Equivalence of germs (of mappings and sets) over k vs that over K
Dmitry Kerner
TL;DR
The paper develops a unified framework to compare equivalence of germ-mappings over a base ring $\Bbbk$ with their counterparts over field extensions $\mathbbm{K}$, including the real/complex and Nash/analytic settings. It proves that, under faithful flatness, equivalence over $\mathbbm{K}$ with unipotent-leading terms implies equivalence over $\Bbbk$, and that over algebraically closed bases the same holds for all extensions; these results extend to scheme-germs via ambient isomorphism of zero sets. The proofs hinge on reducing to tangent-space linear problems using unipotent actions, and then applying Artin–Rees and (Strong) Artin approximation, along with countable polynomial systems to pass from formal to actual equivalences. The work also yields corollaries on families, finite splitting of orbits for finite extensions, and global implications for projective and graded morphisms, with sharpness illustrated by explicit examples. Overall, the results provide robust tools for deformations/unfoldings and for passing equivalence information between base fields and extensions in Singularity Theory and real algebraic geometry.
Abstract
Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence. On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C. These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: * for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k; * for the classical groups of (right/left-right/contact) equivalence of Singularity Theory; * for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic. The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k. Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff 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.