The Stacks project

Lemma 29.51.5. Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $X^0$, resp. $Y^0$ denote the set of generic points of irreducible components of $X$, resp. $Y$. Assume

  1. $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,

  2. either $f$ is quasi-compact or $f$ is separated.

Then there exists a dense open $V \subset Y$ such that $f^{-1}(V) \to V$ is finite.

Proof. Since $Y$ has finitely many irreducible components, we can find a dense open which is a disjoint union of its irreducible components. Thus we may assume $Y$ is irreducible affine with generic point $\eta $. Then the fibre over $\eta $ is finite as $X^0$ is finite.

Assume $f$ is separated and $Y$ irreducible affine. Choose $V \subset Y$ and $U \subset X$ as in Lemma 29.51.1 part (3). Since $f|_ U : U \to V$ is finite, we see that $U \subset f^{-1}(V)$ is closed as well as open (Lemmas 29.41.7 and 29.44.11). Thus $f^{-1}(V) = U \amalg W$ for some open subscheme $W$ of $X$. However, since $U$ contains all the generic points of $X$ we conclude that $W = \emptyset $ as desired.

Assume $f$ is quasi-compact and $Y$ irreducible affine. Then $X$ is quasi-compact, hence there exists a dense open subscheme $U \subset X$ which is separated (Properties, Lemma 28.29.3). Since the set of generic points $X^0$ is finite, we see that $X^0 \subset U$. Thus $\eta \not\in f(X \setminus U)$. Since $X \setminus U \to Y$ is quasi-compact, we conclude that there is a nonempty open $V \subset Y$ such that $f^{-1}(V) \subset U$, see Lemma 29.8.3. After replacing $X$ by $f^{-1}(V)$ and $Y$ by $V$ we reduce to the separated case which we dealt with in the preceding paragraph. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 29.51: Generically finite morphisms

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Your email address will not be published.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