Lemma 10.50.2. Let $K$ be a field. Let $A \subset K$ be a local subring. Then there exists a valuation ring with fraction field $K$ dominating $A$.

Proof. We consider the collection of local subrings of $K$ as a partially ordered set using the relation of domination. Suppose that $\{ A_ i\} _{i \in I}$ is a totally ordered collection of local subrings of $K$. Then $B = \bigcup A_ i$ is a local subring which dominates all of the $A_ i$. Hence by Zorn's Lemma, it suffices to show that if $A \subset K$ is a local ring whose fraction field is not $K$, then there exists a local ring $B \subset K$, $B \not= A$ dominating $A$.

Pick $t \in K$ which is not in the fraction field of $A$. If $t$ is transcendental over $A$, then $A[t] \subset K$ and hence $A[t]_{(t, \mathfrak m)} \subset K$ is a local ring distinct from $A$ dominating $A$. Suppose $t$ is algebraic over $A$. Then for some $a \in A$ the element $at$ is integral over $A$. In this case the subring $A' \subset K$ generated by $A$ and $ta$ is finite over $A$. By Lemma 10.36.17 there exists a prime ideal $\mathfrak m' \subset A'$ lying over $\mathfrak m$. Then $A'_{\mathfrak m'}$ dominates $A$. If $A = A'_{\mathfrak m'}$, then $t$ is in the fraction field of $A$ which we assumed not to be the case. Thus $A \not= A'_{\mathfrak m'}$ as desired. $\square$

Comment #1238 by Jonathan Wise on

This proof does not appear to be complete. In the last paragraph, the element $at$ may be in $A$, so $A' = A$ and does not contradict the maximality of $A$.

Comment #1251 by on

If $A = A'_{\mathfrak m'}$ then $t$ is in the fraction field of $A$ which we assumed not to be the case. I have added this explanation to the proof, see here.

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