Models of group schemes of roots of unity

Mézard, A.; Romagny, M.; Tossici, D.

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 ${\mu }_{p^n,K}$ 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 $W\left(k\right)$, nous étudions en parallèle les modules de Breuil-Kisin des modèles finis plats de ${\mu }_{p^n,K}$, de telle manière que les constructions des groupes de Kummer et des modules de Breuil-Kisin peuvent être comparées. Nous calculons ces objets pour $n\le 3$. Cela nous mène à conjecturer que tous les modèles finis plats de ${\mu }_{p^n,K}$ 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 ${\mu }_{p^n,K}$ 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 $W\left(k\right)$, we provide a parallel study of the Breuil-Kisin modules of finite flat models of ${\mu }_{p^n,K}$, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for $n\le 3$. This leads us to conjecture that all finite flat models of ${\mu }_{p^n,K}$ are Kummer group schemes.

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

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