Structure of Finite-Dimensional Protori
arXiv:1908.04195 · doi:10.3390/axioms8030093
The paper establishes a structure theorem for finite‑dimensional protori (compact connected abelian groups), linking their density, torsion, and divisibility properties to the dual category of finite‑rank torsion‑free abelian groups, and uses this framework to give universal resolutions and lift morphisms to products of periodic locally compact groups and real 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.
arXiv admin note: text overlap with arXiv:1903.08022