Acyclic Calabi–Yau categories
Acyclic Calabi–Yau categories
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DOI：
10.1112/s0010437x08003540
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发表时间：
2006-10
影响因子：
1.8
通讯作者：
B. Keller;I. Reiten
中科院分区：
文献类型：
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作者：
B. Keller;I. Reiten
摘要 我们证明了三角卡拉比-丘范畴的结构定理：代数闭域上的代数 2-卡拉比-丘三角范畴是一个簇范畴，当且仅当它包含一个簇倾斜子范畴，且该子范畴的箭袋没有定向环。我们证明了更高聚类类别的类似特征。作为交换代数的应用，我们证明了在某个 3 维孤立奇点上最大 Cohen-Macaulay 模的稳定范畴是一个簇范畴。这意味着 Iyama 和 Yoshino 首先获得的刚性 Cohen-Macaulay 模块的分类。作为颤动变异组合学的应用，我们证明了簇倾斜对象的自同态代数颤动在表示无限预投影代数的稳定范畴中的非无环性。目前还没有直接的组合证明。在附录中，米歇尔·范登伯格通过诉诸三角轨道范畴的普遍性质，给出了主要定理的另一种证明。
Abstract We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.