Wakamatsu Tilting Modules And Dual Theory

The classical tilting modules are very useful in the representation theory of artinian algebras (see [1] [16] [17] [19] [37]). In order to extend this notion to arbitrary rings, Y. Miyashita defined the notion of tilting modules with finite projective dimension over arbitrary rings in [65]. Furthermore, T. Wakamatsu extended the notion of the classical tilting modules over artinian algebras to that of the generalized tilting modules in [77]. Also, in order to extend the notion of generalized tilting modules over artinian algebras to arbitrary rings, T. Wakamutsu introduced the notion of generalized tilting modules over arbitrary rings, which is usually called Wakamatsu tilting modules. Such a class of module is a generalization of that of modules with finite projective dimension over arbitrary rings in [65] and that of the generalized tilting modules in [77].In the paper, we mainly study some proerties of Wakamatsu tilting modules and dual theory relative to them, we always assume that T_r is a Wakamatsu tilting module, S = End(T_R). Unless stated otherwise, S is a left noetherian ring, R is a right noetherian ring, the modules considered are finitely generated.We divide this paper into four chapters. In chapter 1, we list some symbols and basic concepts which will be used in this paper. Based on the results mentioned above, we get the following results:In chapter 2, we introduce the notion of W_T~k-modules, and characterize Wakamatsu tilting modules with finite injective dimension by using the properties of W_T~k-modules. Finally we apply these results to study extension closed categories of modules.In chapter 3, we give some (necessary and) sufficient conditions on the extension closure of the subcategory T_T~k (R) consisting of T-k-torsionfree modules, and characterize the Wakamatsu tilting _ST_R with the property of (G_k) or (G'_k) by the properties of homologically finite subcategories.In chapter 4, we study Wakamatsu tilting modules with the property of (W~k), and give a necessary and sufficient condition that l.id(_ST) ≤ 1if and only if r.id(T_R) ≤ 1. The basic properties of generalized Gorenstein dimension on modules will be discussed also in this chapter, and finally we give a sufficient condition that left orthogonal dimension and generalized Gorenstein dimension of a module coincide.