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Morava K-theory of orthogonal groups and motives of projective quadrics

Nikita Geldhauser, Andrei Lavrenov, Victor Petrov, Pavel Sechin

Abstract

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.

Morava K-theory of orthogonal groups and motives of projective quadrics

Abstract

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.

Paper Structure

This paper contains 39 sections, 39 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Overview of Morava K-theory
  3. Overview of motives
  4. Overview of results
  5. Morava K-theory of orthogonal groups
  6. Applications to motives
  7. Acknowledgement
  8. Background on oriented cohomology theories
  9. Oriented cohomology theories
  10. Topological filtration
  11. Morava K-theory
  12. Cellular varieties
  13. The Rost nilpotence principle
  14. The Krull--Schmidt Theorem
  15. Background on algebraic groups
  16. ...and 24 more sections

Key Result

Theorem 1.1

As an $\mathbb{F}_2[v_n^{\pm 1}]$-algebra the Morava K-theory of split orthogonal groups can be described explicitly as follows: where $r'=\mathrm{min}\left(2^{n-1},\,\left\lfloor\frac{m+1}{4}\right\rfloor\right)$ and

Theorems & Definitions (74)

  • Theorem 1.1: Theorem \ref{['answer']}
  • Theorem 1.2: Theorem \ref{['result']}
  • Remark
  • Proposition 2.1
  • Theorem : Calmès--Petrov--Zainoulline
  • Remark
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 64 more