Preservation of defect sub-schemes by the action of the tangent sheaf

Let X/S be a noetherian scheme with a coherent image-module M, and TX/S be the relative tangent sheaf acting on M. We give constructive proofs that sub-schemes Y, with defining ideal IY, of points xset membership, variantX where image or Mx is “bad”, are preserved by TX/S, making certain assumptions on X/S. Here bad means one of the following: image is not normal; image has high regularity defect; image does not satisfy Serreʹs condition (Rn); image has high complete intersection defect; image is not Gorenstein; image does not satisfy (Tn); image does not satisfy (Gn); image is not n-Gorenstein; Mx is not free; Mx has high Cohen–Macaulay defect; Mx does not satisfy Serreʹs condition (Sn); Mx has high type. Kodaira–Spencer kernels for syzygies are described, and we give a general form of the assertion that M is locally free in certain cases if it can be acted upon by TX/S.