Cohomological nonvanishing for modules over discrete groups
type: Item
instance of: scholarly article
title: Cohomological nonvanishing for modules over discrete groups
zbMATH Open document ID: 0862.18009
DOI: 10.1016/0022-4049(95)00096-8
published in: Journal of Pure and Applied Algebra
publication date: 1997-04-13T00:00:00Z
review text: Let \(\Gamma\) be a discrete group. A complete resolution over \(\mathbb{Z} \Gamma\) is an acyclic complex \(F_*\) of projective \(\mathbb{Z} \Gamma\)-modules, together with an ordinary projective resolution \(\varepsilon:P_* \to \mathbb{Z}\) such that \(F_*\) and \(P_*\) coincide in sufficiently high dimensions. Let \(F_*\) be a complete resolution over \(\mathbb{Z} \Gamma\) and \(V\) a \(\Gamma\)-module. The Farrell cohomology of \(\Gamma\) with coefficients in \(V\) is defined by the rule \[ \widehat H^* (\Gamma,V) = H^* \bigl(\Hom_{\mathbb{Z} \Gamma} (F_*,V) \bigr). \] Let \({\mathcal X}\) be the class of discrete groups defined as follows: \(\Gamma\in {\mathcal X}\) if there exists a contractible \(\Gamma\)-complex \(X\) with the following properties: (a) \(\Gamma\) includes a normal subgroup \(\Gamma_1\) such that \(\Gamma_1\) acts freely on \(X\) and \(X/ \Gamma_1\) is finite. (b) \(X^H\) is contractible for every finite subgroup \(H\leq \Gamma\). Let \(C^*\) be a cochain \(G\)-complex of finite homological type and \(G\) a finite group. Put \[ \gamma_G (C^*) = |G|\chi \bigl((C^*\otimes \mathbb{Q})^G \bigr)- \chi (C^* \otimes \mathbb{Q}) \] where \(\chi\) is the Euler characteristic. Let \(\Gamma\) be an \({\mathcal X}\)-group, \(X\) an admissible space for \(\Gamma\), \(C^* = C^*(X/ \Gamma_1; \mathbb{Z})\), then \[ \gamma_G(C^*) = |\Gamma: \Gamma_1 |\bigl(\widetilde \chi (\Gamma) - \chi (\Gamma) \bigr), \] the difference between the topological Euler characteristic and the group-theoretical version. Consider \(C^* = \begin{cases} V \quad & \text{for }* =0 \\ 0 \quad & \text{otherwise} \end{cases}\), where \(V\) is a finitely \(\mathbb{Z}\)-generated and \(\mathbb{Z}\)-torsion-free \(G\)-module. Then \[ \gamma_G (C^*) = |G |\text{rank}_\mathbb{Z} V^G-\text{rank}_\mathbb{Z} V. \] Denote this by \(\gamma_G (V)\). In relation the author introduces a numerical invariant \(\gamma_\Gamma(V)\) which determines cohomological nonvanishing under certain hypotheses. The…
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 920031
zbMATH Keywords: Farrell cohomology
zbMATH Keywords: finite homological type
zbMATH Keywords: Euler characteristic
author: Daniel Juan Pineda
reviewed by: L. A. Kurdachenko
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00096-8
OpenAlex ID: W2005579890
cites work: Cohomological restrictions on finite group actions
cites work: Cohomological non-vanishing for modules over \(p\)-groups
cites work: Euler characteristics and cohomology of \(p\)-local discrete groups
cites work: On the vanishing of group cohomology
cites work: Cohomology and group actions
cites work: Q4143495
cites work: Q3240881
cites work: Stable homotopy type of the classifying space of virtually torsion-free groups
cites work: On modules of trivial cohomology over a finite group. I
cites work: The spectrum of an equivariant cohomology ring. I. II
cites work: Modules over finite groups
cites work: Q3882615
cites work: Q3341041
cites work: Minimal resolutions for finite groups
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description: scientific article; zbMATH DE number 920031