Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$.


Let $I$ be the ideal defining the singular locus in Spec $R$. (With the reduced subscheme structure, or defined using minors of a Jacobian matrix, again whatever helps.)


Is it obvious and/or true that any derivation $d:R \to R$, i.e. additive map satisfying the Leibniz rule $d(ab) = a\ db + b\ da$, has $dI \leq I$?


Morally, $d$ is defining an infinitesimal automorphism of Spec $R$, and the singular locus should be preserved by automorphisms. So I would have hoped that there was a mindless proof using the Jacobian, but I haven't found one. As usual, a reference would be even better than a proof.



Is the singular locus ideal preserved by all derivations?
Robert Hart proves in [Hart, R. Derivations on commutative rings. J. London Math. Soc. (2) 8 (1974), 171--175. MR0349654 (50 #2147)] that if $R$ is a finitely generated commutative $k$-algebra, then every $k$-derivation preserves all the Fitting ideals of the module of Kähler differentials $\Omega\_{R/k}$. The first Fitting ideal is the Jacobian ideal.


