Essential Dimension, Symbol Length and $p$-rank

TL;DR

This paper investigates the essential dimension of certain central simple algebras and cohomology groups over fields of characteristic p, establishing bounds based on the p-rank and symbol length, with implications for algebra classification.

Contribution

It provides new lower bounds for the essential dimension of specific central simple algebras and upper bounds for symbol length in Milne-Kato cohomology, linking these to the p-rank of the field.

Findings

Essential dimension of certain algebras is at least +1.

Symbol length in cohomology is bounded by inom rn.

Bounds for essential dimension related to symbol length are established.

Abstract

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^m$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell+1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\operatorname{Br}_{p^m}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^m$. We also prove that the symbol length of the Milne-Kato cohomology group $\operatorname H^{n+1}_{p^m}(F)$ is bounded from above by $\binom rn$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.