The Stacks project

Lemma 29.29.5. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Let $x \in X$ with $s = f(x)$. Then $f$ is quasi-finite at $x$ if and only if $\dim _ x(X_ s) = 0$. In particular, $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$.

Proof. First proof. If $f$ is quasi-finite at $x$ then $\kappa (x)$ is a finite extension of $\kappa (s)$ (by Lemma 29.20.5) and $x$ is isolated in $X_ s$ (by Lemma 29.20.6), hence $\dim _ x(X_ s) = 0$ by Lemma 29.28.1. Conversely, if $\dim _ x(X_ s) = 0$ then by Lemma 29.28.1 we see $\kappa (s) \subset \kappa (x)$ is algebraic and there are no other points of $X_ s$ specializing to $x$. Hence $x$ is closed in its fibre by Lemma 29.20.2 and by Lemma 29.20.6 (3) we conclude that $f$ is quasi-finite at $x$.

Second proof. The fibre $X_ s$ is a scheme locally of finite type over a field, hence locally Noetherian (Lemma 29.15.6). The result now follows from Lemma 29.20.6 and Properties, Lemma 28.10.7. $\square$


Comments (2)

Comment #8421 by Ryo Suzuki on

The proof can be simplified.

By virtue of Lemma 01TH, it is sufficient to prove that: Let be a locally Noetherian scheme, and . Then is isolated in if and only if .

Proof. If is an isolated point, is open (by Definition 06RM). Hence {\rm dim}_x Y = 0 (by Definition 0055). Conversely, if , then there exists an open subset such that . We can take Noetherian because is locally Noetherian. Since is 0-dimensional Noetherian scheme, it is discrete. Hence is isolated point of . q.e.d.

Comment #9045 by on

Thanks very much. I decided to leave in the proof we have now and add yours as a second clearly shorter one. Changes are here.


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Your email address will not be published.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 0397. Beware of the difference between the letter 'O' and the digit '0'.