Scaffolds and generalized integral Galois module structure  [ Échafaudages et structure galoisienne généralisée des entiers ]
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[G] pour G=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[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.

Reçu le : 2014-12-14
Révisé le : 2016-06-17
Accepté le : 2017-07-12
Publié le : 2018-05-03
DOI : https://doi.org/10.5802/aif.3182
Classification : 11S15,  20C11,  16T05,  11R33
Mots clés : Ramification, structure galoisienne, théorie de Hopf–Galois
@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'institut Fourier},
     volume = {68},
     number = {3},
     year = {2018},
     doi = {10.5802/aif.3182},
     language = {en},
     url = {www.numdam.org/item/AIF_2018__68_3_965_0/}
}
Byott, Nigel P.; Childs, Lindsay N.; Elder, G. Griffith. Scaffolds and generalized integral Galois module structure. Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 965-1010. doi : 10.5802/aif.3182. http://www.numdam.org/item/AIF_2018__68_3_965_0/

[1] Aiba, Akira Artin-Schreier extensions and Galois module structure, J. Number Theory, Volume 102 (2003) no. 1, pp. 118-124 | Article | Zbl 1035.11059

[2] Allen, Harry Prince; Sweedler, Moss E. A theory of linear descent based on Hopf algebraic techniques, J. Algebra, Volume 12 (1969), pp. 242-294 | Article | Zbl 0257.1624

[3] Bergé, Anne-Marie Sur l’arithmétique d’une extension diédrale, Ann. Inst. Fourier, Volume 22 (1972) no. 2, pp. 31-59 | Article | Zbl 0223.12105

[4] Bertrandias, Françoise; Bertrandias, Jean-Paul; Ferton, Marie-Josée Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A, Volume 274 (1972), pp. 1388-1391 | Zbl 0235.12008

[5] Bertrandias, Françoise; Ferton, Marie-Josée Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A, Volume 274 (1972), pp. 1330-1333 | Zbl 0235.12008

[6] Bondarko, Mikhail V. Local Leopoldt’s problem for rings of integers in abelian p-extensions of complete discrete valuation fields, Doc. Math., Volume 5 (2000), pp. 657-693 | Zbl 0964.11053

[7] Bondarko, Mikhail V. Local Leopoldt’s problem for ideals in totally ramified p-extensions of complete discrete valuation fields, Algebraic number theory and algebraic geometry (Contemporary Mathematics) Volume 300, American Mathematical Society, Providence, RI, 2002, pp. 27-57 | Article | Zbl 1026.11088

[8] Bondarko, Mikhail V. Leopoldt’s problem for abelian totally ramified extensions of complete discrete valuation fields, Algebra Anal., Volume 18 (2006) no. 5, pp. 99-129 (English transl. in St. Petersbg. Math. J. 18 (2007) no. 5, 757–778) | Zbl 1214.12005

[9] Byott, Nigel P. Galois structure of ideals in wildly ramified abelian p-extensions of a p-adic field, and some applications, J. Théor. Nombres Bordx, Volume 9 (1997) no. 1, pp. 201-219 | Article | Zbl 0889.11040

[10] Byott, Nigel P. On the integral Galois module structure of cyclic extensions of p-adic fields, Q. J. Math., Volume 59 (2008) no. 2, pp. 149-162 | Article | Zbl 1225.11155

[11] Byott, Nigel P. A valuation criterion for normal basis generators of Hopf-Galois extensions on characteristic p, J. Théor. Nombres Bordx, Volume 23 (2011) no. 1, pp. 59-70 | Article | Zbl 1278.11103

[12] Byott, Nigel P.; Elder, G. Griffith A valuation criterion for normal bases in elementary abelian extensions, Bull. Lond. Math. Soc., Volume 39 (2007) no. 5, pp. 705-708 | Article | Zbl 1128.11055

[13] Byott, Nigel P.; Elder, G. Griffith Galois scaffolds and Galois module structure in extensions of characteristic p local fields of degree p 2 , J. Number Theory, Volume 133 (2013) no. 11, pp. 3598-3610 | Article | Zbl 1295.11133

[14] Byott, Nigel P.; Elder, G. Griffith Integral Galois module structure for elementary abelian extensions with a Galois scaffold, Proc. Am. Math. Soc., Volume 142 (2014) no. 11, pp. 3705-3712 | Article | Zbl 1320.11111

[15] Byott, Nigel P.; Elder, G. Griffith Sufficient conditions for large Galois scaffolds, J. Number Theory, Volume 182 (2018), pp. 95-130 | Article | Zbl 06781224

[16] Chase, Stephen U.; Sweedler, Moss E. Hopf algebras and Galois theory, Lecture Notes in Math., Volume 97, Springer, Berlin, 1969 | Zbl 0197.01403

[17] Chellali, Mustapha Erratum on: “Structure of inseparable extensions” by M. E. Sweedler, Int. Math. Forum, Volume 2 (2007) no. 65-68, pp. 3269-3272 | Article | Zbl 1169.12300

[18] Childs, Lindsay; Moss, David J. Hopf algebras and local Galois module theory, Advances in Hopf algebras (Chicago, IL, 1992) (Lecture Notes in Pure and Appl. Math.) Volume 158, Dekker, New York, 1994, pp. 1-24 | Zbl 0826.16035

