Gorenstein投射模与virtually Gorenstein代数

Gorenstein projective modules and virtually Gorenstein algebras are important subjects in representation theory as well as relative homological algebra. These two subjects are closed related to each other. This project is intended to study the Gorenstein projective modules over some algebras and the virtual Gorenstein property of these algebras. More precisely, we will consider some special algebras such as monomial algebras, Koszul algebras and extension algebras of them. We will describe Gorenstein projective modules over these algebras and decide whether they are virtually Gorenstein or not. In particular, we will study and construct a class of noncommutative nonvirtually Gorenstein algebras. The study we carry in the project will help us understand Gorenstein homological algebra better.

Gorenstein投射模与virtually Gorenstein代数均为代数表示论以及相对同调代数的重要研究对象，两者具有非常紧密的联系。本项目拟研究代数上Gorenstein投射模与代数的virtually Gorenstein性质。更精确来说，项目拟研究一些特殊代数，例如单项式代数、Koszul代数以及它们的扩张代数。项目拟具体给出这些代数上Gorenstein投射模的描述，并且同时判断这些代数是否为virtually Gorenstein代数。特别地，项目拟研究并构造一类非交换的非virtually Gorenstein代数。这些问题的解决将有助于我们对Gorenstein同调代数的理解。

Gorenstein投射模与virtually Gorenstein代数是代数表示论与Gorenstein同调理论中的重要研究对象。本项目研究了两个有限维代数的张量积上Gorenstein投射模及其virtually Gorenstein性质。当其中一个代数是Gorenstein代数时，我们给出了两个代数的张量积上Gorenstein投射模的描述，这推广了该领域内的相关结果。特别地，我们考察了根方零代数与对偶数代数的张量积，并由此构造了一类新地非virtually Gorenstein代数。这些结果有助于我们对于Gorenstein同调理论的理解。