理想逼近与函子范畴

项目分成两部分。第一部分，我们将继续研究申请人及其合作者系统建立和发展的理想逼近理论：（1）我们将进一步研究理想逼近理论，并发展方法计算导出范畴和群的模表示论中的ghost数和phantom数；（2）我们将在三角范畴中发展理想逼近理论并研究它的应用；（3）我们将研究理想逼近理论和模型结构之间的联系。在第二部分中，我们将利用我们之前得到的由纯性的三个等价刻画导出的三个粘合来研究函子范畴。具体地说，我们将计算函子范畴中的倾斜和余倾斜对象，研究倾斜理论，计算函子范畴的由稳定函子构成的子范畴生成和上生成的余挠对。

The project is divided to two parts. The aim of the first part is to continue the study of ideal approximation theory that was built systematically by the applicant and his coauthors: (1) we will investigate ideal approximation theory further, and develop methods to study ghost numbers and phantom numbers for derived categories and modular representation theory of finite groups; (2) we will develop the ideal approximation theory in a triangulated category and study its applications; and (3) we will study the connections between ideal approximation theory and model structures. In the second part, we use three recollements which are derived from the three equivalent characterizations of purity to study functor categories. Specifically, we will calculate tilting and cotilting objects, study tilting theory, and calculate cotorsion pairs generated and cogenerated by the subcategories of stable functors in functor categories.