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Let $f_1,...,f_n$ be $n$ germs of holomorphic functions at the origin of$\mathbb{C}^n$ such that $f_i(0)=0$, $1\leq i\leq n$. We give a proof based onthe J. Lipman's theory of residues via Hochschild Homology that the Jacobian of$f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ belongs to theideal generated by $f_1,...,f_n$ if and only if the dimension ot the germ ofcommon zeos of $f_1,...,f_n$ is sttrictly positive. In fact we prove much moregeneral results which are relatives versions of this result replacing the field$\mathbb{C}$ by convenient noetherian rings $\mathbf{A}$ (c.f. Th. 3.1 and Th.3.3). We then show a \L ojasiewicz inequality for the jacobian analogous to theclassical one by S. \L ojasiewicz for the gradient.

Une note \`a propos du Jacobien de $n$ fonctions holomorphes \`a l'origine de $\mathbb{C}^n$