Pairs of Lie-type and large orbits of group actions on filtered modules: (A characteristic-free approach to finite determinacy)

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an “infinitesimal determinacy”, i.e., to the tangent spaces at the orbits of the group action. In this work we formulate a universal, characteristic-free approach to finite determinacy, not necessarily over a field, and for a large class of group actions. We do not restrict to pro-algebraic or Lie groups, rather we introduce the notion of “pairs of (weak) Lie type”, which are groups together with a substitute for the tangent space to the orbit such that the orbit is locally approximated by its tangent space, in a precise sense. This construction may be considered as a kind of replacement of the exponential resp. logarithmic maps. It is of independent interest as it provides a general method to pass from the tangent space to the orbit of a group action in any characteristic. In this generality we establish the “determinacy versus infinitesimal determinacy” criteria, a far reaching generalization of numerous classical and recent results, together with some new applications.