Lemma 41.14.2. Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that
$\pi $ is finite,
$\pi $ is étale,
$\pi ^{-1}(\{ s\} ) = \{ x\} $, and
$\kappa (s) \subset \kappa (x)$ 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. $\square$
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