Generalized Tate Cohomology

<div class="line" id="line-78"> We consider two classes of left R-modules, P and C, such that P &sub; C. If the module M has a P-resolution and a C-resolution then for any module N and n &ge; 0 we de&filig;ne generalized Tate cohomology modules EdxtCn, P (M, N) and show that we get a long exact sequence connecting these modules and the modules Ext n C(M, N) and Ext n P(M, N). When C is the class of Gorenstein projective modules, P is the class of projective modules and when M has a complete resolution we show that the modules EdxtCn,P (M, N) for n &ge; 1 are the usual Tate cohomology modules and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate cohomology EdxtRn (M, N) can be computed using either a complete resolution of M or a complete injective resolution of N. And so,</div><div class="line" id="line-82"> using our dual result, we obtain Avramov and Martsinkovsky&rsquo;s exact sequence under hypotheses di&fflig;erent from theirs.</div>