Vol. 5, No. 1, 2013

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Decomposition of monomial algebras: Applications and algorithms

Janko Böhm, David Eisenbud and Max J. Nitsche

Vol. 5 (2013), 8–14

Considering finite extensions K[A] K[B] of positive affine semigroup rings over a field K we have developed in [Böhm, Eisenbud, Nitsche 2012] 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 [Böhm, Eisenbud, Nitsche 2012], we develop and prove explicit algorithms testing various properties, including being Buchsbaum, CohenMacaulay, Gorenstein, normal, and seminormal. All algorithms are implemented in our Macaulay2 package.

Mathematical Subject Classification 2010
Primary: 13D45
Secondary: 13P99, 13H10
Supplementary material

MonomialAlgebras source code

Received: 15 June 2012
Revised: 27 February 2013
Accepted: 7 April 2013
Janko Böhm
Fachbereich Mathematik
TU Kaiserslautern
D-67663 Kaiserslautern
David Eisenbud
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720-3840
United States
Max J. Nitsche
Max Planck Institute for Mathematics in the Sciences
Inselstrasse 22
D-04103 Leipzig