The University of Southampton
University of Southampton Institutional Repository

Homological finiteness conditions for modules over group algebras

Homological finiteness conditions for modules over group algebras
Homological finiteness conditions for modules over group algebras
We develop a theory of modules of type FP? over group algebras of hierarchically decomposable groups. This class of groups is denoted HF and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an HF-group and M a ZG-module of type FP? then M has finite projective dimension over ZH for all torsion-free subgroups H of G. We also show that if G is an HF-group of type FP? and M is a ZG-module which is ZF-projective for all finite subgroups F of G, then M has finite projective dimension over ZG. Both of these results have as a special case the striking fact that if G is an HF-group of type FP? then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite HF-group of type FP? has finite virtual cohomological dimension.
0024-6107
49-62
Cornick, Jonathan
ac6da931-5c5a-4e49-82e1-81711e06110d
Kropholler, Peter H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Cornick, Jonathan
ac6da931-5c5a-4e49-82e1-81711e06110d
Kropholler, Peter H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4

Cornick, Jonathan and Kropholler, Peter H. (1998) Homological finiteness conditions for modules over group algebras. Journal of the London Mathematical Society, 58 (1), 49-62. (doi:10.1112/S0024610798005729).

Record type: Article

Abstract

We develop a theory of modules of type FP? over group algebras of hierarchically decomposable groups. This class of groups is denoted HF and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an HF-group and M a ZG-module of type FP? then M has finite projective dimension over ZH for all torsion-free subgroups H of G. We also show that if G is an HF-group of type FP? and M is a ZG-module which is ZF-projective for all finite subgroups F of G, then M has finite projective dimension over ZG. Both of these results have as a special case the striking fact that if G is an HF-group of type FP? then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite HF-group of type FP? has finite virtual cohomological dimension.

This record has no associated files available for download.

More information

Published date: August 1998
Organisations: Mathematical Sciences

Identifiers

Local EPrints ID: 349595
URI: http://eprints.soton.ac.uk/id/eprint/349595
ISSN: 0024-6107
PURE UUID: 09078ff2-2cb7-410d-8dac-539d34f98c9c
ORCID for Peter H. Kropholler: ORCID iD orcid.org/0000-0001-5460-1512

Catalogue record

Date deposited: 09 Apr 2013 15:13
Last modified: 15 Mar 2024 03:46

Export record

Altmetrics

Contributors

Author: Jonathan Cornick

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×