Let be an algebraic number field with ring of integers and let be a finite group. We denote by the set of classes in the locally free class group realisable by rings of integers in tamely ramified -Galois -algebras. McCulloh showed that, for every , the set is contained in the so-called Stickelberger subgroup of .
In this paper first we describe the relation between and , where is the kernel of the morphism , induced by the augmentation map . Then, as an example of computation of , we show, just using its definition, that is trivial, when is a cyclic group of order or a dihedral group of order , where is an odd prime number.
Finally we prove that has good functorial behaviour under change of the base field. This has the interesting consequence that, given an algebraic number field , if is a tame Galois -algebra with Galois group and is known to be trivial for some subfield of , then is stably free as an -module.
Soient un corps de nombres d’anneau des entiers et un groupe fini. On note l’ensemble des classes dans le groupe des classes des modules localement libres qui peuvent être obtenues par l’anneau des entiers des K-algèbres galoisiennes modérément ramifiées de groupe de Galois . McCulloh a prouvé que, pour tout , l’ensemble est contenu dans le soi-disant sous-groupe de Stickelberger dans .
Dans ce papier d’abord nous nous focalisons sur la relation entre et , où est le noyau du morphisme , induit par l’augmentation . Puis, comme exemple de calcul du groupe , nous montrons, en utilisant sa définition, que est trivial si est soit un groupe cyclique d’ordre soit un groupe diédral d’ordre , avec premier impair.
Enfin, nous montrons la fonctorialité de par rapport au changement du corps de base. Ceci implique que, soit est un corps de nombres, si est une -algèbre galoisienne modérément ramifiée, de groupe de Galois , et est connu être trivial pour un certain sous-corps de , alors est un -module stablement libre.
Published online:
DOI: 10.5802/pmb.13
Mots-clés : Galois module structure, Realisable classes, Locally free class groups, Fröhlich’s Hom-description of locally free class groups, Stickelberger’s theorem
@article{PMB_2015____69_0, author = {Andrea Siviero}, 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}, language = {en}, url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.13/} }
TY - JOUR AU - Andrea Siviero TI - Realisable classes, Stickelberger subgroup and its behaviour under change of the base field JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2015 SP - 69 EP - 92 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.13/ DO - 10.5802/pmb.13 LA - en ID - PMB_2015____69_0 ER -
%0 Journal Article %A Andrea Siviero %T Realisable classes, Stickelberger subgroup and its behaviour under change of the base field %J Publications mathématiques de Besançon. Algèbre et théorie des nombres %D 2015 %P 69-92 %I Presses universitaires de Franche-Comté %U https://pmb.centre-mersenne.org/articles/10.5802/pmb.13/ %R 10.5802/pmb.13 %G en %F PMB_2015____69_0
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/
[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: -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
Cited by Sources: