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 $\mathrm {K}(n)_{int}^*$ to $A$.

Examples of such theories are algebraic Morava K-theories $\mathrm {K}(nm)_{int}^*$ 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 $\mathrm {K}(n)_{int}^*$ to itself allow us to introduce the gamma filtration on $\mathrm {K}(n)_{int}^*$. 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 $gr^i_\gamma \mathrm {K}(n)_{int}^*$ 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.