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Abstract

Let $f_{1},\ldots,f_{n}$ 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 on J. Lipman's theory of residues via Hochschild homology that the jacobian of $f_{1},\ldots,f_{n}$ belongs to the ideal generated by $f_{1},\ldots,f_{n}$ if and only if the dimension of the germ of common zeros of $f_{1},\ldots,f_{n}$ is strictly positive. In fact, we prove much more general results which are relative versions of this result replacing the field $\mathbb{C}$ by convenient noetherian rings ${\bf A}$ (Ths. 3.1 and 3.3). We then show a /Lojasiewicz inequality for the jacobian analogous to the classical one by S.~/Lojasiewicz for the gradient.