flat-lemma-explain-why-pure-complete Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. If $N$ is $I$-adically complete, then for any $R$-module $M$ and for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R M$ we have $\mathfrak q + I S \not = S$.


sheaves-lemma-compose-f-maps-stalks Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. Let $x \in X$ be a point. The map on stalks $(\varphi \circ \psi)_x : \mathcal{H}_{g(f(x))} \to \mathcal{F}_x$ is the composition $$ \mathcal{H}_{g(f(x))} \xrightarrow{\psi_{f(x)}} \mathcal{G}_{f(x)} \xrightarrow{\varphi_x} \mathcal{F}_x $$


equiv-lemma-equivalence-coherent-over-field \begin{reference} Weak version of the result in \cite{Gabriel} stating that the category of quasi-coherent modules determines the isomorphism class of a scheme. \end{reference} Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated and $Y$ reduced. If there is a $k$-linear equivalence $F : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_Y)$ of categories, then there is an isomorphism $f : Y \to X$ over $k$ and an invertible $\mathcal{O}_Y$-module $\mathcal{L}$ such that $F(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$.


decent-spaces-lemma-residue-field-henselian-local-ring Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The residue field of the henselian local ring of $X$ at $x$ (Definition \ref{definition-henselian-local-ring}) is the residue field of $X$ at $x$ (Definition \ref{definition-residue-field}).


restricted-lemma-Noetherian-adic-finite-type-red Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $\varphi$ is adic the following are equivalent \begin{enumerate} \item $\varphi$ satisfies the condition defined in Lemma \ref{lemma-finite-type-red} and \item $\varphi$ satisfies the condition defined in Lemma \ref{lemma-finite-type}. \end{enumerate}


groupoids-lemma-isomorphism Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $(\mathcal{F}, \alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\alpha$ is an isomorphism.


perfect-lemma-pseudo-coherent-hocolim-with-support Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $K \in D(\mathcal{O}_X)$ supported on $T$. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, and \item $K = \text{hocolim} K_n$ where $K_n$ is perfect, supported on $T$, and $\tau_{\geq -n}K_n \to \tau_{\geq -n}K$ is an isomorphism for all $n$. \end{enumerate}


trace-lemma-additivity Let $K\in DF_{\text{perf}}(\Lambda)$ and $f\in \text{End}_{DF}(K)$. Then $$ \text{Tr}(f|_K) = \sum\nolimits_{p\in \mathbf{Z}} \text{Tr}(f|_{\text{gr}^p K}). $$


morphisms-lemma-noetherian-unramified Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian. Then $f$ is unramified if and only if $f$ is G-unramified.


spaces-pushouts-lemma-colimit-separated-enough Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{I} \to (\Sch/S)_{fppf}$, $i \mapsto X_i$ be a diagram of algebraic spaces over $B$. Assume that \begin{enumerate} \item each $X_i$ is separated over $B$, \item $X = \colim X_i$ exists in the category of algebraic spaces separated over $B$, \item $\coprod X_i \to X$ is surjective, \item if $U \to X$ is an \'etale separated morphism of algebraic spaces and $U_i = X_i \times_X U$, then $U = \colim U_i$ in the category of algebraic spaces separated over $B$, and \item every object $(U_i \to X_i)$ of $\lim X_{i, spaces, \etale}$ with $U_i \to X_i$ separated is of the form $U_i = X_i \times_X U$ for some \'etale separated morphism of algebraic spaces $U \to X$. \end{enumerate} Then $X = \colim X_i$ in the category of all algebraic spaces over $B$.


spaces-morphisms-lemma-unramified-G-unramified Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is G-unramified if and only if $f$ is unramified and locally of finite presentation.


sites-lemma-relocalize-morphism-compare Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$. Let $c : U \to u(V)$ be a morphism. Set $\mathcal{G} = h_V^\#$ and $\mathcal{F} = h_U^\# = f^{-1}h_V^\#$. Let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be the map induced by $c$. Then the diagram of morphisms of topoi of Lemma \ref{lemma-relocalize-morphism} agrees with the diagram of morphisms of topoi of Lemma \ref{lemma-relocalize-morphism-topoi} via the identifications $j_\mathcal{F} = j_U$ and $j_\mathcal{G} = j_V$ of Lemma \ref{lemma-localize-compare}.


stacks-morphisms-lemma-base-change-quasi-finite A base change of a quasi-finite morphism is quasi-finite.


properties-lemma-locally-regular Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is regular. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian and regular. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian and regular. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is regular. \end{enumerate} Moreover, if $X$ is regular then every open subscheme is regular.


spaces-more-morphisms-lemma-two-unramified-morphisms-formally-smooth Let $S$ be a scheme. Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d]^f \\ & Y } $$ be a commutative diagram of algebraic spaces over $S$ where $i$ and $j$ are formally unramified and $f$ is formally smooth. Then the canonical exact sequence $$ 0 \to \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0 $$ of Lemma \ref{lemma-two-unramified-morphisms} is exact and locally split.


