Let be a discrete valuation ring of mixed characteristics , with residue field . Using work of Sekiguchi and Suwa, we construct some finite flat -models of the group scheme of -th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When is perfect and is a complete totally ramified extension of the ring of Witt vectors , we provide a parallel study of the Breuil-Kisin modules of finite flat models of , in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for . This leads us to conjecture that all finite flat models of are Kummer group schemes.
Soit un anneau de valuation discrète de caractéristique mixte , de corps résiduel . Utilisant un travail de Sekiguchi et Suwa, nous construisons des modèles finis plats sur du schéma en groupes des racines -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 est parfait et est une extension complète totalement ramifiée de l’anneau des vecteurs de Witt , nous étudions en parallèle les modules de Breuil-Kisin des modèles finis plats de , de telle manière que les constructions des groupes de Kummer et des modules de Breuil-Kisin peuvent être comparées. Nous calculons ces objets pour . Cela nous mène à conjecturer que tous les modèles finis plats de sont des schémas en groupes de Kummer.
Accepted:
DOI: 10.5802/aif.2784
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
@article{AIF_2013__63_3_1055_0, author = {M\'ezard, A. and Romagny, M. and Tossici, D.}, title = {Models of group schemes of roots of unity}, journal = {Annales de l'Institut Fourier}, pages = {1055--1135}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {3}, year = {2013}, doi = {10.5802/aif.2784}, mrnumber = {3137480}, zbl = {1297.14051}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2784/} }
TY - JOUR AU - Mézard, A. AU - Romagny, M. AU - Tossici, D. TI - Models of group schemes of roots of unity JO - Annales de l'Institut Fourier PY - 2013 SP - 1055 EP - 1135 VL - 63 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2784/ DO - 10.5802/aif.2784 LA - en ID - AIF_2013__63_3_1055_0 ER -
%0 Journal Article %A Mézard, A. %A Romagny, M. %A Tossici, D. %T Models of group schemes of roots of unity %J Annales de l'Institut Fourier %D 2013 %P 1055-1135 %V 63 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2784/ %R 10.5802/aif.2784 %G en %F AIF_2013__63_3_1055_0
Mézard, A.; Romagny, M.; Tossici, D. Models of group schemes of roots of unity. Annales de l'Institut Fourier, Volume 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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