On the Associated Graded ring of Semigroup Algebras

TL;DR

This paper characterizes when the associated graded ring of simplicial affine semigroups is Cohen-Macaulay, using Gröbner bases, and explores Betti numbers and extensions in this context.

Contribution

It provides necessary and sufficient conditions for Cohen-Macaulayness and generalizes homogeneous numerical semigroups to simplicial affine semigroups.

Findings

Criteria for Cohen-Macaulayness using Gröbner bases

Betti numbers of semigroup rings match with associated graded rings

Introduction of nice extensions for simplicial affine semigroups

Abstract

In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gr\"{o}bner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.