A non-commutative analogue of Clausen's view on the id\`{e}le class group

TL;DR

This paper extends Clausen's view on the idèle class group to a non-commutative setting using higher K-theory, simplifying proofs and connecting to class field theory and Hilbert's reciprocity law.

Contribution

It introduces a non-commutative analogue of Clausen's idèle class group for semisimple Q-algebras, generalizing the classical number field case and providing a simpler proof approach.

Findings

Generalizes idèle class group to non-commutative algebras

Simplifies proof using localization theorem

Connects non-commutative class group with class field theory

Abstract

Clausen predicted that Chevalley's id\`{e}le class group of a number field $F$ appears as the first $K$-group of the category of locally compact $F$-vector spaces. This has turned out to be true, and even generalizes to the higher $K$-groups in a suitable sense. We replace $F$ by a semisimple $\mathbb{Q}$-algebra, and obtain Fr\"{o}hlich's non-commutative id\`{e}le class group in an analogous fashion, modulo the reduced norm one elements. Even in the number field case our proof is simpler than the existing one, and based on the localization theorem for percolating subcategories. Finally, using class field theory as input, we interpret Hilbert's reciprocity law (as well as a noncommutative variant) in terms of our results.