Lemma 21.20.11. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}$-modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}'$-modules.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}')}(\mathcal{G}^\bullet , \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{G}^\bullet , \mathcal{I}^\bullet )$ by Modules on Sites, Lemma 18.27.8. $\square$

Comment #2117 by Kestutis Cesnavicius on

In the statement $I^\bullet$ should be $\mathcal{I}^\bullet$.

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