Decomposition of semigroup algebras

TL;DR

This paper introduces a decomposition method for semigroup algebras that enables efficient computation of algebraic properties, including Castelnuovo-Mumford regularity, and confirms the Eisenbud-Goto conjecture in new cases.

Contribution

It provides a novel decomposition approach for semigroup algebras and develops algorithms for computing algebraic invariants, implemented in Macaulay2.

Findings

Decomposition of semigroup rings as modules over subsemigroup rings.

Efficient algorithms for computing Castelnuovo-Mumford regularity.

Confirmation of the Eisenbud-Goto conjecture in new cases.

Abstract

Let A \subseteq B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular we obtain a fast algorithm to compute the Castelnuovo-Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud-Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.