An inductive analytic criterion for flatness

Abstract:

We present a constructive criterion for flatness of a morphism of analytic spaces $\varphi : X \to Y$ (over $\mathbb {K} = \mathbb {R}$ or $\mathbb {C}$) or, more generally, for flatness over $\mathcal {O}_Y$ of a coherent sheaf of $\mathcal {O}_X$-modules $\mathcal {F}$. The criterion is a combination of a simple linear-algebra condition “in codimension zero” and a condition “in codimension one” which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre-dimension of the morphism $\varphi$.