Geometric Auslander criterion for openness of an algebraic morphism

TL;DR

This paper presents an effective geometric criterion to determine when a morphism of schemes over a field is open, based on the presence of vertical components in the fibred power, extending Auslander's flatness criterion.

Contribution

It introduces a new topological criterion for openness of algebraic morphisms, linking geometric properties to fibred powers and generalizing Auslander's flatness criterion.

Findings

Failure of openness is detected by vertical components in the n'th fibred power.

The criterion applies over normal bases of dimension n.

Provides an effective method to verify openness in algebraic geometry.

Abstract

We give an effective criterion for openness of a morphism of schemes of finite type over a field: Over a normal base of dimension n, failure of openness is detected by a vertical component in the n'th fibred power of the morphism. This is a topological analogue of a criterion for flatness that originates with Auslander.