The Stacks project

Lemma 100.4.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

  1. If $f$ is separated, then $f$ is quasi-separated.

  2. If $f$ is DM, then $f$ is quasi-DM.

  3. If $f$ is representable by algebraic spaces, then $f$ is DM.

Proof. To see (1) note that a proper morphism of algebraic spaces is quasi-compact and quasi-separated, see Morphisms of Spaces, Definition 66.40.1. To see (2) note that an unramified morphism of algebraic spaces is locally quasi-finite, see Morphisms of Spaces, Lemma 66.38.7. Finally (3) follows from Lemma 100.3.4. $\square$

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