# Protori and Torsion-Free Abelian Groups

## TL;DR

This paper explores the structure and classification of protori and torsion-free abelian groups using the Resolution Theorem, lattice properties, and non-Archimedean dimension as a key invariant.

## Contribution

It introduces a lattice structure for profinite subgroups of protori, establishes a universal resolution, and defines a non-Archimedean dimension for classification.

## Key findings

- Profinite subgroups form a lattice under intersection and sum.
- Existence of a universal resolution for protori.
- Non-Archimedean dimension as a classification invariant.

## Abstract

The Resolution Theorem for Compact Abelian Groups is applied to show that the profinite subgroups of a finite-dimensional compact connected abelian group (protorus) which induce tori quotients comprise a lattice under intersection (meet) and $+$ (join), facilitating a proof of the existence of a universal resolution. A finite rank torsion-free abelian group $X$ is algebraically isomorphic to a canonical dense subgroup $X_G$ of its Pontryagin dual $G$. A morphism between protori lifts to a product morphism between the universal covers, so morphisms in the category can be studied as pairs of maps: homomorphisms between finitely generated profinite abelian groups and linear maps between finite-dimensional real vector spaces. A concept of non-Archimedean dimension is introduced which acts a useful invariant for classifying protori.