Generalized Tate cohomology

In classical homological algebra one studies objects by taking injective or projective resolutions of an object (usually a module) and then applying an additive functor to the deleted resolution and computing homology. In several areas of Mathematics different kinds of resolutions have proved useful. For example there are the so called relative projective and injective resolutions in group cohomology and there are the resolutions of a sheaf by flabby sheaves in sheaf cohomology. So homological algebra is gradually being enlarged to an area that is now called relative homological algebra.; We will develop some new tools in relative homological algebra but with a special interest in the variety that is called Gorenstein homological algebra and in its connection with Tate cohomology and its generalizations.; We consider two precovering classes of left R-modules P , C such that P⊂C and a left R-module M. For any left R-module N and any n ⩾ 0 we define generalized Tate cohomology modules Ext&d14;nC, PM,N and show that there is a long exact sequence connecting these modules and the modules ExtnCM,N and ExtnPM,N . When P is the class of projective modules and C is the class of Gorenstein projective modules over a left noetherian ring R and when Gor proj dim M < infinity we show that Ext&d14;nC, PM,N are the usual Tate cohomology modules and our sequence gives an exact sequence provided by Avramov and Mastsinkovsky. Then we show that there is a dual result. Using the dual result we prove that over Gorenstein rings Tate cohomology can be computed using either a complete resolution of M or a complete injective resolution of N.; We consider then the question of balance in generalized Tate cohomology. If I,E are two preenveloping classes such that Inj⊂I⊂E then for any left R-modules M, N and for any n ⩾ 1 we define the relative extension modules ExtnE,I M,N by means of a right I - and a right E -resolution of N, and we prove that these modules and the modules ExtnEM,N and ExtnIM,N fit into a long exact sequence. We show that there is a long exact sequence of Ext*E,I M,- associated with a Hom(-, E ) exact sequence 0 → N' → N → N″ → 0 and a long exact sequence of Ext*E,I -,N associated with a Hom(-, E ) exact sequence 0 → M' → M → M″ → 0. Using these properties as well as the dual results for Ext&d14;*C, P-,- we prove that for two pairs of complete hereditary cotorsion theories ( C,L ), ( L,E ) and ( P,M ), ( M,I ) such that P⊂C we have Ext&d14;nC, PM,N ≃ ExtnE,I M,N for any left R modules M, N and for any n ⩾ 1. So in this case there is an occurrence of balance, i.e. the generalized Tate cohomology can be computed either using a left C -resolution and a left P -resolution of M or using a right E -resolution and a right I resolution of N.