dga-lemma-map-into-dual Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left differential graded $A$-module and $N$ is a right differential graded $A$-module, then \begin{align*} \Hom_{\text{Mod}_{(A, \text{d})}}(N, M^\vee) & = \Hom_{\text{Comp}(\mathbf{Z})}(N \otimes_A M, \mathbf{Q}/\mathbf{Z}) \\ & = \text{DifferentialGradedBilinear}_A(N \times M, \mathbf{Q}/\mathbf{Z}) \end{align*}


spaces-morphisms-lemma-intersection-scheme-theoretically-dense Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $U$, $V$ are scheme theoretically dense open subspaces of $X$, then so is $U \cap V$.


obsolete-lemma-gysin-factors-principal Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \dim_\delta(X)$. Let $a \in \Gamma(X, \mathcal{O}_X)$ be a nonzero function. Let $i : D = Z(a) \to X$ be the closed immersion of the zero scheme of $a$. Let $f \in R(X)^*$. In this case $i^*\text{div}_X(f) = 0$ in $A_{n - 2}(D)$.


quot-lemma-hom-closed Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : X' \to X$ be a closed immersion of algebraic spaces over $B$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module and let $\mathcal{G}'$ be a quasi-coherent $\mathcal{O}_{X'}$-module. Then $$ \mathit{Hom}(\mathcal{F}, i_*\mathcal{G}') = \mathit{Hom}(i^*\mathcal{F}, \mathcal{G}') $$ as functors on $(\Sch/B)$.


decent-spaces-lemma-locally-constructible Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E \subset |X|$ be a subset. Then $E$ is \'etale locally constructible (Properties of Spaces, Definition \ref{spaces-properties-definition-locally-constructible}) if and only if $E$ is a locally constructible subset of the topological space $|X|$ (Topology, Definition \ref{topology-definition-constructible}).


spaces-duality-lemma-compare Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Assume $X$ and $Y$ are representable and let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Let $a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ be the right adjoint of $Rf_*$ from Lemma \ref{lemma-twisted-inverse-image}. Let $a_0 : D_\QCoh(\mathcal{O}_{Y_0}) \to D_\QCoh(\mathcal{O}_{X_0})$ be the right adjoint of $Rf_*$ from Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image}. Then  $$ \xymatrix{ D_\QCoh(\mathcal{O}_{X_0}) \ar@{=}[rrrrrr]_{\text{Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_X) \\ D_\QCoh(\mathcal{O}_{Y_0}) \ar[u]^{a_0} \ar@{=}[rrrrrr]^{\text{Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}} & & & & & & D_\QCoh(\mathcal{O}_Y) \ar[u]_a } $$ is commutative.


pione-lemma-affine-etale-over-affine-space Let $k$ be a field of characteristic $p > 0$. Let $X \to \mathbf{A}^n_k$ be an \'etale morphism with $X$ affine. Then there exists a finite \'etale morphism $X \to \mathbf{A}^n_k$.


divisors-lemma-reflexive-depth-2 Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is reflexive, \item for each $x \in X$ one of the following happens \begin{enumerate} \item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module, or \item $\text{depth}(\mathcal{F}_x) \geq 2$. \end{enumerate} \end{enumerate}


more-morphisms-lemma-sum-relatively-finite-presentation \begin{slogan} A direct summand of a module inherits the property of being finitely presented relative to a base. \end{slogan} Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}, \mathcal{F}'$ be quasi-coherent $\mathcal{O}_X$-modules. If $\mathcal{F} \oplus \mathcal{F}'$ is finitely presented relative to $S$, then so are $\mathcal{F}$ and $\mathcal{F}'$.


descent-lemma-faithfully-flat-universally-injective Any faithfully flat ring map is universally injective.


proetale-lemma-closed-immersion-complement-retrocompact-exact Let $i : Z \to X$ be a closed immersion of schemes. If $X \setminus i(Z)$ is a retrocompact open of $X$, then $i_{\proetale, *}$ is exact.


stacks-morphisms-lemma-closed-immersion-proper A closed immersion of algebraic stacks is a proper morphism of algebraic stacks.


dga-lemma-total-complex-tensor-product Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$ be differential graded algebras over $R$. Denote $A^\bullet$, $B^\bullet$ the underlying cochain complexes. As cochain complexes of $R$-modules we have $$ (A \otimes_R B)^\bullet = \text{Tot}(A^\bullet \otimes_R B^\bullet). $$


moduli-curves-lemma-stabilization-morphism Let $g \geq 2$. There is a morphism of algebraic stacks over $\mathbf{Z}$ $$ stabilization : \Curvesstack^{prestable}_g \longrightarrow \overline{\mathcal{M}}_g $$ which sends a prestable family of curves $X \to S$ of genus $g$ to the stable family $Y \to S$ asssociated to it in Lemma \ref{lemma-contract-prestable-to-stable}.


examples-defos-lemma-schemes-morphisms-hull In Example \ref{example-schemes-morphisms} assume $X \to Y$ is a morphism of proper $k$-schemes. Assume $\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor $$ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_{X \to Y}(A))/\cong $$ of isomorphism classes of objects has a hull. If $\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_Y, \mathcal{O}_Y) = 0$, then $F$ is prorepresentable.


