[Paper Review] Finite determinacy of matrices over local rings.II. Tangent modules to the miniversal deformations for group-actions involving the ring automorphisms

Genrich Belitskii, Dmitry Kerner|arXiv (Cornell University)|Apr 21, 2016

Advanced Topics in Algebra20 references8 citations

TL;DR

This paper establishes criteria for finite determinacy of matrices over local rings under group actions involving ring automorphisms (e.g., coordinate changes), extending prior results on $GL(m,R) \times GL(n,R)$-actions. It analyzes the tangent module to the miniversal deformation, $T^1_{(\Sigma,G,A)}$, and derives bounds on its annihilator, providing explicit support conditions that determine finite determinacy for symmetric and skew-symmetric matrices, modules, and forms.

ABSTRACT

We consider matrices with entries in a local ring, Mat(m,n;R). Fix an action of group G on Mat(m,n;R), and a subset of allowed deformations, Σin Mat(m,n;R). The standard question (along the lines of Singularity Theory) is the finite-(Σ,G)-determinacy of matrices. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)}, and their quotient: the tangent module to the miniversal deformation. In particular, the order of determinacy is controlled by the annihilator of this tangent module. Then we have studied this tangent module for the group action GL(m,R) imes GL(n,R) on Mat(m,n;R) and for various natural subgroups of it. These are R-linear group actions. In the current work we study this tangent module for group actions that involve the automorphisms of the ring, or, geometrically, group-actions that involve the local coordinate changes. (These actions are not R-linear.) We obtain various bounds on the support of this module. This gives ready-to-use criteria of determinacy for matrices, (embedded) modules and (skew-)symmetric forms.

Motivation & Objective

• To extend finite determinacy theory to group actions involving ring automorphisms (e.g., local coordinate changes) in addition to linear actions.
• To analyze the tangent module $T^1_{(\Sigma,G,A)}$ for such actions, particularly in the context of symmetric and skew-symmetric matrices.
• To derive explicit criteria for finite determinacy based on the support of $T^1_{(\Sigma,G,A)}$ and its annihilator.
• To generalize Maranda-type results and sufficiency of jets to actions including non-$R$-linear transformations.
• To provide geometric and algebraic conditions—especially on degeneracy loci—for finite determinacy in the presence of coordinate changes.

Proposed method

• Reduces finite $(\Sigma,G)$-determinacy to the study of the tangent module $T^1_{(\Sigma,G,A)} = T_{(\Sigma,A)} / T_{(GA,A)}$.
• Analyzes the support of $T^1_{(\Sigma,G,A)}$ via localization at prime ideals and computation of ranks over residue fields.
• Uses the decomposition of matrices under congruence actions to reduce the problem to invariants like $I_j(A)$ and $\det(A)$.
• Applies results from singularity theory, such as the codimension of congruence orbits, to bound the rank of $T_{(\mathcal{G}_{congr}A,A)}$.
• Relies on localization and generic point analysis to determine when $T^1_{(\Sigma,G,A)}$ is supported only at the origin.
• Employs the condition $\mathfrak{m}^N \subseteq \operatorname{ann}(T^1_{(\Sigma,G,A)})$ as equivalent to finite determinacy.

Experimental results

Research questions

• RQ1 When is a matrix $A \in Mat_{m\times n}(R)$ finitely determinate under a group action that includes ring automorphisms?
• RQ2 How does the support of the tangent module $T^1_{(\Sigma,G,A)}$ determine finite determinacy in the presence of coordinate changes?
• RQ3 What are the geometric conditions on degeneracy loci $V(I_j(A))$ that ensure finite determinacy for symmetric or skew-symmetric matrices?
• RQ4 How does the inclusion of $\operatorname{Der}_{\mathbb{k}}(R)$ in the tangent space affect the structure of $T^1_{(\Sigma,G,A)}$?
• RQ5 Under what conditions is $T^1_{(\Sigma,G,A)}$ supported only at the origin, implying finite determinacy?

Key findings

• For symmetric matrices under $\mathcal{G}_{congr} = GL(m,R) \rtimes \operatorname{Aut}_{\mathbb{k}}(R)$, finite determinacy holds if $\operatorname{ann}(T^1_{(\Sigma,\mathcal{G}_{congr},A)}) = \{0\}$, which occurs when $\operatorname{rank}(\operatorname{Der}_{\mathbb{k}}(R)) < \lfloor m/2 \rfloor$.
• The annihilator of $T^1_{(\Sigma,\mathcal{G}_{congr},A)}$ is supported on the first-order degeneracy locus $\operatorname{Sing}_1(\det(A))$.
• For skew-symmetric matrices, $\operatorname{ann}(T^1_{(\Sigma,\mathcal{G}_{congr},A)})$ is supported on $\operatorname{Sing}_1(I_{m-1}(A))$.
• In the regular, Noetherian, Henselian case, $A \in Mat^{sym}_{m\times m}(R)$ is finitely-$(\Sigma^{sym},\mathcal{G}_{congr})$-determined iff all $V(I_j(A))$ have expected dimension and $V(I_j(A)) \setminus V(I_{j-1}(A))$ are smooth.
• The tangent module $T^1_{(\Sigma,G,A)}$ is generically supported on $\operatorname{Spec}(R)$ if $\operatorname{rank}(\operatorname{Der}_{\mathbb{k}}(R)) < \lfloor m/2 \rfloor$, implying $\operatorname{ann}(T^1_{(\Sigma,G,A)}) = \{0\}$.
• The criterion for finite determinacy is equivalent to $\mathfrak{m}^N \subseteq \operatorname{ann}(T^1_{(\Sigma,G,A)})$, which holds iff $T^1_{(\Sigma,G,A)}$ is supported only at the origin.