半对偶化模及其相关模类的同调性质

Semidualizing module and its homological theory is a main object in the study of relative homological algebra. It has a profound connection with classical homological algebra, Gorenstein homological algebra and (generalized) tilting theory. In this project, we will investigate the relations between the Auslander class, Bass class and the classes of modules with finite C-Gorenstein projective dimension and finite C-Gorenstein injective dimension. Then we will give some equivalent conditions for a semidualizing module to be dualizing with some properties of the dimensions induced by the classes of modules of C-projective, C-Gorenstein projective and their perpendicular classes. Also, we will study the relation between a semidualizing module and its perpendicular class, in particular, we will investigate the relations between a semidualizing module and the Auslander class and Bass class, and finally, we will characterize a semidualizing module by some properties of the Auslander class and Bass class.

半对偶化模及其同调理论是相对同调代数的重要研究对象之一，它与经典同调代数，Gorenstein 同调代数，（广义）倾斜理论都有非常深刻地联系。本项目将以平凡扩张为工具，研究半对偶化模C所对应的Auslander 类和Bass 类与C-Gorenstein 投射维数和C-Gorenstein 内射维数有限的模类的关系，进而利用C-投射模，C-Gorenstein 投射模等模类及其正交类所诱导的维数的性质给出半对偶化模为对偶化模的等价条件。同时，本项目也将研究半对偶化模与其特殊正交类之间的关系，特别是，结合广义倾斜模的相关结论，给出半对偶化模与其Auslander 类和Bass 类的关系，从而利用Auslander 类和Bass 类给出半对偶化模的刻画。

本项目致力于研究与半对偶化模C相关的相对同调代数理论以及Gorenstein同调代数理论。我们研究了C-Gorenstein内射模和C-Gorenstein平坦模所诱导的维数，研究了张量函子关于这两类模的Gorenstein右导出函子，得到了一些关于函子平衡性的结论，并利用这些结论给出了一些半对偶化模成为对偶化模的刻画。我们引入了C-Gorenstein-FP内射模，研究了C-FP内射，C-FP投射以及C-Gorenstein-FP内射模类的性质，证明了这些模类具有某种稳定性，此外，我们利用类似的方法证明了强Gorenstein平坦模范畴以及强Grenstein平坦复形范畴都具有一定的稳定性。