复形范畴中的相对同调与Auslander-Reiten理论

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。

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.