algebraic cobordism



Algebraic cobordism is the bigraded generalized cohomology theory represented by the motivic Thom spectrum MGLMGL. Hence it is the algebraic or motivic analogue of complex cobordism. The (2n,n)(2n,n)-graded part has a geometric description via cobordism classes, at least over fields of characteristic zero.


Let SS be a scheme and MGL SMGL_S the motivic Thom spectrum over SS. Algebraic cobordism is the generalized motivic cohomology theory? MGL S *,*MGL_S^{*,*} represented by MGL SMGL_S:

… formula here …


The (2n,n)-graded part

Let S=Spec(k)S = Spec(k) where kk is a field of characteristic zero. A geometric description of the (2n,n)(2n,n)-graded part of algebraic cobordism was given by Marc Levine and Fabien Morel. More precisely, Levine-Morel constructed the universal oriented cohomology theory Ω *:Sm kCRing *\Omega^* : \Sm_k \to CRing^*. Here oriented signifies the existence of direct image or Gysin homomorphisms for proper morphisms of schemes. This implies the existence ofChern classes for vector bundles.


(Levine-Morel). There is a canonical isomorphism of graded rings

L *Ω *(Spec(k)) \mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k))

where L *\mathbf{L}^* denotes the Lazard ring with an appropriate grading.


(Levine-Morel). Let i:ZXi : Z \hookrightarrow X be a closed immersion of smooth kk-schemes and j:UXj : U \hookrightarrow X the complementary open immersion. There is a canonical exact sequence of graded abelian groups

Ω *d(Z)i *Ω *(X)j *Ω *(U)0, \Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0,

where d=codim(Z,X)d = \codim(Z, X).


(Levine-Morel). Given an embedding kCk \hookrightarrow \mathbf{C}, the canonical homomorphism of graded rings

Ω *(k)MU 2*(pt) \Omega^*(k) \longrightarrow MU^{2*}(pt)

is invertible.


(Levine 2008). The canonical homomorphisms of graded rings

Ω *(X)MGL 2*,*(X) \Omega^*(X) \longrightarrow MGL^{2*,*}(X)

are invertible for all XSm kX \in \Sm_k.


A discussion on MathOverflow:

  • Interdependence between A^1-homotopy theory and algebraic cobordism, MO/36659.

The oriented cohomology theory

A simpler construction was given in

  • M. Levine, R. Pandharipande, Algebraic cobordism revisited, math.AG/0605196

A Borel-Moore homology? version of MGL *,*MGL^{*,*} is considered in

  • Marc Levine, Oriented cohomology, Borel-Moore homology and algebraic cobordism, arXiv.

The comparison with MGL 2*,*MGL^{2*,*} is in

  • Marc Levine, Comparison of cobordism theories, Journal of Algebra, 322(9), 3291-3317, 2009, arXiv.

The construction was extended to derived schemes in the paper

The close connection of algebraic cobordism with K-theory is discussed in

  • José Luis González, Kalle Karu. Universality of K-theory. 2013. arXiv:1301.3815.

An algebraic analogue of h-cobordism:

  • Aravind Asok, Fabien Morel, Smooth varieties up to 𝔸 1\mathbb{A}^1-homotopy and algebraic h-cobordisms, arXiv:0810.0324
Revised on January 23, 2015 11:54:07 by Adeel Khan (