Lemma 20.37.6. 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 complex of $\mathcal{O}_ X$-modules. Then

for all $U \subset X$ open.

Lemma 20.37.6. 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 complex of $\mathcal{O}_ X$-modules. Then

\[ H^0(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ))) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \]

for all $U \subset X$ open.

**Proof.**
We have

\begin{align*} H^0(\Gamma (U, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ))) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(L|_ U, M|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \end{align*}

The first equality is (20.37.0.1). The second equality is true because $\mathcal{I}^\bullet |_ U$ is K-injective by Lemma 20.32.1. $\square$

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