Patching over fields

TL;DR

This paper introduces a new, more elementary patching method for fields and vector spaces, expanding its applications in inverse Galois theory, Brauer groups, and differential modules.

Contribution

It develops a novel patching approach based on fields and vector spaces, simplifying previous methods and broadening their applicability.

Findings

New patching method applicable to fields and vector spaces

Applications to inverse Galois theory for function fields

Extensions to Brauer groups and differential modules

Abstract

We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.