The injectivity of Frobenius acting on cohomology and local cohomology modules (Q1817356)

Let \(R\) be a 2-dimensional normal graded ring over a field of characteristic \(p>0\). In order to describes the tight closure of (0) in \(H^2_{R_+} (R)\), the authors study the condition that the Frobenius map \(F: [H^2_{R_+} (R)]_n \to [H^2_{R_+} (R)]_{pn}\) is injective. This is expressed geometrically as follows: There exists an ample fractional divisor \(D\) on \(X: =\text{Proj} (R)\) such that \(R=R(X,D) =\bigoplus_{n\geq 0} H^0(X, {\mathcal O}_X (nD))\) and then the above map is the induced Frobenius on the cohomology groups \(F_n: H^1(X, {\mathcal O}_X (nD)) \to H^1(X, {\mathcal O}_X (pnD))\). In the case \(n<0\), it is shown that \(F_n\) is injective if \(p\) is greater than a certain bound given by \(X\) and \(D\). Then this is used to show that a rational singularity in characteristic zero is \(F\)-rational for \(p\gg 0\).