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.

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

[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[3] Breuil, C. Schémas en groupes et corps des normes (unpublished manuscript, September 1998)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) (Adv. Stud. Pure Math.), Volume 36, Math. Soc. Japan, 2002, pp. 51-80 | MR | Zbl

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

[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, Mathematical Surveys and Monographs, 80, American Mathematical Society, 2000 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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), Volume 22 (1989) no. 3, pp. 345-375 | Numdam | MR | Zbl

[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), Volume 53 (2001) no. 2, pp. 203-240 | DOI | MR | Zbl

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

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

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

[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 | Zbl

[31] Tossici, D. Models of ${\mu }_{p^2,K}$ over a discrete valuation ring. With an appendix by Xavier Caruso, J. Algebra, Volume 323 (2010) no. 7, pp. 1908-1957 | DOI | MR | Zbl

[32] Underwood, R. $R$-Hopf algebra orders in $K{C}_p^2$, J. Alg., Volume 169 (1994), pp. 418-440 | DOI | MR | Zbl

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