The Stacks project

59.50 Étale cohomology

In the following sections we prove some basic results on étale cohomology. Here is an example of something we know for cohomology of topological spaces which also holds for étale cohomology.

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. $\square$

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 $\mathcal{I}^\bullet |_ V$, $\mathcal{I}^\bullet |_{U \cap V}$. $\square$

Comments (2)

Comment #1257 by Emmanuel Kowalski on

The trailing dots "\ldots" are missing at the end of the long exact sequence in the statement.

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