the category of schemes, see Categories, Definition \\ref{categories-definition-mono-epi}."}
+{"_id": "7809", "title": "injectives-definition-small", "text": "Let $\\mathcal{C}$ be a category, let $I \\subset \\text{Arrows}(\\mathcal{C})$, and let $\\alpha$ be an ordinal. An object $A$ of $\\mathcal{C}$ is said to be {\\it $\\alpha$-small with respect to $I$} if whenever $\\{B_\\beta\\}$ is a system over $\\alpha$ with transition maps in $I$, then the map (\\ref{equation-compare}) is an isomorphism."}
+{"_id": "7810", "title": "injectives-definition-grothendieck-conditions", "text": "Let $\\mathcal{A}$ be an abelian category. We name some conditions \\begin{enumerate} \\item[AB3] $\\mathcal{A}$ has direct sums, \\item[AB4] $\\mathcal{A}$ has AB3 and direct sums are exact, \\item[AB5] $\\mathcal{A}$ has AB3 and filtered colimits are exact. \\end{enumerate} Here are the dual notions \\begin{enumerate} \\item[AB3*] $\\mathcal{A}$ has products, \\item[AB4*] $\\mathcal{A}$ has AB3* and products are exact, \\item[AB5*] $\\mathcal{A}$ has AB3* and filtered limits are exact. \\end{enumerate} We say an object $U$ of $\\mathcal{A}$ is a {\\it generator} if for every $N \\subset M$, $N \\not = M$ in $\\mathcal{A}$ there exists a morphism $U \\to M$ which does not factor through $N$. We say $\\mathcal{A}$ is a {\\it Grothendieck abelian category} if it has AB5 and a generator."}
+{"_id": "7811", "title": "injectives-definition-size", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\\mathcal{A}$. The {\\it size} $|M|$ of $M$ is the cardinality of the set of subobjects of $M$."}
+{"_id": "7849", "title": "brauer-definition-finite", "text": "Let $A$ be a $k$-algebra. We say $A$ is {\\it finite} if $\\dim_k(A) < \\infty$. In this case we write $[A : k] = \\dim_k(A)$."}
+{"_id": "7850", "title": "brauer-definition-skew-field", "text": "A {\\it skew field} is a possibly noncommutative ring with an identity element $1$, with $1 \\not = 0$, in which every nonzero element has a multiplicative inverse."}
+{"_id": "7851", "title": "brauer-definition-simple", "text": "Let $A$ be a $k$-algebra. We say an $A$-module $M$ is {\\it simple} if it is nonzero and the only $A$-submodules are $0$ and $M$. We say $A$ is {\\it simple} if the only two-sided ideals of $A$ are $0$ and $A$."}
+{"_id": "7852", "title": "brauer-definition-central", "text": "A $k$-algebra $A$ is {\\it central} if the center of $A$ is the image of $k \\to A$."}
+{"_id": "7853", "title": "brauer-definition-opposite", "text": "Given a $k$-algebra $A$ we denote $A^{op}$ the $k$-algebra we get by reversing the order of multiplication in $A$. This is called the {\\it opposite algebra}."}
+{"_id": "7854", "title": "brauer-definition-brauer-group", "text": "Let $k$ be a field. The {\\it Brauer group} of $k$ is the abelian group of similarity classes of finite central simple $k$-algebras defined above. Notation $\\text{Br}(k)$."}
+{"_id": "7855", "title": "brauer-definition-splitting", "text": "Let $A$ be a finite central simple $k$-algebra. We say a field extension $k \\subset k'$ {\\it splits} $A$, or $k'$ is a {\\it splitting field} for $A$ if $A \\otimes_k k'$ is a matrix algebra over $k'$."}
+{"_id": "8082", "title": "divisors-definition-associated", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. \\begin{enumerate} \\item We say $x \\in X$ is {\\it associated} to $\\mathcal{F}$ if the maximal ideal $\\mathfrak m_x$ is associated to the $\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$. \\item We denote $\\text{Ass}(\\mathcal{F})$ or $\\text{Ass}_X(\\mathcal{F})$ the set of associated points of $\\mathcal{F}$. \\item The {\\it associated points of $X$} are the associated points of $\\mathcal{O}_X$. \\end{enumerate}"}
+{"_id": "8083", "title": "divisors-definition-embedded", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. \\begin{enumerate} \\item An {\\it embedded associated point} of $\\mathcal{F}$ is an associated point which is not maximal among the associated points of $\\mathcal{F}$, i.e., it is the specialization of another associated point of $\\mathcal{F}$. \\item A point $x$ of $X$ is called an {\\it embedded point} if $x$ is an embedded associated point of $\\mathcal{O}_X$. \\item An {\\it embedded component} of $X$ is an irreducible closed subset $Z = \\overline{\\{x\\}}$ where $x$ is an embedded point of $X$. \\end{enumerate}"}
+{"_id": "8084", "title": "divisors-definition-weakly-associated", "text": "Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. \\begin{enumerate} \\item We say $x \\in X$ is {\\it weakly associated} to $\\mathcal{F}$ if the maximal ideal $\\mathfrak m_x$ is weakly associated to the $\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$. \\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated points of $\\mathcal{F}$. \\item The {\\it weakly associated points of $X$} are the weakly associated points of $\\mathcal{O}_X$. \\end{enumerate}"}
+{"_id": "8085", "title": "divisors-definition-relative-assassin", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The {\\it relative assassin of $\\mathcal{F}$ in $X$ over $S$} is the set $$ \\text{Ass}_{X/S}(\\mathcal{F}) = \\bigcup\\nolimits_{s \\in S} \\text{Ass}_{X_s}(\\mathcal{F}_s) $$ where $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction of $\\mathcal{F}$ to the fibre of $f$ at $s$."}
+{"_id": "8086", "title": "divisors-definition-relative-weak-assassin", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $S$} is the set $$ \\text{WeakAss}_{X/S}(\\mathcal{F}) = \\bigcup\\nolimits_{s \\in S} \\text{WeakAss}(\\mathcal{F}_s) $$ where $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction of $\\mathcal{F}$ to the fibre of $f$ at $s$."}
+{"_id": "8087", "title": "divisors-definition-torsion", "text": "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item We say a local section of $\\mathcal{F}$ is {\\it torsion} if it satisfies the equivalent conditions of Lemma \\ref{lemma-torsion-sections}. \\item We say $\\mathcal{F}$ is {\\it torsion free} if every torsion section of $\\mathcal{F}$ is $0$. \\end{enumerate}"}
+{"_id": "8088", "title": "divisors-definition-reflexive", "text": "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The {\\it reflexive hull} of $\\mathcal{F}$ is the $\\mathcal{O}_X$-module $$ \\mathcal{F}^{**} = \\SheafHom_{\\mathcal{O}_X}( \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X), \\mathcal{O}_X) $$ We say $\\mathcal{F}$ is {\\it reflexive} if the natural map $j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$ is an isomorphism."}
+{"_id": "8089", "title": "divisors-definition-effective-Cartier-divisor", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item A {\\it locally principal closed subscheme} of $S$ is a closed subscheme whose sheaf of ideals is locally generated by a single element. \\item An {\\it effective Cartier divisor} on $S$ is a closed subscheme $D \\subset S$ whose ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_S$ is an invertible $\\mathcal{O}_S$-module. \\end{enumerate}"}
+{"_id": "8090", "title": "divisors-definition-sum-effective-Cartier-divisors", "text": "Let $S$ be a scheme. Given effective Cartier divisors $D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the closed subscheme of $S$ corresponding to the quasi-coherent sheaf of ideals $\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$. We call this the {\\it sum of the effective Cartier divisors $D_1$ and $D_2$}."}
+{"_id": "8091", "title": "divisors-definition-pullback-effective-Cartier-divisor", "text": "Let $f : S' \\to S$ be a morphism of schemes. Let $D \\subset S$ be an effective Cartier divisor. We say the {\\it pullback of $D$ by $f$ is defined} if the closed subscheme $f^{-1}(D) \\subset S'$ is an effective Cartier divisor. In this case we denote it either $f^*D$ or $f^{-1}(D)$ and we call it the {\\it pullback of the effective Cartier divisor}."}
+{"_id": "8092", "title": "divisors-definition-invertible-sheaf-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I}_D$. \\begin{enumerate} \\item The {\\it invertible sheaf $\\mathcal{O}_S(D)$ associated to $D$} is defined by $$ \\mathcal{O}_S(D) = \\SheafHom_{\\mathcal{O}_S}(\\mathcal{I}_D, \\mathcal{O}_S) = \\mathcal{I}_D^{\\otimes -1}. $$ \\item The {\\it canonical section}, usually denoted $1$ or $1_D$, is the global section of $\\mathcal{O}_S(D)$ corresponding to the inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_S$. \\item We write $\\mathcal{O}_S(-D) = \\mathcal{O}_S(D)^{\\otimes -1} = \\mathcal{I}_D$. \\item Given a second effective Cartier divisor $D' \\subset S$ we define $\\mathcal{O}_S(D - D') = \\mathcal{O}_S(D) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(-D')$. \\end{enumerate}"}
+{"_id": "8093", "title": "divisors-definition-regular-section", "text": "Let $(X, \\mathcal{O}_X)$ be a locally ringed space. Let $\\mathcal{L}$ be an invertible sheaf on $X$. A global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a {\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$, $f \\mapsto fs$ is injective."}
+{"_id": "8094", "title": "divisors-definition-zero-scheme-s", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf. Let $s \\in \\Gamma(X, \\mathcal{L})$ be a global section. The {\\it zero scheme} of $s$ is the closed subscheme $Z(s) \\subset X$ defined by the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the map $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$."}
+{"_id": "8095", "title": "divisors-definition-relative-effective-Cartier-divisor", "text": "Let $f : X \\to S$ be a morphism of schemes. A {\\it relative effective Cartier divisor} on $X/S$ is an effective Cartier divisor $D \\subset X$ such that $D \\to S$ is a flat morphism of schemes."}
+{"_id": "8096", "title": "divisors-definition-conormal-sheaf", "text": "Let $f : Z \\to X$ be an immersion. The {\\it conormal algebra $\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $f$} is the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras $\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above."}
+{"_id": "8097", "title": "divisors-definition-normal-cone", "text": "Let $i : Z \\to X$ be an immersion of schemes. The {\\it normal cone $C_ZX$} of $Z$ in $X$ is $$ C_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *}) $$ see Constructions, Definitions \\ref{constructions-definition-cone} and \\ref{constructions-definition-abstract-cone}. The {\\it normal bundle} of $Z$ in $X$ is the vector bundle $$ N_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X})) $$ see Constructions, Definitions \\ref{constructions-definition-vector-bundle} and \\ref{constructions-definition-abstract-vector-bundle}."}
+{"_id": "8098", "title": "divisors-definition-regular-ideal-sheaf", "text": "Let $X$ be a ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$ be a sheaf of ideals. \\begin{enumerate} \\item We say $\\mathcal{J}$ is {\\it regular} if for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open neighbourhood $x \\in U \\subset X$ and a regular sequence $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$ is generated by $f_1, \\ldots, f_r$. \\item We say $\\mathcal{J}$ is {\\it Koszul-regular} if for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open neighbourhood $x \\in U \\subset X$ and a Koszul-regular sequence $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$ is generated by $f_1, \\ldots, f_r$. \\item We say $\\mathcal{J}$ is {\\it $H_1$-regular} if for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open neighbourhood $x \\in U \\subset X$ and a $H_1$-regular sequence $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$ is generated by $f_1, \\ldots, f_r$. \\item We say $\\mathcal{J}$ is {\\it quasi-regular} if for every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open neighbourhood $x \\in U \\subset X$ and a quasi-regular sequence $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$ is generated by $f_1, \\ldots, f_r$. \\end{enumerate}"}
+{"_id": "8099", "title": "divisors-definition-regular-immersion", "text": "Let $i : Z \\to X$ be an immersion of schemes. Choose an open subscheme $U \\subset X$ such that $i$ identifies $Z$ with a closed subscheme of $U$ and denote $\\mathcal{I} \\subset \\mathcal{O}_U$ the corresponding quasi-coherent sheaf of ideals. \\begin{enumerate} \\item We say $i$ is a {\\it regular immersion} if $\\mathcal{I}$ is regular. \\item We say $i$ is a {\\it Koszul-regular immersion} if $\\mathcal{I}$ is Koszul-regular. \\item We say $i$ is a {\\it $H_1$-regular immersion} if $\\mathcal{I}$ is $H_1$-regular. \\item We say $i$ is a {\\it quasi-regular immersion} if $\\mathcal{I}$ is quasi-regular. \\end{enumerate}"}
+{"_id": "8100", "title": "divisors-definition-relative-H1-regular-immersion", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $i : Z \\to X$ be an immersion. \\begin{enumerate} \\item We say $i$ is a {\\it relative quasi-regular immersion} if $Z \\to S$ is flat and $i$ is a quasi-regular immersion. \\item We say $i$ is a {\\it relative $H_1$-regular immersion} if $Z \\to S$ is flat and $i$ is an $H_1$-regular immersion. \\end{enumerate}"}
+{"_id": "8101", "title": "divisors-definition-sheaf-meromorphic-functions", "text": "Let $(X, \\mathcal{O}_X)$ be a locally ringed space. The {\\it sheaf of meromorphic functions on $X$} is the sheaf {\\it $\\mathcal{K}_X$} associated to the presheaf displayed above. A {\\it meromorphic function} on $X$ is a global section of $\\mathcal{K}_X$."}
+{"_id": "8102", "title": "divisors-definition-meromorphic-section", "text": "Let $X$ be a locally ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item We denote $\\mathcal{K}_X(\\mathcal{F})$ the sheaf of $\\mathcal{K}_X$-modules which is the sheafification of the presheaf $U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently $\\mathcal{K}_X(\\mathcal{F}) = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above). \\item A {\\it meromorphic section of $\\mathcal{F}$} is a global section of $\\mathcal{K}_X(\\mathcal{F})$. \\end{enumerate}"}
+{"_id": "8103", "title": "divisors-definition-pullback-meromorphic-sections", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of locally ringed spaces. We say that {\\it pullbacks of meromorphic functions are defined for $f$} if for every pair of open $U \\subset X$, $V \\subset Y$ such that $f(U) \\subset V$, and any section $s \\in \\Gamma(V, \\mathcal{S}_Y)$ the pullback $f^\\sharp(s) \\in \\Gamma(U, \\mathcal{O}_X)$ is an element of $\\Gamma(U, \\mathcal{S}_X)$."}
+{"_id": "8104", "title": "divisors-definition-regular-meromorphic-section", "text": "Let $X$ be a locally ringed space. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. A meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular} if the induced map $\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$ is injective. In other words, $s$ is a regular section of the invertible $\\mathcal{K}_X$-module $\\mathcal{K}_X(\\mathcal{L})$, see Definition \\ref{definition-regular-section}."}
+{"_id": "8105", "title": "divisors-definition-regular-meromorphic-ideal-denominators", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s$ be a regular meromorphic section of $\\mathcal{L}$. The sheaf of ideals $\\mathcal{I}$ constructed in Lemma \\ref{lemma-regular-meromorphic-ideal-denominators} is called the {\\it ideal sheaf of denominators of $s$}."}
+{"_id": "8106", "title": "divisors-definition-Weil-divisor", "text": "Let $X$ be a locally Noetherian integral scheme. \\begin{enumerate} \\item A {\\it prime divisor} is an integral closed subscheme $Z \\subset X$ of codimension $1$. \\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where the sum is over prime divisors of $X$ and the collection $\\{Z \\mid n_Z \\not = 0\\}$ is locally finite (Topology, Definition \\ref{topology-definition-locally-finite}). \\end{enumerate} The group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$."}
+{"_id": "8107", "title": "divisors-definition-order-vanishing", "text": "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$. For every prime divisor $Z \\subset X$ we define the {\\it order of vanishing of $f$ along $Z$} as the integer $$ \\text{ord}_Z(f) = \\text{ord}_{\\mathcal{O}_{X, \\xi}}(f) $$ where the right hand side is the notion of Algebra, Definition \\ref{algebra-definition-ord} and $\\xi$ is the generic point of $Z$."}
+{"_id": "8108", "title": "divisors-definition-principal-divisor", "text": "Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$. The {\\it principal Weil divisor associated to $f$} is the Weil divisor $$ \\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z] $$ where the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in Definition \\ref{definition-order-vanishing}. This makes sense by Lemma \\ref{lemma-divisor-locally-finite}."}
+{"_id": "8109", "title": "divisors-definition-class-group", "text": "Let $X$ be a locally Noetherian integral scheme. The {\\it Weil divisor class group} of $X$ is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: $\\text{Cl}(X)$."}
+{"_id": "8110", "title": "divisors-definition-order-vanishing-meromorphic", "text": "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$ be a regular meromorphic section of $\\mathcal{L}$. For every prime divisor $Z \\subset X$ we define the {\\it order of vanishing of $s$ along $Z$} as the integer $$ \\text{ord}_{Z, \\mathcal{L}}(s) = \\text{ord}_{\\mathcal{O}_{X, \\xi}}(s/s_\\xi) $$ where the right hand side is the notion of Algebra, Definition \\ref{algebra-definition-ord}, $\\xi \\in Z$ is the generic point, and $s_\\xi \\in \\mathcal{L}_\\xi$ is a generator."}
+{"_id": "8111", "title": "divisors-definition-divisor-invertible-sheaf", "text": "Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$ we define the {\\it Weil divisor associated to $s$} as $$ \\text{div}_\\mathcal{L}(s) = \\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X) $$ where the sum is over prime divisors. \\item We define {\\it Weil divisor class associated to $\\mathcal{L}$} as the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$ where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$. This is well defined by Lemma \\ref{lemma-divisor-meromorphic-well-defined}. \\end{enumerate}"}
+{"_id": "8112", "title": "divisors-definition-blow-up", "text": "Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals, and let $Z \\subset X$ be the closed subscheme corresponding to $\\mathcal{I}$, see Schemes, Definition \\ref{schemes-definition-immersion}. The {\\it blowing up of $X$ along $Z$}, or the {\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is the morphism $$ b : \\underline{\\text{Proj}}_X \\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right) \\longrightarrow X $$ The {\\it exceptional divisor} of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup."}
+{"_id": "8113", "title": "divisors-definition-strict-transform", "text": "With $Z \\subset S$ and $f : X \\to S$ as above. \\begin{enumerate} \\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup of $S$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$ by the submodule of sections supported on $\\text{pr}_{S'}^{-1}E$. \\item The {\\it strict transform} of $X$ is the closed subscheme $X' \\subset X \\times_S S'$ cut out by the quasi-coherent ideal of sections of $\\mathcal{O}_{X \\times_S S'}$ supported on $\\text{pr}_{S'}^{-1}E$. \\end{enumerate}"}
+{"_id": "8114", "title": "divisors-definition-admissible-blowup", "text": "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. A morphism $X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a closed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from $U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$."}
+{"_id": "8173", "title": "spaces-definition-relative-representable-property", "text": "With $S$, and $a : F \\to G$ representable as above. Let $\\mathcal{P}$ be a property of morphisms of schemes which \\begin{enumerate} \\item is preserved under any base change, see Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}, and \\item is fppf local on the base, see Descent, Definition \\ref{descent-definition-property-morphisms-local}. \\end{enumerate} In this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $\\xi \\in G(U)$ the resulting morphism of schemes $V_\\xi \\to U$ has property $\\mathcal{P}$."}
+{"_id": "8174", "title": "spaces-definition-algebraic-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. An {\\it algebraic space over $S$} is a presheaf $$ F : (\\Sch/S)^{opp}_{fppf} \\longrightarrow \\textit{Sets} $$ with the following properties \\begin{enumerate} \\item The presheaf $F$ is a sheaf. \\item The diagonal morphism $F  \\to F \\times F$ is representable. \\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$ and a map $h_U \\to F$ which is surjective, and \\'etale. \\end{enumerate}"}
+{"_id": "8175", "title": "spaces-definition-morphism-algebraic-spaces", "text": "Let $F$, $F'$ be algebraic spaces over $S$. A {\\it morphism $f : F \\to F'$ of algebraic spaces over $S$} is a transformation of functors from $F$ to $F'$."}
+{"_id": "8176", "title": "spaces-definition-etale-equivalence-relation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. An {\\it \\'etale equivalence relation} on $U$ over $S$ is an equivalence relation $j : R \\to U \\times_S U$ such that $s, t : R \\to U$ are \\'etale morphisms of schemes."}
+{"_id": "8177", "title": "spaces-definition-presentation", "text": "Let $F$ be an algebraic space over $S$. A {\\it presentation} of $F$ is given by a scheme $U$ over $S$ and an \\'etale equivalence relation $R$ on $U$ over $S$, and a surjective \\'etale morphism $U \\to F$ such that $R = U \\times_F U$."}
+{"_id": "8178", "title": "spaces-definition-immersion", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. \\begin{enumerate} \\item A morphism of algebraic spaces over $S$ is called an {\\it open immersion} if it is representable, and an open immersion in the sense of Definition \\ref{definition-relative-representable-property}. \\item An {\\it open subspace} of $F$ is a subfunctor $F' \\subset F$ such that $F'$ is an algebraic space and $F' \\to F$ is an open immersion. \\item A morphism of algebraic spaces over $S$ is called a {\\it closed immersion} if it is representable, and a closed immersion in the sense of Definition \\ref{definition-relative-representable-property}. \\item A {\\it closed subspace} of $F$ is a subfunctor $F' \\subset F$ such that $F'$ is an algebraic space and $F' \\to F$ is a closed immersion. \\item A morphism of algebraic spaces over $S$ is called an {\\it immersion} if it is representable, and an immersion in the sense of Definition \\ref{definition-relative-representable-property}. \\item A {\\it locally closed subspace} of $F$ is a subfunctor $F' \\subset F$ such that $F'$ is an algebraic space and $F' \\to F$ is an immersion. \\end{enumerate}"}
+{"_id": "8179", "title": "spaces-definition-Zariski-open-covering", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. A {\\it Zariski covering} $\\{F_i \\subset F\\}_{i \\in I}$ of $F$ is given by a set $I$ and a collection of open subspaces $F_i \\subset F$ such that $\\coprod F_i \\to F$ is a surjective map of sheaves."}
+{"_id": "8180", "title": "spaces-definition-small-Zariski-site", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over $S$. A {\\it small Zariski site $F_{Zar}$} of an algebraic space $F$ is one of the sites described above."}
+{"_id": "8181", "title": "spaces-definition-separated", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\\Delta : F \\to F \\times F$ be the diagonal morphism. \\begin{enumerate} \\item We say $F$ is {\\it separated over $S$} if $\\Delta$ is a closed immersion. \\item We say $F$ is {\\it locally separated over $S$}\\footnote{In the literature this often refers to quasi-separated and locally separated algebraic spaces.} if $\\Delta$ is an immersion. \\item We say $F$ is {\\it quasi-separated over $S$} if $\\Delta$ is quasi-compact. \\item We say $F$ is {\\it Zariski locally quasi-separated over $S$}\\footnote{This definition was suggested by B.\\ Conrad.} if there exists a Zariski covering $F = \\bigcup_{i \\in I} F_i$ such that each $F_i$ is quasi-separated. \\end{enumerate}"}
+{"_id": "8182", "title": "spaces-definition-quotient", "text": "Notation $U \\to S$, $G$, $R$ as in Lemma \\ref{lemma-quotient}. If the action of $G$ on $U$ satisfies $(*)$ we say $G$ {\\it acts freely} on the scheme $U$. In this case the algebraic space $U/R$ is denoted $U/G$ and is called the {\\it quotient of $U$ by $G$}."}
+{"_id": "8183", "title": "spaces-definition-base-change", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. \\begin{enumerate} \\item If $F'$ is an algebraic space over $S'$, then the {\\it base change of $F'$ to $S$} is the algebraic space $j^{-1}F'$ described in Lemma \\ref{lemma-change-base-scheme}. We denote it $F'_S$. \\item If $F$ is an algebraic space over $S$, then $F$ {\\it viewed as an algebraic space over $S'$} is the algebraic space $j_!F$ over $S'$ described in Lemma \\ref{lemma-change-base-scheme}. We often simply denote this $F$; if not then we will write $j_!F$. \\end{enumerate}"}
+{"_id": "8346", "title": "topology-definition-separated", "text": "A continuous map $f : X \\to Y$ of topological spaces is called {\\it separated} if and only if the diagonal $\\Delta : X \\to X \\times_Y X$ is a closed map."}
+{"_id": "8347", "title": "topology-definition-base", "text": "Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$ is called a {\\it base for the topology on $X$} or a {\\it basis for the topology on $X$} if the following conditions hold: \\begin{enumerate} \\item Every element $B \\in \\mathcal{B}$ is open in $X$. \\item For every open $U \\subset X$ and every $x \\in U$, there exists an element $B \\in \\mathcal{B}$ such that $x \\in B \\subset U$. \\end{enumerate}"}
+{"_id": "8348", "title": "topology-definition-subbase", "text": "Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$ is called a {\\it subbase for the topology on $X$} or a {\\it subbasis for the topology on $X$} if the finite intersections of elements of $\\mathcal{B}$ form a basis for the topology on $X$."}
+{"_id": "8349", "title": "topology-definition-submersive", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item We say $f$ is a {\\it strict map of topological spaces} if the induced topology and the quotient topology on $f(X)$ agree (see discussion above). \\item We say $f$ is {\\it submersive}\\footnote{This is very different from the notion of a submersion between differential manifolds! It is probably a good idea to use ``strict and surjective'' in stead of ``submersive''.} if $f$ is surjective and strict. \\end{enumerate}"}
+{"_id": "8350", "title": "topology-definition-connected-components", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item We say $X$ is {\\it connected} if $X$ is not empty and whenever $X = T_1 \\amalg T_2$ with $T_i \\subset X$ open and closed, then either $T_1 = \\emptyset$ or $T_2 = \\emptyset$. \\item We say $T \\subset X$ is a {\\it connected component} of $X$ if $T$ is a maximal connected subset of $X$. \\end{enumerate}"}
+{"_id": "8351", "title": "topology-definition-totally-disconnected", "text": "A topological space is {\\it totally disconnected} if the connected components are all singletons."}
+{"_id": "8352", "title": "topology-definition-locally-connected", "text": "A topological space $X$ is called {\\it locally connected} if every point $x \\in X$ has a fundamental system of connected neighbourhoods."}
+{"_id": "8353", "title": "topology-definition-irreducible-components", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item We say $X$ is {\\it irreducible}, if $X$ is not empty, and whenever $X = Z_1 \\cup Z_2$ with $Z_i$ closed, we have $X = Z_1$ or $X = Z_2$. \\item We say $Z \\subset X$ is an {\\it irreducible component} of $X$ if $Z$ is a maximal irreducible subset of $X$. \\end{enumerate}"}
+{"_id": "8354", "title": "topology-definition-generic-point", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item Let $Z \\subset X$ be an irreducible closed subset. A {\\it generic point} of $Z$ is a point $\\xi \\in Z$ such that $Z = \\overline{\\{\\xi\\}}$. \\item The space $X$ is called {\\it Kolmogorov}, if for every $x, x' \\in X$, $x \\not = x'$ there exists a closed subset of $X$ which contains exactly one of the two points. \\item The space $X$ is called {\\it quasi-sober} if every irreducible closed subset has a generic point. \\item The space $X$ is called {\\it sober} if every irreducible closed subset has a unique generic point. \\end{enumerate}"}
+{"_id": "8355", "title": "topology-definition-noetherian", "text": "A topological space is called {\\it Noetherian} if the descending chain condition holds for closed subsets of $X$. A topological space is called {\\it locally Noetherian} if every point has a neighbourhood which is Noetherian."}
+{"_id": "8356", "title": "topology-definition-Krull", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item A {\\it chain of irreducible closed subsets} of $X$ is a sequence $Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$ with $Z_i$ closed irreducible and $Z_i \\not = Z_{i + 1}$ for $i = 0, \\ldots, n - 1$. \\item The {\\it length} of a chain $Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$ of irreducible closed subsets of $X$ is the integer $n$. \\item The {\\it dimension} or more precisely the {\\it Krull dimension} $\\dim(X)$ of $X$ is the element of $\\{-\\infty, 0, 1, 2, 3, \\ldots, \\infty\\}$ defined by the formula: $$ \\dim(X) = \\sup \\{\\text{lengths of chains of irreducible closed subsets}\\} $$ Thus $\\dim(X) = -\\infty$ if and only if $X$ is the empty space. \\item Let $x \\in X$. The {\\it Krull dimension of $X$ at $x$} is defined as $$ \\dim_x(X) = \\min \\{\\dim(U), x\\in U\\subset X\\text{ open}\\} $$ the minimum of $\\dim(U)$ where $U$ runs over the open neighbourhoods of $x$ in $X$. \\end{enumerate}"}
+{"_id": "8357", "title": "topology-definition-equidimensional", "text": "Let $X$ be a topological space. We say that $X$ is {\\it equidimensional} if every irreducible component of $X$ has the same dimension."}
+{"_id": "8358", "title": "topology-definition-codimension", "text": "Let $X$ be a topological space. Let $Y \\subset X$ be an irreducible closed subset. The {\\it codimension} of $Y$ in $X$ is the supremum of the lengths $e$ of chains $$ Y = Y_0 \\subset Y_1 \\subset \\ldots \\subset Y_e \\subset X $$ of irreducible closed subsets in $X$ starting with $Y$. We will denote this $\\text{codim}(Y, X)$."}
+{"_id": "8359", "title": "topology-definition-catenary", "text": "Let $X$ be a topological space. We say $X$ is {\\it catenary} if for every pair of irreducible closed subsets $T \\subset T'$ we have $\\text{codim}(T, T') < \\infty$ and every maximal chain of irreducible closed subsets $$ T = T_0 \\subset T_1 \\subset \\ldots \\subset T_e = T' $$ has the same length (equal to the codimension)."}
+{"_id": "8360", "title": "topology-definition-quasi-compact", "text": "Quasi-compactness. \\begin{enumerate} \\item We say that a topological space $X$ is {\\it quasi-compact} if every open covering of $X$ has a finite refinement. \\item We say that a continuous map $f : X \\to Y$ is {\\it quasi-compact} if the inverse image $f^{-1}(V)$ of every quasi-compact open $V \\subset Y$ is quasi-compact. \\item We say a subset $Z \\subset X$ is {\\it retrocompact} if the inclusion map $Z \\to X$ is quasi-compact. \\end{enumerate}"}
+{"_id": "8361", "title": "topology-definition-locally-quasi-compact", "text": "A topological space $X$ is called {\\it locally quasi-compact}\\footnote{This may not be standard notation. Alternative notions used in the literature are: (1) Every point has some quasi-compact neighbourhood, and (2) Every point has a closed quasi-compact neighbourhood. A scheme has the property that every point has a fundamental system of open quasi-compact neighbourhoods.} if every point has a fundamental system of quasi-compact neighbourhoods."}
+{"_id": "8362", "title": "topology-definition-constructible", "text": "Let $X$ be a topological space. Let $E \\subset X$ be a subset of $X$. \\begin{enumerate} \\item We say $E$ is {\\it constructible}\\footnote{In the second edition of EGA I \\cite{EGA1-second} this was called a ``globally constructible'' set and a the terminology ``constructible'' was used for what we call a locally constructible set.} in $X$ if $E$ is a finite union of subsets of the form $U \\cap V^c$ where $U, V \\subset X$ are open and retrocompact in $X$. \\item We say $E$ is {\\it locally constructible} in $X$ if there exists an open covering $X = \\bigcup V_i$ such that each $E \\cap V_i$ is constructible in $V_i$. \\end{enumerate}"}
+{"_id": "8363", "title": "topology-definition-proper-map", "text": "Let $f : X\\to Y$ be a continuous map between topological spaces. \\begin{enumerate} \\item We say that the map $f$ is {\\it closed} iff the image of every closed subset is closed. \\item We say that the map $f$ is {\\it proper}\\footnote{This is the terminology used in \\cite{Bourbaki}. Usually this is what is called ``universally closed'' in the literature. Thus our notion of proper does not involve any separation conditions.} iff the map $Z \\times X\\to Z \\times Y$ is closed for any topological space $Z$. \\item We say that the map $f$ is {\\it quasi-proper} iff the inverse image $f^{-1}(V)$ of every quasi-compact subset $V \\subset Y$ is quasi-compact. \\item We say that $f$ is {\\it universally closed} iff the map $f': Z \\times_Y X \\to Z$ is closed for any map $g: Z \\to Y$. \\end{enumerate}"}
+{"_id": "8364", "title": "topology-definition-space-jacobson", "text": "Let $X$ be a topological space. Let $X_0$ be the set of closed points of $X$. We say that $X$ is {\\it Jacobson} if every closed subset $Z \\subset X$ is the closure of $Z \\cap X_0$."}
+{"_id": "8365", "title": "topology-definition-specialization", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$, or $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$. Notation: $x' \\leadsto x$. \\item A subset $T \\subset X$ is {\\it stable under specialization} if for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have $x \\in T$. \\item A subset $T \\subset X$ is {\\it stable under generalization} if for all $x \\in T$ and every generalization $x' \\leadsto x$ we have $x' \\in T$. \\end{enumerate}"}
+{"_id": "8366", "title": "topology-definition-lift-specializations", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. \\begin{enumerate} \\item We say that {\\it specializations lift along $f$} or that $f$ is {\\it specializing} if given $y' \\leadsto y$ in $Y$ and any $x'\\in X$ with $f(x') = y'$ there exists a specialization $x' \\leadsto x$ of $x'$ in $X$ such that $f(x) = y$. \\item We say that {\\it generalizations lift along $f$} or that $f$ is {\\it generalizing} if given $y' \\leadsto y$ in $Y$ and any $x\\in X$ with $f(x) = y$ there exists a generalization $x' \\leadsto x$ of $x$ in $X$ such that $f(x') = y'$. \\end{enumerate}"}
+{"_id": "8367", "title": "topology-definition-dimension-function", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item  Let $x, y \\in X$, $x \\not = y$. Suppose $x \\leadsto y$, that is $y$ is a specialization of $x$. We say $y$ is an {\\it immediate specialization} of $x$ if there is no $z \\in X \\setminus \\{x, y\\}$ with $x \\leadsto z$ and $z \\leadsto y$. \\item A map $\\delta : X \\to \\mathbf{Z}$ is called a {\\it dimension function}\\footnote{This is likely nonstandard notation. This notion is usually introduced only for (locally) Noetherian schemes, in which case condition (a) is implied by (b).} if \\begin{enumerate} \\item whenever $x \\leadsto y$ and $x \\not = y$ we have $\\delta(x) > \\delta(y)$, and \\item for every immediate specialization $x \\leadsto y$ in $X$ we have $\\delta(x) = \\delta(y) + 1$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "8368", "title": "topology-definition-nowhere-dense", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item Given a subset $T \\subset X$ the {\\it interior} of $T$ is the largest open subset of $X$ contained in $T$. \\item A subset $T \\subset X$ is called {\\it nowhere dense} if the closure of $T$ has empty interior. \\end{enumerate}"}
+{"_id": "8369", "title": "topology-definition-profinite", "text": "A topological space is {\\it profinite} if it is homeomorphic to a limit of a diagram of finite discrete spaces."}
+{"_id": "8370", "title": "topology-definition-spectral-space", "text": "A topological space $X$ is called {\\it spectral} if it is sober, quasi-compact, the intersection of two quasi-compact opens is quasi-compact, and the collection of quasi-compact opens forms a basis for the topology. A continuous map $f : X \\to Y$ of spectral spaces is called {\\it spectral} if the inverse image of a quasi-compact open is quasi-compact."}
+{"_id": "8371", "title": "topology-definition-extremally-disconnected", "text": "A topological space $X$ is called {\\it extremally disconnected} if the closure of every open subset of $X$ is open."}
+{"_id": "8372", "title": "topology-definition-isolated-point", "text": "Let $X$ be a topological space. We say $x \\in X$ is an {\\it isolated point} of $X$ if $\\{x\\}$ is open in $X$."}
+{"_id": "8373", "title": "topology-definition-paritition", "text": "Let $X$ be a topological space. A {\\it partition} of $X$ is a decomposition $X = \\coprod X_i$ into locally closed subsets $X_i$. The $X_i$ are called the {\\it parts} of the partition. Given two partitions of $X$ we say one {\\it refines} the other if the parts of one are unions of parts of the other."}
+{"_id": "8374", "title": "topology-definition-good-stratification", "text": "Let $X$ be a topological space. A {\\it good stratification} of $X$ is a partition $X = \\coprod X_i$ such that for all $i, j \\in I$ we have $$ X_i \\cap \\overline{X_j} \\not = \\emptyset \\Rightarrow X_i \\subset \\overline{X_j}. $$"}
+{"_id": "8375", "title": "topology-definition-stratification", "text": "Let $X$ be a topological space. A {\\it stratification} of $X$ is given by a partition $X = \\coprod_{i \\in I} X_i$ and a partial ordering on $I$ such that for each $j \\in I$ we have $$ \\overline{X_j} \\subset \\bigcup\\nolimits_{i \\leq j} X_i $$ The parts $X_i$ are called the {\\it strata} of the stratification."}
+{"_id": "8376", "title": "topology-definition-locally-finite", "text": "Let $X$ be a topological space. Let $I$ be a set and for $i \\in I$ let $E_i \\subset X$ be a subset. We say the collection $\\{E_i\\}_{i \\in I}$ is {\\it locally finite} if for all $x \\in X$ there exists an open neighbourhood $U$ of $x$ such that $\\{i \\in I | E_i \\cap U \\not = \\emptyset\\}$ is finite."}
+{"_id": "8377", "title": "topology-definition-topological-group", "text": "A {\\it topological group} is a group $G$ endowed with a topology such that multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$ and inverse $G \\to G$, $x \\mapsto x^{-1}$ are continuous. A {\\it homomorphism of topological groups} is a homomorphism of groups which is continuous."}
+{"_id": "8378", "title": "topology-definition-profinite-group", "text": "A topological group is called a {\\it profinite group} if it satisfies the equivalent conditions of Lemma \\ref{lemma-profinite-group}."}
+{"_id": "8379", "title": "topology-definition-topological-ring", "text": "A {\\it topological ring} is a ring $R$ endowed with a topology such that addition $R \\times R \\to R$, $(x, y) \\mapsto x + y$ and multiplication $R \\times R \\to R$, $(x, y) \\mapsto xy$ are continuous. A {\\it homomorphism of topological rings} is a homomorphism of rings which is continuous."}
+{"_id": "8380", "title": "topology-definition-topological-module", "text": "Let $R$ be a topological ring. A {\\it topological module} is an $R$-module $M$ endowed with a topology such that addition $M \\times M \\to M$ and scalar multiplication $R \\times M \\to M$ are continuous. A {\\it homomorphism of topological modules} is a homomorphism of modules which is continuous."}
+{"_id": "8421", "title": "hypercovering-definition-SR", "text": "Let $\\mathcal{C}$ be a category. We denote $\\text{SR}(\\mathcal{C})$ the category of {\\it semi-representable objects} defined as follows \\begin{enumerate} \\item objects are families of objects $\\{U_i\\}_{i \\in I}$, and \\item morphisms $\\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ are given by a map $\\alpha : I \\to J$ and for each $i \\in I$ a morphism $f_i : U_i \\to V_{\\alpha(i)}$ of $\\mathcal{C}$. \\end{enumerate} Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$. The category of {\\it semi-representable objects over $X$} is the category $\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$."}
+{"_id": "8422", "title": "hypercovering-definition-SR-F", "text": "Let $\\mathcal{C}$ be a category. We denote $F$ the functor {\\it which associates a presheaf to a semi-representable object}. In a formula \\begin{eqnarray*} F : \\text{SR}(\\mathcal{C}) & \\longrightarrow & \\textit{PSh}(\\mathcal{C}) \\\\ \\{U_i\\}_{i \\in I} & \\longmapsto & \\amalg_{i\\in I} h_{U_i} \\end{eqnarray*} where $h_U$ denotes the representable presheaf associated to the object $U$."}
+{"_id": "8423", "title": "hypercovering-definition-covering-SR", "text": "Let $\\mathcal{C}$ be a site. Let $f = (\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ be a morphism in the category $\\text{SR}(\\mathcal{C})$. We say that $f$ is a {\\it covering} if for every $j \\in J$ the family of morphisms $\\{U_i \\to V_j\\}_{i \\in I, \\alpha(i) = j}$ is a covering for the site $\\mathcal{C}$. Let $X$ be an object of $\\mathcal{C}$. A morphism $K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$ is a {\\it covering} if its image in $\\text{SR}(\\mathcal{C})$ is a covering."}
+{"_id": "8424", "title": "hypercovering-definition-hypercovering", "text": "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has fibre products. Let $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$. A {\\it hypercovering of $X$} is a simplicial object $K$ of $\\text{SR}(\\mathcal{C}, X)$ such that \\begin{enumerate} \\item The object $K_0$ is a covering of $X$ for the site $\\mathcal{C}$. \\item For every $n \\geq 0$ the canonical morphism $$ K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1} $$ is a covering in the sense defined above. \\end{enumerate}"}
+{"_id": "8425", "title": "hypercovering-definition-homology", "text": "Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\\textit{PSh}(\\mathcal{C})$. By the above we get a simplicial object $\\mathbf{Z}_K^\\#$ of $\\textit{Ab}(\\mathcal{C})$. We can take its associated complex of abelian presheaves $s(\\mathbf{Z}_K^\\#)$, see Simplicial, Section \\ref{simplicial-section-complexes}. The {\\it homology of $K$} is the homology of the complex of abelian sheaves $s(\\mathbf{Z}_K^\\#)$."}
+{"_id": "8426", "title": "hypercovering-definition-hypercovering-variant", "text": "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has equalizers and fibre products. Let $\\mathcal{G}$ be a presheaf of sets. A {\\it hypercovering of $\\mathcal{G}$} is a simplicial object $K$ of $\\text{SR}(\\mathcal{C})$ endowed with an augmentation $F(K) \\to \\mathcal{G}$ such that \\begin{enumerate} \\item $F(K_0) \\to \\mathcal{G}$ becomes surjective after sheafification, \\item $F(K_1) \\to F(K_0) \\times_\\mathcal{G} F(K_0)$ becomes surjective after sheafification, and \\item $F(K_{n + 1}) \\longrightarrow F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})$ for $n \\geq 1$ becomes surjective after sheafification. \\end{enumerate} We say that a simplicial object $K$ of $\\text{SR}(\\mathcal{C})$ is a {\\it hypercovering} if $K$ is a hypercovering of the final object $*$ of $\\textit{PSh}(\\mathcal{C})$."}
+{"_id": "8481", "title": "algebraic-definition-representable-by-algebraic-space", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. A category fibred in groupoids $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is called {\\it representable by an algebraic space over $S$} if there exists an algebraic space $F$ over $S$ and an equivalence $j : \\mathcal{X} \\to \\mathcal{S}_F$ of categories over $(\\Sch/S)_{fppf}$."}
+{"_id": "8482", "title": "algebraic-definition-representable-by-algebraic-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. A $1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ is called {\\it representable by algebraic spaces} if for any $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $y : (\\Sch/U)_{fppf} \\to \\mathcal{Y}$ the category fibred in groupoids $$ (\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X} $$ over $(\\Sch/U)_{fppf}$ is representable by an algebraic space over $U$."}
+{"_id": "8483", "title": "algebraic-definition-relative-representable-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $f$ is representable by algebraic spaces. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which \\begin{enumerate} \\item is preserved under any base change, and \\item is fppf local on the base, see Descent on Spaces, Definition \\ref{spaces-descent-definition-property-morphisms-local}. \\end{enumerate} In this case we say that $f$ has {\\it property $\\mathcal{P}$} if for every $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $y \\in \\mathcal{Y}_U$ the resulting morphism of algebraic spaces $f_y : F_y \\to U$, see diagram (\\ref{equation-representable-by-algebraic-spaces}), has property $\\mathcal{P}$."}
+{"_id": "8484", "title": "algebraic-definition-algebraic-stack", "text": "Let $S$ be a base scheme contained in $\\Sch_{fppf}$. An {\\it algebraic stack over $S$} is a category $$ p : \\mathcal{X} \\to (\\Sch/S)_{fppf} $$ over $(\\Sch/S)_{fppf}$ with the following properties: \\begin{enumerate} \\item The category $\\mathcal{X}$ is a stack in groupoids over $(\\Sch/S)_{fppf}$. \\item The diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is representable by algebraic spaces. \\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$ and a $1$-morphism $(\\Sch/U)_{fppf} \\to \\mathcal{X}$ which is surjective and smooth\\footnote{In future chapters we will denote this simply $U \\to \\mathcal{X}$ as is customary in the literature. Another good alternative would be to formulate this condition as the existence of a representable category fibred in groupoids $\\mathcal{U}$ and a surjective smooth $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$.}. \\end{enumerate}"}
+{"_id": "8485", "title": "algebraic-definition-deligne-mumford", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$ be an algebraic stack over $S$. We say $\\mathcal{X}$ is a {\\it Deligne-Mumford stack} if there exists a scheme $U$ and a surjective \\'etale morphism $(\\Sch/U)_{fppf} \\to \\mathcal{X}$."}
+{"_id": "8486", "title": "algebraic-definition-morphism-algebraic-stacks", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. The {\\it $2$-category of algebraic stacks over $S$} is the sub $2$-category of the $2$-category of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ (see Categories, Definition \\ref{categories-definition-categories-fibred-in-groupoids-over-C}) defined as follows: \\begin{enumerate} \\item Its objects are those categories fibred in groupoids over $(\\Sch/S)_{fppf}$ which are algebraic stacks over $S$. \\item Its $1$-morphisms $f : \\mathcal{X} \\to \\mathcal{Y}$ are any functors of categories over $(\\Sch/S)_{fppf}$, as in Categories, Definition \\ref{categories-definition-categories-over-C}. \\item Its $2$-morphisms are transformations between functors over $(\\Sch/S)_{fppf}$, as in Categories, Definition \\ref{categories-definition-categories-over-C}. \\end{enumerate}"}
+{"_id": "8487", "title": "algebraic-definition-smooth-groupoid", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. We say $(U, R, s, t, c)$ is a {\\it smooth groupoid}\\footnote{This terminology might be a bit confusing: it does not imply that $[U/R]$ is smooth over anything.} if $s, t : R \\to U$ are smooth morphisms of algebraic spaces."}
+{"_id": "8488", "title": "algebraic-definition-presentation", "text": "Let $\\mathcal{X}$ be an algebraic stack over $S$. A {\\it presentation} of $\\mathcal{X}$ is given by a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces over $S$, and an equivalence $f : [U/R] \\to \\mathcal{X}$."}
+{"_id": "8489", "title": "algebraic-definition-viewed-as", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. If $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is an algebraic stack over $S$, then $\\mathcal{X}$ {\\it viewed as an algebraic stack over $S'$} is the algebraic stack $$ \\mathcal{X} \\longrightarrow (\\Sch/S')_{fppf} $$ gotten by applying construction A of Lemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme} to $\\mathcal{X}$."}
+{"_id": "8490", "title": "algebraic-definition-change-of-base", "text": "Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. Let $\\mathcal{X}'$ be an algebraic stack over $S'$. The {\\it change of base of $\\mathcal{X}'$} is the algebraic space $\\mathcal{X}'_S$ over $S$ described above."}
+{"_id": "8646", "title": "sites-definition-presheaves-sets", "text": "A {\\it presheaf of sets} on $\\mathcal{C}$ is a contravariant functor from $\\mathcal{C}$ to $\\textit{Sets}$. {\\it Morphisms of presheaves} are transformations of functors. The category of presheaves of sets is denoted $\\textit{PSh}(\\mathcal{C})$."}
+{"_id": "8647", "title": "sites-definition-presheaf", "text": "Let $\\mathcal{C}$, $\\mathcal{A}$ be categories. A {\\it presheaf} $\\mathcal{F}$ on $\\mathcal{C}$ with values in $\\mathcal{A}$ is a contravariant functor from $\\mathcal{C}$ to $\\mathcal{A}$, i.e., $\\mathcal{F} : \\mathcal{C}^{opp} \\to \\mathcal{A}$. A {\\it morphism} of presheaves $\\mathcal{F} \\to \\mathcal{G}$ on $\\mathcal{C}$ with values in $\\mathcal{A}$ is a transformation of functors from $\\mathcal{F}$ to $\\mathcal{G}$."}
+{"_id": "8648", "title": "sites-definition-presheaves-injective-surjective", "text": "Let $\\mathcal{C}$ be a category, and let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of presheaves of sets. \\begin{enumerate} \\item We say that $\\varphi$ is {\\it injective} if for every object $U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective. \\item We say that $\\varphi$ is {\\it surjective} if for every object $U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective. \\end{enumerate}"}
+{"_id": "8649", "title": "sites-definition-sub-presheaf", "text": "We say $\\mathcal{F}$ is a {\\it subpresheaf} of $\\mathcal{G}$ if for every object $U \\in \\Ob(\\mathcal{C})$ the set $\\mathcal{F}(U)$ is a subset of $\\mathcal{G}(U)$, compatibly with the restriction mappings."}
+{"_id": "8650", "title": "sites-definition-image", "text": "Notation as in Lemma \\ref{lemma-image}. We say that $\\mathcal{G}'$ is the {\\it image of $\\varphi$}."}
+{"_id": "8651", "title": "sites-definition-family-morphisms-fixed-target", "text": "Let $\\mathcal{C}$ be a category, see Conventions, Section \\ref{conventions-section-categories}. A {\\it family of morphisms with fixed target} in $\\mathcal{C}$ is given by an object $U \\in \\Ob(\\mathcal{C})$, a set $I$ and for each $i\\in I$ a morphism $U_i \\to U$ of $\\mathcal{C}$ with target $U$. We use the notation $\\{U_i \\to U\\}_{i\\in I}$ to indicate this."}
+{"_id": "8652", "title": "sites-definition-site", "text": "A {\\it site}\\footnote{This notation differs from that of \\cite{SGA4}, as explained in the introduction.} is given by a category $\\mathcal{C}$ and a set $\\text{Cov}(\\mathcal{C})$ of families of morphisms with fixed target $\\{U_i \\to U\\}_{i \\in I}$, called {\\it coverings of $\\mathcal{C}$}, satisfying the following axioms \\begin{enumerate} \\item If $V \\to U$ is an isomorphism then $\\{V \\to U\\} \\in \\text{Cov}(\\mathcal{C})$. \\item If $\\{U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ and for each $i$ we have $\\{V_{ij} \\to U_i\\}_{j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$, then $\\{V_{ij} \\to U\\}_{i \\in I, j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$. \\item If $\\{U_i \\to U\\}_{i\\in I}\\in \\text{Cov}(\\mathcal{C})$ and $V \\to U$ is a morphism of $\\mathcal{C}$ then $U_i \\times_U V$ exists for all $i$ and $\\{U_i \\times_U V \\to V \\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "8653", "title": "sites-definition-sheaf-sets", "text": "Let $\\mathcal{C}$ be a site, and let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. We say $\\mathcal{F}$ is a {\\it sheaf} if for every covering $\\{U_i \\to U\\}_{i \\in I} \\in \\text{Cov}(\\mathcal{C})$ the diagram \\begin{equation} \\label{equation-sheaf-condition} \\xymatrix{ \\mathcal{F}(U) \\ar[r] & \\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i) \\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*} & \\prod\\nolimits_{(i_0, i_1) \\in I \\times I} \\mathcal{F}(U_{i_0} \\times_U U_{i_1}) } \\end{equation} represents the first arrow as the equalizer of $\\text{pr}_0^*$ and $\\text{pr}_1^*$."}
+{"_id": "8654", "title": "sites-definition-category-sheaves-sets", "text": "The category {\\it $\\Sh(\\mathcal{C})$} of sheaves of sets is the full subcategory of the category $\\textit{PSh}(\\mathcal{C})$ whose objects are the sheaves of sets."}
+{"_id": "8655", "title": "sites-definition-sheaf", "text": "Let $\\mathcal{C}$ be a site, let $\\mathcal{A}$ be a category and let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$ with values in $\\mathcal{A}$. We say that $\\mathcal{F}$ is a {\\it sheaf} if for all objects $X$ of $\\mathcal{A}$ the presheaf of sets $\\mathcal{F}_X$ (defined above) is a sheaf."}
+{"_id": "8656", "title": "sites-definition-morphism-coverings", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i\\in I}$ be a family of morphisms of $\\mathcal{C}$ with fixed target. Let $\\mathcal{V} = \\{V_j \\to V\\}_{j\\in J}$ be another. \\begin{enumerate} \\item A {\\it morphism of families of maps with fixed target of $\\mathcal{C}$ from  $\\mathcal{U}$ to $\\mathcal{V}$}, or simply a {\\it morphism from $\\mathcal{U}$ to $\\mathcal{V}$} is given by a morphism $U \\to V$, a map of sets $\\alpha : I \\to J$ and for each $i\\in I$ a morphism $U_i \\to V_{\\alpha(i)}$ such that the diagram $$ \\xymatrix{ U_i \\ar[r] \\ar[d] & V_{\\alpha(i)} \\ar[d] \\\\ U \\ar[r] & V } $$ is commutative. \\item In the special case that $U = V$ and $U \\to V$ is the identity we call $\\mathcal{U}$ a {\\it refinement} of the family $\\mathcal{V}$. \\end{enumerate}"}
+{"_id": "8657", "title": "sites-definition-combinatorial-tautological", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and $\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms with fixed target. \\begin{enumerate} \\item  We say $\\mathcal{U}$ and $\\mathcal{V}$ are {\\it combinatorially equivalent} if there exist maps $\\alpha : I \\to J$ and $\\beta : J\\to I$ such that $\\varphi_i = \\psi_{\\alpha(i)}$ and $\\psi_j = \\varphi_{\\beta(j)}$. \\item We say $\\mathcal{U}$ and $\\mathcal{V}$ are {\\it tautologically equivalent} if there exist maps $\\alpha : I \\to J$ and $\\beta : J\\to I$ and for all $i\\in I$ and $j \\in J$ commutative diagrams $$ \\xymatrix{ U_i \\ar[rd] \\ar[rr] & & V_{\\alpha(i)} \\ar[ld] & & V_j \\ar[rd] \\ar[rr] & & U_{\\beta(j)} \\ar[ld] \\\\ & U & & & & U & } $$ with isomorphisms as horizontal arrows. \\end{enumerate}"}
+{"_id": "8658", "title": "sites-definition-separated", "text": "We say that a presheaf of sets $\\mathcal{F}$ on a site $\\mathcal{C}$ is {\\it separated} if, for all coverings of $\\{U_i \\rightarrow U\\}$, the map $\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$ is injective."}
+{"_id": "8659", "title": "sites-definition-associated-sheaf", "text": "Let $\\mathcal{C}$ be a site and let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. The sheaf $\\mathcal{F}^\\# := \\mathcal{F}^{++}$ together with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ is called the {\\it sheaf associated to $\\mathcal{F}$}."}
+{"_id": "8660", "title": "sites-definition-sheaves-injective-surjective", "text": "Let $\\mathcal{C}$ be a site, and let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of sheaves of sets. \\begin{enumerate} \\item We say that $\\varphi$ is {\\it injective} if for every object $U$ of $\\mathcal{C}$ the map $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective. \\item We say that $\\varphi$ is {\\it surjective} if for every object $U$ of $\\mathcal{C}$ and every section $s\\in \\mathcal{G}(U)$ there exists a covering $\\{U_i \\to U\\}$ such that for all $i$ the restriction $s|_{U_i}$ is in the image of $\\varphi : \\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$. \\end{enumerate}"}
+{"_id": "8661", "title": "sites-definition-universal-effective-epimorphisms", "text": "Let $\\mathcal{C}$ be a category. We say that a family $\\{U_i \\to U\\}_{i \\in I}$ is an {\\it effective epimorphism} if all the morphisms $U_i \\to U$ are representable (see Categories, Definition \\ref{categories-definition-representable-morphism}), and for any $X\\in \\Ob(\\mathcal{C})$ the sequence $$ \\xymatrix{ \\Mor_\\mathcal{C}(U, X) \\ar[r] & \\prod\\nolimits_{i \\in I} \\Mor_\\mathcal{C}(U_i, X) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod\\nolimits_{(i, j) \\in I^2} \\Mor_\\mathcal{C}(U_i \\times_U U_j, X) } $$ is an equalizer diagram. We say that a family $\\{U_i \\to U\\}$ is a {\\it universal effective epimorphism} if for any morphism $V \\to U$ the base change $\\{U_i \\times_U V \\to V\\}$ is an effective epimorphism."}
+{"_id": "8662", "title": "sites-definition-weaker-than-canonical", "text": "We say that the topology on a site $\\mathcal{C}$ is {\\it weaker than the canonical topology}, or that the topology is {\\it subcanonical} if all the coverings of $\\mathcal{C}$ are universal effective epimorphisms."}
+{"_id": "8663", "title": "sites-definition-representable-sheaf", "text": "Let $\\mathcal{C}$ be a site whose topology is subcanonical. The Yoneda embedding $h$ (see Categories, Section \\ref{categories-section-opposite}) presents $\\mathcal{C}$ as a full subcategory of the category of sheaves of $\\mathcal{C}$. In this case we call sheaves of the form $h_U$ with $U \\in \\Ob(\\mathcal{C})$ {\\it representable sheaves} on $\\mathcal{C}$. Notation: Sometimes, the representable sheaf $h_U$ associated to $U$ is denoted {\\it $\\underline{U}$}."}
+{"_id": "8664", "title": "sites-definition-continuous", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. A functor $u : \\mathcal{C} \\to \\mathcal{D}$ is called {\\it continuous} if for every $\\{V_i \\to V\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ we have the following \\begin{enumerate} \\item $\\{u(V_i) \\to u(V)\\}_{i\\in I}$ is in $\\text{Cov}(\\mathcal{D})$, and \\item for any morphism $T \\to V$ in $\\mathcal{C}$ the morphism $u(T \\times_V V_i) \\to u(T) \\times_{u(V)} u(V_i)$ is an isomorphism. \\end{enumerate}"}
+{"_id": "8665", "title": "sites-definition-morphism-sites", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. A {\\it morphism of sites} $f : \\mathcal{D} \\to \\mathcal{C}$ is given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ such that the functor $u_s$ is exact."}
+{"_id": "8666", "title": "sites-definition-composition-morphisms-sites", "text": "Let $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let $f : \\mathcal{C}_1 \\to \\mathcal{C}_2$ and $g : \\mathcal{C}_2 \\to \\mathcal{C}_3$ be morphisms of sites given by continuous functors $u : \\mathcal{C}_2 \\to \\mathcal{C}_1$ and $v : \\mathcal{C}_3 \\to \\mathcal{C}_2$. The {\\it composition} $g \\circ f$ is the morphism of sites corresponding to the functor $u \\circ v$."}
+{"_id": "8667", "title": "sites-definition-topos", "text": "A {\\it topos} is the category $\\Sh(\\mathcal{C})$ of sheaves on a site $\\mathcal{C}$. \\begin{enumerate} \\item Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. A {\\it morphism of topoi} $f$ from $\\Sh(\\mathcal{D})$ to $\\Sh(\\mathcal{C})$ is given by a pair of functors $f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ and $f^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ such that \\begin{enumerate} \\item we have $$ \\Mor_{\\Sh(\\mathcal{D})}(f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, f_*\\mathcal{F}) $$ bifunctorially, and \\item the functor $f^{-1}$ commutes with finite limits, i.e., is left exact. \\end{enumerate} \\item Let $\\mathcal{C}$, $\\mathcal{D}$, $\\mathcal{E}$ be sites. Given morphisms of topoi $f :\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ and $g :\\Sh(\\mathcal{E}) \\to \\Sh(\\mathcal{D})$ the {\\it composition $f\\circ g$} is the morphism of topoi defined by the functors $(f \\circ g)_* = f_* \\circ g_*$ and $(f \\circ g)^{-1} = g^{-1} \\circ f^{-1}$. \\end{enumerate}"}
+{"_id": "8668", "title": "sites-definition-quasi-compact", "text": "Let $\\mathcal{C}$ be a site. An object $U$ of $\\mathcal{C}$ is {\\it quasi-compact} if given a covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ there exists another covering $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ and a morphism $\\mathcal{V} \\to \\mathcal{U}$ of families of maps with fixed target given by $\\text{id} : U \\to U$, $\\alpha : J \\to I$, and $V_j \\to U_{\\alpha(j)}$ (see Definition \\ref{definition-morphism-coverings}) such that the image of $\\alpha$ is a finite subset of $I$."}
+{"_id": "8669", "title": "sites-definition-quasi-compact-topos", "text": "An object $\\mathcal{F}$ of a topos $\\Sh(\\mathcal{C})$ is {\\it quasi-compact} if for any surjective map $\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$ of $\\Sh(\\mathcal{C})$ there exists a finite subset $J \\subset I$ such that $\\coprod_{i \\in J} \\mathcal{F}_i \\to \\mathcal{F}$ is surjective. A topos $\\Sh(\\mathcal{C})$ is said to be {\\it quasi-compact} if its final object $*$ is a quasi-compact object."}
+{"_id": "8670", "title": "sites-definition-cocontinuous", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. The functor $u$ is called {\\it cocontinuous} if for every $U \\in \\Ob(\\mathcal{C})$ and every covering $\\{V_j \\to u(U)\\}_{j \\in J}$ of $\\mathcal{D}$ there exists a covering $\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ such that the family of maps $\\{u(U_i) \\to u(U)\\}_{i \\in I}$ refines the covering $\\{V_j \\to u(U)\\}_{j \\in J}$."}
+{"_id": "8671", "title": "sites-definition-localize", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item The site $\\mathcal{C}/U$ is called the {\\it localization of the site $\\mathcal{C}$ at the object $U$}. \\item The morphism of topoi $j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$ is called the {\\it localization morphism}. \\item The functor $j_{U*}$ is called the {\\it direct image functor}. \\item For a sheaf $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf $j_U^{-1}\\mathcal{F}$ is called the {\\it restriction of $\\mathcal{F}$ to $\\mathcal{C}/U$}. \\item For a sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$ the sheaf $j_{U!}\\mathcal{G}$ is called the {\\it extension of $\\mathcal{G}$ by the empty set}. \\end{enumerate}"}
+{"_id": "8672", "title": "sites-definition-special-cocontinuous-functor", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. A {\\it special cocontinuous functor $u$ from $\\mathcal{C}$ to $\\mathcal{D}$} is a cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying the assumptions and conclusions of Lemma \\ref{lemma-equivalence}."}
+{"_id": "8673", "title": "sites-definition-localize-topos", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$. \\begin{enumerate} \\item The topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ is called the {\\it localization of the topos $\\Sh(\\mathcal{C})$ at $\\mathcal{F}$}. \\item The morphism of topoi $j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$ of Lemma \\ref{lemma-localize-topos} is called the {\\it localization morphism}. \\end{enumerate}"}
+{"_id": "8674", "title": "sites-definition-point-topos", "text": "Let $\\mathcal{C}$ be a site. A {\\it point of the topos $\\Sh(\\mathcal{C})$} is a morphism of topoi $p$ from $\\Sh(pt)$ to $\\Sh(\\mathcal{C})$."}
+{"_id": "8675", "title": "sites-definition-point", "text": "Let $\\mathcal{C}$ be a site. A {\\it point $p$ of the site $\\mathcal{C}$} is given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$ such that \\begin{enumerate} \\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ the map $\\coprod u(U_i) \\to u(U)$ is surjective. \\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ and every morphism $V \\to U$ the maps $u(U_i \\times_U V) \\to u(U_i) \\times_{u(U)} u(V)$ are bijective. \\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\mapsto \\mathcal{F}_p$ is left exact. \\end{enumerate}"}
+{"_id": "8676", "title": "sites-definition-pushforward-point", "text": "Let $p$ be a point of the site $\\mathcal{C}$ given by the functor $u$. For a set $E$ we define $p_*E = u^sE$ the sheaf described in Lemma \\ref{lemma-point-pushforward-sheaf} above. We sometimes call this a {\\it skyscraper sheaf}."}
+{"_id": "8677", "title": "sites-definition-2-morphism-topoi", "text": "Let $f, g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be two morphisms of topoi. A {\\it 2-morphism from $f$ to $g$} is given by a transformation of functors $t : f_* \\to g_*$."}
+{"_id": "8678", "title": "sites-definition-morphism-points", "text": "Let $\\mathcal{C}$ be a site. Let $p, p'$ be points of $\\mathcal{C}$ given by functors $u, u' : \\mathcal{C} \\to \\textit{Sets}$. A {\\it morphism $f : p \\to p'$} is given by a transformation of functors $$ f_u : u' \\to u. $$"}
+{"_id": "8679", "title": "sites-definition-enough-points", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item A family of points $\\{p_i\\}_{i\\in I}$ is called {\\it conservative} if every map of sheaves $\\phi : \\mathcal{F} \\to \\mathcal{G}$ which is an isomorphism on all the fibres $\\mathcal{F}_{p_i} \\to \\mathcal{G}_{p_i}$ is an isomorphism. \\item  We say that $\\mathcal{C}$ {\\it has enough points} if there exists a conservative family of points. \\end{enumerate}"}
+{"_id": "8680", "title": "sites-definition-w-contractible", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item We say an object $U$ of $\\mathcal{C}$ is {\\it weakly contractible} if the equivalent conditions of Lemma \\ref{lemma-w-contractible} hold. \\item We say a site has {\\it enough weakly contractible objects} if every object $U$ of $\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$ with $U_i$ weakly contractible for all $i$. \\item More generally, if $P$ is a property of objects of $\\mathcal{C}$ we say that $\\mathcal{C}$ has {\\it enough $P$ objects} if every object $U$ of $\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$ such that $U_i$ has $P$ for all $i$. \\end{enumerate}"}
+{"_id": "8681", "title": "sites-definition-empty", "text": "Let $\\mathcal{C}$ be a site. We say an object $U$ of $\\mathcal{C}$ is {\\it sheaf theoretically empty} if $\\emptyset^\\# \\to h_U^\\#$ is an isomorphism of sheaves."}
+{"_id": "8682", "title": "sites-definition-almost-cocontinuous", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. We say $u$ is {\\it almost cocontinuous} if for every object $U$ of $\\mathcal{C}$ and every covering $\\{V_j \\to u(U)\\}_{j \\in J}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that for each $i$ in $I$ we have at least one of the following two conditions \\begin{enumerate} \\item $u(U_i)$ is sheaf theoretically empty, or \\item the morphism $u(U_i) \\to u(U)$ factors through $V_j$ for some $j \\in J$. \\end{enumerate}"}
+{"_id": "8683", "title": "sites-definition-embedding", "text": "Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. A morphism of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is called an {\\it embedding} if $f_*$ is fully faithful."}
+{"_id": "8684", "title": "sites-definition-subtopos", "text": "Let $\\mathcal{C}$ be a site. A strictly full subcategory $E \\subset \\Sh(\\mathcal{C})$ is a {\\it subtopos} if there exists an embedding of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ such that $E$ is equal to the essential image of the functor $f_*$."}
+{"_id": "8685", "title": "sites-definition-open-subtopos", "text": "Let $\\mathcal{C}$ be a site. A strictly full subcategory $E \\subset \\Sh(\\mathcal{C})$ is an {\\it open subtopos} if there exists a subsheaf $\\mathcal{F}$ of the final object of $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos $\\Sh(\\mathcal{C})/\\mathcal{F}$ described in Lemma \\ref{lemma-open-subtopos}."}
+{"_id": "8686", "title": "sites-definition-closed-subtopos", "text": "Let $\\mathcal{C}$ be a site. A strictly full subcategory $E \\subset \\Sh(\\mathcal{C})$ is an {\\it closed subtopos} if there exists a subsheaf $\\mathcal{F}$ of the final object of $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos described in Lemma \\ref{lemma-closed-subtopos}."}
+{"_id": "8687", "title": "sites-definition-immersion-topoi", "text": "Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi. \\begin{enumerate} \\item We say $f$ is an {\\it open immersion} if $f$ is an embedding and the essential image of $f_*$ is an open subtopos. \\item We say $f$ is a {\\it closed immersion} if $f$ is an embedding and the essential image of $f_*$ is a closed subtopos. \\end{enumerate}"}
+{"_id": "8688", "title": "sites-definition-pushforward-algebraic-structures", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by a functor $u : \\mathcal{C} \\to \\mathcal{D}$. We define the {\\it pushforward} functor for presheaves of algebraic structures by the rule $u^p\\mathcal{F}(U) = \\mathcal{F}(uU)$, and for sheaves of algebraic structures by the same rule, namely $f_*\\mathcal{F}(U) = \\mathcal{F}(uU)$."}
+{"_id": "8689", "title": "sites-definition-global-sections", "text": "The {\\it global sections} of a presheaf of sets $\\mathcal{F}$ over a site $\\mathcal{C}$ is the set $$ \\Gamma(\\mathcal{C}, \\mathcal{F}) = \\Mor_{\\textit{PSh}(\\mathcal{C})}(*, \\mathcal{F}) $$ where $*$ is the final object in the category of presheaves on $\\mathcal{C}$, i.e., the presheaf which associates to every object a singleton."}
+{"_id": "8690", "title": "sites-definition-sieve", "text": "Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$. A {\\it sieve $S$ on $U$} is a subpresheaf $S \\subset h_U$."}
+{"_id": "8691", "title": "sites-definition-sieve-generated", "text": "Let $\\mathcal{C}$ be a category. Given a family of morphisms $\\{f_i : U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ with target $U$ we say the sieve $S$ on $U$ described in Lemma \\ref{lemma-sieves-set} part (\\ref{item-sieve-generated}) is the {\\it sieve  on $U$ generated by the morphisms $f_i$}."}
+{"_id": "8692", "title": "sites-definition-pullback-sieve", "text": "Let $\\mathcal{C}$ be a category. Let $f : V \\to U$ be a morphism of $\\mathcal{C}$. Let $S \\subset h_U$ be a sieve. We define the {\\it pullback of $S$ by $f$} to be the sieve $S \\times_U V$ of $V$ defined by the rule $$ (\\alpha : T \\to V) \\in (S \\times_U V)(T) \\Leftrightarrow (f \\circ \\alpha : T \\to U) \\in S(T) $$"}
+{"_id": "8693", "title": "sites-definition-topology", "text": "Let $\\mathcal{C}$ be a category. A {\\it topology on $\\mathcal{C}$} is given by a rule which assigns to every $U \\in \\Ob(\\mathcal{C})$ a subset $J(U)$ of the set of all sieves on $U$ satisfying the following conditions \\begin{enumerate} \\item For every morphism $f : V \\to U$ in $\\mathcal{C}$, and every element $S \\in J(U)$ the pullback $S \\times_U V$ is an element of $J(V)$. \\item If $S$ and $S'$ are sieves on $U \\in \\Ob(\\mathcal{C})$, if $S \\in J(U)$, and if for all $f \\in S(V)$ the pullback $S' \\times_U V$ belongs to $J(V)$, then $S'$ belongs to $J(U)$. \\item For every $U \\in \\Ob(\\mathcal{C})$ the maximal sieve $S = h_U$ belongs to $J(U)$. \\end{enumerate}"}
+{"_id": "8694", "title": "sites-definition-finer", "text": "Let $\\mathcal{C}$ be a category. Let $J$, $J'$ be two topologies on $\\mathcal{C}$. We say that $J$ is {\\it finer} or {\\it stronger} than $J'$ if and only if for every object $U$ of $\\mathcal{C}$ we have $J'(U) \\subset J(U)$. In this case we also say that $J'$ is {\\it coarser} or {\\it weaker} than $J$."}
+{"_id": "8695", "title": "sites-definition-sheaf-sets-topology", "text": "Let $\\mathcal{C}$ be a category endowed with a topology $J$. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. We say that $\\mathcal{F}$ is a {\\it sheaf} on $\\mathcal{C}$ if for every $U \\in \\Ob(\\mathcal{C})$ and for every covering sieve $S$ of $U$ the canonical map $$ \\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F}) \\longrightarrow \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F}) $$ is bijective."}
+{"_id": "8696", "title": "sites-definition-canonical-topology", "text": "Let $\\mathcal{C}$ be a category. The finest topology on $\\mathcal{C}$ such that all representable presheaves are sheaves, see Lemma \\ref{lemma-topology-presheaves-sheaves}, is called the {\\it canonical topology} of $\\mathcal{C}$."}
+{"_id": "8697", "title": "sites-definition-topology-associated-site", "text": "Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$. The {\\it topology associated to $\\mathcal{C}$} is the topology $J$ constructed in Lemma \\ref{lemma-site-gives-topology} above."}
+{"_id": "8698", "title": "sites-definition-presheaf-separated-topology", "text": "Let $\\mathcal{C}$ be a category. Let $J$ be a topology on $\\mathcal{C}$. We say that a presheaf of sets $\\mathcal{F}$ is {\\it separated} if for every object $U$ and every covering sieve $S$ on $U$ the canonical map $\\mathcal{F}(U) \\to \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$ is injective."}
+{"_id": "8699", "title": "sites-definition-associated-sheaf-topology", "text": "Let $\\mathcal{C}$ be a category endowed with a topology $J$. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. The sheaf $\\mathcal{F}^\\# := LL\\mathcal{F}$ together with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ is called the {\\it sheaf associated to $\\mathcal{F}$}."}
+{"_id": "8700", "title": "sites-definition-point-topology", "text": "Let $\\mathcal{C}$ be a category. Let $J$ be a topology on $\\mathcal{C}$. A {\\it point $p$} of the topology is given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$ such that \\begin{enumerate} \\item For every covering sieve $S$ on $U$ the map $S_p \\to (h_U)_p$ is surjective. \\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$, $\\mathcal{F} \\to \\mathcal{F}_p$ is exact. \\end{enumerate}"}
+{"_id": "8846", "title": "more-etale-definition-f-shriek-separated", "text": "Let $f : X \\to Y$ be a morphism of schemes which is separated (!) and locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. The subsheaf $f_!\\mathcal{F} \\subset f_*\\mathcal{F}$ constructed in Lemma \\ref{lemma-f-shriek-separated} is called the {\\it direct image with compact support}."}
+{"_id": "8847", "title": "more-etale-definition-compact-support", "text": "Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. We let $H^0_c(X, \\mathcal{F}) \\subset H^0(X, \\mathcal{F})$ be the set of sections whose support is proper over $k$. Elements of $H^0_c(X, \\mathcal{F})$ are called {\\it sections with compact support}."}
+{"_id": "8848", "title": "more-etale-definition-f-shriek-lqf", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. We define the {\\it direct image with compact support} to be the functor $$ f_! : \\textit{Ab}(X_\\etale) \\longrightarrow \\textit{Ab}(Y_\\etale) $$ defined by the formula $f_!\\mathcal{F} = (f_{p!}\\mathcal{F})^\\#$, i.e., $f_!\\mathcal{F}$ is the sheafification of the presheaf $f_{p!}\\mathcal{F}$ constructed above."}
+{"_id": "8913", "title": "stacks-properties-definition-points", "text": "Let $\\mathcal{X}$ be an algebraic stack. A {\\it point} of $\\mathcal{X}$ is an equivalence class of morphisms from spectra of fields into $\\mathcal{X}$. The set of points of $\\mathcal{X}$ is denoted $|\\mathcal{X}|$."}
+{"_id": "8914", "title": "stacks-properties-definition-topological-space", "text": "Let $\\mathcal{X}$ be an algebraic stack. The underlying {\\it topological space} of $\\mathcal{X}$ is the set of points $|\\mathcal{X}|$ endowed with the topology constructed in Lemma \\ref{lemma-topology-points}."}
+{"_id": "8915", "title": "stacks-properties-definition-surjective", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is {\\it surjective} if the map $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ of associated topological spaces is surjective."}
+{"_id": "8916", "title": "stacks-properties-definition-quasi-compact", "text": "Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is {\\it quasi-compact} if and only if $|\\mathcal{X}|$ is quasi-compact."}
+{"_id": "8917", "title": "stacks-properties-definition-type-property", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{P}$ be a property of schemes which is local in the smooth topology. We say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$} if any of the equivalent conditions of Lemma \\ref{lemma-type-property} hold."}
+{"_id": "8918", "title": "stacks-properties-definition-property-at-point", "text": "Let $\\mathcal{P}$ be a property of germs of schemes which is smooth local. Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. We say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$ at $x$} if any of the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} holds."}
+{"_id": "8919", "title": "stacks-properties-definition-monomorphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a {\\it monomorphism} if it is representable by algebraic spaces and a monomorphism in the sense of Section \\ref{section-properties-morphisms}."}
+{"_id": "8920", "title": "stacks-properties-definition-immersion", "text": "Immersions. \\begin{enumerate} \\item A morphism of algebraic stacks is called an {\\it open immersion} if it is representable, and an open immersion in the sense of Section \\ref{section-properties-morphisms}. \\item A morphism of algebraic stacks is called a {\\it closed immersion} if it is representable, and a closed immersion in the sense of Section \\ref{section-properties-morphisms}. \\item A morphism of algebraic stacks is called an {\\it immersion} if it is representable, and an immersion in the sense of Section \\ref{section-properties-morphisms}. \\end{enumerate}"}
+{"_id": "8921", "title": "stacks-properties-definition-substacks", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item An {\\it open substack} of $\\mathcal{X}$ is a strictly full subcategory $\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic stack and $\\mathcal{X}' \\to \\mathcal{X}$ is an open immersion. \\item A {\\it closed substack} of $\\mathcal{X}$ is a strictly full subcategory $\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic stack and $\\mathcal{X}' \\to \\mathcal{X}$ is a closed immersion. \\item A {\\it locally closed substack} of $\\mathcal{X}$ is a strictly full subcategory $\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic stack and $\\mathcal{X}' \\to \\mathcal{X}$ is an immersion. \\end{enumerate}"}
+{"_id": "8922", "title": "stacks-properties-definition-reduced-induced-stack", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $Z \\subset |\\mathcal{X}|$ be a closed subset. An {\\it algebraic stack structure on $Z$} is given by a closed substack $\\mathcal{Z}$ of $\\mathcal{X}$ with $|\\mathcal{Z}|$ equal to $Z$. The {\\it reduced induced algebraic stack structure} on $Z$ is the one constructed in Lemma \\ref{lemma-reduced-closed-substack}. The {\\it reduction $\\mathcal{X}_{red}$ of $\\mathcal{X}$} is the reduced induced algebraic stack structure on $|\\mathcal{X}|$."}
+{"_id": "8923", "title": "stacks-properties-definition-residual-gerbe", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. \\begin{enumerate} \\item We say the {\\it residual gerbe of $\\mathcal{X}$ at $x$ exists} if the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-residual-gerbe} hold. \\item If the residual gerbe of $\\mathcal{X}$ at $x$ exists, then the {\\it residual gerbe of $\\mathcal{X}$ at $x$}\\footnote{This clashes with \\cite{LM-B} in spirit, but not in fact. Namely, in Chapter 11 they associate to any point on any quasi-separated algebraic stack a gerbe (not necessarily algebraic) which they call the residual gerbe. We will see in Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-every-point-residual-gerbe} that on a quasi-separated algebraic stack every point has a residual gerbe in our sense which is then equivalent to theirs. For more information on this topic see \\cite[Appendix B]{rydh_etale_devissage}.} is the strictly full subcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$ constructed in Lemma \\ref{lemma-residual-gerbe}. \\end{enumerate}"}
+{"_id": "8924", "title": "stacks-properties-definition-dimension-at-point", "text": "Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$. Let $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$. Let $[U/R] \\to \\mathcal{X}$ be a presentation (Algebraic Stacks, Definition \\ref{algebraic-definition-presentation}) where $U$ is a scheme and let $u \\in U$ be a point that maps to $x$. We define the {\\it dimension of $\\mathcal{X}$ at $x$} to be the element $\\dim_x(\\mathcal{X}) \\in \\mathbf{Z} \\cup \\infty$ such that  $$ \\dim_x(\\mathcal{X}) = \\dim_u(U)-\\dim_{e(u)}(R_u). $$ with notation as in Lemma \\ref{lemma-dimension-at-point-well-defined}."}
+{"_id": "8925", "title": "stacks-properties-definition-dimension", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over $S$. The {\\it dimension} $\\dim(\\mathcal{X})$ of $\\mathcal{X}$ is defined to be $$ \\dim(\\mathcal{X}) = \\sup\\nolimits_{x \\in |\\mathcal{X}|} \\dim_x(\\mathcal{X}) $$"}
+{"_id": "8926", "title": "stacks-properties-definition-number-of-geometric-branches", "text": "Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$. \\begin{enumerate} \\item The {\\it number of geometric branches of $\\mathcal{X}$ at $x$} is either $n \\in \\mathbf{N}$ if the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold for $\\mathcal{P}_n$ defined above, or else $\\infty$. \\item We say $\\mathcal{X}$ is {\\it geometrically unibranch at $x$} if the number of geometric branches of $\\mathcal{X}$ at $x$ is $1$. \\end{enumerate}"}
+{"_id": "8992", "title": "stacks-definition-mor-presheaf", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category, see Categories, Section \\ref{categories-section-fibred-categories}. Given an object $U$ of $\\mathcal{C}$ and objects $x$, $y$ of the fibre category, the {\\it presheaf of morphisms from $x$ to $y$} is the presheaf $$ (f : V \\to U) \\longmapsto \\Mor_{\\mathcal{S}_V}(f^*x, f^*y) $$ described above. It is denoted $\\mathit{Mor}(x, y)$. The subpresheaf $\\mathit{Isom}(x, y)$ whose values over $V$ is the set of isomorphisms $f^*x \\to f^*y$ in the fibre category $\\mathcal{S}_V$ is called the {\\it presheaf of isomorphisms from $x$ to $y$}."}
+{"_id": "8993", "title": "stacks-definition-descent-data", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition \\ref{categories-definition-pullback-functor-fibred-category}. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms of $\\mathcal{C}$. Assume all the fibre products $U_i \\times_U U_j$, and $U_i \\times_U U_j \\times_U U_k$ exist. \\begin{enumerate} \\item A {\\it descent datum $(X_i, \\varphi_{ij})$ in $\\mathcal{S}$ relative to the family $\\{f_i : U_i \\to U\\}$} is given by an object $X_i$ of $\\mathcal{S}_{U_i}$ for each $i \\in I$, an isomorphism $\\varphi_{ij} : \\text{pr}_0^*X_i \\to \\text{pr}_1^*X_j$ in $\\mathcal{S}_{U_i \\times_U U_j}$ for each pair $(i, j) \\in I^2$ such that for every triple of indices $(i, j, k) \\in I^3$ the diagram $$ \\xymatrix{ \\text{pr}_0^*X_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}} \\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & & \\text{pr}_2^*X_k \\\\ & \\text{pr}_1^*X_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} & } $$ in the category $\\mathcal{S}_{U_i \\times_U U_j \\times_U U_k}$ commutes. This is called the {\\it cocycle condition}. \\item A {\\it morphism $\\psi : (X_i, \\varphi_{ij}) \\to (X'_i, \\varphi'_{ij})$ of descent data} is given by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms $\\psi_i : X_i \\to X'_i$ in $\\mathcal{S}_{U_i}$ such that all the diagrams $$ \\xymatrix{ \\text{pr}_0^*X_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i} & \\text{pr}_1^*X_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\ \\text{pr}_0^*X'_i \\ar[r]^{\\varphi'_{ij}} & \\text{pr}_1^*X'_j \\\\ } $$ in the categories $\\mathcal{S}_{U_i \\times_U U_j}$ commute. \\item The category of descent data relative to $\\mathcal{U}$ is denoted $DD(\\mathcal{U})$. \\end{enumerate}"}
+{"_id": "8994", "title": "stacks-definition-pullback-functor", "text": "With $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$, $\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$, $\\alpha : I \\to J$, $h : U \\to V$, and $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback} the functor $$ (Y_j, \\varphi_{jj'}) \\longmapsto (g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}) $$ constructed in that lemma is called the {\\it pullback functor} on descent data."}
+{"_id": "8995", "title": "stacks-definition-effective-descent-datum", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Make a choice of pullbacks as in Categories, Definition \\ref{categories-definition-pullback-functor-fibred-category}. Let $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms with target $U$. Assume all the fibre products $U_i \\times_U U_j$ and $U_i \\times_U U_j \\times_U U_k$ exist. \\begin{enumerate} \\item Given an object $X$ of $\\mathcal{S}_U$ the {\\it trivial descent datum} is the descent datum $(X, \\text{id}_X)$ with respect to the family $\\{\\text{id}_U : U \\to U\\}$. \\item Given an object $X$ of $\\mathcal{S}_U$ we have a {\\it canonical descent datum} on the family of objects $f_i^*X$ by pulling back the trivial descent datum $(X, \\text{id}_X)$ via the obvious map $\\{f_i : U_i \\to U\\} \\to \\{\\text{id}_U : U \\to U\\}$. We denote this descent datum $(f_i^*X, can)$. \\item A descent datum $(X_i, \\varphi_{ij})$ relative to $\\{f_i : U_i \\to U\\}$ is called {\\it effective} if there exists an object $X$ of $\\mathcal{S}_U$ such that $(X_i, \\varphi_{ij})$ is isomorphic to $(f_i^*X, can)$. \\end{enumerate}"}
+{"_id": "8996", "title": "stacks-definition-stack", "text": "Let $\\mathcal{C}$ be a site. A {\\it stack} over $\\mathcal{C}$ is a category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ which satisfies the following conditions: \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category, see Categories, Definition \\ref{categories-definition-fibred-category}, \\item for any $U \\in \\Ob(\\mathcal{C})$ and any $x, y \\in \\mathcal{S}_U$ the presheaf $\\mathit{Mor}(x, y)$ (see Definition \\ref{definition-mor-presheaf}) is a sheaf on the site $\\mathcal{C}/U$, and \\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ of the site $\\mathcal{C}$, any descent datum in $\\mathcal{S}$ relative to $\\mathcal{U}$ is effective. \\end{enumerate}"}
+{"_id": "8997", "title": "stacks-definition-stacks-over-C", "text": "Let $\\mathcal{C}$ be a site. The {\\it $2$-category of stacks over $\\mathcal{C}$} is the sub $2$-category of the $2$-category of fibred categories over $\\mathcal{C}$ (see Categories, Definition \\ref{categories-definition-fibred-categories-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be stacks $p : \\mathcal{S} \\to \\mathcal{C}$. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$ and such that $G$ maps strongly cartesian morphisms to strongly cartesian morphisms. \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "8998", "title": "stacks-definition-stack-in-groupoids", "text": "A {\\it stack in groupoids} over a site $\\mathcal{C}$ is a category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ such that \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids over $\\mathcal{C}$ (see Categories, Definition \\ref{categories-definition-fibred-groupoids}), \\item for all $U \\in \\Ob(\\mathcal{C})$, for all $x, y\\in \\Ob(\\mathcal{S}_U)$ the presheaf $\\mathit{Isom}(x, y)$ is a sheaf on the site $\\mathcal{C}/U$, and \\item for all coverings $\\mathcal{U} = \\{U_i \\to U\\}$ in $\\mathcal{C}$, all descent data $(x_i, \\phi_{ij})$ for $\\mathcal{U}$ are effective. \\end{enumerate}"}
+{"_id": "8999", "title": "stacks-definition-stacks-in-groupoids-over-C", "text": "Let $\\mathcal{C}$ be a site. The {\\it $2$-category of stacks in groupoids over $\\mathcal{C}$} is the sub $2$-category of the $2$-category of stacks over $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be stacks in groupoids $p : \\mathcal{S} \\to \\mathcal{C}$. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$. (Since every morphism is strongly cartesian every functor preserves them.) \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "9000", "title": "stacks-definition-stack-in-sets", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item A {\\it stack in setoids} over $\\mathcal{C}$ is a stack over $\\mathcal{C}$ all of whose fibre categories are setoids. \\item A {\\it stack in sets}, or a {\\it stack in discrete categories} is a stack over $\\mathcal{C}$ all of whose fibre categories are discrete. \\end{enumerate}"}
+{"_id": "9001", "title": "stacks-definition-stacks-in-setoids-over-C", "text": "Let $\\mathcal{C}$ be a site. The {\\it $2$-category of stacks in setoids over $\\mathcal{C}$} is the sub $2$-category of the $2$-category of stacks over $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be stacks in setoids $p : \\mathcal{S} \\to \\mathcal{C}$. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$. (Since every morphism is strongly cartesian every functor preserves them.) \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "9002", "title": "stacks-definition-topology-inherited", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. We say $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ as in Lemma \\ref{lemma-topology-inherited} is the {\\it structure of site on $\\mathcal{S}$ inherited from $\\mathcal{C}$}. We sometimes indicate this by saying that {\\it $\\mathcal{S}$ is endowed with the topology inherited from $\\mathcal{C}$}."}
+{"_id": "9003", "title": "stacks-definition-gerbe", "text": "A {\\it gerbe} over a site $\\mathcal{C}$ is a category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ such that \\begin{enumerate} \\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a stack in groupoids over $\\mathcal{C}$ (see Definition \\ref{definition-stack-in-groupoids}), \\item for $U \\in \\Ob(\\mathcal{C})$ there exists a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that $\\mathcal{S}_{U_i}$ is nonempty, and \\item for $U \\in \\Ob(\\mathcal{C})$ and $x, y \\in \\Ob(\\mathcal{S}_U)$ there exists a covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that $x|_{U_i} \\cong y|_{U_i}$ in $\\mathcal{S}_{U_i}$. \\end{enumerate}"}
+{"_id": "9004", "title": "stacks-definition-gerbe-over-stack-in-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{X}$ and $\\mathcal{Y}$ be stacks in groupoids over $\\mathcal{C}$. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories over $\\mathcal{C}$. We say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$ if the equivalent conditions of Lemma \\ref{lemma-when-gerbe} are satisfied."}
+{"_id": "9005", "title": "stacks-definition-pushforward-stack", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$. Let $\\mathcal{S}$ be a fibred category over $\\mathcal{D}$. In this setting we write {\\it $f_*\\mathcal{S}$} for the fibred category $u^p\\mathcal{S}$ defined above. We say that $f_*\\mathcal{S}$ is the {\\it pushforward of $\\mathcal{S}$ along $f$}."}
+{"_id": "9006", "title": "stacks-definition-pullback-stack", "text": "Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying the hypotheses and conclusions of Sites, Proposition \\ref{sites-proposition-get-morphism}. Let $\\mathcal{S}$ be a stack over $\\mathcal{C}$. In this setting we write {\\it $f^{-1}\\mathcal{S}$} for the stackification of the fibred category $u_p\\mathcal{S}$ over $\\mathcal{D}$ constructed above. We say that $f^{-1}\\mathcal{S}$ is the {\\it pullback of $\\mathcal{S}$ along $f$}."}
+{"_id": "9147", "title": "spaces-simplicial-definition-cartesian-sheaf", "text": "In Situation \\ref{situation-simplicial-site}. \\begin{enumerate} \\item A sheaf $\\mathcal{F}$ of sets or of abelian groups on $\\mathcal{C}$ is {\\it cartesian} if the maps $\\mathcal{F}(\\varphi) : f_\\varphi^{-1}\\mathcal{F}_m \\to \\mathcal{F}_n$ are isomorphisms for all $\\varphi : [m] \\to [n]$. \\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$, then a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is {\\it cartesian} if  the maps $f_\\varphi^*\\mathcal{F}_m \\to \\mathcal{F}_n$ are isomorphisms for all $\\varphi : [m] \\to [n]$. \\item An object $K$ of $D(\\mathcal{C}_{total})$ is {\\it cartesian} if the maps $f_\\varphi^{-1}K_m \\to K_n$ are isomorphisms for all $\\varphi : [m] \\to [n]$. \\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$, then an object $K$ of $D(\\mathcal{O})$ is {\\it cartesian} if the maps $Lf_\\varphi^*K_m \\to K_n$ are isomorphisms for all $\\varphi : [m] \\to [n]$. \\end{enumerate}"}
+{"_id": "9148", "title": "spaces-simplicial-definition-cartesian-derived", "text": "In Situation \\ref{situation-simplicial-site}. A {\\it simplicial system of the derived category} consists of the following data \\begin{enumerate} \\item for every $n$ an object $K_n$ of $D(\\mathcal{C}_n)$, \\item for every $\\varphi : [m] \\to [n]$ a map $K_\\varphi : f_\\varphi^{-1}K_m \\to K_n$ in $D(\\mathcal{C}_n)$ \\end{enumerate} subject to the condition that $$ K_{\\varphi \\circ \\psi} = K_\\varphi \\circ f_\\varphi^{-1}K_\\psi : f_{\\varphi \\circ \\psi}^{-1}K_l = f_\\varphi^{-1} f_\\psi^{-1}K_l \\longrightarrow K_n $$ for any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$. We say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$ are isomorphisms for all $\\varphi$. Given two simplicial systems of the derived category there is an obvious notion of a {\\it morphism of simplicial systems of the derived category}."}
+{"_id": "9149", "title": "spaces-simplicial-definition-cartesian-derived-modules", "text": "In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$. A {\\it simplicial system of the derived category of modules} consists of the following data \\begin{enumerate} \\item for every $n$ an object $K_n$ of $D(\\mathcal{O}_n)$, \\item for every $\\varphi : [m] \\to [n]$ a map $K_\\varphi : Lf_\\varphi^*K_m \\to K_n$ in $D(\\mathcal{O}_n)$ \\end{enumerate} subject to the condition that $$ K_{\\varphi \\circ \\psi} = K_\\varphi \\circ Lf_\\varphi^*K_\\psi : Lf_{\\varphi \\circ \\psi}^*K_l = Lf_\\varphi^* Lf_\\psi^*K_l \\longrightarrow K_n $$ for any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$. We say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$ are isomorphisms for all $\\varphi$. Given two simplicial systems of the derived category there is an obvious notion of a {\\it morphism of simplicial systems of the derived category of modules}."}
+{"_id": "9150", "title": "spaces-simplicial-definition-cartesian-morphism", "text": "Let $a : Y \\to X$ be a morphism of simplicial schemes. We say $a$ is {\\it cartesian}, or that {\\it $Y$ is cartesian over $X$}, if for every morphism $\\varphi : [n] \\to [m]$ of $\\Delta$ the corresponding diagram $$ \\xymatrix{ Y_m \\ar[r]_a \\ar[d]_{Y(\\varphi)} & X_m \\ar[d]^{X(\\varphi)}\\\\ Y_n \\ar[r]^{a} & X_n } $$ is a fibre square in the category of schemes."}
+{"_id": "9151", "title": "spaces-simplicial-definition-fibre-products-simplicial-scheme", "text": "Let $f : X \\to S$ be a morphism of schemes. The {\\it simplicial scheme associated to $f$}, denoted $(X/S)_\\bullet$, is the functor $\\Delta^{opp} \\to \\Sch$, $[n] \\mapsto X \\times_S \\ldots \\times_S X$ described in Simplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}."}
+{"_id": "9184", "title": "examples-stacks-definition-hilbert-d-stack", "text": "We will denote $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ the {\\it degree $d$ finite Hilbert stack of $\\mathcal{X}$ over $\\mathcal{Y}$} constructed above. If $\\mathcal{Y} = S$ we write $\\mathcal{H}_d(\\mathcal{X}) = \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$. If $\\mathcal{X} = \\mathcal{Y} = S$ we denote it $\\mathcal{H}_d$."}
+{"_id": "9269", "title": "models-definition-type", "text": "A {\\it numerical type} $T$ is given by $$ n, m_i, a_{ij}, w_i, g_i $$ where $n \\geq 1$ is an integer and $m_i$, $a_{ij}$, $w_i$, $g_i$ are integers for $1 \\leq i, j \\leq n$ subject to the following conditions \\begin{enumerate} \\item $m_i > 0$, $w_i > 0$, $g_i \\geq 0$, \\item the matrix $A = (a_{ij})$ is symmetric and $a_{ij} \\geq 0$ for $i \\not = j$, \\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$ such that $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$, \\item for each $i$ we have $\\sum_j a_{ij}m_j = 0$, and \\item $w_i | a_{ij}$. \\end{enumerate}"}
+{"_id": "9270", "title": "models-definition-type-equivalent", "text": "We say two numerical types $n, m_i, a_{ij}, w_i, g_i$ and $n', m'_i, a'_{ij}, w'_i, g'_i$ are {\\it equivalent types} if there exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$ such that $m_i = m'_{\\sigma(i)}$, $a_{ij} = a'_{\\sigma(i)\\sigma(j)}$, $w_i = w'_{\\sigma(i)}$, and $g_i = g'_{\\sigma(i)}$."}
+{"_id": "9271", "title": "models-definition-genus", "text": "We say $n, m_i, a_{ij}, w_i, g_i$ is a {\\it numerical type of genus $g$} if $g = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ is the integer from Lemma \\ref{lemma-genus}."}
+{"_id": "9272", "title": "models-definition-type-minus-one", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. We say $i$ is a {\\it $(-1)$-index} if $g_i = 0$ and $a_{ii} = -w_i$."}
+{"_id": "9273", "title": "models-definition-top-genus", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The {\\it topological genus of $T$} is the nonnegative integer $g_{top} = 1 - n + e$ from Lemma \\ref{lemma-top-genus}."}
+{"_id": "9274", "title": "models-definition-type-minimal", "text": "We say the numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$ is {\\it minimal} if there does not exist an $i$ with $g_i = 0$ and $a_{ii} = -w_i$, in other words, if there does not exist a $(-1)$-index."}
+{"_id": "9275", "title": "models-definition-type-minus-two", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$. We say $i$ is a {\\it $(-2)$-index} if $g_i = 0$ and $a_{ii} = -2w_i$."}
+{"_id": "9276", "title": "models-definition-picard-group", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The {\\it Picard group of $T$} is the cokernel of the matrix $(a_{ij}/w_i)$, more precisely $$ \\Pic(T) = \\Coker\\left( \\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n},\\quad e_i \\mapsto \\sum \\frac{a_{ij}}{w_j}e_j \\right) $$ where $e_i$ denotes the $i$th standard basis vector for $\\mathbf{Z}^{\\oplus n}$."}
+{"_id": "9277", "title": "models-definition-minimal-model", "text": "Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. A {\\it minimal model} will be a regular, proper model $X$ for $C$ such that $X$ does not contain an exceptional curve of the first kind (Resolution of Surfaces, Section \\ref{resolve-section-minus-one})."}
+{"_id": "9278", "title": "models-definition-numerical-type-model", "text": "In Situation \\ref{situation-regular-model} the {\\it numerical type associated to $X$} is the numerical type described in Lemma \\ref{lemma-numerical-type-of-model}."}
+{"_id": "9279", "title": "models-definition-semistable", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. We say that $C$ has {\\it semistable reduction} if the equivalent conditions of Lemma \\ref{lemma-semistable} are satisfied."}
+{"_id": "9280", "title": "models-definition-good", "text": "Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$. We say that $C$ has {\\it good reduction} if the equivalent conditions of Lemma \\ref{lemma-good} are satisfied."}
+{"_id": "9339", "title": "spaces-groupoids-definition-equivalence-relation", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $U$ be an algebraic space over $B$. \\begin{enumerate} \\item A {\\it pre-relation} on $U$ over $B$ is any morphism $j : R \\to U \\times_B U$ of algebraic spaces over $B$. In this case we set $t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so that $j = (t, s)$. \\item A {\\it relation} on $U$ over $B$ is a monomorphism $j : R \\to U \\times_B U$ of algebraic spaces over $B$. \\item A {\\it pre-equivalence relation} is a pre-relation $j : R \\to U \\times_B U$ such that the image of $j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for all schemes $T$ over $B$. \\item We say a morphism $R \\to U \\times_B U$ of algebraic spaces over $B$ is an {\\it equivalence relation on $U$ over $B$} if and only if for every $T$ over $B$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$. \\end{enumerate}"}
+{"_id": "9340", "title": "spaces-groupoids-definition-restrict-relation", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $U$ be an algebraic space over $B$. Let $j : R \\to U \\times_B U$ be a pre-relation. Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$. The pre-relation $j' : R' \\to U' \\times_B U'$ of Lemma \\ref{lemma-restrict-relation} is called the {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$. In this situation we sometimes write $R' = R|_{U'}$."}
+{"_id": "9341", "title": "spaces-groupoids-definition-group-space", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. \\begin{enumerate} \\item A {\\it group algebraic space over $B$} is a pair $(G, m)$, where $G$ is an algebraic space over $B$ and $m : G \\times_B G \\to G$ is a morphism of algebraic spaces over $B$ with the following property: For every scheme $T$ over $B$ the pair $(G(T), m)$ is a group. \\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of group algebraic spaces over $B$} is a morphism $\\psi : G \\to G'$ of algebraic spaces over $B$ such that for every $T/B$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism of groups. \\end{enumerate}"}
+{"_id": "9342", "title": "spaces-groupoids-definition-action-group-space", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$. \\begin{enumerate} \\item An {\\it action of $G$ on the algebraic space $X/B$} is a morphism $a : G \\times_B X \\to X$ over $B$ such that for every scheme $T$ over $B$ the map $a : G(T) \\times X(T) \\to X(T)$ defines the structure of a $G(T)$-set on $X(T)$. \\item Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed with an action of $G$. An {\\it equivariant} or more precisely a {\\it $G$-equivariant} morphism $\\psi : X \\to Y$ is a morphism of algebraic spaces over $B$ such that for every $T$ over $B$ the map $\\psi : X(T) \\to Y(T)$ is a morphism of $G(T)$-sets. \\end{enumerate}"}
+{"_id": "9343", "title": "spaces-groupoids-definition-free-action", "text": "Let $B \\to S$, $G \\to B$, and $X \\to B$ as in Definition \\ref{definition-action-group-space}. Let $a : G \\times_B X \\to X$ be an action of $G$ on $X/B$. We say the action is {\\it free} if for every scheme $T$ over $B$ the action $a : G(T) \\times X(T) \\to X(T)$ is a free action of the group $G(T)$ on the set $X(T)$."}
+{"_id": "9344", "title": "spaces-groupoids-definition-pseudo-torsor", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$, and let $a : G \\times_B X \\to X$ be an action of $G$ on $X$. \\begin{enumerate} \\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is {\\it formally principally homogeneous under $G$} if the induced morphism $G \\times_B X \\to X \\times_B X$, $(g, x) \\mapsto (a(g, x), x)$ is an isomorphism. \\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists an $G$-equivariant isomorphism $G \\to X$ over $B$ where $G$ acts on $G$ by left multiplication. \\end{enumerate}"}
+{"_id": "9345", "title": "spaces-groupoids-definition-principal-homogeneous-space", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be a pseudo $G$-torsor over $B$. \\begin{enumerate} \\item We say $X$ is a {\\it principal homogeneous space}, or more precisely a {\\it principal homogeneous $G$-space over $B$} if there exists a fpqc covering\\footnote{The default type of torsor in Groupoids, Definition \\ref{groupoids-definition-principal-homogeneous-space} is a pseudo torsor which is trivial on an fpqc covering. Since $G$, as an algebraic space, can be seen a sheaf of groups there already is a notion of a $G$-torsor which corresponds to fppf-torsor, see Lemma \\ref{lemma-torsor}. Hence we use ``principal homogeneous space'' for a pseudo torsor which is fpqc locally trivial, and we try to avoid using the word torsor in this situation.} $\\{B_i \\to B\\}_{i \\in I}$ such that each $X_{B_i} \\to B_i$ has a section (i.e., is a trivial pseudo $G_{B_i}$-torsor). \\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. We say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a {\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor} if there exists a $\\tau$ covering $\\{B_i \\to B\\}_{i \\in I}$ such that each $X_{B_i} \\to B_i$ has a section. \\item If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is {\\it quasi-isotrivial} if it is a torsor for the \\'etale topology. \\item If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is {\\it locally trivial} if it is a torsor for the Zariski topology. \\end{enumerate}"}
+{"_id": "9346", "title": "spaces-groupoids-definition-equivariant-module", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$, and let $a : G \\times_B X \\to X$ be an action of $G$ on the algebraic space $X$ over $B$. An {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module}, or simply a {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module}, is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_B X}$-module map $$ \\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F} $$ where $\\text{pr}_1 : G \\times_B X \\to X$ is the projection such that \\begin{enumerate} \\item the diagram $$ \\xymatrix{ (1_G \\times a)^*\\text{pr}_2^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} & \\text{pr}_2^*\\mathcal{F} \\\\ (1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] & (m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha} } $$ is a commutative in the category of $\\mathcal{O}_{G \\times_B G \\times_B X}$-modules, and \\item the pullback $$ (e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F} $$ is the identity map. \\end{enumerate} For explanation compare with the relevant diagrams of Equation (\\ref{equation-action})."}
+{"_id": "9347", "title": "spaces-groupoids-definition-groupoid", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. \\begin{enumerate} \\item A {\\it groupoid in algebraic spaces over $B$} is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are algebraic spaces over $B$, and $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$ are morphisms of algebraic spaces over $B$ with the following property: For any scheme $T$ over $B$ the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category. \\item A {\\it morphism $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$} is given by morphisms of algebraic spaces $f : U \\to U'$ and $f : R \\to R'$ over $B$ with the following property:  For any scheme $T$ over $B$ the maps $f$ define a functor from the groupoid category $(U(T), R(T), s, t, c)$ to the groupoid category $(U'(T), R'(T), s', t', c')$. \\end{enumerate}"}
+{"_id": "9348", "title": "spaces-groupoids-definition-groupoid-module", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. A {\\it quasi-coherent module on $(U, R, s, t, c)$} is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module map $$ \\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F} $$ such that \\begin{enumerate} \\item the diagram $$ \\xymatrix{ & \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} & \\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\ \\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\ & \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] & c^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha} } $$ is a commutative in the category of $\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and \\item the pullback $$ e^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F} $$ is the identity map. \\end{enumerate} Compare with the commutative diagrams of Lemma \\ref{lemma-diagram}."}
+{"_id": "9349", "title": "spaces-groupoids-definition-stabilizer-groupoid", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The group algebraic space $j^{-1}(\\Delta_{U/B}) \\to U$ is called the {\\it stabilizer of the groupoid in algebraic spaces $(U, R, s, t, c)$}."}
+{"_id": "9350", "title": "spaces-groupoids-definition-restrict-groupoid", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \\to U$ be a morphism of algebraic spaces over $B$. The morphism of groupoids in algebraic spaces $(U', R', s', t', c') \\to (U, R, s, t, c)$ constructed in Lemma \\ref{lemma-restrict-groupoid} is called the {\\it restriction of $(U, R, s, t, c)$ to $U'$}. We sometime use the notation $R' = R|_{U'}$ in this case."}
+{"_id": "9351", "title": "spaces-groupoids-definition-invariant-open", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over the base $B$. \\begin{enumerate} \\item We say an open subspace $W \\subset U$ is {\\it $R$-invariant} if $t(s^{-1}(W)) \\subset W$. \\item A locally closed subspace $Z \\subset U$ is called {\\it $R$-invariant} if $t^{-1}(Z) = s^{-1}(Z)$ as locally closed subspaces of $R$. \\item A monomorphism of algebraic spaces $T \\to U$ is {\\it $R$-invariant} if $T \\times_{U, t} R = R \\times_{s, U} T$ as algebraic spaces over $R$. \\end{enumerate}"}
+{"_id": "9352", "title": "spaces-groupoids-definition-quotient-sheaf", "text": "Let $B \\to S$ and the pre-relation $j : R \\to U \\times_B U$ be as above. In this setting the {\\it quotient sheaf $U/R$} associated to $j$ is the sheafification of the presheaf (\\ref{equation-quotient-presheaf}) on $(\\Sch/S)_{fppf}$. If $j : R \\to U \\times_B U$ comes from the action of a group algebraic space $G$ over $B$ on $U$ as in Lemma \\ref{lemma-groupoid-from-action} then we denote the quotient sheaf $U/G$."}
+{"_id": "9353", "title": "spaces-groupoids-definition-representable-quotient", "text": "In the situation of Definition \\ref{definition-quotient-sheaf}. We say that the pre-relation $j$ has a {\\it quotient representable by an algebraic space} if the sheaf $U/R$ is an algebraic space. We say that the pre-relation $j$ has a {\\it representable quotient} if the sheaf $U/R$ is representable by a scheme. We will say a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ has a {\\it representable quotient} (resp.\\ {\\it quotient representable by an algebraic space} if the quotient $U/R$ with $j = (t, s)$ is representable (resp.\\ an algebraic space)."}
+{"_id": "9354", "title": "spaces-groupoids-definition-quotient-stack", "text": "Quotient stacks. Let $B \\to S$ be as above. \\begin{enumerate} \\item Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The {\\it quotient stack} $$ p : [U/R] \\longrightarrow (\\Sch/S)_{fppf} $$ of $(U, R, s, t, c)$ is the stackification (see Stacks, Lemma \\ref{stacks-lemma-stackify-groupoids}) of the category fibred in groupoids $[U/_{\\!p}R]$ over $(\\Sch/S)_{fppf}$ associated to (\\ref{equation-quotient-stack}). \\item Let $(G, m)$ be a group algebraic space over $B$. Let $a : G \\times_B X \\to X$ be an action of $G$ on an algebraic space over $B$. The {\\it quotient stack} $$ p : [X/G] \\longrightarrow (\\Sch/S)_{fppf} $$ is the quotient stack associated to the groupoid in algebraic spaces $(X, G \\times_B X, s, t, c)$ over $B$ of Lemma \\ref{lemma-groupoid-from-action}. \\end{enumerate}"}
+{"_id": "9438", "title": "spaces-descent-definition-descent-datum-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. \\begin{enumerate} \\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$ for quasi-coherent sheaves} with respect to the given family is given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $X_i$ for each $i \\in I$, an isomorphism of quasi-coherent $\\mathcal{O}_{X_i \\times_X X_j}$-modules $\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$ for each pair $(i, j) \\in I^2$ such that for every triple of indices $(i, j, k) \\in I^3$ the diagram $$ \\xymatrix{ \\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}} \\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & & \\text{pr}_2^*\\mathcal{F}_k \\\\ & \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} & } $$ of $\\mathcal{O}_{X_i \\times_X X_j \\times_X X_k}$-modules commutes. This is called the {\\it cocycle condition}. \\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to (\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of $\\mathcal{O}_{X_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$ such that all the diagrams $$ \\xymatrix{ \\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i} & \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\ \\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} & \\text{pr}_1^*\\mathcal{F}'_j \\\\ } $$ commute. \\end{enumerate}"}
+{"_id": "9439", "title": "spaces-descent-definition-descent-datum-effective-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_U$-module. We call the unique descent on $\\mathcal{F}$ datum with respect to the covering $\\{U \\to U\\}$ the {\\it trivial descent datum}. \\item The pullback of the trivial descent datum to $\\{U_i \\to U\\}$ is called the {\\it canonical descent datum}. Notation: $(\\mathcal{F}|_{U_i}, can)$. \\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$ for quasi-coherent sheaves with respect to the given family is said to be {\\it effective} if there exists a quasi-coherent sheaf $\\mathcal{F}$ on $U$ such that $(\\mathcal{F}_i, \\varphi_{ij})$ is isomorphic to $(\\mathcal{F}|_{U_i}, can)$. \\end{enumerate}"}
+{"_id": "9440", "title": "spaces-descent-definition-property-morphisms-local", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$. We say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or {\\it $\\tau$ local on the target}, or {\\it local on the base for the $\\tau$-topology} if for any $\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ of algebraic spaces and any morphism of algebraic spaces $f : X \\to Y$ we have $$ f \\text{ has }\\mathcal{P} \\Leftrightarrow \\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}. $$"}
+{"_id": "9441", "title": "spaces-descent-definition-property-morphisms-local-source", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0] smooth, \\linebreak[0] \\etale\\}$. We say $\\mathcal{P}$ is {\\it $\\tau$ local on the source}, or {\\it local on the source for the $\\tau$-topology} if for any morphism $f : X \\to Y$ of algebraic spaces over $S$, and any $\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces we have $$ f \\text{ has }\\mathcal{P} \\Leftrightarrow \\text{each }X_i \\to Y\\text{ has }\\mathcal{P}. $$"}
+{"_id": "9442", "title": "spaces-descent-definition-local-source-target", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. We say $\\mathcal{P}$ is {\\it smooth local on source-and-target} if \\begin{enumerate} \\item (stable under precomposing with smooth maps) if $f : X \\to Y$ is smooth and $g : Y \\to Z$ has $\\mathcal{P}$, then $g \\circ f$ has $\\mathcal{P}$, \\item (stable under smooth base change) if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and \\item (locality) given a morphism $f : X \\to Y$ the following are equivalent \\begin{enumerate} \\item $f$ has $\\mathcal{P}$, \\item for every $x \\in |X|$ there exists a commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with smooth vertical arrows and $u \\in |U|$ with $a(u) = x$ such that $h$ has $\\mathcal{P}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "9443", "title": "spaces-descent-definition-etale-smooth-local-source-target", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. We say $\\mathcal{P}$ is {\\it \\'etale-smooth local on source-and-target} if \\begin{enumerate} \\item (stable under precomposing with \\'etale maps) if $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$, then $g \\circ f$ has $\\mathcal{P}$, \\item (stable under smooth base change) if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and \\item (locality) given a morphism $f : X \\to Y$ the following are equivalent \\begin{enumerate} \\item $f$ has $\\mathcal{P}$, \\item for every $x \\in |X|$ there exists a commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with $b$ smooth and $U \\to X \\times_Y V$ \\'etale and $u \\in |U|$ with $a(u) = x$ such that $h$ has $\\mathcal{P}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "9444", "title": "spaces-descent-definition-descent-datum", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item Let $V \\to Y$ be a morphism of algebraic spaces. A {\\it descent datum for $V/Y/X$} is an isomorphism $\\varphi : V \\times_X Y \\to Y \\times_X V$ of algebraic spaces over $Y \\times_X Y$ satisfying the {\\it cocycle condition} that the diagram $$ \\xymatrix{ V \\times_X Y \\times_X Y \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} & & Y \\times_X Y \\times_X V\\\\ & Y \\times_X Y \\times_X Y \\ar[ru]^{\\varphi_{12}} } $$ commutes (with obvious notation). \\item We also say that the pair $(V/Y, \\varphi)$ is a {\\it descent datum relative to $Y \\to X$}. \\item A {\\it morphism $f : (V/Y, \\varphi) \\to (V'/Y, \\varphi')$ of descent data relative to $Y \\to X$} is a morphism $f : V \\to V'$ of algebraic spaces over $Y$ such that the diagram $$ \\xymatrix{ V \\times_X Y \\ar[r]_{\\varphi} \\ar[d]_{f \\times \\text{id}_Y} & Y \\times_X V \\ar[d]^{\\text{id}_Y \\times f} \\\\ V' \\times_X Y \\ar[r]^{\\varphi'} & Y \\times_X V' } $$ commutes. \\end{enumerate}"}
+{"_id": "9445", "title": "spaces-descent-definition-descent-datum-for-family-of-morphisms", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. \\begin{enumerate} \\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the family $\\{X_i \\to X\\}$} is given by an algebraic space $V_i$ over $X_i$ for each $i \\in I$, an isomorphism $\\varphi_{ij} : V_i \\times_X X_j \\to X_i \\times_X V_j$ of algebraic spaces over $X_i \\times_X X_j$ for each pair $(i, j) \\in I^2$ such that for every triple of indices $(i, j, k) \\in I^3$ the diagram $$ \\xymatrix{ V_i \\times_X X_j \\times_X X_k \\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}} \\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & & X_i \\times_X X_j \\times_X V_k\\\\ & X_i \\times_X V_j \\times_X X_k \\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}} } $$ of algebraic spaces over $X_i \\times_X X_j \\times_X X_k$ commutes (with obvious notation). \\item A {\\it morphism $\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$ of descent data} is given by a family $\\psi = (\\psi_i)_{i \\in I}$ of morphisms $\\psi_i : V_i \\to V'_i$ of algebraic spaces over $X_i$ such that all the diagrams $$ \\xymatrix{ V_i \\times_X X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} & X_i \\times_X V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\ V'_i \\times_X X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_X V'_j } $$ commute. \\end{enumerate}"}
+{"_id": "9446", "title": "spaces-descent-definition-pullback-functor", "text": "With $S, X, X', Y, Y', f, a, a', h$ as in Lemma \\ref{lemma-pullback} the functor $$ (V, \\varphi) \\longmapsto f^*(V, \\varphi) $$ constructed in that lemma is called the {\\it pullback functor} on descent data."}
+{"_id": "9447", "title": "spaces-descent-definition-pullback-functor-family", "text": "With $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$, $\\mathcal{U} = \\{X_i \\to X\\}_{i \\in I}$, $\\alpha : I' \\to I$, $g : X' \\to X$, and $g_i : X'_i \\to X_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback-family} the functor $$ (V_i, \\varphi_{ij}) \\longmapsto (g_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)}) $$ constructed in that lemma is called the {\\it pullback functor} on descent data."}
+{"_id": "9448", "title": "spaces-descent-definition-effective", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item Given an algebraic space $U$ over $X$ we have the {\\it trivial descent datum} of $U$ relative to $\\text{id} : X \\to X$, namely the identity morphism on $U$. \\item By Lemma \\ref{lemma-pullback} we get a {\\it canonical descent datum} on $Y \\times_X U$ relative to $Y \\to X$ by pulling back the trivial descent datum via $f$. We often denote $(Y \\times_X U, can)$ this descent datum. \\item A descent datum $(V, \\varphi)$ relative to $Y/X$ is called {\\it effective} if $(V, \\varphi)$ is isomorphic to the canonical descent datum $(Y \\times_X U, can)$ for some algebraic space $U$ over $X$. \\end{enumerate}"}
+{"_id": "9449", "title": "spaces-descent-definition-effective-family", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. \\begin{enumerate} \\item  Given an algebraic space $U$ over $X$ we have a {\\it canonical descent datum} on the family of algebraic spaces $X_i \\times_X U$ by pulling back the trivial descent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$. We denote this descent datum $(X_i \\times_X U, can)$. \\item A descent datum $(V_i, \\varphi_{ij})$ relative to $\\{X_i \\to S\\}$ is called {\\it effective} if there exists an algebraic space $U$ over $X$ such that $(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_X U, can)$. \\end{enumerate}"}
+{"_id": "9561", "title": "decent-spaces-definition-universally-bounded", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$. Let $f : U \\to X$ be a morphism over $S$. We say the {\\it fibres of $f$ are universally bounded}\\footnote{This is probably nonstandard notation.} if there exists an integer $n$ such that for all fields $k$ and all morphisms $\\Spec(k) \\to X$ the fibre product $\\Spec(k) \\times_X U$ is a finite scheme over $k$ whose degree over $k$ is $\\leq n$."}
+{"_id": "9562", "title": "decent-spaces-definition-very-reasonable", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item We say $X$ is {\\it decent} if for every point $x \\in X$ the equivalent conditions of Lemma \\ref{lemma-UR-finite-above-x} hold, in other words property $(\\gamma)$ of Lemma \\ref{lemma-bounded-fibres} holds. \\item We say $X$ is {\\it reasonable} if the equivalent conditions of Lemma \\ref{lemma-U-universally-bounded} hold, in other words property $(\\delta)$ of Lemma \\ref{lemma-bounded-fibres} holds. \\item We say $X$ is {\\it very reasonable} if the equivalent conditions of Lemma \\ref{lemma-characterize-very-reasonable} hold, i.e., property $(\\epsilon)$ of Lemma \\ref{lemma-bounded-fibres} holds. \\end{enumerate}"}
+{"_id": "9563", "title": "decent-spaces-definition-residue-field", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. The {\\it residue field of $X$ at $x$} is the unique field $\\kappa(x)$ which comes equipped with a monomorphism $\\Spec(\\kappa(x)) \\to X$ representing $x$."}
+{"_id": "9564", "title": "decent-spaces-definition-elemenary-etale-neighbourhood", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in X$ be a point. An {\\it elementary \\'etale neighbourhood} is an \\'etale morphism $(U, u) \\to (X, x)$ where $U$ is a scheme, $u \\in U$ is a point mapping to $x$, and $\\kappa(x) \\to \\kappa(u)$ is an isomorphism. A {\\it morphism of elementary \\'etale neighbourhoods} $(U, u) \\to (U', u')$ is defined as a morphism $U \\to U'$ over $X$ mapping $u$ to $u'$."}
+{"_id": "9565", "title": "decent-spaces-definition-henselian-local-ring", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. The {\\it henselian local ring of $X$ at $x$}, is $$ \\mathcal{O}_{X, x}^h = \\colim \\Gamma(U, \\mathcal{O}_U) $$ where the colimit is over the elementary \\'etale neighbourhoods $(U, u) \\to (X, x)$."}
+{"_id": "9566", "title": "decent-spaces-definition-residual-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The {\\it residual space of $X$ at $x$}\\footnote{This is nonstandard notation.} is the monomorphism $Z_x \\to X$ constructed in Lemma \\ref{lemma-find-singleton-from-point}."}
+{"_id": "9567", "title": "decent-spaces-definition-relative-conditions", "text": "Let $S$ be a scheme. We say an algebraic space $X$ over $S$ {\\it has property $(\\beta)$} if $X$ has the corresponding property of Lemma \\ref{lemma-bounded-fibres}. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. \\begin{enumerate} \\item We say $f$ {\\it has property $(\\beta)$} if for any scheme $T$ and morphism $T \\to Y$ the fibre product $T \\times_Y X$ has property $(\\beta)$. \\item We say $f$ is {\\it decent} if for any scheme $T$ and morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a decent algebraic space. \\item We say $f$ is {\\it reasonable} if for any scheme $T$ and morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a reasonable algebraic space. \\item We say $f$ is {\\it very reasonable} if for any scheme $T$ and morphism $T \\to Y$ the fibre product $T \\times_Y X$ is a very reasonable algebraic space. \\end{enumerate}"}
+{"_id": "9568", "title": "decent-spaces-definition-birational", "text": "Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$. Assume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many irreducible components. We say a morphism $f : X \\to Y$ is {\\it birational} if \\begin{enumerate} \\item $|f|$ induces a bijection between the set of generic points of irreducible components of $|X|$ and the set of generic points of the irreducible components of $|Y|$, and \\item for every generic point $x \\in |X|$ of an irreducible component the local ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is an isomorphism (see clarification below). \\end{enumerate}"}
+{"_id": "9569", "title": "decent-spaces-definition-unibranch", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. We say that $X$ is {\\it unibranch at $x$} if the equivalent conditions of Lemma \\ref{lemma-irreducible-local-ring} hold. We say that $X$ is {\\it unibranch} if $X$ is unibranch at every $x \\in |X|$."}
+{"_id": "9570", "title": "decent-spaces-definition-number-of-geometric-branches", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. The {\\it number of branches of $X$ at $x$} is either $n \\in \\mathbf{N}$ if the equivalent conditions of Lemma \\ref{lemma-nr-branches-local-ring} hold, or else $\\infty$."}
+{"_id": "9571", "title": "decent-spaces-definition-catenary", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. We say $X$ is {\\it catenary} if $|X|$ is catenary (Topology, Definition \\ref{topology-definition-catenary})."}
+{"_id": "9572", "title": "decent-spaces-definition-universally-catenary", "text": "Let $S$ be a scheme. Let $X$ be a decent and locally Noetherian algebraic space over $S$. We say $X$ is {\\it universally catenary} if for every morphism $Y \\to X$ of algebraic spaces which is  locally of finite type and with $Y$ decent, the algebraic space $Y$ is catenary."}
+{"_id": "9670", "title": "groupoids-definition-equivalence-relation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. \\begin{enumerate} \\item A {\\it pre-relation} on $U$ over $S$ is any morphism of schemes $j : R \\to U \\times_S U$. In this case we set $t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so that $j = (t, s)$. \\item A {\\it relation} on $U$ over $S$ is a monomorphism of schemes $j : R \\to U \\times_S U$. \\item A {\\it pre-equivalence relation} is a pre-relation $j : R \\to U \\times_S U$ such that the image of $j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for all $T/S$. \\item We say a morphism $R \\to U \\times_S U$ of schemes is an {\\it equivalence relation on $U$ over $S$} if and only if for every scheme $T$ over $S$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$. \\end{enumerate}"}
+{"_id": "9671", "title": "groupoids-definition-restrict-relation", "text": "Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j : R \\to U \\times_S U$ be a pre-relation. Let $g : U' \\to U$ be a morphism of schemes. The pre-relation $j' : R' \\to U' \\times_S U'$ is called the {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$. In this situation we sometimes write $R' = R|_{U'}$."}
+{"_id": "9672", "title": "groupoids-definition-group-scheme", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item A {\\it group scheme over $S$} is a pair $(G, m)$, where $G$ is a scheme over $S$ and $m : G \\times_S G \\to G$ is a morphism of schemes over $S$ with the following property: For every scheme $T$ over $S$ the pair $(G(T), m)$ is a group. \\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of group schemes over $S$} is a morphism $\\psi : G \\to G'$ of schemes over $S$ such that for every $T/S$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism of groups. \\end{enumerate}"}
+{"_id": "9673", "title": "groupoids-definition-closed-subgroup-scheme", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \\begin{enumerate} \\item A {\\it closed subgroup scheme} of $G$ is a closed subscheme $H \\subset G$ such that $m|_{H \\times_S H}$ factors through $H$ and induces a group scheme structure on $H$ over $S$. \\item An {\\it open subgroup scheme} of $G$ is an open subscheme $G' \\subset G$ such that $m|_{G' \\times_S G'}$ factors through $G'$ and induces a group scheme structure on $G'$ over $S$. \\end{enumerate}"}
+{"_id": "9674", "title": "groupoids-definition-smooth-group-scheme", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \\begin{enumerate} \\item We say $G$ is a {\\it smooth group scheme} if the structure morphism $G \\to S$ is smooth. \\item We say $G$ is a {\\it flat group scheme} if the structure morphism $G \\to S$ is flat. \\item We say $G$ is a {\\it separated group scheme} if the structure morphism $G \\to S$ is separated. \\end{enumerate} Add more as needed."}
+{"_id": "9675", "title": "groupoids-definition-abelian-variety", "text": "Let $k$ be a field. An {\\it abelian variety} is a group scheme over $k$ which is also a proper, geometrically integral variety over $k$."}
+{"_id": "9676", "title": "groupoids-definition-action-group-scheme", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. \\begin{enumerate} \\item An {\\it action of $G$ on the scheme $X/S$} is a morphism $a : G \\times_S X \\to X$ over $S$ such that for every $T/S$ the map $a : G(T) \\times X(T) \\to X(T)$ defines the structure of a $G(T)$-set on $X(T)$. \\item Suppose that $X$, $Y$ are schemes over $S$ each endowed with an action of $G$. An {\\it equivariant} or more precisely a {\\it $G$-equivariant} morphism $\\psi : X \\to Y$ is a morphism of schemes over $S$ such that for every $T/S$ the map $\\psi : X(T) \\to Y(T)$ is a morphism of $G(T)$-sets. \\end{enumerate}"}
+{"_id": "9677", "title": "groupoids-definition-free-action", "text": "Let $S$, $G \\to S$, and $X \\to S$ as in Definition \\ref{definition-action-group-scheme}. Let $a : G \\times_S X \\to X$ be an action of $G$ on $X/S$. We say the action is {\\it free} if for every scheme $T$ over $S$ the action $a : G(T) \\times X(T) \\to X(T)$ is a free action of the group $G(T)$ on the set $X(T)$."}
+{"_id": "9678", "title": "groupoids-definition-pseudo-torsor", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \\times_S X \\to X$ be an action of $G$ on $X$. \\begin{enumerate} \\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is {\\it formally principally homogeneous under $G$} if the induced morphism of schemes $G \\times_S X \\to X \\times_S X$, $(g, x) \\mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$. \\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists an $G$-equivariant isomorphism $G \\to X$ over $S$ where $G$ acts on $G$ by left multiplication. \\end{enumerate}"}
+{"_id": "9679", "title": "groupoids-definition-principal-homogeneous-space", "text": "Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a pseudo $G$-torsor over $S$. \\begin{enumerate} \\item We say $X$ is a {\\it principal homogeneous space} or a {\\it $G$-torsor} if there exists a fpqc covering\\footnote{This means that the default type of torsor is a pseudo torsor which is trivial on an fpqc covering. This is the definition in \\cite[Expos\\'e IV, 6.5]{SGA3}. It is a little bit inconvenient for us as we most often work in the fppf topology.} $\\{S_i \\to S\\}_{i \\in I}$ such that each $X_{S_i} \\to S_i$ has a section (i.e., is a trivial pseudo $G_{S_i}$-torsor). \\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. We say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a {\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor} if there exists a $\\tau$ covering $\\{S_i \\to S\\}_{i \\in I}$ such that each $X_{S_i} \\to S_i$ has a section. \\item If $X$ is a $G$-torsor, then we say that it is {\\it quasi-isotrivial} if it is a torsor for the \\'etale topology. \\item If $X$ is a $G$-torsor, then we say that it is {\\it locally trivial} if it is a torsor for the Zariski topology. \\end{enumerate}"}
+{"_id": "9680", "title": "groupoids-definition-equivariant-module", "text": "Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and let $a : G \\times_S X \\to X$ be an action of the group scheme $G$ on $X/S$. A {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module}, or simply an {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module}, is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_S X}$-module map $$ \\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F} $$ where $\\text{pr}_1 : G \\times_S X \\to X$ is the projection such that \\begin{enumerate} \\item the diagram $$ \\xymatrix{ (1_G \\times a)^*\\text{pr}_1^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} & \\text{pr}_2^*\\mathcal{F} \\\\ (1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] & (m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha} } $$ is a commutative in the category of $\\mathcal{O}_{G \\times_S G \\times_S X}$-modules, and \\item the pullback $$ (e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F} $$ is the identity map. \\end{enumerate} For explanation compare with the relevant diagrams of Equation (\\ref{equation-action})."}
+{"_id": "9681", "title": "groupoids-definition-groupoid", "text": "Let $S$ be a scheme. \\begin{enumerate} \\item A {\\it groupoid scheme over $S$}, or simply a {\\it groupoid over $S$} is a quintuple $(U, R, s, t, c)$ where $U$ and $R$ are schemes over $S$, and $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$ are morphisms of schemes over $S$ with the following property: For any scheme $T$ over $S$ the quintuple $$ (U(T), R(T), s, t, c) $$ is a groupoid category in the sense described above. \\item A {\\it morphism $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ of groupoid schemes over $S$} is given by morphisms of schemes $f : U \\to U'$ and $f : R \\to R'$ with the following property:  For any scheme $T$ over $S$ the maps $f$ define a functor from the groupoid category $(U(T), R(T), s, t, c)$ to the groupoid category $(U'(T), R'(T), s', t', c')$. \\end{enumerate}"}
+{"_id": "9682", "title": "groupoids-definition-groupoid-module", "text": "Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. A {\\it quasi-coherent module on $(U, R, s, t, c)$} is a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module map $$ \\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F} $$ such that \\begin{enumerate} \\item the diagram $$ \\xymatrix{ & \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} & \\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\ \\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\ & \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] & c^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha} } $$ is a commutative in the category of $\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and \\item the pullback $$ e^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F} $$ is the identity map. \\end{enumerate} Compare with the commutative diagrams of Lemma \\ref{lemma-diagram}."}
+{"_id": "9683", "title": "groupoids-definition-stabilizer-groupoid", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The group scheme $j^{-1}(\\Delta_{U/S})\\to U$ is called the {\\it stabilizer of the groupoid scheme $(U, R, s, t, c)$}."}
+{"_id": "9684", "title": "groupoids-definition-restrict-groupoid", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \\to U$ be a morphism of schemes. The morphism of groupoids $(U', R', s', t', c') \\to (U, R, s, t, c)$ constructed in Lemma \\ref{lemma-restrict-groupoid} is called the {\\it restriction of $(U, R, s, t, c)$ to $U'$}. We sometime use the notation $R' = R|_{U'}$ in this case."}
+{"_id": "9685", "title": "groupoids-definition-invariant-open", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$. \\begin{enumerate} \\item A subset $W \\subset U$ is {\\it set-theoretically $R$-invariant} if $t(s^{-1}(W)) \\subset W$. \\item An open $W \\subset U$ is {\\it $R$-invariant} if $t(s^{-1}(W)) \\subset W$. \\item A closed subscheme $Z \\subset U$ is called {\\it $R$-invariant} if $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see Schemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}. \\item A monomorphism of schemes $T \\to U$ is {\\it $R$-invariant} if $T \\times_{U, t} R = R \\times_{s, U} T$ as schemes over $R$. \\end{enumerate}"}
+{"_id": "9686", "title": "groupoids-definition-quotient-sheaf", "text": "Let $\\tau$, $S$, and the pre-relation $j : R \\to U \\times_S U$ be as above. In this setting the {\\it quotient sheaf $U/R$} associated to $j$ is the sheafification of the presheaf (\\ref{equation-quotient-presheaf}) in the $\\tau$-topology. If $j : R \\to U \\times_S U$ comes from the action of a group scheme $G/S$ on $U$ as in Lemma \\ref{lemma-groupoid-from-action} then we sometimes denote the quotient sheaf $U/G$."}
+{"_id": "9687", "title": "groupoids-definition-representable-quotient", "text": "In the situation of Definition \\ref{definition-quotient-sheaf}. We say that the pre-relation $j$ has a {\\it representable quotient} if the sheaf $U/R$ is representable. We will say a groupoid $(U, R, s, t, c)$ has a {\\it representable quotient} if the quotient $U/R$ with $j = (t, s)$ is representable."}
+{"_id": "9688", "title": "groupoids-definition-cartesian-morphism", "text": "Let $S$ be a scheme. Let $f : (U', R', s', t', c') \\to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. We say $f$ is {\\it cartesian}, or that {\\it $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$}, if the diagram $$ \\xymatrix{ R' \\ar[r]_f \\ar[d]_{s'} & R \\ar[d]^s \\\\ U' \\ar[r]^f & U } $$ is a fibre square in the category of schemes. A {\\it morphism of groupoid schemes cartesian over $(U, R, s, t, c)$} is a morphism of groupoid schemes compatible with the structure morphisms towards $(U, R, s, t, c)$."}
+{"_id": "9792", "title": "local-cohomology-definition-cd", "text": "Let $I \\subset A$ be a finitely generated ideal of a ring $A$. The smallest integer $d \\geq -1$ satisfying the equivalent conditions of Lemma \\ref{lemma-cd} is called the {\\it cohomological dimension of $I$ in $A$} and is denoted $\\text{cd}(A, I)$."}
+{"_id": "9793", "title": "local-cohomology-definition-depth-complex", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D^+_{\\textit{Coh}}(A)$. We define the {\\it $I$-depth} of $K$, denoted $\\text{depth}_I(K)$, to be the maximal $m \\in \\mathbf{Z} \\cup \\{\\infty\\}$ such that $H^i_I(K) = 0$ for all $i < m$. If $A$ is local with maximal ideal $\\mathfrak m$ then we call $\\text{depth}_\\mathfrak m(K)$ simply the {\\it depth} of $K$."}
+{"_id": "10594", "title": "more-algebra-definition-stably-free", "text": "Let $R$ be a ring.  \\begin{enumerate} \\item Two modules $M$, $N$ over $R$ are said to be {\\it stably isomorphic} if there exist $n, m \\geq 0$ such that $M \\oplus R^{\\oplus m} \\cong N \\oplus R^{\\oplus n}$ as $R$-modules. \\item A module $M$ is {\\it stably free} if it is stably isomorphic to a free module. \\end{enumerate}"}
+{"_id": "10595", "title": "more-algebra-definition-fitting-ideal", "text": "Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$. The {\\it $k$th Fitting ideal} of $M$ is the ideal $\\text{Fit}_k(M)$ constructed in Lemma \\ref{lemma-fitting-ideal}. Set $\\text{Fit}_{-1}(M) = 0$."}
+{"_id": "10596", "title": "more-algebra-definition-zariski-pair", "text": "A {\\it Zariski pair} is a pair $(A, I)$ such that $I$ is contained in the Jacobson radical of $A$."}
+{"_id": "10597", "title": "more-algebra-definition-henselian-pair", "text": "A {\\it henselian pair} is a pair $(A, I)$ satisfying \\begin{enumerate} \\item $I$ is contained in the Jacobson radical of $A$, and \\item for any monic polynomial $f \\in A[T]$ and factorization $\\overline{f} = g_0h_0$ with $g_0, h_0 \\in A/I[T]$ monic generating the unit ideal in $A/I[T]$, there exists a factorization $f = gh$ in $A[T]$ with $g, h$ monic and $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$. \\end{enumerate}"}
+{"_id": "10598", "title": "more-algebra-definition-absolutely-integrally-closed", "text": "A ring $A$ is {\\it absolutely integrally closed} if every monic $f \\in A[T]$ is a product of linear factors."}
+{"_id": "10599", "title": "more-algebra-definition-auto-ass", "text": "A ring $R$ is said to be {\\it auto-associated} if $R$ is local and its maximal ideal $\\mathfrak m$ is weakly associated to $R$."}
+{"_id": "10600", "title": "more-algebra-definition-torsion", "text": "Let $R$ be a domain. Let $M$ be an $R$-module. \\begin{enumerate} \\item We say an element $x \\in M$ is {\\it torsion} if there exists a nonzero $f \\in R$ such that $fx = 0$. \\item We say $M$ is {\\it torsion free} if the only torsion element of $M$ is $0$. \\end{enumerate}"}
+{"_id": "10601", "title": "more-algebra-definition-reflexive", "text": "Let $R$ be a domain. We say an $R$-module $M$ is {\\it reflexive} if the natural map $$ j : M \\longrightarrow \\Hom_R(\\Hom_R(M, R), R) $$ which sends $m \\in M$ to the map sending $\\varphi \\in \\Hom_R(M, R)$ to $\\varphi(m) \\in R$ is an isomorphism."}
+{"_id": "10602", "title": "more-algebra-definition-reflexive-hull", "text": "Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The module $M^{**} = \\Hom_R(\\Hom_R(M, R), R)$ is called the {\\it reflexive hull} of $M$."}
+{"_id": "10603", "title": "more-algebra-definition-content-ideal", "text": "Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$. If the set of ideals $I$ in $A$ such that $x \\in IM$ has a smallest element, we call it the {\\it content ideal of $x$}."}
+{"_id": "10604", "title": "more-algebra-definition-strict-transform", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal and $a \\in I$. Let $R[\\frac{I}{a}]$ be the affine blowup algebra, see Algebra, Definition \\ref{algebra-definition-blow-up}. Let $M$ be an $R$-module. The {\\it strict transform of $M$ along $R \\to R[\\frac{I}{a}]$} is the $R[\\frac{I}{a}]$-module $$ M' = \\left(M \\otimes_R R[\\textstyle{\\frac{I}{a}}]\\right)/a\\text{-power torsion} $$"}
+{"_id": "10605", "title": "more-algebra-definition-koszul", "text": "Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. The {\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$ is the commutative differential graded algebra defined as follows: \\begin{enumerate} \\item the underlying graded algebra is the exterior algebra $K_\\bullet(\\varphi) = \\wedge(E)$, \\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$ is the unique derivation such that $d(e) = \\varphi(e)$ for all $e \\in E = K_1(\\varphi)$. \\end{enumerate}"}
+{"_id": "10606", "title": "more-algebra-definition-koszul-complex", "text": "Let $R$ be a ring and let $f_1, \\ldots, f_r \\in R$. The {\\it Koszul complex on $f_1, \\ldots, f_r$} is the Koszul complex associated to the map $(f_1, \\ldots, f_r) : R^{\\oplus r} \\to R$. Notation $K_\\bullet(f_\\bullet)$, $K_\\bullet(f_1, \\ldots, f_r)$, $K_\\bullet(R, f_1, \\ldots, f_r)$, or $K_\\bullet(R, f_\\bullet)$."}
+{"_id": "10607", "title": "more-algebra-definition-koszul-regular-sequence", "text": "Let $R$ be a ring. Let $r \\geq 0$ and let $f_1, \\ldots, f_r \\in R$ be a sequence of elements. Let $M$ be an $R$-module. The sequence $f_1, \\ldots, f_r$ is called \\begin{enumerate} \\item {\\it $M$-Koszul-regular} if $H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$ for all $i \\not = 0$, \\item {\\it $M$-$H_1$-regular} if $H_1(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$, \\item {\\it Koszul-regular} if $H_i(K_\\bullet(f_1, \\ldots, f_r)) = 0$ for all $i \\not = 0$, and \\item {\\it $H_1$-regular} if $H_1(K_\\bullet(f_1, \\ldots, f_r)) = 0$. \\end{enumerate}"}
+{"_id": "10608", "title": "more-algebra-definition-regular-ideal", "text": "Let $R$ be a ring and let $I \\subset R$ be an ideal. \\begin{enumerate} \\item We say $I$ is a {\\it regular ideal} if for every $\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ and a regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$ is generated by $f_1, \\ldots, f_r$. \\item We say $I$ is a {\\it Koszul-regular ideal} if for every $\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ and a Koszul-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$ is generated by $f_1, \\ldots, f_r$. \\item We say $I$ is a {\\it $H_1$-regular ideal} if for every $\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ and an $H_1$-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$ is generated by $f_1, \\ldots, f_r$. \\item We say $I$ is a {\\it quasi-regular ideal} if for every $\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ and a quasi-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$ is generated by $f_1, \\ldots, f_r$. \\end{enumerate}"}
+{"_id": "10609", "title": "more-algebra-definition-local-complete-intersection", "text": "A ring map $A \\to B$ is called a {\\it local complete intersection} if it is of finite type and for some (equivalently any) presentation $B = A[x_1, \\ldots, x_n]/I$ the ideal $I$ is Koszul-regular."}
+{"_id": "10610", "title": "more-algebra-definition-topological-ring", "text": "\\begin{reference} \\cite[Sections 7.1 and 7.2]{EGA1} \\end{reference} Let $R$ be a ring and let $M$ be an $R$-module. \\begin{enumerate} \\item We say $R$ is a {\\it topological ring} if $R$ is endowed with a topology such that both addition and multiplication are continuous as maps $R \\times R \\to R$ where $R \\times R$ has the product topology. In this case we say $M$ is a {\\it topological module} if $M$ is endowed with a topology such that addition $M \\times M \\to M$ and scalar multiplication $R \\times M \\to M$ are continuous. \\item A {\\it homomorphism of topological modules} is just a continuous $R$-module map. A {\\it homomorphism of topological rings} is a ring homomorphism which is continuous for the given topologies. \\item We say $M$ is {\\it linearly topologized} if $0$ has a fundamental system of neighbourhoods consisting of submodules. We say $R$ is {\\it linearly topologized} if $0$ has a fundamental system of neighbourhoods consisting of ideals. \\item If $R$ is linearly topologized, we say that $I \\subset R$ is an {\\it ideal of definition} if $I$ is open and if every neighbourhood of $0$ contains $I^n$ for some $n$. \\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-admissible} if $R$ has an ideal of definition. \\item If $R$ is linearly topologized, we say that $R$ is {\\it admissible} if it is pre-admissible and complete\\footnote{By our conventions this includes separated.}. \\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-adic} if there exists an ideal of definition $I$ such that $\\{I^n\\}_{n \\geq 0}$ forms a fundamental system of neighbourhoods of $0$. \\item If $R$ is linearly topologized, we say that $R$ is {\\it adic} if $R$ is pre-adic and complete. \\end{enumerate} Note that a (pre)adic topological ring is the same thing as a (pre)admissible topological ring which has an ideal of definition $I$ such that $I^n$ is open for all $n \\geq 1$."}
+{"_id": "10611", "title": "more-algebra-definition-formally-smooth", "text": "Let $R \\to S$ be a homomorphism of topological rings with $R$ and $S$ linearly topologized. We say $S$ is {\\it formally smooth over $R$} if for every commutative solid diagram $$ \\xymatrix{ S \\ar[r] \\ar@{-->}[rd] & A/J \\\\ R \\ar[r] \\ar[u] & A \\ar[u] } $$ of homomorphisms of topological rings where $A$ is a discrete ring and $J \\subset A$ is an ideal of square zero, a dotted arrow exists which makes the diagram commute."}
+{"_id": "10612", "title": "more-algebra-definition-formally-smooth-adic", "text": "Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset S$ be an ideal. If the equivalent conditions (2)(a) and (2)(b) of Lemma \\ref{lemma-formally-smooth} hold, then we say $R \\to S$ is {\\it formally smooth for the $\\mathfrak n$-adic topology}."}
+{"_id": "10613", "title": "more-algebra-definition-regular", "text": "A ring map $R \\to \\Lambda$ is {\\it regular} if it is flat and for every prime $\\mathfrak p \\subset R$ the fibre ring $$ \\Lambda \\otimes_R \\kappa(\\mathfrak p) = \\Lambda_\\mathfrak p/\\mathfrak p\\Lambda_\\mathfrak p $$ is Noetherian and geometrically regular over $\\kappa(\\mathfrak p)$."}
+{"_id": "10614", "title": "more-algebra-definition-p-basis", "text": "Let $p$ be a prime number. Let $k \\to K$ be an extension of fields of characteristic $p$. Denote $kK^p$ the compositum of $k$ and $K^p$ in $K$. \\begin{enumerate} \\item A subset $\\{x_i\\} \\subset K$ is called {\\it p-independent over $k$} if the elements $x^E = \\prod x_i^{e_i}$ where $0 \\leq e_i < p$ are linearly independent over $kK^p$. \\item A subset $\\{x_i\\}$ of $K$ is called a {\\it p-basis of $K$ over $k$} if the elements $x^E$ form a basis of $K$ over $kK^p$. \\end{enumerate}"}
+{"_id": "10615", "title": "more-algebra-definition-J", "text": "Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$. \\begin{enumerate} \\item We say $R$ is {\\it J-0} if $\\text{Reg}(X)$ contains a nonempty open. \\item We say $R$ is {\\it J-1} if $\\text{Reg}(X)$ is open. \\item We say $R$ is {\\it J-2} if any finite type $R$-algebra is J-1. \\end{enumerate}"}
+{"_id": "10616", "title": "more-algebra-definition-G-ring", "text": "A ring $R$ is called a {\\it G-ring} if $R$ is Noetherian and for every prime $\\mathfrak p$ of $R$ the ring map $R_\\mathfrak p \\to (R_\\mathfrak p)^\\wedge$ is regular."}
+{"_id": "10617", "title": "more-algebra-definition-excellent", "text": "Let $R$ be a ring. \\begin{enumerate} \\item We say $R$ is {\\it quasi-excellent} if $R$ is Noetherian, a G-ring, and J-2. \\item We say $R$ is {\\it excellent} if $R$ is quasi-excellent and universally catenary. \\end{enumerate}"}
+{"_id": "10618", "title": "more-algebra-definition-projective", "text": "Let $R$ be a ring. An $R$-module $J$ is {\\it injective} if and only if the functor $\\Hom_R(-, J) : \\text{Mod}_R \\to \\text{Mod}_R$ is an exact functor."}
+{"_id": "10619", "title": "more-algebra-definition-simple-functors", "text": "Let $R$ be a ring. \\begin{enumerate} \\item For any $R$-module $M$ over $R$ we denote $M^\\vee = \\Hom(M, \\mathbf{Q}/\\mathbf{Z})$ with its natural $R$-module structure. We think of {\\it $M \\mapsto M^\\vee$} as a contravariant functor from the category of $R$-modules to itself. \\item For any $R$-module $M$ we denote $$ F(M) = \\bigoplus\\nolimits_{m \\in M} R[m] $$ the {\\it free module} with basis given by the elements $[m]$ with $m \\in M$. We let $F(M)\\to M$, $\\sum f_i [m_i] \\mapsto \\sum f_i m_i$ be the natural surjection of $R$-modules. We think of $M \\mapsto (F(M) \\to M)$ as a functor from the category of $R$-modules to the category of arrows in $R$-modules. \\end{enumerate}"}
+{"_id": "10620", "title": "more-algebra-definition-K-flat", "text": "Let $R$ be a ring. A complex $K^\\bullet$ is called {\\it K-flat} if for every acyclic complex $M^\\bullet$ the total complex $\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet)$ is acyclic."}
+{"_id": "10621", "title": "more-algebra-definition-derived-tor", "text": "Let $R$ be a ring. Let $M^\\bullet$ be an object of $D(R)$. The {\\it derived tensor product} $$ - \\otimes_R^{\\mathbf{L}} M^\\bullet : D(R) \\longrightarrow D(R) $$ is the exact functor of triangulated categories described above."}
+{"_id": "10622", "title": "more-algebra-definition-tor-independent", "text": "Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. We say $A$ and $B$ are {\\it Tor independent over $R$} if $\\text{Tor}_p^R(A, B) = 0$ for all $p > 0$."}
+{"_id": "10623", "title": "more-algebra-definition-pseudo-coherent", "text": "Let $R$ be a ring. Denote $D(R)$ its derived category. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item An object $K^\\bullet$ of $D(R)$ is {\\it $m$-pseudo-coherent} if there exists a bounded complex $E^\\bullet$ of finite free $R$-modules and a morphism $\\alpha : E^\\bullet \\to K^\\bullet$ such that $H^i(\\alpha)$ is an isomorphism for $i > m$ and $H^m(\\alpha)$ is surjective. \\item An object $K^\\bullet$ of $D(R)$ is {\\it pseudo-coherent} if it is quasi-isomorphic to a bounded above complex of finite free $R$-modules. \\item An $R$-module $M$ is called {\\it $m$-pseudo-coherent} if $M[0]$ is an $m$-pseudo-coherent object of $D(R)$. \\item An $R$-module $M$ is called {\\it pseudo-coherent}\\footnote{This clashes with what is meant by a pseudo-coherent module in \\cite{Bourbaki-CA}.} if $M[0]$ is a pseudo-coherent object of $D(R)$. \\end{enumerate}"}
+{"_id": "10624", "title": "more-algebra-definition-tor-amplitude", "text": "Let $R$ be a ring. Denote $D(R)$ its derived category. Let $a, b \\in \\mathbf{Z}$. \\begin{enumerate} \\item An object $K^\\bullet$ of $D(R)$ has {\\it tor-amplitude in $[a, b]$} if $H^i(K^\\bullet \\otimes_R^\\mathbf{L} M) = 0$ for all $R$-modules $M$ and all $i \\not \\in [a, b]$. \\item An object $K^\\bullet$ of $D(R)$ has {\\it finite tor dimension} if it has tor-amplitude in $[a, b]$ for some $a, b$. \\item An $R$-module $M$ has {\\it tor dimension $\\leq d$} if $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$. \\item An $R$-module $M$ has {\\it finite tor dimension} if $M[0]$ as an object of $D(R)$ has finite tor dimension. \\end{enumerate}"}
+{"_id": "10625", "title": "more-algebra-definition-projective-dimension", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has {\\it finite projective dimension} if $K$ can be represented by a bounded complex of projective modules. We say $K$ as {\\it projective-amplitude in $[a, b]$} if  $K$ is quasi-isomorphic to a complex $$ \\ldots \\to 0 \\to P^a \\to P^{a + 1} \\to \\ldots \\to P^{b - 1} \\to P^b \\to 0 \\to \\ldots $$ where $P^i$ is a projective $R$-module for all $i \\in \\mathbf{Z}$."}
+{"_id": "10626", "title": "more-algebra-definition-injective-dimension", "text": "Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has {\\it finite injective dimension} if $K$ can be represented by a finite complex of injective $R$-modules. We say $K$ has {\\it injective-amplitude in $[a, b]$} if $K$ is isomorphic to a complex $$ \\ldots \\to 0 \\to I^a \\to I^{a + 1} \\to \\ldots \\to I^{b - 1} \\to I^b \\to 0 \\to \\ldots $$ with $I^i$ an injective $R$-module for all $i \\in \\mathbf{Z}$."}
+{"_id": "10627", "title": "more-algebra-definition-near-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is {\\it $I$-projective}\\footnote{This is nonstandard notation.} if the equivalent conditions of Lemma \\ref{lemma-near-projective} hold."}
+{"_id": "10628", "title": "more-algebra-definition-perfect", "text": "Let $R$ be a ring. Denote $D(R)$ the derived category of the abelian category of $R$-modules. \\begin{enumerate} \\item An object $K$ of $D(R)$ is {\\it perfect} if it is quasi-isomorphic to a bounded complex of finite projective $R$-modules. \\item An $R$-module $M$ is {\\it perfect} if $M[0]$ is a perfect object in $D(R)$. \\end{enumerate}"}
+{"_id": "10629", "title": "more-algebra-definition-relatively-finitely-presented", "text": "Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module. We say $M$ is an $A$-module {\\it finitely presented relative to $R$} if the equivalent conditions of Lemma \\ref{lemma-relatively-finitely-presented} hold."}
+{"_id": "10630", "title": "more-algebra-definition-relatively-pseudo-coherent", "text": "Let $R \\to A$ be a finite type ring map. Let $K^\\bullet$ be a complex of $A$-modules. Let $M$ be an $A$-module. Let $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item We say $K^\\bullet$ is {\\it $m$-pseudo-coherent relative to $R$} if the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} hold. \\item We say $K^\\bullet$ is {\\it pseudo-coherent relative to $R$} if $K^\\bullet$ is $m$-pseudo-coherent relative to $R$ for all $m \\in \\mathbf{Z}$. \\item We say $M$ is {\\it $m$-pseudo-coherent relative to $R$} if $M[0]$ is $m$-pseudo-coherent relative to $R$. \\item We say $M$ is {\\it pseudo-coherent relative to $R$} if $M[0]$ is pseudo-coherent relative to $R$. \\end{enumerate}"}
+{"_id": "10631", "title": "more-algebra-definition-pseudo-coherent-perfect", "text": "Let $A \\to B$ be a ring map. \\begin{enumerate} \\item We say $A \\to B$ is a {\\it pseudo-coherent ring map} if it is of finite type and $B$, as a $B$-module, is pseudo-coherent relative to $A$. \\item We say $A \\to B$ is a {\\it perfect ring map} if it is a pseudo-coherent ring map such that $B$ as an $A$-module has finite tor dimension. \\end{enumerate}"}
+{"_id": "10632", "title": "more-algebra-definition-relatively-perfect", "text": "Let $R \\to A$ be a flat ring map of finite presentation. An object $K$ of $D(A)$ is {\\it $R$-perfect} or {\\it perfect relative to $R$} if $K$ is pseudo-coherent (Definition \\ref{definition-pseudo-coherent}) and has finite tor dimension over $R$ (Definition \\ref{definition-tor-amplitude})."}
+{"_id": "10633", "title": "more-algebra-definition-f-power-torsion", "text": "Let $R$ be a ring. Let $M$ be an $R$-module. \\begin{enumerate} \\item Let $I \\subset R$ be an ideal. We say $M$ is an {\\it $I$-power torsion module} if for every $m \\in M$ there exists an $n > 0$ such that $I^n m = 0$. \\item Let $f \\in R$. We say $M$ is {\\it an $f$-power torsion module} if for each $m \\in M$, there exists an $n > 0$ such that $f^n m = 0$. \\end{enumerate}"}
+{"_id": "10634", "title": "more-algebra-definition-derived-complete", "text": "Let $A$ be a ring. Let $K \\in D(A)$. Let $I \\subset A$ be an ideal. We say $K$ is {\\it derived complete with respect to $I$} if for every $f \\in I$ we have $T(K, f) = 0$. If $M$ is an $A$-module, then we say $M$ is {\\it derived complete with respect to $I$} if $M[0] \\in D(A)$ is derived complete with respect to $I$."}
+{"_id": "10635", "title": "more-algebra-definition-weakly-etale", "text": "A ring $A$ is called {\\it absolutely flat} if every $A$-module is flat over $A$. A ring map $A \\to B$ is {\\it weakly \\'etale} or {\\it absolutely flat} if both $A \\to B$ and $B \\otimes_A B \\to B$ are flat."}
+{"_id": "10636", "title": "more-algebra-definition-weak-dimension", "text": "Let $A$ be a ring. Let $d \\geq 0$ be an integer. We say that $A$ has {\\it weak dimension $\\leq d$} if every $A$-module has tor dimension $\\leq d$."}
+{"_id": "10637", "title": "more-algebra-definition-unibranch", "text": "\\begin{reference} \\cite[Chapter 0 (23.2.1)]{EGA4} \\end{reference} Let $A$ be a local ring. We say $A$ is {\\it unibranch} if the reduction $A_{red}$ is a domain and if the integral closure $A'$ of $A_{red}$ in its field of fractions is local. We say $A$ is {\\it geometrically unibranch} if $A$ is unibranch and moreover the residue field of $A'$ is purely inseparable over the residue field of $A$."}
+{"_id": "10638", "title": "more-algebra-definition-number-of-branches", "text": "Let $A$ be a local ring with henselization $A^h$ and strict henselization $A^{sh}$. The {\\it number of branches of $A$} is the number of minimal primes of $A^h$ if finite and $\\infty$ otherwise. The {\\it number of geometric branches of $A$} is the number of minimal primes of $A^{sh}$ if finite and $\\infty$ otherwise."}
+{"_id": "10639", "title": "more-algebra-definition-formally-catenary", "text": "A Noetherian local ring $A$ is {\\it formally catenary} if for every minimal prime $\\mathfrak p \\subset A$ the spectrum of $A^\\wedge/\\mathfrak p A^\\wedge$ is equidimensional."}
+{"_id": "10640", "title": "more-algebra-definition-extension-discrete-valuation-rings", "text": "We say that $A \\to B$ or $A \\subset B$ is an {\\it extension of discrete valuation rings} if $A$ and $B$ are discrete valuation rings and $A \\to B$ is injective and local. In particular, if $\\pi_A$ and $\\pi_B$ are uniformizers of $A$ and $B$, then $\\pi_A = u \\pi_B^e$ for some $e \\geq 1$ and unit $u$ of $B$. The integer $e$ does not depend on the choice of the uniformizers as it is also the unique integer $\\geq 1$ such that $$ \\mathfrak m_A B = \\mathfrak m_B^e $$ The integer $e$ is called the {\\it ramification index} of $B$ over $A$. We say that $B$ is {\\it weakly unramified} over $A$ if $e = 1$. If the extension of residue fields $\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$ is finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we call it the {\\it residual degree} or {\\it residue degree} of the extension $A \\subset B$."}
+{"_id": "10641", "title": "more-algebra-definition-types-of-extensions", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L \\supset K$ be a finite separable extension. With $B$ and $\\mathfrak m_i$, $i = 1, \\ldots, n$ as in Remark \\ref{remark-finite-separable-extension} we say the extension $L/K$ is \\begin{enumerate} \\item {\\it unramified with respect to $A$} if $e_i = 1$ and the extension $\\kappa_A \\subset \\kappa(\\mathfrak m_i)$ is separable for all $i$, \\item {\\it tamely ramified with respect to $A$} if either the characteristic of $\\kappa_A$ is $0$ or the characteristic of $\\kappa_A$ is $p > 0$, the field extensions $\\kappa_A \\subset \\kappa(\\mathfrak m_i)$ are separable, and the ramification indices $e_i$ are prime to $p$, and \\item {\\it totally ramified with respect to $A$} if $n = 1$ and the residue field extension $\\kappa_A \\subset \\kappa(\\mathfrak m_1)$ is trivial. \\end{enumerate} If the discrete valuation ring $A$ is clear from context, then we sometimes say $L/K$ is unramified, totally ramified, or tamely ramified for short."}
+{"_id": "10642", "title": "more-algebra-definition-decomposition-inertia", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m \\subset B$ be a maximal ideal. \\begin{enumerate} \\item The {\\it decomposition group of $\\mathfrak m$} is the subgroup $D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$. \\item The {\\it inertia group of $\\mathfrak m$} is the kernel $I$ of the map $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)$. \\end{enumerate}"}
+{"_id": "10643", "title": "more-algebra-definition-wild-inertia", "text": "With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}. \\begin{enumerate} \\item The {\\it wild inertia group of $\\mathfrak m$} is the subgroup $P$. \\item The {\\it tame inertia group of $\\mathfrak m$} is the quotient $I \\to I_t$. \\end{enumerate} We denote $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$ the surjective map (\\ref{equation-inertia-character}) whose kernel is $P$ and which induces the isomorphism $I_t \\to \\mu_e(\\kappa(\\mathfrak m))$."}
+{"_id": "10644", "title": "more-algebra-definition-mixed", "text": "Let $A$ be a discrete valuation ring. We say $A$ has {\\it mixed characteristic} if the characteristic of the residue field of $A$ is $p > 0$ and the characteristic of the fraction field of $A$ is $0$. In this case we obtain an extension of discrete valuation rings $\\mathbf{Z}_{(p)} \\subset A$ and the {\\it absolute ramification index} of $A$ is the ramification index of this extension."}
+{"_id": "10645", "title": "more-algebra-definition-solution", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. \\begin{enumerate} \\item We say a finite field extension $K \\subset K_1$ is a {\\it weak solution for $A \\subset B$} if all the extensions $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of Remark \\ref{remark-construction} are weakly unramified. \\item We say a finite field extension $K \\subset K_1$ is a {\\it solution for $A \\subset B$} if each extension $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of Remark \\ref{remark-construction} is formally smooth in the $\\mathfrak m_{ij}$-adic topology. \\end{enumerate} We say a solution $K \\subset K_1$ is a {\\it separable solution} if $K \\subset K_1$ is separable."}
+{"_id": "10646", "title": "more-algebra-definition-invertible", "text": "Let $R$ be a ring. An $R$-module $M$ is {\\it invertible} if the functor $$ \\text{Mod}_R \\longrightarrow \\text{Mod}_R,\\quad N \\longmapsto M \\otimes_R N $$ is an equivalence of categories. An invertible $R$-module is said to be {\\it trivial} if it is isomorphic to $A$ as an $A$-module."}
+{"_id": "10647", "title": "more-algebra-definition-extension-valuation-rings", "text": "We say that $A \\to B$ or $A \\subset B$ is an {\\it extension of valuation rings} if $A$ and $B$ are valuation rings and $A \\to B$ is injective and local. Such an extension induces a commutative diagram $$ \\xymatrix{ A \\setminus \\{0\\} \\ar[r] \\ar[d]_v & B \\setminus \\{0\\} \\ar[d]^v \\\\ \\Gamma_A \\ar[r] & \\Gamma_B } $$ where $\\Gamma_A$ and $\\Gamma_B$ are the value groups. We say that $B$ is {\\it weakly unramified} over $A$ if the lower horizontal arrow is a bijection. If the extension of residue fields $\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$ is finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we call it the {\\it residual degree} or {\\it residue degree} of the extension $A \\subset B$."}
+{"_id": "10648", "title": "more-algebra-definition-bezout", "text": "Let $R$ be a domain. \\begin{enumerate} \\item We say $R$ is a {\\it B\\'ezout domain} if every finitely generated ideal of $R$ is principal. \\item We say $R$ is an {\\it elementary divisor domain} if for all $n , m \\geq 1$ and every $n \\times m$ matrix $A$, there exist invertible matrices $U, V$ of size $n \\times n, m \\times m$ such that $$ U A V = \\left( \\begin{matrix} f_1 & 0 & 0 & \\ldots \\\\ 0 & f_2 & 0 & \\ldots \\\\ 0 & 0 & f_3 & \\ldots \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\end{matrix} \\right) $$ with $f_1, \\ldots, f_{\\min(n, m)} \\in R$ and $f_1 | f_2 | \\ldots$. \\end{enumerate}"}
+{"_id": "10734", "title": "etale-definition-unramified-rings", "text": "Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \\to B$ is said to be {\\it unramified homomorphism of local rings} if \\begin{enumerate} \\item $\\mathfrak m_AB = \\mathfrak m_B$, \\item $\\kappa(\\mathfrak m_B)$ is a finite separable extension of $\\kappa(\\mathfrak m_A)$, and \\item $B$ is essentially of finite type over $A$ (this means that $B$ is the localization of a finite type $A$-algebra at a prime). \\end{enumerate}"}
+{"_id": "10735", "title": "etale-definition-unramified-schemes", "text": "(See Morphisms, Definition \\ref{morphisms-definition-unramified} for the definition in the general case.) Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$. \\begin{enumerate} \\item We say $f$ is {\\it unramified at $x$} if $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is an unramified homomorphism of local rings. \\item The morphism $f : X \\to Y$ is said to be {\\it unramified} if it is unramified at all points of $X$. \\end{enumerate}"}
+{"_id": "10736", "title": "etale-definition-flat-rings", "text": "Flatness of modules and rings. \\begin{enumerate} \\item A module $N$ over a ring $A$ is said to be {\\it flat} if the functor $M \\mapsto M \\otimes_A N$ is exact. \\item If this functor is also faithful, we say that $N$ is {\\it faithfully flat} over $A$. \\item A morphism of rings $f : A \\to B$ is said to be {\\it flat (resp. faithfully flat)} if the functor $M \\mapsto M \\otimes_A B$ is exact (resp. faithful and exact). \\end{enumerate}"}
+{"_id": "10737", "title": "etale-definition-flat-schemes", "text": "(See Morphisms, Definition \\ref{morphisms-definition-flat}). Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item Let $x \\in X$. We say $\\mathcal{F}$ is {\\it flat over $Y$ at $x \\in X$} if $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{Y, f(x)}$-module. This uses the map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ to think of $\\mathcal{F}_x$ as a $\\mathcal{O}_{Y, f(x)}$-module. \\item Let $x \\in X$. We say $f$ is {\\it flat at $x \\in X$} if $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat. \\item We say $f$ is {\\it flat} if it is flat at all points of $X$. \\item A morphism $f : X \\to Y$ that is flat and surjective is sometimes said to be {\\it faithfully flat}. \\end{enumerate}"}
+{"_id": "10738", "title": "etale-definition-etale-ring", "text": "Let $A$, $B$ be Noetherian local rings. A local homomorphism $f : A \\to B$ is said to be an {\\it \\'etale homomorphism of local rings} if it is flat and an unramified homomorphism of local rings (please see Definition \\ref{definition-unramified-rings})."}
+{"_id": "10739", "title": "etale-definition-etale-schemes-1", "text": "(See Morphisms, Definition \\ref{morphisms-definition-etale}.) Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type. \\begin{enumerate} \\item Let $x \\in X$. We say $f$ is {\\it \\'etale at $x \\in X$} if $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is an \\'etale homomorphism of local rings. \\item The morphism is said to be {\\it \\'etale} if it is \\'etale at all its points. \\end{enumerate}"}
+{"_id": "10740", "title": "etale-definition-strict-normal-crossings", "text": "Let $X$ be a locally Noetherian scheme. A {\\it strict normal crossings divisor} on $X$ is an effective Cartier divisor $D \\subset X$ such that for every $p \\in D$ the local ring $\\mathcal{O}_{X, p}$ is regular and there exists a regular system of parameters $x_1, \\ldots, x_d \\in \\mathfrak m_p$ and $1 \\leq r \\leq d$ such that $D$ is cut out by $x_1 \\ldots x_r$ in $\\mathcal{O}_{X, p}$."}
+{"_id": "10741", "title": "etale-definition-normal-crossings", "text": "Let $X$ be a locally Noetherian scheme. A {\\it normal crossings divisor} on $X$ is an effective Cartier divisor $D \\subset X$ such that for every $p \\in D$ there exists an \\'etale morphism $U \\to X$ with $p$ in the image and $D \\times_X U$ a strict normal crossings divisor on $U$."}
+{"_id": "10797", "title": "crystalline-definition-divided-power-envelope", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal with $IB \\subset J$. The divided power algebra $(D, \\bar J, \\bar\\gamma)$ constructed in Lemma \\ref{lemma-divided-power-envelope} is called the {\\it divided power envelope of $J$ in $B$ relative to $(A, I, \\gamma)$} and is denoted $D_B(J)$ or $D_{B, \\gamma}(J)$."}
+{"_id": "10798", "title": "crystalline-definition-compatible", "text": "Let $(A, I, \\gamma)$ and $(B, J, \\delta)$ be divided power rings. Let $A \\to B$ be a ring map. We say {\\it $\\delta$ is compatible with $\\gamma$} if there exists a divided power structure $\\bar\\gamma$ on $J + IB$ such that $$ (A, I, \\gamma) \\to (B, J + IB, \\bar \\gamma)\\quad\\text{and}\\quad (B, J, \\delta) \\to (B, J + IB, \\bar \\gamma) $$ are homomorphisms of divided power rings."}
+{"_id": "10799", "title": "crystalline-definition-affine-thickening", "text": "In Situation \\ref{situation-affine}. \\begin{enumerate} \\item A {\\it divided power thickening} of $C$ over $(A, I, \\gamma)$ is a homomorphism of divided power algebras $(A, I, \\gamma) \\to (B, J, \\delta)$ such that $p$ is nilpotent in $B$ and a ring map $C \\to B/J$ such that $$ \\xymatrix{ B \\ar[r] & B/J \\\\ & C \\ar[u] \\\\ A \\ar[uu] \\ar[r] & A/I \\ar[u] } $$ is commutative. \\item A {\\it homomorphism of divided power thickenings} $$ (B, J, \\delta, C \\to B/J) \\longrightarrow (B', J', \\delta', C \\to B'/J') $$ is a homomorphism $\\varphi : B \\to B'$ of divided power $A$-algebras such that $C \\to B/J \\to B'/J'$ is the given map $C \\to B'/J'$. \\item We denote $\\text{CRIS}(C/A, I, \\gamma)$ or simply $\\text{CRIS}(C/A)$ the category of divided power thickenings of $C$ over $(A, I, \\gamma)$. \\item We denote $\\text{Cris}(C/A, I, \\gamma)$ or simply $\\text{Cris}(C/A)$ the full subcategory consisting of $(B, J, \\delta, C \\to B/J)$ such that $C \\to B/J$ is an isomorphism. We often denote such an object $(B \\to C, \\delta)$ with $J = \\Ker(B \\to C)$ being understood. \\end{enumerate}"}
+{"_id": "10800", "title": "crystalline-definition-derivation", "text": "Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring. Let $A \\to B$ be a ring map. Let $M$ be an $B$-module. A {\\it divided power $A$-derivation} into $M$ is a map $\\theta : B \\to M$ which is additive, annihilates the elements of $A$, satisfies the Leibniz rule $\\theta(bb') = b\\theta(b') + b'\\theta(b)$ and satisfies $$ \\theta(\\delta_n(x)) = \\delta_{n - 1}(x)\\theta(x) $$ for all $n \\geq 1$ and all $x \\in J$."}
+{"_id": "10801", "title": "crystalline-definition-divided-power-structure", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals. A {\\it divided power structure $\\gamma$} on $\\mathcal{I}$ is a sequence of maps $\\gamma_n : \\mathcal{I} \\to \\mathcal{I}$, $n \\geq 1$ such that for any object $U$ of $\\mathcal{C}$ the triple $$ (\\mathcal{O}(U), \\mathcal{I}(U), \\gamma) $$ is a divided power ring."}
+{"_id": "10802", "title": "crystalline-definition-divided-power-scheme", "text": "A {\\it divided power scheme} is a triple $(S, \\mathcal{I}, \\gamma)$ where $S$ is a scheme, $\\mathcal{I}$ is a quasi-coherent sheaf of ideals, and $\\gamma$ is a divided power structure on $\\mathcal{I}$. A {\\it morphism of divided power schemes} $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ is a morphism of schemes $f : S \\to S'$ such that $f^{-1}\\mathcal{I}'\\mathcal{O}_S \\subset \\mathcal{I}$ and such that $$ (\\mathcal{O}_{S'}(U'), \\mathcal{I}'(U'), \\gamma') \\longrightarrow (\\mathcal{O}_S(f^{-1}U'), \\mathcal{I}(f^{-1}U'), \\gamma) $$ is a homomorphism of divided power rings for all $U' \\subset S'$ open."}
+{"_id": "10803", "title": "crystalline-definition-divided-power-thickening", "text": "A triple $(U, T, \\gamma)$ as above is called a {\\it divided power thickening} if $U \\to T$ is a thickening."}
+{"_id": "10804", "title": "crystalline-definition-divided-power-thickening-X", "text": "In Situation \\ref{situation-global}. \\begin{enumerate} \\item A {\\it divided power thickening of $X$ relative to $(S, \\mathcal{I}, \\gamma)$} is given by a divided power thickening $(U, T, \\delta)$ over $(S, \\mathcal{I}, \\gamma)$ and an $S$-morphism $U \\to X$. \\item A {\\it morphism of divided power thickenings of $X$ relative to $(S, \\mathcal{I}, \\gamma)$} is defined in the obvious manner. \\end{enumerate} The category of divided power thickenings of $X$ relative to $(S, \\mathcal{I}, \\gamma)$ is denoted $\\text{CRIS}(X/S, \\mathcal{I}, \\gamma)$ or simply $\\text{CRIS}(X/S)$."}
+{"_id": "10805", "title": "crystalline-definition-big-crystalline-site", "text": "In Situation \\ref{situation-global}. \\begin{enumerate} \\item A family of morphisms $\\{(U_i, T_i, \\delta_i) \\to (U, T, \\delta)\\}$ of divided power thickenings of $X/S$ is a {\\it Zariski, \\'etale, smooth, syntomic, or fppf covering} if and only if \\begin{enumerate} \\item $U_i = U \\times_T T_i$ for all $i$ and \\item $\\{T_i \\to T\\}$ is a Zariski, \\'etale, smooth, syntomic, or fppf covering. \\end{enumerate} \\item The {\\it big crystalline site} of $X$ over $(S, \\mathcal{I}, \\gamma)$, is the category $\\text{CRIS}(X/S)$ endowed with the Zariski topology. \\item The topos of sheaves on $\\text{CRIS}(X/S)$ is denoted $(X/S)_{\\text{CRIS}}$ or sometimes $(X/S, \\mathcal{I}, \\gamma)_{\\text{CRIS}}$\\footnote{This clashes with our convention to denote the topos associated to a site $\\mathcal{C}$ by $\\Sh(\\mathcal{C})$.}. \\end{enumerate}"}
+{"_id": "10806", "title": "crystalline-definition-crystalline-site", "text": "In Situation \\ref{situation-global}. \\begin{enumerate} \\item The (small) {\\it crystalline site} of $X$ over $(S, \\mathcal{I}, \\gamma)$, denoted $\\text{Cris}(X/S, \\mathcal{I}, \\gamma)$ or simply $\\text{Cris}(X/S)$ is the full subcategory of $\\text{CRIS}(X/S)$ consisting of those $(U, T, \\delta)$ in $\\text{CRIS}(X/S)$ such that $U \\to X$ is an open immersion. It comes endowed with the Zariski topology. \\item The topos of sheaves on $\\text{Cris}(X/S)$ is denoted $(X/S)_{\\text{cris}}$ or sometimes $(X/S, \\mathcal{I}, \\gamma)_{\\text{cris}}$\\footnote{This clashes with our convention to denote the topos associated to a site $\\mathcal{C}$ by $\\Sh(\\mathcal{C})$.}. \\end{enumerate}"}
+{"_id": "10807", "title": "crystalline-definition-modules", "text": "In Situation \\ref{situation-global}. Let $\\mathcal{C} = \\text{CRIS}(X/S)$ or $\\mathcal{C} = \\text{Cris}(X/S)$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on $\\mathcal{C}$. \\begin{enumerate} \\item We say $\\mathcal{F}$ is {\\it locally quasi-coherent} if for every object $(U, T, \\delta)$ of $\\mathcal{C}$ the restriction $\\mathcal{F}_T$ is a quasi-coherent $\\mathcal{O}_T$-module. \\item We say $\\mathcal{F}$ is {\\it quasi-coherent} if it is quasi-coherent in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local}. \\item We say $\\mathcal{F}$ is a {\\it crystal in $\\mathcal{O}_{X/S}$-modules} if all the comparison maps (\\ref{equation-comparison-modules}) are isomorphisms. \\end{enumerate}"}
+{"_id": "10808", "title": "crystalline-definition-crystal-quasi-coherent-modules", "text": "If $\\mathcal{F}$ satisfies the equivalent conditions of Lemma \\ref{lemma-crystal-quasi-coherent-modules}, then we say that $\\mathcal{F}$ is a {\\it crystal in quasi-coherent modules}. We say that $\\mathcal{F}$ is a {\\it crystal in finite locally free modules} if, in addition, $\\mathcal{F}$ is finite locally free."}
+{"_id": "10809", "title": "crystalline-definition-global-derivation", "text": "In Situation \\ref{situation-global} let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on $\\text{Cris}(X/S)$. An {\\it $S$-derivation $D : \\mathcal{O}_{X/S} \\to \\mathcal{F}$} is a map of sheaves such that for every object $(U, T, \\delta)$ of $\\text{Cris}(X/S)$ the map $$ D : \\Gamma(T, \\mathcal{O}_T) \\longrightarrow \\Gamma(T, \\mathcal{F}) $$ is a divided power $\\Gamma(V, \\mathcal{O}_V)$-derivation where $V \\subset S$ is any open such that $T \\to S$ factors through $V$."}
+{"_id": "10810", "title": "crystalline-definition-F-crystal", "text": "In Situation \\ref{situation-F-crystal} an {\\it $F$-crystal on $X/S$ (relative to $\\sigma$)} is a pair $(\\mathcal{E}, F_\\mathcal{E})$ given by a crystal in finite locally free $\\mathcal{O}_{X/S}$-modules $\\mathcal{E}$ together with a map $$ F_\\mathcal{E} : (F_X)_{\\text{cris}}^*\\mathcal{E} \\longrightarrow \\mathcal{E} $$ An $F$-crystal is called {\\it nondegenerate} if there exists an integer $i \\geq 0$ a map $V : \\mathcal{E} \\to (F_X)_{\\text{cris}}^*\\mathcal{E}$ such that $V \\circ F_{\\mathcal{E}} = p^i \\text{id}$."}
+{"_id": "11141", "title": "varieties-definition-variety", "text": "Let $k$ be a field. A {\\it variety} is a scheme $X$ over $k$ such that $X$ is integral and the structure morphism $X \\to \\Spec(k)$ is separated and of finite type."}
+{"_id": "11142", "title": "varieties-definition-geometrically-reduced", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. \\begin{enumerate} \\item Let $x \\in X$ be a point. We say $X$ is {\\it geometrically reduced at $x$} if for any field extension $k \\subset k'$ and any point $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is reduced. \\item We say $X$ is {\\it geometrically reduced} over $k$ if $X$ is geometrically reduced at every point of $X$. \\end{enumerate}"}
+{"_id": "11143", "title": "varieties-definition-geometrically-connected", "text": "Let $X$ be a scheme over the field $k$. We say $X$ is {\\it geometrically connected} over $k$ if the scheme $X_{k'}$ is connected for every field extension $k'$ of $k$."}
+{"_id": "11144", "title": "varieties-definition-geometrically-irreducible", "text": "Let $X$ be a scheme over the field $k$. We say $X$ is {\\it geometrically irreducible} over $k$ if the scheme $X_{k'}$ is irreducible\\footnote{An irreducible space is nonempty.} for any field extension $k'$ of $k$."}
+{"_id": "11145", "title": "varieties-definition-geometrically-integral", "text": "Let $X$ be a scheme over the field $k$. \\begin{enumerate} \\item Let $x \\in X$. We say $X$ is {\\it geometrically pointwise integral at $x$} if for every field extension $k \\subset k'$ and every $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is integral. \\item We say $X$ is {\\it geometrically pointwise integral} if $X$ is geometrically pointwise integral at every point. \\item We say $X$ is {\\it geometrically integral} over $k$ if the scheme $X_{k'}$ is integral for every field extension $k'$ of $k$. \\end{enumerate}"}
+{"_id": "11146", "title": "varieties-definition-geometrically-normal", "text": "Let $X$ be a scheme over the field $k$. \\begin{enumerate} \\item Let $x \\in X$. We say $X$ is {\\it geometrically normal at $x$} if for every field extension $k \\subset k'$ and every $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is normal. \\item We say $X$ is {\\it geometrically normal} over $k$ if $X$ is geometrically normal at every $x \\in X$. \\end{enumerate}"}
+{"_id": "11147", "title": "varieties-definition-geometrically-regular", "text": "Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. \\begin{enumerate} \\item Let $x \\in X$. We say $X$ is {\\it geometrically regular at $x$} over $k$ if for every finitely generated field extension $k \\subset k'$ and any $x' \\in X_{k'}$ lying over $x$ the local ring $\\mathcal{O}_{X_{k'}, x'}$ is regular. \\item We say $X$ is {\\it geometrically regular over $k$} if $X$ is geometrically regular at all of its points. \\end{enumerate}"}
+{"_id": "11148", "title": "varieties-definition-dual-numbers", "text": "For any ring $R$ the {\\it dual numbers} over $R$ is the $R$-algebra denoted $R[\\epsilon]$. As an $R$-module it is free with basis $1$, $\\epsilon$ and the $R$-algebra structure comes from setting $\\epsilon^2 = 0$."}
+{"_id": "11149", "title": "varieties-definition-tangent-space", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. The set of dotted arrows making (\\ref{equation-tangent-space}) commute with its canonical $\\kappa(x)$-vector space structure is called the {\\it tangent space of $X$ over $S$ at $x$} and we denote it $T_{X/S, x}$. An element of this space is called a {\\it tangent vector} of $X/S$ at $x$."}
+{"_id": "11150", "title": "varieties-definition-algebraic-scheme", "text": "Let $k$ be a field. An {\\it algebraic $k$-scheme} is a scheme $X$ over $k$ such that the structure morphism $X \\to \\Spec(k)$ is of finite type. A {\\it locally algebraic $k$-scheme} is a scheme $X$ over $k$ such that the structure morphism $X \\to \\Spec(k)$ is locally of finite type."}
+{"_id": "11151", "title": "varieties-definition-variety-type", "text": "Let $k$ be a field. Let $X$ be a variety over $k$. \\begin{enumerate} \\item We say $X$ is an {\\it affine variety} if $X$ is an affine scheme. This is equivalent to requiring $X$ to be isomorphic to a closed subscheme of $\\mathbf{A}^n_k$ for some $n$. \\item We say $X$ is a {\\it projective variety} if the structure morphism $X \\to \\Spec(k)$ is projective. By Morphisms, Lemma \\ref{morphisms-lemma-characterize-locally-projective} this is true if and only if $X$ is isomorphic to a closed subscheme of $\\mathbf{P}^n_k$ for some $n$. \\item We say $X$ is a {\\it quasi-projective variety} if the structure morphism $X \\to \\Spec(k)$ is quasi-projective. By Morphisms, Lemma \\ref{morphisms-lemma-characterize-locally-quasi-projective} this is true if and only if $X$ is isomorphic to a locally closed subscheme of $\\mathbf{P}^n_k$ for some $n$. \\item A {\\it proper variety} is a variety such that the morphism $X \\to \\Spec(k)$ is proper. \\item A {\\it smooth variety} is a variety such that the morphism $X \\to \\Spec(k)$ is smooth. \\end{enumerate}"}
+{"_id": "11152", "title": "varieties-definition-euler-characteristic", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. In this situation the {\\it Euler characteristic of $\\mathcal{F}$} is the integer $$ \\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}). $$ For justification of the formula see below."}
+{"_id": "11153", "title": "varieties-definition-regularity", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. We say $\\mathcal{F}$ is {\\it $m$-regular} if $$ H^i(\\mathbf{P}^n_k, \\mathcal{F}(m - i)) = 0 $$ for $i = 1, \\ldots, n$."}
+{"_id": "11154", "title": "varieties-definition-hilbert-polynomial", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$. The function $d \\mapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d))$ is called the {\\it Hilbert polynomial} of $\\mathcal{F}$."}
+{"_id": "11155", "title": "varieties-definition-absolute-frobenius", "text": "Let $p$ be a prime number. Let $X$ be a scheme in characteristic $p$. The {\\it absolute frobenius of $X$} is the morphism $F_X : X \\to X$ given by the identity on the underlying topological space and with $F_X^\\sharp : \\mathcal{O}_X \\to \\mathcal{O}_X$ given by $g \\mapsto g^p$."}
+{"_id": "11156", "title": "varieties-definition-relative-frobenius", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. We define $$ X^{(p)} = X^{(p/S)} = X \\times_{S, F_S} S $$ viewed as a scheme over $S$. Applying Lemma \\ref{lemma-frobenius-endomorphism-identity} we see there is a unique morphism $F_{X/S} : X \\longrightarrow X^{(p)}$ over $S$ fitting into the commutative diagram $$ \\xymatrix{ X \\ar[rr]_{F_{X/S}} \\ar[rrd] \\ar@/^1em/[rrrr]^{F_X} & & X^{(p)} \\ar[rr] \\ar[d] & & X \\ar[d] \\\\ & & S \\ar[rr]^{F_S} & & S } $$ where the right square is cartesian. The morphism $F_{X/S}$ is called the {\\it relative Frobenius morphism of $X/S$}."}
+{"_id": "11157", "title": "varieties-definition-delta-invariant-algebra", "text": "Let $A$ be a reduced Nagata local ring of dimension $1$. The {\\it $\\delta$-invariant of $A$} is $\\text{length}_A(A'/A)$ where $A'$ is as in Lemma \\ref{lemma-pre-delta-invariant}."}
+{"_id": "11158", "title": "varieties-definition-delta-invariant", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $x \\in X$ be a point such that $\\mathcal{O}_{X, x}$ is reduced and $\\dim(\\mathcal{O}_{X, x}) = 1$. The {\\it $\\delta$-invariant of $X$ at $x$} is the $\\delta$-invariant of $\\mathcal{O}_{X, x}$ as defined in Definition \\ref{definition-delta-invariant-algebra}."}
+{"_id": "11159", "title": "varieties-definition-wedge", "text": "Let $A$ and $A_i$, $1 \\leq i \\leq n$ be local rings. We say {\\it $A$ is a wedge of $A_1, \\ldots, A_n$} if there exist isomorphisms $$ \\kappa_{A_1} \\to \\kappa_{A_2} \\to \\ldots \\to \\kappa_{A_n} $$ and $A$ is isomorphic to the ring consisting of $n$-tuples $(a_1, \\ldots, a_n) \\in A_1 \\times \\ldots \\times A_n$ which map to the same element of $\\kappa_{A_n}$."}
+{"_id": "11160", "title": "varieties-definition-curve", "text": "Let $k$ be a field. A {\\it curve} is a variety of dimension $1$ over $k$."}
+{"_id": "11161", "title": "varieties-definition-degree-invertible-sheaf", "text": "Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$ over $k$, and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The {\\it degree} of $\\mathcal{L}$ is defined by $$ \\deg(\\mathcal{L}) = \\chi(X, \\mathcal{L}) - \\chi(X, \\mathcal{O}_X) $$ More generally, if $\\mathcal{E}$ is a locally free sheaf of rank $n$ we define the {\\it degree} of $\\mathcal{E}$ by $$ \\deg(\\mathcal{E}) = \\chi(X, \\mathcal{E}) - n\\chi(X, \\mathcal{O}_X) $$"}
+{"_id": "11162", "title": "varieties-definition-intersection-number", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $i : Z \\to X$ be a closed subscheme of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. We define the {\\it intersection number} $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ as the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial $$ \\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) = \\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z) $$ In the special case that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$ we write $(\\mathcal{L}^d \\cdot Z)$."}
+{"_id": "11163", "title": "varieties-definition-degree", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module. For any closed subscheme the {\\it degree of $Z$ with respect to $\\mathcal{L}$}, denoted $\\deg_\\mathcal{L}(Z)$, is the intersection number $(\\mathcal{L}^d \\cdot Z)$ where $d = \\dim(Z)$."}
+{"_id": "11164", "title": "varieties-definition-embed-dim", "text": "Let $k$ be an algebraically closed field. Let $X$ be a locally algebraic $k$-scheme and let $x \\in X$ be a closed point. The {\\it embedding dimension of $X$ at $x$} is $\\dim_k \\mathfrak m_x/\\mathfrak m_x^2$."}
+{"_id": "11165", "title": "varieties-definition-embedding-dimension", "text": "Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $x \\in X$ be a point. The {\\it embedding dimension of $X/k$ at $x$} is $\\dim_{\\kappa(x)}(T_{X/k, x})$."}
+{"_id": "11244", "title": "cotangent-definition-standard-resolution", "text": "Let $A \\to B$ be a ring map. The {\\it standard resolution of $B$ over $A$} is the augmentation $\\epsilon : P_\\bullet \\to B$ with terms $$ P_0 = A[B],\\quad P_1 = A[A[B]],\\quad \\ldots $$ and maps as constructed above."}
+{"_id": "11245", "title": "cotangent-definition-cotangent-complex-ring-map", "text": "The {\\it cotangent complex} $L_{B/A}$ of a ring map $A \\to B$ is the complex of $B$-modules associated to the simplicial $B$-module $$ \\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} B $$ where $\\epsilon : P_\\bullet \\to B$ is the standard resolution of $B$ over $A$."}
+{"_id": "11246", "title": "cotangent-definition-biderivation", "text": "Let $A \\to B$ be a ring map. Let $M$ be a $(B, B)$-bimodule over $A$. An {\\it $A$-biderivation} is an $A$-linear map $\\lambda : B \\to M$ such that $\\lambda(xy) = x\\lambda(y) + \\lambda(x)y$."}
+{"_id": "11247", "title": "cotangent-definition-atiyah-class", "text": "Let $A \\to B$ be a ring map. Let $M$ be a $B$-module. The map $M \\to L_{B/A} \\otimes_B^\\mathbf{L} M[1]$ in (\\ref{equation-atiyah}) is called the {\\it Atiyah class} of $M$."}
+{"_id": "11248", "title": "cotangent-definition-standard-resolution-sheaves-rings", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. The {\\it standard resolution of $\\mathcal{B}$ over $\\mathcal{A}$} is the augmentation $\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$ with terms $$ \\mathcal{P}_0 = \\mathcal{A}[\\mathcal{B}],\\quad \\mathcal{P}_1 = \\mathcal{A}[\\mathcal{A}[\\mathcal{B}]],\\quad \\ldots $$ and maps as constructed above."}
+{"_id": "11249", "title": "cotangent-definition-cotangent-complex-morphism-sheaves-rings", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. The {\\it cotangent complex} $L_{\\mathcal{B}/\\mathcal{A}}$ is the complex of $\\mathcal{B}$-modules associated to the simplicial module $$ \\Omega_{\\mathcal{P}_\\bullet/\\mathcal{A}} \\otimes_{\\mathcal{P}_\\bullet, \\epsilon} \\mathcal{B} $$ where $\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$ is the standard resolution of $\\mathcal{B}$ over $\\mathcal{A}$. We usually think of $L_{\\mathcal{B}/\\mathcal{A}}$ as an object of $D(\\mathcal{B})$."}
+{"_id": "11250", "title": "cotangent-definition-atiyah-class-general", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{B}$-modules. The map $\\mathcal{F} \\to L_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{F}[1]$ in (\\ref{equation-atiyah-general}) is called the {\\it Atiyah class} of $\\mathcal{F}$."}
+{"_id": "11251", "title": "cotangent-definition-cotangent-complex-morphism-ringed-spaces", "text": "Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of ringed spaces. The {\\it cotangent complex} $L_f$ of $f$ is $L_f = L_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$. We will also use the notation $L_f = L_{X/S} = L_{\\mathcal{O}_X/\\mathcal{O}_S}$."}
+{"_id": "11252", "title": "cotangent-definition-cotangent-complex-morphism-ringed-topoi", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. The {\\it cotangent complex} $L_f$ of $f$ is $L_f = L_{\\mathcal{O}_\\mathcal{C}/f^{-1}\\mathcal{O}_\\mathcal{D}}$. We sometimes write $L_f = L_{\\mathcal{O}_\\mathcal{C}/\\mathcal{O}_\\mathcal{D}}$."}
+{"_id": "11253", "title": "cotangent-definition-cotangent-morphism-schemes", "text": "Let $f : X \\to Y$ be a morphism of schemes. The {\\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of $f$ as a morphism of ringed spaces (Definition \\ref{definition-cotangent-complex-morphism-ringed-spaces})."}
+{"_id": "11254", "title": "cotangent-definition-cotangent-morphism-spaces", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The {\\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of the morphism of ringed topoi $f_{small}$ between the small \\'etale sites of $X$ and $Y$ (see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi} and Definition \\ref{definition-cotangent-complex-morphism-ringed-topoi})."}
+{"_id": "11347", "title": "spaces-cohomology-definition-alternating-cech-complex", "text": "Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of $\\textit{Ab}(X_\\etale)$. The {\\it alternating {\\v C}ech complex}\\footnote{This may be nonstandard notation} $\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F})$ associated to $\\mathcal{F}$ and $f$ is the complex $$ \\Hom(K^0, \\mathcal{F}) \\to \\Hom(K^1, \\mathcal{F}) \\to \\Hom(K^2, \\mathcal{F}) \\to \\ldots $$ with Hom groups computed in $\\textit{Ab}(X_\\etale)$."}
+{"_id": "11348", "title": "spaces-cohomology-definition-coherent", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. A quasi-coherent module $\\mathcal{F}$ on $X$ is called {\\it coherent} if $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module on the site $X_\\etale$ in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local}."}
+{"_id": "11417", "title": "artin-definition-RS", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{Z}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{Z}$ satisfies {\\it condition (RS)} if for every pushout $$ \\xymatrix{ X \\ar[r] \\ar[d] & X' \\ar[d] \\\\ Y \\ar[r] & Y' = Y \\amalg_X X' } $$ in the category of schemes over $S$ where \\begin{enumerate} \\item $X$, $X'$, $Y$, $Y'$ are spectra of local Artinian rings, \\item $X$, $X'$, $Y$, $Y'$ are of finite type over $S$, and \\item $X \\to X'$ (and hence $Y \\to Y'$) is a closed immersion \\end{enumerate} the functor of fibre categories $$ \\mathcal{Z}_{Y'} \\longrightarrow \\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'} $$ is an equivalence of categories."}
+{"_id": "11418", "title": "artin-definition-formal-objects", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. \\begin{enumerate} \\item A {\\it formal object} $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ consists of a Noetherian complete local $S$-algebra $R$, objects $\\xi_n$ of $\\mathcal{X}$ lying over $\\Spec(R/\\mathfrak m_R^n)$, and morphisms $f_n : \\xi_n \\to \\xi_{n + 1}$ of $\\mathcal{X}$ lying over $\\Spec(R/\\mathfrak m^n) \\to \\Spec(R/\\mathfrak m^{n + 1})$ such that $R/\\mathfrak m$ is a field of finite type over $S$. \\item A {\\it morphism of formal objects} $a : \\xi = (R, \\xi_n, f_n) \\to \\eta = (T, \\eta_n, g_n)$ is given by morphisms $a_n : \\xi_n \\to \\eta_n$ such that for every $n$ the diagram $$ \\xymatrix{ \\xi_n \\ar[r]_{f_n} \\ar[d]_{a_n} & \\xi_{n + 1} \\ar[d]^{a_{n + 1}} \\\\ \\eta_n \\ar[r]^{g_n} & \\eta_{n + 1} } $$ is commutative. Applying the functor $p$ we obtain a compatible collection of morphisms $\\Spec(R/\\mathfrak m_R^n) \\to \\Spec(T/\\mathfrak m_T^n)$ and hence a morphism $a_0 : \\Spec(R) \\to \\Spec(T)$ over $S$. We say that $a$ {\\it lies over} $a_0$. \\end{enumerate}"}
+{"_id": "11419", "title": "artin-definition-effective", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. A formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ is called {\\it effective} if it is in the essential image of the functor (\\ref{equation-approximation})."}
+{"_id": "11420", "title": "artin-definition-limit-preserving", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ is {\\it limit preserving} if for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have an equivalence $$ \\colim \\mathcal{X}_{T_i} \\longrightarrow \\mathcal{X}_T $$ of fibre categories."}
+{"_id": "11421", "title": "artin-definition-versal-formal-object", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\xi = (R, \\xi_n, f_n)$ be a formal object. Set $k = R/\\mathfrak m$ and $x_0 = \\xi_1$. We will say that $\\xi$ is {\\it versal} if $\\xi$ as a formal object of $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ (Remark \\ref{remark-formal-objects-match}) is versal in the sense of Formal Deformation Theory, Definition \\ref{formal-defos-definition-versal}."}
+{"_id": "11422", "title": "artin-definition-versal", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $U$ be a scheme locally of finite type over $S$. Let $x$ be an object of $\\mathcal{X}$ lying over $U$. Let $u_0$ be finite type point of $U$. We say $x$ is {\\it versal} at $u_0$ if the morphism $\\hat x$ (\\ref{equation-hat-x}) is smooth, see Formal Deformation Theory, Definition \\ref{formal-defos-definition-smooth-morphism}."}
+{"_id": "11423", "title": "artin-definition-openness-versality", "text": "Let $S$ be a locally Noetherian scheme. \\begin{enumerate} \\item Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ satisfies {\\it openness of versality} if given a scheme $U$ locally of finite type over $S$, an object $x$ of $\\mathcal{X}$ over $U$, and a finite type point $u_0 \\in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \\in U' \\subset U$ such that $x$ is versal at every finite type point of $U'$. \\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $f$ satisfies {\\it openness of versality} if given a scheme $U$ locally of finite type over $S$, an object $y$ of $\\mathcal{Y}$ over $U$, openness of versality holds for $(\\Sch/U)_{fppf} \\times_\\mathcal{Y} \\mathcal{X}$. \\end{enumerate}"}
+{"_id": "11424", "title": "artin-definition-RS-star", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ satisfies {\\it condition (RS*)} if given a fibre product diagram $$ \\xymatrix{ B' \\ar[r] & B \\\\ A' = A \\times_B B' \\ar[u] \\ar[r] & A \\ar[u] } $$ of $S$-algebras, with $B' \\to B$ surjective with square zero kernel, the functor of fibre categories $$ \\mathcal{X}_{\\Spec(A')} \\longrightarrow \\mathcal{X}_{\\Spec(A)} \\times_{\\mathcal{X}_{\\Spec(B)}} \\mathcal{X}_{\\Spec(B')} $$ is an equivalence of categories."}
+{"_id": "11425", "title": "artin-definition-obstruction-theory", "text": "Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. An {\\it obstruction theory} is  given by the following data \\begin{enumerate} \\item for every $S$-algebra $A$ such that $\\Spec(A) \\to S$ maps into an affine open and every object $x$ of $\\mathcal{X}$ over $\\Spec(A)$ an $A$-linear functor $$ \\mathcal{O}_x : \\text{Mod}_A \\to \\text{Mod}_A $$ of {\\it obstruction modules}, \\item for $(x, A)$ as in (1), a ring map $A \\to B$, $M \\in \\text{Mod}_A$, $N \\in \\text{Mod}_B$, and an $A$-linear map $M \\to N$ an induced $A$-linear map $\\mathcal{O}_x(M) \\to \\mathcal{O}_y(N)$ where $y = x|_{\\Spec(B)}$, and \\item for every deformation situation $(x, A' \\to A)$ an {\\it obstruction} element $o_x(A') \\in \\mathcal{O}_x(I)$ where $I = \\Ker(A' \\to A)$. \\end{enumerate} These data are subject to the following conditions \\begin{enumerate} \\item[(i)] the functoriality maps turn the obstruction modules into a functor from the category of triples $(x, A, M)$ to sets, \\item[(ii)] for every morphism of deformation situations $(y, B' \\to B) \\to (x, A' \\to A)$ the element $o_x(A')$ maps to $o_y(B')$, and \\item[(iii)] we have $$ \\text{Lift}(x, A') \\not = \\emptyset \\Leftrightarrow o_x(A') = 0 $$ for every deformation situation $(x, A' \\to A)$. \\end{enumerate}"}
+{"_id": "11426", "title": "artin-definition-naive-obstruction-theory", "text": "Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume that $\\mathcal{X}$ satisfies (RS*). A {\\it naive obstruction theory} is  given by the following data \\begin{enumerate} \\item \\label{item-map} for every $S$-algebra $A$ such that $\\Spec(A) \\to S$ maps into an affine open $\\Spec(\\Lambda) \\subset S$ and every object $x$ of $\\mathcal{X}$ over $\\Spec(A)$ we are given an object $E_x \\in D^-(A)$ and a map $\\xi_x : E \\to \\NL_{A/\\Lambda}$, \\item \\label{item-inf} given $(x, A)$ as in (\\ref{item-map}) there are transformations of functors $$ \\text{Inf}_x( - ) \\to \\Ext^{-1}_A(E_x, -) \\quad\\text{and}\\quad T_x(-) \\to \\Ext^0_A(E_x, -) $$ \\item \\label{item-functoriality} for $(x, A)$ as in (\\ref{item-map}) and a ring map $A \\to B$ setting $y = x|_{\\Spec(B)}$ there is a functoriality map $E_x \\to E_y$ in $D(A)$. \\end{enumerate} These data are subject to the following conditions \\begin{enumerate} \\item[(i)] in the situation of (\\ref{item-functoriality}) the diagram $$ \\xymatrix{ E_y \\ar[r]_{\\xi_y} & \\NL_{B/\\Lambda} \\\\ E_x \\ar[u] \\ar[r]^{\\xi_x} & \\NL_{A/\\Lambda} \\ar[u] } $$ is commutative in $D(A)$, \\item[(ii)] given $(x, A)$ as in (\\ref{item-map}) and $A \\to B \\to C$ setting $y = x|_{\\Spec(B)}$ and $z = x|_{\\Spec(C)}$ the composition of the functoriality maps $E_x \\to E_y$ and $E_y \\to E_z$ is the functoriality map $E_x \\to E_z$, \\item[(iii)] the maps of (\\ref{item-inf}) are isomorphisms compatible with the functoriality maps and the maps of Remark \\ref{remark-functoriality}, \\item[(iv)] the composition $E_x \\to \\NL_{A/\\Lambda} \\to \\Omega_{A/\\Lambda}$ corresponds to the canonical element of $T_x(\\Omega_{A/\\Lambda}) = \\Ext^0(E_x, \\Omega_{A/\\Lambda})$, see Remark \\ref{remark-canonical-element}, \\item[(v)] given a deformation situation $(x, A' \\to A)$ with $I = \\Ker(A' \\to A)$ the composition $E_x \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero in $$ \\Hom_A(E_x, \\NL_{A/\\Lambda}) = \\Ext^0_A(E_x, \\NL_{A/A'}) = \\Ext^1_A(E_x, I) $$ if and only if $x$ lifts to $A'$. \\end{enumerate}"}
+{"_id": "11525", "title": "obsolete-definition-epsilon", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\\dim_\\delta(X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z \\subset X$ be an integral closed subscheme with $\\dim_\\delta(Z) = n - 1$. The {\\it $\\epsilon$-invariant} of this situation is $$ \\epsilon_Z(D_1, D_2) = n_Z \\cdot m_Z $$ where $n_Z$, resp.\\ $m_Z$ is the coefficient of $Z$ in the $(n - 1)$-cycle $[D_1]_{n - 1}$, resp.\\ $[D_2]_{n - 1}$."}
+{"_id": "11526", "title": "obsolete-definition-locally-finite-sum-effective-Cartier-divisors", "text": "Let $X$ be a scheme. Let $\\{D_i\\}_{i \\in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given a function $I \\to \\mathbf{Z}_{\\geq 0}$, $i \\mapsto n_i$. The {\\it sum of the effective Cartier divisors} $D = \\sum n_i D_i$, is the unique effective Cartier divisor $D \\subset X$ such that on any quasi-compact open $U \\subset X$ we have $D|_U = \\sum_{D_i \\cap U \\not = \\emptyset} n_iD_i|_U$ is the sum as in Divisors, Definition \\ref{divisors-definition-sum-effective-Cartier-divisors}."}
+{"_id": "11622", "title": "stacks-sheaves-definition-presheaves", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. \\begin{enumerate} \\item A {\\it presheaf on $\\mathcal{X}$} is a presheaf on the underlying category of $\\mathcal{X}$. \\item A {\\it morphism of presheaves on $\\mathcal{X}$} is a morphism of presheaves on the underlying category of $\\mathcal{X}$. \\end{enumerate} We denote $\\textit{PSh}(\\mathcal{X})$ the category of presheaves on $\\mathcal{X}$."}
+{"_id": "11623", "title": "stacks-sheaves-definition-inherited-topologies", "text": "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item The {\\it associated Zariski site}, denoted $\\mathcal{X}_{Zar}$, is the structure of site on $\\mathcal{X}$ inherited from $(\\Sch/S)_{Zar}$. \\item The {\\it associated \\'etale site}, denoted $\\mathcal{X}_\\etale$, is the structure of site on $\\mathcal{X}$ inherited from $(\\Sch/S)_\\etale$. \\item The {\\it associated smooth site}, denoted $\\mathcal{X}_{smooth}$, is the structure of site on $\\mathcal{X}$ inherited from $(\\Sch/S)_{smooth}$. \\item The {\\it associated syntomic site}, denoted $\\mathcal{X}_{syntomic}$, is the structure of site on $\\mathcal{X}$ inherited from $(\\Sch/S)_{syntomic}$. \\item The {\\it associated fppf site}, denoted $\\mathcal{X}_{fppf}$, is the structure of site on $\\mathcal{X}$ inherited from $(\\Sch/S)_{fppf}$. \\end{enumerate}"}
+{"_id": "11624", "title": "stacks-sheaves-definition-sheaves", "text": "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$. \\begin{enumerate} \\item We say $\\mathcal{F}$ is a {\\it Zariski sheaf}, or a {\\it sheaf for the Zariski topology} if $\\mathcal{F}$ is a sheaf on the associated Zariski site $\\mathcal{X}_{Zar}$. \\item We say $\\mathcal{F}$ is an {\\it \\'etale sheaf}, or a {\\it sheaf for the \\'etale topology} if $\\mathcal{F}$ is a sheaf on the associated \\'etale site $\\mathcal{X}_\\etale$. \\item We say $\\mathcal{F}$ is a {\\it smooth sheaf}, or a {\\it sheaf for the smooth topology} if $\\mathcal{F}$ is a sheaf on the associated smooth site $\\mathcal{X}_{smooth}$. \\item We say $\\mathcal{F}$ is a {\\it syntomic sheaf}, or a {\\it sheaf for the syntomic topology} if $\\mathcal{F}$ is a sheaf on the associated syntomic site $\\mathcal{X}_{syntomic}$. \\item We say $\\mathcal{F}$ is an {\\it fppf sheaf}, or a {\\it sheaf}, or a {\\it sheaf for the fppf topology} if $\\mathcal{F}$ is a sheaf on the associated fppf site $\\mathcal{X}_{fppf}$. \\end{enumerate} A morphism of sheaves is just a morphism of presheaves. We denote these categories of sheaves $\\Sh(\\mathcal{X}_{Zar})$, $\\Sh(\\mathcal{X}_\\etale)$, $\\Sh(\\mathcal{X}_{smooth})$, $\\Sh(\\mathcal{X}_{syntomic})$, and $\\Sh(\\mathcal{X}_{fppf})$."}
+{"_id": "11625", "title": "stacks-sheaves-definition-morphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. We denote $$ f = (f^{-1}, f_*) : \\Sh(\\mathcal{X}_{fppf}) \\longrightarrow \\Sh(\\mathcal{Y}_{fppf}) $$ the {\\it associated morphism of fppf topoi} constructed above. Similarly for the associated Zariski, \\'etale, smooth, and syntomic topoi."}
+{"_id": "11626", "title": "stacks-sheaves-definition-structure-sheaf", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. The {\\it structure sheaf of $\\mathcal{X}$} is the sheaf of rings $\\mathcal{O}_\\mathcal{X} = p^{-1}\\mathcal{O}$."}
+{"_id": "11627", "title": "stacks-sheaves-definition-modules", "text": "Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. \\begin{enumerate} \\item A {\\it presheaf of modules on $\\mathcal{X}$} is a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. The category of presheaves of modules is denoted $\\textit{PMod}(\\mathcal{O}_\\mathcal{X})$. \\item We say a presheaf of modules $\\mathcal{F}$ is an {\\it $\\mathcal{O}_\\mathcal{X}$-module}, or more precisely a {\\it sheaf of $\\mathcal{O}_\\mathcal{X}$-modules} if $\\mathcal{F}$ is an fppf sheaf. The category of $\\mathcal{O}_\\mathcal{X}$-modules is denoted $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$. \\end{enumerate}"}
+{"_id": "11628", "title": "stacks-sheaves-definition-pullback", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$. \\begin{enumerate} \\item The {\\it pullback $x^{-1}\\mathcal{F}$ of $\\mathcal{F}$} is the restriction $\\mathcal{F}|_{(\\mathcal{X}/x)}$ viewed as a presheaf on $(\\Sch/U)_{fppf}$ via the equivalence $\\mathcal{X}/x \\to (\\Sch/U)_{fppf}$ of Lemma \\ref{lemma-localizing}. \\item The {\\it restriction of $\\mathcal{F}$ to $U_\\etale$} is $x^{-1}\\mathcal{F}|_{U_\\etale}$, abusively written $\\mathcal{F}|_{U_\\etale}$. \\end{enumerate}"}
+{"_id": "11629", "title": "stacks-sheaves-definition-quasi-coherent", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. A {\\it quasi-coherent module on $\\mathcal{X}$}, or a {\\it quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module} is a quasi-coherent module on the ringed site $(\\mathcal{X}_{fppf}, \\mathcal{O}_\\mathcal{X})$ as in Modules on Sites, Definition \\ref{sites-modules-definition-site-local}. The category of quasi-coherent sheaves on $\\mathcal{X}$ is denoted $\\QCoh(\\mathcal{O}_\\mathcal{X})$."}
+{"_id": "11630", "title": "stacks-sheaves-definition-locally-quasi-coherent", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. We say $\\mathcal{F}$ is {\\it locally quasi-coherent}\\footnote{This is nonstandard notation.} if $\\mathcal{F}$ is a sheaf for the \\'etale topology and for every object $x$ of $\\mathcal{X}$ the restriction $x^*\\mathcal{F}|_{U_\\etale}$ is a quasi-coherent sheaf. Here $U = p(x)$."}
+{"_id": "11712", "title": "resolve-definition-normalized-blowup", "text": "Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $x \\in X$ be a closed point. The {\\it normalized blowup of $X$ at $x$} is the composition $X'' \\to X' \\to X$ where $X' \\to X$ is the blowup of $X$ in $x$ and $X'' \\to X'$ is the normalization of $X'$."}
+{"_id": "11713", "title": "resolve-definition-reduce-to-rational", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of dimension $2$. \\begin{enumerate} \\item We say $A$ {\\it defines a rational singularity} if for every normal modification $X \\to \\Spec(A)$ we have $H^1(X, \\mathcal{O}_X) = 0$. \\item We say that {\\it reduction to rational singularities is possible for $A$} if the length of the $A$-modules $$ H^1(X, \\mathcal{O}_X) $$ is bounded for all modifications $X \\to \\Spec(A)$ with $X$ normal. \\end{enumerate}"}
+{"_id": "11714", "title": "resolve-definition-resolution", "text": "Let $Y$ be a Noetherian integral scheme. A {\\it resolution of singularities} of $Y$ is a modification $f : X \\to Y$ such that $X$ is regular."}
+{"_id": "11715", "title": "resolve-definition-resolution-surface", "text": "Let $Y$ be a $2$-dimensional Noetherian integral scheme. We say $Y$ has a {\\it resolution of singularities by normalized blowups} if there exists a sequence $$ Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y $$ where \\begin{enumerate} \\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$, \\item $Y_0 \\to Y$ is the normalization, \\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and \\item $Y_n$ is regular. \\end{enumerate}"}
+{"_id": "11719", "title": "exercises-definition-directed-poset", "text": "A {\\it directed set} is a nonempty set $I$ endowed with a preorder $\\leq$ such that given any pair $i, j \\in I$ there exists a $k \\in I$ such that $i \\leq k$ and $j \\leq k$. A {\\it system of rings} over $I$ is given by a ring $A_i$ for each $i \\in I$ and a map of rings $\\varphi_{ij} : A_i \\to A_j$ whenever $i \\leq j$ such that the composition $A_i \\to A_j \\to A_k$ is equal to $A_i \\to A_k$ whenever $i \\leq j \\leq k$."}
+{"_id": "11720", "title": "exercises-definition-colimit", "text": "The ring $A$ constructed in Exercise \\ref{exercise-directed-colimit} is called the {\\it colimit} of the system. Notation $\\colim A_i$."}
+{"_id": "11721", "title": "exercises-definition-finite-presentation", "text": "A module $M$ over $R$ is said to be of {\\it finite presentation} over $R$ if it  is isomorphic to the cokernel of a map of finite free modules $ R^{\\oplus n} \\to R^{\\oplus m}$."}
+{"_id": "11722", "title": "exercises-definition-quasi-compact", "text": "A topological space $X$ is called {\\it quasi-compact} if for any open covering $X = \\bigcup_{i\\in I} U_i$ there is a finite subset $\\{i_1, \\ldots, i_n\\}\\subset I$ such that $X = U_{i_1}\\cup\\ldots U_{i_n}$."}
+{"_id": "11723", "title": "exercises-definition-Hausdorff", "text": "A topological space $X$ is said to verify the separation axiom $T_0$ if for any pair of points $x, y\\in X$, $x\\not = y$ there is an open subset of $X$ containing one but not the other. We say that $X$ is {\\it Hausdorff} if for any pair $x, y\\in X$, $x\\not = y$ there are disjoint open subsets $U, V$ such that $x\\in U$ and $y\\in V$."}
+{"_id": "11724", "title": "exercises-definition-irreducible", "text": "A topological space $X$ is called {\\it irreducible} if $X$ is not empty and if $X = Z_1\\cup Z_2$ with $Z_1, Z_2\\subset X$ closed, then either $Z_1 = X$ or $Z_2 = X$. A subset $T\\subset X$ of a topological space is called {\\it irreducible} if it is an irreducible topological space with the topology induced from $X$. This definition implies $T$ is irreducible if and only if the closure $\\bar T$ of $T$ in $X$ is irreducible."}
+{"_id": "11725", "title": "exercises-definition-generic-point", "text": "A point $x$ of an irreducible topological space $X$ is called a {\\it generic point} of $X$ if $X$ is equal to the closure of the subset $\\{x\\}$."}
+{"_id": "11726", "title": "exercises-definition-Noetherian-space", "text": "A topological space $X$ is called {\\it Noetherian} if any decreasing sequence $Z_1\\supset Z_2 \\supset Z_3\\supset \\ldots$ of closed subsets of $X$ stabilizes. (It is called {\\it Artinian} if any increasing sequence of closed subsets stabilizes.)"}
+{"_id": "11727", "title": "exercises-definition-irreducible-component", "text": "A maximal irreducible subset $T\\subset X$ is called an {\\it irreducible component} of the space $X$. Such an irreducible component of $X$ is automatically a closed subset of $X$."}
+{"_id": "11728", "title": "exercises-definition-closed", "text": "A point $x\\in X$ is called {\\it closed} if $\\overline{\\{x\\}} = \\{ x\\}$. Let $x, y$ be points of $X$. We say that $x$ is a {\\it specialization} of $y$, or that $y$ is a {\\it generalization} of $x$ if $x\\in \\overline{\\{y\\}}$."}
+{"_id": "11729", "title": "exercises-definition-connected-component", "text": "A topological space $X$ is called {\\it connected} if it is nonempty and not the union of two nonempty disjoint open subsets. A {\\it connected component} of $X$ is a maximal connected subset. Any point of $X$ is contained in a connected component of $X$ and any connected component of $X$ is closed in $X$. (But in general a connected component need not be open in $X$.)"}
+{"_id": "11730", "title": "exercises-definition-length", "text": "Let $A$ be a ring. Let $M$ be an $A$-module. The {\\it length} of $M$ as an $R$-module is $$ \\text{length}_A(M) = \\sup \\{ n \\mid \\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M, \\text{ }M_i \\not = M_{i + 1} \\}. $$ In other words, the supremum of the lengths of chains of submodules."}
+{"_id": "11731", "title": "exercises-definition-catenary", "text": "A Noetherian ring $A$ is said to be {\\it catenary} if for any triple of prime ideals ${\\mathfrak p}_1 \\subset {\\mathfrak p}_2 \\subset {\\mathfrak p}_3$ we have $$ ht({\\mathfrak p}_3 / {\\mathfrak p}_1) = ht({\\mathfrak p}_3/{\\mathfrak p}_2) + ht({\\mathfrak p}_2/{\\mathfrak p}_1). $$ Here $ht(\\mathfrak p/\\mathfrak q)$ means the height of $\\mathfrak p/\\mathfrak q$ in the ring $A/\\mathfrak q$. In a formula $$ ht(\\mathfrak p/\\mathfrak q) = \\dim(A_\\mathfrak p/\\mathfrak qA_\\mathfrak p) = \\dim((A/\\mathfrak q)_\\mathfrak p) = \\dim((A/\\mathfrak q)_{\\mathfrak p/\\mathfrak q}) $$ A topological space $X$ is {\\it catenary}, if given $T \\subset T' \\subset X$ with $T$ and $T'$ closed and irreducible, then there exists a maximal chain of irreducible closed subsets $$ T = T_0 \\subset T_1 \\subset \\ldots \\subset T_n = T' $$ and every such chain has the same (finite) length."}
+{"_id": "11732", "title": "exercises-definition-finite-locally-free", "text": "Let $A$ be a ring. Recall that a {\\it finite locally free} $A$-module $M$ is a module such that for every ${\\mathfrak p} \\in \\Spec(A)$ there exists an $f\\in A$, $f \\not \\in {\\mathfrak p}$ such that $M_f$ is a finite free $A_f$-module. We say $M$ is an {\\it invertible module} if $M$ is finite locally free of rank $1$, i.e., for every ${\\mathfrak p} \\in \\Spec(A)$ there exists an $f\\in A$, $f \\not \\in \\mathfrak p$ such that $M_f \\cong A_f$ as an $A_f$-module."}
+{"_id": "11733", "title": "exercises-definition-class-group", "text": "Let $A$ be a ring. The {\\it class group of $A$}, sometimes called the {\\it Picard group of $A$} is the set $\\Pic(A)$ of isomorphism classes of invertible $A$-modules endowed with a group operation defined by tensor product (see Exercise \\ref{exercise-tensor-finite-locally-free})."}
+{"_id": "11734", "title": "exercises-definition-GU-GD", "text": "Let $\\phi : A \\to B$ be a homomorphism of rings. We say that the {\\it going-up theorem} holds for $\\phi$ if the following condition is satisfied: \\begin{itemize} \\item[(GU)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that ${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any $P \\in \\Spec(B)$ lying over ${\\mathfrak p}$, there exists $P'\\in \\Spec(B)$ lying over ${\\mathfrak p}'$ such that $P \\subset P'$. \\end{itemize} Similarly, we say that the {\\it going-down theorem} holds for $\\phi$ if the following condition is satisfied: \\begin{itemize} \\item[(GD)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that ${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any $P' \\in \\Spec(B)$ lying over ${\\mathfrak p}'$, there exists $P\\in \\Spec(B)$ lying over ${\\mathfrak p}$ such that $P \\subset P'$. \\end{itemize}"}
+{"_id": "11735", "title": "exercises-definition-numerical-polynomial", "text": "A {\\it numerical polynomial} is a polynomial $f(x) \\in {\\mathbf Q}[x]$ such that $f(n) \\in {\\mathbf Z}$ for every integer $n$."}
+{"_id": "11736", "title": "exercises-definition-graded-module", "text": "A {\\it graded module} $M$ over a ring $A$ is an $A$-module $M$ endowed with a direct sum decomposition $ \\bigoplus\\nolimits_{n \\in {\\mathbf Z}} M_n $ into $A$-submodules. We will say that $M$ is {\\it locally finite} if all of the $M_n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and that $\\varphi$ is a {\\it Euler-Poincar\\'e function} on finite $A$-modules. This means that for every finitely generated $A$-module $M$ we are given an integer $\\varphi(M) \\in {\\mathbf Z}$ and for every short exact sequence $$ 0 \\longrightarrow M' \\longrightarrow M \\longrightarrow M'' \\longrightarrow 0 $$ we have $\\varphi(M) = \\varphi(M') + \\varphi(M')$. The {\\it Hilbert function} of a locally finite graded module $M$ (with respect to $\\varphi$) is the function $\\chi_\\varphi(M, n) = \\varphi(M_n)$. We say that $M$ has a {\\it Hilbert polynomial} if there is some numerical polynomial $P_\\varphi$ such that $\\chi_\\varphi(M, n) = P_\\varphi(n)$ for all sufficiently large integers $n$."}
+{"_id": "11737", "title": "exercises-definition-graded-algebra", "text": "A {\\it graded $A$-algebra} is a graded $A$-module $B = \\bigoplus_{n \\geq 0} B_n$ together with an $A$-bilinear map $$ B \\times B \\longrightarrow B, \\ (b, b') \\longmapsto bb' $$ that turns $B$ into an $A$-algebra so that $B_n \\cdot B_m \\subset B_{n + m}$. Finally, a {\\it graded module $M$ over a graded $A$-algebra $B$} is given by a graded $A$-module $M$ together with a (compatible) $B$-module structure such that $B_n \\cdot M_d \\subset M_{n + d}$. Now you can define {\\it homomorphisms of graded modules/rings}, {\\it graded submodules}, {\\it graded ideals}, {\\it exact sequences of graded modules}, etc, etc."}
+{"_id": "11738", "title": "exercises-definition-homogeneous-ideal", "text": "Let $R$ be a graded ring. A {\\it homogeneous} ideal is simply an ideal $I \\subset R$ which is also a graded submodule of $R$. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if $f \\in I$ and $$ f = f_0 + f_1 + \\ldots + f_n $$ is the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \\in I$ for each $i$."}
+{"_id": "11739", "title": "exercises-definition-Proj-R", "text": "We define the {\\it homogeneous spectrum $\\text{Proj}(R)$} of the graded ring $R$ to be the set of homogeneous, prime ideals ${\\mathfrak p}$ of $R$ such that $R_{+} \\not \\subset {\\mathfrak p}$. Note that $\\text{Proj}(R)$ is a subset of $\\Spec(R)$ and hence has a natural induced topology."}
+{"_id": "11740", "title": "exercises-definition-Dplus-Vplus", "text": "Let $R = \\oplus_{d \\geq 0} R_d$ be a graded ring, let $f\\in R_d$ and assume that $d \\geq 1$. We define {\\it $R_{(f)}$} to be the subring of $R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and $\\deg(r) = nd$. Furthermore, we define $$ D_{+}(f) = \\{ {\\mathfrak p} \\in \\text{Proj}(R) | f \\not\\in {\\mathfrak p} \\}. $$ Finally, for a homogeneous ideal $I \\subset R$ we define $V_{+}(I) = V(I) \\cap \\text{Proj}(R)$."}
+{"_id": "11741", "title": "exercises-definition-CM", "text": "A Noetherian local ring $A$ is said to be {\\it Cohen-Macaulay} of dimension $d$ if it has dimension $d$ and there exists a system of parameters $x_1, \\ldots, x_d$ for $A$ such that $x_i$ is a nonzerodivisor in $A/(x_1, \\ldots, x_{i-1})$ for $i = 1, \\ldots, d$."}
+{"_id": "11742", "title": "exercises-definition-injective-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. Let $I$ be a filtered object of $\\mathcal{A}$. Assume the filtration on $I$ is finite. We say $I$ is {\\it filtered injective} if each $\\text{gr}^p(I)$ is an injective object of $\\mathcal{A}$."}
+{"_id": "11743", "title": "exercises-definition-finite-filtration-category", "text": "Let $\\mathcal{A}$ be an abelian category. We denote {\\it $\\text{Fil}^f(\\mathcal{A})$} the full subcategory of $\\text{Fil}(\\mathcal{A})$ whose objects consist of those $A \\in \\Ob(\\text{Fil}(\\mathcal{A}))$ whose filtration is finite."}
+{"_id": "11744", "title": "exercises-definition-filtered-quasi-isomorphism", "text": "Let $\\mathcal{A}$ be an abelian category. Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of complexes of $\\text{Fil}(\\mathcal{A})$. We say that $\\alpha$ is a {\\it filtered quasi-isomorphism} if for each $p \\in \\mathbf{Z}$ the morphism $\\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet)$ is a quasi-isomorphism."}
+{"_id": "11745", "title": "exercises-definition-filtered-acyclic", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex of $\\text{Fil}^f(\\mathcal{A})$. We say that $K^\\bullet$ is {\\it filtered acyclic} if for each $p \\in \\mathbf{Z}$ the complex $\\text{gr}^p(K^\\bullet)$ is acyclic."}
+{"_id": "11746", "title": "exercises-definition-integral", "text": "A scheme $X$ is called {\\it integral} if $X$ is nonempty and for every nonempty affine open $U \\subset X$ the ring $\\Gamma(U, \\mathcal{O}_X) = \\mathcal{O}_X(U)$ is a domain."}
+{"_id": "11747", "title": "exercises-definition-dual-numbers", "text": "For any ring $R$ we denote $R[\\epsilon]$ the ring of {\\it dual numbers}. As an $R$-module it is free with basis $1$, $\\epsilon$. The ring structure comes from setting $\\epsilon^2 = 0$."}
+{"_id": "11748", "title": "exercises-definition-tangent-space", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. We dub the set of dotted arrows of Exercise \\ref{exercise-tangent-space-Zariski} the {\\it tangent space of $X$ over $S$} and we denote it $T_{X/S, x}$. An element of this space is called a {\\it tangent vector} of $X/S$ at $x$."}
+{"_id": "11749", "title": "exercises-definition-quasi-coherent", "text": "Let $X$ be a scheme. A sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is {\\it quasi-coherent} if for every affine open $\\Spec(R) = U \\subset X$ the restriction $\\mathcal{F}|_U$ is of the form $\\widetilde M$ for some $R$-module $M$."}
+{"_id": "11750", "title": "exercises-definition-specialization", "text": "Let $X$ be a topological space. Let $x, x' \\in X$. We say $x$ is a {\\it specialization} of $x'$ if and only if $x \\in \\overline{\\{x'\\}}$."}
+{"_id": "11751", "title": "exercises-definition-Noetherian-scheme", "text": "A scheme $X$ is called {\\it locally Noetherian} if and only if for every point $x \\in X$ there exists an affine open $\\Spec(R) = U \\subset X$ such that $R$ is Noetherian. A scheme is {\\it Noetherian} if it is locally Noetherian and quasi-compact."}
+{"_id": "11752", "title": "exercises-definition-coherent", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules. We say $\\mathcal{F}$ is {\\it coherent} if for every point $x \\in X$ there exists an affine open $\\Spec(R) = U \\subset X$ such that $\\mathcal{F}|_U$ is isomorphic to $\\widetilde M$ for some finite $R$-module $M$."}
+{"_id": "11753", "title": "exercises-definition-invertible-sheaf", "text": "Let $X$ be a locally ringed space. An {\\it invertible ${\\mathcal O}_X$-module} on $X$ is a sheaf of ${\\mathcal O}_X$-modules ${\\mathcal L}$ such that every point has an open neighbourhood $U \\subset X$ such that ${\\mathcal L}|_U$ is isomorphic to ${\\mathcal O}_U$ as ${\\mathcal O}_U$-module. We say that ${\\mathcal L}$ is trivial if it is isomorphic to ${\\mathcal O}_X$ as a ${\\mathcal O}_X$-module."}
+{"_id": "11754", "title": "exercises-definition-invertible-module", "text": "Let $R$ be a ring. An {\\it invertible module $M$} is an $R$-module $M$ such that $\\widetilde M$ is an invertible sheaf on the spectrum of $R$. We say $M$ is {\\it trivial} if $M \\cong R$ as an $R$-module."}
+{"_id": "11755", "title": "exercises-definition-picard-group", "text": "Let $X$ be a locally ringed space. The {\\it Picard group of $X$} is the set $\\Pic(X)$ of isomorphism classes of invertible $\\mathcal{O}_X$-modules with addition given by tensor product. See Modules, Definition \\ref{modules-definition-pic}. For a ring $R$ we set $\\Pic(R) = \\Pic(\\Spec(R))$."}
+{"_id": "11756", "title": "exercises-definition-delta", "text": "(Definition of delta.) Suppose that $$ 0 \\to {\\mathcal F}_1 \\to {\\mathcal F}_2 \\to {\\mathcal F}_3 \\to 0 $$ is a short exact sequence of abelian sheaves on any topological space $X$. The boundary map $\\delta : H^0(X, {\\mathcal F}_3) \\to {\\check H}^1(X, {\\mathcal F}_1)$ is defined as follows. Take an element $\\tau \\in H^0(X, {\\mathcal F}_3)$. Choose an open covering ${\\mathcal U} : X = \\bigcup_{i\\in I} U_i$ such that for each $i$ there exists a section $\\tilde \\tau_i \\in {\\mathcal F}_2$ lifting the restriction of $\\tau$ to $U_i$. Then consider the assignment $$ (i_0, i_1) \\longmapsto \\tilde \\tau_{i_0}|_{U_{i_0i_1}} - \\tilde \\tau_{i_1}|_{U_{i_0i_1}}. $$ This is clearly a 1-coboundary in the {\\v C}ech complex ${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. But we observe that (thinking of ${\\mathcal F}_1$ as a subsheaf of ${\\mathcal F}_2$) the RHS always is a section of ${\\mathcal F}_1$ over $U_{i_0i_1}$. Hence we see that the assignment defines a 1-cochain in the complex ${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. The cohomology class of this 1-cochain is by definition {\\it $\\delta(\\tau)$}."}
