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 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.
