The Stacks project

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

  1. each prime ideal $\mathfrak n_ i$ of $S$ lying over the maximal ideal $\mathfrak m$ of $R$ is maximal,

  2. there are finitely many of these, and

  3. $[\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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