有限维数猜想及相关课题

In the representation theory of Artin algebras, the study of various homological dimensions is one of the main topics. Among them, the comparison of the homological dimensions of algebras attracted widespread attention. Such as the comparison of the homological dimensions of algebras and its subalgebras, algebras and its quotient algebras and these algebras have certain equivalent relations, and so on. Recently, we studied the relations between the homological dimensions of two algebras involved in several types of extensions of Artin algebras and the relation among the finitistic dimension of algebras induced by an idempotent under certain conditions. Inspired by the work mentioned above, the research of the program will focus on the following aspects: 1. Study the relation among the finitistic dimension of algebras induced by an idempotent, and try to study the finiteness of the finitistic dimension of CPS-stratified algebras. Based on this work and the relation between the finitistic dimension conjecture and the Auslander-Reiten conjecture , we will also discuss the later conjecture for symmetric algebras. 2. Study the relation between the fininitistic dimension of these algebras involved in a radical-full extension, and then try to study the finiteness of the finitistic dimension of biserial algebras by constructing suitable radical-full extensions. 3. Study the relation between the representation dimension of these algebras involved in a radical-full extension.

在Artin代数的表示理论中，代数的各种同调维数是核心研究内容之一。其中对代数的整体维数、有限维数和表示维数的比较研究吸引了广泛的关注，如研究代数和它的子代数、代数与它的商代数以及有着某种等价关系的两个代数的同调维数之间的关系等。最近，我们对几类代数扩张中两个代数的有限维数之间的关系以及由幂等元诱导的三个代数的有限维数之间的关系进行了研究，并取得了一些初步的研究成果。本项目将在这些研究工作基础上进一步对以下几个问题展开研究：一、在更一般的条件下探求由幂等元诱导的三个代数的有限维数之间的关系，进而研究CPS-析层代数有限维数的有限性，并在此基础上研究对称代数是否满足Auslander-Reiten猜想。二、研究根满扩张中两个代数的有限维数之间的关系，并在此基础上通过构造合适的根满扩张研究双列代数有限维数的有限性；三、研究根满扩张中个代数的表示维数之间的关系。