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 H2R+(R) using the graded module structure of H2R+(R). For this purpose we explore the condition that the Frobenius map F : [H2R+(R)]n → [H2R+(R)]pn induced on graded pieces of H2R+(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 signX(nD)). Then the above map is identified with the induced Frobenius on the cohomology groups Fn : H1(X, script O signX(nD)) → H1(X, script O signX(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 Fn 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.
|Number of pages||15|
|Publication status||Published - 1996 Jul 1|
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