Characteristic-Zero Lifts of Local Galois Extensions

Definitions and mathematical statement

Let $p$ be a prime, let $k$ be an algebraically closed field of characteristic $p$, and let $G$ be a finite group. Write $k[[t]]$ and $k[[z]]$ for formal power series rings. Suppose that a local $G$-Galois extension \[ \iota:k[[t]]\hookrightarrow k[[z]] \] is given. Here the local $G$-Galois terminology is interpreted as a finite inclusion equipped with a faithful continuous action of $G$ by $k$-algebra automorphisms of $k[[z]]$, with \[ k[[z]]^G=\iota(k[[t]]). \] Ramification is allowed.

Does every such extension admit the following lift? There should exist a complete discrete valuation ring $R$ of characteristic zero, a uniformizer $\pi$, an identification $R/(\pi)\cong k$, and a finite injective $R$-algebra homomorphism \[ \Phi:R[[T]]\hookrightarrow R[[Z]], \] together with a faithful continuous action of $G$ by $R$-algebra automorphisms of $R[[Z]]$, such that \[ R[[Z]]^G=\Phi(R[[T]])? \] The required reduction condition is that there exist $k$-algebra isomorphisms \[ \alpha:R[[T]]/\pi R[[T]]\xrightarrow{\sim}k[[t]], \qquad \beta:R[[Z]]/\pi R[[Z]]\xrightarrow{\sim}k[[z]], \] where $\beta$ is $G$-equivariant and, writing $\overline{\Phi}$ for the map induced by $\Phi$, \[ \beta\circ\overline{\Phi}=\iota\circ\alpha. \] The ring $R$ and all lifting data may depend on the given extension.

The supplied formulation places no restriction on the finite group $G$. The positive-characteristic interpretation and the local, possibly ramified meaning of $G$-Galois are made explicit here. The source's universal formulation and its supplied open-status assessment require mathematical review: the supplied statement about cyclic groups alone does not establish the status of the question for arbitrary finite groups.

Short English statement

Given a finite local Galois extension of power series rings over an algebraically closed field of positive characteristic, ask whether it lifts to power series rings over some complete characteristic-zero discrete valuation ring, preserving the specified group action after reduction. No restriction on the finite group is supplied; the universal scope and supplied open label require review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002710$, source problem code catalogue entry, title Characteristic-Zero Lifts of Local Galois Extensions. No individual proposer is identified; the supplied proposed year is 2014.
• Supplied report citation: Valuation Theory and Its Applications (2015), Oberwolfach Reports, pp. 2757--2823, DOI 10.4171/owr/2014/49. Literature-source label: Oberwolfach Report 49/2014. https://doi.org/10.4171/owr/2014/49. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Obus, A generalization of the Oort conjecture. https://arxiv.org/abs/1502.07623.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the local lifting problem is solved for cyclic groups, while no general theorem covers arbitrary finite Galois groups. It consequently describes the extracted universal question as open. This is supplied dated curation, not an independently verified literature assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied record labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This verification claim is reproduced as source metadata and has not been independently checked here.