Finiteness conditions in the stable module category |
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Authors: | Jonathan Cornick Ioannis Emmanouil Peter Kropholler Olympia Talelli |
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Institution: | 1. Department of Mathematics and Computer Science, Queensborough Community College, CUNY 222-05 56th Avenue, Bayside, New York 11364, USA;2. Department of Mathematics, University of Athens, Athens 15784, Greece;3. Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK |
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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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Keywords: | Cohomology of groups Gorenstein projective dimension Stable module category |
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