Oriented cohomology

General Introduction
Examples include algebraic cobordism, algebraic Ktheory, and motivic cohomology.
See also Orientable cohomology, Oriented homology, Oriented BorelMoore 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
Pushforwards 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 LevineMorel 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 quasiprojective, 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:
 An additive functor
 For each projective morphism in of relative codimension , a HM of graded modules
satisfying the following:
 Reasonable behaviour of pushforward wrt identity and compositions.
 For a pullback square given by two transverse morphism , with projective of relative dimension , one has the commutation relation .
 (PB) Let be a rank VB over some . Then is a free module, with a certain explicit basis.
 (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: ladic 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 Kgroup 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.
LevineMorel claims (p. 24) that any oriented bigraded theory (BlochOgus???) 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 PoincarĂ© duality
Something interesting by Merkurjev
Panin on RiemannRoch: K0552
Nenashev on Gysin maps
BorelMoore Functors and Algebraic Oriented Theories , by Mona Mocanasu
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Other Information
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