trace-theorem-deformation-rings

(See \cite[Theorem 3.5]{dJ-conjecture}) Suppose $$ \rho_0: \pi_1(X)\to \text{GL}_n(\mathbf{F}_l) $$ is a continuous, $l\neq p$. Assume \begin{enumerate} \item Conj. holds for $X$, \item $\rho_0 |_{\pi_1(X_{\overline{k}})}$ abs. irred., and \item $l$ does not divide $n$. \end{enumerate} Then the universal deformation ring $R_{\text{univ}}$ of $\rho_0$ is finite flat over $\mathbf{Z}_l$.


trace-lemma-baffling

Let $X$ be a scheme and $g : X \to X$ a morphism. Assume that for all $\varphi : U \to X$ \'etale, there is an isomorphism $$ \xymatrix{ U \ar[rd]_\varphi \ar[rr]^-\sim & & {U \times_{\varphi, X, g} X} \ar[ld]^{\text{pr}_2} \\ & X } $$ functorial in $U$. Then $g$ induces the identity on cohomology (for any sheaf).


trace-lemma-two-actions-agree

In the situation above denote $\alpha : X \to \Spec(k)$ the structure morphism. Consider the stalk $(R^j\alpha_*\mathcal{F})_{\Spec(\bar k)}$ endowed with its natural Galois action as in \'Etale Cohomology, Section \ref{etale-cohomology-section-galois-action-stalks}. Then the identification $$ (R^j\alpha_*\mathcal{F})_{\Spec(\bar k)} \cong H^j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}}) $$ from \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-higher-direct-images} is an isomorphism of $G_k$-modules.


trace-lemma-derived-categories

Morphisms between objects in the derived category. \begin{enumerate} \item Let $I^\bullet \in \text{Comp}^+(\mathcal{A})$ with $I^n$ injective for all $n \in \mathbf{Z}$. Then $$ \Hom_{D(\mathcal{A})}(K^\bullet, I^\bullet) = \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet). $$ \item Let $P^\bullet \in \text{Comp}^-(\mathcal{A})$ with $P^n$ is projective for all $n \in \mathbf{Z}$. Then $$ \Hom_{D(\mathcal{A})}(P^\bullet, K^\bullet) = \Hom_{K(\mathcal{A})}(P^\bullet, K^\bullet). $$ \item If $\mathcal{A}$ has enough injectives and $\mathcal{I} \subset \mathcal{A}$ is the additive subcategory of injectives, then $ D^+(\mathcal{A})\cong K^+(\mathcal{I}) $ (as triangulated categories). \item If $\mathcal{A}$ has enough projectives and $\mathcal{P} \subset \mathcal{A}$ is the additive subcategory of projectives, then $ D^-(\mathcal{A}) \cong K^-(\mathcal{P}). $ \end{enumerate}


trace-lemma-filtered-derived-category

If $\mathcal{A}$ has enough injectives, then $DF^+(\mathcal{A}) \cong K^+(\mathcal{I})$, where $\mathcal{I}$ is the full additive subcategory of $\text{Fil}^f(\mathcal{A})$ consisting of filtered injective objects. Similarly, if $\mathcal{A}$ has enough projectives, then $DF^-(\mathcal{A}) \cong K^+(\mathcal{P})$, where $\mathcal P$ is the full additive subcategory of $\text{Fil}^f(\mathcal{A})$ consisting of filtered projective objects.


trace-lemma-additive-filtered-finite-projective

Let $P \in \text{Fil}^f(\text{Mod}_\Lambda)$ be filtered finite projective, and $f : P \to P$ an endomorphism in $\text{Fil}^f(\text{Mod}_\Lambda)$. Then $$ \text{Tr}(f|_P) = \sum\nolimits_p \text{Tr}(f|_{\text{gr}^p(P)}). $$


trace-lemma-characterize-perfect

Let $\Lambda$ be a left Noetherian ring and $K\in D(\Lambda)$. Then $K$ is perfect if and only if the two following conditions hold: \begin{enumerate} \item $K$ has finite $\text{Tor}$-dimension, and \item for all $i \in \mathbf{Z}$, $H^i(K)$ is a finite $\Lambda$-module. \end{enumerate}


trace-lemma-epsilon

Let $e\in G$ denote the neutral element. The map $$ \begin{matrix} \Lambda[G] & \longrightarrow & \Lambda^{\natural}\\ \sum \lambda_g\cdot g & \longmapsto & \lambda_e \end{matrix} $$ factors through $\Lambda[G]^\natural$. We denote $\varepsilon : \Lambda[G]^\natural\to \Lambda^\natural$ the induced map.


trace-lemma-lambda-trace

Let $f : P\to P$ be an endomorphism of the finite projective $\Lambda[G]$-module $P$. Then $$ \text{Tr}_{\Lambda}(f; P) = \# G \cdot \text{Tr}_\Lambda^G(f; P). $$


trace-lemma-A-module-structure

The map $A\to \Lambda$ defines an $A$-module structure on $\Lambda^\natural$.


trace-lemma-diagonal-action-projective-module

Let $P$ be a finite projective $A[G]$-module and $M$ a $\Lambda[G]$-module, finite projective as a $\Lambda$-module. Then $P \otimes_A M$ is a finite projective $\Lambda[G]$-module, for the structure induced by the diagonal action of $G$.


trace-lemma-multiplicative-trace

With assumptions as in Lemma \ref{lemma-diagonal-action-projective-module}, let $u\in \text{End}_{A[G]}(P)$ and $v\in \text{End}_{\Lambda[G]}(M)$. Then $$ \text{Tr}_\Lambda^G \left(u \otimes v; P \otimes_A M\right) = \text{Tr}_A^G(u; P)\cdot \text{Tr}_\Lambda(v;M). $$


trace-lemma-gamma-z-gamma-trace

Let $P$ be a $\Lambda[\Gamma]$-module, finite and projective as a $\Lambda[G]$-module, and $\gamma \in \Gamma$. Then $$ \text{Tr}_{\Lambda}(\gamma, P) = \# Z_\gamma \cdot \text{Tr}_\Lambda^{Z_\gamma}\left(\gamma, P\right). $$


trace-lemma-weak-trace

Let $P$ be an $A[\Gamma]$-module, finite projective as $A[G]$-module. Let $M$ be a $\Lambda[\Gamma]$-module, finite projective as a $\Lambda$-module. Then $$ \text{Tr}_{\Lambda}^{Z_\gamma}(\gamma, P \otimes_A M) = \text{Tr}_A^{Z_\gamma}(\gamma, P)\cdot \text{Tr}_\Lambda(\gamma, M). $$


trace-lemma-trivial-trace

Let $P$ be a $\Lambda[\Gamma]$-module, finite projective as $\Lambda[G]$-module. Then the coinvariants $P_G = \Lambda \otimes_{\Lambda[G]} P$ form a finite projective $\Lambda$-module, endowed with an action of $\Gamma/G = \mathbf{N}$. Moreover, we have $$ \text{Tr}_\Lambda(1; P_G) = \sum\nolimits'_{\gamma \mapsto 1} \text{Tr}_\Lambda^{Z_\gamma}(\gamma, P) $$ where $\sum_{\gamma\mapsto 1}'$ means taking the sum over the $G$-conjugacy classes in $\Gamma$.


trace-lemma-eventually-constant

Let $\{\mathcal{G}_n\}_{n\geq 1}$ be an inverse system of constructible $\mathbf{Z}/\ell^n\mathbf{Z}$-modules. Suppose that for all $k\geq 1$, the maps $$ \mathcal{G}_{n+1}/\ell^k \mathcal{G}_{n+1}\to \mathcal{G}_n /\ell^k \mathcal{G}_n $$ are isomorphisms for all $n\gg 0$ (where the bound possibly depends on $k$). In other words, assume that the system $\{\mathcal{G}_n/\ell^k\mathcal{G}_n\}_{n\geq 1}$ is eventually constant, and call $\mathcal{F}_k$ the corresponding sheaf. Then the system $\left\{\mathcal{F}_k\right\}_{k\geq 1}$ forms a $\mathbf{Z}_\ell$-sheaf on $X$.


trace-lemma-identify-h2c

There is a canonical isomorphism $$ H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)=(M)_{\pi_1(X_{\overline{k}}, \overline\eta)}(-1) $$ as $\text{Gal}(k^{^{sep}}/k)$-modules.


trace-proposition-compare-filtered-graded

In the situation above, we have $$ \text{gr}^p \circ RT = RT \circ \text{gr}^p $$ where the $RT$ on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram $$ \xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} \ar[d]_{\text{gr}^p} & DF^+(\mathcal{B}) \ar[d]^{\text{gr}^p}\\ D^+(\mathcal{A}) \ar[r]^{RT} & D^+(\mathcal{B}).} $$


trace-proposition-trace-well-defined

Let $K\in D_{perf}(\Lambda)$ and $f\in \text{End}_{D(\Lambda)}(K)$. Then the trace $\text{Tr}(f)\in \Lambda^\natural$ is well defined.


trace-proposition-projective-curve-constructible-cohomology

Let $X$ be a projective curve over a field $k$, $\Lambda$ a finite ring and $K\in D_{ctf}(X, \Lambda)$. Then $R\Gamma(X_{\bar k}, K)\in D_{perf}(\Lambda)$.


trace-proposition-integral-normal-fundamental-group

Let $X$ be an integral normal Noetherian scheme. Let $\overline y\to X$ be an algebraic geometric point lying over the generic point $\eta\in X$. Then $$ \pi_x(X, \overline \eta) = Gal(M/\kappa(\eta)) $$ ($\kappa(\eta)$, function field of $X$) where $$ \kappa(\overline \eta)\supset M\supset \kappa(\eta) = k(X) $$ is the max sub-extension such that for every finite sub extension $M\supset L\supset \kappa(\eta)$ the normalization of $X$ in $L$ is finite \'etale over $X$.


trace-proposition-curve-kpi1

Let $X/k$ as before but $X_{\overline{k}}\neq \mathbf{P}^1_{\overline{k}}$ The functors $ (M, \rho)\mapsto H_c^{2-i}(X_{\overline{k}}, \mathcal{F}_\rho) $ are the left derived functor of $(M, \rho)\mapsto H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)$ so $$ H_c^{2-i}(X_{\overline{k}}, \mathcal{F}_\rho) = H_i(\pi_1(X_{\overline{k}}, \overline \eta), M)(-1) $$ Moreover, there is a derived version, namely $$ R\Gamma_c(X_{\overline{k}}, \mathcal{F}_\rho) = LH_0(\pi_1(X_{\overline{k}}, \overline \eta), M(-1)) = M(-1) \otimes_{\Lambda[[\pi_1(X_{\overline{k}}, \overline \eta)]]}^\mathbf{L} \Lambda $$ in $D(\Lambda[[\widehat{\mathbf{Z}}]])$. Similarly, the functors $(M, \rho)\mapsto H^i(X_{\overline{k}}, \mathcal{F}_\rho)$ are the right derived functor of $(M, \rho)\mapsto M^{\pi_1(X_{\overline{k}}, \overline \eta)}$ so $$ H^i(X_{\overline{k}}, \mathcal{F}_\rho) = H^i(\pi_1(X_{\overline{k}}, \overline \eta), M) $$ Moreover, in this case there is a derived version too.


trace-proposition-cusp-forms-finite

If $\Lambda$ is Noetherian then $C(\Lambda)$ is a finitely generated $\Lambda$-module. Moreover, if $\Lambda$ is a field with prime subfield $\mathbf{F} \subset \Lambda$ then $$ C(\Lambda)=(C(\mathbf{F}))\otimes_{\mathbf{F}}\Lambda $$ compatibly with $T_v$ acting.


