The stable category of Gorenstein-projective modules over a monomial algebra
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| Format: | Preprint |
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2024
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| _version_ | 1866912417651359744 |
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| author | Honma, Takahiro Usui, Satoshi |
| author_facet | Honma, Takahiro Usui, Satoshi |
| contents | Let $Λ$ be an arbitrary monomial algebra. We investigate the stable category $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ of graded Gorenstein-projective $Λ$-modules and the orbit category $\underline{\operatorname{Gproj}}^{\mathbb{Z}} Λ/(1)$ induced by $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ and the degree shift functor $(1)$. We prove that $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ is triangle equivalent to the bounded derived category of a path algebra of Dynkin type $\mathbb{A}$ and that $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ/(1)$ is triangle equivalent to the stable module category of a self-injective Nakayama algebra. Both the path algebra and the self-injective Nakayama algebra will be given explicitly. The latter result provides an explicit description of the stable category of (ungraded) Gorenstein-projective $Λ$-modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04912 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stable category of Gorenstein-projective modules over a monomial algebra Honma, Takahiro Usui, Satoshi Representation Theory 16G50, 16G20, 16G70, 18G80 Let $Λ$ be an arbitrary monomial algebra. We investigate the stable category $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ of graded Gorenstein-projective $Λ$-modules and the orbit category $\underline{\operatorname{Gproj}}^{\mathbb{Z}} Λ/(1)$ induced by $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ and the degree shift functor $(1)$. We prove that $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ$ is triangle equivalent to the bounded derived category of a path algebra of Dynkin type $\mathbb{A}$ and that $\underline{\operatorname{Gproj}}^{\mathbb{Z}}Λ/(1)$ is triangle equivalent to the stable module category of a self-injective Nakayama algebra. Both the path algebra and the self-injective Nakayama algebra will be given explicitly. The latter result provides an explicit description of the stable category of (ungraded) Gorenstein-projective $Λ$-modules. |
| title | The stable category of Gorenstein-projective modules over a monomial algebra |
| topic | Representation Theory 16G50, 16G20, 16G70, 18G80 |
| url | https://arxiv.org/abs/2407.04912 |