Lattice-ordered Abelian Groups and Schauder Bases of Unimodular Fans

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 |Σ| of a fan Σ. A unimodular fan ∆ over |Σ| determines a Schauder basis ofG: its elements are the minimal positive free generators of the pointwise ordered group of ∆-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |Σ|: its maximal cones are the domains of linearity of the elements of 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. 1. Background: -groups and fans We assume familiarity with lattice-ordered Abelian groups (for short, -groups [4, 7]) and fans. By a fan we shall always understand a finite rational polyhedral fan, as defined in [6, 15]. Throughout the paper, N = {1, 2, . . .}, Z is the set of integers, and R is the set of reals. If G is an -group, we let G = {g ∈ G | g ≥ 0}. By -homomorphisms we mean homomorphisms of -groups; the symbol ‘∼= ’ denotes -isomorphism. Kernels of -homomorphisms are precisely -ideals, always denoted by Gothic letters a,m, p, . . .. An -ideal is principal iff it is finitely generated (which for -groups is equivalent to being singly generated). Maximal -ideals are defined in the obvious manner. A finitely generated -group G is Archimedean iff it has no “infinitesimal elements”: thus, whenever 0 < x ≤ y holds, there is n ∈ N such that nx ≤ y; equivalently, the intersection of all maximal -ideals in G is {0}; see [7, 4.1] and [4, 10.2] for details. An -ideal m is maximal in G iff G/m is Archimedean totally ordered. As explained in [4, 13.2.6], Archimedean -groups with a strong order unit (that is, an element u ∈ G such that for every x ∈ G there is n ∈ N with nu ≥ x) are precisely those representable as -groups of real-valued continuous functions over some compact Hausdorff space (operations being defined pointwise); finitely generated -groups may always be endowed with a strong order unit. Received by the editors March 30, 2004 and, in revised form, January 19, 2005. 2000 Mathematics Subject Classification. Primary 06F20, 52B20, 08B30; Secondary 06B25, 55N10.