Bigraded components of F-finite F-modules

Let $A$ be a regular ring containing a field of characteristic $p>0$ and let $R=A[x_1,\dots ,x_m,y_1,\dots ,y_n]$ be standard bigraded over $A$, i.e., $\mathrm{bideg}(A)=(0,0)$, $\mathrm{bideg}(x_i)=(1,0)$ and $\mathrm{bideg}(y_j)=(0,1)$ for all $i$ and $j$. Assume that $M={⨁}_{i,j}{M}_{(i,j)}$ is a bigraded $F_R$-finite, $F_R$-module. We use Lyubeznik’s theory of $F$-finite, $F$-modules from \cite{Lyu-Fmod} to study the bigraded components of $M$. The properties we study include vanishing, rigidity, Bass numbers, associated primes, and injective dimension of the components of $M$. As an application we show that if $(A,\mathfrak{m})$ is regular local ring containing a field of characteristic $p>0$, $R\mathrm{/}I$ is equidimensional, $\mathrm{Bproj}(R\mathrm{/}I)$ is Cohen-Macaulay and non-empty, then $H_I^j(R{)}_{(m,n)}=0$ for all $(m,n)\ge (0,0)$ and all $j>\mathrm{height}I$.