The Stacks project

Lemma 96.22.1. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$ representable by the algebraic space $F$.

  1. If $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_{\acute{e}tale})$, then $\mathcal{I}|_{F_{\acute{e}tale}}$ is injective in $\textit{Ab}(F_{\acute{e}tale})$,

  2. 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. $\square$

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