Transfer principles for Galois cohomology and Serre's conjecture II

TL;DR

This paper develops transfer principles for the cohomological dimension of fields, enabling the construction of specific extensions and applying these to Serre's conjecture II across different characteristics.

Contribution

It introduces new transfer principles for cohomological dimensions and applies them to relate Serre's conjecture II in characteristic zero and positive characteristic.

Findings

Constructed totally ramified extensions with reduced cohomological dimension.

Established algebraic extensions satisfying norm conditions.

Proved Serre's conjecture II in characteristic zero implies the positive characteristic case.

Abstract

In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field $K$ with finite cohomological dimension $\delta$, the two main ones allow to: - construct totally ramified extensions of $K$ with cohomological dimension $\leq \delta - 1$ when $K$ is a complete discrete valuation field; - construct algebraic extensions of $K$ with cohomological dimension $\leq \delta-1$ and satisfying a norm condition. We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic $0$ fields implies Serre's conjecture II for positive characteristic fields.