Prorepresentable Obstructions for Local Action Deformations

Definitions and mathematical statement

Let $\rho$ be the local $G$-action on $k[[t]]$ in the supplied record, and let $N\leq G$. Write \[ B=k[[t]]^N=\{b\in k[[t]]:\rho(n)(b)=b\text{ for every }n\in N\}, \qquad \Theta=\operatorname{Der}_k(k[[t]]),\qquad \Theta^\sharp=\operatorname{Der}_k(B). \] Here $\operatorname{Der}_k(A)$ denotes the module of $k$-linear derivations of a $k$-algebra $A$. The action on derivations is by conjugation, \[ g\cdot\delta=\rho(g)\circ\delta\circ\rho(g)^{-1}, \qquad \Theta^N=\{\delta\in\Theta:n\cdot\delta=\delta\text{ for every }n\in N\}. \] Restriction to $B$ gives a map $\Theta^N\to\Theta^\sharp$. The source's quotient $\Theta^\sharp/\Theta^N$ presupposes that this map identifies $\Theta^N$ with a submodule of $\Theta^\sharp$. Its use of the quotient group $G/N$ also requires $N$ to be normal in $G$, although the supplied statement calls $N$ only a subgroup. Subject to these requirements and the induced $G/N$-module structure, put \[ Q=G/N,\qquad M=\Theta^\sharp/\Theta^N. \] The notation $H^1(Q,M)$ denotes first group cohomology.

Write $D_G$, $D_N^{G/N}$, and $D_{G/N}$ for the deformation functors denoted by these symbols in the source, and write \[ \psi=(\psi_N,\psi_Q):D_G\longrightarrow D_N^{G/N}\times D_{G/N} \] for its stated natural map. The supplied record does not define these functors, their coefficient category, their equivalence relation on deformations, or the superscript in $D_N^{G/N}$. These data must be recovered before this becomes a fully specified deformation problem.

Question, conditional on the missing setup. For every local action and subgroup satisfying the intended source hypotheses, does there exist a pro-representable functor $F$, together with an isomorphism of tangent spaces \[ T_F\cong H^1(Q,M) \] and natural transformations \[ a:D_N^{G/N}\longrightarrow F, \qquad b:D_{G/N}\longrightarrow F, \] such that $a\circ\psi_N=b\circ\psi_Q$ and the resulting natural transformation \[ \widetilde\psi:D_G\longrightarrow D_N^{G/N}\times_F D_{G/N} \] is smooth? The fiber product is understood objectwise: for every coefficient object $A$, \[ \left(D_N^{G/N}\times_F D_{G/N}\right)(A) =\{(x,y):x\in D_N^{G/N}(A),\ y\in D_{G/N}(A),\ a_A(x)=b_A(y)\}. \]

In the usual setting of deformation functors on local Artinian coefficient rings with residue field $k$, pro-representability means representation by continuous local homomorphisms from a complete local coefficient ring with residue field $k$, and $T_F=F(k[\varepsilon]/(\varepsilon^2))$ with its deformation-theoretic vector-space structure. Smoothness has its infinitesimal lifting meaning: writing $P=D_N^{G/N}\times_F D_{G/N}$, for every small extension $A'\twoheadrightarrow A$ in the intended coefficient category, the map \[ D_G(A')\longrightarrow D_G(A)\times_{P(A)}P(A') \] is surjective. A small extension here means a surjection whose kernel is annihilated by the maximal ideal of $A'$. These explanations are conditional on that conventional setting; the record does not specify the coefficient category, the characteristic or other hypotheses on $k$, finiteness conditions on $G$, or continuity and faithfulness conditions on $\rho$. No such additional hypotheses are asserted as supplied facts.

Short English statement

The question asks whether deformations of a local group action map smoothly to pairs of subgroup and quotient deformations that agree in a pro-representable functor $F$, whose tangent space is $H^1(G/N,\Theta^\sharp/\Theta^N)$. The source leaves essential hypotheses and deformation-functor definitions unspecified, so this is a conditional statement requiring mathematical review. Its open status is supplied dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001113$, title Prorepresentable Obstructions for Local Action Deformations, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the proposed year is recorded as 2008.
• Supplied source citation: Mini-Workshop: Group Actions on Curves: Reduction and Lifting (2009), Oberwolfach Reports, pp. 3029--3064, DOI 10.4171/owr/2008/54. Original report and supplied literature-source link: https://doi.org/10.4171/owr/2008/54. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source link, labeled "later local-deformation discussion": https://ems.press/content/serial-article-files/46200?nt=1. No author, title, or more specific bibliographic information for this item is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the analogous smooth fiber-product factorization remains a general open deformation-theoretic question; it says later work analyzes the functors and obstruction spaces without supplying the asserted smooth morphism. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official report, retaining the complete fiber-product smoothness question and tangent-space data. This is a supplied verification claim; no source visit or independent mathematical verification is asserted here.