同伦范畴的 Gorenstein 同调理论和紧生成性

本项目拟研究同伦范畴的 Gorenstein 同调理论。首先将研究同伦范畴中的 Gorenstein 同调性质，进而考虑同伦范畴中的 Gorenstein 对象及与复形范畴、模范畴中 Gorenstein 对象之间的联系；其次将考虑同伦范畴中 Gorenstein 对象所构成的子范畴及与之相关子范畴的紧生成性，基于此，一方面证明一些（预）覆盖与（预）包络的存在性，另一方面运用Brown 表示定理和 Thomason 局部化定理等工具在同伦范畴中建立粘合及范畴等价。本研究对于揭示同伦范畴与复形范畴的区别与联系，探索 Gorenstein 同调理论在同伦范畴中的应用具有重要意义。

The purpose of this project is to investigate Gorenstein homological theory in homotopy categories. Firstly, we will study Gorenstein homological properties in homotopy categories and establish some relationships between Gorenstein objects in homotopy categories, the category of complexes and the category of modules. Secondly, the compactly generatedness of Gorenstein subcategories and related subcategories will be considered. Based on this result, on the one hand, the existence of some (pre)covers and (pre)envelopes will be obtained, on the other hand, some recollements and equivalences of categories will be established using some methods such as the Brown Representability Theorem and Thomason Localization Theorem. The study of this project will play an important role in exploring applications of Gorenstein homological theory in homotopy categories and will reveal the differences and relations of homotopy categories and the category of complexes.