Skip to main content Accessibility help
×
Home

Motivic invariants of p-adic fields

  • Kyle M. Ormsby (a1)

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 BPGLn〉 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.

Copyright

References

Hide All
Bor03.Borghesi, Simone, Algebraic Morava K-theories, Invent. Math. 151 (2003), no. 2, 381413.
Cas86.Cassels, J. W. S., Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986.
DI10.Dugger, Daniel and Isaksen, Daniel C., The motivic Adams spectral sequence, Geom. Topol. 14 (2010), no. 2, 9671014.
DRØ03.Dundas, Bjørn Ian, Röndigs, Oliver, and Østvær, Paul Arne, Motivic functors, Doc. Math. 8 (2003), 489525 (electronic).
Hill.Hill, Michael A., Ext and the motivic steenrod algebra over ℝ, arXiv:0904. 1998.
HK01.Hu, Po and Kriz, Igor, Some remarks on Real and algebraic cobordism, K-Theory 22 (2001), no. 4, 335366.
HKO.Hu, Po, Kriz, Igor, and Ormsby, Kyle M., Convergence of the motivic Adams spectral sequence, J. K-Theory 7 (2011).
HKO10.Hu, Po, Kriz, Igor, and Ormsby, Kyle M., Remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory 7(2011).
Hu03.Hu, Po, S-modules in the category of schemes, Mem. Amer. Math. Soc. 161 (2003), no. 767, viii+125.
IS.Isaksen, Daniel C. and Shkembi, Armira, Motivic connective K-theories and the cohomology of A(1), arXiv:1002.2638.
Jar00.Jardine, J. F., Motivic symmetric spectra, Doc. Math. 5 (2000), 445553 (electronic).
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. 153231.
Mil70.Milnor, John, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318344.
Mil71.Milnor, John, Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press, Princeton, N.J., 1971.
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. 219260.
Mor05.Morel, Fabien, The stable -connectivity theorems, K-Theory 35 (2005), no. 1-2, 168.
MV99.Morel, Fabien and Voevodsky, Vladimir, A1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45143 (2001).
NSØ09.Naumann, Niko, Spitzweck, Markus, and Østvær, Paul Arne, Motivic Landweber exactness, Doc. Math. 14 (2009), 551593.
Orm.Ormsby, Kyle M., The K(1)-local motivic sphere, in preparation.
Orm10.Ormsby, Kyle M., Computations in stable motivic homotopy theory, Ph.D. thesis, University of Michigan, 2010.
Rav86.Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press Inc., Orlando, FL, 1986.
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, 154, Appendix A by Manfred Kolster.
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, 281292.
Vez01.Vezzosi, Gabriele, Brown-Peterson spectra in stable -homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001), 4764.
Voe.Voevodsky, Vladimir, Motivic Eilenberg-MacLane spaces, arXiv:0805.4432.
Voe98.Voevodsky, Vladimir, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), no. Extra Vol. I, 1998, pp. 579604 (electronic).
Voe03a.Voevodsky, Vladimir, Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59104.
Voe03b.Voevodsky, Vladimir, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 157.
Wil75.Wilson, W. Stephen, The Ω-spectrum for Brown-Peterson cohomology. II, Amer. J. Math. 97 (1975), 101123.

Keywords

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