Lemma 28.19.7. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

1. $f$ is locally of finite type, and

2. $f^{-1}(\{ s\} )$ is a finite set.

Then $X_ s$ is a finite discrete topological space, and $f$ is quasi-finite at each point of $X$ lying over $s$.

Proof. Suppose $T$ is a scheme which (a) is locally of finite type over a field $k$, and (b) has finitely many points. Then Lemma 28.15.10 shows $T$ is a Jacobson scheme. A finite Jacobson space is discrete, see Topology, Lemma 5.18.6. Apply this remark to the fibre $X_ s$ which is locally of finite type over $\mathop{\mathrm{Spec}}(\kappa (s))$ to see the first statement. Finally, apply Lemma 28.19.6 to see the second. $\square$

## 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Your email address will not be published.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 02NG. Beware of the difference between the letter 'O' and the digit '0'.