Galois theory of differential schemes

TL;DR

This paper develops a broad Galois theory for differential schemes, extending classical Picard-Vessiot theory and Kolchin's strongly normal theory through categorical and algebraic-geometric methods.

Contribution

It introduces a new Galois framework for differential schemes, generalizing existing theories with novel categorical descent techniques.

Findings

Generalizes Picard-Vessiot theory to differential schemes

Extends Kolchin's strongly normal theory

Uses categorical descent methods in algebraic geometry

Abstract

Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.