The injectivity of Frobenius acting on cohomology and local cohomology modules

Let R be a two-dimensional normal graded ring over a field of characteristic p > 0. We want to describe the tight closure of (0) in the local cohomology module H²_R+(R) using the graded module structure of H²_R+(R). For this purpose we explore the condition that the Frobenius map F : [H²_R+(R)]_n → [H²_R+(R)]_pn induced on graded pieces of H²_R+(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisor D on X = Proj(R) such that R = R(X, D) = ⊕_n≥0H⁰(X, script O sign_X(nD)). Then the above map is identified with the induced Frobenius on the cohomology groups F_n : H¹(X, script O sign_X(nD)) → H¹(X, script O sign_X(pnD)). Our interest is the case n < 0, and in this case, a generalization of Tango's method for integral divisors enables us to show that F_n is injective if p is greater than a certain bound given explicitly by X and nD. This result is useful to study F-rationality of R. The notion of F-rational rings in characteristic p > 0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulo p reduction of a rational singularity in characteristic 0 is F-rational for p ≫ 0. Our result answers to this question affirmatively and also sheds light to behavior of F-rationality in small p.