The Stacks project

Lemma 96.22.3. Let $S$ be a scheme. Consider a $2$-fibre product square

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

of algebraic stacks over $S$. Assume that $f$ is representable by algebraic spaces and that $\mathcal{Y}'$ is representable by an algebraic space $G'$. Then $\mathcal{X}'$ is representable by an algebraic space $F'$ and denoting $f' : F' \to G'$ the induced morphism of algebraic spaces we have

\[ g^{-1}(Rf_*\mathcal{F})|_{G'_{\acute{e}tale}} = Rf'_{small, *}((g')^{-1}\mathcal{F}|_{F'_{\acute{e}tale}}) \]

for any $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$ or in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$

Proof. Follows formally on combining Lemmas 96.21.3 and 96.22.2. $\square$

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