Flatness testing over singular bases
Volume 107 / 2013
Annales Polonici Mathematici 107 (2013), 87-96
MSC: Primary 13C11, 32B99; Secondary 13H99, 13B10, 13P99, 32S45.
DOI: 10.4064/ap107-1-6
Abstract
We show that non-flatness of a morphism $\varphi:X\to Y$ of complex-analytic spaces with a locally irreducible target of dimension $n$ manifests in the existence of vertical components in the $n$-fold fibred power of the pull-back of $\varphi$ to the desingularization of $Y$. An algebraic analogue follows: Let $R$ be a locally (analytically) irreducible finite type $\mathbb C$-algebra and an integral domain of Krull dimension $n$, and let $S$ be a regular $n$-dimensional algebra of finite type over $R$ (but not necessarily a finite $R$-module), such that $\mathop{\rm Spec} S\to\mathop{\rm Spec} R$ is dominant. Then a finite type $R$-algebra $A$ is $R$-flat if and only if $(A^{\otimes^n_R})\otimes_RS$ is a torsion-free $R$-module.
