Lemma 10.122.4. Let $R \to S$ be a finite type ring map. Then $R \to S$ is quasi-finite if and only if for all primes $\mathfrak p \subset R$ the fibre $S \otimes _ R \kappa (\mathfrak p)$ is finite over $\kappa (\mathfrak p)$.

Proof. If the fibres are finite then the map is clearly quasi-finite. For the converse, note that $S \otimes _ R \kappa (\mathfrak p)$ is a $\kappa (\mathfrak p)$-algebra of finite type and of dimension $0$. Hence it is finite over $\kappa (\mathfrak p)$ for example by Lemma 10.115.4. $\square$

Comment #2822 by Dario Weißmann on

Typos in the proof. It sould read "...of finite type and of dimension $0$. Hence it is finite over $\kappa(\mathfrak{p})$ for example by Lemma 10.114.4."

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