平衡对的一些应用

For a given balanced pair in an abelian category, we will first give a suitable definition of tilting object relative to this balanced pair, next we will investigate under what conditions the tilting theorem relative to this balanced pair holds true. On the other hand, we will consider the relative cotorsion pair induced by the relative tilting object and make sure whether or not this relative cotorsion pair is complete and hereditary. Finally, we will study the derived categories relative to the given balanced pair, we will explore whether or not the bounded relative derived category is idempotent complete and has bounded t-structures. This project is aimed to give some interesting results, which will be significant in homological algebra and representation theory of algebras.

对于Abel范畴中一个给定的平衡对，我们首先合理定义相对于此平衡对的相对倾斜对象的概念，接下来研究相对于此平衡对的倾斜定理在何种条件下是成立的。另一方面，我们考虑由相对倾斜对象诱导的相对余挠对并讨论此相对余挠对是否为完备遗传的。最后我们研究相对于一个给定平衡对的相对导出范畴的性质，考察有界的相对导出范畴是否为幂等完备的并讨论它是否具有有界的t-结构。本项目旨在给出一些有意思的结果，这将在同调代数和代数表示论中具有重要的理论意义。