The Stacks project

59.38 Functoriality and sheaves of modules

In this section we are going to reformulate some of the material explained in Descent, Sections 35.8, 35.9, and 35.10 in the setting of étale topologies. Let $f : X \to Y$ be a morphism of schemes. We have seen above, see Sections 59.34, 59.35, and 59.36 that this induces a morphism $f_{small}$ of small étale sites. In Descent, Remark 35.8.4 we have seen that $f$ also induces a natural map

\[ f_{small}^\sharp : \mathcal{O}_{Y_{\acute{e}tale}} \longrightarrow f_{small, *}\mathcal{O}_{X_{\acute{e}tale}} \]

of sheaves of rings on $Y_{\acute{e}tale}$ such that $(f_{small}, f_{small}^\sharp )$ is a morphism of ringed sites. See Modules on Sites, Definition 18.6.1 for the definition of a morphism of ringed sites. Let us just recall here that $f_{small}^\sharp $ is defined by the compatible system of maps

\[ \text{pr}_ V^\sharp : \mathcal{O}(V) \longrightarrow \mathcal{O}(X \times _ Y V) \]

for $V$ varying over the objects of $Y_{\acute{e}tale}$.

It is clear that this construction is compatible with compositions of morphisms of schemes. More precisely, if $f : X \to Y$ and $g : Y \to Z$ are morphisms of schemes, then we have

\[ (g_{small}, g_{small}^\sharp ) \circ (f_{small}, f_{small}^\sharp ) = ((g \circ f)_{small}, (g \circ f)_{small}^\sharp ) \]

as morphisms of ringed topoi. Moreover, by Modules on Sites, Definition 18.13.1 we see that given a morphism $f : X \to Y$ of schemes we get well defined pullback and direct image functors

\begin{align*} f_{small}^* : \textit{Mod}(\mathcal{O}_{Y_{\acute{e}tale}}) \longrightarrow \textit{Mod}(\mathcal{O}_{X_{\acute{e}tale}}), \\ f_{small, *} : \textit{Mod}(\mathcal{O}_{X_{\acute{e}tale}}) \longrightarrow \textit{Mod}(\mathcal{O}_{Y_{\acute{e}tale}}) \end{align*}

which are adjoint in the usual way. If $g : Y \to Z$ is another morphism of schemes, then we have $(g \circ f)_{small}^* = f_{small}^* \circ g_{small}^*$ and $(g \circ f)_{small, *} = g_{small, *} \circ f_{small, *}$ because of what we said about compositions.

There is quite a bit of difference between the category of all $\mathcal{O}_ X$ modules on $X$ and the category between all $\mathcal{O}_{X_{\acute{e}tale}}$-modules on $X_{\acute{e}tale}$. But the results of Descent, Sections 35.8, 35.9, and 35.10 tell us that there is not much difference between considering quasi-coherent modules on $S$ and quasi-coherent modules on $S_{\acute{e}tale}$. (We have already seen this in Theorem 59.17.4 for example.) In particular, if $f : X \to Y$ is any morphism of schemes, then the pullback functors $f_{small}^*$ and $f^*$ match for quasi-coherent sheaves, see Descent, Proposition 35.9.4. Moreover, the same is true for pushforward provided $f$ is quasi-compact and quasi-separated, see Descent, Lemma 35.9.5.

A few words about functoriality of the structure sheaf on big sites. Let $f : X \to Y$ be a morphism of schemes. Choose any of the topologies $\tau \in \{ Zariski,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} $. Then the morphism $f_{big} : (\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau $ becomes a morphism of ringed sites by a map

\[ f_{big}^\sharp : \mathcal{O}_ Y \longrightarrow f_{big, *}\mathcal{O}_ X \]

see Descent, Remark 35.8.4. In fact it is given by the same construction as in the case of small sites explained above.

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