Lemma 10.128.5 (046Z)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.128: More flatness criteria
Lemma 10.128.5 (cite)

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Lemma 10.128.5. Suppose that $R \to S$ is a local ring homomorphism of local rings. Denote $\mathfrak m$ the maximal ideal of $R$ . Suppose

$S$ is essentially of finite presentation over $R$ ,
$S$ is flat over $R$ , and
$f \in S$ is a nonzerodivisor in $S/{\mathfrak m}S$ .

Then $S/fS$ is flat over $R$ , and $f$ is a nonzerodivisor in $S$ .

Proof. Follows directly from Lemma 10.128.4. $\square$

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