{"_id": "11652", "text": "resolve-lemma-equivalence-properties 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 $f$ is separated, proper, finite, if and only if $g_i$ is so for $i = 1, \\ldots, n$."}
{"_id": "5838", "text": "chow-lemma-lci-gysin-product-regular 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 $f$. Then $$ f^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta $$ in $\\CH^*(X) \\otimes \\mathbf{Q}$ where the intersection product is as in Section \\ref{section-intersection-regular}."}
{"_id": "13098", "text": "dga-lemma-restriction-homotopy The functor (\\ref{equation-restriction}) defines an exact functor $K(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})})$ of triangulated categories."}
{"_id": "4956", "text": "spaces-morphisms-lemma-birational Let $S$ be a scheme. Let $X$ and $Y$ be algebraic space over $S$ with $|X|$ and $|Y|$ irreducible. Then $X$ and $Y$ are birational if and only if there are nonempty open subspaces $U \\subset X$ and $V \\subset Y$ which are isomorphic as algebraic spaces over $S$."}
{"_id": "1918", "text": "derived-lemma-unbounded-left-derived Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor of abelian categories. If \\begin{enumerate} \\item every object of $\\mathcal{A}$ is a quotient of an object which is left acyclic for $F$, \\item there exists an integer $n \\geq 0$ such that $L^nF = 0$, \\end{enumerate} Then \\begin{enumerate} \\item $LF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists, \\item any complex consisting of left acyclic objects for $F$ computes $LF$, \\item any complex is the target of a quasi-isomorphism from a complex consisting of left acyclic objects for $F$, \\item for $E \\in D(\\mathcal{A})$ \\begin{enumerate} \\item $H^i(LF(\\tau_{\\leq a + n - 1}E) \\to H^i(LF(E))$ is an isomorphism for $i \\leq a$, \\item $H^i(LF(E)) \\to H^i(LF(\\tau_{\\geq b}E))$ 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 $H^i(LF(E)) = 0$ for $i \\not \\in [a - n + 1, b]$. \\end{enumerate} \\end{enumerate}"}
{"_id": "2894", "text": "dualizing-lemma-well-defined Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings. The functor $\\varphi^!$ is well defined up to isomorphism."}
{"_id": "3413", "text": "formal-defos-theorem-minimal-smooth-prorepresentable-presentations Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal{C}_\\Lambda$. Consider the following conditions \\begin{enumerate} \\item $\\mathcal{F}$ admits a presentation by a normalized smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$, \\item $\\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$, \\item $\\mathcal{F}$ admits a presentation by a minimal smooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$, and \\item $\\mathcal{F}$ satisfies the following conditions \\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} \\end{enumerate} Then (2), (3), (4) are equivalent and are implied by (1). If $k' \\subset k$ is separable, then (1), (2), (3), (4) are all equivalent. Furthermore, the minimal smooth prorepresentable groupoids in functors which provide a presentation of $\\mathcal{F}$ are unique up to isomorphism."}
{"_id": "4693", "text": "stacks-geometry-lemma-dimension-local-ring-pre Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let $U \\to \\mathcal{X}$ be a smooth morphism and let $u \\in U$. Then $$ \\dim(\\mathcal{O}_{U, \\overline{u}}) - \\dim(\\mathcal{O}_{R_u, e(\\overline{u})}) = 2\\dim(\\mathcal{O}_{U, \\overline{u}}) - \\dim(\\mathcal{O}_{R, e(\\overline{u})}) $$ 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": "7880", "text": "divisors-lemma-check-injective-on-weakass Let $X$ be a 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 $x \\not \\in \\text{WeakAss}(\\mathcal{F})$. \\end{enumerate} Then $\\varphi$ is injective."}
{"_id": "3774", "text": "proetale-lemma-compare-cohomology-big-small Let $f : T \\to S$ be a morphism of schemes. For $K$ in $D(S_\\proetale)$ we have $$ H^n_\\proetale(S, \\pi_S^{-1}K) = H^n(S_\\proetale, K) $$ and $$ H^n_\\proetale(T, \\pi_S^{-1}K) = H^n(T_\\proetale, f_{small}^{-1}K). $$ For $M$ in $D((\\Sch/S)_\\proetale)$ we have $$ H^n_\\proetale(T, M) = H^n(T_\\proetale, i_f^{-1}M). $$"}
{"_id": "841", "text": "algebra-lemma-flat-ML-over-ML-ring Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. If $S$ is Mittag-Leffler as an $R$-module, and $M$ is flat and Mittag-Leffler as an $S$-module, then $M$ is Mittag-Leffler as an $R$-module."}
{"_id": "7954", "text": "divisors-lemma-UFD-one-equation-CM Let $R$ be a Noetherian UFD. Let $I \\subset R$ be an ideal such that $R/I$ has no embedded primes and such that every minimal prime over $I$ has height $1$. Then $I = (f)$ for some $f \\in R$."}
{"_id": "7400", "text": "stacks-morphisms-lemma-change-of-base-separated Let $S$ be a scheme. The property of being quasi-DM over $S$, quasi-separated over $S$, or separated over $S$ (see Definition \\ref{definition-absolute-separated}) is stable under change of base scheme, see Algebraic Stacks, Definition \\ref{algebraic-definition-change-of-base}."}
{"_id": "11795", "text": "spaces-duality-lemma-more-base-change In diagram (\\ref{equation-base-change}) assume in addition $g : Y' \\to Y$ is a morphism of affine schemes and $f : X \\to Y$ is proper. 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": "13760", "text": "more-morphisms-lemma-basic-example-pushout Let $A' \\to A$ be a surjection of rings and let $B \\to A$ be a ring map. Let $B' = B \\times_A A'$ be the fibre product of rings. Set $S = \\Spec(A)$, $S' = \\Spec(A')$, $T = \\Spec(B)$, and $T' = \\Spec(B')$. Then $$ \\vcenter{ \\xymatrix{ S \\ar[r]_i \\ar[d]_f & S' \\ar[d]^{f'} \\\\ T \\ar[r]^{i'} & T' } } \\quad\\text{corresponding to}\\quad \\vcenter{ \\xymatrix{ A & A' \\ar[l] \\\\ B \\ar[u] & B' \\ar[l] \\ar[u] } } $$ is a pushout of schemes."}
{"_id": "11475", "text": "obsolete-lemma-cosk-hom-deltak Let $\\mathcal{C}$ be a category with finite coproducts and finite limits. Let $X$ be an object of $\\mathcal{C}$. Let $k \\geq 0$. The canonical map $$ \\Hom(\\Delta[k], X) \\longrightarrow \\text{cosk}_1 \\text{sk}_1 \\Hom(\\Delta[k], X) $$ is an isomorphism."}
{"_id": "916", "text": "algebra-lemma-nonzerodivisor-on-CM Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$. Let $M$ be a finite $R$-module. Let $x \\in \\mathfrak m$ be a nonzerodivisor on $M$. Then $M$ is Cohen-Macaulay if and only if $M/xM$ is Cohen-Macaulay."}
{"_id": "4643", "text": "spaces-limits-lemma-limited-base-change Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated and of finite type. The following are equivalent \\begin{enumerate} \\item The morphism $f$ is proper. \\item For any morphism $Y \\to Z$ 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": "502", "text": "algebra-lemma-silly-normal Let $R$ be a ring. Let $x, y \\in R$ be nonzerodivisors. Let $R[x/y] \\subset R_{xy}$ be the $R$-subalgebra generated by $x/y$, and similarly for the subalgebras $R[y/x]$ and $R[x/y, y/x]$. If $R$ is integrally closed in $R_x$ or $R_y$, then the sequence $$ 0 \\to R \\xrightarrow{(-1, 1)} R[x/y] \\oplus R[y/x] \\xrightarrow{(1, 1)} R[x/y, y/x] \\to 0 $$ is a short exact sequence of $R$-modules."}
{"_id": "13616", "text": "duality-lemma-compactifyable-relative-dualizing 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 (the first component of) a relative dualizing complex for $X$ over $Y$ in the sense of Definition \\ref{definition-relative-dualizing-complex}."}
{"_id": "9401", "text": "spaces-descent-lemma-descending-property-quasi-compact-immersion The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion'' is fpqc local on the base."}
{"_id": "13780", "text": "more-morphisms-lemma-normal-fppf-local-source-and-target The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is normal'' is local in the fppf topology on the target and local in the smooth topology on the source."}
{"_id": "3351", "text": "coherent-lemma-thickening-quasi-affine Let $i : Z \\to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine."}
{"_id": "14741", "text": "descent-lemma-morphism-source-faithfully-flat Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$. Assume $X \\to S$ is surjective and flat. Then the pullback functor of Lemma \\ref{lemma-pullback} is a faithful functor from the category of descent data relative to $X'/S$ to the category of descent data relative to $X/S$."}
{"_id": "13485", "text": "spaces-resolve-lemma-dominate-by-normalized-blowing-up Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$ with $\\dim(X) = 2$. 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": "12588", "text": "constructions-lemma-spec In Situation \\ref{situation-relative-spec}. The functor $F$ is representable by a scheme."}
{"_id": "7554", "text": "stacks-morphisms-lemma-etale-unramified An \\'etale morphism is unramified."}
{"_id": "7364", "text": "sdga-lemma-cohomology-ext Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$ be a differential graded $\\mathcal{O}$-algebra. Let $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module. Let $n \\in \\mathbf{Z}$. We have $$ H^n(\\mathcal{C}, \\mathcal{M}) = \\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{A}, \\mathcal{M}[n]) $$ where on the left hand side we have the cohomology of $\\mathcal{M}$ viewed as a complex of $\\mathcal{O}$-modules."}
{"_id": "4138", "text": "pione-proposition-lefschetz-equivalence 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 $$ \\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2 $$ and that for $x \\in X \\setminus Y$ closed purity holds for $\\mathcal{O}_{X, x}$. Then the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$ is an equivalence. If $X$ or equivalently $Y$ is connected, then $$ \\pi_1(Y, \\overline{y}) \\to \\pi_1(X, \\overline{y}) $$ is an isomorphism for any geometric point $\\overline{y}$ of $Y$."}
{"_id": "2663", "text": "spaces-perfect-lemma-cat-module-support-proper-over-base 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 $\\mathcal{F}$, $\\mathcal{G}$ be finite type, quasi-coherent $\\mathcal{O}_X$-module. \\begin{enumerate} \\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$ are proper over $Y$, 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 $Y$ is locally Noetherian, then the category of coherent $\\mathcal{O}_X$-modules with support proper over $Y$ is a Serre subcategory (Homology, Definition \\ref{homology-definition-serre-subcategory}) of the abelian category of coherent $\\mathcal{O}_X$-modules. \\end{enumerate}"}
{"_id": "12840", "text": "spaces-over-fields-lemma-modification-iso-over-open Let $f : X' \\to X$ be a modification as in Definition \\ref{definition-modification}. There exists a nonempty open $U \\subset X$ such that $f^{-1}(U) \\to U$ is an isomorphism."}
{"_id": "3111", "text": "criteria-lemma-algebraic-morphism-to-algebraic 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 $\\mathcal{Y}$ is an algebraic stack, and \\item $F$ is algebraic (see above), \\end{enumerate} then $\\mathcal{X}$ is an algebraic stack."}
{"_id": "3392", "text": "coherent-lemma-base-change-module-support-proper-over-base 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. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. If the support of $\\mathcal{F}$ is proper over $S$, then the support of $(g')^*\\mathcal{F}$ is proper over $S'$."}
{"_id": "8402", "text": "hypercovering-lemma-cech-spectral-sequence-verdier Let $\\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\\mathcal{F}$ be an abelian sheaf. 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 the global cohomology groups $H^{p + q}(\\mathcal{F})$."}
{"_id": "1257", "text": "algebra-lemma-colimit-formally-unramified 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 unramified, then $S = \\colim_{i \\in I} S_i$ is formally unramified over $R$"}
{"_id": "12444", "text": "topologies-lemma-characterize-sheaf-big Let $S$ be a scheme contained in a big Zariski site $\\Sch_{Zar}$. A sheaf $\\mathcal{F}$ on the big Zariski site $(\\Sch/S)_{Zar}$ is given by the following data: \\begin{enumerate} \\item for every $T/S \\in \\Ob((\\Sch/S)_{Zar})$ a sheaf $\\mathcal{F}_T$ on $T$, \\item for every $f : T' \\to T$ in $(\\Sch/S)_{Zar}$ a map $c_f : f^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$. \\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 $(\\Sch/S)_{Zar}$ the composition $c_g \\circ g^{-1}c_f$ is equal to $c_{f \\circ g}$, and \\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_{Zar}$ is an open immersion then $c_f$ is an isomorphism. \\end{enumerate}"}
{"_id": "6692", "text": "etale-cohomology-lemma-blow-up-square-equivalence 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 $$ \\xymatrix{ E \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\ Z \\ar[r]^i & X } $$ Suppose given \\begin{enumerate} \\item an object $K'$ of $D^+(X'_\\etale)$ with torsion cohomology sheaves, \\item an object $L$ of $D^+(Z_\\etale)$ with torsion cohomology sheaves, and \\item an isomorphism $\\gamma : K'|_E \\to L|_E$. \\end{enumerate} Then there exists an object $K$ of $D^+(X_\\etale)$ and isomorphisms $f : K|_{X'} \\to K'$, $g : K|_Z \\to L$ such that $\\gamma = g|_E \\circ f^{-1}|_E$. Moreover, given \\begin{enumerate} \\item an object $M$ of $D^+(X_\\etale)$ with torsion cohomology sheaves, \\item a morphism $\\alpha : K' \\to M|_{X'}$ of $D(X'_\\etale)$, \\item a morphism $\\beta : L \\to M|_Z$ of $D(Z_\\etale)$, \\end{enumerate} such that $$ \\alpha|_E  = \\beta|_E \\circ \\gamma. $$ Then there exists a morphism $M \\to K$ in $D(X_\\etale)$ whose restriction to $X'$ is $a \\circ f$ and whose restriction to $Z$ is $b \\circ g$."}
{"_id": "2949", "text": "properties-lemma-reduced-is-locally-reduced Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is ``locally reduced'' in the sense of Definition \\ref{definition-locally-P}."}
{"_id": "11698", "text": "resolve-lemma-embedded-resolution Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$ be a proper closed subscheme such that every irreducible component $Y \\subset Z$ of dimension $1$ satisfies the equivalent conditions of Lemma \\ref{lemma-resolve-curve}. Then there exists a sequence $$ X_n \\to \\ldots \\to X_1 \\to X $$ of blowups in closed points such that the inverse image $Z_n$ of $Z$ in $X_n$ is an effective Cartier divisor supported on a strict normal crossings divisor."}
{"_id": "11067", "text": "varieties-lemma-open-in-affine-curve-affine Let $X$ be an affine scheme all of whose local rings are Noetherian of dimension $\\leq 1$. Then any quasi-compact open $U \\subset X$ is affine."}
{"_id": "468", "text": "algebra-lemma-jacobson-prime Let $R$ be a ring. If every prime ideal of $R$ is the intersection of the maximal ideals containing it, then $R$ is Jacobson."}
{"_id": "793", "text": "algebra-lemma-lift-finite-projective-module Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. 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": "9101", "text": "spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple-modules 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 $\\mathcal{C}$. Then we have a canonical isomorphism $$ R\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, La^*E) $$ for $E \\in D(\\mathcal{O}_\\mathcal{C})$."}
{"_id": "179", "text": "spaces-more-morphisms-lemma-geometrically-connected-fibres-towards-normal 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, \\item $Y$ is integral (Spaces over Fields, Definition \\ref{spaces-over-fields-definition-integral-algebraic-space}) with generic point $\\xi$, \\item $Y$ 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 $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $f$ has geometrically connected fibres."}
{"_id": "7909", "text": "divisors-lemma-check-torsion Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is torsion free if and only if $\\mathcal{F}_x$ is a torsion free $\\mathcal{O}_{X, x}$-module for all $x \\in X$."}
{"_id": "12776", "text": "algebraization-lemma-interesting-case-final 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 local ring which has a dualizing complex, \\item all irreducible components of $X$ have the same dimension, \\item the scheme $X \\setminus Y$ is Cohen-Macaulay, \\item $I$ is generated by $d$ elements, \\item $\\dim(X) - \\dim(Z) > d + 2$, and \\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. \\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": "4711", "text": "spaces-morphisms-theorem-chevalley Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every \\'etale locally constructible subset of $|X|$ is an \\'etale locally constructible subset of $|Y|$."}
{"_id": "4691", "text": "stacks-geometry-lemma-dims-of-images Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks, 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 for every finite type point $t \\in |T|$, we have that $\\dim_t( \\mathcal{T}_{f(t)}) \\geq \\dim \\mathcal{T}  - \\dim \\mathcal{Z}$, and there is a non-empty (equivalently, dense) open subset of $|\\mathcal{T}|$ over which equality holds."}
{"_id": "7575", "text": "stacks-morphisms-lemma-base-change-uniqueness The base change of a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion."}
{"_id": "7236", "text": "spaces-chow-lemma-Gm-torsor-divisor-meromorphic-section In Situation \\ref{situation-setup} let $X/B$ be good. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $X$ is integral. Let $s$ be a nonzero meromorphic section of $\\mathcal{L}$. Let $q : T \\to X$ be the morphism of Lemma \\ref{lemma-Gm-torsor}. Then $$ q^*\\text{div}_\\mathcal{L}(s) = \\text{div}_T(q^*(s)) $$ where we view the pullback $q^*(s)$ as a nonzero meromorphic function on $T$ using the isomorphism $q^*\\mathcal{L} \\to \\mathcal{O}_T$"}
{"_id": "9362", "text": "spaces-descent-lemma-finite-locally-free-descends 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 locally free $\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module."}
{"_id": "11502", "text": "obsolete-lemma-henselian Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. The category of Algebraization of Formal Spaces, Equation (\\ref{restricted-equation-modification}) for $A$ is equivalent to the category Algebraization of Formal Spaces, Equation (\\ref{restricted-equation-modification}) for the henselization $A^h$ of $A$."}
{"_id": "13862", "text": "more-morphisms-lemma-Noetherian-approximation-geometrically-connected Let $f : X \\to S$ be a morphism of affine schemes, which is of finite presentation with geometrically connected fibres. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically connected fibres."}
{"_id": "10881", "text": "spaces-pushouts-lemma-glueable Let $(R \\to R', f)$ be a glueing pair, see above. Let $Y$ be an algebraic space over $X$. The following are equivalent \\begin{enumerate} \\item there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ with $Y_i$ affine and $\\Gamma(Y_i, \\mathcal{O}_{Y_i})$ glueable as an $R$-module, \\item for every \\'etale morphism $W \\to Y$ with $W$ affine $\\Gamma(W, \\mathcal{O}_W)$ is a glueable $R$-module. \\end{enumerate}"}
{"_id": "604", "text": "algebra-lemma-geometrically-connected-any-base-change Let $k$ be a field. Let $S$ be a geometrically connected $k$-algebra. Let $R$ be any $k$-algebra. The map $$ R \\longrightarrow R \\otimes_k S $$ induces a bijection on idempotents, and the map $$ \\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R) $$ induces a bijection on connected components."}
{"_id": "685", "text": "algebra-lemma-one-equation Suppose that $R$ is a Noetherian local ring and $x\\in \\mathfrak m$ an element of its maximal ideal. Then $\\dim R \\leq \\dim R/xR + 1$. If $x$ is not contained in any of the minimal primes of $R$ then equality holds. (For example if $x$ is a nonzerodivisor.)"}
{"_id": "8952", "text": "stacks-lemma-stack-in-setoids-equivalent 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 setoids over $\\mathcal{C}$ if and only if $\\mathcal{S}_2$ is a stack in setoids over $\\mathcal{C}$."}
{"_id": "676", "text": "algebra-lemma-d-independent Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. \\begin{enumerate} \\item The degree of the numerical polynomial $\\varphi_{I, M}$ is independent of the ideal of definition $I$. \\item The degree of the numerical polynomial $\\chi_{I, M}$ is independent of the ideal of definition $I$. \\end{enumerate}"}
{"_id": "4784", "text": "spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of algebraic spaces over $S$. Let $V \\subset Y$ be an open subspace such that $V \\to Y$ is quasi-compact. Let $s : V \\to X$ be a morphism such that $f \\circ s = \\text{id}_V$. Let $Y'$ be the scheme theoretic image of $s$. Then $Y' \\to Y$ is an isomorphism over $V$."}
{"_id": "9712", "text": "local-cohomology-lemma-cd-blowup Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $b : X' \\to X = \\Spec(A)$ be the blowing up of $I$. If the fibres of $b$ have dimension $\\leq d - 1$, then $\\text{cd}(A, I) \\leq d$."}
{"_id": "6709", "text": "etale-cohomology-proposition-h-cohomology-structure-sheaf Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\\mathbf{F}_p$. Then $$ H^i((\\Sch/S)_h, \\mathcal{O}^h) = \\colim_F H^i(S, \\mathcal{O}) $$ Here on the left hand side by $\\mathcal{O}^h$ we mean the h sheafification of the structure sheaf."}
{"_id": "14358", "text": "derham-lemma-chern-character There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ over $\\mathbf{Q}$ a ``chern character'' $$ ch^{dR} : K_0(\\textit{Vect}(X)) \\longrightarrow \\prod\\nolimits_{i \\geq 0} H_{dR}^{2i}(X/\\mathbf{Q}) $$ with the following properties \\begin{enumerate} \\item $ch^{dR}$ is a ring map for all $X$, \\item if $f : X' \\to X$ is a morphism of quasi-compact and quasi-separated schemes over $\\mathbf{Q}$, then $f^* \\circ ch^{dR} =  ch^{dR} \\circ f^*$, and \\item given $\\mathcal{L} \\in \\Pic(X)$ we have $ch^{dR}([\\mathcal{L}]) = \\exp(c_1^{dR}(\\mathcal{L}))$. \\end{enumerate}"}
{"_id": "11290", "text": "spaces-cohomology-lemma-sections-with-support-acyclic Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. 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": "4647", "text": "spaces-limits-lemma-refined-valuative-criterion-universally-closed 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, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \\xymatrix{ \\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\ \\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y } $$ 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": "7570", "text": "stacks-morphisms-lemma-scheme-theoretic-image-of-partial-section Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let $\\mathcal{V} \\subset \\mathcal{Y}$ be an open substack such that $\\mathcal{V} \\to \\mathcal{Y}$ is quasi-compact. Let $s : \\mathcal{V} \\to \\mathcal{X}$ be a morphism such that $f \\circ s = \\text{id}_\\mathcal{V}$. Let $\\mathcal{Y}'$ be the scheme theoretic image of $s$. Then $\\mathcal{Y}' \\to \\mathcal{Y}$ is an isomorphism over $\\mathcal{V}$."}
{"_id": "6550", "text": "etale-cohomology-lemma-zero-in-generic-point Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\eta \\in X$ be a generic point of an irreducible component of $X$. \\begin{enumerate} \\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$ whose stalk $\\mathcal{F}_{\\overline{\\eta}}$ is zero. 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 subscheme not containing $\\eta$. \\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf of $\\Lambda$-modules on $X_\\etale$ whose stalk $\\mathcal{F}_{\\overline{\\eta}}$ is zero. 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 subscheme not containing $\\eta$. \\end{enumerate}"}
{"_id": "12736", "text": "algebraization-lemma-punctured-still-connected \\begin{reference} \\cite[Theorem 1.6]{Varbaro} \\end{reference} Let $(A, \\mathfrak m)$ be a Noetherian complete local ring. Let $I$ be a proper ideal of $A$. Set $X = \\Spec(A)$ and $Y = V(I)$. Denote \\begin{enumerate} \\item $d$ the minimal dimension of an irreducible component of $X$, and \\item $c$ the minimal dimension of a closed subset $Z \\subset X$ such that $X \\setminus Z$ is disconnected. \\end{enumerate} Then for $Z \\subset Y$ closed we have $Y \\setminus Z$ is connected if $\\dim(Z) < \\min(c, d - 1) - \\text{cd}(A, I)$. In particular, the punctured spectrum of $A/I$ is connected if $\\text{cd}(A, I) < \\min(c, d - 1)$."}
{"_id": "9548", "text": "decent-spaces-lemma-Jacobson-ft-points-lift-to-closed Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. For $x \\in X_{\\text{ft-pts}}$ and $g : W \\to X$ locally of finite type with $W$ a scheme, if $x \\in \\Im(|g|)$, then there exists a closed point of $W$ mapping to $x$."}
{"_id": "12090", "text": "homology-lemma-differential-objects-ses Let $\\mathcal{A}$ be an abelian category. Let $0 \\to (A, d) \\to (B, d) \\to (C, d) \\to 0$ be a short exact sequence of differential objects. Then we get an exact homology sequence $$ \\ldots \\to H(C, d) \\to H(A, d) \\to H(B, d) \\to H(C, d) \\to \\ldots $$"}
{"_id": "11466", "text": "obsolete-lemma-get-morphism-general Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $t$ be the minimal number of generators for $I$. Let $C$ be a Noetherian $I$-adically complete $A$-algebra. There exists an integer $d \\geq 0$ depending only on $I \\subset A \\to C$ with the following property: given \\begin{enumerate} \\item $c \\geq 0$ and $B$ in Algebraization of Formal Spaces, Equation (\\ref{restricted-equation-C-prime}) such that for $a \\in I^c$ multiplication by $a$ on $\\NL_{B/A}^\\wedge$ is zero in $D(B)$, \\item an integer $n > 2t\\max(c, d)$, \\item an $A/I^n$-algebra map $\\psi_n : B/I^nB \\to C/I^nC$, \\end{enumerate} there exists a map $\\varphi : B \\to C$ of $A$-algebras such that $\\psi_n \\bmod I^{m - c} = \\varphi \\bmod I^{m - c}$ with $m = \\lfloor \\frac{n}{t} \\rfloor$."}
{"_id": "4314", "text": "sites-cohomology-lemma-compare-cohomology-LC With $X \\in \\Ob(\\textit{LC}_{qc})$ and $a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above: \\begin{enumerate} \\item for an abelian sheaf $\\mathcal{F}$ on $X$ we have $H^n(X, \\mathcal{F}) = H^n_{qc}(X, a_X^{-1}\\mathcal{F})$, \\item for $K \\in D^+(X)$ we have $H^n(X, K) = H^n_{qc}(X, a_X^{-1}K)$. \\end{enumerate} For example, if $A$ is an abelian group, then we have $H^n(X, \\underline{A}) = H^n_{qc}(X, \\underline{A})$."}
{"_id": "13225", "text": "modules-lemma-section-direct-sum-quasi-compact Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $I$ be a set. For $i \\in I$, let $\\mathcal{F}_i$ be a sheaf of $\\mathcal{O}_X$-modules. For $U \\subset X$ quasi-compact open the map $$ \\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i(U) \\longrightarrow \\left(\\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i\\right)(U) $$ is bijective."}
{"_id": "9884", "text": "more-algebra-lemma-normal-domain-absolutely-integrally-closed Let $A$ be a normal domain. Then $A$ is absolutely integrally closed if and only if its fraction field is algebraically closed."}
{"_id": "12085", "text": "homology-lemma-filtered-complex Let $\\mathcal{A}$ be an abelian category. Let $A \\to B \\to C$ be a complex of filtered objects of $\\mathcal{A}$. Assume $\\alpha : A \\to B$ and $\\beta : B \\to C$ are strict morphisms of filtered objects. Then $\\text{gr}(\\Ker(\\beta)/\\Im(\\alpha)) = \\Ker(\\text{gr}(\\beta))/\\Im(\\text{gr}(\\alpha)))$."}
{"_id": "955", "text": "algebra-lemma-epimorphism-injective-spec Let $R \\to S$ be an epimorphism of rings. Then \\begin{enumerate} \\item $\\Spec(S) \\to \\Spec(R)$ is injective, and \\item for $\\mathfrak q \\subset S$ lying over $\\mathfrak p \\subset R$ we have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$. \\end{enumerate}"}
{"_id": "6369", "text": "etale-cohomology-theorem-descent-modules If $A \\to B$ is faithfully flat then descent data with respect to $A\\to B$ are effective."}
{"_id": "8899", "text": "stacks-properties-lemma-map-into-reduction Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks. Let $\\mathcal{Z} \\subset \\mathcal{X}$ be a closed substack Assume $\\mathcal{Y}$ is reduced. A morphism $f : \\mathcal{Y} \\to \\mathcal{X}$ factors through $\\mathcal{Z}$ if and only if $f(|\\mathcal{Y}|) \\subset |\\mathcal{Z}|$."}
{"_id": "6909", "text": "stacks-more-morphisms-lemma-make-section 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 $f$ is separated and of finite type, \\item $Y$ is quasi-compact and quasi-separated, \\item $V$ is quasi-compact, and \\item $\\mathcal{X}_V$ is a gerbe over $V$. \\end{enumerate} Then there exists a commutative diagram $$ \\xymatrix{ \\overline{Z} \\ar[rd]_{\\overline{g}} & Z \\ar[l]^j \\ar[d]_g \\ar[r]_h & \\mathcal{X} \\ar[ld]^f \\\\ & Y } $$ with $j$ an open immersion, $\\overline{g}$ and $h$ proper, and such that $|V|$ is contained in the image of $|g|$."}
{"_id": "10104", "text": "more-algebra-lemma-formal-fibres-CM Properties (A), (B), (C), (D), and (E) hold for $P(k \\to R) =$``$R$ is Cohen-Macaulay''."}
{"_id": "8172", "text": "spaces-lemma-viewed-as-properties Let $\\Sch_{fppf}$ be a big fppf site. Let $S \\to S'$ be a morphism of this site. Let $F$ be an algebraic space over $S$. Let $T$ be a scheme over $S$ and let $f : T \\to F$ be a morphism over $S$. Let $f' : T' \\to F'$ be the morphism over $S'$ we get from $f$ by applying the equivalence of categories described in Lemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}. For any property $\\mathcal{P}$ as in Definition \\ref{definition-relative-representable-property} we have $\\mathcal{P}(f') \\Leftrightarrow \\mathcal{P}(f)$."}
{"_id": "11503", "text": "obsolete-lemma-double-dual-rational In Resolution of Surfaces, Situation \\ref{resolve-situation-rational}. Let $M$ be a finite reflexive $A$-module. Let $M \\otimes_A \\mathcal{O}_X$ denote the pullback of the associated $\\mathcal{O}_S$-module. Then $M \\otimes_A \\mathcal{O}_X$ maps onto its double dual."}
{"_id": "11479", "text": "obsolete-lemma-property-higher-rank 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 If $\\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \\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 property $\\mathcal{P}$ holds for $\\mathcal{G}$. \\end{enumerate} \\end{enumerate} Then property $\\mathcal{P}$ holds for every coherent sheaf on $X$."}
{"_id": "351", "text": "algebra-lemma-ideal-in-localization \\begin{slogan} Ideals in the localization of a ring are localizations of ideals. \\end{slogan} Each ideal $I'$ of $S^{-1}A$ takes the form $S^{-1}I$, where one can take $I$ to be the inverse image of $I'$ in $A$."}
{"_id": "3181", "text": "quot-lemma-flat-geometrically-connected-fibres-with-section In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section. Assume that $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$. Then $$ 0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T) \\to 0 $$ is a split exact sequence with splitting given by $\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$."}
{"_id": "8539", "text": "sites-lemma-continuous-with-continuous-left-adjoint A continuous functor of sites which has a continuous left adjoint defines a morphism of sites."}
{"_id": "7674", "text": "schemes-lemma-locally-quasi-compact The underlying topological space of any scheme is locally quasi-compact, see Topology, Definition \\ref{topology-definition-locally-quasi-compact}."}
{"_id": "10449", "text": "more-algebra-lemma-weakly-etale-finite-type Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat. \\begin{enumerate} \\item If $A \\to B$ is of finite type, then $A \\to B$ is unramified. \\item If $A \\to B$ is of finite presentation and flat, then $A \\to B$ is \\'etale. \\end{enumerate} In particular a weakly \\'etale ring map of finite presentation is \\'etale."}
{"_id": "2710", "text": "spaces-perfect-lemma-map-from-pseudo-coherent-to-complex-with-support 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 the complement $U \\subset X$ 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": "2575", "text": "examples-lemma-non-effective-descent-projective There is an etale covering $X\\to S$ of schemes and a descent datum $(V/X,\\varphi)$ relative to $X\\to S$ such that  $V\\to X$ is projective, but the descent datum is not effective in the category of schemes."}
{"_id": "1744", "text": "moduli-lemma-pic-functor-curves-smooth Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to the other assumptions in this section. Then $\\Picardfunctor_{X/B} \\to B$ is smooth."}
{"_id": "13994", "text": "more-morphisms-lemma-descending-property-perfect The property $\\mathcal{P}(f) =$``$f$ is perfect'' is fpqc local on the base."}
{"_id": "4069", "text": "pione-lemma-ses-field Let $k$ be a field with algebraic closure $\\overline{k}$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. If the base change $X_{\\overline{k}}$ is connected, then there is a short exact sequence $$ 1 \\to \\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k)) \\to 1 $$ of profinite topological groups."}
{"_id": "12639", "text": "constructions-lemma-tie-up-psi In Situation \\ref{situation-relative-proj}. Let $(f : T \\to S, d, \\mathcal{L}, \\psi)$ be a quadruple. Let $r_{d, \\mathcal{L}, \\psi} : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ be the associated $S$-morphism. There exists an isomorphism of $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras $$ \\theta : r_{d, \\mathcal{L}, \\psi}^*\\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd) \\right) \\longrightarrow \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n} $$ such that the following diagram commutes $$ \\xymatrix{ \\mathcal{A}^{(d)} \\ar[rr]_-{\\psi}  \\ar[rd]_-{\\psi_{univ}} & & f_*\\left( \\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n} \\right) \\\\  & \\pi_*\\left( \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd) \\right) \\ar[ru]_\\theta } $$ The commutativity of this diagram uniquely determines $\\theta$."}
{"_id": "1273", "text": "algebra-lemma-etale-makes-unramified-closed-at-prime Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$. Assume that $R \\to S$ is of finite type and unramified at $\\mathfrak q$. 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 $$ R' \\otimes_R S = A \\times B $$ \\end{enumerate} with the following properties \\begin{enumerate} \\item $R' \\to A$ is surjective, and \\item $\\mathfrak p'A$ is a prime of $A$ lying over $\\mathfrak p'$ and over $\\mathfrak q$. \\end{enumerate}"}
{"_id": "11474", "text": "obsolete-lemma-equiv Assumptions and notation as in Simplicial, Lemma \\ref{simplicial-lemma-section}. There exists a section $g : U \\to V$ to the morphism $f$ and the composition $g \\circ f$ is homotopy equivalent to the identity on $V$. In particular, the morphism $f$ is a homotopy equivalence."}
{"_id": "2968", "text": "properties-lemma-integral-normal Let $X$ be an integral scheme. Then $X$ is normal if and only if for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is a normal domain."}
{"_id": "14366", "text": "derham-lemma-gysin-transverse-global Let $c \\geq 0$ and $$ \\xymatrix{ Z' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\ Z \\ar[r] & X \\ar[r] & S } $$ satisfy the assumptions of Lemma \\ref{lemma-gysin-transverse} and assume in addition that $X \\to S$ and $X' \\to S'$ are smooth and that $Z \\to X$ and $Z' \\to X'$ are Koszul regular immersions. Then the diagram $$ \\xymatrix{ H^q(Z, \\Omega^p_{Z/S}) \\ar[rr]_-{\\gamma^{p, q}} \\ar[d] & & H^{q + c}(X, \\Omega^{p + c}_{X/S}) \\ar[d] \\\\ H^q(Z', \\Omega^p_{Z'/S'}) \\ar[rr]^{\\gamma^{p, q}} & & H^{q + c}(X', \\Omega^{p + c}_{X'/S'}) } $$ is commutative where $\\gamma^{p, q}$ is as in Remark \\ref{remark-how-to-use}."}
{"_id": "13476", "text": "spaces-resolve-lemma-modification-of-dim-2-is-projective-over-complete If $(A, \\mathfrak m, \\kappa)$ is a complete Noetherian local domain of dimension $2$, then every modification of $\\Spec(A)$ is projective over $A$."}
{"_id": "13324", "text": "modules-lemma-pullback-de-rham-complex Let $f : Y \\to X$ be a continuous map of topological spaces. Let $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\\Omega^\\bullet_{\\mathcal{B}/\\mathcal{A}} = \\Omega^\\bullet_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$ of de Rham complexes."}
{"_id": "47", "text": "spaces-more-morphisms-lemma-open-subspace-thickening Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces over $S$. For any open subspace $U \\subset X$ there exists a unique open subspace $U' \\subset X'$ such that $U = X \\times_{X'} U'$."}
{"_id": "4713", "text": "spaces-morphisms-lemma-trivial-implications Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ is locally separated and $f$ is quasi-separated."}
{"_id": "9314", "text": "spaces-groupoids-lemma-constructing-invariant-opens Let $B \\to S$ as in Section \\ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. \\begin{enumerate} \\item If $s$ and $t$ are open, then for every open $W \\subset U$ the open $s(t^{-1}(W))$ is $R$-invariant. \\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subspaces. \\end{enumerate}"}
{"_id": "7176", "text": "spaces-flat-lemma-relate-zero-affine-push In Situation \\ref{situation-iso} suppose given an affine morphism $g : X \\to X'$. Set $u' = f_*u : f_*\\mathcal{F} \\to f_*\\mathcal{G}$. Then $F_{u, iso} = F_{u', iso}$, $F_{u, inj} = F_{u', inj}$, $F_{u, surj} = F_{u', surj}$, and $F_{u, zero} = F_{u', zero}$."}
{"_id": "2985", "text": "properties-lemma-catenary-local-rings-catenary Let $X$ be a scheme. The following are equivalent \\begin{enumerate} \\item $X$ is catenary, and \\item for any $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is catenary. \\end{enumerate}"}
{"_id": "6264", "text": "curves-lemma-criterion-very-ample-bis 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 $\\mathcal{L}$ is globally generated, \\item $H^1(X, \\mathcal{L}) = 0$, and \\item $\\mathcal{L}$ is ample. \\end{enumerate} Then $\\mathcal{L}^{\\otimes 2}$ is very ample on $X$ over $k$."}
{"_id": "4140", "text": "pione-proposition-injective Let $p$ be a prime number. Let $i : Z \\to X$ be a closed immersion of connected affine schemes over $\\mathbf{F}_p$. For any geometric point $\\overline{z}$ of $Z$ the map $$ \\pi_1(Z, \\overline{z}) \\to \\pi_1(X, \\overline{z}) $$ is injective."}
{"_id": "10209", "text": "more-algebra-lemma-internal-hom-composition Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism $$ R\\Hom_R(L, M) \\otimes_R^\\mathbf{L} R\\Hom_R(K, L) \\longrightarrow R\\Hom_R(K, M) $$ in $D(R)$ functorial in $K, L, M$."}
{"_id": "12233", "text": "categories-lemma-preserve-injective-maps 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 \\begin{enumerate} \\item an injective morphism $M \\to N$ of diagrams of sets over $\\mathcal{I}$ gives rise to an injective map $\\colim M_i \\to \\colim N_i$ of sets, \\item in general the same is not the case for diagrams of abelian groups and their colimits. \\end{enumerate}"}
{"_id": "6021", "text": "flat-lemma-finite-type-flat-at-point-X Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \\in X$ with image $s \\in S$. If $X$ is flat at $x$ over $S$, 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 $V \\to \\Spec(\\mathcal{O}_{S', s'})$ is flat and of finite presentation."}
{"_id": "13011", "text": "spaces-divisors-lemma-dominate-admissible-blowups 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 $b_i : X_i \\to X$, $i = 1, \\ldots, n$ 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": "10930", "text": "varieties-lemma-galois-action-connected-components Let $k$ be a field, with separable algebraic closure $\\overline{k}$. Let $X$ be a scheme over $k$. There is an action $$ \\text{Gal}(\\overline{k}/k)^{opp} \\times \\pi_0(X_{\\overline{k}}) \\longrightarrow \\pi_0(X_{\\overline{k}}) $$ with the following properties: \\begin{enumerate} \\item An element $\\overline{T} \\in \\pi_0(X_{\\overline{k}})$ is fixed by the action if and only if there exists a connected component $T \\subset X$, which is geometrically connected 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 $$ \\xymatrix{ \\text{Gal}(\\overline{k}'/k') \\times \\pi_0(X_{\\overline{k}'}) \\ar[r] \\ar[d] & \\pi_0(X_{\\overline{k}'}) \\ar[d] \\\\ \\text{Gal}(\\overline{k}/k) \\times \\pi_0(X_{\\overline{k}}) \\ar[r] & \\pi_0(X_{\\overline{k}}) } $$ is commutative (where the right vertical arrow is a bijection according to Lemma \\ref{lemma-separably-closed-field-connected-components}). \\end{enumerate}"}
{"_id": "4792", "text": "spaces-morphisms-lemma-universally-injective-representable Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is universally injective (in the sense of Section \\ref{section-representable}) if and only if for all fields $K$ the map $X(K) \\to Y(K)$ is injective."}
{"_id": "7279", "text": "spaces-chow-lemma-spell-out-degree-zero-cycle Let $k$ be a field. Let $X$ be a proper algebraic space 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": "13941", "text": "more-morphisms-lemma-change-hypotheses Suppose that $f : X \\to S$ is locally of finite type, $S$ locally Noetherian, $x \\in X$ a closed point of its fibre $X_s$, and $U \\subset X$ an open subscheme such that $U \\cap X_s = \\emptyset$ and $x \\in \\overline{U}$, then the conclusions of Lemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} hold."}
{"_id": "1743", "text": "moduli-lemma-pic-functor-uniqueness-part Assume the geometric fibres of $X \\to B$ are integral in addition to the other assumptions in this section. Then $\\Picardfunctor_{X/B} \\to B$ is separated."}
{"_id": "5189", "text": "morphisms-lemma-affine-quasi-affine Let $S$ be a scheme. Let $X$ be an affine scheme. A morphism $f : X \\to S$ is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine."}
{"_id": "11379", "text": "artin-lemma-monomorphism Let $S$ be a locally Noetherian scheme. Let $a : F \\to G$ be a transformation of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$. Assume that \\begin{enumerate} \\item $a$ is injective, \\item $F$ satisfies axioms [0], [1], [2], [4], and [5], \\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$, \\item $G$ is an algebraic space locally of finite type over $S$, \\end{enumerate} Then $F$ is an algebraic space."}
{"_id": "7420", "text": "stacks-morphisms-lemma-definition-separated Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Then \\begin{enumerate} \\item $f$ is separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$ are universally closed, and \\item $f$ is quasi-separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$ are quasi-compact. \\item $f$ is quasi-DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$ are locally quasi-finite. \\item $f$ is DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$ are unramified. \\end{enumerate}"}
{"_id": "13951", "text": "more-morphisms-lemma-relative-finite-presentation 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. \\begin{enumerate} \\item If $f$ is locally of finite presentation, then $\\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\\mathcal{F}$ is of finite presentation. \\item The morphism $f$ is locally of finite presentation if and only if $\\mathcal{O}_X$ is of finite presentation relative to $S$. \\end{enumerate}"}
{"_id": "8744", "text": "examples-defos-lemma-power-series-rings-TI In Lemma \\ref{lemma-rings-TI} if $P = k[[x_1, \\ldots, x_n]]/(f)$ for some nonzero $f \\in (x_1, \\ldots, x_n)^2$, then \\begin{enumerate} \\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional if and only if $n = 1$, and \\item $T\\Deformationcategory_P$ is finite dimensional if $$ \\sqrt{(f, \\partial f/\\partial x_1, \\ldots,  \\partial f/\\partial x_n)} = (x_1, \\ldots, x_n) $$ \\end{enumerate}"}
{"_id": "7360", "text": "sdga-lemma-compose-pushforward-hom In the situation above, denote $RT : D(\\mathcal{A}', \\text{d}) \\to D(\\mathcal{B}, \\text{d})$ the right derived extension of (\\ref{equation-pushforward}). Then we have $$ RT(\\mathcal{M}) = Rf_* R\\SheafHom(\\mathcal{N}, \\mathcal{M}) $$ functorially in $\\mathcal{M}$."}
{"_id": "5830", "text": "chow-lemma-section-smooth Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $p : P \\to X$ be a smooth morphism of schemes locally of finite type over $S$ and let $s : X \\to P$ be a section. Then $s$ is a regular immersion and $1 = s^! \\circ p^*$ in $A^*(X)^\\wedge$ where $p^* \\in A^*(P \\to X)^\\wedge$ is the bivariant class of Lemma \\ref{lemma-flat-pullback-bivariant}."}
{"_id": "2149", "text": "cohomology-lemma-mayer-vietoris-cup Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $A = \\Gamma(X, \\mathcal{O}_X)$. Suppose that $X = U \\cup V$ is a union of two open subsets. For objects $K$ and $M$ of $D(\\mathcal{O}_X)$ we have a map of distinguished triangles $$ \\xymatrix{ R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M) \\ar[r] \\ar[d] & R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\ R\\Gamma(X, K) \\otimes_A^\\mathbf{L} (R\\Gamma(U, M) \\oplus R\\Gamma(V, M)) \\ar[r] \\ar[d] & R\\Gamma(U, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\oplus R\\Gamma(V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)) \\ar[d] \\\\ R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(U \\cap V, M) \\ar[r] \\ar[d] & R\\Gamma(U \\cap V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\ R\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M)[1] \\ar[r] & R\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)[1] } $$ where \\begin{enumerate} \\item the horizontal arrows are given by cup product, \\item on the right hand side we have the distinguished triangle of Lemma \\ref{lemma-unbounded-mayer-vietoris} for $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$, and \\item on the left hand side we have the exact functor $R\\Gamma(X, K) \\otimes_A^\\mathbf{L} - $ applied to the distinguished triangle of Lemma \\ref{lemma-unbounded-mayer-vietoris} for $M$. \\end{enumerate}"}
{"_id": "14376", "text": "derham-proposition-de-rham-is-weil Let $k$ be a field of characteristic zero. The functor that sends a smooth projective scheme $X$ over $k$ to $H_{dR}^*(X/k)$ is a Weil cohomology theory in the sense of Weil Cohomology Theories, Definition \\ref{weil-definition-weil-cohomology-theory}."}
{"_id": "6993", "text": "perfect-lemma-perfect-descends-fpqc Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let $E \\in D(\\mathcal{O}_X)$. Then $E$ is perfect if and only if each $Lf_i^*E$ is perfect."}
{"_id": "9948", "text": "more-algebra-lemma-flatten-on-affine-blowup Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$. Let $S$ be a finite type $R$-algebra. Let $M$ be a finite $S$-module. For every valuation ring $A \\subset K$ dominating $R$ there exists an ideal $I \\subset \\mathfrak m$ and a nonzero element $a \\in I$ such that \\begin{enumerate} \\item $I$ is finitely generated, \\item $A$ has center on $R[\\frac{I}{a}]$, \\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$ is not zero, and \\item the strict transform $S_{I, a}$ of $S$ along $R \\to R[\\frac{I}{a}]$ is flat and of finite presentation over $R$, and the strict transform $M_{I, a}$ of $M$ along $R \\to R[\\frac{I}{a}]$ is flat over $R$ and finitely presented over $S_{I, a}$. \\end{enumerate}"}
{"_id": "10770", "text": "crystalline-lemma-describe-omega-small In Situation \\ref{situation-global}. Let $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$. Let $$ (U(1), T(1), \\delta(1)) = (U, T, \\delta) \\times (U, T, \\delta) $$ in $\\text{Cris}(X/S)$. Let $\\mathcal{K} \\subset \\mathcal{O}_{T(1)}$ be the quasi-coherent sheaf of ideals corresponding to the closed immersion $\\Delta : T \\to T(1)$. Then $\\mathcal{K} \\subset \\mathcal{J}_{T(1)}$ is preserved by the divided structure on $\\mathcal{J}_{T(1)}$ and we have $$ (\\Omega_{X/S})_T = \\mathcal{K}/\\mathcal{K}^{[2]} $$"}
{"_id": "9258", "text": "models-lemma-canonical-map-of-pic In Situation \\ref{situation-regular-model} let $T$ be the numerical type associated to $X$. There exists a canonical map $$ \\Pic(C) \\to \\Pic(T) $$ whose kernel is exactly those invertible modules on $C$ which are the restriction of invertible modules $\\mathcal{L}$ on $X$ with $\\deg_{C_i}(\\mathcal{L}|_{C_i}) = 0$ for $i = 1, \\ldots, n$."}
{"_id": "9505", "text": "decent-spaces-lemma-residual-space-regular A reduced, locally Noetherian singleton algebraic space $Z$ is regular."}
{"_id": "15075", "text": "limits-lemma-finite-closed-in-finite-finite-presentation Let $f : X \\to S$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is finite, and \\item $S$ is quasi-compact and quasi-separated. \\end{enumerate} Then there exists a morphism which is finite and of finite presentation $f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$."}
{"_id": "39", "text": "spaces-more-morphisms-lemma-differentials-relative-immersion-section 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 $B$, and assume $i$ (\\'etale locally) has a left inverse. Then the canonical sequence $$ 0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0 $$ of Lemma \\ref{lemma-differentials-relative-immersion} is (\\'etale locally) split exact."}
{"_id": "10695", "text": "etale-theorem-formally-etale Let $f : X \\to S$ be a morphism that is locally of finite presentation. The following are equivalent \\begin{enumerate} \\item $f$ is \\'etale, \\item for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \\subset Y$ defined by square-zero ideals, the natural map $$ \\Mor_S(Y, X) \\longrightarrow \\Mor_S(Y_0, X) $$ is bijective. \\end{enumerate}"}
{"_id": "6674", "text": "etale-cohomology-lemma-compare-cohomology-etale-ph For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$ 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": "11682", "text": "resolve-lemma-blowup-still-good Let $(A, \\mathfrak m, \\kappa)$ be a Nagata local normal domain of dimension $2$. Assume $A$ defines a rational singularity and that the completion $A^\\wedge$ of $A$ is normal. Then \\begin{enumerate} \\item $A^\\wedge$ defines a rational singularity, and \\item if $X \\to \\Spec(A)$ is the blowing up in $\\mathfrak m$, then for a closed point $x \\in X$ the completion $\\mathcal{O}_{X, x}$ is normal. \\end{enumerate}"}
{"_id": "3903", "text": "formal-spaces-lemma-composition-representable The composition of morphisms representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes)."}
{"_id": "7773", "text": "injectives-lemma-G-modules Let $G$ be a topological group. Let $R$ be a ring. The category $\\text{Mod}_{R, G}$ of $R\\text{-}G$-modules, see \\'Etale Cohomology, Definition \\ref{etale-cohomology-definition-G-module-continuous}, has functorial injective hulls. In particular this holds for the category of discrete $G$-modules."}
{"_id": "7367", "text": "sdga-lemma-good-dga Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{B}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra. There exists a quasi-isomorphism of differential graded $\\mathcal{O}$-algebras $(\\mathcal{A}, \\text{d}) \\to (\\mathcal{B}, \\text{d})$ such that $\\mathcal{A}$ is graded flat and K-flat as a complex of $\\mathcal{O}$-modules and such that the same is true after pullback by any morphism of ringed topoi."}
{"_id": "3756", "text": "proetale-lemma-characterize-sheaf-big Let $S$ be a scheme contained in a big pro-\\'etale site $\\Sch_\\proetale$. A sheaf $\\mathcal{F}$ on the big pro-\\'etale site $(\\Sch/S)_\\proetale$ is given by the following data: \\begin{enumerate} \\item for every $T/S \\in \\Ob((\\Sch/S)_\\proetale)$ a sheaf $\\mathcal{F}_T$ on $T_\\proetale$, \\item for every $f : T' \\to T$ in $(\\Sch/S)_\\proetale$ 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 $(\\Sch/S)_\\proetale$ the composition $g_{small}^{-1}c_f \\circ c_g$ is equal to $c_{f \\circ g}$, and \\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\proetale$ is weakly \\'etale then $c_f$ is an isomorphism. \\end{enumerate}"}
{"_id": "13967", "text": "more-morphisms-lemma-rel-n-pseudo-module Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item $\\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$, \\item $\\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, \\item $\\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $\\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$. \\end{enumerate}"}
{"_id": "11515", "text": "obsolete-lemma-dualizing-components In Semistable Reduction, Situation \\ref{models-situation-regular-model} the dualizing module of $C_i$ over $k$ is $$ \\omega_{C_i} = \\omega_X(C_i)|_{C_i} $$ where $\\omega_X$ is as above."}
{"_id": "13370", "text": "defos-lemma-existence-lci If $A \\to B$ is a local complete intersection ring map, then there exists a solution to (\\ref{equation-to-solve})."}
{"_id": "2996", "text": "properties-lemma-characterize-nagata Let $X$ be a locally Noetherian scheme. Then $X$ is Nagata if and only if every integral closed subscheme $Z \\subset X$ is Japanese."}
{"_id": "9302", "text": "spaces-groupoids-lemma-isomorphism Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\\alpha$ is an isomorphism."}
{"_id": "14095", "text": "more-morphisms-lemma-semicontinuous-w Let $f : X \\to Y$ be a locally quasi-finite morphism. Let $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $w$ is upper semi-continuous."}
{"_id": "4066", "text": "pione-lemma-structure-decomposition-separable-closure Let $A$ be a discrete valuation ring with fraction field $K$. Let $K^{sep}$ be a separable closure of $K$. Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$. Let $\\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$. Let $\\mathfrak m = \\mathfrak m^{sep} \\cap A$, let $\\kappa = A/\\mathfrak m$, and let $\\overline{\\kappa} = A^{sep}/\\mathfrak m^{sep}$. Then $\\overline{\\kappa}$ is an algebraic closure of $\\kappa$. Let $G = \\text{Gal}(K^{sep}/K)$, $D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m^{sep}) = \\mathfrak m^{sep}\\}$, and $I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m^{sep} = \\text{id}_{\\kappa(\\mathfrak m^{sep})}\\}$. The decomposition group $D$ fits into a canonical exact sequence $$ 1 \\to I \\to D \\to \\text{Gal}(\\kappa^{sep}/\\kappa) \\to 1 $$ where $\\kappa^{sep} \\subset \\overline{\\kappa}$ is the separable closure of $\\kappa$. 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 exists a canonical surjective map $$ \\theta_{can} : I \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep}) $$ 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 $I_t \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$. \\end{enumerate}"}
{"_id": "11961", "text": "intersection-lemma-compose-flat-pullback Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of varieties. Then $g \\circ f$ is flat and $f^* \\circ g^* = (g \\circ f)^*$ as maps $Z_k(Z) \\to Z_{k + \\dim(X) - \\dim(Z)}(X)$."}
{"_id": "9758", "text": "local-cohomology-lemma-annihilator-frobenius-module Let $p$ be a prime number. Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with $p = 0$ in $A$. Let $M$ be a finite $A$-module such that $M \\otimes_{A, F} A \\cong M$. Then $M$ is finite free."}
{"_id": "1737", "text": "moduli-lemma-pic-existence-part Assume $X \\to B$ is smooth in addition to being proper. Then $\\Picardstack_{X/B} \\to B$ satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-existence})."}
{"_id": "5730", "text": "chow-lemma-decompose-section-formulae In the situation of Lemma \\ref{lemma-decompose-section} assume $Y$ is locally of finite type over $(S, \\delta)$ as in Situation \\ref{situation-setup}. Then we have $i_1^*p^*\\alpha = p_1^*i^*\\alpha$ in $\\CH_k(D_1)$ for all $\\alpha \\in \\CH_k(Y)$."}
{"_id": "11489", "text": "obsolete-lemma-quasi-separated-very-reasonable An algebraic space which is Zariski locally quasi-separated is very reasonable. In particular any quasi-separated algebraic space is very reasonable."}
{"_id": "2621", "text": "bootstrap-lemma-bootstrap-locally-quasi-finite Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Let $X$ be a scheme and let $X \\to F$ be representable by algebraic spaces and locally quasi-finite. Then $X \\to F$ is representable (by schemes)."}
{"_id": "14435", "text": "trace-lemma-switch-l Switching $l$. Let $E$ be a number field. Start with $$ \\rho : \\pi_1(X)\\to SL_2(E_\\lambda) $$ absolutely irreducible continuous, where $\\lambda$ is a place of $E$ not lying above $p$. Then for any second place $\\lambda'$ of $E$ not lying above $p$ there exists a finite extension $E'_{\\lambda'}$ and a absolutely irreducible continuous representation $$ \\rho': \\pi_1(X)\\to SL_2(E'_{\\lambda'}) $$ which is compatible with $\\rho$ in the sense that the characteristic polynomials of all Frobenii are the same."}
{"_id": "14084", "text": "more-morphisms-lemma-descend-quasi-finite-universally-open 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 locally quasi-finite and universally open, and \\item $f_0$ is locally of finite presentation, \\end{enumerate} then there exists an $i \\geq 0$ such that $f_i$ is locally quasi-finite and universally open."}
{"_id": "4681", "text": "stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms-stacky If $f: \\mathcal{U} \\to \\mathcal{X}$ is a smooth morphism of locally Noetherian algebraic stacks, and if $u \\in |\\mathcal{U}|$ with image $x \\in |\\mathcal{X}|$, then $$ \\dim_u (\\mathcal{U}) = \\dim_x(\\mathcal{X}) + \\dim_{u} (\\mathcal{U}_x). $$"}
{"_id": "5769", "text": "chow-lemma-chern-character-dual 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 $ch_i(\\mathcal{E}^\\vee) = (-1)^i ch_i(\\mathcal{E})$ in $A^i(X)$."}
{"_id": "12244", "text": "categories-lemma-initial-essentially-constant Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of cofiltered index categories. If $H$ is initial, then any diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant if and only if $M \\circ H$ is essentially constant."}
{"_id": "2584", "text": "examples-lemma-counter-Grothendieck-existence Counter examples to algebraization of coherent sheaves. \\begin{enumerate} \\item Grothendieck's existence theorem as stated in Cohomology of Schemes, Theorem \\ref{coherent-theorem-grothendieck-existence} is false if we drop the assumption that $X \\to \\Spec(A)$ is separated. \\item The stack of coherent sheaves $\\Cohstack_{X/B}$ of Quot, Theorems \\ref{quot-theorem-coherent-algebraic-general} and \\ref{quot-theorem-coherent-algebraic} is in general not algebraic if we drop the assumption that $X \\to S$ is separated \\item The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$ of Quot, Proposition \\ref{quot-proposition-quot} is not an algebraic space in general if we drop the assumption that $X \\to B$ is separated. \\end{enumerate}"}
{"_id": "10706", "text": "etale-lemma-etale-definition 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 \\'etale at $x$ in the sense of Definition \\ref{definition-etale-schemes-1} if and only if it is \\'etale at $x$ in the sense of Morphisms, Definition \\ref{morphisms-definition-etale}."}
{"_id": "2596", "text": "examples-proposition-quotient-by-finitely-generated-torsion-modules Let $A$ be a Noetherian integral domain. Let $K$ denote its field of fractions. Let $\\text{Mod}_A^{fg}$ denote the category of finitely generated $A$-modules and $\\mathcal{T}^{fg}$ its Serre subcategory of finitely generated torsion modules. Then $\\text{Mod}_A^{fg}/\\mathcal{T}^{fg}$ is canonically equivalent to the category of finite dimensional $K$-vector spaces."}
{"_id": "12977", "text": "spaces-divisors-lemma-relatively-ample-sanity-check 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. Assume $Y$ is a scheme. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/Y$ in the sense of Definition \\ref{definition-relatively-ample}, and \\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X/Y$ in the sense of Morphisms, Definition \\ref{morphisms-definition-relatively-ample}. \\end{enumerate}"}
{"_id": "13104", "text": "dga-lemma-bc-homotopy The functor (\\ref{equation-bc}) defines an exact functor of triangulated categories $K(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})})$."}
{"_id": "2497", "text": "more-groupoids-lemma-invariant-affine-open-around-generic-point-Noetherian Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \\ldots, u_m \\in U$ 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, \\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free. \\end{enumerate}"}
{"_id": "8895", "text": "stacks-properties-lemma-zariski-open-cover-stack-is-scheme 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 a scheme \\end{enumerate} Then $\\mathcal{X}$ is a scheme."}
{"_id": "635", "text": "algebra-lemma-length-localize Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be a multiplicative subset. Then $\\text{length}_R(M) \\geq \\text{length}_{S^{-1}R}(S^{-1}M)$."}
{"_id": "1768", "text": "derived-lemma-products-sums-shifts-triangles Let $\\mathcal{D}$ be a pre-triangulated category. Let $I$ be a set. \\begin{enumerate} \\item Let $X_i$, $i \\in I$ be a family of objects of $\\mathcal{D}$. \\begin{enumerate} \\item If $\\prod X_i$ exists, then $(\\prod X_i)[1] = \\prod X_i[1]$. \\item If $\\bigoplus X_i$ exists, then $(\\bigoplus X_i)[1] = \\bigoplus X_i[1]$. \\end{enumerate} \\item Let $X_i \\to Y_i \\to Z_i \\to X_i[1]$ be a family of distinguished triangles of $\\mathcal{D}$. \\begin{enumerate} \\item If $\\prod X_i$, $\\prod Y_i$, $\\prod Z_i$ exist, then $\\prod X_i \\to \\prod Y_i \\to \\prod Z_i \\to \\prod X_i[1]$ is a distinguished triangle. \\item If $\\bigoplus X_i$, $\\bigoplus Y_i$, $\\bigoplus Z_i$ exist, then $\\bigoplus X_i \\to \\bigoplus Y_i \\to \\bigoplus Z_i \\to \\bigoplus X_i[1]$ is a distinguished triangle. \\end{enumerate} \\end{enumerate}"}
{"_id": "8016", "text": "divisors-lemma-regular-meromorphic-section-exists-noetherian Let $X$ be a locally Noetherian scheme having no embedded points. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then $\\mathcal{L}$ has a regular meromorphic section."}
{"_id": "9541", "text": "decent-spaces-lemma-birational-birational 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 one of the following conditions is satisfied \\begin{enumerate} \\item $f$ is locally of finite type and $Y$ reduced (i.e., integral), \\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."}
{"_id": "9598", "text": "groupoids-lemma-group-scheme-field-countable-affine Let $G$ be a group scheme over a field. There exists an open and closed subscheme $G' \\subset G$ which is a countable union of affines."}
{"_id": "6689", "text": "etale-cohomology-lemma-h-descent-etale-sheaves Let $S$ be a scheme. Then the category fibred in groupoids $$ p : \\mathcal{S} \\longrightarrow (\\Sch/S)_h $$ whose fibre category over $U$ is the category $\\Sh(U_\\etale)$ of sheaves on the small \\'etale site of $U$ is a stack in groupoids."}
{"_id": "13698", "text": "more-morphisms-lemma-immersion-universal-thickening Let $i : Z \\to X$ be an immersion of schemes. 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}, and \\item the conormal sheaf of $i$ in the sense of Morphisms, Definition \\ref{morphisms-definition-conormal-sheaf} agrees with the conormal sheaf of $i$ in the sense of Definition \\ref{definition-universal-thickening}. \\end{enumerate}"}
{"_id": "5977", "text": "flat-theorem-pullback-trivial-fibres Let $p$ be a prime number. Let $Y$ be a quasi-compact and quasi-separated scheme over $\\mathbf{F}_p$. Let $f : X \\to Y$ be a proper, surjective morphism of finite presentation with geometrically connected fibres. Then the functor $$ \\colim_F \\textit{Vect}(Y) \\longrightarrow \\colim_F \\textit{Vect}(X) $$ is fully faithful with essential image described as follows. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module. Assume for all $y \\in Y$ there exists integers $n_y, r_y \\geq 0$ such that $$ F^{n_y, *}\\mathcal{E}|_{X_{y, red}} \\cong \\mathcal{O}_{X_{y, red}}^{\\oplus r_y} $$ Then for some $n \\geq 0$ the $n$th Frobenius power pullback $F^{n, *}\\mathcal{E}$ is the pullback of a finite locally free $\\mathcal{O}_Y$-module."}
{"_id": "9551", "text": "decent-spaces-lemma-punctured-spec Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic space over $S$. Let $x \\in |X|$. Then $$ W = \\{x' \\in |X| : x' \\leadsto x,\\ x' \\not = x\\} $$ is a Noetherian, spectral, sober, Jacobson topological space."}
{"_id": "1669", "text": "dpa-lemma-divided-powers-on-tor Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras. Then there is a canonical structure of a strictly graded commutative $R$-algebra with divided powers on $$ \\operatorname{Tor}_*^R(S, T). $$"}
{"_id": "11449", "text": "obsolete-lemma-helper-finite-type-flat-finite-presentation Let $R$ be a domain with fraction field $K$. Let $S = R[x_1, \\ldots, x_n]$ be a polynomial ring over $R$. Let $M$ be a finite $S$-module. Assume that $M$ is flat over $R$. If for every subring $R \\subset R' \\subset K$, $R \\not = R'$ the module $M \\otimes_R R'$ is finitely presented over $S \\otimes_R R'$, then $M$ is finitely presented over $S$."}
{"_id": "9259", "text": "models-lemma-sequence-torsion In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$ and let $T$ be the numerical type associated to $X$. Let $h \\geq 1$ be an integer prime to $d$. There exists an exact sequence $$ 0 \\to \\Pic(X)[h] \\to \\Pic(C)[h] \\to \\Pic(T)[h] $$"}
{"_id": "12961", "text": "spaces-divisors-lemma-regular-meromorphic-section-exists Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying (a), (b), and (c) of Lemma \\ref{lemma-compute-meromorphic}. Then every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ has a regular meromorphic section."}
{"_id": "8141", "text": "spaces-lemma-product-spaces Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Then $F \\times G$ is an algebraic space, and is a product in the category of algebraic spaces over $S$."}
{"_id": "4327", "text": "sites-cohomology-lemma-section-RHom-over-U Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\\mathcal{O})$. For every object $U$ of $\\mathcal{C}$ we have $$ H^0(U, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U) $$ and we have $H^0(\\mathcal{C}, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O})}(L, M)$."}
{"_id": "4023", "text": "pione-theorem-specialization-map-isomorphism-prime-to-p Let $f : X \\to S$ be a smooth proper morphism with geometrically connected fibres. Let $s' \\leadsto s$ be a specialization. If the characteristic of $\\kappa(s)$ is $p$, then the specialization map $$ sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}}) $$ is surjective and induces an isomorphism $$ \\pi'_1(X_{\\overline{s}'}) \\cong \\pi'_1(X_{\\overline{s}}) $$ of the maximal prime-to-p quotients"}
{"_id": "5320", "text": "morphisms-lemma-differentials-relative-immersion-section Let $i : Z \\to X$ be an immersion of schemes over $S$, and assume $i$ (locally) has a left inverse. Then the canonical sequence $$ 0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0 $$ of Lemma \\ref{lemma-differentials-relative-immersion} is (locally) split exact. In particular, if $s : S \\to X$ is a section of the structure morphism $X \\to S$ then the map $\\mathcal{C}_{S/X} \\to s^*\\Omega_{X/S}$ induced by $\\text{d}_{X/S}$ is an isomorphism."}
{"_id": "11906", "text": "spaces-properties-lemma-isomorphism-ringed-topoi Let $X$, $Y$ be algebraic spaces over $\\mathbf{Z}$. If $$ (g, g^\\sharp) : (\\Sh(X_\\etale), \\mathcal{O}_X) \\longrightarrow (\\Sh(Y_\\etale), \\mathcal{O}_Y) $$ is an isomorphism of ringed topoi, then there exists a unique morphism $f : X \\to Y$ of algebraic spaces such that $(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$ and moreover $f$ is an isomorphism of algebraic spaces."}
{"_id": "5221", "text": "morphisms-lemma-ubiquity-J-2 The following types of schemes are J-2. \\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 Noetherian local ring of dimension $1$. \\item Any scheme locally of finite type over a Nagata ring of dimension $1$. \\item Any scheme locally of finite type over a Dedekind ring of characteristic zero. \\item And so on. \\end{enumerate}"}
{"_id": "13488", "text": "spaces-resolve-lemma-formally-unramified Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let $X$ be an algebraic space 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 $A$, \\item $X \\to \\Spec(A)$ is \\'etale at every point of codimension $0$ in $X$. \\end{enumerate} Then the normalization of $X$ is finite over $X$."}
{"_id": "3818", "text": "proetale-lemma-local-structure-constructible-complex 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 the cohomology sheaves of $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally constant. Then there exists a finite disjoint open covering $X = \\coprod U_i$ and for each $i$ a finite collection of finite projective $\\Lambda^\\wedge$-modules $M_a, \\ldots, M_b$ such that $K|_{U_i}$ is represented by a complex $$ (\\underline{M^a})^\\wedge \\to \\ldots \\to (\\underline{M^b})^\\wedge $$ in $D(U_{i, \\proetale}, \\Lambda)$ for some maps of sheaves of $\\Lambda$-modules $(\\underline{M^i})^\\wedge \\to (\\underline{M^{i + 1}})^\\wedge$."}
{"_id": "4134", "text": "pione-proposition-purity-complete-intersection Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\\geq 3$, then purity holds for $A$ in the sense that any finite \\'etale cover of the punctured spectrum extends."}
{"_id": "7", "text": "stacks-perfect-proposition-derived-pullback-quasi-coherent Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The exact functor $f^*$ induces a commutative diagram $$ \\xymatrix{ D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] & D(\\mathcal{O}_\\mathcal{X}) \\\\ D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] \\ar[u]^{f^*} & D(\\mathcal{O}_\\mathcal{Y}) \\ar[u]^{f^*} } $$ The composition $$ D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\xrightarrow{f^*} D_{\\mathcal{M}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X}) \\xrightarrow{q_\\mathcal{X}} D_\\QCoh(\\mathcal{O}_\\mathcal{X}) $$ is left derivable with respect to the localization $D_{\\mathcal{M}_\\mathcal{Y}}(\\mathcal{O}_\\mathcal{Y}) \\to D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$ and we may define $Lf^*_\\QCoh$ as its left derived functor $$ Lf_\\QCoh^* : D_\\QCoh(\\mathcal{O}_\\mathcal{Y}) \\longrightarrow D_\\QCoh(\\mathcal{O}_\\mathcal{X}) $$ (see Derived Categories, Definitions \\ref{derived-definition-right-derived-functor-defined} and \\ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact and quasi-separated, then $Lf^*_\\QCoh$ and $Rf_{\\QCoh, *}$ satisfy the following adjointness: $$ \\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{X})}(Lf^*_\\QCoh A, B) = \\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{Y})}(A, Rf_{\\QCoh, *}B) $$ for $A \\in D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$ and $B \\in D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X})$."}
{"_id": "10322", "text": "more-algebra-lemma-Mittag-Leffler Let $$ 0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0 $$ be a short exact sequence of inverse systems of abelian groups. If $(A_i)$ and $(C_i)$ are ML, then so is $(B_i)$."}
{"_id": "3926", "text": "formal-spaces-lemma-permanence-property-morphisms Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$ which has the cancellation property. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $g \\circ f$ and $g$ satisfies the equivalent conditions of Lemma \\ref{lemma-property-defines-property-morphisms} then so does $f : X \\to Y$."}
{"_id": "9016", "text": "spaces-simplicial-lemma-restriction-to-components-functorial Let $f : Y \\to X$ be a morphism of simplicial spaces. Then $$ \\xymatrix{ \\Sh(Y_n) \\ar[d] \\ar[r]_{f_n} & \\Sh(X_n) \\ar[d] \\\\ \\Sh(Y_{Zar}) \\ar[r]^{f_{Zar}} & \\Sh(X_{Zar}) } $$ is a commutative diagram of topoi."}
{"_id": "8195", "text": "topology-lemma-from-hausdorff Let $f : X \\to Y$ be continuous map of topological spaces. If $X$ is Hausdorff, then $f$ is separated."}
{"_id": "14048", "text": "more-morphisms-lemma-pushout-finite-type In Situation \\ref{situation-pushout-along-closed-immersion-and-integral} assume $S$ is a locally Noetherian scheme and $X$, $Y$, and $Z$ are locally of finite type over $S$. Then the pushout $Y \\amalg_Z X$ (Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral}) is locally of finite type over $S$."}
