Back to Search
Start Over
On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory
- Source :
- Inventiones mathematicae. 175:435-451
- Publication Year :
- 2008
- Publisher :
- Springer Science and Business Media LLC, 2008.
-
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.<br />LaTeX, 18 pages, uses XY-pic
- Subjects :
- Pure mathematics
Ring (mathematics)
Algebraic cobordism
General Mathematics
14F05
K-Theory and Homology (math.KT)
K-theory
Mathematics::Algebraic Topology
Cohomology
Ground field
Mathematics - Algebraic Geometry
symbols.namesake
55P43
Mathematics::K-Theory and Homology
Mathematics::Category Theory
Euler characteristic
Mathematics - K-Theory and Homology
55N22
FOS: Mathematics
symbols
Sheaf
Complex cobordism
Algebraic Geometry (math.AG)
Mathematics
Subjects
Details
- ISSN :
- 14321297 and 00209910
- Volume :
- 175
- Database :
- OpenAIRE
- Journal :
- Inventiones mathematicae
- Accession number :
- edsair.doi.dedup.....39fa538e780f408d491d8b696e72c9f0