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Essential dimension of cohomology classes via valuation theory

Danny Ofek, Zinovy Reichstein

Abstract

We give a formula for the essential dimension of a cohomology class $α$ in $H^d(K, \mathbb{Q}_p/\mathbb{Z}_p (d))$ when $K$ is a strictly Henselian field. This formula is particularly explicit in the case, where $α$ is a Brauer class (for $d = 2$). As an application of our bound with $d = 3$, we study the essential dimension of exceptional groups by examining the image of the Rost invariant.

Essential dimension of cohomology classes via valuation theory

Abstract

We give a formula for the essential dimension of a cohomology class in when is a strictly Henselian field. This formula is particularly explicit in the case, where is a Brauer class (for ). As an application of our bound with , we study the essential dimension of exceptional groups by examining the image of the Rost invariant.
Paper Structure (18 sections, 26 theorems, 128 equations)

This paper contains 18 sections, 26 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Essential dimension
  4. The norm residue isomorphism
  5. Notational conventions
  6. The homomorphism nu
  7. First properties of $\wedge \nu$
  8. A lower bound on essential dimension
  9. Computing rho
  10. Proof of Theorem \ref{['thm.main']} and Corollary \ref{['cor.infty']}
  11. Proof of Theorem \ref{['thm.brauer']}
  12. Some limitations of Theorem \ref{['thm.brauer']}
  13. A further consequence of Theorem \ref{['thm.main']}
  14. Anisotropic torsors
  15. Proof of Theorem \ref{['thm.rost']} and a remark on spin groups
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(F,\nu)$ be a valued field with value group $\mathbb{Z}^r$. Assume $\nu$ is trivial on a subfield $k\subset F$ with $\operatorname{char} k \neq p$. Let $\alpha \in H^d_p(F)$ and $\omega = \wedge\nu(\alpha)$. Then Moreover, if $(F,\nu)$ is strictly Henselian, then equality holds:

Theorems & Definitions (56)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • ...and 46 more