N-fold module factorizations: triangle equivalences and recollements
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916891722776576 |
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| author | Sun, Yongliang Zhang, Yaohua |
| author_facet | Sun, Yongliang Zhang, Yaohua |
| contents | As an extension of Eisenbud's matrix factorization into the non-commutative realm, X.W. Chen introduced the concept of module factorizations over an arbitrary ring. A theorem of Chen establishes a triangle equivalence between the stable category of module factorizations with Gorenstein projective components and the stable category of Gorenstein projective modules over a quotient ring. In this paper, we introduce $n$-fold module factorizations, which generalize both the commutative $n$-fold matrix factorizations and the non-commutative module factorizations. To adapt triangle equivalences in module factorizations to $n$-fold module factorizations, we identify suitable subcategories of module factorizations and rings for the $n$-analogue. We further provide the $n$-analogue of Chen's theorem on triangle equivalences. Additionally, we study recollements involving the stable categories of higher-fold module factorizations, revealing intriguing recollements within the stable categories of Gorenstein modules of specific matrix subrings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | N-fold module factorizations: triangle equivalences and recollements Sun, Yongliang Zhang, Yaohua Rings and Algebras 16E65, 18G80, 18G65, 18G25 As an extension of Eisenbud's matrix factorization into the non-commutative realm, X.W. Chen introduced the concept of module factorizations over an arbitrary ring. A theorem of Chen establishes a triangle equivalence between the stable category of module factorizations with Gorenstein projective components and the stable category of Gorenstein projective modules over a quotient ring. In this paper, we introduce $n$-fold module factorizations, which generalize both the commutative $n$-fold matrix factorizations and the non-commutative module factorizations. To adapt triangle equivalences in module factorizations to $n$-fold module factorizations, we identify suitable subcategories of module factorizations and rings for the $n$-analogue. We further provide the $n$-analogue of Chen's theorem on triangle equivalences. Additionally, we study recollements involving the stable categories of higher-fold module factorizations, revealing intriguing recollements within the stable categories of Gorenstein modules of specific matrix subrings. |
| title | N-fold module factorizations: triangle equivalences and recollements |
| topic | Rings and Algebras 16E65, 18G80, 18G65, 18G25 |
| url | https://arxiv.org/abs/2406.09655 |