Lifting preprojective algebras to orders and categorifying partial flag varieties
Lifting preprojective algebras to orders and categorifying partial flag varieties
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DOI：
10.2140/ant.2016.10.1527
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发表时间：
2015-03
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通讯作者：
Laurent Demonet;O. Iyama
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文献类型：
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作者：
Laurent Demonet;O. Iyama
我们描述了一个分类的集群代数结构的多齐次坐标环的部分旗品种的任意Dynkin型使用Cohen-Macaulay模的订单。这通过添加缺失的系数完成了Geiss-Leclerc-Schroer的分类。为了实现这一点，对于一个阶$A$和一个幂等元$e \in A$，我们引入了$\operatorname{CM}\nolimits_e A$的一个子范畴$\operatorname{CM}\nolimits_e A$，并研究了它的性质。特别地，在一些温和的假设下，我们对内射$B$-模$Q$构造了正合范畴$（\operatorname{CM}\nolimits_e A）/[Ae] \cong \operatorname{Sub}\nolimits Q$的一个等价，其中$B：= A/（e）$.这些结果推广了Jensen-King-Su关于Grassmannian $\operatorname{Gr}\nolimits_m（\mathbb{C}^n）$的簇代数结构的工作.
We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schroer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)/[Ae] \cong \operatorname{Sub}\nolimits Q$ for an injective $B$-module $Q$ where $B := A/(e)$. These results generalize work by Jensen-King-Su concerning the cluster algebra structure of the Grassmannian $\operatorname{Gr}\nolimits_m(\mathbb{C}^n)$.