余半倾斜理论

发布时间：2019-03-07 12:17

【摘要】：本文主要研究了余半倾斜(cosilting)模和余半倾斜复形的同调性质.具体本文组织如下:1,首先,我们研究了拟倾斜模的对偶—拟余倾斜模。在这一部分中我们证明了所有的拟余倾斜模既是纯内射的又是余有限自同态(cofinendo)的.由此得到,当M是一个拟余倾斜模时,由M余生成的类CogenM总是一个盖类.同时也给出了关于拟余倾斜模的一些特征刻画.作为这部分的主要结果,我们给出了拟余倾斜模的等价类和挠自由类之间的一个一一对应.为后面余半倾斜模的研究提供了一个工具。2,其次,我们引入并研究了余半倾斜复形,余半倾斜模和AIR-余倾斜模.证明了他们三者跟拟余倾斜模之间存在着密不可分的联系.作为大的(即未必有限生成)支撑τ-倾斜模(或者AIR-倾斜模)的对偶-AIR-余倾斜模,我们证明了它具有自己本身特有的属性.例如,我们证明了拟余倾斜模,AIR-余倾斜模和余半倾斜模三者本质上是一致的.反之,作为它们的对偶拟倾斜模,大的支撑τ-倾斜模和半倾斜(silting)模三者却是不一致的.同时我们也证明了余半倾斜模的等价类和两项的余半倾斜复形的等价类之间存在着一个一一对应.我们也给出了余半倾斜模,两项的余半倾斜复形和AIR余倾斜模三者的等价类之间存在着一个一一对应.3,在余倾斜理论中,由所有△-自反模(相对于余倾斜模)组成的类在子模下不是封闭的.为了解决这个问题,F. Mantese在[42]中提出了一个新的概念,遗传的余倾斜模,证明了所有△-自反模(相对于遗传的余倾斜模)组成的类在子模下是封闭的.众所周知,所有的余倾斜模都是余半倾斜模.在这篇论文的最后我们将这一结果推广到余半倾斜理论中,即所有△-自反模(相对于遗传的余半倾斜模)组成的类在子模下是封闭的.首先在第四章中我们给出了关于余半倾斜模的一些相关性质.在这些性质的帮助下,我们获得了一个余半倾斜定理.在第四章的第二节我们介绍了遗传的余半倾斜模,证明了在余半倾斜理论中,所有△-自反模(相对于遗传的余半倾斜模)组成的类在子模下是封闭的.最后我们证明了两个函子Hom_R(-,E(C)/C) οRejc(-)和Ext_R~1(-,C)是自然等价的。
[Abstract]:In this paper, we mainly study the homological properties of cosemi-tilted (cosilting) modules and cosemi-tilted complex. The main contents of this paper are as follows: 1. Firstly, we study the dual-quasi-coskew modules of quasi-tilting modules. In this part, we prove that all quasi-coskew modules are both pure injective and cofinite endomorphism (cofinendo). It is obtained that when M is a quasi-cotilled module, the class CogenM generated by M-cooccurrence is always a cover class. At the same time, some characterizations of quasi-coskew modules are also given. As the main result of this part, we give a one-to-one correspondence between the equivalent class and the torsion free class of quasicotilted modules. In this paper, we provide a tool for the study of the latter half tilting modules. 2. Secondly, we introduce and study the cosemitilting complex, the cooblique modules and the AIR- cotilting modules. It is proved that they are inextricably related to quasi-cooblique modules. As a dual-AIR- cotiled module supporting 蟿-tilting mode (or AIR- tilting mode), we prove that it has its own unique properties. For example, we prove that the quasi-cotilt module, the AIR- cotilt module and the cosemi-tilting module are essentially identical. On the contrary, as their dual quasi-tilting modes, large support 蟿-tilting modes and semi-tilted (silting) modules are inconsistent. At the same time, we also prove that there exists a one-to-one correspondence between the equivalent class of cosemitilting module and the equivalent class of cosemitilted complex of two terms. We also give a one-to-one correspondence between the equivalent classes of the cosemitilting module, the cosemi-tilting complex of two terms and the AIR cotilting module. 3, in the theory of cotilt, there is a one-to-one correspondence between the equivalence classes of the two terms. Classes consisting of all-reflexive modules (relative to cotiled modules) are not closed under submodules. In order to solve this problem, F. Mantese proposed a new concept in [42]. It is proved that the classes of all-reflexive modules (relative to inherited cotilt modules) are closed under submodules. It is well known that all cotilt modules are co-semi-tilting modules. At the end of this paper, we generalize this result to the cosemilateral theory, that is, the classes of all-reflexive modules (relative to the inherited cosemitilted modules) are closed under submodules. In chapter 4, we give some related properties about the cosemilateral modules. With the help of these properties, we obtain a cosemilateral theorem. In the second section of chapter 4, we introduce the cosemi-tilted modules of heredity, and prove that in the cosemitilt theory, the classes of all-reflexive modules (relative to the inherited cosemi-tilting modules) are closed under submodules. Finally, we prove that two functors Hom_R (-, E (C) / C) o Rejc (-) and Ext_R~1 (-, C) are naturally equivalent.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O153.3

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