Bounds for Hilbert Coefficients

Let (A, w) be a noetherian local ring and J an w-primary ideal. Elias [3] proved that depth(G(Jk )) is constant for k ≫ 0 and denoted this number by σ(J). In this paper, we prove the non-positivity for the Hilbert coefficients ei (J) under some conditions for σ(J). In case of J = Q is a parameter ideal, we establish bounds for the Hilbert coefficients of Q in terms of the dimension and the first Hilbert coefficient e 1(Q).