Graded $F$-modules and Local Cohomology

Let $R=k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ let $\text{m}=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ be the maximal ideal generated by the variables, let ${}^{\ast }E$ be the naturally graded injective hull of $R\mathrm{/}\text{m}$ and let ${}^{\ast }E(n)$ be ${}^{\ast }E$ degree shifted downward by $n\mathrm{.}$ We introduce the notion of graded $F$-modules (as a refinement of the notion of $F$-modules) and show that if a graded $F$-module $\text{M}$ has zero-dimensional support, then $\text{M},$ as a graded $R$-module, is isomorphic to a direct sum of a (possibly infinite) number of copies of ${}^{\ast }E(n)\mathrm{.}$ As a consequence, we show that if the functors $T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_s$ and $T$ are defined by $T_j=H_{I_j}^{i_j}(-)$ and $T=T_1\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ T_s,$ where $I_1,\mathrm{.}\mathrm{.}\mathrm{.},I_s$ are homogeneous ideals of $R,$ then as a naturally graded $R$-module, the local cohomology module $H_{\text{m}}^{i_0}(T(R))$ is isomorphic to ${}^{\ast }E(n{)}^c,$ where $c$ is a finite number. If $\text{char}k=0,$ this question is open even for $s=1.$