HRS倾斜理论及相关问题

倾斜理论是代数表示论中的基本理论，并且在代数几何、数学物理中有广泛的应用。HRS倾斜理论推广了基于倾斜模的经典倾斜理论。给定阿贝尔范畴A及其挠对，在A的导出范畴中可构造Happel-Reiten-Smalo(HRS)倾斜范畴B。HRS倾斜范畴的构造仅依赖于阿贝尔范畴及其子范畴，从而有更广的适用范围。范畴A和倾斜范畴B的导出范畴之间存在典范的三角函子，称为实现函子。我们前期的工作已经解决了此实现函子的等价性判别问题。本项目将研究以下两个问题：一，非三角等价的实现函子在HRS-倾斜过程中如何变化，即考虑多次HRS倾斜是否会得到等价的实现函子；二，研究实现函子具体构造问题，期望给出HRS倾斜过程中实现函子的具体构造，从而给出范畴A和B之间更精确的联系。本项目对HRS倾斜理论的研究有利于进一步完善HRS倾斜理论以及相应的silting理论。

Tilting theory plays an important role in representation theory of algebras, which has many appliciatons in algebraic geometry and mathematical physics. HRS tilting theory generalizes the canonical tilting theory by tilting modules. Given an abelian category A and its torsion pair, the corresponding Happel-Reiten-Samlo (HRS) tilting is a new abelian category B in the derived category of A. HRS tilting only depends on an abelian category and one subcategory. This advantage makes HRS tilting having more applications. There exists a canonical functor from the derived categories of B and of A, which is called a realization functor. We have solved the problem when this realization functor is an equivalence. In this program, we focus the following two problems: First, the behavior of a non-equivalent realization functor, more precisely, whether the realization functor becomes an equivalent functor under repeated HRS tilting; Second, we study the construction of a realization functor and expect to give a concrete construction of realization functor. It will provide a more explicit connection between categories A and B. This program will be helpful to complete the HRS tilting theory and corresponding silting theory.