Lemma 10.99.2 (00MF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.99: Criteria for flatness
Lemma 10.99.2 (cite)

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Lemma 10.99.2. Suppose that $R \to S$ is a flat and local ring homomorphism of Noetherian local rings. Denote $\mathfrak m$ the maximal ideal of $R$ . Suppose $f \in S$ 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 10.99.1. $\square$

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