紧生成三角范畴的t-结构及其相关问题

t-structure provides an important approach to construct abelian categories from triangulated categories. It is also an important tool to construct derived equivalences and derived functors. Compactly triangulated categories have arbitrary coproducts and a set of distinguished compact objects. Thus it is possible to apply homotopy limits in the category to construct t-structures. In this program, we plan to study the connections between the t-structures generated by a family of objects of compactly generated categories and its subcategories of compact objects . . In this program, we will study the conditions for a family of objects in a compactly generated triangulated category generating a t-structure. At the same time, we study the behavior of t-structures generated by a family of objects restricted to the subcategories of compact objects. Furthermore, we will consider the conditions for the heart of this t-structure equivalent to a module category. This program will give more theoretic support for the classification of t-structures .

t-结构是从三角范畴构造阿贝尔范畴的重要方法，同时也是研究导出范畴的等价，导出函子的重要工具。紧生成三角范畴具有任意直积和一组紧对象，从而可以利用同伦极限等逼近技巧构造t-结构，这对研究一般的t-结构提供了有利的条件。目前，紧对象生成的t-结构的研究比较成熟，而非紧对象生成t-结构的研究刚刚起步。本项目将具体研究紧生成三角范畴上任意一族对象（紧对象或非紧对象）生成的t-结构和紧子范畴上t-结构之间的对应关系，进一步解决紧生成三角范畴中t-结构的生成问题。. 本项目拟研究紧生成三角范畴中一族对象生成的子范畴成为t-结构的条件。同时，考察由一族对象生成的t-结构在紧子范畴（或者三角子范畴）上的 限制仍是t-结构的条件及两个不同的对象族在生成t-结构何时等价的问题。进一步，研究一族对象生成的t-结构的心等价于模范畴的条件。本项目的研究给三角范畴上t-结构的分类提供重要的依据。