2012/12/31 by Genrich Belitskii, Belitskii, Genrich, Dmitry Kerner +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1212.6894
openalex publication_date 2012/12/31 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
Fix a module M over a local ring R and a group action G on M, not necessarily\nR-linear. To understand how large is the G-orbit of an element z\∈ M one looks\nfor the large submodules of M lying in Gz. We provide the corresponding\n(necessary/sufficient) conditions in terms of the tangent space to the orbit,\nT(Gz,z).\n This question originates from the classical finite determinacy problem of\nSingularity Theory. Our treatment is rather general, in particular we extend\nthe classical criteria of Mather (and many others) to a broad class of rings,\nmodules and group actions.\n When a particular `deformation space' is prescribed, \Σ\⊆ M, the\ndeterminacy question is translated into the properties of the tangent spaces,\nT(Gz,z), T( Si,z), and in particular to the annihilator of their\nquotient.\n