Lemma 59.59.2 (03QU)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.59: Cohomology of a point
Lemma 59.59.2 (cite)

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Lemma 59.59.2. Notation and assumptions as in Lemma 59.59.1. Let $\mathcal{F}$ be an abelian sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ which corresponds to the $G$-module $M$. Then

in $D(\textit{Ab})$ we have a canonical isomorphism $R\Gamma (S, \mathcal{F}) = R\Gamma _ G(M)$,
$H_{\acute{e}tale}^0(S, \mathcal{F}) = M^ G$, and
$H_{\acute{e}tale}^ q(S, \mathcal{F}) = H^ q(G, M)$.

Proof. Combine Lemma 59.59.1 with Lemma 59.56.5. $\square$

The Stacks project

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