Which weakly ramified group actions admit a universal formal deformation?
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 877-902.

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.

Nous considérons une répresentation d’un groupe fini G d’automorphismes d’un anneau de séries formelles k[[t]] sur un corps parfait k de caractéristique positive. Soit D le foncteur associé des déformations formelles en caractéristique mixte. Supposons que l’action de G est faiblement ramifiée, c.-à-d. que le second groupe de ramification est trivial. Exemple : pour une action d’un groupe sur une courbe ordinaire, l’action d’un groupe de ramification sur l’anneau local complèté d’un point quelconque est faiblement ramifiée.

On démontre que les seuls tels foncteurs D qui ne sont pas pro-répresentables se produisent lorsque k est de caractéristique 2 et G est ou bien d’ordre 2, ou bien isomorphe au groupe de Klein. On démontre également que seulement le premier de ces groupes a un foncteur de déformations equicaractéristiques non-pro-répresentable.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2450
Classification: 14B12,  11G20,  14D15
Keywords: Local group action, weak ramification, formal deformation, universality, Nottingham group
Byszewski, Jakub 1; Cornelissen, Gunther 1

1 Universiteit Utrecht Mathematisch Instituut Postbus 80.010 3508 TA Utrecht (Nederland)
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Byszewski, Jakub; Cornelissen, Gunther. Which weakly ramified group actions admit a universal formal deformation?. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 877-902. doi : 10.5802/aif.2450. https://aif.centre-mersenne.org/articles/10.5802/aif.2450/

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