Morava K-theory and Rost invariant

TL;DR

This paper establishes a precise relationship between Morava K-theory motives, Tits algebras, and Rost invariants for inner forms of Borel subgroup varieties, extending previous results to degree 3 invariants.

Contribution

It generalizes Panin's results by linking second Morava K-theory motives with cohomological invariants of degree 3, specifically Tits algebras and Rost invariants.

Findings

Inner forms of Borel varieties have isomorphic motives in Morava K-theory if Tits algebras and Rost invariants match.

Extends the connection between K-theory and algebraic invariants to degree 3 cohomological invariants.

Provides a criterion for motive isomorphism based on cohomological invariants.

Abstract

We prove that inner forms of a variety of Borel subgroups have isomorphic motives with respect to the second Morava K-theory if and only if the corresponding Tits algebras and Rost invariants coincide. This extends Panin's results on interrelationship of K-theory with Tits algebras to the case of cohomological invariants of degree 3.