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Finiteness conditions in the stable module category

Finiteness conditions in the stable module category
Finiteness conditions in the stable module category
We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.
0001-8708
375-400
Cornick, Jonathan
ac6da931-5c5a-4e49-82e1-81711e06110d
Kropholler, Peter H.
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Talelli, Olympia
dde9a122-eceb-441d-9a6c-e18a37f4ac58
Emmanouil, Ioannis
abee50ea-478a-4ba6-899a-a40939ef0a25
Cornick, Jonathan
ac6da931-5c5a-4e49-82e1-81711e06110d
Kropholler, Peter H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Talelli, Olympia
dde9a122-eceb-441d-9a6c-e18a37f4ac58
Emmanouil, Ioannis
abee50ea-478a-4ba6-899a-a40939ef0a25

Cornick, Jonathan, Kropholler, Peter H., Talelli, Olympia and Emmanouil, Ioannis (2014) Finiteness conditions in the stable module category. Advances in Mathematics, 260, 375-400. (doi:10.1016/j.aim.2014.04.006).

Record type: Article

Abstract

We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.

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More information

Accepted/In Press date: 11 April 2014
e-pub ahead of print date: 8 May 2014
Published date: 1 August 2014
Organisations: Mathematical Sciences

Identifiers

Local EPrints ID: 363015
URI: https://eprints.soton.ac.uk/id/eprint/363015
ISSN: 0001-8708
PURE UUID: 7d4b8f2a-fdac-4169-b520-f085eecd43f5
ORCID for Peter H. Kropholler: ORCID iD orcid.org/0000-0001-5460-1512

Catalogue record

Date deposited: 19 Mar 2014 15:25
Last modified: 19 Nov 2019 01:36

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Contributors

Author: Jonathan Cornick
Author: Olympia Talelli
Author: Ioannis Emmanouil

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