  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.
