Decomposition of Semigroup Algebras

Abstract

Let A⊆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.