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

TL;DR

This paper establishes a canonical isomorphism between Voevodsky's algebraic cobordism and Quillen's K-theory, showing how algebraic cobordism can reconstruct algebraic K-theory in a manner analogous to complex cobordism's relation to complex K-theory.

Contribution

It proves a canonical isomorphism between algebraic cobordism and Quillen's K-theory, extending the analogy with complex cobordism and K-theory to the algebraic setting.

Findings

Algebraic cobordism spectrum MGL is used to reconstruct K-theory.

There exists a canonical isomorphism of cohomology theories between MGL and K-theory.

Both theories are oriented and the isomorphism respects orientations.

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. There is a unique ring morphism MGL^{2*,*}(k)--> Z which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic of the structure sheaf of X. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories MGL^{*,*}(X,U) \tensor_{MGL^{2*,*}(k)} Z --> K^{TT}_{- *}(X,U) = K'_{- *}(X-U)} on the category of smooth k-varieties, where K^{TT}_* 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 and the isomorphism above 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.