Lemma 29.20.5 (01TG)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.20: Quasi-finite morphisms
Lemma 29.20.5 (cite)

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Lemma 29.20.5. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Set $s = f(x)$ . If $f$ is quasi-finite at $x$ , then the residue field extension $\kappa (x)/\kappa (s)$ is finite.

Proof. This is clear from Algebra, Definition 10.122.3. $\square$

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