The Stacks project

Lemma 20.41.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $L$ and $M$ be objects of $D(\mathcal{O}_ X)$. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules representing $M$. Let $\mathcal{L}^\bullet $ be a complex of $\mathcal{O}_ X$-modules representing $L$. 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)}(\mathcal{L}^\bullet |_ U, \mathcal{I}^\bullet |_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \end{align*}

The first equality is (20.41.0.1). The second equality is true because $\mathcal{I}^\bullet |_ U$ is K-injective by Lemma 20.32.1. $\square$

Comments (2)

Comment #8624 by nkym on

and are not defined in the statement.

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