奇点范畴与加权射影直线

The singularity categories of algebras and weighted projective lines are important subjects in representation theory of algebras.These two subjects are closely related to each other, both of which recently attract a lot of attention. This project is devoted to the study of homological properties of (graded) singularity categories and the stable categories of vector bundles over weighted projective lines. More precisely, we will study the mutation relation among different tilting objects in some graded singularity categories, and the classification problem of exceptional sequences; we will study Lenzing's problem on weighted projective lines, that is, whether the stable categories of vector bundles dertemine the type of the weighted pojective lines; we will study homological invariants of singularity categories, which will be applied to the study of singular equivalences for some classes of algebras; we will compute the Hochschild-Tate cohomology and study the characteristic map of the singularity category of an algebra; we will study Coxeter groups associated to weighted projective lines, which are supposed to be related to the classification of exceptional sequences in the abelian category of coherent sheaves on weighted projective lines. The study we carry out in this project will help us to understand the inner structures of certain (graded) singularity categories, the categories of coherent sheaves on weighted projective lines and the stable categories of vector bundles.

代数的奇点范畴与加权射影直线均为代数表示论中重要的研究对象，两者有紧密的联系，又同为当前的热门研究课题。本项目拟研究（分次）奇点范畴以及加权射影直线上的向量丛稳定范畴的同调性质。具体来说，项目拟研究具体分次奇点范畴中不同倾斜对象之间的mutation关系以及例外序列的分类问题；研究加权射影直线的Lenzing问题，即，向量丛稳定范畴是否能决定加权射影直线的型；研究奇点范畴的同调不变量，用以判别具体代数类的奇异等价性；研究Hochschild-Tate上同调以及奇点范畴的特征映射；研究加权射影直线的Coxeter群，用以分类凝聚层范畴中的例外序列。这些问题的解决有助于对代数的（分次）奇点范畴、加权射影直线的凝聚层范畴以及向量丛稳定范畴的理解。

代数的奇点范畴与加权射影直线均为代数表示论中重要的研究课题，其研究涉及到Gorenstein同调代数、非交换代数几何、经典奇点理论以及Leavitt路代数表示理论。 我们主要研究具体代数奇点范畴的刻画问题，奇点范畴之间的三角等价问题，以及加权射影直线凝聚层范畴。我们的主要成果如下：1. 利用Leavitt路代数的微分分次导出范畴刻画了根方零代数奇点范畴的完备化范畴；2. 利用代数的左缩进以及Nakyama代数的分解箭图研究了Nakayama代数的奇点范畴；3. 利用代数的同调满态射以及模的自同态代数构造了有限维代数奇点范畴之间的三角函子；4. 利用等变范畴的理论研究了tubular型加权射影直线的凝聚层范畴。上述成果有力推动了具体代数奇点范畴的研究，深化了我们对加权射影直线凝聚层范畴的认知，也为下一步研究奠定了基础。