Une note à propos du jacobien de fonctions holomorphes à l'origine de \mathbb{C}^n
Volume 94 / 2008
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.