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Group Actions on Filtered Modules and Finite Determinacy. Finding Large Submodules in the Orbit by Linearization

C. R. Math. Rep. Acad. Sci. Canada Vol. 38 (4) 2016, pp. 113-155
Vol.38 (4) 2016
Genrich Belitskii; Dmitry Kerner Details
(Received: 2015-07-20 , Revised: 2016-01-27 )
(Received: 2015-07-20 , Revised: 2016-01-27 )

Genrich Belitskii,Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel; e-mail: genrich@math.bgu.ac.il

Dmitry Kerner,Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel; e-mail dmitry.kerner@gmail.com

Abstract/Résumé:

Let \(M\) be a module over a local ring \(R\) and a group action \(G\circlearrowright M\), not necessarily \(R\)-linear. To understand how large is the \(G\)-orbit of an element \(z\in M\) one looks for the large submodules of \(M\) lying in \(Gz\). We provide the corresponding (necessary/sufficient) conditions in terms of the tangent space to the orbit, \(T_{(Gz,z)}\).

This question originates from the classical finite determinacy problem of Singularity Theory. Our treatment is rather general, in particular we extend the classical criteria of Mather (and many others) to a broad class of rings, modules and group actions.

When a particular ‘deformation space’ is prescribed, \(\Sigma\subseteq M\), the determinacy question is translated into the properties of the tangent spaces, \(T_{(Gz,z)}\), \(T_{(\Sigma,z)}\), and in particular to the annihilator of their quotient, \(ann\,{T_{(\Sigma,z)}}/{T_{(Gz,z)}}\).

Etant donné une action d’un groupe sur un module, \(G\circlearrowright M\), et un élément \(z\in M\), on étudie le plus grand sous-module de \(M\) contenu dans l’orbite \(Gz\). On donne des conditions nécessaires et suffisantes décrivant ce module en termes de l’espace tangent a l’orbite, \(T_{(Gz,z)}\). Cela prolonge les critères classiques de la théorie des singularités à une large classe d’anneaux, modules, et actions de groupes.

Keywords: Group actions, finite determinancy, matrix families, matrix singularities, modules over local rings, open orbits, sufficiency of jets

AMS Subject Classification: Deformations of singularities, Canonical forms; reductions; classification, Normal families of functions; mappings, Classification; finite determinacy of map germs, Normal forms 14B07, 15A21, 32A19, 58K40, 58K50

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