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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2304.01370 |
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| _version_ | 1866911121145856000 |
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| author | Cruz, Tiago |
| author_facet | Cruz, Tiago |
| contents | Important correspondences in representation theory can be regarded as restrictions of the Morita--Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita--Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two.
In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators which generalise generators and cogenerators, respectively.
Using these new concepts, we present higher versions of the Morita--Tachikawa correspondence that takes into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimension. These new versions also hold over Noetherian algebras which are finitely generated and projective over a commutative Noetherian ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01370 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Morita-Tachikawa correspondence Cruz, Tiago Representation Theory Rings and Algebras 16E10, 16G10, 16G30 Important correspondences in representation theory can be regarded as restrictions of the Morita--Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita--Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators which generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita--Tachikawa correspondence that takes into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimension. These new versions also hold over Noetherian algebras which are finitely generated and projective over a commutative Noetherian ring. |
| title | Higher Morita-Tachikawa correspondence |
| topic | Representation Theory Rings and Algebras 16E10, 16G10, 16G30 |
| url | https://arxiv.org/abs/2304.01370 |