The Stacks project

Lemma 33.11.1. Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k \subset k'$ be a finitely generated field extension. Then $X$ is locally Noetherian if and only if $X_{k'}$ is locally Noetherian.

Proof. Using Properties, Lemma 28.5.2 we reduce to the case where $X$ is affine, say $X = \mathop{\mathrm{Spec}}(A)$. In this case we have to prove that $A$ is Noetherian if and only if $A_{k'}$ is Noetherian. Since $A \to A_{k'} = k' \otimes _ k A$ is faithfully flat, we see that if $A_{k'}$ is Noetherian, then so is $A$, by Algebra, Lemma 10.164.1. Conversely, if $A$ is Noetherian then $A_{k'}$ is Noetherian by Algebra, Lemma 10.31.8. $\square$


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