Which weakly ramified group actions admit a universal formal deformation?

Jakub Byszewski[1]; Gunther Cornelissen[1]

  • [1] Universiteit Utrecht Mathematisch Instituut Postbus 80.010 3508 TA Utrecht (Nederland)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 3, page 877-902
  • ISSN: 0373-0956

Abstract

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Consider a representation of a finite group G as automorphisms of a power series ring k [ [ t ] ] over a perfect field k of positive characteristic. Let D be the associated formal mixed-characteristic deformation functor. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Example: for a group action on an ordinary curve, the action of a ramification group on the completed local ring of any point is weakly ramified.We prove that the only such D that are not pro-representable occur if k has characteristic two and G is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.

How to cite

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Byszewski, Jakub, and Cornelissen, Gunther. "Which weakly ramified group actions admit a universal formal deformation?." Annales de l’institut Fourier 59.3 (2009): 877-902. <http://eudml.org/doc/10422>.

@article{Byszewski2009,
abstract = {Consider a representation of a finite group $G$ as automorphisms of a power series ring $k[[t ]]$ over a perfect field $k$ of positive characteristic. Let $D$ be the associated formal mixed-characteristic deformation functor. Assume that the action of $G$ is weakly ramified, i.e., the second ramification group is trivial. Example: for a group action on an ordinary curve, the action of a ramification group on the completed local ring of any point is weakly ramified.We prove that the only such $D$ that are not pro-representable occur if $k$ has characteristic two and $G$ is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.},
affiliation = {Universiteit Utrecht Mathematisch Instituut Postbus 80.010 3508 TA Utrecht (Nederland); Universiteit Utrecht Mathematisch Instituut Postbus 80.010 3508 TA Utrecht (Nederland)},
author = {Byszewski, Jakub, Cornelissen, Gunther},
journal = {Annales de l’institut Fourier},
keywords = {Local group action; weak ramification; formal deformation; universality; Nottingham group},
language = {eng},
number = {3},
pages = {877-902},
publisher = {Association des Annales de l’institut Fourier},
title = {Which weakly ramified group actions admit a universal formal deformation?},
url = {http://eudml.org/doc/10422},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Byszewski, Jakub
AU - Cornelissen, Gunther
TI - Which weakly ramified group actions admit a universal formal deformation?
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 3
SP - 877
EP - 902
AB - Consider a representation of a finite group $G$ as automorphisms of a power series ring $k[[t ]]$ over a perfect field $k$ of positive characteristic. Let $D$ be the associated formal mixed-characteristic deformation functor. Assume that the action of $G$ is weakly ramified, i.e., the second ramification group is trivial. Example: for a group action on an ordinary curve, the action of a ramification group on the completed local ring of any point is weakly ramified.We prove that the only such $D$ that are not pro-representable occur if $k$ has characteristic two and $G$ is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.
LA - eng
KW - Local group action; weak ramification; formal deformation; universality; Nottingham group
UR - http://eudml.org/doc/10422
ER -

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