Models of group schemes of roots of unity

A. Mézard[1]; M. Romagny[2]; D. Tossici[3]

  • [1] Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
  • [2] Institut de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
  • [3] Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 1055-1135
  • ISSN: 0373-0956

Abstract

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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 n ≤ 3 . This leads us to conjecture that all finite flat models of μ p n , K are Kummer group schemes.

How to cite

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Mézard, A., Romagny, M., and Tossici, D.. "Models of group schemes of roots of unity." Annales de l’institut Fourier 63.3 (2013): 1055-1135. <http://eudml.org/doc/275525>.

@article{Mézard2013,
abstract = {Let $\mathcal\{O\}_K$ 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 $\mathcal\{O\}_K$-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 $\mathcal\{O\}_K$ 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 $\mu _\{p^n,K\}$, 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.},
affiliation = {Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France; Institut de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy},
author = {Mézard, A., Romagny, M., Tossici, D.},
journal = {Annales de l’institut Fourier},
keywords = {group schemes; roots of unity; Breuil-Kisin module},
language = {eng},
number = {3},
pages = {1055-1135},
publisher = {Association des Annales de l’institut Fourier},
title = {Models of group schemes of roots of unity},
url = {http://eudml.org/doc/275525},
volume = {63},
year = {2013},
}

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
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 1055
EP - 1135
AB - Let $\mathcal{O}_K$ 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 $\mathcal{O}_K$-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 $\mathcal{O}_K$ 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 $\mu _{p^n,K}$, 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.
LA - eng
KW - group schemes; roots of unity; Breuil-Kisin module
UR - http://eudml.org/doc/275525
ER -

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