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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