Lemma 20.42.3 (08DL)—The Stacks project

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Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.42: Internal hom in the derived category
Lemma 20.42.3 (cite)

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Lemma 20.42.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$ . The construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ commutes with restrictions to opens, i.e., for every open $U$ we have $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K|_ U, L|_ U) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)|_ U$ .

Proof. This is clear from the construction and Lemma 20.32.1. $\square$

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