The Stacks project

Lemma 20.34.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. If $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}_ X$-modules, then $\mathcal{H}_ Z(\mathcal{I}^\bullet )$ is K-injective complex of $\mathcal{O}_ X|_ Z$-modules.

Proof. Since $i_* : \textit{Mod}(\mathcal{O}_ X|_ Z) \to \textit{Mod}(\mathcal{O}_ X)$ is exact and left adjoint to $\mathcal{H}_ Z$ (Modules, Lemma 17.13.6) this follows from Derived Categories, Lemma 13.31.9. $\square$

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