Given a coherent ideal sheaf J we construct locally a vector-valued residue current R whose annihilator is precisely the given sheaf. In case J is a complete intersection, R is just the classical Coleff-Herrera product. By means of these currents we can extend various results, previously known for a complete intersection, to general ideal sheaves. Combining with integral formulas we obtain a residue version of the Ehrenpreis-Palamodov fundamental principle. © 2007 Elsevier Masson SAS RÉSUMÉ.-Soit J un faisceau cohérent d'idéaux. Nous construisons localement un courant résiduel R à valeurs vectorielles dont l'annihilateur est J. Au cas où J serait une intersection complète, R est simplement le produit classique de Coleff-Herrera. Ces courants permettent d'étendre au cas général des résultats divers, déjà connus dans le cas d'une intersection complète. En utilisant ces courants résiduels et des formules intégrales, nous obtenons ainsi une version résiduelle du Principe Fondamental d'Ehrenpreis-Palamodov.

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We prove global effective versions of the Briançon-Skoda-Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in C n , and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.

We introduce a calculus for the class ASM (X) of direct images of semi-meromorphic currents on a reduded analytic space X, that extends the classical calculus due to Coleff, Herrera and Passare. Our main result is that each element in this class acts as a kind of multiplication on the sheaf PMX of pseudomeromorphic currents on X. We also prove that ASM (X) as well as PMX and certain subsheaves are closed under the action of holomorphic differential operators and interior multiplication by holomorphic vector fields.