Example

Let $X$, $Y$ be two complex analytic varieties, locally of finite dimension, $f$ a morphism of $X$ into $Y$, and $x$ a point of $X$. Consider the local homomorphism $\rho : \mathscr{O}_{Y, f(x)} \to \mathscr{O}_{X, x}$ associated with $f$. The mapping $d\rho$ is the transpose of the tangent mapping $T_x(f) : T_x(X) \to T_{f(x)}(Y)$. The conditions (i) to (iv) of Proposition 9 are therefore equivalent in this case to the fact that $f$ is a submersion at $x$ (VAR, R, 5.9.1).