  Let $k$ be an infinite field of characteristic $p > 0$ and let $R =
k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon
\text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let
$\mathcal{M}$ be a $F_R$-finite module (in the sense of Lyubeznik
\cite{Lyu-2}). We show that if $r \geq 1$ then the Koszul homology modules
  $H_i(Y_1,\ldots, Y_r; \mathcal{M})$ are $F_{\overline{R}}$-finite modules
where $\overline{R} = R/(Y_1,\ldots, Y_r)$ for $i = 0, \ldots, r$. As an
application if $A$ is a regular ring containing a field of characteristic $p >
0$ and $S = A[X_1,\ldots, X_m]$ is standard graded and $I$ is an arbitrary
graded ideal in $S$ then we give a comprehensive study of graded components of
local cohomology modules $H^i_I(S)$.
  This extends in positive characteristic results we proved in \cite{P}. We
study $H^i_I(S) $ when $A$ is local and prove that if $S/I$ is equidimensional
and $Proj(S/I)$ is Cohen-Macaulay then $H^i_I(S)_n = 0$ for all $n \geq 0$ and
for all $i > \ height \ I$. If $B$ is a equicharacteristic local Noetherian
ring with infinite residue field and with a surjective map $\pi \colon T
\rightarrow B$ where $(T,\mathfrak{n})$ is regular local then we show that the
Koszul cohomology modules $H^j(\mathfrak{n}, H^{\dim T - i}_{\ker \pi }(T))$
depend only on $A, i, j$ and not on $T$ and $\pi$.

Commutative Algebra