Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday
Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday
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DOI：
10.1353/ajm.2015.0019
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发表时间：
2011-04
影响因子：
1.7
通讯作者：
Claire Amiot;O. Iyama;I. Reiten
中科院分区：
文献类型：
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作者：
Claire Amiot;O. Iyama;I. Reiten
根据Auslander的代数McKay对应，简单奇点上Cohen-Macaulay模的稳定范畴是三角形，等价于动态箭图的路代数的$1$-簇范畴(即通过Auslander-Reiten平移作用得到的派生范畴的轨道范畴).本文给出了Gorenstein孤立奇点R上的Cohen-Macaulay模的稳定范畴与有限维代数的广义(高阶)簇范畴之间类似三角等价的系统方法。双模Calabi-Yau代数起着关键作用，它是$R$的高阶Auslander代数，也是$\Lambda$扩张的高阶预射影代数。作为副产品，我们给出了分次Cohen-Macaulay$R$-模的稳定范畴与$\Lambda$的派生范畴之间的三角等价。我们的主要结果特别适用于一类循环商奇点和与二聚体模型相关的某些环仿射三重。
By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.