The Stacks project

Lemma 101.35.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $f$ is étale,

  2. $f$ is DM and for any morphism $V \to \mathcal{Y}$ where $V$ is an algebraic space and any étale morphism $U \to V \times _\mathcal {Y} \mathcal{X}$ where $U$ is an algebraic space, the morphism $U \to V$ is étale,

  3. there exists some surjective, locally of finite presentation, and flat morphism $W \to \mathcal{Y}$ where $W$ is an algebraic space and some surjective étale morphism $T \to W \times _\mathcal {Y} \mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \to W$ is étale.

Proof. Assume (1). Then $f$ is DM and since being étale is preserved by base change, we see that (2) holds.

Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to \mathcal{Y}$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _\mathcal {Y} \mathcal{X}$ (Lemma 101.21.7). Thus we see that (3) holds.

Assume $W \to \mathcal{Y}$ and $T \to W \times _\mathcal {Y} \mathcal{X}$ are as in (3). We first check $f$ is DM. Namely, it suffices to check $W \times _\mathcal {Y} \mathcal{X} \to W$ is DM, see Lemma 101.4.5. By Lemma 101.4.12 it suffices to check $W \times _\mathcal {Y} \mathcal{X}$ is DM. This follows from the existence of $T \to W \times _\mathcal {Y} \mathcal{X}$ by (the easy direction of) Theorem 101.21.6.

Assume $f$ is DM and $W \to \mathcal{Y}$ and $T \to W \times _\mathcal {Y} \mathcal{X}$ are as in (3). Let $V$ be an algebraic space, let $V \to \mathcal{Y}$ be surjective smooth, let $U$ be an algebraic space, and let $U \to V \times _\mathcal {Y} \mathcal{X}$ is surjective and étale (Lemma 101.21.7). We have to check that $U \to V$ is étale. It suffices to prove $U \times _\mathcal {Y} W \to V \times _\mathcal {Y} W$ is étale by Descent on Spaces, Lemma 74.11.28. We may replace $\mathcal{X}, \mathcal{Y}, W, T, U, V$ by $\mathcal{X} \times _\mathcal {Y} W, W, W, T, U \times _\mathcal {Y} W, V \times _\mathcal {Y} W$ (small detail omitted). Thus we may assume that $Y = \mathcal{Y}$ is an algebraic space, there exists an algebraic space $T$ and a surjective étale morphism $T \to \mathcal{X}$ such that $T \to Y$ is étale, and $U$ and $V$ are as before. In this case we know that

\[ U \to V\text{ is étale} \Leftrightarrow \mathcal{X} \to Y\text{ is étale} \Leftrightarrow T \to Y\text{ is étale} \]

by the equivalence of properties (1) and (2) of Lemma 101.34.1 and Definition 101.35.1. This finishes the proof. $\square$

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