Natural bound in Kwiecinski’s criterion for flatness

Abstract:

Kwieciński has proved a geometric criterion for flatness: A morphism $f:X\to Y$ of germs of analytic spaces is not flat if and only if its $i\text {-fold}$ fibre power $f^{\{i\}} :X^{\{i\}}\to Y$ has a vertical component, for some $i$. We show how to bound $i$ using Hironaka’s local flattener: If $f$ is not flat, then $f^{\{d\}}$ has a vertical component, where $d$ is the minimal number of generators of the ideal in ${\mathcal {O}}_{Y}$ of the flattener of $X$.