The Stacks project

Lemma 29.8.5. Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\eta \in S$ be a generic point of an irreducible component of $S$. If $\eta \not\in f(X)$ then there exists an open neighbourhood $V \subset S$ of $\eta $ such that $f^{-1}(V) = \emptyset $.

Proof. Let $Z \subset S$ be the scheme theoretic image of $f$. We have to show that $\eta \not\in Z$. This follows from Lemma 29.6.5 but can also be seen as follows. By Lemma 29.6.3 the morphism $X \to Z$ is dominant, which by Lemma 29.8.3 means all the generic points of all irreducible components of $Z$ are in the image of $X \to Z$. By assumption we see that $\eta \not\in Z$ since $\eta $ would be the generic point of some irreducible component of $Z$ if it were in $Z$. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 29.8: Dominant 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Your email address will not be published.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 02NE. Beware of the difference between the letter 'O' and the digit '0'.