The Lattice of Profinite Subgroups of Protori

TL;DR

This paper explores the structure of profinite subgroups within protori, revealing a lattice framework that facilitates universal resolutions and deepens understanding of their duality and classification.

Contribution

It introduces a lattice structure of profinite subgroups in finite-dimensional protori and constructs a universal resolution in the category of compact abelian groups.

Findings

Profinite subgroups form a lattice under intersection and sum.

A universal resolution for protori is established.

The dual protorus's kernel is characterized as a direct limit of lattice subgroups.

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.