Exercise 6, § 2

Let $U$ be an open neighbourhood of 0 in $\mathbf{C}^n$ and let $f_1, . . . , f_m$ be analytic mappings from U into $\mathbf{C}$. Let $f: U\rightarrow \mathbf{C}^m$ be the mapping $x\mapsto$ $(f_1(x), . . . , f_m(x))$. Now prove the equivalence of the following conditions:

(i) The mapping $f$ is topologically étale at 0 ;

(ii) One has $m=n$ and there exists a neighbourhood V of 0 contained in U such that $f|V$ is injective ;

(iii) The differential of $f$ at 0 is bijective ;

(iv) There exists a neighbourhood V of 0 contained in U such that $f|V$ is an étale morphism of varieties.