Duality on value semigroups

TL;DR

This paper develops a combinatorial duality theory for value semigroups associated with certain curve singularities, extending Cohen-Macaulay duality to a semigroup framework and establishing compatibility between the two dualities.

Contribution

It introduces a new duality for good semigroup ideals that mirrors Cohen-Macaulay duality, providing a combinatorial perspective on singularity theory.

Findings

Establishes a duality on good semigroup ideals

Shows compatibility with Cohen-Macaulay duality

Provides a combinatorial framework for curve singularities

Abstract

We establish a combinatorial counterpart of the Cohen-Macaulay duality on a class of curve singularities which includes algebroid curves. For such singularities the value semigroup and the value semigroup ideals of all fractional ideals satisfy axioms that define so-called good semigroups and good semigroup ideals. We prove that each good semigroup admits a canonical good semigroup ideal which gives rise to a duality on good semigroup ideals. We show that the Cohen-Macaulay duality and our good semigroup duality are compatible under taking values.