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}.


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 $f_*\mathcal{F}$.


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 $D_i$.


stacks-morphisms-lemma-open-immersion-locally-finite-presentation An open immersion is locally of finite presentation.


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 $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.


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 $Y'$.


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 $D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d})$.


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.


constructions-lemma-relative-proj-d In Situation \ref{situation-relative-proj}. The functor $F_d$ is representable by a scheme.


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.


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}


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 $S$.


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}


spaces-descent-lemma-descending-property-universally-submersive The property $\mathcal{P}(f) =$``$f$ is universally submersive'' is fpqc local on the base.


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 $$ \text{pd}_R(E) = \text{pd}_R(R/I) + \text{pd}_{R/I}(E) $$


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 $\mathcal{X}' \subset \bigcup_{i \in I'} \mathcal{X}_i$.


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 $\text{Fit}_r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$.


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 $U, R \in \textit{Spaces}/S$.


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.


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 $1$.


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}


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 $\mathcal{I}$.


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.


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 $\mathcal{X}$.


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 $\QCoh(\mathcal{O}_\mathcal{Y})$.


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 $D(\mathcal{O}_X)$.


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.


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}).


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 $Lf^*$.


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 $\dim(A/\mathfrak p) = d$.


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.


spaces-more-morphisms-lemma-base-change-nodal The property of being at-worst-nodal of relative dimension $1$ is preserved under base change.


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.


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.


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}


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 $X \times_{B, h} T$.


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.


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}


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}


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 $D(\mathcal{O}_U)$.


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 $(U, R, s, t, c)$.


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}


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 $g \circ f$.


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}


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., $Z \cap Y = Z \times_X Y$.


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.


stacks-morphisms-lemma-composition-unramified The composition of unramified morphisms is unramified.


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}


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}.


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 $K \otimes_\mathcal{O}^\mathbf{L} L$.


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 $\Spec(k) \to U/G$.


etale-cohomology-lemma-site-fpqc The collection of fpqc coverings on the category of schemes satisfies the axioms of site.


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 $B = B_i \otimes_{A_i} A$.


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}


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 $\mathcal{F}$.


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.


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}


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.


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 $\Spec(S \otimes_R K) \subset \Spec(S)$.


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.


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 $X$.


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 $\textit{Mod}(\mathcal{O})$.


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.


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.


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.


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 $\textit{Mod}(\mathcal{O})$.


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}


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.


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.


constructions-lemma-projective-space-grassmannian Let $n \geq 1$. There is a canonical isomorphism $\mathbf{G}(n, n + 1) = \mathbf{P}^n_\mathbf{Z}$.


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 $x \in X$.


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.


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}


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.


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 $$ V \longmapsto H^n((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}}) $$


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}


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.


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.


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.


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.


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 $X$.


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 $N^\bullet$.


stacks-morphisms-lemma-composition-proper A composition of proper morphisms is proper.


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 $S'_3$.


stacks-morphisms-lemma-base-change-unramified A base change of an unramified morphism is unramified.


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 $S$.


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}


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}


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 $(X \times_Z Y)_{red} = (X_{red} \times_{Z_{red}} Y_{red})_{red}$.


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 $U = X \setminus D$.


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.


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.


spaces-morphisms-lemma-syntomic-open A syntomic morphism is universally open.


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 $\dim(B_{\mathfrak m_2}) = 2$.


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 $$ \xymatrix{ R \times_{s, U, s} R \times_{s, U, s} R \ar@<3ex>[r]_-{\text{pr}_{12}} \ar@<0ex>[r]_-{\text{pr}_{02}} \ar@<-3ex>[r]_-{\text{pr}_{01}} \ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} & R \times_{s, U, s} R \ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0} \ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} & R \ar[dd]^t \\ \\ R \times_{s, U, t} R \ar@<3ex>[r]_{\text{pr}_1} \ar@<0ex>[r]_c \ar@<-3ex>[r]_{\text{pr}_0} & R \ar@<1ex>[r]_s \ar@<-2ex>[r]_t & U } $$


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.


more-algebra-lemma-quasi-excellent-nagata A quasi-excellent ring is Nagata.


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 $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$.


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.


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 $P$.


etale-proposition-etale-depth Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $\text{depth}(A) = \text{depth}(B)$


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 $X_1$.


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.


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 $\zeta$.


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.


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.


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.


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}


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.


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 $X$.


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 $S/S$.


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 $K'_0(X) = K'_0(R)$.


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 $x \in \text{WeakAss}(f^*\mathcal{G})$.


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 $\mathbf{1} = (\Spec(k), 1, 0)$.


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.


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}


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.


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.