+{"_id": "11757", "title": "exercises-definition-divisor", "text": "Throughout, let $S$ be any scheme and let $X$ be a Noetherian, integral scheme. \\begin{enumerate} \\item A {\\it Weil divisor} on $X$ is a formal linear combination $\\Sigma n_i[Z_i]$ of prime divisors $Z_i$ with integer coefficients. \\item A {\\it prime divisor} is a closed subscheme $Z \\subset X$, which is integral with generic point $\\xi \\in Z$ such that ${\\mathcal O}_{X, \\xi}$ has dimension $1$. We will use the notation ${\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$ when $\\xi \\in Z \\subset X$ is as above. Note that ${\\mathcal O}_{X, Z} \\subset K(X)$ is a subring of the function field of $X$. \\item The {\\it Weil divisor associated to a rational function $f \\in K(X)^\\ast$} is the sum $\\Sigma v_Z(f)[Z]$. Here $v_Z(f)$ is defined as follows \\begin{enumerate} \\item If $f \\in {\\mathcal O}_{X, Z}^\\ast$ then $v_Z(f) = 0$. \\item If $f \\in {\\mathcal O}_{X, Z}$ then $$ v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f)). $$ \\item If $f = \\frac{a}{b}$ with $a, b \\in {\\mathcal O}_{X, Z}$ then $$ v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) - \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)). $$ \\end{enumerate} \\item An {\\it effective Cartier divisor} on a scheme $S$ is a closed subscheme $D \\subset S$ such that every point $d\\in D$ has an affine open neighbourhood $\\Spec(A) = U \\subset S$ in $S$ so that $D \\cap U = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor. \\item The {\\it Weil divisor $[D]$ associated to an effective Cartier divisor $D \\subset X$} of our Noetherian integral scheme $X$ is defined as the sum $\\Sigma v_Z(D)[Z]$ where $v_Z(D)$ is defined as follows \\begin{enumerate} \\item If the generic point $\\xi$ of $Z$ is not in $D$ then $v_Z(D) = 0$. \\item If the generic point $\\xi$ of $Z$ is in $D$ then $$ v_Z(D) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f)) $$ where $f \\in {\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$ is the nonzerodivisor which defines $D$ in an affine neighbourhood of $\\xi$ (as in (4) above). \\end{enumerate} \\item Let $S$ be a scheme. The {\\it sheaf of total quotient rings ${\\mathcal K}_S$} is the sheaf of ${\\mathcal O}_S$-algebras which is the sheafification of the pre-sheaf ${\\mathcal K}'$ defined as follows. For $U \\subset S$ open we set ${\\mathcal K}'(U) = S_U^{-1}{\\mathcal O}_S(U)$ where $S_U \\subset {\\mathcal O}_S(U)$ is the multiplicative subset consisting of sections $f \\in {\\mathcal O}_S(U)$ such that the germ of $f$ in ${\\mathcal O}_{S, u}$ is a nonzerodivisor for every $u\\in U$. In particular the elements of $S_U$ are all nonzerodivisors. Thus ${\\mathcal O}_S$ is a subsheaf of ${\\mathcal K}_S$, and we get a short exact sequence $$ 0 \\to {\\mathcal O}_S^\\ast \\to {\\mathcal K}_S^\\ast \\to {\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast \\to 0. $$ \\item A {\\it Cartier divisor} on a scheme $S$ is a global section of the quotient sheaf ${\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast$. \\item The {\\it Weil divisor associated to a Cartier divisor} $\\tau \\in \\Gamma(X, {\\mathcal K}_X^\\ast/{\\mathcal O}_X^\\ast)$ over our Noetherian integral scheme $X$ is the sum $\\Sigma v_Z(\\tau)[Z]$ where $v_Z(\\tau)$ is defined as by the following recipe \\begin{enumerate} \\item If the germ of $\\tau$ at the generic point $\\xi$ of $Z$ is zero -- in other words the image of $\\tau$ in the stalk $({\\mathcal K}^\\ast/{\\mathcal O}^\\ast)_\\xi$ is ``zero'' -- then $v_Z(\\tau) = 0$. \\item Find an affine open neighbourhood $\\Spec(A) = U \\subset X$ so that $\\tau|_U$ is the image of a section $f \\in {\\mathcal K}(U)$ and moreover $f = a/b$ with $a, b \\in A$. Then we set $$ v_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) - \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)). $$ \\end{enumerate} \\end{enumerate}"}
+{"_id": "11809", "title": "spaces-duality-definition-dualizing-scheme", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. An object $K$ of $D_\\QCoh(\\mathcal{O}_X)$ is called a {\\it dualizing complex} if $K$ satisfies the equivalent conditions of Lemma \\ref{lemma-equivalent-definitions}."}
+{"_id": "11810", "title": "spaces-duality-definition-relative-dualizing-proper-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat morphism of algebraic spaces over $S$ which is of finite presentation. A {\\it relative dualizing complex} for $X/Y$ is a pair $(\\omega_{X/Y}^\\bullet, \\tau)$ consisting of a $Y$-perfect object $\\omega_{X/Y}^\\bullet$ of $D(\\mathcal{O}_X)$ and a map $$ \\tau : Rf_*\\omega_{X/Y}^\\bullet \\longrightarrow \\mathcal{O}_Y $$ such that for any cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ where $Y'$ is an affine scheme the pair $(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$ is isomorphic to the pair $(a'(\\mathcal{O}_{Y'}), \\text{Tr}_{f', \\mathcal{O}_{Y'}})$ studied in Sections \\ref{section-twisted-inverse-image}, \\ref{section-base-change-map}, \\ref{section-base-change-II}, \\ref{section-trace}, \\ref{section-compare-with-pullback}, and \\ref{section-proper-flat}."}
+{"_id": "11922", "title": "spaces-properties-definition-separated", "text": "(Compare Spaces, Definition \\ref{spaces-definition-separated}.) Consider a big fppf site $\\Sch_{fppf} = (\\Sch/\\Spec(\\mathbf{Z}))_{fppf}$. Let $X$ be an algebraic space over $\\Spec(\\mathbf{Z})$. Let $\\Delta : X \\to X \\times X$ be the diagonal morphism. \\begin{enumerate} \\item We say $X$ is {\\it separated} if $\\Delta$ is a closed immersion. \\item We say $X$ is {\\it locally separated}\\footnote{In the literature this often refers to quasi-separated and locally separated algebraic spaces.} if $\\Delta$ is an immersion. \\item We say $X$ is {\\it quasi-separated} if $\\Delta$ is quasi-compact. \\item We say $X$ is {\\it Zariski locally quasi-separated}\\footnote{ This notion was suggested by B.\\ Conrad.} if there exists a Zariski covering $X = \\bigcup_{i \\in I} X_i$ (see Spaces, Definition \\ref{spaces-definition-Zariski-open-covering}) such that each $X_i$ is quasi-separated. \\end{enumerate} Let $S$ is a scheme contained in $\\Sch_{fppf}$, and let $X$ be an algebraic space over $S$. Then we say $X$ is {\\it separated}, {\\it locally separated}, {\\it quasi-separated}, or {\\it Zariski locally quasi-separated} if $X$ viewed as an algebraic space over $\\Spec(\\mathbf{Z})$ (see Spaces, Definition \\ref{spaces-definition-base-change}) has the corresponding property."}
+{"_id": "11923", "title": "spaces-properties-definition-points", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\\it point} of $X$ is an equivalence class of morphisms from spectra of fields into $X$. The set of points of $X$ is denoted $|X|$."}
+{"_id": "11924", "title": "spaces-properties-definition-topological-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The underlying {\\it topological space} of $X$ is the set of points $|X|$ endowed with the topology constructed in Lemma \\ref{lemma-topology-points}."}
+{"_id": "11925", "title": "spaces-properties-definition-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is {\\it quasi-compact} if there exists a surjective \\'etale morphism $U \\to X$ with $U$ quasi-compact."}
+{"_id": "11926", "title": "spaces-properties-definition-type-property", "text": "Let $\\mathcal{P}$ be a property of schemes which is local in the \\'etale topology. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ {\\it has property $\\mathcal{P}$} if any of the equivalent conditions of Lemma \\ref{lemma-type-property} hold."}
+{"_id": "11927", "title": "spaces-properties-definition-property-at-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. Let $\\mathcal{P}$ be a property of germs of schemes which is \\'etale local. We say $X$ {\\it has property $\\mathcal{P}$ at $x$} if any of the equivalent conditions of Lemma \\ref{lemma-local-source-target-at-point} hold."}
+{"_id": "11928", "title": "spaces-properties-definition-locally-constructible", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\subset |X|$ be a subset. We say $E$ is {\\it \\'etale locally constructible} if the equivalent conditions of Lemma \\ref{lemma-locally-constructible} are satisfied."}
+{"_id": "11929", "title": "spaces-properties-definition-dimension-at-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point of $X$. We define the {\\it dimension of $X$ at $x$} to be the element $\\dim_x(X) \\in \\{0, 1, 2, \\ldots, \\infty\\}$ such that $\\dim_x(X) = \\dim_u(U)$ for any (equivalently some) pair $(a : U \\to X, u)$ consisting of an \\'etale morphism $a : U \\to X$ from a scheme to $X$ and a point $u \\in U$ with $a(u) = x$. See Definition \\ref{definition-property-at-point}, Lemma \\ref{lemma-local-source-target-at-point}, and Descent, Lemma \\ref{descent-lemma-dimension-at-point-local}."}
+{"_id": "11930", "title": "spaces-properties-definition-dimension", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it dimension} $\\dim(X)$ of $X$ is defined by the rule $$ \\dim(X) = \\sup\\nolimits_{x \\in |X|} \\dim_x(X) $$"}
+{"_id": "11931", "title": "spaces-properties-definition-dimension-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. The {\\it dimension of the local ring of $X$ at $x$} is the element $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$ satisfying the equivalent conditions of Lemma \\ref{lemma-pre-dimension-local-ring}. In this case we will also say {\\it $x$ is a point of codimension $d$ on $X$}."}
+{"_id": "11932", "title": "spaces-properties-definition-reduced-induced-space", "text": "Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $Z \\subset |X|$ be a closed subset. An {\\it algebraic space structure on $Z$} is given by a closed subspace $Z'$ of $X$ with $|Z'|$ equal to $Z$. The {\\it reduced induced algebraic space structure} on $Z$ is the one constructed in Lemma \\ref{lemma-reduced-closed-subspace}. The {\\it reduction $X_{red}$ of $X$} is the reduced induced algebraic space structure on $|X|$."}
+{"_id": "11933", "title": "spaces-properties-definition-etale", "text": "Let $S$ be a scheme. A morphism $f : X \\to Y$ between algebraic spaces over $S$ is called {\\it \\'etale} if and only if for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme, the composition $f \\circ \\varphi$ is \\'etale also."}
+{"_id": "11934", "title": "spaces-properties-definition-etale-site", "text": "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$, and let $\\Sch_\\etale$ be the corresponding big \\'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The {\\it small \\'etale site $X_\\etale$} of $X$ is defined as follows: \\begin{enumerate} \\item An object of $X_\\etale$ is a morphism $\\varphi : U \\to X$ where $U \\in \\Ob((\\Sch/S)_\\etale)$ is a scheme and $\\varphi$ is an \\'etale morphism, \\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$ is given by a morphism of schemes $\\chi : U \\to U'$ such that $\\varphi = \\varphi' \\circ \\chi$, and \\item a family of morphisms $\\{(U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$ of $X_\\etale$ is a covering if and only if $\\{U_i \\to U\\}_{i \\in I}$ is a covering of $(\\Sch/S)_\\etale$. \\end{enumerate}"}
+{"_id": "11935", "title": "spaces-properties-definition-spaces-etale-site", "text": "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$, and let $\\Sch_\\etale$ be the corresponding big \\'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site {\\it $X_{spaces, \\etale}$} of $X$ is defined as follows: \\begin{enumerate} \\item An object of $X_{spaces, \\etale}$ is a morphism $\\varphi : U \\to X$ where $U$ is an algebraic space over $S$ and $\\varphi$ is an \\'etale morphism of algebraic spaces over $S$, \\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$ of $X_{spaces, \\etale}$ is given by a morphism of algebraic spaces $\\chi : U \\to U'$ such that $\\varphi = \\varphi' \\circ \\chi$, and \\item a family of morphisms $\\{\\varphi_i : (U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$ of $X_{spaces, \\etale}$ is a covering if and only if $|U| = \\bigcup \\varphi_i(|U_i|)$. \\end{enumerate} (As usual we choose a set of coverings of this type, including at least the coverings in $X_\\etale$, as in Sets, Lemma \\ref{sets-lemma-coverings-site} to turn $X_{spaces, \\etale}$ into a site.)"}
+{"_id": "11936", "title": "spaces-properties-definition-etale-topos", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it \\'etale topos} of $X$, or more precisely the {\\it small \\'etale topos} of $X$ is the category $\\Sh(X_\\etale)$ of sheaves of sets on $X_\\etale$."}
+{"_id": "11937", "title": "spaces-properties-definition-f-map", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and let $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$. An {\\it $f$-map $\\varphi : \\mathcal{G} \\to \\mathcal{F}$} is a collection of maps $\\varphi_{(U, V, g)} : \\mathcal{G}(V) \\to \\mathcal{F}(U)$ indexed by commutative diagrams $$ \\xymatrix{ U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\ V \\ar[r] & Y } $$ where $U \\in X_\\etale$, $V \\in Y_\\etale$ such that whenever given an extended diagram $$ \\xymatrix{ U' \\ar[r] \\ar[d]_{g'} & U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\ V' \\ar[r] & V \\ar[r] & Y } $$ with $V' \\to V$ and $U' \\to U$ \\'etale morphisms of schemes the diagram $$ \\xymatrix{ \\mathcal{G}(V) \\ar[rr]_{\\varphi_{(U, V, g)}} \\ar[d]_{\\text{restriction of }\\mathcal{G}} & & \\mathcal{F}(U) \\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\ \\mathcal{G}(V') \\ar[rr]^{\\varphi_{(U', V', g')}} & & \\mathcal{F}(U') } $$ commutes."}
+{"_id": "11938", "title": "spaces-properties-definition-geometric-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item A {\\it geometric point} of $X$ is a morphism $\\overline{x} : \\Spec(k) \\to X$, where $k$ is an algebraically closed field. We often abuse notation and write $\\overline{x} = \\Spec(k)$. \\item For every geometric point $\\overline{x}$ we have the corresponding ``image'' point $x \\in |X|$. We say that $\\overline{x}$ is a {\\it geometric point lying over $x$}. \\end{enumerate}"}
+{"_id": "11939", "title": "spaces-properties-definition-etale-neighbourhood", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$. \\begin{enumerate} \\item An {\\it \\'etale neighborhood} of $\\overline{x}$ of $X$ is a commutative diagram $$ \\xymatrix{ & U \\ar[d]^\\varphi \\\\ {\\bar x} \\ar[r]^{\\bar x} \\ar[ur]^{\\bar u} & X } $$ where $\\varphi$ is an \\'etale morphism of algebraic spaces over $S$. We will use the notation $\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$ to indicate this situation. \\item A {\\it morphism of \\'etale neighborhoods} $(U, \\overline{u}) \\to (U', \\overline{u}')$ is an $X$-morphism $h : U \\to U'$ such that $\\overline{u}' = h \\circ \\overline{u}$. \\end{enumerate}"}
+{"_id": "11940", "title": "spaces-properties-definition-stalk", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a presheaf on $X_\\etale$. Let $\\overline{x}$ be a geometric point of $X$. The {\\it stalk} of $\\mathcal{F}$ at $\\overline{x}$ is $$ \\mathcal{F}_{\\bar x} = \\colim_{(U, \\overline{u})} \\mathcal{F}(U) $$ where $(U, \\overline{u})$ runs over all \\'etale neighborhoods of $\\overline{x}$ in $X$ with $U \\in \\Ob(X_\\etale)$."}
+{"_id": "11941", "title": "spaces-properties-definition-support", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. \\begin{enumerate} \\item The {\\it support of $\\mathcal{F}$} is the set of points $x \\in |X|$ such that $\\mathcal{F}_{\\overline{x}} \\not = 0$ for any (some) geometric point $\\overline{x}$ lying over $x$. \\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section. The {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where $W \\subset U$ is the largest open subset of $U$ on which $\\sigma$ restricts to zero (see Lemma \\ref{lemma-zero-over-image}). \\end{enumerate}"}
+{"_id": "11942", "title": "spaces-properties-definition-structure-sheaf", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it structure sheaf} of $X$ is the sheaf of rings $\\mathcal{O}_X$ on the small \\'etale site $X_\\etale$ described in Lemma \\ref{lemma-sheaf-condition-holds}."}
+{"_id": "11943", "title": "spaces-properties-definition-etale-local-rings", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x}$ be a geometric point of $X$ lying over the point $x \\in |X|$. \\begin{enumerate} \\item The {\\it \\'etale local ring of $X$ at $\\overline{x}$} is the stalk of the structure sheaf $\\mathcal{O}_X$ on $X_\\etale$ at $\\overline{x}$. Notation: $\\mathcal{O}_{X, \\overline{x}}$. \\item The {\\it strict henselization of $X$ at $\\overline{x}$} is the scheme $\\Spec(\\mathcal{O}_{X, \\overline{x}})$. \\end{enumerate}"}
+{"_id": "11944", "title": "spaces-properties-definition-unibranch", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. We say that $X$ is {\\it geometrically unibranch at $x$} if the equivalent conditions of Lemma \\ref{lemma-irreducible-local-ring} hold. We say that $X$ is {\\it geometrically unibranch} if $X$ is geometrically unibranch at every $x \\in |X|$."}
+{"_id": "11945", "title": "spaces-properties-definition-number-of-geometric-branches", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The {\\it number of geometric branches of $X$ at $x$} is either $n \\in \\mathbf{N}$ if the equivalent conditions of Lemma \\ref{lemma-nr-branches-local-ring} hold, or else $\\infty$."}
+{"_id": "11946", "title": "spaces-properties-definition-noetherian", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is {\\it Noetherian} if $X$ is quasi-compact, quasi-separated and locally Noetherian."}
+{"_id": "11947", "title": "spaces-properties-definition-regular-at-point", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. We say {\\it $X$ is regular at $x$} if $\\mathcal{O}_{U, u}$ is a regular local ring for any (equivalently some) pair $(a : U \\to X, u)$ consisting of an \\'etale morphism $a : U \\to X$ from a scheme to $X$ and a point $u \\in U$ with $a(u) = x$."}
+{"_id": "11948", "title": "spaces-properties-definition-quasi-coherent", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\\it quasi-coherent} $\\mathcal{O}_X$-module is a quasi-coherent module on the ringed site $(X_\\etale, \\mathcal{O}_X)$ in the sense of Modules on Sites, Definition \\ref{sites-modules-definition-site-local}. The category of quasi-coherent sheaves on $X$ is denoted $\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "11949", "title": "spaces-properties-definition-locally-projective", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it locally projective} if the equivalent conditions of Lemma \\ref{lemma-locally-projective} are satisfied."}
+{"_id": "12003", "title": "intersection-definition-proper-intersection", "text": "Let $X$ be a nonsingular variety. \\begin{enumerate} \\item Let $W,V \\subset X$ be closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$. We say that $W$ and $V$ {\\it intersect properly} if $\\dim(V \\cap W) \\leq r + s - \\dim(X)$. \\item Let $\\alpha = \\sum n_i [W_i]$ be an $s$-cycle, and $\\beta = \\sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say that $\\alpha$ and $\\beta$ {\\it intersect properly} if $W_i$ and $V_j$ intersect properly for all $i$ and $j$. \\end{enumerate}"}
+{"_id": "12004", "title": "intersection-definition-multiplicity", "text": "In the situation above, if $d \\geq \\dim(\\text{Supp}(M))$, then we set $e_I(M, d)$ equal to $0$ if $d > \\dim(\\text{Supp}(M))$ and equal to $d!$ times the leading coefficient of the numerical polynomial $\\chi_{I, M}$ so that $$ \\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms} $$ The {\\it multiplicity of $M$ for the ideal of definition $I$} is $e_I(M) = e_I(M, \\dim(\\text{Supp}(M)))$."}
+{"_id": "12131", "title": "homology-definition-preadditive", "text": "A category $\\mathcal{A}$ is called {\\it preadditive} if each morphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed with the structure of an abelian group such that the compositions $$ \\Mor(x, y) \\times \\Mor(y, z) \\longrightarrow \\Mor(x, z) $$ are bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of preadditive categories is called {\\it additive} if and only if $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$ is a homomorphism of abelian groups for all $x, y \\in \\Ob(\\mathcal{A})$."}
+{"_id": "12132", "title": "homology-definition-zero-object", "text": "In a preadditive category $\\mathcal{A}$ we call {\\it zero object}, and we denote it $0$ any final and initial object as in Lemma \\ref{lemma-preadditive-zero} above."}
+{"_id": "12133", "title": "homology-definition-direct-sum", "text": "Given a pair of objects $x, y$ in a preadditive category $\\mathcal{A}$, the {\\it direct sum} $x \\oplus y$ of $x$ and $y$ is the direct product $x \\times y$ endowed with the morphisms $i, j, p, q$ as in Lemma \\ref{lemma-preadditive-direct-sum} above."}
+{"_id": "12134", "title": "homology-definition-additive-category", "text": "A category $\\mathcal{A}$ is called {\\it additive} if it is preadditive and finite products exist, in other words it has a zero object and direct sums."}
+{"_id": "12135", "title": "homology-definition-kernel", "text": "Let $\\mathcal{A}$ be a preadditive category. Let $f : x \\to y$ be a morphism. \\begin{enumerate} \\item A {\\it kernel} of $f$ is a morphism $i : z \\to x$ such that (a) $f \\circ i = 0$ and (b) for any $i' : z' \\to x$ such that $f \\circ i' = 0$ there exists a unique morphism $g : z' \\to z$ such that $i' = i \\circ g$. \\item If the kernel of $f$ exists, then we denote this $\\Ker(f) \\to x$. \\item A {\\it cokernel} of $f$ is a morphism $p : y \\to z$ such that (a) $p \\circ f = 0$ and (b) for any $p' : y \\to z'$ such that $p' \\circ f = 0$ there exists a unique morphism $g : z \\to z'$ such that $p' = g \\circ p$. \\item If a cokernel of $f$ exists we denote this $y \\to \\Coker(f)$. \\item If a kernel of $f$ exists, then a {\\it coimage of $f$} is a cokernel for the morphism $\\Ker(f) \\to x$. \\item If a kernel and coimage exist then we denote this $x \\to \\Coim(f)$. \\item If a cokernel of $f$ exists, then the {\\it image of $f$} is a kernel of the morphism $y \\to \\Coker(f)$. \\item If a cokernel and image of $f$ exist then we denote this $\\Im(f) \\to y$. \\end{enumerate}"}
+{"_id": "12136", "title": "homology-definition-karoubian", "text": "Let $\\mathcal{C}$ be a preadditive category. We say $\\mathcal{C}$ is {\\it Karoubian} if every idempotent endomorphism of an object of $\\mathcal{C}$ has a kernel."}
+{"_id": "12137", "title": "homology-definition-abelian-category", "text": "A category $\\mathcal{A}$ is {\\it abelian} if it is additive, if all kernels and cokernels exist, and if the natural map $\\Coim(f) \\to \\Im(f)$ is an isomorphism for all morphisms $f$ of $\\mathcal{A}$."}
+{"_id": "12138", "title": "homology-definition-injective-surjective", "text": "Let $f : x \\to y$ be a morphism in an abelian category. \\begin{enumerate} \\item We say $f$ is {\\it injective} if $\\Ker(f) = 0$. \\item We say $f$ is {\\it surjective} if $\\Coker(f) = 0$. \\end{enumerate} If $x \\to y$ is injective, then we say that $x$ is a {\\it subobject} of $y$ and we use the notation $x \\subset y$. If $x \\to y$ is surjective, then we say that $y$ is a {\\it quotient} of $x$."}
+{"_id": "12139", "title": "homology-definition-exact", "text": "Let $\\mathcal{A}$ be an additive category. We say a sequence of morphisms $$ \\ldots \\to x \\to y \\to z \\to \\ldots $$ in $\\mathcal{A}$ is a {\\it complex} if the composition of any two (drawn) arrows is zero. If $\\mathcal{A}$ is abelian then we say a sequence as above is {\\it exact at $y$} if $\\Im(x \\to y) = \\Ker(y \\to z)$. We say it is {\\it exact} if it is exact at every object. A {\\it short exact sequence} is an exact complex of the form $$ 0 \\to A  \\to B \\to C \\to 0. $$"}
+{"_id": "12140", "title": "homology-definition-ses-split", "text": "Let $\\mathcal{A}$ be an abelian category. Let $i : A \\to B$ and $q : B \\to C$ be morphisms of $\\mathcal{A}$ such that $0 \\to A \\to B \\to C \\to 0$ is a short exact sequence. We say the short exact sequence is {\\it split} if there exist morphisms $j : C \\to B$ and $p : B \\to A$ such that $(B, i, j, p, q)$ is the direct sum of $A$ and $C$."}
+{"_id": "12141", "title": "homology-definition-extension", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A, B \\in \\Ob(\\mathcal{A})$. An {\\it extension $E$ of $B$ by $A$} is a short exact sequence $$ 0 \\to A \\to E \\to B \\to 0. $$ An {\\it morphism of extensions} between two extensions $0 \\to A \\to E \\to B \\to 0$ and $0 \\to A \\to F \\to B \\to 0$ means a morphism $f : E \\to F$ in $\\mathcal{A}$ making the diagram $$ \\xymatrix{ 0 \\ar[r] & A \\ar[r] \\ar[d]^{\\text{id}} & E \\ar[r] \\ar[d]^f & B \\ar[r] \\ar[d]^{\\text{id}} & 0 \\\\ 0 \\ar[r] & A \\ar[r] & F \\ar[r] & B \\ar[r] & 0 } $$ commutative. Thus, the extensions of $B$ by $A$ form a category."}
+{"_id": "12142", "title": "homology-definition-ext-group", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A, B \\in \\Ob(\\mathcal{A})$. The set of isomorphism classes of extensions of $B$ by $A$ is denoted $$ \\Ext_\\mathcal{A}(B, A). $$ This is called the {\\it $\\Ext$-group}."}
+{"_id": "12143", "title": "homology-definition-simple", "text": "Let $\\mathcal{A}$ be an abelian category. An object $A$ of $\\mathcal{A}$ is said to be {\\it simple} if it is nonzero and the only subobjects of $A$ are $0$ and $A$."}
+{"_id": "12144", "title": "homology-definition-Artinian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item We say an object $A$ of $\\mathcal{A}$ is {\\it Artinian} if and only if it satisfies the descending chain condition for subobjects. \\item We say $\\mathcal{A}$ is {\\it Artinian} if every object of $\\mathcal{A}$ is Artinian. \\end{enumerate}"}
+{"_id": "12145", "title": "homology-definition-Noetherian", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item We say an object $A$ of $\\mathcal{A}$ is {\\it Noetherian} if and only if it satisfies the ascending chain condition for subobjects. \\item We say $\\mathcal{A}$ is {\\it Noetherian} if every object of $\\mathcal{A}$ is Noetherian. \\end{enumerate}"}
+{"_id": "12146", "title": "homology-definition-serre-subcategory", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item A {\\it Serre subcategory} of $\\mathcal{A}$ is a nonempty full subcategory $\\mathcal{C}$ of $\\mathcal{A}$ such that given an exact sequence $$ A \\to B \\to C $$ with $A, C \\in \\Ob(\\mathcal{C})$, then also $B \\in \\Ob(\\mathcal{C})$. \\item A {\\it weak Serre subcategory} of $\\mathcal{A}$ is a nonempty full subcategory $\\mathcal{C}$ of $\\mathcal{A}$ such that given an exact sequence $$ A_0 \\to A_1 \\to A_2 \\to A_3 \\to A_4 $$ with $A_0, A_1, A_3, A_4$ in $\\mathcal{C}$, then also $A_2$ in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12147", "title": "homology-definition-kernel-category", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor. Then the full subcategory of objects $C$ of $\\mathcal{A}$ such that $F(C) = 0$ is called the {\\it kernel of the functor $F$}, and is sometimes denoted $\\Ker(F)$."}
+{"_id": "12148", "title": "homology-definition-K-zero", "text": "Let $\\mathcal{A}$ be an abelian category. We denote $K_0(\\mathcal{A})$ the {\\it zeroth $K$-group of $\\mathcal{A}$}. It is the abelian group constructed as follows. Take the free abelian group on the objects on $\\mathcal{A}$ and for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ impose the relation $[B] - [A] - [C] = 0$."}
+{"_id": "12149", "title": "homology-definition-cohomological-delta-functor", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. A {\\it cohomological $\\delta$-functor} or simply a {\\it $\\delta$-functor} from $\\mathcal{A}$ to $\\mathcal{B}$ is given by the following data: \\begin{enumerate} \\item a collection $F^n : \\mathcal{A} \\to \\mathcal{B}$, $n \\geq 0$ of additive functors, and \\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ of $\\mathcal{A}$ a collection $\\delta_{A \\to B \\to C} : F^n(C) \\to F^{n + 1}(A)$, $n \\geq 0$ of morphisms of $\\mathcal{B}$. \\end{enumerate} These data are assumed to satisfy the following axioms \\begin{enumerate} \\item for every short exact sequence as above the sequence $$ \\xymatrix{ 0 \\ar[r] & F^0(A) \\ar[r] & F^0(B) \\ar[r] & F^0(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\  & F^1(A) \\ar[r] & F^1(B) \\ar[r] & F^1(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\  & F^2(A) \\ar[r] & F^2(B) \\ar[r] & \\ldots } $$ is exact, and \\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$ of short exact sequences of $\\mathcal{A}$ the diagrams $$ \\xymatrix{ F^n(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & & F^{n + 1}(A) \\ar[d] \\\\ F^n(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & & F^{n + 1}(A') } $$ are commutative. \\end{enumerate}"}
+{"_id": "12150", "title": "homology-definition-morphism-delta-functors", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let $(F^n, \\delta_F)$ and $(G^n, \\delta_G)$ be $\\delta$-functors from $\\mathcal{A}$ to $\\mathcal{B}$. A {\\it morphism of $\\delta$-functors from $F$ to $G$} is a collection of transformation of functors $t^n : F^n \\to G^n$, $n \\geq 0$ such that for every short exact sequence $0 \\to A \\to B \\to C \\to 0$ of $\\mathcal{A}$ the diagrams $$ \\xymatrix{ F^n(C) \\ar[d]_{t^n} \\ar[rr]_{\\delta_{F, A \\to B \\to C}} & & F^{n + 1}(A) \\ar[d]^{t^{n + 1}} \\\\ G^n(C) \\ar[rr]^{\\delta_{G, A \\to B \\to C}} & & G^{n + 1}(A) } $$ are commutative."}
+{"_id": "12151", "title": "homology-definition-universal-delta-functor", "text": "Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let $F = (F^n, \\delta_F)$ be a $\\delta$-functor from $\\mathcal{A}$ to $\\mathcal{B}$. We say $F$ is a {\\it universal $\\delta$-functor} if and only if for every $\\delta$-functor $G = (G^n, \\delta_G)$ and any morphism of functors $t : F^0 \\to G^0$ there exists a unique morphism of $\\delta$-functors $\\{t^n\\}_{n \\geq 0} : F \\to G$ such that $t = t^0$."}
+{"_id": "12152", "title": "homology-definition-homotopy-equivalent", "text": "Let $\\mathcal{A}$ be an additive category. We say a morphism $a : A_\\bullet \\to B_\\bullet$ is a {\\it homotopy equivalence} if there exists a morphism $b : B_\\bullet \\to A_\\bullet$ such that there exists a homotopy between $a \\circ b$ and $\\text{id}_A$ and there exists a homotopy between $b \\circ a$ and $\\text{id}_B$. If there exists such a morphism between $A_\\bullet$ and $B_\\bullet$, then we say that $A_\\bullet$ and $B_\\bullet$ are {\\it homotopy equivalent}."}
+{"_id": "12153", "title": "homology-definition-quasi-isomorphism", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item A morphism of chain complexes $f : A_\\bullet \\to B_\\bullet$ is called a {\\it quasi-isomorphism} if the induced map $H_i(f) : H_i(A_\\bullet) \\to H_i(B_\\bullet)$ is an isomorphism for all $i \\in \\mathbf{Z}$. \\item A chain complex $A_\\bullet$ is called {\\it acyclic} if all of its homology objects $H_i(A_\\bullet)$ are zero. \\end{enumerate}"}
+{"_id": "12154", "title": "homology-definition-homotopy-equivalent-cochain", "text": "Let $\\mathcal{A}$ be an additive category. We say a morphism $a : A^\\bullet \\to B^\\bullet$ is a {\\it homotopy equivalence} if there exists a morphism $b : B^\\bullet \\to A^\\bullet$ such that there exists a homotopy between $a \\circ b$ and $\\text{id}_A$ and there exists a homotopy between $b \\circ a$ and $\\text{id}_B$. If there exists such a morphism between $A^\\bullet$ and $B^\\bullet$, then we say that $A^\\bullet$ and $B^\\bullet$ are {\\it homotopy equivalent}."}
+{"_id": "12155", "title": "homology-definition-quasi-isomorphism-cochain", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item A morphism of cochain complexes $f : A^\\bullet \\to B^\\bullet$ of $\\mathcal{A}$ is called a {\\it quasi-isomorphism} if the induced maps $H^i(f) : H^i(A^\\bullet) \\to H^i(B^\\bullet)$ is an isomorphism for all $i \\in \\mathbf{Z}$. \\item A cochain complex $A^\\bullet$ is called {\\it acyclic} if all of its cohomology objects $H^i(A^\\bullet)$ are zero. \\end{enumerate}"}
+{"_id": "12156", "title": "homology-definition-shift", "text": "Let $\\mathcal{A}$ be an additive category. Let $A_\\bullet$ be a chain complex with boundary maps $d_{A, n} : A_n \\to A_{n - 1}$. For any $k \\in \\mathbf{Z}$ we define the {\\it $k$-shifted chain complex $A[k]_\\bullet$} as follows: \\begin{enumerate} \\item we set $A[k]_n = A_{n + k}$, and \\item we set $d_{A[k], n} : A[k]_n \\to A[k]_{n - 1}$ equal to $d_{A[k], n} = (-1)^k d_{A, n + k}$. \\end{enumerate} If $f : A_\\bullet \\to B_\\bullet$ is a morphism of chain complexes, then we let $f[k] : A[k]_\\bullet \\to B[k]_\\bullet$ be the morphism of chain complexes with $f[k]_n = f_{k + n}$."}
+{"_id": "12157", "title": "homology-definition-homology-shift", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A_\\bullet$ be a chain complex with boundary maps $d_{A, n} : A_n \\to A_{n - 1}$. For any $k \\in \\mathbf{Z}$ we identify {\\it $H_{i + k}(A_\\bullet) \\rightarrow H_i(A[k]_\\bullet)$} via the identification $A_{i + k} = A[k]_i$."}
+{"_id": "12158", "title": "homology-definition-shift-cochain", "text": "Let $\\mathcal{A}$ be an additive category. Let $A^\\bullet$ be a cochain complex with boundary maps $d_A^n : A^n \\to A^{n + 1}$. For any $k \\in \\mathbf{Z}$ we define the {\\it $k$-shifted cochain complex $A[k]^\\bullet$} as follows: \\begin{enumerate} \\item we set $A[k]^n = A^{n + k}$, and \\item we set $d_{A[k]}^n : A[k]^n \\to A[k]^{n + 1}$ equal to $d_{A[k]}^n = (-1)^k d_A^{n + k}$. \\end{enumerate} If $f : A^\\bullet \\to B^\\bullet$ is a morphism of cochain complexes, then we let $f[k] : A[k]^\\bullet \\to B[k]^\\bullet$ be the morphism of cochain complexes with $f[k]^n = f^{k + n}$."}
+{"_id": "12159", "title": "homology-definition-cohomology-shift", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A^\\bullet$ be a cochain complex with boundary maps $d_A^n : A^n \\to A^{n + 1}$. For any $k \\in \\mathbf{Z}$ we identify {\\it $H^{i + k}(A^\\bullet) \\longrightarrow H^i(A[k]^\\bullet)$} via the identification $A^{i + k} = A[k]^i$."}
+{"_id": "12160", "title": "homology-definition-graded", "text": "Let $\\mathcal{A}$ be an additive category. The {\\it category of graded objects of $\\mathcal{A}$}, denoted $\\text{Gr}(\\mathcal{A})$, is the category with \\begin{enumerate} \\item objects $A = (A^i)$ are families of objects $A^i$, $i \\in \\mathbf{Z}$ of objects of $\\mathcal{A}$, and \\item morphisms $f : A = (A^i) \\to B = (B^i)$ are families of morphisms $f^i : A^i \\to B^i$ of $\\mathcal{A}$. \\end{enumerate}"}
+{"_id": "12161", "title": "homology-definition-graded-shift", "text": "Let $\\mathcal{A}$ be an additive category. If $A = (A^i)$ is a graded object, then the $k$th {\\it shift} $A[k]$ is the graded object with $A[k]^i = A^{k + i}$."}
+{"_id": "12162", "title": "homology-definition-additive-monoidal", "text": "An {\\it additive monoidal category} is an additive category $\\mathcal{A}$ endowed with a monoidal structure $\\otimes, \\phi$ (Categories, Definition \\ref{categories-definition-monoidal-category}) such that $\\otimes$ is an additive functor in each variable."}
+{"_id": "12163", "title": "homology-definition-double-complex", "text": "Let $\\mathcal{A}$ be an additive category. A {\\it double complex} in $\\mathcal{A}$ is given by a system $(\\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\\}_{p, q\\in \\mathbf{Z}})$, where each $A^{p, q}$ is an object of $\\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ are morphisms of $\\mathcal{A}$ such that the following rules hold: \\begin{enumerate} \\item $d_1^{p + 1, q} \\circ d_1^{p, q} = 0$ \\item $d_2^{p, q + 1} \\circ d_2^{p, q} = 0$ \\item $d_1^{p, q + 1} \\circ d_2^{p, q} = d_2^{p + 1, q} \\circ d_1^{p, q}$ \\end{enumerate} for all $p, q \\in \\mathbf{Z}$."}
+{"_id": "12164", "title": "homology-definition-associated-simple-complex", "text": "Let $\\mathcal{A}$ be an additive category. Let $A^{\\bullet, \\bullet}$ be a double complex. The {\\it associated simple complex}, denoted $sA^\\bullet$, also often called the {\\it associated total complex}, denoted $\\text{Tot}(A^{\\bullet, \\bullet})$, is given by $$ sA^n = \\text{Tot}^n(A^{\\bullet, \\bullet}) = \\bigoplus\\nolimits_{n = p + q} A^{p, q} $$ (if it exists) with differential $$ d_{sA^\\bullet}^n = d_{\\text{Tot}(A^{\\bullet, \\bullet})}^n = \\sum\\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q}) $$"}
+{"_id": "12165", "title": "homology-definition-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item A {\\it decreasing filtration} $F$ on an object $A$ is a family $(F^nA)_{n \\in \\mathbf{Z}}$ of subobjects of $A$ such that $$ A \\supset \\ldots \\supset F^nA \\supset F^{n + 1}A \\supset \\ldots \\supset 0 $$ \\item A {\\it filtered object of $\\mathcal{A}$} is pair $(A, F)$ consisting of an object $A$ of $\\mathcal{A}$ and a decreasing filtration $F$ on $A$. \\item A {\\it morphism $(A, F) \\to (B, F)$ of filtered objects} is given by a morphism $\\varphi : A \\to B$ of $\\mathcal{A}$ such that $\\varphi(F^iA) \\subset F^iB$ for all $i \\in \\mathbf{Z}$. \\item The category of filtered objects is denoted $\\text{Fil}(\\mathcal{A})$. \\item Given a filtered object $(A, F)$ and a subobject $X \\subset A$ the {\\it induced filtration} on $X$ is the filtration with $F^nX = X \\cap F^nA$. \\item Given a filtered object $(A, F)$ and a surjection $\\pi : A \\to Y$ the {\\it quotient filtration} is the filtration with $F^nY = \\pi(F^nA)$. \\item A filtration $F$ on an object $A$ is said to be {\\it finite} if there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$. \\item Given a filtered object $(A, F)$ we say $\\bigcap F^iA$ exists if there exists a biggest subobject of $A$ contained in all $F^iA$. We say $\\bigcup F^iA$ exists if there exists a smallest subobject of $A$ containing all $F^iA$. \\item The filtration on a filtered object $(A, F)$ is said to be {\\it separated} if $\\bigcap F^iA = 0$ and {\\it exhaustive} if $\\bigcup F^iA = A$. \\end{enumerate}"}
+{"_id": "12166", "title": "homology-definition-strict", "text": "Let $\\mathcal{A}$ be an abelian category. A morphism $f : A \\to B$ of filtered objects of $\\mathcal{A}$ is said to be {\\it strict} if $f(F^iA) = f(A) \\cap F^iB$ for all $i \\in \\mathbf{Z}$."}
+{"_id": "12167", "title": "homology-definition-spectral-sequence", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item A {\\it spectral sequence in $\\mathcal{A}$} is given by a system $(E_r, d_r)_{r \\geq 1}$ where each $E_r$ is an object of $\\mathcal{A}$, each $d_r : E_r \\to E_r$ is a morphism such that $d_r \\circ d_r = 0$ and $E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$ for $r \\geq 1$. \\item A {\\it morphism of spectral sequences} $f : (E_r, d_r)_{r \\geq 1} \\to (E'_r, d'_r)_{r \\geq 1}$ is given by a family of morphisms $f_r : E_r \\to E'_r$ such that $f_r \\circ d_r = d'_r \\circ f_r$ and such that $f_{r + 1}$ is the morphism induced by $f_r$ via the identifications $E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$ and $E'_{r + 1} = \\Ker(d'_r)/\\Im(d'_r)$. \\end{enumerate}"}
+{"_id": "12168", "title": "homology-definition-limit-spectral-sequence", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \\geq 1}$ be a spectral sequence. \\begin{enumerate} \\item If the subobjects $Z_{\\infty} = \\bigcap Z_r$ and $B_{\\infty} = \\bigcup B_r$ of $E_1$ exist then we define the {\\it limit}\\footnote{This notation is not universally accepted. In some references an additional pair of subobjects $Z_\\infty$ and $B_\\infty$ of $E_1$ such that $0 = B_1 \\subset B_2 \\subset \\ldots \\subset B_\\infty \\subset Z_\\infty \\subset \\ldots \\subset Z_2 \\subset Z_1 = E_1$ is part of the data comprising a spectral sequence!} of the spectral sequence to be the object $E_{\\infty} = Z_{\\infty}/B_{\\infty}$. \\item We say that the spectral sequence {\\it degenerates at $E_r$} if the differentials $d_r, d_{r + 1}, \\ldots$ are all zero. \\end{enumerate}"}
+{"_id": "12169", "title": "homology-definition-exact-couple", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item An {\\it exact couple} is a datum $(A, E, \\alpha, f, g)$ where $A$, $E$ are objects of $\\mathcal{A}$ and $\\alpha$, $f$, $g$ are morphisms as in the following diagram $$ \\xymatrix{ A \\ar[rr]_{\\alpha} & & A \\ar[ld]^g \\\\ & E \\ar[lu]^f & } $$ with the property that the kernel of each arrow is the image of its predecessor. So $\\Ker(\\alpha) = \\Im(f)$, $\\Ker(f) = \\Im(g)$, and $\\Ker(g) = \\Im(\\alpha)$. \\item A {\\it morphism of exact couples} $t : (A, E, \\alpha, f, g) \\to (A', E', \\alpha', f', g')$ is given by morphisms $t_A : A \\to A'$ and $t_E : E \\to E'$ such that $\\alpha' \\circ t_A = t_A \\circ \\alpha$, $f' \\circ t_E = t_A \\circ f$, and $g' \\circ t_A = t_E \\circ g$. \\end{enumerate}"}
+{"_id": "12170", "title": "homology-definition-spectral-sequence-associated-exact-couple", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A, E, \\alpha, f, g)$ be an exact couple. The {\\it spectral sequence associated to the exact couple} is the spectral sequence $(E_r, d_r)_{r \\geq 1}$ with $E_1 = E$, $d_1 = d$, $E_2 = E'$, $d_2 = d' = g' \\circ f'$, $E_3 = E''$, $d_3 = d'' = g'' \\circ f''$, and so on."}
+{"_id": "12171", "title": "homology-definition-differential-object", "text": "Let $\\mathcal{A}$ be an abelian category. A {\\it differential object} of $\\mathcal{A}$ is a pair $(A, d)$ consisting of an object $A$ of $\\mathcal{A}$ endowed with a selfmap $d$ such that $d \\circ d = 0$. A {\\it morphism of differential objects} $(A, d) \\to (B, d)$ is given by a morphism $\\alpha : A \\to B$ such that $d \\circ \\alpha = \\alpha \\circ d$."}
+{"_id": "12172", "title": "homology-definition-differential-object-homology", "text": "For a differential object $(A, d)$ we denote $$ H(A, d) = \\Ker(d)/\\Im(d) $$ its {\\it homology}."}
+{"_id": "12173", "title": "homology-definition-differential-object-selfmap", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(A, d)$ be a differential object of $\\mathcal{A}$. Let $\\alpha : A \\to A$ be an injective selfmap of $A$ which commutes with $d$. The {\\it spectral sequence associated to $(A, d, \\alpha)$} is the spectral sequence $(E_r, d_r)_{r \\geq 0}$ described above."}
+{"_id": "12174", "title": "homology-definition-filtered-differential", "text": "Let $\\mathcal{A}$ be an abelian category. A {\\it filtered differential object} $(K, F, d)$ is a filtered object $(K, F)$ of $\\mathcal{A}$ endowed with an endomorphism $d : (K, F) \\to (K, F)$ whose square is zero: $d \\circ d = 0$."}
+{"_id": "12175", "title": "homology-definition-filtration-cohomology-filtered-differential", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. The {\\it induced filtration} on $H(K, d)$ is the filtration defined by $F^pH(K, d) = \\Im(H(F^pK, d) \\to H(K, d))$."}
+{"_id": "12176", "title": "homology-definition-filtered-differential-ss-converges", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. We say the spectral sequence associated to $(K, F, d)$ \\begin{enumerate} \\item {\\it weakly converges to $H(K)$} if $\\text{gr}H(K) = E_{\\infty}$ via Lemma \\ref{lemma-compute-filtered-cohomology}, \\item {\\it abuts to $H(K)$} if it weakly converges to $H(K)$ and we have $\\bigcap F^pH(K) = 0$ and $\\bigcup F^pH(K) = H(K)$, \\end{enumerate}"}
+{"_id": "12177", "title": "homology-definition-filtered-complex", "text": "Let $\\mathcal{A}$ be an abelian category. A {\\it filtered complex $K^\\bullet$ of $\\mathcal{A}$} is a complex of $\\text{Fil}(\\mathcal{A})$ (see Definition \\ref{definition-filtered})."}
+{"_id": "12178", "title": "homology-definition-filtration-cohomology-filtered-complex", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. The {\\it induced filtration} on $H^n(K^\\bullet)$ is the filtration defined by $F^pH^n(K^\\bullet) = \\Im(H^n(F^pK^\\bullet) \\to H^n(K^\\bullet))$."}
+{"_id": "12179", "title": "homology-definition-bounded-ss", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \\geq r_0}$ be a spectral sequence of bigraded objects of $\\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$. We say such a spectral sequence is \\begin{enumerate} \\item {\\it regular} if for all $p, q \\in \\mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ are zero for $r \\geq b$, \\item {\\it coregular} if for all $p, q \\in \\mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_r^{p - r, q + r - 1} : E_r^{p - r, q + r - 1} \\to E_r^{p, q}$ are zero for $r \\geq b$, \\item {\\it bounded} if for all $n$ there are only a finite number of nonzero $E_{r_0}^{p, n - p}$, \\item {\\it bounded below} if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \\geq b$. \\item {\\it bounded above} if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \\leq b$. \\end{enumerate}"}
+{"_id": "12180", "title": "homology-definition-filtered-complex-ss-converges", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$. We say the spectral sequence associated to $(K^\\bullet, F)$ \\begin{enumerate} \\item {\\it weakly converges to $H^*(K^\\bullet)$} if $\\text{gr}^pH^n(K^\\bullet) = E_{\\infty}^{p, n - p}$ via Lemma \\ref{lemma-compute-cohomology-filtered-complex} for all $p, n \\in \\mathbf{Z}$, \\item {\\it abuts to $H^*(K^\\bullet)$} if it weakly converges to $H^*(K^\\bullet)$ and $\\bigcap_p F^pH^n(K^\\bullet) = 0$ and $\\bigcup_p F^p H^n(K^\\bullet) = H^n(K^\\bullet)$ for all $n$, \\item {\\it converges to $H^*(K^\\bullet)$} if it is regular, abuts to $H^*(K^\\bullet)$, and $H^n(K^\\bullet) = \\lim_p H^n(K^\\bullet)/F^pH^n(K^\\bullet)$. \\end{enumerate}"}
+{"_id": "12181", "title": "homology-definition-ss-double-complex-converge", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^{\\bullet, \\bullet}$ be a double complex. We say the spectral sequence $({}'E_r, {}'d_r)_{r \\geq 0}$ {\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, {\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or {\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$} if Definition \\ref{definition-filtered-complex-ss-converges} applies. Similarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \\geq 0}$ {\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, {\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or {\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$} if Definition \\ref{definition-filtered-complex-ss-converges} applies."}
+{"_id": "12182", "title": "homology-definition-injective", "text": "Let $\\mathcal{A}$ be an abelian category. An object $J \\in \\Ob(\\mathcal{A})$ is called {\\it injective} if for every injection $A \\hookrightarrow B$ and every morphism $A \\to J$ there exists a morphism $B \\to J$ making the following diagram commute $$ \\xymatrix{ A \\ar[r] \\ar[d] & B \\ar@{-->}[ld] \\\\ J & } $$"}
+{"_id": "12183", "title": "homology-definition-enough-injectives", "text": "Let $\\mathcal{A}$ be an abelian category. We say $\\mathcal{A}$ has {\\it enough injectives} if every object $A$ has an injective morphism $A \\to J$ into an injective object $J$."}
+{"_id": "12184", "title": "homology-definition-functorial-injective-embedding", "text": "Let $\\mathcal{A}$ be an abelian category. We say that $\\mathcal{A}$ has {\\it functorial injective embeddings} if there exists a functor $$ J : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A}) $$ such that \\begin{enumerate} \\item $s \\circ J = \\text{id}_\\mathcal{A}$, \\item for any object $A \\in \\Ob(\\mathcal{A})$ the morphism $J(A)$ is injective, and \\item for any object $A \\in \\Ob(\\mathcal{A})$ the object $t(J(A))$ is an injective object of $\\mathcal{A}$. \\end{enumerate} We will denote such a functor by $A \\mapsto (A \\to J(A))$."}
+{"_id": "12185", "title": "homology-definition-projective", "text": "Let $\\mathcal{A}$ be an abelian category. An object $P \\in \\Ob(\\mathcal{A})$ is called {\\it projective} if for every surjection $A \\rightarrow B$ and every morphism $P \\to B$ there exists a morphism $P \\to A$ making the following diagram commute $$ \\xymatrix{ A \\ar[r] & B \\\\ P \\ar@{-->}[u] \\ar[ru] & } $$"}
+{"_id": "12186", "title": "homology-definition-enough-projectives", "text": "Let $\\mathcal{A}$ be an abelian category. We say $\\mathcal{A}$ has {\\it enough projectives} if every object $A$ has an surjective morphism $P \\to A$ from an projective object $P$ onto it."}
+{"_id": "12187", "title": "homology-definition-functorial-projective-surjections", "text": "Let $\\mathcal{A}$ be an abelian category. We say that $\\mathcal{A}$ has {\\it functorial projective surjections} if there exists a functor $$ P : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A}) $$ such that \\begin{enumerate} \\item $t \\circ J = \\text{id}_\\mathcal{A}$, \\item for any object $A \\in \\Ob(\\mathcal{A})$ the morphism $P(A)$ is surjective, and \\item for any object $A \\in \\Ob(\\mathcal{A})$ the object $s(P(A))$ is an projective object of $\\mathcal{A}$. \\end{enumerate} We will denote such a functor by $A \\mapsto (P(A) \\to A)$."}
+{"_id": "12188", "title": "homology-definition-Mittag-Leffler", "text": "Let $\\mathcal{C}$ be an abelian category. We say the inverse system $(A_i)$ satisfies the {\\it Mittag-Leffler condition}, or for short is {\\it ML}, if for every $i$ there exists a $c = c(i) \\geq i$ such that $$ \\Im(A_k \\to A_i) = \\Im(A_c \\to A_i) $$ for all $k \\geq c$."}
+{"_id": "12328", "title": "categories-definition-category", "text": "A {\\it category} $\\mathcal{C}$ consists of the following data: \\begin{enumerate} \\item A set of objects $\\Ob(\\mathcal{C})$. \\item For each pair $x, y \\in \\Ob(\\mathcal{C})$ a set of morphisms $\\Mor_\\mathcal{C}(x, y)$. \\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a composition map $ \\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y) \\to \\Mor_\\mathcal{C}(x, z) $, denoted $(\\phi, \\psi) \\mapsto \\phi \\circ \\psi$. \\end{enumerate} These data are to satisfy the following rules: \\begin{enumerate} \\item For every element $x\\in \\Ob(\\mathcal{C})$ there exists a morphism $\\text{id}_x\\in \\Mor_\\mathcal{C}(x, x)$ such that $\\text{id}_x \\circ \\phi = \\phi$ and $\\psi \\circ \\text{id}_x = \\psi $ whenever these compositions make sense. \\item Composition is associative, i.e., $(\\phi \\circ \\psi) \\circ \\chi = \\phi \\circ ( \\psi \\circ \\chi)$ whenever these compositions make sense. \\end{enumerate}"}
+{"_id": "12329", "title": "categories-definition-isomorphism", "text": "A morphism $\\phi : x \\to y$ is an {\\it isomorphism} of the category $\\mathcal{C}$ if there exists a morphism $\\psi : y \\to x$ such that $\\phi \\circ \\psi = \\text{id}_y$ and $\\psi \\circ \\phi = \\text{id}_x$."}
+{"_id": "12330", "title": "categories-definition-groupoid", "text": "A {\\it groupoid} is a category where every morphism is an isomorphism."}
+{"_id": "12331", "title": "categories-definition-functor", "text": "A {\\it functor} $F : \\mathcal{A} \\to \\mathcal{B}$ between two categories $\\mathcal{A}, \\mathcal{B}$ is given by the following data: \\begin{enumerate} \\item A map $F : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{B})$. \\item For every $x, y \\in \\Ob(\\mathcal{A})$ a map $F : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y))$, denoted $\\phi \\mapsto F(\\phi)$. \\end{enumerate} These data should be compatible with composition and identity morphisms in the following manner: $F(\\phi \\circ \\psi) = F(\\phi) \\circ F(\\psi)$ for a composable pair $(\\phi, \\psi)$ of morphisms of $\\mathcal{A}$ and $F(\\text{id}_x) = \\text{id}_{F(x)}$."}
+{"_id": "12332", "title": "categories-definition-faithful", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. \\begin{enumerate} \\item We say $F$ is {\\it faithful} if for any objects $x, y$ of $\\Ob(\\mathcal{A})$ the map $$ F : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y)) $$ is injective. \\item If these maps are all bijective then $F$ is called {\\it fully faithful}. \\item The functor $F$ is called {\\it essentially surjective} if for any object $y \\in \\Ob(\\mathcal{B})$ there exists an object $x \\in \\Ob(\\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\\mathcal{B}$. \\end{enumerate}"}
+{"_id": "12333", "title": "categories-definition-subcategory", "text": "A {\\it subcategory} of a category $\\mathcal{B}$ is a category $\\mathcal{A}$ whose objects and arrows form subsets of the objects and arrows of $\\mathcal{B}$ and such that source, target and composition in $\\mathcal{A}$ agree with those of $\\mathcal{B}$. We say $\\mathcal{A}$ is a {\\it full subcategory} of $\\mathcal{B}$ if $\\Mor_\\mathcal{A}(x, y) = \\Mor_\\mathcal{B}(x, y)$ for all $x, y \\in \\Ob(\\mathcal{A})$. We say $\\mathcal{A}$ is a {\\it strictly full} subcategory of $\\mathcal{B}$ if it is a full subcategory and given $x \\in \\Ob(\\mathcal{A})$ any object of $\\mathcal{B}$ which is isomorphic to $x$ is also in $\\mathcal{A}$."}
+{"_id": "12334", "title": "categories-definition-transformation-functors", "text": "Let $F, G : \\mathcal{A} \\to \\mathcal{B}$ be functors. A {\\it natural transformation}, or a {\\it morphism of functors} $t : F \\to G$, is a collection $\\{t_x\\}_{x\\in \\Ob(\\mathcal{A})}$ such that \\begin{enumerate} \\item $t_x : F(x) \\to G(x)$ is a morphism in the category $\\mathcal{B}$, and \\item for every morphism $\\phi : x \\to y$ of $\\mathcal{A}$ the following diagram is commutative $$ \\xymatrix{ F(x) \\ar[r]^{t_x} \\ar[d]_{F(\\phi)} & G(x) \\ar[d]^{G(\\phi)} \\\\ F(y) \\ar[r]^{t_y} & G(y) } $$ \\end{enumerate}"}
+{"_id": "12335", "title": "categories-definition-equivalence-categories", "text": "An {\\it equivalence of categories} $F : \\mathcal{A} \\to \\mathcal{B}$ is a functor such that there exists a functor $G : \\mathcal{B} \\to \\mathcal{A}$ such that the compositions $F \\circ G$ and $G \\circ F$ are isomorphic to the identity functors $\\text{id}_\\mathcal{B}$, respectively $\\text{id}_\\mathcal{A}$. In this case we say that $G$ is a {\\it quasi-inverse} to $F$."}
+{"_id": "12336", "title": "categories-definition-product-category", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$ be categories. We define the {\\it product category} $\\mathcal{A} \\times \\mathcal{B}$ to be the category with objects $\\Ob(\\mathcal{A} \\times \\mathcal{B}) = \\Ob(\\mathcal{A}) \\times \\Ob(\\mathcal{B})$ and $$ \\Mor_{\\mathcal{A} \\times \\mathcal{B}}((x, y), (x', y')) := \\Mor_\\mathcal{A}(x, x')\\times \\Mor_\\mathcal{B}(y, y'). $$ Composition is defined componentwise."}
+{"_id": "12337", "title": "categories-definition-opposite", "text": "Given a category $\\mathcal{C}$ the {\\it opposite category} $\\mathcal{C}^{opp}$ is the category with the same objects as $\\mathcal{C}$ but all morphisms reversed."}
+{"_id": "12338", "title": "categories-definition-contravariant", "text": "Let $\\mathcal{C}$, $\\mathcal{S}$ be categories. A {\\it contravariant} functor $F$ from $\\mathcal{C}$ to $\\mathcal{S}$ is a functor $\\mathcal{C}^{opp}\\to \\mathcal{S}$."}
+{"_id": "12339", "title": "categories-definition-presheaf", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item A {\\it presheaf of sets on $\\mathcal{C}$} or simply a {\\it presheaf} is a contravariant functor $F$ from $\\mathcal{C}$ to $\\textit{Sets}$. \\item The category of presheaves is denoted $\\textit{PSh}(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "12340", "title": "categories-definition-representable-functor", "text": "A contravariant functor $F : \\mathcal{C}\\to \\textit{Sets}$ is said to be {\\it representable} if it is isomorphic to the functor of points $h_U$ for some object $U$ of $\\mathcal{C}$."}
+{"_id": "12341", "title": "categories-definition-products", "text": "Let $x, y\\in \\Ob(\\mathcal{C})$. A {\\it product} of $x$ and $y$ is an object $x \\times y \\in \\Ob(\\mathcal{C})$ together with morphisms $p\\in \\Mor_{\\mathcal C}(x \\times y, x)$ and $q\\in\\Mor_{\\mathcal C}(x \\times y, y)$ such that the following universal property holds: for any $w\\in \\Ob(\\mathcal{C})$ and morphisms $\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and $\\beta \\in \\Mor_\\mathcal{C}(w, y)$ there is a unique $\\gamma\\in \\Mor_{\\mathcal C}(w, x \\times y)$ making the diagram $$ \\xymatrix{ w \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\ & & x \\times y \\ar[d]_p \\ar[r]_q & y \\\\ & & x & } $$ commute."}
+{"_id": "12342", "title": "categories-definition-has-products-of-pairs", "text": "We say the category $\\mathcal{C}$ {\\it has products of pairs of objects} if a product $x \\times y$ exists for any $x, y \\in \\Ob(\\mathcal{C})$."}
+{"_id": "12343", "title": "categories-definition-coproducts", "text": "Let $x, y \\in \\Ob(\\mathcal{C})$. A {\\it coproduct}, or {\\it amalgamated sum} of $x$ and $y$ is an object $x \\amalg y \\in \\Ob(\\mathcal{C})$ together with morphisms $i \\in \\Mor_{\\mathcal C}(x, x \\amalg y)$ and $j \\in \\Mor_{\\mathcal C}(y, x \\amalg y)$ such that the following universal property holds: for any $w \\in \\Ob(\\mathcal{C})$ and morphisms $\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and $\\beta \\in \\Mor_\\mathcal{C}(y, w)$ there is a unique $\\gamma \\in \\Mor_{\\mathcal C}(x \\amalg y, w)$ making the diagram $$ \\xymatrix{ & y \\ar[d]^j \\ar[rrdd]^\\beta \\\\ x \\ar[r]^i \\ar[rrrd]_\\alpha & x \\amalg y \\ar@{-->}[rrd]^\\gamma \\\\ & & & w } $$ commute."}
+{"_id": "12344", "title": "categories-definition-has-coproducts-of-pairs", "text": "We say the category $\\mathcal{C}$ {\\it has coproducts of pairs of objects} if a coproduct $x \\amalg y$ exists for any $x, y \\in \\Ob(\\mathcal{C})$."}
+{"_id": "12345", "title": "categories-definition-fibre-products", "text": "Let $x, y, z\\in \\Ob(\\mathcal{C})$, $f\\in \\Mor_\\mathcal{C}(x, y)$ and $g\\in \\Mor_{\\mathcal C}(z, y)$. A {\\it fibre product} of $f$ and $g$ is an object $x \\times_y z\\in \\Ob(\\mathcal{C})$ together with morphisms $p \\in \\Mor_{\\mathcal C}(x \\times_y z, x)$ and $q \\in \\Mor_{\\mathcal C}(x \\times_y z, z)$ making the diagram $$ \\xymatrix{ x \\times_y z \\ar[r]_q \\ar[d]_p & z \\ar[d]^g \\\\ x \\ar[r]^f & y } $$ commute, and such that the following universal property holds: for any $w\\in \\Ob(\\mathcal{C})$ and morphisms $\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and $\\beta \\in \\Mor_\\mathcal{C}(w, z)$ with $f \\circ \\alpha = g \\circ \\beta$ there is a unique $\\gamma \\in \\Mor_{\\mathcal C}(w, x \\times_y z)$ making the diagram $$ \\xymatrix{ w \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\ & & x \\times_y z \\ar[d]^p \\ar[r]_q & z \\ar[d]^g \\\\ & & x \\ar[r]^f & y } $$ commute."}
+{"_id": "12346", "title": "categories-definition-cartesian", "text": "We say a commutative diagram $$ \\xymatrix{ w \\ar[r] \\ar[d] & z \\ar[d] \\\\ x \\ar[r] & y } $$ in a category is {\\it cartesian} if $w$ and the morphisms $w \\to x$ and $w \\to z$ form a fibre product of the morphisms $x \\to y$ and $z \\to y$."}
+{"_id": "12347", "title": "categories-definition-has-fibre-products", "text": "We say the category $\\mathcal{C}$ {\\it has fibre products} if the fibre product exists for any $f\\in \\Mor_{\\mathcal C}(x, y)$ and $g\\in \\Mor_{\\mathcal C}(z, y)$."}
+{"_id": "12348", "title": "categories-definition-representable-morphism", "text": "A morphism $f : x \\to y$ of a category $\\mathcal{C}$ is said to be {\\it representable} if for every morphism $z \\to y$ in $\\mathcal{C}$ the fibre product $x \\times_y z$ exists."}
+{"_id": "12349", "title": "categories-definition-representable-map-presheaves", "text": "Let $\\mathcal{C}$ be a category. Let $F, G : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be functors. We say a morphism $a : F \\to G$ is {\\it representable}, or that {\\it $F$ is relatively representable over $G$}, if for every $U \\in \\Ob(\\mathcal{C})$ and any $\\xi \\in G(U)$ the functor $h_U \\times_G F$ is representable."}
+{"_id": "12350", "title": "categories-definition-pushouts", "text": "Let $x, y, z\\in \\Ob(\\mathcal{C})$, $f\\in \\Mor_\\mathcal{C}(y, x)$ and $g\\in \\Mor_{\\mathcal C}(y, z)$. A {\\it pushout} of $f$ and $g$ is an object $x\\amalg_y z\\in \\Ob(\\mathcal{C})$ together with morphisms $p\\in \\Mor_{\\mathcal C}(x, x\\amalg_y z)$ and $q\\in\\Mor_{\\mathcal C}(z, x\\amalg_y z)$ making the diagram $$ \\xymatrix{ y \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\\\ x \\ar[r]^p & x\\amalg_y z } $$ commute, and such that the following universal property holds: For any $w\\in \\Ob(\\mathcal{C})$ and morphisms $\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and $\\beta \\in \\Mor_\\mathcal{C}(z, w)$ with $\\alpha \\circ f = \\beta \\circ g$ there is a unique $\\gamma\\in \\Mor_{\\mathcal C}(x\\amalg_y z, w)$ making the diagram $$ \\xymatrix{ y \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\ar[rrdd]^\\beta & & \\\\ x \\ar[r]^p \\ar[rrrd]^\\alpha & x \\amalg_y z  \\ar@{-->}[rrd]^\\gamma & & \\\\ & & & w } $$ commute."}
+{"_id": "12351", "title": "categories-definition-cocartesian", "text": "We say a commutative diagram $$ \\xymatrix{ y \\ar[r] \\ar[d] & z \\ar[d] \\\\ x \\ar[r] & w } $$ in a category is {\\it cocartesian} if $w$ and the morphisms $x \\to w$ and $z \\to w$ form a pushout of the morphisms $y \\to x$ and $y \\to z$."}
+{"_id": "12352", "title": "categories-definition-equalizers", "text": "Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$ and that $a, b : X \\to Y$ are morphisms. We say a morphism $e : Z \\to X$ is an {\\it equalizer} for the pair $(a, b)$ if $a \\circ e = b \\circ e$ and if $(Z, e)$ satisfies the following universal property: For every morphism $t : W \\to X$ in $\\mathcal{C}$ such that $a \\circ t = b \\circ t$ there exists a unique morphism $s : W \\to Z$ such that $t = e \\circ s$."}
+{"_id": "12353", "title": "categories-definition-coequalizers", "text": "Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$ and that $a, b : X \\to Y$ are morphisms. We say a morphism $c : Y \\to Z$ is a {\\it coequalizer} for the pair $(a, b)$ if $c \\circ a = c \\circ b$ and if $(Z, c)$ satisfies the following universal property: For every morphism $t : Y \\to W$ in $\\mathcal{C}$ such that $t \\circ a = t \\circ b$ there exists a unique morphism $s : Z \\to W$ such that $t = s \\circ c$."}
+{"_id": "12354", "title": "categories-definition-initial-final", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item An object $x$ of the category $\\mathcal{C}$ is called an {\\it initial} object if for every object $y$ of $\\mathcal{C}$ there is exactly one morphism $x \\to y$. \\item An object $x$ of the category $\\mathcal{C}$ is called a {\\it final} object if for every object $y$ of $\\mathcal{C}$ there is exactly one morphism $y \\to x$. \\end{enumerate}"}
+{"_id": "12355", "title": "categories-definition-mono-epi", "text": "Let $\\mathcal{C}$ be a category and let $f : X \\to Y$ be a morphism of $\\mathcal{C}$. \\begin{enumerate} \\item We say that $f$ is a {\\it monomorphism} if for every object $W$ and every pair of morphisms $a, b : W \\to X$ such that $f \\circ a = f \\circ b$ we have $a = b$. \\item We say that $f$ is an {\\it epimorphism} if for every object $W$ and every pair of morphisms $a, b : Y \\to W$ such that $a \\circ f = b \\circ f$ we have $a = b$. \\end{enumerate}"}
+{"_id": "12356", "title": "categories-definition-limit", "text": "A {\\it limit} of the $\\mathcal{I}$-diagram $M$ in the category $\\mathcal{C}$ is given by an object $\\lim_\\mathcal{I} M$ in $\\mathcal{C}$ together with morphisms $p_i : \\lim_\\mathcal{I} M \\to M_i$ such that \\begin{enumerate} \\item for $\\phi : i \\to i'$ a morphism in $\\mathcal{I}$ we have $p_{i'} =  M(\\phi) \\circ p_i$, and \\item for any object $W$ in $\\mathcal{C}$ and any family of morphisms $q_i : W \\to M_i$ (indexed by $i \\in \\mathcal{I}$) such that for all $\\phi : i \\to i'$ in $\\mathcal{I}$ we have $q_{i'} = M(\\phi) \\circ q_i$ there exists a unique morphism $q : W \\to \\lim_\\mathcal{I} M$ such that $q_i = p_i \\circ q$ for every object $i$ of $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "12357", "title": "categories-definition-colimit", "text": "A {\\it colimit} of the $\\mathcal{I}$-diagram $M$ in the category $\\mathcal{C}$ is given by an object $\\colim_\\mathcal{I} M$ in $\\mathcal{C}$ together with morphisms $s_i : M_i \\to \\colim_\\mathcal{I} M$ such that \\begin{enumerate} \\item for $\\phi : i \\to i'$ a morphism in $\\mathcal{I}$ we have $s_i = s_{i'} \\circ M(\\phi)$, and \\item for any object $W$ in $\\mathcal{C}$ and any family of morphisms $t_i : M_i \\to W$ (indexed by $i \\in \\mathcal{I}$) such that for all $\\phi : i \\to i'$ in $\\mathcal{I}$ we have $t_i = t_{i'} \\circ M(\\phi)$ there exists a unique morphism $t : \\colim_\\mathcal{I} M \\to W$ such that $t_i = t \\circ s_i$ for every object $i$ of $\\mathcal{I}$. \\end{enumerate}"}
+{"_id": "12358", "title": "categories-definition-product", "text": "Suppose that $I$ is a set, and suppose given for every $i \\in I$ an object $M_i$ of the category $\\mathcal{C}$. A {\\it product} $\\prod_{i\\in I} M_i$ is by definition $\\lim_\\mathcal{I} M$ (if it exists) where $\\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects."}
+{"_id": "12359", "title": "categories-definition-coproduct", "text": "Suppose that $I$ is a set, and suppose given for every $i \\in I$ an object $M_i$ of the category $\\mathcal{C}$. A {\\it coproduct} $\\coprod_{i\\in I} M_i$ is by definition $\\colim_\\mathcal{I} M$ (if it exists) where $\\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects."}
+{"_id": "12360", "title": "categories-definition-category-connected", "text": "We say that a category $\\mathcal{I}$ is {\\it connected} if the equivalence relation generated by $x \\sim y \\Leftrightarrow \\Mor_\\mathcal{I}(x, y) \\not = \\emptyset$ has exactly one equivalence class."}
+{"_id": "12361", "title": "categories-definition-cofinal", "text": "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories. We say {\\it $\\mathcal{I}$ is cofinal in $\\mathcal{J}$} or that $H$ is {\\it cofinal} if \\begin{enumerate} \\item for all $y \\in \\Ob(\\mathcal{J})$ there exists a $x \\in \\Ob(\\mathcal{I})$ and a morphism $y \\to H(x)$, and \\item given $y \\in \\Ob(\\mathcal{J})$, $x, x' \\in \\Ob(\\mathcal{I})$ and morphisms $y \\to H(x)$ and $y \\to H(x')$ there exists a sequence of morphisms $$ x = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots \\rightarrow x_{2n} = x' $$ in $\\mathcal{I}$ and morphisms $y \\to H(x_i)$ in $\\mathcal{J}$ such that the diagrams $$ \\xymatrix{ & y \\ar[ld] \\ar[d] \\ar[rd] \\\\ H(x_{2k}) & H(x_{2k + 1}) \\ar[l] \\ar[r] & H(x_{2k + 2}) } $$ commute for $k = 0, \\ldots, n - 1$. \\end{enumerate}"}
+{"_id": "12362", "title": "categories-definition-initial", "text": "Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories. We say {\\it $\\mathcal{I}$ is initial in $\\mathcal{J}$} or that $H$ is {\\it initial} if \\begin{enumerate} \\item for all $y \\in \\Ob(\\mathcal{J})$ there exists a $x \\in \\Ob(\\mathcal{I})$ and a morphism $H(x) \\to y$, \\item for any $y \\in \\Ob(\\mathcal{J})$, $x , x' \\in \\Ob(\\mathcal{I})$ and morphisms $H(x) \\to y$, $H(x') \\to y$ in $\\mathcal{J}$ there exists a sequence of morphisms $$ x = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots \\rightarrow x_{2n} = x' $$ in $\\mathcal{I}$ and morphisms $H(x_i) \\to y$ in $\\mathcal{J}$ such that the diagrams $$ \\xymatrix{ H(x_{2k}) \\ar[rd] & H(x_{2k + 1}) \\ar[l] \\ar[r] \\ar[d] & H(x_{2k + 2}) \\ar[ld] \\\\ & y } $$ commute for $k = 0, \\ldots, n - 1$. \\end{enumerate}"}
+{"_id": "12363", "title": "categories-definition-directed", "text": "We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it directed}, or {\\it filtered} if the following conditions hold: \\begin{enumerate} \\item the category $\\mathcal{I}$ has at least one object, \\item for every pair of objects $x, y$ of $\\mathcal{I}$ there exists an object $z$ and morphisms $x \\to z$, $y \\to z$, and \\item for every pair of objects $x, y$ of $\\mathcal{I}$ and every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$ there exists a morphism $c : y \\to z$ of $\\mathcal{I}$ such that $M(c \\circ a) = M(c \\circ b)$ as morphisms in $\\mathcal{C}$. \\end{enumerate} We say that an index category $\\mathcal{I}$ is {\\it directed}, or {\\it filtered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is filtered (in other words you erase the $M$ in part (3) above)."}
+{"_id": "12364", "title": "categories-definition-codirected", "text": "We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it codirected} or {\\it cofiltered} if the following conditions hold: \\begin{enumerate} \\item the category $\\mathcal{I}$ has at least one object, \\item for every pair of objects $x, y$ of $\\mathcal{I}$ there exists an object $z$ and morphisms $z \\to x$, $z \\to y$, and \\item for every pair of objects $x, y$ of $\\mathcal{I}$ and every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$ there exists a morphism $c : w \\to x$ of $\\mathcal{I}$ such that $M(a \\circ c) = M(b \\circ c)$ as morphisms in $\\mathcal{C}$. \\end{enumerate} We say that an index category $\\mathcal{I}$ is {\\it codirected}, or {\\it cofiltered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is cofiltered (in other words you erase the $M$ in part (3) above)."}
+{"_id": "12365", "title": "categories-definition-directed-set", "text": "Let $I$ be a set and let $\\leq$ be a binary relation on $I$. \\begin{enumerate} \\item We say $\\leq$ is a {\\it preorder} if it is transitive (if $i \\leq j$ and $j \\leq k$ then $i \\leq k$) and reflexive ($i \\leq i$ for all $i \\in I$). \\item A {\\it preordered set} is a set endowed with a preorder. \\item A {\\it directed set} is a preordered set $(I, \\leq)$ such that $I$ is not empty and such that $\\forall i, j \\in I$, there exists $k \\in I$ with $i \\leq k, j \\leq k$. \\item We say $\\leq$ is a {\\it partial order} if it is a preorder which is antisymmetric (if $i \\leq j$ and $j \\leq i$, then $i = j$). \\item A {\\it partially ordered set} is a set endowed with a partial order. \\item A {\\it directed partially ordered set} is a directed set whose ordering is a partial order. \\end{enumerate}"}
+{"_id": "12366", "title": "categories-definition-system-over-poset", "text": "Let $(I, \\leq)$ be a preordered set. Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item A {\\it system over $I$ in $\\mathcal{C}$}, sometimes called a {\\it inductive system over $I$ in $\\mathcal{C}$} is given by objects $M_i$ of $\\mathcal{C}$ and for every $i \\leq i'$ a morphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii} = \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$ whenever $i \\leq i' \\leq i''$. \\item An {\\it inverse system over $I$ in $\\mathcal{C}$}, sometimes called a {\\it projective system over $I$ in $\\mathcal{C}$} is given by objects $M_i$ of $\\mathcal{C}$ and for every $i' \\leq i$ a morphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii} = \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$ whenever $i'' \\leq i' \\leq i$. (Note reversal of inequalities.) \\end{enumerate} We will say $(M_i, f_{ii'})$ is a (inverse) system over $I$ to denote this. The maps $f_{ii'}$ are sometimes called the {\\it transition maps}."}
+{"_id": "12367", "title": "categories-definition-directed-system", "text": "Let $I$ be a preordered set. We say a system (resp.\\ inverse system) $(M_i, f_{ii'})$ is a {\\it directed system} (resp.\\ {\\it directed inverse system}) if $I$ is a directed set (Definition \\ref{definition-directed-set}): $I$ is nonempty and for all $i_1, i_2 \\in I$ there exists $i\\in I$ such that $i_1 \\leq i$ and $i_2 \\leq i$."}
+{"_id": "12368", "title": "categories-definition-essentially-constant-diagram", "text": "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram in a category $\\mathcal{C}$. \\begin{enumerate} \\item Assume the index category $\\mathcal{I}$ is filtered and let $(X, \\{M_i \\to X\\}_i)$ be a cocone for $M$, see Remark \\ref{remark-cones-and-cocones}. We say $M$ is {\\it essentially constant} with {\\it value} $X$ if there exists an $i \\in \\mathcal{I}$ and a morphism $X \\to M_i$ such that \\begin{enumerate} \\item $X \\to M_i \\to X$ is $\\text{id}_X$, and \\item for all $j$ there exist $k$ and morphisms $i \\to k$ and $j \\to k$ such that the morphism $M_j \\to M_k$ equals the composition $M_j \\to X \\to M_i \\to M_k$. \\end{enumerate} \\item Assume the index category $\\mathcal{I}$ is cofiltered and let $(X, \\{X \\to M_i\\}_i)$ be a cone for $M$, see Remark \\ref{remark-cones-and-cocones}. We say $M$ is {\\it essentially constant} with {\\it value} $X$ if there exists an $i \\in \\mathcal{I}$ and a morphism $M_i \\to X$ such that \\begin{enumerate} \\item $X \\to M_i \\to X$ is $\\text{id}_X$, and \\item for all $j$ there exist $k$ and morphisms $k \\to i$ and $k \\to j$ such that the morphism $M_k \\to M_j$ equals the composition $M_k \\to M_i \\to X \\to M_j$. \\end{enumerate} \\end{enumerate} Please keep in mind Lemma \\ref{lemma-essentially-constant-is-limit-colimit} when using this definition."}
+{"_id": "12369", "title": "categories-definition-essentially-constant-system", "text": "Let $\\mathcal{C}$ be a category. A directed system $(M_i, f_{ii'})$ is an {\\it essentially constant system} if $M$ viewed as a functor $I \\to \\mathcal{C}$ defines an essentially constant diagram. A directed inverse system $(M_i, f_{ii'})$ is an {\\it essentially constant inverse system} if $M$ viewed as a functor $I^{opp} \\to \\mathcal{C}$ defines an essentially constant inverse diagram."}
+{"_id": "12370", "title": "categories-definition-exact", "text": "Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor. \\begin{enumerate} \\item Suppose all finite limits exist in $\\mathcal{A}$. We say $F$ is {\\it left exact} if it commutes with all finite limits. \\item Suppose all finite colimits exist in $\\mathcal{A}$. We say $F$ is {\\it right exact} if it commutes with all finite colimits. \\item We say $F$ is {\\it exact} if it is both left and right exact. \\end{enumerate}"}
+{"_id": "12371", "title": "categories-definition-adjoint", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be categories. Let $u : \\mathcal{C} \\to \\mathcal{D}$ and $v : \\mathcal{D} \\to \\mathcal{C}$ be functors. We say that $u$ is a {\\it left adjoint} of $v$, or that $v$ is a {\\it right adjoint} to $u$ if there are bijections $$ \\Mor_\\mathcal{D}(u(X), Y) \\longrightarrow \\Mor_\\mathcal{C}(X, v(Y)) $$ functorial in $X \\in \\Ob(\\mathcal{C})$, and $Y \\in \\Ob(\\mathcal{D})$."}
+{"_id": "12372", "title": "categories-definition-compact-object", "text": "Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.} category. An object $X$ of $\\mathcal{C}$ is called a {\\it categorically compact} if we have $$ \\Mor_\\mathcal{C}(X, \\colim_i M_i) = \\colim_i \\Mor_\\mathcal{C}(X, M_i) $$ for every filtered diagram $M : \\mathcal{I} \\to \\mathcal{C}$ such that $\\colim_i M_i$ exists."}
+{"_id": "12373", "title": "categories-definition-multiplicative-system", "text": "Let $\\mathcal{C}$ be a category. A set of arrows $S$ of $\\mathcal{C}$ is called a {\\it left multiplicative system} if it has the following properties: \\begin{enumerate} \\item[LMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$. \\item[LMS2] Every solid diagram $$ \\xymatrix{ X \\ar[d]_t \\ar[r]_g & Y \\ar@{..>}[d]^s \\\\ Z \\ar@{..>}[r]^f & W } $$ with $t \\in S$ can be completed to a commutative dotted square with $s \\in S$. \\item[LMS3] For every pair of morphisms $f, g : X \\to Y$ and $t \\in S$ with target $X$ such that $f \\circ t = g \\circ t$ there exists a $s \\in S$ with source $Y$ such that $s \\circ f = s \\circ g$. \\end{enumerate} A set of arrows $S$ of $\\mathcal{C}$ is called a {\\it right multiplicative system} if it has the following properties: \\begin{enumerate} \\item[RMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$. \\item[RMS2] Every solid diagram $$ \\xymatrix{ X \\ar@{..>}[d]_t \\ar@{..>}[r]_g & Y \\ar[d]^s \\\\ Z \\ar[r]^f & W } $$ with $s \\in S$ can be completed to a commutative dotted square with $t \\in S$. \\item[RMS3] For every pair of morphisms $f, g : X \\to Y$ and $s \\in S$ with source $Y$ such that $s \\circ f = s \\circ g$ there exists a $t \\in S$ with target $X$ such that $f \\circ t = g \\circ t$. \\end{enumerate} A set of arrows $S$ of $\\mathcal{C}$ is called a {\\it multiplicative system} if it is both a left multiplicative system and a right multiplicative system. In other words, this means that MS1, MS2, MS3 hold, where MS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and MS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1 $=$ MS1.)"}
+{"_id": "12374", "title": "categories-definition-left-localization-as-fraction", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\\mathcal{C}$. Given any morphism $f : X \\to Y'$ in $\\mathcal{C}$ and any morphism $s : Y \\to Y'$ in $S$, we denote by {\\it $s^{-1} f$} the equivalence class of the pair $(f : X \\to Y', s : Y \\to Y')$. This is a morphism from $X$ to $Y$ in $S^{-1} \\mathcal{C}$."}
+{"_id": "12375", "title": "categories-definition-right-localization-as-fraction", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\\mathcal{C}$. Given any morphism $f : X' \\to Y$ in $\\mathcal{C}$ and any morphism $s : X' \\to X$ in $S$, we denote by {\\it $f s^{-1}$} the equivalence class of the pair $(f : X' \\to Y, s : X' \\to X)$. This is a morphism from $X$ to $Y$ in $S^{-1} \\mathcal{C}$."}
+{"_id": "12376", "title": "categories-definition-saturated-multiplicative-system", "text": "Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system. We say $S$ is {\\it saturated} if, in addition to MS1, MS2, MS3, we also have \\begin{enumerate} \\item[MS4] Given three composable morphisms $f, g, h$, if $fg, gh \\in S$, then $g \\in S$. \\end{enumerate}"}
+{"_id": "12377", "title": "categories-definition-horizontal-composition", "text": "Given a diagram as in the left hand side of: $$ \\xymatrix{ \\mathcal{A} \\rtwocell^F_{F'}{t} & \\mathcal{B} \\rtwocell^G_{G'}{s} & \\mathcal{C} } \\text{ gives } \\xymatrix{ \\mathcal{A} \\rrtwocell^{G \\circ F} _{G' \\circ F'}{\\ \\ s \\star t} & & \\mathcal{C} } $$ we define the {\\it horizontal} composition $s \\star t$ to be the transformation of functors ${}_{G'}t \\circ s_F =  s_{F'}\\circ {}_Gt$."}
+{"_id": "12378", "title": "categories-definition-2-category", "text": "A (strict) {\\it $2$-category} $\\mathcal{C}$ consists of the following data \\begin{enumerate} \\item A set of objects $\\Ob(\\mathcal{C})$. \\item For each pair $x, y \\in \\Ob(\\mathcal{C})$ a category $\\Mor_\\mathcal{C}(x, y)$. The objects of $\\Mor_\\mathcal{C}(x, y)$ will be called {\\it $1$-morphisms} and denoted $F : x \\to y$. The morphisms between these $1$-morphisms will be called {\\it $2$-morphisms} and denoted $t : F' \\to F$. The composition of $2$-morphisms in $\\Mor_\\mathcal{C}(x, y)$ will be called {\\it vertical} composition and will be denoted $t \\circ t'$ for $t : F' \\to F$ and $t' : F'' \\to F'$. \\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a functor $$ (\\circ, \\star) : \\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y) \\longrightarrow \\Mor_\\mathcal{C}(x, z). $$ The image of the pair of $1$-morphisms $(F, G)$ on the left hand side will be called the {\\it composition} of $F$ and $G$, and denoted $F\\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will be called the {\\it horizontal} composition and denoted $t \\star s$. \\end{enumerate} These data are to satisfy the following rules: \\begin{enumerate} \\item The set of objects together with the set of $1$-morphisms endowed with composition of $1$-morphisms forms a category. \\item Horizontal composition of $2$-morphisms is associative. \\item The identity $2$-morphism $\\text{id}_{\\text{id}_x}$ of the identity $1$-morphism $\\text{id}_x$ is a unit for horizontal composition. \\end{enumerate}"}
+{"_id": "12379", "title": "categories-definition-sub-2-category", "text": "Let $\\mathcal{C}$ be a $2$-category. A {\\it sub $2$-category} $\\mathcal{C}'$ of $\\mathcal{C}$, is given by a subset $\\Ob(\\mathcal{C}')$ of $\\Ob(\\mathcal{C})$ and sub categories $\\Mor_{\\mathcal{C}'}(x, y)$ of the categories $\\Mor_\\mathcal{C}(x, y)$ for all $x, y \\in \\Ob(\\mathcal{C}')$ such that these, together with the operations $\\circ$ (composition $1$-morphisms), $\\circ$ (vertical composition $2$-morphisms), and $\\star$ (horizontal composition) form a $2$-category."}
+{"_id": "12380", "title": "categories-definition-equivalence", "text": "Two objects $x, y$ of a $2$-category are {\\it equivalent} if there exist $1$-morphisms $F : x \\to y$ and $G : y \\to x$ such that $F \\circ G$ is $2$-isomorphic to $\\text{id}_y$ and $G \\circ F$ is $2$-isomorphic to $\\text{id}_x$."}
+{"_id": "12381", "title": "categories-definition-functor-into-2-category", "text": "Let $\\mathcal{A}$ be a category and let $\\mathcal{C}$ be a $2$-category. \\begin{enumerate} \\item A {\\it functor} from an ordinary category into a $2$-category will ignore the $2$-morphisms unless mentioned otherwise. In other words, it will be a ``usual'' functor into the category formed out of 2-category by forgetting all the 2-morphisms. \\item A {\\it weak functor}, or a {\\it pseudo functor} $\\varphi$ from $\\mathcal{A}$ into the 2-category $\\mathcal{C}$ is given by the following data \\begin{enumerate} \\item a map $\\varphi : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{C})$, \\item for every pair $x, y\\in \\Ob(\\mathcal{A})$, and every morphism $f : x \\to y$ a $1$-morphism $\\varphi(f) : \\varphi(x) \\to \\varphi(y)$, \\item for every $x\\in \\Ob(A)$ a $2$-morphism $\\alpha_x : \\text{id}_{\\varphi(x)} \\to \\varphi(\\text{id}_x)$, and \\item for every pair of composable morphisms $f : x \\to y$, $g : y \\to z$ of $\\mathcal{A}$ a $2$-morphism $\\alpha_{g, f} : \\varphi(g \\circ f) \\to \\varphi(g) \\circ \\varphi(f)$. \\end{enumerate} These data are subject to the following conditions: \\begin{enumerate} \\item the $2$-morphisms $\\alpha_x$ and $\\alpha_{g, f}$ are all isomorphisms, \\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have $\\alpha_{\\text{id}_y, f} = \\alpha_y \\star \\text{id}_{\\varphi(f)}$: $$ \\xymatrix{ \\varphi(x) \\rrtwocell^{\\varphi(f)}_{\\varphi(f)}{\\ \\ \\ \\ \\text{id}_{\\varphi(f)}} & & \\varphi(y) \\rrtwocell^{\\text{id}_{\\varphi(y)}}_{\\varphi(\\text{id}_y)}{\\alpha_y} & & \\varphi(y) } = \\xymatrix{ \\varphi(x) \\rrtwocell^{\\varphi(f)}_{\\varphi(\\text{id}_y) \\circ \\varphi(f)}{\\ \\ \\ \\ \\alpha_{\\text{id}_y, f}} & & \\varphi(y) } $$ \\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have $\\alpha_{f, \\text{id}_x} = \\text{id}_{\\varphi(f)} \\star \\alpha_x$, \\item for any triple of composable morphisms $f : w \\to x$, $g : x \\to y$, and $h : y \\to z$ of $\\mathcal{A}$ we have $$ (\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f}) \\circ \\alpha_{h, g \\circ f} = (\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)}) \\circ \\alpha_{h \\circ g, f} $$ in other words the following diagram with objects $1$-morphisms and arrows $2$-morphisms commutes $$ \\xymatrix{ \\varphi(h \\circ g \\circ f) \\ar[d]_{\\alpha_{h, g \\circ f}} \\ar[rr]_{\\alpha_{h \\circ g, f}} & & \\varphi(h \\circ g) \\circ \\varphi(f) \\ar[d]^{\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)}} \\\\ \\varphi(h) \\circ \\varphi(g \\circ f) \\ar[rr]^{\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f}} & & \\varphi(h) \\circ \\varphi(g) \\circ \\varphi(f) } $$ \\end{enumerate} \\end{enumerate}"}
+{"_id": "12382", "title": "categories-definition-2-1-category", "text": "A (strict) {\\it $(2, 1)$-category} is a $2$-category in which all $2$-morphisms are isomorphisms."}
+{"_id": "12383", "title": "categories-definition-final-object-2-category", "text": "A {\\it final object} of a $(2, 1)$-category $\\mathcal{C}$ is an object $x$ such that \\begin{enumerate} \\item for every $y \\in \\Ob(\\mathcal{C})$ there is a morphism $y \\to x$, and \\item every two morphisms $y \\to x$ are isomorphic by a unique 2-morphism. \\end{enumerate}"}
+{"_id": "12384", "title": "categories-definition-2-fibre-products", "text": "Let $\\mathcal{C}$ be a $(2, 1)$-category. Let $x, y, z\\in \\Ob(\\mathcal{C})$ and $f\\in \\Mor_\\mathcal{C}(x, z)$ and $g\\in \\Mor_{\\mathcal C}(y, z)$. A {\\it 2-fibre product of $f$ and $g$} is a final object in the category of 2-commutative diagrams described above. If a 2-fibre product exists we will denote it $x \\times_z y\\in \\Ob(\\mathcal{C})$, and denote the required morphisms $p\\in \\Mor_{\\mathcal C}(x \\times_z y, x)$ and $q\\in \\Mor_{\\mathcal C}(x \\times_z y, y)$ making the diagram $$ \\xymatrix{ & x \\times_z y \\ar[r]^{p} \\ar[d]_q & x \\ar[d]^{f} \\\\ & y \\ar[r]^{g} & z } $$ 2-commute and we will denote the given invertible 2-morphism exhibiting this by $\\psi : f \\circ p \\to g \\circ q$."}
+{"_id": "12385", "title": "categories-definition-categories-over-C", "text": "Let $\\mathcal{C}$ be a category. The {\\it $2$-category of categories over $\\mathcal{C}$} is the $2$-category defined as follows: \\begin{enumerate} \\item Its objects will be functors $p : \\mathcal{S} \\to \\mathcal{C}$. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$. \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate} In this situation we will denote $$ \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}') $$ the category of $1$-morphisms between $(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$"}
+{"_id": "12386", "title": "categories-definition-fibre-category", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. \\begin{enumerate} \\item The {\\it fibre category} over an object $U\\in \\Ob(\\mathcal{C})$ is the category $\\mathcal{S}_U$ with objects $$ \\Ob(\\mathcal{S}_U) = \\{x\\in \\Ob(\\mathcal{S}) : p(x) = U\\} $$ and morphisms $$ \\Mor_{\\mathcal{S}_U}(x, y) = \\{ \\phi \\in \\Mor_\\mathcal{S}(x, y) : p(\\phi) = \\text{id}_U\\}. $$ \\item A {\\it lift} of an object $U \\in \\Ob(\\mathcal{C})$ is an object $x\\in \\Ob(\\mathcal{S})$ such that $p(x) = U$, i.e., $x\\in \\Ob(\\mathcal{S}_U)$. We will also sometime say that {\\it $x$ lies over $U$}. \\item Similarly, a {\\it lift} of a morphism $f : V \\to U$ in $\\mathcal{C}$ is a morphism $\\phi : y \\to x$ in $\\mathcal{S}$ such that $p(\\phi) = f$. We sometimes say that {\\it $\\phi$ lies over $f$}. \\end{enumerate}"}
+{"_id": "12387", "title": "categories-definition-cartesian-over-C", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. A {\\it strongly cartesian morphism}, or more precisely a {\\it strongly $\\mathcal{C}$-cartesian morphism} is a morphism $\\varphi : y \\to x$ of $\\mathcal{S}$ such that for every $z \\in \\Ob(\\mathcal{S})$ the map $$ \\Mor_\\mathcal{S}(z, y) \\longrightarrow \\Mor_\\mathcal{S}(z, x) \\times_{\\Mor_\\mathcal{C}(p(z), p(x))} \\Mor_\\mathcal{C}(p(z), p(y)), $$ given by $\\psi \\longmapsto (\\varphi \\circ \\psi, p(\\psi))$ is bijective."}
+{"_id": "12388", "title": "categories-definition-fibred-category", "text": "Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$. We say $\\mathcal{S}$ is a {\\it fibred category over $\\mathcal{C}$} if given any $x \\in \\Ob(\\mathcal{S})$ lying over $U \\in \\Ob(\\mathcal{C})$ and any morphism $f : V \\to U$ of $\\mathcal{C}$, there exists a strongly cartesian morphism $f^*x \\to x$ lying over $f$."}
+{"_id": "12389", "title": "categories-definition-pullback-functor-fibred-category", "text": "Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category. \\begin{enumerate} \\item A {\\it choice of pullbacks}\\footnote{This is probably nonstandard terminology. In some texts this is called a ``cleavage''  but it conjures up the wrong image. Maybe a ``cleaving'' would be a better word. A related notion is that of a ``splitting'', but in many texts a ``splitting'' means a choice of pullbacks such that $g^*f^* = (f \\circ g)^*$ for any composable pair of morphisms. Compare also with Definition \\ref{definition-split-fibred-category}.} for $p : \\mathcal{S} \\to \\mathcal{C}$ is given by a choice of a strongly cartesian morphism $f^\\ast x \\to x$ lying over $f$ for any morphism $f: V \\to U$ of $\\mathcal{C}$ and any $x \\in \\Ob(\\mathcal{S}_U)$. \\item Given a choice of pullbacks, for any morphism $f : V \\to U$ of $\\mathcal{C}$ the functor $f^* : \\mathcal{S}_U \\to \\mathcal{S}_V$ described above is called a {\\it pullback functor} (associated to the choices $f^*x \\to x$ made above). \\end{enumerate}"}
+{"_id": "12390", "title": "categories-definition-fibred-categories-over-C", "text": "Let $\\mathcal{C}$ be a category. The {\\it $2$-category of fibred categories over $\\mathcal{C}$} is the sub $2$-category of the $2$-category of categories over $\\mathcal{C}$ (see Definition \\ref{definition-categories-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be fibred categories $p : \\mathcal{S} \\to \\mathcal{C}$. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$ and such that $G$ maps strongly cartesian morphisms to strongly cartesian morphisms. \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate} In this situation we will denote $$ \\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}') $$ the category of $1$-morphisms between $(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$"}
+{"_id": "12391", "title": "categories-definition-inertia-fibred-category", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of fibred categories over $\\mathcal{C}$. The {\\it relative inertia of $\\mathcal{S}$ over $\\mathcal{S}'$} is the fibred category $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$ of Lemma \\ref{lemma-inertia-fibred-category}. \\item By the {\\it inertia fibred category $\\mathcal{I}_\\mathcal{S}$ of $\\mathcal{S}$} we mean $\\mathcal{I}_\\mathcal{S} = \\mathcal{I}_{\\mathcal{S}/\\mathcal{C}}$. \\end{enumerate}"}
+{"_id": "12392", "title": "categories-definition-fibred-groupoids", "text": "Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor. We say that $\\mathcal{S}$ is {\\it fibred in groupoids} over $\\mathcal{C}$ if the following two conditions hold: \\begin{enumerate} \\item For every morphism $f : V \\to U$ in $\\mathcal{C}$ and every lift $x$ of $U$ there is a lift $\\phi : y \\to x$ of $f$ with target $x$. \\item For every pair of morphisms $\\phi : y \\to x$ and $ \\psi : z \\to x$ and any morphism $f : p(z) \\to p(y)$ such that $p(\\phi) \\circ f = p(\\psi)$ there exists a unique lift $\\chi : z \\to y$ of $f$ such that $\\phi \\circ \\chi = \\psi$. \\end{enumerate}"}
+{"_id": "12393", "title": "categories-definition-categories-fibred-in-groupoids-over-C", "text": "Let $\\mathcal{C}$ be a category. The {\\it $2$-category of categories fibred in groupoids over $\\mathcal{C}$} is the sub $2$-category of the $2$-category of fibred categories over $\\mathcal{C}$ (see Definition \\ref{definition-fibred-categories-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be categories $p : \\mathcal{S} \\to \\mathcal{C}$ fibred in groupoids. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "12394", "title": "categories-definition-split-fibred-category", "text": "Let $\\mathcal{C}$ be a category. Suppose that $F : \\mathcal{C}^{opp} \\to \\textit{Cat}$ is a functor to the $2$-category of categories. We will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the fibred category constructed in Example \\ref{example-functor-categories}. A {\\it split fibred category} is a fibred category isomorphic (!) over $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}."}
+{"_id": "12395", "title": "categories-definition-split-category-fibred-in-groupoids", "text": "Let $\\mathcal{C}$ be a category. Suppose that $F : \\mathcal{C}^{opp} \\to \\textit{Groupoids}$ is a functor to the $2$-category of groupoids. We will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the category fibred in groupoids constructed in Example \\ref{example-functor-groupoids}. A {\\it split category fibred in groupoids} is a category fibred in groupoids isomorphic (!) over $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}."}
+{"_id": "12396", "title": "categories-definition-discrete", "text": "A category is called {\\it discrete} if the only morphisms are the identity morphisms."}
+{"_id": "12397", "title": "categories-definition-category-fibred-sets", "text": "Let $\\mathcal{C}$ be a category. A {\\it category fibred in sets}, or a {\\it category fibred in discrete categories} is a category fibred in groupoids all of whose fibre categories are discrete."}
+{"_id": "12398", "title": "categories-definition-categories-fibred-in-sets-over-C", "text": "Let $\\mathcal{C}$ be a category. The {\\it $2$-category of categories fibred in sets over $\\mathcal{C}$} is the sub $2$-category of the category of categories fibred in groupoids over $\\mathcal{C}$ (see Definition \\ref{definition-categories-fibred-in-groupoids-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be categories $p : \\mathcal{S} \\to \\mathcal{C}$ fibred in sets. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "12399", "title": "categories-definition-setoid", "text": "Let us call a category a {\\it setoid}\\footnote{A set on steroids!?} if it is a groupoid where every object has exactly one automorphism: the identity."}
+{"_id": "12400", "title": "categories-definition-category-fibred-setoids", "text": "Let $\\mathcal{C}$ be a category. A {\\it category fibred in setoids} is a category fibred in groupoids all of whose fibre categories are setoids."}
+{"_id": "12401", "title": "categories-definition-categories-fibred-in-setoids-over-C", "text": "Let $\\mathcal{C}$ be a category. The {\\it $2$-category of categories fibred in setoids over $\\mathcal{C}$} is the sub $2$-category of the category of categories fibred in groupoids over $\\mathcal{C}$ (see Definition \\ref{definition-categories-fibred-in-groupoids-over-C}) defined as follows: \\begin{enumerate} \\item Its objects will be categories $p : \\mathcal{S} \\to \\mathcal{C}$ fibred in setoids. \\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that $p' \\circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \\item Its $2$-morphisms $t : G \\to H$ for $G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{S})$. \\end{enumerate}"}
+{"_id": "12402", "title": "categories-definition-representable-fibred-category", "text": "Let $\\mathcal{C}$ be a category. A category fibred in groupoids $p : \\mathcal{S} \\to \\mathcal{C}$ is called {\\it representable} if there exists an object $X$ of $\\mathcal{C}$ and an equivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$ (in the $2$-category of groupoids over $\\mathcal{C}$)."}
+{"_id": "12403", "title": "categories-definition-representable-map-categories-fibred-in-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids over $\\mathcal{C}$. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism. We say $F$ is {\\it representable}, or that {\\it $\\mathcal{X}$ is relatively representable over $\\mathcal{Y}$}, if for every $U \\in \\Ob(\\mathcal{C})$ and any $G : \\mathcal{C}/U \\to \\mathcal{Y}$ the category fibred in groupoids $$ (\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X} \\longrightarrow \\mathcal{C}/U $$ is representable."}
+{"_id": "12404", "title": "categories-definition-monoidal-category", "text": "A triple $(\\mathcal{C}, \\otimes, \\phi)$ where $\\mathcal{C}$ is a category, $\\otimes : \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}$ is a functor, and $\\phi$ is an associativity constraint is called a {\\it monoidal category} if there exists a unit $\\mathbf{1}$."}
+{"_id": "12405", "title": "categories-definition-functor-monoidal-categories", "text": "Let $\\mathcal{C}$ and $\\mathcal{C}'$ be monoidal categories. A {\\it functor of monoidal categories} $F : \\mathcal{C} \\to \\mathcal{C}'$ is given by a functor $F$ as indicated and a natural transformation $$ F(X) \\otimes F(Y) \\to F(X \\otimes Y) $$ such that for all objects $X, Y, Z$ the diagram $$ \\xymatrix{ F(X) \\otimes (F(Y) \\otimes F(Z)) \\ar[r] \\ar[d] & F(X) \\otimes F(Y \\otimes Z) \\ar[r] & F(X \\otimes (Y \\otimes Z)) \\ar[d] \\\\ (F(X) \\otimes F(Y)) \\otimes F(Z) \\ar[r] & F(X \\otimes Y) \\otimes F(Z) \\ar[r] & F((X \\otimes Y) \\otimes Z) } $$ commutes and such that $F(\\mathbf{1})$ is a unit in $\\mathcal{C}'$."}
+{"_id": "12406", "title": "categories-definition-invertible", "text": "Let $\\mathcal{C}$ be a monoidal category. An object $X$ of $\\mathcal{C}$ is called {\\it invertible} if any (or all) of the equivalent conditions of Lemma \\ref{lemma-invertible} hold."}
+{"_id": "12407", "title": "categories-definition-dual", "text": "Given a monoidal category $(\\mathcal{C}, \\otimes, \\phi)$ and an object $X$ a {\\it left dual} is an object $Y$ together with morphisms $\\eta : \\mathbf{1} \\to X \\otimes Y$ and $\\epsilon : Y \\otimes X \\to \\mathbf{1}$ such that the diagrams $$ \\vcenter{ \\xymatrix{ X \\ar[rd]_1 \\ar[r]_-{\\eta \\otimes 1} & X \\otimes Y \\otimes X  \\ar[d]^{1 \\otimes \\epsilon} \\\\ & X } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ Y \\ar[rd]_1 \\ar[r]_-{1 \\otimes \\eta} & Y \\otimes X \\otimes Y  \\ar[d]^{\\epsilon \\otimes 1} \\\\ & Y } } $$ commute. In this situation we say that $X$ is a {\\it right dual} of $Y$."}
+{"_id": "12408", "title": "categories-definition-symmetric-monoidal-category", "text": "A quadruple $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ where $\\mathcal{C}$ is a category, $\\otimes : \\mathcal{C} \\otimes \\mathcal{C} \\to \\mathcal{C}$ is a functor, $\\phi$ is an associativity constraint, and $\\psi$ is a commutativity constraint compatible with $\\phi$ is called a {\\it symmetric monoidal category} if there exists a unit."}
+{"_id": "12409", "title": "categories-definition-functor-symmetric-monoidal-categories", "text": "Let $\\mathcal{C}$ and $\\mathcal{C}'$ be symmetric monoidal categories. A {\\it functor of symmetric monoidal categories} $F : \\mathcal{C} \\to \\mathcal{C}'$ is given by a functor $F$ as indicated and a natural transformation $$ F(X) \\otimes F(Y) \\to F(X \\otimes Y) $$ such that $F$ is a functor of monoidal categories and such that for all objects $X, Y$ the diagram $$ \\xymatrix{ F(X) \\otimes F(Y) \\ar[r] \\ar[d] & F(X \\otimes Y) \\ar[d] \\\\ F(Y) \\otimes F(X) \\ar[r] & F(Y \\otimes X) } $$ commutes."}
+{"_id": "12521", "title": "topologies-definition-zariski-covering", "text": "Let $T$ be a scheme. A {\\it Zariski covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is an open immersion and such that $T = \\bigcup f_i(T_i)$."}
+{"_id": "12522", "title": "topologies-definition-standard-Zariski", "text": "Compare Schemes, Definition \\ref{schemes-definition-standard-covering}. Let $T$ be an affine scheme. A {\\it standard Zariski covering} of $T$ is a Zariski covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j \\to T$ inducing an isomorphism with a standard affine open of $T$."}
+{"_id": "12523", "title": "topologies-definition-big-zariski-site", "text": "A {\\it big Zariski site} is any site $\\Sch_{Zar}$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of Zariski coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category of $\\Sch_{Zar}$ take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item As coverings of $\\Sch_{Zar}$ choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of Zariski coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12524", "title": "topologies-definition-big-small-Zariski", "text": "Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski site containing $S$. \\begin{enumerate} \\item The {\\it big Zariski site of $S$}, denoted $(\\Sch/S)_{Zar}$, is the site $\\Sch_{Zar}/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it small Zariski site of $S$}, which we denote $S_{Zar}$, is the full subcategory of $(\\Sch/S)_{Zar}$ whose objects are those $U/S$ such that $U \\to S$ is an open immersion. A covering of $S_{Zar}$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{Zar}$ with $U \\in \\Ob(S_{Zar})$. \\item The {\\it big affine Zariski site of $S$}, denoted $(\\textit{Aff}/S)_{Zar}$, is the full subcategory of $(\\Sch/S)_{Zar}$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_{Zar}$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{Zar}$ which is a standard Zariski covering. \\end{enumerate}"}
+{"_id": "12525", "title": "topologies-definition-restriction-small-zariski", "text": "In the situation of Lemma \\ref{lemma-at-the-bottom} the functor $i_S^{-1} = \\pi_{S, *}$ is often called the {\\it restriction to the small Zariski site}, and for a sheaf $\\mathcal{F}$ on the big Zariski site we denote $\\mathcal{F}|_{S_{Zar}}$ this restriction."}
+{"_id": "12526", "title": "topologies-definition-etale-covering", "text": "Let $T$ be a scheme. An {\\it \\'etale covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is \\'etale and such that $T = \\bigcup f_i(T_i)$."}
+{"_id": "12527", "title": "topologies-definition-standard-etale", "text": "Let $T$ be an affine scheme. A {\\it standard \\'etale covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine and \\'etale over $T$ and $T = \\bigcup f_j(U_j)$."}
+{"_id": "12528", "title": "topologies-definition-big-etale-site", "text": "A {\\it big \\'etale site} is any site $\\Sch_\\etale$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of \\'etale coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of \\'etale coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12529", "title": "topologies-definition-big-small-etale", "text": "Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale site containing $S$. \\begin{enumerate} \\item The {\\it big \\'etale site of $S$}, denoted $(\\Sch/S)_\\etale$, is the site $\\Sch_\\etale/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it small \\'etale site of $S$}, which we denote $S_\\etale$, is the full subcategory of $(\\Sch/S)_\\etale$ whose objects are those $U/S$ such that $U \\to S$ is \\'etale. A covering of $S_\\etale$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\etale$ with $U \\in \\Ob(S_\\etale)$. \\item The {\\it big affine \\'etale site of $S$}, denoted $(\\textit{Aff}/S)_\\etale$, is the full subcategory of $(\\Sch/S)_\\etale$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_\\etale$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_\\etale$ which is a standard \\'etale covering. \\end{enumerate}"}
+{"_id": "12530", "title": "topologies-definition-restriction-small-etale", "text": "In the situation of Lemma \\ref{lemma-at-the-bottom-etale} the functor $i_S^{-1} = \\pi_{S, *}$ is often called the {\\it restriction to the small \\'etale site}, and for a sheaf $\\mathcal{F}$ on the big \\'etale site we denote $\\mathcal{F}|_{S_\\etale}$ this restriction."}
+{"_id": "12531", "title": "topologies-definition-smooth-covering", "text": "Let $T$ be a scheme. A {\\it smooth covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is smooth and such that $T = \\bigcup f_i(T_i)$."}
+{"_id": "12532", "title": "topologies-definition-standard-smooth", "text": "Let $T$ be an affine scheme. A {\\it standard smooth covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine, $U_j \\to T$ standard smooth and $T = \\bigcup f_j(U_j)$."}
+{"_id": "12533", "title": "topologies-definition-big-smooth-site", "text": "A {\\it big smooth site} is any site $\\Sch_{smooth}$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of smooth coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of smooth coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12534", "title": "topologies-definition-big-small-smooth", "text": "Let $S$ be a scheme. Let $\\Sch_{smooth}$ be a big smooth site containing $S$. \\begin{enumerate} \\item The {\\it big smooth site of $S$}, denoted $(\\Sch/S)_{smooth}$, is the site $\\Sch_{smooth}/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it big affine smooth site of $S$}, denoted $(\\textit{Aff}/S)_{smooth}$, is the full subcategory of $(\\Sch/S)_{smooth}$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_{smooth}$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{smooth}$ which is a standard smooth covering. \\end{enumerate}"}
+{"_id": "12535", "title": "topologies-definition-syntomic-covering", "text": "Let $T$ be a scheme. An {\\it syntomic covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is syntomic and such that $T = \\bigcup f_i(T_i)$."}
+{"_id": "12536", "title": "topologies-definition-standard-syntomic", "text": "Let $T$ be an affine scheme. A {\\it standard syntomic covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine, $U_j \\to T$ standard syntomic and $T = \\bigcup f_j(U_j)$."}
+{"_id": "12537", "title": "topologies-definition-big-syntomic-site", "text": "A {\\it big syntomic site} is any site $\\Sch_{syntomic}$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of syntomic coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of syntomic coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12538", "title": "topologies-definition-big-small-syntomic", "text": "Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic site containing $S$. \\begin{enumerate} \\item The {\\it big syntomic site of $S$}, denoted $(\\Sch/S)_{syntomic}$, is the site $\\Sch_{syntomic}/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it big affine syntomic site of $S$}, denoted $(\\textit{Aff}/S)_{syntomic}$, is the full subcategory of $(\\Sch/S)_{syntomic}$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_{syntomic}$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{syntomic}$ which is a standard syntomic covering. \\end{enumerate}"}
+{"_id": "12539", "title": "topologies-definition-fppf-covering", "text": "Let $T$ be a scheme. An {\\it fppf covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is flat, locally of finite presentation and such that $T = \\bigcup f_i(T_i)$."}
+{"_id": "12540", "title": "topologies-definition-standard-fppf", "text": "Let $T$ be an affine scheme. A {\\it standard fppf covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine, flat and of finite presentation over $T$ and $T = \\bigcup f_j(U_j)$."}
+{"_id": "12541", "title": "topologies-definition-big-fppf-site", "text": "A {\\it big fppf site} is any site $\\Sch_{fppf}$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of fppf coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of fppf coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12542", "title": "topologies-definition-big-small-fppf", "text": "Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$. \\begin{enumerate} \\item The {\\it big fppf site of $S$}, denoted $(\\Sch/S)_{fppf}$, is the site $\\Sch_{fppf}/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it big affine fppf site of $S$}, denoted $(\\textit{Aff}/S)_{fppf}$, is the full subcategory of $(\\Sch/S)_{fppf}$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_{fppf}$ is any covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{fppf}$ which is a standard fppf covering. \\end{enumerate}"}
+{"_id": "12543", "title": "topologies-definition-standard-ph-covering", "text": "Let $T$ be an affine scheme. A {\\it standard ph covering} is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ constructed from a proper surjective morphism $f : U \\to T$ and an affine open covering $U = \\bigcup_{j = 1, \\ldots, m} U_j$ by setting $f_j = f|_{U_j}$."}
+{"_id": "12544", "title": "topologies-definition-ph-covering", "text": "Let $T$ be a scheme. A {\\it ph covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that $f_i$ is locally of finite type and such that for every affine open $U \\subset T$ there exists a standard ph covering $\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family $\\{T_i \\times_T U \\to U\\}_{i \\in I}$."}
+{"_id": "12545", "title": "topologies-definition-big-ph-site", "text": "A {\\it big ph site} is any site $\\Sch_{ph}$ as in Sites, Definition \\ref{sites-definition-site} constructed as follows: \\begin{enumerate} \\item Choose any set of schemes $S_0$, and any set of ph coverings $\\text{Cov}_0$ among these schemes. \\item As underlying category take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$. \\item Choose any set of coverings as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of ph coverings, and the set $\\text{Cov}_0$ chosen above. \\end{enumerate}"}
+{"_id": "12546", "title": "topologies-definition-big-small-ph", "text": "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$. \\begin{enumerate} \\item The {\\it big ph site of $S$}, denoted $(\\Sch/S)_{ph}$, is the site $\\Sch_{ph}/S$ introduced in Sites, Section \\ref{sites-section-localize}. \\item The {\\it big affine ph site of $S$}, denoted $(\\textit{Aff}/S)_{ph}$, is the full subcategory of $(\\Sch/S)_{ph}$ whose objects are affine $U/S$. A covering of $(\\textit{Aff}/S)_{ph}$ is any finite covering $\\{U_i \\to U\\}$ of $(\\Sch/S)_{ph}$ with $U_i$ and $U$ affine. \\end{enumerate}"}
+{"_id": "12547", "title": "topologies-definition-fpqc-covering", "text": "Let $T$ be a scheme. An {\\it fpqc covering of $T$} is a family of morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such that each $f_i$ is flat and such that for every affine open $U \\subset T$ there exists $n \\geq 0$, a map $a : \\{1, \\ldots, n\\} \\to I$ and affine opens $V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$ with $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$."}
+{"_id": "12548", "title": "topologies-definition-standard-fpqc", "text": "Let $T$ be an affine scheme. A {\\it standard fpqc covering} of $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, n}$ with each $U_j$ is affine, flat over $T$ and $T = \\bigcup f_j(U_j)$."}
+{"_id": "12549", "title": "topologies-definition-sheaf-property-fpqc", "text": "Let $F$ be a contravariant functor on the category of schemes with values in sets. \\begin{enumerate} \\item Let $\\{U_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with fixed target. We say that $F$ {\\it satisfies the sheaf property for the given family} if for any collection of elements $\\xi_i \\in F(U_i)$ such that $\\xi_i|_{U_i \\times_T U_j} = \\xi_j|_{U_i \\times_T U_j}$ there exists a unique element $\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$. \\item We say that $F$ {\\it satisfies the sheaf property for the fpqc topology} if it satisfies the sheaf property for any fpqc covering. \\end{enumerate}"}
+{"_id": "12550", "title": "topologies-definition-standard-V-covering", "text": "Let $T$ be an affine scheme. A {\\it standard V covering} is a finite family $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ with $T_j$ affine such that for every morphism $g : \\Spec(V) \\to T$ where $V$ is a valuation ring, there is an extension $V \\subset W$ of valuation rings (More on Algebra, Definition \\ref{more-algebra-definition-extension-valuation-rings}), an index $1 \\leq j \\leq m$, and a commutative diagram $$ \\xymatrix{ \\Spec(W) \\ar[r] \\ar[d] & T_j \\ar[d] \\\\ \\Spec(V) \\ar[r]^g & T } $$"}
+{"_id": "12551", "title": "topologies-definition-V-covering", "text": "Let $T$ be a scheme. A {\\it V covering of $T$} is a family of morphisms $\\{T_i \\to T\\}_{i \\in I}$ of schemes such that for every affine open $U \\subset T$ there exists a standard V covering $\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family $\\{T_i \\times_T U \\to U\\}_{i \\in I}$."}
+{"_id": "12552", "title": "topologies-definition-sheaf-property-V", "text": "Let $F$ be a contravariant functor on the category of schemes with values in sets. We say that $F$ {\\it satisfies the sheaf property for the V topology} if it satisfies the sheaf property for any V covering (see Definition \\ref{definition-sheaf-property-fpqc})."}
+{"_id": "12577", "title": "pic-definition-picard-functor", "text": "Let $\\Sch_{fppf}$ be a big site as in Topologies, Definition \\ref{topologies-definition-big-small-fppf}. Let $f : X \\to S$ be a morphism of this site. The {\\it Picard functor} $\\Picardfunctor_{X/S}$ is the fppf sheafification of the functor $$ (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad T \\longmapsto \\Pic(X_T) $$ If this functor is representable, then we denote $\\underline{\\Picardfunctor}_{X/S}$ a scheme representing it."}
+{"_id": "12578", "title": "pic-definition-genus", "text": "Let $k$ be a field. Let $X$ be a smooth projective geometrically irreducible curve over $k$. The {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$."}
+{"_id": "12655", "title": "constructions-definition-relative-spec", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over $S$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $S$} is the scheme constructed in Lemma \\ref{lemma-glue-relative-spec} which represents the functor $F$ (\\ref{equation-spec}), see Lemma \\ref{lemma-glueing-gives-functor-spec}. We denote it $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$. The ``universal family'' is a morphism of $\\mathcal{O}_S$-algebras $$ \\mathcal{A} \\longrightarrow \\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})} $$"}
+{"_id": "12656", "title": "constructions-definition-affine-n-space", "text": "Let $S$ be a scheme and $n \\geq 0$. The scheme $$ \\mathbf{A}^n_S = \\underline{\\Spec}_S(\\mathcal{O}_S[T_1, \\ldots, T_n]) $$ over $S$ is called {\\it affine $n$-space over $S$}. If $S = \\Spec(R)$ is affine then we also call this {\\it affine $n$-space over $R$} and we denote it $\\mathbf{A}^n_R$."}
+{"_id": "12657", "title": "constructions-definition-vector-bundle", "text": "Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_S$-module\\footnote{The reader may expect here the condition that $\\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with \\cite[II, Definition 1.7.8]{EGA}.}. The {\\it vector bundle associated to $\\mathcal{E}$} is $$ \\mathbf{V}(\\mathcal{E}) = \\underline{\\Spec}_S(\\text{Sym}(\\mathcal{E})). $$"}
+{"_id": "12658", "title": "constructions-definition-abstract-vector-bundle", "text": "Let $S$ be a scheme. A {\\it vector bundle $\\pi : V \\to S$ over $S$} is an affine morphism of schemes such that $\\pi_*\\mathcal{O}_V$ is endowed with the structure of a graded $\\mathcal{O}_S$-algebra $\\pi_*\\mathcal{O}_V = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{E}_n$ such that $\\mathcal{E}_0 = \\mathcal{O}_S$ and such that the maps $$ \\text{Sym}^n(\\mathcal{E}_1) \\longrightarrow \\mathcal{E}_n $$ are isomorphisms for all $n \\geq 0$. A {\\it morphism of vector bundles over $S$} is a morphism $f : V \\to V'$ such that the induced map $$ f^* : \\pi'_*\\mathcal{O}_{V'} \\longrightarrow \\pi_*\\mathcal{O}_V $$ is compatible with the given gradings."}
+{"_id": "12659", "title": "constructions-definition-cone", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Assume that $\\mathcal{O}_S \\to \\mathcal{A}_0$ is an isomorphism\\footnote{Often one imposes the assumption that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{O}_S$. We do not assume this in order to be consistent with \\cite[II, (8.3.1)]{EGA}.}. The {\\it cone associated to $\\mathcal{A}$} or the {\\it affine cone associated to $\\mathcal{A}$} is $$ C(\\mathcal{A}) = \\underline{\\Spec}_S(\\mathcal{A}). $$"}
+{"_id": "12660", "title": "constructions-definition-abstract-cone", "text": "Let $S$ be a scheme. A {\\it cone $\\pi : C \\to S$ over $S$} is an affine morphism of schemes such that $\\pi_*\\mathcal{O}_C$ is endowed with the structure of a graded $\\mathcal{O}_S$-algebra $\\pi_*\\mathcal{O}_C = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{A}_n$ such that $\\mathcal{A}_0 = \\mathcal{O}_S$. A {\\it morphism of cones} from $\\pi : C \\to S$ to $\\pi' : C' \\to S$ is a morphism $f : C \\to C'$ such that the induced map $$ f^* : \\pi'_*\\mathcal{O}_{C'} \\longrightarrow \\pi_*\\mathcal{O}_C $$ is compatible with the given gradings."}
+{"_id": "12661", "title": "constructions-definition-standard-covering", "text": "Let $S$ be a graded ring. Suppose that $D_{+}(f) \\subset \\text{Proj}(S)$ is a standard open. A {\\it standard open covering} of $D_{+}(f)$ is a covering $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$, where $g_1, \\ldots, g_n \\in S$ are homogeneous of positive degree."}
+{"_id": "12662", "title": "constructions-definition-structure-sheaf", "text": "Let $S$ be a graded ring. \\begin{enumerate} \\item The {\\it structure sheaf $\\mathcal{O}_{\\text{Proj}(S)}$ of the homogeneous spectrum of $S$} is the unique sheaf of rings $\\mathcal{O}_{\\text{Proj}(S)}$ which agrees with $\\widetilde S$ on the basis of standard opens. \\item The locally ringed space $(\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$ is called the {\\it homogeneous spectrum} of $S$ and denoted $\\text{Proj}(S)$. \\item The sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules extending $\\widetilde M$ to all opens of $\\text{Proj}(S)$ is called the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules associated to $M$. This sheaf is denoted $\\widetilde M$ as well. \\end{enumerate}"}
+{"_id": "12663", "title": "constructions-definition-twist", "text": "Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$. \\begin{enumerate} \\item We define $\\mathcal{O}_X(n) = \\widetilde{S(n)}$. This is called the $n$th {\\it twist of the structure sheaf of $\\text{Proj}(S)$}. \\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set $\\mathcal{F}(n) = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n)$. \\end{enumerate}"}
+{"_id": "12664", "title": "constructions-definition-projective-space", "text": "The scheme $\\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(\\mathbf{Z}[T_0, \\ldots, T_n])$ is called {\\it projective $n$-space over $\\mathbf{Z}$}. Its base change $\\mathbf{P}^n_S$ to a scheme $S$ is called {\\it projective $n$-space over $S$}. If $R$ is a ring the base change to $\\Spec(R)$ is denoted $\\mathbf{P}^n_R$ and called {\\it projective $n$-space over $R$}."}
+{"_id": "12665", "title": "constructions-definition-relative-proj", "text": "Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-algebras. The {\\it relative homogeneous spectrum of $\\mathcal{A}$ over $S$}, or the {\\it homogeneous spectrum of $\\mathcal{A}$ over $S$}, or the {\\it relative Proj of $\\mathcal{A}$ over $S$} is the scheme constructed in Lemma \\ref{lemma-glue-relative-proj} which represents the functor $F$ (\\ref{equation-proj}), see Lemma \\ref{lemma-glueing-gives-functor-proj}. We denote it $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$."}
+{"_id": "12666", "title": "constructions-definition-projective-bundle", "text": "Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_S$-module\\footnote{The reader may expect here the condition that $\\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with \\cite[II, Definition 4.1.1]{EGA}.}. We denote $$ \\pi : \\mathbf{P}(\\mathcal{E}) = \\underline{\\text{Proj}}_S(\\text{Sym}(\\mathcal{E})) \\longrightarrow S $$ and we call it the {\\it projective bundle associated to $\\mathcal{E}$}. The symbol $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(n)$ indicates the invertible $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}$-module of Lemma \\ref{lemma-apply-relative} and is called the $n$th {\\it twist of the structure sheaf}."}
+{"_id": "12667", "title": "constructions-definition-grassmannian", "text": "Let $0 < k < n$. The scheme $\\mathbf{G}(k, n)$ representing the functor $G(k, n)$ is called {\\it Grassmannian over $\\mathbf{Z}$}. Its base change $\\mathbf{G}(k, n)_S$ to a scheme $S$ is called {\\it Grassmannian over $S$}. If $R$ is a ring the base change to $\\Spec(R)$ is denoted $\\mathbf{G}(k, n)_R$ and called {\\it Grassmannian over $R$}."}
+{"_id": "12803", "title": "algebraization-definition-derived-complete", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals. Let $K \\in D(\\mathcal{O})$. We say that $K$ is {\\it derived complete with respect to $\\mathcal{I}$} if for every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{I}(U)$ the object $T(K|_U, f)$ of $D(\\mathcal{O}_U)$ is zero."}
+{"_id": "12804", "title": "algebraization-definition-algebraizable", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say {\\it $(\\mathcal{F}_n)$ extends to $X$} if there exists an object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction to $U$ is isomorphic to $(\\mathcal{F}_n)$."}
+{"_id": "12805", "title": "algebraization-definition-canonically-algebraizable", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say {\\it $(\\mathcal{F}_n)$ canonically extends to $X$} if the the inverse system $$ \\{\\widetilde{H^0(U, \\mathcal{F}_n)}\\}_{n \\geq 1} $$ in $\\QCoh(\\mathcal{O}_X)$ is pro-isomorphic to an object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$."}
+{"_id": "12806", "title": "algebraization-definition-s-d-inequalities", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers. Let $\\delta^Y_Z$ be as in (\\ref{equation-delta-Z}). We say {\\it $(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities} if for $y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ \\begin{enumerate} \\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) \\not = \\{\\mathfrak m_y^\\wedge\\}$, then $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) \\geq a \\quad\\text{or}\\quad \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b $$ \\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$, then $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a $$ \\end{enumerate} We say {\\it $(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities} if for $y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have $$ \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a \\quad\\text{or}\\quad \\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b $$"}
+{"_id": "12883", "title": "spaces-over-fields-definition-integral-algebraic-space", "text": "Let $S$ be a scheme. We say an algebraic space $X$ over $S$ is {\\it integral} if it is reduced, decent, and $|X|$ is irreducible."}
+{"_id": "12884", "title": "spaces-over-fields-definition-function-field", "text": "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. The {\\it function field}, or the {\\it field of rational functions} of $X$ is the field $R(X)$ of Lemma \\ref{lemma-integral-algebraic-space-rational-functions}."}
+{"_id": "12885", "title": "spaces-over-fields-definition-degree", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be integral algebraic spaces over $S$. Let $f : X \\to Y$ be locally of finite type and dominant. Assume any of the equivalent conditions (1) -- (5) of Lemma \\ref{lemma-finite-degree}. Let $x \\in |X|$ and $y \\in |Y|$ be the generic points. Then the positive integer $$ \\text{deg}(X/Y) = [\\kappa(x) : \\kappa(y)] $$ is called the {\\it degree of $X$ over $Y$}."}
+{"_id": "12886", "title": "spaces-over-fields-definition-Weil-divisor", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. \\begin{enumerate} \\item A {\\it prime divisor} is an integral closed subspace $Z \\subset X$ of codimension $1$, i.e., the generic point of $|Z|$ is a point of codimension $1$ on $X$. \\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where the sum is over prime divisors of $X$ and the collection $\\{|Z| : n_Z \\not = 0\\}$ is locally finite in $|X|$ (Topology, Definition \\ref{topology-definition-locally-finite}). \\end{enumerate} The group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$."}
+{"_id": "12887", "title": "spaces-over-fields-definition-order-vanishing", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \\in R(X)^*$. For every prime divisor $Z \\subset X$ we define the {\\it order of vanishing of $f$ along $Z$} as the integer $$ \\text{ord}_Z(f) = \\text{length}_{\\mathcal{O}_{X, \\xi}^h} (\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) - \\text{length}_{\\mathcal{O}_{X, \\xi}^h} (\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h) $$ where $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors such that the image of $f$ in $Q(\\mathcal{O}_{X, \\xi}^h)$ (Lemma \\ref{lemma-order-vanishing}) is equal to $a/b$. This is well defined by Algebra, Lemma \\ref{algebra-lemma-ord-additive}."}
+{"_id": "12888", "title": "spaces-over-fields-definition-principal-divisor", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \\in R(X)^*$. The {\\it principal Weil divisor associated to $f$} is the Weil divisor $$ \\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z] $$ where the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in Definition \\ref{definition-order-vanishing}. This makes sense by Lemma \\ref{lemma-divisor-locally-finite}."}
+{"_id": "12889", "title": "spaces-over-fields-definition-class-group", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. The {\\it Weil divisor class group} of $X$ is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: $\\text{Cl}(X)$."}
+{"_id": "12890", "title": "spaces-over-fields-definition-order-vanishing-meromorphic", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$ be a regular meromorphic section of $\\mathcal{L}$. For every prime divisor $Z \\subset X$ with generic point $\\xi \\in |Z|$ we define the {\\it order of vanishing of $s$ along $Z$} as the integer $$ \\text{ord}_{Z, \\mathcal{L}}(s) = \\text{length}_{\\mathcal{O}_{X, \\xi}^h} (\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) - \\text{length}_{\\mathcal{O}_{X, \\xi}^h} (\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h) $$ where $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors such that the element $s/s_\\xi$ of $Q(\\mathcal{O}_{X, \\xi}^h)$ constructed above is equal to $a/b$. This is well defined by the above and Algebra, Lemma \\ref{algebra-lemma-ord-additive}."}
+{"_id": "12891", "title": "spaces-over-fields-definition-divisor-invertible-sheaf", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. \\begin{enumerate} \\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$ we define the {\\it Weil divisor associated to $s$} as $$ \\text{div}_\\mathcal{L}(s) = \\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X) $$ where the sum is over prime divisors. This is well defined by Lemma \\ref{lemma-divisor-meromorphic-locally-finite}. \\item We define {\\it Weil divisor class associated to $\\mathcal{L}$} as the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$ where $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$. This is well defined by Lemma \\ref{lemma-divisor-meromorphic-well-defined}. \\end{enumerate}"}
+{"_id": "12892", "title": "spaces-over-fields-definition-modification", "text": "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. A {\\it modification of $X$} is a birational proper morphism $f : X' \\to X$ of algebraic spaces over $S$ with $X'$ integral."}
+{"_id": "12893", "title": "spaces-over-fields-definition-alteration", "text": "Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. An {\\it alteration of $X$} is a proper dominant morphism $f : Y \\to X$ of algebraic spaces over $S$ with $Y$ integral such that $f^{-1}(U) \\to U$ is finite for some nonempty open $U \\subset X$."}
+{"_id": "12894", "title": "spaces-over-fields-definition-geometrically-reduced", "text": "Let $k$ be a field. Let $X$ be an algebraic space over $k$. \\begin{enumerate} \\item Let $x \\in |X|$ be a point. We say $X$ is {\\it geometrically reduced at $x$} if $\\mathcal{O}_{X, \\overline{x}}$ is geometrically reduced over $k$. \\item We say $X$ is {\\it geometrically reduced} over $k$ if $X$ is geometrically reduced at every point of $X$. \\end{enumerate}"}
+{"_id": "12895", "title": "spaces-over-fields-definition-geometrically-connected", "text": "Let $X$ be an algebraic space over the field $k$. We say $X$ is {\\it geometrically connected} over $k$ if the base change $X_{k'}$ is connected for every field extension $k'$ of $k$."}
+{"_id": "12896", "title": "spaces-over-fields-definition-geometrically-irreducible", "text": "Let $k$ be a field.  Let $X$ be a decent algebraic space over $k$. We say $X$ is {\\it geometrically irreducible} if the topological space $|X_{k'}|$ is irreducible\\footnote{An irreducible space is nonempty.} for any field extension $k'$ of $k$."}
+{"_id": "12897", "title": "spaces-over-fields-definition-geometrically-integral", "text": "Let $X$ be an algebraic space over the field $k$. We say $X$ is {\\it geometrically integral} over $k$ if the algebraic space $X_{k'}$ is integral (Definition \\ref{definition-integral-algebraic-space}) for every field extension $k'$ of $k$."}
+{"_id": "12898", "title": "spaces-over-fields-definition-euler-characteristic", "text": "Let $k$ be a field. Let $X$ be a proper algebraic over $k$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. In this situation the {\\it Euler characteristic of $\\mathcal{F}$} is the integer $$ \\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}). $$ For justification of the formula see below."}
+{"_id": "12899", "title": "spaces-over-fields-definition-intersection-number", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $i : Z \\to X$ be a closed subspace of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. We define the {\\it intersection number} $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ as the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial $$ \\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) = \\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z) $$ In the special case that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$ we write $(\\mathcal{L}^d \\cdot Z)$."}
+{"_id": "13013", "title": "spaces-divisors-definition-weakly-associated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \\in |X|$. \\begin{enumerate} \\item We say $x$ is {\\it weakly associated} to $\\mathcal{F}$ if the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-associated} are satisfied. \\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated points of $\\mathcal{F}$. \\item The {\\it weakly associated points of $X$} are the weakly associated points of $\\mathcal{O}_X$. \\end{enumerate} If $X$ is locally Noetherian we will say {\\it $x$ is associated to $\\mathcal{F}$} if and only if $x$ is weakly associated to $\\mathcal{F}$ and we set $\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$. Finally (still assuming $X$ is locally Noetherian), we will say {\\it $x$ is an associated point of $X$} if and only if $x$ is a weakly associated point of $X$."}
+{"_id": "13014", "title": "spaces-divisors-definition-locally-Noetherian-fibre", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $y \\in |Y|$. We say {\\it the fibre of $f$ over $y$ is locally Noetherian} if the equivalent conditions (1), (2), and (3) of Lemma \\ref{lemma-locally-noetherian-fibre} are satisfied. We say {\\it the fibres of $f$ are locally Noetherian} if this holds for every $y \\in |Y|$."}
+{"_id": "13015", "title": "spaces-divisors-definition-relative-weak-assassin", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $Y$} is the set $\\text{WeakAss}_{X/Y}(\\mathcal{F}) \\subset |X|$ consisting of those $x \\in |X|$ such that the equivalent conditions of Lemma \\ref{lemma-relative-assassin} are satisfied. If the fibres of $f$ are locally Noetherian (Definition \\ref{definition-locally-Noetherian-fibre}) then we use the notation $\\text{Ass}_{X/Y}(\\mathcal{F})$."}
+{"_id": "13016", "title": "spaces-divisors-definition-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item A {\\it locally principal closed subspace} of $X$ is a closed subspace whose sheaf of ideals is locally generated by $1$ element. \\item An {\\it effective Cartier divisor} on $X$ is a closed subspace $D \\subset X$ such that the ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_X$ is an invertible $\\mathcal{O}_X$-module. \\end{enumerate}"}
+{"_id": "13017", "title": "spaces-divisors-definition-sum-effective-Cartier-divisors", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Given effective Cartier divisors $D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the closed subspace of $X$ corresponding to the quasi-coherent sheaf of ideals $\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$. We call this the {\\it sum of the effective Cartier divisors $D_1$ and $D_2$}."}
+{"_id": "13018", "title": "spaces-divisors-definition-pullback-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism of algebraic spaces over $S$. Let $D \\subset X$ be an effective Cartier divisor. We say the {\\it pullback of $D$ by $f$ is defined} if the closed subspace $f^{-1}(D) \\subset X'$ is an effective Cartier divisor. In this case we denote it either $f^*D$ or $f^{-1}(D)$ and we call it the {\\it pullback of the effective Cartier divisor}."}
+{"_id": "13019", "title": "spaces-divisors-definition-invertible-sheaf-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I}_D$. \\begin{enumerate} \\item The {\\it invertible sheaf $\\mathcal{O}_X(D)$ associated to $D$} is defined by $$ \\mathcal{O}_X(D) = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}_D, \\mathcal{O}_X) = \\mathcal{I}_D^{\\otimes -1}. $$ \\item The canonical section, usually denoted $1$ or $1_D$, is the global section of $\\mathcal{O}_X(D)$ corresponding to the inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_X$. \\item We write $\\mathcal{O}_X(-D) = \\mathcal{O}_X(D)^{\\otimes -1} = \\mathcal{I}_D$. \\item Given a second effective Cartier divisor $D' \\subset X$ we define $\\mathcal{O}_X(D - D') = \\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(-D')$. \\end{enumerate}"}
+{"_id": "13020", "title": "spaces-divisors-definition-regular-section", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{L}$ be an invertible sheaf on $X$. A global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a {\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$, $f \\mapsto fs$ is injective."}
+{"_id": "13021", "title": "spaces-divisors-definition-zero-scheme-s", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{L}$ be an invertible sheaf. Let $s \\in \\Gamma(X, \\mathcal{L})$. The {\\it zero scheme} of $s$ is the closed subspace $Z(s) \\subset X$ defined by the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the map $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$."}
+{"_id": "13022", "title": "spaces-divisors-definition-relative-effective-Cartier-divisor", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. A {\\it relative effective Cartier divisor} on $X/Y$ is an effective Cartier divisor $D \\subset X$ such that $D \\to Y$ is a flat morphism of algebraic spaces."}
+{"_id": "13023", "title": "spaces-divisors-definition-sheaf-meromorphic-functions", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\\it sheaf of meromorphic functions on $X$} is the sheaf {\\it $\\mathcal{K}_X$} on $X_\\etale$ associated to the presheaf displayed above. A {\\it meromorphic function} on $X$ is a global section of $\\mathcal{K}_X$."}
+{"_id": "13024", "title": "spaces-divisors-definition-meromorphic-section", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules on $X_\\etale$. \\begin{enumerate} \\item We denote $\\mathcal{K}_X(\\mathcal{F})$ the sheaf of $\\mathcal{K}_X$-modules which is the sheafification of the presheaf $U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently $\\mathcal{K}_X(\\mathcal{F}) = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above). \\item A {\\it meromorphic section of $\\mathcal{F}$} is a global section of $\\mathcal{K}_X(\\mathcal{F})$. \\end{enumerate}"}
+{"_id": "13025", "title": "spaces-divisors-definition-pullback-meromorphic-sections", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. We say that {\\it pullbacks of meromorphic functions are defined for $f$} if for every commutative diagram $$ \\xymatrix{ U \\ar[r] \\ar[d] & X \\ar[d] \\\\ V \\ar[r] & Y } $$ with $U \\in X_\\etale$ and $V \\in Y_\\etale$ and any section $s \\in \\mathcal{S}_Y(V)$ the pullback $f^\\sharp(s) \\in \\mathcal{O}_X(U)$ is an element of $\\mathcal{S}_X(U)$."}
+{"_id": "13026", "title": "spaces-divisors-definition-regular-meromorphic-section", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. A meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular} if the induced map $\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$ is injective."}
+{"_id": "13027", "title": "spaces-divisors-definition-relative-proj", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_X$-algebras. The {\\it relative homogeneous spectrum of $\\mathcal{A}$ over $X$}, or the {\\it homogeneous spectrum of $\\mathcal{A}$ over $X$}, or the {\\it relative Proj of $\\mathcal{A}$ over $X$} is the algebraic space $F$ over $X$ of Lemma \\ref{lemma-relative-proj}. We denote it $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$."}
+{"_id": "13028", "title": "spaces-divisors-definition-relatively-ample", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. We say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample}, or {\\it ample on $X/Y$}, or {\\it $f$-ample} if $f : X \\to Y$ is representable and for every morphism $Z \\to Y$ where $Z$ is a scheme, the pullback $\\mathcal{L}_Z$ of $\\mathcal{L}$ to $X_Z = Z \\times_Y X$ is ample on $X_Z/Z$ as in Morphisms, Definition \\ref{morphisms-definition-relatively-ample}."}
+{"_id": "13029", "title": "spaces-divisors-definition-blow-up", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals, and let $Z \\subset X$ be the closed subspace corresponding to $\\mathcal{I}$ (Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-closed-immersion-ideals}). The {\\it blowing up of $X$ along $Z$}, or the {\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is the morphism $$ b : \\underline{\\text{Proj}}_X \\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right) \\longrightarrow X $$ The {\\it exceptional divisor} of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup."}
+{"_id": "13030", "title": "spaces-divisors-definition-strict-transform", "text": "With $Z \\subset B$ and $f : X \\to B$ as above. \\begin{enumerate} \\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup of $B$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$ by the submodule of sections supported on $|\\text{pr}_{B'}^{-1}E|$. \\item The {\\it strict transform} of $X$ is the closed subspace $X' \\subset X \\times_B B'$ cut out by the quasi-coherent ideal of sections of $\\mathcal{O}_{X \\times_B B'}$ supported on $|\\text{pr}_{B'}^{-1}E|$. \\end{enumerate}"}
+{"_id": "13031", "title": "spaces-divisors-definition-admissible-blowup", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \\subset X$ be an open subspace. A morphism $X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a closed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from $U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$."}
+{"_id": "13134", "title": "dga-definition-dga", "text": "Let $R$ be a commutative ring. A {\\it differential graded algebra over $R$} is either \\begin{enumerate} \\item a chain complex $A_\\bullet$ of $R$-modules endowed with $R$-bilinear maps $A_n \\times A_m \\to A_{n + m}$, $(a, b) \\mapsto ab$ such that $$ \\text{d}_{n + m}(ab) = \\text{d}_n(a)b + (-1)^n a\\text{d}_m(b) $$ and such that $\\bigoplus A_n$ becomes an associative and unital $R$-algebra, or \\item a cochain complex $A^\\bullet$ of $R$-modules endowed with $R$-bilinear maps $A^n \\times A^m \\to A^{n + m}$, $(a, b) \\mapsto ab$ such that $$ \\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b) $$ and such that $\\bigoplus A^n$ becomes an associative and unital $R$-algebra. \\end{enumerate}"}
+{"_id": "13135", "title": "dga-definition-homomorphism-dga", "text": "A {\\it homomorphism of differential graded algebras} $f : (A, \\text{d}) \\to (B, \\text{d})$ is an algebra map $f : A \\to B$ compatible with the gradings and $\\text{d}$."}
+{"_id": "13136", "title": "dga-definition-cdga", "text": "A differential graded algebra $(A, \\text{d})$ is {\\it commutative} if $ab = (-1)^{nm}ba$ for $a$ in degree $n$ and $b$ in degree $m$. We say $A$ is {\\it strictly commutative} if in addition $a^2 = 0$ for $\\deg(a)$ odd."}
+{"_id": "13137", "title": "dga-definition-tensor-product", "text": "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$. The {\\it tensor product differential graded algebra} of $A$ and $B$ is the algebra $A \\otimes_R B$ with multiplication defined by $$ (a \\otimes b)(a' \\otimes b') = (-1)^{\\deg(a')\\deg(b)} aa' \\otimes bb' $$ endowed with differential $\\text{d}$ defined by the rule $\\text{d}(a \\otimes b) = \\text{d}(a) \\otimes b + (-1)^m a \\otimes \\text{d}(b)$ where $m = \\deg(a)$."}
+{"_id": "13138", "title": "dga-definition-dgm", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded algebra over $R$. A (right) {\\it differential graded module} $M$ over $A$ is a right $A$-module $M$ which has a grading $M = \\bigoplus M^n$ and a differential $\\text{d}$ such that $M^n A^m \\subset M^{n + m}$, such that $\\text{d}(M^n) \\subset M^{n + 1}$, and such that $$ \\text{d}(ma) = \\text{d}(m)a + (-1)^n m\\text{d}(a) $$ for $a \\in A$ and $m \\in M^n$. A {\\it homomorphism of differential graded modules} $f : M \\to N$ is an $A$-module map compatible with gradings and differentials. The category of (right) differential graded $A$-modules is denoted $\\text{Mod}_{(A, \\text{d})}$."}
+{"_id": "13139", "title": "dga-definition-shift", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $M$ be a differential graded module. For any $k \\in \\mathbf{Z}$ we define the {\\it $k$-shifted module} $M[k]$ as follows \\begin{enumerate} \\item as $A$-module $M[k] = M$, \\item $M[k]^n = M^{n + k}$, \\item $\\text{d}_{M[k]} = (-1)^k\\text{d}_M$. \\end{enumerate} For a morphism $f : M \\to N$ of differential graded $A$-modules we let $f[k] : M[k] \\to N[k]$ be the map equal to $f$ on underlying $A$-modules. This defines a functor $[k] : \\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})}$."}
+{"_id": "13140", "title": "dga-definition-homotopy", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $f, g : M \\to N$ be homomorphisms of differential graded $A$-modules. A {\\it homotopy between $f$ and $g$} is an $A$-module map $h : M \\to N$ such that \\begin{enumerate} \\item $h(M^n) \\subset N^{n - 1}$ for all $n$, and \\item $f(x) - g(x) = \\text{d}_N(h(x)) + h(\\text{d}_M(x))$ for all $x \\in M$. \\end{enumerate} If a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}."}
+{"_id": "13141", "title": "dga-definition-complexes-notation", "text": "Let $(A, \\text{d})$ be a differential graded algebra. The {\\it homotopy category}, denoted $K(\\text{Mod}_{(A, \\text{d})})$, is the category whose objects are the objects of $\\text{Mod}_{(A, \\text{d})}$ and whose morphisms are homotopy classes of homomorphisms of differential graded $A$-modules."}
+{"_id": "13142", "title": "dga-definition-cone", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $f : K \\to L$ be a homomorphism of differential graded $A$-modules. The {\\it cone} of $f$ is the differential graded $A$-module $C(f)$ given by $C(f) = L \\oplus K$ with grading $C(f)^n = L^n \\oplus K^{n + 1}$ and differential $$ d_{C(f)} = \\left( \\begin{matrix} \\text{d}_L & f \\\\ 0 & -\\text{d}_K \\end{matrix} \\right) $$ It comes equipped with canonical morphisms of complexes $i : L \\to C(f)$ and $p : C(f) \\to K[1]$ induced by the obvious maps $L \\to C(f)$ and $C(f) \\to K$."}
+{"_id": "13143", "title": "dga-definition-admissible-ses", "text": "Let $(A, \\text{d})$ be a differential graded algebra. \\begin{enumerate} \\item A homomorphism $K \\to L$ of differential graded $A$-modules is an {\\it admissible monomorphism} if there exists a graded $A$-module map $L \\to K$ which is left inverse to $K \\to L$. \\item A homomorphism $L \\to M$ of differential graded $A$-modules is an {\\it admissible epimorphism} if there exists a graded $A$-module map $M \\to L$ which is right inverse to $L \\to M$. \\item A short exact sequence $0 \\to K \\to L \\to M \\to 0$ of differential graded $A$-modules is an {\\it admissible short exact sequence} if it is split as a sequence of graded $A$-modules. \\end{enumerate}"}
+{"_id": "13144", "title": "dga-definition-distinguished-triangle", "text": "Let $(A, \\text{d})$ be a differential graded algebra. \\begin{enumerate} \\item If $0 \\to K \\to L \\to M \\to 0$ is an admissible short exact sequence of differential graded $A$-modules, then the {\\it triangle associated to $0 \\to K \\to L \\to M \\to 0$} is the triangle  (\\ref{equation-triangle-associated-to-admissible-ses}) of $K(\\text{Mod}_{(A, \\text{d})})$. \\item A triangle of $K(\\text{Mod}_{(A, \\text{d})})$ is called a {\\it distinguished triangle} if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded $A$-modules. \\end{enumerate}"}
+{"_id": "13145", "title": "dga-definition-opposite-dga", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded algebra over $R$. The {\\it opposite differential graded algebra} is the differential graded algebra $(A^{opp}, \\text{d})$ over $R$ where $A^{opp} = A$ as a graded $R$-module, $\\text{d} = \\text{d}$, and multiplication is given by $$ a \\cdot_{opp} b = (-1)^{\\deg(a)\\deg(b)} b a $$ for homogeneous elements $a, b \\in A$."}
+{"_id": "13146", "title": "dga-definition-shift-graded-module", "text": "Let $R$ be a ring. Let $A$ be a $\\mathbf{Z}$-graded $R$-algebra. \\begin{enumerate} \\item Given a right graded $A$-module $M$ we define the {\\it $k$th shifted $A$-module} $M[k]$ as the same as a right $A$-module but with grading $(M[k])^n = M^{n + k}$. \\item Given a left graded $A$-module $M$ we define the {\\it $k$th shifted $A$-module} $M[k]$ as the module with grading $(M[k])^n = M^{n + k}$ and multiplication $A^n \\times (M[k])^m \\to (M[k])^{n + m}$ equal to $(-1)^{nk}$ times the given multiplication $A^n \\times M^{m + k} \\to M^{n + m + k}$. \\end{enumerate}"}
+{"_id": "13147", "title": "dga-definition-unbounded-derived-category", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $\\text{Ac}$ and $\\text{Qis}$ be as in Lemma \\ref{lemma-acyclic}. The {\\it derived category of $(A, \\text{d})$} is the triangulated category $$ D(A, \\text{d}) = K(\\text{Mod}_{(A, \\text{d})})/\\text{Ac} = \\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})}). $$ We denote $H^0 : D(A, \\text{d}) \\to \\text{Mod}_R$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above."}
+{"_id": "13148", "title": "dga-definition-linear-category", "text": "Let $R$ be a ring. An {\\it $R$-linear category $\\mathcal{A}$} is a category where every morphism set is given the structure of an $R$-module and where for $x, y, z \\in \\Ob(\\mathcal{A})$ composition law $$ \\Hom_\\mathcal{A}(y, z) \\times \\Hom_\\mathcal{A}(x, y) \\longrightarrow \\Hom_\\mathcal{A}(x, z) $$ is $R$-bilinear."}
+{"_id": "13149", "title": "dga-definition-functor-linear-categories", "text": "Let $R$ be a ring. A {\\it functor of $R$-linear categories}, or an {\\it $R$-linear} is a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects $x, y$ of $\\mathcal{A}$ the map $F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$ is a homomorphism of $R$-modules."}
+{"_id": "13150", "title": "dga-definition-graded-category", "text": "Let $R$ be a ring. A {\\it graded category $\\mathcal{A}$ over $R$} is a category where every morphism set is given the structure of a graded $R$-module and where for $x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism $$ \\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y) \\longrightarrow \\Hom_\\mathcal{A}(x, z) $$ of graded $R$-modules (i.e., preserving degrees)."}
+{"_id": "13151", "title": "dga-definition-functor-graded-categories", "text": "Let $R$ be a ring. A {\\it functor of graded categories over $R$}, or a {\\it graded functor} is a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects $x, y$ of $\\mathcal{A}$ the map $F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$ is a homomorphism of graded $R$-modules."}
+{"_id": "13152", "title": "dga-definition-H0-of-graded-category", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category over $R$. We let {\\it $\\mathcal{A}^0$} be the category with the same objects as $\\mathcal{A}$ and with $$ \\Hom_{\\mathcal{A}^0}(x, y) = \\Hom^0_\\mathcal{A}(x, y) $$ the degree $0$ graded piece of the graded module of morphisms of $\\mathcal{A}$."}
+{"_id": "13153", "title": "dga-definition-graded-direct-sum", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in Homology, Remark \\ref{homology-remark-direct-sum}) is a {\\it graded direct sum} if $i, j, p, q$ are homogeneous of degree $0$."}
+{"_id": "13154", "title": "dga-definition-dga-category", "text": "Let $R$ be a ring. A {\\it differential graded category $\\mathcal{A}$ over $R$} is a category where every morphism set is given the structure of a differential graded $R$-module and where for $x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism $$ \\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y) \\longrightarrow \\Hom_\\mathcal{A}(x, z) $$ of differential graded $R$-modules."}
+{"_id": "13155", "title": "dga-definition-functor-dga-categories", "text": "Let $R$ be a ring. A {\\it functor of differential graded categories over $R$} is a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects $x, y$ of $\\mathcal{A}$ the map $F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$ is a homomorphism of differential graded $R$-modules."}
+{"_id": "13156", "title": "dga-definition-homotopy-category-of-dga-category", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category over $R$. Then we let \\begin{enumerate} \\item the {\\it category of complexes of $\\mathcal{A}$}\\footnote{This may be nonstandard terminology.} be the category $\\text{Comp}(\\mathcal{A})$ whose objects are the same as the objects of $\\mathcal{A}$ and with $$ \\Hom_{\\text{Comp}(\\mathcal{A})}(x, y) = \\Ker(d : \\Hom^0_\\mathcal{A}(x, y) \\to \\Hom^1_\\mathcal{A}(x, y)) $$ \\item the {\\it homotopy category of $\\mathcal{A}$} be the category $K(\\mathcal{A})$ whose objects are the same as the objects of $\\mathcal{A}$ and with $$ \\Hom_{K(\\mathcal{A})}(x, y) = H^0(\\Hom_\\mathcal{A}(x, y)) $$ \\end{enumerate}"}
+{"_id": "13157", "title": "dga-definition-dg-direct-sum", "text": "Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in Homology, Remark \\ref{homology-remark-direct-sum}) is a {\\it differential graded direct sum} if $i, j, p, q$ are homogeneous of degree $0$ and closed, i.e., $\\text{d}(i) = 0$, etc."}
+{"_id": "13158", "title": "dga-definition-bimodule", "text": "Bimodules. Let $R$ be a ring. \\begin{enumerate} \\item Let $A$ and $B$ be $R$-algebras. An {\\it $(A, B)$-bimodule} is an $R$-module $M$ equippend with $R$-bilinear maps $$ A \\times M \\to M, (a, x) \\mapsto ax \\quad\\text{and}\\quad M \\times B \\to M, (x, b) \\mapsto xb $$ such that the following hold \\begin{enumerate} \\item $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$, \\item $a(xb) = (ax)b$, and \\item $1 x = x = x 1$. \\end{enumerate} \\item Let $A$ and $B$ be $\\mathbf{Z}$-graded $R$-algebras. A {\\it graded $(A, B)$-bimodule} is an $(A, B)$-bimodule $M$ which has a grading $M = \\bigoplus M^n$ such that $A^n M^m \\subset M^{n + m}$ and $M^n B^m \\subset M^{n + m}$. \\item Let $A$ and $B$ be differential graded $R$-algebras. A {\\it differential graded $(A, B)$-bimodule} is a graded $(A, B)$-bimodule which comes equipped with a differential $\\text{d} : M \\to M$ homogeneous of degree $1$ such that $\\text{d}(ax) = \\text{d}(a)x + (-1)^{\\deg(a)}a\\text{d}(x)$ and $\\text{d}(xb) = \\text{d}(x)b + (-1)^{\\deg(x)}x\\text{d}(b)$ for homogeneous elements $a \\in A$, $x \\in M$, $b \\in B$. \\end{enumerate}"}
+{"_id": "13216", "title": "spaces-more-groupoids-definition-split-at-point", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $u \\in |U|$ be a point. \\begin{enumerate} \\item We say $R$ is {\\it strongly split over $u$} if there exists an open subspace $P \\subset R$ such that \\begin{enumerate} \\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \\item $s|_P$, $t|_P$ are finite, and \\item $\\{r \\in |R| : s(r) = u, t(r) = u\\} \\subset |P|$. \\end{enumerate} The choice of such a $P$ will be called a {\\it strong splitting of $R$ over $u$}. \\item We say $R$ is {\\it split over $u$} if there exists an open subspace $P \\subset R$ such that \\begin{enumerate} \\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \\item $s|_P$, $t|_P$ are finite, and \\item $\\{g \\in |G| : g\\text{ maps to }u\\} \\subset |P|$ where $G \\to U$ is the stabilizer. \\end{enumerate} The choice of such a $P$ will be called a {\\it splitting of $R$ over $u$}. \\item We say $R$ is {\\it quasi-split over $u$} if there exists an open subspace $P \\subset R$ such that \\begin{enumerate} \\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a groupoid in algebraic spaces over $B$, \\item $s|_P$, $t|_P$ are finite, and \\item $e(u) \\in |P|$\\footnote{This condition is implied by (a).}. \\end{enumerate} The choice of such a $P$ will be called a {\\it quasi-splitting of $R$ over $u$}. \\end{enumerate}"}
+{"_id": "13332", "title": "modules-definition-globally-generated", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is {\\it generated by global sections} if there exist a set $I$, and global sections $s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$ such that the map $$ \\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X \\longrightarrow \\mathcal{F} $$ which is the map associated to $s_i$ on the summand corresponding to $i$, is surjective. In this case we say that the sections $s_i$ {\\it generate} $\\mathcal{F}$."}
+{"_id": "13333", "title": "modules-definition-generated-by-local-sections", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. Given a set $I$, and local sections $s_i$, $i \\in I$ of $\\mathcal{F}$ we say that the subsheaf $\\mathcal{G}$ of Lemma \\ref{lemma-generated-by-local-sections} above is the {\\it subsheaf generated by the $s_i$}."}
+{"_id": "13334", "title": "modules-definition-support", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item The {\\it support of $\\mathcal{F}$} is the set of points $x \\in X$ such that $\\mathcal{F}_x \\not = 0$. \\item We denote $\\text{Supp}(\\mathcal{F})$ the support of $\\mathcal{F}$. \\item Let $s \\in \\Gamma(X, \\mathcal{F})$ be a global section. The {\\it support of $s$} is the set of points $x \\in X$ such that the image $s_x \\in \\mathcal{F}_x$ of $s$ is not zero. \\end{enumerate}"}
+{"_id": "13335", "title": "modules-definition-locally-generated", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is {\\it locally generated by sections} if for every $x \\in X$ there exists an open neighbourhood $U$ such that $\\mathcal{F}|_U$ is globally generated as a sheaf of $\\mathcal{O}_U$-modules."}
+{"_id": "13336", "title": "modules-definition-finite-type", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is of {\\it finite type} if for every $x \\in X$ there exists an open neighbourhood $U$ such that $\\mathcal{F}|_U$ is generated by finitely many sections."}
+{"_id": "13337", "title": "modules-definition-quasi-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is a {\\it quasi-coherent sheaf of $\\mathcal{O}_X$-modules} if for every point $x \\in X$ there exists an open neighbourhood $x\\in U \\subset X$ such that $\\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$ \\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_U \\longrightarrow \\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_U $$ The category of quasi-coherent $\\mathcal{O}_X$-modules is denoted $\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "13338", "title": "modules-definition-sheaf-associated", "text": "In the situation of Lemma \\ref{lemma-construct-quasi-coherent-sheaves} we say $\\mathcal{F}_M$ is the {\\it sheaf associated to the module $M$ and the ring map $\\alpha$}. If $R = \\Gamma(X, \\mathcal{O}_X)$ and $\\alpha = \\text{id}_R$ we simply say $\\mathcal{F}_M$ is the {\\it sheaf associated to the module $M$}."}
+{"_id": "13339", "title": "modules-definition-finite-presentation", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is of {\\it finite presentation} if for every point $x \\in X$ there exists an open neighbourhood $x\\in U \\subset X$, and  $n, m \\in \\mathbf{N}$ such that $\\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$ \\bigoplus\\nolimits_{j = 1, \\ldots, m} \\mathcal{O}_U \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_U $$"}
+{"_id": "13340", "title": "modules-definition-coherent", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We say that $\\mathcal{F}$ is a {\\it coherent $\\mathcal{O}_X$-module} if the following two conditions hold: \\begin{enumerate} \\item $\\mathcal{F}$ is of finite type, and \\item for every open $U \\subset X$ and every finite collection $s_i \\in \\mathcal{F}(U)$, $i = 1, \\ldots, n$ the kernel of the associated map $\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$ is of finite type. \\end{enumerate} The category of coherent $\\mathcal{O}_X$-modules is denoted $\\textit{Coh}(\\mathcal{O}_X)$."}
+{"_id": "13341", "title": "modules-definition-closed-immersion", "text": "A {\\it closed immersion of ringed spaces}\\footnote{This is nonstandard notation; see discussion above.} is a morphism $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ with the following properties: \\begin{enumerate} \\item The map $i$ is a closed immersion of topological spaces. \\item The associated map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective. Denote the kernel by $\\mathcal{I}$. \\item The $\\mathcal{O}_X$-module $\\mathcal{I}$ is locally generated by sections. \\end{enumerate}"}
+{"_id": "13342", "title": "modules-definition-locally-free", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item We say $\\mathcal{F}$ is {\\it locally free} if for every point $x \\in X$ there exists a set $I$ and an open neighbourhood $x \\in U \\subset X$ such that $\\mathcal{F}|_U$ is isomorphic to $\\bigoplus_{i \\in I} \\mathcal{O}_X|_U$ as an $\\mathcal{O}_X|_U$-module. \\item We say $\\mathcal{F}$ is {\\it finite locally free} if we may choose the index sets $I$ to be finite. \\item We say $\\mathcal{F}$ is {\\it finite locally free of rank $r$} if we may choose the index sets $I$ to have cardinality $r$. \\end{enumerate}"}
+{"_id": "13343", "title": "modules-definition-flat", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. An $\\mathcal{O}_X$-module $\\mathcal{F}$ is {\\it flat} if the functor $$ \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X), \\quad \\mathcal{G} \\mapsto \\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} $$ is exact."}
+{"_id": "13344", "title": "modules-definition-flat-at-point", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $x \\in X$. An $\\mathcal{O}_X$-module $\\mathcal{F}$ is {\\it flat at $x$} if $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{X, x}$-module."}
+{"_id": "13345", "title": "modules-definition-flat-morphism", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $x \\in X$. We say $f$ is {\\it flat at $x$} if the map of rings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat. We say $f$ is {\\it flat} if $f$ is flat at every $x \\in X$."}
+{"_id": "13346", "title": "modules-definition-flat-module", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item We say that $\\mathcal{F}$ is {\\it flat over $Y$ at a point $x \\in X$} if the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{Y, f(x)}$-module. \\item We say that $\\mathcal{F}$ is {\\it flat over $Y$} if $\\mathcal{F}$ is flat over $Y$ at every point $x$ of $X$. \\end{enumerate}"}
+{"_id": "13347", "title": "modules-definition-koszul", "text": "Let $X$ be a ringed space. Let $\\varphi : \\mathcal{E} \\to \\mathcal{O}_X$ be an $\\mathcal{O}_X$-module map. The {\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$ is the sheaf of commutative differential graded algebras defined as follows: \\begin{enumerate} \\item the underlying graded algebra is the exterior algebra $K_\\bullet(\\varphi) = \\wedge(\\mathcal{E})$, \\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$ is the unique derivation such that $d(e) = \\varphi(e)$ for all local sections $e$ of $\\mathcal{E} = K_1(\\varphi)$. \\end{enumerate}"}
+{"_id": "13348", "title": "modules-definition-koszul-complex", "text": "Let $X$ be a ringed space and let $f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$. The {\\it Koszul complex on $f_1, \\ldots, f_r$} is the Koszul complex associated to the map $(f_1, \\ldots, f_n) : \\mathcal{O}_X^{\\oplus n} \\to \\mathcal{O}_X$. Notation $K_\\bullet(\\mathcal{O}_X, f_1, \\ldots, f_n)$, or $K_\\bullet(\\mathcal{O}_X, f_\\bullet)$."}
+{"_id": "13349", "title": "modules-definition-invertible", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. An {\\it invertible $\\mathcal{O}_X$-module} is a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{L}$ such that the functor $$ \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X),\\quad \\mathcal{F} \\longmapsto \\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{F} $$ is an equivalence of categories. We say that $\\mathcal{L}$ is {\\it trivial} if it is isomorphic as an $\\mathcal{O}_X$-module to $\\mathcal{O}_X$."}
+{"_id": "13350", "title": "modules-definition-powers", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given an invertible sheaf $\\mathcal{L}$ on $X$ and $n \\in \\mathbf{Z}$ we define the $n$th {\\it tensor power} $\\mathcal{L}^{\\otimes n}$ of $\\mathcal{L}$ as the image of $\\mathcal{O}_X$ under applying the equivalence $\\mathcal{F} \\mapsto\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ exactly $n$ times."}
+{"_id": "13351", "title": "modules-definition-gamma-star", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Given an invertible sheaf $\\mathcal{L}$ on $X$ we define the {\\it associated graded ring} to be $$ \\Gamma_*(X, \\mathcal{L}) = \\bigoplus\\nolimits_{n \\geq 0} \\Gamma(X, \\mathcal{L}^{\\otimes n}) $$ Given a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set $$ \\Gamma_*(X, \\mathcal{L}, \\mathcal{F}) = \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) $$ which we think of as a graded $\\Gamma_*(X, \\mathcal{L})$-module."}
+{"_id": "13352", "title": "modules-definition-pic", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. The {\\it Picard group} $\\Pic(X)$ of $X$ is the abelian group whose elements are isomorphism classes of invertible $\\mathcal{O}_X$-modules, with addition corresponding to tensor product."}
+{"_id": "13353", "title": "modules-definition-derivation", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be an $\\mathcal{O}_2$-module. A {\\it $\\mathcal{O}_1$-derivation} or more precisely a {\\it $\\varphi$-derivation} into $\\mathcal{F}$ is a map $D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates the image of $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the {\\it Leibniz rule} $$ D(ab) = aD(b) + D(a)b $$ for all $a, b$ local sections of $\\mathcal{O}_2$ (wherever they are both defined). We denote $\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$ the set of $\\varphi$-derivations into $\\mathcal{F}$."}
+{"_id": "13354", "title": "modules-definition-module-differentials", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. The {\\it module of differentials} of $\\varphi$ is the object representing the functor $\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$ which exists by Lemma \\ref{lemma-universal-module}. It is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal $\\varphi$-derivation} is denoted $\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$."}
+{"_id": "13355", "title": "modules-definition-differentials", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of ringed spaces. \\begin{enumerate} \\item Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. An {\\it $S$-derivation} into $\\mathcal{F}$ is a $f^{-1}\\mathcal{O}_S$-derivation, or more precisely a $f^\\sharp$-derivation in the sense of Definition \\ref{definition-derivation}. We denote $\\text{Der}_S(\\mathcal{O}_X, \\mathcal{F})$ the set of $S$-derivations into $\\mathcal{F}$. \\item The {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$} is the module of differentials $\\Omega_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$ endowed with its universal $S$-derivation $\\text{d}_{X/S} : \\mathcal{O}_X \\to \\Omega_{X/S}$. \\end{enumerate}"}
+{"_id": "13356", "title": "modules-definition-differential-operators", "text": "Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $k \\geq 0$ be an integer. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules. A {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$} is an is an $\\mathcal{O}_1$-linear map such that for all local sections $g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $\\mathcal{O}_2$-linear map."}
+{"_id": "13357", "title": "modules-definition-module-principal-parts", "text": "Let $X$ be a topoological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. The module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$ constructed in Lemma \\ref{lemma-module-principal-parts} is called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$."}
+{"_id": "13358", "title": "modules-definition-relative-differential-operators", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of ringed spaces. Let $\\mathcal{F}$ and $\\mathcal{G}$ be $\\mathcal{O}_X$-modules. Let $k \\geq 0$ be an integer. A {\\it differential operator of order $k$ on $X/S$} is a differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ with respect to $f^\\sharp : f^{-1}\\mathcal{O}_S \\to \\mathcal{O}_X$ We denote $\\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G})$ the set of these differential operators."}
+{"_id": "13359", "title": "modules-definition-de-rham-complex", "text": "In the situation above, the {\\it de Rham complex of $\\mathcal{B}$ over $\\mathcal{A}$} is the unique complex $$ \\Omega_{\\mathcal{B}/\\mathcal{A}}^0 \\to \\Omega_{\\mathcal{B}/\\mathcal{A}}^1 \\to \\Omega_{\\mathcal{B}/\\mathcal{A}}^2 \\to \\ldots $$ of sheaves of $\\mathcal{A}$-modules whose differential in degree $0$ is given by $\\text{d} : \\mathcal{B} \\to \\Omega_{\\mathcal{B}/\\mathcal{A}}$ and whose differentials in higher degrees have the following property \\begin{equation} \\label{equation-rule} \\text{d}\\left(b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p\\right) = \\text{d}b_0 \\wedge \\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p \\end{equation} where $b_0, \\ldots, b_p \\in \\mathcal{B}(U)$ are sections over a common open $U \\subset X$."}
+{"_id": "13360", "title": "modules-definition-de-rham-complex-morphism-ringed-spaces", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The {\\it de Rham complex} of $f$ or of $X$ over $Y$ is the complex $$ \\Omega^\\bullet_{X/Y} = \\Omega^\\bullet_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y} $$"}
+{"_id": "13361", "title": "modules-definition-naive-cotangent-complex", "text": "Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings. The {\\it naive cotangent complex} $\\NL_{\\mathcal{B}/\\mathcal{A}}$ is the chain complex (\\ref{equation-naive-cotangent-complex}) $$ \\NL_{\\mathcal{B}/\\mathcal{A}} = \\left(\\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}} \\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right) $$ with $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and $\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}} \\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$ placed in degree $0$."}
+{"_id": "13362", "title": "modules-definition-cotangent-complex-morphism-ringed-topoi", "text": "The {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$ of a morphism of ringed spaces $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is $\\NL_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}$."}
+{"_id": "13423", "title": "defos-definition-strict-morphism-thickenings", "text": "In Situation \\ref{situation-morphism-thickenings} we say that $(f, f')$ is a {\\it strict morphism of thickenings} if the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective."}
+{"_id": "13424", "title": "defos-definition-strict-morphism-thickenings-ringed-topoi", "text": "In Situation \\ref{situation-morphism-thickenings-ringed-topoi} we say that $(f, f')$ is a {\\it strict morphism of thickenings} if the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective."}
+{"_id": "13458", "title": "groupoids-quotients-definition-invariant", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$. We say a morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is {\\it $R$-invariant} if the diagram $$ \\xymatrix{ R \\ar[r]_s \\ar[d]_t & U \\ar[d]^\\phi \\\\ U \\ar[r]^\\phi & X } $$ is commutative. If $j : R \\to U \\times_B U$ comes from the action of a group algebraic space $G$ on $U$ over $B$ as in Groupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-groupoid-from-action}, then we say that $\\phi$ is {\\it $G$-invariant}."}
+{"_id": "13459", "title": "groupoids-quotients-definition-base-change", "text": "In the situation of Lemma \\ref{lemma-base-change-on-invariant} we call $j' : R' \\to U' \\times_B U'$ the {\\it base change} of the pre-relation $j$ to $X'$. We say it is a {\\it flat base change} if $X' \\to X$ is a flat morphism of algebraic spaces."}
+{"_id": "13460", "title": "groupoids-quotients-definition-categorical", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \\to U \\times_B U$ be pre-relation in algebraic spaces over $B$. \\begin{enumerate} \\item We say a morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is a {\\it categorical quotient} if it is $R$-invariant, and for every $R$-invariant morphism $\\psi : U \\to Y$ of algebraic spaces over $B$ there exists a unique morphism $\\chi : X \\to Y$ such that $\\psi = \\phi \\circ \\chi$. \\item Let $\\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$. Assume $U$, $R$ are objects of $\\mathcal{C}$. In this situation we say a morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is a {\\it categorical quotient in $\\mathcal{C}$} if $X \\in \\Ob(\\mathcal{C})$, and $\\phi$ is $R$-invariant, and for every $R$-invariant morphism $\\psi : U \\to Y$ with $Y \\in \\Ob(\\mathcal{C})$ there exists a unique morphism $\\chi : X \\to Y$ such that $\\psi = \\phi \\circ \\chi$. \\item If $B = S$ and $\\mathcal{C}$ is the category of schemes over $S$, then we say $U \\to X$ is a {\\it categorical quotient in the category of schemes}, or simply a {\\it categorical quotient in schemes}. \\end{enumerate}"}
+{"_id": "13461", "title": "groupoids-quotients-definition-universal-categorical", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$ closed under fibre products. Let $j = (t, s) : R \\to U \\times_B U$ be pre-relation in $\\mathcal{C}$, and let $U \\to X$ be an $R$-invariant morphism with $X \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item We say $U \\to X$ is a {\\it universal categorical quotient} in $\\mathcal{C}$ if for every morphism $X' \\to X$ in $\\mathcal{C}$ the morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in $\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$. \\item We say $U \\to X$ is a {\\it uniform categorical quotient} in $\\mathcal{C}$ if for every flat morphism $X' \\to X$ in $\\mathcal{C}$ the morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in $\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$. \\end{enumerate}"}
+{"_id": "13462", "title": "groupoids-quotients-definition-orbit", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. If $u \\in |U|$, then the {\\it orbit}, or more precisely the {\\it $R$-orbit} of $u$ is $$ O_u = \\left\\{ u' \\in |U|\\ : \\begin{matrix} \\exists n \\geq 1, \\ \\exists u_0, \\ldots, u_n \\in |U|\\text{ such that } u_0 = u \\text{ and } u_n = u' \\\\ \\text{and for all }i \\in \\{0, \\ldots, n - 1\\}\\text{ either } u_i = u_{i + 1}\\text{ or } \\\\ \\exists r \\in |R|, \\ s(r) = u_i, t(r) = u_{i + 1} \\text{ or } \\\\ \\exists r \\in |R|, \\ t(r) = u_i, s(r) = u_{i + 1} \\end{matrix} \\right\\} $$"}
+{"_id": "13463", "title": "groupoids-quotients-definition-geometric-orbits", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. Let $\\Spec(k) \\to B$ be a geometric point of $B$. \\begin{enumerate} \\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are {\\it weakly $R$-equivalent} if they are in the same equivalence class for the equivalence relation generated by the relation $j(R(k)) \\subset U(k) \\times U(k)$. \\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are {\\it $R$-equivalent} if for some overfield $k \\subset \\Omega$ the images in $U(\\Omega)$ are weakly $R$-equivalent. \\item The {\\it weak orbit}, or more precisely the {\\it weak $R$-orbit} of $\\overline{u} \\in U(k)$ is set of all elements of $U(k)$ which are weakly $R$-equivalent to $\\overline{u}$. \\item The {\\it orbit}, or more precisely the {\\it $R$-orbit} of $\\overline{u} \\in U(k)$ is set of all elements of $U(k)$ which are $R$-equivalent to $\\overline{u}$. \\end{enumerate}"}
+{"_id": "13464", "title": "groupoids-quotients-definition-set-theoretically-invariant", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. \\begin{enumerate} \\item We say $\\phi : U \\to X$ is {\\it set-theoretically $R$-invariant} if and only if the map $U(k) \\to X(k)$ equalizes the two maps $s, t : R(k) \\to U(k)$ for every algebraically closed field $k$ over $B$. \\item We say $\\phi : U \\to X$ {\\it separates orbits}, or {\\it separates $R$-orbits} if it is set-theoretically $R$-invariant and $\\phi(\\overline{u}) = \\phi(\\overline{u}')$ in $X(k)$ implies that $\\overline{u}, \\overline{u}' \\in U(k)$ are in the same orbit for every algebraically closed field $k$ over $B$. \\end{enumerate}"}
+{"_id": "13465", "title": "groupoids-quotients-definition-set-theoretic-equivalence", "text": "Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation over $B$. \\begin{enumerate} \\item We say $j$ is a {\\it set-theoretic pre-equivalence relation} if for all algebraically closed fields $k$ over $B$ the relation $\\sim_R$ on $U(k)$ defined by $$ \\overline{u} \\sim_R \\overline{u}' \\Leftrightarrow \\begin{matrix} \\exists\\text{ field extension }K/k, \\ \\exists\\ r \\in R(K), \\\\ s(r) = \\overline{u}, \\ t(r) = \\overline{u}' \\end{matrix} $$ is an equivalence relation. \\item We say $j$ is a {\\it set-theoretic equivalence relation} if $j$ is universally injective and a set-theoretic pre-equivalence relation. \\end{enumerate}"}
+{"_id": "13466", "title": "groupoids-quotients-definition-orbit-space", "text": "Let $B \\to S$ as in Section \\ref{section-conventions-notation}. Let $j : R \\to U \\times_B U$ be a pre-relation. We say $\\phi : U \\to X$ is an {\\it orbit space for $R$} if \\begin{enumerate} \\item $\\phi$ is $R$-invariant, \\item $\\phi$ separates $R$-orbits, and \\item $\\phi$ is surjective. \\end{enumerate}"}
+{"_id": "13467", "title": "groupoids-quotients-definition-coarse", "text": "Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation. A morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is called a {\\it coarse quotient} if \\begin{enumerate} \\item $\\phi$ is a categorical quotient, and \\item $\\phi$ is an orbit space. \\end{enumerate} If $S = B$, $U$, $R$ are all schemes, then we say a morphism of schemes $\\phi : U \\to X$ is a {\\it coarse quotient in schemes} if \\begin{enumerate} \\item $\\phi$ is a categorical quotient in schemes, and \\item $\\phi$ is an orbit space. \\end{enumerate}"}
+{"_id": "13468", "title": "groupoids-quotients-definition-topological", "text": "Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation. Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$. \\begin{enumerate} \\item \\label{item-submersive} The morphism $\\phi$ is submersive. \\item \\label{item-invariant-closed} For any $R$-invariant closed subset $Z \\subset |U|$ the image $\\phi(Z)$ is closed in $|X|$. \\item \\label{item-intersect-invariant-closed} Condition (\\ref{item-invariant-closed}) holds and for any pair of $R$-invariant closed subsets $Z_1, Z_2 \\subset |U|$ we have $$ \\phi(Z_1 \\cap Z_2) = \\phi(Z_1) \\cap \\phi(Z_2) $$ \\item The morphism $(t, s) : R \\to U \\times_X U$ is universally submersive. \\label{item-strong} \\end{enumerate} For each of these properties we can also require them to hold after any flat base change, or after any base change, see Definition \\ref{definition-base-change}. In this case we say condition (\\ref{item-submersive}), (\\ref{item-invariant-closed}), (\\ref{item-intersect-invariant-closed}), or (\\ref{item-strong}) holds {\\it uniformly} or {\\it universally}."}
+{"_id": "13469", "title": "groupoids-quotients-definition-functions", "text": "Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation. Let $\\phi : U \\to X$ be an $R$-invariant morphism. Denote $\\phi' = \\phi \\circ s = \\phi \\circ t : R \\to X$. \\begin{enumerate} \\item We denote $(\\phi_*\\mathcal{O}_U)^R$ the $\\mathcal{O}_X$-sub-algebra of $\\phi_*\\mathcal{O}_U$ which is the equalizer of the two maps $$ \\xymatrix{ \\phi_*\\mathcal{O}_U \\ar@<1ex>[rr]^{\\phi_*s^\\sharp} \\ar@<-1ex>[rr]_{\\phi_*t^\\sharp} & & \\phi'_*\\mathcal{O}_R } $$ on $X_\\etale$. We sometimes call this the {\\it sheaf of $R$-invariant functions on $X$}. \\item We say {\\it the functions on $X$ are the $R$-invariant functions on $U$} if the natural map $\\mathcal{O}_X \\to (\\phi_*\\mathcal{O}_U)^R$ is an isomorphism. \\end{enumerate}"}
+{"_id": "13470", "title": "groupoids-quotients-definition-good", "text": "Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation. A morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is called a {\\it good quotient} if \\begin{enumerate} \\item $\\phi$ is invariant, \\item $\\phi$ is affine, \\item $\\phi$ is surjective, \\item condition (\\ref{item-intersect-invariant-closed}) holds universally, and \\item the functions on $X$ are the $R$-invariant functions on $U$. \\end{enumerate}"}
+{"_id": "13471", "title": "groupoids-quotients-definition-geometric", "text": "Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \\to U \\times_B U$ be a pre-relation. A morphism $\\phi : U \\to X$ of algebraic spaces over $B$ is called a {\\it geometric quotient} if \\begin{enumerate} \\item $\\phi$ is an orbit space, \\item condition (\\ref{item-submersive}) holds universally, i.e., $\\phi$ is universally submersive, and \\item the functions on $X$ are the $R$-invariant functions on $U$. \\end{enumerate}"}
+{"_id": "13491", "title": "spaces-resolve-definition-blowup-at-point", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$ be a closed point. By Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-space-closed-point} we can represent $x$ by a closed immersion $i : \\Spec(k) \\to X$. The {\\it blowing up $X' \\to X$ of $X$ at $x$} means the blowing up of $X$ in the closed subspace $Z = i(\\Spec(k)) \\subset X$."}
+{"_id": "13492", "title": "spaces-resolve-definition-normalized-blowup", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ satisfying the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}. Let $x \\in |X|$ be a closed point. The {\\it normalized blowup of $X$ at $x$} is the composition $X'' \\to X' \\to X$ where $X' \\to X$ is the blowup of $X$ at $x$ (Definition \\ref{definition-blowup-at-point}) and $X'' \\to X'$ is the normalization of $X'$."}
+{"_id": "13493", "title": "spaces-resolve-definition-resolution", "text": "Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over $S$. A {\\it resolution of singularities} of $X$ is a modification $f : X \\to Y$ such that $X$ is regular."}
+{"_id": "13494", "title": "spaces-resolve-definition-resolution-surface", "text": "Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral algebraic space over $S$. We say $Y$ has a {\\it resolution of singularities by normalized blowups} if there exists a sequence $$ Y_n \\to X_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y $$ where \\begin{enumerate} \\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$, \\item $Y_0 \\to Y$ is the normalization, \\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and \\item $Y_n$ is regular. \\end{enumerate}"}
+{"_id": "13640", "title": "duality-definition-dualizing-scheme", "text": "Let $X$ be a locally Noetherian scheme. An object $K$ of $D(\\mathcal{O}_X)$ is called a {\\it dualizing complex} if $K$ satisfies the equivalent conditions of Lemma \\ref{lemma-equivalent-definitions}."}
+{"_id": "13641", "title": "duality-definition-good-dualizing", "text": "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing complex on $S$. Let $X$ be a scheme of finite type over $S$. The complex $K$ constructed above is called the {\\it dualizing complex normalized relative to $\\omega_S^\\bullet$} and is denoted $\\omega_X^\\bullet$."}
+{"_id": "13642", "title": "duality-definition-gorenstein", "text": "Let $X$ be a scheme. We say $X$ is {\\it Gorenstein} if $X$ is locally Noetherian and $\\mathcal{O}_{X, x}$ is Gorenstein for all $x \\in X$."}
+{"_id": "13643", "title": "duality-definition-gorenstein-morphism", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \\begin{enumerate} \\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it Gorenstein at $x$} if $f$ is flat at $x$, and the local ring of the scheme $X_y$ at $x$ is Gorenstein. \\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is Gorenstein at every point of $X$. \\end{enumerate}"}
+{"_id": "13644", "title": "duality-definition-relative-dualizing-complex", "text": "Let $X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $W \\subset X \\times_S X$ be any open such that the diagonal $\\Delta_{X/S} : X \\to X \\times_S X$ factors through a closed immersion $\\Delta : X \\to W$. A {\\it relative dualizing complex} is a pair $(K, \\xi)$ consisting of an object $K \\in D(\\mathcal{O}_X)$ and a map $$ \\xi : \\Delta_*\\mathcal{O}_X \\longrightarrow L\\text{pr}_1^*K|_W $$ in $D(\\mathcal{O}_W)$ such that \\begin{enumerate} \\item $K$ is $S$-perfect (Derived Categories of Schemes, Definition \\ref{perfect-definition-relatively-perfect}), and \\item $\\xi$ defines an isomorphism of $\\Delta_*\\mathcal{O}_X$ with $R\\SheafHom_{\\mathcal{O}_W}( \\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$. \\end{enumerate}"}
+{"_id": "14106", "title": "more-morphisms-definition-thickening", "text": "Thickenings. \\begin{enumerate} \\item We say a scheme $X'$ is a {\\it thickening} of a scheme $X$ if $X$ is a closed subscheme of $X'$ and the underlying topological spaces are equal. \\item We say a scheme $X'$ is a {\\it first order thickening} of a scheme $X$ if $X$ is a closed subscheme of $X'$ and the quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero. \\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a {\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that $f'(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed subscheme $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say that $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of thickenings. \\item Let $S$ be a scheme. We similarly define {\\it thickenings over $S$}, and {\\it morphisms of thickenings over $S$}. This means that the schemes $X, X', Y, Y'$ above are schemes over $S$, and that the morphisms $X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $S$. \\end{enumerate}"}
+{"_id": "14107", "title": "more-morphisms-definition-first-order-infinitesimal-neighbourhood", "text": "Let $i : Z \\to X$ be an immersion of schemes. The {\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is the first order thickening $Z \\subset Z'$ over $X$ described above."}
+{"_id": "14108", "title": "more-morphisms-definition-formally-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it formally unramified} if given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of affine schemes over $S$ there exists at most one dotted arrow making the diagram commute."}
+{"_id": "14109", "title": "more-morphisms-definition-universal-thickening", "text": "Let $h : Z \\to X$ be a formally unramified morphism of schemes. \\begin{enumerate} \\item The {\\it universal first order thickening} of $Z$ over $X$ is the thickening $Z \\subset Z'$ constructed in Lemma \\ref{lemma-universal-thickening}. \\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf of $Z$ in its universal first order thickening $Z'$ over $X$. \\end{enumerate} We often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation."}
+{"_id": "14110", "title": "more-morphisms-definition-formally-etale", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it formally \\'etale} if given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of affine schemes over $S$ there exists exactly one dotted arrow making the diagram commute."}
+{"_id": "14111", "title": "more-morphisms-definition-formally-smooth", "text": "Let $f : X \\to S$ be a morphism of schemes. We say $f$ is {\\it formally smooth} if given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of affine schemes over $S$ there exists a dotted arrow making the diagram commute."}
+{"_id": "14112", "title": "more-morphisms-definition-netherlander", "text": "Let $f : X \\to Y$ be a morphism of schemes. The {\\it naive cotangent complex of $f$} is the complex defined in Modules, Definition \\ref{modules-definition-cotangent-complex-morphism-ringed-topoi}. Notation: $\\NL_f$ or $\\NL_{X/Y}$."}
+{"_id": "14113", "title": "more-morphisms-definition-normal", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \\begin{enumerate} \\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it normal at $x$} if $f$ is flat at $x$, and the scheme $X_y$ is geometrically normal at $x$ over $\\kappa(y)$ (see Varieties, Definition \\ref{varieties-definition-geometrically-normal}). \\item We say $f$ is a {\\it normal morphism} if $f$ is normal at every point of $X$. \\end{enumerate}"}
+{"_id": "14114", "title": "more-morphisms-definition-regular", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \\begin{enumerate} \\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it regular at $x$} if $f$ is flat at $x$, and the scheme $X_y$ is geometrically regular at $x$ over $\\kappa(y)$ (see Varieties, Definition \\ref{varieties-definition-geometrically-regular}). \\item We say $f$ is a {\\it regular morphism} if $f$ is regular at every point of $X$. \\end{enumerate}"}
+{"_id": "14115", "title": "more-morphisms-definition-CM", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that all the fibres $X_y$ are locally Noetherian schemes. \\begin{enumerate} \\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$, and the local ring of the scheme $X_y$ at $x$ is Cohen-Macaulay. \\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is Cohen-Macaulay at every point of $X$. \\end{enumerate}"}
+{"_id": "14116", "title": "more-morphisms-definition-etale-neighbourhood", "text": "Let $S$ be a scheme. Let $s \\in S$ be a point. \\begin{enumerate} \\item An {\\it \\'etale neighbourhood of $(S, s)$} is a pair $(U, u)$ together with an \\'etale morphism of schemes $\\varphi : U \\to S$ such that $\\varphi(u) = s$. \\item A {\\it morphism of \\'etale neighbourhoods} $f : (V, v) \\to (U, u)$ of $(S, s)$ is simply a morphism of $S$-schemes $f : V \\to U$ such that $f(v) = u$. \\item An {\\it elementary \\'etale neighbourhood} is an \\'etale neighbourhood $\\varphi : (U, u) \\to (S, s)$ such that $\\kappa(s) = \\kappa(u)$. \\end{enumerate}"}
+{"_id": "14117", "title": "more-morphisms-definition-relatively-finitely-presented-sheaf", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it finitely presented relative to $S$} or {\\it of finite presentation relative to $S$} if there exists an affine open covering $S = \\bigcup V_i$ and for every $i$ an affine open covering $f^{-1}(V_i) = \\bigcup_j U_{ij}$ such that $\\mathcal{F}(U_{ij})$ is a $\\mathcal{O}_X(U_{ij})$-module of finite presentation relative to $\\mathcal{O}_S(V_i)$."}
+{"_id": "14118", "title": "more-morphisms-definition-relative-pseudo-coherence", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$. \\begin{enumerate} \\item We say $E$ is {\\it $m$-pseudo-coherent relative to $S$} if there exists an affine open covering $S = \\bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$ such that the equivalent conditions of Lemma \\ref{lemma-relatively-pseudo-coherent} are satisfied for each of the pairs $(U_{ij} \\to V_i, E|_{U_{ij}})$. \\item We say $E$ is {\\it pseudo-coherent relative to $S$} if $E$ is $m$-pseudo-coherent relative to $S$ for all $m \\in \\mathbf{Z}$. \\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $S$} if $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$. \\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $S$} if $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is pseudo-coherent relative to $S$. \\end{enumerate}"}
+{"_id": "14119", "title": "more-morphisms-definition-pseudo-coherent", "text": "A morphism of schemes $f : X \\to S$ is called {\\it pseudo-coherent} if the equivalent conditions of Lemma \\ref{lemma-pseudo-coherent} are satisfied. In this case we also say that $X$ is pseudo-coherent over $S$."}
+{"_id": "14120", "title": "more-morphisms-definition-perfect", "text": "A morphism of schemes $f : X \\to S$ is called {\\it perfect} if the equivalent conditions of Lemma \\ref{lemma-perfect} are satisfied. In this case we also say that $X$ is perfect over $S$."}
+{"_id": "14121", "title": "more-morphisms-definition-lci", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item Let $x \\in X$. We say that $f$ is {\\it Koszul at $x$} if $f$ is of finite type at $x$ and there exists an open neighbourhood and a factorization of $f|_U$ as $\\pi \\circ i$ where $i : U \\to P$ is a Koszul-regular immersion and $\\pi : P \\to S$ is smooth. \\item We say $f$ is a {\\it Koszul morphism}, or that $f$ is a {\\it local complete intersection morphism} if $f$ is Koszul at every point. \\end{enumerate}"}
+{"_id": "14122", "title": "more-morphisms-definition-weakly-etale", "text": "A morphism of schemes $X \\to Y$ is {\\it weakly \\'etale} or {\\it absolutely flat} if both $X \\to Y$ and the diagonal morphism $X \\to X \\times_Y X$ are flat."}
+{"_id": "14123", "title": "more-morphisms-definition-ind-quasi-affine", "text": "A scheme $X$ is {\\it ind-quasi-affine} if every quasi-compact open of $X$ is quasi-affine. Similarly, a morphism of schemes $X \\to Y$ is {\\it ind-quasi-affine} if $f^{-1}(V)$ is ind-quasi-affine for each affine open $V$ in $Y$."}
+{"_id": "14124", "title": "more-morphisms-definition-affine-stratification", "text": "Let $X$ be a scheme. An {\\it affine stratification} is a locally finite stratification $X = \\coprod_{i \\in I} X_i$ whose strata $X_i$ are affine and such that the inclusion morphisms $X_i \\to X$ are affine."}
+{"_id": "14125", "title": "more-morphisms-definition-affine-stratification-number", "text": "Let $X$ be a nonempty quasi-compact and quasi-separated scheme. The {\\it affine stratification number} is the smallest integer $n \\geq 0$ such that the following equivalent conditions are satisfied \\begin{enumerate} \\item there exists a finite affine stratification $X = \\coprod_{i \\in I} X_i$ where $I$ has length $n$, \\item there exists an affine stratification $X = X_0 \\amalg X_1 \\amalg \\ldots \\amalg X_n$ with index set $\\{0, \\ldots, n\\}$. \\end{enumerate}"}
+{"_id": "14126", "title": "more-morphisms-definition-weighting", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism. A {\\it weighting} or a {\\it pond\\'eration} of $f$ is a map $w : X \\to \\mathbf{Z}$ such that for any diagram $$ \\xymatrix{ X \\ar[d]_f & U \\ar[l]^h \\ar[d]^\\pi \\\\ Y & V \\ar[l]_g } $$ where $V \\to Y$ is \\'etale, $U \\subset X_V$ is open, and $U \\to V$ finite, the function $\\int_\\pi (w \\circ h)$ is locally constant."}
+{"_id": "14277", "title": "sites-modules-definition-free-abelian-presheaf-on", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{G}$ be a presheaf of sets. The {\\it free abelian presheaf} $\\mathbf{Z}_\\mathcal{G}$ on $\\mathcal{G}$ is the abelian presheaf defined by the rule $$ U \\longmapsto \\mathbf{Z}[\\mathcal{G}(U)]. $$ In the special case $\\mathcal{G} = h_X$ of a representable presheaf associated to an object $X$ of $\\mathcal{C}$ we use the notation $\\mathbf{Z}_X = \\mathbf{Z}_{h_X}$. In other words $$ \\mathbf{Z}_X(U) = \\mathbf{Z}[\\Mor_\\mathcal{C}(U, X)]. $$"}
+{"_id": "14278", "title": "sites-modules-definition-free-abelian-sheaf-on", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets. The {\\it free abelian sheaf} $\\mathbf{Z}_\\mathcal{G}^\\#$ on $\\mathcal{G}$ is the abelian sheaf $\\mathbf{Z}_\\mathcal{G}^\\#$ which is the sheafification of the free abelian presheaf on $\\mathcal{G}$. In the special case $\\mathcal{G} = h_X$ of a representable presheaf associated to an object $X$ of $\\mathcal{C}$ we use the notation $\\mathbf{Z}_X^\\#$."}
+{"_id": "14279", "title": "sites-modules-definition-ringed-site", "text": "Ringed sites. \\begin{enumerate} \\item A {\\it ringed site} is a pair $(\\mathcal{C}, \\mathcal{O})$ where $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the {\\it structure sheaf} of the ringed site. \\item Let $(\\mathcal{C}, \\mathcal{O})$, $(\\mathcal{C}', \\mathcal{O}')$ be ringed sites. A {\\it morphism of ringed sites} $$ (f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}) \\longrightarrow (\\mathcal{C}', \\mathcal{O}') $$ is given by a morphism of sites $f : \\mathcal{C} \\to \\mathcal{C}'$ (see Sites, Definition \\ref{sites-definition-morphism-sites}) together with a map of sheaves of rings $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction is the same thing as a map of sheaves of rings $f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$. \\item Let $(f, f^\\sharp) : (\\mathcal{C}_1, \\mathcal{O}_1) \\to (\\mathcal{C}_2, \\mathcal{O}_2)$ and $(g, g^\\sharp) : (\\mathcal{C}_2, \\mathcal{O}_2) \\to (\\mathcal{C}_3, \\mathcal{O}_3)$ be morphisms of ringed sites. Then we define the {\\it composition of morphisms of ringed sites} by the rule $$ (g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp). $$ Here we use composition of morphisms of sites defined in Sites, Definition \\ref{sites-definition-composition-morphisms-sites} and $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of rings $$ \\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2 \\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1 $$ \\end{enumerate}"}
+{"_id": "14280", "title": "sites-modules-definition-ringed-topos", "text": "Ringed topoi. \\begin{enumerate} \\item A {\\it ringed topos} is a pair $(\\Sh(\\mathcal{C}), \\mathcal{O})$ where $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the {\\it structure sheaf} of the ringed topos. \\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$, $(\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi. A {\\it morphism of ringed topoi} $$ (f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{C}'), \\mathcal{O}') $$ is given by a morphism of topoi $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ (see Sites, Definition \\ref{sites-definition-topos}) together with a map of sheaves of rings $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction is the same thing as a map of sheaves of rings $f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$. \\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and $(g, g^\\sharp) : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to (\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of ringed topoi. Then we define the {\\it composition of morphisms of ringed topoi} by the rule $$ (g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp). $$ Here we use composition of morphisms of topoi defined in Sites, Definition \\ref{sites-definition-topos} and $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of rings $$ \\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2 \\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1 $$ \\end{enumerate}"}
+{"_id": "14281", "title": "sites-modules-definition-2-morphism-ringed-topoi", "text": "Let $f, g : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be two morphisms of ringed topoi. A {\\it 2-morphism from $f$ to $g$} is given by a transformation of functors $t : f_* \\to g_*$ such that $$ \\xymatrix{ & \\mathcal{O}_\\mathcal{D} \\ar[ld]_{f^\\sharp} \\ar[rd]^{g^\\sharp} \\\\ f_*\\mathcal{O}_\\mathcal{C} \\ar[rr]^t & & g_*\\mathcal{O}_\\mathcal{C} } $$ is commutative."}
+{"_id": "14282", "title": "sites-modules-definition-presheaf-modules", "text": "Let $\\mathcal{C}$ be a category, and let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$. \\begin{enumerate} \\item A {\\it presheaf of $\\mathcal{O}$-modules} is given by an abelian presheaf $\\mathcal{F}$ together with a map of presheaves of sets $$ \\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F} $$ such that for every object $U$ of $\\mathcal{C}$ the map $\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ defines the structure of an $\\mathcal{O}(U)$-module structure on the abelian group $\\mathcal{F}(U)$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that the diagram $$ \\xymatrix{ \\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} & \\mathcal{F} \\ar[d]^{\\varphi} \\\\ \\mathcal{O} \\times \\mathcal{G} \\ar[r] & \\mathcal{G} } $$ commutes. \\item The set of $\\mathcal{O}$-module morphisms as above is denoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$. \\item The category of presheaves of $\\mathcal{O}$-modules is denoted $\\textit{PMod}(\\mathcal{O})$. \\end{enumerate}"}
+{"_id": "14283", "title": "sites-modules-definition-sheaf-modules", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. \\begin{enumerate} \\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$, see Definition \\ref{definition-presheaf-modules}, such that the underlying presheaf of abelian groups $\\mathcal{F}$ is a sheaf. \\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules} is a morphism of presheaves of $\\mathcal{O}$-modules. \\item Given sheaves of $\\mathcal{O}$-modules $\\mathcal{F}$ and $\\mathcal{G}$ we denote $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$ the set of morphism of sheaves of $\\mathcal{O}$-modules. \\item The category of sheaves of $\\mathcal{O}$-modules is denoted $\\textit{Mod}(\\mathcal{O})$. \\end{enumerate}"}
+{"_id": "14284", "title": "sites-modules-definition-pushforward", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi or ringed sites. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules. We define the {\\it pushforward} of $\\mathcal{F}$ as the sheaf of $\\mathcal{O}_\\mathcal{D}$-modules which as a sheaf of abelian groups equals $f_*\\mathcal{F}$ and with module structure given by the restriction via $f^\\sharp : \\mathcal{O}_\\mathcal{D} \\to f_*\\mathcal{O}_\\mathcal{C}$ of the module structure $$ f_*\\mathcal{O}_\\mathcal{C} \\times f_*\\mathcal{F} \\longrightarrow f_*\\mathcal{F} $$ from Lemma \\ref{lemma-pushforward-module}. \\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules. We define the {\\it pullback} $f^*\\mathcal{G}$ to be the sheaf of $\\mathcal{O}_\\mathcal{C}$-modules defined by the formula $$ f^*\\mathcal{G} = \\mathcal{O}_\\mathcal{C} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}} f^{-1}\\mathcal{G} $$ where the ring map $f^{-1}\\mathcal{O}_\\mathcal{D} \\to \\mathcal{O}_\\mathcal{C}$ is $f^\\sharp$, and where the  module structure is given by Lemma \\ref{lemma-pullback-module}. \\end{enumerate}"}
+{"_id": "14285", "title": "sites-modules-definition-g-shriek", "text": "With $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying (a), (b) above. For $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we define {\\it $g_{p!}\\mathcal{F}$} as the presheaf $$ V \\longmapsto \\colim_{V \\to u(U)} \\mathcal{F}(U) $$ with colimits over $(\\mathcal{I}_V^u)^{opp}$ taken in $\\textit{Ab}$. For $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we set {\\it $g_!\\mathcal{F} = (g_{p!}\\mathcal{F})^\\#$}."}
+{"_id": "14286", "title": "sites-modules-definition-global", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. \\begin{enumerate} \\item We say $\\mathcal{F}$ is a {\\it free $\\mathcal{O}$-module} if $\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module to a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$. \\item We say $\\mathcal{F}$ is {\\it finite free} if $\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module to a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$ with a finite index set $I$. \\item We say $\\mathcal{F}$ is {\\it generated by global sections} if there exists a surjection $$ \\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow \\mathcal{F} $$ from a free $\\mathcal{O}$-module onto $\\mathcal{F}$. \\item Given $r \\geq 0$ we say $\\mathcal{F}$ is {\\it generated by $r$ global sections} if there exists a surjection $\\mathcal{O}^{\\oplus r} \\to \\mathcal{F}$. \\item We say $\\mathcal{F}$ is {\\it generated by finitely many global sections} if it is generated by $r$ global sections for some $r \\geq 0$. \\item We say $\\mathcal{F}$ has a {\\it global presentation} if there exists an exact sequence $$ \\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow \\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow \\mathcal{F} \\longrightarrow 0 $$ of $\\mathcal{O}$-modules. \\item We say $\\mathcal{F}$ has a {\\it global finite presentation} if there exists an exact sequence $$ \\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow \\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow \\mathcal{F} \\longrightarrow 0 $$ of $\\mathcal{O}$-modules with $I$ and $J$ finite sets. \\end{enumerate}"}
+{"_id": "14287", "title": "sites-modules-definition-localize-ringed-site", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$. \\begin{enumerate} \\item The ringed site $(\\mathcal{C}/U, \\mathcal{O}_U)$ is called the {\\it localization of the ringed site $(\\mathcal{C}, \\mathcal{O})$ at the object $U$}. \\item The morphism of ringed topoi $(j_U, j_U^\\sharp) : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ is called the {\\it localization morphism}. \\item The functor $j_{U*} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$ is called the {\\it direct image functor}. \\item For a sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf $j_U^*\\mathcal{F}$ is called the {\\it restriction of $\\mathcal{F}$ to $\\mathcal{C}/U$}. We will sometimes denote it by $\\mathcal{F}|_{\\mathcal{C}/U}$ or even $\\mathcal{F}|_U$. It is described by the simple rule $j_U^*(\\mathcal{F})(X/U) = \\mathcal{F}(X)$. \\item The left adjoint $j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$ of restriction is called {\\it extension by zero}. It exists and is exact by Lemmas \\ref{lemma-extension-by-zero} and \\ref{lemma-extension-by-zero-exact}. \\end{enumerate}"}
+{"_id": "14288", "title": "sites-modules-definition-localize-ringed-topos", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $\\mathcal{F} \\in \\Sh(\\mathcal{C})$. \\begin{enumerate} \\item The ringed topos $(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$ is called the {\\it localization of the ringed topos $(\\Sh(\\mathcal{C}), \\mathcal{O})$ at $\\mathcal{F}$}. \\item The morphism of ringed topoi $(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) : (\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F}) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ of Lemma \\ref{lemma-localize-ringed-topos} is called the {\\it localization morphism}. \\end{enumerate}"}
+{"_id": "14289", "title": "sites-modules-definition-site-local", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. We will freely use the notions defined in Definition \\ref{definition-global}. \\begin{enumerate} \\item We say $\\mathcal{F}$ is {\\it locally free} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a free $\\mathcal{O}_{U_i}$-module. \\item We say $\\mathcal{F}$ is {\\it finite locally free} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a finite free $\\mathcal{O}_{U_i}$-module. \\item We say $\\mathcal{F}$ is {\\it locally generated by sections} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an $\\mathcal{O}_{U_i}$-module generated by global sections. \\item Given $r \\geq 0$ we sat $\\mathcal{F}$ is {\\it locally generated by $r$ sections} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an $\\mathcal{O}_{U_i}$-module generated by $r$ global sections. \\item We say $\\mathcal{F}$ is {\\it of finite type} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an $\\mathcal{O}_{U_i}$-module generated by finitely many global sections. \\item We say $\\mathcal{F}$ is {\\it quasi-coherent} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an $\\mathcal{O}_{U_i}$-module which has a global presentation. \\item We say $\\mathcal{F}$ is {\\it of finite presentation} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an $\\mathcal{O}_{U_i}$-module which has a finite global presentation. \\item We say $\\mathcal{F}$ is {\\it coherent} if and only if $\\mathcal{F}$ is of finite type, and for every object $U$ of $\\mathcal{C}$ and any $s_1, \\ldots, s_n \\in \\mathcal{F}(U)$ the kernel of the map $\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$ is of finite type on $(\\mathcal{C}/U, \\mathcal{O}_U)$. \\end{enumerate}"}
+{"_id": "14290", "title": "sites-modules-definition-flat", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. \\begin{enumerate} \\item A presheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is called {\\it flat} if the functor $$ \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O}), \\quad \\mathcal{G} \\mapsto \\mathcal{G} \\otimes_{p, \\mathcal{O}} \\mathcal{F} $$ is exact. \\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of presheaves of rings is called {\\it flat} if $\\mathcal{O}'$ is flat as a presheaf of $\\mathcal{O}$-modules. \\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings and $\\mathcal{F}$ is a sheaf of $\\mathcal{O}$-modules, then we say $\\mathcal{F}$ is {\\it flat} if the functor $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}), \\quad \\mathcal{G} \\mapsto \\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} $$ is exact. \\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of sheaves of rings on a site is called {\\it flat} if $\\mathcal{O}'$ is flat as a sheaf of $\\mathcal{O}$-modules. \\end{enumerate}"}
+{"_id": "14291", "title": "sites-modules-definition-flat-morphism", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\longrightarrow (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. We say $(f, f^\\sharp)$ is {\\it flat} if the ring map $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$ is flat. We say a morphism of ringed sites is {\\it flat} if the associated morphism of ringed topoi is flat."}
+{"_id": "14292", "title": "sites-modules-definition-flat-module", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a morphism of ringed topoi. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. We say that $\\mathcal{F}$ is {\\it flat over $(\\Sh(\\mathcal{D}), \\mathcal{O}')$} if $\\mathcal{F}$ is flat as an $f^{-1}\\mathcal{O}'$-module."}
+{"_id": "14293", "title": "sites-modules-definition-invertible-sheaf", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. \\begin{enumerate} \\item A finite locally free $\\mathcal{O}$-module $\\mathcal{F}$ is said to have {\\it rank $r$} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ of $U$ such that $\\mathcal{F}|_{U_i}$ is isomorphic to $\\mathcal{O}_{U_i}^{\\oplus r}$ as an $\\mathcal{O}_{U_i}$-module. \\item An $\\mathcal{O}$-module $\\mathcal{L}$ is {\\it invertible} if the functor $$ \\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad \\mathcal{F}  \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{L} $$ is an equivalence. \\item The sheaf {\\it $\\mathcal{O}^*$} is the subsheaf of $\\mathcal{O}$ defined by the rule $$ U \\longmapsto \\mathcal{O}^*(U) = \\{f \\in \\mathcal{O}(U) \\mid \\exists g \\in \\mathcal{O}(U)\\text{ such that }fg = 1\\} $$ It is a sheaf of abelian groups with multiplication as the group law. \\end{enumerate}"}
+{"_id": "14294", "title": "sites-modules-definition-pic", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The {\\it Picard group} $\\Pic(\\mathcal{O})$ of the ringed site is the abelian group whose elements are isomorphism classes of invertible $\\mathcal{O}$-modules, with addition corresponding to tensor product."}
+{"_id": "14295", "title": "sites-modules-definition-derivation", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be an $\\mathcal{O}_2$-module. A {\\it $\\mathcal{O}_1$-derivation} or more precisely a {\\it $\\varphi$-derivation} into $\\mathcal{F}$ is a map $D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates the image of $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the {\\it Leibniz rule} $$ D(ab) = aD(b) + D(a)b $$ for all $a, b$ local sections of $\\mathcal{O}_2$ (wherever they are both defined). We denote $\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$ the set of $\\varphi$-derivations into $\\mathcal{F}$."}
+{"_id": "14296", "title": "sites-modules-definition-module-differentials", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. The {\\it module of differentials} of the ring map $\\varphi$ is the object representing the functor $\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$ which exists by Lemma \\ref{lemma-universal-module}. It is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal $\\varphi$-derivation} is denoted $\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$."}
+{"_id": "14297", "title": "sites-modules-definition-sheaf-differentials", "text": "Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and $Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi. Let $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi. In this situation \\begin{enumerate} \\item for a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules a {\\it $Y$-derivation} $D : \\mathcal{O} \\to \\mathcal{F}$ is just a $f^\\sharp$-derivation, and \\item the {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$} is the module of differentials of $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, see Definition \\ref{definition-module-differentials}. \\end{enumerate} Thus $\\Omega_{X/Y}$ comes equipped with a {\\it universal $Y$-derivation} $\\text{d}_{X/Y} : \\mathcal{O} \\longrightarrow \\Omega_{X/Y}$. We sometimes write $\\Omega_{X/Y} = \\Omega_f$."}
+{"_id": "14298", "title": "sites-modules-definition-differential-operators", "text": "Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $k \\geq 0$ be an integer. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules. A {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$} is an is an $\\mathcal{O}_1$-linear map such that for all local sections $g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $\\mathcal{O}_2$-linear map."}
+{"_id": "14299", "title": "sites-modules-definition-module-principal-parts", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules. The module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$ constructed in Lemma \\ref{lemma-module-principal-parts} is called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$."}
+{"_id": "14300", "title": "sites-modules-definition-naive-cotangent-complex", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $\\mathcal{C}$. The {\\it naive cotangent complex} $\\NL_{\\mathcal{B}/\\mathcal{A}}$ is the chain complex (\\ref{equation-naive-cotangent-complex}) $$ \\NL_{\\mathcal{B}/\\mathcal{A}} = \\left(\\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}} \\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right) $$ with $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and $\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}} \\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$ placed in degree $0$."}
+{"_id": "14301", "title": "sites-modules-definition-cotangent-complex-morphism-ringed-topoi", "text": "Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and $Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi. Let $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi. The {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$ of the given morphism of ringed topoi is $\\NL_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}$. We sometimes write $\\NL_{X/Y} = \\NL_{\\mathcal{O}_X/\\mathcal{O}_Y}$."}
+{"_id": "14302", "title": "sites-modules-definition-locally-ringed", "text": "A ringed site $(\\mathcal{C}, \\mathcal{O})$ is said to be {\\it locally ringed site} if (\\ref{equation-one-is-never-zero}) is an isomorphism, and the equivalent properties of Lemma \\ref{lemma-locally-ringed} are satisfied."}
+{"_id": "14303", "title": "sites-modules-definition-locally-ringed-topos", "text": "A ringed topos $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is said to be {\\it locally ringed} if the underlying ringed site $(\\mathcal{C}, \\mathcal{O})$ is locally ringed."}
+{"_id": "14304", "title": "sites-modules-definition-morphism-locally-ringed-topoi", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Assume $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ and $(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ are locally ringed topoi. We say that $(f, f^\\sharp)$ is a {\\it morphism of locally ringed topoi} if and only if the diagram of sheaves $$ \\xymatrix{ f^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] & \\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\ f^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} & \\mathcal{O}_\\mathcal{C} } $$ (see Lemma \\ref{lemma-locally-ringed-morphism}) is cartesian. If $(f, f^\\sharp)$ is a morphism of ringed sites, then we say that it is a {\\it morphism of locally ringed sites} if the associated morphism of ringed topoi is a morphism of locally ringed topoi."}
+{"_id": "14305", "title": "sites-modules-definition-locally-constant", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc. \\begin{enumerate} \\item We say $\\mathcal{F}$ is a {\\it constant sheaf} of sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc to a constant sheaf $\\underline{E}$ as in Section \\ref{section-constant}. \\item We say $\\mathcal{F}$ is {\\it locally constant} if for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf. \\item If $\\mathcal{F}$ is a sheaf of sets or groups, then we say $\\mathcal{F}$ is {\\it finite locally constant} if the constant values are finite sets or finite groups. \\end{enumerate}"}
+{"_id": "14378", "title": "derham-definition-hodge-filtration", "text": "Let $X \\to S$ be a morphism of schemes. The {\\it Hodge filtration} on $H^n_{dR}(X/S)$ is the filtration with terms $$ F^pH^n_{dR}(X/S) = \\Im\\left(H^n(X, \\sigma_{\\geq p}\\Omega^\\bullet_{X/S}) \\longrightarrow H^n_{dR}(X/S)\\right) $$ where $\\sigma_{\\geq p}\\Omega^\\bullet_{X/S}$ is as in Homology, Section \\ref{homology-section-truncations}."}
+{"_id": "14379", "title": "derham-definition-local-product", "text": "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective Cartier divisor. We say the {\\it de Rham complex of log poles is defined for $Y \\subset X$ over $S$} if for all $y \\in Y$ and local equation $f \\in \\mathcal{O}_{X, y}$ of $Y$ we have \\begin{enumerate} \\item $\\mathcal{O}_{X, y} \\to \\Omega_{X/S, y}$, $g \\mapsto g \\text{d}f$ is a split injection, and \\item $\\Omega^p_{X/S, y}$ is $f$-torsion free for all $p$. \\end{enumerate}"}
+{"_id": "14380", "title": "derham-definition-log-complex", "text": "Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. Then the complex $$ \\Omega^\\bullet_{X/S}(\\log Y) $$ constructed in Lemma \\ref{lemma-log-complex} is the {\\it de Rham complex of log poles for $Y \\subset X$ over $S$}."}
+{"_id": "14443", "title": "trace-definition-geometric-frobenius", "text": "Let $k$ be a finite field with $q = p^f$ elements. Let $X$ be a scheme over $k$. The {\\it geometric frobenius} of $X$ is the morphism $\\pi_X : X \\to X$ over $\\Spec(k)$ which equals $F_X^f$."}
+{"_id": "14444", "title": "trace-definition-arithmetic-frobenius", "text": "The {\\it arithmetic frobenius} is the map $\\text{frob}_k : \\bar k \\to \\bar k$, $x \\mapsto x^q$ of $G_k$."}
+{"_id": "14445", "title": "trace-definition-geometric-frobenius-on-stalk", "text": "If $x \\in X(k)$ is a rational point and $\\bar x : \\Spec(\\bar k) \\to X$ the geometric point lying over $x$, we let $\\pi_x : \\mathcal{F}_{\\bar x} \\to \\mathcal{F}_{\\bar x}$ denote the action by $\\text{frob}_k^{-1}$ and call it the {\\it geometric frobenius}\\footnote{This notation is not standard. This operator is denoted $F_x$ in \\cite{SGA4.5}. We will likely change this notation in the future.}"}
+{"_id": "14446", "title": "trace-definition-trace", "text": "The {\\it trace} of the endomorphism $a$ is the sum of the diagonal entries of a matrix representing it. This defines an additive map $\\text{Tr} : \\text{End}_\\Lambda(\\Lambda^{\\oplus m}) \\to \\Lambda^\\natural$."}
+{"_id": "14447", "title": "trace-definition-derived-functor", "text": "Let $F: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that $\\mathcal{A}$ has enough injectives. We define the {\\it total right derived functor of $F$} as the functor $RF: D^+(\\mathcal{A}) \\to D^+(\\mathcal{B})$ fitting into the diagram $$ \\xymatrix{ D^+(\\mathcal{A}) \\ar[r]^{RF} & D^+(\\mathcal{B}) \\\\ K^+(\\mathcal I) \\ar[u] \\ar[r]^F & K^+(\\mathcal{B}). \\ar[u] } $$ This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let $G: \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor and assume that $\\mathcal{A}$ has enough projectives. We define the {\\it total left derived functor of $G$} as the functor $LG: D^-(\\mathcal{A}) \\to D^-(\\mathcal{B})$ fitting into the diagram $$ \\xymatrix{ D^-(\\mathcal{A}) \\ar[r]^{LG} & D^-(\\mathcal{B}) \\\\ K^-(\\mathcal{P}) \\ar[u] \\ar[r]^G & K^-(\\mathcal{B}). \\ar[u] } $$ This is possible since the left vertical arrow is invertible by the previous lemma."}
+{"_id": "14448", "title": "trace-definition-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item Let $\\text{Fil}(\\mathcal{A})$ be the category of filtered objects $(A, F)$ of $\\mathcal{A}$, where $F$ is a filtration of the form $$ A \\supset \\ldots \\supset F^n A \\supset F^{n+1}A \\supset \\ldots \\supset 0. $$ This is an additive category. \\item We denote $\\text{Fil}^f(\\mathcal{A})$ the full subcategory of $\\text{Fil}(\\mathcal{A})$ whose objects $(A, F)$ have finite filtration. This is also an additive category. \\item An object $I \\in \\text{Fil}^f(\\mathcal{A})$ is called {\\it filtered injective} (respectively {\\it projective}) provided that $\\text{gr}^p(I) = \\text{gr}_F^p(I) = F^pI/F^{p+1}I$ is injective (resp. projective) in $\\mathcal{A}$ for all $p$. \\item The category of complexes $\\text{Comp}(\\text{Fil}^f(\\mathcal{A})) \\supset \\text{Comp}^+(\\text{Fil}^f(\\mathcal{A}))$ and its homotopy category $K(\\text{Fil}^f(\\mathcal{A})) \\supset K^+(\\text{Fil}^f(\\mathcal A))$ are defined as before. \\item A morphism $\\alpha : K^\\bullet \\to L^\\bullet$ of complexes in $\\text{Comp}(\\text{Fil}^f(\\mathcal{A}))$ is called a {\\it filtered quasi-isomorphism} provided that $$ \\text{gr}^p(\\alpha): \\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet) $$ is a quasi-isomorphism for all $p \\in \\mathbf{Z}$. \\item We define $DF(\\mathcal{A})$ (resp. $DF^+(\\mathcal{A})$) by inverting the filtered quasi-isomorphisms in $K(\\text{Fil}^f(\\mathcal{A}))$ (resp. $K^+(\\text{Fil}^f(\\mathcal{A}))$). \\end{enumerate}"}
+{"_id": "14449", "title": "trace-definition-filtered-derived-functors", "text": "Let $T: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that $\\mathcal{A}$ has enough injectives. Define $RT: DF^+(\\mathcal{A}) \\to D F^+(\\mathcal{B})$ to fit in the diagram $$ \\xymatrix{ DF^+(\\mathcal{A}) \\ar[r]^{RT} & DF^+(\\mathcal{B}) \\\\ K^+(\\mathcal{I}) \\ar[u] \\ar[r]^{T \\quad} & K^+(\\text{Fil}^f(\\mathcal{B})). \\ar[u]} $$ This is well-defined by the previous lemma. Let $G: \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor and assume that $\\mathcal{A}$ has enough projectives. Define $LG: DF^+(\\mathcal{A}) \\to DF^+(\\mathcal{B})$ to fit in the diagram $$ \\xymatrix{ DF^-(\\mathcal{A}) \\ar[r]^{LG} & DF^-(\\mathcal{B}) \\\\ K^-(\\mathcal{P}) \\ar[u] \\ar[r]^{G \\quad} & K^-(\\text{Fil}^f(\\mathcal{B})). \\ar[u]} $$ Again, this is well-defined by the previous lemma. The functors $RT$, resp.\\ $LG$, are called the {\\it filtered derived functor} of $T$, resp.\\ $G$."}
+{"_id": "14450", "title": "trace-definition-perfect", "text": "We denote by $K_{perf}(\\Lambda)$ the category whose objects are bounded complexes of finite projective $\\Lambda$-modules, and whose morphisms are morphisms of complexes up to homotopy. The functor $K_{perf}(\\Lambda)\\to D(\\Lambda)$ is fully faithful (Derived Categories, Lemma \\ref{derived-lemma-morphisms-from-projective-complex}). Denote $D_{perf}(\\Lambda)$ its essential image. An object of $D(\\Lambda)$ is called {\\it perfect} if it is in $D_{perf}(\\Lambda)$."}
+{"_id": "14451", "title": "trace-definition-finite-tor-dimension", "text": "Let $\\Lambda$ be a (possibly noncommutative) ring. An object $K\\in D(\\Lambda)$ has {\\it finite $\\text{Tor}$-dimension} if there exist $a, b \\in \\mathbf{Z}$ such that for any right $\\Lambda$-module $N$, we have $H^i(N \\otimes_{\\Lambda}^\\mathbf{L} K) = 0$ for all $i \\not \\in [a, b]$."}
+{"_id": "14452", "title": "trace-definition-global-lefschetz-number", "text": "Let $\\Lambda$ be a finite ring, $X$ a projective curve over a finite field $k$ and $K \\in D_{ctf}(X, \\Lambda)$ (for instance $K = \\underline\\Lambda$). There is a canonical map $c_K : \\pi_X^{-1}K \\to K$, and its base change $c_K|_{X_{\\bar k}}$ induces an action denoted $\\pi_X^*$ on the perfect complex $R\\Gamma(X_{\\bar k}, K|_{X_{\\bar k}})$. The {\\it global Lefschetz number} of $K$ is the trace $\\text{Tr}(\\pi_X^* |_{R\\Gamma(X_{\\bar k}, K)})$ of that action. It is an element of $\\Lambda^\\natural$."}
+{"_id": "14453", "title": "trace-definition-local-lefschetz-number", "text": "With $\\Lambda, X, k, K$ as in Definition \\ref{definition-global-lefschetz-number}. Since $K\\in D_{ctf}(X, \\Lambda)$, for any geometric point $\\bar x$ of $X$, the complex $K_{\\bar x}$ is a perfect complex (in $D_{perf}(\\Lambda)$). As we have seen in Section \\ref{section-frobenii}, the Frobenius $\\pi_X$ acts on $K_{\\bar x}$. The {\\it local Lefschetz number} of $K$ is the sum $$ \\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_X |_{K_{\\overline{x}}}) $$ which is again an element of $\\Lambda^\\natural$."}
+{"_id": "14454", "title": "trace-definition-trace-G", "text": "Let $f : P\\to P$ be an endomorphism of a finite projective $\\Lambda[G]$-module $P$. We define $$ \\text{Tr}_{\\Lambda}^G(f; P) := \\varepsilon\\left(\\text{Tr}_{\\Lambda[G]}(f; P)\\right) $$ to be the {\\it $G$-trace of $f$ on $P$}."}
+{"_id": "14455", "title": "trace-definition-l-adic-sheaf", "text": "Let $X$ be a Noetherian scheme. A {\\it $\\mathbf{Z}_\\ell$-sheaf} on $X$, or simply an {\\it $\\ell$-adic sheaf} $\\mathcal{F}$ is an inverse system $\\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ where \\begin{enumerate} \\item $\\mathcal{F}_n$ is a constructible $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-module on $X_\\etale$, and \\item the transition maps $\\mathcal{F}_{n+1}\\to \\mathcal{F}_n$ induce isomorphisms $\\mathcal{F}_{n+1} \\otimes_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}} \\mathbf{Z}/\\ell^n\\mathbf{Z} \\cong \\mathcal{F}_n$. \\end{enumerate} We say that $\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}_n$ is locally constant. A {\\it morphism} of such is merely a morphism of inverse systems."}
+{"_id": "14456", "title": "trace-definition-torsion-l-adic-sheaf", "text": "A $\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}$ is {\\it torsion} if $\\ell^n : \\mathcal{F} \\to \\mathcal{F}$ is the zero map for some $n$. The abelian category of $\\mathbf{Q}_\\ell$-sheaves on $X$ is the quotient of the abelian category of $\\mathbf{Z}_\\ell$-sheaves by the Serre subcategory of torsion sheaves. In other words, its objects are $\\mathbf{Z}_\\ell$-sheaves on $X$, and if $\\mathcal{F}, \\mathcal{G}$ are two such, then $$ \\Hom_{\\mathbf{Q}_\\ell} \\left(\\mathcal{F}, \\mathcal{G} \\right) = \\Hom_{\\mathbf{Z}_\\ell} \\left(\\mathcal{F}, \\mathcal{G}\\right) \\otimes_{\\mathbf{Z}_\\ell} \\mathbf{Q}_\\ell. $$ We denote by $\\mathcal{F} \\mapsto \\mathcal{F} \\otimes \\mathbf{Q}_\\ell$ the quotient functor (right adjoint to the inclusion). If $\\mathcal{F} = \\mathcal{F}' \\otimes \\mathbf{Q}_\\ell$ where $\\mathcal{F}'$ is a $\\mathbf{Z}_\\ell$-sheaf and $\\bar x$ is a geometric point, then the {\\it stalk} of $\\mathcal{F}$ at $\\bar x$ is $\\mathcal{F}_{\\bar x} = \\mathcal{F}'_{\\bar x} \\otimes \\mathbf{Q}_\\ell$."}
+{"_id": "14457", "title": "trace-definition-cohomology-l-adic", "text": "If $X$ is a separated scheme of finite type over an algebraically closed field $k$ and $\\mathcal{F} = \\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ is a $\\mathbf{Z}_\\ell$-sheaf on $X$, then we define $$ H^i(X, \\mathcal{F}) := \\lim_n H^i(X, \\mathcal{F}_n) \\quad\\text{and}\\quad H_c^i(X, \\mathcal{F}) := \\lim_n H_c^i(X, \\mathcal{F}_n). $$ If $\\mathcal{F} = \\mathcal{F}'\\otimes \\mathbf{Q}_\\ell$ for a $\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}'$ then we set $$ H_c^i(X , \\mathcal{F}) := H_c^i(X, \\mathcal{F}')\\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell. $$ We call these the {\\it $\\ell$-adic cohomology} of $X$ with coefficients $\\mathcal{F}$."}
+{"_id": "14458", "title": "trace-definition-L-function-finite-ring", "text": "Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a constructible flat $\\Lambda$-module on $X_\\etale$. Then we set $$ L(X, \\mathcal{F}) := \\prod\\nolimits_{x\\in |X|} \\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1} \\in \\Lambda [[ T ]] $$ where $|X|$ is the set of closed points of $X$, $\\deg x = [\\kappa(x): k]$ and $\\bar x$ is a geometric point lying over $x$. This definition clearly generalizes to the case where $\\mathcal{F}$ is replaced by a $K \\in D_{ctf}(X, \\Lambda)$. We call this the {\\it $L$-function of $\\mathcal{F}$}."}
+{"_id": "14459", "title": "trace-definition-L-function-l-adic", "text": "Now assume that $\\mathcal{F}$ is a $\\mathbf{Q}_\\ell$-sheaf on $X$. In this case we define $$ L(X, \\mathcal{F}) := \\prod\\nolimits_{x \\in |X|} \\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1} \\in \\mathbf{Q}_\\ell[[T]]. $$ Note that this product converges since there are finitely many points of a given degree. We call this the {\\it $L$-function of $\\mathcal{F}$}."}
+{"_id": "14460", "title": "trace-definition-open", "text": "A subgroup of the form $\\text{Stab}(\\overline y\\in F_{\\overline{x}}(Y))\\subset \\pi_1(X, \\overline{x})$ is called {\\it open}."}
+{"_id": "14461", "title": "trace-definition-unramified", "text": "An {\\it unramified cusp form on $\\text{GL}_2(\\mathbf{A})$ with values in $\\Lambda$}\\footnote{This is likely nonstandard notation.} is a function $$ f : \\text{GL}_2(\\mathbf{A}) \\to \\Lambda $$ such that \\begin{enumerate} \\item $f(x\\gamma) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all $\\gamma\\in \\text{GL}_2(K)$ \\item $f(ux) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all $u\\in \\text{GL}_2(O)$ \\item for all $x\\in \\text{GL}_2(\\mathbf{A})$, $$ \\int_{\\mathbf{A} \\mod K} f \\left(x \\left( \\begin{matrix} 1 & z \\\\ 0 & 1 \\end{matrix} \\right) \\right) dz = 0 $$ see \\cite[Section 4.1]{dJ-conjecture} for an explanation of how to make sense out of this for a general ring $\\Lambda$ in which $p$ is invertible. \\end{enumerate}"}
+{"_id": "14559", "title": "sheaves-definition-presheaf", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which assigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and to each inclusion $V \\subset U$ a map $\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that $\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and whenever $W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves of sets on $X$} is a rule which assigns to each open $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ compatible with restriction maps, i.e., whenever $V \\subset U \\subset X$ are open the diagram $$ \\xymatrix{ \\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} & \\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\ \\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V) } $$ commutes. \\item The category of presheaves of sets on $X$ will be denoted $\\textit{PSh}(X)$. \\end{enumerate}"}
+{"_id": "14560", "title": "sheaves-definition-constant-presheaf", "text": "Let $X$ be a topological space. Let $A$ be a set. The {\\it constant presheaf with value $A$} is the presheaf that assigns the set $A$ to every open $U \\subset X$, and such that all restriction mappings are $\\text{id}_A$."}
+{"_id": "14561", "title": "sheaves-definition-abelian-presheaves", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item A {\\it presheaf of abelian groups on $X$} or an {\\it abelian presheaf over $X$} is a presheaf of sets $\\mathcal{F}$ such that for each open $U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with the structure of an abelian group, and such that all restriction maps $\\rho^U_V$ are homomorphisms of abelian groups, see Lemma \\ref{lemma-abelian-presheaves} above. \\item A {\\it morphism of abelian presheaves over $X$} $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves of sets which induces a homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ for every open $U \\subset X$. \\item The category of presheaves of abelian groups on $X$ is denoted $\\textit{PAb}(X)$. \\end{enumerate}"}
