Singularity Categories of Higher Nakayama Algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910635797774336 |
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| author | Xing, Wei |
| author_facet | Xing, Wei |
| contents | For a higher Nakayama algebra $A$ in the sense of Jasso-Külshammer, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singularity Categories of Higher Nakayama Algebras Xing, Wei Representation Theory For a higher Nakayama algebra $A$ in the sense of Jasso-Külshammer, 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. |
| title | Singularity Categories of Higher Nakayama Algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2306.07006 |