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.

Citation

Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra, 2086-2101. https://doi.org/10.1080/00927872.2018.1529237

• Acceptance Date: Aug 5, 2018
• Publication Date: Feb 22, 2019
• 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