Mathematics – Algebraic Geometry
Scientific paper
2010-02-19
Mathematics
Algebraic Geometry
11 pages; significant changes in the presentation from the previous version. To appear in the Journal of Algebraic Geometry
Scientific paper
Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and \pi_i: X^{d} --> X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \otimes_{i=1}^d \pi_i^*(F) to generic points of Y, for some d greater than or equal to 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.
Avramov Luchezar L.
Iyengar Srikanth B.
No associations
LandOfFree
Detecting flatness over smooth bases does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Detecting flatness over smooth bases, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Detecting flatness over smooth bases will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-363145