2018/05/23 by Sechin, Pavel, Semenov, Nikita
#11E72 #19E15 #20G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1805.09059
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 2n 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 (pn-1)/(p-1) and provide a combinatorial tool to estimate torsion up to codimension pn. 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≤ n.