广义幂级数环理论研究

The generalized power series ring is a common generalization of polynomial rings, Laurent polynomial rings, monoid rings, power series rings and Laurent series rings, etc. The property of this construction is not only depend on the property of its coefficient ring, but also heavily rely on the property of the partially ordered monoid of exponent. Therefore, the research on rings of generalized power series has certain theoretical and application value. In recent years, a great many important results have been obtained in this field. This project will do some further studies on rings of generalized power series and modules over them. The main research content is described as follows: quasi-Armendariz rings relative to a skew generalized power series ring; describe all of the associated primes of a generalized power series module over a skew generalized power series ring in terms of the associated primes of its coefficient module；describe the attached primes of a generalized inverse polynomial module over a skew generalized power series ring in terms of the attached primes of its coefficient module; study the serial property, the Ikeda-Nakayama property and the strong cleanness of skew generalized power series rings, and make a preliminary research on the generalized power series extension for some relative homology properties. We strive to make some innovations in research methods and obtain some distinctive results, which will further enrich and develop the theory and applications of generalized power series rings.

广义幂级数环是多项式环、洛朗多项式环、幺半群环、幂级数环、洛朗级数环等环类的统一推广，其性质不仅取决于其系数环的性质，而且还依赖于作为指数集的偏序幺半群的性质。因此，广义幂级数环的研究有一定的理论和应用价值，近年来，已取得了许多重要的研究成果。本项目将对广义幂级数环及其上模的性质做进一步的研究，主要研究内容为：相对于斜广义幂级数环的拟Armendariz环；广义幂级数模的associated prime理想；广义逆多项式模的attached prime理想；斜广义幂级数环的serial性质、Ikeda-Nakayama性质、强clean性质；初步探索一些相对同调性质的广义幂级数扩张。我们力争在研究方法上有所创新，取得一些有新意的研究成果，进一步地丰富和发展广义幂级数环理论的应用和研究。

本项目中我们研究了广义幂级数环理论和Gorenstein同调理论，主要成果为：(1) 在一定条件下，给出广义逆多项式模的attached prime理想的构造；(2) 把模范畴中几类重要的Gorenstein模--(W,Y,X)-Gorenstein模，VW-Gorenstein模和相对于一个完备、遗传余挠对的Gorenstein投射模的概念推广到了复形范畴中, 分别引入了(W,Y,X)-Gorenstein复形，VW-Gorenstein复形和相对于一个完备、遗传余挠对的Gorenstein投射复形的概念，建立了这些复形与其每个层次上的模的相应Gorenstein性质之间的联系。作为应用，我们把关于(W,Y,X)-Gorenstein模，VW-Gorenstein模和相对于一个完备、遗传余挠对的Gorenstein投射模的稳定性的结果推广到了复形范畴中。这些研究很好地统一了现有文献中关于相关Gorenstein复形的相关研究。(3) 在三角范畴中建立了联系Beligiannis定义的ε-上同调群、Asadallahi-Salarian定义的ε-Tate上同调群和ε-Gorenstein上同调群的Avramov-Martsinkovsky 型正合序列。(4) 我们在交换环上引入并研究了的同调下有界复形的DC-投射维数。 (5) 在交换Noether环上引入并研究了模的基于Tate FC-分解的Tate同调，讨论了这种Tate同调与与之相关的相对同调之间的联系，研究了这种Tate同调的平衡性，并利用这种Tate同调给出了模的FC-投射维数有限性的等价刻画。