Lemma 101.18.3 (06FZ)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.18: Points of finite type
Lemma 101.18.3 (cite)

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Lemma 101.18.3. Let $\mathcal{X}$ be an algebraic stack. We have

$\mathcal{X}_{\text{ft-pts}} = \bigcup \nolimits _{\varphi : U \to \mathcal{X}\text{ smooth}} |\varphi |(U_0)$

where $U_0$ is the set of closed points of $U$ . Here we may let $U$ range over all schemes smooth over $\mathcal{X}$ or over all affine schemes smooth over $\mathcal{X}$ .

Proof. Immediate from Lemma 101.18.1. $\square$

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