Abstract
Let $A \supseteq B$ 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 MONOMIAL ALGEBRAS.
Citation
Janko Böhm. David Eisenbud. Max J. Nitsche. "Decomposition of Semigroup Algebras." Experiment. Math. 21 (4) 385 - 394, 2012.
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