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The Stacks project

Lemma 64.3.7. In the situation above denote \alpha : X \to \mathop{\mathrm{Spec}}(k) the structure morphism. Consider the stalk (R^ j\alpha _*\mathcal{F})_{\mathop{\mathrm{Spec}}(\bar k)} endowed with its natural Galois action as in Étale Cohomology, Section 59.56. Then the identification

(R^ j\alpha _*\mathcal{F})_{\mathop{\mathrm{Spec}}(\bar k)} \cong H^ j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}})

from Étale Cohomology, Theorem 59.53.1 is an isomorphism of G_ k-modules.

Proof. Omitted. \square


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