A d-dimensional extension of a Lemma of Huneke's and formulas for the Hilbert coefficients

A d-dimensional version is given of a 2-dimensional result due to C. Huneke. His result produced a formula relating the length lambda(I-n+1/JI(n)) to the difference P-I (n + 1) - 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) = lambda(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)) greater than or equal to d-1.