The Stacks project

Lemma 33.17.2. Let $f : X \to Y$ be a proper morphism. Let $y \in Y$ be a point such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Assume in addition one of the following conditions is satisfied

  1. for every generic point $\eta $ of an irreducible component of $X$ the field extension $\kappa (\eta )/\kappa (f(\eta ))$ is finite (or algebraic),

  2. for every generic point $\eta $ of an irreducible component of $X$ such that $f(\eta ) \leadsto y$ the field extension $\kappa (\eta )/\kappa (f(\eta ))$ is finite (or algebraic),

  3. $f$ is quasi-finite at every generic point of $X$,

  4. $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$,

  5. add more here.

Then there exists an open neighbourhood $V \subset Y$ of $y$ such that $f^{-1}(V) \to V$ is finite.

Proof. By Lemma 33.17.1 the morphism $f$ is quasi-finite at every point of the fibre $X_ y$. Hence $X_ y$ is a discrete topological space (Morphisms, Lemma 29.20.6). As $f$ is proper the fibre $X_ y$ is quasi-compact, i.e., finite. Thus we can apply Cohomology of Schemes, Lemma 30.21.2 to conclude. $\square$

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Your email address will not be published.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 0AB7. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.17.2 as pdf