trace-proposition-finite-set-frobenii-generate-topologically

There exists a finite set $x_1, \ldots, x_n$ of closed points of $X$ such that set of {\bf all} frobenius elements corresponding to these points topologically generate $\pi_1(X)$.


sheaves-lemma-product-presheaves

Let $X$ be a topological space. The category of presheaves of sets on $X$ has products (see Categories, Definition \ref{categories-definition-product}). Moreover, the set of sections of the product $\mathcal{F} \times \mathcal{G}$ over an open $U$ is the product of the sets of sections of $\mathcal{F}$ and $\mathcal{G}$ over $U$.


sheaves-lemma-abelian-presheaves

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets. Consider the following types of structure on $\mathcal{F}$: \begin{enumerate} \item For every open $U$ the structure of an abelian group on $\mathcal{F}(U)$ such that all restriction maps are abelian group homomorphisms. \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ (see Example \ref{example-singleton-presheaf}) satisfying all the axioms of $+, -, 0$ in a usual abelian group. \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$, a map of presheaves $- : \mathcal{F} \to \mathcal{F}$ and a map $0 : * \to \mathcal{F}$ such that for each open $U \subset X$ the quadruple $(\mathcal{F}(U), +, -, 0)$ is an abelian group, \item A map of presheaves $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ such that for every open $U \subset X$ the map $+ : \mathcal{F}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an abelian group. \end{enumerate} There are natural bijections between the collections of types of data (1) - (4) above.


sheaves-lemma-adjointness-tensor-restrict-presheaves

With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$ \Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other.


sheaves-lemma-sheaf-subset-stalks

Let $\mathcal{F}$ be a sheaf of sets on the topological space $X$. For every open $U \subset X$ the map $$ \mathcal{F}(U) \longrightarrow \prod\nolimits_{x \in U} \mathcal{F}_x $$ is injective.


sheaves-lemma-stalk-abelian-presheaf

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of abelian groups on $X$. There exists a unique structure of an abelian group on $\mathcal{F}_x$ such that for every $U \subset X$ open, $x\in U$ the map $\mathcal{F}(U) \to \mathcal{F}_x$ is a group homomorphism. Moreover, $$ \mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U) $$ holds in the category of abelian groups.


sheaves-lemma-stalk-presheaf-values-in-category

Let $\mathcal{C}$ be a category. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $F$ is faithful, and \item directed colimits exist in $\mathcal{C}$ and $F$ commutes with them. \end{enumerate} Let $X$ be a topological space. Let $x \in X$. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$. Then $$ \mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U) $$ exists in $\mathcal{C}$. Its underlying set is equal to the stalk of the underlying presheaf of sets of $\mathcal{F}$. Furthermore, the construction $\mathcal{F} \mapsto \mathcal{F}_x$ is a functor from the category of presheaves with values in $\mathcal{C}$ to $\mathcal{C}$.


sheaves-lemma-stalk-module

Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $x \in X$. The canonical map $\mathcal{O}_x \times \mathcal{F}_x \to \mathcal{F}_x$ coming from the multiplication map $\mathcal{O} \times \mathcal{F} \to \mathcal{F}$ defines a $\mathcal{O}_x$-module structure on the abelian group $\mathcal{F}_x$.


sheaves-lemma-stalk-tensor-presheaf-modules

Let $X$ be a topological space. Let $\mathcal{O} \to \mathcal{O}'$ be a morphism of presheaves of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $x \in X$. We have $$ \mathcal{F}_x \otimes_{\mathcal{O}_x} \mathcal{O}'_x = (\mathcal{F} \otimes_{p, \mathcal{O}} \mathcal{O}')_x $$ as $\mathcal{O}'_x$-modules.


sheaves-lemma-list-algebraic-structures

The following categories, endowed with the obvious forgetful functor, define types of algebraic structures: \begin{enumerate} \item The category of pointed sets. \item The category of abelian groups. \item The category of groups. \item The category of monoids. \item The category of rings. \item The category of $R$-modules for a fixed ring $R$. \item The category of Lie algebras over a fixed field. \end{enumerate}


sheaves-lemma-properties-algebraic-structures

Let $(\mathcal{C}, F)$ be a type of algebraic structure. \begin{enumerate} \item $\mathcal{C}$ has a final object $0$ and $F(0) = \{ * \}$. \item $\mathcal{C}$ has products and $F(\prod A_i) = \prod F(A_i)$. \item $\mathcal{C}$ has fibre products and $F(A \times_B C) = F(A)\times_{F(B)}F(C)$. \item $\mathcal{C}$ has equalizers, and if $E \to A$ is the equalizer of $a, b : A \to B$, then $F(E) \to F(A)$ is the equalizer of $F(a), F(b) : F(A) \to F(B)$. \item $A \to B$ is a monomorphism if and only if $F(A) \to F(B)$ is injective. \item if $F(a) : F(A) \to F(B)$ is surjective, then $a$ is an epimorphism. \item given $A_1 \to A_2 \to A_3 \to \ldots$, then $\colim A_i$ exists and $F(\colim A_i) = \colim F(A_i)$, and more generally for any filtered colimit. \end{enumerate}


sheaves-lemma-image-contained-in

Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $A, B, C \in \Ob(\mathcal{C})$. Let $f : A \to B$ and $g : C \to B$ be morphisms of $\mathcal{C}$. If $F(g)$ is injective, and $\Im(F(f)) \subset \Im(F(g))$, then $f$ factors as $f = g \circ t$ for some morphism $t : A \to C$.


sheaves-lemma-points-exactness

Let $X$ be a topological space. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of sheaves of sets on $X$. \begin{enumerate} \item The map $\varphi$ is a monomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is injective. \item The map $\varphi$ is an epimorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is surjective. \item The map $\varphi$ is an isomorphism in the category of sheaves if and only if for all $x \in X$ the map $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$ is bijective. \end{enumerate}


sheaves-lemma-characterize-epi-mono

Let $X$ be a topological space. \begin{enumerate} \item Epimorphisms (resp.\ monomorphisms) in the category of presheaves are exactly the surjective (resp.\ injective) maps of presheaves. \item Epimorphisms (resp.\ monomorphisms) in the category of sheaves are exactly the surjective (resp.\ injective) maps of sheaves, and are exactly those maps with are surjective (resp.\ injective) on all the stalks. \item The sheafification of a surjective (resp.\ injective) morphism of presheaves of sets is surjective (resp.\ injective). \end{enumerate}


sheaves-lemma-check-homomorphism-stalks

let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $\mathcal{F}$, $\mathcal{G}$ are sheaves on $X$ with values in $\mathcal{C}$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of the underlying sheaves of sets. If for all points $x \in X$ the map $\mathcal{F}_x \to \mathcal{G}_x$ is a morphism of algebraic structures, then $\varphi$ is a morphism of sheaves of algebraic structures.


sheaves-lemma-sheafification-sheaf

The presheaf $\mathcal{F}^{\#}$ is a sheaf.


sheaves-lemma-stalk-sheafification

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets on $X$. Let $x \in X$. Then $\mathcal{F}_x = \mathcal{F}^\#_x$.


sheaves-lemma-separated-presheaf-into-sheaf

Let $X$ be a topological space. A presheaf $\mathcal{F}$ is separated (see Definition \ref{definition-separated}) if and only if the canonical map $\mathcal{F} \to \mathcal{F}^\#$ is injective.


sheaves-lemma-diagram-fibre-product

Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of sets on $X$. Let $U \subset X$ be open. There is a canonical fibre product diagram $$ \xymatrix{ \mathcal{F}^\#(U) \ar[d] \ar[r] & \Pi(\mathcal{F})(U) \ar[d] \\ \prod_{x \in U} \mathcal{F}_x \ar[r] & \prod_{x \in U} \Pi(\mathcal{F})_x } $$ where the maps are the following: \begin{enumerate} \item The left vertical map has components $\mathcal{F}^\#(U) \to \mathcal{F}^\#_x = \mathcal{F}_x$ where the equality is Lemma \ref{lemma-stalk-sheafification}. \item The top horizontal map comes from the map of presheaves $\mathcal{F} \to \Pi(\mathcal{F})$ described in Section \ref{section-sheafification}. \item The right vertical map has obvious component maps $\Pi(\mathcal{F})(U) \to \Pi(\mathcal{F})_x$. \item The bottom horizontal map has components $\mathcal{F}_x \to \Pi(\mathcal{F})_x$ which come from the map of presheaves $\mathcal{F} \to \Pi(\mathcal{F})$ described in Section \ref{section-sheafification}. \end{enumerate}


sheaves-lemma-sheafify-abelian-presheaf

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. Then there exists a unique structure of abelian sheaf on $\mathcal{F}^\#$ such that $\mathcal{F} \to \mathcal{F}^\#$ is a morphism of abelian presheaves. Moreover, the following adjointness property holds $$ \Mor_{\textit{PAb}(X)}(\mathcal{F}, i(\mathcal{G})) = \Mor_{\textit{Ab}(X)}(\mathcal{F}^\#, \mathcal{G}). $$


sheaves-lemma-sheafification-presheaf-modules

Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf $\mathcal{O}$-modules. Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$. Let $\mathcal{F}^\#$ be the sheafification of $\mathcal{F}$ as a presheaf of abelian groups. There exists a map of sheaves of sets $$ \mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\# $$ which makes the diagram $$ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# } $$ commute and which makes $\mathcal{F}^\#$ into a sheaf of $\mathcal{O}^\#$-modules. In addition, if $\mathcal{G}$ is a sheaf of $\mathcal{O}^\#$-modules, then any morphism of presheaves of $\mathcal{O}$-modules $\mathcal{F} \to \mathcal{G}$ (into the restriction of $\mathcal{G}$ to a $\mathcal{O}$-module) factors uniquely as $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ where $\mathcal{F}^\# \to \mathcal{G}$ is a morphism of $\mathcal{O}^\#$-modules.


sheaves-lemma-adjointness-tensor-restrict

With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection $$ \Hom_{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \Hom_{\mathcal{O}_2}( \mathcal{O}_2 \otimes_{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) $$ In other words, the restriction and change of rings functors are adjoint to each other.


sheaves-lemma-stalk-tensor-sheaf-modules

Let $X$ be a topological space. Let $\mathcal{O} \to \mathcal{O}'$ be a morphism of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf $\mathcal{O}$-modules. Let $x \in X$. We have $$ \mathcal{F}_x \otimes_{\mathcal{O}_x} \mathcal{O}'_x = (\mathcal{F} \otimes_\mathcal{O} \mathcal{O}')_x $$ as $\mathcal{O}'_x$-modules.


sheaves-lemma-pushforward-sheaf

Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$. Then $f_*\mathcal{F}$ is a sheaf on $Y$.


sheaves-lemma-pushforward-composition

Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps of topological spaces. The functors $(g \circ f)_*$ and $g_* \circ f_*$ are equal (on both presheaves and sheaves of sets).


sheaves-lemma-pullback-presheaves

Let $f : X \to Y$ be a continuous map. There exists a functor $f_p : \textit{PSh}(Y) \to \textit{PSh}(X)$ which is left adjoint to $f_*$. For a presheaf $\mathcal{G}$ it is determined by the rule $$ f_p\mathcal{G}(U) = \colim_{f(U) \subset V} \mathcal{G}(V) $$ where the colimit is over the collection of open neighbourhoods $V$ of $f(U)$ in $Y$. The colimits are over directed partially ordered sets. (The restriction mappings of $f_p\mathcal{G}$ are explained in the proof.)


