TY - JOUR
T1 - A NONCOMMUTATIVE ANALOGUE OF CLAUSEN’S VIEW ON THE IDÈLE CLASS GROUP
AU - Braunling, Oliver
AU - Henrard, Ruben
AU - van Roosmalen, Adam Christiaan
N1 - Publisher Copyright:
© The Author(s), 2024.
PY - 2024/4/2
Y1 - 2024/4/2
N2 - 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.
AB - 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.
KW - Hilbert symbol
KW - K-theory
KW - Locally compact modules
KW - exact category
KW - idèle class group
UR - http://www.scopus.com/inward/record.url?scp=85189686170&partnerID=8YFLogxK
U2 - 10.1017/S1474748024000100
DO - 10.1017/S1474748024000100
M3 - Article
AN - SCOPUS:85189686170
SN - 1474-7480
JO - Journal of the Institute of Mathematics of Jussieu
JF - Journal of the Institute of Mathematics of Jussieu
ER -