The Cohen-Macaulay Property of Affine Semigroup Rings in Dimension 2

TL;DR

This paper investigates the Cohen-Macaulay property of certain affine semigroup rings in dimension 2, providing explicit criteria, algorithms, and applications to projective monomial curves.

Contribution

It introduces a numerical criterion for Cohen-Macaulayness when t=2 and offers an algorithm to identify the monomial basis of specific quotient rings.

Findings

Calculated the Hilbert polynomial of the ideal (x^a,y^b)

Provided a numerical criterion for Cohen-Macaulayness when t=2

Developed an algorithm for the monomial basis of R/(x^a,y^b)

Abstract

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion for the ring to be Cohen-Macaulay. Next, we prove a simple numerical criterion for $R$ to be Cohen-Macaulay in the case when $t=2$. We also provide a simple algorithm which identifies the monomial $k$-basis of $R/(x^a,y^b)$. Finally, these simple results are specialized to the case of projective monomial curves in $\mathbb{P}^3$.