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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Natural bound in Kwiecinski's criterion for flatness


Author: Janusz Adamus
Journal: Proc. Amer. Math. Soc. 130 (2002), 3165-3170
MSC (2000): Primary 14B25, 13C11
Published electronically: March 25, 2002
MathSciNet review: 1912993
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Abstract | References | Similar Articles | Additional Information

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 $i$. 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 $X$.


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Additional Information

Janusz Adamus
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: adamus@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06422-5
PII: S 0002-9939(02)06422-5
Keywords: Fibre product, vertical component, local flattener
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: June 11, 2001
Published electronically: March 25, 2002
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society