+{"_id": "14562", "title": "sheaves-definition-presheaf-values-in-category", "text": "Let $X$ be a topological space. Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item A {\\it presheaf $\\mathcal{F}$ on $X$ with values in $\\mathcal{C}$} is given by a rule which assigns to every open $U \\subset X$ an object $\\mathcal{F}(U)$ of $\\mathcal{C}$ and to each inclusion $V \\subset U$ a morphism $\\rho_V^U : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ in $\\mathcal{C}$ such that whenever $W \\subset V \\subset U$ we have $\\rho_W^U = \\rho_W^V \\circ \\rho_V^U$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves with value in $\\mathcal{C}$} is given by a morphism $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ in $\\mathcal{C}$ compatible with restriction morphisms. \\end{enumerate}"}
+{"_id": "14563", "title": "sheaves-definition-underlying-presheaf-sets", "text": "Let $X$ be a topological space. Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a faithful functor. Let $\\mathcal{F}$ be a presheaf on $X$ with values in $\\mathcal{C}$. The presheaf of sets $U \\mapsto F(\\mathcal{F}(U))$ is called the {\\it underlying presheaf of sets of $\\mathcal{F}$}."}
+{"_id": "14564", "title": "sheaves-definition-presheaf-modules", "text": "Let $X$ be a topological space, and let $\\mathcal{O}$ be a presheaf of rings on $X$. \\begin{enumerate} \\item A {\\it presheaf of $\\mathcal{O}$-modules} is given by an abelian presheaf $\\mathcal{F}$ together with a map of presheaves of sets $$ \\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F} $$ such that for every open $U \\subset X$ the map $\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ defines the structure of an $\\mathcal{O}(U)$-module structure on the abelian group $\\mathcal{F}(U)$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that the diagram $$ \\xymatrix{ \\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} & \\mathcal{F} \\ar[d]^{\\varphi} \\\\ \\mathcal{O} \\times \\mathcal{G} \\ar[r] & \\mathcal{G} } $$ commutes. \\item The set of $\\mathcal{O}$-module morphisms as above is denoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$. \\item The category of presheaves of $\\mathcal{O}$-modules is denoted $\\textit{PMod}(\\mathcal{O})$. \\end{enumerate}"}
+{"_id": "14565", "title": "sheaves-definition-sheaf", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf of sets which satisfies the following additional property: Given any open covering $U = \\bigcup_{i \\in I} U_i$ and any collection of sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that $\\forall i, j\\in I$ $$ s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j} $$ there exists a unique section $s \\in \\mathcal{F}(U)$ such that $s_i = s|_{U_i}$ for all $i \\in I$. \\item A {\\it morphism of sheaves of sets} is simply a morphism of presheaves of sets. \\item The category of sheaves of sets on $X$ is denoted $\\Sh(X)$. \\end{enumerate}"}
+{"_id": "14566", "title": "sheaves-definition-constant-sheaf", "text": "Let $X$ be a topological space. Let $A$ be a set. The {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or $\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$ the set of all locally constant maps $U \\to A$ with restriction mappings given by restrictions of functions."}
+{"_id": "14567", "title": "sheaves-definition-abelian-sheaf", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item An {\\it abelian sheaf on $X$} or {\\it sheaf of abelian groups on $X$} is an abelian presheaf on $X$ such that the underlying presheaf of sets is a sheaf. \\item The category of sheaves of abelian groups is denoted $\\textit{Ab}(X)$. \\end{enumerate}"}
+{"_id": "14568", "title": "sheaves-definition-sheaf-values-in-category", "text": "Let $X$ be a topological space. Let $\\mathcal{C}$ be a category with products. A presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$ on $X$ is a {\\it sheaf} if for every open covering the diagram $$ \\xymatrix{ \\mathcal{F}(U) \\ar[r] & \\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\prod\\nolimits_{(i_0, i_1) \\in I \\times I} \\mathcal{F}(U_{i_0} \\cap U_{i_1}) } $$ is an equalizer diagram in the category $\\mathcal{C}$."}
+{"_id": "14569", "title": "sheaves-definition-sheaf-modules", "text": "Let $X$ be a topological space. Let $\\mathcal{O}$ be a sheaf of rings on $X$. \\begin{enumerate} \\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$, see Definition \\ref{definition-presheaf-modules}, such that the underlying presheaf of abelian groups $\\mathcal{F}$ is a sheaf. \\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules} is a morphism of presheaves of $\\mathcal{O}$-modules. \\item Given sheaves of $\\mathcal{O}$-modules $\\mathcal{F}$ and $\\mathcal{G}$ we denote $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$ the set of morphism of sheaves of $\\mathcal{O}$-modules. \\item The category of sheaves of $\\mathcal{O}$-modules is denoted $\\textit{Mod}(\\mathcal{O})$. \\end{enumerate}"}
+{"_id": "14570", "title": "sheaves-definition-separated", "text": "Let $X$ be a topological space. A presheaf of sets $\\mathcal{F}$ on $X$ is {\\it separated} if for every open $U \\subset X$ the map $\\mathcal{F}(U) \\to \\prod_{x \\in U} \\mathcal{F}_x$ is injective."}
+{"_id": "14571", "title": "sheaves-definition-algebraic-structure", "text": "A {\\it type of algebraic structure} is given by a category $\\mathcal{C}$ and a functor $F : \\mathcal{C} \\to \\textit{Sets}$ with the following properties \\begin{enumerate} \\item $F$ is faithful, \\item $\\mathcal{C}$ has limits and $F$ commutes with limits, \\item $\\mathcal{C}$ has filtered colimits and $F$ commutes with them, and \\item $F$ reflects isomorphisms. \\end{enumerate}"}
+{"_id": "14572", "title": "sheaves-definition-injective-surjective", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item A presheaf $\\mathcal{F}$ is called a {\\it subpresheaf} of a presheaf $\\mathcal{G}$ if $\\mathcal{F}(U) \\subset \\mathcal{G}(U)$ for all open $U \\subset X$ such that the restriction maps of $\\mathcal{G}$ induce the restriction maps of $\\mathcal{F}$. If $\\mathcal{F}$ and $\\mathcal{G}$ are sheaves, then $\\mathcal{F}$ is called a {\\it subsheaf} of $\\mathcal{G}$. We sometimes indicate this by the notation $\\mathcal{F} \\subset \\mathcal{G}$. \\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ on $X$ is called {\\it injective} if and only if $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$. \\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ on $X$ is called {\\it surjective} if and only if $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective for all $U$ open in $X$. \\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ on $X$ is called {\\it injective} if and only if $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$. \\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ on $X$ is called {\\it surjective} if and only if for every open $U$ of $X$ and every section $s$ of $\\mathcal{G}(U)$ there exists an open covering $U = \\bigcup U_i$ such that $s|_{U_i}$ is in the image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$ for all $i$. \\end{enumerate}"}
+{"_id": "14573", "title": "sheaves-definition-f-map", "text": "Let $f : X \\to Y$ be a continuous map. Let $\\mathcal{F}$ be a sheaf of sets on $X$ and let $\\mathcal{G}$ be a sheaf of sets on $Y$. An {\\it $f$-map $\\xi : \\mathcal{G} \\to \\mathcal{F}$} is a collection of maps $\\xi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}(V))$ indexed by open subsets $V \\subset Y$ such that $$ \\xymatrix{ \\mathcal{G}(V) \\ar[r]_{\\xi_V} \\ar[d]_{\\text{restriction of }\\mathcal{G}} & \\mathcal{F}(f^{-1}V) \\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\ \\mathcal{G}(V') \\ar[r]^{\\xi_{V'}} & \\mathcal{F}(f^{-1}V') } $$ commutes for all $V' \\subset V \\subset Y$ open."}
+{"_id": "14574", "title": "sheaves-definition-composition-f-maps", "text": "Suppose that $f : X \\to Y$ and $g : Y \\to Z$ are continuous maps of topological spaces. Suppose that $\\mathcal{F}$ is a sheaf on $X$, $\\mathcal{G}$ is a sheaf on $Y$, and $\\mathcal{H}$ is a sheaf on $Z$. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map. Let $\\psi : \\mathcal{H} \\to \\mathcal{G}$ be an $g$-map. The {\\it composition of $\\varphi$ and $\\psi$} is the $(g \\circ f)$-map $\\varphi \\circ \\psi$ defined by the commutativity of the diagrams $$ \\xymatrix{ \\mathcal{H}(W) \\ar[rr]_{(\\varphi \\circ \\psi)_W} \\ar[rd]_{\\psi_W} & & \\mathcal{F}(f^{-1}g^{-1}W) \\\\ & \\mathcal{G}(g^{-1}W) \\ar[ru]_{\\varphi_{g^{-1}W}} } $$"}
+{"_id": "14575", "title": "sheaves-definition-ringed-space", "text": "A {\\it ringed space} is a pair $(X, \\mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\\mathcal{O}_X$ on $X$. A {\\it morphism of ringed spaces} $(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is a pair consisting of a continuous map $f : X \\to Y$ and an $f$-map of sheaves of rings $f^\\sharp : \\mathcal{O}_Y \\to \\mathcal{O}_X$."}
+{"_id": "14576", "title": "sheaves-definition-composition-maps-ringed-spaces", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and $(g, g^\\sharp) : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$ be morphisms of ringed spaces. Then we define the {\\it composition of morphisms of ringed spaces} by the rule $$ (g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp). $$ Here we use composition of $f$-maps defined in Definition \\ref{definition-composition-f-maps}."}
+{"_id": "14577", "title": "sheaves-definition-pushforward", "text": "Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. We define the {\\it pushforward} of $\\mathcal{F}$ as the sheaf of $\\mathcal{O}_Y$-modules which as a sheaf of abelian groups equals $f_*\\mathcal{F}$ and with module structure given by the restriction via $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ of the module structure given in Lemma \\ref{lemma-pushforward-module}. \\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules. We define the {\\it pullback} $f^*\\mathcal{G}$ to be the sheaf of $\\mathcal{O}_X$-modules defined by the formula $$ f^*\\mathcal{G} = \\mathcal{O}_X \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G} $$ where the ring map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$ is the map corresponding to $f^\\sharp$, and where the  module structure is given by Lemma \\ref{lemma-pullback-module}. \\end{enumerate}"}
+{"_id": "14578", "title": "sheaves-definition-skyscraper-sheaf", "text": "Let $X$ be a topological space. \\begin{enumerate} \\item Let $x \\in X$ be a point. Denote $i_x : \\{x\\} \\to X$ the inclusion map. Let $A$ be a set and think of $A$ as a sheaf on the one point space $\\{x\\}$. We call $i_{x, *}A$ the {\\it skyscraper sheaf at $x$ with value $A$}. \\item If in (1) above $A$ is an abelian group then we think of $i_{x, *}A$ as a sheaf of abelian groups on $X$. \\item If in (1) above $A$ is an algebraic structure then we think of $i_{x, *}A$ as a sheaf of algebraic structures. \\item If $(X, \\mathcal{O}_X)$ is a ringed space, then we think of $i_x : \\{x\\} \\to X$ as a morphism of ringed spaces $(\\{x\\}, \\mathcal{O}_{X, x}) \\to (X, \\mathcal{O}_X)$ and if $A$ is a $\\mathcal{O}_{X, x}$-module, then we think of $i_{x, *}A$ as a sheaf of $\\mathcal{O}_X$-modules. \\item We say a sheaf of sets $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there exists an point $x$ of $X$ and a set $A$ such that $\\mathcal{F} \\cong i_{x, *}A$. \\item We say a sheaf of abelian groups $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there exists an point $x$ of $X$ and an abelian group $A$ such that $\\mathcal{F} \\cong i_{x, *}A$ as sheaves of abelian groups. \\item We say a sheaf of algebraic structures $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there exists an point $x$ of $X$ and an algebraic structure $A$ such that $\\mathcal{F} \\cong i_{x, *}A$ as sheaves of algebraic structures. \\item If $(X, \\mathcal{O}_X)$ is a ringed space and $\\mathcal{F}$ is a sheaf of $\\mathcal{O}_X$-modules, then we say $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there exists a point $x \\in X$ and a $\\mathcal{O}_{X, x}$-module $A$ such that $\\mathcal{F} \\cong i_{x, *}A$ as sheaves of $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "14579", "title": "sheaves-definition-presheaf-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. \\begin{enumerate} \\item A {\\it presheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$} is a rule which assigns to each $U \\in \\mathcal{B}$ a set $\\mathcal{F}(U)$ and to each inclusion $V \\subset U$ of elements of $\\mathcal{B}$ a map $\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that $\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$ whenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have $\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves of sets on $\\mathcal{B}$} is a rule which assigns to each element $U \\in \\mathcal{B}$ a map of sets $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ compatible with restriction maps. \\end{enumerate}"}
+{"_id": "14580", "title": "sheaves-definition-sheaf-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. \\begin{enumerate} \\item A {\\it sheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$} is a presheaf of sets on $\\mathcal{B}$ which satisfies the following additional property: Given any $U \\in \\mathcal{B}$, and any covering $U = \\bigcup_{i \\in I} U_i$ with $U_i \\in \\mathcal{B}$, and any coverings $U_i \\cap U_j = \\bigcup_{k \\in I_{ij}} U_{ijk}$ with $U_{ijk} \\in \\mathcal{B}$ the sheaf condition holds: \\begin{itemize} \\item[(**)] For any collection of sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that $\\forall i, j\\in I$, $\\forall k\\in I_{ij}$ $$ s_i|_{U_{ijk}} = s_j|_{U_{ijk}} $$ there exists a unique section $s \\in \\mathcal{F}(U)$ such that $s_i = s|_{U_i}$ for all $i \\in I$. \\end{itemize} \\item A {\\it morphism of sheaves of sets on $\\mathcal{B}$} is simply a morphism of presheaves of sets. \\end{enumerate}"}
+{"_id": "14581", "title": "sheaves-definition-sheaf-structures-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $(\\mathcal{C}, F)$ be a type of algebraic structure. \\begin{enumerate} \\item A {\\it presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$ on $\\mathcal{B}$} is a rule which assigns to each $U \\in \\mathcal{B}$ an object $\\mathcal{F}(U)$ of $\\mathcal{C}$ and to each inclusion $V \\subset U$ of elements of $\\mathcal{B}$ a morphism $\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ in $\\mathcal{C}$ such that $\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$ and whenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have $\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves with values in $\\mathcal{C}$ on $\\mathcal{B}$} is a rule which assigns to each element $U \\in \\mathcal{B}$ a morphism of algebraic structures $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$ compatible with restriction maps. \\item Given a presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$ on $\\mathcal{B}$ we say that $U \\mapsto F(\\mathcal{F}(U))$ is the underlying presheaf of sets. \\item A {\\it sheaf $\\mathcal{F}$ with values in $\\mathcal{C}$ on $\\mathcal{B}$} is a presheaf with values in $\\mathcal{C}$ on $\\mathcal{B}$ whose underlying presheaf of sets is a sheaf. \\end{enumerate}"}
+{"_id": "14582", "title": "sheaves-definition-sheaf-modules-basis", "text": "Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the topology on $X$. Let $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{B}$. \\begin{enumerate} \\item A {\\it presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $\\mathcal{B}$} is a presheaf of abelian groups on $\\mathcal{B}$ together with a morphism of presheaves of sets $\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$ such that for all $U \\in \\mathcal{B}$ the map $\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$ turns the group $\\mathcal{F}(U)$ into an $\\mathcal{O}(U)$-module. \\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of presheaves of $\\mathcal{O}$-modules on $\\mathcal{B}$} is a morphism of abelian presheaves on $\\mathcal{B}$ which induces an $\\mathcal{O}(U)$-module homomorphism $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ for every $U \\in \\mathcal{B}$. \\item Suppose that $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{B}$. A {\\it sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules on $\\mathcal{B}$} is a presheaf of $\\mathcal{O}$-modules on $\\mathcal{B}$ whose underlying presheaf of abelian groups is a sheaf. \\end{enumerate}"}
+{"_id": "14583", "title": "sheaves-definition-restriction", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. \\begin{enumerate} \\item Let $\\mathcal{G}$ be a presheaf of sets, abelian groups or algebraic structures on $X$. The presheaf $j_p\\mathcal{G}$ described in Lemma \\ref{lemma-j-pullback} is called the {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$. \\item Let $\\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or algebraic structures on $X$. The sheaf $j^{-1}\\mathcal{G}$ is called the {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$. \\item If $(X, \\mathcal{O})$ is a ringed space, then the pair $(U, \\mathcal{O}|_U)$ is called the {\\it open subspace of $(X, \\mathcal{O})$ associated to $U$}. \\item If $\\mathcal{G}$ is a presheaf of $\\mathcal{O}$-modules then $\\mathcal{G}|_U$ together with the multiplication map $\\mathcal{O}|_U \\times \\mathcal{G}|_U \\to \\mathcal{G}|_U$ (see Lemma \\ref{lemma-pullback-module}) is called the {\\it restriction of $\\mathcal{G}$ to $U$}. \\end{enumerate}"}
+{"_id": "14584", "title": "sheaves-definition-j-shriek", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a presheaf of sets on $U$. We define the {\\it extension of $\\mathcal{F}$ by the empty set $j_{p!}\\mathcal{F}$} to be the presheaf of sets on $X$ defined by the rule $$ j_{p!}\\mathcal{F}(V) = \\left\\{ \\begin{matrix} \\emptyset & \\text{if} & V \\not \\subset U \\\\ \\mathcal{F}(V) & \\text{if} & V \\subset U \\end{matrix} \\right. $$ with obvious restriction mappings. \\item Let $\\mathcal{F}$ be a sheaf of sets on $U$. We define the {\\it extension of $\\mathcal{F}$ by the empty set $j_!\\mathcal{F}$} to be the sheafification of the presheaf $j_{p!}\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "14585", "title": "sheaves-definition-j-shriek-structures", "text": "Let $X$ be a topological space. Let $j : U \\to X$ be the inclusion of an open subset. \\begin{enumerate} \\item Let $\\mathcal{F}$ be an abelian presheaf on $U$. We define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $0$} to be the abelian presheaf on $X$ defined by the rule $$ j_{p!}\\mathcal{F}(V) = \\left\\{ \\begin{matrix} 0 & \\text{if} & V \\not \\subset U \\\\ \\mathcal{F}(V) & \\text{if} & V \\subset U \\end{matrix} \\right. $$ with obvious restriction mappings. \\item Let $\\mathcal{F}$ be an abelian sheaf on $U$. We define the {\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $0$} to be the sheafification of the abelian presheaf $j_{p!}\\mathcal{F}$. \\item Let $\\mathcal{C}$ be a category having an initial object $e$. Let $\\mathcal{F}$ be a presheaf on $U$ with values in $\\mathcal{C}$. We define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $e$} to be the presheaf on $X$ with values in $\\mathcal{C}$ defined by the rule $$ j_{p!}\\mathcal{F}(V) = \\left\\{ \\begin{matrix} e & \\text{if} & V \\not \\subset U \\\\ \\mathcal{F}(V) & \\text{if} & V \\subset U \\end{matrix} \\right. $$ with obvious restriction mappings. \\item Let $(\\mathcal{C}, F)$ be a type of algebraic structure such that $\\mathcal{C}$ has an initial object $e$. Let $\\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the {\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $e$} to be the sheafification of the presheaf $j_{p!}\\mathcal{F}$ defined above. \\item Let $\\mathcal{O}$ be a presheaf of rings on $X$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}|_U$-modules. In this case we define the {\\it extension by $0$} to be the presheaf of $\\mathcal{O}$-modules which is equal to $j_{p!}\\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\\mathcal{O} \\times j_{p!}\\mathcal{F} \\to j_{p!}\\mathcal{F}$. \\item Let $\\mathcal{O}$ be a sheaf of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}|_U$-modules. In this case we define the {\\it extension by $0$} to be the $\\mathcal{O}$-module which is equal to $j_!\\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\\mathcal{O} \\times j_!\\mathcal{F} \\to j_!\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "14757", "title": "descent-definition-descent-datum-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\{f_i : S_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$. \\begin{enumerate} \\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$ for quasi-coherent sheaves} with respect to the given family is given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $S_i$ for each $i \\in I$, an isomorphism of quasi-coherent $\\mathcal{O}_{S_i \\times_S S_j}$-modules $\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$ for each pair $(i, j) \\in I^2$ such that for every triple of indices $(i, j, k) \\in I^3$ the diagram $$ \\xymatrix{ \\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}} \\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & & \\text{pr}_2^*\\mathcal{F}_k \\\\ & \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} & } $$ of $\\mathcal{O}_{S_i \\times_S S_j \\times_S S_k}$-modules commutes. This is called the {\\it cocycle condition}. \\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to (\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given by a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of $\\mathcal{O}_{S_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$ such that all the diagrams $$ \\xymatrix{ \\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i} & \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\ \\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} & \\text{pr}_1^*\\mathcal{F}'_j \\\\ } $$ commute. \\end{enumerate}"}
+{"_id": "14758", "title": "descent-definition-descent-datum-effective-quasi-coherent", "text": "Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module. We call the unique descent on $\\mathcal{F}$ datum with respect to the covering $\\{S \\to S\\}$ the {\\it trivial descent datum}. \\item The pullback of the trivial descent datum to $\\{S_i \\to S\\}$ is called the {\\it canonical descent datum}. Notation: $(\\mathcal{F}|_{S_i}, can)$. \\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$ for quasi-coherent sheaves with respect to the given covering is said to be {\\it effective} if there exists a quasi-coherent sheaf $\\mathcal{F}$ on $S$ such that $(\\mathcal{F}_i, \\varphi_{ij})$ is isomorphic to $(\\mathcal{F}|_{S_i}, can)$. \\end{enumerate}"}
+{"_id": "14759", "title": "descent-definition-descent-datum-modules", "text": "Let $R \\to A$ be a ring map. \\begin{enumerate} \\item A {\\it descent datum $(N, \\varphi)$ for modules with respect to $R \\to A$} is given by an $A$-module $N$ and an isomorphism of $A \\otimes_R A$-modules $$ \\varphi : N \\otimes_R A \\to A \\otimes_R N $$ such that the {\\it cocycle condition} holds: the diagram of $A \\otimes_R A \\otimes_R A$-module maps $$ \\xymatrix{ N \\otimes_R A \\otimes_R A \\ar[rr]_{\\varphi_{02}} \\ar[rd]_{\\varphi_{01}} & & A \\otimes_R A \\otimes_R N \\\\ & A \\otimes_R N \\otimes_R A \\ar[ru]_{\\varphi_{12}} & } $$ commutes (see below for notation). \\item A {\\it morphism $(N, \\varphi) \\to (N', \\varphi')$ of descent data} is a morphism of $A$-modules $\\psi : N \\to N'$ such that the diagram $$ \\xymatrix{ N \\otimes_R A \\ar[r]_\\varphi \\ar[d]_{\\psi \\otimes \\text{id}_A} & A \\otimes_R N \\ar[d]^{\\text{id}_A \\otimes \\psi} \\\\ N' \\otimes_R A \\ar[r]^{\\varphi'} & A \\otimes_R N' } $$ is commutative. \\end{enumerate}"}
+{"_id": "14760", "title": "descent-definition-descent-datum-effective-module", "text": "Let $R \\to A$ be a ring map. We say a descent datum $(N, \\varphi)$ is {\\it effective} if there exists an $R$-module $M$ and an isomorphism of descent data from $(M \\otimes_R A, can)$ to $(N, \\varphi)$."}
+{"_id": "14761", "title": "descent-definition-split-equalizer", "text": "A {\\it split equalizer} is a diagram (\\ref{equation-equalizer}) with $g_1 \\circ f = g_2 \\circ f$ for which there exist auxiliary morphisms $h : B \\to A$ and $i : C \\to B$ such that \\begin{equation} \\label{equation-split-equalizer-conditions} h \\circ f = 1_A, \\quad f \\circ h = i \\circ g_1, \\quad i \\circ g_2 = 1_B. \\end{equation}"}
+{"_id": "14762", "title": "descent-definition-universally-injective", "text": "A ring map $f: R \\to S$ is {\\it universally injective} if it is universally injective as a morphism in $\\text{Mod}_R$."}
+{"_id": "14763", "title": "descent-definition-C", "text": "Let $R$ be a ring. Define the contravariant functor {\\it $C$} $ : \\text{Mod}_R \\to \\text{Mod}_R$ by setting $$ C(M) = \\Hom_{\\textit{Ab}}(M, \\mathbf{Q}/\\mathbf{Z}), $$ with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$."}
+{"_id": "14764", "title": "descent-definition-effective-descent", "text": "The functor $f^*: \\text{Mod}_R \\to DD_{S/R}$ is called {\\it base extension along $f$}. We say that $f$ is a {\\it descent morphism for modules} if $f^*$ is fully faithful. We say that $f$ is an {\\it effective descent morphism for modules} if $f^*$ is an equivalence of categories."}
+{"_id": "14765", "title": "descent-definition-pushforward", "text": "Define the functor {\\it $f_*$} $: DD_{S/R} \\to \\text{Mod}_R$ by taking $f_*(M, \\theta)$ to be the $R$-submodule of $M$ for which the diagram \\begin{equation} \\label{equation-equalizer-f} \\xymatrix@C=8pc{f_*(M,\\theta) \\ar[r] & M \\ar@<1ex>^{\\theta \\circ (1_M \\otimes  \\delta_0^1)}[r] \\ar@<-1ex>_{1_M \\otimes \\delta_1^1}[r] &  M \\otimes_{S, \\delta_1^1} S_2  } \\end{equation} is an equalizer."}
+{"_id": "14766", "title": "descent-definition-structure-sheaf", "text": "Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Let $S$ be a scheme. Let $\\Sch_\\tau$ be a big site containing $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module. \\begin{enumerate} \\item The {\\it structure sheaf of the big site $(\\Sch/S)_\\tau$} is the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$ which is denoted $\\mathcal{O}$ or $\\mathcal{O}_S$. \\item If $\\tau = \\etale$ the structure sheaf of the small site $S_\\etale$ is the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$ which is denoted $\\mathcal{O}$ or $\\mathcal{O}_S$. \\item The {\\it sheaf of $\\mathcal{O}$-modules associated to $\\mathcal{F}$} on the big site $(\\Sch/S)_\\tau$ is the sheaf of $\\mathcal{O}$-modules $(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$ which is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$). \\item Let $\\tau = \\etale$ (resp.\\ $\\tau = Zariski$). The {\\it sheaf of $\\mathcal{O}$-modules associated to $\\mathcal{F}$} on the small site $S_\\etale$ (resp.\\ $S_{Zar}$) is the sheaf of $\\mathcal{O}$-modules $(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$ which is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$). \\end{enumerate}"}
+{"_id": "14767", "title": "descent-definition-parasitic", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$. \\begin{enumerate} \\item $\\mathcal{F}$ is called {\\it parasitic}\\footnote{This may be nonstandard notation.} if for every flat morphism $U \\to S$ we have $\\mathcal{F}(U) = 0$. \\item $\\mathcal{F}$ is called {\\it parasitic for the $\\tau$-topology} if for every $\\tau$-covering $\\{U_i \\to S\\}_{i \\in I}$ we have $\\mathcal{F}(U_i) = 0$ for all $i$. \\end{enumerate}"}
+{"_id": "14768", "title": "descent-definition-property-local", "text": "Let $\\mathcal{P}$ be a property of schemes. Let $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0] smooth, \\linebreak[0] \\etale, \\linebreak[0] Zariski\\}$. We say $\\mathcal{P}$ is {\\it local in the $\\tau$-topology} if for any $\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ (see Topologies, Section \\ref{topologies-section-procedure}) we have $$ S \\text{ has }\\mathcal{P} \\Leftrightarrow \\text{each }S_i \\text{ has }\\mathcal{P}. $$"}
+{"_id": "14769", "title": "descent-definition-germs", "text": "Germs of schemes. \\begin{enumerate} \\item A pair $(X, x)$ consisting of a scheme $X$ and a point $x \\in X$ is called the {\\it germ of $X$ at $x$}. \\item A {\\it morphism of germs} $f : (X, x) \\to (S, s)$ is an equivalence class of morphisms of schemes $f : U \\to S$ with $f(x) = s$ where $U \\subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$. \\item We define the {\\it composition of morphisms of germs} by composing representatives (this is well defined). \\end{enumerate}"}
+{"_id": "14770", "title": "descent-definition-etale-morphism-germs", "text": "Let $f : (X, x) \\to (S, s)$ be a morphism of germs. We say $f$ is {\\it \\'etale} (resp.\\ {\\it smooth}) if there exists a representative $f : U \\to S$ of $f$ which is an \\'etale morphism (resp.\\ a smooth morphism) of schemes."}
+{"_id": "14771", "title": "descent-definition-local-at-point", "text": "Let $\\mathcal{P}$ be a property of germs of schemes. We say that $\\mathcal{P}$ is {\\it \\'etale local} (resp.\\ {\\it smooth local}) if for any \\'etale (resp.\\ smooth) morphism of germs $(U', u') \\to (U, u)$ we have $\\mathcal{P}(U, u) \\Leftrightarrow \\mathcal{P}(U', u')$."}
+{"_id": "14772", "title": "descent-definition-property-morphisms-local", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$. We say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or {\\it $\\tau$ local on the target}, or {\\it local on the base for the $\\tau$-topology} if for any $\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ (see Topologies, Section \\ref{topologies-section-procedure}) and any morphism of schemes $f : X \\to Y$ over $S$ we have $$ f \\text{ has }\\mathcal{P} \\Leftrightarrow \\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}. $$"}
+{"_id": "14773", "title": "descent-definition-property-morphisms-local-source", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes. Let $\\tau \\in \\{Zariski, \\linebreak[0] fpqc, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. We say $\\mathcal{P}$ is {\\it $\\tau$ local on the source}, or {\\it local on the source for the $\\tau$-topology} if for any morphism of schemes $f : X \\to Y$ over $S$, and any $\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ we have $$ f \\text{ has }\\mathcal{P} \\Leftrightarrow \\text{each }X_i \\to Y\\text{ has }\\mathcal{P}. $$"}
+{"_id": "14774", "title": "descent-definition-local-source-target", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes. We say $\\mathcal{P}$ is {\\it \\'etale local on source-and-target} if \\begin{enumerate} \\item (stable under precomposing with \\'etale maps) if $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$, then $g \\circ f$ has $\\mathcal{P}$, \\item (stable under \\'etale base change) if $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is \\'etale, then the base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and \\item (locality) given a morphism $f : X \\to Y$ the following are equivalent \\begin{enumerate} \\item $f$ has $\\mathcal{P}$, \\item for every $x \\in X$ there exists a commutative diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with \\'etale vertical arrows and $u \\in U$ with $a(u) = x$ such that $h$ has $\\mathcal{P}$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "14775", "title": "descent-definition-local-source-target-at-point", "text": "Let $\\mathcal{Q}$ be a property of morphisms of germs of schemes. We say $\\mathcal{Q}$ is  {\\it \\'etale local on the source-and-target} if for any commutative diagram $$ \\xymatrix{ (U', u') \\ar[d]_a \\ar[r]_{h'} & (V', v') \\ar[d]^b \\\\ (U, u) \\ar[r]^h & (V, v) } $$ of germs with \\'etale vertical arrows we have $\\mathcal{Q}(h) \\Leftrightarrow \\mathcal{Q}(h')$."}
+{"_id": "14776", "title": "descent-definition-descent-datum", "text": "Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item Let $V \\to X$ be a scheme over $X$. A {\\it descent datum for $V/X/S$} is an isomorphism $\\varphi : V \\times_S X \\to X \\times_S V$ of schemes over $X \\times_S X$ satisfying the {\\it cocycle condition} that the diagram $$ \\xymatrix{ V \\times_S X \\times_S X \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} & & X \\times_S X \\times_S V\\\\ & X \\times_S V \\times_S X \\ar[ru]^{\\varphi_{12}} } $$ commutes (with obvious notation). \\item We also say that the pair $(V/X, \\varphi)$ is a {\\it descent datum relative to $X \\to S$}. \\item A {\\it morphism $f : (V/X, \\varphi) \\to (V'/X, \\varphi')$ of descent data relative to $X \\to S$} is a morphism $f : V \\to V'$ of schemes over $X$ such that the diagram $$ \\xymatrix{ V \\times_S X \\ar[r]_{\\varphi} \\ar[d]_{f \\times \\text{id}_X} & X \\times_S V \\ar[d]^{\\text{id}_X \\times f} \\\\ V' \\times_S X \\ar[r]^{\\varphi'} & X \\times_S V' } $$ commutes. \\end{enumerate}"}
+{"_id": "14777", "title": "descent-definition-descent-datum-for-family-of-morphisms", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$. \\begin{enumerate} \\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the family $\\{X_i \\to S\\}$} is given by a scheme $V_i$ over $X_i$ for each $i \\in I$, an isomorphism $\\varphi_{ij} : V_i \\times_S X_j \\to X_i \\times_S V_j$ of schemes over $X_i \\times_S X_j$ for each pair $(i, j) \\in I^2$ such that for every triple of indices $(i, j, k) \\in I^3$ the diagram $$ \\xymatrix{ V_i \\times_S X_j \\times_S X_k \\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}} \\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & & X_i \\times_S X_j \\times_S V_k\\\\ & X_i \\times_S V_j \\times_S X_k \\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}} } $$ of schemes over $X_i \\times_S X_j \\times_S X_k$ commutes (with obvious notation). \\item A {\\it morphism $\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$ of descent data} is given by a family $\\psi = (\\psi_i)_{i \\in I}$ of morphisms of $X_i$-schemes $\\psi_i : V_i \\to V'_i$ such that all the diagrams $$ \\xymatrix{ V_i \\times_S X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} & X_i \\times_S V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\ V'_i \\times_S X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_S V'_j } $$ commute. \\end{enumerate}"}
+{"_id": "14778", "title": "descent-definition-pullback-functor", "text": "With $S, S', X, X', f, a, a', h$ as in Lemma \\ref{lemma-pullback} the functor $$ (V, \\varphi) \\longmapsto f^*(V, \\varphi) $$ constructed in that lemma is called the {\\it pullback functor} on descent data."}
+{"_id": "14779", "title": "descent-definition-pullback-functor-family", "text": "With $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$, $\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, $\\alpha : I \\to J$, $h : S' \\to S$, and $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback-family} the functor $$ (Y_j, \\varphi_{jj'}) \\longmapsto (g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}) $$ constructed in that lemma is called the {\\it pullback functor} on descent data."}
+{"_id": "14780", "title": "descent-definition-effective", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism of schemes. \\begin{enumerate} \\item  Given a scheme $U$ over $S$ we have the {\\it trivial descent datum} of $U$ relative to $\\text{id} : S \\to S$, namely the identity morphism on $U$. \\item By Lemma \\ref{lemma-pullback} we get a {\\it canonical descent datum} on $X \\times_S U$ relative to $X \\to S$ by pulling back the trivial descent datum via $f$. We often denote $(X \\times_S U, can)$ this descent datum. \\item A descent datum $(V, \\varphi)$ relative to $X/S$ is called {\\it effective} if $(V, \\varphi)$ is isomorphic to the canonical descent datum $(X \\times_S U, can)$ for some scheme $U$ over $S$. \\end{enumerate}"}
+{"_id": "14781", "title": "descent-definition-effective-family", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}$ be a family of morphisms with target $S$. \\begin{enumerate} \\item  Given a scheme $U$ over $S$ we have a {\\it canonical descent datum} on the family of schemes $X_i \\times_S U$ by pulling back the trivial descent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$. We denote this descent datum $(X_i \\times_S U, can)$. \\item A descent datum $(V_i, \\varphi_{ij})$ relative to $\\{X_i \\to S\\}$ is called {\\it effective} if there exists a scheme $U$ over $S$ such that $(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_S U, can)$. \\end{enumerate}"}
+{"_id": "14782", "title": "descent-definition-descending-types-morphisms", "text": "Let $\\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\\tau \\in \\{Zariski, fpqc, fppf, \\etale, smooth, syntomic\\}$. We say {\\it morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings} if for any $\\tau$-covering $\\mathcal{U} : \\{U_i \\to S\\}_{i \\in I}$ (see Topologies, Section \\ref{topologies-section-procedure}), any descent datum $(X_i, \\varphi_{ij})$ relative to $\\mathcal{U}$ such that each morphism $X_i \\to U_i$ has property $\\mathcal{P}$ is effective."}
+{"_id": "14910", "title": "simplicial-definition-face-degeneracy", "text": "For any integer $n\\geq 1$, and any $0\\leq j \\leq n$ we let {\\it $\\delta^n_j : [n-1] \\to [n]$} denote the injective order preserving map skipping $j$. For any integer $n\\geq 0$, and any $0\\leq j \\leq n$ we denote {\\it $\\sigma^n_j : [n + 1]  \\to [n]$} the surjective order preserving map with $(\\sigma^n_j)^{-1}(\\{j\\}) = \\{j, j + 1\\}$."}
+{"_id": "14911", "title": "simplicial-definition-simplicial-object", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item A {\\it simplicial object $U$ of $\\mathcal{C}$} is a contravariant functor $U$ from $\\Delta$ to $\\mathcal{C}$, in a formula: $$ U : \\Delta^{opp} \\longrightarrow \\mathcal{C} $$ \\item If $\\mathcal{C}$ is the category of sets, then we call $U$ a {\\it simplicial set}. \\item If $\\mathcal{C}$ is the category of abelian groups, then we call $U$ a {\\it simplicial abelian group}. \\item A {\\it morphism of simplicial objects $U \\to U'$} is a transformation of functors. \\item The {\\it category of simplicial objects of $\\mathcal{C}$} is denoted $\\text{Simp}(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "14912", "title": "simplicial-definition-cosimplicial-object", "text": "Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item A {\\it cosimplicial object $U$ of $\\mathcal{C}$} is a covariant functor $U$ from $\\Delta$ to $\\mathcal{C}$, in a formula: $$ U : \\Delta \\longrightarrow \\mathcal{C} $$ \\item If $\\mathcal{C}$ is the category of sets, then we call $U$ a {\\it cosimplicial set}. \\item If $\\mathcal{C}$ is the category of abelian groups, then we call $U$ a {\\it cosimplicial abelian group}. \\item A {\\it morphism of cosimplicial objects $U \\to U'$} is a transformation of functors. \\item The {\\it category of cosimplicial objects of $\\mathcal{C}$} is denoted $\\text{CoSimp}(\\mathcal{C})$. \\end{enumerate}"}
+{"_id": "14913", "title": "simplicial-definition-product", "text": "Let $\\mathcal{C}$ be a category. Let $U$ and $V$ be simplicial objects of $\\mathcal{C}$. Assume the products $U_n \\times V_n$ exist in $\\mathcal{C}$. The {\\it product of $U$ and $V$} is the simplicial object $U \\times V$ defined as follows: \\begin{enumerate} \\item $(U \\times V)_n = U_n \\times V_n$, \\item $d^n_i = (d^n_i, d^n_i)$, and \\item $s^n_i = (s^n_i, s^n_i)$. \\end{enumerate} In other words, $U \\times V$ is the product of the presheaves $U$ and $V$ on $\\Delta$."}
+{"_id": "14914", "title": "simplicial-definition-fibre-product", "text": "Let $\\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\\mathcal{C}$. Let $a : V \\to U$, $b : W \\to U$ be morphisms. Assume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$. The {\\it fibre product of $V$ and $W$ over $U$} is the simplicial object $V \\times_U W$ defined as follows: \\begin{enumerate} \\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$, \\item $d^n_i = (d^n_i, d^n_i)$, and \\item $s^n_i = (s^n_i, s^n_i)$. \\end{enumerate} In other words, $V \\times_U W$ is the fibre product of the presheaves $V$ and $W$ over the presheaf $U$ on $\\Delta$."}
+{"_id": "14915", "title": "simplicial-definition-push-out", "text": "Let $\\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\\mathcal{C}$. Let $a : U \\to V$, $b : U \\to W$ be morphisms. Assume the pushouts $V_n \\amalg_{U_n} W_n$ exist in $\\mathcal{C}$. The {\\it pushout of $V$ and $W$ over $U$} is the simplicial object $V\\amalg_U W$ defined as follows: \\begin{enumerate} \\item $(V \\amalg_U W)_n = V_n \\amalg_{U_n} W_n$, \\item $d^n_i = (d^n_i, d^n_i)$, and \\item $s^n_i = (s^n_i, s^n_i)$. \\end{enumerate} In other words, $V\\amalg_U W$ is the pushout of the presheaves $V$ and $W$ over the presheaf $U$ on $\\Delta$."}
+{"_id": "14916", "title": "simplicial-definition-product-cosimplicial-objects", "text": "Let $\\mathcal{C}$ be a category. Let $U$ and $V$ be cosimplicial objects of $\\mathcal{C}$. Assume the products $U_n \\times V_n$ exist in $\\mathcal{C}$. The {\\it product of $U$ and $V$} is the cosimplicial object $U \\times V$ defined as follows: \\begin{enumerate} \\item $(U \\times V)_n = U_n \\times V_n$, \\item for any $\\varphi : [n] \\to [m]$ the map $(U \\times V)(\\varphi) : U_n \\times V_n \\to U_m \\times V_m$ is the product $U(\\varphi) \\times V(\\varphi)$. \\end{enumerate}"}
+{"_id": "14917", "title": "simplicial-definition-fibre-product-cosimplicial-objects", "text": "Let $\\mathcal{C}$ be a category. Let $U, V, W$ be cosimplicial objects of $\\mathcal{C}$. Let $a : V \\to U$ and $b : W \\to U$ be morphisms. Assume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$. The {\\it fibre product of $V$ and $W$ over $U$} is the cosimplicial object $V \\times_U W$ defined as follows: \\begin{enumerate} \\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$, \\item for any $\\varphi : [n] \\to [m]$ the map $(V \\times_U W)(\\varphi) : V_n \\times_{U_n} W_n \\to V_m \\times_{U_m} W_m$ is the product $V(\\varphi) \\times_{U(\\varphi)} W(\\varphi)$. \\end{enumerate}"}
+{"_id": "14918", "title": "simplicial-definition-terminology-simplicial-sets", "text": "Let $U$ be a simplicial set. We say $x$ is an {\\it $n$-simplex of $U$} to signify that $x$ is an element of $U_n$. We say that $y$ is the $j$the {\\it face of $x$} to signify that $d^n_jx = y$. We say that $z$ is the $j$th {\\it degeneracy of $x$} if $z = s^n_jx$. A simplex is called {\\it degenerate} if it is the degeneracy of another simplex."}
+{"_id": "14919", "title": "simplicial-definition-truncated-simplicial-object", "text": "An {\\it $n$-truncated simplicial object of $\\mathcal{C}$} is a contravariant functor from $\\Delta_{\\leq n}$ to $\\mathcal{C}$. A {\\it morphism of $n$-truncated simplicial objects} is a transformation of functors. We denote the category of $n$-truncated simplicial objects of $\\mathcal{C}$ by the symbol $\\text{Simp}_n(\\mathcal{C})$."}
+{"_id": "14920", "title": "simplicial-definition-product-with-simplicial-set", "text": "Let $\\mathcal{C}$ be a category such that the coproduct of any two objects of $\\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\\mathcal{C}$. Assume that each $U_n$ is finite nonempty. In this case we define the {\\it product $U \\times V$ of $U$ and $V$} to be the simplicial object of $\\mathcal{C}$ whose $n$th term is the object $$ (U \\times V)_n = \\coprod\\nolimits_{u\\in U_n} V_n $$ with maps for $\\varphi : [m] \\to [n]$ given by the morphism $$ \\coprod\\nolimits_{u\\in U_n} V_n \\longrightarrow \\coprod\\nolimits_{u'\\in U_m} V_m $$ which maps the component $V_n$ corresponding to $u$ to the component $V_m$ corresponding to $u' = U(\\varphi)(u)$ via the morphism $V(\\varphi)$. More loosely, if all of the coproducts displayed above exist (without assuming anything about $\\mathcal{C}$) we will say that the {\\it product $U \\times V$ exists}."}
+{"_id": "14921", "title": "simplicial-definition-hom-deltak-cosimplicial", "text": "Let $\\mathcal{C}$ be a category with finite products. Let $V$ be a cosimplicial object of $\\mathcal{C}$. Let $U$ be a simplicial set such that each $U_n$ is finite nonempty. We define {\\it $\\Hom(U, V)$} to be the cosimplicial object of $\\mathcal{C}$ defined as follows: \\begin{enumerate} \\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$, in other words the unique object of $\\mathcal{C}$ such that its $X$-valued points satisfy $$ \\Mor_\\mathcal{C}(X, \\Hom(U, V)_n) = \\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n)) $$ and \\item for $\\varphi : [m] \\to [n]$ we take the map $\\Hom(U, V)_m \\to \\Hom(U, V)_n$ given by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$ on $X$-valued points as above. \\end{enumerate}"}
+{"_id": "14922", "title": "simplicial-definition-hom-deltak-simplicial", "text": "Let $\\mathcal{C}$ be a category with finite products. Let $V$ be a simplicial object of $\\mathcal{C}$. Let $U$ be a cosimplicial set such that each $U_n$ is finite nonempty. We define {\\it $\\Hom(U, V)$} to be the simplicial object of $\\mathcal{C}$ defined as follows: \\begin{enumerate} \\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$, in other words the unique object of $\\mathcal{C}$ such that its $X$-valued points satisfy $$ \\Mor_\\mathcal{C}(X, \\Hom(U, V)_n) = \\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n)) $$ and \\item for $\\varphi : [m] \\to [n]$ we take the map $\\Hom(U, V)_n \\to \\Hom(U, V)_m$ given by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$ on $X$-valued points as above. \\end{enumerate}"}
+{"_id": "14923", "title": "simplicial-definition-hom-from-simplicial-set", "text": "Let $\\mathcal{C}$ be a category such that the coproduct of any two objects exists. Let $U$ be a simplicial set, with $U_n$ finite nonempty for all $n \\geq 0$. Let $V$ be a simplicial object of $\\mathcal{C}$. We denote {\\it $\\Hom(U, V)$} any simplicial object of $\\mathcal{C}$ such that $$ \\Mor_{\\text{Simp}(\\mathcal{C})}(W, \\Hom(U, V)) = \\Mor_{\\text{Simp}(\\mathcal{C})}(W \\times U, V) $$ functorially in the simplicial object $W$ of $\\mathcal{C}$."}
+{"_id": "14924", "title": "simplicial-definition-split", "text": "Let $\\mathcal{C}$ be a category which admits finite nonempty coproducts. We say a simplicial object $U$ of $\\mathcal{C}$ is {\\it split} if there exist subobjects $N(U_m)$ of $U_m$, $m \\geq 0$ with the property that \\begin{equation} \\label{equation-splitting} \\coprod\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}} N(U_m) \\longrightarrow U_n \\end{equation} is an isomorphism for all $n \\geq 0$. If $U$ is an $r$-truncated simplicial object of $\\mathcal{C}$ then we say $U$ is {\\it split} if there exist subobjects $N(U_m)$ of $U_m$, $r \\geq m \\geq 0$ with the property that (\\ref{equation-splitting}) is an isomorphism for $r \\geq n \\geq 0$."}
+{"_id": "14925", "title": "simplicial-definition-augmentation", "text": "Let $\\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\\mathcal{C}$. An {\\it augmentation $\\epsilon : U \\to X$ of $U$ towards an object $X$ of $\\mathcal{C}$} is a morphism from $U$ into the constant simplicial object $X$."}
+{"_id": "14926", "title": "simplicial-definition-eilenberg-maclane", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object of $\\mathcal{A}$ and let $k$ be an integer $\\geq 0$. The {\\it Eilenberg-Maclane object $K(A, k)$} is given by the object $K(A, k) = i_{k!}U$ which is described in Lemma \\ref{lemma-eilenberg-maclane-object} above."}
+{"_id": "14927", "title": "simplicial-definition-homotopy", "text": "Let $\\mathcal{C}$ be a category having finite coproducts. Suppose that $U$ and $V$ are two simplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be two morphisms. \\begin{enumerate} \\item We say a morphism $$ h : U \\times \\Delta[1] \\longrightarrow V $$ is a {\\it homotopy from $a$ to $b$} if $a = h \\circ e_0$ and $b = h \\circ e_1$. \\item We say the morphisms $a$ and $b$ are {\\it homotopic} or are {\\it in the same homotopy class} if there exists a sequence of morphisms $a = a_0, a_1, \\ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. \\end{enumerate}"}
+{"_id": "14928", "title": "simplicial-definition-homotopy-equivalent", "text": "Let $U$ and $V$ be two simplicial objects of a category $\\mathcal{C}$. We say a morphism $a : U \\to V$ is a {\\it homotopy equivalence} if there exists a morphism $b : V \\to U$ such that $a \\circ b$ is homotopic to $\\text{id}_V$ and $b \\circ a$ is homotopic to $\\text{id}_U$. We say $U$ and $V$ are {\\it homotopy equivalent} if there exists a homotopy equivalence $a : U \\to V$."}
+{"_id": "14929", "title": "simplicial-definition-homotopy-cosimplicial", "text": "Let $\\mathcal{C}$ be a category having finite products. Let $U$ and $V$ be two cosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be two morphisms of cosimplicial objects of $\\mathcal{C}$. \\begin{enumerate} \\item We say a morphism $$ h : U \\longrightarrow \\Hom(\\Delta[1], V) $$ such that $a = e_0 \\circ h$ and $b = e_1 \\circ h$ is a {\\it homotopy from $a$ to $b$}. \\item We say $a$ and $b$ are {\\it homotopic} or are {\\it in the same homotopy class} if there exists a sequence $a = a_0, a_1, \\ldots, a_n = b$ of morphisms from $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists a homotopy from $a_i$ to $a_{i - 1}$ or there exists a homotopy from $a_{i - 1}$ to $a_i$. \\end{enumerate}"}
+{"_id": "14930", "title": "simplicial-definition-trivial-kan", "text": "A map $X \\to Y$ of simplicial sets is called a {\\it trivial Kan fibration} if $X_0 \\to Y_0$ is surjective and for all $n \\geq 1$ and any commutative solid diagram $$ \\xymatrix{ \\partial \\Delta[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y } $$ a dotted arrow exists making the diagram commute."}
+{"_id": "14931", "title": "simplicial-definition-kan", "text": "A map $X \\to Y$ of simplicial sets is called a {\\it Kan fibration} if for all $k, n$ with $1 \\leq n$, $0 \\leq k \\leq n$ and any commutative solid diagram $$ \\xymatrix{ \\Lambda_k[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y } $$ a dotted arrow exists making the diagram commute. A {\\it Kan complex} is a simplicial set $X$ such that $X \\to *$ is a Kan fibration, where $*$ is the constant simplicial set on a singleton."}
+{"_id": "15003", "title": "discriminant-definition-trace-element", "text": "Let $A \\to B$ be a flat quasi-finite map of Noetherian rings. The {\\it trace element} is the unique\\footnote{Uniqueness and existence will be justified in Lemmas \\ref{lemma-trace-unique} and \\ref{lemma-dualizing-tau}.} element $\\tau_{B/A} \\in \\omega_{B/A}$ with the following property: for any Noetherian $A$-algebra $A_1$ such that $B_1 = B \\otimes_A A_1$ comes with a product decomposition $B_1 = C \\times D$ with $A_1 \\to C$ finite the image of $\\tau_{B/A}$ in $\\omega_{C/A_1}$ is $\\text{Trace}_{C/A_1}$. Here we use the base change map (\\ref{equation-bc-dualizing}) and Lemma \\ref{lemma-dualizing-product} to get $\\omega_{B/A} \\to \\omega_{B_1/A_1} \\to \\omega_{C/A_1}$."}
+{"_id": "15004", "title": "discriminant-definition-kahler-different", "text": "Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type. The {\\it K\\\"ahler different} is the $0$th fitting ideal of $\\Omega_{Y/X}$."}
+{"_id": "15005", "title": "discriminant-definition-different", "text": "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $\\omega_{Y/X}$ be the relative dualizing module and let $\\tau_{Y/X} \\in \\Gamma(Y, \\omega_{Y/X})$ be the trace element (Remarks \\ref{remark-relative-dualizing-for-quasi-finite} and \\ref{remark-relative-dualizing-for-flat-quasi-finite}). The annihilator of $$ \\Coker(\\mathcal{O}_Y \\xrightarrow{\\tau_{Y/X}} \\omega_{Y/X}) $$ is the {\\it different} of $Y/X$. It is a coherent ideal $\\mathfrak{D}_f \\subset \\mathcal{O}_Y$."}
+{"_id": "15025", "title": "stacks-limits-definition-limit-preserving", "text": "Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. We say $f$ is {\\it limit preserving} if for every directed limit $U = \\lim U_i$ of affine schemes over $S$ the diagram $$ \\xymatrix{ \\colim \\mathcal{X}_{U_i} \\ar[r] \\ar[d]_f & \\mathcal{X}_U \\ar[d]^f \\\\ \\colim \\mathcal{Y}_{U_i} \\ar[r] & \\mathcal{Y}_U } $$ of fibre categories is $2$-cartesian."}
+{"_id": "302", "title": "spaces-more-morphisms-remark-weakly-radicial", "text": "Let $X \\to Y$ be a morphism of algebraic spaces. For some applications (of radicial morphisms) it is enough to require that for every $\\Spec(K) \\to Y$ where $K$ is a field \\begin{enumerate} \\item the space $|\\Spec(K) \\times_Y X|$ is a singleton, \\item there exists a monomorphism $\\Spec(L) \\to \\Spec(K) \\times_Y X$, and \\item $K \\subset L$ is purely inseparable. \\end{enumerate} If needed later we will may call such a morphism {\\it weakly radicial}. For example if $X \\to Y$ is a surjective weakly radicial morphism then $X(k) \\to Y(k)$ is surjective for every algebraically closed field $k$. Note that the base change $X_{\\overline{\\mathbf{Q}}} \\to \\Spec(\\overline{\\mathbf{Q}})$ of the morphism in Example \\ref{example-universally-injective-not-radicial} is weakly radicial, but not radicial. The analogue of Lemma \\ref{lemma-when-universally-injective-radicial} is that if $X \\to Y$ has property ($\\beta$) and is universally injective, then it is weakly radicial (proof omitted)."}
+{"_id": "303", "title": "spaces-more-morphisms-remark-alternative", "text": "Now that we know that $\\Omega_{X/Y}$ is quasi-coherent we can attempt to construct it in another manner. For example we can use the result of Properties of Spaces, Section \\ref{spaces-properties-section-quasi-coherent-presentation} to construct the sheaf of differentials by glueing. For example if $Y$ is a scheme and if $U \\to X$ is a surjective \\'etale morphism from a scheme towards $X$, then we see that $\\Omega_{U/Y}$ is a quasi-coherent $\\mathcal{O}_U$-module, and since $s, t : R \\to U$ are \\'etale we get an isomorphism $$ \\alpha : s^*\\Omega_{U/Y} \\to \\Omega_{R/Y} \\to t^*\\Omega_{U/Y} $$ by using Morphisms, Lemma \\ref{morphisms-lemma-triangle-differentials-smooth}. You check that this satisfies the cocycle condition and you're done. If $Y$ is not a scheme, then you define $\\Omega_{U/Y}$ as the cokernel of the map $(U \\to Y)^*\\Omega_{Y/S} \\to \\Omega_{U/S}$, and proceed as before. This two step process is a little bit ugly. Another possibility is to glue the sheaves $\\Omega_{U/V}$ for any diagram as in Lemma \\ref{lemma-localize-differentials} but this is not very elegant either. Both approaches will work however, and will give a slightly more elementary construction of the sheaf of differentials."}
+{"_id": "304", "title": "spaces-more-morphisms-remark-topological-invariance-etale-site", "text": "\\begin{reference} Email by Lenny Taelman dated May 1, 2016. \\end{reference} A universal homeomorphism of algebraic spaces need not be representable, see Morphisms of Spaces, Example \\ref{spaces-morphisms-example-universal-homeomorphism}. In fact Theorem \\ref{theorem-topological-invariance} does not hold for universal homeomorphisms. To see this, let $k$ be an algebraically closed field of characteristic $0$ and let $$ \\mathbf{A}^1 \\to X \\to \\mathbf{A}^1 $$ be as in Morphisms of Spaces, Example \\ref{spaces-morphisms-example-universal-homeomorphism}. Recall that the first morphism is \\'etale and identifies $t$ with $-t$ for $t \\in \\mathbf{A}^1_k \\setminus \\{0\\}$ and that the second morphism is our universal homeomorphism. Since $\\mathbf{A}^1_k$ has no nontrivial connected finite \\'etale coverings (because $k$ is algebraically closed of characteristic zero; details omitted), it suffices to construct a nontrivial connected finite \\'etale covering $Y \\to X$. To do this, let $Y$ be the affine line with zero doubled (Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}). Then $Y = Y_1 \\cup Y_2$ with $Y_i = \\mathbf{A}^1_k$ glued along $\\mathbf{A}^1_k \\setminus \\{0\\}$. To define the morphism $Y \\to X$ we use the morphisms $$ Y_1 \\xrightarrow{1} \\mathbf{A}^1_k \\to X \\quad\\text{and}\\quad Y_2 \\xrightarrow{-1} \\mathbf{A}^1_k \\to X. $$ These glue over $Y_1 \\cap Y_2$ by the construction of $X$ and hence define a morphism $Y \\to X$. In fact, we claim that $$ \\xymatrix{ Y \\ar[d] & Y_1 \\amalg Y_2 \\ar[l] \\ar[d] \\\\ X & \\mathbf{A}^1_k \\ar[l] } $$ is a cartesian square. We omit the details; you can use for example Groupoids, Lemma \\ref{groupoids-lemma-criterion-fibre-product}. Since $\\mathbf{A}^1_k \\to X$ is \\'etale and surjective, this proves that $Y \\to X$ is finite \\'etale of degree $2$ which gives the desired example. \\medskip\\noindent More simply, you can argue as follows. The scheme $Y$ has a free action of the group $G = \\{+1, -1\\}$ where $-1$ acts by swapping $Y_1$ and $Y_2$ and changing the sign of the coordinate. Then $X = Y/G$ (see Spaces, Definition \\ref{spaces-definition-quotient}) and hence $Y \\to X$ is finite \\'etale. You can also show directly that there exists a universal homeomorphism $X \\to \\mathbf{A}^1_k$ by using $t \\mapsto t^2$ on affine spaces. In fact, this $X$ is the same as the $X$ above."}
+{"_id": "305", "title": "spaces-more-morphisms-remark-tempting", "text": "It is tempting to think that in the situation of Lemma \\ref{lemma-etale-on-top} we have ``$b$ formally smooth'' $\\Leftrightarrow$ ``$b \\circ a$ formally smooth''. However, this is likely not true in general."}
+{"_id": "306", "title": "spaces-more-morphisms-remark-action-by-derivations", "text": "Assumptions and notation as in Lemma \\ref{lemma-action-by-derivations}. The action of a local section $\\theta$ on $a'$ is sometimes indicated by $\\theta \\cdot a'$. Note that this means nothing else than the fact that $(a')^\\sharp$ and $(\\theta \\cdot a')^\\sharp$ differ by a derivation $D$ which is related to $\\theta$ by Equation (\\ref{equation-D})."}
+{"_id": "307", "title": "spaces-more-morphisms-remark-special-case", "text": "A special case of Lemmas \\ref{lemma-difference-derivation}, \\ref{lemma-action-by-derivations}, \\ref{lemma-sheaf}, and \\ref{lemma-action-sheaf} is where $Y = Y'$. In this case the map $A$ is always zero. The sheaf of Lemma \\ref{lemma-sheaf} is just given by the rule $$ U' \\mapsto \\{a' : U' \\to Y\\text{ over }B\\text{ with } a'|_U = a|_U\\} $$ and we act on this by the sheaf $\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/B}, \\mathcal{C}_{X/X'})$."}
+{"_id": "308", "title": "spaces-more-morphisms-remark-another-special-case", "text": "Another special case of Lemmas \\ref{lemma-difference-derivation}, \\ref{lemma-action-by-derivations}, \\ref{lemma-sheaf}, and \\ref{lemma-action-sheaf} is where $B$ itself is a thickening $Z \\subset Z' = B$ and $Y = Z \\times_{Z'} Y'$. Picture $$ \\xymatrix{ (X \\subset X') \\ar@{..>}[rr]_{(a, ?)} \\ar[rd]_{(g, g')} & & (Y \\subset Y') \\ar[ld]^{(h, h')} \\\\ & (Z \\subset Z') } $$ In this case the map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\\mathcal{C}_{Z/Z'} \\to \\mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \\times_{Z'} Y'$), hence the pullback $g^*\\mathcal{C}_{Z/Z'} = a^*h^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma \\ref{lemma-sheaf} is just given by the rule $$ U' \\mapsto \\{a' : U' \\to Y'\\text{ over }Z'\\text{ with } a'|_U = a|_U\\} $$ and we act on this by the sheaf $\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/Z}, \\mathcal{C}_{X/X'})$."}
+{"_id": "309", "title": "spaces-more-morphisms-remark-chow-Noetherian", "text": "In Lemmas \\ref{lemma-blowup-to-find-embedding} and \\ref{lemma-chow-noetherian} the morphism $g : Z' \\to Y$ is a composition of projective morphisms. Presumably (by the analogue for algebraic spaces of Morphisms, Lemma \\ref{morphisms-lemma-ample-composition}) there exists a $g$-ample invertible sheaf on $Z'$. If we ever need this, then we will state and prove this here."}
+{"_id": "310", "title": "spaces-more-morphisms-remark-inverse-systems-kernel-cokernel-annihilated-by", "text": "Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $\\alpha : (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ be a morphism of $\\textit{Coh}(X, \\mathcal{I})$. Given an affine scheme $U = \\Spec(A)$ and a surjective \\'etale morphism $U \\to X$ denote $I, K \\subset A$ the ideals corresponding to the restrictions $\\mathcal{I}|_U, \\mathcal{K}|_U$. Denote $\\alpha_U : M \\to N$ of finite $A^\\wedge$-modules which corresponds to $\\alpha|_U$ via Cohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-affine}. We claim the following are equivalent \\begin{enumerate} \\item there exists an integer $t \\geq 1$ such that $\\Ker(\\alpha_n)$ and $\\Coker(\\alpha_n)$ are annihilated by $\\mathcal{K}^t$ for all $n \\geq 1$, \\item for any (or some) affine open $\\Spec(A) = U \\subset X$ as above the modules $\\Ker(\\alpha_U)$ and $\\Coker(\\alpha_U)$ are annihilated by $K^t$ for some integer $t \\geq 1$. \\end{enumerate} If these equivalent conditions hold we will say that $\\alpha$ is a {\\it map whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$}. To see the equivalence we refer to Cohomology of Schemes, Remark \\ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}."}
+{"_id": "311", "title": "spaces-more-morphisms-remark-reformulate-existence-theorem", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Picture: $$ \\xymatrix{ X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] & \\ldots & X \\ar[d] \\\\ S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots & S } $$ In this situation we consider systems $(\\mathcal{F}_n, \\varphi_n)$ where \\begin{enumerate} \\item $\\mathcal{F}_n$ is a coherent $\\mathcal{O}_{X_n}$-module, \\item $\\varphi_n : i_n^*\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ is an isomorphism, and \\item $\\text{Supp}(\\mathcal{F}_1)$ is proper over $S_1$. \\end{enumerate} Theorem \\ref{theorem-grothendieck-existence} says that the completion functor $$ \\begin{matrix} \\text{coherent }\\mathcal{O}_X\\text{-modules }\\mathcal{F} \\\\ \\text{with support proper over }A \\end{matrix} \\quad \\longrightarrow \\quad \\begin{matrix} \\text{systems }(\\mathcal{F}_n) \\\\ \\text{as above} \\end{matrix} $$ is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports."}
+{"_id": "312", "title": "spaces-more-morphisms-remark-weaken-separation-axioms-question", "text": "We can ask if in Grothendieck's algebraization theorem (in the form of Lemma \\ref{lemma-algebraize-morphism}), we can get by with weaker separation axioms on the target. Let us be more precise. Let $A$, $I$, $S$, $S_n$, $X$, $Y$, $X_n$, $Y_n$, and $g_n$ be as in the statement of Lemma \\ref{lemma-algebraize-morphism} and assume that \\begin{enumerate} \\item $X \\to S$ is proper, and \\item $Y \\to S$ is locally of finite type. \\end{enumerate} Does there exist a morphism of algebraic spaces $g : X \\to Y$ over $S$ such that $g_n$ is the base change of $g$ to $S_n$? We don't know the answer in general; if you do please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. If $Y \\to S$ is separated, then the result holds by the lemma (there is an immediate reduction to the case where $X$ is finite type over $S$, by choosing a quasi-compact open containing the image of $g_1$). If we only assume $Y \\to S$ is quasi-separated, then the result is true as well. First, as before we may assume $Y$ is quasi-compact as well as quasi-separated. Then we can use either \\cite{Bhatt-Algebraize} or from \\cite{Hall-Rydh-coherent} to algebraize $(g_n)$. Namely, to apply the first reference, we use $$ D_{perf}(X) \\to \\lim D_{perf}(X_n) \\xrightarrow{\\lim Lg_n^*} \\lim D_{perf}(Y_n) = D_{perf}(Y) $$ where the last step uses a Grothendieck existence result for the derived category of the proper algebraic space $Y$ over $R$ (compare with Flatness on Spaces, Remark \\ref{spaces-flat-remark-correct-generality}). The paper cited shows that this arrow determines a morphism $Y \\to X$ as desired. To apply the second reference we use the same argument with coherent modules: $$ \\textit{Coh}(\\mathcal{O}_X) \\to \\lim \\textit{Coh}(\\mathcal{O}_{X_n}) \\xrightarrow{\\lim g_n^*} \\lim \\textit{Coh}(\\mathcal{O}_{Y_n}) = \\textit{Coh}(\\mathcal{O}_Y) $$ where the final equality is a consequence of Grothendieck's existence theorem (Theorem \\ref{theorem-grothendieck-existence}). The second reference tells us that this functor corresponds to a morphism $Y \\to X$ over $R$. If we ever need this generalization we will precisely state and carefully prove the result here."}
+{"_id": "313", "title": "spaces-more-morphisms-remark-match-relative-pseudo-coherence", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of representable algebraic spaces over $S$ which is locally of finite type. Let $f_0 : X_0 \\to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then $f_0$ is locally of finite type. If $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, then $E$ is the pullback of a unique object $E_0$ in $D_\\QCoh(\\mathcal{O}_{X_0})$, see Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}. In this situation the phrase ``$E$ is $m$-pseudo-coherent relative to $Y$'' will be taken to mean ``$E_0$ is $m$-pseudo-coherent relative to $Y_0$'' as defined in More on Morphisms, Section \\ref{more-morphisms-section-relative-pseudo-coherence}."}
+{"_id": "314", "title": "spaces-more-morphisms-remark-compare-L", "text": "The reader may have noticed the similarity between Lemma \\ref{lemma-compute-ext-rel-perfect} and Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-compute-ext}. Indeed, the pseudo-coherent complex $L$ of Lemma \\ref{lemma-compute-ext-rel-perfect} may be characterized as the unique pseudo-coherent complex on $Y$ such that there are functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_Y}(L, \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(K, E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}) $$ compatible with boundary maps for $\\mathcal{F}$ ranging over $\\QCoh(\\mathcal{O}_Y)$. If we ever need this we will formulate a precise result here and give a detailed proof."}
+{"_id": "1556", "title": "algebra-remark-tensor-product-not-exact", "text": "However, tensor product does NOT preserve exact sequences in general. In other words, if $M_1 \\to M_2 \\to M_3$ is exact, then it is not necessarily true that $M_1 \\otimes N \\to M_2 \\otimes N \\to M_3 \\otimes N$ is exact for arbitrary $R$-module $N$."}
+{"_id": "1557", "title": "algebra-remark-flat-module", "text": "For $R$-modules $N$, if the functor $-\\otimes_R N$ is exact, i.e. tensoring with $N$ preserves all exact sequences, then $N$ is said to be {\\it flat} $R$-module. We will discuss this later in Section \\ref{section-flat}."}
+{"_id": "1558", "title": "algebra-remark-fundamental-diagram", "text": "A fundamental commutative diagram associated to a ring map $\\varphi : R \\to S$, a prime $\\mathfrak q \\subset S$ and the corresponding prime $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$ of $R$ is the following $$ \\xymatrix{ \\kappa(\\mathfrak q) = S_{\\mathfrak q}/{\\mathfrak q}S_{\\mathfrak q} & S_{\\mathfrak q} \\ar[l] & S \\ar[r] \\ar[l] & S/\\mathfrak q \\ar[r] & \\kappa(\\mathfrak q) \\\\ \\kappa(\\mathfrak p) \\otimes_R S = S_{\\mathfrak p}/{\\mathfrak p}S_{\\mathfrak p} \\ar[u] & S_{\\mathfrak p} \\ar[u] \\ar[l] & S \\ar[u] \\ar[r] \\ar[l] & S/\\mathfrak pS \\ar[u] \\ar[r] & (R \\setminus \\mathfrak p)^{-1}S/\\mathfrak pS \\ar[u] \\\\ \\kappa(\\mathfrak p) = R_{\\mathfrak p}/{\\mathfrak p}R_{\\mathfrak p} \\ar[u] & R_{\\mathfrak p} \\ar[u] \\ar[l] & R \\ar[u] \\ar[r] \\ar[l] & R/\\mathfrak p \\ar[u] \\ar[r] & \\kappa(\\mathfrak p) \\ar[u] } $$ In this diagram the arrows in the outer left and outer right columns are identical. The horizontal maps induce on the associated spectra always a homeomorphism onto the image. The lower two rows of the diagram make sense without assuming $\\mathfrak q$ exists. The lower squares induce fibre squares of topological spaces. This diagram shows that $\\mathfrak p$ is in the image of the map on Spec if and only if $S \\otimes_R \\kappa(\\mathfrak p)$ is not the zero ring."}
+{"_id": "1559", "title": "algebra-remark-cohen-bound-cardinality", "text": "Let $R$ be a ring. Let $\\kappa$ be an infinite cardinal. By applying Example \\ref{example-oka-family-bound-cardinality} and Proposition \\ref{proposition-oka} we see that any ideal maximal with respect to the property of not being generated by $\\kappa$ elements is prime. This result is not so useful because there exists a ring for which every prime ideal of $R$ can be generated by $\\aleph_0$ elements, but some ideal cannot. Namely, let $k$ be a field, let $T$ be a set whose cardinality is greater than $\\aleph_0$ and let $$ R = k[\\{x_n\\}_{n \\geq 1}, \\{z_{t, n}\\}_{t \\in T, n \\geq 0}]/ (x_n^2, z_{t, n}^2, x_n z_{t, n} - z_{t, n - 1}) $$ This is a local ring with unique prime ideal $\\mathfrak m = (x_n)$. But the ideal $(z_{t, n})$ cannot be generated by countably many elements."}
+{"_id": "1560", "title": "algebra-remark-intersection-powers-ideal", "text": "Lemma \\ref{lemma-intersect-powers-ideal-module-zero} in particular implies that $\\bigcap_n I^n = (0)$ when $I \\subset R$ is a non-unit ideal in a Noetherian local ring $R$. More generally, let $R$ be a Noetherian ring and $I \\subset R$ an ideal. Suppose that $f \\in \\bigcap_{n \\in \\mathbf{N}} I^n$. Then Lemma \\ref{lemma-intersection-powers-ideal-module} says that for every prime ideal $I \\subset \\mathfrak p$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$ such that $f$ maps to zero in $R_g$. In algebraic geometry we express this by saying that ``$f$ is zero in an open neighbourhood of the closed set $V(I)$ of $\\Spec(R)$''."}
+{"_id": "1561", "title": "algebra-remark-period-polynomial", "text": "If $S$ is still Noetherian but $S$ is not generated in degree $1$, then the function associated to a graded $S$-module is a periodic polynomial (i.e., it is a numerical polynomial on the congruence classes of integers modulo $n$ for some $n$)."}
+{"_id": "1562", "title": "algebra-remark-ass-reverse-functorial", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\\Spec(\\varphi)(\\text{Ass}_S(M)) \\supset \\text{Ass}_R(M)$. For example, consider the ring map $R = k \\to S = k[x_1, x_2, x_3, \\ldots]/(x_i^2)$ and $M = S$. Then $\\text{Ass}_R(M)$ is not empty, but $\\text{Ass}_S(S)$ is empty."}
+{"_id": "1563", "title": "algebra-remark-bourbaki", "text": "Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Let $\\mathfrak p$ be a prime of $R$. Then $$ \\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p)) = \\text{Ass}_{S/\\mathfrak pS}(N \\otimes_R \\kappa(\\mathfrak p)) = \\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p)). $$ The first equality by Lemma \\ref{lemma-ass-quotient-ring} and the second by Lemma \\ref{lemma-localize-ass} part (1)."}
+{"_id": "1564", "title": "algebra-remark-weakly-ass-not-functorial", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Then it is not always the case that $\\Spec(\\varphi)(\\text{WeakAss}_S(M)) \\subset \\text{WeakAss}_R(M)$ contrary to the case of associated primes (see Lemma \\ref{lemma-ass-functorial}). An example is to consider the ring map $$ R = k[x_1, x_2, x_3, \\ldots] \\to S = k[x_1, x_2, x_3, \\ldots, y_1, y_2, y_3, \\ldots]/ (x_1y_1, x_2y_2, x_3y_3, \\ldots) $$ and $M = S$. In this case $\\mathfrak q = \\sum x_iS$ is a minimal prime of $S$, hence a weakly associated prime of $M = S$ (see Lemma \\ref{lemma-weakly-ass-minimal-prime-support}). But on the other hand, for any nonzero element of $S$ the annihilator in $R$ is finitely generated, and hence does not have radical equal to $R \\cap \\mathfrak q = (x_1, x_2, x_3, \\ldots)$ (details omitted)."}
+{"_id": "1565", "title": "algebra-remark-ass-functorial", "text": "Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module. Denote $f : \\Spec(S) \\to \\Spec(R)$ the associated map on spectra. Then we have $$ f(\\text{Ass}_S(M)) \\subset \\text{Ass}_R(M) \\subset \\text{WeakAss}_R(M) \\subset f(\\text{WeakAss}_S(M)) $$ see Lemmas \\ref{lemma-ass-functorial}, \\ref{lemma-weakly-ass-reverse-functorial}, and \\ref{lemma-weakly-ass-support}. In general all of the inclusions may be strict, see Remarks \\ref{remark-ass-reverse-functorial} and \\ref{remark-weakly-ass-not-functorial}. If $S$ is Noetherian, then all the inclusions are equalities as the outer two are equal by Lemma \\ref{lemma-ass-weakly-ass}."}
+{"_id": "1566", "title": "algebra-remark-koszul-regular", "text": "In the paper \\cite{Kabele} the author introduces two more regularity conditions for sequences $x_1, \\ldots, x_r$ of elements of a ring $R$. Namely, we say the sequence is {\\it Koszul-regular} if $H_i(K_{\\bullet}(R, x_{\\bullet})) = 0$ for $i \\geq 1$ where $K_{\\bullet}(R, x_{\\bullet})$ is the Koszul complex. The sequence is called {\\it $H_1$-regular} if $H_1(K_{\\bullet}(R, x_{\\bullet})) = 0$. If $R$ is a local ring (possibly non-Noetherian) and the sequence consists of elements of the maximal ideal, then one has the implications regular $\\Rightarrow$ Koszul-regular $\\Rightarrow$ $H_1$-regular $\\Rightarrow$ quasi-regular. By examples the author shows that these implications cannot be reversed in general. We introduce these notions in more detail in More on Algebra, Section \\ref{more-algebra-section-koszul-regular}."}
+{"_id": "1567", "title": "algebra-remark-join-quasi-regular-sequences", "text": "Let $k$ be a field. Consider the ring $$ A = k[x, y, w, z_0, z_1, z_2, \\ldots]/ (y^2z_0 - wx, z_0 - yz_1, z_1 - yz_2, \\ldots) $$ In this ring $x$ is a nonzerodivisor and the image of $y$ in $A/xA$ gives a quasi-regular sequence. But it is not true that $x, y$ is a quasi-regular sequence in $A$ because $(x, y)/(x, y)^2$ isn't free of rank two over $A/(x, y)$ due to the fact that $wx = 0$ in $(x, y)/(x, y)^2$ but $w$ isn't zero in $A/(x, y)$. Hence the analogue of Lemma \\ref{lemma-join-regular-sequences} does not hold for quasi-regular sequences."}
+{"_id": "1568", "title": "algebra-remark-signs-double-complex", "text": "The isomorphism constructed above is the ``correct'' one only up to signs. A good part of homological algebra is concerned with choosing signs for various maps and showing commutativity of diagrams with intervention of suitable signs. For the moment we will simply use the isomorphism as given in the proof above, and worry about signs later."}
+{"_id": "1569", "title": "algebra-remark-curiosity-signs-swap", "text": "An interesting case occurs when $M = N$ in the above. In this case we get a canonical map $\\text{Tor}_i^R(M, M) \\to \\text{Tor}_i^R(M, M)$. Note that this map is not the identity, because even when $i = 0$ this map is not the identity! For example, if $V$ is a vector space of dimension $n$ over a field, then the switch map $V \\otimes_k V \\to V \\otimes_k V$ has $(n^2 + n)/2$ eigenvalues $+1$ and $(n^2-n)/2$ eigenvalues $-1$. In characteristic $2$ it is not even diagonalizable. Note that even changing the sign of the map will not get rid of this."}
+{"_id": "1570", "title": "algebra-remark-Tor-ring-mod-ideal", "text": "The proof of Lemma \\ref{lemma-characterize-flat} actually shows that $$ \\text{Tor}_1^R(M, R/I) = \\Ker(I \\otimes_R M \\to M). $$"}
+{"_id": "1571", "title": "algebra-remark-warning", "text": "It is not true that a finite $R$-module which is $R$-flat is automatically projective. A counter example is where $R = \\mathcal{C}^\\infty(\\mathbf{R})$ is the ring of infinitely differentiable functions on $\\mathbf{R}$, and $M = R_{\\mathfrak m} = R/I$ where $\\mathfrak m = \\{f \\in R \\mid f(0) = 0\\}$ and $I = \\{f \\in R \\mid \\exists \\epsilon, \\epsilon > 0 : f(x) = 0\\ \\forall x, |x| < \\epsilon\\}$."}
+{"_id": "1572", "title": "algebra-remark-flat-ML", "text": "Let $M$ be a flat $R$-module. By Lazard's theorem (Theorem \\ref{theorem-lazard}) we can write $M = \\colim M_i$ as the colimit of a directed system $(M_i, f_{ij})$ where the $M_i$ are free finite $R$-modules. For $M$ to be Mittag-Leffler, it is enough for the inverse system of duals $(\\Hom_R(M_i, R), \\Hom_R(f_{ij}, R))$ to be Mittag-Leffler. This follows from criterion (4) of Proposition \\ref{proposition-ML-characterization} and the fact that for a free finite $R$-module $F$, there is a functorial isomorphism $\\Hom_R(F, R) \\otimes_R N \\cong \\Hom_R(F, N)$ for any $R$-module $N$."}
+{"_id": "1573", "title": "algebra-remark-go-up-ML-modules", "text": "Let $R \\to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module which is Mittag-Leffler as an $R$-module. Then it is in general not the case that $M$ is Mittag-Leffler as an $S$-module. For example suppose that $S$ is the ring of dual numbers over $R$, i.e., $S = R \\oplus R\\epsilon$ with $\\epsilon^2 = 0$. Then an $S$-module consists of an $R$-module $M$ endowed with a square zero $R$-linear endomorphism $\\epsilon : M \\to M$. Now suppose that $M_0$ is an $R$-module which is not Mittag-Leffler. Choose a presentation $F_1 \\xrightarrow{u} F_0 \\to M_0 \\to 0$ with $F_1$ and $F_0$ free $R$-modules. Set $M = F_1 \\oplus F_0$ with $$ \\epsilon = \\left( \\begin{matrix} 0 & 0 \\\\ u & 0 \\end{matrix} \\right) : M \\longrightarrow M. $$ Then $M/\\epsilon M \\cong F_1 \\oplus M_0$ is not Mittag-Leffler over $R = S/\\epsilon S$, hence not Mittag-Leffler over $S$ (see Lemma \\ref{lemma-mod-ideal-ML-modules}). On the other hand, $M/\\epsilon M = M \\otimes_S S/\\epsilon S$ which would be Mittag-Leffler over $S$ if $M$ was, see Lemma \\ref{lemma-tensor-ML-modules}."}
+{"_id": "1574", "title": "algebra-remark-characterize-projective", "text": "Lemma \\ref{lemma-countgen-projective} does not hold without the countable generation assumption.  For example, the $\\mathbf Z$-module $M = \\mathbf{Z}[[x]]$ is flat and Mittag-Leffler but not projective.  It is Mittag-Leffler by Lemma \\ref{lemma-power-series-ML}.  Subgroups of free abelian groups are free, hence a projective $\\mathbf Z$-module is in fact free and so are its submodules.  Thus to show $M$ is not projective it suffices to produce a non-free submodule.  Fix a prime $p$ and consider the submodule $N$ consisting of power series $f(x) = \\sum a_i x^i$ such that for every integer $m \\geq 1$, $p^m$ divides $a_i$ for all but finitely many $i$.  Then $\\sum a_i p^i x^i$ is in $N$ for all $a_i \\in \\mathbf{Z}$, so $N$ is uncountable.  Thus if $N$ were free it would have uncountable rank and the dimension of $N/pN$ over $\\mathbf{Z}/p$ would be uncountable.  This is not true as the elements $x^i \\in N/pN$ for $i \\geq 0$ span $N/pN$."}
+{"_id": "1575", "title": "algebra-remark-matrices-associated-to-elements-epicenter", "text": "Let $R \\to S$ be a ring map. Sometimes the set of elements $g \\in S$ such that $g \\otimes 1 = 1 \\otimes g$ is called the {\\it epicenter} of $S$. It is an $R$-algebra. By the construction of Lemma \\ref{lemma-kernel-difference-projections} we get for each $g$ in the epicenter a matrix factorization $$ (g) = Y X Z $$ with $X \\in \\text{Mat}(n \\times n, R)$, $Y \\in \\text{Mat}(1 \\times n, S)$, and $Z \\in \\text{Mat}(n \\times 1, S)$. Namely, let $x_{i, j}, y_i, z_j$ be as in part (2) of the lemma. Set $X = (x_{i, j})$, let $y$ be the row vector whose entries are the $y_i$ and let $z$ be the column vector whose entries are the $z_j$. With this notation conditions (b) and (c) of Lemma \\ref{lemma-kernel-difference-projections} mean exactly that $Y X \\in \\text{Mat}(1 \\times n, R)$, $X Z \\in \\text{Mat}(n \\times 1, R)$. It turns out to be very convenient to consider the triple of matrices $(X, YX, XZ)$. Given $n \\in \\mathbf{N}$ and a triple $(P, U, V)$ we say that $(P, U, V)$ is a {\\it $n$-triple associated to $g$} if there exists a matrix factorization as above such that $P = X$, $U = YX$ and $V = XZ$."}
+{"_id": "1576", "title": "algebra-remark-resolution-dim-1", "text": "Suppose that $R$ is a $1$-dimensional semi-local Noetherian domain. If there is a maximal ideal $\\mathfrak m \\subset R$ such that $R_{\\mathfrak m}$ is not regular, then we may apply Lemma \\ref{lemma-nonregular-dimension-one} to $(R, \\mathfrak m)$ to get a finite ring extension $R \\subset R_1$. (For example one can do this so that $\\Spec(R_1) \\to \\Spec(R)$ is the blowup of $\\Spec(R)$ in the ideal $\\mathfrak m$.) Of course $R_1$ is a $1$-dimensional semi-local Noetherian domain with the same fraction field as $R$. If $R_1$ is not a regular semi-local ring, then we may repeat the construction to get $R_1 \\subset R_2$. Thus we get a sequence $$ R \\subset R_1 \\subset R_2 \\subset R_3 \\subset \\ldots $$ of finite ring extensions which may stop if $R_n$ is regular for some $n$. Resolution of singularities would be the claim that eventually $R_n$ is indeed regular. In reality this is not the case. Namely, there exists a characteristic $0$ Noetherian local domain $A$ of dimension $1$ whose completion is nonreduced, see \\cite[Proposition 3.1]{Ferrand-Raynaud} or our Examples, Section \\ref{examples-section-local-completion-nonreduced}. For an example in characteristic $p > 0$ see Example \\ref{example-bad-dvr-char-p}. Since the construction of blowing up commutes with completion it is easy to see the sequence never stabilizes. See \\cite{Bennett} for a discussion (mostly in positive characteristic). On the other hand, if the completion of $R$ in all of its maximal ideals is reduced, then the procedure stops (insert future reference here)."}
+{"_id": "1577", "title": "algebra-remark-suitable-systems-limits", "text": "Suppose that $R \\to S$ is a local homomorphism of local rings, which is essentially of finite presentation. Take any system $(\\Lambda, \\leq)$, $R_\\lambda \\to S_\\lambda$ with the properties listed in Lemma \\ref{lemma-limit-essentially-finite-type}. What may happen is that this is the ``wrong'' system, namely, it may happen that property (4) of Lemma \\ref{lemma-limit-essentially-finite-presentation} is not satisfied. Here is an example. Let $k$ be a field. Consider the ring $$ R = k[[z, y_1, y_2, \\ldots]]/(y_i^2 - zy_{i + 1}). $$ Set $S = R/zR$. As system take $\\Lambda = \\mathbf{N}$ and $R_n = k[[z, y_1, \\ldots, y_n]]/(\\{y_i^2 - zy_{i + 1}\\}_{i \\leq n-1})$ and $S_n = R_n/(z, y_n^2)$. All the maps $S_n \\otimes_{R_n} R_{n + 1} \\to S_{n + 1}$ are not localizations (i.e., isomorphisms in this case) since $1 \\otimes y_{n + 1}^2$ maps to zero. If we take instead $S_n' = R_n/zR_n$ then the maps $S'_n \\otimes_{R_n} R_{n + 1} \\to S'_{n + 1}$ are isomorphisms. The moral of this remark is that we do have to be a little careful in choosing the systems."}
+{"_id": "1578", "title": "algebra-remark-functoriality-principal-parts", "text": "Suppose given a commutative diagram of rings $$ \\xymatrix{ B \\ar[r] & B' \\\\ A \\ar[u] \\ar[r] & A' \\ar[u] } $$ a $B$-module $M$, a $B'$-module $M'$, and a $B$-linear map $M \\to M'$. Then we get a compatible system of module maps $$ \\xymatrix{ \\ldots \\ar[r] & P^2_{B'/A'}(M') \\ar[r] & P^1_{B'/A'}(M') \\ar[r] & P^0_{B'/A'}(M') \\\\ \\ldots \\ar[r] & P^2_{B/A}(M) \\ar[r] \\ar[u] & P^1_{B/A}(M) \\ar[r] \\ar[u] & P^0_{B/A}(M) \\ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $P^k_{B/A}(M)$ in terms of generators and relations in the proof of Lemma \\ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}."}
+{"_id": "1579", "title": "algebra-remark-composition-homotopy-equivalent-to-zero", "text": "Let $A \\to B$ and $\\phi : B \\to C$ be ring maps. Then the composition $\\NL_{B/A} \\to \\NL_{C/A} \\to \\NL_{C/B}$ is homotopy equivalent to zero. Namely, this composition is the functoriality of the naive cotangent complex for the square $$ \\xymatrix{ B \\ar[r]_\\phi & C \\\\ A \\ar[r] \\ar[u] & B \\ar[u] } $$ Write $J = \\Ker(B[C] \\to C)$. An explicit homotopy is given by the map $\\Omega_{A[B]/A} \\otimes_A B \\to J/J^2$ which maps the basis element $\\text{d}[b]$ to the class of $[\\phi(b)] - b$ in $J/J^2$."}
+{"_id": "1580", "title": "algebra-remark-lemma-characterize-formally-smooth", "text": "Lemma \\ref{lemma-characterize-formally-smooth} holds more generally whenever $P$ is formally smooth over $R$."}
+{"_id": "1581", "title": "algebra-remark-construct-sh-from-h", "text": "We can also construct $R^{sh}$ from $R^h$. Namely, for any finite separable subextension $\\kappa \\subset \\kappa' \\subset \\kappa^{sep}$ there exists a unique (up to unique isomorphism) finite \\'etale local ring extension $R^h \\subset R^h(\\kappa')$ whose residue field extension reproduces the given extension, see Lemma \\ref{lemma-henselian-cat-finite-etale}. Hence we can set $$ R^{sh} = \\bigcup\\nolimits_{\\kappa \\subset \\kappa' \\subset \\kappa^{sep}} R^h(\\kappa') $$ The arrows in this system, compatible with the arrows on the level of residue fields, exist by Lemma \\ref{lemma-henselian-cat-finite-etale}. This will produce a henselian local ring by Lemma \\ref{lemma-colimit-henselian} since each of the rings $R^h(\\kappa')$ is henselian by Lemma \\ref{lemma-finite-over-henselian}. By construction the residue field extension induced by $R^h \\to R^{sh}$ is the field extension $\\kappa \\subset \\kappa^{sep}$. Hence $R^{sh}$ so constructed is strictly henselian. By Lemma \\ref{lemma-composition-colimit-etale} the $R$-algebra $R^{sh}$ is a colimit of \\'etale $R$-algebras. Hence the uniqueness of Lemma \\ref{lemma-uniqueness-henselian} shows that $R^{sh}$ is the strict henselization."}
+{"_id": "1582", "title": "algebra-remark-Noetherian-complete-local-ring-universally-catenary", "text": "If $k$ is a field then the power series ring $k[[X_1, \\ldots, X_d]]$ is a Noetherian complete local regular ring of dimension $d$. If $\\Lambda$ is a Cohen ring then $\\Lambda[[X_1, \\ldots, X_d]]$ is a complete local Noetherian regular ring of dimension $d + 1$. Hence the Cohen structure theorem implies that any Noetherian complete local ring is a quotient of a regular local ring. In particular we see that a Noetherian complete local ring is universally catenary, see Lemma \\ref{lemma-CM-ring-catenary} and Lemma \\ref{lemma-regular-ring-CM}."}
+{"_id": "1583", "title": "algebra-remark-universally-catenary-does-not-descend", "text": "The property of being ``universally catenary'' does not descend; not even along \\'etale ring maps. In Examples, Section \\ref{examples-section-non-catenary-Noetherian-local} there is a construction of a finite ring map $A \\to B$ with $A$ local Noetherian and not universally catenary, $B$ semi-local with two maximal ideals $\\mathfrak m$, $\\mathfrak n$ with $B_{\\mathfrak m}$ and $B_{\\mathfrak n}$ regular of dimension $2$ and $1$ respectively, and the same residue fields as that of $A$. Moreover, $\\mathfrak m_A$ generates the maximal ideal in both $B_{\\mathfrak m}$ and $B_{\\mathfrak n}$ (so $A \\to B$ is unramified as well as finite). By Lemma \\ref{lemma-etale-makes-unramified-closed} there exists a local \\'etale ring map $A \\to A'$ such that $B \\otimes_A A' = B_1 \\times B_2$ decomposes with $A' \\to B_i$ surjective. This shows that $A'$ has two minimal primes $\\mathfrak q_i$ with $A'/\\mathfrak q_i \\cong B_i$. Since $B_i$ is regular local (since it is \\'etale over either $B_{\\mathfrak m}$ or $B_{\\mathfrak n}$) we conclude that $A'$ is universally catenary."}
+{"_id": "1649", "title": "moduli-curves-remark-boundedness-aut-does-not-work-surfaces", "text": "The boundedness argument in the proof of Lemma \\ref{lemma-curves-diagonal-separated-fp} does not work for moduli of surfaces and in fact, the result is wrong, for example because K3 surfaces over fields can have infinite discrete automorphism groups. The ``reason'' the argument does not work is that on a projective surface $S$ over a field, given ample invertible sheaves $\\mathcal{N}$ and $\\mathcal{L}$ with Hilbert polynomials $Q$ and $P$, there is no a priori bound on the Hilbert polynomial of $\\mathcal{N} \\otimes_{\\mathcal{O}_S} \\mathcal{L}$. In terms of intersection theory, if $H_1$, $H_2$ are ample effective Cartier divisors on $S$, then there is no (upper) bound on the intersection number $H_1 \\cdot H_2$ in terms of $H_1 \\cdot H_1$ and $H_2 \\cdot H_2$."}
+{"_id": "1702", "title": "dpa-remark-forgetful", "text": "The forgetful functor $(A, I, \\gamma) \\mapsto A$ does not commute with colimits. For example, let $$ \\xymatrix{ (B, J, \\delta) \\ar[r] & (B'', J'', \\delta'') \\\\ (A, I, \\gamma) \\ar[r] \\ar[u] & (B', J', \\delta') \\ar[u] } $$ be a pushout in the category of divided power rings. Then in general the map $B \\otimes_A B' \\to B''$ isn't an isomorphism. (It is always surjective.) An explicit example is given by $(A, I, \\gamma) = (\\mathbf{Z}, (0), \\emptyset)$, $(B, J, \\delta) = (\\mathbf{Z}/4\\mathbf{Z}, 2\\mathbf{Z}/4\\mathbf{Z}, \\delta)$, and $(B', J', \\delta') = (\\mathbf{Z}/4\\mathbf{Z}, 2\\mathbf{Z}/4\\mathbf{Z}, \\delta')$ where $\\delta_2(2) = 2$ and $\\delta'_2(2) = 0$ and all higher divided powers equal to zero. Then $(B'', J'', \\delta'') = (\\mathbf{F}_2, (0), \\emptyset)$ which doesn't agree with the tensor product. However, note that it is always true that $$ B''/J'' = B/J \\otimes_{A/I} B'/J' $$ as can be seen from the universal property of the pushout by considering maps into divided power algebras of the form $(C, (0), \\emptyset)$."}
+{"_id": "1703", "title": "dpa-remark-divided-power-polynomial-algebra", "text": "Let $(A, I, \\gamma)$ be a divided power ring. There is a variant of Lemma \\ref{lemma-divided-power-polynomial-algebra} for infinitely many variables. First note that if $s < t$ then there is a canonical map $$ A\\langle x_1, \\ldots, x_s \\rangle \\to A\\langle x_1, \\ldots, x_t\\rangle $$ Hence if $W$ is any set, then we set $$ A\\langle x_w: w \\in W\\rangle = \\colim_{E \\subset W} A\\langle x_e:e \\in E\\rangle $$ (colimit over $E$ finite subset of $W$) with transition maps as above. By the definition of a colimit we see that the universal mapping property of $A\\langle x_w: w \\in W\\rangle$ is completely analogous to the mapping property stated in Lemma \\ref{lemma-divided-power-polynomial-algebra}."}
+{"_id": "1704", "title": "dpa-remark-adjoining-set-of-variables", "text": "We can also adjoin a set (possibly infinite) of exterior or divided power generators in a given degree $d > 0$, rather than just one as in Examples \\ref{example-adjoining-odd} and \\ref{example-adjoining-even}. Namely,  following Remark \\ref{remark-divided-power-polynomial-algebra}: for $(A,\\gamma)$ as above and a set $J$, let $A\\langle T_j:j\\in J\\rangle$ be the directed colimit of the algebras $A\\langle T_j:j\\in S\\rangle$ over all finite subsets $S$ of $J$. It is immediate that this algebra has a unique divided power structure, compatible with the given structure on $A$ and on each generator $T_j$."}
+{"_id": "1705", "title": "dpa-remark-no-good-ci-map", "text": "It appears difficult to define an good notion of ``local complete intersection homomorphisms'' for maps between general Noetherian rings. The reason is that, for a local Noetherian ring $A$, the fibres of $A \\to A^\\wedge$ are not local complete intersection rings. Thus, if $A \\to B$ is a local homomorphism of local Noetherian rings, and the map of completions $A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism in the sense defined above, then $(A_\\mathfrak p)^\\wedge \\to (B_\\mathfrak q)^\\wedge$ is in general {\\bf not} a complete intersection homomorphism in the sense defined above. A solution can be had by working exclusively with excellent Noetherian rings. More generally, one could work with those Noetherian rings whose formal fibres are complete intersections, see \\cite{Rodicio-ci}. We will develop this theory in Dualizing Complexes, Section \\ref{dualizing-section-formal-fibres}."}
+{"_id": "1754", "title": "moduli-remark-quot-numerical", "text": "Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section. Let $I$ be a set and for $i \\in I$ let $E_i \\in D(\\mathcal{O}_X)$ be perfect. Let $P : I \\to \\mathbf{Z}$ be a function. Recall that we have a morphism $$ \\Quotfunctor_{\\mathcal{F}/X/B} \\longrightarrow \\Cohstack_{X/B} $$ which sends the element $\\mathcal{F}_T \\to \\mathcal{Q}$ of $\\Quotfunctor_{\\mathcal{F}/X/B}(T)$ to the object $\\mathcal{Q}$ of $\\Cohstack_{X/B}$ over $T$, see proof of Quot, Proposition \\ref{quot-proposition-quot}. Hence we can form the fibre product diagram $$ \\xymatrix{ \\Quotfunctor^P_{\\mathcal{F}/X/B} \\ar[r] \\ar[d] & \\Cohstack^P_{X/B} \\ar[d] \\\\ \\Quotfunctor_{\\mathcal{F}/X/B} \\ar[r] & \\Cohstack_{X/B} } $$ This is the defining diagram for the algebraic space in the upper left corner. The left vertical arrow is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \\mathbf{Z}$ and $E_i = \\mathcal{L}^{\\otimes i}$ for some invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ (in the last case we sometimes use the notation $\\Quotfunctor^{P, \\mathcal{L}}_{\\mathcal{F}/X/B}$). See Situation \\ref{situation-numerical} and Lemmas \\ref{lemma-open-P} and \\ref{lemma-finite-list-perfect-objects} and Example \\ref{example-hilbert-polynomial}."}
+{"_id": "1755", "title": "moduli-remark-hilb-numerical", "text": "Let $f : X \\to B$ be as in the introduction to this section. Let $I$ be a set and for $i \\in I$ let $E_i \\in D(\\mathcal{O}_X)$ be perfect. Let $P : I \\to \\mathbf{Z}$ be a function. Recall that $\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$, see Quot, Lemma \\ref{quot-lemma-hilb-is-quot}. Thus we can define $$ \\Hilbfunctor^P_{X/B} = \\Quotfunctor^P_{\\mathcal{O}_X/X/B} $$ where $\\Quotfunctor^P_{\\mathcal{O}_X/X/B}$ is as in Remark \\ref{remark-quot-numerical}. The morphism $$ \\Hilbfunctor^P_{X/B} \\longrightarrow \\Hilbfunctor_{X/B} $$ is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \\mathbf{Z}$ and $E_i = \\mathcal{L}^{\\otimes i}$ for some invertible $\\mathcal{O}_X$-module $\\mathcal{L}$. In the last case we sometimes use the notation $\\Hilbfunctor^{P, \\mathcal{L}}_{X/B}$."}
+{"_id": "1756", "title": "moduli-remark-Mor-numerical", "text": "Let $B, X, Y$ be as in the introduction to this section. Let $I$ be a set and for $i \\in I$ let $E_i \\in D(\\mathcal{O}_{Y \\times_B X})$ be perfect. Let $P : I \\to \\mathbf{Z}$ be a function. Recall that $$ \\mathit{Mor}_B(Y, X) \\subset \\Hilbfunctor_{Y \\times_B X/B} $$ is an open subspace, see Quot, Lemma \\ref{quot-lemma-Mor-into-Hilb-open}. Thus we can define $$ \\mathit{Mor}^P_B(Y, X) = \\mathit{Mor}_B(Y, X) \\cap \\Hilbfunctor^P_{Y \\times_B X/B} $$ where $\\Hilbfunctor^P_{Y \\times_B X/B}$ is as in Remark \\ref{remark-hilb-numerical}. The morphism $$ \\mathit{Mor}^P_B(Y, X) \\longrightarrow \\mathit{Mor}_B(Y, X) $$ is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \\mathbf{Z}$, $E_i = \\mathcal{L}^{\\otimes i}$ for some invertible $\\mathcal{O}_{Y \\times_B X}$-module $\\mathcal{L}$. In the last case we sometimes use the notation $\\mathit{Mor}^{P, \\mathcal{L}}_B(Y, X)$."}
+{"_id": "2009", "title": "derived-remark-special-triangles", "text": "Let $\\mathcal{D}$ be an additive category with translation functors $[n]$ as in Definition \\ref{definition-triangle}. Let us call a triangle $(X, Y, Z, f, g, h)$ {\\it special}\\footnote{This is nonstandard notation.} if for every object $W$ of $\\mathcal{D}$ the long sequence of abelian groups $$ \\ldots \\to \\Hom_\\mathcal{D}(W, X) \\to \\Hom_\\mathcal{D}(W, Y) \\to \\Hom_\\mathcal{D}(W, Z) \\to \\Hom_\\mathcal{D}(W, X[1]) \\to \\ldots $$ is exact.  The proof of Lemma \\ref{lemma-third-isomorphism-triangle} shows that if $$ (a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h') $$ is a morphism of special triangles and if two among $a, b, c$ are isomorphisms so is the third. There is a dual statement for {\\it co-special} triangles, i.e., triangles which turn into long exact sequences on applying the functor $\\Hom_\\mathcal{D}(-, W)$. Thus distinguished triangles are special and co-special, but in general there are many more (co-)special triangles, than there are distinguished triangles."}
+{"_id": "2010", "title": "derived-remark-compute-modules", "text": "To see the last displayed equality in the proof above we can argue with elements as follows. We have $s\\pi(l, k, k^{+}) = (l, 0, 0)$. Hence the morphism of the left hand side maps $(l, k, k^{+})$ to $(0, k, k^{+})$. On the other hand $h(l, k, k^{+}) = (0, 0, k)$ and $d(l, k, k^{+}) = (dl, dk + k^{+}, -dk^{+})$. Hence $(dh + hd)(l, k, k^{+}) = d(0, 0, k) + h(dl, dk + k^{+}, -dk^{+}) = (0, k, -dk) + (0, 0, dk + k^{+}) = (0, k, k^{+})$ as desired."}
+{"_id": "2011", "title": "derived-remark-make-commute", "text": "Let $\\mathcal{A}$ be an additive category. Let $0 \\to A_i^\\bullet \\to B_i^\\bullet \\to C_i^\\bullet \\to 0$, $i = 1, 2$ be termwise split exact sequences. Suppose that $a : A_1^\\bullet \\to A_2^\\bullet$, $b : B_1^\\bullet \\to B_2^\\bullet$, and $c : C_1^\\bullet \\to C_2^\\bullet$ are morphisms of complexes such that $$ \\xymatrix{ A_1^\\bullet \\ar[d]_a \\ar[r] & B_1^\\bullet \\ar[r] \\ar[d]_b & C_1^\\bullet \\ar[d]_c \\\\ A_2^\\bullet \\ar[r] & B_2^\\bullet \\ar[r] & C_2^\\bullet } $$ commutes in $K(\\mathcal{A})$. In general, there does {\\bf not} exist a morphism $b' : B_1^\\bullet \\to B_2^\\bullet$ which is homotopic to $b$ such that the diagram above commutes in the category of complexes. Namely, consider Examples, Equation (\\ref{examples-equation-commutes-up-to-homotopy}). If we could replace the middle map there by a homotopic one such that the diagram commutes, then we would have additivity of traces which we do not."}
+{"_id": "2012", "title": "derived-remark-boundedness-conditions-triangulated", "text": "Let $\\mathcal{A}$ be an additive category. Exactly the same proof as the proof of Proposition \\ref{proposition-homotopy-category-triangulated} shows that the categories $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$ are triangulated categories. Namely, the cone of a morphism between bounded (above, below) is bounded (above, below). But we prove below that these are triangulated subcategories of $K(\\mathcal{A})$ which gives another proof."}
+{"_id": "2013", "title": "derived-remark-homotopy-double", "text": "Let $\\mathcal{A}$ be an additive category with countable direct sums. Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes in $\\mathcal{A}$, see Homology, Section \\ref{homology-section-double-complexes}. We can use this category to construct two triangulated categories. \\begin{enumerate} \\item We can consider an object $A^{\\bullet, \\bullet}$ of $\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes as follows $$ \\ldots \\to A^{\\bullet, -1} \\to A^{\\bullet, 0} \\to A^{\\bullet, 1} \\to \\ldots $$ and take the homotopy category $K_{first}(\\text{DoubleComp}(\\mathcal{A}))$ with the corresponding triangulated structure given by Proposition \\ref{proposition-homotopy-category-triangulated}. By Homology, Remark \\ref{homology-remark-double-complex-complex-of-complexes-first} the functor $$ \\text{Tot} : K_{first}(\\text{DoubleComp}(\\mathcal{A})) \\longrightarrow K(\\mathcal{A}) $$ is an exact functor of triangulated categories. \\item We can consider an object $A^{\\bullet, \\bullet}$ of $\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes as follows $$ \\ldots \\to A^{-1, \\bullet} \\to A^{0, \\bullet} \\to A^{1, \\bullet} \\to \\ldots $$ and take the homotopy category $K_{second}(\\text{DoubleComp}(\\mathcal{A}))$ with the corresponding triangulated structure given by Proposition \\ref{proposition-homotopy-category-triangulated}. By Homology, Remark \\ref{homology-remark-double-complex-complex-of-complexes-second} the functor $$ \\text{Tot} : K_{second}(\\text{DoubleComp}(\\mathcal{A})) \\longrightarrow K(\\mathcal{A}) $$ is an exact functor of triangulated categories. \\end{enumerate}"}
+{"_id": "2014", "title": "derived-remark-double-complex-as-tensor-product-of", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$ be additive categories and assume $\\mathcal{C}$ has countable direct sums. Suppose that $$ \\otimes : \\mathcal{A} \\times \\mathcal{B} \\longrightarrow \\mathcal{C}, \\quad (X, Y) \\longmapsto X \\otimes Y $$ is a functor which is bilinear on morphisms. This determines a functor $$ \\text{Comp}(\\mathcal{A}) \\times \\text{Comp}(\\mathcal{B}) \\longrightarrow \\text{DoubleComp}(\\mathcal{C}), \\quad (X^\\bullet, Y^\\bullet) \\longmapsto X^\\bullet \\otimes Y^\\bullet $$ See Homology, Example \\ref{homology-example-double-complex-as-tensor-product-of}. \\begin{enumerate} \\item For a fixed object $X^\\bullet$ of $\\text{Comp}(\\mathcal{A})$ the functor $$ K(\\mathcal{B}) \\longrightarrow K(\\mathcal{C}), \\quad Y^\\bullet \\longmapsto \\text{Tot}(X^\\bullet \\otimes Y^\\bullet) $$ is an exact functor of triangulated categories. \\item For a fixed object $Y^\\bullet$ of $\\text{Comp}(\\mathcal{B})$ the functor $$ K(\\mathcal{A}) \\longrightarrow K(\\mathcal{C}), \\quad X^\\bullet \\longmapsto \\text{Tot}(X^\\bullet \\otimes Y^\\bullet) $$ is an exact functor of triangulated categories. \\end{enumerate} This follows from Remark \\ref{remark-homotopy-double} since the functors $\\text{Comp}(\\mathcal{A}) \\to \\text{DoubleComp}(\\mathcal{C})$, $Y^\\bullet \\mapsto X^\\bullet \\otimes Y^\\bullet$ and $\\text{Comp}(\\mathcal{B}) \\to \\text{DoubleComp}(\\mathcal{C})$, $X^\\bullet \\mapsto X^\\bullet \\otimes Y^\\bullet$ are immediately seen to be compatible with homotopies and termwise split short exact sequences and hence induce exact functors of triangulated categories $$ K(\\mathcal{B}) \\to K_{first}(\\text{DoubleComp}(\\mathcal{C})) \\quad\\text{and}\\quad K(\\mathcal{A}) \\to K_{second}(\\text{DoubleComp}(\\mathcal{C})) $$ Observe that for the first of the two the isomorphism $$ \\text{Tot}(X^\\bullet \\otimes Y^\\bullet[1]) \\cong \\text{Tot}(X^\\bullet \\otimes Y^\\bullet)[1] $$ involves signs (this goes back to the signs chosen in Homology, Remark \\ref{homology-remark-shift-double-complex})."}
+{"_id": "2015", "title": "derived-remark-existence-derived", "text": "In this chapter, we consistently work with ``small'' abelian categories (as is the convention in the Stacks project). For a ``big'' abelian category $\\mathcal{A}$, it isn't clear that the derived category $D(\\mathcal{A})$ exists, because it isn't clear that morphisms in the derived category are sets. In fact, in general they aren't, see Examples, Lemma \\ref{examples-lemma-big-abelian-category}. However, if $\\mathcal{A}$ is a Grothendieck abelian category, and given $K^\\bullet, L^\\bullet$ in $K(\\mathcal{A})$, then by Injectives, Theorem \\ref{injectives-theorem-K-injective-embedding-grothendieck} there exists a quasi-isomorphism $L^\\bullet \\to I^\\bullet$ to a K-injective complex $I^\\bullet$ and Lemma \\ref{lemma-K-injective} shows that $$ \\Hom_{D(\\mathcal{A})}(K^\\bullet, L^\\bullet) = \\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet) $$ which is a set. Some examples of Grothendieck abelian categories are the category of modules over a ring, or more generally the category of sheaves of modules on a ringed site."}
+{"_id": "2016", "title": "derived-remark-truncation-distinguished-triangle", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex of $\\mathcal{A}$. Let $a \\in \\mathbf{Z}$. We claim there is a canonical distinguished triangle $$ \\tau_{\\leq a}K^\\bullet \\to K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet \\to (\\tau_{\\leq a}K^\\bullet)[1] $$ in $D(\\mathcal{A})$. Here we have used the canonical truncation functors $\\tau$ from Homology, Section \\ref{homology-section-truncations}. Namely, we first take the distinguished triangle associated by our $\\delta$-functor (Lemma \\ref{lemma-derived-canonical-delta-functor}) to the short exact sequence of complexes $$ 0 \\to \\tau_{\\leq a}K^\\bullet \\to K^\\bullet \\to K^\\bullet/\\tau_{\\leq a}K^\\bullet \\to 0 $$ Next, we use that the map $K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet$ factors through a quasi-isomorphism $K^\\bullet/\\tau_{\\leq a}K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet$ by the description of cohomology groups in Homology, Section \\ref{homology-section-truncations}. In a similar way we obtain canonical distinguished triangles $$ \\tau_{\\leq a}K^\\bullet \\to \\tau_{\\leq a + 1}K^\\bullet \\to H^{a + 1}(K^\\bullet)[-a-1] \\to (\\tau_{\\leq a}K^\\bullet)[1] $$ and $$ H^a(K^\\bullet)[-a] \\to \\tau_{\\geq a}K^\\bullet \\to \\tau_{\\geq a + 1}K^\\bullet \\to H^a(K^\\bullet)[-a + 1] $$"}
+{"_id": "2017", "title": "derived-remark-easier-proofs", "text": "Let $\\mathcal{A}$ be an abelian category. Using the fact that $K(\\mathcal{A})$ is a triangulated category we may use Lemma \\ref{lemma-acyclic-is-zero} to obtain proofs of some of the lemmas below which are usually proved by chasing through diagrams. Namely, suppose that $\\alpha : K^\\bullet \\to L^\\bullet$ is a quasi-isomorphism of complexes. Then $$ (K^\\bullet, L^\\bullet, C(\\alpha)^\\bullet, \\alpha, i, -p) $$ is a distinguished triangle in $K(\\mathcal{A})$ (Lemma \\ref{lemma-the-same-up-to-isomorphisms}) and $C(\\alpha)^\\bullet$ is an acyclic complex (Lemma \\ref{lemma-acyclic}). Next, let $I^\\bullet$ be a bounded below complex of injective objects. Then $$ \\xymatrix{ \\Hom_{K(\\mathcal{A})}(C(\\alpha)^\\bullet, I^\\bullet) \\ar[r] & \\Hom_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) \\ar[r] & \\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet) \\ar[lld] \\\\ \\Hom_{K(\\mathcal{A})}(C(\\alpha)^\\bullet[-1], I^\\bullet) } $$ is an exact sequence of abelian groups, see Lemma \\ref{lemma-representable-homological}. At this point Lemma \\ref{lemma-acyclic-is-zero} guarantees that the outer two groups are zero and hence $\\Hom_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet) = \\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet)$."}
+{"_id": "2018", "title": "derived-remark-easier-projective", "text": "Let $\\mathcal{A}$ be an abelian category. Suppose that $\\alpha : K^\\bullet \\to L^\\bullet$ is a quasi-isomorphism of complexes. Let $P^\\bullet$ be a bounded above complex of projectives. Then $$ \\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet) \\longrightarrow \\Hom_{K(\\mathcal{A})}(P^\\bullet, L^\\bullet) $$ is an isomorphism. This is dual to Remark \\ref{remark-easier-proofs}."}
+{"_id": "2019", "title": "derived-remark-functorial-ss", "text": "The spectral sequences of Lemma \\ref{lemma-two-ss-complex-functor} are functorial in the complex $K^\\bullet$. This follows from functoriality properties of Cartan-Eilenberg resolutions. On the other hand, they are both examples of a more general spectral sequence which may be associated to a filtered complex of $\\mathcal{A}$. The functoriality will follow from its construction. We will return to this in the section on the filtered derived category, see Remark \\ref{remark-final-functorial}."}
+{"_id": "2020", "title": "derived-remark-big-localization", "text": "Suppose that $\\mathcal{A}$ is a ``big'' abelian category with enough injectives such as the category of abelian groups. In this case we have to be slightly more careful in constructing our resolution functor since we cannot use the axiom of choice with a quantifier ranging over a class. But note that the proof of the lemma does show that any two localization functors are canonically isomorphic. Namely, given quasi-isomorphisms $i : K^\\bullet \\to I^\\bullet$ and $i' : K^\\bullet \\to J^\\bullet$ of a bounded below complex $K^\\bullet$ into bounded below complexes of injectives there exists a unique(!) morphism $a : I^\\bullet \\to J^\\bullet$ in $K^{+}(\\mathcal{I})$ such that $i' = i \\circ a$ as morphisms in $K^{+}(\\mathcal{I})$. Hence the only issue is existence, and we will see how to deal with this in the next section."}
+{"_id": "2021", "title": "derived-remark-match", "text": "Suppose $inj$ is a functor such that $s \\circ inj = \\text{id}$ as in part (2) of Lemma \\ref{lemma-functorial-injective-resolutions}. Write $inj(K^\\bullet) = (i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet))$ as in the proof of that lemma. Suppose $\\alpha : K^\\bullet \\to L^\\bullet$ is a map of bounded below complexes. Consider the map $inj(\\alpha)$ in the category $\\text{InjRes}(\\mathcal{A})$. It induces a commutative diagram $$ \\xymatrix{ K^\\bullet \\ar[rr]^-{\\alpha} \\ar[d]_{i_K} & & L^\\bullet \\ar[d]^{i_L} \\\\ j(K)^\\bullet \\ar[rr]^-{inj(\\alpha)} & & j(L)^\\bullet } $$ of morphisms of complexes. Hence, looking at the proof of Lemma \\ref{lemma-resolution-functor} we see that the functor $j : K^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$ is given by the rule $$ j(\\alpha\\text{ up to homotopy}) = inj(\\alpha)\\text{ up to homotopy}\\in \\Hom_{K^{+}(\\mathcal{I})}(j(K^\\bullet), j(L^\\bullet)) $$ Hence we see that $j$ matches $t \\circ inj$ in this case, i.e., the diagram $$ \\xymatrix{ \\text{Comp}^{+}(\\mathcal{A}) \\ar[rr]_{t \\circ inj} \\ar[rd] & & K^{+}(\\mathcal{I}) \\\\ & K^{+}(\\mathcal{A}) \\ar[ru]_j } $$ is commutative."}
+{"_id": "2022", "title": "derived-remark-big-abelian-category", "text": "Let $\\textit{Mod}(\\mathcal{O}_X)$ be the category of $\\mathcal{O}_X$-modules on a ringed space $(X, \\mathcal{O}_X)$ (or more generally on a ringed site). We will see later that $\\textit{Mod}(\\mathcal{O}_X)$ has enough injectives and in fact functorial injective embeddings, see Injectives, Theorem \\ref{injectives-theorem-sheaves-modules-injectives}. Note that the proof of Lemma \\ref{lemma-into-derived-category} does not apply to $\\textit{Mod}(\\mathcal{O}_X)$. But the proof of Lemma \\ref{lemma-functorial-injective-resolutions} does apply to $\\textit{Mod}(\\mathcal{O}_X)$. Thus we obtain $$ j : K^{+}(\\textit{Mod}(\\mathcal{O}_X)) \\longrightarrow K^{+}(\\mathcal{I}) $$ which is a resolution functor where $\\mathcal{I}$ is the additive category of injective $\\mathcal{O}_X$-modules. This argument also works in the following cases: \\begin{enumerate} \\item The category $\\text{Mod}_R$ of $R$-modules over a ring $R$. \\item The category $\\textit{PMod}(\\mathcal{O})$ of presheaves of $\\mathcal{O}$-modules on a site endowed with a presheaf of rings. \\item The category $\\textit{Mod}(\\mathcal{O})$ of sheaves of $\\mathcal{O}$-modules on a ringed site. \\item Add more here as needed. \\end{enumerate}"}
+{"_id": "2023", "title": "derived-remark-right-derived-functor", "text": "In the situation of Lemma \\ref{lemma-right-derived-functor} we see that we have actually lifted the right derived functor to an exact functor $F \\circ j' : D^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B})$. It is occasionally useful to use such a factorization."}
+{"_id": "2024", "title": "derived-remark-filtered-localization-big", "text": "We can invert the arrow of the lemma only if $\\mathcal{A}$ is a category in our sense, namely if it has a set of objects. However, suppose given a big abelian category $\\mathcal{A}$ with enough injectives, such as $\\textit{Mod}(\\mathcal{O}_X)$ for example. Then for any given set of objects $\\{A_i\\}_{i\\in I}$ there is an abelian subcategory $\\mathcal{A}' \\subset \\mathcal{A}$ containing all of them and having enough injectives, see Sets, Lemma \\ref{sets-lemma-abelian-injectives}. Thus we may use the lemma above for $\\mathcal{A}'$. This essentially means that if we use a set worth of diagrams, etc then we will never run into trouble using the lemma."}
+{"_id": "2025", "title": "derived-remark-final-functorial", "text": "As promised in Remark \\ref{remark-functorial-ss} we discuss the connection of the lemma above with the constructions using Cartan-Eilenberg resolutions. Namely, let $T : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of abelian categories, assume $\\mathcal{A}$ has enough injectives, and let $K^\\bullet$ be a bounded below complex of $\\mathcal{A}$. We give an alternative construction of the spectral sequences ${}'E$ and ${}''E$ of Lemma \\ref{lemma-two-ss-complex-functor}. \\medskip\\noindent First spectral sequence. Consider the ``stupid'' filtration on $K^\\bullet$ obtained by setting $F^p(K^\\bullet) = \\sigma_{\\geq p}(K^\\bullet)$, see Homology, Section \\ref{homology-section-truncations}. Note that this stupid in the sense that $d(F^p(K^\\bullet)) \\subset F^{p + 1}(K^\\bullet)$, compare Homology, Lemma \\ref{homology-lemma-spectral-sequence-filtered-complex-d1}. Note that $\\text{gr}^p(K^\\bullet) = K^p[-p]$ with this filtration. According to Lemma \\ref{lemma-ss-filtered-derived} there is a spectral sequence with $E_1$ term $$ E_1^{p, q} = R^{p + q}T(K^p[-p]) = R^qT(K^p) $$ as in the spectral sequence ${}'E_r$. Observe moreover that the differentials $E_1^{p, q} \\to E_1^{p + 1, q}$ agree with the differentials in $'{}E_1$, see Homology, Lemma \\ref{homology-lemma-spectral-sequence-filtered-complex-d1} part (2) and the description of ${}'d_1$ in the proof of Lemma \\ref{lemma-two-ss-complex-functor}. \\medskip\\noindent Second spectral sequence. Consider the filtration on the complex $K^\\bullet$ obtained by setting $F^p(K^\\bullet) = \\tau_{\\leq -p}(K^\\bullet)$, see Homology, Section \\ref{homology-section-truncations}. The minus sign is necessary to get a decreasing filtration. Note that $\\text{gr}^p(K^\\bullet)$ is quasi-isomorphic to $H^{-p}(K^\\bullet)[p]$ with this filtration. According to Lemma \\ref{lemma-ss-filtered-derived} there is a spectral sequence with $E_1$ term $$ E_1^{p, q} = R^{p + q}T(H^{-p}(K^\\bullet)[p]) = R^{2p + q}T(H^{-p}(K^\\bullet)) = {}''E_2^{i, j} $$ with $i = 2p + q$ and $j = -p$. (This looks unnatural, but note that we could just have well developed the whole theory of filtered complexes using increasing filtrations, with the end result that this then looks natural, but the other one doesn't.) We leave it to the reader to see that the differentials match up. \\medskip\\noindent Actually, given a Cartan-Eilenberg resolution $K^\\bullet \\to I^{\\bullet, \\bullet}$ the induced morphism $K^\\bullet \\to \\text{Tot}(I^{\\bullet, \\bullet})$ into the associated total complex will be a filtered injective resolution for either filtration using suitable filtrations on $\\text{Tot}(I^{\\bullet, \\bullet})$. This can be used to match up the spectral sequences exactly."}
+{"_id": "2026", "title": "derived-remark-uniqueness-derived-colimit", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$. We may think of a derived colimit as an object $K$ of $\\mathcal{D}$ endowed with morphisms $i_n : K_n \\to K$ such that $i_{n + 1} \\circ f_n = i_n$ and such that there exists a morphism $c : K \\to \\bigoplus K_n$ with the property that $$ \\bigoplus K_n \\xrightarrow{1 - f_n} \\bigoplus K_n \\xrightarrow{i_n} K \\xrightarrow{c} \\bigoplus K_n[1] $$ is a distinguished triangle. If $(K', i'_n, c')$ is a second derived colimit, then there exists an isomorphism $\\varphi : K \\to K'$ such that $\\varphi \\circ i_n = i'_n$ and $c' \\circ \\varphi = c$. The existence of $\\varphi$ is TR3 and the fact that $\\varphi$ is an isomorphism is Lemma \\ref{lemma-third-isomorphism-triangle}."}
+{"_id": "2027", "title": "derived-remark-functoriality-derived-colimit", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(a_n) : (K_n, f_n) \\to (L_n, g_n)$ be a morphism of systems of objects of $\\mathcal{D}$. Let $(K, i_n, c)$ be a derived colimit of the first system and let $(L, j_n, d)$ be a derived colimit of the second system with notation as in Remark \\ref{remark-uniqueness-derived-colimit}. Then there exists a morphism $a : K \\to L$ such that $a \\circ i_n = j_n$ and $d \\circ a = (a_n[1]) \\circ c$. This follows from TR3 applied to the defining distinguished triangles."}
+{"_id": "2028", "title": "derived-remark-operations-functor", "text": "Let $F : \\mathcal{T} \\to \\mathcal{T}'$ be an exact functor of triangulated categories. Given a full subcategory $\\mathcal{A}$ of $\\mathcal{T}$ we denote $F(\\mathcal{A})$ the full subcategory of $\\mathcal{T}'$ whose objects consists of all objects $F(A)$ with $A \\in \\Ob(\\mathcal{A})$. We have $$ F(\\mathcal{A}[a, b]) = F(\\mathcal{A})[a, b] $$ $$ F(smd(\\mathcal{A})) \\subset smd(F(\\mathcal{A})), $$ $$ F(add(\\mathcal{A})) \\subset add(F(\\mathcal{A})), $$ $$ F(\\mathcal{A} \\star \\mathcal{B}) \\subset F(\\mathcal{A}) \\star F(\\mathcal{B}), $$ $$ F(\\mathcal{A}^{\\star n}) \\subset F(\\mathcal{A})^{\\star n}. $$ We omit the trivial verifications."}
+{"_id": "2029", "title": "derived-remark-operations-unions", "text": "Let $\\mathcal{T}$ be a triangulated category. Given full subcategories $\\mathcal{A}_1 \\subset \\mathcal{A}_2 \\subset \\mathcal{A}_3 \\subset \\ldots$ and $\\mathcal{B}$ of $\\mathcal{T}$ we have $$ \\left(\\bigcup \\mathcal{A}_i\\right)[a, b] = \\bigcup \\mathcal{A}_i[a, b] $$ $$ smd\\left(\\bigcup \\mathcal{A}_i\\right) = \\bigcup smd(\\mathcal{A}_i), $$ $$ add\\left(\\bigcup \\mathcal{A}_i\\right) = \\bigcup add(\\mathcal{A}_i), $$ $$ \\left(\\bigcup \\mathcal{A}_i\\right) \\star \\mathcal{B} = \\bigcup \\mathcal{A}_i \\star \\mathcal{B}, $$ $$ \\mathcal{B} \\star \\left(\\bigcup \\mathcal{A}_i\\right) = \\bigcup \\mathcal{B} \\star \\mathcal{A}_i, $$ $$ \\left(\\bigcup \\mathcal{A}_i\\right)^{\\star n} = \\bigcup \\mathcal{A}_i^{\\star n}. $$ We omit the trivial verifications."}
+{"_id": "2030", "title": "derived-remark-check-on-generator", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\\mathcal{D}$. Let $T$ be a property of objects of $\\mathcal{D}$. Suppose that \\begin{enumerate} \\item if $K_i \\in D(A)$, $i = 1, \\ldots, r$ with $T(K_i)$ for $i = 1, \\ldots, r$, then $T(\\bigoplus K_i)$, \\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object, \\item if $T(K \\oplus L)$ then $T(K)$ and $T(L)$, and \\item $T(E[n])$ holds for all $n$. \\end{enumerate} Then $T$ holds for all objects of $\\langle E \\rangle$."}
+{"_id": "2261", "title": "cohomology-remark-daniel", "text": "Here is a different approach to the proofs of Lemmas \\ref{lemma-kill-cohomology-class-on-covering} and \\ref{lemma-describe-higher-direct-images} above. Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)$ be the inclusion functor and let $\\#$ be the sheafification functor. Recall that $i_X$ is left exact and $\\#$ is exact. \\begin{enumerate} \\item First prove Lemma \\ref{lemma-include} below which says that the right derived functors of $i_X$ are given by $R^pi_X\\mathcal{F} = \\underline{H}^p(\\mathcal{F})$. Here is another proof: The equality is clear for $p = 0$. Both $(R^pi_X)_{p \\geq 0}$ and $(\\underline{H}^p)_{p \\geq 0}$ are delta functors vanishing on injectives, hence both are universal, hence they are isomorphic. See Homology, Section \\ref{homology-section-cohomological-delta-functor}. \\item A restatement of Lemma \\ref{lemma-kill-cohomology-class-on-covering} is that $(\\underline{H}^p(\\mathcal{F}))^\\# = 0$, $p > 0$ for any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$. To see this is true, use that ${}^\\#$ is exact so $$ (\\underline{H}^p(\\mathcal{F}))^\\# = (R^pi_X\\mathcal{F})^\\# = R^p(\\# \\circ i_X)(\\mathcal{F}) = 0 $$ because $\\# \\circ i_X$ is the identity functor. \\item Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. The presheaf $V \\mapsto H^p(f^{-1}V, \\mathcal{F})$ is equal to $R^p (i_Y \\circ f_*)\\mathcal{F}$. You can prove this by noticing that both give universal delta functors as in the argument of (1) above. Hence Lemma \\ref{lemma-describe-higher-direct-images} says that $R^p f_* \\mathcal{F}= (R^p (i_Y \\circ f_*)\\mathcal{F})^\\#$. Again using that $\\#$ is exact a that $\\# \\circ i_Y$ is the identity functor we see that $$ R^p f_* \\mathcal{F} = R^p(\\# \\circ i_Y \\circ f_*)\\mathcal{F} = (R^p (i_Y \\circ f_*)\\mathcal{F})^\\# $$ as desired. \\end{enumerate}"}
+{"_id": "2262", "title": "cohomology-remark-elucidate-lemma", "text": "Here is a down-to-earth explanation of the meaning of Lemma \\ref{lemma-before-Leray}. It says that given $f : X \\to Y$ and $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_X)$ and given an injective resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ we have $$ \\begin{matrix} R\\Gamma(X, \\mathcal{F}) & \\text{is represented by} & \\Gamma(X, \\mathcal{I}^\\bullet) \\\\ Rf_*\\mathcal{F} & \\text{is represented by} & f_*\\mathcal{I}^\\bullet \\\\ R\\Gamma(Y, Rf_*\\mathcal{F}) & \\text{is represented by} & \\Gamma(Y, f_*\\mathcal{I}^\\bullet) \\end{matrix} $$ the last fact coming from Leray's acyclicity lemma (Derived Categories, Lemma \\ref{derived-lemma-leray-acyclicity}) and Lemma \\ref{lemma-pushforward-injective}. Finally, it combines this with the trivial observation that $$ \\Gamma(X, \\mathcal{I}^\\bullet) = \\Gamma(Y, f_*\\mathcal{I}^\\bullet). $$ to arrive at the commutativity of the diagram of the lemma."}
+{"_id": "2263", "title": "cohomology-remark-Leray-ss-more-structure", "text": "The Leray spectral sequence, the way we proved it in Lemma \\ref{lemma-Leray} is a spectral sequence of $\\Gamma(Y, \\mathcal{O}_Y)$-modules. However, it is quite easy to see that it is in fact a spectral sequence of $\\Gamma(X, \\mathcal{O}_X)$-modules. For example $f$ gives rise to a morphism of ringed spaces $f' :  (X, \\mathcal{O}_X) \\to (Y, f_*\\mathcal{O}_X)$. By Lemma \\ref{lemma-modules-abelian} the terms $E_r^{p, q}$ of the Leray spectral sequence for an $\\mathcal{O}_X$-module $\\mathcal{F}$ and $f$ are identical with those for $\\mathcal{F}$ and $f'$ at least for $r \\geq 2$. Namely, they both agree with the terms of the Leray spectral sequence for $\\mathcal{F}$ as an abelian sheaf. And since $(f_*\\mathcal{O}_X)(Y) = \\mathcal{O}_X(X)$ we see the result. It is often the case that the Leray spectral sequence carries additional structure."}
+{"_id": "2264", "title": "cohomology-remark-explain-arrow", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{G}$ be an $\\mathcal{O}_Y$-module. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $\\varphi$ be an $f$-map from $\\mathcal{G}$ to $\\mathcal{F}$. Choose a resolution $\\mathcal{F} \\to \\mathcal{I}^\\bullet$ by a complex of injective $\\mathcal{O}_X$-modules. Choose resolutions $\\mathcal{G} \\to \\mathcal{J}^\\bullet$ and $f_*\\mathcal{I}^\\bullet \\to (\\mathcal{J}')^\\bullet$ by complexes of injective $\\mathcal{O}_Y$-modules. By Derived Categories, Lemma \\ref{derived-lemma-morphisms-lift} there exists a map of complexes $\\beta$ such that the diagram \\begin{equation} \\label{equation-choice} \\xymatrix{ \\mathcal{G} \\ar[d] \\ar[r] & f_*\\mathcal{F} \\ar[r] & f_*\\mathcal{I}^\\bullet \\ar[d] \\\\ \\mathcal{J}^\\bullet \\ar[rr]^\\beta & & (\\mathcal{J}')^\\bullet } \\end{equation} commutes. Applying global section functors we see that we get a diagram $$ \\xymatrix{  & & \\Gamma(Y, f_*\\mathcal{I}^\\bullet) \\ar[d]_{qis} \\ar@{=}[r] & \\Gamma(X, \\mathcal{I}^\\bullet) \\\\ \\Gamma(Y, \\mathcal{J}^\\bullet) \\ar[rr]^\\beta & & \\Gamma(Y, (\\mathcal{J}')^\\bullet) & } $$ The complex on the bottom left represents $R\\Gamma(Y, \\mathcal{G})$ and the complex on the top right represents $R\\Gamma(X, \\mathcal{F})$. The vertical arrow is a quasi-isomorphism by Lemma \\ref{lemma-before-Leray} which becomes invertible after applying the localization functor $K^{+}(\\mathcal{O}_Y(Y)) \\to D^{+}(\\mathcal{O}_Y(Y))$. The arrow (\\ref{equation-functorial-derived}) is given by the composition of the horizontal map by the inverse of the vertical map."}
+{"_id": "2265", "title": "cohomology-remark-correct-version-base-change-map", "text": "The ``correct'' version of the base change map is the map $$ Lg^* Rf_* \\mathcal{F}^\\bullet \\longrightarrow R(f')_* L(g')^*\\mathcal{F}^\\bullet. $$ The construction of this map involves unbounded complexes, see Remark \\ref{remark-base-change}."}
+{"_id": "2266", "title": "cohomology-remark-compared-ordered-complexes", "text": "This means that if we have two total orderings $<_1$ and $<_2$ on the index set $I$, then we get an isomorphism of complexes $\\tau = \\pi_2 \\circ c_1 : \\check{\\mathcal{C}}_{ord\\text{-}1}(\\mathcal{U}, \\mathcal{F}) \\to \\check{\\mathcal{C}}_{ord\\text{-}2}(\\mathcal{U}, \\mathcal{F})$. It is clear that $$ \\tau(s)_{i_0 \\ldots i_p} = \\text{sign}(\\sigma) s_{i_{\\sigma(0)} \\ldots i_{\\sigma(p)}} $$ where $i_0 <_1 i_1 <_1 \\ldots <_1 i_p$ and $i_{\\sigma(0)} <_2 i_{\\sigma(1)} <_2 \\ldots <_2 i_{\\sigma(p)}$. This is the sense in which the ordered {\\v C}ech complex is independent of the chosen total ordering."}
+{"_id": "2267", "title": "cohomology-remark-locally-finite-sections", "text": "Let $X = \\bigcup_{i \\in I} U_i$ be a locally finite open covering. Denote $j_i : U_i \\to X$ the inclusion map. Suppose that for each $i$ we are given an abelian sheaf $\\mathcal{F}_i$ on $U_i$. Consider the abelian sheaf $\\mathcal{G} = \\bigoplus_{i \\in I} (j_i)_*\\mathcal{F}_i$. Then for $V \\subset X$ open we actually have $$ \\Gamma(V, \\mathcal{G}) = \\prod\\nolimits_{i \\in I} \\mathcal{F}_i(V \\cap U_i). $$ In other words we have $$ \\bigoplus\\nolimits_{i \\in I} (j_i)_*\\mathcal{F}_i = \\prod\\nolimits_{i \\in I} (j_i)_*\\mathcal{F}_i $$ This seems strange until you realize that the direct sum of a collection of sheaves is the sheafification of what you think it should be. See discussion in Modules, Section \\ref{modules-section-kernels}. Thus we conclude that in this case the complex of Lemma \\ref{lemma-covering-resolution} has terms $$ {\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) = \\bigoplus\\nolimits_{i_0 \\ldots i_p} (j_{i_0 \\ldots i_p})_* \\mathcal{F}_{i_0 \\ldots i_p} $$ which is sometimes useful."}
+{"_id": "2268", "title": "cohomology-remark-shift-complex-cech-complex", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering. Let $\\mathcal{F}^\\bullet$ be a bounded below complex of $\\mathcal{O}_X$-modules. Let $b$ be an integer. We claim there is a commutative diagram $$ \\xymatrix{ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))[b] \\ar[r] \\ar[d]_\\gamma & R\\Gamma(X, \\mathcal{F}^\\bullet)[b] \\ar[d] \\\\ \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet[b])) \\ar[r] & R\\Gamma(X, \\mathcal{F}^\\bullet[b]) } $$ in the derived category where the map $\\gamma$ is the map on complexes constructed in Homology, Remark \\ref{homology-remark-shift-double-complex}. This makes sense because the double complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet[b])$ is clearly the same as the double complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet)[0, b]$ introduced in Homology, Remark \\ref{homology-remark-shift-double-complex}. To check that the diagram commutes, we may choose an injective resolution $\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$ as in the proof of Lemma \\ref{lemma-cech-complex-complex}. Chasing diagrams, we see that it suffices to check the diagram commutes when we replace $\\mathcal{F}^\\bullet$ by $\\mathcal{I}^\\bullet$. Then we consider the extended diagram $$ \\xymatrix{ \\Gamma(X, \\mathcal{I}^\\bullet)[b] \\ar[r] \\ar[d] & \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet))[b] \\ar[r] \\ar[d]_\\gamma & R\\Gamma(X, \\mathcal{I}^\\bullet)[b] \\ar[d] \\\\ \\Gamma(X, \\mathcal{I}^\\bullet[b]) \\ar[r] & \\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{I}^\\bullet[b])) \\ar[r] & R\\Gamma(X, \\mathcal{I}^\\bullet[b]) } $$ where the left horizontal arrows are (\\ref{equation-global-sections-to-cech}). Since in this case the horizonal arrows are isomorphisms in the derived category (see proof of Lemma \\ref{lemma-cech-complex-complex}) it suffices to show that the left square commutes. This is true because the map $\\gamma$ uses the sign $1$ on the summands $\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{I}^{q + b})$, see formula in Homology, Remark \\ref{homology-remark-shift-double-complex}."}
+{"_id": "2269", "title": "cohomology-remark-base-change", "text": "The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ is a commutative diagram of ringed spaces. Let $K$ be an object of $D(\\mathcal{O}_X)$. Then there exists a canonical base change map $$ Lg^*Rf_*K \\longrightarrow R(f')_*L(g')^*K $$ in $D(\\mathcal{O}_{S'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \\to L(g')^*K$ Since $L(f')^*Lg^* = L(g')^*Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \\to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \\to K$."}
+{"_id": "2270", "title": "cohomology-remark-compose-base-change", "text": "Consider a commutative diagram $$ \\xymatrix{ X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z' \\ar[r]^m & Z } $$ of ringed spaces. Then the base change maps of Remark \\ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} Lm^* \\circ R(g \\circ f)_* & = Lm^* \\circ Rg_* \\circ Rf_* \\\\ & \\to Rg'_* \\circ Ll^* \\circ Rf_* \\\\ & \\to Rg'_* \\circ Rf'_* \\circ Lk^* \\\\ & = R(g' \\circ f')_* \\circ Lk^* \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "2271", "title": "cohomology-remark-compose-base-change-horizontal", "text": "Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y } $$ of ringed spaces. Then the base change maps of Remark \\ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} L(h \\circ h')^* \\circ Rf_* & = L(h')^* \\circ Lh_* \\circ Rf_* \\\\ & \\to L(h')^* \\circ Rf'_* \\circ Lg^* \\\\ & \\to Rf''_* \\circ L(g')^* \\circ Lg^* \\\\ & = Rf''_* \\circ L(g \\circ g')^* \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "2272", "title": "cohomology-remark-cup-product", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product $$ Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\longrightarrow Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) $$ in $D(\\mathcal{O}_Y)$ for all $K, L$ in $D(\\mathcal{O}_X)$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L) \\to K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L) = Lf^*Rf_*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*Rf_*L$ (Lemma \\ref{lemma-pullback-tensor-product}) with the map $Lf^*Rf_*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*Rf_*L \\to K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ coming from the counit $Lf^* \\circ Rf_* \\to \\text{id}$."}
+{"_id": "2273", "title": "cohomology-remark-spectral-sequence-filtered-object", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a filtered complex of $\\mathcal{O}_X$-modules. If $\\mathcal{F}^\\bullet$ is bounded from below and for each $n$ the filtration on $\\mathcal{F}^n$ is finite, then there is a construction of the spectral sequence in Lemma \\ref{lemma-spectral-sequence-filtered-object} avoiding Injectives, Lemma \\ref{injectives-lemma-K-injective-embedding-filtration}. Namely, by Derived Categories, Lemma \\ref{derived-lemma-right-resolution-by-filtered-injectives} there is a filtered quasi-isomorphism $i : \\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$ of filtered complexes with $\\mathcal{I}^\\bullet$ bounded below, the filtration on $\\mathcal{I}^n$ is finite for all $n$, and with each $\\text{gr}^p\\mathcal{I}^n$ an injective $\\mathcal{O}_X$-module. Then we take the spectral sequence associated to $$ \\Gamma(X, \\mathcal{I}^\\bullet) \\quad\\text{with}\\quad F^p\\Gamma(X, \\mathcal{I}^\\bullet) = \\Gamma(X, F^p\\mathcal{I}^\\bullet) $$ Since cohomology can be computed by evaluating on bounded below complexes of injectives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma \\ref{homology-lemma-biregular-ss-converges}. In fact, this is a special case of the spectral sequence in Derived Categories, Lemma \\ref{derived-lemma-ss-filtered-derived}."}
+{"_id": "2274", "title": "cohomology-remark-cup-with-element-map-total-cohomology", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, M$ be objects of $D(\\mathcal{O}_X)$. Set $A = \\Gamma(X, \\mathcal{O}_X)$. Given $\\xi \\in H^i(X, K)$ we get an associated map $$ \\xi = ``\\xi \\cup -'' : R\\Gamma(X, M)[-i] \\to R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) $$ by representing $\\xi$ as a map $\\xi : A[-i] \\to R\\Gamma(X, K)$ as in the proof of Lemma \\ref{lemma-second-cup-equals-first} and then using the composition $$ R\\Gamma(X, M)[-i] = A[-i] \\otimes_A^\\mathbf{L} R\\Gamma(X, M) \\xrightarrow{\\xi \\otimes 1} R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M) \\to R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) $$ where the second arrow is the global cup product $\\mu$ above. On cohomology this recovers the cup product by $\\xi$ as is clear from Lemma \\ref{lemma-second-cup-equals-first} and its proof."}
+{"_id": "2275", "title": "cohomology-remark-support-cup-product", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the inclusion of a closed subset. Given $K$ and $M$ in $D(\\mathcal{O}_X)$ there is a canonical map $$ K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M) \\longrightarrow R\\mathcal{H}_Z(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) $$ in $D(\\mathcal{O}_X|_Z)$. Here $K|_Z = i^{-1}K$ is the restriction of $K$ to $Z$ viewed as an object of $D(\\mathcal{O}_X|_Z)$. By adjointness of $i_*$ and $R\\mathcal{H}_Z$ of Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support} to construct this map it suffices to produce a canonical map $$ i_*\\left(K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M)\\right) \\longrightarrow K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M $$ To construct this map, we choose a K-injective complex $\\mathcal{I}^\\bullet$ of $\\mathcal{O}_X$-modules representing $M$ and a K-flat complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}_X$-modules representing $K$. Observe that $\\mathcal{K}^\\bullet|_Z$ is a K-flat complex of $\\mathcal{O}_X|_Z$-modules representing $K|_Z$, see Lemma \\ref{lemma-pullback-K-flat}. Hence we need to produce a map of complexes $$ i_*\\text{Tot}\\left( \\mathcal{K}^\\bullet|_Z \\otimes_{\\mathcal{O}_X|_Z} \\mathcal{H}_Z(\\mathcal{I}^\\bullet)\\right) \\longrightarrow \\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{I}^\\bullet) $$ of $\\mathcal{O}_X$-modules. For this it suffices to produce maps $$ i_*(\\mathcal{K}^a|_Z \\otimes_{\\mathcal{O}_X|_Z} \\mathcal{H}_Z(\\mathcal{I}^b)) \\longrightarrow \\mathcal{K}^a \\otimes_{\\mathcal{O}_X} \\mathcal{I}^b $$ Looking at stalks (for example), we see that the left hand side of this formula is equal to $\\mathcal{K}^a \\otimes_{\\mathcal{O}_X} i_*\\mathcal{H}_Z(\\mathcal{I}^b)$ and we can use the inclusion $\\mathcal{H}_Z(\\mathcal{I}^b) \\to \\mathcal{I}^b$ to get our map."}
+{"_id": "2276", "title": "cohomology-remark-support-cup-product-global", "text": "With notation as in Remark \\ref{remark-support-cup-product} we obtain a canonical cup product \\begin{align*} H^a(X, K) \\times H^b_Z(X, M) & = H^a(X, K) \\times H^b(Z, R\\mathcal{H}_Z(M)) \\\\ & \\to H^a(Z, K|_Z) \\times H^b(Z, R\\mathcal{H}_Z(M)) \\\\ & \\to H^{a + b}(Z, K|_Z \\otimes_{\\mathcal{O}_X|_Z}^\\mathbf{L} R\\mathcal{H}_Z(M)) \\\\ & \\to H^{a + b}(Z, R\\mathcal{H}_Z(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)) \\\\ & = H^{a + b}_Z(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\end{align*} Here the equal signs are given by Lemma \\ref{lemma-local-to-global-sections-with-support}, the first arrow is restriction to $Z$, the second arrow is the cup product (Section \\ref{section-cup-product}), and the third arrow is the map from Remark \\ref{remark-support-cup-product}."}
+{"_id": "2277", "title": "cohomology-remark-support-functorial", "text": "Let $f : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$ be a morphism of ringed spaces. Let $Z \\subset X$ be a closed subset and $Z' = f^{-1}(Z)$. Denote $f|_{Z'} : (Z', \\mathcal{O}_{X'}|_{Z'}) \\to (Z, \\mathcal{O}_X|Z)$ be the induced morphism of ringed spaces. For any $K$ in $D(\\mathcal{O}_X)$ there is a canonical map $$ L(f|_{Z'})^*R\\mathcal{H}_Z(K) \\longrightarrow R\\mathcal{H}_{Z'}(Lf^*K) $$ in $D(\\mathcal{O}_{X'}|_{Z'})$. Denote $i : Z \\to X$ and $i' : Z' \\to X'$ the inclusion maps. By Lemma \\ref{lemma-complexes-with-support-on-closed} part (2) applied to $i'$ it is the same thing to give a map $$ i'_* L(f|_{Z'})^* R\\mathcal{H}_Z(K) \\longrightarrow i'_*R\\mathcal{H}_{Z'}(Lf^*K) $$ in $D_{Z'}(\\mathcal{O}_{X'})$. The map of functors $Lf^* \\circ i_* \\to i'_* \\circ L(f|_{Z'})^*$ of Remark \\ref{remark-base-change} is an isomorphism in this case (follows by checking what happens on stalks using that $i_*$ and $i'_*$ are exact and that $\\mathcal{O}_{Z, z} = \\mathcal{O}_{X, z}$ and similarly for $Z'$). Hence it suffices to construct a the top horizonal arrow in the following diagram $$ \\xymatrix{ Lf^* i_* R\\mathcal{H}_Z(K) \\ar[rr] \\ar[rd] & & i'_* R\\mathcal{H}_{Z'}(Lf^*K) \\ar[ld] \\\\ & Lf^*K } $$ The complex $Lf^* i_* R\\mathcal{H}_Z(K)$ is supported on $Z'$. The south-east arrow comes from the adjunction mapping $i_*R\\mathcal{H}_Z(K) \\to K$ (Lemma \\ref{lemma-cohomology-with-support-sheaf-on-support}). Since the adjunction mapping $i'_* R\\mathcal{H}_{Z'}(Lf^*K) \\to Lf^*K$ is universal by Lemma \\ref{lemma-complexes-with-support-on-closed} part (3), we find that the south-east arrow factors uniquely over the south-west arrow and we obtain the desired arrow."}
+{"_id": "2278", "title": "cohomology-remark-discuss-derived-limit", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an inverse system in $D(\\mathcal{O}_X)$. Set $K = R\\lim K_n$. For each $n$ and $m$ let $\\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of $K_n$ and similarly set $\\mathcal{H}^m = H^m(K)$. Let us denote $\\underline{\\mathcal{H}}^m_n$ the presheaf $$ U \\longmapsto \\underline{\\mathcal{H}}^m_n(U) = H^m(U, K_n) $$ Similarly we set $\\underline{\\mathcal{H}}^m(U) = H^m(U, K)$. By Lemma \\ref{lemma-sheafification-cohomology} we see that $\\mathcal{H}^m_n$ is the sheafification of $\\underline{\\mathcal{H}}^m_n$ and $\\mathcal{H}^m$ is the sheafification of $\\underline{\\mathcal{H}}^m$. Here is a diagram $$ \\xymatrix{ K \\ar@{=}[d] & \\underline{\\mathcal{H}}^m \\ar[d] \\ar[r] &  \\mathcal{H}^m \\ar[d] \\\\ R\\lim K_n & \\lim \\underline{\\mathcal{H}}^m_n \\ar[r] &  \\lim \\mathcal{H}^m_n } $$ In general it may not be the case that $\\lim \\mathcal{H}^m_n$ is the sheafification of $\\lim \\underline{\\mathcal{H}}^m_n$. If $U \\subset X$ is an open, then we have short exact sequences \\begin{equation} \\label{equation-ses-Rlim-over-U} 0 \\to R^1\\lim \\underline{\\mathcal{H}}^{m - 1}_n(U) \\to \\underline{\\mathcal{H}}^m(U) \\to \\lim \\underline{\\mathcal{H}}^m_n(U) \\to 0 \\end{equation} by Lemma \\ref{lemma-RGamma-commutes-with-Rlim}."}
+{"_id": "2279", "title": "cohomology-remark-tensor-internal-hom", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. For $K, K', M, M'$ in $D(\\mathcal{O}_X)$ there is a canonical map $$ R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, M') \\longrightarrow R\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M, K' \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M') $$ Namely, by (\\ref{equation-internal-hom}) is the same thing as a map $$ R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, M') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\longrightarrow K' \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M' $$ For this we can first flip the middle two factors (with sign rules as in More on Algebra, Section \\ref{more-algebra-section-sign-rules}) and use the maps $$ R\\SheafHom(K, K') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K \\to K' \\quad\\text{and}\\quad R\\SheafHom(M, M') \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\to M' $$ from Lemma \\ref{lemma-internal-hom-composition} when thinking of $K = R\\SheafHom(\\mathcal{O}_X, K)$ and similarly for $K'$, $M$, and $M'$."}
+{"_id": "2280", "title": "cohomology-remark-projection-formula-for-internal-hom", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\\mathcal{O}_X)$. We claim there is a canonical map $$ Rf_*R\\SheafHom(L, K) \\longrightarrow R\\SheafHom(Rf_*L, Rf_*K) $$ Namely, by (\\ref{equation-internal-hom}) this is the same thing as a map $Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\to Rf_*K$. For this we can use the composition $$ Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\to Rf_*(R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\to Rf_*K $$ where the first arrow is the relative cup product (Remark \\ref{remark-cup-product}) and the second arrow is $Rf_*$ applied to the canonical map $R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L \\to K$ coming from Lemma \\ref{lemma-internal-hom-composition} (with $\\mathcal{O}_X$ in one of the spots)."}
+{"_id": "2281", "title": "cohomology-remark-relative-cup-and-composition", "text": "Let $h : X \\to Y$ be a morphism of ringed spaces. Let $K, L, M$ be objects of $D(\\mathcal{O}_Y)$. The diagram $$ \\xymatrix{ Rf_*R\\SheafHom_{\\mathcal{O}_X}(K, M) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[r] \\ar[d] & Rf_*\\left(R\\SheafHom_{\\mathcal{O}_X}(K, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M\\right) \\ar[d] \\\\ R\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, Rf_*M) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[r] & Rf_*M } $$ is commutative. Here the left vertical arrow comes from Remark \\ref{remark-projection-formula-for-internal-hom}. The top horizontal arrow is Remark \\ref{remark-cup-product}. The other two arrows are instances of the map in Lemma \\ref{lemma-internal-hom-composition} (with one of the entries replaced with $\\mathcal{O}_X$ or $\\mathcal{O}_Y$)."}
+{"_id": "2282", "title": "cohomology-remark-prepare-fancy-base-change", "text": "Let $h : X \\to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\\mathcal{O}_Y)$. We claim there is a canonical map $$ Lh^*R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(Lh^*K, Lh^*L) $$ in $D(\\mathcal{O}_X)$. Namely, by (\\ref{equation-internal-hom}) proved in Lemma \\ref{lemma-internal-hom} such a map is the same thing as a map $$ Lh^*R\\SheafHom(K, L) \\otimes^\\mathbf{L} Lh^*K \\longrightarrow Lh^*L $$ The source of this arrow is $Lh^*(\\SheafHom(K, L) \\otimes^\\mathbf{L} K)$ by Lemma \\ref{lemma-pullback-tensor-product} hence it suffices to construct a canonical map $$ R\\SheafHom(K, L) \\otimes^\\mathbf{L} K \\longrightarrow L. $$ For this we take the arrow corresponding to $$ \\text{id} : R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(K, L) $$ via (\\ref{equation-internal-hom})."}
+{"_id": "2283", "title": "cohomology-remark-fancy-base-change", "text": "Suppose that $$ \\xymatrix{ X' \\ar[r]_h \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ is a commutative diagram of ringed spaces. Let $K, L$ be objects of $D(\\mathcal{O}_X)$. We claim there exists a canonical base change map $$ Lg^*Rf_*R\\SheafHom(K, L) \\longrightarrow R(f')_*R\\SheafHom(Lh^*K, Lh^*L) $$ in $D(\\mathcal{O}_{S'})$. Namely, we take the map adjoint to the composition \\begin{align*} L(f')^*Lg^*Rf_*R\\SheafHom(K, L) & = Lh^*Lf^*Rf_*R\\SheafHom(K, L) \\\\ & \\to Lh^*R\\SheafHom(K, L) \\\\ & \\to R\\SheafHom(Lh^*K, Lh^*L) \\end{align*} where the first arrow uses the adjunction mapping $Lf^*Rf_* \\to \\text{id}$ and the second arrow is the canonical map constructed in Remark \\ref{remark-prepare-fancy-base-change}."}
+{"_id": "2284", "title": "cohomology-remark-uniqueness", "text": "With notation and assumptions as in Lemma \\ref{lemma-uniqueness}. Suppose that $U, V \\in \\mathcal{B}$. Let $\\mathcal{B}'$ be the set of elements of $\\mathcal{B}$ contained in $U \\cap V$. Then $$ (\\{K_{U'}\\}_{U' \\in \\mathcal{B}'}, \\{\\rho_{V'}^{U'}\\}_{V' \\subset U'\\text{ with }U', V' \\in \\mathcal{B}'}) $$ is a system on the ringed space $U \\cap V$ satisfying the assumptions of Lemma \\ref{lemma-uniqueness}. Moreover, both $(K_U|_{U \\cap V}, \\rho^U_{U'})$ and $(K_V|_{U \\cap V}, \\rho^V_{U'})$ are solutions to this system. By the lemma we find a unique isomorphism $$ \\rho_{U, V} : K_U|_{U \\cap V} \\longrightarrow K_V|_{U \\cap V} $$ such that for every $U' \\subset U \\cap V$, $U' \\in \\mathcal{B}$ the diagram $$ \\xymatrix{ K_U|_{U'} \\ar[rr]_{\\rho_{U, V}|_{U'}} \\ar[rd]_{\\rho^U_{U'}} & & K_V|_{U'} \\ar[ld]^{\\rho^V_{U'}} \\\\ & K_{U'} } $$ commutes. Pick a third element $W \\in \\mathcal{B}$. We obtain isomorphisms $\\rho_{U, W} : K_U|_{U \\cap W} \\to K_W|_{U \\cap W}$ and $\\rho_{V, W} : K_U|_{V \\cap W} \\to K_W|_{V \\cap W}$ satisfying similar properties to those of $\\rho_{U, V}$. Finally, we have $$ \\rho_{U, W}|_{U \\cap V \\cap W} = \\rho_{V, W}|_{U \\cap V \\cap W} \\circ \\rho_{U, V}|_{U \\cap V \\cap W} $$ This is true by the uniqueness in the lemma because both sides of the equality are the unique isomorphism compatible with the maps $\\rho^U_{U''}$ and $\\rho^W_{U''}$ for $U'' \\subset U \\cap V \\cap W$, $U'' \\in \\mathcal{B}$. Some minor details omitted. The collection $(K_U, \\rho_{U, V})$ is a descent datum in the derived category for the open covering $\\mathcal{U} : X = \\bigcup_{U \\in \\mathcal{B}} U$ of $X$. In this language we are looking for ``effectiveness of the descent datum'' when we look for the existence of a solution."}
+{"_id": "2285", "title": "cohomology-remark-compatible-with-diagram", "text": "The map (\\ref{equation-projection-formula-map}) is compatible with the base change map of Remark \\ref{remark-base-change} in the following sense. Namely, suppose that $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ is a commutative diagram of ringed spaces.  Let $E \\in D(\\mathcal{O}_X)$ and $K \\in D(\\mathcal{O}_Y)$. Then the diagram $$ \\xymatrix{ Lg^*(Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K) \\ar[r]_p \\ar[d]_t & Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K) \\ar[d]_b \\\\ Lg^*Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} Lg^*K \\ar[d]_b & Rf'_*L(g')^*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K) \\ar[d]_t \\\\ Rf'_*L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} Lg^*K \\ar[rd]_p & Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} L(g')^*Lf^*K) \\ar[d]_c \\\\ & Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{Y'}} L(f')^*Lg^*K) } $$ is commutative. Here arrows labeled $t$ are gotten by an application of Lemma \\ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an application of Remark \\ref{remark-base-change}, arrows labeled $p$ by an application of (\\ref{equation-projection-formula-map}), and $c$ comes from $L(g')^* \\circ Lf^* = L(f')^* \\circ Lg^*$. We omit the verification."}
+{"_id": "2290", "title": "stacks-introduction-remark-diagonal", "text": "We have the formula $S \\times_{\\mathcal{M}_{1, 1}} S' = (S \\times S') \\times_{\\mathcal{M}_{1, 1} \\times \\mathcal{M}_{1, 1}} \\mathcal{M}_{1, 1}$. Hence the key fact is a property of the diagonal $\\Delta_{\\mathcal{M}_{1, 1}}$ of $\\mathcal{M}_{1, 1}$."}
+{"_id": "2291", "title": "stacks-introduction-remark-quotient-stack", "text": "The argument sketched above actually shows that $\\mathcal{M}_{1, 1} = [W/H]$ is a global quotient stack. It is true about 50\\% of the time that an argument proving a moduli stack is algebraic will show that it is a global quotient stack."}
+{"_id": "2448", "title": "restricted-remark-base-change", "text": "Let $\\varphi : A_1 \\to A_2$ be a ring map and let $I_i \\subset A_i$ be ideals such that $\\varphi(I_1^c) \\subset I_2$ for some $c \\geq 1$. This induces ring maps $A_{1, cn} = A_1/I_1^{cn} \\to A_2/I_2^n = A_{2, n}$ for all $n \\geq 1$. Let $\\mathcal{C}_i$ be the category (\\ref{equation-C}) for $(A_i, I_i)$. There is a base change functor \\begin{equation} \\label{equation-base-change-systems} \\mathcal{C}_1 \\longrightarrow \\mathcal{C}_2,\\quad (B_n) \\longmapsto (B_{cn} \\otimes_{A_{1, cn}} A_{2, n}) \\end{equation} Let $\\mathcal{C}_i'$ be the category (\\ref{equation-C-prime}) for $(A_i, I_i)$. If $I_2$ is finitely generated, then there is a base change functor \\begin{equation} \\label{equation-base-change-complete} \\mathcal{C}_1' \\longrightarrow \\mathcal{C}_2',\\quad B \\longmapsto (B \\otimes_{A_1} A_2)^\\wedge \\end{equation} because in this case the completion is complete (Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}). If both $I_1$ and $I_2$ are finitely generated, then the two base change functors agree via the functors (\\ref{equation-from-complete-to-systems}) which are equivalences by Lemma \\ref{lemma-topologically-finite-type}."}
+{"_id": "2449", "title": "restricted-remark-take-bar", "text": "Let $A$ be a Noetherian ring and $I \\subset A$ an ideal. Let $\\mathfrak a \\subset A$ be an ideal. Denote $\\bar A = A/\\mathfrak a$. Let $\\bar I \\subset \\bar A$ be an ideal such that $I^c \\bar A \\subset \\bar I$ and $\\bar I^d \\subset I\\bar A$ for some $c, d \\geq 1$. In this case the base change functor (\\ref{equation-base-change-complete}) for $(A, I)$ to $(\\bar A, \\bar I)$ is given by $B \\mapsto \\bar B = B/\\mathfrak aB$. Namely, we have \\begin{equation} \\label{equation-base-change-to-closed} \\bar B = (B \\otimes_A \\bar A)^\\wedge = (B/\\mathfrak a B)^\\wedge = B/\\mathfrak a B \\end{equation} the last equality because any finite $B$-module is $I$-adically complete by Algebra, Lemma \\ref{algebra-lemma-completion-tensor} and if annihilated by $\\mathfrak a$ also $\\bar I$-adically complete by Algebra, Lemma \\ref{algebra-lemma-change-ideal-completion}."}
+{"_id": "2450", "title": "restricted-remark-linear-approximation", "text": "Let $A$ be a ring and $I \\subset A$ be a finitely generated ideal. Let $C$ be an $I$-adically complete $A$-algebra. Let $\\psi : A[x_1, \\ldots, x_r]^\\wedge \\to C$ be a continuous $A$-algebra map. Suppose given $\\delta_i \\in C$, $i = 1, \\ldots, r$. Then we can consider $$ \\psi' : A[x_1, \\ldots, x_r]^\\wedge \\to C,\\quad x_i \\longmapsto \\psi(x_i) + \\delta_i $$ see Formal Spaces, Remark \\ref{formal-spaces-remark-universal-property}. Then we have $$ \\psi'(g) = \\psi(g) + \\sum \\psi(\\partial g/\\partial x_i)\\delta_i + \\xi $$ with error term $\\xi \\in (\\delta_i\\delta_j)$. This follows by writing $g$ as a power series and working term by term. Convergence is automatic as the coefficients of $g$ tend to zero. Details omitted."}
+{"_id": "2451", "title": "restricted-remark-improve-homomorphism", "text": "Let $A$ be a Noetherian ring and $I \\subset A$ be an ideal. Let $B$ be an object of (\\ref{equation-C-prime}). Let $C$ be an $I$-adically complete $A$-algebra. Let $\\psi_n : B \\to C/I^nC$ be an $A$-algebra homomorphism. The obstruction to lifting $\\psi_n$ to an $A$-algebra homomorphism into $C/I^{2n}C$ is an element $$ o(\\psi_n) \\in \\Ext^1_B(\\NL_{B/A}^\\wedge, I^nC/I^{2n}C) $$ as we will explain. Namely, choose a presentation $B = A[x_1, \\ldots, x_r]^\\wedge/J$. Choose a lift $\\psi : A[x_1, \\ldots, x_r]^\\wedge \\to C$ of $\\psi_n$. Since $\\psi(J) \\subset I^nC$ we get $\\psi(J^2) \\subset I^{2n}C$ and hence we get a $B$-linear homomorphism $$ o(\\psi) : J/J^2 \\longrightarrow I^nC/I^{2n}C, \\quad g \\longmapsto \\psi(g) $$ which of course extends to a $C$-linear map $J/J^2 \\otimes_B C \\to I^nC/I^{2n}C$. Since $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B \\text{d}x_i)$ we get $o(\\psi_n)$ as the image of $o(\\psi)$ by the identification \\begin{align*} & \\Ext^1_B(\\NL_{B/A}^\\wedge, I^nC/I^{2n}C) \\\\ & = \\Coker\\left(\\Hom_B(\\bigoplus B\\text{d}x_i, I^nC/I^{2n}C) \\to \\Hom_B(J/J^2, I^nC/I^{2n}C)\\right) \\end{align*} See More on Algebra, Lemma \\ref{more-algebra-lemma-map-out-of-almost-free} part (1) for the equality. \\medskip\\noindent Suppose that $o(\\psi_n)$ maps to zero in $\\Ext^1_B(\\NL_{B/A}^\\wedge, I^{n'}C/I^{2n'}C)$ for some integer $n'$ with $n > n' > n/2$. We claim that this means we can find an $A$-algebra homomorphism $\\psi'_{2n'} : B \\to C/I^{2n'}C$ which agrees with $\\psi_n$ as maps into $C/I^{n'}C$. The extreme case $n' = n$ explains why we previously said $o(\\psi_n)$ is the obstruction to lifting $\\psi_n$ to $C/I^{2n}C$. Proof of the claim: the hypothesis that $o(\\psi_n)$ maps to zero tells us we can find a $B$-module map $$ h : \\bigoplus B\\text{d}x_i \\longrightarrow I^{n'}C/I^{2n'}C $$ such that $o(\\psi)$ and $h \\circ \\text{d}$ agree as maps into $I^{n'}C/I^{2n'}C$. Say $h(\\text{d}x_i) = \\delta_i \\bmod I^{2n'}C$ for some $\\delta_i \\in I^{n'}C$. Then we look at the map $$ \\psi' : A[x_1, \\ldots, x_r]^\\wedge \\to C,\\quad x_i \\longmapsto \\psi(x_i) - \\delta_i $$ A computation with power series shows that $\\psi'(J) \\subset I^{2n'}C$. Namely, for $g \\in J$ we get $$ \\psi'(g) \\equiv \\psi(g) - \\sum \\psi(\\partial g/\\partial x_i)\\delta_i \\equiv o(\\psi)(g) - (h \\circ \\text{d})(g) \\equiv 0 \\bmod I^{2n'}C $$ See Remark \\ref{remark-linear-approximation} for the first equality. Hence $\\psi'$ induces an $A$-algebra homomorphism $\\psi'_{2n'} : B \\to C/I^{2n'}C$ as desired."}
+{"_id": "2452", "title": "restricted-remark-discussion", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object of (\\ref{equation-C-prime}) which is rig-smooth over $(A, I)$. As far as we know, it is an open question as to whether $B$ is isomorphic to the $I$-adic completion of a finite type $A$-algebra. Here are some things we do know: \\begin{enumerate} \\item If $A$ is a G-ring, then the answer is yes by Proposition \\ref{proposition-approximate}. \\item If $B$ is rig-\\'etale over $(A, I)$, then the answer is yes by Lemma \\ref{lemma-approximate}. \\item If $I$ is principal, then the answer is yes by \\cite[III Theorem 7]{Elkik}. \\item In general there exists an ideal $J = (b_1, \\ldots, b_s) \\subset B$ such that $V(J) \\subset V(IB)$ and such that the $I$-adic completion of each of the affine blowup algebras $B[\\frac{J}{b_i}]$ are isomorphic to the $I$-adic completion of a finite type $A$-algebra. \\end{enumerate} To see the last statement, choose $b_1, \\ldots, b_s$ as in Lemma \\ref{lemma-equivalent-with-artin-smooth} part (4) and use the properties mentioned there to see that Lemma \\ref{lemma-approximate-presentation-rig-smooth} applies to each completion $(B[\\frac{J}{b_i}])^\\wedge$. Part (4) tells us that ``rig-locally a rig-smooth formal algebraic space is the completion of a finite type scheme over $A$'' and it tells us that ``there is an admissible formal blowing up of $\\text{Spf}(B)$ which is affine locally algebraizable''."}
+{"_id": "2453", "title": "restricted-remark-NL-well-defined-topological", "text": "Let $A \\to B$ be an arrow of $\\text{WAdm}^{adic*}$ which is adic and topologically of finite type (see Lemma \\ref{lemma-finite-type}). Write $B = A\\{x_1, \\ldots, x_r\\}/J$. Then we can set\\footnote{In fact, this construction works for arrows of $\\text{WAdm}^{count}$ satisfying the equivalent conditions of Formal Spaces, Lemma \\ref{formal-spaces-lemma-quotient-restricted-power-series}.} $$ \\NL_{B/A}^\\wedge = \\left(J/J^2 \\longrightarrow \\bigoplus B\\text{d}x_i\\right) $$ Exactly as in the proof of Lemma \\ref{lemma-NL-up-to-homotopy} the reader can show that this complex of $B$-modules is well defined up to (unique isomorphism) in the homotopy category $K(B)$. Now, if $A$ is Noetherian and $I \\subset A$ is an ideal of definition, then this construction reproduces the naive cotangent complex of $B$ over $(A, I)$ defined by Equation (\\ref{equation-NL}) in Section \\ref{section-naive-cotangent-complex} simply because $A[x_1, \\ldots, x_n]^\\wedge$ agrees with $A\\{x_1, \\ldots, x_r\\}$ by Formal Spaces, Remark \\ref{formal-spaces-remark-I-adic-completion-and-restricted-power-series}. In particular, we find that, still when $A$ is an adic Noetherian topological ring, the object $\\NL_{B/A}^\\wedge$ is independent of the choice of the ideal of definition $I \\subset A$."}
+{"_id": "2454", "title": "restricted-remark-rig-surjective-more-general", "text": "The condition as formulated in Definition \\ref{definition-rig-surjective} is not right even for morphisms of finite type of locally adic* formal algebraic spaces. For example, if $A = (\\bigcup_{n \\geq 1} k[t^{1/n}])^\\wedge$ where the completion is the $t$-adic completion, then there are no adic morphisms $\\text{Spf}(R) \\to \\text{Spf}(A)$ where $R$ is a complete discrete valuation ring. Thus any morphism $X \\to \\text{Spf}(A)$ would be rig-surjective, but since $A$ is a domain and $t \\in A$ is not zero, we want to think of $A$ as having at least one ``rig-point'', and we do not want to allow $X = \\emptyset$. To cover this particular case, one can consider adic morphisms $$ \\text{Spf}(R) \\longrightarrow Y $$ where $R$ is a valuation ring complete with respect to a principal ideal $J$ whose radical is $\\mathfrak m_R = \\sqrt{J}$. In this case the value group of $R$ can be embedded into $(\\mathbf{R}, +)$ and one obtains the point of view used by Berkovich in defining an analytic space associated to $Y$, see \\cite{Berkovich}. Another approach is championed by Huber. In his theory, one drops the hypothesis that $\\Spec(R/J)$ is a singleton, see \\cite{Huber-continuous-valuations}."}
+{"_id": "2455", "title": "restricted-remark-diagonal-gives-diagonal", "text": "In the situation above consider the diagonal morphisms $\\Delta_f : X' \\to X' \\times_X X'$ and $\\Delta_{f_{/T}} : X'_{/T'} \\to X'_{/T'} \\times_{X_{/T}} X'_{/T'}$. It is easy to see that $$ X'_{/T'} \\times_{X_{/T}} X'_{/T'} = (X' \\times_X X')_{/T''} $$ as subfunctors of $X' \\times_X X'$ where $T'' \\subset |X' \\times_X X'|$ is the inverse image of $T$. Hence we see that $\\Delta_{f_{/T}} = (\\Delta_f)_{/T''}$. We will use this below to show that properties of $\\Delta_f$ are inherited by $\\Delta_{f_{/T}}$."}
+{"_id": "2456", "title": "restricted-remark-compare-formal-modification-artin", "text": "In \\cite[Definition 1.7]{ArtinII} a formal modification is defined as a proper morphism $f : X \\to Y$ of locally Noetherian formal algebraic spaces satisfying the following three conditions\\footnote{We will not completely translate these conditions into the language developed in the Stacks project. We hope nonetheless the discussion here will be useful to the reader.} \\begin{enumerate} \\item[(\\romannumeral1)] the Cramer and Jacobian ideal of $f$ each contain an ideal of definition of $X$, \\item[(\\romannumeral2)] the ideal defining the diagonal map $\\Delta : X \\to X \\times_Y X$ is annihilated by an ideal of definition of $X \\times_Y X$, and \\item[(\\romannumeral3)] any adic morphism $\\text{Spf}(R) \\to Y$ lifts to $\\text{Spf}(R) \\to X$ whenever $R$ is a complete discrete valuation ring. \\end{enumerate} Let us compare these to our list of conditions above. \\medskip\\noindent Ad (\\romannumeral1). Property (\\romannumeral1) agrees with our condition that $f$ be a rig-\\'etale morphism: this follows from Lemma \\ref{lemma-equivalent-with-artin} part (\\ref{item-condition-artin}). \\medskip\\noindent Ad (\\romannumeral2). Assume $f$ is rig-\\'etale. Then $\\Delta_f : X \\to X \\times_Y X$ is rig-\\'etale as a morphism of locally Noetherian formal algebraic spaces which are rig-\\'etale over $X$ (via $\\text{id}_X$ for the first one and via $\\text{pr}_1$ for the second one). See Lemmas \\ref{lemma-base-change-rig-etale} and \\ref{lemma-rig-etale-permanence}. Hence property (\\romannumeral2) agrees with our condition that $\\Delta_f$ be rig-surjective by Lemma \\ref{lemma-closed-immersion-rig-smooth-rig-surjective}. \\medskip\\noindent Ad (\\romannumeral3). Property (\\romannumeral3) does not quite agree with our notion of a rig-surjective morphism, as Artin requires all adic morphisms $\\text{Spf}(R) \\to Y$ to lift to morphisms into $X$ whereas our notion of rig-surjective only asserts the existence of a lift after replacing $R$ by an extension. However, since we already have that $\\Delta_f$ is rig-\\'etale and rig-surjective by (\\romannumeral1) and (\\romannumeral2), these conditions are equivalent by Lemma \\ref{lemma-rig-monomorphism-rig-surjective}."}
+{"_id": "2506", "title": "more-groupoids-remark-local-source-warning", "text": "Warning: Lemma \\ref{lemma-local-source} should be used with care. For example, it applies to $\\mathcal{P}=$``flat'', $\\mathcal{Q}=$``empty'', and $\\mathcal{R}=$``flat and locally of finite presentation''. But given a morphism of schemes $f : X \\to Y$ the largest open $W \\subset X$ such that $f|_W$ is flat is {\\it not} the set of points where $f$ is flat!"}
+{"_id": "2507", "title": "more-groupoids-remark-local-source-apply", "text": "Notwithstanding the warning in Remark \\ref{remark-local-source-warning} there are some cases where Lemma \\ref{lemma-local-source} can be used without causing too much ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list: \\begin{enumerate} \\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and $\\mathcal{P} =$``relative dimension $\\leq d$''. See Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d} and Morphisms, Lemmas \\ref{morphisms-lemma-openness-bounded-dimension-fibres} and \\ref{morphisms-lemma-dimension-fibre-after-base-change}. \\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and $\\mathcal{P} =$``locally quasi-finite''. This is the case $d = 0$ of the previous item, see Morphisms, Lemma \\ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. \\item $\\mathcal{Q} = \\mathcal{R} =$``locally of finite type'', and $\\mathcal{P} =$``unramified''. See Morphisms, Lemmas \\ref{morphisms-lemma-unramified-characterize} and \\ref{morphisms-lemma-set-points-where-fibres-unramified}. \\end{enumerate} What is interesting about the cases listed above is that we do not need to assume that $s, t$ are flat to get a conclusion about the locus where the morphism $h$ has property $\\mathcal{P}$. We continue the list: \\begin{enumerate} \\item[(4)] $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P} =$``flat''. See More on Morphisms, Theorem \\ref{more-morphisms-theorem-openness-flatness} and Lemma \\ref{more-morphisms-lemma-flat-locus-base-change}. \\item[(5)] $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P}=$``Cohen-Macaulay''. See More on Morphisms, Definition \\ref{more-morphisms-definition-CM} and More on Morphisms, Lemmas \\ref{more-morphisms-lemma-base-change-CM} and \\ref{more-morphisms-lemma-flat-finite-presentation-CM-open}. \\item[(6)] $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P}=$``syntomic'' use Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-lci} (the locus is automatically open). \\item[(7)] $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P}=$``smooth''. See Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-smooth} (the locus is automatically open). \\item[(8)] $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P}=$``\\'etale''. See Morphisms, Lemma \\ref{morphisms-lemma-set-points-where-fibres-etale} (the locus is automatically open). \\end{enumerate}"}
+{"_id": "2508", "title": "more-groupoids-remark-warn-dimension-groupoid-on-field", "text": "Warning: Lemma \\ref{lemma-groupoid-on-field-dimension-equal-stabilizer} is wrong without the condition that $s$ and $t$ are locally of finite type. An easy example is to start with the action $$ \\mathbf{G}_{m, \\mathbf{Q}} \\times_{\\mathbf{Q}} \\mathbf{A}^1_{\\mathbf{Q}} \\to \\mathbf{A}^1_{\\mathbf{Q}} $$ and restrict the corresponding groupoid scheme to the generic point of $\\mathbf{A}^1_{\\mathbf{Q}}$. In other words restrict via the morphism $\\Spec(\\mathbf{Q}(x)) \\to \\Spec(\\mathbf{Q}[x]) = \\mathbf{A}^1_{\\mathbf{Q}}$. Then you get a groupoid scheme $(U, R, s, t, c)$ with $U = \\Spec(\\mathbf{Q}(x))$ and $$ R = \\Spec\\left( \\mathbf{Q}(x)[y]\\left[ \\frac{1}{P(xy)}, P \\in \\mathbf{Q}[T], P \\not = 0 \\right] \\right) $$ In this case $\\dim(R) = 1$ and $\\dim(G) = 0$."}
+{"_id": "2598", "title": "examples-remark-reference-existence-regular-nonexcellent-rings", "text": "Non-excellent regular rings whose residue fields have a finite $p$-basis can be constructed even in the function field of $\\mathbb{P}^2_k$, over a  characteristic $p$ field $k = \\overline{k}$. See \\cite[$\\mathsection 4.1$]{DS18}."}
+{"_id": "2599", "title": "examples-remark-specialization", "text": "Here are some remarks: \\begin{enumerate} \\item The presheaves $F$ and $F_n$ are separated presheaves. \\item It turns out that $F$, $F_n$ are not sheaves. \\item One can show that $G$, $G_n$ is actually a sheaf for the fppf topology. \\end{enumerate} We will prove these results if we need them."}
+{"_id": "2600", "title": "examples-remark-contradict-aoki", "text": "Proposition \\ref{proposition-nonalghomstack} contradicts \\cite[Theorem 1.1]{AokiHomStacks}. The problem is the non-effectivity of formal objects for $\\underline{\\Mor}_S(X, [S/A])$. The same problem is mentioned in the Erratum \\cite{AokiHomStacksErr} to \\cite{AokiHomStacks}. Unfortunately, the Erratum goes on to assert that $\\underline{\\Mor}_S(\\mathcal{Y}, \\mathcal{Z})$ is algebraic if $\\mathcal{Z}$ is separated, which also contradicts Proposition \\ref{proposition-nonalghomstack} as $[S/A]$ is separated."}
+{"_id": "2768", "title": "spaces-perfect-remark-match-total-direct-images", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of representable algebraic spaces $X$ and $Y$ over $S$. Let $f_0 : X_0 \\to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then the diagram $$ \\xymatrix{ D_\\QCoh(\\mathcal{O}_{X_0}) \\ar@{=}[rrrrrr]_{\\text{Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_X) \\\\ D_\\QCoh(\\mathcal{O}_{Y_0}) \\ar[u]^{Lf^*_0} \\ar@{=}[rrrrrr]^{\\text{Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_Y) \\ar[u]_{Lf^*} } $$ (Lemma \\ref{lemma-quasi-coherence-pullback} and Derived Categories of Schemes, Lemma \\ref{perfect-lemma-quasi-coherence-pullback}) is commutative. This follows as the equivalences $D_\\QCoh(\\mathcal{O}_{X_0}) \\to D_\\QCoh(\\mathcal{O}_X)$ and $D_\\QCoh(\\mathcal{O}_{Y_0}) \\to D_\\QCoh(\\mathcal{O}_Y)$ of Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} come from pulling back by the (flat) morphisms of ringed sites $\\epsilon : X_\\etale \\to X_{0, Zar}$ and $\\epsilon : Y_\\etale \\to Y_{0, Zar}$ and the diagram of ringed sites $$ \\xymatrix{ X_{0, Zar} \\ar[d]_{f_0} & X_\\etale \\ar[l]^\\epsilon \\ar[d]^f \\\\ Y_{0, Zar} & Y_\\etale \\ar[l]_\\epsilon } $$ is commutative (details omitted). If $f$ is quasi-compact and quasi-separated, equivalently if $f_0$ is quasi-compact and quasi-separated, then we claim $$ \\xymatrix{ D_\\QCoh(\\mathcal{O}_{X_0}) \\ar[d]_{Rf_{0, *}} \\ar@{=}[rrrrrr]_{\\text{Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D_\\QCoh(\\mathcal{O}_{Y_0}) \\ar@{=}[rrrrrr]^{\\text{Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_Y) } $$ (Lemma \\ref{lemma-quasi-coherence-direct-image} and Derived Categories of Schemes, Lemma \\ref{perfect-lemma-quasi-coherence-direct-image}) is commutative as well. This also follows from the commutative diagram of sites displayed above as the proof of Lemma \\ref{lemma-derived-quasi-coherent-small-etale-site} shows that the functor $R\\epsilon_*$ gives the equivalences $D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_{X_0})$ and $D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_{Y_0})$."}
+{"_id": "2769", "title": "spaces-perfect-remark-how-to", "text": "How to choose the collection $\\mathcal{B}$ in Lemma \\ref{lemma-induction-principle-separated}? Here are some examples: \\begin{enumerate} \\item If $X$ is quasi-compact and separated, then we can choose $\\mathcal{B}$ to be the set of quasi-compact and separated objects of $X_{spaces, \\etale}$. Then $X \\in \\mathcal{B}$ and $\\mathcal{B}$ satisfies (1), (2), and (3)(a). With this choice of $\\mathcal{B}$ Lemma \\ref{lemma-induction-principle-separated} reproduces Lemma \\ref{lemma-induction-principle}. \\item If $X$ is quasi-compact with affine diagonal, then we can choose $\\mathcal{B}$ to be the set of objects of $X_{spaces, \\etale}$ which are quasi-compact and have affine diagonal. Again $X \\in \\mathcal{B}$ and $\\mathcal{B}$ satisfies (1), (2), and (3)(a). \\item If $X$ is quasi-compact and quasi-separated, then the smallest subset $\\mathcal{B}$ which contains $X$ and satisfies (1), (2), and (3)(a) is given by the rule $W \\in \\mathcal{B}$ if and only if either $W$ is a quasi-compact open subspace of $X$, or $W$ is a quasi-compact open of an affine object of $X_{spaces, \\etale}$. \\end{enumerate}"}
+{"_id": "2770", "title": "spaces-perfect-remark-addendum", "text": "The proof of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support} shows that $$ R|_W = P \\oplus P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] $$ for some $m \\geq 0$ and $n_j \\geq 0$. Thus the highest degree cohomology sheaf of $R|_W$ equals that of $P$. By repeating the construction for the map $P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] \\to R|_W$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_W$ and $P$ above any given degree."}
+{"_id": "2771", "title": "spaces-perfect-remark-pullback-generator", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Let $E \\in D_\\QCoh(\\mathcal{O}_Y)$ be a generator (see Theorem \\ref{theorem-bondal-van-den-Bergh}). Then the following are equivalent \\begin{enumerate} \\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have $Rf_*K = 0$ if and only if $K = 0$, \\item $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ reflects isomorphisms, and \\item $Lf^*E$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$. \\end{enumerate} The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma \\ref{lemma-affine-morphism}) or if $f$ is an open immersion, or if $f$ is a composition of such."}
+{"_id": "2772", "title": "spaces-perfect-remark-classical-generator", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G$ be a perfect object of $D(\\mathcal{O}_X)$ which is a generator for $D_\\QCoh(\\mathcal{O}_X)$. By Theorem \\ref{theorem-bondal-van-den-Bergh} there is at least one of these. Combining Lemma \\ref{lemma-quasi-coherence-direct-sums} with Proposition \\ref{proposition-compact-is-perfect} and with Derived Categories, Proposition \\ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$."}
+{"_id": "2773", "title": "spaces-perfect-remark-classical-generator-with-support", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$ is quasi-compact. Let $G$ be a perfect object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a generator for $D_{\\QCoh, T}(\\mathcal{O}_X)$. By Lemma \\ref{lemma-generator-with-support} there is at least one of these. Combining the fact that $D_{\\QCoh, T}(\\mathcal{O}_X)$ has direct sums with Lemma \\ref{lemma-compact-is-perfect-with-support} and with Derived Categories, Proposition \\ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf, T}(\\mathcal{O}_X)$."}
+{"_id": "2774", "title": "spaces-perfect-remark-DQCoh-is-Ddga-with-support", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$ is quasi-compact. The analogue of Theorem \\ref{theorem-DQCoh-is-Ddga} holds for $D_{\\QCoh, T}(\\mathcal{O}_X)$. This follows from the exact same argument as in the proof of the theorem, using Lemmas \\ref{lemma-generator-with-support} and \\ref{lemma-compact-is-perfect-with-support} and a variant of Lemma \\ref{lemma-tensor-with-QCoh-complex} with supports. If we ever need this, we will precisely state the result here and give a detailed proof."}
+{"_id": "2775", "title": "spaces-perfect-remark-independence-choice", "text": "Let $X$ be a quasi-compact and quasi-separated algebraic space over a ring $R$. By the construction of the proof of Theorem \\ref{theorem-DQCoh-is-Ddga} there exists a differential graded algebra $(A, \\text{d})$ over $R$ such that $D_\\QCoh(X)$ is $R$-linearly equivalent to $D(A, \\text{d})$ as a triangulated category. One may ask: how unique is $(A, \\text{d})$? The answer is (only) slightly better than just saying that $(A, \\text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \\text{d})$ is a second such pair. Then we have $$ (A, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet) $$ and $$ (B, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, L^\\bullet) $$ for some K-injective complexes $K^\\bullet$ and $L^\\bullet$ of $\\mathcal{O}_X$-modules corresponding to perfect generators of $D_\\QCoh(\\mathcal{O}_X)$. Set $$ \\Omega = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, L^\\bullet) \\quad \\Omega' = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, K^\\bullet) $$ Then $\\Omega$ is a differential graded $B^{opp} \\otimes_R A$-module and $\\Omega'$ is a differential graded $A^{opp} \\otimes_R B$-module. Moreover, the equivalence $$ D(A, \\text{d}) \\to D_\\QCoh(\\mathcal{O}_X) \\to D(B, \\text{d}) $$ is given by the functor $- \\otimes_A^\\mathbf{L} \\Omega'$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark \\ref{dga-remark-hochschild-cohomology}. If we ever need this remark we will provide a precise statement with a detailed proof here."}
+{"_id": "2776", "title": "spaces-perfect-remark-explain-consequence", "text": "Let $S$ be a scheme. Let $(U \\subset X, f : V \\to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $X$, $U$, $V$ are quasi-compact and quasi-separated. By Lemma \\ref{lemma-better-coherator} the functors $DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \\times_X V}$ exist. Moreover, there is a canonical distinguished triangle $$ DQ_X(K) \\to Rj_{U, *}DQ_U(K|_U) \\oplus Rj_{V, *}DQ_V(K|_V) \\to Rj_{U \\times_X V, *}DQ_{U \\times_X V}(K|_{U \\times_X V}) \\to $$ for any $K \\in D(\\mathcal{O}_X)$. This follows by applying the exact functor $DQ_X$ to the distinguished triangle of Lemma \\ref{lemma-exact-sequence-j-star} and using Lemma \\ref{lemma-pushforward-better-coherator} three times."}
+{"_id": "2777", "title": "spaces-perfect-remark-multiplication-map", "text": "With notation as in Lemma \\ref{lemma-affine-morphism-and-hom-out-of-perfect}. The diagram $$ \\xymatrix{ R\\Hom_X(M, Rg'_*L) \\otimes_R^\\mathbf{L} R' \\ar[r] \\ar[d]_\\mu & R\\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \\ar[d]^a \\\\ R\\Hom_X(M, R(g')_*L) \\ar@{=}[r] & R\\Hom_{X'}(L(g')^*M, L) } $$ is commutative where the top horizontal arrow is the map from the lemma, $\\mu$ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \\to L$. The multiplication map is the adjunction map $K' \\otimes_R^\\mathbf{L} R' \\to K'$ for any $K' \\in D(R')$."}
+{"_id": "2778", "title": "spaces-perfect-remark-base-change-of-L", "text": "The pseudo-coherent complex $L$ of part (B) of Lemma \\ref{lemma-compute-ext} is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of schemes we have canonical functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_{Y'}}(Lg^*L, \\mathcal{F}') \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(L(g')^*E, (g')^*\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{F}') $$ for $\\mathcal{F}'$ quasi-coherent on $Y'$. Obsere that we do {\\bf not} use derived pullback on $\\mathcal{G}^\\bullet$ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof."}
+{"_id": "2779", "title": "spaces-perfect-remark-explain-perfect-direct-image", "text": "Let $R$ be a ring. Let $X$ be an algebraic space of finite presentation over $R$. Let $\\mathcal{G}$ be a finitely presented $\\mathcal{O}_X$-module flat over $R$ with support proper over $R$. By Lemma \\ref{lemma-base-change-tensor-perfect} there exists a finite complex of finite projective $R$-modules $M^\\bullet$ such that we have $$ R\\Gamma(X_{R'}, \\mathcal{G}_{R'}) = M^\\bullet \\otimes_R R' $$ functorially in the $R$-algebra $R'$."}
+{"_id": "2934", "title": "dualizing-remark-matlis", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $E$ be an injective hull of $\\kappa$ over $R$. Here is an addendum to Matlis duality: If $N$ is an $\\mathfrak m$-power torsion module and $M = \\Hom_R(N, E)$ is a finite module over the completion of $R$, then $N$ satisfies the descending chain condition. Namely, for any submodules $N'' \\subset N' \\subset N$ with $N'' \\not = N'$, we can find an embedding $\\kappa \\subset N''/N'$ and hence a nonzero map $N' \\to E$ annihilating $N''$ which we can extend to a map $N \\to E$ annihilating $N''$. Thus $N \\supset N' \\mapsto M' = \\Hom_R(N/N', E) \\subset M$ is an inclusion preserving map from submodules of $N$ to submodules of $M$, whence the conclusion."}
+{"_id": "2935", "title": "dualizing-remark-exact-support", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal. Set $B = A/I$. In this case the functor $\\Hom_A(B, -)$ is equal to the functor $$ \\text{Mod}_A \\longrightarrow \\text{Mod}_B,\\quad M \\longmapsto M[I] $$ which sends $M$ to the submodule of $I$-torsion."}
+{"_id": "2936", "title": "dualizing-remark-vanishing-for-arbitrary-modules", "text": "Let $(A, \\mathfrak m)$ and $\\omega_A^\\bullet$ be as in Lemma \\ref{lemma-sitting-in-degrees}. By More on Algebra, Lemma \\ref{more-algebra-lemma-injective-amplitude} we see that $\\omega_A^\\bullet$ has injective-amplitude in $[-d, 0]$ because part (3) of that lemma applies. In particular, for any $A$-module $M$ (not necessarily finite) we have $\\Ext^i_A(M, \\omega_A^\\bullet) = 0$ for $i \\not \\in \\{-d, \\ldots, 0\\}$."}
+{"_id": "2937", "title": "dualizing-remark-specific-injective-hull", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with a normalized dualizing complex $\\omega_A^\\bullet$. By Lemma \\ref{lemma-local-cohomology-of-dualizing} above we see that $R\\Gamma_Z(\\omega_A^\\bullet)$ is an injective hull of the residue field placed in degree $0$. In fact, this gives a ``construction'' or ``realization'' of the injective hull which is slightly more canonical than just picking any old injective hull. Namely, a normalized dualizing complex is unique up to isomorphism, with group of automorphisms the group of units of $A$, whereas an injective hull of $\\kappa$ is unique up to isomorphism, with group of automorphisms the group of units of the completion $A^\\wedge$ of $A$ with respect to $\\mathfrak m$."}
+{"_id": "3089", "title": "properties-remark-normal-connected-irreducible", "text": "Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that $X$ is integral if and only if $X$ is connected, see Lemma \\ref{lemma-normal-locally-Noetherian}. But there exists a connected affine scheme $X$ such that $\\mathcal{O}_{X, x}$ is a domain for all $x \\in X$, but $X$ is not irreducible, see Examples, Section \\ref{examples-section-connected-locally-integral-not-integral}. This example is even a normal scheme (proof omitted), so beware!"}
+{"_id": "3090", "title": "properties-remark-non-integral-Japanese", "text": "In \\cite{Hoobler-finite} a (locally Noetherian) scheme $X$ is called Japanese if for every $x \\in X$ and every associated prime $\\mathfrak p$ of $\\mathcal{O}_{X, x}$ the ring $\\mathcal{O}_{X, x}/\\mathfrak p$ is Japanese. We do not use this definition since there exists a one dimensional Noetherian domain with excellent (in particular Japanese) local rings whose normalization is not finite. See \\cite[Example 1]{Hochster-loci} or \\cite{Heinzer-Levy} or \\cite[Expos\\'e XIX]{Traveaux}. On the other hand, we could circumvent this problem by calling a scheme $X$ Japanese if for every affine open $\\Spec(A) \\subset X$ the ring $A/\\mathfrak p$ is Japanese for every associated prime $\\mathfrak p$ of $A$."}
+{"_id": "3091", "title": "properties-remark-neurotic", "text": "With assumptions and notation of Lemma \\ref{lemma-ample-quasi-coherent}. Denote the displayed map of the lemma by $\\theta_\\mathcal{F}$. Note that the isomorphism $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$ of Lemma \\ref{lemma-ample-gcd-is-one} is just $\\theta_{\\mathcal{L}^{\\otimes n}}$. Consider the multiplication maps $$ \\widetilde{M} \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(n) \\longrightarrow \\widetilde{M(n)} $$ see Constructions, Equation (\\ref{constructions-equation-multiply-more-generally}). Pull this back to $X$ and consider $$ \\xymatrix{ f^*\\widetilde{M} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{O}_Y(n) \\ar[r] \\ar[d]_{\\theta_\\mathcal{F} \\otimes \\theta_{\\mathcal{L}^{\\otimes n}}} & f^*\\widetilde{M(n)} \\ar[d]^{\\theta_{\\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}}} \\\\ \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n} \\ar[r]^{\\text{id}} & \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n} } $$ Here we have used the obvious identification $M(n) = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n})$. This diagram commutes. Proof omitted."}
+{"_id": "3092", "title": "properties-remark-maximal-points-affine", "text": "Lemma \\ref{lemma-maximal-points-affine} above is false if $X$ is not quasi-separated. Here is an example. Take $R = \\mathbf{Q}[x, y_1, y_2, \\ldots]/((x-i)y_i)$. Consider the minimal prime ideal $\\mathfrak p = (y_1, y_2, \\ldots)$ of $R$. Glue two copies of $\\Spec(R)$ along the (not quasi-compact) open $\\Spec(R) \\setminus V(\\mathfrak p)$ to get a scheme $X$ (glueing as in Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}). Then the two maximal points of $X$ corresponding to $\\mathfrak p$ are not contained in a common affine open. The reason is that any open of $\\Spec(R)$ containing $\\mathfrak p$ contains infinitely many of the ``lines'' $x = i$, $y_j = 0$, $j \\not = i$ with parameter $y_i$. Details omitted."}
+{"_id": "3232", "title": "quot-remark-hom-base-change", "text": "In Situation \\ref{situation-hom} let $B' \\to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \\times_B B'$ and denote $\\mathcal{F}'$, $\\mathcal{G}'$ the pullback of $\\mathcal{F}$, $\\mathcal{G}$ to $X'$. Then we obtain a functor $\\mathit{Hom}(\\mathcal{F}', \\mathcal{G}') : (\\Sch/B')^{opp} \\to \\textit{Sets}$ associated to the base change $f' : X' \\to B'$. For a scheme $T$ over $B'$ it is clear that we have $$ \\mathit{Hom}(\\mathcal{F}', \\mathcal{G}')(T) = \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T) $$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \\to B' \\to B$. This trivial remark will occasionally be useful to change the base algebraic space."}
+{"_id": "3233", "title": "quot-remark-coherent-base-change", "text": "In Situation \\ref{situation-coherent} the rule $(T, g, \\mathcal{F}) \\mapsto (T, g)$ defines a $1$-morphism $$ \\Cohstack_{X/B} \\longrightarrow \\mathcal{S}_B $$ of stacks in groupoids (see Lemma \\ref{lemma-coherent-stack}, Algebraic Stacks, Section \\ref{algebraic-section-split}, and Examples of Stacks, Section \\ref{examples-stacks-section-stack-associated-to-sheaf}). Let $B' \\to B$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{S}_{B'} \\to \\mathcal{S}_B$ be the associated $1$-morphism of stacks fibred in sets. Set $X' = X \\times_B B'$. We obtain a stack in groupoids $\\Cohstack_{X'/B'} \\to (\\Sch/S)_{fppf}$ associated to the base change $f' : X' \\to B'$. In this situation the diagram $$ \\vcenter{ \\xymatrix{ \\Cohstack_{X'/B'} \\ar[r] \\ar[d] & \\Cohstack_{X/B} \\ar[d] \\\\ \\mathcal{S}_{B'} \\ar[r] & \\mathcal{S}_B } } \\quad \\begin{matrix} \\text{or in} \\\\ \\text{another} \\\\ \\text{notation} \\end{matrix} \\quad \\vcenter{ \\xymatrix{ \\Cohstack_{X'/B'} \\ar[r] \\ar[d] & \\Cohstack_{X/B} \\ar[d] \\\\ \\Sch/B' \\ar[r] & \\Sch/B } } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space."}
+{"_id": "3234", "title": "quot-remark-q-base-change", "text": "In Situation \\ref{situation-q} let $B' \\to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \\times_B B'$ and denote $\\mathcal{F}'$ the pullback of $\\mathcal{F}$ to $X'$. Thus we have the functor $Q_{\\mathcal{F}'/X'/B'}$ on the category of schemes over $B'$. For a scheme $T$ over $B'$ it is clear that we have $$ Q_{\\mathcal{F}'/X'/B'}(T) = Q_{\\mathcal{F}/X/B}(T) $$ where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \\to B' \\to B$. Similar remarks apply to $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$. These trivial remarks will occasionally be useful to change the base algebraic space."}
+{"_id": "3235", "title": "quot-remark-q-sheaf", "text": "Let $S$ be a scheme, $X$ an algebraic space over $S$, and $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module. Suppose that $\\{f_i : X_i \\to X\\}_{i \\in I}$ is an fpqc covering and for each $i, j \\in I$ we are given an fpqc covering $\\{X_{ijk} \\to X_i \\times_X X_j\\}$. In this situation we have a bijection $$ \\left\\{ \\begin{matrix} \\text{quotients }\\mathcal{F} \\to \\mathcal{Q}\\text{ where } \\\\ \\mathcal{Q}\\text{ is a quasi-coherent }\\\\ \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\text{families of quotients }f_i^*\\mathcal{F} \\to \\mathcal{Q}_i \\text{ where } \\\\ \\mathcal{Q}_i\\text{ is quasi-coherent and } \\mathcal{Q}_i\\text{ and }\\mathcal{Q}_j\\\\ \\text{ restrict to the same quotient on }X_{ijk} \\end{matrix} \\right\\} $$ Namely, let $(f_i^*\\mathcal{F} \\to \\mathcal{Q}_i)_{i \\in I}$ be an element of the right hand side. Then since $\\{X_{ijk} \\to X_i \\times_X X_j\\}$ is an fpqc covering we see that the pullbacks of $\\mathcal{Q}_i$ and $\\mathcal{Q}_j$ restrict to the same quotient of the pullback of $\\mathcal{F}$ to $X_i \\times_X X_j$ (by fully faithfulness in Descent on Spaces, Proposition \\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}). Hence we obtain a descent datum for quasi-coherent modules with respect to $\\{X_i \\to X\\}_{i \\in I}$. By Descent on Spaces, Proposition \\ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we find a map of quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{F} \\to \\mathcal{Q}$ whose restriction to $X_i$ recovers the given maps $f_i^*\\mathcal{F} \\to \\mathcal{Q}_i$. Since the family of morphisms $\\{X_i \\to X\\}$ is jointly surjective and flat, for every point $x \\in |X|$ there exists an $i$ and a point $x_i \\in |X_i|$ mapping to $x$. Note that the induced map on local rings $\\mathcal{O}_{X, \\overline{x}} \\to \\mathcal{O}_{X_i, \\overline{x_i}}$ is faithfully flat, see Morphisms of Spaces, Section \\ref{spaces-morphisms-section-flat}. Thus we see that $\\mathcal{F} \\to \\mathcal{Q}$ is surjective."}
+{"_id": "3236", "title": "quot-remark-q-obs", "text": "In Situation \\ref{situation-q} {\\bf assume} that $\\mathcal{F}$ is flat over $B$. Let $T \\subset T'$ be an first order thickening of schemes over $B$ with ideal sheaf $\\mathcal{J}$. Then $X_T \\subset X_{T'}$ is a first order thickening of algebraic spaces whose ideal sheaf $\\mathcal{I}$ is a quotient of $f_T^*\\mathcal{J}$. We will think of sheaves on $X_{T'}$, resp.\\ $T'$ as sheaves on $X_T$, resp.\\ $T$ using the fundamental equivalence described in More on Morphisms of Spaces, Section \\ref{spaces-more-morphisms-section-thickenings}. Let $$ 0 \\to \\mathcal{K} \\to \\mathcal{F}_T \\to \\mathcal{Q} \\to 0 $$ define an element $x$ of $Q_{\\mathcal{F}/X/B}(T)$. Since $\\mathcal{F}_{T'}$ is flat over $T'$ we have a short exact sequence $$ 0 \\to f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T \\xrightarrow{i} \\mathcal{F}_{T'} \\xrightarrow{\\pi} \\mathcal{F}_T \\to 0 $$ and we have $f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T = \\mathcal{I} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{F}_T$, see Deformation Theory, Lemma \\ref{defos-lemma-deform-module-ringed-topoi}. Let us use the abbreviation $ f_T^*\\mathcal{J} \\otimes_{\\mathcal{O}_{X_T}} \\mathcal{G} = \\mathcal{G} \\otimes_{\\mathcal{O}_T} \\mathcal{J} $ for an $\\mathcal{O}_{X_T}$-module $\\mathcal{G}$. Since $\\mathcal{Q}$ is flat over $T$, we obtain a short exact sequence $$ 0 \\to \\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to \\mathcal{F}_T \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to \\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to \\to 0 $$ Combining the above we obtain an canonical extension $$ 0 \\to \\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to \\pi^{-1}(\\mathcal{K})/i(\\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J}) \\to \\mathcal{K} \\to 0 $$ of $\\mathcal{O}_{X_T}$-modules. This defines a canonical class $$ o_x(T') \\in \\Ext^1_{\\mathcal{O}_{X_T}}(\\mathcal{K}, \\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J}) $$ If $o_x(T')$ is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a $\\mathcal{O}_{X_{T'}}$-submodule $\\mathcal{K}' \\subset \\pi^{-1}(\\mathcal{K})$ sitting in a short exact sequence $0 \\to \\mathcal{K} \\otimes_{\\mathcal{O}_T} \\mathcal{J} \\to \\mathcal{K}' \\to \\mathcal{K} \\to 0$. Then it follows from the lemma reference above that $\\mathcal{Q}' = \\mathcal{F}_{T'}/\\mathcal{K}'$ is a lift of $x$ to an element of $Q_{\\mathcal{F}/X/B}(T')$. Conversely, the reader sees that the existence of a lift implies that $o_x(T')$ is zero. Moreover, if $x \\in Q_{\\mathcal{F}/X/B}^{fp}(T)$, then automatically $x' \\in Q_{\\mathcal{F}/X/B}^{fp}(T')$ by Deformation Theory, Lemma \\ref{defos-lemma-deform-fp-module-ringed-topoi}. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi)."}
+{"_id": "3237", "title": "quot-remark-q-defos", "text": "In Situation \\ref{situation-q} {\\bf assume} that $\\mathcal{F}$ is flat over $B$. We continue the discussion of Remark \\ref{remark-q-obs}. Assume $o_x(T') = 0$. Then we claim that the set of lifts $x' \\in Q_{\\mathcal{F}/X/B}(T')$ is a principal homogeneous space under the group $$ \\Hom_{\\mathcal{O}_{X_T}}(\\mathcal{K}, \\mathcal{Q} \\otimes_{\\mathcal{O}_T} \\mathcal{J}) $$ Namely, given any $\\mathcal{F}_{T'} \\to \\mathcal{Q}'$ flat over $T'$ lifting the quotient $\\mathcal{Q}$ we obtain a commutative diagram with exact rows and columns $$ \\xymatrix{ & 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] \\\\ 0 \\ar[r] & \\mathcal{K} \\otimes \\mathcal{J} \\ar[r] \\ar[d] & \\mathcal{F}_T \\otimes \\mathcal{J} \\ar[r] \\ar[d] & \\mathcal{Q} \\otimes \\mathcal{J} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}' \\ar[r] \\ar[d] & \\mathcal{F}_{T'} \\ar[r] \\ar[d] & \\mathcal{Q}' \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K} \\ar[d] \\ar[r] & \\mathcal{F}_T \\ar[d] \\ar[r] & \\mathcal{Q} \\ar[d] \\ar[r] & 0 \\\\ & 0 & 0 & 0 } $$ (to see this use the observations made in the previous remark). Given a map $\\varphi : \\mathcal{K} \\to \\mathcal{Q} \\otimes \\mathcal{J}$ we can consider the subsheaf $\\mathcal{K}'_\\varphi \\subset \\mathcal{F}_{T'}$ consisting of those local sections $s$ whose image in $\\mathcal{F}_T$ is a local section $k$ of $\\mathcal{K}$ and whose image in $\\mathcal{Q}'$ is the local section $\\varphi(k)$ of $\\mathcal{Q} \\otimes \\mathcal{J}$. Then set $\\mathcal{Q}'_\\varphi = \\mathcal{F}_{T'}/\\mathcal{K}'_\\varphi$. Conversely, any second lift of $x$ corresponds to one of the qotients constructed in this manner. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi)."}
+{"_id": "3238", "title": "quot-remark-quot-via-artins-axioms", "text": "Let $S$ be a Noetherian scheme all of whose local rings are G-rings. Let $X$ be an algebraic space over $S$ whose structure morphism $f : X \\to S$ is of finite presentation and separated. Let $\\mathcal{F}$ be a finitely presented quasi-coherent sheaf on $X$ flat over $S$. In this remark we sketch how one can use Artin's axioms to prove that $\\Quotfunctor_{\\mathcal{F}/X/S}$ is an algebraic space locally of finite presentation over $S$ and avoid using the algebraicity of the stack of coherent sheaves as was done in the proof of Proposition \\ref{proposition-quot}. \\medskip\\noindent We check the conditions listed in Artin's Axioms, Proposition \\ref{artin-proposition-spaces-diagonal-representable}. Representability of the diagonal of $\\Quotfunctor_{\\mathcal{F}/X/S}$ can be seen as follows: suppose we have two quotients $\\mathcal{F}_T \\to \\mathcal{Q}_i$, $i = 1, 2$. Denote $\\mathcal{K}_1$ the kernel of the first one. Then we have to show that the locus of $T$ over which $u : \\mathcal{K}_1 \\to \\mathcal{Q}_2$ becomes zero is representable. This follows for example from Flatness on Spaces, Lemma \\ref{spaces-flat-lemma-F-zero-closed-proper} or from a discussion of the $\\mathit{Hom}$ sheaf earlier in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger) follow from Lemmas \\ref{lemma-quot-sheaf}, \\ref{lemma-q-limit-preserving}, and \\ref{lemma-q-RS-star} (plus some extra work to deal with the properness condition). Axiom [3] (finite dimensionality of tangent spaces) follows from the description of the infinitesimal deformations in Remark \\ref{remark-q-defos} and finiteness of cohomology of coherent sheaves on proper algebraic spaces over fields (Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-proper-pushforward-coherent}). Axiom [4] (effectiveness of formal objects) follows from Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem \\ref{spaces-more-morphisms-theorem-grothendieck-existence}). As usual, the trickiest to verify is axiom [5] (openness of versality). One can for example use the obstruction theory described in Remark \\ref{remark-q-obs} and the description of deformations in Remark \\ref{remark-q-defos} to do this using the criterion in Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory}. Please compare with the second proof of Lemma \\ref{lemma-coherent-defo-thy}."}
+{"_id": "3239", "title": "quot-remark-spaces-base-change", "text": "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$. Let $B\\textit{-Spaces}'_{ft}$ be the category consisting of pairs $(X \\to S, h : S \\to B)$ where $X \\to S$ is an object of $\\Spacesstack'_{ft}$ and $h : S \\to B$ is a morphism. A morphism $(X' \\to S', h') \\to (X \\to S, h)$ in $B\\textit{-Spaces}'_{ft}$ is a morphism $(f, g)$ in $\\Spacesstack'_{ft}$ such that $h \\circ g = h'$. In this situation the diagram $$ \\xymatrix{ B\\textit{-Spaces}'_{ft} \\ar[r] \\ar[d] & \\Spacesstack'_{ft} \\ar[d] \\\\ (\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\\Spacesstack'_{ft}$ to the case of families over a given base algebraic space. Of course, a similar construction works for $B\\textit{-Spaces}'_{fp, flat, proper}$"}
+{"_id": "3240", "title": "quot-remark-spaces-defo-thy", "text": "Lemma \\ref{lemma-spaces-defo-thy} can also be shown using either Artin's Axioms, Lemma \\ref{artin-lemma-dual-openness} (as in the first proof of Lemma \\ref{lemma-coherent-defo-thy}), or using an obstruction theory as in Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory} (as in the second proof of Lemma \\ref{lemma-coherent-defo-thy}). In both cases one uses the deformation and obstruction theory developed in Cotangent, Section \\ref{cotangent-section-deformations-ringed-topoi} to translate the needed properties of deformations and obstructions into $\\Ext$-groups to which Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-compute-ext} can be applied. The second method (using an obstruction theory and therefore using the full cotangent complex) is perhaps the ``standard'' method used in most references."}
+{"_id": "3241", "title": "quot-remark-polarized-base-change", "text": "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$. Let $B\\textit{-Polarized}$ be the category consisting of triples $(X \\to S, \\mathcal{L}, h : S \\to B)$ where $(X \\to S, \\mathcal{L})$ is an object of $\\Polarizedstack$ and $h : S \\to B$ is a morphism. A morphism $(X' \\to S', \\mathcal{L}', h') \\to (X \\to S, \\mathcal{L}, h)$ in $B\\textit{-Polarized}$ is a morphism $(f, g, \\varphi)$ in $\\Polarizedstack$ such that $h \\circ g = h'$. In this situation the diagram $$ \\xymatrix{ B\\textit{-Polarized} \\ar[r] \\ar[d] & \\Polarizedstack \\ar[d] \\\\ (\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\\Polarizedstack$ to the case of families over a given base algebraic space."}
+{"_id": "3242", "title": "quot-remark-polarized-defo-thy", "text": "Lemma \\ref{lemma-polarized-defo-thy} can also be shown using an obstruction theory as in Artin's Axioms, Lemma \\ref{artin-lemma-get-openness-obstruction-theory} (as in the second proof of Lemma \\ref{lemma-coherent-defo-thy}). To do this one has to generalize the deformation and obstruction theory developed in Cotangent, Section \\ref{cotangent-section-deformations-ringed-topoi} to the case of pairs of algebraic spaces and quasi-coherent modules. Another possibility is to use that the $1$-morphism $\\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper}$ is algebraic (Lemma \\ref{lemma-polarized-to-spaces-algebraic}) and the fact that we know openness of versality for the target (Lemma \\ref{lemma-spaces-defo-thy} and Remark \\ref{remark-spaces-defo-thy})."}
+{"_id": "3243", "title": "quot-remark-curves-base-change", "text": "Let $B$ be an algebraic space over $\\Spec(\\mathbf{Z})$. Let $B\\text{-}\\Curvesstack$ be the category consisting of pairs $(X \\to S, h : S \\to B)$ where $X \\to S$ is an object of $\\Curvesstack$ and $h : S \\to B$ is a morphism. A morphism $(X' \\to S', h') \\to (X \\to S, h)$ in $B\\text{-}\\Curvesstack$ is a morphism $(f, g)$ in $\\Curvesstack$ such that $h \\circ g = h'$. In this situation the diagram $$ \\xymatrix{ B\\text{-}\\Curvesstack \\ar[r] \\ar[d] & \\Curvesstack \\ar[d] \\\\ (\\Sch/B)_{fppf} \\ar[r] & \\Sch_{fppf} } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\\Curvesstack$ to the case of families of curves over a given base algebraic space."}
+{"_id": "3244", "title": "quot-remark-alternative-approach-curves", "text": "Consider the $2$-fibre product $$ \\xymatrix{ \\Curvesstack \\times_{\\Spacesstack'_{fp, flat, proper}} \\Polarizedstack \\ar[r] \\ar[d] & \\Polarizedstack \\ar[d] \\\\ \\Curvesstack \\ar[r] & \\Spacesstack'_{fp, flat, proper} } $$ This fibre product parametrized polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. It turns out that the left vertical arrow $$ \\textit{PolarizedCurves} \\longrightarrow \\Curvesstack $$ is algebraic, smooth, and surjective. Namely, this $1$-morphism is algebraic (as base change of the arrow in Lemma \\ref{lemma-polarized-to-spaces-algebraic}), every point is in the image, and there are no obstructions to deforming invertible sheaves on curves (see proof of Lemma \\ref{lemma-curves-existence}). This gives another approach to the algebraicity of $\\Curvesstack$. Namely, by Lemma \\ref{lemma-curves-open-and-closed-in-spaces} we see that $\\textit{PolarizedCurves}$ is an open and closed substack of the algebraic stack $\\Polarizedstack$ and any stack in groupoids which is the target of a smooth algebraic morphism from an algebraic stack is an algebraic stack."}
+{"_id": "3245", "title": "quot-remark-complexes-base-change", "text": "In Situation \\ref{situation-complexes} the rule $(T, g, E) \\mapsto (T, g)$ defines a $1$-morphism $$ \\Complexesstack_{X/B} \\longrightarrow \\mathcal{S}_B $$ of stacks in groupoids (see Lemma \\ref{lemma-complexes-stack}, Algebraic Stacks, Section \\ref{algebraic-section-split}, and Examples of Stacks, Section \\ref{examples-stacks-section-stack-associated-to-sheaf}). Let $B' \\to B$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{S}_{B'} \\to \\mathcal{S}_B$ be the associated $1$-morphism of stacks fibred in sets. Set $X' = X \\times_B B'$. We obtain a stack in groupoids $\\Complexesstack_{X'/B'} \\to (\\Sch/S)_{fppf}$ associated to the base change $f' : X' \\to B'$. In this situation the diagram $$ \\vcenter{ \\xymatrix{ \\Complexesstack_{X'/B'} \\ar[r] \\ar[d] & \\Complexesstack_{X/B} \\ar[d] \\\\ \\mathcal{S}_{B'} \\ar[r] & \\mathcal{S}_B } } \\quad \\begin{matrix} \\text{or in} \\\\ \\text{another} \\\\ \\text{notation} \\end{matrix} \\quad \\vcenter{ \\xymatrix{ \\Complexesstack_{X'/B'} \\ar[r] \\ar[d] & \\Complexesstack_{X/B} \\ar[d] \\\\ \\Sch/B' \\ar[r] & \\Sch/B } } $$ is $2$-fibre product square. This trivial remark will occasionally be useful to change the base algebraic space."}
+{"_id": "3406", "title": "coherent-remark-chow-Noetherian", "text": "In the situation of Chow's Lemma \\ref{lemma-chow-Noetherian}: \\begin{enumerate} \\item The morphism $\\pi$ is actually H-projective (hence projective, see Morphisms, Lemma \\ref{morphisms-lemma-H-projective}) since the morphism $X' \\to \\mathbf{P}^n_S \\times_S X = \\mathbf{P}^n_X$ is a closed immersion (use the fact that $\\pi$ is proper, see Morphisms, Lemma \\ref{morphisms-lemma-image-proper-scheme-closed}). \\item We may assume that $\\pi^{-1}(U)$ is scheme theoretically dense in $X'$. Namely, we can simply replace $X'$ by the scheme theoretic closure of $\\pi^{-1}(U)$. In this case we can think of $U$ as a scheme theoretically dense open subscheme of $X'$. See Morphisms, Section \\ref{morphisms-section-scheme-theoretic-image}. \\item If $X$ is reduced then we may choose $X'$ reduced. This is clear from (2). \\end{enumerate}"}
+{"_id": "3407", "title": "coherent-remark-explain-perfect-direct-image", "text": "A consequence of Lemma \\ref{lemma-perfect-direct-image} is that there exists a finite complex of finite projective $A$-modules $M^\\bullet$ such that we have $$ H^i(X_{A'}, \\mathcal{F}_{A'}) = H^i(M^\\bullet \\otimes_A A') $$ functorially in $A'$. The condition that $\\mathcal{F}$ is flat over $A$ is essential, see \\cite{Hartshorne}."}
+{"_id": "3408", "title": "coherent-remark-inverse-systems-kernel-cokernel-annihilated-by", "text": "Let $X$ be a Noetherian scheme and let $\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$ be quasi-coherent sheaves of ideals. Let $\\alpha : (\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ be a morphism of $\\textit{Coh}(X, \\mathcal{I})$. Given an affine open $\\Spec(A) = U \\subset X$ with $\\mathcal{I}|_U, \\mathcal{K}|_U$ corresponding to ideals $I, K \\subset A$ denote $\\alpha_U : M \\to N$ of finite $A^\\wedge$-modules which corresponds to $\\alpha|_U$ via Lemma \\ref{lemma-inverse-systems-affine}. We claim the following are equivalent \\begin{enumerate} \\item there exists an integer $t \\geq 1$ such that $\\Ker(\\alpha_n)$ and $\\Coker(\\alpha_n)$ are annihilated by $\\mathcal{K}^t$ for all $n \\geq 1$, \\item for any affine open $\\Spec(A) = U \\subset X$ as above the modules $\\Ker(\\alpha_U)$ and $\\Coker(\\alpha_U)$ are annihilated by $K^t$ for some integer $t \\geq 1$, and \\item there exists a finite affine open covering $X = \\bigcup U_i$ such that the conclusion of (2) holds for $\\alpha_{U_i}$. \\end{enumerate} If these equivalent conditions hold we will say that $\\alpha$ is a {\\it map whose kernel and cokernel are annihilated by a power of $\\mathcal{K}$}. To see the equivalence we use the following commutative algebra fact: suppose given an exact sequence $$ 0 \\to T \\to M \\to N \\to Q \\to 0 $$ of $A$-modules with $T$ and $Q$ annihilated by $K^t$ for some ideal $K \\subset A$. Then for every $f, g \\in K^t$ there exists a canonical map $\"fg\": N \\to M$ such that $M \\to N \\to M$ is equal to multiplication by $fg$. Namely, for $y \\in N$ we can pick $x \\in M$ mapping to $fy$ in $N$ and then we can set $\"fg\"(y) = gx$. Thus it is clear that $\\Ker(M/JM \\to N/JN)$ and $\\Coker(M/JM \\to N/JN)$ are annihilated by $K^{2t}$ for any ideal $J \\subset A$. \\medskip\\noindent Applying the commutative algebra fact to $\\alpha_{U_i}$ and $J = I^n$ we see that (3) implies (1). Conversely, suppose (1) holds and $M \\to N$ is equal to $\\alpha_U$. Then there is a $t \\geq 1$ such that $\\Ker(M/I^nM \\to N/I^nN)$ and $\\Coker(M/I^nM \\to N/I^nN)$ are annihilated by $K^t$ for all $n$. We obtain maps $\"fg\" : N/I^nN \\to M/I^nM$ which in the limit induce a map $N \\to M$ as $N$ and $M$ are $I$-adically complete. Since the composition with $N \\to M \\to N$ is multiplication by $fg$ we conclude that $fg$ annihilates $T$ and $Q$. In other words $T$ and $Q$ are annihilated by $K^{2t}$ as desired."}
+{"_id": "3409", "title": "coherent-remark-reformulate-existence-theorem", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a separated morphism of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Picture: $$ \\xymatrix{ X_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] & \\ldots & X \\ar[d] \\\\ S_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots & S } $$ In this situation we consider systems $(\\mathcal{F}_n, \\varphi_n)$ where \\begin{enumerate} \\item $\\mathcal{F}_n$ is a coherent $\\mathcal{O}_{X_n}$-module, \\item $\\varphi_n : i_n^*\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ is an isomorphism, and \\item $\\text{Supp}(\\mathcal{F}_1)$ is proper over $S_1$. \\end{enumerate} Theorem \\ref{theorem-grothendieck-existence} says that the completion functor $$ \\begin{matrix} \\text{coherent }\\mathcal{O}_X\\text{-modules }\\mathcal{F} \\\\ \\text{with support proper over }A \\end{matrix} \\quad \\longrightarrow \\quad \\begin{matrix} \\text{systems }(\\mathcal{F}_n) \\\\ \\text{as above} \\end{matrix} $$ is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports."}
+{"_id": "3544", "title": "formal-defos-remark-predeformation-functor", "text": "We say that a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ is a {\\it predeformation functor} if the associated cofibered set is a predeformation category, i.e.\\ if $F(k)$ is a one element set.  Thus if $\\mathcal{F}$ is a predeformation category, then $\\overline{\\mathcal{F}}$ is a predeformation functor."}
+{"_id": "3545", "title": "formal-defos-remark-localize-cofibered-groupoid", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids, and let $x \\in \\Ob(\\mathcal{F}(k))$.  We denote by $\\mathcal{F}_x$ the category of objects over $x$. An object of $\\mathcal{F}_x$ is an arrow $y \\to x$. A morphism $(y \\to x) \\to (z \\to x)$ in $\\mathcal{F}_x$ is a commutative diagram $$ \\xymatrix{ y \\ar[rr] \\ar[dr] & & z \\ar[dl] \\\\ & x & } $$ There is a forgetful functor $\\mathcal{F}_x \\to \\mathcal{F}$. We define the functor $p_x : \\mathcal{F}_x \\to \\mathcal{C}_\\Lambda$ as the composition $\\mathcal{F}_x \\to \\mathcal{F} \\xrightarrow{p} \\mathcal{C}_\\Lambda$. Then $p_x : \\mathcal{F}_x \\to \\mathcal{C}_\\Lambda$ is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ to a predeformation category at any $x \\in \\Ob(\\mathcal{F}(k))$."}
+{"_id": "3546", "title": "formal-defos-remark-different-sequence-ideals", "text": "Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in groupoids. Suppose that for each $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$ we are given a filtration $\\mathcal{I}_R$ of $R$ by ideals. If $\\mathcal{I}_R$ induces the $\\mathfrak m_R$-adic topology on $R$ for all $R$, then one can define a category $\\widehat{\\mathcal{F}}_\\mathcal{I}$ by mimicking the definition of $\\widehat{\\mathcal{F}}$. This category comes equipped with a morphism $\\widehat{p}_\\mathcal{I} : \\widehat{\\mathcal{F}}_\\mathcal{I} \\to \\widehat{\\mathcal{C}}_\\Lambda$ making it into a category cofibered in groupoids such that $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$ is isomorphic to $\\widehat{\\mathcal{F}}_{\\mathcal{I}_R}(R)$ as defined above. The categories cofibered in groupoids $\\widehat{\\mathcal{F}}_\\mathcal{I}$ and $\\widehat{\\mathcal{F}}$ are equivalent, by using over an object $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$ the equivalence of Lemma \\ref{lemma-formal-objects-different-filtration}."}
+{"_id": "3547", "title": "formal-defos-remark-completion-functor", "text": "Let $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor. Identifying functors with cofibered sets, the completion of $F$ is the functor $\\widehat{F} : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$ given by $\\widehat{F}(S) = \\lim F(S/\\mathfrak{m}_S^{n})$.  This agrees with the definition in Schlessinger's paper \\cite{Sch}."}
+{"_id": "3548", "title": "formal-defos-remark-restrict-completion", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$. We claim that there is a canonical equivalence $$ can : \\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\longrightarrow \\mathcal{F}. $$ Namely, let $A \\in \\Ob(\\mathcal{C}_\\Lambda)$ and let $(A, \\xi_n, f_n)$ be an object of $\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda}(A)$. Since $A$ is Artinian there is a minimal $m \\in \\mathbf{N}$ such that $\\mathfrak m_A^m = 0$. Then $can$ sends $(A, \\xi_n, f_n)$ to $\\xi_m$. This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma \\ref{categories-lemma-equivalence-fibred-categories} because it is an equivalence on all fibre categories by Lemma \\ref{lemma-formal-objects-different-filtration} and the fact that the $\\mathfrak m_A$-adic topology on a local Artinian ring $A$ comes from the zero ideal. We will frequently identify $\\mathcal{F}$ with a full subcategory of $\\widehat{\\mathcal{F}}$ via a quasi-inverse to the functor $can$."}
+{"_id": "3549", "title": "formal-defos-remark-completion-morphism", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Then there is an induced morphism $\\widehat{\\varphi}: \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{G}}$ of categories cofibered in groupoids over $\\widehat{\\mathcal{C}}_\\Lambda$. It sends an object $\\xi = (R, \\xi_n, f_n)$ of $\\widehat{\\mathcal{F}}$ to $(R, \\varphi(\\xi_n), \\varphi(f_n))$, and it sends a morphism $(a_0 : R \\to S, a_n : \\xi_n \\to \\eta_n)$ between objects $\\xi$ and $\\eta$ of $\\widehat{\\mathcal{F}}$ to $(a_0 : R \\to S, \\varphi(a_n) : \\varphi(\\xi_n) \\to \\varphi(\\eta_n))$. Finally, if $t : \\varphi \\to \\varphi'$ is a $2$-morphism between $1$-morphisms $\\varphi, \\varphi': \\mathcal{F} \\to \\mathcal{G}$ of categories cofibred in groupoids, then we obtain a $2$-morphism $\\widehat{t} : \\widehat{\\varphi} \\to \\widehat{\\varphi}'$. Namely, for $\\xi = (R, \\xi_n, f_n)$ as above we set $\\widehat{t}_\\xi = (t_{\\varphi(\\xi_n)})$. Hence completion defines a functor between $2$-categories $$ \\widehat{~} : \\text{Cof}(\\mathcal{C}_\\Lambda) \\longrightarrow \\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda) $$ from the $2$-category of categories cofibred in groupoids over $\\mathcal{C}_\\Lambda$ to the $2$-category of categories cofibred in groupoids over $\\widehat{\\mathcal{C}}_\\Lambda$."}
+{"_id": "3550", "title": "formal-defos-remark-completion-restriction-adjoint", "text": "We claim the completion functor of Remark \\ref{remark-completion-morphism} and the restriction functor $|_{\\mathcal{C}_\\Lambda} : \\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda) \\to \\text{Cof}(\\mathcal{C}_\\Lambda)$ of Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-definition-restricting-base-category}) are ``2-adjoint'' in the following precise sense. Let $\\mathcal{F} \\in \\Ob(\\text{Cof}(\\mathcal{C}_\\Lambda))$ and let $\\mathcal{G} \\in \\Ob(\\text{Cof}(\\widehat{\\mathcal{C}}_\\Lambda))$. Then there is an equivalence of categories $$ \\Phi : \\Mor_{\\mathcal{C}_\\Lambda}( \\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F}) \\longrightarrow \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(\\mathcal{G}, \\widehat{\\mathcal{F}}) $$ To describe this equivalence, we define canonical morphisms $\\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}}$ and $\\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ as follows \\begin{enumerate} \\item Let $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda))$ and let $\\xi$ be an object of the fiber category $\\mathcal{G}(R)$. Choose a pushforward $\\xi \\to \\xi_n$ of $\\xi$ to $R/\\mathfrak m_R^n$ for each $n \\in \\mathbf{N}$, and let $f_n : \\xi_{n + 1} \\to \\xi_n$ be the induced morphism. Then $\\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}}$ sends $\\xi$ to $(R, \\xi_n, f_n)$. \\item This is the equivalence $can : \\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ of Remark \\ref{remark-restrict-completion}. \\end{enumerate} Having said this, the equivalence $\\Phi : \\Mor_{\\mathcal{C}_\\Lambda}( \\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})  \\to \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(\\mathcal{G}, \\widehat{\\mathcal{F}})$ sends a morphism $\\varphi : \\mathcal{G}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ to $$ \\mathcal{G} \\to \\widehat{\\mathcal{G}|_{\\mathcal{C}_\\Lambda}} \\xrightarrow{\\widehat{\\varphi}} \\widehat{\\mathcal{F}} $$ There is a quasi-inverse $\\Psi : \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}( \\mathcal{G}, \\widehat{\\mathcal{F}}) \\to \\Mor_{\\mathcal{C}_\\Lambda}( \\mathcal{G}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F})$ to $\\Phi$ which sends $\\psi : \\mathcal{G} \\to \\widehat{\\mathcal{F}}$ to $$ \\mathcal{G}|_{\\mathcal{C}_\\Lambda} \\xrightarrow{\\psi|_{\\mathcal{C}_\\Lambda}} \\widehat{\\mathcal{F}}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}. $$ We omit the verification that $\\Phi$ and $\\Psi$ are quasi-inverse. We also do not address functoriality of $\\Phi$ (because it would lead into 3-category territory which we want to avoid at all cost)."}
+{"_id": "3551", "title": "formal-defos-remark-completion-restriction-cofset-adjoint", "text": "For a category $\\mathcal{C}$ we denote by $\\text{CofSet}(\\mathcal{C})$ the category of cofibered sets over $\\mathcal{C}$. It is a $1$-category isomorphic the category of functors $\\mathcal{C} \\to \\textit{Sets}$. See Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-convention-cofibered-sets}). The completion and restriction functors restrict to functors $\\widehat{~} : \\text{CofSet}(\\mathcal{C}_\\Lambda) \\to \\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda)$ and $|_{\\mathcal{C}_\\Lambda} : \\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda) \\to \\text{CofSet}(\\mathcal{C}_\\Lambda)$ which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark \\ref{remark-completion-restriction-adjoint} defines a functorial bijection $$ \\Mor_{\\mathcal{C}_\\Lambda}(G|_{\\mathcal{C}_\\Lambda}, F) \\longrightarrow \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}(G, \\widehat{F}) $$ for $F \\in \\Ob(\\text{CofSet}(\\mathcal{C}_\\Lambda))$ and $G \\in \\Ob(\\text{CofSet}(\\widehat{\\mathcal{C}}_\\Lambda))$. Again the map $\\widehat{F}|_{\\mathcal{C}_\\Lambda} \\to F$ is an isomorphism."}
+{"_id": "3552", "title": "formal-defos-remark-restrict-complete-continuous-functor", "text": "Let $G : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$ be a functor that commutes with limits. Then the map $G \\to \\widehat{G|_{\\mathcal{C}_\\Lambda}}$ described in Remark \\ref{remark-completion-restriction-adjoint} is an isomorphism. Indeed, if $S$ is  an object of $\\widehat{\\mathcal{C}}_\\Lambda$, then we have canonical bijections $$ \\widehat{G|_{\\mathcal{C}_\\Lambda}}(S) = \\lim_n G(S/\\mathfrak{m}_S^n) = G(\\lim_n S/\\mathfrak{m}_S^n) = G(S). $$ In particular, if $R$ is an object of $\\widehat{\\mathcal{C}}_\\Lambda$ then $\\underline{R} = \\widehat{\\underline{R}|_{\\mathcal{C}_\\Lambda}}$ because the representable functor $\\underline{R}$ commutes with limits by definition of limits."}
+{"_id": "3553", "title": "formal-defos-remark-formal-objects-yoneda", "text": "Let $R$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$.  It defines a functor $\\underline{R}: \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$ as described in Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-definition-yoneda}). As usual we identify this functor with the associated cofibered set.  If $\\mathcal{F}$ is a cofibered category over $\\mathcal{C}_\\Lambda$, then there is an equivalence of categories \\begin{equation} \\label{equation-formal-objects-maps} \\Mor_{\\mathcal{C}_\\Lambda}( \\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F}) \\longrightarrow \\widehat{\\mathcal{F}}(R). \\end{equation} It is given by the composition $$ \\Mor_{\\mathcal{C}_\\Lambda}( \\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F}) \\xrightarrow{\\Phi} \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}( \\underline{R}, \\widehat{\\mathcal{F}}) \\xrightarrow{\\sim} \\widehat{\\mathcal{F}}(R) $$ where $\\Phi$ is as in Remark \\ref{remark-completion-restriction-adjoint} and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma \\ref{categories-lemma-yoneda-2category}). Explicitly, the equivalence sends a morphism $\\varphi : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ to the formal object $(R, \\varphi(R \\to R/\\mathfrak{m}_R^n), \\varphi(f_n))$ in $\\widehat{\\mathcal{F}}(R)$, where $f_n : R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$ is the projection. \\medskip\\noindent Assume a choice of pushforwards for $\\mathcal{F}$ has been made. Given any $\\xi \\in \\Ob(\\widehat{\\mathcal{F}}(R))$ we construct an explicit $\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ which maps to $\\xi$ under (\\ref{equation-formal-objects-maps}). Namely, say $\\xi = (R, \\xi_n, f_n)$. An object $\\alpha$ in $\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is the same thing as a morphism $\\alpha : R \\to A$ of $\\widehat{\\mathcal{C}}_\\Lambda$ with $A$ Artinian. Let $m \\in \\mathbf{N}$ be minimal such that $\\mathfrak m_A^m = 0$. Then $\\alpha$ factors through a unique $\\alpha_m : R/\\mathfrak m_R^m \\to A$ and we can set $\\underline{\\xi}(\\alpha) = \\alpha_{m, *}\\xi_m$. We omit the description of $\\underline{\\xi}$ on morphisms and we omit the proof that $\\underline{\\xi}$ maps to $\\xi$ via (\\ref{equation-formal-objects-maps}). \\medskip\\noindent Assume a choice of pushforwards for $\\widehat{\\mathcal{F}}$ has been made. In this case the proof of Categories, Lemma \\ref{categories-lemma-yoneda-2category} gives an explicit quasi-inverse $$ \\iota : \\widehat{\\mathcal{F}}(R) \\longrightarrow \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}( \\underline{R}, \\widehat{\\mathcal{F}}) $$ to the 2-Yoneda equivalence which takes $\\xi$ to the morphism $\\iota(\\xi) : \\underline{R} \\to \\widehat{\\mathcal{F}}$ sending $f \\in \\underline{R}(S) = \\Mor_{\\mathcal{C}_\\Lambda}(R, S)$ to $f_*\\xi$. A quasi-inverse to (\\ref{equation-formal-objects-maps}) is then $$ \\widehat{\\mathcal{F}}(R) \\xrightarrow{\\iota} \\Mor_{\\widehat{\\mathcal{C}}_\\Lambda}( \\underline{R}, \\widehat{\\mathcal{F}}) \\xrightarrow{\\Psi} \\Mor_{\\mathcal{C}_\\Lambda}( \\underline{R}|_{\\mathcal{C}_\\Lambda}, \\mathcal{F}) $$ where $\\Psi$ is as in Remark \\ref{remark-completion-restriction-adjoint}. Given $\\xi \\in \\Ob(\\widehat{\\mathcal{F}}(R))$ we have $\\Psi(\\iota(\\xi)) \\cong \\underline{\\xi}$ where $\\underline{\\xi}$ is as in the previous paragraph, because both are mapped to $\\xi$ under the equivalence of categories (\\ref{equation-formal-objects-maps}). Using $\\underline{R} = \\widehat{\\underline{R}|_{\\mathcal{C}_\\Lambda}}$ (see Remark \\ref{remark-restrict-complete-continuous-functor}) and unwinding the definitions of $\\Phi$ and $\\Psi$ we conclude that $\\iota(\\xi)$ is isomorphic to the completion of $\\underline{\\xi}$."}
+{"_id": "3554", "title": "formal-defos-remark-formal-objects-yoneda-map", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$. Let $\\xi = (R, \\xi_n, f_n)$ and $\\eta = (S, \\eta_n, g_n)$ be formal objects of $\\mathcal{F}$. Let $a = (a_n) : \\xi \\to \\eta$ be a morphism of formal objects, i.e., a morphism of $\\widehat{\\mathcal{F}}$. Let $f = \\widehat{p}(a) = a_0 : R \\to S$ be the projection of $a$ in $\\widehat{\\mathcal{C}}_\\Lambda$. Then we obtain a $2$-commutative diagram $$ \\xymatrix{ \\underline{R}|_{\\mathcal{C}_\\Lambda} \\ar[rd]_{\\underline{\\xi}} & & \\underline{S}|_{\\mathcal{C}_\\Lambda} \\ar[ll]^f \\ar[ld]^{\\underline{\\eta}} \\\\ & \\mathcal{F} } $$ where $\\underline{\\xi}$ and $\\underline{\\eta}$ are the morphisms constructed in Remark \\ref{remark-formal-objects-yoneda}. To see this let $\\alpha : S \\to A$ be an object of $\\underline{S}|_{\\mathcal{C}_\\Lambda}$ (see loc.\\ cit.). Let $m \\in \\mathbf{N}$ be minimal such that $\\mathfrak m_A^m = 0$. We get a commutative diagram $$ \\xymatrix{ R \\ar[d]^f \\ar[r] & R/\\mathfrak m_R^m \\ar[d]_{f_m} \\ar[rd]^{\\beta_m} \\\\ S \\ar[r] & S/\\mathfrak m_S^m \\ar[r]^{\\alpha_m} & A } $$ such that the bottom arrows compose to give $\\alpha$. Then $\\underline{\\eta}(\\alpha) = \\alpha_{m, *}\\eta_m$ and $\\underline{\\xi}(\\alpha \\circ f) = \\beta_{m, *}\\xi_m$. The morphism $a_m : \\xi_m \\to \\eta_m$ lies over $f_m$ hence we obtain a canonical morphism $$ \\underline{\\xi}(\\alpha \\circ f) = \\beta_{m, *}\\xi_m \\longrightarrow \\underline{\\eta}(\\alpha) = \\alpha_{m, *}\\eta_m $$ lying over $\\text{id}_A$ such that $$ \\xymatrix{ \\xi_m \\ar[r] \\ar[d]^{a_m} &  \\beta_{m, *}\\xi_m \\ar[d] \\\\ \\eta_m \\ar[r] &  \\alpha_{m, *}\\eta_m } $$ commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors $\\underline{\\xi} \\circ f \\to \\underline{\\eta}$ which witnesses the 2-commutativity of the first diagram of this remark."}
+{"_id": "3555", "title": "formal-defos-remark-spell-out-formal-object", "text": "According to Remark \\ref{remark-formal-objects-yoneda}, giving a formal object $\\xi$ of $\\mathcal{F}$ is equivalent to giving a prorepresentable functor $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ and a morphism $U \\to \\mathcal{F}$."}
+{"_id": "3556", "title": "formal-defos-remark-smoothness-2-categorical", "text": "Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$.  Let $B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.  Choices of pushforwards along $B \\to A$ for objects in the fiber categories $\\mathcal{F}(B)$ and $\\mathcal{G}(B)$ determine functors $\\mathcal{F}(B) \\to \\mathcal{F}(A)$ and $\\mathcal{G}(B) \\to \\mathcal{G}(A)$ fitting into a $2$-commutative diagram $$ \\xymatrix{ \\mathcal{F}(B) \\ar[r]^{\\varphi} \\ar[d] & \\mathcal{G}(B) \\ar[d] \\\\ \\mathcal{F}(A) \\ar[r]^{\\varphi}        & \\mathcal{G}(A) . } $$ Hence there is an induced functor $\\mathcal{F}(B) \\to \\mathcal{F}(A) \\times_{\\mathcal{G}(A)} \\mathcal{G}(B)$.  Unwinding the definitions shows that $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is smooth if and only if this induced functor is essentially surjective whenever $B \\to A$ is surjective (or equivalently, by Lemma \\ref{lemma-smoothness-small-extensions}, whenever $B \\to A$ is a small extension)."}
+{"_id": "3557", "title": "formal-defos-remark-compare-smooth-schlessinger", "text": "The characterization of smooth morphisms in Remark \\ref{remark-smoothness-2-categorical} is analogous to Schlessinger's notion of a smooth morphism of functors, cf.\\ \\cite[Definition 2.2.]{Sch}. In fact, when $\\mathcal{F}$ and $\\mathcal{G}$ are cofibered in sets then our notion is equivalent to Schlessinger's. Namely, in this case let $F, G : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be the corresponding functors, see Remarks \\ref{remarks-cofibered-groupoids} (\\ref{item-convention-cofibered-sets}). Then $F \\to G$ is smooth if and only if for every surjection of rings $B \\to A$ in $\\mathcal{C}_\\Lambda$ the map $F(B) \\to F(A) \\times_{G(A)} G(B)$ is surjective."}
+{"_id": "3558", "title": "formal-defos-remark-smooth-to-iso-classes", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Then the morphism $\\mathcal{F} \\to \\overline{\\mathcal{F}}$ is smooth. Namely, suppose that $f : B \\to A$ is a ring map in $\\mathcal{C}_\\Lambda$. Let $x \\in \\Ob(\\mathcal{F}(A))$ and let $\\overline{y} \\in \\overline{\\mathcal{F}}(B)$ be the isomorphism class of $y \\in \\Ob(\\mathcal{F}(B))$ such that $\\overline{f_*y} = \\overline{x}$. Then we simply take $x' = y$, the implied morphism $x' = y \\to x$ over $B \\to A$, and the equality $\\overline{x'} = \\overline{y}$ as the solution to the problem posed in Definition \\ref{definition-smooth-morphism}."}
+{"_id": "3559", "title": "formal-defos-remark-versal-object", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$, and let $\\xi$ be a formal object of $\\mathcal{F}$.  It follows from the definition of smoothness that versality of $\\xi$ is equivalent to the following condition: If $$ \\xymatrix{ & y \\ar[d] \\\\ \\xi \\ar[r] & x } $$ is a diagram in $\\widehat{\\mathcal{F}}$ such that $y \\to x$ lies over a surjective map $B \\to A$ of Artinian rings (we may assume it is a small extension),  then there exists a morphism $\\xi \\to y$ such that $$ \\xymatrix{ & y \\ar[d] \\\\ \\xi \\ar[r] \\ar[ur] & x } $$ commutes. In particular, the condition that $\\xi$ be versal does not depend on the choices of pushforwards made in the construction of $\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$ in Remark \\ref{remark-formal-objects-yoneda}."}
+{"_id": "3560", "title": "formal-defos-remark-smooth-on-top", "text": "Suppose $\\mathcal{F}$ is a predeformation category admitting a smooth morphism $\\varphi : \\mathcal U \\to \\mathcal{F}$ from a predeformation category $\\mathcal{U}$.  Then by Lemma \\ref{lemma-smooth-morphism-essentially-surjective} $\\varphi$ is essentially surjective, so by Lemma \\ref{lemma-smooth-properties} $p: \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ is smooth if and only if the composition $\\mathcal U \\xrightarrow{\\varphi} \\mathcal{F} \\xrightarrow{p} \\mathcal{C}_\\Lambda$ is smooth, i.e.\\ $\\mathcal{F}$ is smooth if and only if $\\mathcal{U}$ is smooth."}
+{"_id": "3561", "title": "formal-defos-remark-compare-S1-S2-schlessinger", "text": "When $\\mathcal{F}$ is cofibered in sets, conditions (S1) and (S2) are exactly conditions (H1) and (H2) from Schlessinger's paper \\cite{Sch}. Namely, for a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$, conditions (S1) and (S2) state: \\begin{enumerate} \\item [(S1)] If $A_1 \\to A$ and $A_2 \\to A$ are maps in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, then the induced map $F(A_1 \\times_A A_2) \\to F(A_1) \\times_{F(A)} F(A_2)$ is surjective. \\item [(S2)]  If $A \\to k$ is a map in $\\mathcal{C}_\\Lambda$, then the induced map $F(A \\times_k k[\\epsilon]) \\to F(A) \\times_{F(k)} F(k[\\epsilon])$ is bijective. \\end{enumerate} The injectivity of the map $F(A \\times_k k[\\epsilon]) \\to F(A) \\times_{F(k)} F(k[\\epsilon])$ comes from the second part of condition (S2) and the fact that morphisms are identities."}
+{"_id": "3562", "title": "formal-defos-remark-linear-enriched-over-modules", "text": "One can define the notion of an $R$-linearity for any functor between categories enriched over $\\text{Mod}_R$. We made the definition specifically for functors $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$ and $L: \\text{Mod}_R \\to \\text{Mod}_R$ because these are the cases that we have needed so far."}
+{"_id": "3563", "title": "formal-defos-remark-linear-functor", "text": "If $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$ is an $R$-linear functor, then $L$ preserves finite products and sends the zero module to the zero module, see Homology, Lemma \\ref{homology-lemma-additive-additive}. On the other hand, if a functor $\\text{Mod}^{fg}_R \\to \\textit{Sets}$ preserves finite products and sends the zero module to a one element set, then it has a unique lift to a $R$-linear functor, see Lemma \\ref{lemma-linear-functor}."}
+{"_id": "3564", "title": "formal-defos-remark-tangent-space-cofibered-groupoid", "text": "We can globalize the notions of tangent space and differential to arbitrary categories cofibered in groupoids as follows. Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$, and let $x \\in \\Ob(\\mathcal{F}(k))$. As in Remark \\ref{remark-localize-cofibered-groupoid}, we get a predeformation category $\\mathcal{F}_x$. We define $$ T_x\\mathcal{F} = T\\mathcal{F}_x $$ to be the {\\it tangent space of $\\mathcal{F}$ at $x$}. If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$ and $x \\in \\Ob(\\mathcal{F}(k))$, then there is an induced morphism $\\varphi_x: \\mathcal{F}_x \\to \\mathcal{G}_{\\varphi(x)}$. We define the {\\it differential $d_x \\varphi : T_x \\mathcal{F} \\to T_{\\varphi(x)} \\mathcal{G}$ of $\\varphi$ at $x$} to be the map $d \\varphi_x: T \\mathcal{F}_x \\to T \\mathcal{G}_{\\varphi(x)}$. If both $\\mathcal{F}$ and $\\mathcal{G}$ satisfy (S2) then all of these tangent spaces have a natural $k$-vector space structure and all the differentials $d_x \\varphi : T_x \\mathcal{F} \\to T_{\\varphi(x)} \\mathcal{G}$ are $k$-linear (use Lemmas \\ref{lemma-S1-S2-localize} and \\ref{lemma-k-linear-differential})."}
+{"_id": "3565", "title": "formal-defos-remark-compare-schlessinger-H3-pre", "text": "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a predeformation functor satisfying (S1) and (S2). The condition $\\dim_k TF < \\infty$ is precisely condition (H3) from Schlessinger's paper. Recall that (S1) and (S2) correspond to conditions (H1) and (H2), see Remark \\ref{remark-compare-S1-S2-schlessinger}. Thus Lemma \\ref{lemma-versal-object-existence} tells us $$ (H1) + (H2) + (H3) \\Rightarrow \\text{ there exists a versal formal object} $$ for predeformation functors. We will make the link with hulls in Remark \\ref{remark-compare-schlessinger-H3}."}
