A fast flatness testing criterion in characteristic zero

Abstract:

We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let $\varphi :X\to Y$ be a morphism of real or complex analytic spaces and let $\xi$ be a point of $X$. Let $\eta =\varphi (\xi )\in Y$ and let $\sigma :Z\to Y$ be the blowing-up of $Y$ at $\eta$, with $\zeta \in \sigma ^{-1}(\eta )$. Then $\varphi$ is flat at $\xi$ if and only if the pull-back of $\varphi$ by $\sigma$, $X\times _YZ\to Z$ has no torsion at $(\xi ,\zeta )$; i.e., the local ring $\mathcal {O}_{X\times _YZ,(\xi ,\zeta )}$ is a torsion-free ${\mathcal {O}}_{Z,\zeta }$-module. We also prove the corresponding result in the algebraic category over any field of characteristic zero.