Torsion cohomology for solvable groups of finite rank

We define a class U of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that G is a group in U and A a ZG-module. If A is Z-torsion-free and has finite Z-rank, we stipulate a condition on A that guarantees that Hn(G,A) and Hn(G,A) must be finite for n≥0. Moreover, if the underlying abelian group of A is a Černikov group, we identify a similar condition on A that ensures that Hn(G,A) must be a Černikov group for all n≥0.