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For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice$\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrakg$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra$\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomologyof $G/\Gamma$ by using a modification of $G$ associated with a algebraicsub-torus of the Zariski-closure of the image of the adjoint representation.This technique includes the construction due to Guan and developed by the firsttwo authors. In this paper, we also give a Dolbeault version of such techniquefor complex solvmanifolds, i.e. for solvmanifolds endowed with an invariantcomplex structure. We construct a finite dimensional cochain complex whichcomputes the Dolbeault cohomology of a complex solvmanifold $G/\Gamma$ withholomorphic Mostow bundle and we give a construction of a new Lie algebra$\breve {\mathfrak g}$ with a complex structure whose cohomology is isomorphicto the Dolbeault cohomology of $G/\Gamma$.

On de Rham and Dolbeault Cohomology of Solvmanifolds