Local Galois theory in dimension two: Second edition

TL;DR

This paper generalizes Shafarevich's Conjecture to fields of Laurent series in two variables, demonstrating that their absolute Galois groups are semi-free, and extends this to function fields of smooth projective curves over large fields.

Contribution

It proves the semi-freeness of absolute Galois groups for these complex fields, improving upon previous results that only established quasi-freeness.

Findings

Absolute Galois group of Laurent series fields in two variables is semi-free.

Function fields of smooth projective curves over large fields have semi-free Galois groups.

Generalization of Shafarevich's Conjecture to higher-dimensional fields.

Abstract

We prove a generalization of Shafarevich's Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the function field of a smooth projective curve over a large field has semi-free absolute Galois group. In the first edition of this paper it was shown that these groups are quasi-free, which is weaker.