Commuting Hopf-Galois Structures on a Separable Extension

TL;DR

This paper investigates the relationship between commuting Hopf-Galois structures on a separable extension, showing that fractional ideals' freeness over associated orders is equivalent for both structures and analyzing shared properties.

Contribution

It establishes an equivalence in the freeness of fractional ideals over associated orders for commuting Hopf-Galois structures on separable extensions.

Findings

Freeness of fractional ideals over associated orders is equivalent for commuting structures.

Shared properties of associated orders are analyzed.

Results apply to local and global fields in any characteristic.

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.