Singularity Categories of Higher Nakayama Algebras

Abstract

For a higher Nakayama algebra $A$ in the sense of Jasso-K\"{u}lshammer, we show that the singularity category of $A$ is triangulated equivalent to the stable module category of a self-injective higher Nakayama algebra. This generalizes a similar result for usual Nakayama algebras due to Shen. Our proof relies on the existence of $d\mathbb{Z}$-cluster tilting subcategories in the module category of $A$ and the result of Kvamme that each $d\mathbb{Z}$-cluster tilting subcategory of $A$ induces a $d\mathbb{Z}$-cluster tilting subcategory in its singularity category. Moreover, our result provides many concrete examples of the triangulated Auslander-Iyama correspondence introduced by Jasso-Muro, namely, there is a bijective correspondence between the equivalence classes of the singularity categories of $d$-Nakayama algebras with its basic $d\mathbb{Z}$-cluster tilting object and the isomorphism classes of self-injective $(d+1)$-Nakayama algebras.