The Stacks project

Lemma 68.9.1. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I}$ be an injective abelian sheaf on $X_{\acute{e}tale}$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z_{\acute{e}tale}$.

Proof. Observe that for any abelian sheaf $\mathcal{G}$ on $Z_{\acute{e}tale}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _ Z(\mathcal{G}, \mathcal{H}_ Z(\mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits _ X(i_*\mathcal{G}, \mathcal{F}) \]

because after all any section of $i_*\mathcal{G}$ has support in $Z$. Since $i_*$ is exact (Lemma 68.4.1) and as $\mathcal{I}$ is injective on $X_{\acute{e}tale}$ we conclude that $\mathcal{H}_ Z(\mathcal{I})$ is injective on $Z_{\acute{e}tale}$. $\square$


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