Lemma 59.59.1. Let $S = \mathop{\mathrm{Spec}}(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa (s)}$ denote the absolute Galois group. The stalk functor induces an equivalence of categories

$\textit{Ab}(S_{\acute{e}tale}) \longrightarrow \text{Mod}_ G, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}.$

Proof. In Theorem 59.56.3 we have seen the equivalence between sheaves of sets and $G$-sets. The current lemma follows formally from this as an abelian sheaf is just a sheaf of sets endowed with a commutative group law, and a $G$-module is just a $G$-set endowed with a commutative group law. $\square$

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