Orbits of the left-right equivalence of maps in arbitrary characteristic

TL;DR

This paper develops a characteristic-free framework for understanding the orbits of map germs under various equivalences, extending classical results to positive characteristic fields and addressing the more complex left-right equivalence.

Contribution

It establishes the general passage from tangent spaces to group orbits for right, contact, and left-right equivalences in a characteristic-free setting, strengthening classical results.

Findings

Established characteristic-free criteria for tangent space and orbit comparisons.

Extended classical results to arbitrary characteristic fields for left-right equivalence.

Developed the mixed-module structure of the tangent space and analyzed the annihilator ideal.

Abstract

The germs of maps (k^n,o)\to(k^p,o) are traditionally studied up to the right, left-right or contact equivalence. Various questions about the group-orbits are reduced to their tangent spaces. Classically the passage from the tangent spaces to the orbits was done by vector fields integration, hence it was bound to the real/complex-analytic or C^r-category. The purely-algebraic (characteristic-free) approach to the group-orbits of right and contact equivalence has been developed during the last decades. But those methods could not address the (essentially more complicated) left-right equivalence. Moreover, the characteristic-free results (in the right/contact cases) were weaker than those in characteristic zero, because of the (inevitable) pathologies of positive characteristic. In this paper we close these omissions. * We establish the general (characteristic-free) passage from the tangent spaces to the groups orbits for the groups of right, contact and let-right equivalence. Submodules of the tangent spaces ensure (shifted) submodules of the group-orbits. For the left-right equivalence this extends (and strengthens) various classical results of Mather, Gaffney, du Plessis, and others. * A filtration on the space of maps induces the filtration on the group and on the tangent space. We establish the criteria of type "$T_{G^{(j)}}f$ vs $G^{(j)} f$" in their strongest form, for arbitrary base field/ring, provided the characteristic is zero or high for a given map. This brings the "inevitably weaker" results of char>0 to the level of char=0. * As an auxiliary step, important on its own, we develop the mixed-module structure of the tangent space to the left-right group and establish various properties of the annihilator ideal (that defines the instability locus of the map).