The Stacks project

Lemma 65.25.6. 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 65.12.5. Any immersion is locally of finite type, see Lemma 65.23.7. Hence by Lemma 65.25.4 we see $Z_{\text{ft-pts}} \subset X_{\text{ft-pts}} \cap T$. Finally, any nonempty affine scheme $U$ with an ├ętale morphism towards $Z$ has at least one closed point. Hence $Z$ has at least one finite type point by Lemma 65.25.3. The lemma follows. $\square$


Comments (0)


Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06EK. Beware of the difference between the letter 'O' and the digit '0'.