More on Galois cohomology, definability and differential algebraic groups

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.