For every k, the k-torsion freeness of the nth-order Kähler differential module of a hypersurface point is equivalent to the singular locus having codimension at least k+1.
Auslander; Coherent Functors, Proceedings of the Con ference Categorical Alge- bra, La Jolla 1965, Springer Verlag, 1966
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On the $k$-torsion of the module of differentials of order $n$ of hypersurfaces
For every k, the k-torsion freeness of the nth-order Kähler differential module of a hypersurface point is equivalent to the singular locus having codimension at least k+1.