The Stacks project

Lemma 15.56.2. Let $R \to S$ be an epimorphism of rings. Let $I^\bullet $ be a complex of $S$-modules. If $I^\bullet $ is K-injective as a complex of $R$-modules, then $I^\bullet $ is a K-injective complex of $S$-modules.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(R)}(N^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(S)}(N^\bullet , I^\bullet )$ for any complex of $S$-modules $N^\bullet $, see Algebra, Lemma 10.107.14. $\square$

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