Definition 37.34.1. Let $S$ be a scheme. Let $s \in S$ be a point.

An

*étale neighbourhood of $(S, s)$*is a pair $(U, u)$ together with an étale morphism of schemes $\varphi : U \to S$ such that $\varphi (u) = s$.A

*morphism of étale neighbourhoods*$f : (V, v) \to (U, u)$ of $(S, s)$ is simply a morphism of $S$-schemes $f : V \to U$ such that $f(v) = u$.An

*elementary étale neighbourhood*is an étale neighbourhood $\varphi : (U, u) \to (S, s)$ such that $\kappa (s) = \kappa (u)$.

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