Lemma 101.4.16 (06MZ)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.4: Separation axioms
Lemma 101.4.16 (cite)

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Lemma 101.4.16. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ . Assume the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists. If $\mathcal{X}$ is DM, resp. quasi-DM, resp. separated, resp. quasi-separated, then so is $\mathcal{Z}_ x$ .

Proof. This is true because $\mathcal{Z}_ x \to \mathcal{X}$ is a monomorphism hence DM and separated by Lemma 101.4.15. Apply Lemma 101.4.11 to conclude. $\square$

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