Some Discussions On Gorenstein FP-Injective Complexes

As a generalization of Gorenstein FP-injective modules in the category of complexes, Gorenstein FP-injective complexes are introduced and investigated in this thesis. It consists of the following three parts:In the first chapter, we introduce the background and the signification of the study about our main work, and give some necessary definitions and our main results.In chapter 2, we introduce the definition of Gorenstein FP-injective com-plexes and obtain some equivalent characterizations of Gorenstein FP-injective complexes over coherent rings, say let R be left coherent ring and FP-id(RR)<∞, then the complexes C of left R-modules is Gorenstein FP-injective if and only if ExtRi(F, C)=0, for any all FP-injective complexes of left R-modules F and all i≥1. Furthermore, we obtain the relation between Gorenstein FP-injective complexes and Gorenstein FP-injective modules and get the following results, say let R be a left coherent ring R and FP-id(RR)<∞, a complex C of left R-modules is Gorenstein FP-injective if and only if Cmis Gorenstein FP-injective in R-Mod for all m∈Z.Chapter 3 deals with the relation between Gorenstein FP-injective com-plexes and Gorenstein flat complexes by using the complex C+=HomR(C, Q/Z) over a coherent ring. As a consequence, for a left coherent ring R, a complex C of left R-modules is a Gorenstein flat complex if and only if C+ is a Gorenstein FP-injective complex.Meanwhile, we discuss the existence of Gorenstein FP-injective preen-velopes and Gorenstein FP-injective recovers, say let R be an n-FC and per-fect ring, then any complex of left R-modules has a Gorenstein FP-injective preenvelopes and Gorenstein FP-injective recover.