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

TL;DR

This paper explores how Morava K-theories can be used to analyze algebraic groups, revealing new connections between cohomological invariants, motives, and torsion in Chow groups, with implications for algebraic geometry.

Contribution

It introduces novel applications of Morava K-theories to algebraic groups, including detecting triviality of invariants and analyzing torsion in Chow groups.

Findings

Morava K-theory detects triviality of the Rost invariant.

Provides estimates on torsion in Chow groups of quadrics.

Shows splitting criteria for K(n)-motives of varieties.

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 $m\le n$.