Classification of the types for which every Hopf--Galois correspondence is bijective
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916465493409792 |
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| author | Stefanello, Lorenzo Tsang, Cindy |
| author_facet | Stefanello, Lorenzo Tsang, Cindy |
| contents | Let $L/K$ be any finite Galois extension with Galois group $G$. It is known by Chase and Sweedler that the Hopf--Galois correspondence is injective for every Hopf--Galois structure on $L/K$, but it need not be bijective in general. Hopf--Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of $L/K$ in the image of the Hopf--Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups $G$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure on any $G$-Galois extension. In this paper, using a similar approach, we shall classify the groups $N$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure of type $N$ on any Galois extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of the types for which every Hopf--Galois correspondence is bijective Stefanello, Lorenzo Tsang, Cindy Number Theory Group Theory Rings and Algebras 12F10, 16T05, 20N99 Let $L/K$ be any finite Galois extension with Galois group $G$. It is known by Chase and Sweedler that the Hopf--Galois correspondence is injective for every Hopf--Galois structure on $L/K$, but it need not be bijective in general. Hopf--Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of $L/K$ in the image of the Hopf--Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups $G$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure on any $G$-Galois extension. In this paper, using a similar approach, we shall classify the groups $N$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure of type $N$ on any Galois extension. |
| title | Classification of the types for which every Hopf--Galois correspondence is bijective |
| topic | Number Theory Group Theory Rings and Algebras 12F10, 16T05, 20N99 |
| url | https://arxiv.org/abs/2406.15800 |