sheaves-lemma-stalk-pullback-presheaf

Let $f : X \to Y$ be a continuous map. Let $x \in X$. Let $\mathcal{G}$ be a presheaf of sets on $Y$. There is a canonical bijection of stalks $(f_p\mathcal{G})_x = \mathcal{G}_{f(x)}$.


sheaves-lemma-stalk-pullback

Let $x \in X$. Let $\mathcal{G}$ be a sheaf of sets on $Y$. There is a canonical bijection of stalks $(f^{-1}\mathcal{G})_x = \mathcal{G}_{f(x)}$.


sheaves-lemma-pullback-composition

Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps of topological spaces. The functors $(g \circ f)^{-1}$ and $f^{-1} \circ g^{-1}$ are canonically isomorphic. Similarly $(g \circ f)_p \cong f_p \circ g_p$ on presheaves.


sheaves-lemma-f-map

Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. There are canonical bijections between the following three sets: \begin{enumerate} \item The set of maps $\mathcal{G} \to f_*\mathcal{F}$. \item The set of maps $f^{-1}\mathcal{G} \to \mathcal{F}$. \item The set of $f$-maps $\xi : \mathcal{G} \to \mathcal{F}$. \end{enumerate}


sheaves-lemma-pullback-abelian-stalk

Let $f : X \to Y$ be a continuous map. \begin{enumerate} \item Let $\mathcal{G}$ be an abelian presheaf on $Y$. Let $x \in X$. The bijection $\mathcal{G}_{f(x)} \to (f_p\mathcal{G})_x$ of Lemma \ref{lemma-stalk-pullback-presheaf} is an isomorphism of abelian groups. \item Let $\mathcal{G}$ be an abelian sheaf on $Y$. Let $x \in X$. The bijection $\mathcal{G}_{f(x)} \to (f^{-1}\mathcal{G})_x$ of Lemma \ref{lemma-stalk-pullback} is an isomorphism of abelian groups. \end{enumerate}


sheaves-lemma-f-map-sets-algebraic-structures

Let $f : X \to Y$ be a continuous map of topological spaces. Suppose given sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map of underlying sheaves of sets. If for every $V \subset Y$ open the map of sets $\varphi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$ is the effect of a morphism in $\mathcal{C}$ on underlying sets, then $\varphi$ comes from a unique $f$-morphism between sheaves of algebraic structures.


sheaves-lemma-pushforward-presheaf-module

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f_*\mathcal{O} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} $$ which turns $f_*\mathcal{F}$ into a presheaf of $f_*\mathcal{O}$-modules. This construction is functorial in $\mathcal{F}$.


sheaves-lemma-pullback-presheaf-module

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f_p\mathcal{O} \times f_p\mathcal{G} \longrightarrow f_p\mathcal{G} $$ which turns $f_p\mathcal{G}$ into a presheaf of $f_p\mathcal{O}$-modules. This construction is functorial in $\mathcal{G}$.


sheaves-lemma-adjoint-push-pull-presheaves-modules

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $Y$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a presheaf of $f_p\mathcal{O}$-modules. Then $$ \Mor_{\textit{PMod}(f_p\mathcal{O})}(f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PMod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-presheaf-module} and \ref{lemma-pushforward-presheaf-module}, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module via the map $i_\mathcal{O} : \mathcal{O} \to f_*f_p\mathcal{O}$ (defined first in the proof of Lemma \ref{lemma-pullback-presheaves}).


sheaves-lemma-pushforward-module

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. The pushforward $f_*\mathcal{F}$, as defined in Lemma \ref{lemma-pushforward-presheaf-module} is a sheaf of $f_*\mathcal{O}$-modules.


sheaves-lemma-pullback-module

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. There is a natural map of underlying presheaves of sets $$ f^{-1}\mathcal{O} \times f^{-1}\mathcal{G} \longrightarrow f^{-1}\mathcal{G} $$ which turns $f^{-1}\mathcal{G}$ into a sheaf of $f^{-1}\mathcal{O}$-modules.


sheaves-lemma-adjoint-push-pull-modules

Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a sheaf of rings on $Y$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{F}$ be a sheaf of $f^{-1}\mathcal{O}$-modules. Then $$ \Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we think of $f_*\mathcal{F}$ as an $\mathcal{O}$-module by restriction via $\mathcal{O} \to f_*f^{-1}\mathcal{O}$.


sheaves-lemma-adjoint-pullback-pushforward-modules

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. There is a canonical bijection $$ \Hom_{\mathcal{O}_X}(f^*\mathcal{G}, \mathcal{F}) = \Hom_{\mathcal{O}_Y}(\mathcal{G}, f_*\mathcal{F}). $$ In other words: the functor $f^*$ is the left adjoint to $f_*$.


sheaves-lemma-push-pull-composition-modules

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. The functors $(g \circ f)_*$ and $g_* \circ f_*$ are equal. There is a canonical isomorphism of functors $(g \circ f)^* \cong f^* \circ g^*$.


sheaves-lemma-stalk-pullback-modules

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $x \in X$. Then $$ (f^*\mathcal{G})_x = \mathcal{G}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x} $$ as $\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.


sheaves-lemma-skyscraper-stalks

Let $X$ be a topological space, $x \in X$ a point, and $A$ a set. For any point $x' \in X$ the stalk of the skyscraper sheaf at $x$ with value $A$ at $x'$ is $$ (i_{x, *}A)_{x'} = \left\{ \begin{matrix} A & \text{if} & x' \in \overline{\{x\}} \\ \{*\} & \text{if} & x' \not\in \overline{\{x\}} \end{matrix} \right. $$ A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules.


sheaves-lemma-stalk-skyscraper-adjoint

Let $X$ be a topological space, and let $x \in X$ a point. The functors $\mathcal{F} \mapsto \mathcal{F}_x$ and $A \mapsto i_{x, *}A$ are adjoint. In a formula $$ \Mor_{\textit{Sets}}(\mathcal{F}_x, A) = \Mor_{\Sh(X)}(\mathcal{F}, i_{x, *}A). $$ A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have $$ \Hom_{\mathcal{O}_{X, x}}(\mathcal{F}_x, A) = \Hom_{\mathcal{O}_X}(\mathcal{F}, i_{x, *}A). $$


sheaves-lemma-directed-colimits-sections

Let $X$ be a topological space. Let $I$ be a directed set. Let $(\mathcal{F}_i, \varphi_{ii'})$ be a system of sheaves of sets over $I$, see Categories, Section \ref{categories-section-posets-limits}. Let $U \subset X$ be an open subset. Consider the canonical map $$ \Psi : \colim_i \mathcal{F}_i(U) \longrightarrow \left(\colim_i \mathcal{F}_i\right)(U) $$ \begin{enumerate} \item If all the transition maps are injective then $\Psi$ is injective for any open $U$. \item If $U$ is quasi-compact, then $\Psi$ is injective. \item If $U$ is quasi-compact and all the transition maps are injective then $\Psi$ is an isomorphism. \item If $U$ has a cofinal system of open coverings $\mathcal{U} : U = \bigcup_{j\in J} U_j$ with $J$ finite and $U_j \cap U_{j'}$ quasi-compact for all $j, j' \in J$, then $\Psi$ is bijective. \end{enumerate}


sheaves-lemma-compute-pullback-to-limit

In the situation described above, let $i \in \Ob(\mathcal{I})$ and let $\mathcal{G}$ be a sheaf on $X_i$. For $U_i \subset X_i$ quasi-compact open we have $$ p_i^{-1}\mathcal{G}(p_i^{-1}(U_i)) = \colim_{a : j \to i} f_a^{-1}\mathcal{G}(f_a^{-1}(U_i)) $$


sheaves-lemma-descend-opens

In the situation described above, let $i \in \Ob(\mathcal{I})$ and let $U_i \subset X_i$ be a quasi-compact open. Then $$ \colim_{a : j \to i} \mathcal{F}_j(f_a^{-1}(U_i)) = \mathcal{F}(p_i^{-1}(U_i)) $$


sheaves-lemma-cofinal-systems-coverings

With notation as above. For each $U \in \mathcal{B}$, let $C(U) \subset \text{Cov}_\mathcal{B}(U)$ be a cofinal system. For each $U \in \mathcal{B}$, and each $\mathcal{U} : U = \bigcup U_i$ in $C(U)$, let coverings $\mathcal{U}_{ij} : U_i \cap U_j = \bigcup U_{ijk}$, $U_{ijk} \in \mathcal{B}$ be given. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{B}$. The following are equivalent \begin{enumerate} \item The presheaf $\mathcal{F}$ is a sheaf on $\mathcal{B}$. \item For every $U \in \mathcal{B}$ and every covering $\mathcal{U} : U = \bigcup U_i$ in $C(U)$ the sheaf condition $(**)$ holds (for the given coverings $\mathcal{U}_{ij}$). \end{enumerate}


sheaves-lemma-cofinal-systems-coverings-standard-case

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Assume that for every triple $U, U', U'' \in \mathcal{B}$ with $U' \subset U$ and $U'' \subset U$ we have $U' \cap U'' \in \mathcal{B}$. For each $U \in \mathcal{B}$, let $C(U) \subset \text{Cov}_\mathcal{B}(U)$ be a cofinal system. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{B}$. The following are equivalent \begin{enumerate} \item The presheaf $\mathcal{F}$ is a sheaf on $\mathcal{B}$. \item For every $U \in \mathcal{B}$ and every covering $\mathcal{U} : U = \bigcup U_i$ in $C(U)$ and for every family of sections $s_i \in \mathcal{F}(U_i)$ such that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ there exists a unique section $s \in \mathcal{F}(U)$ which restricts to $s_i$ on $U_i$. \end{enumerate}


sheaves-lemma-condition-star-sections

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $U \in \mathcal{B}$. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{B}$. The map $$ \mathcal{F}(U) \to \prod\nolimits_{x \in U} \mathcal{F}_x $$ identifies $\mathcal{F}(U)$ with the elements $(s_x)_{x\in U}$ with the property \begin{itemize} \item[(*)] For any $x \in U$ there exists a $V \in \mathcal{B}$, with $x \in V \subset U$ and a section $\sigma \in \mathcal{F}(V)$ such that for all $y \in V$ we have $s_y = (V, \sigma)$ in $\mathcal{F}_y$. \end{itemize}


sheaves-lemma-extend-off-basis

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{B}$. There exists a unique sheaf of sets $\mathcal{F}^{ext}$ on $X$ such that $\mathcal{F}^{ext}(U) = \mathcal{F}(U)$ for all $U \in \mathcal{B}$ compatibly with the restriction mappings.


sheaves-lemma-restrict-basis-equivalence

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Denote $\Sh(\mathcal{B})$ the category of sheaves on $\mathcal{B}$. There is an equivalence of categories $$ \Sh(X) \longrightarrow \Sh(\mathcal{B}) $$ which assigns to a sheaf on $X$ its restriction to the members of $\mathcal{B}$.


sheaves-lemma-extend-off-basis-structures

Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $\mathcal{B}$. There exists a unique sheaf $\mathcal{F}^{ext}$ with values in $\mathcal{C}$ on $X$ such that $\mathcal{F}^{ext}(U) = \mathcal{F}(U)$ for all $U \in \mathcal{B}$ compatibly with the restriction mappings.


sheaves-lemma-restrict-basis-equivalence-structures

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Denote $\Sh(\mathcal{B}, \mathcal{C})$ the category of sheaves with values in $\mathcal{C}$ on $\mathcal{B}$. There is an equivalence of categories $$ \Sh(X, \mathcal{C}) \longrightarrow \Sh(\mathcal{B}, \mathcal{C}) $$ which assigns to a sheaf on $X$ its restriction to the members of $\mathcal{B}$.


