Torsion pairs via the Ziegler spectrum
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910349566935040 |
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| author | Hügel, Lidia Angeleri Laking, Rosanna Sentieri, Francesco |
| author_facet | Hügel, Lidia Angeleri Laking, Rosanna Sentieri, Francesco |
| contents | We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $τ$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion pairs via the Ziegler spectrum Hügel, Lidia Angeleri Laking, Rosanna Sentieri, Francesco Representation Theory Rings and Algebras 16G10, 16G20, 16E35, 18E40 We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $τ$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring. |
| title | Torsion pairs via the Ziegler spectrum |
| topic | Representation Theory Rings and Algebras 16G10, 16G20, 16E35, 18E40 |
| url | https://arxiv.org/abs/2403.00475 |