Noncommutative Motives, Numerical Equivalence, and Semi-simplicity

In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F ). We prove that NNum(k)F is abelian semi-simple and that Grothendieck’s category Num(k)Q of numerical motives embeds in NNum(k)Q after being factored out by the action of the Tate object. As an application we obtain an alternative proof of Jannsen’s semi-simplicity result, which uses the noncommutative world instead of a Weil cohomology.