Chow motives versus non-commutative motives

TL;DR

This paper formalizes the embedding of Chow motives into non-commutative motives, extending key concepts like finiteness, motivic measures, and zeta functions within the non-commutative framework, revealing new structural insights.

Contribution

It develops a formal embedding of Chow motives into non-commutative motives and applies this to extend finiteness notions, motivic measures, and zeta functions.

Findings

Chow motives can be embedded into non-commutative motives after factoring out Tate actions.

Finiteness notions like Schur and Kimura extend to non-commutative motives.

Gillet-Soule's motivic measure extends to the Grothendieck ring of non-commutative motives.

Abstract

In this article we formalize and enhance Kontsevich's beautiful insight that Chow motives can be embedded into non-commutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the notions of Schur and Kimura finiteness admit an adequate extension to the realm of non-commutative motives; (2) Gillet-Soule's motivic measure admits an extension to the Grothendieck ring of non-commutative motives; (3) certain motivic zeta functions admit an intrinsic construction inside the category of non-commutative motives.