本项目研究Tate-Vogel（上）同调理论及其应用。将揭示Tate-Vogel（上）同调理论与pinched同调代数理论之间的关系，给出计算Tate-Vogel（上）同调的新方法，进而在任意环上研究Tate-Vogel（上）同调的"平衡性"；将深入研究Tate-Vogel同调与Tate同调之间的内在联系，进而给出Tate同调的"泛性质"，并解决Holm猜测"所有的Gorenstein投射模都是Gorenstein平坦的"；将借助Tate-Vogel同调理论研究经典的Depth公式，将这一公式的研究拓展到任意模上（不一定要求其投射维数、完全交维数、Gorenstein维数等有限），进而给出Tate-Vogel同调理论在局部代数中的应用。在本课题的研究中，既有新问题、新理论，又有新思想、新方法。这对于丰富和发展同调代数和交换代数具有重要的意义。

In this project, we study Tate-Vogel (co)homology theory and its applications. We will give some relations between Tate-Vogel (co)homology theory and the pinched homological algebra theory. Then we will provide some new methods for computing Tate-Vogel (co)homology so that we can study balancedness of Tate-Vogel (co)homology over arbitrary associative rings; We will study the relations between Tate-Vogel homology and Tate homology, and then we will prove a universal property for Tate homology and solve Holm's conjecture which shows that all Gorenstein projective modules are Gorenstein flat; We will study the classical depth formula by using Tate-Vogel homology theory so that this formula can be used well for modules that may have infinite projective dimension, complete intersection dimension, Gorenstein dimension, etc.. Then we will give some applications of Tate-Vogel homology theory in local algebra. In this project, there are new questions and theories as well as new ideals and methods. The project is important for enriching and developing Homological Algebra and Commutative Algebra.