Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A
Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A
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发表时间：
2006-10
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通讯作者：
T. Holm;Peter Jørgensen
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作者：
T. Holm;Peter Jørgensen
由于arxiv：Math/0610728和arxiv：Math/0612451中的定理2.2存在问题，预印本arxiv：Math/0610728被撤回。该定理指出，对于具有有限多个不可分解对象的三角范畴，通过寻找Serre函子和悬挂函子的d次方对Auslander-Reiten箭图具有相同作用的最小d，可以组合地计算Calabi-Yau维数。这是错误的，我们感谢Alex Dugas指出了一个反例；有关更多详细信息，请参阅他的论文arxiv：Math/0808.1311的第5节。不幸的是，我们目前还不能提出该定理的修正版本，这意味着我们不能计算具体稳定模范畴的Calabi-Yau维度。由于这些维度是识别具有较高聚类类别的类别所必需的，所以我们目前没有办法实现这样的识别。
The preprints arXiv:math/0610728 and arXiv:math/0612451 are withdrawn due to a problem with Theorem 2.2 in arXiv:math/0610728. The theorem claims that for certain triangulated categories with finitely many indecomposable objects, the Calabi-Yau dimension can be computed combinatorially, by finding the smallest d for which the Serre functor and the d'th power of the suspension functor have the same action on the Auslander-Reiten quiver. This is false, and we are grateful to Alex Dugas for pointing out a counterexample; see Section 5 of his paper arXiv:math/0808.1311 for more details. Unfortunately, we are not presently able to come up with a corrected version of the theorem, and this means that we cannot compute the Calabi-Yau dimensions of concrete stable module categories. Since these dimensions are necessary for identifying the categories with higher cluster categories, we presently have no means to achieve such identifications.