Scaffolds and generalized integral Galois module structure

Abstract

Let L/K be a finite, totally ramified p-extension of complete local fields with residue fields of characteristic p > 0, and let A be a K-algebra acting on L. We define the concept of an A-scaffold on L, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where L/K was Galois and A = K[G] for G = Gal(L/K). When a suitable A-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in A. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with L/K. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois p-extensions in characteristic p. We also apply our results to the non-classical situation where L/K is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power K-Hopf algebra.

Galois scaffolds and semistable extensions

“…It follows from Theorem 4.1 that L/K has a Galois scaffold ({Ψ i }, {λ t }) with precision 1. Since char(K) = p we have Ψ p i = 0 for 1 ≤ i ≤ n. Hence by Theorem A.1(ii) of [3] the extension L/K has a Galois scaffold with precision ∞. Therefore by Corollary 4.3 L/K is stable.…”

“…In [3,Th. 3.1] sufficient conditions are given for an ideal M h L in an extension L/K with a Galois scaffold to be free over its associated order A h .…”

“…As explained on page 974 of [3], it follows from the definition of a Galois scaffold that for every t ∈ Z there is U st ∈ O × K such that the following holds modulo λ t+b(s) M c L :…”

“…For a fractional ideal M h L of L define the associated order is free over A h for a totally ramified extension L/K of degree p n appears to be quite difficult. Two classes of extensions for which a partial answer can be obtained are semistable extensions [2] and extensions with a Galois scaffold [3]. In this note we consider the relation between semistable extensions and Galois scaffolds.…”

Ramified extensions of degree p and their Hopf–Galois module structure

“…One explanation for this progress is that cyclic ramified extensions of degree p naturally possess a scaffold. This is discussed in [BCE14, §4.1], although the definition of scaffold, as presented in [BCE14] in its full generality, may be a challenge to digest. For extensions of degree p however, a very simple sufficiency condition is available: If there is an element x ∈ L with p ∤ v L (x) and an element Ψ ∈ K[G] that "acts like" the derivative d/dx on the K-basis {x i } p−1 i=0 for L over K, there is a scaffold.…”