A numerical characterization of the S₂-ification of a Rees algebra

Let A be a local ring with maximal ideal m. For an arbitrary ideal I of A, we define the generalized Hilbert coefficients j_k(I) \in Z^{k+1} (k=0,1,...,dim A). When the ideal I is m-primary, j_k(I)=(0,...,0,(-1)^k e_k(I)), where e_k(I) is the classical k-th Hilbert coefficient of I. Using these coefficients, we give a numerical characterization of the homogeneous components of the S_2-ification of S=A[It,t^{-1}].