逼近理论与同调方法

逼近的思想源于模的内射包络和投射覆盖的经典理论，它不仅为同调理论提供了新的研究对象及研究工具，而且为同调理论的研究注入了新思想及新方法，并在相关学科中有重要的应用价值。另一方面，同调方法又进一步丰富和促进了逼近理论的发展。本项目旨在利用同调方法研究逼近理论中的一些重要问题。这些问题包括：一般情况下逼近与极小逼近的存在性，特别是左极小平坦逼近、右（极小）FP-内射逼近、右纯内射逼近、纯投射以及有限和局部投射逼近的存在性；Gorenstein同调代数中的一些热点问题； Cohen-Macaulay 环与 Gorenstein 环的结构与分类； 建立新的导出函子，引进新的同调维数和其它新的同调不变量。希望本项目所得结果能统一已知的一些经典结果，进一步丰富逼近和同调理论，解决或部分解决一些著名猜测。

The idea of approximation theory starts from classical works on injective envelopes and projective covers of modules.It provides new objects and new tools as well as new ideas and new methods of research for homoloy theory and has important applications in related fields. On the other hand, homological methods further enrich and promote the development of approximation theory. The aim of this project is to make use of homological methods to study some important questions in approximation theory. These questions include: the existence of (minimal) approximations in general case, in particular, the existence of left minimal flat approxiamations, the existence of right （minimal） FP-injective approximations, the existence of right pure-injective approximations, the existence of pure-projective, finitely and locally projective approximations; some leading topics in Gorenstein homological algebra; the structure and classification of Cohen-Macaulay rings and Gorenstein rings; the establishment of derived functors and the introduction of new homological dimensions and other new homological invariants. It is hoped that the results obtained in this project can unify some classical results, enrich approximation and homology theory and (partially) solve some famous conjectures.