Holmstrom Oriented cohomology

Oriented cohomology

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

category: Computations and Examples

Oriented cohomology

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: http://www.math.uiuc.edu/K-theory/0459. A sequel is in Panin

category: Online References

Oriented cohomology

Panin and Yagunov on Poincare duality

Something interesting by Merkurjev

Panin on Riemann-Roch: http://www.math.uiuc.edu/K-theory/0552

Nenashev on Gysin maps

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

category: Some Research Articles

Oriented cohomology

In characteristic zero, the Chow ring functor is the universal ordinary OCT on $Sm_k$ . 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?

category: Open Problems

Oriented cohomology

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

Definition of transverse morphisms $f: X \to Z$ and $g: Y \to Z$ . Def of additive functor from $Sm_S$ to category of commutative, graded rings with unit: a contravariant functor taking $\emptyset$ to $0$ and disjoint unions to products.

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

An additive functor $V^{op} \to R^*$
For each projective morphism $f: Y \to X$ in $V$ of relative codimension $d$ , a HM of graded $A^*(X)$ -modules $f_*: A^*(Y) \to A^{*+d}(X)$

satisfying the following:

Reasonable behaviour of push-forward wrt identity and compositions.
For a pullback square given by two transverse morphism $f,g$ , with $f$ projective of relative dimension $d$ , one has the commutation relation $g^* f_* = f'_*g'^*$ .
(PB) Let $E \to X$ be a rank $n$ VB over some $X \in V$ . Then $A^*(\mathbb{P}(E))$ is a free $A^*(X)$ -module, with a certain explicit basis.
(EH) Let $E \to X$ be a VB over some $X \in V$ and let $p: U \to X$ be an $E$ -torsor. Then $p^*: A^*(X) \to A^*(U)$ 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 $c_i(E) \in A^i(X), \ i = 0, 1, \ldots n$ , for a rank $n$ vector bundle $E \to X$ . The first Chern class of a tensor product of line bundles is given by a commutative formal group law over $A^*(k)$ .

Example: The Chow ring $X \mapsto CH^*(X)$ is an oriented cohomology theory on $Sm_k$ . We have $CH(k) = \mathbb{Z}$ and the formal group law is the additive FGL.

Example: The Grothendieck group $K_0(X)$ of locally free coherent sheaves is a ring with multiplication induced from tensor product. The functor $X \to K_0(X)[\beta, \beta^{-1}]$ is an oriented cohomology theory. The group law is the multiplicative FGL: $F_m(u, v) = u+v - \beta uv$ .

In characteristic zero, the Chow ring functor is the universal ordinary OCT on $Sm_k$ . 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 $Sm_k$ .

Theorem: Assume $k$ has characteristic zero. Then there exists a universal oriented cohomology theory on $Sm_k$ , denoted by $X \to \Omega^*(X)$ , which we call algebraic cobordism. This universality means what you think it means.

Levine-Morel claims (p. 24) that any oriented bigraded theory $H$ (Bloch-Ogus???) gives an oriented theory, by the formula $X \mapsto \oplus H^{2n, n}(X)$ .

category: Properties

Oriented cohomology

Mona Mocanasu - Northwestern University Title: Push-Forward Maps in Algebraic Oriented Theories Abstract: The existence of a push-forward structure for an algebraic oriented theory on smooth pairs determines its ability to study singular schemes. We describe the needed properties for push-forward maps in a general set-up and discuss the known push-forward structures of the classical theories. Since Chern classes can be constructed from the push-forward maps, this leads to the existence of a Verdier-type theorem for the associated Borel-Moore homology.

category: [Private] Notes