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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