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 `(I) = d, Gd condition and the Artin-Nagata property AN − d−2. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)−HI(n), where J is a general minimal reduction of I, PI(n) and HI(n) are the generalized Hilbert-Samuel polynomial and the generalized Hilbert-Samuel function in the sense of C. Polini and Y. Xie. We then use it to establish formulas to compute the higher generalized Hilbert coefficients of I. As an application, we extend Northcott’s inequality to non m-primary ideals. When equality holds in the generalized Northcott’s inequality, the ideal I enjoys nice properties. Indeed, in this case, we prove that 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.