A fast flatness testing criterion in characteristic zero
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- by Janusz Adamus and Hadi Seyedinejad PDF
- Proc. Amer. Math. Soc. 143 (2015), 2559-2570 Request permission
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.References
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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
- ORCID: 0000-0001-9103-2451
- Email: sseyedin@alumni.uwo.ca
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3326036