{"_id": "4924", "text": "spaces-morphisms-lemma-valuative-criterion-representable Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable. 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": "12982", "text": "spaces-divisors-lemma-vanshing-gives-ample Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$ such that the structure morphism $f : X \\to \\Spec(R)$ is proper. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{L}$ is ample on $X/R$ (Definition \\ref{definition-relatively-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 $p > 0$. \\end{enumerate}"}
{"_id": "241", "text": "spaces-more-morphisms-lemma-lci-permanence Let $S$ be a scheme. Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & Z } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume $Y \\to Z$ is smooth and $X \\to Z$ is a local complete intersection morphism. Then $f : X \\to Y$ is a local complete intersection morphism."}
{"_id": "6493", "text": "etale-cohomology-lemma-central-simple-algebra-pgln \\begin{slogan} Central simple algebras are classified by Galois cohomology of PGL. \\end{slogan} Let $K$ be a field and let $K^{sep}$ be a separable algebraic closure. Then the set of isomorphism classes of central simple algebras of degree $d$ over $K$ is in bijection with the non-abelian cohomology $H^1(\\text{Gal}(K^{sep}/K), \\text{PGL}_d(K^{sep}))$."}
{"_id": "13279", "text": "modules-lemma-flat-resolution-of-flat Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $$ \\ldots \\to \\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F}_0 \\to \\mathcal{Q} \\to 0 $$ be an exact complex of $\\mathcal{O}_X$-modules. If $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}_X$-modules, then for any $\\mathcal{O}_X$-module $\\mathcal{G}$ the complex $$ \\ldots \\to \\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{F}_0 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{Q} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to 0 $$ is exact also."}
{"_id": "10857", "text": "spaces-pushouts-lemma-glue-etale-space-modification 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 $$ Y_{spaces, \\etale} \\longrightarrow Y'_{spaces, \\etale} \\times_{E_{spaces, \\etale}} \\Sh(Z_{spaces, \\etale}) $$ is an equivalence of categories."}
{"_id": "14414", "text": "trace-lemma-when-in-bounded An object $E$ of $D(\\mathcal{A})$ is contained in $D^+(\\mathcal{A})$ if and only if $H^i(E) =0 $ for all $i \\ll 0$. Similar statements hold for $D^-$ and $D^+$."}
{"_id": "2088", "text": "cohomology-lemma-subsheaf-irreducible Let $X$ be an irreducible topological space. Let $\\mathcal{H} \\subset \\underline{\\mathbf{Z}}$ be an abelian subsheaf of the constant sheaf. Then there exists a nonempty open $U \\subset X$ such that $\\mathcal{H}|_U = \\underline{d\\mathbf{Z}}_U$ for some $d \\in \\mathbf{Z}$."}
{"_id": "4717", "text": "spaces-morphisms-lemma-section-immersion Let $S$ be a scheme. Let $f : X \\to T$ be a morphism of algebraic spaces over $S$. Let $s : T \\to X$ be a section of $f$ (in a formula $f \\circ s = \\text{id}_T$). Then \\begin{enumerate} \\item $s$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \\item if $f$ is locally separated, then $s$ is an immersion, \\item if $f$ is separated, then $s$ is a closed immersion, and \\item if $f$ is quasi-separated, then $s$ is quasi-compact. \\end{enumerate}"}
{"_id": "1615", "text": "moduli-curves-lemma-smooth-curves-h0-smooth The morphisms $\\mathcal{M} \\to \\Spec(\\mathbf{Z})$ and $\\mathcal{M}_g \\to \\Spec(\\mathbf{Z})$ are smooth."}
{"_id": "202", "text": "spaces-more-morphisms-lemma-inverse-systems-surjective 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. 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": "5759", "text": "chow-lemma-degree-vector-bundle Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $\\leq 1$. Let $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of constant rank. Then $$ \\deg(\\mathcal{E}) = \\deg(c_1(\\mathcal{E}) \\cap [X]_1) $$ where the left hand side is defined in Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}."}
{"_id": "8611", "text": "sites-lemma-localize-enough Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$. let $\\{(p_i, u_i)\\}_{i\\in I}$ be a family of points of $\\mathcal{C}$. For $x \\in u_i(U)$ let $q_{i, x}$ be the point of $\\mathcal{C}/U$ constructed in Lemma \\ref{lemma-point-localize}. If $\\{p_i\\}$ is a conservative family of points, then $\\{q_{i, x}\\}_{i \\in I, x \\in u_i(U)}$ is a conservative family of points of $\\mathcal{C}/U$. In particular, if $\\mathcal{C}$ has enough points, then so does every localization $\\mathcal{C}/U$."}
{"_id": "7206", "text": "spaces-chow-lemma-delta-is-dimension In Situation \\ref{situation-setup} assume $B$ is Jacobson and that $\\delta(b) = 0$ for every closed point $b$ of $|B|$. Let $X/B$ be good. If $Z \\subset X$ is an integral closed subspace with generic point $\\xi \\in |Z|$, then the following integers are the same: \\begin{enumerate} \\item $\\delta(\\xi) = \\delta_{X/B}(\\xi)$, \\item $\\dim(|Z|)$, \\item $\\text{codim}(\\{z\\}, |Z|)$ for $z \\in |Z|$ closed, \\item the dimension of the local ring of $Z$ at $z$ for $z \\in |Z|$ closed, and \\item $\\dim(\\mathcal{O}_{Z, \\overline{z}})$ for $z \\in |Z|$ closed. \\end{enumerate}"}
{"_id": "759", "text": "algebra-lemma-blowup-dominant Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $a \\in I$. If $a$ is not contained in any minimal prime of $R$, then $\\Spec(R[\\frac{I}{a}]) \\to \\Spec(R)$ has dense image."}
{"_id": "14592", "text": "descent-theorem-descend-algebra-properties If $A \\otimes_R S$ has one of the following properties as an $S$-algebra \\begin{enumerate} \\item[(a)] of finite type; \\item[(b)] of finite presentation; \\item[(c)] formally unramified; \\item[(d)] unramified; \\item[(e)] \\'etale; \\end{enumerate} then so does $A$ as an $R$-algebra (and of course conversely)."}
{"_id": "2188", "text": "cohomology-lemma-internal-hom-diagonal Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L$ in $D(\\mathcal{O}_X)$ there is a canonical morphism $$ K \\longrightarrow R\\SheafHom(L, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) $$ in $D(\\mathcal{O}_X)$ functorial in both $K$ and $L$."}
{"_id": "8818", "text": "more-etale-lemma-f-shriek-separated-direct-sums Let $f : X \\to Y$ be a morphism of schemes which is separated and locally of finite type. Then functor $f_!$ commutes with direct sums."}
{"_id": "12488", "text": "topologies-lemma-fibre-products-ph Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$. The underlying categories of the sites $\\Sch_{ph}$, $(\\Sch/S)_{ph}$, and $(\\textit{Aff}/S)_{ph}$ have fibre products. In each case the obvious functor into the category $\\Sch$ of all schemes commutes with taking fibre products. The category $(\\Sch/S)_{ph}$ has a final object, namely $S/S$."}
{"_id": "7840", "text": "brauer-lemma-dimension-square Let $A$ be a finite central simple algebra over a field $k$. Then $[A : k]$ is a square."}
{"_id": "4389", "text": "sites-cohomology-lemma-left-dual-complex Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}$-modules. If $\\mathcal{F}^\\bullet$ has a left dual in the monoidal category of complexes of $\\mathcal{O}$-modules (Categories, Definition \\ref{categories-definition-dual}) then for every object $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}^\\bullet|_{U_i}$ is strictly perfect and the left dual is as constructed in Example \\ref{example-dual}."}
{"_id": "2974", "text": "properties-lemma-locally-Cohen-Macaulay Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is Cohen-Macaulay. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is Cohen-Macaulay. \\end{enumerate} Moreover, if $X$ is Cohen-Macaulay then every open subscheme is Cohen-Macaulay."}
{"_id": "431", "text": "algebra-lemma-characterize-domain Let $R$ be a ring. \\begin{enumerate} \\item An ideal maximal among the ideals which do not contain a nonzerodivisor is prime. \\item If $R$ is nonzero and every nonzero prime ideal in $R$ contains a nonzerodivisor, then $R$ is a domain. \\end{enumerate}"}
{"_id": "7917", "text": "divisors-lemma-check-reflexive 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 $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module for all $x \\in X$, \\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module for all closed points $x \\in X$. \\end{enumerate}"}
{"_id": "1659", "text": "dpa-lemma-sub-dp-ideal Let $(A, I, \\gamma)$ be a divided power ring. Let $E \\subset I$ be a subset. Then the smallest ideal $J \\subset I$ preserved by $\\gamma$ and containing all $f \\in E$ is the ideal $J$ generated by $\\gamma_n(f)$, $n \\geq 1$, $f \\in E$."}
{"_id": "3326", "text": "coherent-lemma-Cohen-Macaulay-over-regular Let $X$ be a regular scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is Cohen-Macaulay and $\\text{Supp}(\\mathcal{F}) = X$, \\item $\\mathcal{F}$ is finite locally free of rank $> 0$. \\end{enumerate}"}
{"_id": "12565", "text": "pic-lemma-flat-geometrically-connected-fibres-with-section Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}. Assume $f$ has a section $\\sigma$ and that $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism for all $T \\in \\Ob((\\Sch/S)_{fppf})$. Then $$ 0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T) \\to 0 $$ is a split exact sequence with splitting given by $\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$."}
{"_id": "3729", "text": "proetale-lemma-ind-etale-implies Let $A \\to B$ be ind-\\'etale. Then $A \\to B$ is weakly \\'etale (More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale})."}
{"_id": "14597", "text": "descent-lemma-with-section-exact Suppose that $R \\to A$ has a section. Then for any $R$-module $M$ the extended cochain complex (\\ref{equation-extended-complex}) is exact."}
{"_id": "11455", "text": "obsolete-lemma-kill-local Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ be an integer. Assume \\begin{enumerate} \\item $A$ has a dualizing complex, \\item if $\\mathfrak p \\not \\in V(I)$ and $V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, then $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > s$. \\end{enumerate} Then there exists an $n > 0$ and an ideal $J \\subset A$ with $V(J) \\cap V(I) = \\{\\mathfrak m\\}$ such that $JI^n$ annihilates $H^i_\\mathfrak m(M)$ for $i \\leq s$."}
{"_id": "5171", "text": "morphisms-lemma-affine-separated An affine morphism is separated and quasi-compact."}
{"_id": "3789", "text": "proetale-lemma-compare-truncations 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 \\begin{enumerate} \\item in the essential image of $\\epsilon^{-1} :D(X_\\etale, \\Lambda/I) \\to D(X_\\proetale, \\Lambda/I)$, and \\item has tor amplitude in $[a,\\infty)$ for some $a \\in \\mathbf{Z}$, \\end{enumerate} then (1) and (2) hold for $K_n$ as an object of $D(X_\\proetale, \\Lambda/I^n)$."}
{"_id": "8308", "text": "topology-lemma-two-points Let $X$ be a spectral space. Let $x, y \\in X$. Then either there exists a third point specializing to both $x$ and $y$, or there exist disjoint open neighbourhoods containing $x$ and $y$."}
{"_id": "3966", "text": "formal-spaces-lemma-functoriality-etale-site Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$. \\begin{enumerate} \\item There is a continuous functor $Y_{spaces, \\etale} \\to X_{spaces, \\etale}$ which 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 (\\ref{equation-etale-topos}), a morphism of topoi $f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$ whose construction is compatible with compositions. \\end{enumerate}"}
{"_id": "5138", "text": "morphisms-lemma-i-upper-shriek Let $i : Z \\to X$ be a closed immersion of schemes. There is a functor\\footnote{This is likely nonstandard notation.} $i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)"}
{"_id": "13969", "text": "more-morphisms-lemma-complex-relative-pseudo-coherent-modules Let $X \\to S$ be a morphism of schemes which is locally of finite type. 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 relative to $S$ for all $i$. Then $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent relative to $S$."}
{"_id": "3779", "text": "proetale-lemma-classical-point In the situation above the scheme $\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})$ is an object of $X_\\proetale$ and there is a canonical isomorphism $$ \\mathcal{F}(\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})) = \\mathcal{F}_{\\overline{s}} $$ functorial in $\\mathcal{F}$."}
{"_id": "4563", "text": "spaces-limits-lemma-sheafify-finite-presentation-map Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be functors. If $a : F \\to G$ is a transformation which is limit preserving, then the induced transformation of sheaves $F^\\# \\to G^\\#$ is limit preserving."}
{"_id": "1914", "text": "derived-lemma-limit-K-injectives \\begin{slogan} The limit of a ``split'' tower of K-injective complexes is K-injective. \\end{slogan} Let $\\mathcal{A}$ be an abelian category. Let $$ \\ldots \\to I_3^\\bullet \\to I_2^\\bullet \\to I_1^\\bullet $$ be an inverse system of complexes. Assume \\begin{enumerate} \\item each $I_n^\\bullet$ is $K$-injective, \\item each map $I_{n + 1}^m \\to I_n^m$ is a split surjection, \\item the limits $I^m = \\lim I_n^m$ exist. \\end{enumerate} Then the complex $I^\\bullet$ is K-injective."}
{"_id": "144", "text": "spaces-more-morphisms-lemma-base-change-CM 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 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": "5523", "text": "morphisms-lemma-image-nowhere-dense-quasi-finite Let $f : Y \\to X$ be a quasi-finite morphism of schemes. Let $T \\subset Y$ be a closed nowhere dense subset of $Y$. Then $f(T) \\subset X$ is a nowhere dense subset of $X$."}
{"_id": "2353", "text": "restricted-lemma-rig-flat-naive Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$. If $A/I$ is Jacobson for some (equivalently any) ideal of definition $I \\subset A$ and $\\varphi$ is naively rig-flat, then $\\varphi$ is rig-flat."}
{"_id": "10315", "text": "more-algebra-lemma-map-into-Rlim Let $\\mathcal{D}$ be a triangulated category. Let $(K_n)$ be an inverse system of objects of $\\mathcal{D}$. Let $K$ be a derived limit of the system $(K_n)$. Then for every $L$ in $\\mathcal{D}$ we have a short exact sequence $$ 0 \\to R^1\\lim \\Hom_\\mathcal{D}(L, K_n[-1]) \\to \\Hom_\\mathcal{D}(L, K) \\to \\lim \\Hom_\\mathcal{D}(L, K_n) \\to 0 $$"}
{"_id": "13783", "text": "more-morphisms-lemma-regular-fppf-local-source-and-target The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is regular'' is local in the fppf topology on the target and local in the smooth topology on the source."}
{"_id": "9364", "text": "spaces-descent-lemma-finite-over-finite-module 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 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": "13782", "text": "more-morphisms-lemma-smooth-regular A smooth morphism is regular."}
{"_id": "1951", "text": "derived-lemma-uniqueness-maps-postnikov-systems Let $\\mathcal{D}$ be a triangulated category. Given a map (\\ref{equation-map-complexes}) assume we are given Postnikov systems for both complexes. If \\begin{enumerate} \\item $\\Hom(X_i[i], Y'_n[n]) = 0$ for $i = 1, \\ldots, n$, or \\item $\\Hom(Y_n[n], X'_{n - i}[n - i]) = 0$ for $i = 1, \\ldots, n$, or \\item $\\Hom(X_{j - i}[-i + 1], X'_j) = 0$ and $\\Hom(X_j, X'_{j - i}[-i]) = 0$ for $j \\geq i > 0$, \\end{enumerate} then there exists at most one morphism between these Postnikov systems."}
{"_id": "1928", "text": "derived-lemma-enough-K-injectives-Ab4-star Let $\\mathcal{A}$ be an abelian category having enough injectives and exact countable products. Then for every complex there is a quasi-isomorphism to a K-injective complex."}
{"_id": "6594", "text": "etale-cohomology-lemma-base-change-local Consider a cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ and a sheaf $\\mathcal{F}$ on $T_\\etale$. Let $\\{U_i \\to X\\}$ be an \\'etale covering such that $U_i \\to S$ factors as $U_i \\to V_i \\to S$ with $V_i \\to S$ \\'etale and consider the cartesian diagrams $$ \\xymatrix{ U_i \\ar[d]_{f_i} & U_i \\times_X Y \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\ V_i & V_i \\times_S T \\ar[l]_{g_i} } $$ Set $\\mathcal{F}_i = \\mathcal{F}|_{V_i \\times_S T}$. \\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 $\\mathcal{F}$ is an abelian sheaf 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": "2089", "text": "cohomology-lemma-sections-with-support-acyclic Let $i : Z \\to X$ be the inclusion of a closed subset. Let $\\mathcal{I}$ be an injective abelian sheaf on $X$. Then $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf on $Z$."}
{"_id": "5623", "text": "smoothing-lemma-neron-colimit \\begin{slogan} Unramified extensions of DVRs are ind-smooth AKA N\\'eron desingularization \\end{slogan} Let $R \\subset \\Lambda$ be an extension of discrete valuation rings which has ramification index $1$ and induces a separable extension of residue fields and of fraction fields. Then $\\Lambda$ is a filtered colimit of smooth $R$-algebras."}
{"_id": "12475", "text": "topologies-lemma-verify-site-fppf Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf site containing $S$. Then $(\\textit{Aff}/S)_{fppf}$ is a site."}
{"_id": "11904", "text": "spaces-properties-lemma-2-morphism Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $t$ be a $2$-morphism from $(f_{small}, f^\\sharp)$ to itself, see Modules on Sites, Definition \\ref{sites-modules-definition-2-morphism-ringed-topoi}. Then $t = \\text{id}$."}
{"_id": "9126", "text": "spaces-simplicial-lemma-descent-disjoint-union-Artinian-along-fields Let $X \\to S$ be a quasi-compact flat surjective morphism. Let $(V, \\varphi)$ be a descent datum relative to $X \\to S$. If $V$ is a disjoint union of spectra of Artinian rings, then $(V, \\varphi)$ is effective."}
{"_id": "5088", "text": "weil-lemma-negative-cohomology Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If there exists a smooth projective scheme $Y$ over $k$ such that $H^i(Y)$ is nonzero for some $i < 0$, then there exists an equidimensional smooth projective scheme $X$ over $k$ such that the equivalent conditions of Lemma \\ref{lemma-H-0-separable} fail for $X$."}
{"_id": "9469", "text": "decent-spaces-lemma-fun-property-reasonable Let $S$ be a scheme. Let $X$ be a quasi-compact reasonable algebraic space. Then there exists a directed system of quasi-compact and quasi-separated algebraic spaces $X_i$ such that $X = \\colim_i X_i$ (colimit in the category of sheaves)."}
{"_id": "480", "text": "algebra-lemma-conclude-jacobson-Noetherian With notation as above. Assume that $R$ is a Noetherian Jacobson ring. Further assume $R \\to S$ is of finite type. There is a commutative diagram $$ \\xymatrix{ \\text{Constr}(Y) \\ar[r]^{E \\mapsto E_0} \\ar[d]^{E \\mapsto f(E)} & \\text{Constr}(Y_0) \\ar[d]^{E \\mapsto f(E)} \\\\ \\text{Constr}(X) \\ar[r]^{E \\mapsto E_0} & \\text{Constr}(X_0) } $$ where the horizontal arrows are the bijections from Topology, Lemma \\ref{topology-lemma-jacobson-equivalent-constructible}."}
{"_id": "12910", "text": "spaces-divisors-lemma-weakass-pushforward Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Let $y \\in |Y|$ be a point which is not in the image of $|f|$. Then $y$ is not weakly associated to $f_*\\mathcal{F}$."}
{"_id": "6184", "text": "flat-lemma-canonical-blowup-torsion-pd1 Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$. Let $U \\subset X$ be an open such that $\\mathcal{F}|_U = 0$. Then there is a $U$-admissible blowup $$ b : X' \\to X $$ such that $\\mathcal{F}' = b^*\\mathcal{F}$ is equipped with two canonical locally finite filtrations $$ 0 = F^0 \\subset F^1 \\subset F^2 \\subset \\ldots \\subset \\mathcal{F}' \\quad\\text{and}\\quad \\mathcal{F}' = F_1 \\supset F_2 \\supset F_3 \\supset \\ldots \\supset 0 $$ such that for each $n \\geq 1$ there is an effective Cartier divisor $D_n \\subset X'$ with the property that $$ F^i/F^{i - 1} \\quad\\text{and}\\quad F_i/F_{i + 1} $$ are finite locally free of rank $i$ on $D_i$."}
{"_id": "7506", "text": "stacks-morphisms-lemma-open-immersion-locally-finite-presentation An open immersion is locally of finite presentation."}
{"_id": "6602", "text": "etale-cohomology-lemma-base-change-f-star-valuation Consider a cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ Assume that \\begin{enumerate} \\item $f$ is flat and open, \\item the residue fields of $S$ are separably algebraically closed, \\item given an \\'etale morphism $U \\to X$ with $U$ affine we can write $U$ as a finite disjoint union of open subschemes of $X$ (for example if $X$ is a normal integral scheme with separably closed function field), \\item any nonempty open of a fibre $X_s$ of $f$ is connected (for example if $X_s$ is irreducible or empty). \\end{enumerate} Then for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$."}
{"_id": "2662", "text": "spaces-perfect-lemma-base-change-module-support-proper-over-base 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. Let $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module. If the support of $\\mathcal{F}$ is proper over $Y$, then the support of $(g')^*\\mathcal{F}$ is proper over $Y'$."}
{"_id": "7362", "text": "sdga-lemma-compose-hom 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 $$ R\\SheafHom_{\\mathcal{A}''} (\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}', -) = R\\SheafHom_{\\mathcal{A}'}(\\mathcal{N}, R\\SheafHom_{\\mathcal{A}''}(\\mathcal{N}', -)) $$ as functors $D(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$."}
{"_id": "11444", "text": "obsolete-lemma-finite-presentation-module-independent Let $M$ be an $R$-module of finite presentation. For any surjection $\\alpha : R^{\\oplus n} \\to M$ the kernel of $\\alpha$ is a finite $R$-module."}
{"_id": "12635", "text": "constructions-lemma-relative-proj-d In Situation \\ref{situation-relative-proj}. The functor $F_d$ is representable by a scheme."}
{"_id": "9260", "text": "models-lemma-torsion-embeds In Situation \\ref{situation-regular-model} let $h$ be an integer prime to the characteristic of $k$. Then the map $$ \\Pic(X)[h] \\longrightarrow \\Pic((X_k)_{red})[h] $$ is injective."}
{"_id": "1105", "text": "algebra-lemma-limit-finite-presentation Suppose $R \\to S$ is a ring map. Assume that $S$ is of finite presentation 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 $S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$ is an isomorphism. \\end{enumerate}"}
{"_id": "6068", "text": "flat-lemma-finite-type-flat-pure-is-universal Let $f : X \\to S$ be a finite type morphism of schemes. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Assume $\\mathcal{F}$ is flat over $S$. In this case $\\mathcal{F}$ is pure relative to $S$ if and only if $\\mathcal{F}$ is universally pure relative to $S$."}
{"_id": "8959", "text": "stacks-lemma-inertia Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ and $p' : \\mathcal{S}' \\to \\mathcal{C}$ be stacks over the site $\\mathcal{C}$. Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of stacks over $\\mathcal{C}$. \\begin{enumerate} \\item The inertia $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and $\\mathcal{I}_\\mathcal{S}$ are stacks over $\\mathcal{C}$. \\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in groupoids over $\\mathcal{C}$, then so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and $\\mathcal{I}_\\mathcal{S}$. \\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in setoids over $\\mathcal{C}$, then so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and $\\mathcal{I}_\\mathcal{S}$. \\end{enumerate}"}
{"_id": "9385", "text": "spaces-descent-lemma-descending-property-universally-submersive The property $\\mathcal{P}(f) =$``$f$ is universally submersive'' is fpqc local on the base."}
{"_id": "10437", "text": "more-algebra-lemma-additivity-of-pd Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal and let $E$ be a nonzero module over $R/I$. If $R/I$ has finite projective dimension and $E$ has finite projective dimension over $R/I$, then $E$ has finite projective dimension over $R$ and $$ \\text{pd}_R(E) = \\text{pd}_R(R/I) + \\text{pd}_{R/I}(E) $$"}
{"_id": "8893", "text": "stacks-properties-lemma-quasi-compact-finite-subcover Let $\\mathcal X$ be an algebraic stack and $\\mathcal X' \\subset \\mathcal X$ a quasi-compact open substack. Suppose that we have a collection of open substacks $\\mathcal{X}_i \\subset \\mathcal X$  indexed by $i \\in I$ such that $\\mathcal{X}' \\subset \\bigcup_{i \\in I} \\mathcal{X}_i$, where we define the union as in Lemma \\ref{lemma-union-open-substacks}. Then there exists a finite subset $I' \\subset I$ such that $\\mathcal{X}' \\subset \\bigcup_{i \\in I'} \\mathcal{X}_i$."}
{"_id": "12931", "text": "spaces-divisors-lemma-fitting-ideal-generate-locally 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 $x \\in |X|$. Then $\\mathcal{F}$ can be generated by $r$ elements in an \\'etale neighbourhood of $x$ if and only if $\\text{Fit}_r(\\mathcal{F})_{\\overline{x}} = \\mathcal{O}_{X, \\overline{x}}$."}
{"_id": "8478", "text": "algebraic-lemma-fully-faithful Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$. Let $S$ be an object of $\\Sch_{fppf}$. Denote $\\textit{Algebraic-Stacks}/S$ the $2$-category of algebraic stacks over $S$ defined using $\\Sch_{fppf}$. Similarly, denote $\\textit{Algebraic-Stacks}'/S$ the $2$-category of algebraic stacks over $S$ defined using $\\Sch'_{fppf}$. The rule $\\mathcal{X} \\mapsto f^{-1}\\mathcal{X}$ of Lemma \\ref{lemma-change-big-site} defines a functor of $2$-categories $$ \\textit{Algebraic-Stacks}/S \\longrightarrow \\textit{Algebraic-Stacks}'/S $$ which defines equivalences of morphism categories $$ \\Mor_{\\textit{Algebraic-Stacks}/S}(\\mathcal{X}, \\mathcal{Y}) \\longrightarrow \\Mor_{\\textit{Algebraic-Stacks}'/S}(f^{-1}\\mathcal{X}, f^{-1}\\mathcal{Y}) $$ for every objects $\\mathcal{X}, \\mathcal{Y}$ of $\\textit{Algebraic-Stacks}/S$. An object $\\mathcal{X}'$ of $\\textit{Algebraic-Stacks}'/S$ is equivalence to $f^{-1}\\mathcal{X}$ for some $\\mathcal{X}$ in $\\textit{Algebraic-Stacks}/S$ if and only if it has a presentation $\\mathcal{X} = [U'/R']$ with $U', R'$ isomorphic to $f^{-1}U$, $f^{-1}R$ for some $U, R \\in \\textit{Spaces}/S$."}
{"_id": "7365", "text": "sdga-lemma-qis-equivalence Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $\\varphi : \\mathcal{A} \\to \\mathcal{B}$ is a homomorphism of differential graded $\\mathcal{O}$-algebras which induces an isomorphism on cohomology sheaves, then $$ D(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}), \\quad \\mathcal{M} \\longmapsto \\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{B} $$ is an equivalence of categories."}
{"_id": "7261", "text": "spaces-chow-lemma-gysin-bivariant 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}. 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": "4762", "text": "spaces-morphisms-lemma-immersion-permanence Let $S$ be a scheme. Let $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$. \\begin{enumerate} \\item If $Z \\to X$ is representable, locally of finite type, locally quasi-finite, separated, and a monomorphism, then $Z \\to Y$ is representable, locally of finite type, locally quasi-finite, separated, and a monomorphism. \\item If $Z \\to X$ is an immersion and $Y \\to X$ is locally separated, then $Z \\to Y$ is an 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": "3035", "text": "properties-lemma-push-sections-annihilated-by-ideal Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of schemes. 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": "572", "text": "algebra-lemma-perfect A field $k$ is perfect if and only if it is a field of characteristic $0$ or a field of characteristic $p > 0$ such that every element has a $p$th root."}
{"_id": "4690", "text": "stacks-geometry-lemma-monomorphing-a-component-in-of-the-right-dimension Let $\\mathcal{T} \\hookrightarrow \\mathcal{X}$ be a locally of finite type monomorphism of algebraic stacks, with $\\mathcal{X}$ (and thus also $\\mathcal{T}$) being Jacobson, pseudo-catenary, and locally Noetherian. Suppose further that $\\mathcal{T}$ is irreducible of some (finite) dimension $d$, and that $\\mathcal{X}$ is reduced and of dimension less than or equal to $d$. Then there is a non-empty open substack $\\mathcal{V}$ of $\\mathcal{T}$ such that the induced monomorphism $\\mathcal{V} \\hookrightarrow \\mathcal{X}$ is an open immersion which identifies $\\mathcal{V}$ with an open subset of an irreducible component of $\\mathcal{X}$."}
{"_id": "4176", "text": "stacks-cohomology-proposition-smooth-covering-compute-direct-image Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$ be composable morphisms of algebraic stacks. Assume that \\begin{enumerate} \\item $f$ is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and \\item $g$ is quasi-compact and quasi-separated. \\end{enumerate} If $\\mathcal{F}$ is in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ then there is a spectral sequence $$ E_2^{p, q} = R^q(g \\circ f_p)_{\\QCoh, *}f_p^*\\mathcal{F} \\Rightarrow R^{p + q}g_{\\QCoh, *}\\mathcal{F} $$ in $\\QCoh(\\mathcal{O}_\\mathcal{Y})$."}
{"_id": "2133", "text": "cohomology-lemma-compose-cup-product Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and $g : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$ be morphisms of ringed spaces. The relative cup product of Remark \\ref{remark-cup-product} is compatible with compositions in the sense that the diagram $$ \\xymatrix{ R(g \\circ f)_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} R(g \\circ f)_*L \\ar@{=}[rr] \\ar[d] & & Rg_*Rf_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Rg_*Rf_*L \\ar[d] \\\\ R(g \\circ f)_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar@{=}[r] & Rg_*Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) & Rg_*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}  Rf_*L) \\ar[l] } $$ is commutative in $D(\\mathcal{O}_Z)$ for all $K, L$ in $D(\\mathcal{O}_X)$."}
{"_id": "10399", "text": "more-algebra-lemma-eta-second-property 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 map of complexes $$ \\eta_fM^\\bullet \\otimes_A A/fA \\longrightarrow H^\\bullet(M^\\bullet/f) $$ which is a quasi-isomorphism where the right hand side is as constructed above."}
{"_id": "13553", "text": "duality-lemma-pseudo-functor In Situation \\ref{situation-shriek} the constructions of Lemmas \\ref{lemma-shriek-well-defined} and \\ref{lemma-upper-shriek-composition} define a pseudo functor from the category $\\textit{FTS}_S$ into the $2$-category of categories (see Categories, Definition \\ref{categories-definition-functor-into-2-category})."}
{"_id": "10876", "text": "spaces-pushouts-lemma-derived-equivalent In Situation \\ref{situation-formal-glueing} the functor $Rf_*$ induces an equivalence between $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$ and $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$ with quasi-inverse given by $Lf^*$."}
{"_id": "11971", "text": "intersection-lemma-multiplicity-as-a-sum Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of definition. Let $M$ be a finite $A$-module. Let $d \\geq \\dim(\\text{Supp}(M))$. Then $$ e_I(M, d) = \\sum \\text{length}_{A_\\mathfrak p}(M_\\mathfrak p) e_I(A/\\mathfrak p, d) $$ where the sum is over primes $\\mathfrak p \\subset A$ with $\\dim(A/\\mathfrak p) = d$."}
{"_id": "6269", "text": "curves-lemma-plane-curve 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 $Z$ is a curve if and only if $F$ is irreducible."}
{"_id": "275", "text": "spaces-more-morphisms-lemma-base-change-nodal The property of being at-worst-nodal of relative dimension $1$ is preserved under base change."}
{"_id": "8761", "text": "examples-defos-lemma-spaces-hull In Example \\ref{example-spaces} assume $X$ is a proper algebraic space over $k$. 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": "11467", "text": "obsolete-lemma-pullback-K-flat Let $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be a ringed topos. For any complex of $\\mathcal{O}_\\mathcal{C}$-modules $\\mathcal{G}^\\bullet$ there exists a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$ such that $f^*\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}_\\mathcal{D}$-modules for any morphism $f : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ of ringed topoi."}
{"_id": "14014", "text": "more-morphisms-lemma-perfect-NL-lci Let $f : X \\to Y$ be a perfect morphism of locally Noetherian schemes. The following are equivalent \\begin{enumerate} \\item $f$ is a local complete intersection morphism, \\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": "3159", "text": "quot-lemma-extend-isom-to-spaces In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$. We have $$ \\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Isom}(\\mathcal{F}, \\mathcal{G})) = \\{(h, u) \\mid h : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\text{ isomorphism}\\} $$ 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": "199", "text": "spaces-more-morphisms-lemma-chow-finite-type 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 separated morphism of finite type. Then there exists a commutative diagram $$ \\xymatrix{ X \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\overline{X}' \\ar[ld] \\\\ & Y } $$ where $X' \\to X$ is proper surjective, $X' \\to \\overline{X}'$ is an open immersion, and $\\overline{X}' \\to Y$ is proper and representable morphism of algebraic spaces."}
{"_id": "6652", "text": "etale-cohomology-lemma-check-zar In the situation above let $K$ be an object of $D^+(X_{affine, chaotic})$. Then $K$ is in the essential image of the (fully faithful) functor $R\\epsilon_* ; D(X_{affine, Zar}) \\to D(X_{affine, chaotic})$ if and only if the following two conditions hold \\begin{enumerate} \\item $R\\Gamma(\\emptyset, K)$ is zero in $D(\\textit{Ab})$, and \\item if $U = V \\cup W$ with $U, V, W \\subset X$ affine open and $V, W \\subset U$ standard open (Algebra, Definition \\ref{algebra-definition-Zariski-topology}), then the map $c^K_{U, V, W, V \\cap W}$ of Cohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square} is a quasi-isomorphism. \\end{enumerate}"}
{"_id": "13742", "text": "more-morphisms-lemma-lifting-along-artinian Let $f : X \\to S$ be a morphism of schemes. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent: \\begin{enumerate} \\item $f$ is smooth, \\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 small extension of Artinian local rings and $\\beta$ of finite type (!) there exists a dotted arrow making the diagram commute. \\end{enumerate}"}
{"_id": "11234", "text": "cotangent-lemma-etale-localization 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 $L_{X/Y}|_{U_\\etale} = L_{U/V}$ in $D(\\mathcal{O}_U)$."}
{"_id": "9637", "text": "groupoids-lemma-action-groupoid-modules Let $S$ be a scheme. Let $Y$ be a scheme over $S$. Let $(G, m)$ be a group scheme over $Y$. Let $X$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$ be an action of $G$ on $X$ over $Y$. Let $(U, R, s, t, c)$ be the groupoid scheme 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": "7403", "text": "stacks-morphisms-lemma-section-immersion Let $f : \\mathcal{X} \\to \\mathcal{T}$ be a morphism of algebraic stacks. Let $s : \\mathcal{T} \\to \\mathcal{X}$ be a morphism such that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{T}$. Then \\begin{enumerate} \\item $s$ is representable by algebraic spaces and locally of finite type, \\item if $f$ is DM, then $s$ is unramified, \\item if $f$ is quasi-DM, then $s$ is locally quasi-finite, \\item if $f$ is separated, then $s$ is proper, and \\item if $f$ is quasi-separated, then $s$ is quasi-compact and quasi-separated. \\end{enumerate}"}
{"_id": "2340", "text": "restricted-lemma-composition-flat 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 flat, then so is $g \\circ f$."}
{"_id": "9168", "text": "examples-stacks-lemma-stack-fg-quasi-coherent There exists a subcategory $\\QCohstack_{fg, small} \\subset \\QCohstack_{fg}$ with the following properties: \\begin{enumerate} \\item the inclusion functor $\\QCohstack_{fg, small} \\to \\QCohstack_{fg}$ is fully faithful and essentially surjective, and \\item the functor $p_{fg, small} : \\QCohstack_{fg, small} \\to (\\Sch/S)_{fppf}$ turns $\\QCohstack_{fg, small}$ into a stack over $(\\Sch/S)_{fppf}$. \\end{enumerate}"}
{"_id": "4774", "text": "spaces-morphisms-lemma-scheme-theoretic-intersection Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \\subset X$ be closed subspaces. 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 algebraic spaces over $S$, i.e., $Z \\cap Y = Z \\times_X Y$."}
{"_id": "6708", "text": "etale-cohomology-proposition-check-h 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 (\\ref{equation-almost-blow-up-square}) in $(\\Sch/S)_h$ with $X$ affine."}
{"_id": "7552", "text": "stacks-morphisms-lemma-composition-unramified The composition of unramified morphisms is unramified."}
{"_id": "13749", "text": "more-morphisms-lemma-NL-formally-etale 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": "7345", "text": "sdga-lemma-kernel-localization In Definition \\ref{definition-derived-category} the kernel of the localization functor $Q : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to D(\\mathcal{A}, \\text{d})$ is the category $\\text{Ac}$ of Lemma \\ref{lemma-acyclics}."}
{"_id": "4386", "text": "sites-cohomology-lemma-tensor-perfect Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $K, L$ are perfect objects of $D(\\mathcal{O})$, then so is $K \\otimes_\\mathcal{O}^\\mathbf{L} L$."}
{"_id": "8166", "text": "spaces-lemma-quotient-field-map Notation and assumptions as in Lemma \\ref{lemma-quotient}. If $\\Spec(k) \\to U/G$ is a morphism, then there exist \\begin{enumerate} \\item a finite Galois extension $k'/k$, \\item a finite subgroup $H \\subset G$, \\item an isomorphism $H \\to \\text{Gal}(k'/k)$, and \\item an $H$-equivariant morphism $\\Spec(k') \\to U$. \\end{enumerate} Conversely, such data determine a morphism $\\Spec(k) \\to U/G$."}
{"_id": "6399", "text": "etale-cohomology-lemma-site-fpqc The collection of fpqc coverings on the category of schemes satisfies the axioms of site."}
{"_id": "1214", "text": "algebra-lemma-smooth-descends-through-colimit Let $A = \\colim A_i$ be a filtered colimit of rings. Let $A \\to B$ be a smooth ring map. There exists an $i$ and a smooth ring map $A_i \\to B_i$ such that $B = B_i \\otimes_{A_i} A$."}
{"_id": "4404", "text": "sites-cohomology-lemma-perfect-is-compact Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume $\\mathcal{C}$ has the following properties \\begin{enumerate} \\item $\\mathcal{C}$ has a quasi-compact final object $X$, \\item every quasi-compact object of $\\mathcal{C}$ has a cofinal system of coverings which are finite and consist of quasi-compact objects, \\item for a finite covering $\\{U_i \\to U\\}_{i \\in I}$ with $U$, $U_i$ quasi-compact the fibre products $U_i \\times_U U_j$ are quasi-compact. \\end{enumerate} Let $K$ be a perfect object of $D(\\mathcal{O})$. Then \\begin{enumerate} \\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O})$ 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 $(\\mathcal{C}, \\mathcal{O})$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$ for any $\\mathcal{O}$-module $\\mathcal{F}$, then $K$ is a compact object of $D(\\mathcal{O})$. \\end{enumerate}"}
{"_id": "14104", "text": "more-morphisms-proposition-vanishing-affine-stratification-number Let $X$ be a nonempty quasi-compact and quasi-separated scheme with affine stratification number $n$. Then $H^p(X, \\mathcal{F}) = 0$, $p > n$ for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$."}
{"_id": "9555", "text": "decent-spaces-lemma-universally-catenary-scheme Let $S$ be a locally Noetherian and universally catenary scheme. Let $X$ be an algebraic space over $S$ such that $X$ is decent and such that the structure morphism $X \\to S$ is locally of finite type. Then $X$ is catenary."}
{"_id": "6299", "text": "curves-lemma-geometric-genus-normalization Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$. Then \\begin{enumerate} \\item We have $g_{geom}(X/k) = g_{geom}(X_{red}/k)$. \\item If $X' \\to X$ is a birational proper morphism, then $g_{geom}(X'/k) = g_{geom}(X/k)$. \\item If $X^\\nu \\to X$ is the normalization morphism, then $g_{geom}(X^\\nu/k) = g_{geom}(X/k)$. \\end{enumerate}"}
{"_id": "14225", "text": "sites-modules-lemma-pullback-invertible Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi. The pullback $f^*\\mathcal{L}$ of an invertible $\\mathcal{O}_\\mathcal{D}$-module is invertible."}
{"_id": "734", "text": "algebra-lemma-weak-post-bourbaki 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 $$ \\text{WeakAss}_S(N) = \\text{WeakAss}_{S \\otimes_R K}(N \\otimes_R K) $$ via the canonical inclusion $\\Spec(S \\otimes_R K) \\subset \\Spec(S)$."}
{"_id": "8460", "text": "algebraic-lemma-open-fibred-category-is-full 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 and a monomorphism (see Definition \\ref{definition-relative-representable-property} and Descent on Spaces, Lemma \\ref{spaces-descent-lemma-descending-property-monomorphism}). Then $j$ is fully faithful on fibre categories."}
{"_id": "12912", "text": "spaces-divisors-lemma-weakass-reduced Let $S$ be a scheme. Let $X$ be a reduced algebraic space over $S$. Then the weakly associated point of $X$ are exactly the codimension $0$ points of $X$."}
{"_id": "14222", "text": "sites-modules-lemma-towards-constructible-when-serre-subcategory 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. Let $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ be the full subcategory of modules isomorphic to a cokernel as in (\\ref{equation-towards-constructible}). If the kernel of every map of $\\mathcal{O}$-modules of the form $$ \\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i} $$ with $U_i$ and $V_j$ in $\\mathcal{B}$, is in $\\mathcal{A}$, then $\\mathcal{A}$ is weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$."}
{"_id": "7524", "text": "stacks-morphisms-lemma-noetherian-singleton-stack-gerbe Let $\\mathcal{Z}$ be a reduced, locally Noetherian algebraic stack such that $|\\mathcal{Z}|$ is a singleton. Then $\\mathcal{Z}$ is a gerbe over a reduced, locally Noetherian algebraic space $Z$ with $|Z|$ a singleton."}
{"_id": "5166", "text": "morphisms-lemma-surjection-from-quasi-compact Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd]_p & & Y \\ar[dl]^q \\\\ & Z } $$ be a commutative diagram of morphisms of schemes. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact."}
{"_id": "83", "text": "spaces-more-morphisms-lemma-universal-thickening-fibre-product-flat 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 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": "4184", "text": "sites-cohomology-lemma-h1-mod-ab-agree Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$. Let $\\mathcal{F}_{ab}$ denote the underlying sheaf of abelian groups. Then there is a functorial isomorphism $$ H^1(\\mathcal{C}, \\mathcal{F}_{ab}) = H^1(\\mathcal{C}, \\mathcal{F}) $$ where 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": "3203", "text": "quot-lemma-polarized-tangent-space Let $k$ be a field and let $x = (X \\to \\Spec(k), \\mathcal{L})$ be an object of $\\mathcal{X} = \\Polarizedstack$ 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": "10700", "text": "etale-lemma-finitely-many-maps-to-unramified Let $S$ be a Noetherian scheme. Let $X \\to S$ be a quasi-compact unramified morphism. Let $Y \\to S$ be a morphism with $Y$ Noetherian. Then $\\Mor_S(Y, X)$ is a finite set."}
{"_id": "6926", "text": "stacks-more-morphisms-lemma-keel-mori-proper Let $p : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack to an algebraic space. Assume \\begin{enumerate} \\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite, \\item $p$ is proper, and \\item $Y$ is locally Noetherian. \\end{enumerate} Let $f : \\mathcal{X} \\to M$ be the moduli space constructed in Theorem \\ref{theorem-keel-mori}. Then $M \\to Y$ is proper."}
{"_id": "12654", "text": "constructions-lemma-projective-space-grassmannian Let $n \\geq 1$. There is a canonical isomorphism $\\mathbf{G}(n, n + 1) = \\mathbf{P}^n_\\mathbf{Z}$."}
{"_id": "14011", "text": "more-morphisms-lemma-lci-to-regular Let $$ \\xymatrix{ X \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\ & S } $$ be a commutative diagram of morphisms of schemes. Assume $S$ is locally Noetherian, $Y \\to S$ is locally of finite type, $Y$ is regular, and $X \\to S$ is a local complete intersection morphism. Then $f : X \\to Y$ is a local complete intersection morphism and $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$."}
{"_id": "6910", "text": "stacks-more-morphisms-lemma-check-separated-dvr Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume \\begin{enumerate} \\item $\\mathcal{Y}$ is locally Noetherian, \\item $f$ is locally of finite type and quasi-separated, \\item for every commutative diagram $$ \\xymatrix{ \\Spec(K) \\ar[r]_x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\ \\Spec(A) \\ar[r]^y \\ar@{-->}[ru] & \\mathcal{Y} } $$ where $A$ is a discrete valuation ring and $K$ its fraction field and any $2$-arrow $\\gamma : y \\circ j \\to f \\circ x$ the category of dotted arrows (Morphisms of Stacks, Definition \\ref{stacks-morphisms-definition-fill-in-diagram}) is either empty or a setoid with exactly one isomorphism class. \\end{enumerate} Then $f$ is separated."}
{"_id": "8493", "text": "sites-theorem-L-topology Let $\\mathcal{C}$ be a category. Let $J$ be a topology on $\\mathcal{C}$. Let $\\mathcal{F}$ be a presheaf of sets. \\begin{enumerate} \\item The presheaf $L\\mathcal{F}$ is separated. \\item If $\\mathcal{F}$ is separated, then $L\\mathcal{F}$ is a sheaf and the map of presheaves $\\mathcal{F} \\to L\\mathcal{F}$ is injective. \\item If $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to L\\mathcal{F}$ is an isomorphism. \\item The presheaf $LL\\mathcal{F}$ is always a sheaf. \\end{enumerate}"}
{"_id": "1861", "text": "derived-lemma-morphisms-equal-up-to-homotopy-projective Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \\xymatrix{ K^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\ P^\\bullet \\ar[u] \\ar@{-->}[ru]_{\\beta_i} } $$ where $P^\\bullet$ is bounded above and consists of projective objects, and $\\alpha$ is a quasi-isomorphism. Any two morphisms $\\beta_1, \\beta_2$ making the diagram commute up to homotopy are homotopic."}
{"_id": "9020", "text": "spaces-simplicial-lemma-augmentation-pushforward-higher-direct-image Let $X$ be a simplicial space and let $a : X \\to Y$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf on $X_{Zar}$. Then $R^na_*\\mathcal{F}$ is the sheaf associated to the presheaf $$ V \\longmapsto H^n((X \\times_Y V)_{Zar}, \\mathcal{F}|_{(X \\times_Y V)_{Zar}}) $$"}
{"_id": "12001", "text": "intersection-lemma-pullback-and-intersection-product Let $f : X \\to Y$ be a morphism of nonsingular projective varieties. The pullback map on chow groups satisfies: \\begin{enumerate} \\item $f^* : \\CH^*(Y) \\to \\CH^*(X)$ is a ring map, \\item $(g \\circ f)^* = f^* \\circ g^*$ for a composable pair $f, g$, \\item the projection formula holds: $f_*(\\alpha) \\cdot \\beta = f_*( \\alpha \\cdot f^*\\beta)$, and \\item if $f$ is flat then it agrees with the previous definition. \\end{enumerate}"}
{"_id": "2994", "text": "properties-lemma-locally-universally-Japanese Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is universally Japanese. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is universally Japanese. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is universally Japanese. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is universally Japanese. \\end{enumerate} Moreover, if $X$ is universally Japanese then every open subscheme is universally Japanese."}
{"_id": "3776", "text": "proetale-lemma-points-proetale Let $S$ be a scheme. The pro-\\'etale sites $\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and $(\\textit{Aff}/S)_\\proetale$ have enough points."}
{"_id": "14027", "text": "more-morphisms-lemma-go-down Let $X \\to Y \\to Z$ be morphisms of schemes. Assume that $X \\to Y$ is flat and surjective and that $X \\to X \\times_Z X$ is flat. Then $Y \\to Y \\times_Z Y$ is flat."}
{"_id": "10390", "text": "more-algebra-lemma-reduced-derived-complete-complete Let $A$ be a reduced ring derived complete with respect to a finitely generated ideal $I$. Then $A$ is $I$-adically complete."}
{"_id": "5497", "text": "morphisms-lemma-alteration-dimension-general Let $f : X \\to Y$ be a morphism of schemes. Assume that $Y$ is locally Noetherian and $f$ is locally of finite type. Then $$ \\dim(X) \\leq \\dim(Y) + E $$ where $E$ is the supremum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$ where $\\xi$ runs through the generic points of the irreducible components of $X$."}
{"_id": "10135", "text": "more-algebra-lemma-factor-through-K-flat Let $R$ be a ring. Let $a : K^\\bullet \\to L^\\bullet$ be a map of complexes of $R$-modules. If $K^\\bullet$ is K-flat, then there exist a complex $N^\\bullet$ and maps of complexes $b : K^\\bullet \\to N^\\bullet$ and $c : N^\\bullet \\to L^\\bullet$ such that \\begin{enumerate} \\item $N^\\bullet$ is K-flat, \\item $c$ is a quasi-isomorphism, \\item $a$ is homotopic to $c \\circ b$. \\end{enumerate} If the terms of $K^\\bullet$ are flat, then we may choose $N^\\bullet$, $b$, and $c$ such that the same is true for $N^\\bullet$."}
{"_id": "7561", "text": "stacks-morphisms-lemma-composition-proper A composition of proper morphisms is proper."}
{"_id": "13386", "text": "defos-lemma-verify-iv In Situation \\ref{situation-ses-flat-thickenings} the modules $\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules flat over $S'_1$ restricting to $\\mathcal{F}$ on $X$. Their difference (Lemma \\ref{lemma-flat}) is an element $\\theta$ of $\\Ext^1_{\\mathcal{O}_X}( \\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$ whose boundary in $\\Ext^2_{\\mathcal{O}_X}( \\mathcal{F}, f^*\\mathcal{J}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$ equals the obstruction (Lemma \\ref{lemma-flat}) to lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over $S'_3$."}
{"_id": "7553", "text": "stacks-morphisms-lemma-base-change-unramified A base change of an unramified morphism is unramified."}
{"_id": "11442", "text": "obsolete-theorem-coherent-algebraic-general 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 $\\textit{Coh}_{X/B}$ is an algebraic stack over $S$."}
{"_id": "5667", "text": "chow-lemma-delta-is-dimension Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Assume in addition $S$ is a Jacobson scheme, and $\\delta(s) = 0$ for every closed point $s$ of $S$. Let $X$ be locally of finite type over $S$. Let $Z \\subset X$ be an integral closed subscheme and let $\\xi \\in Z$ be its generic point. The following integers are the same: \\begin{enumerate} \\item $\\delta_{X/S}(\\xi)$, \\item $\\dim(Z)$, and \\item $\\dim(\\mathcal{O}_{Z, z})$ where $z$ is a closed point of $Z$. \\end{enumerate}"}
{"_id": "120", "text": "spaces-more-morphisms-lemma-lifting-along-artinian 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. The following are equivalent: \\begin{enumerate} \\item $f$ is smooth, \\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 small extension of Artinian local rings and $\\beta$ of finite type (!) there exists a dotted arrow making the diagram commute. \\end{enumerate}"}
{"_id": "3892", "text": "formal-spaces-lemma-reduction-fibre-products Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms of formal algebraic spaces over $S$. Then $(X \\times_Z Y)_{red} = (X_{red} \\times_{Z_{red}} Y_{red})_{red}$."}
{"_id": "12951", "text": "spaces-divisors-lemma-complement-open-affine-effective-cartier-divisor Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. Let $U \\subset X$ be a dense affine open. Then there exists an effective Cartier divisor $D \\subset X$ with $U = X \\setminus D$."}
{"_id": "3788", "text": "proetale-lemma-compare-locally-constant-derived Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. Let $D_{flc}(X_\\etale, \\Lambda)$, resp.\\ $D_{flc}(X_\\proetale, \\Lambda)$ be the full subcategory of $D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$ consisting of those complexes whose cohomology sheaves are locally constant sheaves of $\\Lambda$-modules of finite type. Then $$ \\epsilon^{-1} : D_{flc}^+(X_\\etale, \\Lambda) \\longrightarrow D_{flc}^+(X_\\proetale, \\Lambda) $$ is an equivalence of categories."}
{"_id": "4633", "text": "spaces-limits-lemma-maximal-ideal Let $\\varphi : X \\to \\Spec(A)$ be a quasi-compact and quasi-separated morphism from an algebraic space to an affine scheme. If $X$ is not a scheme, then there exists an ideal $I \\subset A$ such that the base change $X_{A/I}$ is not a scheme, but for every $I \\subset I'$, $I \\not = I'$ the base change $X_{A/I'}$ is a scheme."}
{"_id": "4885", "text": "spaces-morphisms-lemma-syntomic-open A syntomic morphism is universally open."}
{"_id": "2526", "text": "examples-lemma-Noetherian-Jacobson There exists a Jacobson, universally catenary, Noetherian domain $B$ with maximal ideals $\\mathfrak m_1, \\mathfrak m_2$ such that $\\dim(B_{\\mathfrak m_1}) = 1$ and $\\dim(B_{\\mathfrak m_2}) = 2$."}
{"_id": "9122", "text": "spaces-simplicial-lemma-quasi-coherent-groupoid-R-cartesian Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Let $X$ be the simplicial scheme over $S$ constructed in Lemma \\ref{lemma-groupoid-simplicial}. Let $(R/U)_\\bullet$ be the simplicial scheme associated to $s : R \\to U$, see Definition \\ref{definition-fibre-products-simplicial-scheme}. There exists a cartesian morphism $t_\\bullet : (R/U)_\\bullet \\to X$ of simplicial schemes with low degree morphisms given by $$ \\xymatrix{ R \\times_{s, U, s} R \\times_{s, U, s} R \\ar@<3ex>[r]_-{\\text{pr}_{12}} \\ar@<0ex>[r]_-{\\text{pr}_{02}} \\ar@<-3ex>[r]_-{\\text{pr}_{01}} \\ar[dd]_{(r_0, r_1, r_2) \\mapsto (r_0 \\circ r_1^{-1}, r_1 \\circ r_2^{-1})} & R \\times_{s, U, s} R \\ar@<1ex>[r]_-{\\text{pr}_1} \\ar@<-2ex>[r]_-{\\text{pr}_0} \\ar[dd]_{(r_0, r_1) \\mapsto r_0 \\circ r_1^{-1}} & R \\ar[dd]^t \\\\ \\\\ R \\times_{s, U, t} R \\ar@<3ex>[r]_{\\text{pr}_1} \\ar@<0ex>[r]_c \\ar@<-3ex>[r]_{\\text{pr}_0} & R \\ar@<1ex>[r]_s \\ar@<-2ex>[r]_t & U } $$"}
{"_id": "12317", "text": "categories-lemma-morphisms-representable-fibred-categories Let $\\mathcal{C}$ be a category. Let $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids over $\\mathcal{C}$. Assume that $\\mathcal{X}$, $\\mathcal{Y}$ are representable by objects $X$, $Y$ of $\\mathcal{C}$. Then $$ \\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{X}, \\mathcal{Y}) \\Big/ 2\\text{-isomorphism} = \\Mor_\\mathcal{C}(X, Y) $$ More precisely, given $\\phi : X \\to Y$ there exists a $1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ which induces $\\phi$ on isomorphism classes of objects and which is unique up to unique $2$-isomorphism."}
{"_id": "10108", "text": "more-algebra-lemma-quasi-excellent-nagata A quasi-excellent ring is Nagata."}
{"_id": "14495", "text": "sheaves-lemma-sheafify-universal Let $\\mathcal{F}$ be a presheaf of sets on $X$. Any map $\\mathcal{F} \\to \\mathcal{G}$ into a sheaf of sets factors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$."}
{"_id": "790", "text": "algebra-lemma-characterize-finite-projective-noetherian Let $R$ be a Noetherian ring. Let $P$ be a finite $R$-module. If $\\Ext^1_R(P, M) = 0$ for every finite $R$-module $M$, then $P$ is projective."}
{"_id": "10098", "text": "more-algebra-lemma-map-P-ring-to-completion-P Let $A$ be a $P$-ring where $P$ satisfies (B) and (D). Let $I \\subset A$ be an ideal and let $A^\\wedge$ be the completion of $A$ with respect to $I$. Then the fibres of $A \\to A^\\wedge$ have $P$."}
{"_id": "10728", "text": "etale-proposition-etale-depth Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then $\\text{depth}(A) = \\text{depth}(B)$"}
{"_id": "10888", "text": "spaces-pushouts-lemma-coequalizer-glue In Situation \\ref{situation-coequalizer-glue} let $Y = X' \\amalg Z$ and $R = Y \\times_X Y$ with projections $t, s : R \\to Y$. There exists a coequalizer $X_1$ of $s, t : R \\to Y$ in the category of algebraic spaces over $S$. The morphism $X_1 \\to X$ is a finite universal homeomorphism, an isomorphism over $U$, and $Z \\to X$ lifts to $X_1$."}
{"_id": "6992", "text": "perfect-lemma-pseudo-coherent-descends-fppf Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes. Let $E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each $Lf_i^*E$ is $m$-pseudo-coherent."}
{"_id": "13617", "text": "duality-lemma-relative-dualizing-composition Let $f : Y \\to X$ and $X \\to S$ be morphisms of schemes which are flat and of finite presentation. Let $(K, \\xi)$ and $(M, \\eta)$ be a relative dualizing complex for $X \\to S$ and $Y \\to X$. Set $E = M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lf^*K$. Then $(E, \\zeta)$ is a relative dualizing complex for $Y \\to S$ for a suitable $\\zeta$."}
{"_id": "9886", "text": "more-algebra-lemma-absolutely-integrally-closed-strictly-henselian Let $A$ be absolutely integrally closed. Let $\\mathfrak p \\subset A$ be a prime. Then the local ring $A_\\mathfrak p$ is strictly henselian."}
{"_id": "1820", "text": "derived-lemma-filtered-derived-functors The functors $\\text{gr}^p, \\text{gr}, (\\text{forget }F)$ induce canonical exact functors $$ \\text{gr}^p, \\text{gr}, (\\text{forget }F): DF(\\mathcal{A}) \\longrightarrow D(\\mathcal{A}) $$ which commute with the localization functors."}
{"_id": "10967", "text": "varieties-lemma-quasi-affine-after-field-extension Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-affine for some field extension $k \\subset K$, then $X$ is quasi-affine."}
{"_id": "13126", "text": "dga-lemma-rickard-rings Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. The following are equivalent \\begin{enumerate} \\item there is an $R$-linear equivalence $D(A) \\to D(B)$ of triangulated categories, \\item there exists an object $P$ of $D(B)$ such that \\begin{enumerate} \\item $P$ can be represented by a finite complex of finite projective $B$-modules, \\item if $K \\in D(B)$ with $\\Ext^i_B(P, K) = 0$ for $i \\in \\mathbf{Z}$, then $K = 0$, and \\item $\\Ext^i_B(P, P) = 0$ for $i \\not = 0$ and equal to $A$ for $i= 0$. \\end{enumerate} \\end{enumerate} Moreover, if $B$ is flat as an $R$-module, then this is also equivalent to \\begin{enumerate} \\item[(3)] there exists an $(A, B)$-bimodule $N$ such that $- \\otimes_A^\\mathbf{L} N : D(A) \\to D(B)$ is an equivalence. \\end{enumerate}"}
{"_id": "15123", "text": "limits-lemma-morphism-good-diagram-smooth Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}. If $f$ is smooth, then there exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have $f_i$ is smooth."}
{"_id": "7866", "text": "divisors-lemma-embedded Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then \\begin{enumerate} \\item the generic points of irreducible components of $\\text{Supp}(\\mathcal{F})$ are associated points of $\\mathcal{F}$, and \\item an associated point of $\\mathcal{F}$ is embedded if and only if it is not a generic point of an irreducible component of $\\text{Supp}(\\mathcal{F})$. \\end{enumerate} In particular an embedded point of $X$ is an associated point of $X$ which is not a generic point of an irreducible component of $X$."}
{"_id": "6148", "text": "flat-lemma-fibre-products-h Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$. The underlying categories of the sites $\\Sch_h$, $(\\Sch/S)_h$, and $(\\textit{Aff}/S)_h$ have fibre products. In each case the obvious functor into the category $\\Sch$ of all schemes commutes with taking fibre products. The category $(\\Sch/S)_h$ has a final object, namely $S/S$."}
{"_id": "7098", "text": "perfect-lemma-K-agrees-affine Let $X = \\Spec(R)$ be an affine scheme. Then $K_0(X) = K_0(R)$ and if $R$ is Noetherian then $K'_0(X) = K'_0(R)$."}
{"_id": "12916", "text": "spaces-divisors-lemma-weakly-ass-pullback 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 $x \\in |X|$ and $y = f(x) \\in |Y|$. If \\begin{enumerate} \\item $y \\in \\text{WeakAss}_S(\\mathcal{G})$, \\item $f$ is flat at $x$, and \\item the dimension of the local ring of the fibre of $f$ at $x$ is zero (Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dimension-fibre}), \\end{enumerate} then $x \\in \\text{WeakAss}(f^*\\mathcal{G})$."}
{"_id": "5044", "text": "weil-lemma-motives-monoidal The category $M_k$ with tensor product defined as above is symmetric monoidal with the obvious associativity and commutativity constraints and with unit $\\mathbf{1} = (\\Spec(k), 1, 0)$."}
{"_id": "6032", "text": "flat-lemma-finite-presentation-flat-at-point-X Let $f : X \\to S$ be locally of finite presentation. Let $x \\in X$ with image $s \\in S$. If $f$ is flat at $x$ over $S$, 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 $\\Gamma(X', \\mathcal{O}_{X'})$ is a projective $\\Gamma(S', \\mathcal{O}_{S'})$-module."}
{"_id": "11229", "text": "cotangent-lemma-find-obstruction-ringed-topoi In the situation above we have \\begin{enumerate} \\item There is a canonical element $\\xi \\in \\Ext^2_\\mathcal{O}(L_f, \\mathcal{G})$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (\\ref{equation-to-solve-ringed-topoi}). \\item If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\\Ext^1_\\mathcal{O}(L_f, \\mathcal{G})$. \\item Given a solution $X'$, the set of automorphisms of $X'$ fitting into (\\ref{equation-to-solve-ringed-topoi}) is canonically isomorphic to $\\Ext^0_\\mathcal{O}(L_f, \\mathcal{G})$. \\end{enumerate}"}
{"_id": "7914", "text": "divisors-lemma-isom-depth-2-torsion-free Let $X$ be an integral 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, $\\mathcal{G}$ is torsion free, 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 $\\text{depth}(\\mathcal{F}_x) \\geq 2$. \\end{enumerate} Then $\\varphi$ is an isomorphism."}
{"_id": "5436", "text": "morphisms-lemma-integral-local Let $f : X \\to S$ be a morphism of schemes. The following are equivalent: \\begin{enumerate} \\item The morphism $f$ is integral. \\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 integral. \\item There exists an open covering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is integral. \\end{enumerate} Moreover, if $f$ is integral then for every open subscheme $U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is integral."}
{"_id": "6478", "text": "etale-cohomology-lemma-compute-strangely 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 $X_{i'} \\to X_i$ and $S_{i'} \\to S_i$ 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 $\\mathcal{F}$ be an  abelian sheaf on $X$. Then we have $$ R^pg_*\\mathcal{F} = \\colim_{i \\in I} h_i^{-1}R^pg_{i, *}(f_{i, *}\\mathcal{F}) $$"}
{"_id": "11452", "text": "obsolete-lemma-not-domain Let $(R, \\mathfrak m)$ be a reduced Noetherian local ring of dimension $1$ and let $x \\in \\mathfrak m$ be a nonzerodivisor. Let $\\mathfrak q_1, \\ldots, \\mathfrak q_r$ be the minimal primes of $R$. Then $$ \\text{length}_R(R/(x)) = \\sum\\nolimits_i \\text{ord}_{R/\\mathfrak q_i}(x) $$"}
{"_id": "13587", "text": "duality-lemma-has-dualizing-module-CM-scheme Let $X$ be a locally Noetherian scheme. If there exists a coherent sheaf $\\omega_X$ such that $\\omega_X[0]$ is a dualizing complex on $X$, then $X$ is a Cohen-Macaulay scheme."}
{"_id": "7869", "text": "divisors-lemma-scheme-theoretically-dense-contain-embedded-points Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open subscheme. The following are equivalent \\begin{enumerate} \\item $U$ is scheme theoretically dense in $X$ (Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}), \\item $U$ is dense in $X$ and $U$ contains all embedded points of $X$. \\end{enumerate}"}
{"_id": "1249", "text": "algebra-lemma-etale-under-finite-flat Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism of local rings. Assume $S$ is the localization of an \\'etale ring extension of $R$. Then there exists a finite, finitely presented, faithfully flat ring map $R \\to S'$ such that for every maximal ideal $\\mathfrak m'$ of $S'$ there is a factorization $$ R \\to S \\to S'_{\\mathfrak m'}. $$ of the ring map $R \\to S'_{\\mathfrak m'}$."}
{"_id": "7422", "text": "stacks-morphisms-lemma-characterize-representable-quasi-compact 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 quasi-compact (as in Properties of Stacks, Section \\ref{stacks-properties-section-properties-morphisms}), and \\item for every quasi-compact algebraic stack $\\mathcal{Z}$ and any morphism $\\mathcal{Z} \\to \\mathcal{Y}$ the algebraic stack $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-compact. \\end{enumerate}"}
{"_id": "10844", "text": "spaces-pushouts-theorem-nagata \\begin{reference} \\cite{CLO} \\end{reference} Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X \\to B$ be a separated, finite type morphism. Then $X$ has a compactification over $B$."}
{"_id": "8246", "text": "topology-lemma-lift-covering-of-a-closed Let $X$ be a Hausdorff and locally quasi-compact space. Let $Z \\subset X$ be a quasi-compact (hence closed) subset. 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 we have $Z \\subset \\bigcup V_i$, for all $i$ we have $\\overline{V_i} \\subset U_i$, and we have $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$ for all $(p + 1)$-tuples $i_0, \\ldots, i_p$."}
{"_id": "1281", "text": "algebra-lemma-unramified-over-strictly-henselian Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring. Let $R \\to S$ be an unramified ring map. Then $$ S = A_1 \\times \\ldots \\times A_n \\times B $$ with each $R \\to A_i$ surjective and no prime of $B$ lying over $\\mathfrak m$."}
{"_id": "6601", "text": "etale-cohomology-lemma-base-change-f-star-field Consider the cartesian diagrams of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ Assume that $S$ is the spectrum of a separably closed field. Then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$ for any sheaf $\\mathcal{F}$ on $T_\\etale$."}
{"_id": "14091", "text": "more-morphisms-lemma-weighting-flat-quasi-finite Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is locally quasi-finite, locally of finite presentation, and flat. Then there is a positive weighting $w : X \\to \\mathbf{Z}_{> 0}$ of $f$ given by the rule that sends $x \\in X$ lying over $y \\in Y$ to $$ w(x) = \\text{length}_{\\mathcal{O}_{X, x}} (\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x}) [\\kappa(x) : \\kappa(y)]_i $$ where $[\\kappa' : \\kappa]_i$ is the inseparable degree (Fields, Definition \\ref{fields-definition-insep-degree})."}
{"_id": "6491", "text": "etale-cohomology-lemma-all-modules-quasi-coherent Let $R$ be a local ring of dimension $0$. Let $S = \\Spec(R)$. Then every $\\mathcal{O}_S$-module on $S_\\etale$ is quasi-coherent."}
{"_id": "4053", "text": "pione-lemma-exact-sequence-finite-nr-closed-pts Let $X$ be a scheme. Let $x_1, \\ldots, x_n \\in X$ be a finite number of closed points such that \\begin{enumerate} \\item $U = X \\setminus \\{x_1, \\ldots, x_n\\}$ is connected and is a retrocompact open of $X$, and \\item for each $i$ the punctured spectrum $U_i^{sh}$ of the strict henselization of $\\mathcal{O}_{X, x_i}$ is connected. \\end{enumerate} Then the map $\\pi_1(U) \\to \\pi_1(X)$ is surjective and the kernel is the smallest closed normal subgroup of $\\pi_1(U)$ containing the image of $\\pi_1(U_i^{sh}) \\to \\pi_1(U)$ for $i = 1, \\ldots, n$."}
{"_id": "3112", "text": "criteria-lemma-map-from-algebraic 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 $\\mathcal{X}$ is an algebraic stack and $\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$ is representable by algebraic spaces, then $F$ is algebraic."}
{"_id": "6598", "text": "etale-cohomology-lemma-check-stalks-better Let $f : X \\to S$ be a morphism of schemes. Let $\\overline{x}$ be a geometric point of $X$ with image $\\overline{s}$ in $S$. Let $\\Spec(K) \\to \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$ be a morphism with $K$ a separably closed field. Let $\\mathcal{F}$ be an abelian sheaf on $\\Spec(K)_\\etale$. Let $q \\geq 0$. The following are equivalent \\begin{enumerate} \\item $H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S \\Spec(K), \\mathcal{F}) = H^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S \\Spec(K), \\mathcal{F})$ \\item $H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_{\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})} \\Spec(K), \\mathcal{F}) = H^q(\\Spec(K), \\mathcal{F})$ \\end{enumerate}"}
{"_id": "9267", "text": "models-proposition-bound-picard-group Let $g \\geq 2$. For every numerical type $T$ of genus $g$ and prime number $\\ell > 768g$ we have $$ \\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g $$ where $\\Pic(T)$ is as in Definition \\ref{definition-picard-group}. If $T$ is minimal, then we even have $$ \\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g_{top} \\leq g $$ where $g_{top}$ as in Definition \\ref{definition-top-genus}."}
{"_id": "8051", "text": "divisors-lemma-conormal-sheaf-section-projective-bundle Let $X$ be a scheme. 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": "8739", "text": "examples-defos-lemma-rings-hull In Example \\ref{example-rings} assume $P$ is a finite type $k$-algebra such that $\\Spec(P) \\to \\Spec(k)$ is smooth except at a finite number of points. 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": "225", "text": "spaces-more-morphisms-lemma-flat-base-change-pseudo-coherent A flat base change of a pseudo-coherent morphism is pseudo-coherent."}
{"_id": "13042", "text": "dga-lemma-triangle-independent-splittings 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. The triangle \\begin{equation} \\label{equation-triangle-associated-to-admissible-ses} K \\to L \\to M \\xrightarrow{\\delta} K[1] \\end{equation} with $\\delta$ as in Lemma \\ref{lemma-admissible-ses} is, up to canonical isomorphism in $K(\\text{Mod}_{(A, \\text{d})})$, independent of the choices made in Lemma \\ref{lemma-admissible-ses}."}
{"_id": "9852", "text": "more-algebra-lemma-lift-section-smooth-morphism Let $A$ be a ring, let $I \\subset A$ be an ideal. Consider a commutative diagram $$ \\xymatrix{ B \\ar[rd] \\\\ A \\ar[u] \\ar[r] & A/I } $$ where $B$ is a smooth $A$-algebra. Then there exists an \\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an $A$-algebra map $B \\to A'$ lifting the ring map $B \\to A/I$."}
{"_id": "8832", "text": "more-etale-lemma-lqf-base-change-upper-shriek 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 $Y'_\\etale$ we have $(g')_*(f')^!\\mathcal{F} = f^!g_*\\mathcal{F}$."}
{"_id": "7589", "text": "stacks-morphisms-lemma-composition-lci The composition of local complete intersection morphisms is a local complete intersection."}
{"_id": "15125", "text": "limits-lemma-good-diagram-fibre-product 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 schemes quasi-separated and of finite type over $S$. For each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram $$ \\xymatrix{ X^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\ S \\ar[r] & S_{i_j} } $$ 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 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": "30", "text": "spaces-more-morphisms-lemma-conormal-algebra-functorial-flat 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 cartesian square of algebraic spaces over $S$ 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": "10032", "text": "more-algebra-lemma-formally-smooth-JZ 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. Let $K$ be the residue field of $B$. Then the Jacobi-Zariski sequence for $A \\to B \\to K$ gives an exact sequence $$ 0 \\to H_1(\\NL_{K/A}) \\to \\mathfrak m_B/\\mathfrak m_B^2 \\to \\Omega_{B/A} \\otimes_B K \\to \\Omega_{K/A} \\to 0 $$"}
{"_id": "13176", "text": "spaces-more-groupoids-lemma-property-invariant 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 $\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\\tau$-local on the target (Descent on Spaces, Definition \\ref{spaces-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 subspace such that $s^{-1}(W) \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant (Groupoids in Spaces, Definition \\ref{spaces-groupoids-definition-invariant-open})."}
