Remark 2

Under the hypotheses of cor. 3, the $A$-module $\Omega(A)$ is projective (No. 5, cor. 3 of th. 2), hence free; every derivation of $A$ into $\kappa_A$ therefore lifts to a derivation of $A$. The condition of the statement may therefore be expressed as follows: there exist a generating system $(f_1, \ldots, f_m)$ of $J$ and derivations $D_1, \ldots, D_m$ of $A$ into $\kappa_A$, such that $\det(D_j(f_i)) \neq 0$.