Tate (Co) Homology And Depth

The thesis is mainly about Tate(co)homology theory and its applications in the derived depth formula,stable homology for complexes,generalized Tate cohomology in relative derived categories and depth for triangulated categories,which consists of five chapters.In Chapter 1,some definitions and results needed in this thesis are given.In Chapter 2,we define Tate homology based on complete flat resolutions and study its properties.We get adjointness isomorphisms between this Tate homology and the Tate cohomology.A sufficient condition for the derived depth formula to hold in commutative algebra is obtained.In Chapter 3,we start from Christensen and Jorgensen’s celebrated result:the vanishing of Tate homology,Tor*R(M,N),is a sufficient condition for the derived depth formula to hold for a pair of R-complexes(M,N).We get some sufficient conditions to make Tate homology vanish by studying the stable homology for complexes.Secondly,properties of stable homology for complexes are studied.We show that the vanishing of this homology can detect finiteness of homological dimensions of complexes and reg-ularness of rings.Finally,we introduce the relative stable(co)homolgy by Gorenstein homological modules,discuss their properties and detect the finiteness of the homological dimensions by these functors.In Chapter 4,we introduce and investigate the relative derived categories with respect to X-Gorenstein projective modules and У-Gorenstein injective modules.We get the triangle-equivalence of these relative categories.Generalized Tate cohomology functors with respect to X-Gorenstein projective modules and У-Gorenstein injective modules are defined and the related long exact sequences as well as Avramov-Martsinkovsky type exact sequences about these functors are established.Some applications are given.In Chapter 5,we introduce and study the depth of an object in an R-linear compactly generated triangulated category T admitting small coproducts by local cohomology func-tors and discuss the relationship between depth and dimension.Finally,big cosupport for triangulated categories is studied.