The Stacks project

Lemma 20.21.1. Let $i : Z \to X$ be the inclusion of a closed subset. Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Then $\mathcal{H}_ Z(\mathcal{I})$ is an injective abelian sheaf on $Z$.

Proof. Observe that for any abelian sheaf $\mathcal{G}$ on $Z$ 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 (Modules, Lemma 17.6.1) and $\mathcal{I}$ injective on $X$ we conclude that $\mathcal{H}_ Z(\mathcal{I})$ is injective on $Z$. $\square$


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