丛倾斜范畴及其在同调代数中的应用

丛倾斜理论起源于丛代数的范畴化，是目前十分活跃且备受关注的重要课题。本项目将利用模型结构来研究三角范畴关于丛倾斜范畴的商及其同伦范畴的结构，希望利用模型结构在研究相关的丛倾斜理论时将三角范畴上的"Hom-有限"和"skeletally small"等条件推广到一般情形；利用相对同调代数中的工具研究自内射维数不大于n+1且满足Auslander型条件的Artin代数以及相对于余倾斜模的Auslander-Iyama-Solberg对应；利用函子范畴的理论研究阿贝尔范畴上的预丛倾斜理论以及范畴化的Auslander-Iyama-Solberg对应；利用丛倾斜代数与倾斜代数的关系研究丛倾斜代数的Tate-Hochschild上同调，进一步理解和加深丛倾斜理论在代数表示论和同调代数方面的作用。

Cluster tilting theory is an important project that originated from the categorization of cluster algebras. Now it is very active and has attracted much attention. In this project, we will investigate the structure of the quotient of triangulated categories with respect to cluster tilting categories and its homotopy categories using the tool of model structures. We hope to generalize the conditions "Hom-finite" and "skeletally small" to the general cases when we study the related cluster tilting theory. We will investigate a class of algebras of having self-injective dimension at most n+1 and satisfying Auslander-type condition and study the Auslander-Iyama-Solberg Correspondence relative to a cotilting module using the tools in relative homological algebras. We will study the precluster tilting theory over abelian categories, and obtain the categorified Auslander-Iyama-Solberg Correspondence using the theory of functor categories. We will describe the Tate-Hochschild cohomology of cluster tilted algebras using the relation of cluster tilted algebras and tilted algebras. This project will help us to further understand and develop cluster tilting theory in the representation theory of algebras and homological algebras.