Mittag-Leffler Conditions And Gorenstein Modules

In order to examine the exactness of the inverse limit functor, Grothendieck in 1961 introduced the Mittag-Leffler condition for countable inverse systems. Several years later, Raynaud and Gruson made an intensive study of the connection between the Mittag-leffler condition and the so called Mittag-Leffler module, and they intro-duced the notion of strict Mittag-Leffler conditions. In the past few years, (strict) Mittag-Leffler conditions were successfully employed to solve many different problems in the homological algebra and the representation theory. Recently, Emmanouil stud-ied the relationship between the class of Gorenstein projective modules and the class of Gorenstein flat modules in terms of Mittag-Leffler conditions. Indeed, the translation of the vanishing of Ext into a Mittag-Leffler conditions on certain inverse systems was a key step in solving these problems.In this dissertation, firstly, we study some properties of strict Mittag-leffler con-ditions (modules) and characterize some rings with (strict) Mittag-Leffler conditions. Then some applications in Gorenstein homological algebra are given.This paper is divided into four chapters.In chapter 1, some main results and preliminaries are given.In Chapter 2, we first study some closure properties of Mittag-leffler conditions (modules), it is proved that each pure submodule of a strict Mittag-Leffler module is a locally split submodule. Then we characterize coherent rings, IF rings and perfect rings with strict Mittag-Leffler conditions.In Chapter 3, we study when Gorenstein projective modules are Gorenstein flat by employing Mittag-Leffler conditions. We first obtain some conditions that are e-quivalent to the assertion that all Gorenstein projective modules are Gorenstein flat by strict Mittag-Leffler conditions. Then we give some conditions under which all Gorenstein projective modules are Gorenstein flat by employing the theory of cotor-sion pair. Finally, we investigate the direct limits of Gorenstein projective modules. It is proven that the class of Gorenstein projective modules is closed under direct limits if all modules axe strict Mittag-Leffler over the ground ring R.In Chapter 4, we investigate the dual (Pontryagin dual) of Gorenstein injective modules with strict Mittag-Leffler conditions. We give a condition which is equivalent to the assertion that the dual of Gorenstein injective modules are Gorenstein flat over noetherian rings.