Models of group schemes of roots of unity
[Modèles de schémas en groupes de racines de l’unité]
Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1055-1135.

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 n3. 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 n3. This leads us to conjecture that all finite flat models of μ p n ,K are Kummer group schemes.

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
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
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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. http://www.numdam.org/articles/10.5802/aif.2784/

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