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