Detecting flatness over smooth bases

Luchezar L. Avramov; Srikanth B. Iyengar

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arXiv · 1002.3652

Detecting flatness over smooth bases

Luchezar L. Avramov·Srikanth B. Iyengar

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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.

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Luchezar L. Avramov, Srikanth B. Iyengar. 2010-02-19. Detecting flatness over smooth bases. https://arxiv.org/abs/1002.3652

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