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A note on the Jacobian of \(n\) holomorphic functions at the origin of \({\mathbb C}^n\). (Une note à propos du jacobien de \(n\) fonctions holomorphes à l’origine de \({\mathbb C}^n\).) (French. English summary) Zbl 1154.32010

Let \(f_{1},\ldots ,f_{n}\in \mathcal{O}_{n}\) be \(n\) germs of holomorphic functions at the origin of \(\mathbb{C}^{n},\) such that \(f_{i}(0)=0,\) \(1\leq i\leq n.\) The author gives a proof based on J. Lipman’s theory of residues via Hochschild homology that the Jacobian of \(f_{1},\dots ,f_{n}\) belongs to the ideal \(\left( f_{1},\dots ,f_{n}\right) \mathcal{O}_{n}\) if and only if the dimension of the germ of common zeros of \(f_{1},\dots ,f_{n}\) is strictly positive (results of this type were obtained by S. Spodzieja [Bull. Soc. Sci. Lett. Łódź 39, No. 13 (1989; Zbl 0743.13012)] and W. V. Vasconcelos [Bol. Soc. Mat. Mex., II. Ser. 37, No. 1–2, 549–556 (1992; Zbl 0838.13013)]. In fact, he proves much more general results which are relative versions of the above result replacing \(\mathbb{C} \) by convenient noetherian rings \(A.\) In case the dimension of the germ is zero, the Łojasiewicz type inequality for the Jacobian is also proved.

MSC:

32S05 Local complex singularities
13A10 Radical theory on commutative rings (MSC2000)
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