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

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.