The singularity category as a stable module category
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912563366723584 |
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| author | Chen, Xiao-Wu Wang, Zhengfang |
| author_facet | Chen, Xiao-Wu Wang, Zhengfang |
| contents | We investigate the stabilization $\mathcal{S}$ of the module category over an artinian ring $Λ$ by formally inverting the tensor endofunctor given by the bimodule of relative noncommutative differential $1$-forms. It turns out that $\mathcal{S}$ is a Frobenius abelian category, which is equivalent to the category of finitely presented modules over the zeroth component $L_0$ of the Leavitt ring $L$. It follows that $L_0$ is an FC ring in the sense of Damiano, which is usually not quasi-Frobenius. Moreover, the singularity category of $Λ$ is triangle equivalent to the stable module category over $L_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The singularity category as a stable module category Chen, Xiao-Wu Wang, Zhengfang Representation Theory Commutative Algebra Category Theory Rings and Algebras We investigate the stabilization $\mathcal{S}$ of the module category over an artinian ring $Λ$ by formally inverting the tensor endofunctor given by the bimodule of relative noncommutative differential $1$-forms. It turns out that $\mathcal{S}$ is a Frobenius abelian category, which is equivalent to the category of finitely presented modules over the zeroth component $L_0$ of the Leavitt ring $L$. It follows that $L_0$ is an FC ring in the sense of Damiano, which is usually not quasi-Frobenius. Moreover, the singularity category of $Λ$ is triangle equivalent to the stable module category over $L_0$. |
| title | The singularity category as a stable module category |
| topic | Representation Theory Commutative Algebra Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2509.01056 |