Let be a finite, totally ramified -extension of complete local fields with residue fields of characteristic , and let be a -algebra acting on . We define the concept of an -scaffold on , thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where was Galois and for . When a suitable -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 . 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 . We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois -extensions in characteristic . We also apply our results to the non-classical situation where 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 -Hopf algebra.
Soit une extension finie et totalement ramifiée, de degré une puissance de , de corps locaux complets dont le corps résiduel a caractéristique . Soit une -algèbre qui opère sur . Nous définissons le concept d’un -échafaudage sur . Ceci étend et raffine la notion d’échafaudage galoisien, que nous avons considérée dans plusieurs articles antérieurs, où était une extension galoisienne et pour . Dans le cas où il existe un -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 . 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 . Nous appliquons ces résultats aux extensions galoisiennes biquadratiques de caractéristique 2, et aux extensions totalement et faiblement ramifiées, de degré une puissance de et de caractéristique . Nous appliquons nos résultats aussi à la situation non classique où est une extension finie, purement inséparable, d’exposant quelconque, sur laquelle opère la -algèbre de Hopf des puissances divisées par une dérivation supérieure (mais avec beaucoup d’actions différentes).
Revised:
Accepted:
Published online:
Classification: 11S15, 20C11, 16T05, 11R33
Keywords: Ramification, Galois module structure, Hopf–Galois theory
@article{AIF_2018__68_3_965_0, author = {Byott, Nigel P. and Childs, Lindsay N. and Elder, G. Griffith}, title = {Scaffolds and generalized integral {Galois} module structure}, journal = {Annales de l'Institut Fourier}, pages = {965--1010}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {3}, year = {2018}, doi = {10.5802/aif.3182}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3182/} }
TY - JOUR TI - Scaffolds and generalized integral Galois module structure JO - Annales de l'Institut Fourier PY - 2018 DA - 2018/// SP - 965 EP - 1010 VL - 68 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3182/ UR - https://doi.org/10.5802/aif.3182 DO - 10.5802/aif.3182 LA - en ID - AIF_2018__68_3_965_0 ER -
Byott, Nigel P.; Childs, Lindsay N.; Elder, G. Griffith. Scaffolds and generalized integral Galois module structure. Annales de l'Institut Fourier, Volume 68 (2018) no. 3, pp. 965-1010. doi : 10.5802/aif.3182. https://aif.centre-mersenne.org/articles/10.5802/aif.3182/
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