+{"_id": "3566", "title": "formal-defos-remark-compare-schlessinger-H3", "text": "Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a predeformation functor satisfying (S1) and (S2) and $\\dim_k TF < \\infty$. Recall that these conditions correspond to the conditions (H1), (H2), and (H3) from Schlessinger's paper, see Remark \\ref{remark-compare-schlessinger-H3-pre}. Now, in the classical case (or if $k' \\subset k$ is separable) following Schlessinger we introduce the notion of a hull: a {\\it hull} is a versal formal object $\\xi \\in \\widehat{F}(R)$ such that $d\\underline{\\xi} : T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to TF$ is an isomorphism, i.e., (\\ref{equation-bijective}) holds. Thus Theorem \\ref{theorem-miniversal-object-existence} tells us $$ (H1) + (H2) + (H3) \\Rightarrow \\text{ there exists a hull} $$ in the classical case. In other words, our theorem recovers Schlessinger's theorem on the existence of hulls."}
+{"_id": "3567", "title": "formal-defos-remark-compose-minimal-into-iso-classes", "text": "Let $\\mathcal{F}$ be a predeformation category. Recall that $\\mathcal{F} \\to \\overline{\\mathcal{F}}$ is smooth, see Remark \\ref{remark-smooth-to-iso-classes}. Hence if $\\xi \\in \\widehat{\\mathcal{F}}(R)$ is a versal formal object, then the composition $$ \\underline{R}|_{\\mathcal{C}_\\Lambda} \\longrightarrow \\mathcal{F} \\longrightarrow \\overline{\\mathcal{F}} $$ is smooth (Lemma \\ref{lemma-smooth-properties}) and we conclude that the image $\\overline{\\xi}$ of $\\xi$ in $\\overline{\\mathcal{F}}$ is a versal formal object. If (\\ref{equation-bijective}) holds, then $\\overline{\\xi}$ induces an isomorphism $T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to T\\overline{\\mathcal{F}}$ because $\\mathcal{F} \\to \\overline{\\mathcal{F}}$ identifies tangent spaces. Hence in this case $\\overline{\\xi}$ is a hull for $\\overline{\\mathcal{F}}$, see Remark \\ref{remark-compare-schlessinger-H3}. By Theorem \\ref{theorem-miniversal-object-existence} we can always find such a $\\xi$ if $k' \\subset k$ is separable and $\\mathcal{F}$ is a predeformation category satisfying (S1), (S2), and $\\dim_k T\\mathcal{F} < \\infty$."}
+{"_id": "3568", "title": "formal-defos-remark-compare-schlessinger-H4", "text": "When $\\mathcal{F}$ is cofibered in sets, condition (RS) is exactly condition (H4) from Schlessinger's paper \\cite[Theorem 2.11]{Sch}.  Namely, for a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$, condition (RS) states: If $A_1 \\to A$ and $A_2 \\to A$ are maps in $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, then the induced map $F(A_1 \\times_A A_2) \\to F(A_1) \\times_{F(A)} F(A_2)$ is bijective."}
+{"_id": "3569", "title": "formal-defos-remark-deformation-functor", "text": "We say that a functor $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ is a {\\it deformation functor} if the associated cofibered set is a deformation category, i.e.\\ if $F(k)$ is a one element set and $F$ satisfies (RS). If $\\mathcal{F}$ is a deformation category, then $\\overline{\\mathcal{F}}$ is a predeformation functor but not necessarily a deformation functor, as Lemma \\ref{lemma-RS-associated-functor} shows."}
+{"_id": "3570", "title": "formal-defos-remark-omit-arrow", "text": "When the map $f: A' \\to A$ is clear from the context, we may write $\\textit{Lift}(x, A')$ and $\\text{Lift}(x, A')$ in place of $\\textit{Lift}(x, f)$ and $\\text{Lift}(x, f)$."}
+{"_id": "3571", "title": "formal-defos-remark-tangent-space-lifting", "text": "Let $\\mathcal{F}$ be a category cofibred in groupoids over $\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Let $V$ be a finite dimensional vector space. Then $\\text{Lift}(x_0, k[V])$ is the set of isomorphism classes of $\\mathcal{F}_{x_0}(k[V])$ where $\\mathcal{F}_{x_0}$ is the predeformation category of objects in $\\mathcal{F}$ lying over $x_0$, see Remark \\ref{remark-localize-cofibered-groupoid}. Hence if $\\mathcal{F}$ satisfies (S2), then so does $\\mathcal{F}_{x_0}$ (see Lemma \\ref{lemma-S1-S2-localize}) and by Lemma \\ref{lemma-tangent-space-vector-space} we see that $$ \\text{Lift}(x_0, k[V]) = T\\mathcal{F}_{x_0} \\otimes_k V $$ as $k$-vector spaces."}
+{"_id": "3572", "title": "formal-defos-remark-lift-bijections", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$ satisfying (RS).  Let $$ \\xymatrix{ A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\ A_1 \\ar[r] & A } $$ be a fibre square in $\\mathcal{C}_\\Lambda$ such that either $A_1 \\to A$ or $A_2 \\to A$ is surjective. Let $x \\in \\Ob(\\mathcal{F}(A))$. Given lifts $x_1 \\to x$ and $x_2 \\to x$ of $x$ to $A_1$ and $A_2$, we get by (RS) a lift $x_1 \\times_x x_2 \\to x$ of $x$ to $A_1 \\times_A A_2$. Conversely, by Lemma \\ref{lemma-RS-fiber-square} any lift of $x$ to $A_1 \\times_A A_2$ is of this form. Hence a bijection $$ \\text{Lift}(x, A_1) \\times \\text{Lift}(x, A_2) \\longrightarrow \\text{Lift}(x, A_1 \\times_A A_2). $$ Similarly, if $x_1 \\to x$ is a fixed lifting of $x$ to $A_1$, then there is a bijection $$ \\text{Lift}(x_1, A_1 \\times_A A_2) \\longrightarrow \\text{Lift}(x, A_2). $$ Now let $$ \\xymatrix{ A_1' \\times_A A_2 \\ar[r] \\ar[d] & A_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\ A_1' \\ar[r] & A_1 \\ar[r] & A } $$ be a composition of fibre squares in $\\mathcal{C}_\\Lambda$ with both $A'_1 \\to A_1$ and $A_1 \\to A$ surjective. Let $x_1 \\to x$ be a morphism lying over $A_1 \\to A$. Then by the above we have bijections \\begin{align*} \\text{Lift}(x_1, A_1' \\times_A A_2) & = \\text{Lift}(x_1, A_1') \\times \\text{Lift}(x_1, A_1 \\times_A A_2) \\\\ & = \\text{Lift}(x_1, A_1') \\times \\text{Lift}(x, A_2). \\end{align*}"}
+{"_id": "3573", "title": "formal-defos-remark-free-transitive-action-functorial", "text": "The action of Lemma \\ref{lemma-free-transitive-action} is functorial. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of deformation categories. Let $A' \\to A$ be a surjective ring map whose kernel $I$ is annihilated by $\\mathfrak m_{A'}$. Let $x \\in \\Ob(\\mathcal{F}(A))$. In this situation $\\varphi$ induces the vertical arrows in the following commutative diagram $$ \\xymatrix{ \\text{Lift}(x, A') \\times (T\\mathcal{F} \\otimes_k I) \\ar[d]_{(\\varphi, d\\varphi \\otimes \\text{id}_I)} \\ar[r] & \\text{Lift}(x, A') \\ar[d]^\\varphi \\\\ \\text{Lift}(\\varphi(x), A') \\times (T\\mathcal{G} \\otimes_k I) \\ar[r] & \\text{Lift}(\\varphi(x), A') } $$ The commutativity follows as each of the maps (\\ref{equation-two}), (\\ref{equation-one}), and (\\ref{equation-three}) of the proof of Lemma \\ref{lemma-free-transitive-action} gives rise to a similar commutative diagram."}
+{"_id": "3574", "title": "formal-defos-remark-choice-pushforward-immaterial-infinitesimal-aut", "text": "Up to canonical isomorphism $\\text{Inf}_{x_0}(\\mathcal{F})$ does not depend on the choice of pushforward $x_0 \\to x_0'$ because any two pushforwards are canonically isomorphic. Moreover, if $y_0 \\in \\mathcal{F}(k)$ and $x_0 \\cong y_0$ in $\\mathcal{F}(k)$, then $\\text{Inf}_{x_0}(\\mathcal{F}) \\cong \\text{Inf}_{y_0}(\\mathcal{F})$ where the isomorphism depends (only) on the choice of an isomorphism $x_0 \\to y_0$. In particular, $\\text{Aut}_k(x_0)$ acts on $\\text{Inf}_{x_0}(\\mathcal{F})$."}
+{"_id": "3575", "title": "formal-defos-remark-trivial-aut-point", "text": "Assume $\\mathcal{F}$ is a predeformation category. Then \\begin{enumerate} \\item for $x_0 \\in \\Ob(\\mathcal{F}(k))$ the automorphism group $\\text{Aut}_k(x_0)$ is trivial and hence $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Aut}_{k[\\epsilon]}(x'_0)$, and \\item for $x_0, y_0 \\in \\Ob(\\mathcal{F}(k))$ there is a unique isomorphism $x_0 \\to y_0$ and hence a canonical identification $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Inf}_{y_0}(\\mathcal{F})$. \\end{enumerate} Since $\\mathcal{F}(k)$ is nonempty, choosing $x_0 \\in \\Ob(\\mathcal{F}(k))$ and setting $$ \\text{Inf}(\\mathcal{F}) = \\text{Inf}_{x_0}(\\mathcal{F}) $$ we get a well defined {\\it group of infinitesimal automorphisms of $\\mathcal{F}$}. With this notation we have $\\text{Inf}(\\mathcal{F}_{x_0}) = \\text{Inf}_{x_0}(\\mathcal{F})$. Please compare with the equality $T\\mathcal{F}_{x_0} = T_{x_0}\\mathcal{F}$ in Remark \\ref{remark-tangent-space-cofibered-groupoid}."}
+{"_id": "3576", "title": "formal-defos-remark-infaut-lifting-equalities", "text": "We point out some basic relationships between infinitesimal automorphism groups, liftings, and tangent spaces to automorphism functors. Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Let $x' \\to x$ be a morphism lying over a ring map $A' \\to A$. Then from the definitions we have an equality $$ \\text{Inf}(x'/x) = \\text{Lift}(\\text{id}_x, A') $$ where the liftings are of $\\text{id}_x$ as an object of $\\mathit{Aut}(x')$.  If $x_0 \\in \\Ob(\\mathcal{F}(k))$ and $x'_0$ is the pushforward to $\\mathcal{F}(k[\\epsilon])$, then applying this to $x'_0 \\to x_0$ we get $$ \\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Lift}(\\text{id}_{x_0}, k[\\epsilon]) = T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0), $$ the last equality following directly from the definitions."}
+{"_id": "3577", "title": "formal-defos-remark-confusion-groupoids-in-functors", "text": "A groupoid in functors on $\\mathcal{C}$ amounts to the data of a functor $\\mathcal{C} \\to \\textit{Groupoids}$, and a morphism of groupoids in functors on $\\mathcal{C}$ amounts to a morphism of the corresponding functors $\\mathcal{C} \\to \\textit{Groupoids}$ (where $\\textit{Groupoids}$ is regarded as a 1-category).  However, for our purposes it is more convenient to use the terminology of groupoids in functors. In fact, thinking of a groupoid in functors as the corresponding functor $\\mathcal{C} \\to \\textit{Groupoids}$, or equivalently as the category cofibered in groupoids associated to that functor, can lead to confusion (Remark \\ref{remark-smooth-groupoid-in-functors-warning})."}
+{"_id": "3578", "title": "formal-defos-remark-identity-inverse", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$. There are unique morphisms $e : U \\to R$ and $i : R \\to R$ such that for every object $T$ of $\\mathcal{C}$, $e: U(T) \\to R(T)$ sends $x \\in U(T)$ to the identity morphism on $x$ and $i: R(T) \\to R(T)$ sends $a \\in U(T)$ to the inverse of $a$ in the groupoid category $(U(T), R(T), s, t, c)$. We will sometimes refer to $s$, $t$, $c$, $e$, and $i$ as ``source'', ``target'', ``composition'', ``identity'', and ``inverse''."}
+{"_id": "3579", "title": "formal-defos-remark-reason-existence-coproduct", "text": "Hence a representable groupoid in functors on $\\mathcal{C}$ is given by objects $U$ and $R$ of $\\mathcal{C}$ and morphisms $s, t : U \\to R$ and $c : R \\to R \\amalg_{s, U, t} R$ such that $(\\underline{U}, \\underline{R}, s, t, c)$ satisfies the condition of Definition \\ref{definition-groupoid-in-functors}. The reason for requiring the existence of the pushout $R \\amalg_{s, U, t} R$ is so that the composition morphism $c$ is defined at the level of morphisms in $\\mathcal{C}$. This requirement will always be satisfied below when we consider representable groupoids in functors on $\\widehat{\\mathcal{C}}_\\Lambda$, since by Lemma \\ref{lemma-CLambdahat-pushouts} the category $\\widehat{\\mathcal{C}}_\\Lambda$ admits pushouts."}
+{"_id": "3580", "title": "formal-defos-remark-simplify-terminology", "text": "We will say ``{\\it let $(\\underline{U}, \\underline{R}, s, t, c)$ be a groupoid in functors on $\\mathcal{C}$}'' to mean that we have a representable groupoid in functors. Thus this means that $U$ and $R$ are objects of $\\mathcal{C}$, there are morphisms $s, t : U \\to R$, the pushout $R \\amalg_{s, U, t} R$ exists, there is a morphism $c : R \\to R \\amalg_{s, U, t} R$, and $(\\underline{U}, \\underline{R}, s, t, c)$ is a groupoid in functors on $\\mathcal{C}$."}
+{"_id": "3581", "title": "formal-defos-remark-notation-restriction", "text": "In the situation of Definition \\ref{definition-restricting-groupoids-in-functors}, we often denote $s|_{\\mathcal{C}'}, t|_{\\mathcal{C}'}, c|_{\\mathcal{C}'}$ simply by $s, t, c$."}
+{"_id": "3582", "title": "formal-defos-remark-groupoid-in-functors-complete-restrict", "text": "Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. Then there is a canonical isomorphism $(U, R, s, t, c)^{\\wedge}|_{\\mathcal{C}_\\Lambda} \\cong (U, R, s, t, c)$, see Remark \\ref{remark-restrict-completion}. On the other hand, let $(U, R, s, t, c)$ be a groupoid in functors on $\\widehat{\\mathcal{C}}_\\Lambda$ such that $U, R : \\widehat{\\mathcal{C}}_\\Lambda \\to \\textit{Sets}$ both commute with limits, e.g.\\ if $U, R$ are representable. Then there is a canonical isomorphism $((U, R, s, t, c)|_{\\mathcal{C}_\\Lambda})^{\\wedge} \\cong (U, R, s, t, c)$. This follows from Remark \\ref{remark-restrict-complete-continuous-functor}."}
+{"_id": "3583", "title": "formal-defos-remark-smooth-groupoid-in-functors-warning", "text": "We note that this terminology is potentially confusing: if $(U, R, s, t, c)$ is a smooth groupoid in functors, then the quotient $[U/R]$ need not be a smooth category cofibred in groupoids as defined in Definition \\ref{definition-cofibered-groupoid-projection-smooth}. However smoothness of $(U, R, s, t, c)$ does imply (in fact is equivalent to) smoothness of the quotient morphism $U \\to [U/R]$ as we shall see in Lemma \\ref{lemma-smooth-quotient-morphism}."}
+{"_id": "3584", "title": "formal-defos-remark-smooth-power-series-prorepresentable-smooth-groupoid-in-functors", "text": "Let $(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$ be a prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$. Then $(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$ is smooth if and only if $R_1$ is a power series over $R_0$ via both $s$ and $t$. This follows from Lemma \\ref{lemma-smooth-morphism-power-series}."}
+{"_id": "3585", "title": "formal-defos-remarks-cofibered-groupoids", "text": "Everything about categories fibered in groupoids translates directly to the cofibered setting. The following remarks are meant to fix notation. Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item We often omit the functor $p: \\mathcal{F} \\to \\mathcal{C}$ from the notation. \\item The fiber category over an object $U$ in $\\mathcal{C}$ is denoted by $\\mathcal{F}(U)$. Its objects are those of $\\mathcal{F}$ lying over $U$ and its morphisms are those of $\\mathcal{F}$ lying over $\\text{id}_U$. If $x, y$  are objects of $\\mathcal{F}(U)$, we sometimes write $\\Mor_U(x, y)$ for $\\Mor_{\\mathcal{F}(U)}(x, y)$. \\item The fibre categories $\\mathcal{F}(U)$ are groupoids, see Categories, Lemma \\ref{categories-lemma-fibred-groupoids}. Hence the morphisms in $\\mathcal{F}(U)$ are all isomorphisms. We sometimes write $\\text{Aut}_U(x)$ for $\\Mor_{\\mathcal{F}(U)}(x, x)$. \\item \\label{item-pushforward} Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}$, let $f: U \\to V$ be a morphism in $\\mathcal{C}$, and let $x \\in \\Ob(\\mathcal{F}(U))$. A {\\it pushforward} of $x$ along $f$ is a morphism $x \\to y$ of $\\mathcal{F}$ lying over $f$. A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition \\ref{categories-definition-cartesian-over-C}). We sometimes write $x \\to f_*x$ for ``the'' pushforward of $x$ along $f$. \\item A {\\it choice of pushforwards for $\\mathcal{F}$} is the choice of a pushforward of $x$ along $f$ for every pair $(x, f)$ as above. We can make such a choice of pushforwards for $\\mathcal{F}$ by the axiom of choice. \\item Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}$. Given a choice of pushforwards for $\\mathcal{F}$, there is an associated pseudo-functor $\\mathcal{C} \\to \\textit{Groupoids}$. We will never use this construction so we give no details. \\item \\label{item-cofibered-morphism} A morphism of categories cofibered in groupoids over $\\mathcal{C}$ is a functor commuting with the projections to $\\mathcal{C}$. If $\\mathcal{F}$ and $\\mathcal{F}'$ are categories cofibered in groupoids over $\\mathcal{C}$, we denote the morphisms from $\\mathcal{F}$ to $\\mathcal{F}'$ by $\\Mor_\\mathcal{C}(\\mathcal{F}, \\mathcal{F}')$. \\item \\label{item-definition-cofibered-groupoids-2-category} Categories cofibered in groupoids form a $(2, 1)$-category $\\text{Cof}(\\mathcal{C})$. Its 1-morphisms are the morphisms described in (\\ref{item-cofibered-morphism}). If $p : \\mathcal{F} \\to C$ and $p': \\mathcal{F}' \\to \\mathcal{C}$ are categories cofibered in groupoids and $\\varphi, \\psi : \\mathcal{F} \\to \\mathcal{F}'$ are $1$-morphisms, then a 2-morphism $t : \\varphi \\to \\psi$ is a morphism of functors such that $p'(t_x) = \\text{id}_{p(x)}$ for all $x \\in \\Ob(\\mathcal{F})$. \\item \\label{item-construction-associated-cofibered-groupoid} Let $F : \\mathcal{C} \\to \\textit{Groupoids}$ be a functor. There is a category cofibered in groupoids $\\mathcal{F} \\to \\mathcal{C}$ associated to $F$ as follows. An object of $\\mathcal{F}$ is a pair $(U, x)$ where $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(F(U))$. A morphism $(U, x) \\to (V, y)$ is a pair $(f, a)$ where $f \\in \\Mor_\\mathcal{C}(U, V)$ and $a \\in \\Mor_{F(V)}(F(f)(x), y)$. The functor $\\mathcal{F} \\to \\mathcal{C}$ sends $(U, x)$ to $U$. See Categories, Section \\ref{categories-section-presheaves-groupoids}. \\item \\label{item-associated-functor-isomorphism-classes} Let $\\mathcal{F}$ be cofibered in groupoids over $\\mathcal{C}$. For $U \\in \\Ob(\\mathcal{C})$ set $\\overline{\\mathcal{F}}(U)$ equal to the set of isomorphisms classes of the category $\\mathcal{F}(U)$. If $f : U \\to V$ is a morphism of $\\mathcal{C}$, then we obtain a map of sets $\\overline{\\mathcal{F}}(U) \\to \\overline{\\mathcal{F}}(V)$ by mapping the isomorphism class of $x$ to the isomorphism class of a pushforward $f_*x$ of $x$ see (\\ref{item-pushforward}). Then $\\overline{\\mathcal{F}} : \\mathcal{C} \\to \\textit{Sets}$ is a functor. Similarly, if $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of cofibered categories, we denote by $\\overline{\\varphi}: \\overline{\\mathcal{F}} \\to  \\overline{\\mathcal{G}}$ the associated morphism of functors. \\item \\label{item-convention-cofibered-sets} Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (\\ref{item-construction-associated-cofibered-groupoid}) associates to $F$ a category cofibered in sets. This defines a fully faithful embedding of the category of functors $\\mathcal{C} \\to \\textit{Sets}$ to the category of categories cofibered in groupoids over $\\mathcal{C}$. We identify the category of functors with its image under this embedding. Hence if $F : \\mathcal{C} \\to \\textit{Sets}$ is a functor, we denote the associated category cofibered in sets also by $F$; and if $\\varphi : F \\to G$ is a morphism of functors, we denote still by $\\varphi$ the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section \\ref{categories-section-fibred-in-sets}. \\item \\label{item-definition-yoneda} Let $U$ be an object of $\\mathcal{C}$.  We write $\\underline{U}$ for the functor $\\Mor_\\mathcal{C}(U, -): \\mathcal{C} \\to \\textit{Sets}$.  This defines a fully faithful embedding of $\\mathcal C^{opp}$ into the category of functors $\\mathcal{C} \\to \\textit{Sets}$. Hence, if $f : U \\to V$ is a morphism, we are justified in denoting still by $f$ the induced morphism $\\underline{V} \\to \\underline{U}$, and vice-versa. \\item \\label{item-fibre-product} Fiber products of categories cofibered in groupoids: If $\\mathcal{F} \\to \\mathcal{H}$ and $\\mathcal{G} \\to \\mathcal{H}$ are morphisms of categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$, then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over $\\mathcal{C}_\\Lambda$, as described in Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}. \\item \\label{item-product} Products of categories cofibered in groupoids: If $\\mathcal{F}$ and $\\mathcal{G}$ are categories cofibered in groupoids over $\\mathcal{C}_\\Lambda$ then their product is defined to be the $2$-fiber product $\\mathcal{F} \\times_{\\mathcal{C}_\\Lambda} \\mathcal{G}$ as described in Categories, Lemma \\ref{categories-lemma-2-product-categories-over-C}. \\item \\label{item-definition-restricting-base-category} Restricting the base category: Let $p : \\mathcal{F} \\to \\mathcal{C}$ be a category cofibered in groupoids, and let $\\mathcal{C}'$ be a full subcategory of $\\mathcal{C}$. The restriction $\\mathcal{F}|_{\\mathcal{C}'}$ is the full subcategory of $\\mathcal{F}$ whose objects lie over objects of $\\mathcal{C}'$. It is a category cofibered in groupoids via the functor $p|_{\\mathcal{C}'}: \\mathcal{F}|_{\\mathcal{C}'} \\to \\mathcal{C}'$. \\end{enumerate}"}
+{"_id": "3644", "title": "adequate-remark-settheoretic", "text": "Consider the category $\\textit{Alg}_{fp, A}$ whose objects are $A$-algebras $B$ of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and whose morphisms are $A$-algebra maps. Every $A$-algebra $B$ is a filtered colimit of finitely presented $A$-algebra, i.e., a filtered colimit of objects of $\\textit{Alg}_{fp, A}$. By Lemma \\ref{lemma-adequate-finite-presentation} we conclude every adequate functor $F$ is determined by its restriction to $\\textit{Alg}_{fp, A}$. For some questions we can therefore restrict to functors on $\\textit{Alg}_{fp, A}$. For example, the category of adequate functors does not depend on the choice of the big $\\tau$-site chosen in Section \\ref{section-conventions}."}
+{"_id": "3645", "title": "adequate-remark-linearly-adequate", "text": "Let $A$ be a ring. The proof of Lemma \\ref{lemma-extension-adequate-key} shows that any extension $0 \\to \\underline{M} \\to E \\to L \\to 0$ of module-valued functors on $\\textit{Alg}_A$ with $L$ linearly adequate splits. It uses only the following properties of the module-valued functor $F = \\underline{M}$: \\begin{enumerate} \\item $F(B) \\otimes_B B' \\to F(B')$ is an isomorphism for a flat ring map $B \\to B'$, and \\item $F(C)^{(1)} = F(p_1)(F(B)^{(1)}) \\oplus F(p_2)(F(B)^{(1)})$ where $B = A[x_1, \\ldots, x_n]/(\\sum a_{ij}x_j)$ and $C = A[x_1, \\ldots, x_n, y_1, \\ldots, y_n]/ (\\sum a_{ij}x_j, \\sum a_{ij}y_j)$. \\end{enumerate} These two properties hold for any adequate functor $F$; details omitted. Hence we see that $L$ is a projective object of the abelian category of adequate functors."}
+{"_id": "3646", "title": "adequate-remark-compare", "text": "Let $S$ be a scheme. We have functors $u : \\QCoh(\\mathcal{O}_S) \\to \\textit{Adeq}(\\mathcal{O})$ and $v : \\textit{Adeq}(\\mathcal{O}) \\to \\QCoh(\\mathcal{O}_S)$. Namely, the functor $u : \\mathcal{F} \\mapsto \\mathcal{F}^a$ comes from taking the associated $\\mathcal{O}$-module which is adequate by Lemma \\ref{lemma-adequate-characterize}. Conversely, the functor $v$ comes from restriction $v : \\mathcal{G} \\mapsto \\mathcal{G}|_{S_{Zar}}$, see Lemma \\ref{lemma-same-cohomology-adequate}. Since $\\mathcal{F}^a$ can be described as the pullback of $\\mathcal{F}$ under a morphism of ringed topoi $((\\Sch/S)_\\tau, \\mathcal{O}) \\to (S_{Zar}, \\mathcal{O}_S)$, see Descent, Remark \\ref{descent-remark-change-topologies-ringed-sites} and since restriction is the pushforward we see that $u$ and $v$ are adjoint as follows $$ \\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, v\\mathcal{G}) = \\SheafHom_\\mathcal{O}(u\\mathcal{F}, \\mathcal{G}) $$ where $\\mathcal{O}$ denotes the structure sheaf on the big site. It is immediate from the description that the adjunction mapping $\\mathcal{F} \\to vu\\mathcal{F}$ is an isomorphism for all quasi-coherent sheaves."}
+{"_id": "3647", "title": "adequate-remark-D-adeq-independence-topology", "text": "Let $S$ be a scheme. Let $\\tau, \\tau' \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Denote $\\mathcal{O}_\\tau$, resp.\\ $\\mathcal{O}_{\\tau'}$ the structure sheaf $\\mathcal{O}$ viewed as a sheaf on $(\\Sch/S)_\\tau$, resp.\\ $(\\Sch/S)_{\\tau'}$. Then $D_{\\textit{Adeq}}(\\mathcal{O}_\\tau)$ and $D_{\\textit{Adeq}}(\\mathcal{O}_{\\tau'})$ are canonically isomorphic. This follows from Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules}. Namely, assume $\\tau$ is stronger than the topology $\\tau'$, let $\\mathcal{C} = (\\Sch/S)_{fppf}$, and let $\\mathcal{B}$ the collection of affine schemes over $S$. Assumptions (1) and (2) we've seen above. Assumption (3) is clear and assumption (4) follows from Lemma \\ref{lemma-same-cohomology-adequate}."}
+{"_id": "3648", "title": "adequate-remark-D-adeq-and-D-QCoh", "text": "Let $S$ be a scheme. The morphism $f$ see (\\ref{equation-compare-big-small}) induces adjoint functors $Rf_* : D_{\\textit{Adeq}}(\\mathcal{O}) \\to D_\\QCoh(S)$ and $Lf^* : D_\\QCoh(S) \\to D_{\\textit{Adeq}}(\\mathcal{O})$. Moreover $Rf_* Lf^* \\cong \\text{id}_{D_\\QCoh(S)}$. \\medskip\\noindent We sketch the proof. By Remark \\ref{remark-D-adeq-independence-topology} we may assume the topology $\\tau$ is the Zariski topology. We will use the existence of the unbounded total derived functors $Lf^*$ and $Rf_*$ on $\\mathcal{O}$-modules and their adjointness, see Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-adjoint}. In this case $f_*$ is just the restriction to the subcategory $S_{Zar}$ of $(\\Sch/S)_{Zar}$. Hence it is clear that $Rf_* = f_*$ induces $Rf_* : D_{\\textit{Adeq}}(\\mathcal{O}) \\to D_\\QCoh(S)$. Suppose that $\\mathcal{G}^\\bullet$ is an object of $D_\\QCoh(S)$. We may choose a system $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$ of bounded above complexes of flat $\\mathcal{O}_S$-modules whose transition maps are termwise split injectives and a diagram $$ \\xymatrix{ \\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] & \\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\ \\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] & \\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots } $$ with the properties (1), (2), (3) listed in Derived Categories, Lemma \\ref{derived-lemma-special-direct-system} where $\\mathcal{P}$ is the collection of flat $\\mathcal{O}_S$-modules. Then $Lf^*\\mathcal{G}^\\bullet$ is computed by $\\colim f^*\\mathcal{K}_n^\\bullet$, see Cohomology on Sites, Lemmas \\ref{sites-cohomology-lemma-pullback-K-flat} and \\ref{sites-cohomology-lemma-derived-base-change} (note that our sites have enough points by \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-points-fppf}). We have to see that $H^i(Lf^*\\mathcal{G}^\\bullet) = \\colim H^i(f^*\\mathcal{K}_n^\\bullet)$ is adequate for each $i$. By Lemma \\ref{lemma-abelian-adequate} we conclude that it suffices to show that each $H^i(f^*\\mathcal{K}_n^\\bullet)$ is adequate. \\medskip\\noindent The adequacy of $H^i(f^*\\mathcal{K}_n^\\bullet)$ is local on $S$, hence we may assume that $S = \\Spec(A)$ is affine. Because $S$ is affine $D_\\QCoh(S) = D(\\QCoh(\\mathcal{O}_S))$, see the discussion in Derived Categories of Schemes, Section \\ref{perfect-section-derived-quasi-coherent}. Hence there exists a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{K}_n^\\bullet$ where $\\mathcal{F}^\\bullet$ is a bounded above complex of flat quasi-coherent modules. Then $f^*\\mathcal{F}^\\bullet \\to f^*\\mathcal{K}_n^\\bullet$ is a quasi-isomorphism, and the cohomology sheaves of $f^*\\mathcal{F}^\\bullet$ are adequate. \\medskip\\noindent The final assertion $Rf_* Lf^* \\cong \\text{id}_{D_\\QCoh(S)}$ follows from the explicit description of the functors above. (In plain English: if $\\mathcal{F}$ is quasi-coherent and $p > 0$, then $L_pf^*\\mathcal{F}$ is a parasitic adequate module.)"}
+{"_id": "3649", "title": "adequate-remark-conclusion", "text": "Remark \\ref{remark-D-adeq-and-D-QCoh} above implies we have an equivalence of derived categories $$ D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O}) \\longrightarrow D_\\QCoh(S) $$ where $\\mathcal{C}$ is the category of parasitic adequate modules. Namely, it is clear that $D_\\mathcal{C}(\\mathcal{O})$ is the kernel of $Rf_*$, hence a functor as indicated. For any object $X$ of $D_{\\textit{Adeq}}(\\mathcal{O})$ the map $Lf^*Rf_*X \\to X$ maps to a quasi-isomorphism in $D_\\QCoh(S)$, hence $Lf^*Rf_*X \\to X$ is an isomorphism in $D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O})$. Finally, for $X, Y$ objects of $D_{\\textit{Adeq}}(\\mathcal{O})$ the map $$ Rf_* : \\Hom_{D_{\\textit{Adeq}}(\\mathcal{O})/D_\\mathcal{C}(\\mathcal{O})}(X, Y) \\to \\Hom_{D_\\QCoh(S)}(Rf_*X, Rf_*Y) $$ is bijective as $Lf^*$ gives an inverse (by the remarks above)."}
+{"_id": "3695", "title": "spaces-topologies-remark-change-topologies-ringed", "text": "The sites $(\\textit{Spaces}/X)_\\etale$ and $X_{spaces, \\etale}$ come with structure sheaves. For the small \\'etale site we have seen this in Properties of Spaces, Section \\ref{spaces-properties-section-structure-sheaf}. The structure sheaf $\\mathcal{O}$ on the big \\'etale site $(\\textit{Spaces}/X)_\\etale$ is defined by assigning to an object $U$ the global sections of the structure sheaf of $U$. This makes sense because after all $U$ is an algebraic space itself hence has a structure sheaf. Since $\\mathcal{O}_U$ is a sheaf on the \\'etale site of $U$, the presheaf $\\mathcal{O}$ so defined satisfies the sheaf condition for coverings of $U$, i.e., $\\mathcal{O}$ is a sheaf. We can upgrade the morphisms $i_f$, $\\pi_X$, $i_X$, $f_{small}$, and $f_{big}$ defined above to morphisms of ringed sites, respectively topoi. Let us deal with these one by one. \\begin{enumerate} \\item In Lemma \\ref{lemma-put-in-T-etale} denote $\\mathcal{O}$ the structure sheaf on $(\\textit{Spaces}/X)_\\etale$. We have $(i_f^{-1}\\mathcal{O})(U/Y) = \\mathcal{O}_U(U) = \\mathcal{O}_Y(U)$ by construction. Hence an isomorphism $i_f^\\sharp : i_f^{-1}\\mathcal{O} \\to \\mathcal{O}_Y$. \\item In Lemma \\ref{lemma-at-the-bottom-etale} it was noted that $i_X$ is a special case of $i_f$ with $f = \\text{id}_X$ hence we are back in case (1). \\item In Lemma \\ref{lemma-at-the-bottom-etale} the morphism $\\pi_X$ satisfies $(\\pi_{X, *}\\mathcal{O})(U) = \\mathcal{O}(U) = \\mathcal{O}_X(U)$. Hence we can use this to define $\\pi_X^\\sharp : \\mathcal{O}_X \\to \\pi_{X, *}\\mathcal{O}$. \\item In Lemma \\ref{lemma-morphism-big-small-etale} the extension of $f_{small}$ to a morphism of ringed topoi was discussed in Properties of Spaces, Lemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}. \\item In Lemma \\ref{lemma-morphism-big-small-etale} the functor $f_{big}^{-1}$ is simply the restriction via the inclusion functor $(\\textit{Spaces}/Y)_\\etale \\to (\\textit{Spaces}/X)_\\etale$. Let $\\mathcal{O}_1$ be the structure sheaf on $(\\textit{Spaces}/X)_\\etale$ and let $\\mathcal{O}_2$ be the structure sheaf on $(\\textit{Spaces}/Y)_\\etale$. We obtain a canonical isomorphism $f_{big}^\\sharp : f_{big}^{-1}\\mathcal{O}_1 \\to \\mathcal{O}_2$. \\end{enumerate} Moreover, with these definitions compositions work out correctly too. We omit giving a detailed statement and proof."}
+{"_id": "3845", "title": "proetale-remark-size-w", "text": "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of $A_w$ (Lemma \\ref{lemma-make-w-local}) is at most $\\kappa$. Namely, each $A_E$ has cardinality at most $\\kappa$ and the set of finite subsets of $A$ has cardinality at most $\\kappa$ as well. Thus the result follows as $\\kappa \\otimes \\kappa = \\kappa$, see Sets, Section \\ref{sets-section-cardinals}."}
+{"_id": "3846", "title": "proetale-remark-slightly-stronger", "text": "In each of Lemmas \\ref{lemma-construct}, \\ref{lemma-construct-profinite}, Proposition \\ref{proposition-maps-wich-identify-local-rings}, and Lemma \\ref{lemma-find-Zariski-w-contractible} we find an ind-Zariski ring map with some properties. In the paper \\cite{BS} the authors use the notion of an ind-(Zariski localization) which is a filtered colimit of finite products of principal localizations. It is possible to replace ind-Zariski by ind-(Zariski localization) in each of the results listed above. However, we do not need this and the notion of an ind-Zariski homomorphism of rings as defined here has slightly better formal properties. Moreover, the notion of an ind-Zariski ring map is the natural analogue of the notion of an ind-\\'etale ring map defined in the next section."}
+{"_id": "3847", "title": "proetale-remark-size-T", "text": "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of $T(A)$ is at most $\\kappa$. Namely, each $B_E$ has cardinality at most $\\kappa$ and the index set $I(A)$ has cardinality at most $\\kappa$ as well. Thus the result follows as $\\kappa \\otimes \\kappa = \\kappa$, see Sets, Section \\ref{sets-section-cardinals}. It follows that the ring constructed in the proof of Lemma \\ref{lemma-first-construction} has cardinality at most $\\kappa$ as well."}
+{"_id": "3848", "title": "proetale-remark-first-construction-functorial", "text": "The construction $A \\mapsto T(A)$ is functorial in the following sense: If $A \\to A'$ is a ring map, then we can construct a commutative diagram $$ \\xymatrix{ A \\ar[r] \\ar[d] & T(A) \\ar[d] \\\\ A' \\ar[r] & T(A') } $$ Namely, given $(A \\to A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n))$ in $S(A)$ we can use the ring map $\\varphi : A \\to A'$ to obtain a corresponding element $(A' \\to A'[x_1, \\ldots, x_n]/(f^\\varphi_1, \\ldots, f^\\varphi_n))$ of $S(A')$ where $f^\\varphi$ means the polynomial obtained by applying $\\varphi$ to the coefficients of the polynomial $f$. Moreover, there is a commutative diagram $$ \\xymatrix{ A \\ar[r] \\ar[d] & A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n) \\ar[d] \\\\ A' \\ar[r] & A'[x_1, \\ldots, x_n]/(f^\\varphi_1, \\ldots, f^\\varphi_n) } $$ which is a in the category of rings. For $E \\subset S(A)$ finite, set $E' = \\varphi(E)$ and define $B_E \\to B_{E'}$ in the obvious manner. Taking the colimit gives the desired map $T(A) \\to T(A')$, see Categories, Lemma \\ref{categories-lemma-functorial-colimit}."}
+{"_id": "3849", "title": "proetale-remark-h-limit-preserving", "text": "Let $S$ be a scheme contained in a big site $\\Sch_h$. Let $F$ be a sheaf of sets on $(\\Sch/S)_h$ such that $F(T) = \\colim F(T_i)$ whenever $T = \\lim T_i$ is a directed limit of affine schemes in $(\\Sch/S)_h$. In this situation $F$ extends uniquely to a contravariant functor $F'$ on the category of all schemes over $S$ such that (a) $F'$ satisfies the sheaf property for the h topology and (b) $F'$ is limit preserving. See More on Flatness, Lemma \\ref{flat-lemma-extend-sheaf-h}. In this situation Lemma \\ref{lemma-h-limit-preserving} tells us that $F'$ satisfies the sheaf property for the V topology."}
+{"_id": "3850", "title": "proetale-remark-size-w-contractible", "text": "Let $A$ be a ring. Let $\\kappa$ be an infinite cardinal bigger or equal than the cardinality of $A$. Then the cardinality of the ring $D$ constructed in Proposition \\ref{proposition-find-w-contractible} is at most $$ \\kappa^{2^{2^{2^\\kappa}}}. $$ Namely, the ring map $A \\to D$ is constructed as a composition $$ A \\to A_w = A' \\to C' \\to C \\to D. $$ Here the first three steps of the construction are carried out in the first paragraph of the proof of Lemma \\ref{lemma-get-w-local-algebraic-residue-field-extensions}. For the first step we have $|A_w| \\leq \\kappa$ by Remark \\ref{remark-size-w}. We have $|C'| \\leq \\kappa$ by Remark \\ref{remark-size-T}. Then $|C| \\leq \\kappa$ because $C$ is a localization of $(C')_w$ (it is constructed from $C'$ by an application of Lemma \\ref{lemma-localize-along-closed-profinite} in the proof of Lemma \\ref{lemma-w-local-algebraic-residue-field-extensions}). Thus $C$ has at most $2^\\kappa$ maximal ideals. Finally, the ring map $C \\to D$ identifies local rings and the cardinality of the set of maximal ideals of $D$ is at most $2^{2^{2^\\kappa}}$ by Topology, Remark \\ref{topology-remark-size-projective-cover}. Since $D \\subset \\prod_{\\mathfrak m \\subset D} D_\\mathfrak m$ we see that $D$ has at most the size displayed above."}
+{"_id": "3851", "title": "proetale-remark-extend-to-all", "text": "Let $X$ be a scheme. Because $X_\\proetale$ has enough weakly contractible objects for all $K$ in $D(X_\\proetale)$ we have $K = R\\lim \\tau_{\\geq -n}K$ by Cohomology on Sites, Proposition \\ref{sites-cohomology-proposition-enough-weakly-contractibles}. Since $R\\Gamma$ commutes with $R\\lim$ by Injectives, Lemma \\ref{injectives-lemma-RF-commutes-with-Rlim} we see that $$ R\\Gamma(X, K) = R\\lim R\\Gamma(X, \\tau_{\\geq -n}K) $$ in $D(\\textit{Ab})$. This will sometimes allow us to extend results from bounded below complexes to all complexes."}
+{"_id": "3997", "title": "formal-spaces-remark-mcquillan", "text": "There is a variant of the construction of formal schemes due to McQuillan, see \\cite{McQuillan}. He suggests a slight weakening of the condition of admissibility. Namely, recall that an admissible topological ring is a complete (and separated by our conventions) topological ring $A$ which is linearly topologized such that there exists an ideal of definition: an open ideal $I$ such that any neighbourhood of $0$ contains $I^n$ for some $n \\geq 1$. McQuillan works with what we will call {\\it weakly admissible} topological rings. A weakly admissible topological ring $A$ is a complete (and separated by our conventions) topological ring which is linearly topologized such that there exists an {\\it weak ideal of definition}: an open ideal $I$ such that for all $f \\in I$ we have $f^n \\to 0$ for $n \\to \\infty$. Similarly to the admissible case, if $I$ is a weak ideal of definition and $J \\subset A$ is an open ideal, then $I \\cap J$ is a weak ideal of definition. Thus the weak ideals of definition form a fundamental system of open neighbourhoods of $0$ and one can proceed along much the same route as above to define a larger category of formal schemes based on this notion. The analogues of Lemmas \\ref{lemma-fully-faithful} and \\ref{lemma-formal-scheme-sheaf-fppf} still hold in this setting (with the same proof)."}
+{"_id": "3998", "title": "formal-spaces-remark-sheafification-of-presheaves-in-top", "text": "\\begin{reference} \\cite{Gray} \\end{reference} In this remark we briefly discuss sheafification of presheaves of topological spaces. The exact same arguments work for presheaves of topological abelian groups, topological rings, and topological modules (over a given topological ring). In order to do this in the correct generality let us work over a site $\\mathcal{C}$. The reader who is interested in the case of (pre)sheaves over a topological space $X$ should think of objects of $\\mathcal{C}$ as the opens of $X$, of morphisms of $\\mathcal{C}$ as inclusions of opens, and of coverings in $\\mathcal{C}$ as coverings in $X$, see Sites, Example \\ref{sites-example-site-topological}. Denote $\\Sh(\\mathcal{C}, \\textit{Top})$ the category of sheaves of topological spaces on $\\mathcal{C}$ and denote $\\textit{PSh}(\\mathcal{C}, \\textit{Top})$ the category of presheaves of topological spaces on $\\mathcal{C}$. Let $\\mathcal{F}$ be a presheaf of topological spaces on $\\mathcal{C}$. The sheafification $\\mathcal{F}^\\#$ should satisfy the formula $$ \\Mor_{\\textit{PSh}(\\mathcal{C}, \\textit{Top})}(\\mathcal{F}, \\mathcal{G}) = \\Mor_{\\Sh(\\mathcal{C}, \\textit{Top})}(\\mathcal{F}^\\#, \\mathcal{G}) $$ functorially in $\\mathcal{G}$ from $\\Sh(\\mathcal{C}, \\textit{Top})$. In other words, we are trying to construct the left adjoint to the inclusion functor $\\Sh(\\mathcal{C}, \\textit{Top}) \\to \\textit{PSh}(\\mathcal{C}, \\textit{Top})$. We first claim that $\\Sh(\\mathcal{C}, \\textit{Top})$ has limits and that the inclusion functor commutes with them. Namely, given a category $\\mathcal{I}$ and a functor $i \\mapsto \\mathcal{G}_i$ into $\\Sh(\\mathcal{C}, \\textit{Top})$ we simply define $$ (\\lim \\mathcal{G}_i)(U) = \\lim \\mathcal{G}_i(U) $$ where we take the limit in the category of topological spaces (Topology, Lemma \\ref{topology-lemma-limits}). This defines a sheaf because limits commute with limits (Categories, Lemma \\ref{categories-lemma-colimits-commute}) and in particular products and equalizers (which are the operations used in the sheaf axiom). Finally, a morphism of presheaves from $\\mathcal{F} \\to \\lim \\mathcal{G}_i$ is clearly the same thing as a compatible system of morphisms $\\mathcal{F} \\to \\mathcal{G}_i$. In other words, the object $\\lim \\mathcal{G}_i$ is the limit in the category of presheaves of topological spaces and a fortiori in the category of sheaves of topological spaces. Our second claim is that any morphism of presheaves $\\mathcal{F} \\to \\mathcal{G}$ with $\\mathcal{G}$ an object of $\\Sh(\\mathcal{C}, \\textit{Top})$ factors through a subsheaf $\\mathcal{G}' \\subset \\mathcal{G}$ whose size is bounded. Here we define the {\\it size} $|\\mathcal{H}|$ of a sheaf of topological spaces $\\mathcal{H}$ to be the cardinal $\\sup_{U \\in \\Ob(\\mathcal{C})} |\\mathcal{H}(U)|$. To prove our claim we let $$ \\mathcal{G}'(U) = \\left\\{ \\quad s \\in \\mathcal{G}(U) \\quad \\middle| \\quad \\begin{matrix} \\text{there exists a covering }\\{U_i \\to U\\}_{i \\in I} \\\\ \\text{such that } s|_{U_i} \\in \\Im(\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)) \\end{matrix} \\quad \\right\\} $$ We endow $\\mathcal{G}'(U)$ with the induced topology. Then $\\mathcal{G}'$ is a sheaf of topological spaces (details omitted) and $\\mathcal{G}' \\to \\mathcal{G}$ is a morphism through which the given map $\\mathcal{F} \\to \\mathcal{G}$ factors. Moreover, the size of $\\mathcal{G}'$ is bounded by some cardinal $\\kappa$ depending only on $\\mathcal{C}$ and the presheaf $\\mathcal{F}$ (hint: use that coverings in $\\mathcal{C}$ form a set by our conventions). Putting everything together we see that the assumptions of Categories, Theorem \\ref{categories-theorem-adjoint-functor} are satisfied and we obtain sheafification as the left adjoint of the inclusion functor from sheaves to presheaves. Finally, let $p$ be a point of the site $\\mathcal{C}$ given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$, see Sites, Definition \\ref{sites-definition-point}. For a topological space $M$ the presheaf defined by the rule $$ U \\mapsto \\text{Map}(u(U), M) = \\prod\\nolimits_{x \\in u(U)} M $$ endowed with the product topology is a sheaf of topological spaces. Hence the exact same argument as given in the proof of Sites, Lemma \\ref{sites-lemma-point-pushforward-sheaf} shows that $\\mathcal{F}_p = \\mathcal{F}^\\#_p$, in other words, sheafification commutes with taking stalks at a point."}
+{"_id": "3999", "title": "formal-spaces-remark-compare-with-affine-formal-schemes", "text": "The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA (\\cite{EGA}). To explain this we assume our base scheme is $\\Spec(\\mathbf{Z})$. Let $\\mathfrak X = \\text{Spf}(A)$ be an affine formal scheme. Let $h_\\mathfrak X$ be its functor of points as in Lemma \\ref{lemma-fully-faithful}. Then $h_\\mathfrak X = \\colim h_{\\Spec(A/I)}$ where the colimit is over the collection of ideals of definition of the admissible topological ring $A$. This follows from (\\ref{equation-morphisms-affine-formal-schemes}) when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma \\ref{lemma-formal-scheme-sheaf-fppf}. Thus $h_\\mathfrak X$ is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition \\ref{definition-types-affine-formal-algebraic-space}. Thus Lemma \\ref{lemma-fully-faithful} tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces."}
+{"_id": "4000", "title": "formal-spaces-remark-compare-with-formal-schemes", "text": "Modulo set theoretic issues the category of formal schemes \\`a la EGA (see Section \\ref{section-formal-schemes-EGA}) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is $\\Spec(\\mathbf{Z})$. By Lemma \\ref{lemma-formal-scheme-sheaf-fppf} the functor of points $h_\\mathfrak X$ associated to a formal scheme $\\mathfrak X$ is a sheaf in the fppf topology. By Lemma \\ref{lemma-fully-faithful} the assignment $\\mathfrak X \\mapsto h_\\mathfrak X$ is a fully faithful embedding of the category of formal schemes into the category of fppf sheaves. Given a formal scheme $\\mathfrak X$ we choose an open covering $\\mathfrak X = \\bigcup \\mathfrak X_i$ with $\\mathfrak X_i$ affine formal schemes. Then $h_{\\mathfrak X_i}$ is an affine formal algebraic space by Remark \\ref{remark-compare-with-affine-formal-schemes}. The morphisms $h_{\\mathfrak X_i} \\to h_\\mathfrak X$ are representable and open immersions. Thus $\\{h_{\\mathfrak X_i} \\to h_\\mathfrak X\\}$ is a family as in Definition \\ref{definition-formal-algebraic-space} and we see that $h_\\mathfrak X$ is a formal algebraic space."}
+{"_id": "4001", "title": "formal-spaces-remark-set-theoretic", "text": "Let $S$ be a scheme and let $(\\Sch/S)_{fppf}$ be a big fppf site as in Topologies, Definition \\ref{topologies-definition-big-small-fppf}. As our set theoretic condition on $X$ in Definitions \\ref{definition-affine-formal-algebraic-space} and \\ref{definition-formal-algebraic-space} we take: there exist objects $U, R$ of $(\\Sch/S)_{fppf}$, a morphism $U \\to X$ which is a surjection of fppf sheaves, and a morphism $R \\to U \\times_X U$ which is a surjection of fppf sheaves. In other words, we require our sheaf to be a coequalizer of two maps between representable sheaves. Here are some observations which imply this notion behaves reasonably well: \\begin{enumerate} \\item Suppose $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$ and the system satisfies conditions (1) and (2) of Definition \\ref{definition-affine-formal-algebraic-space}. Then $U = \\coprod_{\\lambda \\in \\Lambda} X_\\lambda \\to X$ is a surjection of fppf sheaves. Moreover, $U \\times_X U$ is a closed subscheme of $U \\times_S U$ by Lemma \\ref{lemma-diagonal-affine-formal-algebraic-space}. Hence if $U$ is representable by an object of $(\\Sch/S)_{fppf}$ then $U \\times_S U$ is too (see Sets, Lemma \\ref{sets-lemma-what-is-in-it}) and the set theoretic condition is satisfied. This is always the case if $\\Lambda$ is countable, see Sets, Lemma \\ref{sets-lemma-what-is-in-it}. \\item Sanity check. Let $\\{X_i \\to X\\}_{i \\in I}$ be as in Definition \\ref{definition-formal-algebraic-space} (with the set theoretic condition as formulated above) and assume that each $X_i$ is actually an affine scheme. Then $X$ is an algebraic space. Namely, if we choose a larger big fppf site $(\\Sch'/S)_{fppf}$ such that $U' = \\coprod X_i$ and $R' = \\coprod X_i \\times_X X_j$ are representable by objects in it, then $X' = U'/R'$ will be an object of the category of algebraic spaces for this choice. Then an application of Spaces, Lemma \\ref{spaces-lemma-fully-faithful} shows that $X$ is an algebraic space for $(\\Sch/S)_{fppf}$. \\item Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition \\ref{definition-formal-algebraic-space}. For each $i$ we can pick $U_i \\in \\Ob((\\Sch/S)_{fppf})$ and $U_i \\to X_i$ which is a surjection of sheaves. Thus if $I$ is not too large (for example countable) then $U = \\coprod U_i \\to X$ is a surjection of sheaves and $U$ is representable by an object of $(\\Sch/S)_{fppf}$. To get $R \\in \\Ob((\\Sch/S)_{fppf})$ surjecting onto $U \\times_X U$ it suffices to assume the diagonal $\\Delta : X \\to X \\times_S X$ is not too wild, for example this always works if the diagonal of $X$ is quasi-compact, i.e., $X$ is quasi-separated. \\end{enumerate}"}
+{"_id": "4002", "title": "formal-spaces-remark-weak-ideals-of-definition", "text": "Let $\\mathfrak X$ be a formal scheme in the sense of McQuillan, see Remark \\ref{remark-mcquillan}. An {\\it weak ideal of definition} for $\\mathfrak X$ is an ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak X$ such that for all $\\mathfrak U \\subset \\mathfrak X$ affine formal open subscheme the ideal $\\mathcal{I}(\\mathfrak U) \\subset \\mathcal{O}_\\mathfrak X(\\mathfrak U)$ is a weak ideal of definition of the weakly admissible topological ring $\\mathcal{O}_\\mathfrak X(\\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. There is a one-to-one correspondence $$ \\{\\text{weak ideals of definition for }\\mathfrak X\\} \\leftrightarrow \\{\\text{thickenings }i : Z \\to h_\\mathfrak X\\text{ as above}\\} $$ This correspondence associates to $\\mathcal{I}$ the scheme $Z = (\\mathfrak X, \\mathcal{O}_\\mathfrak X/\\mathcal{I})$ together with the obvious morphism to $\\mathfrak X$. A {\\it fundamental system of weak ideals of definition} is a collection of weak ideals of definition $\\mathcal{I}_\\lambda$ such that on every affine open formal subscheme $\\mathfrak U \\subset \\mathfrak X$ the ideals $$ I_\\lambda = \\mathcal{I}_\\lambda(\\mathfrak U) \\subset A = \\Gamma(\\mathfrak U, \\mathcal{O}_\\mathfrak X) $$ form a fundamental system of weak ideals of definition of the weakly admissible topological ring $A$. It suffices to check on the members of an affine open covering. We conclude that the formal algebraic space $h_\\mathfrak X$ associated to the McQuillan formal scheme $\\mathfrak X$ is a colimit of schemes as in Lemma \\ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of weak ideals of definition for $\\mathfrak X$."}
+{"_id": "4003", "title": "formal-spaces-remark-ideals-of-definition", "text": "Let $\\mathfrak X$ be a formal scheme \\`a la EGA. An {\\it ideal of definition} for $\\mathfrak X$ is an ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak X$ such that for all $\\mathfrak U \\subset \\mathfrak X$ affine formal open subscheme the ideal $\\mathcal{I}(\\mathfrak U) \\subset \\mathcal{O}_\\mathfrak X(\\mathfrak U)$ is an ideal of definition of the admissible topological ring $\\mathcal{O}_\\mathfrak X(\\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. We do {\\bf not} get the same correspondence between ideals of definition and thickenings $Z \\to h_\\mathfrak X$ as in Remark \\ref{remark-weak-ideals-of-definition}; an example is given in Example \\ref{example-david-hansen}. A {\\it fundamental system of ideals of definition} is a collection of ideals of definition $\\mathcal{I}_\\lambda$ such that on every affine open formal subscheme $\\mathfrak U \\subset \\mathfrak X$ the ideals $$ I_\\lambda = \\mathcal{I}_\\lambda(\\mathfrak U) \\subset A = \\Gamma(\\mathfrak U, \\mathcal{O}_\\mathfrak X) $$ form a fundamental system of ideals of definition of the admissible topological ring $A$. It suffices to check on the members of an affine open covering. Suppose that $\\mathfrak X$ is quasi-compact and that $\\{\\mathcal{I}_\\lambda\\}_{\\lambda \\in \\Lambda}$ is a fundamental system of weak ideals of definition. If $A$ is an admissible topological ring then all sufficiently small open ideals are ideals of definition (namely any open ideal contained in an ideal of definition is an ideal of definition). Thus since we only need to check on the finitely many members of an affine open covering we see that $\\mathcal{I}_\\lambda$ is an ideal of definition for $\\lambda$ sufficiently large. Using the discussion in Remark \\ref{remark-weak-ideals-of-definition} we conclude that the formal algebraic space $h_\\mathfrak X$ associated to the quasi-compact formal scheme $\\mathfrak X$ \\`a la EGA is a colimit of schemes as in Lemma \\ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of ideals of definition for $\\mathfrak X$."}
+{"_id": "4004", "title": "formal-spaces-remark-structure-quasi-compact-quasi-separated", "text": "In this remark we translate the statement and proof of Lemma \\ref{lemma-structure-quasi-compact-quasi-separated} into the language of formal schemes \\`a la EGA. Looking at Remark \\ref{remark-ideals-of-definition} we see that the lemma can be translated as follows \\begin{itemize} \\item[(*)] Every quasi-compact and quasi-separated formal scheme has a fundamental system of ideals of definition. \\end{itemize} To prove this we first use the induction principle (reformulated for quasi-compact and quasi-separated formal schemes) of Cohomology of Schemes, Lemma \\ref{coherent-lemma-induction-principle} to reduce to the following situation: $\\mathfrak X = \\mathfrak U \\cup \\mathfrak V$ with $\\mathfrak U$, $\\mathfrak V$ open formal subschemes, with $\\mathfrak V$ affine, and the result is true for $\\mathfrak U$, $\\mathfrak V$, and $\\mathfrak U \\cap \\mathfrak V$. Pick any ideals of definition $\\mathcal{I} \\subset \\mathcal{O}_\\mathfrak U$ and $\\mathcal{J} \\subset \\mathcal{O}_\\mathfrak V$. By our assumption that we have a fundamental system of ideals of definition on $\\mathfrak U$ and $\\mathfrak V$ and because $\\mathfrak U \\cap \\mathfrak V$ is quasi-compact, we can find ideals of definition $\\mathcal{I}' \\subset \\mathcal{I}$ and $\\mathcal{J}' \\subset \\mathcal{J}$ such that $$ \\mathcal{I}'|_{\\mathfrak U \\cap \\mathfrak V} \\subset \\mathcal{J}|_{\\mathfrak U \\cap \\mathfrak V} \\quad\\text{and}\\quad \\mathcal{J}'|_{\\mathfrak U \\cap \\mathfrak V} \\subset \\mathcal{I}|_{\\mathfrak U \\cap \\mathfrak V} $$ Let $U \\to U' \\to \\mathfrak U$ and $V \\to V' \\to \\mathfrak V$ be the closed immersions determined by the ideals of definition $\\mathcal{I}' \\subset \\mathcal{I} \\subset \\mathcal{O}_\\mathfrak U$ and $\\mathcal{J}' \\subset \\mathcal{J} \\subset \\mathcal{O}_\\mathfrak V$. Let $\\mathfrak U \\cap V$ denote the open subscheme of $V$ whose underlying topological space is that of $\\mathfrak U \\cap \\mathfrak V$. By our choice of $\\mathcal{I}'$ there is a factorization $\\mathfrak U \\cap V \\to U'$. We define similarly $U \\cap \\mathfrak V$ which factors through $V'$. Then we consider $$ Z_U = \\text{scheme theoretic image of } U \\amalg (\\mathfrak U \\cap V) \\longrightarrow U' $$ and $$ Z_V = \\text{scheme theoretic image of } (U \\cap \\mathfrak V) \\amalg V \\longrightarrow V' $$ Since taking scheme theoretic images of quasi-compact morphisms commutes with restriction to opens (Morphisms, Lemma \\ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}) we see that $Z_U \\cap \\mathfrak V = \\mathfrak U \\cap Z_V$. Thus $Z_U$ and $Z_V$ glue to a scheme $Z$ which comes equipped with a morphism $Z \\to \\mathfrak X$. Analogous to the discussion in Remark \\ref{remark-weak-ideals-of-definition} we see that $Z$ corresponds to a weak ideal of definition $\\mathcal{I}_Z \\subset \\mathcal{O}_\\mathfrak X$. Note that $Z_U \\subset U'$ and that $Z_V \\subset V'$. Thus the collection of all $\\mathcal{I}_Z$ constructed in this manner forms a fundamental system of weak ideals of definition. Hence a subfamily gives a fundamental system of ideals of definition, see Remark \\ref{remark-ideals-of-definition}."}
+{"_id": "4005", "title": "formal-spaces-remark-warning", "text": "Lemma \\ref{lemma-representable-affine} is sharp in the following two senses: \\begin{enumerate} \\item If $A$ and $B$ are weakly admissible rings and $\\varphi : A \\to B$ is a continuous map, then $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$ is in general not representable. \\item If $f : Y \\to X$ is a representable morphism of affine formal algebraic spaces and $X = \\text{Spf}(A)$ is McQuillan, then it does not follow that $Y$ is McQuillan. \\end{enumerate} An example for (1) is to take $A = k$ a field (with discrete topology) and $B = k[[t]]$ with the $t$-adic topology. An example for (2) is given in Examples, Section \\ref{examples-section-affine-formal-algebraic-space}."}
+{"_id": "4006", "title": "formal-spaces-remark-warning-completion", "text": "Suppose $X = \\Spec(A)$ and $T \\subset X$ is the zero locus of a finitely generated ideal $I \\subset A$. Let $J = \\sqrt{I}$ be the radical of $I$. Then from the definitions we see that $X_{/T} = \\text{Spf}(A^\\wedge)$ where $A^\\wedge = \\lim A/I^n$ is the $I$-adic completion of $A$. On the other hand, the map $A^\\wedge \\to \\lim A/J^n$ from the $I$-adic completion to the $J$-adic completion can fail to be a ring isomorphisms. As an example let $$ A = \\bigcup\\nolimits_{n \\geq 1} \\mathbf{C}[t^{1/n}] $$ and $I = (t)$. Then $J = \\mathfrak m$ is the maximal ideal of the valuation ring $A$ and $J^2 = J$. Thus the $J$-adic completion of $A$ is $\\mathbf{C}$ whereas the $I$-adic completion is the valuation ring described in Example \\ref{example-david-hansen} (but in particular it is easy to see that $A \\subset A^\\wedge$)."}
+{"_id": "4007", "title": "formal-spaces-remark-variant-adic-star", "text": "Let $P$ be a property of morphisms of $\\textit{WAdm}^{adic*}$. We say $P$ is a {\\it local property} if axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3}) of Situation \\ref{situation-local-property} hold for morphisms of $\\textit{WAdm}^{adic*}$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-property-defines-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$."}
+{"_id": "4008", "title": "formal-spaces-remark-variant-Noetherian", "text": "Let $P$ be a property of morphisms of $\\textit{WAdm}^{Noeth}$. We say $P$ is a {\\it local property} if axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3}), of Situation \\ref{situation-local-property} hold for morphisms of $\\textit{WAdm}^{Noeth}$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-property-defines-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$."}
+{"_id": "4009", "title": "formal-spaces-remark-base-change-variant-adic-star", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see Remark \\ref{remark-variant-adic-star}. We say $P$ is {\\it stable under base change} if given $B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{adic*}$ we have $P(B \\to A) \\Rightarrow P(C \\to A \\widehat{\\otimes}_B C)$. This makes sense as $A \\widehat{\\otimes}_B C$ is an object of $\\textit{WAdm}^{adic*}$ by Lemma \\ref{lemma-completed-tensor-product}. In exactly the same way we obtain a variant of Lemma \\ref{lemma-base-change-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$."}
+{"_id": "4010", "title": "formal-spaces-remark-base-change-variant-Noetherian", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see Remark \\ref{remark-variant-Noetherian}. We say $P$ is {\\it stable under base change} if given $B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$ the property $P(B \\to A)$ implies both that $A \\widehat{\\otimes}_B C$ is adic Noetherian\\footnote{See Lemma \\ref{lemma-completed-tensor-product} for a criterion.} and that $P(C \\to A \\widehat{\\otimes}_B C)$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-base-change-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$."}
+{"_id": "4011", "title": "formal-spaces-remark-base-change-variant-variant-Noetherian", "text": "Let $P$ and $Q$ be local properties of morphisms of $\\textit{WAdm}^{Noeth}$, see Remark \\ref{remark-variant-Noetherian}. We say {\\it $P$ is stable under base change by $Q$} if given $B \\to A$ and $B \\to C$ in $\\textit{WAdm}^{Noeth}$ satisfying $P(B \\to A)$ and $Q(B \\to C)$, then $A \\widehat{\\otimes}_B C$ is adic Noetherian and $P(C \\to A \\widehat{\\otimes}_B C)$ holds. Arguing exactly as in the proof of Lemma \\ref{lemma-base-change-property-morphisms} we obtain the following statement: given morphisms $f : X \\to Y$ and $g : Y \\to Z$ of locally Noetherian formal algebraic spaces over $S$ such that \\begin{enumerate} \\item the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} hold for $f$ and $P$, \\item the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} hold for $g$ and $Q$, \\end{enumerate} then the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} hold for $\\text{pr}_2 : X \\times_Y Z \\to Z$ and $P$."}
+{"_id": "4012", "title": "formal-spaces-remark-composition-variant-adic-star", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see Remark \\ref{remark-variant-adic-star}. We say $P$ is {\\it stable under composition} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{adic*}$ we have $P(B \\to A) \\wedge P(C \\to B) \\Rightarrow P(C \\to A)$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-composition-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$."}
+{"_id": "4013", "title": "formal-spaces-remark-composition-variant-Noetherian", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see Remark \\ref{remark-variant-Noetherian}. We say $P$ is {\\it stable under composition} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{Noeth}$ we have $P(B \\to A) \\wedge P(C \\to B) \\Rightarrow P(C \\to A)$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-composition-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$."}
+{"_id": "4014", "title": "formal-spaces-remark-permanence-variant-adic-star", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{adic*}$, see Remark \\ref{remark-variant-adic-star}. We say $P$ {\\it has the cancellation property} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{adic*}$ we have $P(C \\to A) \\wedge P(C \\to B) \\Rightarrow P(B \\to A)$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-composition-property-morphisms} for morphisms between locally adic* formal algebraic spaces over $S$."}
+{"_id": "4015", "title": "formal-spaces-remark-permanence-variant-Noetherian", "text": "Let $P$ be a local property of morphisms of $\\textit{WAdm}^{Noeth}$, see Remark \\ref{remark-variant-Noetherian}. We say $P$ {\\it has the cancellation property} if given $B \\to A$ and $C \\to B$ in $\\textit{WAdm}^{Noeth}$ we have $P(C \\to B) \\wedge P(C \\to A) \\Rightarrow P(C \\to B)$. In exactly the same way we obtain a variant of Lemma \\ref{lemma-composition-property-morphisms} for morphisms between locally Noetherian formal algebraic spaces over $S$."}
+{"_id": "4016", "title": "formal-spaces-remark-universal-property", "text": "\\begin{reference} \\cite[Chapter 0, 7.5.3]{EGA} \\end{reference} Let $A \\to C$ be a continuous map of complete linearly topologized rings. Then any $A$-algebra map $A[x_1, \\ldots x_r] \\to C$ extends uniquely to a continuous map $A\\{x_1, \\ldots, x_r\\} \\to C$ on restricted power series."}
+{"_id": "4017", "title": "formal-spaces-remark-I-adic-completion-and-restricted-power-series", "text": "Let $A$ be a ring and let $I \\subset A$ be an ideal. If $A$ is $I$-adically complete, then the $I$-adic completion $A[x_1, \\ldots, x_r]^\\wedge$ of $A[x_1, \\ldots, x_r]$ is the restricted power series ring over $A$ as a ring. However, it is not clear that $A[x_1, \\ldots, x_r]^\\wedge$ is $I$-adically complete. We think of the topology on $A\\{x_1, \\ldots, x_r\\}$ as the limit topology (which is always complete) whereas we often think of the topology on $A[x_1, \\ldots, x_r]^\\wedge$ as the $I$-adic topology (not always complete). If $I$ is finitely generated, then $A\\{x_1, \\ldots, x_r\\} = A[x_1, \\ldots, x_r]^\\wedge$ as topological rings, see Algebra, Lemma \\ref{algebra-lemma-hathat-finitely-generated}."}
+{"_id": "4018", "title": "formal-spaces-remark-questions", "text": "Let $A$ be a weakly admissible topological ring and let $(I_\\lambda)$ be a fundamental system of weak ideals of definition. Let $X = \\text{Spf}(A)$, in other words, $X$ is a McQuillan affine formal algebraic space. Let $f : Y \\to X$ be a morphism of affine formal algebraic spaces. In general it will not be true that $Y$ is McQuillan. More specifically, we can ask the following questions: \\begin{enumerate} \\item Assume that $f : Y \\to X$ is a closed immersion. Then $Y$ is McQuillan and $f$ corresponds to a continuous map $\\varphi : A \\to B$ of weakly admissible topological rings which is taut, whose kernel $K \\subset A$ is a closed ideal, and whose image $\\varphi(A)$ is dense in $B$, see Lemma \\ref{lemma-closed-immersion-into-McQuillan}. What conditions on $A$ guarantee that $B = (A/K)^\\wedge$ as in Example \\ref{example-closed-immersion-from-quotient}? \\item What conditions on $A$ guarantee that closed immersions $f : Y \\to X$ correspond to quotients $A/K$ of $A$ by closed ideals, in other words, the corresponding continuous map $\\varphi$ is surjective and open? \\item Suppose that $f : Y \\to X$ is of finite type. Then we get $Y = \\colim \\Spec(B_\\lambda)$ where $(B_\\lambda)$ is an object of $\\mathcal{C}$ by Lemma \\ref{lemma-category-affine-over}. In this case it is true that there exists a fixed integer $r$ such that $B_\\lambda$ is generated by $r$ elements over $A/I_\\lambda$ for all $\\lambda$ (the argument is essentially already given in the proof of (1) $\\Rightarrow$ (2) in Lemma \\ref{lemma-topologically-finite-type-finite-type}). However, it is not clear that the projections $\\lim B_\\lambda \\to B_\\lambda$ are surjective, i.e., it is not clear that $Y$ is McQuillan. Is there an example where $Y$ is not McQuillan? \\item Suppose that $f : Y \\to X$ is of finite type and $Y$ is McQuillan. Then $f$ corresponds to a continuous map $\\varphi : A \\to B$ of weakly admissible topological rings. In fact $\\varphi$ is taut and $B$ is topologically of finite type over $A$, see Lemma \\ref{lemma-topologically-finite-type-finite-type}. In other words, $f$ factors as $$ Y \\longrightarrow \\mathbf{A}^r_X \\longrightarrow X $$ where the first arrow is a closed immersion of McQuillan affine formal algebraic spaces. However, then questions (1) and (2) are in force for $Y \\to \\mathbf{A}^r_X$. \\end{enumerate} Below we will answer these questions when $X$ is countably indexed, i.e., when $A$ has a countable fundamental system of open ideals. If you have answers to these questions in greater generality, or if you have counter examples, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "4019", "title": "formal-spaces-remark-not-enough-sections", "text": "The structure sheaf does not always have ``enough sections''. In Examples, Section \\ref{examples-section-affine-formal-algebraic-space} we have seen that there exist affine formal algebraic spaces which aren't McQuillan and there are even examples whose points are not separated by regular functions."}
+{"_id": "4020", "title": "formal-spaces-remark-bad-quasi-coherent", "text": "Even if the structure sheaf has good properties, this does not mean there is a good theory of quasi-coherent modules. For example, in Examples, Section \\ref{examples-section-nonabelian-QCoh} we have seen that for almost any Noetherian affine formal algebraic spaces the most natural notion of a quasi-coherent module leads to a category of modules which is not abelian."}
+{"_id": "4144", "title": "pione-remark-covering-surjective", "text": "Under the correspondence of Lemma \\ref{lemma-sheaves-point}, the coverings in the small \\'etale site $\\Spec(K)_\\etale$ of $K$ correspond to surjective families of maps in $G\\textit{-Sets}$."}
+{"_id": "4145", "title": "pione-remark-colimits-commute-forgetful", "text": "Let $X$ be a scheme. Consider the natural functors $F_1 : \\textit{F\\'Et}_X \\to \\Sch$ and $F_2 : \\textit{F\\'Et}_X \\to \\Sch/X$. Then \\begin{enumerate} \\item The functors $F_1$ and $F_2$ commute with finite colimits. \\item The functor $F_2$ commutes with finite limits, \\item The functor $F_1$ commutes with connected finite limits, i.e., with equalizers and fibre products. \\end{enumerate} The results on limits are immediate from the discussion in the proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits} and Categories, Lemma \\ref{categories-lemma-connected-limit-over-X}. It is clear that $F_1$ and $F_2$ commute with finite coproducts. By the dual of Categories, Lemma \\ref{categories-lemma-characterize-left-exact} we need to show that $F_1$ and $F_2$ commute with coequalizers. In the proof of Lemma \\ref{lemma-finite-etale-covers-limits-colimits} we saw that coequalizers in $\\textit{F\\'Et}_X$ look \\'etale locally like this $$ \\xymatrix{ \\coprod_{j \\in J} U \\ar@<1ex>[r]^a \\ar@<-1ex>[r]_b & \\coprod_{i \\in I} U \\ar[r] & \\coprod_{t \\in \\text{Coeq}(a, b)} U } $$ which is certainly a coequalizer in the category of schemes. Hence the statement follows from the fact that being a coequalizer is fpqc local as formulate precisely in Descent, Lemma \\ref{descent-lemma-coequalizer-fpqc-local}."}
+{"_id": "4146", "title": "pione-remark-variance", "text": "In the situation of Lemma \\ref{lemma-fundamental-group-Galois-group} let us give a more explicit construction of the isomorphism $\\text{Gal}(K^{sep}/K) \\to \\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})$. Observe that $\\text{Gal}(K^{sep}/K) = \\text{Aut}(\\overline{K}/K)$ as $\\overline{K}$ is the perfection of $K^{sep}$. Since $F_{\\overline{x}}(Y) = \\Mor_X(\\Spec(\\overline{K}), Y)$ we may consider the map $$ \\text{Aut}(\\overline{K}/K) \\times F_{\\overline{x}}(Y) \\to F_{\\overline{x}}(Y), \\quad (\\sigma, \\overline{y}) \\mapsto \\sigma \\cdot \\overline{y} = \\overline{y} \\circ \\Spec(\\sigma) $$ This is an action because $$ \\sigma\\tau \\cdot \\overline{y} = \\overline{y} \\circ \\Spec(\\sigma\\tau) = \\overline{y} \\circ \\Spec(\\tau) \\circ \\Spec(\\sigma) = \\sigma \\cdot (\\tau \\cdot \\overline{y}) $$ The action is functorial in $Y \\in \\textit{F\\'Et}_X$ and we obtain the desired map."}
+{"_id": "4147", "title": "pione-remark-combine", "text": "Let $(A, \\mathfrak m)$ be a complete local Noetherian ring and $f \\in \\mathfrak m$ nonzero. Suppose that $A_f$ is $(S_2)$ and every irreducible component of $\\Spec(A)$ has dimension $\\geq 4$. Then Lemma \\ref{lemma-essentially-surjective-general-better} tells us that the category $$ \\colim\\nolimits_{U' \\subset U\\text{ open, }U_0 \\subset U} \\text{ category of schemes finite \\'etale over }U' $$ is equivalent to the category of schemes finite \\'etale over $U_0$. For example this holds if $A$ is a normal domain of dimension $\\geq 4$!"}
+{"_id": "4180", "title": "stacks-cohomology-remark-bousfield-colocalization", "text": "Let $\\mathcal{X}$ be an algebraic stack. The results of Lemmas \\ref{lemma-adjoint} and \\ref{lemma-adjoint-kernel-parasitic} imply that $$ \\QCoh(\\mathcal{O}_\\mathcal{X}) = \\mathcal{M}_\\mathcal{X} / \\text{Parasitic} \\cap \\mathcal{M}_\\mathcal{X} $$ in words: the category of quasi-coherent modules is the category of locally quasi-coherent modules with the flat base change property divided out by the Serre subcategory consisting of parasitic objects. See Homology, Lemma \\ref{homology-lemma-serre-subcategory-is-kernel}. The existence of the inclusion functor $i : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\mathcal{M}_\\mathcal{X}$ which is left adjoint to the quotient functor means that $\\mathcal{M}_\\mathcal{X} \\to \\QCoh(\\mathcal{O}_\\mathcal{X})$ is a {\\it Bousfield colocalization} or a {\\it right Bousfield localization} (insert future reference here). Our next goal is to show a similar result holds on the level of derived categories."}
+{"_id": "4423", "title": "sites-cohomology-remark-before-Leray", "text": "As a consequence of the results above we find that Derived Categories, Lemma \\ref{derived-lemma-compose-derived-functors} applies to a number of situations. For example, given a morphism $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ of ringed topoi we have $$ R\\Gamma(\\mathcal{D}, Rf_*\\mathcal{F}) = R\\Gamma(\\mathcal{C}, \\mathcal{F}) $$ for any sheaf of $\\mathcal{O}_\\mathcal{C}$-modules $\\mathcal{F}$. Namely, for an injective $\\mathcal{O}_\\mathcal{X}$-module $\\mathcal{I}$ the $\\mathcal{O}_\\mathcal{D}$-module $f_*\\mathcal{I}$ is totally acyclic by Lemma \\ref{lemma-direct-image-injective-sheaf} and a totally acyclic sheaf is acyclic for $\\Gamma(\\mathcal{D}, -)$ by Lemma \\ref{lemma-limp-acyclic}."}
+{"_id": "4424", "title": "sites-cohomology-remark-base-change", "text": "The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{g'} \\ar[d]_{f'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^g & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$. Then there exists a canonical base change map $$ Lg^*Rf_*K \\longrightarrow R(f')_*L(g')^*K $$ in $D(\\mathcal{O}_{\\mathcal{D}'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \\to L(g')^*K$. Since $L(f')^* \\circ Lg^* = L(g')^* \\circ Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \\to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \\to K$."}
+{"_id": "4425", "title": "sites-cohomology-remark-compose-base-change", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{B}'), \\mathcal{O}_{\\mathcal{B}'}) \\ar[r]_k \\ar[d]_{f'} & (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]^l \\ar[d]_{g'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^g \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^m & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\\\ } $$ of ringed topoi. Then the base change maps of Remark \\ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} Lm^* \\circ R(g \\circ f)_* & = Lm^* \\circ Rg_* \\circ Rf_* \\\\ & \\to Rg'_* \\circ Ll^* \\circ Rf_* \\\\ & \\to Rg'_* \\circ Rf'_* \\circ Lk^* \\\\ & = R(g' \\circ f')_* \\circ Lk^* \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "4426", "title": "sites-cohomology-remark-compose-base-change-horizontal", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}''), \\mathcal{O}_{\\mathcal{C}''}) \\ar[r]_{g'} \\ar[d]_{f''} & (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_g \\ar[d]_{f'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{D}''), \\mathcal{O}_{\\mathcal{D}''}) \\ar[r]^{h'} & (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^h & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ of ringed topoi. Then the base change maps of Remark \\ref{remark-base-change} for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} L(h \\circ h')^* \\circ Rf_* & = L(h')^* \\circ Lh^* \\circ Rf_* \\\\ & \\to L(h')^* \\circ Rf'_* \\circ Lg^* \\\\ & \\to Rf''_* \\circ L(g')^* \\circ Lg^* \\\\ & = Rf''_* \\circ L(g \\circ g')^* \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "4427", "title": "sites-cohomology-remark-cup-product", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product $$ Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L \\longrightarrow Rf_*(K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L) $$ in $D(\\mathcal{O}_\\mathcal{D})$ for all $K, L$ in $D(\\mathcal{O}_\\mathcal{C})$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L) \\to K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L) = Lf^*Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} Lf^*Rf_*L$ (Lemma \\ref{lemma-pullback-tensor-product}) with the map $Lf^*Rf_*K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} Lf^*Rf_*L \\to K \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L$ coming from the counit $Lf^* \\circ Rf_* \\to \\text{id}$."}
+{"_id": "4428", "title": "sites-cohomology-remark-discuss-derived-limit", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system in $D(\\mathcal{O})$. Set $K = R\\lim K_n$. For each $n$ and $m$ let $\\mathcal{H}^m_n = H^m(K_n)$ be the $m$th cohomology sheaf of $K_n$ and similarly set $\\mathcal{H}^m = H^m(K)$. Let us denote $\\underline{\\mathcal{H}}^m_n$ the presheaf $$ U \\longmapsto \\underline{\\mathcal{H}}^m_n(U) = H^m(U, K_n) $$ Similarly we set $\\underline{\\mathcal{H}}^m(U) = H^m(U, K)$. By Lemma \\ref{lemma-sheafification-cohomology} we see that $\\mathcal{H}^m_n$ is the sheafification of $\\underline{\\mathcal{H}}^m_n$ and $\\mathcal{H}^m$ is the sheafification of $\\underline{\\mathcal{H}}^m$. Here is a diagram $$ \\xymatrix{ K \\ar@{=}[d] & \\underline{\\mathcal{H}}^m \\ar[d] \\ar[r] &  \\mathcal{H}^m \\ar[d] \\\\ R\\lim K_n & \\lim \\underline{\\mathcal{H}}^m_n \\ar[r] &  \\lim \\mathcal{H}^m_n } $$ In general it may not be the case that $\\lim \\mathcal{H}^m_n$ is the sheafification of $\\lim \\underline{\\mathcal{H}}^m_n$. If $U \\in \\mathcal{C}$, then we have short exact sequences \\begin{equation} \\label{equation-ses-Rlim-over-U} 0 \\to R^1\\lim \\underline{\\mathcal{H}}^{m - 1}_n(U) \\to \\underline{\\mathcal{H}}^m(U) \\to \\lim \\underline{\\mathcal{H}}^m_n(U) \\to 0 \\end{equation} by Lemma \\ref{lemma-RGamma-commutes-with-Rlim}."}
+{"_id": "4429", "title": "sites-cohomology-remark-set-theoretic-LC", "text": "The category $\\textit{LC}$ is a ``big'' category as its objects form a proper class. Similarly, the coverings form a proper class. Let us define the {\\it size} of a topological space $X$ to be the cardinality of the set of points of $X$. Choose a function $Bound$ on cardinals, for example as in Sets, Equation (\\ref{sets-equation-bound}). Finally, let $S_0$ be an initial set of objects of $\\textit{LC}$, for example $S_0 = \\{(\\mathbf{R}, \\text{euclidean topology})\\}$. Exactly as in Sets, Lemma \\ref{sets-lemma-construct-category} we can choose a limit ordinal $\\alpha$ such that $\\textit{LC}_\\alpha = \\textit{LC} \\cap V_\\alpha$ contains $S_0$ and is preserved under all countable limits and colimits which exist in $\\textit{LC}$. Moreover, if $X \\in \\textit{LC}_\\alpha$ and if $Y \\in \\textit{LC}$ and $\\text{size}(Y) \\leq Bound(\\text{size}(X))$, then $Y$ is isomorphic to an object of $\\textit{LC}_\\alpha$. Next, we apply Sets, Lemma \\ref{sets-lemma-coverings-site} to choose set $\\text{Cov}$ of qc covering on $\\textit{LC}_\\alpha$ such that every qc covering in $\\textit{LC}_\\alpha$ is combinatorially equivalent to a covering this set. In this way we obtain a site $(\\textit{LC}_\\alpha, \\text{Cov})$ which we will denote $\\textit{LC}_{qc}$."}
+{"_id": "4430", "title": "sites-cohomology-remark-projection-formula-for-internal-hom", "text": "Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\\mathcal{O}_\\mathcal{C})$. We claim there is a canonical map $$ Rf_*R\\SheafHom(L, K) \\longrightarrow R\\SheafHom(Rf_*L, Rf_*K) $$ Namely, by (\\ref{equation-internal-hom}) this is the same thing as a map $Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L \\to Rf_*K$. For this we can use the composition $$ Rf_*R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*L \\to Rf_*(R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L) \\to Rf_*K $$ where the first arrow is the relative cup product (Remark \\ref{remark-cup-product}) and the second arrow is $Rf_*$ applied to the canonical map $R\\SheafHom(L, K) \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} L \\to K$ coming from Lemma \\ref{lemma-internal-hom-composition} (with $\\mathcal{O}_\\mathcal{C}$ in one of the spots)."}
+{"_id": "4431", "title": "sites-cohomology-remark-prepare-fancy-base-change", "text": "Let $h : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\\mathcal{O}')$. We claim there is a canonical map $$ Lh^*R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(Lh^*K, Lh^*L) $$ in $D(\\mathcal{O})$. Namely, by (\\ref{equation-internal-hom}) proved in Lemma \\ref{lemma-internal-hom} such a map is the same thing as a map $$ Lh^*R\\SheafHom(K, L) \\otimes^\\mathbf{L} Lh^*K \\longrightarrow Lh^*L $$ The source of this arrow is $Lh^*(\\SheafHom(K, L) \\otimes^\\mathbf{L} K)$ by Lemma \\ref{lemma-pullback-tensor-product} hence it suffices to construct a canonical map $$ R\\SheafHom(K, L) \\otimes^\\mathbf{L} K \\longrightarrow L. $$ For this we take the arrow corresponding to $$ \\text{id} : R\\SheafHom(K, L) \\longrightarrow R\\SheafHom(K, L) $$ via (\\ref{equation-internal-hom})."}
+{"_id": "4432", "title": "sites-cohomology-remark-fancy-base-change", "text": "Suppose that $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_h \\ar[d]_{f'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^g & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $K, L$ be objects of $D(\\mathcal{O}_\\mathcal{C})$. We claim there exists a canonical base change map $$ Lg^*Rf_*R\\SheafHom(K, L) \\longrightarrow R(f')_*R\\SheafHom(Lh^*K, Lh^*L) $$ in $D(\\mathcal{O}_{\\mathcal{D}'})$. Namely, we take the map adjoint to the composition \\begin{align*} L(f')^*Lg^*Rf_*R\\SheafHom(K, L) & = Lh^*Lf^*Rf_*R\\SheafHom(K, L) \\\\ & \\to Lh^*R\\SheafHom(K, L) \\\\ & \\to R\\SheafHom(Lh^*K, Lh^*L) \\end{align*} where the first arrow uses the adjunction mapping $Lf^*Rf_* \\to \\text{id}$ and the second arrow is the canonical map constructed in Remark \\ref{remark-prepare-fancy-base-change}."}
+{"_id": "4433", "title": "sites-cohomology-remark-when-derived-shriek-equal", "text": "Warning! Let $u : \\mathcal{C} \\to \\mathcal{D}$, $g$, $\\mathcal{O}_\\mathcal{D}$, and $\\mathcal{O}_\\mathcal{C}$ be as in Lemma \\ref{lemma-existence-derived-lower-shriek}. In general it is {\\bf not} the case that the diagram $$ \\xymatrix{ D(\\mathcal{O}_\\mathcal{C}) \\ar[r]_{Lg_!} \\ar[d]_{forget} & D(\\mathcal{O}_\\mathcal{D}) \\ar[d]^{forget} \\\\ D(\\mathcal{C}) \\ar[r]^{Lg^{Ab}_!} & D(\\mathcal{D}) } $$ commutes where the functor $Lg_!^{Ab}$ is the one constructed in Lemma \\ref{lemma-existence-derived-lower-shriek} but using the constant sheaf $\\mathbf{Z}$ as the structure sheaf on both $\\mathcal{C}$ and $\\mathcal{D}$. In general it isn't even the case that $g_! = g_!^{Ab}$ (see Modules on Sites, Remark \\ref{sites-modules-remark-when-shriek-equal}), but this phenomenon {\\bf can occur even if $g_! = g_!^{Ab}$}! Namely, the construction of $Lg_!$ in the proof of Lemma \\ref{lemma-existence-derived-lower-shriek} shows that $Lg_!$ agrees with $Lg_!^{\\textit{Ab}}$ if and only if the canonical maps $$ Lg^{Ab}_!j_{U!}\\mathcal{O}_U \\longrightarrow j_{u(U)!}\\mathcal{O}_{u(U)} $$ are isomorphisms in $D(\\mathcal{D})$ for all objects $U$ in $\\mathcal{C}$. In general all we can say is that there exists a natural transformation $$ Lg_!^{Ab} \\circ forget \\longrightarrow forget \\circ Lg_! $$"}
+{"_id": "4434", "title": "sites-cohomology-remark-fibred-category", "text": "Assumptions and notation as in Situation \\ref{situation-fibred-category}. Note that setting $\\mathcal{C}' = \\mathcal{D}$ and $u$ equal to the structure functor of $\\mathcal{C}$ gives a situation as in Situation \\ref{situation-morphism-fibred-categories}. Hence Lemma \\ref{lemma-properties-lower-shriek-fibred-category} tells us we have functors  $\\pi_!$, $\\pi_!^{\\textit{Ab}}$, $L\\pi_!$, and $L\\pi_!^{\\textit{Ab}}$ such that $forget \\circ \\pi_! = \\pi_!^{\\textit{Ab}} \\circ forget$ and $forget \\circ L\\pi_! = L\\pi_!^{\\textit{Ab}} \\circ forget$."}
+{"_id": "4435", "title": "sites-cohomology-remark-morphism-fibred-categories", "text": "Assumptions and notation as in Situation \\ref{situation-morphism-fibred-categories}. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$, let $\\mathcal{F}'$ be an abelian sheaf on $\\mathcal{C}'$, and let $t : \\mathcal{F}' \\to g^{-1}\\mathcal{F}$ be a map. Then we obtain a canonical map $$ L\\pi'_!(\\mathcal{F}') \\longrightarrow L\\pi_!(\\mathcal{F}) $$ by using the adjoint $g_!\\mathcal{F}' \\to \\mathcal{F}$ of $t$, the map $Lg_!(\\mathcal{F}') \\to g_!\\mathcal{F}'$, and the equality $L\\pi'_! = L\\pi_! \\circ Lg_!$."}
+{"_id": "4436", "title": "sites-cohomology-remark-map-evaluation-to-derived", "text": "Notation and assumptions as in Example \\ref{example-category-to-point}. Let $\\mathcal{F}^\\bullet$ be a bounded complex of abelian sheaves on $\\mathcal{C}$. For any object $U$ of $\\mathcal{C}$ there is a canonical map $$ \\mathcal{F}^\\bullet(U) \\longrightarrow L\\pi_!(\\mathcal{F}^\\bullet) $$ in $D(\\textit{Ab})$. If $\\mathcal{F}^\\bullet$ is a complex of $\\underline{B}$-modules then this map is in $D(B)$. To prove this, note that we compute $L\\pi_!(\\mathcal{F}^\\bullet)$ by taking a quasi-isomorphism $\\mathcal{P}^\\bullet \\to \\mathcal{F}^\\bullet$ where $\\mathcal{P}^\\bullet$ is a complex of projectives. However, since the topology is chaotic this means that $\\mathcal{P}^\\bullet(U) \\to \\mathcal{F}^\\bullet(U)$ is a quasi-isomorphism hence can be inverted in $D(\\textit{Ab})$, resp.\\ $D(B)$. Composing with the canonical map $\\mathcal{P}^\\bullet(U) \\to \\pi_!(\\mathcal{P}^\\bullet)$ coming from the computation of $\\pi_!$ as a colimit we obtain the desired arrow."}
+{"_id": "4437", "title": "sites-cohomology-remark-simplicial-modules", "text": "Let $\\mathcal{C} = \\Delta$ and let $B$ be any ring. This is a special case of Example \\ref{example-category-to-point} where the assumptions of Lemma \\ref{lemma-compute-by-cosimplicial-resolution} hold. Namely, let $U_\\bullet$ be the cosimplicial object of $\\Delta$ given by the identity functor. To verify the condition we have to show that for $[m] \\in \\Ob(\\Delta)$ the simplicial set $\\Delta[m] : n \\mapsto \\Mor_\\Delta([n], [m])$ is homotopy equivalent to a point. This is explained in Simplicial, Example \\ref{simplicial-example-simplex-contractible}. \\medskip\\noindent In this situation the category $\\textit{Mod}(\\underline{B})$ is just the category of simplicial $B$-modules and the functor $L\\pi_!$ sends a simplicial $B$-module $M_\\bullet$ to its associated complex $s(M_\\bullet)$ of $B$-modules. Thus the results above can be reinterpreted in terms of results on simplicial modules. For example a special case of Lemma \\ref{lemma-eilenberg-zilber} is: if $M_\\bullet$, $M'_\\bullet$ are flat simplicial $B$-modules, then the complex $s(M_\\bullet \\otimes_B M'_\\bullet)$ is quasi-isomorphic to the total complex associated to the double complex $s(M_\\bullet) \\otimes_B s(M'_\\bullet)$. (Hint: use flatness to convert from derived tensor products to usual tensor products.) This is a special case of the Eilenberg-Zilber theorem which can be found in \\cite{Eilenberg-Zilber}."}
+{"_id": "4438", "title": "sites-cohomology-remark-O-homology-B-homology-general", "text": "Let $\\mathcal{C}$ and $B$ be as in Example \\ref{example-category-to-point}. Assume there exists a cosimplicial object as in Lemma \\ref{lemma-compute-by-cosimplicial-resolution}. Let $\\mathcal{O} \\to \\underline{B}$ be a map sheaf of rings on $\\mathcal{C}$ which induces an isomorphism $L\\pi_!\\mathcal{O} \\to L\\pi_!\\underline{B}$. In this case we obtain an exact functor of triangulated categories $$ L\\pi_! : D(\\mathcal{O}) \\longrightarrow D(B) $$ Namely, for any object $K$ of $D(\\mathcal{O})$ we have $L\\pi^{\\textit{Ab}}_!(K) = L\\pi^{\\textit{Ab}}_!(K \\otimes_{\\mathcal{O}}^\\mathbf{L} \\underline{B})$ by Lemma \\ref{lemma-O-homology-qis}. Thus we can define the displayed functor as the composition of $- \\otimes^\\mathbf{L}_\\mathcal{O} \\underline{B}$ with the functor $L\\pi_! : D(\\underline{B}) \\to D(B)$. In other words, we obtain a $B$-module structure on $L\\pi_!(K)$ coming from the (canonical, functorial) identification of $L\\pi_!(K)$ with $L\\pi_!(K \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B})$ of the lemma."}
+{"_id": "4439", "title": "sites-cohomology-remark-homology-augmentation", "text": "Let $\\mathcal{C}$ be a site. Let $\\epsilon : \\mathcal{A}_\\bullet \\to \\mathcal{O}$ be an augmentation (Simplicial, Definition \\ref{simplicial-definition-augmentation}) in the category of sheaves of rings. Assume $\\epsilon$ induces a quasi-isomorphism $s(\\mathcal{A}_\\bullet) \\to \\mathcal{O}$. In this case we obtain an exact functor of triangulated categories $$ L\\pi_! : D(\\mathcal{A}_\\bullet) \\longrightarrow D(\\mathcal{O}) $$ Namely, for any object $K$ of $D(\\mathcal{A}_\\bullet)$ we have $L\\pi_!(K) = L\\pi_!(K \\otimes_{\\mathcal{A}_\\bullet}^\\mathbf{L} \\mathcal{O})$ by Lemma \\ref{lemma-base-change-by-qis}. Thus we can define the displayed functor as the composition of $- \\otimes^\\mathbf{L}_{\\mathcal{A}_\\bullet} \\mathcal{O}$ with the functor $L\\pi_! : D(\\Delta \\times \\mathcal{C}, \\pi^{-1}\\mathcal{O}) \\to D(\\mathcal{O})$ of Remark \\ref{remark-fibred-category}. In other words, we obtain a $\\mathcal{O}$-module structure on $L\\pi_!(K)$ coming from the (canonical, functorial) identification of $L\\pi_!(K)$ with $L\\pi_!(K \\otimes_{\\mathcal{A}_\\bullet}^\\mathbf{L} \\mathcal{O})$ of the lemma."}
+{"_id": "4440", "title": "sites-cohomology-remark-compatible-with-diagram", "text": "The map (\\ref{equation-projection-formula-map}) is compatible with the base change map of Remark \\ref{remark-base-change} in the following sense. Namely, suppose that $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{g'} \\ar[d]_{f'} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^g & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ is a commutative diagram of ringed topoi. Let $E \\in D(\\mathcal{O}_\\mathcal{C})$ and $K \\in D(\\mathcal{O}_\\mathcal{D})$. Then the diagram $$ \\xymatrix{ Lg^*(Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{D}} K) \\ar[r]_p \\ar[d]_t & Lg^*Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K) \\ar[d]_b \\\\ Lg^*Rf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}} Lg^*K \\ar[d]_b & Rf'_*L(g')^*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K) \\ar[d]_t \\\\ Rf'_*L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}} Lg^*K \\ar[rd]_p & Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}} L(g')^*Lf^*K) \\ar[d]_c \\\\ & Rf'_*(L(g')^*E \\otimes^\\mathbf{L}_{\\mathcal{O}_{\\mathcal{D}'}} L(f')^*Lg^*K) } $$ is commutative. Here arrows labeled $t$ are gotten by an application of Lemma \\ref{lemma-pullback-tensor-product}, arrows labeled $b$ by an application of Remark \\ref{remark-base-change}, arrows labeled $p$ by an application of (\\ref{equation-projection-formula-map}), and $c$ comes from $L(g')^* \\circ Lf^* = L(f')^* \\circ Lg^*$. We omit the verification."}
+{"_id": "4663", "title": "spaces-limits-remark-limit-preserving", "text": "Here is an important special case of Proposition \\ref{proposition-characterize-locally-finite-presentation}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is locally of finite presentation over $S$ if and only if $X$, as a functor $(\\Sch/S)^{opp} \\to \\textit{Sets}$, is limit preserving. Compare with Limits, Remark \\ref{limits-remark-limit-preserving}. In fact, we will see in Lemma \\ref{lemma-surjection-is-enough} below that it suffices if the map $$ \\colim X(T_i) \\longrightarrow X(T) $$ is surjective whenever $T = \\lim T_i$ is a directed limit of affine schemes over $S$."}
+{"_id": "4664", "title": "spaces-limits-remark-cannot-embed-in-general", "text": "We have seen in Examples, Section \\ref{examples-section-embedding-affines} that Lemma \\ref{lemma-embedding-into-affine-over-ls-qs} does not hold if we drop the assumption that $X$ be locally separated. This raises the question: Does Lemma \\ref{lemma-embedding-into-affine-over-ls-qs} hold if we drop the assumption that $X$ be quasi-separated? If you know the answer, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "4665", "title": "spaces-limits-remark-finite-type-gives-well-defined-system", "text": "In Situation \\ref{situation-limit-noetherian} Lemmas \\ref{lemma-good-diagram}, \\ref{lemma-limit-from-good-diagram}, and \\ref{lemma-morphism-good-diagram} tell us that the category of algebraic spaces quasi-separated and of finite type over $B$ is equivalent to certain types of inverse systems of algebraic spaces over $(B_i)_{i \\in I}$, namely the ones produced by applying Lemma \\ref{lemma-limit-from-good-diagram} to a diagram of the form (\\ref{equation-good-diagram}). For example, given $X \\to B$ finite type and quasi-separated if we choose two different diagrams $X \\to V_1 \\to B_{i_1}$ and $X \\to V_2 \\to B_{i_2}$ as in (\\ref{equation-good-diagram}), then applying Lemma \\ref{lemma-morphism-good-diagram} to $\\text{id}_X$ (in two directions) we see that the corresponding limit descriptions of $X$ are canonically isomorphic (up to shrinking the directed set $I$). And so on and so forth."}
+{"_id": "4704", "title": "stacks-geometry-remark-upgrade", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. By Artin's Axioms, Lemma \\ref{artin-lemma-effective} our versal formal object in fact comes from a morphism $$ \\Spec(A) \\longrightarrow \\mathcal{X} $$ over $S$. Moreover, the results above each can be upgraded to be compatible with this morphism. Here is a list: \\begin{enumerate} \\item in Lemma \\ref{lemma-versal-ring} the isomorphism $A \\cong A'[[t_1, \\ldots, t_r]]$ or $A' \\cong A[[t_1, \\ldots, t_r]]$ may be chosen compatible with these morphisms, \\item in Lemma \\ref{lemma-versal-ring-field-extension} the homomorphism $A \\to A'$ may be chosen compatible with these morphisms, \\item in Lemma \\ref{lemma-compare-versal-ring-completion} the morphism $\\Spec(\\mathcal{O}_{U, u_0}^\\wedge) \\to \\mathcal{X}$ is the composition of the canonical map $\\Spec(\\mathcal{O}_{U, u_0}^\\wedge) \\to U$ and the given map $U \\to \\mathcal{X}$, \\item in Lemma \\ref{lemma-Artin-approximation-by-smooth-morphism} the isomorphism $\\mathcal{O}_{U, u_0}^\\wedge \\cong A$ may be chosen so $\\Spec(A) \\to \\mathcal{X}$ corresponds to the canonical map in the item above. \\end{enumerate} In each case the statement follows from the fact that our maps are compatible with versal formal elements; we note however that the implied diagrams are $2$-commutative only up to a (noncanonical) choice of a $2$-arrow. Still, this means that the implied map $A' \\to A$ or $A \\to A'$ in (1) is well defined up to formal homotopy, see Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-versal-unique-up-to-homotopy}."}
+{"_id": "4705", "title": "stacks-geometry-remark-groupoid-defo", "text": "In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. By Lemma \\ref{lemma-deformation-category} and Formal Deformation Theory, Theorem \\ref{formal-defos-theorem-presentation-deformation-groupoid} we know that $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ has a presentation by a smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$. Unwinding the definitions, this means we can choose \\begin{enumerate} \\item a Noetherian complete local $\\Lambda$-algebra $A$ with residue field $k$ and a versal formal object $\\xi$ of $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ over $A$, \\item a Noetherian complete local $\\Lambda$-algebra $B$ with residue field $k$ and an isomorphism $$ \\underline{B}|_{\\mathcal{C}_\\Lambda} \\longrightarrow \\underline{A}|_{\\mathcal{C}_\\Lambda} \\times_{\\underline{\\xi}, \\mathcal{F}_{\\mathcal{X}, k, x_0}, \\underline{\\xi}} \\underline{A}|_{\\mathcal{C}_\\Lambda} $$ \\end{enumerate} The projections correspond to formally smooth maps $t : A \\to B$ and $s : A \\to B$ (because $\\xi$ is versal). There is a map $c : B \\to B \\widehat{\\otimes}_{s, A, t} B$ which turns $(A, B, s, t, c)$ into a cogroupoid in the category of Noetherian complete local $\\Lambda$-algebras with residue field $k$ (on prorepresentable functors this map is constructed in Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-presentation-construction}). Finally, the cited theorem tells us that $\\xi$ induces an equivalence $$ [\\underline{A}|_{\\mathcal{C}_\\Lambda} / \\underline{B}|_{\\mathcal{C}_\\Lambda}] \\longrightarrow \\mathcal{F}_{\\mathcal{X}, k, x_0} $$ of groupoids cofibred over $\\mathcal{C}_\\Lambda$. In fact, we also get an equivalence $$ [\\underline{A}/\\underline{B}] \\longrightarrow \\widehat{\\mathcal{F}}_{\\mathcal{X}, k, x_0} $$ of groupoids cofibred over the completed category $\\widehat{\\mathcal{C}}_\\Lambda$ (see discussion in Formal Deformation Theory, Section \\ref{formal-defos-section-prorepresentable-groupoids-in-functors} as to why this works). Of course $A$ is a versal ring to $\\mathcal{X}$ at $x_0$."}
+{"_id": "4706", "title": "stacks-geometry-remark-dimension-algebraic-space", "text": "In general, the dimension of the algebraic space $X$ at a point $x$ may not coincide with the dimension of the underlying topological space $|X|$ at $x$.  E.g.\\ if $k$ is a field of characteristic zero and $X =  \\mathbf{A}^1_k / \\mathbf{Z}$, then $X$ has dimension $1$ (the dimension of $\\mathbf{A}^1_k$) at each of its points, while $|X|$ has the indiscrete topology, and hence is of Krull dimension zero. On the other hand, in Algebraic Spaces, Example \\ref{spaces-example-infinite-product} there is given an example of an algebraic space which is of dimension $0$ at each of its points, while $|X|$ is irreducible of Krull dimension $1$, and admits a generic point (so that the dimension of $|X|$ at any of its points is $1$); see also the discussion of this example in Properties of Spaces, Section \\ref{spaces-properties-section-dimension}. \\medskip\\noindent On the other hand, if $X$ is a {\\it decent} algebraic space, in the sense of Decent Spaces, Definition \\ref{decent-spaces-definition-very-reasonable} (in particular, if $X$ is quasi-separated; see Decent Spaces, Section \\ref{decent-spaces-section-reasonable-decent}) then in fact the dimension of $X$ at $x$ does coincide with the dimension of $|X|$ at $x$; see Decent Spaces, Lemma \\ref{decent-spaces-lemma-dimension-decent-space}."}
+{"_id": "4707", "title": "stacks-geometry-remark-relative-dimension", "text": "(1) One easily verifies (for example, by using the invariance of the relative dimension of locally of finite type morphisms of schemes under base-change; see for example Morphisms, Lemma \\ref{morphisms-lemma-dimension-fibre-after-base-change}) that $\\dim_t(T_x)$ is well-defined, independently of the choices used to compute it. \\medskip\\noindent (2) In the case that $\\mathcal{X}$ is also an algebraic space, it is straightforward to confirm that this definition agrees with the definition of relative dimension given in Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dimension-fibre}."}
+{"_id": "4708", "title": "stacks-geometry-remark-dimension-DM", "text": "For Deligne--Mumford stacks which are suitably decent (e.g.\\ quasi-separated), it will again be the case that $\\dim_x(\\mathcal{X})$ coincides with the topologically defined quantity $\\dim_x |\\mathcal{X}|$.  However, for more general Artin stacks, this will typically not be the case.  For example, if $\\mathcal{X} = [\\mathbf{A}^1/\\mathbf{G}_m]$ (over some field, with the quotient being taken with respect to the usual multiplication action of $\\mathbf{G}_m$ on $\\mathbf{A}^1$), then  $|\\mathcal{X}|$ has two points, one the specialisation of the other (corresponding to the two orbits of $\\mathbf{G}_m$ on $\\mathbf{A}^1$), and hence is of dimension $1$ as a topological space; but $\\dim_x (\\mathcal{X}) = 0$ for both points $x \\in |\\mathcal{X}|$. (An even more extreme example is given by the classifying space $[\\Spec k/\\mathbf{G}_m]$, whose dimension at its unique point is equal to $-1$.)"}
+{"_id": "4709", "title": "stacks-geometry-remark-dimension-tangent-space-well-defined", "text": "Standard manipulations show that $\\dim_t(\\mathcal{T}_x)$ is well-defined, independently of the choices made to compute it."}
+{"_id": "4710", "title": "stacks-geometry-remark-negative-dimension", "text": "We note that in the context of the preceding lemma, it need not be that $\\dim \\mathcal{T} \\geq \\dim \\mathcal{Z}$; this does not contradict the inequality in the statement of the lemma, because the fibres of the morphism $f$ are again algebraic stacks, and so may have negative dimension.  This is illustrated by taking $k$ to be a field, and applying the lemma to the morphism $[\\Spec k/\\mathbf{G}_m] \\to \\Spec k$. \\medskip\\noindent If the morphism $f$ in the statement of the lemma is assumed to be quasi-DM (in the sense of Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-separated}; e.g.\\ morphisms that are representable by algebraic spaces are quasi-DM), then the fibres of the morphism over points of the target are quasi-DM algebraic stacks, and hence are of non-negative dimension.  In this case, the lemma implies that indeed $\\dim \\mathcal{T} \\geq \\dim \\mathcal{Z}$.  In fact, we obtain the following more general result."}
+{"_id": "5029", "title": "spaces-morphisms-remark-immersion", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Since $i$ is a monomorphism we may think of $|Z|$ as a subset of $|X|$; in the rest of this remark we do so. Let $\\partial |Z|$ be the boundary of $|Z|$ in the topological space $|X|$. In a formula $$ \\partial |Z| = \\overline{|Z|} \\setminus |Z|. $$ Let $\\partial Z$ be the reduced closed subspace of $X$ with $|\\partial Z| = \\partial |Z|$ obtained by taking the reduced induced closed subspace structure, see Properties of Spaces, Definition \\ref{spaces-properties-definition-reduced-induced-space}. By construction we see that $|Z|$ is closed in $|X| \\setminus |\\partial Z| = |X \\setminus \\partial Z|$. Hence it is true that any immersion of algebraic spaces can be factored as a closed immersion followed by an open immersion (but not the other way in general, see Morphisms, Example \\ref{morphisms-example-thibaut})."}
+{"_id": "5030", "title": "spaces-morphisms-remark-space-structure-locally-closed-subset", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \\subset |X|$ be a locally closed subset. Let $\\partial T$ be the boundary of $T$ in the topological space $|X|$. In a formula $$ \\partial T = \\overline{T} \\setminus T. $$ Let $U \\subset X$ be the open subspace of $X$ with $|U| = |X| \\setminus \\partial T$, see Properties of Spaces, Lemma \\ref{spaces-properties-lemma-open-subspaces}. Let $Z$ be the reduced closed subspace of $U$ with $|Z| = T$ obtained by taking the reduced induced closed subspace structure, see Properties of Spaces, Definition \\ref{spaces-properties-definition-reduced-induced-space}. By construction $Z \\to U$ is a closed immersion of algebraic spaces and $U \\to X$ is an open immersion, hence $Z \\to X$ is an immersion of algebraic spaces over $S$ (see Spaces, Lemma \\ref{spaces-lemma-composition-immersions}). Note that $Z$ is a reduced algebraic space and that $|Z| = T$ as subsets of $|X|$. We sometimes say $Z$ is the {\\it reduced induced subspace structure} on $T$."}
+{"_id": "5031", "title": "spaces-morphisms-remark-universally-injective-not-separated", "text": "A universally injective morphism of schemes is separated, see Morphisms, Lemma \\ref{morphisms-lemma-universally-injective-separated}. This is not the case for morphisms of algebraic spaces. Namely, the algebraic space $X = \\mathbf{A}^1_k/\\{x \\sim -x \\mid x \\not = 0\\}$ constructed in Spaces, Example \\ref{spaces-example-affine-line-involution} comes equipped with a morphism $X \\to \\mathbf{A}^1_k$ which maps the point with coordinate $x$ to the point with coordinate $x^2$. This is an isomorphism away from $0$, and there is a unique point of $X$ lying above $0$. As $X$ isn't separated this is a universally injective morphism of algebraic spaces which is not separated."}
+{"_id": "5032", "title": "spaces-morphisms-remark-factorization-quasi-compact-quasi-separated", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Then $f$ has a canonical factorization $$ Y \\longrightarrow \\underline{\\Spec}_X(f_*\\mathcal{O}_Y) \\longrightarrow X $$ This makes sense because $f_*\\mathcal{O}_Y$ is quasi-coherent by Lemma \\ref{lemma-pushforward}. The morphism $Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ comes from the canonical $\\mathcal{O}_Y$-algebra map $f^*f_*\\mathcal{O}_Y \\to \\mathcal{O}_Y$ which corresponds to a canonical morphism $Y \\to Y \\times_X \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$ over $Y$ (see Lemma \\ref{lemma-affine-equivalence-algebras}) whence a factorization of $f$ as above."}
+{"_id": "5033", "title": "spaces-morphisms-remark-composition-P", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Suppose that moreover $\\mathcal{P}$ is stable under compositions. Then the class of morphisms of algebraic spaces having property $\\mathcal{P}$ is stable under composition."}
+{"_id": "5034", "title": "spaces-morphisms-remark-base-change-P", "text": "Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale local on the source-and-target. Suppose that moreover $\\mathcal{P}$ is stable under base change. Then the class of morphisms of algebraic spaces having property $\\mathcal{P}$ is stable under base change."}
+{"_id": "5035", "title": "spaces-morphisms-remark-when-apply", "text": "We will apply Lemma \\ref{lemma-local-source-target-global-implies-local} above to all cases listed in Descent, Remark \\ref{descent-remark-list-local-source-target} except ``flat''. In each case we will do this by defining $f$ to have property $\\mathcal{P}$ at $x$ if $f$ has $\\mathcal{P}$ in a neighbourhood of $x$."}
+{"_id": "5117", "title": "weil-remark-transpose", "text": "Let $X$ and $Y$ be smooth projective schemes over $k$. Assume $X$ is equidimensional of dimension $d$ and $Y$ is equidimensional of dimension $e$. Then the isomorphism $X \\times Y \\to Y \\times X$ switching the factors determines an isomorphism $$ \\text{Corr}^r(X, Y) \\longrightarrow \\text{Corr}^{d - e + r}(Y, X),\\quad c \\longmapsto c^t $$ called the {\\it transpose}. It acts on cycles as well as cycle classes. An example which is sometimes useful, is the transpose $[\\Gamma_f]^t = [\\Gamma_f^t]$ of the graph of a morphism $f : Y \\to X$."}
+{"_id": "5118", "title": "weil-remark-lefschetz-tate", "text": "Let $X = \\mathbf{P}^1_k$ and $c_2$ be as in Example \\ref{example-decompose-P1}. In the literature the motive $(X, c_2, 0)$ is sometimes called the {\\it Lefschetz motive} and depending on the reference the notation $L$, $\\mathbf{L}$, $\\mathbf{Q}(-1)$, or $h^2(\\mathbf{P}^1_k)$ may be used to denote it. By Lemma \\ref{lemma-inverse-h2} the Lefschetz motive is isomorphic to $\\mathbf{1}(-1)$. Hence the Lefschetz motive is invertible (Categories, Definition \\ref{categories-definition-invertible}) with inverse $\\mathbf{1}(1)$. The motive $\\mathbf{1}(1)$ is sometimes called the {\\it Tate motive} and depending on the reference the notation $L^{-1}$, $\\mathbf{L}^{-1}$, $\\mathbf{T}$, or $\\mathbf{Q}(1)$ may be used to denote it."}
+{"_id": "5119", "title": "weil-remark-replace-cup-product-classical", "text": "Let $X$ be a smooth projective variety. We obtain maps $$ H^*(X) \\otimes_F H^*(X) \\longrightarrow H^*(X \\times X) \\xrightarrow{\\Delta^*} H^*(X) $$ where the first arrow is as in axiom (B) and $\\Delta^*$ is pullback along the diagonal morphism $\\Delta : X \\to X \\times X$. The composition is the cup product as pullback is an algebra homomorphism and $\\text{pr}_i \\circ \\Delta = \\text{id}$. On the other hand, given cycles $\\alpha, \\beta$ on $X$  the intersection product is defined by the formula $$ \\alpha \\cdot \\beta = \\Delta^!(\\alpha \\times \\beta) $$ In other words, $\\alpha \\cdot \\beta$ is the pullback of the exterior product $\\alpha \\times \\beta$ on $X \\times X$ by the diagonal. Note also that $\\alpha \\times \\beta = \\text{pr}_1^*\\alpha \\cdot \\text{pr}_2^*\\beta$ in $\\CH^*(X \\times X)$ (we omit the proof). Hence, given axiom (C)(a), axiom (C)(c) is equivalent to the statement that $\\gamma$ is compatible with exterior product in the sense that $\\gamma(\\alpha \\times \\beta)$ is equal to $\\text{pr}_1^*\\gamma(\\alpha) \\cup \\text{pr}_2^*\\gamma(\\beta)$. This is how axiom (C)(c) is formulated in \\cite{Kleiman-cycles}."}
+{"_id": "5120", "title": "weil-remark-replace-cup-product", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C)(a). Let $X$ be a smooth projective scheme over $k$. We obtain maps $$ H^*(X) \\otimes_F H^*(X) \\longrightarrow H^*(X \\times X) \\xrightarrow{\\Delta^*} H^*(X) $$ where the first arrow is as in axiom (B) and $\\Delta^*$ is pullback along the diagonal morphism $\\Delta : X \\to X \\times X$. The composition is the cup product as pullback is an algebra homomorphism and $\\text{pr}_i \\circ \\Delta = \\text{id}$. On the other hand, given cycles $\\alpha, \\beta$ on $X$  the intersection product is defined by the formula $$ \\alpha \\cdot \\beta = \\Delta^!(\\alpha \\times \\beta) $$ In other words, $\\alpha \\cdot \\beta$ is the pullback of the exterior product $\\alpha \\times \\beta$ on $X \\times X$ by the diagonal. Note also that $\\alpha \\times \\beta = \\text{pr}_1^*\\alpha \\cdot \\text{pr}_2^*\\beta$ in $\\CH^*(X \\times X)$ (we omit the proof). Hence, given axiom (C)(a), axiom (C)(c) is equivalent to the statement that $\\gamma$ is compatible with exterior product in the sense that $\\gamma(\\alpha \\times \\beta)$ is equal to $\\text{pr}_1^*\\gamma(\\alpha) \\cup \\text{pr}_2^*\\gamma(\\beta)$."}
+{"_id": "5121", "title": "weil-remark-betti-numbers-in-some-sense", "text": "Let $H^*$ be a Weil cohomology theory (Definition \\ref{definition-weil-cohomology-theory}). Let $X$ be a geometrically irreducible smooth projective scheme of dimension $d$ over $k'$ with $k'/k$ a finite separable extension of fields. Suppose that $$ H^0(\\Spec(k')) = F_1 \\times \\ldots \\times F_r $$ for some fields $F_i$. Then we accordingly can write $$ H^*(X) = \\prod\\nolimits_{i = 1, \\ldots, r} H^*(X) \\otimes_{H^0(\\Spec(k'))} F_i $$ Now, our final assumption in Definition \\ref{definition-weil-cohomology-theory} tells us that $H^0(X)$ is free of rank $1$ over $\\prod F_i$. In other words, each of the factors $H^0(X) \\otimes_{H^0(\\Spec(k'))} F_i$ has dimension $1$ over $F_i$. Poincar\\'e duality then tells us that the same is true for cohomology in degree $2d$. What isn't clear however is that the same holds in other degrees. Namely, we don't know that given $0 < n < \\dim(X)$ the integers $$ \\dim_{F_i} H^n(X) \\otimes_{H^0(\\Spec(k'))} F_i $$ are independent of $i$! This question is closely related to the following open question: given an algebraically closed base field $\\overline{k}$, a field of characteristic zero $F$, a classical Weil cohomology theory $H^*$ over $\\overline{k}$ with coefficient field $F$, and a smooth projective variety $X$ over $\\overline{k}$ is it true that the betti numbers of $X$ $$ \\beta_i = \\dim_F H^i(X) $$ are independent of $F$ and the Weil cohomology theory $H^*$?"}
+{"_id": "5122", "title": "weil-remark-trace", "text": "Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Let $X$ be a smooth projective scheme over $k$ which is nonempty and equidimensional of dimension $d$. Combining what was said in the proofs of Lemma \\ref{lemma-poincare-duality} and Homology, Lemma \\ref{homology-lemma-left-dual-graded-vector-spaces} we see that $$ \\gamma([\\Delta]) \\in \\bigoplus\\nolimits_i H^i(X) \\otimes H^{2d - i}(X)(d) $$ defines a perfect duality between $H^i(X)$ and $H^{2d - i}(X)(d)$ for all $i$. In particular, the linear map $\\int_X = \\lambda : H^{2d}(X)(d) \\to F$ of axiom (A6) is unique! We will call the linear map $\\int_X$ the trace map of $X$ from now on."}
+{"_id": "5594", "title": "morphisms-remark-direct-argument", "text": "We can also argue directly that (2) implies (1) in Lemma \\ref{lemma-characterize-affine} above as follows. Assume $S = \\bigcup W_j$ is an affine open covering such that each $f^{-1}(W_j)$ is affine. First argue that $\\mathcal{A} = f_*\\mathcal{O}_X$ is quasi-coherent as in the proof above. Let $\\Spec(R) = V \\subset S$ be affine open. We have to show that $f^{-1}(V)$ is affine. Set $A = \\mathcal{A}(V) = f_*\\mathcal{O}_X(V) = \\mathcal{O}_X(f^{-1}(V))$. By Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} there is a canonical morphism $\\psi : f^{-1}(V) \\to \\Spec(A)$ over $\\Spec(R) = V$. By Schemes, Lemma \\ref{schemes-lemma-good-subcover} there exists an integer $n \\geq 0$, a standard open covering $V = \\bigcup_{i = 1, \\ldots, n} D(h_i)$, $h_i \\in R$, and a map $a : \\{1, \\ldots, n\\} \\to J$ such that each $D(h_i)$ is also a standard open of the affine scheme $W_{a(i)}$. The inverse image of a standard open under a morphism of affine schemes is standard open, see Algebra, Lemma \\ref{algebra-lemma-spec-functorial}. Hence we see that $f^{-1}(D(h_i))$ is a standard open of $f^{-1}(W_{a(i)})$, in particular that $f^{-1}(D(h_i))$ is affine. Because $\\mathcal{A}$ is quasi-coherent we have $A_{h_i} = \\mathcal{A}(D(h_i)) = \\mathcal{O}_X(f^{-1}(D(h_i)))$, so $f^{-1}(D(h_i))$ is the spectrum of $A_{h_i}$. It follows that the morphism $\\psi$ induces an isomorphism of the open $f^{-1}(D(h_i))$ with the open $\\Spec(A_{h_i})$ of $\\Spec(A)$. Since $f^{-1}(V) = \\bigcup f^{-1}(D(h_i))$ and $\\Spec(A) = \\bigcup \\Spec(A_{h_i})$ we win."}
+{"_id": "5595", "title": "morphisms-remark-affine-s-opens-cover-family", "text": "In Properties, Lemma \\ref{properties-lemma-affine-s-opens-cover-quasi-separated} we see that a scheme which has an ample invertible module is separated. This is wrong for schemes having an ample family of invertible modules. Namely, let $X$ be as in Schemes, Example \\ref{schemes-example-affine-space-zero-doubled} with $n = 1$, i.e., the affine line with zero doubled.  We use the notation of that example except that we write $x$ for $x_1$ and $y$ for $y_1$.  There is, for every integer $n$, an invertible sheaf $\\mathcal{L}_n$ on $X$ which is trivial on $X_1$ and $X_2$ and whose transition function $U_{12} \\to U_{21}$ is $f(x) \\mapsto y^n f(y)$.  The global sections of $\\mathcal{L}_n$ are pairs $(f(x), g(y)) \\in k[x] \\oplus k[y]$ such that $y^n f(y) = g(y)$.  The sections $s = (1, y)$ of $\\mathcal{L}_1$ and $t = (x, 1)$ of $\\mathcal{L}_{-1}$ determine an open affine cover because $X_s = X_1$ and $X_t = X_2$.  Therefore $X$ has an ample family of invertible modules but it is not separated."}
+{"_id": "5596", "title": "morphisms-remark-flattening", "text": "The results above are a first step towards more refined flattening techniques for morphisms of schemes. The article \\cite{GruRay} by Raynaud and Gruson contains many wonderful results in this direction."}
+{"_id": "5597", "title": "morphisms-remark-differentials-glue", "text": "The lemma above gives a second way of constructing the module of differentials. Namely, let $f : X \\to S$ be a morphism of schemes. Consider the collection of all affine opens $U \\subset X$ which map into an affine open of $S$. These form a basis for the topology on $X$. Thus it suffices to define $\\Gamma(U, \\Omega_{X/S})$ for such $U$. We simply set $\\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}$ if $A$, $R$ are as in Lemma \\ref{lemma-differentials-affine} above. This works, but it takes somewhat more algebraic preliminaries to construct the restriction mappings and to verify the sheaf condition with this ansatz."}
+{"_id": "5598", "title": "morphisms-remark-differentials-diagonal", "text": "Let $X \\to S$ be a morphism of schemes. According to Lemma \\ref{lemma-differential-product} we have $$ \\Omega_{X \\times_S X/S} = \\text{pr}_1^*\\Omega_{X/S} \\oplus \\text{pr}_2^*\\Omega_{X/S} $$ On the other hand, the diagonal morphism $\\Delta : X \\to X \\times_S X$ is an immersion, which locally has a left inverse. Hence by Lemma \\ref{lemma-differentials-relative-immersion-section} we obtain a canonical short exact sequence $$ 0 \\to \\mathcal{C}_{X/X \\times_S X} \\to \\Omega_{X/S} \\oplus \\Omega_{X/S} \\to \\Omega_{X/S} \\to 0 $$ Note that the right arrow is $(1, 1)$ which is indeed a split surjection. On the other hand, by Lemma \\ref{lemma-differentials-diagonal} we have an identification $\\Omega_{X/S} = \\mathcal{C}_{X/X \\times_S X}$. Because we chose $\\text{d}_{X/S}(f) = s_2(f) - s_1(f)$ in this identification it turns out that the left arrow is the map $(-1, 1)$\\footnote{Namely, the local section $\\text{d}_{X/S}(f) = 1 \\otimes f - f \\otimes 1$ of the ideal sheaf of $\\Delta$ maps via $\\text{d}_{X \\times_S X/X}$ to the local section $1 \\otimes 1 \\otimes 1 \\otimes f - 1 \\otimes f \\otimes 1 \\otimes 1 -1 \\otimes 1 \\otimes f \\otimes 1 + f \\otimes 1 \\otimes 1 \\otimes 1 = \\text{pr}_2^*\\text{d}_{X/S}(f) - \\text{pr}_1^*\\text{d}_{X/S}(f)$.}."}
+{"_id": "5599", "title": "morphisms-remark-base-change-differential-operators", "text": "Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes. Denote $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ the projections. In this remark, given an $\\mathcal{O}_X$-module $\\mathcal{F}$ and an $\\mathcal{O}_Y$-module $\\mathcal{G}$ let us set $$ \\mathcal{F} \\boxtimes \\mathcal{G} = p^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}} q^*\\mathcal{G} $$ Denote $\\mathcal{A}_{X/S}$ the additive category whose objects are quasi-coherent $\\mathcal{O}_X$-modules and whose morphisms are differential operators of finite order on $X/S$. Similarly for $\\mathcal{A}_{Y/S}$ and $\\mathcal{A}_{X \\times_S Y/S}$. The construction of Lemma \\ref{lemma-base-change-differential-operators} determines a functor $$ \\boxtimes :  \\mathcal{A}_{X/S} \\times \\mathcal{A}_{Y/S} \\longrightarrow \\mathcal{A}_{X \\times_S Y/S}, \\quad (\\mathcal{F}, \\mathcal{G}) \\longmapsto \\mathcal{F} \\boxtimes \\mathcal{G} $$ which is bilinear on morphisms. If $X = \\Spec(A)$, $Y = \\Spec(B)$, and $S = \\Spec(R)$, then via the identification of quasi-coherent sheaves with modules this functor is given by $(M, N) \\mapsto M \\otimes_R N$ on objects and sends the morphism $(D, D') : (M, N) \\to (M', N')$ to $D \\otimes D' : M \\otimes_R N \\to M' \\otimes_R N'$."}
+{"_id": "5600", "title": "morphisms-remark-check-val-on-open", "text": "The assumption on uniqueness of the dotted arrows in Lemma \\ref{lemma-refined-valuative-criterion-universally-closed} is necessary (details omitted). Of course, uniqueness is guaranteed if $f$ is separated (Schemes, Lemma \\ref{schemes-lemma-separated-implies-valuative})."}
+{"_id": "5601", "title": "morphisms-remark-generalize-category", "text": "Here is a generalization of the category of irreducible schemes and dominant rational maps. For a scheme $X$ denote $X^0$ the set of points $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 0$, in other words, $X^0$ is the set of generic points of irreducible components of $X$. Then we can consider the category with \\begin{enumerate} \\item objects are schemes $X$ such that every quasi-compact open has finitely many irreducible components, and \\item morphisms from $X$ to $Y$ are rational maps $f : U \\to Y$ from $X$ to $Y$ such that $f(U^0) = Y^0$. \\end{enumerate} If $U \\subset X$ is a dense open of a scheme, then $U^0 \\subset X^0$ need not be an equality, but if $X$ is an object of our category, then this is the case. Thus given two morphisms in our category, the composition is well defined and a morphism in our category."}
+{"_id": "5602", "title": "morphisms-remark-pseudo-morphisms", "text": "There is a variant of Definition \\ref{definition-rational-map} where we consider only those morphism $U \\to Y$ defined on scheme theoretically dense open subschemes $U \\subset X$. We use Lemma \\ref{lemma-intersection-scheme-theoretically-dense} to see that we obtain an equivalence relation. An equivalence class of these is called a {\\it pseudo-morphism from $X$ to $Y$}. If $X$ is reduced the two notions coincide."}
+{"_id": "5603", "title": "morphisms-remark-quasi-finite-finite-over-dense-open", "text": "An alternative to Lemma \\ref{lemma-generically-finite} is the statement that a quasi-finite morphism is finite over a dense open of the target. This will be shown in More on Morphisms, Lemma \\ref{more-morphisms-lemma-quasi-finite-finite-over-dense-open}."}
+{"_id": "5604", "title": "morphisms-remark-definition-generically-finite", "text": "Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type. There are (at least) two properties that we could use to define {\\it generically finite} morphisms. These correspond to whether you want the property to be local on the source or local on the target: \\begin{enumerate} \\item (Local on the target; suggested by Ravi Vakil.) Assume every quasi-compact open of $Y$ has finitely many irreducible components (for example if $Y$ is locally Noetherian). The requirement is that the inverse image of each generic point is finite, see Lemma \\ref{lemma-generically-finite}. \\item (Local on the source.) The requirement is that there exists a dense open $U \\subset X$ such that $U \\to Y$ is locally quasi-finite. \\end{enumerate} In case (1) the requirement can be formulated without the auxiliary condition on $Y$, but probably doesn't give the right notion for general schemes. Property (2) as formulated doesn't imply that the fibres over generic points are finite; however, if $f$ is quasi-compact and $Y$ is as in (1) then it does."}
+{"_id": "5930", "title": "chow-remark-gersten-complex-milnor", "text": "For a field $k$ let us denote $K^M_*(k)$ the quotient of the tensor algebra on $k^*$ divided by the two-sided ideal generated by the elements $x \\otimes 1 - x$. Thus $K^M_0(k) = \\mathbf{Z}$, $K_1^M(k) = k^*$, and $$ K^M_2(k) = k^* \\otimes_\\mathbf{Z} k^* / \\langle x \\otimes 1 - x \\rangle $$ If $(A, \\mathfrak m)$ is a $1$-dimensional Noetherian local domain with fraction field $Q(A)$ and residue field $\\kappa$ there is a tame symbol $$ \\partial_A : K_{i + 1}^M(Q(A)) \\to K_i^M(\\kappa(\\mathfrak m)) $$ You can use the method of Section \\ref{section-tame-symbol} to define these maps, provided you extend the norm map to $K_i^M$ for all $i$. Next, let $X$ be a Noetherian scheme with a dimension function $\\delta$. Then we can use these tame symbols to get the arrows in the following: $$ \\bigoplus\\nolimits_{\\delta(x) = j + 1} K^M_{i + 1}(\\kappa(x)) \\longrightarrow \\bigoplus\\nolimits_{\\delta(x) = j} K^M_i(\\kappa(x)) \\longrightarrow \\bigoplus\\nolimits_{\\delta(x) = j - 1} K^M_{i - 1}(\\kappa(x)) $$ However, it is not clear, if you define the maps as suggested above, that the composition is zero. When $i = 1$ and $j$ arbitrary, this follows from Lemma \\ref{lemma-milnor-gersten-low-degree}. For excellent $X$ this follows from \\cite{Kato-Milnor-K} modulo the verification that Kato's maps are the same as ours."}
+{"_id": "5931", "title": "chow-remark-infinite-sums-rational-equivalences", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Suppose we have infinite collections $\\alpha_i, \\beta_i \\in Z_k(X)$, $i \\in I$ of $k$-cycles on $X$. Suppose that the supports of $\\alpha_i$ and $\\beta_i$ form locally finite collections of closed subsets of $X$ so that $\\sum \\alpha_i$ and $\\sum \\beta_i$ are defined as cycles. Moreover, assume that $\\alpha_i \\sim_{rat} \\beta_i$ for each $i$. Then it is not clear that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$. Namely, the problem is that the rational equivalences may be given by locally finite families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$ but the union $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ may not be locally finite. \\medskip\\noindent In many cases in practice, one has a locally finite family of closed subsets $\\{T_i\\}_{i \\in I}$ such that $\\alpha_i, \\beta_i$ are supported on $T_i$ and such that $\\alpha_i = \\beta_i$ in $\\CH_k(T_i)$, in other words, the families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$ consist of subschemes $W_{i, j} \\subset T_i$. In this case it is true that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$ on $X$, simply because the family $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ is automatically locally finite in this case."}
+{"_id": "5932", "title": "chow-remark-good-cases-K-A", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. We will see later (in Lemma \\ref{lemma-cycles-rational-equivalence-K-group}) that the map $$ \\CH_k(X) \\longrightarrow K_0(\\textit{Coh}_{k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)) $$ of Lemma \\ref{lemma-from-chow-to-K} is injective. Composing with the canonical map $$ K_0(\\textit{Coh}_{k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)) \\longrightarrow K_0(\\textit{Coh}(X)/\\textit{Coh}_{\\leq k - 1}(X)) $$ we obtain a canonical map $$ \\CH_k(X) \\longrightarrow K_0(\\textit{Coh}(X)/\\textit{Coh}_{\\leq k - 1}(X)). $$ We have not been able to find a statement or conjecture in the literature as to whether this map is should be injective or not. It seems reasonable to expect the kernel of this map to be torsion. We will return to this question (insert future reference)."}
+{"_id": "5933", "title": "chow-remark-generalize-to-virtual", "text": "Let $X$ be a scheme locally of finite type over $S$ as in Situation \\ref{situation-setup}. Let $(D, \\mathcal{N}, \\sigma)$ be a triple consisting of a locally principal (Divisors, Definition \\ref{divisors-definition-effective-Cartier-divisor}) closed subscheme $i : D \\to X$, an invertible $\\mathcal{O}_D$-module $\\mathcal{N}$, and a surjection $\\sigma : \\mathcal{N}^{\\otimes -1} \\to i^*\\mathcal{I}_D$ of $\\mathcal{O}_D$-modules\\footnote{This condition assures us that if $D$ is an effective Cartier divisor, then $\\mathcal{N} = \\mathcal{O}_X(D)|_D$.}. Here $\\mathcal{N}$ should be thought of as a {\\it virtual normal bundle of $D$ in $X$}. The construction of $i^* : Z_{k + 1}(X) \\to \\CH_k(D)$ in Definition \\ref{definition-gysin-homomorphism} generalizes to such triples, see Section \\ref{section-gysin-higher-codimension}."}
+{"_id": "5934", "title": "chow-remark-generalize-to-pseudo-divisor", "text": "Let $X$ be a scheme locally of finite type over $S$ as in Situation \\ref{situation-setup}. In \\cite{F} a {\\it pseudo-divisor} on $X$ is defined as a triple $D = (\\mathcal{L}, Z, s)$ where $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, $Z \\subset X$ is a closed subset, and $s \\in \\Gamma(X \\setminus Z, \\mathcal{L})$ is a nowhere vanishing section. Similarly to the above, one can define for every $\\alpha$ in $\\CH_{k + 1}(X)$ a product $D \\cdot \\alpha$ in $\\CH_k(Z \\cap |\\alpha|)$ where $|\\alpha|$ is the support of $\\alpha$."}
+{"_id": "5935", "title": "chow-remark-gysin-on-cycles", "text": "Let $X \\to S$, $\\mathcal{L}$, $s$, $i : D \\to X$ be as in Definition \\ref{definition-gysin-homomorphism} and assume that $\\mathcal{L}|_D \\cong \\mathcal{O}_D$. In this case we can define a canonical map $i^* : Z_{k + 1}(X) \\to Z_k(D)$ on cycles, by requiring that $i^*[W] = 0$ whenever $W \\subset D$ is an integral closed subscheme. The possibility to do this will be useful later on."}
+{"_id": "5936", "title": "chow-remark-pullback-pairs", "text": "Let $f : X' \\to X$ be a morphism of schemes locally of finite type over $S$ as in Situation \\ref{situation-setup}. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. Then we can set $\\mathcal{L}' = f^*\\mathcal{L}$, $s' = f^*s$, and $D' = X' \\times_X D = Z(s')$. This gives a commutative diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ and we can ask for various compatibilities between $i^*$ and $(i')^*$."}
+{"_id": "5937", "title": "chow-remark-when-isomorphism", "text": "We will see later (Lemma \\ref{lemma-vectorbundle}) that if $X$ is a vector bundle of rank $r$ over $Y$ then the pullback map $\\CH_k(Y) \\to \\CH_{k + r}(X)$ is an isomorphism. This is true whenever $X \\to Y$ satisfies the assumptions of Lemma \\ref{lemma-pullback-affine-fibres-surjective}, see \\cite[Lemma 2.2]{Totaro-group}."}
+{"_id": "5938", "title": "chow-remark-restriction-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X \\to Y$ and $Y' \\to Y$ be morphisms of schemes locally of finite type over $S$. Let $X' = Y' \\times_Y X$. Then there is an obvious restriction map $$ A^p(X \\to Y) \\longrightarrow A^p(X' \\to Y'),\\quad c \\longmapsto res(c) $$ obtained by viewing a scheme $Y''$ locally of finite type over $Y'$ as a scheme locally of finite type over $Y$ and settting $res(c) \\cap \\alpha'' = c \\cap \\alpha''$ for any $\\alpha'' \\in \\CH_k(Y'')$. This restriction operation is compatible with compositions in an obvious manner."}
+{"_id": "5939", "title": "chow-remark-bivariant-commute", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. For $i = 1, 2$ let $Z_i \\to X$ be a morphism of schemes locally of finite type. Let $c_i \\in A^{p_i}(Z_i \\to X)$, $i = 1, 2$ be bivariant classes. For any $\\alpha \\in \\CH_k(X)$ we can ask whether $$ c_1 \\cap c_2 \\cap \\alpha = c_2 \\cap c_1 \\cap \\alpha $$ in $\\CH_{k - p_1 - p_2}(Z_1 \\times_X Z_2)$. If this is true and if it holds after any base change by $X' \\to X$ locally of finite type, then we say $c_1$ and $c_2$ {\\it commute}. Of course this is the same thing as saying that $$ res(c_1) \\circ c_2 = res(c_2) \\circ c_1 $$ in $A^{p_1 + p_2}(Z_1 \\times_X Z_2 \\to X)$. Here $res(c_1) \\in A^{p_1}(Z_1 \\times_X Z_2 \\to Z_2)$ is the restriction of $c_1$ as in Remark \\ref{remark-restriction-bivariant}; similarly for $res(c_2)$."}
+{"_id": "5940", "title": "chow-remark-more-general-bivariant", "text": "There is a more general type of bivariant class that doesn't seem to be considered in the literature. Namely, suppose we are given a diagram $$ X \\longrightarrow Z \\longleftarrow Y $$ of schemes locally of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. Let $p \\in \\mathbf{Z}$. Then we can consider a rule $c$ which assigns to every $Z' \\to Z$ locally of finite type maps $$ c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X') $$ for all $k \\in \\mathbf{Z}$ where $X' = X \\times_Z Z'$ and $Y' = Z' \\times_Z Y$ compatible with \\begin{enumerate} \\item proper pushforward if given $Z'' \\to Z'$ proper, \\item flat pullback if given $Z'' \\to Z'$ flat of fixed relative dimension, and \\item gysin maps if given $D' \\subset Z'$ as in Definition \\ref{definition-gysin-homomorphism}. \\end{enumerate} We omit the detailed formulations. Suppose we denote the collection of all such operations $A^p(X \\to Z \\leftarrow Y)$. A simple example of the utility of this concept is when we have a proper morphism $f : X_2 \\to X_1$. Then $f_*$ isn't a bivariant operation in the sense of Definition \\ref{definition-bivariant-class} but it is in the above generalized sense, namely, $f_* \\in A^0(X_1 \\to X_1 \\leftarrow X_2)$."}
+{"_id": "5941", "title": "chow-remark-pullback-cohomology", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : Y' \\to Y$ be a morphism of schemes locally of finite type over $S$. As a special case of Remark \\ref{remark-restriction-bivariant} there is a canonical $\\mathbf{Z}$-algebra map $res : A^*(Y) \\to A^*(Y')$. This map is often denoted $f^*$ in the literature."}
+{"_id": "5942", "title": "chow-remark-ring-loc-classes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $Z \\to X$ be a closed immersion of schemes locally of finite type over $S$ and let $p \\geq 0$. In this setting we define $$ A^{(p)}(Z \\to X) = \\prod\\nolimits_{i \\leq p - 1} A^i(X) \\times \\prod\\nolimits_{i \\geq p} A^i(Z \\to X). $$ Then $A^{(p)}(Z \\to X)$ canonically comes equipped with the structure of a graded algebra. In fact, more generally there is a multiplication $$ A^{(p)}(Z \\to X) \\times A^{(q)}(Z \\to X) \\longrightarrow A^{(\\max(p, q))}(Z \\to X) $$ In order to define these we define maps \\begin{align*} A^i(Z \\to X) \\times A^j(X) & \\to A^{i + j}(Z \\to X) \\\\ A^i(X) \\times A^j(Z \\to X) & \\to A^{i + j}(Z \\to X) \\\\ A^i(Z \\to X) \\times A^j(Z \\to X) & \\to A^{i + j}(Z \\to X) \\end{align*} For the first we use composition of bivariant classes. For the second we use restriction $A^i(X) \\to A^i(Z)$ (Remark \\ref{remark-restriction-bivariant}) and composition $A^i(Z) \\times A^j(Z \\to X) \\to A^{i + j}(Z \\to X)$. For the third, we send $(c, c')$ to $res(c) \\circ c'$ where $res : A^i(Z \\to X) \\to A^i(Z)$ is the restriction map (see Remark \\ref{remark-restriction-bivariant}). We omit the verification that these multiplications are associative in a suitable sense."}
+{"_id": "5943", "title": "chow-remark-res-push", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $Z \\to X$ be a closed immersion of schemes locally of finite type over $S$. Denote $res : A^p(Z \\to X) \\to A^p(Z)$ the restriction map of Remark \\ref{remark-restriction-bivariant}. For $c \\in A^p(Z \\to X)$ we have $res(c) \\cap \\alpha = c \\cap i_*\\alpha$ for $\\alpha \\in \\CH_*(Z)$. Namely $res(c) \\cap \\alpha = c \\cap \\alpha$ and compatibility of $c$ with proper pushforward gives $(Z \\to Z)_*(c \\cap \\alpha) = c \\cap (Z \\to X)_*\\alpha$."}
+{"_id": "5944", "title": "chow-remark-completion-bivariant", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Let $X = \\coprod_{i \\in I} X_i$ and $Y = \\coprod_{j \\in J} Y_j$ be the decomposition of $X$ and $Y$ into their connected components (the connected components are open as $X$ and $Y$ are locally Noetherian, see Topology, Lemma \\ref{topology-lemma-locally-Noetherian-locally-connected} and Properties, Lemma \\ref{properties-lemma-Noetherian-topology}). Let $a(i) \\in J$ be the index such that $f(X_i) \\subset Y_{a(i)}$. Then $A^p(X \\to Y) = \\prod A^p(X_i \\to Y_{a(i)})$ by Lemma \\ref{lemma-disjoint-decomposition-bivariant}. In this setting it is convenient to set $$ A^*(X \\to Y)^\\wedge = \\prod\\nolimits_i A^*(X_i \\to Y_{a(i)}) $$ as a kind of natural completion of the graded $\\mathbf{Z}$-module $A^*(X \\to Y)$ of bivariant classes (we omit specifying the precise sense in which this is a completion). As a special case we set $$ A^*(X)^\\wedge = \\prod A^*(X_i) $$ If $Y \\to Z$ is a second morphism, then the composition $A^*(X \\to Y) \\times A^*(Y \\to Z) \\to A^*(X \\to Z)$ extends to a composition $A^*(X \\to Y)^\\wedge \\times A^*(Y \\to Z)^\\wedge \\to A^*(X \\to Z)^\\wedge$ of completions."}
+{"_id": "5945", "title": "chow-remark-equation-signs", "text": "We could also rewrite equation \\ref{equation-chern-classes} as \\begin{equation} \\label{equation-signs} \\sum\\nolimits_{i = 0}^r c_1(\\mathcal{O}_P(-1))^i \\cap \\pi^*c_{r - i} = 0. \\end{equation} but we find it easier to work with the tautological quotient sheaf $\\mathcal{O}_P(1)$ instead of its dual."}
+{"_id": "5946", "title": "chow-remark-extend-to-finite-locally-free", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. If the rank of $\\mathcal{E}$ is not constant then we can still define the Chern classes of $\\mathcal{E}$. Namely, in this case we can write $$ X = X_0 \\amalg X_1 \\amalg X_2 \\amalg \\ldots $$ where $X_r \\subset X$ is the open and closed subspace where the rank of $\\mathcal{E}$ is $r$. By  Lemma \\ref{lemma-disjoint-decomposition-bivariant} we have $A^p(X) = \\prod A^p(X_r)$. Hence we can define $c_i(\\mathcal{E})$ to be the product of the classes $c_i(\\mathcal{E}|_{X_r})$ in $A^i(X_r)$. Explicitly, if $X' \\to X$ is a morphism locally of finite type, then we obtain by pullback a corresponding decomposition of $X'$ and we find that $$ \\CH_*(X') = \\prod\\nolimits_{r \\geq 0} \\CH_*(X'_r) $$ by our definitions. Then $c_i(\\mathcal{E}) \\in A^i(X)$ is the bivariant class which preserves these direct product decompositions and acts by the already defined operations $c_i(\\mathcal{E}|_{X_r}) \\cap -$ on the factors. Observe that in this setting it may happen that $c_i(\\mathcal{E})$ is nonzero for infinitely many $i$. In this setting we moreover define the ``rank'' of $\\mathcal{E}$ to be the element $r(\\mathcal{E}) \\in A^0(X)$ as the bivariant operation which sends $(\\alpha_r) \\in \\prod \\CH_*(X'_r)$ to $(r\\alpha_r) \\in \\prod \\CH_*(X'_r)$. Note that it is still true that $c_i(\\mathcal{E})$ and $r(\\mathcal{E})$ are in the center of $A^*(X)$."}
+{"_id": "5947", "title": "chow-remark-top-chern-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. In general we write $X = \\coprod X_r$ as in Remark \\ref{remark-extend-to-finite-locally-free}. If only a finite number of the $X_r$ are nonempty, then we can set $$ c_{top}(\\mathcal{E}) = \\sum\\nolimits_r c_r(\\mathcal{E}|_{X_r}) \\in A^*(X) = \\bigoplus A^*(X_r) $$ where the equality is Lemma \\ref{lemma-disjoint-decomposition-bivariant}. If infinitely many $X_r$ are nonempty, we will use the same notation to denote $$ c_{top}(\\mathcal{E}) = \\prod c_r(\\mathcal{E}|_{X_r}) \\in \\prod A^r(X_r) \\subset A^*(X)^\\wedge $$ see Remark \\ref{remark-completion-bivariant} for notation."}
+{"_id": "5948", "title": "chow-remark-fundamental-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-locally-equidimensional}. Let $X = \\coprod X_n$ be the decomposition into open and closed subschemes such that every irreducible component of $X_n$ has $\\delta$-dimension $n$. In this situation we sometimes set $$ [X] = \\sum\\nolimits_n [X_n]_n \\in \\CH^0(X) $$ This class is a kind of ``fundamental class'' of $X$ in Chow theory."}
+{"_id": "5949", "title": "chow-remark-the-proof-shows-more", "text": "The proof of Lemma \\ref{lemma-splitting-principle} shows that the morphism $\\pi : P \\to X$ has the following additional properties: \\begin{enumerate} \\item $\\pi$ is a finite composition of projective space bundles associated to locally free modules of finite constant rank, and \\item for every $\\alpha \\in \\CH_k(X)$ we have $\\alpha = \\pi_*(\\xi_1 \\cap \\ldots \\cap \\xi_d \\cap \\pi^*\\alpha)$ where $\\xi_i$ is the first Chern class of some invertible $\\mathcal{O}_P$-module. \\end{enumerate} The second observation follows from the first and Lemma \\ref{lemma-cap-projective-bundle}. We will add more observations here as needed."}
+{"_id": "5950", "title": "chow-remark-equalities-nonconstant-rank", "text": "The equalities proven above remain true even when we work with finite locally free $\\mathcal{O}_X$-modules whose rank is allowed to be nonconstant. In fact, we can work with polynomials in the rank and the Chern classes as follows. Consider the  graded polynomial ring $\\mathbf{Z}[r, c_1, c_2, c_3, \\ldots]$ where $r$ has degree $0$ and $c_i$ has degree $i$. Let $$ P \\in \\mathbf{Z}[r, c_1, c_2, c_3, \\ldots] $$ be a homogeneous polynomial of degree $p$. Then for any finite locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ on $X$ we can consider $$ P(\\mathcal{E}) = P(r(\\mathcal{E}), c_1(\\mathcal{E}), c_2(\\mathcal{E}), c_3(\\mathcal{E}), \\ldots) \\in A^p(X) $$ see Remark \\ref{remark-extend-to-finite-locally-free} for notation and conventions. To prove relations among these polynomials (for multiple finite locally free modules) we can work locally on $X$ and use the splitting principle as above. For example, we claim that $$ c_2(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{E})) = P(\\mathcal{E}) $$ where $P = 2rc_2 - (r - 1)c_1^2$. Namely, since $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}, \\mathcal{E}) = \\mathcal{E} \\otimes \\mathcal{E}^\\vee$ this follows easily from Lemmas \\ref{lemma-chern-classes-dual} and \\ref{lemma-chern-classes-tensor-product} above by decomposing $X$ into parts where the rank of $\\mathcal{E}$ is constant as in Remark \\ref{remark-extend-to-finite-locally-free}."}
+{"_id": "5951", "title": "chow-remark-extend-chern-character-to-finite-locally-free", "text": "In the discussion above we have defined the Chern character $ch(\\mathcal{E})$ of $\\mathcal{E}$ even if the rank of $\\mathcal{E}$ is not constant. See Remarks \\ref{remark-extend-to-finite-locally-free} and \\ref{remark-equalities-nonconstant-rank}."}
+{"_id": "5952", "title": "chow-remark-splitting-principle-perfect", "text": "The Chern classes of a perfect complex, when defined, satisfy a kind of splitting principle. Namely, suppose that $(S, \\delta), X, E$ are as in Definition \\ref{definition-defined-on-perfect} such that the Chern classes of $E$ are defined. Say we want to prove a relation between the bivariant classes $c_p(E)$, $P_p(E)$, and $ch_p(E)$. To do this, we may choose a bounded complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules representing $E$. Using the splitting principle (Lemma \\ref{lemma-splitting-principle}) we may assume each $\\mathcal{E}^i$ has a filtration whose successive quotients $\\mathcal{L}_{i, j}$ are invertible modules. Settting $x_{i, j} = c_1(\\mathcal{L}_{i, j})$ we see that $$ c(E) = \\prod\\nolimits_{i\\text{ even}} (1 + x_{i, j}) \\prod\\nolimits_{i\\text{ odd}} (1 + x_{i, j})^{-1} $$ and $$ P_p(E) =  \\sum\\nolimits_{i\\text{ even}} (x_{i, j})^p - \\sum\\nolimits_{i\\text{ odd}} (x_{i, j})^p $$ Formally taking the logarithm for the expression for $c(E)$ above we find that $$ \\log(c(E)) = \\sum (-1)^{p - 1}\\frac{P_p(E)}{p} $$ Looking at the construction of the polynomials $P_p$ in Example \\ref{example-power-sum} it follows that $P_p(E)$ is the exact same expression in the Chern classes of $E$ as in the case of vector bundles, in other words, we have \\begin{align*} P_1(E) & = c_1(E), \\\\ P_2(E) & = c_1(E)^2 - 2c_2(E), \\\\ P_3(E) & = c_1(E)^3 - 3c_1(E)c_2(E) + 3c_3(E), \\\\ P_4(E) & = c_1(E)^4 - 4c_1(E)^2c_2(E) + 4c_1(E)c_3(E) + 2c_2(E)^2 - 4c_4(E), \\end{align*} and so on. On the other hand, the bivariant class $P_0(E) = r(E) = ch_0(E)$ cannot be recovered from the Chern class $c(E)$ of $E$; the chern class doesn't know about the rank of the complex."}
+{"_id": "5953", "title": "chow-remark-loc-chern-classes", "text": "In the situation of Definition \\ref{definition-localized-chern} assume $E|_{X \\setminus Z}$ is finite locally free of rank $< p$. In this setting it is convenient to define $$ c^{(p)}(Z \\to X, E) = 1 + c_1(E) + \\ldots + c_{p - 1}(E) + c_p(Z \\to X, E) + c_{p + 1}(Z \\to X, E) + \\ldots $$ as an element of the algebra $A^{(p)}(Z \\to X)$ considered in Remark \\ref{remark-ring-loc-classes}."}
+{"_id": "5954", "title": "chow-remark-gysin-for-immersion", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \\to X$ be an immersion of schemes. In this situation \\begin{enumerate} \\item the conormal sheaf $\\mathcal{C}_{Z/X}$ of $Z$ in $X$ is defined (Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf}), \\item we say a pair consisting of a finite locally free $\\mathcal{O}_Z$-module $\\mathcal{N}$ and a surjection $\\sigma : \\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ is a virtual normal bundle for the immersion $Z \\to X$, \\item choose an open subscheme $U \\subset X$ such that $Z \\to X$ factors through a closed immersion $Z \\to U$ and set $c(Z \\to X, \\mathcal{N}) = c(Z \\to U, \\mathcal{N}) \\circ (U \\to X)^*$. \\end{enumerate} The bivariant class $c(Z \\to X, \\mathcal{N})$ does not depend on the choice of the open subscheme $U$. All of the lemmas have immediate counterparts for this slightly more general construction. We omit the details."}
+{"_id": "5955", "title": "chow-remark-adams-derived", "text": "Let $X$ be a scheme such that $2$ is invertible on $X$. Then the Adams operator $\\psi^2$ can be defined on the $K$-group $K_0(X) = K_0(D_{perf}(\\mathcal{O}_X))$ (Derived Categories of Schemes, Definition \\ref{perfect-definition-K-group}) in a straightforward manner. Namely, given a perfect complex $L$ on $X$ we get an action of the group $\\{\\pm 1\\}$ on $L \\otimes^\\mathbf{L} L$ by switching the factors. Then we can set $$ \\psi^2(L) = [(L \\otimes^\\mathbf{L} L)^+] - [(L \\otimes^\\mathbf{L} L)^-] $$ where $(-)^+$ denotes taking invariants and $(-)^-$ denotes taking anti-invariants (suitably defined). Using exactness of taking invariants and anti-invariants one can argue similarly to the proof of Lemma \\ref{lemma-second-adams-operator} to show that this is well defined. When $2$ is not invertible on $X$ the situation is a good deal more complicated and another approach has to be used."}
+{"_id": "5956", "title": "chow-remark-chern-classes-K", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. The Chern class map defines a canonical map $$ c : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(X) $$ by sending a generator $[\\mathcal{E}]$ on the left hand side to $c(\\mathcal{E}) = 1 + c_1(\\mathcal{E}) + c_2(\\mathcal{E}) + \\ldots$ and extending multiplicatively. Thus $-[\\mathcal{E}]$ is sent to the formal inverse $c(\\mathcal{E})^{-1}$ which is why we have the infinite product on the right hand side. This is well defined by Lemma \\ref{lemma-additivity-chern-classes}."}
+{"_id": "5957", "title": "chow-remark-chern-character-K", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. The Chern character map defines a canonical ring map $$ ch : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(X) \\otimes \\mathbf{Q} $$ by sending a generator $[\\mathcal{E}]$ on the left hand side to $ch(\\mathcal{E})$ and extending additively. This is well defined by Lemma \\ref{lemma-chern-character-additive} and a ring homomorphism by Lemma \\ref{lemma-chern-character-multiplicative}."}
+{"_id": "5958", "title": "chow-remark-perf-Z-cohomology-K", "text": "Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed subscheme. Consider the strictly full, saturated, triangulated subcategory $$ D_{Z, perf}(\\mathcal{O}_X) \\subset D(\\mathcal{O}_X) $$ consisting of perfect complexes of $\\mathcal{O}_X$-modules whose cohomology sheaves are settheoretically supported on $Z$. Denote $\\textit{Coh}_Z(X) \\subset \\textit{Coh}(X)$ the Serre subcategory of coherent $\\mathcal{O}_X$-modules whose set theoretic support is contained in $Z$. Observe that given $E \\in D_{Z, perf}(\\mathcal{O}_X)$ Zariski locally on $X$ only a finite number of the cohomology sheaves $H^i(E)$ are nonzero (and they are all settheoretically supported on $Z$). Hence we can define $$ K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow K_0(\\textit{Coh}_Z(X)) = K'_0(Z) $$ (equality by Lemma \\ref{lemma-K-coherent-supported-on-closed}) by the rule $$ E \\longmapsto [\\bigoplus\\nolimits_{i \\in \\mathbf{Z}} H^{2i}(E)] - [\\bigoplus\\nolimits_{i \\in \\mathbf{Z}} H^{2i + 1}(E)] $$ This works because given a distinguished triangle in $D_{Z, perf}(\\mathcal{O}_X)$ we have a long exact sequence of cohomology sheaves."}
+{"_id": "5959", "title": "chow-remark-perf-Z-regular", "text": "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$ be as in Remark \\ref{remark-perf-Z-cohomology-K}. Assume $X$ is Noetherian regular of finite dimension. Then there is a canonical map $$ K_0(\\textit{Coh}(Z)) \\longrightarrow K_0(D_{Z, perf}(\\mathcal{O}_X)) $$ defined as follows. For any coherent $\\mathcal{O}_Z$-module $\\mathcal{F}$ denote $\\mathcal{F}[0]$ the object of $D(\\mathcal{O}_X)$ which has $\\mathcal{F}$ in degree $0$ and is zero in other degrees. Then $\\mathcal{F}[0]$ is a perfect complex on $X$ by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-regular}. Hence $\\mathcal{F}[0]$ is an object of $D_{Z, perf}(\\mathcal{O}_X)$. On the other hand, given a short exact sequence $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$ of coherent $\\mathcal{O}_Z$-modules we obtain a distinguished triangle $\\mathcal{F}[0] \\to \\mathcal{F}'[0] \\to \\mathcal{F}''[0] \\to \\mathcal{F}[1]$, see Derived Categories, Section \\ref{derived-section-canonical-delta-functor}. This shows that we obtain a map $K_0(\\textit{Coh}(Z)) \\to K_0(D_{Z, perf}(\\mathcal{O}_X))$ by sending $[\\mathcal{F}]$ to $[\\mathcal{F}[0]]$ with apologies for the horrendous notation."}
+{"_id": "5960", "title": "chow-remark-localized-chern-classes-K", "text": "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$ be as in Remark \\ref{remark-perf-Z-cohomology-K}. Assume $X$ is quasi-compact, has the resolution property, and is of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. The localized Chern classes define a canonical map $$ c(Z \\to X, -) : K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow A^0(X) \\times \\prod\\nolimits_{i \\geq 1} A^i(Z \\to X) $$ by sending a generator $[E]$ on the left hand side to $$ c(Z \\to X, E) = 1 + c_1(Z \\to X, E) + c_2(Z \\to X, E) + \\ldots $$ and extending multiplicatively (with product on the right hand side as in Remark \\ref{remark-ring-loc-classes}). This makes sense because by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-resolution-property-perfect-complex} and Definition \\ref{definition-localized-chern} $c_i(Z \\to X, E) $ are defined for all $i \\geq 1$. It is well defined by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-resolution-property-map-perfect-complex} (every map in $D_{Z, perf}(\\mathcal{O}_X)$ can be represented by a map of bounded complexes of finite locally frees) and Lemma \\ref{lemma-additivity-loc-chern-c}."}
+{"_id": "5961", "title": "chow-remark-localized-chern-character-K", "text": "Let $X$, $Z$, $D_{Z, perf}(\\mathcal{O}_X)$ be as in Remark \\ref{remark-perf-Z-cohomology-K}. Assume $X$ is quasi-compact, has the resolution property, and is of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. The localized Chern character defines a canonical additive and multiplicative map $$ ch(Z \\to X, -) : K_0(D_{Z, perf}(\\mathcal{O}_X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} A^i(Z \\to X) $$ by sending a generator $[E]$ on the left hand side to $ch(Z \\to X, E)$ and extending additively. This makes sense because because $ch(Z \\to X, E)$ is defined by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-resolution-property-perfect-complex} and Definition \\ref{definition-localized-chern}. It is well defined by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-resolution-property-map-perfect-complex} (every map in $D_{Z, perf}(\\mathcal{O}_X)$ can be represented by a map of bounded complexes of finite locally frees) and Lemma \\ref{lemma-additivity-loc-chern-P}. The multiplication on $K_0(D_{Z, perf}(X))$ is defined using derived tensor product (Derived Categories of Schemes, Remark \\ref{perfect-remark-perf-Z}) hence $ch(\\alpha \\beta) = ch(\\alpha) ch(\\beta)$ by Lemma \\ref{lemma-loc-chern-tensor-product}."}
+{"_id": "5962", "title": "chow-remark-chern-classes-agree", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$ and assume $X$ is quasi-compact and has the resolution property. With $Z = X$ and notation as in Remarks \\ref{remark-localized-chern-classes-K} and \\ref{remark-localized-chern-character-K} we have $D_{Z, perf}(\\mathcal{O}_X) = D_{perf}(\\mathcal{O}_X)$ and we see that $$ K_0(D_{Z, perf}(\\mathcal{O}_X)) = K_0(D_{perf}(\\mathcal{O}_X)) = K_0(X) $$ see  Derived Categories of Schemes, Definition \\ref{perfect-definition-K-group}. Hence we get $c : K_0(X) \\to \\prod A^i(X)$ and $ch : K_0(X) \\to \\prod A^i(X)$ from Remarks \\ref{remark-localized-chern-classes-K} and \\ref{remark-localized-chern-character-K}. Via the equality $K_0(\\textit{Vect}(X)) = K_0(X)$ of Derived Categories of Schemes, Lemma \\ref{perfect-lemma-K-is-old-K} these maps agree with the maps constructed in Remarks \\ref{remark-chern-classes-K} and \\ref{remark-chern-character-K}."}
+{"_id": "5963", "title": "chow-remark-gysin-chern-classes", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. Assume the gysin map exists for $f$. Then $f^! \\circ c_i(\\mathcal{E}) = c_i(f^*\\mathcal{E}) \\circ f^!$ and similarly for the Chern character, see Lemma \\ref{lemma-lci-gysin-commutes}. If $X$ and $Y$ satisfy the equivalent conditions of Lemma \\ref{lemma-locally-equidimensional} and $Y$ is Cohen-Macaulay (for example), then $f^![Y] = [X]$ by Lemma \\ref{lemma-lci-gysin-easy}. In this case we also get $f^!(c_i(\\mathcal{E}) \\cap [Y]) = c_i(f^*\\mathcal{E}) \\cap [X]$ and similarly for the Chern character."}
+{"_id": "5964", "title": "chow-remark-commuting-exterior", "text": "The upshot of Lemmas \\ref{lemma-chow-cohomology-towards-point} and \\ref{lemma-chow-cohomology-towards-point-commutes} is the following. Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $\\alpha \\in \\CH_*(X)$. Let $Y \\to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \\in A^q(Y \\to Z)$. Then $$ \\alpha \\times (c' \\cap \\beta) = c' \\cap (\\alpha \\times \\beta) $$ in $\\CH_*(X \\times_k Y)$ for any $\\beta \\in \\CH_*(Z)$. Namely, this follows by taking $c = c_\\alpha \\in A^*(X \\to \\Spec(k))$ the bivariant class corresponding to $\\alpha$, see proof of Lemma \\ref{lemma-chow-cohomology-towards-point}."}
+{"_id": "5965", "title": "chow-remark-commuting-exterior-dim-1", "text": "The upshot of Lemmas \\ref{lemma-chow-cohomology-towards-base-dim-1} and \\ref{lemma-chow-cohomology-towards-base-dim-1-commutes} is the following. Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $\\alpha \\in \\CH_*(X)$. Let $Y \\to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \\in A^q(Y \\to Z)$. Then $$ \\alpha \\times (c' \\cap \\beta) = c' \\cap (\\alpha \\times \\beta) $$ in $\\CH_*(X \\times_S Y)$ for any $\\beta \\in \\CH_*(Z)$. Namely, this follows by taking $c = c_\\alpha \\in A^*(X \\to S)$ the bivariant class corresponding to $\\alpha$, see proof of Lemma \\ref{lemma-chow-cohomology-towards-base-dim-1}."}
+{"_id": "5966", "title": "chow-remark-explain-determinant", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring and assume either the characteristic of $\\kappa$ is zero or it is $p$ and $p R = 0$. Let $M_1, \\ldots, M_n$ be finite length $R$-modules. We will show below that there exists an ideal $I \\subset \\mathfrak m$ annihilating $M_i$ for $i = 1, \\ldots, n$ and a section $\\sigma : \\kappa \\to R/I$ of the canonical surjection $R/I \\to \\kappa$. The restriction $M_{i, \\kappa}$ of $M_i$ via $\\sigma$ is a $\\kappa$-vector space of dimension $l_i = \\text{length}_R(M_i)$ and using Lemma \\ref{lemma-determinant-quotient-ring} we see that $$ \\det\\nolimits_\\kappa(M_i) = \\wedge_\\kappa^{l_i}(M_{i, \\kappa}) $$ These isomorphisms are compatible with the isomorphisms $\\gamma_{K \\to M \\to L}$ of Lemma \\ref{lemma-det-exact-sequences} for short exact sequences of finite length $R$-modules annihilated by $I$. The conclusion is that verifying a property of $\\det_\\kappa$ often reduces to verifying corresponding properties of the usual determinant on the category finite dimensional vector spaces. \\medskip\\noindent For $I$ we can take the annihilator (Algebra, Definition \\ref{algebra-definition-annihilator}) of the module $M = \\bigoplus M_i$. In this case we see that $R/I \\subset \\text{End}_R(M)$ hence has finite length. Thus $R/I$ is an Artinian local ring with residue field $\\kappa$. Since an Artinian local ring is complete we see that $R/I$ has a coefficient ring by the Cohen structure theorem (Algebra, Theorem \\ref{algebra-theorem-cohen-structure-theorem}) which is a field by our assumption on $R$."}
+{"_id": "5967", "title": "chow-remark-more-elementary", "text": "Here is a more down to earth description of the determinant introduced above. Let $R$ be a local ring with residue field $\\kappa$. Let $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is exact. Let us abbreviate $I_\\varphi = \\Im(\\varphi)$, $I_\\psi = \\Im(\\psi)$ as above. Assume that $\\text{length}_R(I_\\varphi) = a$ and $\\text{length}_R(I_\\psi) = b$, so that $a + b = \\text{length}_R(M)$ by exactness. Choose admissible sequences $x_1, \\ldots, x_a \\in I_\\varphi$ and $y_1, \\ldots, y_b \\in I_\\psi$ such that the symbol $[x_1, \\ldots, x_a]$ generates $\\det_\\kappa(I_\\varphi)$ and the symbol $[x_1, \\ldots, x_b]$ generates $\\det_\\kappa(I_\\psi)$. Choose $\\tilde x_i \\in M$ such that $\\varphi(\\tilde x_i) = x_i$. Choose $\\tilde y_j \\in M$ such that $\\psi(\\tilde y_j) = y_j$. Then $\\det_\\kappa(M, \\varphi, \\psi)$ is characterized by the equality $$ [x_1, \\ldots, x_a, \\tilde y_1, \\ldots, \\tilde y_b] = (-1)^{ab} \\det\\nolimits_\\kappa(M, \\varphi, \\psi) [y_1, \\ldots, y_b, \\tilde x_1, \\ldots, \\tilde x_a] $$ in $\\det_\\kappa(M)$. This also explains the sign."}
+{"_id": "6224", "title": "flat-remark-finite-presentation", "text": "Note that the $R$-algebras $B_i$ for all $i$ and $A_i$ for $i \\geq 2$ are of finite presentation over $R$. If $S$ is of finite presentation over $R$, then it is also the case that $A_1$ is of finite presentation over $R$. In this case all the ring maps in the complete d\\'evissage are of finite presentation. See Algebra, Lemma \\ref{algebra-lemma-compose-finite-type}. Still assuming $S$ of finite presentation over $R$ the following are equivalent \\begin{enumerate} \\item $M$ is of finite presentation over $S$, \\item $M_1$ is of finite presentation over $A_1$, \\item $M_1$ is of finite presentation over $B_1$, \\item each $M_i$ is of finite presentation both as an $A_i$-module and as a $B_i$-module. \\end{enumerate} The equivalences (1) $\\Leftrightarrow$ (2) and (2) $\\Leftrightarrow$ (3) follow from Algebra, Lemma \\ref{algebra-lemma-finite-finitely-presented-extension}. If $M_1$ is finitely presented, so is $\\Coker(\\alpha_1)$ (see Algebra, Lemma \\ref{algebra-lemma-extension}) and hence $M_2$, etc."}
+{"_id": "6225", "title": "flat-remark-same-notion", "text": "Let $A \\to B$ be a finite type ring map and let $N$ be a finite $B$-module. Let $\\mathfrak q$ be a prime of $B$ lying over the prime $\\mathfrak r$ of $A$. Set $X = \\Spec(B)$, $S = \\Spec(A)$ and $\\mathcal{F} = \\widetilde{N}$ on $X$. Let $x$ be the point corresponding to $\\mathfrak q$ and let $s \\in S$ be the point corresponding to $\\mathfrak p$. Then \\begin{enumerate} \\item if there exists a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$ then there exists a complete d\\'evissage of $N/B/A$ over $\\mathfrak p$, and \\item there exists a complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$ if and only if there exists a complete d\\'evissage of $N/B/A$ at $\\mathfrak q$. \\end{enumerate} There is just a small twist in that we omitted the condition on the relative dimension in the formulation of ``a complete d\\'evissage of $N/B/A$ over $\\mathfrak p$'' which is why the implication in (1) only goes in one direction. The notion of a complete d\\'evissage at $\\mathfrak q$ does have this condition built in. In any case we will only use that existence for $\\mathcal{F}/X/S$ implies the existence for $N/B/A$."}
+{"_id": "6226", "title": "flat-remark-how-in-RG", "text": "Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian} is a key step in the development of results in this chapter. The analogue of this lemma in \\cite{GruRay} is \\cite[I Proposition 3.3.1]{GruRay}: If $R \\to S$ is smooth with geometrically integral fibres, then $S$ is projective as an $R$-module. This is a special case of Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}, but as we will later improve on this lemma anyway, we do not gain much from having a stronger result at this point. We briefly sketch the proof of this as it is given in \\cite{GruRay}. \\begin{enumerate} \\item First reduce to the case where $R$ is Noetherian as above. \\item Since projectivity descends through faithfully flat ring maps, see Algebra, Theorem \\ref{algebra-theorem-ffdescent-projectivity} we may work locally in the fppf topology on $R$, hence we may assume that $R \\to S$ has a section $\\sigma : S \\to R$. (Just by the usual trick of base changing to $S$.) Set $I = \\Ker(S \\to R)$. \\item Localizing a bit more on $R$ we may assume that $I/I^2$ is a free $R$-module and that the completion $S^\\wedge$ of $S$ with respect to $I$ is isomorphic to $R[[t_1, \\ldots, t_n]]$, see Morphisms, Lemma \\ref{morphisms-lemma-section-smooth-morphism}. Here we are using that $R \\to S$ is smooth. \\item To prove that $S$ is projective as an $R$-module, it suffices to prove that $S$ is flat, countably generated and Mittag-Leffler as an $R$-module, see Algebra, Lemma \\ref{algebra-lemma-countgen-projective}. The first two properties are evident. Thus it suffices to prove that $S$ is Mittag-Leffler as an $R$-module. By Algebra, Lemma \\ref{algebra-lemma-power-series-ML} the module $R[[t_1, \\ldots, t_n]]$ is Mittag-Leffler over $R$. Hence Algebra, Lemma \\ref{algebra-lemma-pure-submodule-ML} shows that it suffices to show that the $S \\to S^\\wedge$ is universally injective as a map of $R$-modules. \\item Apply Lemma \\ref{lemma-base-change-universally-flat} to see that $S \\to S^\\wedge$ is $R$-universally injective. Namely, as $R \\to S$ has geometrically integral fibres, any associated point of any fibre ring is just the generic point of the fibre ring which is in the image of $\\Spec(S^\\wedge) \\to \\Spec(S)$. \\end{enumerate} There is an analogy between the proof as sketched just now, and the development of the arguments leading to the proof of Lemma \\ref{lemma-fibres-irreducible-flat-projective-nonnoetherian}. In both a completion plays an essential role, and both times the assumption of having geometrically integral fibres assures one that the map from $S$ to the completion of $S$ is $R$-universally injective."}
+{"_id": "6227", "title": "flat-remark-complete-devissage-flat-finitely-presented-module", "text": "There is a variant of Lemma \\ref{lemma-complete-devissage-flat-finitely-presented-module} where we weaken the flatness condition by assuming only that $N$ is flat at some given prime $\\mathfrak q$ lying over $\\mathfrak r$ but where we strengthen the d\\'evissage condition by assuming the existence of a complete d\\'evissage {\\it at $\\mathfrak q$}. Compare with Lemma \\ref{lemma-complete-devissage-flat-finite-type-module}."}
+{"_id": "6228", "title": "flat-remark-finite-type-flat", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $X \\to S$ is locally of finite type, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and \\item the set of weakly associated points of $S$ is locally finite in $S$. \\end{enumerate} Then $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$ is open in $X$ and $\\mathcal{F}|_U$ is flat over $S$ and locally finitely presented relative to $S$ (see More on Morphisms, Definition \\ref{more-morphisms-definition-relatively-finitely-presented-sheaf}). If we ever need this result in the Stacks project we will convert this remark into a lemma with a proof."}
+{"_id": "6229", "title": "flat-remark-finite-type-flat-algebra", "text": "Let $R \\to S$ be a ring map of finite type. Let $M$ be a finite $S$-module. Assume $\\text{WeakAss}_R(R)$ is finite. Then $$ U = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\} $$ is open in $\\Spec(S)$ and for every $g \\in S$ such that $D(g) \\subset U$ the localization $M_g$ is flat over $R$ and an $S_g$-module finitely presented relative to $R$ (see More on Algebra, Definition \\ref{more-algebra-definition-relatively-finitely-presented}). If we ever need this result in the Stacks project we will convert this remark into a lemma with a proof."}
+{"_id": "6230", "title": "flat-remark-discuss-finite-type", "text": "Let $f : X \\to S$ be a morphism which is locally of finite type and $\\mathcal{F}$ a quasi-coherent finite type $\\mathcal{O}_X$-module. In this case it is still true that (1) and (2) above are equivalent because the proof of Lemma \\ref{lemma-quasi-finite-impurity-elementary} does not use that $f$ is quasi-compact. It is also clear that (3) and (4) are equivalent. However, we don't know if (1) and (3) are equivalent. In this case it may sometimes be more convenient to define purity using the equivalent conditions (3) and (4) as is done in \\cite{GruRay}. On the other hand, for many applications it seems that the correct notion is really that of being universally pure."}
+{"_id": "6231", "title": "flat-remark-flattening-local-scheme-theoretic", "text": "Here is a scheme theoretic reformulation of Theorem \\ref{theorem-flattening-local}. Let $(X, x) \\to (S, s)$ be a morphism of pointed schemes which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Assume $S$ henselian local with closed point $s$. There exists a closed subscheme $Z \\subset S$ with the following property: for any morphism of pointed schemes $(T, t) \\to (S, s)$ the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}_T$ is flat over $T$ at all points of the fibre $X_t$ which map to $x \\in X_s$, and \\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$. \\end{enumerate} Moreover, if $X \\to S$ is of finite presentation at $x$ and $\\mathcal{F}_x$ of finite presentation over $\\mathcal{O}_{X, x}$, then $Z \\to S$ is of finite presentation."}
+{"_id": "6232", "title": "flat-remark-flattening-complete-noetherian", "text": "Tracing the proof of Lemma \\ref{lemma-freebie} to its origins we find a long and winding road. But if we assume that \\begin{enumerate} \\item $f$ is of finite type, \\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, \\item $E = X_s$, and \\item $S$ is the spectrum of a Noetherian complete local ring. \\end{enumerate} then there is a proof relying completely on more elementary algebra as follows: first we reduce to the case where $X$ is affine by taking a finite affine open cover. In this case $Z$ exists by More on Algebra, Lemma \\ref{more-algebra-lemma-flattening-complete-local-universal-property}. The key step in this proof is constructing the closed subscheme $Z$ step by step inside the truncations $\\Spec(\\mathcal{O}_{S, s}/\\mathfrak m_s^n)$. This relies on the fact that flattening stratifications always exist when the base is Artinian, and the fact that $\\mathcal{O}_{S, s} = \\lim \\mathcal{O}_{S, s}/\\mathfrak m_s^n$."}
+{"_id": "6233", "title": "flat-remark-correct-generality", "text": "The result in this section can be generalized. It is probably correct if we only assume $X \\to \\Spec(A)$ to be separated, of finite presentation, and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here."}
+{"_id": "6234", "title": "flat-remark-successive-blowups", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \\subset S$ be a quasi-compact open subscheme. Given a $U$-admissible blowup $S' \\to S$ we denote $X'$ the strict transform of $X$ and $\\mathcal{F}'$ the strict transform of $\\mathcal{F}$ which we think of as a quasi-coherent module on $X'$ (via Divisors, Lemma \\ref{divisors-lemma-strict-transform}). Let $P$ be a property of $\\mathcal{F}/X/S$ which is stable under strict transform (as above) for $U$-admissible blowups. The general problem in this section is: Show (under auxiliary conditions on $\\mathcal{F}/X/S$) there exists a $U$-admissible blowup $S' \\to S$ such that the strict transform $\\mathcal{F}'/X'/S'$ has $P$. \\medskip\\noindent The general strategy will be to use that a composition of $U$-admissible blowups is a $U$-admissible blowup, see Divisors, Lemma \\ref{divisors-lemma-composition-admissible-blowups}. In fact, we will make use of the more precise Divisors, Lemma \\ref{divisors-lemma-composition-finite-type-blowups} and combine it with Divisors, Lemma \\ref{divisors-lemma-strict-transform-composition-blowups}. The result is that it suffices to find a sequence of $U$-admissible blowups $$ S = S_0 \\leftarrow S_1 \\leftarrow \\ldots \\leftarrow S_n $$ such that, setting $\\mathcal{F}_0 = \\mathcal{F}$ and $X_0 = X$ and setting $\\mathcal{F}_i/X_i$ equal to the strict transform of $\\mathcal{F}_{i - 1}/X_{i - 1}$, we arrive at $\\mathcal{F}_n/X_n/S_n$ with property $P$. \\medskip\\noindent In particular, choose a finite type quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_S$ such that $V(\\mathcal{I}) = S \\setminus U$, see Properties, Lemma \\ref{properties-lemma-quasi-coherent-finite-type-ideals}. Let $S' \\to S$ be the blowup in $\\mathcal{I}$ and let $E \\subset S'$ be the exceptional divisor (Divisors, Lemma \\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}). Then we see that we've reduced the problem to the case where there exists an effective Cartier divisor $D \\subset S$ whose support is $X \\setminus U$. In particular we may assume $U$ is scheme theoretically dense in $S$ (Divisors, Lemma \\ref{divisors-lemma-complement-effective-Cartier-divisor}). \\medskip\\noindent Suppose that $P$ is local on $S$: If $S = \\bigcup S_i$ is a finite open covering by quasi-compact opens and $P$ holds for $\\mathcal{F}_{S_i}/X_{S_i}/S_i$ then $P$ holds for $\\mathcal{F}/X/S$. In this case the general problem above is local on $S$ as well, i.e., if given $s \\in S$ we can find a quasi-compact open neighbourhood $W$ of $s$ such that the problem for $\\mathcal{F}_W/X_W/W$ is solvable, then the problem is solvable for $\\mathcal{F}/X/S$. This follows from Divisors, Lemmas \\ref{divisors-lemma-extend-admissible-blowups} and \\ref{divisors-lemma-dominate-admissible-blowups}."}
+{"_id": "6235", "title": "flat-remark-when-you-have-a-complex", "text": "Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object such that $H^i(E)$ is a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$. This property sometimes allows one to reduce questions about $E$ to questions about $H^i(E)$. For example, suppose $$ \\mathcal{E}^a \\xrightarrow{d^a} \\ldots \\xrightarrow{d^{b - 2}} \\mathcal{E}^{b - 1} \\xrightarrow{d^{b - 1}} \\mathcal{E}^b $$ is a bounded complex of finite locally free $\\mathcal{O}_X$-modules representing $E$. Then $\\Im(d^i)$ and $\\Ker(d^i)$ are finite locally free $\\mathcal{O}_X$-modules for all $i$. Namely, suppose by induction we know this for all indices bigger than $i$. Then we can first use the short exact sequence $$ 0 \\to \\Im(d^i) \\to \\Ker(d^{i + 1}) \\to H^{i + 1}(E) \\to 0 $$ and the assumption that $H^{i + 1}(E)$ is perfect of tor dimension $\\leq 1$ to conclude that $\\Im(d^i)$ is finite locally free. The same argument used again for the short exact sequence $$ 0 \\to \\Ker(d^i) \\to \\mathcal{E}^i \\to \\Im(d^i) \\to 0 $$ then gives that $\\Ker(d^i)$ is finite locally free. It follows that the distinguished triangles $$ \\tau_{\\leq k - 1}E \\to \\tau_{\\leq k}E \\to H^k(E)[-k] \\to (\\tau_{\\leq k - 1}E)[1] $$ are represented by the following short exact sequences of bounded complexes of finite locally free modules $$ \\begin{matrix} & & & & & & 0 \\\\ & & & & & & \\downarrow \\\\ \\mathcal{E}^a & \\to & \\ldots & \\to & \\mathcal{E}^{k - 2} & \\to & \\Ker(d^{k - 1}) \\\\ \\downarrow & & & & \\downarrow & & \\downarrow \\\\ \\mathcal{E}^a & \\to & \\ldots & \\to & \\mathcal{E}^{k - 2} & \\to & \\mathcal{E}^{k - 1} & \\to & \\Ker(d^k) \\\\ & & & & & & \\downarrow & & \\downarrow \\\\ & & & & & & \\Im(d^{k - 1}) & \\to & \\Ker(d^k) \\\\ & & & & & & \\downarrow \\\\ & & & & & & 0 \\end{matrix} $$ Here the complexes are the rows and the ``obvious'' zeros are omitted from the display."}
+{"_id": "6236", "title": "flat-remark-Leta", "text": "Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. Let $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. By Cohomology, Lemma \\ref{cohomology-lemma-K-flat-resolution} there exists a quasi-isomorphism $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ such that $\\mathcal{F}^\\bullet$ is a K-flat complex whose terms are flat $\\mathcal{O}_X$-modules. (Even if $\\mathcal{G}^\\bullet$ is a complex of quasi-coherent $\\mathcal{O}_X$-modules, in general $\\mathcal{F}^\\bullet$ will not be so.) It follows that $\\mathcal{F}^i$ is $\\mathcal{I}$-torsion free for all $i$. In this situation we define $$ L\\eta_\\mathcal{I} \\mathcal{G}^\\bullet = \\eta_\\mathcal{I} \\mathcal{F}^\\bullet $$ This is independent of the choice of the K-flat resolution by Lemma \\ref{lemma-eta-qis}. We obtain a functor $L\\eta_\\mathcal{I} : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$. Beware that this functor isn't exact, i.e., does not tranform distinguished triangles into distinguished triangles."}
+{"_id": "6237", "title": "flat-remark-complex-and-divisor-ideal", "text": "In Situation \\ref{situation-complex-and-divisor} for any $i \\in \\mathbf{Z}$ there exists a finite type quasi-coherent sheaf of ideals $\\mathcal{J}_i \\subset \\mathcal{O}_X$ with the following property: for any $U \\subset X$ open such that $\\mathcal{I}|_U$, $\\mathcal{E}^i|_U$, and $\\mathcal{E}^{i + 1}|_U$ are free of ranks $1$, $r_i$, and $r_{i + 1}$, the ideal $\\mathcal{J}_i$ is generated by the $r_i \\times r_i$ minors of the map $$ 1, d^i : \\mathcal{I}\\mathcal{E}^i \\longrightarrow \\mathcal{E}^i \\oplus \\mathcal{I}\\mathcal{E}^{i + 1} $$ with notation as in Section \\ref{section-eta}. By convention we set $\\mathcal{J}_i|_U = \\mathcal{O}_U$ if $r_i = 0$. Observe that $\\mathcal{I}^{r_i}|_U \\subset \\mathcal{J}_i|_U$ in other words, the closed subscheme $V(\\mathcal{J})$ is set theoretically contained in $D$. Formation of the ideal $\\mathcal{J}_i$ commutes with base change by any morphism $f : Y \\to X$ such that the pullback of $D$ by $f$ is defined (Divisors, Definition \\ref{divisors-definition-pullback-effective-Cartier-divisor})."}
+{"_id": "6361", "title": "curves-remark-classical-linear-series", "text": "Let $X$ be a smooth projective curve over an algebraically closed field $k$. We say two effective Cartier divisors $D, D' \\subset X$ are {\\it linearly equivalent} if and only if $\\mathcal{O}_X(D) \\cong \\mathcal{O}_X(D')$ as $\\mathcal{O}_X$-modules. Since $\\Pic(X) = \\text{Cl}(X)$ (Divisors, Lemma \\ref{divisors-lemma-local-rings-UFD-c1-bijective}) we see that $D$ and $D'$ are linearly equivalent if and only if the Weil divisors associated to $D$ and $D'$ define the same element of $\\text{Cl}(X)$. Given an effective Cartier divisor $D \\subset X$ of degree $d$ the {\\it complete linear system} or {\\it complete linear series} $|D|$ of $D$ is the set of effective Cartier divisors $E \\subset X$ which are linearly equivalent to $D$. Another way to say it is that $|D|$ is the set of closed points of the fibre of the morphism $$ \\gamma_d : \\underline{\\Hilbfunctor}^d_{X/k} \\longrightarrow \\underline{\\Picardfunctor}^d_{X/k} $$ (Picard Schemes of Curves, Lemma \\ref{pic-lemma-picard-pieces}) over the closed point corresponding to $\\mathcal{O}_X(D)$. This gives $|D|$ a natural scheme structure and it turns out that $|D| \\cong \\mathbf{P}^m_k$ with $m + 1 = h^0(\\mathcal{O}_X(D))$. In fact, more canonically we have $$ |D| = \\mathbf{P}(H^0(X, \\mathcal{O}_X(D))^\\vee) $$ where $(-)^\\vee$ indicates $k$-linear dual and $\\mathbf{P}$ is as in Constructions, Example \\ref{constructions-example-projective-space}. In this language a {\\it linear system} or a {\\it linear series} on $X$ is a closed subvariety $L \\subset |D|$ which can be cut out by linear equations. If $L$ has dimension $r$, then $L = \\mathbf{P}(V^\\vee)$ where $V \\subset H^0(X, \\mathcal{O}_X(D))$ is a linear subspace of dimension $r + 1$. Thus the classical linear series $L \\subset |D|$ corresponds to the linear series $(\\mathcal{O}_X(D), V)$ as defined above."}
+{"_id": "6362", "title": "curves-remark-rework-duality-locally-free", "text": "Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$. Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}. If $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module with dual $\\mathcal{E}^\\vee$ then we have canonical isomorphisms $$ \\Hom_k(H^{-i}(X, \\mathcal{E}), k) = H^i(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_X^\\bullet) $$ This follows from the lemma and Cohomology, Lemma \\ref{cohomology-lemma-dual-perfect-complex}. If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then we have canonical isomorphisms $$ \\Hom_k(H^{-i}(X, \\mathcal{E}), k) = H^{1 - i}(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\omega_X) $$ by Lemma \\ref{lemma-duality-dim-1-CM}. In particular if $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, then we have $$ \\dim_k H^0(X, \\mathcal{L}) = \\dim_k H^1(X, \\mathcal{L}^{\\otimes -1} \\otimes_{\\mathcal{O}_X} \\omega_X) $$ and $$ \\dim_k H^1(X, \\mathcal{L}) = \\dim_k H^0(X, \\mathcal{L}^{\\otimes -1} \\otimes_{\\mathcal{O}_X} \\omega_X) $$"}
+{"_id": "6363", "title": "curves-remark-genus-higher-dimension", "text": "Suppose that $X$ is a $d$-dimensional proper smooth variety over an algebraically closed field $k$. Then the {\\it arithmetic genus} is often defined as $p_a(X) = (-1)^d(\\chi(X, \\mathcal{O}_X) - 1)$ and the {\\it geometric genus} as $p_g(X) = \\dim_k H^0(X, \\Omega^d_{X/k})$. In this situation the arithmetic genus and the geometric genus no longer agree even though it is still true that $\\omega_X \\cong \\Omega_{X/k}^d$. For example, if $d = 2$, then we have \\begin{align*} p_a(X) - p_g(X) & = h^0(X, \\mathcal{O}_X) - h^1(X, \\mathcal{O}_X) + h^2(X, \\mathcal{O}_X) - 1 - h^0(X, \\Omega^2_{X/k}) \\\\ & = - h^1(X, \\mathcal{O}_X) + h^2(X, \\mathcal{O}_X) - h^0(X, \\omega_X) \\\\ & = - h^1(X, \\mathcal{O}_X) \\end{align*} where $h^i(X, \\mathcal{F}) = \\dim_k H^i(X, \\mathcal{F})$ and where the last equality follows from duality. Hence for a surface the difference $p_g(X) - p_a(X)$ is always nonnegative; it is sometimes called the irregularity of the surface. If $X = C_1 \\times C_2$ is a product of smooth projective curves of genus $g_1$ and $g_2$, then the irregularity is $g_1 + g_2$."}
+{"_id": "6364", "title": "curves-remark-quadratic-extension", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local $k$-algebra. Assume that either $(A, \\mathfrak m, \\kappa)$ is as in Lemma \\ref{lemma-nodal-algebraic}, or $A$ is Nagata as in Lemma \\ref{lemma-2-branches-delta-1}, or $A$ is complete and as in Lemma \\ref{lemma-fitting-ideal}. Then $A$ defines canonically a degree $2$ separable $\\kappa$-algebra $\\kappa'$ as follows \\begin{enumerate} \\item let $q = ax^2 + bxy + cy^2$ be a nondegenerate quadric as in Lemma \\ref{lemma-nodal-algebraic} with coordinates $x, y$ chosen such that $a \\not = 0$ and set $\\kappa' = \\kappa[x]/(ax^2 + bx + c)$, \\item let $A' \\supset A$ be the integral closure of $A$ in its total ring of fractions and set $\\kappa' = A'/\\mathfrak m A'$, or \\item let $\\kappa'$ be the $\\kappa$-algebra such that $\\text{Proj}(\\bigoplus_{n \\geq 0} \\mathfrak m^n/\\mathfrak m^{n + 1}) = \\Spec(\\kappa')$. \\end{enumerate} The equivalence of (1) and (2) was shown in the proof of Lemma \\ref{lemma-2-branches-delta-1}. We omit the equivalence of this with (3). If $X$ is a locally Noetherian $k$-scheme and $x \\in X$ is a point such that $\\mathcal{O}_{X, x} = A$, then (3) shows that $\\Spec(\\kappa') = X^\\nu \\times_X \\Spec(\\kappa)$ where $\\nu : X^\\nu \\to X$ is the normalization morphism."}
+{"_id": "6365", "title": "curves-remark-trivial-quadratic-extension", "text": "Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be as in Remark \\ref{remark-quadratic-extension} and let $\\kappa'/\\kappa$ be the associated separable algebra of degree $2$. Then the following are equivalent \\begin{enumerate} \\item $\\kappa' \\cong \\kappa \\times \\kappa$ as $\\kappa$-algebra, \\item the form $q$ of Lemma \\ref{lemma-nodal-algebraic} can be chosen to be $xy$, \\item $A$ has two branches, \\item the extension $A'/A$ of Lemma \\ref{lemma-2-branches-delta-1} has two maximal ideals, and \\item $A^\\wedge \\cong \\kappa[[x, y]]/(xy)$ as a $k$-algebra. \\end{enumerate} The equivalence between these conditions has been shown in the proof of Lemma \\ref{lemma-2-branches-delta-1}. If $X$ is a locally Noetherian $k$-scheme and $x \\in X$ is a point such that $\\mathcal{O}_{X, x} = A$, then this means exactly that there are two points $x_1, x_2$ of the normalization $X^\\nu$ lying over $x$ and that $\\kappa(x) = \\kappa(x_1) = \\kappa(x_2)$."}
+{"_id": "6761", "title": "etale-cohomology-remark-i-is-j", "text": "In the last statement, it is essential not to forget the case where $i = j$ which is in general a highly nontrivial condition (unlike in the Zariski topology). In fact, frequently important coverings have only one element."}
+{"_id": "6762", "title": "etale-cohomology-remark-empty-covering", "text": "For the empty covering (where $I = \\emptyset$), this implies that $\\mathcal{F}(\\emptyset)$ is an empty product, which is a final object in the corresponding category (a singleton, for both $\\textit{Sets}$ and $\\textit{Ab}$)."}
+{"_id": "6763", "title": "etale-cohomology-remark-fpqc", "text": "The first condition corresponds to fp, which stands for {\\it fid\\`element plat}, faithfully flat in french, and the second to qc, {\\it quasi-compact}. The second part of the first condition is unnecessary when the second condition holds."}
+{"_id": "6764", "title": "etale-cohomology-remark-fpqc-finest", "text": "The fpqc is finer than the Zariski, \\'etale, smooth, syntomic, and fppf topologies. Hence any presheaf satisfying the sheaf condition for the fpqc topology will be a sheaf on the Zariski, \\'etale, smooth, syntomic, and fppf sites. In particular representable presheaves will be sheaves on the \\'etale site of a scheme for example."}
+{"_id": "6765", "title": "etale-cohomology-remark-final-object", "text": "In the case where $\\mathcal{C}$ has a final object, e.g.\\ $S$, it suffices to check the condition of the definition for $U = S$ in the above statement. See Modules on Sites, Lemma \\ref{sites-modules-lemma-local-final-object}."}
+{"_id": "6766", "title": "etale-cohomology-remark-presheaves-no-topology", "text": "Observe that all of the preceding statements are about presheaves so we haven't made use of the topology yet."}
+{"_id": "6767", "title": "etale-cohomology-remark-grothendieck-ss", "text": "This is a Grothendieck spectral sequence for the composition of functors $$ \\textit{Ab}(\\mathcal{C}) \\longrightarrow \\textit{PAb}(\\mathcal{C}) \\xrightarrow{\\check H^0} \\textit{Ab}. $$"}
+{"_id": "6768", "title": "etale-cohomology-remark-refinement", "text": "In the statement of Lemma \\ref{lemma-cech-complex} the covering $\\mathcal{U}$ is a refinement of $\\mathcal{V}$ but not the other way around. Coverings of the form $\\{V \\to U\\}$ do not form an initial subcategory of the category of all coverings of $U$. Yet it is still true that we can compute {\\v C}ech cohomology $\\check H^n(U, \\mathcal{F})$ (which is defined as the colimit over the opposite of the category of coverings $\\mathcal{U}$ of $U$ of the {\\v C}ech cohomology groups of $\\mathcal{F}$ with respect to $\\mathcal{U}$) in terms of the coverings $\\{V \\to U\\}$. We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it."}
+{"_id": "6769", "title": "etale-cohomology-remark-right-derived-global-sections", "text": "Comment on Theorem \\ref{theorem-zariski-fpqc-quasi-coherent}. Since $S$ is a final object in the category $\\mathcal{C}$, the cohomology groups on the right-hand side are merely the right derived functors of the global sections functor. In fact the proof shows that $H^p(U, f^*\\mathcal{F}) = H^p_\\tau(U, \\mathcal{F}^a)$ for any object $f : U \\to S$ of the site $\\mathcal{C}$."}
+{"_id": "6770", "title": "etale-cohomology-remark-constant-locally-constant-maps", "text": "Let $G$ be an abstract group. On any of the sites $(\\Sch/S)_\\tau$ or $S_\\tau$ of Section \\ref{section-big-small} the sheafification $\\underline{G}$ of the constant presheaf associated to $G$ in the {\\it Zariski topology} of the site already gives $$ \\Gamma(U, \\underline{G}) = \\{\\text{Zariski locally constant maps }U \\to G\\} $$ This Zariski sheaf is representable by the group scheme $G_S$ according to Groupoids, Example \\ref{groupoids-example-constant-group}. By Lemma \\ref{lemma-representable-sheaf-fpqc} any representable presheaf satisfies the sheaf condition for the $\\tau$-topology as well, and hence we conclude that the Zariski sheafification $\\underline{G}$ above is also the $\\tau$-sheafification."}
+{"_id": "6771", "title": "etale-cohomology-remark-special-case-fpqc-cohomology-quasi-coherent", "text": "In the terminology introduced above a special case of Theorem \\ref{theorem-zariski-fpqc-quasi-coherent} is $$ H_{fppf}^p(X, \\mathbf{G}_a) = H_\\etale^p(X, \\mathbf{G}_a) = H_{Zar}^p(X, \\mathbf{G}_a) = H^p(X, \\mathcal{O}_X) $$ for all $p \\geq 0$. Moreover, we could use the notation $H^p_{fppf}(X, \\mathcal{O}_X)$ to indicate the cohomology of the structure sheaf on the big fppf site of $X$."}
+{"_id": "6772", "title": "etale-cohomology-remark-no-kummer-sequence-zariski", "text": "Lemma \\ref{lemma-kummer-sequence} is false when ``\\'etale'' is replaced with ``Zariski''. Since the \\'etale topology is coarser than the smooth topology, see Topologies, Lemma \\ref{topologies-lemma-zariski-etale-smooth} it follows that the sequence is also exact in the smooth topology."}
+{"_id": "6773", "title": "etale-cohomology-remark-no-kummer-sequence-smooth-etale-zariski", "text": "Lemma \\ref{lemma-kummer-sequence-syntomic} is false for the smooth, \\'etale, or Zariski topology."}
+{"_id": "6774", "title": "etale-cohomology-remark-etale-between-etale", "text": "Since $U$ and $U'$ are \\'etale over $S$, any $S$-morphism between them is also \\'etale, see Proposition \\ref{proposition-etale-morphisms}. In particular all morphisms of \\'etale neighborhoods are \\'etale."}
+{"_id": "6775", "title": "etale-cohomology-remark-etale-neighbourhoods", "text": "Let $S$ be a scheme and $s \\in S$ a point. In More on Morphisms, Definition \\ref{more-morphisms-definition-etale-neighbourhood} we defined the notion of an \\'etale neighbourhood $(U, u) \\to (S, s)$ of $(S, s)$. If $\\overline{s}$ is a geometric point of $S$ lying over $s$, then any \\'etale neighbourhood $(U, \\overline{u}) \\to (S, \\overline{s})$ gives rise to an \\'etale neighbourhood $(U, u)$ of $(S, s)$ by taking $u \\in U$ to be the unique point of $U$ such that $\\overline{u}$ lies over $u$. Conversely, given an \\'etale neighbourhood $(U, u)$ of $(S, s)$ the residue field extension $\\kappa(s) \\subset \\kappa(u)$ is finite separable (see Proposition \\ref{proposition-etale-morphisms}) and hence we can find an embedding $\\kappa(u) \\subset \\kappa(\\overline{s})$ over $\\kappa(s)$. In other words, we can find a geometric point $\\overline{u}$ of $U$ lying over $u$ such that $(U, \\overline{u})$ is an \\'etale neighbourhood of $(S, \\overline{s})$. We will use these observations to go between the two types of \\'etale neighbourhoods."}
+{"_id": "6776", "title": "etale-cohomology-remark-map-stalks", "text": "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$ and $\\overline{s}' : \\Spec(k') \\to S$ be two geometric points of $S$. A {\\it morphism $a : \\overline{s} \\to \\overline{s}'$ of geometric points} is simply a morphism $a : \\Spec(k) \\to \\Spec(k')$ such that $a \\circ \\overline{s}' = \\overline{s}$. Given such a morphism we obtain a functor from the category of \\'etale neighbourhoods of $\\overline{s}'$ to the category of \\'etale neighbourhoods of $\\overline{s}$ by the rule $(U, \\overline{u}') \\mapsto (U, \\overline{u}' \\circ a)$. Hence we obtain a canonical map $$ \\mathcal{F}_{\\overline{s}'} = \\colim_{(U, \\overline{u}')} \\mathcal{F}(U) \\longrightarrow \\colim_{(U, \\overline{u})} \\mathcal{F}(U) = \\mathcal{F}_{\\overline{s}} $$ from Categories, Lemma \\ref{categories-lemma-functorial-colimit}. Using the description of elements of stalks as triples this maps the element of $\\mathcal{F}_{\\overline{s}'}$ represented by the triple $(U, \\overline{u}', \\sigma)$ to the element of $\\mathcal{F}_{\\overline{s}}$ represented by the triple $(U, \\overline{u}' \\circ a, \\sigma)$. Since the functor above is clearly an equivalence we conclude that this canonical map is an isomorphism of stalk functors. \\medskip\\noindent Let us make sure we have the map of stalks corresponding to $a$ pointing in the correct direction. Note that the above means, according to Sites, Definition \\ref{sites-definition-morphism-points}, that $a$ defines a morphism $a : p \\to p'$ between the points $p, p'$ of the site $S_\\etale$ associated to $\\overline{s}, \\overline{s}'$ by Lemma \\ref{lemma-stalk-gives-point}. There are more general morphisms of points (corresponding to specializations of points of $S$) which we will describe later, and which will not be isomorphisms (insert future reference here)."}
+{"_id": "6777", "title": "etale-cohomology-remark-points-fppf-site", "text": "\\begin{reference} This is discussed in \\cite{Schroeer}. \\end{reference} Let $S = \\Spec(A)$ be an affine scheme. Let $(p, u)$ be a point of the site $(\\textit{Aff}/S)_{fppf}$, see Sites, Sections \\ref{sites-section-points} and \\ref{sites-section-construct-points}. Let $B = \\mathcal{O}_p$ be the stalk of the structure sheaf at the point $p$. Recall that $$ B = \\colim_{(U, x)} \\mathcal{O}(U) = \\colim_{(\\Spec(C), x_C)} C $$ where $x_C \\in u(\\Spec(C))$. It can happen that $\\Spec(B)$ is an object of $(\\textit{Aff}/S)_{fppf}$ and that there is an element $x_B \\in u(\\Spec(B))$ mapping to the compatible system $x_C$. In this case the system of neighbourhoods has an initial object and it follows that $\\mathcal{F}_p = \\mathcal{F}(\\Spec(B))$ for any sheaf $\\mathcal{F}$ on $(\\textit{Aff}/S)_{fppf}$. It is straightforward to see that if $\\mathcal{F} \\mapsto \\mathcal{F}(\\Spec(B))$ defines a point of $\\Sh((\\textit{Aff}/S)_{fppf})$, then $B$ has to be a local $A$-algebra such that for every faithfully flat, finitely presented ring map $B \\to B'$ there is a section $B' \\to B$. Conversely, for any such $A$-algebra $B$ the functor $\\mathcal{F} \\mapsto \\mathcal{F}(\\Spec(B))$ is the stalk functor of a point. Details omitted. It is not clear what a general point of the site $(\\textit{Aff}/S)_{fppf}$ looks like."}
+{"_id": "6778", "title": "etale-cohomology-remark-henselization-Noetherian", "text": "Let $S$ be a scheme. Let $s \\in S$. If $S$ is locally Noetherian then $\\mathcal{O}_{S, s}^h$ is also Noetherian and it has the same completion: $$ \\widehat{\\mathcal{O}_{S, s}} \\cong \\widehat{\\mathcal{O}_{S, s}^h}. $$ In particular, $\\mathcal{O}_{S, s} \\subset \\mathcal{O}_{S, s}^h \\subset \\widehat{\\mathcal{O}_{S, s}}$. The henselization of $\\mathcal{O}_{S, s}$ is in general much smaller than its completion and inherits many of its properties. For example, if $\\mathcal{O}_{S, s}$ is reduced, then so is $\\mathcal{O}_{S, s}^h$, but this is not true for the completion in general. Insert future references here."}
+{"_id": "6779", "title": "etale-cohomology-remark-direct-image-sheaf", "text": "We claim that the direct image of a sheaf is a sheaf. Namely, if $\\{V_j \\to V\\}$ is an \\'etale covering in $Y_\\etale$ then $\\{X \\times_Y V_j \\to X \\times_Y V\\}$ is an \\'etale covering in $X_\\etale$. Hence the sheaf condition for $\\mathcal{F}$ with respect to $\\{X \\times_Y V_i \\to X \\times_Y V\\}$ is equivalent to the sheaf condition for $f_*\\mathcal{F}$ with respect to $\\{V_i \\to V\\}$. Thus if $\\mathcal{F}$ is a sheaf, so is $f_*\\mathcal{F}$."}
+{"_id": "6780", "title": "etale-cohomology-remark-functoriality-general", "text": "More generally, let $\\mathcal{C}_1, \\mathcal{C}_2$ be sites, and assume they have final objects and fibre products. Let $u: \\mathcal{C}_2 \\to \\mathcal{C}_1$ be a functor satisfying: \\begin{enumerate} \\item if $\\{V_i \\to V\\}$ is a covering of $\\mathcal{C}_2$, then $\\{u(V_i) \\to u(V)\\}$ is a covering of $\\mathcal{C}_1$ (we say that $u$ is {\\it continuous}), and \\item $u$ commutes with finite limits (i.e., $u$ is left exact, i.e., $u$ preserves fibre products and final objects). \\end{enumerate} Then one can define $f_*: \\Sh(\\mathcal{C}_1) \\to \\Sh(\\mathcal{C}_2)$ by $ f_* \\mathcal{F}(V) = \\mathcal{F}(u(V))$. Moreover, there exists an exact functor $f^{-1}$ which is left adjoint to $f_*$, see Sites, Definition \\ref{sites-definition-morphism-sites} and Proposition \\ref{sites-proposition-get-morphism}. Warning: It is not enough to require simply that $u$ is continuous and commutes with fibre products in order to get a morphism of topoi."}
+{"_id": "6781", "title": "etale-cohomology-remark-property-C-strong", "text": "Property (C) holds if $f : X \\to Y$ is an open immersion. Namely, if $U \\in \\Ob(X_\\etale)$, then we can view $U$ also as an object of $Y_\\etale$ and $U \\times_Y X = U$. Hence property (C) does not imply that $f_{small, *}$ is exact as this is not the case for open immersions (in general)."}
+{"_id": "6782", "title": "etale-cohomology-remark-affine-inside-equivalence", "text": "In the situation of Theorem \\ref{theorem-topological-invariance} it is also true that $V \\mapsto V_X$ induces an equivalence between those \\'etale morphisms $V \\to Y$ with $V$ affine and those \\'etale morphisms $U \\to X$ with $U$ affine. This follows for example from Limits, Proposition \\ref{limits-proposition-affine}."}
+{"_id": "6783", "title": "etale-cohomology-remark-push-pull-shriek", "text": "In the situation of Lemma \\ref{lemma-monomorphism-big-push-pull} it is true that the canonical map $\\mathcal{F} \\to f_{big}^{-1}f_{big!}\\mathcal{F}$ is an isomorphism for any sheaf of sets $\\mathcal{F}$ on $(\\Sch/X)_\\tau$. The proof is the same. This also holds for sheaves of abelian groups. However, note that the functor $f_{big!}$ for sheaves of abelian groups is defined in Modules on Sites, Section \\ref{sites-modules-section-exactness-lower-shriek} and is in general different from $f_{big!}$ on sheaves of sets. The result for sheaves of abelian groups follows from Modules on Sites, Lemma \\ref{sites-modules-lemma-back-and-forth}."}
+{"_id": "6784", "title": "etale-cohomology-remark-fppf-closed-immersion-not-closed", "text": "In Lemma \\ref{lemma-closed-immersion-pushforward-exact} the case $\\tau = fppf$ is missing. The reason is that given a ring $A$, an ideal $I$ and a faithfully flat, finitely presented ring map $A/I \\to \\overline{B}$, there is no reason to think that one can find {\\it any} flat finitely presented ring map $A \\to B$ with $B/IB \\not = 0$ such that $A/I \\to B/IB$ factors through $\\overline{B}$. Hence the proof of Lemma \\ref{lemma-closed-immersion-almost-cocontinuous} does not work for the fppf topology. In fact it is likely false that $f_{big, *} : \\textit{Ab}((\\Sch/X)_{fppf}) \\to \\textit{Ab}((\\Sch/Y)_{fppf})$ is exact when $f$ is a closed immersion. If you know an example, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "6785", "title": "etale-cohomology-remark-colimit-variant-complexes", "text": "Many of the results above have variants for bounded below complexes, but one has to be careful that the bounds have to be uniform. We explain this in the simplest case. Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$ be a directed set. Let $\\mathcal{F}_i^\\bullet$ be a system over $I$ of complexes of sheaves on $X_\\etale$. Assume there is an integer $a$ such that $\\mathcal{F}_i^n = 0$ for $n < a$ and all $i \\in I$. Then we have $$ H^p_\\etale(X, \\colim \\mathcal{F}_i^\\bullet) = \\colim H^p_\\etale(X, \\mathcal{F}^\\bullet_i) $$ If we ever need this we will state a precise lemma with full proof here."}
+{"_id": "6786", "title": "etale-cohomology-remark-cohomological-descent-finite", "text": "In the situation of Lemma \\ref{lemma-cohomological-descent-finite} if $\\mathcal{G}$ is a sheaf of sets on $Y_\\etale$, then we have $$ \\Gamma(Y, \\mathcal{G}) = \\text{Equalizer}( \\xymatrix{ \\Gamma(X_0, f_0^{-1}\\mathcal{G}) \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\Gamma(X_1, f_1^{-1}\\mathcal{G}) } ) $$ This is proved in exactly the same way, by showing that the sheaf $\\mathcal{G}$ is the equalizer of the two maps $f_{0, *}f_0^{-1}\\mathcal{G} \\to f_{1, *}f_1^{-1}\\mathcal{G}$."}
+{"_id": "6787", "title": "etale-cohomology-remark-every-sheaf-representable", "text": "Another way to state the conclusion of Theorem \\ref{theorem-equivalence-sheaves-point} and Fundamental Groups, Lemma \\ref{pione-lemma-sheaves-point} is to say that every sheaf on $\\Spec(K)_\\etale$ is representable by a scheme $X$ \\'etale over $\\Spec(K)$. This does not mean that every sheaf is representable in the sense of Sites, Definition \\ref{sites-definition-representable-sheaf}. The reason is that in our construction of $\\Spec(K)_\\etale$ we chose a sufficiently large set of schemes \\'etale over $\\Spec(K)$, whereas sheaves on $\\Spec(K)_\\etale$ form a proper class."}
+{"_id": "6788", "title": "etale-cohomology-remark-stalk-pullback", "text": "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$ be a geometric point of $S$. By definition this means that $k$ is algebraically closed. In particular the absolute Galois group of $k$ is trivial. Hence by Theorem \\ref{theorem-equivalence-sheaves-point} the category of sheaves on $\\Spec(k)_\\etale$ is equivalent to the category of sets. The equivalence is given by taking sections over $\\Spec(k)$. This finally provides us with an alternative definition of the stalk functor. Namely, the functor $$ \\Sh(S_\\etale) \\longrightarrow \\textit{Sets}, \\quad \\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}} $$ is isomorphic to the functor $$ \\Sh(S_\\etale) \\longrightarrow \\Sh(\\Spec(k)_\\etale) = \\textit{Sets}, \\quad \\mathcal{F} \\longmapsto \\overline{s}^*\\mathcal{F} $$ To prove this rigorously one can use Lemma \\ref{lemma-stalk-pullback} part (3) with $f = \\overline{s}$. Moreover, having said this the general case of Lemma \\ref{lemma-stalk-pullback} part (3) follows from functoriality of pullbacks."}
+{"_id": "6789", "title": "etale-cohomology-remark-functorial-locally-constant-on-connected", "text": "The equivalences of Lemma \\ref{lemma-locally-constant-on-connected} are compatible with pullbacks. More precisely, suppose $f : Y \\to X$ is a morphism of connected schemes. Let $\\overline{y}$ be geometric point of $Y$ and set $\\overline{x} = f(\\overline{y})$. Then the diagram $$ \\xymatrix{ \\text{finite locally constant sheaves of sets on }Y_\\etale \\ar[r] & \\text{finite }\\pi_1(Y, \\overline{y})\\text{-sets} \\\\ \\text{finite locally constant sheaves of sets on }X_\\etale \\ar[r] \\ar[u]_{f^{-1}} & \\text{finite }\\pi_1(X, \\overline{x})\\text{-sets} \\ar[u] } $$ is commutative, where the vertical arrow on the right comes from the continuous homomorphism $\\pi_1(Y, \\overline{y}) \\to \\pi_1(X, \\overline{x})$ induced by $f$. This follows immediately from the commutative diagram in Fundamental Groups, Theorem \\ref{pione-theorem-fundamental-group}."}
+{"_id": "6790", "title": "etale-cohomology-remark-natural-proof", "text": "The ``natural'' way to prove the previous corollary is to excise $X$ from $\\bar X$. This is possible, we just haven't developed that theory."}
+{"_id": "6791", "title": "etale-cohomology-remark-normalize-H1-Gm", "text": "Let $k$ be an algebraically closed field. Let $n$ be an integer prime to the characteristic of $k$. Recall that $$ \\mathbf{G}_{m, k} = \\mathbf{A}^1_k \\setminus \\{0\\} = \\mathbf{P}^1_k \\setminus \\{0, \\infty\\} $$ We claim there is a canonical isomorphism $$ H^1_\\etale(\\mathbf{G}_{m, k}, \\mu_n) = \\mathbf{Z}/n\\mathbf{Z} $$ What does this mean? This means there is an element $1_k$ in $H^1_\\etale(\\mathbf{G}_{m, k}, \\mu_n)$ such that for every morphism $\\Spec(k') \\to \\Spec(k)$ the pullback map on \\'etale cohomology for the map $\\mathbf{G}_{m, k'} \\to \\mathbf{G}_{m, k}$ maps $1_k$ to $1_{k'}$. (In particular this element is fixed under all automorphisms of $k$.) To see this, consider the $\\mu_{n, \\mathbf{Z}}$-torsor $\\mathbf{G}_{m, \\mathbf{Z}} \\to \\mathbf{G}_{m, \\mathbf{Z}}$, $x \\mapsto x^n$. By the identification of torsors with first cohomology, this pulls back to give our canonical elements $1_k$. Twisting back we see that there are canonical identifications $$ H^1_\\etale(\\mathbf{G}_{m, k}, \\mathbf{Z}/n\\mathbf{Z}) = \\Hom(\\mu_n(k), \\mathbf{Z}/n\\mathbf{Z}), $$ i.e., these isomorphisms are compatible with respect to maps of algebraically closed fields, in particular with respect to automorphisms of $k$."}
+{"_id": "6792", "title": "etale-cohomology-remark-different", "text": "Objects in the derived category $D_{ctf}(X_\\etale, \\Lambda)$ in some sense have better global properties than the perfect objects in  $D(\\mathcal{O}_X)$. Namely, it can happen that a complex of $\\mathcal{O}_X$-modules is locally quasi-isomorphic to a finite complex of finite locally free $\\mathcal{O}_X$-modules, without being globally quasi-isomorphic to a bounded complex of locally free $\\mathcal{O}_X$-modules. The following lemma shows this does not happen for $D_{ctf}$ on a Noetherian scheme."}
+{"_id": "6793", "title": "etale-cohomology-remark-projective-each-degree", "text": "Let $\\Lambda$ be a Noetherian ring. Let $X$ be a scheme. For a bounded complex $K^\\bullet$ of constructible flat $\\Lambda$-modules on $X_\\etale$ each stalk $K^p_{\\overline{x}}$ is a finite projective $\\Lambda$-module. Hence the stalks of the complex are perfect complexes of $\\Lambda$-modules."}
+{"_id": "6794", "title": "etale-cohomology-remark-invariance", "text": "Let $k$ be an algebraically closed field of characteristic $p > 0$. In Section \\ref{section-artin-schreier} we have seen that there is an exact sequence $$ k[x] \\to k[x] \\to H^1_\\etale(\\mathbf{A}^1_k, \\mathbf{Z}/p\\mathbf{Z}) \\to 0 $$ where the first arrow maps $f(x)$ to $f^p - f$. A set of representatives for the cokernel is formed by the polynomials $$ \\sum\\nolimits_{p \\not | n} \\lambda_n x^n $$ with $\\lambda_n \\in k$. (If $k$ is not algebraically closed you have to add some constants to this as well.) In particular when $k' \\supset k$ is an algebraically closed overfield, then the map $$ H^1_\\etale(\\mathbf{A}^1_k, \\mathbf{Z}/p\\mathbf{Z}) \\to H^1_\\etale(\\mathbf{A}^1_{k'}, \\mathbf{Z}/p\\mathbf{Z}) $$ is not an isomorphism in general. In particular, the map $\\pi_1(\\mathbf{A}^1_{k'}) \\to \\pi_1(\\mathbf{A}^1_k)$ between \\'etale fundamental groups (insert future reference here) is not an isomorphism either. Thus the \\'etale homotopy type of the affine line depends on the algebraically closed ground field. From Lemma \\ref{lemma-constant-smooth-statements} above we see that this is a phenomenon which only happens in characteristic $p$ with $p$-power torsion coefficients."}
+{"_id": "6795", "title": "etale-cohomology-remark-base-change-holds", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $n$ be an integer. We will say $BC(f, n, q_0)$ is true if for every commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\ S & S' \\ar[l] & T \\ar[l]_g } $$ with $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and $g$ quasi-compact and quasi-separated, and every abelian sheaf $\\mathcal{F}$ on $T_\\etale$ annihilated by $n$ the base change map $$ (f')^{-1}R^qg_*\\mathcal{F} \\longrightarrow R^qh_*e^{-1}\\mathcal{F} $$ is an isomorphism for $q \\leq q_0$."}
+{"_id": "6796", "title": "etale-cohomology-remark-define-kunneth-map", "text": "Consider a cartesian diagram in the category of schemes: $$ \\xymatrix{ X \\times_S Y \\ar[d]_p \\ar[r]_q \\ar[rd]_c & Y \\ar[d]^g \\\\ X \\ar[r]^f & S } $$ Let $\\Lambda$ be a ring and let $E \\in D(X_\\etale, \\Lambda)$ and $K \\in D(Y_\\etale, \\Lambda)$. Then there is a canonical map $$ Rf_*E \\otimes_\\Lambda^\\mathbf{L} Rg_*K \\longrightarrow Rc_*(p^{-1}E \\otimes_\\Lambda^\\mathbf{L} q^{-1}K) $$ For example we can define this using the canonical maps $Rf_*E \\to Rc_*p^{-1}E$ and $Rg_*K \\to Rc_*q^{-1}K$ and the relative cup product defined in Cohomology on Sites, Remark \\ref{sites-cohomology-remark-cup-product}. Or you can use the adjoint to the map $$ c^{-1}(Rf_*E \\otimes_\\Lambda^\\mathbf{L} Rg_*K) = p^{-1}f^{-1}Rf_*E \\otimes_\\Lambda^\\mathbf{L} q^{-1} g^{-1}Rg_*K \\to p^{-1}E \\otimes_\\Lambda^\\mathbf{L} q^{-1}K $$ which uses the adjunction maps $f^{-1}Rf_*E \\to E$ and $g^{-1}Rg_*K \\to K$."}
+{"_id": "6797", "title": "etale-cohomology-remarks-theorem-modules-exactness", "text": "The results on descent of modules have several applications: \\begin{enumerate} \\item The exactness of the {\\v C}ech complex in positive degrees for the covering $\\{\\Spec(B) \\to \\Spec(A)\\}$ where $A \\to B$ is faithfully flat. This will give some vanishing of cohomology. \\item If $(N, \\varphi)$ is a descent datum with respect to a faithfully flat map $A \\to B$, then the corresponding $A$-module is given by $$ M = \\Ker \\left( \\begin{matrix} N & \\longrightarrow & B \\otimes_A N \\\\ n & \\longmapsto & 1 \\otimes n - \\varphi(n \\otimes 1) \\end{matrix} \\right). $$ See Descent, Proposition \\ref{descent-proposition-descent-module}. \\end{enumerate}"}
+{"_id": "6798", "title": "etale-cohomology-remarks-enough-points", "text": "On points of the geometric sites. \\begin{enumerate} \\item Theorem \\ref{theorem-exactness-stalks} says that the family of points of $S_\\etale$ given by the geometric points of $S$ (Lemma \\ref{lemma-stalk-gives-point}) is conservative, see Sites, Definition \\ref{sites-definition-enough-points}. In particular $S_\\etale$ has enough points. \\item Suppose $\\mathcal{F}$ is a sheaf on the big \\'etale site \\label{item-stalks-big} of $S$. Let $T \\to S$ be an object of the big \\'etale site of $S$, and let $\\overline{t}$ be a geometric point of $T$. Then we define $\\mathcal{F}_{\\overline{t}}$ as the stalk of the restriction $\\mathcal{F}|_{T_\\etale}$ of $\\mathcal{F}$ to the small \\'etale site of $T$. In other words, we can define the stalk of $\\mathcal{F}$ at any geometric point of any scheme $T/S \\in \\Ob((\\Sch/S)_\\etale)$. \\item The big \\'etale site of $S$ also has enough points, by considering all geometric points of all objects of this site, see (\\ref{item-stalks-big}). \\end{enumerate}"}
+{"_id": "6880", "title": "equiv-remark-affine-morphism", "text": "Below we will use that for an affine morphism $h : T \\to S$ we have $h_*\\mathcal{G} \\otimes \\mathcal{H} = h_*(\\mathcal{G} \\otimes h^*\\mathcal{H})$ for $\\mathcal{G} \\in \\QCoh(\\mathcal{O}_T)$ and $\\mathcal{H} \\in \\QCoh(\\mathcal{O}_S)$. This follows immediately on translating into algebra."}
+{"_id": "6881", "title": "equiv-remark-difficult", "text": "If $F, F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to \\mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \\cong F'$; this follows from Proposition \\ref{proposition-siblings-isomorphic} via the argument given in the proof of Theorem \\ref{theorem-fully-faithful}. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma \\ref{lemma-exact-functor-preserving-Coh} it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \\cong F'$ as is shown in \\cite{Noah}. If you have a proof or a counter example in more general situations, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "6933", "title": "stacks-more-morphisms-remark-gerbe-of-lifts", "text": "Consider a diagram $$ \\xymatrix{ W \\ar[d]_x \\\\ \\mathcal{X} \\ar[r] & \\mathcal{X}' } $$ where $\\mathcal{X} \\subset \\mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an algebraic space, and $W \\to \\mathcal{X}$ is smooth. We will construct a category $\\mathcal{C}$ and a functor $$ p : \\mathcal{C} \\longrightarrow W_{spaces, \\etale} $$ (see Properties of Spaces, Definition \\ref{spaces-properties-definition-spaces-etale-site} for notation) as follows. An object of $\\mathcal{C}$ will be a system $(U, U', a, i, x', \\alpha)$ which forms a commutative diagram \\begin{equation} \\label{equation-object} \\vcenter{ \\xymatrix{ U \\ar[d]_a \\ar[r]_i & U' \\ar[dd]^{x'} \\\\ W \\ar[d]_x & \\\\ \\mathcal{X} \\ar[r] & \\mathcal{X}' } } \\end{equation} with commutativity witnessed by the $2$-morphism $\\alpha : x \\circ a \\to x' \\circ i$ such that $U$ and $U'$ are algebraic spaces, $a : U \\to W$ is \\'etale, $x' : U' \\to \\mathcal{X}'$ is smooth, and such that $U = \\mathcal{X} \\times_{\\mathcal{X}'} U'$. In particular $U \\subset U'$ is a thickening. A morphism $$ (U, U', a, i, x', \\alpha) \\to (V, V', b, j, y', \\beta) $$ is given by $(f, f', \\gamma)$ where $f : U \\to V$ is a morphism over $W$, $f' : U' \\to V'$ is a morphism whose restriction to $U$ gives $f$, and $\\gamma : x' \\circ f' \\to y'$ is a $2$-morphism witnessing the commutativity in right triangle of the diagram below \\begin{equation} \\label{equation-morphism} \\vcenter{ \\xymatrix{ & V \\ar[ld]_f \\ar[ldd]^b \\ar[rr]_j & & V' \\ar[ld]_{f'} \\ar[lddd]^{y'} \\\\ U \\ar[d]_a \\ar[rr]_i & & U' \\ar[dd]_{x'} \\\\ W \\ar[d]_x & \\\\ \\mathcal{X} \\ar[rr] & & \\mathcal{X}' } } \\end{equation} Finally, we require that $\\gamma$ is compatible with $\\alpha$ and $\\beta$: in the calculus of $2$-categories of Categories, Sections \\ref{categories-section-formal-cat-cat} and \\ref{categories-section-2-categories} this reads $$ \\beta = (\\gamma \\star \\text{id}_j) \\circ (\\alpha \\star \\text{id}_f) $$ (more succinctly: $\\beta = j^*\\gamma \\circ f^*\\alpha$). Another formulation is that objects are commutative diagrams (\\ref{equation-object}) with some additional properties and morphisms are commutative diagrams (\\ref{equation-morphism}) in the category $\\textit{Spaces}/\\mathcal{X}'$ introduced in Properties of Stacks, Remark \\ref{stacks-properties-remark-representable-over}. This makes it clear that $\\mathcal{C}$ is a category and that the rule $p : \\mathcal{C} \\to W_{spaces, \\etale}$ sending $(U, U', a, i, x', \\alpha)$ to $a : U \\to W$ is a functor."}
+{"_id": "7122", "title": "perfect-remark-support-c-equations", "text": "Let $X$ be a scheme. Let $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$. Denote $Z \\subset X$ the closed subscheme cut out by $f_1, \\ldots, f_c$. For $0 \\leq p < c$ and $1 \\leq i_0 < \\ldots < i_p \\leq c$ we denote $U_{i_0 \\ldots i_p} \\subset X$ the open subscheme where $f_{i_0} \\ldots f_{i_p}$ is invertible. For any $\\mathcal{O}_X$-module $\\mathcal{F}$ we set $$ \\mathcal{F}_{i_0 \\ldots i_p} = (U_{i_0 \\ldots i_p} \\to X)_*(\\mathcal{F}|_{U_{i_0 \\ldots i_p}}) $$ In this situation the {\\it extended alternating {\\v C}ech complex} is the complex of $\\mathcal{O}_X$-modules \\begin{equation} \\label{equation-extended-alternating} 0 \\to \\mathcal{F} \\to \\bigoplus\\nolimits_{i_0} \\mathcal{F}_{i_0} \\to \\ldots \\to \\bigoplus\\nolimits_{i_0 < \\ldots < i_p} \\mathcal{F}_{i_0 \\ldots i_p} \\to \\ldots \\to \\mathcal{F}_{1 \\ldots c} \\to 0 \\end{equation} where $\\mathcal{F}$ is put in degree $0$. The maps are constructed as follows. Given $1 \\leq i_0 < \\ldots < i_{p + 1} \\leq c$ and $0 \\leq j \\leq p + 1$ we have the canonical map $$ \\mathcal{F}_{i_0 \\ldots \\hat i_j \\ldots i_{p + 1}} \\to \\mathcal{F}_{i_0 \\ldots i_p} $$ coming from the inclusion $U_{i_0 \\ldots i_p} \\subset U_{i_0 \\ldots \\hat i_j \\ldots i_{p + 1}}$. The differentials in the extended alternating complex use these canonical maps with sign $(-1)^j$."}
+{"_id": "7123", "title": "perfect-remark-extended-alternating-map-to-support", "text": "Let $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and $\\mathcal{F}$ be as in Remark \\ref{remark-support-c-equations}. Denote $\\mathcal{F}^\\bullet$ the complex (\\ref{equation-extended-alternating}). By Lemma \\ref{lemma-extended-alternating-zero} the cohomology sheaves of $\\mathcal{F}^\\bullet$ are supported on $Z$ hence $\\mathcal{F}^\\bullet$ is an object of $D_Z(\\mathcal{O}_X)$. On the other hand, the equality $\\mathcal{F}^0 = \\mathcal{F}$ determines a canonical map $\\mathcal{F}^\\bullet \\to \\mathcal{F}$ in $D(\\mathcal{O}_X)$. As $i_* \\circ R\\mathcal{H}_Z$ is a right adjoint to the inclusion functor $D_Z(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$, see Cohomology, Lemma \\ref{cohomology-lemma-complexes-with-support-on-closed}, we obtain a canonical commutative diagram $$ \\xymatrix{ \\mathcal{F}^\\bullet \\ar[rd] \\ar[rr] & & \\mathcal{F} \\\\ & i_*R\\mathcal{H}_Z(\\mathcal{F}) \\ar[ru] } $$ in $D(\\mathcal{O}_X)$ functorial in the $\\mathcal{O}_X$-module $\\mathcal{F}$."}
+{"_id": "7124", "title": "perfect-remark-supported-map-c-equations", "text": "With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and $\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}. There is a canonical $\\mathcal{O}_X|_Z$-linear map $$ c_{f_1, \\ldots, f_c} : i^*\\mathcal{F} \\longrightarrow \\mathcal{H}^c_Z(\\mathcal{F}) $$ functorial in $\\mathcal{F}$. Namely, denoting $\\mathcal{F}^\\bullet$ the extended alternating {\\v C}ech complex (\\ref{equation-extended-alternating}) we have the canonical map $\\mathcal{F}^\\bullet \\to i_*R\\mathcal{H}_Z(\\mathcal{F})$ of Remark \\ref{remark-extended-alternating-map-to-support}. This determines a canonical map $$ \\Coker\\left(\\bigoplus \\mathcal{F}_{1 \\ldots \\hat i \\ldots c} \\to \\mathcal{F}_{1 \\ldots c}\\right) \\longrightarrow i_*\\mathcal{H}^c_Z(\\mathcal{F}) $$ on cohomology sheaves in degree $c$. Given a local section $s$ of $\\mathcal{F}$ we can consider the local section $$ \\frac{s}{f_1 \\ldots f_c} $$ of $\\mathcal{F}_{1 \\ldots c}$. The class of this section in the cokernel displayed above depends only on $s$ modulo the image of $(f_1, \\ldots, f_c) : \\mathcal{F}^{\\oplus c} \\to \\mathcal{F}$. Since $i_*i^*\\mathcal{F}$ is equal to the cokernel of $(f_1, \\ldots, f_c) : \\mathcal{F}^{\\oplus c} \\to \\mathcal{F}$ we see that we get an $\\mathcal{O}_X$-module map $i_*i^*\\mathcal{F} \\to i_*\\mathcal{H}_Z^c(\\mathcal{F})$. As $i_*$ is fully faithful we get the map $c_{f_1, \\ldots, f_c}$."}
+{"_id": "7125", "title": "perfect-remark-supported-functorial", "text": "Let $g : X' \\to X$ be a morphism of schemes. Let $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$. Set $f'_i = g^\\sharp(f_i) \\in \\Gamma(X', \\mathcal{O}_{X'})$. Denote $Z \\subset X$, resp.\\ $Z' \\subset X'$ the closed subscheme cut out by $f_1, \\ldots, f_c$, resp.\\ $f'_1, \\ldots, f'_c$. Then $Z' = Z \\times_X X'$. Denote $h : Z' \\to Z$ the induced morphism of schemes. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Set $\\mathcal{F}' = g^*\\mathcal{F}$. In this setting, if $\\mathcal{F}$ is quasi-coherent, then the diagram $$ \\xymatrix{ (i')^{-1}\\mathcal{O}_{X'} \\otimes_{h^{-1}i^{-1}\\mathcal{O}_X} h^{-1}\\mathcal{H}^c_Z(\\mathcal{F}) \\ar[r] & \\mathcal{H}_{Z'}^c(\\mathcal{F}') \\\\ h^*i^*\\mathcal{F} \\ar[r] \\ar[u]_-{c_{f_1, \\ldots, f_c}} & (i')^*\\mathcal{F}' \\ar[u]^-{c_{f'_1, \\ldots, f'_c}} } $$ is commutative where the top horizonal arrow is the map of Cohomology, Remark \\ref{cohomology-remark-support-functorial} on cohomology sheaves in degree $c$. Namely, denote $\\mathcal{F}^\\bullet$, resp.\\ $(\\mathcal{F}')^\\bullet$ the extended alternating {\\v C}ech complex constructed in Remark \\ref{remark-support-c-equations} using $\\mathcal{F}, f_1, \\ldots, f_c$, resp.\\ $\\mathcal{F}', f'_1, \\ldots, f'_c$. Note that $(\\mathcal{F}')^\\bullet = g^*\\mathcal{F}^\\bullet$. Then, without assuming $\\mathcal{F}$ is quasi-coherent, the diagram $$ \\xymatrix{ i'_* L(g|_{Z'})^* R\\mathcal{H}_Z(\\mathcal{F}) \\ar[r] \\ar@{=}[d] & i'_*R\\mathcal{H}_{Z'}(Lg^*\\mathcal{F}) \\ar[d] \\\\ Lg^*i_*R\\mathcal{H}_Z(\\mathcal{F}) & i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}') \\\\ Lg^*(\\mathcal{F}^\\bullet) \\ar[u] \\ar[r] & (\\mathcal{F}')^\\bullet \\ar[u] } $$ is commutative where $g|_{Z'} : (Z', (i')^{-1}\\mathcal{O}_{X'}) \\to (Z, i^{-1}\\mathcal{O}_X)$ is the induced morphism of ringed spaces. Here the top horizontal arrow is given in Cohomology, Remark \\ref{cohomology-remark-support-functorial} as is the explanation for the equal sign. The arrows pointing up are from Remark \\ref{remark-extended-alternating-map-to-support}. The lower horizonal arrow is the map $Lg^*\\mathcal{F}^\\bullet \\to g^*\\mathcal{F}^\\bullet = (\\mathcal{F}')^\\bullet$ and the arrow pointing down is induced by $Lg^*\\mathcal{F} \\to g^*\\mathcal{F} = \\mathcal{F}'$. The diagram commutes because going around the diagram both ways we obtain two arrows $Lg^*\\mathcal{F}^\\bullet \\to i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}')$ whose composition with $i'_*R\\mathcal{H}_{Z'}(\\mathcal{F}') \\to \\mathcal{F}'$ is the canonical map $Lg^*\\mathcal{F}^\\bullet \\to \\mathcal{F}'$. Some details omitted. Now the commutativity of the first diagram follows by looking at this diagram on cohomology sheaves in degree $c$ and using that the construction of the map $i^*\\mathcal{F} \\to \\Coker(\\bigoplus \\mathcal{F}_{1 \\ldots \\hat i \\ldots c} \\to \\mathcal{F}_{1 \\ldots c})$ used in Remark \\ref{remark-supported-map-c-equations} is compatible with pullbacks."}
+{"_id": "7126", "title": "perfect-remark-warning-coherator", "text": "Let $X$ be a quasi-compact scheme with affine diagonal. Even though we know that $D(\\QCoh(\\mathcal{O}_X)) = D_\\QCoh(\\mathcal{O}_X)$ by Proposition \\ref{proposition-quasi-compact-affine-diagonal} strange things can happen and it is easy to make mistakes with this material. One pitfall is to carelessly assume that this equality means derived functors are the same. For example, suppose we have a quasi-compact open $U \\subset X$. Then we can consider the higher right derived functors $$ R^i(\\QCoh)\\Gamma(U, -) : \\QCoh(\\mathcal{O}_X) \\to \\textit{Ab} $$ of the left exact functor $\\Gamma(U, -)$. Since this is a universal $\\delta$-functor, and since the functors $H^i(U, -)$ (defined for all abelian sheaves on $X$) restricted to $\\QCoh(\\mathcal{O}_X)$ form a $\\delta$-functor, we obtain canonical tranformations $$ t^i : R^i(\\QCoh)\\Gamma(U, -) \\to H^i(U, -). $$ These transformations aren't in general isomorphisms even if $X = \\Spec(A)$ is affine! Namely, we have $R^1(\\QCoh)\\Gamma(U, \\widetilde{I}) = 0$ if $I$ an injective $A$-module by construction of right derived functors and the equivalence of $\\QCoh(\\mathcal{O}_X)$ and $\\text{Mod}_A$. But Examples, Lemma \\ref{examples-lemma-nonvanishing} shows there exists $A$, $I$, and $U$ such that $H^1(U, \\widetilde{I}) \\not = 0$."}
+{"_id": "7127", "title": "perfect-remark-addendum", "text": "The proof of Lemma \\ref{lemma-lift-map-from-perfect-complex-with-support} shows that $$ R|_U = P \\oplus P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] $$ for some $m \\geq 0$ and $n_j \\geq 0$. Thus the highest degree cohomology sheaf of $R|_U$ equals that of $P$. By repeating the construction for the map $P^{\\oplus n_1}[1] \\oplus \\ldots \\oplus P^{\\oplus n_m}[m] \\to R|_U$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_U$ and $P$ above any given degree."}
+{"_id": "7128", "title": "perfect-remark-pullback-generator", "text": "Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $E \\in D_\\QCoh(\\mathcal{O}_Y)$ be a generator (see Theorem \\ref{theorem-bondal-van-den-Bergh}). Then the following are equivalent \\begin{enumerate} \\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have $Rf_*K = 0$ if and only if $K = 0$, \\item $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ reflects isomorphisms, and \\item $Lf^*E$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$. \\end{enumerate} The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma \\ref{lemma-affine-morphism}) or if $f$ is an open immersion, or if $f$ is a composition of such. We conclude that \\begin{enumerate} \\item if $X$ is a quasi-affine scheme then $\\mathcal{O}_X$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$, \\item if $X \\subset \\mathbf{P}^n_A$ is a quasi-compact locally closed subscheme, then $\\mathcal{O}_X \\oplus \\mathcal{O}_X(-1) \\oplus \\ldots \\oplus \\mathcal{O}_X(-n)$ is a generator for $D_\\QCoh(\\mathcal{O}_X)$ by Lemma \\ref{lemma-generator-P1}. \\end{enumerate}"}
+{"_id": "7129", "title": "perfect-remark-classical-generator", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $G$ be a perfect object of $D(\\mathcal{O}_X)$ which is a generator for $D_\\QCoh(\\mathcal{O}_X)$. By Theorem \\ref{theorem-bondal-van-den-Bergh} there is at least one of these. Combining Lemma \\ref{lemma-quasi-coherence-direct-sums} with Proposition \\ref{proposition-compact-is-perfect} and with Derived Categories, Proposition \\ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf}(\\mathcal{O}_X)$."}
+{"_id": "7130", "title": "perfect-remark-classical-generator-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. Let $G$ be a perfect object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ which is a generator for $D_{\\QCoh, T}(\\mathcal{O}_X)$. By Lemma \\ref{lemma-generator-with-support} there is at least one of these. Combining the fact that $D_{\\QCoh, T}(\\mathcal{O}_X)$ has direct sums with Lemma \\ref{lemma-compact-is-perfect-with-support} and with Derived Categories, Proposition \\ref{derived-proposition-generator-versus-classical-generator} we see that $G$ is a classical generator for $D_{perf, T}(\\mathcal{O}_X)$."}
+{"_id": "7131", "title": "perfect-remark-DQCoh-is-Ddga-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. The analogue of Theorem \\ref{theorem-DQCoh-is-Ddga} holds for $D_{\\QCoh, T}(\\mathcal{O}_X)$. This follows from the exact same argument as in the proof of the theorem, using Lemmas \\ref{lemma-generator-with-support} and \\ref{lemma-compact-is-perfect-with-support} and a variant of Lemma \\ref{lemma-tensor-with-QCoh-complex} with supports. If we ever need this, we will precisely state the result here and give a detailed proof."}
+{"_id": "7132", "title": "perfect-remark-independence-choice", "text": "Let $X$ be a quasi-compact and quasi-separated scheme over a ring $R$. By the construction of the proof of Theorem \\ref{theorem-DQCoh-is-Ddga} there exists a differential graded algebra $(A, \\text{d})$ over $R$ such that $D_\\QCoh(X)$ is $R$-linearly equivalent to $D(A, \\text{d})$ as a triangulated category. One may ask: how unique is $(A, \\text{d})$? The answer is (only) slightly better than just saying that $(A, \\text{d})$ is well defined up to derived equivalence. Namely, suppose that $(B, \\text{d})$ is a second such pair. Then we have $$ (A, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet) $$ and $$ (B, \\text{d}) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, L^\\bullet) $$ for some K-injective complexes $K^\\bullet$ and $L^\\bullet$ of $\\mathcal{O}_X$-modules corresponding to perfect generators of $D_\\QCoh(\\mathcal{O}_X)$. Set $$ \\Omega = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, L^\\bullet) \\quad \\Omega' = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(L^\\bullet, K^\\bullet) $$ Then $\\Omega$ is a differential graded $B^{opp} \\otimes_R A$-module and $\\Omega'$ is a differential graded $A^{opp} \\otimes_R B$-module. Moreover, the equivalence $$ D(A, \\text{d}) \\to D_\\QCoh(\\mathcal{O}_X) \\to D(B, \\text{d}) $$ is given by the functor $- \\otimes_A^\\mathbf{L} \\Omega'$ and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark \\ref{dga-remark-hochschild-cohomology}. If we ever need this remark we will provide a precise statement with a detailed proof here."}
+{"_id": "7133", "title": "perfect-remark-explain-consequence", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $X = U \\cup V$ with $U$ and $V$ quasi-compact open. By Lemma \\ref{lemma-better-coherator} the functors $DQ_X$, $DQ_U$, $DQ_V$, $DQ_{U \\cap V}$ exist. Moreover, there is a canonical distinguished triangle $$ DQ_X(K) \\to Rj_{U, *}DQ_U(K|_U) \\oplus Rj_{V, *}DQ_V(K|_V) \\to Rj_{U \\cap V, *}DQ_{U \\cap V}(K|_{U \\cap V}) \\to $$ for any $K \\in D(\\mathcal{O}_X)$. This follows by applying the exact functor $DQ_X$ to the distinguished triangle of Cohomology, Lemma \\ref{cohomology-lemma-exact-sequence-j-star} and using Lemma \\ref{lemma-pushforward-better-coherator} three times."}
+{"_id": "7134", "title": "perfect-remark-multiplication-map", "text": "With notation as in Lemma \\ref{lemma-affine-morphism-and-hom-out-of-perfect}. The diagram $$ \\xymatrix{ R\\Hom_X(M, Rg'_*L) \\otimes_R^\\mathbf{L} R' \\ar[r] \\ar[d]_\\mu & R\\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \\ar[d]^a \\\\ R\\Hom_X(M, R(g')_*L) \\ar@{=}[r] & R\\Hom_{X'}(L(g')^*M, L) } $$ is commutative where the top horizontal arrow is the map from the lemma, $\\mu$ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \\to L$. The multiplication map is the adjunction map $K' \\otimes_R^\\mathbf{L} R' \\to K'$ for any $K' \\in D(R')$."}
+{"_id": "7135", "title": "perfect-remark-annoying-compatibility", "text": "Let $S = \\Spec(A)$ be an affine scheme. Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_X$-modules and let $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Let $\\xi \\in H^i(X, \\mathcal{G})$ with pullback $p^*\\xi \\in H^i(X \\times_S Y, p^*\\mathcal{G})$. Then the following diagram is commutative $$ \\xymatrix{ R\\Gamma(X, \\mathcal{F})[-i] \\otimes_A^\\mathbf{L} R\\Gamma(Y, \\mathcal{E}) \\ar[d] \\ar[rr]_-{\\xi \\otimes \\text{id}} & & R\\Gamma(X, \\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) \\otimes_A^\\mathbf{L} R\\Gamma(Y, \\mathcal{E}) \\ar[d] \\\\ R\\Gamma(X \\times_S Y, p^*\\mathcal{F} \\otimes q^*\\mathcal{E})[-i] \\ar[rr]^-{p^*\\xi} & & R\\Gamma(X \\times_S Y, p^*(\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}) \\otimes q^*\\mathcal{E}) } $$ where the unadorned tensor products are over $\\mathcal{O}_{X \\times_S Y}$. The horizontal arrows are from Cohomology, Remark \\ref{cohomology-remark-cup-with-element-map-total-cohomology} and the vertical arrows are (\\ref{equation-kunneth-global}) hence given by pulling back followed by cup product on $X \\times_S Y$. The diagram commutes because the global cup product (on $X \\times_S Y$ with the sheaves $p^*\\mathcal{G}$, $p^*\\mathcal{F}$, and $q^*\\mathcal{E}$) is associative, see Cohomology, Lemma \\ref{cohomology-lemma-cup-product-associative}."}
+{"_id": "7136", "title": "perfect-remark-base-change-of-L", "text": "The pseudo-coherent complex $L$ of part (B) of Lemma \\ref{lemma-compute-ext} is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ of schemes we have canonical functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_{S'}}(Lg^*L, \\mathcal{F}') \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(L(g')^*E, (g')^*\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{F}') $$ for $\\mathcal{F}'$ quasi-coherent on $S'$. Obsere that we do {\\bf not} use derived pullback on $\\mathcal{G}^\\bullet$ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof."}
+{"_id": "7137", "title": "perfect-remark-explain-perfect-direct-image", "text": "Let $R$ be a ring. Let $X$ be a scheme of finite presentation over $R$. Let $\\mathcal{G}$ be a finitely presented $\\mathcal{O}_X$-module flat over $R$ with support proper over $R$. By Lemma \\ref{lemma-base-change-tensor-perfect} there exists a finite complex of finite projective $R$-modules $M^\\bullet$ such that we have $$ R\\Gamma(X_{R'}, \\mathcal{G}_{R'}) = M^\\bullet \\otimes_R R' $$ functorially in the $R$-algebra $R'$."}
+{"_id": "7138", "title": "perfect-remark-compare-L", "text": "The reader may have noticed the similarity between Lemma \\ref{lemma-compute-ext-rel-perfect} and Lemma \\ref{lemma-compute-ext}. Indeed, the pseudo-coherent complex $L$ of Lemma \\ref{lemma-compute-ext-rel-perfect} may be characterized as the unique pseudo-coherent complex on $S$ such that there are functorial isomorphisms $$ \\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) \\longrightarrow \\Ext^i_{\\mathcal{O}_X}(K, E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*\\mathcal{F}) $$ compatible with boundary maps for $\\mathcal{F}$ ranging over $\\QCoh(\\mathcal{O}_S)$. If we ever need this we will formulate a precise result here and give a detailed proof."}
+{"_id": "7139", "title": "perfect-remark-discuss-rel-perfect", "text": "Our Definition \\ref{definition-relatively-perfect} of a relatively perfect complex is equivalent to the one given in \\cite{lieblich-complexes} whenever our definition applies\\footnote{To see this, use Lemma \\ref{lemma-affine-locally-rel-perfect} and More on Algebra, Lemma \\ref{more-algebra-lemma-structure-relatively-perfect}.}. Next, suppose that $f : X \\to S$ is only assumed to be locally of finite type (not necessarily flat, nor locally of finite presentation). The definition in the paper cited above is that $E \\in D(\\mathcal{O}_X)$ is relatively perfect if \\begin{enumerate} \\item[(A)] locally on $X$ the object $E$ should be quasi-isomorphic to a finite complex of $S$-flat, finitely presented $\\mathcal{O}_X$-modules. \\end{enumerate} On the other hand, the natural generalization of our Definition \\ref{definition-relatively-perfect} is \\begin{enumerate} \\item[(B)] $E$ is pseudo-coherent relative to $S$ (More on Morphisms, Definition \\ref{more-morphisms-definition-relative-pseudo-coherence}) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$ (Cohomology, Definition \\ref{cohomology-definition-tor-amplitude}). \\end{enumerate} The advantage of condition (B) is that it clearly defines a triangulated subcategory of $D(\\mathcal{O}_X)$, whereas we suspect this is not the case for condition (A). The advantage of condition (A) is that it is easier to work with in particular in regards to limits."}
+{"_id": "7140", "title": "perfect-remark-K-ring", "text": "Let $X$ be a scheme. The K-group $K_0(X)$ is canonically a commutative ring. Namely, using the derived tensor product $$ \\otimes = \\otimes^\\mathbf{L}_{\\mathcal{O}_X} : D_{perf}(\\mathcal{O}_X) \\times D_{perf}(\\mathcal{O}_X) \\longrightarrow D_{perf}(\\mathcal{O}_X) $$ and Derived Categories, Lemma \\ref{derived-lemma-bilinear-map-K} we obtain a bilinear multiplication. Since $K \\otimes L \\cong L \\otimes K$ we see that this product is commutative. Since $(K \\otimes L) \\otimes M = K \\otimes (L \\otimes M)$ we see that this product is associative. Finally, the unit of $K_0(X)$ is the element $1 = [\\mathcal{O}_X]$. \\medskip\\noindent If $\\textit{Vect}(X)$ and $K_0(\\textit{Vect}(X))$ are as above, then it is clearly the case that $K_0(\\textit{Vect}(X))$ also has a ring structure: if $\\mathcal{E}$ and $\\mathcal{F}$ are finite locally free $\\mathcal{O}_X$-modules, then we set $$ [\\mathcal{E}] \\cdot [\\mathcal{F}] = [\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F}] $$ The reader easily verifies that this indeed defines a bilinear commutative, associative product. Details omitted. The map $$ K_0(\\textit{Vect}(X)) \\longrightarrow K_0(X) $$ constructed above is a ring map with these definitions. \\medskip\\noindent Now assume $X$ is Noetherian. The derived tensor product also produces a map $$ \\otimes = \\otimes^\\mathbf{L}_{\\mathcal{O}_X} : D_{perf}(\\mathcal{O}_X) \\times D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\longrightarrow D^b_{\\textit{Coh}}(\\mathcal{O}_X) $$ Again using Derived Categories, Lemma \\ref{derived-lemma-bilinear-map-K} we obtain a bilinear multiplication $K_0(X) \\times K'_0(X) \\to K'_0(X)$ since $K'_0(X) = K_0(D^b_{\\textit{Coh}}(\\mathcal{O}_X))$ by Lemma \\ref{lemma-Noetherian-Kprime}. The reader easily shows that this gives $K'_0(X)$ the structure of a module over the ring $K_0(X)$."}
