Oriented cohomology
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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:
- An additive functor
- For each projective morphism
in
of relative codimension
, a HM of graded
-modules
satisfying the following:
- Reasonable behaviour of push-forward 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: 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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