Lemma 20.33.1 (08BU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.33: Unbounded Mayer-Vietoris
Lemma 20.33.1 (cite)

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Lemma 20.33.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces. For any object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

$j_{U \cap V!}E|_{U \cap V} \to j_{U!}E|_ U \oplus j_{V!}E|_ V \to E \to j_{U \cap V!}E|_{U \cap V}[1]$

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

Proof. We have seen in Section 20.32 that the restriction functors and the extension by zero functors are computed by just applying the functors to any complex. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_ X$ -modules representing $E$ . The distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 13.12 and especially Lemma 13.12.1) to the short exact sequence of complexes of $\mathcal{O}_ X$ -modules

$0 \to j_{U \cap V!}\mathcal{E}^\bullet |_{U \cap V} \to j_{U!}\mathcal{E}^\bullet |_ U \oplus j_{V!}\mathcal{E}^\bullet |_ V \to \mathcal{E}^\bullet \to 0$

To see this sequence is exact one checks on stalks using Sheaves, Lemma 6.31.8 (computation omitted). $\square$

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