The Lattice of Profinite Subgroups of Protori

Abstract

Compact connected abelian groups, or protori, have intrinsic structural characteristics that present for the entire category. In the case of finite-dimensional torus-free protori, The Resolution Theorem for Compact Abelian Groups sets the stage for demonstrating that the profinite subgroups inducing tori quotients comprise an isogeny class of finitely generated modules over the profinite integers, which is a lattice under intersection (meet) and + (join). The structural results enable the formulation of a universal resolution in the category of protori under morphisms of compact abelian groups. A single profinite subgroup from the lattice in the Resolution Theorem is replaced by the direct limit of the lattice of such subgroups and effects a covering morphism in which the discrete torsion-free Pontryagin dual of the protorus organically emerges as the kernel of the quotient map resolving the protorus. Among other advantages, this enables the study of a finite rank torsion-free abelian group as a canonical subgroup of its dual protorus, with the concomitant topological, analytical, and number-theoretic insights availed by the compact abelian setting.