Sufficient Conditions for Large Galois Scaffolds

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.

Some minor changes to exposition, and references added/updated. To appear in Journal of Number Theory

References in corpus (1)

• Explicit Construction of Self-Dual Integral Normal Bases for the Square-Root of the Inverse Different

Cited by in corpus (1)

• Hopf orders in $K[C_p^3]$ in characteristic $p$