Scaffolds and generalized integral Galois module structure

Byott, Nigel P.; Childs, Lindsay N.; Elder, G. Griffith

Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 965-1010.

Soit $L/K$ une extension finie et totalement ramifiée, de degré une puissance de $p$, de corps locaux complets dont le corps résiduel a caractéristique $p>0$. Soit $A$ une $K$-algèbre qui opère sur $L$. Nous définissons le concept d’un $A$-échafaudage sur $L$. Ceci étend et raffine la notion d’échafaudage galoisien, que nous avons considérée dans plusieurs articles antérieurs, où $L/K$ était une extension galoisienne et $A=K\left[G\right]$ pour $G=\mathrm{Gal}(L/K)$. Dans le cas où il existe un $A$-échafaudage convenable, nous montrons comment résoudre des questions qui généralisent celles de la théorie classique des modules galoisiens des anneaux des entiers. Nous donnons une condition nécessaire et suffisante, qui contient seulement des paramètres numériques, pour qu’un idéal fractionnaire quelconque soit un module libre sur son ordre associé dans $A$. Nous montrons aussi comment déterminer le nombre de générateurs dont on a besoin si l’idéal n’est pas libre, et la dimension d’immersion de l’ordre associé. Dans le cas galoisien, les paramètres numériques sont les nombres de ramification de $L/K$. Nous appliquons ces résultats aux extensions galoisiennes biquadratiques de caractéristique 2, et aux extensions totalement et faiblement ramifiées, de degré une puissance de $p$ et de caractéristique $p$. Nous appliquons nos résultats aussi à la situation non classique où $L/K$ est une extension finie, purement inséparable, d’exposant quelconque, sur laquelle opère la $K$-algèbre de Hopf des puissances divisées par une dérivation supérieure (mais avec beaucoup d’actions différentes).

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 $A=K\left[G\right]$ for $G=\mathrm{Gal}(L/K)$. 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.

Classification : 11S15, 20C11, 16T05, 11R33
Keywords: Ramification, Galois module structure, Hopf–Galois theory
Mot clés : Ramification, structure galoisienne, théorie de Hopf–Galois

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