sheaves-lemma-extend-off-basis-module

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{B}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{B}$. Let $\mathcal{O}^{ext}$ be the sheaf of rings on $X$ extending $\mathcal{O}$ and let $\mathcal{F}^{ext}$ be the abelian sheaf on $X$ extending $\mathcal{F}$, see Lemma \ref{lemma-extend-off-basis-structures}. There exists a canonical map $$ \mathcal{O}^{ext} \times \mathcal{F}^{ext} \longrightarrow \mathcal{F}^{ext} $$ which agrees with the given map over elements of $\mathcal{B}$ and which endows $\mathcal{F}^{ext}$ with the structure of an $\mathcal{O}^{ext}$-module.


sheaves-lemma-restrict-basis-equivalence-modules

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{O}$ be a sheaf of rings on $X$. Denote $\textit{Mod}(\mathcal{O}|_\mathcal{B})$ the category of sheaves of $\mathcal{O}|_\mathcal{B}$-modules on $\mathcal{B}$. There is an equivalence of categories $$ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}|_\mathcal{B}) $$ which assigns to a sheaf of $\mathcal{O}$-modules on $X$ its restriction to the members of $\mathcal{B}$.


sheaves-lemma-f-map-basis-below-structures

Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{C}, F)$ be a type of algebraic structures. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $X$. Let $\mathcal{G}$ be a sheaf with values in $\mathcal{C}$ on $Y$. Let $\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \in \mathcal{B}$ a morphism $$ \varphi_V : \mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V) $$ of $\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition \ref{definition-f-map} and discussion of $f$-maps in Section \ref{section-presheaves-structures-functorial}) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V$ for $V \in \mathcal{B}$.


sheaves-lemma-f-map-basis-below-modules

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \in \mathcal{B}$ a $\mathcal{O}_Y(V)$-module map $$ \varphi_V : \mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V) $$ (where $\mathcal{F}(f^{-1}V)$ has a module structure using $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V)$) compatible with restriction mappings. Then there is a unique $f$-map (see discussion of $f$-maps in Section \ref{section-ringed-spaces-functoriality-modules}) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V$ for $V \in \mathcal{B}$.


sheaves-lemma-f-map-basis-above-and-below-structures

Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{C}, F)$ be a type of algebraic structures. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $X$. Let $\mathcal{G}$ be a sheaf with values in $\mathcal{C}$ on $Y$. Let $\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \in \mathcal{B}_Y$, and $U \in \mathcal{B}_X$ such that $f(U) \subset V$ a morphism $$ \varphi_V^U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U) $$ of $\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition \ref{definition-f-map} and the discussion of $f$-maps in Section \ref{section-presheaves-structures-functorial}) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V^U$ as the composition $$ \mathcal{G}(V) \xrightarrow{\varphi_V} \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restr.}} \mathcal{F}(U) $$ for every pair $(U, V)$ as above.


sheaves-lemma-f-map-basis-above-and-below-modules

Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \in \mathcal{B}_Y$, and $U \in \mathcal{B}_X$ such that $f(U) \subset V$ a $\mathcal{O}_Y(V)$-module map $$ \varphi_V^U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U) $$ compatible with restriction mappings. Here the $\mathcal{O}_Y(V)$-module structure on $\mathcal{F}(U)$ comes from the $\mathcal{O}_X(U)$-module structure via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V) \to \mathcal{O}_X(U)$. Then there is a unique $f$-map of sheaves of modules (see Definition \ref{definition-f-map} and the discussion of $f$-maps in Section \ref{section-ringed-spaces-functoriality-modules}) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V^U$ as the composition $$ \mathcal{G}(V) \xrightarrow{\varphi_V} \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restr.}} \mathcal{F}(U) $$ for every pair $(U, V)$ as above.


sheaves-lemma-j-pullback

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$. \begin{enumerate} \item Let $\mathcal{G}$ be a presheaf of sets on $X$. The presheaf $j_p\mathcal{G}$ (see Section \ref{section-presheaves-functorial}) is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open. \item Let $\mathcal{G}$ be a sheaf of sets on $X$. The sheaf $j^{-1}\mathcal{G}$ is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open. \item For any point $u \in U$ and any sheaf $\mathcal{G}$ on $X$ we have a canonical identification of stalks $$ j^{-1}\mathcal{G}_u = (\mathcal{G}|_U)_u = \mathcal{G}_u. $$ \item On the category of presheaves of $U$ we have $j_pj_* = \text{id}$. \item On the category of sheaves of $U$ we have $j^{-1}j_* = \text{id}$. \end{enumerate} The same description holds for (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures, and (pre)sheaves of modules.


sheaves-lemma-j-shriek

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. \begin{enumerate} \item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}). \item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \Mor_{\Sh(X)}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \Mor_{\Sh(U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$. \item Let $\mathcal{F}$ be a sheaf of sets on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} \emptyset & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$ \item On the category of presheaves of $U$ we have $j_pj_{p!} = \text{id}$. \item On the category of sheaves of $U$ we have $j^{-1}j_! = \text{id}$. \end{enumerate}


sheaves-lemma-j-shriek-abelian

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Consider the functors of restriction and extension by $0$ for abelian (pre)sheaves. \begin{enumerate} \item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}). \item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \Mor_{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \Mor_{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$. \item Let $\mathcal{F}$ be an abelian sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} 0 & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$ \item On the category of abelian presheaves of $U$ we have $j_pj_{p!} = \text{id}$. \item On the category of abelian sheaves of $U$ we have $j^{-1}j_! = \text{id}$. \end{enumerate}


sheaves-lemma-j-shriek-structures

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Consider the functors of restriction and extension by $e$ for (pre)sheaves of algebraic structure defined above. \begin{enumerate} \item The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}). \item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \Mor_{\Sh(X, \mathcal{C})}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(U, \mathcal{C})}(\mathcal{F}, j^{-1}\mathcal{G}) = \Mor_{\Sh(U, \mathcal{C})}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$. \item Let $\mathcal{F}$ be a sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} e & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$ \item On the category of presheaves of algebraic structures on $U$ we have $j_pj_{p!} = \text{id}$. \item On the category of sheaves of algebraic structures on $U$ we have $j^{-1}j_! = \text{id}$. \end{enumerate}


sheaves-lemma-j-shriek-modules

Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_U) \to (X, \mathcal{O})$ be an open subspace. Consider the functors of restriction and extension by $0$ for (pre)sheaves of modules defined above. \begin{enumerate} \item The functor $j_{p!}$ is a left adjoint to restriction, in a formula $$ \Mor_{\textit{PMod}(\mathcal{O})}(j_{p!}\mathcal{F}, \mathcal{G}) = \Mor_{\textit{PMod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$. \item The functor $j_!$ is a left adjoint to restriction, in a formula $$ \Mor_{\textit{Mod}(\mathcal{O})}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\textit{Mod}(\mathcal{O}|_U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$. \item Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} 0 & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$ \item On the category of sheaves of $\mathcal{O}|_U$-modules on $U$ we have $j^{-1}j_! = \text{id}$. \end{enumerate}


sheaves-lemma-equivalence-categories-open

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. The functor $$ j_! : \Sh(U) \longrightarrow \Sh(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = \emptyset$ for all $x \in X \setminus U$.


sheaves-lemma-equivalence-categories-open-abelian

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. The functor $$ j_! : \textit{Ab}(U) \longrightarrow \textit{Ab}(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus U$.


sheaves-lemma-equivalence-categories-open-structures

Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. The functor $$ j_! : \Sh(U, \mathcal{C}) \longrightarrow \Sh(X, \mathcal{C}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = e$ for all $x \in X \setminus U$.


sheaves-lemma-equivalence-categories-open-modules

Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_U) \to (X, \mathcal{O})$ be an open subspace. The functor $$ j_! : \textit{Mod}(\mathcal{O}|_U) \longrightarrow \textit{Mod}(\mathcal{O}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus U$.


sheaves-lemma-stalks-closed-pushforward

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. The stalks of $i_*\mathcal{F}$ are described as follows $$ i_*\mathcal{F}_x = \left\{ \begin{matrix} \{*\} & \text{if} & x \not \in Z \\ \mathcal{F}_x & \text{if} & x \in Z \end{matrix} \right. $$ where $\{*\}$ denotes a singleton set. Moreover, $i^{-1}i_* = \text{id}$ on the category of sheaves of sets on $Z$. Moreover, the same holds for abelian sheaves on $Z$, resp.\ sheaves of algebraic structures on $Z$ where $\{*\}$ has to be replaced by $0$, resp.\ a final object of the category of algebraic structures.


sheaves-lemma-equivalence-categories-closed

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $$ i_* : \Sh(Z) \longrightarrow \Sh(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = \{*\}$ for all $x \in X \setminus Z$.


sheaves-lemma-equivalence-categories-closed-abelian

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $$ i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.


sheaves-lemma-equivalence-categories-closed-structures

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure with final object $0$. The functor $$ i_* : \Sh(Z, \mathcal{C}) \longrightarrow \Sh(X, \mathcal{C}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.


sheaves-lemma-glue-maps

Let $X$ be a topological space. Let $X = \bigcup U_i$ be an open covering. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets on $X$. Given a collection $$ \varphi_i : \mathcal{F}|_{U_i} \longrightarrow \mathcal{G}|_{U_i} $$ of maps of sheaves such that for all $i, j \in I$ the maps $\varphi_i, \varphi_j$ restrict to the same map $\mathcal{F}|_{U_i \cap U_j} \to \mathcal{G}|_{U_i \cap U_j}$ then there exists a unique map of sheaves $$ \varphi : \mathcal{F} \longrightarrow \mathcal{G} $$ whose restriction to each $U_i$ agrees with $\varphi_i$.


sheaves-lemma-glue-sheaves

Let $X$ be a topological space. Let $X = \bigcup_{i\in I} U_i$ be an open covering. Given any glueing data $(\mathcal{F}_i, \varphi_{ij})$ for sheaves of sets with respect to the covering $X = \bigcup U_i$ there exists a sheaf of sets $\mathcal{F}$ on $X$ together with isomorphisms $$ \varphi_i : \mathcal{F}|_{U_i} \to \mathcal{F}_i $$ such that the diagrams $$ \xymatrix{ \mathcal{F}|_{U_i \cap U_j} \ar[r]_{\varphi_i} \ar[d]_{\text{id}} & \mathcal{F}_i|_{U_i \cap U_j} \ar[d]^{\varphi_{ij}} \\ \mathcal{F}|_{U_i \cap U_j} \ar[r]^{\varphi_j} & \mathcal{F}_j|_{U_i \cap U_j} } $$ are commutative.


sheaves-lemma-glue-sheaves-structures

Let $X$ be a topological space. Let $X = \bigcup U_i$ be an open covering. Let $(\mathcal{F}_i, \varphi_{ij})$ be a glueing data of sheaves of abelian groups, resp.\ sheaves of algebraic structures, resp.\ sheaves of $\mathcal{O}$-modules for some sheaf of rings $\mathcal{O}$ on $X$. Then the construction in the proof of Lemma \ref{lemma-glue-sheaves} above leads to a sheaf of abelian groups, resp.\ sheaf of algebraic structures, resp.\ sheaf of $\mathcal{O}$-modules.


sheaves-lemma-mapping-property-glue

Let $X$ be a topological space. Let $X = \bigcup_{i\in I} U_i$ be an open covering. The functor which associates to a sheaf of sets $\mathcal{F}$ the following collection of glueing data $$ (\mathcal{F}|_{U_i}, (\mathcal{F}|_{U_i})|_{U_i \cap U_j} \to (\mathcal{F}|_{U_j})|_{U_i \cap U_j} ) $$ with respect to the covering $X = \bigcup U_i$ defines an equivalence of categories between $\Sh(X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp.\ sheaves of algebraic structures, resp.\ sheaves of $\mathcal{O}$-modules.


