Brauer $p$-dimension of complete discretely valued fields

Abstract:

Let $K$ be a complete discretely valued field of characteristic 0 with residue field $\kappa$ of characteristic $p$. Let $n$ be the $p$-rank of $\kappa$, i.e., $p^n=[\kappa :\kappa ^p]$. It was proved by Parimala and Suresh [Invent. Math. 197 (2014), no. 1, pp. 215–235] that the Brauer $p$-dimension of $K$ lies between $n/2$ and $2n$. For $n\leq 3$, we improve the upper bound to $n+1$ and provide examples to show that our bound is sharp. For $n \leq 2$, we also improve the lower bound to $n$. For general $n$, we construct a family of fields $K_n$ with residue fields of $p$-rank $n$ such that $K_n$ admits a central simple algebra $D_n$ of period $p$ and index $p^{n+1}$. Our sharp lower bounds for $n\leq 2$ and upper bounds for $n\leq 3$ in combination with the nature of these examples motivate us to conjecture that the Brauer $p$-dimension of such fields always lies between $n$ and $n+1$.