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Motivic invariants of p-adic fields

Published online by Cambridge University Press:  19 May 2011

Kyle M. Ormsby
Kyle M. Ormsby
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, [email protected]
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Abstract

We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL〈n〉 over p-adic fields. These spectra interpolate between integral motivic cohomology (n = 0), a connective version of algebraic K-theory (n = 1), and the algebraic Brown-Peterson spectrum (n = ∞). We deduce that, over p-adic fields, the 2-complete BPGL〈n〉 splits over 2-complete BPGL〈0〉, implying that the slice spectral sequence for BPGL collapses.

This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.

Keywords

motivic Adams spectral sequencealgebraic K-theoryalgebraic cobordism
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 597 - 618
Copyright
Copyright © ISOPP 2011

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References

Bor03.Borghesi, Simone, Algebraic Morava K-theories, Invent. Math. 151 (2003), no. 2, 381–413.CrossRefGoogle Scholar
Cas86.Cassels, J. W. S., Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986.CrossRefGoogle Scholar
DI10.Dugger, Daniel and Isaksen, Daniel C., The motivic Adams spectral sequence, Geom. Topol. 14 (2010), no. 2, 967–1014.CrossRefGoogle Scholar
DRØ03.Dundas, Bjørn Ian, Röndigs, Oliver, and Østvær, Paul Arne, Motivic functors, Doc. Math. 8 (2003), 489–525 (electronic).CrossRefGoogle Scholar
Hill.Hill, Michael A., Ext and the motivic steenrod algebra over ℝ, arXiv:0904. 1998.Google Scholar
HK01.Hu, Po and Kriz, Igor, Some remarks on Real and algebraic cobordism, K-Theory 22 (2001), no. 4, 335–366.CrossRefGoogle Scholar
HKO.Hu, Po, Kriz, Igor, and Ormsby, Kyle M., Convergence of the motivic Adams spectral sequence, J. K-Theory 7 (2011).CrossRefGoogle Scholar
HKO10.Hu, Po, Kriz, Igor, and Ormsby, Kyle M., Remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory 7(2011).CrossRefGoogle Scholar
Hu03.Hu, Po, S-modules in the category of schemes, Mem. Amer. Math. Soc. 161 (2003), no. 767, viii+125.Google Scholar
IS.Isaksen, Daniel C. and Shkembi, Armira, Motivic connective K-theories and the cohomology of A(1), arXiv:1002.2638.Google Scholar
Jar00.Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553 (electronic).CrossRefGoogle Scholar
May70.May, J. Peter, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231.Google Scholar
Mil70.Milnor, John, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.CrossRefGoogle Scholar
Mil71.Milnor, John, Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press, Princeton, N.J., 1971.Google Scholar
Mor04.Morel, Fabien, On the motivic π0 of the sphere spectrum, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 219–260.Google Scholar
Mor05.Morel, Fabien, The stable -connectivity theorems, K-Theory 35 (2005), no. 1-2, 1–68.CrossRefGoogle Scholar
MV99.Morel, Fabien and Voevodsky, Vladimir, A1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001).CrossRefGoogle Scholar
NSØ09.Naumann, Niko, Spitzweck, Markus, and Østvær, Paul Arne, Motivic Landweber exactness, Doc. Math. 14 (2009), 551–593.CrossRefGoogle Scholar
Orm.Ormsby, Kyle M., The K(1)-local motivic sphere, in preparation.Google Scholar
Orm10.Ormsby, Kyle M., Computations in stable motivic homotopy theory, Ph.D. thesis, University of Michigan, 2010.Google Scholar
Rav86.Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press Inc., Orlando, FL, 1986.Google Scholar
RW00.Rognes, J. and Weibel, C., Two-primary algebraic K-theory of rings of integers in number fields, J. Amer. Math. Soc. 13 (2000), no. 1, 1–54, Appendix A by Manfred Kolster.CrossRefGoogle Scholar
SØ09.Spitzweck, Markus and Østvær, Paul Arne, The Bott inverted infinite projective space is homotopy algebraic K-theory, Bull. Lond. Math. Soc. 41 (2009), no. 2, 281–292.CrossRefGoogle Scholar
Vez01.Vezzosi, Gabriele, Brown-Peterson spectra in stable -homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001), 47–64.Google Scholar
Voe.Voevodsky, Vladimir, Motivic Eilenberg-MacLane spaces, arXiv:0805.4432.Google Scholar
Voe98.Voevodsky, Vladimir, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 579–604 (electronic).Google Scholar
Voe03a.Voevodsky, Vladimir, Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.CrossRefGoogle Scholar
Voe03b.Voevodsky, Vladimir, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57.CrossRefGoogle Scholar
Wil75.Wilson, W. Stephen, The Ω-spectrum for Brown-Peterson cohomology. II, Amer. J. Math. 97 (1975), 101–123.CrossRefGoogle Scholar