{"_id": "3026", "text": "properties-lemma-application-directed-colimit Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be a quasi-compact open such that $\\mathcal{F}|_U$ is of finite presentation. Then there exists a map of $\\mathcal{O}_X$-modules $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ with (a) $\\mathcal{G}$ of finite presentation, (b) $\\varphi$ is surjective, and (c) $\\varphi|_U$ is an isomorphism."}
{"_id": "13006", "text": "spaces-divisors-lemma-strict-transform-universally-injective 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 $T \\to B$. Then the strict transforms of $\\mathcal{F}_i'$ relative to any blowup $B' \\to B$ form a short exact sequence $0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too."}
{"_id": "13859", "text": "more-morphisms-lemma-Noetherian-approximation-smooth Let $f : X \\to S$ be a morphism of affine schemes, which is smooth. Then there exists a diagram as in Lemma \\ref{lemma-Noetherian-approximation} such that in addition $f_0$ is smooth."}
{"_id": "4689", "text": "stacks-geometry-lemma-dimension-at-finite-type-point If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if $x \\in |\\mathcal{X}|$, then for any open substack $\\mathcal{V}$ of $\\mathcal{X}$ containing $x$, there is a finite type point $x_0 \\in |\\mathcal{V}|$ such that $\\dim_{x_0}(\\mathcal{X}) = \\dim_x(\\mathcal{V})$."}
{"_id": "9999", "text": "more-algebra-lemma-koszul-independence-presentation Let $A \\to B$ be a finite type ring map. If for some presentation $\\alpha : A[x_1, \\ldots, x_n] \\to B$ the kernel $I$ is a Koszul-regular ideal then for any presentation $\\beta : A[y_1, \\ldots, y_m] \\to B$ the kernel $J$ is a Koszul-regular ideal."}
{"_id": "10749", "text": "crystalline-lemma-flat-extension-divided-power-envelope Let $(B, I, \\gamma) \\to (B', I', \\gamma')$ be a homomorphism of divided power rings. Let $I \\subset J \\subset B$ and $I' \\subset J' \\subset B'$ be ideals. Assume \\begin{enumerate} \\item $B/I \\to B'/I'$ is flat, and \\item $J' = JB' + I'$. \\end{enumerate} Then the canonical map $$ D_{B, \\gamma}(J) \\otimes_B B' \\longrightarrow D_{B', \\gamma'}(J') $$ is an isomorphism."}
{"_id": "8343", "text": "topology-lemma-topological-module-limits Let $R$ be a topological ring. The category of topological modules over $R$ has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of $R$-modules."}
{"_id": "13609", "text": "duality-lemma-sanity-check-duality Let $X$ be a proper scheme over a field $k$. Let $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-proper-over-field}. \\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 $\\omega_X$ are as in Lemma \\ref{lemma-duality-proper-over-field} for the morphism $X \\to \\Spec(k')$. \\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-proper-over-field} for the morphism $X_K \\to \\Spec(K)$. \\end{enumerate}"}
{"_id": "3376", "text": "coherent-lemma-torsion-hom-ext Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Any object of $\\textit{Coh}(X, \\mathcal{I})$ which is annihilated by a power of $\\mathcal{I}$ is in the essential image of (\\ref{equation-completion-functor}). Moreover, if $\\mathcal{F}$, $\\mathcal{G}$ are in $\\textit{Coh}(\\mathcal{O}_X)$ and either $\\mathcal{F}$ or $\\mathcal{G}$ is annihilated by a power of $\\mathcal{I}$, then the maps $$ \\xymatrix{ \\Hom_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] & \\Ext_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] \\\\ \\Hom_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge) & \\Ext_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge) } $$ are isomorphisms."}
{"_id": "11807", "text": "spaces-duality-lemma-base-change-relative-dualizing Let $S$ be a scheme. Consider a cartesian square $$ \\xymatrix{ X' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\ Y' \\ar[r]^g & Y } $$ of algebraic spaces over $S$. Assume $X \\to Y$ is proper, flat, and of finite presentation. Let $(\\omega_{X/Y}^\\bullet, \\tau)$ be a relative dualizing complex for $f$. Then $(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$ is a relative dualizing complex for $f'$."}
{"_id": "3621", "text": "adequate-lemma-adequate-by-parasitic Let $S$ be a scheme. The subcategory $\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ of parasitic adequate modules is a Serre subcategory. Moreover, the functor $v$ induces an equivalence of categories $$ \\textit{Adeq}(\\mathcal{O}) / \\mathcal{C} = \\QCoh(\\mathcal{O}_S). $$"}
{"_id": "1772", "text": "derived-lemma-exact-functor-additive An exact functor of pre-triangulated categories is additive."}
{"_id": "11454", "text": "obsolete-lemma-flat-over-gorenstein-gorenstein-fibre Let $A \\to B$ be a flat local homomorphism of Noetherian local rings. If $A$ and $B/\\mathfrak m_A B$ are Gorenstein, then $B$ is Gorenstein."}
{"_id": "6101", "text": "flat-lemma-check-along-closed-fibre Let $S$ be a local scheme with closed point $s$. Let $f : X \\to S$ be 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": "230", "text": "spaces-more-morphisms-lemma-Noetherian-pseudo-coherent Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. If $Y$ is locally Noetherian, then $f$ is pseudo-coherent if and only if $f$ is locally of finite type."}
{"_id": "11799", "text": "spaces-duality-lemma-restriction-compare-with-pullback Suppose we have a diagram (\\ref{equation-base-change}). 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 on Sites, Remark \\ref{sites-cohomology-remark-base-change} and (\\ref{equation-base-change-map})."}
{"_id": "7323", "text": "sdga-lemma-dgm-grothendieck-abelian 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 a Grothendieck abelian category."}
{"_id": "8326", "text": "topology-lemma-one-point-compactification Let $X$ be a Hausdorff, locally quasi-compact space. There exists a map $X \\to X^*$ which identifies $X$ as an open subspace of a quasi-compact Hausdorff space $X^*$ such that $X^* \\setminus X$ is a singleton (one point compactification). In particular, the map $X \\to \\beta(X)$ identifies $X$ with an open subspace of $\\beta(X)$."}
{"_id": "11110", "text": "varieties-lemma-degree-and-det Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$ over $k$, and let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$. Then $$ \\deg(\\mathcal{E}) = \\deg(\\wedge^n(\\mathcal{E})) = \\deg(\\det(\\mathcal{E})) $$"}
{"_id": "11476", "text": "obsolete-lemma-cosk0-hom-deltak Let $\\mathcal{C}$ be a category. Let $X$ be an object of $\\mathcal{C}$ such that the self products $X \\times \\ldots \\times X$ exist. Let $k \\geq 0$ and let $C[k]$ be as in Simplicial, Example \\ref{simplicial-example-simplex-cosimplicial-set}. With notation as in Simplicial, Lemma \\ref{simplicial-lemma-morphism-into-product} the canonical map $$ \\Hom(C[k], X)_1 \\longrightarrow (\\text{cosk}_0 \\text{sk}_0 \\Hom(C[k], X))_1 $$ is identified with the map $$ \\prod\\nolimits_{\\alpha : [k] \\to [1]} X \\longrightarrow X \\times X $$ which is the projection onto the factors where $\\alpha$ is a constant map."}
{"_id": "4874", "text": "spaces-morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation 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 presentation. The open $$ W_n = \\{x \\in |X| \\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\} $$ of Lemma \\ref{lemma-openness-bounded-dimension-fibres} is retrocompact in $|X|$. (See Topology, Definition \\ref{topology-definition-quasi-compact}.)"}
{"_id": "9584", "text": "groupoids-lemma-flat-action-on-group-scheme Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $T$ be a scheme over $S$ and let $\\psi : T \\to G$ be a morphism over $S$. If $T$ is flat over $S$, then the morphism $$ T \\times_S G \\longrightarrow G, \\quad (t, g) \\longmapsto m(\\psi(t), g) $$ is flat. In particular, if $G$ is flat over $S$, then $m : G \\times_S G \\to G$ is flat."}
{"_id": "6595", "text": "etale-cohomology-lemma-base-change-compose Consider a tower of cartesian diagrams of schemes $$ \\xymatrix{ W \\ar[d]_i & Z \\ar[l]^j \\ar[d]^k \\\\ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ Let $K$ in $D(T_\\etale)$. If $$ f^{-1}Rg_*K \\to Rh_*e^{-1}K \\quad\\text{and}\\quad i^{-1}Rh_*e^{-1}K \\to Rj_*k^{-1}e^{-1}K $$ are isomorphisms, then $(f \\circ i)^{-1}Rg_*K \\to Rj_*(e \\circ k)^{-1}K$ is an isomorphism. Similarly, if $\\mathcal{F}$ is an abelian sheaf on $T_\\etale$ and if $$ f^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F} \\quad\\text{and}\\quad i^{-1}R^qh_*e^{-1}\\mathcal{F} \\to R^qj_*k^{-1}e^{-1}\\mathcal{F} $$ are isomorphisms, then $(f \\circ i)^{-1}R^qg_*\\mathcal{F} \\to R^qj_*(e \\circ k)^{-1}\\mathcal{F}$ is an isomorphism."}
{"_id": "4240", "text": "sites-cohomology-lemma-factor-through-K-flat Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. Let $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes of $\\mathcal{O}$-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": "5405", "text": "morphisms-lemma-quasi-projective-permanence Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes. If $g \\circ f$ is quasi-projective and $f$ is quasi-compact\\footnote{This follows if $g$ is quasi-separated by Schemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.}, then $f$ is quasi-projective."}
{"_id": "3659", "text": "spaces-topologies-lemma-morphism-big-small-cartesian-diagram-etale Let $S$ be a scheme. Consider a cartesian diagram $$ \\xymatrix{ Y' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\ X' \\ar[r]^g & X } $$ in $(\\textit{Spaces}/S)_\\etale$. 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": "116", "text": "spaces-more-morphisms-lemma-triangle-differentials-formally-smooth Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$. Assume $f$ is formally smooth. Then $$ 0 \\to f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0 $$ Lemma \\ref{lemma-triangle-differentials} is short exact."}
{"_id": "6831", "text": "equiv-lemma-bounded-fibres Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Let $K_0 \\to K_1 \\to K_2 \\to \\ldots$ be a system of objects of $D_{perf}(\\mathcal{O}_{X \\times Y})$ and $m \\geq 0$ an integer such that \\begin{enumerate} \\item $H^q(K_i)$ is nonzero only for $q \\leq m$, \\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$ the object $$ R\\text{pr}_{2, *}( \\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}^\\mathbf{L} K_n) $$ has vanshing cohomology sheaves in degrees outside $[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$. \\end{enumerate} Then $K_n$ has vanshing cohomology sheaves in degrees outside $[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$. Moreover, if $X$ and $Y$ are smooth over $k$, then for $n$ large enough we find $K_n = K \\oplus C_n$ in $D_{perf}(\\mathcal{O}_{X \\times Y})$ where $K$ has cohomology only indegrees $[-m, m]$ and $C_n$ only in degrees $[-m - n, m - n]$ and the transition maps define isomorphisms between various copies of $K$."}
{"_id": "194", "text": "spaces-more-morphisms-lemma-get-section-after-blowup Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $U \\subset W \\subset Y$ be open subspaces. 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 $Y$ is quasi-compact and quasi-separated, and \\item $U$ and $W$ are quasi-compact. \\end{enumerate} Then there exists a $U$-admissible blowup $b : Y' \\to Y$ and a morphism $s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$."}
{"_id": "9097", "text": "spaces-simplicial-lemma-hypercovering-X-simple-equivalence-bounded 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. Let $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/U)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence $$ D^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/U)_{total}) $$ with quasi-inverse $Ra_*$."}
{"_id": "8192", "text": "topology-lemma-section-closed Let $f : X \\to Y$ be a continuous map of topological spaces. Let $s : Y \\to X$ be a continuous map such that $f \\circ s = \\text{id}_Y$. If $X$ is Hausdorff, then $s(Y)$ is closed."}
{"_id": "13636", "text": "duality-lemma-duality-compact-support-restrict-open With notation as in Lemma \\ref{lemma-duality-compact-support} suppose $U' \\subset U$ is an open subscheme. Then the diagram $$ \\xymatrix{ \\Hom_k(H^i(U, K), k) \\ar[rr] & & H^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet)) \\\\ \\Hom_k(H^i(U', K|_{U'}), k) \\ar[rr] \\ar[u] & & H^{-i}_c(U', R\\SheafHom_{\\mathcal{O}_{U'}}(K, \\omega_{U'/k}^\\bullet)) \\ar[u] } $$ is commutative. Here the horizontal arrows are the isomorphisms of Lemma \\ref{lemma-duality-compact-support}, the vertical arrow on the left is the contragredient to the restriction map $H^i(U, K) \\to H^i(U', K|_{U'})$, and the right vertical arrow is Remark \\ref{remark-covariance-open-lower-shriek} (see discussion before the lemma)."}
{"_id": "4325", "text": "sites-cohomology-lemma-RHom-well-defined Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$ be a quasi-isomorphism of K-injective complexes of $\\mathcal{O}$-modules. Let $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$ be a quasi-isomorphism of complexes of $\\mathcal{O}$-modules. Then $$ \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet) \\longrightarrow \\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet) $$ is a quasi-isomorphism."}
{"_id": "9169", "text": "examples-stacks-lemma-finite-etale-stack The functor $$ p : \\textit{F\\'Et} \\longrightarrow (\\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\\Sch/S)_{fppf}$."}
{"_id": "5821", "text": "chow-lemma-compute-section Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme locally of finite type over $S$. Let $\\mathcal{C}$ be a locally free $\\mathcal{O}_X$-module of rank $r$. Consider the morphisms $$ X = \\underline{\\text{Proj}}_X(\\mathcal{O}_X[T]) \\xrightarrow{i} E = \\underline{\\text{Proj}}_X(\\text{Sym}^*(\\mathcal{C})[T]) \\xrightarrow{\\pi} X $$ Then $c_t(i_*\\mathcal{O}_X) = 0$ for $t = 1, \\ldots, r - 1$ and in $A^0(C \\to E)$ we have $$ p^* \\circ \\pi_* \\circ c_r(i_*\\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^* $$ where $j : C \\to E$ and $p : C \\to X$ are the inclusion and structure morphism of the vector bundle $C = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{C}))$."}
{"_id": "7549", "text": "stacks-morphisms-lemma-open-immersion-etale An open immersion is \\'etale."}
{"_id": "10689", "text": "etale-theorem-structure-unramified Let $f : A \\to B$ be an unramified morphism 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 quotient of a localization of $B'$ at a prime. \\end{enumerate}"}
{"_id": "11648", "text": "resolve-lemma-Nagata-normalized-blowup In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is normal, Nagata, and proper over $X$."}
{"_id": "2822", "text": "dualizing-lemma-neighbourhood-extensions Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a finitely generated ideal such that $A/I \\to B/IB$ is an isomorphism. For $K \\in D_{I^\\infty\\text{-torsion}}(A)$ and $L \\in D(A)$ the map $$ R\\Hom_A(K, L) \\longrightarrow R\\Hom_B(K \\otimes_A B, L \\otimes_A B) $$ is a quasi-isomorphism. In particular, if $M$, $N$ are $A$-modules and $M$ is $I$-power torsion, then the canonical map $$ \\Ext^i_A(M, N) \\longrightarrow \\Ext^i_B(M \\otimes_A B, N \\otimes_A B) $$ is an isomorphism for all $i$."}
{"_id": "11300", "text": "spaces-cohomology-lemma-tensor-hom-coherent 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. 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": "7799", "text": "injectives-lemma-represent-by-filtered-complex-bis In the situation of Lemma \\ref{lemma-represent-by-filtered-complex} assume we have a second inverse system $\\{(E')^i\\}_{i \\in \\mathbf{Z}}$ and a compatible system of maps $(E')^i \\to E$. Then there exists a bi-filtered complex $K^\\bullet$ of $\\mathcal{A}$ such that $K^\\bullet$ represents $E$, $F^iK^\\bullet$ represents $E^i$, and $(F')^iK^\\bullet$ represents $(E')^i$ compatibly with the given maps."}
{"_id": "5067", "text": "weil-lemma-smash-nilpotence \\begin{reference} \\cite{nilpotence} \\end{reference} 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. For $n$ large enough the class of the zero cycle $$ ([x] - [x']) \\times \\ldots \\times ([x] - [x']) \\in \\CH_0(X^n) $$ is torsion."}
{"_id": "1822", "text": "derived-lemma-filtered-bounded-derived Let $\\mathcal{A}$ be an abelian category. The subcategories $\\text{FAc}^{+}(\\mathcal{A})$, $\\text{FAc}^{-}(\\mathcal{A})$, resp.\\ $\\text{FAc}^b(\\mathcal{A})$ are strictly full saturated triangulated subcategories of $K^{+}(\\text{Fil}^f\\mathcal{A})$, $K^{-}(\\text{Fil}^f\\mathcal{A})$, resp.\\ $K^b(\\text{Fil}^f\\mathcal{A})$. The corresponding saturated multiplicative systems (see Lemma \\ref{lemma-operations}) are the sets $\\text{FQis}^{+}(\\mathcal{A})$, $\\text{FQis}^{-}(\\mathcal{A})$, resp.\\ $\\text{FQis}^b(\\mathcal{A})$. \\begin{enumerate} \\item The kernel of the functor $K^{+}(\\text{Fil}^f\\mathcal{A}) \\to DF^{+}(\\mathcal{A})$ is $\\text{FAc}^{+}(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{+}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{+}(\\mathcal{A}) = \\text{FQis}^{+}(\\mathcal{A})^{-1}K^{+}(\\text{Fil}^f\\mathcal{A}) \\longrightarrow DF^{+}(\\mathcal{A}) $$ \\item The kernel of the functor $K^{-}(\\text{Fil}^f\\mathcal{A}) \\to DF^{-}(\\mathcal{A})$ is $\\text{FAc}^{-}(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^{-}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{-}(\\mathcal{A}) = \\text{FQis}^{-}(\\mathcal{A})^{-1}K^{-}(\\text{Fil}^f\\mathcal{A}) \\longrightarrow DF^{-}(\\mathcal{A}) $$ \\item The kernel of the functor $K^b(\\text{Fil}^f\\mathcal{A}) \\to DF^b(\\mathcal{A})$ is $\\text{FAc}^b(\\mathcal{A})$ and this induces an equivalence of triangulated categories $$ K^b(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^b(\\mathcal{A}) = \\text{FQis}^b(\\mathcal{A})^{-1}K^b(\\text{Fil}^f\\mathcal{A}) \\longrightarrow DF^b(\\mathcal{A}) $$ \\end{enumerate}"}
{"_id": "9521", "text": "decent-spaces-lemma-re-characterize-universally-closed Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and decent. (For example if $f$ is representable, or quasi-separated, see Lemma \\ref{lemma-properties-trivial-implications}.) Then $f$ is universally closed if and only if the existence part of the valuative criterion holds."}
{"_id": "8557", "text": "sites-lemma-j-shriek-reflects-injections-surjections Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$. The functor $j_{U!}$ reflects injections and surjections."}
{"_id": "6415", "text": "etale-cohomology-lemma-alternative-zariski Let $S$ be a scheme. Let $S_{affine, Zar}$ denote the full subcategory of $S_{Zar}$ consisting of affine objects. A covering of $S_{affine, Zar}$ will be a standard Zariski covering, see Topologies, Definition \\ref{topologies-definition-standard-Zariski}. Then restriction $$ \\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, Zar}} $$ defines an equivalence of topoi $\\Sh(S_{Zar}) \\cong \\Sh(S_{affine, Zar})$."}
{"_id": "6842", "text": "equiv-lemma-pushforward-invertible Let $f : X \\to Y$ be a finite type separated morphism of schemes with a section $s : Y \\to X$. Let $\\mathcal{F}$ be a finite type quasi-coherent module on $X$, set theoretically supported on $s(Y)$ with $\\mathcal{L} = f_*\\mathcal{F}$ an invertible $\\mathcal{O}_X$-module. If $Y$ is reduced, then $\\mathcal{F} \\cong s_*\\mathcal{L}$."}
{"_id": "6069", "text": "flat-lemma-limit-purity Let $I$ be a directed set. Let $(S_i, g_{ii'})$ be an inverse system of affine schemes over $I$. Set $S = \\lim_i S_i$ and $s \\in S$. Denote $g_i : S \\to S_i$ the projections and set $s_i = g_i(s)$. Suppose that $f : X \\to S$ is a morphism of finite presentation, $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module of finite presentation which is pure along $X_s$ and flat over $S$ at all points of $X_s$. Then there exists an $i \\in I$, a morphism of finite presentation $X_i \\to S_i$, a quasi-coherent $\\mathcal{O}_{X_i}$-module $\\mathcal{F}_i$ of finite presentation which is pure along $(X_i)_{s_i}$ and flat over $S_i$ at all points of $(X_i)_{s_i}$ such that $X \\cong X_i \\times_{S_i} S$ and such that the pullback of $\\mathcal{F}_i$ to $X$ is isomorphic to $\\mathcal{F}$."}
{"_id": "14092", "text": "more-morphisms-lemma-weighting-quasi-finite-Noetherian Let $f : X \\to Y$ be a morphism of schemes. Assume \\begin{enumerate} \\item $f$ is locally quasi-finite, and \\item $Y$ is unibranch and locally Noetherian. \\end{enumerate} Then there is a weighting $w : X \\to \\mathbf{Z}_{\\geq 0}$ given by the rule that sends $x \\in X$ lying over $y \\in Y$ to the ``generic separable degree'' of $\\mathcal{O}_{X, x}^{sh}$ over $\\mathcal{O}_{Y, y}^{sh}$."}
{"_id": "107", "text": "spaces-more-morphisms-lemma-base-change-formally-smooth A base change of a formally smooth morphism is formally smooth."}
{"_id": "1623", "text": "moduli-curves-lemma-prestable-curves-smooth The morphisms $\\Curvesstack^{prestable} \\to \\Spec(\\mathbf{Z})$ and $\\Curvesstack^{prestable}_g \\to \\Spec(\\mathbf{Z})$ are smooth."}
{"_id": "14810", "text": "simplicial-lemma-characterize-cosimplicial-object Let $\\mathcal{C}$ be a category. \\begin{enumerate} \\item Given a cosimplicial object $U$ in $\\mathcal{C}$ we obtain a sequence of objects $U_n = U([n])$ endowed with the morphisms $\\delta^n_j = U(\\delta^n_j) : U_{n - 1} \\to U_n$ and $\\sigma^n_j = U(\\sigma^n_j) : U_{n + 1} \\to U_n$. These morphisms satisfy the relations displayed in Lemma \\ref{lemma-relations-face-degeneracy}. \\item Conversely, given a sequence of objects $U_n$ and morphisms $\\delta^n_j$, $\\sigma^n_j$ satisfying these relations there exists a unique cosimplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$, $\\delta^n_j = U(\\delta^n_j)$, and $\\sigma^n_j = U(\\sigma^n_j)$. \\item A morphism between cosimplicial objects $U$ and $U'$ is given by a family of morphisms $U_n \\to U'_n$ commuting with the morphisms $\\delta^n_j$ and $\\sigma^n_j$. \\end{enumerate}"}
{"_id": "9209", "text": "models-lemma-picard-group-genus-nonpositive Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. If the genus $g$ of $T$ is $\\leq 0$, then $\\Pic(T) = \\mathbf{Z}$."}
{"_id": "9713", "text": "local-cohomology-lemma-support Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. For an $A$-module $M$ the following are equivalent \\begin{enumerate} \\item $H^0_T(M) = M$, and \\item $\\text{Supp}(M) \\subset T$. \\end{enumerate} The category of such $A$-modules is a Serre subcategory of the category $A$-modules closed under direct sums."}
{"_id": "14702", "text": "descent-lemma-characterize-open-immersion 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 \\'etale 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 an open immersion."}
{"_id": "2339", "text": "restricted-lemma-base-change-flat Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$. \\begin{enumerate} \\item If $f$ is flat and $g_{red} : Z_{red} \\to Y_{red}$ is locally of finite type, then the base change $X \\times_Y Z \\to Z$ is flat. \\item If $f$ is flat and locally of finite type, then the base change $X \\times_Y Z \\to Z$ is flat and locally of finite type. \\end{enumerate}"}
{"_id": "11443", "text": "obsolete-theorem-equivalence-sheaves-point 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. Then there is an equivalence of categories $\\Sh(S_\\etale) \\to G\\textit{-Sets}$, $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$."}
{"_id": "9867", "text": "more-algebra-lemma-limits-henselian The property of being Henselian is preserved under limits of pairs. More precisely, let $J$ be a preordered set and let $(A_j, I_j)$ be an inverse system of henselian pairs over $J$. Then $A = \\lim A_j$ equipped with the ideal $I = \\lim I_j$ is a henselian pair $(A, I)$."}
{"_id": "14591", "text": "descent-theorem-descend-module-properties If $M \\otimes_R S$ has one of the following properties as an $S$-module \\begin{enumerate} \\item[(a)] finitely generated; \\item[(b)] finitely presented; \\item[(c)] flat; \\item[(d)] faithfully flat; \\item[(e)] finite projective; \\end{enumerate} then so does $M$ as an $R$-module (and conversely)."}
{"_id": "14065", "text": "more-morphisms-lemma-relative-pseudo-coherent-descends-fppf-base Let $f : X \\to S$ be locally of finite type. Let $\\{S_i \\to S\\}$ be an fppf covering of schemes. Denote $f_i : X_i \\to S_i$ the base change of $f$ and $g_i : X_i \\to X$ the projection. 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 $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$."}
{"_id": "6324", "text": "curves-lemma-nodal-family-fpqc-local-target Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to S\\}$ be an fpqc covering. The following are equivalent \\begin{enumerate} \\item $f$ is at-worst-nodal of relative dimension $1$, \\item each $X \\times_S U_i \\to U_i$ is at-worst-nodal of relative dimension $1$. \\end{enumerate} In other words, being at-worst-nodal of relative dimension $1$ is fpqc local on the target."}
{"_id": "2182", "text": "cohomology-lemma-section-RHom-over-U Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L, M$ be objects of $D(\\mathcal{O}_X)$. For every open $U$ we have $$ H^0(U, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U) $$ and in particular $H^0(X, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O}_X)}(L, M)$."}
{"_id": "3349", "text": "coherent-lemma-section-affine-open-kills-classes 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. Assume that \\begin{enumerate} \\item $X$ is quasi-compact and quasi-separated, and \\item $X_s$ is affine. \\end{enumerate} Then for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and every $p > 0$ and all $\\xi \\in H^p(X, \\mathcal{F})$ there exists an $n \\geq 0$ such that $s^n\\xi = 0$ in $H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$."}
{"_id": "5150", "text": "morphisms-lemma-scheme-theoretic-image-of-partial-section Let $f : X \\to Y$ be a separated morphism of schemes. Let $V \\subset Y$ be a retrocompact open. Let $s : V \\to X$ be a morphism such that $f \\circ s = \\text{id}_V$. Let $Y'$ be the scheme theoretic image of $s$. Then $Y' \\to Y$ is an isomorphism over $V$."}
{"_id": "13047", "text": "dga-lemma-homotopy-category-pre-triangulated Let $(A, \\text{d})$ be a differential graded algebra. The homotopy category $K(\\text{Mod}_{(A, \\text{d})})$ with its natural translation functors and distinguished triangles is a pre-triangulated category."}
{"_id": "12913", "text": "spaces-divisors-lemma-weakly-ass-reverse-functorial 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 we have $$ \\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F})) $$"}
{"_id": "12757", "text": "algebraization-lemma-when-topology 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}). Set $$ \\mathcal{G}_n = \\widetilde{M/I^nM}. $$ If the limit topology on $M$ agrees with the $I$-adic topology, then $\\mathcal{G}_n|_U$ is a coherent $\\mathcal{O}_U$-module and the map of inverse systems $$ (\\mathcal{G}_n|_U) \\longrightarrow (\\mathcal{F}_n) $$ is injective in the abelian category $\\textit{Coh}(U, I\\mathcal{O}_U)$."}
{"_id": "5609", "text": "smoothing-lemma-parse-equation-strictly-standard-one Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$ and write $I = (f_1, \\ldots, f_m)$. Let $a \\in A$. Then (\\ref{equation-strictly-standard-one}) implies there exists an $A$-linear map $\\psi : \\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i \\to A^{\\oplus c}$ such that the composition $$ A^{\\oplus c} \\xrightarrow{(f_1, \\ldots, f_c)} I/I^2 \\xrightarrow{f \\mapsto \\text{d}f} \\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i \\xrightarrow{\\psi} A^{\\oplus c} $$ is multiplication by $a$. Conversely, if such a $\\psi$ exists, then $a^c$ satisfies (\\ref{equation-strictly-standard-one})."}
{"_id": "10725", "text": "etale-lemma-characterize-normal-crossings Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed subscheme. If $X$ is J-2 or Nagata, then following are equivalent \\begin{enumerate} \\item $D$ is a normal crossings divisor in $X$, \\item for every $p \\in D$ the pullback of $D$ to the spectrum of the strict henselization $\\mathcal{O}_{X, p}^{sh}$ is a strict normal crossings divisor. \\end{enumerate}"}
{"_id": "4925", "text": "spaces-morphisms-lemma-finite-separable-enough 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 $f$ satisfies the existence part of the valuative criterion as in Definition \\ref{definition-valuative-criterion}, \\item $f$ satisfies the existence part of the valuative criterion as in Definition \\ref{definition-valuative-criterion} modified by requiring the extension $K \\subset K'$ to be finite separable. \\end{enumerate}"}
{"_id": "1842", "text": "derived-lemma-F-acyclic-ses Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor between abelian categories and assume $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined. Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence of $\\mathcal{A}$. \\begin{enumerate} \\item If $A$ and $C$ are right acyclic for $F$ then so is $B$. \\item If $A$ and $B$ are right acyclic for $F$ then so is $C$. \\item If $B$ and $C$ are right acyclic for $F$ and $F(B) \\to F(C)$ is surjective then $A$ is right acyclic for $F$. \\end{enumerate} In each of the three cases $$ 0 \\to F(A) \\to F(B) \\to F(C) \\to 0 $$ is a short exact sequence of $\\mathcal{B}$."}
{"_id": "11566", "text": "stacks-sheaves-lemma-2-morphisms-presheaves Let $f, g : \\mathcal{X} \\to \\mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $t : f \\to g$ be a $2$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assigned to $t$ there are canonical isomorphisms of functors $$ t^p : g^p \\longrightarrow f^p \\quad\\text{and}\\quad {}_pt : {}_pf \\longrightarrow {}_pg $$ which compatible with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ and with vertical and horizontal composition of $2$-morphisms."}
{"_id": "102", "text": "spaces-more-morphisms-lemma-flatness-morphism-thickenings 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 $X'$ is flat over $B'$, \\item $f$ is flat, \\item $B \\subset B'$ is a finite order thickening, and \\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$. \\end{enumerate} Then $f'$ is flat and $Y'$ is flat over $B'$ at all points in the image of $f'$."}
{"_id": "10731", "text": "etale-proposition-etale-reduced Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then $A$ is reduced if and only if $B$ is so."}
{"_id": "14860", "text": "simplicial-lemma-abelian-limit-skeleta Let $\\mathcal{A}$ be an abelian category. For any simplicial object $V$ of $\\mathcal{A}$ we have $$ V = \\colim_n i_{n!}\\text{sk}_n V $$ where all the transition maps are injections."}
{"_id": "5857", "text": "chow-lemma-identify-chow-for-smooth-dim-1 Let $(S, \\delta)$ be as above. Let $X$ be a smooth scheme over $S$, 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": "6603", "text": "etale-cohomology-lemma-fppf-reduced-fibres-pullback-products Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1} : \\Sh(S_\\etale) \\to \\Sh(X_\\etale)$ commutes with products."}
{"_id": "12623", "text": "constructions-lemma-projective-space-separated Let $S$ be a scheme. The structure morphism $\\mathbf{P}^n_S \\to S$ is \\begin{enumerate} \\item separated, \\item quasi-compact, \\item satisfies the existence and uniqueness parts of the valuative criterion, and \\item universally closed. \\end{enumerate}"}
{"_id": "10216", "text": "more-algebra-lemma-complex-perfect-modules Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of perfect $R$-modules. Then $K^\\bullet$ is a perfect complex."}
{"_id": "12914", "text": "spaces-divisors-lemma-ass-functorial-equal 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. If $X$ is locally Noetherian, then we have $$ \\text{WeakAss}_Y(f_*\\mathcal{F}) = f(\\text{WeakAss}_X(\\mathcal{F})) $$"}
{"_id": "9518", "text": "decent-spaces-lemma-fibre-product-conditions Let $S$ be a scheme. Let $f : X \\to Y$, $g : Z \\to Y$ be morphisms of algebraic spaces over $S$. If $X$ and $Z$ are decent (resp.\\ reasonable, resp.\\ have property  $(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then so does $X \\times_Y Z$."}
{"_id": "14030", "text": "more-morphisms-lemma-reduced-goes-up Let $f : X \\to Y$ be a morphism of schemes. If $Y$ is reduced and $f$ weakly \\'etale, then $X$ is reduced."}
{"_id": "3183", "text": "quot-lemma-compare-pic-with-section In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section. The morphism $\\Picardstack_{X/B, \\sigma} \\to \\Picardstack_{X/B}$ is representable, surjective, and smooth."}
{"_id": "7906", "text": "divisors-lemma-flat-torsion-free Let $X$ be an integral scheme. Any flat quasi-coherent $\\mathcal{O}_X$-module is torsion free."}
{"_id": "2582", "text": "examples-lemma-not-essentially-surjective The canonical map $\\mathcal{X}(S) \\to \\lim \\mathcal{X}(S_n)$ is not essentially surjective."}
{"_id": "136", "text": "spaces-more-morphisms-lemma-morphism-between-flat-Noetherian 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 $X$, $Y$, $Z$ locally Noetherian, \\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 $z$. \\end{enumerate} Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$."}
{"_id": "4675", "text": "stacks-geometry-lemma-multiplicity Let $U_1 \\to \\mathcal{X}$ and $U_2 \\to \\mathcal{X}$ be two smooth morphisms from schemes to a locally Noetherian algebraic stack $\\mathcal{X}$. Let $T_1'$ and $T_2'$ be irreducible components of $|U_1|$ and $|U_2|$ respectively. Assume the closures of the images of $T_1'$ and $T_2'$ are the same irreducible component $T$ of $|\\mathcal{X}|$. Then $m_{T_1', U_1} = m_{T_2', U_2}$."}
{"_id": "14481", "text": "sheaves-lemma-sheaves-structure Suppose the category $\\mathcal{C}$ and the functor $F : \\mathcal{C} \\to \\textit{Sets}$ have the following properties: \\begin{enumerate} \\item $F$ is faithful, \\item $\\mathcal{C}$ has limits and $F$ commutes with them, and \\item the functor $F$ reflects isomorphisms. \\end{enumerate} Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf with values in $\\mathcal{C}$. Then $\\mathcal{F}$ is a sheaf if and only if the underlying presheaf of sets is a sheaf."}
{"_id": "2597", "text": "examples-proposition-quotient-abelian-groups-by-torsion-groups The quotient of the category of abelian groups modulo its Serre subcategory of torsion groups is the category of $\\mathbf{Q}$-vector spaces."}
{"_id": "200", "text": "spaces-more-morphisms-lemma-chow-finite-type-separated 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 quasi-compact. 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 proper surjective morphism and the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion."}
{"_id": "7502", "text": "stacks-morphisms-lemma-finite-presentation-finite-type A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type."}
{"_id": "10426", "text": "more-algebra-lemma-not-awkward \\begin{reference} Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018. \\end{reference} Let $A \\to B$ be a flat homomorphism of Noetherian rings. Let $I \\subset A$ be an ideal. Let $M, N$ be $A$-modules. Set $B_n = B/I^nB$, $M_n = M/I^nM$, $N_n = N/I^nN$. If $M$ is flat over $A$, then we have $$ \\lim \\Ext^i_B(M, N)/I^n \\Ext^i_B(M, N) = \\lim \\Ext^i_{B_n}(M_n, N_n) $$ for all $i \\in \\mathbf{Z}$."}
{"_id": "4869", "text": "spaces-morphisms-lemma-rel-dimension-dimension Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian algebraic spaces over $S$ 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": "12974", "text": "spaces-divisors-lemma-eventual-iso-graded-rings-map-relative-proj 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 $U(\\psi) = Q$, \\item $r_\\psi : Q \\to P$ is an isomorphism, and \\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$ are isomorphisms. \\end{enumerate}"}
