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.
$\square$

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