Let f₁,...,fₙ be n germs of holomorphic functions at the origin of ℂⁿ, such that , 1 ≤ i ≤ n. We give a proof based on J. Lipman’s theory of residues via Hochschild homology that the jacobian of f₁,...,fₙ belongs to the ideal generated by f₁,...,fₙ if and only if the dimension of the germ of common zeros of f₁,...,fₙ is strictly positive. In fact, we prove much more general results which are relative versions of this result replacing the field ℂ by convenient noetherian rings A (Ths. 3.1 and 3.3). We then show a Łojasiewicz inequality for the jacobian analogous to the classical one by S. Łojasiewicz for the gradient.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap94-3-4,
author = {M. Hickel},
title = {Une note a propos du jacobien de n fonctions holomorphes a l'origine de Cn},
journal = {Annales Polonici Mathematici},
volume = {93},
year = {2008},
pages = {245-264},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap94-3-4}
}
M. Hickel. Une note à propos du jacobien de n fonctions holomorphes à l'origine de ℂⁿ. Annales Polonici Mathematici, Tome 93 (2008) pp. 245-264. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap94-3-4/