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Title: | A non-commutative analogue of Clausen's view on the idèle class group | Authors: | Braunling, Oliver HENRARD, Ruben VAN ROOSMALEN, Adam-Christiaan |
Issue Date: | 2024 | Publisher: | CAMBRIDGE UNIV PRESS | Source: | Journal of the Institute of Mathematics of Jussieu, | Status: | Early view | Abstract: | Clausen a predit que le groupe des classes d'ideles de Chevalley d'un corps de nombres F apparait comme le premier K-groupe de la categorie des F-espaces vectoriels localement compacts. Cela s'est avere vrai, et se generalise meme aux groupes K superieurs dans un sens approprie. Nous remplacons F par une $\mathbb {Q}$ -algebre semi-simple, et obtenons le groupe des classes d'ideles noncommutatif de Frohlich de maniere analogue, modulo les elements de norme reduite une. Meme dans le cas du corps de nombres, notre preuve est plus simple que celle existante, et repose sur le theoreme de localisation pour des sous-categories percolees. Enfin, en utilisant la theorie des corps de classes, nous interpretons la loi de reciprocite d'Hilbert (ainsi qu'une variante noncommutative) en termes de nos resultats.Clausen predicted that Chevalley's idele 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 Frohlich's noncommutative idele 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. | Notes: | Braunling, O (corresponding author), Univ Autonoma Madrid, Math, Madrid 28049, Spain. oliver.braunling@gmail.com; ruben.henrard@uhasselt.be; ac.vanroosmalen@xjtlu.edu.cn |
Keywords: | Locally compact modules;K-theory;exact category;idele class group;Hilbert symbol | Document URI: | http://hdl.handle.net/1942/42848 | ISSN: | 1474-7480 | e-ISSN: | 1475-3030 | DOI: | 10.1017/S1474748024000100 | ISI #: | 001195563900001 | Rights: | The Author(s), 2024. Published by Cambridge University Press. | Category: | A1 | Type: | Journal Contribution |
Appears in Collections: | Research publications |
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