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

Abstract (Original language)
Résumé

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.

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.

Article information
Cite this article

Received: 2014-07-14
Published online: 2016-01-24

Zbl

DOI: 10.5802/pmb.13

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

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

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