On the cohomology of finite-dimensional nilpotent groups and Lie rings

We establish vanishing results for the first cohomology group of nilpotent groups and Lie rings when the submodule of invariants is trivial. Our results are obtained within a model-theoretic setting, namely for structures that are definable in a finite-dimensional theory, which encompasses algebraic groups over algebraically closed fields, real semi-algebraic groups, and finite-dimensional Lie algebras over an algebraically or real closed field. Since classical tools - such as computations with spectral sequences and rigidity of the linear dimension - are not available in our setting, we develop an elementary algebraic approach. As applications, we derive a form of Frattini's argument for Cartan subrings and a definable version of Maschke's theorem for actions of definable connected p-divisible abelian groups, with a view toward the ongoing study of soluble finite-dimensional Lie rings.