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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