The Stacks project

Lemma 96.22.2. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over $S$. Assume $\mathcal{X}$, $\mathcal{Y}$ are representable by algebraic spaces $F$, $G$. Denote $f : F \to G$ the induced morphism of algebraic spaces.

  1. For any $\mathcal{F} \in \textit{Ab}(\mathcal{X}_{\acute{e}tale})$ we have

    \[ (Rf_*\mathcal{F})|_{G_{\acute{e}tale}} = Rf_{small, *}(\mathcal{F}|_{F_{\acute{e}tale}}) \]

    in $D(G_{\acute{e}tale})$.

  2. For any object $\mathcal{F}$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ we have

    \[ (Rf_*\mathcal{F})|_{G_{\acute{e}tale}} = Rf_{small, *}(\mathcal{F}|_{F_{\acute{e}tale}}) \]

    in $D(\mathcal{O}_ G)$.

Proof. Part (1) follows immediately from Lemma 96.22.1 and (96.10.3.1) on choosing an injective resolution of $\mathcal{F}$.

Part (2) can be proved as follows. In Lemma 96.10.3 we have seen that $\pi _ G \circ f = f_{small} \circ \pi _ F$ as morphisms of ringed sites. Hence we obtain $R\pi _{G, *} \circ Rf_* = Rf_{small, *} \circ R\pi _{F, *}$ by Cohomology on Sites, Lemma 21.19.2. Since the restriction functors $\pi _{F, *}$ and $\pi _{G, *}$ are exact, we conclude. $\square$


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