Syzygy problems by method of stable categories

我们主要研究射影模链配合物的同伦范畴，因为我们的立场是用同伦范畴的三角形结构来研究稳定范畴。本项目的成果分为以下三组。第一个是关于由扭转对引起的对称性。我们发现了一种具有强对称性的扭转对结构，从而得到了一个新的三角形等价。第二部分是关于单态可表征性（rbm），它测量了对范畴进行三角化的阻碍程度。第三部分是关于正余维模的协同。我们得到了当且仅当环是一个整域时，每个模都等价于某个在一共轭前具有正余维的模。

We mainly studied homotopy categories of chain complexes of projective modules because our standpoint is to use a triangle structure of homotopy categories to investigate stable categories. The results of this project are divided into the following three groups. The first one is on the symmetry caused by torsion pairs. We found a structure of torsion pairs with strong symmetry which leads us to a new triangle equivalence. The second one is on representability by monomorphisms (rbm) which measures the obstruction for the category to be triangulated.The third one is on syzygies of modules with positive codimension. We obtained that every module is equivalent to some module with positive codimension up to first syzygy if and only if the ring is an integral domain.