A fast flatness testing criterion in characteristic zero

Janusz Adamus, Hadi Seyedinejad
2015 Proceedings of the American Mathematical Society  
We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an R-algebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowing-up of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsion-free module over R localized in m. If K is
more » ... zed in m. If K is the field of reals or complex numbers, we give a stronger criterion - without the regularity assumption on R. We also show the corresponding results in the real- and complex-analytic categories.
doi:10.1090/s0002-9939-2015-12463-x fatcat:e4g2jwjnnjfapdcehp2qt7fvnq