Which weakly ramified group actions admit a universal formal deformation?

TL;DR

This paper investigates when weakly ramified group actions on power series rings admit universal formal deformations, showing that non-pro-representability is limited to specific cases in characteristic two.

Contribution

It characterizes conditions under which weakly ramified group actions have universal formal deformations, identifying exceptions in characteristic two with small groups.

Findings

Non-pro-representability occurs only in characteristic two with groups of order two or Klein group.

Only the case with G of order two in characteristic two has a non-pro-representable equicharacteristic deformation.

Most weakly ramified actions admit universal formal deformations, except in specific characteristic two cases.

Abstract

Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve 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.