本项目为同调代数、代数K-理论、（非交换）代数几何、代数表示理论的交叉领域。将建立微分分次代数的Hattori-Stallings迹映射理论，并由此证明Extension conjecture对于有限维初等代数成立；将证明有限Hochschild同调维数为开性质，给出判断代数的Hochschild同调维数有限的退化方法；将揭示monomial代数Hochschild上同调维数为零与Gabriel箭图为树、Hochschild上同调维数有限与整体维数有限之间的等价关系，后者表明Happel问题对于monomial代数是肯定的；将澄清Hochschild（上）同调与cleaving函子之间的关系，为比较代数的Hochschild（上）同调提供cleaving函子方法。

Hochschild (co)homology is one of the main topics in homological algebra, which was applied to many mathematical branches such as group theory, Lie theory, algebraic topology, K-theory, and (noncommutative) algebraic geomrtry. This project is the crossing field of homological algebra, algebraic K-theory, (noncommutative) algebraic geometry and representation theory of algebras. We will establish the theory of Hattori-Stallings trace maps, and apply it to show that the Extension conjecture holds for finite dimensional elementary algebras. We will prove that finite Hochschild homology dimension is open, which will provide a degeneration criterion for an algebra to be of finite Hochschild homology dimension. We will clarify the equivalences between the Hochschild cohomology dimension being zero and the Gabriel quiver being tree, and between finite Hochschild cohomology dimension and finite global dimension, for monomial algebras. The latter implies that the Happel's question has a positive answer for monomial algebras. We will reveal the relationship between Hochschild (co)homology and cleaving functors, which will furnish a cleaving method for comparing the Hochschild (co)homologies of algebras.