The Stacks project

Lemma 59.50.1 (Mayer-Vietoris for étale cohomology). Let $X$ be a scheme. Suppose that $X = U \cup V$ is a union of two opens. For any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ there exists a long exact cohomology sequence

\[ \begin{matrix} 0 \to H^0_{\acute{e}tale}(X, \mathcal{F}) \to H^0_{\acute{e}tale}(U, \mathcal{F}) \oplus H^0_{\acute{e}tale}(V, \mathcal{F}) \to H^0_{\acute{e}tale}(U \cap V, \mathcal{F}) \phantom{\to \ldots } \\ \phantom{0} \to H^1_{\acute{e}tale}(X, \mathcal{F}) \to H^1_{\acute{e}tale}(U, \mathcal{F}) \oplus H^1_{\acute{e}tale}(V, \mathcal{F}) \to H^1_{\acute{e}tale}(U \cap V, \mathcal{F}) \to \ldots \end{matrix} \]

This long exact sequence is functorial in $\mathcal{F}$.

Proof. Observe that if $\mathcal{I}$ is an injective abelian sheaf, then

\[ 0 \to \mathcal{I}(X) \to \mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V) \to 0 \]

is exact. This is true in the first and middle spots as $\mathcal{I}$ is a sheaf. It is true on the right, because $\mathcal{I}(U) \to \mathcal{I}(U \cap V)$ is surjective by Cohomology on Sites, Lemma 21.12.6. Another way to prove it would be to show that the cokernel of the map $\mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V)$ is the first Čech cohomology group of $\mathcal{I}$ with respect to the covering $X = U \cup V$ which vanishes by Lemmas 59.18.7 and 59.19.1. Thus, if $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution, then

\[ 0 \to \mathcal{I}^\bullet (X) \to \mathcal{I}^\bullet (U) \oplus \mathcal{I}^\bullet (V) \to \mathcal{I}^\bullet (U \cap V) \to 0 \]

is a short exact sequence of complexes and the associated long exact cohomology sequence is the sequence of the statement of the lemma. $\square$

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