Graded local cohomology modules with respect to the linked ideals

Abstract

Let $R={\oplus }_{n\in {\mathbb{N}}_0}{R}_n$ be a standard graded ring, $M$ be a finitely generated graded $R$-module and $R_{+}:={\oplus }_{n\in \mathbb{N}}{R}_n$ 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 $H_{R_{+}}^i(M)\ne 0$, 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.

14 pages

Cited by in corpus (1)

• Asymptotic behaviour of graded local cohomology modules via linkage