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arXiv · 1110.3653

Decomposition of semigroup algebras

Janko Boehm·David Eisenbud·Max Joachim Nitsche
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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.

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Janko Boehm, David Eisenbud, Max Joachim Nitsche. 2011-11-10. Decomposition of semigroup algebras. https://doi.org/10.1080/10586458.2012.688376

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