# Contents

## Idea

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

## Definition

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

… formula here …

## Properties

Let $S = Spec(k)$ where $k$ is a field of characteristic zero. A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by Marc Levine and Fabien Morel. More precisely, Levine-Morel constructed the universal oriented cohomology theory $\Omega^* : \Sm_k \to CRing^*$. 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

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

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

###### Theorem

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

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

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

###### Theorem

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

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

is invertible.

###### Theorem

(Levine 2008). The canonical homomorphisms of graded rings

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

are invertible for all $X \in \Sm_k$.

## 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 $MGL^{*,*}$ is considered in

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

The comparison with $MGL^{2*,*}$ 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 $\mathbb{A}^1$-homotopy and algebraic h-cobordisms, arXiv:0810.0324
Revised on January 23, 2015 11:54:07 by Adeel Khan (217.187.67.57)