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

TL;DR

This paper investigates the Brauer p-dimension and absolute Brauer p-dimension of Henselian valued fields, providing conditions and characterizations for these invariants based on residue field properties and Galois group structures.

Contribution

It characterizes the possible sequences of Brauer p-dimensions and absolute Brauer p-dimensions for Henselian fields with specific residue field conditions, extending understanding in valuation theory.

Findings

Determines Brd_p(K) and abrd_p(K) for Henselian fields with residue field restrictions.

Describes the set of possible sequences of abrd_p(E) and Brd_p(E) for certain classes of Henselian fields.

Provides analogous results in positive characteristic cases.

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 $q > 0$.