Group actions on filtered modules and finite determinacy. Finding large submodules in the orbit by linearization

Genrich Belitskii, Dmitry Kerner

Research output: Working paper/PreprintPreprint

Abstract

Fix a module M over a local ring R and a group action G on 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_{(\Si,z)}, and in particular to the annihilator of their quotient.
Original languageEnglish GB
PublisherarXiv:1212.6894 [math.AG]
StatePublished - 2016

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