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The Stacks project

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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