Lemma 101.18.6 (06G2)—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 101.18.6 (cite)

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Lemma 101.18.6. Let $\mathcal{X}$ be an algebraic stack. For any locally closed subset $T \subset |\mathcal{X}|$ we have

\[ T \not= \emptyset \Rightarrow T \cap \mathcal{X}_{\text{ft-pts}} \not= \emptyset . \]

In particular, for any closed subset $T \subset |\mathcal{X}|$ we see that $T \cap \mathcal{X}_{\text{ft-pts}}$ is dense in $T$.

Proof. Let $i : \mathcal{Z} \to \mathcal{X}$ be the reduced induced substack structure on $T$, see Properties of Stacks, Remark 100.10.5. An immersion is locally of finite type, see Lemma 101.17.4. Hence by Lemma 101.18.4 we see $\mathcal{Z}_{\text{ft-pts}} \subset \mathcal{X}_{\text{ft-pts}} \cap T$. Finally, any nonempty affine scheme $U$ with a smooth morphism towards $\mathcal{Z}$ has at least one closed point, hence $\mathcal{Z}$ has at least one finite type point by Lemma 101.18.3. The lemma follows. $\square$

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