Proposition 10.137.13. 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.

Proposition 10.137.13. 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 10.137.8 and Definition 10.136.1. (Note that $\Omega _{S/R}$ is a finitely presented $S$-module if $R \to S$ is of finite presentation, see Lemma 10.130.15.)
$\square$

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