Detecting modules inside the group-orbits R, K, A↻Maps(X,(kp,o)). The case of arbitrary characteristic

The germs of maps (kn,o)→f(kp,o) are traditionally studied up to the right, left-right or contact equivalence. Various questions about the group-orbits G f are transformed to the tangent spaces, TGf. Classically this passage was done by vector fields integration, hence it was bound to the R/C-analytic or Cr-category. In our previous papers we have constructed the characteristic-free approach to this "orbit vs tangent orbit" passage for the groups R,K. This approach was still restricted. E.g. we could not address the (essentially more complicated) A-equivalence. Moreover, the characteristic-free criteria 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 TGf⇝Gf for the groups R, K, A. Thus submodules of TGf ensure submodules of Gf. Even in the classical case of R/C-analytic maps this improves the known criteria. * 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. * We study the mixed module structure of the tangent space TAf and establish several properties of the annihilator ideal Ann[T1Af]. As an immediate application we obtain the relative algebraization of maps, i.e. the R,K,A-extension of the Weierstrass preparation theorem.