Mathematics > Algebraic Geometry
[Submitted on 4 Feb 2008]
Title:Une note à propos du Jacobien de $n$ fonctions holomorphes à l'origine de $\mathbb{C}^n$
View PDFAbstract: 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 on the 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 the ideal generated by $f_1,...,f_n$ if and only if the dimension ot the germ of common zeos of $f_1,...,f_n$ is sttrictly positive. In fact we prove much more general 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 Łojasiewicz inequality for the jacobian analogous to the classical one by S. Łojasiewicz for the gradient.
Submission history
From: Michel Hickel [view email] [via CCSD proxy][v1] Mon, 4 Feb 2008 14:53:34 UTC (18 KB)
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