The Stacks project

Lemma 29.51.4. Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $X^0$, resp. $Y^0$ denote the set of generic points of irreducible components of $X$, resp. $Y$. Let $\eta \in Y^0$. The following are equivalent

  1. $f^{-1}(\{ \eta \} ) \subset X^0$,

  2. $f$ is quasi-finite at all points lying over $\eta $,

  3. $f$ is quasi-finite at all $\xi \in X^0$ lying over $\eta $.

Proof. Condition (1) implies there are no specializations among the points of the fibre $X_\eta $. Hence (2) holds by Lemma 29.20.6. The implication (2) $\Rightarrow $ (3) is immediate. Since $\eta $ is a generic point of $Y$, the generic points of $X_\eta $ are generic points of $X$. Hence (3) and Lemma 29.20.6 imply the generic points of $X_\eta $ are also closed. Thus all points of $X_\eta $ are generic and we see that (1) holds. $\square$

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