descent-theorem-descent

The following conditions are equivalent. \begin{enumerate} \item[(a)] The morphism $f$ is a descent morphism for modules. \item[(b)] The morphism $f$ is an effective descent morphism for modules. \item[(c)] The morphism $f$ is universally injective. \end{enumerate}


descent-lemma-refine-descent-datum

Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ and $\mathcal{V} = \{V_j \to V\}_{j \in J}$ be families of morphisms of schemes with fixed target. Let $(g, \alpha : I \to J, (g_i)) : \mathcal{U} \to \mathcal{V}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. Let $(\mathcal{F}_j, \varphi_{jj'})$ be a descent datum for quasi-coherent sheaves with respect to the family $\{V_j \to V\}_{j \in J}$. Then \begin{enumerate} \item The system $$ \left(g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}\right) $$ is a descent datum with respect to the family $\{U_i \to U\}_{i \in I}$. \item This construction is functorial in the descent datum $(\mathcal{F}_j, \varphi_{jj'})$. \item Given a second morphism $(g', \alpha' : I \to J, (g'_i))$ of families of maps with fixed target with $g = g'$ there exists a functorial isomorphism of descent data $$ (g_i^*\mathcal{F}_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')}) \cong ((g'_i)^*\mathcal{F}_{\alpha'(i)}, (g'_i \times g'_{i'})^*\varphi_{\alpha'(i)\alpha'(i')}). $$ \end{enumerate}


descent-lemma-zariski-descent-effective

Let $S$ be a scheme. Let $S = \bigcup U_i$ be an open covering. Any descent datum on quasi-coherent sheaves for the family $\mathcal{U} = \{U_i \to S\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.


descent-lemma-descent-datum-cosimplicial

Let $R \to A$ be a ring map. Given a descent datum $(N, \varphi)$ we can associate to it a cosimplicial $(A/R)_\bullet$-module $N_\bullet$\footnote{We should really write $(N, \varphi)_\bullet$.} by the rules $N_n = N_{n, n}$ and given $\beta : [n] \to [m]$ setting we define $$ N_\bullet(\beta) = (\varphi^m_{\beta(n)m}) \circ N_{\beta, n} : N_{n, n} \longrightarrow N_{m, m}. $$ This procedure is functorial in the descent datum.


descent-lemma-canonical-descent-datum-cosimplicial

Let $R \to A$ be a ring map. Let $M$ be an $R$-module. The cosimplicial $(A/R)_\bullet$-module associated to the canonical descent datum is isomorphic to the cosimplicial module $(A/R)_\bullet \otimes_R M$.


descent-lemma-ff-exact

Suppose that $R \to A$ is faithfully flat, see Algebra, Definition \ref{algebra-definition-flat}. Then for any $R$-module $M$ the extended cochain complex (\ref{equation-extended-complex}) is exact.


descent-lemma-recognize-effective

Let $R \to A$ be a faithfully flat ring map. Let $(N, \varphi)$ be a descent datum. Then $(N, \varphi)$ is effective if and only if the canonical map $$ A \otimes_R H^0(s(N_\bullet)) \longrightarrow N $$ is an isomorphism.


descent-lemma-descent-descends

Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$ be faithfully flat. Set $A' = R' \otimes_R A$. If all descent data for $R' \to A'$ are effective, then so are all descent data for $R \to A$.


descent-lemma-C-is-faithful

For a ring $R$, the functor $C : \text{Mod}_R \to \text{Mod}_R$ is exact and reflects injections and surjections.


descent-lemma-split-surjection

Let $R$ be a ring. A morphism $f: M \to N$ in $\text{Mod}_R$ is universally injective if and only if $C(f): C(N) \to C(M)$ is a split surjection.


descent-lemma-equalizer-M

For $(M,\theta) \in DD_{S/R}$, the diagram \begin{equation} \label{equation-equalizer-M} \xymatrix@C=8pc{ M \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} & M \otimes_{S, \delta_1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} &  M \otimes_{S, \delta_{12}^1} S_3 } \end{equation} is a split equalizer.


descent-lemma-equalizer-CM

For $(M, \theta) \in DD_{S/R}$, the diagram \begin{equation} \label{equation-coequalizer-CM} \xymatrix@C=8pc{ C(M \otimes_{S, \delta_{12}^1} S_3) \ar@<1ex>[r]^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))} \ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta^2_1)} & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(M). } \end{equation} obtained by applying $C$ to (\ref{equation-equalizer-M}) is a split coequalizer.


descent-lemma-descent-lemma

If $f$ is universally injective, then the diagram \begin{equation} \label{equation-equalizer-f2} \xymatrix@C=8pc{ f_*(M, \theta) \otimes_R S \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} & M \otimes_{S, \delta_1^1} S_2  \ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} & M \otimes_{S, \delta_{12}^1} S_3 } \end{equation} obtained by tensoring (\ref{equation-equalizer-f}) over $R$ with $S$ is an  equalizer.


descent-lemma-flat-to-injective

If $M \in \text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module.


descent-lemma-standard-fpqc-covering

Let $S$ be an affine scheme. Let $\mathcal{U} = \{f_i : U_i \to S\}_{i = 1, \ldots, n}$ be a standard fpqc covering of $S$, see Topologies, Definition \ref{topologies-definition-standard-fpqc}. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{U_i \to S\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.


descent-lemma-galois-descent

Let $k'/k$ be a (finite) Galois extension with Galois group $G$. Let $X$ be a scheme over $k$. The category of quasi-coherent $\mathcal{O}_X$-modules is equivalent to the category of systems $(\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})$ where \begin{enumerate} \item $\mathcal{F}$ is a quasi-coherent module on $X_{k'}$, \item $\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}$ is an isomorphism of modules, \item $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ for all $\sigma, \tau \in G$. \end{enumerate} Here $f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \to X_{k'}$.


descent-lemma-galois-descent-more-general

Let $X \to Y$, $G$, and $f_\sigma : X \to X$ be as above. The category of quasi-coherent $\mathcal{O}_Y$-modules is equivalent to the category of systems $(\mathcal{F}, (\varphi_\sigma)_{\sigma \in G})$ where \begin{enumerate} \item $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module, \item $\varphi_\sigma : \mathcal{F} \to f_\sigma^*\mathcal{F}$ is an isomorphism of modules, \item $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ for all $\sigma, \tau \in G$. \end{enumerate}


descent-lemma-finite-type-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module.


descent-lemma-finite-type-descends-fppf

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of locally ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules. If \begin{enumerate} \item $f$ is open as a map of topological spaces, \item $f$ is surjective and flat, and \item $f^*\mathcal{F}$ is of finite type, \end{enumerate} then $\mathcal{F}$ is of finite type.


descent-lemma-finite-presentation-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is an $\mathcal{O}_{X_i}$-module of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation.


descent-lemma-locally-generated-by-r-sections-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is locally generated by $r$ sections as an $\mathcal{O}_X$-module.


descent-lemma-flat-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a flat $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a flat $\mathcal{O}_X$-module.


descent-lemma-finite-locally-free-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module.


descent-lemma-locally-projective-descends

Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a locally projective $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a locally projective $\mathcal{O}_X$-module.


descent-lemma-finite-over-finite-module

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is a finite morphism. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite type.


descent-lemma-finite-finitely-presented-module

Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is finite and of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite presentation.


descent-lemma-sheaf-condition-holds

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $\tau \in \{Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. The functor defined in (\ref{equation-quasi-coherent-presheaf}) satisfies the sheaf condition with respect to any $\tau$-covering $\{T_i \to T\}_{i \in I}$ of any scheme $T$ over $S$.


descent-lemma-compare-sites

Let $S$ be a scheme. Denote $$ \begin{matrix} \text{id}_{\tau, Zar} & : & (\Sch/S)_\tau \to S_{Zar}, & \tau \in \{Zar, \etale, smooth, syntomic, fppf\} \\ \text{id}_{\tau, \etale} & : & (\Sch/S)_\tau \to S_\etale, & \tau \in \{\etale, smooth, syntomic, fppf\} \\ \text{id}_{small, \etale, Zar} & : & S_\etale \to S_{Zar}, \end{matrix} $$ the morphisms of ringed sites of Remark \ref{remark-change-topologies-ringed}. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_S$-modules which we view a sheaf of $\mathcal{O}$-modules on $S_{Zar}$. Then \begin{enumerate} \item $(\text{id}_{\tau, Zar})^*\mathcal{F}$ is the $\tau$-sheafification of the Zariski sheaf $$ (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}) $$ on $(\Sch/S)_\tau$, and \item $(\text{id}_{small, \etale, Zar})^*\mathcal{F}$ is the \'etale sheafification of the Zariski sheaf $$ (f : T \to S) \longmapsto \Gamma(T, f^*\mathcal{F}) $$ on $S_\etale$. \end{enumerate} Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on $S_\etale$. Then \begin{enumerate} \item[(3)] $(\text{id}_{\tau, \etale})^*\mathcal{G}$ is the $\tau$-sheafification of the \'etale sheaf $$ (f : T \to S) \longmapsto \Gamma(T, f_{small}^*\mathcal{G}) $$ where $f_{small} : T_\etale \to S_\etale$ is the morphism of ringed small \'etale sites of Remark \ref{remark-change-topologies-ringed}. \end{enumerate}


descent-lemma-quasi-coherent-gives-quasi-coherent

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_S$-module. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. \begin{enumerate} \item The sheaf $\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on $(\Sch/S)_\tau$, as defined in Modules on Sites, Definition \ref{sites-modules-definition-site-local}. \item If $\tau = \etale$ (resp.\ $\tau = Zariski$), then the sheaf $\mathcal{F}^a$ is a quasi-coherent $\mathcal{O}$-module on $S_\etale$ (resp.\ $S_{Zar}$) as defined in Modules on Sites, Definition \ref{sites-modules-definition-site-local}. \end{enumerate}


descent-lemma-standard-covering-Cech

Let $S$ be a scheme. Let \begin{enumerate} \item[(a)] $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$ and $\mathcal{C} = (\Sch/S)_\tau$, or \item[(b)] let $\tau = \etale$ and $\mathcal{C} = S_\etale$, or \item[(c)] let $\tau = Zariski$ and $\mathcal{C} = S_{Zar}$. \end{enumerate} Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Let $U \in \Ob(\mathcal{C})$ be affine. Let $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ be a standard affine $\tau$-covering in $\mathcal{C}$. Then \begin{enumerate} \item $\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}$ is a $\tau$-covering of $U$, \item $\mathcal{U}$ is a refinement of $\mathcal{V}$, and \item the induced map on {\v C}ech complexes (Cohomology on Sites, Equation (\ref{sites-cohomology-equation-map-cech-complexes})) $$ \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ is an isomorphism of complexes. \end{enumerate}


descent-lemma-standard-covering-Cech-quasi-coherent

Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Let $\tau$, $\mathcal{C}$, $U$, $\mathcal{U}$ be as in Lemma \ref{lemma-standard-covering-Cech}. Then there is an isomorphism of complexes $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^a) \cong s((A/R)_\bullet \otimes_R M) $$ (see Section \ref{section-descent-modules}) where $R = \Gamma(U, \mathcal{O}_U)$, $M = \Gamma(U, \mathcal{F}^a)$ and $R \to A$ is a faithfully flat ring map. In particular $$ \check{H}^p(\mathcal{U}, \mathcal{F}^a) = 0 $$ for all $p \geq 1$.


