Equidimensionality and regularity

TL;DR

This paper proves that an equidimensional morphism with etale local sections from a regular algebraic space to a normal algebraic space of characteristic zero ensures the regularity and flatness of the base space.

Contribution

It establishes a new criterion linking equidimensionality and etale local sections to the regularity and flatness of the base space in algebraic geometry.

Findings

S is regular under the given conditions

f is flat when the conditions are met

The result applies to algebraic spaces of characteristic zero

Abstract

The existence of an equidimensional morphism f with etale local sections from a regular algebraic space X to a locally noetherian normal algebraic space S of characteristic zero with excellent local rings implies that S is regular and f flat.