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

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