Proposition 10.138.13 (00TN)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.138: Formally smooth maps
Proposition 10.138.13 (cite)

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Proposition 10.138.13. Let $R \to S$ be a ring map. The following are equivalent

$R \to S$ is of finite presentation and formally smooth,
$R \to S$ is smooth.

Proof. Follows from Proposition 10.138.8 and Definition 10.137.1. (Note that $\Omega _{S/R}$ is a finitely presented $S$ -module if $R \to S$ is of finite presentation, see Lemma 10.131.15.) $\square$

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