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Noncommutative motives in positive characteristic and their applications
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Tabuada, Gonçalo (2019) Noncommutative motives in positive characteristic and their applications. Advances in Mathematics, 349 . pp. 648-681. doi:10.1016/j.aim.2019.04.020
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Official URL: http://dx.doi.org/10.1016/j.aim.2019.04.020
Abstract
Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich. Then, we establish a far-reaching noncommutative generalization of the Weil conjectures, originally proven by Dwork and Grothendieck. In the same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originally proven by Hesselholt. As a third main result, we prove that the numerical Grothendieck group of every smooth proper dg category is a finitely generated free abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce the noncommutative motivic Galois (super-)groups and, following an insight of Kontsevich, relate them to their classical commutative counterparts. Finally, we explain how the motivic measure induced by Berthelot's rigid cohomology can be recovered from the theory of noncommutative motives.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science > Mathematics | ||||||||
Journal or Publication Title: | Advances in Mathematics | ||||||||
Publisher: | Academic Press | ||||||||
ISSN: | 0001-8708 | ||||||||
Official Date: | 20 June 2019 | ||||||||
Dates: |
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Volume: | 349 | ||||||||
Page Range: | pp. 648-681 | ||||||||
DOI: | 10.1016/j.aim.2019.04.020 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access |
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