Rational torus-equivariant stable homotopy. II: Algebra of the standard model
type: Item
instance of: scholarly article
title: Rational torus-equivariant stable homotopy. II: Algebra of the standard model
zbMATH Open document ID: 1317.55007
published in: Journal of Pure and Applied Algebra
publication date: 2013-01-30T00:00:00Z
review text: As its title suggests, this is a continuation of earlier work by the author, cf. [J. Pure Appl. Algebra 212, No. 1, 72--98 (2008; Zbl 1128.55011)]. Let \(G=S^{1}\times \dots \times S^{1}\) be a torus of rank~\(r\). In earlier work a certain category \({\mathcal A}(G)\) was introduced and the author together with Shipley showed that it provided an algebraic model for rational \(G\)-equivariant cohomology theories. In fact they showed that the category of differential graded objects of \({\mathcal A}(G)\) is Quillen equivalent to rational \(G\)-spectra. The present paper studies a number of algebraic properties of \({\mathcal A}(G)\). In particular, it is shown that \({\mathcal A}(G)\) has injective dimension equal to \(r\), flatness properties are proved and right adjoints are constructed for the inclusion of \({\mathcal A}(G)\) into certain larger categories, giving concrete constructions of limits in \({\mathcal A}(G)\). The author promises a future systematic study of \({\mathcal A}(G)\) and its similarities to categories of sheaves on projective varieties.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 6132783
zbMATH Keywords: rational equivariant spectra and cohomology theories
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: homological dimension
author: John Greenlees
reviewed by: Andrew Baker
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.02.009
OpenAlex ID: W2078673261
cites work: Rational 𝑆¹-equivariant stable homotopy theory
cites work: Rational torus-equivariant stable homotopy. I: Calculating groups of stable maps
DOI: 10.1016/J.JPAA.2012.02.009
description: scientific article; zbMATH DE number 6132783