Integral representations of finite transformation groups. II: Non-simply- connected spaces
type: Item
instance of: scholarly article
title: Integral representations of finite transformation groups. II: Non-simply- connected spaces
zbMATH Open document ID: 0705.57019
DOI: 10.1016/0022-4049(89)90093-5
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: [This article is reviewed together with the preceding one (see Zbl 0705.57018.] Let R be a commutative ring with unit and let G be a finite group. By a theorem of \textit{L. G. Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] a finitely generated RG-module M is projective if and only if M is RE-projective for all maximal p-elementary subgroups \(E\subseteq G.\) The main result of the first paper is a refinement of Chouinard's theorem for \(M=\bar H^*(X;R)\) where X is a connected G-space with the following properties: X is compact or has finite cohomological dimension mod p for all primes p dividing \(| G|\), and for every maximal p-elementary subgroup \(A\subseteq G\) the cohomology spectral sequence of \(E_{A\times A}X\to BA\) degenerates. The refinement consists in showing that it suffices to test projectivity of M over RC for cyclic subgroups C of prime order in order to obtain the projectivity of M as an RG-module. This is proved using Smith theory to show that the hypothesis of Chouinard's theorem is satisfied. The spectral sequence condition holds if X is a Moore space. The author presents evidence that probably it cannot be altogether dispensed with: For an arbitrary finite group G with non-periodic cohomology he constructs a \({\mathbb{Z}}\)-free, \({\mathbb{Z}}G\)-finitely generated, indecomposable \({\mathbb{Z}}G\)-module M with periodic cohomology which is \({\mathbb{Z}}C\)-projective for all prime order subgroups \(C\subseteq G\), but not \({\mathbb{Z}}G\)-projective. It remains open whether such a counterexample can be realized as the cohomology module of a G-space X; but by the main result it certainly is not isomorphic to the cohomology of a finite- dimensional Moore space with G-action. This provides a host of counterexamples to the generalized Steenrod problem beyond the counterexamples given by \textit{G. Carlsson} [Invent. Math. 64, 171-174 (1981; Zbl 0477.55007)] for \(G={\mathbb{Z}}_ p\times {\mathbb{Z}}_ p\). In fact, the a…
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 55N91
zbMATH DE Number: 4157506
zbMATH Keywords: projective
zbMATH Keywords: G-space
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Smith theory
zbMATH Keywords: Moore space
zbMATH Keywords: Steenrod problem
zbMATH Keywords: universal covering
zbMATH Keywords: Serre homology spectral sequence
zbMATH Keywords: finiteness conditions
author: Amir H. Assadi
reviewed by: Hermann Hähl
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90093-5
OpenAlex ID: W2033147129
cites work: Representations, resolutions and Quillen's dimension theorem
cites work: Actions of finite groups on compact manifolds
cites work: Algebraic invariants for finite group actions. I: Varieties
cites work: Integral representations of finite transformation groups. I
cites work: Varieties in finite transformation groups
cites work: Q5658847
cites work: Q4143495
cites work: The varieties and the cohomology ring of a module
cites work: Q3240881
cites work: Projectivity and relative projectivity over group rings
cites work: Endo-permutation modules over p-groups. II
cites work: Q3866809
cites work: The Euler class and periodicity of group cohomology
cites work: Modules over finite groups
cites work: Q3856855
cites work: Finiteness conditions for CW-complexes. I, II.
cites work: Q5624419
description: scientific article; zbMATH DE number 4157506