A fast flatness testing algorithm in characteristic zero

TL;DR

This paper introduces a rapid, computable criterion for testing flatness in characteristic zero by relating it to torsion properties of modules after blowing-up, with extensions to real and complex analytic categories.

Contribution

It provides a new, efficient method to determine flatness using torsion criteria, applicable in algebraic and analytic contexts, improving computational approaches.

Findings

A criterion for flatness based on torsion after blowing-up.

Extension of the criterion to real and complex analytic categories.

Simplification of flatness testing in characteristic zero.

Abstract

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 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.