Normality and short exact sequences of Hopf-Galois structures. Issue 5 (4th May 2019)

• Record Type: Journal Article
• Authors: Koch, Alan; Kohl, Timothy; Truman, Paul J.; Underwood, Robert
• Is Part Of: Communications in algebra. Volume 47:Issue 5(2019)
• Page Start: 2086
• Page End: 2101
• Publication Date: 2019-05-04
• Subjects: Greither-Pareigis theory -- Hopf-Galois extension -- regular subgroup; 16T05; 20B35; 11S20; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2018.1529237
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Every Hopf-Galois structure on a finite Galois extension K / k where G = G a l ( 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 H N = ( 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 H P ⊆ H N with fixed field F = K H P . 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 H P 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 H P acting on K / F relates to that of the H N 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.