The definition of pure/mixed motive as a correspondence (of suitable smooth schemes) equipped with an algebraic cycle on its tip may be generalized to that of a correspondence carrying instead a cycle in algebraic K-theory (Gillet-Soulé 96, Gillet-Soulé 09).
These might be called Gillet-Soulé motives (Tabuada 13, section 5.3) or just K-motives.
Via the algebraic Chern character these K-motives map to rational pure Chow motives (Tabuada 13, p. 9). Moreover, they faithfully embed into noncommutative motives (Tabuada 13, p. 10).
Henri Gillet and Christophe Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996), 127–176. (arXiv:9507013)
Henri Gillet and Christophe Soulé, Motivic weight complexes for arithmetic varieties J. Algebra 322 (2009), no. 9, 3088–3141.
Gonçalo Tabuada, Chow motives versus noncommutative motives, Journal of Noncommutative Geometry 7:3, 2013, pp. 767–786, arXiv:1103.0200, doi
Grigory Garkusha, Ivan Panin, K-motives of algebraic varieties, arxiv/1108.0375; The triangulated category of K-motives $DK(k)$, arxiv/1302.2163
Last revised on February 10, 2014 at 02:53:15. See the history of this page for a list of all contributions to it.