On the associated graded ring of a semigroup ring

TL;DR

This paper investigates the properties of the associated graded ring of a numerical semigroup ring, focusing on its local cohomology, Buchsbaum condition, and characterizations of Cohen-Macaulayness and Gorensteinness.

Contribution

It provides new descriptions of local cohomology, characterizes when the graded ring is Buchsbaum, and refines criteria for Cohen-Macaulay and Gorenstein properties.

Findings

Describes H^0_M for G(m) and its length when Buchsbaum.

Characterizes when G(m) is Buchsbaum.

Improves existing characterizations of Cohen-Macaulayness and Gorensteinness.

Abstract

Let (R;m) be a numerical semigroup ring. In this paper we study the properties of its associated graded ring G(m). In particular, we describe the H^0_M for G(m) (where M is the homogeneous maximal ideal of G(m)) and we characterize when G(m) is Buchsbaum. Furthermore, we find the length of H^0_M as a G(m)-module, when G(m) is Buchsbaum. In the 3-generated numerical semigroup case, we describe the H^0_M in term of the Apery set of the numerical semigroup associated to R. Finally, we improve two characterizations of the Cohen-Macaulayness and Gorensteinness of G(m) given in [2] and [3], respectively.