Geodesic
$K$-motives of algebraic varieties
Homology, homotopy, and applications, Tome 14 (2012) no. 2, pp. 211-264.

Voir la notice de l'article provenant de la source International Press of Boston

A kind of motivic algebra of spectral categories and modules over them is developed to introduce $K$-motives of algebraic varieties. As an application, bivariant algebraic $K$-theory $K(X,Y)$ as well as bivariant motivic cohomology groups $H^{p,q}(X,Y,\mathbb{Z})$ are defined and studied. We use Grayson’s machinery to produce the Grayson motivic spectral sequence connecting bivariant $K$-theory to bivariant motivic cohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of $K$-motives constructed in the paper. It is also shown that ordinary algebraic $K$-theory is represented by the $K$-motive of the point.
DOI : 10.4310/HHA.2012.v14.n2.a13
Classification : 14F42, 19E08, 55U35
Keywords: motivic homotopy theory, algebraic $K$-theory, spectral category 
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Grigory Garkusha; Ivan Panin. $K$-motives of algebraic varieties. Homology, homotopy, and applications, Tome 14 (2012) no. 2, pp. 211-264. doi : 10.4310/HHA.2012.v14.n2.a13. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2012.v14.n2.a13/

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