The Stacks project

Lemma 66.18.12. 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 $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. If $g$ is ├ętale, then

  1. $f'_{small, *}(\mathcal{F}|_{X'}) = (f_{small, *}\mathcal{F})|_{Y'}$ in $\mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale})$1, and

  2. if $\mathcal{F}$ is an abelian sheaf, then $R^ if'_{small, *}(\mathcal{F}|_{X'}) = (R^ if_{small, *}\mathcal{F})|_{Y'}$.

Proof. Consider the following diagram of functors

\[ \xymatrix{ X'_{spaces, {\acute{e}tale}} \ar[r]_ j & X_{spaces, {\acute{e}tale}} \\ Y'_{spaces, {\acute{e}tale}} \ar[r]^ j \ar[u]^{V' \mapsto V' \times _{Y'} X'} & Y_{spaces, {\acute{e}tale}} \ar[u]_{V \mapsto V \times _ Y X} } \]

The horizontal arrows are localizations and the vertical arrows induce morphisms of sites. Hence the last statement of Sites, Lemma 7.28.1 gives (1). To see (2) apply (1) to an injective resolution of $\mathcal{F}$ and use that restriction is exact and preserves injectives (see Cohomology on Sites, Lemma 21.7.1). $\square$

[1] Also $(f')_{small}^{-1}(\mathcal{G}|_{Y'}) = (f_{small}^{-1}\mathcal{G})|_{X'}$ because of commutativity of the diagram and (

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