Lemma 10.36.21. 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 10.18.5 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 10.31.1. By Lemma 10.36.20 every prime of S \otimes _ R \kappa (\mathfrak p) is a minimal prime. Hence by Lemma 10.31.6 there are at most finitely many. \square
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