{"_id": "14742", "text": "descent-lemma-morphism-source-fpqc-covering Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$. Assume $\\{X \\to S\\}$ is an fpqc covering (for example if $f$ is surjective, flat and quasi-compact). Then the pullback functor of Lemma \\ref{lemma-pullback} is a fully faithful functor from the category of descent data relative to $X'/S$ to the category of descent data relative to $X/S$."}
{"_id": "6859", "text": "equiv-lemma-equivalences-rek 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 $S$. Then the corresponding Fourier-Mukai functors $\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ (Lemma \\ref{lemma-fourier-Mukai-QCoh}) and $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ (Lemma \\ref{lemma-fourier-mukai}) are equivalences."}
{"_id": "13936", "text": "more-morphisms-lemma-horizontal Let $A \\to B$ be a local homomorphism of local rings, and $g \\in \\mathfrak m_B$. Assume \\begin{enumerate} \\item $A$ and $B$ are domains and $A \\subset B$, \\item $B$ is essentially of finite type over $A$, \\item $g$ is not contained in any minimal prime over $\\mathfrak m_AB$, and \\item $\\dim(B/\\mathfrak m_AB) + \\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) = \\text{trdeg}_A(B)$. \\end{enumerate} Then $A \\subset B/gB$, i.e., the generic point of $\\Spec(A)$ is in the image of the morphism $\\Spec(B/gB) \\to \\Spec(A)$."}
{"_id": "6516", "text": "etale-cohomology-lemma-higher-direct-jstar-Gm For any $q \\geq 1$, $R^q j_*\\mathbf{G}_{m, \\eta} = 0$."}
{"_id": "13083", "text": "dga-lemma-analogue-triangle-independent-splittings In Situation \\ref{situation-ABC} let $0 \\to x \\to y \\to z \\to 0$ be an admissible short exact sequence in $\\text{Comp}(\\mathcal{A})$. The triangle $$ \\xymatrix{x\\ar[r] & y\\ar[r] & z\\ar[r]^{\\delta} & x[1]} $$ with $\\delta : z \\to x[1]$ as defined in Lemma \\ref{lemma-get-triangle} is up to canonical isomorphism in $K(\\mathcal{A})$, independent of the choices made in Lemma \\ref{lemma-get-triangle}."}
{"_id": "11879", "text": "spaces-properties-lemma-support-section-closed Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Let $U \\in \\Ob(X_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$. \\begin{enumerate} \\item The support of $\\sigma$ is closed in $|X|$. \\item The support of $\\sigma + \\sigma'$ is contained in the union of the supports of $\\sigma, \\sigma' \\in \\mathcal{F}(X)$. \\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of abelian sheaves on $X_\\etale$, 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": "11468", "text": "obsolete-lemma-Rlim-of-system 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. Let $d \\in \\mathbf{N}$. Assume \\begin{enumerate} \\item $K_n$ is an object of $D^+(\\mathcal{O})$ for all $n$, \\item for $q \\in \\mathbf{Z}$ there exists $n(q)$ such that $H^q(K_{n + 1}) \\to H^q(K_n)$ is an isomorphism for $n \\geq n(q)$, \\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 $H^p(U, H^q(K_n)) = 0$ for $p > d$ and all $q$. \\end{enumerate} Then we have $H^m(R\\lim K_n) = \\lim H^m(K_n)$ for all $m \\in \\mathbf{Z}$."}
{"_id": "12907", "text": "spaces-divisors-lemma-minimal-support-in-weakly-ass-decent 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 $X$ is decent (for example quasi-separated or locally separated), \\item $x \\in \\text{Supp}(\\mathcal{F})$ \\item $x$ is not a specialization of another point in $\\text{Supp}(\\mathcal{F})$. \\end{enumerate} Then $x \\in \\text{WeakAss}(\\mathcal{F})$."}
{"_id": "13787", "text": "more-morphisms-lemma-flat-morphism-from-CM-scheme Let $f : X \\to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \\in X$."}
{"_id": "7910", "text": "divisors-lemma-extension-torsion-free Let $X$ be an integral scheme. 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. If $\\mathcal{F}$ and $\\mathcal{F}''$ are torsion free, then $\\mathcal{F}'$ is torsion free."}
{"_id": "1863", "text": "derived-lemma-projective-resolution-ses Let $\\mathcal{A}$ be an abelian category. Assume $\\mathcal{A}$ has enough projectives. 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 $\\text{Comp}^{+}(\\mathcal{A})$ $$ \\xymatrix{ 0 \\ar[r] & P_1^\\bullet \\ar[r] \\ar[d] & P_2^\\bullet \\ar[r] \\ar[d] & P_3^\\bullet \\ar[r] \\ar[d] & 0 \\\\ 0 \\ar[r] & A^\\bullet \\ar[r] & B^\\bullet \\ar[r] & C^\\bullet \\ar[r] & 0 } $$ where the vertical arrows are projective resolutions and the rows are short exact sequences of complexes. In fact, given any projective resolution $P^\\bullet \\to C^\\bullet$ we may assume $P_3^\\bullet = P^\\bullet$."}
{"_id": "9485", "text": "decent-spaces-lemma-finite-etale-cover-dense-open-scheme 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 scheme, then there exists a dense open subspace of $X$ which is a scheme."}
{"_id": "8048", "text": "divisors-lemma-equation-codim-1-in-projective-space Let $R$ be a UFD. Let $Z \\subset \\mathbf{P}^n_R$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has codimension $1$ in $\\mathbf{P}^n_R$. Then the ideal $I(Z) \\subset R[T_0, \\ldots, T_n]$ corresponding to $Z$ is principal."}
{"_id": "12950", "text": "spaces-divisors-lemma-normal-effective-Cartier-divisor-S1 Let $S$ be a scheme and let $X$ be a locally Noetherian normal algebraic space over $S$. Let $D \\subset X$ be an effective Cartier divisor. Then $D$ is $(S_1)$."}
{"_id": "8833", "text": "more-etale-lemma-lqf-shriek-derived Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. The functors $f_!$ and $f^!$ of Definition \\ref{definition-f-shriek-lqf} and Lemma \\ref{lemma-lqf-f-upper-shriek} induce adjoint functors $f_! : D(X_\\etale) \\to D(Y_\\etale)$ and $Rf^! : D(Y_\\etale) \\to D(X_\\etale)$ on derived categories."}
{"_id": "9131", "text": "spaces-simplicial-lemma-fppf-hypercovering-equivalence-bounded 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 an fppf 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": "1816", "text": "derived-lemma-derived-compare-triangles-split-case Let $\\mathcal{A}$ be an abelian category. Let $$ \\xymatrix{ 0 \\ar[r] & A^\\bullet \\ar[r] & B^\\bullet \\ar[r] & C^\\bullet \\ar[r] & 0 } $$ be a short exact sequences of complexes. Assume this short exact sequence is termwise split. Let $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$ be the distinguished triangle of $K(\\mathcal{A})$ associated to the sequence. The $\\delta$-functor of Lemma \\ref{lemma-derived-canonical-delta-functor} above maps the short exact sequences $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ to a triangle isomorphic to the distinguished triangle $$ (A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta). $$"}
{"_id": "13877", "text": "more-morphisms-lemma-slice-smooth-once-separable-residue-field-extension 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 $x$, \\item the residue field extension $\\kappa(s) \\subset \\kappa(x)$ is separable, and \\item $x$ is not a generic point of $X_s$. \\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": "5339", "text": "morphisms-lemma-two-immersions-smooth 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 and $X \\to Y$ is 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-immersions} is exact."}
{"_id": "11395", "text": "artin-lemma-naive-obstruction-theory-qis Let $S$ and $\\mathcal{X}$ be as in Definition \\ref{definition-naive-obstruction-theory} and let $\\mathcal{X}$ be endowed with a naive obstruction theory. Let $A \\to B$ and $y \\to x$ be as in (\\ref{item-functoriality}). Let $k$ be a $B$-algebra which is a field. Then the functoriality map $E_x \\to E_y$ induces bijections $$ H^i(E_x \\otimes_A^{\\mathbf{L}} k) \\to H^i(E_y \\otimes_B^{\\mathbf{L}} k) $$ for $i = 0, 1$."}
{"_id": "12434", "text": "topologies-lemma-verify-site-Zariski Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski site containing $S$. Both $S_{Zar}$ and $(\\textit{Aff}/S)_{Zar}$ are sites."}
{"_id": "12915", "text": "spaces-divisors-lemma-weakly-associated-finite Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$."}
{"_id": "938", "text": "algebra-lemma-catenary-Noetherian-local Let $(A, \\mathfrak m)$ be a Noetherian local ring. The following are equivalent \\begin{enumerate} \\item $A$ is catenary, and \\item $\\mathfrak p \\mapsto \\dim(A/\\mathfrak p)$ is a dimension function on $\\Spec(A)$. \\end{enumerate}"}
{"_id": "6183", "text": "flat-lemma-blowup-pd1 Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$. Let $U \\subset X$ be a scheme theoretically dense open such that $\\mathcal{F}|_U$ is finite locally free of constant rank $r$. Then there exists a $U$-admissible blowup $b : X' \\to X$ such that there is a canonical short exact sequence $$ 0 \\to \\mathcal{K} \\to b^*\\mathcal{F} \\to \\mathcal{Q} \\to 0 $$ where $\\mathcal{Q}$ is finite locally free of rank $r$ and $\\mathcal{K}$ is a perfect $\\mathcal{O}_X$-module of tor dimension $\\leq 1$ whose restriction to $U$ is zero."}
{"_id": "13482", "text": "spaces-resolve-lemma-dominate-by-blowing-up-in-points 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 $f : Y \\to X$ be a proper morphism of algebraic spaces 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": "11448", "text": "obsolete-lemma-finite-type-flat-over-integral-algebra Let $A \\to B$ be a finite type, flat ring map with $A$ an integral domain. Then $B$ is a finitely presented $A$-algebra."}
{"_id": "233", "text": "spaces-more-morphisms-lemma-flat-finite-presentation-perfect 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 perfect, and \\item $f$ is flat and locally of finite presentation. \\end{enumerate}"}
{"_id": "1808", "text": "derived-lemma-additive-exact-homotopy-category Let $\\mathcal{A}$, $\\mathcal{B}$ be additive categories. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor. The induced functors $$ \\begin{matrix} F : K(\\mathcal{A}) \\longrightarrow K(\\mathcal{B}) \\\\ F : K^{+}(\\mathcal{A}) \\longrightarrow K^{+}(\\mathcal{B}) \\\\ F : K^{-}(\\mathcal{A}) \\longrightarrow K^{-}(\\mathcal{B}) \\\\ F : K^b(\\mathcal{A}) \\longrightarrow K^b(\\mathcal{B}) \\end{matrix} $$ are exact functors of triangulated categories."}
{"_id": "4304", "text": "sites-cohomology-lemma-compare-cohomology-general In Situation \\ref{situation-compare}. Let $X$ be in $\\mathcal{C}$. \\begin{enumerate} \\item for $\\mathcal{F}'$ in $\\mathcal{A}'_X$ we have $H^n_{\\tau'}(X, \\mathcal{F}') = H^n_\\tau(X, \\epsilon_X^{-1}\\mathcal{F}')$, \\item for $K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ we have $H^n_{\\tau'}(X, K') = H^n_\\tau(X, \\epsilon_X^{-1}K')$. \\end{enumerate}"}
{"_id": "14432", "text": "trace-lemma-count-points-projective Let $X$ be a smooth, projective, geometrically irreducible curve over a finite field $k$. Then \\begin{enumerate} \\item the $L$-function $L(X, \\mathbf{Q}_\\ell)$ is a rational function, \\item the eigenvalues $\\alpha_1, \\ldots, \\alpha_{2g}$ of $\\pi_X^*$ on $H^1(X_{\\bar k}, \\mathbf{Q}_\\ell)$ are algebraic integers independent of $\\ell$, \\item the number of rational points of $X$ on $k_n$, where $[k_n : k] = n$, is $$ \\# X(k_n) = 1 - \\sum\\nolimits_{i = 1}^{2g}\\alpha_i^n + q^n, $$ \\item for each $i$, $|\\alpha_i| < q$. \\end{enumerate}"}
{"_id": "2469", "text": "more-groupoids-lemma-groupoid-on-field-separated 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, then $R$ is a separated scheme."}
{"_id": "14068", "text": "more-morphisms-lemma-henselian-relatively-perfect 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 $f : X \\to S$ be 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": "11238", "text": "cotangent-lemma-cotangent-morphism-spaces 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": "13444", "text": "groupoids-quotients-lemma-pre-equivalence-equivalence-relation-points Let $B \\to S$ as in Section \\ref{section-conventions-notation}. Let $j : R \\to U \\times_B U$ be a pre-equivalence relation of algebraic spaces over $B$. Then $$ O_u = \\{u' \\in |U| \\text{ such that } \\exists r \\in |R|, \\ s(r) = u, \\ t(r) = u'\\}. $$"}
{"_id": "42", "text": "spaces-more-morphisms-lemma-finite-type-differentials 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 $\\Omega_{X/Y}$ is a finite type $\\mathcal{O}_X$-module."}
{"_id": "10119", "text": "more-algebra-lemma-K-injective-epimorphism Let $R \\to S$ be an epimorphism of rings. Let $I^\\bullet$ be a complex of $S$-modules. If $I^\\bullet$ is K-injective as a complex of $R$-modules, then $I^\\bullet$ is a K-injective complex of $S$-modules."}
{"_id": "14321", "text": "derham-lemma-cup-product-hodge-graded-commutative Let $p : X \\to S$ be a morphism of schemes. The cup product on $H^*_{Hodge}(X/S)$ is associative and graded commutative."}
{"_id": "10687", "text": "etale-theorem-flat-is-quotient A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology."}
{"_id": "4736", "text": "spaces-morphisms-lemma-characterize-representable-quasi-compact 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 quasi-compact (in the sense of Section \\ref{section-representable}), and \\item for every quasi-compact algebraic space $Z$ and any morphism $Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact. \\end{enumerate}"}
{"_id": "2982", "text": "properties-lemma-dimension-zero \\begin{reference} Email from Ofer Gabber dated June 4, 2016 \\end{reference} Let $X$ be a scheme of dimension zero. The following are equivalent \\begin{enumerate} \\item $X$ is quasi-separated, \\item $X$ is separated, \\item $X$ is Hausdorff, \\item every affine open is closed. \\end{enumerate} In this case the connected components of $X$ are points."}
{"_id": "9021", "text": "spaces-simplicial-lemma-constant-simplicial-space Let $X$ be a topological space. Let $X_\\bullet$ be the constant simplicial topological space with value $X$. The functor $$ X_{\\bullet, Zar} \\longrightarrow X_{Zar},\\quad U \\longmapsto U $$ is continuous and cocontinuous and defines a morphism of topoi $g : \\Sh(X_{\\bullet, Zar}) \\to \\Sh(X)$ as well as a left adjoint $g_!$ to $g^{-1}$. We have \\begin{enumerate} \\item $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial sheaf on $X$, \\item $g_!$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the sheaf $\\mathcal{F}_0$, and \\item $g_*$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the equalizer of the two maps $\\mathcal{F}_0 \\to \\mathcal{F}_1$. \\end{enumerate}"}
{"_id": "13293", "text": "modules-lemma-whole-tensor-algebra-permanence 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 quasi-coherent, then so is each $\\text{T}(\\mathcal{F})$, $\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$. \\item If $\\mathcal{F}$ is locally free, then so is each $\\text{T}(\\mathcal{F})$, $\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$. \\end{enumerate}"}
{"_id": "8844", "text": "more-etale-lemma-derived-lower-shriek-bounded Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ is bounded in the following sense: There exists an integer $N$ such that for $E \\in D(X_\\etale, \\Lambda)$ we have \\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": "14044", "text": "more-morphisms-lemma-descending-property-ind-quasi-affine The property of being ind-quasi-affine is fpqc local on the base."}
{"_id": "6412", "text": "etale-cohomology-lemma-cohomology-enlarge-partial-universe Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$. Let $S$ be a scheme. Let $(\\Sch/S)_\\tau$ and $(\\Sch'/S)_\\tau$ be two big $\\tau$-sites of $S$, and assume that the first is contained in the second. In this case \\begin{enumerate} \\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\tau$ and any object $U$ of $(\\Sch/S)_\\tau$ we have $$ H^p_\\tau(U, \\mathcal{F}'|_{(\\Sch/S)_\\tau}) = H^p_\\tau(U, \\mathcal{F}') $$ 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 $(\\Sch/S)_\\tau$ there is an abelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\tau'$ whose restriction to $(\\Sch/S)_\\tau$ is isomorphic to $\\mathcal{F}$. \\end{enumerate}"}
{"_id": "3805", "text": "proetale-lemma-j-shriek-limits Let $j : U \\to X$ be a quasi-compact open immersion morphism of schemes. The functor $j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$ commutes with limits."}
{"_id": "13008", "text": "spaces-divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free 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}. Assume that $\\mathcal{F}$ is locally free of rank $k$ on $B \\setminus Z_k$. Let $B' \\to B$ be the blowup of $B$ in $Z_k$ and let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$. Then $\\mathcal{F}'$ is locally free of rank $k$."}
{"_id": "3015", "text": "properties-lemma-finite-locally-free-reduced Let $X$ be a reduced scheme. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then the equivalent conditions of Lemma \\ref{lemma-finite-locally-free} are also equivalent to \\begin{enumerate} \\item[(6)] $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type 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": "1799", "text": "derived-lemma-triangle-independent-splittings Let $\\mathcal{A}$ be an additive category. Let $0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$ be termwise split exact sequences as in Definition \\ref{definition-split-ses}. Let $(\\pi')^n$, $(s')^n$ be a second collection of splittings. Denote $\\delta' : C^\\bullet \\longrightarrow A^\\bullet[1]$ the morphism associated to this second set of splittings. Then $$ (1, 1, 1) : (A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta) \\longrightarrow (A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta') $$ is an isomorphism of triangles in $K(\\mathcal{A})$."}
{"_id": "4881", "text": "spaces-morphisms-lemma-base-change-syntomic The base change of a syntomic morphism is syntomic."}
{"_id": "6517", "text": "etale-cohomology-lemma-cohomology-jstar-Gm For all $p \\geq 1$, $H_\\etale^p(X, j_*\\mathbf{G}_{m, \\eta}) = 0$."}
{"_id": "14901", "text": "simplicial-lemma-section Let $f : V \\to U$ be a morphism of simplicial sets. Let $n \\geq 0$ be an integer. Assume \\begin{enumerate} \\item The map $f_i : V_i \\to U_i$ is a bijection for $i < n$. \\item The map $f_n : V_n \\to U_n$ is a surjection. \\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 $f$ is a trivial Kan fibration."}
{"_id": "14032", "text": "more-morphisms-lemma-normal-goes-up Let $f : X \\to Y$ be a morphism of schemes. If $Y$ is a normal scheme and $f$ weakly \\'etale, then $X$ is a normal scheme."}
{"_id": "5893", "text": "chow-lemma-application-herbrand-quotient Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$. Let $M$ be a finite $R$-module, and let $\\psi : M \\to M$ be an $R$-module map. Assume that \\begin{enumerate} \\item $\\Ker(\\psi)$ and $\\Coker(\\psi)$ have finite length, and \\item $\\dim(\\text{Supp}(M)) \\leq 1$. \\end{enumerate} Write $\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$ and denote $f_i \\in \\kappa(\\mathfrak q_i)^*$ the element such that $\\det_{\\kappa(\\mathfrak q_i)}(\\psi_{\\mathfrak q_i}) : \\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i}) \\to \\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i})$ is multiplication by $f_i$. Then we have $$ \\text{length}_R(\\Coker(\\psi)) - \\text{length}_R(\\Ker(\\psi)) = \\sum\\nolimits_{i = 1, \\ldots, t} \\text{ord}_{R/\\mathfrak q_i}(f_i). $$"}
{"_id": "9246", "text": "models-lemma-minimal-model-mapping-property Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$ and genus $> 0$. Let $X$ be the minimal model for $C$ (Lemma \\ref{lemma-minimal-model-unique}). Let $Y$ be a regular proper model for $C$. Then there is a unique morphism of models $Y \\to X$ which is a sequence of contractions of exceptional curves of the first kind."}
{"_id": "5047", "text": "weil-lemma-decompose-P1 In $M_k$ we have $h(\\mathbf{P}^1_k) \\cong \\mathbf{1} \\oplus \\mathbf{1}(-1)$."}
{"_id": "13759", "text": "more-morphisms-lemma-base-change-NL-flat Consider a cartesian diagram of schemes $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\ Y' \\ar[r] & Y } $$ If $X \\to Y$ is flat, then the canonical map $(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism. If in addition $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$ then $L(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism too."}
{"_id": "11483", "text": "obsolete-lemma-separated-fpqc Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a fpqc covering of schemes over $S$. Then the map $$ \\Mor_S(T, X) \\longrightarrow \\prod\\nolimits_{i \\in I} \\Mor_S(T_i, X) $$ is injective."}
{"_id": "1753", "text": "moduli-lemma-complexes-qs-lfp The morphism $\\Complexesstack_{X/B} \\to B$ is quasi-separated and locally of finite presentation."}
{"_id": "4303", "text": "sites-cohomology-lemma-cohomological-descent-general In Situation \\ref{situation-compare}. For any $X$ in $\\mathcal{C}$ the category $\\mathcal{A}_X \\subset \\textit{Ab}(\\mathcal{C}_\\tau/X)$ is a weak Serre subcategory and the functor $$ R\\epsilon_{X, *} : D^+_{\\mathcal{A}_X}(\\mathcal{C}_\\tau/X) \\longrightarrow D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X) $$ is an equivalence with quasi-inverse given by $\\epsilon_X^{-1}$."}
{"_id": "9479", "text": "decent-spaces-lemma-stratify-reasonable Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces $$ \\emptyset = U_{n + 1} \\subset U_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X $$ such that each $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace structure) is a scheme."}
{"_id": "14013", "text": "more-morphisms-lemma-lci-NL Let $f : X \\to Y$ be a local complete intersection homomorphism. Then the naive cotangent complex $\\NL_{X/Y}$ is a perfect object of $D(\\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$."}
{"_id": "6604", "text": "etale-cohomology-lemma-base-change-f-star Let $f : X \\to S$ be a flat morphism of schemes such that for every geometric point $\\overline{x}$ of $X$ the map $$ \\mathcal{O}_{S, f(\\overline{x})}^{sh} \\longrightarrow \\mathcal{O}_{X, \\overline{x}}^{sh} $$ has geometrically connected fibres. Then for every cartesian diagram of schemes $$ \\xymatrix{ X \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\ S & T \\ar[l]_g } $$ with $g$ quasi-compact and quasi-separated we have $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$ for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$."}
{"_id": "4827", "text": "spaces-morphisms-lemma-point-finite-type-monomorphism Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$. The following are equivalent: \\begin{enumerate} \\item $x$ is a finite type point, \\item there exists an algebraic space $Z$ whose underlying topological space $|Z|$ is a singleton, and a morphism $f : Z \\to X$ which is locally of finite type such that $\\{x\\} = |f|(|Z|)$, and \\item there exists an algebraic space $Z$ and a morphism $f : Z \\to X$ with the following properties: \\begin{enumerate} \\item there is a surjective \\'etale morphism $z : \\Spec(k) \\to Z$ where $k$ is a field, \\item $f$ is locally of finite type, \\item $f$ is a monomorphism, and \\item $x = f(z)$. \\end{enumerate} \\end{enumerate}"}
{"_id": "2311", "text": "restricted-lemma-fully-faithfulness Let $A$ be a Noetherian G-ring. Let $I \\subset A$ be an ideal. Let $B, C$ be finite type $A$-algebras. For any $A$-algebra map $\\varphi : B^\\wedge \\to C^\\wedge$ of $I$-adic completions and any $N \\geq 1$ there exist \\begin{enumerate} \\item an \\'etale ring map $C \\to C'$ which induces an isomorphism $C/IC \\to C'/IC'$, \\item an $A$-algebra map $\\varphi : B \\to C'$ \\end{enumerate} such that $\\varphi$ and $\\psi$ agree modulo $I^N$ into $C^\\wedge = (C')^\\wedge$."}
{"_id": "11453", "text": "obsolete-lemma-bound-primes Let $A$ be a Noetherian local normal domain of dimension $2$. For $f \\in \\mathfrak m$ nonzero denote $\\text{div}(f) = \\sum n_i (\\mathfrak p_i)$ the divisor associated to $f$ on the punctured spectrum of $A$. We set $|f| = \\sum n_i$. There exist integers $N$ and $M$ such that $|f + g| \\leq M$ for all $g \\in \\mathfrak m^N$."}
{"_id": "11962", "text": "intersection-lemma-rational-equivalence Let $X$ be a variety. Let $W \\subset X$ be a subvariety of dimension $k + 1$. Let $f \\in \\mathbf{C}(W)^*$ be a nonzero rational function on $W$. Then $\\text{div}_W(f)$ is rationally equivalent to zero on $X$. Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on $X$."}
{"_id": "6367", "text": "etale-cohomology-theorem-enough-injectives The category of abelian sheaves on a site is an abelian category which has enough injectives."}
{"_id": "1742", "text": "moduli-lemma-pic-functor-qs-lfp The morphism $\\Picardfunctor_{X/B} \\to B$ is quasi-separated and locally of finite presentation."}
{"_id": "7518", "text": "stacks-morphisms-lemma-composition-gerbe Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks. 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": "9720", "text": "local-cohomology-lemma-torsion-tensor-product Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable under specialization. Assume $A$ has finite dimension. Then $$ R\\Gamma_T(K) = R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K $$ for $K \\in D(A)$. For $K, L \\in D(A)$ we have $$ R\\Gamma_T(K \\otimes_A^\\mathbf{L} L) = K \\otimes_A^\\mathbf{L} R\\Gamma_T(L) = R\\Gamma_T(K) \\otimes_A^\\mathbf{L} L = R\\Gamma_T(K) \\otimes_A^\\mathbf{L} R\\Gamma_T(L) $$ If $K$ or $L$ is in $D_T(A)$ then so is $K \\otimes_A^\\mathbf{L} L$."}
{"_id": "2370", "text": "restricted-lemma-composition-rig-smooth 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-smooth, then so is $g \\circ f$."}
{"_id": "8891", "text": "stacks-properties-lemma-open-image-substack Let $\\mathcal X$ be an algebraic stack. Let $U$ be an algebraic space and $U \\to \\mathcal X$ a surjective smooth morphism. For an open immersion $V \\hookrightarrow U$, there exists an algebraic stack $\\mathcal Y$, an open immersion $\\mathcal Y \\to \\mathcal X$, and a surjective smooth morphism $V \\to \\mathcal Y$."}
{"_id": "3160", "text": "quot-lemma-coherent-fibred-in-groupoids In Situation \\ref{situation-coherent} the functor $p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$ is fibred in groupoids."}
{"_id": "3259", "text": "spaces-more-cohomology-lemma-compare-cohomology Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Then \\begin{enumerate} \\item For $K$ in $D(X_\\etale)$ we have $H^n_\\etale(X, \\pi_X^{-1}K) = H^n(X_\\etale, K)$. \\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have $H^n_\\etale(X, L\\pi_X^*K) = H^n(X_\\etale, K)$. \\item For $K$ in $D(X_\\etale)$ we have $H^n_\\etale(Y, \\pi_X^{-1}K) = H^n(Y_\\etale, f_{small}^{-1}K)$. \\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have $H^n_\\etale(Y, L\\pi_X^*K) = H^n(Y_\\etale, Lf_{small}^*K)$. \\item For $M$ in $D((\\textit{Spaces}/X)_\\etale)$ we have $H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^{-1}M)$. \\item For $M$ in $D((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$ we have $H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^*M)$. \\end{enumerate}"}
{"_id": "14154", "text": "sites-modules-lemma-adjoint-pull-push-modules 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. 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": "4277", "text": "sites-cohomology-lemma-inverse-limit-complexes Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{F}_n^\\bullet)$ be an inverse system of complexes of $\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$. Suppose given $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ and an integer $n_0$ such that \\begin{enumerate} \\item every object of $\\mathcal{C}$ has a covering whose members are elements of $\\mathcal{B}$, \\item for every $U \\in \\mathcal{B}$ \\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 property), \\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 property), 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": "1586", "text": "moduli-curves-lemma-extend-curves-to-spaces Let $T \\to B$ be a morphism of algebraic spaces. The category $$ \\Mor_B(T, B\\text{-}\\Curvesstack) = \\Mor(T, \\Curvesstack) $$ is the category of families of curves over $T$."}
{"_id": "11785", "text": "spaces-duality-lemma-dualizing-spaces Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be a dualizing complex on $X$. 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": "2427", "text": "restricted-lemma-formal-modifications-locally-algebraic Let $S$ be a scheme. Let $\\mathfrak X' \\to \\mathfrak X$ be a formal modification (Definition \\ref{definition-formal-modification}) of locally Noetherian formal algebraic spaces over $S$. Given \\begin{enumerate} \\item any adic Noetherian topological ring $A$, \\item any adic morphism $\\text{Spf}(A) \\longrightarrow \\mathfrak X$ \\end{enumerate} there exists a proper morphism $X \\to \\Spec(A)$ of algebraic spaces and an isomorphism $$ \\text{Spf}(A) \\times_{\\mathfrak X} \\mathfrak X' \\longrightarrow X_{/Z} $$ over $\\text{Spf}(A)$ of the base change of $\\mathfrak X$ with the formal completion of $X$ along the ``closed fibre'' $Z = X \\times_{\\Spec(A)} \\text{Spf}(A)_{red}$ of $X$ over $A$."}
{"_id": "8341", "text": "topology-lemma-topological-ring-limits The category of topological rings has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of rings."}
{"_id": "2878", "text": "dualizing-lemma-CM Let $A$ be a Noetherian ring. If $A$ has a dualizing complex $\\omega_A^\\bullet$, then $\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Cohen-Macaulay}\\}$ is a dense open subset of $\\Spec(A)$."}
{"_id": "14082", "text": "more-morphisms-lemma-quasi-finite-Noetherian-universally-open Let $f : X \\to Y$ be a morphism of schemes. If \\begin{enumerate} \\item $f$ is locally quasi-finite, \\item $Y$ is unibranch and locally Noetherian, and \\item every irreducible component of $X$ dominates an irreducible component of $Y$, \\end{enumerate} then $f$ is universally open."}
{"_id": "14028", "text": "more-morphisms-lemma-weakly-etale-formally-unramified Let $f : X \\to Y$ be a weakly \\'etale morphism of schemes. Then $f$ is formally unramified, i.e., $\\Omega_{X/Y} = 0$."}
{"_id": "11460", "text": "obsolete-lemma-make-integral-not-in-ideal Let $\\varphi : R \\to S$ be a ring map. Suppose $t \\in S$ satisfies the relation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$. Let $J \\subset S$ be an ideal such that for at least one $i$ we have $\\varphi(a_i) \\not \\in J$. Then there exists a $u \\in S$, $u \\not\\in J$ such that both $u$ and $ut$ are integral over $R$."}
{"_id": "4126", "text": "pione-lemma-extend-covering-general Let $S$ be a quasi-compact and quasi-separated integral normal scheme with generic point $\\eta$. Let $f : X \\to S$ be a quasi-compact and quasi-separated 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": "13457", "text": "groupoids-quotients-lemma-orbit-space-locally-finite-type-over-base Let $B \\to S$ as in Section \\ref{section-conventions-notation}. Let $j = (t, s) : R \\to U \\times_B U$ be a pre-relation. Assume $R, U$ are locally of finite type over $B$. Let $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Then $\\phi$ is an orbit space for $R$ if and only if the natural map $$ U(k)/\\big(\\text{equivalence relation generated by }j(R(k))\\big) \\longrightarrow X(k) $$ is bijective for all algebraically closed fields $k$ over $B$."}
{"_id": "8575", "text": "sites-lemma-localize-cocontinuous-downstairs Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor. Let $V$ be an object of $\\mathcal{D}$. Let ${}^u_V\\mathcal{I}$ be the category introduced in Section \\ref{section-more-functoriality-PSh}. We have a commutative diagram $$ \\vcenter{ \\xymatrix{ \\,_V^u\\mathcal{I} \\ar[r]_j \\ar[d]_{u'} & \\mathcal{C} \\ar[d]^u \\\\ \\mathcal{D}/V \\ar[r]^-{j_V} & \\mathcal{D} } } \\quad\\text{where}\\quad \\begin{matrix} j : (U, \\psi) \\mapsto U \\\\ u' : (U, \\psi) \\mapsto (\\psi : u(U) \\to V) \\end{matrix} $$ Declare a family of morphisms $\\{(U_i, \\psi_i) \\to (U, \\psi)\\}$ of ${}^u_V\\mathcal{I}$ to be a covering if and only if $\\{U_i \\to U\\}$ is a covering in $\\mathcal{C}$. Then \\begin{enumerate} \\item ${}^u_V\\mathcal{I}$ is a site, \\item $j$ is continuous and cocontinuous, \\item $u'$ is cocontinuous, \\item we get a commutative diagram of topoi $$ \\xymatrix{ \\Sh({}^u_V\\mathcal{I}) \\ar[r]_j \\ar[d]_{f'} & \\Sh(\\mathcal{C}) \\ar[d]^f \\\\ \\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} & \\Sh(\\mathcal{D}) } $$ where $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$), and \\item we have $f'_*j^{-1} = j_V^{-1}f_*$. \\end{enumerate}"}
{"_id": "2548", "text": "examples-lemma-perfect-closure-polynomial-ring Let $A = \\mathbb{F}_p[T]$ be the polynomial ring in one variable over $\\mathbb{F}_p$. Let $A_{perf}$  denote the perfect closure of $A$. Then $A \\rightarrow A_{perf}$ is flat and formally unramified, but not formally \\'etale."}
{"_id": "6518", "text": "etale-cohomology-lemma-cohomology-istar-Z For all $q \\geq 1$, $H_\\etale^q(X, \\bigoplus_{x \\in X_0} {i_x}_* \\underline{\\mathbf{Z}}) = 0$."}
{"_id": "19", "text": "spaces-more-morphisms-lemma-check-universally-injective 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, \\item for every \\'etale morphism $V \\to Y$ the map $|X \\times_Y V| \\to |V|$ is injective. \\end{enumerate} Then $f$ is universally injective."}
{"_id": "1189", "text": "algebra-lemma-smooth-independent-presentation Let $R \\to S$ be a ring map of finite presentation. If for some presentation $\\alpha$ of $S$ over $R$ the naive cotangent complex $\\NL(\\alpha)$ is quasi-isomorphic to a finite projective $S$-module placed in degree $0$, then this holds for any presentation."}
{"_id": "6995", "text": "perfect-lemma-finite-push-pseudo-coherent Let $f : X \\to Y$ be a finite morphism of schemes such that $f_*\\mathcal{O}_X$ is pseudo-coherent as an $\\mathcal{O}_Y$-module\\footnote{This means that $f$ is pseudo-coherent, see More on Morphisms, Lemma \\ref{more-morphisms-lemma-finite-pseudo-coherent}.}. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent."}
{"_id": "14247", "text": "sites-modules-lemma-skyscraper-modules-exact Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Let $p$ be a point of the topos $\\Sh(\\mathcal{C})$. \\begin{enumerate} \\item The functor $p_* : \\textit{Mod}(\\mathcal{O}_p) \\to \\textit{Mod}(\\mathcal{O})$, $M \\mapsto p_*M$ is exact. \\item The canonical surjection $p^{-1}p_*M \\to M$ is $\\mathcal{O}_p$-linear. \\item The functorial direct sum decomposition $p^{-1}p_*M = M \\oplus I(M)$ of Lemma \\ref{lemma-skyscraper-exact} is {\\bf not} $\\mathcal{O}_p$-linear in general. \\end{enumerate}"}
{"_id": "9412", "text": "spaces-descent-lemma-descending-fppf-property-locally-separated The property $\\mathcal{P}(f) =$``$f$ is locally separated'' is fppf local on the base."}
{"_id": "13401", "text": "defos-lemma-inf-obs-map-rel-ringed-topoi 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 $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over $\\mathcal{O}_{\\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. There exists an element $$ o(\\varphi) \\in \\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J}) $$ 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": "12882", "text": "spaces-over-fields-lemma-numerical-intersection-effective-Cartier-divisor 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. 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": "13755", "text": "more-morphisms-lemma-smooth-etale-permanence Let $X \\to Y \\to Z$ be morphisms of schemes. Assume $X \\to Z$ smooth and $Y \\to Z$ \\'etale. Then $X \\to Y$ is smooth."}
{"_id": "4685", "text": "stacks-geometry-lemma-catenary-covers If $\\mathcal{X}$ is a pseudo-catenary locally Noetherian algebraic stack, and if $\\mathcal{Y} \\to \\mathcal{X}$ is a locally of finite type morphism, then there exists a smooth surjective morphism $V \\to \\mathcal{Y}$ whose source is a universally catenary scheme; thus $\\mathcal{Y}$ is again pseudo-catenary."}
