Lemma 20.33.5. Let f : X \to Y be a morphism of ringed spaces. Suppose that X = U \cup V is a union of two open subsets. Denote a = f|_ U : U \to Y, b = f|_ V : V \to Y, and c = f|_{U \cap V} : U \cap V \to Y. For every object E of D(\mathcal{O}_ X) there exists a distinguished triangle
Rf_*E \to Ra_*(E|_ U) \oplus Rb_*(E|_ V) \to Rc_*(E|_{U \cap V}) \to Rf_*E[1]
This triangle 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 Rf_*E is computed by f_*\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 f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet |_ U \oplus b_*\mathcal{I}^\bullet |_ V \to c_*\mathcal{I}^\bullet |_{U \cap V} \to 0.
This is a short exact sequence of complexes by Lemma 20.8.3 and the fact that R^1f_*\mathcal{I} = 0 for an injective object \mathcal{I} of \textit{Mod}(\mathcal{O}_ X). The final statement follows from the functoriality of the construction in Injectives, Theorem 19.12.6.
\square
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