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

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 MGLp,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].

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References

  1. Conner, P.E., Floyd, E.E.: The Relation of Cobordism to K-Theories. Lect. Notes Math., vol. 28. Springer, Berlin (1966)

    Google Scholar 

  2. Levine, M., Morel, F.: Algebraic Cobordism. Springer Monograph Series in Mathematics. Springer, Berlin (2007)

  3. Morel, F., Voevodsky, V.: A1-homotopy theory of schemes. Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999)

    MATH  MathSciNet  Article  Google Scholar 

  4. Panin, I., Pimenov, K., Röndigs, O.: On Voevodsky’s algebraic K-theory spectrum. In: The Abel Symposium 2007. Proceedings of the fourth Abel Symposium, Oslo, Norway, August 5–10, 2007. Springer, Berlin. Preprint available via arXiv:0709.3905v2 [math.AG]

  5. Panin, I., Pimenov, K., Röndigs, O.: A universality theorem for Voevodsky’s algebraic cobordism spectrum MGL. To appear in Homology Homotopy Appl. 10(2), (2008), 16 pp. Preprint available via arXiv:0709.4116v1 [math.AG]

  6. Panin, I.: Oriented cohomology theories on algebraic varieties. (After I. Panin and A. Smirnov) K-Theory, 30(3), 265–314 (2003). Special issue in honor of Hyman Bass on his seventieth birthday. Part III

  7. Panin, I.: Riemann–Roch theorems for oriented cohomology. (After I. Panin and A. Smirnov) In: Greenless, J.P.C. (ed.) Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, pp. 261–333. Kluwer Acad. Publ., Dordrecht (2004)

  8. Panin, I.: Push-forwards in oriented cohomology theories of algebraic varieties: II. (After I. Panin and A. Smirnov) Preprint, 83 pp. (2003). Available via http://www.math.uiuc.edu/K-theory/0619/

  9. Panin, I., Yagunov, S.: Rigidity for orientable functors. J. Pure Appl. Algebra 172, 49–77 (2002)

    MATH  MathSciNet  Article  Google Scholar 

  10. Thomason, R., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. In: The Grothendieck Festschrift, 3, pp. 247–436. Birkhäuser, Boston (1990)

    Chapter  Google Scholar 

  11. Vezzosi, G.: Brown–Peterson spectra in stable A1-homotopy theory. Rend. Semin. Mat. Univ. Padova 106, 47–64 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Voevodsky, V.: A1-homotopy theory. Doc. Math., Extra Vol. ICM 1998(I), 579–604 (1998)

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Correspondence to Oliver Röndigs.

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Panin, I., Pimenov, K. & Röndigs, O. On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory. Invent. math. 175, 435–451 (2009). https://doi.org/10.1007/s00222-008-0155-5

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  • DOI: https://doi.org/10.1007/s00222-008-0155-5

Keywords

  • Vector Bundle
  • Chern Class
  • Cohomology Theory
  • Homotopy Category
  • Monoid Homomorphism