[19] Elder, G. Griffith Galois scaffolding in one-dimensional elementary abelian extensions, Proc. Am. Math. Soc., Volume 137 (2009) no. 4, pp. 1193-1203 | Article | Zbl 1222.11140

[20] Elder, G. Griffith A valuation criterion for normal basis generators in local fields of characteristic p, Arch. Math., Volume 94 (2010) no. 1, pp. 43-47 | Article | Zbl 1220.11143

[21] Ferton, Marie-Josée Sur les idéaux d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A, Volume 276 (1973), pp. 1483-1486 | Zbl 0268.12006

[22] Greither, Cornelius; Pareigis, Bodo Hopf Galois theory for separable field extensions, J. Algebra, Volume 106 (1987) no. 1, pp. 239-258 | Article | Zbl 0615.12026

[23] Heiderich, Florian On Hasse-Schmidt rings and module algebras, J. Pure Appl. Algebra, Volume 217 (2013) no. 7, pp. 1303-1315 | Article | Zbl 1314.16022

[24] Huynh, Duc Van Artin–Schreier extensions and generalized associated orders, J. Number Theory, Volume 136 (2014), pp. 28-45 | Article | Zbl 1286.11205

[25] Jacobinski, Heinz Über die Hauptordnung eines Körpers als Gruppenmodul, J. Reine Angew. Math., Volume 213 (1963/1964), pp. 151-164 | Zbl 0124.02303

[26] Johnston, Henri Explicit integral Galois module structure of weakly ramified extensions of local fields, Proc. Am. Math. Soc., Volume 143 (2015) no. 12, pp. 5059-5071 | Article | Zbl 1331.11104

[27] Koch, Alan Hopf Galois structures on primitive purely inseparable extensions, New York J. Math., Volume 20 (2014), pp. 779-797 | Zbl 1307.16028

[28] Koch, Alan Scaffolds and integral Hopf Galois module structure on purely inseparable extensions, New York J. Math., Volume 21 (2015), pp. 73-91 http://nyjm.albany.edu:8000/j/2015/21_73.html | Zbl 1318.16030

[29] Köck, Bernhard Galois structure of Zariski cohomology for weakly ramified covers of curves, Am. J. Math., Volume 126 (2004) no. 5, pp. 1085-1107 | Article | Zbl 1095.14027

[30] Leopoldt, Heinrich-Wolfgang Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math., Volume 201 (1959), pp. 119-149 | Zbl 0098.03403

[31] Marklove, Maria L. Local Galois Module Structure in Characteristic p (2014) (Ph. D. Thesis)

[32] Martel, Bruno Sur l’anneau des entiers d’une extension biquadratique d’un corps 2-adique, C. R. Acad. Sci. Paris Sér. A, Volume 278 (1974), pp. 117-120 | Zbl 0277.12012

[33] Miyata, Yoshimasa On the module structure of rings of integers in 𝔭-adic number fields over associated orders, Math. Proc. Camb. Philos. Soc., Volume 123 (1998) no. 2, pp. 199-212 | Article | Zbl 1073.11525

[34] Montgomery, Susan Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Volume 82, American Mathematical Society, 1993 | Zbl 0793.16029

[35] Noether, Emmy Normalbasis bei Körpern ohne höhere Verzweigung, J. Reine Angew. Math., Volume 167 (1932), pp. 147-152 | Zbl 0003.14601

[36] Riddle, Lawrence Proof of Lucas’s Theorem, 2013 (http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/LucasProof.htm)

[37] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, Volume 67, Springer, New York, 1979 (Translated from the French by Marvin Jay Greenberg) | Zbl 0423.12016

[38] de Smit, Bart; Florence, Mathieu; Thomas, Lara The valuation criterion for normal basis generators, Bull. Lond. Math. Soc., Volume 44 (2012) no. 4, pp. 729-737 | Article | Zbl 1253.11108

[39] de Smit, Bart; Thomas, Lara Local Galois module structure in positive characteristic and continued fractions, Arch. Math., Volume 88 (2007) no. 3, pp. 207-219 | Article | Zbl 1193.11107

[40] Sweedler, Moss E. Structure of inseparable extensions, Ann. Math., Volume 87 (1968), pp. 401-410 (corrigendum in ibid. 89 (1969), 206–207; cf. also [17]) | Article | Zbl 0168.29203

[41] Sweedler, Moss E. Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969 | Zbl 0194.332901

[42] Taylor, Martin J. Formal groups and the Galois module structure of local rings of integers, J. Reine Angew. Math., Volume 358 (1985), pp. 97-103 | Zbl 0582.12008

[43] Thomas, Lara A valuation criterion for normal basis generators in equal positive characteristic, J. Algebra, Volume 320 (2008) no. 10, pp. 3811-3820 | Article | Zbl 1207.11110

[44] Thomas, Lara On the Galois module structure of extensions of local fields, Actes de la Conférence “Fonctions L et Arithmétique” (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres), Laboratoire de Mathématique de Besançon, 2010, pp. 157-194 | Zbl 1223.11135