Lemma 29.51.4 (0BAH)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.51: Generically finite morphisms
Lemma 29.51.4 (cite)

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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

$f^{-1}(\{ \eta \} ) \subset X^0$ ,
$f$ is quasi-finite at all points lying over $\eta$ ,
$f$ 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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