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. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.