三角范畴的相对丛倾斜对象及其自同态代数

The cluster-tilting object is a very important research object in a triangulated category, and now it has become a hot issue in representation theory of algebras. As the completion of the cluster-tilting object, the relative cluster-tilting object also has very good properties and important results, which is closely related to other objects. This project studies the relative cluster-tilting object and its endomorphism algebra, which is divided into the following three parts: .1. This project studies the endomorphism algebra of the relative cluster-tilting object and its representations. We firstly characterize its global dimension, and then classify its modules..2. On the basis of the first part, the relationship between the triangulated category and the module category of the endomorphism algebra of the relative cluster-tilting object is further considered. We firstly construct an equivalence of categories using these two categories, and then find bijections between certain important objects in these two categories..3. As an application of the relative cluster-tilting object, we use it to construct a subfactor category of the triangulated category, and prove that it is still a triangulated category..The project will further enrich cluster-tilting theory.

丛倾斜对象是三角范畴中非常重要的研究对象，现已成为代数表示论研究的热点问题。而相对丛倾斜对象作为丛倾斜对象的完备化，同样有着非常好的性质和重要的结果，并且与其他研究对象也有着极其紧密的联系。本项目将研究三角范畴的相对丛倾斜对象及其自同态代数，具体分为以下三个部分：.1.研究相对丛倾斜对象的自同态代数及其表示。本项目先对它的整体维数进行刻画，再对它的模进行分类。.2.在第一部分的基础上，进一步考虑三角范畴与相对丛倾斜对象的自同态代数的模范畴之间的关系。一是证明这两个范畴之间存在某种等价关系；二是研究这两个范畴中的重要对象之间存在的一一对应关系。.3.作为相对丛倾斜对象的应用，本项目将利用它构造三角范畴的子商范畴，并证明它仍是三角范畴。.本项目的研究将进一步丰富丛倾斜理论。

在本项目中，我们的研究包含两方面内容：一方面，我们给出并证明了相对丛倾斜对象的等价定义，由此进一步给出了ghost丛倾斜子范畴的概念；探讨了ghost丛倾斜子范畴与丛倾斜子范畴等相关概念之间的关系；建立了ghost丛倾斜理论与tau-倾斜理论之间的联系。另一方面，我们研究了一般的加法幂等元半环簇CSr(n,1)的子簇，该结果极大地拓宽了前人的相关工作。