Graded local cohomology modules with respect to the linked ideals

TL;DR

This paper investigates the structure and properties of graded local cohomology modules over a standard graded ring, focusing on linked ideals and their impact on the tameness and components of these modules.

Contribution

It introduces the concept of linkage of ideals over a module and analyzes the tameness of graded local cohomology modules in specific cases, providing new insights into their structure.

Findings

H^i_{\fa}(M) is tame for certain indices i.

Descriptions of graded components when \fa is radically h-licci.

Characterization of local cohomology modules with linked ideals.

Abstract

Let $R=\oplus_{n\in \N_0}R_n$ be a standard graded ring, $M$ be a finitely generated graded $R$-module and $R_+:=\oplus_{n\in \N}R_n$ denotes the irrelevant ideal of $R$. In this paper, considering the new concept of linkage of ideals over a module, we study the graded components $H^i_{\fa}(M)_n$ when $\fa$ is an h-linked ideal over $M$. More precisely, we show that $H^i_{\fa}(M)$ is tame in each of the following cases: \begin{itemize} \item [(i)] $i=f_{\fa}^{R_+}(M)$, the first integer $i$ for which $R_+\nsubseteq \sqrt{0:H^i_{\fa}(M)}$; \item [(ii)] $i=\cd(R_+,M)$, the last integer $i$ for which $H^{i}_{R_+}(M)\neq 0$, and $\fa=\fb+R_+$ where $\fb$ is an h-linked ideal with $R_+$ over $M$. \end{itemize} Also, among other things, we describe the components $H^i_{\fa}(M)_n$ where $\fa$ is radically h-$M$-licci with respect to $R_+$ of length 2.