Cohomology for Picard-Vessiot theory

TL;DR

This paper develops a cohomology framework for Picard-Vessiot theory, linking differential objects to torsors, and applies it to bound the differential splitting degree of certain algebras.

Contribution

It introduces a cohomology theory for Picard-Vessiot theory based on differential Hopf-Galois descent, providing explicit descriptions and new bounds.

Findings

Established a bijective correspondence between differential objects and torsors.

Provided an explicit description of Picard-Vessiot theory in terms of torsors.

Proved a universal bound for the differential splitting degree of differential central simple algebras.

Abstract

We introduce a cohomology theory that classifies differential objects that arise from Picard-Vessiot theory, using the differential Hopf-Galois descent. To do this, we provide an explicit description of Picard-Vessiot theory in terms of differential torsors. We then use this cohomology to give a bijective correspondence between differential objects and differential torsors. As an application, we prove a universal bound for the differential splitting degree of differential central simple algebras.