Applications of the Morava $K$-theory to algebraic groups

Abstract

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields of characteristic zero based on the Morava $K$-theories, which are generalized oriented cohomology theories in the sense of Levine--Morel. We show that the second Morava $K$-theory detects the triviality of the Rost invariant and, more generally, relate the triviality of cohomological invariants and the splitting of Morava motives. We describe the Morava $K$-theory of generalized Rost motives, compute the Morava $K$-theory of some affine varieties, and characterize the powers of the fundamental ideal of the Witt ring with the help of the Morava $K$-theory. Besides, we obtain new estimates on torsion in Chow groups of codimensions up to $2^n$ of quadrics from the $(n+2)$-nd power of the fundamental ideal of the Witt ring. We compute torsion in Chow groups of $K(n)$-split varieties with respect to a prime $p$ in all codimensions up to $\frac{p^n-1}{p-1}$ and provide a combinatorial tool to estimate torsion up to codimension $p^n$. An important role in the proof is played by the gamma filtration on Morava $K$-theories, which gives a conceptual explanation of the nature of the torsion. Furthermore, we show that under some conditions the $K(n)$-motive of a smooth projective variety splits if and only if its $K(m)$-motive splits for all $m\le n$.