We show that non-flatness of a morphism φ:X→ 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 φ to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type 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 Spec S → Spec R is dominant. Then a finite type R-algebra A is R-flat if and only if is a torsion-free R-module.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap107-1-6,
author = {Janusz Adamus and Hadi Seyedinejad},
title = {Flatness testing over singular bases},
journal = {Annales Polonici Mathematici},
volume = {107},
year = {2013},
pages = {87-96},
zbl = {1273.13014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap107-1-6}
}
Janusz Adamus; Hadi Seyedinejad. Flatness testing over singular bases. Annales Polonici Mathematici, Tome 107 (2013) pp. 87-96. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap107-1-6/