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

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