The Stacks project

Lemma 21.35.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. For every object $U$ of $\mathcal{C}$ we have

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

and we have $H^0(\mathcal{C}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(L, M)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}$-modules representing $M$ and a K-flat complex $\mathcal{L}^\bullet $ representing $L$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is K-injective by Lemma 21.34.8. Hence we can compute cohomology over $U$ by simply taking sections over $U$ and the result follows from Lemma 21.34.6. $\square$

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