Triviality of differential Galois cohomologies of linear differential algebraic groups

TL;DR

This paper establishes that the triviality of differential Galois cohomologies for linear differential algebraic groups over a partial differential field K is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed, linking cohomology triviality to field closure properties.

Contribution

It proves the equivalence between the triviality of differential Galois cohomologies and the closure properties of the base field K, connecting cohomology to field theory in differential algebra.

Findings

Triviality of cohomologies is equivalent to K being algebraically closed.

Triviality of cohomologies is equivalent to K being Picard-Vessiot closed.

Triviality of cohomologies is equivalent to K being linearly differentially closed.

Abstract

We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear differential equation with parameters.