An inductive analytic criterion for flatness

TL;DR

This paper introduces a new constructive criterion for flatness of morphisms and coherent sheaves in analytic geometry, combining linear algebra and the Weierstrass preparation theorem to facilitate inductive analysis.

Contribution

It provides a novel inductive criterion for flatness that simplifies checking flatness in analytic spaces using linear algebra and the Weierstrass preparation theorem.

Findings

Provides a linear algebra-based flatness criterion in codimension zero

Extends the criterion to codimension one for inductive reduction

Enables step-by-step flatness verification via fiber dimension reduction

Abstract

We present a constructive criterion for flatness of a morphism of analytic spaces X -> Y or, more generally, for flatness over Y of a coherent sheaf of modules on X. The criterion is a combination of a simple linear-algebra condition "in codimension zero" and a condition "in codimension one" which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre dimension of the morphism.