Cluster categories of formal DG algebras and singularity categories
Cluster categories of formal DG algebras and singularity categories
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DOI：
10.1017/fms.2022.30
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发表时间：
2020-03
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通讯作者：
Norihiro Hanihara
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作者：
Norihiro Hanihara
摘要给定一个负分次Calabi-Yau代数，将其看作微分为零的DG代数，研究了它的簇范畴。我们证明了这个DG代数是符号扭曲的Calabi-Yau，并将其簇范畴实现为一个导出范畴的轨道范畴的三角形化船体和有限维Iwanaga-Gorenstein代数的奇异范畴.沿着，我们给出了两个独立的结果。首先，我们证明了Calabi-Yau代数上凝聚层的导出范畴有一个自然的簇倾斜子范畴，其维数由代数的Calabi-Yau维数和α-不变量决定.其次，我们证明了由DG内函子及其同伦逆得到的两个DG轨道范畴是拟等价的。作为应用，我们证明了一个高表示无限代数的高簇范畴与一个Iwanaga-Gorenstein代数的奇异范畴是三角等价的，并给出了明确的描述.此外，我们表明，我们的研究结果概括的背景下凯勒-Murfet-货车登伯格推导出的轨道类别涉及的AR翻译的平方根。
Abstract Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau and realise its cluster category as a triangulated hull of an orbit category of a derived category and as the singularity category of a finite-dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results that stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by the Calabi-Yau dimension and the a-invariant of the algebra. Second, we prove that two DG orbit categories obtained from a DG endofunctor and its homotopy inverse are quasi-equivalent. As an application, we show that the higher cluster category of a higher representation infinite algebra is triangle equivalent to the singularity category of an Iwanaga-Gorenstein algebra, which is explicitly described. Also, we demonstrate that our results generalise the context of Keller–Murfet–Van den Bergh on the derived orbit category involving a square root of the AR translation.