A d-Dimensional Extension Of A Lemma Of Huneke's And Formulas For The Hilbert Coefficients

Abstract

. A d-dimensional version is given of a 2-dimensional result due to C. Huneke. His result produced a formula relating the length (I n+1 =JI n ) to the difference P I (n + 1) \Gamma H I (n+ 1), where I is primary for the maximal ideal of a 2-dimensional Cohen-Macaulay local ring R, J is a minimal reduction of I, H I (n) = (R=I n ), and P I (n) is the Hilbert-Samuel polynomial of I. We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of I. We also include a characterization, in terms of the Hilbert coefficients of I, of the condition depth(G(I)) d \Gamma 1. 1. Introduction If (R; m) is a d-dimensional (Noetherian) local ring and I is an m-primary ideal of R then let (R=I n ) denote the length of the R-module R=I n . The Hilbert-Samuel function of I is the function H I (n) = (R=I n ). It follows from the theory of Hilbert functions applied to the associated graded ring G(I) = \Phi n0 I n =..