Projective Closure of Semigroup Algebras

TL;DR

This paper characterizes the Cohen-Macaulay and Buchsbaum properties of projective closures of simplicial affine semigroups using Gr"obner bases, and introduces the concept of $k$-lifting to relate these semigroups.

Contribution

It provides new criteria based on Gr"obner bases for properties of semigroup algebras and introduces the $k$-lifting concept for simplicial affine semigroups.

Findings

Characterization of Cohen-Macaulay property via Gr"obner bases.

Criterion for Buchsbaum property using Gr"obner bases.

Introduction and analysis of $k$-lifting for simplicial affine semigroups.

Abstract

This paper investigates the projective closure of simplicial affine semigroups in $\mathbb{N}^{d}$, $d \geq 2$. We present a characterization of the Cohen-Macaulay property for the projective closure of these semigroups using Gr\"{o}bner bases. Additionally, we establish a criterion, based on Gr\"{o}bner bases, for determining the Buchsbaum property of non-Cohen-Macaulay projective closures of numerical semigroup rings. Lastly, we introduce the concept of $k$-lifting for simplicial affine semigroups in $\mathbb{N}^d$, and investigate its relationship with the original simplicial affine semigroup.