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 $$ R^pg_*\mathcal{F} = \colim_{i \in I} h_i^{-1}R^pg_{i, *}(f_{i, *}\mathcal{F}) $$


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 $$ \text{length}_R(R/(x)) = \sum\nolimits_i \text{ord}_{R/\mathfrak q_i}(x) $$


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.


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}


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 $R \to S'_{\mathfrak m'}$.


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}


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 $B$.


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 $i_0, \ldots, i_p$.


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 $\mathfrak m$.


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 $T_\etale$.


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}).


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.


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 $i = 1, \ldots, n$.


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.


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}


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}.


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.


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.


spaces-more-morphisms-lemma-flat-base-change-pseudo-coherent A flat base change of a pseudo-coherent morphism is pseudo-coherent.


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}.


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 $B \to A/I$.


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 $(g')_*(f')^!\mathcal{F} = f^!g_*\mathcal{F}$.


stacks-morphisms-lemma-composition-lci The composition of local complete intersection morphisms is a local complete intersection.


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.


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.


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 $$ 0 \to H_1(\NL_{K/A}) \to \mathfrak m_B/\mathfrak m_B^2 \to \Omega_{B/A} \otimes_B K \to \Omega_{K/A} \to 0 $$


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}).


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.


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.


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.


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 $\dim_{x_0}(\mathcal{X}) = \dim_x(\mathcal{V})$.


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.


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.


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.


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}


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.


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 $f'$.


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 $$ \textit{Adeq}(\mathcal{O}) / \mathcal{C} = \QCoh(\mathcal{O}_S). $$


derived-lemma-exact-functor-additive An exact functor of pre-triangulated categories is additive.


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.


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 $S$.


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.


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}).


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.


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 $\beta(X)$.


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 $$ \deg(\mathcal{E}) = \deg(\wedge^n(\mathcal{E})) = \deg(\det(\mathcal{E})) $$


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.


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}.)


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.


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.


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 $\mathcal{N}^\bullet$.


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.


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 $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.


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.


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 $K$.


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 $f \circ s' = b|_{b^{-1}(W)}$.


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 $Ra_*$.


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.


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).


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.


examples-stacks-lemma-finite-etale-stack The functor $$ p : \textit{F\'Et} \longrightarrow (\Sch/S)_{fppf} $$ defines a stack in groupoids over $(\Sch/S)_{fppf}$.


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 $C = \underline{\Spec}(\text{Sym}^*(\mathcal{C}))$.


stacks-morphisms-lemma-open-immersion-etale An open immersion is \'etale.


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}


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 $X$.


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 $i$.


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.


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.


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.


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}


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.


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.


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 $\Sh(S_{Zar}) \cong \Sh(S_{affine, Zar})$.


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 $\mathcal{F} \cong s_*\mathcal{L}$.


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 $\mathcal{F}$.


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 $\mathcal{O}_{Y, y}^{sh}$.


spaces-more-morphisms-lemma-base-change-formally-smooth A base change of a formally smooth morphism is formally smooth.


moduli-curves-lemma-prestable-curves-smooth The morphisms $\Curvesstack^{prestable} \to \Spec(\mathbf{Z})$ and $\Curvesstack^{prestable}_g \to \Spec(\mathbf{Z})$ are smooth.


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}


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 $\Pic(T) = \mathbf{Z}$.


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.


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.


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}


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 $\Sh(S_\etale) \to G\textit{-Sets}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$.


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 $(A, I)$.


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).


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 $S_i$.


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.


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 $H^0(X, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O}_X)}(L, M)$.


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 $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})$.


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 $V$.


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.


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 $$ \text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F})) $$


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 $\textit{Coh}(U, I\mathcal{O}_U)$.


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}).


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}


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}


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 $\mathcal{B}$.


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.


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 $f'$.


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.


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.


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.


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.


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}


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.


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 $$ \text{WeakAss}_Y(f_*\mathcal{F}) = f(\text{WeakAss}_X(\mathcal{F})) $$


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 $X \times_Y Z$.


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.


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.


divisors-lemma-flat-torsion-free Let $X$ be an integral scheme. Any flat quasi-coherent $\mathcal{O}_X$-module is torsion free.


examples-lemma-not-essentially-surjective The canonical map $\mathcal{X}(S) \to \lim \mathcal{X}(S_n)$ is not essentially surjective.


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 $Z$.


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 $m_{T_1', U_1} = m_{T_2', U_2}$.


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.


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.


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.


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.


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 $i \in \mathbf{Z}$.


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 $\dim_x(X) = \dim_y(Y) + d$.


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}


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 $X/S$.


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.


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 $\Spec(B/gB) \to \Spec(A)$.


etale-cohomology-lemma-higher-direct-jstar-Gm For any $q \geq 1$, $R^q j_*\mathbf{G}_{m, \eta} = 0$.