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Geometric Auslander criterion for flatness
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 1, February 2013
- pp. 125-142
- 10.1353/ajm.2013.0010
- Article
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Our aim is to understand the algebraic notion of flatness in explicit
geometric terms. Let $\varphi: X \to Y$ be a morphism of complex-analytic
spaces, where $Y$ is smooth. We prove that nonflatness of $\varphi$ is
equivalent to a severe discontinuity of the fibres---the existence of a
{\it vertical component} (a local irreducible component at a point of the
source whose image is nowhere-dense in $Y$)---after passage to the
$n$-fold fibred power of $\varphi$, where $n = \dim Y$. Our main theorem
is a more general criterion for flatness over $Y$ of a coherent sheaf of
modules $\cal{F}$ on $X$. In the case that $\varphi$ is a morphism of
complex algebraic varieties, the result implies that the stalk
$\cal{F}_\xi$ of $\cal{F}$ at a point $\xi \in X$ is flat over $R :=
\cal{O}_{Y,\varphi(\xi)}$ if and only if its $n$-fold tensor power is a
torsion-free $R$-module (conjecture of Vasconcelos in the case of
$\Bbb{C}$-algebras).