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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/2510.12433 |
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| _version_ | 1866917012095107072 |
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| author | Godinho, Marina Murphy, Dave |
| author_facet | Godinho, Marina Murphy, Dave |
| contents | The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of a certain DG algebra. This additive equivalence preserves some of the triangulated structure: it commutes with the suspension functor and preserves triangles with at least two indecomposable terms. In the process, we introduce the notion of a linear generator $G$ in a Krull-Schmidt, Hom-finite triangulated category. It turns out that the existence of a linear generator affords a large amount of control over $\mathcal{T}$. For example, it allows us to describe all indecomposable objects in $\mathcal{T}$ in terms of $G$, to determine all triangles of $\mathcal{T}$ with at least two indecomposable objects, and to show that the Rouquier dimension of $\mathcal{T}$ is at most one. Moreover, we prove that there is an additive equivalence (which preserves some of the triangulated structure) between $\mathcal{T}$ and the perfect derived category of a certain DG algebra. Finally, we show that any triangulated category with a linear generator is additively equivalent to a thick subcategory of $\overline{\mathcal{C}}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12433 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Triangulated Categories Admitting Linear Generators Godinho, Marina Murphy, Dave Representation Theory The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of a certain DG algebra. This additive equivalence preserves some of the triangulated structure: it commutes with the suspension functor and preserves triangles with at least two indecomposable terms. In the process, we introduce the notion of a linear generator $G$ in a Krull-Schmidt, Hom-finite triangulated category. It turns out that the existence of a linear generator affords a large amount of control over $\mathcal{T}$. For example, it allows us to describe all indecomposable objects in $\mathcal{T}$ in terms of $G$, to determine all triangles of $\mathcal{T}$ with at least two indecomposable objects, and to show that the Rouquier dimension of $\mathcal{T}$ is at most one. Moreover, we prove that there is an additive equivalence (which preserves some of the triangulated structure) between $\mathcal{T}$ and the perfect derived category of a certain DG algebra. Finally, we show that any triangulated category with a linear generator is additively equivalent to a thick subcategory of $\overline{\mathcal{C}}_n$. |
| title | Triangulated Categories Admitting Linear Generators |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2510.12433 |