Structure of Finite-Dimensional Protori

TL;DR

This paper establishes a comprehensive structure theorem for finite-dimensional protori, revealing the relationships between their algebraic properties and providing universal resolutions and morphism lifting techniques.

Contribution

It introduces a new structure theorem for finite-dimensional protori, connecting their properties with dual categories and providing universal resolutions.

Findings

Parameterization of resolutions by dual categories of torsion-free abelian groups

Existence of a universal resolution independent of subgroup choice

Lifting of morphisms to product morphisms between groups and vector spaces

Abstract

A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.