Some questions about the first derived functor of the inverse limit (Q1903692)

This paper considers towers of groups, i.e. inverse sequences of homomorphisms \(G : G_1 \leftarrow G_2 \leftarrow \dots\) indexed by \(\mathbb{N}\), and gives results about the first right derived functor of the inverse limit \(\varprojlim^1G\). Such questions arose in the study of phantom maps, i.e. pointed maps from a CW-complex such that each restriction to an \(n\)-skeleton is null homotopic. Taking a nilpotent tower \(G\) (a tower of finitely generated nilpotent groups) one sees how \(\varprojlim^1\) data are lost when \(G\) is localized at primes. In the countable abelian case is considered the characterization of \(\varprojlim^1G\) and the description of a \(G\) for a given \(\varprojlim^1G\). Using the Mittag-Leffler property (on stabilization of the sequence of images of the homomorphisms \(G_n \leftarrow G_m)\) one considers the question to know how \(\varprojlim^1G\) depends on the actual group structures of the \(G_n\).