2018/08/19 by Alberto F. Boix, Boix, Alberto F., Gert–Martin Greuel +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1808.06185
openalex publication_date 2018/08/19 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Finite determinacy for mappings has been classically thoroughly studied in\nnumerous scenarios in the real- and complex-analytic category and in the\ndifferentiable case. It means that the map-germ is determined, up to a given\nequivalence relation, by a finite part of its Taylor expansion. The equivalence\nrelation is usually given by a group action and the first step is always to\nreduce the determinacy question to an "infinitesimal determinacy", i.e., to the\ntangent spaces at the orbits of the group action. In this work we formulate a\nuniversal, characteristic-free approach to finite determinacy, not necessarily\nover a field, and for a large class of group actions. We do not restrict to\npro-algebraic or Lie groups, rather we introduce the notion of "pairs of (weak)\nLie type", which are groups together with a substitute for the tangent space to\nthe orbit such that the orbit is locally approximated by its tangent space, in\na precise sense. This construction may be considered as a kind of replacement\nof the exponential resp. logarithmic maps. It is of independent interest as it\nprovides a general method to pass from the tangent space to the orbit of a\ngroup action in any characteristic. In this generality we establish the\n"determinacy versus infinitesimal determinacy" criteria, a far reaching\ngeneralization of numerous classical and recent results, together with some new\napplications.\n