Exterior Powers of Lubin-Tate Groups

S. Mohammad Hadi Hedayatzadeh

doi:10.5802/jtnb.895

S. Mohammad Hadi Hedayatzadeh ¹

¹ Department of Mathematics Mail Code 253-37 California Institute of Technology Pasadena, CA 91125, U.S.A.

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 77-148.

Abstract
Résumé

Let $𝒪$ be the ring of integers of a non-Archimedean local field of characteristic zero and $π$ a fixed uniformizer of $𝒪$ . We prove that the exterior powers of a $π$ -divisible module of dimension at most 1 over a locally Noetherian scheme exist and commute with arbitrary base change. We calculate the height and dimension of the exterior powers in terms of the height of the given $π$ -divisible module. In the case of $p$ -divisible groups, the existence of the exterior powers are proved without any condition on the basis.

Soient $𝒪$ l’anneau des entiers d’un corps local non-archimédien de charactéristique zéro et $π$ une uniformisante de $𝒪$ . On démontre que les puissances extérieures d’un schéma en $𝒪$ -modules $π$ -divisible de dimension au plus 1 sur un schéma de base localement noetherian existent et commutent avec changements de base arbitraires. De même, on calcule la hauteur et la dimension des puissances extérieures en termes de la hauteur du groupe $p$ -divisible ou du schéma en $𝒪$ -modules $π$ -divisible donné. Dans le cas des groupes $p$ -divisibles, on démontre l’existence des puissances extérieures sans aucune hypothèse sur le schéma de base.

MR

DOI: 10.5802/jtnb.895

Classification: 14L05, 14F30

Author's affiliations:

S. Mohammad Hadi Hedayatzadeh ¹

¹ Department of Mathematics Mail Code 253-37 California Institute of Technology Pasadena, CA 91125, U.S.A.

@article{JTNB_2015__27_1_77_0,
     author = {S. Mohammad Hadi Hedayatzadeh},
     title = {Exterior {Powers} of {Lubin-Tate} {Groups}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {77--148},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     doi = {10.5802/jtnb.895},
     mrnumber = {3346966},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.895/}
}

TY  - JOUR
AU  - S. Mohammad Hadi Hedayatzadeh
TI  - Exterior Powers of Lubin-Tate Groups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2015
SP  - 77
EP  - 148
VL  - 27
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.895/
DO  - 10.5802/jtnb.895
LA  - en
ID  - JTNB_2015__27_1_77_0
ER  -

%0 Journal Article
%A S. Mohammad Hadi Hedayatzadeh
%T Exterior Powers of Lubin-Tate Groups
%J Journal de théorie des nombres de Bordeaux
%D 2015
%P 77-148
%V 27
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.895/
%R 10.5802/jtnb.895
%G en
%F JTNB_2015__27_1_77_0

S. Mohammad Hadi Hedayatzadeh. Exterior Powers of Lubin-Tate Groups. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 77-148. doi : 10.5802/jtnb.895. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.895/

References
Cited by

[1] T. Ahsendorf, $𝒪$ -displays and $π$ -divisible formal $𝒪$ -modules, PhD thesis, Universität Bielefeld, 2011.

[2] G.W. Anderson, $t$ -motives, Duke Math. J. 53, 2 (1986), 457–502. | MR | Zbl

[3] V. Buchstaber and A. Lazarev, Dieudonné modules and $p$ -divisible groups associated with Morava $K$ -theory of Eilenberg-Mac Lane spaces, Algebr. Geom. Topol. 7, (2007), 529–564. | MR | Zbl

[4] C.-L. Chai, The group action on the closed fiber of the Lubin-Tate moduli space, Duke Math. J. 82, 3 (1996), 725–754. | MR | Zbl

[5] M. Chen, Le Morphisme Déterminant Pour Les Espaces De Modules De Groupes P-Divisibles, PhD thesis, Université Paris XI, Paris, 2011.

[6] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Hautes Études Sci. Publ. Math. 82, (1995), 5–96 (1996). | MR | Zbl

[7] M. Demazure, Lectures on $p$ -divisible groups, Lecture Notes in Mathematics, 302, Springer-Verlag, Berlin, (1986), Reprint of the 1972 original. | MR | Zbl

[8] M. Demazure and P. Gabriel, Groupes algébriques, Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris, 1970. Avec un appendice ıt Corps de classes local par Michiel Hazewinkel. | Zbl

[9] V.G. Drinfeld, Coverings of $p$ -adic symmetric domains, Funkcional. Anal. i Priložen. 10, 2 (1976), 29–40. | MR | Zbl

[10] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8, (1961), 222. | EuDML | Zbl

[11] M.H. Hedayatzadeh, Exterior powers of Barsotti-Tate groups, Preprint, available at . | arXiv

[12] M.H. Hedayatzadeh, Exterior powers of $π$ -divisible modules over fields, J. Number Theory, 138, (2014), 119–174. | MR

[13] M.J. Hopkins, M. J., and B.H. Gross, Equivariant vector bundles on the Lubin-Tate moduli space, in Topology and representation theory (Evanston, IL, 1992), Contemp. Math. Amer. Math. Soc. 158, Providence, RI, (1994), 23–88. | MR | Zbl

[14] L. Illusie, Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck), Astérisque, 127, (1985), 151–198. Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). | MR | Zbl

[15] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 39, 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2000. | MR | Zbl

[16] H. Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, (1989). Translated from the Japanese by M. Reid. | MR | Zbl

[17] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, 264 Springer-Verlag, Berlin (1972). | MR | Zbl

[18] J.S. Milne, Étale cohomology, Princeton Mathematical Series bf 33, Princeton University Press, Princeton, N.J., (1980). | MR | Zbl

[19] D. Mumford, Bi-extensions of formal groups, in Algebraic Geometry Internat. Colloq., Tata Inst. Fund. Res., (Bombay, 1968). Oxford Univ. Press, London, 1969, 307–322. | MR | Zbl

[20] D.C. Ravenel and W.S. Wilson, The Morava $K$ -theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102, 4 (1980), 691–748. | MR | Zbl

[21] P. Scholze and J. Weinstein, Moduli of $p$ -divisible groups. Preprint, available at http://www.math.uni-bonn.de/people/scholze/. | MR

[22] T. Wedhorn, The dimension of Oort strata of Shimura varieties of PEL-type, in Moduli of abelian varieties (Texel Island, 1999), Progr. Math. 195, Birkhäuser, Basel, (2001), 441–471. | MR | Zbl

[23] J. Weinstein, Formal vector spaces over a local field of positive characteristic, Preprint, available at http://math.bu.edu/people/jsweinst/.

[24] T. Zink, The display of a formal $p$ -divisible group, Astérisque, 278, (2002), 127–248. Cohomologies $p$ -adiques et applications arithmétiques, I. | MR | Zbl

Cited by Sources: