Lemma 20.8.3 (Relative Mayer-Vietoris). 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 $\mathcal{O}_ X$-module $\mathcal{F}$ there exists a long exact sequence
\[ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \oplus b_*(\mathcal{F}|_ V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots \]
This long exact sequence is functorial in $\mathcal{F}$.
Proof.
Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution of $\mathcal{F}$. 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 open $W \subset Y$, and for any $n \geq 0$ the corresponding sequence of groups of sections over $W$
\[ 0 \to \mathcal{I}^ n(f^{-1}(W)) \to \mathcal{I}^ n(U \cap f^{-1}(W)) \oplus \mathcal{I}^ n(V \cap f^{-1}(W)) \to \mathcal{I}^ n(U \cap V \cap f^{-1}(W)) \to 0 \]
was shown to be short exact in the proof of Lemma 20.8.2. 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 $\mathcal{I}^\bullet |_ V$, $\mathcal{I}^\bullet |_{U \cap V}$ see Lemma 20.7.1.
$\square$
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