Title: Equivalence of germs (of mappings and sets) over k vs that over K Show Chinese title

Title: 映射和集合的芽在k上的等价性与在K上的等价性

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. Show Chinese abstract

Abstract: 考虑实解析映射芽，(R^n,o)-> (R^m,o)。 它们可以通过坐标变换成为复解析等价的，但不是实解析等价的。 然而，如果复等价的变换在高阶项下是恒等的，则它意味着实等价。 另一方面，从复解析映射芽 (C^n,o)->(C^m,o) 出发，并取任何域扩张，C 到 K，可以得到：如果两个映射在 K 上等价，则它们在 C 上也等价。 这些（非常有用）性质似乎不太为人所知。 我们在更一般的形式中证明了稍强的性质： * 对于 Maps(X,Y)，其中 X,Y 是基环 k 上的（形式/解析/Nash）概形芽，具有任意奇点； * 对于奇异理论中的经典群（右/左-右/接触）等价； * 对于环的忠实平坦扩张 k -> K。特别是对于任何特征下的域的任意扩张。 当“k 是一个环”时，这对于研究变形/展开很重要。 例如。 它意味着关于域的陈述：如果一个映射族{f_t}在 K 上平凡，则它在 k 上也是平凡的。 关于概形芽的类似陈述（“K 上的同构与 k 上的同构”）通过标准的归约方法得出：“两个映射是接触等价的当且仅当它们的零点集是环境同构的”。 这项研究涉及具有奇异目标的映射的接触等价性，这似乎尚未建立。 我们写下了该理论的相关部分。