Bounds for Hilbert Coefficients

Abstract

Let $(A,\mathfrak{m})$ be a noetherian local ring and $J$ an $\mathfrak{m}$-primary ideal. Elias [3] proved that $\operatorname{depth}(G(J^k))$ is constant for $k \gg 0$ and denoted this number by $\sigma(J)$. In this paper, we prove the non-positivity for the Hilbert coefficients $e_i(J)$ under some conditions for $\sigma(J)$. 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 $e_1(Q)$.