The Structure Of Complexes And Gorenstein Dimensions Of Modules

Gorenstein homological algebra, which is a kind of relative homological algebra, has been investigated by more and more authors since 1969. Foxby, Golod and Vas-concelos independently initiated the study of semidualizing modules (under different names) over a commutative Noetherian ring, White and other people invested them in commutative (possibly non-Noetherian) ring. In 2007, Holm and White gave the defi-nition of semidualizing modules in general rings and got many results with respect to a semidualizing module. Now the homological properties with respect to a semidualizing module is increasing concerned.In this dissertation, we devote to studying the structure of some special complexes with respect to a semidualizing and some Gorenstein homological complexes with re-spect to a semidualizing and some Gorenstein homological dimensions of R-modules. Then we give some applications.The thesis consists of four chapters. Throughout this dissertation but Chapter 4, R denotes a commutative ring with identity. In Chapter 4, R denotes an associative ring with identity unless otherwise stated.In Chapter 1, some main results and preliminaries are given.In Chapter 2, we introduce and study Gc-injective (Gc-projective) complexes. This extends Enochs and Garcia Rozas's notion of Gorenstein injective (Gorenstein projective) complexes. We then show that a complex X is Gc-injective (Gc-projective) if and only if Xm is a Gc-injective (Gc-projective) module for each m∈Z. Finally, we give a relation between Gc-injective dimension of a complex and Gc-injective di-mension of its terms.In Chapter 3, letε(resp., (?)) be a subclass of Bc(R) (resp., Ac(R)) which is closed under isomorphism andεc (resp., (?)c) be the class of R-modules of the form HomR(C, E) (resp., C(?)Q) where E∈ε(resp., Q∈(?)), we devote to investigating the structure ofεc (resp.. (?)c)-complexes and get that a complex X is an (resp., a)εc (resp., (?)c)-complex if and only if X= Hom(C, Y) (resp.. X= C(?) Z) with Y (resp., Z) anε(resp., (?))-complex. And we give some applications to this result and the main result of Chapter 2.In Chapter 4 we first study the properties of strongly Gorenstein flat (resp. Goren-stein FP-injective) modules which are special Gorenstein projective (resp. Gorenstein injective) modules over general rings. By using the obtained properties we prove that the global strongly Gorenstein flat dimension and the global Gorenstein FP-injective dimension of a ring R are identical when R is n-FC or commutative coherent. Finally, we show that if R is a commutative Noetherian ring of finite Krull dimension, then, for any R-module M, SGfdRM= GpdRM, and hence SGfdRM<∞if and only if M<∞, where SGfdRM denotes the strongly Gorenstein flat dimension of M.