+{"_id": "7141", "title": "perfect-remark-pushforward-K", "text": "Let $f : X \\to Y$ be a proper morphism of locally Noetherian schemes. There is a map $$ f_* : K'_0(X) \\longrightarrow K'_0(Y) $$ which sends $[\\mathcal{F}]$ to $$ [\\bigoplus\\nolimits_{i \\geq 0} R^{2i}f_*\\mathcal{F}] - [\\bigoplus\\nolimits_{i \\geq 0} R^{2i + 1}f_*\\mathcal{F}] $$ This is well defined because the sheaves $R^if_*\\mathcal{F}$ are coherent (Cohomology of Schemes, Lemma \\ref{coherent-lemma-locally-projective-pushforward}), because locally only a finite number are nonzero, and because a short exact sequence of coherent sheaves on $X$ produces a long exact sequence of $R^if_*$ on $Y$. If $Y$ is quasi-compact (the only case most often used in practice), then we can rewrite the above as $$ f_*[\\mathcal{F}] = \\sum (-1)^i[R^if_*\\mathcal{F}] = [Rf_*\\mathcal{F}] $$ where we have used the equality $K'_0(Y) = K_0(D^b_{\\textit{Coh}}(Y))$ from Lemma \\ref{lemma-Noetherian-Kprime}."}
+{"_id": "7142", "title": "perfect-remark-perf-Z", "text": "Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme. Consider the strictly full, saturated, triangulated subcategory $$ D_{Z, perf}(\\mathcal{O}_X) \\subset D(\\mathcal{O}_X) $$ consisting of perfect complexes of $\\mathcal{O}_X$-modules whose cohomology sheaves are settheoretically supported on $Z$. The zeroth $K$-group $K_0(D_{Z, perf}(\\mathcal{O}_X))$ of this triangulated category is sometimes denoted $K_Z(X)$ or $K_{0, Z}(X)$. Using derived tensor product exactly as in Remark \\ref{remark-K-ring} we see that $K_0(D_{Z, perf}(\\mathcal{O}_X))$ has a multiplication which is associative and commutative, but in general $K_0(D_{Z, perf}(\\mathcal{O}_X))$ doesn't have a unit."}
+{"_id": "7143", "title": "perfect-remark-functorial-det", "text": "The construction of Lemma \\ref{lemma-determinant-two-term-complexes} is compatible with pullbacks. More precisely, given a morphism $f : X \\to Y$ of schemes and a perfect object $K$ of $D(\\mathcal{O}_Y)$ of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a perfect object $K$ of $D(\\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$ and we have a canonical identification $$ f^*\\det(K) \\longrightarrow \\det(Lf^*K) $$ Moreover, if $K$ has rank $0$, then $\\delta(K)$ pulls back to $\\delta(Lf^*K)$ via this map. This is clear from the affine local construction of the determinant."}
+{"_id": "7205", "title": "spaces-flat-remark-correct-generality", "text": "The result in this section can be generalized. It is probably correct if we only assume $X \\to \\Spec(A)$ to be separated, of finite presentation, and $K_n$ pseudo-coherent relative to $A_n$ supported on a closed subset of $X_n$ proper over $A_n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here."}
+{"_id": "7297", "title": "spaces-chow-remark-sober", "text": "In Situation \\ref{situation-setup} if $X/B$ is good, then $|X|$ is a sober topological space. See Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quasi-separated-sober} or Decent Spaces, Proposition \\ref{decent-spaces-proposition-reasonable-sober}. We will use this without further mention to choose generic points of irreducible closed subsets of $|X|$."}
+{"_id": "7298", "title": "spaces-chow-remark-integral", "text": "In Situation \\ref{situation-setup} if $X/B$ is good, then $X$ is integral (Spaces over Fields, Definition \\ref{spaces-over-fields-definition-integral-algebraic-space}) if and only if $X$ is reduced and $|X|$ is irreducible. Moreover, for any point $\\xi \\in |X|$ there is a unique integral closed subspace $Z \\subset X$ such that $\\xi$ is the generic point of the closed subset $|Z| \\subset |X|$, see Spaces over Fields, Lemma \\ref{spaces-over-fields-lemma-decent-irreducible-closed}."}
+{"_id": "7299", "title": "spaces-chow-remark-irreducible-component", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $Y \\subset X$ be a closed subspace. By Remarks \\ref{remark-sober} and \\ref{remark-integral} there are $1$-to-$1$ correspondences between \\begin{enumerate} \\item irreducible components $T$ of $|Y|$, \\item generic points of irreducible components of $|Y|$, and \\item integral closed subspaces $Z \\subset Y$ with the property that $|Z|$ is an irreducible component of $|Y|$. \\end{enumerate} In this chapter we will call $Z$ as in (3) an {\\it irreducible component of $Y$} and we will call $\\xi \\in |Z|$ its {\\it generic point}."}
+{"_id": "7300", "title": "spaces-chow-remark-residue-field", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Every $x \\in |X|$ can be represented by a (unique) monomorphism $\\Spec(k) \\to X$ where $k$ is a field, see Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-points-monomorphism}. Then $k$ is the {\\it residue field of} $x$ and is denoted $\\kappa(x)$. Recall that $X$ has a dense open subscheme $U \\subset X$ (Properties of Spaces, Proposition \\ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}). If $x \\in U$, then $\\kappa(x)$ agrees with the residue field of $x$ on $U$ as a scheme. See Decent Spaces, Section \\ref{decent-spaces-section-residue-fields-henselian-local-rings}."}
+{"_id": "7301", "title": "spaces-chow-remark-function-field", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is integral. In this case the {\\it function field} $R(X)$ of $X$ is defined and is equal to the residue field of $X$ at its generic point. See Spaces over Fields, Definition \\ref{spaces-over-fields-definition-function-field}. Combining this with Remark \\ref{remark-integral} we find that for any $x \\in X$ the residue field $\\kappa(x)$ is the function field of the unique integral closed subspace $Z \\subset X$ whose generic point is $x$."}
+{"_id": "7302", "title": "spaces-chow-remark-infinite-sums-rational-equivalences", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Suppose we have infinite collections $\\alpha_i, \\beta_i \\in Z_k(X)$, $i \\in I$ of $k$-cycles on $X$. Suppose that the supports of $\\alpha_i$ and $\\beta_i$ form locally finite collections of closed subsets of $X$ so that $\\sum \\alpha_i$ and $\\sum \\beta_i$ are defined as cycles. Moreover, assume that $\\alpha_i \\sim_{rat} \\beta_i$ for each $i$. Then it is not clear that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$. Namely, the problem is that the rational equivalences may be given by locally finite families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$ but the union $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ may not be locally finite. \\medskip\\noindent In many cases in practice, one has a locally finite family of closed subsets $\\{T_i\\}_{i \\in I}$ of $|X|$ such that $\\alpha_i, \\beta_i$ are supported on $T_i$ and such that $\\alpha_i \\sim_{rat} \\beta_i$ ``on'' $T_i$. More precisely, the families $\\{W_{i, j}, f_{i, j} \\in R(W_{i, j})^*\\}_{j \\in J_i}$ consist of integral closed subspaces $W_{i, j}$ with $|W_{i, j}| \\subset T_i$. In this case it is true that $\\sum \\alpha_i \\sim_{rat} \\sum \\beta_i$ on $X$, simply because the family $\\{W_{i, j}\\}_{i \\in I, j\\in J_i}$ is automatically locally finite in this case."}
+{"_id": "7303", "title": "spaces-chow-remark-on-cycles", "text": "Let $S$, $B$, $X$, $\\mathcal{L}$, $s$, $i : D \\to X$ be as in Definition \\ref{definition-gysin-homomorphism} and assume that $\\mathcal{L}|_D \\cong \\mathcal{O}_D$. In this case we can define a canonical map $i^* : Z_{k + 1}(X) \\to Z_k(D)$ on cycles, by requiring that $i^*[W] = 0$ whenever $W \\subset D$. The possibility to do this will be useful later on."}
+{"_id": "7304", "title": "spaces-chow-remark-pullback-pairs", "text": "Let $f : X' \\to X$ be a morphism of good algebraic spaces over $B$ as in Situation \\ref{situation-setup}. Let $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in Definition \\ref{definition-gysin-homomorphism}. Then we can set $\\mathcal{L}' = f^*\\mathcal{L}$, $s' = f^*s$, and $D' = X' \\times_X D = Z(s')$. This gives a commutative diagram $$ \\xymatrix{ D' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\ D \\ar[r]^i & X } $$ and we can ask for various compatibilities between $i^*$ and $(i')^*$."}
+{"_id": "7305", "title": "spaces-chow-remark-pullback-cohomology", "text": "In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of good algebraic spaces over $B$. Then there is a canonical $\\mathbf{Z}$-algebra map $A^*(Y) \\to A^*(X)$. Namely, given $c \\in A^p(Y)$ and $X' \\to X$, then we can let $f^*c$ be defined by the map $c \\cap - : \\CH_k(X') \\to \\CH_{k - p}(X')$ which is given by thinking of $X'$ as an algebraic space over $Y$."}
+{"_id": "7306", "title": "spaces-chow-remark-extend-to-finite-locally-free", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. If the rank of $\\mathcal{E}$ is not constant then we can still define the Chern classes of $\\mathcal{E}$. Namely, in this case we can write $$ X = X_0 \\amalg X_1 \\amalg X_2 \\amalg \\ldots $$ where $X_r \\subset X$ is the open and closed subspace where the rank of $\\mathcal{E}$ is $r$. If $X' \\to X$ is a morphism of good algebraic spaces over $B$, then we obtain by pullback a corresponding decomposition of $X'$ and we find that $$ \\CH_*(X') = \\prod\\nolimits_{r \\geq 0} \\CH_*(X'_r) $$ by our definitions. Then we simply define $c_i(\\mathcal{E})$ to be the bivariant class which preserves these direct product decompositions and acts by the already defined operations $c_i(\\mathcal{E}|_{X_r}) \\cap -$ on the factors. Observe that in this setting it may happen that $c_i(\\mathcal{E})$ is nonzero for infinitely many $i$."}
+{"_id": "7383", "title": "sdga-remark-functoriality-ga", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. We have \\begin{enumerate} \\item Let $\\mathcal{A}$ be a graded $\\mathcal{O}_\\mathcal{C}$-algebra. The multiplication maps of $\\mathcal{A}$ induce multiplication maps $f_*\\mathcal{A}^n \\times f_*\\mathcal{A}^m \\to f_*\\mathcal{A}^{n + m}$ and via $f^\\sharp$ we may view these as $\\mathcal{O}_\\mathcal{D}$-bilinear maps. We will denote $f_*\\mathcal{A}$ the graded $\\mathcal{O}_\\mathcal{D}$-algebra we so obtain. \\item Let $\\mathcal{B}$ be a graded $\\mathcal{O}_\\mathcal{D}$-algebra.  The multiplication maps of $\\mathcal{B}$ induce multiplication maps $f^*\\mathcal{B}^n \\times f^*\\mathcal{B}^m \\to f^*\\mathcal{B}^{n + m}$ and using $f^\\sharp$ we may view these as $\\mathcal{O}_\\mathcal{C}$-bilinear maps. We will denote $f^*\\mathcal{B}$ the graded $\\mathcal{O}_\\mathcal{C}$-algebra we so obtain. \\item The set of homomorphisms $f^*\\mathcal{B} \\to \\mathcal{A}$ of graded $\\mathcal{O}_\\mathcal{C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\\mathcal{B} \\to f_*\\mathcal{A}$ of graded $\\mathcal{O}_\\mathcal{C}$-algebras. \\end{enumerate} Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules."}
+{"_id": "7384", "title": "sdga-remark-functoriality-dga", "text": "Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. \\begin{enumerate} \\item Let $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}_\\mathcal{C}$-algebra. The pushforward will be the differential graded $\\mathcal{O}_\\mathcal{D}$-algebra $(f_*\\mathcal{A}, \\text{d})$ where $f_*\\mathcal{A}$ is as in Remark \\ref{remark-functoriality-ga} and $\\text{d} = f_*\\text{d}$ as maps $f_*\\mathcal{A}^n \\to f_*\\mathcal{A}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied. \\item Let $\\mathcal{B}$ be a differential graded $\\mathcal{O}_\\mathcal{D}$-algebra. The pullback will be the differential graded $\\mathcal{O}_\\mathcal{C}$-algebra $(f^*\\mathcal{B}, \\text{d})$ where $f^*\\mathcal{B}$ is as in Remark \\ref{remark-functoriality-ga} and $\\text{d} = f^*\\text{d}$ as maps $f^*\\mathcal{B}^n \\to f^*\\mathcal{B}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied. \\item The set of homomorphisms $f^*\\mathcal{B} \\to \\mathcal{A}$ of differential graded $\\mathcal{O}_\\mathcal{C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\\mathcal{B} \\to f_*\\mathcal{A}$ of differential graded $\\mathcal{O}_\\mathcal{D}$-algebras. \\end{enumerate} Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules."}
+{"_id": "7385", "title": "sdga-remark-cone-identity", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $C = C(\\text{id}_\\mathcal{A})$ be the cone on the identity map $\\mathcal{A} \\to \\mathcal{A}$ viewed as a map of differential graded $\\mathcal{A}$-modules. Then $$ \\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}}(C, \\mathcal{M}) = \\{(x, y) \\in \\Gamma(\\mathcal{C}, \\mathcal{M}^0) \\times \\Gamma(\\mathcal{C}, \\mathcal{M}^{-1}) \\mid x = \\text{d}(y)\\} $$ where the map from left to right sends $f$ to the pair $(x, y)$ where $x$ is the image of the global section $(0, 1)$ of $C^{-1} = \\mathcal{A}^{-1} \\oplus \\mathcal{A}^0$ and where $y$ is the image of the global section $(1, 0)$ of $C^0 = \\mathcal{A}^0 \\oplus \\mathcal{A}^1$."}
+{"_id": "7386", "title": "sdga-remark-sheaf-graded-sets", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A {\\it sheaf of graded sets} on $\\mathcal{C}$ is a sheaf of sets $\\mathcal{S}$ endowed with a map $\\deg : \\mathcal{S} \\to \\underline{\\mathbf{Z}}$ of sheaves of sets. Let us denote $\\mathcal{O}[\\mathcal{S}]$ the graded $\\mathcal{O}$-module which is the free $\\mathcal{O}$-module on the graded sheaf of sets $\\mathcal{S}$. More precisely, the $n$th graded part of $\\mathcal{O}[\\mathcal{S}]$ is the sheafification of the rule $$ U \\longmapsto \\bigoplus\\nolimits_{s \\in \\mathcal{S}(U),\\ \\deg(s) = n} s \\cdot \\mathcal{O}(U) $$ With zero differential we also may consider this as a differential graded $\\mathcal{O}$-module. Let $\\mathcal{A}$ be a sheaf of graded $\\mathcal{O}$-algebras Then we similarly define $\\mathcal{A}[\\mathcal{S}]$ to be the graded $\\mathcal{A}$-module whose $n$th graded part is the sheafification of the rule $$ U \\longmapsto \\bigoplus\\nolimits_{s \\in \\mathcal{S}(U)} s \\cdot \\mathcal{A}^{n - \\deg(s)}(U) $$ If $\\mathcal{A}$ is a differential graded $\\mathcal{O}$-algebra, the we turn this into a differential graded $\\mathcal{O}$-module by setting $\\text{d}(s) = 0$ for all $s \\in \\mathcal{S}(U)$ and sheafifying."}
+{"_id": "7387", "title": "sdga-remark-why-graded-injective", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a graded injective diffential graded $\\mathcal{A}$-module. Let $$ 0 \\to \\mathcal{M}_1 \\to \\mathcal{M}_2 \\to \\mathcal{M}_3 \\to 0 $$ be a short exact sequence of differential graded $\\mathcal{A}$-modules. Since $\\mathcal{I}$ is graded injective we obtain a short exact sequence of complexes $$ 0 \\to \\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_3, \\mathcal{I}) \\to \\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_2, \\mathcal{I}) \\to \\Hom_{\\text{Mod}^{dg}_{(\\mathcal{A}, \\text{d})}}(\\mathcal{M}_1, \\mathcal{I}) \\to 0 $$ of $\\Gamma(\\mathcal{C}, \\mathcal{O})$-modules. Taking cohomology we obtain a long exact sequence $$ \\xymatrix{ \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_3, \\mathcal{I}) \\ar[d] & \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_3, \\mathcal{I})[1] \\ar[d] \\\\ \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_2, \\mathcal{I}) \\ar[d] & \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_2, \\mathcal{I})[1] \\ar[d] \\\\ \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_1, \\mathcal{I}) \\ar[ruu] & \\Hom_{K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})}(\\mathcal{M}_1, \\mathcal{I})[1] } $$ of groups of homomorphisms in the homotopy category. The point is that we get this even though we didn't assume that our short exact sequence is admissible (so the short exact sequence in general does not define a distinguished triangle in the homotopy category)."}
+{"_id": "7632", "title": "stacks-morphisms-remark-inertia-is-group-in-spaces", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. In Properties of Stacks, Remark \\ref{stacks-properties-remark-representable-over} we have seen that the $2$-category of morphisms $\\mathcal{Z} \\to \\mathcal{X}$ representable by algebraic spaces with target $\\mathcal{X}$ forms a category. In this category the inertia stack of $\\mathcal{X}/\\mathcal{Y}$ is a {\\it group object}. Recall that an object of $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is just a pair $(x, \\alpha)$ where $x$ is an object of $\\mathcal{X}$ and $\\alpha$ is an automorphism of $x$ in the fibre category of $\\mathcal{X}$ that $x$ lives in with $f(\\alpha) = \\text{id}$. The composition $$ c : \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\times_\\mathcal{X} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\longrightarrow \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} $$ is given by the rule on objects $$ ((x, \\alpha), (x', \\alpha'), \\beta) \\mapsto (x, \\alpha \\circ \\beta^{-1} \\circ \\alpha' \\circ \\beta) $$ which makes sense as $\\beta : x \\to x'$ is an isomorphism in the fibre category by our definition of fibre products. The neutral element $e : \\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is given by the functor $x \\mapsto (x, \\text{id}_x)$. We omit the proof that the axioms of a group object hold."}
+{"_id": "7633", "title": "stacks-morphisms-remark-composition", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and stable under composition. Then the property of morphisms of algebraic stacks defined in Definition \\ref{definition-P} is stable under composition. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks having property $\\mathcal{P}$. Choose an algebraic space $W$ and a surjective smooth morphism $W \\to \\mathcal{Z}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$. Finally, choose an algebraic space $U$ and a surjective and smooth morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Then the morphisms $V \\to W$ and $U \\to V$ have property $\\mathcal{P}$ by definition. Whence $U \\to W$ has property $\\mathcal{P}$ as we assumed that $\\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ has property $\\mathcal{P}$."}
+{"_id": "7634", "title": "stacks-morphisms-remark-base-change", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and stable under base change. Then the property of morphisms of algebraic stacks defined in Definition \\ref{definition-P} is stable under base change. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks and assume $f$ has property $\\mathcal{P}$. Choose an algebraic space $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$. Choose an algebraic space $U$ and a surjective smooth morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \\to \\mathcal{Y}' \\times_\\mathcal{Y} V$. Then the morphism $U \\to V$ has property $\\mathcal{P}$ by definition. Whence $V' \\times_V U \\to V'$ has property $\\mathcal{P}$ as we assumed that $\\mathcal{P}$ is stable under base change. Considering the diagram $$ \\xymatrix{ V' \\times_V U \\ar[r] \\ar[d] & \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ V' \\ar[r] & \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ we see that the left top horizontal arrow is smooth and surjective, whence by definition we see that the projection $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ has property $\\mathcal{P}$."}
+{"_id": "7635", "title": "stacks-morphisms-remark-implication", "text": "Let $\\mathcal{P}, \\mathcal{P}'$ be properties of morphisms of algebraic spaces which are smooth local on the source-and-target. Suppose that we have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for morphisms of algebraic spaces. Then we also have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for the properties of morphisms of algebraic stacks defined in Definition \\ref{definition-P} using $\\mathcal{P}$ and $\\mathcal{P}'$. This is clear from the definition."}
+{"_id": "7636", "title": "stacks-morphisms-remark-property-automorphism-groups", "text": "Let $P$ be a property of algebraic spaces over fields which is invariant under ground field extensions. Given an algebraic stack $\\mathcal{X}$ and $x \\in |\\mathcal{X}|$, we say the automorphism group of $\\mathcal{X}$ at $x$ has $P$ if the equivalent conditions of Lemma \\ref{lemma-property-automorphism-groups} are satisfied. For example, we say {\\it the automorphism group of $\\mathcal{X}$ at $x$ is finite}, if $G_x \\to \\Spec(k)$ is finite whenever $x : \\Spec(k) \\to \\mathcal{X}$ is a representative of $x$. Similarly for smooth, proper, etc. (There is clearly an abuse of language going on here, but we believe it will not cause confusion or imprecision.)"}
+{"_id": "7637", "title": "stacks-morphisms-remark-identify-automorphism-groups", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $x \\in |\\mathcal{X}|$ be a point. To indicate the equivalent conditions of Lemma \\ref{lemma-iso-automorphism-groups} are satisfied for $f$ and $x$ in the literature the terminology {\\it $f$ is stabilizer preserving at $x$} or {\\it $f$ is fixed-point reflecting at $x$} is used. We prefer to say {\\it $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$}."}
+{"_id": "7638", "title": "stacks-morphisms-remark-order-type", "text": "We can wonder about the order type of the canonical stratifications which occur as output of the stratifications of type (a) constructed in Lemma \\ref{lemma-every-point-in-a-stratum}. A natural guess is that the well-ordered set $I$ has {\\it cardinality} at most $\\aleph_0$. We have no idea if this is true or false. If you do please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "7639", "title": "stacks-morphisms-remark-etale-smooth-composition", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is \\'etale-smooth local on the source-and-target and stable under composition. Then the property of DM morphisms of algebraic stacks defined in Definition \\ref{definition-etale-smooth-P} is stable under composition. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be DM morphisms of algebraic stacks having property $\\mathcal{P}$. By Lemma \\ref{lemma-composition-separated} the composition $g \\circ f$ is DM. Choose an algebraic space $W$ and a surjective smooth morphism $W \\to \\mathcal{Z}$. Choose an algebraic space $V$ and a surjective \\'etale morphism $V \\to \\mathcal{Y} \\times_\\mathcal{Z} W$ (Lemma \\ref{lemma-DM}). Choose an algebraic space $U$ and a surjective \\'etale morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$. Then the morphisms $V \\to W$ and $U \\to V$ have property $\\mathcal{P}$ by definition. Whence $U \\to W$ has property $\\mathcal{P}$ as we assumed that $\\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ has property $\\mathcal{P}$."}
+{"_id": "7640", "title": "stacks-morphisms-remark-etale-smooth-base-change", "text": "Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is \\'etale-smooth local on the source-and-target and stable under base change. Then the property of DM morphisms of algebraic stacks defined in Definition \\ref{definition-etale-smooth-P} is stable under arbitrary base change. Namely, let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic stacks and $g : \\mathcal{Y}' \\to \\mathcal{Y}$ be a morphism of algebraic stacks and assume $f$ has property $\\mathcal{P}$. Then the base change $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ is a DM morphism by Lemma \\ref{lemma-base-change-separated}. Choose an algebraic space $V$ and a surjective smooth morphism $V \\to \\mathcal{Y}$. Choose an algebraic space $U$ and a surjective \\'etale morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ (Lemma \\ref{lemma-DM}). Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \\to \\mathcal{Y}' \\times_\\mathcal{Y} V$. Then the morphism $U \\to V$ has property $\\mathcal{P}$ by definition. Whence $V' \\times_V U \\to V'$ has property $\\mathcal{P}$ as we assumed that $\\mathcal{P}$ is stable under base change. Considering the diagram $$ \\xymatrix{ V' \\times_V U \\ar[r] \\ar[d] & \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ V' \\ar[r] & \\mathcal{Y}' \\ar[r] & \\mathcal{Y} } $$ we see that the left top horizontal arrow is surjective and $$ V' \\times_V U \\to V' \\times_\\mathcal{Y} (\\mathcal{Y}' \\times_{\\mathcal{Y}'} \\mathcal{X}) = V' \\times_V (\\mathcal{X} \\times_\\mathcal{Y} V) $$ is \\'etale as a base change of $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$, whence by definition we see that the projection $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$ has property $\\mathcal{P}$."}
+{"_id": "7641", "title": "stacks-morphisms-remark-etale-smooth-implication", "text": "Let $\\mathcal{P}, \\mathcal{P}'$ be properties of morphisms of algebraic spaces which are \\'etale-smooth local on the source-and-target. Suppose that we have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for morphisms of algebraic spaces. Then we also have $\\mathcal{P} \\Rightarrow \\mathcal{P}'$ for the properties of morphisms of algebraic stacks defined in Definition \\ref{definition-etale-smooth-P} using $\\mathcal{P}$ and $\\mathcal{P}'$. This is clear from the definition."}
+{"_id": "7642", "title": "stacks-morphisms-remark-get-property-auts-from-diagonal", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Let $U \\to \\mathcal{X}$ be a morphism whose source is an algebraic space. Let $G \\to H$ be the pullback of the morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ to $U$. If $\\Delta_f$ is unramified, \\'etale, etc, so is $G \\to H$. This is true because $$ \\xymatrix{ U \\times_\\mathcal{X} U \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d]^{\\Delta_f} \\\\ U \\times_\\mathcal{Y} U \\ar[r] & \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} } $$ is cartesian and the morphism $G \\to H$ is the base change of the left vertical arrow by the diagonal $U \\to U \\times U$. Compare with the proof of Lemma \\ref{lemma-separated-implies-isom}."}
+{"_id": "7758", "title": "schemes-remark-not-reverse-open-closed", "text": "If $f : X \\to Y$ is an immersion of schemes, then it is in general not possible to factor $f$ as an open immersion followed by a closed immersion. See Morphisms, Example \\ref{morphisms-example-thibaut}."}
+{"_id": "7759", "title": "schemes-remark-intersection-affine-opens", "text": "In general the intersection of two affine opens in $X$ is not affine open. See Example \\ref{example-affine-space-zero-doubled}."}
+{"_id": "7760", "title": "schemes-remark-reduced-induced-locally-closed", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a locally closed subset. In this situation we sometimes also use the phrase ``reduced induced scheme structure on $T$''. It refers to the reduced induced scheme structure from Definition \\ref{definition-reduced-induced-scheme} when we view $T$ as a closed subset of the open subscheme  $X \\setminus \\partial T$ of $X$. Here $\\partial T = \\overline{T} \\setminus T$ is the ``boundary'' of $T$ in the topological space of $X$."}
+{"_id": "7761", "title": "schemes-remark-representable-locally-ringed", "text": "Suppose the functor $F$ is defined on all locally ringed spaces, and if conditions of Lemma \\ref{lemma-glue-functors} are replaced by the following: \\begin{enumerate} \\item $F$ satisfies the sheaf property on the category of locally ringed spaces, \\item there exists a set $I$ and a collection of subfunctors $F_i \\subset F$ such that \\begin{enumerate} \\item each $F_i$ is representable by a scheme, \\item each $F_i \\subset F$ is representable by open immersions on the category of locally ringed spaces, and \\item the collection $(F_i)_{i \\in I}$ covers $F$ as a functor on the category of locally ringed spaces. \\end{enumerate} \\end{enumerate} We leave it to the reader to spell this out further. Then the end result is that the functor $F$ is representable in the category of locally ringed spaces and that the representing object is a scheme."}
+{"_id": "7762", "title": "schemes-remark-fibre-product-schemes-locally-ringed", "text": "Using Remark \\ref{remark-representable-locally-ringed} you can show that the fibre product of morphisms of schemes exists in the category of locally ringed spaces and is a scheme."}
+{"_id": "7763", "title": "schemes-remark-quasi-compact-and-quasi-separated", "text": "The category of quasi-compact and quasi-separated schemes $\\mathcal{C}$ has the following properties. If $X, Y \\in \\Ob(\\mathcal{C})$, then any morphism of schemes $f : X \\to Y$ is quasi-compact and quasi-separated by Lemmas \\ref{lemma-quasi-compact-permanence} and \\ref{lemma-compose-after-separated} with $Z = \\Spec(\\mathbf{Z})$. Moreover, if $X \\to Y$ and $Z \\to Y$ are morphisms $\\mathcal{C}$, then $X \\times_Y Z$ is an object of $\\mathcal{C}$ too. Namely, the projection $X \\times_Y Z \\to Z$ is quasi-compact and quasi-separated as a base change of the morphism $Z \\to Y$, see Lemmas \\ref{lemma-separated-permanence} and \\ref{lemma-quasi-compact-preserved-base-change}. Hence the composition $X \\times_Y Z \\to Z \\to \\Spec(\\mathbf{Z})$ is quasi-compact and quasi-separated, see Lemmas \\ref{lemma-separated-permanence} and \\ref{lemma-composition-quasi-compact}."}
+{"_id": "7812", "title": "injectives-remark-embedding", "text": "The Freyd-Mitchell embedding theorem says there exists a fully faithful exact functor from any abelian category $\\mathcal{A}$ to the category of modules over a ring. Lemma \\ref{lemma-embedding} is not quite as strong. But the result is suitable for the Stacks project as we have to understand sheaves of abelian groups on sites in detail anyway. Moreover, ``diagram chasing'' works in the category of abelian sheaves on $\\mathcal{C}$, for example by working with sections over objects, or by working on the level of stalks using that $\\mathcal{C}$ has enough points. To see how to deduce the Freyd-Mitchell embedding theorem from Lemma \\ref{lemma-embedding} see Remark \\ref{remark-embedding-freyd}."}
+{"_id": "7813", "title": "injectives-remark-embedding-big", "text": "If $\\mathcal{A}$ is a ``big'' abelian category, i.e., if $\\mathcal{A}$ has a class of objects, then Lemma \\ref{lemma-embedding} does not work. In this case, given any set of objects $E \\subset \\Ob(\\mathcal{A})$ there exists an abelian full subcategory $\\mathcal{A}' \\subset \\mathcal{A}$ such that $\\Ob(\\mathcal{A}')$ is a set and $E \\subset \\Ob(\\mathcal{A}')$. Then one can apply Lemma \\ref{lemma-embedding} to $\\mathcal{A}'$. One can use this to prove that results depending on a diagram chase hold in $\\mathcal{A}$."}
+{"_id": "7814", "title": "injectives-remark-embedding-freyd", "text": "Let $\\mathcal{C}$ be a site. Note that $\\textit{Ab}(\\mathcal{C})$ has enough injectives, see Theorem \\ref{theorem-sheaves-injectives}. (In the case that $\\mathcal{C}$ has enough points this is straightforward because  $p_*I$ is an injective sheaf if $I$ is an injective $\\mathbf{Z}$-module and $p$ is a point.) Also, $\\textit{Ab}(\\mathcal{C})$ has a cogenerator (details omitted). Hence Lemma \\ref{lemma-embedding} proves that we have a fully faithful, exact embedding $\\mathcal{A} \\to \\mathcal{B}$ where $\\mathcal{B}$ has a cogenerator and enough injectives. We can apply this to $\\mathcal{A}^{opp}$ and we get a fully faithful exact functor $i : \\mathcal{A} \\to \\mathcal{D} = \\mathcal{B}^{opp}$ where $\\mathcal{D}$ has enough projectives and a generator. Hence $\\mathcal{D}$ has a projective generator $P$. Set $R = \\Mor_\\mathcal{D}(P, P)$. Then $$ \\mathcal{A} \\longrightarrow \\text{Mod}_R, \\quad X \\longmapsto \\Hom_\\mathcal{D}(P, X). $$ One can check this is a fully faithful, exact functor. In other words, one retrieves the Freyd-Mitchell theorem mentioned in Remark \\ref{remark-embedding} above."}
+{"_id": "7815", "title": "injectives-remark-embed-exact-category", "text": "The arguments proving Lemmas \\ref{lemma-site-abelian-category} and \\ref{lemma-embedding} work also for {\\it exact categories}, see \\cite[Appendix A]{Buhler} and \\cite[1.1.4]{BBD}. We quickly review this here and we add more details if we ever need it in the Stacks project. \\medskip\\noindent Let $\\mathcal{A}$ be an additive category. A {\\it kernel-cokernel} pair is a pair $(i, p)$ of morphisms of $\\mathcal{A}$ with $i : A \\to B$, $p : B \\to C$ such that $i$ is the kernel of $p$ and $p$ is the cokernel of $i$. Given a set $\\mathcal{E}$ of kernel-cokernel pairs we say $i : A \\to B$ is an {\\it admissible monomorphism} if $(i, p) \\in \\mathcal{E}$ for some morphism $p$. Similarly we say a morphism $p : B \\to C$ is an {\\it admissible epimorphism} if $(i, p) \\in \\mathcal{E}$ for some morphism $i$. The pair $(\\mathcal{A}, \\mathcal{E})$ is said to be an {\\it exact category} if the following axioms hold \\begin{enumerate} \\item $\\mathcal{E}$ is closed under isomorphisms of kernel-cokernel pairs, \\item for any object $A$ the morphism $1_A$ is both an admissible epimorphism and an admissible monomorphism, \\item admissible monomorphisms are stable under composition, \\item admissible epimorphisms are stable under composition, \\item the push-out of an admissible monomorphism $i : A \\to B$ via any morphism $A \\to A'$ exist and the induced morphism $i' : A' \\to B'$ is an admissible monomorphism, and \\item the base change of an admissible epimorphism $p : B \\to C$ via any morphism $C' \\to C$ exist and the induced morphism $p' : B' \\to C'$ is an admissible epimorphism. \\end{enumerate} Given such a structure let $\\mathcal{C} = (\\mathcal{A}, \\text{Cov})$ where coverings (i.e., elements of $\\text{Cov}$) are given by admissible epimorphisms. The axioms listed above immediately imply that this is a site. Consider the functor $$ F : \\mathcal{A} \\longrightarrow \\textit{Ab}(\\mathcal{C}), \\quad X \\longmapsto h_X $$ exactly as in Lemma \\ref{lemma-embedding}. It turns out that this functor is fully faithful, exact, and reflects exactness. Moreover, any extension of objects in the essential image of $F$ is in the essential image of $F$."}
+{"_id": "7816", "title": "injectives-remark-existence-D", "text": "In the chapter on derived categories we consistently work with ``small'' abelian categories (as is the convention in the Stacks project). For a ``big'' abelian category $\\mathcal{A}$ it isn't clear that the derived category $D(\\mathcal{A})$ exists because it isn't clear that morphisms in the derived category are sets. In general this isn't true, see Examples, Lemma \\ref{examples-lemma-big-abelian-category}. However, if $\\mathcal{A}$ is a Grothendieck abelian category, and given $K^\\bullet, L^\\bullet$ in $K(\\mathcal{A})$, then by Theorem \\ref{theorem-K-injective-embedding-grothendieck} there exists a quasi-isomorphism $L^\\bullet \\to I^\\bullet$ to a K-injective complex $I^\\bullet$ and Derived Categories, Lemma \\ref{derived-lemma-K-injective} shows that $$ \\Hom_{D(\\mathcal{A})}(K^\\bullet, L^\\bullet) = \\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet) $$ which is a set. Some examples of Grothendieck abelian categories are the category of modules over a ring, or more generally the category of sheaves of modules on a ringed site."}
+{"_id": "7817", "title": "injectives-remark-direct-sum-product-derived", "text": "Let $R$ be a ring. Suppose that $M_n$, $n \\in \\mathbf{Z}$ are $R$-modules. Denote $E_n = M_n[-n] \\in D(R)$. We claim that $E = \\bigoplus M_n[-n]$ is {\\it both} the direct sum and the product of the objects $E_n$ in $D(R)$. To see that it is the direct sum, take a look at the proof of Lemma \\ref{lemma-derived-products}. To see that it is the direct product, take injective resolutions $M_n \\to I_n^\\bullet$. By the proof of Lemma \\ref{lemma-derived-products} we have $$ \\prod E_n = \\prod I_n^\\bullet[-n] $$ in $D(R)$. Since products in $\\text{Mod}_R$ are exact, we see that $\\prod I_n^\\bullet$ is quasi-isomorphic to $E$. This works more generally in $D(\\mathcal{A})$ where $\\mathcal{A}$ is a Grothendieck abelian category with Ab4*."}
+{"_id": "7818", "title": "injectives-remark-ext-into-filtered-complex", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $K^\\bullet$ be a filtered complex of $\\mathcal{A}$, see Homology, Definition \\ref{homology-definition-filtered-complex}. For ease of notation denote $K$, $F^pK$, $\\text{gr}^pK$ the object of $D(\\mathcal{A})$ represented by $K^\\bullet$, $F^pK^\\bullet$, $\\text{gr}^pK^\\bullet$. Let $M \\in D(\\mathcal{A})$. Using Lemma \\ref{lemma-K-injective-embedding-filtration} we can construct a spectral sequence $(E_r, d_r)_{r \\geq 1}$ of bigraded objects of $\\mathcal{A}$ with $d_r$ of bidgree $(r, -r + 1)$ and with $$ E_1^{p, q} = \\Ext^{p + q}(M, \\text{gr}^pK) $$ If for every $n$ we have $$ \\Ext^n(M, F^pK) = 0 \\text{ for } p \\gg 0 \\quad\\text{and}\\quad \\Ext^n(M, F^pK) = \\Ext^n(M, K) \\text{ for } p \\ll 0 $$ then the spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$. Namely, choose any complex $M^\\bullet$ representing $M$, choose $j : K^\\bullet \\to J^\\bullet$ as in the lemma, and consider the complex $$ \\Hom^\\bullet(M^\\bullet, I^\\bullet) $$ defined exactly as in More on Algebra, Section \\ref{more-algebra-section-hom-complexes}. Setting $F^p\\Hom^\\bullet(M^\\bullet, I^\\bullet) = \\Hom^\\bullet(M^\\bullet, F^pI^\\bullet)$ we obtain a filtered complex. The spectral sequence of Homology, Section \\ref{homology-section-filtered-complex} has differentials and terms as described above; details omitted. The boundedness and convergence follows from Homology, Lemma \\ref{homology-lemma-ss-converges-trivial}."}
+{"_id": "7819", "title": "injectives-remark-spectral-sequences-ext", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M, K$ be objects of $D(\\mathcal{A})$. For any choice of complex $K^\\bullet$ representing $K$ we can use the filtration $F^pK^\\bullet = \\tau_{\\leq -p}K^\\bullet$ and the discussion in Remark \\ref{remark-ext-into-filtered-complex} to get a spectral sequence with $$ E_1^{p, q} = \\Ext^{2p + q}(M, H^{-p}(K)) $$ This spectral sequence is independent of the choice of complex $K^\\bullet$ representing $K$. After renumbering $p = -j$ and $q = i + 2j$ we find a spectral sequence $(E'_r, d'_r)_{r \\geq 2}$ with $d'_r$ of bidegree $(r, -r + 1)$, with $$ (E'_2)^{i, j} = \\Ext^i(M, H^j(K)) $$ If $M \\in D^-(\\mathcal{A})$ and $K \\in D^+(\\mathcal{A})$ then both $E_r$ and $E'_r$ are bounded and converge to $\\Ext^{p + q}(M, K)$. If we use the filtration $F^pK^\\bullet = \\sigma_{\\geq p}K^\\bullet$ then we get $$ E_1^{p, q} = \\Ext^q(M, K^p) $$ If $M \\in D^-(\\mathcal{A})$ and $K^\\bullet$ is bounded below, then this spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$."}
+{"_id": "7820", "title": "injectives-remark-ext-from-filtered-complex", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $K \\in D(\\mathcal{A})$. Let $M^\\bullet$ be a filtered complex of $\\mathcal{A}$, see Homology, Definition \\ref{homology-definition-filtered-complex}. For ease of notation denote $M$, $M/F^pM$, $\\text{gr}^pM$ the object of $D(\\mathcal{A})$ represented by $M^\\bullet$, $M^\\bullet/F^pM^\\bullet$, $\\text{gr}^pM^\\bullet$. Dually to Remark \\ref{remark-ext-into-filtered-complex} we can construct a spectral sequence $(E_r, d_r)_{r \\geq 1}$ of bigraded objects of $\\mathcal{A}$ with $d_r$ of bidgree $(r, -r + 1)$ and with $$ E_1^{p, q} = \\Ext^{p + q}(\\text{gr}^{-p}M, K) $$ If for every $n$ we have $$ \\Ext^n(M/F^pM, K) = 0 \\text{ for } p \\ll 0 \\quad\\text{and}\\quad \\Ext^n(M/F^pM, K) = \\Ext^n(M, K) \\text{ for } p \\gg 0 $$ then the spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$. Namely, choose a K-injective complex $I^\\bullet$ with injective terms representing $K$, see Theorem \\ref{theorem-K-injective-embedding-grothendieck}. Consider the complex $$ \\Hom^\\bullet(M^\\bullet, I^\\bullet) $$ defined exactly as in More on Algebra, Section \\ref{more-algebra-section-hom-complexes}. Setting $$ F^p\\Hom^\\bullet(M^\\bullet, I^\\bullet) = \\Hom^\\bullet(M^\\bullet/F^{-p + 1}M^\\bullet, I^\\bullet) $$ we obtain a filtered complex (note sign and shift in filtration). The spectral sequence of Homology, Section \\ref{homology-section-filtered-complex} has differentials and terms as described above; details omitted. The boundedness and convergence follows from Homology, Lemma \\ref{homology-lemma-ss-converges-trivial}."}
+{"_id": "7821", "title": "injectives-remark-spectral-sequences-ext-variant", "text": "Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M, K$ be objects of $D(\\mathcal{A})$. For any choice of complex $M^\\bullet$ representing $M$ we can use the filtration $F^pM^\\bullet = \\tau_{\\leq -p}M^\\bullet$ and the discussion in Remark \\ref{remark-ext-into-filtered-complex} to get a spectral sequence with $$ E_1^{p, q} = \\Ext^{2p + q}(H^p(M), K) $$ This spectral sequence is independent of the choice of complex $M^\\bullet$ representing $M$. After renumbering $p = -j$ and $q = i + 2j$ we find a spectral sequence $(E'_r, d'_r)_{r \\geq 2}$ with $d'_r$ of bidegree $(r, -r + 1)$, with $$ (E'_2)^{i, j} = \\Ext^i(H^{-j}(M), K) $$ If $M \\in D^-(\\mathcal{A})$ and $K \\in D^+(\\mathcal{A})$ then $E_r$ and $E'_r$ are bounded and converge to $\\Ext^{p + q}(M, K)$. If we use the filtration $F^pM^\\bullet = \\sigma_{\\geq p}M^\\bullet$ then we get $$ E_1^{p, q} = \\Ext^q(M^{-p}, K) $$ If $K \\in D^+(\\mathcal{A})$ and $M^\\bullet$ is bounded above, then this spectral sequence is bounded and converges to $\\Ext^{p + q}(M, K)$."}
+{"_id": "8115", "title": "divisors-remark-base-change-relative-assassin", "text": "With notation and assumptions as in Lemma \\ref{lemma-base-change-relative-assassin} we see that it is always the case that $(g')^{-1}(\\text{Ass}_{X/S}(\\mathcal{F})) \\supset \\text{Ass}_{X'/S'}(\\mathcal{F}')$. If the morphism $S' \\to S$ is locally quasi-finite, then we actually have $$ (g')^{-1}(\\text{Ass}_{X/S}(\\mathcal{F})) = \\text{Ass}_{X'/S'}(\\mathcal{F}') $$ because in this case the field extensions $\\kappa(s) \\subset \\kappa(s')$ are always finite. In fact, this holds more generally for any morphism $g : S' \\to S$ such that all the field extensions $\\kappa(s) \\subset \\kappa(s')$ are algebraic, because in this case all prime ideals of $\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x)$ are maximal (and minimal) primes, see Algebra, Lemma \\ref{algebra-lemma-integral-over-field}."}
+{"_id": "8116", "title": "divisors-remark-different-reflexive", "text": "If $X$ is a scheme of finite type over a field, then sometimes a different notion of reflexive modules is used (see for example \\cite[bottom of page 5 and Definition 1.1.9]{HL}). This other notion uses $R\\SheafHom$ into a dualizing complex $\\omega_X^\\bullet$ instead of into $\\mathcal{O}_X$ and should probably have a different name because it can be different when $X$ is not Gorenstein. For example, if $X = \\Spec(k[t^3, t^4, t^5])$, then a computation shows the dualizing sheaf $\\omega_X$ is not reflexive in our sense, but it is reflexive in the other sense as $\\omega_X \\to \\SheafHom(\\SheafHom(\\omega_X, \\omega_X), \\omega_X)$ is an isomorphism."}
+{"_id": "8117", "title": "divisors-remark-tensor", "text": "Let $X$ be an integral locally Noetherian scheme. Thanks to Lemma \\ref{lemma-dual-reflexive} we know that the reflexive hull $\\mathcal{F}^{**}$ of a coherent $\\mathcal{O}_X$-module is coherent reflexive. Consider the category $\\mathcal{C}$ of coherent reflexive $\\mathcal{O}_X$-modules. Taking reflexive hulls gives a left adjoint to the inclusion functor $\\mathcal{C} \\to \\textit{Coh}(\\mathcal{O}_X)$. Observe that $\\mathcal{C}$ is an additive category with kernels and cokernels. Namely, given $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ in $\\mathcal{C}$, the usual kernel $\\Ker(\\varphi)$ is reflexive (Lemma \\ref{lemma-sequence-reflexive}) and the reflexive hull $\\Coker(\\varphi)^{**}$ of the usual cokernel is the cokernel in $\\mathcal{C}$. Moreover $\\mathcal{C}$ inherits a tensor product $$ \\mathcal{F} \\otimes_\\mathcal{C} \\mathcal{G} = (\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**} $$ which is associative and symmetric. There is an internal Hom in the sense that for any three objects $\\mathcal{F}, \\mathcal{G}, \\mathcal{H}$ of $\\mathcal{C}$ we have the identity $$ \\Hom_\\mathcal{C}(\\mathcal{F} \\otimes_\\mathcal{C} \\mathcal{G}, \\mathcal{H}) = \\Hom_\\mathcal{C}(\\mathcal{F}, \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H})) $$ see Modules, Lemma \\ref{modules-lemma-internal-hom}. In $\\mathcal{C}$ every object $\\mathcal{F}$ has a {\\it dual object} $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)$. Without further conditions on $X$ it can happen that $$ \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\not \\cong \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\otimes_\\mathcal{C} \\mathcal{G} \\quad\\text{and}\\quad \\mathcal{F} \\otimes_\\mathcal{C} \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\not \\cong \\mathcal{O}_X $$ for $\\mathcal{F}, \\mathcal{G}$ of rank $1$ in $\\mathcal{C}$. To make an example let $X = \\Spec(R)$ where $R$ is as in More on Algebra, Example \\ref{more-algebra-example-ring-not-S2} and let $\\mathcal{F}, \\mathcal{G}$ be the modules corresponding to $M$. Computation omitted."}
+{"_id": "8118", "title": "divisors-remark-ses-regular-section", "text": "Let $X$ be a scheme, $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module, and $s$ a regular section of $\\mathcal{L}$. Then the zero scheme $D = Z(s)$ is an effective Cartier divisor on $X$ and there are short exact sequences $$ 0 \\to \\mathcal{O}_X \\to \\mathcal{L} \\to i_*(\\mathcal{L}|_D) \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X \\to i_*\\mathcal{O}_D \\to 0. $$ Given an effective Cartier divisor $D \\subset X$ using Lemmas \\ref{lemma-characterize-OD} and \\ref{lemma-conormal-effective-Cartier-divisor} we get $$ 0 \\to \\mathcal{O}_X \\to \\mathcal{O}_X(D) \\to i_*(\\mathcal{N}_{D/X}) \\to 0 \\quad\\text{and}\\quad 0 \\to \\mathcal{O}_X(-D) \\to \\mathcal{O}_X \\to i_*(\\mathcal{O}_D) \\to 0 $$"}
+{"_id": "8119", "title": "divisors-remark-affine-punctured-spectrum-standard-proof", "text": "If $(A, \\mathfrak m)$ is a Noetherian local normal domain of dimension $\\geq 2$ and $U$ is the punctured spectrum of $A$, then $\\Gamma(U, \\mathcal{O}_U) = A$. This algebraic version of Hartogs's theorem follows from the fact that $A = \\bigcap_{\\text{height}(\\mathfrak p) = 1} A_\\mathfrak p$ we've seen in Algebra, Lemma \\ref{algebra-lemma-normal-domain-intersection-localizations-height-1}. Thus in this case $U$ cannot be affine (since it would force $\\mathfrak m$ to be a point of $U$). This is often used as the starting point of the proof of Lemma \\ref{lemma-affine-punctured-spec}. To reduce the case of a general Noetherian local ring to this case, we first complete (to get a Nagata local ring), then replace $A$ by $A/\\mathfrak q$ for a suitable minimal prime, and then normalize. Each of these steps does not change the dimension and we obtain a contradiction. You can skip the completion step, but then the normalization in general is not a Noetherian domain. However, it is still a Krull domain of the same dimension (this is proved using Krull-Akizuki) and one can apply the same argument."}
+{"_id": "8120", "title": "divisors-remark-affine-puctured-spectrum-general", "text": "It is not clear how to characterize the non-Noetherian local rings $(A, \\mathfrak m)$ whose punctured spectrum is affine. Such a ring has a finitely generated ideal $I$ with $\\mathfrak m = \\sqrt{I}$. Of course if we can take $I$ generated by $1$ element, then $A$ has an affine puncture spectrum; this gives lots of non-Noetherian examples. Conversely, it follows from the argument in the proof of Lemma \\ref{lemma-affine-punctured-spec} that such a ring cannot possess a nonzerodivisor $f \\in \\mathfrak m$ with $H^0_I(A/fA) = 0$ (so $A$ cannot have a regular sequence of length $2$). Moreover, the same holds for any ring $A'$ which is the target of a local homomorphism of local rings $A \\to A'$ such that $\\mathfrak m_{A'} = \\sqrt{\\mathfrak mA'}$."}
+{"_id": "8121", "title": "divisors-remark-not-always-extra-permanence", "text": "In the situation of Lemma \\ref{lemma-extra-permanence-regular-immersion-noetherian} parts (1), (2), (3) are {\\bf not} equivalent to ``$j \\circ i$ and $j$ are regular immersions at $z$ and $y$''. An example is $X = \\mathbf{A}^1_k = \\Spec(k[x])$, $Y = \\Spec(k[x]/(x^2))$ and $Z = \\Spec(k[x]/(x))$."}
+{"_id": "8122", "title": "divisors-remark-relative-regular-immersion-elements", "text": "The codimension of a relative quasi-regular immersion, if it is constant, does not change after a base change. In fact, if we have a ring map $A \\to B$ and a quasi-regular sequence $f_1, \\ldots, f_r \\in B$ such that $B/(f_1, \\ldots, f_r)$ is flat over $A$, then for any ring map $A \\to A'$ we have a quasi-regular sequence  $f_1 \\otimes 1, \\ldots, f_r \\otimes 1$ in $B' = B \\otimes_A A'$ by More on Algebra, Lemma \\ref{more-algebra-lemma-relative-regular-immersion-algebra} (which was used in the proof of Lemma \\ref{lemma-relative-regular-immersion} above). Now the proof of Lemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood} shows that if $A \\to B$ is flat and locally of finite presentation, then for every prime ideal $\\mathfrak q' \\subset B'$ the sequence $f_1 \\otimes 1, \\ldots, f_r \\otimes 1$ is even a regular sequence in the local ring $B'_{\\mathfrak q'}$."}
+{"_id": "8123", "title": "divisors-remark-structure-sheaf-Xs", "text": "Let $A$ be a Noetherian normal domain. Let $M$ be a rank $1$ finite reflexive $A$-module. Let $s \\in M$ be nonzero. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be the height $1$ primes of $A$ in the support of $M/As$. Then the open $U$ of Lemma \\ref{lemma-structure-sheaf-Xs} is $$ U = \\Spec(A) \\setminus \\left(V(\\mathfrak p_1) \\cup \\ldots \\cup \\mathfrak p_r)\\right) $$ by Lemma \\ref{lemma-Xs-codim-complement}. Moreover, if $M^{[n]}$ denotes the reflexive hull of $M \\otimes_A \\ldots \\otimes_A M$ ($n$-factors), then $$ \\Gamma(U, \\mathcal{O}_U) = \\colim M^{[n]} $$ according to Lemma \\ref{lemma-structure-sheaf-Xs}."}
+{"_id": "8184", "title": "spaces-remark-list-properties-stable-base-change", "text": "Here is a list of properties/types of morphisms which are {\\it stable under arbitrary base change}: \\begin{enumerate} \\item closed, open, and locally closed immersions, see Schemes, Lemma \\ref{schemes-lemma-base-change-immersion}, \\item quasi-compact, see Schemes, Lemma \\ref{schemes-lemma-quasi-compact-preserved-base-change}, \\item universally closed, see Schemes, Definition \\ref{schemes-definition-universally-closed}, \\item (quasi-)separated, see Schemes, Lemma \\ref{schemes-lemma-separated-permanence}, \\item monomorphism, see Schemes, Lemma \\ref{schemes-lemma-base-change-monomorphism} \\item surjective, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-surjective}, \\item universally injective, see Morphisms, Lemma \\ref{morphisms-lemma-universally-injective}, \\item affine, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-affine}, \\item quasi-affine, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-affine}, \\item (locally) of finite type, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-type}, \\item (locally) quasi-finite, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-quasi-finite}, \\item (locally) of finite presentation, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-presentation}, \\item locally of finite type of relative dimension $d$, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-relative-dimension-d}, \\item universally open, see Morphisms, Definition \\ref{morphisms-definition-open}, \\item flat, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-flat}, \\item syntomic, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-syntomic}, \\item smooth, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-smooth}, \\item unramified (resp.\\ G-unramified), see Morphisms, Lemma \\ref{morphisms-lemma-base-change-unramified}, \\item \\'etale, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-etale}, \\item proper, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-proper}, \\item H-projective, see Morphisms, Lemma \\ref{morphisms-lemma-H-projective-base-change}, \\item (locally) projective, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-projective}, \\item finite or integral, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite}, \\item finite locally free, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-finite-locally-free}, \\item universally submersive, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-universally-submersive}, \\item universal homeomorphism, see Morphisms, Lemma \\ref{morphisms-lemma-base-change-universal-homeomorphism}. \\end{enumerate} Add more as needed."}
+{"_id": "8185", "title": "spaces-remark-list-properties-stable-composition", "text": "Of the properties of morphisms which are stable under base change (as listed in Remark \\ref{remark-list-properties-stable-base-change}) the following are also {\\it stable under compositions}: \\begin{enumerate} \\item closed, open and locally closed immersions, see Schemes, Lemma \\ref{schemes-lemma-composition-immersion}, \\item quasi-compact, see Schemes, Lemma \\ref{schemes-lemma-composition-quasi-compact}, \\item universally closed, see Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}, \\item (quasi-)separated, see Schemes, Lemma \\ref{schemes-lemma-separated-permanence}, \\item monomorphism, see Schemes, Lemma \\ref{schemes-lemma-composition-monomorphism}, \\item surjective, see Morphisms, Lemma \\ref{morphisms-lemma-composition-surjective}, \\item universally injective, see Morphisms, Lemma \\ref{morphisms-lemma-composition-universally-injective}, \\item affine, see Morphisms, Lemma \\ref{morphisms-lemma-composition-affine}, \\item quasi-affine, see Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-affine}, \\item (locally) of finite type, see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-type}, \\item (locally) quasi-finite, see Morphisms, Lemma \\ref{morphisms-lemma-composition-quasi-finite}, \\item (locally) of finite presentation, see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-presentation}, \\item universally open, see Morphisms, Lemma \\ref{morphisms-lemma-composition-open}, \\item flat, see Morphisms, Lemma \\ref{morphisms-lemma-composition-flat}, \\item syntomic, see Morphisms, Lemma \\ref{morphisms-lemma-composition-syntomic}, \\item smooth, see Morphisms, Lemma \\ref{morphisms-lemma-composition-smooth}, \\item unramified (resp.\\ G-unramified), see Morphisms, Lemma \\ref{morphisms-lemma-composition-unramified}, \\item \\'etale, see Morphisms, Lemma \\ref{morphisms-lemma-composition-etale}, \\item proper, see Morphisms, Lemma \\ref{morphisms-lemma-composition-proper}, \\item H-projective, see Morphisms, Lemma \\ref{morphisms-lemma-H-projective-composition}, \\item finite or integral, see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite}, \\item finite locally free, see Morphisms, Lemma \\ref{morphisms-lemma-composition-finite-locally-free}, \\item universally submersive, see Morphisms, Lemma \\ref{morphisms-lemma-composition-universally-submersive}, \\item universal homeomorphism, see Morphisms, Lemma \\ref{morphisms-lemma-composition-universal-homeomorphism}. \\end{enumerate} Add more as needed."}
+{"_id": "8186", "title": "spaces-remark-list-properties-fpqc-local-base", "text": "Of the properties mentioned which are stable under base change (as listed in Remark \\ref{remark-list-properties-stable-base-change}) the following are also {\\it fpqc local on the base} (and a fortiori fppf local on the base): \\begin{enumerate} \\item for immersions we have this for \\begin{enumerate} \\item closed immersions, see Descent, Lemma \\ref{descent-lemma-descending-property-closed-immersion}, \\item open immersions, see Descent, Lemma \\ref{descent-lemma-descending-property-open-immersion}, and \\item quasi-compact immersions, see Descent, Lemma \\ref{descent-lemma-descending-property-quasi-compact-immersion}, \\end{enumerate} \\item quasi-compact, see Descent, Lemma \\ref{descent-lemma-descending-property-quasi-compact}, \\item universally closed, see Descent, Lemma \\ref{descent-lemma-descending-property-universally-closed}, \\item (quasi-)separated, see Descent, Lemmas \\ref{descent-lemma-descending-property-quasi-separated}, and \\ref{descent-lemma-descending-property-separated}, \\item monomorphism, see Descent, Lemma \\ref{descent-lemma-descending-property-monomorphism}, \\item surjective, see Descent, Lemma \\ref{descent-lemma-descending-property-surjective}, \\item universally injective, see Descent, Lemma \\ref{descent-lemma-descending-property-universally-injective}, \\item affine, see Descent, Lemma \\ref{descent-lemma-descending-property-affine}, \\item quasi-affine, see Descent, Lemma \\ref{descent-lemma-descending-property-quasi-affine}, \\item (locally) of finite type, see Descent, Lemmas \\ref{descent-lemma-descending-property-locally-finite-type}, and \\ref{descent-lemma-descending-property-finite-type}, \\item (locally) quasi-finite, see Descent, Lemma \\ref{descent-lemma-descending-property-quasi-finite}, \\item (locally) of finite presentation, see Descent, Lemmas \\ref{descent-lemma-descending-property-locally-finite-presentation}, and \\ref{descent-lemma-descending-property-finite-presentation}, \\item locally of finite type of relative dimension $d$, see Descent, Lemma \\ref{descent-lemma-descending-property-relative-dimension-d}, \\item universally open, see Descent, Lemma \\ref{descent-lemma-descending-property-universally-open}, \\item flat, see Descent, Lemma \\ref{descent-lemma-descending-property-flat}, \\item syntomic, see Descent, Lemma \\ref{descent-lemma-descending-property-syntomic}, \\item smooth, see Descent, Lemma \\ref{descent-lemma-descending-property-smooth}, \\item unramified (resp.\\ G-unramified), see Descent, Lemma \\ref{descent-lemma-descending-property-unramified}, \\item \\'etale, see Descent, Lemma \\ref{descent-lemma-descending-property-etale}, \\item proper, see Descent, Lemma \\ref{descent-lemma-descending-property-proper}, \\item finite or integral, see Descent, Lemma \\ref{descent-lemma-descending-property-finite}, \\item finite locally free, see Descent, Lemma \\ref{descent-lemma-descending-property-finite-locally-free}, \\item universally submersive, see Descent, Lemma \\ref{descent-lemma-descending-property-universally-submersive}, \\item universal homeomorphism, see Descent, Lemma \\ref{descent-lemma-descending-property-universal-homeomorphism}. \\end{enumerate} Note that the property of being an ``immersion'' may not be fpqc local on the base, but in Descent, Lemma \\ref{descent-lemma-descending-fppf-property-immersion} we proved that it is fppf local on the base."}
+{"_id": "8187", "title": "spaces-remark-warning", "text": "Consider the property $\\mathcal{P}=$``surjective''. In this case there could be some ambiguity if we say ``let $F \\to G$ be a surjective map''. Namely, we could mean the notion defined in Definition \\ref{definition-relative-representable-property} above, or we could mean a surjective map of presheaves, see Sites, Definition \\ref{sites-definition-presheaves-injective-surjective}, or, if both $F$ and $G$ are sheaves, we could mean a surjective map of sheaves, see Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}, If not mentioned otherwise when discussing morphisms of algebraic spaces we will always mean the first. See Lemma \\ref{lemma-surjective-flat-locally-finite-presentation} for a case where surjectivity implies surjectivity as a map of sheaves."}
+{"_id": "8381", "title": "topology-remark-quasi-components", "text": "\\begin{reference} \\cite[Example 6.1.24]{Engelking} \\end{reference} Let $X$ be a topological space and $x \\in X$. Let $Z \\subset X$ be the connected component of $X$ passing through $x$. Consider the intersection $E$ of all open and closed subsets of $X$ containing $x$. It is clear that $Z \\subset E$. In general $Z \\not = E$. For example, let $X = \\{x, y, z_1, z_2, \\ldots\\}$ with the topology with the following basis of opens, $\\{z_n\\}$, $\\{x, z_n, z_{n + 1}, \\ldots\\}$, and $\\{y, z_n, z_{n + 1}, \\ldots\\}$ for all $n$. Then $Z = \\{x\\}$ and $E = \\{x, y\\}$. We omit the details."}
+{"_id": "8382", "title": "topology-remark-lemma-literature", "text": "Lemma \\ref{lemma-characterize-quasi-compact} is a combination of \\cite[I, p. 75, Lemme 1]{Bourbaki} and \\cite[I, p. 76, Corollaire 1]{Bourbaki}."}
+{"_id": "8383", "title": "topology-remark-proof-literature", "text": "Here are some references to the literature. In \\cite[I, p. 75, Theorem 1]{Bourbaki} you can find: (2) $\\Leftrightarrow$ (4). In \\cite[I, p. 77, Proposition 6]{Bourbaki} you can find: (2) $\\Rightarrow$ (1). Of course, trivially we have (1) $\\Rightarrow$ (4). Thus (1), (2) and (4) are equivalent. Fan Zhou claimed and proved that (3) and (4) are equivalent; let me know if you find a reference in the literature."}
+{"_id": "8384", "title": "topology-remark-obstruction-to-dimension-function", "text": "Combining Lemmas \\ref{lemma-dimension-function-unique} and \\ref{lemma-locally-dimension-function} we see that on a catenary, locally Noetherian, sober topological space the obstruction to having a dimension function is an element of $H^1(X, \\mathbf{Z})$."}
+{"_id": "8385", "title": "topology-remark-size-projective-cover", "text": "Let $X$ be a quasi-compact Hausdorff space. Let $\\kappa$ be an infinite cardinal bigger or equal than the cardinality of $X$. Then the cardinality of the minimal quasi-compact, Hausdorff, extremally disconnected cover $X' \\to X$ (Lemma \\ref{lemma-existence-projective-cover}) is at most $2^{2^\\kappa}$. Namely, choose a subset $S \\subset X'$ mapping bijectively to $X$. By minimality of $X'$ the set $S$ is dense in $X'$. Thus $|X'| \\leq 2^{2^\\kappa}$ by Lemma \\ref{lemma-dense-image}."}
+{"_id": "8386", "title": "topology-remark-locally-finite-stratification", "text": "Given a locally finite stratification $X = \\coprod X_i$ of a topological space $X$, we obtain a family of closed subsets $Z_i = \\bigcup_{j \\leq i} X_j$ of $X$ indexed by $I$ such that $$ Z_i \\cap Z_j = \\bigcup\\nolimits_{k \\leq i, j} Z_k $$ Conversely, given closed subsets $Z_i \\subset X$ indexed by a partially ordered set $I$ such that $X = \\bigcup Z_i$, such that every point has a neighbourhood meeting only finitely many $Z_i$, and such that the displayed formula holds, then we obtain a locally finite stratification of $X$ by setting $X_i = Z_i \\setminus \\bigcup_{j < i} Z_j$."}
+{"_id": "8427", "title": "hypercovering-remark-hypercoverings-really-set", "text": "The lemma does not just say that there is a cofinal system of choices of hypercoverings that is a set, but that really the hypercoverings form a set."}
+{"_id": "8428", "title": "hypercovering-remark-P-covering", "text": "A useful special case of Lemmas \\ref{lemma-add-simplices} and \\ref{lemma-degeneracy-maps-coverings} is the following. Suppose we have a category $\\mathcal{C}$ having fibre products. Let $P \\subset \\text{Arrows}(\\mathcal{C})$ be a subset stable under base change, stable under composition, and containing all isomorphisms. Then one says a {\\it $P$-hypercovering} is an augmentation $a : U \\to X$ from a simplicial object of $\\mathcal{C}$ such that \\begin{enumerate} \\item $U_0 \\to X$ is in $P$, \\item $U_1 \\to U_0 \\times_X U_0$ is in $P$, \\item $U_{n + 1} \\to (\\text{cosk}_n\\text{sk}_n U)_{n + 1}$ is in $P$ for $n \\geq 1$. \\end{enumerate} The category $\\mathcal{C}/X$ has all finite limits, hence the coskeleta used in the formulation above exist (see Categories, Lemma \\ref{categories-lemma-finite-limits-exist}). Then we claim that the morphisms $U_n \\to X$ and $d^n_i : U_n \\to U_{n - 1}$ are in $P$. This follows from the aforementioned lemmas by turning $\\mathcal{C}$ into a site whose coverings are $\\{f : V \\to U\\}$ with $f \\in P$ and taking $K$ given by $K_n = \\{U_n \\to X\\}$."}
+{"_id": "8429", "title": "hypercovering-remark-contractible-category", "text": "Note that the crux of the proof is to use Lemma \\ref{lemma-add-simplices}. This lemma is completely general and does not care about the exact shape of the simplicial sets (as long as they have only finitely many nondegenerate simplices). It seems altogether reasonable to expect a result of the following kind: Given any morphism $a : K \\times \\partial \\Delta[k] \\to L$, with $K$ and $L$ hypercoverings, there exists a morphism of hypercoverings $c : K' \\to K$ and a morphism  $g : K' \\times \\Delta[k] \\to L$ such that $g|_{K' \\times \\partial \\Delta[k]} = a \\circ (c \\times \\text{id}_{\\partial \\Delta[k]})$. In other words, the category of hypercoverings is in a suitable sense contractible."}
+{"_id": "8430", "title": "hypercovering-remark-not-covering-set", "text": "One feature of this description is that if one of the multiple intersections $U_{i_0} \\cap \\ldots \\cap U_{i_{n + 1}}$ is empty then the covering on the right hand side may be the empty covering. Thus it is not automatically the case that the maps $I_{n + 1} \\to (\\text{cosk}_n\\text{sk}_n I)_{n + 1}$ are surjective. This means that the geometric realization of $I$ may be an interesting (non-contractible) space. \\medskip\\noindent In fact, let $I'_n \\subset I_n$ be the subset consisting of those simplices $i \\in I_n$ such that $U_i \\not = \\emptyset$. It is easy to see that $I' \\subset I$ is a subsimplicial set, and that $(I', \\{U_i\\})$ is a hypercovering. Hence we can always refine a hypercovering to a hypercovering where none of the opens $U_i$ is empty."}
+{"_id": "8431", "title": "hypercovering-remark-repackage-into-simplicial-space", "text": "Let us repackage this information in yet another way. Namely, suppose that $(I, \\{U_i\\})$ is a hypercovering of the topological space $X$. Given this data we can construct a simplicial topological space $U_\\bullet$ by setting $$ U_n = \\coprod\\nolimits_{i \\in I_n} U_i, $$ and where for given $\\varphi : [n] \\to [m]$ we let morphisms $U(\\varphi) : U_n \\to U_m$ be the morphism coming from the inclusions $U_i \\subset U_{\\varphi(i)}$ for $i \\in I_n$. This simplicial topological space comes with an augmentation $\\epsilon : U_\\bullet \\to X$. With this morphism the simplicial space $U_\\bullet$ becomes a hypercovering of $X$ along which one has cohomological descent in the sense of \\cite[Expos\\'e Vbis]{SGA4}. In other words, $H^n(U_\\bullet, \\epsilon^*\\mathcal{F}) = H^n(X, \\mathcal{F})$. (Insert future reference here to cohomology over simplicial spaces and cohomological descent formulated in those terms.) Suppose that $\\mathcal{F}$ is an abelian sheaf on $X$. In this case the spectral sequence of Lemma \\ref{lemma-cech-spectral-sequence} becomes the spectral sequence with $E_1$-term $$ E_1^{p, q} = H^q(U_p, \\epsilon_q^*\\mathcal{F}) \\Rightarrow H^{p + q}(U_\\bullet, \\epsilon^*\\mathcal{F}) = H^{p + q}(X, \\mathcal{F}) $$ comparing the total cohomology of $\\epsilon^*\\mathcal{F}$ to the cohomology groups of $\\mathcal{F}$ over the pieces of $U_\\bullet$. (Insert future reference to this spectral sequence here.)"}
+{"_id": "8432", "title": "hypercovering-remark-taking-disjoint-unions", "text": "Let $\\mathcal{C}$ be a site. Let $K$ and $L$ be objects of $\\text{SR}(\\mathcal{C})$. Write $K = \\{U_i\\}_{i \\in I}$ and $L = \\{V_j\\}_{j \\in J}$. Assume $U = \\coprod_{i \\in I} U_i$ and $V = \\coprod_{j \\in J} V_j$ exist. Then we get $$ \\Mor_{\\text{SR}(\\mathcal{C})}(K, L) \\longrightarrow \\Mor_\\mathcal{C}(U, V) $$ as follows. Given $f : K \\to L$ given by $\\alpha : I \\to J$ and $f_i : U_i \\to V_{\\alpha(i)}$ we obtain a transformation of functors $$ \\Mor_\\mathcal{C}(V, -) = \\prod\\nolimits_{j \\in J} \\Mor_\\mathcal{C}(V_j, -) \\to \\prod\\nolimits_{i \\in I} \\Mor_\\mathcal{C}(U_i, -) = \\Mor_\\mathcal{C}(U, -) $$ sending $(g_j)_{j \\in J}$ to $(g_{\\alpha(i)} \\circ f_i)_{i \\in I}$. Hence the Yoneda lemma produces the corresponding map $U \\to V$. Of course, $U \\to V$ maps the summand $U_i$ into the summand $V_{\\alpha(i)}$ via the morphism $f_i$."}
+{"_id": "8433", "title": "hypercovering-remark-take-unions-hypercovering", "text": "Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has fibre products and equalizers and let $K$ be a hypercovering. Write $K_n = \\{U_{n, i}\\}_{i \\in I_n}$. Suppose that \\begin{enumerate} \\item[(a)] $U_n = \\coprod_{i \\in I_n} U_{n, i}$ exists, and \\item[(b)] $\\coprod_{i \\in I_n} h_{U_{n, i}} \\to h_{U_n}$ induces an isomorphism on sheafifications. \\end{enumerate} Then we get another simplicial object $L$ of $\\text{SR}(\\mathcal{C})$ with $L_n = \\{U_n\\}$, see Remark \\ref{remark-taking-disjoint-unions}. Now we claim that $L$ is a hypercovering. To see this we check conditions (1), (2), (3) of Definition \\ref{definition-hypercovering-variant}. Condition (1) follows from (b) and (1) for $K$. Condition (2) follows in exactly the same way. Condition (3) follows because \\begin{align*} F((\\text{cosk}_n \\text{sk}_n L)_{n + 1})^\\# & = ((\\text{cosk}_n \\text{sk}_n F(L)^\\#)_{n + 1}) \\\\ & = ((\\text{cosk}_n \\text{sk}_n F(K)^\\#)_{n + 1}) \\\\ & = F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})^\\# \\end{align*} for $n \\geq 1$ and hence the condition for $K$ implies the condition for $L$ exactly as in (1) and (2). Note that $F$ commutes with connected limits and sheafification is exact proving the first and last equality; the middle equality follows as $F(K)^\\# = F(L)^\\#$ by (b)."}
+{"_id": "8434", "title": "hypercovering-remark-take-unions-hypercovering-X", "text": "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$. Assume $\\mathcal{C}$ has fibre products and let $K$ be a hypercovering of $X$. Write $K_n = \\{U_{n, i}\\}_{i \\in I_n}$. Suppose that \\begin{enumerate} \\item[(a)] $U_n = \\coprod_{i \\in I_n} U_{n, i}$ exists, \\item[(b)] given morphisms $(\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ and $(\\beta, g_k) : \\{W_k\\}_{k \\in K} \\to \\{V_j\\}_{j \\in J}$ in $\\text{SR}(\\mathcal{C})$ such that $U = \\coprod U_i$, $V = \\coprod V_j$, and $W = \\coprod W_j$ exist, then $U \\times_V W = \\coprod_{(i, j, k), \\alpha(i) = j = \\beta(k)} U_i \\times_{V_j} W_k$, \\item[(c)] if $(\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ is a covering in the sense of Definition \\ref{definition-covering-SR} and $U = \\coprod U_i$ and $V = \\coprod V_j$ exist, then the corresponding morphism $U \\to V$ of Remark \\ref{remark-taking-disjoint-unions} is a covering of $\\mathcal{C}$. \\end{enumerate} Then we get another simplicial object $L$ of $\\text{SR}(\\mathcal{C})$ with $L_n = \\{U_n\\}$, see Remark \\ref{remark-taking-disjoint-unions}. Now we claim that $L$ is a hypercovering of $X$. To see this we check conditions (1), (2) of Definition \\ref{definition-hypercovering}. Condition (1) follows from (c) and (1) for $K$ because (1) for $K$ says $K_0 = \\{U_{0, i}\\}_{i \\in I_0}$ is a covering of $\\{X\\}$ in the sense of Definition \\ref{definition-covering-SR}. Condition (2) follows because $\\mathcal{C}/X$ has all finite limits hence $\\text{SR}(\\mathcal{C}/X)$ has all finite limits, and condition (b) says the construction of ``taking disjoint unions'' commutes with these fimite limits. Thus the morphism $$ L_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n L)_{n + 1} $$ is a covering as it is the consequence of applying our ``taking disjoint unions'' functor to the morphism $$ K_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1} $$ which is assumed to be a covering in the sense of Definition \\ref{definition-covering-SR} by (2) for $K$. This makes sense because property (b) in particular assures us that if we start with a finite diagram of semi-representable objects over $X$ for which we can take disjoint unions, then the limit of the diagram in $\\text{SR}(\\mathcal{C}/X)$ still is a semi-representable object over $X$ for which we can take disjoint unions."}
+{"_id": "8491", "title": "algebraic-remark-flat-fp-presentation", "text": "If the morphism $f : \\mathcal{S}_U \\to \\mathcal{X}$ of Lemma \\ref{lemma-stack-presentation} is only assumed surjective, flat and locally of finite presentation, then it will still be the case that $f_{can} : [U/R] \\to \\mathcal{X}$ is an equivalence. In this case the morphisms $s$, $t$ will be flat and locally of finite presentation, but of course not smooth in general."}
+{"_id": "8701", "title": "sites-remark-big-presheaves", "text": "As already pointed out we may consider the category of presheaves with values in any of the ``big'' categories listed in Categories, Remark \\ref{categories-remark-big-categories}. These will be ``big'' categories as well and they will be listed in the above mentioned remark as we go along."}
+{"_id": "8702", "title": "sites-remark-functoriality-presheaves-values", "text": "Suppose that $\\mathcal{A}$ is a category such that any diagram $\\mathcal{I}_Y \\to \\mathcal{A}$ has a colimit in $\\mathcal{A}$. In this case it is clear that there are functors $u^p$ and $u_p$, defined in exactly the same way as above, on the categories of presheaves with values in $\\mathcal{A}$. Moreover, the adjointness of the pair $u^p$ and $u_p$ continues to hold in this setting."}
+{"_id": "8703", "title": "sites-remark-no-big-sites", "text": "(On set theoretic issues -- skip on a first reading.) The main reason for introducing sites is to study the category of sheaves on a site, because it is the generalization of the category of sheaves on a topological space that has been so important in algebraic geometry. In order to avoid thinking about things like ``classes of classes'' and so on, we will not allow sites to be ``big'' categories, in contrast to what we do for categories and $2$-categories. \\medskip\\noindent Suppose that $\\mathcal{C}$ is a category and that $\\text{Cov}(\\mathcal{C})$ is a proper class of coverings satisfying (1), (2) and (3) above. We will not allow this as a site either, mainly because we are going to take limits over coverings. However, there are several natural ways to replace $\\text{Cov}(\\mathcal{C})$ by a set of coverings or a slightly different structure that give rise to the same category of sheaves. For example: \\begin{enumerate} \\item In Sets, Section \\ref{sets-section-coverings-site} we show how to pick a suitable set of coverings that gives the same category of sheaves. \\item Another thing we can do is to take the associated topology (see Definition \\ref{definition-topology-associated-site}). The resulting topology on $\\mathcal{C}$ has the same category of sheaves. Two topologies have the same categories of sheaves if and only if they are equal, see Theorem \\ref{theorem-topology-and-topos}. A topology on a category is given by a choice of sieves on objects. The collection of all possible sieves and even all possible topologies on $\\mathcal{C}$ is a set. \\item We could also slightly modify the notion of a site, see Remark \\ref{remark-shrink-coverings} below, and end up with a canonical set of coverings. \\end{enumerate} Each of these solutions has some minor drawback. For the first, one has to check that constructions later on do not depend on the choice of the set of coverings. For the second, one has to learn about topologies and redo many of the arguments for sites. For the third, see the last sentence of Remark \\ref{remark-shrink-coverings}. \\medskip\\noindent Our approach will be to work with sites as in Definition \\ref{definition-site} above. Given a category $\\mathcal{C}$ with a proper class of coverings as above, we will replace this by a set of coverings producing a site using Sets, Lemma \\ref{sets-lemma-coverings-site}. It is shown in Lemma \\ref{lemma-choice-set-coverings-immaterial} below that the resulting category of sheaves (the topos) is independent of this choice. We leave it to the reader to use one of the other two strategies to deal with these issues if he/she so desires."}
+{"_id": "8704", "title": "sites-remark-sheaf-condition-empty-covering", "text": "If the covering $\\{U_i \\to U\\}_{i \\in I}$ is the empty family (this means that $I = \\emptyset$), then the sheaf condition signifies that $\\mathcal{F}(U) = \\{*\\}$ is a singleton set. This is because in (\\ref{equation-sheaf-condition}) the second and third sets are empty products in the category of sets, which are final objects in the category of sets, hence singletons."}
+{"_id": "8705", "title": "sites-remark-both-refine-same-H0", "text": "In particular this lemma shows that if $\\mathcal{U}$ is a refinement of $\\mathcal{V}$, and if $\\mathcal{V}$ is a refinement of $\\mathcal{U}$, then there is a canonical identification $H^0(\\mathcal{U}, \\mathcal{F}) = H^0(\\mathcal{V}, \\mathcal{F})$."}
+{"_id": "8706", "title": "sites-remark-quasi-continuous", "text": "(Skip on first reading.) Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let us use the definition of tautologically equivalent families of maps, see Definition \\ref{definition-combinatorial-tautological} to (slightly) weaken the conditions defining continuity. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Let us call $u$ {\\it quasi-continuous} if for every $\\mathcal{V} = \\{V_i \\to V\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ we have the following \\begin{enumerate} \\item[(1')] the family of maps $\\{u(V_i) \\to u(V)\\}_{i\\in I}$ is tautologically equivalent to an element of $\\text{Cov}(\\mathcal{D})$, and \\item[(2)] for any morphism $T \\to V$ in $\\mathcal{C}$ the morphism $u(T \\times_V V_i) \\to u(T) \\times_{u(V)} u(V_i)$ is an isomorphism. \\end{enumerate} We are going to see that Lemmas \\ref{lemma-pushforward-sheaf} and \\ref{lemma-adjoint-sheaves} hold in case $u$ is quasi-continuous as well. \\medskip\\noindent We first remark that the morphisms $u(V_i) \\to u(V)$ are representable, since they are isomorphic to representable morphisms (by the first condition). In particular, the family $u(\\mathcal{V}) = \\{u(V_i) \\to u(V)\\}_{i\\in I}$ gives rise to a zeroth {\\v C}ech cohomology group $H^0(u(\\mathcal{V}), \\mathcal{F})$ for any presheaf $\\mathcal{F}$ on $\\mathcal{D}$. Let $\\mathcal{U} = \\{U_j \\to u(V)\\}_{j \\in J}$ be an element of $\\text{Cov}(\\mathcal{D})$ tautologically equivalent to $\\{u(V_i) \\to u(V)\\}_{i \\in I}$. Note that $u(\\mathcal{V})$ is a refinement of $\\mathcal{U}$ and vice versa. Hence by Remark \\ref{remark-both-refine-same-H0} we see that $H^0(u(\\mathcal{V}), \\mathcal{F}) = H^0(\\mathcal{U}, \\mathcal{F})$. In particular, if $\\mathcal{F}$ is a sheaf, then $\\mathcal{F}(u(V)) = H^0(u(\\mathcal{V}), \\mathcal{F})$ because of the sheaf property expressed in terms of zeroth {\\v C}ech cohomology groups. We conclude that $u^p\\mathcal{F}$ is a sheaf if $\\mathcal{F}$ is a sheaf, since $H^0(\\mathcal{V}, u^p\\mathcal{F}) = H^0(u(\\mathcal{V}), \\mathcal{F})$ which we just observed is equal to $\\mathcal{F}(u(V)) = u^p\\mathcal{F}(V)$. Thus Lemma \\ref{lemma-pushforward-sheaf} holds. Lemma \\ref{lemma-adjoint-sheaves} follows immediately."}
+{"_id": "8707", "title": "sites-remark-explain-left-exact", "text": "The conditions of Proposition \\ref{proposition-get-morphism} above are equivalent to saying that $u$ is left exact, i.e., commutes with finite limits. See Categories, Lemmas \\ref{categories-lemma-finite-limits-exist} and \\ref{categories-lemma-characterize-left-exact}. It seems more natural to phrase it in terms of final objects and fibre products since this seems to have more geometric meaning in the examples."}
+{"_id": "8708", "title": "sites-remark-quasi-continuous-morphism-sites", "text": "(Skip on first reading.) Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Analogously to Definition \\ref{definition-morphism-sites} we say that a {\\it quasi-morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$} is given by a quasi-continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ (see Remark \\ref{remark-quasi-continuous}) such that $u_s$ is exact. The analogue of Proposition \\ref{proposition-get-morphism} in this setting is obtained by replacing the word ``continuous'' by the word ``quasi-continuous'', and replacing the word ``morphism'' by ``quasi-morphism''. The proof is literally the same."}
+{"_id": "8709", "title": "sites-remark-pt-topos", "text": "There are many sites that give rise to the topos $\\Sh(pt)$. A useful example is the following. Suppose that $S$ is a set (of sets) which contains at least one nonempty element. Let $\\mathcal{S}$ be the category whose objects are elements of $S$ and whose morphisms are arbitrary set maps. Assume that $\\mathcal{S}$ has fibre products. For example this will be the case if $S = \\mathcal{P}(\\text{infinite set})$ is the power set of any infinite set (exercise in set theory). Make $\\mathcal{S}$ into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section \\ref{sets-section-coverings-site}). We claim that $\\Sh(\\mathcal{S})$ is equivalent to the category of sets. \\medskip\\noindent We first prove this in case $S$ contains $e \\in S$ which is a singleton. In this case, there is an equivalence of topoi $i : \\Sh(pt) \\to \\Sh(\\mathcal{S})$ given by the functors \\begin{equation} \\label{equation-sheaves-pt-sets} i^{-1}\\mathcal{F} = \\mathcal{F}(e), \\quad i_*E = (U \\mapsto \\Mor_{\\textit{Sets}}(U, E)) \\end{equation} Namely, suppose that $\\mathcal{F}$ is a sheaf on $\\mathcal{S}$. For any $U \\in \\Ob(\\mathcal{S}) = S$ we can find a covering $\\{\\varphi_u : e \\to U\\}_{u \\in U}$, where $\\varphi_u$ maps the unique element of $e$ to $u \\in U$. The sheaf condition implies in this case that $\\mathcal{F}(U) = \\prod_{u \\in U} \\mathcal{F}(e)$. In other words $\\mathcal{F}(U) = \\Mor_{\\textit{Sets}}(U, \\mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor $$ i_* : \\textit{Sets} = \\Sh(pt) \\longrightarrow \\Sh(\\mathcal{S}), \\quad E \\longmapsto (U \\mapsto \\Mor_{\\textit{Sets}}(U, E)) $$ is an equivalence of categories, and its inverse is the functor $i^{-1}$ given above. \\medskip\\noindent If $\\mathcal{S}$ does not contain a singleton, then the functor $i_*$ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty $\\tilde e \\in S$ and a map $\\varphi : \\tilde e \\to \\tilde e$ whose image is a singleton. For any sheaf $\\mathcal{F}$ set $$ \\mathcal{F}(e) := \\Im( \\mathcal{F}(\\varphi) : \\mathcal{F}(\\tilde e) \\longrightarrow \\mathcal{F}(\\tilde e) ) $$ and show that this is a quasi-inverse to $i_*$. Details omitted."}
+{"_id": "8710", "title": "sites-remark-morphism-topoi-big", "text": "(Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Although we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose $\\mathcal{C}$ and $\\mathcal{D}$ are given sites. Consider a functor $\\Phi : \\mathcal{C} \\to \\Sh(\\mathcal{D})$. Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on $\\Phi$. \\begin{enumerate} \\item Is it true, given a sheaf $\\mathcal{F}$ on $\\mathcal{D}$, that the rule $U \\mapsto \\Mor_{\\Sh(\\mathcal{D})}(\\Phi(U), \\mathcal{F})$ defines a sheaf on $\\mathcal{C}$? If so, this defines a functor $\\Phi_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$. \\item Is it true that $\\Phi_*$ has a left adjoint? If so, write $\\Phi^{-1}$ for this left adjoint. \\item Is it true that $\\Phi^{-1}$ is exact? \\end{enumerate} If the last question still has the answer ``yes'', then we obtain a morphism of topoi $(\\Phi_*, \\Phi^{-1})$. Moreover, given any morphism of topoi $(f_*, f^{-1})$ we may set $\\Phi(U) = f^{-1}(h_U^\\#)$ and obtain a functor $\\Phi$ as above with $f_* \\cong \\Phi_*$ and $f^{-1} \\cong \\Phi^{-1}$ (compatible with adjoint property). The upshot is that by working with the collection of $\\Phi$ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of $\\Phi$ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of conditions (2) and (3) above; we do this in the case of points of topoi in Section \\ref{section-points}."}
+{"_id": "8711", "title": "sites-remark-quasi-continuous-morphism-topoi", "text": "(Skip on first reading.) Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. A quasi-morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$ (see Remark \\ref{remark-quasi-continuous-morphism-sites}) gives rise to a morphism of topoi $f$ from $\\Sh(\\mathcal{D})$ to $\\Sh(\\mathcal{C})$ exactly as in Lemma \\ref{lemma-morphism-sites-topoi}."}
+{"_id": "8712", "title": "sites-remark-cartesian-cocontinuous", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories. Given morphisms $g : u(U) \\to V$ and $f : W \\to V$ in $\\mathcal{D}$ we can consider the functor $$ \\mathcal{C}^{opp} \\longrightarrow \\textit{Sets},\\quad T \\longmapsto \\Mor_\\mathcal{C}(T, U) \\times_{\\Mor_\\mathcal{D}(u(T), V)} \\Mor_\\mathcal{D}(u(T), W) $$ If this functor is representable, denote $U \\times_{g, V, f} W$ the corresponding object of $\\mathcal{C}$. Assume that $\\mathcal{C}$ and $\\mathcal{D}$ are sites. Consider the property $P$: for every covering $\\{f_j : V_j \\to V\\}$ of $\\mathcal{D}$ and any morphism $g : u(U) \\to V$ we have \\begin{enumerate} \\item $U \\times_{g, V, f_i} V_i$ exists for all $i$, and \\item $\\{U \\times_{g, V, f_i} V_i \\to U\\}$ is a covering of $\\mathcal{C}$. \\end{enumerate} Please note the similarity with the definition of continuous functors. If $u$ has $P$ then $u$ is cocontinuous (details omitted). Many of the cocontinuous functors we will encounter satisfy $P$."}
+{"_id": "8713", "title": "sites-remark-localize-presheaves", "text": "Localization and presheaves. Let $\\mathcal{C}$ be a category. Let $U$ be an object of $\\mathcal{C}$. Strictly speaking the functors $j_U^{-1}$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\\mathcal{C}$ (see Example \\ref{example-indiscrete}). Hence we also obtain a functor $$ j_U^{-1} : \\textit{PSh}(\\mathcal{C}) \\longrightarrow \\textit{PSh}(\\mathcal{C}/U) $$ and functors $$ j_{U*}, j_{U!} : \\textit{PSh}(\\mathcal{C}/U) \\longrightarrow \\textit{PSh}(\\mathcal{C}) $$ which are right, left adjoint to $j_U^{-1}$. By Lemma \\ref{lemma-describe-j-shriek} we see that $j_{U!}\\mathcal{G}$ is the presheaf $$ V \\longmapsto \\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) $$ In addition the functor $j_{U!}$ commutes with fibre products and equalizers."}
+{"_id": "8714", "title": "sites-remark-localization-cartesian-cocontinuous", "text": "Let $\\mathcal{C}$ be a site. Let $U \\to V$ be a morphism of $\\mathcal{C}$. The cocontinuous functors $\\mathcal{C}/U \\to \\mathcal{C}$ and $j : \\mathcal{C}/U \\to \\mathcal{C}/V$ (Lemma \\ref{lemma-relocalize}) satisfy property $P$ of Remark \\ref{remark-cartesian-cocontinuous}. For example, if we have objects $(X/U)$, $(W/V)$, a morphism $g : j(X/U) \\to (W/V)$, and a covering $\\{f_i  : (W_i/V) \\to (W/V)\\}$ then $(X \\times_W W_i/U)$ is an avatar of $(X/U) \\times_{g, (W/V), f_i} (W_i/V)$ and the family $\\{(X \\times_W W_i/U) \\to (X/U)\\}$ is a covering of $\\mathcal{C}/U$."}
+{"_id": "8715", "title": "sites-remark-morphism-topoi-comes-from-morphism-sites", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}. If the site $\\mathcal{D}$ has a final object and fibre products then the functor $u : \\mathcal{D} \\to \\mathcal{C}'$ satisfies all the assumptions of Proposition \\ref{proposition-get-morphism}. Namely, in addition to the properties mentioned in the lemma $u$ also transforms the final object of $\\mathcal{D}$ into the final object of $\\mathcal{C}'$. This is clear from the construction of $u$. Hence, if we first apply Lemmas \\ref{lemma-topos-good-site} to $\\mathcal{D}$ and then Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites} to the resulting morphism of topoi $\\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D}')$ we obtain the following statement: Any morphism of topoi $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ fits into a commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{C}) \\ar[d]_g \\ar[r]_f & \\Sh(\\mathcal{D}) \\ar[d]^e \\\\ \\Sh(\\mathcal{C}') \\ar[r]^{f'} & \\Sh(\\mathcal{D}') } $$ where the following properties hold: \\begin{enumerate} \\item the morphisms $e$ and $g$ are equivalences given by special cocontinuous functors $\\mathcal{C} \\to \\mathcal{C}'$ and $\\mathcal{D} \\to \\mathcal{D}'$, \\item the sites $\\mathcal{C}'$ and $\\mathcal{D}'$ have fibre products, final objects and have subcanonical topologies, \\item the morphism $f' : \\mathcal{C}' \\to \\mathcal{D}'$ comes from a morphism of sites corresponding to a functor $u : \\mathcal{D}' \\to \\mathcal{C}'$ to which Proposition \\ref{proposition-get-morphism} applies, and \\item given any set of sheaves $\\mathcal{F}_i$ (resp.\\ $\\mathcal{G}_j$) on $\\mathcal{C}$ (resp.\\ $\\mathcal{D}$) we may assume each of these is a representable sheaf on $\\mathcal{C}'$ (resp.\\ $\\mathcal{D}'$). \\end{enumerate} It is often useful to replace $\\mathcal{C}$ and $\\mathcal{D}$ by $\\mathcal{C}'$ and $\\mathcal{D}'$."}
+{"_id": "8716", "title": "sites-remark-equivalence-topoi-comes-from-morphism-sites", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites}. Suppose that in addition the original morphism of topoi $\\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ is an equivalence. Then the construction in the proof of Lemma \\ref{lemma-morphism-topoi-comes-from-morphism-sites} gives two functors $$ \\mathcal{C} \\rightarrow \\mathcal{C}' \\leftarrow \\mathcal{D} $$ which are both special cocontinuous functors. Hence in this case we can actually factor the morphism of topoi as a composition $$ \\Sh(\\mathcal{C}) \\rightarrow \\Sh(\\mathcal{C}') = \\Sh(\\mathcal{D}') \\leftarrow \\Sh(\\mathcal{D}) $$ as in Remark \\ref{remark-morphism-topoi-comes-from-morphism-sites}, but with the middle morphism an identity."}
+{"_id": "8717", "title": "sites-remark-improve-proposition-points-limits", "text": "In fact, let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has a final object $X$  and fibre products. Let $p = u: \\mathcal{C} \\to \\textit{Sets}$ be a functor such that \\begin{enumerate} \\item $u(X) = \\{*\\}$ a singleton, and \\item for every pair of morphisms $U \\to W$ and $V \\to W$ with the same target the map $u(U \\times_W V) \\to u(U) \\times_{u(W)} u(V)$ is surjective. \\item for every covering $\\{U_i \\to U\\}$ the map $\\coprod u(U_i) \\to u(U)$ is surjective. \\end{enumerate} Then, in general, $p$ is {\\bf not} a point of $\\mathcal{C}$. An example is the category $\\mathcal{C}$ with two objects $\\{U, X\\}$ and exactly one non-identity arrow, namely $U \\to X$. We endow $\\mathcal{C}$ with the trivial topology, i.e., the only coverings are $\\{U \\to U\\}$ and $\\{X \\to X\\}$. A sheaf $\\mathcal{F}$ is the same thing as a presheaf and consists of a triple $(A, B, A \\to B)$: namely $A = \\mathcal{F}(X)$, $B = \\mathcal{F}(U)$ and $A \\to B$ is the restriction mapping corresponding to $U \\to X$. Note that $U \\times_X U = U$ so fibre products exist. Consider the functor $u = p$ with $u(X) = \\{*\\}$ and $u(U) = \\{*_1, *_2\\}$. This satisfies (1), (2), and (3), but the corresponding stalk functor (\\ref{equation-stalk}) is the functor $$ (A, B, A \\to B) \\longmapsto B \\amalg_A B $$ which isn't exact. Namely, consider $(\\emptyset, \\{1\\}, \\emptyset \\to \\{1\\}) \\to (\\{1\\}, \\{1\\}, \\{1\\} \\to \\{1\\})$ which is an injective map of sheaves, but is transformed into the noninjective map of sets $$ \\{1\\} \\amalg \\{1\\} \\longrightarrow \\{1\\} \\amalg_{\\{1\\}} \\{1\\} $$ by the stalk functor."}
+{"_id": "8718", "title": "sites-remark-not-pushforward", "text": "Warning: The result of Lemma \\ref{lemma-stalk-j-shriek} has no analogue for $j_{U, *}$."}
+{"_id": "8719", "title": "sites-remark-no-pullback-presheaves", "text": "Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor which gives rise to a morphism of sites $\\mathcal{C} \\to \\mathcal{D}$. Note that even in the case of abelian groups we have not defined a pullback functor for presheaves of abelian groups. Since all colimits are representable in the category of abelian groups, we certainly may define a functor $u_p^{ab}$ on abelian presheaves by the same colimits as we have used to define $u_p$ on presheaves of sets. It will also be the case that $u_p^{ab}$ is adjoint to $u^p$ on the categories of abelian presheaves. However, it will not always be the case that $u_p^{ab}$ agrees with $u_p$ on the underlying presheaves of sets."}
+{"_id": "8720", "title": "sites-remark-compose-base-change", "text": "Consider a commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{B}') \\ar[r]_k \\ar[d]_{f'} & \\Sh(\\mathcal{B}) \\ar[d]^f \\\\ \\Sh(\\mathcal{C}') \\ar[r]^l \\ar[d]_{g'} & \\Sh(\\mathcal{C}) \\ar[d]^g \\\\ \\Sh(\\mathcal{D}') \\ar[r]^m & \\Sh(\\mathcal{D}) } $$ of topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} m^{-1} \\circ (g \\circ f)_* & = m^{-1} \\circ g_* \\circ f_* \\\\ & \\to g'_* \\circ l^{-1} \\circ f_* \\\\ & \\to g'_* \\circ f'_* \\circ k^{-1} \\\\ & = (g' \\circ f')_* \\circ k^{-1} \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "8721", "title": "sites-remark-compose-base-change-horizontal", "text": "Consider a commutative diagram $$ \\xymatrix{ \\Sh(\\mathcal{C}'') \\ar[r]_{g'} \\ar[d]_{f''} & \\Sh(\\mathcal{C}') \\ar[r]_g \\ar[d]_{f'} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D}'') \\ar[r]^{h'} & \\Sh(\\mathcal{D}') \\ar[r]^h & \\Sh(\\mathcal{D}) } $$ of ringed topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition \\begin{align*} (h \\circ h')^{-1} \\circ f_* & = (h')^{-1} \\circ h^{-1} \\circ f_* \\\\ & \\to (h')^{-1} \\circ f'_* \\circ g^{-1} \\\\ & \\to f''_* \\circ (g')^{-1} \\circ g^{-1} \\\\ & = f''_* \\circ (g \\circ g')^{-1} \\end{align*} is the base change map for the rectangle. We omit the verification."}
+{"_id": "8722", "title": "sites-remark-enlarge-coverings", "text": "Enlarging the class of coverings. Clearly, if $\\text{Cov}(\\mathcal{C})$ defines the structure of a site on $\\mathcal{C}$ then we may add to $\\mathcal{C}$ any set of families of morphisms with fixed target tautologically equivalent (see Definition \\ref{definition-combinatorial-tautological}) to elements of $\\text{Cov}(\\mathcal{C})$ without changing the topology."}
+{"_id": "8723", "title": "sites-remark-shrink-coverings", "text": "Shrinking the class of coverings. Let $\\mathcal{C}$ be a site. Consider the set $$ \\mathcal{S} = P(\\text{Arrows}(\\mathcal{C})) \\times \\Ob(\\mathcal{C}) $$ where $P(\\text{Arrows}(\\mathcal{C}))$ is the power set of the set of morphisms, i.e., the set of all sets of morphisms. Let $\\mathcal{S}_\\tau \\subset \\mathcal{S}$ be the subset consisting of those $(T, U) \\in \\mathcal{S}$ such that (a) all $\\varphi \\in T$ have target $U$, (b) the collection $\\{\\varphi\\}_{\\varphi \\in T}$ is tautologically equivalent (see Definition \\ref{definition-combinatorial-tautological}) to some covering in $\\text{Cov}(\\mathcal{C})$. Clearly, considering the elements of $\\mathcal{S}_\\tau$ as the coverings, we do not get exactly the notion of a site as defined in Definition \\ref{definition-site}. The structure $(\\mathcal{C}, \\mathcal{S}_\\tau)$ we get satisfies slightly modified conditions. The modified conditions are: \\begin{enumerate} \\item[(0')] $\\text{Cov}(\\mathcal{C}) \\subset P(\\text{Arrows}(\\mathcal{C})) \\times \\Ob(\\mathcal{C})$, \\item[(1')] If $V \\to U$ is an isomorphism then $(\\{V \\to U\\}, U) \\in \\text{Cov}(\\mathcal{C})$. \\item[(2')] If $(T, U) \\in \\text{Cov}(\\mathcal{C})$ and for $f : U' \\to U$ in $T$ we are given $(T_f, U') \\in \\text{Cov}(\\mathcal{C})$, then setting $T' = \\{f \\circ f' \\mid f \\in T,\\ f' \\in T_f\\}$, we get $(T', U) \\in \\text{Cov}(\\mathcal{C})$. \\item[(3')] If $(T, U) \\in \\text{Cov}(\\mathcal{C})$ and $g : V \\to U$ is a morphism of $\\mathcal{C}$ then \\begin{enumerate} \\item $U' \\times_{f, U, g} V$ exists for $f : U' \\to U$ in $T$, and \\item setting $T' = \\{\\text{pr}_2 : U' \\times_{f, U, g} V \\to V \\mid f : U' \\to U \\in T\\}$ for some choice of fibre products we get $(T', V) \\in \\text{Cov}(\\mathcal{C})$. \\end{enumerate} \\end{enumerate} And it is easy to verify that, given a structure satisfying (0') -- (3') above, then after suitably enlarging $\\text{Cov}(\\mathcal{C})$ (compare Sets, Section \\ref{sets-section-coverings-site}) we get a site. Obviously there is little difference between this notion and the actual notion of a site, at least from the point of view of the topology. There are two benefits: because of condition (0') above the coverings automatically form a set, and because of (0') the totality of all structures of this type forms a set as well. The price you pay for this is that you have to keep writing ``tautologically equivalent'' everywhere."}
+{"_id": "8803", "title": "sets-remark-how-to-use-reflection", "text": "The lemma above can also be proved using the reflection principle. However, one has to be careful. Namely, suppose the sentence $\\phi_{scheme}(X)$ expresses the property ``$X$ is a scheme'', then what does the formula $\\phi_{scheme}^{V_\\alpha}(X)$ mean? It is true that the reflection principle says we can find $\\alpha$ such that for all $X \\in V_\\alpha$ we have $\\phi_{scheme}(X) \\leftrightarrow \\phi_{scheme}^{V_\\alpha}(X)$ but this is entirely useless. It is only by combining two such statements that something interesting happens. For example suppose $\\phi_{red}(X, Y)$ expresses the property ``$X$, $Y$ are schemes, and $Y$ is the reduction of $X$'' (see Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}). Suppose we apply the reflection principle to the pair of formulas $\\phi_1(X, Y) = \\phi_{red}(X, Y)$, $\\phi_2(X) = \\exists Y, \\phi_1(X, Y)$. Then it is easy to see that any $\\alpha$ produced by the reflection principle has the property that given $X \\in \\Ob(\\Sch_\\alpha)$ the reduction of $X$ is also an object of $\\Sch_\\alpha$ (left as an exercise)."}
+{"_id": "8804", "title": "sets-remark-what-is-not-in-it", "text": "Let $R$ be a ring. Suppose we consider the ring $\\prod_{\\mathfrak p \\in \\Spec(R)} \\kappa(\\mathfrak p)$. The cardinality of this ring is bounded by $|R|^{2^{|R|}}$, but is not bounded by $|R|^{\\aleph_0}$ in general. For example if $R = \\mathbf{C}[x]$ it is not bounded by $|R|^{\\aleph_0}$ and if $R = \\prod_{n \\in \\mathbf{N}} \\mathbf{F}_2$ it is not bounded by $|R|^{|R|}$. Thus the ``And so on'' of Lemma \\ref{lemma-what-is-in-it} above should be taken with a grain of salt. Of course, if it ever becomes necessary to consider these rings in arguments pertaining to fppf/\\'etale cohomology, then we can change the function $Bound$ above into the function $\\kappa \\mapsto \\kappa^{2^\\kappa}$."}
+{"_id": "8805", "title": "sets-remark-better", "text": "It is likely the case that, for some limit ordinal $\\alpha$, the set of coverings $\\text{Cov}(\\mathcal{C})_\\alpha$ satisfies the conditions of the lemma. This is after all what an application of the reflection principle would appear to give (modulo caveats as described at the end of Section \\ref{section-reflection-principle} and in Remark \\ref{remark-how-to-use-reflection})."}
+{"_id": "8849", "title": "more-etale-remark-covariance-f-shriek-separated", "text": "Let $f : X \\to Y$ be morphism of schemes which is separated and locally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $X' \\subset X$ be an open subscheme. Denote $f' : X' \\to Y$ the restriction of $f$. There is a canonical injective map $$ f'_!(\\mathcal{F}|_{X'}) \\longrightarrow f_!\\mathcal{F} $$ Namely, let $V \\in Y_\\etale$ and consider a section $s' \\in f'_*(\\mathcal{F}|_{X'})(V) = \\mathcal{F}(X' \\times_Y V)$ with support $Z'$ proper over $V$. Then $Z'$ is closed in $X \\times_Y V$ as well, see Cohomology of Schemes, Lemma \\ref{coherent-lemma-functoriality-closed-proper-over-base}. Thus there is a unique section $s \\in \\mathcal{F}(X \\times_Y V) = f_*\\mathcal{F}(V)$ whose restriction to $X' \\times_Y V$ is $s'$ and whose restriction to $X \\times_Y V \\setminus Z'$ is zero, see Lemma \\ref{lemma-section-support-in-locally-closed}. This construction is compatible with restriction maps and hence induces the desired map of sheaves $f'_!(\\mathcal{F}|_{X'}) \\to f_!\\mathcal{F}$ which is clearly injective. By construction we obtain a commutative diagram $$ \\xymatrix{ f'_!(\\mathcal{F}|_{X'}) \\ar[r] \\ar[d] & f_!\\mathcal{F} \\ar[d] \\\\ f'_*(\\mathcal{F}|_{X'}) & f_*\\mathcal{F} \\ar[l] } $$ functorial in $\\mathcal{F}$. It is clear that for $X'' \\subset X'$ open with $f'' = f|_{X''} : X'' \\to Y$ the composition of the canonical maps $f''_!\\mathcal{F}|_{X''} \\to f'_!\\mathcal{F}|_{X'} \\to f_!\\mathcal{F}$ just constructed is the canonical map $f''_!\\mathcal{F}|_{X''} \\to f_!\\mathcal{F}$."}
+{"_id": "8850", "title": "more-etale-remark-covariance-compact-support", "text": "Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Exactly as in Remark \\ref{remark-covariance-f-shriek-separated} there are injective maps $$ H^0_c(X', \\mathcal{F}|_{X'}) \\longrightarrow H^0_c(X, \\mathcal{F}) $$ which turn $H^0_c$ into a ``cosheaf'' on the Zariski site of $X$."}
+{"_id": "8851", "title": "more-etale-remark-f-shriek-base-change-composition", "text": "The isomorphisms between functors constructed above satisfy the following two properties: \\begin{enumerate} \\item Let $f : X \\to Y$, $g : Y \\to Z$, and $h : Z \\to T$ be composable morphisms of schemes which are separated and locally of finite type. Then the diagram $$ \\xymatrix{ (h \\circ g \\circ f)_! \\ar[r] \\ar[d] & (h \\circ g)_! \\circ f_! \\ar[d] \\\\ h_! \\circ (g \\circ f)_! \\ar[r] & h_! \\circ g_! \\circ f_! } $$ commutes where the arrows are those of Lemma \\ref{lemma-f-shriek-composition}. \\item Suppose that we have a diagram of schemes $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_c & X \\ar[d]^f \\\\ Y' \\ar[d]_{g'} \\ar[r]_b & Y \\ar[d]^g \\\\ Z' \\ar[r]^a & Z } $$ with both squares cartesian and $f$ and $g$ separated and locally of finite type. Then the diagram $$ \\xymatrix{ a^{-1} \\circ (g \\circ f)_! \\ar[d] \\ar[rr] & & (g' \\circ f')_! \\circ c^{-1} \\ar[d] \\\\ a^{-1} \\circ g_! \\circ f_! \\ar[r] & g'_! \\circ b^{-1} \\circ f_! \\ar[r] & g'_! \\circ f'_! \\circ c^{-1} } $$ commutes where the horizontal arrows are those of Lemma \\ref{lemma-base-change-f-shriek-separated} the arrows are those of Lemma \\ref{lemma-f-shriek-composition}. \\end{enumerate} Part (1) holds true because we have a similar commutative diagram for pushforwards. Part (2) holds by the very general compatibility of base change maps for pushforwards (Sites, Remark \\ref{sites-remark-compose-base-change}) and the fact that the isomorphisms in Lemmas \\ref{lemma-base-change-f-shriek-separated} and \\ref{lemma-f-shriek-composition} are constructed using the corresponding maps fo pushforwards."}
+{"_id": "8852", "title": "more-etale-remark-covariance-lqf-f-shriek", "text": "Let $f : X \\to Y$ be locally quasi-finite morphism of schemes. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $X' \\subset X$ be an open subscheme and denote $f' : X' \\to Y$ the restriction of $f$. We claim there is a canonical map $$ f'_!(\\mathcal{F}|_{X'}) \\longrightarrow f_!\\mathcal{F} $$ Namely, this map will be the sheafification of a canonical map $$ f'_{p!}(\\mathcal{F}|_{X'}) \\to f_{p!}\\mathcal{F} $$ constructed as follows. Let $V \\in Y_\\etale$ and consider a section $s' = \\sum_{i = 1, \\ldots, n} (Z'_i, s'_i)$ as in (\\ref{equation-formal-sum}) defining an element of $f'_{p!}(\\mathcal{F}|_{X'})(V)$. Then $Z'_i \\subset X'_V$ may also be viewed as a locally closed subscheme of $X_V$ and we have $H_{Z'_i}(\\mathcal{F}|_{X'}) = H_{Z'_i}(\\mathcal{F})$. We will map $s'$ to the exact same sum $s = \\sum_{i = 1, \\ldots, n} (Z'_i, s'_i)$ but now viewed as an element of $f_{p!}\\mathcal{F}(V)$. We omit the verification that this construction is compatible with restriction mappings and functorial in $\\mathcal{F}$. This construction has the following properties: \\begin{enumerate} \\item The maps $f'_{p!}\\mathcal{F}' \\to f_{p!}\\mathcal{F}$ and $f'_!\\mathcal{F}' \\to f_!\\mathcal{F}$ are compatible with the description of stalks given in Lemmas \\ref{lemma-finite-support-stalk} and \\ref{lemma-lqf-f-shriek-stalk}. \\item If $f$ is separated, then the map $f'_{p!}\\mathcal{F}' \\to f_{p!}\\mathcal{F}$ is the same as the map constructed in Remark \\ref{remark-covariance-f-shriek-separated} via the isomorphism in Lemma \\ref{lemma-finite-support-f-shriek-separated}. \\item If $X'' \\subset X'$ is another open, then the composition of $f''_{p!}(\\mathcal{F}|_{X''}) \\to f'_{p!}(\\mathcal{F}|_{X'}) \\to f_{p!}\\mathcal{F}$ is the map $f''_{p!}(\\mathcal{F}|_{X''}) \\to f_{p!}\\mathcal{F}$ for the inclusion $X'' \\subset X$. Sheafifying we conclude the same holds true for $f''_!(\\mathcal{F}|_{X''}) \\to f'_!(\\mathcal{F}|_{X'}) \\to f_!\\mathcal{F}$. \\item The map $f'_!\\mathcal{F}' \\to f_!\\mathcal{F}$ is injective because we can check this on stalks. \\end{enumerate} All of these statements are easily proven by representing elements as finite sums as above and considering what happens to these elements."}
+{"_id": "8853", "title": "more-etale-remark-alternative-lqf-f-shriek", "text": "Lemma \\ref{lemma-lqf-colimit-f-shriek} gives an alternative construction of the functor $f_!$ for locally quasi-finite morphisms $f$. Namely, given a locally quasi-finite morphism $f : X \\to Y$ of schemes we can choose an open covering $X = \\bigcup_{i \\in I} X_i$ such that each $f_i : X_i \\to Y$ is separated. For example choose an affine open covering of $X$. Then we can define $f_!\\mathcal{F}$ as the cokernel of the penultimate map of the complex of the lemma, i.e., $$ f_!\\mathcal{F} = \\Coker\\left( \\bigoplus\\nolimits_{i_0, i_1} f_{i_0i_1, !} \\mathcal{F}|_{X_{i_0i_1}} \\to \\bigoplus\\nolimits_{i_0} f_{i_0, !} \\mathcal{F}|_{X_{i_0}} \\right) $$ where we can use the construction of $f_{i_0, !}$ and $f_{i_0i_1, !}$ in Section \\ref{section-compact-support} because the morphisms $f_{i_0}$ and $f_{i_0 i_1}$ are separated. One can then compute the stalks of $f_!$ (using the separated case, namely Lemma \\ref{lemma-lqf-f-shriek-separated-colimits}) and obtain the result of Lemma \\ref{lemma-lqf-f-shriek-stalk}. Having done so all the other results of this section can be deduced from this as well."}
+{"_id": "8854", "title": "more-etale-remark-construct-map-presheaves-downstairs", "text": "Let $g : Y' \\to Y$ be a morphism of schemes. For an abelian presheaf $\\mathcal{G}'$ on $Y'_\\etale$ let us denote $g_*\\mathcal{G}'$ the presheaf $V \\mapsto \\mathcal{G}'(Y' \\times_Y V)$. If $\\alpha : \\mathcal{G} \\to g_*\\mathcal{G}'$ is a map of abelian presheaves on $Y_\\etale$, then there is a unique map $\\alpha^\\# : \\mathcal{G}^\\# \\to g_*((\\mathcal{G}')^\\#)$ of abelian sheaves on $Y_\\etale$ such that the diagram $$ \\xymatrix{ \\mathcal{G} \\ar[d] \\ar[r]_\\alpha & g_*\\mathcal{G}' \\ar[d] \\\\ \\mathcal{G}^\\# \\ar[r]^-{\\alpha^\\#} & g_*((\\mathcal{G}')^\\#) } $$ is commutative where the vertical maps come from the canonical maps $\\mathcal{G} \\to \\mathcal{G}^\\#$ and $\\mathcal{G}' \\to (\\mathcal{G}')^\\#$. If $\\alpha' : g^{-1}\\mathcal{G}^\\# \\to (\\mathcal{G}')^\\#$ is the map adjoint to $\\alpha^\\#$, then for a geometric point $\\overline{y}' : \\Spec(k) \\to Y'$ with image $\\overline{y} = g \\circ \\overline{y}'$ in $Y$, the map $$ \\alpha'_{\\overline{y}'} : \\mathcal{G}_{\\overline{y}} = (\\mathcal{G}^\\#)_{\\overline{y}} = (g^{-1}\\mathcal{G}^\\#)_{\\overline{y}'} \\longrightarrow (\\mathcal{G}')^\\#_{\\overline{y}'} = \\mathcal{G}'_{\\overline{y}'} $$ is given by mapping the class in the stalk of a section $s$ of $\\mathcal{G}$ over an \\'etale neighbourhood $(V, \\overline{v})$ to the class of the section $\\alpha(s)$ in $g_*\\mathcal{G}'(V) = \\mathcal{G}'(Y' \\times_Y V)$ over the \\'etale neighbourhood $(Y' \\times_Y V, (\\overline{y}', \\overline{v}))$ in the stalk of $\\mathcal{G}'$ at $\\overline{y}'$."}
+{"_id": "8855", "title": "more-etale-remark-pointed-sets", "text": "The material in this section can be generalized to sheaves of pointed sets. Namely, for a site $\\mathcal{C}$ denote $\\Sh^*(\\mathcal{C})$ the category of sheaves of pointed sets. The constructions in this and the preceding section apply, mutatis mutandis, to sheaves of pointed sets. Thus given a locally quasi-finite morphism $f : X \\to Y$ of schemes we obtain an adjoint pair of functors $$ f_! : \\Sh^*(X_\\etale) \\longrightarrow \\Sh^*(Y_\\etale) \\quad\\text{and}\\quad f^! : \\Sh^*(Y_\\etale) \\longrightarrow \\Sh^*(X_\\etale) $$ such that for every geometric point $\\overline{y}$ of $Y$ there are isomorphisms $$ (f_!\\mathcal{F})_{\\overline{y}} = \\coprod\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}} $$ (coproduct taken in the category of pointed sets) functorial in $\\mathcal{F} \\in \\Sh^*(X_\\etale)$ and isomorphisms $$ f^!(\\overline{y}_*S) = \\prod\\nolimits_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*S $$ functorial in the pointed set $S$. If $F : \\textit{Ab}(X_\\etale) \\to \\Sh^*(X_\\etale)$ and $F : \\textit{Ab}(Y_\\etale) \\to \\Sh^*(Y_\\etale)$ denote the forgetful functors, compatibility between the constructions will guarantee the existence of canonical maps $$ f_!F(\\mathcal{F}) \\longrightarrow F(f_!\\mathcal{F}) $$ functorial in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ and $$ F(f^!\\mathcal{G}) \\longrightarrow f^!F(\\mathcal{G}) $$ functorial in $\\mathcal{G} \\in \\textit{Ab}(Y_\\etale)$ which produce the obvious maps on stalks, resp.\\ skyscraper sheaves. In fact, the transformation $F \\circ f^! \\to f^! \\circ F$ is an isomorphism (because $f^!$ commutes with products)."}
+{"_id": "8856", "title": "more-etale-remark-going-around", "text": "Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z } $$ of schemes whose vertical arrows are proper and whose horizontal arrows are separated and locally quasi-finite. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \\begin{matrix} A & B \\\\ C & D \\end{matrix} $$ Then the maps of Lemma \\ref{lemma-shriek-proper-and-open} for the squares are (where we use $Rf_* = f_*$, etc) $$ \\begin{matrix} \\gamma_A : l'_! \\circ f''_* \\to f'_* \\circ k'_! & \\gamma_B : l_! \\circ f'_* \\to f_* \\circ k_! \\\\ \\gamma_C : m'_! \\circ g''_* \\to g'_* \\circ l'_! & \\gamma_D : m_! \\circ g'_* \\to g_* \\circ l_! \\end{matrix} $$ For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further maps $$ \\begin{matrix} \\gamma_{A + B} : (l \\circ l')_! \\circ f''_* \\to f_* \\circ (k \\circ k')_*  \\\\ \\gamma_{C + D} : (m \\circ m')_! \\circ g''_* \\to g_* \\circ (l \\circ l')_! \\\\ \\gamma_{A + C} : m'_! \\circ (g'' \\circ f'')_* \\to (g' \\circ f')_* \\circ k'_! \\\\ \\gamma_{B + D} : m_! \\circ (g' \\circ f')_* \\to (g \\circ f)_* \\circ k_! \\end{matrix} $$ By Lemma \\ref{lemma-shriek-proper-and-open-compose-horizontal} we have $$ \\gamma_{A + B} = \\gamma_B \\circ \\gamma_A, \\quad \\gamma_{C + D} = \\gamma_D \\circ \\gamma_C $$ and by Lemma \\ref{lemma-shriek-proper-and-open-compose} we have $$ \\gamma_{A + C} = \\gamma_A \\circ \\gamma_C, \\quad \\gamma_{B + D} = \\gamma_B \\circ \\gamma_D $$ Here it would be more correct to write $\\gamma_{A + B} = (\\gamma_B \\star \\text{id}_{k'_!}) \\circ (\\text{id}_{l_!} \\star \\gamma_A)$ with notation as in Categories, Section \\ref{categories-section-formal-cat-cat} and similarly for the others. Having said all of this we find (a priori) two transformations $$ m_! \\circ m'_! \\circ g''_* \\circ f''_* \\longrightarrow g_* \\circ f_* \\circ k_! \\circ k'_! $$ namely $$ \\gamma_B \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_C = \\gamma_{B + D} \\circ \\gamma_{A + C} $$ and $$ \\gamma_B \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_C = \\gamma_{A + B} \\circ \\gamma_{C + D} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \\xymatrix{ m_! \\circ g'_* \\circ l'_! \\circ f''_* \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} & g_* \\circ l_! \\circ l'_! \\circ f''_* \\ar[d]^{\\gamma_A} \\\\ m_! \\circ g'_* \\circ f'_* \\circ k'_! \\ar[r]^{\\gamma_D} & g_* \\circ l_! \\circ f'_* \\circ k'_! } $$ commutes. This is true because the squares $A$ and $D$ meet in only one point, more precisely by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \\ref{categories-definition-horizontal-composition}."}
+{"_id": "8927", "title": "stacks-properties-remark-representable-over", "text": "Let $\\mathcal{Y}$ be an algebraic stack. Consider the following $2$-category: \\begin{enumerate} \\item An object is a morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ which is representable by algebraic spaces, \\item a $1$-morphism $(g, \\beta) : (f_1 : \\mathcal{X}_1 \\to \\mathcal{Y}) \\to (f_2 : \\mathcal{X}_2 \\to \\mathcal{Y})$ consists of a morphism $g : \\mathcal{X}_1 \\to \\mathcal{X}_2$ and a $2$-morphism $\\beta : f_1 \\to f_2 \\circ g$, and \\item a $2$-morphism between $(g, \\beta), (g', \\beta') : (f_1 : \\mathcal{X}_1 \\to \\mathcal{Y}) \\to (f_2 : \\mathcal{X}_2 \\to \\mathcal{Y})$ is a $2$-morphism $\\alpha : g \\to g'$ such that $(\\text{id}_{f_2} \\star \\alpha) \\circ \\beta = \\beta'$. \\end{enumerate} Let us denote this $2$-category $\\textit{Spaces}/\\mathcal{Y}$ by analogy with the notation of Topologies on Spaces, Section \\ref{spaces-topologies-section-procedure}. Now we claim that in this $2$-category the morphism categories $$ \\Mor_{\\textit{Spaces}/\\mathcal{Y}}( (f_1 : \\mathcal{X}_1 \\to \\mathcal{Y}), (f_2 : \\mathcal{X}_2 \\to \\mathcal{Y})) $$ are all setoids. Namely, a $2$-morphism $\\alpha$ is a rule which to each object $x_1$ of $\\mathcal{X}_1$ assigns an isomorphism $\\alpha_{x_1} : g(x_1) \\longrightarrow g'(x_1)$ in the relevant fibre category of $\\mathcal{X}_2$ such that the diagram $$ \\xymatrix{ & f_2(x_1) \\ar[ld]_{\\beta_{x_1}} \\ar[rd]^{\\beta'_{x_1}} \\\\ f_2(g(x_1)) \\ar[rr]^{f_2(\\alpha_{x_1})} & & f_2(g'(x_1)) } $$ commutes. But since $f_2$ is faithful (see Algebraic Stacks, Lemma \\ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}) this means that if $\\alpha_{x_1}$ exists, then it is unique! In other words the $2$-category $\\textit{Spaces}/\\mathcal{Y}$ is very close to being a category. Namely, if we replace $1$-morphisms by isomorphism classes of $1$-morphisms we obtain a category. We will often perform this replacement without further mention."}
+{"_id": "8928", "title": "stacks-properties-remark-more-general-presentation", "text": "The result of Lemma \\ref{lemma-points-presentation} can be generalized as follows. Let $\\mathcal{X}$ be an algebraic stack. Let $U$ be an algebraic space and let $f : U \\to \\mathcal{X}$ be a surjective morphism (which makes sense by Section \\ref{section-properties-morphisms}). Let $R = U \\times_\\mathcal{X} U$, let $(U, R, s, t, c)$ be the groupoid in algebraic spaces, and let $f_{can} : [U/R] \\to \\mathcal{X}$ be the canonical morphism as constructed in Algebraic Stacks, Lemma \\ref{algebraic-lemma-map-space-into-stack}. Then the image of $|R| \\to |U| \\times |U|$ is an equivalence relation and $|\\mathcal{X}| = |U|/|R|$. The proof of Lemma \\ref{lemma-points-presentation} works without change. (Of course in general $[U/R]$ is not an algebraic stack, and in general $f_{can}$ is not an isomorphism.)"}
+{"_id": "8929", "title": "stacks-properties-remark-list-properties-local-smooth-topology", "text": "Here is a list of properties which are local for the smooth topology (keep in mind that the fpqc, fppf, and syntomic topologies are stronger than the smooth topology): \\begin{enumerate} \\item locally Noetherian, see Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf}, \\item Jacobson, see Descent, Lemma \\ref{descent-lemma-Jacobson-local-fppf}, \\item locally Noetherian and $(S_k)$, see Descent, Lemma \\ref{descent-lemma-Sk-local-syntomic}, \\item Cohen-Macaulay, see Descent, Lemma \\ref{descent-lemma-CM-local-syntomic}, \\item reduced, see Descent, Lemma \\ref{descent-lemma-reduced-local-smooth}, \\item normal, see Descent, Lemma \\ref{descent-lemma-normal-local-smooth}, \\item locally Noetherian and $(R_k)$, see Descent, Lemma \\ref{descent-lemma-Rk-local-smooth}, \\item regular, see Descent, Lemma \\ref{descent-lemma-regular-local-smooth}, \\item Nagata, see Descent, Lemma \\ref{descent-lemma-Nagata-local-smooth}. \\end{enumerate}"}
+{"_id": "8930", "title": "stacks-properties-remark-local-source-warning", "text": "Warning: Lemma \\ref{lemma-local-source} should be used with care. For example, it applies to $\\mathcal{P}=$``flat'', $\\mathcal{Q}=$``empty'', and $\\mathcal{R}=$``flat and locally of finite presentation''. But given a morphism of algebraic spaces $f : X \\to Y$ the largest open subspace $W \\subset X$ such that $f|_W$ is flat is {\\it not} the set of points where $f$ is flat!"}
+{"_id": "8931", "title": "stacks-properties-remark-local-source-apply", "text": "Notwithstanding the warning in Remark \\ref{remark-local-source-warning} there are some cases where Lemma \\ref{lemma-local-source} can be used without causing ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list: \\begin{enumerate} \\item \\label{item-rel-dim-leq-d} $\\mathcal{Q} = $``locally of finite type'', $\\mathcal{R} = \\emptyset$, and $\\mathcal{P} =$``relative dimension $\\leq d$''. See Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-relative-dimension} and Morphisms of Spaces, Lemmas \\ref{spaces-morphisms-lemma-openness-bounded-dimension-fibres} and \\ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}. \\item \\label{item-loc-quasi-finite} $\\mathcal{Q} =$``locally of finite type'', $\\mathcal{R} = \\emptyset$, and $\\mathcal{P} =$``locally quasi-finite''. This is the case $d = 0$ of the previous item, see Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-quasi-finite-rel-dimension-0}. On the other hand, properties (3) and (4) are spelled out in Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}. \\item \\label{item-unramified} $\\mathcal{Q} = $``locally of finite type'', $\\mathcal{R} = \\emptyset$, and $\\mathcal{P} =$``unramified''. This is Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-where-unramified}. \\item \\label{item-flat} $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P} =$``flat''. See More on Morphisms of Spaces, Theorem \\ref{spaces-more-morphisms-theorem-openness-flatness} and Lemma \\ref{spaces-more-morphisms-lemma-flat-locus-base-change}. Note that here $W(\\mathcal{P}, f)$ is always exactly the set of points where the morphism $f$ is flat because we only consider this open when $f$ has $\\mathcal{Q}$ (see loc.cit.). \\item \\label{item-etale} $\\mathcal{Q} =$``locally of finite presentation'', $\\mathcal{R} =$``flat and locally of finite presentation'', and $\\mathcal{P}=$``\\'etale''. This follows on combining (\\ref{item-unramified}) and (\\ref{item-flat}) because an unramified morphism which is flat and locally of finite presentation is \\'etale, see Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-unramified-flat-lfp-etale}. \\item Add more here as needed (compare with the longer list at More on Groupoids, Remark \\ref{more-groupoids-remark-local-source-apply}). \\end{enumerate}"}
+{"_id": "8932", "title": "stacks-properties-remark-stack-structure-locally-closed-subset", "text": "Let $X$ be an algebraic stack. Let $T \\subset |\\mathcal{X}|$ be a locally closed subset. Let $\\partial T$ be the boundary of $T$ in the topological space $|\\mathcal{X}|$. In a formula $$ \\partial T = \\overline{T} \\setminus T. $$ Let $\\mathcal{U} \\subset \\mathcal{X}$ be the open substack of $X$ with $|\\mathcal{U}| = |\\mathcal{X}| \\setminus \\partial T$, see Lemma \\ref{lemma-open-substacks}. Let $\\mathcal{Z}$ be the reduced closed substack of $\\mathcal{U}$ with $|\\mathcal{Z}| = T$ obtained by taking the reduced induced closed subspace structure, see Definition \\ref{definition-reduced-induced-stack}. By construction $\\mathcal{Z} \\to \\mathcal{U}$ is a closed immersion of algebraic stacks and $\\mathcal{U} \\to \\mathcal{X}$ is an open immersion, hence $\\mathcal{Z} \\to \\mathcal{X}$ is an immersion of algebraic stacks by Lemma \\ref{lemma-composition-immersion}. Note that $\\mathcal{Z}$ is a reduced algebraic stack and that $|\\mathcal{Z}| = T$ as subsets of $|X|$. We sometimes say $\\mathcal{Z}$ is the {\\it reduced induced substack structure} on $T$."}
+{"_id": "8933", "title": "stacks-properties-remark-dimension-empty-stack", "text": "If $\\mathcal{X}$ is a nonempty stack of finite type over a field, then $\\dim(\\mathcal{X})$ is an integer. For an arbitrary locally Noetherian algebraic stack $\\mathcal{X}$, $\\dim(\\mathcal{X})$ is in $Z\\cup \\{\\pm \\infty\\}$, and $\\dim(\\mathcal{X}) = -\\infty$ if and only if $\\mathcal{X}$ is empty."}
+{"_id": "9007", "title": "stacks-remark-alternative", "text": "Suppose that $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids. In this case we can prove Lemma \\ref{lemma-painful} using Categories, Lemma \\ref{categories-lemma-fibred-strict} which says that $\\mathcal{S} \\to \\mathcal{C}$ is equivalent to the category associated to a contravariant functor $F : \\mathcal{C} \\to \\textit{Groupoids}$. In the case of the fibred category associated to $F$ we have $g^* \\circ f^* = (f \\circ g)^*$ on the nose and there is no need to use the maps $\\alpha_{g, f}$. In this case the lemma is (even more) trivial. Of course then one uses that the $\\mathit{Mor}(x, y)$ presheaf is unchanged when passing to an equivalent fibred category which follows from Lemma \\ref{lemma-presheaf-mor-map-fibred-categories}."}
+{"_id": "9008", "title": "stacks-remark-stack-make-small", "text": "(Cutting down a ``big'' stack to get a stack.) Let $\\mathcal{C}$ be a site. Suppose that $p : \\mathcal{S} \\to \\mathcal{C}$ is functor from a ``big'' category to $\\mathcal{C}$, i.e., suppose that the collection of objects of $\\mathcal{S}$ forms a proper class. Finally, suppose that $p : \\mathcal{S} \\to \\mathcal{C}$ satisfies conditions (1), (2), (3) of Definition \\ref{definition-stack}. In general there is no way to replace $p : \\mathcal{S} \\to \\mathcal{C}$ by a equivalent category such that we obtain a stack. The reason is that it can happen that a fibre categories $\\mathcal{S}_U$ may have a proper class of isomorphism classes of objects. On the other hand, suppose that \\begin{enumerate} \\item[(4)] for every $U \\in \\Ob(\\mathcal{C})$ there exists a set $S_U \\subset \\Ob(\\mathcal{S}_U)$ such that every object of $\\mathcal{S}_U$ is isomorphic in $\\mathcal{S}_U$ to an element of $S_U$. \\end{enumerate} In this case we can find a full subcategory $\\mathcal{S}_{small}$ of $\\mathcal{S}$ such that, setting $p_{small} = p|_{\\mathcal{S}_{small}}$, we have \\begin{enumerate} \\item[(a)] the functor $p_{small} : \\mathcal{S}_{small} \\to \\mathcal{C}$ defines a stack, and \\item[(b)] the inclusion $\\mathcal{S}_{small} \\to \\mathcal{S}$ is fully faithful and essentially surjective. \\end{enumerate} (Hint: For every $U \\in \\Ob(\\mathcal{C})$ let $\\alpha(U)$ denote the smallest ordinal such that $\\Ob(\\mathcal{S}_U) \\cap V_{\\alpha(U)}$ surjects onto the set of isomorphism classes of $\\mathcal{S}_U$, and set $\\alpha = \\sup_{U \\in \\Ob(\\mathcal{C})} \\alpha(U)$. Then take $\\Ob(\\mathcal{S}_{small}) = \\Ob(\\mathcal{S}) \\cap V_\\alpha$. For notation used see Sets, Section \\ref{sets-section-sets-hierarchy}.)"}
+{"_id": "9009", "title": "stacks-remarks-definition-descent-datum", "text": "Two remarks on Definition \\ref{definition-descent-data} are in order. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. Let $\\{f_i : U_i \\to U\\}_{i \\in I}$, and $(X_i, \\varphi_{ij})$ be as in Definition \\ref{definition-descent-data}. \\begin{enumerate} \\item There is a diagonal morphism $\\Delta : U_i \\to U_i \\times_U U_i$. We can pull back $\\varphi_{ii}$ via this morphism to get an automorphism $\\Delta^\\ast \\varphi_{ii} \\in \\text{Aut}_{U_i}(x_i)$. On pulling back the cocycle condition for the triple $(i, i, i)$ by $\\Delta_{123} : U_i \\to U_i \\times_U U_i \\times_U U_i$ we deduce that $\\Delta^\\ast \\varphi_{ii} \\circ \\Delta^\\ast \\varphi_{ii} = \\Delta^\\ast \\varphi_{ii}$; thus $\\Delta^\\ast \\varphi_{ii} = \\text{id}_{x_i}$. \\item There is a morphism $\\Delta_{13}: U_i \\times_U U_j \\to U_i \\times_U U_j \\times_U U_i$ and we can pull back the cocycle condition for the triple $(i, j, i)$ to get the identity $(\\sigma^\\ast \\varphi_{ji}) \\circ \\varphi_{ij} = \\text{id}_{\\text{pr}_0^\\ast x_i}$, where $\\sigma : U_i \\times_U U_j \\to U_j \\times_U U_i$ is the switching morphism. \\end{enumerate}"}
+{"_id": "9152", "title": "spaces-simplicial-remark-augmentation-site", "text": "In Situation \\ref{situation-simplicial-site} an {\\it augmentation $a_0$ towards a site $\\mathcal{D}$} will mean \\begin{enumerate} \\item[(A)] $a_0 : \\mathcal{C}_0 \\to \\mathcal{D}$ is a morphism of sites given by a continuous functor $u_0 : \\mathcal{D} \\to \\mathcal{C}_0$ such that for all $\\varphi, \\psi : [0] \\to [n]$ we have $u_\\varphi \\circ u_0 = u_\\psi \\circ u_0$. \\item[(B)] $a_0 : \\Sh(\\mathcal{C}_0) \\to \\Sh(\\mathcal{D})$ is a morphism of topoi given by a cocontinuous functor $u_0 : \\mathcal{C}_0 \\to \\mathcal{D}$ such that for all $\\varphi, \\psi : [0] \\to [n]$ we have $u_0 \\circ u_\\varphi = u_0 \\circ u_\\psi$. \\end{enumerate}"}
+{"_id": "9153", "title": "spaces-simplicial-remark-morphism-simplicial-sites", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in Situation \\ref{situation-simplicial-site}. A {\\it morphism $h$ between simplicial sites} will mean \\begin{enumerate} \\item[(A)] Morphisms of sites $h_n : \\mathcal{C}_n \\to \\mathcal{C}'_n$ such that $f'_\\varphi \\circ h_n = h_m \\circ f_\\varphi$ as morphisms of sites for all $\\varphi : [m] \\to [n]$. \\item[(B)] Cocontinuous functors $v_n : \\mathcal{C}_n \\to \\mathcal{C}'_n$ inducing morphisms of topoi $h_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}'_n)$ such that $u'_\\varphi \\circ v_n = v_m \\circ u_\\varphi$ as functors for all $\\varphi : [m] \\to [n]$. \\end{enumerate} In both cases we have $f'_\\varphi \\circ h_n = h_m \\circ f_\\varphi$ as morphisms of topoi, see Sites, Lemma \\ref{sites-lemma-composition-cocontinuous} for case B and Sites, Definition \\ref{sites-definition-composition-morphisms-sites} for case A."}
+{"_id": "9154", "title": "spaces-simplicial-remark-morphism-augmentation-simplicial-sites", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in Situation \\ref{situation-simplicial-site}. Let $a_0$, resp.\\ $a'_0$ be an augmentation towards a site $\\mathcal{D}$, resp.\\ $\\mathcal{D}'$ as in Remark \\ref{remark-augmentation-site}. Let $h$ be a morphism between simplicial sites as in Remark \\ref{remark-morphism-simplicial-sites}. We say a morphism of topoi $h_{-1} : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{D}')$ is {\\it compatible with $h$, $a_0$, $a'_0$} if \\begin{enumerate} \\item[(A)] $h_{-1}$ comes from a morphism of sites $h_{-1} : \\mathcal{D} \\to \\mathcal{D}'$ such that $a'_0 \\circ h_0 = h_{-1} \\circ a_0$ as morphisms of sites. \\item[(B)] $h_{-1}$ comes from a cocontinuous functor $v_{-1} : \\mathcal{D} \\to \\mathcal{D}'$ such that $u'_0 \\circ v_0 = v_{-1} \\circ u_0$ as functors. \\end{enumerate} In both cases we have $a'_0 \\circ h_0 = h_{-1} \\circ a_0$ as morphisms of topoi, see Sites, Lemma \\ref{sites-lemma-composition-cocontinuous} for case B and Sites, Definition \\ref{sites-definition-composition-morphisms-sites} for case A."}
+{"_id": "9155", "title": "spaces-simplicial-remark-morphism-simplicial-sites-modules", "text": "Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and $\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ and $\\mathcal{O}'$ be a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{C}'_{total}$. We will say that $(h, h^\\sharp)$ is a {\\it morphism between ringed simplicial sites} if $h$ is a morphism between simplicial sites as in Remark \\ref{remark-morphism-simplicial-sites} and $h^\\sharp : h_{total}^{-1}\\mathcal{O}' \\to \\mathcal{O}$ or equivalently $h^\\sharp : \\mathcal{O}' \\to h_{total, *}\\mathcal{O}$ is a homomorphism of sheaves of rings."}
+{"_id": "9156", "title": "spaces-simplicial-remark-warning-cartesian-modules", "text": "Lemma \\ref{lemma-Serre-subcat-cartesian-modules} notwithstanding, it can happen that the category of cartesian $\\mathcal{O}$-modules is abelian without being a Serre subcategory of $\\textit{Mod}(\\mathcal{O})$. Namely, suppose that we only know that $f_{\\delta_1^1}$ and $f_{\\delta_0^1}$ are flat. Then it follows easily from Lemma \\ref{lemma-characterize-cartesian-modules} that the category of cartesian $\\mathcal{O}$-modules is abelian. But if $f_{\\delta_0^2}$ is not flat (for example), there is no reason for the inclusion functor from the category of cartesian $\\mathcal{O}$-modules to all $\\mathcal{O}$-modules to be exact."}
+{"_id": "9157", "title": "spaces-simplicial-remark-semi-representable-over-object", "text": "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$. The category $\\text{SR}(\\mathcal{C}, X)$ of {\\it semi-representable objects over $X$} is defined by the formula $\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$. See Hypercoverings, Definition \\ref{hypercovering-definition-SR}. Thus we may apply the above discussion to the site $\\mathcal{C}/X$. Briefly, the constructions above give \\begin{enumerate} \\item a site $\\mathcal{C}/K$ for $K$ in $\\text{SR}(\\mathcal{C}, X)$, \\item a decomposition $\\Sh(\\mathcal{C}/K) = \\prod \\Sh(\\mathcal{C}/U_i)$ if $K = \\{U_i/X\\}$, \\item a localization functor $j : \\mathcal{C}/K \\to \\mathcal{C}/X$, \\item a morphism $f : \\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C}/L)$ for $f : K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$. \\end{enumerate} All results of this section hold in this situation by replacing $\\mathcal{C}$ everywhere by $\\mathcal{C}/X$."}
+{"_id": "9158", "title": "spaces-simplicial-remark-semi-representable-ringed", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings on $\\mathcal{C}$. In this case, for any semi-representable object $K$ of $\\mathcal{C}$ the site $\\mathcal{C}/K$ is a ringed site with sheaf of rings $\\mathcal{O}_K = j^{-1}\\mathcal{O}_\\mathcal{C}$. The constructions above give \\begin{enumerate} \\item a ringed site $(\\mathcal{C}/K, \\mathcal{O}_K)$ for $K$ in $\\text{SR}(\\mathcal{C})$, \\item a decomposition $\\textit{Mod}(\\mathcal{O}_K) = \\prod \\textit{Mod}(\\mathcal{O}_{U_i})$ if $K = \\{U_i\\}$, \\item a localization morphism $j : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ of ringed topoi, \\item a morphism $f : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to (\\Sh(\\mathcal{C}/L), \\mathcal{O}_L)$ of ringed topoi for $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$. \\end{enumerate} Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint $$ j_! : \\textit{Mod}(\\mathcal{O}_K) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C}), $$ which in terms of the product decomposition given in (2) sends $(\\mathcal{F}_i)_{i \\in I}$ to $\\bigoplus j_{i, !}\\mathcal{F}_i$. Similarly, given $f : K \\to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \\textit{Mod}(\\mathcal{O}_K) \\to \\textit{Mod}(\\mathcal{O}_L)$. Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\\mathcal{O}_K = j^{-1}\\mathcal{O}_\\mathcal{C}$ and $\\mathcal{O}_K = f^{-1}\\mathcal{O}_L$)."}
+{"_id": "9159", "title": "spaces-simplicial-remark-semi-representable-ringed-over-object", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings on $\\mathcal{C}$. Let $X \\in \\Ob(\\mathcal{C})$ and denote $\\mathcal{O}_X = \\mathcal{O}_\\mathcal{C}|_{\\mathcal{C}/U}$. Then we can combine the constructions given in Remarks \\ref{remark-semi-representable-over-object} and \\ref{remark-semi-representable-ringed} to get \\begin{enumerate} \\item a ringed site $(\\mathcal{C}/K, \\mathcal{O}_K)$ for $K$ in $\\text{SR}(\\mathcal{C}, X)$, \\item a decomposition $\\textit{Mod}(\\mathcal{O}_K) = \\prod \\textit{Mod}(\\mathcal{O}_{U_i})$ if $K = \\{U_i\\}$, \\item a localization morphism $j : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to (\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$ of ringed topoi, \\item a morphism $f : (\\Sh(\\mathcal{C}/K), \\mathcal{O}_K) \\to (\\Sh(\\mathcal{C}/L), \\mathcal{O}_L)$ of ringed topoi for $f : K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$. \\end{enumerate} Of course all of the results mentioned in Remark \\ref{remark-semi-representable-ringed} hold in this setting as well."}
+{"_id": "9160", "title": "spaces-simplicial-remark-augmentation-over-object", "text": "Let $\\mathcal{C}$ be a site. Let $X \\in \\Ob(\\mathcal{C})$. Recall that we have a category $\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$ of semi-representable objects over $X$, see Remark \\ref{remark-semi-representable-over-object}. We may apply the above discussion to the site $\\mathcal{C}/X$. Briefly, the constructions above give \\begin{enumerate} \\item a site $(\\mathcal{C}/K)_{total}$ for a simplicial $K$ object of $\\text{SR}(\\mathcal{C}, X)$, \\item a localization functor $j_{total} : (\\mathcal{C}/K)_{total} \\to \\mathcal{C}/X$, \\item localization functors $j_n : \\mathcal{C}/K_n \\to \\mathcal{C}/X$, \\item a morphism of topoi $a : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/X)$, \\item morphisms of topoi $a_n : \\Sh(\\mathcal{C}/K_n) \\to \\Sh(\\mathcal{C}/X)$, \\item a functor $a^{Sh}_! : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/X)$ left adjoint to $a^{-1}$, and \\item a functor $a_! : \\textit{Ab}((\\mathcal{C}/K)_{total}) \\to \\textit{Ab}(\\mathcal{C}/X)$ left adjoint to $a^{-1}$. \\end{enumerate} All of the results of this section hold in this setting. To prove this one replaces the site $\\mathcal{C}$ everywhere by $\\mathcal{C}/X$."}
+{"_id": "9161", "title": "spaces-simplicial-remark-augmentation-ringed", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Given a simplicial semi-representable object $K$ of $\\mathcal{C}$ we set $\\mathcal{O} = a^{-1}\\mathcal{O}_\\mathcal{C}$, where $a$ is as in Lemmas \\ref{lemma-augmentation-simplicial-semi-representable} and \\ref{lemma-comparison}. The constructions above, keeping track of the sheaves of rings as in Remark \\ref{remark-semi-representable-ringed}, give \\begin{enumerate} \\item a ringed site $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ for a simplicial $K$ object of $\\text{SR}(\\mathcal{C})$, \\item a morphism of ringed topoi $a : (\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$, \\item morphisms of ringed topoi $a_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$, \\item a functor $a_! : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ left adjoint to $a^*$. \\end{enumerate} The functor $a_!$ exists (but in general is not exact) because $a^{-1}\\mathcal{O}_\\mathcal{C} = \\mathcal{O}$ and we can replace the use of Modules on Sites, Lemma \\ref{sites-modules-lemma-g-shriek-adjoint} in the proof of Lemma \\ref{lemma-comparison} by Modules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules}. As discussed in Remark \\ref{remark-semi-representable-ringed} there are exact functors $a_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ left adjoint to $a_n^*$. Consequently, the morphisms $a$ and $a_n$ are flat. Remark \\ref{remark-semi-representable-ringed} implies the morphism of ringed topoi $f_\\varphi : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}/K_m), \\mathcal{O}_m)$ for $\\varphi : [m] \\to [n]$ is flat and there exists an exact functor $f_{\\varphi !} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O}_m)$ left adjoint to $f_\\varphi^*$. This in turn implies that for the flat morphism of ringed topoi $g_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to (\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O})$ the functor $g_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O})$ left adjoint to $g_n^*$ is exact, see Lemma \\ref{lemma-exactness-g-shriek-modules}."}
+{"_id": "9162", "title": "spaces-simplicial-remark-augmentation-ringed-over-object", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $X \\in \\Ob(\\mathcal{C})$ and denote $\\mathcal{O}_X = \\mathcal{O}_\\mathcal{C}|_{\\mathcal{C}/X}$. Then we can combine the constructions given in Remarks \\ref{remark-augmentation-over-object} and \\ref{remark-augmentation-ringed} to get \\begin{enumerate} \\item a ringed site $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ for a simplicial $K$ object of $\\text{SR}(\\mathcal{C}, X)$, \\item a morphism of ringed topoi $a : (\\Sh((\\mathcal{C}/K)_{total}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$, \\item morphisms of ringed topoi $a_n : (\\Sh(\\mathcal{C}/K_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}/X), \\mathcal{O}_X)$, \\item a functor $a_! : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_X)$ left adjoint to $a^*$. \\end{enumerate} Of course, all the results mentioned in Remark \\ref{remark-augmentation-ringed} hold in this setting as well."}
+{"_id": "9163", "title": "spaces-simplicial-remark-compare-cohomology-hypercovering-presheaf", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets on $\\mathcal{C}$. If $\\mathcal{C}$ has equalizers and fibre products, then we've defined the notion of a hypercovering of $\\mathcal{G}$ in Hypercoverings, Definition \\ref{hypercovering-definition-hypercovering-variant}. We claim that all the results in this section have a valid counterpart in this setting. To see this, define the localization $\\mathcal{C}/\\mathcal{G}$ of $\\mathcal{C}$ at $\\mathcal{G}$ exactly as in Sites, Lemma \\ref{sites-lemma-localize-topos-site} (which is stated only for sheaves; the topos $\\Sh(\\mathcal{C}/\\mathcal{G})$ is equal to the localization of the topos $\\Sh(\\mathcal{C})$ at the sheaf $\\mathcal{G}^\\#$). Then the reader easily shows that the site $\\mathcal{C}/\\mathcal{G}$ has fibre products and equalizers and that a hypercovering of $\\mathcal{G}$ in $\\mathcal{C}$ is the same thing as a hypercovering for the site $\\mathcal{C}/\\mathcal{G}$. Hence replacing the site $\\mathcal{C}$ by $\\mathcal{C}/\\mathcal{G}$ in the lemmas on hypercoverings above we obtain proofs of the corresponding results for hypercoverings of $\\mathcal{G}$. Example: for a hypercovering $K$ of $\\mathcal{G}$ we have $$ R\\Gamma(\\mathcal{C}/\\mathcal{G}, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E) $$ for $E \\in D^+(\\mathcal{C}/\\mathcal{G})$ where $a : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C}/\\mathcal{G})$ is the canonical augmentation. This is Lemma \\ref{lemma-compare-cohomology-hypercovering}. Let $R\\Gamma(\\mathcal{G}, -) : D(\\mathcal{C}) \\to D(\\textit{Ab})$ be defined as the derived functor of the functor $H^0(\\mathcal{G}, -) = H^0(\\mathcal{G}^\\#, -)$ discussed in Hypercoverings, Section \\ref{hypercovering-section-hypercoverings-verdier} and Cohomology on Sites, Section \\ref{sites-cohomology-section-limp}. We have $$ R\\Gamma(\\mathcal{G}, E) = R\\Gamma(\\mathcal{C}/\\mathcal{G}, j^{-1}E) $$ by the analogue of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-of-open} for the localization fuctor $j : \\mathcal{C}/\\mathcal{G} \\to \\mathcal{C}$. Putting everything together we obtain $$ R\\Gamma(\\mathcal{G}, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}j^{-1}E) = R\\Gamma((\\mathcal{C}/K)_{total}, g^{-1}E) $$ for $E \\in D^+(\\mathcal{C})$ where $g : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C})$ is the composition of $a$ and $j$."}
+{"_id": "9185", "title": "examples-stacks-remark-higher-rank", "text": "Note that the whole discussion in this section works if we want to consider those quasi-coherent sheaves which are locally generated by at most $\\kappa$ sections, for some infinite cardinal $\\kappa$, e.g., $\\kappa = \\aleph_0$."}
+{"_id": "9186", "title": "examples-stacks-remark-stack-spaces", "text": "Ignoring set theoretical difficulties\\footnote{The difficulty is not that $\\Spacesstack$ is a proper class, since by our definition of an algebraic space over $S$ there is only a set worth of isomorphism classes of algebraic spaces over $S$. It is rather that arbitrary disjoint unions of algebraic spaces may end up being too large, hence lie outside of our chosen ``partial universe'' of sets.} $\\Spacesstack$ also satisfies descent for objects and hence is a stack. Namely, we have to show that given \\begin{enumerate} \\item an fppf covering $\\{U_i \\to U\\}_{i \\in I}$, \\item for each $i \\in I$ an algebraic space $X_i/U_i$, and \\item for each $i, j \\in I$ an isomorphism $\\varphi_{ij} : X_i \\times_U U_j \\to U_i \\times_U X_j$ of algebraic spaces over $U_i \\times_U U_j$ satisfying the cocycle condition over $U_i \\times_U U_j \\times_U U_k$, \\end{enumerate} there exists an algebraic space $X/U$ and isomorphisms $X_{U_i} \\cong X_i$ over $U_i$ recovering the isomorphisms $\\varphi_{ij}$. First, note that by Sites, Lemma \\ref{sites-lemma-glue-sheaves} there exists a sheaf $X$ on $(\\Sch/U)_{fppf}$ recovering the $X_i$ and the $\\varphi_{ij}$. Then by Bootstrap, Lemma \\ref{bootstrap-lemma-locally-algebraic-space} we see that $X$ is an algebraic space (if we ignore the set theoretic condition of that lemma). We will use this argument in the next section to show that if we consider only algebraic spaces of finite type, then we obtain a stack."}
+{"_id": "9187", "title": "examples-stacks-remark-higher-cardinality-spaces", "text": "Note that the whole discussion in this section works if we want to consider those algebraic spaces $X/U$ which are locally of finite type such that the inverse image in $X$ of an affine open of $U$ can be covered by countably many affines. If needed we can also introduce the notion of a morphism of $\\kappa$-type (meaning some bound on the number of generators of ring extensions and some bound on the cardinality of the affines over a given affine in the base) where $\\kappa$ is a cardinal, and then we can produce a stack $$ \\Spacesstack_\\kappa \\longrightarrow (\\Sch/S)_{fppf} $$ in exactly the same manner as above (provided we make sure that $\\Sch$ is large enough depending on $\\kappa$)."}
+{"_id": "9188", "title": "examples-stacks-remark-principal-stack-in-groupoids", "text": "We conjecture that up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ p : G\\textit{-Principal} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$. This would follow if one could show that given \\begin{enumerate} \\item a covering $\\{U_i \\to U\\}_{i \\in I}$ of $(\\Sch/S)_{fppf}$, \\item an group algebraic space $H$ over $U$, \\item for every $i$ a principal homogeneous $H_{U_i}$-space $X_i$ over $U_i$, and \\item $H$-equivariant isomorphisms $\\varphi_{ij} : X_{i, U_i \\times_U U_j} \\to X_{j, U_i \\times_U U_j}$ satisfying the cocycle condition, \\end{enumerate} there exists a principal homogeneous $H$-space $X$ over $U$ which recovers $(X_i, \\varphi_{ij})$. The technique of the proof of Bootstrap, Lemma \\ref{bootstrap-lemma-descent-torsor} reduces this to a set theoretical question, so the reader who ignores set theoretical questions will ``know'' that the result is true. In \\url{https://math.columbia.edu/~dejong/wordpress/?p=591} there is a suggestion as to how to approach this problem."}
+{"_id": "9189", "title": "examples-stacks-remark-X-mod-G-group", "text": "Let $S$ be a scheme. Let $G$ be an abstract group. Let $X$ be an algebraic space over $S$. Let $G \\to \\text{Aut}_S(X)$ be a group homomorphism. In this setting we can define $[[X/G]]$ similarly to the above as follows: \\begin{enumerate} \\item An object of $[[X/G]]$ consists of a triple $(U, P, \\varphi : P \\to X)$ where \\begin{enumerate} \\item $U$ is an object of $(\\Sch/S)_{fppf}$, \\item $P$ is a sheaf on $(\\Sch/U)_{fppf}$ which comes with an action of $G$ that turns it into a torsor under the constant sheaf with value $G$, and \\item $\\varphi : P \\to X$ is a $G$-equivariant map of sheaves. \\end{enumerate} \\item A morphism $(f, g) : (U, P, \\varphi) \\to (U', P', \\varphi')$ is given by a morphism of schemes $f : T \\to T'$ and a $G$-equivariant isomorphism $g : P \\to f^{-1}P'$ such that $\\varphi = \\varphi' \\circ g$. \\end{enumerate} In exactly the same manner as above we obtain a functor $$ [[X/G]] \\longrightarrow (\\Sch/S)_{fppf} $$ which turns $[[X/G]]$ into a stack in groupoids over $(\\Sch/S)_{fppf}$. The constant sheaf $\\underline{G}$ is (provided the cardinality of $G$ is not too large) representable by $G_S$ on $(\\Sch/S)_{fppf}$ and this version of $[[X/G]]$ is equivalent to the stack $[[X/G_S]]$ introduced above."}
+{"_id": "9281", "title": "models-remark-genus-equality", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a minimal numerical type with $n > 1$. Equality $g = g_{top}$ can hold in Lemma \\ref{lemma-genus-nonnegative}. For example, if $m_i = w_i = 1$ and $g_i = 0$ for all $i$ and $a_{ij} \\in \\{0, 1\\}$ for $i < j$."}
+{"_id": "9282", "title": "models-remark-numerical-type-not-from-model", "text": "Not every numerical type comes from a model for the silly reason that there exist numerical types whose genus is negative. There exist a minimal numerical types of positive genus which are not the numerical type associated to a model (over some dvr) of a smooth projective geometrically irreducible curve (over the fraction field of the dvr). A simple example is $n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 6$, $g_1 = 1$. Namely, in this case the special fibre $X_k$ would not be geometrically connected because it would live over an extension $\\kappa$ of $k$ of degree $6$. This is a contradiction with the fact that the generic fibre is geometrically connected (see More on Morphisms, Lemma \\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}). Similarly, $n = 2$, $m_1 = m_2 = 1$, $-a_{11} = -a_{22} = a_{12} = a_{21} = 6$, $w_1 = w_2 = 6$, $g_1 = g_2 = 1$ would be an example for the same reason (details omitted). But if the gcd of the $w_i$ is $1$ we do not have an example."}
+{"_id": "9283", "title": "models-remark-genus-change", "text": "In the situation of Lemma \\ref{lemma-blowdown-regular-model} we can also say exactly how the genus $g_i$ of $C_i$ and the genus $g'_i$ of $C'_i$ are related. The formula is $$ g'_i = \\frac{w_i}{w'_i}(g_i - 1) + 1 + \\frac{(C_i \\cdot C_n)^2 - w_n(C_i \\cdot C_n)}{2w'_iw_n} $$ where $w_i = [\\kappa_i : k]$, $w_n = [\\kappa_n : k]$, and $w'_i = [\\kappa'_i : k]$. To prove this we consider the short exact sequence $$ 0 \\to \\mathcal{O}_{X'}(-C'_i) \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_{C'_i} \\to 0 $$ and its pullback to $X$ which reads $$ 0 \\to \\mathcal{O}_X(-C'_i - e_iC_n) \\to \\mathcal{O}_X \\to \\mathcal{O}_{C_i + e_i C_n} \\to 0 $$ with $e_i$ as in the proof of Lemma \\ref{lemma-blowdown-regular-model}. Since $Rf_*f^*\\mathcal{L} = \\mathcal{L}$ for any invertible module $\\mathcal{L}$ on $X'$ (details omitted), we conclude that $$ Rf_*\\mathcal{O}_{C_i + e_i C_n} = \\mathcal{O}_{C'_i} $$ as complexes of coherent sheaves on $X'_k$. Hence both sides have the same Euler characteristic and this agrees with the Euler characteristic of $\\mathcal{O}_{C_i + e_i C_n}$ on $X_k$. Using the exact sequence $$ 0 \\to \\mathcal{O}_{C_i + e_i C_n} \\to \\mathcal{O}_{C_i} \\oplus \\mathcal{O}_{e_iC_n} \\to \\mathcal{O}_{C_i \\cap e_iC_n} \\to 0 $$ and further filtering $\\mathcal{O}_{e_iC_n}$ (details omitted) we find $$ \\chi(\\mathcal{O}_{C'_i}) = \\chi(\\mathcal{O}_{C_i}) - {e_i + 1 \\choose 2}(C_n \\cdot C_n) - e_i(C_i \\cdot C_n) $$ Since $e_i = -(C_i \\cdot C_n)/(C_n \\cdot C_n)$ and $(C_n \\cdot C_n) = -w_n$ this leads to the formula stated at the start of this remark. If we ever need this we will formulate this as a lemma and provide a detailed proof."}
+{"_id": "9284", "title": "models-remark-compare-contractions", "text": "Let $f : X \\to X'$ be as in Lemma \\ref{lemma-blowdown-regular-model}. Let $n, m_i, a_{ij}, w_i, g_i$ be the numerical type associated to $X$ and let $n', m'_i, a'_{ij}, w'_i, g'_i$ be the numerical type associated to $X'$. It is clear from Lemma \\ref{lemma-blowdown-regular-model} and Remark \\ref{remark-genus-change} that this agrees with the contraction of numerical types in Lemma \\ref{lemma-contract} except for the value of $w'_i$. In the geometric situation $w'_i$ is some positive integer dividing both $w_i$ and $w_n$. In the numerical case we chose $w'_i$ to be the largest possible integer dividing $w_i$ such that $g'_i$ (as given by the formula) is an integer. This works well in the numerical setting in that it helps compare the Picard groups of the numerical types, see Lemma \\ref{lemma-contract-picard-group} (although only injectivity is every used in the following and this injectivity works as well for smaller $w'_i$)."}
+{"_id": "9285", "title": "models-remark-improving-bound", "text": "Results in the literature suggest that one can improve the bound given in the statement of Theorem \\ref{theorem-semistable-reduction}. For example, in \\cite{DM} it is shown that semistable reduction of $C$ and its Jacobian are the same thing if the residue field is perfect and presumably this is true for general residue fields as well. For an abelian variety we have semistable reduction if the action of Galois on the $\\ell$-torsion is trivial for any $\\ell \\geq 3$ not equal to the residue characteristic. Thus we can presumably choose $\\ell = 5$ in the formula (\\ref{equation-bound}) for $B_g$ (but the proof would take a lot more work; if we ever need this we will make a precise statement and provide a proof here)."}
+{"_id": "9355", "title": "spaces-groupoids-remark-quotient-variant", "text": "A variant of the construction above would have been to sheafify the functor $$ \\begin{matrix} (\\textit{Spaces}/B)^{opp}_{fppf} & \\longrightarrow & \\textit{Sets}, \\\\ X & \\longmapsto  & U(X)/\\sim_X \\end{matrix} $$ where now $\\sim_X \\subset U(X) \\times U(X)$ is the equivalence relation generated by the image of $j : R(X) \\to U(X) \\times U(X)$. Here of course $U(X) = \\Mor_B(X, U)$ and $R(X) = \\Mor_B(X, R)$. In fact, the result would have been the same, via the identifications of (insert future reference in Topologies of Spaces here)."}
+{"_id": "9356", "title": "spaces-groupoids-remark-fundamental-square", "text": "In future chapters we will use the ambiguous notation where instead of writing $\\mathcal{S}_X$ for the stack in sets associated to $X$ we simply write $X$. Using this notation the diagram of Lemma \\ref{lemma-quotient-stack-2-arrow} becomes the familiar diagram $$ \\xymatrix{ R \\ar[r]_s \\ar[d]_t & U \\ar[d]^\\pi \\\\ U \\ar[r]^-\\pi & [U/R] } $$ In the following sections we will show that this diagram has many good properties. In particular we will show that it is a $2$-fibre product (Section \\ref{section-quotient-stack-2-cartesian}) and that it is close to being a $2$-coequalizer of $s$ and $t$ (Section \\ref{section-quotient-stacks-2-coequalize})."}
+{"_id": "9450", "title": "spaces-descent-remark-list-local-source-target", "text": "Using Lemma \\ref{lemma-smooth-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is smooth local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma \\ref{lemma-smooth-local-source-target} is trivial to check, and we omit it. Here is the list: \\begin{enumerate} \\item flat, see Lemmas \\ref{lemma-flat-fpqc-local-source} and \\ref{lemma-descending-property-flat}, \\item locally of finite presentation, see Lemmas \\ref{lemma-locally-finite-presentation-fppf-local-source} and \\ref{lemma-descending-property-locally-finite-presentation}, \\item locally finite type, see Lemmas \\ref{lemma-locally-finite-type-fppf-local-source} and \\ref{lemma-descending-property-locally-finite-type}, \\item universally open, see Lemmas \\ref{lemma-universally-open-fppf-local-source} and \\ref{lemma-descending-property-universally-open}, \\item syntomic, see Lemmas \\ref{lemma-syntomic-syntomic-local-source} and \\ref{lemma-descending-property-syntomic}, \\item smooth, see Lemmas \\ref{lemma-smooth-smooth-local-source} and \\ref{lemma-descending-property-smooth}, \\item add more here as needed. \\end{enumerate}"}
+{"_id": "9451", "title": "spaces-descent-remark-list-etale-smooth-local-source-target", "text": "Using Lemma \\ref{lemma-etale-smooth-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is etale local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma \\ref{lemma-etale-smooth-local-source-target} is trivial to check, and we omit it. Here is the list: \\begin{enumerate} \\item \\'etale, see Lemmas \\ref{lemma-etale-etale-local-source} and \\ref{lemma-descending-property-etale}, \\item locally quasi-finite, see Lemmas \\ref{lemma-locally-quasi-finite-etale-local-source} and \\ref{lemma-descending-property-quasi-finite}, \\item unramified, see Lemmas \\ref{lemma-unramified-etale-local-source} and \\ref{lemma-descending-property-unramified}, and \\item add more here as needed. \\end{enumerate} Of course any property listed in Remark \\ref{remark-list-local-source-target} is a fortiori an example that could be listed here."}
+{"_id": "9452", "title": "spaces-descent-remark-easier", "text": "Let $S$ be a scheme. Let $Y \\to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \\varphi)$ be a descent datum relative to $Y \\to X$. We may think of the isomorphism $\\varphi$ as an isomorphism $$ (Y \\times_X Y) \\times_{\\text{pr}_0, Y} V \\longrightarrow (Y \\times_X Y) \\times_{\\text{pr}_1, Y} V $$ of algebraic spaces over $Y \\times_X Y$. So loosely speaking one may think of $\\varphi$ as a map $\\varphi : \\text{pr}_0^*V \\to \\text{pr}_1^*V$\\footnote{Unfortunately, we have chosen the ``wrong'' direction for our arrow here. In Definitions \\ref{definition-descent-datum} and \\ref{definition-descent-datum-for-family-of-morphisms} we should have the opposite direction to what was done in Definition \\ref{definition-descent-datum-quasi-coherent} by the general principle that ``functions'' and ``spaces'' are dual.}. The cocycle condition then says that $\\text{pr}_{02}^*\\varphi = \\text{pr}_{12}^*\\varphi \\circ \\text{pr}_{01}^*\\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves."}
+{"_id": "9453", "title": "spaces-descent-remark-easier-family", "text": "Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$. Let $(V_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to X\\}$. We may think of the isomorphisms $\\varphi_{ij}$ as isomorphisms $$ (X_i \\times_X X_j) \\times_{\\text{pr}_0, X_i} V_i \\longrightarrow (X_i \\times_X X_j) \\times_{\\text{pr}_1, X_j} V_j $$ of algebraic spaces over $X_i \\times_X X_j$. So loosely speaking one may think of $\\varphi_{ij}$ as an isomorphism $\\text{pr}_0^*V_i \\to \\text{pr}_1^*V_j$ over $X_i \\times_X X_j$. The cocycle condition then says that $\\text{pr}_{02}^*\\varphi_{ik} = \\text{pr}_{12}^*\\varphi_{jk} \\circ \\text{pr}_{01}^*\\varphi_{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves."}
+{"_id": "9573", "title": "decent-spaces-remark-recall", "text": "Before we give the proof of the next lemma let us recall some facts about \\'etale morphisms of schemes: \\begin{enumerate} \\item An \\'etale morphism is flat and hence generalizations lift along an \\'etale morphism (Morphisms, Lemmas \\ref{morphisms-lemma-etale-flat} and \\ref{morphisms-lemma-generalizations-lift-flat}). \\item An \\'etale morphism is unramified, an unramified morphism is locally quasi-finite, hence fibres are discrete (Morphisms, Lemmas \\ref{morphisms-lemma-flat-unramified-etale}, \\ref{morphisms-lemma-unramified-quasi-finite}, and \\ref{morphisms-lemma-quasi-finite-at-point-characterize}). \\item A quasi-compact \\'etale morphism is quasi-finite and in particular has finite fibres (Morphisms, Lemmas \\ref{morphisms-lemma-quasi-finite-locally-quasi-compact} and \\ref{morphisms-lemma-quasi-finite}). \\item An \\'etale scheme over a field $k$ is a disjoint union of spectra of finite separable field extension of $k$ (Morphisms, Lemma \\ref{morphisms-lemma-etale-over-field}). \\end{enumerate} For a general discussion of \\'etale morphisms, please see \\'Etale Morphisms, Section \\ref{etale-section-etale-morphisms}."}
+{"_id": "9574", "title": "decent-spaces-remark-reasonable", "text": "Reasonable algebraic spaces are technically easier to work with than very reasonable algebraic spaces. For example, if $X \\to Y$ is a quasi-compact \\'etale surjective morphism of algebraic spaces and $X$ is reasonable, then so is $Y$, see Lemma \\ref{lemma-descent-conditions} but we don't know if this is true for the property ``very reasonable''. Below we give another technical property enjoyed by reasonable algebraic spaces."}
+{"_id": "9575", "title": "decent-spaces-remark-functoriality-henselian-local-ring", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \\in |X|$ with image $y \\in |Y|$. Choose an elementary \\'etale neighbourhood $(V, v) \\to (Y, y)$ (possible by Lemma \\ref{lemma-decent-space-elementary-etale-neighbourhood}). Then $V \\times_Y X$ is an algebraic space \\'etale over $X$ which has a unique point $x'$ mapping to $x$ in $X$ and to $v$ in $V$. (Details omitted; use that all points can be represented by monomorphisms from spectra of fields.) Choose an elementary \\'etale neighbourhood $(U, u) \\to (V \\times_Y X, x')$. Then we obtain the following commutative diagram $$ \\xymatrix{ \\Spec(\\mathcal{O}_{X, \\overline{x}}) \\ar[r] \\ar[d] & \\Spec(\\mathcal{O}_{X, x}^h) \\ar[r] \\ar[d] & \\Spec(\\mathcal{O}_{U, u}) \\ar[r] \\ar[d] & U \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(\\mathcal{O}_{Y, \\overline{y}}) \\ar[r] & \\Spec(\\mathcal{O}_{Y, y}^h) \\ar[r] & \\Spec(\\mathcal{O}_{V, v}) \\ar[r] & V \\ar[r] & Y } $$ This comes from the identifications $\\mathcal{O}_{X, \\overline{x}} = \\mathcal{O}_{U, u}^{sh}$, $\\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h$, $\\mathcal{O}_{Y, \\overline{y}} = \\mathcal{O}_{V, v}^{sh}$, $\\mathcal{O}_{Y, y}^h = \\mathcal{O}_{V, v}^h$ see in Lemma \\ref{lemma-describe-henselian-local-ring} and Properties of Spaces, Lemma \\ref{spaces-properties-lemma-describe-etale-local-ring} and the functoriality of the (strict) henselization discussed in Algebra, Sections \\ref{algebra-section-ind-etale} and \\ref{algebra-section-henselization}."}
+{"_id": "9576", "title": "decent-spaces-remark-one-point-decent-scheme", "text": "We will see in Limits of Spaces, Lemma \\ref{spaces-limits-lemma-reduction-scheme} that an algebraic space whose reduction is a scheme is a scheme."}
+{"_id": "9577", "title": "decent-spaces-remark-very-reasonable", "text": "An informal description of the properties $(\\beta)$, decent, reasonable, very reasonable was given in Section \\ref{section-reasonable-decent}. A morphism has one of these properties if (very) loosely speaking the fibres of the morphism have the corresponding properties. Being decent is useful to prove things about specializations of points on $|X|$. Being reasonable is a bit stronger and technically quite easy to work with."}
+{"_id": "9689", "title": "groupoids-remark-warning-group-scheme-geometrically-irreducible", "text": "Warning: The result of Lemma \\ref{lemma-group-scheme-field-geometrically-irreducible} does not mean that every irreducible component of $G/k$ is geometrically irreducible. For example the group scheme $\\mu_{3, \\mathbf{Q}} = \\Spec(\\mathbf{Q}[x]/(x^3 - 1))$ over $\\mathbf{Q}$ has two irreducible components corresponding to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over $\\Spec(\\mathbf{Q})$."}
+{"_id": "9690", "title": "groupoids-remark-easy", "text": "If $G$ is a group scheme over a field, is there always a quasi-compact open and closed subgroup scheme? By Proposition \\ref{proposition-connected-component} this question is only interesting if $G$ has infinitely many connected components (geometrically)."}
+{"_id": "9691", "title": "groupoids-remark-when-reduced", "text": "Any group scheme over a field of characteristic $0$ is reduced, see \\cite[I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4]{Perrin-thesis} and also \\cite[Proposition 4.2.8]{Perrin}. This was a question raised in \\cite[page 80]{Oort}. We have seen in Lemma \\ref{lemma-group-scheme-characteristic-zero-smooth} that this holds when the group scheme is locally of finite type."}
+{"_id": "9692", "title": "groupoids-remark-reduced-smooth-not-true-general", "text": "Let $k$ be a field of characteristic $p > 0$. Let $\\alpha \\in k$ be an element which is not a $p$th power. The closed subgroup scheme $$ G = V(x^p + \\alpha y^p) \\subset \\mathbf{G}_{a, k}^2 $$ is reduced and irreducible but not smooth (not even normal)."}
+{"_id": "9693", "title": "groupoids-remark-fun-with-torsors", "text": "Let $(G, m)$ be a group scheme over the scheme $S$. In this situation we have the following natural types of questions: \\begin{enumerate} \\item If $X \\to S$ is a pseudo $G$-torsor and $X \\to S$ is surjective, then is $X$ necessarily a $G$-torsor? \\item Is every $\\underline{G}$-torsor on $(\\Sch/S)_{fppf}$ representable? In other words, does every $\\underline{G}$-torsor come from a fppf $G$-torsor? \\item Is every $G$-torsor an fppf (resp.\\ smooth, resp.\\ \\'etale, resp.\\ Zariski) torsor? \\end{enumerate} In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on $G \\to S$. For example: If $S$ is the spectrum of a field, then the answer to (1) is yes because then $\\{X \\to S\\}$ is a fpqc covering trivializing $X$. If $G \\to S$ is affine, then the answer to (2) is yes (insert future reference here). If $G = \\text{GL}_{n, S}$ then the answer to (3) is yes and in fact any $\\text{GL}_{n, S}$-torsor is locally trivial (insert future reference here)."}
+{"_id": "9794", "title": "local-cohomology-remark-upshot", "text": "Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. The upshot of the discussion above is that $R\\Gamma_T : D^+(A) \\to D_T^+(A)$ is the right adjoint to the inclusion functor $D_T^+(A) \\to D^+(A)$. If $\\dim(A) < \\infty$, then $R\\Gamma_T : D(A) \\to D_T(A)$ is the right adjoint to the inclusion functor $D_T(A) \\to D(A)$. In both cases we have $$ H^i_T(K) = H^i(R\\Gamma_T(K)) = R^iH^0_T(K) = \\colim_{Z \\subset T\\text{ closed}} H^i_Z(K) $$ This follows by combining Lemmas \\ref{lemma-adjoint}, \\ref{lemma-adjoint-ext}, \\ref{lemma-equal-plus}, and \\ref{lemma-equal-full}."}
+{"_id": "9795", "title": "local-cohomology-remark-closure", "text": "Let $j : U \\to X$ be an open immersion of locally Noetherian schemes. Let $x \\in U$. Let $i_x : W_x \\to U$ be the integral closed subscheme with generic point $x$ and let $\\overline{\\{x\\}}$ be the closure in $X$. Then we have a commutative diagram $$ \\xymatrix{ W_x \\ar[d]_{i_x} \\ar[r]_{j'} & \\overline{\\{x\\}} \\ar[d]^i \\\\ U \\ar[r]^j & X } $$ We have $j_*i_{x, *}\\mathcal{O}_{W_x} = i_*j'_*\\mathcal{O}_{W_x}$. As the left vertical arrow is a closed immersion we see that $j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent if and only if $j'_*\\mathcal{O}_{W_x}$ is coherent."}
+{"_id": "9796", "title": "local-cohomology-remark-no-finiteness-pushforward", "text": "Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion of an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module. If there exists an $x \\in \\text{Ass}(\\mathcal{F})$ and $z \\in Z \\cap \\overline{\\{x\\}}$ such that $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\leq 1$, then $j_*\\mathcal{F}$ is not coherent. To prove this we can do a flat base change to the spectrum of $\\mathcal{O}_{X, z}$. Let $X' = \\overline{\\{x\\}}$. The assumption implies $\\mathcal{O}_{X' \\cap U} \\subset \\mathcal{F}$. Thus it suffices to see that $j_*\\mathcal{O}_{X' \\cap U}$ is not coherent. This is clear because $X' = \\{x, z\\}$, hence $j_*\\mathcal{O}_{X' \\cap U}$ corresponds to $\\kappa(x)$ as an $\\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not a closed point. \\medskip\\noindent In fact, the converse of Lemma \\ref{lemma-sharp-finiteness-pushforward} holds true: given an open immersion $j : U \\to X$ of integral Noetherian schemes and there exists a $z \\in X \\setminus U$ and an associated prime $\\mathfrak p$ of the completion $\\mathcal{O}_{X, z}^\\wedge$ with $\\dim(\\mathcal{O}_{X, z}^\\wedge/\\mathfrak p) = 1$, then $j_*\\mathcal{O}_U$ is not coherent. Namely, you can pass to the local ring, you can enlarge $U$ to the punctured spectrum, you can pass to the completion, and then the argument above gives the nonfiniteness."}
+{"_id": "9797", "title": "local-cohomology-remark-astute-reader", "text": "The astute reader will have realized that we can get away with a slightly weaker condition on the formal fibres of the local rings of $A$. Namely, in the situation of Theorem \\ref{theorem-finiteness} assume $A$ is universally catenary but make no assumptions on the formal fibres. Suppose we have an $n$ and we want to prove that $H^i_Z(M)$ are finite for $i \\leq n$. Then the exact same proof shows that it suffices that $s_{A, I}(M) > n$ and that the formal fibres of local rings of $A$ are $(S_n)$. On the other hand, if we want to show that $H^s_Z(M)$ is not finite where $s = s_{A, I}(M)$, then our arguments prove this if the formal fibres are $(S_{s - 1})$."}
+{"_id": "9798", "title": "local-cohomology-remark-higher-order-operators", "text": "We can upgrade Lemmas \\ref{lemma-derivation} and \\ref{lemma-etale-derivation} to include higher order differential operators. If we ever need this we will state and prove a precise lemma here."}
+{"_id": "9799", "title": "local-cohomology-remark-better-bound", "text": "The paper \\cite{AHS} shows, besides many other things, that if $A$ is local, then Proposition \\ref{proposition-uniform-artin-rees} also holds with $e = t$ replaced by $e = \\dim(A)$. Looking at Lemma \\ref{lemma-cd-sequence-Koszul} it is natural to ask whether Proposition \\ref{proposition-uniform-artin-rees} holds with $e = t$ replaced with $e = \\text{cd}(A, I)$. We don't know."}
+{"_id": "9800", "title": "local-cohomology-remark-strict-pro-isomorphism", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Say $I = (f_1, \\ldots, f_r)$. Denote $K_n^\\bullet$ the Koszul complex on $f_1^n, \\ldots, f_r^n$ as in More on Algebra, Situation \\ref{more-algebra-situation-koszul} and denote $K_n \\in D(A)$ the corresponding object. Let $M^\\bullet$ be a bounded complex of finite $A$-modules and denote $M \\in D(A)$ the corresponding object. Consider the following inverse systems in $D(A)$: \\begin{enumerate} \\item $M^\\bullet/I^nM^\\bullet$, i.e., the complex whose terms are $M^i/I^nM^i$, \\item $M \\otimes_A^\\mathbf{L} A/I^n$, \\item $M \\otimes_A^\\mathbf{L} K_n$, and \\item $M \\otimes_P^\\mathbf{L} P/J^n$ (see below). \\end{enumerate} All of these inverse systems are isomorphic as pro-objects: the isomorphism between (2) and (3) follows from More on Algebra, Lemma \\ref{more-algebra-lemma-sequence-Koszul-complexes}. The isomorphism between (1) and (2) is given in More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion-plain-completion}. For the last one, see below. \\medskip\\noindent However, we can ask if these isomorphisms of pro-systems are ``strict''; this terminology and question is related to the discussion in \\cite[pages 61, 62]{quillenhomology}. Namely, given a category $\\mathcal{C}$ we can define a ``strict pro-category'' whose objects are inverse systems $(X_n)$ and whose morphisms $(X_n) \\to (Y_n)$ are given by tuples $(c, \\varphi_n)$ consisting of a $c \\geq 0$ and morphisms $\\varphi_n : X_n \\to Y_{n - c}$ for all $n \\geq c$ satisfying an obvious compatibility condition and up to a certain equivalence (given essentially by increasing $c$). Then we ask whether the above inverse systems are isomorphic in this strict pro-category. \\medskip\\noindent This clearly cannot be the case for (1) and (3) even when $M = A[0]$. Namely, the system $H^0(K_n) = A/(f_1^n, \\ldots, f_r^n)$ is not strictly pro-isomorphic in the category of modules to the system $A/I^n$ in general. For example, if we take $A = \\mathbf{Z}[x_1, \\ldots, x_r]$ and $f_i = x_i$, then $H^0(K_n)$ is not annihilated by $I^{r(n - 1)}$.\\footnote{Of course, we can ask whether these pro-systems are isomorphic in a category whose objects are inverse systems and where maps are given by tuples $(r, c, \\varphi_n)$ consisting of $r \\geq 1$, $c \\geq 0$ and maps $\\varphi_n : X_{rn} \\to Y_{n - c}$ for $n \\geq c$.} \\medskip\\noindent It turns out that the results above show that the natural map from (2) to (1) discussed in More on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion-plain-completion} is a strict pro-isomorphism. We will sketch the proof. Using standard arguments involving stupid truncations, we first reduce to the case where $M^\\bullet$ is given by a single finite $A$-module $M$ placed in degree $0$. Pick $N, c \\geq 0$ as in Proposition \\ref{proposition-uniform-artin-rees}. The proposition implies that for $n \\geq N$ we get factorizations $$ M \\otimes_A^\\mathbf{L} A/I^n \\to \\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^n) \\to M \\otimes_A^\\mathbf{L} A/I^{n - c} $$ of the transition maps in the system (2). On the other hand, by More on Algebra, Lemma \\ref{more-algebra-lemma-tor-strictly-pro-zero}, we can find another constant $c' = c'(M) \\geq 0$ such that the maps $\\text{Tor}_i^A(M, A/I^{n'}) \\to \\text{Tor}_i(M, A/I^{n' - c'})$ are zero for $i = 1, 2, \\ldots, t$ and $n' \\geq c'$. Then it follows from Derived Categories, Lemma \\ref{derived-lemma-trick-vanishing-composition} that the map $$ \\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^{n + tc'}) \\to \\tau_{\\geq -t}(M \\otimes_A^\\mathbf{L} A/I^n) $$ factors through $M \\otimes_A^\\mathbf{L}A/I^{n + tc'} \\to M/I^{n + tc'}M$. Combined with the previous result we obtain a factorization $$ M \\otimes_A^\\mathbf{L}A/I^{n + tc'} \\to M/I^{n + tc'}M \\to M \\otimes_A^\\mathbf{L} A/I^{n - c} $$ which gives us what we want. If we ever need this result, we will carefully state it and provide a detailed proof. \\medskip\\noindent For number (4) suppose we have a Noetherian ring $P$, a ring homomorphism $P \\to A$, and an ideal $J \\subset P$ such that $I = JA$. By More on Algebra, Section \\ref{more-algebra-section-derived-base-change} we get a functor $M \\otimes_P^\\mathbf{L} - : D(P) \\to D(A)$ and we get an inverse system $M \\otimes_P^\\mathbf{L} P/J^n$ in $D(A)$ as in (4). If $P$ is Noetherian, then the system in (4) is pro-isomorphic to the system in (1) because we can compare with Koszul complexes. If $P \\to A$ is finite, then the system (4) is strictly pro-isomorphic to the system (2) because the inverse system $A \\otimes_P^\\mathbf{L} P/J^n$ is strictly pro-isomorphic to the inverse system $A/I^n$ (by the discussion above) and because we have $$ M \\otimes_P^\\mathbf{L} P/J^n = M \\otimes_A^\\mathbf{L} (A \\otimes_P^\\mathbf{L} P/J^n) $$ by More on Algebra, Lemma \\ref{more-algebra-lemma-derived-base-change}. \\medskip\\noindent A standard example in (4) is to take $P = \\mathbf{Z}[x_1, \\ldots, x_r]$, the map $P \\to A$ sending $x_i$ to $f_i$, and $J = (x_1, \\ldots, x_r)$. In this case one shows that $$ M \\otimes_P^\\mathbf{L} P/J^n = M \\otimes_{A[x_1, \\ldots, x_r]}^\\mathbf{L} A[x_1, \\ldots, x_r]/(x_1, \\ldots, x_r)^n $$ and we reduce to one of the cases discussed above (although this case is strictly easier as $A[x_1, \\ldots, x_r]/(x_1, \\ldots, x_r)^n$ has tor dimension at most $r$ for all $n$ and hence the step using Proposition \\ref{proposition-uniform-artin-rees} can be avoided). This case is discussed in the proof of \\cite[Proposition 3.5.1]{BS}."}
+{"_id": "9801", "title": "local-cohomology-remark-duals", "text": "Given a pair $(M, n)$ consisting of an integer $n \\geq 0$ and a finite $A/I^n$-module $M$ we set $M^\\vee = \\Hom_{A/I^n}(M, A/I^n)$. Given a pair $(\\mathcal{F}, n)$ consisting of an integer $n$ and a coherent $\\mathcal{O}_{Y_n}$-module $\\mathcal{F}$ we set $$ \\mathcal{F}^\\vee = \\SheafHom_{\\mathcal{O}_{Y_n}}(\\mathcal{F}, \\mathcal{O}_{Y_n}) $$ Given $(M, n)$ as above, there is a canonical map $$ can : p^*(M^\\vee) \\longrightarrow (p^*M)^\\vee $$ Namely, if we choose a presentation $(A/I^n)^{\\oplus s} \\to (A/I^n)^{\\oplus r} \\to M \\to 0$ then we obtain a presentation $\\mathcal{O}_{Y_n}^{\\oplus s} \\to \\mathcal{O}_{Y_n}^{\\oplus r} \\to p^*M \\to 0$. Taking duals we obtain exact sequences $$ 0 \\to M^\\vee \\to (A/I^n)^{\\oplus r} \\to (A/I^n)^{\\oplus s} $$ and $$ 0 \\to (p^*M)^\\vee \\to \\mathcal{O}_{Y_n}^{\\oplus r} \\to \\mathcal{O}_{Y_n}^{\\oplus s} $$ Pulling back the first sequence by $p$ we find the desired map $can$. The construction of this map is functorial in the finite $A/I^n$-module $M$. The kernel and cokernel of $can$ are scheme theoretically supported on $Y_c$ if $M$ is an $(A, n, c)$-module. Namely, in that case for $a \\in I^c$ the map $a : M \\to M$ factors through a finite free $A/I^n$-module for which $can$ is an isomorphism. Hence $a$ annihilates the kernel and cokernel of $can$."}
+{"_id": "10649", "title": "more-algebra-remark-relative-modules-over-fibre-product", "text": "In Situation \\ref{situation-relative-module-over-fibre-product}. Assume $B' \\to D'$ is of finite presentation and suppose we are given a $D'$-module $L'$. We claim there is a bijective correspondence between \\begin{enumerate} \\item surjections of $D'$-modules $L' \\to Q'$ with $Q'$ of finite presentation over $D'$ and flat over $B'$, and \\item pairs of surjections of modules $(L' \\otimes_{D'} D \\to Q_1, L' \\otimes_{D'} C' \\to Q_2)$ with \\begin{enumerate} \\item $Q_1$ of finite presentation over $D$ and flat over $B$, \\item $Q_2$ of finite presentation over $C'$ and flat over $A'$, \\item $Q_1 \\otimes_D C = Q_2 \\otimes_{C'} C$ as quotients of $L' \\otimes_{D'} C$. \\end{enumerate} \\end{enumerate} The correspondence between these is given by $Q \\mapsto (Q_1, Q_2)$ with $Q_1 = Q \\otimes_{D'} D$ and $Q_2 = Q \\otimes_{D'} C'$. And for the converse we use $Q = Q_1 \\times_{Q_{12}} Q_2$ where $Q_{12}$ the common quotient $Q_1 \\otimes_D C = Q_2 \\otimes_{C'} C$ of $L' \\otimes_{D'} C$. As quotient map we use $$ L' \\longrightarrow (L' \\otimes_{D'} D) \\times_{(L' \\otimes_{D'} C)} (L' \\otimes_{D'} C') \\longrightarrow Q_1 \\times_{Q_{12}} Q_2 = Q $$ where the first arrow is surjective by Lemma \\ref{lemma-module-over-fibre-product-bis} and the second by Lemma \\ref{lemma-surjection-module-over-fibre-product}. The claim follows by Lemmas \\ref{lemma-relative-flat-module-over-fibre-product} and \\ref{lemma-relative-finitely-presented-module-over-fibre-product}."}
+{"_id": "10650", "title": "more-algebra-remark-when-does-condition-hold", "text": "Let $R$ be a ring. When does $R$ satisfy the condition mentioned in Lemmas \\ref{lemma-flat-finite-type-finite-presentation-local-module}, \\ref{lemma-flat-finite-type-finite-presentation-local}, and \\ref{lemma-flat-graded-finite-type-finite-presentation}? This holds if \\begin{enumerate} \\item $R$ is local, \\item $R$ is Noetherian, \\item $R$ is a domain, \\item $R$ is a reduced ring with finitely many minimal primes, or \\item $R$ has finitely many weakly associated primes, see Algebra, Lemma \\ref{algebra-lemma-zero-at-weakly-ass-zero}. \\end{enumerate} Thus these lemmas hold in all cases listed above."}
+{"_id": "10651", "title": "more-algebra-remark-what-does-it-mean", "text": "The assertion of Lemma \\ref{lemma-lift-fs} is quite strong. Namely, suppose that we have a diagram $$ \\xymatrix{ & B \\\\ A \\ar[r] & A' \\ar[u] } $$ of local homomorphisms of Noetherian complete local rings where $A \\to A'$ induces an isomorphism of residue fields $k = A/\\mathfrak m_A = A'/\\mathfrak m_{A'}$ and with $B \\otimes_{A'} k$ formally smooth over $k$. Then we can extend this to a commutative diagram $$ \\xymatrix{ C \\ar[r] & B \\\\ A \\ar[r] \\ar[u] & A' \\ar[u] } $$ of local homomorphisms of Noetherian complete local rings where $A \\to C$ is formally smooth in the $\\mathfrak m_C$-adic topology and where $C \\otimes_A k \\cong B \\otimes_{A'} k$. Namely, pick $A \\to C$ as in Lemma \\ref{lemma-lift-fs} lifting $B \\otimes_{A'} k$ over $k$. By formal smoothness we can find the arrow $C \\to B$, see Lemma \\ref{lemma-lift-continuous}. Denote $C \\otimes_A^\\wedge A'$ the completion of $C \\otimes_A A'$ with respect to the ideal $C \\otimes_A \\mathfrak m_{A'}$. Note that $C \\otimes_A^\\wedge A'$ is a Noetherian complete local ring (see Algebra, Lemma \\ref{algebra-lemma-completion-Noetherian}) which is flat over $A'$ (see Algebra, Lemma \\ref{algebra-lemma-flat-module-powers}). We have moreover \\begin{enumerate} \\item $C \\otimes_A^\\wedge A' \\to B$ is surjective, \\item if $A \\to A'$ is surjective, then $C \\to B$ is surjective, \\item if $A \\to A'$ is finite, then $C \\to B$ is finite, and \\item if $A' \\to B$ is flat, then $C \\otimes_A^\\wedge A' \\cong B$. \\end{enumerate} Namely, by Nakayama's lemma for nilpotent ideals (see Algebra, Lemma \\ref{algebra-lemma-NAK}) we see that $C \\otimes_A k \\cong B \\otimes_{A'} k$ implies that $C \\otimes_A A'/\\mathfrak m_{A'}^n \\to B/\\mathfrak m_{A'}^nB$ is surjective for all $n$. This proves (1). Parts (2) and (3) follow from part (1). Part (4) follows from Algebra, Lemma \\ref{algebra-lemma-mod-injective}."}
+{"_id": "10652", "title": "more-algebra-remark-G-does-not-survive-completion", "text": "Let $R$ be a G-ring and let $I \\subset R$ be an ideal. In general it is not the case that the $I$-adic completion $R^\\wedge$ is a G-ring. An example was given by Nishimura in \\cite{Nishimura}. A generalization and, in some sense, clarification of this example can be found in the last section of \\cite{Dumitrescu}."}
+{"_id": "10653", "title": "more-algebra-remark-P-resolution", "text": "In fact, we can do better than Lemma \\ref{lemma-K-flat-resolution}. Namely, we can find a quasi-isomorphism $P^\\bullet \\to M^\\bullet$ where $P^\\bullet$ is a complex of $R$-modules endowed with a filtration $$ 0 = F_{-1}P^\\bullet \\subset F_0P^\\bullet \\subset F_1P^\\bullet \\subset \\ldots \\subset P^\\bullet $$ by subcomplexes such that \\begin{enumerate} \\item $P^\\bullet = \\bigcup F_pP^\\bullet$, \\item the inclusions $F_iP^\\bullet \\to F_{i + 1}P^\\bullet$ are termwise split injections, \\item the quotients $F_{i + 1}P^\\bullet/F_iP^\\bullet$ are isomorphic to direct sums of shifts $R[k]$ (as complexes, so differentials are zero). \\end{enumerate} This will be shown in Differential Graded Algebra, Lemma \\ref{dga-lemma-resolve}. Moreover, given such a complex we obtain a distinguished triangle $$ \\bigoplus F_iP^\\bullet \\to \\bigoplus F_iP^\\bullet \\to M^\\bullet \\to \\bigoplus F_iP^\\bullet[1] $$ in $D(R)$. Using this we can sometimes reduce statements about general complexes to statements about $R[k]$ (this of course only works if the statement is preserved under taking direct sums). More precisely, let $T$ be a property of objects of $D(R)$. Suppose that \\begin{enumerate} \\item if $K_i \\in D(R)$, $i \\in I$ is a family of objects with $T(K_i)$ for all $i \\in I$, then $T(\\bigoplus K_i)$, \\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object, \\item $T(R[k])$ holds for all $k$. \\end{enumerate} Then $T$ holds for all objects of $D(R)$."}
+{"_id": "10654", "title": "more-algebra-remark-warning-compute-base-change", "text": "Let $R \\to A$ be a ring map, and let $N$ and $N'$ be $A$-modules. Denote $N_R$ and $N'_R$ the restriction of $N$ and $N'$ to $R$-modules, see Algebra, Section \\ref{algebra-section-base-change}. In this situation, the objects $N_R \\otimes_R^\\mathbf{L} N'$ and $N \\otimes_R^\\mathbf{L} N'_R$ of $D(A)$ are in general not isomorphic! In other words, one has to pay careful attention as to which of the two sides is being used to provide the $A$-module structure. \\medskip\\noindent For a specific example, set $R = k[x, y]$, $A = R/(xy)$, $N = R/(x)$ and $N' = A = R/(xy)$. The resolution $0 \\to R \\xrightarrow{xy} R \\to N'_R \\to 0$ shows that $N \\otimes_R^\\mathbf{L} N'_R = N[1] \\oplus N$ in $D(A)$. The resolution $0 \\to R \\xrightarrow{x} R \\to N_R \\to 0$ shows that $N_R \\otimes_R^\\mathbf{L} N'$ is represented by the complex $A \\xrightarrow{x} A$. To see these two complexes are not isomorphic, one can show that the second complex is not isomorphic in $D(A)$ to the direct sum of its cohomology groups, or one can show that the first complex is not a perfect object of $D(A)$ whereas the second one is. Some details omitted."}
+{"_id": "10655", "title": "more-algebra-remark-sign-explanation", "text": "In the yoga of super vector spaces the sign used in the proof of Lemma \\ref{lemma-evaluate-and-more} above can be explained as follows. A super vector space is just a finite dimensional vector space $V$ which comes with a direct sum decomposition $V = V^+ \\oplus V^-$. Here we think of the elements of $V^+$ as the even elements and the elements of $V^-$ as the odd ones. Given two super vector spaces $V$ and $W$ we set $$ (V \\otimes W)^+ = (V^+ \\otimes W^+) \\oplus (V^- \\otimes W^-) $$ and similarly for the odd part. In the category of super vector spaces the isomorphism $$ \\psi : V \\otimes W \\longrightarrow W \\otimes V $$ is defined to be the usual one, except that on the summand $V^- \\otimes W^-$ we use the negative of the usual identification. In this way we obtain a symmetric monoidal category, see Categories, Section \\ref{categories-section-monoidal}. An object $V$ of the category of super vector spaces has a left dual which we denote $V^\\vee$ which comes equipped with an identity $\\eta : \\mathbf{1} \\to V \\otimes V^\\vee$ and an evaluation map $\\epsilon : V^\\vee \\otimes V \\to \\mathbf{1}$ which induce canonical isomorphisms $\\Hom(V, W) = W \\otimes V^\\vee$ and $\\Hom(V^\\vee, U) = V \\otimes U$, see Categories, Lemma \\ref{categories-lemma-left-dual}. Given three super vector spaces $U$, $V$, $W$ we can try to construct the analogue $$ c : \\Hom(V, W) \\otimes U \\longrightarrow \\Hom(\\Hom(U, V), W) $$ of the maps $c_{p, r, s}$ which occur in the lemma above. Using the formulae given above (which do not involve signs) this becomes a map $$ W \\otimes V^\\vee \\otimes U \\longrightarrow W \\otimes (V \\otimes U^\\vee)^\\vee = W \\otimes (U^\\vee)^\\vee \\otimes V^\\vee $$ To find this arrow in a canonical fashion we need to do two things: \\begin{enumerate} \\item we need to use the commutativity constraint $\\psi : V^\\vee \\otimes U \\to U \\otimes V^\\vee$ which introduces a sign on $(V^\\vee)^- \\otimes U^-$, and \\item we need to use the canonical isomorphism $U \\to (U^\\vee)^\\vee$ which comes from the identification of $U^\\vee$ as the {\\bf right} dual of $U$ using $\\psi$ as in Categories, Lemma \\ref{categories-lemma-dual-symmetric}. This differs from the usual identification by $-1$ on the odd part of $U$. \\end{enumerate} Part (1) explains the sign $(-1)^{qr}$ in the proof of the lemma and part (2) explains the sign $(-1)^r$ in the proof of the lemma."}
+{"_id": "10656", "title": "more-algebra-remark-smoothness-ext-1-zero", "text": "The following two statements follow from Lemma \\ref{lemma-ext-1-zero}, Algebra, Definition \\ref{algebra-definition-smooth}, and Algebra, Proposition \\ref{algebra-proposition-characterize-formally-smooth}. \\begin{enumerate} \\item A ring map $A \\to B$ is smooth if and only if $A \\to B$ is of finite presentation and $\\Ext^1_B(\\NL_{B/A}, N) = 0$ for every $B$-module $N$. \\item A ring map $A \\to B$ is formally smooth if and only if $\\Ext^1_B(\\NL_{B/A}, N) = 0$ for every $B$-module $N$. \\end{enumerate}"}
+{"_id": "10657", "title": "more-algebra-remark-Rlim-cohomology", "text": "Consider the category $\\mathbf{N}$ whose objects are natural numbers and whose morphisms are unique arrows $i \\to j$ if $j \\geq i$. Endow $\\mathbf{N}$ with the chaotic topology (Sites, Example \\ref{sites-example-indiscrete}) so that a sheaf $\\mathcal{F}$ is the same thing as an inverse system $$ \\mathcal{F}_1 \\leftarrow \\mathcal{F}_2 \\leftarrow \\mathcal{F}_3 \\leftarrow \\ldots $$ of sets over $\\mathbf{N}$. Note that $\\Gamma(\\mathbf{N}, \\mathcal{F}) = \\lim \\mathcal{F}_n$. For an inverse system of abelian groups $\\mathcal{F}_n$ we have $$ R^p\\lim \\mathcal{F}_n = H^p(\\mathbf{N}, \\mathcal{F}) $$ because both sides are the higher right derived functors of $\\mathcal{F} \\mapsto \\lim \\mathcal{F}_n = H^0(\\mathbf{N}, \\mathcal{F})$. Thus the existence of $R\\lim$ also follows from the general material in Cohomology on Sites, Sections \\ref{sites-cohomology-section-cohomology-sheaves} and \\ref{sites-cohomology-section-unbounded}."}
+{"_id": "10658", "title": "more-algebra-remark-compare-derived-limit", "text": "Let $(K_n)$ be an inverse system of objects of $D(\\textit{Ab})$. Let $K = R\\lim K_n$ be a derived limit of this system (see Derived Categories, Section \\ref{derived-section-derived-limit}). Such a derived limit exists because $D(\\textit{Ab})$ has countable products (Derived Categories, Lemma \\ref{derived-lemma-products}). By Lemma \\ref{lemma-lift-to-system-complexes-Ab} we can also lift $(K_n)$ to an object $M$ of $D(\\mathbf{N})$. Then $K \\cong R\\lim M$ where $R\\lim$ is the functor (\\ref{equation-Rlim}) because $R\\lim M$ is also a derived limit of the system $(K_n)$ by Lemma \\ref{lemma-distinguished-triangle-Rlim}. Thus, although there may be many isomorphism classes of lifts $M$ of the system $(K_n)$, the isomorphism type of $R\\lim M$ is independent of the choice because it is isomorphic to the derived limit $K = R\\lim K_n$ of the system. Thus we may apply results on $R\\lim$ proved in this section to derived limits. For example, for every $p \\in \\mathbf{Z}$ there is a canonical short exact sequence $$ 0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(K) \\to \\lim H^p(K_n) \\to 0 $$ because we may apply Lemma \\ref{lemma-distinguished-triangle-Rlim} to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma \\ref{lemma-distinguished-triangle-Rlim} to the (defining) distinguished triangle $K \\to \\prod K_n \\to \\prod K_n \\to K[1]$."}
+{"_id": "10659", "title": "more-algebra-remark-Rlim-cohomology-modules", "text": "This remark is a continuation of Remark \\ref{remark-Rlim-cohomology}. A sheaf of rings on $\\mathbf{N}$ is just an inverse system of rings $(A_n)$. A sheaf of modules over $(A_n)$ is exactly the same thing as an object of the category $\\textit{Mod}(\\mathbf{N}, (A_n))$ defined above. The derived functor $R\\lim$ of Lemma \\ref{lemma-compute-Rlim-modules} is simply $R\\Gamma(\\mathbf{N}, -)$ from the derived category of modules to the derived category of modules over the global sections of the structure sheaf. It is true in general that cohomology of groups and modules agree, see Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-cohomology-modules-abelian-agree}."}
+{"_id": "10660", "title": "more-algebra-remark-how-unique", "text": "With assumptions as in Lemma \\ref{lemma-lift-to-system-complexes}. A priori there are many isomorphism classes of objects $M$ of $D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ which give rise to the system $(K_n, \\varphi_n)$ of the lemma. For each such $M$ we can consider the complex $R\\lim M \\in D(A)$ where $A = \\lim A_n$. By Lemma \\ref{lemma-distinguished-triangle-Rlim-modules} we see that $R\\lim M$ is a derived limit of the inverse system $(K_n)$ of $D(A)$. Hence we see that the isomorphism class of $R\\lim M$ in $D(A)$ is independent of the choices made in constructing $M$. In particular, we may apply results on $R\\lim$ proved in this section to derived limits of inverse systems in $D(A)$. For example, for every $p \\in \\mathbf{Z}$ there is a canonical short exact sequence $$ 0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(R\\lim K_n) \\to \\lim H^p(K_n) \\to 0 $$ because we may apply Lemma \\ref{lemma-distinguished-triangle-Rlim-modules} to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma \\ref{lemma-distinguished-triangle-Rlim-modules} to the (defining) distinguished triangle $R\\lim K_n \\to \\prod K_n \\to \\prod K_n \\to (R\\lim K_n)[1]$ of the derived limit."}
+{"_id": "10661", "title": "more-algebra-remark-constructing-tensor-with-limits-functorially", "text": "Let $A$ be a ring. Let $(E_n)$ be an inverse system of objects of $D(A)$. We've seen above that a derived limit $R\\lim E_n$ exists. Thus for every object $K$ of $D(A)$ also the derived limit $R\\lim( K \\otimes_A^\\mathbf{L} E_n )$ exists. It turns out that we can construct these derived limits functorially in $K$ and obtain an exact functor $$ R\\lim(- \\otimes_A^\\mathbf{L} E_n) : D(A) \\longrightarrow D(A) $$ of triangulated categories. Namely, we first lift $(E_n)$ to an object $E$ of $D(\\mathbf{N}, A)$, see Lemma \\ref{lemma-lift-to-system-complexes}. (The functor will depend on the choice of this lift.) Next, observe that there is a ``diagonal'' or ``constant'' functor $$ \\Delta : D(A) \\longrightarrow D(\\mathbf{N}, A) $$ mapping the complex $K^\\bullet$ to the constant inverse system of complexes with value $K^\\bullet$. Then we simply define $$ R\\lim(K \\otimes_A^\\mathbf{L} E_n) = R\\lim(\\Delta(K)\\otimes^\\mathbf{L} E) $$ where on the right hand side we use the functor $R\\lim$ of Lemma \\ref{lemma-compute-Rlim-modules} and the functor $- \\otimes^\\mathbf{L} -$ of Lemma \\ref{lemma-derived-tensor-product-systems}."}
+{"_id": "10662", "title": "more-algebra-remark-glueing-data", "text": "In this remark we define a category of glueing data. Let $R \\to S$ be a ring map. Let $f_1, \\ldots, f_t \\in R$ and $I = (f_1, \\ldots, f_t)$. Consider the category $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ as the category whose \\begin{enumerate} \\item objects are systems $(M', M_i, \\alpha_i, \\alpha_{ij})$, where $M'$ is an $S$-module, $M_i$ is an $R_{f_i}$-module, $\\alpha_i : (M')_{f_i} \\to M_i \\otimes_R S$ is an isomorphism, and $\\alpha_{ij} : (M_i)_{f_j} \\to (M_j)_{f_i}$ are isomorphisms such that \\begin{enumerate} \\item $\\alpha_{ij} \\circ \\alpha_i = \\alpha_j$ as maps $(M')_{f_if_j} \\to (M_j)_{f_i}$, and \\item $\\alpha_{jk} \\circ \\alpha_{ij} = \\alpha_{ik}$ as maps $(M_i)_{f_jf_k} \\to (M_k)_{f_if_j}$ (cocycle condition). \\end{enumerate} \\item morphisms $(M', M_i, \\alpha_i, \\alpha_{ij}) \\to (N', N_i, \\beta_i, \\beta_{ij})$ are given by maps $\\varphi' : M' \\to N'$ and $\\varphi_i : M_i \\to N_i$ compatible with the given maps $\\alpha_i, \\beta_i, \\alpha_{ij}, \\beta_{ij}$. \\end{enumerate} There is a canonical functor $$ \\text{Can} : \\text{Mod}_R \\longrightarrow \\text{Glue}(R \\to S, f_1, \\ldots, f_t), \\quad M \\longmapsto (M \\otimes_R S, M_{f_i}, \\text{can}_i, \\text{can}_{ij}) $$ where $\\text{can}_i : (M \\otimes_R S)_{f_i} \\to M_{f_i} \\otimes_R S$ and $\\text{can}_{ij} : (M_{f_i})_{f_j} \\to (M_{f_j})_{f_i}$ are the canonical isomorphisms. For any object $\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$ of the category $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ we define $$ H^0(\\mathbf{M}) = \\{(m', m_i) \\mid \\alpha_i(m') = m_i \\otimes 1, \\alpha_{ij}(m_i) = m_j\\} $$ in other words defined by the exact sequence $$ 0 \\to H^0(\\mathbf{M}) \\to M' \\times \\prod M_i \\to \\prod M'_{f_i} \\times \\prod (M_i)_{f_j} $$ similar to (\\ref{equation-glueing-complex}). We think of $H^0(\\mathbf{M})$ as an $R$-module. Thus we also get a functor $$ H^0 : \\text{Glue}(R \\to S, f_1, \\ldots, f_t) \\longrightarrow \\text{Mod}_R $$ Our next goal is to show that the functors $\\text{Can}$ and $H^0$ are sometimes quasi-inverse to each other."}
+{"_id": "10663", "title": "more-algebra-remark-formal-glueing-algebras", "text": "The equivalences of Proposition \\ref{proposition-equivalence}, Theorem \\ref{theorem-formal-glueing}, and Proposition \\ref{proposition-formal-glueing} preserve properties of modules. For example if $M$ corresponds to $\\mathbf{M} = (M', M_i, \\alpha_i, \\alpha_{ij})$ then $M$ is finite, or finitely presented, or flat, or projective over $R$ if and only if $M'$ and $M_i$ have the corresponding property over $S$ and $R_{f_i}$. This follows from the fact that $R \\to S \\times \\prod R_{f_i}$ is faithfully flat and descend and ascent of these properties along faithfully flat maps, see Algebra, Lemma \\ref{algebra-lemma-descend-properties-modules} and Theorem \\ref{algebra-theorem-ffdescent-projectivity}. These functors also preserve the $\\otimes$-structures on either side. Thus, it defines equivalences of various categories built out of the pair $(\\text{Mod}_R, \\otimes)$, such as the category of algebras."}
+{"_id": "10664", "title": "more-algebra-remark-topological-analogue", "text": "Given a differential manifold $X$ with a compact closed submanifold $Z$ having complement $U$, specifying a sheaf on $X$ is the same as specifying a sheaf on $U$, a sheaf on an unspecified tubular neighbourhood $T$ of $Z$ in $X$, and an isomorphism between the two resulting sheaves along $T \\cap U$. Tubular neighbourhoods do not exist in algebraic geometry as such, but results such as Proposition \\ref{proposition-equivalence}, Theorem \\ref{theorem-formal-glueing}, and Proposition \\ref{proposition-formal-glueing} allow us to work with formal neighbourhoods instead."}
+{"_id": "10665", "title": "more-algebra-remark-not-descent", "text": "While $R \\to R_f$ is always flat, $R \\to R^\\wedge$ is typically not flat unless $R$ is Noetherian (see Algebra, Lemma \\ref{algebra-lemma-completion-flat} and the discussion in Examples, Section \\ref{examples-section-nonflat}). Consequently, we cannot in general apply faithfully flat descent as discussed in Descent, Section \\ref{descent-section-descent-modules} to the morphism $R \\to R^\\wedge \\oplus R_f$. Moreover, even in the Noetherian case, the usual definition of a descent datum for this morphism refers to the ring $R^\\wedge \\otimes_R R^\\wedge$, which we will avoid considering in this section."}
+{"_id": "10666", "title": "more-algebra-remark-BL-special-case", "text": "Suppose that $f$ is a nonzerodivisor. Then Algebra, Lemma \\ref{algebra-lemma-completion-differ-by-torsion} shows that $f$ is a nonzerodivisor in $R^\\wedge$. Hence $(R, f)$ is a glueing pair."}
+{"_id": "10667", "title": "more-algebra-remark-noetherian-case", "text": "If $R \\to R^\\wedge$ is flat, then for each positive integer $n$ tensoring the sequence $0 \\to R[f^n] \\to R \\to R$ with $R^\\wedge$ gives the sequence $0 \\to R[f^n] \\otimes_R R^\\wedge \\to R^\\wedge \\to R^\\wedge$. Combined with Lemma \\ref{lemma-torsion-completion} we conclude that $R[f^n] \\to R^\\wedge[f^n]$ is an isomorphism. Thus $(R, f)$ is a glueing pair. This holds in particular if $R$ is Noetherian, see Algebra, Lemma \\ref{algebra-lemma-completion-flat}."}
+{"_id": "10668", "title": "more-algebra-remark-glueable", "text": "Let $(R \\to R', f)$ be a glueing pair and let $M$ be an $R$-module. Here are some  observations which can be used to determine whether $M$ is glueable for $(R \\to R', f)$. \\begin{enumerate} \\item By Lemma \\ref{lemma-same-f-torsion-module} we see that $M$ is glueable for $(R \\to R^\\wedge, f)$ if and only if $M[f^\\infty] \\to M \\otimes_R R^\\wedge$ is injective. This holds if $M[f] \\to M^\\wedge$ is injective, i.e., when $M[f] \\cap \\bigcap_{n = 1}^\\infty f^n M = 0$. \\item If $\\text{Tor}_1^R(M, R'_f) = 0$, then $M$ is glueable for $(R \\to R', f)$ (use Algebra, Lemma \\ref{algebra-lemma-long-exact-sequence-tor}). This is equivalent to saying that $\\text{Tor}_1^R(M, R')$ is $f$-power torsion. In particular, any flat $R$-module is glueable for $(R \\to R', f)$. \\item If $R \\to R'$ is flat, then $\\text{Tor}_1^R(M, R') = 0$ for every $R$-module so every $R$-module is glueable for $(R \\to R', f)$. This holds in particular when $R$ is Noetherian and $R' = R^\\wedge$, see Algebra, Lemma \\ref{algebra-lemma-completion-flat} \\end{enumerate}"}
+{"_id": "10669", "title": "more-algebra-remark-what-you-get-for-general-modules", "text": "Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module that is not necessarily glueable for $(R \\to R', f)$. Setting $M' = M \\otimes_R R'$ and $M_1 = M_f$ we obtain the glueing datum $\\text{Can}(M) = (M', M_1, \\text{can})$. Then $\\tilde M = H^0(M', M_1, \\text{can})$ is an $R$-module that is glueable for $(R \\to R', f)$ and the canonical map $M \\to \\tilde M$ gives isomorphisms $M \\otimes_R R' \\to \\tilde M \\otimes_R R'$ and $M_f \\to \\tilde M_f$, see Theorem \\ref{theorem-BL-glueing}. From the exactness of the sequences $$ M \\to (M \\otimes_R R' )\\oplus M_f \\to M \\otimes_R (R')_f  \\to 0 $$ and $$ 0 \\to \\tilde M \\to (\\tilde M \\otimes_R R') \\oplus \\tilde M_f \\to \\tilde M \\otimes_R  (R')_f \\to 0 $$ we conclude that the map $M \\to \\tilde M$ is surjective."}
+{"_id": "10670", "title": "more-algebra-remark-compare-BL", "text": "In \\cite{Beauville-Laszlo} it is assumed that $f$ is a nonzerodivisor in $R$ and $R' = R^\\wedge$, which gives a glueing pair by Lemma \\ref{lemma-same-f-torsion}. Even in this setting Theorem \\ref{theorem-BL-glueing} says something new: the results of \\cite{Beauville-Laszlo} only apply to modules on which $f$ is a nonzerodivisor (and hence glueable in our sense, see Lemma \\ref{lemma-same-f-torsion-module}). Lemma \\ref{lemma-BL-properties} also provides a slight extension of the results of  \\cite{Beauville-Laszlo}: not only can we allow $M$ to have nonzero $f$-power torsion, we do not even require it to be glueable."}
+{"_id": "10671", "title": "more-algebra-remark-derived-completion", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. The left adjoint to the inclusion functor $D_{comp}(A, I) \\to D(A)$ which exists by Lemma \\ref{lemma-derived-completion} is called the {\\it derived completion}. To indicate this we will say ``let $K^\\wedge$ be the derived completion of $K$''. Please keep in mind that the unit of the adjunction is a functorial map $K \\to K^\\wedge$."}
+{"_id": "10672", "title": "more-algebra-remark-Leta", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Suppose that $M^\\bullet$ is a complex of $A$-modules. By Lemma \\ref{lemma-K-flat-resolution} we can choose a quasi-isomorphism $K^\\bullet \\to M^\\bullet$ such that $K^\\bullet$ is K-flat and consists of flat $A$-modules. In particular, $f$ is a nonzerodivisor on $K^i$ for all $i$ and we have $\\eta_fK^\\bullet$ defined above. In this situation we define $$ L\\eta_f M^\\bullet = \\eta_fK^\\bullet $$ This is independent of the choice of the K-flat resolution by Lemma \\ref{lemma-eta-qis}. We obtain a functor $L\\eta_f : D(A) \\to D(A)$. Beware that this functor isn't exact, i.e., does not tranform distinguished triangles into distinguished triangles."}
+{"_id": "10673", "title": "more-algebra-remark-eta-BZ", "text": "Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a complex of $A$-modules such that $f$ is a nonzerodivisor on all $M^i$. For every $i$ set $\\overline{M}^i = M^i/fM^i$. Denote $B^i \\subset Z^i \\subset \\overline{M}^i$ the boundaries and cocycles for the differentials on the complex $\\overline{M}^\\bullet = M^\\bullet \\otimes_A A/fA$. We claim that there exists a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & B^{i + 1} \\ar[r] \\ar@{=}[d] & B^{i + 1} \\oplus B^i \\ar[r] \\ar[d]^{s, s'} & B^i \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & B^{i + 1} \\ar[r]^-s & (\\eta_fM)^i /f(\\eta_fM)^i \\ar[r]^-t & Z^i \\ar[r] & 0 } $$ with exact rows. Here are the constructions of the maps \\begin{enumerate} \\item If $x \\in (\\eta_fM)^i$ then $x = f^ix'$ with $d^i(x') = 0$ in $\\overline{M}^{i + 1}$. Hence we can define the map $t$ by sending $x$ to the class of $x'$. \\item If $y \\in M^{i + 1}$ has class $\\overline{y}$ in $B^{i + 1} \\subset \\overline{M}^{i + 1}$ then we can write $y = fy' + d^i(x)$ for $y' \\in M^{i + 1}$ and $x \\in M^i$. Hence we can define the map $s$ sending $\\overline{y}$ to the class of $f^{i + 1}x$ in $(\\eta_fM)^i /f(\\eta_fM)^i$; we omit the verification that this is well defined. \\item If $x \\in M^i$ has class $\\overline{x}$ in $B^i \\subset \\overline{M}^i$ then we can write $x = fx' + d^{i - 1}(z)$ for $x' \\in M^i$ and $z \\in M^{i - 1}$. We define the map $s'$ by sending $\\overline{x}$ to the class of $f^i d^{i - 1}(z)$ in $(\\eta_fM)^i/f(\\eta_fM)^i$. This is well defined because if $fx' + d^{i - 1}(z) = 0$, then $f^ix'$ is in $(\\eta_fM)^i$ and consequently $f^id^{i - 1}(z)$ is in $f(\\eta_fM)^i$. \\end{enumerate} We omit the verification that the lower row in the displayed diagram is a short exact sequence of modules. It is immediately clear from these constructions that we have commutative diagrams $$ \\xymatrix{ B^{i + 1} \\oplus B^i \\ar[d]^{s, s'} \\ar[r] & B^{i + 2} \\oplus B^{i + 1} \\ar[d]^{s, s'} \\\\ (\\eta_fM)^i /f(\\eta_fM)^i \\ar[r] & (\\eta_fM)^{i + 1} /f(\\eta_fM)^{i + 1} } $$ where the upper horizontal arrow is given by the identification of the summands $B^{i + 1}$ in source and target. In other words, we have found an acyclic subcomplex of $\\eta_fM^\\bullet / f(\\eta_fM^\\bullet) = \\eta_fM^\\bullet \\otimes_A A/fA$ and the quotient by this subcomplex is a complex whose terms $Z^i/B^i$ are the cohomology modules of the complex $\\overline{M}^\\bullet = M^\\bullet \\otimes_A A/fA$."}
+{"_id": "10674", "title": "more-algebra-remark-weird-systems", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Set $A_n = A/I^n$ for $n \\geq 1$. Consider the following category: \\begin{enumerate} \\item An object is a sequence $\\{E_n\\}_{n \\geq 1}$ where $E_n$ is a finite $A_n$-module. \\item A morphism $\\{E_n\\} \\to \\{E'_n\\}$ is given by maps $$ \\varphi_n : I^cE_n \\longrightarrow E'_n/E'_n[I^c] \\quad\\text{for }n \\geq c $$ where $E'_n[I^c]$ is the torsion submodule (Section \\ref{section-torsion}) up to equivalence: we say $(c, \\varphi_n)$ is the same as $(c + 1, \\overline{\\varphi}_n)$ where $\\overline{\\varphi}_n : I^{c + 1}E_n \\longrightarrow E'_n/E'_n[I^{c + 1}]$ is the induced map. \\end{enumerate} Composition of $(c, \\varphi_n) : \\{E_n\\} \\to \\{E'_n\\}$ and $(c', \\varphi'_n) : \\{E'_n\\} \\to \\{E''_n\\}$ is defined by the obvious compositions $$ I^{c + c'}E_n \\to I^{c'}E'_n/E'_n[I^{c}] \\to E''_n/E''_n[I^{c + c'}] $$ for $n \\geq c + c'$. We omit the verification that this is a category."}
+{"_id": "10675", "title": "more-algebra-remark-awkward", "text": "The awkwardness in the statement of Lemma \\ref{lemma-dejong-kollar-kovacs} is partly due to the fact that there are no obvious maps between the modules $\\Ext^i_{A_n}(M_n, N_n)$ for varying $n$. What we may conclude from the lemma is that there exists a $c \\geq 0$ such that for $m \\gg n \\gg 0$ there are (canonical) maps $$ I^c\\Ext^i_{A_n}(M_m, N_m)/I^n\\Ext^i_{A_n}(M_m, N_m) \\to \\Ext^i_{A_n}(M_n, N_n)/\\Ext^i_{A_n}(M_n, N_n)[I^c] $$ whose kernel and cokernel are annihilated by $I^c$. This is the (weak) sense in which we get a system of modules."}
+{"_id": "10676", "title": "more-algebra-remark-finite-separable-extension", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite separable field extension. Let $B \\subset L$ be the integral closure of $A$ in $L$. Picture: $$ \\xymatrix{ B \\ar[r] & L \\\\ A \\ar[u] \\ar[r] & K \\ar[u] } $$ By Algebra, Lemma \\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension} the ring extension $A \\subset B$ is finite, hence $B$ is Noetherian. By Algebra, Lemma \\ref{algebra-lemma-integral-sub-dim-equal} the dimension of $B$ is $1$, hence $B$ is a Dedekind domain, see Algebra, Lemma \\ref{algebra-lemma-characterize-Dedekind}. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$ be the maximal ideals of $B$ (i.e., the primes lying over $\\mathfrak m_A$). We obtain extensions of discrete valuation rings $$ A \\subset B_{\\mathfrak m_i} $$ and hence ramification indices $e_i$ and residue degrees $f_i$. We have $$ [L : K] = \\sum\\nolimits_{i = 1, \\ldots, n} e_i f_i $$ by Algebra, Lemma \\ref{algebra-lemma-finite-extension-dim-1} applied to a uniformizer in $A$. We observe that $n = 1$ if $A$ is henselian (by Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}), e.g.\\ if $A$ is complete."}
+{"_id": "10677", "title": "more-algebra-remark-tower-of-rings", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension. Let $\\mathfrak m \\subset B$ be a maximal ideal of the integral closure of $A$ in $L$. Let $$ P \\subset I \\subset D \\subset G $$ be the wild inertia, inertia, decomposition group of $\\mathfrak m$. Consider the diagram $$ \\xymatrix{ \\mathfrak m \\ar@{-}[d] \\ar@{-}[r] & \\mathfrak m^P \\ar@{-}[d] \\ar@{-}[r] & \\mathfrak m^I \\ar@{-}[d] \\ar@{-}[r] & \\mathfrak m^D \\ar@{-}[d] \\ar@{-}[r] & A \\cap \\mathfrak m \\ar@{-}[d] \\\\ B & B^P \\ar[l] & B^I \\ar[l] & B^D \\ar[l] & A \\ar[l] } $$ Observe that $B^P, B^I, B^D$ are the integral closures of $A$ in the fields $L^P$, $L^I$, $L^D$. Thus we also see that $B^P$ is the integral closure of $B^I$ in $L^P$ and so on. Observe that $\\mathfrak m^P = \\mathfrak m \\cap B^P$, $\\mathfrak m^I = \\mathfrak m \\cap B^I$, and $\\mathfrak m^D = \\mathfrak m \\cap B^D$. Hence the top line of the diagram corresponds to the images of $\\mathfrak m \\in \\Spec(B)$ under the induced maps of spectra. Having said all of this we have the following \\begin{enumerate} \\item the extension $L^I/L^D$ is Galois with group $D/I$, \\item the extension $L^P/L^I$ is Galois with group $I_t = I/P$, \\item the extension $L^P/L^D$ is Galois with group $D/P$, \\item $\\mathfrak m^I$ is the unique prime of $B^I$ lying over $\\mathfrak m^D$, \\item $\\mathfrak m^P$ is the unique prime of $B^P$ lying over $\\mathfrak m^I$, \\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^P$, \\item $\\mathfrak m^P$ is the unique prime of $B^P$ lying over $\\mathfrak m^D$, \\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^I$, \\item $\\mathfrak m$ is the unique prime of $B$ lying over $\\mathfrak m^D$, \\item $A \\to B^D_{\\mathfrak m^D}$ is \\'etale and induces a trivial residue field extension, \\item $B^D_{\\mathfrak m^D} \\to B^I_{\\mathfrak m^I}$ is \\'etale and induces a Galois extension of residue fields with Galois group $D/I$, \\item $A \\to B^I_{\\mathfrak m^I}$ is \\'etale, \\item $B^I_{\\mathfrak m^I} \\to B^P_{\\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a trivial residue field extension, \\item $B^D_{\\mathfrak m^D} \\to B^P_{\\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a separable residue field extension, \\item $A \\to B^P_{\\mathfrak m^P}$ has ramification index $|I/P|$ prime to $p$ and induces a separable residue field extension. \\end{enumerate} Statements (1), (2), and (3) are immediate from Galois theory (Fields, Section \\ref{fields-section-galois-theory}) and Lemma \\ref{lemma-galois-inertia}. Statements (4) -- (9) are clear from Lemma \\ref{lemma-galois}. Part (12) is Lemma \\ref{lemma-inertial-invariants-unramified}. Since we have the factorization $A \\to B^D_{\\mathfrak m^D} \\to B^I_{\\mathfrak m^I}$ we obtain the \\'etaleness in (10) and (11) as a consequence. The residue field extension in (10) must be trivial because it is separable and $D/I$ maps onto $\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)$ as shown in Lemma \\ref{lemma-galois-galois}. The same argument provides the proof of the statement on residue fields in (11). To see (13), (14), and (15) it suffices to prove (13). By the above, the extension $L^P/L^I$ is Galois with a cyclic Galois group of order prime to $p$, the prime $\\mathfrak m^P$ is the unique prime lying over $\\mathfrak m^I$ and the action of $I/P$ on the residue field is trivial. Thus we can apply Lemma \\ref{lemma-galois-inertia} to this extension and the discrete valuation ring $B^I_{\\mathfrak m^I}$ to see that (13) holds."}
+{"_id": "10678", "title": "more-algebra-remark-canonical-inertia-character", "text": "In order to use the inertia character $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$ for infinite Galois extensions, it is convenient to scale it. Let $A, K, L, B, \\mathfrak m, G, P, I, D, e, \\theta$ be as in Lemma \\ref{lemma-galois-inertia} and Definition \\ref{definition-wild-inertia}. Then $e = q |I_t|$ with $q$ is a power of the characteristic $p$ of $\\kappa(\\mathfrak m)$ if positive or $1$ if zero. Note that $\\mu_e(\\kappa(\\mathfrak m)) = \\mu_{|I_t|}(\\kappa(\\mathfrak m))$ because the characteristic of $\\kappa(\\mathfrak m)$ is $p$. Consider the map $$ \\theta_{can} = q\\theta : I \\longrightarrow \\mu_{|I_t|}(\\kappa(\\mathfrak m)) $$ This map induces an isomorphism $\\theta_{can} : I_t \\to \\mu_{|I_t|}(\\kappa(\\mathfrak m))$. We have $\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$ for $\\tau \\in D$ and $\\sigma \\in I$ by Lemma \\ref{lemma-inertia-character}. Finally, if $M/L$ is an extension such that $M/K$ is Galois and $\\mathfrak m'$ is a prime of the integral closure of $A$ in $M$ lying over $\\mathfrak m$, then we get the commutative diagram $$ \\xymatrix{ I' \\ar[r]_-{\\theta'_{can}} \\ar[d] & \\mu_{|I'_t|}(\\kappa(\\mathfrak m')) \\ar[d]^{(-)^{|I'_t|/|I_t|}} \\\\ I \\ar[r]^-{\\theta_{can}} & \\mu_{|I_t|}(\\kappa(\\mathfrak m)) } $$ by Lemma \\ref{lemma-compare-inertia}."}
+{"_id": "10679", "title": "more-algebra-remark-construction", "text": "Let $A \\to B$ be an extension of discrete valuation rings with fraction fields $K \\subset L$. Let $K \\subset K_1$ be a finite extension of fields. Let $A_1 \\subset K_1$ be the integral closure of $A$ in $K_1$. On the other hand, let $L_1 = (L \\otimes_K K_1)_{red}$. Then $L_1$ is a nonempty finite product of finite field extensions of $L$. Let $B_1$ be the integral closure of $B$ in $L_1$. We obtain compatible commutative diagrams $$ \\vcenter{ \\xymatrix{ L \\ar[r] & L_1 \\\\ K \\ar[u] \\ar[r] & K_1 \\ar[u] } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ B \\ar[r] & B_1 \\\\ A \\ar[u] \\ar[r] & A_1 \\ar[u] } } $$ In this situation we have the following \\begin{enumerate} \\item By Algebra, Lemma \\ref{algebra-lemma-integral-closure-Dedekind} the ring $A_1$ is a Dedekind domain and $B_1$ is a finite product of Dedekind domains. \\item Note that $L \\otimes_K K_1 = (B \\otimes_A A_1)_\\pi$ where $\\pi \\in A$ is a uniformizer and that $\\pi$ is a nonzerodivisor on $B \\otimes_A A_1$.  Thus the ring map $B \\otimes_A A_1 \\to B_1$ is integral with kernel consisting of nilpotent elements. Hence $\\Spec(B_1) \\to \\Spec(B \\otimes_A A_1)$ is surjective on spectra (Algebra, Lemma \\ref{algebra-lemma-integral-overring-surjective}). The map $\\Spec(B \\otimes_A A_1) \\to \\Spec(A_1)$ is surjective as $A_1/\\mathfrak m_A A_1 \\to B/\\mathfrak m_AB \\otimes_{\\kappa_A} A_1/\\mathfrak m_A A_1$ is an injective ring map with $A_1/\\mathfrak m_A A_1$ Artinian. We conclude that $\\Spec(B_1) \\to \\Spec(A_1)$ is surjective. \\item Let $\\mathfrak m_i$, $i = 1, \\ldots n$ with $n \\geq 1$ be the maximal ideals of $A_1$. For each $i = 1, \\ldots, n$ let $\\mathfrak m_{ij}$, $j = 1, \\ldots, m_i$ with $m_i \\geq 1$ be the maximal ideals of $B_1$ lying over $\\mathfrak m_i$. We obtain diagrams $$ \\xymatrix{ B \\ar[r] & (B_1)_{\\mathfrak m_{ij}} \\\\ A \\ar[u] \\ar[r] & (A_1)_{\\mathfrak m_i} \\ar[u] } $$ of extensions of discrete valuation rings. \\item If $A$ is henselian (for example complete), then $A_1$ is a discrete valuation ring, i.e., $n = 1$. Namely, $A_1$ is a union of finite extensions of $A$ which are domains, hence local by Algebra, Lemma \\ref{algebra-lemma-finite-over-henselian}. \\item If $B$ is henselian (for example complete), then $B_1$ is a product of discrete valuation rings, i.e., $m_i = 1$ for $i = 1, \\ldots, n$. \\item If $K \\subset K_1$ is purely inseparable, then $A_1$ and $B_1$ are both discrete valuation rings, i.e., $n = 1$ and $m_1 = 1$. This is true because for every $b \\in B_1$ a $p$-power power of $b$ is in $B$, hence $B_1$ can only have one maximal ideal. \\item If $K \\subset K_1$ is finite separable, then $L_1 = L \\otimes_K K_1$ and is a finite product of finite separable extensions too. Hence $A \\subset A_1$ and $B \\subset B_1$ are finite by Algebra, Lemma \\ref{algebra-lemma-Noetherian-normal-domain-finite-separable-extension}. \\item If $A$ is Nagata, then $A \\subset A_1$ is finite. \\item If $B$ is Nagata, then $B \\subset B_1$ is finite. \\end{enumerate}"}
+{"_id": "10680", "title": "more-algebra-remark-determinant-as-socle", "text": "Let $R$ be a ring. Let $M$ be a finite projective $R$-module. Then we can consider the graded commutative $R$-algebra exterior algebra $\\wedge^*_R(M)$ on $M$ over $R$. A formula for $\\det(M)$ is that $\\det(M) \\subset \\wedge^*_R(M)$ is the annihilator of $M \\subset \\wedge^*_R(M)$. This is sometimes useful as it does not refer to the decomposition of $R$ into a product. Of course, to prove this satisfies the desired properties one has to either decompose $R$ into a product (as above), or one has to look at the localizations at primes of $R$."}
+{"_id": "10742", "title": "etale-remark-technicality-needed", "text": "The result on ``reducedness'' does not hold with a weaker definition of \\'etale local ring maps $A \\to B$ where one drops the assumption that $B$ is essentially of finite type over $A$. Namely, it can happen that a Noetherian local domain $A$ has nonreduced completion $A^\\wedge$, see Examples, Section \\ref{examples-section-local-completion-nonreduced}. But the ring map $A \\to A^\\wedge$ is flat, and $\\mathfrak m_AA^\\wedge$ is the maximal ideal of $A^\\wedge$ and of course $A$ and $A^\\wedge$ have the same residue fields. This is why it is important to consider this notion only for ring extensions which are essentially of finite type (or essentially of finite presentation if $A$ is not Noetherian)."}
+{"_id": "10811", "title": "crystalline-remark-completed-affine-site", "text": "In Situation \\ref{situation-affine} we denote $\\text{Cris}^\\wedge(C/A)$ the category whose objects are pairs $(B \\to C, \\delta)$ such that \\begin{enumerate} \\item $B$ is a $p$-adically complete $A$-algebra, \\item $B \\to C$ is a surjection of $A$-algebras, \\item $\\delta$ is a divided power structure on $\\Ker(B \\to C)$, \\item $A \\to B$ is a homomorphism of divided power rings. \\end{enumerate} Morphisms are defined as in Definition \\ref{definition-affine-thickening}. Then $\\text{Cris}(C/A) \\subset \\text{Cris}^\\wedge(C/A)$ is the full subcategory consisting of those $B$ such that $p$ is nilpotent in $B$. Conversely, any object $(B \\to C, \\delta)$ of $\\text{Cris}^\\wedge(C/A)$ is equal to the limit $$ (B \\to C, \\delta) = \\lim_e (B/p^eB \\to C, \\delta) $$ where for $e \\gg 0$ the object $(B/p^eB \\to C, \\delta)$ lies in $\\text{Cris}(C/A)$, see Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}. In particular, we see that $\\text{Cris}^\\wedge(C/A)$ is a full subcategory of the category of pro-objects of $\\text{Cris}(C/A)$, see Categories, Remark \\ref{categories-remark-pro-category}."}
+{"_id": "10812", "title": "crystalline-remark-filtration-differentials", "text": "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power structure on $B$. The universal module $\\Omega_{B/A, \\delta}$ comes with a little bit of extra structure, namely the $B$-submodule $N$ of $\\Omega_{B/A, \\delta}$ generated by $\\text{d}_{B/A, \\delta}(J)$. In terms of the isomorphism given in Lemma \\ref{lemma-diagonal-and-differentials} this corresponds to the image of $K \\cap J(1)$ in $\\Omega_{B/A, \\delta}$. Consider the $A$-algebra $D = B \\oplus \\Omega^1_{B/A, \\delta}$ with ideal $\\bar J = J \\oplus N$ and divided powers $\\bar \\delta$ as in the proof of the lemma. Then $(D, \\bar J, \\bar \\delta)$ is a divided power ring and the two maps $B \\to D$ given by $b \\mapsto b$ and $b \\mapsto b + \\text{d}_{B/A, \\delta}(b)$ are homomorphisms of divided power rings over $A$. Moreover, $N$ is the smallest submodule of $\\Omega_{B/A, \\delta}$ such that this is true."}
+{"_id": "10813", "title": "crystalline-remark-divided-powers-de-rham-complex", "text": "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power structure on $B$. Set $\\Omega_{B/A, \\delta}^i = \\wedge^i_B \\Omega_{B/A, \\delta}$ where $\\Omega_{B/A, \\delta}$ is the target of the universal divided power $A$-derivation $\\text{d} = \\text{d}_{B/A} : B \\to \\Omega_{B/A, \\delta}$. Note that $\\Omega_{B/A, \\delta}$ is the quotient of $\\Omega_{B/A}$ by the $B$-submodule generated by the elements $\\text{d}\\delta_n(x) - \\delta_{n - 1}(x)\\text{d}x$ for $x \\in J$. We claim Algebra, Lemma \\ref{algebra-lemma-de-rham-complex} applies. To see this it suffices to verify the elements $\\text{d}\\delta_n(x) - \\delta_{n - 1}(x)\\text{d}x$ of $\\Omega_B$ are mapped to zero in $\\Omega^2_{B/A, \\delta}$. We observe that $$ \\text{d}(\\delta_{n - 1}(x)) \\wedge \\text{d}x = \\delta_{n - 2}(x) \\text{d}x \\wedge \\text{d}x = 0 $$ in $\\Omega^2_{B/A, \\delta}$ as desired. Hence we obtain a {\\it divided power de Rham complex} $$ \\Omega^0_{B/A, \\delta} \\to \\Omega^1_{B/A, \\delta} \\to \\Omega^2_{B/A, \\delta} \\to \\ldots $$ which will play an important role in the sequel."}
+{"_id": "10814", "title": "crystalline-remark-connection", "text": "Let $A \\to B$ be a ring map. Let $\\Omega_{B/A} \\to \\Omega$ be a quotient satisfying the assumptions of Algebra, Lemma \\ref{algebra-lemma-de-rham-complex}. Let $M$ be a $B$-module. A {\\it connection} is an additive map $$ \\nabla : M \\longrightarrow M \\otimes_B \\Omega $$ such that $\\nabla(bm) = b \\nabla(m) + m \\otimes \\text{d}b$ for $b \\in B$ and $m \\in M$. In this situation we can define maps $$ \\nabla : M \\otimes_B \\Omega^i \\longrightarrow M \\otimes_B \\Omega^{i + 1} $$ by the rule $\\nabla(m \\otimes \\omega) = \\nabla(m) \\wedge \\omega + m \\otimes \\text{d}\\omega$. This works because if $b \\in B$, then \\begin{align*} \\nabla(bm \\otimes \\omega) - \\nabla(m \\otimes b\\omega) & = \\nabla(bm) \\wedge \\omega + bm \\otimes \\text{d}\\omega - \\nabla(m) \\wedge b\\omega - m \\otimes \\text{d}(b\\omega) \\\\ & = b\\nabla(m) \\wedge \\omega + m \\otimes \\text{d}b \\wedge \\omega + bm \\otimes \\text{d}\\omega \\\\ & \\ \\ \\ \\ \\ \\ - b\\nabla(m) \\wedge \\omega - bm \\otimes \\text{d}(\\omega) - m \\otimes \\text{d}b \\wedge \\omega = 0 \\end{align*} As is customary we say the connection is {\\it integrable} if and only if the composition $$ M \\xrightarrow{\\nabla} M \\otimes_B \\Omega^1 \\xrightarrow{\\nabla} M \\otimes_B \\Omega^2 $$ is zero. In this case we obtain a complex $$ M \\xrightarrow{\\nabla} M \\otimes_B \\Omega^1 \\xrightarrow{\\nabla} M \\otimes_B \\Omega^2 \\xrightarrow{\\nabla} M \\otimes_B \\Omega^3 \\xrightarrow{\\nabla} M \\otimes_B \\Omega^4 \\to \\ldots $$ which is called the de Rham complex of the connection."}
+{"_id": "10815", "title": "crystalline-remark-base-change-connection", "text": "Consider a commutative diagram of rings $$ \\xymatrix{ B \\ar[r]_\\varphi & B' \\\\ A \\ar[u] \\ar[r] & A' \\ar[u] } $$ Let $\\Omega_{B/A} \\to \\Omega$ and $\\Omega_{B'/A'} \\to \\Omega'$ be quotients satisfying the assumptions of Algebra, Lemma \\ref{algebra-lemma-de-rham-complex}. Assume there is a map $\\varphi : \\Omega \\to \\Omega'$ which fits into a commutative diagram $$ \\xymatrix{ \\Omega_{B/A} \\ar[r] \\ar[d] & \\Omega_{B'/A'} \\ar[d] \\\\ \\Omega \\ar[r]^{\\varphi} & \\Omega' } $$ where the top horizontal arrow is the canonical map $\\Omega_{B/A} \\to \\Omega_{B'/A'}$ induced by $\\varphi : B \\to B'$. In this situation, given any pair $(M, \\nabla)$ where $M$ is a $B$-module and $\\nabla : M \\to M \\otimes_B \\Omega$ is a connection we obtain a {\\it base change} $(M \\otimes_B B', \\nabla')$ where $$ \\nabla' : M \\otimes_B B' \\longrightarrow (M \\otimes_B B') \\otimes_{B'} \\Omega' = M \\otimes_B \\Omega' $$ is defined by the rule $$ \\nabla'(m \\otimes b') = \\sum m_i \\otimes b'\\text{d}\\varphi(b_i) + m \\otimes \\text{d}b'  $$ if $\\nabla(m) = \\sum m_i \\otimes \\text{d}b_i$. If $\\nabla$ is integrable, then so is $\\nabla'$, and in this case there is a canonical map of de Rham complexes (Remark \\ref{remark-connection}) \\begin{equation} \\label{equation-base-change-map-complexes} M \\otimes_B \\Omega^\\bullet \\longrightarrow (M \\otimes_B B') \\otimes_{B'} (\\Omega')^\\bullet = M \\otimes_B (\\Omega')^\\bullet \\end{equation} which maps $m \\otimes \\eta$ to $m \\otimes \\varphi(\\eta)$."}
+{"_id": "10816", "title": "crystalline-remark-functoriality-big-cris", "text": "Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ be a morphism of divided power schemes over $\\mathbf{Z}_{(p)}$. Set $S_0 = V(\\mathcal{I})$ and $S'_0 = V(\\mathcal{I}')$. Let $$ \\xymatrix{ X \\ar[r]_f \\ar[d] & Y \\ar[d] \\\\ S_0 \\ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. Then we get a continuous and cocontinuous functor $$ \\text{CRIS}(X/S) \\longrightarrow \\text{CRIS}(Y/S') $$ by letting $(U, T, \\delta)$ correspond to $(U, T, \\delta)$ with $U \\to X \\to Y$ as the $S'$-morphism from $U$ to $Y$. Hence we get a morphism of topoi $$ f_{\\text{CRIS}} : (X/S)_{\\text{CRIS}} \\longrightarrow (Y/S')_{\\text{CRIS}} $$ see Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}."}
+{"_id": "10817", "title": "crystalline-remark-compare-big-zariski", "text": "In Situation \\ref{situation-global}. The functor (\\ref{equation-forget}) is cocontinuous (details omitted) and commutes with products and fibred products (Lemma \\ref{lemma-divided-power-thickening-fibre-products}). Hence we obtain a morphism of topoi $$ U_{X/S} : (X/S)_{\\text{CRIS}} \\longrightarrow \\Sh((\\Sch/X)_{Zar}) $$ from the big crystalline topos of $X/S$ to the big Zariski topos of $X$. See Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}."}
+{"_id": "10818", "title": "crystalline-remark-big-structure-morphism", "text": "In Situation \\ref{situation-global}. Consider the closed subscheme $S_0 = V(\\mathcal{I}) \\subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition \\ref{definition-divided-power-thickening-X} with $S_0$ instead of $X$. Hence we get a site $\\text{CRIS}(S_0/S)$. If $f : X \\to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of morphisms of ringed topoi $$ \\xymatrix{ (X/S)_{\\text{CRIS}} \\ar[r]_{f_{\\text{CRIS}}} \\ar[d]_{U_{X/S}} & (S_0/S)_{\\text{CRIS}} \\ar[d]^{U_{S_0/S}} \\\\ \\Sh((\\Sch/X)_{Zar}) \\ar[r]^{f_{big}} & \\Sh((\\Sch/S_0)_{Zar}) \\ar[rd] \\\\ & & \\Sh((\\Sch/S)_{Zar}) } $$ by Remark \\ref{remark-functoriality-big-cris}. We think of the composition $(X/S)_{\\text{CRIS}} \\to \\Sh((\\Sch/S)_{Zar})$ as the structure morphism of the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism $$ (X/S)_{\\text{CRIS}} \\longrightarrow \\Sh((\\Sch/S)_{Zar}) $$ is defined as we can take the lower route through the diagram above. Thus it is the morphism of topoi corresponding to the cocontinuous functor $\\text{CRIS}(X/S) \\to (\\Sch/S)_{Zar}$ given by the rule $(U, T, \\delta)/S \\mapsto U/S$, see Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}."}
+{"_id": "10819", "title": "crystalline-remark-compatibilities-big-cris", "text": "The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark \\ref{remark-functoriality-big-cris} we obtain a commutative diagram of ringed topoi $$ \\xymatrix{ (X/S)_{\\text{CRIS}} \\ar[d] \\ar[r] & (Y/S')_{\\text{CRIS}} \\ar[d] \\\\ \\Sh((\\Sch/S)_{Zar}) \\ar[r] & \\Sh((\\Sch/S')_{Zar}) } $$ where the vertical arrows are the structure morphisms."}
+{"_id": "10820", "title": "crystalline-remark-functoriality-cris", "text": "Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ be a morphism of divided power schemes over $\\mathbf{Z}_{(p)}$. Let $$ \\xymatrix{ X \\ar[r]_f \\ar[d] & Y \\ar[d] \\\\ S_0 \\ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. By analogy with Topologies, Lemma \\ref{topologies-lemma-morphism-big-small} we define $$ f_{\\text{cris}} : (X/S)_{\\text{cris}} \\longrightarrow (Y/S')_{\\text{cris}} $$ by the formula $f_{\\text{cris}} = \\pi_Y \\circ f_{\\text{CRIS}} \\circ i_X$ where $i_X$ and $\\pi_Y$ are as in Lemma \\ref{lemma-compare-big-small} for $X$ and $Y$ and where $f_{\\text{CRIS}}$ is as in Remark \\ref{remark-functoriality-big-cris}."}
+{"_id": "10821", "title": "crystalline-remark-compare-zariski", "text": "In Situation \\ref{situation-global}. The functor (\\ref{equation-forget-small}) is continuous, cocontinuous, and commutes with products and fibred products. Hence we obtain a morphism of topoi $$ u_{X/S} : (X/S)_{\\text{cris}} \\longrightarrow \\Sh(X_{Zar}) $$ relating the small crystalline topos of $X/S$ with the small Zariski topos of $X$. See Sites, Section \\ref{sites-section-cocontinuous-morphism-topoi}."}
+{"_id": "10822", "title": "crystalline-remark-structure-morphism", "text": "In Situation \\ref{situation-global}. Consider the closed subscheme $S_0 = V(\\mathcal{I}) \\subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition \\ref{definition-divided-power-thickening-X} with $S_0$ instead of $X$. Hence we get a site $\\text{Cris}(S_0/S)$. If $f : X \\to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of ringed topoi $$ \\xymatrix{ (X/S)_{\\text{cris}} \\ar[r]_{f_{\\text{cris}}} \\ar[d]_{u_{X/S}} & (S_0/S)_{\\text{cris}} \\ar[d]^{u_{S_0/S}} \\\\ \\Sh(X_{Zar}) \\ar[r]^{f_{small}} & \\Sh(S_{0, Zar}) \\ar[rd] \\\\ & & \\Sh(S_{Zar}) } $$ see Remark \\ref{remark-functoriality-cris}. We think of the composition $(X/S)_{\\text{cris}} \\to \\Sh(S_{Zar})$ as the structure morphism of the crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism $$ \\tau_{X/S} : (X/S)_{\\text{cris}} \\longrightarrow \\Sh(S_{Zar}) $$ is defined as we can take the lower route through the diagram above."}
+{"_id": "10823", "title": "crystalline-remark-compatibilities", "text": "The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark \\ref{remark-functoriality-cris} we obtain a commutative diagram of ringed topoi $$ \\xymatrix{ (X/S)_{\\text{cris}} \\ar[d] \\ar[r] & (Y/S')_{\\text{cris}} \\ar[d] \\\\ \\Sh((\\Sch/S)_{Zar}) \\ar[r] & \\Sh((\\Sch/S')_{Zar}) } $$ where the vertical arrows are the structure morphisms."}
+{"_id": "10824", "title": "crystalline-remark-crystal", "text": "To formulate the general notion of a crystal we use the language of stacks and strongly cartesian morphisms, see Stacks, Definition \\ref{stacks-definition-stack} and Categories, Definition \\ref{categories-definition-cartesian-over-C}. In Situation \\ref{situation-global} let $p : \\mathcal{C} \\to \\text{Cris}(X/S)$ be a stack. A {\\it crystal in objects of $\\mathcal{C}$ on $X$ relative to $S$} is a {\\it cartesian section} $\\sigma : \\text{Cris}(X/S) \\to \\mathcal{C}$, i.e., a functor $\\sigma$ such that $p \\circ \\sigma = \\text{id}$ and such that $\\sigma(f)$ is strongly cartesian for all morphisms $f$ of $\\text{Cris}(X/S)$. Similarly for the big crystalline site."}
+{"_id": "10825", "title": "crystalline-remark-first-order-thickening", "text": "In Situation \\ref{situation-global}. Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$. Write $\\Omega_{T/S, \\delta} = (\\Omega_{X/S})_T$, see Lemma \\ref{lemma-module-of-differentials}. We explicitly describe a first order thickening $T'$ of $T$. Namely, set $$ \\mathcal{O}_{T'} = \\mathcal{O}_T \\oplus \\Omega_{T/S, \\delta} $$ with algebra structure such that $\\Omega_{T/S, \\delta}$ is an ideal of square zero. Let $\\mathcal{J} \\subset \\mathcal{O}_T$ be the ideal sheaf of the closed immersion $U \\to T$. Set $\\mathcal{J}' = \\mathcal{J} \\oplus \\Omega_{T/S, \\delta}$. Define a divided power structure on $\\mathcal{J}'$ by setting $$ \\delta_n'(f, \\omega) = (\\delta_n(f), \\delta_{n - 1}(f)\\omega), $$ see Lemma \\ref{lemma-divided-power-first-order-thickening}. There are two ring maps $$ p_0, p_1 : \\mathcal{O}_T \\to \\mathcal{O}_{T'} $$ The first is given by $f \\mapsto (f, 0)$ and the second by $f \\mapsto (f, \\text{d}_{T/S, \\delta}f)$. Note that both are compatible with the divided power structures on $\\mathcal{J}$ and $\\mathcal{J}'$ and so is the quotient map $\\mathcal{O}_{T'} \\to \\mathcal{O}_T$. Thus we get an object $(U, T', \\delta')$ of $\\text{Cris}(X/S)$ and a commutative diagram $$ \\xymatrix{ & T \\ar[ld]_{\\text{id}} \\ar[d]^i \\ar[rd]^{\\text{id}} \\\\ T & T' \\ar[l]_{p_0} \\ar[r]^{p_1} & T } $$ of $\\text{Cris}(X/S)$ such that $i$ is a first order thickening whose ideal sheaf is identified with $\\Omega_{T/S, \\delta}$ and such that $p_1^* - p_0^* : \\mathcal{O}_T \\to \\mathcal{O}_{T'}$ is identified with the universal derivation $\\text{d}_{T/S, \\delta}$ composed with the inclusion $\\Omega_{T/S, \\delta} \\to \\mathcal{O}_{T'}$."}
