相对于半对偶模的Gorenstein同调维数与覆盖包络理论

This project will be based on the relative homological algebra theory. Through the study of the existence of the C-Gorenstein projective precovers over the rings which are wider than commutative Neotherian rings with dualizing modules, the C-Gorenstein injective envelopes and the C-Gorenstein flat covers over any associative ring and the C-Gorenstein projective covers over perfect rings, to probe into the correctness of partial conclusions of "Holm’s Series Conjectures". At the same time the project will also study the relative homological algebra in the category of complexes. Recently, a special attention is paid to the C-Gorenstein projective, injective and flat dimensions of homological bounded complexes which have nice functorial descriptions. Based on it, we will continue study the characterizations of C-Gorenstein homological dimensions of unbounded complexes, and consider the relationships between of these dimensions and those already existed dimensions for complexes. The aim of which is to offer evidence for mastering all the relative homological invariant of complexes and giving some homological characterizations of rings described by complexes, and then enrich and develop the relative homological algebra theory.

本项目将立足于相对同调代数理论，通过研究比具有对偶模的交换Neother环更广泛的一类环上模的C-Gorenstein投射预覆盖、任意结合环上模的C-Gorenstein内射包络与C-Gorenstein平坦覆盖、完全环上任意模的C-Gorenstein投射覆盖的存在性，探讨“Holm系列猜想”中部分结论的正确性。同时本项目还将进行复形范畴中相对同调理论的研究。近期我们对同调有界复形考虑了C-Gorenstein投射、内射与平坦维数有限性的函子刻画，我们将以此为基础继续研究任意复形的C-Gorenstein投射、内射与平坦维数的刻画，并给出其与复形已有Gorenstein同调维数之间的关系，从而为掌握复形的各种相对同调不变量、给出环由复形表述的相对同调性质提供依据，由此进一步丰富和发展相对同调代数理论。

本项目主要借助复形范畴中的覆盖包络理论研究相对于半对偶模的Gorenstein同调维数理论。主要研究成果有：（1）研究了相对于半对偶模C的复形的有限Gorenstein同调维数。特别考虑了同调有界复形的C-Gorenstein同调维数的函子刻画。进而利用C-Gorenstein同调模的分解给出了关于C-Gorenstein同调维数的稳定性结论。（2）介绍并研究了相对于半对偶模C 的 Ding投射模，扩充了已有的Foxby等价图。推广了Gillespie 与 Ding, Mao等人的工作。（3）研究了复形的Ding同调维数，利用此维数给出了一些古典环类的新刻画。