## Abstract

Quillen’s algebraic *K*-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field *k* the algebraic cobordism P^{1}-spectrum MGL of Voevodsky is considered as a commutative P^{1}-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

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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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