Defect Invariant Nakayama Algebras
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929369292734464 |
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| author | Sen, Emre Todorov, Gordana Zhu, Shijie |
| author_facet | Sen, Emre Todorov, Gordana Zhu, Shijie |
| contents | We show that for a given Nakayama algebra $Θ$, there exist countably many cyclic Nakayama algebras $Λ_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $Λ_i$ is isomorphic to $Θ$ and we describe those algebras $Λ_i$. We show, among these algebras, there exists a unique algebra $Λ$ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both $Θ$ and $Λ$. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00254 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Defect Invariant Nakayama Algebras Sen, Emre Todorov, Gordana Zhu, Shijie Representation Theory Rings and Algebras 16E05 16E10 16G20 We show that for a given Nakayama algebra $Θ$, there exist countably many cyclic Nakayama algebras $Λ_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $Λ_i$ is isomorphic to $Θ$ and we describe those algebras $Λ_i$. We show, among these algebras, there exists a unique algebra $Λ$ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both $Θ$ and $Λ$. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra. |
| title | Defect Invariant Nakayama Algebras |
| topic | Representation Theory Rings and Algebras 16E05 16E10 16G20 |
| url | https://arxiv.org/abs/2406.00254 |