Lemma 29.20.7 (02NG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.20: Quasi-finite morphisms
Lemma 29.20.7 (cite)

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Lemma 29.20.7. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$ . Assume that

$f$ is locally of finite type, and
$f^{-1}(\{ s\} )$ is a finite set.

Then $X_ s$ is a finite discrete topological space, and $f$ is quasi-finite at each point of $X$ lying over $s$ .

Proof. Suppose $T$ is a scheme which (a) is locally of finite type over a field $k$ , and (b) has finitely many points. Then Lemma 29.16.10 shows $T$ is a Jacobson scheme. A finite Jacobson space is discrete, see Topology, Lemma 5.18.6. Apply this remark to the fibre $X_ s$ which is locally of finite type over $\mathop{\mathrm{Spec}}(\kappa (s))$ to see the first statement. Finally, apply Lemma 29.20.6 to see the second. $\square$

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