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Open AccessArticle

Structure of Finite-Dimensional Protori

University of Hawaii, 874 Dillingham Blvd., Honolulu, HI 96817, USA
Axioms 2019, 8(3), 93; https://doi.org/10.3390/axioms8030093
Received: 26 June 2019 / Revised: 30 July 2019 / Accepted: 30 July 2019 / Published: 1 August 2019
(This article belongs to the Collection Topological Groups)
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. View Full-Text
Keywords: compact abelian group; torus; torus-free; periodic; protorus; profinite subgroup; torus quotient; torsion-free abelian group; finite rank compact abelian group; torus; torus-free; periodic; protorus; profinite subgroup; torus quotient; torsion-free abelian group; finite rank
MDPI and ACS Style

Lewis, W. Structure of Finite-Dimensional Protori. Axioms 2019, 8, 93.

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