Rational cohomology of algebraic solvable groups

If G is an affine algebraic group over a field F, and M is a finite-dimensional Fvector space, then M is a rational G-module if G acts on A4 via a morphism of algebraic groups over F: G p→AutF(M). An infinite-dimensional F-vector space M is a rational G-module if it is the union UiMi of finite-dimensional G-stable vector spaces M, such that the G-action on each of them is rational. In [S], Hochschild developed the foundations of rational cohomology, i.e., cohomology H*rat(G, M) in the category of rational G-modules. The most recent applications of rational cohomology (e.g. [4] and [6]) seem to be mainly restricted to groups defined over fields of nonzero characteristic. In this paper we will utilize the rational cohomology groups of algebraic solvable groups defined over the rational numbers Q. Our goal is to prove, for algebraic solvable G and for trivial Q-coefficients, an analog (Theorem 2.23) of the following theorem of Mostow [11,8.1] and Van Est [11,3.6.1]: If G is a connected simply connected real solvable Lie group and D is a discrete cocompact subgroup such that Adg(G) and Adg(D) have the same algebraic hulls, then the Lie algebra cohomology H*(gR, R) is isomorphic to the group cohomology H*(D), R) (trivial R-coefficients in both cases)