Purity of branch and critical locus

To a dominant morphism $X/S \to Y/S$ of Nœtherian 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 $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ 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.

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