Purity of branch and critical locus

To a dominant morphism $X\mathrm{/}S\to Y\mathrm{/}S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X\mathrm{/}Y}\subset B_{X\mathrm{/}Y}$ of the critical locus in the branch locus of $X\mathrm{/}Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials ${\mathrm{\Omega }}_{X\mathrm{/}Y}$, ${\mathrm{\Omega }}_{X\mathrm{/}S}$, and ${\mathrm{\Omega }}_{Y\mathrm{/}S}$ that imply bounds on the codimensions of $C_{X\mathrm{/}Y}$ and $B_{X\mathrm{/}Y}$. 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.