When the associated graded ring of a semigroup ring is Complete Intersection

TL;DR

This paper investigates when the associated graded ring of a numerical semigroup ring is a complete intersection, introducing new concepts of beta-rectangular and gamma-rectangular Apéry sets to characterize this property.

Contribution

It introduces and characterizes beta-rectangular and gamma-rectangular Apéry sets, providing new criteria for the associated graded ring to be a complete intersection.

Findings

Beta-rectangular and gamma-rectangular Apéry sets are fundamental concepts.

Provides four equivalent conditions for the associated graded ring to be a complete intersection.

Offers a new characterization linking Apéry sets to the complete intersection property.

Abstract

Let (R, m) be the semigroup ring associated to a numerical semigroup S. In this paper we study the property of its associated graded ring G(m) to be Complete Intersection. In particular, we introduce and characterise beta-rectangular and gamma-rectangular Ap\'ery sets, which will be the fundamental concepts of the paper and will provide, respectively, a sufficient condition and a characterisation for G(m) to be Complete Intersection. Then we use these notions to give four equivalent conditions for G(m) in order to be Complete Intersection.