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algebraic cobordism

Contents

Idea

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.

Definition

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 …

Properties

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.

Theorem

(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.

Theorem

(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).

Theorem

(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.

Theorem

(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.

References

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 (217.187.67.57)