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Rost's degree formula
In algebraic geometry, the Rost degree formula, which was conjectured by V.Voevodsky and in generalized versions by M. Rost, relates special characteristic numbers of two smooth projective varities, between which a rational morphism exists.

Generalizations
Generalized degree formula was proved by M.Levine and F.Morel in the course of their work on algebraic cobordism.

Generalized degree formula
In algebraic geometry, the generalized degree formula, proved by M.Levine and F.Morel, expresses any element of the group of algebraic cobordisms in terms of it's degree (as an element of the Lazard ring) and positively graded cobordisms.

Statement
First, we need to introduce the notion of degree of a morphism which is a generalization of the degree of a finite morphism.

Let A be an oriented cohomology theory (in algebro-geometric sense) and let it satisfy the generically constant property (saying that the natural morphism $$ p^*_K:A(k)\rightarrow A(K) $$ is isomorphism for any field extension K/k).

Then for any irreducible finite type k-scheme X define the degree map $$\deg: A(X) \rightarrow A(k) $$ to be
 * $$\deg = (p^*_{k(X)})^{-1}\circ i^*_{k(X)},$$

where $$ i_{k(X)}: \mathrm{Spec } k(X) \rightarrow X $$ is the generic point of X.

Now let X be a finite type scheme over a field k of characteristic zero. For simplicity assume it to be irreducible, and denote by $$\tilde{X}\rightarrow X$$ its resolution of singularities.

Let $$A_*$$ be an oriented Borel-Moore weak homology theory satisfying the generically constant and the localization properties. Let $$\alpha\in A_*(X)$$ be any element.

Then
 * $$\alpha - \deg(\alpha)\cdot [\tilde{X}\rightarrow X] = \sum_i v_i [Z_i \rightarrow X], $$

where $$v_i$$ are alements of $$A_*(k)$$ and $$Z_i$$ are smooth schemes of positive codimension in X.

Applications
Let's deduce Rost's degree formula from it. Denote by S_R any homological Steenrod operation. And recall that it exists on the level of algebraic cobordisms (and commutes with the projection to Chow groups).

Consider the identity element of the algebraic cobordisms on X. Its pushforward to the point gives the element [X] of the Lazard ring, which projects to zero in Chow groups. On the other hand if apply the Steenrod operation and then push it forward to the point we should also get 0 in Chow groups modulo p (where the Steenrod operation is defined). If the degree of the operation is equal to the dimension of X, S_R([X])=f_*(S_R(1_X)) is a characteristic number of X, and we get that it is divisible by p. Denote by s_R(X)= S_R(X)/p.

Now let f be a morphism between two smooth projective varieties of the same dimension. Generalized projective formula applied to f says that [f] = (deg f)1_X + \sum u_i v_i, where deg f is a number, a degree of the map, and u_i are negatively graded elements of the Lazard ring, v_i are positively graded cobordisms on X. Do the same thing as above. Applying any Steenrod operation to this equality and then pushing forward to the point, we get that S_R(Y) = (deg f)S_R(X) + ..., and we now explain that ... is divided by pn_X, where n_X is the least common divisor of degrees of the points on X. Dividing by p, we get the Rost's degree formula.

Oriented cohomology theory (algebraic geometry)
In algebraic geometry, oriented cohomology theories are an analogue of complex oriented cohomology theories. The axioms are not though fixed, and thus one might speak of oriented cohomology theories in different sense. Though there is a list of axioms which allows for the universal oriented theory of algebraic coboridms.

Thom classes, Chern classes and pushforwards
One might think of an orientation on a cohomology theory as given by either the structure of Thom classes or Chern classes for vector bundles or the structure of pushforward for a projective morphisms. I. Panin and A. Smirnov showed that under reasonable assumptions these are the same.

Here is a list of axioms for different structure of orientation and the connection between them.

History
They appeared in a preprint by I. Panin and A. Smirnov, which explained how different notions of orientation fit together and where the generalization of the Riemann-Roch theorem to this setting was proved. M. Levine started constructing the universal oriented theory, and finished this work with F. Morel. This theory is called algebraic cobordism. A. Vishik has developed framework, allowing to work with what he called 'theories of rational type', and what appeared to be Levine-Morel's free theories, i.e. factors of algebraic cobordisms.