Models of group schemes of roots of unity
Annales de l'Institut Fourier, Volume 63 (2013) no. 3, p. 1055-1135

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.

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.

DOI : https://doi.org/10.5802/aif.2784
Classification:  14L15
Keywords: group schemes, roots of unity, Breuil-Kisin module
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     pages = {1055-1135},
     doi = {10.5802/aif.2784},
     mrnumber = {3137480},
     zbl = {1297.14051},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2013__63_3_1055_0/

[1] Abbes, A.; Saito, T. Ramification of local fields with imperfect residue fields I, Amer. J. Math., Tome 124 (2002), pp. 879-920 | Article | MR 1925338 | Zbl 1084.11064

[2] Abramovich, D.; Romagny, M. Moduli of Galois covers in mixed characteristics (to appear in Algebra and Number Theory)

[3] Breuil, C. Schémas en groupes et corps des normes (unpublished manuscript, September 1998)

[4] Breuil, C. Integral p-adic Hodge Theory, Algebraic geometry 2000, Azumino (Hotaka), Math. Soc. Japan (Adv. Stud. Pure Math.) Tome 36 (2002), pp. 51-80 | MR 1971512 | Zbl 1046.11085

[5] Byott, N. P. Cleft extensions of Hopf algebras, Proc. London Math. Soc., Tome 67 (1993), pp. 227-307 | MR 1220775 | Zbl 0795.16026

[6] Caruso, X. Estimation des dimensions de certaines variétés de Kisin (preprint, arXiv:1005.2394)

[7] Childs, L. N. Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, American Mathematical Society, Mathematical Surveys and Monographs, Tome 80 (2000) | MR 1767499 | Zbl 0944.11038

[8] Childs, L. N.; Underwood, R. G. Cyclic Hopf orders defined by isogenies of formal groups, Amer. J. of Math., Tome 125 (2003), pp. 1295-1334 | Article | MR 2018662 | Zbl 1041.16026

[9] Eisenbud, D. Commutative algebra with a view toward algebraic geometry, Springer-Verlag, Graduate Texts in Math., Tome 150 (1995) | MR 1322960 | Zbl 0819.13001

[10] Fargues, L. La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math., Tome 645 (2010), pp. 1-39 | Article | MR 2673421 | Zbl 1199.14015

[11] Fontaine, J.-M. Représentations p-adiques des corps locaux I, The Grothendieck Festschrift, Vol. II, Birkhäuser (Progr. Math.) Tome 87 (1990), pp. 249-309 | MR 1106901 | Zbl 0575.14038

[12] Greither, C. Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z., Tome 210 (1992), pp. 37-67 | Article | MR 1161169 | Zbl 0737.11038

[13] Greither, C.; Childs, L. p-Elementary group schemes-constructions and Raynaud’s Theory, Hopf algebra, Polynomial formal Groups and Raynaud Orders, Mem. Amer. Soc., Tome 136 (1998), pp. 91-118 | Zbl 0983.14032

[14] Imai, N. On the connected components of moduli spaces of finite flat models (to appear in Amer. J. Math) | Zbl 1205.14025

[15] Katz, N.; Mazur, B. Arithmetic moduli of elliptic curves, Princeton University Press, Annals of Mathematics Studies, Tome 108 (1985) | MR 772569 | Zbl 0576.14026

[16] Kisin, M. Crystalline representations and F-crystals, Algebraic geometry and number theory, Birkhäuser (Progr. Math.) Tome 253 (2006), pp. 459-496 | MR 2263197 | Zbl 1184.11052

[17] Kisin, M. Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), Tome 170 (2009) no. 3, pp. 1085-1180 | Article | MR 2600871 | Zbl 1201.14034

[18] Larson, R. Hopf algebra orders determined by group valuations, J. Algebra, Tome 38 (1976) no. 2, pp. 414-452 | Article | MR 404413 | Zbl 0407.20007

[19] Lau, E. A relation between Dieudonné displays and crystalline Dieudonné theory (preprint, arXiv:1006.2720)

[20] Liu, T. The correspondence between Barsotti-Tate groups and Kisin modules when p=2 (preprint, (2011))

[21] Manin, Y. I. Cubic forms. Algebra, geometry, arithmetic, North-Holland (1986) | MR 833513 | Zbl 0582.14010

[22] Mézard, A.; Romagny, M.; Tossici, D. Sekiguchi-Suwa Theory revisited (preprint, (2011))

[23] Romagny, M. Effective models of group schemes (to appear in the Journal of Algebraic Geometry)

[24] Sekiguchi, T.; Oort, N. F. Avd Suwa On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4), Tome 22 (1989) no. 3, pp. 345-375 | Numdam | MR 1011987 | Zbl 0714.14024

[25] Sekiguchi, T.; Suwa, N. On the unified Kummer-Artin-Schreier-Witt Theory (no. 111 in the preprint series of the Laboratoire de Mathématiques Pures de Bordeaux (1999))

[26] Sekiguchi, T.; Suwa, N. A note on extensions of algebraic and formal groups. IV. Kummer-Artin-Schreier-Witt theory of degree p 2 , Tohoku Math. J. (2), Tome 53 (2001) no. 2, pp. 203-240 | Article | MR 1829979 | Zbl 1073.14546

[27] Serre, J.-P. Corps Locaux, Hermann (1980) | MR 354618

[28] Smith, J. D. H. An introduction to quasigroups and their representations, Chapman & Hall, Studies in Advanced Mathematics (2007) | MR 2268350 | Zbl 1122.20035

[29] Tate, J.; Oort, F. Group schemes of prime order, Ann. Sci. Ec. Norm. Sup., Tome 3 (1970), pp. 1-21 | Numdam | MR 265368 | Zbl 0195.50801

[30] Tossici, D. Effective models and extension of torsors over a discrete valuation ring of unequal characteristic, Int. Math. Res. Not. IMRN (2008) (Art. ID rnn111, 68 pp) | MR 2448085 | Zbl 1194.14069

[31] Tossici, D. Models of μ p 2 ,K over a discrete valuation ring. With an appendix by Xavier Caruso, J. Algebra, Tome 323 (2010) no. 7, pp. 1908-1957 | Article | MR 2594655 | Zbl 1193.14059

[32] Underwood, R. R-Hopf algebra orders in KC p 2 , J. Alg., Tome 169 (1994), pp. 418-440 | Article | MR 1297158 | Zbl 0820.16036

[33] Waterhouse, W.; Weisfeiler, B. One-dimensional affine group schemes, J. Algebra, Tome 66 (1980) no. 2, pp. 550-568 | Article | MR 593611 | Zbl 0452.14013