Semigroups of valuations dominating local domains

TL;DR

This paper introduces a new criterion for semigroups to be valuation semigroups dominating local domains, demonstrating that existing necessary conditions are not sufficient through specific counterexamples.

Contribution

It provides a novel criterion for valuation semigroups and constructs counterexamples showing the insufficiency of previous necessary conditions.

Findings

New criterion for semigroups of valuations

Counterexamples of semigroups not realizable as valuation semigroups

Shows limitations of classical necessary conditions

Abstract

A new criterion is given for a semigroup to be the semigroup of a valuation dominating an equicharacteristic local domain. The criterion is used to construct examples of well ordered subsemigroups of the positive rational numbers which are of ordinal type omega, but are not the value semigroup of a valuation on an equicharacteristic noetherian local domain. This shows that the necessary conditions on value semigroups given in Appendix 3 to Zariski and Samuel's book ``Commutative Algebra'' are not sufficient.