Lemma 101.50.3 (0GWD)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.50: Integral algebraic stacks
Lemma 101.50.3 (cite)

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Lemma 101.50.3. Let $\mathcal{X}$ be an algebraic stack which is reduced and quasi-separated and whose associated topological space $|\mathcal{X}|$ is irreducible. Then $\mathcal{X}$ is integral.

Proof. If $\mathcal{X}$ is quasi-separated, then $\mathcal{X}$ is decent by Lemma 101.48.2. If $\mathcal{X}$ is quasi-separated, then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-compact, hence $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is quasi-compact as the base change of $\Delta$ by $\Delta$ , see Lemma 101.7.3. Thus we see that all the hypotheses of Definition 101.50.1 hold (and we also see that we may replace “quasi-separated” by “ $\Delta _\mathcal {X}$ is quasi-compact”). $\square$

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