Inhabitants of interesting subsets of the Bousfield lattice
type: Item
instance of: scholarly article
title: Inhabitants of interesting subsets of the Bousfield lattice
zbMATH Open document ID: 1426.55021
author: A. D. Brooke-Taylor
author: Birgit Richter
author: Benedikt Löwe
published in: Journal of Pure and Applied Algebra
publication date: 2018-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1702.03245
full work available at URL: dichotomy.final.pdf
review text: \textit{A. K. Bousfield} [Comment. Math. Helv. 54, 368--377 (1979; Zbl 0421.55002)] introduced an equivalence relation on the stable homotopy category by declaring two spectra to be equivalent when they determine the same classes of acyclic spectra. This equivalence relation has also become known as Bousfield equivalence. Its class of equivalence classes \(\mathbf{B}\) forms a set, and inclusion of classes of acyclic spectra induces the structure of a lattice on \(\mathbf{B}\). Bousfield [loc. cit.] and \textit{M. Hovey} and \textit{J. H. Palmieri} [Contemp. Math. 239, 175--196 (1999; Zbl 0947.55014)] introduced certain subsets of \(\mathbf{BA} \subseteq \mathbf{cBA}\subseteq \mathbf{DL} \subseteq \mathbf{B}\) of the Bousfield lattice. For example, \( \mathbf{DL}\) is the distributive lattice of all Bousfield classes \(\langle X \rangle\) with the property \(\langle X \rangle \wedge \langle X \rangle = \langle X \rangle\) and \(\mathbf{cBA}\) is the complete Boolean algebra of closed classes. Working \(p\)-locally with respect to a prime \(p\), the paper under review provides lower bounds for the size of \( \mathbf{DL}\setminus \mathbf{cBA}\) and \( \mathbf{cBA}\setminus \mathbf{BA}\). As an application, the authors show that the Bousfield class of the Eilenberg--Mac Lane spectrum \(H\mathbb F_p\) lies in \( \mathbf{DL}\setminus \mathbf{cBA}\).
reviewed by: Steffen Sagave
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 6859160
zbMATH Keywords: stable homotopy category
zbMATH Keywords: Bousfield equivalence
MaRDI profile type: Publication
OpenAlex ID: W2588994260
arXiv ID: 1702.03245
cites work: The Boolean algebra of spectra
cites work: The localization of spectra with respect to homology
cites work: Ohkawa’s theorem: There is a set of Bousfield classes
cites work: Q4327004
cites work: Q4934305
cites work: The injective hull of homotopy types with respect to generalized homology functors
cites work: Localization with Respect to Certain Periodic Homology Theories
Wikidata QID: Q59882927
DOI: 10.1016/J.JPAA.2017.09.012
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description: scientific article