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Natural bound in Kwieciński’s criterion for flatness. (English) Zbl 1052.14009

Summary: M. Kwieciński [Manuscr. Math. 97, 163–173 (1998; Zbl 0956.32019)] 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 \(i\)-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 \(X\).

MSC:

14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
13C11 Injective and flat modules and ideals in commutative rings

Citations:

Zbl 0956.32019
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References:

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