The Stacks project

Lemma 15.104.13. Let $A \to B$ be a ring map such that $B \otimes _ A B \to B$ is flat.

  1. If $A \to B$ is of finite type, then $A \to B$ is unramified.

  2. If $A \to B$ is of finite presentation and flat, then $A \to B$ is étale.

In particular a weakly étale ring map of finite presentation is étale.

Proof. Part (1) follows from Lemma 15.104.12 and Algebra, Definition 10.151.1. Part (2) follows from part (1) and Algebra, Lemma 10.151.8. $\square$

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