Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

TL;DR

This paper investigates the finite determinacy of matrices over local rings under group actions, providing criteria based on tangent modules for various types of deformations, which advances understanding in Singularity Theory.

Contribution

It develops criteria for finite determinacy of matrices under group actions by analyzing tangent modules, extending previous work to new classes of deformations and subgroups.

Findings

Criteria for determinacy of module deformations

Explicit tangent module calculations for GL actions

Applications to symmetric and filtered modules

Abstract

We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteq Mat(m,n,R). The standard question of Singularity Theory is the finite-(\Sigma,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces to \Sigma and to the orbit, T_{(\Sigma,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. In this work we study this tangent module for the group action GL(m,R)\times GL(n,R) on Mat(m,n,R) and various natural subgroups of it. We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules etc.