Castelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Ideals

Let $A$ be a noetherian local ring of dimension $d \geq 1$ and $\operatorname{depth}(A) \geq d-1$. In this paper, we study the non-positivity for the Hilbert coefficients of parameter ideals in the ring $A$. Moreover, we establish a bound for the Castelnuovo-Mumford regularity of associated graded ring of $A$ with respect to parameter ideal in terms of the first Hilbert coefficient and the dimension.