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The Stacks project

Lemma 59.50.2 (Relative Mayer-Vietoris). Let f : X \to Y be a morphism of schemes. Suppose that X = U \cup V is a union of two open subschemes. 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 abelian sheaf \mathcal{F} on X_{\acute{e}tale} there exists a long exact sequence

0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \oplus b_*(\mathcal{F}|_ V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots

on Y_{\acute{e}tale}. This long exact sequence is functorial in \mathcal{F}.

Proof. Let \mathcal{F} \to \mathcal{I}^\bullet be an injective resolution of \mathcal{F} on X_{\acute{e}tale}. We claim that we get a 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.

Namely, for any W in Y_{\acute{e}tale}, and for any n \geq 0 the corresponding sequence of groups of sections over W

0 \to \mathcal{I}^ n(W \times _ Y X) \to \mathcal{I}^ n(W \times _ Y U) \oplus \mathcal{I}^ n(W \times _ Y V) \to \mathcal{I}^ n(W \times _ Y (U \cap V)) \to 0

was shown to be short exact in the proof of Lemma 59.50.1. The lemma follows by taking cohomology sheaves and using the fact that \mathcal{I}^\bullet |_ U is an injective resolution of \mathcal{F}|_ U and similarly for \mathcal{I}^\bullet |_ V, \mathcal{I}^\bullet |_{U \cap V}. \square


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