On #-F Complexes

A complex C is said to be #-injective (resp.,#-projective) if it is a complex of injective (resp. projective) modules, that is all terms Ci are injective (resp., projective) R-modules for i∈Z (see, e.g., [5]).#-injective (resp.,#-projective) complexes, which are used to study the injective (resp., projective) dimension of homologically bounded below (resp., bounded above) complexes, plays an very important role in the classical Hyper Homological Algebra. According to [11], a complex C is called Gorenstein injective if there exists an exact sequence of complexes such that each Ii is an injective complex, C= Ker(I0→I1) and Hom(E,-) exacts the sequence for any injective complex E. Dually, one can give the definition of Gorenstein projective complexes. In [11], the authors proved that Gorenstein injective (resp., Gorenstein projective) complexes are actually the com-plexes of Gorenstein injective (resp., Gorenstein projective) modules over a Gorenstein ring. Recently, Yang [46] showed that the above results hold over arbitrary rings (also see Example 5.4.10). In this dissertation, we introduce and study a general concept of #-F complexes for a given class F of R-modules:let F be a class of R-modules, a complex X is called a #-F complex if it is a complex of modules in F, that is all terms Xi are in F for any i∈Z.The thesis consists of five chapters.In Chapter 1, some main results and preliminaries are given.In Chapter 2, some characterizations of #-F complexes are given. The existence of #F-(pre)envelopes and #F-(pre)covers of complexes is discussed, and as a conse-quence, some known results are recovered. Some relations between #F-(pre)covers of a complex X and F-(pre) covers of the R-modules Xi are also studied.Chapter 3 is devoted to the study of #-injective complexes and #L-dimensions of complexes, where #L is the class of #-injective complexes. In Chapter 4, we introduce and study Kaplansky classes of complexes. We give some results by which one can construct a large number of Kaplansky classes of com-plexes. We also give some relations between Kaplansky classes of complexes and co-torsion pairs.In Chapter 5, we study some relations between W-Gorenstein complexes and W-Gorenstein modules for a given class W of modules, where W is the class of exact complexes with all cycle modules in W. In particular, we show that W-Gorenstein complexes are actually the complexes of W-Gorenstein modules whenever W⊥W. We also indicate when the cycle modules of a W-Gorenstein complexes are W-Gorenstein modules.