An inductive analytic criterion for flatness
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- by Janusz Adamus, Edward Bierstone and Pierre D. Milman
- Proc. Amer. Math. Soc. 140 (2012), 3703-3713
- DOI: https://doi.org/10.1090/S0002-9939-2012-11211-0
- Published electronically: March 6, 2012
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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 $\varphi$.References
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Bibliographic 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
- 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
- © Copyright 2012 American Mathematical Society
- 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
- MathSciNet review: 2944711