Lemma 101.4.14. Let \mathcal{X} be an algebraic stack. Let W be an algebraic space, and let f : W \to \mathcal{X} be a surjective, flat, locally finitely presented morphism.
If f is unramified (i.e., étale, i.e., \mathcal{X} is Deligne-Mumford), then \mathcal{X} is DM.
If f is locally quasi-finite, then \mathcal{X} is quasi-DM.
Proof.
Note that if f is unramified, then it is étale by Morphisms of Spaces, Lemma 67.39.12. This explains the parenthetical remark in (1). Assume f is unramified (resp. locally quasi-finite). We have to show that \Delta _\mathcal {X} : \mathcal{X} \to \mathcal{X} \times \mathcal{X} is unramified (resp. locally quasi-finite). Note that W \times W \to \mathcal{X} \times \mathcal{X} is also surjective, flat, and locally of finite presentation. Hence it suffices to show that
W \times _{\mathcal{X} \times \mathcal{X}, \Delta _\mathcal {X}} \mathcal{X} = W \times _\mathcal {X} W \longrightarrow W \times W
is unramified (resp. locally quasi-finite), see Properties of Stacks, Lemma 100.3.3. By assumption the morphism \text{pr}_ i : W \times _\mathcal {X} W \to W is unramified (resp. locally quasi-finite). Hence the displayed arrow is unramified (resp. locally quasi-finite) by Morphisms of Spaces, Lemma 67.38.11 (resp. Morphisms of Spaces, Lemma 67.27.8).
\square
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