Noncommutative motives in positive characteristic and their applications
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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 overkis abelian semi-simple, as conjecturedby Kontsevich. Then, we establish a far-reaching noncommutative generaliza-tion of the Weil conjectures, originally proved by Dwork andGrothendieck. Inthe same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originallyproven by Hesselholt. As a third main result, we prove that the numericalGrothendieck group of every smooth proper dg category is a finitely generatedfree abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce the noncommutative motivic Galois (super-)groupsand, following aninsight 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 noncommutativemotives.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Advances in mathematics
Tabuada, Gonçalo. “Noncommutative motives in positive characteristic and their applications.” Advances in mathematics, vol. 349, 2019, pp. 648-681 © 2019 The Author