Lemma 29.36.14. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent

1. The morphism $f$ is étale at $x$.

2. There exist an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_ U : U \to V$ is standard étale (see Definition 29.36.1).

Proof. Follows from the definitions and Algebra, Proposition 10.144.4. $\square$

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