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Lattice-ordered Abelian groups and Schauder bases of unimodular fans, II

Author: Vincenzo Marra
Journal: Trans. Amer. Math. Soc. 365 (2013), 2545-2568
MSC (2010): Primary 06F20, 52B20, 08B30; Secondary 06B25, 55N10
Published electronically: January 17, 2013
MathSciNet review: 3020108
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Abstract: Unimodular fans are central to toric algebraic geometry, where they correspond to non-singular toric varieties. The Schauder bases mentioned in the title may be described as the standard bases of the free $ \mathbb{Z}$-module of support functions (=invariant Cartier divisors) of a unimodular (a.k.a. regular) fan. An abstract, purely algebraic version of Schauder bases was investigated in the first part of the present paper, with motivations coming from the theory of lattice-ordered Abelian groups. The main result obtained there is that such abstract Schauder bases can be characterised in terms of the maximal spectral space of lattice-ordered Abelian groups, with no reference to polyhedral geometry. The results in the present paper will show that abstract Schauder bases can in fact be characterised in the language of lattice-ordered groups by means of an elementary algebraic notion that we call regularity, with no reference to either maximal spectral spaces or to polyhedral geometry. We prove that finitely generated projective lattice-ordered Abelian groups are precisely the lattice-ordered Abelian groups that have a finite, regular set of positive generators. This theorem complements Beynon's well-known 1977 result that the finitely generated projective lattice-ordered Abelian groups are precisely the finitely presented ones; and the core of the proof consists in showing that finite, regular sets of positive generators are the same thing as abstract Schauder bases. We give three applications of the main result. First, we establish a necessary and sufficient criterion for the lattice-group isomorphism of two lattice-ordered Abelian groups with finite, regular sets of positive generators. Next, we classify in elementary terms (i.e. without reference to spectral spaces) all finitely generated projective lattice-ordered Abelian groups whose maximal spectrum is a closed topological surface. Finally, we show how to explicitly construct $ \mathbb{Z}$-module bases of any finitely generated projective lattice-ordered Abelian group.

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Additional Information

Vincenzo Marra
Affiliation: Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano, via Comelico 39/41, I-20135 Milano, Italy
Address at time of publication: Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, via Cesare Saldini 50, I-20133 Milano, Italy

Keywords: Lattice-ordered Abelian group, free $ℓ$-group, projective $ℓ$-group, unimodular fan, regular fan, maximal spectral space, pairwise disjointness, linear independence, Euler characteristic, $\mathbb{Z}$-module basis.
Received by editor(s): July 13, 2011
Received by editor(s) in revised form: September 8, 2011
Published electronically: January 17, 2013
Dedicated: Dedicated to A.M.W. Glass on the occasion of his retirement.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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