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

S. Huckaba

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))\geq d-1.