A new formula for the classical dominant dimension using bimodules
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911121446797312 |
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| author | Cruz, Tiago Marczinzik, René |
| author_facet | Cruz, Tiago Marczinzik, René |
| contents | We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new formula for the classical dominant dimension using bimodules Cruz, Tiago Marczinzik, René Representation Theory Rings and Algebras 16D20, 16E10 (Primary) 16D40 (Secondary) We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata. |
| title | A new formula for the classical dominant dimension using bimodules |
| topic | Representation Theory Rings and Algebras 16D20, 16E10 (Primary) 16D40 (Secondary) |
| url | https://arxiv.org/abs/2508.18398 |