Group actions on matrices over local rings. Annihilators of T^1-modules for the groups \mathcal{G}_{lr} , \mathcal{G}_{congr}

Abstract

We consider matrices with entries in a local ring, Mat(R). Fix a group action, G on Mat(R), and a subset of allowed deformations, \Sigma. The traditional objects of study in Singularity Theory and Algebraic Geometry are the tangent spaces T_{(\Sigma,A)}, T_{(GA,A)}, and their quotient, the tangent module to the miniversal deformation, T^1_{(\Sigma,G,A)}. This module plays the key role in various deformation problems, e.g., deformations of maps, of modules, of (skew-)symmetric forms. In particular, the first question is to determine the support/annihilator of this tangent module. In [Belitski-Kerner.1] we have studied this tangent module for various R-linear group actions. In the current work we study the support of the module T^1_{(\Sigma,G,A)} for group actions that involve automorphisms of the ring. (Geometrically, these are group actions that involve the local coordinate changes.) We obtain various bounds on localizations of T^1_{(\Sigma,G,A)} and compute the radical of the annihilator of T^1_{(\Sigma,G,A)}, i.e., the set-theoretic support. This brings the definition of an (apparently new) type of singular locus, the "essential singular locus" of a map/sub-scheme. It reflects the "unexpected" singularities of a subscheme, ignoring those imposed by the singularities of the ambient space. Unlike the classical singular locus (defined by a Fitting ideal of the module of differentials) the essential is defined by the annihilator ideal of the module of derivations.