+{"_id": "10826", "title": "crystalline-remark-second-order-thickening", "text": "In Situation \\ref{situation-global}. Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$. Write $\\Omega_{T/S, \\delta} = (\\Omega_{X/S})_T$, see Lemma \\ref{lemma-module-of-differentials}. We also write $\\Omega^2_{T/S, \\delta}$ for its second exterior power. We explicitly describe a second order thickening $T''$ of $T$. Namely, set $$ \\mathcal{O}_{T''} = \\mathcal{O}_T \\oplus \\Omega_{T/S, \\delta} \\oplus \\Omega_{T/S, \\delta} \\oplus \\Omega^2_{T/S, \\delta} $$ with algebra structure defined in the following way $$ (f, \\omega_1, \\omega_2, \\eta) \\cdot (f', \\omega_1', \\omega_2', \\eta') = (ff', f\\omega_1' + f'\\omega_1, f\\omega_2' + f'\\omega_2, f\\eta' + f'\\eta + \\omega_1 \\wedge \\omega_2' + \\omega_1' \\wedge \\omega_2). $$ Let $\\mathcal{J} \\subset \\mathcal{O}_T$ be the ideal sheaf of the closed immersion $U \\to T$. Let $\\mathcal{J}''$ be the inverse image of $\\mathcal{J}$ under the projection $\\mathcal{O}_{T''} \\to \\mathcal{O}_T$. Define a divided power structure on $\\mathcal{J}''$ by setting $$ \\delta_n''(f, \\omega_1, \\omega_2, \\eta) = (\\delta_n(f), \\delta_{n - 1}(f)\\omega_1, \\delta_{n - 1}(f)\\omega_2, \\delta_{n - 1}(f)\\eta + \\delta_{n - 2}(f)\\omega_1 \\wedge \\omega_2) $$ see Lemma \\ref{lemma-divided-power-second-order-thickening}. There are three ring maps $q_0, q_1, q_2 : \\mathcal{O}_T \\to \\mathcal{O}_{T''}$ given by \\begin{align*} q_0(f) & = (f, 0, 0, 0), \\\\ q_1(f) & = (f, \\text{d}f, 0, 0), \\\\ q_2(f) & = (f, \\text{d}f, \\text{d}f, 0) \\end{align*} where $\\text{d} = \\text{d}_{T/S, \\delta}$. Note that all three are compatible with the divided power structures on $\\mathcal{J}$ and $\\mathcal{J}''$. There are three ring maps $q_{01}, q_{12}, q_{02} : \\mathcal{O}_{T'} \\to \\mathcal{O}_{T''}$ where $\\mathcal{O}_{T'}$ is as in Remark \\ref{remark-first-order-thickening}. Namely, set \\begin{align*} q_{01}(f, \\omega) & = (f, \\omega, 0, 0), \\\\ q_{12}(f, \\omega) & = (f, \\text{d}f, \\omega, \\text{d}\\omega), \\\\ q_{02}(f, \\omega) & = (f, \\omega, \\omega, 0) \\end{align*} These are also compatible with the given divided power structures. Let's do the verifications for $q_{12}$: Note that $q_{12}$ is a ring homomorphism as \\begin{align*} q_{12}(f, \\omega)q_{12}(g, \\eta) & = (f, \\text{d}f, \\omega, \\text{d}\\omega)(g, \\text{d}g, \\eta, \\text{d}\\eta) \\\\ & = (fg, f\\text{d}g + g \\text{d}f, f\\eta + g\\omega, f\\text{d}\\eta + g\\text{d}\\omega + \\text{d}f \\wedge \\eta + \\text{d}g \\wedge \\omega) \\\\ & = q_{12}(fg, f\\eta + g\\omega) = q_{12}((f, \\omega)(g, \\eta)) \\end{align*} Note that $q_{12}$ is compatible with divided powers because \\begin{align*} \\delta_n''(q_{12}(f, \\omega)) & = \\delta_n''((f, \\text{d}f, \\omega, \\text{d}\\omega)) \\\\ & = (\\delta_n(f), \\delta_{n - 1}(f)\\text{d}f, \\delta_{n - 1}(f)\\omega, \\delta_{n - 1}(f)\\text{d}\\omega + \\delta_{n - 2}(f)\\text{d}(f) \\wedge \\omega) \\\\ & = q_{12}((\\delta_n(f), \\delta_{n - 1}(f)\\omega)) = q_{12}(\\delta'_n(f, \\omega)) \\end{align*} The verifications for $q_{01}$ and $q_{02}$ are easier. Note that $q_0 = q_{01} \\circ p_0$, $q_1 = q_{01} \\circ p_1$, $q_1 = q_{12} \\circ p_0$, $q_2 = q_{12} \\circ p_1$, $q_0 = q_{02} \\circ p_0$, and $q_2 = q_{02} \\circ p_1$. Thus $(U, T'', \\delta'')$ is an object of $\\text{Cris}(X/S)$ and we get morphisms $$ \\xymatrix{ T'' \\ar@<2ex>[r] \\ar@<0ex>[r] \\ar@<-2ex>[r] & T' \\ar@<1ex>[r] \\ar@<-1ex>[r] & T } $$ of $\\text{Cris}(X/S)$ satisfying the relations described above. In applications we will use $q_i : T'' \\to T$ and $q_{ij} : T'' \\to T'$ to denote the morphisms associated to the ring maps described above."}
+{"_id": "10827", "title": "crystalline-remark-equivalence-more-general", "text": "The equivalence of Proposition \\ref{proposition-crystals-on-affine} holds if we start with a surjection $P \\to C$ where $P/A$ satisfies the strong lifting property of Algebra, Lemma \\ref{algebra-lemma-smooth-strong-lift}. To prove this we can argue as in the proof of Lemma \\ref{lemma-crystals-on-affine-smooth}. (Details will be added here if we ever need this.) Presumably there is also a direct proof of this result, but the advantage of using polynomial rings is that the rings $D(n)$ are $p$-adic completions of divided power polynomial rings and the algebra is simplified."}
+{"_id": "10828", "title": "crystalline-remark-vanishing", "text": "The proof of Proposition \\ref{proposition-compare-with-de-Rham} shows that the conclusion $$ Ru_{X/S, *}(\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0 $$ for $i > 0$ is true for any $\\mathcal{O}_{X/S}$-module $\\mathcal{F}$ which satisfies conditions (1) and (2) of Proposition \\ref{proposition-compute-cohomology}. This applies to the following non-crystals: $\\Omega^i_{X/S}$ for all $i$, and any sheaf of the form $\\underline{\\mathcal{F}}$, where $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module. In particular, it applies to the sheaf $\\underline{\\mathcal{O}_X} = \\underline{\\mathbf{G}_a}$. But note that we need something like Lemma \\ref{lemma-automatic-connection} to produce a de Rham complex which requires $\\mathcal{F}$ to be a crystal. Hence (currently) the collection of sheaves of modules for which the full statement of Proposition \\ref{proposition-compare-with-de-Rham} holds is exactly the category of crystals in quasi-coherent modules."}
+{"_id": "10829", "title": "crystalline-remark-compute-direct-image", "text": "Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ be a morphism of divided power schemes over $\\mathbf{Z}_{(p)}$. Let $$ \\xymatrix{ X \\ar[r]_f \\ar[d] & X' \\ar[d] \\\\ S_0 \\ar[r] & S'_0 } $$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\\mathcal{F}$ be an $\\mathcal{O}_{X/S}$-module on $\\text{Cris}(X/S)$. Then $Rf_{\\text{cris}, *}\\mathcal{F}$ can be computed as follows. \\medskip\\noindent Given an object $(U', T', \\delta')$ of $\\text{Cris}(X'/S')$ set $U = X \\times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \\delta)$ the divided power scheme over $S$ such that $$ \\xymatrix{ T \\ar[r] \\ar[d] & T' \\ar[d] \\\\ S \\ar[r] & S' } $$ is cartesian in the category of divided power schemes, see Lemma \\ref{lemma-fibre-product}. There is an induced morphism $U \\to T_0$ and we obtain a morphism $(U/T)_{\\text{cris}} \\to (X/S)_{\\text{cris}}$, see Remark \\ref{remark-functoriality-cris}. Let $\\mathcal{F}_U$ be the pullback of $\\mathcal{F}$. Let $\\tau_{U/T} : (U/T)_{\\text{cris}} \\to T_{Zar}$ be the structure morphism. Then we have \\begin{equation} \\label{equation-identify-pushforward} \\left(Rf_{\\text{cris}, *}\\mathcal{F}\\right)_{T'} = R(T \\to T')_*\\left(R\\tau_{U/T, *} \\mathcal{F}_U \\right) \\end{equation} where the left hand side is the restriction (see Section \\ref{section-sheaves}). \\medskip\\noindent Hints: First, show that $\\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma \\ref{sites-lemma-localize-topos-site}) of $\\text{Cris}(X/S)$ at the sheaf of sets $f_{\\text{cris}}^{-1}h_{(U', T', \\delta')}$. Next, reduce the statement to the case where $\\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\\mathcal{O}_{X/S}$-module is an injective $\\mathcal{O}_{U/T}$-module on $\\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward."}
+{"_id": "10830", "title": "crystalline-remark-mayer-vietoris", "text": "In the situation of Remark \\ref{remark-compute-direct-image} suppose we have an open covering $X = X' \\cup X''$. Denote $X''' = X' \\cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$ to $X'$, $X''$, and $X'''$. Moreover, let $\\mathcal{F}'$, $\\mathcal{F}''$, and $\\mathcal{F}'''$ be the restriction of $\\mathcal{F}$ to the crystalline sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle $$ Rf_{\\text{cris}, *}\\mathcal{F} \\longrightarrow Rf'_{\\text{cris}, *}\\mathcal{F}' \\oplus Rf''_{\\text{cris}, *}\\mathcal{F}'' \\longrightarrow Rf'''_{\\text{cris}, *}\\mathcal{F}''' \\longrightarrow Rf_{\\text{cris}, *}\\mathcal{F}[1] $$ in $D(\\mathcal{O}_{X'/S'})$. \\medskip\\noindent Hints: This is a formal consequence of the fact that the subcategories $\\text{Cris}(X'/S)$, $\\text{Cris}(X''/S)$, $\\text{Cris}(X'''/S)$ correspond to open subobjects of the final sheaf on $\\text{Cris}(X/S)$ and that the last is the intersection of the first two."}
+{"_id": "10831", "title": "crystalline-remark-cech-complex", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $A$ a $\\mathbf{Z}_{(p)}$-algebra. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a separated\\footnote{This assumption is not strictly necessary, as using hypercoverings the construction of the remark can be extended to the general case.} scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules. \\medskip\\noindent Choose an affine open covering $X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$ of $X$. Write $U_\\lambda = \\Spec(C_\\lambda)$. Choose a polynomial algebra $P_\\lambda$ over $A$ and a surjection $P_\\lambda \\to C_\\lambda$. Having fixed these choices we can construct a {\\v C}ech complex which computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$. \\medskip\\noindent Given $n \\geq 0$ and $\\lambda_0, \\ldots, \\lambda_n \\in \\Lambda$ write $U_{\\lambda_0 \\ldots \\lambda_n} = U_{\\lambda_0} \\cap \\ldots \\cap U_{\\lambda_n}$. This is an affine scheme by assumption. Write $U_{\\lambda_0 \\ldots \\lambda_n} = \\Spec(C_{\\lambda_0 \\ldots \\lambda_n})$. Set $$ P_{\\lambda_0 \\ldots \\lambda_n} = P_{\\lambda_0} \\otimes_A \\ldots \\otimes_A P_{\\lambda_n} $$ which comes with a canonical surjection onto $C_{\\lambda_0 \\ldots \\lambda_n}$. Denote the kernel $J_{\\lambda_0 \\ldots \\lambda_n}$ and set $D_{\\lambda_0 \\ldots \\lambda_n}$ the $p$-adically completed divided power envelope of $J_{\\lambda_0 \\ldots \\lambda_n}$ in $P_{\\lambda_0 \\ldots \\lambda_n}$ relative to $\\gamma$. Let $M_{\\lambda_0 \\ldots \\lambda_n}$ be the $P_{\\lambda_0 \\ldots \\lambda_n}$-module corresponding to the restriction of $\\mathcal{F}$ to $\\text{Cris}(U_{\\lambda_0 \\ldots \\lambda_n}/S)$ via Proposition \\ref{proposition-crystals-on-affine}. By construction we obtain a cosimplicial divided power ring $D(*)$ having in degree $n$ the ring $$ D(n) = \\prod\\nolimits_{\\lambda_0 \\ldots \\lambda_n} D_{\\lambda_0 \\ldots \\lambda_n} $$ (use that divided power envelopes are functorial and the trivial cosimplicial structure on the ring $P(*)$ defined similarly). Since $M_{\\lambda_0 \\ldots \\lambda_n}$ is the ``value'' of $\\mathcal{F}$ on the objects $\\Spec(D_{\\lambda_0 \\ldots \\lambda_n})$ we see that $M(*)$ defined by the rule $$ M(n) = \\prod\\nolimits_{\\lambda_0 \\ldots \\lambda_n} M_{\\lambda_0 \\ldots \\lambda_n} $$ forms a cosimplicial $D(*)$-module. Now we claim that we have $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = s(M(*)) $$ Here $s(-)$ denotes the cochain complex associated to a cosimplicial module (see Simplicial, Section \\ref{simplicial-section-dold-kan-cosimplicial}). \\medskip\\noindent Hints: The proof of this is similar to the proof of Proposition \\ref{proposition-compute-cohomology} (in particular the result holds for any module satisfying the assumptions of that proposition)."}
+{"_id": "10832", "title": "crystalline-remark-alternating-cech-complex", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $A$ a $\\mathbf{Z}_{(p)}$-algebra. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a separated quasi-compact scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules. \\medskip\\noindent Choose a finite affine open covering $X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$ of $X$ and a total ordering on $\\Lambda$. Write $U_\\lambda = \\Spec(C_\\lambda)$. Choose a polynomial algebra $P_\\lambda$ over $A$ and a surjection $P_\\lambda \\to C_\\lambda$. Having fixed these choices we can construct an alternating {\\v C}ech complex which computes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$. \\medskip\\noindent We are going to use the notation introduced in Remark \\ref{remark-cech-complex}. Denote $\\Omega_{\\lambda_0 \\ldots \\lambda_n}$ the $p$-adically completed module of differentials of $D_{\\lambda_0 \\ldots \\lambda_n}$ over $A$ compatible with the divided power structure. Let $\\nabla$ be the integrable connection on $M_{\\lambda_0 \\ldots \\lambda_n}$ coming from Proposition \\ref{proposition-crystals-on-affine}. Consider the double complex $M^{\\bullet, \\bullet}$ with terms $$ M^{n, m} = \\bigoplus\\nolimits_{\\lambda_0 < \\ldots < \\lambda_n} M_{\\lambda_0 \\ldots \\lambda_n} \\otimes^\\wedge_{D_{\\lambda_0 \\ldots \\lambda_n}} \\Omega^m_{D_{\\lambda_0 \\ldots \\lambda_n}}. $$ For the differential $d_1$ (increasing $n$) we use the usual {\\v C}ech differential and for the differential $d_2$ we use the connection, i.e., the differential of the de Rham complex. We claim that $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = \\text{Tot}(M^{\\bullet, \\bullet}) $$ Here $\\text{Tot}(-)$ denotes the total complex associated to a double complex, see Homology, Definition \\ref{homology-definition-associated-simple-complex}. \\medskip\\noindent Hints: We have $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\Gamma(\\text{Cris}(X/S), \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega_{X/S}^\\bullet) $$ by Proposition \\ref{proposition-compare-with-de-Rham}. The right hand side of the formula is simply the alternating {\\v C}ech complex for the covering $X = \\bigcup_{\\lambda \\in \\Lambda} U_\\lambda$ (which induces an open covering of the final sheaf of $\\text{Cris}(X/S)$) and the complex $\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega_{X/S}^\\bullet$, see Proposition \\ref{proposition-compute-cohomology-crystal}. Now the result follows from a general result in cohomology on sites, namely that the alternating {\\v C}ech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here)."}
+{"_id": "10833", "title": "crystalline-remark-quasi-coherent", "text": "In the situation of Remark \\ref{remark-compute-direct-image} assume that $S \\to S'$ is quasi-compact and quasi-separated and that $X \\to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules $\\mathcal{F}$ the sheaves $R^if_{\\text{cris}, *}\\mathcal{F}$ are locally quasi-coherent. \\medskip\\noindent Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\\mathcal{O}_{T'}$-modules, where $(U', T', \\delta')$ is any object of $\\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (\\ref{equation-identify-pushforward}), the fact that $T \\to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \\to S'$), and Cohomology of Schemes, Lemma \\ref{coherent-lemma-quasi-coherence-higher-direct-images} to see that it suffices to show that the sheaves $R^i\\tau_{U/T, *}\\mathcal{F}_U$ are quasi-coherent. Note that $U \\to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas \\ref{schemes-lemma-quasi-compact-permanence} and \\ref{schemes-lemma-quasi-compact-permanence}. \\medskip\\noindent This reduces us to proving that $R^i\\tau_{X/S, *}\\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\\tau_{X/S}$ is the structure morphism, see Remark \\ref{remark-structure-morphism}. We may work locally on $S$, hence we may assume $S$ affine (see Lemma \\ref{lemma-localize}). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark \\ref{remark-mayer-vietoris}) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma \\ref{coherent-lemma-quasi-coherence-higher-direct-images}). Say $X = \\Spec(C)$ and $S = \\Spec(A)$ so that $(A, I, \\gamma)$ and $A \\to C$ are as in Situation \\ref{situation-affine}. Choose a polynomial algebra $P$ over $A$ and a surjection $P \\to C$ as in Section \\ref{section-quasi-coherent-crystals}. Let $(M, \\nabla)$ be the module corresponding to $\\mathcal{F}$, see Proposition \\ref{proposition-crystals-on-affine}. Applying  Proposition \\ref{proposition-compute-cohomology-crystal} we see that $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is represented by $M \\otimes_D \\Omega_D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \\Spec(A)$. Suppose that $g \\in A$. Then we conclude that similarly $R\\Gamma(\\text{Cris}(X_g/S_g), \\mathcal{F})$ is computed by $M_g \\otimes_{D_g} \\Omega_{D_g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules."}
+{"_id": "10834", "title": "crystalline-remark-bounded-cohomology", "text": "In the situation of Remark \\ref{remark-compute-direct-image} assume that $S \\to S'$ is quasi-compact and quasi-separated and that $X \\to S_0$ is of finite type and quasi-separated. Then there exists an integer $i_0$ such that for any crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules $\\mathcal{F}$ we have $R^if_{\\text{cris}, *}\\mathcal{F} = 0$ for all $i > i_0$. \\medskip\\noindent Hints: Arguing as in Remark \\ref{remark-quasi-coherent} (using Cohomology of Schemes, Lemma \\ref{coherent-lemma-quasi-coherence-higher-direct-images}) we reduce to proving that $H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i \\gg 0$ in the situation of Proposition \\ref{proposition-compute-cohomology-crystal} when $C$ is a finite type algebra over $A$. This is clear as we can choose a finite polynomial algebra and we see that $\\Omega^i_D = 0$ for $i \\gg 0$."}
+{"_id": "10835", "title": "crystalline-remark-bounded-cohomology-over-point", "text": "In Situation \\ref{situation-global} let $\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules. Assume that $S_0$ has a unique point and that $X \\to S_0$ is of finite presentation. \\begin{enumerate} \\item If $\\dim X = d$ and $X/S_0$ has embedding dimension $e$, then $H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i > d + e$. \\item If $X$ is separated and can be covered by $q$ affines, and $X/S_0$ has embedding dimension $e$, then $H^i(\\text{Cris}(X/S), \\mathcal{F}) = 0$ for $i > q + e$. \\end{enumerate} Hints: In case (1) we can use that $$ H^i(\\text{Cris}(X/S), \\mathcal{F}) = H^i(X_{Zar}, Ru_{X/S, *}\\mathcal{F}) $$ and that $Ru_{X/S, *}\\mathcal{F}$ is locally calculated by a de Rham complex constructed using an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (see Lemma \\ref{lemma-compute-cohomology-crystal-smooth}). These de Rham complexes are zero in all degrees $> e$. Hence (1) follows from Cohomology, Proposition \\ref{cohomology-proposition-vanishing-Noetherian}. In case (2) we use the alternating {\\v C}ech complex (see Remark \\ref{remark-alternating-cech-complex}) to reduce to the case $X$ affine. In the affine case we prove the result using the de Rham complex associated to an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (it takes some work to construct such a thing)."}
+{"_id": "10836", "title": "crystalline-remark-base-change", "text": "In the situation of Remark \\ref{remark-compute-direct-image} assume $S = \\Spec(A)$ and $S' = \\Spec(A')$ are affine. Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'/S'}$-module. Let $\\mathcal{F}$ be the pullback of $\\mathcal{F}'$. Then there is a canonical base change map $$ L(S' \\to S)^*R\\tau_{X'/S', *}\\mathcal{F}' \\longrightarrow R\\tau_{X/S, *}\\mathcal{F} $$ where $\\tau_{X/S}$ and $\\tau_{X'/S'}$ are the structure morphisms, see Remark \\ref{remark-structure-morphism}. On global sections this gives a base change map \\begin{equation} \\label{equation-base-change-map} R\\Gamma(\\text{Cris}(X'/S'), \\mathcal{F}') \\otimes^\\mathbf{L}_{A'} A \\longrightarrow R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) \\end{equation} in $D(A)$. \\medskip\\noindent Hint: Compose the very general base change map of Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change} with the canonical map $Lf_{\\text{cris}}^*\\mathcal{F}' \\to f_{\\text{cris}}^*\\mathcal{F}' = \\mathcal{F}$."}
+{"_id": "10837", "title": "crystalline-remark-base-change-isomorphism", "text": "The map (\\ref{equation-base-change-map}) is an isomorphism provided all of the following conditions are satisfied: \\begin{enumerate} \\item $p$ is nilpotent in $A'$, \\item $\\mathcal{F}'$ is a crystal in quasi-coherent $\\mathcal{O}_{X'/S'}$-modules, \\item $X' \\to S'_0$ is a quasi-compact, quasi-separated morphism, \\item $X = X' \\times_{S'_0} S_0$, \\item $\\mathcal{F}'$ is a flat $\\mathcal{O}_{X'/S'}$-module, \\item $X' \\to S'_0$ is a local complete intersection morphism (see More on Morphisms, Definition \\ref{more-morphisms-definition-lci}; this holds for example if $X' \\to S'_0$ is syntomic or smooth), \\item $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition \\ref{more-algebra-definition-tor-independent}; this holds for example if either $S_0 \\to S'_0$ or $X' \\to S'_0$ is flat). \\end{enumerate} Hints: Condition (1) means that in the arguments below $p$-adic completion does nothing and can be ignored. Using condition (3) and Mayer Vietoris (see Remark \\ref{remark-mayer-vietoris}) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \\Spec(C')$ and we are given a presentation $C' = A'/I'[x_1, \\ldots, x_n]/(\\bar f'_1, \\ldots, \\bar f'_c)$ where $\\bar f'_1, \\ldots, \\bar f'_c$ is a Koszul-regular sequence in $A'/I'$. (This means that smooth locally $\\bar f'_1, \\ldots, \\bar f'_c$ forms a regular sequence, see More on Algebra, Lemma \\ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}.) We choose a lift of $\\bar f'_i$ to an element $f'_i \\in A'[x_1, \\ldots, x_n]$. By (4) we see that $X = \\Spec(C)$ with $C = A/I[x_1, \\ldots, x_n]/(\\bar f_1, \\ldots, \\bar f_c)$ where $f_i \\in A[x_1, \\ldots, x_n]$ is the image of $f'_i$. By property (7) we see that $\\bar f_1, \\ldots, \\bar f_c$ is a Koszul-regular sequence in $A/I[x_1, \\ldots, x_n]$. The divided power envelope of $I'A'[x_1, \\ldots, x_n] + (f'_1, \\ldots, f'_c)$ in $A'[x_1, \\ldots, x_n]$ relative to $\\gamma'$ is $$ D' = A'[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c \\rangle/(\\xi_i - f'_i) $$ see Lemma \\ref{lemma-describe-divided-power-envelope}. Then you check that $\\xi_1 - f'_1, \\ldots, \\xi_n - f'_n$ is a Koszul-regular sequence in the ring $A'[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle$. Similarly the divided power envelope of $IA[x_1, \\ldots, x_n] + (f_1, \\ldots, f_c)$ in $A[x_1, \\ldots, x_n]$ relative to $\\gamma$ is $$ D = A[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle/(\\xi_i - f_i) $$ and $\\xi_1 - f_1, \\ldots, \\xi_n - f_n$ is a Koszul-regular sequence in the ring $A[x_1, \\ldots, x_n]\\langle \\xi_1, \\ldots, \\xi_c\\rangle$. It follows that $D' \\otimes_{A'}^\\mathbf{L} A = D$. Condition (2) implies $\\mathcal{F}'$ corresponds to a pair $(M', \\nabla)$ consisting of a $D'$-module with connection, see Proposition \\ref{proposition-crystals-on-affine}. Then $M = M' \\otimes_{D'} D$ corresponds to the pullback $\\mathcal{F}$. By assumption (5) we see that $M'$ is a flat $D'$-module, hence $$ M = M' \\otimes_{D'} D = M' \\otimes_{D'} D' \\otimes_{A'}^\\mathbf{L} A = M' \\otimes_{A'}^\\mathbf{L} A $$ Since the modules of differentials $\\Omega_{D'}$ and $\\Omega_D$ (as defined in Section \\ref{section-quasi-coherent-crystals}) are free $D'$-modules on the same generators we see that $$ M \\otimes_D \\Omega^\\bullet_D = M' \\otimes_{D'} \\Omega^\\bullet_{D'} \\otimes_{D'} D = M' \\otimes_{D'} \\Omega^\\bullet_{D'} \\otimes_{A'}^\\mathbf{L} A $$ which proves what we want by Proposition \\ref{proposition-compute-cohomology-crystal}."}
+{"_id": "10838", "title": "crystalline-remark-rlim", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $A$ an algebra over $\\mathbf{Z}_{(p)}$ with $p$ nilpotent in $A/I$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a scheme over $S_0$ with $p$ locally nilpotent on $X$. Let $\\mathcal{F}$ be any $\\mathcal{O}_{X/S}$-module. For $e \\gg 0$ we have $(p^e) \\subset I$ is preserved by $\\gamma$, see Divided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}. Set $S_e = \\Spec(A/p^eA)$ for $e \\gg 0$. Then $\\text{Cris}(X/S_e)$ is a full subcategory of $\\text{Cris}(X/S)$ and we denote $\\mathcal{F}_e$ the restriction of $\\mathcal{F}$ to $\\text{Cris}(X/S_e)$. Then $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\lim_e R\\Gamma(\\text{Cris}(X/S_e), \\mathcal{F}_e) $$ \\medskip\\noindent Hints: Suffices to prove this for $\\mathcal{F}$ injective. In this case the sheaves $\\mathcal{F}_e$ are injective modules too, the transition maps $\\Gamma(\\mathcal{F}_{e + 1}) \\to \\Gamma(\\mathcal{F}_e)$ are surjective, and we have $\\Gamma(\\mathcal{F}) = \\lim_e \\Gamma(\\mathcal{F}_e)$ because any object of $\\text{Cris}(X/S)$ is locally an object of one of the categories $\\text{Cris}(X/S_e)$ by definition of $\\text{Cris}(X/S)$."}
+{"_id": "10839", "title": "crystalline-remark-comparison", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $Y$ be a smooth scheme over $S$ and set $X = Y \\times_S S_0$. Let $\\mathcal{F}$ be a crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules. Then \\begin{enumerate} \\item $\\gamma$ extends to a divided power structure on the ideal of $X$ in $Y$ so that $(X, Y, \\gamma)$ is an object of $\\text{Cris}(X/S)$, \\item the restriction $\\mathcal{F}_Y$ (see Section \\ref{section-sheaves}) comes endowed with a canonical integrable connection $\\nabla : \\mathcal{F}_Y \\to \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega_{Y/S}$, and \\item we have $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\Gamma(Y, \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega^\\bullet_{Y/S}) $$ in $D(A)$. \\end{enumerate} Hints: See Divided Power Algebra, Lemma \\ref{dpa-lemma-gamma-extends} for (1). See Lemma \\ref{lemma-automatic-connection} for (2). For Part (3) note that there is a map, see (\\ref{equation-restriction}). This map is an isomorphism when $X$ is affine, see Lemma \\ref{lemma-compute-cohomology-crystal-smooth}. This shows that $Ru_{X/S, *}\\mathcal{F}$ and $\\mathcal{F}_Y \\otimes \\Omega^\\bullet_{Y/S}$ are quasi-isomorphic as complexes on $Y_{Zar} = X_{Zar}$. Since $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\Gamma(X_{Zar}, Ru_{X/S, *}\\mathcal{F})$ the result follows."}
+{"_id": "10840", "title": "crystalline-remark-perfect", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\\mathcal{O}_{X/S}$-modules. Then $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is a perfect object of $D(A)$. \\medskip\\noindent Hints: By Remark \\ref{remark-base-change-isomorphism} we have $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) \\otimes_A^\\mathbf{L} A/I \\cong R\\Gamma(\\text{Cris}(X/S_0), \\mathcal{F}|_{\\text{Cris}(X/S_0)}) $$ By Remark \\ref{remark-comparison} we have $$ R\\Gamma(\\text{Cris}(X/S_0), \\mathcal{F}|_{\\text{Cris}(X/S_0)}) = R\\Gamma(X, \\mathcal{F}_X \\otimes \\Omega^\\bullet_{X/S_0}) $$ Using the stupid filtration on the de Rham complex we see that the last displayed complex is perfect in $D(A/I)$ as soon as the complexes $$ R\\Gamma(X, \\mathcal{F}_X \\otimes \\Omega^q_{X/S_0}) $$ are perfect complexes in $D(A/I)$, see More on Algebra, Lemma \\ref{more-algebra-lemma-two-out-of-three-perfect}. This is true by standard arguments in coherent cohomology using that $\\mathcal{F}_X \\otimes \\Omega^q_{X/S_0}$ is a finite locally free sheaf and $X \\to S_0$ is proper and flat (insert future reference here). Applying More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal} we see that $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) \\otimes_A^\\mathbf{L} A/I^n $$ is a perfect object of $D(A/I^n)$ for all $n$. This isn't quite enough unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent by our assumption that $p$ is nilpotent, see Divided Power Algebra, Lemma \\ref{dpa-lemma-nil}, we cannot conclude that $I^n = 0$ for some $n$. A counter example is $\\mathbf{F}_p\\langle x \\rangle$. To prove it in general when $\\mathcal{F} = \\mathcal{O}_{X/S}$ the argument of \\url{https://math.columbia.edu/~dejong/wordpress/?p=2227} works. When the coefficients $\\mathcal{F}$ are non-trivial the argument of \\cite{Faltings-very} seems to be as follows. Reduce to the case $pA = 0$ by More on Algebra, Lemma \\ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}. In this case the Frobenius map $A \\to A$, $a \\mapsto a^p$ factors as $A \\to A/I \\xrightarrow{\\varphi} A$ (as $x^p = 0$ for $x \\in I$). Set $X^{(1)} = X \\otimes_{A/I, \\varphi} A$. The absolute Frobenius morphism of $X$ factors through a morphism $F_X : X \\to X^{(1)}$ (a kind of relative Frobenius). Affine locally if $X = \\Spec(C)$ then $X^{(1)} = \\Spec( C \\otimes_{A/I, \\varphi} A)$ and $F_X$ corresponds to $C \\otimes_{A/I, \\varphi} A \\to C$, $c \\otimes a \\mapsto c^pa$. This defines morphisms of ringed topoi $$ (X/S)_{\\text{cris}} \\xrightarrow{(F_X)_{\\text{cris}}} (X^{(1)}/S)_{\\text{cris}} \\xrightarrow{u_{X^{(1)}/S}} \\Sh(X^{(1)}_{Zar}) $$ whose composition is denoted $\\text{Frob}_X$. One then shows that $R\\text{Frob}_{X, *}\\mathcal{F}$ is representable by a perfect complex of $\\mathcal{O}_{X^{(1)}}$-modules(!) by a local calculation."}
+{"_id": "10841", "title": "crystalline-remark-complete-perfect", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\\mathcal{O}_{X/S}$-modules. Then $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is a perfect object of $D(A)$. \\medskip\\noindent Hints: We know that $K = R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$ is the derived limit $K = R\\lim K_e$ of the cohomologies over $A/p^eA$, see Remark \\ref{remark-rlim}. Each $K_e$ is a perfect complex of $D(A/p^eA)$ by Remark \\ref{remark-perfect}. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma \\ref{more-algebra-lemma-Rlim-perfect-gives-complete}."}
+{"_id": "10842", "title": "crystalline-remark-complete-comparison", "text": "Let $p$ be a prime number. Let $(A, I, \\gamma)$ be a divided power ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \\Spec(A)$ and $S_0 = \\Spec(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set $X = Y \\times_S S_0$. Let $\\mathcal{F}$ be a finite type crystal in quasi-coherent $\\mathcal{O}_{X/S}$-modules. Then \\begin{enumerate} \\item there exists a coherent $\\mathcal{O}_Y$-module $\\mathcal{F}_Y$ endowed with integrable connection $$ \\nabla : \\mathcal{F}_Y \\longrightarrow \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega_{Y/S} $$ such that $\\mathcal{F}_Y/p^e\\mathcal{F}_Y$ is the module with connection over $A/p^eA$ found in Remark \\ref{remark-comparison}, and \\item we have $$ R\\Gamma(\\text{Cris}(X/S), \\mathcal{F}) = R\\Gamma(Y, \\mathcal{F}_Y \\otimes_{\\mathcal{O}_Y} \\Omega^\\bullet_{Y/S}) $$ in $D(A)$. \\end{enumerate} Hints: The existence of $\\mathcal{F}_Y$ is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as $R\\lim$ of the versions modulo $p^e$ (see Remark \\ref{remark-rlim} for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem \\ref{coherent-theorem-formal-functions} for the right hand side). Each of the versions modulo $p^e$ are isomorphic by Remark \\ref{remark-comparison}."}
+{"_id": "10843", "title": "crystalline-remark-F-crystal-variants", "text": "Let $(\\mathcal{E}, F)$ be an $F$-crystal as in Definition \\ref{definition-F-crystal}. In the literature the nondegeneracy condition is often part of the definition of an $F$-crystal. Moreover, often it is also assumed that $F \\circ V = p^n\\text{id}$. What is needed for the result below is that there exists an integer $j \\geq 0$ such that $\\Ker(F)$ and $\\Coker(F)$ are killed by $p^j$. If the rank of $\\mathcal{E}$ is bounded (for example if $X$ is quasi-compact), then both of these conditions follow from the nondegeneracy condition as formulated in the definition. Namely, suppose $R$ is a ring, $r \\geq 1$ is an integer and $K, L \\in \\text{Mat}(r \\times r, R)$ are matrices with $K L = p^i 1_{r \\times r}$. Then $\\det(K)\\det(L) = p^{ri}$.  Let $L'$ be the adjugate matrix of $L$, i.e., $L' L = L L' = \\det(L)$. Set $K' = p^{ri} K$ and $j = ri + i$. Then we have $K' L = p^j 1_{r \\times r}$ as $K L = p^i$ and $$ L K' = L K \\det(L) \\det(M) = L K L L' \\det(M) = L p^i L' \\det(M) = p^j 1_{r \\times r} $$ It follows that if $V$ is as in Definition \\ref{definition-F-crystal} then setting $V' = p^N V$ where $N > i \\cdot \\text{rank}(\\mathcal{E})$ we get $V' \\circ F = p^{N + i}$ and $F \\circ V' = p^{N + i}$."}
+{"_id": "10899", "title": "spaces-pushouts-remark-essentially-constant", "text": "The meaning of Lemma \\ref{lemma-essentially-constant} is the system $X_1 \\to X_2 \\to X_3 \\to \\ldots$ is essentially constant with value $X$. See Categories, Definition \\ref{categories-definition-essentially-constant-diagram}."}
+{"_id": "11166", "title": "varieties-remark-exact-sequence-induction", "text": "Let $k$ be an infinite field. Let $n \\geq 1$. Given a finite number of coherent modules $\\mathcal{F}_i$ on $\\mathbf{P}^n_k$ we can choose a single $s \\in \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$ such that the statement of Lemma \\ref{lemma-exact-sequence-induction} works for each of them. To prove this, just apply the lemma to $\\bigoplus \\mathcal{F}_i$."}
+{"_id": "11167", "title": "varieties-remark-exact-sequence-induction-cohomology", "text": "In the situation of Lemmas \\ref{lemma-hyperplane} and \\ref{lemma-exact-sequence-induction} we have $H \\cong \\mathbf{P}^{n - 1}_k$ with Serre twists $\\mathcal{O}_H(d) = i^*\\mathcal{O}_{\\mathbf{P}^n_k}(d)$. For every $d \\in \\mathbf{Z}$ we have a short exact sequence $$ 0 \\to \\mathcal{F}(d - 1) \\to \\mathcal{F}(d) \\to i_*(\\mathcal{G}(d)) \\to 0 $$ Namely, tensoring by $\\mathcal{O}_{\\mathbf{P}^n_k}(d)$ is an exact functor and by the projection formula (Cohomology, Lemma \\ref{cohomology-lemma-projection-formula}) we have $i_*(\\mathcal{G}(d)) = i_*\\mathcal{G} \\otimes \\mathcal{O}_{\\mathbf{P}^n_k}(d)$. We obtain corresponding long exact sequences $$ H^i(\\mathbf{P}^n_k, \\mathcal{F}(d - 1)) \\to H^i(\\mathbf{P}^n_k, \\mathcal{F}(d)) \\to H^i(H, \\mathcal{G}(d)) \\to H^{i + 1}(\\mathbf{P}^n_k, \\mathcal{F}(d - 1)) $$ This follows from the above and the fact that we have $H^i(\\mathbf{P}^n_k, i_*\\mathcal{G}(d)) = H^i(H, \\mathcal{G}(d))$ by Cohomology of Schemes, Lemma \\ref{coherent-lemma-relative-affine-cohomology} (closed immersions are affine)."}
+{"_id": "11168", "title": "varieties-remark-n-fold-relative-frobenius", "text": "Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. For $n \\geq 1$ $$ X^{(p^n)} = X^{(p^n/S)} = X \\times_{S, F_S^n} S $$ viewed as a scheme over $S$. Observe that $X \\mapsto X^{(p^n)}$ is a functor. Applying Lemma \\ref{lemma-frobenius-endomorphism-identity} we see $F_{X/S, n} = (F_X^n, \\text{id}_S) : X \\longrightarrow X^{(p^n)}$ is a morphism over $S$ fitting into the commutative diagram $$ \\xymatrix{ X \\ar[rr]_{F_{X/S, n}} \\ar[rrd] \\ar@/^1em/[rrrr]^{F_X^n} & & X^{(p^n)} \\ar[rr] \\ar[d] & & X \\ar[d] \\\\ & & S \\ar[rr]^{F_S^n} & & S } $$ where the right square is cartesian. The morphism $F_{X/S, n}$ is sometimes called the {\\it $n$-fold relative Frobenius morphism of $X/S$}. This makes sense because we have the formula $$ F_{X/S, n} = F_{X^{(p^{n - 1})}/S} \\circ \\ldots \\circ F_{X^{(p)}/S} \\circ F_{X/S} $$ which shows that $F_{X/S, n}$ is the composition of $n$ relative Frobenii. Since we have $$ F_{X^{(p^m)}/S} = F_{X^{(p^{m - 1})}/S}^{(p)} = \\ldots = F_{X/S}^{(p^m)} $$ (details omitted) we get also that $$ F_{X/S, n} = F_{X/S}^{(p^{n - 1})} \\circ \\ldots \\circ F_{X/S}^{(p)} \\circ F_{X/S} $$"}
+{"_id": "11169", "title": "varieties-remark-conductor", "text": "Let $A$ be a reduced ring. Let $I, J$ be ideals of $A$ such that $V(I) \\cup V(J) = \\Spec(A)$. Set $B = A/J$. Then $I \\to IB$ is an isomorphism of $A$-modules. Namely, we have $IB = I + J/J = I/(I \\cap J)$ and $I \\cap J$ is zero because $A$ is reduced and $\\Spec(A) = V(I) \\cup V(J) = V(I \\cap J)$. Thus for any projective $A$-module $P$ we also have $IP = I(P/JP)$."}
+{"_id": "11170", "title": "varieties-remark-useless-generalization", "text": "In fact, if $X$ is a scheme whose reduction is a Noetherian separated scheme of dimension $1$, then $X$ has an ample invertible sheaf. The argument to prove this is the same as the proof of Proposition \\ref{proposition-dim-1-noetherian-separated-has-ample} except one uses Limits, Lemma \\ref{limits-lemma-ample-on-reduction} instead of Cohomology of Schemes, Lemma \\ref{coherent-lemma-ample-on-reduction}."}
+{"_id": "11255", "title": "cotangent-remark-variant-cotangent-complex", "text": "Let $A \\to B$ be a ring map. Let $\\mathcal{A}$ be the category of arrows $\\psi : C \\to B$ of $A$-algebras and let $\\mathcal{S}$ be the category of maps $E \\to B$ where $E$ is a set. There are adjoint functors $V : \\mathcal{A} \\to \\mathcal{S}$ (the forgetful functor) and $U : \\mathcal{S} \\to \\mathcal{A}$ which sends $E \\to B$ to $A[E] \\to B$. Let $X_\\bullet$ be the simplicial object of $\\text{Fun}(\\mathcal{A}, \\mathcal{A})$ constructed in Simplicial, Section \\ref{simplicial-section-standard}. The diagram $$ \\xymatrix{ \\mathcal{A} \\ar[d] \\ar[r] & \\mathcal{S} \\ar@<1ex>[l] \\ar[d] \\\\ \\textit{Alg}_A \\ar[r] & \\textit{Sets} \\ar@<1ex>[l] } $$ commutes. It follows that $X_\\bullet(\\text{id}_B : B \\to B)$ is equal to the standard resolution of $B$ over $A$."}
+{"_id": "11256", "title": "cotangent-remark-resolution", "text": "Let $A \\to B$ be any ring map. Let us call an augmented simplicial $A$-algebra $\\epsilon : P_\\bullet \\to B$ a {\\it resolution of $B$ over $A$} if each $P_n$ is a polynomial algebra and $\\epsilon$ is a trivial Kan fibration of simplicial sets. If $P_\\bullet \\to B$ is an augmentation of a simplicial $A$-algebra with each $P_n$ a polynomial algebra surjecting onto $B$, then the following are equivalent \\begin{enumerate} \\item $\\epsilon : P_\\bullet \\to B$ is a resolution of $B$ over $A$, \\item $\\epsilon : P_\\bullet \\to B$ is a quasi-isomorphism on associated complexes, \\item $\\epsilon : P_\\bullet \\to B$ induces a homotopy equivalence of simplicial sets. \\end{enumerate} To see this use Simplicial, Lemmas \\ref{simplicial-lemma-trivial-kan-homotopy}, \\ref{simplicial-lemma-homotopy-equivalence}, and \\ref{simplicial-lemma-qis-simplicial-abelian-groups}. A resolution $P_\\bullet$ of $B$ over $A$ gives a cosimplicial object $U_\\bullet$ of $\\mathcal{C}_{B/A}$ as in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution} and it follows that $$ L\\pi_!\\mathcal{F} = \\mathcal{F}(P_\\bullet) $$ functorially in $\\mathcal{F}$, see Lemma \\ref{lemma-identify-pi-shriek}. The (formal part of the) proof of Proposition \\ref{proposition-polynomial} shows that resolutions exist. We also have seen in the first proof of Lemma \\ref{lemma-pi-shriek-standard} that the standard resolution of $B$ over $A$ is a resolution (so that this terminology doesn't lead to a conflict). However, the argument in the proof of Proposition \\ref{proposition-polynomial} shows the existence of resolutions without appealing to the simplicial computations in Simplicial, Section \\ref{simplicial-section-standard}. Moreover, for {\\it any} choice of resolution we have a canonical isomorphism $$ L_{B/A} = \\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} B $$ in $D(B)$ by Lemma \\ref{lemma-compute-cotangent-complex}. The freedom to choose an arbitrary resolution can be quite useful."}
+{"_id": "11257", "title": "cotangent-remark-homotopy-triangle", "text": "Suppose that we are given a square (\\ref{equation-commutative-square}) such that there exists an arrow $\\kappa : B \\to A'$ making the diagram commute: $$ \\xymatrix{ B \\ar[r]_\\beta \\ar[rd]_\\kappa & B' \\\\ A \\ar[u] \\ar[r]^\\alpha & A' \\ar[u] } $$ In this case we claim the functoriality map $P_\\bullet \\to P'_\\bullet$ is homotopic to the composition $P_\\bullet \\to B \\to A' \\to P'_\\bullet$. Namely, using $\\kappa$ the functoriality map factors as $$ P_\\bullet \\to P_{A'/A', \\bullet} \\to P'_\\bullet $$ where $P_{A'/A', \\bullet}$ is the standard resolution of $A'$ over $A'$. Since $A'$ is the polynomial algebra on the empty set over $A'$ we see from Simplicial, Lemma \\ref{simplicial-lemma-standard-simplicial-homotopy} that the augmentation $\\epsilon_{A'/A'} : P_{A'/A', \\bullet} \\to A'$ is a homotopy equivalence of simplicial rings. Observe that the homotopy inverse map $c : A' \\to P_{A'/A', \\bullet}$ constructed in the proof of that lemma is just the structure morphism, hence we conclude what we want because the two compositions $$ \\xymatrix{ P_\\bullet \\ar[r] & P_{A'/A', \\bullet} \\ar@<1ex>[rr]^{\\text{id}} \\ar@<-1ex>[rr]_{c \\circ \\epsilon_{A'/A'}} & & P_{A'/A', \\bullet} \\ar[r] & P'_\\bullet } $$ are the two maps discussed above and these are homotopic (Simplicial, Remark \\ref{simplicial-remark-homotopy-pre-post-compose}). Since the second map $P_\\bullet \\to P'_\\bullet$ induces the zero map $\\Omega_{P_\\bullet/A} \\to \\Omega_{P'_\\bullet/A'}$ we conclude that the functoriality map $L_{B/A} \\to L_{B'/A'}$ is homotopic to zero in this case."}
+{"_id": "11258", "title": "cotangent-remark-triangle", "text": "We sketch an alternative, perhaps simpler, proof of the existence of the fundamental triangle. Let $A \\to B \\to C$ be ring maps and assume that $B \\to C$ is injective. Let $P_\\bullet \\to B$ be the standard resolution of $B$ over $A$ and let $Q_\\bullet \\to C$ be the standard resolution of $C$ over $B$. Picture $$ \\xymatrix{ P_\\bullet : & A[A[A[B]]] \\ar[d] \\ar@<2ex>[r] \\ar@<0ex>[r] \\ar@<-2ex>[r] & A[A[B]] \\ar[d] \\ar@<1ex>[r] \\ar@<-1ex>[r] \\ar@<1ex>[l] \\ar@<-1ex>[l] & A[B] \\ar[d] \\ar@<0ex>[l] \\ar[r] & B \\\\ Q_\\bullet : & A[A[A[C]]] \\ar@<2ex>[r] \\ar@<0ex>[r] \\ar@<-2ex>[r] & A[A[C]] \\ar@<1ex>[r] \\ar@<-1ex>[r] \\ar@<1ex>[l] \\ar@<-1ex>[l] & A[C] \\ar@<0ex>[l] \\ar[r] & C } $$ Observe that since $B \\to C$ is injective, the ring $Q_n$ is a polynomial algebra over $P_n$ for all $n$. Hence we obtain a cosimplicial object in $\\mathcal{C}_{C/B/A}$ (beware reversal arrows). Now set $\\overline{Q}_\\bullet = Q_\\bullet \\otimes_{P_\\bullet} B$. The key to the proof of Proposition \\ref{proposition-triangle} is to show that $\\overline{Q}_\\bullet$ is a resolution of $C$ over $B$. This follows from Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-O-homology-qis} applied to $\\mathcal{C} = \\Delta$, $\\mathcal{O} = P_\\bullet$, $\\mathcal{O}' = B$, and $\\mathcal{F} = Q_\\bullet$ (this uses that $Q_n$ is flat over $P_n$; see Cohomology on Sites, Remark \\ref{sites-cohomology-remark-simplicial-modules} to relate simplicial modules to sheaves). The key fact implies that the distinguished triangle of Proposition \\ref{proposition-triangle} is the distinguished triangle associated to the short exact sequence of simplicial $C$-modules $$ 0 \\to \\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet} C \\to \\Omega_{Q_\\bullet/A} \\otimes_{Q_\\bullet} C \\to \\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C \\to 0 $$ which is deduced from the short exact sequences $0 \\to \\Omega_{P_n/A} \\otimes_{P_n} Q_n \\to \\Omega_{Q_n/A} \\to \\Omega_{Q_n/P_n} \\to 0$ of Algebra, Lemma \\ref{algebra-lemma-ses-formally-smooth}. Namely, by Remark \\ref{remark-resolution} and the key fact the complex on the right hand side represents $L_{C/B}$ in $D(C)$. \\medskip\\noindent If $B \\to C$ is not injective, then we can use the above to get a fundamental triangle for $A \\to B \\to B \\times C$. Since $L_{B \\times C/B} \\to L_{B/B} \\oplus L_{C/B}$ and $L_{B \\times C/A} \\to L_{B/A} \\oplus L_{C/A}$ are quasi-isomorphism in $D(B \\times C)$ (Lemma \\ref{lemma-cotangent-complex-product}) this induces the desired distinguished triangle in $D(C)$ by tensoring with the flat ring map $B \\times C \\to C$."}
+{"_id": "11259", "title": "cotangent-remark-explicit-map", "text": "Let $A \\to B \\to C$ be ring maps with $B \\to C$ injective. Recall the notation $P_\\bullet$, $Q_\\bullet$, $\\overline{Q}_\\bullet$ of Remark \\ref{remark-triangle}. Let $R_\\bullet$ be the standard resolution of $C$ over $B$. In this remark we explain how to get the canonical identification of $\\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C$ with $L_{C/B} = \\Omega_{R_\\bullet/B} \\otimes_{R_\\bullet} C$. Let $S_\\bullet \\to B$ be the standard resolution of $B$ over $B$. Note that the functoriality map $S_\\bullet \\to R_\\bullet$ identifies $R_n$ as a polynomial algebra over $S_n$ because $B \\to C$ is injective. For example in degree $0$ we have the map $B[B] \\to B[C]$, in degree $1$ the map $B[B[B]] \\to B[B[C]]$, and so on. Thus $\\overline{R}_\\bullet = R_\\bullet \\otimes_{S_\\bullet} B$ is a simplicial polynomial algebra over $B$ as well and it follows (as in Remark \\ref{remark-triangle}) from Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-O-homology-qis} that $\\overline{R}_\\bullet \\to C$ is a resolution. Since we have a commutative diagram $$ \\xymatrix{ Q_\\bullet \\ar[r] & R_\\bullet \\\\ P_\\bullet \\ar[u] \\ar[r] & S_\\bullet \\ar[u] \\ar[r] & B } $$ we obtain a canonical map $\\overline{Q}_\\bullet = Q_\\bullet \\otimes_{P_\\bullet} B \\to \\overline{R}_\\bullet$. Thus the maps $$ L_{C/B} = \\Omega_{R_\\bullet/B} \\otimes_{R_\\bullet} C \\longrightarrow \\Omega_{\\overline{R}_\\bullet/B} \\otimes_{\\overline{R}_\\bullet} C \\longleftarrow \\Omega_{\\overline{Q}_\\bullet/B} \\otimes_{\\overline{Q}_\\bullet} C $$ are quasi-isomorphisms (Remark \\ref{remark-resolution}) and composing one with the inverse of the other gives the desired identification."}
+{"_id": "11260", "title": "cotangent-remark-make-map", "text": "Let $A \\to B$ be a ring map.  Working on $\\mathcal{C}_{B/A}$ as in Section \\ref{section-compute-L-pi-shriek} let $\\mathcal{J} \\subset \\mathcal{O}$ be the kernel of $\\mathcal{O} \\to \\underline{B}$. Note that $L\\pi_!(\\mathcal{J}) = 0$ by Lemma \\ref{lemma-apply-O-B-comparison}. Set $\\Omega =  \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$ so that $L_{B/A} = L\\pi_!(\\Omega)$ by Lemma \\ref{lemma-compute-cotangent-complex}. It follows that $L\\pi_!(\\mathcal{J} \\to \\Omega) = L\\pi_!(\\Omega) = L_{B/A}$. Thus, for any object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$ we obtain a map \\begin{equation} \\label{equation-comparison-map-A} (J \\to \\Omega_{P/A} \\otimes_P B) \\longrightarrow L_{B/A} \\end{equation} where $J = \\Ker(P \\to B)$ in $D(A)$, see Cohomology on Sites, Remark \\ref{sites-cohomology-remark-map-evaluation-to-derived}. Continuing in this manner, note that $L\\pi_!(\\mathcal{J} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B}) = L\\pi_!(\\mathcal{J}) = 0$ by Lemma \\ref{lemma-O-homology-B-homology}. Since $\\text{Tor}_0^\\mathcal{O}(\\mathcal{J}, \\underline{B}) = \\mathcal{J}/\\mathcal{J}^2$ the spectral sequence $$ H_p(\\mathcal{C}_{B/A}, \\text{Tor}_q^\\mathcal{O}(\\mathcal{J}, \\underline{B})) \\Rightarrow  H_{p + q}(\\mathcal{C}_{B/A}, \\mathcal{J} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B}) = 0 $$ (dual of Derived Categories, Lemma \\ref{derived-lemma-two-ss-complex-functor}) implies that $H_0(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) = 0$ and $H_1(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) = 0$. It follows that the complex of $\\underline{B}$-modules $\\mathcal{J}/\\mathcal{J}^2 \\to \\Omega$ satisfies $\\tau_{\\geq -1}L\\pi_!(\\mathcal{J}/\\mathcal{J}^2 \\to \\Omega) = \\tau_{\\geq -1}L_{B/A}$. Thus, for any object $U = (P \\to B)$ of $\\mathcal{C}_{B/A}$ we obtain a map \\begin{equation} \\label{equation-comparison-map} (J/J^2 \\to \\Omega_{P/A} \\otimes_P B) \\longrightarrow \\tau_{\\geq -1}L_{B/A} \\end{equation} in $D(B)$, see Cohomology on Sites, Remark \\ref{sites-cohomology-remark-map-evaluation-to-derived}."}
+{"_id": "11261", "title": "cotangent-remark-explicit-comparison-map", "text": "We can make the comparison map of Lemma \\ref{lemma-relation-with-naive-cotangent-complex} explicit in the following way. Let $P_\\bullet$ be the standard resolution of $B$ over $A$. Let $I = \\Ker(A[B] \\to B)$. Recall that $P_0 = A[B]$. The map of the lemma is given by the commutative diagram $$ \\xymatrix{ L_{B/A} \\ar[d] & \\ldots \\ar[r] & \\Omega_{P_2/A} \\otimes_{P_2} B \\ar[r] \\ar[d] & \\Omega_{P_1/A} \\otimes_{P_1} B \\ar[r] \\ar[d] & \\Omega_{P_0/A} \\otimes_{P_0} B \\ar[d] \\\\ \\NL_{B/A} & \\ldots \\ar[r] & 0 \\ar[r] &  I/I^2 \\ar[r] & \\Omega_{P_0/A} \\otimes_{P_0} B } $$ We construct the downward arrow with target $I/I^2$ by sending $\\text{d}f \\otimes b$ to the class of $(d_0(f) - d_1(f))b$ in $I/I^2$. Here $d_i : P_1 \\to P_0$, $i = 0, 1$ are the two face maps of the simplicial structure. This makes sense as $d_0 - d_1$ maps $P_1$ into $I = \\Ker(P_0 \\to B)$. We omit the verification that this rule is well defined. Our map is compatible with the differential $\\Omega_{P_1/A} \\otimes_{P_1} B \\to \\Omega_{P_0/A} \\otimes_{P_0} B$ as this differential maps $\\text{d}f \\otimes b$ to $\\text{d}(d_0(f) - d_1(f)) \\otimes b$. Moreover, the differential $\\Omega_{P_2/A} \\otimes_{P_2} B \\to \\Omega_{P_1/A} \\otimes_{P_1} B$ maps $\\text{d}f \\otimes b$ to $\\text{d}(d_0(f) - d_1(f) + d_2(f)) \\otimes b$ which are annihilated by our downward arrow. Hence a map of complexes. We omit the verification that this is the same as the map of Lemma \\ref{lemma-relation-with-naive-cotangent-complex}."}
+{"_id": "11262", "title": "cotangent-remark-surjection", "text": "Adopt notation as in Remark \\ref{remark-make-map}. The arguments given there show that the differential $$ H_2(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) \\longrightarrow H_0(\\mathcal{C}_{B/A}, \\text{Tor}_1^\\mathcal{O}(\\mathcal{J}, \\underline{B})) $$ of the spectral sequence is an isomorphism. Let $\\mathcal{C}'_{B/A}$ denote the full subcategory of $\\mathcal{C}_{B/A}$ consisting of surjective maps $P \\to B$. The agreement of the cotangent complex with the naive cotangent complex (Lemma \\ref{lemma-relation-with-naive-cotangent-complex}) shows that we have an exact sequence of sheaves $$ 0 \\to \\underline{H_1(L_{B/A})} \\to \\mathcal{J}/\\mathcal{J}^2 \\xrightarrow{\\text{d}} \\Omega \\to \\underline{H_2(L_{B/A})} \\to 0 $$ on $\\mathcal{C}'_{B/A}$. It follows that $\\Ker(d)$ and $\\Coker(d)$ on the whole category $\\mathcal{C}_{B/A}$ have vanishing higher homology groups, since these are computed by the homology groups of constant simplicial abelian groups by Lemma \\ref{lemma-identify-pi-shriek}. Hence we conclude that $$ H_n(\\mathcal{C}_{B/A}, \\mathcal{J}/\\mathcal{J}^2) \\to H_n(L_{B/A}) $$ is an isomorphism for all $n \\geq 2$. Combined with the remark above we obtain the formula $H_2(L_{B/A}) = H_0(\\mathcal{C}_{B/A}, \\text{Tor}_1^\\mathcal{O}(\\mathcal{J}, \\underline{B}))$."}
+{"_id": "11263", "title": "cotangent-remark-first-homology-symmetric-power", "text": "In the situation of Lemma \\ref{lemma-vanishing-symmetric-powers} one can show that $H_k(\\mathcal{C}, \\text{Sym}^k(\\mathcal{F})) = \\wedge^k_B(H_1(\\mathcal{C}, \\mathcal{F}))$. Namely, it can be deduced from the proof that $H_k(\\mathcal{C}, \\text{Sym}^k(\\mathcal{F}))$ is the $S_k$-coinvariants of $$ H^{-k}(L\\pi_!(\\mathcal{F}) \\otimes_B^\\mathbf{L} L\\pi_!(\\mathcal{F}) \\otimes_B^\\mathbf{L} \\ldots \\otimes_B^\\mathbf{L} L\\pi_!(\\mathcal{F})) = H_1(\\mathcal{C}, \\mathcal{F})^{\\otimes k} $$ Thus our claim is that this action is given by the usual action of $S_k$ on the tensor product multiplied by the sign character. To prove this one has to work through the sign conventions in the definition of the total complex associated to a multi-complex. We omit the verification."}
+{"_id": "11264", "title": "cotangent-remark-elucidate-ss", "text": "In the situation of Theorem \\ref{theorem-quillen-spectral-sequence} let $I = \\Ker(A \\to B)$. Then $H^{-1}(L_{B/A}) = H_1(\\mathcal{C}_{B/A}, \\Omega) = I/I^2$, see Lemma \\ref{lemma-surjection}. Hence $H_k(\\mathcal{C}_{B/A}, \\text{Sym}^k(\\Omega)) = \\wedge^k_B(I/I^2)$ by Remark \\ref{remark-first-homology-symmetric-power}. Thus the $E_1$-page looks like $$ \\begin{matrix} B \\\\ 0 \\\\ 0 & I/I^2 \\\\ 0 & H^{-2}(L_{B/A}) \\\\ 0 & H^{-3}(L_{B/A}) & \\wedge^2(I/I^2) \\\\ 0 & H^{-4}(L_{B/A}) & H_3(\\mathcal{C}_{B/A}, \\text{Sym}^2(\\Omega)) \\\\ 0 & H^{-5}(L_{B/A}) & H_4(\\mathcal{C}_{B/A}, \\text{Sym}^2(\\Omega)) & \\wedge^3(I/I^2) \\end{matrix} $$ with horizontal differential. Thus we obtain edge maps $\\text{Tor}_i^A(B, B) \\to H^{-i}(L_{B/A})$, $i > 0$ and $\\wedge^i_B(I/I^2) \\to \\text{Tor}_i^A(B, B)$. Finally, we have $\\text{Tor}_1^A(B, B) = I/I^2$ and there is a five term exact sequence $$ \\text{Tor}_3^A(B, B) \\to H^{-3}(L_{B/A}) \\to \\wedge^2_B(I/I^2) \\to \\text{Tor}_2^A(B, B) \\to H^{-2}(L_{B/A}) \\to 0 $$ of low degree terms."}
+{"_id": "11265", "title": "cotangent-remark-elucidate-degree-two", "text": "Let $A \\to B$ be a ring map. Let $P_\\bullet$ be a resolution of $B$ over $A$ (Remark \\ref{remark-resolution}). Set $J_n = \\Ker(P_n \\to B)$. Note that $$ \\text{Tor}_2^{P_n}(B, B) =  \\text{Tor}_1^{P_n}(J_n, B) = \\Ker(J_n \\otimes_{P_n} J_n \\to J_n^2). $$ Hence $H_2(L_{B/A})$ is canonically equal to $$ \\Coker(\\text{Tor}_2^{P_1}(B, B) \\to \\text{Tor}_2^{P_0}(B, B)) $$ by Remark \\ref{remark-surjection}. To make this more explicit we choose $P_2$, $P_1$, $P_0$ as in Example \\ref{example-resolution-length-two}. We claim that $$ \\text{Tor}_2^{P_1}(B, B) = \\wedge^2(\\bigoplus\\nolimits_{t \\in T} B)\\ \\oplus \\ \\bigoplus\\nolimits_{t \\in T} J_0\\ \\oplus \\ \\text{Tor}_2^{P_0}(B, B) $$ Namely, the basis elements $x_t \\wedge x_{t'}$ of the first summand corresponds to the element $x_t \\otimes x_{t'} - x_{t'} \\otimes x_t$ of $J_1 \\otimes_{P_1} J_1$. For $f \\in J_0$ the element $x_t \\otimes f$ of the second summand corresponds to the element $x_t \\otimes s_0(f) - s_0(f) \\otimes x_t$ of $J_1 \\otimes_{P_1} J_1$. Finally, the map $\\text{Tor}_2^{P_0}(B, B) \\to \\text{Tor}_2^{P_1}(B, B)$ is given by $s_0$. The map $d_0 - d_1 : \\text{Tor}_2^{P_1}(B, B) \\to \\text{Tor}_2^{P_0}(B, B)$ is zero on the last summand, maps $x_t \\otimes f$ to $f \\otimes f_t - f_t \\otimes f$, and maps $x_t \\wedge x_{t'}$ to $f_t \\otimes f_{t'} - f_{t'} \\otimes f_t$. All in all we conclude that there is an exact sequence $$ \\wedge^2_B(J_0/J_0^2) \\to \\text{Tor}_2^{P_0}(B, B) \\to H^{-2}(L_{B/A}) \\to 0 $$ In this way we obtain a direct proof of a consequence of Quillen's spectral sequence discussed in Remark \\ref{remark-elucidate-ss}."}
+{"_id": "11266", "title": "cotangent-remark-functoriality-lichtenbaum-schlessinger", "text": "Consider a commutative square $$ \\xymatrix{ A' \\ar[r] & B' \\\\ A \\ar[u] \\ar[r] & B \\ar[u] } $$ of ring maps. Choose a factorization $$ \\xymatrix{ A' \\ar[r] & P' \\ar[r] & B' \\\\ A \\ar[u] \\ar[r] & P \\ar[u] \\ar[r] & B \\ar[u] } $$ with $P$ a polynomial algebra over $A$ and $P'$ a polynomial algebra over $A'$. Choose generators $f_t$, $t \\in T$ for $\\Ker(P \\to B)$. For $t \\in T$ denote $f'_t$ the image of $f_t$ in $P'$. Choose $f'_s \\in P'$ such that the elements $f'_t$ for $t \\in T' = T \\amalg S$ generate the kernel of $P' \\to B'$. Set $F = \\bigoplus_{t \\in T} P$ and $F' = \\bigoplus_{t' \\in T'} P'$. Let $Rel = \\Ker(F \\to P)$ and $Rel' = \\Ker(F' \\to P')$ where the maps are given by multiplication by $f_t$, resp.\\ $f'_t$ on the coordinates. Finally, set $TrivRel$, resp.\\ $TrivRel'$ equal to the submodule of $Rel$, resp.\\ $TrivRel$ generated by the elements $(\\ldots, f_{t'}, 0, \\ldots, 0, -f_t, 0, \\ldots)$ for $t, t' \\in T$, resp.\\ $T'$. Having made these choices we obtain a canonical commutative diagram $$ \\xymatrix{ L' : & Rel'/TrivRel' \\ar[r] & F' \\otimes_{P'} B' \\ar[r] & \\Omega_{P'/A'} \\otimes_{P'} B' \\\\ L : \\ar[u] & Rel/TrivRel \\ar[r] \\ar[u] & F \\otimes_P B \\ar[r] \\ar[u] & \\Omega_{P/A} \\otimes_P B \\ar[u] } $$ Moreover, tracing through the choices made in the proof of Lemma \\ref{lemma-compare-higher} the reader sees that one obtains a commutative diagram $$ \\xymatrix{ L_{B'/A'} \\ar[r] & L' \\\\ L_{B/A} \\ar[r] \\ar[u] & L \\ar[u] } $$"}
+{"_id": "11267", "title": "cotangent-remark-map-sections-over-U", "text": "It is clear from the proof of Lemma \\ref{lemma-compute-L-morphism-sheaves-rings} that for any $U \\in \\Ob(\\mathcal{C})$ there is a canonical map $L_{\\mathcal{B}(U)/\\mathcal{A}(U)} \\to L_{\\mathcal{B}/\\mathcal{A}}(U)$ of complexes of $\\mathcal{B}(U)$-modules. Moreover, these maps are compatible with restriction maps and the complex $L_{\\mathcal{B}/\\mathcal{A}}$ is the sheafification of the rule $U \\mapsto L_{\\mathcal{B}(U)/\\mathcal{A}(U)}$."}
+{"_id": "11268", "title": "cotangent-remark-compute-L-pi-shriek", "text": "In the situation above, for every $U \\subset X$ open let $P_{\\bullet, U}$ be the standard resolution of $\\mathcal{O}_X(U)$ over $\\Lambda$. Set $\\mathbf{A}_{n, U} = \\Spec(P_{n, U})$. Then $\\mathbf{A}_{\\bullet, U}$ is a cosimplicial object of the fibre category $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ of $\\mathcal{C}_{X/\\Lambda}$ over $U$. Moreover, as discussed in Remark \\ref{remark-resolution} we have that $\\mathbf{A}_{\\bullet, U}$ is a cosimplicial object of $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ as in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Since the construction $U \\mapsto \\mathbf{A}_{\\bullet, U}$ is functorial in $U$, given any (abelian) sheaf $\\mathcal{F}$ on $\\mathcal{C}_{X/\\Lambda}$ we obtain a complex of presheaves $$ U \\longmapsto \\mathcal{F}(\\mathbf{A}_{\\bullet, U}) $$ whose cohomology groups compute the homology of $\\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek} that the sheafification computes $L_n\\pi_!(\\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \\mapsto \\mathcal{F}(\\mathbf{A}_{n, U})$ computes $L\\pi_!(\\mathcal{F})$."}
+{"_id": "11269", "title": "cotangent-remark-compute-L-pi-shriek-spaces", "text": "In the situation above, for every object $U \\to X$ of $X_\\etale$ let $P_{\\bullet, U}$ be the standard resolution of $\\mathcal{O}_X(U)$ over $\\Lambda$. Set $\\mathbf{A}_{n, U} = \\Spec(P_{n, U})$. Then $\\mathbf{A}_{\\bullet, U}$ is a cosimplicial object of the fibre category $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ of $\\mathcal{C}_{X/\\Lambda}$ over $U$. Moreover, as discussed in Remark \\ref{remark-resolution} we have that $\\mathbf{A}_{\\bullet, U}$ is a cosimplicial object of $\\mathcal{C}_{\\mathcal{O}_X(U)/\\Lambda}$ as in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}. Since the construction $U \\mapsto \\mathbf{A}_{\\bullet, U}$ is functorial in $U$, given any (abelian) sheaf $\\mathcal{F}$ on $\\mathcal{C}_{X/\\Lambda}$ we obtain a complex of presheaves $$ U \\longmapsto \\mathcal{F}(\\mathbf{A}_{\\bullet, U}) $$ whose cohomology groups compute the homology of $\\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compute-left-derived-pi-shriek} that the sheafification computes $L_n\\pi_!(\\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \\mapsto \\mathcal{F}(\\mathbf{A}_{n, U})$ computes $L\\pi_!(\\mathcal{F})$."}
+{"_id": "11349", "title": "spaces-cohomology-remark-variant", "text": "In Lemmas \\ref{lemma-check-separated-dvr} and \\ref{lemma-check-proper-dvr} it suffices to consider complete discrete valuation rings. To be precise in Lemma \\ref{lemma-check-separated-dvr} we can replace condition (3) by the following condition: Given any commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y } $$ where $A$ is a complete discrete valuation ring with fraction field $K$ there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma \\ref{lemma-check-separated-dvr} (3) the completion $A^\\wedge$ is a discrete valuation ring (More on Algebra, Lemma \\ref{more-algebra-lemma-completion-dvr}) and the uniqueness of the arrow $\\Spec(A^\\wedge) \\to X$ implies the uniqueness of the arrow $\\Spec(A) \\to X$ for example by Properties of Spaces, Proposition \\ref{spaces-properties-proposition-sheaf-fpqc}. Similarly in Lemma \\ref{lemma-check-proper-dvr} we can replace condition (3) by the following condition: Given any commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\ \\Spec(A) \\ar[r] & Y } $$ where $A$ is a complete discrete valuation ring with fraction field $K$ there exists an extension $A \\subset A'$ of complete discrete valuation rings inducing a fraction field extension $K \\subset K'$ such that there exists a unique arrow $\\Spec(A') \\to X$ making the diagram $$ \\xymatrix{ \\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\ \\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y } $$ commute. Namely, given any diagram as in Lemma \\ref{lemma-check-proper-dvr} part (3) the existence of any commutative diagram $$ \\xymatrix{ \\Spec(L) \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\ \\Spec(B) \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y } $$ for {\\it any} extension $A \\subset B$ of discrete valuation rings will imply there exists an arrow $\\Spec(A) \\to X$ fitting into the diagram. This was shown in Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-push-down-solution}. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field."}
+{"_id": "11427", "title": "artin-remark-deformation-category-implies", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a field of finite type over $S$ and $x_0$ an object of $\\mathcal{X}$ over $k$. Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be as in (\\ref{equation-predeformation-category}). If $\\mathcal{F}$ is a deformation category, i.e., if $\\mathcal{F}$ satisfies the Rim-Schlessinger condition (RS), then we see that $\\mathcal{F}$ satisfies Schlessinger's conditions (S1) and (S2) by Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-RS-implies-S1-S2}. Let $\\overline{\\mathcal{F}}$ be the functor of isomorphism classes, see Formal Deformation Theory, Remarks \\ref{formal-defos-remarks-cofibered-groupoids} (\\ref{formal-defos-item-associated-functor-isomorphism-classes}). Then $\\overline{\\mathcal{F}}$ satisfies (S1) and (S2) as well, see Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-S1-S2-associated-functor}. This holds in particular in the situation of Lemma \\ref{lemma-deformation-category}."}
+{"_id": "11428", "title": "artin-remark-formal-objects-match", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\xi = (R, \\xi_n, f_n)$ be a formal object. Set $k = R/\\mathfrak m$ and $x_0 = \\xi_1$. The formal object $\\xi$ defines a formal object $\\xi$ of the predeformation category $\\mathcal{F}_{\\mathcal{X}, k, x_0}$. This follows immediately from Definition \\ref{definition-formal-objects} above, Formal Deformation Theory, Definition \\ref{formal-defos-definition-formal-objects}, and our construction of the predeformation category $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ in Section \\ref{section-predeformation-categories}."}
+{"_id": "11429", "title": "artin-remark-strong-effectiveness", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume we have \\begin{enumerate} \\item an affine open $\\Spec(\\Lambda) \\subset S$, \\item an inverse system $(R_n)$ of $\\Lambda$-algebras with surjective transition maps whose kernels are locally nilpotent, \\item a system $(\\xi_n)$ of objects of $\\mathcal{X}$ lying over the system $(\\Spec(R_n))$. \\end{enumerate} In this situation, set $R = \\lim R_n$. We say that $(\\xi_n)$ is {\\it effective} if there exists an object $\\xi$ of $\\mathcal{X}$ over $\\Spec(R)$ whose restriction to $\\Spec(R_n)$ gives the system $(\\xi_n)$."}
+{"_id": "11430", "title": "artin-remark-trade-openness-versality-diagonal-with-strong-effectiveness", "text": "There is a way to deduce openness of versality of the diagonal of an category fibred in groupoids from a strong formal effectiveness axiom. Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume \\begin{enumerate} \\item $\\Delta_\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ has (RS*), \\item $\\mathcal{X}$ is limit preserving, \\item given an inverse system $(R_n)$ of $S$-algebras as in Remark \\ref{remark-strong-effectiveness} where $\\Ker(R_m \\to R_n)$ is an ideal of square zero for all $m \\geq n$ the functor $$ \\mathcal{X}_{\\Spec(\\lim R_n)} \\longrightarrow \\lim_n \\mathcal{X}_{\\Spec(R_n)} $$ is fully faithful. \\end{enumerate} Then $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ satisfies openness of versality. This follows by applying Lemma \\ref{lemma-SGE-implies-openness-versality} to fibre products of the form $\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}, y} (\\Sch/V)_{fppf}$ for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\\mathcal{X} \\times \\mathcal{X}$ over $V$. If we ever need this, we will change this remark into a lemma and provide a detailed proof."}
+{"_id": "11431", "title": "artin-remark-functoriality", "text": "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}. Suppose $A \\to B$ is a ring map and $y = x|_{\\Spec(B)}$. Let $M \\in \\text{Mod}_A$, $N \\in \\text{Mod}_B$ and let $M \\to N$ an $A$-linear map. Then there are canonical maps $\\text{Inf}_x(M) \\to \\text{Inf}_y(N)$ and $T_x(M) \\to T_y(N)$ simply because there is a pullback functor $$ \\textit{Lift}(x, A[M]) \\to \\textit{Lift}(y, B[N]) $$ coming from the ring map $A[M] \\to B[N]$. Similarly, given a morphism of deformation situations $(y, B' \\to B) \\to (x, A' \\to A)$ we obtain a pullback functor $\\textit{Lift}(x, A') \\to \\textit{Lift}(y, B')$. Since the construction of the action, the addition, and the scalar multiplication on $\\text{Inf}_x$ and $T_x$ use only morphisms in the categories of lifts (see proof of Formal Deformation Theory, Lemma \\ref{formal-defos-lemma-linear-functor}) we see that the constructions above are functorial. In other words we obtain $A$-linear maps $$ \\text{Inf}_x(M) \\to \\text{Inf}_y(N) \\quad\\text{and}\\quad T_x(M) \\to T_y(N) $$ such that the diagrams $$ \\vcenter{ \\xymatrix{ \\text{Inf}_y(J) \\ar[r] & \\text{Inf}(y'/y) \\\\ \\text{Inf}_x(I) \\ar[r] \\ar[u] & \\text{Inf}(x'/x) \\ar[u] } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ T_y(J) \\times \\text{Lift}(y, B') \\ar[r] & \\text{Lift}(y, B') \\\\ T_x(I) \\times \\text{Lift}(x, A') \\ar[r] \\ar[u] & \\text{Lift}(x, A') \\ar[u] } } $$ commute. Here $I = \\Ker(A' \\to A)$, $J = \\Ker(B' \\to B)$, $x'$ is a lift of $x$ to $A'$ (which may not always exist) and $y' = x'|_{\\Spec(B')}$."}
+{"_id": "11432", "title": "artin-remark-automorphisms", "text": "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}. Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition map $$ \\text{Inf}(x'/x) \\times \\Mor_{\\textit{Lift}(x, A')}(x', x'') \\times \\text{Inf}(x''/x) \\longrightarrow \\Mor_{\\textit{Lift}(x, A')}(x', x''). $$ Since $\\textit{Lift}(x, A')$ is a groupoid, if $\\Mor_{\\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines a simply transitive left action of $\\text{Inf}(x'/x)$ on $\\Mor_{\\textit{Lift}(x, A')}(x', x'')$ and a simply transitive right action by $\\text{Inf}(x''/x)$. Now the lemma says that $\\text{Inf}(x'/x) = \\text{Inf}_x(I) = \\text{Inf}(x''/x)$. We claim that the two actions described above agree via these identifications. Namely, either $x' \\not \\cong x''$ in which the claim is clear, or $x' \\cong x''$ and in that case we may assume that $x'' = x'$ in which case the result follows from the fact that $\\text{Inf}(x'/x)$ is commutative. In particular, we obtain a well defined action $$ \\text{Inf}_x(I) \\times \\Mor_{\\textit{Lift}(x, A')}(x', x'') \\longrightarrow \\Mor_{\\textit{Lift}(x, A')}(x', x'') $$ which is simply transitive as soon as $\\Mor_{\\textit{Lift}(x, A')}(x', x'')$ is nonempty."}
+{"_id": "11433", "title": "artin-remark-short-exact-sequence-thickenings", "text": "Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $A$ be an $S$-algebra. There is a notion of a {\\it short exact sequence} $$ (x, A_1' \\to A) \\to (x, A_2' \\to A) \\to (x, A_3' \\to A) $$ of deformation situations: we ask the corresponding maps between the kernels $I_i = \\Ker(A_i' \\to A)$ give a short exact sequence $$ 0 \\to I_3 \\to I_2 \\to I_1 \\to 0 $$ of $A$-modules. Note that in this case the map $A_3' \\to A_1'$ factors through $A$, hence there is a canonical isomorphism $A_1' = A[I_1]$."}
+{"_id": "11434", "title": "artin-remark-compare-deformation-spaces", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ has (RS*). Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of $\\mathcal{X}$ over $\\Spec(k)$. Then we have equalities of $k$-vector spaces $$ T\\mathcal{F}_{\\mathcal{X}, k, x_0} = T_{x_0}(k) \\quad\\text{and}\\quad \\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) = \\text{Inf}_{x_0}(k) $$ where the spaces on the left hand side of the equality signs are given in (\\ref{equation-tangent-space}) and (\\ref{equation-infinitesimal-automorphisms}) and the spaces on the right hand side are given by Lemma \\ref{lemma-properties-lift-RS-star}."}
+{"_id": "11435", "title": "artin-remark-canonical-element", "text": "Assumptions and notation as in Lemma \\ref{lemma-properties-lift-RS-star}. Choose an affine open $\\Spec(\\Lambda) \\subset S$ such that $\\Spec(A) \\to S$ corresponds to a ring map $\\Lambda \\to A$. Consider the ring map $$ A \\longrightarrow A[\\Omega_{A/\\Lambda}], \\quad a \\longmapsto (a, \\text{d}_{A/\\Lambda}(a)) $$ Pulling back $x$ along the corresponding morphism $\\Spec(A[\\Omega_{A/\\Lambda}]) \\to \\Spec(A)$ we obtain a deformation $x_{can}$ of $x$ over $A[\\Omega_{A/\\Lambda}]$. We call this the {\\it canonical element} $$ x_{can} \\in T_x(\\Omega_{A/\\Lambda}) = \\text{Lift}(x, A[\\Omega_{A/\\Lambda}]). $$ Next, assume that $\\Lambda$ is Noetherian and $\\Lambda \\to A$ is of finite type. Let $k = \\kappa(\\mathfrak p)$ be a residue field at a finite type point $u_0$ of $U = \\Spec(A)$. Let $x_0 = x|_{u_0}$. By (RS*) and the fact that $A[k] = A \\times_k k[k]$ the space $T_x(k)$ is the tangent space to the deformation functor $\\mathcal{F}_{\\mathcal{X}, k, x_0}$. Via $$ T\\mathcal{F}_{U, k, u_0} = \\text{Der}_\\Lambda(A, k) = \\Hom_A(\\Omega_{A/\\Lambda}, k) $$ (see Formal Deformation Theory, Example \\ref{formal-defos-example-tangent-space-prorepresentable-functor}) and functoriality of $T_x$ the canonical element produces the map on tangent spaces induced by the object $x$ over $U$. Namely, $\\theta \\in T\\mathcal{F}_{U, k, u_0}$ maps to $T_x(\\theta)(x_{can})$ in $T_x(k) = T\\mathcal{F}_{\\mathcal{X}, k, x_0}$."}
+{"_id": "11436", "title": "artin-remark-canonical-isomorphism", "text": "Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra such that $\\Spec(A) \\to S$ maps into an affine open and let $x, y$ be objects of $\\mathcal{X}$ over $\\Spec(A)$. Further, let $A \\to B$ be a ring map and let $\\alpha : x|_{\\Spec(B)} \\to y|_{\\Spec(B)}$ be a morphism of $\\mathcal{X}$ over $\\Spec(B)$. Consider the ring map $$ B \\longrightarrow B[\\Omega_{B/A}], \\quad b \\longmapsto (b, \\text{d}_{B/A}(b)) $$ Pulling back $\\alpha$ along the corresponding morphism $\\Spec(B[\\Omega_{B/A}]) \\to \\Spec(B)$ we obtain a morphism $\\alpha_{can}$ between the pullbacks of $x$ and $y$ over $B[\\Omega_{B/A}]$. On the other hand, we can pullback $\\alpha$ by the morphism $\\Spec(B[\\Omega_{B/A}]) \\to \\Spec(B)$ corresponding to the injection of $B$ into the first summand of $B[\\Omega_{B/A}]$. By the discussion of Remark \\ref{remark-automorphisms} we can take the difference $$ \\varphi(x, y, \\alpha) = \\alpha_{can} - \\alpha|_{\\Spec(B[\\Omega_{B/A}])} \\in \\text{Inf}_{x|_{\\Spec(B)}}(\\Omega_{B/A}). $$ We will call this the {\\it canonical automorphism}. It depends on all the ingredients $A$, $x$, $y$, $A \\to B$ and $\\alpha$."}
+{"_id": "11437", "title": "artin-remark-no-fibre-products", "text": "The site $(\\textit{Noetherian}/S)_\\tau$ does not have fibre products. Hence we have to be careful in working with sheaves. For example, the continuous inclusion functor $(\\textit{Noetherian}/S)_\\tau \\to (\\Sch/S)_\\tau$ does not define a morphism of sites. See Examples, Section \\ref{examples-section-sheaves-locally-Noetherian} for an example in case $\\tau = fppf$."}
+{"_id": "11438", "title": "artin-remark-G-rings", "text": "In particular, we cannot prove that the desired result is true for every Situation \\ref{situation-contractions} because we will need to assume the local rings of $S$ are G-rings. If you can prove the result in general or if you have a counter example, please let us know at \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "11439", "title": "artin-remark-how-to-think-compatibility", "text": "In Situation \\ref{situation-contractions} let $V$ be a locally Noetherian scheme over $S$. Let $(Z, u', \\hat x)$ be a triple satisfying (1), (2), and (3) above. We want to explain a way to think about the compatibility condition (4). It will not be mathematically precise as we are going use a fictitious category $\\textit{An}_S$ of analytic spaces over $S$ and a fictitious analytification functor $$ \\left\\{ \\begin{matrix} \\text{locally Noetherian formal} \\\\ \\text{algebraic spaces over }S \\end{matrix} \\right\\} \\longrightarrow \\textit{An}_S, \\quad\\quad Y \\longmapsto Y^{an} $$ For example if $Y = \\text{Spf}(k[[t]])$ over $S = \\Spec(k)$, then $Y^{an}$ should be thought of as an open unit disc. If $Y = \\Spec(k)$, then $Y^{an}$ is a single point. The category $\\textit{An}_S$ should have open and closed immersions and we should be able to take the open complement of a closed. Given $Y$ the morphism $Y_{red} \\to Y$ should induces a closed immersion $Y_{red}^{an} \\to Y^{an}$. We set $Y^{rig} = Y^{an} \\setminus Y_{red}^{an}$ equal to its open complement. If $Y$ is an algebraic space and if $Z \\subset Y$ is closed, then the morphism $Y_{/Z} \\to Y$ should induce an open immersion $Y_{/Z}^{an} \\to Y^{an}$ which in turn should induce an open immersion $$ can : (Y_{/Z})^{rig} \\longrightarrow (Y \\setminus Z)^{an} $$ Also, given a formal modification $g : Y' \\to Y$ of locally Noetherian formal algebraic spaces, the induced morphism $g^{rig} : (Y')^{rig} \\to Y^{rig}$ should be an isomorphism. Given $\\text{An}_S$ and the analytification functor, we can consider the requirement that $$ \\xymatrix{ (V_{/Z})^{rig} \\ar[rr]_{can} \\ar[d]_{(g^{rig})^{-1} \\circ \\hat x^{an}} & & (V \\setminus Z)^{an} \\ar[d]^{(u')^{an}} \\\\ (X'_{/T'})^{rig} \\ar[rr]^{can} & & (X' \\setminus T')^{an} } $$ commutes. This makes sense as $g^{rig} : (X'_{T'})^{rig} \\to W^{rig}$ is an isomorphism and $U' = X' \\setminus T'$. Finally, under some assumptions of faithfulness of the analytification functor, this requirement will be equivalent to the compatibility condition formulated above. We hope this will motivate the reader to think of the compatibility of $u'$ and $\\hat x$ as the requirement that some maps be equal, rather than asking for the existence of a certain commutative diagram."}
+{"_id": "11440", "title": "artin-remark-diagonal", "text": "In Situation \\ref{situation-contractions}. Let $V$ be a locally Noetherian scheme over $S$. Let $(Z_i, u'_i, \\hat x_i) \\in F(V)$ for $i = 1, 2$. Let $V'_i \\to V$, $\\hat x'_i$ and $x'_i$ witness the compatibility between $u'_i$ and $\\hat x_i$ for $i = 1, 2$. \\medskip\\noindent Set $V' = V'_1 \\times_V V'_2$. Let $E' \\to V'$ denote the equalizer of the morphisms $$ V' \\to V'_1 \\xrightarrow{x'_1} X' \\quad\\text{and}\\quad V' \\to V'_2 \\xrightarrow{x'_2} X' $$ Set $Z = Z_1 \\cap Z_2$. Let $E_W \\to V_{/Z}$ be the equalizer of the morphisms $$ V_{/Z} \\to V_{/Z_1} \\xrightarrow{\\hat x_1} W \\quad\\text{and}\\quad V_{/Z} \\to V_{/Z_2} \\xrightarrow{\\hat x_2} W $$ Observe that $E' \\to V$ is separated and locally of finite type and that $E_W$ is a locally Noetherian formal algebraic space separated over $V$. The compatibilities between the various morphisms involved show that \\begin{enumerate} \\item $\\Im(E' \\to V) \\cap (Z_1 \\cup Z_2)$ is contained in $Z = Z_1 \\cap Z_2$,  \\item the morphism $E' \\times_V (V \\setminus Z) \\to V \\setminus Z$ is a monomorphism and is equal to the equalizer of the restrictions of $u'_1$ and $u'_2$ to $V \\setminus (Z_1 \\cup Z_2)$, \\item the morphism $E'_{/Z} \\to V_{/Z}$ factors through $E_W$ and the diagram $$ \\xymatrix{ E'_{/Z} \\ar[r] \\ar[d] & X'_{/T'} \\ar[d]^g \\\\ E_W \\ar[r] & W } $$ is cartesian. In particular, the morphism $E'_{/Z} \\to E_W$ is a formal modification as the base change of $g$, \\item $E'$, $(E' \\to V)^{-1}Z$, and $E'_{/Z} \\to E_W$ is a triple as in Situation \\ref{situation-contractions} with base scheme the locally Noetherian scheme $V$, \\item given a morphism $\\varphi : A \\to V$ of locally Noetherian schemes, the following are equivalent \\begin{enumerate} \\item $(Z_1, u'_1, \\hat x_1)$ and $(Z_2, u'_2, \\hat x_2)$ restrict to the same element of $F(A)$, \\item $A \\setminus \\varphi^{-1}(Z) \\to V \\setminus Z$ factors through $E' \\times_V (V \\setminus Z)$ and $A_{/\\varphi^{-1}(Z)} \\to V_{/Z}$ factors through $E_W$. \\end{enumerate} \\end{enumerate} We conclude, using Lemmas \\ref{lemma-solution} and \\ref{lemma-functor-is-solution}, that if there is a solution $E \\to V$ for the triple in (4), then $E$ represents $F \\times_{\\Delta, F \\times F} V$ on the category of locally Noetherian schemes over $V$."}
+{"_id": "11441", "title": "artin-remark-separated-needed", "text": "The proof of Theorem \\ref{theorem-contractions} uses that $X'$ and $W$ are separated over $S$ in two places. First, the proof uses this in showing $\\Delta : F \\to F \\times F$ is representable by algebraic spaces. This use of the assumption can be entirely avoided by proving that $\\Delta$ is representable by applying the theorem in the separated case to the triples $E'$, $(E' \\to V)^{-1}Z$, and $E'_{/Z} \\to E_W$ found in Remark \\ref{remark-diagonal} (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma \\ref{lemma-formal-object-effective} is the only place in our proof of Theorem \\ref{theorem-contractions} where we really need to use that $X' \\to S$ is separated. The reader checks that we use the assumption only to obtain the morphism $x' : V' \\to X'$. The existence of $x'$ can be shown, using results in the literature, if $X' \\to S$ is quasi-separated, see More on Morphisms of Spaces, Remark \\ref{spaces-more-morphisms-remark-weaken-separation-axioms-question}. We conclude the theorem holds as stated with ``separated'' replaced by ``quasi-separated''. If we ever need this we will precisely state and carefully prove this here."}
+{"_id": "11527", "title": "obsolete-remark-composition-of-adjoints-isomorphic-to-identity", "text": "The information which used to be contained in this remark is now subsumed in the combination of Categories, Lemmas \\ref{categories-lemma-adjoint-fully-faithful} and \\ref{categories-lemma-left-adjoint-composed-fully-faithful}."}
+{"_id": "11528", "title": "obsolete-remark-weak-serre-subcategory", "text": "The following remarks are obsolete as they are subsumed in Homology, Lemmas \\ref{homology-lemma-biregular-ss-converges} and \\ref{homology-lemma-first-quadrant-ss}. Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{C} \\subset \\mathcal{A}$ be a weak Serre subcategory (see Homology, Definition \\ref{homology-definition-serre-subcategory}). Suppose that $K^{\\bullet, \\bullet}$ is a double complex to which Homology, Lemma \\ref{homology-lemma-first-quadrant-ss} applies such that for some $r \\geq 0$ all the objects ${}'E_r^{p, q}$ belong to $\\mathcal{C}$. Then all the cohomology groups $H^n(sK^\\bullet)$ belong to $\\mathcal{C}$. Namely, the assumptions imply that the kernels and images of ${}'d_r^{p, q}$ are in $\\mathcal{C}$. Whereupon we see that each ${}'E_{r + 1}^{p, q}$ is in $\\mathcal{C}$. By induction we see that each ${}'E_\\infty^{p, q}$ is in $\\mathcal{C}$. Hence each $H^n(sK^\\bullet)$ has a finite filtration whose subquotients are in $\\mathcal{C}$. Using that $\\mathcal{C}$ is closed under extensions we conclude that $H^n(sK^\\bullet)$ is in $\\mathcal{C}$ as claimed. The same result holds for the second spectral sequence associated to $K^{\\bullet, \\bullet}$. Similarly, if $(K^\\bullet, F)$ is a filtered complex to which Homology, Lemma \\ref{homology-lemma-biregular-ss-converges} applies and for some $r \\geq 0$ all the objects $E_r^{p, q}$ belong to $\\mathcal{C}$, then each $H^n(K^\\bullet)$ is an object of $\\mathcal{C}$."}
+{"_id": "11529", "title": "obsolete-remark-projective-resolution", "text": "Let $R$ be a ring. For any set $S$ we let $F(S)$ denote the free $R$-module on $S$. Then any left $R$-module has the following two step resolution $$ F(M \\times M) \\oplus F(R \\times M) \\to F(M) \\to M \\to 0. $$ The first map is given by the rule $$ [m_1, m_2] \\oplus [r, m] \\mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m]. $$"}
+{"_id": "11530", "title": "obsolete-remark-section-colimits", "text": "This reference/tag used to refer to a Section in the chapter Smoothing Ring Maps, but the material has since been subsumed in Algebra, Section \\ref{algebra-section-colimits-flat}."}
+{"_id": "11531", "title": "obsolete-remark-algebra", "text": "Let $R$ be a ring. Suppose that we have $F \\in R[X, Y]_d$ and $G \\in R[X, Y]_e$ such that, setting $S = R[X, Y]/(F)$ we have (1) $S_n$ is finite locally free of rank $d$ for all $n \\geq d$, and (2) multiplication by $G$ defines isomorphisms $S_n \\to S_{n + e}$ for all $n \\geq d$. In this case we may define a finite, locally free $R$-algebra $A$ as follows: \\begin{enumerate} \\item as an $R$-module $A = S_{ed}$, and \\item multiplication $A \\times A \\to A$ is given by the rule that $H_1 H_2 = H_3$ if and only if $G^d H_3 = H_1 H_2$ in $S_{2ed}$. \\end{enumerate} This makes sense because multiplication by $G^d$ induces a bijective map $S_{de} \\to S_{2de}$. It is easy to see that this defines a ring structure. Note the confusing fact that the element $G^d$ defines the unit element of the ring $A$."}
+{"_id": "11532", "title": "obsolete-remark-equation-derivatives", "text": "This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition \\ref{restricted-proposition-approximate} which became unused because of a rearrangement of the material."}
+{"_id": "11533", "title": "obsolete-remark-equation-ci", "text": "This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition \\ref{restricted-proposition-approximate} which became unused because of a rearrangement of the material."}
+{"_id": "11534", "title": "obsolete-remark-equation-in-ideal", "text": "This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition \\ref{restricted-proposition-approximate} which became unused because of a rearrangement of the material."}
+{"_id": "11535", "title": "obsolete-remark-equation-derivatives-analogue", "text": "This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition \\ref{restricted-proposition-approximate} which became unused because of a rearrangement of the material."}
+{"_id": "11536", "title": "obsolete-remark-equation-go-down", "text": "This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Lemma \\ref{restricted-lemma-lift-approximation} which became unused because of a rearrangement of the material."}
+{"_id": "11537", "title": "obsolete-remark-from-shriek-to-star", "text": "Let $U$ be an object of $\\mathcal{C}$. For any abelian sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$ one may wonder whether there is a canonical map $$ c : j_{U!}\\mathcal{G} \\longrightarrow j_{U*}\\mathcal{G} $$ To construct such a thing is the same as constructing a map $j_U^{-1}j_{U!}\\mathcal{G} \\to \\mathcal{G}$. Note that restriction commutes with sheafification. Thus we can use the presheaf of Modules on Sites, Lemma \\ref{sites-modules-lemma-extension-by-zero}. Hence it suffices to define for $V/U$ a map $$ \\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) \\longrightarrow \\mathcal{G}(V/U) $$ compatible with restrictions. It looks like we can take the which is zero on all summands except for the one where $\\varphi$ is the structure morphism $\\varphi_0 : V \\to U$ where we take $1$. However, this isn't compatible with restriction mappings: namely, if $\\alpha : V' \\to V$ is a morphism of $\\mathcal{C}$, then denote $V'/U$ the object of $\\mathcal{C}/U$ with structure morphism $\\varphi'_0 = \\varphi_0 \\circ \\alpha$. We need to check that the diagram $$ \\xymatrix{ \\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) \\ar[d] \\ar[r] & \\mathcal{G}(V/U) \\ar[d] \\\\ \\bigoplus\\nolimits_{\\varphi' \\in \\Mor_\\mathcal{C}(V', U)} \\mathcal{G}(V' \\xrightarrow{\\varphi'} U) \\ar[r] & \\mathcal{G}(V'/U) } $$ commutes. The problem here is that there may be a morphism $\\varphi : V \\to U$ different from $\\varphi_0$ such that $\\varphi \\circ \\alpha = \\varphi'_0$. Thus the left vertical arrow will send the summand corresponding to $\\varphi$ into the summand on which the lower horizontal arrow is equal to $1$ and almost surely the diagram doesn't commute."}
+{"_id": "11538", "title": "obsolete-remark-pullback-K-flat", "text": "This remark used to discuss what we know about pullbacks of K-flat complexes being K-flat or not, but is now obsoleted by Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-pullback-K-flat}."}
+{"_id": "11539", "title": "obsolete-remark-cohomology-topics", "text": "This tag used to refer to a section of the chapter on cohomology listing topics to be treated."}
+{"_id": "11540", "title": "obsolete-remark-sites-cohomology-topics", "text": "This tag used to refer to a section of the chapter on cohomology listing topics to be treated."}
+{"_id": "11541", "title": "obsolete-remark-V-implies-C", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}."}
+{"_id": "11542", "title": "obsolete-remark-V-implies-cohomology", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}."}
+{"_id": "11543", "title": "obsolete-remark-induction-step-V-C", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-compare-qc-zar}."}
+{"_id": "11544", "title": "obsolete-remark-V-implies-C-etale-fppf", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}."}
+{"_id": "11545", "title": "obsolete-remark-V-implies-cohomology-etale-fppf", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}."}
+{"_id": "11546", "title": "obsolete-remark-induction-step-V-C-etale-fppf", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-fppf-etale}."}
+{"_id": "11547", "title": "obsolete-remark-V-implies-C-etale-ph", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}."}
+{"_id": "11548", "title": "obsolete-remark-V-implies-cohomology-etale-ph", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}."}
+{"_id": "11549", "title": "obsolete-remark-V-implies-cohomology-etale-ph-extra", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-extra-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}."}
+{"_id": "11550", "title": "obsolete-remark-make-class-zero", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-make-class-zero-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}."}
+{"_id": "11551", "title": "obsolete-remark-induction-step-V-C-etale-ph", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-ph-etale}."}
+{"_id": "11552", "title": "obsolete-remark-V-implies-C-etale-h", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}."}
+{"_id": "11553", "title": "obsolete-remark-V-implies-cohomology-etale-h", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}."}
+{"_id": "11554", "title": "obsolete-remark-V-implies-cohomology-etale-h-extra", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-V-implies-cohomology-extra-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}."}
+{"_id": "11555", "title": "obsolete-remark-induction-step-V-C-etale-h", "text": "This tag used to refer to the special case of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-induction-step-V-C-general} pertaining to the situation described in \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-compare-h-etale}."}
+{"_id": "11556", "title": "obsolete-remark-how-used", "text": "This tag used to be in the chapter on \\'etale cohomology, but is no longer suitable there because of a reorganization. The content of the tag was the following: \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-when-ctf} can be used to prove that if $f : X \\to Y$ is a separated, finite type morphism of schemes and $Y$ is Noetherian, then $Rf_!$ induces a functor $D_{ctf}(X_\\etale, \\Lambda) \\to D_{ctf}(Y_\\etale, \\Lambda)$. An example of this argument, when $Y$ is the spectrum of a field and $X$ is a curve is given in The Trace Formula, Proposition \\ref{trace-proposition-projective-curve-constructible-cohomology}."}
+{"_id": "11557", "title": "obsolete-remark-proof-works-when", "text": "This remark used to discuss to what extend the original proof of Lemma \\ref{lemma-sheaf-fpqc-quasi-separated} (of December 18, 2009) generalizes."}
+{"_id": "11558", "title": "obsolete-remark-very-reasonable-Zariski-locally-quasi-separated", "text": "Very reasonable algebraic spaces form a strictly larger collection than Zariski locally quasi-separated algebraic spaces. Consider an algebraic space of the form $X = [U/G]$ (see Spaces, Definition \\ref{spaces-definition-quotient}) where $G$ is a finite group acting without fixed points on a non-quasi-separated scheme $U$. Namely, in this case $U \\times_X U = U \\times G$ and clearly both projections to $U$ are quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal $U \\times_X U \\to U \\times U$ is not quasi-compact, hence this algebraic space is not quasi-separated. Now, take $U$ the infinite affine space over a field $k$ of characteristic $\\not = 2$ with zero doubled, see Schemes, Example \\ref{schemes-example-not-quasi-separated}. Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \\{+1, -1\\}$, and let $-1$ act by $-1$ on all coordinates, and by switching $0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally quasi-separated (details omitted)."}
+{"_id": "11559", "title": "obsolete-remark-different-topologies", "text": "We obtain a second topology $\\tau_Y$ on $\\mathcal{C}_{X/Y}$ by taking the topology inherited from $Y_{Zar}$. There is a third topology $\\tau_{X \\to Y}$ where a family of morphisms $\\{(U_i \\to A_i) \\to (U \\to A)\\}$ is a covering if and only if $U = \\bigcup U_i$, $V = \\bigcup V_i$ and $A_i \\cong V_i \\times_V A$. This is the topology inherited from the topology on the site $(X/Y)_{Zar}$ whose underlying category is the category of pairs $(U, V)$ as in Lemma \\ref{lemma-category-fibred} part (3). The coverings of $(X/Y)_{Zar}$ are families $\\{(U_i, V_i) \\to (U, V)\\}$ such that $U = \\bigcup U_i$ and $V = \\bigcup V_i$. There are morphisms of topoi $$ \\xymatrix{ \\Sh(\\mathcal{C}_{X/Y}) = \\Sh(\\mathcal{C}_{X/Y}, \\tau_X) & \\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y}) \\ar[l] \\ar[r] & \\Sh(\\mathcal{C}_{X/Y}, \\tau_Y) } $$ (recall that $\\tau_X$ is our ``default'' topology). The pullback functors for these arrows are sheafification and pushforward is the identity on underlying presheaves. The diagram of topoi $$ \\xymatrix{ \\Sh(X_{Zar}) \\ar[d]^f & \\Sh(\\mathcal{C}_{X/Y}) \\ar[l]^\\pi & \\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y}) \\ar[l] \\ar[d] \\\\ \\Sh(Y_{Zar}) & & \\Sh(\\mathcal{C}_{X/Y}, \\tau_Y) \\ar[ll] } $$ is {\\bf not} commutative. Namely, the pullback of a nonzero abelian sheaf on $Y$ is a nonzero abelian sheaf on $(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y})$, but we can certainly find examples where such a sheaf pulls back to zero on $X$. Note that any presheaf $\\mathcal{F}$ on $Y_{Zar}$ gives a sheaf $\\underline{\\mathcal{F}}$ on $\\mathcal{C}_{Y/X}$ by the rule which assigns to $(U \\to A/V)$ the set $\\mathcal{F}(V)$. Even if $\\mathcal{F}$ happens to be a sheaf it isn't true in general that $\\underline{\\mathcal{F}} = \\pi^{-1}f^{-1}\\mathcal{F}$. This is related to the noncommutativity of the diagram above, as we can describe $\\underline{\\mathcal{F}}$ as the pushforward of the pullback of $\\mathcal{F}$ to $\\Sh(\\mathcal{C}_{X/Y}, \\tau_{X \\to Y})$ via the lower horizontal and right vertical arrows. An example is the sheaf $\\underline{\\mathcal{O}}_Y$. But what is true is that there is a map $\\underline{\\mathcal{F}} \\to \\pi^{-1}f^{-1}\\mathcal{F}$ which is transformed (as we shall see later) into an isomorphism after applying $\\pi_!$."}
+{"_id": "11560", "title": "obsolete-remark-construction-E", "text": "Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces over $S$. Let $U$ be another algebraic space over $B$. Denote $q : X \\times_B U \\to U$ the second projection. Consider the distinguished triangle $$ Lq^*L_{U/B} \\to L_{X \\times_B U/B} \\to E \\to Lq^*L_{U/B}[1] $$ of Cotangent, Section \\ref{cotangent-section-fibre-product}. For any sheaf $\\mathcal{F}$ of $\\mathcal{O}_{X \\times_B U}$-modules we have the Atiyah class $$ \\mathcal{F} \\to L_{X \\times_B U/B} \\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} \\mathcal{F}[1] $$ see Cotangent, Section \\ref{cotangent-section-atiyah-general}. We can compose this with the map to $E$ and choose a distinguished triangle $$ E(\\mathcal{F}) \\to \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} E[1] \\to E(\\mathcal{F})[1] $$ in $D(\\mathcal{O}_{X \\times_B U})$. By construction the Atiyah class lifts to a map $$ e_\\mathcal{F} : E(\\mathcal{F}) \\longrightarrow Lq^*L_{U/B} \\otimes_{\\mathcal{O}_{X \\times_B U}}^\\mathbf{L} \\mathcal{F}[1] $$ fitting into a morphism of distinguished triangles $$ \\xymatrix{ \\mathcal{F} \\otimes^\\mathbf{L} Lq^*L_{U/B}[1] \\ar[r] & \\mathcal{F} \\otimes^\\mathbf{L} L_{X \\times_B U/B}[1] \\ar[r] & \\mathcal{F} \\otimes^\\mathbf{L} E[1] \\\\ E(\\mathcal{F}) \\ar[r] \\ar[u]^{e_\\mathcal{F}} & \\mathcal{F} \\ar[r] \\ar[u]^{Atiyah} & \\mathcal{F} \\otimes^\\mathbf{L} E[1] \\ar[u]^{=} } $$ Given $S, B, X, f, U, \\mathcal{F}$ we fix a choice of $E(\\mathcal{F})$ and $e_\\mathcal{F}$."}
+{"_id": "11561", "title": "obsolete-remark-construction-ob", "text": "With notation as in Remark \\ref{remark-construction-E} let $i : U \\to U'$ be a first order thickening of $U$ over $B$. Let $\\mathcal{I} \\subset \\mathcal{O}_{U'}$ be the quasi-coherent sheaf of ideals cutting out $B$ in $B'$. The fundamental triangle $$ Li^*L_{U'/B} \\to L_{U/B} \\to L_{U/U'} \\to Li^*L_{U'/B}[1] $$ together with the map $L_{U/U'} \\to \\mathcal{I}[1]$ determine a map $e_{U'} : L_{U/B} \\to \\mathcal{I}[1]$. Combined with the map $e_\\mathcal{F}$ of the previous remark we obtain $$ (\\text{id}_\\mathcal{F} \\otimes Lq^*e_{U'}) \\cup e_\\mathcal{F} : E(\\mathcal{F}) \\longrightarrow \\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}} q^*\\mathcal{I}[2] $$ (we have also composed with the map from the derived tensor product to the usual tensor product). In other words, we obtain an element $$ \\xi_{U'} \\in \\Ext^2_{\\mathcal{O}_{X \\times_B U}}( E(\\mathcal{F}), \\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_B U}} q^*\\mathcal{I}) $$"}
+{"_id": "11562", "title": "obsolete-remark-not-true-not-quasi-compact", "text": "This remark used to say that it wasn't clear whether the arrows of Chow Homology, Lemma \\ref{chow-lemma-cycles-k-group} were isomorphisms in general. However, we've now found a proof of this fact."}
+{"_id": "11563", "title": "obsolete-remark-tangent-spaces", "text": "You got here because of a duplicate tag. Please see Formal Deformation Theory, Section \\ref{formal-defos-section-tangent-spaces} for the actual content."}
+{"_id": "11564", "title": "obsolete-remark-examples-formal-defos", "text": "This tag used to point to a section describing several examples of deformation problems. These now each have their own section. See Deformation Problems, Sections \\ref{examples-defos-section-finite-projective-modules}, \\ref{examples-defos-section-representations}, \\ref{examples-defos-section-continuous-representations}, and \\ref{examples-defos-section-graded-algebras}."}
+{"_id": "11631", "title": "stacks-sheaves-remark-ambiguity", "text": "We only use this notation when the symbol $\\mathcal{X}$ refers to a category fibred in groupoids, and not a scheme, an algebraic space, etc. In this way we will avoid confusion with the small \\'etale site of a scheme, or algebraic space which is denoted $X_\\etale$ (in which case we use a roman capital instead of a calligraphic one)."}
+{"_id": "11632", "title": "stacks-sheaves-remark-flat", "text": "In the situation of Lemma \\ref{lemma-functoriality-structure-sheaf} the morphism of ringed topoi $f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$ is flat as is clear from the equality $f^{-1}\\mathcal{O}_\\mathcal{X} = \\mathcal{O}_\\mathcal{Y}$. This is a bit counter intuitive, for example because a closed immersion of algebraic stacks is typically not flat (as a morphism of algebraic stacks). However, exactly the same thing happens when taking a closed immersion $i : X \\to Y$ of schemes: in this case the associated morphism of big $\\tau$-sites $i : (\\Sch/X)_\\tau \\to (\\Sch/Y)_\\tau$ also is flat."}
+{"_id": "11633", "title": "stacks-sheaves-remark-cech-complex-presheaves", "text": "We can define the complex $\\mathcal{K}^\\bullet(f, \\mathcal{F})$ also if $\\mathcal{F}$ is a presheaf, only we cannot use the reference to Sites, Section \\ref{sites-section-pullback} to define the pullback maps. To explain the pullback maps, suppose given a commutative diagram $$ \\xymatrix{ \\mathcal{V} \\ar[rd]_g \\ar[rr]_h & &  \\mathcal{U} \\ar[ld]^f \\\\ & \\mathcal{X} } $$ of categories fibred in groupoids over $(\\Sch/S)_{fppf}$ and a presheaf $\\mathcal{G}$ on $\\mathcal{U}$ we can define the pullback map $f_*\\mathcal{G} \\to g_*h^{-1}\\mathcal{G}$ as the composition $$ f_*\\mathcal{G} \\longrightarrow f_*h_*h^{-1}\\mathcal{G} = g_*h^{-1}\\mathcal{G} $$ where the map comes from the adjunction map $\\mathcal{G} \\to h_*h^{-1}\\mathcal{G}$. This works because in our situation the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with their counter parts on sheaves). See Sections \\ref{section-presheaves} and \\ref{section-sheaves}."}
+{"_id": "11634", "title": "stacks-sheaves-remark-cech-complex-sections", "text": "Let us ``compute'' the value of the relative {\\v C}ech complex on an object $x$ of $\\mathcal{X}$. Say $p(x) = U$. Consider the $2$-fibre product diagram (which serves to introduce the notation $g : \\mathcal{V} \\to \\mathcal{Y}$) $$ \\xymatrix{ \\mathcal{V} \\ar@{=}[r] \\ar[d]_g & (\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U} \\ar[r] \\ar[d] & \\mathcal{U} \\ar[d]^f \\\\ \\mathcal{Y} \\ar@{=}[r] & (\\Sch/U)_{fppf} \\ar[r]^-x & \\mathcal{X} } $$ Note that the morphism $\\mathcal{V}_n \\to \\mathcal{U}_n$ of the proof of Lemma \\ref{lemma-generalities} induces an equivalence $\\mathcal{V}_n = (\\Sch/U)_{fppf} \\times_{x, \\mathcal{X}} \\mathcal{U}_n$. Hence we see from (\\ref{equation-pushforward}) that $$ \\Gamma(x, \\mathcal{K}^\\bullet(f, \\mathcal{F})) = \\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, x^{-1}\\mathcal{F}) $$ In words: The value of the relative {\\v C}ech complex on an object $x$ of $\\mathcal{X}$ is the {\\v C}ech complex of the base change of $f$ to $\\mathcal{X}/x \\cong (\\Sch/U)_{fppf}$. This implies for example that Lemma \\ref{lemma-homotopy} implies Lemma \\ref{lemma-homotopy-sheafified} and more generally that results on the (usual) {\\v C}ech complex imply results for the relative {\\v C}ech complex."}
+{"_id": "11716", "title": "resolve-remark-compare-Garel", "text": "Let $\\mathbf{F}_p \\subset \\Lambda \\subset R \\subset S$ and $\\text{Tr}$ be as in Lemma \\ref{lemma-trace-higher}. By de Rham Cohomology, Proposition \\ref{derham-proposition-Garel} there is a canonical map of complexes $$ \\Theta_{S/R} : \\Omega_{S/\\Lambda}^\\bullet \\longrightarrow \\Omega_{R/\\Lambda}^\\bullet $$ The computation in de Rham Cohomology, Example \\ref{derham-example-Garel} shows that $\\Theta_{S/R}(x^i \\text{d}x) = \\text{Tr}_x(x^i\\text{d}x)$ for all $i$. Since $\\text{Trace}_{S/R} = \\Theta^0_{S/R}$ is identically zero and since $$ \\Theta_{S/R}(a \\wedge b) = a \\wedge \\Theta_{S/R}(b) $$ for $a \\in \\Omega^i_{R/\\Lambda}$ and $b \\in \\Omega^j_{S/\\Lambda}$ it follows that $\\text{Tr} = \\Theta_{S/R}$. The advantage of using $\\text{Tr}$ is that it is a good deal more elementary to construct."}
+{"_id": "11717", "title": "resolve-remark-dualizing-setup", "text": "Let $X$ be an integral Noetherian normal scheme of dimension $2$. In this case the following are equivalent \\begin{enumerate} \\item $X$ has a dualizing complex $\\omega_X^\\bullet$, \\item there is a coherent $\\mathcal{O}_X$-module $\\omega_X$ such that $\\omega_X[n]$ is a dualizing complex, where $n$ can be any integer. \\end{enumerate} This follows from the fact that $X$ is Cohen-Macaulay (Properties, Lemma \\ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}) and Duality for Schemes, Lemma \\ref{duality-lemma-dualizing-module-CM-scheme}. In this situation we will say that $\\omega_X$ is a {\\it dualizing module} in accordance with Duality for Schemes, Section \\ref{duality-section-dualizing-module}. In particular, when $A$ is a Noetherian normal local domain of dimension $2$, then we say {\\it $A$ has a dualizing module $\\omega_A$} if the above is true. In this case, if $X \\to \\Spec(A)$ is a normal modification, then $X$ has a dualizing module too, see Duality for Schemes, Example \\ref{duality-example-proper-over-local}. In this situation we always denote $\\omega_X$ the dualizing module normalized with respect to $\\omega_A$, i.e., such that $\\omega_X[2]$ is the dualizing complex normalized relative to $\\omega_A[2]$. See Duality for Schemes, Section \\ref{duality-section-glue}."}
+{"_id": "11718", "title": "resolve-remark-pic-blowup", "text": "Let $b : X \\to X'$ be the contraction of an exceptional curve of the first kind $E \\subset X$. From Lemma \\ref{lemma-pic-blowup} we obtain an identification $$ \\Pic(X) = \\Pic(X') \\oplus \\mathbf{Z} $$ where $\\mathcal{L}$ corresponds to the pair $(\\mathcal{L}', n)$ if and only if $\\mathcal{L} = (b^*\\mathcal{L}')(-nE)$, i.e., $\\mathcal{L}(nE) = b^*\\mathcal{L}'$. In fact the proof of Lemma \\ref{lemma-pic-blowup} shows that $\\mathcal{L}' = b_*\\mathcal{L}(nE)$. Of course the assignment $\\mathcal{L} \\mapsto \\mathcal{L}'$ is a group homomorphism."}
+{"_id": "11758", "title": "exercises-remark-simple-geometric", "text": "Of course the idea of this exercise is to find a simple argument in each case rather than applying a ``big'' theorem. Nonetheless it is good to be guided by general principles."}
+{"_id": "11759", "title": "exercises-remark-flat-not-projective", "text": "If $M$ is of finite presentation and flat over $A$, then $M$ is projective over $A$. Thus your example will have to involve a ring $A$ which is not Noetherian. I know of an example where $A$ is the ring of ${\\mathcal C}^\\infty$-functions on ${\\mathbf R}$."}
+{"_id": "11760", "title": "exercises-remark-flat-given-residue-field-extension-general", "text": "The same result holds for arbitrary field extensions $k \\subset K$."}
+{"_id": "11761", "title": "exercises-remark-not-hausdorff", "text": "Usually the word compact is reserved for quasi-compact and Hausdorff spaces."}
+{"_id": "11762", "title": "exercises-remark-singularities", "text": "A singularity on a curve over a field $k$ is called an ordinary double point if the complete local ring of the curve at the point is of the form $k'[[x, y]]/(f)$, where (a) $k'$ is a finite separable extension of $k$, (b) the initial term of $f$ has degree two, i.e., it looks like $q = ax^2 + bxy + cy^2$ for some $a, b, c\\in k'$ not all zero, and (c) $q$ is a nondegenerate quadratic form over $k'$ (in char 2 this means that $b$ is not zero). In general there is one isomorphism class of such rings for each isomorphism class of pairs $(k', q)$."}
+{"_id": "11763", "title": "exercises-remark-HNSS", "text": "Let $k$ be a field. Then for every integer $n\\in {\\mathbf N}$ and every maximal ideal ${\\mathfrak m} \\subset k[x_1, \\ldots, x_n]$ the quotient $k[x_1, \\ldots, x_n]/{\\mathfrak m}$ is a finite field extension of $k$. This will be shown later in the course. Of course (please check this) it implies a similar statement for maximal ideals of finitely generated $k$-algebras. The exercise above proves it in the case $k = {\\mathbf C}$."}
+{"_id": "11764", "title": "exercises-remark-Hilbert-Nullstellensatz", "text": "This is the Hilbert Nullstellensatz. Namely it says that the closed subsets of $\\Spec(k[x_1, \\ldots, x_n])$ (which correspond to radical ideals by a previous exercise) are determined by the closed points contained in them."}
+{"_id": "11765", "title": "exercises-remark-cover", "text": "In algebraic geometric language this means that the property of ``being finitely generated'' or ``being flat'' is local for the Zariski topology (in a suitable sense). You can also show this for the property ``being of finite presentation''."}
+{"_id": "11766", "title": "exercises-remark-elimination-theory", "text": "Finding the image as above usually is done by using elimination theory."}
+{"_id": "11767", "title": "exercises-remark-continuous-proj-spec", "text": "There is a continuous map $ \\text{Proj}(R) \\longrightarrow \\Spec(R_0) $."}
+{"_id": "11768", "title": "exercises-remark-CM-dim-1-embedding-dim-2", "text": "This suggests that a local Noetherian Cohen-Macaulay ring of dimension 1 and embedding dimension 2 is of the form $B/FB$, where $B$ is a 2-dimensional regular local ring. This is more or less true (under suitable ``niceness'' properties of the ring)."}
+{"_id": "11769", "title": "exercises-remark-strange-fp", "text": "Let $h \\in {\\mathbf Z}[y]$ be a monic polynomial of degree $d$. Then: \\begin{enumerate} \\item The map $A = {\\mathbf Z}[x] \\to B ={\\mathbf Z}[y]$, $x \\mapsto h$ is finite locally free of rank $d$. \\item For all primes $p$ the map $A_p = {\\mathbf F}_p[x]\\to B_p = {\\mathbf F}_p[y]$, $y \\mapsto h(y) \\bmod p$ is finite locally free of rank $d$. \\end{enumerate}"}
+{"_id": "11770", "title": "exercises-remark-direct-sum-stalk-abelian", "text": "Let $X$ be a topological space. In the category of abelian sheaves the direct sum of a family of sheaves $\\{{\\mathcal F}_i\\}_{i\\in I}$ is the sheaf associated to the presheaf $U \\mapsto \\oplus {\\mathcal F}_i(U)$. Consequently the stalk of the direct sum at a point $x$ is the direct sum of the stalks of the ${\\mathcal F}_i$ at $x$."}
+{"_id": "11771", "title": "exercises-remark-open-immersion", "text": "When $(X, {\\mathcal O}_X)$ is a ringed space and $U \\subset X$ is an open subset then $(U, {\\mathcal O}_X|_U)$ is a ringed space. Notation: ${\\mathcal O}_U = {\\mathcal O}_X|_U$. There is a canonical morphisms of ringed spaces $$ j : (U, {\\mathcal O}_U) \\longrightarrow (X, {\\mathcal O}_X). $$ If $(X, {\\mathcal O}_X)$ is a locally ringed space, so is $(U, {\\mathcal O}_U)$ and $j$ is a morphism of locally ringed spaces. If $(X, {\\mathcal O}_X)$ is a scheme so is $(U, {\\mathcal O}_U)$ and $j$ is a morphism of schemes. We say that $(U, {\\mathcal O}_U)$ is an {\\it open subscheme} of $(X, {\\mathcal O}_X)$ and that $j$ is an {\\it open immersion}. More generally, any morphism $j' : (V, {\\mathcal O}_V) \\to (X, {\\mathcal O}_X)$ that is {\\it isomorphic} to a morphism $j : (U, {\\mathcal O}_U) \\to (X, {\\mathcal O}_X)$ as above is called an open immersion."}
+{"_id": "11772", "title": "exercises-remark-separated-base-fibre-product-affines-affine", "text": "It turns out this cannot happen with $S$ separated. Do you know why?"}
+{"_id": "11773", "title": "exercises-remark-affine-dimension", "text": "If your scheme is affine then dimension is the same as the Krull dimension of the underlying ring. So you can use last semesters results to compute dimension."}
+{"_id": "11774", "title": "exercises-remark-tsen", "text": "Exercise \\ref{exercise-has-rational-section} is a special case of ``Tsen's theorem''. Exercise \\ref{exercise-no-section-curve} shows that the method is limited to low degree equations (conics when the base and fibre have dimension 1)."}
+{"_id": "11775", "title": "exercises-remark-interpretation-skolem-noether", "text": "The interpretation of the results of Exercise \\ref{exercise-for-number-theorists} and \\ref{exercise-quasi-section} is that given the morphism $X \\to S$ all of whose fibres are nonempty, there exists a finite surjective morphism $S' \\to S$ such that the base change $X_{S'} \\to S'$ does have a section. This is not a general fact, but it holds if the base is the spectrum of a dedekind ring with finite residue fields at closed points, and the morphism $X \\to S$ is flat with geometrically irreducible generic fibre. See Exercise \\ref{exercise-no-quasi-section} below for an example where it doesn't work."}
+{"_id": "11776", "title": "exercises-remark-tangent-space-relative", "text": "Exercise \\ref{exercise-compute-TS} explains why it is necessary to consider the tangent space of $X$ over $S$ to get a good notion."}
+{"_id": "11777", "title": "exercises-remark-extend-off-open", "text": "If $U \\to X$ is a quasi-compact immersion then any quasi-coherent sheaf on $U$ is the restriction of a quasi-coherent sheaf on $X$. If $X$ is a Noetherian scheme, and $U \\subset X$ is open, then any coherent sheaf on $U$ is the restriction of a coherent sheaf on $X$. Of course the exercise above is easier, and shouldn't use these general facts."}
+{"_id": "11778", "title": "exercises-remark-fitting-omega-not-sings", "text": "The $k$th Fitting ideal of $\\Omega_{X/S}$ is commonly used to define the singular scheme of the morphism $X \\to S$ when $X$ has relative dimension $k$ over $S$. But as part (a) shows, you have to be careful doing this when your family does not have ``constant'' fibre dimension, e.g., when it is not flat. As part (b) shows, flatness doesn't guarantee it works either (and yes this is a flat family). In ``good cases'' -- such as in (c) -- for families of curves you expect the $0$-th Fitting ideal to be zero and the $1$st Fitting ideal to define (scheme-theoretically) the singular locus."}
+{"_id": "11779", "title": "exercises-remark-invertible-projective-space", "text": "Let $k$ be a field. Let $\\mathbf{P}^2_k = \\text{Proj}(k[X_0, X_1, X_2])$. Any invertible sheaf on $\\mathbf{P}^2_k$ is isomorphic to $\\mathcal{O}_{\\mathbf{P}^2_k}(n)$ for some $n \\in \\mathbf{Z}$. Recall that $$ \\Gamma(\\mathbf{P}^2_k, \\mathcal{O}_{\\mathbf{P}^2_k}(n)) = k[X_0, X_1, X_2]_n $$ is the degree $n$ part of the polynomial ring. For a quasi-coherent sheaf $\\mathcal{F}$ on $\\mathbf{P}^2_k$ set $\\mathcal{F}(n) = \\mathcal{F} \\otimes_{\\mathcal{O}_{\\mathbf{P}^2_k}} \\mathcal{O}_{\\mathbf{P}^2_k}(n)$ as usual."}
+{"_id": "11780", "title": "exercises-remark-recall-dimension-theory", "text": "Freely use the following facts on dimension theory (and add more if you need more). \\begin{enumerate} \\item The dimension of a scheme is the supremum of the length of chains of irreducible closed subsets. \\item The dimension of a finite type scheme over a field is the maximum of the dimensions of its affine opens. \\item The dimension of a Noetherian scheme is the maximum of the dimensions of its irreducible components. \\item The dimension of an affine scheme coincides with the dimension of the corresponding ring. \\item Let $k$ be a field and let $A$ be a finite type $k$-algebra. If $A$ is a domain, and $x \\not = 0$, then $\\dim(A) = \\dim(A/xA) + 1$. \\end{enumerate}"}
+{"_id": "11781", "title": "exercises-remark-chi", "text": "Given a projective scheme $X$ over a field $k$ and a coherent sheaf $\\mathcal{F}$ on $X$ we set $$ \\chi(X, \\mathcal{F}) = \\sum\\nolimits_{i \\geq 0} (-1)^i\\dim_k H^i(X, \\mathcal{F}). $$"}
+{"_id": "11782", "title": "exercises-remarks-divisors", "text": "Here are some trivial remarks. \\begin{enumerate} \\item On a Noetherian integral scheme $X$ the sheaf ${\\mathcal K}_X$ is constant with value the function field $K(X)$. \\item To make sense out of the definitions above one needs to show that $$ \\text{length}_{\\mathcal O}({\\mathcal O}/(ab)) = \\text{length}_{\\mathcal O}({\\mathcal O}/(a)) + \\text{length}_{\\mathcal O}({\\mathcal O}/(b)) $$ for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional local domain ${\\mathcal O}$. This will be done in the lectures. \\end{enumerate}"}
+{"_id": "11811", "title": "spaces-duality-remark-iso-on-RSheafHom", "text": "In the situation of Lemma \\ref{lemma-iso-on-RSheafHom} we have $$ DQ_Y(Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) = Rf_* DQ_X(R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) $$ by Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-pushforward-better-coherator}. Thus if $R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\in D_\\QCoh(\\mathcal{O}_X)$, then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow. On the other hand, if we know that $R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) \\in D_\\QCoh(\\mathcal{O}_Y)$, then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow. If both are true then we see that (\\ref{equation-sheafy-trace}) is an isomorphism. Combining this with Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-quasi-coherence-internal-hom} we see that $Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if \\begin{enumerate} \\item $L$ and $Rf_*L$ are perfect, or \\item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent. \\end{enumerate} For (2) we use that $a(K)$ is bounded below if $K$ is bounded below, see Lemma \\ref{lemma-twisted-inverse-image-bounded-below}."}
+{"_id": "11812", "title": "spaces-duality-remark-going-around", "text": "Let $S$ be a scheme. Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z } $$ of quasi-compact and quasi-separated algebraic spaces over $S$ where all squares are cartesian and where $(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are Tor independent pairs of maps. Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the right adjoints of Lemma \\ref{lemma-twisted-inverse-image} for $f$, $f'$, $f''$, $g$, $g'$, $g''$. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \\begin{matrix} A & B \\\\ C & D \\end{matrix} $$ Then the maps (\\ref{equation-base-change-map}) for the squares are (where we use $k^* = Lk^*$, etc) $$ \\begin{matrix} \\gamma_A : (k')^* \\circ a' \\to a'' \\circ (l')^* & \\gamma_B : k^* \\circ a \\to a' \\circ l^* \\\\ \\gamma_C : (l')^* \\circ b' \\to b'' \\circ (m')^* & \\gamma_D : l^* \\circ b \\to b' \\circ m^* \\end{matrix} $$ For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further base change maps $$ \\begin{matrix} \\gamma_{A + B} : (k \\circ k')^* \\circ a \\to a'' \\circ (l \\circ l')^* \\\\ \\gamma_{C + D} : (l \\circ l')^* \\circ b \\to b'' \\circ (m \\circ m')^* \\\\ \\gamma_{A + C} : (k')^* \\circ (a' \\circ b') \\to (a'' \\circ b'') \\circ (m')^* \\\\ \\gamma_{A + C} : k^* \\circ (a \\circ b) \\to (a' \\circ b') \\circ m^* \\end{matrix} $$ By Lemma \\ref{lemma-compose-base-change-maps-horizontal} we have $$ \\gamma_{A + B} = \\gamma_A \\circ \\gamma_B, \\quad \\gamma_{C + D} = \\gamma_C \\circ \\gamma_D $$ and by Lemma \\ref{lemma-compose-base-change-maps} we have $$ \\gamma_{A + C} = \\gamma_C \\circ \\gamma_A, \\quad \\gamma_{B + D} = \\gamma_D \\circ \\gamma_B $$ Here it would be more correct to write $\\gamma_{A + B} = (\\gamma_A \\star \\text{id}_{l^*}) \\circ (\\text{id}_{(k')^*} \\star \\gamma_B)$ with notation as in Categories, Section \\ref{categories-section-formal-cat-cat} and similarly for the others. However, we continue the abuse of notation used in the proofs of Lemmas \\ref{lemma-compose-base-change-maps} and \\ref{lemma-compose-base-change-maps-horizontal} of dropping $\\star$ products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Having said all of this we find (a priori) two transformations $$ (k')^* \\circ k^* \\circ a \\circ b \\longrightarrow a'' \\circ b'' \\circ (m')^* \\circ m^* $$ namely $$ \\gamma_C \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_B = \\gamma_{A + C} \\circ \\gamma_{B + D} $$ and $$ \\gamma_C \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_B = \\gamma_{C + D} \\circ \\gamma_{A + B} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \\xymatrix{ (k')^* \\circ a' \\circ l^* \\circ b \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} & (k')^* \\circ a' \\circ b' \\circ m^* \\ar[d]^{\\gamma_A} \\\\ a'' \\circ (l')^* \\circ l^* \\circ b \\ar[r]^{\\gamma_D} & a'' \\circ (l')^* \\circ b' \\circ m^* } $$ commutes. This is true by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \\ref{categories-definition-horizontal-composition}."}
+{"_id": "11950", "title": "spaces-properties-remark-list-properties-local-etale-topology", "text": "Here is a list of properties which are local for the \\'etale topology (keep in mind that the fpqc, fppf, syntomic, and smooth topologies are stronger than the \\'etale topology): \\begin{enumerate} \\item locally Noetherian, see Descent, Lemma \\ref{descent-lemma-Noetherian-local-fppf}, \\item Jacobson, see Descent, Lemma \\ref{descent-lemma-Jacobson-local-fppf}, \\item locally Noetherian and $(S_k)$, see Descent, Lemma \\ref{descent-lemma-Sk-local-syntomic}, \\item Cohen-Macaulay, see Descent, Lemma \\ref{descent-lemma-CM-local-syntomic}, \\item Gorenstein, see Duality for Schemes, Lemma \\ref{duality-lemma-gorenstein-local-syntomic}, \\item reduced, see Descent, Lemma \\ref{descent-lemma-reduced-local-smooth}, \\item normal, see Descent, Lemma \\ref{descent-lemma-normal-local-smooth}, \\item locally Noetherian and $(R_k)$, see Descent, Lemma \\ref{descent-lemma-Rk-local-smooth}, \\item regular, see Descent, Lemma \\ref{descent-lemma-regular-local-smooth}, \\item Nagata, see Descent, Lemma \\ref{descent-lemma-Nagata-local-smooth}. \\end{enumerate}"}
+{"_id": "11951", "title": "spaces-properties-remark-list-properties-local-ring-local-etale-topology", "text": "Let $P$ be a property of local rings. Assume that for any \\'etale ring map $A \\to B$ and $\\mathfrak q$ is a prime of $B$ lying over the prime $\\mathfrak p$ of $A$, then $P(A_\\mathfrak p) \\Leftrightarrow P(B_\\mathfrak q)$. Then we obtain an \\'etale local property of germs $(U, u)$ of schemes by setting $\\mathcal{P}(U, u) = P(\\mathcal{O}_{U, u})$. In this situation we will use the terminology ``the local ring of $X$ at $x$ has $P$'' to mean $X$ has property $\\mathcal{P}$ at $x$. Here is a list of such properties $P$: \\begin{enumerate} \\item Noetherian, see More on Algebra, Lemma \\ref{more-algebra-lemma-Noetherian-etale-extension}, \\item dimension $d$, see More on Algebra, Lemma \\ref{more-algebra-lemma-dimension-etale-extension}, \\item regular, see More on Algebra, Lemma \\ref{more-algebra-lemma-regular-etale-extension}, \\item discrete valuation ring, follows from (2), (3), and Algebra, Lemma \\ref{algebra-lemma-characterize-dvr}, \\item reduced, see More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-reduced}, \\item normal, see More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-normal}, \\item Noetherian and depth $k$, see More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-depth}, \\item Noetherian and Cohen-Macaulay, see More on Algebra, Lemma \\ref{more-algebra-lemma-henselization-CM}, \\item Noetherian and Gorenstein, see Dualizing Complexes, Lemma \\ref{dualizing-lemma-flat-under-gorenstein}. \\end{enumerate} There are more properties for which this holds, for example G-ring and Nagata. If we every need these we will add them here as well as references to detailed proofs of the corresponding algebra facts."}
+{"_id": "11952", "title": "spaces-properties-remark-cannot-decide-yet", "text": "Lemma \\ref{lemma-point-like-spaces} holds for decent algebraic spaces, see Decent Spaces, Lemma \\ref{decent-spaces-lemma-decent-point-like-spaces}. In fact a decent algebraic space with one point is a scheme, see Decent Spaces, Lemma \\ref{decent-spaces-lemma-when-field}. This also holds when $X$ is locally separated, because a locally separated algebraic space is decent, see Decent Spaces, Lemma \\ref{decent-spaces-lemma-locally-separated-decent}."}
+{"_id": "11953", "title": "spaces-properties-remark-explain-equivalence", "text": "Let us explain the meaning of Lemma \\ref{lemma-compare-etale-sites}. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a sheaf on the small \\'etale site $X_\\etale$ of $X$. The lemma says that there exists a unique sheaf $\\mathcal{F}'$ on $X_{spaces, \\etale}$ which restricts back to $\\mathcal{F}$ on the subcategory $X_\\etale$. If $U \\to X$ is an \\'etale morphism of algebraic spaces, then how do we compute $\\mathcal{F}'(U)$? Well, by definition of an algebraic space there exists a scheme $U'$ and a surjective \\'etale morphism $U' \\to U$. Then $\\{U' \\to U\\}$ is a covering in $X_{spaces, \\etale}$ and hence we get an equalizer diagram $$ \\xymatrix{ \\mathcal{F}'(U) \\ar[r] & \\mathcal{F}(U') \\ar@<1ex>[r] \\ar@<-1ex>[r] & \\mathcal{F}(U' \\times_U U'). } $$ Note that $U' \\times_U U'$ is a scheme, and hence we may write $\\mathcal{F}$ and not $\\mathcal{F}'$. Thus we see how to compute $\\mathcal{F}'$ when given the sheaf $\\mathcal{F}$."}
+{"_id": "11954", "title": "spaces-properties-remark-stalk-pullback", "text": "This remark is the analogue of \\'Etale Cohomology, Remark \\ref{etale-cohomology-remark-stalk-pullback}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\overline{x} : \\Spec(k) \\to X$ be a geometric point of $X$. By \\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-equivalence-sheaves-point} the category of sheaves on $\\Spec(k)_\\etale$ is equivalent to the category of sets (by taking a sheaf to its global sections). Hence it follows from Lemma \\ref{lemma-stalk-pullback} part (4) applied to the morphism $\\overline{x}$ that the functor $$ \\Sh(X_\\etale) \\longrightarrow \\textit{Sets}, \\quad \\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{x}} $$ is isomorphic to the functor $$ \\Sh(X_\\etale) \\longrightarrow \\Sh(\\Spec(k)_\\etale) = \\textit{Sets}, \\quad \\mathcal{F} \\longmapsto \\overline{x}^*\\mathcal{F} $$ Hence we may view the stalk functors as pullback functors along geometric morphisms (and not just some abstract morphisms of topoi as in the result of Lemma \\ref{lemma-stalk-gives-point})."}
+{"_id": "11955", "title": "spaces-properties-remark-map-stalks", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. We claim that for any pair of geometric points $\\overline{x}$ and $\\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\\overline{x}''$ so that there exists a commutative diagram $$ \\xymatrix{ \\overline{x}'' \\ar[r] \\ar[d] \\ar[rd]^{\\overline{x}''} & \\overline{x}' \\ar[d]^{\\overline{x}'} \\\\ \\overline{x} \\ar[r]^{\\overline{x}} & X. } $$ Since the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ is given by pullback along the morphism $\\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks."}
+{"_id": "12005", "title": "intersection-remark-trivial-generalization", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $I \\subset A$ be an ideal. The following are equivalent \\begin{enumerate} \\item $I' = I + \\text{Ann}(M)$ is an ideal of definition (Algebra, Definition \\ref{algebra-definition-ideal-definition}), \\item the image $\\overline{I}$ of $I$ in $\\overline{A} = A/\\text{Ann}(M)$ is an ideal of definition, \\item $\\text{Supp}(M/IM) \\subset \\{\\mathfrak m\\}$, \\item $\\dim(\\text{Supp}(M/IM)) \\leq 0$, and \\item $\\text{length}_A(M/IM) < \\infty$. \\end{enumerate} This follows from Algebra, Lemma \\ref{algebra-lemma-support-point} (details omitted). If this is the case we have $M/I^nM = M/(I')^nM$ for all $n$ and $M/I^nM = M/\\overline{I}^nM$ for all $n$ if $M$ is viewed as an $\\overline{A}$-module. Thus we can define $$ \\chi_{I, M}(n) = \\text{length}_A(M/I^nM) = \\sum\\nolimits_{p = 0, \\ldots, n - 1} \\text{length}_A(I^pM/I^{p + 1}M) $$ and we get $$ \\chi_{I, M}(n) = \\chi_{I', M}(n) = \\chi_{\\overline{I}, M}(n) $$ for all $n$ by the equalities above. All the results of Algebra, Section \\ref{algebra-section-Noetherian-local} and all the results in this section, have analogues in this setting. In particular we can define multiplicities $e_I(M, d)$ for $d \\geq \\dim(\\text{Supp}(M))$ and we have $$ \\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms} $$ as in the case where $I$ is an ideal of definition."}
+{"_id": "12006", "title": "intersection-remark-Serre-conjectures", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring. Let $M$ and $N$ be nonzero finite $A$-modules such that $M \\otimes_A N$ is supported in $\\{\\mathfrak m\\}$. Then $$ \\chi(M, N) = \\sum (-1)^i \\text{length}_A \\text{Tor}_i^A(M, N) $$ is finite. Let $r = \\dim(\\text{Supp}(M))$ and $s = \\dim(\\text{Supp}(N))$. In \\cite{Serre_algebre_locale} it is shown that $r + s \\leq \\dim(A)$ and the following conjectures are made: \\begin{enumerate} \\item if $r + s < \\dim(A)$, then $\\chi(M, N) = 0$, and \\item if $r + s = \\dim(A)$, then $\\chi(M, N) > 0$. \\end{enumerate} The arguments that prove Lemma \\ref{lemma-tor-sheaf} and Proposition \\ref{proposition-positivity} can be leveraged (as is done in Serre's text) to show that (1) and (2) are true if $A$ contains a field. Currently, conjecture (1) is known in general and it is known that $\\chi(M, N) \\geq 0$ in general (Gabber). Positivity is, as far as we know, still an open problem."}
+{"_id": "12007", "title": "intersection-remark-quasi-projective", "text": "Lemma \\ref{lemma-moving-move} and Theorem \\ref{theorem-well-defined} also hold for nonsingular quasi-projective varieties with the same proof. The only change is that one needs to prove the following version of the moving Lemma \\ref{lemma-moving}: Let  $X \\subset \\mathbf{P}^N$ be a closed subvariety. Let $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let $X^{reg} \\subset X$ be the open subset of nonsingular points. Let $Z \\subset X^{reg}$ be a closed subvariety of dimension $d$ and $T_i \\subset X^{reg}$, $i \\in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \\subset \\mathbf{P}^N$ such that $C$ intersects $X$ properly and such that $$ (C \\cdot X)|_{X^{reg}} = Z + \\sum\\nolimits_{j \\in J} m_j Z_j $$ where $Z_j \\subset X^{reg}$ are irreducible of dimension $d$, distinct from $Z$, and $$ \\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i) $$ with strict inequality if $Z$ does not intersect $T_i$ properly in $X^{reg}$."}
+{"_id": "12189", "title": "homology-remark-direct-sum", "text": "Note that the proof of Lemma \\ref{lemma-preadditive-direct-sum} shows that given $p$ and $q$ the morphisms $i$, $j$ are uniquely determined by the rules $p \\circ i = \\text{id}_x$, $q \\circ j = \\text{id}_y$, $p \\circ j = 0$, $q \\circ i = 0$. Moreover, we automatically have $i \\circ p + j \\circ q = \\text{id}_{x \\oplus y}$. Similarly, given $i$, $j$ the morphisms $p$ and $q$ are uniquely determined. Finally, given objects $x, y, z$ and morphisms $i : x \\to z$, $j : y \\to z$, $p : z \\to x$ and $q : z \\to y$ such that $p \\circ i = \\text{id}_x$, $q \\circ j = \\text{id}_y$, $p \\circ j = 0$, $q \\circ i = 0$ and $i \\circ p + j \\circ q = \\text{id}_z$, then $z$ is the direct sum of $x$ and $y$ with the four morphisms equal to $i, j, p, q$."}
+{"_id": "12190", "title": "homology-remark-direct-sums-not-exact", "text": "There are abelian categories $\\mathcal{A}$ having countable direct sums but where countable direct sums are not exact. An example is the opposite of the category of abelian sheaves on $\\mathbf{R}$. Namely, the category of abelian sheaves on $\\mathbf{R}$ has countable products, but countable products are not exact. For such a category the functor $\\text{Gr}(\\mathcal{A}) \\to \\mathcal{A}$, $(A^i) \\mapsto \\bigoplus A^i$ described above is not exact. It is still true that $\\text{Gr}(\\mathcal{A})$ is equivalent to the category of graded objects $(A, k)$ of $\\mathcal{A}$, but the kernel in the category of graded objects of a map $\\varphi : (A, k) \\to (B, k)$ is not equal to $\\Ker(\\varphi)$ endowed with a direct sum decomposition, but rather it is the direct sum of the kernels of the maps $k^iA \\to k^iB$."}
+{"_id": "12191", "title": "homology-remark-triple-complex", "text": "Let $\\mathcal{A}$ be an additive category. Let $A^{\\bullet, \\bullet, \\bullet}$ be a triple complex. The associated total complex is the complex with terms $$ \\text{Tot}^n(A^{\\bullet, \\bullet, \\bullet}) = \\bigoplus\\nolimits_{p + q + r = n} A^{p, q, r} $$ and differential $$ d^n_{\\text{Tot}(A^{\\bullet, \\bullet, \\bullet})} = \\sum\\nolimits_{p + q + r = n} d_1^{p, q, r} + (-1)^pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r} $$ With this definition a simple calculation shows that the associated total complex is equal to $$ \\text{Tot}(A^{\\bullet, \\bullet, \\bullet}) = \\text{Tot}(\\text{Tot}_{12}(A^{\\bullet, \\bullet, \\bullet})) = \\text{Tot}(\\text{Tot}_{23}(A^{\\bullet, \\bullet, \\bullet})) $$ In other words, we can either first combine the first two of the variables and then combine sum of those with the last, or we can first combine the last two variables and then combine the first with the sum of the last two."}
+{"_id": "12192", "title": "homology-remark-shift-double-complex", "text": "Let $\\mathcal{A}$ be an additive category. Let $A^{\\bullet, \\bullet}$ be a double complex with differentials $d_1^{p, q}$ and $d_2^{p, q}$. Denote $A^{\\bullet, \\bullet}[a, b]$ the double complex with $$ (A^{\\bullet, \\bullet}[a, b])^{p, q} = A^{p + a, q + b} $$ and differentials $$ d_{A^{\\bullet, \\bullet}[a, b], 1}^{p, q} = (-1)^a d_1^{p + a, q + b} \\quad\\text{and}\\quad d_{A^{\\bullet, \\bullet}[a, b], 2}^{p, q} = (-1)^b d_2^{p + a, q + b} $$ In this situation there is a well defined isomorphism $$ \\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[a + b] \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet}[a, b]) $$ which in degree $n$ is given by the map $$ \\xymatrix{ (\\text{Tot}(A^{\\bullet, \\bullet})[a + b])^n = \\bigoplus_{p + q = n + a + b} A^{p, q} \\ar[d]^{\\epsilon(p, q, a, b)\\text{id}_{A^{p, q}}} \\\\ \\text{Tot}(A^{\\bullet, \\bullet}[a, b])^n = \\bigoplus_{p' + q' = n} A^{p' + a, q' + b} } $$ for some sign $\\epsilon(p, q, a, b)$. Of course the summand $A^{p, q}$ maps to the summand $A^{p' + a, q' + b}$ when $p = p' + a$ and $q = q' + b$. To figure out the conditions on these signs observe that on the source we have $$ d|_{A^{p, q}} = (-1)^{a + b}\\left(d_1^{p, q} + (-1)^pd_2^{p, q}\\right) $$ whereas on the target we have $$ d|_{A^{p' + a, q' + b}} = (-1)^ad_1^{p' + a, q' + b} + (-1)^{p'}(-1)^bd_2^{p' + a, q' + b} $$ Thus our constraints are that $$ (-1)^a \\epsilon(p, q, a, b) = \\epsilon(p + 1, q, a, b)(-1)^{a + b} \\Leftrightarrow \\epsilon(p + 1, q, a, b) = (-1)^b \\epsilon(p, q, a, b) $$ and $$ (-1)^{p' + b}\\epsilon(p, q, a, b) = \\epsilon(p, q + 1, a, b) (-1)^{a + b + p} \\Leftrightarrow \\epsilon(p, q, a, b) = \\epsilon(p, q + 1, a, b) $$ Thus we choose $\\epsilon(p, q, a, b) = (-1)^{pb}$."}
+{"_id": "12193", "title": "homology-remark-double-complex-complex-of-complexes-first", "text": "Let $\\mathcal{A}$ be an additive category with countable direct sums. Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes. We can consider an object $A^{\\bullet, \\bullet}$ of $\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes as follows $$ \\ldots \\to A^{\\bullet, -1} \\to A^{\\bullet, 0} \\to A^{\\bullet, 1} \\to \\ldots $$ For the variant where we switch the role of the indices, see Remark \\ref{remark-double-complex-complex-of-complexes-second}. In this remark we show that taking the associated total complex is compatible with all the structures on complexes we have studied in the chapter so far. \\medskip\\noindent First, observe that the shift functor on double complexes viewed as complexes of complexes in the manner given above is the functor $[0, 1]$ defined in Remark \\ref{remark-shift-double-complex}. By Remark \\ref{remark-shift-double-complex} the functor $$ \\text{Tot} : \\text{DoubleComp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{A}) $$ is compatible with shift functors, in the sense that we have a functorial isomorphism $\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to \\text{Tot}(A^{\\bullet, \\bullet}[0, 1])$. \\medskip\\noindent Second, if $$ f, g : A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} $$ are homotopic when $f$ and $g$ are viewed as morphisms of complexes of complexes in the manner given above, then $$ \\text{Tot}(f), \\text{Tot}(g) : \\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet}) $$ are homotopic maps of complexes. Indeed, let $h = (h^q)$ be a homotopy between $f$ and $g$. If we denote $h^{p, q} : A^{p, q} \\to B^{p, q - 1}$ the component in degree $p$ of $h^q$, then this means that $$ f^{p, q} - g^{p, q} = d_2^{p, q - 1} \\circ h^{p, q} + h^{p, q + 1} \\circ d_2^{p, q} $$ The fact that $h^q : A^{\\bullet, q} \\to B^{\\bullet, q - 1}$ is a map of complexes means that $$ d_1^{p, q - 1} \\circ h^{p, q} = h^{p + 1, q} \\circ d_1^{p, q} $$ Let us define $h' = ((h')^n)$ the homotopy given by the maps $(h')^n : \\text{Tot}^n(A^{\\bullet, \\bullet}) \\to \\text{Tot}^{n - 1}(B^{\\bullet, \\bullet})$ using $(-1)^ph^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$. Then we see that $$ d_{\\text{Tot}(B^{\\bullet, \\bullet})} \\circ h' + h' \\circ d_{\\text{Tot}(A^{\\bullet, \\bullet})} $$ restricted to the summand $A^{p, q}$ is equal to $$ d_1^{p, q - 1} \\circ (-1)^p h^{p, q} + (-1)^p d_2^{p, q - 1} \\circ (-1)^p h^{p, q} + (-1)^{p + 1} h^{p + 1, q} \\circ d_1^{p, q} + (-1)^p h^{p, q + 1} \\circ (-1)^p d_2^{p, q} $$ which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above. This proves the second compatibility. \\medskip\\noindent Third, suppose that in the paragraph above we have $f = g$. Then the assignment $h \\leadsto h'$ above is compatible with the identification of Lemma \\ref{lemma-homotopy-shift-cochain}. More precisely, if we view $h$ as a morphism of complexes of complexes $A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}[0, -1]$ via this lemma then $$ \\text{Tot}(A^{\\bullet, \\bullet}) \\xrightarrow{\\text{Tot}(h)} \\text{Tot}(B^{\\bullet, \\bullet}[0, -1]) \\xrightarrow{\\gamma^{-1}} \\text{Tot}(B^{\\bullet, \\bullet})[-1] $$ is equal to $h'$ viewed as a morphism of complexes via the lemma. Here $\\gamma$ is the identification of Remark \\ref{remark-shift-double-complex}. The verification of this third point is immediate. \\medskip\\noindent Fourth, let $$ 0 \\to A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} \\to C^{\\bullet, \\bullet} \\to 0 $$ be a complex of double complexes and suppose we are given splittings $s^q : C^{\\bullet, q} \\to B^{\\bullet, q}$ and $\\pi^q : B^{\\bullet, q} \\to A^{\\bullet, q}$ of this as in Lemma \\ref{lemma-ses-termwise-split-cochain} when we view double complexes as complexes of complexes in the manner given above. This on the one hand produces a map $$ \\delta : C^{\\bullet, \\bullet} \\longrightarrow A^{\\bullet, \\bullet}[0, 1] $$ by the procedure in Lemma \\ref{lemma-ses-termwise-split-cochain}. On the other hand taking $\\text{Tot}$ we obtain a complex $$ 0 \\to \\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet}) \\to \\text{Tot}(C^{\\bullet, \\bullet}) \\to 0 $$ which is termwise split (see below) and hence comes with a morphism $$ \\delta' : \\text{Tot}(C^{\\bullet, \\bullet}) \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet})[1] $$ well defined up to homotopy by Lemmas \\ref{lemma-ses-termwise-split-cochain} and \\ref{lemma-ses-termwise-split-homotopy-cochain}. Claim: these maps agree in the sense that $$ \\text{Tot}(C^{\\bullet, \\bullet}) \\xrightarrow{\\text{Tot}(\\delta)} \\text{Tot}(A^{\\bullet, \\bullet}[0, 1]) \\xrightarrow{\\gamma^{-1}} \\text{Tot}(A^{\\bullet, \\bullet})[1] $$ is equal to $\\delta'$ where $\\gamma$ is as in Remark \\ref{remark-shift-double-complex}. To see this denote $s^{p, q} : C^{p, q} \\to B^{\\bullet, q}$ and $\\pi^{p, q} : B^{p, q} \\to A^{p, q}$ the components of $s^q$ and $\\pi^q$. As splittings $(s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to \\text{Tot}^n(B^{\\bullet, \\bullet})$ and $(\\pi')^n : \\text{Tot}^n(B^{\\bullet, \\bullet}) \\to \\text{Tot}^n(A^{\\bullet, \\bullet})$ we use the maps whose components are $s^{p, q}$ and $\\pi^{p, q}$ for $p + q = n$. We recall that $$ (\\delta')^n = (\\pi')^{n + 1} \\circ d_{\\text{Tot}(B^{\\bullet, \\bullet})}^n \\circ (s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to \\text{Tot}^{n + 1}(A^{\\bullet, \\bullet}) $$ The restriction of this to the summand $C^{p, q}$ is equal to $$ \\pi^{p + 1, q} \\circ d_1^{p, q} \\circ s^{p, q} + \\pi^{p, q + 1} \\circ (-1)^p d_2^{p, q} \\circ s^{p, q} = \\pi^{p, q + 1} \\circ (-1)^p d_2^{p, q} \\circ s^{p, q} $$ The equality holds because $s^q$ is a morphism of complexes (with $d_1$ as differential) and because $\\pi^{p + 1, q} \\circ s^{p + 1, q} = 0$ as $s$ and $\\pi$ correspond to a direct sum decomposition of $B$ in every bidegree. On the other hand, for $\\delta$ we have $$ \\delta^q =  \\pi^q \\circ d_2 \\circ s^q : C^{\\bullet, q} \\to A^{\\bullet, q + 1} $$ whose restriction to the summand $C^{p, q}$ is equal to $\\pi^{p, q + 1} \\circ d_2^{p, q} \\circ s^{p, q}$. The difference in signs is exactly canceled out by the sign of $(-1)^p$ in the isomorphism $\\gamma$ and the fourth claim is proven."}
+{"_id": "12194", "title": "homology-remark-double-complex-complex-of-complexes-second", "text": "Let $\\mathcal{A}$ be an additive category with countable direct sums. Let $\\text{DoubleComp}(\\mathcal{A})$ denote the category of double complexes. We can consider an object $A^{\\bullet, \\bullet}$ of $\\text{DoubleComp}(\\mathcal{A})$ as a complex of complexes as follows $$ \\ldots \\to A^{-1, \\bullet} \\to A^{0, \\bullet} \\to A^{1, \\bullet} \\to \\ldots $$ For the variant where we switch the role of the indices, see Remark \\ref{remark-double-complex-complex-of-complexes-first}. In this remark we show that taking the associated total complex is compatible with all the structures on complexes we have studied in the chapter so far. \\medskip\\noindent First, observe that the shift functor on double complexes viewed as complexes of complexes in the manner given above is the functor $[1, 0]$ defined in Remark \\ref{remark-shift-double-complex}. By Remark \\ref{remark-shift-double-complex} the functor $$ \\text{Tot} : \\text{DoubleComp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{A}) $$ is compatible with shift functors, in the sense that we have a functorial isomorphism $\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to \\text{Tot}(A^{\\bullet, \\bullet}[1, 0])$. \\medskip\\noindent Second, if $$ f, g : A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} $$ are homotopic when $f$ and $g$ are viewed as morphisms of complexes of complexes in the manner given above, then $$ \\text{Tot}(f), \\text{Tot}(g) : \\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet}) $$ are homotopic maps of complexes. Indeed, let $h = (h^p)$ be a homotopy between $f$ and $g$. If we denote $h^{p, q} : A^{p, q} \\to B^{p - 1, q}$ the component in degree $p$ of $h^q$, then this means that $$ f^{p, q} - g^{p, q} = d_1^{p - 1, q} \\circ h^{p, q} + h^{p + 1, q} \\circ d_1^{p, q} $$ The fact that $h^p : A^{p, \\bullet} \\to B^{p - 1, \\bullet}$ is a map of complexes means that $$ d_2^{p - 1, q} \\circ h^{p, q} = h^{p, q + 1} \\circ d_2^{p, q} $$ Let us define $h' = ((h')^n)$ the homotopy given by the maps $(h')^n : \\text{Tot}^n(A^{\\bullet, \\bullet}) \\to \\text{Tot}^{n - 1}(B^{\\bullet, \\bullet})$ using $h^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$. Then we see that $$ d_{\\text{Tot}(B^{\\bullet, \\bullet})} \\circ h' + h' \\circ d_{\\text{Tot}(A^{\\bullet, \\bullet})} $$ restricted to the summand $A^{p, q}$ is equal to $$ d_1^{p - 1, q} \\circ h^{p, q} + (-1)^{p - 1} d_2^{p - 1, q} \\circ h^{p, q} + h^{p + 1, q} \\circ d_1^{p, q} + h^{p, q + 1} \\circ (-1)^p d_2^{p, q} $$ which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above. This proves the second compatibility. \\medskip\\noindent Third, suppose that in the paragraph above we have $f = g$. Then the assignment $h \\leadsto h'$ above is compatible with the identification of Lemma \\ref{lemma-homotopy-shift-cochain}. More precisely, if we view $h$ as a morphism of complexes of complexes $A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet}[-1, 0]$ via this lemma then $$ \\text{Tot}(A^{\\bullet, \\bullet}) \\xrightarrow{\\text{Tot}(h)} \\text{Tot}(B^{\\bullet, \\bullet}[-1, 0]) \\xrightarrow{\\gamma^{-1}} \\text{Tot}(B^{\\bullet, \\bullet})[-1] $$ is equal to $h'$ viewed as a morphism of complexes via the lemma. Here $\\gamma$ is the identification of Remark \\ref{remark-shift-double-complex}. The verification of this third point is immediate. \\medskip\\noindent Fourth, let $$ 0 \\to A^{\\bullet, \\bullet} \\to B^{\\bullet, \\bullet} \\to C^{\\bullet, \\bullet} \\to 0 $$ be a complex of double complexes and suppose we are given splittings $s^p : C^{p, \\bullet} \\to B^{p, \\bullet}$ and $\\pi^p : B^{p, \\bullet} \\to A^{p, \\bullet}$ of this as in Lemma \\ref{lemma-ses-termwise-split-cochain} when we view double complexes as complexes of complexes in the manner given above. This on the one hand produces a map $$ \\delta : C^{\\bullet, \\bullet} \\longrightarrow A^{\\bullet, \\bullet}[0, 1] $$ by the procedure in Lemma \\ref{lemma-ses-termwise-split-cochain}. On the other hand taking $\\text{Tot}$ we obtain a complex $$ 0 \\to \\text{Tot}(A^{\\bullet, \\bullet}) \\to \\text{Tot}(B^{\\bullet, \\bullet}) \\to \\text{Tot}(C^{\\bullet, \\bullet}) \\to 0 $$ which is termwise split (see below) and hence comes with a morphism $$ \\delta' : \\text{Tot}(C^{\\bullet, \\bullet}) \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet})[1] $$ well defined up to homotopy by Lemmas \\ref{lemma-ses-termwise-split-cochain} and \\ref{lemma-ses-termwise-split-homotopy-cochain}. Claim: these maps agree in the sense that $$ \\text{Tot}(C^{\\bullet, \\bullet}) \\xrightarrow{\\text{Tot}(\\delta)} \\text{Tot}(A^{\\bullet, \\bullet}[1, 0]) \\xrightarrow{\\gamma^{-1}} \\text{Tot}(A^{\\bullet, \\bullet})[1] $$ is equal to $\\delta'$ where $\\gamma$ is as in Remark \\ref{remark-shift-double-complex}. To see this denote $s^{p, q} : C^{p, q} \\to B^{\\bullet, q}$ and $\\pi^{p, q} : B^{p, q} \\to A^{p, q}$ the components of $s^q$ and $\\pi^q$. As splittings $(s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to \\text{Tot}^n(B^{\\bullet, \\bullet})$ and $(\\pi')^n : \\text{Tot}^n(B^{\\bullet, \\bullet}) \\to \\text{Tot}^n(A^{\\bullet, \\bullet})$ we use the maps whose components are $s^{p, q}$ and $\\pi^{p, q}$ for $p + q = n$. We recall that $$ (\\delta')^n = (\\pi')^{n + 1} \\circ d_{\\text{Tot}(B^{\\bullet, \\bullet})}^n \\circ (s')^n : \\text{Tot}^n(C^{\\bullet, \\bullet}) \\to \\text{Tot}^{n + 1}(A^{\\bullet, \\bullet}) $$ The restriction of this to the summand $C^{p, q}$ is equal to $$ \\pi^{p + 1, q} \\circ d_1^{p, q} \\circ s^{p, q} + \\pi^{p, q + 1} \\circ (-1)^p d_2^{p, q} \\circ s^{p, q} = \\pi^{p + 1, q} \\circ d_1^{p, q} \\circ s^{p, q} $$ The equality holds because $s^p$ is a morphism of complexes (with $d_2$ as differential) and because $\\pi^{p, q + 1} \\circ s^{p, q + 1} = 0$ as $s$ and $\\pi$ correspond to a direct sum decomposition of $B$ in every bidegree. On the other hand, for $\\delta$ we have $$ \\delta^p =  \\pi^p \\circ d_1 \\circ s^p : C^{p, \\bullet} \\to A^{p + 1, \\bullet} $$ whose restriction to the summand $C^{p, q}$ is equal to $\\pi^{p + 1, q} \\circ d_1^{p, q} \\circ s^{p, q}$. Thus we get the same as before which matches with the fact that the isomorphism $\\gamma : \\text{Tot}(A^{\\bullet, \\bullet})[1] \\to \\text{Tot}(A^{\\bullet, bullet}[1, 0])$ is defined without the intervetion of signs."}
+{"_id": "12195", "title": "homology-remark-allow-translation-functors", "text": "It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accomodate this (and to get around certain technical issues) we introduce the following notion. Let $\\mathcal{A}$ be an abelian category. Let $(T_r)_{r \\geq 1}$ be a sequence of {\\it translation} or {\\it shift} functors, i.e., $T_r : \\mathcal{A} \\to \\mathcal{A}$ is an isomorphism of categories. In this setting a {\\it spectral sequence} is given by a system $(E_r, d_r)_{r \\geq 1}$ where each $E_r$ is an object of $\\mathcal{A}$, each $d_r : E_r \\to T_rE_r$ is a morphism such that $T_rd_r \\circ d_r = 0$ so that $$ \\xymatrix{ \\ldots \\ar[r] & T_r^{-1}E_r \\ar[r]^-{T_r^{-1}d_r} & E_r \\ar[r]^-{d_r} & T_rE_r \\ar[r]^{T_r d_r} & T_r^2E_r \\ar[r] & \\ldots } $$ is a complex and $E_{r + 1} = \\Ker(d_r)/\\Im(T_r^{-1}d_r)$ for $r \\geq 1$. It is clear what a {\\it morphism of spectral sequences} means in this setting. In this setting we can still define $$ 0 = B_1 \\subset B_2 \\subset \\ldots \\subset B_r \\subset \\ldots \\subset Z_r \\subset \\ldots \\subset Z_2 \\subset Z_1 = E_1 $$ and $Z_\\infty$ and $B_\\infty$ (if they exist) as above."}
+{"_id": "12196", "title": "homology-remark-shifted-exact-couple", "text": "Let $\\mathcal{A}$ be an abelian category. Let $S, T : \\mathcal{A} \\to \\mathcal{A}$ be shift functors, i.e., isomorphisms of categories. We will indicate the $n$-fold compositions by $S^nA$ and $T^nA$ for $A \\in \\Ob(\\mathcal{A})$ and $n \\in \\mathbf{Z}$. In this situation an {\\it exact couple} is a datum $(A, E, \\alpha, f, g)$ where $A$, $E$ are objects of $\\mathcal{A}$ and $\\alpha : A \\to T^{-1}A$, $f : E \\to A$, $g : A \\to SE$ are morphisms such that $$ \\xymatrix{ TE \\ar[r]^-{Tf} & TA \\ar[r]^-{T\\alpha} & A \\ar[r]^-{g} & SE \\ar[r]^{Sf} & SA } $$ is an exact complex. Let's visualize this as follows $$ \\xymatrix{ TA \\ar[rrr]_{T\\alpha} & & & A \\ar[ld]^g \\ar[rrr]_\\alpha & & & T^{-1}A \\ar[ld]^{T^{-1}g} \\\\ & TE \\ar[lu]^{Tf} \\ar@{..}[r] & SE & & E \\ar[lu]^f \\ar@{..}[r] & T^{-1}SE } $$ We set $d = g \\circ f : E \\to SE$. Then $d \\circ S^{-1}d = g \\circ f \\circ S^{-1}g \\circ S^{-1}f = 0$ because $f \\circ S^{-1}g = 0$. Set $E' = \\Ker(d)/\\Im(S^{-1}d)$. Set $A' = \\Im(T\\alpha)$. Let $\\alpha' : A' \\to T^{-1}A'$ induced by $\\alpha$. Let $f' : E' \\to A'$ be induced by $f$ which works because $f(\\Ker(d)) \\subset \\Ker(g) = \\Im(T\\alpha)$. Finally, let $g' : A' \\to TSE'$ induced by ``$Tg \\circ (T\\alpha)^{-1}$''\\footnote{This works because $TSE' = \\Ker(TSd)/\\Im(Td)$ and $Tg(\\Ker(T\\alpha)) = Tg(\\Im(Tf)) = \\Im(T(d))$ and $TS(d)(\\Im(Tg)) = \\Im(TSg \\circ TSf \\circ Tg) = 0$.}. \\medskip\\noindent In exactly the same way as above we find \\begin{enumerate} \\item $\\Ker(d) = f^{-1}(\\Ker(g)) = f^{-1}(\\Im(T\\alpha))$, \\item $\\Im(d) = g(\\Im(f)) = g(\\Ker(\\alpha))$, \\item $(A', E', \\alpha', f', g')$ is an exact couple for the shift functors $TS$ and $T$. \\end{enumerate} We obtain a spectral sequence (as in Remark \\ref{remark-allow-translation-functors}) with $E_1 = E$, $E_2 = E'$, etc, with $d_r : E_r \\to T^{r - 1}SE_r$ for all $r \\geq 1$. Lemma \\ref{lemma-spectral-sequence-associated-exact-couple} tells us that $$ SB_{r + 1} = g(\\Ker(T^{-r + 1}\\alpha \\circ \\ldots \\circ T^{-1}\\alpha \\circ \\alpha)) $$ and $$ Z_{r + 1} = f^{-1}(\\Im(T\\alpha \\circ T^2\\alpha \\circ \\ldots \\circ T^r\\alpha)) $$ in this situation. The description of the map $d_{r + 1}$ is similar to that given in the lemma. (It may be easier to use these explicit descriptions to prove one gets a spectral sequence from such an exact couple.)"}
+{"_id": "12197", "title": "homology-remark-differential-object-selfmap", "text": "Let $\\mathcal{A}$ be an abelian category and let $S, T : \\mathcal{A} \\to \\mathcal{A}$ be shift functors, i.e., isomorphisms of categories. Assume that $TS = ST$ as functors. Consider pairs $(A, d)$ consisting of an object $A$ of $\\mathcal{A}$ and a morphism $d : A \\to SA$ such that $d \\circ S^{-1}d = 0$. The category of these objects is abelian. We define $H(A, d) = \\Ker(d)/\\Im(S^{-1}d)$ and we observe that $H(SA, Sd) = SH(A, d)$ (canonical isomorphism). Given a short exact sequence $$ 0 \\to (A, d) \\to (B, d) \\to (C, d) \\to 0 $$ we obtain a long exact homology sequence $$ \\ldots \\to S^{-1}H(C, d) \\to H(A, d) \\to H(B, d) \\to H(C, d) \\to SH(A, d) \\to \\ldots $$ (note the shifts in the boundary maps). Since $ST = TS$ the functor $T$ defines a shift functor on pairs by setting $T(A, d) = (TA, Td)$. Next, let $\\alpha : (A, d) \\to T^{-1}(A, d)$ be injective with cokernel $(Q, d)$. Then we get an exact couple as in Remark \\ref{remark-shifted-exact-couple} with shift functors $TS$ and $T$ given by $$ (H(A, d), S^{-1}H(Q, d), \\overline{\\alpha}, f, g) $$ where $\\overline{\\alpha} : H(A, d) \\to T^{-1}H(A, d)$ is induced by $\\alpha$, the map $f : S^{-1}H(Q, d) \\to H(A, d)$ is the boundary map and $g : H(A, d) \\to TH(Q, d) = TS(S^{-1}H(Q, d))$ is induced by the quotient map $A \\to TQ$. Thus we get a spectral sequence as above with $E_1 = S^{-1}H(Q, d)$ and differentials $d_r : E_r \\to T^rSE_r$. As above we set $E_0 = S^{-1}Q$ and $d_0 : E_0 \\to SE_0$ given by $S^{-1}d : S^{-1}Q \\to Q$. If according to our conventions we define $B_r \\subset Z_r \\subset E_0$, then we have for $r \\geq 1$ that \\begin{enumerate} \\item $SB_r$ is the image of $$ (T^{-r + 1}\\alpha \\circ \\ldots \\circ T^{-1}\\alpha)^{-1} \\Im(T^{-r}S^{-1}d) $$ under the natural map $T^{-1}A \\to Q$, \\item $Z_r$ is the image of $$ (S^{-1}T^{-1}d)^{-1} \\Im(\\alpha \\circ \\ldots \\circ T^{r - 1}\\alpha) $$ under the natural map $S^{-1}T^{-1}A \\to S^{-1}Q$. \\end{enumerate} The differentials can be described as follows: if $x \\in Z_r$, then pick $x' \\in S^{-1}T^{-1}A$ mapping to $x$. Then $S^{-1}T^{-1}d(x')$ is $(\\alpha \\circ \\ldots \\circ T^{r - 1}\\alpha)(y)$ for some $y \\in T^{r - 1}A$. Then $d_r(x) \\in T^rSE_r$ is represented by the class of the image of $y$ in $T^rSE_0 = T^rQ$ modulo $T^rSB_r$."}
+{"_id": "12198", "title": "homology-remark-need-left-exactness", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be as in Lemma \\ref{lemma-adjoint-preserve-injectives}. In the presence of assumption (1) assumption (2) is equivalent to requiring that $v$ is exact. Moreover, condition (2) is necessary. Here is an example. Let $A \\to B$ be a ring map. Let $u : \\text{Mod}_B \\to \\text{Mod}_A$ be $u(N) = N_A$ and let $v : \\text{Mod}_A \\to \\text{Mod}_B$ be $v(M) = M \\otimes_A B$. Then $u$ is right adjoint to $v$, and $u$ is exact and $v$ is right exact, but $v$ does not transform injective maps into injective maps in general (i.e., $v$ is not left exact). Moreover, it is {\\bf not} the case that $u$ transforms injective $B$-modules into injective $A$-modules. For example, if $A = \\mathbf{Z}$ and $B = \\mathbf{Z}/p\\mathbf{Z}$, then the injective $B$-module $\\mathbf{Z}/p\\mathbf{Z}$ is not an injective $\\mathbf{Z}$-module. In fact, the lemma applies to this example if and only if the ring map $A \\to B$ is flat."}
+{"_id": "12199", "title": "homology-remark-faithfulness-needed", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, $u : \\mathcal{A} \\to \\mathcal{B}$ and $v : \\mathcal{B} \\to \\mathcal{A}$ be as In Lemma \\ref{lemma-adjoint-enough-injectives}. In the presence of conditions (1) and (2) condition (4) is equivalent to $v$ being faithful. Moreover, condition (4) is needed. An example is to consider the case where the functors $u$ and $v$ are both the zero functor."}
+{"_id": "12410", "title": "categories-remark-big-categories", "text": "Big categories. In some texts a category is allowed to have a proper class of objects. We will allow this as well in these notes but only in the following list of cases (to be updated as we go along). In particular, when we say: ``Let $\\mathcal{C}$ be a category'' then it is understood that $\\Ob(\\mathcal{C})$ is a set. \\begin{enumerate} \\item The category $\\textit{Sets}$ of sets. \\item The category $\\textit{Ab}$ of abelian groups. \\item The category $\\textit{Groups}$ of groups. \\item Given a group $G$ the category $G\\textit{-Sets}$ of sets with a left $G$-action. \\item Given a ring $R$ the category $\\text{Mod}_R$ of $R$-modules. \\item Given a field $k$ the category of vector spaces over $k$. \\item The category of rings. \\item The category of schemes. \\item The category $\\textit{Top}$ of topological spaces. \\item Given a topological space $X$ the category $\\textit{PSh}(X)$ of presheaves of sets over $X$. \\item Given a topological space $X$ the category $\\Sh(X)$ of sheaves of sets over $X$. \\item Given a topological space $X$ the category $\\textit{PAb}(X)$ of presheaves of abelian groups over $X$. \\item Given a topological space $X$ the category $\\textit{Ab}(X)$ of sheaves of abelian groups over $X$. \\item Given a small category $\\mathcal{C}$ the category of functors from $\\mathcal{C}$ to $\\textit{Sets}$. \\item Given a category $\\mathcal{C}$ the category of presheaves of sets over $\\mathcal{C}$. \\item Given a site $\\mathcal{C}$ the category of sheaves of sets over $\\mathcal{C}$. \\end{enumerate} One of the reason to enumerate these here is to try and avoid working with something like the ``collection'' of ``big'' categories which would be like working with the collection of all classes which I think definitively is a meta-mathematical object."}
+{"_id": "12411", "title": "categories-remark-unique-identity", "text": "It follows directly from the definition that any two identity morphisms of an object $x$ of $\\mathcal{A}$ are the same. Thus we may and will speak of {\\it the} identity morphism $\\text{id}_x$ of $x$."}
+{"_id": "12412", "title": "categories-remark-functor-into-sets", "text": "Suppose that $\\mathcal{A}$ is a category. A functor $F$ from $\\mathcal{A}$ to $\\textit{Sets}$ is a mathematical object (i.e., it is a set not a class or a formula of set theory, see Sets, Section \\ref{sets-section-sets-everything}) even though the category of sets is ``big''. Namely, the range of $F$ on objects will be a set $F(\\Ob(\\mathcal{A}))$ and then we may think of $F$ as a functor between $\\mathcal{A}$ and the full subcategory of the category of sets whose objects are elements of $F(\\Ob(\\mathcal{A}))$."}
+{"_id": "12413", "title": "categories-remark-functors-sets-sets", "text": "This is one instance where the same thing does not hold if $\\mathcal{A}$ is a ``big'' category. For example consider functors $\\textit{Sets} \\to \\textit{Sets}$. As we have currently defined it such a functor is a class and not a set. In other words, it is given by a formula in set theory (with some variables equal to specified sets)! It is not a good idea to try to consider all possible formulae of set theory as part of the definition of a mathematical object. The same problem presents itself when considering sheaves on the category of schemes for example. We will come back to this point later."}
+{"_id": "12414", "title": "categories-remark-diagram-small", "text": "The index category of a (co)limit will never be allowed to have a proper class of objects. In this project it means that it cannot be one of the categories listed in Remark \\ref{remark-big-categories}"}
+{"_id": "12415", "title": "categories-remark-limit-colim", "text": "We often write $\\lim_i M_i$, $\\colim_i M_i$, $\\lim_{i\\in \\mathcal{I}} M_i$, or $\\colim_{i\\in \\mathcal{I}} M_i$ instead of the versions indexed by $\\mathcal{I}$. Using this notation, and using the description of limits and colimits of sets in Section \\ref{section-limit-sets} below, we can say the following. Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. \\begin{enumerate} \\item The object $\\lim_i M_i$ if it exists satisfies the following property $$ \\Mor_\\mathcal{C}(W, \\lim_i M_i) = \\lim_i \\Mor_\\mathcal{C}(W, M_i) $$ where the limit on the right takes place in the category of sets. \\item The object $\\colim_i M_i$ if it exists satisfies the following property $$ \\Mor_\\mathcal{C}(\\colim_i M_i, W) = \\lim_{i\\in \\mathcal{I}^\\text{opp}} \\Mor_\\mathcal{C}(M_i, W) $$ where on the right we have the limit over the opposite category with value in the category of sets. \\end{enumerate} By the Yoneda lemma (and its dual) this formula completely determines the limit, respectively the colimit."}
+{"_id": "12416", "title": "categories-remark-cones-and-cocones", "text": "Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram. In this setting a {\\it cone} for $M$ is given by an object $W$ and a family of morphisms $q_i : W \\to M_i$, $i \\in \\Ob(\\mathcal{I})$ such that for all morphisms $\\phi : i \\to i'$ of $\\mathcal{I}$ the diagram $$ \\xymatrix{ & W \\ar[dl]_{q_i} \\ar[dr]^{q_{i'}} \\\\ M_i \\ar[rr]^{M(\\phi)} & & M_{i'} } $$ is commutative. The collection of cones forms a category with an obvious notion of morphisms. Clearly, the limit of $M$, if it exists, is a final object in the category of cones. Dually, a {\\it cocone} for $M$ is given by an object $W$ and a family of morphisms $t_i : M_i \\to W$ such that for all morphisms $\\phi : i \\to i'$ in $\\mathcal{I}$ the diagram $$ \\xymatrix{ M_i \\ar[rr]^{M(\\phi)} \\ar[dr]_{t_i} & & M_{i'} \\ar[dl]^{t_{i'}} \\\\ & W } $$ commutes. The collection of cocones forms a category with an obvious notion of morphisms. Similarly to the above the colimit of $M$ exists if and only if the category of cocones has an initial object."}
+{"_id": "12417", "title": "categories-remark-preorder-versus-partial-order", "text": "Let $I$ be a preordered set. From $I$ we can construct a canonical partially ordered set $\\overline{I}$ and an order preserving map $\\pi : I \\to \\overline{I}$. Namely, we can define an equivalence relation $\\sim$ on $I$ by the rule $$ i \\sim j \\Leftrightarrow (i \\leq j\\text{ and }j \\leq i). $$ We set $\\overline{I} = I/\\sim$ and we let $\\pi : I \\to \\overline{I}$ be the quotient map. Finally, $\\overline{I}$ comes with a unique partial ordering such that $\\pi(i) \\leq \\pi(j) \\Leftrightarrow i \\leq j$. Observe that if $I$ is a directed set, then $\\overline{I}$ is a directed partially ordered set. Given an (inverse) system $N$ over $\\overline{I}$ we obtain an (inverse) system $M$ over $I$ by setting $M_i = N_{\\pi(i)}$. This construction defines a functor between the category of inverse systems over $I$ and $\\overline{I}$. In fact, this is an equivalence. The reason is that if $i \\sim j$, then for any system $M$ over $I$ the maps $M_i \\to M_j$ and $M_j \\to M_i$ are mutually inverse isomorphisms. More precisely, choosing a section $s : \\overline{I} \\to I$ of $\\pi$ a quasi-inverse of the functor above sends $M$ to $N$ with $N_{\\overline{i}} = M_{s(\\overline{i})}$. Finally, this correspondence is compatible with colimits of systems: if $M$ and $N$ are related as above and if either $\\colim_{\\overline{I}} N$ or $\\colim_I M$ exists then so does the other and $\\colim_{\\overline{I}} N = \\colim_I M$. Similar results hold for inverse systems and limits of inverse systems."}
+{"_id": "12418", "title": "categories-remark-trick-needed", "text": "Note that a finite directed set $(I, \\geq)$ always has a greatest object $i_\\infty$. Hence any colimit of a system $(M_i, f_{ii'})$ over such a set is trivial in the sense that the colimit equals $M_{i_\\infty}$. In contrast, a colimit indexed by a finite filtered category need not be trivial. For instance, let $\\mathcal{I}$ be the category with a single object $i$ and a single non-trivial morphism $e$ satisfying $e = e \\circ e$. The colimit of a diagram $M : \\mathcal{I} \\to Sets$ is the image of the idempotent $M(e)$. This illustrates that something like the trick of passing to $\\mathcal{I}\\times \\omega$ in the proof of Lemma \\ref{lemma-directed-category-system} is essential."}
+{"_id": "12419", "title": "categories-remark-ind-category", "text": "Let $\\mathcal{C}$ be a category. There exists a big category $\\text{Ind-}\\mathcal{C}$ of {\\it ind-objects of} $\\mathcal{C}$. Namely, if $F : \\mathcal{I} \\to \\mathcal{C}$ and $G : \\mathcal{J} \\to \\mathcal{C}$ are filtered diagrams in $\\mathcal{C}$, then we can define $$ \\Mor_{\\text{Ind-}\\mathcal{C}}(F, G) = \\lim_i \\colim_j \\Mor_\\mathcal{C}(F(i), G(j)). $$ There is a canonical functor $\\mathcal{C} \\to \\text{Ind-}\\mathcal{C}$ which maps $X$ to the {\\it constant system} on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here."}
+{"_id": "12420", "title": "categories-remark-pro-category", "text": "Let $\\mathcal{C}$ be a category. There exists a big category $\\text{Pro-}\\mathcal{C}$ of {\\it pro-objects} of $\\mathcal{C}$. Namely, if $F : \\mathcal{I} \\to \\mathcal{C}$ and $G : \\mathcal{J} \\to \\mathcal{C}$ are cofiltered diagrams in $\\mathcal{C}$, then we can define $$ \\Mor_{\\text{Pro-}\\mathcal{C}}(F, G) = \\lim_j \\colim_i \\Mor_\\mathcal{C}(F(i), G(j)). $$ There is a canonical functor $\\mathcal{C} \\to \\text{Pro-}\\mathcal{C}$ which maps $X$ to the {\\it constant system} on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here."}
+{"_id": "12421", "title": "categories-remark-pro-category-copresheaves", "text": "Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{I} \\to \\mathcal{C}$ and $G : \\mathcal{J} \\to \\mathcal{C}$ be cofiltered diagrams in $\\mathcal{C}$. Consider the functors $A, B : \\mathcal{C} \\to \\textit{Sets}$ defined by $$ A(X) = \\colim_i \\Mor_\\mathcal{C}(F(i), X) \\quad\\text{and}\\quad B(X) = \\colim_j \\Mor_\\mathcal{C}(G(j), X) $$ We claim that a morphism of pro-systems from $F$ to $G$ is the same thing as a transformation of functors $t : B \\to A$. Namely, given $t$ we can apply $t$ to the class of $\\text{id}_{G(j)}$ in $B(G(j))$ to get a compatible system of elements $\\xi_j \\in A(G(j)) = \\colim_i \\Mor_\\mathcal{C}(F(i), G(j))$ which is exactly our definition of a morphism in $\\text{Pro-}\\mathcal{C}$ in Remark \\ref{remark-pro-category}. We omit the construction of a transformation $B \\to A$ given a morphism of pro-objects from $F$ to $G$."}
+{"_id": "12422", "title": "categories-remark-how-to-use-it", "text": "The lemma above is often used to construct the free something on something. For example the free abelian group on a set, the free group on a set, etc. The idea, say in the case of the free group on a set $E$ is to consider the functor $$ F : \\textit{Groups} \\to \\textit{Sets},\\quad G \\longmapsto \\text{Map}(E, G) $$ This functor commutes with limits. As our family of objects we can take a family $E \\to G_i$ consisting of groups $G_i$ of cardinality at most $\\max(\\aleph_0, |E|)$ and set maps $E \\to G_i$ such that every isomorphism class of such a structure occurs at least once. Namely, if $E \\to G$ is a map from $E$ to a group $G$, then the subgroup $G'$ generated by the image has cardinality at most $\\max(\\aleph_0, |E|)$. The lemma tells us the functor is representable, hence there exists a group $F_E$ such that $\\Mor_{\\textit{Groups}}(F_E, G) = \\text{Map}(E, G)$. In particular, the identity morphism of $F_E$ corresponds to a map $E \\to F_E$ and one can show that $F_E$ is generated by the image without imposing any relations. \\medskip\\noindent Another typical application is that we can use the lemma to construct colimits once it is known that limits exist. We illustrate it using the category of topological spaces which has limits by Topology, Lemma \\ref{topology-lemma-limits}. Namely, suppose that $\\mathcal{I} \\to \\textit{Top}$, $i \\mapsto X_i$ is a functor. Then we can consider $$ F : \\textit{Top} \\longrightarrow \\textit{Sets},\\quad Y \\longmapsto \\lim_\\mathcal{I} \\Mor_{\\textit{Top}}(X_i, Y) $$ This functor commutes with limits. Moreover, given any topological space $Y$ and an element $(\\varphi_i : X_i \\to Y)$ of $F(Y)$, there is a subspace $Y' \\subset Y$ of cardinality at most $|\\coprod X_i|$ such that the morphisms $\\varphi_i$ map into $Y'$. Namely, we can take the induced topology on the union of the images of the $\\varphi_i$. Thus it is clear that the hypotheses of the lemma are satisfied and we find a topological space $X$ representing the functor $F$, which precisely means that $X$ is the colimit of the diagram $i \\mapsto X_i$."}
+{"_id": "12423", "title": "categories-remark-motivation-localization", "text": "The motivation for the construction of $S^{-1} \\mathcal{C}$ is to ``force'' the morphisms in $S$ to be invertible by artificially creating inverses to them (at the cost of some existing morphisms possibly becoming identified with each other). This is similar to the localization of a commutative ring at a multiplicative subset, and more generally to the localization of a noncommutative ring at a right denominator set (see \\cite[Section 10A]{Lam}). This is more than just a similarity: The construction of $S^{-1} \\mathcal{C}$ (or, more precisely, its version for additive categories $\\mathcal{C}$) actually generalizes the latter type of localization. Namely, a noncommutative ring can be viewed as a pre-additive category with a single object (the morphisms being the elements of the ring); a multiplicative subset of this ring then becomes a set $S$ of morphisms satisfying LMS1 (aka RMS1). Then, the conditions RMS2 and RMS3 for this category and this subset $S$ translate into the two conditions (``right permutable'' and ``right reversible'') of a right denominator set (and similarly for LMS and left denominator sets), and $S^{-1} \\mathcal{C}$ (with a properly defined additive structure) is the one-object category corresponding to the localization of the ring."}
+{"_id": "12424", "title": "categories-remark-left-localization-morphisms-colimit", "text": "Let $\\mathcal{C}$ be a category. Let $S$ be a left multiplicative system. Given an object $Y$ of $\\mathcal{C}$ we denote $Y/S$ the category whose objects are $s : Y \\to Y'$ with $s \\in S$ and whose morphisms are commutative diagrams $$ \\xymatrix{ & Y \\ar[ld]_s \\ar[rd]^t & \\\\ Y' \\ar[rr]^a & & Y'' } $$ where $a : Y' \\to Y''$ is arbitrary. We claim that the category $Y/S$ is filtered (see Definition \\ref{definition-directed}). Namely, LMS1 implies that $\\text{id}_Y : Y \\to Y$ is in $Y/S$; hence $Y/S$ is nonempty. LMS2 implies that given $s_1 : Y \\to Y_1$ and $s_2 : Y \\to Y_2$ we can find a diagram $$ \\xymatrix{ Y \\ar[d]_{s_1} \\ar[r]_{s_2} & Y_2 \\ar[d]^t \\\\ Y_1 \\ar[r]^a & Y_3 } $$ with $t \\in S$. Hence $s_1 : Y \\to Y_1$ and $s_2 : Y \\to Y_2$ both have maps to $t \\circ s_2 : Y \\to Y_3$ in $Y/S$. Finally, given two morphisms $a, b$ from $s_1 : Y \\to Y_1$ to $s_2 : Y \\to Y_2$ in $Y/S$ we see that $a \\circ s_1 = b \\circ s_1$; hence by LMS3 there exists a $t : Y_2 \\to Y_3$ in $S$ such that $t \\circ a = t \\circ b$. Now the combined results of Lemmas \\ref{lemma-morphisms-left-localization} and \\ref{lemma-equality-morphisms-left-localization} tell us that \\begin{equation} \\label{equation-left-localization-morphisms-colimit} \\Mor_{S^{-1}\\mathcal{C}}(X, Y) = \\colim_{(s : Y \\to Y') \\in Y/S} \\Mor_\\mathcal{C}(X, Y') \\end{equation} This formula expressing morphism sets in $S^{-1}\\mathcal{C}$ as a filtered colimit of morphism sets in $\\mathcal{C}$ is occasionally useful."}
+{"_id": "12425", "title": "categories-remark-right-localization-morphisms-colimit", "text": "Let $\\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. Given an object $X$ of $\\mathcal{C}$ we denote $S/X$ the category whose objects are $s : X' \\to X$ with $s \\in S$ and whose morphisms are commutative diagrams $$ \\xymatrix{ X' \\ar[rd]_s \\ar[rr]_a & & X'' \\ar[ld]^t \\\\ & X } $$ where $a : X' \\to X''$ is arbitrary. The category $S/X$ is cofiltered (see Definition \\ref{definition-codirected}). (This is dual to the corresponding statement in Remark \\ref{remark-left-localization-morphisms-colimit}.) Now the combined results of Lemmas \\ref{lemma-morphisms-right-localization} and \\ref{lemma-equality-morphisms-right-localization} tell us that \\begin{equation} \\label{equation-right-localization-morphisms-colimit} \\Mor_{S^{-1}\\mathcal{C}}(X, Y) = \\colim_{(s : X' \\to X) \\in (S/X)^{opp}} \\Mor_\\mathcal{C}(X', Y) \\end{equation} This formula expressing morphisms in $S^{-1}\\mathcal{C}$ as a filtered colimit of morphisms in $\\mathcal{C}$ is occasionally useful."}
+{"_id": "12426", "title": "categories-remark-big-2-categories", "text": "Big $2$-categories. In many texts a $2$-category is allowed to have a class of objects (but hopefully a ``class of classes'' is not allowed). We will allow these ``big'' $2$-categories as well, but only in the following list of cases (to be updated as we go along): \\begin{enumerate} \\item The $2$-category of categories $\\textit{Cat}$. \\item The $(2, 1)$-category of categories $\\textit{Cat}$. \\item The $2$-category of groupoids $\\textit{Groupoids}$; this is a $(2, 1)$-category. \\item The $2$-category of fibred categories over a fixed category. \\item The $(2, 1)$-category of fibred categories over a fixed category. \\end{enumerate} See Definition \\ref{definition-2-1-category}. Note that in each case the class of objects of the $2$-category $\\mathcal{C}$ is a proper class, but for all objects $x, y \\in \\Ob(C)$ the category $\\Mor_\\mathcal{C}(x, y)$ is ``small'' (according to our conventions)."}
+{"_id": "12427", "title": "categories-remark-other-2-categories", "text": "Thus there are variants of the construction of Example \\ref{example-2-1-category-of-categories} above where we look at the $2$-category of groupoids, or categories fibred in groupoids over a fixed category, or stacks. And so on."}
+{"_id": "12428", "title": "categories-remark-other-description-2-fibre-product", "text": "Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be categories. Let $F : \\mathcal{A} \\to \\mathcal{C}$ and $G : \\mathcal{B} \\to \\mathcal{C}$ be functors. Another, slightly more symmetrical, construction of a $2$-fibre product $\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$ is as follows. An object is a quintuple $(A, B, C, a, b)$ where $A, B, C$ are objects of $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ and where $a : F(A) \\to C$ and $b : G(B) \\to C$ are isomorphisms. A morphism $(A, B, C, a, b) \\to (A', B', C', a', b')$ is given by a triple of morphisms $A \\to A', B \\to B', C \\to C'$ compatible with the morphisms $a, b, a', b'$. We can prove directly that this leads to a $2$-fibre product. However, it is easier to observe that the functor $(A, B, C, a, b) \\mapsto (A, B, b^{-1} \\circ a)$ gives an equivalence from the category of quintuples to the category constructed in Example \\ref{example-2-fibre-product-categories}."}
+{"_id": "12429", "title": "categories-remark-alternative-fibred-groupoids-strict", "text": "We can use the $2$-Yoneda lemma to give an alternative proof of Lemma \\ref{lemma-fibred-groupoids-strict}. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids. We define a contravariant functor $F$ from $\\mathcal{C}$ to the category of groupoids as follows: for $U\\in \\Ob(\\mathcal{C})$ let $$ F(U) = \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S}). $$ If $f : U \\to V$ the induced functor $\\mathcal{C}/U \\to \\mathcal{C}/V$ induces the morphism $F(f) : F(V) \\to F(U)$. Clearly $F$ is a functor. Let $\\mathcal{S}'$ be the associated category fibred in groupoids from Example \\ref{example-functor-groupoids}. There is an obvious functor $G : \\mathcal{S}' \\to \\mathcal{S}$ over $\\mathcal{C}$ given by taking the pair $(U, x)$, where $U \\in \\Ob(\\mathcal{C})$ and $x \\in F(U)$, to $x(\\text{id}_U) \\in \\mathcal{S}$.  Now Lemma \\ref{lemma-yoneda-2category} implies that for each $U$, $$ G_U : \\mathcal{S}'_U = F(U)= \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S}) \\to \\mathcal{S}_U $$ is an equivalence, and thus $G$ is an equivalence between $\\mathcal{S}$ and $\\mathcal{S}'$ by Lemma \\ref{lemma-equivalence-fibred-categories}."}
+{"_id": "12430", "title": "categories-remark-left-dual-adjoint", "text": "Lemma \\ref{lemma-left-dual} says in particular that $Z \\mapsto Z \\otimes Y$ is the right adjoint of $Z' \\mapsto Z' \\otimes X$. In particular, uniqueness of adjoint functors guarantees that a left dual of $X$, if it exists, is unique up to unique isomorphism. Conversely, assume the functor $Z \\mapsto Z \\otimes Y$ is a right adjoint of the functor $Z' \\mapsto Z' \\otimes X$, i.e., we're given a bijection $$ \\Mor(Z' \\otimes X, Z) \\longrightarrow \\Mor(Z', Z \\otimes Y) $$ functorial in both $Z$ and $Z'$. The unit of the adjunction produces maps $$ \\eta_Z : Z \\to Z \\otimes X \\otimes Y $$ functorial in $Z$ and the counit of the adjoint produces maps $$ \\epsilon_{Z'} : Z' \\otimes Y \\otimes X \\to Z' $$ functorial in $Z'$. In particular, we find $\\eta = \\eta_\\mathbf{1} : \\mathbf{1} \\to X \\otimes Y$ and $\\epsilon = \\epsilon_\\mathbf{1} : Y \\otimes X \\to \\mathbf{1}$. As an exercise in the relationship between units, counits, and the adjunction isomorphism, the reader can show that we have $$ (\\epsilon \\otimes \\text{id}_Y) \\circ \\eta_Y = \\text{id}_Y \\quad\\text{and}\\quad \\epsilon_X \\circ (\\eta \\otimes \\text{id}_X) = \\text{id}_X $$ However, this isn't enough to show that $(\\epsilon \\otimes \\text{id}_Y) \\circ (\\text{id}_Y \\otimes \\eta) = \\text{id}_Y$ and $(\\text{id}_X \\otimes \\epsilon) \\circ (\\eta \\otimes \\text{id}_X) = \\text{id}_X$, because we don't know in general that $\\eta_Y = \\text{id}_Y \\otimes \\eta$ and we don't know that $\\epsilon_X = \\epsilon \\otimes \\text{id}_X$. For this it would suffice to know that our adjunction isomorphism has the following property: for every $W, Z, Z'$ the diagram $$ \\xymatrix{ \\Mor(Z' \\otimes X, Z) \\ar[r] \\ar[d]_{\\text{id}_W \\otimes -} & \\Mor(Z', Z \\otimes Y) \\ar[d]^{\\text{id}_W \\otimes -} \\\\ \\Mor(W \\otimes Z' \\otimes X, W \\otimes Z) \\ar[r] & \\Mor(W \\otimes Z', W \\otimes Z \\otimes Y) } $$ If this holds, we will say {\\it the adjunction is compatible with the given tensor structure}. Thus the requirement that $Z \\mapsto Z \\otimes Y$ be the right adjoint of $Z' \\mapsto Z' \\otimes X$ compatible with the given tensor structure is an equivalent formulation of the property of being a left dual."}
+{"_id": "12553", "title": "topologies-remark-choice-sites", "text": "Take any category $\\Sch_\\alpha$ constructed as in Sets, Lemma \\ref{sets-lemma-construct-category} starting with the set of schemes $\\{X, Y, S\\}$. Choose any set of coverings $\\text{Cov}_{fppf}$ on $\\Sch_\\alpha$ as in Sets, Lemma \\ref{sets-lemma-coverings-site} starting with the category $\\Sch_\\alpha$ and the class of fppf coverings. Let $\\Sch_{fppf}$ denote the big fppf site so obtained. Next, for $\\tau \\in \\{Zariski, \\etale, smooth, syntomic\\}$ let $\\Sch_\\tau$ have the same underlying category as $\\Sch_{fppf}$ with coverings $\\text{Cov}_\\tau \\subset \\text{Cov}_{fppf}$ simply the subset of $\\tau$-coverings. It is straightforward to check that this gives rise to a big site $\\Sch_\\tau$."}
+{"_id": "12579", "title": "pic-remark-when-proposition-applies", "text": "Let $f : X \\to S$ be a morphism of schemes. The assumption of Proposition \\ref{proposition-hilb-d-representable} and hence the conclusion holds in each of the following cases: \\begin{enumerate} \\item $X$ is quasi-affine, \\item $f$ is quasi-affine, \\item $f$ is quasi-projective, \\item $f$ is locally projective, \\item there exists an ample invertible sheaf on $X$, \\item there exists an $f$-ample invertible sheaf on $X$, and \\item there exists an $f$-very ample invertible sheaf on $X$. \\end{enumerate} Namely, in each of these cases, every finite set of points of a fibre $X_s$ is contained in a quasi-compact open $U$ of $X$ which comes with an ample invertible sheaf, is isomorphic to an open of an affine scheme, or is isomorphic to an open of $\\text{Proj}$ of a graded ring (in each case this follows by unwinding the definitions). Thus the existence of suitable affine opens by Properties, Lemma \\ref{properties-lemma-ample-finite-set-in-affine}."}
+{"_id": "12580", "title": "pic-remark-universal-object-hilb-d", "text": "Let $X$ be a geometrically irreducible smooth proper curve over a field $k$ as in Proposition \\ref{proposition-hilb-d}. Let $d \\geq 0$. The universal closed object is a relatively effective divisor $$ D_{univ} \\subset \\underline{\\Hilbfunctor}^{d + 1}_{X/k} \\times_k X $$ over $\\underline{\\Hilbfunctor}^{d + 1}_{X/k}$ by Lemma \\ref{lemma-divisors-on-curves}. In fact, $D_{univ}$ is isomorphic as a scheme to $\\underline{\\Hilbfunctor}^d_{X/k} \\times_k X$, see proof of Lemma \\ref{lemma-universal-object}. In particular, $D_{univ}$ is an effective Cartier divisor and we obtain an invertible module $\\mathcal{O}(D_{univ})$. If $[D] \\in \\underline{\\Hilbfunctor}^{d + 1}_{X/k}$ denotes the $k$-rational point corresponding to the effective Cartier divisor $D \\subset X$ of degree $d + 1$, then the restriction of $\\mathcal{O}(D_{univ})$ to the fibre $[D] \\times X$ is $\\mathcal{O}_X(D)$."}
+{"_id": "12668", "title": "constructions-remark-relative-glueing-functorial", "text": "There is a functoriality property for the constructions explained in Lemmas \\ref{lemma-relative-glueing} and \\ref{lemma-relative-glueing-sheaves}. Namely, suppose given two collections of data $(f_U : X_U \\to U, \\rho^U_V)$ and $(g_U : Y_U \\to U, \\sigma^U_V)$ as in Lemma \\ref{lemma-relative-glueing}. Suppose for every $U \\in \\mathcal{B}$ given a morphism $h_U : X_U \\to Y_U$ over $U$ compatible with the restrictions $\\rho^U_V$ and $\\sigma^U_V$. Functoriality means that this gives rise to a morphism of schemes $h : X \\to Y$ over $S$ restricting back to the morphisms $h_U$, where $f : X \\to S$ is obtained from the datum $(f_U : X_U \\to U, \\rho^U_V)$ and $g : Y \\to S$ is obtained from the datum $(g_U : Y_U \\to U, \\sigma^U_V)$. \\medskip\\noindent Similarly, suppose given two collections of data $(f_U : X_U \\to U, \\mathcal{F}_U, \\rho^U_V, \\theta^U_V)$ and $(g_U : Y_U \\to U, \\mathcal{G}_U, \\sigma^U_V, \\eta^U_V)$ as in Lemma \\ref{lemma-relative-glueing-sheaves}. Suppose for every $U \\in \\mathcal{B}$ given a morphism $h_U : X_U \\to Y_U$ over $U$ compatible with the restrictions $\\rho^U_V$ and $\\sigma^U_V$, and a morphism $\\tau_U : h_U^*\\mathcal{G}_U \\to \\mathcal{F}_U$ compatible with the maps $\\theta^U_V$ and $\\eta^U_V$. Functoriality means that these give rise to a morphism of schemes $h : X \\to Y$ over $S$ restricting back to the morphisms $h_U$, and a morphism $h^*\\mathcal{G} \\to \\mathcal{F}$ restricting back to the maps $h_U$ where $(f : X \\to S, \\mathcal{F})$ is obtained from the datum $(f_U : X_U \\to U, \\mathcal{F}_U, \\rho^U_V, \\theta^U_V)$ and where $(g : Y \\to S, \\mathcal{G})$ is obtained from the datum $(g_U : Y_U \\to U, \\mathcal{G}_U, \\sigma^U_V, \\eta^U_V)$. \\medskip\\noindent We omit the verifications and we omit a suitable formulation of ``equivalence of categories'' between relative glueing data and relative objects."}
+{"_id": "12669", "title": "constructions-remark-global-sections-not-isomorphism", "text": "The map from $M_0$ to the global sections of $\\widetilde M$ is generally far from being an isomorphism. A trivial example is to take $S = k[x, y, z]$ with $1 = \\deg(x) = \\deg(y) = \\deg(z)$ (or any number of variables) and to take $M = S/(x^{100}, y^{100}, z^{100})$. It is easy to see that $\\widetilde M = 0$, but $M_0 = k$."}
+{"_id": "12670", "title": "constructions-remark-not-isomorphism", "text": "In general the map constructed in Lemma \\ref{lemma-widetilde-tensor} above is not an isomorphism. Here is an example. Let $k$ be a field. Let $S = k[x, y, z]$ with $k$ in degree $0$ and $\\deg(x) = 1$, $\\deg(y) = 2$, $\\deg(z) = 3$. Let $M = S(1)$ and $N = S(2)$, see Algebra, Section \\ref{algebra-section-graded} for notation. Then $M \\otimes_S N = S(3)$. Note that \\begin{eqnarray*} S_z & = & k[x, y, z, 1/z] \\\\ S_{(z)} & = & k[x^3/z, xy/z, y^3/z^2] \\cong k[u, v, w]/(uw - v^3) \\\\ M_{(z)} & = & S_{(z)} \\cdot x + S_{(z)} \\cdot y^2/z \\subset S_z \\\\ N_{(z)} & = & S_{(z)} \\cdot y + S_{(z)} \\cdot x^2 \\subset S_z \\\\ S(3)_{(z)} & = & S_{(z)} \\cdot z \\subset S_z \\end{eqnarray*} Consider the maximal ideal $\\mathfrak m = (u, v, w) \\subset S_{(z)}$. It is not hard to see that both $M_{(z)}/\\mathfrak mM_{(z)}$ and $N_{(z)}/\\mathfrak mN_{(z)}$ have dimension $2$ over $\\kappa(\\mathfrak m)$. But $S(3)_{(z)}/\\mathfrak mS(3)_{(z)}$ has dimension $1$. Thus the map $M_{(z)} \\otimes N_{(z)} \\to S(3)_{(z)}$ is not an isomorphism."}
+{"_id": "12671", "title": "constructions-remark-missing-finite-type", "text": "What's missing in the list of properties above? Well to be sure the property of being of finite type. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. (Another property which is missing is ``smoothness''. And I'm sure there are many more you can think of.)"}
+{"_id": "12672", "title": "constructions-remark-not-in-invertible-locus", "text": "Assumptions as in Lemma \\ref{lemma-invertible-map-into-proj} above. The image of the morphism $r_{\\mathcal{L}, \\psi}$ need not be contained in the locus where the sheaf $\\mathcal{O}_X(1)$ is invertible. Here is an example. Let $k$ be a field. Let $S = k[A, B, C]$ graded by $\\deg(A) = 1$, $\\deg(B) = 2$, $\\deg(C) = 3$. Set $X = \\text{Proj}(S)$. Let $T = \\mathbf{P}^2_k = \\text{Proj}(k[X_0, X_1, X_2])$. Recall that $\\mathcal{L} = \\mathcal{O}_T(1)$ is invertible and that $\\mathcal{O}_T(n) = \\mathcal{L}^{\\otimes n}$. Consider the composition $\\psi$ of the maps $$ S \\to k[X_0, X_1, X_2] \\to \\Gamma_*(T, \\mathcal{L}). $$ Here the first map is $A \\mapsto X_0$, $B \\mapsto X_1^2$, $C \\mapsto X_2^3$ and the second map is (\\ref{equation-global-sections}). By the lemma this corresponds to a morphism $r_{\\mathcal{L}, \\psi} : T \\to X = \\text{Proj}(S)$ which is easily seen to be surjective. On the other hand, in Remark \\ref{remark-not-isomorphism} we showed that the sheaf $\\mathcal{O}_X(1)$ is not invertible at all points of $X$."}
+{"_id": "12807", "title": "algebraization-remark-compare-with-completion", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. Let $K \\mapsto K^\\wedge$ be the derived completion functor of Proposition \\ref{proposition-derived-completion}. For any $n \\geq 1$ the object $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$ is derived complete as it is annihilated by powers of local sections of $\\mathcal{I}$. Hence there is a canonical factorization $$ K \\to K^\\wedge \\to K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n $$ of the canonical map $K \\to K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$. These maps are compatible for varying $n$ and we obtain a comparison map $$ K^\\wedge \\longrightarrow R\\lim \\left(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n\\right) $$ The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism."}
+{"_id": "12808", "title": "algebraization-remark-localization-and-completion", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. Let $K \\mapsto K^\\wedge$ be the derived completion functor of Proposition \\ref{proposition-derived-completion}. It follows from the construction in the proof of the proposition that $K^\\wedge|_U$ is the derived completion of $K|_U$ for any $U \\in \\Ob(\\mathcal{C})$. But we can also prove this as follows. From the definition of derived complete objects it follows that $K^\\wedge|_U$ is derived complete. Thus we obtain a canonical map $a : (K|_U)^\\wedge \\to K^\\wedge|_U$. On the other hand, if $E$ is a derived complete object of $D(\\mathcal{O}_U)$, then $Rj_*E$ is a derived complete object of $D(\\mathcal{O})$ by Lemma \\ref{lemma-pushforward-derived-complete}. Here $j$ is the localization morphism (Modules on Sites, Section \\ref{sites-modules-section-localize}). Hence we also obtain a canonical map $b : K^\\wedge \\to Rj_*((K|_U)^\\wedge)$. We omit the (formal) verification that the adjoint of $b$ is the inverse of $a$."}
+{"_id": "12809", "title": "algebraization-remark-completed-tensor-product", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals.  Denote $K \\mapsto K^\\wedge$ the adjoint of Proposition \\ref{proposition-derived-completion}. Then we set $$ K \\otimes^\\wedge_\\mathcal{O} L = (K \\otimes_\\mathcal{O}^\\mathbf{L} L)^\\wedge $$ This {\\it completed tensor product} defines a functor $D_{comp}(\\mathcal{O}) \\times D_{comp}(\\mathcal{O}) \\to D_{comp}(\\mathcal{O})$ such that we have $$ \\Hom_{D_{comp}(\\mathcal{O})}(K, R\\SheafHom_\\mathcal{O}(L, M)) = \\Hom_{D_{comp}(\\mathcal{O})}(K \\otimes_\\mathcal{O}^\\wedge L, M) $$ for $K, L, M \\in D_{comp}(\\mathcal{O})$. Note that $R\\SheafHom_\\mathcal{O}(L, M) \\in D_{comp}(\\mathcal{O})$ by Lemma \\ref{lemma-derived-complete-internal-hom}."}
+{"_id": "12810", "title": "algebraization-remark-local-calculation-derived-completion", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. Let $K \\mapsto K^\\wedge$ be the derived completion of Proposition \\ref{proposition-derived-completion}. Let $U \\in \\Ob(\\mathcal{C})$ be an object such that $\\mathcal{I}$ is generated as an ideal sheaf by $f_1, \\ldots, f_r \\in \\mathcal{I}(U)$. Set $A = \\mathcal{O}(U)$ and $I = (f_1, \\ldots, f_r) \\subset A$. Warning: it may not be the case that $I = \\mathcal{I}(U)$. Then we have $$ R\\Gamma(U, K^\\wedge) = R\\Gamma(U, K)^\\wedge $$ where the right hand side is the derived completion of the object $R\\Gamma(U, K)$ of $D(A)$ with respect to $I$. This is true because derived completion commutes with localization (Remark \\ref{remark-localization-and-completion}) and Lemma \\ref{lemma-formal-functions-general}."}
+{"_id": "12811", "title": "algebraization-remark-references", "text": "Here are some references to discussions of related material the literature. It seems that a ``derived formal functions theorem'' for proper maps goes back to \\cite[Theorem 6.3.1]{lurie-thesis}. There is the discussion in \\cite{dag12}, especially Chapter 4 which discusses the affine story, see More on Algebra, Section \\ref{more-algebra-section-derived-completion}. In \\cite[Section 2.9]{G-R} one finds a discussion of proper base change and derived completion using (ind) coherent modules. An analogue of (\\ref{equation-formal-functions}) for complexes of quasi-coherent modules can be found as \\cite[Theorem 6.5]{HL-P}"}
+{"_id": "12812", "title": "algebraization-remark-interesting-case-variant", "text": "In Lemma \\ref{lemma-algebraization-principal-variant} if $A$ is universally catenary with Cohen-Macaulay formal fibres (for example if $A$ has a dualizing complex), then the condition that $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$ are finite $A$-modules, is equivalent with $$ \\text{depth}((A/f)_\\mathfrak q) + \\dim((A/\\mathfrak q)_\\mathfrak p) > 2 $$ for all $\\mathfrak q \\in V(f) \\setminus V(\\mathfrak a)$ and $\\mathfrak p \\in V(\\mathfrak q) \\cap V(\\mathfrak a)$ by Local Cohomology, Theorem \\ref{local-cohomology-theorem-finiteness}. \\medskip\\noindent For example, if $A/fA$ is $(S_2)$ and if every irreducible component of $Z = V(\\mathfrak a)$ has codimension $\\geq 3$ in $Y = \\Spec(A/fA)$, then we get the finiteness of $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$. This should be contrasted with the slightly weaker conditions found in Lemma \\ref{lemma-algebraization-principal} (see also Remark \\ref{remark-interesting-case})."}
+{"_id": "12813", "title": "algebraization-remark-discussion", "text": "Let $Y$ be a Noetherian scheme and let $Z \\subset Y$ be a closed subset. By Lemma \\ref{lemma-discussion} we have $$ \\delta_Z(y) \\leq \\min \\left\\{ k \\middle| \\begin{matrix} \\text{ there exist specializations in }Y \\\\ y_0 \\leftarrow y'_0 \\rightarrow y_1 \\leftarrow y'_1 \\rightarrow \\ldots \\leftarrow y'_{k - 1} \\rightarrow y_k = y \\\\ \\text{ with }y_0 \\in Z\\text{ and }y_i' \\leadsto y_i \\text{ immediate} \\end{matrix} \\right\\} $$ We claim that if $Y$ is of finite type over a field, then equality holds. If we ever need this result we will formulate a precise result and prove it here. However, in general if we define $\\delta_Z$ by the right hand side of this inequality, then we don't know if Lemma \\ref{lemma-change-distance-function} remains true."}
+{"_id": "12814", "title": "algebraization-remark-interesting-case", "text": "Let $(A, \\mathfrak m)$ be a complete Noetherian normal local domain of dimension $\\geq 4$ and let $f \\in \\mathfrak m$ be nonzero. Then assumptions (1), (2), (3), (5), and (6) of Lemma \\ref{lemma-algebraization-principal} are satisfied. Thus vectorbundles on the formal completion of $U$ along $U \\cap V(f)$ can be algebraized. In Lemma \\ref{lemma-algebraization-principal-bis} we will generalize this to more general coherent formal modules; please also compare with Remark \\ref{remark-interesting-case-bis}."}
+{"_id": "12815", "title": "algebraization-remark-interesting-case-bis", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to $f \\in \\mathfrak m$. Let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, f\\mathcal{O}_U)$ where $U$ is the punctured spectrum of $A$. Set $Y = V(f) \\subset X = \\Spec(A)$. If for $y \\in U \\cap V(f)$ closed in $U$, i.e., with $\\dim(\\overline{\\{y\\}}) = 1$, we assume the $\\mathcal{O}_{X, y}^\\wedge$-module $\\mathcal{F}_y^\\wedge$ satisfies the following two conditions \\begin{enumerate} \\item $\\mathcal{F}_y^\\wedge[1/f]$ is $(S_2)$ as a $\\mathcal{O}_{X, y}^\\wedge[1/f]$-module, and \\item for $\\mathfrak p \\in \\text{Ass}(\\mathcal{F}_y^\\wedge[1/f])$ we have $\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) \\geq 3$. \\end{enumerate} Then $(\\mathcal{F}_n)$ is the completion of a coherent module on $U$. This follows from Lemmas \\ref{lemma-algebraization-principal-bis} and \\ref{lemma-unwinding-conditions}."}
+{"_id": "12816", "title": "algebraization-remark-question", "text": "We are unable to prove or disprove the analogue of Proposition \\ref{proposition-d-generators} where the assumption that $I$ has $d$ generators is replaced with the assumption $\\text{cd}(A, I) \\leq d$. If you know a proof or have a counter example, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. Another obvious question is to what extend the conditions in Proposition \\ref{proposition-d-generators} are necessary."}
+{"_id": "12817", "title": "algebraization-remark-interesting-case-ter", "text": "In the situation of Proposition \\ref{proposition-algebraization-regular-sequence} if we assume $A$ has a dualizing complex, then the condition that $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$ are finite is equivalent to $$ \\text{depth}(\\mathcal{F}_{1, y}) + \\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z}) > 2 $$ for all $y \\in U \\cap Y$ and $z \\in Z \\cap \\overline{\\{y\\}}$. See Local Cohomology, Lemma \\ref{local-cohomology-lemma-finiteness-Rjstar}. This holds for example if $\\mathcal{F}_1$ is a finite locally free $\\mathcal{O}_{U \\cap Y}$-module, $Y$ is $(S_2)$, and $\\text{codim}(Z', Y') \\geq 3$ for every pair of irreducible components $Y'$ of $Y$, $Z'$ of $Z$ with $Z' \\subset Y'$."}
+{"_id": "12818", "title": "algebraization-remark-interesting-case-quater", "text": "Proposition \\ref{proposition-algebraization-flat} is a local version of \\cite[Theorem 2.10 (i)]{Baranovsky}. It is straightforward to deduce the global results from the local one; we will sketch the argument. Namely, suppose $(R, \\mathfrak m)$ is a complete Noetherian local ring and $X \\to \\Spec(R)$ is a proper morphism. For $n \\geq 1$ set $X_n = X \\times_{\\Spec(R)} \\Spec(R/\\mathfrak m^n)$. Let $Z \\subset X_1$ be a closed subset of the special fibre. Set $U = X \\setminus Z$ and denote $j : U \\to X$ the inclusion morphism. Suppose given an object $$ (\\mathcal{F}_n) \\text{ of } \\textit{Coh}(U, \\mathfrak m\\mathcal{O}_U) $$ which is flat over $R$ in the sense that $\\mathcal{F}_n$ is flat over $R/\\mathfrak m^n$ for all $n$. Assume that $j_*\\mathcal{F}_1$ and $R^1j_*\\mathcal{F}_1$ are coherent modules. Then affine locally on $X$ we get a canonical extension of $(\\mathcal{F}_n)$ by Proposition \\ref{proposition-algebraization-flat} and formation of this extension commutes with localization (by Lemma \\ref{lemma-algebraization-principal-variant}). Thus we get a canonical global object $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, \\mathfrak m\\mathcal{O}_X)$ whose restriction of $U$ is $(\\mathcal{F}_n)$. By Grothendieck's existence theorem (Cohomology of Schemes, Proposition \\ref{coherent-proposition-existence-proper}) we see there exists a coherent $\\mathcal{O}_X$-module $\\mathcal{G}$ whose completion is $(\\mathcal{G}_n)$. In this way we see that $(\\mathcal{F}_n)$ is algebraizable, i.e., it is the completion of a coherent $\\mathcal{O}_U$-module. \\medskip\\noindent We add that the coherence of $j_*\\mathcal{F}_1$ and $R^1j_*\\mathcal{F}_1$ is a condition on the special fibre. Namely, if we denote $j_1 : U_1 \\to X_1$ the special fibre of $j : U \\to X$, then we can think of $\\mathcal{F}_1$ as a coherent sheaf on $U_1$ and we have $j_*\\mathcal{F}_1 = j_{1, *}\\mathcal{F}_1$ and $R^1j_*\\mathcal{F}_1 = R^1j_{1, *}\\mathcal{F}_1$. Hence for example if $X_1$ is $(S_2)$ and irreducible, we have $\\dim(X_1) - \\dim(Z) \\geq 3$, and $\\mathcal{F}_1$ is a locally free $\\mathcal{O}_{U_1}$-module, then $j_{1, *}\\mathcal{F}_1$ and $R^1j_{1, *}\\mathcal{F}_1$ are coherent modules."}
+{"_id": "12819", "title": "algebraization-remark-compare-SGA2", "text": "In SGA2 we find the following result. Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$. Assume $A$ is a quotient of a regular ring, the element $f$ is a nonzerodivisor, and \\begin{enumerate} \\item[(a)] if $\\mathfrak p \\subset A$ is a prime ideal with $\\dim(A/\\mathfrak p) = 1$, then $\\text{depth}(A_\\mathfrak p) \\geq 2$, and \\item[(b)] $\\text{depth}(A/fA) \\geq 3$, or equivalently $\\text{depth}(A) \\geq 4$. \\end{enumerate} Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. Then the map $$ \\Pic(U) \\to \\Pic(U_0) $$ is injective. This is \\cite[Exposee XI, Lemma 3.16]{SGA2}\\footnote{Condition (a) follows from condition (b), see Algebra, Lemma \\ref{algebra-lemma-depth-localization}.}. This result from SGA2 follows from Proposition \\ref{proposition-injective-pic} because \\begin{enumerate} \\item a quotient of a regular ring has a dualizing complex (see Dualizing Complexes, Lemma \\ref{dualizing-lemma-regular-gorenstein} and Proposition \\ref{dualizing-proposition-dualizing-essentially-finite-type}), and \\item if $\\text{depth}(A) \\geq 4$ then $\\text{depth}(A_\\mathfrak p) \\geq 2$ for all primes $\\mathfrak p$ with $\\dim(A/\\mathfrak p) = 2$, see Algebra, Lemma \\ref{algebra-lemma-depth-localization}. \\end{enumerate}"}
+{"_id": "12820", "title": "algebraization-remark-surjective-Pic-second", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$. The conclusion of Lemma \\ref{lemma-surjective-Pic-first} holds if we assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor, \\item one of the following is true \\begin{enumerate} \\item $A_f$ is $(S_2)$ and for $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ minimal we have $\\dim(A/\\mathfrak p) \\geq 4$, or \\item if $\\mathfrak p \\not \\in V(f)$ and $V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$. \\end{enumerate} \\item $H^3_{\\mathfrak m}(A/fA) = 0$. \\end{enumerate} The proof is exactly the same as the proof of Lemma \\ref{lemma-surjective-Pic-first} using Lemma \\ref{lemma-equivalence-better} instead of Lemma \\ref{lemma-equivalence}. Two points need to be made here: (a) it seems hard to find examples where one knows $H^3_{\\mathfrak m}(A/fA) = 0$ without assuming $\\text{depth}(A/fA) \\geq 4$, and (b) the proof of Lemma \\ref{lemma-equivalence-better} is a good deal harder than the proof of Lemma \\ref{lemma-equivalence}."}
+{"_id": "12900", "title": "spaces-over-fields-remark-alternate-proof-scheme-codim-1", "text": "Here is a sketch of a proof of Lemma \\ref{lemma-codim-1-point-in-schematic-locus} which avoids using More on Groupoids, Lemma \\ref{more-groupoids-lemma-find-affine-codimension-1}. \\medskip\\noindent Step 1. We may assume $X$ is a reduced Noetherian separated algebraic space (for example by Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian} or by Limits of Spaces, Lemma \\ref{spaces-limits-lemma-reduction-scheme}) and we may choose a finite surjective morphism $Y \\to X$ where $Y$ is a Noetherian scheme (by Limits of Spaces, Proposition \\ref{spaces-limits-proposition-there-is-a-scheme-finite-over}). \\medskip\\noindent Step 2. After replacing $X$ by an open neighbourhood of $x$, there exists a birational finite morphism $X' \\to X$ and a closed subscheme $Y' \\subset X' \\times_X Y$ such that $Y' \\to X'$ is surjective finite locally free. Namely, because $X$ is reduced there is a dense open subspace $U \\subset X$ over which $Y$ is flat (Morphisms of Spaces, Proposition \\ref{spaces-morphisms-proposition-generic-flatness-reduced}). Then we can choose a $U$-admissible blowup $b : \\tilde X \\to X$ such that the strict transform $\\tilde Y$ of $Y$ is flat over $\\tilde X$, see More on Morphisms of Spaces, Lemma \\ref{spaces-more-morphisms-lemma-flat-after-blowing-up}. (An alternative is to use Hilbert schemes if one wants to avoid using the result on blowups). Then we let $X' \\subset \\tilde X$ be the scheme theoretic closure of $b^{-1}(U)$ and $Y' = X' \\times_{\\tilde X} \\tilde Y$. Since $x$ is a codimension $1$ point, we see that $X' \\to X$ is finite over a neighbourhood of $x$ (Lemma \\ref{lemma-finite-in-codim-1}). \\medskip\\noindent Step 3. After shrinking $X$ to a smaller neighbourhood of $x$ we get that $X'$ is a scheme. This holds because $Y'$ is a scheme and $Y' \\to X'$ being finite locally free and because every finite set of codimension $1$ points of $Y'$ is contained in an affine open. Use Properties of Spaces, Proposition \\ref{spaces-properties-proposition-finite-flat-equivalence-global} and Varieties, Proposition \\ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}. \\medskip\\noindent Step 4. There exists an affine open $W' \\subset X'$ containing all points lying over $x$ which is the inverse image of an open subspace of $X$. To prove this let $Z \\subset X$ be the closure of the set of points where $X' \\to X$ is not an isomorphism. We may assume $x \\in Z$ otherwise we are already done. Then $x$ is a generic point of an irreducible component of $Z$ and after shrinking $X$ we may assume $Z$ is an affine scheme (Lemma \\ref{lemma-generic-point-in-schematic-locus}). Then the inverse image $Z' \\subset X'$ is an affine scheme as well. Say $x_1, \\ldots, x_n \\in Z'$ are the points mapping to $x$. Then we can find an affine open $W'$ in $X'$ whose intersection with $Z'$ is the inverse image of a principal open of $Z$ containing $x$. Namely, we first pick an affine open $W' \\subset X'$ containing $x_1, \\ldots, x_n$ using Varieties, Proposition \\ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}. Then we pick a principal open $D(f) \\subset Z$ containing $x$ whose inverse image $D(f|_{Z'})$ is contained in $W' \\cap Z'$. Then we pick $f' \\in \\Gamma(W', \\mathcal{O}_{W'})$ restricting to $f|_{Z'}$ and we replace $W'$ by $D(f') \\subset W'$. Since $X' \\to X$ is an isomorphism away from $Z' \\to Z$ the choice of $W'$ guarantees that the image $W \\subset X$ of $W'$ is open with inverse image $W'$ in $X'$. \\medskip\\noindent Step 5. Then $W' \\to W$ is a finite surjective morphism and $W$ is a scheme by Cohomology of Spaces, Lemma \\ref{spaces-cohomology-lemma-image-affine-finite-morphism-affine-Noetherian} and the proof is complete."}
+{"_id": "12901", "title": "spaces-over-fields-remark-when-does-the-argument-work", "text": "Let $k$ be finite field. Let $K \\supset k$ be a geometrically irreducible field extension. Then $K$ is the limit of geometrically irreducible finite type $k$-algebras $A$. Given $A$ the estimates of Lang and Weil \\cite{LW}, show that for $n \\gg 0$ there exists an $k$-algebra homomorphism $A \\to k'$ with $k'/k$ of degree $n$. Analyzing the argument given in the proof of Lemma \\ref{lemma-scheme-after-purely-transcendental-base-change} we see that if $X$ is a quasi-separated algebraic space over $k$ and $X_K$ is a scheme, then $X$ is a scheme. If we ever need this result we will precisely formulate it and prove it here."}
+{"_id": "13159", "title": "dga-remark-evaluation-map-left", "text": "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M$ be a left differential graded $A$-module. Let $N^\\bullet$ be a complex of $R$-modules. The constructions above produce a right differential graded $A$-module $\\Hom(M, N^\\bullet)$ and then a leftt differential graded $A$-module $\\Hom(\\Hom(M, N^\\bullet), N^\\bullet)$. We claim there is an evaluation map $$ ev : M \\longrightarrow \\Hom(\\Hom(M, N^\\bullet), N^\\bullet) $$ in the category of left differential graded $A$-modules. To define it, by Lemma \\ref{lemma-characterize-hom} it suffices to construct an $A$-bilinear pairing $$ \\Hom(M, N^\\bullet) \\times M \\longrightarrow N^\\bullet $$ compatible with grading and differentials. For this we take $$ (f, x) \\longmapsto f(x) $$ We leave it to the reader to verify this is compatible with grading, differentials, and $A$-bilinear. The map $ev$ on underlying complexes of $R$-modules is More on Algebra, Item (\\ref{more-algebra-item-evaluation})."}
+{"_id": "13160", "title": "dga-remark-evaluation-map-right", "text": "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M$ be a right differential graded $A$-module. Let $N^\\bullet$ be a complex of $R$-modules. The constructions above produce a left differential graded $A$-module $\\Hom(M, N^\\bullet)$ and then a right differential graded $A$-module $\\Hom(\\Hom(M, N^\\bullet), N^\\bullet)$. We claim there is an evaluation map $$ ev : M \\longrightarrow \\Hom(\\Hom(M, N^\\bullet), N^\\bullet) $$ in the category of right differential graded $A$-modules. To define it, by Lemma \\ref{lemma-characterize-hom} it suffices to construct an $A$-bilinear pairing $$ M \\times \\Hom(M, N^\\bullet) \\longrightarrow N^\\bullet $$ compatible with grading and differentials. For this we take $$ (x, f) \\longmapsto (-1)^{\\deg(x)\\deg(f)}f(x) $$ We leave it to the reader to verify this is compatible with grading, differentials, and $A$-bilinear. The map $ev$ on underlying complexes of $R$-modules is More on Algebra, Item (\\ref{more-algebra-item-evaluation})."}
+{"_id": "13161", "title": "dga-remark-shift-dual", "text": "Let $R$ be a ring. Let $A$ be a differential graded $R$-algebra. Let $M^\\bullet$ and $N^\\bullet$ be complexes of $R$-modules. Let $k \\in \\mathbf{Z}$ and consider the isomorphism $$ \\Hom^\\bullet(M^\\bullet, N^\\bullet)[-k] \\longrightarrow \\Hom^\\bullet(M^\\bullet[k], N^\\bullet) $$ of complexes of $R$-modules defined in More on Algebra, Item (\\ref{more-algebra-item-shift-hom}). If $M^\\bullet$ has the structure of a left, resp.\\ right differential graded $A$-module, then this is a map of right, resp.\\ left differential graded $A$-modules (with the module structures as defined in this section). We omit the verification; we warn the reader that the $A$-module structure on the shift of a left graded $A$-module is defined using a sign, see Definition \\ref{definition-shift-graded-module}."}
+{"_id": "13162", "title": "dga-remark-P-resolution", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra. Using P-resolutions we can sometimes reduce statements about general objects of $D(A, \\text{d})$ to statements about $A[k]$. Namely, let $T$ be a property of objects of $D(A, \\text{d})$ and assume that \\begin{enumerate} \\item if $K_i$, $i \\in I$ is a family of objects of $D(A, \\text{d})$ and $T(K_i)$ holds for all $i \\in I$, then $T(\\bigoplus K_i)$, \\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle of $D(A, \\text{d})$ and $T$ holds for two, then $T$ holds for the third object, and \\item $T(A[k])$ holds for all $k \\in \\mathbf{Z}$. \\end{enumerate} Then $T$ holds for all objects of $D(A, \\text{d})$. This is clear from Lemmas \\ref{lemma-property-P-sequence} and \\ref{lemma-resolve}."}
+{"_id": "13163", "title": "dga-remark-graded-shift-functors", "text": "Let $R$ be a ring. Let $\\mathcal{D}$ be an $R$-linear category endowed with a collection of $R$-linear functors $[n] : \\mathcal{D} \\to \\mathcal{D}$, $x \\mapsto x[n]$ indexed by $n \\in \\mathbf{Z}$ such that $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_\\mathcal{D}$ (equality as functors). This allows us to construct a graded category $\\mathcal{D}^{gr}$ over $R$ with the same objects of $\\mathcal{D}$ setting $$ \\Hom_{\\mathcal{D}^{gr}}(x, y) = \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Hom_\\mathcal{D}(x, y[n]) $$ for $x, y$ in $\\mathcal{D}$. Observe that $(\\mathcal{D}^{gr})^0 = \\mathcal{D}$ (see Definition \\ref{definition-H0-of-graded-category}). Moreover, the graded category $\\mathcal{D}^{gr}$ inherits $R$-linear graded functors $[n]$ satisfying $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_{\\mathcal{D}^{gr}}$ with the property that $$ \\Hom_{\\mathcal{D}^{gr}}(x, y[n]) = \\Hom_{\\mathcal{D}^{gr}}(x, y)[n] $$ as graded $R$-modules compatible with composition of morphisms. \\medskip\\noindent Conversely, suppose given a graded category $\\mathcal{A}$ over $R$ endowed with a collection of $R$-linear graded functors $[n]$ satisfying $[n] \\circ [m] = [n + m]$ and $[0] = \\text{id}_\\mathcal{A}$ which are moreover equipped with isomorphisms $$ \\Hom_\\mathcal{A}(x, y[n]) = \\Hom_\\mathcal{A}(x, y)[n] $$ as graded $R$-modules compatible with composition of morphisms. Then the reader easily shows that $\\mathcal{A} = (\\mathcal{A}^0)^{gr}$. \\medskip\\noindent Here are two examples of the relationship $\\mathcal{D} \\leftrightarrow \\mathcal{A}$ we established above: \\begin{enumerate} \\item Let $\\mathcal{B}$ be an additive category. If $\\mathcal{D} = \\text{Gr}(\\mathcal{B})$, then $\\mathcal{A} = \\text{Gr}^{gr}(\\mathcal{B})$ as in Example \\ref{example-graded-category-graded-objects}. \\item If $A$ is a graded ring and $\\mathcal{D} = \\text{Mod}_A$ is the category of graded right $A$-modules, then $\\mathcal{A} = \\text{Mod}^{gr}_A$, see Example \\ref{example-gm-gr-cat}. \\end{enumerate}"}
+{"_id": "13164", "title": "dga-remark-shift-tensor-no-sign", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $M$ be a right differential graded $A$-module. Then for every $k \\in \\mathbf{Z}$ there is an isomorphism $$ (M \\otimes_A N)[k] \\longrightarrow M[k] \\otimes_A N $$ of right differential graded $B$-modules defined without the intervention of signs, see More on Algebra, Section \\ref{more-algebra-section-sign-rules}."}
+{"_id": "13165", "title": "dga-remark-shift-hom-no-sign", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a right differential graded $B$-module. Then for every $k \\in \\mathbf{Z}$ there is an isomorphism $$ \\Hom_{\\text{Mod}^{gr}_B}(N, N')[k] \\longrightarrow \\Hom_{\\text{Mod}^{gr}_B}(N, N'[k]) $$ of right differential graded $A$-modules defined without the intervention of signs, see More on Algebra, Section \\ref{more-algebra-section-sign-rules}."}
+{"_id": "13166", "title": "dga-remark-source-graded-projective", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Is there a characterization of those differential graded $A$-modules $P$ for which we have $$ \\Hom_{K(A, \\text{d})}(P, M) =  \\Hom_{D(A, \\text{d})}(P, M) $$ for all differential graded $A$-modules $M$? Let $\\mathcal{D} \\subset K(A, \\text{d})$ be the full subcategory whose objects are the objects $P$ satisfying the above. Then $\\mathcal{D}$ is a strictly full saturated triangulated subcategory of $K(A, \\text{d})$. If $P$ is projective as a graded $A$-module, then to see where $P$ is an object of $\\mathcal{D}$ it is enough to check that $\\Hom_{K(A, \\text{d})}(P, M) = 0$ whenever $M$ is acyclic. However, in general it is not enough to assume that $P$ is projective as a graded $A$-module. Example: take $A = R = k[\\epsilon]$ where $k$ is a field and $k[\\epsilon] = k[x]/(x^2)$ is the ring of dual numbers. Let $P$ be the object with $P^n = R$ for all $n \\in \\mathbf{Z}$ and differential given by multiplication by $\\epsilon$. Then $\\text{id}_P \\in \\Hom_{K(A, \\text{d})}(P, P)$ is a nonzero element but $P$ is acyclic."}
+{"_id": "13167", "title": "dga-remark-graded-projective-is-compact", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let us say a differential graded $A$-module $M$ is {\\it finite} if $M$ is generated, as a right $A$-module, by finitely many elements. If $P$ is a differential graded $A$-module which is finite graded projective, then we can ask: Does $P$ give a compact object of $D(A, \\text{d})$? Presumably, this is not true in general, but we do not know a counter example. However, if $P$ is also an object of the category $\\mathcal{D}$ of Remark \\ref{remark-source-graded-projective}, then this is the case (this follows from the fact that direct sums in $D(A, \\text{d})$ are given by direct sums of modules; details omitted)."}
+{"_id": "13168", "title": "dga-remark-tilting-equivalence", "text": "In Lemma \\ref{lemma-tilting-equivalence} we can replace condition (2) by the condition that $N$ is a classical generator for $D_{compact}(B, d)$, see Derived Categories, Proposition \\ref{derived-proposition-generator-versus-classical-generator}. Moreover, if we knew that $R\\Hom(N, B)$ is a compact object of $D(A, \\text{d})$, then it suffices to check that $N$ is a weak generator for $D_{compact}(B, \\text{d})$. We omit the proof; we will add it here if we ever need it in the Stacks project."}
+{"_id": "13169", "title": "dga-remark-lift-equivalence-to-dga", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras. Suppose given an $R$-linear equivalence $$ F : D(A, \\text{d}) \\longrightarrow D(B, \\text{d}) $$ of triangulated categories. Set $N = F(A)$. Then $N$ is a differential graded $B$-module. Since $F$ is an equivalence and $A$ is a compact object of $D(A, \\text{d})$, we conclude that $N$ is a compact object of $D(B, \\text{d})$. Since $A$ generates $D(A, \\text{d})$ and $F$ is an equivalence, we see that $N$ generates $D(B, \\text{d})$. Finally, $H^k(A) = \\Hom_{D(A, \\text{d})}(A, A[k])$ and as $F$ an equivalence we see that $F$ induces an isomorphism $H^k(A) = \\Hom_{D(B, \\text{d})}(N, N[k])$ for all $k$. In order to conclude that there is an equivalence $D(A, \\text{d}) \\longrightarrow D(B, \\text{d})$ which arises from the construction in Lemma \\ref{lemma-tilting-equivalence} all we need is a left $A$-module structure on $N$ compatible with derivation and commuting with the given right $B$-module structure. In fact, it suffices to do this after replacing $N$ by a quasi-isomorphic differential graded $B$-module. The module structure can be constructed in certain cases. For example, if we assume that $F$ can be lifted to a differential graded functor $$ F^{dg} : \\text{Mod}^{dg}_{(A, \\text{d})} \\longrightarrow \\text{Mod}^{dg}_{(B, \\text{d})} $$ (for notation see Example \\ref{example-dgm-dg-cat}) between the associated differential graded categories, then this holds. Another case is discussed in the proposition below."}
+{"_id": "13170", "title": "dga-remark-rickard", "text": "Let $A, B, F, N$ be as in Proposition \\ref{proposition-rickard}. It is not clear that $F$ and the functor $G(-) = - \\otimes_A^\\mathbf{L} N$ are isomorphic. By construction there is an isomorphism $N = G(A) \\to F(A)$ in $D(B, \\text{d})$. It is straightforward to extend this to a functorial isomorphism $G(M) \\to F(M)$ for $M$ is a differential graded $A$-module which is graded projective (e.g., a sum of shifts of $A$). Then one can conclude that $G(M) \\cong F(M)$ when $M$ is a cone of a map between such modules. We don't know whether more is true in general."}
+{"_id": "13171", "title": "dga-remark-centers", "text": "Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If $D(A)$ and $D(B)$ are equivalent as $R$-linear triangulated categories, then the centers of $A$ and $B$ are isomorphic as $R$-algebras. In particular, if $A$ and $B$ are commutative, then $A \\cong B$. The rather tricky proof can be found in \\cite[Proposition 9.2]{Rickard} or \\cite[Proposition 6.3.2]{KZ}. Another approach might be to use Hochschild cohomology (see remark below)."}
+{"_id": "13172", "title": "dga-remark-hochschild-cohomology", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras which are derived equivalent, i.e., such that there exists an $R$-linear equivalence $D(A, \\text{d}) \\to D(B, \\text{d})$ of triangulated categories. We would like to show that certain invariants of $(A, \\text{d})$ and $(B, \\text{d})$ coincide. In many situations one has more control of the situation. For example, it may happen that there is an equivalence of the form $$ - \\otimes_A \\Omega : D(A, \\text{d}) \\longrightarrow D(B, \\text{d}) $$ for some differential graded $(A, B)$-bimodule $\\Omega$ (this happens in the situation of Proposition \\ref{proposition-rickard} and is often true if the equivalence comes from a geometric construction). If also the quasi-inverse of our functor is given as $$ - \\otimes_B^\\mathbf{L} \\Omega' : D(B, \\text{d}) \\longrightarrow D(A, \\text{d}) $$ for a differential graded $(B, A)$-bimodule $\\Omega'$ (and as before such a module $\\Omega'$ often exists in practice). In this case we can consider the functor $$ D(A^{opp} \\otimes_R A, \\text{d}) \\longrightarrow D(B^{opp} \\otimes_R B, \\text{d}),\\quad M \\longmapsto \\Omega' \\otimes^\\mathbf{L}_A M \\otimes_A^\\mathbf{L} \\Omega $$ on derived categories of bimodules (use Lemma \\ref{lemma-bimodule-over-tensor} to turn bimodules into right modules). Observe that this functor sends the $(A, A)$-bimodule $A$ to the $(B, B)$-bimodule $B$. Under suitable conditions (e.g., flatness of $A$, $B$, $\\Omega$ over $R$, etc) this functor will be an equivalence as well. If this is the case, then it follows that we have isomorphisms of Hochschild cohomology groups $$ HH^i(A, \\text{d}) = \\Hom_{D(A^{opp} \\otimes_R A, \\text{d})}(A, A[i]) \\longrightarrow \\Hom_{D(B^{opp} \\otimes_R B, \\text{d})}(B, B[i]) = HH^i(B, \\text{d}). $$ For example, if $A = H^0(A)$, then $HH^0(A, \\text{d})$ is equal to the center of $A$, and this gives a conceptual proof of the result mentioned in Remark \\ref{remark-centers}. If we ever need this remark we will provide a precise statement with a detailed proof here."}
+{"_id": "13217", "title": "spaces-more-groupoids-remark-finite-monoid", "text": "Let $f : X \\to Y$ be a separated morphism of algebraic spaces. The sheaf $(X/Y)_{fin}$ comes with a natural map $(X/Y)_{fin} \\to Y$ by mapping the pair $(a, Z) \\in (X/Y)_{fin}(T)$ to the element $a \\in Y(T)$. We can use Lemma \\ref{lemma-finite-separated} to define operations $$ \\star_i : (X/Y)_{fin} \\times_Y (X/Y)_{fin} \\longrightarrow (X/Y)_{fin} $$ by the rules \\begin{align*} \\star_1 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\cup Z_2) \\\\ \\star_2 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\cap Z_2) \\\\ \\star_3 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_1 \\setminus Z_2) \\\\ \\star_4 : ((a, Z_1), (a, Z_2)) & \\longmapsto (a, Z_2 \\setminus Z_1). \\end{align*} The reason this works is that $Z_1 \\cap Z_2$ is both open and closed inside $Z_1$ and $Z_2$ (which also implies that $Z_1 \\cup Z_2$ is the disjoint union of the other three pieces). Thus we can think of $(X/Y)_{fin}$ as an $\\mathbf{F}_2$-algebra (without unit) over $Y$ with multiplication given by $ss' = \\star_2(s, s')$, and addition given by $$ s + s' = \\star_1(\\star_3(s, s'), \\star_4(s, s')) $$ which boils down to taking the symmetric difference. Note that in this sheaf of algebras $0 = (1_Y, \\emptyset)$ and that indeed $s + s = 0$ for any local section $s$. If $f : X \\to Y$ is finite, then this algebra has a unit namely $1 = (1_Y, X)$ and $\\star_3(s, s') = s(1 + s')$, and $\\star_4(s, s') = (1 + s)s'$."}
+{"_id": "13218", "title": "spaces-more-groupoids-remark-finite-quasi-finite-separated-morphism-schemes", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite morphism of schemes. In this case the sheaf $(X/Y)_{fin}$ is closely related to the sheaf $f_!\\mathbf{F}_2$ (insert future reference here) on $Y_\\etale$. Namely, if $V \\to Y$ is \\'etale, and $s \\in \\Gamma(V, f_!\\mathbf{F}_2)$, then $s \\in \\Gamma(V \\times_Y X, \\mathbf{F}_2)$ is a section with proper support $Z = \\text{Supp}(s)$ over $V$. Since $f$ is also locally quasi-finite we see that the projection $Z \\to V$ is actually finite. Since the support of a section of a constant abelian sheaf is open we see that the pair $(V \\to Y, \\text{Supp}(s))$ satisfies \\ref{equation-finite-conditions}. In fact, $f_!\\mathbf{F}_2 \\cong (X/Y)_{fin}|_{Y_\\etale}$ in this case which also explains the $\\mathbf{F}_2$-algebra structure introduced in Remark \\ref{remark-finite-monoid}."}
+{"_id": "13219", "title": "spaces-more-groupoids-remark-warning", "text": "The condition that $f$ be separated cannot be dropped from Proposition \\ref{proposition-finite-algebraic-space}. An example is to take $X$ the affine line with zero doubled, see Schemes, Example \\ref{schemes-example-affine-space-zero-doubled}, $Y = \\mathbf{A}^1_k$ the affine line, and $X \\to Y$ the obvious map. Recall that over $0 \\in Y$ there are two points $0_1$ and $0_2$ in $X$. Thus $(X/Y)_{fin}$ has four points over $0$, namely $\\emptyset, \\{0_1\\}, \\{0_2\\}, \\{0_1, 0_2\\}$. Of these four points only three can be lifted to an open subscheme of $U \\times_Y X$ finite over $U$ for $U \\to Y$ \\'etale, namely $\\emptyset, \\{0_1\\}, \\{0_2\\}$. This shows that $(X/Y)_{fin}$ if representable by an algebraic space is not \\'etale over $Y$. Similar arguments show that $(X/Y)_{fin}$ is really not an algebraic space. Details omitted."}
+{"_id": "13220", "title": "spaces-more-groupoids-remark-not-scheme", "text": "Let $Y = \\mathbf{A}^1_{\\mathbf{R}}$ be the affine line over the real numbers, and let $X = \\Spec(\\mathbf{C})$ mapping to the $\\mathbf{R}$-rational point $0$ in $Y$. In this case the morphism $f : X \\to Y$ is finite, but it is not the case that $(X/Y)_{fin}$ is a scheme. Namely, one can show that in this case the algebraic space $(X/Y)_{fin}$ is isomorphic to the algebraic space of Spaces, Example \\ref{spaces-example-non-representable-descent} associated to the extension $\\mathbf{R} \\subset \\mathbf{C}$. Thus it is really necessary to leave the category of schemes in order to represent the sheaf $(X/Y)_{fin}$, even when $f$ is a finite morphism."}
+{"_id": "13363", "title": "modules-remark-sections-support-in-closed", "text": "Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset. Let $\\mathcal{F}$ be an abelian sheaf on $X$. For $U \\subset X$ open set $$ \\mathcal{H}_Z(\\mathcal{F})(U) = \\{s \\in \\mathcal{F}(U) \\mid \\text{ the support of }s\\text{ is contained in }Z \\cap U\\} $$ Then $\\mathcal{H}_Z(\\mathcal{F})$ is an abelian subsheaf of $\\mathcal{F}$. It is the largest abelian subsheaf of $\\mathcal{F}$ whose support is contained in $Z$. By Lemma \\ref{lemma-i-star-exact} we may (and we do) view $\\mathcal{H}_Z(\\mathcal{F})$ as an abelian sheaf on $Z$. In this way we obtain a left exact functor $$ \\textit{Ab}(X) \\longrightarrow \\textit{Ab}(Z),\\quad \\mathcal{F} \\longmapsto \\mathcal{H}_Z(\\mathcal{F}) \\text{ viewed as abelian sheaf on }Z $$ All of the statements made above follow directly from Lemma \\ref{lemma-support-section-closed}."}
+{"_id": "13364", "title": "modules-remark-i-star-right-adjoint", "text": "In Sheaves, Remark \\ref{sheaves-remark-i-star-not-exact} we showed that $i_*$ as a functor on the categories of sheaves of sets does not have a right adjoint simply because it is not exact. However, it is very close to being true, in fact, the functor $i_*$ is exact on sheaves of pointed sets, sections with support in $Z$ can be defined for sheaves of pointed sets, and $\\mathcal{H}_Z$ makes sense and is a right adjoint to $i_*$."}
+{"_id": "13365", "title": "modules-remark-infinite-direct-sum-quasi-coherent-not", "text": "Warning: It is not true in general that an infinite direct sum of quasi-coherent $\\mathcal{O}_X$-modules is quasi-coherent. For more esoteric behaviour of quasi-coherent modules see Example \\ref{example-quasi-coherent}."}
+{"_id": "13366", "title": "modules-remark-condition-necessary", "text": "In the lemma above some condition beyond the condition that $X$ is quasi-compact is necessary. See Sheaves, Example \\ref{sheaves-example-conditions-needed-colimit}."}
+{"_id": "13367", "title": "modules-remark-sections-support-in-closed-modules", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$ be a closed subset. For an $\\mathcal{O}_X$-module $\\mathcal{F}$ we can consider the {\\it submodule of sections with support in $Z$}, denoted $\\mathcal{H}_Z(\\mathcal{F})$, defined by the rule $$ \\mathcal{H}_Z(\\mathcal{F})(U) = \\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset U \\cap Z\\} $$ Observe that $\\mathcal{H}_Z(\\mathcal{F})(U)$ is a module over $\\mathcal{O}_X(U)$, i.e., $\\mathcal{H}_Z(\\mathcal{F})$ is an $\\mathcal{O}_X$-module. By construction $\\mathcal{H}_Z(\\mathcal{F})$ is the largest $\\mathcal{O}_X$-submodule of $\\mathcal{F}$ whose support is contained in $Z$. Applying Lemma \\ref{lemma-i-star-equivalence} to the morphism of ringed spaces $(Z, \\mathcal{O}_X|_Z) \\to (X, \\mathcal{O}_X)$ we may (and we do) view $\\mathcal{H}_Z(\\mathcal{F})$ as an $\\mathcal{O}_X|_Z$-module on $Z$. Thus we obtain a functor $$ \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X|_Z), \\quad \\mathcal{F} \\longmapsto \\mathcal{H}_Z(\\mathcal{F}) \\text{ viewed as an }\\mathcal{O}_X|_Z\\text{-module on }Z $$ This functor is left exact, but in general not exact. All of the statements made above follow directly from Lemma \\ref{lemma-support-section-closed}. Clearly the construction is compatible with the construction in Remark \\ref{remark-sections-support-in-closed}."}
+{"_id": "13368", "title": "modules-remark-functoriality-principal-parts", "text": "Let $X$ be a topological space. Suppose given a commutative diagram of sheaves of rings $$ \\xymatrix{ \\mathcal{B} \\ar[r] & \\mathcal{B}' \\\\ \\mathcal{A} \\ar[u] \\ar[r] & \\mathcal{A}' \\ar[u] } $$ on $X$, a $\\mathcal{B}$-module $\\mathcal{F}$, a $\\mathcal{B}'$-module $\\mathcal{F}'$, and a $\\mathcal{B}$-linear map $\\mathcal{F} \\to \\mathcal{F}'$. Then we get a compatible system of module maps $$ \\xymatrix{ \\ldots \\ar[r] & \\mathcal{P}^2_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] & \\mathcal{P}^1_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] & \\mathcal{P}^0_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\\\ \\ldots \\ar[r] & \\mathcal{P}^2_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] & \\mathcal{P}^1_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] & \\mathcal{P}^0_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $\\mathcal{P}^k_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{M})$ in terms of (local) generators and relations in the proof of Lemma \\ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}."}
+{"_id": "13425", "title": "defos-remark-trivial-thickening", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. A first order thickening $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ is said to be {\\it trivial} if there exists a morphism of ringed spaces $\\pi : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$ which is a left inverse to $i$. The choice of such a morphism $\\pi$ is called a {\\it trivialization} of the first order thickening. Given $\\pi$ we obtain a splitting \\begin{equation} \\label{equation-splitting} \\mathcal{O}_{X'} = \\mathcal{O}_X \\oplus \\mathcal{I} \\end{equation} as sheaves of algebras on $X$ by using $\\pi^\\sharp$ to split the surjection $\\mathcal{O}_{X'} \\to \\mathcal{O}_X$. Conversely, such a splitting determines a morphism $\\pi$. The category of trivialized first order thickenings of $(X, \\mathcal{O}_X)$ is equivalent to the category of  $\\mathcal{O}_X$-modules."}
+{"_id": "13426", "title": "defos-remark-trivial-extension", "text": "Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a trivial first order thickening of ringed spaces and let $\\pi : (X', \\mathcal{O}_{X'}) \\to (X, \\mathcal{O}_X)$ be a trivialization. Then given any triple $(\\mathcal{F}, \\mathcal{K}, c)$ consisting of a pair of $\\mathcal{O}_X$-modules and a map $c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$ we may set $$ \\mathcal{F}'_{c, triv} = \\mathcal{F} \\oplus \\mathcal{K} $$ and use the splitting (\\ref{equation-splitting}) associated to $\\pi$ and the map $c$ to define the $\\mathcal{O}_{X'}$-module structure and obtain an extension (\\ref{equation-extension}). We will call $\\mathcal{F}'_{c, triv}$ the {\\it trivial extension} of $\\mathcal{F}$ by $\\mathcal{K}$ corresponding to $c$ and the trivialization $\\pi$. Given any extension $\\mathcal{F}'$ as in (\\ref{equation-extension}) we can use $\\pi^\\sharp : \\mathcal{O}_X \\to \\mathcal{O}_{X'}$ to think of $\\mathcal{F}'$ as an $\\mathcal{O}_X$-module extension, hence a class $\\xi_{\\mathcal{F}'}$ in $\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})$. Lemma \\ref{lemma-inf-ext} assures that $\\mathcal{F}' \\mapsto \\xi_{\\mathcal{F}'}$ induces a bijection $$ \\left\\{ \\begin{matrix} \\text{isomorphism classes of extensions}\\\\ \\mathcal{F}'\\text{ as in (\\ref{equation-extension}) with }c = c_{\\mathcal{F}'} \\end{matrix} \\right\\} \\longrightarrow \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}) $$ Moreover, the trivial extension $\\mathcal{F}'_{c, triv}$ maps to the zero class."}
+{"_id": "13427", "title": "defos-remark-extension-functorial", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(X, \\mathcal{O}_X) \\to (X'_i, \\mathcal{O}_{X'_i})$, $i = 1, 2$ be first order thickenings with ideal sheaves $\\mathcal{I}_i$. Let $h : (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2})$ be a morphism of first order thickenings of $(X, \\mathcal{O}_X)$. Picture $$ \\xymatrix{ & (X, \\mathcal{O}_X) \\ar[ld] \\ar[rd] & \\\\ (X'_1, \\mathcal{O}_{X'_1}) \\ar[rr]^h & &  (X'_2, \\mathcal{O}_{X'_2}) } $$ Observe that $h^\\sharp : \\mathcal{O}_{X'_2} \\to \\mathcal{O}_{X'_1}$ in particular induces an $\\mathcal{O}_X$-module map $\\mathcal{I}_2 \\to \\mathcal{I}_1$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}_i$. Assume furthermore given a map of $\\mathcal{O}_X$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$ such that $$ \\xymatrix{ \\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } $$ is commutative. Then there is a canonical functoriality $$ \\left\\{ \\begin{matrix} \\mathcal{F}'_2\\text{ as in (\\ref{equation-extension}) with }\\\\ c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2 \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\mathcal{F}'_1\\text{ as in (\\ref{equation-extension}) with }\\\\ c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1 \\end{matrix} \\right\\} $$ Namely, thinking of all sheaves $\\mathcal{O}_X$, $\\mathcal{O}_{X'_i}$, $\\mathcal{F}$, $\\mathcal{K}_i$, etc as sheaves on $X$, we set given $\\mathcal{F}'_2$ the sheaf $\\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2 \\ar[r] \\ar[d] & \\mathcal{F}'_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar@{=}[d] \\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}_1 \\ar[r] & \\mathcal{F}'_1 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ We omit the construction of the $\\mathcal{O}_{X'_1}$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$)."}
