The Stacks project

Lemma 10.164.6. Let $R \to S$ be a ring map. Assume that

  1. $R \to S$ is faithfully flat, and

  2. $S$ is Noetherian and has property $(R_ k)$.

Then $R$ is Noetherian and has property $(R_ k)$.

Proof. We have already seen that (1) and (2) imply that $R$ is Noetherian, see Lemma 10.164.1. Let $\mathfrak p \subset R$ be a prime ideal and assume $\dim (R_{\mathfrak p}) \leq k$. Choose a prime $\mathfrak q \subset S$ lying over $\mathfrak p$ which corresponds to a minimal prime of the fibre ring $S \otimes _ R \kappa (\mathfrak p)$. Then $A = R_{\mathfrak p} \to S_{\mathfrak q} = B$ is a flat local ring homomorphism of Noetherian local rings with $\mathfrak m_ AB$ an ideal of definition of $B$. Hence $\dim (A) = \dim (B)$ (Lemma 10.112.7). As $S$ has $(R_ k)$ we conclude that $B$ is a regular local ring. By Lemma 10.110.9 we conclude that $A$ is regular. $\square$

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