Detecting flatness over smooth bases

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.