+{"_id": "13428", "title": "defos-remark-trivial-extension-functorial", "text": "Let $(X, \\mathcal{O}_X)$, $(X, \\mathcal{O}_X) \\to (X'_i, \\mathcal{O}_{X'_i})$, $\\mathcal{I}_i$, and $h : (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2})$ be as in Remark \\ref{remark-extension-functorial}. Assume that we are given trivializations $\\pi_i : X'_i \\to X$ such that $\\pi_1 = h \\circ \\pi_2$. In other words, assume $h$ is a morphism of trivialized first order thickening of $(X, \\mathcal{O}_X)$. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}_i$. Assume furthermore given a map of $\\mathcal{O}_X$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$ such that $$ \\xymatrix{ \\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } $$ is commutative. In this situation the construction of Remark \\ref{remark-trivial-extension} induces a commutative diagram $$ \\xymatrix{ \\{\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension}) with } c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2\\} \\ar[d] \\ar[rr] & & \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_2) \\ar[d] \\\\ \\{\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension}) with } c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1\\} \\ar[rr] & & \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1) } $$ where the vertical map on the right is given by functoriality of $\\Ext$ and the map $\\mathcal{K}_2 \\to \\mathcal{K}_1$ and the vertical map on the left is the one from Remark \\ref{remark-extension-functorial}."}
+{"_id": "13429", "title": "defos-remark-short-exact-sequence-thickenings", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. We define a sequence of morphisms of first order thickenings $$ (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2}) \\to (X'_3, \\mathcal{O}_{X'_3}) $$ of $(X, \\mathcal{O}_X)$ to be a {\\it complex} if the corresponding maps between the ideal sheaves $\\mathcal{I}_i$ give a complex of $\\mathcal{O}_X$-modules $\\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(X'_1, \\mathcal{O}_{X'_1}) \\to (X_3', \\mathcal{O}_{X'_3})$ factors through $(X, \\mathcal{O}_X) \\to (X'_3, \\mathcal{O}_{X'_3})$, i.e., the first order thickening $(X'_1, \\mathcal{O}_{X'_1})$ of $(X, \\mathcal{O}_X)$ is trivial and comes with a canonical trivialization $\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$. \\medskip\\noindent We say a sequence of morphisms of first order thickenings $$ (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2}) \\to (X'_3, \\mathcal{O}_{X'_3}) $$ of $(X, \\mathcal{O}_X)$ is {\\it a short exact sequence} if the corresponding maps between ideal sheaves is a short exact sequence $$ 0 \\to \\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1 \\to 0 $$ of $\\mathcal{O}_X$-modules."}
+{"_id": "13430", "title": "defos-remark-complex-thickenings-and-ses-modules", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Let $$ (X'_1, \\mathcal{O}_{X'_1}) \\to (X'_2, \\mathcal{O}_{X'_2}) \\to (X'_3, \\mathcal{O}_{X'_3}) $$ be a complex first order thickenings of $(X, \\mathcal{O}_X)$, see Remark \\ref{remark-short-exact-sequence-thickenings}. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of an $\\mathcal{O}_X$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}_i$. Assume given a short exact sequence of $\\mathcal{O}_X$-modules $$ 0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0 $$ such that $$ \\vcenter{ \\xymatrix{ \\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ \\mathcal{I}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]_-{c_3} \\ar[d] & \\mathcal{K}_3 \\ar[d] \\\\ \\mathcal{I}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\ar[r]^-{c_2} & \\mathcal{K}_2 } } $$ are commutative. Finally, assume given an extension $$ 0 \\to \\mathcal{K}_2 \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0 $$ as in (\\ref{equation-extension}) with $\\mathcal{K} = \\mathcal{K}_2$ of $\\mathcal{O}_{X'_2}$-modules with $c_{\\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark \\ref{remark-extension-functorial} to obtain an extension $\\mathcal{F}'_1$ on $X'_1$ (we'll describe $\\mathcal{F}'_1$ in this special case below). By Remark \\ref{remark-trivial-extension} using the canonical splitting $\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$ of Remark \\ref{remark-short-exact-sequence-thickenings} we obtain $\\xi_{\\mathcal{F}'_1} \\in \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1)$. Finally, we have the obstruction $$ o(\\mathcal{F}, \\mathcal{K}_3, c_3) \\in \\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_3) $$ see Lemma \\ref{lemma-inf-obs-ext}. In this situation we {\\bf claim} that the canonical map $$ \\partial : \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_1) \\longrightarrow \\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K}_3) $$ coming from the short exact sequence $0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0$ sends $\\xi_{\\mathcal{F}'_1}$ to the obstruction class $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$. \\medskip\\noindent To prove this claim choose an embedding $j : \\mathcal{K}_3 \\to \\mathcal{K}$ where $\\mathcal{K}$ is an injective $\\mathcal{O}_X$-module. We can lift $j$ to a map $j' : \\mathcal{K}_2 \\to \\mathcal{K}$. Set $\\mathcal{E}'_2 = j'_*\\mathcal{F}'_2$ equal to the pushout of $\\mathcal{F}'_2$ by $j'$ so that $c_{\\mathcal{E}'_2} = j' \\circ c_2$. Picture: $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2 \\ar[r] \\ar[d]_{j'} & \\mathcal{F}'_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K} \\ar[r] & \\mathcal{E}'_2 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ Set $\\mathcal{E}'_3 = \\mathcal{E}'_2$ but viewed as an $\\mathcal{O}_{X'_3}$-module via $\\mathcal{O}_{X'_3} \\to \\mathcal{O}_{X'_2}$. Then $c_{\\mathcal{E}'_3} = j \\circ c_3$. The proof of Lemma \\ref{lemma-inf-obs-ext} constructs $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\\mathcal{O}_X$-modules $$ 0 \\to \\mathcal{K}/\\mathcal{K}_3 \\to \\mathcal{E}'_3/\\mathcal{K}_3 \\to \\mathcal{F} \\to 0 $$ On the other hand, note that $\\mathcal{F}'_1 = \\mathcal{F}'_2/\\mathcal{K}_3$ hence the class $\\xi_{\\mathcal{F}'_1}$ is the class of the extension $$ 0 \\to \\mathcal{K}_2/\\mathcal{K}_3 \\to \\mathcal{F}'_2/\\mathcal{K}_3 \\to \\mathcal{F} \\to 0 $$ seen as a sequence of $\\mathcal{O}_X$-modules using $\\pi^\\sharp$ where $\\pi : (X'_1, \\mathcal{O}_{X'_1}) \\to (X, \\mathcal{O}_X)$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2/\\mathcal{K}_3 \\ar[r] \\ar[d] & \\mathcal{F}'_2/\\mathcal{K}_3 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}/\\mathcal{K}_3 \\ar[r] & \\mathcal{E}'_3/\\mathcal{K}_3 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ which is $\\mathcal{O}_X$-linear (with the $\\mathcal{O}_X$-module structures given above)."}
+{"_id": "13431", "title": "defos-remark-trivial-thickening-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. A first order thickening $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ is said to be {\\it trivial} if there exists a morphism of ringed topoi $\\pi : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ which is a left inverse to $i$. The choice of such a morphism $\\pi$ is called a {\\it trivialization} of the first order thickening. Given $\\pi$ we obtain a splitting \\begin{equation} \\label{equation-splitting-ringed-topoi} \\mathcal{O}' = \\mathcal{O} \\oplus \\mathcal{I} \\end{equation} as sheaves of algebras on $\\mathcal{C}$ by using $\\pi^\\sharp$ to split the surjection $\\mathcal{O}' \\to \\mathcal{O}$. Conversely, such a splitting determines a morphism $\\pi$. The category of trivialized first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is equivalent to the category of  $\\mathcal{O}$-modules."}
+{"_id": "13432", "title": "defos-remark-trivial-extension-ringed-topoi", "text": "Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$ be a trivial first order thickening of ringed topoi and let $\\pi : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ be a trivialization. Then given any triple $(\\mathcal{F}, \\mathcal{K}, c)$ consisting of a pair of $\\mathcal{O}$-modules and a map $c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$ we may set $$ \\mathcal{F}'_{c, triv} = \\mathcal{F} \\oplus \\mathcal{K} $$ and use the splitting (\\ref{equation-splitting-ringed-topoi}) associated to $\\pi$ and the map $c$ to define the $\\mathcal{O}'$-module structure and obtain an extension (\\ref{equation-extension-ringed-topoi}). We will call $\\mathcal{F}'_{c, triv}$ the {\\it trivial extension} of $\\mathcal{F}$ by $\\mathcal{K}$ corresponding to $c$ and the trivialization $\\pi$. Given any extension $\\mathcal{F}'$ as in (\\ref{equation-extension-ringed-topoi}) we can use $\\pi^\\sharp : \\mathcal{O} \\to \\mathcal{O}'$ to think of $\\mathcal{F}'$ as an $\\mathcal{O}$-module extension, hence a class $\\xi_{\\mathcal{F}'}$ in $\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})$. Lemma \\ref{lemma-inf-ext-ringed-topoi} assures that $\\mathcal{F}' \\mapsto \\xi_{\\mathcal{F}'}$ induces a bijection $$ \\left\\{ \\begin{matrix} \\text{isomorphism classes of extensions}\\\\ \\mathcal{F}'\\text{ as in (\\ref{equation-extension-ringed-topoi}) with } c = c_{\\mathcal{F}'} \\end{matrix} \\right\\} \\longrightarrow \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}) $$ Moreover, the trivial extension $\\mathcal{F}'_{c, triv}$ maps to the zero class."}
+{"_id": "13433", "title": "defos-remark-extension-functorial-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$, $i = 1, 2$ be first order thickenings with ideal sheaves $\\mathcal{I}_i$. Let $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$ be a morphism of first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$. Picture $$ \\xymatrix{ & (\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[ld] \\ar[rd] & \\\\ (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\ar[rr]^h & &  (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) } $$ Observe that $h^\\sharp : \\mathcal{O}'_2 \\to \\mathcal{O}'_1$ in particular induces an $\\mathcal{O}$-module map $\\mathcal{I}_2 \\to \\mathcal{I}_1$. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}_i$. Assume furthermore given a map of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$ such that $$ \\xymatrix{ \\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } $$ is commutative. Then there is a canonical functoriality $$ \\left\\{ \\begin{matrix} \\mathcal{F}'_2\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }\\\\ c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2 \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} \\mathcal{F}'_1\\text{ as in (\\ref{equation-extension-ringed-topoi}) with }\\\\ c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1 \\end{matrix} \\right\\} $$ Namely, thinking of all sheaves $\\mathcal{O}$, $\\mathcal{O}'_i$, $\\mathcal{F}$, $\\mathcal{K}_i$, etc as sheaves on $\\mathcal{C}$, we set given $\\mathcal{F}'_2$ the sheaf $\\mathcal{F}'_1$ equal to the pushout, i.e., fitting into the following diagram of extensions $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2 \\ar[r] \\ar[d] & \\mathcal{F}'_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar@{=}[d] \\ar[r] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}_1 \\ar[r] & \\mathcal{F}'_1 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ We omit the construction of the $\\mathcal{O}'_1$-module structure on the pushout (this uses the commutativity of the diagram involving $c_1$ and $c_2$)."}
+{"_id": "13434", "title": "defos-remark-trivial-extension-functorial-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$, $(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$, $\\mathcal{I}_i$, and $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$ be as in Remark \\ref{remark-extension-functorial-ringed-topoi}. Assume that we are given trivializations $\\pi_i : (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ such that $\\pi_1 = h \\circ \\pi_2$. In other words, assume $h$ is a morphism of trivialized first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}_i$. Assume furthermore given a map of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$ such that $$ \\xymatrix{ \\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } $$ is commutative. In this situation the construction of Remark \\ref{remark-trivial-extension-ringed-topoi} induces a commutative diagram $$ \\xymatrix{ \\{\\mathcal{F}'_2\\text{ as in (\\ref{equation-extension-ringed-topoi}) with } c_2 = c_{\\mathcal{F}'_2}\\text{ and }\\mathcal{K} = \\mathcal{K}_2\\} \\ar[d] \\ar[rr] & & \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_2) \\ar[d] \\\\ \\{\\mathcal{F}'_1\\text{ as in (\\ref{equation-extension-ringed-topoi}) with } c_1 = c_{\\mathcal{F}'_1}\\text{ and }\\mathcal{K} = \\mathcal{K}_1\\} \\ar[rr] & & \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1) } $$ where the vertical map on the right is given by functoriality of $\\Ext$ and the map $\\mathcal{K}_2 \\to \\mathcal{K}_1$ and the vertical map on the left is the one from Remark \\ref{remark-extension-functorial-ringed-topoi}."}
+{"_id": "13435", "title": "defos-remark-obstruction-extension-functorial-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$, $(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_i), \\mathcal{O}'_i)$, $\\mathcal{I}_i$, and $h : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2)$ be as in Remark \\ref{remark-extension-functorial-ringed-topoi}. Observe that $h^\\sharp : \\mathcal{O}'_2 \\to \\mathcal{O}'_1$ in particular induces an $\\mathcal{O}$-module map $\\mathcal{I}_2 \\to \\mathcal{I}_1$. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}_i$. Assume furthermore given a map of $\\mathcal{O}$-modules $\\mathcal{K}_2 \\to \\mathcal{K}_1$ such that $$ \\xymatrix{ \\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } $$ is commutative. Then we {\\bf claim} the map $$ \\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_2) \\longrightarrow \\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1) $$ sends $o(\\mathcal{F}, \\mathcal{K}_2, c_2)$ to $o(\\mathcal{F}, \\mathcal{K}_1, c_1)$. \\medskip\\noindent To prove this claim choose an embedding $j_2 : \\mathcal{K}_2 \\to \\mathcal{K}_2'$ where $\\mathcal{K}_2'$ is an injective $\\mathcal{O}$-module. As in the proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} we can choose an extension of $\\mathcal{O}_2$-modules $$ 0 \\to \\mathcal{K}_2' \\to \\mathcal{E}_2 \\to \\mathcal{F} \\to 0 $$ such that $c_{\\mathcal{E}_2} = j_2 \\circ c_2$. The proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} constructs $o(\\mathcal{F}, \\mathcal{K}_2, c_2)$ as the Yoneda extension class (in the sense of Derived Categories, Section \\ref{derived-section-ext}) of the exact sequence of $\\mathcal{O}$-modules $$ 0 \\to \\mathcal{K}_2 \\to \\mathcal{K}_2' \\to \\mathcal{E}_2/\\mathcal{K}_2 \\to \\mathcal{F} \\to 0 $$ Let $\\mathcal{K}_1'$ be the cokernel of $\\mathcal{K}_2 \\to \\mathcal{K}_1 \\oplus \\mathcal{K}_2'$. There is an injection $j_1 : \\mathcal{K}_1 \\to \\mathcal{K}_1'$ and a map $\\mathcal{K}_2' \\to \\mathcal{K}_1'$ forming a commutative square. We form the pushout: $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2' \\ar[r] \\ar[d] & \\mathcal{E}_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}_1' \\ar[r] & \\mathcal{E}_1 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ There is a canonical $\\mathcal{O}_1$-module structure on $\\mathcal{E}_1$ and for this structure we have $c_{\\mathcal{E}_1} = j_1 \\circ c_1$ (this uses the commutativity of the diagram involving $c_1$ and $c_2$ above). The procedure of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} tells us that $o(\\mathcal{F}, \\mathcal{K}_1, c_1)$ is the Yoneda extension class of the exact sequence of $\\mathcal{O}$-modules $$ 0 \\to \\mathcal{K}_1 \\to \\mathcal{K}_1' \\to \\mathcal{E}_1/\\mathcal{K}_1 \\to \\mathcal{F} \\to 0 $$ Since we have maps of exact sequences $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2 \\ar[d] \\ar[r] & \\mathcal{K}_2' \\ar[d] \\ar[r] & \\mathcal{E}_2/\\mathcal{K}_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar@{=}[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}_2 \\ar[r] & \\mathcal{K}_2' \\ar[r] & \\mathcal{E}_2/\\mathcal{K}_2 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ we conclude that the claim is true."}
+{"_id": "13436", "title": "defos-remark-short-exact-sequence-thickenings-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. We define a sequence of morphisms of first order thickenings $$ (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to (\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3) $$ of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ to be a {\\it complex} if the corresponding maps between the ideal sheaves $\\mathcal{I}_i$ give a complex of $\\mathcal{O}$-modules $\\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)$ factors through $(\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3)$, i.e., the first order thickening $(\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1)$ of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is trivial and comes with a canonical trivialization $\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$. \\medskip\\noindent We say a sequence of morphisms of first order thickenings $$ (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to (\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3) $$ of $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is {\\it a short exact sequence} if the corresponding maps between ideal sheaves is a short exact sequence $$ 0 \\to \\mathcal{I}_3 \\to \\mathcal{I}_2 \\to \\mathcal{I}_1 \\to 0 $$ of $\\mathcal{O}$-modules."}
+{"_id": "13437", "title": "defos-remark-complex-thickenings-and-ses-modules-ringed-topoi", "text": "Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module. Let $$ (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{D}_2), \\mathcal{O}'_2) \\to (\\Sh(\\mathcal{D}_3), \\mathcal{O}'_3) $$ be a complex first order thickenings of $(\\Sh(\\mathcal{C}), \\mathcal{O})$, see Remark \\ref{remark-short-exact-sequence-thickenings-ringed-topoi}. Let $(\\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of an $\\mathcal{O}$-module $\\mathcal{K}_i$ and a map $c_i : \\mathcal{I}_i \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}_i$. Assume given a short exact sequence of $\\mathcal{O}$-modules $$ 0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0 $$ such that $$ \\vcenter{ \\xymatrix{ \\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_2} \\ar[d] & \\mathcal{K}_2 \\ar[d] \\\\ \\mathcal{I}_1 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-{c_1} & \\mathcal{K}_1 } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ \\mathcal{I}_3 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]_-{c_3} \\ar[d] & \\mathcal{K}_3 \\ar[d] \\\\ \\mathcal{I}_2 \\otimes_\\mathcal{O} \\mathcal{F} \\ar[r]^-{c_2} & \\mathcal{K}_2 } } $$ are commutative. Finally, assume given an extension $$ 0 \\to \\mathcal{K}_2 \\to \\mathcal{F}'_2 \\to \\mathcal{F} \\to 0 $$ as in (\\ref{equation-extension-ringed-topoi}) with $\\mathcal{K} = \\mathcal{K}_2$ of $\\mathcal{O}'_2$-modules with $c_{\\mathcal{F}'_2} = c_2$. In this situation we can apply the functoriality of Remark \\ref{remark-extension-functorial-ringed-topoi} to obtain an extension $\\mathcal{F}'_1$ of $\\mathcal{O}'_1$-modules (we'll describe $\\mathcal{F}'_1$ in this special case below). By Remark \\ref{remark-trivial-extension-ringed-topoi} using the canonical splitting $\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ of Remark \\ref{remark-short-exact-sequence-thickenings-ringed-topoi} we obtain $\\xi_{\\mathcal{F}'_1} \\in \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1)$. Finally, we have the obstruction $$ o(\\mathcal{F}, \\mathcal{K}_3, c_3) \\in \\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_3) $$ see Lemma \\ref{lemma-inf-obs-ext-ringed-topoi}. In this situation we {\\bf claim} that the canonical map $$ \\partial : \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_1) \\longrightarrow \\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K}_3) $$ coming from the short exact sequence $0 \\to \\mathcal{K}_3 \\to \\mathcal{K}_2 \\to \\mathcal{K}_1 \\to 0$ sends $\\xi_{\\mathcal{F}'_1}$ to the obstruction class $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$. \\medskip\\noindent To prove this claim choose an embedding $j : \\mathcal{K}_3 \\to \\mathcal{K}$ where $\\mathcal{K}$ is an injective $\\mathcal{O}$-module. We can lift $j$ to a map $j' : \\mathcal{K}_2 \\to \\mathcal{K}$. Set $\\mathcal{E}'_2 = j'_*\\mathcal{F}'_2$ equal to the pushout of $\\mathcal{F}'_2$ by $j'$ so that $c_{\\mathcal{E}'_2} = j' \\circ c_2$. Picture: $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2 \\ar[r] \\ar[d]_{j'} & \\mathcal{F}'_2 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K} \\ar[r] & \\mathcal{E}'_2 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ Set $\\mathcal{E}'_3 = \\mathcal{E}'_2$ but viewed as an $\\mathcal{O}'_3$-module via $\\mathcal{O}'_3 \\to \\mathcal{O}'_2$. Then $c_{\\mathcal{E}'_3} = j \\circ c_3$. The proof of Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} constructs $o(\\mathcal{F}, \\mathcal{K}_3, c_3)$ as the boundary of the class of the extension of $\\mathcal{O}$-modules $$ 0 \\to \\mathcal{K}/\\mathcal{K}_3 \\to \\mathcal{E}'_3/\\mathcal{K}_3 \\to \\mathcal{F} \\to 0 $$ On the other hand, note that $\\mathcal{F}'_1 = \\mathcal{F}'_2/\\mathcal{K}_3$ hence the class $\\xi_{\\mathcal{F}'_1}$ is the class of the extension $$ 0 \\to \\mathcal{K}_2/\\mathcal{K}_3 \\to \\mathcal{F}'_2/\\mathcal{K}_3 \\to \\mathcal{F} \\to 0 $$ seen as a sequence of $\\mathcal{O}$-modules using $\\pi^\\sharp$ where $\\pi : (\\Sh(\\mathcal{D}_1), \\mathcal{O}'_1) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$ is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram $$ \\xymatrix{ 0 \\ar[r] & \\mathcal{K}_2/\\mathcal{K}_3 \\ar[r] \\ar[d] & \\mathcal{F}'_2/\\mathcal{K}_3 \\ar[r] \\ar[d] & \\mathcal{F} \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & \\mathcal{K}/\\mathcal{K}_3 \\ar[r] & \\mathcal{E}'_3/\\mathcal{K}_3 \\ar[r] & \\mathcal{F} \\ar[r] & 0 } $$ which is $\\mathcal{O}$-linear (with the $\\mathcal{O}$-module structures given above)."}
+{"_id": "13645", "title": "duality-remark-iso-on-RSheafHom", "text": "In the situation of Lemma \\ref{lemma-iso-on-RSheafHom} we have $$ DQ_Y(Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) = Rf_* DQ_X(R\\SheafHom_{\\mathcal{O}_X}(L, a(K))) $$ by Derived Categories of Schemes, Lemma \\ref{perfect-lemma-pushforward-better-coherator}. Thus if $R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\in D_\\QCoh(\\mathcal{O}_X)$, then we can ``erase'' the $DQ_Y$ on the left hand side of the arrow. On the other hand, if we know that $R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K) \\in D_\\QCoh(\\mathcal{O}_Y)$, then we can ``erase'' the $DQ_Y$ from the right hand side of the arrow. If both are true then we see that (\\ref{equation-sheafy-trace}) is an isomorphism. Combining this with Derived Categories of Schemes, Lemma \\ref{perfect-lemma-quasi-coherence-internal-hom} we see that $Rf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K)) \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)$ is an isomorphism if \\begin{enumerate} \\item $L$ and $Rf_*L$ are perfect, or \\item $K$ is bounded below and $L$ and $Rf_*L$ are pseudo-coherent. \\end{enumerate} For (2) we use that $a(K)$ is bounded below if $K$ is bounded below, see Lemma \\ref{lemma-twisted-inverse-image-bounded-below}."}
+{"_id": "13646", "title": "duality-remark-going-around", "text": "Consider a commutative diagram $$ \\xymatrix{ X'' \\ar[r]_{k'} \\ar[d]_{f''} & X' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y'' \\ar[r]^{l'} \\ar[d]_{g''} & Y' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\ Z'' \\ar[r]^{m'} & Z' \\ar[r]^m & Z } $$ of quasi-compact and quasi-separated schemes where all squares are cartesian and where $(f, l)$, $(g, m)$, $(f', l')$, $(g', m')$ are Tor independent pairs of maps. Let $a$, $a'$, $a''$, $b$, $b'$, $b''$ be the right adjoints of Lemma \\ref{lemma-twisted-inverse-image} for $f$, $f'$, $f''$, $g$, $g'$, $g''$. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows $$ \\begin{matrix} A & B \\\\ C & D \\end{matrix} $$ Then the maps (\\ref{equation-base-change-map}) for the squares are (where we use $k^* = Lk^*$, etc) $$ \\begin{matrix} \\gamma_A : (k')^* \\circ a' \\to a'' \\circ (l')^* & \\gamma_B : k^* \\circ a \\to a' \\circ l^* \\\\ \\gamma_C : (l')^* \\circ b' \\to b'' \\circ (m')^* & \\gamma_D : l^* \\circ b \\to b' \\circ m^* \\end{matrix} $$ For the $2 \\times 1$ and $1 \\times 2$ rectangles we have four further base change maps $$ \\begin{matrix} \\gamma_{A + B} : (k \\circ k')^* \\circ a \\to a'' \\circ (l \\circ l')^* \\\\ \\gamma_{C + D} : (l \\circ l')^* \\circ b \\to b'' \\circ (m \\circ m')^* \\\\ \\gamma_{A + C} : (k')^* \\circ (a' \\circ b') \\to (a'' \\circ b'') \\circ (m')^* \\\\ \\gamma_{B + D} : k^* \\circ (a \\circ b) \\to (a' \\circ b') \\circ m^* \\end{matrix} $$ By Lemma \\ref{lemma-compose-base-change-maps-horizontal} we have $$ \\gamma_{A + B} = \\gamma_A \\circ \\gamma_B, \\quad \\gamma_{C + D} = \\gamma_C \\circ \\gamma_D $$ and by Lemma \\ref{lemma-compose-base-change-maps} we have $$ \\gamma_{A + C} = \\gamma_C \\circ \\gamma_A, \\quad \\gamma_{B + D} = \\gamma_D \\circ \\gamma_B $$ Here it would be more correct to write $\\gamma_{A + B} = (\\gamma_A \\star \\text{id}_{l^*}) \\circ (\\text{id}_{(k')^*} \\star \\gamma_B)$ with notation as in Categories, Section \\ref{categories-section-formal-cat-cat} and similarly for the others. However, we continue the abuse of notation used in the proofs of Lemmas \\ref{lemma-compose-base-change-maps} and \\ref{lemma-compose-base-change-maps-horizontal} of dropping $\\star$ products with identities as one can figure out which ones to add as long as the source and target of the transformation is known. Having said all of this we find (a priori) two transformations $$ (k')^* \\circ k^* \\circ a \\circ b \\longrightarrow a'' \\circ b'' \\circ (m')^* \\circ m^* $$ namely $$ \\gamma_C \\circ \\gamma_A \\circ \\gamma_D \\circ \\gamma_B = \\gamma_{A + C} \\circ \\gamma_{B + D} $$ and $$ \\gamma_C \\circ \\gamma_D \\circ \\gamma_A \\circ \\gamma_B = \\gamma_{C + D} \\circ \\gamma_{A + B} $$ The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that $$ \\xymatrix{ (k')^* \\circ a' \\circ l^* \\circ b \\ar[r]_{\\gamma_D} \\ar[d]_{\\gamma_A} & (k')^* \\circ a' \\circ b' \\circ m^* \\ar[d]^{\\gamma_A} \\\\ a'' \\circ (l')^* \\circ l^* \\circ b \\ar[r]^{\\gamma_D} & a'' \\circ (l')^* \\circ b' \\circ m^* } $$ commutes. This is true by Categories, Lemma \\ref{categories-lemma-properties-2-cat-cats} or more simply the discussion preceding Categories, Definition \\ref{categories-definition-horizontal-composition}."}
+{"_id": "13647", "title": "duality-remark-check-over-affines", "text": "Consider a cartesian diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of quasi-compact and quasi-separated schemes with $(g, f)$ Tor independent. Let $V \\subset Y$ and $V' \\subset Y'$ be affine opens with $g(V') \\subset V$. Form the cartesian diagrams $$ \\vcenter{ \\xymatrix{ U \\ar[r] \\ar[d] & X \\ar[d] \\\\ V \\ar[r] & Y } } \\quad\\text{and}\\quad \\vcenter{ \\xymatrix{ U' \\ar[r] \\ar[d] & X' \\ar[d] \\\\ V' \\ar[r] & Y' } } $$ Assume (\\ref{equation-sheafy}) with respect to $K$ and the first diagram and (\\ref{equation-sheafy}) with respect to $Lg^*K$ and the second diagram are isomorphisms. Then the restriction of the base change map (\\ref{equation-base-change-map}) $$ L(g')^*a(K) \\longrightarrow a'(Lg^*K) $$ to $U'$ is isomorphic to the base change map (\\ref{equation-base-change-map}) for $K|_V$ and the cartesian diagram $$ \\xymatrix{ U' \\ar[r] \\ar[d] & U \\ar[d] \\\\ V' \\ar[r] & V } $$ This follows from the fact that (\\ref{equation-sheafy}) is a special case of the base change map (\\ref{equation-base-change-map}) and that the base change maps compose correctly if we stack squares horizontally, see Lemma \\ref{lemma-compose-base-change-maps-horizontal}. Thus in order to check the base change map restricted to $U'$ is an isomorphism it suffices to work with the last diagram."}
+{"_id": "13648", "title": "duality-remark-trace-map-finite", "text": "If $f : Y \\to X$ is a finite morphism of Noetherian schemes, then the diagram $$ \\xymatrix{ Rf_*a(K) \\ar[r]_-{\\text{Tr}_{f, K}} \\ar@{=}[d] & K \\ar@{=}[d] \\\\ R\\SheafHom_{\\mathcal{O}_X}(f_*\\mathcal{O}_Y, K) \\ar[r] & K } $$ is commutative for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$. This follows from Lemma \\ref{lemma-finite-twisted}. The lower horizontal arrow is induced by the map $\\mathcal{O}_X \\to f_*\\mathcal{O}_Y$ and the upper horizontal arrow is the trace map discussed in Section \\ref{section-trace}."}
+{"_id": "13649", "title": "duality-remark-relative-dualizing-complex", "text": "Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$ be a proper, flat morphism of finite presentation. Let $a$ be the adjoint of Lemma \\ref{lemma-twisted-inverse-image} for $f$. In this situation, $\\omega_{X/Y}^\\bullet = a(\\mathcal{O}_Y)$ is sometimes called the {\\it relative dualizing complex}. By Lemma \\ref{lemma-compare-with-pullback-flat-proper} there is a functorial isomorphism $a(K) = Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet$ for $K \\in D_\\QCoh(\\mathcal{O}_Y)$. Moreover, the trace map $$ \\text{Tr}_{f, \\mathcal{O}_Y} : Rf_*\\omega_{X/Y}^\\bullet \\to \\mathcal{O}_Y $$ of Section \\ref{section-trace} induces the trace map for all $K$ in $D_\\QCoh(\\mathcal{O}_Y)$. More precisely the diagram $$ \\xymatrix{ Rf_*a(K) \\ar[rrr]_{\\text{Tr}_{f, K}} \\ar@{=}[d] & & & K \\ar@{=}[d] \\\\ Rf_*(Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet) \\ar@{=}[r] & K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\omega_{X/Y}^\\bullet \\ar[rr]^-{\\text{id}_K \\otimes \\text{Tr}_{f, \\mathcal{O}_Y}} & & K } $$ where the equality on the lower right is Derived Categories of Schemes, Lemma \\ref{perfect-lemma-cohomology-base-change}. If $g : Y' \\to Y$ is a morphism of quasi-compact and quasi-separated schemes and $X' = Y' \\times_Y X$, then by Lemma \\ref{lemma-proper-flat-base-change} we have $\\omega_{X'/Y'}^\\bullet = L(g')^*\\omega_{X/Y}^\\bullet$ where $g' : X' \\to X$ is the projection and by Lemma \\ref{lemma-trace-map-and-base-change} the trace map $$ \\text{Tr}_{f', \\mathcal{O}_{Y'}} : Rf'_*\\omega_{X'/Y'}^\\bullet \\to \\mathcal{O}_{Y'} $$ for $f' : X' \\to Y'$ is the base change of $\\text{Tr}_{f, \\mathcal{O}_Y}$ via the base change isomorphism."}
+{"_id": "13650", "title": "duality-remark-relative-dualizing-complex-relative-cup-product", "text": "Let $f : X \\to Y$, $\\omega^\\bullet_{X/Y}$, and $\\text{Tr}_{f, \\mathcal{O}_Y}$ be as in Remark \\ref{remark-relative-dualizing-complex}. Let $K$ and $M$ be in $D_\\QCoh(\\mathcal{O}_X)$ with $M$ pseudo-coherent (for example perfect). Suppose given a map $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M \\to \\omega^\\bullet_{X/Y}$ which corresponds to an isomorphism $K \\to R\\SheafHom_{\\mathcal{O}_X}(M, \\omega^\\bullet_{X/Y})$ via Cohomology, Equation (\\ref{cohomology-equation-internal-hom}). Then the relative cup product (Cohomology, Remark \\ref{cohomology-remark-cup-product}) $$ Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\to Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\to Rf_*\\omega^\\bullet_{X/Y} \\xrightarrow{\\text{Tr}_{f, \\mathcal{O}_Y}} \\mathcal{O}_Y $$ determines an isomorphism $Rf_*K \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*M, \\mathcal{O}_Y)$. Namely, since $\\omega^\\bullet_{X/Y} = a(\\mathcal{O}_Y)$ the canonical map (\\ref{equation-sheafy-trace}) $$ Rf_*R\\SheafHom_{\\mathcal{O}_X}(M, \\omega^\\bullet_{X/Y}) \\to R\\SheafHom_{\\mathcal{O}_Y}(Rf_*M, \\mathcal{O}_Y) $$ is an isomorphism by Lemma \\ref{lemma-iso-on-RSheafHom} and Remark \\ref{remark-iso-on-RSheafHom} and the fact that $M$ and $Rf_*M$ are pseudo-coherent, see Derived Categories of Schemes, Lemma \\ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general}. To see that the relative cup product induces this isomorphism use the commutativity of the diagram in Cohomology, Remark \\ref{cohomology-remark-relative-cup-and-composition}."}
+{"_id": "13651", "title": "duality-remark-van-den-bergh", "text": "Lemma \\ref{lemma-van-den-bergh} means our relative dualizing complex is {\\it rigid} in a sense analogous to the notion introduced in \\cite{vdB-rigid}. Namely, since the functor on the right of (\\ref{equation-rigid}) is ``quadratic'' in $\\omega_{X/Y}^\\bullet$ and the functor on the left of (\\ref{equation-rigid}) is ``linear'' this ``pins down'' the complex $\\omega_{X/Y}^\\bullet$ to some extent. There is an approach to duality theory using ``rigid'' (relative) dualizing complexes, see for example \\cite{Neeman-rigid}, \\cite{Yekutieli-rigid}, and \\cite{Yekutieli-Zhang}. We will return to this in Section \\ref{section-relative-dualizing-complexes}."}
+{"_id": "13652", "title": "duality-remark-local-calculation-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Using the lemmas above we can compute $f^!$ locally as follows. Suppose that we are given affine opens $$ \\xymatrix{ U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\ V \\ar[r]^i & Y } $$ Since $j^! \\circ f^! = g^! \\circ i^!$ (Lemma \\ref{lemma-upper-shriek-composition}) and since $j^!$ and $i^!$ are given by restriction (Lemma \\ref{lemma-shriek-open-immersion}) we see that $$ (f^!E)|_U = g^!(E|_V) $$ for any $E \\in D^+_\\QCoh(\\mathcal{O}_X)$. Write $U = \\Spec(A)$ and $V = \\Spec(R)$ and let $\\varphi : R \\to A$ be the finite type ring map corresponding to $g$. Choose a presentation $A = P/I$ where $P = R[x_1, \\ldots, x_n]$ is a polynomial algebra in $n$ variables over $R$. Choose an object $K \\in D^+(R)$ corresponding to $E|_V$ (Derived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded}). Then we claim that $f^!E|_U$ corresponds to $$ \\varphi^!(K) = R\\Hom(A, K \\otimes_R^\\mathbf{L} P)[n] $$ where $R\\Hom(A, -) : D(P) \\to D(A)$ is the functor of Dualizing Complexes, Section \\ref{dualizing-section-trivial} and where $\\varphi^! : D(R) \\to D(A)$ is the functor of Dualizing Complexes, Section \\ref{dualizing-section-relative-dualizing-complex-algebraic}. Namely, the choice of presentation gives a factorization $$ U \\rightarrow \\mathbf{A}^n_V \\to \\mathbf{A}^{n - 1}_V \\to \\ldots \\to \\mathbf{A}^1_V \\to V $$ Applying Lemma \\ref{lemma-shriek-affine-line} exactly $n$ times we see that $(\\mathbf{A}^n_V \\to V)^!(E|_V)$ corresponds to $K \\otimes_R^\\mathbf{L} P[n]$. By Lemmas \\ref{lemma-sheaf-with-exact-support-quasi-coherent} and \\ref{lemma-shriek-closed-immersion} the last step corresponds to applying $R\\Hom(A, -)$."}
+{"_id": "13653", "title": "duality-remark-independent-omega-S", "text": "Let $S$ be a Noetherian scheme which has a dualizing complex. Let $f : X \\to Y$ be a morphism of schemes of finite type over $S$. Then the functor $$ f_{new}^! : D^+_{Coh}(\\mathcal{O}_Y) \\to D^+_{Coh}(\\mathcal{O}_X) $$ is independent of the choice of the dualizing complex $\\omega_S^\\bullet$ up to canonical isomorphism. We sketch the proof. Any second dualizing complex is of the form $\\omega_S^\\bullet \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{L}$ where $\\mathcal{L}$ is an invertible object of $D(\\mathcal{O}_S)$, see Lemma \\ref{lemma-dualizing-unique-schemes}. For any separated morphism $p : U \\to S$ of finite type we have $p^!(\\omega_S^\\bullet \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{L}) = p^!(\\omega_S^\\bullet) \\otimes^\\mathbf{L}_{\\mathcal{O}_U} Lp^*\\mathcal{L}$ by Lemma \\ref{lemma-compare-with-pullback-perfect}. Hence, if $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ are the dualizing complexes normalized relative to $\\omega_S^\\bullet$ we see that $\\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}$ and $\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}$ are the dualizing complexes normalized relative to $\\omega_S^\\bullet \\otimes_{\\mathcal{O}_S}^\\mathbf{L} \\mathcal{L}$ (where $a : X \\to S$ and $b : Y \\to S$ are the structure morphisms). Then the result follows as \\begin{align*} & R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K, \\omega_Y^\\bullet \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}), \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\ & = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R(\\SheafHom_{\\mathcal{O}_Y}(K, \\omega_Y^\\bullet) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lb^*\\mathcal{L}), \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\ & = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K, \\omega_Y^\\bullet) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} La^*\\mathcal{L}) \\\\ & = R\\SheafHom_{\\mathcal{O}_X}(Lf^*R\\SheafHom_{\\mathcal{O}_Y}(K, \\omega_Y^\\bullet), \\omega_X^\\bullet) \\end{align*} for $K \\in D^+_{Coh}(\\mathcal{O}_Y)$. The last equality because $La^*\\mathcal{L}$ is invertible in $D(\\mathcal{O}_X)$."}
+{"_id": "13654", "title": "duality-remark-dualizing-finite", "text": "Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing complex. Let $f : X \\to Y$ be a finite morphism between schemes of finite type over $S$. Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes normalized relative to $\\omega_S^\\bullet$. Then we have $$ f_*\\omega_X^\\bullet = R\\SheafHom(f_*\\mathcal{O}_X, \\omega_Y^\\bullet) $$ in $D_\\QCoh^+(f_*\\mathcal{O}_X)$ by Lemmas \\ref{lemma-finite-twisted} and \\ref{lemma-proper-map-good-dualizing-complex} and the trace map of Example \\ref{example-trace-proper} is the map $$ \\text{Tr}_f : Rf_*\\omega_X^\\bullet = f_*\\omega_X^\\bullet = R\\SheafHom(f_*\\mathcal{O}_X, \\omega_Y^\\bullet) \\longrightarrow \\omega_Y^\\bullet $$ which often goes under the name ``evaluation at $1$''."}
+{"_id": "13655", "title": "duality-remark-relative-dualizing-complex-shriek", "text": "Let $f : X \\to Y$ be a flat proper morphism of finite type schemes over a pair $(S, \\omega_S^\\bullet)$ as in Situation \\ref{situation-dualizing}. The relative dualizing complex (Remark \\ref{remark-relative-dualizing-complex}) is $\\omega_{X/Y}^\\bullet = a(\\mathcal{O}_Y)$. By Lemma \\ref{lemma-proper-map-good-dualizing-complex} we have the first canonical isomorphism in $$ \\omega_X^\\bullet = a(\\omega_Y^\\bullet) = Lf^*\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}^\\bullet $$ in $D(\\mathcal{O}_X)$. The second canonical isomorphism follows from the discussion in Remark \\ref{remark-relative-dualizing-complex}."}
+{"_id": "13656", "title": "duality-remark-the-same-is-true", "text": "Let $S$ be a Noetherian scheme endowed with a dualizing complex $\\omega_S^\\bullet$. In this case Lemmas \\ref{lemma-shriek}, \\ref{lemma-flat-shriek}, \\ref{lemma-flat-quasi-finite-shriek}, and \\ref{lemma-CM-shriek} are true for any morphism $f : X \\to Y$ of finite type schemes over $S$ but with $f^!$ replaced by $f_{new}^!$. This is clear because in each case the proof reduces immediately to the affine case and then $f^! = f_{new}^!$ by Lemma \\ref{lemma-duality-bootstrap}."}
+{"_id": "13657", "title": "duality-remark-CM-morphism-compare-dualizing", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Assume $f$ is a Cohen-Macaulay morphism of relative dimension $d$. Let $\\omega_{X/Y} = H^{-d}(f^!\\mathcal{O}_Y)$ be the unique nonzero cohomology sheaf of $f^!\\mathcal{O}_Y$, see Lemma \\ref{lemma-CM-shriek}. Then there is a canonical isomorphism $$ f^!K = Lf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}[d] $$ for $K \\in D^+_\\QCoh(\\mathcal{O}_Y)$, see Lemma \\ref{lemma-perfect-comparison-shriek}. In particular, if $S$ has a dualizing complex $\\omega_S^\\bullet$, $\\omega_Y^\\bullet = (Y \\to S)^!\\omega_S^\\bullet$, and $\\omega_X^\\bullet = (X \\to S)^!\\omega_S^\\bullet$ then we have $$ \\omega_X^\\bullet = Lf^*\\omega_Y^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\omega_{X/Y}[d] $$ Thus if further $X$ and $Y$ are connected and Cohen-Macaulay and if $\\omega_Y$ and $\\omega_X$ denote the unique nonzero cohomology sheaves of $\\omega_Y^\\bullet$ and $\\omega_X^\\bullet$, then we have $$ \\omega_X = f^*\\omega_Y \\otimes_{\\mathcal{O}_X} \\omega_{X/Y}. $$ Similar results hold for $X$ and $Y$ arbitrary finite type schemes over $S$ (i.e., not necessarily separated over $S$) with dualizing complexes normalized with respect to $\\omega_S^\\bullet$ as in Section \\ref{section-glue}."}
+{"_id": "13658", "title": "duality-remark-duality-proper-over-field", "text": "Let $k$, $X$, and $\\omega_X^\\bullet$ be as in Lemma \\ref{lemma-duality-proper-over-field}. The identity on the complex $\\omega_X^\\bullet$ corresponds, via the functorial isomorphism in part (5), to a map $$ t : H^0(X, \\omega_X^\\bullet) \\longrightarrow k $$ For an arbitrary $K$ in $D_\\QCoh(\\mathcal{O}_X)$ the identification $\\Hom(K, \\omega_X^\\bullet)$ with $H^0(X, K)^\\vee$ in part (5) corresponds to the pairing $$ \\Hom_X(K, \\omega_X^\\bullet) \\times H^0(X, K) \\longrightarrow k,\\quad (\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta)) $$ This follows from the functoriality of the isomorphisms in (5). Similarly for any $i \\in \\mathbf{Z}$ we get the pairing $$ \\Ext^i_X(K, \\omega_X^\\bullet) \\times H^{-i}(X, K) \\longrightarrow k,\\quad (\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta)) $$ Here we think of $\\alpha$ as a morphism $K[-i] \\to \\omega_X^\\bullet$ and $\\beta$ as an element of $H^0(X, K[-i])$ in order to define $\\alpha(\\beta)$. Observe that if $K$ is general, then we only know that this pairing is nondegenerate on one side: the pairing induces an isomorphism of $\\Hom_X(K, \\omega_X^\\bullet)$, resp.\\ $\\Ext^i_X(K, \\omega_X^\\bullet)$ with the $k$-linear dual of $H^0(X, K)$, resp.\\ $H^{-i}(X, K)$ but in general not vice versa. If $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, then $\\Hom_X(K, \\omega_X^\\bullet)$, $\\Ext_X(K, \\omega_X^\\bullet)$, $H^0(X, K)$, and $H^i(X, K)$ are finite dimensional $k$-vector spaces (by Derived Categories of Schemes, Lemmas \\ref{perfect-lemma-coherent-internal-hom} and \\ref{perfect-lemma-direct-image-coherent-bdd-below}) and the pairings are perfect in the usual sense."}
+{"_id": "13659", "title": "duality-remark-coherent-duality-proper-over-field", "text": "We continue the discussion in Remark \\ref{remark-duality-proper-over-field} and we use the same notation $k$, $X$, $\\omega_X^\\bullet$, and $t$. If $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module we obtain perfect pairings $$ \\langle -, - \\rangle : \\Ext^i_X(\\mathcal{F}, \\omega_X^\\bullet) \\times H^{-i}(X,\\mathcal{F}) \\longrightarrow k,\\quad (\\alpha, \\beta) \\longmapsto t(\\alpha(\\beta)) $$ of finite dimensional $k$-vector spaces. These pairings satisfy the following (obvious) functoriality: if $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a homomorphism of coherent $\\mathcal{O}_X$-modules, then we have $$ \\langle \\alpha \\circ \\varphi, \\beta \\rangle = \\langle \\alpha, \\varphi(\\beta) \\rangle $$ for $\\alpha \\in \\Ext^i_X(\\mathcal{G}, \\omega_X^\\bullet)$ and $\\beta \\in H^{-i}(X, \\mathcal{F})$. In other words, the $k$-linear map $\\Ext^i_X(\\mathcal{G}, \\omega_X^\\bullet) \\to \\Ext^i_X(\\mathcal{F}, \\omega_X^\\bullet)$ induced by $\\varphi$ is, via the pairings, the $k$-linear dual of the $k$-linear map $H^{-i}(X, \\mathcal{F}) \\to H^{-i}(X, \\mathcal{G})$ induced by $\\varphi$. Formulated in this manner, this still works if $\\varphi$ is a homomorphism of quasi-coherent $\\mathcal{O}_X$-modules."}
+{"_id": "13660", "title": "duality-remark-rework-duality-locally-free-CM", "text": "Let $X$ be a proper Cohen-Macaulay scheme over a field $k$ which is equidimensional of dimension $d$. Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-proper-over-field}. By Lemma \\ref{lemma-duality-proper-over-field-CM} we have $\\omega_X^\\bullet = \\omega_X[d]$. Let $t : H^d(X, \\omega_X) \\to k$ be the map of Remark \\ref{remark-duality-proper-over-field}. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module with dual $\\mathcal{E}^\\vee$. Then we have perfect pairings $$ H^i(X, \\omega_X \\otimes_{\\mathcal{O}_X} \\mathcal{E}^\\vee) \\times H^{d - i}(X, \\mathcal{E}) \\longrightarrow k,\\quad (\\xi, \\eta) \\longmapsto t(1 \\otimes \\epsilon)(\\xi \\cup \\eta)) $$ where $\\cup$ is the cup-product and $\\epsilon : \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\mathcal{E} \\to \\mathcal{O}_X$ is the evaluation map. This is a special case of Lemma \\ref{lemma-duality-proper-over-field-perfect}."}
+{"_id": "13661", "title": "duality-remark-relative-dualizing-complex-bis", "text": "Let $X \\to S$ be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma \\ref{lemma-existence-relative-dualizing} there exists a relative dualizing complex $(\\omega_{X/S}^\\bullet, \\xi)$ in the sense of Definition \\ref{definition-relative-dualizing-complex}. Consider any morphism $g : S' \\to S$ where $S'$ is quasi-compact and quasi-separated (for example an affine open of $S$). By Lemma \\ref{lemma-base-change-relative-dualizing} we see that $(L(g')^*\\omega_{X/S}^\\bullet, L(g')^*\\xi)$ is a relative dualizing complex for the base change $f' : X' \\to S'$ in the sense of Definition \\ref{definition-relative-dualizing-complex}. Let $\\omega_{X'/S'}^\\bullet$ be the relative dualizing complex for $X' \\to S'$ in the sense of Remark \\ref{remark-relative-dualizing-complex}. Combining Lemmas \\ref{lemma-flat-proper-relative-dualizing} and \\ref{lemma-uniqueness-relative-dualizing} we see that there is a unique isomorphism $$ \\omega_{X'/S'}^\\bullet \\longrightarrow L(g')^*\\omega_{X/S}^\\bullet $$ compatible with (\\ref{equation-pre-rigid}) and $L(g')^*\\xi$. These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over $S$ and the base change isomorphisms of Lemma \\ref{lemma-proper-flat-base-change} (if we ever need this compatibility we will carefully state and prove it here)."}
+{"_id": "13662", "title": "duality-remark-extension-by-zero", "text": "Let $j : U \\to X$ be an open immersion of Noetherian schemes. Sending $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ to a Deligne system whose restriction to $U$ is $K$ determines a functor $$ Rj_! : D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\longrightarrow \\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X) $$ which is ``exact'' by Lemma \\ref{lemma-extension-by-zero-triangle} and which is ``left adjoint'' to the functor $j^* : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ by Lemma \\ref{lemma-lift-map}."}
+{"_id": "13663", "title": "duality-remark-extension-by-zero-linear-pro-system", "text": "Let $(A_n)$ and $(B_n)$ be inverse systems of a category $\\mathcal{C}$. Let us say a linear-pro-morphism from $(A_n)$ to $(B_n)$ is given by a compatible family of morphisms $\\varphi_n : A_{cn + d} \\to B_n$ for all $n \\geq 1$ for some fixed integers $c, d \\geq 1$. We'll say $(\\varphi_n : A_{cn + d} \\to B_n)$ and $(\\psi_n : A_{c'n + d'} \\to B_n)$ determine the same morphism if there exist $c'' \\geq \\max(c, c')$ and $d'' \\geq \\max(d, d')$ such that the two induced morphisms $A_{c'' n + d''} \\to B_n$ are the same for all $n$. It seems likely that Deligne systems $(K_n)$ with given value on $U$ are well defined up to linear-pro-isomorphisms. If we ever need this we will carefully formulate and prove this here."}
+{"_id": "13664", "title": "duality-remark-covariance-open-j-lower-shriek", "text": "Let $X \\supset U \\supset U'$ be open subschemes of a Noetherian scheme $X$. Denote $j : U \\to X$ and $j' : U' \\to X$ the inclusion morphisms. We claim there is a canonical map $$ Rj'_!(K|_{U'}) \\longrightarrow Rj_!K $$ functorial for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$. Namely, by Lemma \\ref{lemma-lift-map} we have for any $L$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ the map \\begin{align*} \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)}(Rj_!K, L) & = \\Hom_U(K, L|_U) \\\\ & \\to \\Hom_{U'}(K|_{U'}, L|_{U'}) \\\\ & = \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)}(Rj'_!(K|_{U'}), L) \\end{align*} functorial in $L$ and $K'$. The functoriality in $L$ shows by Categories, Remark \\ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rj'_!(K|_{U'}) \\to Rj_!K$ which is functorial in $K$ by the functoriality of the arrow above in $K$. \\medskip\\noindent Here is an explicit construction of this arrow. Namely, suppose that $\\mathcal{F}^\\bullet$ is a bounded complex of coherent $\\mathcal{O}_X$-modules whose restriction to $U$ represents $K$ in the derived category. We have seen in the proof of Lemma \\ref{lemma-extension-by-zero} that such a complex always exists. Let $\\mathcal{I}$, resp.\\ $\\mathcal{I}'$ be a quasi-coherent sheaf of ideals on $X$ with $V(\\mathcal{I}) = X \\setminus U$, resp.\\ $V(\\mathcal{I}') = X \\setminus U'$. After replacing $\\mathcal{I}$ by $\\mathcal{I} + \\mathcal{I}'$ we may assume $\\mathcal{I}' \\subset \\mathcal{I}$. By construction $Rj_!K$, resp.\\ $Rj'_!(K|_{U'})$ is represented by the inverse system $(K_n)$, resp.\\ $(K'_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ with $$ K_n = \\mathcal{I}^n\\mathcal{F}^\\bullet \\quad\\text{resp.}\\quad K'_n = (\\mathcal{I}')^n\\mathcal{F}^\\bullet $$ Clearly the map constructed above is given by the maps $K'_n \\to K_n$ coming from the inclusions $(\\mathcal{I}')^n \\subset \\mathcal{I}^n$."}
+{"_id": "13665", "title": "duality-remark-compose-inverse-systems", "text": "Let $\\mathcal{C}$ be a category. Suppose given an inverse system $$ \\ldots \\xrightarrow{\\alpha_4} (M_{3, n}) \\xrightarrow{\\alpha_3} (M_{2, n}) \\xrightarrow{\\alpha_2} (M_{1, n}) $$ of inverse systems in the category of pro-objects of $\\mathcal{C}$. In other words, the arrows $\\alpha_i$ are morphisms of pro-objects. By Categories, Example \\ref{categories-example-pro-morphism-inverse-systems} we can represent each $\\alpha_i$ by a pair $(m_i, a_i)$ where $m_i : \\mathbf{N} \\to \\mathbf{N}$ is an increasing function and $a_{i, n} : M_{i, m_i(n)} \\to M_{i - 1, n}$ is a morphism of $\\mathcal{C}$ making the diagrams $$ \\xymatrix{ \\ldots \\ar[r] & M_{i, m_i(3)} \\ar[d]^{a_{i, 3}} \\ar[r] & M_{i, m_i(2)} \\ar[d]^{a_{i, 2}} \\ar[r] & M_{i, m_i(1)} \\ar[d]^{a_{i, 1}} \\\\ \\ldots \\ar[r] & M_{i - 1, 3} \\ar[r] & M_{i - 1, 2} \\ar[r] & M_{i - 1, 1} } $$ commute. By replacing $m_i(n)$ by $\\max(n, m_i(n))$ and adjusting the morphisms $a_i(n)$ accordingly (as in the example referenced) we may assume that $m_i(n) \\geq n$. In this situation consider the inverse system $$ \\ldots \\to M_{4, m_4(m_3(m_2(4)))} \\to M_{3, m_3(m_2(3))} \\to M_{2, m_2(2)} \\to M_{1, 1} $$ with general term $$ M_k = M_{k, m_k(m_{k - 1}(\\ldots (m_2(k))\\ldots))} $$ For any object $N$ of $\\mathcal{C}$ we have $$ \\colim_i \\colim_n \\Mor_\\mathcal{C}(M_{i, n}, N) = \\colim_k \\Mor_\\mathcal{C}(M_k, N)  $$ We omit the details. In other words, we see that the inverse system $(M_k)$ has the property $$ \\colim_i \\Mor_{\\text{Pro-}\\mathcal{C}}((M_{i, n}), N) = \\Mor_{\\text{Pro-}\\mathcal{C}}((M_k), N) $$ This property determines the inverse system $(M_k)$ up to pro-isomorphism by the discussion in Categories, Remark \\ref{categories-remark-pro-category-copresheaves}. In this way we can turn certain inverse systems in $\\text{Pro-}\\mathcal{C}$ into pro-objects with countable index categories."}
+{"_id": "13666", "title": "duality-remark-composition-lower-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of $\\textit{FTS}_S$. Let us define the composition $$ Rg_! \\circ Rf_! : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\longrightarrow \\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z) $$ Namely, by the very construction of $Rf_!$ for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ the output $Rf_!K$ is the pro-isomorphism class of an inverse system $(M_n)$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$. Then, since $Rg_!$ is constructed similarly, we see that $$ \\ldots \\to Rg_!M_3 \\to Rg_!M_2 \\to Rg_!M_1 $$ is an inverse system of $\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$. By the discussion in Remark \\ref{remark-compose-inverse-systems} there is a unique pro-isomorphism class, which we will denote $Rg_! Rf_! K$, of inverse systems in $D^b_{\\textit{Coh}}(\\mathcal{O}_Z)$ such that $$ \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z)}(Rg_!Rf_!K, L) = \\colim_n \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Z)}(Rg_!M_n, L) $$ We omit the discussion necessary to see that this construction is functorial in $K$ as it will immediately follow from the next lemma."}
+{"_id": "13667", "title": "duality-remark-covariance-open-lower-shriek", "text": "In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$ and let $U \\subset X$ be an open. Set $g = f|_U : U \\to Y$. Then there is a canonical morphism $$ Rg_!(K|_U) \\longrightarrow Rf_!K $$ functorial in $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ which can be defined in at least 3 ways. \\begin{enumerate} \\item Denote $i : U \\to X$ the inclusion morphism. We have $Rg_! = Rf_! \\circ Ri_!$ by Lemma \\ref{lemma-composition-lower-shriek} and we can use $Rf_!$ applied to the map $Ri_!(K|_U) \\to K$ which is a special case of Remark \\ref{remark-covariance-open-j-lower-shriek}. \\item Choose a compactification $j : X \\to \\overline{X}$ of $X$ over $Y$ with structure morphism $\\overline{f} : \\overline{X} \\to Y$. Set $j' = j \\circ i : U \\to \\overline{X}$. We can use that $Rf_! = R\\overline{f}_* \\circ Rj_!$ and $Rg_! = R\\overline{f}_* \\circ Rj'_!$ and we can use $R\\overline{f}_*$ applied to the map $Rj'_!(K|_U) \\to Rj_!K$ of Remark \\ref{remark-covariance-open-j-lower-shriek}. \\item We can use \\begin{align*} \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L) & = \\Hom_X(K, f^!L) \\\\ & \\to \\Hom_U(K|_U, f^!L|_U) \\\\ & = \\Hom_U(K|_U, g^!L) \\\\ & = \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rg_!(K|_U), L) \\end{align*} functorial in $L$ and $K$. Here we have used Proposition \\ref{proposition-duality-compactly-supported} twice and the construction of upper shriek functors which shows that $g^! = i^* \\circ f^!$. The functoriality in $L$ shows by Categories, Remark \\ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rg_!(K|_U) \\to Rf_!K$ in $\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$. \\end{enumerate} Each of these three constructions gives the same arrow; we omit the details."}
+{"_id": "13668", "title": "duality-remark-covariance-etale-lower-shriek", "text": "Let us generalize the covariance of compactly supported cohomology given in Remark \\ref{remark-covariance-open-lower-shriek} to \\'etale morphisms. Namely, in Situation \\ref{situation-shriek} suppose given a commutative diagram $$ \\xymatrix{ U \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\ & Y } $$ of $\\textit{FTS}_S$ with $h$ \\'etale. Then there is a canonical morphism $$ Rg_!(h^*K) \\longrightarrow Rf_!K $$ functorial in $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. We define this transformation using the sequence of maps \\begin{align*} \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L) & = \\Hom_X(K, f^!L) \\\\ & \\to \\Hom_U(h^*K, h^*(f^!L)) \\\\ & = \\Hom_U(h^*K, h^!f^!L) \\\\ & = \\Hom_U(h^*K, g^!L) \\\\ & = \\Hom_{\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rg_!(h^*K), L) \\end{align*} functorial in $L$ and $K$. Here we have used Proposition \\ref{proposition-duality-compactly-supported} twice, we have used the equality $h^* = h^!$ of Lemma \\ref{lemma-shriek-etale}, and we have used the equality $h^! \\circ f^! = g^!$ of Lemma \\ref{lemma-upper-shriek-composition}. The functoriality in $L$ shows by Categories, Remark \\ref{categories-remark-pro-category-copresheaves} that we obtain a canonical map $Rg_!(h^*K) \\to Rf_!K$ in $\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$."}
+{"_id": "13669", "title": "duality-remark-covariance-lower-shriek", "text": "In Remarks \\ref{remark-covariance-open-lower-shriek} and \\ref{remark-covariance-etale-lower-shriek} we have seen that the construction of compactly supported cohomology is covariant with respect to open immersions and \\'etale morphisms. In fact, the correct generality is that given a commutative diagram $$ \\xymatrix{ U \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\ & Y } $$ of $\\textit{FTS}_S$ with $h$ flat and quasi-finite there exists a canonical transformation $$ Rg_! \\circ h^* \\longrightarrow Rf_! $$ As in Remark \\ref{remark-covariance-etale-lower-shriek} this map can be constructed using a transformation of functors $h^* \\to h^!$ on $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$. Recall that $h^!K = h^*K \\otimes \\omega_{U/X}$ where $\\omega_{U/X} = h^!\\mathcal{O}_X$ is the relative dualizing sheaf of the flat quasi-finite morphism $h$ (see Lemmas \\ref{lemma-perfect-comparison-shriek} and \\ref{lemma-flat-quasi-finite-shriek}). Recall that $\\omega_{U/X}$ is the same as the relative dualizing module which will be constructed in Discriminants, Remark \\ref{discriminant-remark-relative-dualizing-for-quasi-finite} by Discriminants, Lemma \\ref{discriminant-lemma-compare-dualizing}. Thus we can use the trace element $\\tau_{U/X} : \\mathcal{O}_U \\to \\omega_{U/X}$ which will be constructed in Discriminants, Remark \\ref{discriminant-remark-relative-dualizing-for-flat-quasi-finite} to define our transformation. If we ever need this, we will precisely formulate and prove the result here."}
+{"_id": "14127", "title": "more-morphisms-remark-action-by-derivations", "text": "Assumptions and notation as in Lemma \\ref{lemma-action-by-derivations}. The action of a local section $\\theta$ on $a'$ is sometimes indicated by $\\theta \\cdot a'$. Note that this means nothing else than the fact that $(a')^\\sharp$ and $(\\theta \\cdot a')^\\sharp$ differ by a derivation $D$ which is related to $\\theta$ by Equation (\\ref{equation-D})."}
+{"_id": "14128", "title": "more-morphisms-remark-special-case", "text": "A special case of Lemmas \\ref{lemma-difference-derivation}, \\ref{lemma-action-by-derivations}, \\ref{lemma-sheaf}, and \\ref{lemma-action-sheaf} is where $Y = Y'$. In this case the map $A$ is always zero. The sheaf of Lemma \\ref{lemma-sheaf} is just given by the rule $$ U' \\mapsto \\{a' : U' \\to Y\\text{ over }S\\text{ with } a'|_U = a|_U\\} $$ and we act on this by the sheaf $\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/S}, \\mathcal{C}_{X/X'})$."}
+{"_id": "14129", "title": "more-morphisms-remark-another-special-case", "text": "Another special case of Lemmas \\ref{lemma-difference-derivation}, \\ref{lemma-action-by-derivations}, \\ref{lemma-sheaf}, and \\ref{lemma-action-sheaf} is where $S$ itself is a thickening $Z \\subset Z' = S$ and $Y = Z \\times_{Z'} Y'$. Picture $$ \\xymatrix{ (X \\subset X') \\ar@{..>}[rr]_{(a, ?)} \\ar[rd]_{(g, g')} & & (Y \\subset Y') \\ar[ld]^{(h, h')} \\\\ & (Z \\subset Z') } $$ In this case the map $A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\\mathcal{C}_{Z/Z'} \\to \\mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \\times_{Z'} Y'$), hence the pullback $g^*\\mathcal{C}_{Z/Z'} = a^*h^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\\mathcal{C}_{Z/Z'} \\to a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma \\ref{lemma-sheaf} is just given by the rule $$ U' \\mapsto \\{a' : U' \\to Y'\\text{ over }Z'\\text{ with } a'|_U = a|_U\\} $$ and we act on this by the sheaf $\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/Z}, \\mathcal{C}_{X/X'})$."}
+{"_id": "14130", "title": "more-morphisms-remark-tiny-improvement", "text": "Lemma \\ref{lemma-action-by-derivations-etale-localization} can be improved in the following way. Suppose that we have commutative diagrams as in Lemma \\ref{lemma-action-by-derivations-etale-localization} but we do not assume that $X_2 \\to X_1$ and $S_2 \\to S_1$ are \\'etale. Next, suppose we have $\\theta_1 : a_1^*\\Omega_{X_1/S_1} \\to \\mathcal{C}_{T_1/T'_1}$ and $\\theta_2 : a_2^*\\Omega_{X_2/S_2} \\to \\mathcal{C}_{T_2/T'_2}$ such that $$ \\xymatrix{ f_*\\mathcal{O}_{X_2} \\ar[rr]_{f_*D_2} & & f_*a_{2, *}\\mathcal{C}_{T_2/T_2'} \\\\ \\mathcal{O}_{X_1} \\ar[rr]^{D_1} \\ar[u]^{f^\\sharp} & & a_{1, *}\\mathcal{C}_{T_1/T_1'} \\ar[u]_{\\text{induced by }(h')^\\sharp} } $$ is commutative where $D_i$ corresponds to $\\theta_i$ as in Equation (\\ref{equation-D}). Then we have the conclusion of Lemma \\ref{lemma-action-by-derivations-etale-localization}. The importance of the condition that both $X_2 \\to X_1$ and $S_2 \\to S_1$ are \\'etale is that it allows us to construct a $\\theta_2$ from $\\theta_1$."}
+{"_id": "14131", "title": "more-morphisms-remark-geometric-proof-bertini-irreducible", "text": "Let us sketch a ``geometric'' proof of a special case of Lemma \\ref{lemma-bertini-irreducible}. Namely, say $k$ is an algebraically closed field and $X \\subset \\mathbf{P}^n_k$ is smooth and irreducible of dimension $\\geq 2$. Then we claim there is a hyperplane $H \\subset \\mathbf{P}^n_k$ such that $X \\cap H$ is smooth and irreducible. Namely, by Varieties, Lemma \\ref{varieties-lemma-bertini} for a general $v \\in V = kT_0 \\oplus \\ldots \\oplus kT_n$ the corresponding hyperplane section $X \\cap H_v$ is smooth. On the other hand, by Enriques-Severi-Zariski the scheme $X \\cap H_v$ is connected, see Varieties, Lemma \\ref{varieties-lemma-connectedness-ample-divisor}. Hence $X \\cap H_v$ is smooth and irreducible."}
+{"_id": "14132", "title": "more-morphisms-remark-necessary-condition-slice-smooth", "text": "The second condition in Lemma \\ref{lemma-slice-smooth-once} is necessary even if $x$ is a closed point of a positive dimensional fibre. An example is the following: Let $k$ be a field of characteristic $p > 0$ which is imperfect. Let $a \\in k$ be an element which is not a $p$th power. Let $\\mathfrak m = (x, y^p - a) \\subset k[x, y]$. This corresponds to a closed point $w$ of $X = \\mathbf{A}^2_k$. Set $S = \\mathbf{A}^1_k$ and let $f : X \\to S$ be the morphism corresponding to $k[x] \\to k[x, y]$. Then there does not exist any commutative diagram $$ \\xymatrix{ S' \\ar[rr]_h \\ar[rd]_g & & X \\ar[ld]^f \\\\ & S } $$ with $g$ \\'etale and $w$ in the image of $h$. This is clear as the residue field extension $\\kappa(f(w)) \\subset \\kappa(w)$ is purely inseparable, but for any $s' \\in S'$ with $g(s') = f(w)$ the extension $\\kappa(f(w)) \\subset \\kappa(s')$ would be separable."}
+{"_id": "14133", "title": "more-morphisms-remark-topologies", "text": "In terms of topologies Lemmas \\ref{lemma-dominate-etale-neighbourhood-finite-flat} and \\ref{lemma-dominate-etale-affine-finite-flat} mean the following. Let $S$ be any scheme. Let $\\{f_i : U_i \\to S\\}$ be an \\'etale covering of $S$. There exists a Zariski open covering $S = \\bigcup V_j$, for each $j$ a finite locally free, surjective morphism $W_j \\to V_j$, and for each $j$ a Zariski open covering $\\{W_{j, k} \\to W_j\\}$ such that the family $\\{W_{j, k} \\to S\\}$ refines the given \\'etale covering $\\{f_i : U_i \\to S\\}$. What does this mean in practice? Well, for example, suppose we have a descent problem which we know how to solve for Zariski coverings and for fppf coverings of the form $\\{\\pi : T \\to S\\}$ with $\\pi$ finite locally free and surjective. Then this descent problem has an affirmative answer for \\'etale coverings as well. This trick was used by Gabber in his proof that $\\text{Br}(X) = \\text{Br}'(X)$ for an affine scheme $X$, see \\cite{Hoobler}."}
+{"_id": "14134", "title": "more-morphisms-remark-change-topologies", "text": "As a consequence of Lemma \\ref{lemma-fppf-ph} we obtain a comparison morphism $$ \\epsilon : (\\Sch/S)_{ph} \\longrightarrow (\\Sch/S)_{fppf} $$ This is the morphism of sites given by the identity functor on underlying categories (with suitable choices of sites as in Topologies, Remark \\ref{topologies-remark-choice-sites}). The functor $\\epsilon_*$ is the identity on underlying presheaves and the functor $\\epsilon^{-1}$ associated to an fppf sheaf its ph sheafification. By composition we can in addition compare the ph topology with the syntomic, smooth, \\'etale, and Zariski topologies."}
+{"_id": "14135", "title": "more-morphisms-remark-full-specialization-sequence", "text": "The proof of Lemma \\ref{lemma-closed-point-nearby-fibre} actually shows that there exists a sequence of specializations $$ x \\leadsto x_1 \\leadsto x_2 \\leadsto \\ldots \\leadsto x_d \\leadsto x' $$ where all $x_i$ are in the fibre $X_s$, each specialization is immediate, and $x_d$ is a closed point of $X_s$. The integer $d = \\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\dim(\\overline{\\{x\\}})$ where the closure is taken in $X_s$. Moreover, the points $x_i$ can be chosen to avoid any closed subset of $X_s$ which does not contain the point $x$."}
+{"_id": "14136", "title": "more-morphisms-remark-alternative-closed-point-nearby-fibre", "text": "We can use Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open} or its variant Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} to give an alternative proof of Lemma \\ref{lemma-closed-point-nearby-fibre} in case $S$ is locally Noetherian. Here is a rough sketch. Namely, first replace $S$ by the spectrum of the local ring at $s'$. Then we may use induction on $\\dim(S)$. The case $\\dim(S) = 0$ is trivial because then $s' = s$. Replace $X$ by the reduced induced scheme structure on $\\overline{\\{x\\}}$. Apply Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} to $X \\to S$ and $x' \\mapsto s'$ and any nonempty open $U \\subset X$ containing $x$. This gives us a closed subscheme $x' \\in Z \\subset X$ a point $z \\in Z$ such that $Z \\to S$ is quasi-finite at $x'$ and such that $f(z) \\not = s'$. Then $z$ is a closed point of $X_{f(z)}$, and $z \\leadsto x'$. As $f(z) \\not = s'$ we see $\\dim(\\mathcal{O}_{S, f(z)}) < \\dim(S)$. Since $x$ is the generic point of $X$ we see $x \\leadsto z$, hence $s = f(x) \\leadsto f(z)$. Apply the induction hypothesis to $s \\leadsto f(z)$ and $z \\mapsto f(z)$ to win."}
+{"_id": "14137", "title": "more-morphisms-remark-perfect-permanence", "text": "It is not true that a morphism between schemes $X, Y$ perfect over a base $S$ is perfect. An example is $S = \\Spec(k)$, $X = \\Spec(k)$, $Y = \\Spec(k[x]/(x^2)$ and $X \\to Y$ the unique $S$-morphism."}
+{"_id": "14306", "title": "sites-modules-remark-no-extension", "text": "In general the functor $g_!$ cannot be extended to categories of modules in case $g$ is (part of) a morphism of ringed topoi. Namely, given any ring map $A \\to B$ the functor $M \\mapsto B \\otimes_A M$ has a right adjoint (restriction) but not in general a left adjoint (because its existence would imply that $A \\to B$ is flat). We will see in Section \\ref{section-localize} below that it is possible to define $j_!$ on sheaves of modules in the case of a localization of sites. We will discuss this in greater generality in Section \\ref{section-lower-shriek-modules} below."}
+{"_id": "14307", "title": "sites-modules-remark-localize-shriek-equal", "text": "In the situation of Lemma \\ref{lemma-extension-by-zero} the diagram $$ \\xymatrix{ \\textit{Mod}(\\mathcal{O}_U) \\ar[r]_{j_{U!}} \\ar[d]_{forget} & \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[d]^{forget} \\\\ \\textit{Ab}(\\mathcal{C}/U) \\ar[r]^{j^{Ab}_{U!}} & \\textit{Ab}(\\mathcal{C}) } $$ commutes. This is clear from the explicit description of the functor $j_{U!}$ in the lemma."}
+{"_id": "14308", "title": "sites-modules-remark-localize-presheaves", "text": "Localization and presheaves of modules; see Sites, Remark \\ref{sites-remark-localize-presheaves}. Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of rings. Let $U$ be an object of $\\mathcal{C}$. Strictly speaking the functors $j_U^*$, $j_{U*}$ and $j_{U!}$ have not been defined for presheaves of $\\mathcal{O}$-modules. But of course, we can think of a presheaf as a sheaf for the chaotic topology on $\\mathcal{C}$ (see Sites, Examples \\ref{sites-example-indiscrete}). Hence we also obtain a functor $$ j_U^* : \\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O}_U) $$ and functors $$ j_{U*}, j_{U!} : \\textit{PMod}(\\mathcal{O}_U) \\longrightarrow \\textit{PMod}(\\mathcal{O}) $$ which are right, left adjoint to $j_U^*$. Inspecting the proof of Lemma \\ref{lemma-extension-by-zero} we see that $j_{U!}\\mathcal{G}$ is the presheaf $$ V \\longmapsto \\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)} \\mathcal{G}(V \\xrightarrow{\\varphi} U) $$ In addition the functor $j_{U!}$ is exact (by Lemma \\ref{lemma-extension-by-zero-exact} in the case of the discrete topologies). Moreover, if $\\mathcal{C}$ is actually a site, and $\\mathcal{O}$ is actually a sheaf of rings, then the diagram $$ \\xymatrix{ \\textit{Mod}(\\mathcal{O}_U) \\ar[r]_{j_{U!}} \\ar[d]_{forget} & \\textit{Mod}(\\mathcal{O}) \\\\ \\textit{PMod}(\\mathcal{O}_U) \\ar[r]^{j_{U!}} & \\textit{PMod}(\\mathcal{O}) \\ar[u]_{(\\ )^\\#} } $$ commutes."}
+{"_id": "14309", "title": "sites-modules-remark-j-shriek-tensor", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{C}$ and consider the localization morphism $j : \\Sh(\\mathcal{C})/\\mathcal{F} \\to \\Sh(\\mathcal{C})$. See Sites, Definition \\ref{sites-definition-localize-topos}. We claim that (a) $j_!\\mathbf{Z} = \\mathbf{Z}_\\mathcal{F}^\\#$ and (b) $j_!(j^{-1}\\mathcal{H}) = j_!\\mathbf{Z} \\otimes_\\mathbf{Z} \\mathcal{H}$ for any abelian sheaf $\\mathcal{H}$ on $\\mathcal{C}$. Let $\\mathcal{G}$ be an abelian on $\\mathcal{C}$. Part (a) follows from the Yoneda lemma because $$ \\Hom(j_!\\mathbf{Z}, \\mathcal{G}) = \\Hom(\\mathbf{Z}, j^{-1}\\mathcal{G}) = \\Hom(\\mathbf{Z}_\\mathcal{F}^\\#, \\mathcal{G}) $$ where the second equality holds because both sides of the equality evaluate to the set of maps from $\\mathcal{F} \\to \\mathcal{G}$ viewed as an abelian group. For (b) we use the Yoneda lemma and \\begin{align*} \\Hom(j_!(j^{-1}\\mathcal{H}), \\mathcal{G}) & = \\Hom(j^{-1}\\mathcal{H}, j^{-1}\\mathcal{G}) \\\\ & = \\Hom(\\mathbf{Z}, \\SheafHom(j^{-1}\\mathcal{H}, j^{-1}\\mathcal{G})) \\\\ & = \\Hom(\\mathbf{Z}, j^{-1}\\SheafHom(\\mathcal{H}, \\mathcal{G})) \\\\ & = \\Hom(j_!\\mathbf{Z}, \\SheafHom(\\mathcal{H}, \\mathcal{G})) \\\\ & = \\Hom(j_!\\mathbf{Z} \\otimes_\\mathbf{Z} \\mathcal{H}, \\mathcal{G}) \\end{align*} Here we use adjunction, the fact that taking $\\SheafHom$ commutes with localization, and Lemma \\ref{lemma-internal-hom-adjoint-tensor}."}
+{"_id": "14310", "title": "sites-modules-remark-functoriality-principal-parts", "text": "Let $\\mathcal{C}$ be a site. Suppose given a commutative diagram of sheaves of rings $$ \\xymatrix{ \\mathcal{B} \\ar[r] & \\mathcal{B}' \\\\ \\mathcal{A} \\ar[u] \\ar[r] & \\mathcal{A}' \\ar[u] } $$ a $\\mathcal{B}$-module $\\mathcal{F}$, a $\\mathcal{B}'$-module $\\mathcal{F}'$, and a $\\mathcal{B}$-linear map $\\mathcal{F} \\to \\mathcal{F}'$. Then we get a compatible system of module maps $$ \\xymatrix{ \\ldots \\ar[r] & \\mathcal{P}^2_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] & \\mathcal{P}^1_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\ar[r] & \\mathcal{P}^0_{\\mathcal{B}'/\\mathcal{A}'}(\\mathcal{F}') \\\\ \\ldots \\ar[r] & \\mathcal{P}^2_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] & \\mathcal{P}^1_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[r] \\ar[u] & \\mathcal{P}^0_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{F}) \\ar[u] } $$ These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $\\mathcal{P}^k_{\\mathcal{B}/\\mathcal{A}}(\\mathcal{M})$ in terms of (local) generators and relations in the proof of Lemma \\ref{lemma-module-principal-parts} but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma \\ref{lemma-sequence-of-principal-parts}."}
+{"_id": "14311", "title": "sites-modules-remark-not-pushforward", "text": "Warning: The result of Lemma \\ref{lemma-stalk-j-shriek} has no analogue for $j_{U, *}$."}
+{"_id": "14312", "title": "sites-modules-remark-when-shriek-equal", "text": "Warning! Let $u : \\mathcal{C} \\to \\mathcal{D}$, $g$, $\\mathcal{O}_\\mathcal{D}$, and $\\mathcal{O}_\\mathcal{C}$ be as in Lemma \\ref{lemma-lower-shriek-modules}. In general it is {\\bf not} the case that the diagram $$ \\xymatrix{ \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[r]_{g_!} \\ar[d]_{forget} & \\textit{Mod}(\\mathcal{O}_\\mathcal{D}) \\ar[d]^{forget} \\\\ \\textit{Ab}(\\mathcal{C}) \\ar[r]^{g^{Ab}_!} & \\textit{Ab}(\\mathcal{D}) } $$ commutes (here $g^{Ab}_!$ is the one from Lemma \\ref{lemma-g-shriek-adjoint}). There is a transformation of functors $$ g_!^{Ab} \\circ forget \\longrightarrow forget \\circ g_! $$ From the proof of Lemma \\ref{lemma-lower-shriek-modules} we see that this is an isomorphism if and only if $g^{Ab}_!j_{U!}\\mathcal{O}_U \\to g_!j_{U!}\\mathcal{O}_U$ is an isomorphism for all objects $U$ of $\\mathcal{C}$. Since we have $g_!j_{U!}\\mathcal{O}_U = j_{u(U)!}\\mathcal{O}_{u(U)}$ this holds if and only if $$ g^{Ab}_!j_{U!}\\mathcal{O}_U \\longrightarrow j_{u(U)!}\\mathcal{O}_{u(U)} $$ is an isomorphism for all objects $U$ of $\\mathcal{C}$. Note that for such a $U$ we obtain a commutative diagram $$ \\xymatrix{ \\mathcal{C}/U \\ar[r]_-{u'} \\ar[d]_{j_U} & \\mathcal{D}/u(U) \\ar[d]^{j_{u(U)}} \\\\ \\mathcal{C} \\ar[r]^u & \\mathcal{D} } $$ of cocontinuous functors of sites, see Sites, Lemma \\ref{sites-lemma-localize-cocontinuous} and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where $g' : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{D}/u(U))$ is the morphism of topoi induced by the cocontinuous functor $u'$. Hence we see that $g_! = g^{Ab}_!$ if the canonical map \\begin{equation} \\label{equation-compare-on-localizations} (g')^{Ab}_!\\mathcal{O}_U \\longrightarrow \\mathcal{O}_{u(U)} \\end{equation} is an isomorphism for all objects $U$ of $\\mathcal{C}$."}
+{"_id": "14381", "title": "derham-remark-relative-cup-product", "text": "Let $p : X \\to S$ be a morphism of schemes. Then we can think of $\\Omega^\\bullet_{X/S}$ as a sheaf of differential graded $p^{-1}\\mathcal{O}_S$-algebras, see Differential Graded Sheaves, Definition \\ref{sdga-definition-dga}. In particular, the discussion in Differential Graded Sheaves, Section \\ref{sdga-section-misc} applies. For example, this means that for any commutative diagram $$ \\xymatrix{ X \\ar[d]_p \\ar[r]_f & Y \\ar[d]^q \\\\ S \\ar[r]^h & T } $$ of schemes there is a canonical relative cup product $$ \\mu : Rf_*\\Omega^\\bullet_{X/S} \\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/S} \\longrightarrow Rf_*\\Omega^\\bullet_{X/S} $$ in $D(Y, q^{-1}\\mathcal{O}_T)$ which is associative and which on cohomology reproduces the cup product discussed above."}
+{"_id": "14382", "title": "derham-remark-cup-product-as-a-map", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\xi \\in H_{dR}^n(X/S)$. According to the discussion Differential Graded Sheaves, Section \\ref{sdga-section-misc} there exists a canonical morphism $$ \\xi' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[n] $$ in $D(f^{-1}\\mathcal{O}_S)$ uniquely characterized by (1) and (2) of the following list of properties: \\begin{enumerate} \\item $\\xi'$ can be lifted to a map in the derived category of right differential graded $\\Omega^\\bullet_{X/S}$-modules, and \\item $\\xi'(1) = \\xi$ in $H^0(X, \\Omega^\\bullet_{X/S}[n]) = H^n_{dR}(X/S)$, \\item the map $\\xi'$ sends $\\eta \\in H^m_{dR}(X/S)$ to $\\xi \\cup \\eta$ in $H^{n + m}_{dR}(X/S)$, \\item the construction of $\\xi'$ commutes with restrictions to opens: for $U \\subset X$ open the restriction $\\xi'|_U$ is the map corresponding to the image $\\xi|_U \\in H^n_{dR}(U/S)$, \\item for any diagram as in Remark \\ref{remark-relative-cup-product} we obtain a commutative diagram $$ \\xymatrix{ Rf_*\\Omega^\\bullet_{X/S} \\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/S} \\ar[d]_{\\xi' \\otimes \\text{id}} \\ar[r]_-\\mu & Rf_*\\Omega^\\bullet_{X/S} \\ar[d]^{\\xi'} \\\\ Rf_*\\Omega^\\bullet_{X/S}[n] \\otimes_{q^{-1}\\mathcal{O}_T}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/S} \\ar[r]^-\\mu & Rf_*\\Omega^\\bullet_{X/S}[n] } $$ in $D(Y, q^{-1}\\mathcal{O}_T)$. \\end{enumerate}"}
+{"_id": "14383", "title": "derham-remark-truncations", "text": "Here is a reformulation of the calculations above in more abstract terms. Let $p : X \\to S$ be a morphism of schemes. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. If we view $\\text{d}\\log$ as a map $$ \\mathcal{O}_X^*[-1] \\to \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S} $$ then using $\\Pic(X) = H^1(X, \\mathcal{O}_X^*)$ as above we find a cohomology class $$ \\gamma_1(\\mathcal{L}) \\in H^2(X, \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S}) $$ The image of $\\gamma_1(\\mathcal{L})$ under the map $\\sigma_{\\geq 1}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}$ recovers $c_1^{dR}(\\mathcal{L})$. In particular we see that $c_1^{dR}(\\mathcal{L}) \\in F^1H^2_{dR}(X/S)$, see Section \\ref{section-hodge-filtration}. The image of $\\gamma_1(\\mathcal{L})$ under the map $\\sigma_{\\geq 1}\\Omega^\\bullet_{X/S} \\to \\Omega^1_{X/S}[-1]$ recovers $c_1^{Hodge}(\\mathcal{L})$. Taking the cup product (see Section \\ref{section-hodge-filtration}) we obtain $$ \\xi = \\gamma_1(\\mathcal{L}_1) \\cup \\ldots \\cup \\gamma_1(\\mathcal{L}_a) \\in H^{2a}(X, \\sigma_{\\geq a}\\Omega^\\bullet_{X/S}) $$ The commutative diagrams in Section \\ref{section-hodge-filtration} show that $\\xi$ is mapped to $c_1^{dR}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{dR}(\\mathcal{L}_a)$ in $H^{2a}_{dR}(X/S)$ by the map $\\sigma_{\\geq a}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}$. Also, it follows $c_1^{dR}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{dR}(\\mathcal{L}_a)$ is contained in $F^a H^{2a}_{dR}(X/S)$. Similarly, the map $\\sigma_{\\geq a}\\Omega^\\bullet_{X/S} \\to \\Omega^a_{X/S}[-a]$ sends $\\xi$ to $c_1^{Hodge}(\\mathcal{L}_1) \\cup \\ldots \\cup c_1^{Hodge}(\\mathcal{L}_a)$ in $H^a(X, \\Omega^a_{X/S})$."}
+{"_id": "14384", "title": "derham-remark-log-forms", "text": "Let $p : X \\to S$ be a morphism of schemes. For $i > 0$ denote $\\Omega^i_{X/S, log} \\subset \\Omega^i_{X/S}$ the abelian subsheaf generated by local sections of the form $$ \\text{d}\\log(u_1) \\wedge \\ldots \\wedge \\text{d}\\log(u_i) $$ where $u_1, \\ldots, u_n$ are invertible local sections of $\\mathcal{O}_X$. For $i = 0$ the subsheaf $\\Omega^0_{X/S, log} \\subset \\mathcal{O}_X$ is the image of $\\mathbf{Z} \\to \\mathcal{O}_X$. For every $i \\geq 0$ we have a map of complexes $$ \\Omega^i_{X/S, log}[-i] \\longrightarrow \\Omega^\\bullet_{X/S} $$ because the derivative of a logarithmic form is zero. Moreover, wedging logarithmic forms gives another, hence we find bilinear maps $$ \\wedge :  \\Omega^i_{X/S, log} \\times \\Omega^j_{X/S, log} \\longrightarrow \\Omega^{i + j}_{X/S, log} $$ compatible with (\\ref{equation-wedge}) and the maps above. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Using the map of abelian sheaves $\\text{d}\\log : \\mathcal{O}_X^* \\to \\Omega^1_{X/S, log}$ and the identification $\\Pic(X) = H^1(X, \\mathcal{O}_X^*)$ we find a canonical cohomology class $$ \\tilde \\gamma_1(\\mathcal{L}) \\in H^1(X, \\Omega^1_{X/S, log}) $$ These classes have the following properties \\begin{enumerate} \\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical map $\\Omega^1_{X/S, log}[-1] \\to \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S}$ sends $\\tilde \\gamma_1(\\mathcal{L})$ to the class $\\gamma_1(\\mathcal{L}) \\in  H^2(X, \\sigma_{\\geq 1}\\Omega^\\bullet_{X/S})$ of Remark \\ref{remark-truncations}, \\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical map $\\Omega^1_{X/S, log}[-1] \\to \\Omega^\\bullet_{X/S}$ sends $\\tilde \\gamma_1(\\mathcal{L})$ to $c_1^{dR}(\\mathcal{L})$ in $H^2_{dR}(X/S)$, \\item the image of $\\tilde \\gamma_1(\\mathcal{L})$ under the canonical map $\\Omega^1_{X/S, log} \\to \\Omega^1_{X/S}$ sends $\\tilde \\gamma_1(\\mathcal{L})$ to $c_1^{Hodge}(\\mathcal{L})$ in $H^1(X, \\Omega^1_{X/S})$, \\item the construction of these classes is compatible with pullbacks, \\item add more here. \\end{enumerate}"}
+{"_id": "14385", "title": "derham-remark-de-rham-complex-graded", "text": "Let $G$ be an abelian monoid written additively with neutral element $0$. Let $R \\to A$ be a ring map and assume $A$ comes with a grading $A = \\bigoplus_{g \\in G} A_g$ by $R$-modules such that $R$ maps into $A_0$ and $A_g \\cdot A_{g'} \\subset A_{g + g'}$. Then the module of differentials comes with a grading $$ \\Omega_{A/R} = \\bigoplus\\nolimits_{g \\in G} \\Omega_{A/R, g} $$ where $\\Omega_{A/R, g}$ is the $R$-submodule of $\\Omega_{A/R}$ generated by $a_0 \\text{d}a_1$ with $a_i \\in A_{g_i}$ such that $g = g_0 + g_1$. Similarly, we obtain $$ \\Omega^p_{A/R} = \\bigoplus\\nolimits_{g \\in G} \\Omega^p_{A/R, g} $$ where $\\Omega^p_{A/R, g}$ is the $R$-submodule of $\\Omega^p_{A/R}$ generated by $a_0 \\text{d}a_1 \\wedge \\ldots \\wedge \\text{d}a_p$ with $a_i \\in A_{g_i}$ such that $g = g_0 + g_1 + \\ldots + g_p$. Of course the differentials preserve the grading and the wedge product is compatible with the gradings in the obvious manner."}
+{"_id": "14386", "title": "derham-remark-gauss-manin", "text": "In Lemma \\ref{lemma-spectral-sequence-smooth} consider the cohomology sheaves $$ \\mathcal{H}^q_{dR}(X/Y) = H^q(Rf_*\\Omega^\\bullet_{X/Y})) $$ If $f$ is proper in addition to being smooth and $S$ is a scheme over $\\mathbf{Q}$ then $\\mathcal{H}^q_{dR}(X/Y)$ is finite locally free (insert future reference here). If we only assume $\\mathcal{H}^q_{dR}(X/Y)$ are flat $\\mathcal{O}_Y$-modules, then we obtain (tiny argument omitted) $$ E_1^{p, q} = \\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) $$ and the differentials in the spectral sequence are maps $$ d_1^{p, q} : \\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) \\longrightarrow \\Omega^{p + 1}_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) $$ In particular, for $p = 0$ we obtain a map $d_1^{0, q} : \\mathcal{H}^q_{dR}(X/Y) \\to \\Omega^1_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y)$ which turns out to be an integrable connection $\\nabla$ (insert future reference here) and the complex $$ \\mathcal{H}^q_{dR}(X/Y) \\to \\Omega^1_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) \\to \\Omega^2_{Y/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{H}^q_{dR}(X/Y) \\to \\ldots $$ with differentials given by $d_1^{\\bullet, q}$ is the de Rham complex of $\\nabla$. The connection $\\nabla$ is known as the {\\it Gauss-Manin connection}."}
+{"_id": "14387", "title": "derham-remark-projective-space-bundle-formula", "text": "In the situation of Proposition \\ref{proposition-projective-space-bundle-formula} we get moreover that the map $$ \\tilde \\xi : \\bigoplus\\nolimits_{t = 0, \\ldots, r - 1} \\Omega^\\bullet_{X/S}[-2t] \\longrightarrow Rp_*\\Omega^\\bullet_{P/S} $$ is an isomorphism in $D(X, (X \\to S)^{-1}\\mathcal{O}_X)$ as follows immediately from the application of Proposition \\ref{proposition-global-generation-on-fibres}. Note that the arrow for $t = 0$ is simply the canonical map $c_{P/X} : \\Omega^\\bullet_{X/S} \\to Rp_*\\Omega^\\bullet_{P/S}$ of Section \\ref{section-de-rham-complex}. In fact, we can pin down this map further in this particular case. Namely, consider the canonical map $$ \\xi' : \\Omega^\\bullet_{P/S} \\to \\Omega^\\bullet_{P/S}[2] $$ of Remark \\ref{remark-cup-product-as-a-map} corresponding to $c_1^{dR}(\\mathcal{O}_P(1))$. Then $$ \\xi'[2(t - 1)] \\circ \\ldots \\circ \\xi'[2] \\circ \\xi' :  \\Omega^\\bullet_{P/S} \\to \\Omega^\\bullet_{P/S}[2t] $$ is the map of Remark \\ref{remark-cup-product-as-a-map} corresponding to $c_1^{dR}(\\mathcal{O}_P(1))^t$. Tracing through the choices made in the proof of Proposition \\ref{proposition-global-generation-on-fibres} we find the value $$ \\tilde \\xi|_{\\Omega^\\bullet_{X/S}[-2t]} = Rp_*\\xi'[-2] \\circ \\ldots \\circ Rp_*\\xi'[-2(t - 1)] \\circ Rp_*\\xi'[-2t] \\circ c_{P/X}[-2t] $$ for the restriction of our isomorphism to the summand $\\Omega^\\bullet_{X/S}[-2t]$. This has the following simple consequence we will use below: let $$ M = \\bigoplus\\nolimits_{t = 1, \\ldots, r - 1} \\Omega^\\bullet_{X/S}[-2t] \\quad\\text{and}\\quad K = \\bigoplus\\nolimits_{t = 0, \\ldots, r - 2} \\Omega^\\bullet_{X/S}[-2t] $$ viewed as subcomplexes of the source of the arrow $\\tilde \\xi$. It follows formally from the discussion above that $$ c_{P/X} \\oplus \\tilde \\xi|_M : \\Omega^\\bullet_{X/S} \\oplus M \\longrightarrow Rp_*\\Omega^\\bullet_{P/S} $$ is an isomorphism and that the diagram $$ \\xymatrix{ K \\ar[d]_{\\tilde \\xi|_K} \\ar[r]_{\\text{id}} & M[2] \\ar[d]^{(\\tilde \\xi|_M)[2]} \\\\ Rp_*\\Omega^\\bullet_{P/S} \\ar[r]^{Rp_*\\xi'} & Rp_*\\Omega^\\bullet_{P/S}[2] } $$ commutes where $\\text{id} : K \\to M[2]$ identifies the summand corresponding to $t$ in the deomposition of $K$ to the summand corresponding to $t + 1$ in the decomposition of $M$."}
+{"_id": "14388", "title": "derham-remark-check-log-completion-1", "text": "Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \\subset X$ be an effective Cartier divisor. If the map $$ \\mathcal{O}_{X, y}^\\wedge \\longrightarrow \\mathcal{O}_{Y, y}^\\wedge $$ has a section for all $y \\in Y$, then the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. If we ever need this result we will formulate a precise statement and add a proof here."}
+{"_id": "14389", "title": "derham-remark-check-log-completion-2", "text": "Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite type over $S$. Let $Y \\subset X$ be an effective Cartier divisor. If for every $y \\in Y$ we can find a diagram of schemes over $S$ $$ X \\xleftarrow{\\varphi} U \\xrightarrow{\\psi} V $$ with $\\varphi$ \\'etale and $\\psi|_{\\varphi^{-1}(Y)} : \\varphi^{-1}(Y) \\to V$ \\'etale, then the de Rham complex of log poles is defined for $Y \\subset X$ over $S$. A special case is when the pair $(X, Y)$ \\'etale locally looks like $(V \\times \\mathbf{A}^1, V \\times \\{0\\})$. If we ever need this result we will formulate a precise statement and add a proof here."}
+{"_id": "14390", "title": "derham-remark-local-description", "text": "Let $A$ be a ring. Let $P = A[x_1, \\ldots, x_n]$. Let $f_1, \\ldots, f_n \\in P$ and set $B = P/(f_1, \\ldots, f_n)$. Assume $A \\to B$ is quasi-finite. Then $B$ is a relative global complete intersection over $A$ (Algebra, Definition \\ref{algebra-definition-relative-global-complete-intersection}) and $(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2$ is free with generators the classes $\\overline{f}_i$ by Algebra, Lemma \\ref{algebra-lemma-relative-global-complete-intersection-conormal}. Consider the following diagram $$ \\xymatrix{ \\Omega_{A/\\mathbf{Z}} \\otimes_A B \\ar[r] & \\Omega_{P/\\mathbf{Z}} \\otimes_P B \\ar[r] & \\Omega_{P/A} \\otimes_P B \\\\ & (f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2 \\ar[u] \\ar@{=}[r] & (f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2 \\ar[u] } $$ The right column represents $\\NL_{B/A}$ in $D(B)$ hence has cohomology $\\Omega_{B/A}$ in degree $0$. The top row is the split short exact sequence $0 \\to \\Omega_{A/\\mathbf{Z}} \\otimes_A B \\to \\Omega_{P/\\mathbf{Z}} \\otimes_P B \\to \\Omega_{P/A} \\otimes_P B \\to 0$. The middle column has cohomology $\\Omega_{B/\\mathbf{Z}}$ in degree $0$ by Algebra, Lemma \\ref{algebra-lemma-differential-seq}. Thus by Lemma \\ref{lemma-funny-map} we obtain canonical $B$-module maps $$ \\Omega^p_{B/\\mathbf{Z}} \\longrightarrow \\Omega^p_{A/\\mathbf{Z}} \\otimes_A \\det(\\NL_{B/A}) $$ whose composition with $\\Omega^p_{A/\\mathbf{Z}} \\to \\Omega^p_{B/\\mathbf{Z}}$ is multiplication by $\\delta(\\NL_{B/A})$."}
+{"_id": "14391", "title": "derham-remark-splitting-principle", "text": "The analogues of Weil Cohomology Theories, Lemmas \\ref{weil-lemma-splitting-principle} (splitting principle) and \\ref{weil-lemma-chern-classes-E-tensor-L} (chern classes of tensor products) hold for de Rham Chern classes on quasi-compact and quasi-separated schemes. This is clear as we've shown in the proof of Lemma \\ref{lemma-chern-classes} that all the axioms of Weil Cohomology Theories, Section \\ref{weil-section-chern} are satisfied."}
+{"_id": "14392", "title": "derham-remark-hodge-cohomology-is-weil", "text": "In exactly the same manner as above one can show that Hodge cohomology $X \\mapsto H_{Hodge}^*(X/k)$ equipped with $c_1^{Hodge}$ determines a Weil cohomology theory. If we ever need this, we will precisely formulate and prove this here. This leads to the following amusing consequence: If the betti numbers of a Weil cohomology theory are independent of the chosen Weil cohomology theory (over our field $k$ of characteristic $0$), then the Hodge-to-de Rham spectral sequence degenerates at $E_1$! Of course, the degeneration of the Hodge-to-de Rham spectral sequence is known (see for example \\cite{Deligne-Illusie} for a marvelous algebraic proof), but it is by no means an easy result! This suggests that proving the independence of betti numbers is a hard problem as well and as far as we know is still an open problem. See Weil Cohomology Theories, Remark \\ref{weil-remark-betti-numbers-in-some-sense} for a related question."}
+{"_id": "14393", "title": "derham-remark-gysin-equations", "text": "Let $X \\to S$ be a morphism of schemes. Let $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$. Let $Z \\subset X$ be the closed subscheme cut out by $f_1, \\ldots, f_c$. Below we will study the {\\it gysin map} \\begin{equation} \\label{equation-gysin} \\gamma^p_{f_1, \\ldots, f_c} : \\Omega^p_{Z/S} \\longrightarrow \\mathcal{H}_Z^c(\\Omega^{p + c}_{X/S}) \\end{equation} defined as follows. Given a local section $\\omega$ of $\\Omega^p_{Z/S}$ which is the restriction of a section $\\tilde \\omega$ of $\\Omega^p_{X/S}$ we set $$ \\gamma^p_{f_1, \\ldots, f_c}(\\omega) = c_{f_1, \\ldots, f_c}(\\tilde \\omega|_Z) \\wedge \\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_c $$ where $c_{f_1, \\ldots, f_c} : \\Omega^p_{X/S} \\otimes \\mathcal{O}_Z \\to \\mathcal{H}_Z^c(\\Omega^p_{X/S})$ is the map constructed in Derived Categories of Schemes, Remark \\ref{perfect-remark-supported-map-c-equations}. This is well defined: given $\\omega$ we can change our choice of $\\tilde \\omega$ by elements of the form $\\sum f_i \\omega'_i + \\sum \\text{d}(f_i) \\wedge \\omega''_i$ which are mapped to zero by the construction."}
+{"_id": "14394", "title": "derham-remark-how-to-use", "text": "Let $X \\to S$, $i : Z \\to X$, and $c \\geq 0$ be as in Lemma \\ref{lemma-gysin-global}. Let $p \\geq 0$ and assume that $\\mathcal{H}^i_Z(\\Omega^{p + c}_{X/S}) = 0$ for $i = 0, \\ldots, c - 1$. This vanishing holds if $X \\to S$ is smooth and $Z \\to X$ is a Koszul regular immersion, see Derived Categories of Schemes, Lemma \\ref{perfect-lemma-supported-vanishing}. Then we obtain a map $$ \\gamma^{p, q} : H^q(Z, \\Omega^p_{Z/S}) \\longrightarrow H^{q + c}(X, \\Omega^{p + c}_{X/S}) $$ by first using $\\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})$ to map into $$ H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})) = H^q(Z, R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})[c]) = H^q(X, i_*R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S})[c]) $$ and then using the adjunction map $i_*R\\mathcal{H}_Z(\\Omega^{p + c}_{X/S}) \\to \\Omega^{p + c}_{X/S}$ to continue on to the desired Hodge cohomology module."}
+{"_id": "14462", "title": "trace-remark-may-be-confusing", "text": "It may or may not be the case that $F^f_U$ equals $\\pi_U$."}
+{"_id": "14463", "title": "trace-remark-compute-degree-lifting", "text": "The computation of the degrees can be done by lifting (in some obvious sense) to characteristic 0 and considering the situation with complex coefficients. This method almost never works, since lifting is in general impossible for schemes which are not projective space."}
+{"_id": "14464", "title": "trace-remarks-derived-categories", "text": "Notes on derived categories. \\begin{enumerate} \\item There are some set-theoretical problems when $\\mathcal{A}$ is somewhat arbitrary, which we will happily disregard. \\item The categories $K(A)$ and $D(A)$ are endowed with the structure of a triangulated category. \\item The categories $\\text{Comp}(\\mathcal{A})$ and $K(\\mathcal{A})$ can also be defined when $\\mathcal{A}$ is an additive category. \\end{enumerate}"}
+{"_id": "14465", "title": "trace-remark-cohomology-of-derived-functor", "text": "In these cases, it is true that $R^iF(K^\\bullet) = H^i(RF(K^\\bullet))$, where the left hand side is defined to be $i$th homology of the complex $F(K^\\bullet)$."}
+{"_id": "14466", "title": "trace-remark-variant", "text": "A variant of this lemma is to consider a Noetherian scheme $X$ and the category $D_{perf}(\\mathcal{O}_X)$ of complexes which are locally quasi-isomorphic to a finite complex of finite locally free $\\mathcal{O}_X$-modules. Objects $K$ of $D_{perf}(\\mathcal{O}_X)$ can be characterized by having coherent cohomology sheaves and bounded tor dimension."}
+{"_id": "14467", "title": "trace-remark-content-trivial-trace", "text": "Let us try to illustrate the content of the formula of Lemma \\ref{lemma-weak-trace}. Suppose that $\\Lambda$, viewed as a trivial $\\Gamma$-module, admits a finite resolution $ 0\\to P_r\\to \\ldots \\to P_1 \\to P_0\\to \\Lambda\\to 0 $ by some $\\Lambda[\\Gamma]$-modules $P_i$ which are finite and projective as $\\Lambda[G]$-modules. In that case $$ H_*\\left(\\left(P_\\bullet\\right)_G\\right) = \\text{Tor}_*^{\\Lambda[G]}\\left(\\Lambda, \\Lambda\\right) = H_*(G, \\Lambda) $$ and $$ \\text{Tr}_\\Lambda^{Z_\\gamma}\\left(\\gamma, P_\\bullet\\right) =\\frac{1}{\\# Z_\\gamma}\\text{Tr}_\\Lambda(\\gamma, P_\\bullet)=\\frac{1}{\\# Z_\\gamma}\\text{Tr}(\\gamma, \\Lambda) = \\frac{1}{\\# Z_\\gamma}. $$ Therefore, Lemma \\ref{lemma-weak-trace} says $$ \\text{Tr}_\\Lambda (1 , P_G) = \\text{Tr}\\left(1 |_{H_*(G, \\Lambda)}\\right) = {\\sum_{\\gamma\\mapsto 1}}'\\frac{1}{\\# Z_\\gamma}. $$ This can be interpreted as a point count on the stack $BG$. If $\\Lambda = \\mathbf{F}_\\ell$ with $\\ell$ prime to $\\# G$, then $H_*(G, \\Lambda)$ is $\\mathbf{F}_\\ell$ in degree 0 (and $0$ in other degrees) and the formula reads $$ 1 = \\sum\\nolimits_{ \\frac{\\sigma\\text{-conjugacy}}{\\text{classes}\\langle\\gamma\\rangle} } \\frac{1}{\\# Z_\\gamma} \\mod \\ell. $$ This is in some sense a ``trivial'' trace formula for $G$. Later we will see that (\\ref{equation-trace-formula}) can in some cases be viewed as a highly nontrivial trace formula for a certain type of group, see Section \\ref{section-abstract-trace-formula}."}
+{"_id": "14468", "title": "trace-remark-on-trace-formula-again", "text": "Remarks on Theorem \\ref{theorem-trace-formula-again}. \\begin{enumerate} \\item This formula holds in any dimension. By a d\\'evissage lemma (which uses proper base change etc.) it reduces to the current statement -- in that generality. \\item The complex $R\\Gamma_c(X_{\\bar k}, K)$ is defined by choosing an open immersion $j : X \\hookrightarrow \\bar X$ with $\\bar X$ projective over $k$ of dimension at most 1 and setting $$ R\\Gamma_c(X_{\\bar k}, K) := R\\Gamma(\\bar X_{\\bar k}, j_!K). $$ This is independent of the choice of $\\bar X$ follows from (insert reference here). We define $H^i_c(X_{\\bar k}, K)$ to be the $i$th cohomology group of $R\\Gamma_c(X_{\\bar k}, K)$. \\end{enumerate}"}
+{"_id": "14469", "title": "trace-remark-stronger", "text": "Even though all we did are reductions and mostly algebra, the trace formula Theorem \\ref{theorem-trace-formula-again} is much stronger than Weil's geometric trace formula (Theorem \\ref{theorem-weil-trace-formula}) because it applies to coefficient systems (sheaves), not merely constant coefficients."}
+{"_id": "14470", "title": "trace-remark-stalk-l-adic-sheaf", "text": "If $\\mathcal{F} = \\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ is a $\\mathbf{Z}_\\ell$-sheaf on $X$ and $\\bar x$ is a geometric point then $M_n = \\left\\{\\mathcal{F}_{n, \\bar x}\\right\\}$ is an inverse system of finite $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-modules such that $M_{n+1}\\to M_n$ is surjective and $M_n = M_{n+1}/\\ell^n M_{n+1}$. It follows that $$ M = \\lim_n M_n = \\lim \\mathcal{F}_{n, \\bar x} $$ is a finite $\\mathbf{Z}_\\ell$-module. Indeed, $M/\\ell M= M_1$ is finite over $\\mathbf{F}_\\ell$, so by Nakayama $M$ is finite over $\\mathbf{Z}_\\ell$. Therefore, $M\\cong \\mathbf{Z}_\\ell^{\\oplus r} \\oplus \\oplus_{i = 1}^t \\mathbf{Z}_\\ell/\\ell^{e_i}\\mathbf{Z}_\\ell$ for some $r, t\\geq 0$, $e_i\\geq 1$. The module $M = \\mathcal{F}_{\\bar x}$ is called the {\\it stalk} of $\\mathcal{F}$ at $\\bar x$."}
+{"_id": "14471", "title": "trace-remark-torsion-stalks", "text": "Since a $\\mathbf{Z}_\\ell$-sheaf is only defined on a Noetherian scheme, it is torsion if and only if its stalks are torsion."}
+{"_id": "14472", "title": "trace-remark-T", "text": "Intuitively, $T$ should be thought of as $T = t^f$ where $p^f = \\# k$. The definitions are then independent of the size of the ground field."}
+{"_id": "14473", "title": "trace-remark-which-is-harder", "text": "Since we have only developed some theory of traces and not of determinants, Theorem \\ref{theorem-A} is harder to prove than Theorem \\ref{theorem-B}. We will only prove the latter, for the former see \\cite{SGA4.5}. Observe also that there is no version of this theorem more general for $\\mathbf{Z}_\\ell$ coefficients since there is no $\\ell$-torsion."}
+{"_id": "14474", "title": "trace-remark-projective", "text": "Thus we conclude that if $X$ is also projective then we have functorially in the representation $\\rho$ the identifications $$ H^0(X_{\\overline{k}}, \\mathcal{F}_\\rho) = M^{\\pi_1(X_{\\overline{k}}, \\overline\\eta)} $$ and $$ H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho) = M_{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}(-1) $$ Of course if $X$ is not projective, then $H^0_c(X_{\\overline{k}}, \\mathcal{F}_\\rho) = 0$."}
+{"_id": "14475", "title": "trace-remark-poincare-groups", "text": "By the proposition and Trivial duality then you get $$ H^{2-i}_c(X_{\\overline{k}}, \\mathcal{F}_\\rho) \\times H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho^\\wedge(1)) \\to \\mathbf{Q}/\\mathbf{Z} $$ a perfect pairing. If $X$ is projective then this is Poincare duality."}
+{"_id": "14476", "title": "trace-remark-abstract-trace-formula", "text": "Here are some observations concerning this notion. \\begin{enumerate} \\item If modeling projective curves then we can use cohomology and we don't need factor $q^n$. \\item The only examples I know are $\\Gamma = \\pi_1(X, \\overline \\eta)$ where $X$ is smooth, geometrically irreducible and $K(\\pi, 1)$ over finite field. In this case $q = (\\# k)^{\\dim X}$. Modulo the proposition, we proved this for curves in this course. \\item Given the integer $q$ then the sets $S_d$ are uniquely determined. (You can multiple $q$ by an integer $m$ and then replace $S_d$ by $m^d$ copies of $S_d$ without changing the formula.) \\end{enumerate}"}
+{"_id": "14477", "title": "trace-remark-lafforgue", "text": "We now have, thanks to Lafforgue and many other mathematicians, complete theorems like this two above for $\\text{GL}_n$ and allowing ramification! In other words, the full global Langlands correspondence for $\\text{GL}_n$ is known for function fields of curves over finite fields. At the same time this does not mean there aren't a lot of interesting questions left to answer about the fundamental groups of curves over finite fields, as we shall see below."}
+{"_id": "14586", "title": "sheaves-remark-confusion", "text": "There is always a bit of confusion as to whether it is necessary to say something about the set of sections of a sheaf over the empty set $\\emptyset \\subset X$. It is necessary, and we already did if you read the definition right. Namely, note that the empty set is covered by the empty open covering, and hence the ``collection of sections $s_i$'' from the definition above actually form an element of the empty product which is the final object of the category the sheaf has values in. In other words, if you read the definition right you automatically deduce that $\\mathcal{F}(\\emptyset) = \\textit{a final object}$, which in the case of a sheaf of sets is a singleton. If you do not like this argument, then you can just require that $\\mathcal{F}(\\emptyset) = \\{*\\}$. \\medskip\\noindent In particular, this condition will then ensure that if $U, V \\subset X$ are open and {\\it disjoint} then $$ \\mathcal{F}(U \\cup V) = \\mathcal{F}(U) \\times \\mathcal{F}(V). $$ (Because the fibre product over a final object is a product.)"}
+{"_id": "14587", "title": "sheaves-remark-j-shriek-not-exact", "text": "Let $j : U \\to X$ be an open immersion of topological spaces as above. Let $x \\in X$, $x \\not \\in U$. Let $\\mathcal{F}$ be a sheaf of sets on $U$. Then $j_!\\mathcal{F}_x = \\emptyset$ by Lemma \\ref{lemma-j-shriek}. Hence $j_!$ does not transform a final object of $\\Sh(U)$ into a final object of $\\Sh(X)$ unless $U = X$. According to our conventions in Categories, Section \\ref{categories-section-exact-functor} this means that the functor $j_!$ is not left exact as a functor between the categories of sheaves of sets. It will be shown later that $j_!$ on abelian sheaves is exact, see Modules, Lemma \\ref{modules-lemma-j-shriek-exact}."}
+{"_id": "14588", "title": "sheaves-remark-i-star-not-exact", "text": "Let $i : Z \\to X$ be a closed immersion of topological spaces as above. Let $x \\in X$, $x \\not \\in Z$. Let $\\mathcal{F}$ be a sheaf of sets on $Z$. Then $(i_*\\mathcal{F})_x = \\{ * \\}$ by Lemma \\ref{lemma-stalks-closed-pushforward}. Hence if $\\mathcal{F} = * \\amalg *$, where $*$ is the singleton sheaf, then $i_*\\mathcal{F}_x = \\{*\\} \\not = i_*(*)_x \\amalg i_*(*)_x$ because the latter is a two point set. According to our conventions in Categories, Section \\ref{categories-section-exact-functor} this means that the functor $i_*$ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma \\ref{categories-lemma-exact-adjoint}. \\medskip\\noindent On the other hand, we will see later (see Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint}) that $i_*$ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on $X$ the sheaf of sections supported in $Z$."}
+{"_id": "14589", "title": "sheaves-remark-closed-immersion-spaces", "text": "We have not discussed the relationship between closed immersions and ringed spaces. This is because the notion of a closed immersion of ringed spaces is best discussed in the setting of quasi-coherent sheaves, see Modules, Section \\ref{modules-section-closed-immersion}."}
+{"_id": "14783", "title": "descent-remark-standard-covering", "text": "Let $R$ be a ring. Let $f_1, \\ldots, f_n\\in R$ generate the unit ideal. The ring $A = \\prod_i R_{f_i}$ is a faithfully flat $R$-algebra. We remark that the cosimplicial ring $(A/R)_\\bullet$ has the following ring in degree $n$: $$ \\prod\\nolimits_{i_0, \\ldots, i_n} R_{f_{i_0}\\ldots f_{i_n}} $$ Hence the results above recover Algebra, Lemmas \\ref{algebra-lemma-standard-covering}, \\ref{algebra-lemma-cover-module} and \\ref{algebra-lemma-glue-modules}. But the results above actually say more because of exactness in higher degrees. Namely, it implies that {\\v C}ech cohomology of quasi-coherent sheaves on affines is trivial. Thus we get a second proof of Cohomology of Schemes, Lemma \\ref{coherent-lemma-cech-cohomology-quasi-coherent-trivial}."}
+{"_id": "14784", "title": "descent-remark-homotopy-equivalent-cosimplicial-algebras", "text": "Let $R$ be a ring. Let $A_\\bullet$ be a cosimplicial $R$-algebra. In this setting a descent datum corresponds to an cosimplicial $A_\\bullet$-module $M_\\bullet$ with the property that for every $n, m \\geq 0$ and every $\\varphi : [n] \\to [m]$ the map $M(\\varphi) : M_n \\to M_m$ induces an isomorphism $$ M_n \\otimes_{A_n, A(\\varphi)} A_m \\longrightarrow M_m. $$ Let us call such a cosimplicial module a {\\it cartesian module}. In this setting, the proof of Proposition \\ref{proposition-descent-module} can be split in the following steps \\begin{enumerate} \\item If $R \\to R'$ and $R \\to A$ are faithfully flat, then descent data for $A/R$ are effective if descent data for $(R' \\otimes_R A)/R'$ are effective. \\item Let $A$ be an $R$-algebra. Descent data for $A/R$ correspond to cartesian $(A/R)_\\bullet$-modules. \\item If $R \\to A$ has a section then $(A/R)_\\bullet$ is homotopy equivalent to $R$, the constant cosimplicial $R$-algebra with value $R$. \\item If $A_\\bullet \\to B_\\bullet$ is a homotopy equivalence of cosimplicial $R$-algebras then the functor $M_\\bullet \\mapsto M_\\bullet \\otimes_{A_\\bullet} B_\\bullet$ induces an equivalence of categories between cartesian $A_\\bullet$-modules and cartesian $B_\\bullet$-modules. \\end{enumerate} For (1) see Lemma \\ref{lemma-descent-descends}. Part (2) uses Lemma \\ref{lemma-descent-datum-cosimplicial}. Part (3) we have seen in the proof of Lemma \\ref{lemma-with-section-exact} (it relies on Simplicial, Lemma \\ref{simplicial-lemma-push-outs-simplicial-object-w-section}). Moreover, part (4) is a triviality if you think about it right!"}
+{"_id": "14785", "title": "descent-remark-reflects", "text": "Any functor $F : \\mathcal{A} \\to \\mathcal{B}$ of abelian categories which is exact and takes nonzero objects to nonzero objects reflects injections and surjections. Namely, exactness implies that $F$ preserves kernels and cokernels (compare with Homology, Section \\ref{homology-section-functors}). For example, if $f : R \\to S$ is a  faithfully flat ring homomorphism, then $\\bullet \\otimes_R S: \\text{Mod}_R \\to \\text{Mod}_S$ has these properties."}
+{"_id": "14786", "title": "descent-remark-adjunction", "text": "We will use frequently the standard adjunction between $\\Hom$ and tensor  product, in the form of the natural isomorphism of contravariant functors \\begin{equation} \\label{equation-adjunction} C(\\bullet_1 \\otimes_R \\bullet_2) \\cong \\Hom_R(\\bullet_1, C(\\bullet_2)):  \\text{Mod}_R \\times \\text{Mod}_R \\to \\text{Mod}_R \\end{equation} taking $f: M_1 \\otimes_R M_2 \\to \\mathbf{Q}/\\mathbf{Z}$ to the map $m_1 \\mapsto  (m_2 \\mapsto f(m_1 \\otimes m_2))$. See Algebra, Lemma \\ref{algebra-lemma-hom-from-tensor-product-variant}. A corollary of this observation is that if $$ \\xymatrix@C=9pc{ C(M) \\ar@<1ex>[r] \\ar@<-1ex>[r] & C(N) \\ar[r] & C(P) } $$ is a split coequalizer diagram in $\\text{Mod}_R$, then so is $$ \\xymatrix@C=9pc{ C(M \\otimes_R Q) \\ar@<1ex>[r] \\ar@<-1ex>[r] & C(N \\otimes_R Q) \\ar[r] & C(P  \\otimes_R Q) } $$ for any $Q \\in \\text{Mod}_R$."}
+{"_id": "14787", "title": "descent-remark-functorial-splitting", "text": "Let $f: M \\to N$ be a universally injective morphism in $\\text{Mod}_R$. By  choosing a splitting $g$ of $C(f)$, we may construct a functorial splitting of $C(1_P \\otimes f)$  for each $P \\in \\text{Mod}_R$. Namely, by (\\ref{equation-adjunction}) this amounts to splitting $\\Hom_R(P,  C(f))$  functorially in $P$, and this is achieved by the map $g \\circ \\bullet$."}
+{"_id": "14788", "title": "descent-remark-descent-lemma", "text": "If $f$ is a split injection in $\\text{Mod}_R$, one can simplify the argument by  splitting $f$ directly, without using $C$. Things are even simpler if $f$ is faithfully flat; in this  case, the conclusion of Lemma \\ref{lemma-descent-lemma}  is immediate because tensoring over $R$ with $S$ preserves all equalizers."}
+{"_id": "14789", "title": "descent-remark-when-locally-split", "text": "It would make things easier to have a faithfully flat ring homomorphism $g: R \\to T$ for which $T \\to S \\otimes_R T$ has some  extra structure. For instance, if one could ensure that $T \\to S \\otimes_R T$ is split in  $\\textit{Rings}$, then it would follow that every property of a module or algebra which is stable  under base extension and which descends along faithfully flat morphisms also descends along  universally injective morphisms. An obvious guess would be to find $g$ for which $T$ is not only faithfully flat  but also injective in $\\text{Mod}_R$, but even for $R = \\mathbf{Z}$ no such homomorphism can exist."}
+{"_id": "14790", "title": "descent-remark-locally-free-descends", "text": "Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that $R$ is a ring and that $M$ is a projective $R$-module which is a countable direct sum $M = \\bigoplus L_n$ of rank 1 locally free modules, but not locally free, see Examples, Lemma \\ref{examples-lemma-projective-not-locally-free}. Then $M$ becomes free on making the faithfully flat base change $$ R \\longrightarrow \\bigoplus\\nolimits_{m \\geq 1} \\bigoplus\\nolimits_{(i_1, \\ldots, i_m) \\in \\mathbf{Z}^{\\oplus m}} L_1^{\\otimes i_1} \\otimes_R \\ldots \\otimes_R L_m^{\\otimes i_m} $$ But we don't know what happens for fppf coverings. In other words, we don't know the answer to the following question: Suppose $A \\to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \\otimes_A B$ is free. Is $M$ a locally free $A$-module? It turns out that if $A$ is Noetherian, then the answer is yes. This follows from the results of \\cite{Bass}. But in general we don't know the answer. If you know the answer, or have a reference, please email \\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}."}
+{"_id": "14791", "title": "descent-remark-Zariski-site-space", "text": "In Topologies, Lemma \\ref{topologies-lemma-Zariski-usual} we have seen that the small Zariski site of a scheme $S$ is equivalent to $S$ as a topological space in the sense that the categories of sheaves are naturally equivalent. Now that $S_{Zar}$ is also endowed with a structure sheaf $\\mathcal{O}$ we see that sheaves of modules on the ringed site $(S_{Zar}, \\mathcal{O})$ agree with sheaves of modules on the ringed space $(S, \\mathcal{O}_S)$."}
+{"_id": "14792", "title": "descent-remark-change-topologies-ringed", "text": "Let $f : T \\to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section \\ref{topologies-section-change-topologies} becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\\Sch/T)_\\tau \\to (\\Sch/S)_{\\tau'}$, or $f_{sites} : (\\Sch/S)_\\tau \\to S_{\\tau'}$ is given by the continuous functor $S'/S \\mapsto T \\times_S S'/S$. Hence, given $S'/S$ we let $$ f_{sites}^\\sharp : \\mathcal{O}(S'/S) \\longrightarrow f_{sites, *}\\mathcal{O}(S'/S) = \\mathcal{O}(S \\times_S S'/T) $$ be the usual map $\\text{pr}_{S'}^\\sharp : \\mathcal{O}(S') \\to \\mathcal{O}(T \\times_S S')$. Similarly, the morphism $i_f : \\Sh(T_\\tau) \\to \\Sh((\\Sch/S)_\\tau)$ for $\\tau \\in \\{Zar, \\etale\\}$, see Topologies, Lemmas \\ref{topologies-lemma-put-in-T} and \\ref{topologies-lemma-put-in-T-etale}, becomes a morphism of ringed topoi because $i_f^{-1}\\mathcal{O} = \\mathcal{O}$. Here are some special cases: \\begin{enumerate} \\item The morphism of big sites $f_{big} : (\\Sch/X)_{fppf} \\to (\\Sch/Y)_{fppf}$, becomes a morphism of ringed sites $$ (f_{big}, f_{big}^\\sharp) : ((\\Sch/X)_{fppf}, \\mathcal{O}_X) \\longrightarrow ((\\Sch/Y)_{fppf}, \\mathcal{O}_Y) $$ as in Modules on Sites, Definition \\ref{sites-modules-definition-ringed-site}. Similarly for the big syntomic, smooth, \\'etale and Zariski sites. \\item The morphism of small sites $f_{small} : X_\\etale \\to Y_\\etale$ becomes a morphism of ringed sites $$ (f_{small}, f_{small}^\\sharp) : (X_\\etale, \\mathcal{O}_X) \\longrightarrow (Y_\\etale, \\mathcal{O}_Y) $$ as in Modules on Sites, Definition \\ref{sites-modules-definition-ringed-site}. Similarly for the small Zariski site. \\end{enumerate}"}
+{"_id": "14793", "title": "descent-remark-change-topologies-ringed-sites", "text": "Remark \\ref{remark-change-topologies-ringed} and Lemma \\ref{lemma-compare-sites} have the following applications: \\begin{enumerate} \\item Let $S$ be a scheme. The construction $\\mathcal{F} \\mapsto \\mathcal{F}^a$ is the pullback under the morphism of ringed sites $\\text{id}_{\\tau, Zar} : ((\\Sch/S)_\\tau, \\mathcal{O}) \\to (S_{Zar}, \\mathcal{O})$ or the morphism $\\text{id}_{small, \\etale, Zar} : (S_\\etale, \\mathcal{O}) \\to (S_{Zar}, \\mathcal{O})$. \\item Let $f : X \\to Y$ be a morphism of schemes. For any of the morphisms $f_{sites}$ of ringed sites of Remark \\ref{remark-change-topologies-ringed} we have $$ (f^*\\mathcal{F})^a = f_{sites}^*\\mathcal{F}^a. $$ This follows from (1) and the fact that pullbacks are compatible with compositions of morphisms of ringed sites, see Modules on Sites, Lemma \\ref{sites-modules-lemma-push-pull-composition-modules}. \\end{enumerate}"}
+{"_id": "14794", "title": "descent-remark-smooth-permanence", "text": "With the assumptions (1) and $p$ smooth in Lemma \\ref{lemma-smooth-permanence} it is not automatically the case that $X \\to Y$ is smooth. A counter example is $S = \\Spec(k)$, $X = \\Spec(k[s])$, $Y = \\Spec(k[t])$ and $f$ given by $t \\mapsto s^2$. But see also Lemma \\ref{lemma-syntomic-permanence} for some information on the structure of $f$."}
+{"_id": "14795", "title": "descent-remark-descending-properties-standard", "text": "In Lemma \\ref{lemma-descending-properties} above if $\\tau = smooth$ then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$."}
+{"_id": "14796", "title": "descent-remark-descending-properties-morphisms-standard", "text": "(This is a repeat of Remark \\ref{remark-descending-properties-standard} above.) In Lemma \\ref{lemma-descending-properties-morphisms} above if $\\tau = smooth$ then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$."}
+{"_id": "14797", "title": "descent-remark-properties-morphisms-local-source-standard", "text": "(This is a repeat of Remarks \\ref{remark-descending-properties-standard} and \\ref{remark-descending-properties-morphisms-standard} above.) In Lemma \\ref{lemma-properties-morphisms-local-source} above if $\\tau = smooth$ then in condition (4) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\\tau = syntomic$ or $\\tau = \\etale$."}
+{"_id": "14798", "title": "descent-remark-list-local-source-target", "text": "Using Lemma \\ref{lemma-etale-local-source-target} and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are \\'etale local on the source-and-target. In each case we list the lemma which implies the property is \\'etale local on the source and the lemma which implies the property is \\'etale local on the target. In each case the third assumption of Lemma \\ref{lemma-etale-local-source-target} is trivial to check, and we omit it. Here is the list: \\begin{enumerate} \\item flat, see Lemmas \\ref{lemma-flat-fpqc-local-source} and \\ref{lemma-descending-property-flat}, \\item locally of finite presentation, see Lemmas \\ref{lemma-locally-finite-presentation-fppf-local-source} and \\ref{lemma-descending-property-locally-finite-presentation}, \\item locally finite type, see Lemmas \\ref{lemma-locally-finite-type-fppf-local-source} and \\ref{lemma-descending-property-locally-finite-type}, \\item universally open, see Lemmas \\ref{lemma-universally-open-fppf-local-source} and \\ref{lemma-descending-property-universally-open}, \\item syntomic, see Lemmas \\ref{lemma-syntomic-syntomic-local-source} and \\ref{lemma-descending-property-syntomic}, \\item smooth, see Lemmas \\ref{lemma-smooth-smooth-local-source} and \\ref{lemma-descending-property-smooth}, \\item \\'etale, see Lemmas \\ref{lemma-etale-etale-local-source} and \\ref{lemma-descending-property-etale}, \\item locally quasi-finite, see Lemmas \\ref{lemma-locally-quasi-finite-etale-local-source} and \\ref{lemma-descending-property-quasi-finite}, \\item unramified, see Lemmas \\ref{lemma-unramified-etale-local-source} and \\ref{lemma-descending-property-unramified}, \\item G-unramified, see Lemmas \\ref{lemma-unramified-etale-local-source} and \\ref{lemma-descending-property-unramified}, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "14799", "title": "descent-remark-compare-definitions", "text": "At this point we have three possible definitions of what it means for a property $\\mathcal{P}$ of morphisms to be ``\\'etale local on the source and target'': \\begin{enumerate} \\item[(ST)] $\\mathcal{P}$ is \\'etale local on the source and $\\mathcal{P}$ is \\'etale local on the target, \\item[(DM)] (the definition in the paper \\cite[Page 100]{DM} by Deligne and Mumford) for every diagram $$ \\xymatrix{ U \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\ X \\ar[r]^f & Y } $$ with surjective \\'etale vertical arrows we have $\\mathcal{P}(h) \\Leftrightarrow \\mathcal{P}(f)$, and \\item[(SP)] $\\mathcal{P}$ is \\'etale local on the source-and-target. \\end{enumerate} In this section we have seen that (SP) $\\Rightarrow$ (DM) $\\Rightarrow$ (ST). The Examples \\ref{example-silly-one} and \\ref{example-silly-two} show that neither implication can be reversed. Finally, Lemma \\ref{lemma-etale-local-source-target} shows that the difference disappears when looking at properties of morphisms which are stable under postcomposing with open immersions, which in practice will always be the case."}
+{"_id": "14800", "title": "descent-remark-easier", "text": "Let $X \\to S$ be a morphism of schemes. Let $(V/X, \\varphi)$ be a descent datum relative to $X \\to S$. We may think of the isomorphism $\\varphi$ as an isomorphism $$ (X \\times_S X) \\times_{\\text{pr}_0, X} V \\longrightarrow (X \\times_S X) \\times_{\\text{pr}_1, X} V $$ of schemes over $X \\times_S X$. So loosely speaking one may think of $\\varphi$ as a map $\\varphi : \\text{pr}_0^*V \\to \\text{pr}_1^*V$\\footnote{Unfortunately, we have chosen the ``wrong'' direction for our arrow here. In Definitions \\ref{definition-descent-datum} and \\ref{definition-descent-datum-for-family-of-morphisms} we should have the opposite direction to what was done in Definition \\ref{definition-descent-datum-quasi-coherent} by the general principle that ``functions'' and ``spaces'' are dual.}. The cocycle condition then says that $\\text{pr}_{02}^*\\varphi = \\text{pr}_{12}^*\\varphi \\circ \\text{pr}_{01}^*\\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves."}
+{"_id": "14801", "title": "descent-remark-easier-family", "text": "Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$. Let $(V_i, \\varphi_{ij})$ be a descent datum relative to $\\{X_i \\to S\\}$. We may think of the isomorphisms $\\varphi_{ij}$ as isomorphisms $$ (X_i \\times_S X_j) \\times_{\\text{pr}_0, X_i} V_i \\longrightarrow (X_i \\times_S X_j) \\times_{\\text{pr}_1, X_j} V_j $$ of schemes over $X_i \\times_S X_j$. So loosely speaking one may think of $\\varphi_{ij}$ as an isomorphism $\\text{pr}_0^*V_i \\to \\text{pr}_1^*V_j$ over $X_i \\times_S X_j$. The cocycle condition then says that $\\text{pr}_{02}^*\\varphi_{ik} = \\text{pr}_{12}^*\\varphi_{jk} \\circ \\text{pr}_{01}^*\\varphi_{ij}$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves."}
+{"_id": "14802", "title": "descent-remark-morphisms-of-schemes-satisfy-fpqc-descent", "text": "Lemma \\ref{lemma-refine-coverings-fully-faithful} says that morphisms of schemes satisfy fpqc descent. In other words, given a scheme $S$ and schemes $X$, $Y$ over $S$ the functor $$ (\\Sch/S)^{opp} \\longrightarrow \\textit{Sets}, \\quad T \\longmapsto \\Mor_T(X_T, Y_T) $$ satisfies the sheaf condition for the fpqc topology. The simplest case of this is the following. Suppose that $T \\to S$ is a surjective flat morphism of affines. Let $\\psi_0 : X_T \\to Y_T$ be a morphism of schemes over $T$ which is compatible with the canonical descent data. Then there exists a unique morphism $\\psi : X \\to Y$ whose base change to $T$ is $\\psi_0$. In fact this special case follows in a straightforward manner from Lemma \\ref{lemma-fully-faithful}. And, in turn, that lemma is a formal consequence of the following two facts: (a) the base change functor by a faithfully flat morphism is faithful, see Lemma \\ref{lemma-ff-base-change-faithful} and (b) a scheme satisfies the sheaf condition for the fpqc topology, see Lemma \\ref{lemma-fpqc-universal-effective-epimorphisms}."}
+{"_id": "14803", "title": "descent-remark-what-product-means", "text": "In the statement of Lemma \\ref{lemma-descent-data-sheaves} the condition that $h_{S_i} \\times F$ is representable is equivalent to the condition that the restriction of $F$ to $(\\Sch/S_i)_\\tau$ is representable."}
+{"_id": "14932", "title": "simplicial-remark-relations", "text": "By abuse of notation we sometimes write $d_i : U_n \\to U_{n - 1}$ instead of $d^n_i$, and similarly for $s_i : U_n \\to U_{n + 1}$. The relations among the morphisms $d^n_i$ and $s^n_i$ may be expressed as follows: \\begin{enumerate} \\item If $i < j$, then $d_i \\circ d_j = d_{j - 1} \\circ d_i$. \\item If $i < j$, then $d_i \\circ s_j = s_{j - 1} \\circ d_i$. \\item We have $\\text{id} = d_j \\circ s_j = d_{j + 1} \\circ s_j$. \\item If $i > j + 1$, then $d_i \\circ s_j = s_j \\circ d_{i - 1}$. \\item If $i \\leq j$, then $s_i \\circ s_j = s_{j + 1} \\circ s_i$. \\end{enumerate} This means that whenever the compositions on both the left and the right are defined then the corresponding equality should hold."}
+{"_id": "14933", "title": "simplicial-remark-relations-cosimplicial", "text": "By abuse of notation we sometimes write $\\delta_i : U_{n - 1} \\to U_n$ instead of $\\delta^n_i$, and similarly for $\\sigma_i : U_{n + 1} \\to U_n$. The relations among the morphisms $\\delta^n_i$ and $\\sigma^n_i$ may be expressed as follows: \\begin{enumerate} \\item If $i < j$, then $\\delta_j \\circ \\delta_i = \\delta_i \\circ \\delta_{j - 1}$. \\item If $i < j$, then $\\sigma_j \\circ \\delta_i = \\delta_i \\circ \\sigma_{j - 1}$. \\item We have $\\text{id} = \\sigma_j \\circ \\delta_j = \\sigma_j \\circ \\delta_{j + 1}$. \\item If $i > j + 1$, then $\\sigma_j \\circ \\delta_i = \\delta_{i - 1} \\circ \\sigma_j$. \\item If $i \\leq j$, then $\\sigma_j \\circ \\sigma_i = \\sigma_i \\circ \\sigma_{j + 1}$. \\end{enumerate} This means that whenever the compositions on both the left and the right are defined then the corresponding equality should hold."}
+{"_id": "14934", "title": "simplicial-remark-explicit-face-degeneracy", "text": "Let $U$, and $U_{n + 1}$ be as in Lemma \\ref{lemma-work-out}. On $T$-valued points we can easily describe the face and degeneracy maps of $\\tilde U$. Explicitly, the maps $d^{n + 1}_i : U_{n + 1} \\to U_n$ are given by $$ (f_0, \\ldots, f_{n + 1}) \\longmapsto f_i. $$ And the maps $s^n_j : U_n \\to U_{n + 1}$ are given by \\begin{eqnarray*} f & \\longmapsto & ( s^{n - 1}_{j - 1} \\circ d^{n - 1}_0 \\circ f, \\\\ & & s^{n - 1}_{j - 1} \\circ d^{n - 1}_1 \\circ f, \\\\ & & \\ldots\\\\ & & s^{n - 1}_{j - 1} \\circ d^{n - 1}_{j - 1} \\circ f, \\\\ & & f, \\\\ & & f, \\\\ & & s^{n - 1}_j \\circ d^{n - 1}_{j + 1} \\circ f, \\\\ & & s^{n - 1}_j \\circ d^{n - 1}_{j + 2} \\circ f, \\\\ & & \\ldots\\\\ & & s^{n - 1}_j \\circ d^{n - 1}_n \\circ f ) \\end{eqnarray*} where we leave it to the reader to verify that the RHS is an element of the displayed set of Lemma \\ref{lemma-work-out}. For $n = 0$ there is one map, namely $f \\mapsto (f, f)$. For $n = 1$ there are two maps, namely $f \\mapsto (f, f, s_0d_1f)$ and $f \\mapsto (s_0d_0f, f, f)$. For $n = 2$ there are three maps, namely $f \\mapsto (f, f, s_0d_1f, s_0d_2f)$, $f \\mapsto (s_0d_0f, f, f, s_1d_2f)$, and $f \\mapsto (s_1d_0f, s_1d_1f, f, f)$. And so on and so forth."}
+{"_id": "14935", "title": "simplicial-remark-cosk-simplicial-sets", "text": "The construction of Lemma \\ref{lemma-work-out} above in the case of simplicial sets is the following. Given an $n$-truncated simplicial set $U$, we make a canonical $(n + 1)$-truncated simplicial set $\\tilde U$ as follows. We add a set of $(n + 1)$-simplices $U_{n + 1}$ by the formula of the lemma. Namely, an element of $U_{n + 1}$ is a numbered collection of $(f_0, \\ldots, f_{n + 1})$ of $n$-simplices, with the property that they glue as they would in a $(n + 1)$-simplex. In other words, the $i$th face of $f_j$ is the $(j-1)$st face of $f_i$ for $i < j$. Geometrically it is obvious how to define the face and degeneracy maps for $\\tilde U$. If $V$ is an $(n + 1)$-truncated simplicial set, then its $(n + 1)$-simplices give rise to compatible collections of $n$-simplices $(f_0, \\ldots, f_{n + 1})$ with $f_i \\in V_n$. Hence there is a natural map $\\Mor(\\text{sk}_nV, U) \\to \\Mor(V, \\tilde U)$ which is inverse to the canonical restriction mapping the other way. \\medskip\\noindent Also, it is enough to do the combinatorics of the construction in the case of truncated simplicial sets. Namely, for any object $T$ of the category $\\mathcal{C}$, and any $n$-truncated simplicial object $U$ of $\\mathcal{C}$ we can consider the $n$-truncated simplicial set $\\Mor(T, U)$. We may apply the construction to this, and take its set of $(n + 1)$-simplices, and require this to be representable. This is a good way to think about the result of Lemma \\ref{lemma-work-out}."}
+{"_id": "14936", "title": "simplicial-remark-inductive-coskeleton", "text": "{\\it Inductive construction of coskeleta.} Suppose that $\\mathcal{C}$ is a category with finite limits. Suppose that $U$ is an $m$-truncated simplicial object in $\\mathcal{C}$. Then we can inductively construct $n$-truncated objects $U^n$ as follows: \\begin{enumerate} \\item To start, set $U^m = U$. \\item Given $U^n$ for $n \\geq m$ set $U^{n + 1} = \\tilde U^n$, where $\\tilde U^n$ is constructed from $U^n$ as in Lemma \\ref{lemma-work-out}. \\end{enumerate} Since the construction of Lemma \\ref{lemma-work-out} has the property that it leaves the $n$-skeleton of $U^n$ unchanged, we can then define $\\text{cosk}_m U$ to be the simplicial object with $(\\text{cosk}_m U)_n = U^n_n = U^{n + 1}_n = \\ldots$. And it follows formally from Lemma \\ref{lemma-work-out} that $U^n$ satisfies the formula $$ \\Mor_{\\text{Simp}_n(\\mathcal{C})}(V, U^n) = \\Mor_{\\text{Simp}_m(\\mathcal{C})}(\\text{sk}_mV, U) $$ for all $n \\geq m$. It also then follows formally from this that $$ \\Mor_{\\text{Simp}(\\mathcal{C})}(V, \\text{cosk}_mU) = \\Mor_{\\text{Simp}_m(\\mathcal{C})}(\\text{sk}_mV, U) $$ with $\\text{cosk}_mU$ chosen as above."}
+{"_id": "14937", "title": "simplicial-remark-existence-cosk", "text": "We do not need all finite limits in order to be able to define the coskeleton functors. Here are some remarks \\begin{enumerate} \\item We have seen in Examples \\ref{example-cosk0} that if $\\mathcal{C}$ has products of pairs of objects then $\\text{cosk}_0$ exists. \\item For $k > 0$ the functor $\\text{cosk}_k$ exists if $\\mathcal{C}$ has finite connected limits. \\end{enumerate} This is clear from the inductive procedure of constructing coskeleta (Remarks \\ref{remark-cosk-simplicial-sets} and \\ref{remark-inductive-coskeleton}) but it also follows from the fact that the categories $(\\Delta/[n])_{\\leq k}$ for $k \\geq 1$ and $n \\geq k + 1$ used in Lemma \\ref{lemma-existence-cosk} are connected. Observe that we do not need the categories for $n \\leq k$ by Lemma \\ref{lemma-trivial-cosk} or Lemma \\ref{lemma-recover-cosk}. (As $k$ gets higher the categories $(\\Delta/[n])_{\\leq k}$ for $k \\geq 1$ and $n \\geq k + 1$ are more and more connected in a topological sense.)"}
+{"_id": "14938", "title": "simplicial-remark-augmentation", "text": "Let $\\mathcal{C}$ be a category with fibre products. Let $V$ be a simplicial object. Let $\\epsilon : V \\to X$ be an augmentation. Let $U$ be the simplicial object whose $n$th term is the $(n + 1)$st fibred product of $V_0$ over $X$. By a simple combination of Lemmas \\ref{lemma-augmentation-howto} and \\ref{lemma-cosk-minus-one} we  obtain a canonical morphism $V \\to U$."}
+{"_id": "14939", "title": "simplicial-remark-sk-literature", "text": "In some texts the composite functor $$ \\text{Simp}(\\mathcal{C}) \\xrightarrow{\\text{sk}_m} \\text{Simp}_m(\\mathcal{C}) \\xrightarrow{i_{m!}} \\text{Simp}(\\mathcal{C}) $$ is denoted $\\text{sk}_m$. This makes sense for simplicial sets, because then Lemma \\ref{lemma-n-skeleton-sets} says that $i_{m!} \\text{sk}_m V$ is just the sub simplicial set of $V$ consisting of all $i$-simplices of $V$, $i \\leq m$ and their degeneracies. In those texts it is also customary to denote the composition $$ \\text{Simp}(\\mathcal{C}) \\xrightarrow{\\text{sk}_m} \\text{Simp}_m(\\mathcal{C}) \\xrightarrow{\\text{cosk}_m} \\text{Simp}(\\mathcal{C}) $$ by $\\text{cosk}_m$."}
+{"_id": "14940", "title": "simplicial-remark-degenerate-subcomplex", "text": "In the situation of Lemma \\ref{lemma-decompose-associated-complexes} the subcomplex $D(U) \\subset s(U)$ can also be defined as the subcomplex with terms $$ D(U)_n = \\Im\\left( \\bigoplus\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}, \\ m < n} U_m \\xrightarrow{\\bigoplus U(\\varphi)} U_n\\right) $$ Namely, since $U_m$ is the direct sum of the subobject $N(U_m)$ and the images of $N(U_k)$ for surjections $[m] \\to [k]$ with $k < m$ this is clearly the same as the definition of $D(U)_n$ given in the proof of Lemma \\ref{lemma-decompose-associated-complexes}. Thus we see that if $U$ is a simplicial abelian group, then elements of $D(U)_n$ are exactly the sums of degenerate $n$-simplices."}
+{"_id": "14941", "title": "simplicial-remark-homotopy-better", "text": "Let $\\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be simplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{C}$. A {\\it homotopy from $a$ to $b$} is given by morphisms\\footnote{In the literature, often the maps $h_{n + 1, i} \\circ s_i : U_n \\to V_{n + 1}$ are used instead of the maps $h_{n, i}$. Of course the relations these maps satisfy are different from the ones in Lemma \\ref{lemma-relations-homotopy}.} $h_{n, i} : U_n \\to V_n$, for $n \\geq 0$, $i = 0, \\ldots, n + 1$ satisfying the relations of Lemma \\ref{lemma-relations-homotopy}. As in Definition \\ref{definition-homotopy} we say the morphisms $a$ and $b$ are {\\it homotopic} if there exists a sequence of morphisms $a = a_0, a_1, \\ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. Clearly, if $F : \\mathcal{C} \\to \\mathcal{C}'$ is any functor and $\\{h_{n, i}\\}$ is a homotopy from $a$ to $b$, then $\\{F(h_{n, i})\\}$ is a homotopy from $F(a)$ to $F(b)$. Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$ are homotopic. Since the lemma says that the newer notion is the same as the old one in case finite coproduct exist, we deduce in particular that functors preserve the original notion whenever both categories have finite coproducts."}
+{"_id": "14942", "title": "simplicial-remark-homotopy-pre-post-compose", "text": "Let $\\mathcal{C}$ be any category. Suppose two morphisms $a, a' : U \\to V$ of simplicial objects are homotopic. Then for any morphism $b : V \\to W$ the two maps $b \\circ a, b \\circ a' : U \\to W$ are homotopic. Similarly, for any morphism $c : X \\to U$ the two maps $a \\circ c, a' \\circ c : X \\to V$ are homotopic. In fact the maps $b \\circ a \\circ c, b \\circ a' \\circ c : X \\to W$ are homotopic. Namely, if the maps $h_{n, i} : U_n \\to V_n$ define a homotopy from $a$ to $a'$ then the maps $b \\circ h_{n, i} \\circ c$ define a homotopy from $b \\circ a \\circ c$ to $b \\circ a' \\circ c$. In this way we see that we obtain a new category $\\text{hSimp}(\\mathcal{C})$ with the same objects as $\\text{Simp}(\\mathcal{C})$ but whose morphisms are homotopy classes of of morphisms of $\\text{Simp}(\\mathcal{C})$. Thus there is a canonical functor $$ \\text{Simp}(\\mathcal{C}) \\longrightarrow \\text{hSimp}(\\mathcal{C}) $$ which is essentially surjective and surjective on sets of morphisms."}
+{"_id": "14943", "title": "simplicial-remark-homotopy-cosimplicial-better", "text": "Let $\\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be cosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be morphisms of cosimplicial objects of $\\mathcal{C}$. A {\\it homotopy from $a$ to $b$} is given by morphisms $h_{n, \\alpha} : U_n \\to V_n$, for $n \\geq 0$, $\\alpha \\in \\Delta[1]_n$ satisfying (\\ref{equation-property-homotopy-cosimplicial}) for all morphisms $f$ of $\\Delta$ and such that $a_n = h_{n, 0 : [n] \\to [1]}$ and $b_n = h_{n, 1 : [n] \\to [1]}$ for all $n \\geq 0$. As in Definition \\ref{definition-homotopy-cosimplicial} we say the morphisms $a$ and $b$ are {\\it homotopic} if there exists a sequence of morphisms $a = a_0, a_1, \\ldots, a_n = b$ from $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists a homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy from $a_i$ to $a_{i - 1}$. Clearly, if $F : \\mathcal{C} \\to \\mathcal{C}'$ is any functor and $\\{h_{n, i}\\}$ is a homotopy from $a$ to $b$, then $\\{F(h_{n, i})\\}$ is a homotopy from $F(a)$ to $F(b)$. Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$ are homotopic. This new notion is the same as the old one in case finite products exist. We deduce in particular that functors preserve the original notion whenever both categories have finite products."}
+{"_id": "15006", "title": "discriminant-remark-relative-dualizing-for-quasi-finite", "text": "Let $f : Y \\to X$ be a locally quasi-finite morphism of locally Noetherian schemes. It is clear from Lemma \\ref{lemma-localize-dualizing} that there is a unique coherent $\\mathcal{O}_Y$-module $\\omega_{Y/X}$ on $Y$ such that for every pair of affine opens $\\Spec(B) = V \\subset Y$, $\\Spec(A) = U \\subset X$ with $f(V) \\subset U$ there is a canonical isomorphism $$ H^0(V, \\omega_{Y/X}) = \\omega_{B/A} $$ and where these isomorphisms are compatible with restriction maps."}
+{"_id": "15007", "title": "discriminant-remark-relative-dualizing-for-flat-quasi-finite", "text": "Let $f : Y \\to X$ be a flat locally quasi-finite morphism of locally Noetherian schemes. Let $\\omega_{Y/X}$ be as in Remark \\ref{remark-relative-dualizing-for-quasi-finite}. It is clear from the uniqueness, existence, and compatibility with localization of trace elements (Lemmas \\ref{lemma-trace-unique}, \\ref{lemma-dualizing-tau}, and \\ref{lemma-trace-base-change}) that there exists a global section $$ \\tau_{Y/X} \\in \\Gamma(Y, \\omega_{Y/X}) $$ such that for every pair of affine opens $\\Spec(B) = V \\subset Y$, $\\Spec(A) = U \\subset X$ with $f(V) \\subset U$ that element $\\tau_{Y/X}$ maps to $\\tau_{B/A}$ under the canonical isomorphism $H^0(V, \\omega_{Y/X}) = \\omega_{B/A}$."}
+{"_id": "15008", "title": "discriminant-remark-construction-pairing", "text": "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings. Let $J$ be the annihilator of $\\Ker(B \\otimes_A B \\to B)$. There is a canonical $B$-bilinear pairing \\begin{equation} \\label{equation-pairing-noether} \\omega_{B/A} \\times J \\longrightarrow B \\end{equation} defined as follows. Choose a factorization $A \\to B' \\to B$ with $A \\to B'$ finite and $B' \\to B$ inducing an open immersion of spectra. Let $J'$ be the annihilator of $\\Ker(B' \\otimes_A B' \\to B')$. We first define $$ \\Hom_A(B', A) \\times J' \\longrightarrow B',\\quad (\\lambda, \\sum b_i \\otimes c_i) \\longmapsto \\sum \\lambda(b_i)c_i $$ This is $B'$-bilinear exactly because for $\\xi \\in J'$ and $b \\in B'$ we have $(b \\otimes 1)\\xi = (1 \\otimes b)\\xi$. By Lemma \\ref{lemma-noether-different-localization} and the fact that $\\omega_{B/A} = \\Hom_A(B', A) \\otimes_{B'} B$ we can extend this to a $B$-bilinear pairing as displayed above."}
+{"_id": "15009", "title": "discriminant-remark-universal-finite-syntomic-smooth-top", "text": "Let $\\pi_d : Y_d \\to X_d$ be as in Example \\ref{example-universal-finite-syntomic}. Let $U_d \\subset X_d$ be the maximal open over which $V_d = \\pi_d^{-1}(U_d)$ is finite syntomic as in Lemma \\ref{lemma-universal-finite-syntomic}. Then it is also true that $V_d$ is smooth over $\\mathbf{Z}$. (Of course the morphism $V_d \\to U_d$ is not smooth when $d \\geq 2$.) Arguing as in the proof of Lemma \\ref{lemma-universal-finite-syntomic-smooth} this corresponds to the following deformation problem: given a small extension $C' \\to C$ and a finite syntomic $C$-algebra $B$ with a section $B \\to C$, find a finite syntomic $C'$-algebra $B'$ and a section $B' \\to C'$ whose tensor product with $C$ recovers $B \\to C$. By Lemma \\ref{lemma-syntomic-finite} we may write $B = C[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ as a relative global complete intersection. After a change of coordinates with may assume $x_1, \\ldots, x_n$ are in the kernel of $B \\to C$. Then the polynomials $f_i$ have vanishing constant terms. Choose any lifts $f'_i \\in C'[x_1, \\ldots, x_n]$ of $f_i$ with vanishing constant terms. Then  $B' = C'[x_1, \\ldots, x_n]/(f'_1, \\ldots, f'_n)$ with section $B' \\to C'$ sending $x_i$ to zero works."}
+{"_id": "15010", "title": "discriminant-remark-local-description-delta", "text": "Let $Y \\to X$ be a locally quasi-finite syntomic morphism of schemes. What does the pair $(\\det(\\NL_{Y/X}), \\delta(\\NL_{Y/X}))$ look like locally? Choose affine opens $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ with $f(V) \\subset U$ and an integer $n$ and $f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$. Then $$ \\NL_{B/A} = \\left( (f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2 \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} B \\text{d} x_i\\right) $$ and $(f_1, \\ldots, f_n)/(f_1, \\ldots, f_n)^2$ is free with generators the classes $\\overline{f}_i$. See proof of Lemma \\ref{lemma-syntomic-quasi-finite}. Thus $\\det(L_{B/A})$ is free on the generator $$ \\text{d}x_1 \\wedge \\ldots \\wedge \\text{d}x_n \\otimes (\\overline{f}_1 \\wedge \\ldots \\wedge \\overline{f}_n)^{\\otimes -1} $$ and the section $\\delta(\\NL_{B/A})$ is the element $$ \\delta(\\NL_{B/A}) = \\det(\\partial f_j/ \\partial x_i) \\cdot \\text{d}x_1 \\wedge \\ldots \\wedge \\text{d}x_n \\otimes (\\overline{f}_1 \\wedge \\ldots \\wedge \\overline{f}_n)^{\\otimes -1} $$ by definition."}
+{"_id": "15011", "title": "discriminant-remark-different-generalization", "text": "We can generalize Definition \\ref{definition-different}. Suppose that $f : Y \\to X$ is a quasi-finite morphism of Noetherian schemes with the following properties \\begin{enumerate} \\item the open $V \\subset Y$ where $f$ is flat contains $\\text{Ass}(\\mathcal{O}_Y)$ and $f^{-1}(\\text{Ass}(\\mathcal{O}_X))$, \\item the trace element $\\tau_{V/X}$ comes from a section $\\tau \\in \\Gamma(Y, \\omega_{Y/X})$. \\end{enumerate} Condition (1) implies that $V$ contains the associated points of $\\omega_{Y/X}$ by Lemma \\ref{lemma-dualizing-associated-primes}. In particular, $\\tau$ is unique if it exists (Divisors, Lemma \\ref{divisors-lemma-restriction-injective-open-contains-ass}). Given $\\tau$ we can define the different $\\mathfrak{D}_f$ as the annihilator of $\\Coker(\\tau : \\mathcal{O}_Y \\to \\omega_{Y/X})$. This agrees with the Dedekind different in many cases (Lemma \\ref{lemma-agree-dedekind}). However, for non-flat maps between non-normal rings, this generalization no longer measures ramification of the morphism, see Example \\ref{example-no-different}."}
+{"_id": "15012", "title": "discriminant-remark-collect-results-qf-gorenstein", "text": "Let $f : Y \\to X$ be a quasi-finite Gorenstein morphism of Noetherian schemes. Let $\\mathfrak D_f \\subset \\mathcal{O}_Y$ be the different and let $R \\subset Y$ be the closed subscheme cut out by $\\mathfrak D_f$. Then we have \\begin{enumerate} \\item $\\mathfrak D_f$ is a locally principal ideal, \\item $R$ is a locally principal closed subscheme, \\item $\\mathfrak D_f$ is affine locally the same as the Noether different, \\item formation of $R$ commutes with base change, \\item if $f$ is finite, then the norm of $R$ is the discriminant of $f$, and \\item if $f$ is \\'etale in the associated points of $Y$, then $R$ is an effective Cartier divisor and $\\omega_{Y/X} = \\mathcal{O}_Y(R)$. \\end{enumerate} This follows from Lemmas \\ref{lemma-flat-gorenstein-agree-noether}, \\ref{lemma-base-change-different}, and \\ref{lemma-norm-different-is-discriminant}."}
+{"_id": "15013", "title": "discriminant-remark-collect-results-qf-gorenstein-two", "text": "Let $S$ be a Noetherian scheme endowed with a dualizing complex $\\omega_S^\\bullet$. Let $f : Y \\to X$ be a quasi-finite Gorenstein morphism of compactifyable schemes over $S$. Assume moreover $Y$ and $X$ Cohen-Macaulay and $f$ \\'etale at the generic points of $Y$. Then we can combine Duality for Schemes, Remark \\ref{duality-remark-CM-morphism-compare-dualizing} and Remark \\ref{remark-collect-results-qf-gorenstein} to see that we have a canonical isomorphism $$ \\omega_Y = f^*\\omega_X \\otimes_{\\mathcal{O}_Y} \\omega_{Y/X} = f^*\\omega_X \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(R) $$ of $\\mathcal{O}_Y$-modules. If further $f$ is finite, then the isomorphism $\\mathcal{O}_Y(R) = \\omega_{Y/X}$ comes from the global section $\\tau_{Y/X} \\in H^0(Y, \\omega_{Y/X})$ which corresponds via duality to the map $\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$, see Lemma \\ref{lemma-compare-trace}."}
+{"_id": "15130", "title": "limits-remark-limit-preserving", "text": "Let $S$ be a scheme. Let us say that a functor $F : (\\Sch/S)^{opp} \\to \\textit{Sets}$ is {\\it limit preserving} if for every directed inverse system $\\{T_i\\}_{i \\in I}$ of affine schemes with limit $T$ we have $F(T) = \\colim_i F(T_i)$. Let $X$ be a scheme over $S$, and let $h_X : (\\Sch/S)^{opp} \\to \\textit{Sets}$ be its functor of points, see Schemes, Section \\ref{schemes-section-representable}. In this terminology Proposition \\ref{proposition-characterize-locally-finite-presentation} says that a scheme $X$ is locally of finite presentation over $S$ if and only if $h_X$ is limit preserving."}
+{"_id": "15131", "title": "limits-remark-cannot-do-better", "text": "We cannot do better than this if we do not assume more on $S$ and the morphism $f : X \\to S$. For example, in general it will not be possible to find a {\\it closed} immersion $X \\to X'$ as in the lemma. The reason is that this would imply that $f$ is quasi-compact which may not be the case. An example is to take $S$ to be infinite dimensional affine space with $0$ doubled and $X$ to be one of the two infinite dimensional affine spaces."}
+{"_id": "15132", "title": "limits-remark-more-general-modification", "text": "The lemma above can be generalized as follows. Let $S$ be a scheme and let $T \\subset S$ be a closed subset. Assume there exists a cofinal system of open neighbourhoods $T \\subset W_i$ such that (1) $W_i \\setminus T$ is quasi-compact and (2) $W_i \\subset W_j$ is an affine morphism. Then $W = \\lim W_i$ is a scheme which contains $T$ as a closed subscheme. Set $U = X \\setminus T$ and $V = W \\setminus T$. Then the base change functor $$ \\left\\{ \\begin{matrix} f : X \\to S\\text{ of finite presentation} \\\\ f^{-1}(U) \\to U\\text{ is an isomorphism} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} g : Y \\to W\\text{ of finite presentation} \\\\ g^{-1}(V) \\to V\\text{ is an isomorphism} \\end{matrix} \\right\\} $$ is an equivalence of categories. If we ever need this we will change this remark into a lemma and provide a detailed proof."}
+{"_id": "15133", "title": "limits-remark-finite-type-gives-well-defined-system", "text": "In Situation \\ref{situation-limit-noetherian} Lemmas \\ref{lemma-good-diagram}, \\ref{lemma-limit-from-good-diagram}, and \\ref{lemma-morphism-good-diagram} tell us that the category of schemes quasi-separated and of finite type over $S$ is equivalent to certain types of inverse systems of schemes over $(S_i)_{i \\in I}$, namely the ones produced by applying Lemma \\ref{lemma-limit-from-good-diagram} to a diagram of the form (\\ref{equation-good-diagram}). For example, given $X \\to S$ finite type and quasi-separated if we choose two different diagrams $X \\to V_1 \\to S_{i_1}$ and $X \\to V_2 \\to S_{i_2}$ as in (\\ref{equation-good-diagram}), then applying Lemma \\ref{lemma-morphism-good-diagram} to $\\text{id}_X$ (in two directions) we see that the corresponding limit descriptions of $X$ are canonically isomorphic (up to shrinking the directed set $I$). And so on and so forth."}
+{"_id": "20", "title": "spaces-more-morphisms-lemma-flat-case", "text": "A flat monomorphism of algebraic spaces is representable by schemes."}
+{"_id": "22", "title": "spaces-more-morphisms-lemma-flat-surjective-monomorphism", "text": "A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism."}
+{"_id": "31", "title": "spaces-more-morphisms-lemma-match-modules-differentials", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $f_{small} : X_\\etale \\to Y_\\etale$ be the associated morphism of small \\'etale sites, see Descent, Remark \\ref{descent-remark-change-topologies-ringed}. Then there is a canonical isomorphism $$ (\\Omega_{X/Y})^a = \\Omega_{X_\\etale/Y_\\etale} $$ compatible with universal derivations. Here the first module is the sheaf on $X_\\etale$ associated to the quasi-coherent $\\mathcal{O}_X$-module $\\Omega_{X/Y}$, see Morphisms, Definition \\ref{morphisms-definition-sheaf-differentials}, and the second module is the one from Modules on Sites, Definition \\ref{sites-modules-definition-module-differentials}."}
+{"_id": "37", "title": "spaces-more-morphisms-lemma-immersion-differentials", "text": "Let $S$ be a scheme. If $X \\to Y$ is an immersion of algebraic spaces over $S$ then $\\Omega_{X/S}$ is zero."}
+{"_id": "79", "title": "spaces-more-morphisms-lemma-immersion-universal-thickening", "text": "Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Then \\begin{enumerate} \\item $i$ is formally unramified, \\item the universal first order thickening of $Z$ over $X$ is the first order infinitesimal neighbourhood of $Z$ in $X$ of Definition \\ref{definition-first-order-infinitesimal-neighbourhood}, \\item the conormal sheaf of $i$ in the sense of Definition \\ref{definition-conormal-sheaf} agrees with the conormal sheaf of $i$ in the sense of Definition \\ref{definition-universal-thickening}. \\end{enumerate}"}
+{"_id": "88", "title": "spaces-more-morphisms-lemma-transitivity-conormal-unramified", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be formally unramified morphisms of algebraic spaces over $S$. \\begin{enumerate} \\item If $Z \\subset Z'$ is the universal first order thickening of $Z$ over $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$ over $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is the universal first order thickening of $Z$ over $Y$. \\item There is a canonical exact sequence $$ i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ where the maps come from Lemma \\ref{lemma-universal-thickening-functorial} and $i : Z \\to Y$ is the first morphism. \\end{enumerate}"}
+{"_id": "94", "title": "spaces-more-morphisms-lemma-unramified-flat-formally-etale", "text": "An unramified flat morphism is formally \\'etale."}
+{"_id": "115", "title": "spaces-more-morphisms-lemma-descending-property-formally-smooth", "text": "The property $\\mathcal{P}(f) =$``$f$ is formally smooth'' is fpqc local on the base."}
+{"_id": "127", "title": "spaces-more-morphisms-lemma-NL-formally-etale", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is formally \\'etale, \\item $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$. \\end{enumerate}"}
+{"_id": "128", "title": "spaces-more-morphisms-lemma-NL-smooth", "text": "Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent \\begin{enumerate} \\item $f$ is smooth, and \\item $f$ is locally of finite presentation, $H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$ is finite locally free. \\end{enumerate}"}
+{"_id": "131", "title": "spaces-more-morphisms-lemma-morphism-between-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $Z$, \\item $X$ is flat over $Z$, \\item for every $z \\in |Z|$ the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$, and \\item $Y$ is locally of finite type over $Z$. \\end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$."}
+{"_id": "143", "title": "spaces-more-morphisms-lemma-flat-morphism-from-CM", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\\mathcal{O}_{Y, f(\\overline{x})}$ is Cohen-Macaulay for all $x \\in |X|$."}
+{"_id": "150", "title": "spaces-more-morphisms-lemma-flat-morphism-from-gorenstein", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$. If $X$ is Gorenstein, then $f$ is Gorenstein and $\\mathcal{O}_{Y, f(\\overline{x})}$ is Gorenstein for all $x \\in |X|$."}
+{"_id": "151", "title": "spaces-more-morphisms-lemma-base-change-gorenstein", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. Let $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$ be the base change of $f$. Let $x' \\in |X'|$ be a point with image $x \\in |X|$. \\begin{enumerate} \\item If $f$ is Gorenstein at $x$, then $f' : X' \\to Y'$ is Gorenstein at $x'$. \\item If $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at $x'$, then $f$ is Gorenstein at $x$. \\end{enumerate}"}
+{"_id": "166", "title": "spaces-more-morphisms-lemma-universally-catenary-dimension-function", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\\delta : |B| \\to \\mathbf{Z}$ be a function. Assume $B$ is decent, locally Noetherian, and universally catenary and $\\delta$ is a dimension function. If $X$ is a decent algebraic space over $B$ whose structure morphism $f : X \\to B$ is locally of finite type we define $\\delta_X : |X| \\to \\mathbf{Z}$ by the rule $$ \\delta_X(x) = \\delta(f(x)) + \\text{transcendence degreeof }x/f(x) $$ (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dimension-fibre}). Then $\\delta_X$ is a dimension function."}
+{"_id": "168", "title": "spaces-more-morphisms-lemma-etale-splits-off-just-one-quasi-finite-part", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $x \\in |X|$ with image $y \\in |Y|$. Assume that \\begin{enumerate} \\item $f$ is locally of finite type, \\item $f$ is separated, and \\item $f$ is quasi-finite at $x$. \\end{enumerate} Then there exists an \\'etale morphism $(U, u) \\to (Y, y)$ of pointed algebraic spaces and a decomposition $$ U \\times_Y X = W \\amalg V $$ into open and closed subspaces such that the morphism $V \\to U$ is finite and there exists a point $v \\in |V|$ which maps to $x$ in $|X|$ and $u$ in $|U|$."}
+{"_id": "210", "title": "spaces-more-morphisms-lemma-algebraize-formal-algebraic-space-finite-over-proper", "text": "Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X \\to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \\geq 1$ we set $X_n = X \\times_S S_n$. Suppose given a commutative diagram $$ \\xymatrix{ Y_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\ X_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots } $$ of algebraic spaces with cartesian squares. Assume that \\begin{enumerate} \\item $Y_1 \\to X_1$ is a finite morphism, and \\item $Y_1 \\to S_1$ is proper. \\end{enumerate} Then there exists a finite morphism of algebraic spaces $Y \\to X$ such that $Y_n = Y \\times_S S_n$ for all $n \\geq 1$. Moreover, $Y$ is proper over $S$."}
+{"_id": "217", "title": "spaces-more-morphisms-lemma-flat-base-change-regular-immersion", "text": "\\begin{slogan} Regular immersions are stable under flat base change. \\end{slogan} Let $S$ be a scheme. Let $i : Z \\to X$ be a Koszul-regular, $H_1$-regular, or quasi-regular immersion of algebraic spaces over $S$. Let $X' \\to X$ be a flat morphism of algebraic spaces over $S$. Then the base change $i' : Z \\times_X X' \\to X'$ is a Koszul-regular, $H_1$-regular, or quasi-regular immersion."}
+{"_id": "219", "title": "spaces-more-morphisms-lemma-transitivity-conormal-quasi-regular", "text": "Let $S$ be a scheme. Let $Z \\to Y \\to X$ be immersions of algebraic spaces over $S$. Assume that $Z \\to Y$ is $H_1$-regular. Then the canonical sequence of Lemma \\ref{lemma-transitivity-conormal} $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ is exact and (\\'etale) locally split."}
+{"_id": "220", "title": "spaces-more-morphisms-lemma-composition-regular-immersion", "text": "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of algebraic spaces over $S$. \\begin{enumerate} \\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$. \\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$. \\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular immersion, then $j \\circ i$ is a quasi-regular immersion. \\end{enumerate}"}
+{"_id": "221", "title": "spaces-more-morphisms-lemma-permanence-regular-immersion", "text": "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of algebraic spaces over $S$. Assume that the sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ of Lemma \\ref{lemma-transitivity-conormal} is exact and locally split. \\begin{enumerate} \\item If $j \\circ i$ is a quasi-regular immersion, so is $i$. \\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$. \\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$. \\end{enumerate}"}
+{"_id": "222", "title": "spaces-more-morphisms-lemma-extra-permanence-regular-immersion-noetherian", "text": "Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of algebraic spaces over $S$. Assume $X$ is locally Noetherian. The following are equivalent \\begin{enumerate} \\item $i$ and $j$ are Koszul regular immersions, \\item $i$ and $j \\circ i$ are Koszul regular immersions, \\item $j \\circ i$ is a Koszul regular immersion and the conormal sequence $$ 0 \\to i^*\\mathcal{C}_{Y/X} \\to \\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Y} \\to 0 $$ is exact and locally split. \\end{enumerate}"}
+{"_id": "226", "title": "spaces-more-morphisms-lemma-composition-pseudo-coherent", "text": "A composition of pseudo-coherent morphisms is pseudo-coherent."}
+{"_id": "228", "title": "spaces-more-morphisms-lemma-flat-finite-presentation-pseudo-coherent", "text": "A flat morphism which is locally of finite presentation is pseudo-coherent."}
+{"_id": "229", "title": "spaces-more-morphisms-lemma-permanence-pseudo-coherent", "text": "Let $f : X \\to Y$ be a morphism of algebraic spaces pseudo-coherent over a base algebraic space $B$. Then $f$ is pseudo-coherent."}
+{"_id": "231", "title": "spaces-more-morphisms-lemma-flat-base-change-perfect", "text": "A flat base change of a perfect morphism is perfect."}
+{"_id": "232", "title": "spaces-more-morphisms-lemma-composition-perfect", "text": "A composition of perfect morphisms is perfect."}
+{"_id": "235", "title": "spaces-more-morphisms-lemma-lci", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Let $P$ be an algebraic space smooth over $Y$. Let $U \\to X$ be an \\'etale morphism of algebraic spaces and let $i : U \\to P$ an immersion of algebraic spaces over $Y$. Picture: $$ \\xymatrix{ X \\ar[rd] & U \\ar[l] \\ar[d] \\ar[r]_i & P \\ar[ld] \\\\ & Y } $$ Then $i$ is a Koszul-regular immersion of algebraic spaces."}
+{"_id": "236", "title": "spaces-more-morphisms-lemma-lci-properties", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Then \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $f$ is pseudo-coherent, and \\item $f$ is perfect. \\end{enumerate}"}
+{"_id": "240", "title": "spaces-more-morphisms-lemma-regular-immersion-lci", "text": "Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces over $S$ is a local complete intersection morphism."}
+{"_id": "242", "title": "spaces-more-morphisms-lemma-descending-property-lci", "text": "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is fpqc local on the base."}
+{"_id": "243", "title": "spaces-more-morphisms-lemma-lci-syntomic-local-source", "text": "The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism'' is syntomic local on the source."}
+{"_id": "260", "title": "spaces-more-morphisms-lemma-triangulated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. The full subcategory of $D(\\mathcal{O}_X)$ consisting of $Y$-perfect objects is a saturated\\footnote{Derived Categories, Definition \\ref{derived-definition-saturated}.} triangulated subcategory."}
+{"_id": "356", "title": "algebra-lemma-characterize-finitely-presented-module-hom", "text": "Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $N$ is a finitely presented $R$-module, \\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map $\\colim \\Hom_R(N, M_i) \\to \\Hom_R(K, M)$ is bijective. \\end{enumerate}"}
+{"_id": "398", "title": "algebra-lemma-characterize-local-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. Assume $R$ and $S$ are local rings. The following are equivalent: \\begin{enumerate} \\item $\\varphi$ is a local ring map, \\item $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$, and \\item $\\varphi^{-1}(\\mathfrak m_S) = \\mathfrak m_R$. \\item For any $x \\in R$, if $\\varphi(x)$ is invertible in $S$, then $x$ is invertible in $R$. \\end{enumerate}"}
+{"_id": "400", "title": "algebra-lemma-surjective-on-spec-units", "text": "Let $\\varphi : R \\to S$ be a ring map such that the induced map $\\Spec(S) \\to \\Spec(R)$ is surjective. Then an element $x \\in R$ is a unit if and only if $\\varphi(x) \\in S$ is a unit."}
+{"_id": "408", "title": "algebra-lemma-closed-union-connected-components", "text": "Let $R$ be a ring. Let $T \\subset \\Spec(R)$ be a subset of the spectrum. The following are equivalent \\begin{enumerate} \\item $T$ is closed and is a union of connected components of $\\Spec(R)$, \\item $T$ is an intersection of open and closed subsets of $\\Spec(R)$, and \\item $T = V(I)$ where $I \\subset R$ is an ideal generated by idempotents. \\end{enumerate} Moreover, the ideal in (3) if it exists is unique."}
+{"_id": "430", "title": "algebra-lemma-primes-principal", "text": "Let $R$ be a ring. \\begin{enumerate} \\item An ideal $I \\subset R$ maximal with respect to not being principal is prime. \\item If every prime ideal of $R$ is principal, then every ideal of $R$ is principal. \\end{enumerate}"}
+{"_id": "465", "title": "algebra-lemma-invert-closed-split", "text": "Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset. Assume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$ is closed. If $R$ is Noetherian, or $\\Spec(R)$ is a Noetherian topological space, or $S$ is finitely generated as a monoid, then $R \\cong S^{-1}R \\times R'$ for some ring $R'$."}
+{"_id": "478", "title": "algebra-lemma-corollary-jacobson", "text": "Any finite type algebra over $\\mathbf{Z}$ is Jacobson."}
+{"_id": "601", "title": "algebra-lemma-geometrically-connected", "text": "Let $k$ be a field. Let $R$ be a $k$-algebra. The following are equivalent \\begin{enumerate} \\item for every field extension $k \\subset k'$ the spectrum of $R \\otimes_k k'$ is connected, and \\item for every finite separable field extension $k \\subset k'$ the spectrum of $R \\otimes_k k'$ is connected. \\end{enumerate}"}
+{"_id": "606", "title": "algebra-lemma-characterize-geometrically-integral", "text": "Let $k$ be a field. Let $S$ be a $k$-algebra. The following are equivalent \\begin{enumerate} \\item $S$ is geometrically integral over $k$, \\item for every finite extension $k'/k$ of fields the ring $S \\otimes_k k'$ is a domain, \\item $S \\otimes_k \\overline{k}$ is a domain where $\\overline{k}$ is the algebraic closure of $k$. \\end{enumerate}"}
+{"_id": "655", "title": "algebra-lemma-K0-and-K0prime-Artinian-local", "text": "Let $R$ be a local Artinian ring. There is a commutative diagram $$ \\xymatrix{ K_0(R) \\ar[rr] \\ar[d]_{\\text{rank}_R} & & K'_0(R) \\ar[d]^{\\text{length}_R} \\\\ \\mathbf{Z} \\ar[rr]^{\\text{length}_R(R)} & & \\mathbf{Z} } $$ where the vertical maps are isomorphisms by Lemmas \\ref{lemma-K0prime-Artinian} and \\ref{lemma-K0-local}."}
+{"_id": "658", "title": "algebra-lemma-integral-closure-graded", "text": "Let $R \\to S$ be a homomorphism of graded rings. Let $S' \\subset S$ be the integral closure of $R$ in $S$. Then $$ S' = \\bigoplus\\nolimits_{d \\geq 0} S' \\cap S_d, $$ i.e., $S'$ is a graded $R$-subalgebra of $S$."}
+{"_id": "684", "title": "algebra-lemma-minimal-over-r", "text": "Let $R$ be a Noetherian ring. Let $f_1, \\ldots, f_r \\in R$. \\begin{enumerate} \\item If $\\mathfrak p$ is minimal over $(f_1, \\ldots, f_r)$ then the height of $\\mathfrak p$ is $\\leq r$. \\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and $\\mathfrak q$ is minimal over $(\\mathfrak p, f_1, \\ldots, f_r)$, then every chain of primes between $\\mathfrak p$ and $\\mathfrak q$ has length at most $r$. \\end{enumerate}"}
+{"_id": "688", "title": "algebra-lemma-Noetherian-finite-nr-primes", "text": "A Noetherian ring with finitely many primes has dimension $\\leq 1$."}
+{"_id": "697", "title": "algebra-lemma-ses-dimension", "text": "Let $R$ be a Noetherian ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence of finite $R$-modules. Then $\\max\\{\\dim(\\text{Supp}(M')), \\dim(\\text{Supp}(M''))\\} = \\dim(\\text{Supp}(M))$."}
+{"_id": "730", "title": "algebra-lemma-weakly-ass-quotient-ring", "text": "Let $R$ be a ring. Let $I$ be an ideal. Let $M$ be an $R/I$-module. Via the canonical injection $\\Spec(R/I) \\to \\Spec(R)$ we have $\\text{WeakAss}_{R/I}(M) = \\text{WeakAss}_R(M)$."}
+{"_id": "784", "title": "algebra-lemma-tor-left-right", "text": "Let $R$ be a ring. For any $i \\geq 0$ the functors $\\text{Mod}_R \\times \\text{Mod}_R \\to \\text{Mod}_R$, $(M, N) \\mapsto \\text{Tor}_i^R(M, N)$ and $(M, N) \\mapsto \\text{Tor}_i^R(N, M)$ are canonically isomorphic."}
+{"_id": "838", "title": "algebra-lemma-quotient-module-ML", "text": "Let $M_1 \\to M_2 \\to M_3 \\to 0$ be an exact sequence of $R$-modules. If $M_1$ is finitely generated and $M_2$ is Mittag-Leffler, then $M_3$ is Mittag-Leffler."}
+{"_id": "861", "title": "algebra-lemma-hathat", "text": "\\begin{reference} Taken from an unpublished note of Lenstra and de Smit. \\end{reference} Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module. Denote $K_n = \\Ker(M^\\wedge \\to M/I^nM)$. Then $M^\\wedge$ is $I$-adically complete if and only if $K_n$ is equal to $I^nM^\\wedge$ for all $n \\geq 1$."}
+{"_id": "905", "title": "algebra-lemma-artinian-variant-local-criterion-flatness", "text": "Let $R$ be an Artinian local ring. Let $M$ be an $R$-module. Let $I \\subset R$ be a proper ideal. The following are equivalent \\begin{enumerate} \\item $M$ is flat over $R$, and \\item $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$. \\end{enumerate}"}
+{"_id": "910", "title": "algebra-lemma-exact-artinian-local", "text": "In Situation \\ref{situation-complex}. Let $R$ be a Artinian local ring. Suppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$ is exact at $R^{n_e}, \\ldots, R^{n_1}$. Then the complex is isomorphic to a direct sum of trivial complexes."}
+{"_id": "958", "title": "algebra-lemma-epimorphism-cardinality", "text": "Let $R \\to S$ be an epimorphism of rings. Then the cardinality of $S$ is at most the cardinality of $R$. In a formula: $|S| \\leq |R|$."}
+{"_id": "1012", "title": "algebra-lemma-dimension-graded", "text": "Let $k$ be a field. Let $S$ be a finitely generated graded algebra over $k$. Assume $S_0 = k$. Let $P(T) \\in \\mathbf{Q}[T]$ be the polynomial such that $\\dim(S_d) = P(d)$ for all $d \\gg 0$. See Proposition \\ref{proposition-graded-hilbert-polynomial}. Then \\begin{enumerate} \\item The irrelevant ideal $S_{+}$ is a maximal ideal $\\mathfrak m$. \\item Any minimal prime of $S$ is a homogeneous ideal and is contained in $S_{+} = \\mathfrak m$. \\item We have $\\dim(S) = \\deg(P) + 1 = \\dim_x\\Spec(S)$ (with the convention that $\\deg(0) = -1$) where $x$ is the point corresponding to the maximal ideal $S_{+} = \\mathfrak m$. \\item The Hilbert function of the local ring $R = S_{\\mathfrak m}$ is equal to the Hilbert function of $S$. \\end{enumerate}"}
+{"_id": "1104", "title": "algebra-lemma-limit-finite-type", "text": "Suppose $R \\to S$ is a ring map. Assume that $S$ is of finite type over $R$. Then there exists a directed set $(\\Lambda, \\leq)$, and a system of ring maps $R_\\lambda \\to S_\\lambda$ such that \\begin{enumerate} \\item The colimit of the system $R_\\lambda \\to S_\\lambda$ is equal to $R \\to S$. \\item Each $R_\\lambda$ is of finite type over $\\mathbf{Z}$. \\item Each $S_\\lambda$ is of finite type over $R_\\lambda$. \\item For each $\\lambda \\leq \\mu$ the map $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ presents $S_\\mu$ as a quotient of $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$. \\end{enumerate}"}
+{"_id": "1113", "title": "algebra-lemma-variant-local-criterion-flatness-general", "text": "Let $R \\to S$ be a local homomorphism of local rings. Let $I \\not = R$ be an ideal in $R$. Let $M$ be an $S$-module. Assume \\begin{enumerate} \\item $S$ is essentially of finite presentation over $R$, \\item $M$ is of finite presentation over $S$, \\item $\\text{Tor}_1^R(M, R/I) = 0$, and \\item $M/IM$ is flat over $R/I$. \\end{enumerate} Then $M$ is flat over $R$."}
+{"_id": "1158", "title": "algebra-lemma-NL-of-localization", "text": "If $S \\subset A$ is a multiplicative subset of $A$, then $\\NL_{S^{-1}A/A}$ is homotopy equivalent to the zero complex."}
+{"_id": "1175", "title": "algebra-lemma-syntomic-descends", "text": "\\begin{slogan} Being syntomic is fpqc local on the base. \\end{slogan} Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map. Set $S' = R'\\otimes_R S$. Then $R \\to S$ is syntomic if and only if $R' \\to S'$ is syntomic."}
+{"_id": "1190", "title": "algebra-lemma-localize-smooth", "text": "Let $R \\to S$ be a smooth ring map. Any localization $S_g$ is smooth over $R$. If $f \\in R$ maps to an invertible element of $S$, then $R_f \\to S$ is smooth."}
+{"_id": "1220", "title": "algebra-lemma-section-smooth", "text": "\\begin{slogan} If $R$ is a summand of $S$ and $S$ is smooth over $R$, then the $I$-adic completion of $S$ is often a power series over $R$ where $I$ is the kernel of the projection map from $S$ to $R$. \\end{slogan} Let $\\varphi : R \\to S$ be a smooth ring map. Let $\\sigma : S \\to R$ be a left inverse to $\\varphi$. Set $I = \\Ker(\\sigma)$. Then \\begin{enumerate} \\item $I/I^2$ is a finite locally free $R$-module, and \\item if $I/I^2$ is free, then $S^\\wedge \\cong R[[t_1, \\ldots, t_d]]$ as $R$-algebras, where $S^\\wedge$ is the $I$-adic completion of $S$. \\end{enumerate}"}
+{"_id": "1296", "title": "algebra-lemma-henselian-functorial-prepare", "text": "Let $R \\to S$ be a local map of local rings. Let $S \\to S^h$ be the henselization. Let $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$ be a prime of $A$ lying over $\\mathfrak m_R$ such that $R/\\mathfrak m_R \\cong \\kappa(\\mathfrak q)$. Then there exists a unique morphism of rings $f : A \\to S^h$ fitting into the commutative diagram $$ \\xymatrix{ A \\ar[r]_f & S^h \\\\ R \\ar[u] \\ar[r] & S \\ar[u] } $$ such that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$."}
+{"_id": "1597", "title": "moduli-curves-lemma-pre-genus-in-CM-1", "text": "We have $\\Curvesstack^{h0, 1} \\subset \\Curvesstack^{CM, 1}$ as open substacks of $\\Curvesstack$."}
+{"_id": "1612", "title": "moduli-curves-lemma-smooth-curves-smooth", "text": "The morphism $\\Curvesstack^{smooth} \\to \\Spec(\\mathbf{Z})$ is smooth."}
+{"_id": "1620", "title": "moduli-curves-lemma-gorenstein-dualizing", "text": "Let $X \\to S$ be a family of curves with Gorenstein fibres equidimensional of dimension $1$ (Lemma \\ref{lemma-gorenstein-1-curves}). Then the relative dualizing sheaf $\\omega_{X/S}$ is an invertible $\\mathcal{O}_X$-module whose formation commutes with arbitrary base change."}
+{"_id": "1626", "title": "moduli-curves-lemma-semistable-one-piece-per-genus", "text": "There is a decomposition into open and closed substacks $$ \\Curvesstack^{semistable} = \\coprod\\nolimits_{g \\geq 1} \\Curvesstack^{semistable}_g $$ where each $\\Curvesstack^{semistable}_g$ is characterized as follows: \\begin{enumerate} \\item given a family of curves $f : X \\to S$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $S \\to \\Curvesstack$ factors through $\\Curvesstack^{semistable}_g$, \\item $X \\to S$ is a semistable family of curves and $R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$, \\end{enumerate} \\item given $X$ a scheme proper over a field $k$ with $\\dim(X) \\leq 1$ the following are equivalent \\begin{enumerate} \\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors through $\\Curvesstack^{semistable}_g$, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$ has no rational tail, \\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $\\omega_{X_s}^{\\otimes m}$ is globally generated for $m \\geq 2$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "1627", "title": "moduli-curves-lemma-semistable-curves-smooth", "text": "The morphisms $\\Curvesstack^{semistable} \\to \\Spec(\\mathbf{Z})$ and $\\Curvesstack^{semistable}_g \\to \\Spec(\\mathbf{Z})$ are smooth."}
+{"_id": "1662", "title": "dpa-lemma-need-only-gamma-p", "text": "Let $p$ be a prime number. Let $A$ be a ring such that every integer $n$ not divisible by $p$ is invertible, i.e., $A$ is a $\\mathbf{Z}_{(p)}$-algebra. Let $I \\subset A$ be an ideal. Two divided power structures $\\gamma, \\gamma'$ on $I$ are equal if and only if $\\gamma_p = \\gamma'_p$. Moreover, given a map $\\delta : I \\to I$ such that \\begin{enumerate} \\item $p!\\delta(x) = x^p$ for all $x \\in I$, \\item $\\delta(ax) = a^p\\delta(x)$ for all $a \\in A$, $x \\in I$, and \\item $\\delta(x + y) = \\delta(x) + \\sum\\nolimits_{i + j = p, i,j \\geq 1} \\frac{1}{i!j!} x^i y^j + \\delta(y)$ for all $x, y \\in I$, \\end{enumerate} then there exists a unique divided power structure $\\gamma$ on $I$ such that $\\gamma_p = \\delta$."}
+{"_id": "1680", "title": "dpa-lemma-check-lci-at-maximal-ideals", "text": "Let $A$ be a Noetherian ring. Then $A$ is a local complete intersection if and only if $A_\\mathfrak m$ is a complete intersection for every maximal ideal $\\mathfrak m$ of $A$."}
+{"_id": "1686", "title": "dpa-lemma-avramov-map-finite-type", "text": "Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map such that the image of $\\Spec(B) \\to \\Spec(A)$ contains all closed points of $\\Spec(A)$. Then the following are equivalent \\begin{enumerate} \\item $B$ is a complete intersection and $A \\to B$ has finite tor dimension, \\item $A$ is a complete intersection and $A \\to B$ is a local complete intersection in the sense of More on Algebra, Definition \\ref{more-algebra-definition-local-complete-intersection}. \\end{enumerate}"}
+{"_id": "1708", "title": "moduli-lemma-coherent-existence-part", "text": "Assume $X \\to B$ is proper as well as of finite presentation. Then $\\Cohstack_{X/B} \\to B$ satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-existence})."}
+{"_id": "1711", "title": "moduli-lemma-coherent-closed", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Then the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of Lemma \\ref{lemma-coherent-functorial} is a closed immersion."}
+{"_id": "1729", "title": "moduli-lemma-hilb-s-lfp", "text": "The morphism $\\Hilbfunctor_{X/B} \\to B$ is separated and locally of finite presentation."}
+{"_id": "1730", "title": "moduli-lemma-hilb-existence-part", "text": "Assume $X \\to B$ is proper as well as of finite presentation. Then $\\Hilbfunctor_{X/B} \\to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-valuative-criterion})."}
+{"_id": "1732", "title": "moduli-lemma-hilb-closed", "text": "Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\\pi$ induces a closed immersion $\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$."}
+{"_id": "1761", "title": "derived-lemma-third-map-idempotent", "text": "Let $\\mathcal{D}$ be a pre-triangulated category. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle. If $$ \\xymatrix{ Z \\ar[r]_h \\ar[d]_c & X[1] \\ar[d]^{a[1]} \\\\ Z \\ar[r]^h & X[1] } $$ is commutative and $a^2 = a$, $c^2 = c$, then there exists a morphism $b : Y \\to Y$ with $b^2 = b$ such that $(a, b, c)$ is an endomorphism of the triangle $(X, Y, Z, f, g, h)$."}
+{"_id": "1830", "title": "derived-lemma-direct-sum-computes", "text": "Assumptions and notation as in Situation \\ref{situation-derived-functor}. Let $X, Y$ be objects of $\\mathcal{D}$. If $X \\oplus Y$ computes $RF$, then $X$ and $Y$ compute $RF$. Similarly for $LF$."}
+{"_id": "1834", "title": "derived-lemma-irrelevant", "text": "In Situation \\ref{situation-classical}. \\begin{enumerate} \\item Let $X$ be an object of $K^{+}(\\mathcal{A})$. The right derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$ is defined at $X$ if and only if the right derived functor of $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic. \\item Let $X$ be an object of $K^{+}(\\mathcal{A})$. Then $X$ computes the right derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$ if and only if $X$ computes the right derived functor of $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$. \\item Let $X$ be an object of $K^{-}(\\mathcal{A})$. The left derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$ is defined at $X$ if and only if the left derived functor of $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic. \\item Let $X$ be an object of $K^{-}(\\mathcal{A})$. Then $X$ computes the left derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$ if and only if $X$ computes the left derived functor of $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$. \\end{enumerate}"}
+{"_id": "1857", "title": "derived-lemma-cohomology-bounded-above", "text": "Let $\\mathcal{A}$ be an abelian category. Let $K^\\bullet$ be a complex of $\\mathcal{A}$. \\begin{enumerate} \\item If $K^\\bullet$ has a projective resolution then $H^n(K^\\bullet) = 0$ for $n \\gg 0$. \\item If $H^n(K^\\bullet) = 0$ for $n \\gg 0$ then there exists a quasi-isomorphism $L^\\bullet \\to K^\\bullet$ with $L^\\bullet$ bounded above. \\end{enumerate}"}
+{"_id": "1890", "title": "derived-lemma-filtered-localization-functor", "text": "Let $\\mathcal{A}$ be an abelian category with enough injectives. Let $\\mathcal{I}^f \\subset \\text{Fil}^f(\\mathcal{A})$ denote the strictly full additive subcategory whose objects are the filtered injective objects. The canonical functor $$ K^{+}(\\mathcal{I}^f) \\longrightarrow DF^{+}(\\mathcal{A}) $$ is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Furthermore the diagrams $$ \\xymatrix{ K^{+}(\\mathcal{I}^f) \\ar[d]_{\\text{gr}^p} \\ar[r] & DF^{+}(\\mathcal{A}) \\ar[d]_{\\text{gr}^p} \\\\ K^{+}(\\mathcal{I}) \\ar[r] & D^{+}(\\mathcal{A}) } \\quad \\xymatrix{ K^{+}(\\mathcal{I}^f) \\ar[d]^{\\text{forget }F} \\ar[r] & DF^{+}(\\mathcal{A}) \\ar[d]^{\\text{forget }F} \\\\ K^{+}(\\mathcal{I}) \\ar[r] & D^{+}(\\mathcal{A}) } $$ are commutative, where $\\mathcal{I} \\subset \\mathcal{A}$ is the strictly full additive subcategory whose objects are the injective objects."}
+{"_id": "1913", "title": "derived-lemma-enough-K-injectives-implies", "text": "Let $\\mathcal{A}$ be an abelian category. Assume every complex has a quasi-isomorphism towards a K-injective complex. Then any exact functor $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ of triangulated categories has a right derived functor $$ RF : D(\\mathcal{A}) \\longrightarrow \\mathcal{D}' $$ and $RF(I^\\bullet) = F(I^\\bullet)$ for K-injective complexes $I^\\bullet$."}
+{"_id": "1938", "title": "derived-lemma-classical-generator-generator", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\\mathcal{D}$. If $E$ is a classical generator of $\\mathcal{D}$, then $E$ is a generator."}
+{"_id": "1963", "title": "derived-proposition-derived-category", "text": "Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough injectives. Denote $\\mathcal{I} \\subset \\mathcal{A}$ the strictly full additive subcategory whose objects are the injective objects of $\\mathcal{A}$. The functor $$ K^{+}(\\mathcal{I}) \\longrightarrow D^{+}(\\mathcal{A}) $$ is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories."}
+{"_id": "2087", "title": "cohomology-lemma-vanishing-generated-one-section", "text": "\\begin{reference} This is a special case of \\cite[Proposition 3.6.1]{Tohoku}. \\end{reference} Let $X$ be a topological space. Let $d \\geq 0$ be an integer. Assume \\begin{enumerate} \\item $X$ is quasi-compact, \\item the quasi-compact opens form a basis for $X$, and \\item the intersection of two quasi-compact opens is quasi-compact. \\item $H^p(X, j_!\\underline{\\mathbf{Z}}_U) = 0$ for all $p > d$ and any quasi-compact open $j : U \\to X$. \\end{enumerate} Then $H^p(X, \\mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\\mathcal{F}$ on $X$."}
+{"_id": "2091", "title": "cohomology-lemma-proper-base-change-spectral", "text": "Let $f : X \\to Y$ be a spectral map of spectral spaces. Let $y \\in Y$. Let $E \\subset Y$ be the set of points specializing to $y$. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Then $(R^pf_*\\mathcal{F})_y = H^p(f^{-1}(E), \\mathcal{F}|_{f^{-1}(E)})$."}
+{"_id": "2107", "title": "cohomology-lemma-K-flat-two-out-of-three-ses", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes such that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}_X$-modules. If two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third."}
+{"_id": "2171", "title": "cohomology-lemma-inverse-limit-complexes", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n^\\bullet)$ be an inverse system of complexes of $\\mathcal{O}_X$-modules. Let $m \\in \\mathbf{Z}$. Assume there exist a set $\\mathcal{B}$ of open subsets of $X$ and an integer $n_0$ such that \\begin{enumerate} \\item every open in $X$ has a covering whose members are elements of $\\mathcal{B}$, \\item for every $U \\in \\mathcal{B}$ \\begin{enumerate} \\item the systems of abelian groups $\\mathcal{F}_n^{m - 2}(U)$ and $\\mathcal{F}_n^{m - 1}(U)$ have vanishing $R^1\\lim$ (for example these have the Mittag-Leffler condition), \\item the system of abelian groups $H^{m - 1}(\\mathcal{F}_n^\\bullet(U))$ has vanishing $R^1\\lim$ (for example it has the Mittag-Leffler condition), and \\item we have $H^m(\\mathcal{F}_n^\\bullet(U)) = H^m(\\mathcal{F}_{n_0}^\\bullet(U))$ for all $n \\geq n_0$. \\end{enumerate} \\end{enumerate} Then the maps $H^m(\\mathcal{F}^\\bullet) \\to \\lim H^m(\\mathcal{F}_n^\\bullet) \\to H^m(\\mathcal{F}_{n_0}^\\bullet)$ are isomorphisms of sheaves where $\\mathcal{F}^\\bullet = \\lim \\mathcal{F}_n^\\bullet$ is the termwise inverse limit."}
+{"_id": "2180", "title": "cohomology-lemma-RHom-well-defined", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$ be a quasi-isomorphism of K-injective complexes of $\\mathcal{O}_X$-modules. Let $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$ be a quasi-isomorphism of complexes of $\\mathcal{O}_X$-modules. Then $$ \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet) \\longrightarrow \\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet) $$ is a quasi-isomorphism."}
+{"_id": "2213", "title": "cohomology-lemma-n-pseudo-module", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is $0$-pseudo-coherent if and only if $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and \\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is $(-1)$-pseudo-coherent if and only if $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation. \\end{enumerate}"}
+{"_id": "2227", "title": "cohomology-lemma-tensor-perfect", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. If $K, L$ are perfect objects of $D(\\mathcal{O}_X)$, then so is $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$."}
+{"_id": "2231", "title": "cohomology-lemma-left-dual-complex", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules. If $\\mathcal{F}^\\bullet$ has a left dual in the monoidal category of complexes of $\\mathcal{O}_X$-modules (Categories, Definition \\ref{categories-definition-dual}) then $\\mathcal{F}^\\bullet$ is a locally bounded complex whose terms are locally direct summands of finite free $\\mathcal{O}_X$-modules and the left dual is as constructed in Example \\ref{example-dual}."}
+{"_id": "2241", "title": "cohomology-lemma-perfect-is-compact", "text": "Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties: \\begin{enumerate} \\item $X$ is quasi-compact, \\item there exists a basis of quasi-compact open subsets, and \\item the intersection of any two quasi-compact opens is quasi-compact. \\end{enumerate} Let $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then \\begin{enumerate} \\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O}_X)$ in the following sense: if $M = \\bigoplus_{i \\in I} M_i$ is bounded below, then $\\Hom(K, M) = \\bigoplus_{i \\in I} \\Hom(K, M_i)$. \\item[(b)] If $X$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$, then $K$ is a compact object of $D(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "2329", "title": "restricted-lemma-base-change-finite-type-red", "text": "Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in Lemma \\ref{lemma-finite-type-red}. Then $P$ is stable under base change (Formal Spaces, Situation \\ref{formal-spaces-situation-base-change-local-property})."}
+{"_id": "2346", "title": "restricted-lemma-rig-closed-point-in-image", "text": "Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\\mathfrak p \\subset A$ be rig-closed. Let $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ be the ideals of topologically nilpotent elements. If $\\varphi$ is flat, then the following are equivalent \\begin{enumerate} \\item the maximal ideal of $A/\\mathfrak p$ is in the image of $\\Spec(B/\\mathfrak b) \\to \\Spec(A/\\mathfrak a)$, \\item there exists a rig-closed prime ideal $\\mathfrak q \\subset B$ such that $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$. \\end{enumerate} and if so then $\\varphi$, $\\mathfrak p$, and $\\mathfrak q$ satisfy the conclusions of Lemma \\ref{lemma-rig-closed-point-relative}."}
+{"_id": "2359", "title": "restricted-lemma-rig-flat-morphisms", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is rig-flat, \\item for every commutative diagram $$ \\xymatrix{ U \\ar[d] \\ar[r] & V \\ar[d] \\\\ X \\ar[r] & Y } $$ with $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$ representable by algebraic spaces and \\'etale, the morphism $U \\to V$ corresponds to a rig-flat map in $\\textit{WAdm}^{Noeth}$, \\item there exists a covering $\\{Y_j \\to Y\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} such that each $X_{ji} \\to Y_j$  corresponds to a rig-flat map in $\\textit{WAdm}^{Noeth}$, and \\item there exist a covering $\\{X_i \\to X\\}$ as in Formal Spaces, Definition \\ref{formal-spaces-definition-formal-algebraic-space} and for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \\to Y$ is representable by algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds to a rig-flat map in $\\textit{WAdm}^{Noeth}$. \\end{enumerate}"}
+{"_id": "2360", "title": "restricted-lemma-base-change-rig-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-flat and $g$ is locally of finite type, then the base change $X \\times_Y Z \\to Z$ is rig-flat."}
+{"_id": "2361", "title": "restricted-lemma-composition-rig-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are rig-flat, then so is $g \\circ f$."}
+{"_id": "2371", "title": "restricted-lemma-rig-smooth-rig-flat-morphism", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. If $f$ is rig-smooth, then $f$ is rig-flat."}
+{"_id": "2378", "title": "restricted-lemma-rig-etale-finite-type", "text": "A rig-\\'etale morphism of locally Noetherian formal algebraic spaces is locally of finite type."}
+{"_id": "2429", "title": "restricted-lemma-Noetherian-local-ring-properties", "text": "Notation and assumptions as in Lemma \\ref{lemma-Noetherian-local-ring}. Let $f : X' \\to \\Spec(A)$ correspond to $g : Y' \\to \\Spec(B)$ via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if $g$ is so."}
+{"_id": "2431", "title": "restricted-lemma-dominate-by-admissible-blowup", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $f : X \\to S$ be an object of (\\ref{equation-modification}). Then there exists a $U$-admissible blowup $S' \\to S$ which dominates $X$."}
+{"_id": "2474", "title": "more-groupoids-lemma-groupoid-on-field-translate-open", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $U$ is the spectrum of a field, $W \\subset R$ is open, and $Z \\to R$ is a morphism of schemes, then the image of the composition $Z \\times_{s, U, t} W \\to R \\times_{s, U, t} R \\to R$ is open."}
+{"_id": "2478", "title": "more-groupoids-lemma-groupoid-characteristic-zero-smooth", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \\begin{enumerate} \\item $U = \\Spec(k)$ with $k$ a field, \\item $s, t$ are locally of finite type, and \\item the characteristic of $k$ is zero. \\end{enumerate} Then $s, t : R \\to U$ are smooth."}
+{"_id": "2479", "title": "more-groupoids-lemma-reduced-group-scheme-perfect-field-characteristic-p-smooth", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume \\begin{enumerate} \\item $U = \\Spec(k)$ with $k$ a field, \\item $s, t$ are locally of finite type, \\item $R$ is reduced, and \\item $k$ is perfect. \\end{enumerate} Then $s, t : R \\to U$ are smooth."}
+{"_id": "2487", "title": "more-groupoids-lemma-quasi-compact-groupoid-on-field-image", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field and assume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact). Let $Z \\subset U \\times_S U$ be the scheme theoretic image (see Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}) of $j = (t, s) : R \\to U \\times_S U$. Then $$ (U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z, \\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z}) $$ is a groupoid scheme over $S$."}
+{"_id": "2517", "title": "examples-lemma-countable-coherent", "text": "Let $R$ be a countable ring. Then $R$ is coherent if and only if $R^\\mathbf{N}$ is a flat $R$-module."}
+{"_id": "2519", "title": "examples-lemma-almost-integral-when-powerseries-flat", "text": "Let $R$ be a domain with fraction field $K$. If $R[[x]]$ is flat over $R[x]$, then $R$ is normal if and only if $R$ is completely normal (Algebra, Definition \\ref{algebra-definition-almost-integral})."}
+{"_id": "2520", "title": "examples-lemma-completion-polynomial-ring-not-flat-bis", "text": "If $R$ is a valuation ring of dimension $> 1$, then $R[[x]]$ is flat over $R$ but not flat over $R[x]$."}
+{"_id": "2535", "title": "examples-lemma-ideal-generated-by-idempotents-projective", "text": "Let $R$ be a ring. Let $I \\subset R$ be an ideal generated by a countable collection of idempotents. Then $I$ is projective as an $R$-module."}
+{"_id": "2550", "title": "examples-lemma-excellent-regular-local-rings", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of characteristic $p > 0$. Suppose $[\\kappa : \\kappa^p] < \\infty$. Then $A$ is excellent if and only if $A \\to A^\\wedge$ is formally \\'etale."}
+{"_id": "2569", "title": "examples-lemma-sum-is-product", "text": "Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{F}_n$ is a collection of locally quasi-coherent sheaves with the flat base change property on $\\mathcal{X}$, then $\\oplus_n \\mathcal{F}_n[n] \\to \\prod_n \\mathcal{F}_n[n]$ is an isomorphism in $D(\\mathcal{O}_\\mathcal{X})$."}
+{"_id": "2573", "title": "examples-lemma-nonadditivity-of-trace", "text": "There exists a ring $R$, a distinguished triangle $(K, L, M, \\alpha, \\beta, \\gamma)$ in the homotopy category $K(R)$, and an endomorphism $(a, b, c)$ of this distinguished triangle, such that $K$, $L$, $M$ are perfect complexes and $\\text{Tr}_K(a) + \\text{Tr}_M(c) \\not = \\text{Tr}_L(b)$."}
+{"_id": "2579", "title": "examples-lemma-proper-spaces-not-algebraic", "text": "The stack in groupoids $$ p'_{fp, flat, proper} : \\Spacesstack'_{fp, flat, proper} \\longrightarrow \\Sch_{fppf} $$ whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$ (see Quot, Section \\ref{quot-section-stack-of-spaces}) is not an algebraic stack."}
+{"_id": "2592", "title": "examples-lemma-Z-not-quasi-compact", "text": "The scheme $\\Spec(\\mathbf{Z})$ is not quasi-compact in the canonical topology on the category of schemes."}
+{"_id": "2606", "title": "bootstrap-lemma-representable-by-spaces-transformation-diagonal", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Let $a : F \\to G$ be representable by algebraic spaces. Then $\\Delta_{F/G} : F \\to F \\times_G F$ is representable by algebraic spaces."}
+{"_id": "2636", "title": "bootstrap-lemma-spaces-smooth-locally-representable", "text": "Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object of $\\Sch_\\alpha$. Let $$ F : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets} $$ be a presheaf with the following properties: \\begin{enumerate} \\item $F$ is a sheaf for the \\'etale topology, \\item there exists an algebraic space $U$ over $S$ and a map $U \\to F$ which is representable by algebraic spaces, surjective, and smooth. \\end{enumerate} Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}."}
+{"_id": "2685", "title": "spaces-perfect-lemma-direct-image-coherator", "text": "Let $S$ be a scheme and let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ are quasi-compact and have affine diagonal. Then, denoting $$ \\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y)) $$ the right derived functor of $f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ the diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ is commutative."}
+{"_id": "2688", "title": "spaces-perfect-lemma-flat-pullback-injective-quasi-coherent", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over $S$. If $\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$, then $f_*\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "2714", "title": "spaces-perfect-lemma-pseudo-coherent-hocolim-with-support", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is quasi-compact. Let $K \\in D(\\mathcal{O}_X)$ supported on $T$. The following are equivalent \\begin{enumerate} \\item $K$ is pseudo-coherent, and \\item $K = \\text{hocolim} K_n$ where $K_n$ is perfect, supported on $T$, and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$. \\end{enumerate}"}
+{"_id": "2745", "title": "spaces-perfect-lemma-chi-locally-constant", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \\in D(\\mathcal{O}_X)$ be perfect. The function $$ \\chi_E : |X| \\longrightarrow \\mathbf{Z},\\quad x \\longmapsto \\sum (-1)^i \\beta_i(x) $$ is locally constant on $X$."}
+{"_id": "2784", "title": "dualizing-lemma-product-injectives", "text": "Let $R$ be a ring. Any product of injective $R$-modules is injective."}
+{"_id": "2794", "title": "dualizing-lemma-projective-covers-local", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Any finite $R$-module has a projective cover."}
+{"_id": "2795", "title": "dualizing-lemma-injective-hull", "text": "Let $R$ be a ring. Any $R$-module has an injective hull."}
+{"_id": "2893", "title": "dualizing-lemma-formal-fibres-lci", "text": "Properties (A), (B), (C), (D), and (E) of More on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres} hold for $P(k \\to R) =$``$R$ is a local complete intersection''. See Divided Power Algebra, Definition \\ref{dpa-definition-lci}."}
+{"_id": "2984", "title": "properties-lemma-catenary-dimension-function", "text": "Let $S$ be a locally Noetherian scheme. The following are equivalent: \\begin{enumerate} \\item $S$ is catenary, and \\item locally in the Zariski topology there exists a dimension function on $S$ (see Topology, Definition \\ref{topology-definition-dimension-function}). \\end{enumerate}"}
+{"_id": "2986", "title": "properties-lemma-scheme-regular-iff-all-Rk", "text": "Let $X$ be a locally Noetherian scheme. Then $X$ is regular if and only if $X$ has $(R_k)$ for all $k \\geq 0$."}
+{"_id": "2997", "title": "properties-lemma-nagata-universally-Japanese", "text": "Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item $X$ is Nagata, and \\item $X$ is locally Noetherian and universally Japanese. \\end{enumerate}"}
+{"_id": "2999", "title": "properties-lemma-geometrically-unibranch", "text": "\\begin{reference} Compare with \\cite[Proposition 2.3]{Etale-coverings} \\end{reference} Let $X$ be a Noetherian scheme. The following are equivalent \\begin{enumerate} \\item $X$ is geometrically unibranch (Definition \\ref{definition-unibranch}), \\item for every point $x \\in X$ which is not the generic point of an irreducible component of $X$, the punctured spectrum of the strict henselization $\\mathcal{O}_{X, x}^{sh}$ is connected. \\end{enumerate}"}
+{"_id": "3006", "title": "properties-lemma-section-maps-backwards", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$ be a section. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $X$ is quasi-compact and $\\mathcal{F}$ is of finite type, then (\\ref{equation-hom-invert-s}) is injective, and \\item if $X$ is quasi-compact and quasi-separated and $\\mathcal{F}$ is of finite presentation, then (\\ref{equation-hom-invert-s}) is bijective. \\end{enumerate}"}
+{"_id": "3064", "title": "properties-lemma-quasi-affine-invertible-nonvanishing-section", "text": "Let $X$ be a quasi-affine scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $E \\subset W \\subset X$ with $E$ finite and $W$ open. Then there exists an $s \\in \\Gamma(X, \\mathcal{L})$ such that $X_s$ is affine and $E \\subset X_s \\subset W$."}
+{"_id": "3096", "title": "criteria-lemma-stack-in-setoids-descent", "text": "Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be stacks in groupoids over $(\\Sch/S)_{fppf}$. Suppose that $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Z} \\to \\mathcal{Y}$ are $1$-morphisms. If \\begin{enumerate} \\item $\\mathcal{Y}$, $\\mathcal{Z}$ are representable by algebraic spaces $Y$, $Z$ over $S$, \\item the associated morphism of algebraic spaces $Y \\to Z$ is surjective, flat and locally of finite presentation, and \\item $\\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{X}$ is a stack in setoids, \\end{enumerate} then $\\mathcal{X}$ is a stack in setoids."}
+{"_id": "3121", "title": "criteria-lemma-etale-covering-restriction-of-scalars", "text": "Let $S$ be a scheme. Let $X' \\to X \\to Z \\to B$ be morphisms of algebraic spaces over $S$. Assume \\begin{enumerate} \\item $X' \\to X$ is \\'etale, and \\item $Z \\to B$ is finite locally free. \\end{enumerate} Then $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is representable by algebraic spaces and \\'etale. If $X' \\to X$ is also surjective, then $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is surjective."}
+{"_id": "3126", "title": "criteria-lemma-etale-map-hilbert", "text": "In the situation of Lemma \\ref{lemma-map-hilbert}. Assume that $G$, $H$ are representable by algebraic spaces and \\'etale. Then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by algebraic spaces and \\'etale. If also $H$ is surjective and the induced functor $\\mathcal{X}' \\to \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$ is surjective, then $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective."}
+{"_id": "3148", "title": "quot-theorem-polarized-algebraic", "text": "The stack $\\Polarizedstack$ (Situation \\ref{situation-polarized}) is algebraic. In fact, for any algebraic space $B$ the stack $B\\textit{-Polarized}$ (Remark \\ref{remark-polarized-base-change}) is algebraic."}
+{"_id": "3149", "title": "quot-theorem-curves-algebraic", "text": "\\begin{reference} See \\cite[Proposition 3.3, page 8]{dJHS} and \\cite[Appendix B by Jack Hall, Theorem B.1]{Smyth}. \\end{reference} The stack $\\Curvesstack$ (Situation \\ref{situation-curves}) is algebraic. In fact, for any algebraic space $B$ the stack $B\\text{-}\\Curvesstack$ (Remark \\ref{remark-curves-base-change}) is algebraic."}
+{"_id": "3150", "title": "quot-theorem-complexes-algebraic", "text": "\\begin{reference} \\cite{lieblich-complexes} \\end{reference} Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is proper, flat, and of finite presentation. Then $\\Complexesstack_{X/B}$ is an algebraic stack over $S$."}
+{"_id": "3154", "title": "quot-lemma-hom-limits", "text": "In Situation \\ref{situation-hom}. If $\\mathcal{F}$ is of finite presentation and $f$ is quasi-compact and quasi-separated, then $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is limit preserving."}
+{"_id": "3158", "title": "quot-lemma-isom-sheaf", "text": "In Situation \\ref{situation-hom} the functor $\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$  satisfies the sheaf property for the fpqc topology."}
+{"_id": "3171", "title": "quot-lemma-q-sheaf-in-X", "text": "In Situation \\ref{situation-q} let $\\{X_i \\to X\\}_{i \\in I}$ be an fpqc covering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$ be an fpqc covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$ the pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. For every scheme $T$ over $B$ the diagram $$ \\xymatrix{ Q_{\\mathcal{F}/X/B}(T) \\ar[r] & \\prod\\nolimits_i Q_{\\mathcal{F}_i/X_i/B}(T) \\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*} & \\prod\\nolimits_{i, j, k} Q_{\\mathcal{F}_{ijk}/X_{ijk}/B}(T) } $$ presents the first arrow as the equalizer of the other two. The same is true for the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$."}
+{"_id": "3182", "title": "quot-lemma-pic-with-section-stack", "text": "In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section. Then $\\Picardstack_{X/B, \\sigma}$ as defined above is a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "3207", "title": "quot-lemma-curves-fibred-in-groupoids", "text": "The category $\\Curvesstack$ is fibred in groupoids over $\\Sch_{fppf}$."}
+{"_id": "3208", "title": "quot-lemma-curves-stack", "text": "The category $\\Curvesstack$ is a stack in groupoids over $\\Sch_{fppf}$."}
+{"_id": "3212", "title": "quot-lemma-curves-tangent-space", "text": "Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of $\\mathcal{X} = \\Curvesstack$ over $\\Spec(k)$. \\begin{enumerate} \\item If $k$ is of finite type over $\\mathbf{Z}$, then the vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$ (see Artin's Axioms, Section \\ref{artin-section-tangent-spaces}) are finite dimensional, and \\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$ (see Artin's Axioms, Section \\ref{artin-section-inf}) are finite dimensional. \\end{enumerate}"}
+{"_id": "3250", "title": "spaces-more-cohomology-lemma-h0-proper-over-henselian-local", "text": "Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \\to \\Spec(A)$ be a proper morphism. Let $X_0 \\subset X$ be the fibre of $f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we have $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$."}
+{"_id": "3252", "title": "spaces-more-cohomology-lemma-proper-pushforward-stalk", "text": "Let $S$ be a scheme. Let $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$. Let $\\overline{x} \\to X$ be a geometric point. For any sheaf $\\mathcal{F}$ on $Y_\\etale$ the canonical map $$ (f_*\\mathcal{F})_{\\overline{x}} \\longrightarrow \\Gamma(Y_{\\overline{x}}, \\mathcal{F}_{\\overline{x}}) $$ is bijective."}
+{"_id": "3254", "title": "spaces-more-cohomology-lemma-proper-base-change-stalk", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic spaces. Let $\\overline{y} \\to Y$ be a geometric point. \\begin{enumerate} \\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $(R^nf_*\\mathcal{F})_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$. \\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have $(R^nf_*E)_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, E_{\\overline{y}})$. \\end{enumerate}"}
+{"_id": "3255", "title": "spaces-more-cohomology-lemma-base-change-separably-closed", "text": "Let $k \\subset k'$ be an extension of separably closed fields. Let $X$ be a proper algebraic space over $k$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^q_\\etale(X, \\mathcal{F}) \\to H^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \\geq 0$."}
+{"_id": "3264", "title": "spaces-more-cohomology-lemma-compare-cohomology-etale-fppf", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above: \\begin{enumerate} \\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$ for an abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$. \\end{enumerate} Example: if $A$ is an abelian group, then $H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$."}
+{"_id": "3267", "title": "spaces-more-cohomology-lemma-finite-push-pull-fppf-etale", "text": "In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is finite, then $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\\etale)$."}
+{"_id": "3272", "title": "spaces-more-cohomology-lemma-cohomological-descent-etale-fppf-modules-unbounded", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \\in D_\\QCoh(\\mathcal{O}_X)$ the maps $$ L\\pi_X^*K \\longrightarrow R\\epsilon_{X, *}(La_X^*\\mathcal{F}) \\quad\\text{and}\\quad K \\longrightarrow Ra_{X, *}(La_X^*K) $$ are isomorphisms. Here $a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ is as above."}
+{"_id": "3275", "title": "spaces-more-cohomology-lemma-compare-cohomology-etale-ph", "text": "Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$ as above: \\begin{enumerate} \\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{ph}(X, a_X^{-1}\\mathcal{F})$ for a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $H^q(X_\\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves \\end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\\etale(X, \\underline{A}) = H^q_{ph}(X, \\underline{A})$."}
+{"_id": "3325", "title": "coherent-lemma-hom-into-S2", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{G}$ has property $(S_1)$, then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_1)$. \\item If $\\mathcal{G}$ has property $(S_2)$, then $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_2)$. \\end{enumerate}"}
+{"_id": "3358", "title": "coherent-lemma-cohomology-powers-ideal-times-sheaf", "text": "Given a morphism of schemes $f : X \\to Y$, a quasi-coherent sheaf $\\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Y$. Assume $Y$ locally Noetherian, $f$ proper, and $\\mathcal{F}$ coherent. Then $$ \\mathcal{M} = \\bigoplus\\nolimits_{n \\geq 0} R^pf_*(\\mathcal{I}^n\\mathcal{F}) $$ is a graded $\\mathcal{A} = \\bigoplus_{n \\geq 0} \\mathcal{I}^n$-module which is quasi-coherent and of finite type."}
+{"_id": "3393", "title": "coherent-lemma-cat-module-support-proper-over-base", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$, $\\mathcal{G}$ be finite type, quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$ are proper over $S$, then the same is true for $\\mathcal{F} \\oplus \\mathcal{G}$, for any extension of $\\mathcal{G}$ by $\\mathcal{F}$, for $\\Im(u)$ and $\\Coker(u)$ given any $\\mathcal{O}_X$-module map $u : \\mathcal{F} \\to \\mathcal{G}$, and for any quasi-coherent quotient of $\\mathcal{F}$ or $\\mathcal{G}$. \\item If $S$ is locally Noetherian, then the category of coherent $\\mathcal{O}_X$-modules with support proper over $S$ is a Serre subcategory (Homology, Definition \\ref{homology-definition-serre-subcategory}) of the abelian category of coherent $\\mathcal{O}_X$-modules. \\end{enumerate}"}
+{"_id": "3439", "title": "formal-defos-lemma-S1-small-extensions", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal C_\\Lambda$. Then $\\mathcal{F}$ satisfies (S1) if the condition of (S1) is assumed to hold only when $A_2 \\to A$ is a small extension."}
+{"_id": "3463", "title": "formal-defos-lemma-miniversal-object-unique", "text": "Let $\\mathcal{F}$ be a predeformation category. Let $\\xi$ be a versal formal object of $\\mathcal{F}$ such that (\\ref{equation-bijective-orbits}) holds. Then $\\xi$ is a minimal versal formal object. In particular, such $\\xi$ are unique up to isomorphism."}
+{"_id": "3477", "title": "formal-defos-lemma-infaut-vector-space", "text": "Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\text{Inf}_{x_0}(\\mathcal{F})$ is equal as a set to $T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$, and so has a natural $k$-vector space structure such that addition agrees with composition of automorphisms."}
+{"_id": "3485", "title": "formal-defos-lemma-map-between-smooth", "text": "Let $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$ be maps of categories cofibred in groupoids over $\\mathcal{C}_\\Lambda$. If \\begin{enumerate} \\item $\\mathcal{F}$, $\\mathcal{G}$ are deformation categories \\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and \\item $\\mathcal{F} \\to \\mathcal{H}$ is smooth. \\end{enumerate} Then $\\mathcal{F} \\to \\mathcal{G}$ is smooth."}
+{"_id": "3501", "title": "formal-defos-lemma-homotopy", "text": "Being formally homotopic is an equivalence relation on sets of morphisms in $\\widehat{\\mathcal{C}}_\\Lambda$."}
+{"_id": "3622", "title": "adequate-lemma-direct-image-parasitic-adequate", "text": "Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes. For any parasitic adequate $\\mathcal{O}_T$-module on $(\\Sch/T)_\\tau$ the pushforward $f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$ are parasitic adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$."}
+{"_id": "3624", "title": "adequate-lemma-describe-Dplus-adequate", "text": "Let $U = \\Spec(A)$ be an affine scheme. The bounded below derived category $D^+(\\textit{Adeq}(\\mathcal{O}))$ is the localization of $K^+(\\QCoh(\\mathcal{O}_U))$ at the multiplicative subset of universal quasi-isomorphisms."}
+{"_id": "3636", "title": "adequate-lemma-big-ext", "text": "Let $U = \\Spec(A)$ be an affine scheme. Let $M$, $N$ be $A$-modules. For all $i$ we have a canonical isomorphism $$ \\Ext^i_{\\textit{Mod}(\\mathcal{O})}(M^a, N^a) = \\text{Pext}^i_A(M, N) $$ functorial in $M$ and $N$."}
+{"_id": "3658", "title": "spaces-topologies-lemma-composition-etale", "text": "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$ in $(\\textit{Spaces}/S)_\\etale$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and $g_{small} \\circ f_{small} = (g \\circ f)_{small}$."}
+{"_id": "3669", "title": "spaces-topologies-lemma-composition-fppf", "text": "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$ of algebraic spaces over $S$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$."}
+{"_id": "3672", "title": "spaces-topologies-lemma-ph", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \\begin{enumerate} \\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$ is a ph covering of $X$. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering and for each $i$ we have a ph covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then $\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a ph covering. \\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering and $X' \\to X$ is a morphism of algebraic spaces then $\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a ph covering. \\end{enumerate}"}
+{"_id": "3675", "title": "spaces-topologies-lemma-composition-ph", "text": "Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$ of algebraic spaces over $S$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$."}
+{"_id": "3677", "title": "spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc", "text": "Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, \\'etale or Zariski covering is an fpqc covering."}
+{"_id": "3696", "title": "proetale-theorem-proper-base-change", "text": "Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be a morphism of schemes giving rise to the base change diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal such that $\\Lambda/I$ is torsion. Let $K$ be an object of $D(X_\\proetale)$ such that \\begin{enumerate} \\item $K$ is derived complete, and \\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ is bounded below with cohomology sheaves coming from $X_\\etale$, \\item $\\Lambda/I^n$ is a perfect $\\Lambda$-module\\footnote{This assumption can be removed if $K$ is a constructible complex, see \\cite{BS}.}. \\end{enumerate} Then the base change map $$ Lg_{comp}^*Rf_*K \\longrightarrow Rf'_*L(g')^*_{comp}K $$ is an isomorphism."}
+{"_id": "3747", "title": "proetale-lemma-verify-site-proetale", "text": "Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site containing $S$. Both $S_\\proetale$ and $(\\textit{Aff}/S)_\\proetale$ are sites."}
+{"_id": "3754", "title": "proetale-lemma-composition-proetale", "text": "Given schemes $X$, $Y$, $Y$ in $\\Sch_\\proetale$ and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and $g_{small} \\circ f_{small} = (g \\circ f)_{small}$."}
+{"_id": "3768", "title": "proetale-lemma-enough-compact-proetale", "text": "Let $S$ be a scheme. Let $\\Lambda$ be a ring. \\begin{enumerate} \\item $D(S_\\proetale)$ is compactly generated, \\item $D(S_\\proetale, \\Lambda)$ is compactly generated, \\item $D(S_\\proetale, \\mathcal{A})$ is compactly generated for any sheaf of rings $\\mathcal{A}$ on $S_\\proetale$, \\item $D((\\Sch/S)_\\proetale)$ is compactly generated, \\item $D((\\Sch/S)_\\proetale, \\Lambda)$ is compactly generated, and \\item $D((\\Sch/S)_\\proetale, \\mathcal{A})$ is compactly generated for any sheaf of rings $\\mathcal{A}$ on $(\\Sch/S)_\\proetale$, \\end{enumerate}"}
+{"_id": "3771", "title": "proetale-lemma-describe-pullback", "text": "Let $S$ be a scheme. Consider the morphism $$ \\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale $$ of Lemma \\ref{lemma-at-the-bottom}. Let $\\mathcal{F}$ be a sheaf on $S_\\proetale$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule $$ (\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\proetale, f_{small}^{-1}\\mathcal{F}) $$ where $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings."}
+{"_id": "3773", "title": "proetale-lemma-compare-higher-direct-image", "text": "Let $f : T \\to S$ be a morphism of schemes. For $K$ in $D((\\Sch/T)_\\proetale)$ we have $$ (Rf_{big, *}K)|_{S_\\proetale} = Rf_{small, *}(K|_{T_\\proetale}) $$ in $D(S_\\proetale)$. More generally, let $S' \\in \\Ob((\\Sch/S)_\\proetale)$ with structure morphism $g : S' \\to S$. Consider the fibre product $$ \\xymatrix{ T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ Then for $K$ in $D((\\Sch/T)_\\proetale)$ we have $$ i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K) $$ in $D(S'_\\proetale)$ and $$ g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K) $$ in $D((\\Sch/S')_\\proetale)$."}
+{"_id": "3777", "title": "proetale-lemma-cofinal-etale", "text": "Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$ be a geometric point. The category of pro-\\'etale neighbourhoods of $\\overline{s}$ is cofiltered."}
+{"_id": "3784", "title": "proetale-lemma-compare-cohomology", "text": "Let $X$ be a scheme. \\begin{enumerate} \\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have $$ H^i(X_\\etale, \\mathcal{F}) = H^i(X_\\proetale, \\epsilon^{-1}\\mathcal{F}) $$ for all $i$. \\item For $K \\in D^+(X_\\etale)$ we have $$ R\\Gamma(X_\\etale, K) = R\\Gamma(X_\\proetale, \\epsilon^{-1}K) $$ \\end{enumerate}"}
+{"_id": "3792", "title": "proetale-lemma-more-general-point", "text": "Let $k$ be a field. Let $G = \\text{Gal}(k^{sep}/k)$ be its absolute Galois group. Further,  \\begin{enumerate} \\item let $M$ be a profinite abelian group with a continuous $G$-action, or \\item let $\\Lambda$ be a Noetherian ring and $I \\subset \\Lambda$ an ideal an let $M$ be an $I$-adically complete $\\Lambda$-module with continuous $G$-action. \\end{enumerate} Then there is a canonical sheaf $\\underline{M}^\\wedge$ on $\\Spec(k)_\\proetale$ associated to $M$ such that $$ H^i(\\Spec(k), \\underline{M}^\\wedge) = H^i_{cont}(G, M) $$ as abelian groups or $\\Lambda$-modules."}
+{"_id": "3803", "title": "proetale-lemma-open-immersion", "text": "Let $j : U \\to X$ be an open immersion of schemes. Then $\\text{id} \\cong j^{-1}j_!$ and $j^{-1}j_* \\cong \\text{id}$ and the functors $j_!$ and $j_*$ are fully faithful."}
+{"_id": "3814", "title": "proetale-lemma-Noetherian-adic-constructible", "text": "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a constructible $\\Lambda$-sheaf on $X_\\proetale$. Then $\\mathcal{F}$ is adic constructible."}
+{"_id": "3822", "title": "proetale-lemma-proetale-cohomology-independent-partial-universe", "text": "Let $S$ be a scheme. Let $S_\\proetale \\subset S_\\proetale'$ be two small pro-\\'etale sites of $S$ as constructed in Definition \\ref{definition-big-small-proetale}. Then the inclusion functor satisfies the assumptions of  Sites, Lemma \\ref{sites-lemma-bigger-site}. Hence there exist morphisms of topoi $$ \\xymatrix{ \\Sh(S_\\proetale) \\ar[r]^g & \\Sh(S_\\proetale') \\ar[r]^f & \\Sh(S_\\proetale) } $$ whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover, \\begin{enumerate} \\item for $\\mathcal{F}' \\in \\textit{Ab}(S_\\proetale')$ we have $H^p(S_\\proetale', \\mathcal{F}') = H^p(S_\\proetale, g^{-1}\\mathcal{F}')$, \\item for $\\mathcal{F} \\in \\textit{Ab}(S_\\proetale)$ we have $$ H^p(S_\\proetale, \\mathcal{F}) = H^p(S_\\proetale', g_*\\mathcal{F}) = H^p(S_\\proetale', f^{-1}\\mathcal{F}). $$ \\end{enumerate}"}
+{"_id": "3824", "title": "proetale-lemma-cohomology-enlarge-partial-universe", "text": "Let $S$ be a scheme. Let $(\\Sch/S)_\\proetale$ and $(\\Sch'/S)_\\proetale$ be two big pro-\\'etale sites of $S$ as in Definition \\ref{definition-big-small-proetale}. Assume that the first is contained in the second. In this case \\begin{enumerate} \\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\proetale$ and any object $U$ of $(\\Sch/S)_\\proetale$ we have $$ H^p(U, \\mathcal{F}'|_{(\\Sch/S)_\\proetale}) = H^p(U, \\mathcal{F}') $$ In words: the cohomology of $\\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\\mathcal{F}'$ restricted to the smaller site over $U$. \\item for any abelian sheaf $\\mathcal{F}$ on $(\\Sch/S)_\\proetale$ there is an abelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\proetale'$ whose restriction to $(\\Sch/S)_\\proetale$ is isomorphic to $\\mathcal{F}$. \\end{enumerate}"}
+{"_id": "3829", "title": "proetale-proposition-Noetherian-adic-constructible", "text": "Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of $D_{cons}(X, \\Lambda)$. Then $K$ is adic constructible (Definition \\ref{definition-adic-constructible})."}
+{"_id": "3858", "title": "formal-spaces-lemma-completion-adic-star", "text": "Let $R$ be a topological ring. Let $M$ be a topological $R$-module. Let $I \\subset R$ be a finitely generated ideal. Assume $M$ has an open submodule whose topology is $I$-adic. Then $M^\\wedge$ has an open submodule whose topology is $I$-adic and we have $M^\\wedge/I^n M^\\wedge = M/I^nM$ for all $n \\geq 1$."}
+{"_id": "3859", "title": "formal-spaces-lemma-weakly-admissible-henselian", "text": "Let $A$ be a weakly admissible topological ring. Let $I \\subset A$ be a weak ideal of definition. Then $(A, I)$ is a henselian pair."}
+{"_id": "3861", "title": "formal-spaces-lemma-taut-weakly-admissible", "text": "Let $\\varphi : A \\to B$ be a continuous map of weakly admissible topological rings. The following are equivalent \\begin{enumerate} \\item $\\varphi$ is taut, \\item for every weak ideal of definition $I \\subset A$ the closure of $\\varphi(I)B$ is a weak ideal of definition of $B$ and these form a fundamental system of weak ideals of definition of $B$. \\end{enumerate}"}
+{"_id": "3869", "title": "formal-spaces-lemma-characterize-affine-formal-algebraic-space", "text": "Let $S$ be a scheme. Let $X$ be a sheaf on $(\\Sch/S)_{fppf}$. Then $X$ is an affine formal algebraic space if and only if the following hold \\begin{enumerate} \\item any morphism $U \\to X$ where $U$ is an affine scheme over $S$ factors through a morphism $T \\to X$ which is representable and a thickening with $T$ an affine scheme over $S$, and \\item a set theoretic condition as in Remark \\ref{remark-set-theoretic}. \\end{enumerate}"}
+{"_id": "3888", "title": "formal-spaces-lemma-completion-countably-indexed", "text": "\\begin{reference} Email by Ofer Gabber of September 11, 2014. \\end{reference} Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$. Let $T \\subset X$ be a closed subscheme. \\begin{enumerate} \\item If the formal completion $X_{/T}$ is countably indexed and there exist countably many $f_1, f_2, f_3, \\ldots \\in A$ such that $T = V(f_1, f_2, f_3, \\ldots)$, then $X_{/T}$ is adic*. \\item The conclusion of (1) is wrong if we omit the assumption that $T$ can be cut out by countably many functions in $X$. \\end{enumerate}"}
+{"_id": "3904", "title": "formal-spaces-lemma-base-change-representable", "text": "A base change of a morphism representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes)."}
+{"_id": "3908", "title": "formal-spaces-lemma-representable-by-algebraic-spaces", "text": "Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$. Let $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space."}
+{"_id": "3930", "title": "formal-spaces-lemma-characterize-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is of finite type, \\item $f$ is representable by algebraic spaces and is of finite type in the sense of Bootstrap, Definition \\ref{bootstrap-definition-property-transformation}. \\end{enumerate}"}
+{"_id": "3933", "title": "formal-spaces-lemma-permanence-finite-type", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is locally of finite type, then $f : X \\to Y$ is locally of finite type."}
+{"_id": "3957", "title": "formal-spaces-lemma-composition-separated", "text": "All of the separation axioms listed in Definition \\ref{definition-separated-morphism} are stable under composition of morphisms."}
+{"_id": "3972", "title": "formal-spaces-lemma-structure-sheaf", "text": "Every formal algebraic space has a structure sheaf."}
+{"_id": "3973", "title": "formal-spaces-lemma-higher-vanishing-structure-sheaf", "text": "If $X$ is a countably indexed affine formal algebraic space, then we have $H^n(X_\\etale, \\mathcal{O}_X) = 0$ for $n > 0$."}
+{"_id": "4022", "title": "pione-theorem-global", "text": "Let $Y$ be an excellent regular scheme over a field. Let $f : X \\to Y$ be a finite type morphism of schemes with $X$ normal. Let $V \\subset X$ be the maximal open subscheme where $f$ is \\'etale. Then the inclusion morphism $V \\to X$ is affine."}
+{"_id": "4063", "title": "pione-lemma-normal-pione-quotient-inertia", "text": "Let $X$ be a normal integral scheme with function field $K$. With notation as above, the following three subgroups of $\\text{Gal}(K^{sep}/K) = \\pi_1(\\Spec(K))$ are equal \\begin{enumerate} \\item the kernel of the surjection $\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X)$, \\item the smallest normal closed subgroup containing $I_y$ for all $y \\in X^{sep}$, and \\item the smallest normal closed subgroup containing $\\text{Gal}(K^{sep}/K_x^{sh})$ for all $x \\in  X$. \\end{enumerate}"}
+{"_id": "4064", "title": "pione-lemma-unramified", "text": "Let $X$ be an integral normal scheme with function field $K$. Let $L/K$ be a finite extension. Let $Y \\to X$ be the normalization of $X$ in $L$. The following are equivalent \\begin{enumerate} \\item $X$ is unramified in $L$ as defined in Section \\ref{section-normal}, \\item $Y \\to X$ is an unramified morphism of schemes, \\item $Y \\to X$ is an \\'etale morphism of schemes, \\item $Y \\to X$ is a finite \\'etale morphism of schemes, \\item for $x \\in X$ the projection $Y \\times_X \\Spec(\\mathcal{O}_{X, x}) \\to \\Spec(\\mathcal{O}_{X, x})$ is unramified, \\item same as in (5) but with $\\mathcal{O}_{X, x}^h$, \\item same as in (5) but with $\\mathcal{O}_{X, x}^{sh}$, \\item for $x \\in X$ the scheme theoretic fibre $Y_x$ is \\'etale over $x$ of degree $\\geq [L : K]$. \\end{enumerate} If $L/K$ is Galois with Galois group $G$, then these are also equivalent to \\begin{enumerate} \\item[(9)] for $y \\in Y$ the group $I_y = \\{g \\in G \\mid g(y) = y\\text{ and } g \\bmod \\mathfrak m_y = \\text{id}_{\\kappa(y)}\\}$ is trivial. \\end{enumerate}"}
+{"_id": "4072", "title": "pione-lemma-specialization-map-composition", "text": "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres. Let $s'' \\leadsto s' \\leadsto s$ be specializations of points of $S$. A composition of specialization maps $\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$ is a specialization map $\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$."}
+{"_id": "4073", "title": "pione-lemma-specialization-map-valuation-ring", "text": "Let $f : X \\to S$ be a proper morphism with geometrically connected fibres. Let $s' \\leadsto s$ be a specialization of points of $S$ and let $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$ be a specialization map. Then there exists a strictly henselian valuation ring $R$ over $S$ with algebraically closed fraction field such that $sp$ is isomorphic to $sp_R$ defined above."}
+{"_id": "4081", "title": "pione-lemma-restriction-fully-faithful-general-special", "text": "Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme. Let $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$ ordered by reverse inclusion. Assume one of the following holds \\begin{enumerate} \\item $X$ is quasi-affine and $$ \\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n}) $$ is an isomorphism, or \\item $X$ has an ample invertible module $\\mathcal{L}$ and $$ \\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m}) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n}) $$ is an isomorphism for all $m \\gg 0$, or \\item for every $V \\in \\mathcal{V}$ and every finite locally free $\\mathcal{O}_V$-module $\\mathcal{E}$ the map $$ \\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'}) \\longrightarrow \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n}) $$ is an isomorphism. \\end{enumerate} Then the functor $$ \\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y $$ is fully faithful."}
+{"_id": "4082", "title": "pione-lemma-pushout-along-closed-immersion-and-integral", "text": "In More on Morphisms, Situation \\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral}, for example if $Z \\to Y$ and $Z \\to X$ are closed immersions of schemes, there is an equivalence of categories $$ \\textit{F\\'Et}_{Y \\amalg_Z X} \\longrightarrow \\textit{F\\'Et}_Y \\times_{\\textit{F\\'Et}_Z} \\textit{F\\'Et}_X $$"}
+{"_id": "4093", "title": "pione-lemma-lift-simple", "text": "In Situation \\ref{situation-local-lefschetz}. Let $V$ be finite \\'etale over $U$. Assume \\begin{enumerate} \\item[(a)] $A$ has a dualizing complex, \\item[(b)] the pair $(A, (f))$ is henselian, \\item[(c)] one of the following is true \\begin{enumerate} \\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 3$, or \\item[(ii)] for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ we have $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$. \\end{enumerate} \\item[(d)] $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$ for some $Y_0 \\to X_0$ finite \\'etale. \\end{enumerate} Then $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale."}
+{"_id": "4102", "title": "pione-lemma-faithful-general", "text": "In Situation \\ref{situation-local-lefschetz}. Let $U' \\subset U$ be open and contain $U_0$. Assume for $\\mathfrak p \\subset A$ minimal with $\\mathfrak p \\in U'$, $\\mathfrak p \\not \\in U_0$ we have $\\dim(A/\\mathfrak p) \\geq 2$. Then $$ \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad V' \\longmapsto V_0 = V' \\times_{U'} U_0 $$ is a faithful functor. Moreover, there exists a $U'$ satisfying the assumption and any smaller open $U'' \\subset U'$ containing $U_0$ also satisfies this assumption. In particular, the restriction functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0} $$ is faithful."}
+{"_id": "4103", "title": "pione-lemma-fully-faithful-general-better", "text": "In Situation \\ref{situation-local-lefschetz} assume \\begin{enumerate} \\item $A$ has a dualizing complex and is $f$-adically complete, \\item every irreducible component of $X$ not contained in $X_0$ has dimension $\\geq 3$. \\end{enumerate} Then the restriction functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0} $$ is fully faithful."}
+{"_id": "4104", "title": "pione-lemma-fully-faithful-general", "text": "In Situation \\ref{situation-local-lefschetz} assume \\begin{enumerate} \\item $A$ is $f$-adically complete, \\item $f$ is a nonzerodivisor. \\item $H^1_\\mathfrak m(A/fA)$ is a finite $A$-module. \\end{enumerate} Then the restriction functor $$ \\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0} $$ is fully faithful."}
+{"_id": "4150", "title": "stacks-cohomology-lemma-general-pushforward-fppf", "text": "Let $\\mathcal{M}$ be a rule which associates to every algebraic stack $\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ such that \\begin{enumerate} \\item $\\mathcal{M}_\\mathcal{X}$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ for all algebraic stacks $\\mathcal{X}$, \\item for a smooth morphism of algebraic stacks $f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps $\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$, \\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth morphisms of algebraic stacks with $|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is in $\\mathcal{M}_\\mathcal{X}$ if and only if $f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and \\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic stacks and $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$ into $\\mathcal{M}_\\mathcal{X}$. \\end{enumerate} Then for any quasi-compact and quasi-separated morphism  $f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$ into $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in fppf topology.)"}
+{"_id": "4172", "title": "stacks-cohomology-lemma-quasi-coherent-weak-serre", "text": "Let $\\mathcal{X}$ be an algebraic stack. \\begin{enumerate} \\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$. \\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$ is a weak Serre subcategory of $\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$. \\end{enumerate}"}
+{"_id": "4175", "title": "stacks-cohomology-proposition-smooth-covering-compute-cohomology", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a morphism of algebraic stacks. Assume $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module. Then there is a spectral sequence $$ E_2^{p, q} = H^q(\\mathcal{U}_p, f_p^*\\mathcal{F}) \\Rightarrow H^{p + q}(\\mathcal{X}, \\mathcal{F}) $$ where $f_p$ is the morphism $\\mathcal{U} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} \\mathcal{U} \\to \\mathcal{X}$ ($p + 1$ factors)."}
+{"_id": "4213", "title": "sites-cohomology-lemma-compute-cohomology-on-sheaf-sets", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be a presheaf of sets on $\\mathcal{C}$. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module and denote $\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then $H^p(K, \\mathcal{F}) = H^p(K, \\mathcal{F}_{ab})$."}
+{"_id": "4315", "title": "sites-cohomology-lemma-cup-compatible-with-naive", "text": "In the situation above the following diagram commutes $$ \\xymatrix{ f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} f_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d] & Rf_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L} Rf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\ \\text{Tot}( f_*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{D}} f_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} & Rf_*(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L} \\mathcal{M}^\\bullet) \\ar[d] \\\\ f_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}} \\mathcal{M}^\\bullet) \\ar[r] & Rf_*\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}} \\mathcal{M}^\\bullet) } $$"}
+{"_id": "4334", "title": "sites-cohomology-lemma-internal-hom-diagonal", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L$ in $D(\\mathcal{O})$ there is a canonical morphism $$ K \\longrightarrow R\\SheafHom(L, K \\otimes_\\mathcal{O}^\\mathbf{L} L) $$ in $D(\\mathcal{O})$ functorial in both $K$ and $L$."}
+{"_id": "4340", "title": "sites-cohomology-lemma-special-square-cocontinuous", "text": "Assume given a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ of ringed topoi. Assume \\begin{enumerate} \\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors $u$, $u'$, $v$, and $v'$ as in Sites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}, \\item $v \\circ u' = u \\circ v'$, \\item $v$ and $v'$ are continuous as well as cocontinuous, \\item for any object $V'$ of $\\mathcal{D}'$ the functor ${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$ given by $v$ is cofinal, \\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$ and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$, and \\item $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$ is exact\\footnote{Holds if fibre products and equalizers exist in $\\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.}. \\end{enumerate} Then we have $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$."}
