The Stacks project

Lemma 10.150.2. Let $R \to S$ be a ring map of finite presentation. The following are equivalent:

  1. $R \to S$ is formally étale,

  2. $R \to S$ is étale.

Proof. Assume that $R \to S$ is formally étale. Then $R \to S$ is smooth by Proposition 10.138.13. By Lemma 10.148.2 we have $\Omega _{S/R} = 0$. Hence $R \to S$ is étale by definition.

Assume that $R \to S$ is étale. Then $R \to S$ is formally smooth by Proposition 10.138.13. By Lemma 10.148.2 it is formally unramified. Hence $R \to S$ is formally étale. $\square$

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