Cluster structures for the A8$A_\infty$ singularity
Cluster structures for the A8$A_\infty$ singularity
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A8$A_infty$ 奇点的簇结构
DOI：
10.1112/jlms.12735
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
August J
中科院分区：
文献类型：
--
作者：
August J
研究了A∞$A\infty$曲线奇点上的Z$\mathbb {Z}$-分次极大Cohen-Macaulay（MCM）模范畴C2 $\mathcal {C} 2 $，证明了它具有无限型A$A$簇组合.特别地，我们证明了这个Frobenius范畴（或一个合适的子范畴）稳定等价于Holm-Jørgensen，Fisher和Paquette-Yıldırım的无限型A$A$簇范畴。因此，C2\mathcal {C}_2$有集群倾斜子范畴，由（完成的）∞\infty$-gon的某些三角形建模。我们使用Frobenius结构来进一步扩展这一点，以考虑极大几乎刚性子范畴，并证明这些子范畴及其突变表现出完整的∞$\infty$‐gon的组合学。
We study a category C2$\mathcal {C}_2$ of Z$\mathbb {Z}$‐graded maximal Cohen‐Macaulay (MCM) modules over the A∞$A_\infty$ curve singularity and demonstrate that it has infinite type A$A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type A$A$ cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, C2$\mathcal {C}_2$ has cluster tilting subcategories modelled by certain triangulations of the (completed) ∞$\infty$‐gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed ∞$\infty$‐gon.