Scaffolds and integral Hopf Galois module structure on purely inseparable extensions

TL;DR

This paper develops a framework using Hopf Galois scaffolds to analyze purely inseparable extensions of local fields, leading to a Hopf algebra analogue of the Normal Basis Theorem and criteria for fractional ideals to be free over their orders.

Contribution

It introduces the concept of Hopf Galois scaffolds for purely inseparable extensions and applies this to characterize when fractional ideals are free over their associated orders.

Findings

Existence of scaffolds depends on the action of the Hopf algebra.

Established a Hopf Galois analogue of the Normal Basis Theorem.

Provided criteria for fractional ideals to be free over their orders.

Abstract

Let $p$ be prime. Let $L/K$ be a finite, totally ramified, purely inseparable extension of local fields, $\left[ L:K\right] =p^{n},\;n\geq2.$ It is known that $L/K$ is Hopf Galois for numerous Hopf algebras $H,$ each of which can act on the extension in numerous ways. For a certain collection of such $H$ we construct "Hopf Galois scaffolds" which allow us to obtain a Hopf analogue to the Normal Basis Theorem for $L/K.$ The existence of a scaffold structure depends on the chosen action of $H$ on $L.$ We apply the theory of scaffolds to describe when the fractional ideals of $L$ are free over their associated orders in $H.$