Skip to main content Accessibility help
×
Home

ON THE MOTIVIC SPECTRAL SEQUENCE

  • Grigory Garkusha (a1) and Ivan Panin (a2) (a3)

Abstract

It is shown that the Grayson tower for $K$ -theory of smooth algebraic varieties is isomorphic to the slice tower of $S^{1}$ -spectra. We also extend the Grayson tower to bispectra, and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$ -theory spectrum $\mathit{KGL}$ . This solves Suslin’s problem about these two spectral sequences in the affirmative.

Copyright

References

Hide All
1. Beilinson, A., Letter to Christophe Soulé, 1982, K-theory Preprint archives 694.
2. Bloch, S. and Lichtenbaum, S., A spectral sequence for motivic cohomology, 1995, K-theory Preprint archives 62.
3. Dundas, B. I., Levine, M., Østvær, P. A., Röndigs, O. and Voevodsky, V., Motivic Homotopy Theory, Universitext (Springer-Verlag, Berlin, 2007).
4. Friedlander, E. M. and Suslin, A., The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), 773875.
5. Garkusha, G. and Panin, I., K-motives of algebraic varieties, Homology, Homotopy Appl. (2) 14 (2012), 211264.
6. Garkusha, G. and Panin, I., The triangulated category of K-motives DK - eff (k), J. K-Theory (1) 14 (2014), 103137.
7. Grayson, D., Higher algebraic K-theory II [after Daniel Quillen], in Algebraic K-theory, Evanston 1976, Lecture Notes in Mathematics, Volume 551, pp. 217240 (Springer-Verlag, Berlin, Heidelberg, New York, 1976).
8. Grayson, D., Adams operations on higher K-theory, K-Theory 6 (1992), 97111.
9. Grayson, D., Weight filtrations via commuting automorphisms, K-Theory 9 (1995), 139172.
10. Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra (1) 165 (2001), 63127.
11. Hovey, M., Shipley, B. and Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149208.
12. Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445552.
13. Kahn, B. and Levine, M., Motives of Azumaya algebras, J. Inst. Math. Jussieu (3) 9 (2010), 481599.
14. Levine, M., Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom. (2) 10 (2001), 299363.
15. Levine, M., The homotopy coniveau filtration, J. Topol. 1 (2008), 217267.
16. Levine, M., The slice filtration and Grothendieck–Witt groups, Pure Appl. Math. Q. (4) 7 (2011), 15431584.
17. Morel, F., The stable A1 -connectivity theorems, K-theory 35 (2006), 168.
18. Morel, F. and Voevodsky, V., A1 -homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143.
19. Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. (1) 9 (1996), 205236.
20. Panin, I., Pimenov, K. and Röndigs, O., On Voevodsky’s algebraic K-theory spectrum BGL , Abel. Symp. Proc. 4 (2009), 279330.
21. Podkopaev, O., The equivalence of Grayson and Friedlander–Suslin spectral sequences, Preprint, 2013, arXiv:1309.7597.
22. Riou, J., Catégorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math. France 135 (2007), 495547.
23. Riou, J., Algebraic K-theory, A1 -homotopy and Riemann–Roch theorems, J. Topol. 3 (2010), 229264.
24. Rognes, J., Introduction to Higher Algebraic K-theory, Lecture Notes 2010, available at folk.uio.no/rognes/kurs/mat9570v10/akt.pdf.
25. Schwede, S., An untitled book project about symmetric spectra, available at www.math.uni-bonn.de/∼schwede (version v3.0/April 2012).
26. Suslin, A., On the Grayson spectral sequence, Tr. Mat. Inst. Steklova 241 (2003), 218253. (Russian). English transl. in Proc. Steklov Inst. Math. (2) 241 (2003), 202–237.
27. Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, in The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Science Series C: Mathematics, Physics and Science, Volume 548, pp. 117189 (Kluwer Academic Publishers, Dordrecht, 2000).
28. Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift III, Progress in Mathematics, Volume 88, pp. 247435 (Birkhäuser, Boston, 1990).
29. Voevodsky, V., A1 -Homotopy theory, Doc. Math. ICM(1) (1998), 417442.
30. Voevodsky, V., Open problems in the motivic stable homotopy theory. I, in Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), International Press Lecture Series, Volume 3, pp. 334 (International Press, Somerville, MA, 2002).
31. Voevodsky, V., A possible new approach to the motivic spectral sequence for algebraic K-theory, in Recent Progress in Homotopy Theory (Baltimore, MD, 2000), Contemporary Mathematics, Volume 293, pp. 371379 (American Mathematical Society, Providence, RI, 2002).
32. Voevodsky, V., Cancellation theorem, Doc. Math. Extra Volume in honor of A. Suslin (2010), 671–685.
33. Waldhausen, F., Algebraic K-theory of spaces, in Algebraic and Geometric Topology, Proceedings Conference, New Brunswick/USA 1983, Lecture Notes in Mathematics, Volume 1126, pp. 318419 (Springer-Verlag, Berlin, Heidelberg, New York, 1985).
34. Walker, M. E., Motivic cohomology and the K-theory of automorphisms, PhD thesis, University of Illinois at Urbana-Champaign, 1996.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

ON THE MOTIVIC SPECTRAL SEQUENCE

  • Grigory Garkusha (a1) and Ivan Panin (a2) (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed