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Regular version of the site

Working paper

Chern classes from Morava K-theories to p^n-typical oriented theories

Sechin P.
Generalizing the definition of Cartier, we introduce pn-typical formal group laws over ℤ(p)-algebras. An oriented cohomology theory in the sense of Levin-Morel is called pn-typical if its corresponding formal group law is p^n-typical. The main result of the paper is the construction of 'Chern classes' from the algebraic n-th Morava K-theory to every p^n-typical oriented cohomology theory.  If the coefficient ring of a p^n-typical theory is a free ℤ(p)-module we also prove that these Chern classes freely generate all operations to it. Examples of such theories are algebraic mn-th Morava K-theories K(nm)∗ for all m∈ℕ and CH∗⊗ℤ(p) (operations to Chow groups were studied in a previous paper). The universal pn-typical oriented theory is BP{n}∗=BP∗/(v_j,j∤n) which coefficient ring is also a free ℤ_(p)-module.  Chern classes from the n-th algebraic Morava K-theory K(n)∗ to itself allow us to introduce the gamma filtration on K(n)∗. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on K0. The major difference from the classical case is that Chern classes from the graded factors gr^i_γ K(n)^∗ to CH^i⊗ℤ_(p) are surjective for i≤p^n. For some projective homogeneous varieties this allows to estimate p-torsion in Chow groups of codimension up to pn.