Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

TL;DR

This paper establishes a precise criterion for when the valuation ring of certain elementary abelian p-extensions in characteristic p is free over its associated order, confirming a conjecture about Galois scaffolds' utility in Galois module theory.

Contribution

It provides a necessary and sufficient condition for the valuation ring to be free over its associated order in the context of elementary abelian p-extensions with Galois scaffolds, linking to prior results in cyclic Kummer extensions.

Findings

Condition matches Miyata's criterion for cyclic Kummer extensions

Valuation ring is free over the associated order under the given condition

Supports the utility of Galois scaffolds in Galois module questions

Abstract

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.