加权射影线上的倾斜理论

On tilting theory over weighted projective lines

本文是加权射影线上凝聚层相关范畴中倾斜理论的一些工作综述,主要分为3个部分:1)讨论加权射影线上凝聚层范畴到相应典范代数上有限生成模范畴倾斜过程中“丢失部分”的结构,证明当权型为(2,2,n)时,“丢失部分”是阿贝尔范畴;2)给出权型为(2,2,2,2)的加权射影线上向量丛稳定范畴的一个典范倾斜对象,并证明不存在所有不可分解直和项都是秩为2的向量丛构成的典范倾斜对象;3)利用丛倾斜理论,构造亏格为1且权为3的加权射影线上向量丛稳定范畴的tubular倾斜对象,以及实现权型为(2,2,2,2)的加权射影线上凝聚层范畴及其导出范畴中倾斜对象自同态代数的完全分类.

In this paper,we summarize some of our works on the tilting theory in the category of coherent sheaves over weighted projective lines.Primarily,our summary is divided into three parts:(1)investigating the structure of the“missing part”from the category of coherent sheaves on a weighted projective line to the category of finitely generated right modules on the associated canonical algebra,and proving that,for the weighted projective line of type(2,2,n),the“missing part”carries an abelian structure;(2)finding a canonical tilting object for the stable category of vector bundles on a weighted projective line of type(2,2,2,2),and showing that there does not exist any tilting object consisting only of vector bundles of rank two such that its endomorphism algebra is a canonical algebra;(3)based on the cluster tilting theory,constructing tubular tilting objects in the stable category of vector bundles on a weighted projective line of weight triple and genus one,and classifying the endomorphism algebras of tilting objects in the category of coherent sheaves on a weighted projective line of type(2,2,2,2)and the associated bounded derived category.