Analytic residue theory in the non-complete intersection case

In previous work of the authors and their collaborators (see, e.g., Progr. Math. 114 (1993)) it was shown how the equivalence of several constructions of residue currents associated to complete intersection families of (germs of) holomorphic functions in C-n could be profitably used to solve algebra ic problems like effective versions of the Nullstellensatz. In this work, t he authors explain how such ideas can be transposed to the non-complete int ersection situation, leading to an explicit way to construct a Green curren t attached to a purely dimensional cycle in P-n. This construction extends a previous result of the authors done in the complete intersection case. Wh en the cycle is defined over Q, they give a closed expression for the analy tic contribution in the definition of its logarithmic height (as the residu e at lambda = 0 of a zeta -function attached to a system of generators of t he ideal which defines the cycle). They also introduce an extension of the Cauchy-Weil division process and apply it in order to make explicit the mem bership of the Jacobian determinant of n elements f(i) is an element of O-n , j = 1,...,n (which fail to define a regular sequence) in the ideal (f(1), ...,f(n)).