On the algebraic invariants of certain affine semigroup algebras

TL;DR

This paper investigates the algebraic properties of semigroup algebras generated by specific affine semigroups in two dimensions, establishing their Cohen-Macaulay and Koszul properties, and explicitly computing their resolutions for small k.

Contribution

It provides explicit descriptions of the algebraic invariants and resolutions of a family of affine semigroup algebras, including their Cohen-Macaulay, Gorenstein, and Koszul properties, for the first time.

Findings

Proves $k[S_{a,d,k}]$ is Cohen-Macaulay for all k.

Shows $k[S_{a,d,k}]$ is Gorenstein if and only if k=2.

Explicitly computes syzygies, resolutions, and Hilbert series for k=2,3,4.

Abstract

Let $a$ and $d$ be two linearly independent vectors in $\mathbb{N}^2$, over the field of rational numbers. For a positive integer $k \geq 2$, consider the sequence $a, a+d, \ldots, a+kd$ such that the affine semigroup $S_{a,d,k} = \langle a, a+d, \ldots, a+kd \rangle$ is minimally generated by this sequence. We study the properties of affine semigroup algebra $k[S_{a,d,k}]$ associated to this semigroup. We prove that $k[S_{a,d,k}]$ is always Cohen-Macaulay and it is Gorenstein if and only if $k=2$. For $k=2,3,4$, we explicitly compute the syzygies, minimal graded free resolution and Hilbert series of $k[S_{a,d,k}].$ We also give a minimal generating set and a Gr\"{o}bner basis of the defining ideal of $k[S_{a,d,k}].$ Consequently, we prove that $k[S_{a,d,k}]$ is Koszul. Finally, we prove that the Castelnuovo-Mumford regularity of $k[S_{a,d,k}]$ is $1$ for any $a,d,k.$