Purity of branch and critical locus

TL;DR

This paper investigates the relationship between the critical and branch loci in dominant morphisms of schemes, providing new bounds on their codimensions that extend classical purity results to broader classes of morphisms.

Contribution

It introduces conditions on modules of differentials that imply bounds on the codimensions of critical and branch loci, generalizing classical purity theorems.

Findings

Bounds on codimensions of critical and branch loci established

Conditions on modules of differentials derived for various morphisms

Generalization of classical purity results to wider morphism classes

Abstract

To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms.