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On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory
| Venue: | Invent. Math |
| Citations: | 7 - 3 self |
BibTeX
@TECHREPORT{Panin_onthe,
author = {I. Panin and K. Pimenov and O. Röndigs},
title = {On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory},
institution = {Invent. Math},
year = {}
}
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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 = ⊕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 φ: MGL 0 (k) → Z which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic χ(X,OX) of the structure sheaf OX. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories
Keyphrases
voevodsky algebraic cobordism smooth projective k-variety commutative 1-ring spectrum algebraic cobordism euler characteristic graded theory algebraic k-theory class mgl unique ring morphism cohomology theory ground field structure sheaf ox bigraded theory mgl main result state algebraic cobordism 1-spectrum mgl canonical grade
