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An inductive analytic criterion for flatness


Authors: Janusz Adamus, Edward Bierstone and Pierre D. Milman
Journal: Proc. Amer. Math. Soc. 140 (2012), 3703-3713
MSC (2010): Primary 13C11, 32B99; Secondary 14B25
DOI: https://doi.org/10.1090/S0002-9939-2012-11211-0
Published electronically: March 6, 2012
MathSciNet review: 2944711
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Abstract: We present a constructive criterion for flatness of a morphism of analytic spaces $ \varphi : X \to Y$ (over $ \mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$) or, more generally, for flatness over $ \mathcal {O}_Y$ of a coherent sheaf of $ \mathcal {O}_X$-modules $ \mathcal {F}$. The criterion is a combination of a simple linear-algebra condition ``in codimension zero'' and a condition ``in codimension one'' which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre-dimension of the morphism $ \varphi $.


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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, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
Email: jadamus@uwo.ca

Edward Bierstone
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1 – and – Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email: bierston@fields.utoronto.ca

Pierre D. Milman
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email: milman@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11211-0
Keywords: Flat, Weierstrass preparation, local flattener, generic flatness
Received by editor(s): January 10, 2011
Received by editor(s) in revised form: April 25, 2011
Published electronically: March 6, 2012
Additional Notes: This research was partially supported by Natural Sciences and Engineering Research Council of Canada Discovery Grant OGP 355418-2008, Polish Ministry of Science Discovery Grant NN201 540538 (first author), and by NSERC Discovery Grants OGP 0009070 (second author) and OGP 0008949 (third author)
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society

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