Lemma 10.35.21. Suppose $R \to S$ is finite. Then the fibres of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ are finite.

Proof. By the discussion in Remark 10.16.8 the fibres are the spectra of the rings $S \otimes _ R \kappa (\mathfrak p)$. As $R \to S$ is finite, these fibre rings are finite over $\kappa (\mathfrak p)$ hence Noetherian by Lemma 10.30.1. By Lemma 10.35.20 every prime of $S \otimes _ R \kappa (\mathfrak p)$ is a minimal prime. Hence by Lemma 10.30.6 there are at most finitely many. $\square$

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