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

The germs of maps (kⁿ,o)→f(k^p,o) are traditionally studied up to the right (R), left-right (A) or contact (K) equivalence. Various questions about the group-orbits Gf (for G one of R,K,A) are reduced to their tangent spaces, T_Gf. Classically the passage T_Gf⇝Gf was done by vector fields integration, hence it was bound to the R/C-analytic or C^r-category. The purely-algebraic (characteristic-free) approach to the group-orbits of R,K has been developed during the last decades. But those methods could not address the (essentially more complicated) A-equivalence. Moreover, the characteristic-free results (for R,K) 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 T_Gf⇝Gf for the groups R,K,A. Submodules of T_Gf ensure (shifted) submodules of Gf. For the A-equivalence this extends (and strengthens) various classical results of Mather, Gaffney, du Plessis, and others. Given a filtration on the space of maps, one has the filtration on the group, G^(∙), and on the tangent space, TG_(∙). We establish the criteria of type “TG_(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 f. 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 T_Af and establish various properties of the annihilator ideal a_A, defining the instability locus of the map.