Brauer $p$-dimension and Kato's Swan Conductor

TL;DR

This paper explores the relationship between Kato's Swan conductor and the Brauer p-dimension in characteristic p fields, focusing on henselian discretely valued and semi-global fields, with implications for function fields of algebraic curves.

Contribution

It introduces a new approach using Gersten-type sequences to analyze ramification and provides partial results on Brauer p-dimension for specific function fields.

Findings

Established bounds on Brauer p-dimension for semi-global fields.

Analyzed ramification behavior using Gersten-type sequences.

Provided partial results for function fields with good reduction.

Abstract

We use Kato's Swan conductor to study the Brauer $p$-dimension of fields of characteristic $p>0$. We mainly investigate two types of fields: henselian discretely valued fields and semi-global fields. While investigating the Brauer $p$-dimension of semi-global fields, we use a Gersten-type sequence to analyse the ramification behavior of a Brauer class in a $2$-dimensional regular local ring. Using this result, we give a partial result on the Brauer $p$-dimension of function fields of algebraic curves over $\bar{k}((t))$ with good reduction.