## 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.

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.

Original language | English GB |
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Publisher | arXiv:1904.10790 [math.AC] |

State | Published - 2019 |