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Natural bound in Kwiecinski's criterion for flatness

Authors:	Janusz Adamus

Institution:	Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Abstract:	Kwiecinski 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 . We show how to bound using Hironaka's local flattener: If is not flat, then $f^{\{d\}}$ has a vertical component, where is the minimal number of generators of the ideal in ${\mathcal{O}}_{Y}$ of the flattener of .

Keywords:	Fibre product vertical component local flattener

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