Koszul homology of $F$-finite module and applications

Let $k$ be an infinite field of characteristic $p>0$ and let $R=k[Y_1,\dots ,Y_d]$ (or $R=k[[Y_1,\dots ,Y_d]]$). Let $F\text{\hspace{0.17em}⁣}:\text{Mod}(R)\to \text{Mod}(R)$ 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 $r\ge 1$ then the Koszul homology modules $H_i(Y_1,\dots ,Y_r;\mathcal{M})$ are $F_{\bar{R}}$-finite modules where $\bar{R}=R\mathrm{/}(Y_1,\dots ,Y_r)$ for $i=0,\dots ,r$. As an application if $A$ is a regular ring containing a field of characteristic $p>0$ and $S=A[X_1,\dots ,X_m]$ 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 $S\mathrm{/}I$ is equidimensional and $Proj(S\mathrm{/}I)$ is Cohen-Macaulay then $H_I^i(S{)}_n=0$ for all $n\ge 0$ and for all $i>heightI$. If $B$ is a equicharacteristic local Noetherian ring with infinite residue field and with a surjective map $\pi \text{\hspace{0.17em}⁣}:T\to B$ where $(T,\mathfrak{n})$ is regular local then we show that the Koszul cohomology modules $H^j(\mathfrak{n},H_{\ker \pi }^{\dim T-i}(T))$ depend only on $A,i,j$ and not on $T$ and $\pi$.