A generalization of Ohkawa's theorem
type: Item
instance of: scholarly article
zbMATH Open document ID: 1300.55007
DOI: 10.1112/S0010437X13007616
author: Carles Casacuberta
author: Javier J. Gutiérrez
author: Jiří Rosický
published in: Compositio Mathematica
publication date: 2014-08-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.6395
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 6333835
zbMATH Keywords: homotopy
zbMATH Keywords: model category
zbMATH Keywords: acyclicity
zbMATH Keywords: Bousfield class
zbMATH Keywords: localization
MaRDI profile type: Publication
arXiv ID: 1203.6395
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OpenAlex ID: W3104968866
title: A generalization of Ohkawa's theorem
review text: In the homotopy category of spectra, one can consider the Bousfield class of a spectrum \(E\), which consists of all spectra \(X\) such that \(E\wedge X=0\); it can also be described as the \(E_*\)-spectra, where \(E_*\) is the reduced homology theory associated to \(E\). A result of Ohkawa says that the collection of all possible Bousfield classes in the homotopy category of spectra forms a set, rather than a proper class. The question of whether Bousfield classes form a set in the derived category of a commutative ring was shown in certain cases by Neeman and by Dwyer-Palmieri, and eventually was established for the general case by Stevenson and Iyengar-Krause.NEWLINENEWLINEIn this paper, the authors look for a common framework for both the homotopy category of spectra and the derived category of a commutative ring. They prove that an analogue of Ohkawa's theorem holds in the very general case of a monoidal, combinatorial, pointed model category on which is defined a functor which behaves sufficiently like a homology theory to define Bousfield classes in the model category. In doing so, they establish several new examples, such as the homotopy category of modules over a commutative ring spectrum and the stable motivic homotopy category over a Noetherian scheme of finite Krull dimension, for which the Bousfield classes form a set.
reviewed by: Julie Bergner
description: scientific article; zbMATH DE number 6333835