Gorenstein同调代数中若干问题的研究

In recent years, Gorenstein homological algebra has has drawn wide attention and has been studied extensively. This project will continue to carry out the following researches on Gorenstein homological algebra: (1) We will introduce a class of generalized Gorenstein modules to unify the related researches on several important Gorenstein modules; the generalized Gorenstein complexes, the corresponding concept of the generalized Gorenstein modules in the category of complexes, will also be investigated. We will characterize generalized Gorenstein properties of complexes via the generalized Gorenstein properties of modules by establishing the relationship between the generalized Gorenstein complexes and the generalized Gorenstein property of modules of all its terms. (2) Corresponding to the concept of differential modules, a notion of differential objects will be introduced in triangulated categories, and Gorenstein homological theory for differential objects in triangulated categories will be established. (3) The pure Gorenstein homological theory will be established systematically in the category of modules, the category of complexes and triangulated categories, respectively. The study of this project, on the one hand, can further enrich and develop Gorenstein homological theory, on the other hand, can further clarify the internal relations and essential differences among the category of modules, the category of complexes and triangulated categories.

近年来，Gorenstein同调代数得到了极大的关注和研究，本项目将继续开展Gorenstein同调代数的以下研究：(1)引入一类广义Gorenstein模，统一关于几类重要Gorenstein模的相关研究；考察这类广义Gorenstein模在复形范畴中的对应概念-广义Gorenstein复形，通过建立广义Gorenstein复形与其各个层次上模的广义Gorenstein性质之间的联系，利用模的广义Gorenstein性质研究复形的广义Gorenstein性质；(2)对应于微分模的概念，在三角范畴中引入微分对象的概念，建立三角范畴中微分对象的Gorenstein同调理论；(3)系统建立模范畴、复形范畴和三角范畴中的纯Gorenstein同调理论。本项目的研究，一方面可以进一步地丰富和发展Gorenstein同调理论，另一方面可以进一步地阐明模范畴、复形范畴和三角范畴的内在联系和本质区别。

本项目中，我们首先引入并研究了(V,W,Y,X)-Gorenstein模(复形)，大量关于各类Gorenstein模(复形)的研究结果得到了统一和推广，其中V,W,Y,X是四个模类。其次，研究了正合范畴的整体Gorenstein维数和正合范畴中的Gorenstein上同调。第三，研究了相对于X-Gorenstein投射模和Y-Gorenstein内射模的相对上同调、Tate上同调，建立了连接相对上同调、绝对上同调和Tate上同调的AM-型正合序列。第四，分别给出了Gorenstein投射模类和Gorenstein平坦模类是Cotilting类的环的等价刻画和Gorenstein内射模类是Tilting类的环的等价刻画。作为应用，给出了Gorenstein环、Ding-Chen环、Prüfer整环和Dedekind整环的一些新刻画。第五，引入了Slightly (m,n)-凝聚环的概念，证明了R是左Slightly (m,n)-凝聚环当且仅当(m,n)-投射左R-模类和(m,n)-内射左R-模类构成的余挠对是遗传的，当且仅当R是左 (m,n)-凝聚环并且(m,n)-平坦右R-模类和(m,n)-余挠右R-模类构成的余挠对是遗传的。作为应用，我们在Slightly (m,n)-凝聚环上给出了模的(m,n)-同调维数和环的(m,n)-整体维数的许多类似于模的和环的经典同调维数的刻画。最后，引入了相对于半对偶双模的X-内射模和X平坦模的概念,利用这两类模给出了X-凝聚环的一些等价刻画，这里X是一些有限表示右R-模的类，一些经典结果得到了统一和推广。本项目进一步丰富和发展了相对同调论的研究。