We provide some proofs of generation of derived bounded and unbounded categories of chain complexes of groups in Kropholler’s hierarchy in terms of the classes of modules induced up from subgroups that are at a lower level in the hierarchy compared to the big group. We formulate and use some generation results from the module category for this. The treatment is fairly straight-forward. We use these results to show that stable module categories for a large class of infinite groups, as defined by Mazza and Symonds in [14], are well-generated which is a generalization of the analogous result for finite groups whose stable module categories are compactly generated.