The Stacks project

Lemma 15.56.3. Let $A \to B$ be a ring map. If $I^\bullet $ is a K-injective complex of $A$-modules, then $\mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )$ is a K-injective complex of $B$-modules.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(B)}(N^\bullet , \mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(A)}(N^\bullet , I^\bullet )$ by Algebra, Lemma 10.14.4. $\square$

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