A characterization of cohomological dimension for a big class of groups

It was shown by Cornick and Kropholler (1998) in [5] that if a group G is in h F then the finitistic dimension of Z G , findim Z G , is equal to the supremum of the projective dimensions of the Z G -modules M such that pd Z H M < ∞ for all finite subgroups H of G. Here we show that a locally h F...

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Bibliographic Details
Published in:Journal of algebra 2011-01, Vol.326 (1), p.238-244
Main Author: Talelli, Olympia
Format: Article
Language:English
Subjects:
Classifying space for proper actions
Cohomological dimension
Elementary abelian groups
Finitistic dimension
Periodic cohomology
Projective length of injectives
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Summary:It was shown by Cornick and Kropholler (1998) in [5] that if a group G is in h F then the finitistic dimension of Z G , findim Z G , is equal to the supremum of the projective dimensions of the Z G -modules M such that pd Z H M < ∞ for all finite subgroups H of G. Here we show that a locally h F -group G is of type Φ if and only if findim Z G < ∞ . A group G is of type Φ if for any Z G -module M, pd Z G M < ∞ if and only if pd Z H M < ∞ for every finite subgroup H of G. Type Φ was proposed in Talelli (2007) [18] as an algebraic characterization for those groups G that admit a finite dimensional model for E ̲ G . We also deduce a number of corollaries for locally h F -groups.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2010.01.021