Rational Polyhedra and Projective Lattice-ordered Abelian Groups with Order Unit

An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-group is finitely presented, the converse does not hold in general. Classical algebraic topology (á la Whitehead) will be combined in this paper with the W lodarczyk-Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital l-groups.