Scaffolds and generalized integral Galois module structure

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

doi:10.5802/aif.3182

Byott, Nigel P.¹; Childs, Lindsay N.²; Elder, G. Griffith ³

¹ Department of Mathematics University of Exeter Exeter, EX4 4QF (UK)
² Department of Mathematics and Statistics University at Albany Albany, NY 12222 (USA)
³ Department of Mathematics University of Nebraska at Omaha Omaha NE 68182-0243 (USA)

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

Abstract
Résumé

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.

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).

Received: 2014-12-15
Revised: 2016-06-18
Accepted: 2017-07-13
Published online: 2018-05-04

DOI: 10.5802/aif.3182

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

Author's affiliations:

Byott, Nigel P. ¹; Childs, Lindsay N. ²; Elder, G. Griffith ³

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
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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/

References
Cited by

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

[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 | DOI | Zbl

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

[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

[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

[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

[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 | DOI | Zbl

[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

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

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

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

[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 | DOI | Zbl

[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

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

[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 | DOI | Zbl

[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

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

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

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

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

[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 | DOI | Zbl

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

[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

[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 | DOI | Zbl

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

[31] Marklove, Maria L. Local Galois Module Structure in Characteristic $p$ , University of Exeter (U.K.) (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

[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 | DOI | Zbl

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

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

[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, 67, Springer, New York, 1979 (Translated from the French by Marvin Jay Greenberg) | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

[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] | DOI | Zbl

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

[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

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

[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

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