Normality and Short Exact Sequences of Hopf-Galois Structures

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