Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Let $R={⨁}_{n\ge 0}{R}_n$, $\text{fa}\supseteq {⨁}_{n>0}{R}_n$ and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of graded generalized local cohomology modules. First of all, we show that for all $i\ge 0$, $H_{\text{fa}}^i(M,N{)}_n$, the $n$-th graded component of the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\text{fa}$, vanishes for all $n\gg 0$. Furthermore, some sufficient conditions are proposed to satisfy the equality $\sup{\en(H^i_{\fa}(M, N))| i\geq 0}= \sup{\en(H^i_{R_+}(M, N))| i\geq 0}$. Some sufficient conditions are also proposed for tameness of $H_{\text{fa}}^i(M,N)$ such that $i=f_{\text{fa}}^{R_{+}}(M,N)$ or $i={\text{cd}}_{\text{fa}}(M,N)$, where $f_{\text{fa}}^{R_{+}}(M,N)$ and ${\text{cd}}_{\text{fa}}(M,N)$ denote the $R_{+}$-finiteness dimension and the cohomological dimension of $M$ and $N$ with respect to $\text{fa}$, respectively. We finally consider the Artinian property of some submodules and quotient modules of $H_{\text{fa}}^j(M,N)$, where $j$ is the first or last non-minimax level of $H_{\text{fa}}^i(M,N)$.