Sufficient Conditions for Large Galois Scaffolds

TL;DR

This paper establishes sufficient conditions for the existence of Galois scaffolds in finite Galois p-extensions of local fields, facilitating the study of their module structure and enabling new constructions of Hopf orders.

Contribution

It provides general sufficient conditions for Galois scaffold existence applicable to any Galois p-group and extends previous results to characteristic zero, with applications to Hopf order construction.

Findings

Conditions for Galois scaffold existence in general Galois p-extensions.

Extension of Elder's result from characteristic p to characteristic 0.

Construction of new Hopf orders over valuation rings.

Abstract

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and for $K$ of either possible characteristic (0 or $p$), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [G.G. Elder, Proc. A.M.S. 137 (2009), 1193-1203] from characteristic $p$ to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring $\mathfrak{O}_K$ that lie in $K[G]$ for $G$ an elementary abelian $p$-group.