# Oriented cohomology

• ## General Introduction

Examples include algebraic cobordism, algebraic K-theory, and motivic cohomology.

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• ## Online References

See references under Algebraic cobordism. The main reference is the book by Levine and Morel

Push-forwards in oriented cohomology theories of algebraic varieties, by Ivan Panin and Alexander Smirnov: K0459. A sequel is in Panin

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• ## Properties

(Following Levine-Morel p. 14) Fix a base scheme . Write for separated schemes of finite type over . Write for the full subcat of smooth quasiprojective -schemes. A full subcat of is admissible if it satisfies: 1. It contains and the empty scheme. 2. If and is quasi-projective, then . 3. It is closed under products and disjoint unions.

Definition of transverse morphisms and . Def of additive functor from to category of commutative, graded rings with unit: a contravariant functor taking to and disjoint unions to products.

Let be admissible. An oriented cohomology theory on is given by:

2. For each projective morphism in of relative codimension , a HM of graded -modules

satisfying the following:

1. Reasonable behaviour of push-forward wrt identity and compositions.
2. For a pullback square given by two transverse morphism , with projective of relative dimension , one has the commutation relation .
3. (PB) Let be a rank VB over some . Then is a free -module, with a certain explicit basis.
4. (EH) Let be a VB over some and let be an -torsor. Then is an isomorphism.

The abbreviations for points 3 and 4 stand for Projective Bundle formula and Extended Homotopy property.

Now suppose the base scheme is a field. Can use Grothendieck's method to define Chern classes , for a rank vector bundle . The first Chern class of a tensor product of line bundles is given by a commutative formal group law over .

Example: The Chow ring is an oriented cohomology theory on . We have and the formal group law is the additive FGL.

Example: The Grothendieck group of locally free coherent sheaves is a ring with multiplication induced from tensor product. The functor is an oriented cohomology theory. The group law is the multiplicative FGL: .

In characteristic zero, the Chow ring functor is the universal ordinary OCT on . A rational analogue holds over any field. Examples of ordinary cohomology theories: l-adic cohomology, de Rham cohomology over a field of char zero, the even part of Betti cohomology associated to a complex embedding of the base field. In some sense the universality of the Chow ring explains the cycle class map in all these theories.

Over any field, the K-group functor described above is the universal multiplicative and periodic OCT on .

Theorem: Assume has characteristic zero. Then there exists a universal oriented cohomology theory on , denoted by , which we call algebraic cobordism. This universality means what you think it means.

Levine-Morel claims (p. 24) that any oriented bigraded theory (Bloch-Ogus???) gives an oriented theory, by the formula .

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• ## Open Problems

In characteristic zero, the Chow ring functor is the universal ordinary OCT on . Conjecture: This holds over any field.

Would it make sense to define algebraic cobordism over a field of char zero? Is resolution of singularities the problem?

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• ## Computations and Examples

Oriented Cohomology and Motivic Decompositions of Relative Cellular Spaces , by Alexander Nenashev and Kirill Zainoulline

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• ## Some Research Articles

Panin and Yagunov on PoincarĂ© duality

Something interesting by Merkurjev

Panin on Riemann-Roch: K0552

Nenashev on Gysin maps

Borel-Moore Functors and Algebraic Oriented Theories , by Mona Mocanasu

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