Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T18:02:39.136Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on motivic homotopy theory over algebraically closed fields

Published online by Cambridge University Press:  21 January 2010

Po Hu ,
Igor Kriz  and
Kyle Ormsby
Po Hu
Affiliation:
Department of Mathematics, Wayne State University, U.S.A., [email protected]
Igor Kriz
Affiliation:
Department of Mathematics, Wayne State University, U.S.A., [email protected]
Kyle Ormsby
Affiliation:
Department of Mathematics, Wayne State University, U.S.A., [email protected]
Get access

Abstract

We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

Keywords

Motivic homotopy theoryAdams spectral sequencesJ -homomorphism
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 1 , February 2011 , pp. 55 - 89
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adams, J.F.: Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, 1974Google Scholar
2.Adams, J.F.: On the groups J(X). I, Topology 2 (1963) 181–195CrossRefGoogle Scholar
3.Adams, J.F.: On the groups J(X). II, Topology 3 (1965) 137–171CrossRefGoogle Scholar
4.Adams, J.F.: On the groups J(X). IV, Topology 5 (1966) 21–71, correction: Topology 7 (1968) 331CrossRefGoogle Scholar
5.Bousfield, A.K., The localization of spectra with respect to homology, Topology 18 (1979) 257–281CrossRefGoogle Scholar
6.Borghesi, S.: Algebraic Morava K-theories, Invent. Math. 151 (2003) 381–413CrossRefGoogle Scholar
7.Dugger, D., Isaksen, D.: Motivic cell structures, Alg. Geom. Top. 5 (2005) 615–652CrossRefGoogle Scholar
8.Dugger, D., Isaksen, D.C.: The motivic Adams spectral sequence, arXiv:0901.1632Google Scholar
9.Hornbostel, J.: -representability of hermitian K-theory and Witt groups, Topology 44 (2005) 661–687CrossRefGoogle Scholar
10.Hu, P., Kriz, I.: Some remarks on Real and algebraic cobordism, K-theory 22 (2001) 335–366CrossRefGoogle Scholar
11.Hu, P., Kriz, I.: Appendix to P.Hu: On the Picard group of the stable 1-homotopy category, Topology 44 (2005) 609–640CrossRefGoogle Scholar
12.Isaksen, D.: Private communicationGoogle Scholar
13.Karoubi, M.: Relations between algebraic K-theory and Hermitian K-theory, Proceedings of the Luminy conference on algebraic K-theory (Luminy 1983), J. Pure Appl. Alg., vol. 34, 1984, pp. 259–263Google Scholar
14.Levine, M., Morel, F.: Algebraic Cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007Google Scholar
15.Levine, M., Morel, F.: Cobordisme algebrique I, C.R. Acad.Sci. Paris Ser. I Math. 332 (2001) 723–728CrossRefGoogle Scholar
16.Levine, M., Morel, F.: Cobordisme algebrique II, C.R. Acad.Sci. Paris Ser. I Math. 332 (2001) 815–820CrossRefGoogle Scholar
17.Morel, F.: Suite spectrale d'Adams et invariants cohomologiques des formes quadratiques C.R. Acad.Sci. Paris Ser. I Math. 328 (1999) 963–968CrossRefGoogle Scholar
18.Morel, F.: On the motivic stable π0 of the sphere spectrum, Axiomatic, Enriched and Motivic Homotopy Theory, pp. 219–260, 2004 Kluwer Academic PublishersCrossRefGoogle Scholar
19.Morel, F., Voevodsky, V.: 1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45–143CrossRefGoogle Scholar
20.Nakaoka, M.: Cohomology modp of symmetric products of spheres I,II, J. Inst. Poly. Osaka City Univ. 9 (1958) 1–18, 10 (1959) 67–89Google Scholar
21.Quillen, D.: On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972) 552–586CrossRefGoogle Scholar
22.Ravenel, D.C.: Complex cobordism and the stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986Google Scholar
23.Suslin, A.: On the K-theory of algebraically closed fields, Invent. Math. 73 (1983) 241–245CrossRefGoogle Scholar
24.Voevodsky, V.: On the zero slice of the sphere spectrum, T. Mat. Inst. Steklova 246 (2004) 106–115, Proc. Steklov Inst. Math. 246 (2004), 93–102Google Scholar
25.Voevodsky, V.: Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Etudes Sci. 998 (2003) 1–57CrossRefGoogle Scholar
26.Voevodsky, V.: Motivic Eilenberg-MacLane spaces, preprint, UIUC K-theory archive, 846Google Scholar
27.Voevodsky, V.: Unstable motivic homotopy categories in Nisnevich and cdh-topologies, preprint, UIUC K-theory archive, 444Google Scholar