The Stacks project

Lemma 103.8.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is quasi-compact, quasi-separated, and representable by algebraic spaces. Let $\mathcal{F}$ be in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Then for an object $y : V \to \mathcal{Y}$ of $\mathcal{Y}$ we have

\[ (R^ if_*\mathcal{F})|_{V_{\acute{e}tale}} = R^ if'_{small, *}(\mathcal{F}|_{U_{\acute{e}tale}}) \]

where $f' : U = V \times _\mathcal {Y} \mathcal{X} \to V$ is the base change of $f$.

Proof. By Sheaves on Stacks, Lemma 96.21.3 we can reduce to the case where $\mathcal{X}$ is represented by $U$ and $\mathcal{Y}$ is represented by $V$. Of course this also uses that the pullback of $\mathcal{F}$ to $U$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_ U)$ by Proposition 103.8.1. Then the result follows from Sheaves on Stacks, Lemma 96.22.2 and the fact that $R^ if_*$ may be computed in the étale topology by Proposition 103.8.1. $\square$

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