The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of Noncommutative Geometry


Full-Text PDF (277 KB) | Metadata | Table of Contents | JNCG summary
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