Lemma 10.164.4 (07NG)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.164: Descending properties
Lemma 10.164.4 (cite)

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Lemma 10.164.4. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and
$S$ is a regular ring.

Then $R$ is a regular ring.

Proof. We see that $R$ is Noetherian by Lemma 10.164.1. Let $\mathfrak p \subset R$ be a prime. Choose a prime $\mathfrak q \subset S$ lying over $\mathfrak p$ . Then Lemma 10.110.9 applies to $R_\mathfrak p \to S_\mathfrak q$ and we conclude that $R_\mathfrak p$ is regular. Since $\mathfrak p$ was arbitrary we see $R$ is regular. $\square$

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