Realisable classes, Stickelberger subgroup and its behaviour under change of the base field

Andrea Siviero

doi:10.5802/pmb.13

Andrea Siviero

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 69-92.

Résumé
Abstract

Soient $K$ un corps de nombres d’anneau des entiers $O_{K}$ et $G$ un groupe fini. On note $R (O_{K} [G])$ l’ensemble des classes dans le groupe des classes des modules localement libres $Cl (O_{K} [G])$ qui peuvent être obtenues par l’anneau des entiers des K-algèbres galoisiennes modérément ramifiées de groupe de Galois $G$ . McCulloh a prouvé que, pour tout $G$ , l’ensemble $R (O_{K} [G])$ est contenu dans le soi-disant sous-groupe de Stickelberger $St (O_{K} [G])$ dans $Cl (O_{K} [G])$ .

Dans ce papier d’abord nous nous focalisons sur la relation entre $St (O_{K} [G])$ et ${Cl}^{\circ} (O_{K} [G])$ , où ${Cl}^{\circ} (O_{K} [G])$ est le noyau du morphisme $Cl (O_{K} [G]) ⟶ Cl (O_{K})$ , induit par l’augmentation $O_{K} [G] ⟶ O_{K}$ . Puis, comme exemple de calcul du groupe $St (O_{K} [G])$ , nous montrons, en utilisant sa définition, que $St (ℤ [G])$ est trivial si $G$ est soit un groupe cyclique d’ordre $p$ soit un groupe diédral d’ordre $2 p$ , avec $p$ premier impair.

Enfin, nous montrons la fonctorialité de $St (O_{K} [G])$ par rapport au changement du corps de base. Ceci implique que, soit $L$ est un corps de nombres, si $N$ est une $L$ -algèbre galoisienne modérément ramifiée, de groupe de Galois $G$ , et $St (O_{K} [G])$ est connu être trivial pour un certain sous-corps $K$ de $L$ , alors $O_{N}$ est un $O_{K} [G]$ -module stablement libre.

Let $K$ be an algebraic number field with ring of integers $O_{K}$ and let $G$ be a finite group. We denote by $R (O_{K} [G])$ the set of classes in the locally free class group $Cl (O_{K} [G])$ realisable by rings of integers in tamely ramified $G$ -Galois $K$ -algebras. McCulloh showed that, for every $G$ , the set $R (O_{K} [G])$ is contained in the so-called Stickelberger subgroup $St (O_{K} [G])$ of $Cl (O_{K} [G])$ .

In this paper first we describe the relation between $St (O_{K} [G])$ and ${Cl}^{\circ} (O_{K} [G])$ , where ${Cl}^{\circ} (O_{K} [G])$ is the kernel of the morphism $Cl (O_{K} [G]) ⟶ Cl (O_{K})$ , induced by the augmentation map $O_{K} [G] ⟶ O_{K}$ . Then, as an example of computation of $St (O_{K} [G])$ , we show, just using its definition, that $St (ℤ [G])$ is trivial, when $G$ is a cyclic group of order $p$ or a dihedral group of order $2 p$ , where $p$ is an odd prime number.

Finally we prove that $St (O_{K} [G])$ has good functorial behaviour under change of the base field. This has the interesting consequence that, given an algebraic number field $L$ , if $N$ is a tame Galois $L$ -algebra with Galois group $G$ and $St (O_{K} [G])$ is known to be trivial for some subfield $K$ of $L$ , then $O_{N}$ is stably free as an $O_{K} [G]$ -module.

Reçu le : 2014-07-14
Publié le : 2016-01-24

Zbl

DOI : 10.5802/pmb.13

Classification : 11R33, 11R04, 11R18, 11R29, 11R32, 11R65
Mots clés : Galois module structure, Realisable classes, Locally free class groups, Fröhlich’s Hom-description of locally free class groups, Stickelberger’s theorem

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     title = {Realisable classes, {Stickelberger} subgroup and its behaviour under change of the base field},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {69--92},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2015},
     doi = {10.5802/pmb.13},
     zbl = {1414.11152},
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     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.13/}
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Andrea Siviero. Realisable classes, Stickelberger subgroup and its behaviour under change of the base field. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 69-92. doi : 10.5802/pmb.13. https://pmb.centre-mersenne.org/articles/10.5802/pmb.13/

Bibliographie
Cité par

[AC15] A. Agboola and L. McCulloh, On the relative Galois module structure of rings of integers in tame extensions, Available at: http://arxiv.org/abs/1410.4829. | DOI | MR | Zbl

[BS13] N. P. Byott and B. Sodaïgui, Realizable Galois module classes over the group ring for non abelian extensions, Ann. Inst. Fourier (Grenoble), 63 (1):303–371, 2013. | DOI | MR | Zbl

[BS05a] N. P. Byott and B. Sodaïgui, Galois module structure for extensions of degree 8: realizable classes over the group ring, J. Number Theory, 112 (1):1–19, 2005. | DOI | MR | Zbl

[BS05b] N. P. Byott and B. Sodaïgui, Realizable Galois module classes for tetrahedral extensions, Compos. Math., 141 (3):573–582, 2005. | DOI | MR | Zbl

[CR81] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1981. With applications to finite groups and orders, A Wiley-Interscience Publication. | Zbl

[CR87] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1987. With applications to finite groups and orders, A Wiley-Interscience Publication. | Zbl

[Lan94] S. Lang, Algebraic number theory, volume 110 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994. | DOI | Zbl

[McC77] L. R. McCulloh, A Stickelberger condition on Galois module structure for Kummer extensions of prime degree, In Algebraic number fields: $L$ -functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages 561–588. Academic Press, London, 1977.

[McC83] L. R. McCulloh, Galois module structure of elementary abelian extensions, J. Algebra, 82 (1):102–134, 1983. | DOI | MR | Zbl

[McC87] L. R. McCulloh, Galois module structure of abelian extensions, J. Reine Angew. Math., 375/376:259–306, 1987. | DOI | MR | Zbl

[McC] L. R. McCulloh, From galois module classes to steinitz classes. arXiv:1207.5702 - Informal report given in Oberwolfach in February 2002.

[Rei03] I. Reiner, Maximal orders, volume 28 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original, with a foreword by M. J. Taylor. | Zbl

[Rim59] D. S. Rim, Modules over finite groups, Ann. of Math. (2), 69:700–712, 1959. | DOI | MR | Zbl

[Ser94] J. P. Serre, Cohomologie galoisienne, volume 5 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, fifth edition, 1994. | DOI

[Siv13] A. Siviero, Class invariants for tame Galois algebras. PhD thesis, Université Bordeaux I - Universiteit Leiden, 2013. Available at: http://tel.archives-ouvertes.fr/tel-00847787.

[Swa62] R. G. Swan, Projective modules over group rings and maximal orders Ann. of Math. (2), 76:55–61, 1962. | DOI | MR | Zbl

[Tay81] M. J. Taylor, On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math., 63(1):41–79, 1981. | DOI | Zbl

[Was97] L. C. Washington, Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997. | DOI | Zbl

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