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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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
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:	Date Event 20 June 2019 Published 23 April 2019 Available 2 April 2019 Accepted
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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