Protori and Torsion-Free Abelian Groups

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.