The Stacks project

33.11 Change of fields and locally Noetherian schemes

Let $X$ a locally Noetherian scheme over a field $k$. It is not always that case that $X_{k'}$ is locally Noetherian too. For example if $X = \mathop{\mathrm{Spec}}(\overline{\mathbf{Q}})$ and $k = \mathbf{Q}$, then $X_{\overline{\mathbf{Q}}}$ is the spectrum of $\overline{\mathbf{Q}} \otimes _{\mathbf{Q}} \overline{\mathbf{Q}}$ which is not Noetherian. (Hint: It has too many idempotents). But if we only base change using finitely generated field extensions then the Noetherian property is preserved. (Or if $X$ is locally of finite type over $k$, since this property is preserved under base change.)

Lemma 33.11.1. Let $k$ be a field. Let $X$ be a scheme over $k$. Let $k'/k$ 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. $\square$