An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Brüstle, Thomas Hanson, Eric J. Roy, Sunny Schiffler, Ralf |
| author_facet | Brüstle, Thomas Hanson, Eric J. Roy, Sunny Schiffler, Ralf |
| contents | Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid $A$-modules are in bijection with the triangulations of the polygon $S.$
In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal{E}_\diamond$ on $\text{mod}\,A$ such that the maximal almost rigid $A$-modules in the usual exact structure are exactly the maximal rigid $A$-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal{E}_\diamond$ translates into a theory of maximal almost rigid modules in the usual exact structure.
As an application, we show that with the exact structure $\mathcal{E}_\diamond$, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu.
We also discuss generalizations to quivers of type $\mathbb{D}$ and gentle algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04627 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$ Brüstle, Thomas Hanson, Eric J. Roy, Sunny Schiffler, Ralf Representation Theory Category Theory 16G20, 18G25 Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid $A$-modules are in bijection with the triangulations of the polygon $S.$ In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal{E}_\diamond$ on $\text{mod}\,A$ such that the maximal almost rigid $A$-modules in the usual exact structure are exactly the maximal rigid $A$-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal{E}_\diamond$ translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure $\mathcal{E}_\diamond$, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type $\mathbb{D}$ and gentle algebras. |
| title | An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$ |
| topic | Representation Theory Category Theory 16G20, 18G25 |
| url | https://arxiv.org/abs/2410.04627 |