Lemma 15.44.1. If $A \to B$ is an étale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $A_{\mathfrak p}$ is Noetherian if and only if $B_{\mathfrak q}$ is Noetherian.

Proof. Since $A_\mathfrak p \to B_\mathfrak q$ is faithfully flat we see that $B_\mathfrak q$ Noetherian implies that $A_\mathfrak p$ is Noetherian, see Algebra, Lemma 10.164.1. Conversely, if $A_\mathfrak p$ is Noetherian, then $B_\mathfrak q$ is Noetherian as it is a localization of a finite type $A_\mathfrak p$-algebra. $\square$

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