Decomposition of Monomial Algebras: Applications and Algorithms
Janko Böhm, David Eisenbud, and Max Joachim Nitsche
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Submission date: 11. Jun. 2012
published in: Journal of software for algebra and geometry : Macaulay2, 5 (2013), p. 8-14
DOI number (of the published article): 10.2140/jsag.2013.5.8
MSC-Numbers: 13D45, 13P99, 13H10
Keywords and phrases: Affine semigroup rings, Buchsbaum, Cohen-Macaulay, Gorenstein, normal, seminormal, Castelnuovo-Mumford regularity, Computational commutative algebra
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Considering finite extensions K[A] ⊆ K[B] of positive affine semigroup rings over a field K we have developed in  an algorithm to decompose K[B] as a direct sum of monomial ideals in K[A]. By computing the regularity of homogeneous semigroup rings from the decomposition we have confirmed the Eisenbud-Goto conjecture in a range of new cases not tractable by standard methods. Here we first illustrate this technique and its implementation in our Macaulay2 package MonomialAlgebrasby computing the decomposition and the regularity step by step for an explicit example. We then focus on ring-theoretic properties of simplicial semigroup rings. From the characterizations given in  we develop and prove explicit algorithms testing properties like Buchsbaum, Cohen-Macaulay, Gorenstein, normal, and seminormal, all of which imply the Eisenbud-Goto conjecture. All algorithms are implemented in our Macaulay2 package.