本项目将研究Cartan-Eilenberg Gorenstein平坦复形和Gorenstein平坦复形的性质，阐明与Gorenstein平坦模之间的关系；研究一般环上复形的Gorenstein投射覆盖、Gorenstein内射包络和Gorenstein平坦覆盖的存在性；给定一模类L，就与其相关的复形类(每个层次上的模都在L中的复形类，每个层次上的模都在L中的正合复形类，及每个循环都在L中的正合复形类等)研究覆盖与包络的存在性，以及由模范畴中的余挠对所诱导的复形范畴中的余挠对的完备性，即Gillespie猜想，由此展开对复形的相对同调性质的研究，并与复形的Tate(上)同调联系起来，建立连接绝对同调、相对同调和Tate(上)同调的Avramov-Martsinkovsky型正合序列，揭示复形范畴中区别于模范畴的本质特征。力争在这些问题的研究中取得本质性进展，将进一步丰富和发展相对同调代数。

This project will study the properties of Cartan-Eilenberg Gorenstein flat complexes and Gorenstein flat complexes, and clarify the relationship between Gorenstein flat modules. The project studies the existence of Gorenstein projective covers, Gorenstein injective envelopes and Gorenstein flat covers for every complex over an arbitrary ring; Given a class of modules L, as far as the classes of complexes associated with L concerned, the existence of covers and envelopes are studied, such as the class of all complexes with each term in L, the class of all exact complexes with each term in L, the class of all complexes with each cycle in L and so on. At the same time, the project studies the completeness of the cotorsion pairs in the category of complexes induced by the cotorsion pair in the category of modules, namely Gillespie conjecture, thus expand to study the relative homological properties of complexes as well as its links with Tate (co)homology of complexes, trying to establish the Avramov-Martsinkovsky type exact sequence connecting the Tate (co)homology, the absolute homology and the relative homology in the category of complexes, and to reveal the essential characteristics of the category of complexes different from the category of modules. We strive to make substantive progress in the study of these issues, and to further enrich and develop the theory of relative homological algebra.