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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lattice-ordered Abelian groups and Schauder bases of unimodular fans


Authors: Corrado Manara, Vincenzo Marra and Daniele Mundici
Journal: Trans. Amer. Math. Soc. 359 (2007), 1593-1604
MSC (2000): Primary 06F20, 52B20, 08B30; Secondary 06B25, 55N10
Published electronically: October 16, 2006
MathSciNet review: 2272142
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Abstract: Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group $ G$ in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support $ \vert\Sigma\vert$ of a fan $ \Sigma$. A unimodular fan $ \Delta$ over $ \vert\Sigma\vert$ determines a Schauder basis of $ G$: its elements are the minimal positive free generators of the pointwise ordered group of $ \Delta$-linear support functions. Conversely, a Schauder basis $ \mathbf{H}$ of $ G$ determines a unimodular fan over $ \vert\Sigma\vert$: its maximal cones are the domains of linearity of the elements of $ \mathbf{H}$. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, $ G$ is finitely generated projective iff it can be presented by a purely lattice-theoretical word.


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Additional Information

Corrado Manara
Affiliation: Via Pellicioli 10, 24127 Bergamo, Italy
Email: corrado.manara@gmail.com

Vincenzo Marra
Affiliation: Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano, via Comelico 39/41, I-20135 Milano, Italy
Email: marra@dico.unimi.it

Daniele Mundici
Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/A, I-50134 Firenze, Italy
Email: mundici@math.unifi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03935-3
PII: S 0002-9947(06)03935-3
Keywords: Lattice-ordered Abelian group, unimodular fan, projective $\ell$-group, De Concini-Procesi starring, singular homology group
Received by editor(s): March 30, 2004
Received by editor(s) in revised form: January 19, 2005
Published electronically: October 16, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.