本文選題：τ-剛性模　切入點(diǎn)：投射模　出處：《南京信息工程大學(xué)》2017年碩士論文

【摘要】：傾斜理論是代數(shù)表示論的重要工具之一,它起源于反射函子,傾斜模的第一個(gè)公理是由Brenner和Butler提出,現(xiàn)在我們廣泛接受的是由Happel和Ringel提出的.傾斜理論的主要思想是當(dāng)表示論中的一個(gè)代數(shù)A很難直接去研究時(shí)可以用另一個(gè)簡(jiǎn)單的代數(shù)B來代替A,從而使問題簡(jiǎn)單化.通過構(gòu)造傾斜模M得到一些重要結(jié)果,近期一些代數(shù)學(xué)者通過推廣經(jīng)典的傾斜理論得到τ-傾斜理論.注意到任何一個(gè)τ-傾斜模都是一些不可分解的τ-剛性模的直和.因此,我們只要找到代數(shù)上的不可分解的τ-剛性模就可以確定它的τ-傾斜模.本文通過對(duì)τ-剛性模進(jìn)行研究,得到一些初步的結(jié)果,主要工作如下:(1) τ-剛性模與投射模.給出了某類特殊的代數(shù)上利用單模構(gòu)造不可分解τ-剛性模的方法.并由此得出所有τ-剛性模是投射模的根平方為零的本原的不可分解代數(shù)是局部代數(shù).進(jìn)一步得出給定一個(gè)本原不可分解的代數(shù)A,如果A的所有的τ-剛性模都是投射模,則它是局部代數(shù).(2) τ-剛性模與余傾斜模.對(duì)于任意一個(gè)本原的不可分解的有限維代數(shù)B的內(nèi)射模DB是τ-剛性模當(dāng)且僅當(dāng)B的自內(nèi)射維數(shù)小于或等于1.然后再給出例子說明存在一類代數(shù)B滿足它的所有的不可分解內(nèi)射模是τ-剛性模但DB不是τ-剛性模.接著再給出余傾斜模與τ-剛性模之間的一些關(guān)系.(3)傾斜代數(shù)上的τ-剛性模.利用傾斜定理給出了傾斜代數(shù)上投射維數(shù)小于等于1的不可分解τ-剛性模的刻畫.
[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.
【學(xué)位授予單位】：南京信息工程大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153.3

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 謝宗真;張孝金;;所有τ-剛性模是投射模的代數(shù)[J];山東大學(xué)學(xué)報(bào)(理學(xué)版);2016年02期

2 張孝金;張?zhí)?;根平方為零的Nakayama代數(shù)上的τ-傾斜模[J];南京大學(xué)學(xué)報(bào)(數(shù)學(xué)半年刊);2013年02期

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本文編號(hào)：1669907