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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
32/2012

Decomposition of Monomial Algebras: Applications and Algorithms

Janko Böhm, David Eisenbud and Max Joachim Nitsche

Abstract

Considering finite extensions $K[A]\subseteq K[B]$ of positive affine semigroup rings over a field $K$ we have developed in [1] 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 [MonomialAlgebras] by 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 [1] 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.

Received:
11.06.12
Published:
11.06.12
MSC Codes:
13D45, 13P99, 13H10
Keywords:
Affine semigroup rings, Buchsbaum, Cohen-Macaulay, Gorenstein, normal, seminormal, Castelnuovo-Mumford regularity, Computational commutative algebra

Related publications

inJournal
2013 Journal Open Access
Janko Böhm, David Eisenbud and Max Joachim Bernd Nitsche

Decomposition of monomial algebras : applications and algorithms

In: Journal of software for algebra and geometry, 5 (2013), pp. 8-14