Cohen-Macaulay Auslander 代数

This proposal is about the intersection of representation theory of algebras and algebraic geometry. We will describe the structure of the Cohen-Macaulay Auslander algebras of the (gentle) cluster-tilted algebras, and also study their properties related to representation theory, especially their rigid modules. Furthermore, we will study the quiver Grassmannians of the cluster-tilted algebras, namely, use the quiver Grassmannians of the Cohen-Macaulay Auslander algebras of the cluster-tilted algebras to realize the desingularization of the quiver Grassmannians of the cluster-tilted algebras, especially the Gorenstein projective modules.

本项目是代数表示论和代数几何的交叉课题。我们拟刻画丛倾斜代数（特别是 gentle 丛倾斜代数）的 Cohen-Macaulay Auslander 代数的结构和表示论基本性质，特别是刻画 rigid 模的性质。在此基础上，我们将利用丛倾斜代数的 Cohen-Macaulay Auslander 代数的 quiver Grassmannian 实现丛倾斜代数的 quiver Grassmannian 的奇点消解。

本项目是代数表示论和代数几何的交叉课题。我们主要刻画了gentle代数，skewed-gentle代数的Cohen-Macaulay Auslander代数，证明了gentle代数的Cohen-Macaulay Auslander代数是gentle代数，gentle代数是表示有限的当且仅当它的Cohen-Macaulay Auslander代数是表示有限型的，gentle代数的不可分解模由维数向量唯一决定，则它的Cohen-Macaulay Auslander代数上的不可分解模也由维数向量唯一决定，skewed-gentle代数的Cohen-Macaulay Auslander代数是对应gentle代数的Cohen-Macaulay Auslander代数的skewed-gentle代数。在奇点消解方面，对1-Gorenstein gentle代数（包括gentle丛倾斜代数），我们利用它的Cohen-Macaulay Auslander代数quiver Grassmannians实现了gentle代数上quiver Grassmannians的奇点消解。