Normality and short exact sequences of Hopf-Galois structures

Abstract

Every Hopf-Galois structure on a finite Galois extension K/k where corresponds uniquely to a regular subgroup , normalized by , in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is . For a given such N we consider the Hopf-Galois structure arising from a subgroup that is also normalized by . This subgroup gives rise to a Hopf sub-algebra with fixed field . 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 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.