本质维数、符号长度与$p$ -秩
Essential Dimension, Symbol Length and $p$-rank
摘要：
证明了当$k$是完备的时，域$F$上的次数为$p^{\ell m}$且指数为$p$的中心单代数的本质维数至少为$\ell+1$.我们通过观察$F$的$p$ -秩限制了$\operatorname{Br}_{p^m}(F)$中的符号长度，并且存在度为$p^{\ell m}$和指数为$p^m$的不可分解的$p$ -代数来做到这一点。我们还证明了Milne-Kato上同调群$\operatorname H^{n+1}_{p^m}(F)$的符号长度由$\binom rn$从上有界，其中$r$是域的$p$ -秩，并提供了给定符号长度的Brauer类的本质维数的上界和下界。
英文摘要：
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.
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