Gorenstein同调与逼近理论

在Artin代数R中引入满足Auslander条件的模的概念。利用满足Auslander条件的模的性质来研究R何时满足Auslander条件，得到Auslander-Gorenstein代数的等价刻画。在Auslander条件下，研究有限生成Gorenstein投射模和∞-合冲模的同调性质，分别建立由有限生成Gorenstein投射模组成的子范畴和由有限生成∞-合冲模组成的子范畴的反变有限性与R是Gorenstein代数之间的联系。引入广义本质Gorenstein代数的概念，利用Gorenstein投射模，Gorenstein内射模和相关模类的性质以及Morita等价理论来刻画广义本质Gorenstein代数。在这些工作的基础上，研究Auslander-Gorenstein猜想和在广义本质Gorenstein代数中研究Gorenstein对称猜想。争取在这些问题的研究中取得本质性的进展。

Let R be an Artin algebra. We will introduce the notion of modules satisfying the Auslander condition. By using the properties of modules satisfying the Auslander condition, we will study when R satisfies the Auslander condition, and then give some equivalent characterizations of Auslander-Gorenstein algebras. Under the Auslander condition, we will establish the connection between the contravariant finiteness of the subcategory consisting of finitely generated Gorenstein projective modules (∞-syzygy modules respectively) and that R being Gorenstein. We will introduce the notion of generalized virtually Gorenstein algebras. By using the properties of Gorenstein projective modules, Gorenstein injective modules and related modules as well as the Morita equivalent theory, we will characterize generalized virtually Gorenstein algebras. Based on these, we will study the Auslander-Gorenstein conjecture and the Gorenstein symmetric conjecture over generalized virtually Gorenstein algebras. We try to make essential progress on these problems.