【作者】 张欣；

【导师】 姚海楼；

【作者基本信息】 北京工业大学 ， 基础数学， 2005， 硕士

【摘要】 本文研究的主要内容是余代数上的余倾斜余模。余代数的概念来源于对代数的进一步拓展。1976 年,J.A.Green将余代数的概念引进并初步研究了其结构。1977 年, M.Takeuchi 又给出了余张量积和co hom函子的概念和相应的性质,完善了余代数的结构。代数中的许多概念被引入到了余代数中,并相应得到了许多研究成果。本文就是借鉴这种思想,把代数中倾斜模的概念和性质引入到余代数余模范畴中。本文的第2 章首先借鉴了R.R.Colby 等人的思想,对应代数中的倾斜模的概念定义了余代数上的余倾斜余模。接着研究了余倾斜余模的一些性质。主要给出了双C ? 余模U 和双B? 余模U 相互对称的结论。并利用四个函子-（?）_CU,Tor₁ C （-, U）, Co hom_B（U ,-）和Ext_B¹ （U ,-）的核以及挠理论等知识,得到了有关余倾斜余模的性质的一些结论,其中把经典的倾斜定理的部分结果推广到了余代数中来。在第3 章,本文讨论了余倾斜余模的结构特征,证明了每个余倾斜余模都可以写成不可分解的且两两非同构的余模的直和形式。进而每个余倾斜余模包含所有的内射不可分解模作为其直和项。并利用quiver 和路余代数等知识构造了路余代数中余倾斜余模的几个具体例子。而本文的第1 章给出了全文所涉及到的基本概念,并对文章的背景加以介绍。本文的研究结果把代数中的倾斜模概念及其理论引入到了余代数中来,丰富了余代数的内容,使余代数的结构得到进一步的完善,也为以后余代数的进一步发展打下了基础。更多还原

【Abstract】 The purpose of this paper is to investigate cotilting comodules for coalgebras. The notation of coalgebras came from the further development of algebras. In 1976, J.A.Green introduced the conception of coalgebras and studied the fundamental structure. In 1977, M.Takeuchi presented the conceptions of cotensor and Cohom functor as well as their properties that perfected the structure of coalgebras. A lot of knowledge of algebras have been guided into coalgebras. We make use of this idea and obtain the concept of cotilting comodules for coalgebras. In chapter 2, we firstly make use of the idea from R.R.colby etc. and introduce the concept of cotilting comodules for coalgebras. Secondly we invertigate the properties of such a comodule. We show that C-comodule U and B-comodule U are symmetric. We obtain several results about the properties of cotilting comodule by using the kernel of functors -（?）_CU, Tor₁^C （U.-）, Co hom_B（U ,-）and （ ）Ext_B¹（U ,-） and the knowledge of torsion pair. We discuss a duality associated with the Theorem of Brenner-Butler, and we show it hold partially. In chapter 3, We discuss the characterization of cotilting comodule and we prove that every cotilting comodule is a direct sum of indecomposable and pairwise non-isomorphic comodules,and we prove every cotilting comodule contains all injective indecomposables as direct summands. Using the knowledge of quivers and the path coalgebras, We give some examples of cotilting comodule for coalgebras. In chapter 1, we give some fundamental notations, which will be used in the sequel, and introduce the backgrounds of the thesis. The main results of this article is that we introduce the concept and the properties of cotilting comodules for coalgebras. These enriched the contents of coalgebras, make the structures of coalgebras furtherly perfect, and lay a foundation for the further development of the coalgebra.更多还原