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Essential Dimension of Central Simple Algebras when the Characteristic is Bad

Adam Chapman, Kelly McKinnie

TL;DR

This survey addresses the essential dimension of central simple algebras in bad characteristic, where the field characteristic $p$ divides the algebra degree. It synthesizes foundational definitions, cohomological frameworks (notably Kato–Milne and Izhboldin groups), and a broad set of lower and upper bounds for $ ext{Alg}_{d,e}$, including sharp results for degrees $4$ and $8$ and for sequences of linked cyclic algebras. A key contribution is an improved upper bound $ ext{ed}( ext{Alg}_{8,2})\ ext{≤ }8$ in characteristic $2$, derived via descent arguments that avoid prior reliance on $ ext{ed}_p((oldsymbol{Z}/poldsymbol{Z})^{ imes n})$. The paper also ties essential dimension to linear algebraic groups, via $ ext{ed}( ext{Alg}_{n,n})= ext{ed}( ext{PGL}_n)$ and related equalities, and discusses the Dec$_{p^m,n}$ and LCA functors, highlighting both established results and open questions in the bad-characteristic regime.

Abstract

This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose characteristic divides the degree of the algebras under discussion. Upper and lower bounds as well as a few precise evaluations of this dimension are included.

Essential Dimension of Central Simple Algebras when the Characteristic is Bad

TL;DR

This survey addresses the essential dimension of central simple algebras in bad characteristic, where the field characteristic divides the algebra degree. It synthesizes foundational definitions, cohomological frameworks (notably Kato–Milne and Izhboldin groups), and a broad set of lower and upper bounds for , including sharp results for degrees and and for sequences of linked cyclic algebras. A key contribution is an improved upper bound in characteristic , derived via descent arguments that avoid prior reliance on . The paper also ties essential dimension to linear algebraic groups, via and related equalities, and discusses the Dec and LCA functors, highlighting both established results and open questions in the bad-characteristic regime.

Abstract

This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose characteristic divides the degree of the algebras under discussion. Upper and lower bounds as well as a few precise evaluations of this dimension are included.

Paper Structure

This paper contains 11 sections, 8 theorems, 16 equations, 1 table.

Table of Contents

  1. Introduction
  2. Central Simple Algebras
  3. Finitely Generated Fields
  4. Algebras of Degree 4
  5. Algebras of Degree 8 and Exponent 2
  6. Sequences of Linked Cyclic Algebras
  7. The $\operatorname{Dec}_{p^m,n}$ functor
  8. Cohomology Groups
  9. Algebras of prime degree
  10. Algebras of Prime Exponent
  11. Linear Algebraic Groups

Key Result

Lemma 5.1

Suppose $F$ is a field of $\operatorname{char}(F)=2$, $K=F[i : i^2+i=\alpha]$ a separable quadratic field extension of $F$ and $Q=[a,bi+c)_{2,K}$ a quaternion algebra over $K$ with trivial corestriction to $F$, where $a,b,c \in F$. Then $Q$ descends to $F$.

Theorems & Definitions (17)

  • Lemma 5.1: cf. Baek:2011
  • proof
  • Theorem 5.2
  • proof
  • Corollary 6.1
  • proof
  • Proposition 6.3
  • proof
  • Theorem 7.1
  • proof
  • ...and 7 more