A map (k^n,o)-> (k^p,o) with no critical point at the origin can be rectified to a
linear map. Maps with critical points have rich structure and are studied up to the
groups of right/left-right/contact equivalence.
The group orbits are complicated and are traditionally studied via their tangent
space. This transition is classically done by vector fields integration, thus binding
the theory to the real/complex case.
I will present the new approach to this subject. One studies the maps of germs of
Noetherian schemes, in any characteristic. The corresponding groups of equivalence
admit `good' tangent spaces. The submodules of the tangent spaces lead to submodules
of the group orbits. This allows to bring these maps to `convenient' forms. For
example, we get the (relative) finite determinacy, and accordingly the (relative)
algebraization of maps/ideals/modules.
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