Sufficient Conditions for Large Galois Scaffolds

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 [BCE]. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] 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 OK that lie in K[G] for G an elementary abelian p-group.