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
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)