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The triangulated category of K-motives

Published online by Cambridge University Press:  12 May 2014

Grigory Garkusha  and
Ivan Panin
Grigory Garkusha
Affiliation:
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United [email protected]
Ivan Panin
Affiliation:
St. Petersburg Branch of V. A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, [email protected] St. Petersburg State University, Department of Mathematics and Mechanics, Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, [email protected]
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Abstract

For any perfect field k a triangulated category of K-motives is constructed in the style of Voevodsky's construction of the category . To each smooth k-variety X the K-motive is associated in the category and

where pt = Spec(k) and K(X) is Quillen's K-theory of X.

Keywords

Motivic homotopy theoryalgebraic K-theoryspectral categories14F4219E0855U35
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 1 , August 2014 , pp. 103 - 137
Copyright
Copyright © ISOPP 2014 

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References

1.Friedlander, E. M., Suslin, A., The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875.Google Scholar
2.Garkusha, G., Panin, I., K-motives of algebraic varieties, Homology Homotopy Appl. 14(2) (2012), 211–264.Google Scholar
3.Garkusha, G., Panin, I., On the motivic spectral sequence, preprint arXiv:1210.2242.CrossRefGoogle Scholar
4.Geisser, T., Hesselholt, L., Topological cyclic homology of schemes, in Algebraic K-theory (Seattle, WA, 1997), Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 41–87.Google Scholar
5.Grayson, D., Weight filtrations via commuting automorphisms, K-Theory 9 (1995), 139–172.CrossRefGoogle Scholar
6.Hovey, M., Shipley, B., Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.Google Scholar
7.Keller, B., Derived categories and their uses, In Handbook of Algebra, vol. 1, North-Holland, Amsterdam, 1996, pp. 671–701.Google Scholar
8.Knizel, A., Homotopy invariant presheaves with Kor-transfers, MSc Diploma, St. Petersburg State University, 2012.Google Scholar
9.Morel, F., The stable 1-connectivity theorems, K-Theory 35 (2006), 1–68.Google Scholar
10.Morel, F., Voevodsky, V., 1-homotopy theory of schemes, Publ. Math. IHES 90 (1999), 45–143.Google Scholar
11.Schwede, S., An untitled book project about symmetric spectra, available at www.math.uni-bonn.de/schwede (version v3.0/April 2012).Google Scholar
12.Schwede, S., Shipley, B., Stable model categories are categories of modules, Topology 42(1) (2003), 103–153.Google Scholar
13.Suslin, A., Voevodsky, V., Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht (2000), pp. 117–189.Google Scholar
14.Thomason, R. W., Trobaugh, T., Higher algebraic K-theory ofschemes and of derived categories, The Grothendieck Festschrift III, Progress in Mathematics 88, Birkhäuser, 1990, pp. 247–435.Google Scholar
15.Voevodsky, V., Cohomological theory of presheaves with transfers, in Cycles, Transfers and Motivic Homology Theories (Voevodsky, V., Suslin, A. and Friedlander, E., eds.), Annals of Math. Studies, Princeton University Press, 2000.Google Scholar
16.Voevodsky, V., Triangulated category of motives over a field, in Cycles, Transfers and Motivic Homology Theories (Voevodsky, V., Suslin, A. and Friedlander, E., eds.), Annals of Math. Studies, Princeton University Press, 2000.Google Scholar
17.Waldhausen, F., Algebraic K-theory of spaces, In Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lecture Notes in Mathematics 1126, Springer-Verlag, 1985, pp. 318–419.Google Scholar
18.Walker, M. E., Motivic cohomology and the K-theory ofautomorphisms, PhD Thesis, University of Illinois at Urbana-Champaign, 1996.Google Scholar