τ-刚性模、局部代数和倾斜代数

发布时间：2018-03-26 22:31

本文选题：τ-刚性模　切入点：投射模　出处：《南京信息工程大学》2017年硕士论文

【摘要】：倾斜理论是代数表示论的重要工具之一,它起源于反射函子,倾斜模的第一个公理是由Brenner和Butler提出,现在我们广泛接受的是由Happel和Ringel提出的.倾斜理论的主要思想是当表示论中的一个代数A很难直接去研究时可以用另一个简单的代数B来代替A,从而使问题简单化.通过构造倾斜模M得到一些重要结果,近期一些代数学者通过推广经典的倾斜理论得到τ-倾斜理论.注意到任何一个τ-倾斜模都是一些不可分解的τ-刚性模的直和.因此,我们只要找到代数上的不可分解的τ-刚性模就可以确定它的τ-倾斜模.本文通过对τ-刚性模进行研究,得到一些初步的结果,主要工作如下:(1) τ-刚性模与投射模.给出了某类特殊的代数上利用单模构造不可分解τ-刚性模的方法.并由此得出所有τ-刚性模是投射模的根平方为零的本原的不可分解代数是局部代数.进一步得出给定一个本原不可分解的代数A,如果A的所有的τ-刚性模都是投射模,则它是局部代数.(2) τ-刚性模与余倾斜模.对于任意一个本原的不可分解的有限维代数B的内射模DB是τ-刚性模当且仅当B的自内射维数小于或等于1.然后再给出例子说明存在一类代数B满足它的所有的不可分解内射模是τ-刚性模但DB不是τ-刚性模.接着再给出余倾斜模与τ-刚性模之间的一些关系.(3)倾斜代数上的τ-刚性模.利用倾斜定理给出了倾斜代数上投射维数小于等于1的不可分解τ-刚性模的刻画.
[Abstract]:Tilt theory is one of the important tools of algebraic representation theory. It originates from reflection functor. The first axiom of tilting mode is proposed by Brenner and Butler. The main idea of tilt theory is that when one algebra A in representation theory is difficult to study directly, another simple algebra B can be used instead of A, thus making the problem simple. Some important results are obtained by constructing the inclined module M. Recently, some algebraic scholars obtained 蟿 -tilt theory by extending the classical tilting theory. It is noted that any 蟿 -tilting module is the direct sum of some indecomposable 蟿 -rigid modules. If we find the indecomposable 蟿 -rigid module on algebra, we can determine its 蟿 -tilted module. In this paper, we obtain some preliminary results by studying 蟿 -rigid module. The main work is as follows: 1) 蟿 -rigid modules and projective modules. The method of constructing indecomposable 蟿 -rigid modules by using a kind of special algebras is given, and it is obtained that all 蟿 -rigid modules are primitive whose root square of projective modules is zero. Given a primitive indecomposable algebra A, if all 蟿 -rigid modules of A are projective modules, The injective module DB of any primitive indecomposable finite dimensional algebra B is a 蟿 -rigid module if and only if the self-injective dimension of B is less than or equal to 1. The example shows that all indecomposable injective modules of a class of algebras B are 蟿 -rigid modules but DB is not 蟿 -rigid modules. Then, some relations between cotilting modules and 蟿 -rigid modules are given. By using the tilting theorem, we give the characterization of indecomposable 蟿 -rigid modules with projective dimension less than or equal to 1 on tilting algebras.
【学位授予单位】：南京信息工程大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3

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