Purity of branch and critical locus

To a dominant morphism X/S→Y/S of Nœtherian integral S-schemes one has the inclusion CX/Y⊂BX/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 ΩX/Y, ΩX/S, and ΩY/S that imply bounds on the codimensions of CX/Y and BX/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 Waerdenʼs purity for birational morphisms.