Finite-Dimensional Protori Are Adelic Tori

Abstract

We show the category of finite-dimensional compact connected abelian groups (finite-dimensional protori) {\it is} the category of {\it adelic tori}, for which there is a complete Lie theory: for each adelic torus $G$ there is a proper short exact sequence $0\ra\q^n\ra \mathcal{L}(G)\xrightarrow{\exp} G\ra 0$ with $\exp$ the {\it adelic exponential}. Extending the definition of nonarchimedean dimension to apply to number fields, the capstone result is the fact that all number fields have nonarchimedean dimension 1.

8 pages. Submitted to Journal of Lie Theory. "Finite-Dimensional Protori Are Adelic Tori" builds on the work of E.L. Lady (one of my thesis advisers) on the Grothendieck ring of discrete finite rank torsion-free abelian groups under quasi-homomorphism (e.g., three foundational papers on the Grothendieck ring in the Journal of Algebra in the 1980's)