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(mxn)(R). Fix a group action, G (sic) Mat(mxn)(R), and a subset of allowed deformations, Sigma subset of Mat(mxn)(R). The standard question in 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 T-(Sigma,T-A), T-(GA,T-A), and their quotient, the tangent module to the miniversal deformation, T-(Sigma,G,A)(1) = T-(Sigma,T-A)/T(G (A,A)) . In particular, the order of determinacy is controlled by the annihilator of this tangent module, ann(T-(Sigma,G,A)(1)). In this work we study this tangent module for the group action GL(m, R) x GL(n, R) (sic) Mat(mxn)(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. (C) 2018 Elsevier B.V. All rights reserved.