CPS-析层代数的同调维数及相关课题

The homological dimension of algebras is one of the main objects in the study of the representation theory of Artin algebras. Among them, the finitistic dimension has a close relation with several important homological conjectures. Thus, the study of the finitistic dimension has attracted a wide range of attention. Let e be an idempotent in an Artin algebra A. It is an interesting topic to discuss the relation among the finitistic dimension of A, eAe and A/AeA. Recently, under the condition that AeA is a stratifying ideal with finite projective dimension, we have proved that the finiteness of the finitistic dimension of eAe and A/AeA implies that of the finitistic dimension of A. The program aims to discuss the relation among the finitistic dimension of A, eAe and A/AeA in more general settings. The research will focus on the following three aspects: 1. study the finiteness of the finitistic dimension of A under the condition that the finitistic dimension of eAe and A/AeA are finite, and further study the finiteness of the finitistic dimension of CPS-stratified algebras; 2. study the finiteness of the finitistic dimension of eAe under the condition that the finitistic dimension of A is finite; 3. consider the Auslander -Reiten conjecture for A under the condition that it is true for eAe and A/AeA.

代数的同调维数是Artin代数的表示理论中主要的研究对象之一。其中，代数的有限维数与表示论中其它几个重要的同调猜想有着紧密的联系。因此，对代数的有限维数的研究是表示论中一个十分活跃的研究课题。设A是Artin代数，e是A中的幂等元，对代数A、eAe 和A/AeA的有限维数之间关系的研究吸引了广泛的关注。最近，我们证明了当AeA是A的析层理想且投射维数有限时，eAe和A/AeA的有限维数有限意味着代数A的有限维数也有限。本项目力图在更一般的条件下，进一步探求A、eAe 和A/AeA的有限维数之间的关系，并围绕以下三个主要问题展开研究：1. 当eAe 和A/AeA的有限维数有限时，研究A的有限维数的有限性，进而研究CPS-析层代数的有限维数的有限性；2. 当A的有限维数有限时，研究eAe的有限维数的有限性；3. 当eAe和A/AeA满足AR猜想时，研究A是否也满足AR猜想。

有限维数猜想是Artin代数的表示理论中一个著名的猜想，研究该猜想的方法也是多种多样。本项目主要研究由幂等元诱导的三个代数的有限维数之间的关系，同时也试图利用该方法对表示论中另一个猜想Auslander-Reiten猜想展开研究，并取得了一些较好的成果。该项目共完成SCI论文两篇，其中一篇已被《Journal of Algebra》收录，另一篇已被《Communicaitons in Algebra》录用待发表。