The Stacks project

Lemma 41.14.2. Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

  1. $\pi $ is finite,

  2. $\pi $ is étale,

  3. $\pi ^{-1}(\{ s\} ) = \{ x\} $, and

  4. $\kappa (s) \subset \kappa (x)$ is purely inseparable1.

Then there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism.

Proof. By Lemma 41.7.3 there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is a closed immersion. But a morphism which is étale and a closed immersion is an open immersion (for example by Theorem 41.14.1). Hence after shrinking $U$ we obtain an isomorphism. $\square$

[1] In view of condition (2) this is equivalent to $\kappa (s) = \kappa (x)$.

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