Structure of Finite-Dimensional Protori

Wayne Lewis

doi:10.3390/axioms8030093

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.

Highlights

All compact groups are finite-dimensional and all torsion-free groups have finite rank.We carry out a study of the structure of compact, connected abelian groups, or protori
The main results are a Structure Theorem for Protori (Theorem 1), a universal resolution for a protorus (Corollary 6), a structural result on the lattice of compact open subgroups of zero-dimensional subgroups of a protorus under a natural locally compact topology (Proposition 6), and a lifting theorem for morphisms of protori (Theorem 2), which facilitates a reduction to a decoupled analysis of morphisms between periodic LCA groups
Applying the structure theory developed in Theorem 1, we derive a Structure Theorem for Morphisms (Theorem 2) a new result stating that a morphism f : G Ñ H of protori with duals X

Summary

Introduction

All compact groups are finite-dimensional and all torsion-free groups have finite rank. The main results are a Structure Theorem for Protori (Theorem 1), a universal resolution for a protorus (Corollary 6), a structural result on the lattice of compact open subgroups of zero-dimensional subgroups of a protorus under a natural locally compact topology (Proposition 6), and a lifting theorem for morphisms of protori (Theorem 2), which facilitates a reduction to a decoupled analysis of morphisms between periodic LCA groups. Applying the structure theory developed in Theorem 1, we derive a Structure Theorem for Morphisms (Theorem 2) a new result stating that a morphism f : G Ñ H of protori with duals X pXand Y lifts to a product map between minimal divisible locally compact covers f |∆pf L : ∆.

Background

Structure of Protori

Addition defines a surjective morphism

Applications