Generalized Hilbert coefficients and Northcott's inequality

Let R be a Cohen-Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread ℓ(I)=d, satisfies the G_d condition and the weak Artin-Nagata property ANd-2-. We provide a formula relating the length λ(Iⁿ⁺¹/JIⁿ) to the difference P_I(n)-H_I(n), where J is a general minimal reduction of I, P_I(n) and H_I(n) are respectively the generalized Hilbert-Samuel polynomial and the generalized Hilbert-Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I. As an application, we extend Northcott's inequality to non-m-primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen-Macaulay. We also recover results of G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j₁(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.