The abelianization of inverse limits of groups

TL;DR

This paper explores how the process of abelianization interacts with inverse limits of groups, showing under certain conditions that the abelianization of a limit relates closely to the limit of abelianizations, with the kernel being a cotorsion group.

Contribution

It establishes conditions under which the abelianization of a group limit maps onto the limit of abelianizations, characterizing the kernel as a cotorsion group and analyzing its Ulm length.

Findings

The natural map from abelianization of a limit to the limit of abelianizations is surjective.

The kernel of this map is a cotorsion group.

Ulm length of the kernel does not exceed  for countable products.

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 $G:\mathcal{T}\longrightarrow\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}(\lim_{t\in\mathcal{T}}G_t)\longrightarrow\lim_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ 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 $\aleph_1$.