On Balance For Relative Homology Of Modules Over A Virtually Gorenstein Ring

In this thesis,we study over a virtually Gorenstein ring the relative homology and Tate homology of modules with respect to the Gorenstein projection subcategory and the Gorenstein flat subcategory.In chapter 1,we introduce the background and the main results of the thesis,and give some basic definitions and facts needed in the later chapters.In chapter 2,we define,based on proper χ-resolutions,the relative homology functor Tor*χM=Hi(X(?)R-)(resp.,Tor*Mχ=Hi(-(?)RX))of an arbitarary module K,where χ is a precovering subcategory.In particular,we show that over a virtually Gorenstein ring,there exist natural isomorphisms TorngpM(K,L)≌TornMgF(K,L),TorngpM(K,L)≌TotnMgp(K,L)and TorngFM(K,L)≌TornMgF(K,L)for each in-teger n and any module L.Here,the symbol gp denotes the subcategory of all Gorenstein projective modules,the symbol gF the subcategory of all Gorenstein flat modules.Above results show that over a virtually Gorenstein ring R,the ten-sor product functor-(?)R-is left balanced by gp×gp,gF×gF and gp×gF,respectively.In chapter 3,we define over a virtually Gorenstein ring,the generalized Tate homology functor(?)*gpM(-,-)and Tor*gFM(-,-).Avramov-Martsinkovsky type exact sequences,which are constructed to connect such Tate homology functors and relative homology functors are given.Let M and N are any two modules.In particular,we show that over a virtually Gorenstein ring,there exist natural isomorphisms(?)mgpM(M,N)≌(?)nMgp(M,N),(?)ngFM(M,N)≌(?)nMgF(M,N) and(?)ngFM(M,N)≌(?)nMgp(M,N)for each integer n.