Theoretical Reseach On Tate Homology Of Several Modules

In this paper,Tate homology TorFC of modules admitting Tate FC-resolutions is investigated firstly.In particular,a long exact sequence connecting TorFC,TorFC and TorGFC is established.As applications,the vanishing and the balance of this Tate homol-ogy are proved.Secondly,Tate homology theory in an abelian category is investigated.Tate homology functors TorWA and TorAV of objects are defined,and an A-M exact se-quence of Tate W-homology(Tate V-homology)of objects is given.Finally,the notion of complete y-resolutions of objects is introduced for a cotorsion pair(X,y),and some properties of the corresponding Gorenstein objects are studied.Based on this,the defi-nition of Tate homology relative to cotorsion pair is given,and the corresponding A-M type exact sequence is established.