Commutative Hopf-Galois module structure of tame 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 $H$.

arXiv comment15 pages