TY - JOUR

T1 - Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

AU - Belitskii, Genrich

AU - Kerner, Dmitry

N1 - Funding Information:
D.K. was supported by the Israel Science Foundation (grant No. 844/14).
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We consider matrices with entries in a local ring, Matm×n(R). Fix a group action, G⥁Matm×n(R), and a subset of allowed deformations, Σ⊆Matm×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)⥁Matm×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.

AB - We consider matrices with entries in a local ring, Matm×n(R). Fix a group action, G⥁Matm×n(R), and a subset of allowed deformations, Σ⊆Matm×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)⥁Matm×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.

UR - http://www.scopus.com/inward/record.url?scp=85048940051&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2018.06.007

DO - 10.1016/j.jpaa.2018.06.007

M3 - Article

AN - SCOPUS:85048940051

SN - 0022-4049

VL - 223

SP - 1288

EP - 1321

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 3

ER -