Detecting flatness over smooth bases

Abstract

Given an essentially finite type morphism of schemes f : X → Y f\colon X\to Y and a positive integer d d , let f { d } : X { d } → Y f^{\{d\}}\colon X^{\{d\}}\to Y denote the natural map from the d d -fold fiber product X { d } = X × Y ⋯ × Y X X^{\{d\}}= X\times _{Y}\cdots \times _{Y}X and π i : X { d } → X \pi _i\colon X^{\{d\}}\to X the i i th canonical projection. When Y Y is smooth over a field and F \mathcal F is a coherent sheaf on X X , it is proved that F \mathcal F is flat over Y Y if (and only if) f { d } f^{\{d\}} maps the associated points of ⨂ i = 1 d π i ∗ F {\bigotimes _{i=1}^d}\pi _i^*{\mathcal F} to generic points of Y Y , for some d ≥ dim ⁡ Y 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.