Free differential Galois groups
Commutative Algebra
2022-03-22 v3
Abstract
We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat's freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over the rationals. This is the first known case of the twenty year old conjecture.
Keywords
Cite
@article{arxiv.1904.07806,
title = {Free differential Galois groups},
author = {Annette Bachmayr and David Harbater and Julia Hartmann and Michael Wibmer},
journal= {arXiv preprint arXiv:1904.07806},
year = {2022}
}
Comments
16 pages. Version of Oct. 20, 2020. Expanded the discussion after Definition 2.4; added an assumption on not being finite type after the proof of Lemma 3.2