Lemma 15.56.1. Let $R \to S$ be a flat ring map. If $I^\bullet $ is a K-injective complex of $S$-modules, then $I^\bullet $ is K-injective as a complex of $R$-modules.

**Proof.**
This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(R)}(M^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(S)}(M^\bullet \otimes _ R S, I^\bullet )$ by Algebra, Lemma 10.14.3 and the fact that tensoring with $S$ is exact.
$\square$

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