On Hilbert-Samuel Coefficients of Graded Local Cohomology Modules

Abstract

Let $R= \oplus _{n\in \mathbb {N}_{0}}R_{n}$ be a homogeneous Noetherian ring with local base ring $(R_{0},\mathfrak {m}_{0})$. Let $R_{+}= \oplus _{n\in \mathbb {N}}R_{n}$ denote the irrelevant ideal of R and let $M=\oplus _{n\in \mathbb {Z}}M_{n}$ be a finitely generated graded R-module. In this paper, we extend the results of Brodmann et al. (Proc. Amer. Math. Soc. 131, 2977–2985, 2003) and Brodmann and Rohrer (Proc. Amer. Math. Soc. 193, 987–993, 2005) when $\dim (R_{0})=2$. Actually, we show that the Hilbert-Samuel coefficient $e_{1}({\mathfrak {q}_{0}},H_{R_{+}}^{i}(M)_{n})$ has asymptotic behavior for all n ≪ 0 and also we establish in certain cases the asymptotic behavior of the Hilbert-Samuel coefficient $e_{2}({\mathfrak {q}_{0}},H_{R_{+}}^{i}(M)_{n})$ for all n ≪ 0, where $H_{R_{+}}^{i}(M)_{n}$ is the n-th graded component of the local cohomology $H_{R_{+}}^{i}(M)$ and $\mathfrak {q}_{0}$ is an $\mathfrak {m}_{0}$-primary ideal of R.