Lemma 15.42.1 (07QK)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.42: Ascending properties along regular ring maps
Lemma 15.42.1 (cite)

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Lemma 15.42.1. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi$ is regular,
$S$ is Noetherian, and
$R$ is Noetherian and reduced.

Then $S$ is reduced.

Proof. For Noetherian rings being reduced is the same as having properties $(S_1)$ and $(R_0)$ , see Algebra, Lemma 10.157.3. Hence we may apply Algebra, Lemmas 10.163.4 and 10.163.5. $\square$

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