Formal prime ideals of infinite value and their algebraic resolution

TL;DR

This paper extends the resolution of prime ideals of infinite value from rank 1 valuations to valuations of arbitrary rank in local domains over fields of characteristic zero, addressing a complex algebraic problem.

Contribution

It generalizes the resolution of prime ideals of infinite value to valuations of any rank, building on prior work limited to rank 1 cases.

Findings

Resolution of prime ideals of infinite value for arbitrary rank valuations.

Identification of conditions under which prime ideals of infinite value can be resolved.

Extension of existing methods to higher-rank valuation scenarios.

Abstract

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 $\hat R$, 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 $\hat R$ that leads to this obstruction. We extend their result on the resolution of prime ideals of infinite value to valuations of arbitrary rank.