The Stacks project

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 ├ętale, 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. $\square$

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

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