Flatness testing over singular bases

Janusz Adamus, and Hadi Seyedinejad. "Flatness testing over singular bases." Annales Polonici Mathematici 107.1 (2013): 87-96. <http://eudml.org/doc/280866>.

@article{JanuszAdamus2013,
abstract = {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 $(A^\{⊗i^n_R\})⊗_R S$ is a torsion-free R-module.},
author = {Janusz Adamus, Hadi Seyedinejad},
journal = {Annales Polonici Mathematici},
keywords = {flat; fibered power; vertical component},
language = {eng},
number = {1},
pages = {87-96},
title = {Flatness testing over singular bases},
url = {http://eudml.org/doc/280866},
volume = {107},
year = {2013},
}

TY - JOUR
AU - Janusz Adamus
AU - Hadi Seyedinejad
TI - Flatness testing over singular bases
JO - Annales Polonici Mathematici
PY - 2013
VL - 107
IS - 1
SP - 87
EP - 96
AB - 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 $(A^{⊗i^n_R})⊗_R S$ is a torsion-free R-module.
LA - eng
KW - flat; fibered power; vertical component
UR - http://eudml.org/doc/280866
ER -