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

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 $p\mathrm{\ne }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(\widehat{K})$, under restrictions on the residue field $\widehat{K}$, such as the condition abrd${}_p(\widehat{K})=0$. It describes the set ${\mathrm{\Sigma }}_0$ of sequences ${\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{d}}_p(E),{\mathrm{B}\mathrm{r}\mathrm{d}}_p(E)$, $p \in \mathbb P$,where$\mathbb P$isthesetofprimenumbersand$E$ runs across the class of Henselian fields with char$(\widehat{E})=0$ and a projective absolute Galois group ${\mathcal{G}}_{\widehat{E}}$. Specifically, ${\mathrm{\Sigma }}_0$ contains a sequence $a_p,b_p\in \mathbb{N}\cup \{0,\mathrm{\infty }\}$, $p \in \mathbb P$,whenever$a_{2} \le 2b_{2}$and$a_{p} \ge b_{p}$,foreach$p$. Similar results are obtained in characteristic $q>0$.