The Stacks project

Lemma 101.36.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $f$ is étale, and

  2. $f$ is locally of finite presentation, flat, and unramified,

  3. $f$ is locally of finite presentation, flat, and its diagonal is étale.

Proof. The equivalence of (2) and (3) follows immediately from Lemma 101.36.9. Thus in each case the morphism $f$ is DM. Then we can choose Then we can choose algebraic spaces $U$, $V$, a smooth surjective morphism $V \to \mathcal{Y}$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). To finish the proof we have to show that $U \to V$ is étale if and only if it is locally of finite presentation, flat, and unramified. This follows from Morphisms of Spaces, Lemma 67.39.12 (and the more trivial Morphisms of Spaces, Lemmas 67.39.10, 67.39.8, and 67.39.7). $\square$

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