Some properties of generalized local cohomology modules

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. We prove that if $R$ is local with maximal ideal $\fm$ and $ M\otimes_R N$ is of finite length then $H_{\fm}^t(M,N)$ is of finite length for all $t\geq 0$ and $l_R(H_{\fm}^t (M,N))\leq \sum_{i=0}^t l_R(\Ext_R^i(M,H_{\fm}^{t-i}(N)))$. This yields, $l_R(H_{\fm}^t(M,N))=l_R(\Ext_R^t(M,N))$. Additionally, we show that $\Ext_R^i(R/{\fa},N)$ is Artinian for all $ i\leq t$ if and only if $H_{\fa}^i(M,N)$ is Artinian for all $i\leq t$. Moreover, we show that whenever $\dim (R/{\fa})=0$ then $H_{\fa}^t(M,N)$ is Artinian for all $t \geq 0$.