The Picard-Vessiot theory, constrained cohomology, and linear differential algebraic groups

TL;DR

This paper characterizes algebraically closed and Picard-Vessiot closed differential fields using differential Galois cohomology and explores applications to parameterized Picard-Vessiot theory.

Contribution

It establishes a new criterion linking differential field properties with the triviality of cohomology groups for linear differential algebraic groups.

Findings

Differential fields are algebraically closed and Picard-Vessiot closed iff certain cohomology groups are trivial.

Provides a new cohomological criterion for Picard-Vessiot closure.

Applications to parameterized Picard-Vessiot theory are demonstrated.

Abstract

We prove that a differential field K is algebraically closed and Picard-Vessiot closed if and only if the differential Galois cohomology group H^1_\partial(K,G) is trivial for any linear differential algebraic group G over K. We give an application to the parameterized Picard-Vessiot theory.