Valuation Semigroups of Two Dimensional Local Rings

TL;DR

This paper characterizes when a semigroup can be realized as the valuation semigroup of a two-dimensional local ring, providing new examples and a generalization of existing theorems to include residue field extensions.

Contribution

It offers a necessary and sufficient condition for semigroups and field extensions to correspond to valuations dominating two-dimensional regular local rings, extending Spivakovsky's theorem.

Findings

Provides a complete characterization of valuation semigroups in this setting.

Constructs surprising examples of such semigroups.

Generalizes existing theorems to include residue field extensions.

Abstract

We consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian local domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valaution dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension.