复形范畴中的相对同调与Auslander-Reiten理论

In the project, we choose the suitable class W of modules to construct complexes with kinds of properties, such as #-W complexes, exact #-W complexes, DG-W complexes, Cartan-Eilenberg-W complexes and so on, we will study approximation properties of these classes of complexes and their left or right orthogonal classes, solve some hot-spot homological issues in the category of modules at present (for example, the conjecture of Gorenstein flat covers); we will introduce some homological dimensions of complexes, characterize them by using functors Ext and Tor, discuss the difference and relation between these dimensions of complexes and corresponding ones of modules coming from complexes, and give characterizations of homological properties of rings described by complexes; we will also study Auslander-Reiten theory by combining relatively homological theory, we will establish Auslander-Reiten formulas and existence theorems of almost split sequences (triangles) in the categories of complexes and its subcategories, exact categories of complexes, homotopy categories and categories of N-complexes; we will give characterizations of almost split sequences with respect to generalized functors Ext, and find out differences or connections between such sequences and usual almost split sequences. The project is of great importance for understanding algebraic properties of complexes, for studying homological properties of rings described by complexes, and for further enriching and developing homological algebra and representation theory.

选取适当的模类W构造具有各种性质的复形，包括#-W复形、正合#-W复形、DG-W复形、Cartan-Eilenberg-W复形等，研究这些复形类及其左右正交类的逼近性质，解决当前模范畴中的一些热点同调问题（如：Gorenstein平坦覆盖猜想）；引入复形的多种同调维数，利用Ext和Tor函子刻画并研究这些维数和构成复形的模的相应同调维数的区别和联系，给出环的由复形表述的一些同调性质的刻画；结合相对同调理论研究Auslander-Reiten理论，在复形范畴及其子范畴、复形的正合范畴、同伦范畴和N-复形范畴中建立Auslander-Reiten公式和几乎可裂序列（三角）的存在性结果；给出相对于广义Ext函子的几乎可裂序列的刻画以及它们和一般几乎可裂序列之间的关系。本项目对于理解复形的代数性质，研究环的由复形表述的同调性质，进一步丰富和发展同调代数与表示理论具有重要意义。

本项目研究了由投射模和Gorenstein投射模构成的复形的性质，由此给出了quasi-Frobenius环整体维数为零的一个充分必要条件；引入了模的局部同态上的CI维数，研究了这种维数与局部同态上的G维数、CI维数、和投射维数的关系；在交换Noetherian环上, 研究了相对于半对偶化模的稳定同调模，给出了模的同调维数有限的函子刻画；给出了Ding投射和Ding内射复形的结构刻画，改进了James Gillespie的相关结果；引入了Clean-正合以及Clean-导出范畴的概念,分别给出了Clean-短正合列和Clean-正合复形的等价刻画，研究了Clean-导出范畴的性质；研究了内射模构成的复形类的逼近理论，证明了在Noether环上，内射模构成的复形类和其左正交类构成完备的内射余挠对；推广和统一了Gorenstein平坦模的的概念。