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 $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}⟶\mathcal{G}rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}(\underset{t\in \mathcal{T}}{\lim }{G}_t)⟶\underset{t\in \mathcal{T}}{\lim }\mathrm{Ab}(G_t)$$ 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 ${\aleph }_1$.

To appear in Israel Journal of Mathematics