Some questions about the first derived functor of the inverse limit
type: Item
instance of: scholarly article
title: Some questions about the first derived functor of the inverse limit
zbMATH Open document ID: 0837.55005
DOI: 10.1016/0022-4049(94)00107-T
author: C. A. McGibbon
author: Richard John Steiner
published in: Journal of Pure and Applied Algebra
publication date: 1996-05-21T00:00:00Z
review text: 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 \(G_n\).
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 18A30
zbMATH DE Number: 825318
zbMATH Keywords: tower of groups
zbMATH Keywords: derived functor
zbMATH Keywords: inverse limit
zbMATH Keywords: phantom map
reviewed by: Georges Hoff
MaRDI profile type: Publication
cites work: Homotopy limits, completions and localizations
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description: scientific article; zbMATH DE number 825318