The Stacks project

65.26 Sheaves of modules on algebraic spaces

If $X$ is an algebraic space, then a sheaf of modules on $X$ is a sheaf of $\mathcal{O}_ X$-modules on the small étale site of $X$ where $\mathcal{O}_ X$ is the structure sheaf of $X$. The category of sheaves of modules is denoted $\textit{Mod}(\mathcal{O}_ X)$.

Given a morphism $f : X \to Y$ of algebraic spaces, by Lemma 65.21.3 we get a morphism of ringed topoi and hence by Modules on Sites, Definition 18.13.1 we get well defined pullback and direct image functors

65.26.0.1
\begin{equation} \label{spaces-properties-equation-push-pull} f^* : \textit{Mod}(\mathcal{O}_ Y) \longrightarrow \textit{Mod}(\mathcal{O}_ X), \quad f_* : \textit{Mod}(\mathcal{O}_ X) \longrightarrow \textit{Mod}(\mathcal{O}_ Y) \end{equation}

which are adjoint in the usual way. If $g : Y \to Z$ is another morphism of algebraic spaces over $S$, then we have $(g \circ f)^* = f^* \circ g^*$ and $(g \circ f)_* = g_* \circ f_*$ simply because the morphisms of ringed topoi compose in the corresponding way (by the lemma).

Lemma 65.26.1. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces over $S$. Then $f^{-1}\mathcal{O}_ Y = \mathcal{O}_ X$, and $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$ for any sheaf of $\mathcal{O}_ Y$-modules $\mathcal{G}$. In particular, $f^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$ is exact.

Proof. By the description of inverse image in Lemma 65.18.11 and the definition of the structure sheaves it is clear that $f_{small}^{-1}\mathcal{O}_ Y = \mathcal{O}_ X$. Since the pullback

\[ f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \otimes _{f_{small}^{-1}\mathcal{O}_ Y} \mathcal{O}_ X \]

by definition we conclude that $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$. The exactness is clear because $f_{small}^{-1}$ is exact, as $f_{small}$ is a morphism of topoi. $\square$

We continue our abuse of notation introduced in Equation (65.18.11.1) by writing

65.26.1.1
\begin{equation} \label{spaces-properties-equation-restrict-modules} \mathcal{G}|_{X_{\acute{e}tale}} = f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \end{equation}

in the situation of the lemma above. We will discuss this in a more technical fashion in Section 65.27.

Lemma 65.26.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r] \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian square of algebraic spaces over $S$. Let $\mathcal{F} \in \textit{Mod}(\mathcal{O}_ X)$. If $g$ is étale, then $f'_*(\mathcal{F}|_{X'}) = (f_*\mathcal{F})|_{Y'}$1 and $R^ if'_*(\mathcal{F}|_{X'}) = (R^ if_*\mathcal{F})|_{Y'}$ in $\textit{Mod}(\mathcal{O}_{Y'})$.

Proof. This is a reformulation of Lemma 65.18.12 in the case of modules. $\square$

Lemma 65.26.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A sheaf $\mathcal{F}$ of $\mathcal{O}_ X$-modules is given by the following data:

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ a sheaf $\mathcal{F}_ U$ of $\mathcal{O}_ U$-modules on $U_{\acute{e}tale}$,

  2. for every $f : U' \to U$ in $X_{\acute{e}tale}$ an isomorphism $c_ f : f_{small}^*\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^*c_ f$ is equal to $c_{f \circ g}$.

Proof. Combine Lemmas 65.26.1 and 65.18.13, and use the fact that any morphism between objects of $X_{\acute{e}tale}$ is an étale morphism of schemes. $\square$

[1] Also $(f')^*(\mathcal{G}|_{Y'}) = (f^*\mathcal{G})|_{X'}$ by commutativity of the diagram and (65.26.1.1)

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