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

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Let X / S be a noetherian scheme with a coherent O X -module M, and T X / S be the relative tangent sheaf acting on M. We give constructive proofs that sub-schemes Y, with defining ideal I Y , of points x ∈ X where O x or M x is “bad”, are preserved by T X / S , making certain assumptions on X / S . Here bad means one of the following: O x is not normal; O x has high regularity defect; O x does not satisfy Serre's condition ( R n ) ; O x has high complete intersection defect; O x is not Gorenstein; O x does not satisfy ( T n ) ; O x does not satisfy ( G n ) ; O x is not n-Gorenstein; M x is not free; M x has high Cohen–Macaulay defect; M x does not satisfy Serre's condition ( S n ) ; M x 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 T X / S .