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The Stacks project

Lemma 101.4.3. 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 67.40.1. To see (2) note that an unramified morphism of algebraic spaces is locally quasi-finite, see Morphisms of Spaces, Lemma 67.38.7. Finally (3) follows from Lemma 101.3.4. \square


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