Tate-Vogel And Relative Cohomlogies

Tate cohomology originated from the study of representations of finite groups. It was created in1950s, based on Tate’s observation that the ZG-module Z with the trivial action admits a complete projective resolution. As an attempt to generalize the theory to arbitrary groups, Tate-Vogel cohomology was developed by Vogel, Goichot, Mislin, Benson and Carlson. On the other hand, the study of relative homological algebra focuses on the issue of dimensions and relative cohomology functors based on left resolutions or right resolutions that are constructed via precovers or preenvelopes. The most important relationship between Tatc-Vogcl cohomology and relative coho-mology is the Avramov-Martsinkovsky exact sequence, which connects relative and Tate-Vogel cohomologies via a long exact sequence. This dissertation is devoted to studying Tate-Vogel cohomology using flats instead of projectives and some relative homological dimensions using relative cohomology functors. Some applications are given.This paper is divided into four chapters.In Chapter1, some main results and preliminaries are given.In Chapter2, we study Tate-Vogel and relative cohomlogies of complexes based on flats. When R is right coherent, we give general techniques for computing those cohomologies of complexes of finite Gorenstein flat dimension. Moreover, we get an Avramov-Martsinkovsky exact sequence for complexes of finite Gorenstein flat dimen-sion, connecting relative and Tate-Vogel cohomologies via a long exact sequence.In Chapter3, we study torsionfree and divisible dimensions in terms of right derived functors of-(?)-. We also investigate the cotorsion pair cogenerated by the class of cyclic torsionfree right R-modules. As applications, some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings are given. In Chapter4, we investigate duality pairs induced by C-Gorenstein projective modules. It is proven that a commutative Noetherian R is Artinian if and only if (gPc,gLc) is a duality pair if and only if(gLc, gPc) is a duality pair and M+∈gLc whenever M∈gPc.As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.