descent-lemma-equivalence-quasi-coherent-limits

Let $S$ be a scheme. Let $\tau \in \{Zariski, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. The functors $$ \QCoh(\mathcal{O}_S) \longrightarrow \textit{Mod}((\Sch/S)_\tau, \mathcal{O}) \quad\text{and}\quad \QCoh(\mathcal{O}_S) \longrightarrow \textit{Mod}(S_\tau, \mathcal{O}) $$ defined by the rule $\mathcal{F} \mapsto \mathcal{F}^a$ seen in Proposition \ref{proposition-equivalence-quasi-coherent} are \begin{enumerate} \item fully faithful, \item compatible with direct sums, \item compatible with colimits, \item right exact, \item exact as a functor $\QCoh(\mathcal{O}_S) \to \textit{Mod}(S_\etale, \mathcal{O})$, \item {\bf not} exact as a functor $\QCoh(\mathcal{O}_S) \to \textit{Mod}((\Sch/S)_\tau, \mathcal{O})$ in general, \item given two quasi-coherent $\mathcal{O}_S$-modules $\mathcal{F}$, $\mathcal{G}$ we have $(\mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G})^a = \mathcal{F}^a \otimes_\mathcal{O} \mathcal{G}^a$, \item given two quasi-coherent $\mathcal{O}_S$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation we have $(\SheafHom_{\mathcal{O}_S}(\mathcal{F}, \mathcal{G}))^a = \SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G}^a)$, and \item given a short exact sequence $0 \to \mathcal{F}_1^a \to \mathcal{E} \to \mathcal{F}_2^a \to 0$ of $\mathcal{O}$-modules then $\mathcal{E}$ is quasi-coherent\footnote{Warning: This is misleading. See part (6).}, i.e., $\mathcal{E}$ is in the essential image of the functor. \end{enumerate}


descent-lemma-higher-direct-images-small-etale

Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \'etale or Zariski topology, there are canonical isomorphisms $R^if_{small, *}(\mathcal{F}^a) = (R^if_*\mathcal{F})^a$.


descent-lemma-cohomology-parasitic

Let $S$ be a scheme. Let $\tau \in \{Zar, \etale, smooth, syntomic, fppf\}$. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules on $(\Sch/S)_\tau$. \begin{enumerate} \item If $\mathcal{G}$ is parasitic for the $\tau$-topology, then $H^p_\tau(U, \mathcal{G}) = 0$ for every $U$ open in $S$, resp.\ \'etale over $S$, resp.\ smooth over $S$, resp.\ syntomic over $S$, resp.\ flat and locally of finite presentation over $S$. \item If $\mathcal{G}$ is parasitic then $H^p_\tau(U, \mathcal{G}) = 0$ for every $U$ flat over $S$. \end{enumerate}


descent-lemma-direct-image-parasitic

Let $f : T \to S$ be a morphism of schemes. For any parasitic $\mathcal{O}$-module on $(\Sch/T)_\tau$ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^if_*\mathcal{F}$ are parasitic $\mathcal{O}$-modules on $(\Sch/S)_\tau$.


descent-lemma-quasi-coherent-and-flat-base-change

Let $S$ be a scheme. Let $\tau \in \{Zar, \etale\}$. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on $(\Sch/S)_{fppf}$ such that \begin{enumerate} \item $\mathcal{G}|_{S_\tau}$ is quasi-coherent, and \item for every flat, locally finitely presented morphism $g : U \to S$ the canonical map $g_{\tau, small}^*(\mathcal{G}|_{S_\tau}) \to \mathcal{G}|_{U_\tau}$ is an isomorphism. \end{enumerate} Then $H^p(U, \mathcal{G}) = H^p(U, \mathcal{G}|_{U_\tau})$ for every $U$ flat and locally of finite presentation over $S$.


descent-lemma-equiv-fibre-product

For a scheme $X$ denote $|X|$ the underlying set. Let $f : X \to S$ be a morphism of schemes. Then $$ |X \times_S X| \to |X| \times_{|S|} |X| $$ is surjective.


descent-lemma-universal-effective-epimorphism-affine

Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of affine schemes. The following are equivalent \begin{enumerate} \item for any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $$ \Gamma(X, \mathcal{F}) = \text{Equalizer}\left( \xymatrix{ \prod\nolimits_{i \in I} \Gamma(X_i, f_i^*\mathcal{F}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{i, j \in I} \Gamma(X_i \times_X X_j, (f_i \times f_j)^*\mathcal{F}) } \right) $$ \item $\{f_i : X_i \to X\}_{i \in I}$ is a universal effective epimorphism (Sites, Definition \ref{sites-definition-universal-effective-epimorphisms}) in the category of affine schemes. \end{enumerate}


descent-lemma-universal-effective-epimorphism-surjective

Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of schemes. \begin{enumerate} \item If the family is universal effective epimorphism in the category of schemes, then $\coprod f_i$ is surjective. \item If $X$ and $X_i$ are affine and the family is a universal effective epimorphism in the category of affine schemes, then $\coprod f_i$ is surjective. \end{enumerate}


descent-lemma-check-universal-effective-epimorphism-affine

Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of schemes. If for every morphism $Y \to X$ with $Y$ affine the family of base changes $g_i : Y_i \to Y$ forms an effective epimorphism, then the family of $f_i$ forms a universally effective epimorphism in the category of schemes.


descent-lemma-universal-effective-epimorphism

Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of affine schemes. Assume the equivalent assumption of Lemma \ref{lemma-universal-effective-epimorphism-affine} hold and that moreover for any morphism of affines $Y \to X$ the map $$ \coprod X_i \times_X Y \longrightarrow Y $$ is a submersive map of topological spaces (Topology, Definition \ref{topology-definition-submersive}). Then our family of morphisms is a universal effective epimorphism in the category of schemes.


descent-lemma-open-fpqc-covering

Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering. Suppose that for each $i$ we have an open subset $W_i \subset T_i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)$ as open subsets of $T_i \times_T T_j$. Then there exists a unique open subset $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.


descent-lemma-fpqc-universal-effective-epimorphisms

Let $\{T_i \to T\}$ be an fpqc covering, see Topologies, Definition \ref{topologies-definition-fpqc-covering}. Then $\{T_i \to T\}$ is a universal effective epimorphism in the category of schemes, see Sites, Definition \ref{sites-definition-universal-effective-epimorphisms}. In other words, every representable functor on the category of schemes satisfies the sheaf condition for the fpqc topology, see Topologies, Definition \ref{topologies-definition-sheaf-property-fpqc}.


descent-lemma-coequalizer-fpqc-local

Consider schemes $X, Y, Z$ and morphisms $a, b : X \to Y$ and a morphism $c : Y \to Z$ with $c \circ a = c \circ b$. Set $d = c \circ a = c \circ b$. If there exists an fpqc covering $\{Z_i \to Z\}$ such that \begin{enumerate} \item for all $i$ the morphism $Y \times_{c, Z} Z_i \to Z_i$ is the coequalizer of $(a, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$ and $(b, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$, and \item for all $i$ and $i'$ the morphism $Y \times_{c, Z} (Z_i \times_Z Z_{i'}) \to (Z_i \times_Z Z_{i'})$ is the coequalizer of $(a, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to Y \times_{c, Z} (Z_i \times_Z Z_{i'})$ and $(b, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to Y \times_{c, Z} (Z_i \times_Z Z_{i'})$ \end{enumerate} then $c$ is the coequalizer of $a$ and $b$.


descent-lemma-flat-finitely-presented-permanence-algebra

Let $R \to A \to B$ be ring maps. Assume $R \to B$ is of finite presentation and $A \to B$ faithfully flat and of finite presentation. Then $R \to A$ is of finite presentation.


descent-lemma-finite-type-local-source-fppf-algebra

Let $R \to A \to B$ be ring maps. Assume $R \to B$ is of finite type and $A \to B$ faithfully flat and of finite presentation. Then $R \to A$ is of finite type.


descent-lemma-flat-finitely-presented-permanence

\begin{reference} \cite[IV, 17.7.5 (i) and (ii)]{EGA}. \end{reference} Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $f$ is surjective, flat and locally of finite presentation and assume that $p$ is locally of finite presentation (resp.\ locally of finite type). Then $q$ is locally of finite presentation (resp.\ locally of finite type).


descent-lemma-syntomic-smooth-etale-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective, and syntomic (resp.\ smooth, resp.\ \'etale), \item $p$ is syntomic (resp.\ smooth, resp.\ \'etale). \end{enumerate} Then $q$ is syntomic (resp.\ smooth, resp.\ \'etale).


descent-lemma-smooth-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective, flat, and locally of finite presentation, \item $p$ is smooth (resp.\ \'etale). \end{enumerate} Then $q$ is smooth (resp.\ \'etale).


descent-lemma-syntomic-permanence

Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that \begin{enumerate} \item $f$ is surjective, flat, and locally of finite presentation, \item $p$ is syntomic. \end{enumerate} Then both $q$ and $f$ are syntomic.


descent-lemma-curiosity

Let $X \to Y \to Z$ be morphism of schemes. Let $P$ be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that $X \to Z$ has $P$ and that $\{X \to Y\}$ can be refined by an fppf covering of $Y$. Then $Y \to Z$ is $P$.


descent-lemma-descending-properties

Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Assume that \begin{enumerate} \item the property is local in the Zariski topology, \item for any morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, \'etale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, property $\mathcal{P}$ holds for $S'$ if property $\mathcal{P}$ holds for $S$, and \item for any surjective morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, \'etale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, property $\mathcal{P}$ holds for $S$ if property $\mathcal{P}$ holds for $S'$. \end{enumerate} Then $\mathcal{P}$ is $\tau$ local on the base.


descent-lemma-Noetherian-local-fppf

The property $\mathcal{P}(S) =$``$S$ is locally Noetherian'' is local in the fppf topology.


descent-lemma-Jacobson-local-fppf

The property $\mathcal{P}(S) =$``$S$ is Jacobson'' is local in the fppf topology.


descent-lemma-locally-finite-nr-irred-local-fppf

The property $\mathcal{P}(S) =$``every quasi-compact open of $S$ has a finite number of irreducible components'' is local in the fppf topology.


descent-lemma-Sk-local-syntomic

The property $\mathcal{P}(S) =$``$S$ is locally Noetherian and $(S_k)$'' is local in the syntomic topology.


descent-lemma-CM-local-syntomic

The property $\mathcal{P}(S) =$``$S$ is Cohen-Macaulay'' is local in the syntomic topology.


descent-lemma-reduced-local-smooth

The property $\mathcal{P}(S) =$``$S$ is reduced'' is local in the smooth topology.


descent-lemma-normal-local-smooth

\begin{slogan} Normality is local in the smooth topology. \end{slogan} The property $\mathcal{P}(S) =$``$S$ is normal'' is local in the smooth topology.


descent-lemma-Rk-local-smooth

The property $\mathcal{P}(S) =$``$S$ is locally Noetherian and $(R_k)$'' is local in the smooth topology.


descent-lemma-regular-local-smooth

The property $\mathcal{P}(S) =$``$S$ is regular'' is local in the smooth topology.


descent-lemma-Nagata-local-smooth

The property $\mathcal{P}(S) =$``$S$ is Nagata'' is local in the smooth topology.


descent-lemma-descend-reduced

If $f : X \to Y$ is a flat and surjective morphism of schemes and $X$ is reduced, then $Y$ is reduced.


descent-lemma-descend-regular

Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular.


