Lemma 21.34.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules. Let $\mathcal{L}^\bullet $ be a K-flat complex of $\mathcal{O}$-modules. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}$-modules.
Proof. Namely, if $\mathcal{K}^\bullet $ is an acyclic complex of $\mathcal{O}$-modules, then
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) & = H^0(\Gamma (\mathcal{C}, \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 (\mathcal{C}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\text{Tot}( \mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ))) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}( \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ) \\ & = 0 \end{align*}
The first equality by (21.34.0.2). The second equality by Lemma 21.34.1. The third equality by (21.34.0.2). The final equality because $\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet )$ is acyclic because $\mathcal{L}^\bullet $ is K-flat (Definition 21.17.2) and because $\mathcal{I}^\bullet $ is K-injective. $\square$
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