The Stacks project

Lemma 29.8.4. Let $f : X \to S$ be a quasi-compact dominant morphism of schemes. Let $g : S' \to S$ be a morphism of schemes and denote $f' : X' \to S'$ the base change of $f$ by $g$. If generalizations lift along $g$, then $f'$ is dominant.

Proof. Observe that $f'$ is quasi-compact by Schemes, Lemma 26.19.3. Let $\eta ' \in S'$ be the generic point of an irreducible component of $S'$. If generalizations lift along $g$, then $\eta = g(\eta ')$ is the generic point of an irreducible component of $S$. By Lemma 29.8.3 we see that $\eta $ is in the image of $f$. Hence $\eta '$ is in the image of $f'$ by Schemes, Lemma 26.17.5. It follows that $f'$ is dominant by Lemma 29.8.3. $\square$

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Your email address will not be published.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 0H3F. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.8.4 as pdf