descent-lemma-dimension-at-point-local

Let $f : U \to V$ be an \'etale morphism of schemes. Let $u \in U$ and $v = f(u)$. Then $\dim_u(U) = \dim_v(V)$.


descent-lemma-dimension-local-ring-local

Let $f : U \to V$ be an \'etale morphism of schemes. Let $u \in U$ and $v = f(u)$. Then $\dim(\mathcal{O}_{U, u}) = \dim(\mathcal{O}_{V, v})$.


descent-lemma-pullback-property-local-target

Let $\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$ local on the target. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $Y' \to Y$ which is flat, resp.\ flat and locally of finite presentation, resp.\ syntomic, resp.\ \'etale, resp.\ an open immersion, the base change $f' : Y' \times_Y X \to Y'$ of $f$ has property $\mathcal{P}$.


descent-lemma-largest-open-of-the-base

Let $\tau \in \{fppf, syntomic, smooth, \etale\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$ local on the target. For any morphism of schemes $f : X \to Y$ there exists a largest open $W(f) \subset Y$ such that the restriction $X_{W(f)} \to W(f)$ has $\mathcal{P}$. Moreover, \begin{enumerate} \item if $g : Y' \to Y$ is flat and locally of finite presentation, syntomic, smooth, or \'etale and the base change $f' : X_{Y'} \to Y'$ has $\mathcal{P}$, then $g(Y') \subset W(f)$, \item if $g : Y' \to Y$ is flat and locally of finite presentation, syntomic, smooth, or \'etale, then $W(f') = g^{-1}(W(f))$, and \item if $\{g_i : Y_i \to Y\}$ is a $\tau$-covering, then $g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$ by $Y_i \to Y$. \end{enumerate}


descent-lemma-descending-properties-morphisms

Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{fpqc, fppf, \etale, smooth, syntomic\}$. Assume that \begin{enumerate} \item the property is preserved under flat, flat and locally of finite presentation, \'etale, smooth, or syntomic base change depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic (compare with Schemes, Definition \ref{schemes-definition-preserved-by-base-change}), \item the property is Zariski local on the base. \item for any surjective morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, \'etale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, and any morphism of schemes $f : X \to S$ property $\mathcal{P}$ holds for $f$ if property $\mathcal{P}$ holds for the base change $f' : X' = S' \times_S X \to S'$. \end{enumerate} Then $\mathcal{P}$ is $\tau$ local on the base.


descent-lemma-descending-property-quasi-compact

The property $\mathcal{P}(f) =$``$f$ is quasi-compact'' is fpqc local on the base.


descent-lemma-descending-property-quasi-separated

The property $\mathcal{P}(f) =$``$f$ is quasi-separated'' is fpqc local on the base.


descent-lemma-descending-property-universally-closed

The property $\mathcal{P}(f) =$``$f$ is universally closed'' is fpqc local on the base.


descent-lemma-descending-property-universally-open

The property $\mathcal{P}(f) =$``$f$ is universally open'' is fpqc local on the base.


descent-lemma-descending-property-universally-submersive

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


descent-lemma-descending-property-separated

The property $\mathcal{P}(f) =$``$f$ is separated'' is fpqc local on the base.


descent-lemma-descending-property-surjective

The property $\mathcal{P}(f) =$``$f$ is surjective'' is fpqc local on the base.


descent-lemma-descending-property-universally-injective

The property $\mathcal{P}(f) =$``$f$ is universally injective'' is fpqc local on the base.


descent-lemma-descending-property-universal-homeomorphism

The property $\mathcal{P}(f) =$``$f$ is a universal homeomorphism'' is fpqc local on the base.


descent-lemma-descending-property-locally-finite-type

The property $\mathcal{P}(f) =$``$f$ is locally of finite type'' is fpqc local on the base.


descent-lemma-descending-property-locally-finite-presentation

The property $\mathcal{P}(f) =$``$f$ is locally of finite presentation'' is fpqc local on the base.


descent-lemma-descending-property-finite-type

The property $\mathcal{P}(f) =$``$f$ is of finite type'' is fpqc local on the base.


descent-lemma-descending-property-finite-presentation

The property $\mathcal{P}(f) =$``$f$ is of finite presentation'' is fpqc local on the base.


descent-lemma-descending-property-proper

The property $\mathcal{P}(f) =$``$f$ is proper'' is fpqc local on the base.


descent-lemma-descending-property-flat

The property $\mathcal{P}(f) =$``$f$ is flat'' is fpqc local on the base.


descent-lemma-descending-property-open-immersion

The property $\mathcal{P}(f) =$``$f$ is an open immersion'' is fpqc local on the base.


descent-lemma-descending-property-isomorphism

The property $\mathcal{P}(f) =$``$f$ is an isomorphism'' is fpqc local on the base.


descent-lemma-descending-property-affine

The property $\mathcal{P}(f) =$``$f$ is affine'' is fpqc local on the base.


descent-lemma-descending-property-closed-immersion

The property $\mathcal{P}(f) =$``$f$ is a closed immersion'' is fpqc local on the base.


descent-lemma-descending-property-quasi-affine

The property $\mathcal{P}(f) =$``$f$ is quasi-affine'' is fpqc local on the base.


descent-lemma-descending-property-quasi-compact-immersion

The property $\mathcal{P}(f) =$``$f$ is a quasi-compact immersion'' is fpqc local on the base.


descent-lemma-descending-property-integral

The property $\mathcal{P}(f) =$``$f$ is integral'' is fpqc local on the base.


descent-lemma-descending-property-finite

The property $\mathcal{P}(f) =$``$f$ is finite'' is fpqc local on the base.


descent-lemma-descending-property-quasi-finite

The properties $\mathcal{P}(f) =$``$f$ is locally quasi-finite'' and $\mathcal{P}(f) =$``$f$ is quasi-finite'' are fpqc local on the base.


descent-lemma-descending-property-relative-dimension-d

The property $\mathcal{P}(f) =$``$f$ is locally of finite type of relative dimension $d$'' is fpqc local on the base.


descent-lemma-descending-property-syntomic

The property $\mathcal{P}(f) =$``$f$ is syntomic'' is fpqc local on the base.


descent-lemma-descending-property-smooth

The property $\mathcal{P}(f) =$``$f$ is smooth'' is fpqc local on the base.


descent-lemma-descending-property-unramified

The property $\mathcal{P}(f) =$``$f$ is unramified'' is fpqc local on the base. The property $\mathcal{P}(f) =$``$f$ is G-unramified'' is fpqc local on the base.


descent-lemma-descending-property-etale

The property $\mathcal{P}(f) =$``$f$ is \'etale'' is fpqc local on the base.


descent-lemma-descending-property-finite-locally-free

The property $\mathcal{P}(f) =$``$f$ is finite locally free'' is fpqc local on the base. Let $d \geq 0$. The property $\mathcal{P}(f) =$``$f$ is finite locally free of degree $d$'' is fpqc local on the base.


descent-lemma-descending-property-monomorphism

The property $\mathcal{P}(f) =$``$f$ is a monomorphism'' is fpqc local on the base.


descent-lemma-descending-property-regular-immersion

The properties \begin{enumerate} \item[] $\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion'', \item[] $\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion'', and \item[] $\mathcal{P}(f) =$``$f$ is a quasi-regular immersion'' \end{enumerate} are fpqc local on the base.


descent-lemma-descending-fppf-property-immersion

The property $\mathcal{P}(f) =$``$f$ is an immersion'' is fppf local on the base.


descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism

Let $f : X \to Y$ be a flat, quasi-compact, surjective monomorphism. Then f is an isomorphism.


descent-lemma-universally-injective-etale-open-immersion

A universally injective \'etale morphism is an open immersion.


descent-lemma-flat-universally-injective

Let $f : X \to Y$ be a morphism of schemes. Let $X^0$ denote the set of generic points of irreducible components of $X$. If \begin{enumerate} \item $f$ is flat and separated, \item for $\xi \in X^0$ we have $\kappa(f(\xi)) = \kappa(\xi)$, and \item if $\xi, \xi' \in X^0$, $\xi \not = \xi'$, then $f(\xi) \not = f(\xi')$, \end{enumerate} then $f$ is universally injective.


descent-lemma-descending-property-proper-over-base

Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $Z$ be a closed subset of $X$. If there exists an fpqc covering $\{Y_i \to Y\}$ such that the inverse image $Z_i \subset Y_i \times_Y X$ is proper over $Y_i$ (Cohomology of Schemes, Definition \ref{coherent-definition-proper-over-base}) then $Z$ is proper over $Y$.


descent-lemma-descending-property-ample

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\{g_i : S_i \to S\}_{i \in I}$ be an fpqc covering. Let $f_i : X_i \to S_i$ be the base change of $f$ and let $\mathcal{L}_i$ be the pullback of $\mathcal{L}$ to $X_i$. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/S$, and \item $\mathcal{L}_i$ is ample on $X_i/S_i$ for every $i \in I$. \end{enumerate}


descent-lemma-precompose-property-local-source

Let $\tau \in \{fpqc, fppf, syntomic, smooth, \etale, Zariski\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$ local on the source. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $a : X' \to X$ which is flat, resp.\ flat and locally of finite presentation, resp.\ syntomic, resp.\ \'etale, resp.\ an open immersion, the composition $f \circ a : X' \to Y$ has property $\mathcal{P}$.


descent-lemma-largest-open-of-the-source

Let $\tau \in \{fppf, syntomic, smooth, \etale\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$ local on the source. For any morphism of schemes $f : X \to Y$ there exists a largest open $W(f) \subset X$ such that the restriction $f|_{W(f)} : W(f) \to Y$ has $\mathcal{P}$. Moreover, if $g : X' \to X$ is flat and locally of finite presentation, syntomic, smooth, or \'etale and $f' = f \circ g : X' \to Y$, then $g^{-1}(W(f)) = W(f')$.


descent-lemma-properties-morphisms-local-source

Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{fpqc, \linebreak[0] fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Assume that \begin{enumerate} \item the property is preserved under precomposing with flat, flat locally of finite presentation, \'etale, smooth or syntomic morphisms depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, \item the property is Zariski local on the source, \item the property is Zariski local on the target, \item for any morphism of affine schemes $X \to Y$, and any surjective morphism of affine schemes $X' \to X$ which is flat, flat of finite presentation, \'etale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, \'etale, smooth, or syntomic, property $\mathcal{P}$ holds for $f$ if property $\mathcal{P}$ holds for the composition $f' : X' \to Y$. \end{enumerate} Then $\mathcal{P}$ is $\tau$ local on the source.


descent-lemma-flat-fpqc-local-source

The property $\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source.


descent-lemma-injective-local-rings-fpqc-local-source

Then property $\mathcal{P}(f : X \to Y)=$``for every $x \in X$ the map of local rings $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is injective'' is fpqc local on the source.


descent-lemma-locally-finite-presentation-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is locally of finite presentation'' is fppf local on the source.


descent-lemma-locally-finite-type-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is locally of finite type'' is fppf local on the source.


descent-lemma-open-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is open'' is fppf local on the source.


descent-lemma-universally-open-fppf-local-source

The property $\mathcal{P}(f)=$``$f$ is universally open'' is fppf local on the source.


descent-lemma-syntomic-syntomic-local-source

The property $\mathcal{P}(f)=$``$f$ is syntomic'' is syntomic local on the source.


descent-lemma-smooth-smooth-local-source

The property $\mathcal{P}(f)=$``$f$ is smooth'' is smooth local on the source.


descent-lemma-etale-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is \'etale'' is \'etale local on the source.


descent-lemma-locally-quasi-finite-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is locally quasi-finite'' is \'etale local on the source.


