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Inventiones mathematicae

, Volume 175, Issue 2, pp 435–451 | Cite as

On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory

  • Ivan Panin
  • Konstantin Pimenov
  • Oliver Röndigs
Article

Abstract

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commutative P1-ring spectrum. Setting \(\mathrm{MGL}^i = \bigoplus_{p-2q =i}\mathrm{MGL}^{p,q}\) we regard the bigraded theory MGL p,q as just a graded theory. There is a unique ring morphism \(\phi\colon\mathrm{MGL}^0(k)\to\mathbb{Z}\) which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic \(\chi(X, \mathcal{O}_X)\) of the structure sheaf \(\mathcal{O}_X\). Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories
$$\varphi\colon\mathrm{MGL}^{\ast}(X,X-Z) \otimes_{\mathrm{MGL}^{0}(k)} \mathbb{Z} \xrightarrow{\cong} \mathrm{K}_{- *}(X \ on \ Z)= \mathrm{K}^{\prime}_{- *}(Z) $$
on the category \(\mathcal{S}m\mathcal{O}p/S\) in the sense of [6], where K*(X on Z) is Thomason–Trobaugh K-theory and K * is Quillen’s K-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented in the sense of [6] and ϕ respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism [1].

Keywords

Vector Bundle Chern Class Cohomology Theory Homotopy Category Monoid Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Ivan Panin
    • 1
    • 2
  • Konstantin Pimenov
    • 2
  • Oliver Röndigs
    • 3
  1. 1.Universität Bielefeld, SFB 701BielefeldGermany
  2. 2.Steklov Institute of MathematicsSt. PetersburgRussia
  3. 3.Institut für MathematikUniversität OsnabrückOsnabrückGermany

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