The Stacks project

Lemma 99.4.14. 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.

  1. If $f$ is unramified (i.e., ├ętale, i.e., $\mathcal{X}$ is Deligne-Mumford), then $\mathcal{X}$ is DM.

  2. If $f$ is locally quasi-finite, then $\mathcal{X}$ is quasi-DM.

Proof. Note that if $f$ is unramified, then it is ├ętale by Morphisms of Spaces, Lemma 65.39.12. 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 98.3.3. 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 65.38.11 (resp. Morphisms of Spaces, Lemma 65.27.8). $\square$


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