Chow motives versus noncommutative motives

Abstract

In this article we formalize and enhance Kontsevich’s beautiful insight that Chow motives can be embedded into noncommutative 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 noncommutative motives; (2) Gillet–Soulé’s motivic measure admits an extension to the Grothendieck ring of noncommutative motives; (3) certain motivic zeta functions admit an intrinsic construction inside the category of noncommutative motives.