On Hilbert-Samuel coefficients of graded local cohomology modules (Q6187531)

Let \(R = R_0[R_1]\) be a commutative Noetherian graded ring, where \((R_0, \mathfrak{m}_0)\) is a local ring. Let \(R_+\) denote the irrelevant ideal of \(R\), \(q_0\) be an \(\mathfrak{m}_0\)-primary ideal of \(R_0\), and \(M\) be a finitely generated \(\mathbb{Z}\)-graded \(R\)-module. It is known that each of the local cohomology modules \[ \text{H}_{R_+}^i(M) = \underset{n}{\varinjlim}\ \text{Ext}_R^i\left(\frac{R}{(R_+)^n}, M\right); \quad i \in \mathbb{N}_0 \] has a \(\mathbb{Z}\)-grading in a natural way, and the \(R_0\)-module \(\text{H}_{R_+}^i(M)_n\) is finitely generated for all \(n \in \mathbb{Z}\), with \(\text{H}_{R_+}^i(M)_n = 0\) for all \(n \gg 0\). Thus, the study of the asymptotic behavior of \(\text{H}_{R_+}^i(M)_n\) for \(n \ll 0\) is of great interest. This paper focuses on examining the asymptotic behavior of the Hilbert-Samuel coefficients \(e_1(q_0,\text{H}_{R_+}^i(M)_n)\) and \(e_2(q_0, \text{H}_{R_+}^i(M)_n)\) for \(n \ll 0\). Let \(T\) be a finitely generated \(R_0\)-module. It is known that there exists a polynomial \(P \in \mathbb{Q}[X]\) of degree \(d = \dim_{R_0} T\) such that \(\ell_{R_0}(T/q_0^{n+1}T) = P_T^{q_0}(n)\) for all \(n \gg 0\). This polynomial is called the \textit{Hilbert-Samuel polynomial of \(T\) with respect to \(q_0\)}. There are integers \(e_0(q_0,T), e_1(q_0,T), \dots, e_d(q_0,T)\) such that \[ P_T^{q_0}(n) = \sum \limits_{i=0}^d (-1)^i e_i(q_0,T) \binom{n+d-i}{d-i}. \] The integer \(e_i(q_0,T)\) is called \textit{the \(i\)th Hilbert-Samuel coefficient of \(T\) with respect to \(q_0\)}. Let \(i\in \mathbb{N}_0\) and \(\dim R_0 = 2 = \dim_{R_0}(\text{H}_{R_+}^i(M)_n)\) for all \(n \ll 0\). Let \(q_0=(x,y)\) and \begin{itemize} \item[(i)] \(x\) and \(y\) form an \(\mathfrak{m}_0\)-filter regular sequence with respect to \(M\) and with respect to \(\text{H}_{R_+}^i(M)_n\) for all \(n \ll 0\). \item[(ii)] \(x\) is a parameter for \(R_0\) and a superficial element of \(q_0\) with respect to \(\text{H}_{R_+}^i(M)_n\) for all \(n \ll 0\). \end{itemize} Theorem 4.6 of this paper asserts that there exists a polynomial \(P \in \mathbb{Q}[X]\) of degree less than \(i\) such that \(e_1(q_0,\text{H}_{R_+}^i(M)_n) = P(n)\) for all \(n \ll 0\). Additionally, under some extra assumptions, Theorem 4.10 of this paper establishes a numerical polynomial upper bound for \(e_2(q_0,\text{H}_{R_+}^i(M)_n)\) for all \(n\ll 0\). These results improve several related results in the literature.