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| Format: | Preprint |
| Veröffentlicht: |
2022
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| Online-Zugang: | https://arxiv.org/abs/2210.09344 |
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| _version_ | 1866917854810472448 |
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| author | Cruz, Tiago |
| author_facet | Cruz, Tiago |
| contents | In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. These are used to establish precise connections between Ringel duality, split quasi-hereditary covers and double centraliser properties.
These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring and they are shown to behave nicely under change of rings techniques. It turns out that relative codominant dimension with respect to a summand of a characteristic tilting module is a useful tool to construct quasi-hereditary covers of Noetherian algebras and measure their quality. In particular, this homological invariant is used to construct split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of $q$-Schur algebras. Combining techniques of cover theory with relative dominant dimension theory we obtain a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09344 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On split quasi-hereditary covers and Ringel duality Cruz, Tiago Representation Theory Rings and Algebras 16E30, 16G30, 13E10, 20G43, 17B10 In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. These are used to establish precise connections between Ringel duality, split quasi-hereditary covers and double centraliser properties. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring and they are shown to behave nicely under change of rings techniques. It turns out that relative codominant dimension with respect to a summand of a characteristic tilting module is a useful tool to construct quasi-hereditary covers of Noetherian algebras and measure their quality. In particular, this homological invariant is used to construct split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of $q$-Schur algebras. Combining techniques of cover theory with relative dominant dimension theory we obtain a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. |
| title | On split quasi-hereditary covers and Ringel duality |
| topic | Representation Theory Rings and Algebras 16E30, 16G30, 13E10, 20G43, 17B10 |
| url | https://arxiv.org/abs/2210.09344 |