Truman, PJ, Kohl, T, Koch, A and Underwood, R (2019) Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra.

[img] Text
tarp-ses.pdf - Accepted Version
Restricted to Repository staff only until 22 February 2020.
Available under License Creative Commons Attribution Non-commercial.

Download (209kB)


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 λ(G) ≤ B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is HN = (K[N])λ(G). For a given such N we consider the Hopf-Galois structure arising from a subgroup P ⊳ N that is also normalized by λ(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.

Item Type: Article
Additional Information: This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Taylor and Francis at Please refer to any applicable terms of use of the publisher.
Uncontrolled Keywords: Hopf-Galois extension, Greither-Pareigis theory, regular subgroup, MSC 16T05,20B35,11S20
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Maths
Depositing User: Symplectic
Date Deposited: 20 Aug 2018 13:37
Last Modified: 25 Feb 2019 16:24

Actions (login required)

View Item View Item