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.
$\square$

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