Maryam Jahangiri, Azadeh Nadali, Khadijeh Sayyari: Graded local cohomology modules with respect to the linked ideals, 15-29

Abstract:

Let $R=\oplus_{n\in {\Bbb N}_0}R_n$ be a standard graded ring, $M$ be a finitely generated graded $R$-module and $R_+:=\oplus_{n\in {\Bbb N}}R_n$ 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 $H^i_{\mathfrak{a}}(M)_n$ when ${\mathfrak{a}}$ is an h-linked ideal over $M$. More precisely, we show that $H^i_{\mathfrak{a}}(M)$ is tame in each of the following cases:

(i)	$i={f_{\mathfrak{a}}^{R_+}}(M)$, the first integer $i$ for which $R_+\nsubseteq \sqrt{0:H^i_{\mathfrak{a}}(M)}$;
(ii)	$i={\rm cd}(R_+,M)$, the last integer $i$ for which $H^{i}_{R_+}(M)\neq 0$, and ${\mathfrak{a}}={\mathfrak{b}}+R_+$ where ${\mathfrak{b}}$ is an h-linked ideal with $R_+$ over $M$.

Also, among other things, we describe the components $H^i_{\mathfrak{a}}(M)_n$ where ${\mathfrak{a}}$ is radically h-$M$-licci with respect to $R_+$ of length 2.

Key Words: Graded local cohomology modules, linkage of ideals, finiteness dimension, cohomological dimension, relative Cohen-Macaulay.

2010 Mathematics Subject Classification: Primary 13D45, 13A02, 13C40.

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