Bass numbers of graded components of local cohomology modules

TL;DR

This paper investigates the asymptotic behavior of Bass numbers of graded components of local cohomology modules over standard graded rings, providing insights into their structure in various algebraic contexts.

Contribution

It offers new results on the asymptotic properties of Bass numbers for local cohomology modules in different settings, including regular rings and Cohen-Macaulay modules.

Findings

Bass numbers stabilize or exhibit predictable patterns asymptotically.

Explicit descriptions of Bass numbers are obtained for regular rings and Cohen-Macaulay modules.

Results contribute to understanding the structure of local cohomology modules in graded algebra.

Abstract

Let $R=\bigoplus_{n\in \NN_0}R_n$ be a standard graded ring, $R_+=\bigoplus_{n\in \NN}R_n$ its irrelevant ideal, and $M$ a finitely generated graded $R$-module. In this paper, we study the asymptotic behavior of the sequence $\{\mu^i(\p_0, H^j_{R_+}(M)_n)\}_{n\in \Z}$ of Bass numbers of graded components of local cohomology modules with respect to an ideal $\p_0\in \Spec(R_0)$ in each of the following cases: (1) $i=0$ or $i= 1$ and $j\leq f_{R_+}(M)$, (2) $R_0$ is regular, $i= \hei(\p_0)$ or $i= \hei(\p_0)- 1$ and $j= \cd_{R_+}(M)$, (3) $M$ is relative Cohen-Macaulay with respect to $R_+$. Here, $\cd_{R_+}(M)$ and $f_{R_+}(M)$ denote the cohomological dimension and finiteness dimension of $M$ with respect to $R_+$, respectively.