Gorenstein Derived Category And The Relate Researches

In the past thirty five years, the theory and the use of triangulated cate-gories has enjoyed a vigorous development; in the past forty years, many algebrastructures and representations have been constructed in the representation the-ory of algebras; and since 1965, the relative homological algebra, especially theGorenstein homological algebra, has been developed to an advanced level. Thisthesis for Ph.D degree is to study Gorenstein homological algebra by using thetheory of triangulated categories and the methods in the representation theoryof algebras. The main results obtained are as follows.In Chapter 2, we introduce and study the Gorenstein derived category andGorenstein singularity category, which should be taken as a natural developmentin Gorenstein homological algebra. Among the others, we1. give the relation between the Gorenstein derived category and the derivedcategory;2. prove that the homotopy category Kb(GP) of bounded complexes ofGorenstein projective objects is a triangulated subcategory of Dgbp(A); and thatthe bounded Gorenstein derived category of a Gorenstein ring, or of a finitedimensional algebra over a field, is triangle-equivalent to the homotopy categoryof the upper bounded complexes of Gorenstein projective modules with finitenon-zero cohomological groups;3. interpret the Gorenstein derived functor as Hom functor in the corre-sponding bounded Gorenstein derived category.4. prove that a ring is Gorenstein if and only if its Gorenstein singular-ity categories are zero; and in general, we measure how far a ring is from theGorensteinness by embedding the stable category of a Frobenius category intothe corresponding Gorenstein singularity category.The works in 2 and 3 are mainly generalizations of the corresponding resultsin the derived category; while the ones in 1 and 4 seems to be completely new. In Chapter 3, we give concrete constructions of Gorenstein projective mod-ules by using the technique of one-point extensions. We describe the form ofstrongly completeΛ-projective resolutions; In particular, we give the specificform of strongly complete projective resolutions of one-point extension. Thisdescription compares the strongly Gorenstein projective modules of one-pointextensions with the ones of the original algebras. Because Gorenstein projec-tive modules are direct summands of strongly Gorenstein projective modules, weobtain inductively the construction and rich examples of Gorenstein projectivemodules.Note that the Gorensteinness has been generalized to a triangulated cate-gory C with a proper class of trianglesξ. In Chapter 4, we make further studyand obtain some new results in this direction. Using the functorξxtGP(ξ)(-,-),ξ-Gprojective resolutions, and Schanuel classes relative to GP(ξ), we give threeequivalent definitions for theξ-Gprojective dimension of X∈C; introduce thenotions ofξ-tilting class, and then give a suffcient and necessary condition torecognizeξ-n-Gorenstein triangulated category; introduce the notions of stronglyξ-Gorenstein objects, and prove thatΔ-Gprojective is necessarily a direct sum-mand of a stronglyΔ-Gprojective object.The classical Auslander-Reiten transpose is constructed via projective mod-ules. In chapter 5, we introduce the relative transpose via Gorenstein projectivemodules, and hence generalize some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version.