Decomposition of semigroup algebras

Details Decomposition of semigroup algebras Decomposition of semigroup algebras

2011-10-17

arxiv.org/abs/1110.3653v2

Mathematics

Commutative Algebra

12 pages, 2 figures, minor revisions. Package may be downloaded at http://www.math.uni-sb.de/ag/schreyer/jb/Macaulay2/Monomial

Scientific paper

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.

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