In this section we collect some results on comparing cohomology defined using stacks and using algebraic spaces.
Lemma 96.22.1. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$ representable by the algebraic space $F$.
If $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$, then $\mathcal{I}|_{F_{\acute{e}tale}}$ is injective in $\textit{Ab}(F_{\acute{e}tale})$,
If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$, then $\mathcal{I}^\bullet |_{F_{\acute{e}tale}}$ is a K-injective complex in $\textit{Ab}(F_{\acute{e}tale})$.
The same does not hold for modules.
Proof. This follows formally from the fact that the restriction functor $\pi _{F, *} = i_ F^{-1}$ (see Lemma 96.10.1) is right adjoint to the exact functor $\pi _ F^{-1}$, see Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9. To see that the lemma does not hold for modules, we refer the reader to Étale Cohomology, Lemma 59.99.1. $\square$
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.
For any $\mathcal{F} \in \textit{Ab}(\mathcal{X}_{\acute{e}tale})$ we have
in $D(G_{\acute{e}tale})$.
For any object $\mathcal{F}$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ we have
in $D(\mathcal{O}_ G)$.
\[ (Rf_*\mathcal{F})|_{G_{\acute{e}tale}} = Rf_{small, *}(\mathcal{F}|_{F_{\acute{e}tale}}) \]
\[ (Rf_*\mathcal{F})|_{G_{\acute{e}tale}} = Rf_{small, *}(\mathcal{F}|_{F_{\acute{e}tale}}) \]
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. $\square$
Lemma 96.22.3. Let $S$ be a scheme. Consider a $2$-fibre product square 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 for any $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$ or in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$
\[ \xymatrix{ \mathcal{X}' \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r]^ g & \mathcal{Y} } \]
\[ g^{-1}(Rf_*\mathcal{F})|_{G'_{\acute{e}tale}} = Rf'_{small, *}((g')^{-1}\mathcal{F}|_{F'_{\acute{e}tale}}) \]
Proof. Follows formally on combining Lemmas 96.21.3 and 96.22.2. $\square$
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