Group Schemes over artinian rings and Applications

@article{Berbec2009,
abstract = {Let $ n $ be a positive integer and $ A^\{\prime\} $ a complete characteristic zero discrete valuation ring with maximal ideal $ \mathfrak\{m\} $, absolute ramification index $ e<p-1 $ and perfect residue field $ k $ of characteristic $ p>2 $. In this paper we classify smooth finite dimensional formal $ p $-faithful groups over $ A_\{n\}^\{\prime\}=A^\{\prime\}/\mathfrak\{m\}^\{n\}A^\{\prime\} $, i.e. groups on which the “multiplication by $ p $” morphism is faithfully flat, in particular $ p $-divisible groups. As applications, we prove that $ p $-divisible groups over $ k $, and the morphisms between them, lift canonically to $ A^\{\prime\}/pA^\{\prime\} $, and we study liftings to characteristic zero of certain connected $ p $-divisible groups of dimension $ d $ and height $ h $ over $ k=\overline\{k\} $, with $ d $ and $ h $ coprime. When $e=1$, we classify finite flat group schemes over $ A^\{\prime\}/p^\{2\}A^\{\prime\} $ of $ p $-power order and prove that a finite flat group scheme over $ A^\{\prime\}/p^\{n\}A^\{\prime\} $ of $ p $-power order, having flat $p^i$-torsion for every $i\ge 1$, lifts to $A^\{\prime\}$.},
affiliation = {University of California at Berkeley Department of Mathematics Berkeley, CA 94720 (USA)},
author = {Berbec, Ioan},
journal = {Annales de l’institut Fourier},
keywords = {Group scheme; $p$-divisible group; almost canonical lifting; group scheme; -divisible group},
language = {eng},
number = {6},
pages = {2371-2427},
publisher = {Association des Annales de l’institut Fourier},
title = {Group Schemes over artinian rings and Applications},
url = {http://eudml.org/doc/10458},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Berbec, Ioan
TI - Group Schemes over artinian rings and Applications
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2371
EP - 2427
AB - Let $ n $ be a positive integer and $ A^{\prime} $ a complete characteristic zero discrete valuation ring with maximal ideal $ \mathfrak{m} $, absolute ramification index $ e<p-1 $ and perfect residue field $ k $ of characteristic $ p>2 $. In this paper we classify smooth finite dimensional formal $ p $-faithful groups over $ A_{n}^{\prime}=A^{\prime}/\mathfrak{m}^{n}A^{\prime} $, i.e. groups on which the “multiplication by $ p $” morphism is faithfully flat, in particular $ p $-divisible groups. As applications, we prove that $ p $-divisible groups over $ k $, and the morphisms between them, lift canonically to $ A^{\prime}/pA^{\prime} $, and we study liftings to characteristic zero of certain connected $ p $-divisible groups of dimension $ d $ and height $ h $ over $ k=\overline{k} $, with $ d $ and $ h $ coprime. When $e=1$, we classify finite flat group schemes over $ A^{\prime}/p^{2}A^{\prime} $ of $ p $-power order and prove that a finite flat group scheme over $ A^{\prime}/p^{n}A^{\prime} $ of $ p $-power order, having flat $p^i$-torsion for every $i\ge 1$, lifts to $A^{\prime}$.
LA - eng
KW - Group scheme; $p$-divisible group; almost canonical lifting; group scheme; -divisible group
UR - http://eudml.org/doc/10458
ER -