三角范畴在相对同调理论中的应用

发布时间：2018-05-28 00:26

本文选题：三角范畴 + 同伦范畴　；参考：《南京大学》2016年博士论文

【摘要】：在Abelian范畴中,对于具有某种投射(或内射)性质的对象的研究是相对同调代数的主要课题.现在经典同调代数的研究已经广泛的应用了三角范畴的工具,Bousfield的局部化理论是其基础之一,而这一理论实质上给出三角范畴的一个粘合[45].另一方面,在同伦范畴中,由投射和内射对象诱导的粘合给出复形的同调分解[54],进而可以引入并研究复形的同调维数.在本文的第二章,我们研究了一个平衡对如何给出同伦范畴的一个粘合.在第三章,我们考虑了由纯性诱导的平衡对在同伦范畴中的表现,定义并研究了复形的纯同调维数.Auslander-Reiten理论是Artin代数表示理论研究的有力工具.而这一理论已经被广泛的应用到三角范畴的研究中,在第四章,我们证明了Artin代数上的有限表现模的有界同伦范畴存在Auslander-Reiten三角,并且给出一些应用.在Gorenstein环上,关于有限生成Gorenstein投射模的奇点等价被很多学者所关心.我们试图将这一等价的条件放到最弱.在第五章,我们将这一等价在任意环上对于任意的Gorenstein投射模给出证明.
[Abstract]:In the Abelian category, the study of objects with certain projective (or injective) properties is the main subject of relative homology algebra. Now the research of classical homology algebra has widely applied the tool of triangular category. The localization theory of Bousfield is one of its foundations, and this theory essentially gives a binding of the triangle category. 45]. on the other hand, in the homotopy category, the homotopy decomposition [54] is given by the adhesion induced by projective and injective objects, and then the homology dimension of the complex can be introduced and studied. In the second chapter of this paper, we study how a balance gives a bonding of the homotopy category. In the third chapter, we consider the pure induced leveling. The.Auslander-Reiten theory is a powerful tool for the study of Artin algebra representation theory, which has been widely applied to the study of trigonometric categories. In the fourth chapter, we prove the bounded homotopy category of finite expression modules on Artin algebras. In the Auslander-Reiten triangle, and give some applications. On the Gorenstein ring, the odd point equivalence of the finite generating Gorenstein projective modules is concerned by many scholars. We try to put this equivalent condition to the weakest. In the fifth chapter, we give a proof of this equivalence to any Gorenstein projective module on any ring.
【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】：O154.1

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