The Stacks project

A proper morphism is finite in a neighbourhood of a finite fiber.

Lemma 29.21.2. (For a more general version see More on Morphisms, Lemma 36.39.2.) Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume

  1. $S$ is locally Noetherian,

  2. $f$ is proper, and

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

Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{ s\} )$ by Morphisms, Lemma 28.19.7. By Morphisms, Lemma 28.53.2 the set of points at which $f$ is quasi-finite is an open $U \subset X$. Let $Z = X \setminus U$. Then $s \not\in f(Z)$. Since $f$ is proper the set $f(Z) \subset S$ is closed. Choose any open neighbourhood $V \subset S$ of $s$ with $Z \cap V = \emptyset $. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 28.19.9), hence has finite fibres (Morphisms, Lemma 28.19.10), hence is finite by Lemma 29.21.1. $\square$


Comments (1)

Comment #854 by Olivier BENOIST on

Suggested slogan: A proper morphism is finite in a neighbourhood of a finite fiber.


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Your email address will not be published.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