The Stacks project

Lemma 10.163.5. Let $\varphi : R \to S$ be a ring map. Assume

  1. $R$ is Noetherian,

  2. $S$ is Noetherian

  3. $\varphi $ is flat,

  4. the fibre rings $S \otimes _ R \kappa (\mathfrak p)$ have property $(R_ k)$, and

  5. $R$ has property $(R_ k)$.

Then $S$ has property $(R_ k)$.

Proof. Let $\mathfrak q$ be a prime of $S$ lying over a prime $\mathfrak p$ of $R$. Assume that $\dim (S_{\mathfrak q}) \leq k$. Since $\dim (S_{\mathfrak q}) = \dim (R_{\mathfrak p}) + \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q})$ by Lemma 10.112.7 we see that $\dim (R_{\mathfrak p}) \leq k$ and $\dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) \leq k$. Hence $R_{\mathfrak p}$ and $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ are regular by assumption. It follows that $S_{\mathfrak q}$ is regular by Lemma 10.112.8. $\square$

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