Lemma 20.34.6 (0G72)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.34: Cohomology with support in a closed subset, II
Lemma 20.34.6 (cite)

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Lemma 20.34.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Denote $j : U = X \setminus Z \to X$ the inclusion of the complement. There is a distinguished triangle

$i_*R\mathcal{H}_ Z(K) \to K \to Rj_*(K|_ U) \to i_*R\mathcal{H}_ Z(K)[1]$

in $D(\mathcal{O}_ X)$ functorial for $K$ in $D(\mathcal{O}_ X)$ .

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ all of whose terms are injective $\mathcal{O}_ X$ -modules representing $K$ . See Section 20.28. Recall that $\mathcal{I}^\bullet |_ U$ is a K-injective complex of $\mathcal{O}_ U$ -modules, see Lemma 20.32.1. Hence each of the derived functors in the distinguished triangle is gotten by applying the underlying functor to $\mathcal{I}^\bullet$ . Hence it suffices to prove that for an injective $\mathcal{O}_ X$ -module $\mathcal{I}$ we have a short exact sequence

$0 \to i_*\mathcal{H}_ Z(\mathcal{I}) \to \mathcal{I} \to j_*(\mathcal{I}|_ U) \to 0$

This follows from Lemma 20.8.1 and the definitions. $\square$

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