This paper introduces a variant of the derived series, and uses it to show that if \(G\) is an amenable group and \(\phi:G\to\Gamma\) is a homomorphism with kernel in Strebel’s homologically defined class \(D(R)\), where \(R\) is a prime field or \(\mathbb{Z}\leq{R}\leq\mathbb{Q}\), then the \(L^2\)-Betti numbers and \(L^2\)-signature defects of closed manifolds \(M\) over \(G\) are invariants of \(R\Gamma\)-cobordism. The bulk of the paper is algebraic, reviewing various notions of localization (Bousfield, Vogel, Cohn), defining the \(R\Gamma\)-local derived series, and establishing its key properties of functoriality and injectivity. (Earlier work in this direction assumed that \(G\) is PTFA, i.e., solvable, with torsion free abelian sections, and used the Harvey derived series, which is based on Ore localization and is not functorial in the sense considered here.) The main result is applied to give an analogue of a theorem of Chang and Weinberger, on \((4k-1)\)-manifolds (with \(k>1\)) which are simple homotopy equivalent and tangentially equivalent but not homology cobordant, and to give a similar result for 3-manifolds homology equivalent to a generalized quaternionic spherical space form.