范畴代数是代数表示论与有限群模表示理论中的重要研究对象，代数的Gorenstein同调性质与代数的分层理论之间有着密切联系，均为当前的热门研究课题。本项目拟研究交换环上的有限范畴代数，特别地，研究其上的Gorenstein同调性质与分层结构。具体来说，项目拟研究交换环上有限范畴代数成为Gorenstein代数的充要条件、有限Gorenstein范畴代数的Gorenstein同调维数以及Gorenstein投射模范畴；研究交换环上有限范畴代数的分层结构，特别是拟遗传结构，并进一步研究拟遗传有限范畴代数的Ringel对偶以及如何通过滤链范畴来描述Gorenstein投射模范畴。这些问题的解决将有助于我们对范畴代数、Gorenstein同调代数以及拟遗传代数的理解。

Category algebras are important subjects in representation theory of algebras and modular representation theory of finite groups, and Gorenstein homological properties and stratification theory of algebras are closely related to each other, both of which recently attract a lot of attention. This project is devoted to the study of finite category algebras over commutative rings, in particular, we study of Gorenstein homological properties and stratification theory of these algebras. More precisely, we will study a sufficient and necessary condition for when a finite category algebra over a commutative ring is Gorenstein; we will study the Gorenstein homological dimension of Gorenstein finite category algebras and the category of Gorenstein-projective modules; we will study the stratified structure, the quasi-hereditary structure in particular, of finite category algebras over commutative rings; we will study the Ringel dual of quasi-hereditary finite category algebras. We will study the category of Gorenstein-projective modules described via filtration categories. The study we carry out in this project will help us to understand the category algebra, Gorenstein homological algebra and quasi-hereditary algebra.