Document Type : Research Paper

Author

• Ali Fathi

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$, and let $N$ be a non-zero finitely generated $R$-module of dimension $d$. Assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M, N$ with respect to $\mathfrak a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$ is zero for all $i>p+d$ and the top generalized local cohomology $\operatorname{H}^{p+d}_{\mathfrak a}(M, N)$ is Artinian. In this paper, we study the vanishing and attached primes of $\operatorname{H}^{p+d}_{\mathfrak a}(M, N)$. Also, we prove that if $\operatorname{H}^{p+r}_{\mathfrak a}(M, R/\mathfrak p)=0$ for a fixed non-negative integer $r$ and for all $\mathfrak p$ in $\operatorname {supp}_R(N)$, then $\operatorname{H}^{p+s}_{\mathfrak a}(M, N)=0$ for all $s\geq r$ and so $c<p+r$. We deduce that if $p\leq c$, then
$$c=p+\min\{t\in\mathbb N_0: \operatorname{H}^{p+t}_{\mathfrak a}(M, R/\mathfrak p)=0 \text{ for all } \mathfrak p\in\operatorname {supp}_R(N)\}-1.$$
Also, we prove that, for each $i$ with $p\leq i\leq c$, there exists $\mathfrak p_i$ in $\operatorname{supp}_R(N)$ such that $\operatorname{H}^{i}_{\mathfrak a}(M, R/\mathfrak p_i)\neq 0$.

Keywords

• Generalized local cohomology module
• cohomological dimension
• coassociated prime ideal
• attached prime ideal