The Stacks project

Lemma 35.23.9. Let $E/k$ be a field extension. Then a morphism $X \to Y$ over $k$ is dominant if and only if the pullback $X_ E \to Y_ E$ is dominant.

Proof. By Morphisms, Lemma 29.23.4, the morphism $\mathop{\mathrm{Spec}}(E) \to \mathop{\mathrm{Spec}}(k)$ is universally open. So $Y_ E \to Y$ is open. Therefore, if $X \to Y$ is dominant, Morphisms, Lemma 29.8.5 gives that $X_ E \to Y_ E$ is dominant. Conversely, suppose that $X_ E \to Y_ E$ is dominant. The morphism $Y_ E \to Y$ is surjective by Morphisms, Lemma 29.9.4, and so the composition $X_ E \to Y_ E \to Y$ is dominant. This is also the composition $X_ E \to X \to Y$, and so $X \to Y$ is dominant. $\square$

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