  We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the
deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact
solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d +
\lambda\omega$, where $\omega$ is a closed 1-form. This cohomology naturally
arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$
with a completely solvable Lie algebra $\mathfrak{g}$ and a cocompact lattice
$\Gamma \subset G$ the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$
coincides with the cohomology $H^*_{\lambda \omega}(\mathfrak{g})$ of the Lie
algebra $\mathfrak{g}$ associated with the one-dimensional representation
$\rho_{\lambda \omega}: \mathfrak{g} \to {\mathbb K}, \rho_{\lambda
\omega}(\xi) = \lambda \omega(\xi)$. Moreover $H^*_{\lambda \omega}(G/\Gamma,
{\mathbb C})$ is non-trivial if and only if $-\lambda [\omega]$ belongs to the
finite subset $\{0\} \cup \tilde \Omega_{\mathfrak{g}}$ in $H^1(G/\Gamma,
{\mathbb C})$ well defined in terms of $\mathfrak{g}$.

Differential Geometry;Rings and Algebras