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A fast flatness testing criterion in characteristic zero


Authors: Janusz Adamus and Hadi Seyedinejad
Journal: Proc. Amer. Math. Soc. 143 (2015), 2559-2570
MSC (2010): Primary 32B99, 13C11, 14B25, 14P99, 26E05, 32H99, 32S45, 13B10, 13P99
DOI: https://doi.org/10.1090/S0002-9939-2015-12463-X
Published electronically: February 11, 2015
MathSciNet review: 3326036
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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.


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Additional Information

Janusz Adamus
Affiliation: Department of Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7—and—Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland
Email: jadamus@uwo.ca

Hadi Seyedinejad
Affiliation: Department of Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7
Address at time of publication: Department of Mathematical Sciences, University of Kashan, Kashan, Iran
Email: sseyedin@alumni.uwo.ca

DOI: https://doi.org/10.1090/S0002-9939-2015-12463-X
Keywords: Flat, open, torsion-free, fibred product, vertical component
Received by editor(s): October 15, 2013
Received by editor(s) in revised form: January 20, 2014, and January 30, 2014
Published electronically: February 11, 2015
Additional Notes: The first author’s research was partially supported by the Natural Sciences and Engineering Research Council of Canada
Communicated by: Lev Borisov
Article copyright: © Copyright 2015 American Mathematical Society