The Stacks project

Lemma 21.20.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The restriction of a K-injective complex of $\mathcal{O}$-modules to $\mathcal{C}/U$ is a K-injective complex of $\mathcal{O}_ U$-modules.

Proof. Follows immediately from Derived Categories, Lemma 13.31.9 and the fact that the restriction functor has the exact left adjoint $j_!$. See discussion above. $\square$

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