The Stacks project

Proposition 41.19.5. Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an étale homomorphism of local rings. Then $A$ is reduced if and only if $B$ is so.

Proof. It is clear from the faithful flatness of $A \to B$ that if $B$ is reduced, so is $A$. See also Algebra, Lemma 10.164.2. Conversely, assume $A$ is reduced. By assumption $B$ is a localization of a finite type $A$-algebra $B'$ at some prime $\mathfrak q$. After replacing $B'$ by a localization we may assume that $B'$ is étale over $A$, see Lemma 41.11.2. Then we see that Algebra, Lemma 10.163.7 applies to $A \to B'$ and $B'$ is reduced. Hence $B$ is reduced. $\square$

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