Vanishing theorems and Brauer-Hasse-Noether exact sequences for the cohomology of higher-dimensional fields

TL;DR

This paper proves vanishing theorems and establishes Brauer-Hasse-Noether type exact sequences for the Galois cohomology of higher-dimensional local fields over finite, p-adic, or number fields, advancing understanding of their arithmetic properties.

Contribution

It introduces new vanishing theorems and exact sequences for the cohomology of higher-dimensional fields, extending classical results to more complex field structures.

Findings

Vanishing theorems for Galois cohomology groups over higher-dimensional fields.

Exact sequences analogous to Brauer-Hasse-Noether for these fields.

Identification of non-vanishing cohomology groups in specific cases.

Abstract

Let $k$ be a finite field, a $p$-adic field or a number field. Let $K$ be a finite extension of the Laurent series field in $m$ variables $k((x_1,...,x_m))$ or, more generally, a finite extension of the field of rational functions $k((x_1,...,x_m))(y_1,...,y_n)$. When $r$ is an integer, we consider the Galois module $\mathbb{Q}/\mathbb{Z}(r)$ over $K$ and we prove several vanishing theorems for its cohomology. In the particular case when $K$ is a finite extension of the Laurent series field in two variables $k((x_1,x_2))$, we also prove exact sequences that play the role of the Brauer-Hasse-Noether exact sequence for the field $K$ and that involve some of the cohomology groups of $\mathbb{Q}/\mathbb{Z}(r)$ which do not vanish.