Geometric Auslander criterion for flatness

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let $\varphi: X \to Y$ be a morphism of complex-analytic spaces, where $Y$ is smooth. We prove that nonflatness of $\varphi$ is equivalent to a severe discontinuity of the fibres---the existence of a {\it vertical component} (a local irreducible component at a point of the source whose image is nowhere-dense in $Y$)---after passage to the $n$-fold fibred power of $\varphi$, where $n = \dim Y$. Our main theorem is a more general criterion for flatness over $Y$ of a coherent sheaf of modules $\cal{F}$ on $X$. In the case that $\varphi$ is a morphism of complex algebraic varieties, the result implies that the stalk $\cal{F}_\xi$ of $\cal{F}$ at a point $\xi \in X$ is flat over $R := \cal{O}_{Y,\varphi(\xi)}$ if and only if its $n$-fold tensor power is a torsion-free $R$-module (conjecture of Vasconcelos in the case of $\Bbb{C}$-algebras)