Morphisms whose image contains the generic points are dominant

Lemma 29.8.3. Let $f : X \to S$ be a quasi-compact morphism of schemes. Then $f$ is dominant if and only if for every irreducible component $Z \subset S$ the generic point of $Z$ is in the image of $f$.

Proof. Let $V \subset S$ be an affine open. Because $f$ is quasi-compact we may choose finitely many affine opens $U_ i \subset f^{-1}(V)$, $i = 1, \ldots , n$ covering $f^{-1}(V)$. Consider the morphism of affines

$f' : \coprod \nolimits _{i = 1, \ldots , n} U_ i \longrightarrow V.$

A disjoint union of affines is affine, see Schemes, Lemma 26.6.8. Generic points of irreducible components of $V$ are exactly the generic points of the irreducible components of $S$ that meet $V$. Also, $f$ is dominant if and only if $f'$ is dominant no matter what choices of $V, n, U_ i$ we make above. Thus we have reduced the lemma to the case of a morphism of affine schemes. The affine case is Algebra, Lemma 10.30.6. $\square$

Comment #1655 by Matthieu Romagny on

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