Normality and short exact sequences of Hopf-Galois structures

Abstract

Every Hopf-Galois structure on a finite Galois extension K/k where $G=Gal(K/k)$ corresponds uniquely to a regular subgroup $N\le B=\text{Perm}(G)$, normalized by $\lambda (G)\le B$, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is $H_{\mathrm{N}}={(K[N])}^{\lambda (G)}$. For a given such N we consider the Hopf-Galois structure arising from a subgroup $P⊲N$ that is also normalized by $\lambda (G)$. This subgroup gives rise to a Hopf sub-algebra $H_{\mathrm{P}}\subseteq H_{\mathrm{N}}$ 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 H_P to $F{\otimes }_{\mathrm{k}}{H}_{\mathrm{P}}$ 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.

Keywords:

• Greither-Pareigis theory
• Hopf-Galois extension
• regular subgroup

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T05; 20B35; 11S20