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Decomposition of semigroup algebras
Janko Böhm, David Eisenbud and Max Joachim Nitsche
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].