APPLICATIONS OF THE MORAVA K-THEORY TO ALGEBRAIC GROUPS

Sechin, Pavel und Semenov, Nikita (2021): APPLICATIONS OF THE MORAVA K-THEORY TO ALGEBRAIC GROUPS. In: Annales Scientifiques De L Ecole Normale Superieure, Vol. 54, No. 4: pp. 945-990

Abstract

In this article we discuss an approach to cohomological invariants of algebraic groups based on the Morava K-theories. 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 compute the Morava K-theory of generalized Rost motives and 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 quadrics and investigate torsion in Chow groups of K(n)-split varieties. An important role in the proofs 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 if the K(n)-motive of a smooth projective variety splits, then its K(m)-motive splits for all m <= n.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0012-9593
• Language: English