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Motivic strict ring models for $ K$-theory


Authors: Oliver Röndigs, Markus Spitzweck and Paul Arne Østvær
Journal: Proc. Amer. Math. Soc. 138 (2010), 3509-3520
MSC (2010): Primary 14F42, 55P43; Secondary 19E08
Published electronically: May 10, 2010
MathSciNet review: 2661551
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Abstract: It is shown that the $ K$-theory of every noetherian base scheme of finite Krull dimension is represented by a commutative strict ring object in the setting of motivic stable homotopy theory. The adjective `strict' is used here in order to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. An analogous topological result follows by running the same type of arguments as in the motivic setting.


References [Enhancements On Off] (What's this?)

  • 1. D.-C. Cisinski, F. Déglise.
    Triangulated categories of motives.
    arXiv:0912.2110.
  • 2. Bjørn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær, Motivic functors, Doc. Math. 8 (2003), 489–525 (electronic). MR 2029171
  • 3. David Gepner and Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic 𝐾-theory, Doc. Math. 14 (2009), 359–396. MR 2540697
  • 4. J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553 (electronic). MR 1787949
  • 5. Fabien Morel and Vladimir Voevodsky, 𝐴¹-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
  • 6. N. Naumann, M. Spitzweck, P. A. Østvær.
    Motivic Landweber exactness.
    Doc. Math. 14:551-593 (electronic), 2009.
  • 7. N. Naumann, M. Spitzweck, P. A. Østvær.
    Chern classes, $ {K}$-theory and Landweber exactness over nonregular base schemes,
    in Motives and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch, Fields Institute Communications, Vol. 56, 307-317, AMS, Providence, RI, 2009.
  • 8. I. Panin, K. Pimenov, O. Röndigs.
    On Voevodsky's algebraic $ K$-theory spectrum,
    in Algebraic Topology, Abel Symposium 2007, 279-330, Springer-Verlag, Berlin, 2009.
  • 9. Oliver Röndigs and Paul Arne Østvær, Motives and modules over motivic cohomology, C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 751–754 (English, with English and French summaries). MR 2227753, 10.1016/j.crma.2006.03.013
  • 10. Oliver Röndigs and Paul Arne Østvær, Modules over motivic cohomology, Adv. Math. 219 (2008), no. 2, 689–727. MR 2435654, 10.1016/j.aim.2008.05.013
  • 11. S. Schwede.
    An untitled book project about symmetric spectra.
    Available on the author's homepage, http://www.math.uni-bonn.de/Tschwede.
  • 12. Markus Spitzweck and Paul Arne Østvær, The Bott inverted infinite projective space is homotopy algebraic 𝐾-theory, Bull. Lond. Math. Soc. 41 (2009), no. 2, 281–292. MR 2496504, 10.1112/blms/bdn124
  • 13. M. Spitzweck, P. A. Østvær.
    A Bott inverted model for equivariant unitary topological $ {K}$-theory.
    To appear in Math. Scand.
  • 14. Vladimir Voevodsky, 𝐀¹-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604 (electronic). MR 1648048

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Additional Information

Oliver Röndigs
Affiliation: Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany
Email: oroendig@math.uni-osnabrueck.de

Markus Spitzweck
Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email: markussp@math.uio.no

Paul Arne Østvær
Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email: paularne@math.uio.no

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10394-3
Received by editor(s): October 13, 2009
Received by editor(s) in revised form: January 19, 2010
Published electronically: May 10, 2010
Communicated by: Brooke Shipley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.