Graded F-modules and local cohomology (Q2903277)

Following the theory of F-modules developed by \textit{G. Lyubeznik} [J. Reine Angew. Math. 491, 65--130 (1997; Zbl 0904.13003)], the author defines the notion of graded F-modules over a polynomial ring \((R,\mathfrak{m})=k[x_1,\dots,x_n]\) wherein \(k\) is a field of positive characteristic. In the main theorem of the paper, he shows that a graded F-module whose support is of dimension zero is a direct sum of copies of \(^*E(k)(n)\) -- the injective hull of \(k\) shifted downward by the fix integer \(n\). As a corollary to the main theorem, it is shown that if \(I\) is a homogeneous ideal of \(R\), then for each \(i,j\) there is a homogeneous isomorphism \(H^i_{\mathfrak{m}}(H^j_I(R))\simeq^*E(k)(n)^c\) as graded \(R-\)modules. The proof of the latter is based on the already known facts in the non-graded version given in the above reference.