# Journal of Noncommutative Geometry

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**Volume 9, Issue 3, 2015, pp. 851–875**

**DOI: 10.4171/JNCG/210**

Published online: 2015-10-29

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

Gonçalo Tabuada^{[1]}(1) Massachusetts Institute of Technology, Cambridge, USA

In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of ${\bf A}^1$-homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's $K$-theory ($KV$) 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 étale $K$-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.

*Keywords: *$\mathbf A^1$ homotopy, noncommutative motives, algebraic K-theory, periodic cyclic homology, homotopy Chern characters, noncommutative algebraic geometry

Tabuada Gonçalo: $\mathbf A^1$-homotopy theory of noncommutative motives. *J. Noncommut. Geom.* 9 (2015), 851-875. doi: 10.4171/JNCG/210