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*(XonZ) 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].