Orbits of the left-right equivalence of maps in arbitrary characteristic
Dmitry Kerner
TL;DR
This work develops a characteristic-free framework for left-right equivalence of map germs by connecting tangent-space data to full group orbits for the groups $\mathscr R$, $\mathscr K$, and $\mathscr A$. It introduces $\mathscr G$-implicit function theorems, establishes filtration-based orbit criteria, and analyzes the extended tangent space $T_{\mathscr A}f$ with its annihilator $\mathfrak a_{\mathscr A}$, including finiteness results under finite singularity type. The methods integrate arc-analytic techniques, Weierstraß division, Baker-Campbell-Hausdorff expansions, and Artin approximation, together with Kostant-Rosenlicht theory for unipotent groups, to obtain new determinacy and orbit-closure results in arbitrary characteristic. The framework unifies and extends Mather-type determinacy results, providing robust algebraic tools for singularity theory beyond characteristic zero and enabling new applications in algebraic geometry and computational settings.
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).