1 results
Essential Dimension, Symbol Length and
$p$-rank
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 04 February 2020, pp. 882-890
- Print publication:
- December 2020
-
- Article
- Export citation
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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.
