Categories for Grassmannian Cluster Algebras of Infinite Rank
Categories for Grassmannian Cluster Algebras of Infinite Rank
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无限阶格拉斯曼簇代数的范畴
DOI：
10.1093/imrn/rnad004
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发表时间：
2023
影响因子：
1
通讯作者：
August J
中科院分区：
文献类型：
--
作者：
August J
我们构造了无限秩的格拉斯曼范畴，提供了由詹森，金和苏引入的格拉斯曼簇范畴的一个无限模拟。每个无限秩的Grassmannian范畴都是某个超曲面奇点上的分次极大Cohen-Macaulay模范畴。我们证明了无限秩的Grassmannian范畴的兰金泛自由模与相应的无限秩Grassmannian簇代数的Plücker坐标成双射.此外，这种双射是结构保持的，因为它将范畴中的刚性与普吕克坐标的相容性联系起来。沿着这一思路，我们得到了一个计算无限秩Grassmannian范畴中任意两个兰金秩的一般自由模之间的-空间维数的组合公式。
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 rankin 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-spaces between any two generically free modules of rankin the Grassmannian category of infinite rank.