Equivalence of germs (of mappings and sets) over k versus that over K

Abstract Consider ‐analytic mapping‐germs, . They can be equivalent (by coordinate changes) ‐analytically, but not ‐analytically. However, if the transformation of ‐equivalence is modulo higher order terms, then it implies the ‐equivalence. On the other hand, starting from ‐analytic map‐germs , and taking any field extension , one has: if , then . These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: for , where are (formal/analytic/‐Nash) germs of spaces, with arbitrary singularities, over a base ring for the classical groups of (right/left–right/contact) equivalence of singularity theory; for faithfully flat extensions of rings . In particular, for arbitrary extensions of fields. The case “ is a ring” is important for the study of deformations/unfoldings. For example, it implies the statement for field extensions: if a family of ‐maps is ‐trivial, then it is also ‐trivial. Similar statements for germs of spaces (“isomorphism over vs. isomorphism over ”) follow by the standard reduction “Two maps are contact equivalent if and only if their zero sets are ambient isomorphic.” This study involves the contact equivalence of maps with singular targets, which seems to be not well established. We write down the relevant part of this theory.