A^1-homotopy theory of noncommutative motives

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In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of A(1)-homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi Villamayor's K-theory (K V) and Weibel's homotopy K-theory (KH). As an application, we obtain a complete classification of all the natural transformations defined on KV,KH. This leads to a streamlined construction of Weibel's homotopy Chern character from KV to periodic cyclic homology. Along the way we extend Dwyer Friedlander's etale K-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.
Original languageEnglish
Pages (from-to)851-875
JournalJournal Of Noncommutative Geometry
Issue number3
Publication statusPublished - 2015


  • A1 homotopy
  • noncommutative motives
  • algebraic K-theory
  • periodic cyclic homology
  • homotopy Chern characters
  • noncommutative algebraic geometry


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