Castelnuovo-Mumford regularity and Hilbert coefficients of parameter ideals (Q2332977)

Let \(R\) be a commutative Noetherian local ring with identity. Let \(Q\) be a parameter ideal of \(R\) and \(d:=\dim R\). The main aim of this paper is to provide an upper bound for the Castelnuovo-Mumford regularity of the associated graded ring of \(R\) with respect to \(Q\). Let \(P_Q(x)\in \mathbb{Q}[x]\) be the \textit{Hilbert-Samuel polynomial} of \(R\) with respect to \(Q\). So, there are integers \(e_0(Q), \dots, e_d(Q)\) such that \[P_Q(n)=\sum_{i=0}^{d} (-1)^i\binom{n+d-i-1}{d-i}e_i(Q)\] for all \(n\gg 0\). The integers \(e_i(Q)\) are called \textit{Hilbert coefficients} of \(Q\). Let \(G(Q):=\bigoplus\limits_{i\geq 0} Q^i/Q^{i+1}\) denote the \textit{associated graded ring} of \(R\) with respect to \(Q\). The \textit{Castelnuovo-Mumford regularity} of \(G(Q)\) is defined by \[ \text{reg}(G(Q)):=\max \{a_i(G(Q))+i| \ i\geq 0\}, \] where \(a_i(G(Q))\) is the supremum of the integers \(n\) such that \(\text{H}_{G(Q)_+}^i(G(Q))_n\neq 0\). Assume that \(d\geq 1\) and \(\text{depth R}\geq d-1\). The main result of this paper asserts that \[ \text{reg}(G(Q))\leq \begin{cases} \max\{-e_1(Q)-1,0\} \ \ \ \text{if} \ d=1, & \\ \max\{\left(-4e_1(Q)\right)^{(d-1)!}+e_1(Q)-1,0\} \ \ \ \text{if} \ d\geq 2. \end{cases} \]