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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