  Recently, Cochran and Harvey defined torsion-free derived series of groups
and proved an injectivity theorem on the associated torsion-free quotients. We
show that there is a universal construction which extends such an injectivity
theorem to an isomorphism theorem. Our result relates injectivity theorems to a
certain homology localization of groups. In order to give a concrete
combinatorial description and existence proof of the necessary homology
localization, we introduce a new version of algebraic closures of groups with
coefficients by considering a certain type of equations.
