Pavel Sechin, Morava K-theory pure motives with applications to quadrics

Morava K-theories K(n) are cohomology theories that have graded fields of positive characteristic as coefficient rings and that are obtained from algebraic cobordism of Levine-Morel by change of coefficients. Pure motives with respect to K(n) fit in-between Chow motives and K0-motives (with p-localized or p-torsion coefficients), e.g. allowing to transfer K(n)-decompositions to K(m)-decompositions whenever m is strictly less than n. Thus, it might be a reasonable approach in the study of motivic decompositions to start with K(1)-motives (i.e. more or less K0-motives)and continue to K(2)-, K(3)-motives and so on, eventually arriving to Chow-motives. On the other hand we formulate a conjectural principle that connects the splitting of K(n)-motive with the triviality of cohomological invariants of degrees less than n 1. I plan to outline the proof of this principle for quadrics and explain its consequences for Chow groups of quadrics lying in powers of the fundamental ideal in the Witt ring. The talk is mostl