Lemma 65.18.12. 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 étale, then

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

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).
$\square$

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