Models of group schemes of roots of unity

Mézard, A.; Romagny, M.; Tossici, D.

doi:10.5802/aif.2784

Models of group schemes of roots of unity
[Modèles de schémas en groupes de racines de l’unité]

Mézard, A.¹ ; Romagny, M.² ; Tossici, D.³

¹ Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
² Institut de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
³ Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1055-1135.

Résumé
Abstract

Soit $𝒪_{K}$ un anneau de valuation discrète de caractéristique mixte $(0, p)$ , de corps résiduel $k$ . Utilisant un travail de Sekiguchi et Suwa, nous construisons des modèles finis plats sur $𝒪_{K}$ du schéma en groupes $μ_{p^{n}, K}$ des racines $p^{n}$ -ièmes de l’unité, que nous appelons schémas en groupes de Kummer. Nous développons soigneusement le cadre général et les propriétés algébriques de cette construction. Lorsque $k$ est parfait et $𝒪_{K}$ est une extension complète totalement ramifiée de l’anneau des vecteurs de Witt $W (k)$ , nous étudions en parallèle les modules de Breuil-Kisin des modèles finis plats de $μ_{p^{n}, K}$ , de telle manière que les constructions des groupes de Kummer et des modules de Breuil-Kisin peuvent être comparées. Nous calculons ces objets pour $n \leq 3$ . Cela nous mène à conjecturer que tous les modèles finis plats de $μ_{p^{n}, K}$ sont des schémas en groupes de Kummer.

Let $𝒪_{K}$ be a discrete valuation ring of mixed characteristics $(0, p)$ , with residue field $k$ . Using work of Sekiguchi and Suwa, we construct some finite flat $𝒪_{K}$ -models of the group scheme $μ_{p^{n}, K}$ of $p^{n}$ -th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and $𝒪_{K}$ is a complete totally ramified extension of the ring of Witt vectors $W (k)$ , we provide a parallel study of the Breuil-Kisin modules of finite flat models of $μ_{p^{n}, K}$ , in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for $n \leq 3$ . This leads us to conjecture that all finite flat models of $μ_{p^{n}, K}$ are Kummer group schemes.

Reçu le : 2011-05-04
Accepté le : 2012-02-21

MR Zbl

DOI : 10.5802/aif.2784

Classification : 14L15
Keywords: group schemes, roots of unity, Breuil-Kisin module
Mot clés : schéma en groupes, racines de l’unité, module de Breuil-Kisin

Affiliations des auteurs :

Mézard, A. ¹ ; Romagny, M. ² ; Tossici, D. ³

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     title = {Models of group schemes of roots of unity},
     journal = {Annales de l'Institut Fourier},
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Mézard, A.; Romagny, M.; Tossici, D. Models of group schemes of roots of unity. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1055-1135. doi : 10.5802/aif.2784. https://aif.centre-mersenne.org/articles/10.5802/aif.2784/

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