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

S. Huckaba<P>

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.