Commutative Hopf-Galois module structure of tame extensions

TL;DR

This paper investigates the module structure of fractional ideals in tame Galois extensions of p-adic and number fields, establishing conditions under which these ideals are free or locally free over their associated orders in commutative Hopf algebras.

Contribution

It proves that fractional ideals are free over their associated orders in tame Galois extensions and extends this to almost classically Galois and abelian extensions, clarifying module structures in these cases.

Findings

Fractional ideals are free over their associated orders in tame Galois extensions.

This freeness extends to almost classically Galois extensions.

Ambiguous fractional ideals are locally free in abelian extensions.

Abstract

We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L $ is locally free over its associated order in $ H $.