The Cosilting Theory

This thesis is mainly focused on the homological properties of cosilting modules and cosilting complexes. The thesis is organised as follow.1. Firstly, we introduce and study the duality of quasi-tilting modules - quasi-cotilting modules. We prove that all quasi-cotilting modules are pure-injective and cofinendo. It follows that the class CogenM is always a covering class whenever M is a quasi-cotilting module. Some characterizations of quasi-cotilting modules are given. As a main result, we prove that there is a bijective correspondence between the equivalent classes of quasi-cotilting modules and torsion-free covering classes.2. Secondly, we introduce three notions of cosilting modules, cosilting com-plexes and AIR cctilting modules. We prove that there are close connection between them and quasi-cotilting modules. For AIR-cotilting modules, which are dual of large support τ-tilting modules (or AIR-tilting modules), we prove that it has its own prop-erties. For instance, we show that AIR-cotilting modules, quasi-cotilting modules and cosilting modules are the same. In contrast to the fact that large support τ-tilting modules, quasi-tilting modules and silting modules are not always same. At the same time, we prove that there are bijections between equivalent class of cosilting modules,two-terms cosilting complexes and AIR-cotilting modules.3. In the cotilting theory, the class consisting of △-reflexive modules with respect to a cotilting module is not closed under submodules. In order to solve this problem,F. Mantese introduced a new notion, hereditary cotilting module in [42] and proved that the class consisting of △-reflexive modules with respect to a hereditary cotilting module is closed under submodules. It is well known that every cotilting module is cosilting. At the last of this thesis, we extend this result to the cosilting theory. Firstly,we introduce the three classes of right R-modules with respect to a cosilting module and study some related properties in chapter 4. With the help of these properties, we obtain a Cosilting Theorem. Next, we introduce a new notion, called hereditary cosilt-ing module, and prove that the class consisting of △-reflexive modules with respect to a hereditary cosilting module is closed under submodules. Finally, we obtain the fact that the functor Hom_R(-, E(C)/C)ο Rejc(-) is naturally equivalent to the functor Ext_R~1(--,C).