Generalized Hilbert coefficients and Northcott's inequality

Abstract

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 Gd condition and the weak Artin–Nagata property ANd−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 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.