Lemma 10.163.10 (07NF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.163: Ascending properties
Lemma 10.163.10 (cite)

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Regularity ascends along smooth maps of rings.

Lemma 10.163.10. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi$ is smooth,
$R$ is a regular ring.

Then $S$ is regular.

Proof. This follows from Lemma 10.163.5 applied for all $(R_ k)$ using Lemma 10.140.3 to see that the hypotheses are satisfied. $\square$

Comments (1)

Comment #857 by Bhargav Bhatt on July 26, 2014 at 16:23

Suggested slogan: Regularity ascends along smooth maps of rings.

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