Quiver表示范畴的同调理论与导出范畴

Quiver表示理论在环论、同调代数、李代数、代数几何以及群与代数的表示理论中有着重要的应用。本项目拟研究阿贝尔范畴的quiver表示范畴中的同调理论。首先，研究quiver表示范畴中的经典同调性质、Gorenstein同调性质和余挠理论等同调性质。其次，借助余挠理论等同调性质探讨quiver表示范畴及其导出(同伦)范畴中的粘合、三角等价与紧生成性。最后，通过粘合、三角等价与紧生成性研究覆盖(包络)、余挠理论和Gorenstein同调性质等同调性质，计算和分析各种同调不变量，进而将其应用到环与代数上。这些研究将为揭示quiver表示范畴中同调理论的本质提供新的思路和方法，对探索quiver表示的应用具有重要意义。

The theory of quiver representations has important applications in ring theory, homological algebra, Lie algebra, algebraic geometry and the representation theory of groups and algebras. The purpose of this project is to investigate homological theory in the category of quiver representations with values in an abelian category. Firstly, we will study homological property such as classical homological property, Gorenstein homological property and cotorsion theory in the category of quiver representations. Secondly, using homological property such as cotorsion theory, we will study recollements, triangle equivalences and compactly generatedness of the category of quiver representations and its derived(homotopy) category. Finally, we will consider homological property such as covers(envelopes), cotorsion theory and Gorenstein homological property, calculate and analyze various homological invariants by means of recollements, triangle equivalences and compactly generatedness, and apply them to rings and algebras. These research will provide new ideas and methods in revealing the nature of homological theory in the category of quiver representations, will also play an important role in exploring applications of quiver representations.