Lemma 10.164.2 (033F)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.164: Descending properties
Lemma 10.164.2 (cite)

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Lemma 10.164.2. Let $R \to S$ be a ring map. Assume that

$R \to S$ is faithfully flat, and
$S$ is reduced.

Then $R$ is reduced.

Proof. This is clear as $R \to S$ is injective. $\square$

Comments (1)

Comment #873 by Dylan on August 02, 2014 at 08:28

Suggested slogan: If you're covered (faithfully and flatly) by a reduced guy, then you're a reduced guy.

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