Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

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Following an insight of Kontsevich, we prove that the quotient of Voevodsky's category of geometric mixed motives DM by the endofunctor -Q(1)[2] embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show also that this embedding is compatible with the one between pure motives. As an application, we obtain a precise relation between the Picard groups Pic(-), the Grothendieck groups, the Schur-finitenss, and the Kimura-finitenss of the categories DM and KMM. In particular, the quotient of Pic(DM) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). Along the way, we relate KMM with Morel-Voevodsky's stable A1-homotopy category, recover the twisted algebraic K-theory of Kahn-Levine from KMM, and extend Elmendorf-Mandell's foundational work on multicategories to a broader setting
Original languageUnknown
Pages (from-to)506-545
JournalAdvances In Mathematics
Issue numberNA
Publication statusPublished - 1 Jan 2014

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