本项目主要研究复形的 Tate-Vogel 上同调与相对上同调。首先，我们拟从模范畴中满足一定条件的对偶对与 AB-序对出发，寻求合适的余挠对并定义相应的复形的同调维数。其次，以余挠对为工具，构造复形范畴中的 Tate-Vogel 上同调函子，研究复形的同调维数以及与有限维猜测相关的问题，并比较复形的 Gorenstein 平坦维数与平坦维数、Ding-投射维数与投射维数以及 Ding-内射维数与内射维数。最后，利用 Tate-Vogel 上同调理论，研究复形的相对上同调，并探讨其与 Tate-Vogel 上同调之间的联系。本项目的研究对于丰富和发展复形的同调理论具有重要意义。

The main object of this project is to study Tate-Vogel cohomology and relative cohomology for complexes. First of all, we will find appropriate cotorsion pairs and define the corresponding homological dimensions of complexes based on duality pairs and AB-contexts with some additional conditions in the category of modules. Secondly, we will take cotorsion pairs as a tool to construct Tate-Vogel cohomology functors in the category of complexes and study the homological dimensions of complexes and some problems related to the Finitistic Dimension Conjecture. Furthermore, we will compare Gorenstein flat (resp., Ding-projective, Ding-injective) dimensions and flat (resp., projective, injective) dimensions for complexes. Finally, we will study the relative cohomology of complexes by Tate-Vogel cohomology theory. The connections between the relative cohomology and the Tate-Vogel cohomology will be considered. The study of this project will play an important role in the homology theory of complexes.