Noncommutative motives, numerical equivalence, and semi-simplicity

Abstract

Making use of Hochschild homology, we introduce the correct 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 into 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 celebrate semi-simplicity result, which uses the noncommutative world instead of a classical Weil cohomology.

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