正合结构与倾斜理论

发布时间：2018-03-14 20:55

本文选题：阿贝尔范畴　切入点：正合范畴　出处：《南京大学》2016年博士论文　论文类型：学位论文

【摘要】：正合范畴在同调代数,代数表示论,代数几何,数学物理等学科中有至关重要的作用.正合范畴最早是Quillen在1973年提出的.另一方面,倾斜理论起源于有限维代数的表示理论,该理论最基本的工作由Bernstein-Gelfand-Ponomarev在1973年提出的,之后被Brenner和Butler在1980年进行了推广.长久以来,正合范畴和倾斜理论得到了许多人的关注,使得该理论极大地促进了同调代数和代数表示理论的发展.在本文中,我们主要考察了以下三个方面：阿贝尔范畴中的正合结构,张量积函子的相对左导出函子以及函子范畴中的倾斜理论.全文一共分为四章.第一章主要给出了研究背景和主要结果.第二章在一个具有小的Ext群的阿贝尔范畴(?)中,首先证明了平衡对,使得(?)具有足够的F-投射和F-内射对象的ExtA1(-,-)的子函子F以及使得(?)具有足够的ε-投射和ε-内射对象的Quillen正合结构ε这三者之间的一一对应.然后在该条件下,我们得到了一个在正合语境下的Wakamatsu引理的加强版本,也证明了满的预盖使其核在他们的右ε一正交类里的ε-分解子范畴和单的预包络使其余核在他们的左ε一正交类里的ε一余分解子范畴这二者是互相唯一决定的.最后我们应用这些结果到模范畴中,构造了一些新的(预)包络类,(预)盖类以及一些完备可遗传的ε一余挠对.第三章介绍了在模范畴中的相对左导出函子Torn(F,F')(-,-)的概念,统一了其他相关的左导出函子.然后我们根据该相对左导出函子给出了计算模的F-分解维数的准则.我们也构造了一些关于正合结构ε的完备可遗传的余挠对,并得到一些应用.第四章介绍了在函子范畴中n-倾斜对象和n-余倾斜对象的概念,并且分别给出了n-倾斜对象(类)和n-余倾斜对象(类)的等价刻画.
[Abstract]:The exact category plays an important role in homology algebra, algebraic representation theory, algebraic geometry, mathematics and physics. The exact category was first proposed by Quillen in 1973. On the other hand, the tilting theory originated from the representation theory of finite dimensional algebra. The most basic work of this theory was put forward by Bernstein-Gelfand-Ponomarev in 1973, and then extended by Brenner and Butler in 1980. In this paper, we mainly study the following three aspects: the exact structure in Abelian category. The relative left derived functor of tensor product functor and the tilting theory in the category of functors are divided into four chapters. The first chapter gives the research background and main results. ), the first proof is that the equilibrium pair is such that? ) A subfunctor F with sufficient F- projection and F- injective object ExtA1 + -) and such that? ) A one-to-one correspondence between the Quillen exact structure 蔚 with sufficient 蔚 -projective objects and 蔚 -injective object. Then, we obtain an enhanced version of the Wakamatsu Lemma in an exact context. It is also proved that the full precover such that the kernels in their right 蔚 -orthogonal class are 蔚 -decomposed subcategories and the simple preenvelope so that the remaining kernels in their left 蔚 -orthogonal class have 蔚 -cofactorization subcategories are mutually unique determinants. Finally, we apply these results to the category of modules. In this paper, we construct some new (pre-) envelop classes and complete heritable 蔚 -cotorsion pairs. In Chapter 3, we introduce the concept of relative left derived functor Tornn (FNF) -FFN- (-) in the category of modules, and the following results are obtained: (1) in this paper, we introduce the concept of the relative left derived functor (Tornn) in the category of modules. In this paper, we unify other related left derived functors. Then we give the criteria for calculating the F-decomposition dimension of the modules according to the relative left derived functors. We also construct some complete hereditary cotorsion pairs for the exact structure 蔚. In chapter 4th, the concepts of n- tilted object and n- cotilting object in functional subcategory are introduced, and the equivalent characterizations of n- tilted object (class) and n-cotilt object (class) are given respectively.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O153.3

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