  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.
