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

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 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 Z-module bases of any finitely generated projective lattice-ordered Abelian group