  We introduce new techniques to study the differential complexes associated to
tube structures on $M \times \mathbb{T}^m$ of corank $m$, in which $M$ is a
compact manifold and $\mathbb{T}^m$ is the $m$-torus. By systematically
employing partial Fourier series, for complex tube structures, we completely
characterize global solvability, in a given degree, in terms of a weak form of
hypoellipticity, thus generalizing existing results and providing a broad
answer to an open problem proposed by Hounie and Zugliani (2017). We also
obtain new results on the finiteness of the cohomology spaces in intermediate
degrees. In the case of real tube structures, we extend an isomorphism for the
cohomology spaces originally obtained by Dattori da Silva and Meziani (2016) in
the case $M = \mathbb{T}^n$. Moreover, we establish necessary and sufficient
conditions for the differential operator to have closed range in the first
degree.
