Noncommutative motives, numerical equivalence, and semi-simplicity

Matilde Marcolli; Goncalo Tabuada

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arXiv · 1105.2950

Noncommutative motives, numerical equivalence, and semi-simplicity

Matilde Marcolli·Goncalo Tabuada

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Abstract

In this article we further the study of the relationship between pure motives and noncommutative motives. 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.

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BibTeX RIS

Matilde Marcolli, Goncalo Tabuada. 2011-05-15. Noncommutative motives, numerical equivalence, and semi-simplicity. https://arxiv.org/abs/1105.2950

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