[Paper Review] Group actions on filtered modules and finite determinacy. Finding large submodules in the orbit by linearization

Genrich Belitskii, Dmitry Kerner|arXiv (Cornell University)|Dec 31, 2012

Rings, Modules, and Algebras7 citations

TL;DR

This paper establishes necessary and sufficient conditions for finite determinacy of group actions on filtered modules over local rings by analyzing the tangent space to the orbit, $T_{(Gz,z)}$, and its relationship with the deformation space $Σ$. The key result shows that if $M_j \subseteq T_{(G^{(1)}z,z)}$, then the orbit $Gz$ contains an open neighborhood $\{z\} + M_j$, generalizing classical finite determinacy criteria to broad classes of rings, modules, and group actions via linearization and Artin-type approximation.

ABSTRACT

Fix a module M over a local ring R and a group action G on M, not necessarily R-linear. To understand how large is the G-orbit of an element z\in M one looks for the large submodules of M lying in Gz. We provide the corresponding (necessary/sufficient) conditions in terms of the tangent space to the orbit, T_{(Gz,z)}. This question originates from the classical finite determinacy problem of Singularity Theory. Our treatment is rather general, in particular we extend the classical criteria of Mather (and many others) to a broad class of rings, modules and group actions. When a particular `deformation space' is prescribed, Σ\subseteq M, the determinacy question is translated into the properties of the tangent spaces, T_{(Gz,z)}, T_{(\Si,z)}, and in particular to the annihilator of their quotient.

Motivation & Objective

• To generalize classical finite determinacy criteria in singularity theory to arbitrary local rings, filtered modules, and non-linear group actions.
• To determine when the $G$-orbit of an element $z \in M$ contains an open neighborhood $\{z\} + M_j$ in the filtration topology.
• To provide necessary and sufficient conditions for finite $G$-determinacy in terms of the tangent space $T_{(Gz,z)}$ and its interaction with the deformation subspace $\Sigma \subseteq M$.
• To extend the theory to families of deformations using Henselian rings and relative group actions, ensuring orbit containment for one-parameter families.

Proposed method

• Analyzes the tangent space $T_{(Gz,z)}$ to the orbit $Gz$ at $z$ as the key object for determining orbit largeness.
• Uses linearization of the group action to reduce the problem to studying $T_{(G^{(1)}z,z)}$, the tangent space at the identity of the stabilizer group.
• Applies Artin-type approximation theorems to lift formal solutions in the completion $\widehat{R}$ to solutions in $R$, ensuring the orbit contains $\{z\} + M_j$.
• Establishes that $M_j \subseteq T_{(G^{(1)}z,z)}$ implies $\{z\} + M_j \subseteq Gz$ via Nakayama’s lemma in the filtered setting.
• Considers one-parameter families $z(t) \in SM$ with $S = \mathbb{k}[[t]]$ or $S = \mathbb{k}\{t\}$, and proves that if $z(t) \in \{z\} + T_{(Gz,z)}$, then $z(t) \in Gz$ via relative group actions $G_S \subset GL_S(SM)$.
• Uses the completion of the orbit and the structure of $\widehat{G}^{(1)}\hat{z}$ to relate $T_{(\widehat{G}^{(1)}z,z)}$ to $\widehat{M}_j$, enabling the use of Theorem 4.6 and Lemma 3.19.

Experimental results

Research questions

• RQ1 Under what conditions does the $G$-orbit of $z \in M$ contain an open neighborhood $\{z\} + M_j$ in the filtration topology?
• RQ2 How can the finite determinacy of $z$ with respect to a $G$-invariant deformation subspace $\Sigma \subseteq M$ be characterized using tangent space data?
• RQ3 What is the role of the annihilator of the quotient $T_{(\Sigma,z)} / T_{(Gz,z)}$ in determining finite determinacy?
• RQ4 Can one-parameter families $z(t) \in SM$ that lie pointwise in $\{z\} + T_{(Gz,z)}$ be lifted to elements of the orbit $Gz$?
• RQ5 To what extent can the condition $T_{(G^{(1)}z,z)} \subseteq T_{(M,z)} \approx M$ be weakened while preserving the conclusion $M_j \subseteq T_{(G^{(1)}z,z)}$?

Key findings

• If $M_j \subseteq T_{(G^{(1)}z,z)}$, then $\{z\} + M_j \subseteq Gz$, meaning the orbit contains an open neighborhood of $z$ in the filtration topology.
• The condition $M_j \subseteq T_{(G^{(1)}z,z)}$ is both necessary and sufficient for $z$ to be $k$-determined with $k = j$, generalizing classical criteria.
• For unipotent group actions and Henselian rings $S$, if $z(t) \in SM_j$ and $z(t) \in \{z\} + T_{(Gz,z)}$, then $z(t) \in Gz$, implying orbit containment for one-parameter families.
• The use of Artin-type approximation allows lifting formal solutions in $\widehat{R}$ to solutions in $R$, ensuring $\{z\} + M_j \subseteq G^{(1)}z$.
• The completion of the orbit satisfies $\{z\} + \widehat{M}_j \subseteq \widehat{G}^{(1)}\hat{z}$, which implies $\widehat{M}_j \subseteq \overline{T_{(\widehat{G}^{(1)}z,z)}}$, linking completion and tangent space structures.
• Even when $T_{(G^{(1)}z,z)} \cap M_\infty$ is a submodule (e.g., trivial), it is not sufficient to guarantee $M_j \subseteq T_{(G^{(1)}z,z)}$, as shown by counterexamples with $R = \mathbb{k}[[x]]$ and $M = R$.