More on Galois cohomology, definability and differential algebraic groups

TL;DR

This paper explores Galois cohomology within model theory, linking forms of definable groups to cohomology sets, and proves a countability result for constrained cohomology in differential algebraic groups over certain fields.

Contribution

It introduces a cohomological twisting method to analyze fibers in cohomology sequences and establishes a countability result for the first constrained cohomology set in differential algebraic groups.

Findings

Connected forms of definable groups to first cohomology sets.

Developed a twisting cohomology method for analyzing fibers.

Proved the first constrained cohomology set is countable over certain fields.

Abstract

We make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired on Serre's algebraic twisting) to describe arbitrary fibres in cohomology sequences -- yielding a useful finiteness result on cohomology sets. Applied to the special case of differential fields and Kolchin's constrained cohomology, we prove that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable.