Relative Cohomology and Generalized Tate Cohomology

We compare relative cohomology theories arising from using different proper resolutions of modules. Criteria for the vanishing of such distinctions are given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties admitted by the two different cohomology theories are actually equivalent in some cases. As applications, we recover many results obtained earlier in various contexts. At last we investigate derived functors with respect to the Auslander and Bass classes.