Local-global Galois theory of arithmetic function fields

TL;DR

This paper explores the connection between local and global Galois groups of function fields over complete discretely valued fields, establishing conditions for descent and injectivity, and applying these to local-global principles and algebraic van Kampen theorems.

Contribution

It provides necessary and sufficient conditions for local extensions to descend globally and for the local Galois group to embed into the global Galois group, along with applications to local-global principles.

Findings

Conditions for local extensions to descend globally

Criteria for local Galois groups to inject into global groups

A local-global principle for the index of varieties

Abstract

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.