Une note à propos du jacobien de n fonctions holomorphes à l'origine de ℂⁿ
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)....