The Stacks project

Lemma 10.119.13. Let $R$ be a Noetherian local domain with fraction field $K$. Assume that $R$ is not a field. Let $L/K$ be a finitely generated field extension. Then there exists discrete valuation ring $A$ with fraction field $L$ which dominates $R$.

Proof. If $L$ is not finite over $K$ choose a transcendence basis $x_1, \ldots , x_ r$ of $L$ over $K$ and replace $R$ by $R[x_1, \ldots , x_ r]$ localized at the maximal ideal generated by $\mathfrak m_ R$ and $x_1, \ldots , x_ r$. Thus we may assume $K \subset L$ finite.

By Lemma 10.119.1 we may assume $\dim (R) = 1$.

Let $A \subset L$ be the integral closure of $R$ in $L$. By Lemma 10.119.12 this is Noetherian. By Lemma 10.36.17 there is a prime ideal $\mathfrak q \subset A$ lying over the maximal ideal of $R$. By Lemma 10.119.7 the ring $A_{\mathfrak q}$ is a discrete valuation ring dominating $R$ as desired. $\square$

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