Sheaves of Abelian l-groups

Abstract

Keimel was first to associate a spectrum with an abelian l-group G. An alternative construction of a spectrum is proposed here. Its points are the prime l-ideals of G, i.e., they are the points of Keimel’s spectrum plus the improper prime l-ideal G. The spectrum is equipped with the inverse topology, which is different from the one used by Keimel. The new spectrum is denoted by Spec_inv(G) and is called the inverse spectrum of G. There is a natural construction of a sheaf of l-groups on Spec_inv(G), which is called the structure sheaf of G on Spec_inv(G). The inverse spectrum and its structure sheaf first appeared in Rump and Yang (Bull Lond Math Soc 40:263–273, 2008) via the Jaffard-Ohm Theorem. We present an elementary and direct construction. The relationship between an l-group H and an l-subgroup G is analyzed via the structure sheaves on the inverse spectra. The structure sheaves are used to give a sheaf-theoretic presentation of the valuation theory of a field. The group of Cartier divisors of an integral affine scheme is identified canonically with an l-group. Special attention is paid to Prüfer domains, where the ties between rings and l-groups are particularly close.