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

TL;DR

This paper proves a criterion relating the Jacobian of holomorphic functions at the origin in ^n to the dimension of their common zero set, using Hochschild homology and residues, and extends the result to noetherian rings.

Contribution

It introduces a new proof based on Hochschild homology for the Jacobian ideal criterion and generalizes it to noetherian rings, also establishing a Lojasiewicz inequality for the Jacobian.

Findings

Jacobian belongs to the ideal generated by functions iff zero set dimension is positive

Generalization of the result to noetherian rings

Establishment of a Lojasiewicz inequality for the Jacobian

Abstract

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 \L ojasiewicz inequality for the jacobian analogous to the classical one by S. \L ojasiewicz for the gradient.