A higher dimensional Auslander-Iyama-Solberg correspondence
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929335542218752 |
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| author | Cruz, Tiago Psaroudakis, Chrysostomos |
| author_facet | Cruz, Tiago Psaroudakis, Chrysostomos |
| contents | In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between $n$-minimal Auslander-Gorenstein algebras and $n$-precluster tilting modules. If $A$ is an $n$-minimal Auslander-Gorenstein algebra, then the pair $(A,P)$ is a relative $(n+1)$-Auslander-Gorenstein pair in the sense of the authors, where $P$ is the minimal faithful projective-injective left $A$-module.
We establish a higher dimensional Auslander-Iyama-Solberg, where $P$ is replaced by any self-orthogonal module $Q$ having finite projective and injective dimension. This new correspondence provides a bijection between relative Auslander--Gorenstein pairs and a new class of objects that generalise precluster tilting modules. This way, we obtain a new correspondence coming from the modular representation theory of general linear groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A higher dimensional Auslander-Iyama-Solberg correspondence Cruz, Tiago Psaroudakis, Chrysostomos Representation Theory Rings and Algebras 16G10, 16E10, 16E65 (Primary) 16G50, 20G43, 16G20, 16S50, 18G25 (Secondary) In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between $n$-minimal Auslander-Gorenstein algebras and $n$-precluster tilting modules. If $A$ is an $n$-minimal Auslander-Gorenstein algebra, then the pair $(A,P)$ is a relative $(n+1)$-Auslander-Gorenstein pair in the sense of the authors, where $P$ is the minimal faithful projective-injective left $A$-module. We establish a higher dimensional Auslander-Iyama-Solberg, where $P$ is replaced by any self-orthogonal module $Q$ having finite projective and injective dimension. This new correspondence provides a bijection between relative Auslander--Gorenstein pairs and a new class of objects that generalise precluster tilting modules. This way, we obtain a new correspondence coming from the modular representation theory of general linear groups. |
| title | A higher dimensional Auslander-Iyama-Solberg correspondence |
| topic | Representation Theory Rings and Algebras 16G10, 16E10, 16E65 (Primary) 16G50, 20G43, 16G20, 16S50, 18G25 (Secondary) |
| url | https://arxiv.org/abs/2405.02736 |