Unfoldings of maps, the first results on stable maps, and results of Mather-Yau/Gaffney-Hauser type in arbitrary characteristic

TL;DR

This paper extends classical results on stable maps and unfoldings from real/complex cases to arbitrary characteristic fields, providing criteria for triviality, versality, stability, and classification of map-germs.

Contribution

It generalizes fundamental theorems on map-germ stability and unfoldings to arbitrary fields, including criteria for triviality, versality, and classification.

Findings

Separable unfoldings are locally trivial iff infinitesimally trivial.

Unfoldings are locally versal iff infinitesimally versal.

Stable maps are determined by their local algebras.

Abstract

Consider the (formal/analytic/algebraic) map-germs Maps(X,(k^p,o)). Let G be the group of right/contact/left-right transformations. I extend the following (classical) results from the real/complex-analytic case to the case of arbitrary field k. * A separable unfolding is locally trivial iff it is infinitesimally trivial. * An unfolding is locally versal iff it is infinitesimally versal. * The criterion of factorization of map-germs in zero characteristic. * Criteria of trivialization of unfoldings over affine base. * Fibration of K-orbits into A-orbits. * A map is locally stable iff it is infinitesimally stable. * Stable maps are unfodings of their genotypes. * Stable maps are determined by their local algebras. * Results of Mather-Yau/Scherk/Gaffney-Hauser type. How does the module T^1_G f, or related algebras, determine the G-equivalence type of f?