Lemma 20.33.4. Let (X, \mathcal{O}_ X) be a ringed space. Suppose that X = U \cup V is a union of two open subsets. For an object E of D(\mathcal{O}_ X) we have a distinguished triangle
R\Gamma (X, E) \to R\Gamma (U, E) \oplus R\Gamma (V, E) \to R\Gamma (U \cap V, E) \to R\Gamma (X, E)[1]
and in particular a long exact cohomology sequence
\ldots \to H^ n(X, E) \to H^ n(U, E) \oplus H^0(V, E) \to H^ n(U \cap V, E) \to H^{n + 1}(X, E) \to \ldots
The construction of the distinguished triangle and the long exact sequence is functorial in E.
Proof.
Choose a K-injective complex \mathcal{I}^\bullet representing E. We may assume \mathcal{I}^ n is an injective object of \textit{Mod}(\mathcal{O}_ X) for all n, see Injectives, Theorem 19.12.6. Then R\Gamma (X, E) is computed by \Gamma (X, \mathcal{I}^\bullet ). Similarly for U, V, and U \cap V by Lemma 20.32.1. Hence 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
0 \to \mathcal{I}^\bullet (X) \to \mathcal{I}^\bullet (U) \oplus \mathcal{I}^\bullet (V) \to \mathcal{I}^\bullet (U \cap V) \to 0.
We have seen this is a short exact sequence in the proof of Lemma 20.8.2. The final statement follows from the functoriality of the construction in Injectives, Theorem 19.12.6.
\square
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