cotangent-lemma-fibre-product-tor-independent In the situation above, if $X$ and $Y$ are Tor independent over $B$, then the object $E$ in (\ref{equation-fibre-product}) is zero. In this case we have $$ L_{X \times_B Y/B} = Lp^*L_{X/B} \oplus Lq^*L_{Y/B} $$


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


algebra-lemma-one-equation-module Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and $f \in \mathfrak m$ an element of the maximal ideal of $R$. Then $$ \dim(\text{Supp}(M/fM)) \leq \dim(\text{Supp}(M)) \leq \dim(\text{Supp}(M/fM)) + 1 $$ If $f$ is not in any of the minimal primes of the support of $M$ (for example if $f$ is a nonzerodivisor on $M$), then equality holds for the right inequality.


sites-modules-lemma-constructions-invertible Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. \begin{enumerate} \item If $\mathcal{L}$, $\mathcal{N}$ are invertible $\mathcal{O}$-modules, then so is $\mathcal{L} \otimes_\mathcal{O} \mathcal{N}$. \item If $\mathcal{L}$ is an invertible $\mathcal{O}$-module, then so is $\SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O})$ and the evaluation map $\mathcal{L} \otimes_\mathcal{O} \SheafHom_\mathcal{O}(\mathcal{L}, \mathcal{O}) \to \mathcal{O}$ is an isomorphism. \end{enumerate}


proetale-lemma-quasi-compact-quasi-separated-commutes-direct-sums Let $X$ be a quasi-compact and quasi-separated scheme. The functor $R\Gamma(X, -) : D^+(X_\proetale) \to D(\textit{Ab})$ commutes with direct sums and homotopy colimits.


categories-lemma-connected-colimit-under-X Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$. Let $M : \mathcal{I} \to X/\mathcal{C}$ be a diagram in the category of objects under $X$. If the index category $\mathcal{I}$ is connected and the colimit of $M$ exists in $X/\mathcal{C}$, then the colimit of the composition $\mathcal{I} \to X/\mathcal{C} \to \mathcal{C}$ exists and is the same.


spaces-morphisms-lemma-flat-is-flat-at-all-points Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if $f$ is flat at all points of $|X|$.


moduli-curves-theorem-stable-smooth-proper Let $g \geq 2$. The algebraic stack $\overline{\mathcal{M}}_g$ is a Deligne-Mumford stack, proper and smooth over $\Spec(\mathbf{Z})$. Moreover, the locus $\mathcal{M}_g$ parametrizing smooth curves is a dense open substack.


more-algebra-lemma-hom-complex-K-flat Let $R$ be a ring. Let $P^\bullet$ be a bounded above complex of projective $R$-modules. Let $K^\bullet$ be a K-flat complex of $R$-modules. If $P^\bullet$ is a perfect object of $D(R)$, then $\Hom^\bullet(P^\bullet, K^\bullet)$ is K-flat and represents $R\Hom_R(P^\bullet, K^\bullet)$.


varieties-lemma-image-connected-component \begin{reference} Email from Ofer Gabber dated June 4, 2016 \end{reference} Let $k \subset K$ be an extension of fields. Let $X$ be a scheme over $k$. Denote $p : X_K \to X$ the projection morphism. Let $\overline{T} \subset X_K$ be a connected component. Then $p(\overline{T})$ is a connected component of $X$.


spaces-cohomology-lemma-surjective-finite-morphism-ample Let $R$ be a Noetherian ring. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $f$ is finite and surjective. The following are equivalent \begin{enumerate} \item $X$ is a scheme and $\mathcal{L}$ is ample, and \item $Y$ is a scheme and $f^*\mathcal{L}$ is ample. \end{enumerate}


spaces-divisors-lemma-finite-ass Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \cap W$ is finite for every quasi-compact open $W \subset |X|$.


sites-lemma-push-pull-good-case Suppose the functor $u : \mathcal{C} \to \mathcal{D}$ satisfies the hypotheses of Proposition \ref{proposition-get-morphism}, and hence gives rise to a morphism of sites $f : \mathcal{D} \to \mathcal{C}$. In this case the pullback functor $f^{-1}$ (resp.\ $u_p$) and the pushforward functor $f_*$ (resp. $u^p$) extend to an adjoint pair of functors on the categories of sheaves (resp.\ presheaves)  of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp.\ presheaf) of sets.


more-morphisms-lemma-normalized-base-change-with-reduced-fibre-over-curve-separable Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Assume the field extensions $K \subset \kappa(\eta)$ are separable for every generic point $\eta$ of an irreducible component of $X$. Then there exists a finite separable extension $L/K$ such that in the diagram $$ \xymatrix{ Y \ar[rd]_g \ar[r]_-\nu & X \times_S T \ar[d] \ar[r] & X \ar[d]_f \\ & T \ar[r] & S } $$ the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\Spec(L)$ and $\nu : Y \to X \times_S T$ is the normalization.


spaces-cohomology-lemma-cohomology-with-support-sheaf-on-support Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{G}$ be an injective abelian sheaf on $Z_\etale$. Then $\mathcal{H}^p_Z(i_*\mathcal{G}) = 0$ for $p > 0$.