Lemma 10.164.5. Let $R \to S$ be a ring map. Assume that
$R \to S$ is faithfully flat, and
$S$ is Noetherian and has property $(S_ k)$.
Then $R$ is Noetherian and has property $(S_ k)$.
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. 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) and $\text{depth}(A) = \text{depth}(B)$ (Lemma 10.163.2). Hence since $B$ has $(S_ k)$ we see that $A$ has $(S_ k)$. $\square$
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