{"_id": "11396", "text": "artin-lemma-naive-obstruction-theory-gives-openness Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}^{opp}$ be a category fibred in groupoids. Assume that $\\mathcal{X}$ satisfies (RS*) and that $\\mathcal{X}$ has a naive obstruction theory. Then openness of versality holds for $\\mathcal{X}$ provided the complexes $E_x$ of Definition \\ref{definition-naive-obstruction-theory} have finitely generated cohomology groups for pairs $(A, x)$ where $A$ is of finite type over $S$."}
{"_id": "12479", "text": "topologies-lemma-composition-fppf Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{fppf}$ and morphisms $f : X \\to Y$, $g : Y \\to Z$ we have $g_{big} \\circ f_{big} = (g \\circ f)_{big}$."}
{"_id": "6655", "text": "etale-cohomology-lemma-compare-cohomology Let $f : T \\to S$ be a morphism of schemes. Then \\begin{enumerate} \\item For $K$ in $D(S_\\etale)$ we have $H^n_\\etale(S, \\pi_S^{-1}K) = H^n(S_\\etale, K)$. \\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have $H^n_\\etale(S, L\\pi_S^*K) = H^n(S_\\etale, K)$. \\item For $K$ in $D(S_\\etale)$ we have $H^n_\\etale(T, \\pi_S^{-1}K) = H^n(T_\\etale, f_{small}^{-1}K)$. \\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have $H^n_\\etale(T, L\\pi_S^*K) = H^n(T_\\etale, Lf_{small}^*K)$. \\item For $M$ in $D((\\Sch/S)_\\etale)$ we have $H^n_\\etale(T, M) = H^n(T_\\etale, i_f^{-1}M)$. \\item For $M$ in $D((\\Sch/S)_\\etale, \\mathcal{O})$ we have $H^n_\\etale(T, M) = H^n(T_\\etale, i_f^*M)$. \\end{enumerate}"}
{"_id": "13873", "text": "more-morphisms-lemma-nr-branches-fibre Let $X \\to S$ be a morphism of schemes and $x \\in X$ a point with image $s$. Then \\begin{enumerate} \\item the number of branches of the fibre $X_s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_s$ passing through $u$ taken over elementary \\'etale neighbourhoods $(U, u) \\to (X, x)$, \\item the number of geometric branches of the fibre $X_s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_s$ passing through $u$ taken over \\'etale neighbourhoods $(U, u) \\to (X, x)$, \\item the fibre $X_s$ 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 the fibre $U_s$ 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 $U_s$ passing through $u$. \\end{enumerate}"}
{"_id": "12487", "text": "topologies-lemma-verify-site-ph Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$. Then $(\\textit{Aff}/S)_{ph}$ is a site."}
{"_id": "13486", "text": "spaces-resolve-lemma-iso-completions Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated maximal ideal $\\mathfrak m$. Let $X$ be a decent algebraic space 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 map $\\mathcal{O}_{X, p}^h \\to \\mathcal{O}_{Y, q}^h$ of henselian local rings induces an isomorphism on completions."}
{"_id": "3633", "text": "adequate-lemma-pure-injective-resolutions Let $A$ be a ring. \\begin{enumerate} \\item Any $A$-module has a pure injective resolution. \\end{enumerate} Let $M \\to N$ be a map of $A$-modules. Let $M \\to M^\\bullet$ be a universally exact resolution and let $N \\to I^\\bullet$ be a pure injective resolution. \\begin{enumerate} \\item[(2)] There exists a map of complexes $M^\\bullet \\to I^\\bullet$ inducing the given map $$ M = \\Ker(M^0 \\to M^1) \\to \\Ker(I^0 \\to I^1) = N $$ \\item[(3)] two maps $\\alpha, \\beta : M^\\bullet \\to I^\\bullet$ inducing the same map $M \\to N$ are homotopic. \\end{enumerate}"}
{"_id": "6562", "text": "etale-cohomology-lemma-pullback-ctf Let $\\Lambda$ be a Noetherian ring. Let $f : X \\to Y$ be a morphism of schemes. If $K \\in D_{ctf}(Y_\\etale, \\Lambda)$ then $Lf^*K \\in D_{ctf}(X_\\etale, \\Lambda)$."}
{"_id": "7177", "text": "spaces-flat-lemma-flat 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": "12780", "text": "algebraization-lemma-well-defined-chi-triple The quantity $\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ in (\\ref{equation-chi-triple}) does not depend on the choice of $\\mathcal{F}', \\alpha', \\alpha'_0$ as in Lemma \\ref{lemma-prepare-chi-triple}."}
{"_id": "9593", "text": "groupoids-lemma-immersion-group-schemes-closed-over-field Let $i : G' \\to G$ be an immersion of group schemes over a field $k$. Then $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme of $G$."}
{"_id": "6421", "text": "etale-cohomology-lemma-describe-h1-mun Let $S$ be a scheme. There is a canonical identification $$ H_\\etale^1(S, \\mu_n) = \\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above} $$ if $n$ is invertible on $S$. In general we have $$ H_{fppf}^1(S, \\mu_n) = \\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above}. $$ The same result holds with fppf replaced by syntomic."}
{"_id": "6467", "text": "etale-cohomology-lemma-exactness-lower-shriek Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Let $f : X \\to Y$ be a morphism of schemes. Let $$ f_{big} : \\Sh((\\Sch/X)_\\tau) \\longrightarrow \\Sh((\\Sch/Y)_\\tau) $$ be the corresponding morphism of topoi as in Topologies, Lemma \\ref{topologies-lemma-morphism-big}, \\ref{topologies-lemma-morphism-big-etale}, \\ref{topologies-lemma-morphism-big-smooth}, \\ref{topologies-lemma-morphism-big-syntomic}, or \\ref{topologies-lemma-morphism-big-fppf}. \\begin{enumerate} \\item The functor $f_{big}^{-1} : \\textit{Ab}((\\Sch/Y)_\\tau) \\to \\textit{Ab}((\\Sch/X)_\\tau)$ has a left adjoint $$ f_{big!} : \\textit{Ab}((\\Sch/X)_\\tau) \\to \\textit{Ab}((\\Sch/Y)_\\tau) $$ which is exact. \\item The functor $f_{big}^* : \\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O}) \\to \\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O})$ has a left adjoint $$ f_{big!} : \\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O}) \\to \\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O}) $$ which is exact. \\end{enumerate} Moreover, the two functors $f_{big!}$ agree on underlying sheaves of abelian groups."}
{"_id": "12925", "text": "spaces-divisors-lemma-base-change-relative-assassin-quasi-finite With notation and assumptions as in Lemma \\ref{lemma-base-change-relative-assassin}. Assume $g$ is locally quasi-finite, or more generally that for every $y' \\in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$. Then $\\text{Ass}_{X'/Y'}(\\mathcal{F}')$ is the inverse image of $\\text{Ass}_{X/Y}(\\mathcal{F})$."}
{"_id": "12321", "text": "categories-lemma-spell-out-representable-map-stack-in-groupoids 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. If $F$ is representable then every one of the functors $$ F_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U $$ between fibre categories is faithful."}
{"_id": "14516", "text": "sheaves-lemma-adjoint-pull-push-presheaves-modules 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. Let $\\mathcal{G}$ be a presheaf of $f_*\\mathcal{O}$-modules. Then $$ \\Mor_{\\textit{PMod}(\\mathcal{O})}( \\mathcal{O} \\otimes_{p, f_pf_*\\mathcal{O}} f_p\\mathcal{G}, \\mathcal{F}) = \\Mor_{\\textit{PMod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}). $$ Here we use Lemmas \\ref{lemma-pullback-presheaf-module} and \\ref{lemma-pushforward-presheaf-module}, and we use the map $c_\\mathcal{O} : f_pf_*\\mathcal{O} \\to \\mathcal{O}$ in the definition of the tensor product."}
{"_id": "12456", "text": "topologies-lemma-composition-etale Given schemes $X$, $Y$, $Y$ in $\\Sch_\\etale$ 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": "13478", "text": "spaces-resolve-lemma-equivalence-properties 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 $f$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, if and only if $g_i$ is so for $i = 1, \\ldots, n$."}
{"_id": "13307", "text": "modules-lemma-determinant-as-socle Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a flat and finitely presented $\\mathcal{O}_X$-module. Denote $$ \\det(\\mathcal{F}) \\subset \\wedge^*_{\\mathcal{O}_X}(\\mathcal{F}) $$ the annihilator of $\\mathcal{F} \\subset \\wedge^*_{\\mathcal{O}_X}(\\mathcal{F})$. Then $\\det(\\mathcal{F})$ is an invertible $\\mathcal{O}_X$-module."}
{"_id": "14338", "text": "derham-lemma-multiplication-log Let $p : X \\to S$ 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 $S$. \\begin{enumerate} \\item The maps $\\wedge : \\Omega^p_{X/S} \\times \\Omega^q_{X/S} \\to \\Omega^{p + q}_{X/S}$ extend uniquely to $\\mathcal{O}_X$-bilinear maps $$ \\wedge : \\Omega^p_{X/S}(\\log Y) \\times \\Omega^q_{X/S}(\\log Y) \\to \\Omega^{p + q}_{X/S}(\\log Y) $$ satisfying the Leibniz rule $ \\text{d}(\\omega \\wedge \\eta) = \\text{d}(\\omega) \\wedge \\eta + (-1)^{\\deg(\\omega)} \\omega \\wedge \\text{d}(\\eta)$, \\item with multiplication as in (1) the map $\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}(\\log(Y)$ is a homomorphism of differential graded $\\mathcal{O}_S$-algebras, \\item via the maps in (1) we have $\\Omega^p_{X/S}(\\log Y) = \\wedge^p(\\Omega^1_{X/S}(\\log Y))$, and \\item the map $\\text{Res} : \\Omega^\\bullet_{X/S}(\\log Y) \\to \\Omega^\\bullet_{Y/S}[-1]$ satisfies $$ \\text{Res}(\\omega \\wedge \\eta) = \\text{Res}(\\omega) \\wedge \\eta|_Y + (-1)^{\\deg(\\omega)} \\omega|_Y \\wedge \\text{Res}(\\eta) $$ \\end{enumerate}"}
{"_id": "13950", "text": "more-morphisms-lemma-relative-finite-presentation-characterize 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. The following are equivalent \\begin{enumerate} \\item $\\mathcal{F}$ is of finite presentation relative to $S$, \\item for every affine opens $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$ the $\\mathcal{O}_X(U)$-module $\\mathcal{F}(U)$ is finitely presented relative to $\\mathcal{O}_S(V)$. \\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 $\\mathcal{F}|_U$ is of finite presentation relative to $V$."}
{"_id": "7901", "text": "divisors-lemma-zero-fitting-ideal-omega-unramified Let $f : X \\to S$ be a morphism of schemes which is locally of finite type. The closed subscheme $Z \\subset X$ cut out by the $0$th fitting ideal of $\\Omega_{X/S}$ is exactly the set of points where $f$ is not unramified."}
{"_id": "6555", "text": "etale-cohomology-lemma-restrict-and-shriek-from-etale-c Let $\\Lambda$ be a Noetherian ring. If $j : U \\to X$ is an \\'etale morphism of schemes, then \\begin{enumerate} \\item $K|_U \\in D_c(U_\\etale, \\Lambda)$ if $K \\in D_c(X_\\etale, \\Lambda)$, and \\item $j_!M \\in D_c(X_\\etale, \\Lambda)$ if $M \\in D_c(U_\\etale, \\Lambda)$ and the morphism $j$ is quasi-compact and quasi-separated. \\end{enumerate}"}
{"_id": "10112", "text": "more-algebra-lemma-characterize-injective Let $R$ be a ring. Let $J$ be an $R$-module. The following are equivalent \\begin{enumerate} \\item $J$ is injective, \\item $\\Ext^1_R(M, J) = 0$ for every $R$-module $M$. \\end{enumerate}"}
{"_id": "6020", "text": "flat-lemma-finite-type-flat-along-fibre-variant 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 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 flat over $\\mathcal{O}_{S', s'}$."}
{"_id": "6811", "text": "equiv-lemma-trace-map \\begin{reference} The proof given here follows the argument given in \\cite[Remark 3.4]{MS} \\end{reference} Let $f : X' \\to X$ be a proper birational morphism of integral Noetherian schemes with $X$ regular. The map $\\mathcal{O}_X \\to Rf_*\\mathcal{O}_{X'}$ canonically splits in $D(\\mathcal{O}_X)$."}
{"_id": "6140", "text": "flat-lemma-Noetherian-h-covering Let $X$ be a Noetherian scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a finite family of finite type morphisms. The following are equivalent \\begin{enumerate} \\item $\\coprod_{i \\in I} X_i \\to X$ is universally submersive (Morphisms, Definition \\ref{morphisms-definition-submersive}), and \\item $\\{X_i \\to X\\}_{i \\in I}$ is an h covering. \\end{enumerate}"}
{"_id": "11064", "text": "varieties-lemma-good-intersection 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 $K \\to L$ be an integral ring map. Then there exist $a \\in \\mathfrak m_A$ and $x \\in K$ which map to the same element of $L$ such that $\\mathfrak m_A = \\sqrt{(a)}$."}
{"_id": "12599", "text": "constructions-lemma-structure-morphism-proj Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ has a canonical morphism towards the affine scheme $\\Spec(S_0)$, agreeing with the map on topological spaces coming from Algebra, Definition \\ref{algebra-definition-proj}."}
{"_id": "11504", "text": "obsolete-lemma-good-blowing-up Let $b : X' \\to X$ be the blowing up of a smooth projective scheme over a field $k$ 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 $K_0(Z)$. Then $[Lb^*\\mathcal{O}_Z] = [\\mathcal{O}_E] \\cdot \\alpha''$ in $K_0(X')$ for some $\\alpha'' \\in K_0(X')$."}
{"_id": "5704", "text": "chow-lemma-flat-pullback-divisor-invertible-sheaf 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 $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module. Let $f : X \\to Y$ be a flat morphism of relative dimension $r$. Let $\\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ f^*(c_1(\\mathcal{L}) \\cap [Y]) = c_1(f^*\\mathcal{L}) \\cap [X] $$ in $\\CH_{n + r - 1}(X)$."}
{"_id": "10686", "text": "etale-theorem-flat-map-open Let $Y$ be a locally Noetherian scheme. Let $f : X \\to Y$ be a morphism which is flat and locally of finite type. Then $f$ is (universally) open."}
{"_id": "5831", "text": "chow-lemma-lci-gysin-well-defined 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 $S$. The bivariant class $f^!$ is independent of the choice of the factorization $f = g \\circ i$ with $g$ smooth (provided one exists)."}
{"_id": "11522", "text": "obsolete-lemma-tangent-and-inf We have the following canonical $k$-vector space identifications: \\begin{enumerate} \\item In Deformation Problems, Example \\ref{examples-defos-example-finite-projective-modules} if $x_0 = (k, V)$, then $T_{x_0}\\mathcal{F} = (0)$ and $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{End}_k(V)$ are finite dimensional. \\item In Deformation Problems, Example \\ref{examples-defos-example-representations} if $x_0 = (k, V, \\rho_0)$, then $T_{x_0}\\mathcal{F} = \\Ext^1_{k[\\Gamma]}(V, V) = H^1(\\Gamma, \\text{End}_k(V))$ and $\\text{Inf}_{x_0}(\\mathcal{F}) = H^0(\\Gamma, \\text{End}_k(V))$ are finite dimensional if $\\Gamma$ is finitely generated. \\item In Deformation Problems, Example \\ref{examples-defos-example-continuous-representations} if $x_0 = (k, V, \\rho_0)$, then $T_{x_0}\\mathcal{F} = H^1_{cont}(\\Gamma, \\text{End}_k(V))$ and $\\text{Inf}_{x_0}(\\mathcal{F}) = H^0_{cont}(\\Gamma, \\text{End}_k(V))$ are finite dimensional if $\\Gamma$ is topologically finitely generated. \\item In Deformation Problems, Example \\ref{examples-defos-example-graded-algebras} if $x_0 = (k, P)$, then $T_{x_0}\\mathcal{F}$ and $\\text{Inf}_{x_0}(\\mathcal{F}) = \\text{Der}_k(P, P)$ are finite dimensional if $P$ is finitely generated over $k$. \\end{enumerate}"}
{"_id": "7493", "text": "stacks-morphisms-lemma-quasi-finite-permanence 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-finite and $g$ is quasi-separated and quasi-DM then $f$ is quasi-finite."}
{"_id": "13953", "text": "more-morphisms-lemma-base-change-relative-finite-presentation 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. Let $S' \\to S$ be a morphism of schemes, set $X' = X \\times_S S'$ and denote $\\mathcal{F}'$ the pullback of $\\mathcal{F}$ to $X'$. If $\\mathcal{F}$ is of finite presentation relative to $S$, then $\\mathcal{F}'$ is of finite presentation relative to $S'$."}
{"_id": "4982", "text": "spaces-morphisms-proposition-generic-flatness-reduced-quasi-separated 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 quasi-separated, \\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 and of finite presentation and such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of finite presentation over $\\mathcal{O}_{X_W}$."}
{"_id": "832", "text": "algebra-lemma-ML-also Let $R$ be a ring and $M$ an $R$-module. Then $M$ is Mittag-Leffler if and only if for every finite free $R$-module $F$ and module map $f: F \\to M$, there exists a finitely presented $R$-module $Q$ and a module map $g : F \\to Q$ 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 $N$."}
{"_id": "10218", "text": "more-algebra-lemma-perfect-push-perfect Let $A \\to B$ be a ring map. Assume that $B$ is perfect as an $A$-module. Let $K^\\bullet$ be a perfect complex of $B$-modules. Then $K^\\bullet$ is perfect as a complex of $A$-modules."}
{"_id": "6049", "text": "flat-lemma-explain-why-pure-ML Let $R$ be a henselian local ring with maximal ideal $\\mathfrak m$. Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is countably generated and Mittag-Leffler as an $R$-module. 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": "12572", "text": "pic-lemma-pic-descends Let $k$ be a field. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $k$-rational point, then for any Galois extension $k'/k$ we have $$ \\Pic(X) = \\Pic(X_{k'})^{\\text{Gal}(k'/k)} $$ Moreover the action of $\\text{Gal}(k'/k)$ on $\\Pic(X_{k'})$ is continuous."}
{"_id": "7436", "text": "stacks-morphisms-lemma-composition-integral Compositions of integral, resp.\\ finite morphisms of algebraic stacks are integral, resp.\\ finite."}
{"_id": "8645", "text": "sites-proposition-sheafification-adjoint-topology Let $\\mathcal{C}$ be a category endowed with a topology. Let $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$. 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": "8982", "text": "stacks-lemma-stack-in-groupoids-pushforward Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites. Let $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack in groupoids over $\\mathcal{D}$. Then $u^p\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$."}
{"_id": "10388", "text": "more-algebra-lemma-derived-complete-henselian Let $A$ be a ring derived complete with respect to an ideal $I$. Then $(A, I)$ is a henselian pair."}
{"_id": "2468", "text": "more-groupoids-lemma-groupoid-on-field-open-multiplication 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, then the composition morphism $c : R \\times_{s, U, t} R \\to R$ is open."}
{"_id": "4163", "text": "stacks-cohomology-lemma-relative-leray Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$ be quasi-compact and quasi-separated morphisms of algebraic stacks. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then there exists a spectral sequence with $E_2$-page $$ E_2^{p, q} = R^pg_{\\QCoh, *}(R^qf_{\\QCoh, *}\\mathcal{F}) $$ converging to $R^{p + q}(g \\circ f)_{\\QCoh, *}\\mathcal{F}$."}
{"_id": "7891", "text": "divisors-lemma-relative-weak-assassin-assassin-finite-type 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. Then $\\text{WeakAss}_{X/S}(\\mathcal{F}) = \\text{Ass}_{X/S}(\\mathcal{F})$."}
{"_id": "8751", "text": "examples-defos-lemma-local-ring In Example \\ref{example-schemes} let $X$ be a scheme over $k$ Let $p \\in X$ be a point. With $\\Deformationcategory_{\\mathcal{O}_{X, p}}$ as in Example \\ref{example-rings} there is a natural functor $$ \\Deformationcategory_X \\longrightarrow \\Deformationcategory_{\\mathcal{O}_{X, p}} $$ of deformation categories."}
{"_id": "4199", "text": "sites-cohomology-lemma-cech-cohomology Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$. There is a transformation $$ \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -) \\longrightarrow R\\Gamma(U, -) $$ of functors $\\textit{Ab}(\\mathcal{C}) \\to D^{+}(\\mathbf{Z})$. In particular this gives a transformation of functors $\\check{H}^p(U, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for $\\mathcal{F}$ ranging over $\\textit{Ab}(\\mathcal{C})$."}
{"_id": "7559", "text": "stacks-morphisms-lemma-characterize-etale Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \\begin{enumerate} \\item $f$ is \\'etale, and \\item $f$ is locally of finite presentation, flat, and unramified, \\item $f$ is locally of finite presentation, flat, and its diagonal is \\'etale. \\end{enumerate}"}
{"_id": "13929", "text": "more-morphisms-lemma-category-quasi-projective Let $S$ be a scheme which has an ample invertible sheaf. Let $\\text{QP}_S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma \\ref{lemma-quasi-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 $\\text{QP}_S \\to \\text{QP}_{S'}$, \\item if $X \\in \\text{QP}_S$ and $Y \\in \\text{QP}_X$, then $Y \\in \\text{QP}_S$, \\item the category $\\text{QP}_S$ is closed under fibre products, \\item the category $\\text{QP}_S$ is closed under finite disjoint unions, \\item if $X \\to S$ is projective, then $X \\in \\text{QP}_S$, \\item if $X \\to S$ is quasi-affine of finite type, then $X$ is in $\\text{QP}_S$, \\item if $X \\to S$ is quasi-finite and separated, then $X \\in \\text{QP}_S$, \\item if $X \\to S$ is a quasi-compact immersion, then $X \\in \\text{QP}_S$, \\item add more here. \\end{enumerate}"}
{"_id": "12230", "text": "categories-lemma-preserve-products 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 $y \\to z$. Then \\begin{enumerate} \\item If $M$ and $N$ are diagrams of sets over $\\mathcal{I}$, then $\\colim (M_i \\times N_i) \\to \\colim M_i \\times \\colim N_i$ is surjective, \\item in general colimits of diagrams of sets over $\\mathcal{I}$ do not commute with finite nonempty products. \\end{enumerate}"}
{"_id": "10277", "text": "more-algebra-lemma-pull-relative-pseudo-coherent-module Let $R \\to A \\to B$ be finite type ring maps. Let $m \\in \\mathbf{Z}$. Let $M$ be an $A$-module. Assume $B$ is flat over $A$ and $B$ as a $B$-module is pseudo-coherent relative to $A$. If $M$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$, then $M \\otimes_A B$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$."}
{"_id": "6520", "text": "etale-cohomology-lemma-pullback-on-h2-curve Let $\\pi : X \\to Y$ be a nonconstant morphism of smooth projective curves over an algebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$. The map $$ \\pi^* : H^2_\\etale(Y, \\mu_n) \\longrightarrow H^2_\\etale(X, \\mu_n) $$ is given by multiplication by the degree of $\\pi$."}
{"_id": "6377", "text": "etale-cohomology-theorem-quasi-finite-etale-locally Let $A\\to B$ be finite type ring map and $\\mathfrak p \\subset A$ a prime ideal. Then there exist an \\'etale ring map $A \\to A'$ and a prime $\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$ such that \\begin{enumerate} \\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$, \\item $ B \\otimes_A A' = B_1\\times \\ldots \\times B_r \\times C$, \\item $ A'\\to B_i$ is finite and there exists a unique prime $q_i\\subset B_i$ lying over $\\mathfrak p'$, and \\item all irreducible components of the fibre $\\Spec(C \\otimes_{A'} \\kappa(\\mathfrak p'))$ of $C$ over $\\mathfrak p'$ have dimension at least 1. \\end{enumerate}"}
{"_id": "9520", "text": "decent-spaces-lemma-relative-conditions-local Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{P} \\in \\{(\\beta), decent, reasonable, very\\ reasonable\\}$. The following are equivalent \\begin{enumerate} \\item $f$ 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 $\\mathcal{P}$, and \\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each morphism $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$. \\end{enumerate} If $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$, then this is also equivalent to \\begin{enumerate} \\item[(5)] 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 $\\mathcal{P}$. \\end{enumerate}"}
{"_id": "7318", "text": "sdga-lemma-extension-by-zero-dg In the situation above we have $$ \\Hom_{\\text{Mod}_{(\\mathcal{A}, \\text{d})}^{dg}}( j_!\\mathcal{M}, \\mathcal{N}) = \\Hom_{\\text{Mod}_{(\\mathcal{A}_U, \\text{d})}^{dg}}( \\mathcal{M}, j^*\\mathcal{N}) $$"}
{"_id": "6830", "text": "equiv-lemma-boundedness Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $\\mathcal{A}$ be an abelian category. Let $H : D_{perf}(\\mathcal{O}_X) \\to \\mathcal{A}$ be a homological functor (Derived Categories, Definition \\ref{derived-definition-homological}) such that for all $K$ in $D_{perf}(\\mathcal{O}_X)$ the object $H^i(K)$ is nonzero for only a finite number of $i \\in \\mathbf{Z}$. Then there exists an integer $m \\geq 1$ such that $H^i(\\mathcal{F}) = 0$ for any coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and $i \\not \\in [-m, m]$. Similarly for cohomological functors."}
{"_id": "852", "text": "algebra-lemma-universally-injective-submodule-powerseries Let $R$ be a Noetherian ring and let $M$ be a $R$-module.  Suppose $M$ is a direct sum of countably generated $R$-modules, and suppose there is a universally injective map $M \\to R[[t_1, \\ldots, t_n]]$ for some $n$. Then $M$ is projective."}
{"_id": "7918", "text": "divisors-lemma-flat-pullback-reflexive Let $f : X \\to Y$ be a flat morphism of integral locally Noetherian schemes. Let $\\mathcal{G}$ be a coherent reflexive $\\mathcal{O}_Y$-module. Then $f^*\\mathcal{G}$ is a coherent reflexive $\\mathcal{O}_X$-module."}
{"_id": "4318", "text": "sites-cohomology-lemma-compose-cup-product Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and $f' : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$ be morphisms of ringed topoi. The relative cup product of Remark \\ref{remark-cup-product} is compatible with compositions in the sense that the diagram $$ \\xymatrix{ R(f' \\circ f)_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} R(f' \\circ f)_*L \\ar@{=}[rr] \\ar[d] & & Rf'_*Rf_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} Rf'_*Rf_*L \\ar[d] \\\\ R(f' \\circ f)_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar@{=}[r] & Rf'_*Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) & Rf'_*(Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}  Rf_*L) \\ar[l] } $$ is commutative in $D(\\mathcal{O}'')$ for all $K, L$ in $D(\\mathcal{O})$."}
{"_id": "5215", "text": "morphisms-lemma-universally-catenary-local-rings-universally-catenary Let $S$ be a locally Noetherian scheme. The following are equivalent: \\begin{enumerate} \\item $S$ is universally catenary, and \\item all local rings $\\mathcal{O}_{S, s}$ of $S$ are universally catenary. \\end{enumerate}"}
{"_id": "10730", "text": "etale-proposition-etale-regular Let $A$, $B$ be Noetherian local rings. Let $f : A \\to B$ be an \\'etale homomorphism of local rings. Then $A$ is regular if and only if $B$ is so."}
{"_id": "10682", "text": "etale-theorem-formally-unramified Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type. Then the following are equivalent: \\begin{enumerate} \\item $f$ is unramified, \\item the morphism $f$ is formally unramified: for any affine $S$-scheme $T$ and subscheme $T_0$ of $T$ defined by a square-zero ideal, the natural map $$ \\Hom_S(T, X) \\longrightarrow \\Hom_S(T_0, X) $$ is injective. \\end{enumerate}"}
{"_id": "9251", "text": "models-lemma-numerical-type-rational-point In Situation \\ref{situation-regular-model} assume $C$ has a $K$-rational point. Then \\begin{enumerate} \\item $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k$ over $k$, \\item if $x \\in C_i$, then $H^0(C_i, \\mathcal{O}_{C_i}) = k$ and $m_i = 1$, and \\item $H^0(X_k, \\mathcal{O}_{X_k}) = k$ and $X_k$ has genus equal to the genus of $C$. \\end{enumerate}"}
{"_id": "7689", "text": "schemes-lemma-fibre-products The category of schemes has a final object, products and fibre products. In other words, the category of schemes has finite limits, see Categories, Lemma \\ref{categories-lemma-finite-limits-exist}."}
{"_id": "2381", "text": "restricted-lemma-composition-rig-etale 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-\\'etale, then so is $g \\circ f$."}
{"_id": "1676", "text": "dpa-lemma-ci-well-defined Let $(A, \\mathfrak m)$ be a Noetherian complete local ring. The following are equivalent \\begin{enumerate} \\item for every surjection of local rings $R \\to A$ with $R$ a regular local ring, the kernel of $R \\to A$ is generated by a regular sequence, and \\item for some surjection of local rings $R \\to A$ with $R$ a regular local ring, the kernel of $R \\to A$ is generated by a regular sequence. \\end{enumerate}"}
{"_id": "10513", "text": "more-algebra-lemma-permanence-tame Let $A$ be a discrete valuation ring with fraction field $K$. \\begin{enumerate} \\item If $L/K$ is a finite separable extension which is tamely ramified with respect to $A$, then there exists a Galois extension $M/K$ containing $L$ which is tamely ramified with respect to $A$. \\item If $L_1/K$, $L_2/K$ are finite separable extensions which are tamely ramified with respect to $A$, then there exists a a finite separable extension $L/K$ which is tamely ramified with respect to $A$ containing $L_1$ and $L_2$. \\end{enumerate}"}
{"_id": "216", "text": "spaces-more-morphisms-lemma-regular-immersion-noetherian Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Assume $X$ is locally Noetherian. Then $i$ is Koszul-regular $\\Leftrightarrow$ $i$ is $H_1$-regular $\\Leftrightarrow$ $i$ is quasi-regular."}
{"_id": "10193", "text": "more-algebra-lemma-factor-through-finite-projective Let $R$ be a ring. Let $\\varphi : M \\to N$ be an $R$-module map. If $\\varphi$ factors through a projective module and $M$ is a finite $R$-module, then $\\varphi$ factors through a finite projective module."}
{"_id": "2603", "text": "bootstrap-lemma-morphism-spaces-is-representable-by-spaces Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is representable by algebraic spaces."}
{"_id": "4083", "text": "pione-lemma-faithful In Situation \\ref{situation-local-lefschetz}. Assume one of the following holds \\begin{enumerate} \\item $\\dim(A/\\mathfrak p) \\geq 2$ for every minimal prime $\\mathfrak p \\subset A$ with $f \\not \\in \\mathfrak p$, or \\item every connected component of $U$ meets $U_0$. \\end{enumerate} 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."}
{"_id": "3218", "text": "quot-lemma-complexes-fibred-in-groupoids In Situation \\ref{situation-complexes} the functor $p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$ is fibred in groupoids."}
{"_id": "4445", "text": "fields-lemma-field-semi-simple Every exact sequence of modules over a field splits."}
{"_id": "4162", "text": "stacks-cohomology-lemma-leray Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then there exists a spectral sequence with $E_2$-page $$ E_2^{p, q} = H^p(\\mathcal{Y}, R^qf_{\\QCoh, *}\\mathcal{F}) $$ converging to $H^{p + q}(\\mathcal{X}, \\mathcal{F})$."}
{"_id": "6261", "text": "curves-lemma-globally-generated-curve In Situation \\ref{situation-Cohen-Macaulay-curve} assume that $X$ is integral. Let $0 \\to \\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0$ be a short exact sequence of coherent $\\mathcal{O}_X$-modules with $\\mathcal{F}$ torsion free, $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$, and $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$. Then $\\mathcal{F}$ is globally generated."}
{"_id": "11472", "text": "obsolete-lemma-lqf-f-shriek-composition Let $f : X \\to Y$ and $g : Y \\to Z$ be composable locally quasi-finite morphisms of schemes. Then $g_! \\circ f_! = (g \\circ f)_!$ and $f^! \\circ g^! = (g \\circ f)^!$."}
{"_id": "10875", "text": "spaces-pushouts-lemma-reverse-commutes-with-flat-base-change In Situation \\ref{situation-formal-glueing}. Let $X' \\to X$ be a flat morphism of algebraic spaces. Set $Z' = X' \\times_X Z$ and $Y' = X' \\times_X Y$. The pullbacks $\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_{X'})$ and $\\QCoh(Y \\to X, Z) \\to \\QCoh(Y' \\to X', Z')$ are compatible with the functors (\\ref{equation-reverse}) and \\ref{equation-formal-glueing-modules})."}
{"_id": "6561", "text": "etale-cohomology-lemma-restrict-and-shriek-from-etale-ctf Let $\\Lambda$ be a Noetherian ring. If $j : U \\to X$ is an \\'etale morphism of schemes, then \\begin{enumerate} \\item $K|_U \\in D_{ctf}(U_\\etale, \\Lambda)$ if $K \\in D_{ctf}(X_\\etale, \\Lambda)$, and \\item $j_!M \\in D_{ctf}(X_\\etale, \\Lambda)$ if $M \\in D_{ctf}(U_\\etale, \\Lambda)$ and the morphism $j$ is quasi-compact and quasi-separated. \\end{enumerate}"}
{"_id": "11458", "text": "obsolete-lemma-change-equation-multiply Let $R$ be a ring and let $\\varphi : R[x] \\to S$ be a ring map. Let $t \\in S$. If $t$ is integral over $R[x]$, then there exists an $\\ell \\geq 0$ such that for every $a \\in R$ the element $\\varphi(a)^\\ell t$ is integral over $\\varphi_a : R[y] \\to S$, defined by $y \\mapsto \\varphi(ax)$ and $r \\mapsto \\varphi(r)$ for $r\\in R$."}
{"_id": "13472", "text": "spaces-resolve-theorem-resolve Let $S$ be a scheme. Let $Y$ be a two dimensional integral Noetherian algebraic space over $S$. 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 and $Y^\\nu$ has finitely many singular points $y_1, \\ldots, y_m \\in |Y|$ such that the completions of the henselian local rings $\\mathcal{O}_{Y^\\nu, y_i}^h$ are normal. \\end{enumerate}"}
{"_id": "5661", "text": "chow-lemma-tame-symbol The tame symbol (\\ref{equation-tame-symbol}) satisfies (\\ref{item-bilinear-better}), (\\ref{item-skew-better}), (\\ref{item-normalization}), (\\ref{item-1-x-better}) and hence gives a map $\\partial_A : Q(A)^* \\times Q(A)^* \\to \\kappa(\\mathfrak m)^*$ satisfying (\\ref{item-bilinear}), (\\ref{item-skew}), (\\ref{item-1-x})."}
{"_id": "372", "text": "algebra-lemma-tensor-algebra-localization Let $R$ be a ring and let $S \\subset R$ be a multiplicative subset. Then $S^{-1}T_R(M) = T_{S^{-1}R}(S^{-1}M)$ for any $R$-module $M$. Similar for symmetric and exterior algebras."}
{"_id": "93", "text": "spaces-more-morphisms-lemma-characterize-formally-etale 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 $f$ is formally unramified and the universal first order thickening of $X$ over $Y$ is equal to $X$, \\item $f$ is formally unramified and $\\mathcal{C}_{X/Y} = 0$, and \\item $\\Omega_{X/Y} = 0$ and $\\mathcal{C}_{X/Y} = 0$. \\end{enumerate}"}
{"_id": "11494", "text": "obsolete-lemma-infinite-sequence 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 pairwise distinct finite type points such that $x$ is not versal at $u_n$ for all $n$. After replacing $u_n$ by a subsequence, 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": "10691", "text": "etale-theorem-smooth-etale-over-n-space Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$. If $\\varphi$ is smooth at $x$, then there exist an integer $n \\geq 0$ and affine opens $V \\subset Y$ and $U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$ such that there exists a commutative diagram $$ \\xymatrix{ X \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi & \\mathbf{A}^n_R \\ar[d] \\ar@{=}[r] &  \\Spec(R[x_1, \\ldots, x_n]) \\ar[dl] \\\\ Y & V \\ar[l] \\ar@{=}[r] & \\Spec(R) } $$ where $\\pi$ is \\'etale."}
{"_id": "5109", "text": "weil-proposition-weil-cohomology-theory-classical Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A classical Weil cohomology theory is the same thing as a $\\mathbf{Q}$-linear functor $$ G : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces} $$ of symmetric monoidal categories together with an isomorphism $F[2] \\to G(\\mathbf{1}(1))$ of graded $F$-vector spaces such that in addition \\begin{enumerate} \\item $G(h(X))$ lives in nonnegative degrees, and \\item $\\dim_F G^0(h(X)) = 1$ \\end{enumerate} for any smooth projective variety $X$."}
{"_id": "4088", "text": "pione-lemma-fully-faithful In Situation \\ref{situation-local-lefschetz} assume $A$ has depth $\\geq 3$ and $A$ is henselian or more generally $(A, (f))$ is a henselian pair. Then the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$ is fully faithful."}
{"_id": "2325", "text": "restricted-lemma-base-change-finite-type Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in Lemma \\ref{lemma-finite-type}. Then $P$ is stable under base change as defined in Formal Spaces, Remark \\ref{formal-spaces-remark-base-change-variant-adic-star}."}
{"_id": "7555", "text": "stacks-morphisms-lemma-immersion-unramified An immersion is unramified."}
{"_id": "2524", "text": "examples-lemma-base-change-regular-sequence There exists a local homomorphism of local rings $A \\to B$ and a regular sequence $x, y$ in the maximal ideal of $B$ such that $B/(x, y)$ is flat over $A$, but such that the images $\\overline{x}, \\overline{y}$ of $x, y$ in $B/\\mathfrak m_AB$ do not form a regular sequence, nor even a Koszul-regular sequence."}
