Abstract representation theory via coherent Auslander-Reiten diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917059933241344 |
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| author | Sánchez, Álvaro |
| author_facet | Sánchez, Álvaro |
| contents | We provide a general method to study representations of quivers over abstract stable homotopy theories (e.g. arbitrary rings, schemes, dg algebras, or ring spectra) in terms of Auslander-Reiten diagrams. For a finite acyclic quiver $Q$ and a stable $\infty$-category $\mathcal{C}$, we prove an abstract equivalence of the representations $\mathcal{C}^Q$ with a certain mesh $\infty$-category $\mathcal{C}^{\mathbb{Z}Q,\, \mathrm{mesh}}$ of representations of the repetitive quiver $\mathbb{Z}Q$, that we build inductively using abstract reflection functors. This allows to produce, from the symmetries of the Auslander-Reiten quiver, universal autoequivalences of representations $\mathcal{C}^Q$ in any stable $\infty$-category $\mathcal{C}$, which are the elements of the spectral Picard group of $Q$. In particular, we get abstract versions of key functors in classical representation theory -- e.g. reflection functors, the Auslander-Reiten translation, the Serre functor, etc. Moreover, for representations of trees this enables us to realize the whole derived Picard group over a field as a factor of the spectral Picard group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abstract representation theory via coherent Auslander-Reiten diagrams Sánchez, Álvaro Representation Theory Algebraic Topology Category Theory 55U35, 16G20 (Primary), 18G80, 55P43 (Secondary) We provide a general method to study representations of quivers over abstract stable homotopy theories (e.g. arbitrary rings, schemes, dg algebras, or ring spectra) in terms of Auslander-Reiten diagrams. For a finite acyclic quiver $Q$ and a stable $\infty$-category $\mathcal{C}$, we prove an abstract equivalence of the representations $\mathcal{C}^Q$ with a certain mesh $\infty$-category $\mathcal{C}^{\mathbb{Z}Q,\, \mathrm{mesh}}$ of representations of the repetitive quiver $\mathbb{Z}Q$, that we build inductively using abstract reflection functors. This allows to produce, from the symmetries of the Auslander-Reiten quiver, universal autoequivalences of representations $\mathcal{C}^Q$ in any stable $\infty$-category $\mathcal{C}$, which are the elements of the spectral Picard group of $Q$. In particular, we get abstract versions of key functors in classical representation theory -- e.g. reflection functors, the Auslander-Reiten translation, the Serre functor, etc. Moreover, for representations of trees this enables us to realize the whole derived Picard group over a field as a factor of the spectral Picard group. |
| title | Abstract representation theory via coherent Auslander-Reiten diagrams |
| topic | Representation Theory Algebraic Topology Category Theory 55U35, 16G20 (Primary), 18G80, 55P43 (Secondary) |
| url | https://arxiv.org/abs/2511.02731 |