同伦范畴与紧生成子

发布时间：2017-12-28 11:33

本文关键词：同伦范畴与紧生成子　出处：《中国科学技术大学》2016年博士论文　论文类型：学位论文

【摘要】：这是一篇研究同伦范畴的博士学位论文,主要包含以下两个方面的内容。1.对于不含sink点的有限箭图Q及其所对应的有限维根方零代数A,我们研究A的内射模正合复形的同伦范畴Kac(A-Inj)。该同伦范畴是紧生成的三角范畴。在本文第三章,我们引入了Q的内射Leavitt复形,并证明了内射Leavitt复形是正合的。通过构造内射Leavitt复形的子复形滤链,我们得到了第三章的主要结果:(1)设Q为不含sink点的有限箭图。则Q的内射Leavitt复形是同伦范畴K_(ac)(A-Inj)的紧生成子。在本文第四章,我们首先回顾了Leavitt路代数、微分分次代数以及微分分次模的基本定义。我们赋予了内射Leavitt复形某个Leavitt路代数模结构,使得其为微分分次双模,然后证明了:(2)设Q为不含sink点的有限箭图。则Q的内射Leavitt复形是微分分次右拟平衡双模。特别地,我们得到:内射Leavitt复形的微分分次自同态代数拟同构于Leavitt路代数。此时,Leavitt路代数是自然Z-分次代数且被视为微分为零的微分分次代数。这给出了Leavitt路代数新的同调刻画。2.我们考虑上三角矩阵环的同伦范畴。设R,S是任意的两个含幺环,RMS为只-S-双模且R[M]为上三角矩阵环(?)。在本文第五章,我们引入了上三角矩阵环的下零同伦复形的概念,并证明了:(1)对于上三角矩阵环,存在同伦范畴之间的ladder,其高度为2。我们明确地写出了该三角范畴粘合的所有三角函子。通过对三角函子做左导出和右导出,我们得到了上三角矩阵环导出范畴的三角范畴粘合。在本文第六章,我们有如下结论。(2)设R[M]为上三角矩阵环。假设MS是平坦右S-模,则存在如下局部化序列:若S为单环,则R[M]为R关于M的单点扩张。根据上述结论可得:单点扩张保持内射模正合复形的同伦范畴。若R为单环,则R[M]为S关于M的单点余扩张。我们利用同伦极小复形证明了:单点余扩张保持内射模正合复形的同伦范畴。
[Abstract]:This is a doctoral dissertation on the category of homotopy, which mainly contains the following two aspects. 1. for the finite arrow graph Q without sink points and their corresponding finite dimensional root null algebra A, we study the homotopy category Kac (A-Inj) of the injective modules of A. The homotopy category is a triangulated category which is tightly generated. In the third chapter of this paper, we introduce the Q's injective Leavitt complex, and prove that the injective Leavitt complex is a coincidence. By constructing the subcomplex filter chain of an injective Leavitt complex, we get the main results of third chapters: (1) set Q to be a finite arrow with no sink point. The internal injective Leavitt complex of Q is the compact generation of the homotopy category K_ (AC) (A-Inj). In the fourth chapter of this paper, we first review the basic definitions of Leavitt road algebra, differential sub algebra and differential sub modules. We give the injective Leavitt complex, a Leavitt path algebra module structure, and make it a differential graded double module. Then we prove: (2) Q is a finite quiver without sink points. The internal injective Leavitt complex of Q is a differential and graded right quasi equilibrium double mode. In particular, we obtain that the differential fractional automorphism algebra of an injective Leavitt complex is quasi isomorphic to the Leavitt path algebra. At this point, the Leavitt path algebra is a fractional algebra of a natural Z- subalgebra which is regarded as a zero differential. This gives a new homological characterization of Leavitt's path algebra. 2. we consider the homotopy category of the triangular matrix ring. Set R, S is arbitrary two unitary rings, RMS is only -S- double mode and R[M] is the upper triangular matrix ring (?). In the fifth chapter, we introduce the concept of the lower zero homotopy complex of the upper triangular matrix ring, and prove that: (1) for the upper triangular matrix ring, there is a homotopy category between ladder, whose height is 2. We explicitly write all the triangulated category of bonding triangle functor. The left and right of the triangle derived derived functors, we obtain a triangulated category of upper triangular matrix ring derived category of adhesive. In the sixth chapter of this article, we have the following conclusions. (2) set R[M] as the upper triangular matrix ring. Assuming that MS is a flat right S- module, there is a sequence of localization as follows: if S is a single ring, then R[M] is a single point expansion of R on M. According to the above conclusion, it is obtained that the single point expansion keeps the homotopy category of the injective mode complex. If R is a single ring, then R[M] is a single point Yu Kuozhang of S about M. We use the homotopy minimal complex to prove that the single point residual expansion keeps the homotopy category of the injective module.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O154.1

【相似文献】