Lemma 10.119.10. Let R \to S be a homomorphism of domains inducing an injection of fraction fields K \subset L. If R is Noetherian local of dimension 1 and [L : K] < \infty then
each prime ideal \mathfrak n_ i of S lying over the maximal ideal \mathfrak m of R is maximal,
there are finitely many of these, and
[\kappa (\mathfrak n_ i) : \kappa (\mathfrak m)] < \infty for each i.
Proof. Pick x \in \mathfrak m nonzero. Apply Lemma 10.119.9 to the submodule S \subset L \cong K^{\oplus n} where n = [L : K]. Thus the ring S/xS has finite length over R. It follows that S/\mathfrak m S has finite length over \kappa (\mathfrak m). In other words, \dim _{\kappa (\mathfrak m)} S/\mathfrak m S is finite (Lemma 10.52.6). Thus S/\mathfrak mS is Artinian (Lemma 10.53.2). The structural results on Artinian rings implies parts (1) and (2), see for example Lemma 10.53.6. Part (3) is implied by the finiteness established above. \square
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