Decomposition of Monomial Algebras: Applications and Algorithms

TL;DR

This paper introduces algorithms for decomposing monomial algebras and analyzing their properties, enabling verification of the Eisenbud-Goto conjecture in new cases and providing practical tools via Macaulay2.

Contribution

It develops explicit algorithms for decomposing monomial algebras and testing key ring-theoretic properties, with implementations in Macaulay2.

Findings

Confirmed the Eisenbud-Goto conjecture in new cases

Provided algorithms for properties like Cohen-Macaulay and Gorenstein

Implemented tools in Macaulay2 package

Abstract

Considering finite extensions K[A] \subseteq K[B] of positive affine semigroup rings over a field K we have developed in [1] an algorithm to decompose K[B] as a direct sum of monomial ideals in K[A]. By computing the regularity of homogeneous semigroup rings from the decomposition we have confirmed the Eisenbud-Goto conjecture in a range of new cases not tractable by standard methods. Here we first illustrate this technique and its implementation in our Macaulay2 package MonomialAlgebras by computing the decomposition and the regularity step by step for an explicit example. We then focus on ring-theoretic properties of simplicial semigroup rings. From the characterizations given in [1] we develop and prove explicit algorithms testing properties like Buchsbaum, Cohen-Macaulay, Gorenstein, normal, and seminormal, all of which imply the Eisenbud-Goto conjecture. All algorithms are implemented in our Macaulay2 package.