Commutative algebra for cohomology rings of virtual duality groups
type: Item
instance of: scholarly article
title: Commutative algebra for cohomology rings of virtual duality groups
zbMATH Open document ID: 0901.20040
published in: Journal of Algebra
publication date: 1998-09-28T00:00:00Z
review text: \textit{D. J. Benson} and \textit{J. F. Carlson} [Trans. Am. Math. Soc. 342, No. 2, 447-488 (1994; Zbl 0816.20044)]\ and \textit{D. J. Benson} and \textit{J. P. C. Greenless} [J. Pure Appl. Algebra 122, No. 1-2, 41-53 (1997; Zbl 0886.57022)]\ have studied homological duality properties for finite groups. Here the authors consider a virtual duality group \(\Gamma\) and show that the cohomology ring \(H^*(\Gamma)\) has strong ring theoretic duality properties, which combine the familiar homological duality of the (duality) subgroups of \(\Gamma\) of finite index with the duality enjoyed by the finite quotients. Both these two different types of duality may be obtained as special cases of the authors' result. The main theorem of the paper -- too complicated to be repeated here -- is expressed in terms of a spectral sequence involving the Grothendieck \(J\)-local cohomology where \(J\) is the ideal of positive degree elements in \(H^*(\Gamma)\). The authors give both a purely algebraic approach which applies to the situation over an arbitrary commutative Noetherian ring, and a topological approach for those virtual duality groups for which the duality is realized topologically (arithmetic groups, mapping class groups). The latter approach applies also to more general situations involving Lie groups and to generalized cohomology theories.
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 22D35
Mathematics Subject Classification ID: 55M05
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 1038262
zbMATH Keywords: cohomology of groups
zbMATH Keywords: duality groups
zbMATH Keywords: Poincaré duality
zbMATH Keywords: homological duality
zbMATH Keywords: cohomology rings
zbMATH Keywords: spectral sequences
author: David Benson
author: John Greenlees
reviewed by: Urs Stammbach
MaRDI profile type: Publication
full work available at URL: jabr.1996.6944
OpenAlex ID: W2048780145
cites work: Q3999421
cites work: Q3997902
cites work: Complexity and multiple complexes
cites work: Projective Resolutions and Poincare Duality Complexes
cites work: Functional Equations for Poincaré Series in Group Cohomology
cites work: Commutative algebra for cohomology rings of classifying spaces of compact Lie groups
cites work: Q4129638
cites work: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts
cites work: Corners and arithmetic groups (Appendice: Arrondissement des varietes a coins par A. Douady et L. Herault)
cites work: Q3708977
cites work: Commutative algebra in stable homotopy theory and a completion theorem
cites work: Q4896542
cites work: Q4716283
cites work: Algebras over equivariant sphere spectra
cites work: \(K\)-homology of universal spaces and local cohomology of the representation ring
cites work: Commutative algebra in group cohomology
cites work: Q4896543
cites work: Q4896544
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: The virtual cohomological dimension of the mapping class group of an orientable surface
cites work: Q3910741
cites work: On groups of type \((FP)_\infty\)
cites work: Equivariant stable homotopy theory. With contributions by J. E. McClure
cites work: The cohomology of \(SL_3(Z)\)
cites work: Homologically trivial actions on chain complexes and finiteness properties of infinite groups
DOI: 10.1006/JABR.1996.6944
description: scientific article; zbMATH DE number 1038262