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

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

2007-09-26

arxiv.org/abs/0709.4124v1

Mathematics

Algebraic Geometry

LaTeX, 18 pages, uses XY-pic

Scientific paper

10.1007/s00222-008-0155-5

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.

Affiliated with

Also associated with

No associations

Rating

On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-694738