Lemma 101.4.3 (050E)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.4: Separation axioms
Lemma 101.4.3 (cite)

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Lemma 101.4.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

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