Singularity categories via higher McKay quivers with potential

Abstract

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

arXiv comment16 pages; v2: minor changes; v3: references added, readability improved; v4: published in Selecta Mathematica. New Series

References in corpus (5)

• Formal completions and idempotent completions of triangulated categories of singularities
• On two examples by Iyama and Yoshino
• Frobenius categories, Gorenstein algebras and rational surface singularities
• K-theory and the singularity category of quotient singularities
• Singularity categories via the derived quotient