论文标题
Grassmannian集群无限等级的类别
Categories for Grassmannian cluster algebras of infinite rank
论文作者
论文摘要
我们构建了格拉曼尼亚类别的无限等级类别，提供了詹森，国王和苏介绍的格拉曼尼亚群集类别的无限类似物。每个无限等级类别的硕士类别都被视为在某个高度表面奇点上分级最大Cohen-Macaulay模块的类别。我们表明，格拉曼尼亚无限等级类别中的一般免费模块$ 1 $与适当的Grassmannian群集代数的无限等级的Plücker坐标进行了培养。此外，这种两者的培养是保存结构，因为它与Plücker坐标的兼容性相关联。在此过程中，我们开发了一个组合公式，以计算任何两个级别的自由模块$ \ mathrm {ext}^1 $ spaces在格拉曼尼亚无限等级类别中的任何两个一般的自由模块$ 1 $之间。
We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen-Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.
