Lemma 67.25.6 (06EK)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.25: Points of finite type
Lemma 67.25.6 (cite)

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Lemma 67.25.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . For any locally closed subset $T \subset |X|$ we have

$T \not= \emptyset \Rightarrow T \cap X_{\text{ft-pts}} \not= \emptyset .$

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

Proof. Let $i : Z \to X$ be the reduced induce subspace structure on $T$ , see Remark 67.12.5. Any immersion is locally of finite type, see Lemma 67.23.7. Hence by Lemma 67.25.4 we see $Z_{\text{ft-pts}} \subset X_{\text{ft-pts}} \cap T$ . Finally, any nonempty affine scheme $U$ with an étale morphism towards $Z$ has at least one closed point. Hence $Z$ has at least one finite type point by Lemma 67.25.3. The lemma follows. $\square$

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