The Stacks project

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

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

  2. $R \to S$ 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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