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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.03693 |
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| _version_ | 1866913927182417920 |
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| author | Caranguay-Mainguez, Jhony F. Rizzo, Pedro Velez-Marulanda, Jose A. |
| author_facet | Caranguay-Mainguez, Jhony F. Rizzo, Pedro Velez-Marulanda, Jose A. |
| contents | Let $\mathbf{k}$ be a field of any characteristic and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $V$ is a finite dimensional right $Λ$-module that lies in the mouth of a stable homogeneous tube $\mathfrak{T}$ of the Auslander-Reiten quiver $Λ$ with $\underline{\mathrm{End}}_Λ(V)$ a division ring, then $V$ has a versal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[\![t]\!]$. As consequence we obtain that if $\mathbf{k}$ is algebraically closed, $Λ$ is a symmetric special biserial $\mathbf{k}$-algebra and $V$ is a band $Λ$-module with $\underline{\mathrm{End}}_Λ(V) \cong \mathbf{k}$ that lies in the mouth of its homogeneous tube, then $R(Λ,V)$ is universal and isomorphic to $\mathbf{k}[\![t]\!]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On universal deformation rings and stable homogeneous tubes Caranguay-Mainguez, Jhony F. Rizzo, Pedro Velez-Marulanda, Jose A. Representation Theory Let $\mathbf{k}$ be a field of any characteristic and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $V$ is a finite dimensional right $Λ$-module that lies in the mouth of a stable homogeneous tube $\mathfrak{T}$ of the Auslander-Reiten quiver $Λ$ with $\underline{\mathrm{End}}_Λ(V)$ a division ring, then $V$ has a versal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[\![t]\!]$. As consequence we obtain that if $\mathbf{k}$ is algebraically closed, $Λ$ is a symmetric special biserial $\mathbf{k}$-algebra and $V$ is a band $Λ$-module with $\underline{\mathrm{End}}_Λ(V) \cong \mathbf{k}$ that lies in the mouth of its homogeneous tube, then $R(Λ,V)$ is universal and isomorphic to $\mathbf{k}[\![t]\!]$. |
| title | On universal deformation rings and stable homogeneous tubes |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2507.03693 |