Lemma 20.41.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules. Let $\mathcal{L}^\bullet $ be a K-flat complex of $\mathcal{O}_ X$-modules. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}_ X$-modules.

**Proof.**
Namely, if $\mathcal{K}^\bullet $ is an acyclic complex of $\mathcal{O}_ X$-modules, then

The first equality by (20.41.0.1). The second equality by Lemma 20.41.1. The third equality by (20.41.0.1). The final equality because $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )$ is acyclic because $\mathcal{L}^\bullet $ is K-flat (Definition 20.26.2) and because $\mathcal{I}^\bullet $ is K-injective. $\square$

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