环的凝聚性与复形的相对同调理论

Complexes of modules over rings are fundamental to spectral sequences and derived categories, which are powerful tools in homological algebra. Simultaneously, they can be regarded as a generalization of modules. In recent years, there are increasingly many people who investigate the categories of complexes rather than just adopt complexes as a tool. In this procedure, coherence of rings often plays an important role. In fact, there are a larg amount of known results are based on the coherence of rings and some classical results can be generalized to the case that the base ring is coherent. Therefore, we concentrate on complexes of modules, which are the main object of this project, combining with the consideration on coherence of rings. On one hand, some results in the category of modules will be generalized to that of complexes. On the other hand, hyperhomology theory will be developed by some tools such as covers, envelopes, spectral sequences and derived categories. Meanwhile, some properties of rings will be characterized by complexes of modules over them. Some results will also be generalized to wider instances. The present project contains investigations on coherence of rings, covers (envelopes) and homological dimensions of complexes as well as interior structure and endomorpism rings of complexes. This will enrich the existing theory on relative homology and hyperhomology and provide concrete examples and motivation for the study of those more general categories.

在同调代数中，模的复形是构造谱序列和导出范畴这两大同调工具的基础，同时它还可以看成模的推广。近年来，越来越多的人开始研究复形范畴本身，而不再是仅仅把它当成一个工具。在此过程中，环的凝聚性常常扮演着重要的角色，许多已知的结论都是基于环的凝聚性的，而且一些经典的结果也可以推广到凝聚环的情形。因此，本项目以模的复形为研究对象，结合环的凝聚性，一方面把模范畴中的有关结果推广或应用到复形范畴，另一方面，利用覆盖、包络、谱序列、导出范畴等工具，发展超同调理论，同时利用复形刻画环的有关性质，并将其中的部分结果推广到更一般的Abel范畴。本项目的主要研究内容包括环的凝聚性、复形的覆盖（包络）与同调维数、复形的内部结构与自同态环等。这将进一步丰富现有的相对同调和超同调理论，为研究更一般的范畴提供具体的例子和理论源泉。

模与复形是代数学中重要的研究对象和工具. 人们利用这些工具得到了一些特殊的环与代数的结构、表示以及同调性质等方面丰硕成果. 在向更一般的情形推广的过程中, 环的凝聚性往往起到关键的作用. 本项目围绕环的凝聚性和复形的相对同调理论展开研究. 主要成果包括以下几个方面: (1)在一些特殊的凝聚环上建立了相对于半对偶模的Tate同调Gorenstein同调理论; (2)把模范畴中的有关概念结果推广到复形范畴, 给出了几类重要的复形(如W-Gorenstein复形和Gorenstein FP-内射复形)的性质和等价刻画; (3)研究了自同态环为除环的模以及自同态环为unit-正则环的模性质与结构, 并利用这些模刻画了rudimentary环和V-环; (4)把模与复形范畴中的有关概念和结果推广到更一般的范畴, 提出了相对于一个子范畴的相对导出范畴和相对于一个cotorsion triple的倾斜子范畴的概念. 这些成果为研究环与代数的结构与表示以及Abel范畴和三角范畴提供了新的思路和工具.