本文選題：阿貝爾范疇　切入點(diǎn)：正合范疇　出處：《南京大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：正合范疇在同調(diào)代數(shù),代數(shù)表示論,代數(shù)幾何,數(shù)學(xué)物理等學(xué)科中有至關(guān)重要的作用.正合范疇最早是Quillen在1973年提出的.另一方面,傾斜理論起源于有限維代數(shù)的表示理論,該理論最基本的工作由Bernstein-Gelfand-Ponomarev在1973年提出的,之后被Brenner和Butler在1980年進(jìn)行了推廣.長(zhǎng)久以來,正合范疇和傾斜理論得到了許多人的關(guān)注,使得該理論極大地促進(jìn)了同調(diào)代數(shù)和代數(shù)表示理論的發(fā)展.在本文中,我們主要考察了以下三個(gè)方面：阿貝爾范疇中的正合結(jié)構(gòu),張量積函子的相對(duì)左導(dǎo)出函子以及函子范疇中的傾斜理論.全文一共分為四章.第一章主要給出了研究背景和主要結(jié)果.第二章在一個(gè)具有小的Ext群的阿貝爾范疇(?)中,首先證明了平衡對(duì),使得(?)具有足夠的F-投射和F-內(nèi)射對(duì)象的ExtA1(-,-)的子函子F以及使得(?)具有足夠的ε-投射和ε-內(nèi)射對(duì)象的Quillen正合結(jié)構(gòu)ε這三者之間的一一對(duì)應(yīng).然后在該條件下,我們得到了一個(gè)在正合語境下的Wakamatsu引理的加強(qiáng)版本,也證明了滿的預(yù)蓋使其核在他們的右ε一正交類里的ε-分解子范疇和單的預(yù)包絡(luò)使其余核在他們的左ε一正交類里的ε一余分解子范疇這二者是互相唯一決定的.最后我們應(yīng)用這些結(jié)果到模范疇中,構(gòu)造了一些新的(預(yù))包絡(luò)類,(預(yù))蓋類以及一些完備可遺傳的ε一余撓對(duì).第三章介紹了在模范疇中的相對(duì)左導(dǎo)出函子Torn(F,F')(-,-)的概念,統(tǒng)一了其他相關(guān)的左導(dǎo)出函子.然后我們根據(jù)該相對(duì)左導(dǎo)出函子給出了計(jì)算模的F-分解維數(shù)的準(zhǔn)則.我們也構(gòu)造了一些關(guān)于正合結(jié)構(gòu)ε的完備可遺傳的余撓對(duì),并得到一些應(yīng)用.第四章介紹了在函子范疇中n-傾斜對(duì)象和n-余傾斜對(duì)象的概念,并且分別給出了n-傾斜對(duì)象(類)和n-余傾斜對(duì)象(類)的等價(jià)刻畫.
[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.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O153.3

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本文編號(hào)：1612862