descent-lemma-unramified-etale-local-source

The property $\mathcal{P}(f)=$``$f$ is unramified'' is \'etale local on the source. The property $\mathcal{P}(f)=$``$f$ is G-unramified'' is \'etale local on the source.


descent-lemma-local-source-target-implies

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on source-and-target. Then \begin{enumerate} \item $\mathcal{P}$ is \'etale local on the source, \item $\mathcal{P}$ is \'etale local on the target, \item $\mathcal{P}$ is stable under postcomposing with \'etale morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is \'etale, then $g \circ f$ has $\mathcal{P}$, and \item $\mathcal{P}$ has a permanence property: given $f : X \to Y$ and $g : Y \to Z$ \'etale such that $g \circ f$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$. \end{enumerate}


descent-lemma-local-source-target-characterize

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on source-and-target. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item[(a)] $f$ has property $\mathcal{P}$, \item[(b)] for every $x \in X$ there exists an \'etale morphism of germs $a : (U, u) \to (X, x)$, an \'etale morphism $b : V \to Y$, and a morphism $h : U \to V$ such that $f \circ a = b \circ h$ and $h$ has $\mathcal{P}$, \item[(c)] for any commutative diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ with $a$, $b$ \'etale the morphism $h$ has $\mathcal{P}$, \item[(d)] for some diagram as in (c) with $a : U \to X$ surjective $h$ has $\mathcal{P}$, \item[(e)] there exists an \'etale covering $\{Y_i \to Y\}_{i \in I}$ such that each base change $Y_i \times_Y X \to Y_i$ has $\mathcal{P}$, \item[(f)] there exists an \'etale covering $\{X_i \to X\}_{i \in I}$ such that each composition $X_i \to Y$ has $\mathcal{P}$, \item[(g)] there exists an \'etale covering $\{Y_i \to Y\}_{i \in I}$ and for each $i \in I$ an \'etale covering $\{X_{ij} \to Y_i \times_Y X\}_{j \in J_i}$ such that each morphism $X_{ij} \to Y_i$ has $\mathcal{P}$. \end{enumerate}


descent-lemma-etale-local-source-target

Let $\mathcal{P}$ be a property of morphisms of schemes. Assume \begin{enumerate} \item $\mathcal{P}$ is \'etale local on the source, \item $\mathcal{P}$ is \'etale local on the target, and \item $\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \to Y$ has $\mathcal{P}$ and $Y \subset Z$ is an open subscheme then $X \to Z$ has $\mathcal{P}$. \end{enumerate} Then $\mathcal{P}$ is \'etale local on the source-and-target.


descent-lemma-etale-etale-local-source-target

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the source-and-target. Given a commutative diagram of schemes $$ \vcenter{ \xymatrix{ X' \ar[d]_{g'} \ar[r]_{f'} & Y' \ar[d]^g \\ X \ar[r]^f & Y } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x' \ar[d] \ar[r] & y' \ar[d] \\ x \ar[r] & y } } $$ such that $g'$ is \'etale at $x'$ and $g$ is \'etale at $y'$, then $x \in W(f) \Leftrightarrow x' \in W(f')$ where $W(-)$ is as in Lemma \ref{lemma-largest-open-of-the-source}.


descent-lemma-orbits

Let $k$ be a field. Let $n \geq 2$. For $1 \leq i, j \leq n$ with $i \not = j$ and $d \geq 0$ denote $T_{i, j, d}$ the automorphism of $\mathbf{A}^n_k$ given in coordinates by $$ (x_1, \ldots, x_n) \longmapsto (x_1, \ldots, x_{i - 1}, x_i + x_j^d, x_{i + 1}, \ldots, x_n) $$ Let $W \subset \mathbf{A}^n_k$ be a nonempty open subscheme such that $T_{i, j, d}(W) = W$ for all $i, j, d$ as above. Then either $W = \mathbf{A}^n_k$ or the characteristic of $k$ is $p > 0$ and $\mathbf{A}^n_k \setminus W$ is a finite set of closed points whose coordinates are algebraic over $\mathbf{F}_p$.


descent-lemma-etale-tau-local-source-target

Let $\mathcal{P}$ be a property of morphisms of schemes. Assume \begin{enumerate} \item $\mathcal{P}$ is \'etale local on the source, \item $\mathcal{P}$ is smooth local on the target, \item $\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \to Y$ has $\mathcal{P}$ and $Y \subset Z$ is an open subscheme then $X \to Z$ has $\mathcal{P}$. \end{enumerate} Given a commutative diagram of schemes $$ \vcenter{ \xymatrix{ X' \ar[d]_{g'} \ar[r]_{f'} & Y' \ar[d]^g \\ X \ar[r]^f & Y } } \quad\text{with points}\quad \vcenter{ \xymatrix{ x' \ar[d] \ar[r] & y' \ar[d] \\ x \ar[r] & y } } $$ such that $g$ is smooth $y'$ and $X' \to X \times_Y Y'$ is \'etale at $x'$, then $x \in W(f) \Leftrightarrow x' \in W(f')$ where $W(-)$ is as in Lemma \ref{lemma-largest-open-of-the-source}.


descent-lemma-local-source-target-global-implies-local

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on the source-and-target. Consider the property $\mathcal{Q}$ of morphisms of germs defined by the rule $$ \mathcal{Q}((X, x) \to (S, s)) \Leftrightarrow \text{there exists a representative }U \to S \text{ which has }\mathcal{P} $$ Then $\mathcal{Q}$ is \'etale local on the source-and-target as in Definition \ref{definition-local-source-target-at-point}.


descent-lemma-local-source-target-local-implies-global

Let $\mathcal{P}$ be a property of morphisms of schemes which is \'etale local on source-and-target. Let $Q$ be the associated property of morphisms of germs, see Lemma \ref{lemma-local-source-target-global-implies-local}. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ has property $\mathcal{P}$, and \item for every $x \in X$ the morphism of germs $(X, x) \to (Y, f(x))$ has property $\mathcal{Q}$. \end{enumerate}


descent-lemma-flat-at-point

The property of morphisms of germs $$ \mathcal{P}((X, x) \to (S, s)) = \mathcal{O}_{S, s} \to \mathcal{O}_{X, x}\text{ is flat} $$ is \'etale local on the source-and-target.


descent-lemma-etale-on-fiber

Consider a commutative diagram of morphisms of schemes $$ \xymatrix{ U' \ar[r] \ar[d] & V' \ar[d] \\ U \ar[r] & V } $$ with \'etale vertical arrows and a point $v' \in V'$ mapping to $v \in V$. Then the morphism of fibres $U'_{v'} \to U_v$ is \'etale.


descent-lemma-dimension-local-ring-fibre

Let $d \in \{0, 1, 2, \ldots, \infty\}$. The property of morphisms of germs $$ \mathcal{P}_d((X, x) \to (S, s)) = \text{the local ring } \mathcal{O}_{X_s, x} \text{ of the fibre has dimension }d $$ is \'etale local on the source-and-target.


descent-lemma-transcendence-degree-at-point

Let $r \in \{0, 1, 2, \ldots, \infty\}$. The property of morphisms of germs $$ \mathcal{P}_r((X, x) \to (S, s)) \Leftrightarrow \text{trdeg}_{\kappa(s)} \kappa(x) = r $$ is \'etale local on the source-and-target.


descent-lemma-dimension-at-point

Let $d \in \{0, 1, 2, \ldots, \infty\}$. The property of morphisms of germs $$ \mathcal{P}_d((X, x) \to (S, s)) \Leftrightarrow \dim_x (X_s) = d $$ is \'etale local on the source-and-target.


descent-lemma-family-is-one

Let $S$ be a scheme. Let $\{X_i \to S\}_{i \in I}$ be a family of morphisms with target $S$. Set $X = \coprod_{i \in I} X_i$, and consider it as an $S$-scheme. There is a canonical equivalence of categories $$ \begin{matrix} \text{category of descent data } \\ \text{relative to the family } \{X_i \to S\}_{i \in I} \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix} $$ which maps $(V_i, \varphi_{ij})$ to $(V, \varphi)$ with $V = \coprod_{i\in I} V_i$ and $\varphi = \coprod \varphi_{ij}$.


descent-lemma-pullback

Pullback of descent data for schemes over schemes. \begin{enumerate} \item Let $$ \xymatrix{ X' \ar[r]_f \ar[d]_{a'} & X \ar[d]^a \\ S' \ar[r]^h & S } $$ be a commutative diagram of morphisms of schemes. The construction $$ (V \to X, \varphi) \longmapsto f^*(V \to X, \varphi) = (V' \to X', \varphi') $$ where $V' = X' \times_X V$ and where $\varphi'$ is defined as the composition $$ \xymatrix{ V' \times_{S'} X' \ar@{=}[r] & (X' \times_X V) \times_{S'} X' \ar@{=}[r] & (X' \times_{S'} X') \times_{X \times_S X} (V \times_S X) \ar[d]^{\text{id} \times \varphi} \\ X' \times_{S'} V' \ar@{=}[r] & X' \times_{S'} (X' \times_X V) & (X' \times_{S'} X') \times_{X \times_S X} (X \times_S V) \ar@{=}[l] } $$ defines a functor from the category of descent data relative to $X \to S$ to the category of descent data relative to $X' \to S'$. \item Given two morphisms $f_i : X' \to X$, $i = 0, 1$ making the diagram commute the functors $f_0^*$ and $f_1^*$ are canonically isomorphic. \end{enumerate}


descent-lemma-pullback-family

Let $\mathcal{U} = \{U_i \to S'\}_{i \in I}$ and $\mathcal{V} = \{V_j \to S\}_{j \in J}$ be families of morphisms with fixed target. Let $\alpha : I \to J$, $h : S' \to S$ and $g_i : U_i \to V_{\alpha(i)}$ be a morphism of families of maps with fixed target, see Sites, Definition \ref{sites-definition-morphism-coverings}. \begin{enumerate} \item Let $(Y_j, \varphi_{jj'})$ be a descent datum relative to the family $\{V_j \to S'\}$. The system $$ \left( g_i^*Y_{\alpha(i)}, (g_i \times g_{i'})^*\varphi_{\alpha(i)\alpha(i')} \right) $$ (with notation as in Remark \ref{remark-easier-family}) is a descent datum relative to $\mathcal{V}$. \item This construction defines a functor between descent data relative to $\mathcal{U}$ and descent data relative to $\mathcal{V}$. \item Given a second $\alpha' : I \to J$, $h' : S' \to S$ and $g'_i : U_i \to V_{\alpha'(i)}$ morphism of families of maps with fixed target, then if $h = h'$ the two resulting functors between descent data are canonically isomorphic. \item These functors agree, via Lemma \ref{lemma-family-is-one}, with the pullback functors constructed in Lemma \ref{lemma-pullback}. \end{enumerate}


descent-lemma-surjective-flat-epi

A surjective and flat morphism is an epimorphism in the category of schemes.


descent-lemma-ff-base-change-faithful

Let $h : S' \to S$ be a surjective, flat morphism of schemes. The base change functor $$ \Sch/S \longrightarrow \Sch/S', \quad X \longmapsto S' \times_S X $$ is faithful.


descent-lemma-faithful

In the situation of Lemma \ref{lemma-pullback} assume that $f : X' \to X$ is surjective and flat. Then the pullback functor is faithful.


descent-lemma-fully-faithful

In the situation of Lemma \ref{lemma-pullback} assume \begin{enumerate} \item $\{f : X' \to X\}$ is an fpqc covering (for example if $f$ is surjective, flat, and quasi-compact), and \item $S = S'$. \end{enumerate} Then the pullback functor is fully faithful.