# Detecting flatness over smooth bases - Mathematics > Algebraic Geometry

Abstract: Given an essentially finite type morphism of schemes f: X -> Y and apositive integer d, let f^{d}: X^{d} -> Y denote the natural map from thed-fold fiber product, X^{d}, of X over Y and \pi i: X^{d} -> X the i-thcanonical projection. When Y smooth over a field and F is a coherent sheaf onX, it is proved that F is flat over Y if and only if f^{d} maps theassociated points of the tensor product sheaf \otimes {i=1}^d \pi i^*F togeneric points of Y, for some d greater than or equal to dim Y. The equivalentstatement in commutative algebra is an analog-but not a consequence-of aclassical criterion of Auslander and Lichtenbaum for the freeness of finitelygenerated modules over regular local rings.

Author: Luchezar L. Avramov, Srikanth B. Iyengar

Source: https://arxiv.org/