Noncommutative motives in positive characteristic and their applications

TL;DR

This paper develops a noncommutative motive theory in positive characteristic, proving semi-simplicity, generalizing Weil conjectures, and connecting to classical Galois groups and cohomology, with broad implications for algebraic geometry.

Contribution

It proves the semi-simplicity of noncommutative numerical motives, generalizes classical conjectures, and introduces noncommutative motivic Galois groups, extending the scope of motive theory.

Findings

Category of noncommutative numerical motives is abelian semi-simple.

Established a noncommutative generalization of the Weil conjectures.

Numerical Grothendieck groups of smooth proper dg categories are finitely generated free abelian groups.

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 proved 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.