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 P-1-spectrum MGL of Voevodsky is considered as a commutative P-1-ring spectrum. Setting MGL(i) = circle plus(p-2q=i) MGL(p,q) we regard the bigraded theory MGL(p,q) as just a graded theory. There is a unique ring morphism phi: MGL(0)(k) -> Z which sends the class [X](MGL) of a smooth projective k-variety X to the Euler characteristic chi(X, O-X) of the structure sheaf OX. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories phi: MGL* (X, X - Z) circle times(MGL0(k)) Z ->(congruent to) K-* (X on Z) = K-*'(Z) on the category smOp/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. 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].