模型结构与相对同调

The main purpose of this project is to study model structures and relative homology. First of all, we set up the corresponding model structures by constructing the cotorsion pairs based on a properly chosen class of special modules (or objects). Secondly, by making full use of the theory of model structures, we investigate relative cohomology functors and homological dimensions in the category of complexes, and establish the Avramov-Martsinkovsky sequence connecting absolute and relative cohomology functors. Finally, we explore the existence of (pre)envelopes and (pre)covers in the functor category and determine the corresponding relations between homological properties in the functor category and their counterparts in the module category. It is expected that the results obtained in this project will be helpful to solve some homological conjectures.

本项目主要研究模型结构与相对同调。首先，我们适当选取特殊性质的模（或对象）类，通过构造相应的余挠对，建立对应的模型结构。其次，充分利用模型结构理论，研究复形范畴中的相对上同调函子及同调维数，建立连接复形的经典上同调函子和相对上同调函子的 Avramov-Martsinkovsky 型正合序列。最后，探讨函子范畴中（预）包络和（预）覆盖的存在性，确定函子范畴与模范畴中相应同调性质的对应关系。希望本项目所得结果有助于一些同调猜测的解决。

本课题部分解决了Enochs猜测，作为应用，解决了凝聚环用FP-内射模的预覆盖的存在性刻画的公开问题。利用模型结构理论成功地构造了形式三角矩阵环上的余挠对，由此研究了相应的同伦范畴之间的粘合，同时研究了形式三角矩阵环上的Gorenstein同调理论。我们还研究了phantom包络和Ext-phantom覆盖的内在联系，并刻画了相关环类。本项目所得主要结果进一步丰富了模型结构与同调理论的研究。