Bass numbers of graded components of local cohomology modules

Let $R={⨁}_{n\in {\text{NN}}_0}{R}_n$ be a standard graded ring, $R_{+}={⨁}_{n\in \text{NN}}{R}_n$ its irrelevant ideal, and $M$ a finitely generated graded $R$-module. In this paper, we study the asymptotic behavior of the sequence $\{{\mu }^i({\text{p}}_0,H_{R_{+}}^j(M{)}_n){\}}_{n\in \mathbb{Z}}$ of Bass numbers of graded components of local cohomology modules with respect to an ideal ${\text{p}}_0\in \text{Spec}(R_0)$ in each of the following cases: (1) $i=0$ or $i=1$ and $j\le f_{R_{+}}(M)$, (2) $R_0$ is regular, $i=\text{hei}({\text{p}}_0)$ or $i=\text{hei}({\text{p}}_0)-1$ and $j={\text{cd}}_{R_{+}}(M)$, (3) $M$ is relative Cohen-Macaulay with respect to $R_{+}$. Here, ${\text{cd}}_{R_{+}}(M)$ and $f_{R_{+}}(M)$ denote the cohomological dimension and finiteness dimension of $M$ with respect to $R_{+}$, respectively.