The abelianization of inverse limits of groups

Abstract

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all small colimits. In this paper we investigate the relation between the abelianization of a limit of groups and the limit of their abelianizations. We show that if T is a countable directed poset and G: T → Grp is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$Ab(\mathop {lim}\limits_{t \in T} {G_t}) \to \mathop {lim}\limits_{t \in T} Ab({G_t})$$ is surjective, and its kernel is a cotorsion group. In the special case of a countable product of groups, we show that the Ulm length of the kernel does not exceed ℵ1.