Lemma 101.4.15 (06MY)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.4: Separation axioms
Lemma 101.4.15 (cite)

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Lemma 101.4.15. A monomorphism of algebraic stacks is separated and DM. The same is true for immersions of algebraic stacks.

Proof. If $f : \mathcal{X} \to \mathcal{Y}$ is a monomorphism of algebraic stacks, then $\Delta _ f$ is an isomorphism, see Properties of Stacks, Lemma 100.8.4. Since an isomorphism of algebraic spaces is proper and unramified we see that $f$ is separated and DM. The second assertion follows from the first as an immersion is a monomorphism, see Properties of Stacks, Lemma 100.9.5. $\square$

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