+{"_id": "4341", "title": "sites-cohomology-lemma-special-square-continuous", "text": "Consider a commutative diagram $$ \\xymatrix{ (\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'} \\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\ (\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} & (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) } $$ of ringed topoi and suppose we have functors $$ \\xymatrix{ \\mathcal{C}' \\ar[r]_{v'} & \\mathcal{C} \\\\ \\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} & \\mathcal{D} \\ar[u]_u } $$ such that (with notation as in Sites, Sections \\ref{sites-section-morphism-sites} and \\ref{sites-section-cocontinuous-morphism-topoi}) we have \\begin{enumerate} \\item $u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$, \\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$, \\item $u \\circ v = v' \\circ u'$, \\item $v$ and $v'$ are continuous as well as cocontinuous, and \\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$ and $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$. \\end{enumerate} Then $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors $D^+(\\mathcal{O}_\\mathcal{C}) \\to D^+(\\mathcal{O}_{\\mathcal{D}'})$. If in addition \\begin{enumerate} \\item[(6)] $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$ is exact\\footnote{Holds if fibre products and equalizers exist in $\\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.}, \\end{enumerate} then $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$."}
+{"_id": "4387", "title": "sites-cohomology-lemma-summands-perfect", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $K \\oplus L$ is a perfect object of $D(\\mathcal{O})$, then so are $K$ and $L$."}
+{"_id": "4395", "title": "sites-cohomology-lemma-invertible-derived", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\\mathcal{O})$. The following are equivalent \\begin{enumerate} \\item $M$ is invertible in $D(\\mathcal{O})$, see Categories, Definition \\ref{categories-definition-invertible}, and \\item there is a locally finite\\footnote{This means that for every object $U$ of $\\mathcal{C}$ there is a covering $\\{U_i \\to U\\}$ such that for every $i$ the sheaf $\\mathcal{O}_n|_{U_i}$ is nonzero for only a finite number of $n$.} direct product decomposition $$ \\mathcal{O} = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n $$ and for each $n$ there is an invertible $\\mathcal{O}_n$-module $\\mathcal{H}^n$ (Modules on Sites, Definition \\ref{sites-modules-definition-invertible-sheaf}) and $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O})$. \\end{enumerate} If (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O})$. If $(\\mathcal{C}, \\mathcal{O})$ is a locally ringed site these condition are also equivalent to \\begin{enumerate} \\item[(3)] for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ and for each $i$ an integer $n_i$ such that $M|_{U_i}$ is represented by an invertible $\\mathcal{O}_{U_i}$-module placed in degree $n_i$. \\end{enumerate}"}
+{"_id": "4399", "title": "sites-cohomology-lemma-compact-objects-if-enough-qc", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume every object of $\\mathcal{C}$ has a covering by quasi-compact objects. Then every compact object of $D(\\mathcal{O})$ is a direct summand in $D(\\mathcal{O})$ of a finite complex whose terms are finite direct sums of $\\mathcal{O}$-modules of the form $j_!\\mathcal{O}_U$ where $U$ is a quasi-compact object of $\\mathcal{C}$."}
+{"_id": "4454", "title": "fields-lemma-algebraic-elements", "text": "Let $E/k$ be a field extension. Then the elements of $E$ algebraic over $k$ form a subextension of $E/k$."}
+{"_id": "4462", "title": "fields-lemma-algebraic-closures-isomorphic", "text": "Any two algebraic closures of a field are isomorphic."}
+{"_id": "4574", "title": "spaces-limits-lemma-descend-section", "text": "Notation and assumptions as in Situation \\ref{situation-descent}. Suppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$. Set $\\mathcal{F}_i = f_{0i}^*\\mathcal{F}_0$ for $i \\geq 0$ and set $\\mathcal{F} = f_0^*\\mathcal{F}_0$. Then $$ \\Gamma(X, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(X_i, \\mathcal{F}_i) $$"}
+{"_id": "4582", "title": "spaces-limits-lemma-eventually-affine", "text": "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \\begin{enumerate} \\item $Y$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item $X \\to Y$ affine. \\end{enumerate} Then $X_i \\to Y$ is affine for $i$ large enough."}
+{"_id": "4583", "title": "spaces-limits-lemma-eventually-finite", "text": "Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume \\begin{enumerate} \\item $Y$ quasi-compact and quasi-separated, \\item $X_i$ quasi-compact and quasi-separated, \\item the transition morphisms $X_{i'} \\to X_i$ are finite, \\item $X_i \\to Y$ locally of finite type \\item $X \\to Y$ integral. \\end{enumerate} Then $X_i \\to Y$ is finite for $i$ large enough."}
+{"_id": "4589", "title": "spaces-limits-lemma-descend-affine", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If $f$ is affine, then $f_i$ is affine for some $i \\geq 0$."}
+{"_id": "4594", "title": "spaces-limits-lemma-descend-monomorphism", "text": "Notation and assumptions as in Situation \\ref{situation-descent-property}. If \\begin{enumerate} \\item $f$ is a monomorphism, \\item $f_0$ is locally of finite type, \\end{enumerate} then $f_i$ is a monomorphism for some $i \\geq 0$."}
+{"_id": "4623", "title": "spaces-limits-lemma-push-sections-annihilated-by-ideal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a quasi-coherent sheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the subsheaf of sections annihilated by $f^{-1}\\mathcal{I}\\mathcal{O}_X$. Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf of sections annihilated by $\\mathcal{I}$."}
+{"_id": "4625", "title": "spaces-limits-lemma-push-sections-supported-on-closed-subset", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $T \\subset |Y|$ be a closed subset. Assume $|Y| \\setminus T$ corresponds to an open subspace $V \\subset Y$ such that $V \\to Y$ is quasi-compact. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$ be the subsheaf of sections supported on $|f|^{-1}T$. Then $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf of sections supported on $T$."}
+{"_id": "4630", "title": "spaces-limits-lemma-minimal-closed-subspace", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. If $X$ is not a scheme, then there exists a closed subspace $Z \\subset X$ such that $Z$ is not a scheme, but every proper closed subspace $Z' \\subset Z$ is a scheme."}
+{"_id": "4631", "title": "spaces-limits-lemma-minimal-nonscheme", "text": "Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume that every proper closed subspace $Z \\subset X$ is a scheme, but $X$ is not a scheme. Then $X$ is reduced and irreducible."}
+{"_id": "4636", "title": "spaces-limits-lemma-glueing-near-point", "text": "Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open. Let $s \\in S$ be a point in the complement of $U$. With $V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is an equivalence of categories $$ \\colim_{s \\in U' \\supset U\\text{ open}} FP_{U'} \\longrightarrow FP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})} $$ where $FP_T$  is the category of algebraic spaces of finite presentation over $T$."}
+{"_id": "4640", "title": "spaces-limits-lemma-modifications", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$ be a closed point such that $U = X \\setminus \\{x\\} \\to X$ is quasi-compact. With $V = \\Spec(\\mathcal{O}_{X, x}^h) \\setminus \\{\\mathfrak m_x^h\\}$ the base change functor $$ \\left\\{ \\begin{matrix} f : Y \\to X\\text{ of finite presentation} \\\\ f^{-1}(U) \\to U\\text{ is an isomorphism} \\end{matrix} \\right\\} \\longrightarrow \\left\\{ \\begin{matrix} g : Y \\to \\Spec(\\mathcal{O}_{X, x}^h)\\text{ of finite presentation} \\\\ g^{-1}(V) \\to V\\text{ is an isomorphism} \\end{matrix} \\right\\} $$ is an equivalence of categories."}
+{"_id": "4644", "title": "spaces-limits-lemma-reach-point-closure-Noetherian", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ finite type and $Y$ locally Noetherian. Let $y \\in |Y|$ be a point in the closure of the image of $|f|$. Then there exists a commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\ \\Spec(A) \\ar[r] & Y } $$ where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume that the point $x \\in |X|$ corresponding to $\\Spec(K) \\to X$ is a codimension $0$ point\\footnote{See discussion in Properties of Spaces, Section \\ref{spaces-properties-section-generic-points}.} and that $K$ is the residue field of a point on a scheme \\'etale over $X$."}
+{"_id": "4677", "title": "stacks-geometry-lemma-branches-multiplicity", "text": "Let $\\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$ with image $s \\in S$. If $\\mathcal{O}_{S, s}$ is a G-ring, then the map of Lemma \\ref{lemma-branches} preserves multiplicities."}
+{"_id": "4679", "title": "stacks-geometry-lemma-dimension-for-stacks", "text": "If $\\mathcal{X}$ is a locally Noetherian algebraic stack and $x \\in |\\mathcal{X}|$. Let $U \\to \\mathcal{X}$ be a smooth morphism from an algebraic space to $\\mathcal{X}$, let $u$ be any point of $|U|$ mapping to $x$. Then we have $$ \\dim_x(\\mathcal{X}) =  \\dim_u(U) - \\dim_{u}(U_x) $$ where the relative dimension $\\dim_u(U_x)$ is defined by Definition \\ref{definition-relative-dimension} and the dimension of $\\mathcal{X}$ at $x$ is as in Properties of Stacks, Definition \\ref{stacks-properties-definition-dimension-at-point}."}
+{"_id": "4688", "title": "stacks-geometry-lemma-dimension-via-components", "text": "If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if $x \\in |\\mathcal{X}|$, then $\\dim_x(\\mathcal{X}) = \\sup_{\\mathcal{T}} \\{ \\dim_x(\\mathcal{T}) \\} $, where $\\mathcal{T}$ runs over all the irreducible components of $|\\mathcal{X}|$ passing through $x$ (endowed with their induced reduced structure)."}
+{"_id": "4695", "title": "stacks-geometry-lemma-dimension-formula", "text": "Suppose that $\\mathcal{X}$ is an algebraic stack, locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$ be a morphism where $k$ is a field of finite type over $S$. Represent $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in Remark \\ref{remark-groupoid-defo} by a cogroupoid $(A, B, s, t, c)$ of Noetherian complete local $S$-algebras with residue field $k$. Then $$ \\text{the dimension of the local ring of }\\mathcal{X}\\text{ at }x_0 = 2\\dim A - \\dim B $$"}
+{"_id": "4723", "title": "spaces-morphisms-lemma-match-separated", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. \\begin{enumerate} \\item The morphism $f$ is locally separated. \\item The morphism $f$ is (quasi-)separated in the sense of Definition \\ref{definition-separated} above if and only if $f$ is (quasi-)separated in the sense of Section \\ref{section-representable}. \\end{enumerate} In particular, if $f : X \\to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition \\ref{definition-separated} if and only if $f$ is (quasi-)separated as a morphism of schemes."}
+{"_id": "4755", "title": "spaces-morphisms-lemma-monomorphism-local", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \\begin{enumerate} \\item $f$ is a monomorphism, \\item for every scheme $Z$ and morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism, \\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism, \\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$ such that the base change $V \\times_Y X \\to V$ is a monomorphism, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is a monomorphism. \\end{enumerate}"}
+{"_id": "4787", "title": "spaces-morphisms-lemma-scheme-theoretically-dense-quasi-compact", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \\subset X$ be an open subspace. If $U \\to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$."}
+{"_id": "4797", "title": "spaces-morphisms-lemma-affine-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is affine (in the sense of Section \\ref{section-representable}) if and only if for all affine schemes $Z$ and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine."}
+{"_id": "4806", "title": "spaces-morphisms-lemma-quasi-affine-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is quasi-affine (in the sense of Section \\ref{section-representable}) if and only if for all affine schemes $Z$ and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is quasi-affine."}
+{"_id": "4829", "title": "spaces-morphisms-lemma-ubiquity-nagata", "text": "The following types of algebraic spaces are Nagata. \\begin{enumerate} \\item Any algebraic space locally of finite type over a Nagata scheme. \\item Any algebraic space locally of finite type over a field. \\item Any algebraic space locally of finite type over a Noetherian complete local ring. \\item Any algebraic space locally of finite type over $\\mathbf{Z}$. \\item Any algebraic space locally of finite type over a Dedekind ring of characteristic zero. \\item And so on. \\end{enumerate}"}
+{"_id": "4850", "title": "spaces-morphisms-lemma-inverse-image-constructible", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $E \\subset |Y|$ be a subset. If $E$ is \\'etale locally constructible in $Y$, then $f^{-1}(E)$ is \\'etale locally constructible in $X$."}
+{"_id": "4856", "title": "spaces-morphisms-lemma-fpqc-quotient-topology", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a flat, quasi-compact, surjective morphism of algebraic spaces over $S$. A subset $T \\subset |Y|$ is open (resp.\\ closed) if and only $f^{-1}(|T|)$ is open (resp.\\ closed) in $|X|$. In other words $f$ is submersive, and in fact universally submersive."}
+{"_id": "4866", "title": "spaces-morphisms-lemma-pf-flat-module-open", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Assume $f$ locally finite presentation, $\\mathcal{F}$ of finite type, $X = \\text{Supp}(\\mathcal{F})$, and $\\mathcal{F}$ flat over $Y$. Then $f$ is universally open."}
+{"_id": "4880", "title": "spaces-morphisms-lemma-composition-syntomic", "text": "The composition of syntomic morphisms is syntomic."}
+{"_id": "4939", "title": "spaces-morphisms-lemma-integral-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is integral, resp.\\ finite (in the sense of Section \\ref{section-representable}), if and only if for all affine schemes $Z$ and morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine and integral, resp.\\ finite, over $Z$."}
+{"_id": "4950", "title": "spaces-morphisms-lemma-finite-locally-free-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is finite locally free (in the sense of Section \\ref{section-representable}) if and only if $f$ is affine and the sheaf $f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_Y$-module."}
+{"_id": "4955", "title": "spaces-morphisms-lemma-rational-map-from-reduced-to-separated", "text": "Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$. Assume $X$ is reduced and $Y$ is separated over $S$. Let $\\varphi$ be a rational map from $X$ to $Y$ with domain of definition $U \\subset X$. Then there exists a unique morphism $f : U \\to Y$ of algebraic spaces representing $\\varphi$."}
+{"_id": "4965", "title": "spaces-morphisms-lemma-nagata-normalization-finite", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $Y$ is Nagata, \\item $f$ is quasi-separated of finite type, \\item $X$ is reduced. \\end{enumerate} Then the normalization $\\nu : Y' \\to Y$ of $Y$ in $X$ is finite."}
+{"_id": "4968", "title": "spaces-morphisms-lemma-normalization-reduced", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-prepare-normalization}. The normalization morphism $\\nu$ factors through the reduction $X_{red}$ and $X^\\nu \\to X_{red}$ is the normalization of $X_{red}$."}
+{"_id": "4976", "title": "spaces-morphisms-lemma-universal-homeomorphism-representable", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is a universal homeomorphism (in the sense of Section \\ref{section-representable}) if and only if for every morphism of algebraic spaces $Z \\to Y$ the base change map $Z \\times_Y X \\to Z$ induces a homeomorphism $|Z \\times_Y X| \\to |Z|$."}
+{"_id": "5042", "title": "weil-lemma-motives", "text": "The category $M_k$ whose objects are motives over $k$ and morphisms are morphisms of motives over $k$ is a $\\mathbf{Q}$-linear category. There is a contravariant functor $$ h : \\{\\text{smooth projective schemes over }k\\} \\longrightarrow M_k $$ defined by $h(X) = (X, 1, 0)$ and $h(f) = [\\Gamma_f]$."}
+{"_id": "5043", "title": "weil-lemma-Karoubian", "text": "The category $M_k$ is Karoubian."}
+{"_id": "5081", "title": "weil-lemma-square-diagonal", "text": "Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme over $k$ which is equidimensional of dimension $d$. We have $$ \\sum\\nolimits_i (-1)^i\\dim_F H^i(X) = \\deg(\\Delta \\cdot \\Delta) = \\deg(c_d(\\mathcal{T}_{X/k})) $$"}
+{"_id": "5110", "title": "weil-proposition-weil-cohomology-theory", "text": "Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a $1$-to-$1$ correspondence between the following \\begin{enumerate} \\item data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and \\item $\\mathbf{Q}$-linear symmetric monoidal functors $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ such that $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$. \\end{enumerate}"}
+{"_id": "5129", "title": "morphisms-lemma-composition-closed-immersion", "text": "A composition of closed immersions is a closed immersion."}
+{"_id": "5151", "title": "morphisms-lemma-scheme-theoretically-dense-quasi-compact", "text": "Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. If the inclusion morphism $U \\to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$."}
+{"_id": "5183", "title": "morphisms-lemma-affine-diagonal", "text": "Let $X$ be a scheme such that for every point $x \\in X$ there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and a global section $s \\in \\Gamma(X, \\mathcal{L})$ such that $x \\in X_s$ and $X_s$ is affine. Then the diagonal of $X$ is an affine morphism."}
+{"_id": "5216", "title": "morphisms-lemma-catenary-check-irreducible", "text": "Let $S$ be a locally Noetherian scheme. Then $S$ is universally catenary if and only if the irreducible components of $S$ are universally catenary."}
+{"_id": "5256", "title": "morphisms-lemma-descent-quasi-compact", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $g : Y' \\to Y$ be open and surjective such that the base change $f' : X' \\to Y'$ is quasi-compact. Then $f$ is quasi-compact."}
+{"_id": "5276", "title": "morphisms-lemma-finite-flat-is-finite-locally-free", "text": "Let $X$ be a scheme. The following are equivalent \\begin{enumerate} \\item every finite flat quasi-coherent $\\mathcal{O}_X$-module is finite locally free, and \\item every closed subset $Z \\subset X$ which is closed under generalizations is open. \\end{enumerate}"}
+{"_id": "5463", "title": "morphisms-lemma-universal-homeo-iso-if-invert-p", "text": "Let $p$ be a prime number. Let $A \\to B$ be a ring map which induces an isomorphism $A[1/p] \\to B[1/p]$ (for example if $p$ is nilpotent in $A$). The following are equivalent \\begin{enumerate} \\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and \\item the kernel of $A \\to B$ is a locally nilpotent ideal and for every $b \\in B$ there exists a $p$-power $q$ with $qb$ and $b^q$ in the image of $A \\to B$. \\end{enumerate}"}
+{"_id": "5482", "title": "morphisms-lemma-birational-dominant", "text": "Let $f : X \\to Y$ be a morphism of schemes having finitely many irreducible components. If $f$ is birational then $f$ is dominant."}
+{"_id": "5660", "title": "chow-lemma-well-defined-tame-symbol", "text": "The formula (\\ref{equation-tame-symbol}) determines a well defined element of $\\kappa(\\mathfrak m)^*$. In other words, the right hand side does not depend on the choice of the local factorizations or the choice of $B$."}
+{"_id": "5697", "title": "chow-lemma-coherent-sheaf-cross-p1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{F}$ be a coherent sheaf on $X \\times \\mathbf{P}^1$. Let $i_0, i_\\infty : X \\to X \\times \\mathbf{P}^1$ be the closed immersion such that $i_t(x) = (x, t)$. Denote $\\mathcal{F}_0 = i_0^*\\mathcal{F}$ and $\\mathcal{F}_\\infty = i_\\infty^*\\mathcal{F}$. Assume \\begin{enumerate} \\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$, \\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F}_0)) \\leq k$, $\\dim_\\delta(\\text{Supp}(\\mathcal{F}_\\infty)) \\leq k$, and \\item for any embedded associated point $\\xi$ of $\\mathcal{F}$ either $\\xi \\not \\in (X \\times \\mathbf{P}^1)_0 \\cup (X \\times \\mathbf{P}^1)_\\infty$ or $\\delta(\\xi) < k$. \\end{enumerate} Then $[\\mathcal{F}_0]_k \\sim_{rat} [\\mathcal{F}_\\infty]_k$ as $k$-cycles on $X$."}
+{"_id": "5698", "title": "chow-lemma-Serre-subcategories", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. The categories $\\textit{Coh}_{\\leq k}(X)$ are Serre subcategories of the abelian category $\\textit{Coh}(X)$."}
+{"_id": "5738", "title": "chow-lemma-factors-through-rational-equivalence", "text": "\\begin{reference} Very weak form of \\cite[Theorem 17.1]{F} \\end{reference} Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \\in \\mathbf{Z}$. Suppose given a rule which assigns to every locally of finite type morphism $Y' \\to Y$ and every $k$ a map $$ c \\cap - : Z_k(Y') \\longrightarrow \\CH_{k - p}(X') $$ where $Y' = X' \\times_X Y$, satisfying condition (3) of Definition \\ref{definition-bivariant-class} whenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then $c \\cap -$ factors through rational equivalence."}
+{"_id": "5739", "title": "chow-lemma-bivariant-weaker", "text": "\\begin{reference} Weak form of \\cite[Theorem 17.1]{F} \\end{reference} Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \\in \\mathbf{Z}$. Suppose given a rule which assigns to every locally of finite type morphism $Y' \\to Y$ and every $k$ a map $$ c \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X') $$ where $Y' = X' \\times_X Y$, satisfying conditions (1), (2) of Definition \\ref{definition-bivariant-class} and condition (3) whenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then $c \\cap -$ is a bivariant class."}
+{"_id": "5745", "title": "chow-lemma-first-chern-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \\dim_\\delta(X)$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The first Chern class of $\\mathcal{L}$ on $X$ of Definition \\ref{definition-chern-classes} is equal to the Weil divisor associated to $\\mathcal{L}$ by Definition \\ref{definition-divisor-invertible-sheaf}."}
+{"_id": "5758", "title": "chow-lemma-spell-out-degree-zero-cycle", "text": "Let $k$ be a field. Let $X$ be proper over $k$. Let $\\alpha = \\sum n_i[Z_i]$ be in $Z_0(X)$. Then $$ \\deg(\\alpha) = \\sum n_i\\deg(Z_i) $$ where $\\deg(Z_i)$ is the degree of $Z_i \\to \\Spec(k)$, i.e., $\\deg(Z_i) = \\dim_k \\Gamma(Z_i, \\mathcal{O}_{Z_i})$."}
+{"_id": "5770", "title": "chow-lemma-pre-derived-chern-class", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$ be an object such that there exists a finite complex $\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules representing $E$. Then $c(\\mathcal{E}^\\bullet) \\in A^*(X)$, $ch(\\mathcal{E}^\\bullet) \\in A^*(X) \\otimes \\mathbf{Q}$, and $P_p(\\mathcal{E}^\\bullet) \\in A^p(X)$ are independent of the choice of the complex."}
+{"_id": "5773", "title": "chow-lemma-chern-classes-perfect-dual", "text": "In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes are defined. Then $c_i(E^\\vee) = (-1)^i c_i(E)$, $P_i(E^\\vee) = (-1)^iP_i(E)$, and $ch_i(E^\\vee) = (-1)^ich_i(E)$ in $A^i(X)$."}
+{"_id": "5784", "title": "chow-lemma-silly-tensor-invertible", "text": "In Lemma \\ref{lemma-silly} assume $E_2$ has constant rank $0$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $$ c'_i(E_2 \\otimes \\mathcal{L}) = \\sum\\nolimits_{j = 0}^i \\binom{- i + j}{j} c'_{i - j}(E_2) c_1(\\mathcal{L})^j $$"}
+{"_id": "5797", "title": "chow-lemma-independent-loc-chern", "text": "The localized class constructed above is independent of choices."}
+{"_id": "5806", "title": "chow-lemma-loc-chern-classes-commute", "text": "In the situation of Definition \\ref{definition-localized-chern} assume $P_p(Z \\to X, E)$, resp.\\ $c_p(Z \\to X, E)$ is defined. Let $Y \\to X$ be locally of finite type and $c \\in A^*(Y \\to X)$. Then $$ P_p(Z \\to X, E) \\circ c = c \\circ P_p(Z \\to X, E), $$ respectively $$ c_p(Z \\to X, E) \\circ c = c \\circ c_p(Z \\to X, E) $$ in $A^*(Y \\times_X Z \\to X)$."}
+{"_id": "5826", "title": "chow-lemma-adams-and-chern", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. If $\\psi^2$ is as in Lemma \\ref{lemma-second-adams-operator} and $c$ and $ch$ are as in Remarks \\ref{remark-chern-classes-K} and \\ref{remark-chern-character-K} then we have $c_i(\\psi^2(\\alpha)) = 2^i c_i(\\alpha)$ and $ch_i(\\psi^2(\\alpha)) = 2^i ch_i(\\alpha)$ for all $\\alpha \\in K_0(\\textit{Vect}(X))$."}
+{"_id": "5839", "title": "chow-lemma-projection-formula-regular", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map is defined for $f$ and that $f$ is proper. Then $$ f_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta $$ in $\\CH^*(Y) \\otimes \\mathbf{Q}$ where the intersection product is as in Section \\ref{section-intersection-regular}."}
+{"_id": "5841", "title": "chow-lemma-exterior-product-well-defined", "text": "The map $\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m}(X \\times_k Y)$ is well defined."}
+{"_id": "5851", "title": "chow-lemma-intersection-regular-smooth", "text": "Let $k$ be a field. Let $X$ be a smooth scheme over $k$ which is quasi-compact and has affine diagonal. Then the intersection product on $\\CH^*(X)$ constructed in this section agrees after tensoring with $\\mathbf{Q}$ with the intersection product constructed in Section \\ref{section-intersection-regular}."}
+{"_id": "5856", "title": "chow-lemma-associative-dim-1", "text": "The product defined above is associative. More precisely, with $(S, \\delta)$ as above, let $X$ be smooth over $S$, let $Y, Z, W$ be schemes locally of finite type over $X$, let $\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$. Then $(\\alpha \\cdot \\beta) \\cdot \\gamma = \\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$."}
+{"_id": "5881", "title": "chow-lemma-symbol-defined", "text": "Let $A$ be a Noetherian local ring. Let $a, b \\in A$. \\begin{enumerate} \\item If $M$ is a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$, then $\\text{length}_A(M/abM) < \\infty$ and $(M/abM, a, b)$ is a $(2, 1)$-periodic exact complex. \\item If $a, b$ are nonzerodivisors and $\\dim(A) = 1$ then $\\text{length}_A(A/(ab)) < \\infty$ and $(A/(ab), a, b)$ is a $(2, 1)$-periodic exact complex. \\end{enumerate} In particular, in these cases $\\det_\\kappa(M/abM, a, b) \\in \\kappa^*$, resp.\\ $\\det_\\kappa(A/(ab), a, b) \\in \\kappa^*$ are defined."}
+{"_id": "5888", "title": "chow-lemma-symbol-is-usual-tame-symbol", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. For nonzero $x, y \\in K$ we have $$ d_A(x, y) = (-1)^{\\text{ord}_A(x)\\text{ord}_A(y)} \\frac{x^{\\text{ord}_A(y)}}{y^{\\text{ord}_A(x)}} \\bmod \\mathfrak m_A, $$ in other words the symbol is equal to the usual tame symbol."}
+{"_id": "5890", "title": "chow-lemma-symbol-is-steinberg", "text": "Let $A$ be a Noetherian local domain of dimension $1$ with fraction field $K$. For $x \\in K \\setminus \\{0, 1\\}$ we have $$ d_A(x, 1 -x) = 1 $$"}
+{"_id": "5894", "title": "chow-lemma-secondary-ramification", "text": "\\begin{reference} When $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}. \\end{reference} Let $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$. Let $f, g \\in K^*$. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height $1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an element of $A^*_{\\mathfrak q}$. Then we have $$ \\sum\\nolimits_{i = 1, \\ldots, t} \\text{ord}_{A/\\mathfrak q_i}(d_{A_{\\mathfrak q_i}}(f, g)) = 0 $$ We can also write this as $$ \\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1} \\text{ord}_{A/\\mathfrak q}(d_{A_{\\mathfrak q}}(f, g)) = 0 $$ since at any height one prime $\\mathfrak q$ of $A$ where $f, g \\in A^*_{\\mathfrak q}$ we have $d_{A_{\\mathfrak q}}(f, g) = 1$ by Lemma \\ref{lemma-symbol-when-one-is-a-unit}."}
+{"_id": "5899", "title": "chow-lemma-coherent-sheaf-cap-c1", "text": "Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$. Then the element $$ [\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}] - [\\mathcal{F}] \\in K_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)) $$ lies in the subgroup $B_k(X)$ of Lemma \\ref{lemma-cycles-rational-equivalence-K-group} and maps to the element $c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}$ via the map $B_k(X) \\to \\CH_k(X)$."}
+{"_id": "5973", "title": "flat-theorem-existence", "text": "In Situation \\ref{situation-existence} there exists a finitely presented $\\mathcal{O}_X$-module $\\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\\varphi_n$."}
+{"_id": "5974", "title": "flat-theorem-existence-derived", "text": "In Situation \\ref{situation-existence-derived} there exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$ such that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\\varphi_n$."}
+{"_id": "6023", "title": "flat-lemma-finite-type-flat-at-point-local-X", "text": "Let $f : X \\to S$ be a morphism which is locally of finite type. Let $x \\in X$ with image $s \\in S$. If $f$ is flat at $x$ over $S$, then $\\mathcal{O}_{X, x}$ is essentially of finite presentation over $\\mathcal{O}_{S, s}$."}
+{"_id": "6033", "title": "flat-lemma-finite-presentation-flat-along-fibre-X", "text": "Let $f : X \\to S$ be of finite presentation. Let $s \\in S$. If $X$ is flat over $S$ at all points of $X_s$, then there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and a commutative diagram of schemes $$ \\xymatrix{ X \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\ S & S' \\ar[l] } $$ with $g$ \\'etale, $X_s \\subset g(X')$, such that $X'$, $S'$ are affine, and such that $\\Gamma(X', \\mathcal{O}_{X'})$ is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module."}
+{"_id": "6041", "title": "flat-lemma-weak-bourbaki", "text": "Let $R \\to S$ be a ring map which is essentially of finite type. Let $N$ be a localization of a finite $S$-module flat over $R$. Let $M$ be an $R$-module. Then $$ \\text{WeakAss}_S(M \\otimes_R N) = \\bigcup\\nolimits_{\\mathfrak p \\in \\text{WeakAss}_R(M)} \\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p)) $$"}
+{"_id": "6062", "title": "flat-lemma-quasi-finite-pure", "text": "Let $f : X \\to S$ be a separated, finite type morphism of schemes. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. Assume that $\\text{Supp}(\\mathcal{F}_s)$ is finite for every $s \\in S$. Then the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is pure relative to $S$, \\item the scheme theoretic support of $\\mathcal{F}$ is finite over $S$, and \\item $\\mathcal{F}$ is universally pure relative to $S$. \\end{enumerate} In particular, given a quasi-finite separated morphism $X \\to S$ we see that $X$ is pure relative to $S$ if and only if $X \\to S$ is finite."}
+{"_id": "6063", "title": "flat-lemma-flat-geometrically-integral-fibres-pure", "text": "Let $f : X \\to S$ be a finite type, flat morphism of schemes with geometrically integral fibres. Then $X$ is universally pure over $S$."}
+{"_id": "6083", "title": "flat-lemma-flat", "text": "In Situation \\ref{situation-flat}. \\begin{enumerate} \\item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology. \\item If $f$ is quasi-compact and locally of finite presentation and $\\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving. \\end{enumerate}"}
+{"_id": "6085", "title": "flat-lemma-flattening-stratification-artinian", "text": "Let $S$ be the spectrum of an Artinian ring. For any scheme $X$ over $S$, and any quasi-coherent $\\mathcal{O}_X$-module there exists a universal flattening. In fact the universal flattening is given by a closed immersion $S' \\to S$, and hence is a flattening stratification for $\\mathcal{F}$ as well."}
+{"_id": "6090", "title": "flat-lemma-Noetherian-finite-type-injective-into-flat-mod-m", "text": "If in Situation \\ref{situation-mod-injective} the ring $A$ is Noetherian then the lemma holds."}
+{"_id": "6094", "title": "flat-lemma-valuation-ring-finite-type-injective-into-flat-mod-m", "text": "If in Situation \\ref{situation-mod-injective} the ring $A$ is a valuation ring then the lemma holds."}
+{"_id": "6127", "title": "flat-lemma-get-section-after-blowup", "text": "Let $S$ be a scheme. Let $U \\subset W \\subset S$ be open subschemes. Let $f : X \\to W$ be a morphism and let $s : U \\to X$ be a morphism such that $f \\circ s = \\text{id}_U$. Assume \\begin{enumerate} \\item $f$ is proper, \\item $S$ is quasi-compact and quasi-separated, and \\item $U$ and $W$ are quasi-compact. \\end{enumerate} Then there exists a $U$-admissible blowup $b : S' \\to S$ and a morphism $s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$."}
+{"_id": "6131", "title": "flat-lemma-invert-right-multiplicative-system", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. The functor $(X, \\overline{X}) \\mapsto X$ defines an equivalence from the category of compactifications localized (Categories, Lemma \\ref{categories-lemma-right-localization}) at the right multiplicative system of Lemma \\ref{lemma-right-multiplicative-system} to the category of compactifyable schemes over $S$."}
+{"_id": "6144", "title": "flat-lemma-refine-by-h", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms such that $f_i$ is locally of finite presentation for all $i$. The following are equivalent \\begin{enumerate} \\item $\\{T_i \\to T\\}_{i \\in I}$ is an h covering, \\item there is an h covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and \\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is an h covering. \\end{enumerate}"}
+{"_id": "6147", "title": "flat-lemma-verify-site-h", "text": "Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$. Then $(\\textit{Aff}/S)_h$ is a site."}
+{"_id": "6149", "title": "flat-lemma-affine-big-site-h", "text": "Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$. The functor $(\\textit{Aff}/S)_h \\to (\\Sch/S)_h$ is cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_h)$ to $\\Sh((\\Sch/S)_h)$."}
+{"_id": "6152", "title": "flat-lemma-composition-h", "text": "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_h$ and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$."}
+{"_id": "6155", "title": "flat-lemma-blow-up-square-ph", "text": "Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$, see Topologies, Definition \\ref{topologies-definition-big-small-ph}. Then for any blow up square (\\ref{equation-blow-up-square}) in the category $(\\Sch/S)_{ph}$ the diagram $$ \\xymatrix{ \\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\ \\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l] } $$ is cartesian in the category of sets."}
+{"_id": "6156", "title": "flat-lemma-thickening-ph", "text": "Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$ as in Topologies, Definition \\ref{topologies-definition-big-small-ph}. Let $X \\to X'$ be a morphism of $(\\Sch/S)_{ph}$ which is a thickening. Then $\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective."}
+{"_id": "6169", "title": "flat-lemma-h-sheaf-lim-F", "text": "Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$. Let $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}. The rule $$ \\mathcal{F}(X) = \\lim_F \\Gamma(X, \\mathcal{O}_X) $$ defines a sheaf on $(\\Sch/S)_h$."}
+{"_id": "6185", "title": "flat-lemma-blowup-map-pd1", "text": "Let $X$ be a scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a homorphism of perfect $\\mathcal{O}_X$-modules of tor dimension $\\leq 1$. Let $U \\subset X$ be a scheme theoretically dense open such that $\\mathcal{F}|_U = 0$ and $\\mathcal{G}|_U = 0$. Then there is a $U$-admissible blowup $b : X' \\to X$ such that the kernel, image, and cokernel of $b^*\\varphi$ are perfect $\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$."}
+{"_id": "6203", "title": "flat-proposition-finite-type-injective-into-flat-mod-m", "text": "Let $A \\to B$ be a local ring homomorphism of local rings which is essentially of finite type. Let $M$ be a flat $A$-module, $N$ a finite $B$-module and $u : N \\to M$ an $A$-module map such that $\\overline{u} : N/\\mathfrak m_AN \\to M/\\mathfrak m_AM$ is injective. Then $u$ is $A$-universally injective, $N$ is of finite presentation over $B$, and $N$ is flat over $A$."}
+{"_id": "6243", "title": "curves-lemma-extend-to-completion", "text": "Let $k$ be a field. Let $X \\to Y$ be a morphism of varieties with $Y$ proper and $X$ a curve. There exists a factorization $X \\to \\overline{X} \\to Y$ where $X \\to \\overline{X}$ is an open immersion and $\\overline{X}$ is a projective curve."}
+{"_id": "6244", "title": "curves-lemma-nonsingular-model-smooth", "text": "Let $k$ be a field. Let $X$ be a curve and let $Y$ be the nonsingular projective model of $X$. If $k$ is perfect, then $Y$ is a smooth projective curve."}
+{"_id": "6268", "title": "curves-lemma-equation-plane-curve", "text": "Let $Z \\subset \\mathbf{P}^2_k$ be a closed subscheme which is equidimensional of dimension $1$ and has no embedded points (equivalently $Z$ is Cohen-Macaulay). Then the ideal $I(Z) \\subset k[T_0, T_1, T_2]$ corresponding to $Z$ is principal."}
+{"_id": "6275", "title": "curves-lemma-genus-zero-singular", "text": "Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$. Assume $X$ is singular and has genus $0$. Then there exists a diagram $$ \\xymatrix{ x' \\ar[d] \\ar[r] & X' \\ar[d]^\\nu \\ar[r] & \\Spec(k') \\ar[d] \\\\ x \\ar[r] & X \\ar[r] & \\Spec(k) } $$ where \\begin{enumerate} \\item $k'/k$ is a nontrivial finite extension, \\item $X' \\cong \\mathbf{P}^1_{k'}$, \\item $x'$ is a $k'$-rational point of $X'$, \\item $x$ is a $k$-rational point of $X$, \\item $X' \\setminus \\{x'\\} \\to X \\setminus \\{x\\}$ is an isomorphism, \\item $0 \\to \\mathcal{O}_X \\to \\nu_*\\mathcal{O}_{X'} \\to k'/k \\to 0$ is a short exact sequence where $k'/k = \\kappa(x')/\\kappa(x)$ indicates the skyscraper sheaf on the point $x$. \\end{enumerate}"}
+{"_id": "6279", "title": "curves-lemma-rhe", "text": "Notation and assumptions as in Lemma \\ref{lemma-rh}. For a closed point $x \\in X$ let $d_x$ be the multiplicity of $x$ in $R$. Then $$ 2g_X - 2 = (2g_Y - 2) \\deg(f) + \\sum\\nolimits d_x [\\kappa(x) : k] $$ Moreover, we have the following results \\begin{enumerate} \\item $d_x = \\text{length}_{\\mathcal{O}_{X, x}}(\\Omega_{X/Y, x})$, \\item $d_x \\geq e_x - 1$ where $e_x$ is the ramification index of $\\mathcal{O}_{X, x}$ over $\\mathcal{O}_{Y, y}$, \\item $d_x = e_x - 1$ if and only if $\\mathcal{O}_{X, x}$ is tamely ramified over $\\mathcal{O}_{Y, y}$. \\end{enumerate}"}
+{"_id": "6280", "title": "curves-lemma-dominated-by-smooth", "text": "Let $k$ be a field. Let $f : X \\to Y$ be a surjective morphism of curves over $k$. If $X$ is smooth over $k$ and $Y$ is normal, then $Y$ is smooth over $k$."}
+{"_id": "6284", "title": "curves-lemma-purely-inseparable-smooth-genus", "text": "Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. Assume \\begin{enumerate} \\item $X$ is smooth, \\item $H^0(X, \\mathcal{O}_X) = k$, \\item $k(X)/k(Y)$ is purely inseparable. \\end{enumerate} Then $Y$ is smooth, $H^0(Y, \\mathcal{O}_Y) = k$, and the genus of $Y$ is equal to the genus of $X$."}
+{"_id": "6287", "title": "curves-lemma-ramification-to-algebraic-closure", "text": "Let $X$ be a smooth curve over a field $k$. Let $\\overline{x} \\in X_{\\overline{k}}$ be a closed point with image $x \\in X$. The ramification index of $\\mathcal{O}_{X, x} \\subset \\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$ is the inseparable degree of $\\kappa(x)/k$."}
+{"_id": "6288", "title": "curves-lemma-complete-local-ring-pushout", "text": "In the situation above, let $Z = \\Spec(k')$ where $k'$ is a field and $Z' = \\Spec(k'_1 \\times \\ldots \\times k'_n)$ with $k'_i/k'$ finite extensions of fields. Let $x \\in X$ be the image of $Z \\to X$ and $x'_i \\in X'$ the image of $\\Spec(k'_i) \\to X'$. Then we have a fibre product diagram $$ \\xymatrix{ \\prod\\nolimits_{i = 1, \\ldots, n} k'_i & \\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_{X', x'_i}^\\wedge \\ar[l] \\\\ k' \\ar[u] & \\mathcal{O}_{X, x}^\\wedge \\ar[u] \\ar[l] } $$ where the horizontal arrows are given by the maps to the residue fields."}
+{"_id": "6295", "title": "curves-lemma-multicross-gorenstein-is-nodal", "text": "Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \\in X$ be a multicross singularity (Definition \\ref{definition-multicross}). If $X$ is Gorenstein, then $x$ is a node."}
+{"_id": "6303", "title": "curves-lemma-bound-torsion-simple", "text": "Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Let $\\ell$ be a prime number invertible in $k$. Then $$ \\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell] \\leq \\dim_k H^1(X, \\mathcal{O}_X) + g_{geom}(X/k) $$ where $g_{geom}(X/k)$ is as defined above."}
+{"_id": "6331", "title": "curves-lemma-degree-more-than-2g-1-and-Z", "text": "In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and has genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a nonempty $0$-dimensional closed subscheme. If $\\deg(\\mathcal{L}) \\geq 2g - 1 + \\deg(Z)$, then $\\mathcal{L}$ is globally generated and $H^0(X, \\mathcal{L}) \\to H^0(X, \\mathcal{L}|_Z)$ is surjective."}
+{"_id": "6371", "title": "etale-cohomology-theorem-cech-derived", "text": "On $\\textit{PAb}(\\mathcal{C})$ the functors $\\check{H}^p(\\mathcal{U}, -)$ are the right derived functors of $\\check{H}^0(\\mathcal{U}, -)$."}
+{"_id": "6375", "title": "etale-cohomology-theorem-standard-etale", "text": "A ring map $A \\to B$ is \\'etale at a prime $\\mathfrak q$ if and only if there exists $g \\in B$, $g \\not \\in \\mathfrak q$ such that $B_g$ is standard \\'etale over $A$."}
+{"_id": "6378", "title": "etale-cohomology-theorem-hensel", "text": "Complete local rings are henselian."}
+{"_id": "6391", "title": "etale-cohomology-theorem-fundamental-exact-sequence", "text": "There is a short exact sequence of \\'etale sheaves on $X$ $$ 0 \\longrightarrow \\mathbf{G}_{m, X} \\longrightarrow j_* \\mathbf{G}_{m, \\eta} \\longrightarrow \\bigoplus\\nolimits_{x \\in X_0} {i_x}_* \\underline{\\mathbf{Z}} \\longrightarrow 0. $$"}
+{"_id": "6398", "title": "etale-cohomology-lemma-yoneda", "text": "\\begin{slogan} Morphisms between objects are in bijection with natural transformations between the functors they represent. \\end{slogan} Let $\\mathcal{C}$ be a category, and $X, Y \\in \\Ob(\\mathcal{C})$. There is a natural bijection $$ \\begin{matrix} \\Mor_\\mathcal{C}(X, Y) & \\longrightarrow & \\Mor_{\\textit{PSh}(\\mathcal{C})} (h_X, h_Y) \\\\ \\psi & \\longmapsto & h_\\psi = \\psi \\circ - : h_X \\to h_Y. \\end{matrix} $$"}
+{"_id": "6402", "title": "etale-cohomology-lemma-algebra-descent", "text": "If $A \\to B$ is faithfully flat, then the complex $(B/A)_\\bullet$ is exact in positive degrees, and $H^0((B/A)_\\bullet) = A$."}
+{"_id": "6404", "title": "etale-cohomology-lemma-cech-presheaves", "text": "The functor $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)$ is exact on the category $\\textit{PAb}(\\mathcal{C})$."}
+{"_id": "6405", "title": "etale-cohomology-lemma-yoneda-presheaf", "text": "For any presheaf $\\mathcal{F}$ on a category $\\mathcal{C}$ there is a functorial isomorphism $$ \\Hom_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F}) = \\mathcal{F}(U). $$"}
+{"_id": "6406", "title": "etale-cohomology-lemma-cech-complex-describe", "text": "The {\\v C}ech complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ can be described explicitly as follows \\begin{eqnarray*} \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) & = & \\left( \\prod_{i_0 \\in I} \\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{U_{i_0}}, \\mathcal{F}) \\to \\prod_{i_0, i_1 \\in I} \\Hom_{\\textit{PAb}(\\mathcal{C})}( \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}}, \\mathcal{F}) \\to \\ldots \\right) \\\\ & = & \\Hom_{\\textit{PAb}(\\mathcal{C})}\\left( \\left( \\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow \\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow \\ldots \\right), \\mathcal{F}\\right) \\end{eqnarray*}"}
+{"_id": "6407", "title": "etale-cohomology-lemma-exact", "text": "The complex of abelian presheaves \\begin{align*} \\mathbf{Z}_\\mathcal{U}^\\bullet \\quad : \\quad \\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow \\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow \\bigoplus_{i_0, i_1, i_2 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}} \\leftarrow \\ldots \\end{align*} is exact in all degrees except $0$ in $\\textit{PAb}(\\mathcal{C})$."}
+{"_id": "6410", "title": "etale-cohomology-lemma-tau-affine", "text": "Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$. Any $\\tau$-covering of an affine scheme can be refined by a standard $\\tau$-covering."}
+{"_id": "6414", "title": "etale-cohomology-lemma-etale-topos-independent-partial-universe", "text": "Let $S$ be a scheme. The \\'etale topos of $S$ is independent (up to canonical equivalence) of the construction of the small \\'etale site in Definition \\ref{definition-tau-site}."}
+{"_id": "6418", "title": "etale-cohomology-lemma-etale-fpqc", "text": "Any \\'etale covering is an fpqc covering."}
+{"_id": "6430", "title": "etale-cohomology-lemma-support-section-closed", "text": "Let $S$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$. Let $U \\in \\Ob(S_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$. \\begin{enumerate} \\item The support of $\\sigma$ is closed in $U$. \\item The support of $\\sigma + \\sigma'$ is contained in the union of the supports of $\\sigma, \\sigma' \\in \\mathcal{F}(U)$. \\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of abelian sheaves on $S_\\etale$, then the support of $\\varphi(\\sigma)$ is contained in the support of $\\sigma \\in \\mathcal{F}(U)$. \\item The support of $\\mathcal{F}$ is the union of the images of the supports of all local sections of $\\mathcal{F}$. \\item If $\\mathcal{F} \\to \\mathcal{G}$ is surjective then the support of $\\mathcal{G}$ is a subset of the support of $\\mathcal{F}$. \\item If $\\mathcal{F} \\to \\mathcal{G}$ is injective then the support of $\\mathcal{F}$ is a subset of the support of $\\mathcal{G}$. \\end{enumerate}"}
+{"_id": "6434", "title": "etale-cohomology-lemma-describe-henselization", "text": "Let $S$ be a scheme. Let $s \\in S$. Then we have $$ \\mathcal{O}_{S, s}^h = \\colim_{(U, u)} \\mathcal{O}(U) $$ where the colimit is over the filtered category of \\'etale neighbourhoods $(U, u)$ of $(S, s)$ such that $\\kappa(s) = \\kappa(u)$."}
+{"_id": "6440", "title": "etale-cohomology-lemma-sections-upstairs-submersive", "text": "Let $S$ be a scheme. Let $f : X \\to S$ be a morphism such that \\begin{enumerate} \\item $f$ is submersive, and \\item the geometric fibres of $f$ are connected. \\end{enumerate} Let $\\mathcal{F}$ be a sheaf on $S_\\etale$. Then $\\Gamma(S, \\mathcal{F}) = \\Gamma(X, f^{-1}_{small}\\mathcal{F})$."}
+{"_id": "6458", "title": "etale-cohomology-lemma-integral-universally-injective", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume that $f$ is universally injective and integral (for example a closed immersion). Then \\begin{enumerate} \\item $f_{small, *} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ reflects injections and surjections, \\item $f_{small, *} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ commutes with pushouts and coequalizers (and more generally finite connected colimits), \\item $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves), \\item the map $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$ is surjective for any sheaf (of sets or of abelian groups) $\\mathcal{F}$ on $X_\\etale$, \\item the functor $f_{small, *}$ is faithful (on sheaves of sets and on abelian sheaves), \\item $f_{small, *} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$ is exact, and \\item the functor $Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$ is almost cocontinuous. \\end{enumerate}"}
+{"_id": "6485", "title": "etale-cohomology-lemma-modules-abelian", "text": "Let $G$ be a topological group. Let $R$ be a ring. For every $i \\geq 0$ the diagram $$ \\xymatrix{ \\text{Mod}_{R, G} \\ar[rr]_{H^i(G, -)} \\ar[d] & & \\text{Mod}_R \\ar[d] \\\\ \\text{Mod}_G \\ar[rr]^{H^i(G, -)} & & \\textit{Ab} } $$ whose vertical arrows are the forgetful functors is commutative."}
+{"_id": "6492", "title": "etale-cohomology-lemma-brauer-inverse", "text": "Let $A$ be a finite central simple algebra over $K$. Then $$ \\begin{matrix} A \\otimes_K A^{opp} & \\longrightarrow & \\text{End}_K(A) \\\\ \\ a \\otimes a' & \\longmapsto & (x \\mapsto a x a') \\end{matrix} $$ is an isomorphism of algebras over $K$."}
+{"_id": "6495", "title": "etale-cohomology-lemma-annihilated-by-degree", "text": "\\begin{reference} Argument taken from \\cite{Saltman-torsion}. \\end{reference} Let $S$ be a scheme. Let $\\mathcal{A}$ be an Azumaya algebra which is locally free of rank $d^2$ over $S$. Then the class of $\\mathcal{A}$ in the Brauer group of $S$ is annihilated by $d$."}
+{"_id": "6528", "title": "etale-cohomology-lemma-constructible-constructible", "text": "Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$ be a sheaf of sets, abelian groups, $\\Lambda$-modules (with $\\Lambda$ Noetherian) on $X_\\etale$. If there exist constructible locally closed subschemes $T_i \\subset X$ such that (a) $X = \\bigcup T_j$ and (b) $\\mathcal{F}|_{T_j}$ is constructible, then $\\mathcal{F}$ is constructible."}
+{"_id": "6567", "title": "etale-cohomology-lemma-cohomology-with-support-sheaf-on-support", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. Let $\\mathcal{G}$ be an injective abelian sheaf on $Z_\\etale$. Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$."}
+{"_id": "6569", "title": "etale-cohomology-lemma-complexes-with-support-on-closed", "text": "Let $i : Z \\to X$ be a closed immersion of schemes. The map $Ri_{small, *} = i_{small, *} : D(Z_\\etale) \\to D(X_\\etale)$ induces an equivalence $D(Z_\\etale) \\to D_Z(X_\\etale)$ with quasi-inverse $$ i_{small}^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_Z(X_\\etale)} $$"}
+{"_id": "6572", "title": "etale-cohomology-lemma-affine-only-closed-points", "text": "Let $S$ be an affine scheme such that (1) all points are closed, and (2) all residue fields are separably algebraically closed. Then for any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have $H^i(S_\\etale, \\mathcal{F}) = 0$ for $i > 0$."}
+{"_id": "6597", "title": "etale-cohomology-lemma-base-change-f-star-general-stalks", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ where $g : T \\to S$ is quasi-compact and quasi-separated. Let $\\mathcal{F}$ be an abelian sheaf on $T_\\etale$. Let $q \\geq 0$. The following are equivalent \\begin{enumerate} \\item For every geometric point $\\overline{x}$ of $X$ with image $\\overline{s} = f(\\overline{x})$ we have $$ H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S T, \\mathcal{F}) = H^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S T, \\mathcal{F}) $$ \\item $f^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "6628", "title": "etale-cohomology-lemma-base-change-separably-closed", "text": "Let $K/k$ be an extension of separably closed fields. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$. Then the map $H^q_\\etale(X, \\mathcal{F}) \\to H^q_\\etale(X_K, \\mathcal{F}|_{X_K})$ is an isomorphism for $q \\geq 0$."}
+{"_id": "6651", "title": "etale-cohomology-lemma-kunneth", "text": "Let $k$ be a separably closed field. Let $X$ and $Y$ be finite type schemes over $k$. Let $n \\geq 1$ be an integer invertible in $k$. Then for $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and $K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ we have $$ R\\Gamma(X \\times_{\\Spec(k)} Y, \\text{pr}_1^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} \\text{pr}_2^{-1}K ) = R\\Gamma(X, E) \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} R\\Gamma(Y, K) $$"}
+{"_id": "6665", "title": "etale-cohomology-lemma-compare-cohomology-etale-fppf", "text": "For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above: \\begin{enumerate} \\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$ for an abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$. \\end{enumerate} Example: if $A$ is an abelian group, then $H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$."}
+{"_id": "6667", "title": "etale-cohomology-lemma-cohomological-descent-etale-fppf-modules", "text": "Let $S$ be a scheme. For $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_S$-module on $S_\\etale$ the maps $$ \\pi_S^*\\mathcal{F} \\longrightarrow R\\epsilon_{S, *}(a_S^*\\mathcal{F}) \\quad\\text{and}\\quad \\mathcal{F} \\longrightarrow Ra_{S, *}(a_S^*\\mathcal{F}) $$ are isomorphisms with $a_S : \\Sh((\\Sch/S)_{fppf}) \\to \\Sh(S_\\etale)$ as above."}
+{"_id": "6670", "title": "etale-cohomology-lemma-proper-push-pull-ph-etale", "text": "In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have $a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$."}
+{"_id": "6677", "title": "etale-cohomology-lemma-proper-push-pull-h-etale", "text": "In Lemma \\ref{lemma-push-pull-h-etale} if $f$ is proper, then we have $a_Y^{-1} \\circ f_{small, *} = f_{big, h, *} \\circ a_X^{-1}$."}
+{"_id": "6681", "title": "etale-cohomology-lemma-compare-cohomology-etale-h", "text": "For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$ as above: \\begin{enumerate} \\item $H^q(X_\\etale, \\mathcal{F}) = H^q_h(X, a_X^{-1}\\mathcal{F})$ for a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$, \\item $H^q(X_\\etale, K) = H^q_h(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves. \\end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\\etale(X, \\underline{A}) = H^q_h(X, \\underline{A})$."}
+{"_id": "6687", "title": "etale-cohomology-lemma-glue-etale-sheaf-fppf", "text": "Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes. The functor $$ \\Sh(X_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\} $$ is an equivalence of categories."}
+{"_id": "6691", "title": "etale-cohomology-lemma-blow-up-square-etale-cohomology", "text": "Let $X$ be a scheme and let $K \\in D^+(X_\\etale)$ have torsion cohomology sheaves. Let $Z \\subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square $$ \\xymatrix{ E \\ar[d] \\ar[r] & X' \\ar[d]^b \\\\ Z \\ar[r] & X } $$ Then there is a canonical long exact sequence $$ H^p_\\etale(X, K) \\to H^p_\\etale(X', K|_{X'}) \\oplus H^p_\\etale(Z, K|_Z) \\to H^p_\\etale(E, K|_E) \\to H^{p + 1}_\\etale(X, K) $$"}
+{"_id": "6698", "title": "etale-cohomology-proposition-cohomology-restrict-small-site", "text": "Let $S$ be a scheme and $\\mathcal{F}$ an abelian sheaf on $(\\Sch/S)_\\etale$. Then $\\mathcal{F}|_{S_\\etale}$ is a sheaf on $S_\\etale$ and $$ H^p_\\etale(S, \\mathcal{F}|_{S_\\etale}) = H^p_\\etale(S, \\mathcal{F}) $$ for all $p \\geq 0$."}
+{"_id": "6799", "title": "equiv-theorem-bondal-van-den-bergh", "text": "\\begin{reference} \\cite[Theorem A.1]{BvdB} \\end{reference} Let $X$ be a projective scheme over a field $k$. Let $F : D_{perf}(\\mathcal{O}_X)^{opp} \\to \\text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k F(E[n]) < \\infty $$ for all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \\mapsto \\Hom_X(E, K)$ for some $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$."}
+{"_id": "6801", "title": "equiv-theorem-countable", "text": "\\begin{reference} Slight improvement of \\cite{AT} \\end{reference} Let $K$ be an algebraically closed field. Let $\\mathbf{X}$ be a smooth proper scheme over $K$. There are at most countably many isomorphism classes of smooth proper schemes $\\mathbf{Y}$ over $K$ which are derived equivalent to $\\mathbf{X}$."}
+{"_id": "6803", "title": "equiv-lemma-Serre-functor", "text": "In the situation of Definition \\ref{definition-Serre-functor}. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories."}
+{"_id": "6804", "title": "equiv-lemma-Serre-functor-Gorenstein-proper", "text": "Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein. Consider the complex $\\omega_X^\\bullet$ of Duality for Schemes, Lemmas \\ref{duality-lemma-duality-proper-over-field}. Then the functor $$ S : D_{perf}(\\mathcal{O}_X) \\longrightarrow D_{perf}(\\mathcal{O}_X),\\quad K \\longmapsto S(K) = \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K $$ is a Serre functor."}
+{"_id": "6823", "title": "equiv-lemma-on-product-general", "text": "Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated schemes over $R$ having the resolution property. For any finite type quasi-coherent $\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$ there exist a surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$ where $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module and $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module."}
+{"_id": "6927", "title": "stacks-more-morphisms-proposition-affine-smooth-lift-to-first-order", "text": "\\begin{reference} Email of Matthew Emerton dated April 27, 2016. \\end{reference} Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a first order thickening of algebraic stacks. Let $W$ be an affine scheme and let $W \\to \\mathcal{X}$ be a smooth morphism. Then there exists a cartesian diagram $$ \\xymatrix{ W \\ar[d] \\ar[r] & W' \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{X}' } $$ with $W' \\to \\mathcal{X}'$ smooth and $W'$ affine."}
+{"_id": "6955", "title": "perfect-lemma-support-quasi-coherent", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is a retrocompact open of $X$. Let $i : T \\to X$ be the inclusion. \\begin{enumerate} \\item For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ we have $i_*R\\mathcal{H}_T(E)$ in $D_{\\QCoh, T}(\\mathcal{O}_X)$. \\item The functor $i_* \\circ R\\mathcal{H}_T : D_\\QCoh(\\mathcal{O}_X) \\to D_{\\QCoh, T}(\\mathcal{O}_X)$ is right adjoint to the inclusion functor $D_{\\QCoh, T}(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_X)$. \\end{enumerate}"}
+{"_id": "6956", "title": "perfect-lemma-support-direct-sums", "text": "Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that $X \\setminus T$ is a retrocompact open of $X$. Then for a family of objects $E_i$, $i \\in I$ of $D_\\QCoh(\\mathcal{O}_X)$ we have $R\\mathcal{H}_T(\\bigoplus E_i) = \\bigoplus R\\mathcal{H}_T(E_i)$."}
+{"_id": "6962", "title": "perfect-lemma-supported-map-global", "text": "Let $X$ be a scheme. Let $Z \\to X$ be a closed immersion of finite presentation whose conormal sheaf $\\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map $$ c : \\wedge^c(\\mathcal{C}_{Z/X})^\\vee \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{F} \\longrightarrow \\mathcal{H}_Z^c(\\mathcal{F}) $$ functorial in the quasi-coherent module $\\mathcal{F}$."}
+{"_id": "6967", "title": "perfect-lemma-direct-image-coherator", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $X$ and $Y$ are quasi-compact and have affine diagonal. Then, denoting $$ \\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y)) $$ the right derived functor of $f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$ the diagram $$ \\xymatrix{ D(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\ D(\\QCoh(\\mathcal{O}_Y)) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) } $$ is commutative."}
+{"_id": "7037", "title": "perfect-lemma-kunneth-Ext", "text": "In the situation above, assume $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor independent over $S$. If $K$ is perfect, $K' \\in D_\\QCoh(\\mathcal{O}_X)$, $M$ is perfect, and $M' \\in D_\\QCoh(\\mathcal{O}_Y)$, then (\\ref{equation-kunneth-ext}) is an isomorphism."}
+{"_id": "7056", "title": "perfect-lemma-pullback-and-limits", "text": "Let $R$ be a ring. Let $X$ be a scheme and let $f : X \\to \\Spec(R)$ be proper, flat, and of finite presentation. Let $(M_n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map $$ \\mathcal{O}_X \\otimes_R (\\lim M_n) \\longrightarrow \\lim \\mathcal{O}_X \\otimes_R M_n $$ induces an isomorphism from the source to $DQ_X$ applied to the target."}
+{"_id": "7112", "title": "perfect-proposition-perfect-resolution-property", "text": "Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Denote \\begin{enumerate} \\item $\\mathcal{A}$ the additive category of finite locally free $\\mathcal{O}_X$-modules, \\item $K^b(\\mathcal{A})$ the homotopy category of bounded complexes in $\\mathcal{A}$, see Derived Categories, Section \\ref{derived-section-homotopy}, and \\item $D_{perf}(\\mathcal{O}_X)$ the strictly full, saturated, triangulated subcategory of $D(\\mathcal{O}_X)$ consisting of perfect objects. \\end{enumerate} With this notation the obvious functor $$ K^b(\\mathcal{A}) \\longrightarrow D_{perf}(\\mathcal{O}_X) $$ is an exact functor of trianglated categories which factors through an equivalence $S^{-1}K^b(\\mathcal{A}) \\to D_{perf}(\\mathcal{O}_X)$ of triangulated categories where $S$ is the saturated multiplicative system of quasi-isomorphisms in $K^b(\\mathcal{A})$."}
+{"_id": "7146", "title": "spaces-flat-theorem-flattening-map", "text": "In Situation \\ref{situation-iso} assume \\begin{enumerate} \\item $f$ is of finite presentation, \\item $\\mathcal{F}$ is of finite presentation, flat over $B$, and pure relative to $B$, and \\item $u$ is surjective. \\end{enumerate} Then $F_{iso}$ is representable by a closed immersion $Z \\to B$. Moreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is of finite presentation."}
+{"_id": "7159", "title": "spaces-flat-lemma-open-in-fibre-where-flat", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type. Let $y \\in |Y|$ and $F = f^{-1}(\\{y\\}) \\subset |X|$. Then the set $$ \\{x \\in F \\mid \\mathcal{F} \\text{ flat over }Y\\text{ at }x\\} $$ is open in $F$."}
+{"_id": "7163", "title": "spaces-flat-lemma-check-along-closed-fibre", "text": "Let $S$ be a local scheme with closed point $s$. Let $f : X \\to S$ be a morphism from an algebraic space $X$ to $S$ which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Assume that \\begin{enumerate} \\item every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ specializes to a point of the closed fibre $X_s$\\footnote{For example this holds if $f$ is finite type and $\\mathcal{F}$ is pure along $X_s$, or if $f$ is proper.}, \\item $\\mathcal{F}$ is flat over $S$ at every point of $X_s$. \\end{enumerate} Then $\\mathcal{F}$ is flat over $S$."}
+{"_id": "7180", "title": "spaces-flat-lemma-F-zero-somewhat-closed-points", "text": "In Situation \\ref{situation-somewhat-closed}. Let $T \\subset S$ be a subset. Let $s \\in S$ be in the closure of $T$. For $t \\in T$, let $u_t$ be the pullback of $u$ to $X_t$ and let $u_s$ be the pullback of $u$ to $X_s$. If $X$ is locally of finite presentation over $S$, $\\mathcal{G}$ is of finite presentation\\footnote{It would suffice if $X$ is locally of finite type over $S$ and $\\mathcal{G}$ is finitely presented relative to $S$, but this notion hasn't yet been defined in the setting of algebraic spaces. The definition for schemes is given in More on Morphisms, Section \\ref{more-morphisms-section-finite-type-finite-presentation}.}, and $u_t = 0$ for all $t \\in T$, then $u_s = 0$."}
+{"_id": "7183", "title": "spaces-flat-lemma-F-iso-closed", "text": "In Situation \\ref{situation-iso}. Assume \\begin{enumerate} \\item $f$ is locally of finite presentation, \\item $\\mathcal{F}$ is locally of finite presentation and flat over $B$, \\item the support of $\\mathcal{F}$ is proper over $B$, and \\item $u$ is surjective. \\end{enumerate} Then the functor $F_{iso}$ is an algebraic space and $F_{iso} \\to B$ is a closed immersion. If $\\mathcal{G}$ is of finite presentation, then $F_{iso} \\to B$ is of finite presentation."}
+{"_id": "7185", "title": "spaces-flat-lemma-freebie", "text": "Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \\to S$ be a morphism of algebraic spaces which is locally of finite type. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $E \\subset |X_s|$ be a subset. There exists a closed subscheme $Z \\subset S$ with the following property: for any morphism of pointed schemes $(T, t) \\to (S, s)$ the following are equivalent \\begin{enumerate} \\item $\\mathcal{F}_T$ is flat over $T$ at all points of $|X_t|$ which map to a point of $E \\subset |X_s|$, and \\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$. \\end{enumerate} Moreover, if $X \\to S$ is locally of finite presentation, $\\mathcal{F}$ is of finite presentation, and $E \\subset |X_s|$ is closed and quasi-compact, then $Z \\to S$ is of finite presentation."}
+{"_id": "7213", "title": "spaces-chow-lemma-additivity-sheaf-cycle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$ be a short exact sequence of coherent $\\mathcal{O}_X$-modules. Assume that the $\\delta$-dimension of the supports of $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ are $\\leq k$. Then $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$."}
+{"_id": "7226", "title": "spaces-chow-lemma-finite-flat", "text": "In Situation \\ref{situation-setup} let $X, Y/B$ be good. Let $f : X \\to Y$ be a finite locally free morphism of degree $d$ (see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-finite-locally-free}). Then $f$ is both proper and flat of relative dimension $0$, and $$ f_*f^*\\alpha = d\\alpha $$ for every $\\alpha \\in Z_k(Y)$."}
+{"_id": "7267", "title": "spaces-chow-lemma-vectorbundle", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $$ p : E = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E})) \\longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$."}
+{"_id": "7270", "title": "spaces-chow-lemma-cap-commutative-chern", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. Then $c_j(\\mathcal{L}) \\in A^j(X)$ commutes with every element $c \\in A^p(X)$. In particular, if $\\mathcal{F}$ is a second locally free $\\mathcal{O}_X$-module on $X$ of rank $s$, then $$ c_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha = c_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha $$ as elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$."}
+{"_id": "7274", "title": "spaces-chow-lemma-additivity-chern-classes", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Suppose that $\\mathcal{E}$ sits in an exact sequence $$ 0 \\to \\mathcal{E}_1 \\to \\mathcal{E} \\to \\mathcal{E}_2 \\to 0 $$ of finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$. The total Chern classes satisfy $$ c(\\mathcal{E}) = c(\\mathcal{E}_1) c(\\mathcal{E}_2) $$ in $A^*(X)$."}
+{"_id": "7278", "title": "spaces-chow-lemma-chern-classes-tensor-product", "text": "In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free $\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have $$ c_1(\\mathcal{E} \\otimes \\mathcal{F}) = r c_1(\\mathcal{F}) + s c_1(\\mathcal{E}) $$ $$ c_2(\\mathcal{E} \\otimes \\mathcal{F}) = r^2 c_2(\\mathcal{F}) + rs c_1(\\mathcal{F})c_1(\\mathcal{E}) + s^2 c_2(\\mathcal{E}) $$ and so on (see proof)."}
+{"_id": "7280", "title": "spaces-chow-lemma-degrees-and-numerical-intersections", "text": "Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $Z \\subset X$ be a closed subspace of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules. Then $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) = \\deg( c_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_1) \\cap [Z]_d) $$ where the left hand side is defined in Spaces over Fields, Definition \\ref{spaces-over-fields-definition-intersection-number}."}
+{"_id": "7327", "title": "sdga-lemma-good-quotient", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module. There exists a homomorphism $\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules with the following properties \\begin{enumerate} \\item $\\mathcal{P} \\to \\mathcal{M}$ is surjective, \\item $\\Ker(\\text{d}_\\mathcal{P}) \\to \\Ker(\\text{d}_\\mathcal{M})$ is surjective, and \\item $\\mathcal{P}$ is good. \\end{enumerate}"}
+{"_id": "7350", "title": "sdga-lemma-homotopy-colimit", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}_n$ be a system of differential graded $\\mathcal{A}$-modules. Then the derived colimit $\\text{hocolim} \\mathcal{M}_n$ in $D(\\mathcal{A}, \\text{d})$ is represented by the differential graded module $\\colim \\mathcal{M}_n$."}
+{"_id": "7368", "title": "sdga-proposition-homotopy-category-triangulated", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$. The homotopy category $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ is a triangulated category where \\begin{enumerate} \\item the shift functors are those constructed in Section \\ref{section-shift-dg}, \\item the distinghuished triangles are those triangles in $K(\\text{Mod}_{(\\mathcal{A}, \\text{d})})$ which are isomorphic as a triangle to a triangle $$ \\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N} \\xrightarrow{\\delta} \\mathcal{K}[1],\\quad\\quad \\delta = \\pi \\circ \\text{d}_\\mathcal{L} \\circ s $$ constructed from an admissible short exact sequence $0 \\to \\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N} \\to 0$ in $\\text{Mod}_{(\\mathcal{A}, \\text{d})}$ above. \\end{enumerate}"}
+{"_id": "7434", "title": "stacks-morphisms-lemma-composition-affine", "text": "Compositions of affine morphisms of algebraic stacks are affine."}
+{"_id": "7469", "title": "stacks-morphisms-lemma-finite-type-points-surjective-morphism", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type and surjective, then $f(\\mathcal{X}_{\\text{ft-pts}}) = \\mathcal{Y}_{\\text{ft-pts}}$."}
+{"_id": "7484", "title": "stacks-morphisms-lemma-representable-by-spaces-quasi-finite", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is representable by algebraic spaces. The following are equivalent \\begin{enumerate} \\item $f$ is locally quasi-finite (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item $f$ is locally of finite type and for every morphism $\\Spec(k) \\to \\mathcal{Y}$ where $k$ is a field the space $|\\Spec(k) \\times_\\mathcal{Y} \\mathcal{X}|$ is discrete. \\end{enumerate}"}
+{"_id": "7489", "title": "stacks-morphisms-lemma-characterize-locally-quasi-finite", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is locally quasi-finite, \\item $f$ is quasi-DM and for any morphism $V \\to \\mathcal{Y}$ with $V$ an algebraic space and any locally quasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where $U$ is an algebraic space the morphism $U \\to V$ is locally quasi-finite, \\item for any morphism $V \\to \\mathcal{Y}$ from an algebraic space $V$ there exists a surjective, flat, locally finitely presented, and locally quasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where $U$ is an algebraic space such that $U \\to V$ is locally quasi-finite, \\item there exists algebraic spaces $U$, $V$, a surjective, flat, and locally of finite presentation morphism $V \\to \\mathcal{Y}$, and a morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite such that $U \\to V$ is locally quasi-finite. \\end{enumerate}"}
+{"_id": "7491", "title": "stacks-morphisms-lemma-composition-quasi-finite", "text": "The composition of quasi-finite morphisms is quasi-finite."}
+{"_id": "7520", "title": "stacks-morphisms-lemma-gerbe-with-section", "text": "Let $\\pi : \\mathcal{X} \\to U$ be a morphism from an algebraic stack to an algebraic space and let $x : U \\to \\mathcal{X}$ be a section of $\\pi$. Set $G = \\mathit{Isom}_\\mathcal{X}(x, x)$, see Definition \\ref{definition-isom}. If $\\mathcal{X}$ is a gerbe over $U$, then \\begin{enumerate} \\item there is a canonical equivalence of stacks in groupoids $$ x_{can} : [U/G] \\longrightarrow \\mathcal{X}. $$ where $[U/G]$ is the quotient stack for the trivial action of $G$ on $U$, \\item $G \\to U$ is flat and locally of finite presentation, and \\item $U \\to \\mathcal{X}$ is surjective, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7530", "title": "stacks-morphisms-lemma-spectral-qcqs", "text": "Let $\\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack. Then $|\\mathcal{X}|$ is a spectral topological space."}
+{"_id": "7531", "title": "stacks-morphisms-lemma-sober-qs", "text": "Let $\\mathcal{X}$ be an algebraic stack whose diagonal is quasi-compact (for example if $\\mathcal{X}$ is quasi-separated). Then there is an open covering $|\\mathcal{X}| = \\bigcup U_i$ with $U_i$ spectral. In particular $|\\mathcal{X}|$ is a sober topological space."}
+{"_id": "7538", "title": "stacks-morphisms-lemma-etale-local-quasi-DM-at-x-inertia", "text": "Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is quasi-DM with separated diagonal (equivalently $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and separated). Let $x \\in |\\mathcal{X}|$. Assume $x$ can be represented by a quasi-compact morphism $\\Spec(k) \\to \\mathcal{X}$. Then there exists a morphism of algebraic stacks $$ g : \\mathcal{U} \\longrightarrow \\mathcal{X} $$ with the following properties \\begin{enumerate} \\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and $g$ induces an isomorphism between the residual gerbes at $u$ and $x$, \\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and \\'etale, \\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid scheme with $U$, $R$ affine, and $s, t$ finite, flat, and locally of finite presentation. \\end{enumerate}"}
+{"_id": "7547", "title": "stacks-morphisms-lemma-composition-etale", "text": "The composition of \\'etale morphisms is \\'etale."}
+{"_id": "7550", "title": "stacks-morphisms-lemma-etale", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is \\'etale, \\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$ where $V$ is an algebraic space and any \\'etale morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space, the morphism $U \\to V$ is \\'etale, \\item there exists some surjective, locally of finite presentation, and flat morphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some surjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \\to W$ is \\'etale. \\end{enumerate}"}
+{"_id": "7556", "title": "stacks-morphisms-lemma-unramified", "text": "Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is unramified, \\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$ where $V$ is an algebraic space and any \\'etale morphism $U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space, the morphism $U \\to V$ is unramified, \\item there exists some surjective, locally of finite presentation, and flat morphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some surjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \\to W$ is unramified. \\end{enumerate}"}
+{"_id": "7557", "title": "stacks-morphisms-lemma-permanence-unramified", "text": "Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is unramified and $\\mathcal{Y} \\to \\mathcal{Z}$ is DM, then $\\mathcal{X} \\to \\mathcal{Y}$ is unramified."}
+{"_id": "7560", "title": "stacks-morphisms-lemma-base-change-proper", "text": "A base change of a proper morphism is proper."}
+{"_id": "7576", "title": "stacks-morphisms-lemma-composition-uniqueness", "text": "The composition of morphisms of algebraic stacks which satisfy the uniqueness part of the valuative criterion is another morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion."}
+{"_id": "7590", "title": "stacks-morphisms-lemma-flat-base-change-lci", "text": "A flat base change of a local complete intersection morphism is a local complete intersection morphism."}
+{"_id": "7666", "title": "schemes-lemma-colimit-quasi-coherent", "text": "Let $X = \\Spec(R)$ be an affine scheme. The direct sum of an arbitrary collection of quasi-coherent sheaves on $X$ is quasi-coherent. The same holds for colimits."}
+{"_id": "7713", "title": "schemes-lemma-section-immersion", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $s : S \\to X$ be a section of $f$ (in a formula $f \\circ s = \\text{id}_S$). Then $s$ is an immersion. If $f$ is separated then $s$ is a closed immersion. If $f$ is quasi-separated, then $s$ is quasi-compact."}
+{"_id": "7718", "title": "schemes-lemma-curiosity", "text": "Let $f : X \\to S$ be a morphism. Assume $f$ is separated and $S$ is a separated scheme. Suppose $U \\subset X$ and $V \\subset X$ are affine. Then $U \\cap V$ is affine (and a closed subscheme of $U \\times V$)."}
+{"_id": "7725", "title": "schemes-lemma-injective-points", "text": "Let $j : X \\to Y$ be a morphism of schemes. If $j$ is injective on points, then $j$ is separated."}
+{"_id": "7836", "title": "brauer-lemma-base-change", "text": "Let $A$ be a finite central simple algebra over $k$. Let $k \\subset k'$ be a field extension. Then $A' = A \\otimes_k k'$ is a finite central simple algebra over $k'$."}
+{"_id": "7841", "title": "brauer-lemma-automorphism-inner", "text": "Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is inner. In particular, any automorphism of $\\text{Mat}(n \\times n, k)$ is inner."}
+{"_id": "7846", "title": "brauer-lemma-splitting-field-degree", "text": "Consider a finite central skew field $K$ over $k$. Let $d^2 = [K : k]$. For any finite splitting field $k'$ for $K$ the degree $[k' : k]$ is divisible by $d$."}
+{"_id": "7889", "title": "divisors-lemma-relative-assassin-affine-open", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $U \\subset X$ and $V \\subset S$ be affine opens with $f(U) \\subset V$. Write $U = \\Spec(A)$, $V = \\Spec(R)$, and set $M = \\Gamma(U, \\mathcal{F})$. Let $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime. Then $$ \\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Rightarrow x \\in \\text{Ass}_{X/S}(\\mathcal{F}) $$ If all fibres $X_s$ of $f$ are locally Noetherian, then $\\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Leftrightarrow x \\in \\text{Ass}_{X/S}(\\mathcal{F})$ for all pairs $(\\mathfrak p, x)$ as above."}
+{"_id": "7905", "title": "divisors-lemma-torsion", "text": "Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The torsion sections of $\\mathcal{F}$ form a quasi-coherent $\\mathcal{O}_X$-submodule $\\mathcal{F}_{tors} \\subset \\mathcal{F}$. The quotient module $\\mathcal{F}/\\mathcal{F}_{tors}$ is torsion free."}
+{"_id": "7907", "title": "divisors-lemma-flat-pullback-torsion", "text": "Let $f : X \\to Y$ be a flat morphism of integral schemes. Let $\\mathcal{G}$ be a torsion free quasi-coherent $\\mathcal{O}_Y$-module. Then $f^*\\mathcal{G}$ is a torsion free $\\mathcal{O}_X$-module."}
+{"_id": "7916", "title": "divisors-lemma-reflexive-torsion-free", "text": "Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If $\\mathcal{F}$ is reflexive, then $\\mathcal{F}$ is torsion free. \\item The map $j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$ is injective if and only if $\\mathcal{F}$ is torsion free \\end{enumerate}"}
+{"_id": "7945", "title": "divisors-lemma-regular-section-associated-points", "text": "Let $X$ be a locally Noetherian scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Then $s$ is a regular section if and only if $s$ does not vanish in the associated points of $X$."}
+{"_id": "7976", "title": "divisors-lemma-flat-relative-Cartier-divisor", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $D \\subset X$ be a relative effective Cartier divisor. If $f$ is locally of finite presentation, then there exists an open subscheme $U \\subset X$ such that $D \\subset U$ and such that $f|_U : U \\to S$ is flat."}
+{"_id": "7977", "title": "divisors-lemma-michael-artin", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $D \\subset X$ be a relative effective Cartier divisor on $X/S$. If $f$ is flat at all points of $X \\setminus D$, then $f$ is flat."}
+{"_id": "7997", "title": "divisors-lemma-koszul-regular-smooth-locally-regular", "text": "Let $i : Z \\to X$ be a Koszul regular closed immersion. Then there exists a surjective smooth morphism $X' \\to X$ such that the base change $i' : Z \\times_X X' \\to X'$ of $i$ is a regular immersion."}
+{"_id": "8002", "title": "divisors-lemma-flat-relative-H1-regular", "text": "Let $X \\to S$ be a morphism of schemes. Let $Z \\to X$ be a relative $H_1$-regular immersion. Assume $X \\to S$ is locally of finite presentation. Then \\begin{enumerate} \\item there exists an open subscheme $U \\subset X$ such that $Z \\subset U$ and such that $U \\to S$ is flat, and \\item $Z \\to X$ is a regular immersion and the same remains true after any base change. \\end{enumerate}"}
+{"_id": "8004", "title": "divisors-lemma-section-smooth-regular-immersion", "text": "Let $f : X \\to S$ be a smooth morphism of schemes. Let $\\sigma : S \\to X$ be a section of $f$. Then $\\sigma$ is a regular immersion."}
+{"_id": "8014", "title": "divisors-lemma-locally-Noetherian-K", "text": "Let $X$ be a locally Noetherian scheme. \\begin{enumerate} \\item For any $x \\in X$ we have $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$ is the set of nonzerodivisors, and hence $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$. \\item For any affine open $U \\subset X$ the ring $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$. \\end{enumerate}"}
+{"_id": "8019", "title": "divisors-lemma-reduced-normalization", "text": "Let $X$ be a scheme. Assume $X$ is reduced and any quasi-compact open $U \\subset X$ has a finite number of irreducible components. Then the normalization morphism $\\nu : X^\\nu \\to X$ is the morphism $$ \\underline{\\Spec}_X(\\mathcal{O}') \\longrightarrow X $$ where $\\mathcal{O}' \\subset \\mathcal{K}_X$ is the integral closure of $\\mathcal{O}_X$ in the sheaf of meromorphic functions."}
+{"_id": "8021", "title": "divisors-lemma-regular-meromorphic-section-exists", "text": "Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. In each of the following cases $\\mathcal{L}$ has a regular meromorphic section: \\begin{enumerate} \\item $X$ is integral, \\item $X$ is reduced and any quasi-compact open has a finite number of irreducible components, \\item $X$ is locally Noetherian and has no embedded points. \\end{enumerate}"}
+{"_id": "8035", "title": "divisors-lemma-reflexive-normal", "text": "Let $X$ be an integral locally Noetherian normal scheme. For $\\mathcal{F}$ and $\\mathcal{G}$ coherent reflexive $\\mathcal{O}_X$-modules the map $$ (\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**} \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) $$ is an isomorphism. The rule $\\mathcal{F}, \\mathcal{G} \\mapsto (\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**}$ defines an abelian group law on the set of isomorphism classes of rank $1$ coherent reflexive $\\mathcal{O}_X$-modules."}
+{"_id": "8036", "title": "divisors-lemma-normal-class-group", "text": "Let $X$ be an integral locally Noetherian normal scheme. The group of rank $1$ coherent reflexive $\\mathcal{O}_X$-modules is isomorphic to the Weil divisor class group $\\text{Cl}(X)$ of $X$."}
+{"_id": "8039", "title": "divisors-lemma-affine-Xs", "text": "Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}. The following are equivalent \\begin{enumerate} \\item the inclusion morphism $j : U \\to X$ is affine, and \\item for every $x \\in X \\setminus U$ there is an $n > 0$ such that $s^n \\in \\mathfrak m_x \\mathcal{F}^{[n]}_x$. \\end{enumerate}"}
+{"_id": "8146", "title": "spaces-lemma-algebraic-space-coproduct-sheaves", "text": "Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ be an algebraic space over $S$. Given a set $I$ and sheaves $F_i$ on $\\Ob((\\Sch/S)_{fppf})$, if $F \\cong \\coprod_{i\\in I} F_i$ as sheaves, then each $F_i$ is an algebraic space over $S$."}
+{"_id": "8243", "title": "topology-lemma-baire-category-locally-compact", "text": "Let $X$ be a locally quasi-compact Hausdorff space. Let $U_n \\subset X$, $n \\geq 1$ be dense open subsets. Then $\\bigcap_{n \\geq 1} U_n$ is dense in $X$."}
+{"_id": "8336", "title": "topology-lemma-noetherian-partition-refined-by-stratification", "text": "Let $X$ be a Noetherian topological space. Any finite partition of $X$ can be refined by a finite good stratification."}
+{"_id": "8342", "title": "topology-lemma-topological-ring-colimits", "text": "The category of topological rings has colimits and colimits commute with the forgetful functor to the category of rings."}
+{"_id": "8344", "title": "topology-lemma-topological-module-colimits", "text": "Let $R$ be a topological ring. The category of topological modules over $R$ has colimits and colimits commute with the forgetful functor to the category of modules over $R$."}
+{"_id": "8406", "title": "hypercovering-lemma-covering-sheaf", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $n \\geq 0$ be an integer. Let $u : \\mathcal{F} \\to F(K_n)$ be a morphism of presheaves which becomes surjective on sheafification. Then there exists a morphism of hypercoverings $f: L \\to K$ such that $F(f_n) : F(L_n) \\to F(K_n)$ factors through $u$."}
+{"_id": "8411", "title": "hypercovering-lemma-homotopy", "text": "Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. Let $K, L$ be hypercoverings of $X$. Let $a, b : K \\to L$ be morphisms of hypercoverings. There exists a morphism of hypercoverings $c : K' \\to K$ such that $a \\circ c$ is homotopic to $b \\circ c$."}
+{"_id": "8420", "title": "hypercovering-proposition-cohomology-hypercoverings", "text": "Let $\\mathcal{C}$ be a site with fibre products and products of pairs. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$. Let $i \\geq 0$. Then \\begin{enumerate} \\item for every $\\xi \\in H^i(\\mathcal{F})$ there exists a hypercovering $K$ such that $\\xi$ is in the image of the canonical map $\\check{H}^i(K, \\mathcal{F}) \\to H^i(\\mathcal{F})$, and \\item if $K, L$ are hypercoverings and $\\xi_K \\in \\check{H}^i(K, \\mathcal{F})$, $\\xi_L \\in \\check{H}^i(L, \\mathcal{F})$ are elements mapping to the same element of $H^i(\\mathcal{F})$, then there exists a hypercovering $M$ and morphisms $M \\to K$ and $M \\to L$ such that $\\xi_K$ and $\\xi_L$ map to the same element of $\\check{H}^i(M, \\mathcal{F})$. \\end{enumerate} In other words, modulo set theoretical issues, the cohomology groups of $\\mathcal{F}$ on $\\mathcal{C}$ are the colimit of the {\\v C}ech cohomology groups of $\\mathcal{F}$ over all hypercoverings."}
+{"_id": "8440", "title": "algebraic-lemma-product-representable-transformations", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$ be representable $1$-morphisms. Then $$ f_1 \\times f_2 : \\mathcal{X}_1 \\times \\mathcal{X}_2 \\longrightarrow \\mathcal{Y}_1 \\times \\mathcal{Y}_2 $$ is a representable $1$-morphism."}
+{"_id": "8451", "title": "algebraic-lemma-get-a-stack", "text": "\\begin{reference} Lemma in an email of Matthew Emerton dated June 15, 2016 \\end{reference} Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X} \\to \\mathcal{Z}$ is representable by algebraic spaces and $\\mathcal{Y}$ is a stack in groupoids, then $\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}$ is a stack in groupoids."}
+{"_id": "8457", "title": "algebraic-lemma-descent-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property}. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \\xymatrix{ \\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\ \\mathcal{Z} \\ar[r]^g & \\mathcal{Y} } $$ Assume that for every scheme $U$ and object $x$ of $\\mathcal{Y}_U$, there exists an fppf covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential image of the functor $g : \\mathcal{Z}_{U_i} \\to \\mathcal{Y}_{U_i}$. In this case, if $f'$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$."}
+{"_id": "8458", "title": "algebraic-lemma-product-representable-transformations-property", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{P}$ be a property as in Definition \\ref{definition-relative-representable-property} which is stable under composition. Let $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over $(\\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$ be $1$-morphisms representable by algebraic spaces. If $f_1$ and $f_2$ have property $\\mathcal{P}$ so does $ f_1 \\times f_2 : \\mathcal{X}_1 \\times \\mathcal{X}_2 \\to \\mathcal{Y}_1 \\times \\mathcal{Y}_2 $."}
+{"_id": "8465", "title": "algebraic-lemma-product-spaces", "text": "Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$. Then $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$ is an algebraic stack, and is a product in the $2$-category of algebraic stacks over $S$."}
+{"_id": "8584", "title": "sites-lemma-localize-pushforward", "text": "In the situation of Lemma \\ref{lemma-localize-topos}, the functor $j_{\\mathcal{F}, *}$ is the one associates to $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the sheaf $$ U \\longmapsto \\{\\alpha : \\mathcal{F}|_U \\to \\mathcal{G}|_U \\text{ such that } \\alpha \\text{ is a right inverse to }\\varphi|_U \\}. $$"}
+{"_id": "8613", "title": "sites-lemma-check-morphism-sites", "text": "Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites. Let $\\{(q_i, v_i)\\}_{i\\in I}$ be a conservative family of points of $\\mathcal{D}$. If each functor $u_i = v_i \\circ u$ defines a point of $\\mathcal{C}$, then $u$ defines a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$."}
+{"_id": "8621", "title": "sites-lemma-characterize-empty", "text": "Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$. The following are equivalent: \\begin{enumerate} \\item $U$ is sheaf theoretically empty, \\item $\\mathcal{F}(U)$ is a singleton for each sheaf $\\mathcal{F}$, \\item $\\emptyset^\\#(U)$ is a singleton, \\item $\\emptyset^\\#(U)$ is nonempty, and \\item the empty family is a covering of $U$ in $\\mathcal{C}$. \\end{enumerate} Moreover, if $U$ is sheaf theoretically empty, then for any morphism $U' \\to U$ of $\\mathcal{C}$ the object $U'$ is sheaf theoretically empty."}
+{"_id": "8638", "title": "sites-lemma-finer-topology", "text": "Assumption and notation as in Theorem \\ref{theorem-topology-and-topos}. Then $J \\subset J'$ if and only if every sheaf for the topology $J'$ is a sheaf for the topology $J$."}
+{"_id": "8639", "title": "sites-proposition-sheaves-on-group", "text": "The functors $\\mathcal{F} \\mapsto \\mathcal{F}({}_GG)$ and $S \\mapsto \\mathcal{F}_S$ define quasi-inverse equivalences between $\\Sh(\\mathcal{T}_G)$ and $G\\textit{-Sets}$."}
+{"_id": "8731", "title": "examples-defos-lemma-continuous-representations-hull", "text": "In Example \\ref{example-continuous-representations} assume $\\Gamma$ is topologically finitely generated. Let $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation. Assume $\\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad A \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong $$ of isomorphism classes of objects has a hull. If $H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is prorepresentable."}
+{"_id": "8734", "title": "examples-defos-lemma-graded-algebras-hull", "text": "In Example \\ref{example-graded-algebras} assume $P$ is a finitely generated graded $k$-algebra. Assume $\\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad A \\longmapsto \\Ob(\\Deformationcategory_P(A))/\\cong $$ of isomorphism classes of objects has a hull."}
+{"_id": "8742", "title": "examples-defos-lemma-strict-henselization", "text": "In Example \\ref{example-rings} let $P$ be a $k$-algebra. Assume $P$ is a local ring and let $P^{sh}$ be a strict henselization of $P$. There is a natural functor $$ \\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^{sh}} $$ of deformation categories."}
+{"_id": "8748", "title": "examples-defos-lemma-schemes-hull", "text": "In Example \\ref{example-schemes} assume $X$ is a proper $k$-scheme. Assume $\\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad A \\longmapsto \\Ob(\\Deformationcategory_X(A))/\\cong $$ of isomorphism classes of objects has a hull. If $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$, then $F$ is prorepresentable."}
+{"_id": "8784", "title": "examples-defos-lemma-smoothing-curve-isolated-lci", "text": "Let $k$ be a field and let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$, \\item $X$ is a local complete intersection over $k$, and \\item $X \\to \\Spec(k)$ is smooth except at finitely many points. \\end{enumerate} Then there exists a flat, separated, finite type morphism $Y \\to \\Spec(k[[t]])$ whose generic fibre is smooth and whose special fibre is isomorphic to $X$."}
+{"_id": "8809", "title": "more-etale-lemma-separated-etale-shriek", "text": "Let $j : U \\to X$ be a separated \\'etale morphism. Let $\\mathcal{F}$ be an abelian sheaf on $U_\\etale$. The image of the injective map $j_!\\mathcal{F} \\to j_*\\mathcal{F}$ of \\'Etale Cohomology, Lemma \\ref{etale-cohomology-lemma-shriek-into-star-separated-etale} is the subsheaf of Lemma \\ref{lemma-f-shriek-separated}."}
+{"_id": "8842", "title": "more-etale-lemma-base-change-shriek", "text": "Consider a cartesian square $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of quasi-compact and quasi-separated schemes with $f$ separated and of finite type. Then there is a canonical isomorphism $$ g^{-1} \\circ Rf_! \\to Rf'_! \\circ (g')^{-1} $$ Moreover, these isomorphisms are compatible with the isomorphisms of Lemma \\ref{lemma-shriek-composition}."}
+{"_id": "8845", "title": "more-etale-lemma-upper-shriek-derived", "text": "Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$ has a right adjoint $Rf^! : D(Y_\\etale, \\Lambda) \\to D(X_\\etale, \\Lambda)$."}
+{"_id": "8878", "title": "stacks-properties-lemma-composition-monomorphism", "text": "Compositions of monomorphisms of algebraic stacks are monomorphisms."}
+{"_id": "8906", "title": "stacks-properties-lemma-residual-gerbe-regular", "text": "A reduced, locally Noetherian algebraic stack $\\mathcal{Z}$ such that $|\\mathcal{Z}|$ is a singleton is regular."}
+{"_id": "8947", "title": "stacks-lemma-stack-gives-stack-groupoids", "text": "Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack. Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ be the category fibred in groupoids associated to $\\mathcal{S}$ constructed in Categories, Lemma \\ref{categories-lemma-fibred-gives-fibred-groupoids}. Then $p' : \\mathcal{S}' \\to \\mathcal{C}$ is a stack in groupoids."}
+{"_id": "8985", "title": "stacks-lemma-fibred-groupoids-category-pullback", "text": "With notation and assumptions as in Lemma \\ref{lemma-fibred-category-pullback}. If $\\mathcal{S}$ is fibred in groupoids, then $u_p\\mathcal{S}$ is fibred in groupoids."}
+{"_id": "8990", "title": "stacks-lemma-when-localization-stack", "text": "Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. Then $j_U : \\mathcal{C}/U \\to \\mathcal{C}$ is a stack over $\\mathcal{C}$ if and only if $h_U$ is a sheaf."}
+{"_id": "9051", "title": "spaces-simplicial-lemma-augmentation-spectral-sequence-modules", "text": "With notation as above for any $K$ in $D^+(\\mathcal{O})$ there is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$ with $$ E_1^{p, q} = R^qa_{p, *} K_p $$ converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$."}
+{"_id": "9057", "title": "spaces-simplicial-lemma-derived-cartesian-modules", "text": "In Situation \\ref{situation-simplicial-site}. \\begin{enumerate} \\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only if $H^q(K)$ is a cartesian abelian sheaf for all $q$. \\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ such that the morphisms $f_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1})$ are flat. Then an object $K$ of $D(\\mathcal{O})$ is cartesian if and only if $H^q(K)$ is a cartesian $\\mathcal{O}$-module for all $q$. \\end{enumerate}"}
+{"_id": "9070", "title": "spaces-simplicial-lemma-has-P", "text": "Let $\\mathcal{C}$ be a site. \\begin{enumerate} \\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor $j : \\mathcal{C}/K \\to \\mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}. \\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$ the functor $v : \\mathcal{C}/K \\to \\mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}. \\end{enumerate}"}
+{"_id": "9096", "title": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above. Then we have a canonical isomorphism $$ R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, a^{-1}E) $$ for $E \\in D(\\mathcal{C}/X)$."}
+{"_id": "9102", "title": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded-modules", "text": "Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\\mathcal{C}$. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ denote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence $$ D^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O}) $$ with quasi-inverse $Ra_*$."}
+{"_id": "9106", "title": "spaces-simplicial-lemma-bbd-unbounded-glueing", "text": "In Situation \\ref{situation-locally-given}. Assume \\begin{enumerate} \\item $\\mathcal{C}$ has equalizers and fibre products, \\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$ given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$ such that \\begin{enumerate} \\item $\\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them, \\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$ and $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$, \\item every object of $\\mathcal{D}$ has a covering whose members are objects of $\\mathcal{B}$, \\end{enumerate} \\item all negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero, and \\item there exist weak Serre subcategories $\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$ for all $U \\in \\Ob(\\mathcal{C})$ satisfying conditions (\\ref{item-restriction}), (\\ref{item-local}), and (\\ref{item-bounded-dimension}), \\item $E_U \\in D_{\\mathcal{A}_U}(\\mathcal{O}_U)$. \\end{enumerate} Then there exists a solution unique up to unique isomorphism."}
+{"_id": "9111", "title": "spaces-simplicial-lemma-proper-hypercovering-equivalence-bounded", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. Let $\\mathcal{A} \\subset \\textit{Ab}(U_{Zar})$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence $$ D^+(X) \\longrightarrow D_\\mathcal{A}^+(U_{Zar}) $$ with quasi-inverse $Ra_*$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in Lemma \\ref{lemma-augmentation}."}
+{"_id": "9112", "title": "spaces-simplicial-lemma-spectral-sequence-proper-hypercovering", "text": "Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf on $X$. Let $\\mathcal{F}_n$ be the pullback to $U_n$. If $U$ is a proper hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q(U_p, \\mathcal{F}_p) $$ converging to $H^{p + q}(X, \\mathcal{F})$."}
+{"_id": "9118", "title": "spaces-simplicial-lemma-adjoint-functors-cartesian-modules", "text": "Let $f : V \\to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are flat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated. Then $f^*$ and $f_*$ form an adjoint pair of functors between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$."}
+{"_id": "9119", "title": "spaces-simplicial-lemma-cartesian-modules-with-section", "text": "Let $f : X \\to S$ be a morphism of schemes which has a section\\footnote{In fact, it would be enough to assume that $f$ has fpqc locally on $S$ a section, since we have descent of quasi-coherent modules by Descent, Section \\ref{descent-section-fpqc-descent-quasi-coherent}.}. Let $(X/S)_\\bullet$ be the simplicial scheme associated to $X \\to S$, see Definition \\ref{definition-fibre-products-simplicial-scheme}. Then pullback defines an equivalence between the category of quasi-coherent $\\mathcal{O}_S$-modules and the category of quasi-coherent modules on $((X/S)_\\bullet)_{Zar}$."}
+{"_id": "9121", "title": "spaces-simplicial-lemma-quasi-coherent-groupoid-simplicial", "text": "Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $X$ be the simplicial scheme over $S$ constructed in Lemma \\ref{lemma-groupoid-simplicial}. Then the category of quasi-coherent modules on $(U, R, s, t, c)$ is equivalent to the category of quasi-coherent modules on $X_{Zar}$."}
+{"_id": "9132", "title": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$. If $U$ is an fppf hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p) $$ converging to $H^{p + q}_\\etale(X, \\mathcal{F})$."}
+{"_id": "9138", "title": "spaces-simplicial-lemma-spectral-sequence-fppf-hypercovering-modules", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$. If $U$ is an fppf hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p) $$ converging to $H^{p + q}_\\etale(X, \\mathcal{F})$."}
+{"_id": "9145", "title": "spaces-simplicial-lemma-ph-hypercovering-equivalence-bounded", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. Let $\\mathcal{A} \\subset \\textit{Ab}(U_\\etale)$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence $$ D^+(X_\\etale) \\longrightarrow D_\\mathcal{A}^+(U_\\etale) $$ with quasi-inverse $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$ is as in Section \\ref{section-simplicial-algebraic-spaces}."}
+{"_id": "9146", "title": "spaces-simplicial-lemma-spectral-sequence-ph-hypercovering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$. If $U$ is a ph hypercovering of $X$, then there exists a canonical spectral sequence $$ E_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p) $$ converging to $H^{p + q}_\\etale(X, \\mathcal{F})$."}
+{"_id": "9165", "title": "examples-stacks-lemma-stack-of-quasi-coherent-sheaves", "text": "The functor $p : \\QCohstack \\to (\\Sch/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition \\ref{stacks-definition-stack}."}
+{"_id": "9173", "title": "examples-stacks-lemma-stack-ft-spaces", "text": "There exists a subcategory $\\Spacesstack_{ft, small} \\subset \\Spacesstack_{ft}$ with the following properties: \\begin{enumerate} \\item the inclusion functor $\\Spacesstack_{ft, small} \\to \\Spacesstack_{ft}$ is fully faithful and essentially surjective, and \\item the functor $p_{ft, small} : \\Spacesstack_{ft, small} \\to (\\Sch/S)_{fppf}$ turns $\\Spacesstack_{ft, small}$ into a stack over $(\\Sch/S)_{fppf}$. \\end{enumerate}"}
+{"_id": "9175", "title": "examples-stacks-lemma-variant-torsors-sheaf-stack-in-groupoids", "text": "Up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ p : \\mathcal{G}/\\mathcal{B}\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9178", "title": "examples-stacks-lemma-group-quotient-stack-in-groupoids", "text": "Up to a replacement as in Stacks, Remark \\ref{stacks-remark-stack-make-small} the functor $$ p : [[X/G]] \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9179", "title": "examples-stacks-lemma-classifying-stacks", "text": "\\begin{slogan} The classifying stack of a group scheme or group algebraic space. \\end{slogan} Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then the stacks in groupoids $$ [B/G],\\quad [[B/G]],\\quad G\\textit{-Torsors},\\quad \\mathcal{G}/\\mathcal{B}\\textit{-Torsors} $$ are all canonically equivalent. If $G \\to B$ is flat and locally of finite presentation, then these are also equivalent to $G\\textit{-Principal}$."}
+{"_id": "9181", "title": "examples-stacks-lemma-hilbert-d-stack", "text": "The category $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ endowed with the functor $p$ above defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
+{"_id": "9206", "title": "models-lemma-picard-rank-1", "text": "Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The Picard group of $T$ is a finitely generated abelian group of rank $1$."}
+{"_id": "9252", "title": "models-lemma-genus-reduction-smaller", "text": "In Situation \\ref{situation-regular-model} assume $X$ is a minimal model, $\\gcd(m_1, \\ldots, m_n) = 1$, and $H^0((X_k)_{red}, \\mathcal{O}) = k$. Then the map $$ H^1(X_k, \\mathcal{O}_{X_k}) \\to H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}}) $$ is surjective and has a nontrivial kernel as soon as $(X_k)_{red} \\not = X_k$."}
+{"_id": "9254", "title": "models-lemma-equality-genus-reduction-bigger-than", "text": "If equality holds in Lemma \\ref{lemma-genus-reduction-bigger-than} then \\begin{enumerate} \\item the unique irreducible component of $X_k$ containing $x$ is a smooth projective geometrically irreducible curve over $k$, \\item if $C \\subset X_k$ is another irreducible component, then $\\kappa = H^0(C, \\mathcal{O}_C)$ is a finite separable extension of $k$, $C$ has a $\\kappa$-rational point, and $C$ is smooth over $\\kappa$ \\end{enumerate}"}
+{"_id": "9300", "title": "spaces-groupoids-lemma-diagram-pull", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. The diagram \\begin{equation} \\label{equation-pull} \\xymatrix{ R \\times_{t, U, t} R \\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_0 \\times c \\circ (i, 1)} & R \\ar[r]^t \\ar[d]^{\\text{id}_R} & U \\ar[d]^{\\text{id}_U} \\\\ R \\times_{s, U, t} R \\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} & R \\ar[r]^t \\ar[d]^s & U \\\\ R \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t & U } \\end{equation} is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian."}
+{"_id": "9304", "title": "spaces-groupoids-lemma-colimits", "text": "Let $B \\to S$ be as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits."}
+{"_id": "9309", "title": "spaces-groupoids-lemma-action-groupoid-modules", "text": "Let $B \\to S$ as in Section \\ref{section-notation}. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces constructed in Lemma \\ref{lemma-groupoid-from-action}. The rule $(\\mathcal{F}, \\alpha) \\mapsto (\\mathcal{F}, \\alpha)$ defines an equivalence of categories between $G$-equivariant $\\mathcal{O}_X$-modules and the category of quasi-coherent modules on $(U, R, s, t, c)$."}
+{"_id": "9318", "title": "spaces-groupoids-lemma-quotient-groupoid-restrict", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \\to U$ a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$. The map of quotient sheaves $$ U'/R' \\longrightarrow U/R $$ is injective. If the composition $$ \\xymatrix{ U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h & R \\ar[r]_s & U } $$ is a surjection of fppf sheaves then the map is bijective. This holds for example if $\\{h : U' \\times_{g, U, t} R \\to U\\}$ is an $fppf$-covering, or if $U' \\to U$ is a surjection of sheaves, or if $\\{g : U' \\to U\\}$ is a covering in the fppf topology."}
+{"_id": "9359", "title": "spaces-descent-lemma-finite-type-descends", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module."}
+{"_id": "9361", "title": "spaces-descent-lemma-flat-descends", "text": "Let $X$ be an algebraic space over a scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that each $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module."}
+{"_id": "9372", "title": "spaces-descent-lemma-syntomic-permanence", "text": "Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & B } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that \\begin{enumerate} \\item $f$ is surjective, flat, and locally of finite presentation, \\item $p$ is syntomic. \\end{enumerate} Then both $q$ and $f$ are syntomic."}
+{"_id": "9388", "title": "spaces-descent-lemma-descending-property-universal-homeomorphism", "text": "The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism'' is fpqc local on the base."}
+{"_id": "9409", "title": "spaces-descent-lemma-descending-property-finite-locally-free", "text": "The property $\\mathcal{P}(f) =$``$f$ is finite locally free'' is fpqc local on the base."}
+{"_id": "9420", "title": "spaces-descent-lemma-open-fppf-local-source", "text": "The property $\\mathcal{P}(f)=$``$f$ is open'' is fppf local on the source."}
+{"_id": "9539", "title": "decent-spaces-lemma-birational-dominant", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational then $f$ is dominant."}
+{"_id": "9540", "title": "decent-spaces-lemma-birational-generic-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $y \\in Y$ is the generic point of an irreducible component, then the base change $X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$ is an isomorphism."}
+{"_id": "9546", "title": "decent-spaces-lemma-normalization-normal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be an integral birational morphism of decent algebraic spaces over $S$ which have finitely many irreducible components. Then there exists a factorization $Y^\\nu \\to X \\to Y$ and $Y^\\nu \\to X$ is the normalization of $X$."}
+{"_id": "9550", "title": "decent-spaces-lemma-decent-Jacobson", "text": "Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Then $X$ is Jacobson if and only if $|X|$ is Jacobson."}
+{"_id": "9556", "title": "decent-spaces-lemma-universally-catenary", "text": "Let $S$ be a scheme. Let $X$ be a decent, locally Noetherian, and universally catenary algebraic space over $S$. Then any decent algebraic space locally of finite type over $X$ is universally catenary."}
+{"_id": "9604", "title": "groupoids-lemma-algebraic-center", "text": "Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then the center of $G$ is a closed subgroup scheme of $G$."}
+{"_id": "9633", "title": "groupoids-lemma-colimit-finite-type", "text": "Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that \\begin{enumerate} \\item $U$, $R$ are affine, \\item there exist $e_i \\in \\mathcal{O}_R(R)$ such that every element $g \\in \\mathcal{O}_R(R)$ can be uniquely written as $\\sum s^*(f_i)e_i$ for some $f_i \\in \\mathcal{O}_U(U)$. \\end{enumerate} Then every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$ is a filtered colimit of finite type quasi-coherent modules."}
+{"_id": "9666", "title": "groupoids-lemma-descend-along-finite-quasi-projective", "text": "Let $X \\to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that one of the following holds \\begin{enumerate} \\item $V \\to X$ is projective, \\item $V \\to X$ is quasi-projective, \\item there exists an ample invertible sheaf on $V$, \\item there exists an $X$-ample invertible sheaf on $V$, \\item there exists an $X$-very ample invertible sheaf on $V$. \\end{enumerate} Then any descent datum on $V/X/Y$ is effective."}
+{"_id": "9732", "title": "local-cohomology-lemma-kollar-finiteness-H1-local", "text": "Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Set $Z = V(I)$. Let $M$ be a finite $A$-module. The following are equivalent \\begin{enumerate} \\item $H^1_Z(M)$ is a finite $A$-module, and \\item for all $\\mathfrak p \\in \\text{Ass}(M)$, $\\mathfrak p \\not \\in Z$ and all $\\mathfrak q \\in V(\\mathfrak p + I)$ the completion of $(A/\\mathfrak p)_\\mathfrak q$ does not have associated primes of dimension $1$. \\end{enumerate}"}
+{"_id": "9734", "title": "local-cohomology-lemma-ideal-depth-function", "text": "Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $\\mathfrak p \\in V(I)$ be a prime ideal. Assume $e = \\text{depth}_{IA_\\mathfrak p}(M_\\mathfrak p) < \\infty$. Then there exists a nonempty open $U \\subset V(\\mathfrak p)$ such that $\\text{depth}_{IA_\\mathfrak q}(M_\\mathfrak q) \\geq e$ for all $\\mathfrak q \\in U$."}
+{"_id": "9736", "title": "local-cohomology-lemma-finite-nr-points-next-S", "text": "Let $X$ be a Noetherian scheme with dualizing complex $\\omega_X^\\bullet$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $k \\geq 0$ be an integer. Assume $\\mathcal{F}$ is $(S_k)$. Then there is a finite number of points $x \\in X$ such that $$ \\text{depth}(\\mathcal{F}_x) = k \\quad\\text{and}\\quad \\dim(\\text{Supp}(\\mathcal{F}_x)) > k $$"}
+{"_id": "9749", "title": "local-cohomology-lemma-make-S2-along-Z", "text": "Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume $X$ is universally catenary and the formal fibres of local rings have $(S_1)$. Then there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$ of coherent $\\mathcal{O}_X$-modules such that \\begin{enumerate} \\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism for $x \\in X \\setminus Z$, \\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective and $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) = 1$ for $x \\in Z$ such that there exists an immediate specialization $x' \\leadsto x$ with $x' \\not \\in Z$ and $x' \\in \\text{Ass}(\\mathcal{F})$, \\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$ for the remaining $x \\in Z$. \\end{enumerate} If $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(Z) \\subset Y$."}
+{"_id": "9767", "title": "local-cohomology-lemma-frobenius", "text": "Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism. Let $I \\subset A$ be a finitely generated ideal. Set $Z = V(I)$. There exists an isomorphism $R\\Gamma_Z(A) \\otimes_{A, F}^\\mathbf{L} A \\cong R\\Gamma_Z(A)$."}
+{"_id": "9769", "title": "local-cohomology-lemma-etale-frobenius", "text": "Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism. If $V \\to \\Spec(A)$ is quasi-compact, quasi-separated, and \\'etale, then there exists an isomorphism $R\\Gamma(V, \\mathcal{O}_V) \\otimes_{A, F}^\\mathbf{L} A \\cong R\\Gamma(V, \\mathcal{O}_V)$."}
+{"_id": "9789", "title": "local-cohomology-proposition-finiteness-complex", "text": "Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Let $s \\in \\mathbf{Z}$. Let $K \\in D_{\\textit{Coh}}^+(A)$. The following are equivalent \\begin{enumerate} \\item $H^i_T(K)$ is a finite $A$-module for $i \\leq s$, and \\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$ we have $$ \\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > s $$ \\end{enumerate}"}
+{"_id": "9845", "title": "more-algebra-lemma-lift-factorization", "text": "Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a monic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a factorization of $f$ in $A/I[x]$ and assume that  $\\overline{g}$, $\\overline{h}$ generate the unit ideal in $A/I[x]$. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ lifting the given factorization over $A/I$."}
+{"_id": "9873", "title": "more-algebra-lemma-henselization-local-ring", "text": "\\begin{slogan} Compatibility henselization of pairs and of local rings. \\end{slogan} The functor of Lemma \\ref{lemma-henselization} associates to a local ring $(A, \\mathfrak m)$ its henselization."}
+{"_id": "9897", "title": "more-algebra-lemma-flat-descent-flat-at-primes-over", "text": "Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal. Let $M$ be an $S$-module. Let $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal such that \\begin{enumerate} \\item the map $V(I') \\to V(I)$ induced by $\\Spec(R') \\to \\Spec(R)$ is surjective, and \\item $R'_{\\mathfrak p'}$ is flat over $R$ for all primes $\\mathfrak p' \\in V(I')$. \\end{enumerate} If (\\ref{equation-flat-at-primes-over}) holds for $(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then (\\ref{equation-flat-at-primes-over}) holds for $(R \\to S, I, M)$."}
+{"_id": "9899", "title": "more-algebra-lemma-base-change-flat-at-primes", "text": "In Situation \\ref{situation-flattening-general} let $R' \\to R''$ be an $R$-algebra map. Let $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals. If (\\ref{equation-flat-at-primes}) holds for $(R', I')$, then (\\ref{equation-flat-at-primes}) holds for $(R'', I'')$."}
+{"_id": "9902", "title": "more-algebra-lemma-flat-module-powers-variant", "text": "In Situation \\ref{situation-flattening-general}. Let $I \\subset R$ be an ideal. Assume \\begin{enumerate} \\item $R$ is a Noetherian ring, \\item $S$ is a Noetherian ring, \\item $M$ is a finite $S$-module, and \\item for each $n \\geq 1$ and any prime $\\mathfrak q \\in V(J + IS)$ the module $(M/I^n M)_{\\mathfrak q}$ is flat over $R/I^n$. \\end{enumerate} Then (\\ref{equation-flat-at-primes}) holds for $(R, I)$, i.e., for every prime $\\mathfrak q \\in V(J + IS)$ the localization $M_{\\mathfrak q}$ is flat over $R$."}
+{"_id": "9911", "title": "more-algebra-lemma-descent-flatness-injective-integral", "text": "Let $R \\to S$ be an injective integral ring map. Let $M$ be a finitely presented module over $R[x_1, \\ldots, x_n]$. If $M \\otimes_R S$ is flat over $S$, then $M$ is flat over $R$."}
+{"_id": "10038", "title": "more-algebra-lemma-regular-field-extension", "text": "Let $k \\subset K$ be a field extension. Then $k \\to K$ is a regular ring map if and only if $K$ is a separable field extension of $k$."}
+{"_id": "10048", "title": "more-algebra-lemma-completion-normal", "text": "Let $A$ be a Noetherian local ring. If $A^\\wedge$ is normal, then so is $A$."}
+{"_id": "10086", "title": "more-algebra-lemma-check-G-ring", "text": "Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if for every finite free ring map $R \\to S$ the formal fibres of $S$ are regular rings."}
+{"_id": "10091", "title": "more-algebra-lemma-henselian-local-limit-G-rings", "text": "Let $(A, \\mathfrak m)$ be a henselian local ring. Then $A$ is a filtered colimit of a system of henselian local G-rings with local transition maps."}
+{"_id": "10105", "title": "more-algebra-lemma-formal-fibres-Rk", "text": "Fix $n \\geq 0$. Properties (A), (B), (C), (D), and (E) hold for $P(k \\to R) =$``$R \\otimes_k k'$ has $(R_n)$ for all finite extensions $k'/k$''."}
+{"_id": "10116", "title": "more-algebra-lemma-JM-injective", "text": "Let $R$ be a ring. For every $R$-module $M$ the $R$-module $J(M)$ is injective."}
+{"_id": "10225", "title": "more-algebra-lemma-colim-and-lim-of-duals", "text": "\\begin{slogan} Trivial duality for systems of perfect objects. \\end{slogan} Let $A$ be a ring. Let $(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(A)$. Let $K = \\text{hocolim} K_n$ be the derived colimit (Derived Categories, Definition \\ref{derived-definition-derived-colimit}). Then for any object $E$ of $D(A)$ we have $$ R\\Hom_A(K, E) = R\\lim E \\otimes^\\mathbf{L}_A K_n^\\vee $$ where $(K_n^\\vee)$ is the inverse system of dual perfect complexes."}
+{"_id": "10237", "title": "more-algebra-lemma-isolate-a-cohomology-group", "text": "Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let $K^\\bullet$ be a pseudo-coherent complex of $R$-modules. Assume that for some $i \\in \\mathbf{Z}$ the maps $$ H^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p) \\longrightarrow H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) \\quad\\text{and}\\quad H^{i - 1}(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p) \\longrightarrow H^{i - 1}(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) $$ are surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that \\begin{enumerate} \\item $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect object of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$, \\item $H^i(K^\\bullet)_f$ is a finite free $R_f$-module, and \\item there is a canonical direct sum decomposition $$ K^\\bullet \\otimes_R R_f \\cong \\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f) \\oplus H^i(K^\\bullet)_f[-i] \\oplus \\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f) $$ in $D(R_f)$. \\end{enumerate}"}
+{"_id": "10240", "title": "more-algebra-lemma-split-using-ext-zero", "text": "Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$. Assume $\\Ext^{-a}_R(K, M) = 0$ for any $R$-module $M$. Then there is a unique direct sum decomposition $K \\cong \\tau_{\\leq a - 1}K \\oplus \\tau_{\\geq a + 1}K$ and $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$."}
+{"_id": "10272", "title": "more-algebra-lemma-complex-relative-pseudo-coherent-modules", "text": "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded above complex of $A$-modules such that $K^i$ is $(m - i)$-pseudo-coherent relative to $R$ for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent relative to $R$. In particular, if $K^\\bullet$ is a bounded above complex of $A$-modules pseudo-coherent relative to $R$, then $K^\\bullet$ is pseudo-coherent relative to $R$."}
+{"_id": "10273", "title": "more-algebra-lemma-cohomology-relative-pseudo-coherent", "text": "Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet \\in D^{-}(A)$ such that $H^i(K^\\bullet)$ is $(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$ for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$."}
+{"_id": "10316", "title": "more-algebra-lemma-map-from-hocolim", "text": "Let $\\mathcal{D}$ be a triangulated category. Let $(K_n)$ be a system of objects of $\\mathcal{D}$. Let $K$ be a derived colimit of the system $(K_n)$. Then for every $L$ in $\\mathcal{D}$ we have a short exact sequence $$ 0 \\to R^1\\lim \\Hom_\\mathcal{D}(K_n, L[-1]) \\to \\Hom_\\mathcal{D}(K, L) \\to \\lim \\Hom_\\mathcal{D}(K_n, L) \\to 0 $$"}
+{"_id": "10356", "title": "more-algebra-lemma-cover-spec", "text": "Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map which induces an isomorphism $R/fR \\to R'/fR'$. The map $\\Spec(R') \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective. For example, the map $\\Spec(R^\\wedge) \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective."}
+{"_id": "10379", "title": "more-algebra-lemma-derived-complete-zero-bis", "text": "\\begin{slogan} Derived Nakayama \\end{slogan} \\begin{reference} A related result is \\cite[Proposition 6.5]{Dwyer-Greenlees}. The derived Nakayama lemma can for example be found in Bhatt's 3rd lecture on Prismatic cohomology at Columbia University in Fall 2018 as Section 2 property (2). Leonid Positselski proposed a proof in \\url{https://mathoverflow.net/a/331501}. However, we follow the proof suggested by Anonymous in the comments. \\end{reference} Let $I$ be a finitely generated ideal of a ring $A$. Let $K$ be a derived complete object of $D(A)$. If $K \\otimes_A^\\mathbf{L} A/I = 0$, then $K = 0$."}
+{"_id": "10381", "title": "more-algebra-lemma-derived-completion-spectral-sequence", "text": "Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal. Let $K^\\bullet$ be a filtered complex of $A$-modules. There exists a canonical spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded derived complete $A$-modules with $d_r$ of bidegree $(r, -r + 1)$ and with $$ E_1^{p, q} = H^{p + q}((\\text{gr}^pK^\\bullet)^\\wedge) $$ If the filtration on each $K^n$ is finite, then the spectral sequence is bounded and converges to $H^*((K^\\bullet)^\\wedge)$."}
+{"_id": "10383", "title": "more-algebra-lemma-restriction-derived-complete-equivalence", "text": "Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal. If $A \\to B$ is flat and $A/I \\cong B/IB$, then the restriction functor $D(B) \\to D(A)$ induces an equivalence $D_{comp}(B, IB) \\to D_{comp}(A, I)$."}
+{"_id": "10385", "title": "more-algebra-lemma-when-does-it-work", "text": "Let $A$ be a ring and $f \\in A$. Set $I = (f)$. In this situation we have the naive derived completion $K \\mapsto K' = R\\lim (K \\otimes_A^\\mathbf{L} A/f^nA)$ and the derived completion $$ K \\mapsto K^\\wedge = R\\lim (K \\otimes_A^\\mathbf{L} (A \\xrightarrow{f^n} A)) $$ of Lemma \\ref{lemma-derived-completion-koszul}. The natural transformation of functors $K^\\wedge \\to K'$ is an isomorphism if and only if the $f$-power torsion of $A$ is bounded."}
+{"_id": "10386", "title": "more-algebra-lemma-torsion-and-derived-complete", "text": "Let $I$ be a finitely generated ideal in a ring $A$. Let $M$ be a derived complete $A$-module. If $M$ is an $I$-power torsion module, then $I^nM = 0$ for some $n$."}
+{"_id": "10394", "title": "more-algebra-lemma-derived-complete-finite", "text": "Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be a derived complete $A$-module. If $M/IM$ is a finite $A/I$-module, then $M = \\lim M/I^nM$ and $M$ is a finite $A^\\wedge$-module."}
+{"_id": "10396", "title": "more-algebra-lemma-derived-completion-tensor-finite", "text": "Let $I$ be an ideal in a Noetherian ring $A$. Let ${}^\\wedge$ denote derived completion with respect to $I$. Let $K \\in D^-(A)$. \\begin{enumerate} \\item If $M$ is a finite $A$-module, then $(K \\otimes_A^\\mathbf{L} M)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} M$. \\item If $L \\in D(A)$ is pseudo-coherent, then $(K \\otimes_A^\\mathbf{L} L)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} L$. \\end{enumerate}"}
+{"_id": "10411", "title": "more-algebra-lemma-Rlim-gives-complete", "text": "Let $A$ be a ring and $I \\subset A$ an ideal. Suppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$ in $D(A/I^{n + 1})$. If \\begin{enumerate} \\item $A$ is Noetherian, \\item $K_1$ is bounded above, and \\item the maps induce isomorphisms $K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$, \\end{enumerate} then $K = R\\lim K_n$ is a derived complete object of $D^-(A)$ and $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$."}
+{"_id": "10477", "title": "more-algebra-lemma-completion-disconnected", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. The punctured spectrum of $A^\\wedge$ is disconnected if and only if the punctured spectrum of $A^h$ is disconnected."}
+{"_id": "10483", "title": "more-algebra-lemma-geometrically-normal-fibres-universally-catenary", "text": "\\begin{reference} \\cite[Corollary 2.3]{Heinzer-Rotthaus-Wiegand} \\end{reference} Let $(A, \\mathfrak m)$ be a Noetherian local ring with geometrically normal formal fibres. Then \\begin{enumerate} \\item $A^h$ is universally catenary, and \\item if $A$ is unibranch (for example normal), then $A$ is universally catenary. \\end{enumerate}"}
+{"_id": "10497", "title": "more-algebra-lemma-composition-unramified", "text": "Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be finite separable extensions. Let $B$ be the integral closure of $A$ in $L$. If $L/K$ is unramified with respect to $A$ and $M/L$ is unramified with respect to $B_\\mathfrak m$ for every maximal ideal $\\mathfrak m$ of $B$, then $M/K$ is unramified with respect to $A$."}
+{"_id": "10562", "title": "more-algebra-lemma-unimodular-vector", "text": "Let $R$ be a B\\'ezout domain. Let $n \\geq 1$ and $f_1, \\ldots, f_n \\in R$ generate the unit ideal. There exists an invertible $n \\times n$ matrix in $R$ whose first row is $f_1 \\ldots f_n$."}
+{"_id": "10572", "title": "more-algebra-lemma-symmetric-monoidal-derived", "text": "Let $R$ be a ring. The derived category $D(R)$ of $R$ is a symmetric monoidal category with tensor product given by derived tensor product and associativity and commutativity constraints as in Section \\ref{section-sign-rules}."}
+{"_id": "10586", "title": "more-algebra-proposition-regular-strong-generator", "text": "Let $R$ be a Noetherian ring. The following are equivalent \\begin{enumerate} \\item $R$ is regular of finite dimension, \\item $D_{perf}(R)$ has a strong generator, and \\item $R$ is a strong generator for $D_{perf}(R)$. \\end{enumerate}"}
+{"_id": "10593", "title": "more-algebra-proposition-propdimd", "text": "\\begin{reference} \\cite[Lemma 3.14]{Artin-Lipman} has this result without the assumption that the ring is catenary \\end{reference} Let $R$ be a catenary Noetherian local normal domain. Let $J \\subset R$ be a radical ideal. Then there exists a nonzero element $f \\in J$ such that $R/fR$ is reduced."}
+{"_id": "10684", "title": "etale-theorem-flatness-grothendieck", "text": "Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be a local homomorphism. If $M$ is a finite $B$-module that is flat as an $A$-module, and $t \\in \\mathfrak m_B$ is an element such that multiplication by $t$ is injective on $M/\\mathfrak m_AM$, then $M/tM$ is also $A$-flat."}
+{"_id": "10685", "title": "etale-theorem-flat-open", "text": "Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be a morphism which is locally of finite type. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The set of points in $X$ where $\\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$."}
+{"_id": "10688", "title": "etale-theorem-structure-etale", "text": "Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then there exist $f, g \\in A[t]$ such that \\begin{enumerate} \\item $B' = A[t]_g/(f)$ is standard \\'etale -- see (a) and (b) above, and \\item $B$ is isomorphic to a localization of $B'$ at a prime. \\end{enumerate}"}
+{"_id": "10690", "title": "etale-theorem-geometric-structure", "text": "Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$. Let $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$. If $\\varphi$ is \\'etale at $x$, then there exist exists an affine open $U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$ such that we have the following diagram $$ \\xymatrix{ X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-j & \\Spec(R[t]_{f'}/(f)) \\ar[d] \\\\ Y & V \\ar[l] \\ar@{=}[r] & \\Spec(R) } $$ where $j$ is an open immersion, and $f \\in R[t]$ is monic."}
+{"_id": "10723", "title": "etale-lemma-smooth-pullback-normal-crossings", "text": "\\begin{slogan} Pullback of a normal crossings divisor by a smooth morphism is a normal crossings divisor. \\end{slogan} Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a normal crossings divisor. If $f : Y \\to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a normal crossings divisor on $Y$."}
+{"_id": "10732", "title": "etale-proposition-etale-normal", "text": "Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then $A$ is a normal domain if and only if $B$ is so."}
+{"_id": "10743", "title": "crystalline-theorem-cohomology-F-crystal", "text": "In Situation \\ref{situation-F-crystal} let $(\\mathcal{E}, F_\\mathcal{E})$ be a nondegenerate $F$-crystal. Assume $A$ is a $p$-adically complete Noetherian ring and that $X \\to S_0$ is proper smooth. Then the canonical map $$ F_\\mathcal{E} \\circ (F_X)_{\\text{cris}}^* : R\\Gamma(\\text{Cris}(X/S), \\mathcal{E}) \\otimes^\\mathbf{L}_{A, \\sigma} A \\longrightarrow R\\Gamma(\\text{Cris}(X/S), \\mathcal{E}) $$ becomes an isomorphism after inverting $p$."}
+{"_id": "10745", "title": "crystalline-lemma-divided-power-envelop-quotient", "text": "Let $(A, I, \\gamma)$ be a divided power ring. Let $\\varphi : B' \\to B$ be a surjection of $A$-algebras with kernel $K$. Let $IB \\subset J \\subset B$ be an ideal. Let $J' \\subset B'$ be the inverse image of $J$. Write $D_{B', \\gamma}(J') = (D', \\bar J', \\bar\\gamma)$. Then $D_{B, \\gamma}(J) = (D'/K', \\bar J'/K', \\bar\\gamma)$ where $K'$ is the ideal generated by the elements $\\bar\\gamma_n(k)$ for $n \\geq 1$ and $k \\in K$."}
+{"_id": "10760", "title": "crystalline-lemma-differentials-completion", "text": "Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power structure on $B$. Let $p$ be a prime number. Assume that $A$ is a $\\mathbf{Z}_{(p)}$-algebra and that $p$ is nilpotent in $B/J$. Then we have $$ \\lim_e \\Omega_{B_e/A, \\bar\\delta} = \\lim_e \\Omega_{B/A, \\delta}/p^e\\Omega_{B/A, \\delta} = \\lim_e \\Omega_{B^\\wedge/A, \\delta^\\wedge}/p^e \\Omega_{B^\\wedge/A, \\delta^\\wedge} $$ see proof for notation and explanation."}
+{"_id": "10763", "title": "crystalline-lemma-divided-power-thickening-base-change-flat", "text": "In Situation \\ref{situation-global}. Let $$ \\xymatrix{ (U_3, T_3, \\delta_3) \\ar[d] \\ar[r] & (U_2, T_2, \\delta_2) \\ar[d] \\\\ (U_1, T_1, \\delta_1) \\ar[r] & (U, T, \\delta) } $$ be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \\mathcal{I}, \\gamma)$. If $T_2 \\to T$ is flat and $U_2 = T_2 \\times_T U$, then $T_3 = T_1 \\times_T T_2$ (as schemes)."}
+{"_id": "10854", "title": "spaces-pushouts-lemma-glue-etale-sheaf-fppf", "text": "Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of algebraic spaces over $S$. The functor $$ \\Sh(X_\\etale) \\longrightarrow \\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\} $$ is an equivalence of categories."}
+{"_id": "10865", "title": "spaces-pushouts-lemma-pushout-along-closed-immersion-and-integral", "text": "In More on Morphisms, Situation \\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral} let $Y \\amalg_Z X$ be the pushout in the category of schemes (More on Morphisms, Proposition \\ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral}). Then $Y \\amalg_Z X$ is also a pushout in the category of algebraic spaces over $S$."}
+{"_id": "10880", "title": "spaces-pushouts-lemma-fully-faithful-on-separated", "text": "In Situation \\ref{situation-formal-glueing} the functor (\\ref{equation-formal-glueing-spaces}) is fully faithful on algebraic spaces separated over $X$. More precisely, it induces a bijection $$ \\Mor_X(X'_1, X'_2) \\longrightarrow \\Mor_{\\textit{Spaces}(Y \\to X, Z)}(F(X'_1), F(X'_2)) $$ whenever $X'_2 \\to X$ is separated."}
+{"_id": "10885", "title": "spaces-pushouts-lemma-glueing-ff", "text": "Let $(R \\to R', f)$ be a glueing pair, see above. The functor (\\ref{equation-beauville-laszlo-glueing-spaces}) is fully faithful on the full subcategory of algebraic spaces $Y/X$ which are (a) glueable for $(R \\to R', f)$ and (b) have affine diagonal $Y \\to Y \\times_X Y$."}
+{"_id": "10886", "title": "spaces-pushouts-lemma-glueing-quasi-affines", "text": "Let $(R \\to R', f)$ be a glueing pair, see above. Any object $(V, V', Y')$ of $\\textit{Spaces}(U \\leftarrow U' \\to X')$ with $V$, $V'$, $Y'$ quasi-affine is isomorphic to the image under the functor (\\ref{equation-beauville-laszlo-glueing-spaces}) of a separated algebraic space $Y$ over $X$."}
+{"_id": "10902", "title": "varieties-lemma-birational-varieties", "text": "Let $X$ and $Y$ be varieties over a field $k$. The following are equivalent \\begin{enumerate} \\item $X$ and $Y$ are birational varieties, \\item the function fields $k(X)$ and $k(Y)$ are isomorphic, \\item there exist nonempty opens of $X$ and $Y$ which are isomorphic as varieties, \\item there exists an open $U \\subset X$ and a birational morphism $U \\to Y$ of varieties. \\end{enumerate}"}
+{"_id": "10913", "title": "varieties-lemma-Noetherian-geometrically-reduced-at-point", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \\in X$. Assume $X$ locally Noetherian and geometrically reduced at $x$. Then there exists an open neighbourhood $U \\subset X$ of $x$ which is geometrically reduced over $k$."}
+{"_id": "10931", "title": "varieties-lemma-galois-action-connected-components-continuous", "text": "Let $k$ be a field, with separable algebraic closure $\\overline{k}$. Let $X$ be a scheme over $k$. Assume \\begin{enumerate} \\item $X$ is quasi-compact, and \\item the connected components of $X_{\\overline{k}}$ are open. \\end{enumerate} Then \\begin{enumerate} \\item[(a)] $\\pi_0(X_{\\overline{k}})$ is finite, and \\item[(b)] the action of $\\text{Gal}(\\overline{k}/k)$ on $\\pi_0(X_{\\overline{k}})$ is continuous. \\end{enumerate} Moreover, assumptions (1) and (2) are satisfied when $X$ is of finite type over $k$."}
+{"_id": "10941", "title": "varieties-lemma-irreducible-dense-rational-points-geometrically-irreducible", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X$ is irreducible and has a dense set of $k$-rational points, then $X$ is geometrically irreducible."}
+{"_id": "10970", "title": "varieties-lemma-projective-after-field-extension", "text": "Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is projective over $K$ for some field extension $k \\subset K$, then $X$ is projective over $k$."}
+{"_id": "10971", "title": "varieties-lemma-tangent-space", "text": "The set of dotted arrows making (\\ref{equation-tangent-space}) commute has a canonical $\\kappa(x)$-vector space structure."}
+{"_id": "10980", "title": "varieties-lemma-noether-normalization", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$. Let $V \\subset S$ be an affine open neighbourhood of $s$. If $f$ is locally of finite type and $\\dim_x(X_s) = d$, then there exists an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a factorization $$ U \\xrightarrow{\\pi} \\mathbf{A}^d_V \\to V $$ of $f|_U : U \\to V$ such that $\\pi$ is quasi-finite."}
+{"_id": "10984", "title": "varieties-lemma-immersion-into-affine", "text": "Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$. Let $V \\subset S$ be an affine open neighbourhood of $s$. If $f$ is locally of finite type and $$ r = \\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) = \\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) = \\dim_{\\kappa(x)} T_{X/S, x} $$ then there exist \\begin{enumerate} \\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a factorization $$ U \\xrightarrow{j} \\mathbf{A}^{r + 1}_V \\to V $$ of $f|_U$ such that $j$ is an immersion, or \\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a factorization $$ U \\xrightarrow{j} D \\to V $$ of $f|_U$ such that $j$ is a closed immersion and $D \\to V$ is smooth of relative dimension $r$. \\end{enumerate}"}
+{"_id": "10996", "title": "varieties-lemma-generated-by-dim-plus-1-sections", "text": "Let $k$ be an infinite field. Let $X$ be an algebraic $k$-scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $V \\to \\Gamma(X, \\mathcal{L})$ be a linear map of $k$-vector spaces whose image generates $\\mathcal{L}$. Then there exists a subspace $W \\subset V$ with $\\dim_k(W) \\leq \\dim(X) + 1$ which generates $\\mathcal{L}$."}
+{"_id": "11022", "title": "varieties-lemma-units-variety", "text": "Let $k$ be a field. Let $X$ be a variety over $k$. The group $\\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group provided at least one of the following conditions holds: \\begin{enumerate} \\item $k$ is integrally closed in $\\Gamma(X, \\mathcal{O}_X)$, \\item $k$ is algebraically closed in $k(X)$, \\item $X$ is geometrically integral over $k$, or \\item $k$ is the ``intersection'' of the field extensions $k \\subset \\kappa(x)$ where $x$ runs over the closed points of $x$. \\end{enumerate}"}
+{"_id": "11045", "title": "varieties-lemma-hilbert-polynomial-H0", "text": "Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$ with Hilbert polynomial $P \\in \\mathbf{Q}[t]$. Then $$ P(d) = \\dim_k H^0(\\mathbf{P}^n_k, \\mathcal{F}(d)) $$ for all $d \\gg 0$."}
+{"_id": "11055", "title": "varieties-lemma-glue-valuation-ring", "text": "In Situation \\ref{situation-glue} assume that $B$ is a valuation ring. Then for every unit $u$ of $A$ either $u \\in R$ or $u^{-1} \\in R$."}
+{"_id": "11065", "title": "varieties-lemma-localization-semi-local", "text": "Let $A$ be a ring. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be a finite set of a primes of $A$. Let $S = A \\setminus \\bigcup \\mathfrak p_i$. Then $S$ is a multiplicative system and $S^{-1}A$ is a semi-local ring whose maximal ideals correspond to the maximal elements of the set $\\{\\mathfrak p_i\\}$."}
+{"_id": "11068", "title": "varieties-lemma-complement-codim-1-closed-points", "text": "Let $X$ be a scheme. Let $U \\subset X$ be an open. Assume \\begin{enumerate} \\item $U$ is a retrocompact open of $X$, \\item $X \\setminus U$ is discrete, and \\item for $x \\in X \\setminus U$ the local ring $\\mathcal{O}_{X, x}$ is Noetherian of dimension $\\leq 1$. \\end{enumerate} Then (1) there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and a section $s$ such that $U = X_s$ and (2) the map $\\Pic(X) \\to \\Pic(U)$ is surjective."}
+{"_id": "11076", "title": "varieties-lemma-surjection-on-pic-quasi-finite", "text": "Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ is Noetherian of dimension $\\leq 1$, $f$ is finite, and there exists a dense open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is a closed immersion. Then every invertible $\\mathcal{O}_X$-module is the pullback of an invertible $\\mathcal{O}_Y$-module."}
+{"_id": "11103", "title": "varieties-lemma-dim-1-nonproper-affine", "text": "Let $X$ be a separated scheme of finite type over $k$. If $\\dim(X) \\leq 1$ and no irreducible component of $X$ is proper of dimension $1$, then $X$ is affine."}
+{"_id": "11210", "title": "cotangent-lemma-mod-regular-sequence", "text": "Let $A \\to B$ be a surjective ring map whose kernel $I$ is generated by a Koszul-regular sequence (for example a regular sequence). Then $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$."}
+{"_id": "11214", "title": "cotangent-lemma-find-obstruction", "text": "In the situation above we have \\begin{enumerate} \\item There is a canonical element $\\xi \\in \\Ext^2_B(L_{B/A}, N)$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (\\ref{equation-to-solve}). \\item If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_B(L_{B/A}, N)$. \\item Given a solution $B'$, the set of automorphisms of $B'$ fitting into (\\ref{equation-to-solve}) is canonically isomorphic to $\\Ext^0_B(L_{B/A}, N)$. \\end{enumerate}"}
+{"_id": "11225", "title": "cotangent-lemma-find-obstruction-ringed-spaces", "text": "In the situation above we have \\begin{enumerate} \\item There is a canonical element $\\xi \\in \\Ext^2_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (\\ref{equation-to-solve-ringed-spaces}). \\item If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$. \\item Given a solution $X'$, the set of automorphisms of $X'$ fitting into (\\ref{equation-to-solve-ringed-spaces}) is canonically isomorphic to $\\Ext^0_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$. \\end{enumerate}"}
+{"_id": "11233", "title": "cotangent-lemma-cotangent-morphism-schemes", "text": "In the situation above there is a canonical isomorphism $$ L_{X/\\Lambda} =  L\\pi_!(Li^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) = L\\pi_!(i^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) = L\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}} \\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X) $$ in $D(\\mathcal{O}_X)$."}
+{"_id": "11243", "title": "cotangent-proposition-cotangent-complex-local-complete-intersection", "text": "Let $A \\to B$ be a local complete intersection map. Then $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$."}
+{"_id": "11301", "title": "spaces-cohomology-lemma-local-isomorphism", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Let $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism of $\\mathcal{O}_X$-modules. Let $\\overline{x}$ be a geometric point of $X$ lying over $x \\in |X|$. \\begin{enumerate} \\item If $\\mathcal{F}_{\\overline{x}} = 0$ then there exists an open neighbourhood $X' \\subset X$ of $x$ such that $\\mathcal{F}|_{X'} = 0$. \\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to \\mathcal{F}_{\\overline{x}}$ is injective, then there exists an open neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is injective. \\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to \\mathcal{F}_{\\overline{x}}$ is surjective, then there exists an open neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is surjective. \\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to \\mathcal{F}_{\\overline{x}}$ is bijective, then there exists an open neighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "11351", "title": "artin-lemma-predeformation-category", "text": "The functor $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ defined above is a predeformation category."}
+{"_id": "11369", "title": "artin-lemma-versal-implies-smooth", "text": "Let $S$ be a locally Noetherian scheme. Let $f : U \\to V$ be a morphism of schemes locally of finite type over $S$. Let $u_0 \\in U$ be a finite type point. The following are equivalent \\begin{enumerate} \\item $f$ is smooth at $u_0$, \\item $f$ viewed as an object of $(\\Sch/V)_{fppf}$ over $U$ is versal at $u_0$. \\end{enumerate}"}
+{"_id": "11381", "title": "artin-lemma-algebraic-stack-RS-star", "text": "Let $\\mathcal{X}$ be an algebraic stack over a base $S$. Then $\\mathcal{X}$ satisfies (RS*)."}
+{"_id": "11415", "title": "artin-proposition-second-diagonal-representable", "text": "Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Assume that \\begin{enumerate} \\item $\\Delta_\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$ is representable by algebraic spaces, \\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section \\ref{section-axioms}), \\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$. \\end{enumerate} Then $\\mathcal{X}$ is an algebraic stack."}
+{"_id": "11445", "title": "obsolete-lemma-p-ring-map", "text": "Let $\\varphi : R \\to S$ be a ring map. If \\begin{enumerate} \\item for any $x \\in S$ there exists $n > 0$ such that $x^n$ is in the image of $\\varphi$, and \\item for any $x \\in \\Ker(\\varphi)$ there exists $n > 0$ such that $x^n = 0$, \\end{enumerate} then $\\varphi$ induces a homeomorphism on spectra. Given a prime number $p$ such that \\begin{enumerate} \\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such that there exists an $n > 0$ with $x^{p^n} \\in \\varphi(R)$ and $p^nx \\in \\varphi(R)$, and \\item[(b)] the kernel of $\\varphi$ is generated by nilpotent elements, \\end{enumerate} then (1) and (2) hold, and for any ring map $R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ also satisfies (a), (b), (1), and (2) and in particular induces a homeomorphism on spectra."}
+{"_id": "11447", "title": "obsolete-lemma-spec-localization-first", "text": "Let $S$ be a multiplicative set of $A$. Then the map $$ f: \\Spec(S^{-1}A)\\longrightarrow \\Spec(A) $$ induced by the canonical ring map $A \\to S^{-1}A$ is a homeomorphism onto its image and $\\Im(f) = \\{ \\mathfrak p \\in \\Spec(A) : \\mathfrak p\\cap S = \\emptyset \\}$."}
+{"_id": "11451", "title": "obsolete-lemma-faithfully-flat-injective", "text": "If $R \\to S$ is a faithfully flat ring map then for every $R$-module $M$ the map $M \\to S \\otimes_R M$, $x \\mapsto 1 \\otimes x$ is injective."}
+{"_id": "11456", "title": "obsolete-lemma-algebraize-local-cohomology-bis-bis", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ and $d$ be integers. Assume \\begin{enumerate} \\item[(a)] $A$ has a dualizing complex, \\item[(b)] $\\text{cd}(A, I) \\leq d$, \\item[(c)] if $\\mathfrak p \\not \\in V(I)$ then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ or $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s$. \\end{enumerate} Then the assumptions of Algebraic and Formal Geometry, Lemma \\ref{algebraization-lemma-algebraize-local-cohomology-bis} hold for $A, I, \\mathfrak m, M$ and $H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M/I^nM)$ is an isomorphism for $i \\leq s$ and these modules are annihilated by a power of $I$."}
+{"_id": "11457", "title": "obsolete-lemma-combine-one", "text": "In Algebraic and Formal Geometry, Situation \\ref{algebraization-situation-bootstrap} we have $H^s_\\mathfrak a(M) = \\lim H^s_\\mathfrak a(M/I^nM)$."}
+{"_id": "11465", "title": "obsolete-lemma-formally-smooth-smooth", "text": "Let $R$ be a ring. Let $S$ be a $R$-algebra. If $S$ is of finite presentation and formally smooth over $R$ then $S$ is smooth over $R$."}
+{"_id": "11469", "title": "obsolete-lemma-trivialities-cohomological-descent-abelian", "text": "In Simplicial Spaces, Situation \\ref{spaces-simplicial-situation-simplicial-site} let $a_0$ be an augmentation towards a site $\\mathcal{D}$ as in Simplicial Spaces, Remark \\ref{spaces-simplicial-remark-augmentation-site}. Suppose given strictly full weak Serre subcategories $$ \\mathcal{A} \\subset \\textit{Ab}(\\mathcal{D}),\\quad \\mathcal{A}_n \\subset \\textit{Ab}(\\mathcal{C}_n) $$ Then \\begin{enumerate} \\item[(1)] the collection of abelian sheaves $\\mathcal{F}$ on $\\mathcal{C}_{total}$ whose restriction to $\\mathcal{C}_n$ is in $\\mathcal{A}_n$ for all $n$ is a strictly full weak Serre subcategory $\\mathcal{A}_{total} \\subset \\textit{Ab}(\\mathcal{C}_{total})$. \\end{enumerate} If $a_n^{-1}$ sends $\\mathcal{A}$ into $\\mathcal{A}_n$ for all $n$, then \\begin{enumerate} \\item[(2)] $a^{-1}$ sends $\\mathcal{A}$ into $\\mathcal{A}_{total}$ and \\item[(3)] $a^{-1}$ sends $D_\\mathcal{A}(\\mathcal{D})$ into $D_{\\mathcal{A}_{total}}(\\mathcal{C}_{total})$. \\end{enumerate} If $R^qa_{n, *}$ sends $\\mathcal{A}_n$ into $\\mathcal{A}$ for all $n, q$, then \\begin{enumerate} \\item[(4)] $R^qa_*$ sends $\\mathcal{A}_{total}$ into $\\mathcal{A}$ for all $q$, and \\item[(5)] $Ra_*$ sends $D_{\\mathcal{A}_{total}}^+(\\mathcal{C}_{total})$ into $D_\\mathcal{A}^+(\\mathcal{D})$. \\end{enumerate}"}
+{"_id": "11470", "title": "obsolete-lemma-glue-f-upper-shriek", "text": "Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. There exists a unique functor $f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$ such that \\begin{enumerate} \\item for any open $j : U \\to X$ with $f \\circ j$ separated there is a canonical isomorphism $j^! \\circ f^! = (f \\circ j)^!$, and \\item these isomorphisms for $U \\subset U' \\subset X$ are compatible with the isomorphisms in More \\'Etale Cohomology, Lemma \\ref{more-etale-lemma-upper-shriek-restriction}. \\end{enumerate}"}
+{"_id": "11471", "title": "obsolete-lemma-lqf-f-upper-shriek-stalk", "text": "Let $f : X \\to Y$ be a morphism of schemes which is locally quasi-finite. For an abelian group $A$ and a geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have $f^!(\\overline{y}_*A) = \\prod\\nolimits_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*A$."}
+{"_id": "11473", "title": "obsolete-lemma-P-not-preserved-base-change", "text": "Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras. Let $N$ be a differential graded $(A, B)$-bimodule with property (P). Let $M$ be a differential graded $A$-module with property (P). Then $Q = M \\otimes_A N$ is a differential graded $B$-module which represents $M \\otimes_A^\\mathbf{L} N$ in $D(B)$ and which has a filtration $$ 0 = F_{-1}Q \\subset F_0Q \\subset F_1Q \\subset \\ldots \\subset Q $$ by differential graded submodules such that $Q = \\bigcup F_pQ$, the inclusions $F_iQ \\to F_{i + 1}Q$ are admissible monomorphisms, the quotients $F_{i + 1}Q/F_iQ$ are isomorphic as differential graded $B$-modules to a direct sum of $(A \\otimes_R B)[k]$."}
+{"_id": "11477", "title": "obsolete-lemma-stein-projective", "text": "Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $X \\subset \\mathbf{P}^n_R$ be a closed subscheme. Assume that $R = \\Gamma(X, \\mathcal{O}_X)$. Then the special fibre $X_k$ is geometrically connected."}
+{"_id": "11481", "title": "obsolete-lemma-bound-degree-in-nbhd-generic-point", "text": "Let $f : X \\to Y$ be a morphism schemes. Assume \\begin{enumerate} \\item $X$ and $Y$ are integral schemes, \\item $f$ is locally of finite type and dominant, \\item $f$ is either quasi-compact or separated, \\item $f$ is generically finite, i.e., one of (1) -- (5) of Morphisms, Lemma \\ref{morphisms-lemma-finite-degree} holds. \\end{enumerate} Then there is a nonempty open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is finite locally free of degree $\\deg(X/Y)$. In particular, the degrees of the fibres of $f^{-1}(V) \\to V$ are bounded by $\\deg(X/Y)$."}
+{"_id": "11482", "title": "obsolete-lemma-factors-through-quotient", "text": "Let $S = \\Spec(R)$ be an affine scheme. Let $X$ be an algebraic space over $S$. Let $q_i : \\mathcal{F} \\to \\mathcal{Q}_i$, $i = 1, 2$ be surjective maps of quasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{Q}_1$ flat over $S$. Let $T \\to S$ be a quasi-compact morphism of schemes such that there exists a factorization $$ \\xymatrix{ & \\mathcal{F}_T \\ar[rd]^{q_{2, T}} \\ar[ld]_{q_{1, T}} \\\\ \\mathcal{Q}_{1, T} & & \\mathcal{Q}_{2, T} \\ar@{..>}[ll] } $$ Then exists a closed subscheme $Z \\subset S$ such that (a) $T \\to S$ factors through $Z$ and (b) $q_{1, Z}$ factors through $q_{2, Z}$. If $\\Ker(q_2)$ is a finite type $\\mathcal{O}_X$-module and $X$ quasi-compact, then we can take $Z \\to S$ of finite presentation."}
+{"_id": "11484", "title": "obsolete-lemma-sheaf-fpqc-open-covering", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $X = \\bigcup_{j \\in J} X_j$ be a Zariski covering, see Spaces, Definition \\ref{spaces-definition-Zariski-open-covering}. If each $X_j$ satisfies the sheaf property for the fpqc topology then $X$ satisfies the sheaf property for the fpqc topology."}
+{"_id": "11486", "title": "obsolete-lemma-reasonable-kolmogorov", "text": "Let $S$ be a scheme. Let $X$ be a reasonable algebraic space over $S$. Then $|X|$ is Kolmogorov (see Topology, Definition \\ref{topology-definition-generic-point})."}
+{"_id": "11488", "title": "obsolete-lemma-very-reasonable-Zariski-local", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a Zariski open covering $X = \\bigcup X_i$ such that each $X_i$ is very reasonable, then $X$ is very reasonable."}
+{"_id": "11490", "title": "obsolete-lemma-representable-very-reasonable", "text": "Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $Y \\to X$ be a representable morphism. If $X$ is very reasonable, so is $Y$."}
+{"_id": "11491", "title": "obsolete-lemma-very-reasonable-quasi-compact-pieces", "text": "Let $S$ be a scheme. Let $X$ be a very reasonable algebraic space over $S$. There exists a set of schemes $U_i$ and morphisms $U_i \\to X$ such that \\begin{enumerate} \\item each $U_i$ is a quasi-compact scheme, \\item each $U_i \\to X$ is \\'etale, \\item both projections $U_i \\times_X U_i \\to U_i$ are quasi-compact, and \\item the morphism $\\coprod U_i \\to X$ is surjective (and \\'etale). \\end{enumerate}"}
+{"_id": "11492", "title": "obsolete-lemma-vanishing-surjective", "text": "In Cohomology of Spaces, Situation \\ref{spaces-cohomology-situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is surjective."}
+{"_id": "11493", "title": "obsolete-lemma-vanishing-universally-closed", "text": "In Cohomology of Spaces, Situation \\ref{spaces-cohomology-situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is universally closed."}
+{"_id": "11507", "title": "obsolete-lemma-blowing-up-denominators", "text": "Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Assume $X$ integral with $\\dim_\\delta(X) = n$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$. Let $U \\subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\\mathcal{L}$ over $U$. There exists a projective morphism $$ \\pi : X' \\longrightarrow X $$ such that \\begin{enumerate} \\item $X'$ is integral, \\item $\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is an isomorphism, \\item there exist effective Cartier divisors $D, E \\subset X'$ such that $$ \\pi^*\\mathcal{L} = \\mathcal{O}_{X'}(D - E), $$ \\item the meromorphic section $s$ corresponds, via the isomorphism above, to the meromorphic section $1_D \\otimes (1_E)^{-1}$ (see Divisors, Definition \\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor}), \\item we have $$ \\pi_*([D]_{n - 1} - [E]_{n - 1}) = \\text{div}_\\mathcal{L}(s) $$ in $Z_{n - 1}(X)$. \\end{enumerate}"}
+{"_id": "11516", "title": "obsolete-lemma-directed-colimit-finite-type", "text": "Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules."}
+{"_id": "11517", "title": "obsolete-lemma-points-monomorphism", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The map $\\{\\Spec(k) \\to X \\text{ monomorphism}\\} \\to |X|$ is injective."}
+{"_id": "11518", "title": "obsolete-lemma-locally-ringed-space-direct-summand-free", "text": "Let $X$ be a locally ringed space. A direct summand of a finite free $\\mathcal{O}_X$-module is finite locally free."}
+{"_id": "11519", "title": "obsolete-lemma-characterize-injective", "text": "Let $R$ be a ring. Let $E$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $E$ is an injective $R$-module, and \\item given an ideal $I \\subset R$ and a module map $\\varphi : I \\to E$ there exists an extension of $\\varphi$ to an $R$-module map $R \\to E$. \\end{enumerate}"}
+{"_id": "11520", "title": "obsolete-lemma-periodic-length", "text": "Let $R$ be a local ring. \\begin{enumerate} \\item If $(M, N, \\varphi, \\psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \\varphi, \\psi) = \\text{length}_R(M) - \\text{length}_R(N)$. \\item If $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \\varphi, \\psi) = 0$. \\item Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \\to (M_1, N_1, \\varphi_1, \\psi_1) \\to (M_2, N_2, \\varphi_2, \\psi_2) \\to (M_3, N_3, \\varphi_3, \\psi_3) \\to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \\varphi_2, \\psi_2) = e_R(M_1, N_1, \\varphi_1, \\psi_1) + e_R(M_3, N_3, \\varphi_3, \\psi_3). $$ \\end{enumerate}"}
+{"_id": "11521", "title": "obsolete-lemma-examples-have-RS", "text": "Deformation Problems, Examples \\ref{examples-defos-example-finite-projective-modules}, \\ref{examples-defos-example-representations}, \\ref{examples-defos-example-continuous-representations}, and \\ref{examples-defos-example-graded-algebras} satisfy the Rim-Schlessinger condition (RS)."}
+{"_id": "11577", "title": "stacks-sheaves-lemma-enough-points", "text": "Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. The site $\\mathcal{X}_\\tau$ has enough points."}
+{"_id": "11605", "title": "stacks-sheaves-lemma-cech-to-cohomology-modules", "text": "Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is an object of $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$, \\item for every object $x$ of $\\mathcal{X}$ there exists a covering $\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic to $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, \\item the category $\\mathcal{U}$ has equalizers, and \\item the functor $f$ is faithful. \\end{enumerate} Then there is a first quadrant spectral sequence of $\\Gamma(\\mathcal{O}_\\mathcal{X})$-modules $$ E_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^*\\mathcal{F}) \\Rightarrow H^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F}) $$ converging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology."}
+{"_id": "11618", "title": "stacks-sheaves-proposition-coherator", "text": "Let $\\mathcal{X}$ be an algebraic stack over $S$. \\begin{enumerate} \\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is a Grothendieck abelian category. Consequently, $\\QCoh(\\mathcal{O}_\\mathcal{X})$ has enough injectives and all limits. \\item The inclusion functor $\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ has a right adjoint\\footnote{This functor is sometimes called the {\\it coherator}.} $$ Q : \\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\longrightarrow \\QCoh(\\mathcal{O}_\\mathcal{X}) $$ such that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping $Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism. \\end{enumerate}"}
+{"_id": "11650", "title": "resolve-lemma-extend-rational-map-normalized-blowing-up", "text": "Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is Noetherian, Nagata, and has dimension $2$. Let $Y$ be a proper scheme over $S$. Given an $S$-rational map $f : U \\to Y$ from $X$ to $Y$ there exists a sequence $$ X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 \\to X $$ and an $S$-morphism $f_n : X_n \\to Y$ such that $X_0 \\to X$ is the normalization, $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed point, and $f_n$ and $f$ agree."}
+{"_id": "11651", "title": "resolve-lemma-equivalence", "text": "The functor $F$ (\\ref{equation-equivalence}) is an equivalence."}
+{"_id": "11677", "title": "resolve-lemma-sequence-blowups", "text": "Let $X$ be a locally Noetherian scheme. Let $$ (X, p) = (X_0, p_0) \\leftarrow (X_1, p_1) \\leftarrow (X_2, p_2) \\leftarrow (X_3, p_3) \\leftarrow \\ldots $$ be a sequence of blowups such that \\begin{enumerate} \\item $p_i$ is closed, maps to $p_{i - 1}$, and $\\kappa(p_i) = \\kappa(p_{i - 1})$, \\item there exists an $x_1 \\in \\mathfrak m_p$ whose image in $\\mathfrak m_{p_i}$, $i > 0$ defines the exceptional divisor $E_i \\subset X_i$. \\end{enumerate} Then the sequence is obtained from a nonsingular arc $a : T \\to X$ as above."}
+{"_id": "11686", "title": "resolve-lemma-issquare", "text": "Let $\\kappa$ be a field. Let $I \\subset \\kappa[x, y]$ be an ideal. Let $$ a + b x + c y + d x^2 + exy + f y^2 \\in I^2 $$ for some $a, b, c, d, e, f \\in k$ not all zero. If the colength of $I$ in $\\kappa[x, y]$ is $> 1$, then $a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2$ for some $j, g, h, i \\in \\kappa$."}
+{"_id": "11786", "title": "spaces-duality-lemma-dualizing-unique-spaces", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ for some invertible object $L$ of $D(\\mathcal{O}_X)$."}
+{"_id": "11787", "title": "spaces-duality-lemma-dimension-function-scheme", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian quasi-separated algebraic space over $S$. Let $\\omega_X^\\bullet$ be a dualizing complex on $X$. Then $X$ the function $|X| \\to \\mathbf{Z}$ defined by $$ x \\longmapsto \\delta(x)\\text{ such that } \\omega_{X, \\overline{x}}^\\bullet[-\\delta(x)] \\text{ is a normalized dualizing complex over } \\mathcal{O}_{X, \\overline{x}} $$ is a dimension function on $|X|$."}
+{"_id": "11801", "title": "spaces-duality-lemma-compare-with-pullback-flat-proper", "text": "Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. The map (\\ref{equation-compare-with-pullback}) is an isomorphism for every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "11806", "title": "spaces-duality-lemma-existence-relative-dualizing", "text": "Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. There exists a relative dualizing complex $(\\omega_{X/Y}^\\bullet, \\tau)$."}
+{"_id": "11817", "title": "spaces-properties-lemma-characterize-separated", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent \\begin{enumerate} \\item $X$ is a separated algebraic space, \\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine the fibre product $U \\times_X V$ is affine and $$ \\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V) \\longrightarrow \\mathcal{O}(U \\times_X V) $$ is surjective. \\end{enumerate}"}
+{"_id": "11828", "title": "spaces-properties-lemma-finite-disjoint-quasi-compact", "text": "A finite disjoint union of quasi-compact algebraic spaces is a quasi-compact algebraic space."}
+{"_id": "11888", "title": "spaces-properties-lemma-reduced-local-ring", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. The following are equivalent \\begin{enumerate} \\item the local ring of $X$ at $x$ is reduced (Remark \\ref{remark-list-properties-local-ring-local-etale-topology}), \\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for some geometric point $\\overline{x}$ lying over $x$, and \\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for any geometric point $\\overline{x}$ lying over $x$. \\end{enumerate}"}
+{"_id": "11895", "title": "spaces-properties-lemma-regular-at-x", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$ be a point. The following are equivalent \\begin{enumerate} \\item $X$ is regular at $x$, and \\item the \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is regular for any (equivalently some) geometric point $\\overline{x}$ lying over $x$. \\end{enumerate}"}
+{"_id": "11968", "title": "intersection-lemma-compute-tor-nonsingular", "text": "Let $X$ be a nonsingular variety. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. The $\\mathcal{O}_X$-module $\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is coherent, has stalk at $x$ equal to $\\text{Tor}_p^{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)$, is supported on $\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G})$, and is nonzero only for $p \\in \\{0, \\ldots, \\dim(X)\\}$."}
+{"_id": "11980", "title": "intersection-lemma-exterior-product-rational-equivalence", "text": "Let $X$ and $Y$ be varieties. Let $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$. If $\\alpha \\sim_{rat} 0$ or $\\beta \\sim_{rat} 0$, then $\\alpha \\times \\beta \\sim_{rat} 0$."}
+{"_id": "11981", "title": "intersection-lemma-exterior-product", "text": "Let $X$ and $Y$ be nonsingular varieties. Let $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$. Then \\begin{enumerate} \\item $\\text{pr}_Y^*(\\beta) = [X] \\times \\beta$ and $\\text{pr}_X^*(\\alpha) = \\alpha \\times [Y]$, \\item $\\alpha \\times [Y]$ and $[X]\\times \\beta$ intersect properly on $X\\times Y$, and \\item we have $\\alpha \\times \\beta = (\\alpha \\times [Y])\\cdot ([X]\\times\\beta) = pr_Y^*(\\alpha) \\cdot pr_X^*(\\beta)$ in $Z_{r + s}(X \\times Y)$. \\end{enumerate}"}
+{"_id": "11987", "title": "intersection-lemma-flat-pullback-and-intersections", "text": "Let $f : X \\to Y$ be a flat morphism of nonsingular varieties. Let $\\alpha$ be a $r$-cycle on $Y$ and $\\beta$ an $s$-cycle on $Y$. Assume that $\\alpha$ and $\\beta$ intersect properly. Then $f^*\\alpha$ and $f^*\\beta$ intersect properly and $f^*( \\alpha \\cdot \\beta ) = f^*\\alpha \\cdot f^*\\beta$."}
+{"_id": "12079", "title": "homology-lemma-composition-strict", "text": "Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$, $g : B \\to C$ be strict morphisms of filtered objects. \\begin{enumerate} \\item In general the composition $g \\circ f$ is not strict. \\item If $g$ is injective, then $g \\circ f$ is strict. \\item If $f$ is surjective, then $g \\circ f$ is strict. \\end{enumerate}"}
+{"_id": "12082", "title": "homology-lemma-fibre-product-filtered", "text": "Let $\\mathcal{A}$ be an abelian category. Let $A, B, C \\in \\text{Fil}(\\mathcal{A})$. Let $f : B \\to A$ and $g : C \\to A$ be morphisms. Then there exists a fibre product $$ \\xymatrix{ B \\times_A C \\ar[r]_{g'} \\ar[d]_{f'} & B \\ar[d]^f \\\\ C \\ar[r]^g & A } $$ in $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$."}
+{"_id": "12092", "title": "homology-lemma-spectral-sequence-filtered-differential-d1", "text": "Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$. The spectral sequence $(E_r, d_r)_{r \\geq 0}$ associated to $(K, F, d)$ has $$ d_1^p : E_1^p = H(\\text{gr}^pK) \\longrightarrow H(\\text{gr}^{p + 1}K) = E_1^{p + 1} $$ equal to the boundary map in homology associated to the short exact sequence of differential objects $$ 0 \\to \\text{gr}^{p + 1}K \\to F^pK/F^{p + 2}K \\to \\text{gr}^pK \\to 0. $$"}
+{"_id": "12108", "title": "homology-lemma-right-resolution-gives-qis", "text": "Let $M^\\bullet$ be a complex of abelian groups. Let $$ 0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots $$ be an exact complex of complexes of abelian groups. Set $A^{p, q} = A_p^q$ to obtain a double complex. Then the map $M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$ induced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism."}
+{"_id": "12120", "title": "homology-lemma-essentially-constant-into-karoubian", "text": "Let $\\mathcal{I}$ be a category, let $\\mathcal{A}$ be a pre-additive Karoubian category, and let $M : \\mathcal{I} \\to \\mathcal{A}$ be a diagram. \\begin{enumerate} \\item Assume $\\mathcal{I}$ is filtered. The following are equivalent \\begin{enumerate} \\item $M$ is essentially constant, \\item $X = \\colim M$ exists and there exists a cofinal filtered subcategory $\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$ such that $X_{i'}$ maps isomorphically to $X$ and $Z_{i'}$ to zero in $M_{i''}$ for some $i' \\to i''$ in $\\mathcal{I}'$. \\end{enumerate} \\item Assume $\\mathcal{I}$ is cofiltered. The following are equivalent \\begin{enumerate} \\item $M$ is essentially constant, \\item $X = \\lim M$ exists and there exists an initial cofiltered subcategory $\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$ such that $X$ maps isomorphically to $X_{i'}$ and $M_{i''} \\to Z_{i'}$ is zero for some $i'' \\to i'$ in $\\mathcal{I}'$. \\end{enumerate} \\end{enumerate}"}
+{"_id": "12220", "title": "categories-lemma-product-with-connected", "text": "Let $\\mathcal{I}$ and $\\mathcal{J}$ be a categories and denote $p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection. If $\\mathcal{I}$ is connected, then for a diagram $M : \\mathcal{J} \\to \\mathcal{C}$ the colimit $\\colim_\\mathcal{J} M$ exists if and only if $\\colim_{\\mathcal{I} \\times \\mathcal{J}} M \\circ p$ exists and if so these colimits are equal."}
+{"_id": "12225", "title": "categories-lemma-push-outs-coequalizers-exist", "text": "Let $\\mathcal{C}$ be a category. The following are equivalent: \\begin{enumerate} \\item Connected finite colimits exist in $\\mathcal{C}$. \\item Coequalizers and pushouts exist in $\\mathcal{C}$. \\end{enumerate}"}
+{"_id": "12266", "title": "categories-lemma-right-localization-diagram", "text": "Let $\\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of $\\mathcal{C}$ and if $$ \\xymatrix{ Q(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\ Q(Y) \\ar[r]^b & Q(Y') } $$ is a commutative diagram in $S^{-1}\\mathcal{C}$, then there exists a morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative diagram $$ \\xymatrix{ X \\ar[d]_f & X'' \\ar[l]^s \\ar[d]^{f''} \\ar[r]_g & X' \\ar[d]^{f'} \\\\ Y & Y'' \\ar[l]_t \\ar[r]^h & Y' } $$ in $\\mathcal{C}$ with $s, t \\in S$ and $a = gs^{-1}$, $b = ht^{-1}$."}
+{"_id": "12300", "title": "categories-lemma-inertia-fibred-groupoids", "text": "Let $\\mathcal{C}$ be a category. If $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids, then so is the inertia fibred category $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{C}$."}
+{"_id": "12320", "title": "categories-lemma-prepare-representable-map-stack-in-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids over $\\mathcal{C}$. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism. Let $G : \\mathcal{C}/U \\to \\mathcal{Y}$ be a $1$-morphism. Then $$ (\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X} \\longrightarrow \\mathcal{C}/U $$ is a category fibred in groupoids."}
+{"_id": "12322", "title": "categories-lemma-criterion-representable-map-stack-in-groupoids", "text": "Let $\\mathcal{C}$ be a category. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids over $\\mathcal{C}$. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism. Make a choice of pullbacks for $\\mathcal{Y}$. Assume \\begin{enumerate} \\item each functor $F_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$ between fibre categories is faithful, and \\item for each $U$ and each $y \\in \\mathcal{Y}_U$ the presheaf $$ (f : V \\to U) \\longmapsto \\{(x, \\phi) \\mid x \\in \\mathcal{X}_V, \\phi : f^*y \\to F(x)\\}/\\cong $$ is a representable presheaf on $\\mathcal{C}/U$. \\end{enumerate} Then $F$ is representable."}
+{"_id": "12442", "title": "topologies-lemma-composition", "text": "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{Zar}$ and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and $g_{small} \\circ f_{small} = (g \\circ f)_{small}$."}
+{"_id": "12443", "title": "topologies-lemma-morphism-big-small-cartesian-diagram", "text": "Let $\\Sch_{Zar}$ be a big Zariski site. Consider a cartesian diagram $$ \\xymatrix{ T' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\ S' \\ar[r]^g & S } $$ in $\\Sch_{Zar}$. Then $i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$ and $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$."}
+{"_id": "12489", "title": "topologies-lemma-affine-big-site-ph", "text": "Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$. The functor $(\\textit{Aff}/S)_{ph} \\to (\\Sch/S)_{ph}$ is cocontinuous and induces an equivalence of topoi from $\\Sh((\\textit{Aff}/S)_{ph})$ to $\\Sh((\\Sch/S)_{ph})$."}
+{"_id": "12492", "title": "topologies-lemma-composition-ph", "text": "Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{ph}$ and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$."}
+{"_id": "12501", "title": "topologies-lemma-fpqc-covering-affines-mapping-in", "text": "Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of morphisms of schemes with target $T$. Assume that \\begin{enumerate} \\item each $f_i$ is flat, and \\item every affine scheme $Z$ and morphism $h : Z \\to T$ there exists a standard fpqc covering $\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ which refines the family $\\{T_i \\times_T Z \\to Z\\}_{i \\in I}$. \\end{enumerate} Then $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$."}
+{"_id": "12555", "title": "pic-lemma-hilb-d-limit-preserving", "text": "Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is of finite presentation, then the functor $\\Hilbfunctor^d_{X/S}$ is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving})."}
+{"_id": "12653", "title": "constructions-lemma-gkn-representable", "text": "Let $0 < k < n$. The functor $G(k, n)$ of (\\ref{equation-gkn}) is representable by a scheme."}
+{"_id": "12698", "title": "algebraization-lemma-restriction-derived-complete", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals. The inverse image of $D_{comp}(\\mathcal{O}, \\mathcal{I})$ under the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$ is $D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}')$."}
+{"_id": "12705", "title": "algebraization-lemma-restriction-derived-complete-equivalence", "text": "Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a homomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals. If $\\mathcal{O} \\to \\mathcal{O}'$ is flat and $\\mathcal{O}/\\mathcal{I} \\cong \\mathcal{O}'/\\mathcal{I}\\mathcal{O}'$, then the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$ induces an equivalence $D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}') \\to D_{comp}(\\mathcal{O}, \\mathcal{I})$."}
+{"_id": "12747", "title": "algebraization-lemma-fully-faithful-very-general", "text": "Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be the set of open subschemes of $U$ containing $U \\cap V(I)$ ordered by reverse inclusion. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent $\\mathcal{O}_V$-modules for some $V \\in \\mathcal{V}$. The map $$ \\colim_{V' \\geq V} \\Hom_V(\\mathcal{G}|_{V'}, \\mathcal{F}|_{V'}) \\longrightarrow \\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge) $$ is bijective if the following assumptions hold: \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$, $\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$ and $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$. \\end{enumerate}"}
+{"_id": "12756", "title": "algebraization-lemma-when-ML", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If the inverse system $H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler, then the canonical maps $$ \\widetilde{M/I^nM}|_U \\to \\mathcal{F}_n $$ are surjective for all $n$ where $M$ is as in (\\ref{equation-guess})."}
+{"_id": "12765", "title": "algebraization-lemma-construct-unique", "text": "In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$ and $d \\geq 1$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete, has a dualizing complex, and $\\text{cd}(A, I) \\leq d$, \\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module, \\item $(\\mathcal{F}_n)$ satisfies the strict $(1, 1 + d)$-inequalities. \\end{enumerate} Then there exists a unique coherent $\\mathcal{O}_U$-module $\\mathcal{F}$ whose completion is $(\\mathcal{F}_n)$ such that for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$ we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$."}
+{"_id": "12783", "title": "algebraization-lemma-injective-torsion-in-pic", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$ be a nonzerodivisor and assume that $\\text{depth}(A/fA) \\geq 2$, or equivalently $\\text{depth}(A) \\geq 3$. Let $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. The map $$ \\Pic(U) \\to \\Pic(U_0) $$ is injective on torsion."}
+{"_id": "12792", "title": "algebraization-proposition-application-higher", "text": "Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$. Let $\\mathcal{F}$ be a coherent module on $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $s \\geq 0$. Assume \\begin{enumerate} \\item $A$ is $I$-adically complete and has a dualizing complex, \\item if $x \\in U \\setminus V(I)$ then $\\text{depth}(\\mathcal{F}_x) > s$ or $$ \\text{depth}(\\mathcal{F}_x) + \\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + s + 1 $$ for all $z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$, \\item one of the following conditions holds: \\begin{enumerate} \\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$ is $(S_{s + 1})$, or \\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In the sense that the difference of the maximal and minimal values on $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}. \\end{enumerate} \\end{enumerate} Then the maps $$ H^i(U, \\mathcal{F}) \\longrightarrow \\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F}) $$ are isomorphisms for $i < s$. Moreover we have an isomorphism $$ \\colim H^s(V, \\mathcal{F}) \\longrightarrow \\lim H^s(U, \\mathcal{F}/I^n\\mathcal{F}) $$ where the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$."}
+{"_id": "12799", "title": "algebraization-proposition-trivial-local-pic-complete-intersection", "text": "Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\\geq 4$, then the Picard group of the punctured spectrum of $A$ is trivial."}
+{"_id": "12833", "title": "spaces-over-fields-lemma-order-vanishing-agrees", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \\in R(X)^*$. If the prime divisor $Z \\subset X$ meets the schematic locus of $X$, then the order of vanishing $\\text{ord}_Z(f)$ of Definition \\ref{definition-order-vanishing} agrees with the order of vanishing of Divisors, Definition \\ref{divisors-definition-order-vanishing}."}
+{"_id": "12839", "title": "spaces-over-fields-lemma-normal-c1-injective", "text": "Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. If $X$ is normal, then the map (\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$ is injective."}
+{"_id": "12879", "title": "spaces-over-fields-lemma-intersection-number-additive", "text": "In the situation of Definition \\ref{definition-intersection-number} the intersection number $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$, then we have $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) = (\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) + (\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z) $$"}
+{"_id": "12880", "title": "spaces-over-fields-lemma-intersection-number-in-terms-of-components", "text": "In the situation of Definition \\ref{definition-intersection-number} let $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let $m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}} (\\mathcal{O}_{Z, \\overline{x}_i})$ where $\\overline{x}_i$ is a geometric generic point of $Z_i$. Then $$ (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) = \\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i) $$"}
+{"_id": "12903", "title": "spaces-divisors-lemma-weakly-ass-support", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\text{WeakAss}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$."}
+{"_id": "12909", "title": "spaces-divisors-lemma-restriction-injective-open-contains-weakly-ass", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $U \\to X$ is an \\'etale morphism such that $\\text{WeakAss}(\\mathcal{F}) \\subset \\Im(|U| \\to |X|)$, then $\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective."}
+{"_id": "12918", "title": "spaces-divisors-lemma-finite-flat-weak-assassin-up-down", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a finite flat morphism of algebraic spaces. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Let $x \\in |X|$ be a point with image $y \\in |Y|$. Then $$ x \\in \\text{WeakAss}(g^*\\mathcal{G}) \\Leftrightarrow y \\in \\text{WeakAss}(\\mathcal{G}) $$"}
+{"_id": "12921", "title": "spaces-divisors-lemma-locally-finite-type-locally-Noetherian-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then  the fibres of $f$ are locally Noetherian."}
+{"_id": "12923", "title": "spaces-divisors-lemma-bourbaki", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. Assume \\begin{enumerate} \\item $\\mathcal{F}$ is flat over $Y$, \\item $X$ and $Y$ are locally Noetherian, and \\item the fibres of $f$ are locally Noetherian. \\end{enumerate} Then $$ \\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) = \\{x \\in \\text{Ass}_{X/Y}(\\mathcal{F})\\text{ such that } f(x) \\in \\text{Ass}_Y(\\mathcal{G}) \\} $$"}
+{"_id": "12928", "title": "spaces-divisors-lemma-base-change-fitting-ideal", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_Y$-module. Then $f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_X = \\text{Fit}_i(f^*\\mathcal{F})$."}
+{"_id": "12932", "title": "spaces-divisors-lemma-fitting-ideal-finite-locally-free", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. Let $r \\geq 0$. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is finite locally free of rank $r$ \\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and $\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_X$, and \\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and $\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_X$ for $k \\geq r$. \\end{enumerate}"}
+{"_id": "12934", "title": "spaces-divisors-lemma-finite-presentation-module", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation. Let $X = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset \\ldots$ be as in Lemma \\ref{lemma-locally-free-rank-r-pullback}. Set $X_r = Z_{r - 1} \\setminus Z_r$. Then $X' = \\coprod_{r \\geq 0} X_r$ represents the functor $$ F_{flat} : \\Sch/X \\longrightarrow \\textit{Sets},\\quad\\quad T \\longmapsto \\left\\{ \\begin{matrix} \\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\ \\emptyset & \\text{otherwise} \\end{matrix} \\right. $$ Moreover, $\\mathcal{F}|_{X_r}$ is locally free of rank $r$ and the morphisms $X_r \\to X$ and $X' \\to X$ are of finite presentation."}
+{"_id": "12938", "title": "spaces-divisors-lemma-effective-Cartier-makes-dimension-drop", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \\subset X$ be an effective Cartier divisor. Let $x \\in |D|$. If $\\dim_x(X) < \\infty$, then $\\dim_x(D) < \\dim_x(X)$."}
+{"_id": "12942", "title": "spaces-divisors-lemma-pullback-effective-Cartier-defined", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $D \\subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases: \\begin{enumerate} \\item $f(x) \\not \\in |D|$ for any weakly associated point $x$ of $X$, \\item $f$ is flat, and \\item add more here as needed. \\end{enumerate}"}
+{"_id": "12949", "title": "spaces-divisors-lemma-effective-Cartier-divisor-Sk", "text": "Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. Let $D \\subset X$ be an effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$."}
+{"_id": "12963", "title": "spaces-divisors-lemma-twists-of-structure-sheaf", "text": "In Situation \\ref{situation-relative-proj}. The relative Proj comes equipped with a quasi-coherent sheaf of $\\mathbf{Z}$-graded algebras $\\bigoplus_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n)$ and a canonical homomorphism of graded algebras $$ \\psi : \\pi^*\\mathcal{A} \\longrightarrow \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n) $$ whose base change to any scheme over $X$ agrees with Constructions, Lemma \\ref{constructions-lemma-glue-relative-proj-twists}."}
+{"_id": "12966", "title": "spaces-divisors-lemma-relative-proj-quasi-compact", "text": "In Situation \\ref{situation-relative-proj}. If one of the following holds \\begin{enumerate} \\item $\\mathcal{A}$ is of finite type as a sheaf of $\\mathcal{A}_0$-algebras, \\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an $\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type $\\mathcal{A}_0$-module, \\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule $\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that $\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$, \\end{enumerate} then $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is quasi-compact."}
+{"_id": "12967", "title": "spaces-divisors-lemma-relative-proj-finite-type", "text": "In Situation \\ref{situation-relative-proj}. If $\\mathcal{A}$ is of finite type as a sheaf of $\\mathcal{O}_X$-algebras, then $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is of finite type."}
+{"_id": "12968", "title": "spaces-divisors-lemma-relative-proj-universally-closed", "text": "In Situation \\ref{situation-relative-proj}. If $\\mathcal{O}_X \\to \\mathcal{A}_0$ is an integral algebra map\\footnote{In other words, the integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}_0$, see Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-integral-closure}, equals $\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra, then $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is universally closed."}
+{"_id": "12969", "title": "spaces-divisors-lemma-relative-proj-proper", "text": "In Situation \\ref{situation-relative-proj}. The following conditions are equivalent \\begin{enumerate} \\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module and $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra, \\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module  and $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_X$-algebra. \\end{enumerate} If these conditions hold, then $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is proper."}
+{"_id": "12994", "title": "spaces-divisors-lemma-blow-up-reduced-space", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\\mathcal{I}$ is reduced."}
+{"_id": "12995", "title": "spaces-divisors-lemma-blowup-finite-nr-irreducibles", "text": "Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \\to X$ be the blowup of $X$ in a closed subspace. If $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization} then so does $X'$."}
+{"_id": "13004", "title": "spaces-divisors-lemma-strict-transform-different-centers", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \\subset B$ be a closed subspace. Let $D \\subset B$ be an effective Cartier divisor. Let $Z' \\subset B$ be the closed subspace cut out by the product of the ideal sheaves of $Z$ and $D$. Let $B' \\to B$ be the blowup of $B$ in $Z$. \\begin{enumerate} \\item The blowup of $B$ in $Z'$ is isomorphic to $B' \\to B$. \\item Let $f : X \\to B$ be a morphism of algebraic spaces and let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If the subsheaf of $\\mathcal{F}$ of sections supported on $|f^{-1}D|$ is zero, then the strict transform of $\\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\\mathcal{F}$ relative to the blowing up of $B$ in $Z'$. \\end{enumerate}"}
+{"_id": "13007", "title": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal", "text": "Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_B$-module. Let $Z_k \\subset S$ be the closed subscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}. Let $B' \\to B$ be the blowup of $B$ in $Z_k$ and let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$. Then $\\mathcal{F}'$ can locally be generated by $\\leq k$ sections."}
+{"_id": "13034", "title": "dga-lemma-compose-homotopy", "text": "Let $(A, \\text{d})$ be a differential graded algebra. Let $f, g : L \\to M$ be homomorphisms of differential graded $A$-modules. Suppose given further homomorphisms $a : K \\to L$, and $c : M \\to N$. If $h : L \\to M$ is an $A$-module map which defines a homotopy between $f$ and $g$, then $c \\circ h \\circ a$ defines a homotopy between $c \\circ f \\circ a$ and $c \\circ g \\circ a$."}
+{"_id": "13053", "title": "dga-lemma-characterize-hom-other-side", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra. Let $M$ be a right differential graded $A$-module and let $M'$ be a left differential graded $A$-module. Let $N^\\bullet$ be a complex of $R$-modules. Then we have $$ \\Hom_{\\text{left diff graded }A\\text{-modules}}(M', \\Hom(M, N^\\bullet)) = \\Hom_{\\text{Comp}(R)}(M \\otimes_A M', N^\\bullet) $$ where $M \\otimes_A M'$ is viewed as a complex of $R$-modules as in Section \\ref{section-tensor-product}."}
+{"_id": "13091", "title": "dga-lemma-what-makes-a-bimodule-dg", "text": "Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras over $R$. Let $M$ be a right differential graded $B$-module. There is a $1$-to-$1$ correspondence between $(A, B)$-bimodule structures on $M$ compatible with the given differential graded $B$-module structure and homomorphisms $$ A \\longrightarrow \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, M) $$ of differential graded $R$-algebras."}
+{"_id": "13109", "title": "dga-lemma-base-change-K-flat", "text": "Let $R \\to R'$ be a ring map. Let $(A, \\text{d})$ be a differential graded $R$-algebra. Let $(A', \\text{d})$ be the base change, i.e., $A' = A \\otimes_R R'$. If $A$ is K-flat as a complex of $R$-modules, then \\begin{enumerate} \\item $- \\otimes_A^\\mathbf{L} A' : D(A, \\text{d}) \\to D(A', \\text{d})$ is equal to the right derived functor of $$ K(A, \\text{d}) \\longrightarrow K(A', \\text{d}),\\quad M \\longmapsto M \\otimes_R R' $$ \\item the diagram $$ \\xymatrix{ D(A, \\text{d}) \\ar[r]_{- \\otimes_A^\\mathbf{L} A'} \\ar[d]_{restriction} & D(A', \\text{d}) \\ar[d]^{restriction} \\\\ D(R) \\ar[r]^{- \\otimes_R^\\mathbf{L} R'} & D(R') } $$ commutes, and \\item if $M$ is K-flat as a complex of $R$-modules, then the differential graded $A'$-module $M \\otimes_R R'$ represents $M \\otimes_A^\\mathbf{L} A'$. \\end{enumerate}"}
+{"_id": "13128", "title": "dga-lemma-compose-tensor-functors-tor", "text": "Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and $(C, \\text{d})$ be differential graded $R$-algebras. Assume $A \\otimes_R C$ represents $A \\otimes^\\mathbf{L}_R C$ in $D(R)$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-bimodule. Then the composition $$ \\xymatrix{ D(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & & D(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & & D(C, \\text{d}) } $$ is isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for some differential graded $(A, C)$-bimodule $N''$."}
+{"_id": "13185", "title": "spaces-more-groupoids-lemma-groupoid-on-field-geometrically-irreducible", "text": "In Situation \\ref{situation-groupoid-on-field} the algebraic space $R$ is geometrically unibranch. In Situation \\ref{situation-group-over-field} the algebraic space $G$ is geometrically unibranch."}
+{"_id": "13251", "title": "modules-lemma-quasi-coherent-limit-finite-presentation", "text": "Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $R = \\Gamma(X, \\mathcal{O}_X)$. Let $M$ be an $R$-module. The $\\mathcal{O}_X$-module $\\mathcal{F}_M$ associated to $M$ is a directed colimit of finitely presented $\\mathcal{O}_X$-modules."}
+{"_id": "13258", "title": "modules-lemma-i-star-quasi-coherent", "text": "Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a closed immersion of ringed spaces. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module. Then $i_*\\mathcal{F}$ is locally on $X$ the cokernel of a map of quasi-coherent $\\mathcal{O}_X$-modules."}
+{"_id": "13289", "title": "modules-lemma-stalk-tensor-algebra", "text": "In the situation described above. Let $x \\in X$. There are canonical isomorphisms of $\\mathcal{O}_{X, x}$-modules $\\text{T}(\\mathcal{F})_x = \\text{T}(\\mathcal{F}_x)$, $\\text{Sym}(\\mathcal{F})_x = \\text{Sym}(\\mathcal{F}_x)$, and $\\wedge(\\mathcal{F})_x = \\wedge(\\mathcal{F}_x)$."}
+{"_id": "13301", "title": "modules-lemma-pullback-invertible", "text": "Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\\mathcal{L}$ of an invertible $\\mathcal{O}_Y$-module is invertible."}
+{"_id": "13316", "title": "modules-lemma-differential-seq", "text": "In Lemma \\ref{lemma-functoriality-differentials} suppose that $\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel $\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that $\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact sequence of $\\mathcal{O}'_2$-modules $$ \\mathcal{I}/\\mathcal{I}^2 \\longrightarrow \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2 \\longrightarrow \\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1} \\longrightarrow 0 $$ The leftmost map is characterized by the rule that a local section $f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$."}
+{"_id": "13326", "title": "modules-lemma-differentials-relative-de-rham-complex-order-1", "text": "Let $f : X \\to Y$ be a morphism of ringed spaces. The differentials $\\text{d} : \\Omega^i_{X/Y}  \\to \\Omega^{i + 1}_{X/Y}$ are differential operators of order $1$ on $X/Y$."}
+{"_id": "13382", "title": "defos-lemma-inf-obs-map-rel", "text": "Let $(f, f')$ be a morphism of first order thickenings as in Situation \\ref{situation-morphism-thickenings}. Let $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map. Assume that $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over $S'$ and that $(f, f')$ is a strict morphism of thickenings. There exists an element $$ o(\\varphi) \\in  \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J}) $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$."}
+{"_id": "13453", "title": "groupoids-quotients-lemma-set-theoretic-equivalence-geometric", "text": "In the situation of Definition \\ref{definition-set-theoretic-equivalence}. The following are equivalent: \\begin{enumerate} \\item The morphism $j$ is a set-theoretic equivalence relation. \\item The morphism $j$ is universally injective and $j(|R|) \\subset |U \\times_B U|$ contains the image of $|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}). \\item For every algebraically closed field $k$ over $B$ of sufficiently large cardinality the map $j : R(k) \\to U(k) \\times U(k)$ is injective and its image is an equivalence relation. \\end{enumerate} If $j$ is decent, or locally separated, or quasi-separated these are also equivalent to \\begin{enumerate} \\item[(4)] For every algebraically closed field $k$ over $B$ the map $j : R(k) \\to U(k) \\times U(k)$ is injective and its image is an equivalence relation. \\end{enumerate}"}
+{"_id": "13456", "title": "groupoids-quotients-lemma-orbit-space", "text": "Let $B \\to S$ as in Section \\ref{section-conventions-notation}. Let $j : R \\to U \\times_B U$ be a set-theoretic pre-equivalence relation. A morphism $\\phi : U \\to X$ is an orbit space for $R$ if and only if \\begin{enumerate} \\item $\\phi \\circ s = \\phi \\circ t$, i.e., $\\phi$ is invariant, \\item the induced morphism $(t, s) : R \\to U \\times_X U$ is surjective, and \\item the morphism $\\phi : U \\to X$ is surjective. \\end{enumerate} This characterization applies for example if $j$ is a pre-equivalence relation, or comes from a groupoid in algebraic spaces over $B$, or comes from the action of a group algebraic space over $B$ on $U$."}
+{"_id": "13473", "title": "spaces-resolve-lemma-modification", "text": "Let $(A, \\mathfrak m, \\kappa)$ be a $2$-dimensional Noetherian local domain such that $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ is a normal scheme. Then any modification $f : X \\to \\Spec(A)$ is a morphism as in (\\ref{equation-modification})."}
+{"_id": "13483", "title": "spaces-resolve-lemma-Nagata-normalized-blowup", "text": "In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is a normal Nagata algebraic space proper over $X$."}
+{"_id": "13495", "title": "duality-theorem-lichtenbaum", "text": "Let $X$ be a nonempty separated scheme of finite type over a field $k$. Let $d = \\dim(X)$. The following are equivalent \\begin{enumerate} \\item $H^d(X, \\mathcal{F}) = 0$ for all coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$ on $X$, \\item $H^d(X, \\mathcal{F}) = 0$ for all quasi-coherent $\\mathcal{O}_X$-modules $\\mathcal{F}$ on $X$, and \\item no irreducible component $X' \\subset X$ of dimension $d$ is proper over $k$. \\end{enumerate}"}
+{"_id": "13542", "title": "duality-lemma-proper-perfect-base-change", "text": "Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes. Let $g : Y' \\to Y$ be a morphism with $Y'$ Noetherian. If $X$ and $Y'$ are tor independent over $Y$, then the base change map (\\ref{equation-base-change-map}) is an isomorphism for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$."}
+{"_id": "13556", "title": "duality-lemma-restrict-before-or-after", "text": "In Situation \\ref{situation-shriek} let $$ \\xymatrix{ U \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\ V \\ar[r]^{j'} & Y } $$ be a commutative diagram of $\\textit{FTS}_S$ where $j$ and $j'$ are open immersions. Then $j^* \\circ f^! = g^! \\circ (j')^*$ as functors $D^+_\\QCoh(\\mathcal{O}_Y) \\to D^+(\\mathcal{O}_U)$."}
+{"_id": "13570", "title": "duality-lemma-good-dualizing-independence-covering", "text": "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\\mathcal{U}$, $\\mathcal{V}$ be two finite open coverings of $X$ by schemes separated over $S$. If there exists a dualizing complex normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then there exists a dualizing complex normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{V}$ and these complexes are canonically isomorphic."}
+{"_id": "13571", "title": "duality-lemma-existence-good-dualizing", "text": "In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type over $S$ and let $\\mathcal{U}$ be a finite open covering of $X$ by schemes separated over $S$. Then there exists a dualizing complex normalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$."}
+{"_id": "13573", "title": "duality-lemma-open-immersion-good-dualizing-complex", "text": "Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}. Let $j : X \\to Y$ be an open immersion of schemes of finite type over $S$. Let $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes normalized relative to $\\omega_S^\\bullet$. Then there is a canonical isomorphism $\\omega_X^\\bullet = \\omega_Y^\\bullet|_X$."}
+{"_id": "13604", "title": "duality-lemma-gorenstein-local-source-and-target", "text": "The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is Gorenstein'' is local in the fppf topology on the target and local in the syntomic topology on the source."}
+{"_id": "13632", "title": "duality-lemma-lower-shriek-well-defined", "text": "The functor $Rf_!$ is, up to isomorphism, independent of the choice of the compactification."}
+{"_id": "13681", "title": "more-morphisms-lemma-descending-property-thickening", "text": "The property of being a thickening is fpqc local. Similarly for first order thickenings."}
+{"_id": "13694", "title": "more-morphisms-lemma-affine-formally-unramified", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally unramified if and only if $\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally unramified ring map."}
+{"_id": "13708", "title": "more-morphisms-lemma-formally-etale-not-affine", "text": "If $f : X \\to S$ is a formally \\'etale morphism, then given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\ S & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of schemes over $S$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition \\ref{definition-formally-etale} the condition that $T$ be affine may be dropped."}
+{"_id": "13739", "title": "more-morphisms-lemma-differentials-formally-unramified-formally-smooth", "text": "Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$. Assume that $Z$ is formally smooth over $S$. Then the canonical exact sequence $$ 0 \\to \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0 $$ of Lemma \\ref{lemma-universally-unramified-differentials-sequence} is short exact."}
+{"_id": "13757", "title": "more-morphisms-lemma-base-change-NL", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ The canonical map $(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ induces an isomorphism on $H^0$ and a surjection on $H^{-1}$."}
+{"_id": "13758", "title": "more-morphisms-lemma-flat-base-change-NL", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ X' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ If $Y' \\to Y$ is flat, then the canonical map $(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism."}
+{"_id": "13770", "title": "more-morphisms-lemma-base-change-flatness-fibres", "text": "Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$. Assume \\begin{enumerate} \\item $X$ is locally of finite presentation over $S$, \\item $X$ is flat over $S$, and \\item $Y$ is locally of finite type over $S$. \\end{enumerate} Then the set $$ U = \\{x \\in X \\mid X\\text{ flat at }x \\text{ over }Y\\}. $$ is open in $X$ and its formation commutes with arbitrary base change."}
+{"_id": "13785", "title": "more-morphisms-lemma-CM-dimension", "text": "Let $f : X \\to Y$ be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point $x$ in $X$ with image $y$ in $Y$, $$ \\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y), $$ where $X_y$ denotes the fiber over $y$."}
+{"_id": "13835", "title": "more-morphisms-lemma-connected-along-section-good", "text": "Let $f : X \\to Y$, $s : Y \\to X$ be as in Situation \\ref{situation-connected-along-section}. Assume $f$ of finite type. Let $y \\in Y$ be a point. Then there exists a nonempty open $V \\subset \\overline{\\{y\\}}$ such that the inverse image of $X^0$ in the base change $X_V$ is open and closed in $X_V$."}
+{"_id": "13837", "title": "more-morphisms-lemma-connected-along-section-open-neighbourhood", "text": "Let $f : X \\to Y$, $s : Y \\to X$ be as in Situation \\ref{situation-connected-along-section}. Let $y \\in Y$ be a point. Assume \\begin{enumerate} \\item $f$ is of finite presentation and flat, and \\item the fibre $X_y$ is geometrically reduced. \\end{enumerate} Then $X^0$ is a neighbourhood of $X^0_y$ in $X$."}
+{"_id": "13849", "title": "more-morphisms-lemma-trivial-on-fibres", "text": "Let $f : X \\to S$ be a flat, proper morphism of finite presentation such that $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this remains true after arbitrary base change. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $\\mathcal{E}|_{X_s}$ is isomorphic to $\\mathcal{O}_{X_s}^{\\oplus r_s}$ for all $s \\in S$, and \\item $S$ is reduced. \\end{enumerate} Then $\\mathcal{E} = f^*\\mathcal{N}$ for some finite locally free $\\mathcal{O}_S$-module $\\mathcal{N}$."}
+{"_id": "13864", "title": "more-morphisms-lemma-Noetherian-approximation-standard-syntomic", "text": "Let $f : X \\to S$ be a morphism of affine schemes, which is standard syntomic (see Morphisms, Definition \\ref{morphisms-definition-syntomic}). Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ is standard syntomic."}
+{"_id": "13889", "title": "more-morphisms-lemma-relative-isomorphism-approximation", "text": "With notation an assumptions as in Lemma \\ref{lemma-relative-map-approximation} assume that $\\varphi$ induces an isomorphism on completions. Then we can choose our diagram such that $f$ is \\'etale."}
+{"_id": "13902", "title": "more-morphisms-lemma-quasi-finite-separated-pass-through-finite-addendum", "text": "With notation and hypotheses as in Lemma \\ref{lemma-quasi-finite-separated-pass-through-finite}. Assume moreover that $f$ is locally of finite presentation. Then we can choose the factorization such that $T$ is finite and of finite presentation over $S$."}
+{"_id": "13930", "title": "more-morphisms-lemma-integral-over-quasi-affine", "text": "Let $X$ be a quasi-affine scheme. Let $f : U \\to X$ be an integral morphism. Then $U$ is quasi-affine and the diagram $$ \\xymatrix{ U \\ar[r] \\ar[d] & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\ X \\ar[r] & \\Spec(\\Gamma(X, \\mathcal{O}_X)) } $$ is cartesian."}
+{"_id": "13937", "title": "more-morphisms-lemma-equality-dimensions", "text": "Let $A \\to B$ be a local homomorphism of local rings. Assume \\begin{enumerate} \\item $A$ and $B$ are domains and $A \\subset B$, \\item $B$ is essentially of finite type over $A$, and \\item $B$ is flat over $A$. \\end{enumerate} Then we have $$ \\dim(B/\\mathfrak m_AB) + \\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) = \\text{trdeg}_A(B). $$"}
+{"_id": "13939", "title": "more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open", "text": "Let $\\varphi : A \\to B$ be a local ring map of local rings. Let $V \\subset \\Spec(B)$ be an open subscheme which contains at least one prime not lying over $\\mathfrak m_A$. Assume $A$ is Noetherian, $\\varphi$ essentially of finite type, and $A/\\mathfrak m_A \\subset B/\\mathfrak m_B$ is finite. Then there exists a $\\mathfrak q \\in V$, $\\mathfrak m_A \\not = \\mathfrak q \\cap A$ such that $A \\to B/\\mathfrak q$ is the localization of a quasi-finite ring map."}
+{"_id": "13947", "title": "more-morphisms-lemma-proper-flat-geom-red", "text": "Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is proper, flat, and of finite presentation, and \\item the geometric fibres of $f$ are reduced. \\end{enumerate} Then the function $n_{X/S} : S \\to \\mathbf{Z}$ counting the numbers of geometric connected components of fibres of $f$ is locally constant."}
+{"_id": "13952", "title": "more-morphisms-lemma-finite-morphism-relative-finite-presentation", "text": "Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\\pi_*\\mathcal{F}$ is of finite presentation relative to $S$."}
+{"_id": "13954", "title": "more-morphisms-lemma-pull-relative-finite-presentation", "text": "Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite type. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module. If $f : X \\to Y$ is locally of finite presentation and $\\mathcal{G}$ of finite presentation relative to $S$, then $f^*\\mathcal{G}$ is of finite presentation relative to $S$."}
+{"_id": "13955", "title": "more-morphisms-lemma-composition-relative-finite-presentation", "text": "Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite type. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. If $Y \\to S$ is locally of finite presentation and $\\mathcal{F}$ is of finite presentation relative to $Y$, then $\\mathcal{F}$ is of finite presentation relative to $S$."}
+{"_id": "13956", "title": "more-morphisms-lemma-ses-relatively-finite-presentation", "text": "Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{F}'' \\to 0$ be a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules. \\begin{enumerate} \\item If $\\mathcal{F}', \\mathcal{F}''$ are finitely presented relative to $S$, then so is $\\mathcal{F}$. \\item If $\\mathcal{F}'$ is a finite type $\\mathcal{O}_X$-module and $\\mathcal{F}$ is finitely presented relative to $S$, then $\\mathcal{F}''$ is finitely presented relative to $S$. \\end{enumerate}"}
+{"_id": "13959", "title": "more-morphisms-lemma-relative-pseudo-coherence", "text": "Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. If $E$ in $D(\\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$, then $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$. If $E$ is pseudo-coherent relative to $S$, then $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$."}
+{"_id": "13964", "title": "more-morphisms-lemma-closed-morphism-relative-pseudo-coherence", "text": "Let $i : X \\to Y$ morphism of schemes locally of finite type over a base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed subset of $Y$. Let $E$ be an object of $D(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$."}
+{"_id": "13970", "title": "more-morphisms-lemma-cohomology-relative-pseudo-coherent", "text": "Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$. If $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent relative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$."}
+{"_id": "13983", "title": "more-morphisms-lemma-descending-property-pseudo-coherent", "text": "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is fpqc local on the base."}
+{"_id": "13985", "title": "more-morphisms-lemma-pseudo-coherent-syntomic-local-source", "text": "The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent'' is syntomic local on the source."}
+{"_id": "13996", "title": "more-morphisms-lemma-perfect-closed-immersion-perfect-direct-image", "text": "Let $i : Z \\to X$ be a perfect closed immersion of schemes. Then $i_*\\mathcal{O}_X$ is a perfect $\\mathcal{O}_X$-module, i.e., it is a perfect object of $D(\\mathcal{O}_X)$."}
+{"_id": "13998", "title": "more-morphisms-lemma-perfect-fppf-local-source", "text": "The property $\\mathcal{P}(f) =$``$f$ is perfect'' is fppf local on the source."}
+{"_id": "13999", "title": "more-morphisms-lemma-factor-regular-immersion", "text": "Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. Assume \\begin{enumerate} \\item $X$ is locally Noetherian, \\item $j \\circ i$ is a regular immersion, and \\item $i$ is perfect. \\end{enumerate} Then $i$ and $j$ are regular immersions."}
+{"_id": "14000", "title": "more-morphisms-lemma-koszul-independence-factorization", "text": "Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$. Let $u \\in U$. Let $i : U \\to P$, $i' : U \\to P'$ be immersions over $S$. Assume $P$ and $P'$ smooth over $S$. Then the following are equivalent \\begin{enumerate} \\item $i$ is a Koszul-regular immersion in a neighbourhood of $x$, and \\item $i'$ is a Koszul-regular immersion in a neighbourhood of $x$. \\end{enumerate}"}
+{"_id": "14035", "title": "more-morphisms-lemma-relative-frobenius-weakly-etale", "text": "Let $U \\to X$ be a weakly \\'etale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \\to U \\times_{X, F_X} X$ is an isomorphism."}
+{"_id": "14047", "title": "more-morphisms-lemma-pushout-separated", "text": "In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}. If $X$ and $Y$ are separated, then the pushout $Y \\amalg_Z X$ (Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral}) is separated. Same with ``separated over $S$'', ``quasi-separated'', and ``quasi-separated over $S$''."}
+{"_id": "14077", "title": "more-morphisms-lemma-improve-stratification", "text": "Let $X$ be a scheme. Let $X = \\coprod_{i \\in I} X_i$ be a finite affine stratification. There exists an affine stratification with index set $\\{0, \\ldots, n\\}$ where $n$ is the length of $I$."}
+{"_id": "14078", "title": "more-morphisms-lemma-qc-affine-stratification", "text": "Let $X$ be a scheme. The following are equivalent \\begin{enumerate} \\item $X$ has a finite affine stratification, and \\item $X$ is quasi-compact and quasi-separated. \\end{enumerate}"}
+{"_id": "14079", "title": "more-morphisms-lemma-affine-stratification-number-bound", "text": "Let $X$ be a separated scheme which has an open covering by $n + 1$ affines. Then the affine stratification number of $X$ is at most $n$."}
+{"_id": "14080", "title": "more-morphisms-lemma-affine-stratification-number-bound-Noetherian", "text": "Let $X$ be a Noetherian scheme of dimension $\\infty > d \\geq 0$. Then the affine stratification number of $X$ is at most $d$."}
+{"_id": "14087", "title": "more-morphisms-lemma-large-open", "text": "Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open morphism of schemes. Let $n_{X/Y}$ be as in Lemma \\ref{lemma-count-geometric-fibres}. If $n_{X/Y}$ attains a maximum $d < \\infty$, then the set $$ Y_d = \\{y \\in Y \\mid n_{X/Y}(y) = d\\} $$ is open in $Y$ and the morphism $f^{-1}(Y_d) \\to Y_d$ is finite."}
+{"_id": "14105", "title": "more-morphisms-proposition-asn-weighting", "text": "Let $f : X \\to Y$ be a surjective quasi-finite morphism of schemes. Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $X$ affine and $Y$ separated\\footnote{It suffices if the diagonal of $Y$ is affine.}. Then the affine stratification number of $Y$ is at most the number of distinct values of $\\int_f w$."}
+{"_id": "14190", "title": "sites-modules-lemma-tensor-product-permanence", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules. \\begin{enumerate} \\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are finite locally free, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated by sections, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent, then $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$ is coherent. \\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$. \\end{enumerate}"}
+{"_id": "14227", "title": "sites-modules-lemma-pic-set", "text": "Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. There exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$ such that each invertible module on $(\\mathcal{C}, \\mathcal{O})$ is isomorphic to exactly one of the $\\mathcal{L}_i$."}
+{"_id": "14318", "title": "derham-lemma-finite-de-Rham", "text": "Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \\Spec(A)$. Then $H^i_{dR}(X/S)$ is a finite $A$-module for all $i$."}
+{"_id": "14324", "title": "derham-lemma-kunneth-de-rham-relative", "text": "Assume $X \\to S$ and $Y \\to S$ are smooth and quasi-compact and the morphisms $X \\to X \\times_S X$ and $Y \\to Y \\times_S Y$ are affine. Then the relative cup product $$ Ra_*\\Omega^\\bullet_{X/S} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Rb_*\\Omega^\\bullet_{Y/S} \\longrightarrow Rf_*\\Omega^\\bullet_{X \\times_S Y/S} $$ is an isomorphism in $D(\\mathcal{O}_S)$."}
+{"_id": "14349", "title": "derham-lemma-splitting-on-omega-a", "text": "With notation as in Lemma \\ref{lemma-blowup} for $a \\geq 0$ there is a unique arrow $Rb_*\\Omega^a_{X'/S} \\to \\Omega^a_{X/S}$ in $D(\\mathcal{O}_X)$ whose composition with $\\Omega^a_{X/S} \\to Rb_*\\Omega^a_{X'/S}$ is the identity on $\\Omega^a_{X/S}$."}
+{"_id": "14364", "title": "derham-lemma-gysin-differential-hodge", "text": "Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}. Assume $X \\to S$ is smooth and $Z \\to X$ Koszul regular. The gysin maps $\\gamma^{p, q}$ are compatible with the de Rham differentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$."}
+{"_id": "14365", "title": "derham-lemma-gysin-projection-global", "text": "Let $X \\to S$, $i : Z \\to X$, and $c \\geq 0$ be as in Lemma \\ref{lemma-gysin-global}. Assume $X \\to S$ smooth and $Z \\to X$ Koszul regular. Given $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have $\\gamma^{p, q}(\\alpha|_Z) = \\alpha \\cup \\gamma^{0, 0}(1)$ in $H^{q + c}(X, \\Omega^{p + c}_{X/S})$ with $\\gamma^{a, b}$ as in Remark \\ref{remark-how-to-use}."}
+{"_id": "14367", "title": "derham-lemma-class-of-a-point", "text": "Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$ of dimension $d$. Let $Z \\subset X$ be the reduced closed subscheme consisting of a single $k$-rational point $x$. Then the image of $1 \\in k = H^0(Z, \\mathcal{O}_Z) = H^0(Z, \\Omega^0_{Z/k})$ by the map $H^0(Z, \\Omega^0_{Z/k}) \\to H^d(X, \\Omega^d_{X/k})$ of Remark \\ref{remark-how-to-use} is nonzero."}
+{"_id": "14377", "title": "derham-proposition-relative-poincare-duality", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a proper smooth morphism of schemes all of whose fibres are nonempty and equidimensional of dimension $d$. There exists an $\\mathcal{O}_S$-module map $$ t : R^{2d}f_*\\Omega^\\bullet_{X/S} \\longrightarrow \\mathcal{O}_S $$ unique up to precomposing by multiplication by a unit of $H^0(X, \\mathcal{O}_X)$ with the following property: the pairing $$ Rf_*\\Omega^\\bullet_{X/S} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/S}[2d] \\longrightarrow \\mathcal{O}_S, \\quad (\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi') $$ is a perfect pairing of perfect complexes on $S$."}
+{"_id": "14412", "title": "trace-lemma-sheaf-over-finite-field-has-frobenius-descent", "text": "Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then there are canonical isomorphisms $\\pi_X^{-1} \\mathcal{F} \\cong \\mathcal{F}$ and $\\mathcal{F} \\cong {\\pi_X}_*\\mathcal{F}$."}
+{"_id": "14420", "title": "trace-lemma-weil-mod", "text": "Consider the situation of Theorem \\ref{theorem-weil-trace-formula} and let $\\ell$ be a prime number invertible in $k$. Then $$ \\sum\\nolimits_{i = 0}^2 (-1)^i \\text{Tr}(\\varphi^* |_{H^i (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})}) = V(\\varphi) \\mod \\ell^n. $$"}
+{"_id": "14430", "title": "trace-lemma-l-adic-abelian", "text": "The category of $\\mathbf{Z}_\\ell$-sheaves on $X$ is abelian."}
+{"_id": "14431", "title": "trace-lemma-piece-together", "text": "Suppose we have $K_n\\in D_{perf}(\\mathbf{Z}/\\ell^n\\mathbf{Z})$, $\\pi_n : K_n\\to K_n$ and isomorphisms $\\varphi_n : K_{n+1} \\otimes^\\mathbf{L}_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}} \\mathbf{Z}/\\ell^n\\mathbf{Z} \\to K_n$ compatible with $\\pi_{n+1}$ and $\\pi_n$. Then \\begin{enumerate} \\item the elements $t_n = \\text{Tr}(\\pi_n |_{K_n})\\in \\mathbf{Z}/\\ell^n\\mathbf{Z}$ form an element $t_\\infty = \\{t_n\\}$ of $\\mathbf{Z}_\\ell$, \\item the $\\mathbf{Z}_\\ell$-module $H_\\infty^i = \\lim_n H^i(k_n)$ is finite and is nonzero for finitely many $i$ only, and \\item the operators $H^i(\\pi_n): H^i(K_n)\\to H^i(K_n)$ are compatible and define $\\pi_\\infty^i : H_\\infty^i\\to H_\\infty^i$ satisfying $$ \\sum (-1)^i \\text{Tr}( \\pi_\\infty^i |_{H_\\infty^i \\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell}) = t_\\infty. $$ \\end{enumerate}"}
+{"_id": "14434", "title": "trace-lemma-eigenvalues-algebraic", "text": "Algebraicity of eigenvalues. If $\\Lambda$ is a field then the eigenvalues $t_v$ for $f\\in C(\\Lambda)$ are algebraic over the prime subfield $\\mathbf{F} \\subset \\Lambda$."}
+{"_id": "14499", "title": "sheaves-lemma-sheafify-presheaf-structures", "text": "Let $X$ be a topological space. Let $(\\mathcal{C}, F)$ be a type of algebraic structure. Let $\\mathcal{F}$ be a presheaf with values in $\\mathcal{C}$ on $X$. Then there exists a sheaf $\\mathcal{F}^\\#$ with values in $\\mathcal{C}$ and a morphism $\\mathcal{F} \\to \\mathcal{F}^\\#$ of presheaves with values in $\\mathcal{C}$ with the following properties: \\begin{enumerate} \\item The map $\\mathcal{F} \\to \\mathcal{F}^\\#$ identifies the underlying sheaf of sets of $\\mathcal{F}^\\#$ with the sheafification of the underlying presheaf of sets of $\\mathcal{F}$. \\item For any morphism $\\mathcal{F} \\to \\mathcal{G}$, where $\\mathcal{G}$ is a sheaf with values in $\\mathcal{C}$ there exists a unique factorization $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$. \\end{enumerate}"}
+{"_id": "14520", "title": "sheaves-lemma-adjoint-pull-push-modules", "text": "Let $f : X \\to Y$ be a continuous map of topological spaces. Let $\\mathcal{O}$ be a sheaf of rings on $X$. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules. Let $\\mathcal{G}$ be a sheaf of $f_*\\mathcal{O}$-modules. Then $$ \\Mor_{\\textit{Mod}(\\mathcal{O})}( \\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}). $$ Here we use Lemmas \\ref{lemma-pullback-module} and \\ref{lemma-pushforward-module}, and we use the canonical map $f^{-1}f_*\\mathcal{O} \\to \\mathcal{O}$ in the definition of the tensor product."}
+{"_id": "14606", "title": "descent-lemma-equalizer-S", "text": "The diagram \\begin{equation} \\label{equation-equalizer-S} \\xymatrix@C=8pc{ S_1 \\ar[r]^{\\delta^1_1} & S_2 \\ar@<1ex>[r]^{\\delta^2_2} \\ar@<-1ex>[r]_{\\delta^2_1} &  S_3 } \\end{equation} is a split equalizer."}
+{"_id": "14626", "title": "descent-lemma-equivalence-quasi-coherent-properties", "text": "Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Let $\\mathcal{P}$ be one of the properties of modules\\footnote{The list is: free, finite free, generated by global sections, generated by $r$ global sections, generated by finitely many global sections, having a global presentation, having a global finite presentation, locally free, finite locally free, locally generated by sections, locally generated by $r$ sections, finite type, of finite presentation, coherent, or flat.} defined in Modules on Sites, Definitions \\ref{sites-modules-definition-global}, \\ref{sites-modules-definition-site-local}, and \\ref{sites-modules-definition-flat}. The equivalences of categories $$ \\QCoh(\\mathcal{O}_S) \\longrightarrow \\QCoh((\\Sch/S)_\\tau, \\mathcal{O}) \\quad\\text{and}\\quad \\QCoh(\\mathcal{O}_S) \\longrightarrow \\QCoh(S_\\tau, \\mathcal{O}) $$ defined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in Proposition \\ref{proposition-equivalence-quasi-coherent} have the property $$ \\mathcal{F}\\text{ has }\\mathcal{P} \\Leftrightarrow \\mathcal{F}^a\\text{ has }\\mathcal{P}\\text{ as an }\\mathcal{O}\\text{-module} $$ except (possibly) when $\\mathcal{P}$ is ``locally free'' or ``coherent''. If $\\mathcal{P}=$``coherent'' the equivalence holds for $\\QCoh(\\mathcal{O}_S) \\to \\QCoh(S_\\tau, \\mathcal{O})$ when $S$ is locally Noetherian and $\\tau$ is Zariski or \\'etale."}
+{"_id": "14662", "title": "descent-lemma-regular-local-ring-local", "text": "Let $f : U \\to V$ be an \\'etale morphism of schemes. Let $u \\in U$ and $v = f(u)$. Then $\\mathcal{O}_{U, u}$ is a regular local ring if and only if $\\mathcal{O}_{V, v}$ is a regular local ring."}
+{"_id": "14740", "title": "descent-lemma-morphism-with-section-equivalence", "text": "Let $f : X' \\to X$ be a morphism of schemes over a base scheme $S$. Assume there exists a morphism $g : X \\to X'$ over $S$, for example if $f$ has a section. Then the pullback functor of Lemma \\ref{lemma-pullback} defines an equivalence of categories between the category of descent data relative to $X/S$ and $X'/S$."}
+{"_id": "14833", "title": "simplicial-lemma-N-d-in-N", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object in $\\mathcal{A}$. Let $N(U_m)$ as in Lemma \\ref{lemma-splitting-abelian-category} above. Then $d^m_m(N(U_m)) \\subset N(U_{m - 1})$."}
+{"_id": "14869", "title": "simplicial-lemma-N-faithful", "text": "Let $\\mathcal{A}$ be an abelian category. The functor $N$ is faithful, and reflects isomorphisms, injections and surjections."}
+{"_id": "14871", "title": "simplicial-lemma-dual-dold-kan", "text": "Let $\\mathcal{A}$ be an abelian category. \\begin{enumerate} \\item The functor $s : \\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$ is exact. \\item The maps $s(U)^n \\to Q(U)^n$ define a morphism of cochain complexes. \\item There exists a functorial direct sum decomposition $s(U) = D(U) \\oplus Q(U)$ in $\\text{CoCh}_{\\geq 0}(\\mathcal{A})$. \\item The functor $Q$ is exact. \\item The morphism of complexes $s(U) \\to Q(U)$ is a quasi-isomorphism. \\item The functor $U \\mapsto Q(U)^\\bullet$ defines an equivalence of categories $\\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$. \\end{enumerate}"}
+{"_id": "14873", "title": "simplicial-lemma-contractible", "text": "Let $\\mathcal{C}$ be a category with finite coproducts. Let $U$ be a simplicial object of $\\mathcal{C}$. Consider the maps $e_1, e_0 : U \\to U \\times \\Delta[1]$, and $\\pi : U \\times \\Delta[1] \\to U$, see Lemma \\ref{lemma-back-to-U}. \\begin{enumerate} \\item We have $\\pi \\circ e_1 = \\pi \\circ e_0 = \\text{id}_U$, and \\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$, and $e_0 \\circ \\pi$ are homotopic. \\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$, and $e_1 \\circ \\pi$ are homotopic. \\end{enumerate}"}
+{"_id": "14874", "title": "simplicial-lemma-fibre-products-simplicial-object-w-section", "text": "Let $f : Y \\to X$ be a morphism of a category $\\mathcal{C}$ with fibre products. Assume $f$ has a section $s$. Consider the simplicial object $U$ constructed in Example \\ref{example-fibre-products-simplicial-object} starting with $f$. The morphism $U \\to U$ which in each degree is the self map $(s \\circ f)^{n + 1}$ of $Y \\times_X \\ldots \\times_X Y$ given by $s \\circ f$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant simplicial object $X$."}
+{"_id": "14880", "title": "simplicial-lemma-push-outs-simplicial-object-w-section", "text": "Let $f : X \\to Y$ be a morphism of a category $\\mathcal{C}$ with pushouts. Assume there is a morphism $s : Y \\to X$ with $s \\circ f = \\text{id}_X$. Consider the cosimplicial object $U$ constructed in Example \\ref{example-push-outs-simplicial-object} starting with $f$. The morphism $U \\to U$ which in each degree is the self map of $Y \\amalg_X \\ldots \\amalg_X Y$ given by $f \\circ s$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant cosimplicial object $X$."}
+{"_id": "14885", "title": "simplicial-lemma-backwards-homotopy", "text": "Let $\\mathcal{A}$ be an abelian category. Let $U$, $V$ be simplicial objects of $\\mathcal{A}$. Let $a, b : U \\to V$ be a pair of morphisms. Assume the corresponding maps of chain complexes $N(a), N(b) : N(U) \\to N(V)$ are homotopic by a homotopy $\\{N_n : N(U)_n \\to N(V)_{n + 1}\\}$. Then there exists a homotopy from $a$ to $b$ as in Definition \\ref{definition-homotopy}. Moreover, one can choose the homotopy $h : U \\times \\Delta[1] \\to V$ such that $N_n = N(h)_n$ where $N(h)$ is the homotopy coming from $h$ as in Section \\ref{section-homotopy-abelian}."}
+{"_id": "14893", "title": "simplicial-lemma-kan-base-change", "text": "Let $f : X \\to Y$ be a Kan fibration of simplicial sets. Let $Y' \\to Y$ be a morphism of simplicial sets. Then $X \\times_Y Y' \\to Y'$ is a Kan fibration."}
+{"_id": "14898", "title": "simplicial-lemma-surjection-simplicial-abelian-groups-kan", "text": "Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups which is termwise surjective. Then $f$ is a Kan fibration of simplicial sets."}
+{"_id": "14902", "title": "simplicial-lemma-homotopy", "text": "Let $f^0, f^1 : V \\to U$ be maps of simplicial sets. Let $n \\geq 0$ be an integer. Assume \\begin{enumerate} \\item The maps $f^j_i : V_i \\to U_i$, $j = 0, 1$ are equal for $i < n$. \\item The canonical morphism $U \\to \\text{cosk}_n \\text{sk}_n U$ is an isomorphism. \\item The canonical morphism $V \\to \\text{cosk}_n \\text{sk}_n V$ is an isomorphism. \\end{enumerate} Then $f^0$ is homotopic to $f^1$."}
+{"_id": "14954", "title": "discriminant-lemma-compare-dualizing-algebraic", "text": "Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings. Let $\\omega_{B/A}^\\bullet \\in D(B)$ be the algebraic relative dualizing complex discussed in Dualizing Complexes, Section \\ref{dualizing-section-relative-dualizing-complexes-Noetherian}. Then there is a (nonunique) isomorphism $\\omega_{B/A} = H^0(\\omega_{B/A}^\\bullet)$."}
+{"_id": "14968", "title": "discriminant-lemma-noether-different", "text": "Let $A \\to B$ be a finite type ring map. Let $\\mathfrak{D} \\subset B$ be the Noether different. Then $V(\\mathfrak{D})$ is the set of primes $\\mathfrak q \\subset B$ such that $A \\to B$ is not unramified at $\\mathfrak q$."}
+{"_id": "14969", "title": "discriminant-lemma-base-change-kahler-different", "text": "Consider a cartesian diagram of schemes $$ \\xymatrix{ Y' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\ X' \\ar[r]^g & X } $$ with $f$ locally of finite type. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$ be the closed subscheme cut out by the K\\\"ahler different of $f$, resp.\\ $f'$. Then $Y' \\to Y$ induces an isomorphism $R' \\to R \\times_Y Y'$."}
+{"_id": "14970", "title": "discriminant-lemma-kahler-different", "text": "Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type. Let $R \\subset Y$ be the closed subscheme defined by the K\\\"ahler different. Then $R \\subset Y$ is exactly the set of points where $f$ is not unramified."}
+{"_id": "14975", "title": "discriminant-lemma-flat-agree-dedekind", "text": "Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes. Let $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$ be affine open subschemes with $f(V) \\subset U$. If the Dedekind different of $A \\to B$ is defined, then $$ \\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}_{B/A}} $$ as coherent ideal sheaves on $V$."}
+{"_id": "15037", "title": "limits-lemma-descend-section", "text": "In Situation \\ref{situation-descent}. Suppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$. Set $\\mathcal{F}_i = f_{i0}^*\\mathcal{F}_0$ for $i \\geq 0$ and set $\\mathcal{F} = f_0^*\\mathcal{F}_0$. Then $$ \\Gamma(S, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(S_i, \\mathcal{F}_i) $$"}
+{"_id": "15088", "title": "limits-lemma-chow-EGA", "text": "Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$ be a separated morphism of finite type. Assume that $X$ has finitely many irreducible components. Then there exists an $n \\geq 0$ and a diagram $$ \\xymatrix{ X \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\ & S & } $$ where $X' \\to \\mathbf{P}^n_S$ is an immersion, and $\\pi : X' \\to X$ is proper and surjective. Moreover, there exists an open dense subscheme $U \\subset X$ such that $\\pi^{-1}(U) \\to U$ is an isomorphism of schemes."}
+{"_id": "15103", "title": "limits-lemma-refined-valuative-criterion-universally-closed", "text": "Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper."}
+{"_id": "15111", "title": "limits-lemma-proper-top-cohomology-finite-presentation", "text": "Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume \\begin{enumerate} \\item $f$ is a proper morphism of finite presentation all of whose fibres have dimension $\\leq d$, \\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation. \\end{enumerate} Then $R^df_*\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation."}
+{"_id": "15124", "title": "limits-lemma-morphism-good-diagram-proper", "text": "Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have $f_i$ is proper."}