论文标题
正式DG代数和奇异性类别的群集类别
Cluster categories of formal DG algebras and singularity categories
论文作者
论文摘要
鉴于呈负分级的calabi-yau代数，我们将其视为DG代数，差异化并研究其群集类别。我们表明，该DG代数是标志性的Calabi-Yau，并将其集群类别视为衍生类别类别的轨道类别的三角形船体，并且是有限的Iwanaga-Gorenstein代数的有限尺寸尺寸类别。一路上，我们给出了两个结果。首先，我们表明，在卡拉比远代数上的连贯滑轮的派生类别具有自然的集群倾斜子类别，其尺寸由Calabi-yau尺寸确定，并且是代数的$ invariant。其次，我们证明从DG内函数获得的两个DG轨道类别及其同质逆逆逆逆逆轨道类别是准等效的。作为一个应用，我们表明，较高表示的较高群集类别无限代数是三角形的，等于iwanaga-gorenstein代数的奇异性类别，该类别是明确描述的。另外，我们证明我们的结果概括了凯勒（Keller）的上下文 - 摩尔菲特（Murfet） - van den bergh在涉及AR翻译平方根的派生轨道类别上。
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 realize 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 which 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. Secondly, 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 generalize the context of Keller--Murfet--Van den Bergh on the derived orbit category involving a square root of the AR translation.
