{
  "id": "1805.09059",
  "version": "v1",
  "published": "2018-05-23T11:16:13.000Z",
  "updated": "2018-05-23T11:16:13.000Z",
  "title": "Applications of the Morava $K$-theory to algebraic groups",
  "authors": [
    "Pavel Sechin",
    "Nikita Semenov"
  ],
  "comment": "36 pages",
  "categories": [
    "math.AG",
    "math.KT"
  ],
  "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 $m\\le n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-23T11:16:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G15",
      "11E72",
      "19E15"
    ],
    "keywords": [
      "algebraic groups",
      "applications",
      "chow groups",
      "fundamental ideal",
      "codimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}