Cluster-tilted algebras are Gorenstein and stably Calabi–Yau
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau
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DOI：
10.1016/j.aim.2006.07.013
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发表时间：
2005-12
影响因子：
1.7
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
证明了在2-Calabi-Yau三角范畴中，每个簇倾斜子范畴都是Gorenstein的，它的每个簇生成的内射维数至多为1。我们证明了它的Cohen-Macaulay模的稳定范畴是3-Calabi-Yau。我们特别推出，集群倾斜代数是Gorenstein的维数最多为1，遗传，如果他们是有限的整体维数。我们的结果也适用于稳定的（！）极大刚性模的自同态环[Christof Geithmic，Bernard Leclerc，Jan Schröer，Rigid modules over preprojective algebras，arXiv：math.RT/0503324，Invent.数学、出版中]。此外，我们还证明了非稳定自同态环上相对3-Calabi-Yau对偶的一个一般结果。这加强和推广了在[Christof Geiglit，Bernard Leclerc，Jan Schröer，Rigid modules over preprojective algebras，arXiv：math.RT/0503324，Invent.数学、在印刷中]用于简单模块。最后，我们将2-Calabi-Yau范畴的相对Calabi-Yau对偶的结果推广到d-Calabi-Yau范畴。我们展示了如何产生许多d-簇倾斜代数的例子。
We prove that in a 2-Calabi–Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen–Macaulay modules is 3-Calabi–Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi–Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi–Yau duality from 2-Calabi–Yau to d-Calabi–Yau categories. We show how to produce many examples of d-cluster tilted algebras.