Some Topics in Gorenstein Homological Algebra

Using the class of finitely generated Gorenstein projective modules, Avramov and Martsinkovsky defined Gorenstein cohomology modules for finitely generated modules over noetherian rings. They also extended the definition of Tate cohomology and they showed that the Tate cohomology measures the ”difference” between the absolute and the relative Gorenstein cohomology. We extend their ideas: given two classes of modules P and C such that P ⊂ C, we define generalized Tate cohomology modules with respect to these classes and show that there is an exact sequence connecting these modules and the relative cohomology modules computed by means of P and respectively C resolutions. We prove that the generalized Tate cohomology with respect to the class of projective and that of Gorenstein projective modules is the usual Tate cohomology and that our exact sequence becomes Avramov‐Martsinkovsky’s exact sequence in this case. We also show that we have balance in generalized Tate cohomology