Example 11

\* Let S and X be two complex analytic spaces and let $f : X \to S$ be a morphism. We say that $f$ is flat at a point $x$ of X if $\mathscr{O}_{X,x}$, considered as $\mathscr{O}_{S,f(x)}$-module by means of the homomorphism $f^* : \mathscr{O}_{S,f(x)} \to \mathscr{O}_{X,x}$, is flat. The set of points of X where $f$ is flat is an open set of X, and the restriction of $f$ to this open set is an open mapping. If X and S are connected analytic varieties of finite dimension, $f$ is flat (at every point of X) if and only if $f(X)$ is open in S and the fibres $f^{-1}(s)$, for $s \in f(X)$, all have the same dimension. \*