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

Abstract

We study non-additive operations from algebraic Morava K-theories to oriented cohomology theories in algebraic geometry. For oriented cohomology theory $A$ that has a {$p^n$}-typical formal group law over a $\mathbb{Z}_{(p)}$-algebra we construct `Chern classes' from the algebraic $n$-th Morava K-theory with $p$-local coefficients to $A$. If the coefficient ring of $A$ is a free $\mathbb{Z}_{(p)}$-module we also prove that these Chern classes freely generate all operations from $\mathrm{K}(n)$ to $A$. Examples of such theories are algebraic Morava K-theories $\mathrm{K}(nm)^*$ for all $m\in\mathbb{N}$ and Chow groups with $p$-local coefficients. The universal $p^n$-typical oriented theory is $BP\{n\}^*$ whose coefficient ring is also a free $\mathbb{Z}_{(p)}$-module. Chern classes from the $n$-th algebraic Morava K-theory $\mathrm{K}(n)$ to itself allow us to introduce the gamma filtration on $\mathrm{K}(n)$. 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 $\mathrm{K}_0$. The major difference from the classical case is that Chern classes from the graded factors $gr^i_\gamma \mathrm{K}(n)^*$ to Chow groups with p-local coefficients are surjective for $i\le p^n$, which allows to estimate $p$-torsion in Chow groups of codimension up to $p^n$ of some varieties.