Bousfield localization as an algebraic closure of groups
type: Item
instance of: scholarly article
title: Bousfield localization as an algebraic closure of groups
zbMATH Open document ID: 0684.55011
DOI: 10.1007/BF02765889
author: Emmanuel D. Farjoun
author: S. Shelah
author: Kent E. Orr
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The notions of HR-local group and the HR-localization of a group arose in A. K. Bousfield's work on homology localization of spaces. (Here R is a subring of the rationals.) \textit{A. K. Bousfield} [Mem. Am. Math. Soc. 186 (1977; Zbl 0364.20058)] constructs the HR-localization of a group via certain natural transfinite towers which stabilize appropriately and then proceeds to show, for example, that HR-localization coincides with R- completion for certain groups (e.g., polycyclic-by-finite). The present paper provides an alternative characterization of HR-localness in terms of the unique solvability of certain (\(\Gamma\)-)systems of equations defined over the group in question. Furthermore, adjoining the roots of all such \(\Gamma\)-systems to the group (with appropriate relations provided by the \(\Gamma\)-systems) essentially constructs the HR- localization of the group. The authors apply their result, in particular, to derive bounds on the size of the HZ-localization.
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4121022
zbMATH Keywords: HR-localization
zbMATH Keywords: homology localization
zbMATH Keywords: R-completion
reviewed by: John Oprea
MaRDI profile type: Publication
cites work: Homological localization towers for groups and Π-modules
cites work: Homotopy limits, completions and localizations
cites work: Q3769080
full work available at URL: https://doi.org/10.1007/bf02765889
OpenAlex ID: W2059141828
description: scientific article; zbMATH DE number 4121022