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 inseparable

^{1}.

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

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