Detecting flatness over smooth bases

TL;DR

This paper characterizes flatness of coherent sheaves over smooth bases using fiber products and associated points, providing a geometric criterion linked to classical algebraic conditions.

Contribution

It establishes a new geometric criterion for flatness over smooth schemes, connecting fiber product associated points to flatness, extending classical algebraic results.

Findings

Flatness characterized by associated points mapping to generic points

Criterion applies for some d ≥ dim Y

Links geometric flatness to algebraic freeness conditions

Abstract

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.