A NONCOMMUTATIVE ANALOGUE OF CLAUSEN’S VIEW ON THE IDÈLE CLASS GROUP

Oliver Braunling, Ruben Henrard, Adam Christiaan van Roosmalen

Research output: Contribution to journalArticlepeer-review

Abstract

Clausen predicted that Chevalley’s idèle 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 Q-algebra and obtain Fröhlich’s noncommutative idèle 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.

Original languageEnglish
JournalJournal of the Institute of Mathematics of Jussieu
Early online date2 Apr 2024
DOIs
Publication statusE-pub ahead of print - 2 Apr 2024

Keywords

  • Hilbert symbol
  • K-theory
  • Locally compact modules
  • exact category
  • idèle class group

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