Decomposition of semigroup algebras
Janko Böhm, David Eisenbud, and Max Joachim Nitsche
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Submission date: 18. Oct. 2011 (revised version: November 2011)
published in: Experimental mathematics, 21 (2012) 4, p. 385-394
DOI number (of the published article): 10.1080/10586458.2012.688376
MSC-Numbers: 13D45, 13P99, 13H10
Keywords and phrases: Semigroup rings, Castelnuovo-Mumford regularity, Eisenbud-Goto conjecture, Computational commutative algebra
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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.