本项目为同调代数与非交换代数、非交换代数几何的交叉领域。拟从代数表示论的角度来研究代数的形式光滑性问题。根据模-相对Hochschild上同调对代数的形式光滑性的同调刻划，通过对模-相对Hochschild上同调的探讨，研究代数的导出等价、代数的模范畴的recollement以及代数的导出范畴的recollement，它们分别与代数的形式光滑性之间的内在联系；通过形式光滑双模对形式光滑代数的刻划，利用形式光滑代数构造新的形式光滑代数。在发展代数的形式光滑性理论的同时，促进同调代数、非交换代数、非交换代数几何等相关领域的发展，具有重要的科学意义及研究价值。

This project is a crossed field of homological algebras, noncommutative algebras and noncommutative algebraic geometry. We will investigate the formal smoothness of algebras in the point of view of the representation theory of algebras. Based on the discussion of module-relative -Hochschild cohomology, we study the relations between the formal smoothness of algebras and derived equivalences, between the formal smoothness of and recollements of module categories of algebras, between the formal smoothness and recollements of derived categories of algebras, respectively. Moreover, we try to construct new formally smooth algebra according to a given formally smooth algebra. With the development of the formal smoothness of algebras, we promote a few relevant developments in the fields of homological algebras, noncommutative algebras and noncommutative algebraic geometry, which has important scientific significance and research values.