Oriented cohomology

• General Introduction

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

  See also Orientable cohomology, Oriented homology, Oriented Borel-Moore homology, Algebraic cobordism

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• Search results

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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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• Paper References

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• Definition

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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:

  1. An additive functor
  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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• Standard theorems

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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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• Connections to Number Theory

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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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• History and Applications

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• Some Research Articles

  Panin and Yagunov on Poincaré 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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• Other Information

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• General Introduction
• Search results
• Online References
• Paper References
• Definition
• Properties
• Standard theorems
• Open Problems
• Connections to Number Theory
• Computations and Examples
• History and Applications
• Some Research Articles
• Other Information
• Comments Posted
• Comments