Lemma 10.122.4. Let $R \to S$ be a finite type ring map and $\mathfrak p$ be a prime ideal of $R$. Then the following are equivalent:
$R \to S$ is quasi-finite at all primes of $S$ lying over $\mathfrak p$,
$S \otimes _ R \kappa (\mathfrak p)$ is a finite $\kappa (\mathfrak p)$-algebra, and
$\mathop{\mathrm{Spec}}(S \otimes _ R \kappa (\mathfrak p))$ is a finite set.
Proof. Condition (1) says the topology on $\mathop{\mathrm{Spec}}(S \otimes _ R \kappa (\mathfrak p))$ is discrete (as every point is open). Hence the equivalence of (1), (2), (3) follows from Lemma 10.61.3. $\square$
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