JournalsjncgVol. 9, No. 3pp. 851–875

A1\mathbf A^1-homotopy theory of noncommutative motives

  • Gonçalo Tabuada

    Massachusetts Institute of Technology, Cambridge, USA
$\mathbf A^1$-homotopy theory of noncommutative motives cover
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In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of A1{\bf A}^1-homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's KK-theory (KVKV) and Weibel's homotopy KK-theory (KHKH). As an application, we obtain a complete classification of all the natural transformations defined on KV,KHKV, KH. This leads to a streamlined construction of Weibel's homotopy Chern character from KVKV to periodic cyclic homology. Along the way we extend Dwyer–Friedlander's étale KK-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.

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Gonçalo Tabuada, A1\mathbf A^1-homotopy theory of noncommutative motives. J. Noncommut. Geom. 9 (2015), no. 3, pp. 851–875

DOI 10.4171/JNCG/210