The Stacks project

Lemma 37.64.9. Let $f : X \to Y$ be a morphism of schemes. Then $X \to Y$ is weakly étale in each of the following cases

  1. $X \to Y$ is a flat monomorphism,

  2. $X \to Y$ is an open immersion,

  3. $X \to Y$ is flat and unramified,

  4. $X \to Y$ is étale.

Proof. If (1) holds, then $\Delta _{X/Y}$ is an isomorphism (Schemes, Lemma 26.23.2), hence certainly $f$ is weakly étale. Case (2) is a special case of (1). The diagonal of an unramified morphism is an open immersion (Morphisms, Lemma 29.35.13), hence flat. Thus a flat unramified morphism is weakly étale. An étale morphism is flat and unramified (Morphisms, Lemma 29.36.5), hence (4) follows from (3). $\square$

Comments (4)

Comment #3425 by Shane on

What does monomorphism mean in (1)? Certainly not injective on the underlying topological spaces.

Comment #3426 by Shane on

What does monomorphism mean in (1)? Certainly not injective on the underlying topological spaces.

Comment #3441 by Samir Canning on

@Shane. In a category with fibered products, a monomorphism is equivalent to the diagonal being an isomorphism. For example, you can see Tag 26.23.2. Maybe it would be useful to reference this tag in the proof?

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