Alan Koch
Normality and Short Exact Sequences of Hopf-Galois Structures
Koch, Alan; Kohl, Timothy; Truman, Paul J.; Underwood, Robert
Abstract
Every Hopf-Galois structure on a finite Galois extension K/k where G = Gal(K/k) corresponds uniquely to a regular subgroup N = B = Perm(G), normalized by ?(G) = B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is HN = (K[N])?(G). For a given such N we consider the Hopf-Galois structure arising from a subgroup P ? N that is also normalized by ?(G). This subgroup gives rise to a Hopf sub-algebra HP ? HN with fixed field F = KHP . By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension K/F where the action arises by base changing HP to F ?k HP which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.
Acceptance Date | Aug 5, 2018 |
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Publication Date | Feb 22, 2019 |
Publicly Available Date | Mar 28, 2024 |
Journal | Communications in Algebra |
Print ISSN | 0092-7872 |
Publisher | Taylor and Francis |
Pages | 2086-2101 |
DOI | https://doi.org/10.1080/00927872.2018.1529237 |
Keywords | Hopf-Galois extension, Greither-Pareigis theory, regular subgroup, MSC 16T05,20B35,11S20 |
Publisher URL | https://doi.org/10.1080/00927872.2018.1529237 |
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Publisher Licence URL
https://creativecommons.org/licenses/by-nc/4.0/
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