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On Voevodsky's Algebraic K-Theory Spectrum

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Algebraic Topology

Part of the book series: Abel Symposia ((ABEL,volume 4))

Abstract

Under a certain normalization assumption we prove that the P1-spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec.(Z). Following an idea of Voevodsky, we equip the P1-spectrum BGL with the structure of a commutative P1-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec.(Z). For an arbitrary Noetherian scheme S of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on BGL. This monoidal structure is relevant for our proof of the motivic Conner–Floyd theorem (Panin et al., Invent Math 175:435–451, 2008). It has also been used to obtain a motivic version of Snaith’s theorem (Gepner and Snaith, arXiv:0712.2817v1 [math.AG]).

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Author information

Authors and Affiliations

  1. Steklov Institute of Mathematics, St. Petersburg, Russia

    Ivan Panin

  2. Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany

    Konstantin Pimenov

  3. Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany

    Oliver Röndigs

Authors
  1. Ivan Panin
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  2. Konstantin Pimenov
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  3. Oliver Röndigs
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Corresponding author

Correspondence to Oliver Röndigs .

Editor information

Editors and Affiliations

  1. Universitet NTNU, Norges Teknisk Naturvitenskap., Trondheim, 7491, Norway

    Nils Baas

  2. College of Letters, Arts & Sciences, University of Southern California, S. Vermont Ave. 3620, Los Angeles, 90089-2532, U.S.A.

    Eric M. Friedlander

  3. University of Oslo, Department of Mathematics, Oslo, 0316, Norway

    Björn Jahren

  4. University of Oslo, Department of Mathematics, Oslo, 0316, Norway

    Paul Arne Østvær

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Panin, I., Pimenov, K., Röndigs, O. (2009). On Voevodsky's Algebraic K-Theory Spectrum. In: Baas, N., Friedlander, E., Jahren, B., Østvær, P. (eds) Algebraic Topology. Abel Symposia, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01200-6_10

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