  We show that non-flatness of a morphism f of complex-analytic spaces with a
locally irreducible target Y of dimension n manifests in the existence of
vertical components in the n-fold fibred power of the pull-back of f to the
desingularization of Y. An algebraic analogue follows: Let R be a locally
(analytically) irreducible finite type complex-algebra and an integral domain
of Krull dimension n, and let S be a regular n-dimensional algebra of finite
type over R (but not necessarily a finite R-module), such that the induced
morphism of spectra is dominant. Then a finite type R-algebra A is R-flat if
and only if the tensor product of S with the n-fold tensor power of A over R is
a torsion-free R-module.
