Essential Dimension, Symbol Length and $p$-rank

Chapman, Adam ; McKinnie, Kelly

Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 882-890

Résumé

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 $\text{Br}_{p^{m}}(F)$ 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 Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ 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.

Détail

DOI : 10.4153/S0008439520000119

Mots-clés : essential dimension, symbol length, p-rank, fields of positive characteristic, Brauer group, central simple algebra, Kato–Milne cohomology