范畴代数的Gorenstein同调性质

发布时间：2018-04-03 09:13

本文选题：范畴代数　切入点：有限EI范畴　出处：《中国科学技术大学》2017年博士论文

【摘要】：本博士论文研究有限EI范畴代数的Gorenstein同调性质.具体来说,我们研究了有限EI范畴代数的Gorenstein性、有限EI范畴代数上Gorenstein投射模的张量积、有限EI范畴代数上平凡模的极大Cohen-Macaulay逼近以及有限EI范畴代数的奇点范畴的谱等相关内容.论文的具体安排如下:在第一章中,我们首先回顾了范畴代数、张量三角范畴、奇点范畴以及Goren-stein 同调代数的历史起源与发展现况.然后,我们介绍了论文的主要结果.最后,我们简要地介绍了论文的结构.在第二章中,我们介绍了范畴代数、Gorenstein同调代数以及三角矩阵环的基本定义和已知结果.特别地,我们给出了上三角矩阵环上的投射模、内射模以及Gorenstein投射模的具体刻画.然后,我们介绍了后面章节将要用到的一些工具.在第三章中,我们首先回顾了上三角矩阵环的定义和基本性质.然后,我们观察到有限EI范畴代数与某个上三角矩阵代数同构.从而,我们以上三角矩阵代数为工具给出了有限EI范畴代数成为Gorenstein代数的充分必要条件:有限EI范畴代数是Gorenstein代数当且仅当所给定的有限EI范畴为有限投射的.在这里,我们引入了新的概念:投射范畴.最后,我们给出了有限自由EI范畴的等价刻画,并给出了有限EI范畴代数是1-Gorenstein代数的充分必要条件:有限EI范畴代数是1-Gorenstein代数当且仅当所给定的有限EI范畴为有限投射自由EI范畴.在第四章中,我们首先给出了有限EI范畴代数上Gorenstein投射模张量封闭的等价刻画:有限EI范畴代数上Gorenstein投射模张量封闭当且仅当其上的投射模张量封闭.然后,我们给出了有限投射的EI范畴代数上Gorenstein投射模张量封闭的一个必要条件:所给定的有限投射的EI范畴中的每个态射均为单射.最后,我们证明了有限投射的自由EI范畴代数上Gorenstein投射模张量封闭的一个充分必要条件为所给定的范畴中的每个态射均为单射.在第五章中,我们首先回顾了有限EI范畴代数的模范畴与某个函子范畴等价,并由此等价将有限EI范畴代数的模范畴与此函子范畴视为一致.然后,在有限自由EI范畴的条件下,我们具体构造了函子E,并且证明了:若所给有限自由EI范畴还是投射的,则我们所构造的函子E,看作范畴代数上的模,是Gorenstein投射模.我们还给出了平凡模成为Gorenstein投射模的条件.最后,我们得到了本章的主要结果:上述所构造的函子E是范畴代数上平凡模的极大Cohen-Macaulay 逼近.我们所构造的函子E是 Gorenstein 投射模范畴的稳定范畴的张量单位.在第六章中,我们考察了 Gorenstein范畴代数的奇点范畴的谱.我们回顾了Schur函子的概念,并用它来描述限制函子.从而,我们观察到,限制函子是Verdier商函子.我们利用Verdier商函子重新证明了徐斐关于范畴代数上的模范畴的有界导出范畴的谱的刻画.利用Verdier商函子,我们给出了关于Gorenstein范畴代数的奇点范畴的谱的类似的刻画.
[Abstract]:Gorenstein homological properties of this doctoral dissertation research category EI algebra. Specifically, we studied the Gorenstein of finite category EI algebra, EI algebra Gorenstein finite projective tensor approximation, maximal Cohen-Macaulay trivial module finite EI algebra and EI algebra in the category of finite category singularity spectrum and other related content. The arrangement of this paper are as follows: in the first chapter, we first reviewed the category of algebra, tensor triangulated category, in singularity category Goren-stein homological algebra and historical origin and development. Then, we introduce the main results of this paper. Finally, we briefly introduce the structure of the thesis. In the second chapter, we introduce the category Gorenstein algebra, homological algebra and triangular matrix rings of basic definitions and known results. In particular, we give a projective upper triangular matrix rings, Injective modules and specific characterizations of Gorenstein projective modules. Then, we introduce some tools that will be used in later chapters. In the third chapter, we first review the definition of upper triangular matrix rings and basic properties. Then, we observed the finite EI algebra and a category three angle matrix algebra isomorphism. Thus, we over triangular matrix algebra presents a necessary and sufficient condition for Gorenstein algebras of finite EI algebra to be a category: limited category of EI algebra is Gorenstein algebra if and only if the given finite EI category for finite projective. Here, we introduce a new concept: the project category. Finally, we give an equivalent characterization of Co. free EI category, and given the limited EI algebra is a sufficient and necessary condition for 1-Gorenstein algebras, EI algebras are 1-Gorenstein algebras of finite category if and only if the given category of EI Co. Finite projective free EI category. In the fourth chapter, we first give the equivalent characterizations of finite EI algebras category Gorenstein projective tensor closure: finite EI category algebra Gorenstein projective tensor closed IFF closed on the projective tensor. Then, we give a necessary condition of EI algebra the finite projective Gorenstein projective tensor closure: every morphism category EI finite projective given in all single shot. Finally, we prove that every morphism category and a necessary and sufficient condition of free EI algebra on finite projective Gorenstein projective tensor closed as given in the is a single shot. In the fifth chapter, we first review of a category of finite EI algebra and a functor category is equivalent to the category, and thus the model domain equivalent finite EI algebras and the category of the functor category as Consistent. Then, in the EI category under the condition of limited freedom, we construct the functor E, and prove that if the limited free EI category or projection, we constructed the E functor, as the category of algebraic model, Gorenstein projective modules. We also give the ordinary mode become Gorenstein projective modules. Finally, we obtain the main results of this chapter: the structure of the functor E is maximal Cohen-Macaulay approximation trivial module category algebra. We construct the functor E is a tensor category Gorenstein projection unit stable module category. In the sixth chapter, we study the category of singularities Gorenstein algebra the spectrum. We review the concept of Schur functors, and use it to describe the restriction functor. Thus, we observed that the restriction functor Verdier functor. We use Verdier to prove Xu Fei about taking functor A category of the category of algebraic characterization of bounded derived category of the spectrum. By Verdier quotient functor, we give a description of the singularity category of a Gorenstein category algebra spectrum is similar.

【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】：2017
【分类号】：O154.1

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