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Characteristic-independence of Betti numbers of graph ideals

Katzman, M. (2006) Characteristic-independence of Betti numbers of graph ideals. Journal of Combinatorial Theory Series A, 113 (3). pp. 435-454. ISSN 0097-3165

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Abstract

In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.

In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field. (c) 2005 Elsevier Inc. All rights reserved.

Item Type: Article
Copyright, Publisher and Additional Information: © 2006 Elsevier. This is an author produced version of a paper subsequently published in Journal of Combinatorial Theory Series A. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: graph ideal; Betti numbers; Stanley-Reisner rings
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 19 Nov 2009 15:30
Last Modified: 08 Feb 2013 16:59
Published Version: http://dx.doi.org/10.1016/j.jcta.2005.04.005
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.jcta.2005.04.005
URI: http://eprints.whiterose.ac.uk/id/eprint/10162

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