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
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) & = H^0(\Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )))) \\ & = H^0(\Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ))) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}( \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ) \\ & = 0 \end{align*}
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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