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Motivic connective K-theories and the cohomology of A(1)

Published online by Cambridge University Press:  24 May 2011

Daniel C. Isaksen  and
Armira Shkembi
Daniel C. Isaksen
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USAisaksen@math.wayne.edu
Armira Shkembi
Affiliation:
Mathematics and Sciences Department, St. Leo University, St. Leo, FL 33574, USAarmira.shkembi@saintleo.edu
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Abstract

We make some computations in stable motivic homotopy theory over Spec ℂ, completed at 2. Using homotopy fixed points and the algebraic K-theory spectrum, we construct over ℂ a motivic analogue of the real K-theory spectrum KO. We also establish a theory of motivic connective covers over ℂ to obtain a motivic version of ko. We establish an Adams spectral sequence for computing motivic ko-homology. The E2-term of this spectral sequence involves Ext groups over the subalgebra A(1) of the motivic Steenrod algebra. We make several explicit computations of these E2-terms in interesting special cases.

Keywords

Motivic stable homotopy theoryAdams spectral sequenceconnective K-theorySteenrod algebra
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 619 - 661
Copyright
Copyright © ISOPP 2011

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References

1.Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics, Univ. of Chicago Press, 1974.Google Scholar
2.Bousfield, A. K., The localization of spectra with respect to homology, Topology 18 (1979) 257–281.CrossRefGoogle Scholar
3.Davis, D., Generalized homology and the generalized vector field problem, Quart. J. Math. Oxford 25 (1974) 169–193.CrossRefGoogle Scholar
4.Dugger, D. and Isaksen, D. C., Topological hypercovers and 1-realizations, Math. Z. 246 (2004) 667–689.CrossRefGoogle Scholar
5.Dugger, D. and Isaksen, D. C., The Hopf condition for bilinear forms over arbitrary fields, Ann. of Math. 165 (2007) 943–964.CrossRefGoogle Scholar
6.Dugger, D., and Isaksen, D. C., Motivic cell structures, Algebr. Geom. Topol. 5 (2005) 615–652.CrossRefGoogle Scholar
7.Dugger, D. and Isaksen, D. C., The motivic Adams spectral sequence, Geom. Topol. 14 (2010) 967–1014.CrossRefGoogle Scholar
8.Dundas, B. I., Levine, M., Østvær, P. A., Röndigs, O., and Voevodsky, V., Motivic homotopy theory, Lectures from the Summer School held in Nordfjordeid, August 2002, Universitext, Springer-Verlag 2007.Google Scholar
9.Dundas, B. I., Röndigs, O., and Østvær, P. A., Motivic functors, Doc. Math. 8 (2003) 489–525.CrossRefGoogle Scholar
10.Gitler, S., Mahowald, M., and Milgram, R. J., The nonimmersion problem for RPn and higher-order cohomology operations, Proc. Nat. Acad. Sci. U.S.A. 60 (1968) 432–437.CrossRefGoogle Scholar
11.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc., 2003.Google Scholar
12.Hornbostel, J., A 1-representability of Hermitian K-theory and Witt groups, Topology 44 (2005) 661–687.CrossRefGoogle Scholar
13.Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63–127.CrossRefGoogle Scholar
14.Hu, P., S-modules in the category of schemes, Mem. Amer. Math. Soc. 161 (2003).Google Scholar
15.Hu, P., Kriz, I., and Ormsby, K., Some remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory 7 (2011).CrossRefGoogle Scholar
16.Hu, P., Kriz, I., and Ormsby, K., Equivariant and Real motivic stable homotopy theory, preprint, 2010, K-theory preprint archives.Google Scholar
17.Isaksen, D. C., Flasque model structures for simplicial presheaves, K-Theory 36 (2005) 371–395.CrossRefGoogle Scholar
18.Isaksen, D. C., The cohomology of motivic A(2), Homotopy Homology Appl. 11 (2009) 251–274.CrossRefGoogle Scholar
19.Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000) 445–553.CrossRefGoogle Scholar
20.Kobal, D.K-theory, Hermitian K-theory and the Karoubi tower, K-Theory 17 (1999) 113–140.CrossRefGoogle Scholar
21.May, J. P., E∞ ring spaces and E∞ ring spectra, with contributions by Quinn, Frank, Ray, Nigel, and Tornehave, Jørgen, Lecture Notes in Mathematics 577, Springer-Verlag, 1977.Google Scholar
22.Morel, F. and Voevodsky, V., 1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45–143.CrossRefGoogle Scholar
23.Østvær, P. A., Etale descent for real number fields, Topology 42 (2003) 197–225.CrossRefGoogle Scholar
24.Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres, second edition, Amer. Math. Soc. Chelsea, 2004.CrossRefGoogle Scholar
25.Röndigs, O., Spitzweck, M., and Østvær, P. A., Motivic strict ring models for K-theory, Proc. Amer. Math. Soc. 138 (2010) 3509–3520.CrossRefGoogle Scholar
26.Suslin, A. A., On the K-theory of local fields, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), J. Pure Appl. Algebra 34 (1984) 301–318.CrossRefGoogle Scholar
27.Voevodsky, V., Motivic Eilenberg-Maclane spaces, preprint, 2008, arXiv:0805.4432.Google Scholar
28.Voevodsky, V., Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003) 1–57.CrossRefGoogle Scholar
29.Voevodsky, V., Motivic cohomology with ℤ/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003) 59–104.CrossRefGoogle Scholar
30.Wilson, W. S., The Ω-spectrum for Brown-Peterson Cohomology II, Amer. J. Math. 97 (1975) 101–123.CrossRefGoogle Scholar