Triviality of differential Galois cohomology of linear differential algebraic groups

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.

Generalized Picard–Vessiot extensions and differential Galois cohomology

“…Finally we will mention, as requested by the referee, a nice related result [14], which we became aware of after the current paper was submitted, and which does in a sense give a satisfactory extension of [18] to the case of several commuting derivations, although we did not check the proofs in detail. The content of the main result of [14], using notation of the current paper is that the following are equivalent, in the context of differential fields (K, ∆) where ∆ = {∂ 1 , . .…”

“…So this says that the main theorem of [18] does extend naturally to several commuting derivations, modulo the case of principal homogeneous spaces for differential algebraic subgroups of the additive group. The proof (of (1) implies (2)) in [14] essentially follows the general line of the inductive argument of [18], reducing to the cases where G is finite, a connected subgroup of the multiplicative group, a connected subgroup of the additive group, or noncommutative simple. The additional assumption in (1) deals with the third case and allows the arguments to go through (with some additional complications and use of results in the literature).…”

“…In the same section we will give a positive result (in several derivations) around triviality of H 1 ∆ and closure under "generalized strongly normal extensions of linear type". We will also mention in Section 4, a recent result of Minchenko and Ovchinnikov [14], written after the current paper was submitted for publication, which says in effect that definable subgroups of the additive group present the only obstruction to generalizing [18] to the case m > 1.…”

Generalized Picard-Vessiot extensions and differential Galois cohomology

“…Finally we will mention, as requested by the referee, a nice related result [14], which we became aware of after the current paper was submitted, and which does in a sense give a satisfactory extension of [17] to the case of several commuting derivations, although we did not check the proofs in detail. The content of the main result of [14], using notation of the current paper is that the following are equivalent, in the context of differential fields (K, ∆) where ∆ = {∂ 1 , . .…”

“…In the same section we will give a positive result (in several derivations) around triviality of H 1 ∆ and closure under "generalized strongly normal extensions of linear type". We will also mention in Section 4, a recent result of Minchenko and Ovchinnikov [14], written after the current paper was submitted for publication, which says in effect that definable subgroups of the additive group present the only obstruction to generalizing [17] to the case m > 1.…”