The injectivity of Frobenius acting on cohomology and local cohomology modules

Abstract

LetR be a two-dimensional normal graded ring over a field of characteristicp>0. We want to describe the tight closure of (O) in the local cohomology moduleH R+ 2 (R) using the graded module structure ofH R+ 2 (R). For this purpose we explore the condition that the Frobenius mapF: [H R+ 2 (R)]n→[H R+ 2 (R)]pninduced on graded pieces ofH R+ 2 (R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorD onX=Proj (R) such thatR=R (X, D)= ⊕ n≥0H0(X O X (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups Open image in new window Our interest is the casen 0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulop reduction of a rational signularity in characteristic 0 isF-rational forp≫0. Our result answers to this question affirmatively and also sheds light to behavior ofF-rationality in smallp.