【作者】 张必成；

【导师】 施武杰；

【作者基本信息】 苏州大学 ， 应用数学， 2006， 博士

【摘要】 古典倾斜模在Artin代数的表示理论中是非常有用的（见[1][16][17][19][37]）。为了把Artin代数上的古典倾斜模推广到任意环上，Y. Miyashita在[65]定义了任意环上的具有有限投射维数的倾斜模，这种模推广了Artin代数上的古典倾斜模的概念。T. Wakamatsu在[77]把Artin代数上的古典倾斜模进一步推广到Artin代数上的广义倾斜模。为了把Artin代数上的广义倾斜模推广到任意环上，T. Wakamatsu在[78]引进了任意环上的广义倾斜模的概念，即我们通常所说的Wakamatsu倾斜模。这种模类是[65]中任意环上的具有有限投射维数的倾斜模和[77]中广义倾斜模的推广。 本论文我们主要研究某些Wakamatsu倾斜模和与之相关的对偶理论。我们总假定TR是Wakamatsu倾斜模，S=End（TR）。除非例外说明，S是左Noether环，R是右Noether环，涉及的模均指有限生成模。本文共分四章，第一章主要介绍本文常用的概念。本文在前人的基础上，讨论了以下内容： 第二章引进了WT-k模的概念，对具有有限内射维数的Wakamatsu倾斜模进行了刻画。最后给出了这些结果在扩张闭模范畴中的应用。 第三章给出了T-k-挠自由模范畴TTk（R）扩张闭的一些充分条件和充要条件，并用同调有限子范畴的性质对具有性质（Gk）或性质（G′k）的Wakamatsu倾斜模STR进行了刻画。 第四章我们研究了具有性质（Wk）的Wakamatsu倾斜模，给出了l.id（ST）≤1当且仅当r.id（TR）≤1的一个充分必要条件。模的广义Gorenstein维数的基本性质在本章也将给予讨论，最后给出了模的左正交维数等于广义Gorenstein维数的一个充分条件。更多还原

【Abstract】 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.更多还原