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

M. Hickel (2008)

Annales Polonici Mathematici

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Let f₁,...,fₙ be n germs of holomorphic functions at the origin of ℂⁿ, such that ${f}_{i}\left(0\right)=0$, 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....