Generalized Injectivity And Generalized Projectivity Of Complexes

This thesis consists of five chapters.In chapter one, we introduce some notions, backgrounds and some known facts needed in the sequel, and finally, list the main results of this thesis.In chapter two, we extend the notions of copure injective modules and copure flat modules to these of copure injective complexes and copure flat complexes. We study the relationships between copure injective (resp., copure flat) complexes and copure injective (resp., copure flat) modules. It is proved that a complex C of left R-modules is copure injective (resp., copure flat) if and only if every term Ci is copure injective (resp., copure flat) in R-Mod, for all i∈Z. Some characterizations of copure injective complexes and copure flat complexes are also obtained.In chapter three, first, we introduce the definition of strongly P-projective modules, and give some characterizations of strongly P-projective modules. Second, we extend the notion of P-projective modules to that of P-projective complexes. We study the relationships between P-projective complexes and P-projective module. It is proved that a complex C is P-projective if and only if every Ci is P-projective for all i∈Z. Some characterizations of P-projective complexes are also obtained.In chapter four, we extend the notion of (m, n)-injective modules to that of (m, n)-injective complexes. Some characterizations of (m, n)-injective com-plexes are obtained. Second, we introduce and investigate#-I(m,n) complexes. It is proved that each complex has a#-I(m,n) cover over a left (m, n)-coherent ring. Some characterizations of#-I(m,n) complexes also obtained.In chapter five, first, the properties of DG-injective complexes and DG-projective complexes are studied and investigated. Then we define a new right derived functors ExtDgn(-,-) using the DG-projective precover resolution of the first variable or DG-injective preenvelope resolution of the second vari-able. Some properties of ExtDgn(-,-) are obtained. We get some equivalen-t characterizations of DG-injective and DG-projecitve complexes. We prove that if ε-rD(c)≤1orε-lD(c)≤1,then there are natural isomorphisms ExtDgn(-,-)≌Extn(-,-).