Liftable D_4-Covers

TL;DR

This paper proves that certain group actions on local power series rings in characteristic 2, specifically D_4-actions, can always be lifted to characteristic 0, introducing a new family called supersimple D_4-actions.

Contribution

It establishes the existence of liftable D_4-actions in characteristic 2 and introduces supersimple D_4-actions that can always be lifted to characteristic 0.

Findings

Liftable D_4-actions exist in characteristic 2.

Supersimple D_4-actions can always be lifted to characteristic 0.

The paper provides a new family of liftable actions in the context of local Galois covers.

Abstract

Let k be an algebraically closed field of characteristic p and let G be a subgroup of Aut(k[[t]]) be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G inside Aut(R[[t]]) which reduces to the given action, i.e. a lift to characteristic 0. We show that liftable actions exists in the case that G = D_4 and p = 2. In fact we introduce a family, the supersimple D_4 -actions, which can always be lifted to characteristic 0.