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Homological Finiteness Conditions for Modules Over Group Algebras
Author(s) -
Cornick Jonathan,
Kropholler Peter H.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005729
Subject(s) - mathematics , cohomological dimension , torsion (gastropod) , countable set , pure mathematics , type (biology) , dimension (graph theory) , group (periodic table) , finite group , discrete mathematics , combinatorics , cohomology , physics , medicine , ecology , surgery , quantum mechanics , biology
We develop a theory of modules of type FP ∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted h F and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h F‐group and M a Z G ‐module of type FP ∞ then M has finite projective dimension over Z H for all torsion‐free subgroups H of G . We also show that if G is an h F‐group of type FP ∞ and M is a Z G ‐module which is Z F ‐projective for all finite subgroups F of G , then M has finite projective dimension over Z G . Both of these results have as a special case the striking fact that if G is an h F‐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 h F‐group of type FP ∞ has finite virtual cohomological dimension.