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ESSENTIAL DIMENSION, SYMBOL LENGTH AND -RANK
Canadian Mathematical Bulletin ( IF 0.6 ) Pub Date : 2020-02-04 , DOI: 10.4153/s0008439520000119
Adam Chapman , Kelly McKinnie

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.

中文翻译:

基本尺寸、符号长度和等级

我们证明,在包含特征 $p$ 的基域 $k$ 的域 $F$ 上,阶 $p^{\ell m}$ 和指数 $p^m$ 的中心简单代数的本质维数至少为 $ \ell+1$ 当 $k$ 是完美的。我们通过观察 $F$ 的 $p$-rank 限制了 $\operatorname{Br}_{p^m}(F)$ 中的符号长度并且存在不可分解的 $p$-阶代数 $ p^{\ell m}$ 和指数 $p^m$。我们还证明了 Milne-Kato 上同调群 $\operatorname H^{n+1}_{p^m}(F)$ 的符号长度以 $\binom rn$ 为界,其中 $r$ 是$p$-rank 字段,并为给定符号长度的 Brauer 类的基本维度提供上限和下限。
更新日期:2020-02-04
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