相对同调代数中的Gorenstein FP-内射性及应用研究

In this project, we will study Gorenstein homological algebra in the categories of modules and complexes, especially the homological properties of Gorenstein FP-injective modules. Firstly, we study the Holm’ s Conjecture, such as the existence of Gorenstein FP-injective covers for an arbitrary module over coherent rings, the relations between the weak Gorenstein global dimension and the global Gorenstein FP-injective dimension of a coherent ring, the characterization of a noetherian ring by using of Gorenstein FP-injective modules, and that every Gorenstein projective module is Gorenstein flat over an associative ring. Then we shall establish Gorenstein FP-injective complexes in terms of the Gorenstein FP-injectivity of modules. In order to elaborate the inner connection and essential differences between the category of modules and the category of complexes, we will study the Gorenstein FP-injectivity of modules and complexes by contrast；Finally, we will discuss the study of how to link up Gorenstein homological modules with the classical derived functor. Thus we can expand the application sphere of Gorenstein homological algebra, so that enrich and develope the theory of Gorenstein homological algebra.

本项目将研究模与复形范畴中的Gorenstein同调理论，特别是Gorenstein FP-内射模的同调性质.研究凝聚环上任意模的Gorenstein FP-内射覆盖的存在性、凝聚环的弱Gorenstein整体维数与整体Gorenstein FP-内射维数的关系、利用Gorenstein FP-内射模来刻画Noether环及任意结合环上每个Gorenstein投射模都是Gorenstein平坦的等“Holm系列猜想”；利用Gorenstein FP-内射模的性质建立Gorenstein FP-内射复形的同调性质，通过对模范畴和模的复形范畴中的Gorenstein FP-内射性的对比研究，阐明模范畴和复形范畴的内在联系和本质区别；探讨怎样把Gorenstein同调模与经典导出函子联合起来研究，拓广Gorenstein同调理论的应用范围，由此进一步丰富和发展Gorenstein同调代数理论．

本项研究内容包括：模范畴中的Gorenstein FP-内射性的研究，泛Gorenstein同调方法及应用研究，相对模类与同调维数的研究。在数学天元基金的资助下，本项研究取得了一系列的成果：讨论了一类特殊的Gorenstein FP-内射模－强GorensteinFP-内射模及其众多性质, 证明了每个Gorenstein FP-内射模都是一个强Gorenstein FP-内射模的直和项；对任一正整数n，研究了一类广义形式的强Gorenstein FP-内射模，即n-强Gorenstein FP-内射模，并利用该模类刻画了n-强Gorenstein Von Neumann正则环；利用泛Gorenstein同调方法引入了弱Gorenstein投射、内射与平坦模的概念，推广了一些已知结论，部分解决了Holm提出的一个公开问题，即证明了当环R的整体维数r.IFD(R)＜∞时，每个Gorenstein投射左R-模都是Gorenstein平坦的；利用Gorenstein内射模类和Tor函子引入了GI-平坦模的概念，进而刻画了模与环的GI-平坦维数；研究了Gorenstein平坦模类的Ext-右正交类，即Gorenstein余挠模，引入了模与环的Gorenstein余挠维数的概念，并刻画了凝聚环的弱Gorenstein整体维数；此外，还讨论了n-FC环R上的Gorenstein投射模、Gorenstein平坦模和强Gorenstein平坦模之间的关系。