Commuting Hopf-Galois Structures on a Separable Extension

Abstract

Let $L/K$ be a finite separable extension of local or global fields in any characteristic, let $H_1,H_2$ be two Hopf algebras giving Hopf-Galois structures on the extension, and suppose that the actions of $H_1,H_2$ on $L$ commute. We show that a fractional ideal $\mathfrak{B}$ of $L$ is free over its associated order in $H_1$ if and only if it is free over its associated order in $H_2$. We also study which properties these associated orders share.