Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\widehat{R}$, the completion of $R$. When the rank of $\nu$ is $1$, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$, there is no natural ideal in $\widehat{R}$ that...