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

We consider matrices with entries in a local ring, Mat_m×n(R). Fix a group action, G⥁Mat_m×n(R), and a subset of allowed deformations, Σ⊆Mat_m×n(R). The standard question in Singularity Theory is the finite-(Σ,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 T_(Σ,A), T_(GA,A), and their quotient, the tangent module to the miniversal deformation, [Figure presented]. In particular, the order of determinacy is controlled by the annihilator of this tangent module, ann(T_(Σ,G,A) ¹). In this work we study this tangent module for the group action GL(m,R)×GL(n,R)⥁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 and others.