On Brauer $p$-dimensions and absolute Brauer $p$-dimensions of Henselian fields

Abstract

This paper determines the Brauer $p$-dimension Brd$_{p}(K)$ and the absolute Brauer $p$-dimension abrd$_{p}(K)$ of a Henselian valued field $(K, v)$, for a prime $p \neq {\rm char}(\widehat K)$, under restrictions on the residue field $\widehat K$, such as the condition abrd$_{p}(\widehat K) = 0$. It describes the set $\Sigma_{0}$ of sequences ${\rm abrd}_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, where $\mathbb P$ is the set of prime numbers and $E$ runs across the class of Henselian fields with char$(\widehat E) = 0$ and a projective absolute Galois group $\mathcal{G}_{\widehat E}$. Specifically, $\Sigma_{0}$ contains a sequence $a_{p}, b_{p} \in \mathbb N \cup \{0, \infty \}$, $p \in \mathbb P$, whenever $a_{2} \le 2b_{2}$ and $a_{p} \ge b_{p}$, for each $p$. Similar results are obtained in characteristic $q > 0$.