三角范畴的Gorenstein同调理论和Brown可表示性

本项目主要研究三角范畴的Gorenstein同调理论，并以Brown可表示性为工具，建立三角范畴与Abel范畴中的Gorenstein同调理论之间的联系。令 E 表示三角的真类，其作用相当于正合范畴中短正合序列的类。我们将研究三角范畴的E-Gorenstein同调维数，构造E-Gorenstein仿真塔和E-Gorenstein胞腔塔，从Gorenstein同调代数的视角揭示三角范畴的拓扑性质和相对同调性质；研究三角范畴的E-Tate上同调理论，建立连接E-上同调群、E-Gorenstein上同调群和E-Tate上同调群的Avramov-Martsinkovsky型正合序列；以Brown可表示性为工具，考察三角范畴与模范畴中的Gorenstein投射、内射对象以及相应的同调维数之间的联系，探索Gorenstein同调理论在稳定同伦范畴、导出范畴等一些重要的三角范畴中的应用。

The main purpose of this project is to study Gorenstein homological theory for triangulated categories, and build relationship for Gorenstein homological theories between triangulated categories and abelian categories by applying the Brown representability. Let E denote the proper class of triangles, which is an analogous of the class of short exact sequences in exact categories. We will study E-Gorenstein homological dimensions for triangulated categories and consider the construction of E-Gorenstein phantom tower and E-Gorenstein cellular tower, in order to reveal topological and relative homological properties of triangulated categories from the perspective of Gorenstein homological algebra. Moreover, we will study E-Tate cohomology, and establish an Avramov-Martsinkovsky type exact sequence which relates E-cohomology, E-Gorenstein cohomology and E-Tate cohomology groups. Taking the Brown representability as a tool, we will consider the connections of Gorenstein projective, injective objects and corresponding homological dimensions in triangulated categories and the category of modules, and explore applications of Gorenstein homological theory in some important triangulated categories such as stable homotopy category and derived category.