Effectively categorical abelian groups

We study effective categoricity of computable abelian groups of the form ⊕i∈ωH, where H is a subgroup of (Q,+). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite.We study Δn0-categoricity in this class of groups, for n>1. We introduce a new algebraic concept of S-independence which is a generalization of the well-known notion of p-independence. We develop the theory of S-independent sets. We apply these techniques to show that every homogeneous completely decomposable group is Δ30-categorical.We prove that a homogeneous completely decomposable group of infinite rank is Δ20-categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semi-low complement (within the set of all primes).We apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.