Maximal almost rigid modules over gentle algebras
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909301302362112 |
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| author | Barnard, Emily Simoes, Raquel Coelho Gunawan, Emily Schiffler, Ralf |
| author_facet | Barnard, Emily Simoes, Raquel Coelho Gunawan, Emily Schiffler, Ralf |
| contents | We study maximal almost rigid modules over a gentle algebra $A$. We prove that the number of indecomposable direct summands of every maximal almost rigid $A$-module is equal to the sum of the number of vertices and the number of arrows of the Gabriel quiver of $A$. Moreover, the algebra $A$, considered as an $A$-module, can be completed to a maximal almost rigid module in a unique way.
Gentle algebras are precisely the tiling algebras of surfaces with marked points. We show that the (permissible) triangulations of the surface of $A$ are in bijection with the maximal almost rigid $A$-modules.
Furthermore, we study the endomorphism algebra $C=\text{End}_A T$ of a maximal almost rigid module $T$. We construct a fully faithful functor $G\colon \text{mod}\,A\to \text{mod}\, \overline{A}$ into the module category of a bigger gentle algebra $\overline{A}$ and show that $G$ maps maximal almost rigid $A$-modules to tilting $\overline{A}$-modules. In particular, $C$ and $\overline{A}$ are derived equivalent and $C$ is gentle.
After giving a geometric realization of the functor $G$, we obtain a tiling $G(\mathbf{T})$ of the surface of $\overline{A}$ as the image of the triangulation $\mathbf{T}$ corresponding to $T$. We then show that the tiling algebra of $G(\mathbf{T})$ is $C$. Moreover, the tiling algebra of $\mathbf{T}$ is obtained algebraically from $C$ as the tensor algebra with respect to the $C$-bimodule $\text{Ext}_C^2(DC,C)$, which also is fundamental in cluster-tilting theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal almost rigid modules over gentle algebras Barnard, Emily Simoes, Raquel Coelho Gunawan, Emily Schiffler, Ralf Representation Theory Combinatorics Primary 05E10, 16G20, Secondary 16G70, 13F60 We study maximal almost rigid modules over a gentle algebra $A$. We prove that the number of indecomposable direct summands of every maximal almost rigid $A$-module is equal to the sum of the number of vertices and the number of arrows of the Gabriel quiver of $A$. Moreover, the algebra $A$, considered as an $A$-module, can be completed to a maximal almost rigid module in a unique way. Gentle algebras are precisely the tiling algebras of surfaces with marked points. We show that the (permissible) triangulations of the surface of $A$ are in bijection with the maximal almost rigid $A$-modules. Furthermore, we study the endomorphism algebra $C=\text{End}_A T$ of a maximal almost rigid module $T$. We construct a fully faithful functor $G\colon \text{mod}\,A\to \text{mod}\, \overline{A}$ into the module category of a bigger gentle algebra $\overline{A}$ and show that $G$ maps maximal almost rigid $A$-modules to tilting $\overline{A}$-modules. In particular, $C$ and $\overline{A}$ are derived equivalent and $C$ is gentle. After giving a geometric realization of the functor $G$, we obtain a tiling $G(\mathbf{T})$ of the surface of $\overline{A}$ as the image of the triangulation $\mathbf{T}$ corresponding to $T$. We then show that the tiling algebra of $G(\mathbf{T})$ is $C$. Moreover, the tiling algebra of $\mathbf{T}$ is obtained algebraically from $C$ as the tensor algebra with respect to the $C$-bimodule $\text{Ext}_C^2(DC,C)$, which also is fundamental in cluster-tilting theory. |
| title | Maximal almost rigid modules over gentle algebras |
| topic | Representation Theory Combinatorics Primary 05E10, 16G20, Secondary 16G70, 13F60 |
| url | https://arxiv.org/abs/2408.16904 |