Cluster categories of formal DG algebras and singularity categories
Forum of Mathematics, Sigma, Tome 10 (2022)
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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.
@article{10_1017_fms_2022_30,
author = {Norihiro Hanihara},
title = {Cluster categories of formal {DG} algebras and singularity categories},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {10},
year = {2022},
doi = {10.1017/fms.2022.30},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.30/}
}
Norihiro Hanihara. Cluster categories of formal DG algebras and singularity categories. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.30
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