Finite generation of the numerical Grothendieck group

TL;DR

This paper proves that the numerical Grothendieck group of smooth proper dg categories over a finite field is finitely generated and free, and establishes semi-simplicity and rationality properties of noncommutative motives and zeta functions.

Contribution

It demonstrates the finite generation and freeness of the numerical Grothendieck group using topological cyclic homology and noncommutative motives, confirming conjectures by Kontsevich.

Findings

Numerical Grothendieck groups are finitely generated free abelian groups.

The category of noncommutative numerical motives is abelian semi-simple.

Zeta functions of endomorphisms are rational and satisfy a functional equation.

Abstract

Let k be a finite base field. In this note, making use of topological periodic cyclic homology and of the theory of noncommutative motives, we prove that the numerical Grothendieck group of every smooth proper dg k-linear category is a finitely generated free abelian group. Along the way, we prove moreover that the category of noncommutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich. Furthermore, we show that the zeta functions of endomorphisms of noncommutative Chow motives are rational and satisfy a functional equation.