The Stacks project

Lemma 29.8.4. Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\eta \in S$ be a generic point of an irreducible component of $S$. If $\eta \not\in f(X)$ then there exists an open neighbourhood $V \subset S$ of $\eta $ such that $f^{-1}(V) = \emptyset $.

Proof. Let $Z \subset S$ be the scheme theoretic image of $f$. We have to show that $\eta \not\in Z$. This follows from Lemma 29.6.5 but can also be seen as follows. By Lemma 29.6.3 the morphism $X \to Z$ is dominant, which by Lemma 29.8.3 means all the generic points of all irreducible components of $Z$ are in the image of $X \to Z$. By assumption we see that $\eta \not\in Z$ since $\eta $ would be the generic point of some irreducible component of $Z$ if it were in $Z$. $\square$


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