Projective Closure of Semigroup Algebras

Abstract

This paper investigates the projective closure of simplicial affine semigroups in ${\mathbb{N}}^d$, $d\ge 2$. We present a characterization of the Cohen-Macaulay property for the projective closure of these semigroups using Gröbner bases. Additionally, we establish a criterion, based on Gröbner 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.