Lemma 37.64.12 (0950)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.64: Weakly étale morphisms
Lemma 37.64.12 (cite)

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Lemma 37.64.12. Let $f : X \to Y$ be a morphism of schemes. If $Y$ is a normal scheme and $f$ weakly étale, then $X$ is a normal scheme.

Proof. By More on Algebra, Lemma 15.45.6 a scheme $S$ is normal if and only if for all $s \in S$ the strict henselization of $\mathcal{O}_{S, s}$ is a normal domain. Hence the lemma follows from Lemma 37.64.11. $\square$

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