Flatness testing over singular bases

TL;DR

This paper characterizes the non-flatness of morphisms between complex-analytic spaces and algebraic structures through the existence of vertical components and torsion properties in tensor products, linking geometric and algebraic flatness criteria.

Contribution

It establishes a new criterion for flatness using vertical components in fibred powers and torsion-freeness in tensor products, bridging complex-analytic and algebraic perspectives.

Findings

Non-flatness manifests as vertical components in fibred powers.

Flatness is characterized by torsion-freeness in tensor products.

Provides an algebraic analogue of a geometric flatness criterion.

Abstract

We show that non-flatness of a morphism f of complex-analytic spaces with a locally irreducible target Y of dimension n manifests in the existence of vertical components in the n-fold fibred power of the pull-back of f to the desingularization of Y. An algebraic analogue follows: Let R be a locally (analytically) irreducible finite type complex-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 the induced morphism of spectra is dominant. Then a finite type R-algebra A is R-flat if and only if the tensor product of S with the n-fold tensor power of A over R is a torsion-free R-module.