Hilbert Coefficients and the Depths of Associated Graded Rings

This work was motivated in part by the following general question:given an ideal I in a Cohen–Macaulay (abbreviated to CM)local ring R such that dim R/I=0, what information about I andits associated graded ring can be obtained from the Hilbertfunction and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel)function of I, we mean the function H_I(n)=(R/Iⁿ) for all n1,where denotes length. Samuel 23] showed that for large valuesof n, the function H_I(n) coincides with a polynomial P_I(n) ofdegree d=dim R. This polynomial is referred to as the Hilbert,or Hilbert–Samuel, polynomial of I. The Hilbert polynomialis often written in the form where e₀(I), ..., e_d(I) are integers uniquely determined byI. These integers are known as the Hilbert coefficients of I.