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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
67/2011

Decomposition of semigroup algebras

Janko Böhm, David Eisenbud and Max Joachim Nitsche

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].

Received:
Oct 18, 2011
Published:
Oct 18, 2011
MSC Codes:
13D45, 13P99, 13H10
Keywords:
Semigroup rings, Castelnuovo-Mumford regularity, Eisenbud-Goto conjecture, Computational commutative algebra

Related publications

inJournal
2012 Repository Open Access
Janko Böhm, David Eisenbud and Max Joachim Bernd Nitsche

Decomposition of semigroup algebras

In: Experimental mathematics, 21 (2012) 4, pp. 385-394