{"_id": "7877", "text": "divisors-lemma-minimal-support-in-weakly-ass Let $X$ be a 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 $x \\in \\text{WeakAss}(\\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is weakly associated to $\\mathcal{O}_X$."}
{"_id": "10748", "text": "crystalline-lemma-flat-base-change-divided-power-envelope Let $(A, I, \\gamma)$ be a divided power ring. Let $B \\to B'$ be a homomorphism of $A$-algebras. Assume that \\begin{enumerate} \\item $B/IB \\to B'/IB'$ is flat, and \\item $\\text{Tor}_1^B(B', B/IB) = 0$. \\end{enumerate} Then for any ideal $IB \\subset J \\subset B$ the canonical map $$ D_B(J) \\otimes_B B' \\longrightarrow D_{B'}(JB') $$ is an isomorphism."}
{"_id": "8841", "text": "more-etale-lemma-pseudo-functor Let $f : X \\to Y$, $g : Y \\to Z$, $h : Z \\to T$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then the diagram $$ \\xymatrix{ Rh_! \\circ Rg_! \\circ Rf_! \\ar[r]_{\\gamma_C} \\ar[d]^{\\gamma_A} & R(h \\circ g)_! \\circ Rf_! \\ar[d]_{\\gamma_{A + B}} \\\\ Rh_! \\circ R(g \\circ f)_! \\ar[r]^{\\gamma_{B + C}} & R(h \\circ g \\circ f)_! } $$ of isomorphisms of Lemma \\ref{lemma-shriek-composition} commutes (for the meaning of the $\\gamma$'s see proof)."}
{"_id": "3778", "text": "proetale-lemma-geometric-lift-to-cover Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$. Let $\\mathcal{U} = \\{\\varphi_i : S_i \\to S\\}_{i\\in I}$ be a pro-\\'etale covering. Then there exist $i \\in I$ and geometric point $\\overline{s}_i$ of $S_i$ mapping to $\\overline{s}$."}
{"_id": "7269", "text": "spaces-chow-lemma-first-chern-class In Situation \\ref{situation-setup} let $X/B$ be good. 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 bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}."}
{"_id": "3266", "text": "spaces-more-cohomology-lemma-proper-push-pull-fppf-etale In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have \\begin{enumerate} \\item $a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$, and \\item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves. \\end{enumerate}"}
{"_id": "10163", "text": "more-algebra-lemma-characterize-pseudo-coherent-colimit-ext Let $R$ be a ring. Let $K \\in D^-(R)$. Let $m \\in \\mathbf{Z}$. Then $K$ is $m$-pseudo-coherent if and only if for any filtered colimit $M = \\colim M_i$ of $R$-modules we have $\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and $\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective."}
{"_id": "12711", "text": "algebraization-lemma-formal-functions Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $X$ be a Noetherian scheme over $A$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume that $H^p(X, \\mathcal{F})$ is a finite $A$-module for all $p$. Then there are short exact sequences $$ 0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}/I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})^\\wedge \\to \\lim H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\to 0 $$ of $A$-modules where $H^p(X, \\mathcal{F})^\\wedge$ is the usual $I$-adic completion. If $f$ is proper, then the $R^1\\lim$ term is zero."}
{"_id": "4773", "text": "spaces-morphisms-lemma-i-upper-shriek Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. There is a functor\\footnote{This is likely nonstandard notation.} $i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)"}
{"_id": "15116", "text": "limits-lemma-glueing-near-multiple-closed-points 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 schemes of finite presentation over the scheme $T$."}
{"_id": "8017", "text": "divisors-lemma-make-maps-regular-section Suppose given \\begin{enumerate} \\item $X$ a locally Noetherian scheme, \\item $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module, \\item $s$ a regular meromorphic section of $\\mathcal{L}$, and \\item $\\mathcal{F}$ coherent on $X$ without embedded associated points and $\\text{Supp}(\\mathcal{F}) = X$. \\end{enumerate} Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of denominators of $s$. Let $T \\subset X$ be the union of the supports of $\\mathcal{O}_X/\\mathcal{I}$ and $\\mathcal{L}/s(\\mathcal{I})$ which is a nowhere dense closed subset $T \\subset X$ according to Lemma \\ref{lemma-regular-meromorphic-ideal-denominators}. Then there are canonical injective maps $$ 1 : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F}, \\quad s : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X}\\mathcal{L} $$ whose cokernels are supported on $T$."}
{"_id": "3120", "text": "criteria-lemma-etale-base-change-restriction-of-scalars Let $S$ be a scheme. Let $X \\to Z \\to B$ and $B' \\to B$ be morphisms of algebraic spaces over $S$. Set $Z' = B' \\times_B Z$ and $X' = B' \\times_B X$. Then $$ \\text{Res}_{Z'/B'}(X') = B' \\times_B \\text{Res}_{Z/B}(X) $$ in $\\Sh((\\Sch/S)_{fppf})$."}
{"_id": "9885", "text": "more-algebra-lemma-construct-absolute-integral-closure For any ring $A$ there exists an extension $A \\subset B$ such that \\begin{enumerate} \\item $B$ is a filtered colimit of finite free $A$-algebras, \\item $B$ is free as an $A$-module, and \\item $B$ is absolutely integrally closed. \\end{enumerate}"}
{"_id": "2392", "text": "restricted-lemma-flat-rig-surjective Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $B$ be an $I$-adically complete $A$-algebra. Assume that \\begin{enumerate} \\item the $I$-torsion in $A$ is $0$, \\item $A/I^n \\to B/I^nB$ is flat and of finite type for all $n$. \\end{enumerate} Then $\\text{Spf}(B) \\to \\text{Spf}(A)$ is rig-surjective if and only if $A/I \\to B/IB$ is faithfully flat."}
{"_id": "1876", "text": "derived-lemma-j-is-exact Let $\\mathcal{A}$ be an abelian category with enough injectives. Any resolution functor $j : K^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$ is exact."}
{"_id": "7346", "text": "sdga-lemma-H0-over-D In Definition \\ref{definition-derived-category} the functor $H^0 : K(\\text{Mod}_{(\\mathcal{A}, \\text{d})}) \\to \\textit{Mod}(\\mathcal{O})$ factors through a homological functor $H^0 : D(\\mathcal{A}, \\text{d}) \\to \\textit{Mod}(\\mathcal{O})$."}
{"_id": "4285", "text": "sites-cohomology-lemma-square-triangle-general Let $\\mathcal{C}$ be a site. Consider a commutative diagram $$ \\xymatrix{ \\mathcal{D} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[d] \\\\ \\mathcal{E} \\ar[r] & \\mathcal{G} } $$ of presheaves of sets on $\\mathcal{C}$ and assume that \\begin{enumerate} \\item $\\mathcal{G}^\\# = \\mathcal{E}^\\# \\amalg_{\\mathcal{D}^\\#} \\mathcal{F}^\\#$, and \\item $\\mathcal{D}^\\# \\to \\mathcal{F}^\\#$ is injective. \\end{enumerate} Then there is a canonical distinguished triangle $$ R\\Gamma(\\mathcal{G}, K) \\to R\\Gamma(\\mathcal{E}, K) \\oplus R\\Gamma(\\mathcal{F}, K) \\to R\\Gamma(\\mathcal{D}, K) \\to R\\Gamma(\\mathcal{G}, K)[1] $$ functorial in $K \\in D(\\mathcal{C})$ where $R\\Gamma(\\mathcal{G}, -)$ is the cohomology discussed in Section \\ref{section-limp}."}
{"_id": "7078", "text": "perfect-lemma-triangulated Let $f : X \\to S$ be a morphism of schemes which is flat and locally of finite presentation. The full subcategory of $D(\\mathcal{O}_X)$ consisting of $S$-perfect objects is a saturated\\footnote{Derived Categories, Definition \\ref{derived-definition-saturated}.} triangulated subcategory."}
{"_id": "11322", "text": "spaces-cohomology-lemma-Noetherian-h1-zero Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Assume that for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$. Then $X$ is an affine scheme."}
{"_id": "6058", "text": "flat-lemma-Noetherian-base-change 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 $s \\in S$. If $\\mathcal{O}_{S, s}$ is Noetherian then $\\mathcal{F}$ is pure along $X_s$ if and only if $\\mathcal{F}$ is universally pure along $X_s$."}
{"_id": "13836", "text": "more-morphisms-lemma-connected-along-section-locally-constructible Let $f : X \\to Y$, $s : Y \\to X$ be as in Situation \\ref{situation-connected-along-section}. If $f$ is of finite presentation then $X^0$ is locally constructible in $X$."}
{"_id": "10220", "text": "more-algebra-lemma-flat-base-change-perfect Let $A \\to B$ be a flat ring map. Let $M$ be a perfect $A$-module. Then $M \\otimes_A B$ is a perfect $B$-module."}
{"_id": "10250", "text": "more-algebra-lemma-perfect-modulo-two-ideals Let $R$ be a ring. Let $I, J \\subset R$ be ideals. 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 $K \\otimes_R^\\mathbf{L} R/J$ is perfect in $D(R/J)$. \\end{enumerate} Then $K \\otimes_R^\\mathbf{L} R/IJ$ is perfect in $D(R/IJ)$."}
{"_id": "13834", "text": "more-morphisms-lemma-base-change-connected-along-section Let $f : X \\to Y$, $s : Y \\to X$ be as in Situation \\ref{situation-connected-along-section}. If $g : Y' \\to Y$ is any morphism, consider the base change diagram $$ \\xymatrix{ X' \\ar[r]_{g'} \\ar[d]^{f'} & X \\ar[d]_f \\\\ Y' \\ar@/^1pc/[u]^{s'} \\ar[r]^g & Y \\ar@/_1pc/[u]_s } $$ so that we obtain $(X')^0 \\subset X'$. Then $(X')^0 = (g')^{-1}(X^0)$."}
{"_id": "10986", "text": "varieties-lemma-dominate-valuation-ring-dimension-fibres Let $f : X \\to \\Spec(R)$ be a morphism from an irreducible scheme to the spectrum of a valuation ring. If $f$ is locally of finite type and surjective, then the special fibre is equidimensional of dimension equal to the dimension of the generic fibre."}
{"_id": "2461", "text": "more-groupoids-lemma-property-G-invariant Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \\to U$ be its stabilizer group scheme. Let $\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$. Let $\\mathcal{P}$ be a property of morphisms 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 subscheme such that $G_W \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids, Definition \\ref{groupoids-definition-invariant-open})."}
{"_id": "191", "text": "spaces-more-morphisms-lemma-finite-after-blowing-up Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $B$. Let $X$ be an algebraic space over $S$. Let $U \\subset B$ be a quasi-compact open subspace. Assume \\begin{enumerate} \\item $X \\to B$ is proper, and \\item $X_U \\to U$ is finite locally free. \\end{enumerate} Then there exists a $U$-admissible blowup $B' \\to B$ such that the strict transform of $X$ is finite locally free over $B'$."}
{"_id": "7661", "text": "schemes-lemma-widetilde-constructions Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ be an affine scheme. There are canonical isomorphisms \\begin{enumerate} \\item $ \\widetilde{M \\otimes_R N} \\cong \\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N $, see Modules, Section \\ref{modules-section-tensor-product}. \\item $ \\widetilde{\\text{T}^n(M)} \\cong \\text{T}^n(\\widetilde M) $, $ \\widetilde{\\text{Sym}^n(M)} \\cong \\text{Sym}^n(\\widetilde M) $, and $ \\widetilde{\\wedge^n(M)} \\cong \\wedge^n(\\widetilde M) $, see Modules, Section \\ref{modules-section-symmetric-exterior}. \\item if $M$ is a finitely presented $R$-module, then $ \\SheafHom_{\\mathcal{O}_X}(\\widetilde M, \\widetilde N) \\cong \\widetilde{\\Hom_R(M,  N)} $, see Modules, Section \\ref{modules-section-internal-hom}. \\end{enumerate}"}
{"_id": "5338", "text": "morphisms-lemma-differentials-relative-immersion-smooth Let $i : Z \\to X$ be an immersion of schemes over $S$. Assume that $Z$ is smooth over $S$. Then the canonical exact sequence $$ 0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0 $$ of Lemma \\ref{lemma-differentials-relative-immersion} is short exact."}
{"_id": "14012", "text": "more-morphisms-lemma-perfect-conormal-free-lci Let $i : X \\to Y$ be an immersion. If \\begin{enumerate} \\item $i$ is perfect, \\item $Y$ is locally Noetherian, and \\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, \\end{enumerate} then $i$ is a regular immersion."}
{"_id": "14184", "text": "sites-modules-lemma-special-locally-free Any of the properties (1) -- (8) of Definition \\ref{definition-site-local} is intrinsic (see discussion in Section \\ref{section-intrinsic})."}
{"_id": "14840", "text": "simplicial-lemma-cosk-up Let $\\mathcal{C}$ be a category which has finite limits. \\begin{enumerate} \\item For every $n$ the functor $\\text{sk}_n : \\text{Simp}(\\mathcal{C}) \\to \\text{Simp}_n(\\mathcal{C})$ has a right adjoint $\\text{cosk}_n$. \\item For every $n' \\geq n$ the functor $\\text{sk}_n : \\text{Simp}_{n'}(\\mathcal{C}) \\to \\text{Simp}_n(\\mathcal{C})$ has a right adjoint, namely $\\text{sk}_{n'}\\text{cosk}_n$. \\item For every $m \\geq n \\geq 0$ and every $n$-truncated simplicial object $U$ of $\\mathcal{C}$ we have $\\text{cosk}_m \\text{sk}_m \\text{cosk}_n U = \\text{cosk}_n U$. \\item If $U$ is a simplicial object of $\\mathcal{C}$ such that the canonical map $U \\to \\text{cosk}_n \\text{sk}_nU$ is an isomorphism for some $n \\geq 0$, then the canonical map $U \\to \\text{cosk}_m \\text{sk}_mU$ is an isomorphism for all $m \\geq n$. \\end{enumerate}"}
{"_id": "13633", "text": "duality-lemma-compactly-supported-triangle In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. Let $$ K \\to L \\to M \\to K[1] $$ be a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Then there exists an inverse system of distinguished triangles $$ K_n \\to L_n \\to M_n \\to K_n[1] $$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ such that the pro-systems $(K_n)$, $(L_n)$, and $(M_n)$ give $Rf_!K$, $Rf_!L$, and $Rf_!M$."}
{"_id": "781", "text": "algebra-lemma-tor-welldefined 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 that $G_\\bullet$ 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 $$ H_i(\\alpha) : H_i(F_\\bullet \\otimes_R N) \\longrightarrow H_i(G_\\bullet \\otimes_R N) $$ 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": "13451", "text": "groupoids-quotients-lemma-set-theoretically-invariant-invariant-when-reduced In the situation of Definition \\ref{definition-set-theoretically-invariant}. Let $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$. Assume \\begin{enumerate} \\item $\\phi$ is set-theoretically $R$-invariant, \\item $R$ is reduced, and \\item $X$ is locally separated over $B$. \\end{enumerate} Then $\\phi$ is $R$-invariant."}
{"_id": "9034", "text": "spaces-simplicial-lemma-restriction-injective-to-component-site-module In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ such that $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for all $\\varphi : [n] \\to [m]$. If $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then $\\mathcal{I}_n$ is injective in $\\textit{Mod}(\\mathcal{O}_n)$."}
{"_id": "6055", "text": "flat-lemma-pure-along-X-s In Situation \\ref{situation-pre-pure} the following are equivalent \\begin{enumerate} \\item there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$ of $\\mathcal{F}$ above $s$ where $S^h$ is the henselization of $S$ at $s$, \\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ such that $(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood, and \\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$ such that $T \\to S$ is quasi-finite at $t$. \\end{enumerate}"}
{"_id": "13474", "text": "spaces-resolve-lemma-closed-immersion-on-fibre Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring. Let $g : X \\to Y$ be a morphism in the category (\\ref{equation-modification}). If the induced morphism $X_\\kappa \\to Y_\\kappa$ of special fibres is a closed immersion, then $g$ is a closed immersion."}
{"_id": "7665", "text": "schemes-lemma-kernel-cokernel-quasi-coherent Let $X = \\Spec(R)$ be an affine scheme. Kernels and cokernels of maps of quasi-coherent $\\mathcal{O}_X$-modules are quasi-coherent."}
{"_id": "2757", "text": "spaces-perfect-proposition-Noetherian Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. 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": "90", "text": "spaces-more-morphisms-lemma-formally-etale-not-affine Let $S$ be a scheme. Let $f : X \\to Y$ be a formally \\'etale morphism of algebraic spaces over $S$. Then given any solid commutative diagram $$ \\xymatrix{ X \\ar[d]_f & T \\ar[d]^i \\ar[l]_a \\\\ Y & T' \\ar[l] \\ar@{-->}[lu] } $$ where $T \\subset T'$ is a first order thickening of algebraic spaces over $Y$ 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": "3623", "text": "adequate-lemma-quotient-easy Let $S$ be a scheme. Let $\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ denote the full subcategory consisting of parasitic adequate modules. Then $$ D(\\textit{Adeq}(\\mathcal{O}))/D_\\mathcal{C}(\\textit{Adeq}(\\mathcal{O})) = D(\\QCoh(\\mathcal{O}_S)) $$ and similarly for the bounded versions."}
{"_id": "6286", "text": "curves-lemma-point-over-separable-extension Let $k$ be a field. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \\mathcal{O}_X) = k$ and genus $g \\geq 2$. Then there exists a closed point $x \\in X$ with $\\kappa(x)/k$ separable of degree $\\leq 2g - 2$."}
{"_id": "4135", "text": "pione-proposition-purity-smooth-over-depth2 Let $A \\to B$ be a local homomorphism of local Noetherian rings. Assume $A$ has depth $\\geq 2$, $A \\to B$ is formally smooth for the $\\mathfrak m_B$-adic topology, and $\\dim(B) > \\dim(A)$. For any open $V \\subset Y = \\Spec(B)$ which contains \\begin{enumerate} \\item any prime $\\mathfrak q \\subset B$ such that $\\mathfrak q \\cap A \\not = \\mathfrak m_A$, \\item the prime $\\mathfrak m_A B$ \\end{enumerate} the functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_V$ is an equivalence. In particular purity holds for $B$."}
{"_id": "12911", "text": "spaces-divisors-lemma-check-injective-on-weakass Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. 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 $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$ is injective, or \\item $x \\not \\in \\text{WeakAss}(\\mathcal{F})$. \\end{enumerate} Then $\\varphi$ is injective."}
{"_id": "1919", "text": "derived-lemma-hocolim-subsequence Let $\\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$. Let $n_1 < n_2 < n_3 < \\ldots$ be a sequence of integers. Assume $\\bigoplus K_n$ and $\\bigoplus K_{n_i}$ exist. Then there exists an isomorphism $\\text{hocolim} K_{n_i} \\to \\text{hocolim} K_n$ such that $$ \\xymatrix{ K_{n_i} \\ar[r] \\ar[d]_{\\text{id}} & \\text{hocolim} K_{n_i} \\ar[d] \\\\ K_{n_i} \\ar[r] & \\text{hocolim} K_n } $$ commutes for all $i$."}
{"_id": "8404", "text": "hypercovering-lemma-product-hypercoverings Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\\mathcal{C}$. If $K, L$ are hypercoverings of $X$, then $K \\times L$ is a hypercovering of $X$."}
{"_id": "117", "text": "spaces-more-morphisms-lemma-differentials-formally-unramified-formally-smooth 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 $B$. Assume that $Z$ is formally smooth over $B$. Then the canonical exact sequence $$ 0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0 $$ of Lemma \\ref{lemma-universally-unramified-differentials-sequence} is short exact."}
{"_id": "6556", "text": "etale-cohomology-lemma-pullback-c Let $\\Lambda$ be a Noetherian ring. Let $f : X \\to Y$ be a morphism of schemes. If $K \\in D_c(Y_\\etale, \\Lambda)$ then $Lf^*K \\in D_c(X_\\etale, \\Lambda)$."}
{"_id": "11463", "text": "obsolete-lemma-P1-localize Let $R$ be a ring. Let $F(X, Y) \\in R[X, Y]$ be homogeneous of degree $d$. Let $S = R[X, Y]/(F)$ as a graded ring. Let $\\mathfrak p \\subset R$ be a prime such that some coefficient of $F$ is not in $\\mathfrak p$. There exists an $f \\in R$ $f \\not\\in \\mathfrak p$, an integer $e$, and a $G \\in R[X, Y]_e$ such that multiplication by $G$ induces isomorphisms $(S_n)_f \\to (S_{n + e})_f$ for all $n \\geq d$."}
{"_id": "8305", "text": "topology-lemma-spectral-if-continuous-wrt-constructible-top Let $X$ and $Y$ be spectral spaces. Let $f : X \\to Y$ be a continuous map. Then $f$ is spectral if and only if $f$ is continuous in the constructible topology."}
{"_id": "12614", "text": "constructions-lemma-localization-map-proj With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above. Assume there exists a $g \\in A_0$ such that $\\psi$ induces an isomorphism $A_g \\to B$. Then $U(\\psi) = Y$, $r_\\psi : Y \\to X$ is an open immersion which induces an isomorphism of $Y$ with the inverse image of $D(g) \\subset \\Spec(A_0)$. Moreover the map $\\theta$ is an isomorphism."}
{"_id": "6047", "text": "flat-lemma-explain-why-pure-complete Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $R \\to S$ be a ring map. Let $N$ be an $S$-module. If $N$ is $I$-adically complete, 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 $\\mathfrak q + I S \\not = S$."}
{"_id": "14510", "text": "sheaves-lemma-compose-f-maps-stalks 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. Let $x \\in X$ be a point. The map on stalks $(\\varphi \\circ \\psi)_x : \\mathcal{H}_{g(f(x))} \\to \\mathcal{F}_x$ is the composition $$ \\mathcal{H}_{g(f(x))} \\xrightarrow{\\psi_{f(x)}} \\mathcal{G}_{f(x)} \\xrightarrow{\\varphi_x} \\mathcal{F}_x $$"}
{"_id": "6843", "text": "equiv-lemma-equivalence-coherent-over-field \\begin{reference} Weak version of the result in \\cite{Gabriel} stating that the category of quasi-coherent modules determines the isomorphism class of a scheme. \\end{reference} Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated and $Y$ reduced. If there is a $k$-linear equivalence $F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$ of categories, then there is an isomorphism $f : Y \\to X$ over $k$ and an invertible $\\mathcal{O}_Y$-module $\\mathcal{L}$ such that $F(\\mathcal{F}) = f^*\\mathcal{F} \\otimes \\mathcal{L}$."}
{"_id": "9492", "text": "decent-spaces-lemma-residue-field-henselian-local-ring Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \\in |X|$. The residue field of the henselian local ring of $X$ at $x$ (Definition \\ref{definition-henselian-local-ring}) is the residue field of $X$ at $x$ (Definition \\ref{definition-residue-field})."}
{"_id": "2333", "text": "restricted-lemma-Noetherian-adic-finite-type-red Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$. If $\\varphi$ is adic the following are equivalent \\begin{enumerate} \\item $\\varphi$ satisfies the condition defined in Lemma \\ref{lemma-finite-type-red} and \\item $\\varphi$ satisfies the condition defined in Lemma \\ref{lemma-finite-type}. \\end{enumerate}"}
{"_id": "9625", "text": "groupoids-lemma-isomorphism Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\\alpha$ is an isomorphism."}
{"_id": "7020", "text": "perfect-lemma-pseudo-coherent-hocolim-with-support 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": "14417", "text": "trace-lemma-additivity Let $K\\in DF_{\\text{perf}}(\\Lambda)$ and $f\\in \\text{End}_{DF}(K)$. Then $$ \\text{Tr}(f|_K) = \\sum\\nolimits_{p\\in \\mathbf{Z}} \\text{Tr}(f|_{\\text{gr}^p K}). $$"}
{"_id": "5347", "text": "morphisms-lemma-noetherian-unramified Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian. Then $f$ is unramified if and only if $f$ is G-unramified."}
{"_id": "10848", "text": "spaces-pushouts-lemma-colimit-separated-enough 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 $B$. Assume that \\begin{enumerate} \\item each $X_i$ is separated over $B$, \\item $X = \\colim X_i$ exists in the category of algebraic spaces separated over $B$, \\item $\\coprod X_i \\to X$ is surjective, \\item if $U \\to X$ is an \\'etale separated morphism of algebraic spaces and $U_i = X_i \\times_X U$, then $U = \\colim U_i$ in the category of algebraic spaces separated over $B$, and \\item every object $(U_i \\to X_i)$ of $\\lim X_{i, spaces, \\etale}$ with $U_i \\to X_i$ separated is of the form $U_i = X_i \\times_X U$ for some \\'etale separated morphism of algebraic spaces $U \\to X$. \\end{enumerate} Then $X = \\colim X_i$ in the category of all algebraic spaces over $B$."}
{"_id": "4895", "text": "spaces-morphisms-lemma-unramified-G-unramified Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is G-unramified if and only if $f$ is unramified and locally of finite presentation."}
{"_id": "8592", "text": "sites-lemma-relocalize-morphism-compare 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 $\\mathcal{D}$. Let $c : U \\to u(V)$ be a morphism. Set $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$. Let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be the map induced by $c$. Then the diagram of morphisms of topoi of Lemma \\ref{lemma-relocalize-morphism} agrees with the diagram of morphisms of topoi of Lemma \\ref{lemma-relocalize-morphism-topoi} via the identifications $j_\\mathcal{F} = j_U$ and $j_\\mathcal{G} = j_V$ of Lemma \\ref{lemma-localize-compare}."}
{"_id": "7492", "text": "stacks-morphisms-lemma-base-change-quasi-finite A base change of a quasi-finite morphism is quasi-finite."}
{"_id": "2976", "text": "properties-lemma-locally-regular Let $X$ be a scheme. The following are equivalent: \\begin{enumerate} \\item The scheme $X$ is regular. \\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is Noetherian and regular. \\item There exists an affine open covering $X = \\bigcup U_i$ such that each $\\mathcal{O}_X(U_i)$ is Noetherian and regular. \\item There exists an open covering $X = \\bigcup X_j$ such that each open subscheme $X_j$ is regular. \\end{enumerate} Moreover, if $X$ is regular then every open subscheme is regular."}
{"_id": "118", "text": "spaces-more-morphisms-lemma-two-unramified-morphisms-formally-smooth Let $S$ be a scheme. Let $$ \\xymatrix{ Z \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\ & Y } $$ be a commutative diagram of algebraic spaces over $S$ 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": "13057", "text": "dga-lemma-map-into-dual Let $(A, \\text{d})$ be a differential graded algebra. If $M$ is a left differential graded $A$-module and $N$ is a right differential graded $A$-module, then \\begin{align*} \\Hom_{\\text{Mod}_{(A, \\text{d})}}(N, M^\\vee) & = \\Hom_{\\text{Comp}(\\mathbf{Z})}(N \\otimes_A M, \\mathbf{Q}/\\mathbf{Z}) \\\\ & = \\text{DifferentialGradedBilinear}_A(N \\times M, \\mathbf{Q}/\\mathbf{Z}) \\end{align*}"}
{"_id": "4789", "text": "spaces-morphisms-lemma-intersection-scheme-theoretically-dense Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $U$, $V$ are scheme theoretically dense open subspaces of $X$, then so is $U \\cap V$."}
{"_id": "11505", "text": "obsolete-lemma-gysin-factors-principal Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \\dim_\\delta(X)$. Let $a \\in \\Gamma(X, \\mathcal{O}_X)$ be a nonzero function. Let $i : D = Z(a) \\to X$ be the closed immersion of the zero scheme of $a$. Let $f \\in R(X)^*$. In this case $i^*\\text{div}_X(f) = 0$ in $A_{n - 2}(D)$."}
{"_id": "3155", "text": "quot-lemma-hom-closed Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : X' \\to X$ be a closed immersion of algebraic spaces over $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module and let $\\mathcal{G}'$ be a quasi-coherent $\\mathcal{O}_{X'}$-module. Then $$ \\mathit{Hom}(\\mathcal{F}, i_*\\mathcal{G}') = \\mathit{Hom}(i^*\\mathcal{F}, \\mathcal{G}') $$ as functors on $(\\Sch/B)$."}
{"_id": "9481", "text": "decent-spaces-lemma-locally-constructible Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E \\subset |X|$ be a subset. Then $E$ is \\'etale locally constructible (Properties of Spaces, Definition \\ref{spaces-properties-definition-locally-constructible}) if and only if $E$ is a locally constructible subset of the topological space $|X|$ (Topology, Definition \\ref{topology-definition-constructible})."}
{"_id": "11808", "text": "spaces-duality-lemma-compare Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Assume $X$ and $Y$ are representable and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ be the right adjoint of $Rf_*$ from Lemma \\ref{lemma-twisted-inverse-image}. Let $a_0 : D_\\QCoh(\\mathcal{O}_{Y_0}) \\to D_\\QCoh(\\mathcal{O}_{X_0})$ be the right adjoint of $Rf_*$ from Duality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}. Then  $$ \\xymatrix{ D_\\QCoh(\\mathcal{O}_{X_0}) \\ar@{=}[rrrrrr]_{\\text{Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_X) \\\\ D_\\QCoh(\\mathcal{O}_{Y_0}) \\ar[u]^{a_0} \\ar@{=}[rrrrrr]^{\\text{Derived Categories of Spaces, Lemma \\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\\QCoh(\\mathcal{O}_Y) \\ar[u]_a } $$ is commutative."}
{"_id": "4128", "text": "pione-lemma-affine-etale-over-affine-space Let $k$ be a field of characteristic $p > 0$. Let $X \\to \\mathbf{A}^n_k$ be an \\'etale morphism with $X$ affine. Then there exists a finite \\'etale morphism $X \\to \\mathbf{A}^n_k$."}
{"_id": "7921", "text": "divisors-lemma-reflexive-depth-2 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 each $x \\in X$ one of the following happens \\begin{enumerate} \\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module, or \\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$. \\end{enumerate} \\end{enumerate}"}
{"_id": "13957", "text": "more-morphisms-lemma-sum-relatively-finite-presentation \\begin{slogan} A direct summand of a module inherits the property of being finitely presented relative to a base. \\end{slogan} Let $X \\to S$ be a morphism of schemes which is locally of finite type. Let $\\mathcal{F}, \\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules. If $\\mathcal{F} \\oplus \\mathcal{F}'$ is finitely presented relative to $S$, then so are $\\mathcal{F}$ and $\\mathcal{F}'$."}
{"_id": "14601", "text": "descent-lemma-faithfully-flat-universally-injective Any faithfully flat ring map is universally injective."}
{"_id": "3800", "text": "proetale-lemma-closed-immersion-complement-retrocompact-exact Let $i : Z \\to X$ be a closed immersion of schemes. If $X \\setminus i(Z)$ is a retrocompact open of $X$, then $i_{\\proetale, *}$ is exact."}
{"_id": "7562", "text": "stacks-morphisms-lemma-closed-immersion-proper A closed immersion of algebraic stacks is a proper morphism of algebraic stacks."}
{"_id": "13032", "text": "dga-lemma-total-complex-tensor-product Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$. Denote $A^\\bullet$, $B^\\bullet$ the underlying cochain complexes. As cochain complexes of $R$-modules we have $$ (A \\otimes_R B)^\\bullet = \\text{Tot}(A^\\bullet \\otimes_R B^\\bullet). $$"}
{"_id": "1638", "text": "moduli-curves-lemma-stabilization-morphism Let $g \\geq 2$. There is a morphism of algebraic stacks over $\\mathbf{Z}$ $$ stabilization : \\Curvesstack^{prestable}_g \\longrightarrow \\overline{\\mathcal{M}}_g $$ which sends a prestable family of curves $X \\to S$ of genus $g$ to the stable family $Y \\to S$ asssociated to it in Lemma \\ref{lemma-contract-prestable-to-stable}."}
{"_id": "8756", "text": "examples-defos-lemma-schemes-morphisms-hull In Example \\ref{example-schemes-morphisms} assume $X \\to Y$ is a morphism of proper $k$-schemes. Assume $\\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad A \\longmapsto \\Ob(\\Deformationcategory_{X \\to Y}(A))/\\cong $$ of isomorphism classes of objects has a hull. If $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = \\text{Der}_k(\\mathcal{O}_Y, \\mathcal{O}_Y) = 0$, then $F$ is prorepresentable."}
{"_id": "11239", "text": "cotangent-lemma-fibre-product-tor-independent In the situation above, if $X$ and $Y$ are Tor independent over $B$, then the object $E$ in (\\ref{equation-fibre-product}) is zero. In this case we have $$ L_{X \\times_B Y/B} = Lp^*L_{X/B} \\oplus Lq^*L_{Y/B} $$"}
{"_id": "13730", "text": "more-morphisms-lemma-base-change-formally-smooth A base change of a formally smooth morphism is formally smooth."}
{"_id": "705", "text": "algebra-lemma-one-equation-module Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and $f \\in \\mathfrak m$ an element of the maximal ideal of $R$. Then $$ \\dim(\\text{Supp}(M/fM)) \\leq \\dim(\\text{Supp}(M)) \\leq \\dim(\\text{Supp}(M/fM)) + 1 $$ If $f$ is not in any of the minimal primes of the support of $M$ (for example if $f$ is a nonzerodivisor on $M$), then equality holds for the right inequality."}
{"_id": "14226", "text": "sites-modules-lemma-constructions-invertible Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed space. \\begin{enumerate} \\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible $\\mathcal{O}$-modules, then so is $\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N}$. \\item If $\\mathcal{L}$ is an invertible $\\mathcal{O}$-module, then so is $\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$ and the evaluation map $\\mathcal{L} \\otimes_\\mathcal{O} \\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O}) \\to \\mathcal{O}$ is an isomorphism. \\end{enumerate}"}
{"_id": "3767", "text": "proetale-lemma-quasi-compact-quasi-separated-commutes-direct-sums Let $X$ be a quasi-compact and quasi-separated scheme. The functor $R\\Gamma(X, -) : D^+(X_\\proetale) \\to D(\\textit{Ab})$ commutes with direct sums and homotopy colimits."}
{"_id": "12216", "text": "categories-lemma-connected-colimit-under-X Let $\\mathcal{C}$ be a category. Let $X$ be an object of $\\mathcal{C}$. Let $M : \\mathcal{I} \\to X/\\mathcal{C}$ be a diagram in the category of objects under $X$. If the index category $\\mathcal{I}$ is connected and the colimit of $M$ exists in $X/\\mathcal{C}$, then the colimit of the composition $\\mathcal{I} \\to X/\\mathcal{C} \\to \\mathcal{C}$ exists and is the same."}
{"_id": "4851", "text": "spaces-morphisms-lemma-flat-is-flat-at-all-points 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 $f$ is flat at all points of $|X|$."}
{"_id": "1585", "text": "moduli-curves-theorem-stable-smooth-proper Let $g \\geq 2$. The algebraic stack $\\overline{\\mathcal{M}}_g$ is a Deligne-Mumford stack, proper and smooth over $\\Spec(\\mathbf{Z})$. Moreover, the locus $\\mathcal{M}_g$ parametrizing smooth curves is a dense open substack."}
{"_id": "10416", "text": "more-algebra-lemma-hom-complex-K-flat Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex of projective $R$-modules. Let $K^\\bullet$ be a K-flat complex of $R$-modules. If $P^\\bullet$ is a perfect object of $D(R)$, then $\\Hom^\\bullet(P^\\bullet, K^\\bullet)$ is K-flat and represents $R\\Hom_R(P^\\bullet, K^\\bullet)$."}
{"_id": "10929", "text": "varieties-lemma-image-connected-component \\begin{reference} Email from Ofer Gabber dated June 4, 2016 \\end{reference} Let $k \\subset K$ be an extension of fields. Let $X$ be a scheme over $k$. Denote $p : X_K \\to X$ the projection morphism. Let $\\overline{T} \\subset X_K$ be a connected component. Then $p(\\overline{T})$ is a connected component of $X$."}
{"_id": "11335", "text": "spaces-cohomology-lemma-surjective-finite-morphism-ample Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of algebraic spaces proper over $R$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective. The following are equivalent \\begin{enumerate} \\item $X$ is a scheme and $\\mathcal{L}$ is ample, and \\item $Y$ is a scheme and $f^*\\mathcal{L}$ is ample. \\end{enumerate}"}
{"_id": "12908", "text": "spaces-divisors-lemma-finite-ass 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. Then $\\text{Ass}(\\mathcal{F}) \\cap W$ is finite for every quasi-compact open $W \\subset |X|$."}
{"_id": "8628", "text": "sites-lemma-push-pull-good-case Suppose the functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfies the hypotheses of Proposition \\ref{proposition-get-morphism}, and hence gives rise to a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$. In this case the pullback functor $f^{-1}$ (resp.\\ $u_p$) and the pushforward functor $f_*$ (resp. $u^p$) extend to an adjoint pair of functors on the categories of sheaves (resp.\\ presheaves)  of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp.\\ presheaf) of sets."}
{"_id": "14038", "text": "more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve-separable 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 $1$. Assume the field extensions $K \\subset \\kappa(\\eta)$ are separable for every generic point $\\eta$ of an irreducible component of $X$. Then there exists a finite separable 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": "11291", "text": "spaces-cohomology-lemma-cohomology-with-support-sheaf-on-support Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$. Let $\\mathcal{G}$ be an injective abelian sheaf on $Z_\\etale$. Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$."}