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Paper   IPM / M / 11459
School of Mathematics
  Title:   Periodic flat resolutions and periodicity in group (co)homology
  Author(s): 
1.  J. Asadollahi
2.  Sh. Salarian (A. Hajizamani)
  Status:   Published
  Journal: Forum Math.
  Year:  2010
  Pages:   doi: 10.1515/FORM.2011.060
  Supported by:  IPM
  Abstract:
Using the notion of flat covers and proper flat resolutions, we study modules with periodic flat resolutions. It follows that equivalently, we may study modules with periodic homology. We specialize our results to the category of modules over integral group ring \mathbbZΓ, where Γ is an arbitrary group. Among other results, we show that if a group Γ is in a certain class of groups then Γ has periodic homology of period q after some steps with the periodicity isomorphisms of homology groups induced by the cap product with an element in Hq(Γ,C), where C is the cotorsion envelope of the trivial Γ-module \mathbbZ, if and only if it has periodic cohomology of period q after some steps with the periodicity isomorphisms of cohomology groups induced by the cup product with an element in Hq(Γ,\mathbbZ).

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