Lemma 10.31.7. Let $R \to S$ be a ring map. Let $R \to R'$ be of finite type. If $S$ is Noetherian, then the base change $S' = R' \otimes _ R S$ is Noetherian.

**Proof.**
By Lemma 10.14.2 finite type is stable under base change. Thus $S \to S'$ is of finite type. Since $S$ is Noetherian we can apply Lemma 10.31.1.
$\square$

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