Some properties of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring, $\text{fa}$ 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 $\text{fm}$ and $M{\otimes }_{R}N$ is of finite length then $H_{\text{fm}}^t(M,N)$ is of finite length for all $t\ge 0$ and $l_R(H_{\text{fm}}^t(M,N))\le {\sum }_{i=0}^t{l}_R({\text{Ext}}_R^i(M,H_{\text{fm}}^{t-i}(N)))$. This yields, $l_R(H_{\text{fm}}^t(M,N))=l_R({\text{Ext}}_R^t(M,N))$. Additionally, we show that ${\text{Ext}}_R^i(R\mathrm{/}\text{fa},N)$ is Artinian for all $i\le t$ if and only if $H_{\text{fa}}^i(M,N)$ is Artinian for all $i\le t$. Moreover, we show that whenever $\dim (R\mathrm{/}\text{fa})=0$ then $H_{\text{fa}}^t(M,N)$ is Artinian for all $t\ge 0$.