Tate cohomology over fairly general rings
  Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings.   This paper improves the theory further by giving a new definition that works over more general rings, specifically, those with a dualizing complex.   The new definition of Tate cohomology retains the desirable properties established by Avramov and Martsinkovsky. Notably, there is a long exact sequence connecting it to ordinary Ext groups. 
Peter Jorgensen