Cluster categories of formal DG algebras and singularity categories

Abstract Given a negatively graded Calabi-Yau algebra, we regard it as DG algebra with vanishing differentials and study its cluster category. We show that this is sign-twisted realise category triangulated hull of an orbit derived the singularity finite-dimensional Iwanaga-Gorenstein algebra. Along way, give two results stand on their own. First, coherent sheaves over has natural tilting subcategory whose dimension determined by -invariant Second, prove categories obtained from endofunctor homotopy inverse are quasi-equivalent. As application, higher representation infinite triangle equivalent to which explicitly described. Also, demonstrate our generalise context Keller–Murfet–Van den Bergh involving square root AR translation.