导出范畴是Grothendieck和Verdier在代数几何中引入的，通过众多著名数学家的发展，现已成为当今代数几何和代数表示论等领域中不可缺少的关键方法之一。本项目将利用导出范畴，稳定范畴等一系列同调代数的方法，并借鉴组合和几何方法，对分次导出等价，Phi-分次代数的Koszul对偶理论、导出等价构造以及与Morita型稳定等价的转换关系、稳定范畴的Morita理论及相关不变量等热点，前沿问题开展深层次的系统性研究，并将这些研究结果应用于Broue猜想以及代数群，李代数等领域中出现的结合代数类的研究中，有十分重要意义。

Derived categories were introduced by Grothendieck and Verdier in algebraic geometry. Due to the excellent works of many international leading mathematicians, the method of derived categories is nowadays one of the most important methods in representation theory and geometry. We shall use derived categories, stable categories, together with methods from combinatorics and geometry, to investigate collaboratively and systematically the following important and hot topics: graded dervied equivalences and Koszul duality, construction of derived equivalences and its relationship with stable equivalences of Morita type,and study the Morita theory for stable categories. This project will significantly benefit the study of Koszul duality, the Broue Conjecture and the study of many algebras arising from algebraic gropus and lie theory.