ABSTRACT
One of the main features of Gorenstein homological algebra is its strong connection with Tate cohomology. This chapter considers derived functors of Hom defined by using right projective resolutions, right Gorenstein projective resolutions, and right complete projective resolutions (complete projective resolvents), and discusses balance results and shows that there exist exact sequences connecting these derived functors. It investigates the invariants called Tate-Betti and Tate- Bass numbers and relates the Tate-Betti and Tate-Bass numbers of a module M with those associated to its Matlis dual, Mv. The chapter introduces and examines a notion of generalized Tate cohomology, and its connections with the usual Tate cohomology. It also considers Tate homology and its generalizations, and gives sufficient conditions in order to have balance in generalized Tate (co)homology.
