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  • Namely, just as if $k$ is a field, then there is an equivalence of categories between continuous transitive $G$-sets ($G$ the absolute Galois group) and algebraic extensions of $k$, there is an equivalence of categories between finite \'etale maps $R o S$ ($R$ a fixed ring, say a domain) and finite continuous $G$-sets for $G$ the \'etale fundamental group.
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