2014/02/18 by Gonçalo Tabuada, Tabuada, Goncalo
Mathematics · #14A22 #14C15 #14C22 #18F25 #18G55 #19E08 #19E15 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.1402.4438
openalex publication_date 2014/02/18 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Following an insight of Kontsevich, we prove that the quotient of Voevodsky's\ncategory of geometric mixed motives DM by the endofunctor -Q(1)[2] embeds\nfully-faithfully into Kontsevich's category of noncommutative mixed motives\nKMM. We show also that this embedding is compatible with the one between pure\nmotives. As an application, we obtain a precise relation between the Picard\ngroups Pic(-), the Grothendieck groups, the Schur-finitenss, and the\nKimura-finitenss of the categories DM and KMM. In particular, the quotient of\nPic(DM) by the subgroup of Tate twists Q(i)[2i] injects into Pic(KMM). Along\nthe way, we relate KMM with Morel-Voevodsky's stable A1-homotopy category,\nrecover the twisted algebraic K-theory of Kahn-Levine from KMM, and extend\nElmendorf-Mandell's foundational work on multicategories to a broader setting.\n