Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Detecting flatness over smooth bases


Authors: Luchezar L. Avramov and Srikanth B. Iyengar
Journal: J. Algebraic Geom. 22 (2013), 35-47
DOI: https://doi.org/10.1090/S1056-3911-2012-00579-7
Published electronically: May 21, 2012
MathSciNet review: 2993046
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Abstract | References | Additional Information

Abstract: Given an essentially finite type morphism of schemes $ f\colon X\to Y$ and a positive integer $ d$, let $ f^{\{d\}}\colon X^{\{d\}}\to Y$ denote the natural map from the $ d$-fold fiber product $ X^{\{d\}}= X\times _{Y}\cdots \times _{Y}X$ and $ \pi _i\colon X^{\{d\}}\to X$ the $ i$th canonical projection. When $ Y$ is smooth over a field and $ \mathcal F$ is a coherent sheaf on $ X$, it is proved that $ \mathcal F$ is flat over $ Y$ if (and only if) $ f^{\{d\}}$ maps the associated points of $ {\bigotimes _{i=1}^d}\pi _i^*{\mathcal F}$ to generic points of $ Y$, for some $ d\ge \dim Y$. The equivalent statement in commutative algebra is an analog--but not a consequence--of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.


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Additional Information

Luchezar L. Avramov
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: avramov@math.unl.edu

Srikanth B. Iyengar
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: iyengar@math.unl.edu

DOI: https://doi.org/10.1090/S1056-3911-2012-00579-7
Received by editor(s): March 24, 2010
Received by editor(s) in revised form: November 22, 2010
Published electronically: May 21, 2012
Additional Notes: Research partly supported by NSF grants DMS-0803082 (LLA) and DMS-0903493 (SBI)

American Mathematical Society